Photorefractive Volume Holography in Art

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Photorefractive Volume Holography in Artificial Neural Networks
Citation
Brady, David Jones
(1990)
Photorefractive Volume Holography in Artificial Neural Networks.
Dissertation (Ph.D.), California Institute of Technology.
doi:10.7907/1YB6-SE42.
Abstract
This thesis describes the use of volume holography to implement large-scale linear transformations on distributed optical fields. Such transformations are useful in the construction of hardware for artificial neural networks. The reconstruction of multiple grating holograms in layers of thin transparencies and in continuous volume media is considered and conditions under which such holograms may be used for linear transformations are derived. The control of the nature of the transformation implemented using fractal sampling grids is reviewed and the impact of such sampling grids on the energy efficiency of the overall system is considered. Information storage in volume holograms is shown to require multiple exposures and the impact of multiple exposures on linear hologram formations in saturable media and photorefractive materials is considered. It is shown for both types of media that the overall diffraction efficiency of a recorded hologram must decrease with the square of the rank of the transformation implemented. A theory for hologram formation in photorefractive materials with multiple trapping species is developed and compared with experimental results. The impact of multiple species and fixing mechanisms on linear hologram formation is evaluated. A method for refreshing the diffraction efficiency of photorefractive holograms in adaptive systems is described and demonstrated. The construction of thick holograms for linear transformations in waveguides is considered. A novel method for controlling such holograms is described and demonstrated. Learning in holographic neural networks is considered and two experimental holographic neural systems are described. The relative strengths of optical and electronic technologies for implementations of neural interconnections are considered.
Item Type:
Thesis (Dissertation (Ph.D.))
Subject Keywords:
artificial neural networks; barium titanate; holography; lithium niobate
Degree Grantor:
California Institute of Technology
Division:
Engineering and Applied Science
Major Option:
Applied Physics
Thesis Availability:
Public (worldwide access)
Research Advisor(s):
Psaltis, Demetri (advisor)
Bellan, Paul Murray (co-advisor)
Thesis Committee:
Psaltis, Demetri (chair)
Kimble, H. Jeff
Hopfield, John J.
Vahala, Kerry J.
Bellan, Paul Murray
Abu-Mostafa, Yaser S.
Bridges, William B.
Defense Date:
19 December 1989
Non-Caltech Author Email:
dbrady (AT) duke.edu
Funders:
Funding Agency
Grant Number
Office of Naval Research (ONR)
UNSPECIFIED
Aerojet
UNSPECIFIED
ARCS Foundation
UNSPECIFIED
Record Number:
CaltechETD:etd-05022006-155139
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DOI:
10.7907/1YB6-SE42
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PHOTOREFRACTIVE VOLUME HOLOGRAPHY
IN ARTIFICIAL NEURAL NETWORKS

Thesis by
David Jones Brady

In Partial Fulfillment of the Requirements
for the Degree of
Doctor of Philosophy

California Institute of Technology
Pasadena, California
1990
(Submitted December 19, 1989)

@ 1990

David Jones Brady

Ill

ACKNOWLEDGEMENTS
Professor Demetri Psaltis has been a patient teacher, a wise advisor and a
loyal friend. Although I have not always been able to follow where he has led
me, he has never led me astray. I would like to thank him for his support. I am
sorry that I fouled him so hard and so often on the basket ball court.
The work described in this thesis reflects collaboration and discussion with
Dr.

Ken Hsu, Dr.

Xiang-Guang Gu, R. Scott Hudson, Dr.

David Kagan,

Daniel Raguin, Dr. Jeffery Yu, Sidney Li, and Dr. Kelvin Wagner. My stay
at Caltech has been made more pleasant and my work easier as a result of the
generous friendships of Mark Neifeld, Dr. Nabeel Riza, Cheol Hoon Park, Dr.
Robert Snapp, Dr. Fai Ho Mok, Dr. John Hong, Alan Yamamura, Charles Stirk,
Chaunyi Ji, Steven Lin, Francis Ho, Seiji Kobayashi, Subrata Rakshit, and Yong
Qiao. I would like to thank Su McKinley and Helen Carrier for the kind and
efficient manner in which they provide administrative support. When I began
my work with this group, Dr. Gabriel Sirat told me that it was natural for the
experienced members to help the new members. I hope that I have returned to
the group some portion of the experience Dr. Sirat lent me.
The first three years of my studies at Caltech were supported by the Office of
Naval Research and the American Society for Engineering Education through the
ONR/ ASEE Graduate Fellowship Program. Support in later years was provided
by the Aerojet Corporation through the Caltech Program in Advanced Technologies and by the ARCS Foundation. I am deeply grateful to these organizations.
I am also grateful to Ratnakar Neurgaonkar and Rockwell International, who
supplied the SBN used in some of my experiments.

I cannot convey the depth of my appreciation for the unwavering support of
Verl and Norma Brady, but I proudly acknowledge their help and humbly thank
them for it. My wife, Rachael Alexandra, and my daughter, Katherine Anne,
have also been generous. I will not say that I did this for them, for they might
have been as happy if I had stayed home more. I thank them, however, for
understanding.

ABSTRACT
This thesis describes the use of volume holography to implement large-scale
linear transformations on distributed optical fields.

Such transformations are

useful in the construction of hardware for artificial neural networks. The reconstruction of multiple grating holograms in layers of thin transparencies and in
continuous volume media is considered and conditions under which such holograms may be used for linear transformations are derived. The control of the nature of the transformation implemented using fractal sampling grids is reviewed
and the impact of such sampling grids on the energy efficiency of the overall
system is considered. Information storage in volume holograms is shown to require multiple exposures and the impact of multiple exposures on linear hologram
formations in saturable media and photorefractive materials is considered. It is
shown for both types of media that the overall diffraction efficiency of a recorded
hologram must decrease with the square of the rank of the transformation implemented. A theory for hologram formation in photorefractive materials with multiple trapping species is developed and compared with experimental results. The
impact of multiple species and fixing mechanisms on linear hologram formation
is evaluated. A method for refreshing the diffraction efficiency of photorefractive
holograms in adaptive systems is described and demonstrated. The construction
of thick holograms for linear transformations in waveguides is considered. A novel
method for controlling such holograms is described and demonstrated. Learning
in holographic neural networks is considered and two experimental holographic
neural systems are described. The relative strengths of optical and electronic
technologies for implementations of neural interconnections are considered.

TABLE OF CONTENTS
Acknowledgements . . . .

111

Technical Acknowledgemets

Abstract

....

Vlll

Table of Contents

1. Introduction

1.1 The allure of volume holography

1.2 A brief history of volume holography

1.3 Optics and artificial neural networks

1.4 Thesis overview . . . . . . . . .

2. The reconstruction of volume holograms

2.1 Introduction

. . . . . . .

2.2 A cascade of thin holograms

2.2.1 A phase-locked cascade of sinusoidal gratings

2.2.2 Cascaded sinusoidal gratings with variable amplitude and phase 44
2.2.3 A cascaded hologram interconnection network . .
2.3 Coupled wave theory for many beam volume holograms

2.4 Arbitrary linear transformations in volume interconnection networks 68
3. Formation and control of volume holograms
3.1 Introduction

. . . . . . . . . . .

3.2 Control of a cascade of thin holograms.

3.3 Formation of volume holograms in linear media

3.4 Volume holograms in saturable media

....

3.5 Control of volume holograms with polychromatic signals

103

Vll

4. Formation of photorefractive volume holograms

110

. . . . . . . . . . .

110

4.2 Dynamics of the photorefractive effect

111

4.3 Linear formation of photorefractive holograms

119

4.4 Multiple active species . . . . . . . . . .

131

4.5 Periodically refreshed photorefractive holograms

150

4.6 Fixing mechanisms and linear hologram formation

158

4.1 Introduction

5. Holographic interconnections in waveguides

160

. . . . . . . . . . .

160

5.2 An integrated vector matrix multiplier

161

5.3 Formation of integrated volume holograms

170

5.4 Experimental results

184

5.5 Applications

190

5.1 Introduction

6. Learning in optical neural networks
6.1 Introduction

. . . . . . . .

193
193

6.2 A photorefractive outer-product memory

199

6.3 A photorefractive perceptron

205

6.4 Optical versus electronic networks

216

References

224

1. INTRODUCTION
1.1

THE ALLURE OF VOLUME HOLOGRAPHY

In the sense that it offers, within the limits of its resolution, the possibility
of complete knowledge of and control over the properties of a region of space,
volume holography represents the ultimate technique in the quest to understand
and manipulate the electromagnetic properties of the physical world. One can
imagine using this technique to construct arbitrary three-dimensional objects.
One might imagine constructing such objects using a "brick by brick" approach,
where volume elements are stacked layer by layer to form the object. Holographic
techniques, on the other hand, form a portion of every volume element, or "voxel,"
in each processing step. A hologram is formed by interference between "control
beams." While this thesis deals exclusively with optical holograms, the control
beams might in general lie in any portion of the electromagnetic spectrum or
might even consist of acoustic or particle beams. Continuing our analogy with
masonry, holography is a technique for forming an object by injecting streams of
clay and mortar whose superposition is the desired stack of bricks. The change
in each voxel in each processing step is related via a linear transformation to the
amplitudes of all the control beams.
Obviously, if the physical parameters of each voxel are independent then any
method for controlling the volume must have at least one degree of freedom for
each voxel. Thus, the number of degrees of freedom in a set of holographic control
beams cannot be less than the number of degrees of freedom used in the brick by
brick approach, i.e., one degree of freedom per brick. The potential advantage
of holography lies in the fact that it is not necessary to reach into the volume to

form or evaluate a hologram. The volume is controlled and evaluated using only
boundary conditions. While this is a crucial advantage in the control process,
removing the need to individually probe each voxel is particularly significant in
evaluating the state of a volume, since for typical materials no nondestructive
alternative exists even in principle.
The potential of volume holography for information storage was recognized
from its conception. van Heerden was first to emphasize this possibility and noted
that the number of bits which can be stored in a volume is proportional to the
resolution of the holographic system. For optical holography, this limit suggests
the possibility of storing up to 10 12 bits per cm3 • This possibility represents "the
allure of volume holography." While many workers have been drawn to holography by this possibility, the promise of holograms for high density information
storage remains largely unfulfilled. The principle reasons why volume holograms
have failed to contribute are first, that the natural parallel format of the inputs
and outputs to a hologram are not well matched to the sequential format of
information in traditional information processing devices, second, that competing information storage technologies have developed at a breakneck pace, and
third, that supporting optical technologies and devices have not been sufficiently
advanced for large-scale holographic systems.
Since volume holography has existed for 30 years, most of a professional lifetime, it is easy to lose perspective and to downplay its applicability. It should be
recalled, however, that information science itself is not much older than holography. Since information science is likely just reaching its paleozoic era, it seems
unwise to predict the future of information storage hardware based on the experience of the last 30 years. The integrated electronic microchip, also an invention

of thirty years, has just over the last decade begun to exert its influence as a
component of everyday devices.

Over the next 30 years, the implications of

this invention in the control of ordinary devices and the construction of exotic
hardware will likely continue to multiply in unforeseen ways.
The competitors of volume holography in the area of large-scale information
storage technologies have changed dramatically over the last decade. Planar electronic memories have grown in scale to the point where they are able to challenge
mechanically accessed magnetic memories in speed sensitive applications. The
resolution and speed of magnetic memories has continued to improve as fabrication technologies, materials and magnetic heads have become more advanced.
Planar optical memories, which where roughly contemporaries of holographic
memories in conception, have begun to play a major role. The history of planar
optical memories is particularly illuminating since it shows that even a technology
which is basically a good idea from its inception is adopted only as the original
idea is refined and the needs of the overall system evolve to a compatible state.
While these competing technologies are fundamentally planar, the advantage of
volume holography lies in the possibility of 3-D access rather than 3-D storage.
Planar media can be rolled into 3-D, as is done with magnetic tape and optical
film. The fact that the readout is still planar makes this approach excruciatingly
slow, however. Magnetic disks are also typically stacked, but require a separate
readout system for each layer. As is shown in some detail in chapter 2, a volume
hologram can be viewed as a stack of optical disks read out in parallel with beams
generated on a single boundary. In fact, implementation of volume storage using
a stack of optical disks may be a practical approach.
Volume holographic information storage is ripe for a fresh look because of

the potential it offers for large scale parallel interconnections in parallel computing systems. Novel computing architectures are of increasing interest because
traditional architectures seem to be getting better at the sort of ~umerically intensive problems computers have historically handled well rather than adding
new sorts of problems, such as abstract pattern recognition, to their repertoire.
These novel architectures are typically based on parallel consideration of spatial or temporal relationships which map easily onto the parallel nature of bulk
optical systems. The most intensely parallel systems are artificial distributed
neural processors. Examples of such neural systems are used to explore the use
of volume holographic storage in this thesis.
While the need for parallel adaptive hardware is the primary driving force in
the renaissance of interest in volume holographic storage, developments in optical technology have also played a role. Advances in coherent sources and spatial
light modulators have continued at a steady pace for some time. Further dramatic improvements in laser sources seem assured in light of recent developments
in solid state laser technology. High power, high efficiency, narrow line solid state
sources should in the near future eliminate the troublesome need for gas lasers in
holographic systems. The future of spatial light modulators is less assured, but
promising devices are under development and, as shown by the experimental systems described in this thesis, interesting holographic systems can be constructed
using commercially available devices. While the development of holographic materials has also proceeded at a steady pace, advances in the understanding of
currently available materials has probably been of more significance over the last
decade. While fundamental improvements in the properties of photorefractive
materials seem unlikely, it is possible that new holographic media with exciting

capabilities will come out of research into organic materials.

1.2

A BRIEF HISTORY OF VOLUME HOLOGRAPHY

For the purposes of this thesis, holography can be viewed as a technology first,
for changing the properties of sensitive media in proportion to the intensity of a
set of control beams and second, for probing the properties of media using similar
beams. The historic basis of the first task lies in conventional photography. The
historic basis of the second task lies in scattering phenomena used to determine
the internal structure of volumes of unknown composition. Since the relationship
between holography and scattering phenomena is less commonly emphasized, this
section is biased somewhat toward the history of scattering phenomena. Ballistic
scattering, i.e., throwing a projectile into the volume and measuring the direction
at which it bounces out, is the simplest technique. When the contents of the
volume are complex, information can be gathered more efficiently if complex
excitations are used as probes. The properties of the volume are determined
by their effect on the diffraction of the probes. Early in this century, electron
and x-ray scattering were used to probe the atomic structure of materials. Of
particular interest is the x-ray microscope developed by Bragg, which used a
two-step process to magnify and image 2-D projections of crystal structures [2].
Unfortunately, the range of materials to which Bragg's technique could be applied
was limited by the fact that he had no means of determining the phase of the
scattered signals.
The problem of determining the phase of scattered signals, at least at optical
frequencies, was solved by Gabor [3], who later coined the term holography to
describe his discovery. Gabor's initial interest was the application to electron mi-

croscopy of the two-step imaging technique. This technique is based on recording
a transparency proportional to the scattered signal. When this transparency is
illuminated, the recording field is recreated at a magnification proportional to the
ratio of reconstructing and recording wavelengths. Ideally, the transmittance of
the transparency is proportional to the field of the diffracted x-rays. In Bragg's
x-ray system, however, only the intensity of the scattered signal could be detected
and used to form the transparency. For this reason, Bragg's technique could not
be used to form images of crystals where this phase information was needed.
Gabor was able to overcome this problem by recording the phase information in
the interference between the probe and the diffracted light. Since a crystal is
known to be a lattice of identical unit cells, a priori knowledge about the nature
of crystals can in many cases be used to make the problem of determining crystal
structures using information from scattered x-rays tractable without resorting to
Gabor's method [4]. Scattering from an optical volume hologram can be regarded
as the optical analog of x-ray crystallography, but a volume hologram need not
be regular or periodic. Evaluation of a volume hologram thus must rely critically
on holographic methods.
In line with the application he envisioned, the method Gabor proposed for
hologram formation had the probe and the diffracted signals co-propagating.
This colinear alignment made it difficult to extract the image from noise in reconstructed holograms. This problem was overcome in 1962 with the development
of two-beam off-axis holography by Leith and Upatnieks [5, 6]. While we don't
say much here about the relative advantages of the off-axis geometry, we should
note that the systems described in this thesis use this geometry because of the
analytic simplification offered by the separation of the probes and the scattered

signals and because the media used in our experiments generally have modulation transfer characteristics which are maximal at relatively high spatial carrier
frequencies.
A new form of holography invented by Denisyuk [7] improved the image
quality of single beam holograms using a reflective geometry based on a method
for making color photographs developed by Lippman [8].

In contrast to the

planar slice of the interference pattern provided by Gabor's method, the grating
wavevectors in the recording interference pattern in Denisyuk's holograms were
normal to the surface of the holographic emulsion. In order to record a large
number of fringes, the emulsion in Denisyuk's holograms was much thicker than
the wavelength of the recording light. The holograms were thus volume in nature
and evinced the primary attribute of volume holograms, Bragg sensitivity. The
image quality was improved in these holograms because the reconstructing beam
was not Bragg matched to the conjugate of the object.
Independently, van Heerden also considered holograms in volume media and
estimated their storage capacity [l]. The advent of volume, as opposed to thin,
holograms necessitated new wavecoupling theories. Of course, similar theories
had long existed to describe X-ray diffraction and acousto-optics effects. The
early extensions to volume holography were scattering theories based on the
assumption of weak coupling. Phase modulated volume holograms can support
very high diffraction efficiencies, however. The first theory to adequately describe
such effects was the coupled wave theory developed by Kogelnik [9]. The early
history of volume holography and the many theories for holographic coupling
which have since arisen are reviewed by Collier et al. [10], by Russell [11] and by
Solymar and Cooke [12].

In this thesis we are primarily concerned with holograms in photorefractive
media, which are electro-optic material in which phase holograms may be formed
by the displacement of photo-generated charge. Photorefractive effects, first observed in lithium niobate and lithium tantalate [13), have been observed in a
wide range of electro-optic materials.

The ·principle families of materials are

ferro-electric crystals such as lithium niobate, barium titantate [14] , and strontium barium niobate [15), silinites such as the oxides of bismuth with silicon,
germanium or titanium [16), and semiconductors such as gallium arsenide and
indium phosphide [17, 18].
A number of theories have been used to describe the formation of photorefractive holograms. The most commonly used theory is the band transport model
developed by Kuktarev [19). This theory is reviewed in chapter 4. More detailed
reviews and overviews of current topics photorefractive research are provided in
[20, 21).
As discussed in section 1.1, the possibility of using holography for mass information storage was recognized from its inception. Early attempts to realize this
potential may be dichotomized into techniques for conventional digital storage
and associative memory techniques. Of course, optical data storage on planar
media is well established, both in analog form on film and, more recently, in digital form on disks. With the exception of imaging radars, such systems have not
involved holographic techniques, although recently holographic storage on optical
disks has been investigated [22). Digital storage in volume media is complicated
by the fact that it is not a simple task to take an independent look at the information stored in a single voxel. Early proposals for information storage in a volume
generally were "page oriented," which meant that image holograms were multi-

plexed in the volume at different carrier frequencies or spatial positions. Each
stored page could be reconstructed by reading the hologram with the appropriate reference [23, 25, 24, 26]. Multiple image holograms must be superposed at
each spatial position if the capacity of a volume system is to exceed the storage
capacity of easily controlled planar media. The principle difficulties encountered
in constructing page oriented memories were the loss of dynamic range when
multiple holograms are recorded [27] and the lack of computing systems which
could make good use of data in 2-D blocks. The first of these problems also arises
in the systems considered in this thesis and is considered in some detail below.
The second problem does not arise in neural processors, where the spatially distributed signals are not only tolerated, but are in fact necessary. In spite of these
problems, Staebler, Burke, Phillips and Amodei were able to record as many as
500 superposed plane wave holograms in LiNb03 [28]. While the experimental

work in this study was very impressive, the assumption was made in the analysis
that fixing was of use in multiple hologram storage and a fixing technique was
used in the experiments. The assumption that fixing increases the storage capacity is dubious for reasons explained in chapter 4. Recently, Mok, Tackitt and
Stoll have recorded 500 image bearing holograms in LiNbOa with very uniform
diffraction efficiency using the multiple exposure technique outlined in chapter 4

[29].
Early proposals for holographic associative memories were based on the ability of a hologram to reconstruct a complete signal from a partial reference [30,
31]. Of course, this principle is not strictly valid, since a limited region on a plane

cannot have as many degrees of freedom as are available across the whole plane,
but with feedback or pre-processing methods this difficulty can be overcome. In

recent years several systems have been proposed which incorporate such refinements. These systems fall into the realm of optical neural processors and are
reviewed in the next section. Most optical associative memories of this sort are
based on the ability of a thin hologram to act as a spatial filter. This ability is
compromised to some extent by the Bragg selectivity of volume holograms. The
work described in much of this thesis can be viewed as an attempt to combine
the storage capacity of page oriented memories with the associative capabilities
of optical associative memories.

1.3

OPTICS AND ARTIFICIAL NEURAL NETWORKS

Advances in the understanding of biological neural networks and the continued ineffectiveness of traditional information processing systems in pattern
recognition applications have led in recent years to a dramatic resurgence of interest in artificial neural systems [32-36]. The structure of these systems is to a
large extent biologically motivated. The reason neurobiology is viewed as a source
for ideas in the design of computing systems lies in the empirical distinction that
exists between the numerical problems which modern digital computers handle
well and the pattern extraction problems at which humans and other animals
excel. Interest in optical neural computers is based on the expectation that this
difference in capabilities is due to basic hardware differences. In some instances,
particularly in the case of sensory systems, it is possible to construct circuits that
are reasonably accurate replicas of biological systems [37]. More often, however,
we only attempt to extract basic properties that may be evident in the nervous
system and use this information to guide the design of computers. The hope is
that if we are successful in identifying relevant properties, then the neural anal-

ogy will be a positive input to the process of designing a computer even though
we do not have a detailed understanding of the circuitry of the nervous system.
The systems considered in this thesis are based on the assumption that the
potential advantages of neural systems lie the following aspects of their architecture:
1. Parallelism. There are perhaps 10 12 neurons in the human brain. At the

simplest level of abstraction, each neuron may be viewed as an independent
processor. Each processor operates asynchronously on its input signals.
Typicaily, many processors will be simultaneously active. In conventional
systems, in contrast, only a very few processors are simultaneously active.
2. Dense Interconnections. A neuron in the brain typically receives inputs
from several hundred to several thousand other units to produce its own
output which is broadcast to roughly the same number of units.

It is

widely believed that the synaptic strengths serve as part of the memory of
the system.
3. Learning. Humans learn to perform most tasks by exposure to the problem
and practice at solving it. Conventional computers, in contrast, generally
learn by algorithmic instruction. Based on this distinction, the design of
neural systems is inverse to the design of logical systems in the sense that
the system designer tries to match the neural network hardware to the problem he wishes to solve but expects the neural software, i.e., the synaptic
strengths, to adapt itself to the specifics of the problem. The only software
which the designer specifies is the "learning algorithm" which controls the
adaptation of the synapses. In conventional machines the hardware is de-

signed to be general purpose and the system designer specifies software to
adapt his machine to problems of interest. The traditional view that the
adapting a general purpose machine is easier than building special purpose
hardware is correct for traditional numerical computational problems, but
is not likely to apply to pattern recognition problems which have historically
been difficult to solve with simple algorithms.
The reason optical technology has been considered for applications in neural
hardware is that dense interconnections can be implemented in optics with relative ease. The concept that holography could be used to mimic brain functions
occurred in van Heerden's original paper on volume holography and in several of
the early papers on optical associative memories [1,30, 31]. While the conceptions of the mechanisms behind brain functions which are currently popular are
substantially different from that assumed by van Heerden, the basic idea that
the linear interconnection capabilities of optical holography makes it a suitable
technology for artificial neural interconnections remains sound. More recently a
second attraction for the inclusion of optics in neural systems has arisen, which
is that dynamic holography, such as occurs in photorefractive crystals, provides
a very natural basis for implementation of neural learning algorithms.

In step with the revival of interest in artificial neural systems, interest in optical neural computing has grown dramatically in recent years. Implementations
of Hopfield style networks were proposed by Psaltis and Farhat [38] in 1985. The
first proposed implementation was based on an incoherent vector-matrix multiplier [40] and was not holographic in nature. A second architecture proposed
for processing two-dimensional signals was based on more traditional holographic
associative principles but again used thin storage media. Since 1985 a large num-

her of optical implementations of artificial neural networks have been proposed
[41- 71]. In these systems, the activity of the neurons is represented by the optical field or intensity in some spatial mode. The modes are interconnected by
transformations stored in either holographic or image plane optical elements and
a nonlinear electronic or optical device is used to threshold the activity of each
mode to generate feedback and output signals. The principle differences between
the various systems involve the nature of the light sources and modulators used,
the choice of thin or thick interconnection media, and the choice of a thresholding
element. Of course, the neural algorithm being implemented often plays a key
role in determining the choices taken.
This thesis is limited mostly to a consideration of the capabilities of volume
holographic elements as media for neural-style interconnections. We rely in particular on photorefractive media. These media have been used in resonator style
memories [41, 52,44, 45] and in linear interconnection networks [50, 51, 56, 57,
59]. The second approach is taken here. Since we wish our results to be as general as possible, very little consideration is given to specific neural models. In
fact, after reviewing the framework for volume holograms in neural systems in
this chapter, we do not consider neural systems again until chapter 6.
The various components of a pair of "neurons" and their interconnection
hardware is shown conceptually in Fig. l.l(a).

Each neuron receives inputs

through the synapses on its dendrites, it processes these inputs in some fashion, and then it broadcasts the result on the axon where it is picked up by the
dendrites of other neurons. Except at the conceptual level, we do not claim
any biological authenticity for this model. The diagram in Fig. l.l(b) shows the
holographic analog of the two neurons in Fig. l.l(a). The neurons are nonlinear

SYNAPS

Figure 1.1( a). Conceptual structure of a neural system. Neurons are nonlinear
nodes which receive inputs from other nodes via a set of "dendrites."

Each

neuron broadcasts its output along an "axon." Connections between axons and
dendrites are made through "synapses."

devices which detect and emit light. The activity of each neuron is coded in the
amplitude or intensity of the signal it emits. The input into each neuron is sensed
by a detector. Interconnections between neurons are made by holographic gratings. The strength of the connection between a pair of neurons depends on the
modulation depth of the grating which connects them. We can draw some direct
analogies between the component structure of a neuron and its optical simulation.
The output light beam serves the role of the axon, broadcasting the signal from
each neuron. The holographic grating plays the role of the synapse, directing the

signal from one neuron to the next and the optical pathways through which light
is transferred from the hologram to the detector area of the neuron is analogous
to the dendrites. As a group, the optical source and detector, and the circuits
that process the detected signal are reminiscent of the soma of the neuron.

LIGHT

SOURCE

HOLOGRAM

Figure 1.l(b). Optical analog of a neural system. A light source broadcasts the
activity of each neuron into an optical mode which corresponds to the axon for
that neuron. The synapses correspond to holograms which diffract light into optical modes incident on detectors at each neuron. The diffracted modes correspond
to dendrites.

Due to the photolithographic techniques which are used to build them, arrays
of opto-electronic devices are generally constrained to planes. This constraint
leads to architectures for optical neural computers such as that shown in Fig. 1.2.
A plane of sources at the input is connected to a plane of detectors at the output
via a volume hologram. Light sources on a training plane are used in concert
with the input plane to control the hologram. To form a general neural network,

Input
plo.ne

tro.1n1ng
plo.ne

output
plo.ne

Fourier
lens

voluMe
hologro.M

f"ourler
lens

Figure 1.2. Optical neural computer architecture.

this basic module can be cascaded and feedback channels can be added. By
fortunate coincidence, this layered architecture is also employed in most models
for artificial neural nets. The holographic element in this system implements a
linear transformation between the input and output planes. The formation and
control of this element is the principle theme of this thesis.
Since we do not consider the planar elements in detail, it is useful at this
point to briefly review the devices which are available or under development for
this purpose. These devices can be classified according to the nonlinearities they
implement and the amount of gain they provide. Transmissive devices, such as
liquid crystal displays, magneto-optic light valves [72), and etalon arrays [73,
74, 75), provide no gain. Liquid crystal displays and magneto-optic devices are
typically electrically controlled, though optical control is possible. Signal dependent nonlinearities are implemented in these devices by performing electrical
operations on detected optical signals and using the resulting electrical signals to
drive the device. Etalon arrays, on the other hand, are optically controlled. The

nonlinearity implemented by these devices is determined by the architecture and
detailed physics of the materials used. In order to use these devices in a multilayer network, optical gain must be provided elsewhere in the optical architecture.
Blocking layer devices, such as liquid crystal light valves [76] and multichannel
spatial light modulators [77], provide gains of 10 6 - 10 9 by isolating the output
of the device from the input. Control passes from the input plane to the output
plane via photogenerated charge. Essentially, these devices can be regarded as a
cascade of a detector array, a simple nonlinearity, and an electrically-controlled
transmissive modulator. Recently, a third class of modulators has been developed
in which detectors and optical sources are integrated monolithically [78]. Since
they involve active sources and integrated devices, these devices can in principle
provide complex nonlinear functionality and gain. While the experiments described in this thesis use liquid crystal light valves, we do not mean to argue that
these devices are superior to other types. All of the above devices have potential
applications in neural interconnection systems. For our purposes it is enough to
note that devices adequate for the systems described here exist now and that
potentially superior devices are under development.

