Magnified x-ray phase imaging using asymmetric Bragg reflection: Experiment and theory

Peter Modregger,1,* Daniel Lübbert,2,1 Peter Schäfer,1 and Rolf Köhler1

1Institut für Physik, Humboldt-Universität zu Berlin, Newtonstr. 15, D-12489 Berlin, Germany2Institut für Synchrotronstrahlung, Forschungszentrum Karlsruhe, D-76344 Eggenstein-Leopoldshafen, Germany

�Received 14 March 2006; revised manuscript received 16 June 2006; published 22 August 2006�

X-ray imaging using asymmetric Bragg reflection in the hard x-ray regime opens the way to improve thespatial resolution limit below 1 �m by magnifying the image before detection, simultaneously providing astrong phase contrast. A theoretical formalism of the imaging process is established. Based on this algorithm,numerical simulations are performed and demonstrate that both Fresnel propagation and Bragg diffractioncontribute to contrast formation. The achievable resolution of this technique is investigated theoretically; theresults obtained can be used to improve future experimental setups. Furthermore, the minimum detectablephase gradient is estimated, for comparison with other phase sensitive imaging techniques. Results frombiological objects demonstrate that the technique is viable for imaging both in two and three dimensions.Refraction contrast images are extracted from experimental projection images by an algorithm similar todiffraction-enhanced imaging �DEI�, and used to achieve three-dimensional tomographic reconstruction.

DOI: 10.1103/PhysRevB.74.054107 PACS number�s�: 87.59.�e, 87.57.Ce, 61.10.�i

I. INTRODUCTION

Contrast enhancement in x-ray imaging can be achievedby exploiting phase contrast in addition to absorption con-trast. This is particularly advantageous in the case of lightorganic, weakly absorbing objects. One of the most oftenused techniques for experimentally obtaining phase contrastis Fresnel propagation or in-line holography.1–4 A principalalternative is analyzer-based imaging, also known asdiffraction-enhanced imaging �DEI�.5

DEI is based on the combination of transmission of anx-ray wave field through the object under study �radiogra-phy� with subsequent Bragg diffraction at an analyzer crys-tal. Owing to the narrow angular range of Bragg diffraction,the analyzer efficiently selects rays locally deviated at theobject by specific angular amounts due to phase gradientswithin the sample, thus enhancing image contrast.

This technique has proven useful for the investigation oflaser-fusion targets,6 for x-ray diffraction topography,7 andbiological and medical imaging.8 Advanced analysis tech-niques have been developed5,8–10 which combine several im-ages taken at different angular positions of the analyzer crys-tal. In this way, the effects of x-ray absorption, refraction,and small-angle scattering can be experimentally separatedand a set of complementary images with specific contrastfeatures is obtained.

While the standard DEI setup is based on symmetric re-flection from the analyzer surface, asymmetric reflectionsmake it possible to simultaneously realize imagemagnification.6 By using two consecutive reflections from apair of analyzers, magnification can be achieved in both im-age dimensions.7,9,11–13 For a given analyzer surface orienta-tion, the magnification factor uniquely depends on the x-rayphoton energy. Synchrotron radiation is advantageous, sinceits energy tunability gives one the flexibility to widely varythe magnification by small changes in energy.

Asymmetric-reflection DEI thus opens the way to phase-contrast imaging with sub-micrometer spatial resolution. Itrepresents a path out of a conflict encountered in direct x-ray

imaging: Particularly when using state-of-the-art CCD cam-eras, spatial resolution improvements usually require reduc-ing the converter screen thickness, at the expense of detec-tion efficiency. By exploiting post-transmission imagemagnification, commercial CCD cameras with moderatepixel sizes but high sensitivity remain usable even for appli-cations requiring highest spatial resolution.

In this way, magnified x-ray imaging allows one to simul-taneously realize submicrometer resolution and phase con-trast. While the magnification factor is seemingly unlimited,the actual resolution is determined by a complex interplay ofseveral concurring factors. For a precise determination ofresolution limits, a complete theory of the imaging process istherefore mandatory; resolution cannot simply be equatedwith geometrical quantities such as the projected x-ray pen-etration depth in the analyzer.

In this article, we present an instrumental realization ofthe principle of magnified imaging—the “BraggMagnifier”—optimized in view of high spatial resolution. Inthe theoretical part, a comprehensive description of the im-age formation process is developed, including wave propa-gation behind the object �Fresnel diffraction� and reflectionat the analyzer crystals �Bragg diffraction�. Based on thistheory, the achievable spatial resolution is estimated numeri-cally for several different scenarios and model objects. Theresults reveal significant differences between the limitingcases of absorption objects and phase objects. Moreover, im-plications for further instrumental optimization are deduced.The phase sensitivity of the technique is quantified. An ex-ample of an experimental measurement performed on bio-logical objects will be shown, including results of three-dimensional imaging after tomographic reconstruction.

