101098rsta20010972

Phase recovery and lensless imaging byiterative methods in optical X-ray

and electron direg raction

By J C H Spence1 U Weierstall1 a n d M Howells2

1Department of Physics and Astronomy Arizona State UniversityTempe AZ 85287-1504 USA

2Advanced Light Source Lawrence Berkeley Laboratory1 Cyclotron Road Berkeley CA 94720 USA

Published online 28 March 2002

Thomas Youngrsquos quantitative analysis of interference enotects provided the conshy denceneeded to revive the wave theory of light and shy rmly established the concept of phasein optics Phase plays a similarly fundamental role in matterwave interferometryfor which the shy eld-emission electron microscope provides ideal instrumentation Thewaveparticle duality is vividly demonstrated by experimental `Youngrsquos fringesrsquo usingcoherent electron beams under conditions in which the regight time is less than the timebetween particle emission A brief historical review is given of electron interferom-etry and holography including the AharonovBohm enotect and the electron Sagnacinterferometer The simultaneous development of phase-contrast imaging at subnano-metre spatial resolution has greatly deepened our understanding of atomic processesin biology materials science and condensed-matter physics while electron hologra-phy has become a routine tool for the mapping of electrostatic and magnetic shy elds inmaterials on a nanometre scale The encoding of phase information in scattered far-shy eld intensities is discussed and non-interferometric non-crystallographic methodsfor phase retrieval are reviewed in relationship to electron holography Examples ofphase measurement and dinotraction-limited imaging using the hybrid inputoutputiterative algorithm are given including simulations for soft X-ray imaging and newexperimental results for coherent electron and visible-light scattering Image recon-struction is demonstrated from experimental electron and visible-light Fraunhoferdinotraction patterns The prospects this provides for lensless imaging using parti-cles for which no lenses exist (such as neutrons condensates coherent atom beamsand X-rays) are discussed These new interactions can be expected to provide newinformation perhaps for example in biology with the advantage of less damage tosamples

Keywords direg raction phase recovery lensless imagingphase problem coherent imaging

One contribution of 15 to a special Theme Issue Interference 200 years after Thomas Youngrsquo sdiscoveriesrsquo

Phil Trans R Soc Lond A (2002) 360 875895

875

creg 2002 The Royal Society

876 J C H Spence U Weierstall and M Howells

1 Introduction

It was the quantitative analysis of interference enotects which distinguished the workof the great polymath Thomas Young from that of his predecessors and so gaveFresnel and others the conshy dence to later develop the wave theory of light fully(Young 1804) Whereas Huygens and Grimaldi had earlier advocated wave theoriesagainst the authority of Newton Young was the shy rst to analyse both his new andNewtonrsquos old experimental data using the correct path dinoterences at slits and edgesand to deduce from them both the wavelength and frequency of light (Young usedRoemerrsquos remarkably accurate estimate of the speed of light made a century earlier)In this way the concept of phase was shy rmly established within optics

But the waveparticle duality remained In 1909 G I Taylor wrote `Sir J JThomson has suggested that if the light in a dinotraction experiment were so greatlyreduced that only a few indivisible units of energy should occur on a Huygenszone at once then the ordinary phenomena of dinotraction would be modishy edrsquo Tay-lorrsquos experiments aimed at understanding this dualism examined the shadow-edgesof a needle (Taylor 1909) using a gas regame illuminating a slit as a coherent sourcesmoked-glass attenuators and exposure times of up to three months The modernequivalent of this experiment for matter waves has been shown by Tonomura (1993)and can also be found as shy gure 3 of the accompanying article by Lichte (2002) Thestatistical build-up of an electron `two-slitrsquo interference pattern (actually obtainedusing an electrostatic biprism) is observed under conditions in which each electronarrives at one pixel of the detector long before the next leaves the source Diracrsquoscomment that `each photon interferes only with itselfrsquo clearly also applies here toelectrons under these experimental conditions Localization of the particle at thesource prevents a determination of which hole the electron went through consis-tent with both the uncertainty principle and the Van CittertZernike theorem Thistheorem ensures that the spatial coherence width span the two holes In this caseuncertainty is synonymous with coherence As discussed further below these twoconcepts are also linked through the deshy nition of beam degeneracy

This extension of Thomas Youngrsquos work on interference of light to interferenceof massive particles lies at the heart of much modern physics since it requires theassignment of a phase to the matterwave wavefunction That idea played a crucialrole in the development of quantum mechanics with the argument that for a single-valued wave function phase may change only by 2nordm around any closed loop Thisargument was used to justify the quantization of electron orbitals in Bohrrsquos atomictheory the quantitative success of which (for hydrogen spectra) again gave conshy denceto the developers of formal quantum mechanics It arises in Diracrsquos quantizationcondition for magnetic monopole strength (Dirac 1931) and in the quantization ofmagnetic regux in superconductors amongst other places and so may be consideredto be one of the more important ideas of 20th-century physics

In this paper we are concerned with the encoding and retrieval of the phase ofelectron and optical waveshy elds from simple scattering experiments for the purposeof image formation Because of the bright coherent shy eld-emission electron sourcesavailable the sophisticated design regexibility and convenience of modern transmissionelectron microscopes (TEMs) much of the most signishy cant matterwave interferom-etry has been undertaken using the high-quality coherent electron beam availablein a TEM Under this heading we should include modern phase-contrast atomic-

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 877

resolution imaging the probe-formation process in scanning electron microscopyand electron crystallography For example the principle of coherent detection orig-inally suggested by Schiske in 1968 has since been extensively applied to allowatomic-resolution images to be synthesized from a through-focal series of electronmicrographs (for a historical review see Saxton (1994)) In this way by emacr cientlydeconvolving the instrument response controlling noise and compensating for lossof information due to partial coherence resolution has been extended beyond theconventional Scherzer point resolution limit out to the information limit set by elec-tronic instabilities (Coene et al 1992) This limit currently stands at slightly below1 A (OrsquoKeefe et al 2001) which will soon open up new possibilities in materialsscience for understanding atomic processes in matter especially at interfaces and inglasses and for the discovery of useful new nanostructures Atomic-resolution TEMprovided the shy rst images of the fullerene `buckyballsrsquo (Iijima 1980) and the discoveryof carbon nanotubes (Iijima 1991) For biology although resolution may be limitedby other factors such as variations in the conformation of molecules radiation dam-age and the bending of monomolecular layers the impact of this approach has beenconsiderable Using two-dimensional crystals it has revealed the detailed mechanismof photosynthesis in cell membranes (Henderson et al 1990 see also Subramaniamet al 2002) For macromolecules such as the ribosome responsible for protein pro-duction (`life itself rsquo) the cryomicroscopy method was the shy rst to reveal the shape(at 1 nm resolution) of large molecules which cannot be crystallized (Frank 1996)Despite recent limited success in crystallizing membrane proteins for X-ray workelectron cryomicroscopy remains vital for assisting in phasing X-ray data conshy rmingthe heavy-atom positions assumed in X-ray multi-wavelength anomalous dinotractionanalysis and in showing how smaller proteins of known structure shy t together andinteract

2 Electron interferometry and holography

More directly related to Youngrsquos work is modern electron interferometry and holog-raphy Following the 1927 discovery of electron dinotraction (Davisson amp Germer 1927Thomson amp Reid 1927) in conshy rmation of de Brogliersquos hypothesis observations ofhelium-atom dinotraction by crystals soon after provided further conshy rmation of thematterwave hypothesis (Esterman amp Stern 1930) The observation of Fresnel fringesin an electron microscope (EM) by Boersch (1943) provided further evidence of wave-like properties|and the realization that these patterns encoded phase informationwhich might be reconstructed|led to the birth of holography (Gabor 1949) (SinceGabor was unable to solve the twin-image problem presented by these in-line holo-grams Boersch with some justishy cation remained convinced that he had invented`holographyrsquo and viewed the term as an unnecessary acronym for an interferencepattern (T Mulvey 1996 personal communication)) Early partly successful exper-iments (Haine amp Mulvey 1952) used the `transmission methodrsquo based on the real-ization (currently being rediscovered by the electron-microscopy community) thatan out-of-focus image is an in-line hologram Gaborrsquos original aim was to improvethe resolution of EM images using holography to expose the phase of the wavefrontaberrations and so to deblur the images Reconstructions were obtained in 1951using a monochromatized arc lamp at 1 nm resolution (before the invention of laseror useful computers) However EM images were limited in resolution by mechanical

Phil Trans R Soc Lond A (2002)

878 J C H Spence U Weierstall and M Howells

and electronic instabilities rather than aberrations until the early 1960s Only in1995 was `Gaborrsquos dreamrsquo fully realized when onot-axis EM holograms were shy rst usedto extend resolution beyond the limit of 0198 nm set by aberrations toward the`super-resolutionrsquo limit set by electronic instabilities of 01 nm (Orchowski et al 1995)

The earliest beam-splitters for electrons used either Bragg dinotraction in a thincrystal in the transmission geometry (Marton 1952) or an electrostatic biprism(Moellenstedt amp Duker 1955) which has become the favoured choice today Thesubsequent development of the shy eld-emission electron source (Crewe et al 1968) atthe end of the 1960s was of the greatest importance for the shy eld since it provideda source of nanometre dimensions and sub-eV energy spread with a brightness perunit bandwidth greater than current-generation synchrotrons (Qian et al 1993)By combining Moellenstedtrsquos biprism with Crewersquos shy eld emission gun modern EMinstruments using beam energies of a few hundred keV are capable of recordingin a few seconds images showing hundreds of Fresnel fringes with a lateral coher-ence width of perhaps 01 m m and a longitudinal coherence length of several hundredangstroms They thus provide for interferometry a beam of extremely high qualityby the standards of accelerator physics

The refractive index for electrons in a magnetic shy eld was shy rst derived using Fer-matrsquos principle in 1948 (Ehrenberg amp Siday 1948) The AharonovBohm (AB) enotect(Aharonov amp Bohm 1959) subsequently demonstrated observable enotects along opti-cal paths in which shy elds (but not potentials) are zero suggesting that potentialsare more fundamental than shy elds (This idea had apparently already been suggestedby Maxwell) More recent electron-interferometry experiments include for examplethe observation of regux quantization using the AB enotect (Tonomura 1988) the scalar(electrostatic) AB enotect with electrons (Matteucci et al 1992) and the electronSagnac interferometer (Hasselbach amp Nicklaus 1993) From this has grown the shy eldof electron holography now widely used to map out on a nanometre scale internalmagnetic and electrostatic shy elds in materials from digital recording media to ferro-electrics semiconductors and superconducting vortices (for reviews see Voelkl 1998Lichte 1988 Tonomura 1993)

A dinoterent class of electron-interference experiments depends on the chromaticcoherence of electron sources and their second-order coherence properties For thefree particle beams used for electron interferometry a degeneracy parameter macr maybe deshy ned to indicate the degree of crowding in phase space a unity value indicatestwo electrons of opposite spin per cell with cell dimensions deshy ned by the uncertaintyprinciple (Components of centK are deshy ned by energy spread and beam divergencecomponents of centr deshy ne the coherence patch and longitudinal coherence length) Itis readily shown (Spence et al 1994) that macr is equal to the ratio of actual sourcebrightness to Langmuirrsquos theoretical maximum value and that macr may be measuredfrom the dimensions of the coherence patch and the source energy spread For a coldtungsten nano-tip shy eld emitter we measure macr = 10iexcl4 (Spence et al 1994) whereasa value of about unity is found for undulators at synchrotrons (For an optical laserin which any number of indistinguishable bosons may occupy a state macr = 1015)Coulomb repulsion between electrons limits the value of macr Nevertheless the bright-ness of electron shy eld emitters exceeds that of current synchrotron undulators (Qianet al 1993) The enotects of photon bunching seen in the Hanbury Brown and Twissexperiment (Loudon 1976) and the suggestion that electrons should `antibunchrsquo for

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 879

large values of macr were made at an early stage by Purcell (1956) In a quantum pic-ture we imagine wave packets overlapping coherently along the propagation axisby chance giving non-classical (non-random) results for the probability of detectingparticles between wave packets For this to occur however the temporal coherenceof the source must exceed the time between particle emissions so that the particlesare in the same quantum state This requires the source degeneracy macr for electronsto approach unity It is therefore not surprising that despite considerable enotortin our own laboratory and elsewhere antibunching has yet to be observed in freecoherent electron beams In condensed matter however these conditions are readilyachieved and fermion antibunching has recently been observed for the shy rst time ina two-dimensional electron gas in the quantum Hall regime (Henny et al 1999) Areview of other novel enotects expected in multiparticle fermion interferometry can befound in Silverman (1993)

3 Phase in imaging iterative methods for phase measurement

An image is a two-dimensional near-shy eld waveshy eld which may be real or complexIt may be obtained by Fourier transform from far-shy eld scattered amplitudes if thephases of these are known Holography encodes this phase information emacr cientlythrough the use of a reference wave and Fourier transform (FT) holography (whichuses a point reference in the same plane and to one side of the object) is par-ticularly enotective in separating the real and virtual twin images after the recon-struction process This involves only illumination of the hologram with the originalreference wave Phase information is encoded in the displacements of the `Youngrsquosfringesrsquo which result in this geometry (The method has been extended to allow a`complicated or extended referencersquo (Szoke 1997) such as a known fragment of amolecule and thus may be considered as an extension of the `heavy-atomrsquo methodof crystallography) Since this experimental inversion (reconstruction) of FT holo-grams to images produces a unique result it should not surprise us that an iterativesearch algorithm the hybrid inputoutput (HIO) algorithm (Gerchberg amp Saxton1972 Fienup 1982) applied to the same geometry should perform well Remarkablyhowever it has been found that the algorithm also converges for complex objectsconsisting of two separated parts neither of which is known but which generate anappreciable fringe modulation in the scattering pattern A knowledge of the `supportrsquoof the object (and of the sign of the scattering density) is found to be sumacr cient forinversion the support is the area within which the object transmission function T (r)is non-zero (or unknown) and so may take the form for example of the Youngrsquosfringe pinhole mask Dinoterent `unknownrsquo objects shy ll each hole The inversion givenshy nite known support has been shown to be `almost alwaysrsquo unique (Bruck amp Sodin1979) An error metric which depends only on the known values of the supportand Fourier modulus monitors progress toward this unique solution In summarywe have recently come to understand that this experimental geometry of ThomasYoung now onoters the possibility of lensless imaging using any of the radiations forwhich lenses do not exist such as coherent atom beams X-rays and neutrons Thecomplementary arrangement|two compact complex objects lying on a transpar-ent membrane|also works For a dinoterent non-interferometric approach to phasemeasurement akin to the through-focus method mentioned in the introduction andbased on the `transport of intensityrsquo equations see Paganin amp Nugent (1998) This

Phil Trans R Soc Lond A (2002)

880 J C H Spence U Weierstall and M Howells

method which requires intensity measurements across two planes downstream ofthe object has been used for example to consider a non-interferometric variant ofthe AB enotect (Paganin 2001) (For a phase object a single plane is sumacr cient sinceunit intensity can be assumed everywhere in the object plane) The phase vorticeswhich occur at zeros of intensity have also attracted considerable interest (Allen etal 2001) So far it has been applied to cases using collimated illumination only sothat resolution is limited by detector pixel size however there seems no reason notto apply it to the case of diverging illumination which provides magnishy cation anda resolution limit imposed by the source size

Despite many simulations few experimental tests of the HIO algorithm haveappeared The algorithm iterates between real and reciprocal space (related by aFourier transform) imposing known information in each domain The moduli ofthe scattered amplitudes I(u) are assumed known and in real space the objectfunction T (r) is set to its known values outside the support (Here r and u aretwo-dimensional vectors T (r) is the ratio of the complex exit waveshy eld (the in-focusimage) to the incident wave function often assumed planar) The support need notalways be known very accurately and an estimate can be obtained from the 50contour of the autocorrelation function which is given by the Fourier transform ofI(u) Finally knowledge of the sign of the real (and possibly imaginary) parts of Tmay be used The introduction of a small amount of `feedbackrsquo (Fienup 1987) distin-guishes the HIO algorithm from the error-reduction algorithm (Gerchberg amp Saxton1972) Convergence is found to be improved further in three-dimensions (Miao et al 2001) For real objects (such as the charge density which dinotracts X-rays) a singleobject is sumacr cient and the computational boundary used in the iterations may bea dinoterent shape from the physical support which it includes When viewed as anoptimization process the algorithm appears to be capable of shy nding the thousands ofunknown parameters (the phases in a typical image) in the presence of noise withoutthe usual problems of local minima and stagnation It thus appears to solve a verylarge global optimization problem despite the application of a non-convex constraint(the known Fourier moduli) (A convex set of solutions is one for which all pointslying on any line joining any two points within the N -dimensional set lie inside theset a kidney-shaped set in two dimensions is non-convex N is the number of pixelsin the image If a unique solution can be shown to exist (as in this case) iterationbetween the boundaries of two convex sets guarantees convergence to this solutionwithout stagnation at local minima Hence convex constraints such as symmetryknown support or sign are sought in optimization problems) While inversion fromknown phase information is a convex process leading to a unique result (Lindaas etal 1998) inversion from intensity alone is non-convex so that local minima mayformally be expected in approaches based on numerical optimization (Stark 1987)The HIO algorithm has been analysed in considerable detail (Fienup 1982 Paganin2001) including treatments based on Bregman projections onto convex sets (Stark1987 Bauschke et al 2002) and those going back to Sayre (1952) which start fromthe realization that Bragg sampling undersamples the autocorrelation function ofa molecule in a crystal (Miao et al 2001) The number of Fourier series equationsis then compared with the number of unknowns as the support is made a progres-sively smaller fraction of the total dinotracting area The lowest spatial frequency inthe dinotraction pattern corresponds to a distance wider than the support Using this`oversamplingrsquo approach soft X-ray images of lithographed symbols were recently

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Phase recovery and lensless imaging 881

reconstructed from far-shy eld scattered intensities in a lensless scattering experiment(Miao et al 1999) Spatial resolution is limited only by noise and wavelength in theseexperiments For technical reasons however the reconstruction was assisted by low-resolution optical images which provided the low-order spatial frequencies and theobject transmission function was of a special mask-like form having only two (com-plex) transmission values whose signs were known We have therefore conductedboth simulations and experiments for phase measurement in scattering of X-rayscoherent visible light and coherent electron beams to evaluate this HIO algorithmfor more general objects

4 Simulations for X-rays

Simulations have been undertaken for objects consisting of an array of randomlypositioned pinholes to determine if this image could be reconstructed from its softX-ray dinotraction pattern If so these holes could later be shy lled with structures ofinterest for reconstruction The area occupied by pinholes has been limited to thecoherence width of the beam by a 1 m m aperture This aperture will not fall exactly inthe plane of the pinholes therefore the wavefront at the pinholes will have undergoneFresnel dinotraction at the aperture and so will be complex Figure 1 shows the resultsof these simulations assuming a wavelength of 25 nm for the incident soft X-raybeam The pinholes are 10 nm in diameter After 150 iterations the HIO algorithmhas converged without stagnation to an image which is related to the object byinversion through the centre The modulus of the estimated object function was takenat each iteration (modulus constraint) since the object is assumed real Only a roughestimate of the size of the aperture is needed the computational support conditionapplied in the iterations is a triangle larger than the aperture The unimportantinversion ambiguity cannot be resolved by the HIO algorithm since an invertedimage our real computational support and the circular aperture all have the sameFourier modulus The Fresnel propagation makes this strictly a complex objecthowever we obtain an accurate reconstruction of the holes in the shy lm despite the useof a positivity (modulus) constraint in object space The phase of the complex wavetransmitted through the pinholes is not reconstructed due to the imposed positivity

5 Experiments with light

Coherent laser optical phase retrieval experiments have been performed using theHIO algorithm for several objects The phase was successfully retrieved for severaldinoterent support constraints and object types So far only opaque support con-straints (apertures) have been used since these present fewer problems with exces-sive central beam intensity (blooming) The source was a frequency-stabilized HeNe laser ( para = 6328 nm) collimated to provide the planar wavefronts needed at thesample for real objects (shy gure 2) Dinotracted light was focused with a lens onto aliquid-nitrogen-cooled charge-coupled device (CCD) camera and the digital imagereconstructed using the HIO algorithm in the Digital Micrograph script language

The shy rst object was a resolution test object with transparent numbers and linesin an opaque chrome shy lm on glass The illuminated area was limited by a thin300 m m diameter aperture in contact with the chrome shy lm Assuming the glass isregat to within less than para =10 across the illuminated area the chrome mask acts as a

Phil Trans R Soc Lond A (2002)

882 J C H Spence U Weierstall and M Howells

(a) (b)

(d )(c)

Figure 1 Reconstruction of simulated pinholes illuminated by a Fresnel direg racted X-ray wave-macreld (out-of-focus physical support) (a) Simulated random array of pinholes illuminated bywave amplitude which has been Fresnel propagated 200 nm from an aperture to the pinholesWavelength para = 25 nm pinhole size 10 nm aperture diameter 1 m m (b) 200 nm out-of-focuswavemacreld from 1 m m aperture incident on pinholes (c) The Fourier transform of (a) is a speckledAiryrsquo s disc Modulus used as input for the HIO algorithm (d) Reconstructed pinholes after 150iterations with modulus constraint (To enforce positivity in real space only the modulus ofthe pixel values inside the support area is retained during each iteration) A large triangularcomputational support was used The image is inverted this is an ambiguity which cannot beresolved by the HIO algorithm

real binary object so that the dinotraction pattern should be symmetric A coherentoptical image was also recorded with the camera moved into the image plane ofthe lens This is shown in shy gure 3a The camera was then moved to the back focalplane of the lens and the dinotraction pattern intensity I(u) was recorded (shy gure 3b)The autocorrelation FT I(u v) was used to determine the approximate size of theobject and a circular support half the size of the autocorrelation was used in the

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Phase recovery and lensless imaging 883

f

double polarizer to adjust intensity

collimator

lens diffraction pattern

CCD detectorHendashNe laser= 6328 nml

object with support mask

Figure 2 Schematic of the arrangement used for optical experiments

(a) (b)300 m m

Figure 3 (a) Coherent optical image of optical target transparent letters and bars in an oth-erwise opaque chrome macrlm on glass A 300 m m aperture acts as the support (HeNe laserpara = 6328 nm) (b) Coherent laser optical direg raction pattern of object shown in (a)

computer The HIO algorithm was used to recover the phases of the object exitwave and the algorithm always converged after about 50 iterations with dinoterentinitial random phase sets (shy gure 4a) A larger triangular support constraint wasalso used to recover the image as shown in shy gure 4b The normalized root-mean-squared error in the object domain (Fienup 1987) (a measure of dinoterence betweenthe current image estimate and the known pixel values outside the support) did notdrop below 02 probably due to imperfections in the dinotractogram eg noise andlight reregections from the entrance glass surface of the CCD camera which createasymmetry in the dinotraction pattern (see shy gure 4b above centre) Despite theseproblems the reconstructed image looks almost identical to the coherent image inshy gure 3a

A strong phase object (introducing phase shifts greater than ordm =2) was made usinga support consisting of two separated areas (two apertures of 300 m m diameter sep-

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884 J C H Spence U Weierstall and M Howells

(a) (b)

Figure 4 (a) Modulus of reconstructed image wave using a tight circular support constraint(b) Modulus of reconstructed image wave using a loose triangular support constraint A positivityconstraint was used in real space

mica

no mica(a) (b)

Figure 5 (a) Coherent laser optical image of phase object with two-hole support The edgesof two mica plates are visible across the hole on the right (b) Laser optical direg raction patternobtained by merging three patterns with direg erent exposure times to avoid saturation of thecentral beam

arated by 350 m m) One hole was covered with two 5 m m thick mica sheets whichoverlap in one region The expected phase shift is calculated to be 9 ordm The coherentoptical image is shown in shy gure 5a The mica is transparent except for the edgeswhere Fresnel contrast enotects are visible

The resulting dinotraction pattern is shown in shy gure 5b The central beam intensityin the dinotraction pattern from this transparent object was too strong to be recordedsimultaneously with the weaker high spatial frequencies so that several exposureswere needed All exposures were merged to one pattern in which saturated pixelswere replaced by scaled pixels from the lower exposure patterns The computational

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 885

black = -

white = +

(a) (b)

(c)

Figure 6 (a) Modulus and (b) phase of reconstructed image wave (c) phase reconstructed fromshifted direg raction pattern shows equiphase contour lines Phase shift by mica ordm No sign ormodulus constraint

support size and orientation were derived from the autocorrelation The complexHIO-phase retrieval algorithm was used to recover the image wave phase and modulusshown in shy gure 6 The phase shift by the mica plates in the reconstruction is ordm which since the algorithm reconstructs phase modulo 2 ordm is in good agreement withour independent measurement of 9 ordm based on the measured thickness and knownrefractive index of mica Thus the HIO algorithm is capable of quantitative phasemeasurement

Experiments with a transparent physical support have been attempted using alimiting aperture upstream of the sample but have not been successful so far becauseof the high intensity in the centre of the dinotraction pattern which causes bloomingin the CCD chip This problem persists for three-dimensional objects however athree-quadrant mask in the object plane has been proposed to address this dimacr culty(Sayre et al 1998) (Beam stops cannot be placed near enough to the CCD toavoid generating additional fringes) Experiments using the known intensity envelopeof a focused beam as the support have also not been successful for such complexobjects there appears to be a need for sharp physical support edges perhaps requiredto generate reference edge waves (For real objects there is no such requirement)However a coherent probe spanning two `knownrsquo holes works well in simulation

Previous experiments have been based on objects of simple type such as binary

Phil Trans R Soc Lond A (2002)

886 J C H Spence U Weierstall and M Howells

(a)

(b)

Figure 7 (a) Coherent laser optical image of cheek cells in two-hole support (300 m m apertures)About seven cells clump together in the upper hole Only Fresnel edge contrast is visible dueto defocus the cell nuclei are visible as black dots (b) Laser optical direg raction pattern mergedfrom four direg erent exposures to avoid saturation of the central beam

objects which contain much less information than greyscale objects Experimentswith human cheek cells which serve as more complicated greyscale phase objectshave therefore been performed The cheek cells were positioned on a glass slide insidetwo 300 m m apertures separated by 300 m m The apertures act as the opaque supportconsisting of two separated parts as needed for reconstruction of complex objectsFigure 7a shows the coherent optical image of the slightly defocused cheek cells asformed with a lens Only Fresnel edge contrast is visible while the nuclei producesome absorption Figure 7b shows the optical dinotraction pattern merged from fourdinoterent exposures The size and orientation of the support were estimated fromthe autocorrelation The complex reconstruction of phase and amplitude after about50 HIO iterations using the support constraint is shown in shy gure 8 The cells arevisible with large-area phase contrast in the phase image and the phase shift of thecells is about ordm In the reconstructed amplitude only edge contrast is visible as inthe optical image

For a three-dimensional object extended along the optical axis the question arisesas to which plane in the object is reconstructed given that the detected intensity liesat inshy nity and contains contributions from scattering at every point along the axisin the object space A simulation to clarify this is shown in shy gure 9 The simulatedobject consists of two planes separated by 1 mm in the z-direction Plane 1 has a holecontaining a complex object (the number 1) while plane 2 has two holes containingcomplex objects (numbers 2 and 3) The object exit wave calculated on plane 2

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 887

(a) (b)

Figure 8 (a) Modulus and (b) phase of reconstructed image wave The modulus shows only edgecontrast whereas the phase shows the cells in large-area phase contrast The phase shift due tothe cells is about ordm No modulus or sign constraint

(shy gure 9a) shows plane 1 out of focus The modulus of the Fourier transform (Fouriermodulus) of this image wave is used as input for the HIO algorithm In the inversiondinoterent shapes of the supports used at each plane shown inset can distinguish thetwo object planes If a plane is assumed unknown a larger support hole is usedthat contains the physical support Depending on which support plane is assumed asknown (inset) in the iteration (plane 1 support known and plane 2 support unknownor vice versa) the other plane is always reconstructed out of focus by +1 or iexcl 1 mm(shy gure 9c d) That means that in the case of known support on plane 1 the algorithmreconstructs the wave as seen through an imaging lens with out-of-focus images ofthe holes on plane 2 As expected the depth of focus for interactive inversion is givenby the expression for an ideal lens which imposes a dinotraction limit equal to that ofthe highest spatial frequency recorded in the dinotraction pattern

To verify this experimentally a pin was mounted 12 mm upstream of a pair of300 m m apertures The coherent optical image is shown in shy gure 10a A Fresnel fringesurrounds the pin since the lens is focused on the apertures The image was recoveredfrom the recorded dinotraction pattern intensity (shy gure 10b) Again three dinotractionpatterns with dinoterent exposure times had to be merged to avoid pixel saturationand blooming The reconstructed image modulus is shown in shy gure 11a where the

Phil Trans R Soc Lond A (2002)

888 J C H Spence U Weierstall and M Howells

illuminationdirection

plane 1

1 mm

(a) (b)

(c) (d)

Figure 9 (a) Schematic view of a simulated object consisting of two planes separated by 1 mm inthe direction of light propagation Plane 1 has a hole containing a complex object (number 1) andplane 2 has two holes containing complex objects (numbers 2 and 3) (b) Modulus of complexexit wave on plane 2 (c) Iterative reconstruction with plane 2 support assumed known (thesupport shape used is shown in the top right-hand corner) Plane 1 is out of focus by 1 mm(d) Iterative reconstruction with plane 1 support assumed known (the support shape is shownin the top right-hand corner) Plane 2 is out of focus by iexcl1 mm

pin appears out of focus (Fresnel fringe barely visible) because the aperture shapewas imposed as support The pin can be brought into focus by using the appropriateFresnel propagator with centz = 12 mm and para = 6328 nm taking the apertures outof focus (see shy gure 11b) Thus as expected the reconstructed plane is always thatplane in which the support condition is imposed where the algorithm shy nds a uniquesolution subject to the given boundary condition Propagation to other planes maybe performed subsequently since the complex waveshy eld is obtained subject to thedepth-of-focus limitation The relative sideways shift of the pin inside the aperturebetween shy gure 10a and shy gure 11a is due to a slight unintentional tilt of the object(pin plus pinhole) between recording the image and the dinotraction pattern

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 889

(a) (b)

Figure 10 (a) Coherent laser optical image of pin in front of aperture (b) Laser optical direg ractionpattern obtained by merging three patterns with direg erent exposure times to avoid saturation ofthe central beam

(a) (b)

Figure 11 (a) Reconstructed modulus Only the hole with the out-of-focus pin is shown(b) Numerical back propagation of reconstructed complex wavemacreld by 12 mm to plane of pinPin appears in focus Support constraint only

6 Experiments with coherent electrons

Using a special support mask made by electron-beam lithography experiments havealso been undertaken using the HIO algorithm to retrieve the phases from experimen-tal transmission electron dinotraction patterns (Weierstall et al 2002) Images mightthen be reconstructed that are free of electron lens aberrations The support maskcontains hole pairs of various sizes with pairs separated by 1 m m The separationbetween holes of a pair is ca 50 nm less than the coherence width of the illuminationbeam The holes irregular in shape form the object we wish to reconstruct in futurework we plan to introduce nanoscale objects into the holes Figure 12a below showsthe experimental dinotraction pattern from two holes shown in shy gure 12c obtained

Phil Trans R Soc Lond A (2002)

890 J C H Spence U Weierstall and M Howells

20 nm

(a) (b)

(c)

Figure 12 (a) Experimental electron direg raction pattern from two nanoscale holes in an opaquemask (shown in (c)) recorded using a macreld-emission electron microscope (CM200 FEG at 40 kV1k pound1k CCD with 24 m m pixels camera length 611 m) Youngrsquo s fringes from the two holes arevisible these fringes encode phase information (b) Reconstruction of image of holes from thedireg raction pattern (a) after 100 iterations of the HIO algorithm Modulus constraint applied inimage space with large triangular computational support The hole shape is comparable with theTEM image in (c) (c) Conventional TEM image of holes that created the direg raction pattern (a)

with a Philips CM 200 shy eld-emission electron microscope at 40 kV The acceleratingvoltage was lowered to 40 kV to make the multi-layered support opaque ModulatedYoungrsquos fringes are clearly seen

The resulting image after 100 HIO iterations is shown in shy gure 12b The holesreconstructed from the dinotraction pattern modulus look similar to the TEM imageof the holes in shy gure 12c but the resolution of the reconstruction is lower thanthe normal TEM image resolution High spatial frequency information from thehole edge is missing in the experimental dinotraction pattern due perhaps to drift ofthe imaging lenses during the exposure time

Table 1 shows a summary chart of all the conditions so far examined with HIO sim-

Phil Trans R Soc Lond A (2002)

Phase

recovery

an

dlen

slessim

agin

g891

Table 1 Summary chart showing direg erent conditions for phase recovery with HIO algorithm

( CArsquo denotes the complex algorithm (uses direg racted intensity and support constraint in real space) RIPrsquo denotes the complex algorithm withpositivity constraint of Re ordf Im ordf (uses direg racted intensity and support constraint in real space negative Re ordf and Im ordf inside the supportarea are set positive) Mrsquo denotes the modulus constraint on ordf in real space (uses direg racted intensity and support constraint in real spacecomplex numbers inside the support area are replaced by their modulus to enforce positivity) Prsquo denotes the phase constraint on ordf in realspace (uses direg racted intensity and support constraint in real space the phase is extracted and retained from complex numbers inside thesupport area modulus is extracted and set to 1 inside the support area))

general pure strong pure weakrsquo real weak-phasecomplex object phase object phase object (amplitude) object withordf = A exp(iiquest ) ordf = exp(i iquest (r)) iquest lt ordm =2 object absorption

known two-holephysical support

CA (laser experimentpin cheek cells)

CA P (laserexperiment mica)

RIP CA M RIP CA RIP CA

known one-holephysical support

CA only for certainsupport shapes(eg triangle withsharp edges)

CA slowconvergence Pworks better

RIP CA M RIP CA (laserexperimentnumbers)

RIP CA

unknown two-or-more-hole physicalsupport loose(eg triangle compu-tational support)

| | RIP M ( TEM experimentholes) (laserexperiment numbers)(planned X-ray micapinholes)

RIP (Miao X-rayexperimenttransparent support)

unknown one-holephysical support(eg triangle compu-tational support)

| | RIP only for holewith low symmetryand sharp edgesslow convergence

M RIP only for holewith low symmetryand sharp edgesslow convergence

RIP only for holewith low symmetryand sharp edgesslow convergence

Phil

Tra

ns

R

Soc

L

on

d

A(2

002)

892 J C H Spence U Weierstall and M Howells

ulations and experiments The enotectiveness of dinoterent constraints in real space fordinoterent physical and computational supports is compared The physical supportis the support which exists in the experiment and which may or may not be accu-rately known The computational support is the support used in the HIO iterationsThe table shows experiments that have been performed in italics The shy rst columnshows dinoterent possibilities for the physical support the most favourable at the topthe worst at the bottom The computational support and known physical supportare identical The computational support used with the unknown physical supportis larger than the physical support Reconstructions with one-hole physical supportand loose computational support are only possible with weak or real objects andsupport holes with sharp edges and low symmetry and convergence are very slow insimulations A known two-or-more-hole support proved to be most favourable in allcases

7 Discussion and conclusions direg ractive imaging with other particles

In this paper we have reviewed the implications of Thomas Youngrsquos experiment formatter waves and in particular for electron interferometry By taking advantage ofthe very high quality of the coherent electron beam available in the modern shy eld-emission electron microscope a number of important results (such as the AharonovBohm enotect and the electron Sagnac enotect) have been established and the shy eldof atomic-resolution transmission electron microscopy phase-contrast imaging andelectron holography opened up All these methods continue to produce a large sci-entishy c pay-onot in microstructural materials characterization and discovery and inbiology A new lensless imaging method is then reviewed which may be understoodin part by analogy with electron holography This HIO algorithm uses an iterativeapproach to assign phases to dinotraction pattern intensities To evaluate its perfor-mance we give experimental examples of image reconstruction with both coherentelectrons and light together with simulations for the soft X-ray case Thus we haveshown experimentally that the HIO algorithm is capable of solving the phase problemfor scattering from non-periodic objects and of making quantitative measurements ofphase (modulo 2ordm ) in good agreement with independent measurements The beam-splitting arrangement of an interferometer is not required and dinotraction-limitedlensless imaging seems possible as a result This may open the way for imaging withthe many particles for which no lenses exist such as neutrons ballistic electrons insolids at low temperature or neutral atoms and ions either as free coherent beamsor condensates (Bright coherent sources have recently appeared for many of these(Pauly 2000)) The dinotering interactions of these particles may provide useful newprobes for chemistry materials science and biology For example the reduced radia-tion damage enotects of imaging organic material with low-energy electrons neutronsand helium atoms have recently been compared (Spence et al 1999) and HIO sim-ulations have been completed for neutron imaging of a diatom (Spence et al 2001)The very small refractive index of organic matter for neutrons limits resolution andcontrast for thin samples For low-energy ions or metastable helium strong samplebeam interactions must be considered Damage from neutral helium-atom imagingof molecules should be negligible however penetration is also negligible (producinga proshy le image) Detectors for coherent atom beams still present serious problemssince area detectors are inemacr cient For biology the weak atom interaction is desirable

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 893

to avoid damage however a strong interaction at the detector is also needed so thationization before detection must be arranged

For real objects we have seen that the resolution in images recovered by the HIOalgorithm is largely independent of the accuracy with which the support is knownThe addition of noise anotects the probability that the unique solution will be found (asmeasured by the error metric) rather than the resolution which is near dinotraction-limited For complex objects our experience indicates that a much more accurateknowledge of the physical support boundary is needed approaching the scale ofthe resolution sought The tight support and two-hole geometry (see Fienup (1987)for other geometries) needed for complex objects (complex refractive index) willbecome increasingly accessible with the development of new lithographic techniquesfor support construction This can be combined with independent characterization ofthe support shape and the scaling information provided by the known autocorrelationof the support Experimentally favourable cases for imaging with nanoscale resolutionat present require loose support which table 1 shows greatly favours real objectssuch as the charge density which dinotracts hard X-rays at energies well away fromabsorption edges or very thin samples for high-energy electrons Unfortunately beamcoherence is more readily obtained for soft X-rays

Using modern lithographic techniques it is becoming increasingly possible toestablish the experimental conditions of known support required by the HIO algo-rithm both for free particle beams and for electrons in solids at low temperaturesThus for example we have seen the algorithm used to determine the phase of wavefunctions for electrons in a one-dimensional YBCO Josephson junction using TEMimages of the structure to deshy ne the required support (boundary conditions) (Car-mody et al 2000) Given the fundamental role that phase has played historically inphysics and which Thomas Young was amongst the shy rst to appreciate any techniquecapable of exposing phase can be expected to yield many surprises in physics in thefuture

Supported by ARO award DAAD190010500 We thank Dr Q Chen for the use of macrgure 12 (fromWeierstall et al 2002)

References

Aharonov Y amp Bohm D 1959 Signimacrcance of the electromagnetic potentials in the quantumtheory Phys Rev 115 485

Allen L J Faulkner H M L Nugent K A Oxley M P amp Paganin D 2001 Phase retrievalfrom images in the presence of macrrst-order vortices Phys Rev E 63 037602

Bauschke H H et al 2002 (In the press)

Boersch H 1943 Fresnelsche Beugung im Elektronenmikroskop Phys Z 44 202211

Bruck Y amp Sodin L 1979 On the ambiguity of the image-reconstruction problem Opt Com-mun 30 304308

Carmody M Moeckly B H Merkle K amp Marks L 2000 Spatial variation of the current ingrain boundary Josephson junctions J Appl Phys 87 24542459

Coene W Janssen G Op de Beek M amp van Dyck D 1992 Phase retrieval through focusvariation for ultra-resolution in macreld emission transmission electron microscopy Phys RevLett 69 37433746

Crewe A V Eggenberger D N Wall J amp Welter L M 1968 Electron gun using a macreldemission source Rev Scient Instrum 39 576

Phil Trans R Soc Lond A (2002)

876 J C H Spence U Weierstall and M Howells

1 Introduction

It was the quantitative analysis of interference enotects which distinguished the workof the great polymath Thomas Young from that of his predecessors and so gaveFresnel and others the conshy dence to later develop the wave theory of light fully(Young 1804) Whereas Huygens and Grimaldi had earlier advocated wave theoriesagainst the authority of Newton Young was the shy rst to analyse both his new andNewtonrsquos old experimental data using the correct path dinoterences at slits and edgesand to deduce from them both the wavelength and frequency of light (Young usedRoemerrsquos remarkably accurate estimate of the speed of light made a century earlier)In this way the concept of phase was shy rmly established within optics

But the waveparticle duality remained In 1909 G I Taylor wrote `Sir J JThomson has suggested that if the light in a dinotraction experiment were so greatlyreduced that only a few indivisible units of energy should occur on a Huygenszone at once then the ordinary phenomena of dinotraction would be modishy edrsquo Tay-lorrsquos experiments aimed at understanding this dualism examined the shadow-edgesof a needle (Taylor 1909) using a gas regame illuminating a slit as a coherent sourcesmoked-glass attenuators and exposure times of up to three months The modernequivalent of this experiment for matter waves has been shown by Tonomura (1993)and can also be found as shy gure 3 of the accompanying article by Lichte (2002) Thestatistical build-up of an electron `two-slitrsquo interference pattern (actually obtainedusing an electrostatic biprism) is observed under conditions in which each electronarrives at one pixel of the detector long before the next leaves the source Diracrsquoscomment that `each photon interferes only with itselfrsquo clearly also applies here toelectrons under these experimental conditions Localization of the particle at thesource prevents a determination of which hole the electron went through consis-tent with both the uncertainty principle and the Van CittertZernike theorem Thistheorem ensures that the spatial coherence width span the two holes In this caseuncertainty is synonymous with coherence As discussed further below these twoconcepts are also linked through the deshy nition of beam degeneracy

This extension of Thomas Youngrsquos work on interference of light to interferenceof massive particles lies at the heart of much modern physics since it requires theassignment of a phase to the matterwave wavefunction That idea played a crucialrole in the development of quantum mechanics with the argument that for a single-valued wave function phase may change only by 2nordm around any closed loop Thisargument was used to justify the quantization of electron orbitals in Bohrrsquos atomictheory the quantitative success of which (for hydrogen spectra) again gave conshy denceto the developers of formal quantum mechanics It arises in Diracrsquos quantizationcondition for magnetic monopole strength (Dirac 1931) and in the quantization ofmagnetic regux in superconductors amongst other places and so may be consideredto be one of the more important ideas of 20th-century physics

In this paper we are concerned with the encoding and retrieval of the phase ofelectron and optical waveshy elds from simple scattering experiments for the purposeof image formation Because of the bright coherent shy eld-emission electron sourcesavailable the sophisticated design regexibility and convenience of modern transmissionelectron microscopes (TEMs) much of the most signishy cant matterwave interferom-etry has been undertaken using the high-quality coherent electron beam availablein a TEM Under this heading we should include modern phase-contrast atomic-

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 877

resolution imaging the probe-formation process in scanning electron microscopyand electron crystallography For example the principle of coherent detection orig-inally suggested by Schiske in 1968 has since been extensively applied to allowatomic-resolution images to be synthesized from a through-focal series of electronmicrographs (for a historical review see Saxton (1994)) In this way by emacr cientlydeconvolving the instrument response controlling noise and compensating for lossof information due to partial coherence resolution has been extended beyond theconventional Scherzer point resolution limit out to the information limit set by elec-tronic instabilities (Coene et al 1992) This limit currently stands at slightly below1 A (OrsquoKeefe et al 2001) which will soon open up new possibilities in materialsscience for understanding atomic processes in matter especially at interfaces and inglasses and for the discovery of useful new nanostructures Atomic-resolution TEMprovided the shy rst images of the fullerene `buckyballsrsquo (Iijima 1980) and the discoveryof carbon nanotubes (Iijima 1991) For biology although resolution may be limitedby other factors such as variations in the conformation of molecules radiation dam-age and the bending of monomolecular layers the impact of this approach has beenconsiderable Using two-dimensional crystals it has revealed the detailed mechanismof photosynthesis in cell membranes (Henderson et al 1990 see also Subramaniamet al 2002) For macromolecules such as the ribosome responsible for protein pro-duction (`life itself rsquo) the cryomicroscopy method was the shy rst to reveal the shape(at 1 nm resolution) of large molecules which cannot be crystallized (Frank 1996)Despite recent limited success in crystallizing membrane proteins for X-ray workelectron cryomicroscopy remains vital for assisting in phasing X-ray data conshy rmingthe heavy-atom positions assumed in X-ray multi-wavelength anomalous dinotractionanalysis and in showing how smaller proteins of known structure shy t together andinteract

2 Electron interferometry and holography

More directly related to Youngrsquos work is modern electron interferometry and holog-raphy Following the 1927 discovery of electron dinotraction (Davisson amp Germer 1927Thomson amp Reid 1927) in conshy rmation of de Brogliersquos hypothesis observations ofhelium-atom dinotraction by crystals soon after provided further conshy rmation of thematterwave hypothesis (Esterman amp Stern 1930) The observation of Fresnel fringesin an electron microscope (EM) by Boersch (1943) provided further evidence of wave-like properties|and the realization that these patterns encoded phase informationwhich might be reconstructed|led to the birth of holography (Gabor 1949) (SinceGabor was unable to solve the twin-image problem presented by these in-line holo-grams Boersch with some justishy cation remained convinced that he had invented`holographyrsquo and viewed the term as an unnecessary acronym for an interferencepattern (T Mulvey 1996 personal communication)) Early partly successful exper-iments (Haine amp Mulvey 1952) used the `transmission methodrsquo based on the real-ization (currently being rediscovered by the electron-microscopy community) thatan out-of-focus image is an in-line hologram Gaborrsquos original aim was to improvethe resolution of EM images using holography to expose the phase of the wavefrontaberrations and so to deblur the images Reconstructions were obtained in 1951using a monochromatized arc lamp at 1 nm resolution (before the invention of laseror useful computers) However EM images were limited in resolution by mechanical