1.4

THESIS OVERVIEW

The primary goal of this thesis is to demonstrate methods for using volume
holograms to implement adaptive linear transformations. Two problems are encountered in attempting to use volume holograms in this manner. First, the
number of degrees of freedom which can be used to implement a transformation
in a volume is insufficient to perform an arbitrary linear transformation on arbitrary optical fields. This problem can be addressed by appropriately constraining

the degrees of freedom of the readout field. Methods for designing appropriate
constraints are presented in a recent thesis by Xiang-Guang Gu [79) and are
briefly mentioned in chapter 2 of the present thesis. The second problem is that
the number of degrees of freedom in a monochromatic optical field is insufficient
to control a volume hologram. This problem is addressed by making multiple
exposures to form holograms. The implications of this second problem are the
the topic of this thesis.
Chapters 2 and 3 develop a theory for the formation, control and reconstruction of volume holograms for linear interconnections. As described in section 1.2,
a number of authors have considered the formation and reconstruction of volume
holograms. The analysis presented here differs from previous analyses in that
many grating holograms are strongly emphasized. The conditions assumed in
the analysis are such as to keep the holograms in a linear first-order regime. We
show in these chapters that by applying discrete Fourier analysis we can illuminate a number of surprisingly simple characteristics of volume holograms. The
simplicity is surprising because rigorous analysis of even single grating holograms
is well known as a messy and difficult prospect. The many grating problem is
easy in contrast because the modulation depths of the individual gratings are
extraordinarily small. A second limitation on the many grating problem which
is used to make the analysis tractable is the limitation of the spatial bandwidth
of the gratings to a small convex region in the allowed grating space.
In order to develop the application of discrete Fourier analysis to volume
holographic transformations in as simple a manner as possible, much of chapter 2 is concerned with the analysis of Bragg diffraction from a cascade of thin
holograms. Information storage was first proposed in such a system by Pohl

m 1974 [80]. Pohl suggested that the information stored in different layers be
discriminated using pulses and time of flight measurements rather than the holographic methods suggested in chapter 2, however. A another approach suggested
by Thaxter and Kestigian [81], consisted of layers of thin SBN holograms. The
gratings stored in a layer could be masked and unmasked electronically, allowing
independent holographic readout of each plane. More recently, Johnson and Tanguay have suggested information storage in a holographic cascade [84] and have
analyzed this system using a beam propagation method [83, 82]. To the author's
knowledge, however, a detailed consideration of this problem such as that given
in chapter 2 has not been previously published. Interesting new results which
this approach yields include the relationships between "distributed" and "local"
holograms derived in subsection 2.2.2 and various constraints on many beam
interconnection networks derived in subsection 2.2.3. The system is shown to exhibit diffraction characteristics similar to those of continuous volume holograms
and is used to describe many of these characteristics. The analysis in section 2.2
is based on a lecture given to the Fourier optics class at Caltech in the spring of
1988. The third section of chapter 3 presents coupled wave theory in a discrete
multiple grating context. While the perspective from the many grating point
of view using discrete Fourier methods is not conventional, the results in this
section are trivial extensions of previous studies, particularly Kogelnik's [9]. The
final section describes what we mean by an "arbitrary linear transformation" and
describes methods for constraining the input fields to achieve such transformations. The fractal sampling grids described in [50, 85, 70, 86, 79] are reviewed. A
new result describing the impact of sampling grids on the diffraction efficiency of
interconnections systems is presented in this section.

Chapter 3 describes the use of optical fields to control both cascades of
thin holograms and true volume holograms. While the determination of threedimensional structure from holographic data has been an area of active interest
for some time [88], the control of three-dimensional structures with holography
has not received similar attention. In principle, control is achieved by simply
inverting the scattering process. In practice, the lack of sufficient boundary conditions to do this in one step leads to constraints on the nature of the structures
which can be recorded. The key points in this chapter are that multiple exposures are needed to control a hologram and that it is possible to construct an
arbitrary volume hologram in a linear holographic medium if the total number
of degrees of freedom in the recording exposures is equal to the total number of
degrees of freedom in the recorded hologram. As an example of real holographic
media, hologram formation in saturable materials is considered in section 3.3.
We find that it is possible to form holograms linearly in a saturable material if
an appropriate sacrifice of the total dynamic range of the material is made in
each exposure. The final section of chapter 3 considers the use of polychromatic
light to control volume holograms, the advantage of such control being that a
polychromatic field can specify a volume hologram in one exposure. We find,
however, that this advantage is likely to be negated by the difficulty of controlling a polychromatic field. Some of the results developed in this chapter were
first presented in [87].
Chapter 4 applies the holographic control techniques of chapter 3 to photorefractive materials. These materials are of particular interest because semipermanent high efficiency phase gratings can be recorded in them using reasonable optical powers. We review the formation of photorefractive holograms with

a special emphasis on many grating holograms in the first two sections of the
chapter. The third section considers a method for linearizing hologram formation in photorefractive media and describes experimental results based on this
method. Similar exposure methods were found by Bl¢tekjaer [27) and by Burke
and Sheng [89]. Our results, derived directly from the band transport theory,
are slightly different than those in [27, 89]. Strasser et al. (90] present a fourth
variant, but the applicability of this version is clouded by the fact that the analysis used to derive it lacks internal self-consistency. A different exposure method
based on a stocastic approach was described by McRuer et al. [91]. This section
is based on results presented in [57, 59, 92, 93]. Figure 4.1 in this section was
generated by Sidney Li. We find that the cost to the dynamic range exacted by
linear multiple hologram storage is identical to the cost to the dynamic range in
saturable materials. A discussion, new to this thesis, of the advantages of hologram formation in the presence of a large applied field is presented. In section
4.4 we consider the impact of multiple trapping species on hologram formation
and describe experimental results which show the effects of multiple species in
LiNb03. We explain our experimental results using a band transport theory for
multiple species materials developed earlier by Valley [94]. In order to explain
our results it is necessary to change the boundary conditions Valley assumed in
his analysis. The development of appropriate boundary conditions and the solution of the transport equations in the presence of an applied field form the major
new contributions of this section. Figures 4.9 and 4.10 in this section were generated by Dr. David Kagan. The principal advantage of photorefractive materials
for holographic interconnects is that photorefractive holograms can be recorded
dynamically in situ. In an adaptive recording environment it is not expected

that the holographic recording schedule will be optimal. In section 4.5 we suggest a method for using copying techniques to restore the diffraction efficiency
of a hologram in an adaptive recording environment and describe experimental
results demonstrating this method. This method was described by Brady, Hsu,
and Psaltis in [92, 93). The experiments described in this section were performed
jointly by the author and by Dr. Ken Hsu. The final section of chapter 4 briefly
considers the impact of fixing mechanisms on photorefractive storage.

It is interesting to note that both of the key problems in implementing holographic interconnections are avoided by thick holograms in waveguides. Since
the inputs and outputs to a planar waveguided are one-dimensional, a twodimensional waveguide supports sufficient degrees of freedom to implement an
arbitrary linear transformation on its inputs.

A waveguide hologram can be

controlled in a single exposure by a 2-D control field incident from above the
waveguide. In chapter 5 we consider holograms for linear interconnections in
optical waveguides. The key result of the chapter is a method for specifying the
matrix for an integrated vector matrix multiplier in one exposure using one SLM.
We present experimental results in support of this method. A number of previous
studies have been made of holograms in waveguides. A review of photorefractive
holograms in waveguides is presented in [95). The significant new contribution of
chapter 5 is the development of a method for control of complex holograms from
above the waveguide. Simple chirped gratings have previously been recorded and
etched on waveguides from above the guided plane [96). Some of the analysis in
chapter 5 was first presented in [97].
Chapter 6 returns to the use of volume holograms in artificial neural networks.
The first section considers neural learning algorithms and their implementation

in optical networks. Sections 6.2 and 6.3 describe two experimental systems, an
outer-product based associative storage system and an adaptive optical perceptron. Section 6.2 is based on work presented in [98, 70). The perceptron system
of section 6.3 was previously considered in [60, 59, 57, 99). Perceptron learning
has since be implemented by Yoshinaga et al. [68), by Hong et al. [100), and by
Paek et al. [101]. Chapter 6 concludes with a consideration of the relative advantages of optical and electronic interconnection technologies for implementations
of neural processors. Some of the results described in section 6.4 were presented
in [102).

2. THE RECONSTRUCTION
OF VOLUME HOLOGRAMS
2.1

INTRODUCTION

While it may seem strange to describe hologram reconstruction before we
deal with hologram formation, this approach is taken because an understanding
of how information is extracted from a volume is needed to guide us in later
chapters when we consider how to store information in it. Rather than review
the many previous studies which have considered the reconstruction of volume
holograms, we focus in this chapter on perspectives on the problem which will
be particularly useful to our application. The second section of this chapter
considers the reconstruction of a cascade of thin holograms. This system, which
is fascinating in its own right, is analyzed because it is conceptually simpler than
a continuous volume hologram and yet it exhibits quite similar properties. We do
not mean to argue that the systems are totally equivalent, but an understanding
of the properties of a cascade of thin holograms provides good intuition into
the properties of volume holograms. This analogy will be particularly useful
in chapter 3, where we consider the control of volume holograms. The third
section of the present chapter considers the reconstruction of multiple grating
volume holograms from the point of view of coupled wave theory. Multiple grating
holograms are emphasized because the information storage capacity of a hologram
in signal processing applications is proportional to the number of gratings the
hologram can support. The final section of this chapter uses the results of the
second and third sections to describe techniques by which volume holograms may
be used to implement arbitrary linear transformations.

Three basic techniques have been applied to the problem of calculating the
effect of volume holograms on their read out fields. Prior to the development of
coupled wave theory, integration of the scattered field under the Born approximation was the most popular technique for analyzing volume diffraction [7,1]. This
approach views the diffracted field as arising from weak diffraction from the undepleted read out beam. Integrating the diffracted signals from each voxel yields
an expression for the total diffracted field. This intuitively satisfying approach is
tractable only for weak coupling, when the Born approximation is appropriate.
At high diffraction efficiencies this approximation fails. More recently, Langbein
and Lederer have developed a more sophisticated version of this approach using Green's functions [103). The second technique involves a decomposition of
the boundary conditions imposed by the incident field into eigenmodes of the
holographically perturbed medium (104- 106]. This technique allows for rigorous
solutions in cases where the eigenmodes can be found. Describing the eigenmodes
is very difficult in all but the simplest cases. The third technique is coupled wave
theory [9]. This technique involves solving a set of coupled differential equations generated by substituting spatially modulated versions of the eigenmodes
of the unperturbed system into the holographically perturbed differential wave
equation.
In addition to the three basic categories described above, volume holographic
coupling theories may be classified according to how they treat optical polarization and anisotropy. These issues can be dealt with within the general framework
of the methods described above. While polarization effects may prove useful in
the detailed design of information processing systems, such effects do not add
substantially to the capabilities of volume holograms at the level of abstraction

employed in much of this thesis. For this reason, polarization effects are not
considered here. Excellent reviews of theories dealing with the reconstruction of
volume holograms are provided in [107, 11, 12].
The analysis of a cascade of thin holograms presented in the next section is
based on a discrete analogy to the eigenmodal approach. As discussed in chapter
1, this analysis is similar to the beam propagation method of [83, 82]. Because
coupled wave theory is the most convenient method for analyzing the information
processing systems described later in this thesis, the third section of this chapter
concentrates on this approach.

2.2

A CASCADE OF THIN HOLOGRAMS

INCIDENT
FIELD

Figure 2.1. A thin hologram.

By "thin hologram" we mean a two-dimensional mask with transmittance
function t( x, y ). A typical thin holographic system is sketched in Fig. 2.1. Under

Kichhoff boundary conditions [108], the optical field at the plane of the hologram,
the plane z = 0 in Fig. 2.1, is the field incident on the hologram from the negative
z axis multiplied by t(x,y). Given the boundary conditions, the field at all points

in the positive z half-space can be calculated. A volume hologram differs from a
thin one in that its extent along the z axis is not negligible. Intuitively, a volume
hologram can be regarded as a stack of thin holograms. Such a stack is sketched
in Fig. 2.2. The stack consists of N cascaded transparencies. The transparencies
lie in parallel planes spaced along the z axis by the distance l. The x and y axes
lie in the plane of the first transparency. In this section we consider this stack in
some detail. We find that a cascade of thin holograms exhibits the characteristic
features of volume holograms, i.e., Bragg diffraction and increasing diffraction
efficiency with increasing overall thickness.
Absorption based volume holograms can have only limited diffraction efficiency [27] and are generally ill-suited to multiply exposed applications. For this
reason, we assume throughout this thesis that we are dealing with holograms
based primarily on perturbations to the real part of the index of refraction. Such
"phase holograms" may be formed in thin media using, for example, gelatin or
thermoplastic films. In our cascade of thin hoiograms we assume that the optical
phase delay at (x, y) on the n th hologram is given by the function transmittance of the n th hologram is

(2.1)

We consider three functional forms for

INCIDENT
FIELD

Figure 2.2. A cascade of thin holograms.
in the cascade is a sinusoidal grating, i.e.,

(2.2)

jJ is a position vector in the x-y plane and .K is the wavevector of the grating.
The wavelength, orientation, amplitude and phase of the grating are the same
in all of the holograms in the cascade. Our goal in considering this case is to
determine the conditions under which the cascade implements a linear connection
from a read out mode to exactly one diffracted mode. If these conditions hold,
then when more complex holograms are stored in each layer of the cascade we
can associate the component of the recorded holograms which is described by
Eq. (2.2) with the connection between specific pairs of modes. We assume that

these conditions hold in our analysis of the second two cases. The second case
we consider is
n

= mn sin( l{ · jJ + <;n)

(2.3)

In this case, each hologram is a grating with_ the same grating wavevector, but
the amplitudes and phases of the gratings vary. Analysis of this case illustrates
the following important concepts:
1. If mn is constant over n and l<;n I grows linearly with n then the cascade

implements a linear connection between an input mode propagating with
wavevector kr and an output mode propagating with wavevector ks if and
only if

(2.4)
where z is a unit vector along the z axis. It is important to note that I{
is confined to the x-y plane. If these conditions are satisfied, we say that
the "distributed hologram" which connects the r th input mode to the s th
output mode is recorded in the cascade of thin holograms. In the same
vein, each thin transparency is a "local hologram."
2. We refer to

(2.5)
as "the grating wavevector" of the corresponding distributed hologram.
Similarly, we refer to I< as the grating wavevector of the local holograms.
Two distributed grating wavevectors, 1{9 and 1{9 , are called "distinct" if
there exists no pair of read out and diffracted modes which are coupled by
both grating wavevectors.

3. For fixed K, there exist exactly N mutually distinct grating wavevectors.
This means that there exist exactly N distinct distributed holograms.
4. Returning to derive a transformation which relates the amplitudes and phases of the
distributed holograms stored in the cascade with the amplitudes and phases
of the local holograms. Since each distributed hologram corresponds to a
specific connection between pairs of modes, knowing the amplitudes and
phases of the distributed holograms allows us to express the transformation
implemented by the cascade.
The third case we consider is
N-1

=LL mngsin(Kg(K) · (p+Nlz)).

(2.6)

g=O

In this case, the local holograms are a superposition of the gratings corresponding
to all the local grating wavevectors in a suitable range with arbitrary amplitude
and phase. The range of the local grating wavevectors is constrained by the
fact that we limit ourselves to considering interconnections between a set of read
out modes controlled by an input SLM and a set of diffracted modes incident
on an output detector array.

As suggested by Eq. (2.6), the transformation

implemented by the cascade in this case can be described by decomposing the
transmittances of the local holograms in terms of the distributed holograms. In
summary, our goals in this section are:
1. To derive, in subsection 2.2.1, the sense in which a distributed grating may

be considered a linear connection between two modes,

2. To develop, in subsection 2.2.2, the correspondence between the local transmittances and the distributed holograms, and
3. To describe, in subsection 2.2.3, how multiple distributed holograms can
be superposed to form an interconnection network.
2.2.1

A phase-locked cascade of sinusoidal gratings

Suppose that the amplitude transmittance of each transparency in a cascade
18

(2.7)
Such transparencies may be formed by perturbing the index of refraction, n( x, y),
of a dielectric sheet of thickness 8 such that

n(x,y) = 1 + (n 0 -1)(1 + a:sin(u 0 x)),

(2.8)

where n 0 is the unperturbed index of refraction, and -1 ~ a: ::; 1. The phase
delay at ( x, y) experienced by a monochromatic field of wavelength ..\ propagating
through the sheet is

(2.9)
where k0

= 21r/..\ and ¢ = n k 8.

Discarding the constant term in (x,y),

we find that this phase delay yields the transmittance of Eq. (2. 7) if we define

= a:(1 - ;J

0 •

To maintain the analogy with the usual volume holographic

case, we will assume that a: is very small, specifically that. a:~ o(l/N). To insure
that propagation through each transparency can be considered in the geometric
regime, we assume that ¢ 0 is of order 1. This implies that m ~ o(l/ N).

We wish to determine the field diffracted from the system of Fig. 2.2 when
a plane wave of the form
(2.10)
is incident. When U0 (x, z) is incident on a thin transparency with transmittance

t( x) placed at z = O, the field at z = 0 on the positive side of the transparency is
(2.11)
By expanding t( x) in the Fourier series
00

L Jq(m)ejquox,

t(x) =

(2.12)

q=-oo

where Jq(m) is the lh order Bessel function of the first kind [109], Eq. (2.11) can
be rewritten
u+ = Ao

L Jq(m)ej(kosin0+quo)x_
00

(2.13)

q=-oo

Eq. (2.13) is a boundary condition from which we can calculate the field in the
positive half space. If we assume, as is reasonable for the present problem, that
we are interested in the field for z less than the size of the aperture of the thin
hologram, then it can be shown that each Fourier component of Eq. (2.13) gives
rise to a plane wave. The field in this regime is
00

) A oL....J
~ J qme
( ) {fU +( x,z=

(2.14)

q=-oo

where
(2.15)

The qth order diffracted off of the first transparency produces the field

(2.16)

on the positive side of the second transparency. This boundary condition produces plane waves propagating at the spatial frequencies

k}q') = J1f(q)l 2 - (k~q) + q'u 0 ) 2

(2.17)

k~q') = k~q) + q'u 0 •
Substituting from Eq. (2.15), Eq. (2.17) becomes

(2.18)

Thus the second transparency maps the spectrum of plane waves described by
Eq. (2.15) back onto itself. Since the x components of the wavevectors of these
plane waves always consist of harmonics of u 0 on the carrier k 0 sin 8, the field at
any z may be specified by a vector of the form

U1
-O(z) =

U_1
U_q

where Uq is the field amplitude of the plane wave with wavevector f

(2.19)

Each transparency implements the transformation

Ji(m)

lo(m)

J1(m)

J_1(m)

Jo(m) · Ji(m)

(2.20)

on U(z). Propagation from one transparency to the next implements an additional transformation of the form

(2.21)

P-q

where

(2.22)
After N transparencies the field is

(2.23)
where U(O) is the field incident on the first transparency.
We can expand U(O) in the eigenvectors of the transformation

(2.24)
such that

(2.25)

where
(2.26)
and ll'.n is a constal:!-t for each n. Then

(2.27)

Of course, for T1 as given by Eq. (2.20), i.e., when T1 is of infinite dimension, this
approach is unreasonable. Fortunately, it is possible to dramatically curtail the
dimensionality of T 1 by considering Bragg matching constraints and the scaling
of the various diffracted orders with m.
Suppose, for example, that we wish to couple light from the zeroth order to
the qth order. This might be accomplished either by direct diffraction via the
qth harmonic of the local gratings or by diffraction through several intermediate

orders. Consider light diffracted into the q1h order via C intermediate orders.
The phase of the field diffracted into the q1h order by the ( C + 1 lh transparency
IS

<.pc

= L k1i) l,

(2.28)

i=l

where ki is the wavevector of the ith intermediate order. The phase of the field
which is undiffracted by the first transparency and which then follows this path
to the qth order at the ( C + 2) th transparency is
'-PC+I = k 0 COS 0[ +

L kz l.
(i)

(2.29)

{i}

Assuming that the individual contribution of each transparency is small, the
amplitude of the field diffracted onto the qth order can grow only if each addition

to the field is in phase with the existing field. Since the change in the phase of
the existing field from the ( C + 1 )1h transparency to the ( C + 2) th transparency
is k1q)z, the field diffracted by the (C + l) th transparency is in phase with that
diffracted by the (C + 2)1h transparency if r..pc + k1q)z = r..pc+ 1 , or, equivalently, if

IT Pi= Po IT Pi·

(2.30)

Thus, independent of the intermediate steps, the zeroth and qth orders can be
coupled effectively only if pq = Po•

J±q(m) is of order lql in m [110]. Thus the signal diffracted directly from the
zeroth order to the qth is proportional to mlql. A signal diffracted from the zeroth
order through intermediate orders to the lh order is of order Il{i} mlil :5 mlql .
This means that even with proper phase matching, higher order diffraction will
be very weak if m is small. Of course, if an intermediate order, r, could grow
significantly before coupling to the q th order then this problem could be overcome.
r-r,L •

.i..uis wou.1

d reqmre
• s1n1u11,aneous
· 1;,.
pnase
matcn1ng rrom
tne zerotn oraer to tne r·+..h

order and from the r th order to the qth order. Since phase matching is transitive,
the zeroth order would also be phase matched to the qth order.
In order to analyze the phase matching requirements in more detail, consider the case in which the readout beam is undepleted. Of course, if strong
coupling occurs this condition will ultimately be violated. Assuming, however,
that diffraction occurs gradually, an analysis of the undepleted regime allows us
to quantify the phase matching requirements. The field diffracted into the q th
order at the n th transparency under the undepleted assumption is proportional

to p~ Jq( m )A 0 • After N transparencies the field in the qth order is

Uq(N) = LPf-np~Jq(m)A 0
n=l

N [eit/Jq + ei(N+l)t/Jql
.t/J

=Jq(m)AoPq

1 - eJ q

smy

!£=1
!!±1.
[sin Nt/J

= Jq(m)AoPo

(2.31)

approximately zero unless

(2.32)

for some integer c. If this condition is satisfied, Uq(N) grows linearly with N.
Substituting kiq) from Eq. (2.15), we find

(2.33)

Note that 'lpq = 0 when
. 0

Sln

q>-..
= -2A'

(2.34)

where A = 21r /u 0 • Eq. (2.34) is the Bragg condition which appears in volume
holography, x-ray diffraction, and various other fields. For volume holograms,
Eq. (2.34) is the only condition under which exact phase matching can occur.
For the cascade of thin holograms, additional solutions may be generated by
substituting Eq. (2.33) into Eq. (2.32) . If we assume that k0 l ~ 1 and cos 0 ~ 1,

ORDER
+2
+1
-1

Figure 2.3. Phase matched orders for sin{) = ->i./2A, "'f = 2, c = 0, c1 = 1. The
z-component of the wavevectors for the +2 and -1 order modes is 21r / I less than
the z-component of the wavevectors for modes O and +1.
these solutions take the form

q.X

smB = - 2A + q[·

(2.35)

These solutions are precluded in a volume hologram because Eq. (2.32) must be
satisfied for arbitrary l in a continuous medium. Note that in some cases {) can
be found such that Eq. (2.35) is simultaneously satisfied for several values of q

Given 0, c and q satisfying Eq. (2.35), it can be shown that the q'

order where

(2.36)

and,= 2A2 / >.l, is also phase matched at 0. c and c' are integer constants. This
condition is illustrated for sin0 = ->./2A and , = 2 in Fig. 2.3. As shown in
the figure, Eq. (2.35) is satisfied when the optical path along the z axis between
transparencies for the qth order differs by an an integral multiple of 271" from
the optical path for the 0 th order. For the parameters selected, this condition is
satisfied for q = ±1, 2.

It is possible to simultaneously couple from the zeroth order mode to more
than one higher order mode in two ways. The first method, true phase matching
based on identical phase shifts between transparencies, has no analog in continuous volume holography. This is the method described in the previous paragraph.
A second means by which coupling can occur to more than one diffracted mode
arises when the cascade is not "thick" enough to prevent coupling between adjacent modes. The simplest case arises when 1P-I - 1P1 < 211"/N. If 'lp1 = 0
then,

1P-1 = kol cos 0(1 -

u~l
k 0 COS 0'

1- 2( - - - 2 ~ - - -

ko COS 0

(2.37)

where we assume A ~ >.. The +1 and -1 orders are both phase matched if

1P-1 < 271" / N, which is the case if
2A2
NZ< Tcos0.

This case corresponds to the Raman-Nath regime for volume holography.

(2.38)

Diffraction from the zeroth order to the first order is of particular interest.
We assume N is large enough that Eq. (2.38) is violated and that sin 0 = -A/2A.

In this case only the first diffracted order and associated solutions of Eq. (2.36)
are phase matched. If I is greater than 2 then all such solutions are at least two
orders away from the zeroth and first orders and can thus be ignored in light
of the small size of m. The appropriateness of discarding higher orders can be
verified by considering the fact that the total power in all these higher orders
cannot be greater than [N :Z:::~ 2 mq]2 ~ N 2 m 4 • The transformation T may in
this case be collapsed to the form

(2.39)

where PI = p 0 = p. The eigenvalues of this matrix are

(2.40)

J1fml

. l cos (J ±.J arctan J
1k
A±= e 0
om

(2.41)

where we make use of the fact that to first order in m

(2.42)

The eigenvectors corresponding to A± are

(2.43)

Assuming that

(2.44)

il(O)= (;,)
we find

(2.45)

and

(2.46)

Thus
ejNarctan

..,

U(N) = Aop

JJ1fmj -e -jNarclan JJl(mj )
Om

jNarclan

~l(mj +e -jNarclan Nmj
Om

~m~))

11
_ A el.ko IN cos 8 ( sin(N arctan 1 om

cos(N arctan

(2.47)

~:f;:~) ·

To lowest order in m,
sin(Nm) \

U(N) = A,.eikolNcos8
.:
I.
, ,
( cos (1vm)}
-2-

(2.48)

The intensity of the light in the zeroth order mode after the cascade is

The intensity of the light in the first order mode is IA0 12 sin 2 ( Nt). If Nm = 1r
the diffraction efficiency of the system reaches 100 percent to first order in m.
Since we would like the diffraction to be linear, we will assume that m is much
less than 7r / N, in which case the diffracted field is linear in m and the diffraction

DIFFRACTED
BEAM

READOUT
BEAM

Figure 2.4. Read out geometry in the general case.
efficiency for the intensity is approximately N 2 m 2 / 4. These results are, of course,
quite similar to results describing two beam coupling in volume holography.
The observant reader will have noticed that while we promised to consider
an arbitrary local grating vector in the x-y plane in this subsection, we have thus
far considered only a grating wavevector oriented along the x-axis. Since we can
always define an x axis parallel to the local grating wavevector, the problem we
have thus far considered does, in fact, correspond to the general case, except that
the propagation vector of the read out field need not lie in the same plane as the
grating wavevector and z. To analyze the most general case we must add a y
component to f(O). This situation may be analyzed most simply if we define (,
the angle between the projection of f(O) in the z-y plane and the z axis. The
geometry of the read out beam is sketched in Fig. 2.4. The incident field can be
expressed as

Uo(x, y, z) = Aoeiko(zcos9cos (+xsin9+ycos9sin().

(2.49)

f - fJ is the angle between f< 0) and the x axis. If we define new coordinate axes,
x', y', and z', to correspond to the original axes rotated about the x axis by -(,
U0 may be expressed as

(2.50)
Since x' is identical to x, the description of the transmittance of the local holograms given by Eq. (2.12) is unchanged in the new coordinate system. This means
that, just as was shown above, the holograms in the cascade diffract the incident
field into a spectrum of plane waves which can be described by a vector U(z)
with components Uq, where Uq is the amplitude of the plane wave propagating
with wavevector
k~?) = ✓k~ - (k 0 sinfJ + qu 0 )2
k(q)
x'

(2.51)

. fJ + qu •
= k sin

In the x, y, z coordinate system the wavevector of the qth plane wave is
kiq) = cos ( ✓k~ - (k 0 sinfJ + qu 0 ) 2
k1q) = ko sin fJ cos ( + qu 0

(2.52)

k~q) = sin ( ✓k~ - (k 0 sin 0 + qu 0 ) 2 •
Just as above, coupling from the zeroth order mode to the qth order can be strong
only if the change in phase between transparencies for the two modes differs by an
integral product of 271". Since oblique incidence reduces the optical path between
transparencies along the z axes the phase matching condition given by Eq. (2.35),
must be modified. The new condition is

q>.

smB = -2A + qlcos(

(2.53)

Note that the phase matching condition for c = 0 1s unchanged. If sin f) =

-q>-./2A, phase matching is maintained as the holographic system is rotated

about the grating wavevector.

