DEVELOPMENTS IN PHOTOREFRACTIVE
TWO-BEAM COUPLING SYSTEMS
by
Valéria B. Damião
B.A., University of Brasília, Brazil, 1990
M.S., University of Colorado, USA, 1995

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillmentof the requirements for the degree ofDoctor of Philosophy
Department of Electrical & Computer Engineering
2000

This thesis entitled:
Developments in Photorefractive Two-Beam Coupling Systems
written by Valéria B. Damião
has been approved for the
Department of Electrical & Computer Enginneering___________________________________
Dr. Dana Z. Anderson___________________________________
Dr. Kelvin Wagner

Date_______________________

The final copy of this thesis has been examined by the signatories, and wefind that both the content and the form meet acceptable presentation
standards fo scholarly work in the above mentioned discipline.

Damião, Valéria B. (Ph.D. Electrical Engineering)
Developments in Photorefractive Two-Beam Coupling Systems.
Thesis directed by Dr. Dana Z. Anderson

Photorefractive systems hold the promise for improving the performance ofphotonic signal processing applications. This thesis describes several developments
pertaining to such systems, in particular to those employing two-beam coupling.
The utility of photorefractive systems is stymied by the lack of a componentoriented technology. Conventionally, an entire optical table, full of mirror mounts andother optical components, is required to build a photorefractive system. To address
this shortfall, two-beam coupling modules were designed and fabricated, whose ports
were standard multimode fiber optic connectors instead of free-space propagating
beams, dispensing with the need for time-consuming alignment and adjustments.
These modules employ lenses to couple light into and out of our chosenphotorefractive medium, crystalline barium titanate. We demonstrate the utility of themodular concept by implementing an autotuning filter constructed solely by thesemodules.
For even greater integration of the two-beam coupling units, we present a
novel spherical crystal geometry which incorporates the functions of both the lenses
and of the photorefractive medium in a single element. While investigating these
spherical crystals, we observed internal whispering-gallery mode oscillations. Theseoscillating patterns adopted simple polygonal shapes depending on the pump entry
and crystal-axis angles. We modeled the occurrence conditions of the lowest orderiii

triangle pattern by assuming a single two-beam coupling interaction, using standard
plane-wave theory.
Also, as part of a higher-level photonic system, we required a carrier
suppressor to remove unwanted correlation between different signal-containing
beams. By use of two-beam coupling in the novelty-filter configuration, we obtained
more than 70 dB of carrier suppression, to our knowledge, the highest reported in theliterature. The use of a simplified, geometrical interpretation of an operator theory
for two-beam coupling, which provided the proper choice of parameters needed for
perfect suppression, was crucial in obtaining this result.
Finally, the full formalism of the operator theory for two-beam coupling is
introduced. This formalism provides the freedom of representing the information tobest suit the information-processing problem at hand. A closed-form solution forcomplex coupling is presented for the case of a single spatial mode interaction, the“2-by-2” case. The already reported solutions for pure-real coupling, corresponding
to energy transfer only, and that for pure-imaginary coupling, phase transfer only, aregiven as special cases of the 2-by-2 complex coupling solution.

iv

A meus queridos pais, pela constante e incansável dedicação.

ACKNOWLEDGMENTS

I gratefully acknowledge the following:First, Prof. Dana Anderson, for the opportunity to work on a well equippedlaboratory and for the financial support (with the exception of the last semester)
without which this Ph.D. would not have been possible. I am grateful for the mostsalient, eye-opening, of his teachings: the essentiality of socio-political andcommunication skills for a successful career.
Most of all, Bruce Tiemann, my partner and unofficial experimental advisor,
for his dedication in teaching me the workings of nature in so many different aspects
and for helping me develop a set of laboratory skills of fundamental importance on
the work developed in this thesis.
Specially, Vladimir Schkunov and Ben Thompson, for their patience inelucidating numerous theoretical questions, and Prof. Kelvin Wagner for kindly
listening and providing academic counseling.
Finally, all colleagues who participated in one way or another in my learning
experience. In particular, Leslie Czaia, for preparing several of the crystals used inour experiments, for patiently sharing her talents in crystal processing techniques, andfor guiding us through the hidden nooks of our laboratories.

vi

CONTENTS

Chapter
1. Introduction
1.1. Motivation ...............................................................................................