Phil Trans R Soc Lond A (2002)

878 J C H Spence U Weierstall and M Howells

and electronic instabilities rather than aberrations until the early 1960s Only in1995 was `Gaborrsquos dreamrsquo fully realized when onot-axis EM holograms were shy rst usedto extend resolution beyond the limit of 0198 nm set by aberrations toward the`super-resolutionrsquo limit set by electronic instabilities of 01 nm (Orchowski et al 1995)

The earliest beam-splitters for electrons used either Bragg dinotraction in a thincrystal in the transmission geometry (Marton 1952) or an electrostatic biprism(Moellenstedt amp Duker 1955) which has become the favoured choice today Thesubsequent development of the shy eld-emission electron source (Crewe et al 1968) atthe end of the 1960s was of the greatest importance for the shy eld since it provideda source of nanometre dimensions and sub-eV energy spread with a brightness perunit bandwidth greater than current-generation synchrotrons (Qian et al 1993)By combining Moellenstedtrsquos biprism with Crewersquos shy eld emission gun modern EMinstruments using beam energies of a few hundred keV are capable of recordingin a few seconds images showing hundreds of Fresnel fringes with a lateral coher-ence width of perhaps 01 m m and a longitudinal coherence length of several hundredangstroms They thus provide for interferometry a beam of extremely high qualityby the standards of accelerator physics

The refractive index for electrons in a magnetic shy eld was shy rst derived using Fer-matrsquos principle in 1948 (Ehrenberg amp Siday 1948) The AharonovBohm (AB) enotect(Aharonov amp Bohm 1959) subsequently demonstrated observable enotects along opti-cal paths in which shy elds (but not potentials) are zero suggesting that potentialsare more fundamental than shy elds (This idea had apparently already been suggestedby Maxwell) More recent electron-interferometry experiments include for examplethe observation of regux quantization using the AB enotect (Tonomura 1988) the scalar(electrostatic) AB enotect with electrons (Matteucci et al 1992) and the electronSagnac interferometer (Hasselbach amp Nicklaus 1993) From this has grown the shy eldof electron holography now widely used to map out on a nanometre scale internalmagnetic and electrostatic shy elds in materials from digital recording media to ferro-electrics semiconductors and superconducting vortices (for reviews see Voelkl 1998Lichte 1988 Tonomura 1993)

A dinoterent class of electron-interference experiments depends on the chromaticcoherence of electron sources and their second-order coherence properties For thefree particle beams used for electron interferometry a degeneracy parameter macr maybe deshy ned to indicate the degree of crowding in phase space a unity value indicatestwo electrons of opposite spin per cell with cell dimensions deshy ned by the uncertaintyprinciple (Components of centK are deshy ned by energy spread and beam divergencecomponents of centr deshy ne the coherence patch and longitudinal coherence length) Itis readily shown (Spence et al 1994) that macr is equal to the ratio of actual sourcebrightness to Langmuirrsquos theoretical maximum value and that macr may be measuredfrom the dimensions of the coherence patch and the source energy spread For a coldtungsten nano-tip shy eld emitter we measure macr = 10iexcl4 (Spence et al 1994) whereasa value of about unity is found for undulators at synchrotrons (For an optical laserin which any number of indistinguishable bosons may occupy a state macr = 1015)Coulomb repulsion between electrons limits the value of macr Nevertheless the bright-ness of electron shy eld emitters exceeds that of current synchrotron undulators (Qianet al 1993) The enotects of photon bunching seen in the Hanbury Brown and Twissexperiment (Loudon 1976) and the suggestion that electrons should `antibunchrsquo for

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 879

large values of macr were made at an early stage by Purcell (1956) In a quantum pic-ture we imagine wave packets overlapping coherently along the propagation axisby chance giving non-classical (non-random) results for the probability of detectingparticles between wave packets For this to occur however the temporal coherenceof the source must exceed the time between particle emissions so that the particlesare in the same quantum state This requires the source degeneracy macr for electronsto approach unity It is therefore not surprising that despite considerable enotortin our own laboratory and elsewhere antibunching has yet to be observed in freecoherent electron beams In condensed matter however these conditions are readilyachieved and fermion antibunching has recently been observed for the shy rst time ina two-dimensional electron gas in the quantum Hall regime (Henny et al 1999) Areview of other novel enotects expected in multiparticle fermion interferometry can befound in Silverman (1993)

3 Phase in imaging iterative methods for phase measurement

An image is a two-dimensional near-shy eld waveshy eld which may be real or complexIt may be obtained by Fourier transform from far-shy eld scattered amplitudes if thephases of these are known Holography encodes this phase information emacr cientlythrough the use of a reference wave and Fourier transform (FT) holography (whichuses a point reference in the same plane and to one side of the object) is par-ticularly enotective in separating the real and virtual twin images after the recon-struction process This involves only illumination of the hologram with the originalreference wave Phase information is encoded in the displacements of the `Youngrsquosfringesrsquo which result in this geometry (The method has been extended to allow a`complicated or extended referencersquo (Szoke 1997) such as a known fragment of amolecule and thus may be considered as an extension of the `heavy-atomrsquo methodof crystallography) Since this experimental inversion (reconstruction) of FT holo-grams to images produces a unique result it should not surprise us that an iterativesearch algorithm the hybrid inputoutput (HIO) algorithm (Gerchberg amp Saxton1972 Fienup 1982) applied to the same geometry should perform well Remarkablyhowever it has been found that the algorithm also converges for complex objectsconsisting of two separated parts neither of which is known but which generate anappreciable fringe modulation in the scattering pattern A knowledge of the `supportrsquoof the object (and of the sign of the scattering density) is found to be sumacr cient forinversion the support is the area within which the object transmission function T (r)is non-zero (or unknown) and so may take the form for example of the Youngrsquosfringe pinhole mask Dinoterent `unknownrsquo objects shy ll each hole The inversion givenshy nite known support has been shown to be `almost alwaysrsquo unique (Bruck amp Sodin1979) An error metric which depends only on the known values of the supportand Fourier modulus monitors progress toward this unique solution In summarywe have recently come to understand that this experimental geometry of ThomasYoung now onoters the possibility of lensless imaging using any of the radiations forwhich lenses do not exist such as coherent atom beams X-rays and neutrons Thecomplementary arrangement|two compact complex objects lying on a transpar-ent membrane|also works For a dinoterent non-interferometric approach to phasemeasurement akin to the through-focus method mentioned in the introduction andbased on the `transport of intensityrsquo equations see Paganin amp Nugent (1998) This

Phil Trans R Soc Lond A (2002)

880 J C H Spence U Weierstall and M Howells

method which requires intensity measurements across two planes downstream ofthe object has been used for example to consider a non-interferometric variant ofthe AB enotect (Paganin 2001) (For a phase object a single plane is sumacr cient sinceunit intensity can be assumed everywhere in the object plane) The phase vorticeswhich occur at zeros of intensity have also attracted considerable interest (Allen etal 2001) So far it has been applied to cases using collimated illumination only sothat resolution is limited by detector pixel size however there seems no reason notto apply it to the case of diverging illumination which provides magnishy cation anda resolution limit imposed by the source size

Despite many simulations few experimental tests of the HIO algorithm haveappeared The algorithm iterates between real and reciprocal space (related by aFourier transform) imposing known information in each domain The moduli ofthe scattered amplitudes I(u) are assumed known and in real space the objectfunction T (r) is set to its known values outside the support (Here r and u aretwo-dimensional vectors T (r) is the ratio of the complex exit waveshy eld (the in-focusimage) to the incident wave function often assumed planar) The support need notalways be known very accurately and an estimate can be obtained from the 50contour of the autocorrelation function which is given by the Fourier transform ofI(u) Finally knowledge of the sign of the real (and possibly imaginary) parts of Tmay be used The introduction of a small amount of `feedbackrsquo (Fienup 1987) distin-guishes the HIO algorithm from the error-reduction algorithm (Gerchberg amp Saxton1972) Convergence is found to be improved further in three-dimensions (Miao et al 2001) For real objects (such as the charge density which dinotracts X-rays) a singleobject is sumacr cient and the computational boundary used in the iterations may bea dinoterent shape from the physical support which it includes When viewed as anoptimization process the algorithm appears to be capable of shy nding the thousands ofunknown parameters (the phases in a typical image) in the presence of noise withoutthe usual problems of local minima and stagnation It thus appears to solve a verylarge global optimization problem despite the application of a non-convex constraint(the known Fourier moduli) (A convex set of solutions is one for which all pointslying on any line joining any two points within the N -dimensional set lie inside theset a kidney-shaped set in two dimensions is non-convex N is the number of pixelsin the image If a unique solution can be shown to exist (as in this case) iterationbetween the boundaries of two convex sets guarantees convergence to this solutionwithout stagnation at local minima Hence convex constraints such as symmetryknown support or sign are sought in optimization problems) While inversion fromknown phase information is a convex process leading to a unique result (Lindaas etal 1998) inversion from intensity alone is non-convex so that local minima mayformally be expected in approaches based on numerical optimization (Stark 1987)The HIO algorithm has been analysed in considerable detail (Fienup 1982 Paganin2001) including treatments based on Bregman projections onto convex sets (Stark1987 Bauschke et al 2002) and those going back to Sayre (1952) which start fromthe realization that Bragg sampling undersamples the autocorrelation function ofa molecule in a crystal (Miao et al 2001) The number of Fourier series equationsis then compared with the number of unknowns as the support is made a progres-sively smaller fraction of the total dinotracting area The lowest spatial frequency inthe dinotraction pattern corresponds to a distance wider than the support Using this`oversamplingrsquo approach soft X-ray images of lithographed symbols were recently

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 881

reconstructed from far-shy eld scattered intensities in a lensless scattering experiment(Miao et al 1999) Spatial resolution is limited only by noise and wavelength in theseexperiments For technical reasons however the reconstruction was assisted by low-resolution optical images which provided the low-order spatial frequencies and theobject transmission function was of a special mask-like form having only two (com-plex) transmission values whose signs were known We have therefore conductedboth simulations and experiments for phase measurement in scattering of X-rayscoherent visible light and coherent electron beams to evaluate this HIO algorithmfor more general objects

4 Simulations for X-rays

Simulations have been undertaken for objects consisting of an array of randomlypositioned pinholes to determine if this image could be reconstructed from its softX-ray dinotraction pattern If so these holes could later be shy lled with structures ofinterest for reconstruction The area occupied by pinholes has been limited to thecoherence width of the beam by a 1 m m aperture This aperture will not fall exactly inthe plane of the pinholes therefore the wavefront at the pinholes will have undergoneFresnel dinotraction at the aperture and so will be complex Figure 1 shows the resultsof these simulations assuming a wavelength of 25 nm for the incident soft X-raybeam The pinholes are 10 nm in diameter After 150 iterations the HIO algorithmhas converged without stagnation to an image which is related to the object byinversion through the centre The modulus of the estimated object function was takenat each iteration (modulus constraint) since the object is assumed real Only a roughestimate of the size of the aperture is needed the computational support conditionapplied in the iterations is a triangle larger than the aperture The unimportantinversion ambiguity cannot be resolved by the HIO algorithm since an invertedimage our real computational support and the circular aperture all have the sameFourier modulus The Fresnel propagation makes this strictly a complex objecthowever we obtain an accurate reconstruction of the holes in the shy lm despite the useof a positivity (modulus) constraint in object space The phase of the complex wavetransmitted through the pinholes is not reconstructed due to the imposed positivity

5 Experiments with light

Coherent laser optical phase retrieval experiments have been performed using theHIO algorithm for several objects The phase was successfully retrieved for severaldinoterent support constraints and object types So far only opaque support con-straints (apertures) have been used since these present fewer problems with exces-sive central beam intensity (blooming) The source was a frequency-stabilized HeNe laser ( para = 6328 nm) collimated to provide the planar wavefronts needed at thesample for real objects (shy gure 2) Dinotracted light was focused with a lens onto aliquid-nitrogen-cooled charge-coupled device (CCD) camera and the digital imagereconstructed using the HIO algorithm in the Digital Micrograph script language

The shy rst object was a resolution test object with transparent numbers and linesin an opaque chrome shy lm on glass The illuminated area was limited by a thin300 m m diameter aperture in contact with the chrome shy lm Assuming the glass isregat to within less than para =10 across the illuminated area the chrome mask acts as a

Phil Trans R Soc Lond A (2002)

882 J C H Spence U Weierstall and M Howells

(a) (b)

(d )(c)

Figure 1 Reconstruction of simulated pinholes illuminated by a Fresnel direg racted X-ray wave-macreld (out-of-focus physical support) (a) Simulated random array of pinholes illuminated bywave amplitude which has been Fresnel propagated 200 nm from an aperture to the pinholesWavelength para = 25 nm pinhole size 10 nm aperture diameter 1 m m (b) 200 nm out-of-focuswavemacreld from 1 m m aperture incident on pinholes (c) The Fourier transform of (a) is a speckledAiryrsquo s disc Modulus used as input for the HIO algorithm (d) Reconstructed pinholes after 150iterations with modulus constraint (To enforce positivity in real space only the modulus ofthe pixel values inside the support area is retained during each iteration) A large triangularcomputational support was used The image is inverted this is an ambiguity which cannot beresolved by the HIO algorithm

real binary object so that the dinotraction pattern should be symmetric A coherentoptical image was also recorded with the camera moved into the image plane ofthe lens This is shown in shy gure 3a The camera was then moved to the back focalplane of the lens and the dinotraction pattern intensity I(u) was recorded (shy gure 3b)The autocorrelation FT I(u v) was used to determine the approximate size of theobject and a circular support half the size of the autocorrelation was used in the

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 883

f

double polarizer to adjust intensity

collimator

lens diffraction pattern

CCD detectorHendashNe laser= 6328 nml

object with support mask

Figure 2 Schematic of the arrangement used for optical experiments

(a) (b)300 m m

Figure 3 (a) Coherent optical image of optical target transparent letters and bars in an oth-erwise opaque chrome macrlm on glass A 300 m m aperture acts as the support (HeNe laserpara = 6328 nm) (b) Coherent laser optical direg raction pattern of object shown in (a)

computer The HIO algorithm was used to recover the phases of the object exitwave and the algorithm always converged after about 50 iterations with dinoterentinitial random phase sets (shy gure 4a) A larger triangular support constraint wasalso used to recover the image as shown in shy gure 4b The normalized root-mean-squared error in the object domain (Fienup 1987) (a measure of dinoterence betweenthe current image estimate and the known pixel values outside the support) did notdrop below 02 probably due to imperfections in the dinotractogram eg noise andlight reregections from the entrance glass surface of the CCD camera which createasymmetry in the dinotraction pattern (see shy gure 4b above centre) Despite theseproblems the reconstructed image looks almost identical to the coherent image inshy gure 3a

A strong phase object (introducing phase shifts greater than ordm =2) was made usinga support consisting of two separated areas (two apertures of 300 m m diameter sep-

Phil Trans R Soc Lond A (2002)

884 J C H Spence U Weierstall and M Howells

(a) (b)

Figure 4 (a) Modulus of reconstructed image wave using a tight circular support constraint(b) Modulus of reconstructed image wave using a loose triangular support constraint A positivityconstraint was used in real space

mica

no mica(a) (b)

Figure 5 (a) Coherent laser optical image of phase object with two-hole support The edgesof two mica plates are visible across the hole on the right (b) Laser optical direg raction patternobtained by merging three patterns with direg erent exposure times to avoid saturation of thecentral beam

arated by 350 m m) One hole was covered with two 5 m m thick mica sheets whichoverlap in one region The expected phase shift is calculated to be 9 ordm The coherentoptical image is shown in shy gure 5a The mica is transparent except for the edgeswhere Fresnel contrast enotects are visible

The resulting dinotraction pattern is shown in shy gure 5b The central beam intensityin the dinotraction pattern from this transparent object was too strong to be recordedsimultaneously with the weaker high spatial frequencies so that several exposureswere needed All exposures were merged to one pattern in which saturated pixelswere replaced by scaled pixels from the lower exposure patterns The computational

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 885

black = -

white = +

(a) (b)

(c)

Figure 6 (a) Modulus and (b) phase of reconstructed image wave (c) phase reconstructed fromshifted direg raction pattern shows equiphase contour lines Phase shift by mica ordm No sign ormodulus constraint

support size and orientation were derived from the autocorrelation The complexHIO-phase retrieval algorithm was used to recover the image wave phase and modulusshown in shy gure 6 The phase shift by the mica plates in the reconstruction is ordm which since the algorithm reconstructs phase modulo 2 ordm is in good agreement withour independent measurement of 9 ordm based on the measured thickness and knownrefractive index of mica Thus the HIO algorithm is capable of quantitative phasemeasurement

Experiments with a transparent physical support have been attempted using alimiting aperture upstream of the sample but have not been successful so far becauseof the high intensity in the centre of the dinotraction pattern which causes bloomingin the CCD chip This problem persists for three-dimensional objects however athree-quadrant mask in the object plane has been proposed to address this dimacr culty(Sayre et al 1998) (Beam stops cannot be placed near enough to the CCD toavoid generating additional fringes) Experiments using the known intensity envelopeof a focused beam as the support have also not been successful for such complexobjects there appears to be a need for sharp physical support edges perhaps requiredto generate reference edge waves (For real objects there is no such requirement)However a coherent probe spanning two `knownrsquo holes works well in simulation

Previous experiments have been based on objects of simple type such as binary

Phil Trans R Soc Lond A (2002)

886 J C H Spence U Weierstall and M Howells

(a)

(b)

Figure 7 (a) Coherent laser optical image of cheek cells in two-hole support (300 m m apertures)About seven cells clump together in the upper hole Only Fresnel edge contrast is visible dueto defocus the cell nuclei are visible as black dots (b) Laser optical direg raction pattern mergedfrom four direg erent exposures to avoid saturation of the central beam

objects which contain much less information than greyscale objects Experimentswith human cheek cells which serve as more complicated greyscale phase objectshave therefore been performed The cheek cells were positioned on a glass slide insidetwo 300 m m apertures separated by 300 m m The apertures act as the opaque supportconsisting of two separated parts as needed for reconstruction of complex objectsFigure 7a shows the coherent optical image of the slightly defocused cheek cells asformed with a lens Only Fresnel edge contrast is visible while the nuclei producesome absorption Figure 7b shows the optical dinotraction pattern merged from fourdinoterent exposures The size and orientation of the support were estimated fromthe autocorrelation The complex reconstruction of phase and amplitude after about50 HIO iterations using the support constraint is shown in shy gure 8 The cells arevisible with large-area phase contrast in the phase image and the phase shift of thecells is about ordm In the reconstructed amplitude only edge contrast is visible as inthe optical image

For a three-dimensional object extended along the optical axis the question arisesas to which plane in the object is reconstructed given that the detected intensity liesat inshy nity and contains contributions from scattering at every point along the axisin the object space A simulation to clarify this is shown in shy gure 9 The simulatedobject consists of two planes separated by 1 mm in the z-direction Plane 1 has a holecontaining a complex object (the number 1) while plane 2 has two holes containingcomplex objects (numbers 2 and 3) The object exit wave calculated on plane 2

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 887

(a) (b)

Figure 8 (a) Modulus and (b) phase of reconstructed image wave The modulus shows only edgecontrast whereas the phase shows the cells in large-area phase contrast The phase shift due tothe cells is about ordm No modulus or sign constraint

(shy gure 9a) shows plane 1 out of focus The modulus of the Fourier transform (Fouriermodulus) of this image wave is used as input for the HIO algorithm In the inversiondinoterent shapes of the supports used at each plane shown inset can distinguish thetwo object planes If a plane is assumed unknown a larger support hole is usedthat contains the physical support Depending on which support plane is assumed asknown (inset) in the iteration (plane 1 support known and plane 2 support unknownor vice versa) the other plane is always reconstructed out of focus by +1 or iexcl 1 mm(shy gure 9c d) That means that in the case of known support on plane 1 the algorithmreconstructs the wave as seen through an imaging lens with out-of-focus images ofthe holes on plane 2 As expected the depth of focus for interactive inversion is givenby the expression for an ideal lens which imposes a dinotraction limit equal to that ofthe highest spatial frequency recorded in the dinotraction pattern

To verify this experimentally a pin was mounted 12 mm upstream of a pair of300 m m apertures The coherent optical image is shown in shy gure 10a A Fresnel fringesurrounds the pin since the lens is focused on the apertures The image was recoveredfrom the recorded dinotraction pattern intensity (shy gure 10b) Again three dinotractionpatterns with dinoterent exposure times had to be merged to avoid pixel saturationand blooming The reconstructed image modulus is shown in shy gure 11a where the

Phil Trans R Soc Lond A (2002)

888 J C H Spence U Weierstall and M Howells

illuminationdirection

plane 1

1 mm

(a) (b)

(c) (d)

Figure 9 (a) Schematic view of a simulated object consisting of two planes separated by 1 mm inthe direction of light propagation Plane 1 has a hole containing a complex object (number 1) andplane 2 has two holes containing complex objects (numbers 2 and 3) (b) Modulus of complexexit wave on plane 2 (c) Iterative reconstruction with plane 2 support assumed known (thesupport shape used is shown in the top right-hand corner) Plane 1 is out of focus by 1 mm(d) Iterative reconstruction with plane 1 support assumed known (the support shape is shownin the top right-hand corner) Plane 2 is out of focus by iexcl1 mm

pin appears out of focus (Fresnel fringe barely visible) because the aperture shapewas imposed as support The pin can be brought into focus by using the appropriateFresnel propagator with centz = 12 mm and para = 6328 nm taking the apertures outof focus (see shy gure 11b) Thus as expected the reconstructed plane is always thatplane in which the support condition is imposed where the algorithm shy nds a uniquesolution subject to the given boundary condition Propagation to other planes maybe performed subsequently since the complex waveshy eld is obtained subject to thedepth-of-focus limitation The relative sideways shift of the pin inside the aperturebetween shy gure 10a and shy gure 11a is due to a slight unintentional tilt of the object(pin plus pinhole) between recording the image and the dinotraction pattern

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 889

(a) (b)

Figure 10 (a) Coherent laser optical image of pin in front of aperture (b) Laser optical direg ractionpattern obtained by merging three patterns with direg erent exposure times to avoid saturation ofthe central beam

(a) (b)

Figure 11 (a) Reconstructed modulus Only the hole with the out-of-focus pin is shown(b) Numerical back propagation of reconstructed complex wavemacreld by 12 mm to plane of pinPin appears in focus Support constraint only

6 Experiments with coherent electrons

Using a special support mask made by electron-beam lithography experiments havealso been undertaken using the HIO algorithm to retrieve the phases from experimen-tal transmission electron dinotraction patterns (Weierstall et al 2002) Images mightthen be reconstructed that are free of electron lens aberrations The support maskcontains hole pairs of various sizes with pairs separated by 1 m m The separationbetween holes of a pair is ca 50 nm less than the coherence width of the illuminationbeam The holes irregular in shape form the object we wish to reconstruct in futurework we plan to introduce nanoscale objects into the holes Figure 12a below showsthe experimental dinotraction pattern from two holes shown in shy gure 12c obtained

Phil Trans R Soc Lond A (2002)

890 J C H Spence U Weierstall and M Howells

20 nm

(a) (b)

(c)

Figure 12 (a) Experimental electron direg raction pattern from two nanoscale holes in an opaquemask (shown in (c)) recorded using a macreld-emission electron microscope (CM200 FEG at 40 kV1k pound1k CCD with 24 m m pixels camera length 611 m) Youngrsquo s fringes from the two holes arevisible these fringes encode phase information (b) Reconstruction of image of holes from thedireg raction pattern (a) after 100 iterations of the HIO algorithm Modulus constraint applied inimage space with large triangular computational support The hole shape is comparable with theTEM image in (c) (c) Conventional TEM image of holes that created the direg raction pattern (a)

with a Philips CM 200 shy eld-emission electron microscope at 40 kV The acceleratingvoltage was lowered to 40 kV to make the multi-layered support opaque ModulatedYoungrsquos fringes are clearly seen

The resulting image after 100 HIO iterations is shown in shy gure 12b The holesreconstructed from the dinotraction pattern modulus look similar to the TEM imageof the holes in shy gure 12c but the resolution of the reconstruction is lower thanthe normal TEM image resolution High spatial frequency information from thehole edge is missing in the experimental dinotraction pattern due perhaps to drift ofthe imaging lenses during the exposure time

Table 1 shows a summary chart of all the conditions so far examined with HIO sim-

Phil Trans R Soc Lond A (2002)

Phase

recovery

an

dlen

slessim

agin

g891

Table 1 Summary chart showing direg erent conditions for phase recovery with HIO algorithm

( CArsquo denotes the complex algorithm (uses direg racted intensity and support constraint in real space) RIPrsquo denotes the complex algorithm withpositivity constraint of Re ordf Im ordf (uses direg racted intensity and support constraint in real space negative Re ordf and Im ordf inside the supportarea are set positive) Mrsquo denotes the modulus constraint on ordf in real space (uses direg racted intensity and support constraint in real spacecomplex numbers inside the support area are replaced by their modulus to enforce positivity) Prsquo denotes the phase constraint on ordf in realspace (uses direg racted intensity and support constraint in real space the phase is extracted and retained from complex numbers inside thesupport area modulus is extracted and set to 1 inside the support area))

general pure strong pure weakrsquo real weak-phasecomplex object phase object phase object (amplitude) object withordf = A exp(iiquest ) ordf = exp(i iquest (r)) iquest lt ordm =2 object absorption

known two-holephysical support

CA (laser experimentpin cheek cells)

CA P (laserexperiment mica)

RIP CA M RIP CA RIP CA

known one-holephysical support

CA only for certainsupport shapes(eg triangle withsharp edges)

CA slowconvergence Pworks better

RIP CA M RIP CA (laserexperimentnumbers)

RIP CA

unknown two-or-more-hole physicalsupport loose(eg triangle compu-tational support)

| | RIP M ( TEM experimentholes) (laserexperiment numbers)(planned X-ray micapinholes)

RIP (Miao X-rayexperimenttransparent support)

unknown one-holephysical support(eg triangle compu-tational support)

| | RIP only for holewith low symmetryand sharp edgesslow convergence

M RIP only for holewith low symmetryand sharp edgesslow convergence

RIP only for holewith low symmetryand sharp edgesslow convergence

Phil

Tra

ns

R

Soc

L

on

d

A(2

002)

892 J C H Spence U Weierstall and M Howells

ulations and experiments The enotectiveness of dinoterent constraints in real space fordinoterent physical and computational supports is compared The physical supportis the support which exists in the experiment and which may or may not be accu-rately known The computational support is the support used in the HIO iterationsThe table shows experiments that have been performed in italics The shy rst columnshows dinoterent possibilities for the physical support the most favourable at the topthe worst at the bottom The computational support and known physical supportare identical The computational support used with the unknown physical supportis larger than the physical support Reconstructions with one-hole physical supportand loose computational support are only possible with weak or real objects andsupport holes with sharp edges and low symmetry and convergence are very slow insimulations A known two-or-more-hole support proved to be most favourable in allcases

7 Discussion and conclusions direg ractive imaging with other particles

In this paper we have reviewed the implications of Thomas Youngrsquos experiment formatter waves and in particular for electron interferometry By taking advantage ofthe very high quality of the coherent electron beam available in the modern shy eld-emission electron microscope a number of important results (such as the AharonovBohm enotect and the electron Sagnac enotect) have been established and the shy eldof atomic-resolution transmission electron microscopy phase-contrast imaging andelectron holography opened up All these methods continue to produce a large sci-entishy c pay-onot in microstructural materials characterization and discovery and inbiology A new lensless imaging method is then reviewed which may be understoodin part by analogy with electron holography This HIO algorithm uses an iterativeapproach to assign phases to dinotraction pattern intensities To evaluate its perfor-mance we give experimental examples of image reconstruction with both coherentelectrons and light together with simulations for the soft X-ray case Thus we haveshown experimentally that the HIO algorithm is capable of solving the phase problemfor scattering from non-periodic objects and of making quantitative measurements ofphase (modulo 2ordm ) in good agreement with independent measurements The beam-splitting arrangement of an interferometer is not required and dinotraction-limitedlensless imaging seems possible as a result This may open the way for imaging withthe many particles for which no lenses exist such as neutrons ballistic electrons insolids at low temperature or neutral atoms and ions either as free coherent beamsor condensates (Bright coherent sources have recently appeared for many of these(Pauly 2000)) The dinotering interactions of these particles may provide useful newprobes for chemistry materials science and biology For example the reduced radia-tion damage enotects of imaging organic material with low-energy electrons neutronsand helium atoms have recently been compared (Spence et al 1999) and HIO sim-ulations have been completed for neutron imaging of a diatom (Spence et al 2001)The very small refractive index of organic matter for neutrons limits resolution andcontrast for thin samples For low-energy ions or metastable helium strong samplebeam interactions must be considered Damage from neutral helium-atom imagingof molecules should be negligible however penetration is also negligible (producinga proshy le image) Detectors for coherent atom beams still present serious problemssince area detectors are inemacr cient For biology the weak atom interaction is desirable

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 893

to avoid damage however a strong interaction at the detector is also needed so thationization before detection must be arranged

For real objects we have seen that the resolution in images recovered by the HIOalgorithm is largely independent of the accuracy with which the support is knownThe addition of noise anotects the probability that the unique solution will be found (asmeasured by the error metric) rather than the resolution which is near dinotraction-limited For complex objects our experience indicates that a much more accurateknowledge of the physical support boundary is needed approaching the scale ofthe resolution sought The tight support and two-hole geometry (see Fienup (1987)for other geometries) needed for complex objects (complex refractive index) willbecome increasingly accessible with the development of new lithographic techniquesfor support construction This can be combined with independent characterization ofthe support shape and the scaling information provided by the known autocorrelationof the support Experimentally favourable cases for imaging with nanoscale resolutionat present require loose support which table 1 shows greatly favours real objectssuch as the charge density which dinotracts hard X-rays at energies well away fromabsorption edges or very thin samples for high-energy electrons Unfortunately beamcoherence is more readily obtained for soft X-rays

Using modern lithographic techniques it is becoming increasingly possible toestablish the experimental conditions of known support required by the HIO algo-rithm both for free particle beams and for electrons in solids at low temperaturesThus for example we have seen the algorithm used to determine the phase of wavefunctions for electrons in a one-dimensional YBCO Josephson junction using TEMimages of the structure to deshy ne the required support (boundary conditions) (Car-mody et al 2000) Given the fundamental role that phase has played historically inphysics and which Thomas Young was amongst the shy rst to appreciate any techniquecapable of exposing phase can be expected to yield many surprises in physics in thefuture

Supported by ARO award DAAD190010500 We thank Dr Q Chen for the use of macrgure 12 (fromWeierstall et al 2002)

References

Aharonov Y amp Bohm D 1959 Signimacrcance of the electromagnetic potentials in the quantumtheory Phys Rev 115 485

Allen L J Faulkner H M L Nugent K A Oxley M P amp Paganin D 2001 Phase retrievalfrom images in the presence of macrrst-order vortices Phys Rev E 63 037602

Bauschke H H et al 2002 (In the press)

Boersch H 1943 Fresnelsche Beugung im Elektronenmikroskop Phys Z 44 202211

Bruck Y amp Sodin L 1979 On the ambiguity of the image-reconstruction problem Opt Com-mun 30 304308

Carmody M Moeckly B H Merkle K amp Marks L 2000 Spatial variation of the current ingrain boundary Josephson junctions J Appl Phys 87 24542459

Coene W Janssen G Op de Beek M amp van Dyck D 1992 Phase retrieval through focusvariation for ultra-resolution in macreld emission transmission electron microscopy Phys RevLett 69 37433746

Crewe A V Eggenberger D N Wall J amp Welter L M 1968 Electron gun using a macreldemission source Rev Scient Instrum 39 576

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 877

resolution imaging the probe-formation process in scanning electron microscopyand electron crystallography For example the principle of coherent detection orig-inally suggested by Schiske in 1968 has since been extensively applied to allowatomic-resolution images to be synthesized from a through-focal series of electronmicrographs (for a historical review see Saxton (1994)) In this way by emacr cientlydeconvolving the instrument response controlling noise and compensating for lossof information due to partial coherence resolution has been extended beyond theconventional Scherzer point resolution limit out to the information limit set by elec-tronic instabilities (Coene et al 1992) This limit currently stands at slightly below1 A (OrsquoKeefe et al 2001) which will soon open up new possibilities in materialsscience for understanding atomic processes in matter especially at interfaces and inglasses and for the discovery of useful new nanostructures Atomic-resolution TEMprovided the shy rst images of the fullerene `buckyballsrsquo (Iijima 1980) and the discoveryof carbon nanotubes (Iijima 1991) For biology although resolution may be limitedby other factors such as variations in the conformation of molecules radiation dam-age and the bending of monomolecular layers the impact of this approach has beenconsiderable Using two-dimensional crystals it has revealed the detailed mechanismof photosynthesis in cell membranes (Henderson et al 1990 see also Subramaniamet al 2002) For macromolecules such as the ribosome responsible for protein pro-duction (`life itself rsquo) the cryomicroscopy method was the shy rst to reveal the shape(at 1 nm resolution) of large molecules which cannot be crystallized (Frank 1996)Despite recent limited success in crystallizing membrane proteins for X-ray workelectron cryomicroscopy remains vital for assisting in phasing X-ray data conshy rmingthe heavy-atom positions assumed in X-ray multi-wavelength anomalous dinotractionanalysis and in showing how smaller proteins of known structure shy t together andinteract

2 Electron interferometry and holography

More directly related to Youngrsquos work is modern electron interferometry and holog-raphy Following the 1927 discovery of electron dinotraction (Davisson amp Germer 1927Thomson amp Reid 1927) in conshy rmation of de Brogliersquos hypothesis observations ofhelium-atom dinotraction by crystals soon after provided further conshy rmation of thematterwave hypothesis (Esterman amp Stern 1930) The observation of Fresnel fringesin an electron microscope (EM) by Boersch (1943) provided further evidence of wave-like properties|and the realization that these patterns encoded phase informationwhich might be reconstructed|led to the birth of holography (Gabor 1949) (SinceGabor was unable to solve the twin-image problem presented by these in-line holo-grams Boersch with some justishy cation remained convinced that he had invented`holographyrsquo and viewed the term as an unnecessary acronym for an interferencepattern (T Mulvey 1996 personal communication)) Early partly successful exper-iments (Haine amp Mulvey 1952) used the `transmission methodrsquo based on the real-ization (currently being rediscovered by the electron-microscopy community) thatan out-of-focus image is an in-line hologram Gaborrsquos original aim was to improvethe resolution of EM images using holography to expose the phase of the wavefrontaberrations and so to deblur the images Reconstructions were obtained in 1951using a monochromatized arc lamp at 1 nm resolution (before the invention of laseror useful computers) However EM images were limited in resolution by mechanical

Phil Trans R Soc Lond A (2002)

878 J C H Spence U Weierstall and M Howells

and electronic instabilities rather than aberrations until the early 1960s Only in1995 was `Gaborrsquos dreamrsquo fully realized when onot-axis EM holograms were shy rst usedto extend resolution beyond the limit of 0198 nm set by aberrations toward the`super-resolutionrsquo limit set by electronic instabilities of 01 nm (Orchowski et al 1995)

The earliest beam-splitters for electrons used either Bragg dinotraction in a thincrystal in the transmission geometry (Marton 1952) or an electrostatic biprism(Moellenstedt amp Duker 1955) which has become the favoured choice today Thesubsequent development of the shy eld-emission electron source (Crewe et al 1968) atthe end of the 1960s was of the greatest importance for the shy eld since it provideda source of nanometre dimensions and sub-eV energy spread with a brightness perunit bandwidth greater than current-generation synchrotrons (Qian et al 1993)By combining Moellenstedtrsquos biprism with Crewersquos shy eld emission gun modern EMinstruments using beam energies of a few hundred keV are capable of recordingin a few seconds images showing hundreds of Fresnel fringes with a lateral coher-ence width of perhaps 01 m m and a longitudinal coherence length of several hundredangstroms They thus provide for interferometry a beam of extremely high qualityby the standards of accelerator physics

The refractive index for electrons in a magnetic shy eld was shy rst derived using Fer-matrsquos principle in 1948 (Ehrenberg amp Siday 1948) The AharonovBohm (AB) enotect(Aharonov amp Bohm 1959) subsequently demonstrated observable enotects along opti-cal paths in which shy elds (but not potentials) are zero suggesting that potentialsare more fundamental than shy elds (This idea had apparently already been suggestedby Maxwell) More recent electron-interferometry experiments include for examplethe observation of regux quantization using the AB enotect (Tonomura 1988) the scalar(electrostatic) AB enotect with electrons (Matteucci et al 1992) and the electronSagnac interferometer (Hasselbach amp Nicklaus 1993) From this has grown the shy eldof electron holography now widely used to map out on a nanometre scale internalmagnetic and electrostatic shy elds in materials from digital recording media to ferro-electrics semiconductors and superconducting vortices (for reviews see Voelkl 1998Lichte 1988 Tonomura 1993)

A dinoterent class of electron-interference experiments depends on the chromaticcoherence of electron sources and their second-order coherence properties For thefree particle beams used for electron interferometry a degeneracy parameter macr maybe deshy ned to indicate the degree of crowding in phase space a unity value indicatestwo electrons of opposite spin per cell with cell dimensions deshy ned by the uncertaintyprinciple (Components of centK are deshy ned by energy spread and beam divergencecomponents of centr deshy ne the coherence patch and longitudinal coherence length) Itis readily shown (Spence et al 1994) that macr is equal to the ratio of actual sourcebrightness to Langmuirrsquos theoretical maximum value and that macr may be measuredfrom the dimensions of the coherence patch and the source energy spread For a coldtungsten nano-tip shy eld emitter we measure macr = 10iexcl4 (Spence et al 1994) whereasa value of about unity is found for undulators at synchrotrons (For an optical laserin which any number of indistinguishable bosons may occupy a state macr = 1015)Coulomb repulsion between electrons limits the value of macr Nevertheless the bright-ness of electron shy eld emitters exceeds that of current synchrotron undulators (Qianet al 1993) The enotects of photon bunching seen in the Hanbury Brown and Twissexperiment (Loudon 1976) and the suggestion that electrons should `antibunchrsquo for

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 879

large values of macr were made at an early stage by Purcell (1956) In a quantum pic-ture we imagine wave packets overlapping coherently along the propagation axisby chance giving non-classical (non-random) results for the probability of detectingparticles between wave packets For this to occur however the temporal coherenceof the source must exceed the time between particle emissions so that the particlesare in the same quantum state This requires the source degeneracy macr for electronsto approach unity It is therefore not surprising that despite considerable enotortin our own laboratory and elsewhere antibunching has yet to be observed in freecoherent electron beams In condensed matter however these conditions are readilyachieved and fermion antibunching has recently been observed for the shy rst time ina two-dimensional electron gas in the quantum Hall regime (Henny et al 1999) Areview of other novel enotects expected in multiparticle fermion interferometry can befound in Silverman (1993)

3 Phase in imaging iterative methods for phase measurement

An image is a two-dimensional near-shy eld waveshy eld which may be real or complexIt may be obtained by Fourier transform from far-shy eld scattered amplitudes if thephases of these are known Holography encodes this phase information emacr cientlythrough the use of a reference wave and Fourier transform (FT) holography (whichuses a point reference in the same plane and to one side of the object) is par-ticularly enotective in separating the real and virtual twin images after the recon-struction process This involves only illumination of the hologram with the originalreference wave Phase information is encoded in the displacements of the `Youngrsquosfringesrsquo which result in this geometry (The method has been extended to allow a`complicated or extended referencersquo (Szoke 1997) such as a known fragment of amolecule and thus may be considered as an extension of the `heavy-atomrsquo methodof crystallography) Since this experimental inversion (reconstruction) of FT holo-grams to images produces a unique result it should not surprise us that an iterativesearch algorithm the hybrid inputoutput (HIO) algorithm (Gerchberg amp Saxton1972 Fienup 1982) applied to the same geometry should perform well Remarkablyhowever it has been found that the algorithm also converges for complex objectsconsisting of two separated parts neither of which is known but which generate anappreciable fringe modulation in the scattering pattern A knowledge of the `supportrsquoof the object (and of the sign of the scattering density) is found to be sumacr cient forinversion the support is the area within which the object transmission function T (r)is non-zero (or unknown) and so may take the form for example of the Youngrsquosfringe pinhole mask Dinoterent `unknownrsquo objects shy ll each hole The inversion givenshy nite known support has been shown to be `almost alwaysrsquo unique (Bruck amp Sodin1979) An error metric which depends only on the known values of the supportand Fourier modulus monitors progress toward this unique solution In summarywe have recently come to understand that this experimental geometry of ThomasYoung now onoters the possibility of lensless imaging using any of the radiations forwhich lenses do not exist such as coherent atom beams X-rays and neutrons Thecomplementary arrangement|two compact complex objects lying on a transpar-ent membrane|also works For a dinoterent non-interferometric approach to phasemeasurement akin to the through-focus method mentioned in the introduction andbased on the `transport of intensityrsquo equations see Paganin amp Nugent (1998) This

Phil Trans R Soc Lond A (2002)

880 J C H Spence U Weierstall and M Howells

method which requires intensity measurements across two planes downstream ofthe object has been used for example to consider a non-interferometric variant ofthe AB enotect (Paganin 2001) (For a phase object a single plane is sumacr cient sinceunit intensity can be assumed everywhere in the object plane) The phase vorticeswhich occur at zeros of intensity have also attracted considerable interest (Allen etal 2001) So far it has been applied to cases using collimated illumination only sothat resolution is limited by detector pixel size however there seems no reason notto apply it to the case of diverging illumination which provides magnishy cation anda resolution limit imposed by the source size

Despite many simulations few experimental tests of the HIO algorithm haveappeared The algorithm iterates between real and reciprocal space (related by aFourier transform) imposing known information in each domain The moduli ofthe scattered amplitudes I(u) are assumed known and in real space the objectfunction T (r) is set to its known values outside the support (Here r and u aretwo-dimensional vectors T (r) is the ratio of the complex exit waveshy eld (the in-focusimage) to the incident wave function often assumed planar) The support need notalways be known very accurately and an estimate can be obtained from the 50contour of the autocorrelation function which is given by the Fourier transform ofI(u) Finally knowledge of the sign of the real (and possibly imaginary) parts of Tmay be used The introduction of a small amount of `feedbackrsquo (Fienup 1987) distin-guishes the HIO algorithm from the error-reduction algorithm (Gerchberg amp Saxton1972) Convergence is found to be improved further in three-dimensions (Miao et al 2001) For real objects (such as the charge density which dinotracts X-rays) a singleobject is sumacr cient and the computational boundary used in the iterations may bea dinoterent shape from the physical support which it includes When viewed as anoptimization process the algorithm appears to be capable of shy nding the thousands ofunknown parameters (the phases in a typical image) in the presence of noise withoutthe usual problems of local minima and stagnation It thus appears to solve a verylarge global optimization problem despite the application of a non-convex constraint(the known Fourier moduli) (A convex set of solutions is one for which all pointslying on any line joining any two points within the N -dimensional set lie inside theset a kidney-shaped set in two dimensions is non-convex N is the number of pixelsin the image If a unique solution can be shown to exist (as in this case) iterationbetween the boundaries of two convex sets guarantees convergence to this solutionwithout stagnation at local minima Hence convex constraints such as symmetryknown support or sign are sought in optimization problems) While inversion fromknown phase information is a convex process leading to a unique result (Lindaas etal 1998) inversion from intensity alone is non-convex so that local minima mayformally be expected in approaches based on numerical optimization (Stark 1987)The HIO algorithm has been analysed in considerable detail (Fienup 1982 Paganin2001) including treatments based on Bregman projections onto convex sets (Stark1987 Bauschke et al 2002) and those going back to Sayre (1952) which start fromthe realization that Bragg sampling undersamples the autocorrelation function ofa molecule in a crystal (Miao et al 2001) The number of Fourier series equationsis then compared with the number of unknowns as the support is made a progres-sively smaller fraction of the total dinotracting area The lowest spatial frequency inthe dinotraction pattern corresponds to a distance wider than the support Using this`oversamplingrsquo approach soft X-ray images of lithographed symbols were recently

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 881

reconstructed from far-shy eld scattered intensities in a lensless scattering experiment(Miao et al 1999) Spatial resolution is limited only by noise and wavelength in theseexperiments For technical reasons however the reconstruction was assisted by low-resolution optical images which provided the low-order spatial frequencies and theobject transmission function was of a special mask-like form having only two (com-plex) transmission values whose signs were known We have therefore conductedboth simulations and experiments for phase measurement in scattering of X-rayscoherent visible light and coherent electron beams to evaluate this HIO algorithmfor more general objects