This fact results in "shift invariance" normal

to the grating wavevector. The effects of this property on information storage
systems is discussed below.
In this subsection we have considered diffraction from a cascade of sinusoidal
gratings with locked phases and amplitudes. While we have assumed that the
local grating wavevector lies along the x axis, we have considered an arbitrary
incidence angle for the read out field. The case of arbitrary grating orientation
and arbitrary incidence can be solved by simple rotation of the coordinate axes.
We have found that if Nl is large enough that Eq. (2.38) is violated, 1 > 2, and

Nm ~ 1r then the cascade couples a Bragg matched incident beam to only one
diffracted order. In this case, the cascade of thin local holograms implements
a "distributed hologram" which connects a specific class of inputs to specific
outputs. Due to shift invariance normal to the grating wavevector, the number
of inputs which can be diffracted by the system is large. In section 2.4 we consider
methods for constraining the inputs and outputs so that the connection formed
by a distributed hologram can be associated with a specific pair of input and
output modes.
2.2.2

A cascade of sinusoidal gratings with variable amplitude and phase

In this subsection we consider the case

(2.54)

We can determine the diffraction properties of this system by writing the perturbation at each hologram in terms of the perturbations associated with holograms

with known diffraction properties. Consider, for example, the hologram which
couples

(2.55)
and

(2.56)
where lkrl =

lksl = k

0 •

This hologram is formed by interfering the two signals

in a linear medium, where by "linear medium" we mean a material in which
the index of refraction changes in proportion to the recording intensity. The
recording intensity in this case is

(2.57)
where Kg = kr - ks and cp RS -

/2 is the difference between the phases of R 0

and S0 • We may assume without loss of generality that 'PRS = 0. If we use this
intensity to record the local holograms in a cascade, the index of refraction at
the n th transparency becomes

n(x, y) = 1 + (n 0 - 1)(1 + a sin(K • jJ + nKg))),

(2.58)

where I{ is the component of Kg in the x-y plane and Kgz is the z component
of K9 . The transmittance of the n th hologram is

= L Jq( m )ejq(K•p+nK zl).
00

(2.59)

q=-oo

Eq. (2.59) is equivalent to Eq. (2.7) except that the phase of the grating on the
transparencies increases linearly with n.

As described in subsection 2.2.1, coupling from one mode to another is possible only if the field diffracted at each transparency is in phase with the field
which is already in the diffracted order. With the additional phase shift at each
hologram the phase matching condition given by Eq. (2.32) becomes
(2.60)
For first order coupling between R( r) and S( r), 'l/J1 = Kgzl and Eq. (2.60) is
satisfied.

Eq. (2.60) may also be satisfied for higher order beams, but such

coupling is very weak if the conditions for two beam coupling described in the
previous subsection are satisfied. For the rest of this section we assume that the
amplitudes of the the holograms are small enough that we need only consider
first order couplings.

In a single transparency, a grating wavevector couples an incident beam with
wavevector f(i) to a diffracted beam with wavevector f(i) + K. According to
Eq. (2.60), incident light with wavevector f(i) is diffracted by a cascade of thin
holograms to light propagating with the wavevector

fU)

= f(i) +I<+ (Kz + 21rc/l)z

(2.61)

This diffraction can occur only if f U) corresponds to a propagating mode for
the wavelength of the read out light. The endpoints of the wavevectors of the
propagating modes for a given wavelength lie on a spheroid ( or two connected
spheroids in the anisotropic case) known as the wave normal surface. For a homogeneous isotropic system the normal surface is a sphere of radius k. According
to Eq. (2.61 ), a pair of modes is coupled if their wavevectors are separated on
the normal surface by ]{9 + (21rc/l)z.

As defined in the introduction to this section, two different grating wavevectors are distinct if they do not couple the same pairs of optical beams. Eq. (2.60)
allows us to quantize this definition. If a pair of modes satisfies Eq. (2.60) for
the grating wavevector Kg, then the equation cannot be satisfied for the grating
wavevector Kg, if
(2.62)
for all integral values of c. The maximum number of vectors l{g for which this
condition can be satisfied is N. A set of N wavevectors which are mutually
distinct in the sense of Eq. (2.62) is given by

(2.63)

where n is an integer between O and N - 1. This set of grating wavevectors is
sketched on the normal surface in Fig. 2.5. By construction, no two of these
grating wavevectors can couple the same pair of beams.
At this point it is useful to clarify the distinction between "local" and "distributed" holograms. By the n th local hologram we mean the transmittance of
the n th transparency. By the n' th distributed hologram we mean the component
of the transmittances of the local holograms which is described by

(2.64)

The N local holograms can be described in terms the N distributed holograms, as
we now show. Suppose that the N distinct distributed holograms have been linearly superposed in the cascaded local hologram system. Let an, and (n, represent

Figure 2.5. Wave normal surface and grating wavevectors. l / A is assumed to be

2.
the amplitude and phase with which the n'

distributed hologram is recorded.

Ignoring higher order terms under the approximations derived in subsection 2.2.1,
the local hologram stored in the n th transparency is
N'

. (K.... ... 21rnn
I" ))
tn = L_,; an 1 + Jm sm
· p + N + 1:,n' ,

(2.65)

n'=l

where m is a constant of order 1/N. We relate the amplitudes of the local
and distributed holograms by comparing Eq. (2.65) with Eq. (2.54). Letting
bn = -j(mn/m)ei,;;n, where mn and ~n are the amplitude and phase of the n th

local hologram, we find

B=PA,

(2.66)

where

b1
-+

(2.67)

b2
bN
aoeiCo \
a1 eiC1

....

a2eiC2

(2.68)

aNeiCN
and

·2,r

•4,r

-2,r(N-1)

e17:r

·4,r

·8,r

e17:r

el_N_

-2rr(N-1)

e17:r

-4,r(N-1)

el-N-

• 4,r(N-1)

(2.69)

-2,r(~-1)2 )

Pis a Vandermonde matrix. Matrices of this form are known to be nonsingular.

By inverting P, A corresponding to arbitrary B can be found.
To determine the diffraction properties of an arbitrary series of gratings with
spatial frequency K, we first use t(x, y) as expressed in Eq. (2.54) to determine B.
--1

We then find A using A= P

B. A readout signal can then be decomposed into

plane wave components phase matched to each of the N distributed holograms
described by A. As described in subsection 2.2.1, each distributed hologram

50
corresponds to a linear connection from an input mode to an output mode. The
strength of the connection between the fields is anm, The diffraction efficiency
of the connection is lanl 2 (Nm) 2 /4.
2.2.3

A cascaded hologram interconnection network

\\OUTPUT

"J PLANE

Figure 2.6. Thin hologram cascade optical interconnection system.

As described in the introduction to this section, we consider the case
N-1

=LL

ffing

sin(Kg(K) · (p + Nlz)).

(2.70)

g=O

in this subsection. Since we are primarily interested in systems in which a very
large number of distributed holograms are recorded for use as linear interconnections, the range of distributed grating wavevectors K 9 which we wish to consider

is limited by the architecture of the interconnection system. A typical architecture for such a system is sketched in Fig. 2.6. A cascade of thin holograms is
addressed by light generated on an input 2-D spatial light modulator (SLM). The
thin holograms are spaced by l along the z axis. The origin of the z axis is in
the center of the first hologram, which defines the x-y plane. The input plane
lies in the x 1 y 1 plane. The z 1 direction is normal to the input plane. The primed
coordinate system corresponds to the unprimed coordinate system rotated by an
angle B about they axis. Readout signals are generated by a 2-D pixel array on
the input plane. The pixels are arranged on a regular 2-D grid with a separation
between pixels of .6.. Each pixel occupies an area 82 • The signals generated by
the pixel array are collimated by a lens of focal length F. The lens lies in a plane
parallel to the x' -y' plane. The holograms diffract light from the input signals
onto a set of output signals. The output signals which exit the final thin hologram
are focused onto a detector array by a second lens, also of focal length F. The
output plane lies in the x" -y 11 plane. The z 11 direction is normal to this plane.
The double primed coordinate system corresponds to the unprimed coordinate
system rotated by an angle -8 about the y axis. The detectors are also arranged
on a 2-D grid and spaced by fl. The apertures of the holograms, the SLM, and
the detector array are of length L on a side.

If >-./8 is smaller than L/F, then the field generated in the volume by a single
pixel is approximately a plane wave. The amount by which >-.F/8L must exceed
one depends on the effective aperture of the holograms, which depends on B. If

L = Nl, for example, it can be shown that the plane wave approximation holds
if

(2. 71)
The plane wave generated by the pixel at x' = i' .6., y 1 = j' .6. propagates with the
wavevector

k....i'j' = i k 0 .6. A' + j kF .6. YA, + k~o [l -

-rx

i'ko.6.
= (- cos B - k [l

+ (

(i1.6.) 2 (j' .6.) 2] ½Al

- F

(i'.6.)2
(j'.6.)2]½
. B)Ax + j'ko.6.A
sm
-y

(2.72)

i' k .6.
i' .6. 2
j' .6. 2 l
-jsin 0 + k 1 - ( F) - (F) ] cos 0) z

0 [

Simlarly, the signal which is detected at the pixel at x" = i" .6., y" = j" .6. propagates with the wavevector

'"k A
.,, A
z'"k-ow.AxA" + J ow. ,,
(J-.,, w..)2]
½z,,
k....i"j" = - - y + k0 [1 - (z- w..)2
. B)Ax+---y
j"ko.6.A
= ( -i"ko.6.
- c o s B + k [l - (i".6.)2
- - - (j".6.)2]½
-sm
i" k .6.
i" .6. 2
j" .6. 2 l
+ ( - ; sin0+k0 [1-(F) -(F) ] 2 cos0)2

(2.73)

A distributed sinusoidal grating with wavevector

(2.74)
diffracts light from the i' j'

input pixel to the i" j"

output pixel.

The ( i', j') th pixel is not the only input pixel which is phase matched to the
grating wavevector Ki'j'i"j"· Recalling Eq. (2.60), we find that any incident plane
wave whose wavevector, kr satisfies

(2.75)

will be phase matched. [kr - Ki'j'i"j"]t is the component of kr - Ki'j'i"j" in the

x-y plane. Eq. (2. 75) can be transformed to

(2.76)
Thus, any input corresponding to a wavevector which is displaced from ki'j' in a
direction orthogonal to Ki'j'i"j" will be phase matched to read out the (i'j'i"j 11 )1h
grating.
Assuming that the curvature of the normal surface over the bandwidth accessed by the input and output planes is small we can visualize the gratings
between the input and output planes with the help of Fig. 2.7. The figure shows
the end points on the wave normal surface of the wavevectors of the input and
output signals. We assume that the angular bandwidth of the signal is small
enough that each set of signals may be represented by a N' x N' grid of points,
where N' = L / ~- The separation between the two grids, 2k 0 sin 0, is assume to
be much larger than the widths of the grids and is not shown to scale in the figure. (2N' - 1 )2 different planar wavevectors can be drawn between points on the
two grids, as shown in the figure. The out-of-plane component of a wavevector
between two points depends on the position of the two points.
Two grating wavevectors with the same transverse components are distinct
if no kr exists which satisfies Eq. (2.76) for both gratings. By substituting the
corresponding input wavevectors into Eq. (2.76), it can be shown that the two
grating wavevectors sketched in Fig. 2.8 are distinct if

(2. 77)
where terms greater than first order in ~/ F are discarded. Since the grating
wavevectors sketched in Fig. 2.8 have the greatest inclination to the horizontal

yrx

FOURIER REPRESENTATION
OF THE
INPUT CHANNELS

FOURIER REPRESENTATION
OF THE
OUTPUT CHANNELS

....,:,,,

2k 0 sin()

Figure 2. 7. Planar representation of grating wavevectors.

FOURIER REPRESENTATION
OF THE
INPUT CHANNELS

FOURIER REPRESENTATION
OF THE
OUTPUT CHANNELS

2k0 sin8

Figure 2.8. Grating wavevectors with maximal inclination.

Cl
Cl

of any pair of wavevectors between the two grids, it can be shown a distributed
grating cannot be phase matched to more than one pixel in each row of the input
grid if Eq. (2. 77) is satisfied. (We assume that N' ~ N.) Combining this result
with Eq. (2. 71 ), we require

>-.F

(2.78)

8N b.l

A lower bound on the number of distinct distributed gratings the system can
support is provided by Fig. 2.9. Each of the grating wavevectors shown in this
figure corresponds to a distinct grating. The number of such grating wavevectors
is 2N'

N' 2 •

Suppose that the transmittance of the n th transparency is a linear superposition of the transmittances corresponding to the distinct distributed gratings
between light from the input and output planes.

For the gratings shown in

Fig. 2.9, we can write such a superposition as

-2-

N 1 -1

N 1-1
-2-

N 1-1

a.,i -N'-1
.,, .,/1 + jmsinf i?:,
N'-1
--z J '
i - --z J

·11 .,, •

i'-_N 1 -1 i"--N'-1 1.,, __ N -3

N 1 -1
-2-

·,_

N 1 -1

N 1 -1
-2-

.,,_

N 1 -1

1p....

+ nlzJ')')

N 1 -1
-2-

·11_

N 1 -1

a-,N'-1
.,, .,,(1 + jmsin(J?.,N'-1
.,, .,, • (jJ + nlz))),
i -i J
i -z J

z ---2- z ---2-J ---2-

(2. 79)
where we have agam discarded higher order terms. Note tn(iJ) could also be
expressed in terms of the (2N' - 1)2 distinct local grating wavevectors in the
plane of each hologram. The components of the N x (2N' - 1)2 local gratings
can be related to the components of the distributed gratings using the results of
subsection 2.2.2.

FOURIER REPRESENTATION
OF THE
OUTPUT CHANNELS

FOURIER REPRESENTATION
OF THE
INPUT CHANNELS

<:11
-l

2k0 sin0

Figure 2.9. Distinct gratings in the thick case.

The read out field may be written

;;'\ ~ ~ R
U( r1
= L...t L...t i'j'e k-,• J l•T '
i'

(2.80)