01

1.2. Two-beam coupling: an overview............................................................ 06
2. Two-beam coupling modules
2.1. Introduction..............................................................................................

15

2.2. The multimode fiber optics interconnections........................................... 17
2.3. Two-beam coupling modules: designs and results................................... 19
2.3.1. The original design........................................................................ 21
2.3.1.1. The alignment proceedure......................................................

23

2.3.1.2. Results....................................................................................

24

2.3.2. The next generation.......................................................................

26

2.3.2.1. A new base design.................................................................. 26
2.3.2.2. Brewster-cut crystal................................................................ 28
2.3.2.3. Graded-index lenses...............................................................

29

2.3.2.4. Buffered fibers........................................................................ 30
2.3.2.5. Results for the new generation...............................................

31

2.4. Discussion and future design.................................................................... 32
2.5. An application example: the auto-tuning filter......................................... 36vii

2.6. Conclusions............................................................................................... 43
3. Barium titanate spheres and spherical disks
3.1. Introduction............................................................................................... 44
3.2. Preparation................................................................................................ 45
3.3. Two-beam coupling with spheres............................................................. 46
3.4. Whispering-gallery mode patterns............................................................ 48
3.4.1. A family of patterns........................................................................ 49
3.4.2. A theory for the triangle occurrence regions.................................. 54
3.4.3. Unidirectional oscillation............................................................... 57
3.5. Conclusions............................................................................................... 57
4. Carrier suppression with two-beam coupling
4.1. Introduction............................................................................................... 59
4.2. A geometrical picture of two-beam coupling..........................................

63

4.3. The carrier suppressor............................................................................... 71
4.4. Future Work.............................................................................................. 78
5. An operator approach to two-beam coupling:the 2-by-2 complex-coupling case
5.1. Introduction............................................................................................... 85
5.2. The operator formulation.......................................................................... 88
5.3. A closed-form solution to the 2-by-2 case................................................ 93
Conclusions........................................................................................................... 104
Bibliography......................................................................................................... 110

viii

FIGURES

Figure
1.1.

Two-beam coupling diagram................................................................ 08

1.2.

The photorefractive effect.................................................................... 09

1.3.

Normalized intensity plots of gain and loss beam pairs...................... 12

1.4.

Spatial structure of fanning light.......................................................... 13

2.1.

Two-beam coupling module................................................................. 17

2.2.

Schematic with top and side view of two-beam coupling module....... 21

2.3.

Cross-section of a collet with an aspheric lens..................................... 22

2.4.

Setup of the alignment of the input collets........................................... 24

2.5.

Details of the base design..................................................................... 27

2.6.

Top view of Brewster-cut crystal......................................................... 29

2.7.

Cross section of new lens-ferrule assembly......................................... 30

2.8.

Representation of total displacement and tilt errors between lenses.... 34

2.9.

Suggestion for modular base design..................................................... 36

2.10.

Autotuning filter in two different architectures.................................... 38

2.11.

Experimental setup for the autotuning filter with reflexive coupling... 40

2.12.

Input and output intensity spectra for the autotuning filter.................. 42

3.1.

The “beach ball” pattern of a spherical photorefractive crystal........... 46

3.2.

Two-beam coupling between two pairs of fibers................................. 47

ix

3.3.

Diagrams of minimum interbeam angles.............................................. 50

3.4.

Experimental setup for observing oscillating patterns.......................... 52

3.5.

Oscillation patterns inside a barium titanate spherical disk.................. 53

3.6.

Schematic showing the geometrical relationship betweenψ and φ....... 55

3.7.