4 Simulations for X-rays

Simulations have been undertaken for objects consisting of an array of randomlypositioned pinholes to determine if this image could be reconstructed from its softX-ray dinotraction pattern If so these holes could later be shy lled with structures ofinterest for reconstruction The area occupied by pinholes has been limited to thecoherence width of the beam by a 1 m m aperture This aperture will not fall exactly inthe plane of the pinholes therefore the wavefront at the pinholes will have undergoneFresnel dinotraction at the aperture and so will be complex Figure 1 shows the resultsof these simulations assuming a wavelength of 25 nm for the incident soft X-raybeam The pinholes are 10 nm in diameter After 150 iterations the HIO algorithmhas converged without stagnation to an image which is related to the object byinversion through the centre The modulus of the estimated object function was takenat each iteration (modulus constraint) since the object is assumed real Only a roughestimate of the size of the aperture is needed the computational support conditionapplied in the iterations is a triangle larger than the aperture The unimportantinversion ambiguity cannot be resolved by the HIO algorithm since an invertedimage our real computational support and the circular aperture all have the sameFourier modulus The Fresnel propagation makes this strictly a complex objecthowever we obtain an accurate reconstruction of the holes in the shy lm despite the useof a positivity (modulus) constraint in object space The phase of the complex wavetransmitted through the pinholes is not reconstructed due to the imposed positivity

5 Experiments with light

Coherent laser optical phase retrieval experiments have been performed using theHIO algorithm for several objects The phase was successfully retrieved for severaldinoterent support constraints and object types So far only opaque support con-straints (apertures) have been used since these present fewer problems with exces-sive central beam intensity (blooming) The source was a frequency-stabilized HeNe laser ( para = 6328 nm) collimated to provide the planar wavefronts needed at thesample for real objects (shy gure 2) Dinotracted light was focused with a lens onto aliquid-nitrogen-cooled charge-coupled device (CCD) camera and the digital imagereconstructed using the HIO algorithm in the Digital Micrograph script language

The shy rst object was a resolution test object with transparent numbers and linesin an opaque chrome shy lm on glass The illuminated area was limited by a thin300 m m diameter aperture in contact with the chrome shy lm Assuming the glass isregat to within less than para =10 across the illuminated area the chrome mask acts as a

Phil Trans R Soc Lond A (2002)

882 J C H Spence U Weierstall and M Howells

(a) (b)

(d )(c)

Figure 1 Reconstruction of simulated pinholes illuminated by a Fresnel direg racted X-ray wave-macreld (out-of-focus physical support) (a) Simulated random array of pinholes illuminated bywave amplitude which has been Fresnel propagated 200 nm from an aperture to the pinholesWavelength para = 25 nm pinhole size 10 nm aperture diameter 1 m m (b) 200 nm out-of-focuswavemacreld from 1 m m aperture incident on pinholes (c) The Fourier transform of (a) is a speckledAiryrsquo s disc Modulus used as input for the HIO algorithm (d) Reconstructed pinholes after 150iterations with modulus constraint (To enforce positivity in real space only the modulus ofthe pixel values inside the support area is retained during each iteration) A large triangularcomputational support was used The image is inverted this is an ambiguity which cannot beresolved by the HIO algorithm

real binary object so that the dinotraction pattern should be symmetric A coherentoptical image was also recorded with the camera moved into the image plane ofthe lens This is shown in shy gure 3a The camera was then moved to the back focalplane of the lens and the dinotraction pattern intensity I(u) was recorded (shy gure 3b)The autocorrelation FT I(u v) was used to determine the approximate size of theobject and a circular support half the size of the autocorrelation was used in the

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 883

f

double polarizer to adjust intensity

collimator

lens diffraction pattern

CCD detectorHendashNe laser= 6328 nml

object with support mask

Figure 2 Schematic of the arrangement used for optical experiments

(a) (b)300 m m

Figure 3 (a) Coherent optical image of optical target transparent letters and bars in an oth-erwise opaque chrome macrlm on glass A 300 m m aperture acts as the support (HeNe laserpara = 6328 nm) (b) Coherent laser optical direg raction pattern of object shown in (a)

computer The HIO algorithm was used to recover the phases of the object exitwave and the algorithm always converged after about 50 iterations with dinoterentinitial random phase sets (shy gure 4a) A larger triangular support constraint wasalso used to recover the image as shown in shy gure 4b The normalized root-mean-squared error in the object domain (Fienup 1987) (a measure of dinoterence betweenthe current image estimate and the known pixel values outside the support) did notdrop below 02 probably due to imperfections in the dinotractogram eg noise andlight reregections from the entrance glass surface of the CCD camera which createasymmetry in the dinotraction pattern (see shy gure 4b above centre) Despite theseproblems the reconstructed image looks almost identical to the coherent image inshy gure 3a

A strong phase object (introducing phase shifts greater than ordm =2) was made usinga support consisting of two separated areas (two apertures of 300 m m diameter sep-

Phil Trans R Soc Lond A (2002)

884 J C H Spence U Weierstall and M Howells

(a) (b)

Figure 4 (a) Modulus of reconstructed image wave using a tight circular support constraint(b) Modulus of reconstructed image wave using a loose triangular support constraint A positivityconstraint was used in real space

mica

no mica(a) (b)

Figure 5 (a) Coherent laser optical image of phase object with two-hole support The edgesof two mica plates are visible across the hole on the right (b) Laser optical direg raction patternobtained by merging three patterns with direg erent exposure times to avoid saturation of thecentral beam

arated by 350 m m) One hole was covered with two 5 m m thick mica sheets whichoverlap in one region The expected phase shift is calculated to be 9 ordm The coherentoptical image is shown in shy gure 5a The mica is transparent except for the edgeswhere Fresnel contrast enotects are visible

The resulting dinotraction pattern is shown in shy gure 5b The central beam intensityin the dinotraction pattern from this transparent object was too strong to be recordedsimultaneously with the weaker high spatial frequencies so that several exposureswere needed All exposures were merged to one pattern in which saturated pixelswere replaced by scaled pixels from the lower exposure patterns The computational

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 885

black = -

white = +

(a) (b)

(c)

Figure 6 (a) Modulus and (b) phase of reconstructed image wave (c) phase reconstructed fromshifted direg raction pattern shows equiphase contour lines Phase shift by mica ordm No sign ormodulus constraint

support size and orientation were derived from the autocorrelation The complexHIO-phase retrieval algorithm was used to recover the image wave phase and modulusshown in shy gure 6 The phase shift by the mica plates in the reconstruction is ordm which since the algorithm reconstructs phase modulo 2 ordm is in good agreement withour independent measurement of 9 ordm based on the measured thickness and knownrefractive index of mica Thus the HIO algorithm is capable of quantitative phasemeasurement

Experiments with a transparent physical support have been attempted using alimiting aperture upstream of the sample but have not been successful so far becauseof the high intensity in the centre of the dinotraction pattern which causes bloomingin the CCD chip This problem persists for three-dimensional objects however athree-quadrant mask in the object plane has been proposed to address this dimacr culty(Sayre et al 1998) (Beam stops cannot be placed near enough to the CCD toavoid generating additional fringes) Experiments using the known intensity envelopeof a focused beam as the support have also not been successful for such complexobjects there appears to be a need for sharp physical support edges perhaps requiredto generate reference edge waves (For real objects there is no such requirement)However a coherent probe spanning two `knownrsquo holes works well in simulation

Previous experiments have been based on objects of simple type such as binary

Phil Trans R Soc Lond A (2002)

886 J C H Spence U Weierstall and M Howells

(a)

(b)

Figure 7 (a) Coherent laser optical image of cheek cells in two-hole support (300 m m apertures)About seven cells clump together in the upper hole Only Fresnel edge contrast is visible dueto defocus the cell nuclei are visible as black dots (b) Laser optical direg raction pattern mergedfrom four direg erent exposures to avoid saturation of the central beam

objects which contain much less information than greyscale objects Experimentswith human cheek cells which serve as more complicated greyscale phase objectshave therefore been performed The cheek cells were positioned on a glass slide insidetwo 300 m m apertures separated by 300 m m The apertures act as the opaque supportconsisting of two separated parts as needed for reconstruction of complex objectsFigure 7a shows the coherent optical image of the slightly defocused cheek cells asformed with a lens Only Fresnel edge contrast is visible while the nuclei producesome absorption Figure 7b shows the optical dinotraction pattern merged from fourdinoterent exposures The size and orientation of the support were estimated fromthe autocorrelation The complex reconstruction of phase and amplitude after about50 HIO iterations using the support constraint is shown in shy gure 8 The cells arevisible with large-area phase contrast in the phase image and the phase shift of thecells is about ordm In the reconstructed amplitude only edge contrast is visible as inthe optical image

For a three-dimensional object extended along the optical axis the question arisesas to which plane in the object is reconstructed given that the detected intensity liesat inshy nity and contains contributions from scattering at every point along the axisin the object space A simulation to clarify this is shown in shy gure 9 The simulatedobject consists of two planes separated by 1 mm in the z-direction Plane 1 has a holecontaining a complex object (the number 1) while plane 2 has two holes containingcomplex objects (numbers 2 and 3) The object exit wave calculated on plane 2

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 887

(a) (b)

Figure 8 (a) Modulus and (b) phase of reconstructed image wave The modulus shows only edgecontrast whereas the phase shows the cells in large-area phase contrast The phase shift due tothe cells is about ordm No modulus or sign constraint

(shy gure 9a) shows plane 1 out of focus The modulus of the Fourier transform (Fouriermodulus) of this image wave is used as input for the HIO algorithm In the inversiondinoterent shapes of the supports used at each plane shown inset can distinguish thetwo object planes If a plane is assumed unknown a larger support hole is usedthat contains the physical support Depending on which support plane is assumed asknown (inset) in the iteration (plane 1 support known and plane 2 support unknownor vice versa) the other plane is always reconstructed out of focus by +1 or iexcl 1 mm(shy gure 9c d) That means that in the case of known support on plane 1 the algorithmreconstructs the wave as seen through an imaging lens with out-of-focus images ofthe holes on plane 2 As expected the depth of focus for interactive inversion is givenby the expression for an ideal lens which imposes a dinotraction limit equal to that ofthe highest spatial frequency recorded in the dinotraction pattern

To verify this experimentally a pin was mounted 12 mm upstream of a pair of300 m m apertures The coherent optical image is shown in shy gure 10a A Fresnel fringesurrounds the pin since the lens is focused on the apertures The image was recoveredfrom the recorded dinotraction pattern intensity (shy gure 10b) Again three dinotractionpatterns with dinoterent exposure times had to be merged to avoid pixel saturationand blooming The reconstructed image modulus is shown in shy gure 11a where the

Phil Trans R Soc Lond A (2002)

888 J C H Spence U Weierstall and M Howells

illuminationdirection

plane 1

1 mm

(a) (b)

(c) (d)

Figure 9 (a) Schematic view of a simulated object consisting of two planes separated by 1 mm inthe direction of light propagation Plane 1 has a hole containing a complex object (number 1) andplane 2 has two holes containing complex objects (numbers 2 and 3) (b) Modulus of complexexit wave on plane 2 (c) Iterative reconstruction with plane 2 support assumed known (thesupport shape used is shown in the top right-hand corner) Plane 1 is out of focus by 1 mm(d) Iterative reconstruction with plane 1 support assumed known (the support shape is shownin the top right-hand corner) Plane 2 is out of focus by iexcl1 mm

pin appears out of focus (Fresnel fringe barely visible) because the aperture shapewas imposed as support The pin can be brought into focus by using the appropriateFresnel propagator with centz = 12 mm and para = 6328 nm taking the apertures outof focus (see shy gure 11b) Thus as expected the reconstructed plane is always thatplane in which the support condition is imposed where the algorithm shy nds a uniquesolution subject to the given boundary condition Propagation to other planes maybe performed subsequently since the complex waveshy eld is obtained subject to thedepth-of-focus limitation The relative sideways shift of the pin inside the aperturebetween shy gure 10a and shy gure 11a is due to a slight unintentional tilt of the object(pin plus pinhole) between recording the image and the dinotraction pattern

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 889

(a) (b)

Figure 10 (a) Coherent laser optical image of pin in front of aperture (b) Laser optical direg ractionpattern obtained by merging three patterns with direg erent exposure times to avoid saturation ofthe central beam

(a) (b)

Figure 11 (a) Reconstructed modulus Only the hole with the out-of-focus pin is shown(b) Numerical back propagation of reconstructed complex wavemacreld by 12 mm to plane of pinPin appears in focus Support constraint only

6 Experiments with coherent electrons

Using a special support mask made by electron-beam lithography experiments havealso been undertaken using the HIO algorithm to retrieve the phases from experimen-tal transmission electron dinotraction patterns (Weierstall et al 2002) Images mightthen be reconstructed that are free of electron lens aberrations The support maskcontains hole pairs of various sizes with pairs separated by 1 m m The separationbetween holes of a pair is ca 50 nm less than the coherence width of the illuminationbeam The holes irregular in shape form the object we wish to reconstruct in futurework we plan to introduce nanoscale objects into the holes Figure 12a below showsthe experimental dinotraction pattern from two holes shown in shy gure 12c obtained

Phil Trans R Soc Lond A (2002)

890 J C H Spence U Weierstall and M Howells

20 nm

(a) (b)

(c)

Figure 12 (a) Experimental electron direg raction pattern from two nanoscale holes in an opaquemask (shown in (c)) recorded using a macreld-emission electron microscope (CM200 FEG at 40 kV1k pound1k CCD with 24 m m pixels camera length 611 m) Youngrsquo s fringes from the two holes arevisible these fringes encode phase information (b) Reconstruction of image of holes from thedireg raction pattern (a) after 100 iterations of the HIO algorithm Modulus constraint applied inimage space with large triangular computational support The hole shape is comparable with theTEM image in (c) (c) Conventional TEM image of holes that created the direg raction pattern (a)

with a Philips CM 200 shy eld-emission electron microscope at 40 kV The acceleratingvoltage was lowered to 40 kV to make the multi-layered support opaque ModulatedYoungrsquos fringes are clearly seen

The resulting image after 100 HIO iterations is shown in shy gure 12b The holesreconstructed from the dinotraction pattern modulus look similar to the TEM imageof the holes in shy gure 12c but the resolution of the reconstruction is lower thanthe normal TEM image resolution High spatial frequency information from thehole edge is missing in the experimental dinotraction pattern due perhaps to drift ofthe imaging lenses during the exposure time

Table 1 shows a summary chart of all the conditions so far examined with HIO sim-

Phil Trans R Soc Lond A (2002)

Phase

recovery

an

dlen

slessim

agin

g891

Table 1 Summary chart showing direg erent conditions for phase recovery with HIO algorithm

( CArsquo denotes the complex algorithm (uses direg racted intensity and support constraint in real space) RIPrsquo denotes the complex algorithm withpositivity constraint of Re ordf Im ordf (uses direg racted intensity and support constraint in real space negative Re ordf and Im ordf inside the supportarea are set positive) Mrsquo denotes the modulus constraint on ordf in real space (uses direg racted intensity and support constraint in real spacecomplex numbers inside the support area are replaced by their modulus to enforce positivity) Prsquo denotes the phase constraint on ordf in realspace (uses direg racted intensity and support constraint in real space the phase is extracted and retained from complex numbers inside thesupport area modulus is extracted and set to 1 inside the support area))

general pure strong pure weakrsquo real weak-phasecomplex object phase object phase object (amplitude) object withordf = A exp(iiquest ) ordf = exp(i iquest (r)) iquest lt ordm =2 object absorption

known two-holephysical support

CA (laser experimentpin cheek cells)

CA P (laserexperiment mica)

RIP CA M RIP CA RIP CA

known one-holephysical support

CA only for certainsupport shapes(eg triangle withsharp edges)

CA slowconvergence Pworks better

RIP CA M RIP CA (laserexperimentnumbers)

RIP CA

unknown two-or-more-hole physicalsupport loose(eg triangle compu-tational support)

| | RIP M ( TEM experimentholes) (laserexperiment numbers)(planned X-ray micapinholes)

RIP (Miao X-rayexperimenttransparent support)

unknown one-holephysical support(eg triangle compu-tational support)

| | RIP only for holewith low symmetryand sharp edgesslow convergence

M RIP only for holewith low symmetryand sharp edgesslow convergence

RIP only for holewith low symmetryand sharp edgesslow convergence

Phil

Tra

ns

R

Soc

L

on

d

A(2

002)

892 J C H Spence U Weierstall and M Howells

ulations and experiments The enotectiveness of dinoterent constraints in real space fordinoterent physical and computational supports is compared The physical supportis the support which exists in the experiment and which may or may not be accu-rately known The computational support is the support used in the HIO iterationsThe table shows experiments that have been performed in italics The shy rst columnshows dinoterent possibilities for the physical support the most favourable at the topthe worst at the bottom The computational support and known physical supportare identical The computational support used with the unknown physical supportis larger than the physical support Reconstructions with one-hole physical supportand loose computational support are only possible with weak or real objects andsupport holes with sharp edges and low symmetry and convergence are very slow insimulations A known two-or-more-hole support proved to be most favourable in allcases

7 Discussion and conclusions direg ractive imaging with other particles

In this paper we have reviewed the implications of Thomas Youngrsquos experiment formatter waves and in particular for electron interferometry By taking advantage ofthe very high quality of the coherent electron beam available in the modern shy eld-emission electron microscope a number of important results (such as the AharonovBohm enotect and the electron Sagnac enotect) have been established and the shy eldof atomic-resolution transmission electron microscopy phase-contrast imaging andelectron holography opened up All these methods continue to produce a large sci-entishy c pay-onot in microstructural materials characterization and discovery and inbiology A new lensless imaging method is then reviewed which may be understoodin part by analogy with electron holography This HIO algorithm uses an iterativeapproach to assign phases to dinotraction pattern intensities To evaluate its perfor-mance we give experimental examples of image reconstruction with both coherentelectrons and light together with simulations for the soft X-ray case Thus we haveshown experimentally that the HIO algorithm is capable of solving the phase problemfor scattering from non-periodic objects and of making quantitative measurements ofphase (modulo 2ordm ) in good agreement with independent measurements The beam-splitting arrangement of an interferometer is not required and dinotraction-limitedlensless imaging seems possible as a result This may open the way for imaging withthe many particles for which no lenses exist such as neutrons ballistic electrons insolids at low temperature or neutral atoms and ions either as free coherent beamsor condensates (Bright coherent sources have recently appeared for many of these(Pauly 2000)) The dinotering interactions of these particles may provide useful newprobes for chemistry materials science and biology For example the reduced radia-tion damage enotects of imaging organic material with low-energy electrons neutronsand helium atoms have recently been compared (Spence et al 1999) and HIO sim-ulations have been completed for neutron imaging of a diatom (Spence et al 2001)The very small refractive index of organic matter for neutrons limits resolution andcontrast for thin samples For low-energy ions or metastable helium strong samplebeam interactions must be considered Damage from neutral helium-atom imagingof molecules should be negligible however penetration is also negligible (producinga proshy le image) Detectors for coherent atom beams still present serious problemssince area detectors are inemacr cient For biology the weak atom interaction is desirable

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 893

to avoid damage however a strong interaction at the detector is also needed so thationization before detection must be arranged

For real objects we have seen that the resolution in images recovered by the HIOalgorithm is largely independent of the accuracy with which the support is knownThe addition of noise anotects the probability that the unique solution will be found (asmeasured by the error metric) rather than the resolution which is near dinotraction-limited For complex objects our experience indicates that a much more accurateknowledge of the physical support boundary is needed approaching the scale ofthe resolution sought The tight support and two-hole geometry (see Fienup (1987)for other geometries) needed for complex objects (complex refractive index) willbecome increasingly accessible with the development of new lithographic techniquesfor support construction This can be combined with independent characterization ofthe support shape and the scaling information provided by the known autocorrelationof the support Experimentally favourable cases for imaging with nanoscale resolutionat present require loose support which table 1 shows greatly favours real objectssuch as the charge density which dinotracts hard X-rays at energies well away fromabsorption edges or very thin samples for high-energy electrons Unfortunately beamcoherence is more readily obtained for soft X-rays

Using modern lithographic techniques it is becoming increasingly possible toestablish the experimental conditions of known support required by the HIO algo-rithm both for free particle beams and for electrons in solids at low temperaturesThus for example we have seen the algorithm used to determine the phase of wavefunctions for electrons in a one-dimensional YBCO Josephson junction using TEMimages of the structure to deshy ne the required support (boundary conditions) (Car-mody et al 2000) Given the fundamental role that phase has played historically inphysics and which Thomas Young was amongst the shy rst to appreciate any techniquecapable of exposing phase can be expected to yield many surprises in physics in thefuture

Supported by ARO award DAAD190010500 We thank Dr Q Chen for the use of macrgure 12 (fromWeierstall et al 2002)

References

Aharonov Y amp Bohm D 1959 Signimacrcance of the electromagnetic potentials in the quantumtheory Phys Rev 115 485

Allen L J Faulkner H M L Nugent K A Oxley M P amp Paganin D 2001 Phase retrievalfrom images in the presence of macrrst-order vortices Phys Rev E 63 037602

Bauschke H H et al 2002 (In the press)

Boersch H 1943 Fresnelsche Beugung im Elektronenmikroskop Phys Z 44 202211

Bruck Y amp Sodin L 1979 On the ambiguity of the image-reconstruction problem Opt Com-mun 30 304308

Carmody M Moeckly B H Merkle K amp Marks L 2000 Spatial variation of the current ingrain boundary Josephson junctions J Appl Phys 87 24542459

Coene W Janssen G Op de Beek M amp van Dyck D 1992 Phase retrieval through focusvariation for ultra-resolution in macreld emission transmission electron microscopy Phys RevLett 69 37433746

Crewe A V Eggenberger D N Wall J amp Welter L M 1968 Electron gun using a macreldemission source Rev Scient Instrum 39 576

Phil Trans R Soc Lond A (2002)

878 J C H Spence U Weierstall and M Howells

and electronic instabilities rather than aberrations until the early 1960s Only in1995 was `Gaborrsquos dreamrsquo fully realized when onot-axis EM holograms were shy rst usedto extend resolution beyond the limit of 0198 nm set by aberrations toward the`super-resolutionrsquo limit set by electronic instabilities of 01 nm (Orchowski et al 1995)

The earliest beam-splitters for electrons used either Bragg dinotraction in a thincrystal in the transmission geometry (Marton 1952) or an electrostatic biprism(Moellenstedt amp Duker 1955) which has become the favoured choice today Thesubsequent development of the shy eld-emission electron source (Crewe et al 1968) atthe end of the 1960s was of the greatest importance for the shy eld since it provideda source of nanometre dimensions and sub-eV energy spread with a brightness perunit bandwidth greater than current-generation synchrotrons (Qian et al 1993)By combining Moellenstedtrsquos biprism with Crewersquos shy eld emission gun modern EMinstruments using beam energies of a few hundred keV are capable of recordingin a few seconds images showing hundreds of Fresnel fringes with a lateral coher-ence width of perhaps 01 m m and a longitudinal coherence length of several hundredangstroms They thus provide for interferometry a beam of extremely high qualityby the standards of accelerator physics

The refractive index for electrons in a magnetic shy eld was shy rst derived using Fer-matrsquos principle in 1948 (Ehrenberg amp Siday 1948) The AharonovBohm (AB) enotect(Aharonov amp Bohm 1959) subsequently demonstrated observable enotects along opti-cal paths in which shy elds (but not potentials) are zero suggesting that potentialsare more fundamental than shy elds (This idea had apparently already been suggestedby Maxwell) More recent electron-interferometry experiments include for examplethe observation of regux quantization using the AB enotect (Tonomura 1988) the scalar(electrostatic) AB enotect with electrons (Matteucci et al 1992) and the electronSagnac interferometer (Hasselbach amp Nicklaus 1993) From this has grown the shy eldof electron holography now widely used to map out on a nanometre scale internalmagnetic and electrostatic shy elds in materials from digital recording media to ferro-electrics semiconductors and superconducting vortices (for reviews see Voelkl 1998Lichte 1988 Tonomura 1993)

A dinoterent class of electron-interference experiments depends on the chromaticcoherence of electron sources and their second-order coherence properties For thefree particle beams used for electron interferometry a degeneracy parameter macr maybe deshy ned to indicate the degree of crowding in phase space a unity value indicatestwo electrons of opposite spin per cell with cell dimensions deshy ned by the uncertaintyprinciple (Components of centK are deshy ned by energy spread and beam divergencecomponents of centr deshy ne the coherence patch and longitudinal coherence length) Itis readily shown (Spence et al 1994) that macr is equal to the ratio of actual sourcebrightness to Langmuirrsquos theoretical maximum value and that macr may be measuredfrom the dimensions of the coherence patch and the source energy spread For a coldtungsten nano-tip shy eld emitter we measure macr = 10iexcl4 (Spence et al 1994) whereasa value of about unity is found for undulators at synchrotrons (For an optical laserin which any number of indistinguishable bosons may occupy a state macr = 1015)Coulomb repulsion between electrons limits the value of macr Nevertheless the bright-ness of electron shy eld emitters exceeds that of current synchrotron undulators (Qianet al 1993) The enotects of photon bunching seen in the Hanbury Brown and Twissexperiment (Loudon 1976) and the suggestion that electrons should `antibunchrsquo for

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 879

large values of macr were made at an early stage by Purcell (1956) In a quantum pic-ture we imagine wave packets overlapping coherently along the propagation axisby chance giving non-classical (non-random) results for the probability of detectingparticles between wave packets For this to occur however the temporal coherenceof the source must exceed the time between particle emissions so that the particlesare in the same quantum state This requires the source degeneracy macr for electronsto approach unity It is therefore not surprising that despite considerable enotortin our own laboratory and elsewhere antibunching has yet to be observed in freecoherent electron beams In condensed matter however these conditions are readilyachieved and fermion antibunching has recently been observed for the shy rst time ina two-dimensional electron gas in the quantum Hall regime (Henny et al 1999) Areview of other novel enotects expected in multiparticle fermion interferometry can befound in Silverman (1993)

3 Phase in imaging iterative methods for phase measurement

An image is a two-dimensional near-shy eld waveshy eld which may be real or complexIt may be obtained by Fourier transform from far-shy eld scattered amplitudes if thephases of these are known Holography encodes this phase information emacr cientlythrough the use of a reference wave and Fourier transform (FT) holography (whichuses a point reference in the same plane and to one side of the object) is par-ticularly enotective in separating the real and virtual twin images after the recon-struction process This involves only illumination of the hologram with the originalreference wave Phase information is encoded in the displacements of the `Youngrsquosfringesrsquo which result in this geometry (The method has been extended to allow a`complicated or extended referencersquo (Szoke 1997) such as a known fragment of amolecule and thus may be considered as an extension of the `heavy-atomrsquo methodof crystallography) Since this experimental inversion (reconstruction) of FT holo-grams to images produces a unique result it should not surprise us that an iterativesearch algorithm the hybrid inputoutput (HIO) algorithm (Gerchberg amp Saxton1972 Fienup 1982) applied to the same geometry should perform well Remarkablyhowever it has been found that the algorithm also converges for complex objectsconsisting of two separated parts neither of which is known but which generate anappreciable fringe modulation in the scattering pattern A knowledge of the `supportrsquoof the object (and of the sign of the scattering density) is found to be sumacr cient forinversion the support is the area within which the object transmission function T (r)is non-zero (or unknown) and so may take the form for example of the Youngrsquosfringe pinhole mask Dinoterent `unknownrsquo objects shy ll each hole The inversion givenshy nite known support has been shown to be `almost alwaysrsquo unique (Bruck amp Sodin1979) An error metric which depends only on the known values of the supportand Fourier modulus monitors progress toward this unique solution In summarywe have recently come to understand that this experimental geometry of ThomasYoung now onoters the possibility of lensless imaging using any of the radiations forwhich lenses do not exist such as coherent atom beams X-rays and neutrons Thecomplementary arrangement|two compact complex objects lying on a transpar-ent membrane|also works For a dinoterent non-interferometric approach to phasemeasurement akin to the through-focus method mentioned in the introduction andbased on the `transport of intensityrsquo equations see Paganin amp Nugent (1998) This

Phil Trans R Soc Lond A (2002)

880 J C H Spence U Weierstall and M Howells

method which requires intensity measurements across two planes downstream ofthe object has been used for example to consider a non-interferometric variant ofthe AB enotect (Paganin 2001) (For a phase object a single plane is sumacr cient sinceunit intensity can be assumed everywhere in the object plane) The phase vorticeswhich occur at zeros of intensity have also attracted considerable interest (Allen etal 2001) So far it has been applied to cases using collimated illumination only sothat resolution is limited by detector pixel size however there seems no reason notto apply it to the case of diverging illumination which provides magnishy cation anda resolution limit imposed by the source size

Despite many simulations few experimental tests of the HIO algorithm haveappeared The algorithm iterates between real and reciprocal space (related by aFourier transform) imposing known information in each domain The moduli ofthe scattered amplitudes I(u) are assumed known and in real space the objectfunction T (r) is set to its known values outside the support (Here r and u aretwo-dimensional vectors T (r) is the ratio of the complex exit waveshy eld (the in-focusimage) to the incident wave function often assumed planar) The support need notalways be known very accurately and an estimate can be obtained from the 50contour of the autocorrelation function which is given by the Fourier transform ofI(u) Finally knowledge of the sign of the real (and possibly imaginary) parts of Tmay be used The introduction of a small amount of `feedbackrsquo (Fienup 1987) distin-guishes the HIO algorithm from the error-reduction algorithm (Gerchberg amp Saxton1972) Convergence is found to be improved further in three-dimensions (Miao et al 2001) For real objects (such as the charge density which dinotracts X-rays) a singleobject is sumacr cient and the computational boundary used in the iterations may bea dinoterent shape from the physical support which it includes When viewed as anoptimization process the algorithm appears to be capable of shy nding the thousands ofunknown parameters (the phases in a typical image) in the presence of noise withoutthe usual problems of local minima and stagnation It thus appears to solve a verylarge global optimization problem despite the application of a non-convex constraint(the known Fourier moduli) (A convex set of solutions is one for which all pointslying on any line joining any two points within the N -dimensional set lie inside theset a kidney-shaped set in two dimensions is non-convex N is the number of pixelsin the image If a unique solution can be shown to exist (as in this case) iterationbetween the boundaries of two convex sets guarantees convergence to this solutionwithout stagnation at local minima Hence convex constraints such as symmetryknown support or sign are sought in optimization problems) While inversion fromknown phase information is a convex process leading to a unique result (Lindaas etal 1998) inversion from intensity alone is non-convex so that local minima mayformally be expected in approaches based on numerical optimization (Stark 1987)The HIO algorithm has been analysed in considerable detail (Fienup 1982 Paganin2001) including treatments based on Bregman projections onto convex sets (Stark1987 Bauschke et al 2002) and those going back to Sayre (1952) which start fromthe realization that Bragg sampling undersamples the autocorrelation function ofa molecule in a crystal (Miao et al 2001) The number of Fourier series equationsis then compared with the number of unknowns as the support is made a progres-sively smaller fraction of the total dinotracting area The lowest spatial frequency inthe dinotraction pattern corresponds to a distance wider than the support Using this`oversamplingrsquo approach soft X-ray images of lithographed symbols were recently

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 881

reconstructed from far-shy eld scattered intensities in a lensless scattering experiment(Miao et al 1999) Spatial resolution is limited only by noise and wavelength in theseexperiments For technical reasons however the reconstruction was assisted by low-resolution optical images which provided the low-order spatial frequencies and theobject transmission function was of a special mask-like form having only two (com-plex) transmission values whose signs were known We have therefore conductedboth simulations and experiments for phase measurement in scattering of X-rayscoherent visible light and coherent electron beams to evaluate this HIO algorithmfor more general objects

4 Simulations for X-rays

Simulations have been undertaken for objects consisting of an array of randomlypositioned pinholes to determine if this image could be reconstructed from its softX-ray dinotraction pattern If so these holes could later be shy lled with structures ofinterest for reconstruction The area occupied by pinholes has been limited to thecoherence width of the beam by a 1 m m aperture This aperture will not fall exactly inthe plane of the pinholes therefore the wavefront at the pinholes will have undergoneFresnel dinotraction at the aperture and so will be complex Figure 1 shows the resultsof these simulations assuming a wavelength of 25 nm for the incident soft X-raybeam The pinholes are 10 nm in diameter After 150 iterations the HIO algorithmhas converged without stagnation to an image which is related to the object byinversion through the centre The modulus of the estimated object function was takenat each iteration (modulus constraint) since the object is assumed real Only a roughestimate of the size of the aperture is needed the computational support conditionapplied in the iterations is a triangle larger than the aperture The unimportantinversion ambiguity cannot be resolved by the HIO algorithm since an invertedimage our real computational support and the circular aperture all have the sameFourier modulus The Fresnel propagation makes this strictly a complex objecthowever we obtain an accurate reconstruction of the holes in the shy lm despite the useof a positivity (modulus) constraint in object space The phase of the complex wavetransmitted through the pinholes is not reconstructed due to the imposed positivity

5 Experiments with light

Coherent laser optical phase retrieval experiments have been performed using theHIO algorithm for several objects The phase was successfully retrieved for severaldinoterent support constraints and object types So far only opaque support con-straints (apertures) have been used since these present fewer problems with exces-sive central beam intensity (blooming) The source was a frequency-stabilized HeNe laser ( para = 6328 nm) collimated to provide the planar wavefronts needed at thesample for real objects (shy gure 2) Dinotracted light was focused with a lens onto aliquid-nitrogen-cooled charge-coupled device (CCD) camera and the digital imagereconstructed using the HIO algorithm in the Digital Micrograph script language

The shy rst object was a resolution test object with transparent numbers and linesin an opaque chrome shy lm on glass The illuminated area was limited by a thin300 m m diameter aperture in contact with the chrome shy lm Assuming the glass isregat to within less than para =10 across the illuminated area the chrome mask acts as a

Phil Trans R Soc Lond A (2002)

882 J C H Spence U Weierstall and M Howells

(a) (b)

(d )(c)

Figure 1 Reconstruction of simulated pinholes illuminated by a Fresnel direg racted X-ray wave-macreld (out-of-focus physical support) (a) Simulated random array of pinholes illuminated bywave amplitude which has been Fresnel propagated 200 nm from an aperture to the pinholesWavelength para = 25 nm pinhole size 10 nm aperture diameter 1 m m (b) 200 nm out-of-focuswavemacreld from 1 m m aperture incident on pinholes (c) The Fourier transform of (a) is a speckledAiryrsquo s disc Modulus used as input for the HIO algorithm (d) Reconstructed pinholes after 150iterations with modulus constraint (To enforce positivity in real space only the modulus ofthe pixel values inside the support area is retained during each iteration) A large triangularcomputational support was used The image is inverted this is an ambiguity which cannot beresolved by the HIO algorithm

real binary object so that the dinotraction pattern should be symmetric A coherentoptical image was also recorded with the camera moved into the image plane ofthe lens This is shown in shy gure 3a The camera was then moved to the back focalplane of the lens and the dinotraction pattern intensity I(u) was recorded (shy gure 3b)The autocorrelation FT I(u v) was used to determine the approximate size of theobject and a circular support half the size of the autocorrelation was used in the

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 883

f

double polarizer to adjust intensity

collimator

lens diffraction pattern

CCD detectorHendashNe laser= 6328 nml

object with support mask

Figure 2 Schematic of the arrangement used for optical experiments

(a) (b)300 m m

Figure 3 (a) Coherent optical image of optical target transparent letters and bars in an oth-erwise opaque chrome macrlm on glass A 300 m m aperture acts as the support (HeNe laserpara = 6328 nm) (b) Coherent laser optical direg raction pattern of object shown in (a)

computer The HIO algorithm was used to recover the phases of the object exitwave and the algorithm always converged after about 50 iterations with dinoterentinitial random phase sets (shy gure 4a) A larger triangular support constraint wasalso used to recover the image as shown in shy gure 4b The normalized root-mean-squared error in the object domain (Fienup 1987) (a measure of dinoterence betweenthe current image estimate and the known pixel values outside the support) did notdrop below 02 probably due to imperfections in the dinotractogram eg noise andlight reregections from the entrance glass surface of the CCD camera which createasymmetry in the dinotraction pattern (see shy gure 4b above centre) Despite theseproblems the reconstructed image looks almost identical to the coherent image inshy gure 3a

A strong phase object (introducing phase shifts greater than ordm =2) was made usinga support consisting of two separated areas (two apertures of 300 m m diameter sep-

Phil Trans R Soc Lond A (2002)

884 J C H Spence U Weierstall and M Howells

(a) (b)

Figure 4 (a) Modulus of reconstructed image wave using a tight circular support constraint(b) Modulus of reconstructed image wave using a loose triangular support constraint A positivityconstraint was used in real space

mica

no mica(a) (b)

Figure 5 (a) Coherent laser optical image of phase object with two-hole support The edgesof two mica plates are visible across the hole on the right (b) Laser optical direg raction patternobtained by merging three patterns with direg erent exposure times to avoid saturation of thecentral beam

arated by 350 m m) One hole was covered with two 5 m m thick mica sheets whichoverlap in one region The expected phase shift is calculated to be 9 ordm The coherentoptical image is shown in shy gure 5a The mica is transparent except for the edgeswhere Fresnel contrast enotects are visible

The resulting dinotraction pattern is shown in shy gure 5b The central beam intensityin the dinotraction pattern from this transparent object was too strong to be recordedsimultaneously with the weaker high spatial frequencies so that several exposureswere needed All exposures were merged to one pattern in which saturated pixelswere replaced by scaled pixels from the lower exposure patterns The computational

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 885

black = -

white = +

(a) (b)

(c)

Figure 6 (a) Modulus and (b) phase of reconstructed image wave (c) phase reconstructed fromshifted direg raction pattern shows equiphase contour lines Phase shift by mica ordm No sign ormodulus constraint

support size and orientation were derived from the autocorrelation The complexHIO-phase retrieval algorithm was used to recover the image wave phase and modulusshown in shy gure 6 The phase shift by the mica plates in the reconstruction is ordm which since the algorithm reconstructs phase modulo 2 ordm is in good agreement withour independent measurement of 9 ordm based on the measured thickness and knownrefractive index of mica Thus the HIO algorithm is capable of quantitative phasemeasurement

Experiments with a transparent physical support have been attempted using alimiting aperture upstream of the sample but have not been successful so far becauseof the high intensity in the centre of the dinotraction pattern which causes bloomingin the CCD chip This problem persists for three-dimensional objects however athree-quadrant mask in the object plane has been proposed to address this dimacr culty(Sayre et al 1998) (Beam stops cannot be placed near enough to the CCD toavoid generating additional fringes) Experiments using the known intensity envelopeof a focused beam as the support have also not been successful for such complexobjects there appears to be a need for sharp physical support edges perhaps requiredto generate reference edge waves (For real objects there is no such requirement)However a coherent probe spanning two `knownrsquo holes works well in simulation

Previous experiments have been based on objects of simple type such as binary

Phil Trans R Soc Lond A (2002)

886 J C H Spence U Weierstall and M Howells

(a)

(b)

Figure 7 (a) Coherent laser optical image of cheek cells in two-hole support (300 m m apertures)About seven cells clump together in the upper hole Only Fresnel edge contrast is visible dueto defocus the cell nuclei are visible as black dots (b) Laser optical direg raction pattern mergedfrom four direg erent exposures to avoid saturation of the central beam

objects which contain much less information than greyscale objects Experimentswith human cheek cells which serve as more complicated greyscale phase objectshave therefore been performed The cheek cells were positioned on a glass slide insidetwo 300 m m apertures separated by 300 m m The apertures act as the opaque supportconsisting of two separated parts as needed for reconstruction of complex objectsFigure 7a shows the coherent optical image of the slightly defocused cheek cells asformed with a lens Only Fresnel edge contrast is visible while the nuclei producesome absorption Figure 7b shows the optical dinotraction pattern merged from fourdinoterent exposures The size and orientation of the support were estimated fromthe autocorrelation The complex reconstruction of phase and amplitude after about50 HIO iterations using the support constraint is shown in shy gure 8 The cells arevisible with large-area phase contrast in the phase image and the phase shift of thecells is about ordm In the reconstructed amplitude only edge contrast is visible as inthe optical image

For a three-dimensional object extended along the optical axis the question arisesas to which plane in the object is reconstructed given that the detected intensity liesat inshy nity and contains contributions from scattering at every point along the axisin the object space A simulation to clarify this is shown in shy gure 9 The simulatedobject consists of two planes separated by 1 mm in the z-direction Plane 1 has a holecontaining a complex object (the number 1) while plane 2 has two holes containingcomplex objects (numbers 2 and 3) The object exit wave calculated on plane 2

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 887

(a) (b)

Figure 8 (a) Modulus and (b) phase of reconstructed image wave The modulus shows only edgecontrast whereas the phase shows the cells in large-area phase contrast The phase shift due tothe cells is about ordm No modulus or sign constraint

(shy gure 9a) shows plane 1 out of focus The modulus of the Fourier transform (Fouriermodulus) of this image wave is used as input for the HIO algorithm In the inversiondinoterent shapes of the supports used at each plane shown inset can distinguish thetwo object planes If a plane is assumed unknown a larger support hole is usedthat contains the physical support Depending on which support plane is assumed asknown (inset) in the iteration (plane 1 support known and plane 2 support unknownor vice versa) the other plane is always reconstructed out of focus by +1 or iexcl 1 mm(shy gure 9c d) That means that in the case of known support on plane 1 the algorithmreconstructs the wave as seen through an imaging lens with out-of-focus images ofthe holes on plane 2 As expected the depth of focus for interactive inversion is givenby the expression for an ideal lens which imposes a dinotraction limit equal to that ofthe highest spatial frequency recorded in the dinotraction pattern

To verify this experimentally a pin was mounted 12 mm upstream of a pair of300 m m apertures The coherent optical image is shown in shy gure 10a A Fresnel fringesurrounds the pin since the lens is focused on the apertures The image was recoveredfrom the recorded dinotraction pattern intensity (shy gure 10b) Again three dinotractionpatterns with dinoterent exposure times had to be merged to avoid pixel saturationand blooming The reconstructed image modulus is shown in shy gure 11a where the

Phil Trans R Soc Lond A (2002)

888 J C H Spence U Weierstall and M Howells

illuminationdirection

plane 1

1 mm

(a) (b)

(c) (d)

Figure 9 (a) Schematic view of a simulated object consisting of two planes separated by 1 mm inthe direction of light propagation Plane 1 has a hole containing a complex object (number 1) andplane 2 has two holes containing complex objects (numbers 2 and 3) (b) Modulus of complexexit wave on plane 2 (c) Iterative reconstruction with plane 2 support assumed known (thesupport shape used is shown in the top right-hand corner) Plane 1 is out of focus by 1 mm(d) Iterative reconstruction with plane 1 support assumed known (the support shape is shownin the top right-hand corner) Plane 2 is out of focus by iexcl1 mm

pin appears out of focus (Fresnel fringe barely visible) because the aperture shapewas imposed as support The pin can be brought into focus by using the appropriateFresnel propagator with centz = 12 mm and para = 6328 nm taking the apertures outof focus (see shy gure 11b) Thus as expected the reconstructed plane is always thatplane in which the support condition is imposed where the algorithm shy nds a uniquesolution subject to the given boundary condition Propagation to other planes maybe performed subsequently since the complex waveshy eld is obtained subject to thedepth-of-focus limitation The relative sideways shift of the pin inside the aperturebetween shy gure 10a and shy gure 11a is due to a slight unintentional tilt of the object(pin plus pinhole) between recording the image and the dinotraction pattern

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Phase recovery and lensless imaging 889

(a) (b)

Figure 10 (a) Coherent laser optical image of pin in front of aperture (b) Laser optical direg ractionpattern obtained by merging three patterns with direg erent exposure times to avoid saturation ofthe central beam

(a) (b)

Figure 11 (a) Reconstructed modulus Only the hole with the out-of-focus pin is shown(b) Numerical back propagation of reconstructed complex wavemacreld by 12 mm to plane of pinPin appears in focus Support constraint only

6 Experiments with coherent electrons

Using a special support mask made by electron-beam lithography experiments havealso been undertaken using the HIO algorithm to retrieve the phases from experimen-tal transmission electron dinotraction patterns (Weierstall et al 2002) Images mightthen be reconstructed that are free of electron lens aberrations The support maskcontains hole pairs of various sizes with pairs separated by 1 m m The separationbetween holes of a pair is ca 50 nm less than the coherence width of the illuminationbeam The holes irregular in shape form the object we wish to reconstruct in futurework we plan to introduce nanoscale objects into the holes Figure 12a below showsthe experimental dinotraction pattern from two holes shown in shy gure 12c obtained

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890 J C H Spence U Weierstall and M Howells

20 nm

(a) (b)

(c)

Figure 12 (a) Experimental electron direg raction pattern from two nanoscale holes in an opaquemask (shown in (c)) recorded using a macreld-emission electron microscope (CM200 FEG at 40 kV1k pound1k CCD with 24 m m pixels camera length 611 m) Youngrsquo s fringes from the two holes arevisible these fringes encode phase information (b) Reconstruction of image of holes from thedireg raction pattern (a) after 100 iterations of the HIO algorithm Modulus constraint applied inimage space with large triangular computational support The hole shape is comparable with theTEM image in (c) (c) Conventional TEM image of holes that created the direg raction pattern (a)

with a Philips CM 200 shy eld-emission electron microscope at 40 kV The acceleratingvoltage was lowered to 40 kV to make the multi-layered support opaque ModulatedYoungrsquos fringes are clearly seen

The resulting image after 100 HIO iterations is shown in shy gure 12b The holesreconstructed from the dinotraction pattern modulus look similar to the TEM imageof the holes in shy gure 12c but the resolution of the reconstruction is lower thanthe normal TEM image resolution High spatial frequency information from thehole edge is missing in the experimental dinotraction pattern due perhaps to drift ofthe imaging lenses during the exposure time

