Computer-generated holography

Holography is a technique originally invented by Hungarian physicist Dennis Gabor (1900–1979) to improve the resolving power on electron microscopes. An object is illuminated with a coherent (usually monochromatic) light beam; the scattered light is brought to interference with a reference beam of the same source, recording the interference pattern. CGH as defined in the introduction has broadly three tasks:

1. Computation of the virtual scattered wavefront
2. Encoding the wavefront data, preparing it for display
3. Reconstruction: Modulating the interference pattern onto a coherent light beam by technological means, to transport it to the user observing the hologram.

Note that it is not always justified to make a strict distinction between these steps; however it helps the discussion to structure it in this way.

Computer generated holograms offer important advantages over optical holograms since there is no need for a real object. Because of this breakthrough, a three-dimensional display was expected when the first algorithms were reported at 1966.{{Cite journal|last1 = Brown |first1=Byron R.|last2 = Lohmann |first2=Adolf W.|title = Complex spatial filtering with binary masks|journal = Applied Optics|volume = 5|issue=6|pages = 967–9|date = 1966|doi = 10.1364/AO.5.000967|pmid=20048989|bibcode=1966ApOpt...5..967B}}

Unfortunately, the researchers soon realized that there are noticeable lower and upper bounds in terms of computational speed and image quality and fidelity respectively. Wavefront calculations are computationally very intensive; even with modern mathematical techniques and high-end computing equipment, real-time computation is tricky. There are many different methods for calculating the interference pattern for a CGH.

In the following 25 years, many methods for computer-generated holograms were proposed in the fields of holographic information and computational reduction as well as in computational and quantization techniques.{{Cite journal|author=L.B. Lesem|author2=P.M. Hirsch|author3=J.A. Jordan|name-list-style=amp|title=Computer synthesis of holograms for 3-D display|journal= Communications of the ACM |volume=11|issue=10|pages=661–674|date=1968|doi=10.1145/364096.364111|s2cid=18707299 |doi-access=free}}{{Cite journal|doi=10.1147/rd.132.0150|author=L.B. Lesem|author2=P.M. Hirsch|author3=J.A. Jordan|name-list-style=amp|title=The Kinοform: A New Wavefront Reconstruction Device|journal=IBM Journal of Research and Development |volume=13|issue=2|pages=150–155|date=1969|url=http://www.research.ibm.com/journal/rd/132/lesem.pdf}}{{Cite journal|author=W.H. Lee|s2cid=15902468|title=Sampled Fourier Transform Hologram Generated by Computer|journal=Appl. Opt. |volume=9|pages=639–643|date=1970|doi=10.1364/AO.9.000639|pmid=20076253|issue=3}}{{Cite journal|author=D. Leseberg|author2=O. Bryngdahl|name-list-style=amp|title=Computer-generated rainbow holograms|journal=Appl. Opt. |volume=23|issue=14|pages=2441–2447|date=1984|doi=10.1364/AO.23.002441|pmid=18213016|bibcode=1984ApOpt..23.2441L}}{{Cite journal|author=F. Wyrowski|author2=R. Hauck|author3=O. Bryngdahl|name-list-style=amp|title=Computer-generated holography: hologram repetition and phase manipulation|journal=J. Opt. Soc. Am. A |volume=4|issue=4|pages=694–698|date=1987|doi=10.1364/JOSAA.4.000694|bibcode=1987JOSAA...4..694W}}{{Cite journal|author=D. Leseberg|author2=C. Frère|name-list-style=amp|title=Computer-generated holograms of 3-D objects composed of tilted planar segments|journal=Appl. Opt. |volume=27|issue=14|pages=3020–3024|date=1988|pmid=20531880|doi=10.1364/AO.27.003020|bibcode=1988ApOpt..27.3020L}}{{cite journal |last1=Clark |first1=Matthew |title=Two-dimensional, three-dimensional, and gray-scale images reconstructed from computer-generated holograms designed by use of a direct-search method |journal=Applied Optics |date=1 September 1999 |volume=38 |issue=25 |pages=5331–5337 |doi=10.1364/ao.38.005331|pmid=18324035 |bibcode=1999ApOpt..38.5331C }} The algorithms can be categorized in two main concepts: Fourier transform holograms and point source holograms.