II. THEORY OF IMAGE FORMATION

In this section we will discuss the image formation of theBragg magnifier sketched in Fig. 1: A monochromatic waveis transmitted through the sample and diffracted twice by two

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asymmetrically cut analyzer crystals, thus magnifying theimage in both dimensions.

Several attempts have been reported in the literature toestimate the spatial resolution limit in analyzer-based imag-ing, but due to the intricate influences of free-space propaga-tion and Bragg reflection the phase was not adequately dealtwith. Therefore, we first derive the diffraction integral for theBragg magnifier which takes the mentioned effects fully intoaccount and then analyze it paying special attention to themaximum achievable spatial resolution and the minimum de-tectable phase variation present in the beam after transmis-sion. In this first step we will assume perfect monochromac-ity and collimation of the incident beam. Effects of disperionand divergence will be briefly discussed in Secs. VII andVIII. Since only relative intensities are of interest, globalphase terms and constant coefficients will be omitted.

A. Contrast formation

In the following we will describe, the wave fields D0 �be-tween sample and first analyzer� D1 �between first and sec-ond analyzer� and D2 �after second analyzer� �see Fig. 1�. D0is given by the transmission of the monochromatic incidentplane wave Din, propagating in the z direction through thesample described by its complex refractive index n�x ,y ,z�:

D0�x,y� = Din exp�ik� dzn�x,y,z�� , �1�

where k is the modulus of the wave vector. Obviously, mul-tiple scattering and effects of propagation within the sample

are neglected here, as commonly done in the literature.

B. Fresnel approximation

Applying three-dimensional Fourier transform to the three

wave fields Di�ri� leads to their Fourier presentations D̃i�ki�:

Di�r� =� d3kiD̃i�ki�eiki·ri. �2�

We will now demonstrate that in case of monochromaticwaves Fresnel propagation �i.e., free-space propagation ofthe x-ray beam after transmission through the sample� is al-ready described by Eq. �2�. A similar discussion in Fourierspace leading to the same result can be found in Ref. 14

The monochromaticity restricts the possible wave vectorski to those fulfilling �ki�=

�c with � the angular frequency and

c the vacuum speed of light, and therefore the correspondingthree-dimensional wave field is only nonzero on the surfaceof a sphere with radius �k�. Expressing the wave vector ki bydeviations qi from a reference vector k0i the latter pointing inmain beam direction, written in components

ki = k0i + qxiexi + qyieyi + qziezi �3�

the small angle approximation qzi� �ki�—this is Fresnel ap-proximation in Fourier space—can be performed by conclud-ing

ki2 = k0i

2 + qxi2 + qyi

2 + qzi2 + 2k0iqzi Þ qzi � −

1

2ki�qxi

2 + qyi2 � .

�4�

Due to the monochromaticity, the argument of the integral�2� can be multiplied by a Dirac delta function �D��k�− �

c�

and the corresponding sphere can be replaced by a rotationparabola at k0i, yielding

Di�ri� =� dqxidqyiD̂i�qxi,qyi�ei�qxixi+qyiyi−�zi/2ki��qxi2 +qyi

2 �,

�5�

where D̂i�qxi ,qyi� is the two-dimensional Fourier representa-tion of the amplitude in the xy plane at zi=0. Equation �5� isexactly the Fresnel integral in Fourier space.

C. Diffraction integral of the Bragg magnifier

The dynamical theory of diffraction15 connects a singlewave before the reflection to a single wave after the reflec-tion, where the wave vectors are related by

ki = ki−1 + gi + �i�ki−1�n̂i �i = 1,2� . �6�

gi is the reciprocal lattice vector of the reflection i, �i�ki−1� isthe deviation parameter and n̂i are the normals of the ana-lyzer surfaces. The wave amplitudes themselves relate in thefollowing way:

D̃i�ki� = D̃i−1�ki−1�R̂i�ki−1�e−i�i�ki−1�n̂irio �i = 1,2� , �7�

where R̂i is the reflectivity of the ith reflection, which can becalculated from dynamical theory of x-ray diffraction.15

FIG. 1. Scheme of the used coordinate systems �i. i=0,1 ,2refers to the coordinate system before reflection, after the first re-flection, and after both reflections, respectively. The unit vectors ofeach coordinate system are denoted exi, eyi, and ezi the latter beingparallel to the corresponding main beam direction k0i. Position vec-tors in each coordinate system are noted as ri. The distance betweenthe origins of �0 and �1 is d1 �sample to analyzer distance�, the onebetween �1 and �2 is d2. The lines at the sides on the lateralsurfaces of the crystals indicate the orientation of the lattice planes.