Table 1 shows a summary chart of all the conditions so far examined with HIO sim-

Phil Trans R Soc Lond A (2002)

Phase

recovery

an

dlen

slessim

agin

g891

Table 1 Summary chart showing direg erent conditions for phase recovery with HIO algorithm

( CArsquo denotes the complex algorithm (uses direg racted intensity and support constraint in real space) RIPrsquo denotes the complex algorithm withpositivity constraint of Re ordf Im ordf (uses direg racted intensity and support constraint in real space negative Re ordf and Im ordf inside the supportarea are set positive) Mrsquo denotes the modulus constraint on ordf in real space (uses direg racted intensity and support constraint in real spacecomplex numbers inside the support area are replaced by their modulus to enforce positivity) Prsquo denotes the phase constraint on ordf in realspace (uses direg racted intensity and support constraint in real space the phase is extracted and retained from complex numbers inside thesupport area modulus is extracted and set to 1 inside the support area))

general pure strong pure weakrsquo real weak-phasecomplex object phase object phase object (amplitude) object withordf = A exp(iiquest ) ordf = exp(i iquest (r)) iquest lt ordm =2 object absorption

known two-holephysical support

CA (laser experimentpin cheek cells)

CA P (laserexperiment mica)

RIP CA M RIP CA RIP CA

known one-holephysical support

CA only for certainsupport shapes(eg triangle withsharp edges)

CA slowconvergence Pworks better

RIP CA M RIP CA (laserexperimentnumbers)

RIP CA

unknown two-or-more-hole physicalsupport loose(eg triangle compu-tational support)

| | RIP M ( TEM experimentholes) (laserexperiment numbers)(planned X-ray micapinholes)

RIP (Miao X-rayexperimenttransparent support)

unknown one-holephysical support(eg triangle compu-tational support)

| | RIP only for holewith low symmetryand sharp edgesslow convergence

M RIP only for holewith low symmetryand sharp edgesslow convergence

RIP only for holewith low symmetryand sharp edgesslow convergence

Phil

Tra

ns

R

Soc

L

on

d

A(2

002)

892 J C H Spence U Weierstall and M Howells

ulations and experiments The enotectiveness of dinoterent constraints in real space fordinoterent physical and computational supports is compared The physical supportis the support which exists in the experiment and which may or may not be accu-rately known The computational support is the support used in the HIO iterationsThe table shows experiments that have been performed in italics The shy rst columnshows dinoterent possibilities for the physical support the most favourable at the topthe worst at the bottom The computational support and known physical supportare identical The computational support used with the unknown physical supportis larger than the physical support Reconstructions with one-hole physical supportand loose computational support are only possible with weak or real objects andsupport holes with sharp edges and low symmetry and convergence are very slow insimulations A known two-or-more-hole support proved to be most favourable in allcases

7 Discussion and conclusions direg ractive imaging with other particles

In this paper we have reviewed the implications of Thomas Youngrsquos experiment formatter waves and in particular for electron interferometry By taking advantage ofthe very high quality of the coherent electron beam available in the modern shy eld-emission electron microscope a number of important results (such as the AharonovBohm enotect and the electron Sagnac enotect) have been established and the shy eldof atomic-resolution transmission electron microscopy phase-contrast imaging andelectron holography opened up All these methods continue to produce a large sci-entishy c pay-onot in microstructural materials characterization and discovery and inbiology A new lensless imaging method is then reviewed which may be understoodin part by analogy with electron holography This HIO algorithm uses an iterativeapproach to assign phases to dinotraction pattern intensities To evaluate its perfor-mance we give experimental examples of image reconstruction with both coherentelectrons and light together with simulations for the soft X-ray case Thus we haveshown experimentally that the HIO algorithm is capable of solving the phase problemfor scattering from non-periodic objects and of making quantitative measurements ofphase (modulo 2ordm ) in good agreement with independent measurements The beam-splitting arrangement of an interferometer is not required and dinotraction-limitedlensless imaging seems possible as a result This may open the way for imaging withthe many particles for which no lenses exist such as neutrons ballistic electrons insolids at low temperature or neutral atoms and ions either as free coherent beamsor condensates (Bright coherent sources have recently appeared for many of these(Pauly 2000)) The dinotering interactions of these particles may provide useful newprobes for chemistry materials science and biology For example the reduced radia-tion damage enotects of imaging organic material with low-energy electrons neutronsand helium atoms have recently been compared (Spence et al 1999) and HIO sim-ulations have been completed for neutron imaging of a diatom (Spence et al 2001)The very small refractive index of organic matter for neutrons limits resolution andcontrast for thin samples For low-energy ions or metastable helium strong samplebeam interactions must be considered Damage from neutral helium-atom imagingof molecules should be negligible however penetration is also negligible (producinga proshy le image) Detectors for coherent atom beams still present serious problemssince area detectors are inemacr cient For biology the weak atom interaction is desirable

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 893

to avoid damage however a strong interaction at the detector is also needed so thationization before detection must be arranged

For real objects we have seen that the resolution in images recovered by the HIOalgorithm is largely independent of the accuracy with which the support is knownThe addition of noise anotects the probability that the unique solution will be found (asmeasured by the error metric) rather than the resolution which is near dinotraction-limited For complex objects our experience indicates that a much more accurateknowledge of the physical support boundary is needed approaching the scale ofthe resolution sought The tight support and two-hole geometry (see Fienup (1987)for other geometries) needed for complex objects (complex refractive index) willbecome increasingly accessible with the development of new lithographic techniquesfor support construction This can be combined with independent characterization ofthe support shape and the scaling information provided by the known autocorrelationof the support Experimentally favourable cases for imaging with nanoscale resolutionat present require loose support which table 1 shows greatly favours real objectssuch as the charge density which dinotracts hard X-rays at energies well away fromabsorption edges or very thin samples for high-energy electrons Unfortunately beamcoherence is more readily obtained for soft X-rays

Using modern lithographic techniques it is becoming increasingly possible toestablish the experimental conditions of known support required by the HIO algo-rithm both for free particle beams and for electrons in solids at low temperaturesThus for example we have seen the algorithm used to determine the phase of wavefunctions for electrons in a one-dimensional YBCO Josephson junction using TEMimages of the structure to deshy ne the required support (boundary conditions) (Car-mody et al 2000) Given the fundamental role that phase has played historically inphysics and which Thomas Young was amongst the shy rst to appreciate any techniquecapable of exposing phase can be expected to yield many surprises in physics in thefuture

Supported by ARO award DAAD190010500 We thank Dr Q Chen for the use of macrgure 12 (fromWeierstall et al 2002)

References

Aharonov Y amp Bohm D 1959 Signimacrcance of the electromagnetic potentials in the quantumtheory Phys Rev 115 485

Allen L J Faulkner H M L Nugent K A Oxley M P amp Paganin D 2001 Phase retrievalfrom images in the presence of macrrst-order vortices Phys Rev E 63 037602

Bauschke H H et al 2002 (In the press)

Boersch H 1943 Fresnelsche Beugung im Elektronenmikroskop Phys Z 44 202211

Bruck Y amp Sodin L 1979 On the ambiguity of the image-reconstruction problem Opt Com-mun 30 304308

Carmody M Moeckly B H Merkle K amp Marks L 2000 Spatial variation of the current ingrain boundary Josephson junctions J Appl Phys 87 24542459

Coene W Janssen G Op de Beek M amp van Dyck D 1992 Phase retrieval through focusvariation for ultra-resolution in macreld emission transmission electron microscopy Phys RevLett 69 37433746

Crewe A V Eggenberger D N Wall J amp Welter L M 1968 Electron gun using a macreldemission source Rev Scient Instrum 39 576

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 879

large values of macr were made at an early stage by Purcell (1956) In a quantum pic-ture we imagine wave packets overlapping coherently along the propagation axisby chance giving non-classical (non-random) results for the probability of detectingparticles between wave packets For this to occur however the temporal coherenceof the source must exceed the time between particle emissions so that the particlesare in the same quantum state This requires the source degeneracy macr for electronsto approach unity It is therefore not surprising that despite considerable enotortin our own laboratory and elsewhere antibunching has yet to be observed in freecoherent electron beams In condensed matter however these conditions are readilyachieved and fermion antibunching has recently been observed for the shy rst time ina two-dimensional electron gas in the quantum Hall regime (Henny et al 1999) Areview of other novel enotects expected in multiparticle fermion interferometry can befound in Silverman (1993)

3 Phase in imaging iterative methods for phase measurement

An image is a two-dimensional near-shy eld waveshy eld which may be real or complexIt may be obtained by Fourier transform from far-shy eld scattered amplitudes if thephases of these are known Holography encodes this phase information emacr cientlythrough the use of a reference wave and Fourier transform (FT) holography (whichuses a point reference in the same plane and to one side of the object) is par-ticularly enotective in separating the real and virtual twin images after the recon-struction process This involves only illumination of the hologram with the originalreference wave Phase information is encoded in the displacements of the `Youngrsquosfringesrsquo which result in this geometry (The method has been extended to allow a`complicated or extended referencersquo (Szoke 1997) such as a known fragment of amolecule and thus may be considered as an extension of the `heavy-atomrsquo methodof crystallography) Since this experimental inversion (reconstruction) of FT holo-grams to images produces a unique result it should not surprise us that an iterativesearch algorithm the hybrid inputoutput (HIO) algorithm (Gerchberg amp Saxton1972 Fienup 1982) applied to the same geometry should perform well Remarkablyhowever it has been found that the algorithm also converges for complex objectsconsisting of two separated parts neither of which is known but which generate anappreciable fringe modulation in the scattering pattern A knowledge of the `supportrsquoof the object (and of the sign of the scattering density) is found to be sumacr cient forinversion the support is the area within which the object transmission function T (r)is non-zero (or unknown) and so may take the form for example of the Youngrsquosfringe pinhole mask Dinoterent `unknownrsquo objects shy ll each hole The inversion givenshy nite known support has been shown to be `almost alwaysrsquo unique (Bruck amp Sodin1979) An error metric which depends only on the known values of the supportand Fourier modulus monitors progress toward this unique solution In summarywe have recently come to understand that this experimental geometry of ThomasYoung now onoters the possibility of lensless imaging using any of the radiations forwhich lenses do not exist such as coherent atom beams X-rays and neutrons Thecomplementary arrangement|two compact complex objects lying on a transpar-ent membrane|also works For a dinoterent non-interferometric approach to phasemeasurement akin to the through-focus method mentioned in the introduction andbased on the `transport of intensityrsquo equations see Paganin amp Nugent (1998) This

Phil Trans R Soc Lond A (2002)

880 J C H Spence U Weierstall and M Howells

method which requires intensity measurements across two planes downstream ofthe object has been used for example to consider a non-interferometric variant ofthe AB enotect (Paganin 2001) (For a phase object a single plane is sumacr cient sinceunit intensity can be assumed everywhere in the object plane) The phase vorticeswhich occur at zeros of intensity have also attracted considerable interest (Allen etal 2001) So far it has been applied to cases using collimated illumination only sothat resolution is limited by detector pixel size however there seems no reason notto apply it to the case of diverging illumination which provides magnishy cation anda resolution limit imposed by the source size

Despite many simulations few experimental tests of the HIO algorithm haveappeared The algorithm iterates between real and reciprocal space (related by aFourier transform) imposing known information in each domain The moduli ofthe scattered amplitudes I(u) are assumed known and in real space the objectfunction T (r) is set to its known values outside the support (Here r and u aretwo-dimensional vectors T (r) is the ratio of the complex exit waveshy eld (the in-focusimage) to the incident wave function often assumed planar) The support need notalways be known very accurately and an estimate can be obtained from the 50contour of the autocorrelation function which is given by the Fourier transform ofI(u) Finally knowledge of the sign of the real (and possibly imaginary) parts of Tmay be used The introduction of a small amount of `feedbackrsquo (Fienup 1987) distin-guishes the HIO algorithm from the error-reduction algorithm (Gerchberg amp Saxton1972) Convergence is found to be improved further in three-dimensions (Miao et al 2001) For real objects (such as the charge density which dinotracts X-rays) a singleobject is sumacr cient and the computational boundary used in the iterations may bea dinoterent shape from the physical support which it includes When viewed as anoptimization process the algorithm appears to be capable of shy nding the thousands ofunknown parameters (the phases in a typical image) in the presence of noise withoutthe usual problems of local minima and stagnation It thus appears to solve a verylarge global optimization problem despite the application of a non-convex constraint(the known Fourier moduli) (A convex set of solutions is one for which all pointslying on any line joining any two points within the N -dimensional set lie inside theset a kidney-shaped set in two dimensions is non-convex N is the number of pixelsin the image If a unique solution can be shown to exist (as in this case) iterationbetween the boundaries of two convex sets guarantees convergence to this solutionwithout stagnation at local minima Hence convex constraints such as symmetryknown support or sign are sought in optimization problems) While inversion fromknown phase information is a convex process leading to a unique result (Lindaas etal 1998) inversion from intensity alone is non-convex so that local minima mayformally be expected in approaches based on numerical optimization (Stark 1987)The HIO algorithm has been analysed in considerable detail (Fienup 1982 Paganin2001) including treatments based on Bregman projections onto convex sets (Stark1987 Bauschke et al 2002) and those going back to Sayre (1952) which start fromthe realization that Bragg sampling undersamples the autocorrelation function ofa molecule in a crystal (Miao et al 2001) The number of Fourier series equationsis then compared with the number of unknowns as the support is made a progres-sively smaller fraction of the total dinotracting area The lowest spatial frequency inthe dinotraction pattern corresponds to a distance wider than the support Using this`oversamplingrsquo approach soft X-ray images of lithographed symbols were recently

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 881

reconstructed from far-shy eld scattered intensities in a lensless scattering experiment(Miao et al 1999) Spatial resolution is limited only by noise and wavelength in theseexperiments For technical reasons however the reconstruction was assisted by low-resolution optical images which provided the low-order spatial frequencies and theobject transmission function was of a special mask-like form having only two (com-plex) transmission values whose signs were known We have therefore conductedboth simulations and experiments for phase measurement in scattering of X-rayscoherent visible light and coherent electron beams to evaluate this HIO algorithmfor more general objects

4 Simulations for X-rays

Simulations have been undertaken for objects consisting of an array of randomlypositioned pinholes to determine if this image could be reconstructed from its softX-ray dinotraction pattern If so these holes could later be shy lled with structures ofinterest for reconstruction The area occupied by pinholes has been limited to thecoherence width of the beam by a 1 m m aperture This aperture will not fall exactly inthe plane of the pinholes therefore the wavefront at the pinholes will have undergoneFresnel dinotraction at the aperture and so will be complex Figure 1 shows the resultsof these simulations assuming a wavelength of 25 nm for the incident soft X-raybeam The pinholes are 10 nm in diameter After 150 iterations the HIO algorithmhas converged without stagnation to an image which is related to the object byinversion through the centre The modulus of the estimated object function was takenat each iteration (modulus constraint) since the object is assumed real Only a roughestimate of the size of the aperture is needed the computational support conditionapplied in the iterations is a triangle larger than the aperture The unimportantinversion ambiguity cannot be resolved by the HIO algorithm since an invertedimage our real computational support and the circular aperture all have the sameFourier modulus The Fresnel propagation makes this strictly a complex objecthowever we obtain an accurate reconstruction of the holes in the shy lm despite the useof a positivity (modulus) constraint in object space The phase of the complex wavetransmitted through the pinholes is not reconstructed due to the imposed positivity

5 Experiments with light

Coherent laser optical phase retrieval experiments have been performed using theHIO algorithm for several objects The phase was successfully retrieved for severaldinoterent support constraints and object types So far only opaque support con-straints (apertures) have been used since these present fewer problems with exces-sive central beam intensity (blooming) The source was a frequency-stabilized HeNe laser ( para = 6328 nm) collimated to provide the planar wavefronts needed at thesample for real objects (shy gure 2) Dinotracted light was focused with a lens onto aliquid-nitrogen-cooled charge-coupled device (CCD) camera and the digital imagereconstructed using the HIO algorithm in the Digital Micrograph script language

The shy rst object was a resolution test object with transparent numbers and linesin an opaque chrome shy lm on glass The illuminated area was limited by a thin300 m m diameter aperture in contact with the chrome shy lm Assuming the glass isregat to within less than para =10 across the illuminated area the chrome mask acts as a

Phil Trans R Soc Lond A (2002)

882 J C H Spence U Weierstall and M Howells

(a) (b)

(d )(c)

Figure 1 Reconstruction of simulated pinholes illuminated by a Fresnel direg racted X-ray wave-macreld (out-of-focus physical support) (a) Simulated random array of pinholes illuminated bywave amplitude which has been Fresnel propagated 200 nm from an aperture to the pinholesWavelength para = 25 nm pinhole size 10 nm aperture diameter 1 m m (b) 200 nm out-of-focuswavemacreld from 1 m m aperture incident on pinholes (c) The Fourier transform of (a) is a speckledAiryrsquo s disc Modulus used as input for the HIO algorithm (d) Reconstructed pinholes after 150iterations with modulus constraint (To enforce positivity in real space only the modulus ofthe pixel values inside the support area is retained during each iteration) A large triangularcomputational support was used The image is inverted this is an ambiguity which cannot beresolved by the HIO algorithm

real binary object so that the dinotraction pattern should be symmetric A coherentoptical image was also recorded with the camera moved into the image plane ofthe lens This is shown in shy gure 3a The camera was then moved to the back focalplane of the lens and the dinotraction pattern intensity I(u) was recorded (shy gure 3b)The autocorrelation FT I(u v) was used to determine the approximate size of theobject and a circular support half the size of the autocorrelation was used in the

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 883

f

double polarizer to adjust intensity

collimator

lens diffraction pattern

CCD detectorHendashNe laser= 6328 nml

object with support mask

Figure 2 Schematic of the arrangement used for optical experiments

(a) (b)300 m m

Figure 3 (a) Coherent optical image of optical target transparent letters and bars in an oth-erwise opaque chrome macrlm on glass A 300 m m aperture acts as the support (HeNe laserpara = 6328 nm) (b) Coherent laser optical direg raction pattern of object shown in (a)

computer The HIO algorithm was used to recover the phases of the object exitwave and the algorithm always converged after about 50 iterations with dinoterentinitial random phase sets (shy gure 4a) A larger triangular support constraint wasalso used to recover the image as shown in shy gure 4b The normalized root-mean-squared error in the object domain (Fienup 1987) (a measure of dinoterence betweenthe current image estimate and the known pixel values outside the support) did notdrop below 02 probably due to imperfections in the dinotractogram eg noise andlight reregections from the entrance glass surface of the CCD camera which createasymmetry in the dinotraction pattern (see shy gure 4b above centre) Despite theseproblems the reconstructed image looks almost identical to the coherent image inshy gure 3a

A strong phase object (introducing phase shifts greater than ordm =2) was made usinga support consisting of two separated areas (two apertures of 300 m m diameter sep-

Phil Trans R Soc Lond A (2002)

884 J C H Spence U Weierstall and M Howells

(a) (b)

Figure 4 (a) Modulus of reconstructed image wave using a tight circular support constraint(b) Modulus of reconstructed image wave using a loose triangular support constraint A positivityconstraint was used in real space

mica

no mica(a) (b)

Figure 5 (a) Coherent laser optical image of phase object with two-hole support The edgesof two mica plates are visible across the hole on the right (b) Laser optical direg raction patternobtained by merging three patterns with direg erent exposure times to avoid saturation of thecentral beam

arated by 350 m m) One hole was covered with two 5 m m thick mica sheets whichoverlap in one region The expected phase shift is calculated to be 9 ordm The coherentoptical image is shown in shy gure 5a The mica is transparent except for the edgeswhere Fresnel contrast enotects are visible

The resulting dinotraction pattern is shown in shy gure 5b The central beam intensityin the dinotraction pattern from this transparent object was too strong to be recordedsimultaneously with the weaker high spatial frequencies so that several exposureswere needed All exposures were merged to one pattern in which saturated pixelswere replaced by scaled pixels from the lower exposure patterns The computational

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 885

black = -

white = +

(a) (b)

(c)

Figure 6 (a) Modulus and (b) phase of reconstructed image wave (c) phase reconstructed fromshifted direg raction pattern shows equiphase contour lines Phase shift by mica ordm No sign ormodulus constraint

support size and orientation were derived from the autocorrelation The complexHIO-phase retrieval algorithm was used to recover the image wave phase and modulusshown in shy gure 6 The phase shift by the mica plates in the reconstruction is ordm which since the algorithm reconstructs phase modulo 2 ordm is in good agreement withour independent measurement of 9 ordm based on the measured thickness and knownrefractive index of mica Thus the HIO algorithm is capable of quantitative phasemeasurement

Experiments with a transparent physical support have been attempted using alimiting aperture upstream of the sample but have not been successful so far becauseof the high intensity in the centre of the dinotraction pattern which causes bloomingin the CCD chip This problem persists for three-dimensional objects however athree-quadrant mask in the object plane has been proposed to address this dimacr culty(Sayre et al 1998) (Beam stops cannot be placed near enough to the CCD toavoid generating additional fringes) Experiments using the known intensity envelopeof a focused beam as the support have also not been successful for such complexobjects there appears to be a need for sharp physical support edges perhaps requiredto generate reference edge waves (For real objects there is no such requirement)However a coherent probe spanning two `knownrsquo holes works well in simulation

Previous experiments have been based on objects of simple type such as binary

Phil Trans R Soc Lond A (2002)

886 J C H Spence U Weierstall and M Howells

(a)

(b)

Figure 7 (a) Coherent laser optical image of cheek cells in two-hole support (300 m m apertures)About seven cells clump together in the upper hole Only Fresnel edge contrast is visible dueto defocus the cell nuclei are visible as black dots (b) Laser optical direg raction pattern mergedfrom four direg erent exposures to avoid saturation of the central beam

objects which contain much less information than greyscale objects Experimentswith human cheek cells which serve as more complicated greyscale phase objectshave therefore been performed The cheek cells were positioned on a glass slide insidetwo 300 m m apertures separated by 300 m m The apertures act as the opaque supportconsisting of two separated parts as needed for reconstruction of complex objectsFigure 7a shows the coherent optical image of the slightly defocused cheek cells asformed with a lens Only Fresnel edge contrast is visible while the nuclei producesome absorption Figure 7b shows the optical dinotraction pattern merged from fourdinoterent exposures The size and orientation of the support were estimated fromthe autocorrelation The complex reconstruction of phase and amplitude after about50 HIO iterations using the support constraint is shown in shy gure 8 The cells arevisible with large-area phase contrast in the phase image and the phase shift of thecells is about ordm In the reconstructed amplitude only edge contrast is visible as inthe optical image

For a three-dimensional object extended along the optical axis the question arisesas to which plane in the object is reconstructed given that the detected intensity liesat inshy nity and contains contributions from scattering at every point along the axisin the object space A simulation to clarify this is shown in shy gure 9 The simulatedobject consists of two planes separated by 1 mm in the z-direction Plane 1 has a holecontaining a complex object (the number 1) while plane 2 has two holes containingcomplex objects (numbers 2 and 3) The object exit wave calculated on plane 2

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 887

(a) (b)

Figure 8 (a) Modulus and (b) phase of reconstructed image wave The modulus shows only edgecontrast whereas the phase shows the cells in large-area phase contrast The phase shift due tothe cells is about ordm No modulus or sign constraint

(shy gure 9a) shows plane 1 out of focus The modulus of the Fourier transform (Fouriermodulus) of this image wave is used as input for the HIO algorithm In the inversiondinoterent shapes of the supports used at each plane shown inset can distinguish thetwo object planes If a plane is assumed unknown a larger support hole is usedthat contains the physical support Depending on which support plane is assumed asknown (inset) in the iteration (plane 1 support known and plane 2 support unknownor vice versa) the other plane is always reconstructed out of focus by +1 or iexcl 1 mm(shy gure 9c d) That means that in the case of known support on plane 1 the algorithmreconstructs the wave as seen through an imaging lens with out-of-focus images ofthe holes on plane 2 As expected the depth of focus for interactive inversion is givenby the expression for an ideal lens which imposes a dinotraction limit equal to that ofthe highest spatial frequency recorded in the dinotraction pattern

To verify this experimentally a pin was mounted 12 mm upstream of a pair of300 m m apertures The coherent optical image is shown in shy gure 10a A Fresnel fringesurrounds the pin since the lens is focused on the apertures The image was recoveredfrom the recorded dinotraction pattern intensity (shy gure 10b) Again three dinotractionpatterns with dinoterent exposure times had to be merged to avoid pixel saturationand blooming The reconstructed image modulus is shown in shy gure 11a where the

Phil Trans R Soc Lond A (2002)

888 J C H Spence U Weierstall and M Howells

illuminationdirection

plane 1

1 mm

(a) (b)

(c) (d)

Figure 9 (a) Schematic view of a simulated object consisting of two planes separated by 1 mm inthe direction of light propagation Plane 1 has a hole containing a complex object (number 1) andplane 2 has two holes containing complex objects (numbers 2 and 3) (b) Modulus of complexexit wave on plane 2 (c) Iterative reconstruction with plane 2 support assumed known (thesupport shape used is shown in the top right-hand corner) Plane 1 is out of focus by 1 mm(d) Iterative reconstruction with plane 1 support assumed known (the support shape is shownin the top right-hand corner) Plane 2 is out of focus by iexcl1 mm

pin appears out of focus (Fresnel fringe barely visible) because the aperture shapewas imposed as support The pin can be brought into focus by using the appropriateFresnel propagator with centz = 12 mm and para = 6328 nm taking the apertures outof focus (see shy gure 11b) Thus as expected the reconstructed plane is always thatplane in which the support condition is imposed where the algorithm shy nds a uniquesolution subject to the given boundary condition Propagation to other planes maybe performed subsequently since the complex waveshy eld is obtained subject to thedepth-of-focus limitation The relative sideways shift of the pin inside the aperturebetween shy gure 10a and shy gure 11a is due to a slight unintentional tilt of the object(pin plus pinhole) between recording the image and the dinotraction pattern

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 889

(a) (b)

Figure 10 (a) Coherent laser optical image of pin in front of aperture (b) Laser optical direg ractionpattern obtained by merging three patterns with direg erent exposure times to avoid saturation ofthe central beam

(a) (b)

Figure 11 (a) Reconstructed modulus Only the hole with the out-of-focus pin is shown(b) Numerical back propagation of reconstructed complex wavemacreld by 12 mm to plane of pinPin appears in focus Support constraint only

6 Experiments with coherent electrons

Using a special support mask made by electron-beam lithography experiments havealso been undertaken using the HIO algorithm to retrieve the phases from experimen-tal transmission electron dinotraction patterns (Weierstall et al 2002) Images mightthen be reconstructed that are free of electron lens aberrations The support maskcontains hole pairs of various sizes with pairs separated by 1 m m The separationbetween holes of a pair is ca 50 nm less than the coherence width of the illuminationbeam The holes irregular in shape form the object we wish to reconstruct in futurework we plan to introduce nanoscale objects into the holes Figure 12a below showsthe experimental dinotraction pattern from two holes shown in shy gure 12c obtained

Phil Trans R Soc Lond A (2002)

890 J C H Spence U Weierstall and M Howells

20 nm

(a) (b)

(c)

Figure 12 (a) Experimental electron direg raction pattern from two nanoscale holes in an opaquemask (shown in (c)) recorded using a macreld-emission electron microscope (CM200 FEG at 40 kV1k pound1k CCD with 24 m m pixels camera length 611 m) Youngrsquo s fringes from the two holes arevisible these fringes encode phase information (b) Reconstruction of image of holes from thedireg raction pattern (a) after 100 iterations of the HIO algorithm Modulus constraint applied inimage space with large triangular computational support The hole shape is comparable with theTEM image in (c) (c) Conventional TEM image of holes that created the direg raction pattern (a)

with a Philips CM 200 shy eld-emission electron microscope at 40 kV The acceleratingvoltage was lowered to 40 kV to make the multi-layered support opaque ModulatedYoungrsquos fringes are clearly seen

The resulting image after 100 HIO iterations is shown in shy gure 12b The holesreconstructed from the dinotraction pattern modulus look similar to the TEM imageof the holes in shy gure 12c but the resolution of the reconstruction is lower thanthe normal TEM image resolution High spatial frequency information from thehole edge is missing in the experimental dinotraction pattern due perhaps to drift ofthe imaging lenses during the exposure time

Table 1 shows a summary chart of all the conditions so far examined with HIO sim-

Phil Trans R Soc Lond A (2002)

Phase

recovery

an

dlen

slessim

agin

g891

Table 1 Summary chart showing direg erent conditions for phase recovery with HIO algorithm

( CArsquo denotes the complex algorithm (uses direg racted intensity and support constraint in real space) RIPrsquo denotes the complex algorithm withpositivity constraint of Re ordf Im ordf (uses direg racted intensity and support constraint in real space negative Re ordf and Im ordf inside the supportarea are set positive) Mrsquo denotes the modulus constraint on ordf in real space (uses direg racted intensity and support constraint in real spacecomplex numbers inside the support area are replaced by their modulus to enforce positivity) Prsquo denotes the phase constraint on ordf in realspace (uses direg racted intensity and support constraint in real space the phase is extracted and retained from complex numbers inside thesupport area modulus is extracted and set to 1 inside the support area))

general pure strong pure weakrsquo real weak-phasecomplex object phase object phase object (amplitude) object withordf = A exp(iiquest ) ordf = exp(i iquest (r)) iquest lt ordm =2 object absorption

known two-holephysical support

CA (laser experimentpin cheek cells)

CA P (laserexperiment mica)

RIP CA M RIP CA RIP CA

known one-holephysical support

CA only for certainsupport shapes(eg triangle withsharp edges)

CA slowconvergence Pworks better

RIP CA M RIP CA (laserexperimentnumbers)

RIP CA

unknown two-or-more-hole physicalsupport loose(eg triangle compu-tational support)

| | RIP M ( TEM experimentholes) (laserexperiment numbers)(planned X-ray micapinholes)

RIP (Miao X-rayexperimenttransparent support)

unknown one-holephysical support(eg triangle compu-tational support)

| | RIP only for holewith low symmetryand sharp edgesslow convergence

M RIP only for holewith low symmetryand sharp edgesslow convergence

RIP only for holewith low symmetryand sharp edgesslow convergence

Phil

Tra

ns

R

Soc

L

on

d

A(2

002)

892 J C H Spence U Weierstall and M Howells

ulations and experiments The enotectiveness of dinoterent constraints in real space fordinoterent physical and computational supports is compared The physical supportis the support which exists in the experiment and which may or may not be accu-rately known The computational support is the support used in the HIO iterationsThe table shows experiments that have been performed in italics The shy rst columnshows dinoterent possibilities for the physical support the most favourable at the topthe worst at the bottom The computational support and known physical supportare identical The computational support used with the unknown physical supportis larger than the physical support Reconstructions with one-hole physical supportand loose computational support are only possible with weak or real objects andsupport holes with sharp edges and low symmetry and convergence are very slow insimulations A known two-or-more-hole support proved to be most favourable in allcases

7 Discussion and conclusions direg ractive imaging with other particles

In this paper we have reviewed the implications of Thomas Youngrsquos experiment formatter waves and in particular for electron interferometry By taking advantage ofthe very high quality of the coherent electron beam available in the modern shy eld-emission electron microscope a number of important results (such as the AharonovBohm enotect and the electron Sagnac enotect) have been established and the shy eldof atomic-resolution transmission electron microscopy phase-contrast imaging andelectron holography opened up All these methods continue to produce a large sci-entishy c pay-onot in microstructural materials characterization and discovery and inbiology A new lensless imaging method is then reviewed which may be understoodin part by analogy with electron holography This HIO algorithm uses an iterativeapproach to assign phases to dinotraction pattern intensities To evaluate its perfor-mance we give experimental examples of image reconstruction with both coherentelectrons and light together with simulations for the soft X-ray case Thus we haveshown experimentally that the HIO algorithm is capable of solving the phase problemfor scattering from non-periodic objects and of making quantitative measurements ofphase (modulo 2ordm ) in good agreement with independent measurements The beam-splitting arrangement of an interferometer is not required and dinotraction-limitedlensless imaging seems possible as a result This may open the way for imaging withthe many particles for which no lenses exist such as neutrons ballistic electrons insolids at low temperature or neutral atoms and ions either as free coherent beamsor condensates (Bright coherent sources have recently appeared for many of these(Pauly 2000)) The dinotering interactions of these particles may provide useful newprobes for chemistry materials science and biology For example the reduced radia-tion damage enotects of imaging organic material with low-energy electrons neutronsand helium atoms have recently been compared (Spence et al 1999) and HIO sim-ulations have been completed for neutron imaging of a diatom (Spence et al 2001)The very small refractive index of organic matter for neutrons limits resolution andcontrast for thin samples For low-energy ions or metastable helium strong samplebeam interactions must be considered Damage from neutral helium-atom imagingof molecules should be negligible however penetration is also negligible (producinga proshy le image) Detectors for coherent atom beams still present serious problemssince area detectors are inemacr cient For biology the weak atom interaction is desirable

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 893

to avoid damage however a strong interaction at the detector is also needed so thationization before detection must be arranged

For real objects we have seen that the resolution in images recovered by the HIOalgorithm is largely independent of the accuracy with which the support is knownThe addition of noise anotects the probability that the unique solution will be found (asmeasured by the error metric) rather than the resolution which is near dinotraction-limited For complex objects our experience indicates that a much more accurateknowledge of the physical support boundary is needed approaching the scale ofthe resolution sought The tight support and two-hole geometry (see Fienup (1987)for other geometries) needed for complex objects (complex refractive index) willbecome increasingly accessible with the development of new lithographic techniquesfor support construction This can be combined with independent characterization ofthe support shape and the scaling information provided by the known autocorrelationof the support Experimentally favourable cases for imaging with nanoscale resolutionat present require loose support which table 1 shows greatly favours real objectssuch as the charge density which dinotracts hard X-rays at energies well away fromabsorption edges or very thin samples for high-energy electrons Unfortunately beamcoherence is more readily obtained for soft X-rays

Using modern lithographic techniques it is becoming increasingly possible toestablish the experimental conditions of known support required by the HIO algo-rithm both for free particle beams and for electrons in solids at low temperaturesThus for example we have seen the algorithm used to determine the phase of wavefunctions for electrons in a one-dimensional YBCO Josephson junction using TEMimages of the structure to deshy ne the required support (boundary conditions) (Car-mody et al 2000) Given the fundamental role that phase has played historically inphysics and which Thomas Young was amongst the shy rst to appreciate any techniquecapable of exposing phase can be expected to yield many surprises in physics in thefuture

Supported by ARO award DAAD190010500 We thank Dr Q Chen for the use of macrgure 12 (fromWeierstall et al 2002)

References

Aharonov Y amp Bohm D 1959 Signimacrcance of the electromagnetic potentials in the quantumtheory Phys Rev 115 485

Allen L J Faulkner H M L Nugent K A Oxley M P amp Paganin D 2001 Phase retrievalfrom images in the presence of macrrst-order vortices Phys Rev E 63 037602

Bauschke H H et al 2002 (In the press)

Boersch H 1943 Fresnelsche Beugung im Elektronenmikroskop Phys Z 44 202211

Bruck Y amp Sodin L 1979 On the ambiguity of the image-reconstruction problem Opt Com-mun 30 304308

Carmody M Moeckly B H Merkle K amp Marks L 2000 Spatial variation of the current ingrain boundary Josephson junctions J Appl Phys 87 24542459

Coene W Janssen G Op de Beek M amp van Dyck D 1992 Phase retrieval through focusvariation for ultra-resolution in macreld emission transmission electron microscopy Phys RevLett 69 37433746

Crewe A V Eggenberger D N Wall J amp Welter L M 1968 Electron gun using a macreldemission source Rev Scient Instrum 39 576

Phil Trans R Soc Lond A (2002)

880 J C H Spence U Weierstall and M Howells

method which requires intensity measurements across two planes downstream ofthe object has been used for example to consider a non-interferometric variant ofthe AB enotect (Paganin 2001) (For a phase object a single plane is sumacr cient sinceunit intensity can be assumed everywhere in the object plane) The phase vorticeswhich occur at zeros of intensity have also attracted considerable interest (Allen etal 2001) So far it has been applied to cases using collimated illumination only sothat resolution is limited by detector pixel size however there seems no reason notto apply it to the case of diverging illumination which provides magnishy cation anda resolution limit imposed by the source size

Despite many simulations few experimental tests of the HIO algorithm haveappeared The algorithm iterates between real and reciprocal space (related by aFourier transform) imposing known information in each domain The moduli ofthe scattered amplitudes I(u) are assumed known and in real space the objectfunction T (r) is set to its known values outside the support (Here r and u aretwo-dimensional vectors T (r) is the ratio of the complex exit waveshy eld (the in-focusimage) to the incident wave function often assumed planar) The support need notalways be known very accurately and an estimate can be obtained from the 50contour of the autocorrelation function which is given by the Fourier transform ofI(u) Finally knowledge of the sign of the real (and possibly imaginary) parts of Tmay be used The introduction of a small amount of `feedbackrsquo (Fienup 1987) distin-guishes the HIO algorithm from the error-reduction algorithm (Gerchberg amp Saxton1972) Convergence is found to be improved further in three-dimensions (Miao et al 2001) For real objects (such as the charge density which dinotracts X-rays) a singleobject is sumacr cient and the computational boundary used in the iterations may bea dinoterent shape from the physical support which it includes When viewed as anoptimization process the algorithm appears to be capable of shy nding the thousands ofunknown parameters (the phases in a typical image) in the presence of noise withoutthe usual problems of local minima and stagnation It thus appears to solve a verylarge global optimization problem despite the application of a non-convex constraint(the known Fourier moduli) (A convex set of solutions is one for which all pointslying on any line joining any two points within the N -dimensional set lie inside theset a kidney-shaped set in two dimensions is non-convex N is the number of pixelsin the image If a unique solution can be shown to exist (as in this case) iterationbetween the boundaries of two convex sets guarantees convergence to this solutionwithout stagnation at local minima Hence convex constraints such as symmetryknown support or sign are sought in optimization problems) While inversion fromknown phase information is a convex process leading to a unique result (Lindaas etal 1998) inversion from intensity alone is non-convex so that local minima mayformally be expected in approaches based on numerical optimization (Stark 1987)The HIO algorithm has been analysed in considerable detail (Fienup 1982 Paganin2001) including treatments based on Bregman projections onto convex sets (Stark1987 Bauschke et al 2002) and those going back to Sayre (1952) which start fromthe realization that Bragg sampling undersamples the autocorrelation function ofa molecule in a crystal (Miao et al 2001) The number of Fourier series equationsis then compared with the number of unknowns as the support is made a progres-sively smaller fraction of the total dinotracting area The lowest spatial frequency inthe dinotraction pattern corresponds to a distance wider than the support Using this`oversamplingrsquo approach soft X-ray images of lithographed symbols were recently

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 881

reconstructed from far-shy eld scattered intensities in a lensless scattering experiment(Miao et al 1999) Spatial resolution is limited only by noise and wavelength in theseexperiments For technical reasons however the reconstruction was assisted by low-resolution optical images which provided the low-order spatial frequencies and theobject transmission function was of a special mask-like form having only two (com-plex) transmission values whose signs were known We have therefore conductedboth simulations and experiments for phase measurement in scattering of X-rayscoherent visible light and coherent electron beams to evaluate this HIO algorithmfor more general objects

4 Simulations for X-rays

Simulations have been undertaken for objects consisting of an array of randomlypositioned pinholes to determine if this image could be reconstructed from its softX-ray dinotraction pattern If so these holes could later be shy lled with structures ofinterest for reconstruction The area occupied by pinholes has been limited to thecoherence width of the beam by a 1 m m aperture This aperture will not fall exactly inthe plane of the pinholes therefore the wavefront at the pinholes will have undergoneFresnel dinotraction at the aperture and so will be complex Figure 1 shows the resultsof these simulations assuming a wavelength of 25 nm for the incident soft X-raybeam The pinholes are 10 nm in diameter After 150 iterations the HIO algorithmhas converged without stagnation to an image which is related to the object byinversion through the centre The modulus of the estimated object function was takenat each iteration (modulus constraint) since the object is assumed real Only a roughestimate of the size of the aperture is needed the computational support conditionapplied in the iterations is a triangle larger than the aperture The unimportantinversion ambiguity cannot be resolved by the HIO algorithm since an invertedimage our real computational support and the circular aperture all have the sameFourier modulus The Fresnel propagation makes this strictly a complex objecthowever we obtain an accurate reconstruction of the holes in the shy lm despite the useof a positivity (modulus) constraint in object space The phase of the complex wavetransmitted through the pinholes is not reconstructed due to the imposed positivity

5 Experiments with light

Coherent laser optical phase retrieval experiments have been performed using theHIO algorithm for several objects The phase was successfully retrieved for severaldinoterent support constraints and object types So far only opaque support con-straints (apertures) have been used since these present fewer problems with exces-sive central beam intensity (blooming) The source was a frequency-stabilized HeNe laser ( para = 6328 nm) collimated to provide the planar wavefronts needed at thesample for real objects (shy gure 2) Dinotracted light was focused with a lens onto aliquid-nitrogen-cooled charge-coupled device (CCD) camera and the digital imagereconstructed using the HIO algorithm in the Digital Micrograph script language

The shy rst object was a resolution test object with transparent numbers and linesin an opaque chrome shy lm on glass The illuminated area was limited by a thin300 m m diameter aperture in contact with the chrome shy lm Assuming the glass isregat to within less than para =10 across the illuminated area the chrome mask acts as a

Phil Trans R Soc Lond A (2002)

882 J C H Spence U Weierstall and M Howells

(a) (b)

(d )(c)

Figure 1 Reconstruction of simulated pinholes illuminated by a Fresnel direg racted X-ray wave-macreld (out-of-focus physical support) (a) Simulated random array of pinholes illuminated bywave amplitude which has been Fresnel propagated 200 nm from an aperture to the pinholesWavelength para = 25 nm pinhole size 10 nm aperture diameter 1 m m (b) 200 nm out-of-focuswavemacreld from 1 m m aperture incident on pinholes (c) The Fourier transform of (a) is a speckledAiryrsquo s disc Modulus used as input for the HIO algorithm (d) Reconstructed pinholes after 150iterations with modulus constraint (To enforce positivity in real space only the modulus ofthe pixel values inside the support area is retained during each iteration) A large triangularcomputational support was used The image is inverted this is an ambiguity which cannot beresolved by the HIO algorithm

real binary object so that the dinotraction pattern should be symmetric A coherentoptical image was also recorded with the camera moved into the image plane ofthe lens This is shown in shy gure 3a The camera was then moved to the back focalplane of the lens and the dinotraction pattern intensity I(u) was recorded (shy gure 3b)The autocorrelation FT I(u v) was used to determine the approximate size of theobject and a circular support half the size of the autocorrelation was used in the

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 883

f

double polarizer to adjust intensity

collimator

lens diffraction pattern

CCD detectorHendashNe laser= 6328 nml

object with support mask

Figure 2 Schematic of the arrangement used for optical experiments

(a) (b)300 m m

Figure 3 (a) Coherent optical image of optical target transparent letters and bars in an oth-erwise opaque chrome macrlm on glass A 300 m m aperture acts as the support (HeNe laserpara = 6328 nm) (b) Coherent laser optical direg raction pattern of object shown in (a)

computer The HIO algorithm was used to recover the phases of the object exitwave and the algorithm always converged after about 50 iterations with dinoterentinitial random phase sets (shy gure 4a) A larger triangular support constraint wasalso used to recover the image as shown in shy gure 4b The normalized root-mean-squared error in the object domain (Fienup 1987) (a measure of dinoterence betweenthe current image estimate and the known pixel values outside the support) did notdrop below 02 probably due to imperfections in the dinotractogram eg noise andlight reregections from the entrance glass surface of the CCD camera which createasymmetry in the dinotraction pattern (see shy gure 4b above centre) Despite theseproblems the reconstructed image looks almost identical to the coherent image inshy gure 3a

A strong phase object (introducing phase shifts greater than ordm =2) was made usinga support consisting of two separated areas (two apertures of 300 m m diameter sep-

Phil Trans R Soc Lond A (2002)

884 J C H Spence U Weierstall and M Howells

(a) (b)

Figure 4 (a) Modulus of reconstructed image wave using a tight circular support constraint(b) Modulus of reconstructed image wave using a loose triangular support constraint A positivityconstraint was used in real space

mica

no mica(a) (b)

Figure 5 (a) Coherent laser optical image of phase object with two-hole support The edgesof two mica plates are visible across the hole on the right (b) Laser optical direg raction patternobtained by merging three patterns with direg erent exposure times to avoid saturation of thecentral beam

arated by 350 m m) One hole was covered with two 5 m m thick mica sheets whichoverlap in one region The expected phase shift is calculated to be 9 ordm The coherentoptical image is shown in shy gure 5a The mica is transparent except for the edgeswhere Fresnel contrast enotects are visible

The resulting dinotraction pattern is shown in shy gure 5b The central beam intensityin the dinotraction pattern from this transparent object was too strong to be recordedsimultaneously with the weaker high spatial frequencies so that several exposureswere needed All exposures were merged to one pattern in which saturated pixelswere replaced by scaled pixels from the lower exposure patterns The computational

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 885

black = -

white = +

(a) (b)

(c)