Triangle occurrence regions.................................................................. 56

4.1.

Schematic of the optically-smart antenna array.................................... 62

4.2.

Carrier suppression via a two-beam coupling interaction..................... 65

4.3.

Carrier and sideband vectors on plus-minus coordinate system........... 68

4.4.

Output vs. input modulation strength................................................... 70

4.5.

Gain for suppressing the carrier as a function of input intesity ratio... 71

4.6.

Ressonant circuit of the electrooptic modulator................................... 72

4.7.

Picture and setup of the carrier suppressor........................................... 74

4.8.

Measured output carrier as a function of modulation index................. 76

4.9.

Experimental output vs. input modulation strength.............................. 77

4.10.

Channelized electrooptic modulation and carrier suppression............. 80

4.11.

Picture and setup of carrier suppressor with channelized modulator... 81

4.12.

Output intensity of the carrier, signal and second order component.... 83

5.1.

Representation of

5.2.

Plot of the evolution of the density matrix .......................................... 99

5.3.

Evolution of the density matrix for real and imaginary coupling........ 101

in

1

2

-coordinate space.................................. 95

3xCHAPTER 1:
INTRODUCTION

1.1.

MOTIVATION

The inherently parallel nature of optical information processing, coupled with
its typically higher bandwidth, confers a potential advantage over conventional
electronically-based processing, such as digital signal processing. Photorefractive
media store real-time holograms which have memory and can provide high optical
gain (20 cm-1) at moderately low intensities on the order of W/cm2, characteristics
which are useful in the implementation of optical information processing
architectures. Photorefractive-based systems can be subdivided in three maincategories: optical computing, image processing and signal processing. Examples of
optical computing systems include associative memories and correlators [Gabor, '69;
Khoury, '94; Neifeld, '93; Staebler, '75; Yu, '94], neural networks [Denz, '99; Gu, '93;
Psaltis, '88], and reconfigurable array interconnects [Anderson, '87; Marrakchi, '90;
Weiss, '89]. Image processing systems include functions such as image amplification[Hong, '90], image thresholding [Klein, '86a; Sayano, '88], edge enhancement[Feinberg, '80; Joseph, '92], amplifying spatial filtering [Chang, '90; Fainman, '86],
novelty filters [Anderson, '89; Khoury, '91; Uesu, '95]. Narrow-band filters [Hong,'93; Rakuljic, '93] and optical heterodyning [Hamel de Monchenault, '88; Khoury,'93] are examples of signal processing systems. Photorefractive systems applications
have been reviewed in [Gunter, '88].