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Standard dynamical diffraction theory is valid for a coordi-nate system with its origin on the surface of the crystal andtherefore the additional phase factor e−i�i�ki−1�n̂irio in Eq. �7�takes an arbitrary origin ri0 into account. If d1 is the vectorlinking the sample origin and the first analyzer surface andd2 is the vector linking the first analyzer surface with thesecond, then r10=d1 and r20=d1+d2 are valid. Using theabove equations and with r2=r−d1−d2 it is easy to showthat

D2�r2� =� d3k2D̃0�k0�R̂1�k0�R̂2�k1�ei�k2·r2+k1·d2+k0·d1�

�8�

is valid.Using substitution we will rewrite Eq. �8� to components

of the original coordinate system �qx0, qy0, and qz0�, i.e., thecoordinate system for the wave field before any reflection.For this purpose we will first connect the wave vectors be-fore and after the first reflection. The reference wave vectorsk00 and k10 defined in the same way as the reference vectorin Eq. �3� are connected by k10=k00+g1+�01n̂1 with �01=�1�k00�. Using the components of k1 and k0 similar to Eq.�3� and Eq. �6� leads to

qx1ex1 + qy1ey1 + qz1ez1 = qx0ex0 + qy0ey0 + qz0ez0

+ ��1 − �01�n̂1. �9�

By multiplying with the vector n̂1, which is perpendicular ton̂1 and lies in the xz plane we get

�10�

The use of the Fresnel approximation, implying qxi�qzi, jus-tifies the first order approximation qx1�− 1

m1qx0, where m1 is

the magnification factor of the first reflection. Consequently,qz1=− 1

2k � 1m1

2 qx02 +qy0

2 � holds �Eq. �4�. By inserting this intoEq. �10� the second order approximation

qx1 � −1

m1qx0 − m̃1qz0 + m̂1

1

2k� 1

m12qx0

2 + qy02 � �11�

is obtained. With similar reasoning there is further the trivialresult

qy1 = qy0 �12�

which was already used in Eq. �11�. Presuming that the twomagnification directions are perpendicular to each other �i.e.,avoiding shear effects� an analogous discussion for the sec-ond reflection leads to

qy2 � −1

m2qy0 − m̃2qz1 + m̂2

1

2k� 1

m12qx0

2 +1

m22qy0

2 ��13�

and

qx2 = qx1. �14�

Inserting Eqs. �4� and �11�–�14� in Eq. �8�, taking into ac-count that ez�i−1� di for i=1,2 holds, and performing theFresnel approximation as discussed above the diffraction in-tegral of the Bragg magnifier is obtained:

D2�r2� =� dqx0dqy0D̂0�qx0,qy0�R̂1�qx0

k�R̂2�qy0

k�

exp�i�−1

m1qx0rx2 −

1

m2qy0ry2

+m̃1rx2 − d1

2k�qx0

2 + qy02 �

+m̂1rx2 + m̃2ry2 − d2

2k� 1

m12qx0

2 + qy02 �

+m̂2ry2 − rz2

2k� 1

m12qx0

2 +1

m22qy0

2 � � . �15�

As deviations from main beam directions are in the order ofqi�10−4 it is justified to use the standard coplanar approxi-

mation of diffraction theory: R̂i�ki−1�� R̂i� q�x,y�

k�.

Equation �15� describes the most general case of imagingwith the Bragg magnifier and can be simplified for realisticdistances di and magnification factors above m�10 to

D2�r2� =� dqx0dqy0D̂0�qx0,qy0�R̂1�qx0

k�R̂2�qy0

k�

exp�i�−1

m1qx0rx2 −

1

m2qy0ry2

+m̃1rx2 − d1

2k�qx0

2 + qy02 �

+m̂1rx2 + m̃2ry2 − d2

2kqy0

2 � . �16�

For simplicity we will restrict the discussion in the followingto one dimensional objects, thus neglecting the second re-flection and the free space propagation after the first reflec-tion. It is also convenient to rewrite the components of r2 tocomponents of r0 defined in the original coordinate systemrx2=−m1rx0=−m1x and rz0= m̃1rx2−d1=z. Now the input am-plitude can be written as D0�x ,y�=D0�x�, the indices x0 canbe omitted and Eq. �16� becomes a one-dimensional integralyielding

Dout�x� =� dqD̂0�q�R̂�q

k�ei�qx−z/2kq2� �17�

directly on the first analyzer surface �note the implicit depen-dence z=z�x�. The restriction to one dimensional objects isthe correct description of the imaging process for conven-tional DEI and a good approximation for vertical or horizon-tal lines of the image in the case of the Bragg Magnifier.Equation �17� was also motivated by Spal in Ref. 16.

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D. Numerical simulation

In order to understand the imaging process we will useforward simulation in the sense that the diffraction integral�17� will be calculated numerically for several types of testamplitudes D0 and the results will be discussed.

