digital holography: 30 years
TRANSCRIPT
L. YaroslavskyDept. of Interdisciplinary Studies,
Faculty of Engineering, Tel Aviv University, Tel Aviv, Israel
DIGITALDIGITALHOLOGRAPHY:HOLOGRAPHY:
30 YEARS 30 YEARS
TICSP Seminar, Tampere, Finland
Oct. 8, 2002
Outline
•Digital holography in a historical perspective
•Digital holography: What and for What
•Computer reconstruction of holograms and digitalholographic cameras
•Computer simulation of holographic imaging
•Computer generated holograms and optical informationprocessing
•Computer generated display holograms
•Scientific and technological problems of digital holography
Denis Gabor. Holography, 1947-48.
From D. Gabor’s Nobel Lectrure: “...The response of the optical industry to this was sodisappointing that we did not publish a paper on it until 11 years later, in 1966 (5). Around 1955holography went into a long hibernation.
The revival came suddenly and explosively in 1963, with the publication of the first successfullaser” holograms by Emmett N. Leith and Juris Upatnieks of the University of Michigan, AnnArbor. Their success was due not only to the laser, but to the long theoretical preparation ofEmmett Leith, which started in 1955. This was unknown to me and to the world, because Leithapplied his ideas first to the problem of the “side-looking radar” which at that time was classified.This was in fact two-dimensional holography with electromagnetic waves, a counterpart of electronholography. When the laser became available, in 1962, Leith and Upatnieks could at once produceresults far superior to mine”.
Object
Source ofcoherent light
Mirror
Recordingmedium
Object beam
Reference beam
Leith-Upatnieks’s method for recording andreconstructing hologram.
Virtualobject
(“real”)
Source ofcoherent
light
Mirror
Hologram
Objectbeam
Referencebeam
Virtual object(“imaginary”)
Scattered reference beam
Schematic diagram of hologram recording
Object
Objectbeam Photographic
plate Mirror
Laserbeam
Referencebeam
Hologram
Virtualobject
Whitelight
source
Hologram playback
Reflection (Denisyuk type) hologram (1962)
1947-48 -D. Gabor, Microscopy by reconstructed waveforms, 1, Proc. Royal Society, A197,454-487, 1949
1961-62 - D. Gabor, Light and information, in: Progress in Optics, 1, ed. By E. Wolf,Amsterdam, 1961, pp. 109-153- E. N. Leith, J. Upatnieks, New techniques in wavefront reconstruction, JOSA,51, 1469-1473, 1961- Yu. N. Denisyuk, Photographic reconstruction of the optical properties of anobject in its own scattered radiation field, Dokl. Akad. Nauk SSSR, 144,1275-1279, 1962
1966-67 - B. R. Brown, A. Lohmann, Complex spatial filtering with binary masks, Appl.Optics, 5, No. 6, 967-969, 1966- T. S. Huang, B. Prasada, Considerations on the generation and processing ofholograms by digital computers, MIT/RLE Quat. Prog. Rep. 81, Apr. 15, 1966, pp.199-205- A. W. Lohmann, D. Paris, Binary Fraunhofer holograms generated by computer, Appl. Optics, 6, No. 10, 1739-1748, 1967- J. W. Goodman, R. W. Lawrence, Digital image formation from electronicallydetected holograms, Appl. Phys. Lett., 11, pp. 77-79, Aug. 1, 1967,
1971-72 - D. Gabor is awarded Nobel Prize in Physics (1971) "for his invention anddevelopment of the holographic method"- T. Huang, “Digital Holography”, Proc. of IEEE, 59, 1335-1346, 1971.- M.A. Kronrod, N.S. Merzlyakov, L.P. Yaroslavsky, Computer Synthesis ofTransparency Holograms, Soviet Physics-Technical Physics, v. 13, 1972, p. 414 -418.- M.A. Kronrod, N.S. Merzlyakov, L.P. Yaroslavsky, Reconstruction of a Hologramwith a Computer, Soviet Physics-Technical Physics, v. 17, no. 2, 1972, p. 419 - 420.