1

The emphasis in our research group is to perform information processing by
use of photorefractive oscillating architectures which we call “rings”, such as the flipflop [Anderson, '91], the bistable ring [Lininger, '90], the demultiplexer [Saffman,'91], and the feature extractor [Anderson, '92]. These demonstrations are described as“self-organized”, for the reasons that no external programming of the photorefractive
ring is needed, nor is any a priori information about the input signals, except for therequirement that they be both temporally and spatially orthogonal. The dynamics of aring oscillator generate real-time volume holograms in the photorefractive crystals
which we call gratings. The competition dynamics between the generated gratings
allow for different algorithms, such as separating temporally uncorrelated signals in
different rings in the demultiplexer, or extracting the principal component within the
input signals in the feature extractor.
This thesis focuses upon a fundamental functional unit in photorefractive
processing: the two-beam coupling interaction. It is the main functional unit fromwhich the ring systems are built, and it is also used in many of the more generalapplications mentioned above, such as in image amplifiers, novelty filters, notch
filters, and others.
The next couple chapters in this thesis are motivated by the importance ofdeveloping a more component oriented technology for photorefractive functions, as it
will significantly contribute to making photorefractive system testing and building
more practical. Its widespread use in photorefractive systems justifies the choice of
two-beam coupling as the example-function for testing such technology. Naturally,
the same concept can be used for other photorefractive functions as well, such as2phase conjugation or four-wave mixing. The role that the development of such acomponent technology could represent to photorefractive systems can be illustrated
by drawing an analogy with electronic systems, where systems can be assembled by
the use of chips with a wide variety of function, canceling the need of building thesystem from the transistor level. For this purpose, in chapter 2, we introduce a
modular design, where each module is pre-aligned to perform a particular
photorefractive function, and the modules are interconnected by multimode opticalfibers. Such technology allows systems containing several modules to be easily
assembled and re-configured without the need of any system alignment. Each module
is built for a specific desired function – a two-beam coupling module built for gain
will be optimized differently than one built for novelty filtering. In the end ofchapter 2, we give an application example using such modules. There we show how
their use allowed a quick and easy verification of the change in behavior of a ringsystem when an extra module is inserted into its feedback loop, sparing the need fortedious realignment on the conventional optical table.
In chapter 3, we introduce the use of spherical photorefractive crystals. Their
original purpose was in developing more compact two-beam coupling units. Due tothe curvature and high refractive index of the crystal, no other optical elements arenecessary for these modules except for the input and output fibers and the crystal
itself. A built example of such module is shown herein. Interestingly, when testing
the spherical crystals, we found that a single pump beam generated oscillating
patterns within the crystal. These patterns arise from total internal reflections within
the crystal, forming closed-path oscillating modes, known as whispering-gallery3modes. A spherical disk, an equatorial slice of the sphere containing the crystal’s
uniaxis, the “c-axis”, was produced to allow better visualization of the patterns. The
occurrence of the strongest mode, the triangle pattern, was mapped as a function ofpump input and c-axis angles. A simple theory, assuming pumping of the modes bymeans of a single two-beam coupling interaction, resulted in good qualitative
agreement with the experimentally obtained mapping. Due to the lack of a phaseconjugate of the pump-beam, we believe that the oscillations are unidirectional,
meaning that these structures are miniature analogous of unidirectional ringoscillators pumped by a two-beam coupling interaction, which is the architecture used
for several of our signal processing systems, such as the feature extractor.
As mentioned above, the ring processors developed in our laboratories
typically require that the input signals be temporally and spatially orthogonal.
However, when using electrooptic modulators to impose a temporal signal upon
different optical beams, the signals in these beams will have a correlation term due tothe common carrier. Therefore, for the ring processor to function properly, it becomes
necessary to remove the optical carrier from each signal beam. This is the case for asystem being currently developed, the “optically-smart antenna array”, which we
describe in more detail in chapter 4. This system contains a feature extractor,
currently renamed as the “autotuning filter”, whose purpose is to select the strongest
signal within the several signals presented at its input. Because the optical signals inthis system are generated by using an electrooptic modulator, for the reason justexplained above, a carrier suppressor is needed in front of the filter’s input. Onceagain, two-beam coupling proves its versatility by performing adaptive carrier

4

suppression when used in a novelty-filter configuration. Carrier suppression of morethen 70 dB is demonstrated. Such results were possible due to the development of an
operator formalism which is the subject of the last chapter. In chapter 4, we provide a
simplified, geometrical picture of this operator theory applied to the carrier
suppression problem. This geometrical picture provided a greater intuitive
understanding of the suppression mechanism, and allowed us to set the parameters
involved in the carrier suppressor, such as input intensity ratio of the beams and themodulation strength.
Finally, the last chapter describes the operator algebra formulation for twobeam coupling. In 1979, Kuktarev published a theoretical model for beam coupling
which employed coupled wave theory [Kogelnik, '69] taking into account the effectof self-diffraction of light induced gratings. [Kukhtarev, '79] The equations presented
in the next section are based on the plane-wave approximation to this theory,
considered the foundation for two-beam coupling modeling. However, as the numberof plane-waves interfering in the crystal increases, this theory becomes increasingly
more difficult and awkward to deal with. A generalized operator algebraic theory, forwhich the input beams can be cast not only on a plane-wave basis, but on any set oforthogonal spatial bases is then indicated. The operator theory presented in chapter 5
treats the photorefractive medium as a black box described by a coupling operator T,
which acts upon an input field vector to produce the output field vector,E(out)=T E(in). The field vectors are represented in whichever basis is better suited
to the information processing task at hand. Due to this freedom in representing theinformation, the operator theory bridges a gap between the physics of two-beam5