where Ri'j' is the amplitude of the signal generated by the (i'j') th input pixel.
Since we are interested in distributed holograms which are weak enough to remain
in the linear regime, we can compute the diffracted field by forming a linear
superposition of the fields diffracted from each incident plane wave. We assume
that the incident waves are undepleted. In this case, the diffracted field is
~~~~
·11•r
U '( r;;'\1 = L...t
L...t L...t L...t mai'j'i"j" R i'j'e '£.,,
•1 •
i'

(2.81)

The amplitude of the field incident on the ( i", j" /h output pixel is

Si"j"

=LL mai'j'i"j"Ri'j'•
i'

(2.82)

Each of the N' parameters ai'j'i"j" corresponds to one of the 2N' -N' grating
amplitudes. Obviously, since ~ N' degrees of freedom are used to implement

connections, the transformation between the input and output fields de-

scribed by Eq. (2.82) is a highly constrained form of linear transformation. The
implementation of less constrained transformations with this system is explored
in section 2.4.
The nature of the transformation implemented by Eq. (2.82) is clearer in
the special case when the separation between the input and output grids in the
Fourier space is large enough that all the gratings are effectively parallel to the
x axis.

In this case, all the inputs in each column of· the input grid will be

phase matched to the same gratings and the transformation implemented by
the system will be shift invariant along the y axis. The condition under which
this case holds can be visualized using Fig. 2.10. The two gratings sketched
in this figure, i<-N'-I N'-I N'-1 o and K-N'-1 o N'-1 -N'-I, exhibit the greatest
' '
inclination with respect to the x axis of any pair of gratings from points in the
same column with identical planar components. If these two gratings are not
distinct then no pair of gratings with identical planar components originating
from the same input column can be distinct. The line normal to the gratings
passing through the origin of the upper grating is shown as a dashed line in
Fig. 2.10. If the origin of the lower grating is closer to this line than its nearest
neighbor in the adjacent column of inputs, then the two gratings are not distinct.
Simple geometry shows this to be the case if
sin20 >

N'L
F.

(2.83)

If we want N' = 100, this constraint forces F/L to be at least 25. If Eq. (2.83)
is satisfied, then all the signals from a given input column are diffracted onto
the output grid by the same gratings and signals in different columns are not
diffracted by the same gratings. Eq. (2.82) becomes
(2.84)

in this case. The field distribution on the i"

column of the output array is a sum

over i' of the correlations integrated over y of the field distributions on the i' th
column of the input array with ai'i"jd• In spite of the simplicity of this result, this
situation is not generally desirable because satisfying Eq. (2.83) severely restricts
the scale of N'.

F1JURIER REPRESENTATmN
Df" THE
lJUTPUT CHANNELS

F1lURitR REPRESENTATIIJN
[If' Tl£
INPUT CHANNEL.~

• • • Zl
' I
• • • • I,.
• • • • II
• • • I,
• • • I'· • I

----

:t--------2k0 sin 0

I• • • • •

• •

T.
.--

Figure 2.10. Condition for vertical degeneracies.

• •
• • •
• • •
• • •

• I

Before leaving this section, we review some of the approximations we have
made and consider the system which satisfies them. It is reasonable to assume
that b = 3-\, ~ = 10-\, NZ= L, and F = 5L. If,\= 1 µm and L = 1 cm then

N' = N = L/(10,\) = 103 . Substituting in Eq. (2. 78) yields 28° > (} > 4°. (} is
further constrained by the assumption made after Eq. (2.36) that 1 > 2. In the
present case this assumption implies 0 < 9°. This last assumption was made to
insure that only first order diffraction occurs with measurable efficiency. If higher
order diffracted fields do not overlap the output fields this assumption may not
be necessary. If these conditions are met, the analysis leading up to Eq. (2.82) is
valid. The number of distinct gratings which can be stored in the holograms is

~ 109.
2.3

COUPLED WAVE THEORY FOR MANY BEAM VOLUME HOLOGRAMS

We now turn our attention to diffraction in a continuous volume hologram in
which a perturbation, 6€, to the optical permittivity, €, has been recorded. In
general, € and 6€ are two-dimensional tensors. Our goal in this section is to use
coupled wave theory to find the optical fields diffracted by a volume hologram
given boundary conditions and 6€. We first consider the case

6€ = m sin(J{ • r).

(2.85)

Just as in the previous section, we derive conditions under which a sinusoidal
grating implements a linear connection between two modes and we consider the
formation of interconnection networks based on superposition of grating interconnections. We are able to show that the same principles of Bragg matching
and rotational degeneracy as were derived in the previous section apply to the

volume case. In the final section of this chapter we consider methods for blocking
degenerate interconnections and thereby allow us to implement arbitrary linear
transformations on distributed fields.
The optical fields in a volume satisfy the Maxwell equations

...

ail

"vxE=-8t

. . an

"vxH= 8t

(2.86)

V-D=O
V-B=O
and the material equations

B=µi1
jj = (E + 6.t)E.

(2.87)

Using the identity
(2.88)
and assuming a monochromatic field at frequency w = k 0 / #, Eqs. (2.86) yield
(2.89)

Using Poisson's equation and assuming an isotropic medium, we find
(2.90)
To first order in the perturbation,
(2.91)
This term is often dropped from the coupled wave equation under the assumption
that the polarization of the field is perpendicular to the grating wavevector. In

most situations this geometry is desired to maximize the modulation depth of
recorded holograms. In photorefractive holograms, however, the polarization is
often parallel to the grating wavevector due to the need to adjust the geometry to
the available electro-optic coefficients. In this case this term cannot necessarily
be dropped.
Suppose that 6€ = m sin(.K • f'). Just as in the previous section, a read out
beam with wavevector kr is coupled to a diffracted beam with wavevector
(2.92)
For this coupling to occur with high efficiency, ks must lie on the normal surface.
When ks is on the normal surface we say that kr is Bragg matched to read out
the grating. We define the mismatch parameter
(2.93)
in order to include in our analysis the possibility that the read out beam is
misaligned. The coupled wave approach consists of assuming that
(2.94)
where er and es are unit vectors along the polarizations of the corresponding
plane waves. We can now solve for E(r) by substituting Eq. (2.94) in Eq. (2.89).
Substituting Eq. (2.94) into Eq. (2.91) we find

V(V · E) = -€-l

[j; R(z)er. J{[(kr + K)ei(kr+K),f' + (kr - K)ei(kr-K)•r]

S(z)es · I<[(ks + I<)ej(k.+K)•r + (ks - I<)ej(k.-K)•r]

+ c.c.]
(2.95)

The components of Eq. (2.95) which depend on (kr +I<) and (ks-K) correspond
to higher order disturbances. Since these wavevectors are unlikely to lie on the
normal surface, These terms can be ignored in the typical case. The term which
depends on ( kr - K) ~ k8 oscillates with the same spatial dependence as the
diffracted mode but is polarized orthogonally to the propagating mode and thus
cannot contribute to the diffracted field. Similarly the term which depends on

(ks + K) oscillates with the spatial dependence of the read out mode along a
polarization orthogonal to the polarization of the mode and thus cannot affect
the field in that mode. We know that these two terms are orthogonal to the
polarizations of the modes because they are polarized along the propagation
directions of the corresponding beams. In isotropic media the polarization of the
fields is orthogonal to the propagation direction and the terms in Eq. (2.95) can
be dropped from the wave equation.
Separating Fourier components of the remaining terms in Eq. (2.89) we find

(2.96)

and

(2.97)
If the amplitude of the perturbation is small compared to the dielectric constant

we may assume that the length scale for coupling between the two beams is much
greater than the optical wavelength. In this case the second order derivatives of
Rand Sare much smaller than k[r,s]Z times the first derivatives. The assumption
that this is the case is known as the "slowly varying envelope approximation." If

S(0) = 0, the solution to these equations under this approximation is

R(0)eie[cos( J v 2 + e2) -

R(z)][ S(z) - [

j sin( Jv

- r,;;:eieR(O)sin(v?+[i)
vc

+e2)]]
(2.98)

where

(2.99)
and
(2.100)
This result appears as Eq. ( 42) in [9].
As in the previous section, we assume that v is small enough that diffraction
remains in the linear regime. Linearity is assured if

Jv2 + e2) ~ 1.
Jv2 + e2

sin(

(2.101)

For small v, this condition implies

(2.102)

l{)j < 1rksz
Lz '

(2.103)

where Lz is the thickness of the hologram. Eq. (2.103) specifies the range of read
out angles over which phase matching is maintained. The analogous result for
cascaded thin holograms was given by Eq. (2.60).

)-------

Figure 2.11. Volume holographic interconnection system.
Consider the system of Fig. 2.6 with a volume hologram substituted for the
cascade of thin holograms. Such a system is sketched in Fig. 2.11. The system
is again addressed by light from a spatial light modulator in the x' -y' plane and
the diffracted signal is detected by a detector array in the x" -y" plane. The
parameters~, S, L, 0, and N' are all as defined in subsection 2.2.3. The optical
wavevectors in the readout and diffracted fields are exactly as described above,
as are the corresponding grating wavevectors.
In Eq. (2. 77) we presented a condition for a thin transparency cascade interconnection network which ensured that pixels in different columns of the same
row could not be diffracted by the same grating wfvevector. A similar condition
can be derived for a volume hologram interconnection network by requiring that
0 be large enough that light from a given input pixel cannot be diffracted by a

grating which diffracts light from an adjacent pixel in the same row. A read out

beam generated at a pixel in column (i' ± 1) is not diffracted by Ki'j'i"j" for all

j', i 11 and j" if lt9(i'+l)I > 1rksz/Lz, where

t9(i'+l) = ko - lk(i'+l)j' - Ki'j'i"j"I

(2.104)

= -2k 2 ~ sin20

We have discarded second order terms in/::;./ F. Thus, gratings corresponding to
different i' are distinct if
(2.105)
Eq. (2.105) is the volume holographic analog of Eq. (2. 77).
The distinction between local and distributed holograms which we developed
for the thin hologram cascade is not needed in considering continuous volume
holograms. A connection in a continuous volume hologram is made by a Fourier
component of the perturbation which forms the hologram. The Fourier expansion
of this perturbation can be written
N 1 -1

N 1 -1

N 1 -1
-2-

-2-

N 1 -1

N 1 -1
-2-

-2-

N 1 -1

N 1 -3

·,_
.,,_
.,,_
I ---2- I ---2- J ---2-

a.,i -N'-1
.,, .,, exp(K.,
-- , J
a -N'-1
--a.,, J.,, • rl
'J

(2.106)

N 1 -1

a.,I -N'2-1- ,.,,J.,, exp(J{I., -N'-1
.,, .,, • r1.
2-i J
' )

N 1 -1 ·11
N 1 -1 ·11
N 1 -1
I =--2- I =--2- J =--2·1

This expression is the continuous analog of Eq. (2.79). With the substitution
of 6f. for tn(P), the analysis of subsection 2.2.3 applies to a volume holographic
interconnection network exactly as to a cascaded thin hologram network. In
particular, Eq. (2.82) also describes the output signals for the volume system.
The condition under which a volume hologram implements the transformation

given by Eq. (2.84) is identical to Eq. (2.83). In section 2.4 we explore in more
detail the the nature of the transformations which can be implemented with both
the volume system and the thin hologram cascade.
We saw in subsection 2.2.2 that a local planar grating in a cascade of N
thin transparencies corresponds to N distinct distributed gratings. In a volume
hologram, the same rule of thumb holds true, except that the effective N can be
much larger because the resolution along z in a volume system approaches A, as
opposed to l in the cascade. As an example of an interconnection network based
on continuous media, assume that 8 = 3,\, D. = 3,\, Lz = L, F = 5L, ,\ = lµm
and L = 1 cm. The improvement in .6. over the cascade is possible because of
the higher resolution in z. Note that the parameter '"Y, which limited 0 in the
thin case, does not apply in the volume case. The range allowed for 0 using these
parameters is 28° > 0 > 12°. Under these conditions approximately 1011 distinct
gratings may be stored in the volume.

2.4

ARBITRARY LINEAR TRANSFORMATIONS IN VOLUME INTERCONNECTION

NETWORKS

By an arbitrary linear transformation we mean a transformation which maps
an input containing N1 degrees of freedom into an output containing N2 degrees
of freedom using N1N2 independent weights. Unfortunately, volume holography
cannot be used to implement arbitrary linear transformations between arbitrary
incident and diffracted fields. This is because N'

degrees of freedom, corre-

sponding in this case to N 14 distinct gratings, would be needed to implement an
arbitrary linear transformation between the N'

components of the input field

and the N' components of the output field. As we have seen, only o( N 3 ) distinct

gratings wavevectors connect the two field distributions. However, by limiting
the degrees of freedom in the input and output fields, it is possible to design systems which implement arbitrary linear transformations between the input and
output field distributions. In this section we discuss two approaches to designing
such systems, one method based on the "fractal sampling grids" and a second
method based on holograms between orthogonal spatial modes.
A linear transformation may be represented as a vector-matrix multiplication. The input and output vectors in the present case correspond to the input
and output fields. The amplitude of each Fourier component of the field is represented by a corresponding vector component. The matrix components represent
the grating amplitudes. As we have seen, if the input and output fields are unconstrained, the components of the grating matrix which transforms them are
highly correlated. Assuming that the amplitude of each of the gratings is an
independent variable, a volume holographic transformation may be arbitrary if
each grating is used to form exactly one connection between an input and an
output component.
A sampling grid is an overlay which selects active pixels from the input or output planes. A transformation which the holograpr...ic system implements between
the active pixels is arbitrary if none of the grating components of the hologram
implements a connection between two different pairs of pixels. In designing a
sampling grid, the goal is to select a maximal number of pixels for a given ratio
of input nodes to output nodes such that an arbitrary linear transformation is
still possible. The number of components in the interconnection matrix is equal
to the product of the number of input and output points used. Based on the
constraint on the scale of the matrix mentioned above, we will be satisfied with

a sampling grid if this product is o(N' ). A simple example of a sampling grid
which meets this criterion is afforded by Fig. 2.9. If only the top and bottom
rows of the input plane are used for reading out the hologram and if the bottom row of the ou'tput plane is not used, each grating is used to form exactly
one connection. The input vector is of dimension 2N'. The output vector is
of dimension N'(N' - 1). The interconnection matrix consists of 2N'\N' - 1)

independent components. This number is N' less than the maximu~ because
the N' 2 gratings which connect the top row of the input to the bottom row of
the output must be discarded due to the fact that the gratings which connect
the top row of the input to the top row of the output also connect the bottom
row of the input to the bottom row of the output.
In many applications it is desirable for the dimension of the input and output
vectors to be nearly the same. This is especially true if. transformations are to
be cascaded, as in multilayer neural networks. A simple grid derived from the
asymmetric grid of the previous paragraph which balances the number of pixels
on the input and output planes is shown in Fig. 2.12. The input plane consists

of ..,/Ni' rows of N' active points. The rows are spaced by ..,/Ni' inactive rows.
The output plane also consists of ..,/Ni' rows of N' active points. The rows are
grouped together in the center of the plane. The number of active points on the

input plane and on the output plane is N' 2 • The interconnection matrix consists

of N' independent components.
The "fractal" nature of sampling grids arises from the fact that a grid appropriate for a value of N' may be used to generate a grid for a larger value of N'.
This is done by substituting a grid which meets the design criteria onto its own

INPUT SAMPLING GRID

OUTPUT SAMPLING GRID

...

......

........
..
....

........

....

-:i
......

Figure 2.12. Symmetric sampling grid. N' = 64.

active nodes. It c~n be shown that this new grid, of dimension N' also meets the

design criteria. This means that given one grid which allows o( N' ) connections,
a suitable grid may be found for larger scale systems. The grids which are generated display the "self similar" property of fractal patterns. Many variations of
suitable fractal sampling grids exist. In particular, grids which uniformly sample
the coupled planes and grids with variable ratios of input and output points have
been derived. We define the dimension of the input plane to be

(2.107)

and the dimension of the output plane to be
d _ logN2

2 - logN''

(2.108)

where N1 and N2 are the number of active points on the input and output sampling grids, respectively. It can be shown that sampling grids can be constructed
with any desired dimensions subject to the following constraints [79):

(2.109)

Thorough reviews of the use of fractal sampling grids are presented in [85, 79,

70).
When a sampling grid is used to mask the output detector array, some of the
energy diffracted from the active inputs will be diffracted onto unused portions
of the output array and will be wasted. We can explore the impact of this fact

Figure 2.13(a). Example of input plane of a fractal sampling grid.

Figure 2.13(b). Example of output plane of a fractal sampling grid.

on the overall energy efficiency of the system as a function of the dimensions of
the input and output grids using sampling grids of the sort sketched in Fig. 2.13.
These grids consist of rows of N' input pixels spaced in x by s1 - 1 unused rows
and rows of N' output pixels separated by s2 - 1 unused rows. The number of

active pixels on the input grid is N1 = N' / s1. The input dimension is
logs1
d1 = 2- logN'.

The number of active pixels on the output grid is N2

(2.110)

N' / s2. The output

dimension is
logs2
d2 = 2- logN'.

Figure 2.14. Geometry of sampling grid.

(2.111)

As sketched in Fig. 2.14, they component of the grating wavevector between
• 1.m the J·1th ac t·1ve mpu
t row an d the i·11th p1xe
l.mt h e J·11th active
the i·1th p1xe
output

row 1s

ko.6.
[K~i'j'i"j" ]y = (J., s1 - J.,, s2 ) F

(2.112)

Letting Q = F[I

Q = J·I s1 - J·II s2.

(2.113)

Given integers Q, s1 and s2, a simple theorem in number theory [111] tells us
that there exist integers j' and j" which satisfy Eq. (2.113) if and only if d
divides Q with no remainder, where dis the greatest common divisor of s 1 and
s2.

Assuming that this condition is satisfied, the theorem further states that all

solutions for j' and j 11 are described by
•/

·I

J =Jo+

(S2)
-;JP

+ (S1)
J =JO
d P,
•II

(2.114)

•fl

where (j' 0 , ji) is a solution and p is an arbitrary integer. If

(2.115)

then at most one of these solutions is such that j' and j" are both between

-(N' - 1)/2 and (N' - 1)/2. This means that if Eq. (2.115) is satisfied the
th
. b etween t h e J·1th mput
• umque,
separat 10n
row an d the J'" output row 1s
i.e.,
no

two different pairs of active input and output rows are separated by the same
distance in y. Since we have assumed sin 8 large enough that a grating cannot be

phase matched to more than one input in a row, a grating cannot couple more
than one active input-output pair if Eq. (2.115) is satisfied.
The number of independent connections which can be made between the
active inputs and ~utputs on these grids is

(2.116)
where we have substituted from Eq. (2.115). For the grids shown in Fig. 2.13
s1

= 10, d1 = 1.53, s2 = 13, d1 = 1.47, d = l, N' = 130, and S ~ 2.2 x 106 . As a

second example, grids with s1 = 26, d1 = 1.33, s2 = 5, d2 = 1.67, and d = l are
shown in Fig. 2.15. N' and Sare the same as in Fig. 2.13.
We now turn our attention to calculating the ratio of the total power diffracted
to active pixels on the output grid to the total power diffracted to inactive pix-

els. Suppose that the recorded hologram is described by the matrix W. Since we

are using fractal grids to break all the degeneracies, every component of W may
be assumed to be independent. Diffraction from the input plane to the output

plane may be regarded as the multiplication of an input vector IR) by W. The
amplitudes of the diffracted modes can be represented by the vector IS), where

IS)= WIR).

(2.117)

Each component of the vectors IR) and IS) corresponds to the field at a specific

pixel. Using singular value decomposition W can be expressed

w= L /3q ISq) (Rql'

(2.118)

q=l

where the vectors !Sq) are N2 dimensional and orthonormal, the vectors IRq) are

N1 dimensional and orthonormal, /3q is a constant and r is the rank of W. An

·················· ............................................................ , ................................................... .

Figure 2.15(a). Example of input plane of a fractal sampling grid.

Figure 2.15(b). Example of output plane of a fractal sampling grid.

arbitrary input vector IR) can be decomposed in basis vectors which include the
vectors used in the singular value decomposition as a subset.
N1

IR) =

O'.p IRP) .

(2.119)

p=l

IR 1::

The orthonormal vectors Rr+ 1 ) .. ·

span the null space of

W. The field

on the active output pixels can similarly be expressed

IS) =

L aq/3q ISq).

(2.120)

q=l

The power diffracted onto the active pixels of the output array is

Po= (SIS)=

L laql 1/3ql

(2.121)

q=l

Each distinct grating stored in the hologram corresponds to a component of
the matrix

W. In addition to diffracting light from a pixel in the j' th active input

row to a pixel in the j''

active output row, a grating between these rows also

diffracts light from one active pixel in the other N' / s1 -1 active rows to an unused
output pixel. The fields diffracted onto the unused pixels add incoherently. The
total power diffracted onto the unused output pixels is

(2.122)

where IRi~j' 12 is the mean over i' and j' of the activity at the i' j'

input pixel

and w(i'j')(i"j") is the strength of the connection between the (i'j 1 )1h input pixel

and the (i"j 11 )1h output pixel. Assuming that (RIR) = 1, IRi~j'l 2 = sifN' .

Substituting from Eq. (2.118) and using the orthonormality of ISq) and IRq), we
find that
(2.123)
•1

·11

i ,J ,s

'II

Thus,
(2.124)

The ratio of the power diffracted onto the active pixels of the output to power
diffracted onto unused pixels is

Eq laql 2 l,Bql 2 N' 2 E;=l laql 2
Pu :=::! (N' - s1) Eq l,Bql 2 :=::! (N' - s1)r '
Po

(2.125)

where we assume that l,Bql is approximately constant over q. If the input field is

completely random then (laql 2 ) = sif N' and

(2.126)

where we have assumed that s1s2

3. In terms of the

dimension of the output grid, d2,

Pu = (N'2-d2 - 1).
Po

(2.127)

For random inputs, P0 / Pu is less than or equal to 1 for all s2 except s2 = 1, which
corresponds to the entire output plane being active. A plot of P0 / Pu verses d2 is
shown in Fig. 2.16.

102 -----~---,----.-------.----.---r-----.--.----7

101

100

10-1

_..... -

10' ~---------

10-3

L ·• 1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

Figure 2.16.
values of N 1•

fu- verses d2 for a random input signal. Plots are shown for various

In the applications described later in this thesis, IR) is typically close to a
specific IRq), in which case laql is near one and
N'2

-----(N' - s1)r

(2.128)

r must be less than or equal to the lesser of N1 and N2. Generally, it is desirable

to make r as large as possible, since r corresponds to the number of different input
vectors which yield a strong output. It is also desirable to keep P0 / Pu as large
as possible, since the diffraction efficiency from the active inputs to the active
outputs cannot exceed this ratio. If we assume that r is as large as possible, i.e.,
r = min(N1, N2), then Po/ Pu decreases monotonically as d1 increases. Assuming

Pa
Pu = !ft - l = N 12 -d2 - 1

(2.129)

If we want to increase the number of degrees of freedom in the input we must

increase d1. Particular significance may be associated with the input dimension
for which the total power diffracted onto the active points for an eigenvector
input is equal to the total power diffracted to unused output pixels. The value
of d1 for which this is the case is
log2
d1p = 1 + logN'

(2.130)

A plot of dip verses N' is shown in Fig. 2.17.
Sampling grids are not a unique solution to the problem of removing degeneracies from volume holographic interconnection systems. Other sets of spatial

1.5
1.45

1.4
1.35
1.3

-0

1.25
1.2
1.15

1.1

1.05

100

200

300

400

500
N'

Figure 2.17. d 1 P verses log N' .

600

700

800

900

1000

modes, such as the spherical waves suggested in [56], could also yield independent
connections. The number of independent connections cannot, however, exceed
the bounds derived in this chapter. This is because any connection between
modes may be described by a linear superposition of distinct gratings. While the
question of whether or not other sets of modes can have greater energy efficiency
than fractal sampling grids has not been completely resolved, whenever the input
is multimode degeneracies will cause some fraction of the read out energy to be
lost to inactive areas of the output plane if such areas exist.

3. FORMATION AND CONTROL
OF VOLUME HOLOGRAMS
3.1

INTRODUCTION

In this chapter we consider techniques for controlling the physical parameters
of 3-D systems using light. The principle difficulty encountered in the control
of such systems is that there is no simple means of reaching into a volume to
specify the parameters of a single volume element, or voxel, without affecting the
parameters of other voxels. Methods which can be used to overcome this problem
include layered growth using lithographic and epitaxial techniques and critical
control techniques in which control signals pass through the entire volume but
effect only a single voxel. Critical control techniques are especially appropriate
to specify parameters using multiphoton processes [112, 113], in which case the
control signals may consist of beams at different wavelengths which cross only in
one voxel or focusing beams which exceed critical intensity only in one voxel (114].
In this chapter we discuss a third approach based on holographic techniques. The
principle difference between holographic control and other techniques is that in
holographic systems there is no one-to-one relationship between the state of a
specific control signal and the state of a specific voxel. The control signals in an
optical holographic system correspond to fields on the boundary of the system.
The state of a given voxel is a function of the fields generated in it over time.
The relationship between the final states of the voxels and the control signals
is specified by a global transformation in space and time. To drive the voxels
to some desired state, this transformation must be inverted to find appropriate
control signals. The advantages of this approach are that it is much easier to

implement than layered growth and that it is much less material dependent than
critical control.
We define the number of degrees of freedom in a system to be the number
of independent variables which are needed to specify its state. If we set one
parameter at each point in the volume, the number of degrees of freedom stored
in a volume hologram is equal to the number of voxels. Obviously, a set of control
signals cannot independently specify more degrees of freedom in a system then
are present in the control signals themselves. Volume holographic control is based
on specifying fields on the boundaries of the volume. These fields contain one
degree of freedom for each resolution element, or pixel, on the boundary. In a
monochromatic optical system, the resolution of the signals used to control and
evaluate the state of a volume is lower bounded by .A. The number of pixels with
independently specifiable states on the aperture of such a system is o(A/ .X 2 ),
where A is the aperture of the system and .A is the wavelength. The number of
voxels which can be independently determined in such a system is o(V/ .X 3 ) [l, 115,
85). However, these voxels cannot be independently specified unless the number
of degrees of freedom in the control signals is raised to V / .A 3 • This can be done
by changing the boundary conditions in time or by using multiple wavelengths
in the control signals. In discrete steps, at least V / .XA independent exposures or
wavelengths are needed to control the volume. The case .XA > V corresponds to
thin holograms, which can be specified in one exposure.

In the following three sections of this chapter we consider the control of
volume holograms using multiple exposures. The second section considers the
formation of arbitrary linear holograms in the cascade of thin transparencies,
where by "arbitrary", we mean that all the degrees of freedom of the holographic

system are independently specified, and by "linear" we mean that the transmittance of each transparency is proportional to the energy distribution which
passed through it during the recording process. We assume that the system is
controlled by light incident through the first transparency. This system is analyzed because it allows us to illustrate from an almost intuitive level both why
multiple exposures are needed and how such exposures can be used to created
an arbitrary hologram. The results of this section are extended to continuous
volume holograms in linear media in the third section. Our goals in the third
section are to describe more precisely what we mean by "arbitrary" and "linear"
in the context of continuous media and to show how arbitrary holograms can be
recorded in such media. In the fourth section we consider saturable holographic
media. Since such media respond linearly only over a limited range of exposures,
the dynamic range of the holograms which can be stored in them is limited. The
principle result of this section is that the diffraction efficiency of a hologram stored
in a saturable medium is inversely proportional to R 2 , where R is the rank of the
hologram, i.e., the minimum number of exposures needed to record it. The final
section of this chapter considers the control of holograms using polychromatic
signals. Polychromatic control allows a hologram to be made in one exposure.
The diffraction efficiency of a polychromatically controlled hologram could, in
principle, be as high as R- 1 . However, the difficultly of generating appropriate
polychromatic control signals makes this potential difficult to realize.

3.2

CONTROL OF A CASCADE OF THIN HOLOGRAMS

In thin holography, a hologram is simply a faithful recording of a spatially
modulated intensity pattern. In a cascaded system, once the recording image on
the first transparency is known, the image on each successive transparency can
be determined using a propagation operator. This means that if we know the
information being stored in one hologram, we know the information being stored
in all the holograms. To store independent information in each hologram, it is
necessary to vary the image over several exposures.
As an example of this process, suppose that we wish to record a grating with
wavevector I< in each of N cascaded holograms. The amplitude and phase of the
grating are to be independently set at each hologram. We record the grating in
a series of N' exposures using the interference pattern between the plane waves

(3.1)
and

(3.2)
to record a grating in each hologram in the n'

exposure. Since we are limiting

ourselves to recording gratings with wavevector I{, we require that the component

of km' - ksn' in the x-y plane be equal to K for all n. In propagating from one
hologram to the next the phases of Rn' and Sn' change by [krn,]zl and [ksn']zl,
respectively. Thus, the spatially modulated terms in the intensity pattern on the

n th transparency during the n'

exposure are

(3.3)

and t;n,, where Kn' = [km' - ksn']z. As defined above, by "linear" medium we
mean a holographic material in which the transmittance of each transparency is
proportional to the energy distribution used to record it. If this is the case, the
spatially modulated part of the transmittance of the n th transparency after N'
exposures is the real part of
N'

tn(P) = Linn'•

(3.4)

n'=I

Defining the N-dimensional vector H with tn(P) as its n th component, Eq. (3.4)
can be expressed

s1e-i1

ei2K1l

ei2K2l

ei2K3l

ei2KN,l

s2e-i 1h

ei3K1l

ei3K2l

ej3K3l

ej3KN,l

s3e-i3

ei(N-I)K1l

ei(N-I)K2l

ei(N-I)K3l

ei(N-I)KN,l

SN,e~jN 1 )

(3.5)
Suppose that we are given H.

We can invert Eq.(3.5) to determine the

is nonsingular. We saw in section 2.2.2 that N distinct values of Kn' exist. If we
assume, as in that section, that Kn,l = 2(n' - l)'rr/N then Eq. (3.5) becomes

....

·2,r

·4,r

-2(N-l)rr

eJ 7l

eJ7f

·3,r

·6,r

eJ7f

•(N-1),r

eJ7f

·2(N-l),r

s1e-i1

s2e-i2

-3(N-1)1r

s3e-i 3

eJ
eJ

•(N-1) 2 ,r

(3.6)

SN,e-iN,

This matrix in this equation is of the Vandermonde form and is known to be
nonsingular [116]. This equation can thus be used to derive an exposure scheme

to store an arbitrary ii. Note that we have assumed that the number of inputoutput beam pairs separated by K at the input aperture of the cascade is greater
than or equal to the number of layer, N. If N exceeds the range of N' then it
would be impossible to make enough distinct exposures through the aperture to
control all N plates. In the remainder of this chapter we assume that N = N'.

//
INPUT

PLANE

TRAINI~

PLANE

Ill

y\
~ PLANE

OUTPUT

Figure 3.1. Control system for a thin hologram cascade.

Inverting the matrix in Eq. (3.6) provides us with a technique for devising a set of N exposures which will form an arbitrary set of gratings with the
wavevector l{ in the cascaded system. There are of order N 2 possible vectors

K which we can form in each thin hologram. The gratings corresponding to
each of these wavevectors can be recorded in parallel in the system of Fig. 2.6,
which is repeated as Fig. 3.1. Within the spatial bandwidth of the distributed

holograms, this approach allows us to arbitrarily specify the transmittances of
the transparencies in the cascade in o( N) steps. This is equivalent to arbitrarily
specifying the amplitudes and phases of the 2N 3 - N 2 distributed holographic
gratings which can be stored in this system. Fig. 3.1 is the same as Fig. 2.6 except that a training plane has been added to generate the signals incident on the
output plane. The interference pattern between signals generated on the input
plane and signals generated on the training plane creates a hologram on each of
the cascaded transparencies.
Suppose that only the top and bottom rows of pixels on the input plane
are active, as in Fig. 2.9.

Since each distributed hologram couples only one

input in this case, it is trivially seen that the amplitudes and phases of the
distributed holograms can be specified in 2N exposures. This is done by recording
in sequence the connections from each input to all the outputs to which it is
connected. During an exposure, a single input pixel is active and the training
pixels corresponding to the outputs to which it is connected are active with