Figure 6 (a) Modulus and (b) phase of reconstructed image wave (c) phase reconstructed fromshifted direg raction pattern shows equiphase contour lines Phase shift by mica ordm No sign ormodulus constraint

support size and orientation were derived from the autocorrelation The complexHIO-phase retrieval algorithm was used to recover the image wave phase and modulusshown in shy gure 6 The phase shift by the mica plates in the reconstruction is ordm which since the algorithm reconstructs phase modulo 2 ordm is in good agreement withour independent measurement of 9 ordm based on the measured thickness and knownrefractive index of mica Thus the HIO algorithm is capable of quantitative phasemeasurement

Experiments with a transparent physical support have been attempted using alimiting aperture upstream of the sample but have not been successful so far becauseof the high intensity in the centre of the dinotraction pattern which causes bloomingin the CCD chip This problem persists for three-dimensional objects however athree-quadrant mask in the object plane has been proposed to address this dimacr culty(Sayre et al 1998) (Beam stops cannot be placed near enough to the CCD toavoid generating additional fringes) Experiments using the known intensity envelopeof a focused beam as the support have also not been successful for such complexobjects there appears to be a need for sharp physical support edges perhaps requiredto generate reference edge waves (For real objects there is no such requirement)However a coherent probe spanning two `knownrsquo holes works well in simulation

Previous experiments have been based on objects of simple type such as binary

Phil Trans R Soc Lond A (2002)

886 J C H Spence U Weierstall and M Howells

(a)

(b)

Figure 7 (a) Coherent laser optical image of cheek cells in two-hole support (300 m m apertures)About seven cells clump together in the upper hole Only Fresnel edge contrast is visible dueto defocus the cell nuclei are visible as black dots (b) Laser optical direg raction pattern mergedfrom four direg erent exposures to avoid saturation of the central beam

objects which contain much less information than greyscale objects Experimentswith human cheek cells which serve as more complicated greyscale phase objectshave therefore been performed The cheek cells were positioned on a glass slide insidetwo 300 m m apertures separated by 300 m m The apertures act as the opaque supportconsisting of two separated parts as needed for reconstruction of complex objectsFigure 7a shows the coherent optical image of the slightly defocused cheek cells asformed with a lens Only Fresnel edge contrast is visible while the nuclei producesome absorption Figure 7b shows the optical dinotraction pattern merged from fourdinoterent exposures The size and orientation of the support were estimated fromthe autocorrelation The complex reconstruction of phase and amplitude after about50 HIO iterations using the support constraint is shown in shy gure 8 The cells arevisible with large-area phase contrast in the phase image and the phase shift of thecells is about ordm In the reconstructed amplitude only edge contrast is visible as inthe optical image

For a three-dimensional object extended along the optical axis the question arisesas to which plane in the object is reconstructed given that the detected intensity liesat inshy nity and contains contributions from scattering at every point along the axisin the object space A simulation to clarify this is shown in shy gure 9 The simulatedobject consists of two planes separated by 1 mm in the z-direction Plane 1 has a holecontaining a complex object (the number 1) while plane 2 has two holes containingcomplex objects (numbers 2 and 3) The object exit wave calculated on plane 2

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 887

(a) (b)

Figure 8 (a) Modulus and (b) phase of reconstructed image wave The modulus shows only edgecontrast whereas the phase shows the cells in large-area phase contrast The phase shift due tothe cells is about ordm No modulus or sign constraint

(shy gure 9a) shows plane 1 out of focus The modulus of the Fourier transform (Fouriermodulus) of this image wave is used as input for the HIO algorithm In the inversiondinoterent shapes of the supports used at each plane shown inset can distinguish thetwo object planes If a plane is assumed unknown a larger support hole is usedthat contains the physical support Depending on which support plane is assumed asknown (inset) in the iteration (plane 1 support known and plane 2 support unknownor vice versa) the other plane is always reconstructed out of focus by +1 or iexcl 1 mm(shy gure 9c d) That means that in the case of known support on plane 1 the algorithmreconstructs the wave as seen through an imaging lens with out-of-focus images ofthe holes on plane 2 As expected the depth of focus for interactive inversion is givenby the expression for an ideal lens which imposes a dinotraction limit equal to that ofthe highest spatial frequency recorded in the dinotraction pattern

To verify this experimentally a pin was mounted 12 mm upstream of a pair of300 m m apertures The coherent optical image is shown in shy gure 10a A Fresnel fringesurrounds the pin since the lens is focused on the apertures The image was recoveredfrom the recorded dinotraction pattern intensity (shy gure 10b) Again three dinotractionpatterns with dinoterent exposure times had to be merged to avoid pixel saturationand blooming The reconstructed image modulus is shown in shy gure 11a where the

Phil Trans R Soc Lond A (2002)

888 J C H Spence U Weierstall and M Howells

illuminationdirection

plane 1

1 mm

(a) (b)

(c) (d)

Figure 9 (a) Schematic view of a simulated object consisting of two planes separated by 1 mm inthe direction of light propagation Plane 1 has a hole containing a complex object (number 1) andplane 2 has two holes containing complex objects (numbers 2 and 3) (b) Modulus of complexexit wave on plane 2 (c) Iterative reconstruction with plane 2 support assumed known (thesupport shape used is shown in the top right-hand corner) Plane 1 is out of focus by 1 mm(d) Iterative reconstruction with plane 1 support assumed known (the support shape is shownin the top right-hand corner) Plane 2 is out of focus by iexcl1 mm

pin appears out of focus (Fresnel fringe barely visible) because the aperture shapewas imposed as support The pin can be brought into focus by using the appropriateFresnel propagator with centz = 12 mm and para = 6328 nm taking the apertures outof focus (see shy gure 11b) Thus as expected the reconstructed plane is always thatplane in which the support condition is imposed where the algorithm shy nds a uniquesolution subject to the given boundary condition Propagation to other planes maybe performed subsequently since the complex waveshy eld is obtained subject to thedepth-of-focus limitation The relative sideways shift of the pin inside the aperturebetween shy gure 10a and shy gure 11a is due to a slight unintentional tilt of the object(pin plus pinhole) between recording the image and the dinotraction pattern

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 889

(a) (b)

Figure 10 (a) Coherent laser optical image of pin in front of aperture (b) Laser optical direg ractionpattern obtained by merging three patterns with direg erent exposure times to avoid saturation ofthe central beam

(a) (b)

Figure 11 (a) Reconstructed modulus Only the hole with the out-of-focus pin is shown(b) Numerical back propagation of reconstructed complex wavemacreld by 12 mm to plane of pinPin appears in focus Support constraint only

6 Experiments with coherent electrons

Using a special support mask made by electron-beam lithography experiments havealso been undertaken using the HIO algorithm to retrieve the phases from experimen-tal transmission electron dinotraction patterns (Weierstall et al 2002) Images mightthen be reconstructed that are free of electron lens aberrations The support maskcontains hole pairs of various sizes with pairs separated by 1 m m The separationbetween holes of a pair is ca 50 nm less than the coherence width of the illuminationbeam The holes irregular in shape form the object we wish to reconstruct in futurework we plan to introduce nanoscale objects into the holes Figure 12a below showsthe experimental dinotraction pattern from two holes shown in shy gure 12c obtained

Phil Trans R Soc Lond A (2002)

890 J C H Spence U Weierstall and M Howells

20 nm

(a) (b)

(c)

Figure 12 (a) Experimental electron direg raction pattern from two nanoscale holes in an opaquemask (shown in (c)) recorded using a macreld-emission electron microscope (CM200 FEG at 40 kV1k pound1k CCD with 24 m m pixels camera length 611 m) Youngrsquo s fringes from the two holes arevisible these fringes encode phase information (b) Reconstruction of image of holes from thedireg raction pattern (a) after 100 iterations of the HIO algorithm Modulus constraint applied inimage space with large triangular computational support The hole shape is comparable with theTEM image in (c) (c) Conventional TEM image of holes that created the direg raction pattern (a)

with a Philips CM 200 shy eld-emission electron microscope at 40 kV The acceleratingvoltage was lowered to 40 kV to make the multi-layered support opaque ModulatedYoungrsquos fringes are clearly seen

The resulting image after 100 HIO iterations is shown in shy gure 12b The holesreconstructed from the dinotraction pattern modulus look similar to the TEM imageof the holes in shy gure 12c but the resolution of the reconstruction is lower thanthe normal TEM image resolution High spatial frequency information from thehole edge is missing in the experimental dinotraction pattern due perhaps to drift ofthe imaging lenses during the exposure time

Table 1 shows a summary chart of all the conditions so far examined with HIO sim-

Phil Trans R Soc Lond A (2002)

Phase

recovery

an

dlen

slessim

agin

g891

Table 1 Summary chart showing direg erent conditions for phase recovery with HIO algorithm

( CArsquo denotes the complex algorithm (uses direg racted intensity and support constraint in real space) RIPrsquo denotes the complex algorithm withpositivity constraint of Re ordf Im ordf (uses direg racted intensity and support constraint in real space negative Re ordf and Im ordf inside the supportarea are set positive) Mrsquo denotes the modulus constraint on ordf in real space (uses direg racted intensity and support constraint in real spacecomplex numbers inside the support area are replaced by their modulus to enforce positivity) Prsquo denotes the phase constraint on ordf in realspace (uses direg racted intensity and support constraint in real space the phase is extracted and retained from complex numbers inside thesupport area modulus is extracted and set to 1 inside the support area))

general pure strong pure weakrsquo real weak-phasecomplex object phase object phase object (amplitude) object withordf = A exp(iiquest ) ordf = exp(i iquest (r)) iquest lt ordm =2 object absorption

known two-holephysical support

CA (laser experimentpin cheek cells)

CA P (laserexperiment mica)

RIP CA M RIP CA RIP CA

known one-holephysical support

CA only for certainsupport shapes(eg triangle withsharp edges)

CA slowconvergence Pworks better

RIP CA M RIP CA (laserexperimentnumbers)

RIP CA

unknown two-or-more-hole physicalsupport loose(eg triangle compu-tational support)

| | RIP M ( TEM experimentholes) (laserexperiment numbers)(planned X-ray micapinholes)

RIP (Miao X-rayexperimenttransparent support)

unknown one-holephysical support(eg triangle compu-tational support)

| | RIP only for holewith low symmetryand sharp edgesslow convergence

M RIP only for holewith low symmetryand sharp edgesslow convergence

RIP only for holewith low symmetryand sharp edgesslow convergence

Phil

Tra

ns

R

Soc

L

on

d

A(2

002)

892 J C H Spence U Weierstall and M Howells

ulations and experiments The enotectiveness of dinoterent constraints in real space fordinoterent physical and computational supports is compared The physical supportis the support which exists in the experiment and which may or may not be accu-rately known The computational support is the support used in the HIO iterationsThe table shows experiments that have been performed in italics The shy rst columnshows dinoterent possibilities for the physical support the most favourable at the topthe worst at the bottom The computational support and known physical supportare identical The computational support used with the unknown physical supportis larger than the physical support Reconstructions with one-hole physical supportand loose computational support are only possible with weak or real objects andsupport holes with sharp edges and low symmetry and convergence are very slow insimulations A known two-or-more-hole support proved to be most favourable in allcases

7 Discussion and conclusions direg ractive imaging with other particles

In this paper we have reviewed the implications of Thomas Youngrsquos experiment formatter waves and in particular for electron interferometry By taking advantage ofthe very high quality of the coherent electron beam available in the modern shy eld-emission electron microscope a number of important results (such as the AharonovBohm enotect and the electron Sagnac enotect) have been established and the shy eldof atomic-resolution transmission electron microscopy phase-contrast imaging andelectron holography opened up All these methods continue to produce a large sci-entishy c pay-onot in microstructural materials characterization and discovery and inbiology A new lensless imaging method is then reviewed which may be understoodin part by analogy with electron holography This HIO algorithm uses an iterativeapproach to assign phases to dinotraction pattern intensities To evaluate its perfor-mance we give experimental examples of image reconstruction with both coherentelectrons and light together with simulations for the soft X-ray case Thus we haveshown experimentally that the HIO algorithm is capable of solving the phase problemfor scattering from non-periodic objects and of making quantitative measurements ofphase (modulo 2ordm ) in good agreement with independent measurements The beam-splitting arrangement of an interferometer is not required and dinotraction-limitedlensless imaging seems possible as a result This may open the way for imaging withthe many particles for which no lenses exist such as neutrons ballistic electrons insolids at low temperature or neutral atoms and ions either as free coherent beamsor condensates (Bright coherent sources have recently appeared for many of these(Pauly 2000)) The dinotering interactions of these particles may provide useful newprobes for chemistry materials science and biology For example the reduced radia-tion damage enotects of imaging organic material with low-energy electrons neutronsand helium atoms have recently been compared (Spence et al 1999) and HIO sim-ulations have been completed for neutron imaging of a diatom (Spence et al 2001)The very small refractive index of organic matter for neutrons limits resolution andcontrast for thin samples For low-energy ions or metastable helium strong samplebeam interactions must be considered Damage from neutral helium-atom imagingof molecules should be negligible however penetration is also negligible (producinga proshy le image) Detectors for coherent atom beams still present serious problemssince area detectors are inemacr cient For biology the weak atom interaction is desirable

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 893

to avoid damage however a strong interaction at the detector is also needed so thationization before detection must be arranged

For real objects we have seen that the resolution in images recovered by the HIOalgorithm is largely independent of the accuracy with which the support is knownThe addition of noise anotects the probability that the unique solution will be found (asmeasured by the error metric) rather than the resolution which is near dinotraction-limited For complex objects our experience indicates that a much more accurateknowledge of the physical support boundary is needed approaching the scale ofthe resolution sought The tight support and two-hole geometry (see Fienup (1987)for other geometries) needed for complex objects (complex refractive index) willbecome increasingly accessible with the development of new lithographic techniquesfor support construction This can be combined with independent characterization ofthe support shape and the scaling information provided by the known autocorrelationof the support Experimentally favourable cases for imaging with nanoscale resolutionat present require loose support which table 1 shows greatly favours real objectssuch as the charge density which dinotracts hard X-rays at energies well away fromabsorption edges or very thin samples for high-energy electrons Unfortunately beamcoherence is more readily obtained for soft X-rays

Using modern lithographic techniques it is becoming increasingly possible toestablish the experimental conditions of known support required by the HIO algo-rithm both for free particle beams and for electrons in solids at low temperaturesThus for example we have seen the algorithm used to determine the phase of wavefunctions for electrons in a one-dimensional YBCO Josephson junction using TEMimages of the structure to deshy ne the required support (boundary conditions) (Car-mody et al 2000) Given the fundamental role that phase has played historically inphysics and which Thomas Young was amongst the shy rst to appreciate any techniquecapable of exposing phase can be expected to yield many surprises in physics in thefuture

Supported by ARO award DAAD190010500 We thank Dr Q Chen for the use of macrgure 12 (fromWeierstall et al 2002)

References

Aharonov Y amp Bohm D 1959 Signimacrcance of the electromagnetic potentials in the quantumtheory Phys Rev 115 485

Allen L J Faulkner H M L Nugent K A Oxley M P amp Paganin D 2001 Phase retrievalfrom images in the presence of macrrst-order vortices Phys Rev E 63 037602

Bauschke H H et al 2002 (In the press)

Boersch H 1943 Fresnelsche Beugung im Elektronenmikroskop Phys Z 44 202211

Bruck Y amp Sodin L 1979 On the ambiguity of the image-reconstruction problem Opt Com-mun 30 304308

Carmody M Moeckly B H Merkle K amp Marks L 2000 Spatial variation of the current ingrain boundary Josephson junctions J Appl Phys 87 24542459

Coene W Janssen G Op de Beek M amp van Dyck D 1992 Phase retrieval through focusvariation for ultra-resolution in macreld emission transmission electron microscopy Phys RevLett 69 37433746

Crewe A V Eggenberger D N Wall J amp Welter L M 1968 Electron gun using a macreldemission source Rev Scient Instrum 39 576

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 881

reconstructed from far-shy eld scattered intensities in a lensless scattering experiment(Miao et al 1999) Spatial resolution is limited only by noise and wavelength in theseexperiments For technical reasons however the reconstruction was assisted by low-resolution optical images which provided the low-order spatial frequencies and theobject transmission function was of a special mask-like form having only two (com-plex) transmission values whose signs were known We have therefore conductedboth simulations and experiments for phase measurement in scattering of X-rayscoherent visible light and coherent electron beams to evaluate this HIO algorithmfor more general objects

4 Simulations for X-rays

Simulations have been undertaken for objects consisting of an array of randomlypositioned pinholes to determine if this image could be reconstructed from its softX-ray dinotraction pattern If so these holes could later be shy lled with structures ofinterest for reconstruction The area occupied by pinholes has been limited to thecoherence width of the beam by a 1 m m aperture This aperture will not fall exactly inthe plane of the pinholes therefore the wavefront at the pinholes will have undergoneFresnel dinotraction at the aperture and so will be complex Figure 1 shows the resultsof these simulations assuming a wavelength of 25 nm for the incident soft X-raybeam The pinholes are 10 nm in diameter After 150 iterations the HIO algorithmhas converged without stagnation to an image which is related to the object byinversion through the centre The modulus of the estimated object function was takenat each iteration (modulus constraint) since the object is assumed real Only a roughestimate of the size of the aperture is needed the computational support conditionapplied in the iterations is a triangle larger than the aperture The unimportantinversion ambiguity cannot be resolved by the HIO algorithm since an invertedimage our real computational support and the circular aperture all have the sameFourier modulus The Fresnel propagation makes this strictly a complex objecthowever we obtain an accurate reconstruction of the holes in the shy lm despite the useof a positivity (modulus) constraint in object space The phase of the complex wavetransmitted through the pinholes is not reconstructed due to the imposed positivity

5 Experiments with light

Coherent laser optical phase retrieval experiments have been performed using theHIO algorithm for several objects The phase was successfully retrieved for severaldinoterent support constraints and object types So far only opaque support con-straints (apertures) have been used since these present fewer problems with exces-sive central beam intensity (blooming) The source was a frequency-stabilized HeNe laser ( para = 6328 nm) collimated to provide the planar wavefronts needed at thesample for real objects (shy gure 2) Dinotracted light was focused with a lens onto aliquid-nitrogen-cooled charge-coupled device (CCD) camera and the digital imagereconstructed using the HIO algorithm in the Digital Micrograph script language

The shy rst object was a resolution test object with transparent numbers and linesin an opaque chrome shy lm on glass The illuminated area was limited by a thin300 m m diameter aperture in contact with the chrome shy lm Assuming the glass isregat to within less than para =10 across the illuminated area the chrome mask acts as a

Phil Trans R Soc Lond A (2002)

882 J C H Spence U Weierstall and M Howells

(a) (b)

(d )(c)

Figure 1 Reconstruction of simulated pinholes illuminated by a Fresnel direg racted X-ray wave-macreld (out-of-focus physical support) (a) Simulated random array of pinholes illuminated bywave amplitude which has been Fresnel propagated 200 nm from an aperture to the pinholesWavelength para = 25 nm pinhole size 10 nm aperture diameter 1 m m (b) 200 nm out-of-focuswavemacreld from 1 m m aperture incident on pinholes (c) The Fourier transform of (a) is a speckledAiryrsquo s disc Modulus used as input for the HIO algorithm (d) Reconstructed pinholes after 150iterations with modulus constraint (To enforce positivity in real space only the modulus ofthe pixel values inside the support area is retained during each iteration) A large triangularcomputational support was used The image is inverted this is an ambiguity which cannot beresolved by the HIO algorithm

real binary object so that the dinotraction pattern should be symmetric A coherentoptical image was also recorded with the camera moved into the image plane ofthe lens This is shown in shy gure 3a The camera was then moved to the back focalplane of the lens and the dinotraction pattern intensity I(u) was recorded (shy gure 3b)The autocorrelation FT I(u v) was used to determine the approximate size of theobject and a circular support half the size of the autocorrelation was used in the

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 883

f

double polarizer to adjust intensity

collimator

lens diffraction pattern

CCD detectorHendashNe laser= 6328 nml

object with support mask

Figure 2 Schematic of the arrangement used for optical experiments

(a) (b)300 m m

Figure 3 (a) Coherent optical image of optical target transparent letters and bars in an oth-erwise opaque chrome macrlm on glass A 300 m m aperture acts as the support (HeNe laserpara = 6328 nm) (b) Coherent laser optical direg raction pattern of object shown in (a)

computer The HIO algorithm was used to recover the phases of the object exitwave and the algorithm always converged after about 50 iterations with dinoterentinitial random phase sets (shy gure 4a) A larger triangular support constraint wasalso used to recover the image as shown in shy gure 4b The normalized root-mean-squared error in the object domain (Fienup 1987) (a measure of dinoterence betweenthe current image estimate and the known pixel values outside the support) did notdrop below 02 probably due to imperfections in the dinotractogram eg noise andlight reregections from the entrance glass surface of the CCD camera which createasymmetry in the dinotraction pattern (see shy gure 4b above centre) Despite theseproblems the reconstructed image looks almost identical to the coherent image inshy gure 3a

A strong phase object (introducing phase shifts greater than ordm =2) was made usinga support consisting of two separated areas (two apertures of 300 m m diameter sep-

Phil Trans R Soc Lond A (2002)

884 J C H Spence U Weierstall and M Howells

(a) (b)

Figure 4 (a) Modulus of reconstructed image wave using a tight circular support constraint(b) Modulus of reconstructed image wave using a loose triangular support constraint A positivityconstraint was used in real space

mica

no mica(a) (b)

Figure 5 (a) Coherent laser optical image of phase object with two-hole support The edgesof two mica plates are visible across the hole on the right (b) Laser optical direg raction patternobtained by merging three patterns with direg erent exposure times to avoid saturation of thecentral beam

arated by 350 m m) One hole was covered with two 5 m m thick mica sheets whichoverlap in one region The expected phase shift is calculated to be 9 ordm The coherentoptical image is shown in shy gure 5a The mica is transparent except for the edgeswhere Fresnel contrast enotects are visible

The resulting dinotraction pattern is shown in shy gure 5b The central beam intensityin the dinotraction pattern from this transparent object was too strong to be recordedsimultaneously with the weaker high spatial frequencies so that several exposureswere needed All exposures were merged to one pattern in which saturated pixelswere replaced by scaled pixels from the lower exposure patterns The computational

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 885

black = -

white = +

(a) (b)

(c)

Figure 6 (a) Modulus and (b) phase of reconstructed image wave (c) phase reconstructed fromshifted direg raction pattern shows equiphase contour lines Phase shift by mica ordm No sign ormodulus constraint

support size and orientation were derived from the autocorrelation The complexHIO-phase retrieval algorithm was used to recover the image wave phase and modulusshown in shy gure 6 The phase shift by the mica plates in the reconstruction is ordm which since the algorithm reconstructs phase modulo 2 ordm is in good agreement withour independent measurement of 9 ordm based on the measured thickness and knownrefractive index of mica Thus the HIO algorithm is capable of quantitative phasemeasurement

Experiments with a transparent physical support have been attempted using alimiting aperture upstream of the sample but have not been successful so far becauseof the high intensity in the centre of the dinotraction pattern which causes bloomingin the CCD chip This problem persists for three-dimensional objects however athree-quadrant mask in the object plane has been proposed to address this dimacr culty(Sayre et al 1998) (Beam stops cannot be placed near enough to the CCD toavoid generating additional fringes) Experiments using the known intensity envelopeof a focused beam as the support have also not been successful for such complexobjects there appears to be a need for sharp physical support edges perhaps requiredto generate reference edge waves (For real objects there is no such requirement)However a coherent probe spanning two `knownrsquo holes works well in simulation

Previous experiments have been based on objects of simple type such as binary

Phil Trans R Soc Lond A (2002)

886 J C H Spence U Weierstall and M Howells

(a)

(b)

Figure 7 (a) Coherent laser optical image of cheek cells in two-hole support (300 m m apertures)About seven cells clump together in the upper hole Only Fresnel edge contrast is visible dueto defocus the cell nuclei are visible as black dots (b) Laser optical direg raction pattern mergedfrom four direg erent exposures to avoid saturation of the central beam

objects which contain much less information than greyscale objects Experimentswith human cheek cells which serve as more complicated greyscale phase objectshave therefore been performed The cheek cells were positioned on a glass slide insidetwo 300 m m apertures separated by 300 m m The apertures act as the opaque supportconsisting of two separated parts as needed for reconstruction of complex objectsFigure 7a shows the coherent optical image of the slightly defocused cheek cells asformed with a lens Only Fresnel edge contrast is visible while the nuclei producesome absorption Figure 7b shows the optical dinotraction pattern merged from fourdinoterent exposures The size and orientation of the support were estimated fromthe autocorrelation The complex reconstruction of phase and amplitude after about50 HIO iterations using the support constraint is shown in shy gure 8 The cells arevisible with large-area phase contrast in the phase image and the phase shift of thecells is about ordm In the reconstructed amplitude only edge contrast is visible as inthe optical image

For a three-dimensional object extended along the optical axis the question arisesas to which plane in the object is reconstructed given that the detected intensity liesat inshy nity and contains contributions from scattering at every point along the axisin the object space A simulation to clarify this is shown in shy gure 9 The simulatedobject consists of two planes separated by 1 mm in the z-direction Plane 1 has a holecontaining a complex object (the number 1) while plane 2 has two holes containingcomplex objects (numbers 2 and 3) The object exit wave calculated on plane 2

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 887

(a) (b)

Figure 8 (a) Modulus and (b) phase of reconstructed image wave The modulus shows only edgecontrast whereas the phase shows the cells in large-area phase contrast The phase shift due tothe cells is about ordm No modulus or sign constraint

(shy gure 9a) shows plane 1 out of focus The modulus of the Fourier transform (Fouriermodulus) of this image wave is used as input for the HIO algorithm In the inversiondinoterent shapes of the supports used at each plane shown inset can distinguish thetwo object planes If a plane is assumed unknown a larger support hole is usedthat contains the physical support Depending on which support plane is assumed asknown (inset) in the iteration (plane 1 support known and plane 2 support unknownor vice versa) the other plane is always reconstructed out of focus by +1 or iexcl 1 mm(shy gure 9c d) That means that in the case of known support on plane 1 the algorithmreconstructs the wave as seen through an imaging lens with out-of-focus images ofthe holes on plane 2 As expected the depth of focus for interactive inversion is givenby the expression for an ideal lens which imposes a dinotraction limit equal to that ofthe highest spatial frequency recorded in the dinotraction pattern

To verify this experimentally a pin was mounted 12 mm upstream of a pair of300 m m apertures The coherent optical image is shown in shy gure 10a A Fresnel fringesurrounds the pin since the lens is focused on the apertures The image was recoveredfrom the recorded dinotraction pattern intensity (shy gure 10b) Again three dinotractionpatterns with dinoterent exposure times had to be merged to avoid pixel saturationand blooming The reconstructed image modulus is shown in shy gure 11a where the

Phil Trans R Soc Lond A (2002)

888 J C H Spence U Weierstall and M Howells

illuminationdirection

plane 1

1 mm

(a) (b)

(c) (d)

Figure 9 (a) Schematic view of a simulated object consisting of two planes separated by 1 mm inthe direction of light propagation Plane 1 has a hole containing a complex object (number 1) andplane 2 has two holes containing complex objects (numbers 2 and 3) (b) Modulus of complexexit wave on plane 2 (c) Iterative reconstruction with plane 2 support assumed known (thesupport shape used is shown in the top right-hand corner) Plane 1 is out of focus by 1 mm(d) Iterative reconstruction with plane 1 support assumed known (the support shape is shownin the top right-hand corner) Plane 2 is out of focus by iexcl1 mm

pin appears out of focus (Fresnel fringe barely visible) because the aperture shapewas imposed as support The pin can be brought into focus by using the appropriateFresnel propagator with centz = 12 mm and para = 6328 nm taking the apertures outof focus (see shy gure 11b) Thus as expected the reconstructed plane is always thatplane in which the support condition is imposed where the algorithm shy nds a uniquesolution subject to the given boundary condition Propagation to other planes maybe performed subsequently since the complex waveshy eld is obtained subject to thedepth-of-focus limitation The relative sideways shift of the pin inside the aperturebetween shy gure 10a and shy gure 11a is due to a slight unintentional tilt of the object(pin plus pinhole) between recording the image and the dinotraction pattern

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 889

(a) (b)

Figure 10 (a) Coherent laser optical image of pin in front of aperture (b) Laser optical direg ractionpattern obtained by merging three patterns with direg erent exposure times to avoid saturation ofthe central beam

(a) (b)

Figure 11 (a) Reconstructed modulus Only the hole with the out-of-focus pin is shown(b) Numerical back propagation of reconstructed complex wavemacreld by 12 mm to plane of pinPin appears in focus Support constraint only

6 Experiments with coherent electrons

Using a special support mask made by electron-beam lithography experiments havealso been undertaken using the HIO algorithm to retrieve the phases from experimen-tal transmission electron dinotraction patterns (Weierstall et al 2002) Images mightthen be reconstructed that are free of electron lens aberrations The support maskcontains hole pairs of various sizes with pairs separated by 1 m m The separationbetween holes of a pair is ca 50 nm less than the coherence width of the illuminationbeam The holes irregular in shape form the object we wish to reconstruct in futurework we plan to introduce nanoscale objects into the holes Figure 12a below showsthe experimental dinotraction pattern from two holes shown in shy gure 12c obtained

Phil Trans R Soc Lond A (2002)

890 J C H Spence U Weierstall and M Howells

20 nm

(a) (b)

(c)

Figure 12 (a) Experimental electron direg raction pattern from two nanoscale holes in an opaquemask (shown in (c)) recorded using a macreld-emission electron microscope (CM200 FEG at 40 kV1k pound1k CCD with 24 m m pixels camera length 611 m) Youngrsquo s fringes from the two holes arevisible these fringes encode phase information (b) Reconstruction of image of holes from thedireg raction pattern (a) after 100 iterations of the HIO algorithm Modulus constraint applied inimage space with large triangular computational support The hole shape is comparable with theTEM image in (c) (c) Conventional TEM image of holes that created the direg raction pattern (a)

with a Philips CM 200 shy eld-emission electron microscope at 40 kV The acceleratingvoltage was lowered to 40 kV to make the multi-layered support opaque ModulatedYoungrsquos fringes are clearly seen

The resulting image after 100 HIO iterations is shown in shy gure 12b The holesreconstructed from the dinotraction pattern modulus look similar to the TEM imageof the holes in shy gure 12c but the resolution of the reconstruction is lower thanthe normal TEM image resolution High spatial frequency information from thehole edge is missing in the experimental dinotraction pattern due perhaps to drift ofthe imaging lenses during the exposure time

Table 1 shows a summary chart of all the conditions so far examined with HIO sim-

Phil Trans R Soc Lond A (2002)

Phase

recovery

an

dlen

slessim

agin

g891

Table 1 Summary chart showing direg erent conditions for phase recovery with HIO algorithm

( CArsquo denotes the complex algorithm (uses direg racted intensity and support constraint in real space) RIPrsquo denotes the complex algorithm withpositivity constraint of Re ordf Im ordf (uses direg racted intensity and support constraint in real space negative Re ordf and Im ordf inside the supportarea are set positive) Mrsquo denotes the modulus constraint on ordf in real space (uses direg racted intensity and support constraint in real spacecomplex numbers inside the support area are replaced by their modulus to enforce positivity) Prsquo denotes the phase constraint on ordf in realspace (uses direg racted intensity and support constraint in real space the phase is extracted and retained from complex numbers inside thesupport area modulus is extracted and set to 1 inside the support area))

general pure strong pure weakrsquo real weak-phasecomplex object phase object phase object (amplitude) object withordf = A exp(iiquest ) ordf = exp(i iquest (r)) iquest lt ordm =2 object absorption

known two-holephysical support

CA (laser experimentpin cheek cells)

CA P (laserexperiment mica)

RIP CA M RIP CA RIP CA

known one-holephysical support

CA only for certainsupport shapes(eg triangle withsharp edges)

CA slowconvergence Pworks better

RIP CA M RIP CA (laserexperimentnumbers)

RIP CA

unknown two-or-more-hole physicalsupport loose(eg triangle compu-tational support)

| | RIP M ( TEM experimentholes) (laserexperiment numbers)(planned X-ray micapinholes)

RIP (Miao X-rayexperimenttransparent support)

unknown one-holephysical support(eg triangle compu-tational support)

| | RIP only for holewith low symmetryand sharp edgesslow convergence

M RIP only for holewith low symmetryand sharp edgesslow convergence

RIP only for holewith low symmetryand sharp edgesslow convergence

Phil

Tra

ns

R

Soc

L

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892 J C H Spence U Weierstall and M Howells

ulations and experiments The enotectiveness of dinoterent constraints in real space fordinoterent physical and computational supports is compared The physical supportis the support which exists in the experiment and which may or may not be accu-rately known The computational support is the support used in the HIO iterationsThe table shows experiments that have been performed in italics The shy rst columnshows dinoterent possibilities for the physical support the most favourable at the topthe worst at the bottom The computational support and known physical supportare identical The computational support used with the unknown physical supportis larger than the physical support Reconstructions with one-hole physical supportand loose computational support are only possible with weak or real objects andsupport holes with sharp edges and low symmetry and convergence are very slow insimulations A known two-or-more-hole support proved to be most favourable in allcases

7 Discussion and conclusions direg ractive imaging with other particles

In this paper we have reviewed the implications of Thomas Youngrsquos experiment formatter waves and in particular for electron interferometry By taking advantage ofthe very high quality of the coherent electron beam available in the modern shy eld-emission electron microscope a number of important results (such as the AharonovBohm enotect and the electron Sagnac enotect) have been established and the shy eldof atomic-resolution transmission electron microscopy phase-contrast imaging andelectron holography opened up All these methods continue to produce a large sci-entishy c pay-onot in microstructural materials characterization and discovery and inbiology A new lensless imaging method is then reviewed which may be understoodin part by analogy with electron holography This HIO algorithm uses an iterativeapproach to assign phases to dinotraction pattern intensities To evaluate its perfor-mance we give experimental examples of image reconstruction with both coherentelectrons and light together with simulations for the soft X-ray case Thus we haveshown experimentally that the HIO algorithm is capable of solving the phase problemfor scattering from non-periodic objects and of making quantitative measurements ofphase (modulo 2ordm ) in good agreement with independent measurements The beam-splitting arrangement of an interferometer is not required and dinotraction-limitedlensless imaging seems possible as a result This may open the way for imaging withthe many particles for which no lenses exist such as neutrons ballistic electrons insolids at low temperature or neutral atoms and ions either as free coherent beamsor condensates (Bright coherent sources have recently appeared for many of these(Pauly 2000)) The dinotering interactions of these particles may provide useful newprobes for chemistry materials science and biology For example the reduced radia-tion damage enotects of imaging organic material with low-energy electrons neutronsand helium atoms have recently been compared (Spence et al 1999) and HIO sim-ulations have been completed for neutron imaging of a diatom (Spence et al 2001)The very small refractive index of organic matter for neutrons limits resolution andcontrast for thin samples For low-energy ions or metastable helium strong samplebeam interactions must be considered Damage from neutral helium-atom imagingof molecules should be negligible however penetration is also negligible (producinga proshy le image) Detectors for coherent atom beams still present serious problemssince area detectors are inemacr cient For biology the weak atom interaction is desirable

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 893

to avoid damage however a strong interaction at the detector is also needed so thationization before detection must be arranged

For real objects we have seen that the resolution in images recovered by the HIOalgorithm is largely independent of the accuracy with which the support is knownThe addition of noise anotects the probability that the unique solution will be found (asmeasured by the error metric) rather than the resolution which is near dinotraction-limited For complex objects our experience indicates that a much more accurateknowledge of the physical support boundary is needed approaching the scale ofthe resolution sought The tight support and two-hole geometry (see Fienup (1987)for other geometries) needed for complex objects (complex refractive index) willbecome increasingly accessible with the development of new lithographic techniquesfor support construction This can be combined with independent characterization ofthe support shape and the scaling information provided by the known autocorrelationof the support Experimentally favourable cases for imaging with nanoscale resolutionat present require loose support which table 1 shows greatly favours real objectssuch as the charge density which dinotracts hard X-rays at energies well away fromabsorption edges or very thin samples for high-energy electrons Unfortunately beamcoherence is more readily obtained for soft X-rays

Using modern lithographic techniques it is becoming increasingly possible toestablish the experimental conditions of known support required by the HIO algo-rithm both for free particle beams and for electrons in solids at low temperaturesThus for example we have seen the algorithm used to determine the phase of wavefunctions for electrons in a one-dimensional YBCO Josephson junction using TEMimages of the structure to deshy ne the required support (boundary conditions) (Car-mody et al 2000) Given the fundamental role that phase has played historically inphysics and which Thomas Young was amongst the shy rst to appreciate any techniquecapable of exposing phase can be expected to yield many surprises in physics in thefuture

Supported by ARO award DAAD190010500 We thank Dr Q Chen for the use of macrgure 12 (fromWeierstall et al 2002)

References

Aharonov Y amp Bohm D 1959 Signimacrcance of the electromagnetic potentials in the quantumtheory Phys Rev 115 485

Allen L J Faulkner H M L Nugent K A Oxley M P amp Paganin D 2001 Phase retrievalfrom images in the presence of macrrst-order vortices Phys Rev E 63 037602

Bauschke H H et al 2002 (In the press)

Boersch H 1943 Fresnelsche Beugung im Elektronenmikroskop Phys Z 44 202211

Bruck Y amp Sodin L 1979 On the ambiguity of the image-reconstruction problem Opt Com-mun 30 304308

Carmody M Moeckly B H Merkle K amp Marks L 2000 Spatial variation of the current ingrain boundary Josephson junctions J Appl Phys 87 24542459

Coene W Janssen G Op de Beek M amp van Dyck D 1992 Phase retrieval through focusvariation for ultra-resolution in macreld emission transmission electron microscopy Phys RevLett 69 37433746

Crewe A V Eggenberger D N Wall J amp Welter L M 1968 Electron gun using a macreldemission source Rev Scient Instrum 39 576

Phil Trans R Soc Lond A (2002)

882 J C H Spence U Weierstall and M Howells

(a) (b)

(d )(c)

Figure 1 Reconstruction of simulated pinholes illuminated by a Fresnel direg racted X-ray wave-macreld (out-of-focus physical support) (a) Simulated random array of pinholes illuminated bywave amplitude which has been Fresnel propagated 200 nm from an aperture to the pinholesWavelength para = 25 nm pinhole size 10 nm aperture diameter 1 m m (b) 200 nm out-of-focuswavemacreld from 1 m m aperture incident on pinholes (c) The Fourier transform of (a) is a speckledAiryrsquo s disc Modulus used as input for the HIO algorithm (d) Reconstructed pinholes after 150iterations with modulus constraint (To enforce positivity in real space only the modulus ofthe pixel values inside the support area is retained during each iteration) A large triangularcomputational support was used The image is inverted this is an ambiguity which cannot beresolved by the HIO algorithm

real binary object so that the dinotraction pattern should be symmetric A coherentoptical image was also recorded with the camera moved into the image plane ofthe lens This is shown in shy gure 3a The camera was then moved to the back focalplane of the lens and the dinotraction pattern intensity I(u) was recorded (shy gure 3b)The autocorrelation FT I(u v) was used to determine the approximate size of theobject and a circular support half the size of the autocorrelation was used in the

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 883

f

double polarizer to adjust intensity

collimator

lens diffraction pattern

CCD detectorHendashNe laser= 6328 nml

object with support mask

Figure 2 Schematic of the arrangement used for optical experiments

(a) (b)300 m m

Figure 3 (a) Coherent optical image of optical target transparent letters and bars in an oth-erwise opaque chrome macrlm on glass A 300 m m aperture acts as the support (HeNe laserpara = 6328 nm) (b) Coherent laser optical direg raction pattern of object shown in (a)

computer The HIO algorithm was used to recover the phases of the object exitwave and the algorithm always converged after about 50 iterations with dinoterentinitial random phase sets (shy gure 4a) A larger triangular support constraint wasalso used to recover the image as shown in shy gure 4b The normalized root-mean-squared error in the object domain (Fienup 1987) (a measure of dinoterence betweenthe current image estimate and the known pixel values outside the support) did notdrop below 02 probably due to imperfections in the dinotractogram eg noise andlight reregections from the entrance glass surface of the CCD camera which createasymmetry in the dinotraction pattern (see shy gure 4b above centre) Despite theseproblems the reconstructed image looks almost identical to the coherent image inshy gure 3a

A strong phase object (introducing phase shifts greater than ordm =2) was made usinga support consisting of two separated areas (two apertures of 300 m m diameter sep-

Phil Trans R Soc Lond A (2002)

884 J C H Spence U Weierstall and M Howells

(a) (b)

Figure 4 (a) Modulus of reconstructed image wave using a tight circular support constraint(b) Modulus of reconstructed image wave using a loose triangular support constraint A positivityconstraint was used in real space

mica

no mica(a) (b)

Figure 5 (a) Coherent laser optical image of phase object with two-hole support The edgesof two mica plates are visible across the hole on the right (b) Laser optical direg raction patternobtained by merging three patterns with direg erent exposure times to avoid saturation of thecentral beam

arated by 350 m m) One hole was covered with two 5 m m thick mica sheets whichoverlap in one region The expected phase shift is calculated to be 9 ordm The coherentoptical image is shown in shy gure 5a The mica is transparent except for the edgeswhere Fresnel contrast enotects are visible

The resulting dinotraction pattern is shown in shy gure 5b The central beam intensityin the dinotraction pattern from this transparent object was too strong to be recordedsimultaneously with the weaker high spatial frequencies so that several exposureswere needed All exposures were merged to one pattern in which saturated pixelswere replaced by scaled pixels from the lower exposure patterns The computational

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 885

black = -

white = +

(a) (b)

(c)

Figure 6 (a) Modulus and (b) phase of reconstructed image wave (c) phase reconstructed fromshifted direg raction pattern shows equiphase contour lines Phase shift by mica ordm No sign ormodulus constraint

support size and orientation were derived from the autocorrelation The complexHIO-phase retrieval algorithm was used to recover the image wave phase and modulusshown in shy gure 6 The phase shift by the mica plates in the reconstruction is ordm which since the algorithm reconstructs phase modulo 2 ordm is in good agreement withour independent measurement of 9 ordm based on the measured thickness and knownrefractive index of mica Thus the HIO algorithm is capable of quantitative phasemeasurement

Experiments with a transparent physical support have been attempted using alimiting aperture upstream of the sample but have not been successful so far becauseof the high intensity in the centre of the dinotraction pattern which causes bloomingin the CCD chip This problem persists for three-dimensional objects however athree-quadrant mask in the object plane has been proposed to address this dimacr culty(Sayre et al 1998) (Beam stops cannot be placed near enough to the CCD toavoid generating additional fringes) Experiments using the known intensity envelopeof a focused beam as the support have also not been successful for such complexobjects there appears to be a need for sharp physical support edges perhaps requiredto generate reference edge waves (For real objects there is no such requirement)However a coherent probe spanning two `knownrsquo holes works well in simulation

Previous experiments have been based on objects of simple type such as binary

Phil Trans R Soc Lond A (2002)

886 J C H Spence U Weierstall and M Howells

(a)

(b)

Figure 7 (a) Coherent laser optical image of cheek cells in two-hole support (300 m m apertures)About seven cells clump together in the upper hole Only Fresnel edge contrast is visible dueto defocus the cell nuclei are visible as black dots (b) Laser optical direg raction pattern mergedfrom four direg erent exposures to avoid saturation of the central beam

objects which contain much less information than greyscale objects Experimentswith human cheek cells which serve as more complicated greyscale phase objectshave therefore been performed The cheek cells were positioned on a glass slide insidetwo 300 m m apertures separated by 300 m m The apertures act as the opaque supportconsisting of two separated parts as needed for reconstruction of complex objectsFigure 7a shows the coherent optical image of the slightly defocused cheek cells asformed with a lens Only Fresnel edge contrast is visible while the nuclei producesome absorption Figure 7b shows the optical dinotraction pattern merged from fourdinoterent exposures The size and orientation of the support were estimated fromthe autocorrelation The complex reconstruction of phase and amplitude after about50 HIO iterations using the support constraint is shown in shy gure 8 The cells arevisible with large-area phase contrast in the phase image and the phase shift of thecells is about ordm In the reconstructed amplitude only edge contrast is visible as inthe optical image

For a three-dimensional object extended along the optical axis the question arisesas to which plane in the object is reconstructed given that the detected intensity liesat inshy nity and contains contributions from scattering at every point along the axisin the object space A simulation to clarify this is shown in shy gure 9 The simulatedobject consists of two planes separated by 1 mm in the z-direction Plane 1 has a holecontaining a complex object (the number 1) while plane 2 has two holes containingcomplex objects (numbers 2 and 3) The object exit wave calculated on plane 2

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 887

(a) (b)

Figure 8 (a) Modulus and (b) phase of reconstructed image wave The modulus shows only edgecontrast whereas the phase shows the cells in large-area phase contrast The phase shift due tothe cells is about ordm No modulus or sign constraint

(shy gure 9a) shows plane 1 out of focus The modulus of the Fourier transform (Fouriermodulus) of this image wave is used as input for the HIO algorithm In the inversiondinoterent shapes of the supports used at each plane shown inset can distinguish thetwo object planes If a plane is assumed unknown a larger support hole is usedthat contains the physical support Depending on which support plane is assumed asknown (inset) in the iteration (plane 1 support known and plane 2 support unknownor vice versa) the other plane is always reconstructed out of focus by +1 or iexcl 1 mm(shy gure 9c d) That means that in the case of known support on plane 1 the algorithmreconstructs the wave as seen through an imaging lens with out-of-focus images ofthe holes on plane 2 As expected the depth of focus for interactive inversion is givenby the expression for an ideal lens which imposes a dinotraction limit equal to that ofthe highest spatial frequency recorded in the dinotraction pattern