One of the more prevalent methods that can be used to generate phase-only holograms is the Gerchberg-Saxton (GS) algorithm.{{Cite journal |last1=Memmolo |first1=Pasquale |last2=Miccio |first2=Lisa |last3=Merola |first3=Francesco |last4=Paciello |first4=Antonio |last5=Embrione |first5=Valerio |last6=Fusco |first6=Sabato |last7=Ferraro |first7=Pietro |last8=Antonio Netti |first8=Paolo |date=2014-01-01 |title=Investigation on specific solutions of Gerchberg–Saxton algorithm |url=https://www.sciencedirect.com/science/article/pii/S0143816613001942 |journal=Optics and Lasers in Engineering |volume=52 |pages=206–211 |doi=10.1016/j.optlaseng.2013.06.008 |bibcode=2014OptLE..52..206M |issn=0143-8166|url-access=subscription }}{{Cite journal |last1=Liu |first1=Shujian |last2=Takaki |first2=Yasuhiro |date=January 2020 |title=Optimization of Phase-Only Computer-Generated Holograms Based on the Gradient Descent Method |journal=Applied Sciences |language=en |volume=10 |issue=12 |pages=4283 |doi=10.3390/app10124283 |doi-access=free |issn=2076-3417}}

In the first one, the Fourier transformation is used to simulate the propagation of each plane of depth of the object to the hologram plane. The Fourier transformation concept was first introduced by Byron R. Brown and Adolf W. Lohmann with the detour phase method leading to cell oriented holograms. A coding technique suggested by Burch{{Cite journal|author=J.J. Burch|title=A Computer Algorithm for the Synthesis of Spatial Frequency Filters|journal=Proceedings of the IEEE|volume=55|issue=4|pages=599–601|date=1967|doi=10.1109/PROC.1967.5620}} replaced the cell oriented holograms by point holograms and made this kind of computer generated holograms more attractive.

In a Fourier Transform hologram the reconstruction of the image occurs in the far field. This is usually achieved by using the Fourier transforming properties of a positive lens for reconstruction. So there are two steps in this process: computing the light field in the far observer plane, and then Fourier transforming this field back to the lens plane. These holograms are called Fourier Based Holograms. First CGHs based on the Fourier transform could reconstruct only 2D images. Brown and Lohmann{{Cite journal|doi=10.1147/rd.132.0160|author=B.R. Brown|author2=A.W. Lohmann|name-list-style=amp|title=Computer-generated Binary Holograms|journal=IBM Journal of Research and Development|volume=13|issue=2|pages=160–168|date=1969|url=http://www.loreti.it/Download/PDF/CGH/ibmrd1302D.pdf|access-date=2009-06-17|archive-date=2012-02-24|archive-url=https://web.archive.org/web/20120224150548/http://www.loreti.it/Download/PDF/CGH/ibmrd1302D.pdf|url-status=dead}} introduced a technique to calculate computer generated holograms of 3D objects. Calculation of the light propagation from three-dimensional objects is performed according to the usual parabolic approximation to the Fresnel-Kirchhoff diffraction integral. The wavefront to be reconstructed by the hologram is, therefore, the superposition of the Fourier transforms of each object plane in depth, modified by a quadratic phase factor.

File:PS CGH.gif

The second computational strategy is based on the point source concept, where the object is broken down in self-luminous points. An elementary hologram is calculated for every point source and the final hologram is synthesized by superimposing all the elementary holograms. This concept has been first reported by Waters

{{Cite journal

|author=J.P.Waters

|title=Holographic Image synthesis utilizing theoretical methods

|journal=Appl. Phys. Lett.

|volume=9

|issue=11

|pages=405–407

|date=1968

|url=http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=APPLAB000009000011000405000001&idtype=cvips&gifs=yes

|doi=10.1063/1.1754630

|doi-access=free

}} whose major assumption originated with Rogers

{{Cite journal

|author=G.L. Rogers

|journal=Nature

|volume=166

|pages=237

|date=1950

|title=Gabor diffraction microscopy: the hologram as a generalized zone-plate

|pmid=15439257

|doi=10.1038/166237a0

|issue=4214

|bibcode=1950Natur.166..237R

|doi-access=free

}} who recognized that a Fresnel zone plate could be considered a special case of the hologram proposed by Gabor. But, as far as most of the object points were non-zero, the computational complexity of the point-source concept was much higher than in the Fourier transformation concept. Some researchers tried to overcome this drawback by predefining and storing all the possible elementary holograms using special data storage techniques

{{Cite journal

|author=M. Lucente

|title=Interactive computation of holograms using a look-up table

|journal=Journal of Electronic Imaging

|volume=2

|pages=28–34

|date=1993

|doi = 10.1117/12.133376

|citeseerx=10.1.1.51.4513

|bibcode=1993JEI.....2...28L

}} because of the huge capacity that is needed in this case, others by using special hardware.{{Cite journal

|display-authors=4|author=T. Ito|author2=K. Yoshida|author3=S. Takahashi|author4=T. Yabe|author5=T. Kunugi

|title=Special-purpose computer for holography HORN-2

|journal=Comput. Phys. Commun.