coupling and its theoretical-computational representation, leading to a deeper
intuitive understanding of the beam coupling phenomena. In fact, this is exemplified
in chapter 4, section 4.4, where a geometrical picture of the operator formalism led tothe prediction of carrier suppression features by simply graphically representing thefield vectors and studying the action of T upon these vectors. In the last chapter, we
present the analytical solution for the coupling matrix T for the case of one spatial
mode in each of the two input beams. We present the “complex-coupling” solution,
which corresponds to the general case consisting of a combination of intensity
coupling (real case) and phase coupling (imaginary case).
Before proceeding to chapter 2, in the next section we present a briefsummary of two-beam coupling, for those readers who are not familiar with theprocess.

1.2.

TWO-BEAM COUPLING: AN OVERVIEWThis overview is just that, an overview, where an effort is made to only cover
the main aspects of two-beam coupling necessary for the understanding of thefollowing chapters. We explain the physical phenomenon behind the two-beamcoupling process and introduce the two-beam coupling equations. For more detailed
accounts we refer the reader to [Solymar, '96; Yeh, '93].
Two-beam coupling is a nonlinear optical interaction that can transfer energy
or phase information between two overlapping beams in a photorefractive medium.

6

The two-beam coupling medium of this thesis is always an electrically unbiased,
open-circuit, barium titanate (BaTiO 3) crystal, where the charge transport is primarily
due to diffusion, resulting in a pure energy transfer process [Buse, '97]. Barium
titanate displays high two-beam coupling gain over a wide wavelength range.
(Barium titanate is a perovskite ferroelectric material belonging to the 4mmcrystallographic group symmetry. It’s photorefractive properties are widely available
in the literature. [Klein, '86b; Mazur, '99].)
A two-beam coupling diagram is shown in figure 1.1. Two beams cross inside
a crystal forming an interference pattern, which is sinusoidal if the beams are
assumed to be plane waves. The bright regions of the interference pattern can excite
charge carriers from impurity donors, typically iron ions (Fe2+ and Fe3+) in barium
titanate, to the conduction band. In this model we assume the only charge carriers areelectrons. The electrons in the conduction band are free to migrate whereas theionized donors are fixed to the crystal lattice. Electrons may experience severaltransitions as they migrate, in other words, they are re-combined with the ionizeddonors and re-excited to the conduction band. Because the excitation of the electron
is proportional to the light intensity, more electrons are generated in the brightregions as opposed to the dark regions of the interference pattern. This causes anaverage migration or diffusion of electrons to the dark regions. This charge separation
results in a charge density function ( ) with the same periodicity of the interference
pattern, as shown in figure 1.2. The charge density and the resulting space-charge
field (ESC) are related by Poisson’s equation, ∇⋅ ESC = , which introduces a 90°
phase shift between

and ESC.

7

Barium titanate displays a strong linear electrooptic effect, also known as
Pockel’s effect, which generates a refraction index grating in the medium whichmimics the space-charge field. The optical fields which originally recorded thisgrating will now scatter from it, i.e., the fields experience self-diffraction. The 90°phase shift between the index grating and the interference pattern causes thediffracted beams to interfere constructively in one output direction (along the “gainbeam”) and destructively in the other output direction (along the “loss beam”). This
results in an energy transfer between the beams. The direction of energy transfer isdetermined by the crystal’s optical axis. We designate the beam that loses energy as
the “loss beam”, and the one that gains energy, the “gain beam”. For a given purpose,
sometimes the gain beam is the pump, sometimes it is the signal. Typically it is easierto follow the pump and signal paths, though the gain and loss never change.