To verify this experimentally a pin was mounted 12 mm upstream of a pair of300 m m apertures The coherent optical image is shown in shy gure 10a A Fresnel fringesurrounds the pin since the lens is focused on the apertures The image was recoveredfrom the recorded dinotraction pattern intensity (shy gure 10b) Again three dinotractionpatterns with dinoterent exposure times had to be merged to avoid pixel saturationand blooming The reconstructed image modulus is shown in shy gure 11a where the

Phil Trans R Soc Lond A (2002)

888 J C H Spence U Weierstall and M Howells

illuminationdirection

plane 1

1 mm

(a) (b)

(c) (d)

Figure 9 (a) Schematic view of a simulated object consisting of two planes separated by 1 mm inthe direction of light propagation Plane 1 has a hole containing a complex object (number 1) andplane 2 has two holes containing complex objects (numbers 2 and 3) (b) Modulus of complexexit wave on plane 2 (c) Iterative reconstruction with plane 2 support assumed known (thesupport shape used is shown in the top right-hand corner) Plane 1 is out of focus by 1 mm(d) Iterative reconstruction with plane 1 support assumed known (the support shape is shownin the top right-hand corner) Plane 2 is out of focus by iexcl1 mm

pin appears out of focus (Fresnel fringe barely visible) because the aperture shapewas imposed as support The pin can be brought into focus by using the appropriateFresnel propagator with centz = 12 mm and para = 6328 nm taking the apertures outof focus (see shy gure 11b) Thus as expected the reconstructed plane is always thatplane in which the support condition is imposed where the algorithm shy nds a uniquesolution subject to the given boundary condition Propagation to other planes maybe performed subsequently since the complex waveshy eld is obtained subject to thedepth-of-focus limitation The relative sideways shift of the pin inside the aperturebetween shy gure 10a and shy gure 11a is due to a slight unintentional tilt of the object(pin plus pinhole) between recording the image and the dinotraction pattern

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 889

(a) (b)

Figure 10 (a) Coherent laser optical image of pin in front of aperture (b) Laser optical direg ractionpattern obtained by merging three patterns with direg erent exposure times to avoid saturation ofthe central beam

(a) (b)

Figure 11 (a) Reconstructed modulus Only the hole with the out-of-focus pin is shown(b) Numerical back propagation of reconstructed complex wavemacreld by 12 mm to plane of pinPin appears in focus Support constraint only

6 Experiments with coherent electrons

Using a special support mask made by electron-beam lithography experiments havealso been undertaken using the HIO algorithm to retrieve the phases from experimen-tal transmission electron dinotraction patterns (Weierstall et al 2002) Images mightthen be reconstructed that are free of electron lens aberrations The support maskcontains hole pairs of various sizes with pairs separated by 1 m m The separationbetween holes of a pair is ca 50 nm less than the coherence width of the illuminationbeam The holes irregular in shape form the object we wish to reconstruct in futurework we plan to introduce nanoscale objects into the holes Figure 12a below showsthe experimental dinotraction pattern from two holes shown in shy gure 12c obtained

Phil Trans R Soc Lond A (2002)

890 J C H Spence U Weierstall and M Howells

20 nm

(a) (b)

(c)

Figure 12 (a) Experimental electron direg raction pattern from two nanoscale holes in an opaquemask (shown in (c)) recorded using a macreld-emission electron microscope (CM200 FEG at 40 kV1k pound1k CCD with 24 m m pixels camera length 611 m) Youngrsquo s fringes from the two holes arevisible these fringes encode phase information (b) Reconstruction of image of holes from thedireg raction pattern (a) after 100 iterations of the HIO algorithm Modulus constraint applied inimage space with large triangular computational support The hole shape is comparable with theTEM image in (c) (c) Conventional TEM image of holes that created the direg raction pattern (a)

with a Philips CM 200 shy eld-emission electron microscope at 40 kV The acceleratingvoltage was lowered to 40 kV to make the multi-layered support opaque ModulatedYoungrsquos fringes are clearly seen

The resulting image after 100 HIO iterations is shown in shy gure 12b The holesreconstructed from the dinotraction pattern modulus look similar to the TEM imageof the holes in shy gure 12c but the resolution of the reconstruction is lower thanthe normal TEM image resolution High spatial frequency information from thehole edge is missing in the experimental dinotraction pattern due perhaps to drift ofthe imaging lenses during the exposure time

Table 1 shows a summary chart of all the conditions so far examined with HIO sim-

Phil Trans R Soc Lond A (2002)

Phase

recovery

an

dlen

slessim

agin

g891

Table 1 Summary chart showing direg erent conditions for phase recovery with HIO algorithm

( CArsquo denotes the complex algorithm (uses direg racted intensity and support constraint in real space) RIPrsquo denotes the complex algorithm withpositivity constraint of Re ordf Im ordf (uses direg racted intensity and support constraint in real space negative Re ordf and Im ordf inside the supportarea are set positive) Mrsquo denotes the modulus constraint on ordf in real space (uses direg racted intensity and support constraint in real spacecomplex numbers inside the support area are replaced by their modulus to enforce positivity) Prsquo denotes the phase constraint on ordf in realspace (uses direg racted intensity and support constraint in real space the phase is extracted and retained from complex numbers inside thesupport area modulus is extracted and set to 1 inside the support area))

general pure strong pure weakrsquo real weak-phasecomplex object phase object phase object (amplitude) object withordf = A exp(iiquest ) ordf = exp(i iquest (r)) iquest lt ordm =2 object absorption

known two-holephysical support

CA (laser experimentpin cheek cells)

CA P (laserexperiment mica)

RIP CA M RIP CA RIP CA

known one-holephysical support

CA only for certainsupport shapes(eg triangle withsharp edges)

CA slowconvergence Pworks better

RIP CA M RIP CA (laserexperimentnumbers)

RIP CA

unknown two-or-more-hole physicalsupport loose(eg triangle compu-tational support)

| | RIP M ( TEM experimentholes) (laserexperiment numbers)(planned X-ray micapinholes)

RIP (Miao X-rayexperimenttransparent support)

unknown one-holephysical support(eg triangle compu-tational support)

| | RIP only for holewith low symmetryand sharp edgesslow convergence

M RIP only for holewith low symmetryand sharp edgesslow convergence

RIP only for holewith low symmetryand sharp edgesslow convergence

Phil

Tra

ns

R

Soc

L

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892 J C H Spence U Weierstall and M Howells

ulations and experiments The enotectiveness of dinoterent constraints in real space fordinoterent physical and computational supports is compared The physical supportis the support which exists in the experiment and which may or may not be accu-rately known The computational support is the support used in the HIO iterationsThe table shows experiments that have been performed in italics The shy rst columnshows dinoterent possibilities for the physical support the most favourable at the topthe worst at the bottom The computational support and known physical supportare identical The computational support used with the unknown physical supportis larger than the physical support Reconstructions with one-hole physical supportand loose computational support are only possible with weak or real objects andsupport holes with sharp edges and low symmetry and convergence are very slow insimulations A known two-or-more-hole support proved to be most favourable in allcases

7 Discussion and conclusions direg ractive imaging with other particles

In this paper we have reviewed the implications of Thomas Youngrsquos experiment formatter waves and in particular for electron interferometry By taking advantage ofthe very high quality of the coherent electron beam available in the modern shy eld-emission electron microscope a number of important results (such as the AharonovBohm enotect and the electron Sagnac enotect) have been established and the shy eldof atomic-resolution transmission electron microscopy phase-contrast imaging andelectron holography opened up All these methods continue to produce a large sci-entishy c pay-onot in microstructural materials characterization and discovery and inbiology A new lensless imaging method is then reviewed which may be understoodin part by analogy with electron holography This HIO algorithm uses an iterativeapproach to assign phases to dinotraction pattern intensities To evaluate its perfor-mance we give experimental examples of image reconstruction with both coherentelectrons and light together with simulations for the soft X-ray case Thus we haveshown experimentally that the HIO algorithm is capable of solving the phase problemfor scattering from non-periodic objects and of making quantitative measurements ofphase (modulo 2ordm ) in good agreement with independent measurements The beam-splitting arrangement of an interferometer is not required and dinotraction-limitedlensless imaging seems possible as a result This may open the way for imaging withthe many particles for which no lenses exist such as neutrons ballistic electrons insolids at low temperature or neutral atoms and ions either as free coherent beamsor condensates (Bright coherent sources have recently appeared for many of these(Pauly 2000)) The dinotering interactions of these particles may provide useful newprobes for chemistry materials science and biology For example the reduced radia-tion damage enotects of imaging organic material with low-energy electrons neutronsand helium atoms have recently been compared (Spence et al 1999) and HIO sim-ulations have been completed for neutron imaging of a diatom (Spence et al 2001)The very small refractive index of organic matter for neutrons limits resolution andcontrast for thin samples For low-energy ions or metastable helium strong samplebeam interactions must be considered Damage from neutral helium-atom imagingof molecules should be negligible however penetration is also negligible (producinga proshy le image) Detectors for coherent atom beams still present serious problemssince area detectors are inemacr cient For biology the weak atom interaction is desirable

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 893

to avoid damage however a strong interaction at the detector is also needed so thationization before detection must be arranged

For real objects we have seen that the resolution in images recovered by the HIOalgorithm is largely independent of the accuracy with which the support is knownThe addition of noise anotects the probability that the unique solution will be found (asmeasured by the error metric) rather than the resolution which is near dinotraction-limited For complex objects our experience indicates that a much more accurateknowledge of the physical support boundary is needed approaching the scale ofthe resolution sought The tight support and two-hole geometry (see Fienup (1987)for other geometries) needed for complex objects (complex refractive index) willbecome increasingly accessible with the development of new lithographic techniquesfor support construction This can be combined with independent characterization ofthe support shape and the scaling information provided by the known autocorrelationof the support Experimentally favourable cases for imaging with nanoscale resolutionat present require loose support which table 1 shows greatly favours real objectssuch as the charge density which dinotracts hard X-rays at energies well away fromabsorption edges or very thin samples for high-energy electrons Unfortunately beamcoherence is more readily obtained for soft X-rays

Using modern lithographic techniques it is becoming increasingly possible toestablish the experimental conditions of known support required by the HIO algo-rithm both for free particle beams and for electrons in solids at low temperaturesThus for example we have seen the algorithm used to determine the phase of wavefunctions for electrons in a one-dimensional YBCO Josephson junction using TEMimages of the structure to deshy ne the required support (boundary conditions) (Car-mody et al 2000) Given the fundamental role that phase has played historically inphysics and which Thomas Young was amongst the shy rst to appreciate any techniquecapable of exposing phase can be expected to yield many surprises in physics in thefuture

Supported by ARO award DAAD190010500 We thank Dr Q Chen for the use of macrgure 12 (fromWeierstall et al 2002)

References

Aharonov Y amp Bohm D 1959 Signimacrcance of the electromagnetic potentials in the quantumtheory Phys Rev 115 485

Allen L J Faulkner H M L Nugent K A Oxley M P amp Paganin D 2001 Phase retrievalfrom images in the presence of macrrst-order vortices Phys Rev E 63 037602

Bauschke H H et al 2002 (In the press)

Boersch H 1943 Fresnelsche Beugung im Elektronenmikroskop Phys Z 44 202211

Bruck Y amp Sodin L 1979 On the ambiguity of the image-reconstruction problem Opt Com-mun 30 304308

Carmody M Moeckly B H Merkle K amp Marks L 2000 Spatial variation of the current ingrain boundary Josephson junctions J Appl Phys 87 24542459

Coene W Janssen G Op de Beek M amp van Dyck D 1992 Phase retrieval through focusvariation for ultra-resolution in macreld emission transmission electron microscopy Phys RevLett 69 37433746

Crewe A V Eggenberger D N Wall J amp Welter L M 1968 Electron gun using a macreldemission source Rev Scient Instrum 39 576

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 883

f

double polarizer to adjust intensity

collimator

lens diffraction pattern

CCD detectorHendashNe laser= 6328 nml

object with support mask

Figure 2 Schematic of the arrangement used for optical experiments

(a) (b)300 m m

Figure 3 (a) Coherent optical image of optical target transparent letters and bars in an oth-erwise opaque chrome macrlm on glass A 300 m m aperture acts as the support (HeNe laserpara = 6328 nm) (b) Coherent laser optical direg raction pattern of object shown in (a)

computer The HIO algorithm was used to recover the phases of the object exitwave and the algorithm always converged after about 50 iterations with dinoterentinitial random phase sets (shy gure 4a) A larger triangular support constraint wasalso used to recover the image as shown in shy gure 4b The normalized root-mean-squared error in the object domain (Fienup 1987) (a measure of dinoterence betweenthe current image estimate and the known pixel values outside the support) did notdrop below 02 probably due to imperfections in the dinotractogram eg noise andlight reregections from the entrance glass surface of the CCD camera which createasymmetry in the dinotraction pattern (see shy gure 4b above centre) Despite theseproblems the reconstructed image looks almost identical to the coherent image inshy gure 3a

A strong phase object (introducing phase shifts greater than ordm =2) was made usinga support consisting of two separated areas (two apertures of 300 m m diameter sep-

Phil Trans R Soc Lond A (2002)

884 J C H Spence U Weierstall and M Howells

(a) (b)

Figure 4 (a) Modulus of reconstructed image wave using a tight circular support constraint(b) Modulus of reconstructed image wave using a loose triangular support constraint A positivityconstraint was used in real space

mica

no mica(a) (b)

Figure 5 (a) Coherent laser optical image of phase object with two-hole support The edgesof two mica plates are visible across the hole on the right (b) Laser optical direg raction patternobtained by merging three patterns with direg erent exposure times to avoid saturation of thecentral beam

arated by 350 m m) One hole was covered with two 5 m m thick mica sheets whichoverlap in one region The expected phase shift is calculated to be 9 ordm The coherentoptical image is shown in shy gure 5a The mica is transparent except for the edgeswhere Fresnel contrast enotects are visible

The resulting dinotraction pattern is shown in shy gure 5b The central beam intensityin the dinotraction pattern from this transparent object was too strong to be recordedsimultaneously with the weaker high spatial frequencies so that several exposureswere needed All exposures were merged to one pattern in which saturated pixelswere replaced by scaled pixels from the lower exposure patterns The computational

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 885

black = -

white = +

(a) (b)

(c)

Figure 6 (a) Modulus and (b) phase of reconstructed image wave (c) phase reconstructed fromshifted direg raction pattern shows equiphase contour lines Phase shift by mica ordm No sign ormodulus constraint

support size and orientation were derived from the autocorrelation The complexHIO-phase retrieval algorithm was used to recover the image wave phase and modulusshown in shy gure 6 The phase shift by the mica plates in the reconstruction is ordm which since the algorithm reconstructs phase modulo 2 ordm is in good agreement withour independent measurement of 9 ordm based on the measured thickness and knownrefractive index of mica Thus the HIO algorithm is capable of quantitative phasemeasurement

Experiments with a transparent physical support have been attempted using alimiting aperture upstream of the sample but have not been successful so far becauseof the high intensity in the centre of the dinotraction pattern which causes bloomingin the CCD chip This problem persists for three-dimensional objects however athree-quadrant mask in the object plane has been proposed to address this dimacr culty(Sayre et al 1998) (Beam stops cannot be placed near enough to the CCD toavoid generating additional fringes) Experiments using the known intensity envelopeof a focused beam as the support have also not been successful for such complexobjects there appears to be a need for sharp physical support edges perhaps requiredto generate reference edge waves (For real objects there is no such requirement)However a coherent probe spanning two `knownrsquo holes works well in simulation

Previous experiments have been based on objects of simple type such as binary

Phil Trans R Soc Lond A (2002)

886 J C H Spence U Weierstall and M Howells

(a)

(b)

Figure 7 (a) Coherent laser optical image of cheek cells in two-hole support (300 m m apertures)About seven cells clump together in the upper hole Only Fresnel edge contrast is visible dueto defocus the cell nuclei are visible as black dots (b) Laser optical direg raction pattern mergedfrom four direg erent exposures to avoid saturation of the central beam

objects which contain much less information than greyscale objects Experimentswith human cheek cells which serve as more complicated greyscale phase objectshave therefore been performed The cheek cells were positioned on a glass slide insidetwo 300 m m apertures separated by 300 m m The apertures act as the opaque supportconsisting of two separated parts as needed for reconstruction of complex objectsFigure 7a shows the coherent optical image of the slightly defocused cheek cells asformed with a lens Only Fresnel edge contrast is visible while the nuclei producesome absorption Figure 7b shows the optical dinotraction pattern merged from fourdinoterent exposures The size and orientation of the support were estimated fromthe autocorrelation The complex reconstruction of phase and amplitude after about50 HIO iterations using the support constraint is shown in shy gure 8 The cells arevisible with large-area phase contrast in the phase image and the phase shift of thecells is about ordm In the reconstructed amplitude only edge contrast is visible as inthe optical image

For a three-dimensional object extended along the optical axis the question arisesas to which plane in the object is reconstructed given that the detected intensity liesat inshy nity and contains contributions from scattering at every point along the axisin the object space A simulation to clarify this is shown in shy gure 9 The simulatedobject consists of two planes separated by 1 mm in the z-direction Plane 1 has a holecontaining a complex object (the number 1) while plane 2 has two holes containingcomplex objects (numbers 2 and 3) The object exit wave calculated on plane 2

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 887

(a) (b)

Figure 8 (a) Modulus and (b) phase of reconstructed image wave The modulus shows only edgecontrast whereas the phase shows the cells in large-area phase contrast The phase shift due tothe cells is about ordm No modulus or sign constraint

(shy gure 9a) shows plane 1 out of focus The modulus of the Fourier transform (Fouriermodulus) of this image wave is used as input for the HIO algorithm In the inversiondinoterent shapes of the supports used at each plane shown inset can distinguish thetwo object planes If a plane is assumed unknown a larger support hole is usedthat contains the physical support Depending on which support plane is assumed asknown (inset) in the iteration (plane 1 support known and plane 2 support unknownor vice versa) the other plane is always reconstructed out of focus by +1 or iexcl 1 mm(shy gure 9c d) That means that in the case of known support on plane 1 the algorithmreconstructs the wave as seen through an imaging lens with out-of-focus images ofthe holes on plane 2 As expected the depth of focus for interactive inversion is givenby the expression for an ideal lens which imposes a dinotraction limit equal to that ofthe highest spatial frequency recorded in the dinotraction pattern

To verify this experimentally a pin was mounted 12 mm upstream of a pair of300 m m apertures The coherent optical image is shown in shy gure 10a A Fresnel fringesurrounds the pin since the lens is focused on the apertures The image was recoveredfrom the recorded dinotraction pattern intensity (shy gure 10b) Again three dinotractionpatterns with dinoterent exposure times had to be merged to avoid pixel saturationand blooming The reconstructed image modulus is shown in shy gure 11a where the

Phil Trans R Soc Lond A (2002)

888 J C H Spence U Weierstall and M Howells

illuminationdirection

plane 1

1 mm

(a) (b)

(c) (d)

Figure 9 (a) Schematic view of a simulated object consisting of two planes separated by 1 mm inthe direction of light propagation Plane 1 has a hole containing a complex object (number 1) andplane 2 has two holes containing complex objects (numbers 2 and 3) (b) Modulus of complexexit wave on plane 2 (c) Iterative reconstruction with plane 2 support assumed known (thesupport shape used is shown in the top right-hand corner) Plane 1 is out of focus by 1 mm(d) Iterative reconstruction with plane 1 support assumed known (the support shape is shownin the top right-hand corner) Plane 2 is out of focus by iexcl1 mm

pin appears out of focus (Fresnel fringe barely visible) because the aperture shapewas imposed as support The pin can be brought into focus by using the appropriateFresnel propagator with centz = 12 mm and para = 6328 nm taking the apertures outof focus (see shy gure 11b) Thus as expected the reconstructed plane is always thatplane in which the support condition is imposed where the algorithm shy nds a uniquesolution subject to the given boundary condition Propagation to other planes maybe performed subsequently since the complex waveshy eld is obtained subject to thedepth-of-focus limitation The relative sideways shift of the pin inside the aperturebetween shy gure 10a and shy gure 11a is due to a slight unintentional tilt of the object(pin plus pinhole) between recording the image and the dinotraction pattern

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 889

(a) (b)

Figure 10 (a) Coherent laser optical image of pin in front of aperture (b) Laser optical direg ractionpattern obtained by merging three patterns with direg erent exposure times to avoid saturation ofthe central beam

(a) (b)

Figure 11 (a) Reconstructed modulus Only the hole with the out-of-focus pin is shown(b) Numerical back propagation of reconstructed complex wavemacreld by 12 mm to plane of pinPin appears in focus Support constraint only

6 Experiments with coherent electrons

Using a special support mask made by electron-beam lithography experiments havealso been undertaken using the HIO algorithm to retrieve the phases from experimen-tal transmission electron dinotraction patterns (Weierstall et al 2002) Images mightthen be reconstructed that are free of electron lens aberrations The support maskcontains hole pairs of various sizes with pairs separated by 1 m m The separationbetween holes of a pair is ca 50 nm less than the coherence width of the illuminationbeam The holes irregular in shape form the object we wish to reconstruct in futurework we plan to introduce nanoscale objects into the holes Figure 12a below showsthe experimental dinotraction pattern from two holes shown in shy gure 12c obtained

Phil Trans R Soc Lond A (2002)

890 J C H Spence U Weierstall and M Howells

20 nm

(a) (b)

(c)

Figure 12 (a) Experimental electron direg raction pattern from two nanoscale holes in an opaquemask (shown in (c)) recorded using a macreld-emission electron microscope (CM200 FEG at 40 kV1k pound1k CCD with 24 m m pixels camera length 611 m) Youngrsquo s fringes from the two holes arevisible these fringes encode phase information (b) Reconstruction of image of holes from thedireg raction pattern (a) after 100 iterations of the HIO algorithm Modulus constraint applied inimage space with large triangular computational support The hole shape is comparable with theTEM image in (c) (c) Conventional TEM image of holes that created the direg raction pattern (a)

with a Philips CM 200 shy eld-emission electron microscope at 40 kV The acceleratingvoltage was lowered to 40 kV to make the multi-layered support opaque ModulatedYoungrsquos fringes are clearly seen

The resulting image after 100 HIO iterations is shown in shy gure 12b The holesreconstructed from the dinotraction pattern modulus look similar to the TEM imageof the holes in shy gure 12c but the resolution of the reconstruction is lower thanthe normal TEM image resolution High spatial frequency information from thehole edge is missing in the experimental dinotraction pattern due perhaps to drift ofthe imaging lenses during the exposure time

Table 1 shows a summary chart of all the conditions so far examined with HIO sim-

Phil Trans R Soc Lond A (2002)

Phase

recovery

an

dlen

slessim

agin

g891

Table 1 Summary chart showing direg erent conditions for phase recovery with HIO algorithm

( CArsquo denotes the complex algorithm (uses direg racted intensity and support constraint in real space) RIPrsquo denotes the complex algorithm withpositivity constraint of Re ordf Im ordf (uses direg racted intensity and support constraint in real space negative Re ordf and Im ordf inside the supportarea are set positive) Mrsquo denotes the modulus constraint on ordf in real space (uses direg racted intensity and support constraint in real spacecomplex numbers inside the support area are replaced by their modulus to enforce positivity) Prsquo denotes the phase constraint on ordf in realspace (uses direg racted intensity and support constraint in real space the phase is extracted and retained from complex numbers inside thesupport area modulus is extracted and set to 1 inside the support area))

general pure strong pure weakrsquo real weak-phasecomplex object phase object phase object (amplitude) object withordf = A exp(iiquest ) ordf = exp(i iquest (r)) iquest lt ordm =2 object absorption

known two-holephysical support

CA (laser experimentpin cheek cells)

CA P (laserexperiment mica)

RIP CA M RIP CA RIP CA

known one-holephysical support

CA only for certainsupport shapes(eg triangle withsharp edges)

CA slowconvergence Pworks better

RIP CA M RIP CA (laserexperimentnumbers)

RIP CA

unknown two-or-more-hole physicalsupport loose(eg triangle compu-tational support)

| | RIP M ( TEM experimentholes) (laserexperiment numbers)(planned X-ray micapinholes)

RIP (Miao X-rayexperimenttransparent support)

unknown one-holephysical support(eg triangle compu-tational support)

| | RIP only for holewith low symmetryand sharp edgesslow convergence

M RIP only for holewith low symmetryand sharp edgesslow convergence

RIP only for holewith low symmetryand sharp edgesslow convergence

Phil

Tra

ns

R

Soc

L

on

d

A(2

002)

892 J C H Spence U Weierstall and M Howells

ulations and experiments The enotectiveness of dinoterent constraints in real space fordinoterent physical and computational supports is compared The physical supportis the support which exists in the experiment and which may or may not be accu-rately known The computational support is the support used in the HIO iterationsThe table shows experiments that have been performed in italics The shy rst columnshows dinoterent possibilities for the physical support the most favourable at the topthe worst at the bottom The computational support and known physical supportare identical The computational support used with the unknown physical supportis larger than the physical support Reconstructions with one-hole physical supportand loose computational support are only possible with weak or real objects andsupport holes with sharp edges and low symmetry and convergence are very slow insimulations A known two-or-more-hole support proved to be most favourable in allcases

7 Discussion and conclusions direg ractive imaging with other particles

In this paper we have reviewed the implications of Thomas Youngrsquos experiment formatter waves and in particular for electron interferometry By taking advantage ofthe very high quality of the coherent electron beam available in the modern shy eld-emission electron microscope a number of important results (such as the AharonovBohm enotect and the electron Sagnac enotect) have been established and the shy eldof atomic-resolution transmission electron microscopy phase-contrast imaging andelectron holography opened up All these methods continue to produce a large sci-entishy c pay-onot in microstructural materials characterization and discovery and inbiology A new lensless imaging method is then reviewed which may be understoodin part by analogy with electron holography This HIO algorithm uses an iterativeapproach to assign phases to dinotraction pattern intensities To evaluate its perfor-mance we give experimental examples of image reconstruction with both coherentelectrons and light together with simulations for the soft X-ray case Thus we haveshown experimentally that the HIO algorithm is capable of solving the phase problemfor scattering from non-periodic objects and of making quantitative measurements ofphase (modulo 2ordm ) in good agreement with independent measurements The beam-splitting arrangement of an interferometer is not required and dinotraction-limitedlensless imaging seems possible as a result This may open the way for imaging withthe many particles for which no lenses exist such as neutrons ballistic electrons insolids at low temperature or neutral atoms and ions either as free coherent beamsor condensates (Bright coherent sources have recently appeared for many of these(Pauly 2000)) The dinotering interactions of these particles may provide useful newprobes for chemistry materials science and biology For example the reduced radia-tion damage enotects of imaging organic material with low-energy electrons neutronsand helium atoms have recently been compared (Spence et al 1999) and HIO sim-ulations have been completed for neutron imaging of a diatom (Spence et al 2001)The very small refractive index of organic matter for neutrons limits resolution andcontrast for thin samples For low-energy ions or metastable helium strong samplebeam interactions must be considered Damage from neutral helium-atom imagingof molecules should be negligible however penetration is also negligible (producinga proshy le image) Detectors for coherent atom beams still present serious problemssince area detectors are inemacr cient For biology the weak atom interaction is desirable

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 893

to avoid damage however a strong interaction at the detector is also needed so thationization before detection must be arranged

For real objects we have seen that the resolution in images recovered by the HIOalgorithm is largely independent of the accuracy with which the support is knownThe addition of noise anotects the probability that the unique solution will be found (asmeasured by the error metric) rather than the resolution which is near dinotraction-limited For complex objects our experience indicates that a much more accurateknowledge of the physical support boundary is needed approaching the scale ofthe resolution sought The tight support and two-hole geometry (see Fienup (1987)for other geometries) needed for complex objects (complex refractive index) willbecome increasingly accessible with the development of new lithographic techniquesfor support construction This can be combined with independent characterization ofthe support shape and the scaling information provided by the known autocorrelationof the support Experimentally favourable cases for imaging with nanoscale resolutionat present require loose support which table 1 shows greatly favours real objectssuch as the charge density which dinotracts hard X-rays at energies well away fromabsorption edges or very thin samples for high-energy electrons Unfortunately beamcoherence is more readily obtained for soft X-rays

Using modern lithographic techniques it is becoming increasingly possible toestablish the experimental conditions of known support required by the HIO algo-rithm both for free particle beams and for electrons in solids at low temperaturesThus for example we have seen the algorithm used to determine the phase of wavefunctions for electrons in a one-dimensional YBCO Josephson junction using TEMimages of the structure to deshy ne the required support (boundary conditions) (Car-mody et al 2000) Given the fundamental role that phase has played historically inphysics and which Thomas Young was amongst the shy rst to appreciate any techniquecapable of exposing phase can be expected to yield many surprises in physics in thefuture

Supported by ARO award DAAD190010500 We thank Dr Q Chen for the use of macrgure 12 (fromWeierstall et al 2002)

References

Aharonov Y amp Bohm D 1959 Signimacrcance of the electromagnetic potentials in the quantumtheory Phys Rev 115 485

Allen L J Faulkner H M L Nugent K A Oxley M P amp Paganin D 2001 Phase retrievalfrom images in the presence of macrrst-order vortices Phys Rev E 63 037602

Bauschke H H et al 2002 (In the press)

Boersch H 1943 Fresnelsche Beugung im Elektronenmikroskop Phys Z 44 202211

Bruck Y amp Sodin L 1979 On the ambiguity of the image-reconstruction problem Opt Com-mun 30 304308

Carmody M Moeckly B H Merkle K amp Marks L 2000 Spatial variation of the current ingrain boundary Josephson junctions J Appl Phys 87 24542459

Coene W Janssen G Op de Beek M amp van Dyck D 1992 Phase retrieval through focusvariation for ultra-resolution in macreld emission transmission electron microscopy Phys RevLett 69 37433746

Crewe A V Eggenberger D N Wall J amp Welter L M 1968 Electron gun using a macreldemission source Rev Scient Instrum 39 576

Phil Trans R Soc Lond A (2002)

884 J C H Spence U Weierstall and M Howells

(a) (b)

Figure 4 (a) Modulus of reconstructed image wave using a tight circular support constraint(b) Modulus of reconstructed image wave using a loose triangular support constraint A positivityconstraint was used in real space

mica

no mica(a) (b)

Figure 5 (a) Coherent laser optical image of phase object with two-hole support The edgesof two mica plates are visible across the hole on the right (b) Laser optical direg raction patternobtained by merging three patterns with direg erent exposure times to avoid saturation of thecentral beam

arated by 350 m m) One hole was covered with two 5 m m thick mica sheets whichoverlap in one region The expected phase shift is calculated to be 9 ordm The coherentoptical image is shown in shy gure 5a The mica is transparent except for the edgeswhere Fresnel contrast enotects are visible

The resulting dinotraction pattern is shown in shy gure 5b The central beam intensityin the dinotraction pattern from this transparent object was too strong to be recordedsimultaneously with the weaker high spatial frequencies so that several exposureswere needed All exposures were merged to one pattern in which saturated pixelswere replaced by scaled pixels from the lower exposure patterns The computational

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 885

black = -

white = +

(a) (b)

(c)

Figure 6 (a) Modulus and (b) phase of reconstructed image wave (c) phase reconstructed fromshifted direg raction pattern shows equiphase contour lines Phase shift by mica ordm No sign ormodulus constraint

support size and orientation were derived from the autocorrelation The complexHIO-phase retrieval algorithm was used to recover the image wave phase and modulusshown in shy gure 6 The phase shift by the mica plates in the reconstruction is ordm which since the algorithm reconstructs phase modulo 2 ordm is in good agreement withour independent measurement of 9 ordm based on the measured thickness and knownrefractive index of mica Thus the HIO algorithm is capable of quantitative phasemeasurement

Experiments with a transparent physical support have been attempted using alimiting aperture upstream of the sample but have not been successful so far becauseof the high intensity in the centre of the dinotraction pattern which causes bloomingin the CCD chip This problem persists for three-dimensional objects however athree-quadrant mask in the object plane has been proposed to address this dimacr culty(Sayre et al 1998) (Beam stops cannot be placed near enough to the CCD toavoid generating additional fringes) Experiments using the known intensity envelopeof a focused beam as the support have also not been successful for such complexobjects there appears to be a need for sharp physical support edges perhaps requiredto generate reference edge waves (For real objects there is no such requirement)However a coherent probe spanning two `knownrsquo holes works well in simulation

Previous experiments have been based on objects of simple type such as binary

Phil Trans R Soc Lond A (2002)

886 J C H Spence U Weierstall and M Howells

(a)

(b)

Figure 7 (a) Coherent laser optical image of cheek cells in two-hole support (300 m m apertures)About seven cells clump together in the upper hole Only Fresnel edge contrast is visible dueto defocus the cell nuclei are visible as black dots (b) Laser optical direg raction pattern mergedfrom four direg erent exposures to avoid saturation of the central beam

objects which contain much less information than greyscale objects Experimentswith human cheek cells which serve as more complicated greyscale phase objectshave therefore been performed The cheek cells were positioned on a glass slide insidetwo 300 m m apertures separated by 300 m m The apertures act as the opaque supportconsisting of two separated parts as needed for reconstruction of complex objectsFigure 7a shows the coherent optical image of the slightly defocused cheek cells asformed with a lens Only Fresnel edge contrast is visible while the nuclei producesome absorption Figure 7b shows the optical dinotraction pattern merged from fourdinoterent exposures The size and orientation of the support were estimated fromthe autocorrelation The complex reconstruction of phase and amplitude after about50 HIO iterations using the support constraint is shown in shy gure 8 The cells arevisible with large-area phase contrast in the phase image and the phase shift of thecells is about ordm In the reconstructed amplitude only edge contrast is visible as inthe optical image

For a three-dimensional object extended along the optical axis the question arisesas to which plane in the object is reconstructed given that the detected intensity liesat inshy nity and contains contributions from scattering at every point along the axisin the object space A simulation to clarify this is shown in shy gure 9 The simulatedobject consists of two planes separated by 1 mm in the z-direction Plane 1 has a holecontaining a complex object (the number 1) while plane 2 has two holes containingcomplex objects (numbers 2 and 3) The object exit wave calculated on plane 2

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 887

(a) (b)

Figure 8 (a) Modulus and (b) phase of reconstructed image wave The modulus shows only edgecontrast whereas the phase shows the cells in large-area phase contrast The phase shift due tothe cells is about ordm No modulus or sign constraint

(shy gure 9a) shows plane 1 out of focus The modulus of the Fourier transform (Fouriermodulus) of this image wave is used as input for the HIO algorithm In the inversiondinoterent shapes of the supports used at each plane shown inset can distinguish thetwo object planes If a plane is assumed unknown a larger support hole is usedthat contains the physical support Depending on which support plane is assumed asknown (inset) in the iteration (plane 1 support known and plane 2 support unknownor vice versa) the other plane is always reconstructed out of focus by +1 or iexcl 1 mm(shy gure 9c d) That means that in the case of known support on plane 1 the algorithmreconstructs the wave as seen through an imaging lens with out-of-focus images ofthe holes on plane 2 As expected the depth of focus for interactive inversion is givenby the expression for an ideal lens which imposes a dinotraction limit equal to that ofthe highest spatial frequency recorded in the dinotraction pattern

To verify this experimentally a pin was mounted 12 mm upstream of a pair of300 m m apertures The coherent optical image is shown in shy gure 10a A Fresnel fringesurrounds the pin since the lens is focused on the apertures The image was recoveredfrom the recorded dinotraction pattern intensity (shy gure 10b) Again three dinotractionpatterns with dinoterent exposure times had to be merged to avoid pixel saturationand blooming The reconstructed image modulus is shown in shy gure 11a where the

Phil Trans R Soc Lond A (2002)

888 J C H Spence U Weierstall and M Howells

illuminationdirection

plane 1

1 mm

(a) (b)

(c) (d)

Figure 9 (a) Schematic view of a simulated object consisting of two planes separated by 1 mm inthe direction of light propagation Plane 1 has a hole containing a complex object (number 1) andplane 2 has two holes containing complex objects (numbers 2 and 3) (b) Modulus of complexexit wave on plane 2 (c) Iterative reconstruction with plane 2 support assumed known (thesupport shape used is shown in the top right-hand corner) Plane 1 is out of focus by 1 mm(d) Iterative reconstruction with plane 1 support assumed known (the support shape is shownin the top right-hand corner) Plane 2 is out of focus by iexcl1 mm

pin appears out of focus (Fresnel fringe barely visible) because the aperture shapewas imposed as support The pin can be brought into focus by using the appropriateFresnel propagator with centz = 12 mm and para = 6328 nm taking the apertures outof focus (see shy gure 11b) Thus as expected the reconstructed plane is always thatplane in which the support condition is imposed where the algorithm shy nds a uniquesolution subject to the given boundary condition Propagation to other planes maybe performed subsequently since the complex waveshy eld is obtained subject to thedepth-of-focus limitation The relative sideways shift of the pin inside the aperturebetween shy gure 10a and shy gure 11a is due to a slight unintentional tilt of the object(pin plus pinhole) between recording the image and the dinotraction pattern

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 889

(a) (b)

Figure 10 (a) Coherent laser optical image of pin in front of aperture (b) Laser optical direg ractionpattern obtained by merging three patterns with direg erent exposure times to avoid saturation ofthe central beam

(a) (b)

Figure 11 (a) Reconstructed modulus Only the hole with the out-of-focus pin is shown(b) Numerical back propagation of reconstructed complex wavemacreld by 12 mm to plane of pinPin appears in focus Support constraint only

6 Experiments with coherent electrons

Using a special support mask made by electron-beam lithography experiments havealso been undertaken using the HIO algorithm to retrieve the phases from experimen-tal transmission electron dinotraction patterns (Weierstall et al 2002) Images mightthen be reconstructed that are free of electron lens aberrations The support maskcontains hole pairs of various sizes with pairs separated by 1 m m The separationbetween holes of a pair is ca 50 nm less than the coherence width of the illuminationbeam The holes irregular in shape form the object we wish to reconstruct in futurework we plan to introduce nanoscale objects into the holes Figure 12a below showsthe experimental dinotraction pattern from two holes shown in shy gure 12c obtained

Phil Trans R Soc Lond A (2002)

890 J C H Spence U Weierstall and M Howells

20 nm

(a) (b)

(c)

Figure 12 (a) Experimental electron direg raction pattern from two nanoscale holes in an opaquemask (shown in (c)) recorded using a macreld-emission electron microscope (CM200 FEG at 40 kV1k pound1k CCD with 24 m m pixels camera length 611 m) Youngrsquo s fringes from the two holes arevisible these fringes encode phase information (b) Reconstruction of image of holes from thedireg raction pattern (a) after 100 iterations of the HIO algorithm Modulus constraint applied inimage space with large triangular computational support The hole shape is comparable with theTEM image in (c) (c) Conventional TEM image of holes that created the direg raction pattern (a)

with a Philips CM 200 shy eld-emission electron microscope at 40 kV The acceleratingvoltage was lowered to 40 kV to make the multi-layered support opaque ModulatedYoungrsquos fringes are clearly seen

The resulting image after 100 HIO iterations is shown in shy gure 12b The holesreconstructed from the dinotraction pattern modulus look similar to the TEM imageof the holes in shy gure 12c but the resolution of the reconstruction is lower thanthe normal TEM image resolution High spatial frequency information from thehole edge is missing in the experimental dinotraction pattern due perhaps to drift ofthe imaging lenses during the exposure time

Table 1 shows a summary chart of all the conditions so far examined with HIO sim-

Phil Trans R Soc Lond A (2002)

Phase

recovery

an

dlen

slessim

agin

g891

Table 1 Summary chart showing direg erent conditions for phase recovery with HIO algorithm

( CArsquo denotes the complex algorithm (uses direg racted intensity and support constraint in real space) RIPrsquo denotes the complex algorithm withpositivity constraint of Re ordf Im ordf (uses direg racted intensity and support constraint in real space negative Re ordf and Im ordf inside the supportarea are set positive) Mrsquo denotes the modulus constraint on ordf in real space (uses direg racted intensity and support constraint in real spacecomplex numbers inside the support area are replaced by their modulus to enforce positivity) Prsquo denotes the phase constraint on ordf in realspace (uses direg racted intensity and support constraint in real space the phase is extracted and retained from complex numbers inside thesupport area modulus is extracted and set to 1 inside the support area))

general pure strong pure weakrsquo real weak-phasecomplex object phase object phase object (amplitude) object withordf = A exp(iiquest ) ordf = exp(i iquest (r)) iquest lt ordm =2 object absorption

known two-holephysical support

CA (laser experimentpin cheek cells)

CA P (laserexperiment mica)

RIP CA M RIP CA RIP CA

known one-holephysical support

CA only for certainsupport shapes(eg triangle withsharp edges)

CA slowconvergence Pworks better

RIP CA M RIP CA (laserexperimentnumbers)

RIP CA

unknown two-or-more-hole physicalsupport loose(eg triangle compu-tational support)

| | RIP M ( TEM experimentholes) (laserexperiment numbers)(planned X-ray micapinholes)

RIP (Miao X-rayexperimenttransparent support)

unknown one-holephysical support(eg triangle compu-tational support)

| | RIP only for holewith low symmetryand sharp edgesslow convergence

M RIP only for holewith low symmetryand sharp edgesslow convergence

RIP only for holewith low symmetryand sharp edgesslow convergence

Phil

Tra

ns

R

Soc

L

on

d

A(2

002)

892 J C H Spence U Weierstall and M Howells

ulations and experiments The enotectiveness of dinoterent constraints in real space fordinoterent physical and computational supports is compared The physical supportis the support which exists in the experiment and which may or may not be accu-rately known The computational support is the support used in the HIO iterationsThe table shows experiments that have been performed in italics The shy rst columnshows dinoterent possibilities for the physical support the most favourable at the topthe worst at the bottom The computational support and known physical supportare identical The computational support used with the unknown physical supportis larger than the physical support Reconstructions with one-hole physical supportand loose computational support are only possible with weak or real objects andsupport holes with sharp edges and low symmetry and convergence are very slow insimulations A known two-or-more-hole support proved to be most favourable in allcases

7 Discussion and conclusions direg ractive imaging with other particles

In this paper we have reviewed the implications of Thomas Youngrsquos experiment formatter waves and in particular for electron interferometry By taking advantage ofthe very high quality of the coherent electron beam available in the modern shy eld-emission electron microscope a number of important results (such as the AharonovBohm enotect and the electron Sagnac enotect) have been established and the shy eldof atomic-resolution transmission electron microscopy phase-contrast imaging andelectron holography opened up All these methods continue to produce a large sci-entishy c pay-onot in microstructural materials characterization and discovery and inbiology A new lensless imaging method is then reviewed which may be understoodin part by analogy with electron holography This HIO algorithm uses an iterativeapproach to assign phases to dinotraction pattern intensities To evaluate its perfor-mance we give experimental examples of image reconstruction with both coherentelectrons and light together with simulations for the soft X-ray case Thus we haveshown experimentally that the HIO algorithm is capable of solving the phase problemfor scattering from non-periodic objects and of making quantitative measurements ofphase (modulo 2ordm ) in good agreement with independent measurements The beam-splitting arrangement of an interferometer is not required and dinotraction-limitedlensless imaging seems possible as a result This may open the way for imaging withthe many particles for which no lenses exist such as neutrons ballistic electrons insolids at low temperature or neutral atoms and ions either as free coherent beamsor condensates (Bright coherent sources have recently appeared for many of these(Pauly 2000)) The dinotering interactions of these particles may provide useful newprobes for chemistry materials science and biology For example the reduced radia-tion damage enotects of imaging organic material with low-energy electrons neutronsand helium atoms have recently been compared (Spence et al 1999) and HIO sim-ulations have been completed for neutron imaging of a diatom (Spence et al 2001)The very small refractive index of organic matter for neutrons limits resolution andcontrast for thin samples For low-energy ions or metastable helium strong samplebeam interactions must be considered Damage from neutral helium-atom imagingof molecules should be negligible however penetration is also negligible (producinga proshy le image) Detectors for coherent atom beams still present serious problemssince area detectors are inemacr cient For biology the weak atom interaction is desirable

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 893

to avoid damage however a strong interaction at the detector is also needed so thationization before detection must be arranged

For real objects we have seen that the resolution in images recovered by the HIOalgorithm is largely independent of the accuracy with which the support is knownThe addition of noise anotects the probability that the unique solution will be found (asmeasured by the error metric) rather than the resolution which is near dinotraction-limited For complex objects our experience indicates that a much more accurateknowledge of the physical support boundary is needed approaching the scale ofthe resolution sought The tight support and two-hole geometry (see Fienup (1987)for other geometries) needed for complex objects (complex refractive index) willbecome increasingly accessible with the development of new lithographic techniquesfor support construction This can be combined with independent characterization ofthe support shape and the scaling information provided by the known autocorrelationof the support Experimentally favourable cases for imaging with nanoscale resolutionat present require loose support which table 1 shows greatly favours real objectssuch as the charge density which dinotracts hard X-rays at energies well away fromabsorption edges or very thin samples for high-energy electrons Unfortunately beamcoherence is more readily obtained for soft X-rays