|volume=93

|issue=1|pages=13–20

|date=1996

|doi=10.1016/0010-4655(95)00125-5

|bibcode=1996CoPhC..93...13I

}}

In the point-source concept the major problem is the trade-off between data storage capacity and computational speed. In particular, algorithms that increase computational speed usually have much greater data storage requirements while algorithms that reduce data storage requirements have high computational complexity

{{Cite journal

|author=H. Yang|author2=E. S. Kim

|title=Waveform-decomposition-based algorithm for horizontal parallax-only-display computer-generated holograms

|journal=Opt. Lett.

|volume=21

|pages=510–512

|date=1996

|doi=10.1364/OL.21.000510

|pmid=19865455

|issue=7

|bibcode=1996OptL...21..510Y

}}

{{Cite journal

|author=J. L. Juárez-Peréz|author2=A. Olivares- Peréz|author3=L. R. Berriel-Valdos|name-list-style=amp

|title=Nonredundant calculations for creating Fresnel holograms

|journal=Appl. Opt.

|volume=36

|pages=7437–7443

|date=1997

|doi=10.1364/AO.36.007437

|pmid=18264254

|issue=29

}}

{{Cite journal

|author=H. Yoshikawa|author2=S. Iwase|author3=T. Oneda|name-list-style=amp

|title=Fast Computation of Fresnel Holograms employing Difference

|journal=Optical Review

|volume=8

|issue=5

|pages=331–335

|date=2001

|doi = 10.1007/s10043-001-0331-y

|bibcode=2001OptRv...8..331Y

}} (though some optimizations are possible{{Cite journal

|doi=10.1063/1.168429

|author=A. D. Stein

|author2=Z. Wang

|author3=J. S. Leigh, Jr.

|title=Computer-generated holograms: A simplified ray-tracing approach

|journal=Computers in Physics

|volume=6

|issue=4

|pages=389–393

|date=1992

|url=http://www.aip.org/cip/vol6.htm

|bibcode=1992ComPh...6..389S

|access-date=2010-09-14

|archive-url=https://web.archive.org/web/20100201052304/http://www.aip.org/cip/vol6.htm

|archive-date=2010-02-01

|url-status=dead

|url-access=subscription

}}).

Another concept which leads to point source CGHs is the ray tracing method. Ray tracing is perhaps the simplest method of computer generated holography to visualize. Essentially, the path length difference between the distance a virtual "reference beam" and a virtual "object beam" have to travel is calculated; this will give the relative phase of the scattered object beam.

Over the last three decades, both concepts have made remarkable progress improving computational speed and image quality. However, some technical restraints, like computation and storage capacity, still burden digital holography which makes real-time applications almost impossible with current standard computer hardware.

Once it is known what the scattered wavefront of the object looks like or how it may be computed, it must be fixed on a spatial light modulator (SLM), abusing this term to include not only LCD displays or similar devices, but also films and masks. Basically, there are different types of SLMs available: Pure phase modulators (retarding the illuminating wave), pure amplitude modulators (blocking the illumination light), polarization modulators (influencing the polarization state of light)

{{Cite journal

|author = M. Nakajima|author2 = H. Komatsu|author3 = Y. Mitsuhashi|author4 = T. Morikawa

|journal = Appl. Opt.

|pages = 1030–1033

|title = Computer generated polarization holograms: phase recording by polarization effect in photodichroic materials

|volume = 15

|issue = 4|date = 1976

|doi=10.1364/ao.15.001030|pmid = 20165114|bibcode = 1976ApOpt..15.1030N}} and SLMs which have the capability of combined phase/amplitude modulation.

{{Cite book

|author=W. Lauterborn|author2=T. Kurz

|title=Coherent Optics

|publisher=Springer

|edition=2nd

|date=2002

|isbn=978-3-540-43933-2}}

In the case of pure phase or amplitude modulation, clearly quality losses are unavoidable. Early forms of pure amplitude holograms were simply printed in black and white, meaning that the amplitude had to be encoded with one bit of depth only.

Similarly, the kinoform is a pure-phase encoding invented at IBM in the early days of CGH.

{{Cite journal

|author=L. B. Lesem|author2=P. M. Hirsch|author3=J. A. Jordan, Jr.

|title=The Kinoform: A New Wavefront Reconstruction Device

|journal= IBM Journal of Research and Development

|volume=13

|pages=150–155

|date=1969

|url=http://domino.watson.ibm.com/tchjr/journalindex.nsf/0b9bc46ed06cbac1852565e6006fe1a0/ecd8ca89a481416c85256bfa00684019?OpenDocument

|issue=2

|doi=10.1147/rd.132.0150|url-access=subscription

}}

Even if a fully complex phase/amplitude modulation would be ideal, a pure phase or pure amplitude solution is normally preferred because it is much easier to implement technologically. Nevertheless, for the creation of complicated light distribution simultaneous modulation of amplitude and phase is reasonable. So far two different approaches for amplitude-phase-modulation have been implemented. One is based on phase-only or amplitude-only modulation and consecutive spatial filtering,

{{Cite journal

|author = V. Arrizon|author2 = G. Mendez|author3 = D. Sanchez-de-La-Llave

|journal = Opt. Express

|issue= 20

|pages = 7913–7927

|title = Accurate encoding of arbitrary complex fields with amplitude-only liquid crystal spatial light modulators

|volume = 13

|date = 2005

|doi=10.1364/opex.13.007913

|pmid = 19498821|bibcode = 2005OExpr..13.7913A|doi-access = free}} the other one is based on polarization holograms with variable orientation and magnitude of local birefringence.