amplitudes and phases proportional to the amplitudes and phases desired in the
associated connections. The distributed holograms cannot be recorded in fewer
than 2N exposures because the rank of the transformation being recorded is

2N. Since the rank of the transformation cannot be lowered in this case without
sacrificing some of the distributed holograms, this is the minimal number of
exposures needed to specify all the distributed holograms between the input and
the output planes.

3.3

FORMATION OF VOLUME HOLOGRAMS IN LINEAR MEDIA

Jiu:;
tJ

INPU~

PLA~

)---

~ PLANE

OUTPUT

TRAININ~
PLANE

Figure 3.2. Volume holographic interconnection system

A hologram in a continuous medium is a perturbation, A( r) to the optical
properties of the volume. A medium is "linear" if a hologram can be formed in
it in which is proportional to the exposure. In terms of the recording intensity,

I(r, t), A(r) in a linear medium is
(3.7)
where A 0 and E 0 are constants. In this section we consider once again the system
of Fig. 2.10, which is repeated as Fig. 3.2, and we argue that a perturbation A( r)

which contains a maximal number independently specified connections can be
linearly recorded in o( N) exposures in this system.
As explained in chapter 2, the field in the volume of the hologram is

-+ t)
~ of, (t) eJ°k' I 3 /•T +L....t'l-'i"j"
~ ,/..
(t) eJ°k-11
U( r,
=L....t'Pi'j'
' 3·11•T•
i'j'
i" i"

(3.8)

This field creates an intensity pattern in the volume which is proportional to

IU(r, t)l 2 . Assuming a linear medium, a hologram forms which is proportional to
the temporal integral of this intensity pattern. We are primarily interested in the
components of the hologram which couple beams from SLM1 onto beams from
SLM2 and vice versa. Thus we will ignore the low spatial frequency components

of IU( r, t)1 2 and consider the recorded perturbation to be

(3.9)

The maximum number of independent Fourier components in A( r) corresponds to
the number of distinct grating wavevectors appearing in Eq. (3.9). The wavevector of the grating formed between light from the i'j'

input pixel and the i 11 j 11th

output pixel is

(3.10)

+[(,Bi'j' - 1i"j") cos 0 + (Pi" + Ui') sin 0]2.
where Ui' = 2i 17r/L, Vj', Pi" and qp, are similarly defined, ,Bi'j' = k-u;,/2k-vJ,/2k
and ,i"j" = k- pf,,/2k-q},,/2k. In a well designed optical system, i.e., a system
where the input aperture is somewhat less than the focal length, the quadratic

terms do not contribute significantly to the magnitude of Ki'j'i"j"• In this case,

Let G correspond to the number of distinct values Ki'j'i"j" can assume. By the
number of "distinct values" of Ki'j'i"j" we mean the number of points separated
by at least 1r/L in the range of I{i'j'i"j"• Ui', Vj', Pi", and qj'' assume N = L/~ =

L 2 /(>-.F) values between -1r(N - 1)/L and 1r(N - 1)/L, where we assume that
~ is equal to the resolution of the optical system, >-.F/L.

Vj' - qj" can assume 2N -

For given Ui' and Pi",

1 values. Thus, G is equal to 2N - 1 times S, the

number of distinct values that (Ui' - Pi" )cos0x + (Ui' +Pi") sin Bz can assume.
Letting

e= (Ui' - Pi" )cos0 and ( = (Ui' +Pi") sin 0, we can estimate Sas
(3.12)

Substituting~ into Eq. (2.90), we find sin20 > 1/4 and G > (2N -l)(N -1) 2 ~

2N 3 •
In order to arbitrarily specify the connections between the inputs and the
outputs in the system of Fig. 3.2, the discrepancy between the number of distinct
gratings, 2N 3 , and the number of degrees of freedom in a single exposure, 2N 2 ,
must be recovered by varying the recording fields in time. The simplest method
for doing this is to use discrete exposures, in which case,

(3.13)

where tn is the exposure time for the n th exposure and Ne is the total number of
exposures. The 2N 3 distinct gratings in the system of Fig. 3.2 may be controlled

m 2N exposures usmg a training plane and the top and bottom rows of the
input plane, as described in the previous section. However, if the active pixels
of the input and output planes are constrained to lie on sampling grids, it is
difficult to produce holograms with coherent fan-in if the hologram is controlled
by pixels which will be inactive during reconstruction. We saw in chapter 2 that
a hologram which interconnects appropriate sampling grids may be represented

by a matrix, W. W can be represented as the sum of Router-products

W =

L /'n 1¢>n) (VJnl,

(3.14)

n=l

R is the rank of the matrix. We can record the hologram corresponding to Win R
exposures by setting ¢>i•j1(n) and VJi"j"(n) equal to the corresponding components

In this section we have shown that holograms can be recorded in linear media
which utilize almost all the degrees of freedom of the volume. The number of
exposures needed to form such holograms is equal to the rank of the transformation implemented. In the following section and in chapter 4 we consider the
costs that the need to make multiple exposures imposes on the dynamic range of
holograms recorded in real media.

3.4

VOLUME HOLOGRAMS IN SATURABLE MEDIA

Real holographic media perform linearly only over a limited range of exposures. The disadvantage of using multiple exposures to record a hologram in
such materials is that the dynamic range of the information bearing component
of the holographic perturbation is limited by the number of exposures. In this

section we consider the simplest form of real holographic media, i.e., media which
respond linearly up to a saturation exposure. Photographic film belongs to this
class of materials, at least in the sense that it can often be assumed that film
responds linearly up to a certain exposure Emax. Information is recorded in saturable media by converting some specially prepared light sensitive resource to an
insensitive form. In film the resource is silver halide grains, which are converted
into metallic silver. Once the exposure exceeds that for which almost all the light
sensitive material is converted, further exposure does not affect the medium.
Consider a hologram recorded in a saturable medium using Ne exposures.
The recording architecture is assumed to be that of Fig. 3.2. The field during
the n th exposure is

L VJ(n).,.,ejk,,

U (n)( rr) -_

I J

3 ,.f'

+ L jk,11 11 .f'
.,,.,,e

i'j'

(3.15)

i"j"

The corresponding intensity is

i'j' p';4•'

i'j'

q';4j'

(n)VJ •, •, .,, .,, e
+LL
i' j' i 11 j 11
i'j' i 11 j 11
IJ

•, •,

.,,

(3.16)
We wish to record a hologram which is a linear superposition of the intensity
patterns in all the exposures. This is the case in a saturable material is the
exposure time is constant over all exposures and the total exposure is less than

the saturation value. In this case,

Aoto ~ ["'\:""" "'\:""" •1.(n) -+-(n) * j(k;, J ,-k;11J 11),r
A( r;;'I_
1 - - - L- L- L- Eo n= 1 l·,J·, I·,,J·,, ' J p q

(3.17)

where t 0 is the exposure time.
To ensure that the hologram is recorded linearly, we require that
Ne

L J(n)(r) < Emax·

(3.18)

n=l

at almost all r. If the Fourier components of the recording field are not correlated,
then both the mean and the standard deviation of the intensity are given by
(3.19)
i'j'

i" j"

Statistics of this sort are well known in speckle patterns [11 7]. Since [( n) ( r) is
very unlikely to exceed a few times its standard deviation, we can ensure that
hologram formation occurs linearly by choosing t O such that

(3.20)
where a is a number between zero and one. Substituting into Eq. (3.17) yields

(3.21)
·,·,
1111e -j(k,,, ,-k,11, 11)•r]
ip (n)*+LL
i'j' i 11 j 11
t J

p q

where Amax = (AoEmax/ E 0 ). Amax is the maximum amplitude of the perturbation which can be recorded linearly.

Our goal in this section is to discover how the effective amplitude of A( r)
scales with the rank of the hologram being recorded. Since we have seen that the
sampling grids of Fig. 2.9 can be used to record a hologram in a minimal number
of exposures, we can assume without loss of generality that sampling grids are
used to record the hologram of Eq. (3.21). In this case, the transformation implemented by the hologram can be expressed as a sum of outer-products. Writing
Eq. (3.21) in this form yields the transformation matrix

(3.22)

The choice of the vectors ln) and 11/in) used to record Wis not unique. However,
as reflected by the sum over the mean recording intensities in the denominator
of Eq. (3.22), the choice of the recording vectors does affect the amplitude of the
recorded hologram. In general, we would like to use the recording vectors which
yield the maximal amplitude in the recorded hologram.
Orthonormal bases {lun)} and {lvn)} can be found in the input and output
spaces, respectively, by singular value decomposition such that
R'

w= r 2: f3n lun) (vnl,

(3.23)

n=l

where R is the rank of W, f3n is a positive real number, where r is a positive real
scaling factor [118]. The inclusion of r in this expression allows us to impose the
normalization condition

I: f3n = l.

(3.24)

n=l

An arbitrary set of recording vectors ln) and 11/in) can be expressed in terms of

100

sum over the bases vectors lun) and lvn), respectively, as

lef>n) =

L Snn' lun,)
n'

j?j,n) =

(3.25)

L Tnn' lvn')
n'

Substituting in Eq. (3.22) yields

(3.26)

For specific vectors in the orthonormal recording bases jup) and lvp), Eq. (3.23)
yields
(3.27)
According to Eq. (3.26) ·

(3.28)

Equating these results we find

(3.29)

Summing over p, Eq. (3.29) becomes

(3.30)

To maximize the amplitude of the recorded hologram, we pick the coefficients
Tmm'

and Smm' which maximize the scaling factor, r.

This maximization is

101

achieved by comparing terms in the numerator of the right-hand side of Eq. (3.30)
with terms in the same parameters in the denominator. If we maximize the ratios
of terms in the numerator with their corresponding terms in the denominator,
then the value of the overall fraction is maximized. Thus we wish to choose values
of rmm' and Smm' which maximize the fraction

From the triangle inequality we know that the maximum of this ratio occurs when

= Smm'· We also know that a solution for suitable recording vectors with
coefficients rmm' and Smm' which satisfy this condition exists, namely rmm' =
rmm'

Smm'

= bmm'• Assuming rmm' = 8mm',
f = Wo.

(3.31)

Thus, when the hologram is recorded with the maximal strength in its components,
R'

W= ~o ~ f3n lun) (vnl.

(3.32)

n=l

This transformation can be recorded between the vectors lun) and lvn) if the
recording time of the n th exposure is proportional to f3n. When the outer-products
between the basis vectors are all stored with equal weight, the normalization
condition yields f3n = 1/ Rand
R'

W= ;; L lun) (vnl •

(3.33)

n=l

The diffraction efficiency, i.e., the ratio of the diffracted intensity to the
undiffracted intensity, when a pattern li,L,) on the input plane reconstructs the

102

hologram is

(1/JI wrw 11/J)
T/1/J =
(1/Jl1/J)

(3.34)

Assuming that 11/J) has no component in the nul space of W, 11/J) can be decomposed as 11/J) = 'I:,~ rn lvn) so that

(3.35)

r, is maximal when rm = 1, where f3m is the largest singular value of W. Our

purpose in using a volume rather than a thin medium is to store as many different
associations as possible, we are not typically interested in holograms where one
singular value is much larger than the others.

If all the singular values are

approximately equal, then the maximum diffraction efficiency is achieved when
any basis vector is used to reconstruct the hologram. When lvn) is active on the
input plane the diffraction efficiency is

T/n

'f/max

= T/max /3n ~ W°'

(3.36)

where we have again applied the normalization condition on f3nIn summary, arbitrary volume holograms can be recorded linearly in saturable
materials by dividing the dynamic range of the material equally between the exposures needed to record the hologram. The hologram is recorded with maximal
strength if the orthogonal basis vectors generated by singular value decomposition are used to record it. Assuming that each exposure is weighted equally, the
maximum diffraction efficiency of the hologram falls with the square of the rank

103

of the transformation recorded, R. If our goal is to store associations between orthogonal patterns, this means that the diffraction efficiency when R associations
are recorded is R 2 times less than the diffraction efficiency when 1 association is
recorded. Because various techniques, such as placing the hologram in an cavity,
can be used to improve the diffraction efficiency of a hologram, the fact that the
amplitude of the perturbation falls with R is more significant in the sense that
it limits the dynamic range, and thus the storage capacity, of the hologram. In
chapter 4 we show that a similar loss of dynamic range tinder multiple exposure
occurs in photorefractive materials, which are not saturable in the sense described
above. The rest of this chapter is devoted to considering holograms formed using
multiple independent wavelengths instead of multiple exposures.

3.5

CONTROL OF VOLUME HOLOGRAMS WITH POLYCHROMATIC SIGNALS

As an alternative to multiple exposure, a hologram may be controlled in a
single exposure using polychromatic light. Since the number of degrees of freedom in a single wavelength component of a polychromatic field cannot exceed C,
as derived above, at least R = G / C distinct wavelengths are needed to control
an arbitrary hologram. In this section we show that a hologram recorded in Ne
exposures can be recorded in one exposure using signals carried independently on

Ne wavelengths. Polychromatic control is potentially superior to monochromatic
multiply-exposed control because coherence between different wavelength components could yield greater modulation depth and thus better dynamic range in the
perturbation. Unfortunately, the difficulty of simultaneously and independently
specifying signals in many wavelengths may outweigh the potential advantages
of polychromatic control in practical systems.

104

INPUT

>---

~ PLANE

OUTPUT

TRAINING
HOLOGRAM

Figure 3.3. Polychromatically controlled volume interconnect system.
Consider polychromatic control of the system of Fig. 3.2.

The SLMs of

Fig. 3.2 must be replaced by devices capable of separately controlling each wavelength component of the control beams. One means of constructing such devices
is to use Ne distinct SLMs, each illuminated by a different wavelength, to generate the input and training signals. The input and training planes would consist
of superimposed images of the signals on all the corresponding SLMs. A second
approach would be to replace the input and training SLMs with volume holograms which store an appropriate image at each wavelength. Each hologram is
simultaneously reconstructed by Ne beams with different wavelengths. The reconstructed fields of different wavelengths must, of course, be independent. A

105

sketch of a system incorporating such holograms is shown in Fig. 3.3. Supposing
for the moment that one of these methods can be used to generate appropriate
images, we assume that a field of the form given in Eq. (3.15) is generated in the
volume at each wavelength. The total field in the volume is

U(r, t) =

t [L
n=l

,f,;,;J•)ej(ki/"'-,-w<•>t) +

L ,f,;"j"(n),i(k,,;;,,,.,_,_J•>t)] '
i" j"

i' j'

(3.37)
.... (n)

where ki'j'

( )

. . th

and w n are the wavevector of the i 1J 1

component and the fre-

quency for the n th wavelength. The perturbation that forms in the volume is

A(r) = ~:

j IU(r, t)l dt.

(3.38)

If T ~ ( w(n) - w(n')

)-l for all n' f. n, then interference patterns due to each

wavelength add incoherently. In this case

(3.39)

where J(n)(r) is exactly as written in Eq. (3.16) with the proviso that n now
corresponds to the n th wavelength rather than the n th exposure. Assuming that
the statistics for the field amplitudes are the same as in the previous section, the
requirement that the exposure remain less than Emax forces T to be less than or
equal to the value given in Eq. (3.18). This means that hologram formation using

Ne independently controlled mutually incoherent wavelengths yields exactly the
perturbation described by Eq. (3.21) and that, in particular, the dependence of
the modulation terms in A( r) on Ne is exactly the same as for control using
multiple exposures.

106

The modulation depth of the signal terms in A( r) might be increased if the
integration time T were short enough that modulation terms could form between
different wavelengths. Suppose that w(m) = w0 + mw, where w0 ~ w. In this
case,
AoT ~ ~ - -wT(m-m') 2 sin ( wT(17;-m'))

A(f') =-E
~ ~ e J

w T( m-m')

m=l m'=l

') * .

~ ~ .t•.('"!' ) .t. ( m
[ ~ ~ 'l-'z'J' 'f'p'q'
i'j' p'q'

- (m)

eJ(k;'j'

-kp'q'

)•r

(m )

+ L L

'II 'II

'}*

p q

(3.40)

illjll pllqll

+ LL

'Ip •(·,mJ·,)

+L L

'Ip .,m·,

i' j' i 11 j 11

(m)

-k;u , ·11

)•r

-J

i' j' i" j 11

(m )

} -J(k;,
. - ., (m ) -k;11- •11 (m) )•r
( '}* (.,,m.,,e

i J

' J

where we have discarded integrals over e±2iwaT under the assumption that w 0 T ~
1. If NewT < 1 then
. (wT(m-m'))

. wT( m-m') 2 Sln

e-J

wT(m - m')

~ 1

(3.41)

for all m', m. To ensure that each wavelength represents an independent recording
channel we must require that lk},7,) - f~:;,') I ~ 21r / L. Assuming that the propagation directions are nearly colinear, this is equivalent to assuming IlkJ,7,)1-lkJ,7,') 11 ~

21r/L. Since lkf,7,)1 = (w 0 +mw)/c, this constraint implies that w ~ 21rc/L.
To find a set of mode amplitudes which can be used to record a given hologram, we set the terms in each grating wavevector of Eq. (3.40) equal to the
corresponding terms in the Fourier representation of the desired value of A( r)

107

and solve for the mode amplitudes, i.e., we generate a set of equations constraining the mode amplitudes by setting

L ip~,7)<1>*(k~;'J) - Kg)
k;;'l/
equal to the coefficient of Fourier component of A(r) with wavevector Kg, By

*(k~;'J) - Rg) we mean the coefficient of the output mode propagating with
wavevector k},j,)-i?_g, The difference between monochromatic and polychromatic
recording with cross-wavelength coherence is that in the coherent polychromatic
case,

"""'(m)

....

can be nonzero even when ki'j' - Kg does not lie on the same normal surface as

k};'J). In this sense, an effective thickness is added to the normal surface and the
control space becomes three-dimensional.
The three-dimensional nature of the control space with coherent polychromatic control allows us to record an arbitrary holographic transformation using
a single reference beam. A grating with wavevector l{9 is controlled by the interference pattern formed by the reference and signal field separated from the
reference by J{g in the three-dimensional control space. Since the amplitude and
phase of the reference are fixed, the relative phase and amplitude of the grating
correspond to the phase and amplitude of the signal field. The modulation depth
for gratings recorded by this approach is maximum when the total intensity in
all the signal fields is equal to the reference field. In this case, the mean intensity
of a signal, Is is Ne times less than the reference intensity, Ir. The amplitude

of a signal field is ,JN;, times less than the amplitude of the reference field. The
mean modulation depth, m, for a grating in the interference pattern is

(3.42)

In both the saturable media considered in this chapter and in the photorefractive media considered in chapter 4 the maximum amplitude of a perturbation
formed in a single exposure is linearly proportional to the modulation depth of
the recording interference pattern. With this fact in mind, Eq. (3.42) suggests
that coherent polychromatic control would allow us to record an arbitrary volume
hologram with a fall off in amplitude proportional to the square root of the rank
of the hologram and a fall off in diffraction efficiency proportional to the rank of
the hologram. The diffraction efficiency would thus be R times better than for
the same hologram stored using multiple exposures.
The principle difficulty involved in recording coherently with polychromatic
light is the problem of limiting T. If we assume that Ne = 1000, L = 1 cm and

A= 1 µm, then the conditions which follow Eq. (3.40) imply that

(3.43)

One method of limiting T to this range would be to use a holographic medium
which is excited in a two-step process. If the hologram were sensitized by a
short pulse during recording with the signals of Eq. (3.37) then the integration
time could be suitably limited. An alternative approach which is not material
dependent is to assume that the recording signals themselves are riding on such
a pulse. Unfortunately, it would be extremely difficult to generate Ne ~ 1000

109

pulses shifted coherently in frequency with respect to one another and spatially
modulated in 2-D. The prime candidate for controlling such a set of pulses, a
volume hologram, cannot perform the task in a straight forward way because the
temporal bandwidth of the pulses must be wider than the Bragg sensitivity of the
hologram, resulting in extreme dispersion problems. Thus, while the combination
of temporal and spatial signals to control volume holograms is of considerable
interest, pulsed control seems unlikely to directly challenge sequential control in
the near term.

110

4. FORMATION OF PHOTOREFRACTIVE
VOLUME HOLOGRAMS
4.1

INTRODUCTION

Photorefractive holograms consist of photogenerated space charge distributions in electro-optic materials. The charge patterns modulate the index of refraction of the material via the electro-optic effect. The formation of photorefractive
holograms may be described by dynamical equations coupling the free carrier
densities, the local trap densities, the current density, the space charge field, and
the optical fields. Physical models which yield suitable dynamical equations have
been developed by Kukhtarev et al. [19], Moharam et al. [119], and Feinberg et

al. [120].

In this thesis we consider information storage in photorefractive crystals.
In order to effectively control the information recording process, we would like
hologram formation to occur as linearly as possible. In this chapter we consider Kukhtarev's band transport model with particular emphasis on situations
in which the holographic component of the charge pattern grows linearly with
exposure. In chapter 3 we argued that information storage in volume holograms
requires multiple exposures and we developed an exposure procedure for recording
holograms which implement arbitrary transformations between sampled planes.
Here, we develop an exposure schedule which allows us to apply the results of
chapter 3 to hologram formation in photorefractive crystals.
The second section of this chapter is a review of the single active trapping
species, single carrier version of the band transport model. Solutions to the band
transport equations have been analyzed in great detail by many authors [20, 21].

111

While the analysis in this chapter mainly follows previous solutions, the emphasis
on many grating holograms is not common in the photorefractive literature. We
do not consider dynamic beam coupling effects in our discussion. Such effects can
be suppressed by writing polarization switching gratings or by writing in the drift
dominated regime. In the third section of this chapter we consider cases in which
the formation of photorefractive holograms may be linearized. The key to being
able to record linearly is an exposure schedule which uniformly weights each
exposure. The fourth section considers hologram formation in the presence of
multiple photoactive trapping species. Multiple species lead to multiple response
times in the recording process, which complicates the design of an exposure
schedule. In the fifth section we consider a copying technique for improving the
diffraction efficiency of linearly recorded photorefractive holograms in adaptive
systems. The final section considers the impact of fixing mechanisms on hologram
formation.

4.2

DYNAMICS OF THE PHOTOREFRACTIVE EFFECT

In the simplest case, a photorefractive hologram forms as a result of charge
migration via a single carrier from neutral local trapping sites to ionized sites of
the same species. The dynamics of hologram formation in this case are described
by the charge continuity equation

(4.1)
the charge generation rate equation

(4.2)

112

the current equation

(4.3)
and the Poisson equation

( 4.4)

where
n is the excited carrier number density.

ND is the donor species number density.

N"Jj is the ionized donor number density.

NA is the number density of ionized acceptors.
J is the current density.

E is the quasi-static electric field.
e is the electron charge.

/3 is the donor thermal ionization rate.
sf is the photoionization rate.

I is the illuminating intensity.
, is the carrier-trap recombination rate.
µ is the carrier mobility.

kB is Boltzman's constant.

T is the temperature.

113

iph is the photovoltaic current.

is the static permittivity.

While we assume for the present that the acceptors are not photoionizable, Ni.
must be nonzero in order for the amplitude of the spatial variation in

NiJ to be

nonzero.
We are agam interested in forming holograms in the system sketched in
Fig. 3.3. As described in chapters 2 and 3, the optical field is generated on
a pair of SLMs. For reasons described below we will assume that cross-gratings
recorded between signals generated on the same SLM may be ignored. In order
to simplify our analysis, we will assume that the ratio of the total intensity generated on the input SLM to the total intensity generated on the training SLM is
m 0 ~ 1. The recording intensity is

(4.5)

ffiij

and 'Pij are the modulation depth and phase of the fringe pattern between

the i th input beam and the jlh training beam. J0 is the spatial mean of the
recording intensity. The recording intensity can, of course, be affected by selfdiffraction effects, in which case the wave equation for the optical field is coupled
with the band transport equation. We will assume, however, that such effects
are suppressed. In practice, suppression is often possible by judicious choice of
the recording geometry, the beam polarizations, and the applied field.
We can expand the spatial and temporal dependence of n, Njj, E, and J, in

114

Fourier series in harmonics of the wavevectors Kij:
00

I: I:

n(r,t) =no+

N"t,(r, t) = Njjo +

(4.7)

I: I:

E(r,t) = Eo +

Poo=-oo pll=-oo
#0

f(r,t) =

Poo=- 00 pll=-oo

r JL--Pij.Kij•T

(4.8)

JPe JL--Pij.Kij•T

(4.9)

•J

PN1N2=-oo

1: + I: I:
P#O

(4.6)

•1

PN1N2=-oo

where the ij th component of pis Pij• Substituting the Fourier expansions into
the band transport equations and matching zeroth order Fourier components we
find

dn 0 = dN"t, 0
dt
dt

(4.10)

dN"t, O
dt =({3 + sI0 )(ND - N Do) - 1 n 0 N Do

~ [ 1 I mij N+
jcp;
1 I mii N+
-jcp,
- ~ 28 02 D(p;j=-l)e J + 28 02 D(p;j=l)e

(4.11)

+ ,n(p;j=l)Nb(p;j=-1) + ,n(p;j=-I)Nb(p;j=I)],
fo = eµnoEo + 'ii-aolo +

L eµn(p;j=-I)E(p;j=l) + L eµn(p;j=l)E(p;j=-1) (4.12)
ZJ

where we have noted that the magnitude of qth order Fourier components will be
of qth order in the modulation depths and we have kept terms to second order.

115

E is an applied field. We have assumed that the photovoltaic current is linear

in the illuminating intensity and the absorption, a. °K is the Glass constant [121].

In deriving the above equations we have made use of the fact that the first
order terms in the Fourier expansion are of first order in the modulation depth
of their driving terms. While the second order terms in the above equations
are small compared to the zeroth order terms, the sum in Eq. (4.11) may scale
with the zeroth order terms. If the phases of the Fourier components in the
intensity pattern are uncorrelated and all the Fourier components obey the same
statistics, then this sum is unlikely to exceed Ng times the variance of the squared
modulation terms, m 0 / Ng. Ng is the number of distinct Fourier components in
the recording intensity. This sum can be neglected only if we make the assumption
that m 0 ~ 1.
The "quasi-steady approximation" [122] consists of the assumption that the
charge excitation and recombination dynamics are much faster than the charge
migration effects which cause the buildup of the space charge field. Under this
approximation the carrier density n may be assumed to respond instantaneously
to the current state of the illuminating field and the trap density. Stated in
this way the approximation differs slightly from the traditional quasi-steady approximation, which has been taken to imply only that the spatial mean of the
carrier density responds instantaneously to the illuminating intensity and is constant during illumination. The distinction between the two cases is developed
in section 4.4. For the present problem, the approximation consists of discarding the higher order terms in the driving equations for the zeroth order Fourier
coefficients and assuming that the time derivatives of these coefficients are zero.

116

Under this approximation,

(4.14)

(4.15)

(4.16)
Typically the recombination rate is much greater than the charge excitation rate,
in which case NiJ O ~ Ni,_ and

( 4.17)

The first order Fourier components are described by differential equations
in the odd order Fourier coefficients. The third and higher order terms can be
dropped relative to the first order terms using exactly the same approximation as
was used to drop the second order terms in the driving equations for the zeroth
order components. Let Xij,I represent the first order Fourier coefficient of the
parameter x for the grating wavevector Kij. To first order in modulation depth,
the band transport equations for the first order Fourier coefficients corresponding
to the iih component of the illuminating intensity are

dND+ IJ,
· · 1 + ;· RZJ·· · f IJ,· 1
dt

dn · · 1 _
_z_J,_
dt

(4.18)

dND+..
m··
ZJ,l -- s 1o_.!1..
.. N+
__d_t..;;..;....
2 e )/{)ij(ND - N+
Do ) - s IN+
o Dij,1 - ,no N+
Dij,I - ,n,3,l
Do (419)

( 4.20)

117

( 4.21)

where we assume that the thermal excitation rate is much less than the photoexictation rate and we define the diffusion field Ed,ij

= kB T Kij / e. The Fourier

coefficients for the absorption can be related to the trap densities using the relationship

a= shwV(ND - N'jj),

( 4.22)

where V is the volume of the crystal. Thus,

Nn+··1

+ = Oo
IJ,
= snwVNn-·1
+ .
"J,
(Nn-Nno)

Oijl

(4.23)

The quasi-steady approximation, as stated above, amounts to an assumption that n is a "slave" variable. This approximation allows us to assume that
dnij,i/dt = O, in which case Eqs. (4.18)-(4.21) can be reduced to a first order

differential equation in the first order Fourier coefficient for the field. Since there
are no driving terms for the field normRl to the grating wavevector, we assume
that each first order Fourier coefficient is parallel to the corresponding grating
wavevector. In this case, we can substitute the scalar Eij,1 for Eij,1• Solving for
the field in the above equations we find

dEij- '1
Ttdt

3·,f'l••E
= - E tJ,··1 + m··e
.,..,, s
IJ

(4.24)

where
( 4.25)

and
( 4.26)

where

Eq is the saturation limit for the space charge field based on the maximum
amplitude of the ionized trapping species.
Assuming constant mean intensity, the solution to Eq. (4.24) is

( 4.27)

If the intensity is not constant, the solution is complicated by the intensity dependence of the effective photovoltaic field. In materials with no photovoltaic
field or in which NA~ Nn, Eq. (4.24) can be recast as a differential equation in
the mean exposure by defining p = I 0 t. In this case,

(4.28)

where 'Yt = I 0 rt, Eqs. (4.27) and ( 4.28) form the basis of our discussion of the
linear formation of many grating photorefractive holograms. Note that the time
constant for diffusion dominated holograms is function of the grating frequency.

119

In the interconnect architectures considered here this fact is unlikely to create
problems .because the bandwidth of the interconnections is typically small compared to the mean grating frequency.

4.3

LINEAR FORMATION OF PHOTOREFRACTIVE HOLOGRAMS

Our goal in this section is to determine the effect of storing multiple grating
volume holograms in photorefractive media. The analysis is very similar the
analysis used in section 3.4 to consider holograms in saturable media. Our first
task in this section is to establish the link between the space charge field of the
previous section and the holographic perturbation analyzed in section 2.3. Once
we have shown the correspondence, we revert back to the abstract representation
of the perturbation which was used in chapter 3.
The space charge field modulates the index of refraction of a photorefractive
crystal via the linear electro-optic effect. This effect is described by electro-optic
coefficients, rijk, which are traditionally defined in terms of the perturbation to
the impermeability tensor, f/,

( 4.29)
where Ek is the component of the space charge field along the k th axis. The
corresponding perturbation to the permittivity tensor is

(4.30)

We saw in chapter 2 that for low diffraction efficiencies the field diffracted by a
hologram is linearly proportional to e2 •.6.ee1 where e1 and e2 are the polarizations

120

of the coupled beams. The field diffracted by the grating with wavevector i?ij is
proportional to -€;ffreffEij,I where by definition

(4.31)

A plot of (n 3 reff /2) 2 as a function of the angle between i?ij and the c axis for
BaTiO 3 is shown in Fig. 4.1. Both beam polarizations are assumed to lie in the
plane of incidence.
In a holographic interconnection system such as that sketched in Fig. 3.3, the
spatial bandwidth of the input and training signals may be assumed to be small
enough that the effective coupling constant does not vary over the bandwidth
used in grating space. Small variations in the effective coupling constant might
be compensated by adaptive feedback in neural network applications. Typically,
the angle between the input and training planes is selected to position the volume
used in grating space near a maxima of the effective coupling constant.
Assuming that the effective coupling constant is uniform over the region of
interest in grating space, the connection between an input and an output pixel is
completely described by the amplitude of the first Fourier component of the space
charge field of the corresponding grating. Thus, a one-to-one correspondence
exists between the amplitudes of the first order Fourier components of the fields
and the Fourier coefficients of the perturbation A( r) discussed in chapter 3. Using
the formalism developed in chapter 3, we see from Eq. ( 4.27) that for constant
mean recording intensity the analog of Eq. (3.13) for photorefractive holography

121

2.50 -,--,~..,.....,,-,--....-,.......,..-r-........---,--r-r-,-~~.,...-y~~~-,---,~..,.....,,..,......-,......,......-,....,..--,-,....,..,
2.00
1.50
1.00

0.00
-0.50
-1.00
-1.50
-2.00
- 2.50 +-'-~~-i-~~~........~~-+-~~--;.~~~-1--~~-t-~~....................~~--1
0.00
100.00
200.00
250.00
50.00
150.00
300.00
350.00
400.00

direction

Figure 4.1.

Effective coupling constant versus the angle between the grating

wavevector and the c axis in BaTi03.

122
18

( 4.32)

where E 0 = l 0 Tt. The exponential decay of previously recorded signals as more
information is recorded creates a "window" in exposure such that after a certain
characteristic exposure previously stored information is lost.
Consider a hologram recorded in discrete exposures. During each individual
exposure the recording signals are constant. In this case Eq ( 4.32) becomes

+ c.c.
( 4.33)
where tn is the time of the n th exposure, Tn =

I::;:\ tn,, and Ne is the total

number of exposures recorded.

In order to form a hologram which is a linear sum of the recording signals
over all the exposures, tn must be selected such that the factor

fn = (1 - exp(-tn/rt)) exp(- ~ tn' /rt)
n'>n
is constant for all n. This may be achieved in a relatively simple manner if Tt is
real. From Eq. (4.25) we see that this is the case when E 0 = Eph = 0. Choosing

tn = Tt log (

l+(n-l)x)
l+ ( n-2 ) x ,

( 4.34)

123

where X = l - exp(-tif rt), yields

(1 - e

.=!n.
Tt

- '\""Ne
t ITt
L.,n 1=n+1 n

= ------

1 + (Ne - l)x

( 4.35)

and