The numerical integration is done in the following way.For a given sample to analyzer distance d1, Eq. �17� ismerely the inverse Fourier transform of the function

D̂0�q�R̂�q /k�exp�−i z2kq2�. This can be conveniently computed

with fast Fourier transform for several distances d1. By dis-playing the resulting intensity distribution as a function ofthe back-projected position on sample x and of the distancez, an “intensity landscape” is obtained �see Fig. 8�. If just theintensity distribution along the analyzer surface is of interestlinear interpolation to the position of the analyzer is appliedto the intensity landscape �indicated by the black line in Fig.8�b�. We have used 214 up to 217 sample points in Fourierspace in order to avoid numerical artifacts.

Different types of test amplitudes corresponding to in-creasingly realistic model samples will be discussed in thesubsequent sections. We will start with a � function as testamplitude in order to estimate the spatial resolution limit �ascommonly done�, demonstrate the influence of Bragg reflec-tion on Fresnel diffraction at the example of a rectangularshaped amplitude and, finally, use a Gauss shaped modelsample to investigate the dependence of the observable con-trast on the sample to analyzer distance.

The reflection curve R̂�q /k� was calculated according todynamical theory with the same parameters for all subse-quent simulations: Si-224, polarization, and a magnifica-tion factor of 40 at a photon energy of 8.048 keV, which isthe first reflection of our experimental setup �see Fig. 9�.

III. RESPONSE FUNCTION

Generally, a diffraction integral describing an imagingprocess can be analyzed by means of linear system theory.17

In this section we will work out the similarities and the limi-tations of the applicability of linear system theory to ourcase. In order to do so, we will analyze Eq. �17� in two steps.First we set the sample to analyzer distance z to zero �i.e.,neglecting the effects of Fresnel propagation�. Using a�-shaped input amplitude, being the description of a single

object point, and its well known Fourier transform �D̂0�q�= 1

2� in Eq. �17� we obtain

R�x� =� dqR̂�q

k�eiqx �18�

which is exactly the inverse Fourier transform of the reflec-tion curve. The function R�x� is called influence function andits physical interpretation is as follows.18 A point source po-sitioned directly on the crystal surface would imply a com-plex amplitude distribution after reflection described by theinfluence function �see Fig. 2�.

Using again a �-shaped input amplitude, now takingFresnel propagation to z�0 into account, the response func-tion �RF� for the Bragg magnifier is obtained

RF�x� =� dqR̂�q

k�ei�qx−�z/2k�q2. �19�

The similarity to the amplitude spread function �ASF� inlinear system theory is striking but misleading. In this theorythe output amplitude Dout is equal to the convolution of theinput amplitude D0 and the ASF: Dout=D0 � ASF. Accordingto the convolution theorem, the Fourier transform of Dout isthe product of the individual Fourier transforms of D0 and

the ASF, respectively, yielding: D̂out= D̂0 ·ASF. But due tothe dependence of z�x� on x formula �17� cannot be writtenas a simple convolution in direct space.

Figure 2 shows the numerically calculated response func-tion for three different sample to analyzer distances andstrongly emphasizes the necessity to take Fresnel propaga-tion into account. Note that only the modulus of the RF isshown, not its phase distribution, so that the width of the firstfringe does not necessarily correspond to the spatial resolu-tion. This would only be true in the absence of any phasevariations.

IV. RESOLUTION OF AN ABSORPTION OBJECT

The Sparrow criterion19 states that two image points aredistinguishable if an intensity minimum exists between them.The criterion is therefore applicable for both coherent andincoherent illumination. The Sparrow criterion was appliedin the following way. The amplitude in direct space of twoimage points separated by the distance x0 is represented by

0 2 4 6 8 101e−5

1e−4

1e−3

1e−2

1e−1

1

position on sample [mm]

|RF

|2 [arb

. uni

ts]

z=0mm

z=5mm

z=20mm

FIG. 2. �Color online� Numerical calculation of complex re-sponse functions: semilogarithmic plot of the squared modulus ofRF�x�. The reflection curve was calculated according to dynamicaltheory for the Si-224 reflection � polarization� at an x-ray photonenergy of 8.048 keV and a magnification factor of 40. The curve forz=0 mm is identical with the squared modulus of the influencefunction R�x�. The x axis has been scaled to take backprojectiononto the sample into account. The oscillations �for z=0 mm� areso-called Kato fringes due to interference effects upon reflection.The relative intensity of the second maximum is about 0.3% of thefirst, so that the first Kato fringe may be visible in experimentalimages. With increasing distance the additional effects of Fresnelpropagation become more visible.

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D0�x�=��x�+��x−x0�. The resulting intensity distributionshows either one maximum �i.e., the Sparrow criterion is notfulfilled� or it shows two maxima corresponding to the twoobject points and a minimum between them. For the desiredparameters �i.e., chosen reflection and sample to analyzerdistance� the distance between the two object points x0 wasvaried until the minimum separating the two maxima hadcontrast of 5% compared to the lower one of the twomaxima.