Using modern lithographic techniques it is becoming increasingly possible toestablish the experimental conditions of known support required by the HIO algo-rithm both for free particle beams and for electrons in solids at low temperaturesThus for example we have seen the algorithm used to determine the phase of wavefunctions for electrons in a one-dimensional YBCO Josephson junction using TEMimages of the structure to deshy ne the required support (boundary conditions) (Car-mody et al 2000) Given the fundamental role that phase has played historically inphysics and which Thomas Young was amongst the shy rst to appreciate any techniquecapable of exposing phase can be expected to yield many surprises in physics in thefuture

Supported by ARO award DAAD190010500 We thank Dr Q Chen for the use of macrgure 12 (fromWeierstall et al 2002)

References

Aharonov Y amp Bohm D 1959 Signimacrcance of the electromagnetic potentials in the quantumtheory Phys Rev 115 485

Allen L J Faulkner H M L Nugent K A Oxley M P amp Paganin D 2001 Phase retrievalfrom images in the presence of macrrst-order vortices Phys Rev E 63 037602

Bauschke H H et al 2002 (In the press)

Boersch H 1943 Fresnelsche Beugung im Elektronenmikroskop Phys Z 44 202211

Bruck Y amp Sodin L 1979 On the ambiguity of the image-reconstruction problem Opt Com-mun 30 304308

Carmody M Moeckly B H Merkle K amp Marks L 2000 Spatial variation of the current ingrain boundary Josephson junctions J Appl Phys 87 24542459

Coene W Janssen G Op de Beek M amp van Dyck D 1992 Phase retrieval through focusvariation for ultra-resolution in macreld emission transmission electron microscopy Phys RevLett 69 37433746

Crewe A V Eggenberger D N Wall J amp Welter L M 1968 Electron gun using a macreldemission source Rev Scient Instrum 39 576

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 885

black = -

white = +

(a) (b)

(c)

Figure 6 (a) Modulus and (b) phase of reconstructed image wave (c) phase reconstructed fromshifted direg raction pattern shows equiphase contour lines Phase shift by mica ordm No sign ormodulus constraint

support size and orientation were derived from the autocorrelation The complexHIO-phase retrieval algorithm was used to recover the image wave phase and modulusshown in shy gure 6 The phase shift by the mica plates in the reconstruction is ordm which since the algorithm reconstructs phase modulo 2 ordm is in good agreement withour independent measurement of 9 ordm based on the measured thickness and knownrefractive index of mica Thus the HIO algorithm is capable of quantitative phasemeasurement

Experiments with a transparent physical support have been attempted using alimiting aperture upstream of the sample but have not been successful so far becauseof the high intensity in the centre of the dinotraction pattern which causes bloomingin the CCD chip This problem persists for three-dimensional objects however athree-quadrant mask in the object plane has been proposed to address this dimacr culty(Sayre et al 1998) (Beam stops cannot be placed near enough to the CCD toavoid generating additional fringes) Experiments using the known intensity envelopeof a focused beam as the support have also not been successful for such complexobjects there appears to be a need for sharp physical support edges perhaps requiredto generate reference edge waves (For real objects there is no such requirement)However a coherent probe spanning two `knownrsquo holes works well in simulation

Previous experiments have been based on objects of simple type such as binary

Phil Trans R Soc Lond A (2002)

886 J C H Spence U Weierstall and M Howells

(a)

(b)

Figure 7 (a) Coherent laser optical image of cheek cells in two-hole support (300 m m apertures)About seven cells clump together in the upper hole Only Fresnel edge contrast is visible dueto defocus the cell nuclei are visible as black dots (b) Laser optical direg raction pattern mergedfrom four direg erent exposures to avoid saturation of the central beam

objects which contain much less information than greyscale objects Experimentswith human cheek cells which serve as more complicated greyscale phase objectshave therefore been performed The cheek cells were positioned on a glass slide insidetwo 300 m m apertures separated by 300 m m The apertures act as the opaque supportconsisting of two separated parts as needed for reconstruction of complex objectsFigure 7a shows the coherent optical image of the slightly defocused cheek cells asformed with a lens Only Fresnel edge contrast is visible while the nuclei producesome absorption Figure 7b shows the optical dinotraction pattern merged from fourdinoterent exposures The size and orientation of the support were estimated fromthe autocorrelation The complex reconstruction of phase and amplitude after about50 HIO iterations using the support constraint is shown in shy gure 8 The cells arevisible with large-area phase contrast in the phase image and the phase shift of thecells is about ordm In the reconstructed amplitude only edge contrast is visible as inthe optical image

For a three-dimensional object extended along the optical axis the question arisesas to which plane in the object is reconstructed given that the detected intensity liesat inshy nity and contains contributions from scattering at every point along the axisin the object space A simulation to clarify this is shown in shy gure 9 The simulatedobject consists of two planes separated by 1 mm in the z-direction Plane 1 has a holecontaining a complex object (the number 1) while plane 2 has two holes containingcomplex objects (numbers 2 and 3) The object exit wave calculated on plane 2

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 887

(a) (b)

Figure 8 (a) Modulus and (b) phase of reconstructed image wave The modulus shows only edgecontrast whereas the phase shows the cells in large-area phase contrast The phase shift due tothe cells is about ordm No modulus or sign constraint

(shy gure 9a) shows plane 1 out of focus The modulus of the Fourier transform (Fouriermodulus) of this image wave is used as input for the HIO algorithm In the inversiondinoterent shapes of the supports used at each plane shown inset can distinguish thetwo object planes If a plane is assumed unknown a larger support hole is usedthat contains the physical support Depending on which support plane is assumed asknown (inset) in the iteration (plane 1 support known and plane 2 support unknownor vice versa) the other plane is always reconstructed out of focus by +1 or iexcl 1 mm(shy gure 9c d) That means that in the case of known support on plane 1 the algorithmreconstructs the wave as seen through an imaging lens with out-of-focus images ofthe holes on plane 2 As expected the depth of focus for interactive inversion is givenby the expression for an ideal lens which imposes a dinotraction limit equal to that ofthe highest spatial frequency recorded in the dinotraction pattern

To verify this experimentally a pin was mounted 12 mm upstream of a pair of300 m m apertures The coherent optical image is shown in shy gure 10a A Fresnel fringesurrounds the pin since the lens is focused on the apertures The image was recoveredfrom the recorded dinotraction pattern intensity (shy gure 10b) Again three dinotractionpatterns with dinoterent exposure times had to be merged to avoid pixel saturationand blooming The reconstructed image modulus is shown in shy gure 11a where the

Phil Trans R Soc Lond A (2002)

888 J C H Spence U Weierstall and M Howells

illuminationdirection

plane 1

1 mm

(a) (b)

(c) (d)

Figure 9 (a) Schematic view of a simulated object consisting of two planes separated by 1 mm inthe direction of light propagation Plane 1 has a hole containing a complex object (number 1) andplane 2 has two holes containing complex objects (numbers 2 and 3) (b) Modulus of complexexit wave on plane 2 (c) Iterative reconstruction with plane 2 support assumed known (thesupport shape used is shown in the top right-hand corner) Plane 1 is out of focus by 1 mm(d) Iterative reconstruction with plane 1 support assumed known (the support shape is shownin the top right-hand corner) Plane 2 is out of focus by iexcl1 mm

pin appears out of focus (Fresnel fringe barely visible) because the aperture shapewas imposed as support The pin can be brought into focus by using the appropriateFresnel propagator with centz = 12 mm and para = 6328 nm taking the apertures outof focus (see shy gure 11b) Thus as expected the reconstructed plane is always thatplane in which the support condition is imposed where the algorithm shy nds a uniquesolution subject to the given boundary condition Propagation to other planes maybe performed subsequently since the complex waveshy eld is obtained subject to thedepth-of-focus limitation The relative sideways shift of the pin inside the aperturebetween shy gure 10a and shy gure 11a is due to a slight unintentional tilt of the object(pin plus pinhole) between recording the image and the dinotraction pattern

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 889

(a) (b)

Figure 10 (a) Coherent laser optical image of pin in front of aperture (b) Laser optical direg ractionpattern obtained by merging three patterns with direg erent exposure times to avoid saturation ofthe central beam

(a) (b)

Figure 11 (a) Reconstructed modulus Only the hole with the out-of-focus pin is shown(b) Numerical back propagation of reconstructed complex wavemacreld by 12 mm to plane of pinPin appears in focus Support constraint only

6 Experiments with coherent electrons

Using a special support mask made by electron-beam lithography experiments havealso been undertaken using the HIO algorithm to retrieve the phases from experimen-tal transmission electron dinotraction patterns (Weierstall et al 2002) Images mightthen be reconstructed that are free of electron lens aberrations The support maskcontains hole pairs of various sizes with pairs separated by 1 m m The separationbetween holes of a pair is ca 50 nm less than the coherence width of the illuminationbeam The holes irregular in shape form the object we wish to reconstruct in futurework we plan to introduce nanoscale objects into the holes Figure 12a below showsthe experimental dinotraction pattern from two holes shown in shy gure 12c obtained

Phil Trans R Soc Lond A (2002)

890 J C H Spence U Weierstall and M Howells

20 nm

(a) (b)

(c)

Figure 12 (a) Experimental electron direg raction pattern from two nanoscale holes in an opaquemask (shown in (c)) recorded using a macreld-emission electron microscope (CM200 FEG at 40 kV1k pound1k CCD with 24 m m pixels camera length 611 m) Youngrsquo s fringes from the two holes arevisible these fringes encode phase information (b) Reconstruction of image of holes from thedireg raction pattern (a) after 100 iterations of the HIO algorithm Modulus constraint applied inimage space with large triangular computational support The hole shape is comparable with theTEM image in (c) (c) Conventional TEM image of holes that created the direg raction pattern (a)

with a Philips CM 200 shy eld-emission electron microscope at 40 kV The acceleratingvoltage was lowered to 40 kV to make the multi-layered support opaque ModulatedYoungrsquos fringes are clearly seen

The resulting image after 100 HIO iterations is shown in shy gure 12b The holesreconstructed from the dinotraction pattern modulus look similar to the TEM imageof the holes in shy gure 12c but the resolution of the reconstruction is lower thanthe normal TEM image resolution High spatial frequency information from thehole edge is missing in the experimental dinotraction pattern due perhaps to drift ofthe imaging lenses during the exposure time

Table 1 shows a summary chart of all the conditions so far examined with HIO sim-

Phil Trans R Soc Lond A (2002)

Phase

recovery

an

dlen

slessim

agin

g891

Table 1 Summary chart showing direg erent conditions for phase recovery with HIO algorithm

( CArsquo denotes the complex algorithm (uses direg racted intensity and support constraint in real space) RIPrsquo denotes the complex algorithm withpositivity constraint of Re ordf Im ordf (uses direg racted intensity and support constraint in real space negative Re ordf and Im ordf inside the supportarea are set positive) Mrsquo denotes the modulus constraint on ordf in real space (uses direg racted intensity and support constraint in real spacecomplex numbers inside the support area are replaced by their modulus to enforce positivity) Prsquo denotes the phase constraint on ordf in realspace (uses direg racted intensity and support constraint in real space the phase is extracted and retained from complex numbers inside thesupport area modulus is extracted and set to 1 inside the support area))

general pure strong pure weakrsquo real weak-phasecomplex object phase object phase object (amplitude) object withordf = A exp(iiquest ) ordf = exp(i iquest (r)) iquest lt ordm =2 object absorption

known two-holephysical support

CA (laser experimentpin cheek cells)

CA P (laserexperiment mica)

RIP CA M RIP CA RIP CA

known one-holephysical support

CA only for certainsupport shapes(eg triangle withsharp edges)

CA slowconvergence Pworks better

RIP CA M RIP CA (laserexperimentnumbers)

RIP CA

unknown two-or-more-hole physicalsupport loose(eg triangle compu-tational support)

| | RIP M ( TEM experimentholes) (laserexperiment numbers)(planned X-ray micapinholes)

RIP (Miao X-rayexperimenttransparent support)

unknown one-holephysical support(eg triangle compu-tational support)

| | RIP only for holewith low symmetryand sharp edgesslow convergence

M RIP only for holewith low symmetryand sharp edgesslow convergence

RIP only for holewith low symmetryand sharp edgesslow convergence

Phil

Tra

ns

R

Soc

L

on

d

A(2

002)

892 J C H Spence U Weierstall and M Howells

ulations and experiments The enotectiveness of dinoterent constraints in real space fordinoterent physical and computational supports is compared The physical supportis the support which exists in the experiment and which may or may not be accu-rately known The computational support is the support used in the HIO iterationsThe table shows experiments that have been performed in italics The shy rst columnshows dinoterent possibilities for the physical support the most favourable at the topthe worst at the bottom The computational support and known physical supportare identical The computational support used with the unknown physical supportis larger than the physical support Reconstructions with one-hole physical supportand loose computational support are only possible with weak or real objects andsupport holes with sharp edges and low symmetry and convergence are very slow insimulations A known two-or-more-hole support proved to be most favourable in allcases

7 Discussion and conclusions direg ractive imaging with other particles

In this paper we have reviewed the implications of Thomas Youngrsquos experiment formatter waves and in particular for electron interferometry By taking advantage ofthe very high quality of the coherent electron beam available in the modern shy eld-emission electron microscope a number of important results (such as the AharonovBohm enotect and the electron Sagnac enotect) have been established and the shy eldof atomic-resolution transmission electron microscopy phase-contrast imaging andelectron holography opened up All these methods continue to produce a large sci-entishy c pay-onot in microstructural materials characterization and discovery and inbiology A new lensless imaging method is then reviewed which may be understoodin part by analogy with electron holography This HIO algorithm uses an iterativeapproach to assign phases to dinotraction pattern intensities To evaluate its perfor-mance we give experimental examples of image reconstruction with both coherentelectrons and light together with simulations for the soft X-ray case Thus we haveshown experimentally that the HIO algorithm is capable of solving the phase problemfor scattering from non-periodic objects and of making quantitative measurements ofphase (modulo 2ordm ) in good agreement with independent measurements The beam-splitting arrangement of an interferometer is not required and dinotraction-limitedlensless imaging seems possible as a result This may open the way for imaging withthe many particles for which no lenses exist such as neutrons ballistic electrons insolids at low temperature or neutral atoms and ions either as free coherent beamsor condensates (Bright coherent sources have recently appeared for many of these(Pauly 2000)) The dinotering interactions of these particles may provide useful newprobes for chemistry materials science and biology For example the reduced radia-tion damage enotects of imaging organic material with low-energy electrons neutronsand helium atoms have recently been compared (Spence et al 1999) and HIO sim-ulations have been completed for neutron imaging of a diatom (Spence et al 2001)The very small refractive index of organic matter for neutrons limits resolution andcontrast for thin samples For low-energy ions or metastable helium strong samplebeam interactions must be considered Damage from neutral helium-atom imagingof molecules should be negligible however penetration is also negligible (producinga proshy le image) Detectors for coherent atom beams still present serious problemssince area detectors are inemacr cient For biology the weak atom interaction is desirable

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 893

to avoid damage however a strong interaction at the detector is also needed so thationization before detection must be arranged

For real objects we have seen that the resolution in images recovered by the HIOalgorithm is largely independent of the accuracy with which the support is knownThe addition of noise anotects the probability that the unique solution will be found (asmeasured by the error metric) rather than the resolution which is near dinotraction-limited For complex objects our experience indicates that a much more accurateknowledge of the physical support boundary is needed approaching the scale ofthe resolution sought The tight support and two-hole geometry (see Fienup (1987)for other geometries) needed for complex objects (complex refractive index) willbecome increasingly accessible with the development of new lithographic techniquesfor support construction This can be combined with independent characterization ofthe support shape and the scaling information provided by the known autocorrelationof the support Experimentally favourable cases for imaging with nanoscale resolutionat present require loose support which table 1 shows greatly favours real objectssuch as the charge density which dinotracts hard X-rays at energies well away fromabsorption edges or very thin samples for high-energy electrons Unfortunately beamcoherence is more readily obtained for soft X-rays

Using modern lithographic techniques it is becoming increasingly possible toestablish the experimental conditions of known support required by the HIO algo-rithm both for free particle beams and for electrons in solids at low temperaturesThus for example we have seen the algorithm used to determine the phase of wavefunctions for electrons in a one-dimensional YBCO Josephson junction using TEMimages of the structure to deshy ne the required support (boundary conditions) (Car-mody et al 2000) Given the fundamental role that phase has played historically inphysics and which Thomas Young was amongst the shy rst to appreciate any techniquecapable of exposing phase can be expected to yield many surprises in physics in thefuture

Supported by ARO award DAAD190010500 We thank Dr Q Chen for the use of macrgure 12 (fromWeierstall et al 2002)

References

Aharonov Y amp Bohm D 1959 Signimacrcance of the electromagnetic potentials in the quantumtheory Phys Rev 115 485

Allen L J Faulkner H M L Nugent K A Oxley M P amp Paganin D 2001 Phase retrievalfrom images in the presence of macrrst-order vortices Phys Rev E 63 037602

Bauschke H H et al 2002 (In the press)

Boersch H 1943 Fresnelsche Beugung im Elektronenmikroskop Phys Z 44 202211

Bruck Y amp Sodin L 1979 On the ambiguity of the image-reconstruction problem Opt Com-mun 30 304308

Carmody M Moeckly B H Merkle K amp Marks L 2000 Spatial variation of the current ingrain boundary Josephson junctions J Appl Phys 87 24542459

Coene W Janssen G Op de Beek M amp van Dyck D 1992 Phase retrieval through focusvariation for ultra-resolution in macreld emission transmission electron microscopy Phys RevLett 69 37433746

Crewe A V Eggenberger D N Wall J amp Welter L M 1968 Electron gun using a macreldemission source Rev Scient Instrum 39 576

Phil Trans R Soc Lond A (2002)

886 J C H Spence U Weierstall and M Howells

(a)

(b)

Figure 7 (a) Coherent laser optical image of cheek cells in two-hole support (300 m m apertures)About seven cells clump together in the upper hole Only Fresnel edge contrast is visible dueto defocus the cell nuclei are visible as black dots (b) Laser optical direg raction pattern mergedfrom four direg erent exposures to avoid saturation of the central beam

objects which contain much less information than greyscale objects Experimentswith human cheek cells which serve as more complicated greyscale phase objectshave therefore been performed The cheek cells were positioned on a glass slide insidetwo 300 m m apertures separated by 300 m m The apertures act as the opaque supportconsisting of two separated parts as needed for reconstruction of complex objectsFigure 7a shows the coherent optical image of the slightly defocused cheek cells asformed with a lens Only Fresnel edge contrast is visible while the nuclei producesome absorption Figure 7b shows the optical dinotraction pattern merged from fourdinoterent exposures The size and orientation of the support were estimated fromthe autocorrelation The complex reconstruction of phase and amplitude after about50 HIO iterations using the support constraint is shown in shy gure 8 The cells arevisible with large-area phase contrast in the phase image and the phase shift of thecells is about ordm In the reconstructed amplitude only edge contrast is visible as inthe optical image

For a three-dimensional object extended along the optical axis the question arisesas to which plane in the object is reconstructed given that the detected intensity liesat inshy nity and contains contributions from scattering at every point along the axisin the object space A simulation to clarify this is shown in shy gure 9 The simulatedobject consists of two planes separated by 1 mm in the z-direction Plane 1 has a holecontaining a complex object (the number 1) while plane 2 has two holes containingcomplex objects (numbers 2 and 3) The object exit wave calculated on plane 2

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 887

(a) (b)

Figure 8 (a) Modulus and (b) phase of reconstructed image wave The modulus shows only edgecontrast whereas the phase shows the cells in large-area phase contrast The phase shift due tothe cells is about ordm No modulus or sign constraint

(shy gure 9a) shows plane 1 out of focus The modulus of the Fourier transform (Fouriermodulus) of this image wave is used as input for the HIO algorithm In the inversiondinoterent shapes of the supports used at each plane shown inset can distinguish thetwo object planes If a plane is assumed unknown a larger support hole is usedthat contains the physical support Depending on which support plane is assumed asknown (inset) in the iteration (plane 1 support known and plane 2 support unknownor vice versa) the other plane is always reconstructed out of focus by +1 or iexcl 1 mm(shy gure 9c d) That means that in the case of known support on plane 1 the algorithmreconstructs the wave as seen through an imaging lens with out-of-focus images ofthe holes on plane 2 As expected the depth of focus for interactive inversion is givenby the expression for an ideal lens which imposes a dinotraction limit equal to that ofthe highest spatial frequency recorded in the dinotraction pattern

To verify this experimentally a pin was mounted 12 mm upstream of a pair of300 m m apertures The coherent optical image is shown in shy gure 10a A Fresnel fringesurrounds the pin since the lens is focused on the apertures The image was recoveredfrom the recorded dinotraction pattern intensity (shy gure 10b) Again three dinotractionpatterns with dinoterent exposure times had to be merged to avoid pixel saturationand blooming The reconstructed image modulus is shown in shy gure 11a where the

Phil Trans R Soc Lond A (2002)

888 J C H Spence U Weierstall and M Howells

illuminationdirection

plane 1

1 mm

(a) (b)

(c) (d)

Figure 9 (a) Schematic view of a simulated object consisting of two planes separated by 1 mm inthe direction of light propagation Plane 1 has a hole containing a complex object (number 1) andplane 2 has two holes containing complex objects (numbers 2 and 3) (b) Modulus of complexexit wave on plane 2 (c) Iterative reconstruction with plane 2 support assumed known (thesupport shape used is shown in the top right-hand corner) Plane 1 is out of focus by 1 mm(d) Iterative reconstruction with plane 1 support assumed known (the support shape is shownin the top right-hand corner) Plane 2 is out of focus by iexcl1 mm

pin appears out of focus (Fresnel fringe barely visible) because the aperture shapewas imposed as support The pin can be brought into focus by using the appropriateFresnel propagator with centz = 12 mm and para = 6328 nm taking the apertures outof focus (see shy gure 11b) Thus as expected the reconstructed plane is always thatplane in which the support condition is imposed where the algorithm shy nds a uniquesolution subject to the given boundary condition Propagation to other planes maybe performed subsequently since the complex waveshy eld is obtained subject to thedepth-of-focus limitation The relative sideways shift of the pin inside the aperturebetween shy gure 10a and shy gure 11a is due to a slight unintentional tilt of the object(pin plus pinhole) between recording the image and the dinotraction pattern

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 889

(a) (b)

Figure 10 (a) Coherent laser optical image of pin in front of aperture (b) Laser optical direg ractionpattern obtained by merging three patterns with direg erent exposure times to avoid saturation ofthe central beam

(a) (b)

Figure 11 (a) Reconstructed modulus Only the hole with the out-of-focus pin is shown(b) Numerical back propagation of reconstructed complex wavemacreld by 12 mm to plane of pinPin appears in focus Support constraint only

6 Experiments with coherent electrons

Using a special support mask made by electron-beam lithography experiments havealso been undertaken using the HIO algorithm to retrieve the phases from experimen-tal transmission electron dinotraction patterns (Weierstall et al 2002) Images mightthen be reconstructed that are free of electron lens aberrations The support maskcontains hole pairs of various sizes with pairs separated by 1 m m The separationbetween holes of a pair is ca 50 nm less than the coherence width of the illuminationbeam The holes irregular in shape form the object we wish to reconstruct in futurework we plan to introduce nanoscale objects into the holes Figure 12a below showsthe experimental dinotraction pattern from two holes shown in shy gure 12c obtained

Phil Trans R Soc Lond A (2002)

890 J C H Spence U Weierstall and M Howells

20 nm

(a) (b)

(c)

Figure 12 (a) Experimental electron direg raction pattern from two nanoscale holes in an opaquemask (shown in (c)) recorded using a macreld-emission electron microscope (CM200 FEG at 40 kV1k pound1k CCD with 24 m m pixels camera length 611 m) Youngrsquo s fringes from the two holes arevisible these fringes encode phase information (b) Reconstruction of image of holes from thedireg raction pattern (a) after 100 iterations of the HIO algorithm Modulus constraint applied inimage space with large triangular computational support The hole shape is comparable with theTEM image in (c) (c) Conventional TEM image of holes that created the direg raction pattern (a)

with a Philips CM 200 shy eld-emission electron microscope at 40 kV The acceleratingvoltage was lowered to 40 kV to make the multi-layered support opaque ModulatedYoungrsquos fringes are clearly seen

The resulting image after 100 HIO iterations is shown in shy gure 12b The holesreconstructed from the dinotraction pattern modulus look similar to the TEM imageof the holes in shy gure 12c but the resolution of the reconstruction is lower thanthe normal TEM image resolution High spatial frequency information from thehole edge is missing in the experimental dinotraction pattern due perhaps to drift ofthe imaging lenses during the exposure time

Table 1 shows a summary chart of all the conditions so far examined with HIO sim-

Phil Trans R Soc Lond A (2002)

Phase

recovery

an

dlen

slessim

agin

g891

Table 1 Summary chart showing direg erent conditions for phase recovery with HIO algorithm

( CArsquo denotes the complex algorithm (uses direg racted intensity and support constraint in real space) RIPrsquo denotes the complex algorithm withpositivity constraint of Re ordf Im ordf (uses direg racted intensity and support constraint in real space negative Re ordf and Im ordf inside the supportarea are set positive) Mrsquo denotes the modulus constraint on ordf in real space (uses direg racted intensity and support constraint in real spacecomplex numbers inside the support area are replaced by their modulus to enforce positivity) Prsquo denotes the phase constraint on ordf in realspace (uses direg racted intensity and support constraint in real space the phase is extracted and retained from complex numbers inside thesupport area modulus is extracted and set to 1 inside the support area))

general pure strong pure weakrsquo real weak-phasecomplex object phase object phase object (amplitude) object withordf = A exp(iiquest ) ordf = exp(i iquest (r)) iquest lt ordm =2 object absorption

known two-holephysical support

CA (laser experimentpin cheek cells)

CA P (laserexperiment mica)

RIP CA M RIP CA RIP CA

known one-holephysical support

CA only for certainsupport shapes(eg triangle withsharp edges)

CA slowconvergence Pworks better

RIP CA M RIP CA (laserexperimentnumbers)

RIP CA

unknown two-or-more-hole physicalsupport loose(eg triangle compu-tational support)

| | RIP M ( TEM experimentholes) (laserexperiment numbers)(planned X-ray micapinholes)

RIP (Miao X-rayexperimenttransparent support)

unknown one-holephysical support(eg triangle compu-tational support)

| | RIP only for holewith low symmetryand sharp edgesslow convergence

M RIP only for holewith low symmetryand sharp edgesslow convergence

RIP only for holewith low symmetryand sharp edgesslow convergence

Phil

Tra

ns

R

Soc

L

on

d

A(2

002)

892 J C H Spence U Weierstall and M Howells

ulations and experiments The enotectiveness of dinoterent constraints in real space fordinoterent physical and computational supports is compared The physical supportis the support which exists in the experiment and which may or may not be accu-rately known The computational support is the support used in the HIO iterationsThe table shows experiments that have been performed in italics The shy rst columnshows dinoterent possibilities for the physical support the most favourable at the topthe worst at the bottom The computational support and known physical supportare identical The computational support used with the unknown physical supportis larger than the physical support Reconstructions with one-hole physical supportand loose computational support are only possible with weak or real objects andsupport holes with sharp edges and low symmetry and convergence are very slow insimulations A known two-or-more-hole support proved to be most favourable in allcases

7 Discussion and conclusions direg ractive imaging with other particles

In this paper we have reviewed the implications of Thomas Youngrsquos experiment formatter waves and in particular for electron interferometry By taking advantage ofthe very high quality of the coherent electron beam available in the modern shy eld-emission electron microscope a number of important results (such as the AharonovBohm enotect and the electron Sagnac enotect) have been established and the shy eldof atomic-resolution transmission electron microscopy phase-contrast imaging andelectron holography opened up All these methods continue to produce a large sci-entishy c pay-onot in microstructural materials characterization and discovery and inbiology A new lensless imaging method is then reviewed which may be understoodin part by analogy with electron holography This HIO algorithm uses an iterativeapproach to assign phases to dinotraction pattern intensities To evaluate its perfor-mance we give experimental examples of image reconstruction with both coherentelectrons and light together with simulations for the soft X-ray case Thus we haveshown experimentally that the HIO algorithm is capable of solving the phase problemfor scattering from non-periodic objects and of making quantitative measurements ofphase (modulo 2ordm ) in good agreement with independent measurements The beam-splitting arrangement of an interferometer is not required and dinotraction-limitedlensless imaging seems possible as a result This may open the way for imaging withthe many particles for which no lenses exist such as neutrons ballistic electrons insolids at low temperature or neutral atoms and ions either as free coherent beamsor condensates (Bright coherent sources have recently appeared for many of these(Pauly 2000)) The dinotering interactions of these particles may provide useful newprobes for chemistry materials science and biology For example the reduced radia-tion damage enotects of imaging organic material with low-energy electrons neutronsand helium atoms have recently been compared (Spence et al 1999) and HIO sim-ulations have been completed for neutron imaging of a diatom (Spence et al 2001)The very small refractive index of organic matter for neutrons limits resolution andcontrast for thin samples For low-energy ions or metastable helium strong samplebeam interactions must be considered Damage from neutral helium-atom imagingof molecules should be negligible however penetration is also negligible (producinga proshy le image) Detectors for coherent atom beams still present serious problemssince area detectors are inemacr cient For biology the weak atom interaction is desirable

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 893

to avoid damage however a strong interaction at the detector is also needed so thationization before detection must be arranged

For real objects we have seen that the resolution in images recovered by the HIOalgorithm is largely independent of the accuracy with which the support is knownThe addition of noise anotects the probability that the unique solution will be found (asmeasured by the error metric) rather than the resolution which is near dinotraction-limited For complex objects our experience indicates that a much more accurateknowledge of the physical support boundary is needed approaching the scale ofthe resolution sought The tight support and two-hole geometry (see Fienup (1987)for other geometries) needed for complex objects (complex refractive index) willbecome increasingly accessible with the development of new lithographic techniquesfor support construction This can be combined with independent characterization ofthe support shape and the scaling information provided by the known autocorrelationof the support Experimentally favourable cases for imaging with nanoscale resolutionat present require loose support which table 1 shows greatly favours real objectssuch as the charge density which dinotracts hard X-rays at energies well away fromabsorption edges or very thin samples for high-energy electrons Unfortunately beamcoherence is more readily obtained for soft X-rays

Using modern lithographic techniques it is becoming increasingly possible toestablish the experimental conditions of known support required by the HIO algo-rithm both for free particle beams and for electrons in solids at low temperaturesThus for example we have seen the algorithm used to determine the phase of wavefunctions for electrons in a one-dimensional YBCO Josephson junction using TEMimages of the structure to deshy ne the required support (boundary conditions) (Car-mody et al 2000) Given the fundamental role that phase has played historically inphysics and which Thomas Young was amongst the shy rst to appreciate any techniquecapable of exposing phase can be expected to yield many surprises in physics in thefuture

Supported by ARO award DAAD190010500 We thank Dr Q Chen for the use of macrgure 12 (fromWeierstall et al 2002)

References

Aharonov Y amp Bohm D 1959 Signimacrcance of the electromagnetic potentials in the quantumtheory Phys Rev 115 485

Allen L J Faulkner H M L Nugent K A Oxley M P amp Paganin D 2001 Phase retrievalfrom images in the presence of macrrst-order vortices Phys Rev E 63 037602

Bauschke H H et al 2002 (In the press)

Boersch H 1943 Fresnelsche Beugung im Elektronenmikroskop Phys Z 44 202211

Bruck Y amp Sodin L 1979 On the ambiguity of the image-reconstruction problem Opt Com-mun 30 304308

Carmody M Moeckly B H Merkle K amp Marks L 2000 Spatial variation of the current ingrain boundary Josephson junctions J Appl Phys 87 24542459

Coene W Janssen G Op de Beek M amp van Dyck D 1992 Phase retrieval through focusvariation for ultra-resolution in macreld emission transmission electron microscopy Phys RevLett 69 37433746

Crewe A V Eggenberger D N Wall J amp Welter L M 1968 Electron gun using a macreldemission source Rev Scient Instrum 39 576

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 887

(a) (b)

Figure 8 (a) Modulus and (b) phase of reconstructed image wave The modulus shows only edgecontrast whereas the phase shows the cells in large-area phase contrast The phase shift due tothe cells is about ordm No modulus or sign constraint

(shy gure 9a) shows plane 1 out of focus The modulus of the Fourier transform (Fouriermodulus) of this image wave is used as input for the HIO algorithm In the inversiondinoterent shapes of the supports used at each plane shown inset can distinguish thetwo object planes If a plane is assumed unknown a larger support hole is usedthat contains the physical support Depending on which support plane is assumed asknown (inset) in the iteration (plane 1 support known and plane 2 support unknownor vice versa) the other plane is always reconstructed out of focus by +1 or iexcl 1 mm(shy gure 9c d) That means that in the case of known support on plane 1 the algorithmreconstructs the wave as seen through an imaging lens with out-of-focus images ofthe holes on plane 2 As expected the depth of focus for interactive inversion is givenby the expression for an ideal lens which imposes a dinotraction limit equal to that ofthe highest spatial frequency recorded in the dinotraction pattern

To verify this experimentally a pin was mounted 12 mm upstream of a pair of300 m m apertures The coherent optical image is shown in shy gure 10a A Fresnel fringesurrounds the pin since the lens is focused on the apertures The image was recoveredfrom the recorded dinotraction pattern intensity (shy gure 10b) Again three dinotractionpatterns with dinoterent exposure times had to be merged to avoid pixel saturationand blooming The reconstructed image modulus is shown in shy gure 11a where the

Phil Trans R Soc Lond A (2002)

888 J C H Spence U Weierstall and M Howells

illuminationdirection

plane 1

1 mm

(a) (b)

(c) (d)

Figure 9 (a) Schematic view of a simulated object consisting of two planes separated by 1 mm inthe direction of light propagation Plane 1 has a hole containing a complex object (number 1) andplane 2 has two holes containing complex objects (numbers 2 and 3) (b) Modulus of complexexit wave on plane 2 (c) Iterative reconstruction with plane 2 support assumed known (thesupport shape used is shown in the top right-hand corner) Plane 1 is out of focus by 1 mm(d) Iterative reconstruction with plane 1 support assumed known (the support shape is shownin the top right-hand corner) Plane 2 is out of focus by iexcl1 mm

pin appears out of focus (Fresnel fringe barely visible) because the aperture shapewas imposed as support The pin can be brought into focus by using the appropriateFresnel propagator with centz = 12 mm and para = 6328 nm taking the apertures outof focus (see shy gure 11b) Thus as expected the reconstructed plane is always thatplane in which the support condition is imposed where the algorithm shy nds a uniquesolution subject to the given boundary condition Propagation to other planes maybe performed subsequently since the complex waveshy eld is obtained subject to thedepth-of-focus limitation The relative sideways shift of the pin inside the aperturebetween shy gure 10a and shy gure 11a is due to a slight unintentional tilt of the object(pin plus pinhole) between recording the image and the dinotraction pattern

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 889

(a) (b)

Figure 10 (a) Coherent laser optical image of pin in front of aperture (b) Laser optical direg ractionpattern obtained by merging three patterns with direg erent exposure times to avoid saturation ofthe central beam

(a) (b)

Figure 11 (a) Reconstructed modulus Only the hole with the out-of-focus pin is shown(b) Numerical back propagation of reconstructed complex wavemacreld by 12 mm to plane of pinPin appears in focus Support constraint only

6 Experiments with coherent electrons

Using a special support mask made by electron-beam lithography experiments havealso been undertaken using the HIO algorithm to retrieve the phases from experimen-tal transmission electron dinotraction patterns (Weierstall et al 2002) Images mightthen be reconstructed that are free of electron lens aberrations The support maskcontains hole pairs of various sizes with pairs separated by 1 m m The separationbetween holes of a pair is ca 50 nm less than the coherence width of the illuminationbeam The holes irregular in shape form the object we wish to reconstruct in futurework we plan to introduce nanoscale objects into the holes Figure 12a below showsthe experimental dinotraction pattern from two holes shown in shy gure 12c obtained

Phil Trans R Soc Lond A (2002)

890 J C H Spence U Weierstall and M Howells

20 nm

(a) (b)

(c)

Figure 12 (a) Experimental electron direg raction pattern from two nanoscale holes in an opaquemask (shown in (c)) recorded using a macreld-emission electron microscope (CM200 FEG at 40 kV1k pound1k CCD with 24 m m pixels camera length 611 m) Youngrsquo s fringes from the two holes arevisible these fringes encode phase information (b) Reconstruction of image of holes from thedireg raction pattern (a) after 100 iterations of the HIO algorithm Modulus constraint applied inimage space with large triangular computational support The hole shape is comparable with theTEM image in (c) (c) Conventional TEM image of holes that created the direg raction pattern (a)

with a Philips CM 200 shy eld-emission electron microscope at 40 kV The acceleratingvoltage was lowered to 40 kV to make the multi-layered support opaque ModulatedYoungrsquos fringes are clearly seen

The resulting image after 100 HIO iterations is shown in shy gure 12b The holesreconstructed from the dinotraction pattern modulus look similar to the TEM imageof the holes in shy gure 12c but the resolution of the reconstruction is lower thanthe normal TEM image resolution High spatial frequency information from thehole edge is missing in the experimental dinotraction pattern due perhaps to drift ofthe imaging lenses during the exposure time

Table 1 shows a summary chart of all the conditions so far examined with HIO sim-

Phil Trans R Soc Lond A (2002)

Phase

recovery

an

dlen

slessim

agin

g891

Table 1 Summary chart showing direg erent conditions for phase recovery with HIO algorithm

( CArsquo denotes the complex algorithm (uses direg racted intensity and support constraint in real space) RIPrsquo denotes the complex algorithm withpositivity constraint of Re ordf Im ordf (uses direg racted intensity and support constraint in real space negative Re ordf and Im ordf inside the supportarea are set positive) Mrsquo denotes the modulus constraint on ordf in real space (uses direg racted intensity and support constraint in real spacecomplex numbers inside the support area are replaced by their modulus to enforce positivity) Prsquo denotes the phase constraint on ordf in realspace (uses direg racted intensity and support constraint in real space the phase is extracted and retained from complex numbers inside thesupport area modulus is extracted and set to 1 inside the support area))

general pure strong pure weakrsquo real weak-phasecomplex object phase object phase object (amplitude) object withordf = A exp(iiquest ) ordf = exp(i iquest (r)) iquest lt ordm =2 object absorption

known two-holephysical support

CA (laser experimentpin cheek cells)

CA P (laserexperiment mica)

RIP CA M RIP CA RIP CA

known one-holephysical support

CA only for certainsupport shapes(eg triangle withsharp edges)

CA slowconvergence Pworks better

RIP CA M RIP CA (laserexperimentnumbers)

RIP CA

unknown two-or-more-hole physicalsupport loose(eg triangle compu-tational support)

| | RIP M ( TEM experimentholes) (laserexperiment numbers)(planned X-ray micapinholes)

RIP (Miao X-rayexperimenttransparent support)

unknown one-holephysical support(eg triangle compu-tational support)

| | RIP only for holewith low symmetryand sharp edgesslow convergence

M RIP only for holewith low symmetryand sharp edgesslow convergence

RIP only for holewith low symmetryand sharp edgesslow convergence

Phil

Tra

ns

R

Soc

L

on

d

A(2

002)

892 J C H Spence U Weierstall and M Howells

ulations and experiments The enotectiveness of dinoterent constraints in real space fordinoterent physical and computational supports is compared The physical supportis the support which exists in the experiment and which may or may not be accu-rately known The computational support is the support used in the HIO iterationsThe table shows experiments that have been performed in italics The shy rst columnshows dinoterent possibilities for the physical support the most favourable at the topthe worst at the bottom The computational support and known physical supportare identical The computational support used with the unknown physical supportis larger than the physical support Reconstructions with one-hole physical supportand loose computational support are only possible with weak or real objects andsupport holes with sharp edges and low symmetry and convergence are very slow insimulations A known two-or-more-hole support proved to be most favourable in allcases

7 Discussion and conclusions direg ractive imaging with other particles

In this paper we have reviewed the implications of Thomas Youngrsquos experiment formatter waves and in particular for electron interferometry By taking advantage ofthe very high quality of the coherent electron beam available in the modern shy eld-emission electron microscope a number of important results (such as the AharonovBohm enotect and the electron Sagnac enotect) have been established and the shy eldof atomic-resolution transmission electron microscopy phase-contrast imaging andelectron holography opened up All these methods continue to produce a large sci-entishy c pay-onot in microstructural materials characterization and discovery and inbiology A new lensless imaging method is then reviewed which may be understoodin part by analogy with electron holography This HIO algorithm uses an iterativeapproach to assign phases to dinotraction pattern intensities To evaluate its perfor-mance we give experimental examples of image reconstruction with both coherentelectrons and light together with simulations for the soft X-ray case Thus we haveshown experimentally that the HIO algorithm is capable of solving the phase problemfor scattering from non-periodic objects and of making quantitative measurements ofphase (modulo 2ordm ) in good agreement with independent measurements The beam-splitting arrangement of an interferometer is not required and dinotraction-limitedlensless imaging seems possible as a result This may open the way for imaging withthe many particles for which no lenses exist such as neutrons ballistic electrons insolids at low temperature or neutral atoms and ions either as free coherent beamsor condensates (Bright coherent sources have recently appeared for many of these(Pauly 2000)) The dinotering interactions of these particles may provide useful newprobes for chemistry materials science and biology For example the reduced radia-tion damage enotects of imaging organic material with low-energy electrons neutronsand helium atoms have recently been compared (Spence et al 1999) and HIO sim-ulations have been completed for neutron imaging of a diatom (Spence et al 2001)The very small refractive index of organic matter for neutrons limits resolution andcontrast for thin samples For low-energy ions or metastable helium strong samplebeam interactions must be considered Damage from neutral helium-atom imagingof molecules should be negligible however penetration is also negligible (producinga proshy le image) Detectors for coherent atom beams still present serious problemssince area detectors are inemacr cient For biology the weak atom interaction is desirable

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 893

to avoid damage however a strong interaction at the detector is also needed so thationization before detection must be arranged

For real objects we have seen that the resolution in images recovered by the HIOalgorithm is largely independent of the accuracy with which the support is knownThe addition of noise anotects the probability that the unique solution will be found (asmeasured by the error metric) rather than the resolution which is near dinotraction-limited For complex objects our experience indicates that a much more accurateknowledge of the physical support boundary is needed approaching the scale ofthe resolution sought The tight support and two-hole geometry (see Fienup (1987)for other geometries) needed for complex objects (complex refractive index) willbecome increasingly accessible with the development of new lithographic techniquesfor support construction This can be combined with independent characterization ofthe support shape and the scaling information provided by the known autocorrelationof the support Experimentally favourable cases for imaging with nanoscale resolutionat present require loose support which table 1 shows greatly favours real objectssuch as the charge density which dinotracts hard X-rays at energies well away fromabsorption edges or very thin samples for high-energy electrons Unfortunately beamcoherence is more readily obtained for soft X-rays

Using modern lithographic techniques it is becoming increasingly possible toestablish the experimental conditions of known support required by the HIO algo-rithm both for free particle beams and for electrons in solids at low temperaturesThus for example we have seen the algorithm used to determine the phase of wavefunctions for electrons in a one-dimensional YBCO Josephson junction using TEMimages of the structure to deshy ne the required support (boundary conditions) (Car-mody et al 2000) Given the fundamental role that phase has played historically inphysics and which Thomas Young was amongst the shy rst to appreciate any techniquecapable of exposing phase can be expected to yield many surprises in physics in thefuture

Supported by ARO award DAAD190010500 We thank Dr Q Chen for the use of macrgure 12 (fromWeierstall et al 2002)

References

Aharonov Y amp Bohm D 1959 Signimacrcance of the electromagnetic potentials in the quantumtheory Phys Rev 115 485

Allen L J Faulkner H M L Nugent K A Oxley M P amp Paganin D 2001 Phase retrievalfrom images in the presence of macrrst-order vortices Phys Rev E 63 037602

Bauschke H H et al 2002 (In the press)

Boersch H 1943 Fresnelsche Beugung im Elektronenmikroskop Phys Z 44 202211

Bruck Y amp Sodin L 1979 On the ambiguity of the image-reconstruction problem Opt Com-mun 30 304308

Carmody M Moeckly B H Merkle K amp Marks L 2000 Spatial variation of the current ingrain boundary Josephson junctions J Appl Phys 87 24542459

Coene W Janssen G Op de Beek M amp van Dyck D 1992 Phase retrieval through focusvariation for ultra-resolution in macreld emission transmission electron microscopy Phys RevLett 69 37433746

Crewe A V Eggenberger D N Wall J amp Welter L M 1968 Electron gun using a macreldemission source Rev Scient Instrum 39 576

Phil Trans R Soc Lond A (2002)

888 J C H Spence U Weierstall and M Howells

illuminationdirection

plane 1

1 mm

(a) (b)

(c) (d)

Figure 9 (a) Schematic view of a simulated object consisting of two planes separated by 1 mm inthe direction of light propagation Plane 1 has a hole containing a complex object (number 1) andplane 2 has two holes containing complex objects (numbers 2 and 3) (b) Modulus of complexexit wave on plane 2 (c) Iterative reconstruction with plane 2 support assumed known (thesupport shape used is shown in the top right-hand corner) Plane 1 is out of focus by 1 mm(d) Iterative reconstruction with plane 1 support assumed known (the support shape is shownin the top right-hand corner) Plane 2 is out of focus by iexcl1 mm

pin appears out of focus (Fresnel fringe barely visible) because the aperture shapewas imposed as support The pin can be brought into focus by using the appropriateFresnel propagator with centz = 12 mm and para = 6328 nm taking the apertures outof focus (see shy gure 11b) Thus as expected the reconstructed plane is always thatplane in which the support condition is imposed where the algorithm shy nds a uniquesolution subject to the given boundary condition Propagation to other planes maybe performed subsequently since the complex waveshy eld is obtained subject to thedepth-of-focus limitation The relative sideways shift of the pin inside the aperturebetween shy gure 10a and shy gure 11a is due to a slight unintentional tilt of the object(pin plus pinhole) between recording the image and the dinotraction pattern