A( r J = I (l
;;'I

~[~ ~ ,1..

)·'·*

( ) j(k-,• 3 ,-k-11
( 36)
3 ,, )•r] +
c.c. 4 .

~ ~ ~ + (Ne _ l) X ) n=
., ., .,, .,,
1 J

A particularly simple result occurs when t1 ~ Tt, in which case x ~ l. By
following this exposure schedule, the formation of photorefractive holograms becomes linear in the sense discussed in chapter 3. Using the formalism developed
there, an arbitrary volume hologram can be recorded in a crystal. Unfortunately,
this linearization of the recording process exacts a cost to the dynamic range of
the recorded hologram, as evidenced by the dependence of the amplitude of the
perturbation on N; 1 . Recall that the same inverse dependence on the number
of exposures recorded was found in chapter 3 for saturable media. In chapter 3
this factor was found to cause a R- 2 fall off in the maximum diffraction efficiency
which can be achieved for a given interconnect pattern. Since the perturbation of
Eq. ( 4.36) is linear in modulation depth, as was the perturbation in section 3.4,
the analysis of that section applies to the present case. In particular, the maximum diffraction efficiency of a rank R hologram recorded in this way is at least

R- 2 times less than the maximum diffraction efficiency for a rank 1 hologram.
If uncorrelated information is stored in each exposure, the inverse dependence of the amplitude of the perturbation on the number of exposures results in
a fall off in diffraction efficiency of photorefractive hologram proportional to the
square of the number of exposures. This behavior is demonstrated in Fig. 4.2,

124

where the power diffracted from plane wave holograms recorded in a cermm
doped strontium barium niobate (SBN) crystal is shown. The holograms were
written with grating wavevectors approximately along the c axis of the crystal.
The writing beams and the reconstruction beams were extraordinarily polarized
plane waves with a wavelength of 514 nm. Each hologram was recorded between
two plane waves. The ratio of the intensities of the two beams was 100 to 1.
The weaker beam propagated toward the c axis. The thickness of the crystal
along the optical path was 2 mm. The power diffracted by each of two holograms
written sequentially to equal diffraction efficiency is one quarter the saturation
diffracted power from a single plane wave hologram. The power diffracted by
each of three holograms written to equal diffraction efficiency, shown on a vertical scale stretched by a ratio of 5/2, is one ninth the saturation diffraction
efficiency for a single hologram. The power diffracted per hologram relative to
the saturation diffracted power for a single hologram when 111 holograms were
recorded using this approach is shown in Fig. 4.3. The each spot in this figure
is approximately 104 times weaker in intensity than the spot which forms for the
single exposure hologram of Fig. 4.2. The characteristic exposure for this crystal, '"'It, was 119 mJ /cm 2 gratings written with 1400 lines/mm. Inverse square
dependence of the diffraction efficiency of recorded holograms in more complex
holographic recording systems is demonstrated in chapter 6.

If Tt is complex, the additional degree of freedom provided by shifting the
phase between signals from the input and training planes must be employed to

en be a global phase shift of the fringe pattern
between input and output plane in the n th exposure. tn and en are selected to
achieve linear recording. Let

125

Figure 4.2. Diffracted power per hologram for one, two and three holograms.

126

Figure 4.3.

Focused spots diffracted from 111 plane wave holograms written

sequentially to equal diffraction efficiencies.

127

make

fn = eien(l - e-tnTe exp(-jwrtn)) exp(- 2:)tn 1/Te+ jwrtn,))
n'>n

(4.37)

constant for all n. 1/re and Wr are the real and imaginary parts of 1/rt, respectively. In order to maximize the diffraction efficiency of the recorded hologram,
t1 should be selected such that 11 - exp(-ti/re) exp(-jwet1)I is maximal. If
WrTe ~ 1 the maximum occurs at t1 ~ 7r /wr. For all values of Wr the maximum

occurs for t1 < 7r / Wr.

6 can be selected arbitrarily. Given t1 and 6, tn and

ln may be discovered using a recursive process based on the requirement that
f n+ 1 = f n. This approach yields

( 4.38)

Separating the real and complex parts of Eq. ( 4.38) yields two transcendental
equations in the two unknowns tn+I and en+l· These equations can be solved
numericaliy without difficuity.
To compare hologram recording with a complex time constant with the result
obtained in Eq. ( 4.36), we would like to estimate tn as n grows large. Since the
effect of a nonzero imaginary component of Tt early in the writing process is to
increase the rate at which the perturbation grows, we find that the exposure time
for the n th hologram is reduced by the complex term. For real values of Tt, tn
asymptotically approaches rt/n. If we postulate that tn = Te/n - En for complex
Tt, then by substituting in the constraint lfnl 2 = lfn+11 2 we find that En decreases

monotonically as n increases. Asymptotically we find that tn approaches re/n.

128

This behavior is shown in Fig. 4.4, which is a plot of tn versus n for various
values of WrTe- As expected the curves merge asymptotically.
The value of lfnl 2 when Ne ~ l holograms are recorded in a system with a
complex time constant is

(4.39)

Substituting in Eq. ( 4.33) we find

Clearly, critical importance is attached to the size of Wrre. Eq. ( 4.25) yields

( 4.41)

We have neglected photovoltaic terms for simplicity. Since WrTe scales with the
applied field, we may assume for large WrTe that the applied field is much greater
than the diffusion field. Maximizing lwrrel with respect to E 0 11/ Eq yields

lwrrelmax = -1 1~-€ - ~-j.
eµ
1c

(4.42)

when
(4.43)
Note that E 0 must exceed Eq to make WrTe large. When this is the case, the
amplitude of the first Fourier component saturates at Eq. According to Eq. ( 4.42),

129

10·1

t... 10-2

"'
8.
..:

10·3

10 ◄

10•5 .___.__..........__._..............__.....___._..................~ - - - ~ ~ ~ - - - ~ - 100
10 1
102
103
10◄
Exposure Number

Figure 4.4. Exposure time versus exposure number for various values of WTrExposure time is found by solving Eq. ( 4.38).

130

the imaginary component of the response time is likely to be large in materials
where c is large. Precise estimation of the maximum value of WrTe which can be
achieved is hampered by the lack of good estimates for 1 . Maximum values of 50
to 100 high permittivity materials such as SBN and BaTi03 are not unreasonable,
however. In this case, applying a driving field in excess of the maximum field
that the charge density can support may still increase the storage capacity of a
crystal for multiple hologram storage.
Consider once more our assumption that the field diffracted by a grating is
linear in the grating amplitude. We saw in section 2.3 that the field diffracted
by a volume hologram is linear in the sine of the product of the perturbation
amplitude and the thickness of the hologram. Our analysis in this section and in
section 3.3 has shown that the perturbation amplitude in photorefractives and
in saturable materials must fall at least linearly with the rank of the hologram
being recorded. The sine dependence of the diffracted field on the perturbation
amplitude means that the diffraction efficiency does does not begin to fall with
the square of the rank of the hologram until the rank is large enough that the
perturbation amplitude is much less than the amplitude needed approach 100%
diffraction efficiency, Arno- In photorefractive materials, the maximum perturbation amplitude can be well in excess of Arno [125].
Fundamental limitations on the dynamic range of photorefractive media due
to the finite trap density force the amplitude of the perturbation to fall linearly
with \/R, [59]. If

(4.44)
then these limitations, rather than the control problem, limit the amplitude of the

131

perturbation. For Ron the range of 1000 to 10,000, Eq. (4.44) can be satisfied
using currently available materials. This means that the techniques described
in this section should allow us to record holograms of rank up to 10,000 with
diffraction efficiency near 10-4 , the same diffraction efficiency as occurs to each
point in Fig. 4.3. Depending on the numerical aperture of the input and output
planes, storing holograms of such high rank allows us to store from 10 9-10 11
independently-controlled distinct gratings per cm3 in photorefractive media. In
neural applications, these holograms would store 10 3-10 5 associations between
images.

4.4

MULTIPLE ACTIVE SPECIES

Typically, a photorefractive crystal is heavily doped with a single impurity
during growth. As we saw in section 4.2, however, even when only one trapping
species is photoactive, a second species is needed to allow the active species to be
partially ionized before a hologram is recorded. Partial ionization is necessary for
vacant trapping sites to be available for displaced carriers. In the most general
case, more than one trapping species is photoactive. For each active species, a
charge generation equation describing the ionization and recombination dynamics for that species must be added to the band transport equations. When the
equations are linearized, the system is found to have the same number of characteristic time constants as independent charge generation equations. In this
section we describe experimental results which show evidence of multiple time
constants for hologram formation in a LiNb03 crystal, we review a simple model
for hologram formation in a material with two active species and we discuss the
effect of multiple time constants on linear hologram formation.

132

A system appropriate for recording the dynamics of hologram formation and
erasure in photorefractive materials is sketched in Fig. 4.5. A hologram is written
by a pair of writing beams for a specified time. The diffraction efficiency of the
hologram is monitored continuously using a longer wavelength probe beam. At
the end of the recording time the writing beams are switched off and an erase
beam with the same total intensity as the writing beams incident at a non-Bragg
matched angle is switched on. The probe continues to monitor the diffraction
efficiency as the grating decays.

Ar
ERASE
BEAM

Figure 4.5. Experimental system for recording the dynamics of hologram formation and erasure.

The single trapping species theory outlined in the previous section predicts

133

a growth in the diffraction efficiency of the probe proportional to

during the writing phase and decay proportional to exp(-2t/re) during the erasure phase. In experiments with iron doped LiNb03 crystals we have observed
dramatically different results.

A plot of an experimental curve is shown

Fig. 4.6. The transition from writing to erasing occurs after the hologram has
been exposed for 2000 seconds and is marked by a vertical line. The key thing to
notice is the rise in diffracted power which occurs at the beginning of the erasure
process. The crystal used in this experiment was doped with 0.015% Fe in the
melt and was reduced after poling. The writing intensity for the plot shown was
50 mW /cm 2 • The grating spacing was 2315 lines/mm. The grating vector was

along the z axis. The writing and erasing beams were ordinarily polarized to
reduce beam coupling effects. The probe was extra-ordinarily polarized. The
ratio between the intensities of the writing beams was 10 to 1.
Previously, increases in diffraction efficiency during the erasure process have
been associated with self-reinforcement effects during Bragg matched erasure
[124] and back-coupling effects in very high diffraction efficiency holograms [125].

The first of these effects is suppressed in the present experiment by the use of
a non-Bragg matched erasure beam. The second effect does not apply because
the diffraction efficiency did not exceed 0.005. The behavior observed in this
experiment can be explained by the presense in the crystal of two separate grating
components which develop with different time constants and whose relative phase
is nonzero. At saturation the two components are out of phase. When erasure
begins, the more quickly decaying component is erased and the slowly decaying

134

xI0-3

0.9
0.8

....0

0.7

;:I:

0..

-0

0.6

c,j

0.5

-0

.?:;

0.4

.:;

....

0.3

500

1000

1500

2000

time (seconds)

Figure 4.6_ Diffracted probe power versus exposure time.

2500

3000

135

component is "unmasked." This two time constant theory can be supported by
comparing erasure data for holograms written for a short time with erasure data
for holograms written to saturation. Fig. 4. 7 is a log plot of experimental results
under the experimental conditions described above. The upper curve shows the
decay of a grating which was recorded for 2000 seconds and then erased for 1735
seconds. The data shown represents the diffracted power from 1735 seconds of
erasure to 2035 seconds. The lower curve shows the decay of a grating which was
written for 10 seconds and then erased for 300 seconds. The key thing to note is
that the slope of the lower curve, corresponding to the time constant with which
the diffraction efficiency decays, is much more negative than the slope of the
upper curve. This behavior is expected if two grating components evolve with
distinct time constants. For short exposure and erasure times, relatively more
of the quicker grating component is recorded. When the hologram is erased, the
decay curve reflects the presence of this component. In a saturated hologram,
after erasure has been underway for some time the faster grating component is
much reduced compared to the slow component and the decay is dominated by
the slower decay rate.
Band transport equations for the simplest case of hologram formation in
the presence of multiple trapping species were developed by Valley in 1983 [94].
Valley's theory was used to explain indications of multiple time constant decay
found in holograms recorded in BSO by Mullen [126]. Similar results have since
been found in LiNb03 [127] and GaAs [128]. Valley's theory consists essentially
of the assumption that Ni_ is also a photoactive parameter. Note that in addition
to adding more trapping species to the dynamical theory, a second carrier species
could also be added. Since the carrier distributions are typically slaved to the

136

-12.6
-12.8
-13

...

?;:

-13.2

0.
"O

:-:l

-13.4

...

~ -13.6

.::: -13.8

...

bl)

.52

-14
-14.2
-14.4
-14.6

100

150

200

250

300

time (seconds)

Figure 4. 7. Log relative diffracted power versus erase time for a hologram written
and erased for a long time ( upper curve) and immediately upon erasure for a
hologram written for a short time (lower curve).

137

more slowly evolving variables, however, the presense of multiple carriers alone
does not add characteristic time constants to the dynamical behavior. For this
reason we limit ourselves to a single carrier here. Multiple carrier theories are
described in [129]. The band transport equations in this case are

aN;,.

aND

......

V ·J

at+ -at- - -at- = -e-

(4.45)

aN+
at = (f3n + snl)(Nn - Njj) - ,nnNjj

(4.46)

( 4.47)

( 4.48)

( 4.49)
These equations reduce to the single species equations in the limit as f3A, SA --t 0.
Suppose that

I(r) = I 0 (l + m cos(I< · r)).

(4.50)

Linearizing using the same techniques as in the previous section and again making the quasi-steady approximation yields conditions for the zeroth order charge
distributions
( 4.51)

(4.52)

(4.53)
where nn

snl /,n and nA = sAI /,A· Given nn and nA, these equations

138

can be solved for n 0 , NJ 0 and NA.a· The dynamics for the first order Fourier
components are described by

dn 1
dt

dNf> 1
dt

dN;_ i
jI< Ji
dt +-e-

(4.54)

( 4.55)

( 4.56)

( 4.57)

( 4.58)
where we have assumed that the applied field and i!, are parallel to K and we
have defined K = IKI.

In the previous section we used the quasi-steady approximation to reduce
the second order dynamical equations to a first order equation. The solution to
this equation involved only one boundary condition, the initial state of the space
charge field. Eqs.( 4.54)-( 4.58), in contrast, represent a third order dynamical
system. In Valley's analysis, the time constants of a third order equation for

the development of the space charge field are determined. The system is then
reduced to second order based on the discovery that one of the time constants
is much smaller than the other two. In order to specify the dynamics of the
second order system it is necessary to specify two boundary conditions, typically
the initial state of the field and the initial time derivative of the field amplitude.

139

Unfortunately, Valley's approach leads to an ad hoc assumption of
( 4.59)
where 'idi = Ej eµn~, for the initial condition of the derivative of the field when
erasure begins. This condition is inconsistent with our empirical results, in which
the derivative of the amplitude of the field is initially positive.
This difficulty points out an inconsistency in the the quasi-steady approximation as it has commonly been applied in the literature. The approximation has
been used to imply that the zeroth order of the carrier density responds instantaneously to optical illumination, but the same assumption has not been applied
to higher order Fourier components of the carrier density. When the approximation is applied in this manner, a fast time constant corresponding to the carrier
response time is found when the first order equations are solved. If the approximation is applied consistently, however, all time derivatives of the carrier density
should be set to zero. In this case, the carrier density is said to be "slaved"
by the slowly varying parameters [130]. In this case, Eqs. (4.54)-(4.58) reduce
immediately to a second order system. More importantly, boundary conditions
for the field and its derivative can be obtained directly by assuming continuity in
the slowly varying parameters and instantaneous response in the carrier density
to ambient conditions.

If we set the time derivative of the carrier density to zero, Eqs. ( 4.54)-( 4.58)
can be reduced to three independent equations in the dynamic variables

Ni>

and Ni_ 1 and in the parameter n1. These equations are

dNiJ 1 = msnlo d{ En_ Njj 1 _ ~
ryn

( 4.60)

140

dNA 1 =_ms Alo cK EA_ NA 1 + n1
dt

TJA

(4.61)

and

(4.62)

where the various parameters are defined below and where we have eliminated
the current density by differentiating Eq. (4.58) and comparing the result with
Eq. (4.54). This yields

(4.63)
We can solve Eqs. (4.60)-(4.62) for N't 1 (t) and NA 1 (t) if we are able to specify
the initial condition of these variables. For writing in a material where no grating
exists, N"}j 1 (t = 0) = NA 1 (t = 0) = 0. When we erase a pre-recorded grating, we
may assume that N°t 1 (t) and NA 1 (t) remain continuous even when the driving
terms are discontinuous.
Since it is the dynrunical behavior of the space charge field which is of interest
for holographic applications, we solve Eqs. ( 4.60)-( 4.62) and Eq. (4.58) for the
field. This yields

(4.64)

where

_1 = T3 ((-1- + _1_) (-1- _ j-1) + _1_ + _1_ + _1_ +-1-)
T1
1"/A
1"/D
1"diff
1"dr
7D1"di
7DTJA
TATdi
7ATJD
(4.65)

141

_1 -T3 (_1_(_1__ j_l)+--1-+_._1_)

Tf -

TJA'TJD

Tdiff

Tdr

TJATdiTD

( 4.66)

TJDTdiTA

1 .1 1 1
-=---J-+-+T3
Tdiff
Tdr
TD
TA

( 4.67)

(4.68)

= (,DNjj

TA= (,A(NA -

TJD

1 is the recombination time for species D.

N.,1 0 ) ) - 1 is the recombination time for species A.

= (sDlo + ,Dno)- 1 is the inverse of the sum of the photoproduction

rate and the ion recombination rate for species D.
TJA

= (sAio + ,Ano)- 1 is the inverse of the sum of the photoproduction

rate an.cl the ion recombination rate for species A.
Tdi

= eµeno is the dielectric relaxation time.

Tdiff

Tdr

= KZµekBT is the diffusion time.

= (µKE

0 )-

1 is the drift time.

Assuming that the field and its first derivative are continuous, the solution

142

to Eq. ( 4.64) is

( 4.69)

where
(4.70)

If there is no grating at t = 0, E 1 (0) = 0. Solving for the initial condition
for the first derivative of the field using Eqs. (4.58) and (4.60)-( 4.62) under the
assumption that N"f>/t = 0) = NA 1 (t = 0) = 0 yields

In the experiments described above, a hoiogram is recorded at constant modulation depth for a fixed time, t 0 • The modulation depth is then switched to
zero and the hologram is erased. Switching the modulation depth to zero causes
a discontinuity in the time derivative of the field amplitude. This discontinuity
must be explicitly accounted for in our solution to the second order system. The
field amplitude during recording is

(4.72)

143

where dEifdtlo is given by Eq. (4.71). During erasure the field amplitude is

where the initial condition for the field on erasure is found by assuming that its
value is continuous when we switch from writing to erasing. The initial condition
of the derivative of the field is found by solving Eqs. ( 4.58) and ( 4.60)-( 4.62)
under the assumption that the ionized trap densities are continuous. In the case
of a hologram written to saturation this approach yields

for the derivative at the start of the erasure cycle. The derivative at the beginning
of the erasure cycle is the negative of the derivative at the beginning of the write
cycle because at saturation the derivative due to the homogeneous equation is
just sufficient to cancel the driving terms. When the driving terms are turned
off the derivative is equal to the negative of their contribution to the derivative.
The initial derivative on writing contains only the driving terms.

The diffraction efficiency of a recorded hologram increases when erasure begins if the initial value of dlE11 2 / dt is positive. If we write to saturation then this
initial value can be evaluated using Eq. (4.74):

144

dlE11 2
dt

dE*

*dE

= mEsdtlse + mE dt lse
2 2
= _ IT3j m [?R{T]} [(E; + EJ)(snloµKEn + sAloµKEA)

ryA

X (snloµKEn

ryn

+ BAloµKEA)

+ IEphl\

Tdi

TJATD

TJDTA

)( ~ + ~ )]
TD

+ ~{Ti E h(Eo _ jEd)} [(snloµKEn + sAioµKEA)( ~ + ~)

'TJA

+ (snloµKEn + SAioµKEA)(

TJD

TJATD

)]]

TJDTA

(4.75)
Note that if Eph is zero then the sign of the derivative in Eq. (4. 75) must be negative, since the term inside the large bracket which does not contain Eph contains
only squared values and positive constants. If Eph is not zero then whether or
not the diffraction efficiency increases on erasure depends on the effective phase
of the photovoltaic field, on the relative doping densities of the trapping species
and on the relative magnitudes of the applied field and the effective diffusion
and photovoltaic fields. Sensitivity to the magnitude of the photovoltaic field has
been observed in our experiments. Fig. 4.8 is an experimental writing and erasure curve. The writing intensity in this case is 100 mW/ cm2 , twice the intensity
of Fig. 4.6. The exposure time is 1000 sec., half as iong as in Fig. 4.6. Note that
the bump on erasure is much less pronounced in this case.
The simulated diffraction efficiency during the recording and erasure process
when the photovoltaic field is 180° out of phase with the illumination pattern is
shown in Fig. 4.9. We relied as much as possible on published parameter values
for LiNb03 in generating this curve. The effective photovoltaic field was assumed
to be 1.2 kilovolts/ cm at a recording intensity of 56mW. The key parameter values

145

xl0-4
4.5

....

3.5

0..

-0

t,
ro

:;::

2.5

-0

.:::

....

1.5

0.5

time (seconds)

Figure 4.8. Experimental writing and erasure curve in Iron doped LiNb03 under
the same conditions as in Fig. 4.6, except that the recording and erasing intensity
is doubled and the exposure time is halved.

146

used are listed in table 4.1. Note that the simulation shows a hump in the initial
stages of writing, which we do not observe in our experiments. This hump occurs
because the initial derivative of the field is out of phase with the final state of the
field. This derivative eventually switches sign and the field amplitude, but not
the ionized trap densities, passes through zero before reaching saturation. In the
simulation shown in Fig. 4.10 the photovoltaic term is assumed to be 112° out
of phase with the recording interference pattern. Since the initial derivative of
the field is not exactly 180° out of phase in this case, the magnitude of the field
does not pass through zero during the writting cycle. A characteristic hump is
observed in the erasure cycle. A qualitatively similar experimental curve plotted
as a dashed line is superposed on the simulation in Fig. 4.10. The writing intensity
for the experimental curve was 50 mW/ cm2 at 488 nm. The spatial frequency of
the recorded grating was 2315 lines/mm.
Table 4.1: Parameters for Figs. 4.9 and 4.10.

µ(V-l sec. -lcm. 2)

'Yd

'Ya

0.8

10-15

10-14

1019

1016

While our theoretical results lead us to conclude that the qualitative behavior observed in our experiments can be expected of materials in which multiple
species are active, it is not reasonable to try and fit theoretical parameters on the
basis of our experiments. This is because the problem of fitting curves with multiple exponential terms is extremely ill-posed and because we have no evidence
excluding the possibility that more than two species of traps and more than one
carrier are active in our crystal. A detailed analysis of hologram formation in
our crystals would have to be based principally on non-holographic data.
We now turn to the question of linear hologram formation in materials with

147

1.2

...
Q.)

0.8

0..
-0
Q.)

tl2"'
:a

0.6

Q.)

0.4

0.2

500

1000

1500

2000

2500

3000

time (seconds)

Figure 4.9. Simulation of recording and erasure in a multiple species environment.
The photovoltaic field is assumed to be 180° out of phase with the fringe pattern
of the recording intensity.

148

1.4

1.2

..,....

:le

..,

0.8

"'
.t:
......
:.a..,

0.6

-0

-~;,;
...

0.4

0.2

500

1500

2000

2500

3000

time (seconds)

Figure 4.10. Simulation of recording and erasure process with a phase shift of

112° in the photovoltaic field relative to the phase of the illumination pattern.
The solid line is a simulation and the dashed line is the experimental curve of
Fig. 4.6.

149

multiple trapping species. Since the state of the field depends on multiple variables in these materials, it is no longer the case that fields which are of the same
amplitude at one time will remain equal during erasure. In fact, a pair of field
amplitudes cannot be made to follow the same path in time in this system unless
their exposure history is exactly the same. Of course, if one time constant is
much longer than the other, the exposure cycle for a single time constant can be
followed and the shorter time constant can be ignored since it does not effect the
magnitude after substantial erasure. If the two time constants are more nearly
equal, a method suggested by McRuer et al. for the single species case could
be applied to record holograms linearly [91]. Under this model each exposure
lasts for a very short time compared to the response times. Associated pairs of
images are recorded in a repeating sequence until the system reaches saturation.
The dynamics of hologram formation in a material containing T active trapping
species is described by a set of T coupled equations similar to Eqs. ( 4.60) and
(4.61 ). These equations can be expressed in vector form as

dN =... ...
--= QN +D.
dt
- '

(4. 76)

where the i th component of N is Ni, the first Fourier component of the number density of the ionized sites for the ith species. Q is a matrix describing the
coupling and D contains the driving terms for the growth of the grating. The
component of the recorded hologram which corresponds to information recorded
in the qth exposure cycle is recorded for a time 8, erased for a time (Ne -1)8 and
then again recorded. This sequence repeats until the system reaches reaches an
equilibrium state. At equilibrium, the amount by which the charge pattern corresponding to the lh exposure decays in making ( Ne - 1) uncorrelated exposures

150

is exactly canceled by the amount by which this charge pattern is increased by
the qth exposure in each cycle. If 8 is very small, the decay of the information
stored by the qth exposure due to the uncorrelated exposures ca~ be expressed
as b..eN = (Ne - l)8QN. The growth of the grating due to the qth exposure is

b..rN = 8( QN + D). At saturation, b..e + b..r = 0, which implies

--1

Q jj

N=--Ne

(4. 77)

Since we know from the Poisson equation that the amplitude of the space charge
field is linear in the trapping densities, we find once again that the amplitude of
the perturbation for Ne independent associated pairs of exposure patterns is Ne
times less than amplitude for one exposure.
The dynamics of hologram control in the presence of multiple species can be
substantially more complicated than the single trap case. Various other recording
schemes using resonances between the time constants and phase shifts in the
recording beams are interesting topics for future investigation.

4.5

PERIODICALLY REFRESHED PHOTOREFRACTIVE HOLOGRAMS

In this section we describe a system in which part of the decrease in the
diffraction efficiency of a multiply exposed hologram is recovered by periodic
copying between two holographic media. Palais and Wise [131) have previously
demonstrated that copying a weak hologram can dramatically increase the diffraction efficiency of the hologram. Johnson et. al. [132] have applied this technique
to improve the diffraction efficiency of multiply exposed silver halide films. In
this section we demonstrate that periodic copying between two dynamic media

151

improves the diffraction efficiency by a factor of Ne. A related result is that
periodic copying between two holograms results in a stable diffraction efficiency
when an indefinitely long sequence of exposures is performed. Long sequences of
exposures are used in adaptive holographic systems, such as those described in
chapter 6.
The architecture of the system is shown in Fig. 4.11. A series of holograms
between a reference plane wave, Rl, and a set of signal beams are recorded in a
SBN:Ce crystal. Shutters S4 and S5 are closed during this operation. We use
plane waves as signal beams in our experiments. Different beams are generated
by rotation of the mirror RM. Let Ir and Is represent the intensities of the
reference and signal beams, respectively. The diffraction efficiency of the recorded
holograms is monitored by illuminating the crystal with the phase conjugated
reference, Rl *. The path of the diffracted read out beam to the output CCD
is shown as a dashed line in the figure. A self-pumped BaTi03 phase conjugate
mirror is used to generate the conjugate wave. In addition to providing automatic
alignment of the conjugate beam, the PCM compensates for phase distortions due
to imperfections in the SBN. When the diffraction efficiency of the holograms
becomes unacceptably low, the recorded holograms are copied from the SBN
to a second holographic medium, which in our experiments is a thermoplastic
plate. The thermoplastic hologram is formed using the light diffracted by the
SBN hologram and reference wave, R2. Shutters S2 and S4 are closed. The
hologram written on the thermoplastic plate is copied back to the SBN using
reference beams Rl and R2* (see Fig. 4.11). The intensity of R2* is selected to
make the intensity of the signal beam diffracted from the thermoplastic equal
to the original signal intensity, ls. Shutters Sl and S5 are closed during this

152

- -►
.1..

--0----►

- - - ~...._ __.I g~~era

_ RM

Thermo-plastic
Plate

BaTi03
PERIODICAILY REFRESHED RECORDING SYSTEM

Figure 4.11. Experimental layout for a periodic refreshing system.

153

step. The result is a rejuvenated hologram of each of the signal beams in the
SBN. At this point, we begin adding a new series of holograms to those already
stored in the SBN. When the diffraction efficiency of the holograms again falls
unacceptably low, we repeat the copying process.
We begin the experiment by recording a series of N1 holograms in the SBN
following the schedule of Eq. ( 4.34) for x = 1. After the exposure sequence, the
amplitude of the space charge for each hologram is An= A 0 /M1. If N1 is large
enough, then the diffraction efficiency per hologram is small and the diffraction
efficiency of the n th hologram, 'f/n, is proportional to IAn 12 • This implies that

'f/n = 1Jo/N1 2 , where 'f/o is proportional to IA 0 j2 • After N1 exposures, we copy the
holograms summed in the SBN onto the thermoplastic. We then copy back to
the SBN as described above. The amplitude of the saturation space charge in
a photorefractive hologram is proportional to the modulation depth with which
it is recorded. Since the intensity of the signal beam reconstructed from the
thermoplastic hologram is equal to ls, the intensity of each of the N1 equally
recorded components is Is/N1. The modulation depth for each component is
the ratio of ✓Irls/N1 to the total recording intensity, (Ir + Is).

Since this

modulation depth is -JN1 times less than the modulation depth for one hologram,
the amplitude of the space charge grating for each component is An= A 0 j,Jii 1 .

'f/n is thus 1J0 /N1. The total diffraction efficiency, 1JT = N11Jn, is restored to the
saturation value of 'f/o.
We now begin recording a new series of holograms in the SBN. The first
hologram of each new series is written until its amplitude is equal to the partially
erased amplitude of the copied holograms. Let A~) and ijn be the amplitude and
exposure time of the n th hologram recorded in the SBN in the jlh recording cycle.

154

We would like ijn to be such that AW) is constant for all j and m. Assuming
that this is the case, we use an argument similar to that described in the previous
paragraph to conclude that AW = A 0 / ✓"5:.,{;;._; Ni at the start of the j th recording
cycle, i.e., just after the hologram is copied for the (j-1 )th time. Nj is the number
of exposures we make in the j1h cycle. The perturbation in the crystal at the
start of the j th recording cycle is
j-1 Ni

4>(r)

=LL

A~) Jf)(r)

l=l n=l
j-1 N 1

LL J!l)(r),

(4.78)

✓Lr:} N, l=l n=l

where Jf) (r) is a normalized distribution which describes the spatial structure of
the n th hologram in the j1h recording cycle. The amplitude of the first hologram
in the j1h recording cycle is equal to the partially erased amplitude of the copied
holograms if

( 4. 79)

Letting xU) = l - exp(-t11al), Eq. (4.79) yields x(j) = 1 - ( ✓"5:.,{;;._f N1/(l +

✓"5:.,{;;._11 Ni)). Letting x = X(j) in the exposure schedule of Eq. (4.34) we make Nj
exposures in the j1h cycle to record "5:.,{= 1 Ni holograms with equal amplitudes.
In order to maintain a constant diffraction efficiency from the thermoplastic, Nj
is selected such that the total diffraction efficiency of the summed hologram on
the SBN falls back to its value after the first N1 exposures, i.e., 'rfo/N1. Each
time Ne = L., Ni holograms are copied back and forth, the diffraction efficiency
for each hologram is restored to 1 /Ne.

155

>- 0,5

.·=1t..

1----t

I\ . ·.
I \

1----t

u 0.1

Li...
Li... 0.05

\\ •,'#•

I\ ...

,,
~-I . '·,
I \
\ I \\ I\
\ I
\ I \
,,
,1 ~
# ,1

D 0.01
I0,005
1----t

Ct::

Li...
Li... 0.001
1----t

NUMBER OF EXPOSURES

Figure 4.12. Log relative diffraction efficiency per hologram versus log number
of exposures.

156

Fig. 4.12 is a log-log plot of the mean diffraction diffraction efficiency per
hologram relative to the diffraction efficiency for a single exposure versus the
number of exposures. The figure shows experimental results for recording up to
25 holograms in a SBN:Ce crystal. The diffracted phase conjugate reference for
each hologram was monitored by the CCD shown in Fig. 4.11 . The solid line in
Fig. 4.12 corresponds to the theoretical N; 2 decay in the diffraction efficiency
per hologram. The *S are experimental data points for the mean diffracted power
of holograms stored according to this schedule. The dashed line shows the theoretical path followed by the diffraction efficiency per hologram with periodic