∆n

Loss
beam

c

θ

z

π−α
Gain
beam
BaTiO3Figure 1.1. Two-beam coupling diagram. Loss beam transfers energy
to gain beam. The direction of energy transfer is determined by the caxis direction and the sign of the charge carriers. Higher gray level
saturation implies brightness of the beams.

8

IInterference intensity patternzρCharge densityz
ESC

Space-charge fieldz∆nRefractive index gratingz

Figure 1.2. The photorefractive effect. The intensity pattern from twointerfering beams create a charge separation with the same periodicity
and shape as the intensity variation. The associated space-charge field,
which is shifted by 90° when no external fields are applied, induces anindex grating in the medium via the linear electrooptic effect (Pockel’s
effect).
In steady-state, the energy transfer process is represented by the following
two-beam coupling differential equations with respect to the propagation direction z:

9 dA1
 dz = G A2

∗
 dA2
 dz = −G A1[1.1]

where G represents the grating strength:

G=

2

1
2 I0

2

A1A2* ,and I0 = A1 + A2 is the total intensity. The gain coefficient[1.2]

depends upon thematerial properties of the crystal and upon the geometrical arrangement of the setup,
and is given by:=2
reff , where reff = −2 no2ne2 r42 Esc ( , ) cos sin(2 ) .
n cos

In the expression above, reff is the effective electrooptic coefficient,wavelength in air of the input beams,[1.3]

is theis the half-angle between the two beams,

isthe angle between the beams’s bisector and the c-axis, and no and ne, the ordinary andextraordinary index of refraction (no ≈ 2.437, ne ≈ 2.365). The reff depends on thepolarization state of the fields. In this treatment we only take into account the main
component of barium titanate’s electrooptic tensor (r42=1640 at low frequencies),
since the other components are more than one order of magnitude smaller. For

10extraordinary fields, this term is proportional to cos sin2 . Additionally, the spacecharge field is a function of both

and .
kT
iK B
q
ESC =
K2
1+ 2
kD

It depends on

through the wave vector K = 4 n sin , and on[1.4]

through the Debye

screening length ΛD, which in turn depends on the effective dielectric constant∈ =∈// cos2+∈⊥ sin2, where ∈/ / and ∈⊥ are the DC dielectric constant for anelectric field parallel and perpendicular to the c-axis, respectively. Barium titanate
displays a high dielectric anisotropy with ∈/ /= 135 and ∈⊥ = 3600 .
Solving the above equations, we get the following expressions for theintensities of the gain-beam I1 and the loss-beam I2:

1+m I1 (z) = I1 (0) 1 + me − z−1 I (z) = I (0) 1 + m
2
2z
1+ m−1 e[1.5]

where m is the input intensity ratio I2(0)/ I1(0).
Using the above equations, we plot in figure 1.3 the gain and loss beam
intensities as they propagate through the crystal, for different values of the gain
coefficient, 5, 10 and 20 cm-1, and a input intensity ratio of 100. As expected, energy
transfer occurs earlier for higher gain coefficients.

11Energy transfer in two-beam coupling
1
0.8
0.6
0.4
0.2
0
0

0.5

1

1.5

2

Interaction length (cm)Figure 1.3. Normalized intensity plots of gain and loss beam pairs,
I1(z) and Is(z) for γ=5 cm-1 (solid curve), 10 cm-1 (dashed curve), and20 cm-1 (dash-dot curve). The input intensity ratio is set at 100.
Besides the gain, two other parameters are important for evaluating coupling
process. The first one is losses, which can be divided into passive losses and active
losses. Passive losses are due to surface reflections and absorption in the crystal.
Absorption is not taken into account in the above expressions. To account for it, one
simply multiplies equations [1.5] by e- z, where α is the absorption coefficient. The
curves in figure 1.3 would then be weighed by an exponential decay as they
propagate in z. The active losses are due to scattered light from impurities and defects
from the crystal, which are amplified by the photorefractive gain, forming what iscalled “fanning”. A picture of fanning simulation is shown in figure 1.4, after thework from [Zozulya, '95]. The losses due to fanning may be reduced by aligning the
crystal in the direction of the beam fan, but the trade-off is reduced output signal
dynamic range.