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 889

(a) (b)

Figure 10 (a) Coherent laser optical image of pin in front of aperture (b) Laser optical direg ractionpattern obtained by merging three patterns with direg erent exposure times to avoid saturation ofthe central beam

(a) (b)

Figure 11 (a) Reconstructed modulus Only the hole with the out-of-focus pin is shown(b) Numerical back propagation of reconstructed complex wavemacreld by 12 mm to plane of pinPin appears in focus Support constraint only

6 Experiments with coherent electrons

Using a special support mask made by electron-beam lithography experiments havealso been undertaken using the HIO algorithm to retrieve the phases from experimen-tal transmission electron dinotraction patterns (Weierstall et al 2002) Images mightthen be reconstructed that are free of electron lens aberrations The support maskcontains hole pairs of various sizes with pairs separated by 1 m m The separationbetween holes of a pair is ca 50 nm less than the coherence width of the illuminationbeam The holes irregular in shape form the object we wish to reconstruct in futurework we plan to introduce nanoscale objects into the holes Figure 12a below showsthe experimental dinotraction pattern from two holes shown in shy gure 12c obtained

Phil Trans R Soc Lond A (2002)

890 J C H Spence U Weierstall and M Howells

20 nm

(a) (b)

(c)

Figure 12 (a) Experimental electron direg raction pattern from two nanoscale holes in an opaquemask (shown in (c)) recorded using a macreld-emission electron microscope (CM200 FEG at 40 kV1k pound1k CCD with 24 m m pixels camera length 611 m) Youngrsquo s fringes from the two holes arevisible these fringes encode phase information (b) Reconstruction of image of holes from thedireg raction pattern (a) after 100 iterations of the HIO algorithm Modulus constraint applied inimage space with large triangular computational support The hole shape is comparable with theTEM image in (c) (c) Conventional TEM image of holes that created the direg raction pattern (a)

with a Philips CM 200 shy eld-emission electron microscope at 40 kV The acceleratingvoltage was lowered to 40 kV to make the multi-layered support opaque ModulatedYoungrsquos fringes are clearly seen

The resulting image after 100 HIO iterations is shown in shy gure 12b The holesreconstructed from the dinotraction pattern modulus look similar to the TEM imageof the holes in shy gure 12c but the resolution of the reconstruction is lower thanthe normal TEM image resolution High spatial frequency information from thehole edge is missing in the experimental dinotraction pattern due perhaps to drift ofthe imaging lenses during the exposure time

Table 1 shows a summary chart of all the conditions so far examined with HIO sim-

Phil Trans R Soc Lond A (2002)

Phase

recovery

an

dlen

slessim

agin

g891

Table 1 Summary chart showing direg erent conditions for phase recovery with HIO algorithm

( CArsquo denotes the complex algorithm (uses direg racted intensity and support constraint in real space) RIPrsquo denotes the complex algorithm withpositivity constraint of Re ordf Im ordf (uses direg racted intensity and support constraint in real space negative Re ordf and Im ordf inside the supportarea are set positive) Mrsquo denotes the modulus constraint on ordf in real space (uses direg racted intensity and support constraint in real spacecomplex numbers inside the support area are replaced by their modulus to enforce positivity) Prsquo denotes the phase constraint on ordf in realspace (uses direg racted intensity and support constraint in real space the phase is extracted and retained from complex numbers inside thesupport area modulus is extracted and set to 1 inside the support area))

general pure strong pure weakrsquo real weak-phasecomplex object phase object phase object (amplitude) object withordf = A exp(iiquest ) ordf = exp(i iquest (r)) iquest lt ordm =2 object absorption

known two-holephysical support

CA (laser experimentpin cheek cells)

CA P (laserexperiment mica)

RIP CA M RIP CA RIP CA

known one-holephysical support

CA only for certainsupport shapes(eg triangle withsharp edges)

CA slowconvergence Pworks better

RIP CA M RIP CA (laserexperimentnumbers)

RIP CA

unknown two-or-more-hole physicalsupport loose(eg triangle compu-tational support)

| | RIP M ( TEM experimentholes) (laserexperiment numbers)(planned X-ray micapinholes)

RIP (Miao X-rayexperimenttransparent support)

unknown one-holephysical support(eg triangle compu-tational support)

| | RIP only for holewith low symmetryand sharp edgesslow convergence

M RIP only for holewith low symmetryand sharp edgesslow convergence

RIP only for holewith low symmetryand sharp edgesslow convergence

Phil

Tra

ns

R

Soc

L

on

d

A(2

002)

892 J C H Spence U Weierstall and M Howells

ulations and experiments The enotectiveness of dinoterent constraints in real space fordinoterent physical and computational supports is compared The physical supportis the support which exists in the experiment and which may or may not be accu-rately known The computational support is the support used in the HIO iterationsThe table shows experiments that have been performed in italics The shy rst columnshows dinoterent possibilities for the physical support the most favourable at the topthe worst at the bottom The computational support and known physical supportare identical The computational support used with the unknown physical supportis larger than the physical support Reconstructions with one-hole physical supportand loose computational support are only possible with weak or real objects andsupport holes with sharp edges and low symmetry and convergence are very slow insimulations A known two-or-more-hole support proved to be most favourable in allcases

7 Discussion and conclusions direg ractive imaging with other particles

In this paper we have reviewed the implications of Thomas Youngrsquos experiment formatter waves and in particular for electron interferometry By taking advantage ofthe very high quality of the coherent electron beam available in the modern shy eld-emission electron microscope a number of important results (such as the AharonovBohm enotect and the electron Sagnac enotect) have been established and the shy eldof atomic-resolution transmission electron microscopy phase-contrast imaging andelectron holography opened up All these methods continue to produce a large sci-entishy c pay-onot in microstructural materials characterization and discovery and inbiology A new lensless imaging method is then reviewed which may be understoodin part by analogy with electron holography This HIO algorithm uses an iterativeapproach to assign phases to dinotraction pattern intensities To evaluate its perfor-mance we give experimental examples of image reconstruction with both coherentelectrons and light together with simulations for the soft X-ray case Thus we haveshown experimentally that the HIO algorithm is capable of solving the phase problemfor scattering from non-periodic objects and of making quantitative measurements ofphase (modulo 2ordm ) in good agreement with independent measurements The beam-splitting arrangement of an interferometer is not required and dinotraction-limitedlensless imaging seems possible as a result This may open the way for imaging withthe many particles for which no lenses exist such as neutrons ballistic electrons insolids at low temperature or neutral atoms and ions either as free coherent beamsor condensates (Bright coherent sources have recently appeared for many of these(Pauly 2000)) The dinotering interactions of these particles may provide useful newprobes for chemistry materials science and biology For example the reduced radia-tion damage enotects of imaging organic material with low-energy electrons neutronsand helium atoms have recently been compared (Spence et al 1999) and HIO sim-ulations have been completed for neutron imaging of a diatom (Spence et al 2001)The very small refractive index of organic matter for neutrons limits resolution andcontrast for thin samples For low-energy ions or metastable helium strong samplebeam interactions must be considered Damage from neutral helium-atom imagingof molecules should be negligible however penetration is also negligible (producinga proshy le image) Detectors for coherent atom beams still present serious problemssince area detectors are inemacr cient For biology the weak atom interaction is desirable

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 893

to avoid damage however a strong interaction at the detector is also needed so thationization before detection must be arranged

For real objects we have seen that the resolution in images recovered by the HIOalgorithm is largely independent of the accuracy with which the support is knownThe addition of noise anotects the probability that the unique solution will be found (asmeasured by the error metric) rather than the resolution which is near dinotraction-limited For complex objects our experience indicates that a much more accurateknowledge of the physical support boundary is needed approaching the scale ofthe resolution sought The tight support and two-hole geometry (see Fienup (1987)for other geometries) needed for complex objects (complex refractive index) willbecome increasingly accessible with the development of new lithographic techniquesfor support construction This can be combined with independent characterization ofthe support shape and the scaling information provided by the known autocorrelationof the support Experimentally favourable cases for imaging with nanoscale resolutionat present require loose support which table 1 shows greatly favours real objectssuch as the charge density which dinotracts hard X-rays at energies well away fromabsorption edges or very thin samples for high-energy electrons Unfortunately beamcoherence is more readily obtained for soft X-rays

Using modern lithographic techniques it is becoming increasingly possible toestablish the experimental conditions of known support required by the HIO algo-rithm both for free particle beams and for electrons in solids at low temperaturesThus for example we have seen the algorithm used to determine the phase of wavefunctions for electrons in a one-dimensional YBCO Josephson junction using TEMimages of the structure to deshy ne the required support (boundary conditions) (Car-mody et al 2000) Given the fundamental role that phase has played historically inphysics and which Thomas Young was amongst the shy rst to appreciate any techniquecapable of exposing phase can be expected to yield many surprises in physics in thefuture

Supported by ARO award DAAD190010500 We thank Dr Q Chen for the use of macrgure 12 (fromWeierstall et al 2002)

References

Aharonov Y amp Bohm D 1959 Signimacrcance of the electromagnetic potentials in the quantumtheory Phys Rev 115 485

Allen L J Faulkner H M L Nugent K A Oxley M P amp Paganin D 2001 Phase retrievalfrom images in the presence of macrrst-order vortices Phys Rev E 63 037602

Bauschke H H et al 2002 (In the press)

Boersch H 1943 Fresnelsche Beugung im Elektronenmikroskop Phys Z 44 202211

Bruck Y amp Sodin L 1979 On the ambiguity of the image-reconstruction problem Opt Com-mun 30 304308

Carmody M Moeckly B H Merkle K amp Marks L 2000 Spatial variation of the current ingrain boundary Josephson junctions J Appl Phys 87 24542459

Coene W Janssen G Op de Beek M amp van Dyck D 1992 Phase retrieval through focusvariation for ultra-resolution in macreld emission transmission electron microscopy Phys RevLett 69 37433746

Crewe A V Eggenberger D N Wall J amp Welter L M 1968 Electron gun using a macreldemission source Rev Scient Instrum 39 576

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 889

(a) (b)

Figure 10 (a) Coherent laser optical image of pin in front of aperture (b) Laser optical direg ractionpattern obtained by merging three patterns with direg erent exposure times to avoid saturation ofthe central beam

(a) (b)

Figure 11 (a) Reconstructed modulus Only the hole with the out-of-focus pin is shown(b) Numerical back propagation of reconstructed complex wavemacreld by 12 mm to plane of pinPin appears in focus Support constraint only

6 Experiments with coherent electrons

Using a special support mask made by electron-beam lithography experiments havealso been undertaken using the HIO algorithm to retrieve the phases from experimen-tal transmission electron dinotraction patterns (Weierstall et al 2002) Images mightthen be reconstructed that are free of electron lens aberrations The support maskcontains hole pairs of various sizes with pairs separated by 1 m m The separationbetween holes of a pair is ca 50 nm less than the coherence width of the illuminationbeam The holes irregular in shape form the object we wish to reconstruct in futurework we plan to introduce nanoscale objects into the holes Figure 12a below showsthe experimental dinotraction pattern from two holes shown in shy gure 12c obtained

Phil Trans R Soc Lond A (2002)

890 J C H Spence U Weierstall and M Howells

20 nm

(a) (b)

(c)

Figure 12 (a) Experimental electron direg raction pattern from two nanoscale holes in an opaquemask (shown in (c)) recorded using a macreld-emission electron microscope (CM200 FEG at 40 kV1k pound1k CCD with 24 m m pixels camera length 611 m) Youngrsquo s fringes from the two holes arevisible these fringes encode phase information (b) Reconstruction of image of holes from thedireg raction pattern (a) after 100 iterations of the HIO algorithm Modulus constraint applied inimage space with large triangular computational support The hole shape is comparable with theTEM image in (c) (c) Conventional TEM image of holes that created the direg raction pattern (a)

with a Philips CM 200 shy eld-emission electron microscope at 40 kV The acceleratingvoltage was lowered to 40 kV to make the multi-layered support opaque ModulatedYoungrsquos fringes are clearly seen

The resulting image after 100 HIO iterations is shown in shy gure 12b The holesreconstructed from the dinotraction pattern modulus look similar to the TEM imageof the holes in shy gure 12c but the resolution of the reconstruction is lower thanthe normal TEM image resolution High spatial frequency information from thehole edge is missing in the experimental dinotraction pattern due perhaps to drift ofthe imaging lenses during the exposure time

Table 1 shows a summary chart of all the conditions so far examined with HIO sim-

Phil Trans R Soc Lond A (2002)

Phase

recovery

an

dlen

slessim

agin

g891

Table 1 Summary chart showing direg erent conditions for phase recovery with HIO algorithm

( CArsquo denotes the complex algorithm (uses direg racted intensity and support constraint in real space) RIPrsquo denotes the complex algorithm withpositivity constraint of Re ordf Im ordf (uses direg racted intensity and support constraint in real space negative Re ordf and Im ordf inside the supportarea are set positive) Mrsquo denotes the modulus constraint on ordf in real space (uses direg racted intensity and support constraint in real spacecomplex numbers inside the support area are replaced by their modulus to enforce positivity) Prsquo denotes the phase constraint on ordf in realspace (uses direg racted intensity and support constraint in real space the phase is extracted and retained from complex numbers inside thesupport area modulus is extracted and set to 1 inside the support area))

general pure strong pure weakrsquo real weak-phasecomplex object phase object phase object (amplitude) object withordf = A exp(iiquest ) ordf = exp(i iquest (r)) iquest lt ordm =2 object absorption

known two-holephysical support

CA (laser experimentpin cheek cells)

CA P (laserexperiment mica)

RIP CA M RIP CA RIP CA

known one-holephysical support

CA only for certainsupport shapes(eg triangle withsharp edges)

CA slowconvergence Pworks better

RIP CA M RIP CA (laserexperimentnumbers)

RIP CA

unknown two-or-more-hole physicalsupport loose(eg triangle compu-tational support)

| | RIP M ( TEM experimentholes) (laserexperiment numbers)(planned X-ray micapinholes)

RIP (Miao X-rayexperimenttransparent support)

unknown one-holephysical support(eg triangle compu-tational support)

| | RIP only for holewith low symmetryand sharp edgesslow convergence

M RIP only for holewith low symmetryand sharp edgesslow convergence

RIP only for holewith low symmetryand sharp edgesslow convergence

Phil

Tra

ns

R

Soc

L

on

d

A(2

002)

892 J C H Spence U Weierstall and M Howells

ulations and experiments The enotectiveness of dinoterent constraints in real space fordinoterent physical and computational supports is compared The physical supportis the support which exists in the experiment and which may or may not be accu-rately known The computational support is the support used in the HIO iterationsThe table shows experiments that have been performed in italics The shy rst columnshows dinoterent possibilities for the physical support the most favourable at the topthe worst at the bottom The computational support and known physical supportare identical The computational support used with the unknown physical supportis larger than the physical support Reconstructions with one-hole physical supportand loose computational support are only possible with weak or real objects andsupport holes with sharp edges and low symmetry and convergence are very slow insimulations A known two-or-more-hole support proved to be most favourable in allcases

7 Discussion and conclusions direg ractive imaging with other particles

In this paper we have reviewed the implications of Thomas Youngrsquos experiment formatter waves and in particular for electron interferometry By taking advantage ofthe very high quality of the coherent electron beam available in the modern shy eld-emission electron microscope a number of important results (such as the AharonovBohm enotect and the electron Sagnac enotect) have been established and the shy eldof atomic-resolution transmission electron microscopy phase-contrast imaging andelectron holography opened up All these methods continue to produce a large sci-entishy c pay-onot in microstructural materials characterization and discovery and inbiology A new lensless imaging method is then reviewed which may be understoodin part by analogy with electron holography This HIO algorithm uses an iterativeapproach to assign phases to dinotraction pattern intensities To evaluate its perfor-mance we give experimental examples of image reconstruction with both coherentelectrons and light together with simulations for the soft X-ray case Thus we haveshown experimentally that the HIO algorithm is capable of solving the phase problemfor scattering from non-periodic objects and of making quantitative measurements ofphase (modulo 2ordm ) in good agreement with independent measurements The beam-splitting arrangement of an interferometer is not required and dinotraction-limitedlensless imaging seems possible as a result This may open the way for imaging withthe many particles for which no lenses exist such as neutrons ballistic electrons insolids at low temperature or neutral atoms and ions either as free coherent beamsor condensates (Bright coherent sources have recently appeared for many of these(Pauly 2000)) The dinotering interactions of these particles may provide useful newprobes for chemistry materials science and biology For example the reduced radia-tion damage enotects of imaging organic material with low-energy electrons neutronsand helium atoms have recently been compared (Spence et al 1999) and HIO sim-ulations have been completed for neutron imaging of a diatom (Spence et al 2001)The very small refractive index of organic matter for neutrons limits resolution andcontrast for thin samples For low-energy ions or metastable helium strong samplebeam interactions must be considered Damage from neutral helium-atom imagingof molecules should be negligible however penetration is also negligible (producinga proshy le image) Detectors for coherent atom beams still present serious problemssince area detectors are inemacr cient For biology the weak atom interaction is desirable

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 893

to avoid damage however a strong interaction at the detector is also needed so thationization before detection must be arranged

For real objects we have seen that the resolution in images recovered by the HIOalgorithm is largely independent of the accuracy with which the support is knownThe addition of noise anotects the probability that the unique solution will be found (asmeasured by the error metric) rather than the resolution which is near dinotraction-limited For complex objects our experience indicates that a much more accurateknowledge of the physical support boundary is needed approaching the scale ofthe resolution sought The tight support and two-hole geometry (see Fienup (1987)for other geometries) needed for complex objects (complex refractive index) willbecome increasingly accessible with the development of new lithographic techniquesfor support construction This can be combined with independent characterization ofthe support shape and the scaling information provided by the known autocorrelationof the support Experimentally favourable cases for imaging with nanoscale resolutionat present require loose support which table 1 shows greatly favours real objectssuch as the charge density which dinotracts hard X-rays at energies well away fromabsorption edges or very thin samples for high-energy electrons Unfortunately beamcoherence is more readily obtained for soft X-rays

Using modern lithographic techniques it is becoming increasingly possible toestablish the experimental conditions of known support required by the HIO algo-rithm both for free particle beams and for electrons in solids at low temperaturesThus for example we have seen the algorithm used to determine the phase of wavefunctions for electrons in a one-dimensional YBCO Josephson junction using TEMimages of the structure to deshy ne the required support (boundary conditions) (Car-mody et al 2000) Given the fundamental role that phase has played historically inphysics and which Thomas Young was amongst the shy rst to appreciate any techniquecapable of exposing phase can be expected to yield many surprises in physics in thefuture

Supported by ARO award DAAD190010500 We thank Dr Q Chen for the use of macrgure 12 (fromWeierstall et al 2002)

References

Aharonov Y amp Bohm D 1959 Signimacrcance of the electromagnetic potentials in the quantumtheory Phys Rev 115 485

Allen L J Faulkner H M L Nugent K A Oxley M P amp Paganin D 2001 Phase retrievalfrom images in the presence of macrrst-order vortices Phys Rev E 63 037602

Bauschke H H et al 2002 (In the press)

Boersch H 1943 Fresnelsche Beugung im Elektronenmikroskop Phys Z 44 202211

Bruck Y amp Sodin L 1979 On the ambiguity of the image-reconstruction problem Opt Com-mun 30 304308

Carmody M Moeckly B H Merkle K amp Marks L 2000 Spatial variation of the current ingrain boundary Josephson junctions J Appl Phys 87 24542459

Coene W Janssen G Op de Beek M amp van Dyck D 1992 Phase retrieval through focusvariation for ultra-resolution in macreld emission transmission electron microscopy Phys RevLett 69 37433746

Crewe A V Eggenberger D N Wall J amp Welter L M 1968 Electron gun using a macreldemission source Rev Scient Instrum 39 576

Phil Trans R Soc Lond A (2002)

890 J C H Spence U Weierstall and M Howells

20 nm

(a) (b)

(c)

Figure 12 (a) Experimental electron direg raction pattern from two nanoscale holes in an opaquemask (shown in (c)) recorded using a macreld-emission electron microscope (CM200 FEG at 40 kV1k pound1k CCD with 24 m m pixels camera length 611 m) Youngrsquo s fringes from the two holes arevisible these fringes encode phase information (b) Reconstruction of image of holes from thedireg raction pattern (a) after 100 iterations of the HIO algorithm Modulus constraint applied inimage space with large triangular computational support The hole shape is comparable with theTEM image in (c) (c) Conventional TEM image of holes that created the direg raction pattern (a)

with a Philips CM 200 shy eld-emission electron microscope at 40 kV The acceleratingvoltage was lowered to 40 kV to make the multi-layered support opaque ModulatedYoungrsquos fringes are clearly seen

The resulting image after 100 HIO iterations is shown in shy gure 12b The holesreconstructed from the dinotraction pattern modulus look similar to the TEM imageof the holes in shy gure 12c but the resolution of the reconstruction is lower thanthe normal TEM image resolution High spatial frequency information from thehole edge is missing in the experimental dinotraction pattern due perhaps to drift ofthe imaging lenses during the exposure time

Table 1 shows a summary chart of all the conditions so far examined with HIO sim-

Phil Trans R Soc Lond A (2002)

Phase

recovery

an

dlen

slessim

agin

g891

Table 1 Summary chart showing direg erent conditions for phase recovery with HIO algorithm

( CArsquo denotes the complex algorithm (uses direg racted intensity and support constraint in real space) RIPrsquo denotes the complex algorithm withpositivity constraint of Re ordf Im ordf (uses direg racted intensity and support constraint in real space negative Re ordf and Im ordf inside the supportarea are set positive) Mrsquo denotes the modulus constraint on ordf in real space (uses direg racted intensity and support constraint in real spacecomplex numbers inside the support area are replaced by their modulus to enforce positivity) Prsquo denotes the phase constraint on ordf in realspace (uses direg racted intensity and support constraint in real space the phase is extracted and retained from complex numbers inside thesupport area modulus is extracted and set to 1 inside the support area))

general pure strong pure weakrsquo real weak-phasecomplex object phase object phase object (amplitude) object withordf = A exp(iiquest ) ordf = exp(i iquest (r)) iquest lt ordm =2 object absorption

known two-holephysical support

CA (laser experimentpin cheek cells)

CA P (laserexperiment mica)

RIP CA M RIP CA RIP CA

known one-holephysical support

CA only for certainsupport shapes(eg triangle withsharp edges)

CA slowconvergence Pworks better

RIP CA M RIP CA (laserexperimentnumbers)

RIP CA

unknown two-or-more-hole physicalsupport loose(eg triangle compu-tational support)

| | RIP M ( TEM experimentholes) (laserexperiment numbers)(planned X-ray micapinholes)

RIP (Miao X-rayexperimenttransparent support)

unknown one-holephysical support(eg triangle compu-tational support)

| | RIP only for holewith low symmetryand sharp edgesslow convergence

M RIP only for holewith low symmetryand sharp edgesslow convergence

RIP only for holewith low symmetryand sharp edgesslow convergence

Phil

Tra

ns

R

Soc

L

on

d

A(2

002)

892 J C H Spence U Weierstall and M Howells

ulations and experiments The enotectiveness of dinoterent constraints in real space fordinoterent physical and computational supports is compared The physical supportis the support which exists in the experiment and which may or may not be accu-rately known The computational support is the support used in the HIO iterationsThe table shows experiments that have been performed in italics The shy rst columnshows dinoterent possibilities for the physical support the most favourable at the topthe worst at the bottom The computational support and known physical supportare identical The computational support used with the unknown physical supportis larger than the physical support Reconstructions with one-hole physical supportand loose computational support are only possible with weak or real objects andsupport holes with sharp edges and low symmetry and convergence are very slow insimulations A known two-or-more-hole support proved to be most favourable in allcases

7 Discussion and conclusions direg ractive imaging with other particles

In this paper we have reviewed the implications of Thomas Youngrsquos experiment formatter waves and in particular for electron interferometry By taking advantage ofthe very high quality of the coherent electron beam available in the modern shy eld-emission electron microscope a number of important results (such as the AharonovBohm enotect and the electron Sagnac enotect) have been established and the shy eldof atomic-resolution transmission electron microscopy phase-contrast imaging andelectron holography opened up All these methods continue to produce a large sci-entishy c pay-onot in microstructural materials characterization and discovery and inbiology A new lensless imaging method is then reviewed which may be understoodin part by analogy with electron holography This HIO algorithm uses an iterativeapproach to assign phases to dinotraction pattern intensities To evaluate its perfor-mance we give experimental examples of image reconstruction with both coherentelectrons and light together with simulations for the soft X-ray case Thus we haveshown experimentally that the HIO algorithm is capable of solving the phase problemfor scattering from non-periodic objects and of making quantitative measurements ofphase (modulo 2ordm ) in good agreement with independent measurements The beam-splitting arrangement of an interferometer is not required and dinotraction-limitedlensless imaging seems possible as a result This may open the way for imaging withthe many particles for which no lenses exist such as neutrons ballistic electrons insolids at low temperature or neutral atoms and ions either as free coherent beamsor condensates (Bright coherent sources have recently appeared for many of these(Pauly 2000)) The dinotering interactions of these particles may provide useful newprobes for chemistry materials science and biology For example the reduced radia-tion damage enotects of imaging organic material with low-energy electrons neutronsand helium atoms have recently been compared (Spence et al 1999) and HIO sim-ulations have been completed for neutron imaging of a diatom (Spence et al 2001)The very small refractive index of organic matter for neutrons limits resolution andcontrast for thin samples For low-energy ions or metastable helium strong samplebeam interactions must be considered Damage from neutral helium-atom imagingof molecules should be negligible however penetration is also negligible (producinga proshy le image) Detectors for coherent atom beams still present serious problemssince area detectors are inemacr cient For biology the weak atom interaction is desirable

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 893

to avoid damage however a strong interaction at the detector is also needed so thationization before detection must be arranged

For real objects we have seen that the resolution in images recovered by the HIOalgorithm is largely independent of the accuracy with which the support is knownThe addition of noise anotects the probability that the unique solution will be found (asmeasured by the error metric) rather than the resolution which is near dinotraction-limited For complex objects our experience indicates that a much more accurateknowledge of the physical support boundary is needed approaching the scale ofthe resolution sought The tight support and two-hole geometry (see Fienup (1987)for other geometries) needed for complex objects (complex refractive index) willbecome increasingly accessible with the development of new lithographic techniquesfor support construction This can be combined with independent characterization ofthe support shape and the scaling information provided by the known autocorrelationof the support Experimentally favourable cases for imaging with nanoscale resolutionat present require loose support which table 1 shows greatly favours real objectssuch as the charge density which dinotracts hard X-rays at energies well away fromabsorption edges or very thin samples for high-energy electrons Unfortunately beamcoherence is more readily obtained for soft X-rays

Using modern lithographic techniques it is becoming increasingly possible toestablish the experimental conditions of known support required by the HIO algo-rithm both for free particle beams and for electrons in solids at low temperaturesThus for example we have seen the algorithm used to determine the phase of wavefunctions for electrons in a one-dimensional YBCO Josephson junction using TEMimages of the structure to deshy ne the required support (boundary conditions) (Car-mody et al 2000) Given the fundamental role that phase has played historically inphysics and which Thomas Young was amongst the shy rst to appreciate any techniquecapable of exposing phase can be expected to yield many surprises in physics in thefuture

Supported by ARO award DAAD190010500 We thank Dr Q Chen for the use of macrgure 12 (fromWeierstall et al 2002)

References

Aharonov Y amp Bohm D 1959 Signimacrcance of the electromagnetic potentials in the quantumtheory Phys Rev 115 485

Allen L J Faulkner H M L Nugent K A Oxley M P amp Paganin D 2001 Phase retrievalfrom images in the presence of macrrst-order vortices Phys Rev E 63 037602

Bauschke H H et al 2002 (In the press)

Boersch H 1943 Fresnelsche Beugung im Elektronenmikroskop Phys Z 44 202211

Bruck Y amp Sodin L 1979 On the ambiguity of the image-reconstruction problem Opt Com-mun 30 304308

Carmody M Moeckly B H Merkle K amp Marks L 2000 Spatial variation of the current ingrain boundary Josephson junctions J Appl Phys 87 24542459

Coene W Janssen G Op de Beek M amp van Dyck D 1992 Phase retrieval through focusvariation for ultra-resolution in macreld emission transmission electron microscopy Phys RevLett 69 37433746

Crewe A V Eggenberger D N Wall J amp Welter L M 1968 Electron gun using a macreldemission source Rev Scient Instrum 39 576

Phil Trans R Soc Lond A (2002)

Phase

recovery

an

dlen

slessim

agin

g891

Table 1 Summary chart showing direg erent conditions for phase recovery with HIO algorithm

( CArsquo denotes the complex algorithm (uses direg racted intensity and support constraint in real space) RIPrsquo denotes the complex algorithm withpositivity constraint of Re ordf Im ordf (uses direg racted intensity and support constraint in real space negative Re ordf and Im ordf inside the supportarea are set positive) Mrsquo denotes the modulus constraint on ordf in real space (uses direg racted intensity and support constraint in real spacecomplex numbers inside the support area are replaced by their modulus to enforce positivity) Prsquo denotes the phase constraint on ordf in realspace (uses direg racted intensity and support constraint in real space the phase is extracted and retained from complex numbers inside thesupport area modulus is extracted and set to 1 inside the support area))

general pure strong pure weakrsquo real weak-phasecomplex object phase object phase object (amplitude) object withordf = A exp(iiquest ) ordf = exp(i iquest (r)) iquest lt ordm =2 object absorption

known two-holephysical support

CA (laser experimentpin cheek cells)

CA P (laserexperiment mica)

RIP CA M RIP CA RIP CA

known one-holephysical support

CA only for certainsupport shapes(eg triangle withsharp edges)

CA slowconvergence Pworks better

RIP CA M RIP CA (laserexperimentnumbers)

RIP CA

unknown two-or-more-hole physicalsupport loose(eg triangle compu-tational support)

| | RIP M ( TEM experimentholes) (laserexperiment numbers)(planned X-ray micapinholes)

RIP (Miao X-rayexperimenttransparent support)

unknown one-holephysical support(eg triangle compu-tational support)

| | RIP only for holewith low symmetryand sharp edgesslow convergence

M RIP only for holewith low symmetryand sharp edgesslow convergence

RIP only for holewith low symmetryand sharp edgesslow convergence

Phil

Tra

ns

R

Soc

L

on

d

A(2

002)

892 J C H Spence U Weierstall and M Howells

ulations and experiments The enotectiveness of dinoterent constraints in real space fordinoterent physical and computational supports is compared The physical supportis the support which exists in the experiment and which may or may not be accu-rately known The computational support is the support used in the HIO iterationsThe table shows experiments that have been performed in italics The shy rst columnshows dinoterent possibilities for the physical support the most favourable at the topthe worst at the bottom The computational support and known physical supportare identical The computational support used with the unknown physical supportis larger than the physical support Reconstructions with one-hole physical supportand loose computational support are only possible with weak or real objects andsupport holes with sharp edges and low symmetry and convergence are very slow insimulations A known two-or-more-hole support proved to be most favourable in allcases

7 Discussion and conclusions direg ractive imaging with other particles

In this paper we have reviewed the implications of Thomas Youngrsquos experiment formatter waves and in particular for electron interferometry By taking advantage ofthe very high quality of the coherent electron beam available in the modern shy eld-emission electron microscope a number of important results (such as the AharonovBohm enotect and the electron Sagnac enotect) have been established and the shy eldof atomic-resolution transmission electron microscopy phase-contrast imaging andelectron holography opened up All these methods continue to produce a large sci-entishy c pay-onot in microstructural materials characterization and discovery and inbiology A new lensless imaging method is then reviewed which may be understoodin part by analogy with electron holography This HIO algorithm uses an iterativeapproach to assign phases to dinotraction pattern intensities To evaluate its perfor-mance we give experimental examples of image reconstruction with both coherentelectrons and light together with simulations for the soft X-ray case Thus we haveshown experimentally that the HIO algorithm is capable of solving the phase problemfor scattering from non-periodic objects and of making quantitative measurements ofphase (modulo 2ordm ) in good agreement with independent measurements The beam-splitting arrangement of an interferometer is not required and dinotraction-limitedlensless imaging seems possible as a result This may open the way for imaging withthe many particles for which no lenses exist such as neutrons ballistic electrons insolids at low temperature or neutral atoms and ions either as free coherent beamsor condensates (Bright coherent sources have recently appeared for many of these(Pauly 2000)) The dinotering interactions of these particles may provide useful newprobes for chemistry materials science and biology For example the reduced radia-tion damage enotects of imaging organic material with low-energy electrons neutronsand helium atoms have recently been compared (Spence et al 1999) and HIO sim-ulations have been completed for neutron imaging of a diatom (Spence et al 2001)The very small refractive index of organic matter for neutrons limits resolution andcontrast for thin samples For low-energy ions or metastable helium strong samplebeam interactions must be considered Damage from neutral helium-atom imagingof molecules should be negligible however penetration is also negligible (producinga proshy le image) Detectors for coherent atom beams still present serious problemssince area detectors are inemacr cient For biology the weak atom interaction is desirable

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 893

to avoid damage however a strong interaction at the detector is also needed so thationization before detection must be arranged

For real objects we have seen that the resolution in images recovered by the HIOalgorithm is largely independent of the accuracy with which the support is knownThe addition of noise anotects the probability that the unique solution will be found (asmeasured by the error metric) rather than the resolution which is near dinotraction-limited For complex objects our experience indicates that a much more accurateknowledge of the physical support boundary is needed approaching the scale ofthe resolution sought The tight support and two-hole geometry (see Fienup (1987)for other geometries) needed for complex objects (complex refractive index) willbecome increasingly accessible with the development of new lithographic techniquesfor support construction This can be combined with independent characterization ofthe support shape and the scaling information provided by the known autocorrelationof the support Experimentally favourable cases for imaging with nanoscale resolutionat present require loose support which table 1 shows greatly favours real objectssuch as the charge density which dinotracts hard X-rays at energies well away fromabsorption edges or very thin samples for high-energy electrons Unfortunately beamcoherence is more readily obtained for soft X-rays

Using modern lithographic techniques it is becoming increasingly possible toestablish the experimental conditions of known support required by the HIO algo-rithm both for free particle beams and for electrons in solids at low temperaturesThus for example we have seen the algorithm used to determine the phase of wavefunctions for electrons in a one-dimensional YBCO Josephson junction using TEMimages of the structure to deshy ne the required support (boundary conditions) (Car-mody et al 2000) Given the fundamental role that phase has played historically inphysics and which Thomas Young was amongst the shy rst to appreciate any techniquecapable of exposing phase can be expected to yield many surprises in physics in thefuture

Supported by ARO award DAAD190010500 We thank Dr Q Chen for the use of macrgure 12 (fromWeierstall et al 2002)

References

Aharonov Y amp Bohm D 1959 Signimacrcance of the electromagnetic potentials in the quantumtheory Phys Rev 115 485

Allen L J Faulkner H M L Nugent K A Oxley M P amp Paganin D 2001 Phase retrievalfrom images in the presence of macrrst-order vortices Phys Rev E 63 037602

Bauschke H H et al 2002 (In the press)

Boersch H 1943 Fresnelsche Beugung im Elektronenmikroskop Phys Z 44 202211

Bruck Y amp Sodin L 1979 On the ambiguity of the image-reconstruction problem Opt Com-mun 30 304308

Carmody M Moeckly B H Merkle K amp Marks L 2000 Spatial variation of the current ingrain boundary Josephson junctions J Appl Phys 87 24542459

Coene W Janssen G Op de Beek M amp van Dyck D 1992 Phase retrieval through focusvariation for ultra-resolution in macreld emission transmission electron microscopy Phys RevLett 69 37433746

Crewe A V Eggenberger D N Wall J amp Welter L M 1968 Electron gun using a macreldemission source Rev Scient Instrum 39 576

Phil Trans R Soc Lond A (2002)

892 J C H Spence U Weierstall and M Howells

ulations and experiments The enotectiveness of dinoterent constraints in real space fordinoterent physical and computational supports is compared The physical supportis the support which exists in the experiment and which may or may not be accu-rately known The computational support is the support used in the HIO iterationsThe table shows experiments that have been performed in italics The shy rst columnshows dinoterent possibilities for the physical support the most favourable at the topthe worst at the bottom The computational support and known physical supportare identical The computational support used with the unknown physical supportis larger than the physical support Reconstructions with one-hole physical supportand loose computational support are only possible with weak or real objects andsupport holes with sharp edges and low symmetry and convergence are very slow insimulations A known two-or-more-hole support proved to be most favourable in allcases

7 Discussion and conclusions direg ractive imaging with other particles

In this paper we have reviewed the implications of Thomas Youngrsquos experiment formatter waves and in particular for electron interferometry By taking advantage ofthe very high quality of the coherent electron beam available in the modern shy eld-emission electron microscope a number of important results (such as the AharonovBohm enotect and the electron Sagnac enotect) have been established and the shy eldof atomic-resolution transmission electron microscopy phase-contrast imaging andelectron holography opened up All these methods continue to produce a large sci-entishy c pay-onot in microstructural materials characterization and discovery and inbiology A new lensless imaging method is then reviewed which may be understoodin part by analogy with electron holography This HIO algorithm uses an iterativeapproach to assign phases to dinotraction pattern intensities To evaluate its perfor-mance we give experimental examples of image reconstruction with both coherentelectrons and light together with simulations for the soft X-ray case Thus we haveshown experimentally that the HIO algorithm is capable of solving the phase problemfor scattering from non-periodic objects and of making quantitative measurements ofphase (modulo 2ordm ) in good agreement with independent measurements The beam-splitting arrangement of an interferometer is not required and dinotraction-limitedlensless imaging seems possible as a result This may open the way for imaging withthe many particles for which no lenses exist such as neutrons ballistic electrons insolids at low temperature or neutral atoms and ions either as free coherent beamsor condensates (Bright coherent sources have recently appeared for many of these(Pauly 2000)) The dinotering interactions of these particles may provide useful newprobes for chemistry materials science and biology For example the reduced radia-tion damage enotects of imaging organic material with low-energy electrons neutronsand helium atoms have recently been compared (Spence et al 1999) and HIO sim-ulations have been completed for neutron imaging of a diatom (Spence et al 2001)The very small refractive index of organic matter for neutrons limits resolution andcontrast for thin samples For low-energy ions or metastable helium strong samplebeam interactions must be considered Damage from neutral helium-atom imagingof molecules should be negligible however penetration is also negligible (producinga proshy le image) Detectors for coherent atom beams still present serious problemssince area detectors are inemacr cient For biology the weak atom interaction is desirable

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 893

to avoid damage however a strong interaction at the detector is also needed so thationization before detection must be arranged

For real objects we have seen that the resolution in images recovered by the HIOalgorithm is largely independent of the accuracy with which the support is knownThe addition of noise anotects the probability that the unique solution will be found (asmeasured by the error metric) rather than the resolution which is near dinotraction-limited For complex objects our experience indicates that a much more accurateknowledge of the physical support boundary is needed approaching the scale ofthe resolution sought The tight support and two-hole geometry (see Fienup (1987)for other geometries) needed for complex objects (complex refractive index) willbecome increasingly accessible with the development of new lithographic techniquesfor support construction This can be combined with independent characterization ofthe support shape and the scaling information provided by the known autocorrelationof the support Experimentally favourable cases for imaging with nanoscale resolutionat present require loose support which table 1 shows greatly favours real objectssuch as the charge density which dinotracts hard X-rays at energies well away fromabsorption edges or very thin samples for high-energy electrons Unfortunately beamcoherence is more readily obtained for soft X-rays

Using modern lithographic techniques it is becoming increasingly possible toestablish the experimental conditions of known support required by the HIO algo-rithm both for free particle beams and for electrons in solids at low temperaturesThus for example we have seen the algorithm used to determine the phase of wavefunctions for electrons in a one-dimensional YBCO Josephson junction using TEMimages of the structure to deshy ne the required support (boundary conditions) (Car-mody et al 2000) Given the fundamental role that phase has played historically inphysics and which Thomas Young was amongst the shy rst to appreciate any techniquecapable of exposing phase can be expected to yield many surprises in physics in thefuture

Supported by ARO award DAAD190010500 We thank Dr Q Chen for the use of macrgure 12 (fromWeierstall et al 2002)

References

Aharonov Y amp Bohm D 1959 Signimacrcance of the electromagnetic potentials in the quantumtheory Phys Rev 115 485

Allen L J Faulkner H M L Nugent K A Oxley M P amp Paganin D 2001 Phase retrievalfrom images in the presence of macrrst-order vortices Phys Rev E 63 037602

Bauschke H H et al 2002 (In the press)

Boersch H 1943 Fresnelsche Beugung im Elektronenmikroskop Phys Z 44 202211

Bruck Y amp Sodin L 1979 On the ambiguity of the image-reconstruction problem Opt Com-mun 30 304308

Carmody M Moeckly B H Merkle K amp Marks L 2000 Spatial variation of the current ingrain boundary Josephson junctions J Appl Phys 87 24542459

Coene W Janssen G Op de Beek M amp van Dyck D 1992 Phase retrieval through focusvariation for ultra-resolution in macreld emission transmission electron microscopy Phys RevLett 69 37433746

Crewe A V Eggenberger D N Wall J amp Welter L M 1968 Electron gun using a macreldemission source Rev Scient Instrum 39 576

Phil Trans R Soc Lond A (2002)

Phase recovery and lensless imaging 893

to avoid damage however a strong interaction at the detector is also needed so thationization before detection must be arranged

For real objects we have seen that the resolution in images recovered by the HIOalgorithm is largely independent of the accuracy with which the support is knownThe addition of noise anotects the probability that the unique solution will be found (asmeasured by the error metric) rather than the resolution which is near dinotraction-limited For complex objects our experience indicates that a much more accurateknowledge of the physical support boundary is needed approaching the scale ofthe resolution sought The tight support and two-hole geometry (see Fienup (1987)for other geometries) needed for complex objects (complex refractive index) willbecome increasingly accessible with the development of new lithographic techniquesfor support construction This can be combined with independent characterization ofthe support shape and the scaling information provided by the known autocorrelationof the support Experimentally favourable cases for imaging with nanoscale resolutionat present require loose support which table 1 shows greatly favours real objectssuch as the charge density which dinotracts hard X-rays at energies well away fromabsorption edges or very thin samples for high-energy electrons Unfortunately beamcoherence is more readily obtained for soft X-rays

Using modern lithographic techniques it is becoming increasingly possible toestablish the experimental conditions of known support required by the HIO algo-rithm both for free particle beams and for electrons in solids at low temperaturesThus for example we have seen the algorithm used to determine the phase of wavefunctions for electrons in a one-dimensional YBCO Josephson junction using TEMimages of the structure to deshy ne the required support (boundary conditions) (Car-mody et al 2000) Given the fundamental role that phase has played historically inphysics and which Thomas Young was amongst the shy rst to appreciate any techniquecapable of exposing phase can be expected to yield many surprises in physics in thefuture

Supported by ARO award DAAD190010500 We thank Dr Q Chen for the use of macrgure 12 (fromWeierstall et al 2002)

References

Aharonov Y amp Bohm D 1959 Signimacrcance of the electromagnetic potentials in the quantumtheory Phys Rev 115 485

Allen L J Faulkner H M L Nugent K A Oxley M P amp Paganin D 2001 Phase retrievalfrom images in the presence of macrrst-order vortices Phys Rev E 63 037602

Bauschke H H et al 2002 (In the press)

Boersch H 1943 Fresnelsche Beugung im Elektronenmikroskop Phys Z 44 202211

Bruck Y amp Sodin L 1979 On the ambiguity of the image-reconstruction problem Opt Com-mun 30 304308

Carmody M Moeckly B H Merkle K amp Marks L 2000 Spatial variation of the current ingrain boundary Josephson junctions J Appl Phys 87 24542459

Coene W Janssen G Op de Beek M amp van Dyck D 1992 Phase retrieval through focusvariation for ultra-resolution in macreld emission transmission electron microscopy Phys RevLett 69 37433746

Crewe A V Eggenberger D N Wall J amp Welter L M 1968 Electron gun using a macreldemission source Rev Scient Instrum 39 576

Phil Trans R Soc Lond A (2002)

894 J C H Spence U Weierstall and M Howells

Davisson C J amp Germer L H 1927 Phys Rev 30 705

Dirac P 1931 Quantum singularities in the electromagnetic macreld Proc R Soc Lond A 13360

Ehrenberg W amp Siday R E 1948 The refractive index in electron optics and the principles ofdynamics Proc Phys Soc B 62 8

Esterman I amp Stern O 1930 Z Phys 61 95

Fienup J R 1982 Phase retrieval algorithms|a comparison Appl Opt 21 27582769

Fienup J R 1987 Reconstruction of a complex-valued object from the modulus of its Fourier-transform using a support constraint J Opt Soc Am A 4 118123

Frank J 1996 Three dimensional electron microscopy of macromolecular assemblies Academic

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Lichte H 2002 Electron interference mystery and reality Phil Trans R Soc Lond A 360897920

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Phil Trans R Soc Lond A (2002)