copying when N1 = 5. The # s show experimental data points found when this
approach was followed. The dotted line corresponds to a decrease in diffraction
efficiency proportional to N- 1 . Thermoplastic holograms were made after 5, 10
and 15 exposures.
One might like to implement periodically refreshed storage using two photorefractive media. Unfortunately, copying between two photorefractive crystals
is complicated by the decay of the hologram being copied while the second hologram is recorded. Suppose that we wish to copy a hologram between two identical
photorefractive crystals. As we read out the first hologram, the light intensity
diffracted from it decays exponentially at a rate proportional to the read out
intensity. The rate at which the second hologram builds up is determined by the
overall intensity incident on it. If the reference beam for the second hologram is
too bright then the modulation depth is too weak; if it is too low then the second
hologram does not build up sufficiently before the first hologram decays. If we set
the intensity of the reference beam such that the modulation depth is constant at
the second hologram, then the overall diffraction efficiency of the copied hologram

157

can exceed the efficiency of the original only if the strength of the perturbation
produced in the crystal by a single exposure, exceeds the strength required to
achieve (in principle) 100% diffraction efficiency. This can be accomplished by
selecting a crystal with the appropriate combination of electro-optic coefficient,
dielectric constant, index, and thickness. For example, the critical thickness for
holograms using the rs 1 electro-optic coefficient in barium titanate is approximately 2 mm. Since crystals of such thickness are readily available, it should be
possible to extend the copying technique to an all photorefractive system.
We have described periodic copying between two holograms using two key
simplifications: the holographic media are planar, which implies that they can
be copied in one exposure, and there is a fixed number of exposures M that
need to be done. The implementation of learning algorithms in optical neural
networks is an important class of problems that requires an arbitrary number
of exposures. From Eq. (4. 78) we see that the perturbation that accumulates
in the crystal is simply the average of all the exposures, independent of their
number. If the individual exposures are statistically independent, then the sum
in Eq. ( 4. 78) will grow in proportion to ,JM. This will precisely counterbalance
the N- 1 / 2 factor in Eq. (4. 78) yielding a steady state diffraction efficiency that is
independent of M. If the individual exposures are correlated ( a case we have not
analyzed in the section), then the sum can grow faster than ,JM. In this case
the reduction in diffraction efficiency due to an additional exposure would have
been less that what we calculated. The copying process recovers this smaller loss
in diffraction efficiency and the steady state overall efficiency remains unchanged
and independent of M.
The extension of periodically refreshed recording to holograms utilizing vol-

158

ume degrees of freedom is complicated by the fact that a volume hologram cannot
in general be evaluated or copied in a single exposure. If the complete hologram
is not known a priori, i.e., if we wish to record holograms adaptively, it is not possible to ensure that the minimal number of exposures will be used in recording.
The copying technique described here allows us to recover the loss in diffraction
efficiency due to exposing a hologram more than R times, where R is the rank of
the hologram. In our experiments R was limited to 1 by our use of a single reference beam. For holograms of higher rank, R references would be required. Using
appropriate reference signals to copy a stored hologram results in an increase in
the diffraction efficiency of each component of the hologram by the factor Ne/ R 2 ,
compared to the efficiency obtained without copying, where Ne is the number of
exposures used to form the original hologram. In the simplest case the steady
state efficiency is again independent of Ne.

4.6

FIXING MECHANISMS AND LINEAR HOLOGRAM FORMATION

Before leaving the topic of linear volume hologram formation in photorefractive materials, we briefly consider the impact of fixing techniques. Thermal fixing
has been observed in LiNb03 [133, 28), where the fixing mechanism is associated
with hydrogen migration [134]. Electrical fixing has been observed in BaTi03
and SBN [135, 136, 81], where the mechanism was associated with domain reversal. Fixing has been achieved in LiNb03 by heating the crystal. Within a
certain temperature range it was found that the ion conductivity is greater than
the carrier conductivity and ions migrate to cancel the photorefractive grating.
When the crystal is exposed to read out light, the photorefractive grating is
partially erased and the ion grating is unmasked. Since the ion grating is not

159

light sensitive, the resulting hologram remains until the hologram is reheated. In
SBN fixing has been achieved by heating and by applying a strong field. The
suspected mechanism is domain reversal to cancel the photorefractive grating.
As in LiNb03, the photorefractive grating is erased under read out illumination.
Since the domains are not light sensitive, the resulting hologram remains until
the crystal is repoled.
While such fixing mechanisms may play an important role in stablizing holographic interconnection systems, they are not generally of use in increasing the
storage capacity of photorefractive holograms. The reason is that, if the fixing
is not permanent, previously fixed holograms will decay as newly recorded holograms are fixed. This decay will result in the same N; 1 fall off in field amplitude
as was found above without fixing. If the fixing mechanism is permanent, it will
also be saturable and the N; 1 fall off for saturable effects will come into play.
Thus, our conclusion with regard to fixing mechanisms is that, while they are of
substantial interest in hologram stablization, they need not play a crucial role in
our consideration of hologram formation.

160

5. HOLOGRAPHIC
INTERCONNECTIONS IN WAVEGUIDES
5.1

INTRODUCTION

Many of the difficulties which we have encountered in considering the control of bulk volume holograms are avoided by thick holograms in waveguides.
While the scale of the interconnections systems which can be implemented in
waveguides is relatively small, holograms in waveguides are of interest because
they are relatively easy to control and because they may be interfaced with integrated opto-electronic devices. Previous uses of thick holograms in waveguides
have included grating couplers and distributed feedback lasers [137). Holograms
for dynamic applications have also been considered, especially in photorefractive crystals. A review of work on photorefractive holograms in waveguides is
presented by Wood et al. [95].
The potential for information storage in integrated volume holograms was

considered by JarJ1son [138], ,vho shov;ed that the number of degrees of freedom which can be stored in a planar waveguide hologram scales with the area
of the hologram divided by the square of the guided wavelength. In this chapter, we consider integrated holograms for large-scale linear transformations. We
rederive Jannson's result in this context and describe a novel method for recording holograms in a waveguide using unguided light. The use of unguided light
dramatically simplifies the problem of forming appropriate holograms with high
dynamic range. We present experimental results for holograms in photorefractive
waveguides formed by titanium indiffusion in LiNb03.

161

A thick hologram in a waveguide may be regarded as a vector-matrix multiplier mapping an input vector corresponding to the incident field to an output
vector corresponding to the diffracted field via a matrix represented by the interconnecting hologram. In the second section of this chapter we develop this
analogy in detail and derive basic relationships for this system. The third section describes methods for controlling a hologram in the waveguide with unguided
light. The fourth section describes an experimental demonstration of multiple
grating holograms written with unguided light and reconstructed with guided
light. The concluding section briefly considers applications for integrated volume
holograms.

5.2

AN INTEGRATED VECTOR-MATRIX MULTIPLIER

An architecture for an integrated optical vector-matrix multiplier is shown in
Fig. 5.1 [97]. Light from each of N1 input channel waveguides is coupled into a
slab waveguide and collimated by an integrated lens. The collimated beams are
diffracted by a volume hologram distributed over an area A of the slab waveguide.
The diffracted signals are focused by a second integrated lens into a set of N z
output channel waveguides. In this section we derive conditions such that a
unique holographic grating couples light from each input channel to each output
channel. If the gratings are weak enough that the hologram does not deplete the
input beams, the fraction of the lth input which is diffracted to the m th output
is linear in the corresponding grating amplitude and the diffraction of light from
the inputs to the outputs may be regarded as a vector-matrix multiplication.
As shown Fig. 5.1, we define the z axis to be the principle axis of propagation in the waveguide. The x axis is normal to the plane of the waveguide and

162

OUTPUT CHANNELS

INPUT
CHANNELS

SUBSTRATE

Figure 5.1. Integrated optical vector-matrix multiplier architecture.

the y axis is transverse to the z axis in the plane of the waveguide. The origin
of the coordinate system is at the entrance face of the hologram at the center

of the waveguide. \Ve assume that the \vaveguide has a boundary vvith air at
x = -d/2 and a substrate boundary at x = d/2. In the absence of a holographic

perturbation, the regions x < -d/2, -d/2 < x < d/2 and d/2 < x are homogeneous and isotropic. The indices of refraction of these regions are n1, n2 and n3,
respectively. We assume that n1 < n3 < n2. The guided modes of this system
have been analyzed by a number of workers [139, 137]. In considering the system
of Fig. 5.1, the most important results of this analysis are that the dependence
of the guided modes on x is separable from the dependence on z and y and that
the wave normal curve for the longitudinal components of the guided modes is a

163

circle. We will assume that d and (n1, n2, n3) are such that the slab waveguide
in Fig. 5.1 supports only one mode in x.
This system is most easily visualized in the Fourier domain. The signal from
the [th input channel is transformed into a plane wave propagating in a unique
direction, with wavevector f(l). Similarly, the signal incident on the m th output
channel is associated with the wavevector f(m). The hologram is a perturbation,

-6.n( r), to the index of the refraction of the waveguiding material. We can expand
-6.n( r) in a Fourier series:
-6.n( r) =

L KgejKg•i\

(5.1)

where Kg and Kg are the amplitude and the wavevector of the g1h Fourier component. Although only one index is shown, the sum in Eq. (5.1) is over three
dimensions in the space containing Kg. Since the purpose of the hologram is
to couple guided modes, we will not be interested in Fourier components corresponding to Kg with large components in the out-of plane (x) direction. Note
that -6.n( r) = 0 in the region x < -d/2.
At a given wavelength, the longitudinal components of the wavevectors of the
guided modes of a planar waveguide are constrained to lie on a set of wave normal
curves in the guiding plane. In a single mode homogeneous isotropic waveguide,
these curves take the form of a circle of radius k = 21meff / >., where >. is the
free space wavelength of the guided light and neff is the effective index of the
waveguide. A set of input and output wavevectors on this wave normal circle is
sketched in Fig. 5.2. Each of these wavevectors corresponds to an eigenmode of
the waveguide. In the presence of a holographic perturbation, the plane waves

164

OUTPUT
\JAVEVECT □ RS

INPUT
\J AVEVECT □ RS

Figure 5.2. Optical wavevectors of the coupled guided beams on the wave normal
curve. The connecting grating wavevectors are shown as dashed lines.

165

corresponding to these wavevectors cease to be eigenmodes. If the perturbation
is weak, fields in the waveguide can be described in terms of the plane wave
eigenmodes weighted by slowly varying amplitude functions. These amplitude
functions are determined using coupled wave theory [9). The coupling strength
between the 1th input and the m th output is proportional to the amplitude of the
Fourier component of f:ln with wavevector
(5.2)
In Fig. 5.2, this condition means that the wavevector of the grating which couples
an input to an output must join the endpoints of the coupled optical wavevectors.
A set of grating wavevectors coupling the fields represented in the figure is shown
by the dashed lines.
Let

of the incident and diffracted fields. a is the angle between adjacent plane wave
components. The angle between the wavevector corresponding to the 1th input
channel and the z axis is -(

input is
f(l) = k cos(

+ la )z - k sin(

(5.3)

~ k[cos() - lasin()]z - k[sin()

+ lacos()]y,

where x, y and, z are unit vectors along the corresponding axes and we have made
the paraxial approximation; la ~ 1. The angle between the wavevector corresponding to the m th output channel and the z axis is

wavevector is
f(m)

= k cos(

(5.4)
~ k[cos() - masin()]z

+ k[sin() + macos()]y.

166

The field, E( r), in the waveguide may be represented by a weighted sum of
the incident and diffracted fields. We represent the x dependence of the guided
modes by the normalized transverse field distribution £(x ). Coupling from the
input modes to the output modes results in a z dependence in the amplitudes of
the fields in each mode. Letting W1(z) and m(z) represent the amplitudes of the
fields corresponding to the 1th and m th input and output channels, respectively,
the field in the waveguide is

E( r) = £( x)

L W1(z )eff(l).Pez + £( x) L
(5.5)

where pis the position vector in the y-z plane er and em represent the polarization
vectors of the corresponding modes. In the absence of a hologram '111 and m
are independent of z. The effect of the holographic coupling is determined by
substituting E( r) into the wave equation

\7 x \7 x E - n2( r)k 2 E = 0,

(5.6)

where k = 2rr / A and n( r) is the index of refraction. Applying the slowly varying
envelope approximation and keeping only the first order in .6..n, this approach
yields

(5.7)

where n 0 is the index distribution in the absence of the perturbation. The next
step in coupled mode analysis is to match terms at identical spatial frequencies,

167

which allows us to convert Eq. (5.6) into a set of coupled linear equations, with
each equation describing how one of the modes is coupled to the rest.

The

difficulty in this case arises from the fact that the wavevectors of the guided
modes are confined to the y - z plane, whereas the grating wavevectors can point
in any direction (notice the distinction between p and f' in Eq. (5.7)). When
the recorded grating has a strong component in the x direction, guided modes
can only be coupled to radiation or substrate modes. Since we are interested
in coupling guided modes to one another, we must keep the grating wavevector
approximately confined in the y - z plane. If the guided modes are tightly
confined, Eq. ( 5. 7) can be limited to the vicinity of the guiding region. Integrating
Eq. (5. 7) across the waveguide in the x direction, we obtain:

The coupling terms due to a given Kg are significant only if IKgxl is small compared to 41r / d. We will assume that this condition is satisfied for ~11 K corresponding to nonzero Kg, In this case, sin(Kgxd/2)/Kgxd/2 ~ 1. Eq. (5.8)
must hold independently for each distinct Fourier component. If we multiply
through the equation by e-fi?s)•P and integrate over the area of the hologram,
the harmonic dependence, e-ff(r),iJ, of each term in Eq. (5.8) is replaced by

(5.9)
where !:).krs = fand z, Ly and Lz. The separation in Fourier space between the wavevectors of the

168

input and output modes is chosen such that the integral of Eq. (5.9) effectively
vanishes if r =J. s. This implies that
,\

,\

a>------,

..

21meJJLy cos 21rneff Lz sm

(5.10)

In this case, multiplying both sides of Eq. (5.8) by e-/f
(5.11)

where fl.Kglm = Kg - (f<1) - f(m)). Note that we assume that Kg is zero for Kg
which would cross-couple the outputs. The coupling term between the l th output
mode and m th input mode is strong only if sin(AKglm(y)Ly/2) /(fl.Kglm(y)/2) and

sin(!::,,.Kglm(z)Lz/2) /(!::,,.Kglm(z)/2) are nonvanishing. This is the case if
....

IAKglm(y)

Tr

I < Ly

(5.12)

and
....

Tr

jAKglm(z) I < Lz

(5.13)

For the spacing between modes given by Eq. (5.10), these conditions are satisfied
only if I{g = I<1m = f(!) - f(m). Discarding terms for which this condition is not
satisfied, Eq. (5.11) becomes

(5.14)

where "'Im is the amplitude of the grating at spatial frequency Kim. A similar
equation may be derived to describe coupling from the output modes to the input
modes.

169

Since we are interested in using the system of Fig. 5.1 to implement linear
transformations, we assume that the diffraction efficiency of the hologram is very
weak. In this case, the input signals may be assumed to be undepleted by the
hologram. Let Him= -jLzn2k 2 "'1mfkim)_ The solution to Eq. (5.14) under the
assumption of undepleted inputs is

(5.15)

where [~]m = m(Lz), rih = w,(O) and we assume that m(O) = 0. Since each
component of H corresponds to the an independent grating, the hologram may
be used to implement an arbitrary linear transformation.
Assuming a diffraction limited system, the spatial bandwidth of the input and
output fields is B = 21r Ly cos/ >.J, where f is the focal length of the integrated
lenses. The number of modes which may be used in the input and output fields, R,
is equal B divided by the separation in Fourier space between modes, 21rneJJO'./ >..
Substituting for a from Eq. (5.10) and assuming that Lz sin = Ly cos¢>, R =

Asin¢>cos¢>/>.J, where A= LzLy. The number of connections a hologram in the
plane can make between channels is

(5.16)

If f 2 is of the same order as A, R 2 scales as A/>. 2 [138]. Assuming Ly cos 0/ f = l,

A= lcm 2 >. = .5µm, and= 1r/6, R 2 = 10 8 •

170

5.3

FORMATION OF INTEGRATED VOLUME HOLOGRAMS

In this section we consider methods for recording a hologram in a slab
waveguide using unguided light incident from above the waveguide ( along
the x axis of Fig. 5.1). Recording with unguided light yields two advantages
over recording with guided light. First, unguided recording avoids unintended
perturbations to the waveguide. If the waveguide is sensitive to the the guided
beams it is difficult to avoid crosstalk between input beams and between output
beams and damage to the waveguide outside of the holographic region. With
unguided control, the unintended gratings which give rise to crosstalk are not
recorded.

The problem of damage to the waveguide can be overcome if the

waveguide is not sensitive to the guided beams. In photorefractive materials, for
example, there is a threshold wavelength beyond which holograms may no longer
be written. This allows us to use choose a guided wavelength which does not
damage the waveguide while using a shorter unguided wavelength to control the
hologram. The second advantage of unguided control is that it allows us to control
the hologram in the guided plane with a single exposure. In contrast, multiple
exposures are needed to form an arbitrary hologram with guided recording beams.
As we saw in previous chapters, recording with multiple exposures is in general
undesirable because it greatly complicates dynamic control of the hologram and
because it is difficult to maintain a large dynamic range in multiply exposed
holograms.
We consider two algorithms for recording a hologram in the guided plane. In
the first case each of the N1N2 gratings in the plane is formed using a distinct
pair of control beams. In the second case the gratings are formed using a single

171

reference beam and N1N2 beams generated by a spatial light modulator (SLM).
The advantage of the first approach is that all the gratings may be recorded
in the plane with no out-of-plane components. The second
approach
sacrifices
precision in the spatial orientation of the recorded gratings in order to simplify
the generation of the recording beams and improve the modulation depth with
which each grating is recorded.
The distinction between the two recording techniques can be clarified using
a simple graphical technique. We have seen that for a grating with wavevector

Kg, the Bragg condition, Eq. (5.2), specifies the coupled optical wavevectors, f(l)
and f(m), uniquely in the guided plane. Eq. (5.2) may, however, be satisfied for
a given Kg by many pairs of unguided optical wavevectors. Consider the threedimensional wave normal surface sketched in Fig. 5.3. We assume for simplicity
that this surface is a sphere. Any pair of points on the surface which are joined
by Kg are Bragg matched to the grating. As sketched in Fig. 5.3, the locus of
such pairs is a cone whose axis is parallel to the grating wavevector. The cone
intersects the normal surface in two parallel circles separated from each other by
the grating wavevector.
Consider a 2-D version of Fig. 5.3 formed by projecting the wave normal
surface onto the guided plane. Such a figure is sketched in Fig. 5.4 . The circle
in this figure is the wave normal curve in the guided plane. The two parallel
lines are the projections of the circles on the cone of Fig. 5.3 . We will refer to
these lines as the "degeneracy lines" of Kg. The degeneracy lines are separated
by Kg, The optical wavevectors in the plane which are coupled by the grating
lie on the wave normal curve at the points where it intersects the degeneracy
lines. The backward propagating conjugates of the coupled pair, shown in the

172

--·--------

Figure 5.3. Wave normal surface showing a grating wavevector and its associated
degeneracy cone.

173

Figure 5.4. Wave normal surface projected onto a plane.

174

figure as dashed lines, are also coupled. Fig. 5.4 is equivalent to Fig. 5.3 in the
sense that there is a one-to-one correspondence between points on the interior of
the wave normal curve in Fig. 5.4 and points on the hemisphere of the normal
surface above the guided plane. Thus, each point on the interior of the wave
normal curve corresponds to an unguided beam incident from above the plane.
Each point on the wave normal curve corresponds to a guided beam.
We are now prepared to consider in more detail the two recording methods
mentioned above. In the first case, we record Kg using a pair of wavevectors separated by Kg on opposite degeneracy lines. As an example, Fig. 5.5(a) shows the
projection in the guided plane of a pair of unguided optical wavevectors which
could be used to record the grating of Fig. 5.4. Of course, our goal is to simultaneously record many gratings in the plane. Fig. 5.5(b) shows a set of grating
wavevectors ( as dotted lines) and the degeneracy lines associated with them. For
simplicity, the optical wavevectors coupled by these gratings are not shown. To
record these gratings from above the plane, we must select an appropriate pair of
optical signals for each pair of degeneracy lines. Cross-talk between the recording
signals need not concern us because gratings between optical wavevectors which
are not on opposite degeneracy lines will not in general be Bragg matched to
couple the signals in the plane. In Fig. 5.5( c) we have translated the grating
wavevectors of Fig. 5.5(b) along the associated degeneracy lines. Each endpoint
of a grating wavevector corresponds to an unguided optical wavevector which is
to be used to control that grating.
This recording method is unattractive for two reasons.

First, smce each

grating is written by a separate pair of control beams, 2N1N2 distinct signals must
be controlled independently. Second, the modulation depth of the fringe pattern

175

Figure 5.5(a). Projection in the guided plane of the normal surface and a pair of
unguided propagating wavevectors. The grating wavevector and the degeneracy
lines in the plane are also shown.

176

Figure 5.5(b ). Set of grating wavevectors and their associated degeneracy lines.
The grating wavevectors interconnect three input guided channels with three
output channels.

177

Figure 5.5( c). Grating wavevectors of Fig. 5.5(b) displaced along the degeneracy
lines.

178

corresponding to each grating is weak when all N1N2 gratings are recorded with
separate pairs of control beams. A much stronger modulation depth is achieved
if all N1N2 gratings are written using a single reference beam [97]. However,
a grating can be written from above the plane only when both of the writing
beams lie on its degeneracy lines. Thus, a set of grating wavevectors may be
recorded with a single reference beam only if there exists a point which lies on a
degeneracy line of each and every the grating. Since knowledge of two wavevectors
on a degeneracy line, i.e., a reference and an input, uniquely specifies a grating
wavevector, such an intersection point is impossible if there is fan-in or fan-out in
the interconnection matrix. The second recording method we consider overcomes
this dilemma by accepting a tilt out of the guided plane in the recording gratings.
Such a tilt is allowable because the readout beams are confined to a thin guiding
region. As ,ve saw in Eq. (5.8), the component of the grating wavevector out
of the guiding plane may be as large as 1r / d before the coupling strength of an
otherwise Bragg matched grating is seriously affected. Since the waveguide may
be only a few wavelengths thick, guided beams may reconstruct gratings with
fairly sizable out-of-plane components.

If we are not concerned about Bragg mismatch out of the plane of the waveguide, we can record the interconnection pattern of Fig. 5.5(b) with any group
of optical signals which record gratings with planar components matched to the
desired gratings. The projection in the guided plane of the endpoints of the
wavevectors of such a set of signals is similar to similar to Fig. 5.5( c) in the sense
that there exists a line between a pair of endpoints which corresponds to each
of the desired grating wavevectors. If we do not worry about Bragg mismatch
out of the plane, however, signals which record a grating need not lie on its

179

.,\ II I"'

If

, I
I I
I I
I /
I I

\ I
\ I
\ I
\ \

\\ fl
\'
\I

81

II

" II"
\I
\l 11

Figure 5.6. Projection in the guided plane of the endpoints of a set of unguided
recording beams for recording the interconnection matrix of Fig. 5.2 using a single
reference.

180

degeneracy lines. Since we would like to record all the gratings using a single
reference, we consider the set of recording beams with wavevectors corresponding to the endpoints shown in Fig. 5.6. The grating wavevectors shown in this
figure correspond to the interconnection pattern of Fig. 5.5(b) with an endpoint
of each the grating wavevectors shifted onto a single reference point. Since the
grating wavevectors no longer join points on their degeneracy lines, the actual
gratings recorded between the reference beam and the signal beams have out-ofplane components. However, if the out-of-plane component of the actual grating
wavevector recorded for each desired grating wavevector is less than 1r / d, then
the gratings recorded using the beams suggested by Fig. 5.6 will implement the
interconnection pattern shown in Fig. 5.5(b ). The remainder of this section is
dedicated to showing that this constraint on the out-of-plane components can be
satisfied for gratings recorded in a specific recording architecture.
An architecture for recording a hologram in the waveguide using a single
reference is shown in Fig. 5.7. The hologram is formed between the Fourier
transform of the signal recorded on a spatial light modulator and a plane wave
reference.

The SL:t-.1 consists of a 2-D array of S independently controllable

pixels. Each pixel controls a single grating in the waveguiding plane. We refer
to the pixel that controls the (lm )1h grating as the ( lm )ih pixel. The SLM lies
in the y'-z' plane. The optical axis of the SLM-Fourier lens system is along x'.
The optical axis for light propagating in the waveguide is along z. x is normal
to the waveguide and y is transverse to the propagation direction in the plane
of the waveguide. The x' y' z' coordinate system corresponds to the xyz system
rotated by an angle -0 about z. The reference beam is assumed to propagate
in a direction normal to z and at an angle 0 with respect to x. The geometry in

181

UREFERENCE

OUTPUT CHANNELS

INPUT CHANNELS

Figure 5. 7. Architecture for control of guided volume holograms with a single
reference and SLM

the plane is the same as in Fig. 5.1.
The cross sectional area at the plane of the waveguide of a beam of light
collimated from one pixel of the SLM is >.~F 2 / 8, where 8 is the area of a single
pixel, Ar is the wavelength of the recording light, and Fis the focal length of the
Fourier lens. If
(5.17)

where A is the area of the holographic interaction region in the waveguide, then
the light generated by a pixel centered at (y', z') is to a good approximation a

182

plane wave propagating with the wavevector
....

k = kr(1 -

= [kr(1 -

(U 2

+ V 2)
k2

2 T

(u2 + v2)

2 T2

) cos 0

+ [u cos 0 - kr(1 where u =

)x' + uy' + vz'

+ u sin 0]x

(u2 +v2)

2 T2

) sin 0]y

(5.18)

+ vz,

k '
k '
....
!:.!/-,
v = 7,
amd kr = f,:.
Let k1m
be the wavevector for light

collimated from the ( lm )th pixel. Ulm and VJm correspond to u and v for the

(lm )th pixel. Let Him represent the optical field at the lm th pixel . The field due
to the SLM near the plane of the waveguide may be written

U(x, Y, z) =eikr(xcos8-ysin8)

(5.19)

The reference field is R = eikr(xcosB+ysinB)_ The interference pattern between the
signal and reference fields is

XL

.k (
. 8)
Hime -J r xcos -ysm

( u2

Im

+t12

21<~

Im

ej(u1m(ycos8+xsin8)+vimZ)

lm

+ c.c.
(5.20)
Assuming that the holographic perturbation is linear in I( x, y, z ), the amplitude
of the Fourier component of the perturbation at spatial frequency
....

Kim =( UJm sin 0 -

cos 0

( u2

lm

+ v2 )

2kr

Im

)x

-(2krsin0-uimcos0-sin0
is proportional to Him.

(u2

+ v2 )

lm kr lm )y+v1mz

(5.21)

183

Suppose that we wish to control the interconnection pattern of Fig. 5.2 using
the architecture of Fig. 5. 7. To control the lm th grating with the lm th pixel we
must select u1m, vim, and B such that the components of I<1m in the guided plane
given by Eq. (5.21) are equal to Idiscarding second order terms we find kr sin B = k sin
cos
Ulm

= (l + m )ak--n
cos

(5.22)

17

V/m

= (l - m )ak sin
I<1m and
Eq. (5.18) we see that the lm th grating controlled by the field on a pixel centered
at (Yzm, Z1m) = ((l + m)aF(A,r cos/>.. cos B),

u- m)aF(>..r/ >..) sin

).

Gratings

which couple the 1th input to the output channels are controlled by the pixels on
the line

z'

= -tan
- -
- -B + 21a

F).r cos ---.
>. cos B

(5.23)

Gratings which couple the from input channels to the m th output are controlled
by pixels on the line

y' -

z'
F>..r cos 2ma
n·
tan cos 0
.,\ cos u

(5.24)

To ensure that the single reference exposure method does not violate the
Bragg condition out of the plane, we require that the x component of Eq. (5.21)
be less than 1r / d:

(5.25)
Let r be the radius of the active area of the SLM. Eq. (5.25) is satisfied for all l

184

and m if
(5.26)
Assuming that tan2 8 ~ (krd), Eq. (5.26) simplifies to

(5.27)

The number of gratings in the guided plane that we can control using the SLM
is S = 1rr 2 /8. Substituting from Eq. (5.27) we find

(5.28)

Substituting from Eq. (5.17), we see that Eq. (5.28) is satisfied if

(5.29)

S differs from the number of gratings we can distinguish in the plane, given as

R 2 in Eq. (5.16) , only by the factor >.. 2 / Ard,

5.4

EXPERIMENTAL RESULTS

In our experiments the SLM and lens of Fig. 5. 7 are simulated by a squarewave grating. The experimental setup is sketched in Fig. 5.8. The grating lies in
the y' -z' plane. The transmittance of the grating is a one-dimensional function,
t(e), which may be described by a Fourier series over harmonics of a fundamental

frequency I{1(, where { is a unit vector in the y' -z' plane. Assuming that a

185

GRATING

z' ~//REFERENCE

y'/4'

(£

Figure 5.8. Experimental out-of-plane recording system.
plane wave propagating along the positive x' axis is incident on the grating, the
wavevector of the n th harmonic of the optical field diffracted from the grating is

(5.30)

Comparing Eq. (5.30) with Eq. (5.18), we define Un= nI<1(f;' and Vn = nI<1{z'.
The field transmitted by the grating is

'°'

U( x,y,z ) = L.....t H ne jp<-;.).f'.

(5.31)

A hologram is formed in the waveguiding plane between this field and the reference beam R = eikr(xcosB+ysinB). The amplitude of the Fourier component of

186

this hologram at spatial frequency Kn = p0i) - kr ( cos Bx + sin By) is proportional

The component of the hologram at spatial frequency Ito the m th output only if Kn = Kim• Substituting Un and Vn from Eq. (5.27)
into Eq. (5.19) and solving for l and m we find K(n) = I<1m when

nKf (
cos B sin , )
l = - - cos,--+ - ak
cos sin
(5.32)

nK1(
cosB
sin,)
m = - - cos,-- - - ak
cos
where,= cos- 1

(! · y'). If tan,= -cosBtan then l = 0 and the n th harmonic

controls the coupling from the 0th input channel to the m th output channel,
where mis given by Eq. (5.32). { is parallel to the line of Eq. (5.23) in this case.
Similarly, if tan 1 = cos Btan from the output channel. corresponding to m = 0 to the lth input channel.
Fig. 5.9 is a sketch of the grating wavevectors formed in the guided plane
by the five lowest order Fourier components of a recording grating at various
rotations, 1 , of the grating in the y' -z' plane. A section of the wave normal
curve and the positions of the five grating wavevectors recorded on the normal
surface are shown for each value of,. In the case shown

¾= i.5, and

¥, = 0.02. At 1 = -tan- 1 (cosBtandifferent outputs.
We have recorded photorefractive holograms in single mode titanium indiffused waveguides on nominally pure y-cut LiNb03 substrates. Holograms were
recorded from above the waveguide using the 514 nm light from an Ar+ laser.
Guided 633 nm HeN e light was used to reconstruct the holograms. The ratio of

187

[:,I
II

Ii
ii,

-20 -15 -10 -5

10

15

20

Figure 5.9. Grating wavevectors in the plane verses rotation of the recording
grating.

188

Ch)

Figure 5.10. Diffracted HeNe signals at the output of a waveguide. The diffracting hologram was formed using unguided Ar+ control beams.

189

the guided HeNe wavelength to the Ar+ wavelength in the crystal was~ 1.5. The
area of slab waveguide used to write the holograms was ~ lcm 2 • As in Fig. 5.8,
holograms were formed between a plane wave reference and light diffracted from
a grating. For the experiments described here, we used a Ronchi grating with
100 lines/inch. By rotating the grating in the y 1-z1 plane, we were able to observe variations in the pattern of grating wavevectors stored in the plane similar
to those shown in Fig. 5.9. For
input was diffracted. The diffraction pattern end-coupled out of the waveguide
for this input is shown in Fig. 5.l0(a). The 14 lowest diffracted orders are visible.
(The angles inside the waveguide are given. The angle between the Ar+ beams
incident on the substrate was 5°. Small angles were used because the sample we
used is not guiding along y.)
We have assumed that the amplitude of the hologram we write in the waveguide is proportional to the recording intensity. This is not quite the case for photorefractive holograms, which are linear in the modulation depth of the recording
fringe pattern. If the intensity is fairly uniform, however, the distinction need
