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