Figure 3 shows the result of using the Sparrow criterion todetermine the spatial resolution of the Bragg magnifier. It isobvious from the figure that the resolution limit varies byhalf an order of magnitude in the given interval of sample toanalyzer distance �1 mm to 1 cm�. According to these re-sults, optimum resolution is predicted for the smallest dis-tances. It is therefore advantageous to construct instrumentalrealizations of the Bragg magnifier as compact as possible, tofully exploit these optimum conditions. The vertical axis ofFig. 3 shows that resolutions of few micrometers are easilyaccessible with the Bragg magnifier technique, and evenresolutions well in the submicrometer regime �0.2–0.5 �m�are within reach if some care is taken to optimize the instru-mental setup.

V. RESOLUTION OF A PHASE OBJECT

The spatial resolution limit in the case of phase objectscan be dealt with by introducing a phase difference �� be-tween the two object points �i.e., D0�x�=��x�+��x−x0�ei��.Let us first consider the case of a phase difference ��=�.Obviously, the corresponding response functions of the twoobjects also differ in phase by a factor of �. Then the coher-ent superposition of the two always has a minimum,19 re-gardless of the distance between the two object points. Con-sequently, the Sparrow criterion would state that the imagepoints are always separated, which would imply an infiniteresolution limit.

Naturally, an increasing phase difference between twoneighboring object points becomes increasingly unrealisticas the distance between them decreases. The phase relationof a real world sample will therefore be located somewherebetween the two extreme cases of pure absorption and purephase objects.

Figure 4 shows the dependence on the relative phase dif-ferences between two object points of the correspondingresolution for a fixed sample to analzyer distance. In verygood approximation the resolution improves linearly withphase difference. Consequently, for a given sample the reso-lution with phase contrast is superior to the case of pureabsorption contrast.

For future comparison with other techniques we now es-timate the minimum detectable phase difference. The angulardeviation � of the local propagation direction is related tothe derivative of the local phase ��x� of a wave front by20

� = −�

2�

d�

dx�x� . �20�

If an intensity contrast of 10% is considered to be well de-tectable in experiment, Eq. �20� and a linear approximationof the reflection curve �see Fig. 5� can be used to estimate theminimum detectable phase variation to be ��

�x �0.06 rad�m. For

in-line holography, the best value found in the literature sofar is ��

�x �1 rad 4�m .

VI. NUMERICAL EXAMPLES

A. Pure absorption objects

The effects of Fresnel diffraction on the propagation of awave are commonly demonstrated at the example of diffrac-tion at an edge. It is well known from light optics that oscil-lations, called Fresnel fringes, occur in the geometrically il-

1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

sample to analyzer distance [mm]

reso

lutio

n [m

m]

left flank (a)top position (b)right flank (c)

FIG. 3. �Color online� Maximum achievable spatial resolutionof the Bragg magnifier for pure absorption objects as a function ofthe sample to analyzer distance for Si-224 and three different work-ing points on the analyzer rocking curve �left, right flank, and topposition; see Fig. 5�.

0 20 40 60 80 100 120 140 160 1800

0.1

0.2

0.3

0.4

0.5

Df [degree]

reso

lutio

n [m

m]

z=1mm (top)z=5mm (top)z=5mm (right flank)

FIG. 4. �Color online� Spatial resolution of the Bragg magnifieras a function of the phase difference between two object points fordifferent cases. Obviously, the resolution improves almost linearlywith the phase difference.

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luminated area �see Fig. 6 dashed line�. It is furthermore wellknown from x-ray topography that Bragg reflection from theanalyzer crystals introduces an second fringe system, Katofringes, visible as oscillations in Fig. 2 �z=0 mm line�. Thesuperposition of both fringe types might be expected to resultin a very complex situation with multiple fringe systems af-

fecting image quality. However, this will turn out to bewrong.

A rectangular shaped, pure absorption input amplitude D0with an aperture of 20 �m was used to investigate the effectsof Bragg reflection on Fresnel diffraction. Figure 6 comparesthe observable intensity distributions with and without re-flection. The surprising result is that Fresnel fringes arestrongly attenuated by the reflection. For explanation, con-sider that fast oscillations in direct space �i.e., Fresnelfringes� correspond to high frequencies in Fourier space.High frequencies in Fourier space are suppressed by the re-flection curve and therefore cannot pass the analyzer crystal.Therefore, the observable intensity distribution is smoothed,thus eliminating parts of the fringe systems.

B. Phase objects

The results of the preceding sections will now be vali-dated for a realistic test sample. In order to facilitate easyinterpretation a Gauss shaped test sample was chosen. Theinput amplitude in direct space D0�x� was calculated accord-ing to formula �1� with the complex refractive index of amor-phous carbon �modeling biological samples�. The numeri-cally calculated Fourier transform of the input amplitude

D̂0�q� was used to compute the diffraction integral �17�.The Gauss function of the test sample �indicated by the

shaded area in Fig. 7� had a variance of 1 �m and a thick-ness of 5 �m, thus providing an �intensity� absorption con-trast of about 0.4%, a maximum phase variation of about� /2 and maximum phase variation of about 0.6 rad

�m �i.e., tentimes the minimum detectable phase variation estimatedabove�.