not concern us here. Under the assumption of linear recording, the strength of a
connection in the plane is proportional to Him• This is confirmed in Fig. 5. lO(b)
and (c). Fig. 5.lO(b) shows the squared spectrum of the Ronchi grating used
to make the hologram of Fig. 5.lO(a). Fig. 5.lO(c) shows, on a slightly different
horizontal scale, the power in the central spots of Fig. 5.lO(a). The noise in the
base lines of Fig. 5.10 is the dark noise of a CCD camera.

190

5.5

APPLICATIONS

Volume holograms in waveguides are applicable to systems which require fixed
or adaptive linear transformations. While the out-of-plane control method described here provides a simple method for producing integrated holograms which
fixed transformations, the use of this method for dynamic control of connections
in a plane is also promising. Such a dynamically controlled vector-matrix multiplier could find application in switching networks or in adaptive artificial neural
networks.
Semiconducting photorefractive materials are especially attractive for dynamic implementations of integrated volume holography. This is because photorefractive response times in these materials are relatively fast and, at least in the
case of GaAs, integrated technologies are well developed. Using GaAs, we may
expect to be able to monolithically construct architectures such as that shown
in Fig. 5.11. In this system an array of laser diodes is dynamically connected to
an array of photodetectors using a photorefractive hologram with out-of-plane
control. Since photorefractive effects arise on fairly short time scale in GaAs
[18], one could expect to reconfigure the interconnection matrix in this device in
a few microseconds. Depending on the number of input channels which could
be integrated, from 104 to 10 8 weighted interconnections could be stored in this
device. One problem with integrating a large number of input channels might be
the integration of more than 100 lasers on a single chip. This problem could be
overcome by fanning out the laser outputs to feed several input channels, each
channel being controlled by a simple modulator. The principle difficulty arising
in the fabrication of this device is the well known problem of monolithically inte-

191

~ATIAL LIGHT
---------------·
MODULATOR

:::>

C:

------------

REfERENCE

BEAM

FOURIER
LENS

DETECTOR

ARRAY

w'AVEGUIDING LAYER
CLADDING LAYER
Go.As SUBSTRATE

Figure 5.11. Monolithic integration of active devices and photorefractive
waveguides in GaAs.

192

grating active and linear waveguiding regions. While considerable further work
is needed to overcome this difficulty, many grating integrated holograms clearly
offer interesting new possibilities to holographic information processing.

193

6. LEARNING IN OPTICAL NEURAL NETWORKS
6.1

INTRODUCTION

The potential advantages of neural computation over alternative computational strategies lie in the ability of neural machines to "learn" to solve problems.

In contrast to conventional computers, where the system· designer must have detailed knowledge of the algorithm used to solve the problem, a neural computer
adapts itself to solving a given task without specific instructions. The detailed
programming of a neural machine consists of specifying a learning algorithm to
control the adaptation of the network. The network learns to produce the appropriate response to a class of inputs by being presented with a sufficient number of
examples. The presentation of these examples causes the strength of the connections between neurons that comprise the network to be modified according to the
specifics of the learning algorithm. A successful learning procedure will result in
a trained network that responds correctly to the examples it has seen previously
and also to other inputs that are in some sense similar to the known patterns.
When we consider a physical realization of a neural network model, we have
two options in incorporating learning capability. The first is to build a network
with fixed but initially programmable connections. An auxiliary conventional
computer is used to "learn" the correct values of the connection strengths. Once
learning is complete the network is programmed by the computer. While this approach may be reasonable for some applications, the second alternative, a system
with continuously modifiable connections is potentially much more powerful.
Both of these options can be implemented in volume holographic systems.

In chapter 3, we saw how a known linear transformation can be recorded in a

194

hologram. This procedure could be used to load a set of precalculated connection
strengths into a network. In the second section of this chapter, we describe an
experimental system for loading a prescribed set of weights into a hologram.
Since the weights are never changed after programming, the only measure of
the network's performance is the size of the network which can be constructed
at reasonable cost to operate with reasonable speed.

Due to the constraints

on the diffraction efficiency and dynamic range of multiply exposed holograms
described in section 2.4, section 3.4, and section 4.3, the potential of holographic
systems for implementation of large scale interconnection networks is somewhat
less overwhelming than it might seem at first glance. A comparison of optical and
electronic neural interconnections is presented in the last section of this chapter.
For dynamic learning applications, however, the ease with which learning can be
implemented in holographic systems substantially improves the attractiveness of
optical systems.
Most models of dynamic neural learning are based on Hebb's conjecture that
if a pair of neurons are consistently simultaneously active then strength of the
connection between them should increase [140]. In dynamic holographic systems,
the connection between a pair of modes grows in proportion to the activity of
the two modes. This fundamental correspondence between Hebbian learning and
hologram formation is perhaps the most compelling motivation for the development of holographic neural systems. In the third section of this chapter, an
experimental dynamic photorefractive learning system is described. Ultimately,
more complex learning systems could be implemented by separating the short
term behavior of the holographic system, controlled by the learning rule, from
the long term behavior, controlled by the periodic refreshing technique described

195

in section 4.5. This approach would allow direct linear implementation of learning processes with arbitrarily many exposures while avoiding unduly large losses
in the dynamic range of the stored hologram.
It is convenient to separate the wide range of learning algorithms that have

been discussed in the literature into three categories: prescribed learning, error
driven learning and self organization. We draw the distinction between these
algorithms with the aid of Fig. 6.1, where a general network is drawn with the
vector~( k) as its input and '}L( k) the output at the k th iteration (or time interval).
The vector£( k) is used to represent the activity of the internal units and Wij( k) is
the connection strength between the i th and the jlh unit. Let ~(m), m = 1 ... M,
be a set of specified input vectors and let

lm) be the responses which the network

must produce for each of these input vectors.
A prescribed learning algorithm calculates the strength of each weight simply
as a function of the vectors ~(m) and '}f_(m):
.. W SJ

f·IJ·(x(m)
y(m))
l_

m=l. .. M

(6.1)

This type of procedure is relatively simple ( "easy learning"). It is perhaps the
most sensible approach in a single layer network. The widely used outer-product
algorithm [34, 35) is an example of this type of learning algorithm, as are some
schemes which utilize the pseudoinverse (34,141, 142]. Despite its simplicity, prescribed learning is limited in several important respects. First, while prescribed
learning is well understood for single layer systems, the existing algorithms for
two layers are largely localized representations; each input ~(m) activates a single
internal neuron (143, 144, 145). Moreover, the entire learning procedure usually has to be completed a priori. This last limitation is not encountered in the

196

x(k)==>

===>

y(k)

Figure 6.1. General neural network architecture.

Input
Neurons

Output
Neurons

Figure 6.2. Two layer network with lateral inhibition. Connections ending with
an open circle are inhibitory.

197

simplest form of prescribed learning, the outer-product rule:

Wij

= ~ X~m)Y)m)

(6.2)

m=l

In this case new memories may be programmed by simply adding the outerproducts of new samples to the weight matrix. Note that once the interconnection matrix has been determined by a prescribed learning algorithm, it may be
expressed in the form of a sum of at most N outer-products, where N is the total
number of neurons in each layer. As we saw in chapters 2 and 3, volume holograms record interconnections matrices represented by sums of outer-products
in a very natural way. Thus, matrices which can be expressed in this form are
particularly simple to implement in optics [38, 39, 42, 43, 49].
Error driven learning is distinguished by the fact that the output of the
system, 'J!_(k), is monitored and compared to the desired response 'J!..(m). In each
learning step, an incremental change, l::iWij is made to the weight between the

i th and j1h neuron to reduce the error.

(6.3)

The l::iWij is calculated from the vectors ;r(m) and 'J!..(m) and the current setting
of the weight matrix Wrs(k) (from which the state of the entire network can be
calculated). The perceptron [146] and adaline [147] algorithms are examples of
error driven learning for single layer networks. Interest in such learning algorithms has been renewed recently by the development of procedures suitable for
multilayered networks [148, 33, 149]. Error driven algorithms ("hard learning")
are more difficult to implement than prescribed learning since they require a

198

large number of iterations before errors can be reduced to sufficiently low levels.
In multilayered systems, however, this type of learning can provide an effective
mechanism for matching the available resources ( connections and neurons) to the
requirements of the problem. In optical realizations error driven algorithms are
more difficult to implement than prescribed approaches due to the need for dynamically modifiable interconnections and the incorporation of an optical system
that monitors the performance and causes the necessary changes in the weights.
While this problem could be avoided by performing learning off line in computer
simulations and recording the optimized interconnection matrix as in prescribed
learning, this approach has the disadvantage that once again the matrix is fixed

a priori, thus preventing the network from being adaptive.
In the case of self organizing learning algorithms we require not that the
specified inputs produce a particular response but rather that they satisfy a
general restriction, often imposed by the structure of the network itself. Since
there is no a priori expected response, the learning rule for self organizing systems
is simply

(6.4)
This type of learning procedure can be useful, for instance, at intermediate levels
of a network where the purpose is not to elicit an external response but rather to
generate appropriate internal representations of the information that is presented
as input to the network. There is a broad range of self organizing algorithms,
the simplest of which is probably lateral inhibition to enforce grandmother cell
representations [34, 150). The objective of the learning procedure is to have each
distinct pattern in an input set of neurons activate a single neuron in a second

199

set. In the architecture shown in Fig. 6.2 this is accomplished via inhibitory
connections between the neurons in the second set. Once a particular neuron
in the second layer is partially turned on for a specific pattern it prevents the
connections to the other neurons in the second set from assuming values that
will result in activity at more than one neuron. The details of the dynamics of
such procedures can be quite complex ( e.g., [151]), as can corresponding optical
implementations. An advantageous feature of optics in connection with self organization is that global training signals, such as fixed lateral inhibition between
all the neurons in a given layer, can easily be broadcast with optical beams.
While the linearization techniques described earlier in this thesis make it possible to implement any learning scheme in volume holography, little is gained if
the system used to control the hologram must perform more complex computations than the hologram itself. For this reason, it is generally desirable for
mapping between the abstract learning algorithm and its physical realization be
as direct as possible. In the following two sections, we describe experimental
holographic neural systems which implement learning in a very direct manner.
The first system implements prescribed outer-product storage. The second implements perceptron style learning.

6.2

A PHOTOREFRACTIVE OUTER-PRODUCT MEMORY

In chapters 3 and 4 we saw how connections between a large number of pro-

cessing nodes can be formed in a volume hologram. The simplest demonstration
of neural processing using photorefractive connections is an outer-product style
memory based on the exposure schedule described in chapter 4. We have used
the sampling grids shown in Fig. 6.3 to construct an outer-product style asso-

200

•••••••••••••••••••••••••••••••••••••••••••••••••••••••••• .... ••••••o•••••••••••••••••••••••••••••••••

••• •••• • ••i1•••••••••• •i..&ii • • • • 1 •• •••• • •• • ••• • 1ee e •••••• ee••• •••••••••• ••• 1 •• ••••• ••• 8 ••••• ••••• ••••••

....................................................................................................
•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••eeeee:.••••;.

....................................................................................................

Figure 6.3. Sampling grids for outer-product memory. The input consists of 10
tightly packed rows of 100 pixels, as shown at top. The output consists of 5 rows
of 100 pixels spaced by 10 unused rows, as shown below.

201

OUTPUT
PLANE

TRAINING

PLANE

FOURIER

LENS

VOLUME
HOLOGRAM

FOURIER
LENS

Figure 6.4. Photorefractive outer-product memory architecture.

ciative interconnect system. The experimental system is sketched in 6.4. The
images were introduced into the system using a computer-controlled liquid crystal television monitor read out by an argon ion laser operating at a wavelength
of 488 nm. A liquid crystal light valve was placed at the input plane to improve
the contrast of the images before they were introduced to the holographic system. The holographic medium was a cesium doped strontium (0.6) barium (0.4)
niobate crystal.
Volume holograms between associated patterns on the input and output grids
were formed in sequence. Because the patterns were constrained to lie on sampling grids, the transformation stored in each hologram was a true outer-product

202

of the associated patterns. The m th associated pair of patterns was recorded with
an exposure Em given by

Em= E 0 log(m - l ),

(6.5)

where E 0 is the saturation exposure. The first associated pair was recorded to
saturation. In our crystal, E 0 ~ 200 mJ / cm2 • As described in chapter 4, this
exposure schedule was necessary to insure that the hologram stored in the volume
was a linear sum of the outer-products of each of the associated input-output

The patterns stored in the system consisted of random code-name pairs such
as those shown in Fig. 6.5. The names were arranged on 10 x 100 pixel arrays.
The random codes were arranged on five 100 pixel rows spaced by 11 unused rows.
In an effort to increase the orthogonality of the recorded images, a piezo-electric
mirror was used to add a random phase to the active pixels for each name.
We stored up to 20 image pairs using this system. Between recording cycles,
the fidelity with which previously recorded images could be reconstructed was
evaluated by introducing stored names at the input and detecting reconstructed
codes on a CCD camera at the output plane. For a small number of exposures, the
fidelity of the reconstruction (recall) was good, confirming the effectiveness of the
sampling grids at ensuring independent interconnections. As more names were
stored it became increasing difficult to reconstruct previously recorded patterns.
This was partly due to correlations between the stored patterns, but was chiefly
a result of a decrease in the diffraction efficiency per stored pattern inherent in
the multiple exposure process.

203

Figure 6.5. Examples of pairs of associated images.

204

RELATIVE DIFFRACTION VS. M - NAME 5

0.2

cc:

-0.2
-0.4

-0.6

-0.8

-1

1¥

§ -1.2
-1.4
-1.6

0.5

0.6

0.7

0.8

0.9

1.1

1.2

1.3

....
1 A

1.5

LOG NUMBER OF HOLOGRAMS

Figure 6.6. Log diffraction efficiency for the 5th association vs. log number of
associations. The slope of the least squares fit is -1.95

205

As predicted in chapter 4, we observed a fall-off in the diffraction efficiencies
of the stored holograms as more holograms were loaded into the system. Due to
the fact that the stored patterns were not orthogonal, the diffraction efficiencies
of the different stored images did not degrade uniformly. We found decay rates
versus the log of the number of stored patterns which varied from -1.2 to just
over -2. This fall off in diffraction efficiency is illustrated in Fig. 6.6, which shows
the efficiency with which one of the patterns is reconstructed vs the number of
patterns stored on a log-log scale. The slope of the experimental best-fit line
is -1.95. The theory derived in chapter 4 predicts a slope of -2. This decrease
in diffraction efficiency under multiple exposure will limit the number of images
which can be stored in currently available photorefractive materials to the range
of several hundred to one thousand [125).

6.3

A PHOTOREFRACTIVE PERCEPTRON

The perceptron [146, 36] is an example of a supervised adaptive neural model
which can be implemented using photorefractive crystals. A perceptron consists
of a set of input neurons with activities described by a vector x which drive a
single output neuron via a weight vector w. The activity of the output vector is
high if and only if w·x > w 0 where w 0 is a fixed threshold level. A perceptron can
be trained to separate a set of input vectors into two classes by various methods,
the simplest of which involves updating the weight vector according to

w( n + 1) = w( n) + ax
(6.6)

w0 (n+l)=wo(n)+a
where w( n) is the state of the weight vector at the discrete time n when x is
presented for classification. a is zero if x is correctly classified and 1 (-1) if

206

x is misclassified in the low (high) state.

If training vectors from a set { x}

are presented in sequence, Eq. (6.6) is known to converge on a weight vector
implementing an arbitrary prescribed dichotomy if such a weight vector exists.
One means of implementing a perceptron in a photorefractive system would
be to update each interconnection in series as prescribed by Eq. (6.6). This
approach has two disadvantages. The first is that the saturable nature of the
photorefractive response limits the range of Wi to (-w 8 at, Wsat) for coherent systems and (0, Wsat) for incoherent systems. The weights determined by Eq. (6.6)
may not be guaranteed to lie within these bounds. The second disadvantage to
this approach is that each weight must be updated independently. In order to
update the weights in this way we would need to detect the value of each weight
and generate optical beams specifically to change that weight by the prescribed
amount.

In a volume holographic implementation with spatially multiplexed

weights it is not possible to change the weights independently.
These problems can be avoided by modifying the learning procedure to conform more closely to the dynamics of hologram formation. This can be done
particularly simply in an incoherent system. An architecture for an incoherent
photorefractive perceptron is shown schematically in Fig. 6. 7. Much of the complexity of the systems described in the previous section is avoided in this system
so that we can concentrate on the use of photorefractive dynamics in learning.
The input to the system, x, corresponds to a two-dimensional pattern recorded
from a video monitor onto a liquid crystal light valve. The light valve transfers
this pattern onto a laser beam. This beam is split into two paths which cross in
a photorefractive crystal. The light propagating along each path is focused such
that an image of the input pattern is formed on the crystal. The images along

207

lo.ser

'-------'rt

PB

LCLV TV

ul.

BS$-

coMputer

Xto.l

P L2 D

,B-0-··8···-

Figure 6.7. Photorefractive perceptron. PB is a polarizing beam splitter. 11 and
12 are imaging lenses. WP is a quarter waveplate. PM is a piezoelectric mirror.
Pis a polarizer. Dis a detector. Solid lines show electronic control. Dashed lines
show the optical path.

208

both paths are of the same size and are superimposed on the crystal. The intensity diffracted from one of the two paths onto the other by a hologram stored in
the crystal is isolated by a polarizer and spatially integrated by

a single output

detector. The thresholded output of this detector corresponds to the output of
a perceptron. The fact that the connections in this system are stored locally in
the image plane of the input allows us to very simply control each connection
independently. This is at a cost, however, of the loss of the high connection densities achieved by using the entire volume of the storage medium in the Fourier
domain.
The i th component of the input to this system corresponds to the intensity in
the i th pixel of the input pattern. The interconnection strength, Wi, between the

i th input and the output neuron corresponds to the diffraction efficiency of the
hologram taking one path into the other at the i th pixel of the image plane. In
analogy with Eq. (6.6), Wi may be updated by exposing the crystal with the input
along both paths. If the modulation depth between the light in the two paths is
high then where Xi is high Wi is increased. If the modulation depth is low between
the two paths then where Xi is high Wi is reduced. The modulation depth between
two optical beams can be adjusted by a variety of simple mechanisms. In Fig. 6.7
we choose to control m(t) using a mirror mounted on a piezoelectric crystal.
By varying the frequency and the amplitude of oscillations in the piezoelectric
crystal we can electronically set both m(t) and (t) over a continuous range
without changing the intensity in the optical beams or interrupting readout of
the system.
We have implemented the architecture of Fig. 6. 7 using a SBN60:Ce crystal.
We used the 488 nm line of an argon ion laser to record holograms in this crystal.

209

Most of the patterns considered were laid out on 10 x 10 grids of pixels, thus
allowing 100 input channels. Ultimately, the number of channels which may be
achieved using this architecture is limited by the number of pixels which may
be imaged onto the crystal with a depth of focus sufficient to isolate each pixel
along the length of the crystal.
Using the variation on the perceptron learning algorithm described below
with fixed exposure times ~tr and ~te for recording and erasing, we were able
to correctly classify various sets of input patterns. An example of such a set is
shown in Fig. 6.8. In one training sequence, we grouped patterns 1 and 2 together
with a high output and patterns 3 and 4 together with a low output. After all
four patterns had been presented four times, the system gave the correct output
for all patterns. The weights stored in the crystal were corrected seven times,
four times by recording and three by erasing. Fig. 6.9(a) shows the output of the
detector as pattern 1 is recorded in the second learning cycle. The dashed line
in this figure corresponds to the threshold level. Fig. 6.9(b) shows the output of
the detector as pattern 3 is erased in the second learning cycle.
Applying the results of chapter 4 to this system we find that

(6.7)
Two problems prevent the use of the exposure schedule in this system. The first
is that the assumption of approximately constant intensity in each exposure is
violated in an incoherent image plane system. The second is that, while the
perceptron algorithm is known to converge, the number of training steps needed
to reach convergence can be very large. This second problem could be corrected
by using the periodic copying technique described in chapter 4.

210

.. ..
-1

1111

..

.,...._

Figure 6.8. Training patterns.

....

C:

C:

.p
.p
0.

+>
+>
a.

QI

QI

4-

--

4-

.p
::,
0.
op
::,

op

::,
0.
op
::,

seconds
Figure 6.9. Detector output during trroning.

seconds

........

212

In lieu of using an exposure schedule in our simple perceptron we assume
that t( s) = 6.te in cycles in which erasure occurs and t( s) = 6.tr in cycles in
which writing occurs. Since r is inversely proportional to the optical intensity,
we can express ¼at each pixel of the input as axi. Letting mi(t) = 0 when the
n th training vector yields too high an output, we find from Eq. (6.7) that the i th

component of the weight vector is updated according to
(6.8)
Let mi(t) = m when the n th training vector yields too low an output, we find
that the the ith component of n + l th weight vector is
Wi(n

+ 1) =e-2~trax;wi(n) + m2Wsat(l - e-~trax;)2
(6.9)

Note that, as one might intuitively wish, Wi decreases monotonically with Xi
if the output for xis too high and increases monotonically with Xi if the output
for x is too low. Eqs. (6.8) and (6.9) may be greatly simplified if we assume
that Xi takes on only the values O and 1. Since the output power incident on
the detector may be arbitrarily renormalized, we may also assume without loss
of generality that m 2 w 8 at = 1. Under these assumptions Eq. (6.8) may be recast
in the form

(6.10)
where 'Ye = e-~tea. Eq. (6.9) becomes

(6.11)

213

...
...

:::,

C.

is

20

,~

~~

. ur

~J

60

80

ll

V'J

100

120

PRESENTATION NUMBER

Figure 6.10. Learning curve from simulations.

214

20

40

60

80

100

120

140

160

PRESENTATION NUMBER

Figure 6.11. Experimental learning curve.

180

200

215

We assumed both in our simulations and in our experiments that an acceptable value for w 0 could be guessed a priori. This assumption is unnecessary if
one of the stored weights is used as a threshold. An example ~f convergence
in simulations run under this training algorithm is shown in Fig. 6.10. In this
figure we plot fJ x (w(n) · x(n) - Wo) as a function of n. fJ is 1 (-1) if x(n) is in

n+ (f2_). The system has converged when a cycle through all stored vectors
yields all positive outputs. This example involves ten randomly selected and
classified training vectors drawn from a hundred-dimensional space. n+ and n_
each contain five vectors. The training vectors are presented twelve times in sequence before a solution vector was found. In the thirteenth sequence through
the vectors h( w( n) • x< n) - w 0 ) > 0 for all n, indicating that a solution has been
obtained. 1 was 0.9 and the weights were randomly initialized. A learning curve
under similar circumstances in the experimental system is shown in Fig. 6.11. In
this case nineteen cycles were necessary to converge on a solution vector. Most
of the discrepancy between the experimental system and the simulations arises
from the difficulty involved setting the learning parameters and initial conditions
to be identical. The key trend to notice in both figures is that the amplitude
and frequency of negative outputs decreases with each cycle until convergence
is reached. We found that it was relatively easy to achieve convergence both in
simulations and in the experimental system.
A final point to notice about Fig. 6.10 and Fig. 6.11 is that convergence is
achieved even though the exposure involved in each training cycle is large. We
know from Eq. (6.7) that the effect of a given exposure on the final state of Wi
is decreased by at least a factor of 1 q, where q is the number of exposures made
after the exposure of interest. Since 1 q ~ l for q > 10, information recorded

216

early in the training sequences is erased from the crystal before convergence is
reached. The early exposures drive the system toward a region of weight space
from which convergence is achieved, rather than store information.

6.4

OPTICAL VERSUS ELECTRONIC NETWORKS

In "electronic" implementations of neural networks, neurons and the channels

which interconnect them are integrated on the same 2-D surface. In large scale
networks the area devoted to the interconnections is much greater than the area
devoted to the processing nodes [152, 153]. In "optical" implementations neurons are again arranged on a plane, but the addition of light sources, modulators
and detectors allows internodal communication to be done optically through the
third dimension. The advantages of electronic neural processors are that they
can be constructed using conventional processing techniques, typical device sizes
are small and device speeds are respectable. The advantages of the optical implementations are that no expensive lithographic area need be devoted to linear
devices and wiring and adaptation can be implemented in a direct and simple
manner [102]. In this section we compare optical and electronic implementations as a means of putting the techniques described in this thesis in perspective
and outlining situations in which holographic optics may be of use in massively
parallel processing.
A number of previous studies have considered the relative advantages of optical and electronical systems. Barakat and Reif derive a constraint on the scale
of problems which an optical system in a volume V can resolve in time T [154].
Their analysis is based on an analogy with similar constraints which have been
derived relating the integrated area of VLSI circuits and T to the scale of the

217

problem the circuit can resolve. In this thesis we are interested in a much more
limited class of problems than was considered by Barakat and Reif. The optical systems we consider are also considerably less powerful, since we allow only
changes to the linear properties of each voxel. We have assumed that each layer
of the optical system performs the largest possible linear transformation in each
time step.
Studies closer to the systems of interest in this thesis were completed by
Goodman et al. [155], Feldman et al. [156), Wu et al. [157], and Miller [158).
These studies consider the relative energy efficiencies of optical and electronic
interconnections. For our purposes, the key result of these analyses is that the
power in communicating a bit of information is lower bounded by

CV 2
P=--,

2r

(6.12)

where C is the capacitance charged to bring the detecting device to its threshold
voltage V. T is the time used to communicate the bit.
Consider the following question: What are the maximum size and speed for
a network with c connections per neuron implemented using a chip of area A
dissipating power P if (a) the neural interconnections are purely electronic or
(b) the neural interconnections are optical. We answer this question for three

different types of connectivity:

(1) The input to each neuron is a weighted sum of the activities of its c nearest
neighbors, as is the case in the volume holographic interconnection systems
described in this thesis.

218

(2a) Each neuron receives unweighted the activities of its c nearest neighbors.
The electronic implementation uses an independent wire for each connection.

(2b) Each neuron receives unweighted the activities of its c nearest neighbors.
The electronic implementation uses active "axons" to broadcast the activity
of each neuron.

Let Rs be the ratio of the number of neurons in the optical network and the
number of neurons in the electronic network, Rr be the ratio of the response time
of the optical net and the response time of the electronic net, and R1 =

t be the

ratio of the rates at which interconnections are made in the two networks. PA is
the ratio between the area of a single processing element in the optical network,

8;, and the area, 8;, of a single processing element in the electronic network. p E is
the ratio of the energy required to charge a single element of the optical network
and the energy required to charge a single element of the electronic network. 'T/o
describes the maximum diffraction efficiency of a single holographic grating in
the optical system.
In cases (1) and (2b), the electronic layout can be done using a crossbar
network. In this case each connection takes up one unit of area on the chip.
Sketches of appropriate layouts for the electronic circuits are shown in Fig. 6.12
and Fig. 6.13. In the optical case only active devices are integrated on the chip.
Thus, Rs in these two cases is PAC. In case (2a), the area of the connections between neurons grows as cz as the scale of the chip grows, as is seen in Fig. 6.14. In
this case, Rs is PA cz. In every case, the capacitance of the system is proportional
to the area of the chip, which is fixed. As we saw in this chapter and in chapters

219

Figure 6.12. Electronic layout for case (1).

220

Figure 6.13. Electronic layout for case (2b ).

221

Figure 6.14. Electronic layout for case (2a).

222

3 and 4, however, the diffraction efficiency of weighted volume holographic interconnections scales inversely with c2 • This implies that Rr = PEC 2 /r, 0 in case 1,
where r, 0 is the ratio of the maximum perturbation in a hologram to the perturbation which causes diffraction efficiency 1. Our results are summarized in the
following table.
OPTICAL VS. ELECTRONIC NETWORKS
Case

2a

2b

Rs

PAC

PAC2

PAC

Rr

PEC2

l!.1l

l!.1l

7/o

7/o

7/o

R1

&.'!12. &.r, c2
PE C

PE

i!.d.r,
PE o

The significance of these results depends on the relative costs of integrated
area and power and on the size of the scaling parameters. Obviously, if integrated area is the key parameter, the advantage goes to optics when large scale
interconnections are required. For networks of of sufficient size and interconnection density, optical connections yield advantages in both speed and size in cases
(2a) and (2b ). These systems forgo what may be one of the key advantages of
optics, however, which is that holography allows us to implement Hebbian style
adaptation directly in hardware. This advantage is achieved in case (1) at a cost
of loss in dynamic range, or diffraction efficiency.
This problem brings us back to where we began this thesis, with the the hope
volume holography offers for implementing very large linear transformations, if
a means can be found to control the hologram without wasting all its dynamic
range. In this thesis, we have demonstrated this problem by experimentally documenting the decay in diffraction efficiency of multiply exposed holograms. We

223

have shown that 770 can be made larger than 1 in photorefractive crystals by increasing the imaginary component of the time constant. We should note that in
certain crystals 17 0 is much greater than one in any event [125). We have considered copying techniques which allow us to keep adaptively recorded holograms
at the c- 2 decay rate over arbitrarily long exposure cycles. We have considered the possibility that the scaling may be reduced to c- 1 using polychromatic
control. And we have shown how this problem can be overcome in integrated
architectures.
With these results in mind, it is my belief that Rr for case (1) can be made
to remain within one or two orders of magnitude of 1 out to the maximum
connectivity which can be supported in a single aperture. If this goal is achieved
the long postulated advantages of optical holography in signal processing will be
accessible.

224

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