Holography and Digital Holography:a historical perspective
Digital holography:•Computer synthesis of holograms anddiffractive optical elements
•Computer reconstruction of holograms andinterferograms
•Computer simulation of holographic imaging
Digital holography reflects, in the most purifiedway, informational pith and marrow ofholography which motivated two the mostfamous inventors in holography , D. Gabor andYu.N. Denisyuk
New qualities that are brought to optical informationsystems by digital computers and processors:
• Flexibility and adaptability.The most substantial advantage of digital computers as compared with analog electronic and opticalinformation processing devices is that no hardware modifications are necessary to reprogram digitalcomputers to solving different tasks. With the same hardware, one can build an arbitrary problemsolver by simply selecting or designing an appropriate code for the computer. This feature makesdigital computers also an ideal vehicle for processing optical signals adaptively since, with the help ofcomputers, they can adapt rapidly and easily to varying signals, tasks and end user requirements.
• Digital computers integrated into optical information processing systems enable them to perform arbitrary signal transformations
• Acquiring and processing quantitative data contained in optical signals, and integrating optical systems into other informational systems and networks is most natural when data are handled in digital form.
In the same way as in economics money are general equivalent, digital signals are general equivalent ininformation handling. A digital signal within the computer that represents an optical one is, so to say,purified information carried by the optical signal and deprived of its physical integument. Thanksto its universal nature, the digital signal is an ideal means for integrating different informationalsystems.
Applications of digital holography
• Optical metrology and non-destructivetesting
• Optical information handling and processing
• Information display: 3-D television andvirtual reality
Hologramsensor
Preprocessingof digitizedhologram
Imagereconstruction(DFT/DFrT)
Imageprocessing
Hologram
Analog-to-digital
conversion
Outputimage
Computer
Digital Holography: Digital Reconstruction of Holograms
HologramFourier PlaneFirst focal plane Second focal plane
Reconstruction of Fresnel Holograms (Equivalent optical setup)
hologr_reconstr_fastmovie(N);
Laser
Collimator
Beam spatialfilter
Lens
Microscope
Object table
DigitalPhoto-graphiccamera
Computer
One of the main drawbacks ofmicroscopy: the higher is the spatialresolution, the lower is depth of focus.
This problem can be resolved byholography.Holography is capable of recording 3-Dinformation. Optical reconstruction isthen possible with visual 3-D observation.Drawbacks of optical holography:
-Intermediate step(photographic developmentof holograms) is needed.-Quantitative 3-D analysisrequires bringing inadditional facilities
Radical solution: optical holography withhologram recording by electron means(digital photographic cameras) and digitalreconstruction of holograms. This is theprinciple of digital holographicmicroscopy.
Digital Holographic/Interferometric Microscopy
Phase
Intensity
Computer
Reconstruction
Adopted from: F. Dubois, O. Monnom, C.Yourassowsky, L.-C. Legros, Appl. Optics, v. 14,No. 14. pp. 2621-2626
Onion peel in aninterferometric microscope
3-D holographic microscopy: an example
Speckle noise in coherent imaging:Monte Carlo simulation
Hologram
Hologram sensorMeasuring hologramorthogonal/amplitude-phasecomponents:
- Limitation of thehologram size
- Limitation of thehologram componentdynamic range
- Hologram signalquantization
Reconstruction ofthe hologram
Reconstructed image
Diffuselyreflecting
object
Reflectedwave front
Speckle noise due to limitation of the hologram size
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.2
0.4
0.6
0.8
1
GrLv 1/82/83/84/85/86/87/88/8
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.2
0.4
0.6
0.8
1
SzLim 10.90.80.70.60.50.40.30.2
Hol
ogra
m (a
nten
na) s
ize
limita
tion
Image brightnessHologram (antenna) size limitation degree Reconstructed image mean value
Spec
kle
cont
rast
Speckle noise due to hologram signal dynamic rangelimitation
Image brightness
0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5 NoLim3.53,2532.82.652.52Intvl 1
Spec
kle
cont
rast
Reconstructed image mean value
1 1.5 2 2.5 3 3.5
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5 GrLv 1/82/83/84/85/86/87/88/8
Dyn
amic
rang
e lim
itatio
n de
gree
Dynamic range (in units of hologram,orthogonal component standard deviation)
Speckle noise due to hologram signal quantization
0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
NoQuanQ=645648403224168
10 20 30 40 50 60
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4 GrLv 8/87/86/85/84/83/82/81/8
Image brightness
Dyn
amic
rang
e lim
itatio
n de
gree
Number of hologram orthogonalcomponent quantization levels
Spec
kle
cont
rast
Reconstructed image mean value
Hologram signal quantization optimization:P-th law quantization
17 33 49 65 81 97 113 1290
0.05
0.1
0.15
0.2
Number of quantization levels
Orth. component quantization: Speckle contrast for P-th law quantization
P=10.750.50.30.2
Mathematicalmodel of the
object
Complexamplitude of
objectwave field
Computation ofmathematicalhologram:-Fourier Transform-Fresnel Transform-Composition ofspherical waves
Codingcomputergeneratedhologram
for recording
Recordingcomputergeneratedhologram
Digital-to-analogconversion
Hologramusage model
Computer
Digital Holography:
Synthesis of Holograms and Diffractive Optical elements
Additive representationof complex
amplitude of the wavefield: (Re+iIm)
Exponential representation ofcomplex amplitude of the wave field
(Magn . Exp(iPhase))
Phase media
Binary media
Amplitude media
Multiple phase method
Double phase method
Explicit spatial carriermethods
2-D symplex representation
Representation by orthogonaland bi-orthogonal
components
Coding by “symmetrization”
Binary media
Phase media
Combined phase-amplitude media
“Phase-detour” method
“Kinoform”
“On axis” holograms
Methods for recording computer generated holograms
Computer generated holograms
Binary CGH
Gray scale CGH
Reconstructed image
Opto-electronic correlators with CGH
F F
1-stFourier
lens
Parallellaserlightbeam
Inputimage
Template
TVcamera
Correlation plane
F F
2-ndFourier
lens
SpatialLight modulator
Computer
Jointspectrum
plane
Input image
Computer controlled nonlinearmedia and reflective matched
filterCorrelation
output
Parabolicmirror
Nonlinear computer controlled opticalcorrelator with a parabolic mirror
Joint transform correlator
Target location with electro-optical correlators
Computer generated display holograms
Chinese “magic mirrors
Types of computer generated displayholograms
• Holograms of objects represented as a set of points
• Fourier holograms of plane objects
• Fresnel holograms of objects specified in multipleplanes
• Composite stereo Fourier holograms
• “Programmed diffuser” Fourier holograms