{{Cite journal

|author = M. Fratz|author2 = P. Fischer|author3 = D. M. Giel

|s2cid = 5726900|journal = Opt. Lett.

|issue = 23

|pages = 3659–3661

|title = Full phase and amplitude control in computer-generated holography

|volume = 34

|date = 2009

|doi=10.1364/ol.34.003659

|pmid = 19953153|bibcode = 2009OptL...34.3659F| url=http://publica.fraunhofer.de/documents/N-110879.html |url-access = subscription}} Holograms with a constraint, such as phase-only or amplitude-only, may be computed via algorithms such as the Gerchberg-Saxton algorithm or more general optimisation algorithms such as direct search, simulated annealing

{{Cite journal

|author = P. J. Christopher |author2 = A. Kadis |author3 = G. S. D. Gordon |author4 = T. D. Wilkinson

|journal = Computer Physics Communications

|issue= 108139

|title = HoloGen: An open-source toolbox for high-speed hologram generation

|volume = 270

|date = 2022

|page = 108139 |doi=10.1016/j.cpc.2021.108139

| issn=0010-4655

| arxiv=2008.12214|bibcode = 2022CoPhC.27008139C |s2cid = 221340546 }} or stochastic gradient descent using, for example, TensorFlow.{{Citation |last=G. S. D. Gordon |title=gsdgordon/hologramGenerationTensorflow |date=2020-04-21 |url=https://github.com/gsdgordon/hologramGenerationTensorflow |access-date=2024-01-20}}

The third (technical) issue is beam modulation and actual wavefront reconstruction. Masks may be printed, resulting often in a grained pattern structure since most printers can make only dots (although very small ones). Films may be developed by laser exposure. Holographic displays are currently yet a challenge ({{as of|2008|lc=y}}), although successful prototypes have been built. An ideal display for computer generated holograms would consist of pixels smaller than a wavelength of light with adjustable phase and brightness. Such displays have been called phased array optics.

{{cite book

| author = Wowk B

| author-link = Brian Wowk

| chapter = Phased Array Optics

| title = Molecular Speculations on Global Abundance

| editor = BC Crandall

| publisher = MIT Press

| date = 1996

| pages = [https://archive.org/details/nanotechnology00bccr/page/147 147–160]

| isbn = 978-0-262-03237-7

| chapter-url = http://www.phased-array.com/1996-Book-Chapter.html

| access-date = 2007-02-18

| url-access = registration

| url = https://archive.org/details/nanotechnology00bccr/page/147

}} Further progress in nanotechnology is required to build them.

Currently, several companies and university departments are researching on the field of CGH devices:

• VividQ{{cite web|url=http://www.vivid-q.com|title=VividQ Home|work=vivid-q.com}} provides software for real-time CGH devices, allowing for the generation of images with over 200 depth layers using standard computing power
• MIT Media Lab{{cite web|url=http://alumni.media.mit.edu/~lucente/holo/holovideo.html|title=The Holovideo Page by Mark Lucente|work=mit.edu}} has developed the "Holovideo" CGH display
• SeeReal Technologies have prototyped a CGH display
• Cortical Cafe CGH Kit{{cite web|url=http://corticalcafe.com/software_onlineCGHinstructions.htm|title=CorticalCafe Free Desktop Software|work=corticalcafe.com}} is a CGH related hobbyist site with instructions, source code, and a web-application for CGH creation.

{{Unreferenced section|date=September 2018}}

Recently computer-generated holography has been extended in its usage beyond light optics, and applied in generating structured electron wavefunctions with a desired amplitude and phase profile. The computer generated holograms are designed by the interference of a target wave with a reference wave, which could be, e.g. a plane-like wave slightly tilted in one direction. The holographic diffractive optical elements used are usually constructed out of thin membranes of materials such as silicon nitride.

{{Reflist}}

• Ekberg M., Larsson M., Hård S. (1990). "Multilevel Phase Holograms Manufactured by Electron-Beam Lithography". Opt. Lett. (OSA) 15 (10): 568-569. 0146-9592/90/100568-02$2.00/0

{{Display technology}}

Category:Holography