−10 −5 0 5 10 15 20 250.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Da [angsec]

refle

ctio

n cu

rve

|R(D

a)|

(a)

(b)

(c)

FIG. 5. �Color online� Reflection curve of the Si-224 reflection � polarization� at 8.048 keV, calculated according to dynamicaltheory. The three marked working points �a�,�b�,�c�, indicating theposition of the main beam direction on the reflection curve, corre-spond to the curves of Fig. 3. The minimum detectable phase varia-tion is estimated from the angular deviation corresponding with theminimal detectable intensity contrast.

−15 −10 −5 0 5 10 150

0.2

0.4

0.6

0.8

1

1.2

position on sample [mm]

inte

nsity

[arb

. uni

ts]

without reflection

with reflection

FIG. 6. �Color online� Attenuation of Fresnel fringes due toreflection. The figure shows the numerically calculated intensitydistribution of a rectangular shaped input amplitude �20 �m width�before �thin line� and after reflection �thick lines� at a sample toanalyzer distance of 10 mm. The symmetric fringes appearing in theintensitiy distribution before reflection are due to Fresnel propaga-tion. The asymmetric fringes appearing after reflection are stronglyinfluenced by Kato fringes. The attenuation of Fresnel fringes afterreflection is well visible.

−5 0 50

5

10

thic

knes

of t

he s

ampl

e [µ

m]

−5 0 50

0.5

1

1.5

2

position on sample [µm]

inte

nsity

[arb

. uni

ts]

int. without reflectionint. with reflection

FIG. 7. �Color online� Intensity distribution for Gauss shapedsample �amorphous carbon, width =1 �m, shown as the shadedarea� before and after the reflection, calculated for a distance of10 mm from the sample. Note that the intensity profile before re-flection represents a focused beam, due to similarity of the sampleprofile with usual refractive x-ray lenses. The profile after reflectioncorresponds to a cut from the intensity landscape of Fig. 8�b� alongthe line indicated.

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Figure 7 shows the calculated intensity distribution alongthe analyzer surface in the medium sample to analyzer dis-tance z=10 mm for the case of Fresnel propagation �in-lineholography; dashed line� and after the Si-224 reflection witha magnification factor of 40 �imaging with the Bragg mag-nifier; solid line�.

As expected the intensity distribution without reflection isproportional to the Laplacian of the phase21 whereas withreflection it is roughly proportional to the gradient of thephase8,22 resulting in the typical dark-bright contrast of ana-lyzer based imaging. For the given parameters the contrast ofimaging with the Bragg magnifier is about twice as strong asfor in-line holography. Furthermore, paying special attentionto the slopes of the intensity distributions, smaller phasevariations are detectable after reflection indicating thatanalyzer-based imaging is more sensitive to phase variationsthan in-line holography and thus more sensitive to densityvariations present in the sample.

We conclude this section with a comparison of the inten-sity landscapes calculated for free space propagation and af-ter reflection �Fig. 8�, using the same parameters as in Fig. 7.The focussing behavior visible in Fig. 8�a� is due to thespecial shape of the test sample which roughly correspondsto refractive x-ray lenses. The typical dark-bright contrast foranalyzer based imaging �Fig. 8�b� is preserved during thepropagation, underlining the small influence of Fresnel dif-fraction. Realizing that the strongest dark-bright contrast at adistance of 100 mm in Fig. 8�b� is the desired contrast, it caneven be stated that the attenuation of Fresnel fringes afterreflection holds true for the case of phase objects.

Comparing both imaging techniques it can be stated thatthe contrast of imaging with the Bragg magnifier is compa-rable with or even �at small distances� superior to the con-trast of in-line holography. It is also striking that the contrastas a function of distance is approximately constant, thus al-lowing one to reduce sample to analyzer distance in order toimprove the spatial resolution �as discussed above� withoutloosing contrast. Furthermore, by comparing Figs. 8�a� and8�b� it is striking that the divergence is reduced for the casewith reflection, thus making the resolution dependence ondistance for the Bragg magnifier superior to in-line hologra-phy.

VII. EXPERIMENTAL EXAMPLE

The experiment was carried out at the ID-02-01 beamline�“BAMline”� of the Bessy Synchrotron, Berlin. The x-raysource was a 7 T wavelength shifter. The imaging systemwas implemented in a very compact setup �the “Bragg mag-nifier box,” Fig. 9� and was tested in the laboratory beforebeing transferred to the beamline as a turnkey instrument. InFig. 9 the x-ray beam enters from the right through a systemof slits and then hits the sample, which is mounted on avertical rotation axis on top of a translation stage. Aftertransmission through the sample, the beam is diffracted intwo mutually perpendicular directions: vertically by the firstanalyzer crystal �Si-224 reflection� and horizontally by thesecond analyzer �Si-004 reflection�. At the x-ray energy used�Cu-K , �=1.54 Å, from a double-crystal monochromator�,

the scattering angle of the first reflection was close to 90°—ageometric precondition for realizing a short distance betweenthe two analyzer crystals. Both crystals were asymmetricallycut, with asymmetries chosen in such a way that the magni-fication factors in both image dimensions were 40. The tiltangles of both analyzers were adjusted to realize shear-freeimaging conditions, as verified with the help of a rectangulargrid test object.