Mark Lucente, Interactive three-dimensional holographic displays:seeing the future in depth, From special issue of SIGGRAPH's``Computer Graphics'' publication on ``Current, New, andEmerging Display Systems'', May 1997. See alsohttp://www.media.mit.edu/groups/spi/M2.html-L.P. Jaroslavski, N.S. Merzlyakov,
Stereoscopic Approach to 3-D Display UsingComputer-Generated Holograms, AppliedOptics, v.16, No. 8, 1977, p. 2034.-L.P. Jaroslavski, N.S. Merzlyakov, ,Information Display Using the Methods ofDigitalHolography, Computer Graphics andImage Processing, v. 8, No.1, 1979, p. 1-29
3-D holographic display
Programmed diffuser method for synthesis ofdisplay Fourier holograms: Generating correlated diffuser
∑
2-D Random numbergenerator
Real(FFT)
Hard limiter1
-1Input
2-D Random numbergenerator
Real(FFT)
Hard limiter1
-1Input
Diffuserdirectivity
pattern
Binaryrandomnumber
generatorBRN=[0,1]Prob(1)=Prob(0)=
0.5
BRN 1-BRN
i
Random diffuser
Real part Imaginary part
Programmed diffuser method for synthesis ofdisplay Fourier holograms
HologramObject
reflectivitydistribution
A(x,y)
Object 3-Dshape z(x,y)
Object’ssurface
directivitypattern
Object’s wavefield complexamplitude
A(x,y)exp[i2π(z(x,y)+PrDiff(x,y))] FFT
3-Dobject’smodel
Programmed diffusergenerator
PrDiff(x,y)
Reconstructedimages
progr_diff_fast_movie.m
Imitation of 3D viewing by means of CGHs withprogrammed diffuser
Basic scientific problems of digital holography
- Digital representation of
optical signals and
transforms
Wave front propagation transforms
f
xa(x)
D
Input plane Output plane
( ) ( )
( )22
222exp
fxD
fxDixa
−+
−+λ
πKirchhof equation:
( ) ( )
( )
( )∫−+
−+
=X
dxfxD
fxDi
xaf22
222exp
λπ
α
For D>>max|x-f| (“near zone” propagation), Fresnelapproximation
If exp(iπx2/D2) 1 and , (“far zone” propagation)Fraunhofer approximation:
( ) ( )∫
−≅
X
dxD
xfixafλ
πα 2exp( ) ( ) ( )∫
−≅
X
dxDfxixaf
λπα
2exp
Discrete Representation of Optical Transforms:Discrete Fourier Transforms
x
a(x) Sampled signal
x∆u
f
( )fαSampled signal spectrum
f∆
v
( ) ( )( )xukxaxaN
kreconstr_signk ∆+−= ∑
−
=
1
0ϕ
( ) ( )( )fvrffN
kreconstr_spnr ∆+−= ∑
−
=
1
0ϕαα
Continuos signal
Continuous signalspectrum
( ) ( ) ( )dxfxiexpxaf ∫∞
∞−
= πα 2 ( )( )∑−
=
++
=1
021 N
kk
v,ur N
vrukiexpaN
πα
Fourier integral Shifted DFT (canonic form)
fx/N ∆∆= 1
Signal and spectrum sampling
( )∑−
=
+
=
1
0221 N
kk
v,ur N
rukiexpNkviexpa
Nππα
( )∑−
=
+
−=
1
0221 N
kk
v,ur N
vrkiexpNruiexpa
Nππα
Direct and inverse Shifted DFTs (reduced form)
L.P. Yaroslavsky, Shifted Discrete Fourier Transforms, In: Digital Signal Processing, Ed. by V. Cappellini, and A. G.Constantinides, Avademic Press, London, 1980, p. 69- 74.
Discrete Representation of Optical Transforms:Discrete Fresnel Transforms
0 50 100 150 200 2500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
q=0
q=0.0001
q=0.0002
q=0.0004
q=0.0008
q=0.0016
( ){ }∑−
=
+−−=1
0
2, //exp1 N
kk
wr Nwrkia
Nκκπακ
( ){ }∑−
=
+−=1
0
2, //exp1 N
kr
wk Nwrki
Na κκπακκ
( ) ( )∑−
=
−=
1
0
2 2expexp1;;N
k Nkriqki
NrqNfrinc ππ
Absolute values of function for N=256 anddifferent “focusing” parameter q
( )rqNfrinc ;;
Integral Fresnel Transform:
( ) ( ) ( )∫∞
∞−
−−= dx
Dfxixaf
2
exp πα
Discrete Fresnel Transforms
Signal and its transform discretizationwith shift basis functions( ) ( )( )[ ]{ }xxukxsincxk ∆∆+−= /πφ
( ) ( )( )[ ]{ }ffvrfsincfr ∆∆+−= /πχ
2/1 fDxN ∆∆= ( ) 2/1/ xf ∆∆=κ
w is the overall shift w=uκ-v/κ
- Holographic sensor correction
- Image perfection, restoration, enhancement and quantification
Fast computational algorithms:
•FFTs and Pruned FFTs
•Linear and nonlinear
recursive algorithms
- Real time 3-D holographic display and TV
- Methods for coding and recording CGH
- New media for recording CGH
- Opto-electronic computers
www.eng.tau.ac.il/~yaro