After being reflected twice, the �laterally expanded� beamleft the instrument through the front-side circular opening,which was used to mount a CCD camera. The detector was acommercial Bruker-AXS Smart Apex 2 camera with40962 pixels and a nominal pixel size of 15 �m. It providedseveral binning modes, of which 11 binning �i.e., no bin-ning� was used for two-dimensional imaging �effective pixelsize 0.375 �m�, and 22 binning was used for the tomog-raphic scans �effective pixel size 0.75 �m� to speed up ac-quisition. The setup had a total field of view of 1.5 mm.Tomographic scans were taken by recording series of up to720 magnified projection images upon sample rotationaround the vertical axis.

FIG. 8. �Color online� Comparison of the intensity landscapes incase of free space propagation �a� and after reflection �b�. The testsample was Gauss shaped amorphous carbon, see Fig. 7. The ob-servable intensity is calculated in dependence on the backprojectedposition on sample �horizontal axis� and on the distance betweensample and analyzer �vertical axis�. The line in �b� indicates theposition of the analyzer surface of Fig. 7.

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Figure 10 shows a magnified two-dimensional projectionimage of a spider leg, used as a model object. The analyzerswere tuned to the flank of their rocking curve to yield maxi-mum phase contrast, as obvious from the characteristicdouble-contrast �dark-bright� visible at each of the lateralfeatures �“hair”� of the spider leg. While the use of phasecontrast enhances image contrast and improves spatial reso-lution �see Sec. V�, it can, in principle, pose a problem forthree-dimensional tomographic reconstruction. Conventionalalgorithms and available software for 3D reconstruction areusually based on pure absorption contrast. Application to se-ries of projection images showing strong phase contrast, asin Fig. 10, is therefore considered a purely tentative proce-dure. However, effects of dispersion and divergence reducethe visibility of Kato and Fresnel fringes in the experimentalimages, thus reducing the quantitative information contentsin the images but improving the qualitative resemblance be-tween sample and image. For the simple algorithm used herefor tentative 3D reconstruction this even has the beneficialeffect of reducing artefacts during reconstruction. The resultin Fig. 11 demonstrates that a very reasonable 3D represen-tation of the spider leg can be obtained, giving a clear imageof both the overall external shape and the inner structure ofthe object.

Further optimization of the data evaluation and image re-construction procedure will be performed by developing adedicated tomographic reconstruction algorithm for phasecontrast imaging. An approach based on interleaved samplerotation and analyzer rocking scans has already been taken.

Moreover, we are presently performing more detailedstudies on the quantitative influence of the effects of beam

FIG. 9. �Color online� Technical sketch of the Bragg magnifier.The path of x rays �incident from the right� is indicated by brightlines, the position of the sample by the darker line. The ring near theend of the x-ray path indicates the position of the CCD camera. Forreasons of clarity some components of the instrument have beenomitted from the picture.

100mm

FIG. 10. Experimental projection image of a spider leg acquiredat a working point on the left flank of the first analyzer reflectioncurve. Note the dark-bright double contrast at the vertical structures�“hair”�, which is a clear evidence of phase contrast. Also note thatthis image is less strongly affected by Fresnel and Kato fringes thanmight be expected from theory, due to both the complex inner struc-ture of the object and remaining divergence and dispersion of theincident beam.

FIG. 11. �Color online� Three-dimensional reconstruction ofphase contrast tomographic data of the spider leg. Data evaluationwas performed by applying the DEI algorithm �Ref. 5� at eachindividual sample rotation position prior to reconstruction. The cubein the lower right corner has a edge length of 100 �m.

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divergence and dispersion on image quality. The approach isbased on incoherent superposition of several partial imagesgenerated by the different components of the wave field. Pre-liminary results show that under our experimental conditionsdispersion is the dominating influence reducing the expectedcontrast by about a factor of 2, but preserving the qualitativephase contrast distribution.

VIII. DISCUSSION

The theoretical framework presented above covers wavepropagation over the entire spatial range from sample to de-tector. Experimental results show that magnification in bothimage dimensions has been achieved independently, with noadditional shear betwen the x and y dimensions �see Refs. 7and 11�. This was one main goal in instrument design, andindicates that the two analyzer crystals are well alignedwithin two mutally perpendicular diffraction planes. Contrastsimulations for the case of a simplified one-dimensional ob-ject and a single analyzer crystal show the influence of inter-ference phenomena related to propagation in space and todiffraction. Up to now these effects were not taken fully intoaccount in the literature, although they are quite essential forthe discussion of spatial resolution. These simulations dem-onstrate that phase sensitivity of “Bragg magnification” issuperior to that of conventional in-line holography. This ad-vantage holds in particular for comparatively short distancesbetween sample and analyzer crystals. Regarding resolutionthere is a trade-off in view of angular acceptance of the ana-lyzer crystals: large angular acceptance �low index reflec-tions, comparatively long wavelengths� results in a small halfwidth of the influence function �good spatial resolution indiffraction�, but also in a low phase sensitivity and a consid-erable spread of the signal during propagation. Conversely, asmall angle of acceptance leads to a more extended influencefunction but also to higher phase sensitivity and a lowerspread of the signal during propagation. Therefore, the Braggmagnifier setup presented here was constructed with the fol-lowing compromise in mind: short sample to analyzer dis-tances and medium acceptance angles. This gives an ex-tended range of good spatial resolution, i.e., allows for largefields of view. Using magnification factors of 40 and abovethe field of view after magnification is well adapted to cur-rent high-performance CCD detectors with spatial resolu-tions in the range of few ten micrometers.

The calculations given above were performed for the caseof incident monochromatic plane waves. The influence ofadditional dispersion and divergence of the incoming beamon contrast will be the subject of a forthcoming paper. Pre-liminary results show that as long as their combined effect�angular spread due to divergence and dispersion� does notconsiderably exceed the angular acceptance of the analyzer,the phase contrast features are essentially preserved, eventhough contrast is quantitatively reduced as compared to theplane wave case. Propagation contrast is slightly less sensi-tive to divergence. Even in the case of predominance ofpropagation contrast the use of magnifier crystals proves use-ful as the acceptance angle cuts a part of the interferencepattern of propagation contrast of a single object and elimi-

nates far reaching interference phenomena. In the given ex-perimental case the analyzer acceptance angles were roughlycomparable to the angular width corresponding to the finitebandwidth. In other words, nearly the whole radiation im-pinging on the sample contributed to image formation. Thisfurther contributed to the efficiency of the entire imagingprocess, in addition to the fact that thanks to magnification acomparatively thick luminescence layer with high x-ray ab-sorption efficiency could be used in the CCD camera, whichalso allowed for comparatively short exposure times at agood pixel resolution �0.75 or 0.375 �m at the sample plane�and a high dynamic range.

IX. CONCLUSION

In conclusion, we have established a theory of the processof image formation in analyzer-based magnified x-ray imag-ing. The theory takes into account the processes of bothFresnel propagation and Bragg reflection.

Numerical simulations were performed based on theabove theory. A set of numerical model objects was used toanalyze the spatial resolution achievable with the imagingsetup. One result from simulations is that the influences ofsimultaneously occurring Fresnel and Kato fringes dampeach other, thus yielding a better overall spatial resolutionthan expected from a naive theory. The simulations furthershow that spatial resolution is optimal for a very compactinstrumental setup.

Comparison of different model objects shows that theachievable resolution is indeed superior for phase contrast ascompared to pure absorption objects. This further underlinesthe attractiveness of exploiting phase contrast mechanismsfor microstructural investigations.

In view of the above findings, the experimental instru-ment �“Bragg magnifier box”� was designed as compact aspossible, realizing a very short distance between the two ana-lyzer crystals for optimum spatial resolution. Two dimen-sional image magnification by factors of 40 was achieved,with pixel sizes of 0.75 and 0.375 �m.

Simultaneously, strong phase contrast was achieved, asevidenced, e.g., by double-contrast features in projection im-ages. The series of magnified radiographic images provedsuitable for 3D tomographic reconstruction using conven-tional �absorption-based� reconstruction software, yielding3D data sets with submicrometer voxel sizes which give ac-cess to the full external and internal structure of the objectinvestigated.

Future developments will include a full 3D reconstructionalgorithm simultaneously also fully taking into accountphase contrast. Combined tomographic and DEI scans willprove decisive for quantitative phase analysis in two- andthree-dimensional imaging.

ACKNOWLEDGMENTS

The results in this paper were obtained with the help ofmany colleagues. We would like to thank Jane Richter andRainer Schurbert �HU�, Heinrich Riesemeier and JürgenGoebbels �BAM�, Astrid Haibel and Alexander Rack �HMI�,

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and Oskar Paris �MPI-KG Golm� for technical assistance,experimental support, and discussions. Furthermore, we areindebted to Lukas Helfen, Vaclav Holy, Tilo Baumbach, andJürgen Härtwig for fruitful discussions, to the ESRF scien-

tific computing group �Claudio Ferrero, Alessandro Mirone�for their PYHST tomographic reconstruction software, as wellas to a group of Summer Students at the HU-Berlin for helpwith data analysis.

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