12Figure 1.4. Spatial structure of fanning light inside a crystal.

The gratings in a photorefractive crystal adapts to the average input signals
integrated over the time constant (τ) of the crystal. The inverse time constant ofbarium titanate is typically between 1-100 Hz. One might question whether such aslow response doesn’t significantly limit the applicability of the photorefractive
processors. In some applications, such as phase-conjugation, reconfigurable
interconnections and in some image processing systems, a fast photorefractive
response is desirable. However, for our applications the slow response time is not alimiting factor, but in fact actually advantageous as it allows signals which are merely
1/τ apart in frequency to be distinguished by the system. In some of theseapplications, the slow time constant results in long term statistical correlations whichallow the system to distinguish even between signals with overlapping bandwidths.
For instance, the signals could consist of speech. The speech from a particular person
may vary on a short-time scale, however, on a longer time scale a person’s speech ismore correlated with itself than with another’s, which is the key property that allows
the speakers to be separated [Min, '88]. On the other end, the upper frequency limit

13

for the signals in the photorefractive processors is given by Bragg matchingconsiderations and is a few gigahertz for a 1 cm grating. This limit is inversely
proportional to the grating length.

14

CHAPTER 2:
TWO-BEAM-COUPLING MODULES

2.1.

INTRODUCTION

The development of photorefractive processing systems, consisting of two or
more photorefractive elements, is stymied by the lack of a well-developed component
technology which would allow for rapid proof-of-concepts and testing of systems
without the burden of extensive, time-consuming element-by-element alignment, as is
the case of conventional photorefractive systems.
As a first step towards the development of photorefractive component
technology, we developed modular photorefractive units that could be easilyinterconnected by the use of multimode, optical fibers. The main idea is to have thephotorefractive crystal within a module and its associated fiber optic elements
optimized for a specific photorefractive function and then permanently fixed in place.
The modules are connectorized, and may be interconnected with multimode fibers inany desired configuration. Thus, the assembly of a complete optical system neverentails alignment of beams through crystals or other optical elements which are partof each photorefractive function. This means that, with a collection of appropriate
modules in hand, a complex system can be quickly assembled or reconfigured.

15

The geometry of each module can be optimized to perform any of thecommon wave-mixing effects that have found their way into conceptual and realprocessing systems, such as two-beam coupling gain [Khuktarev,'79], novelty
filtering[Anderson,'89;Horowitz,'91a],

Golomb,'84;Feinberg,'83],

self-

and

four-wave

mixing[Cronin-mutually-pumped

phase

conjugation[Feinberg,'82;Sharp,'94;Yeh,'92],

photorefractive

oscillation[Anderson,'92;White,'82;Yeh,'85], and more. Because it is employed ubiquitously inthe photorefractive systems developed in our laboratory, two-beam coupling is thefunction of choice to test the modular concept.
To this end we have built several two-beam coupling photorefractive
modules, such as the one shown in figure 2.1. This chapter discusses the developmentof these two-beam coupling modules, starting with design considerations, followedby the alignment procedure and the resulting characterization parameters. In the lastsection of the chapter we test the performance of the modules when these areinterconnected to form a photorefractive system, namely an auto-tuning filter, thefunctioning of which is explained in that section.

16

Figure 2.1. Two-beam coupling module. The picture shows a ringoscillator which is obtained by simply using an optical fiber to connect
the output signal port (gain port) to the input signal port. Another fiber
provides the input pump.

2.2.

THE MULTIMODE FIBER OPTICS INTERCONNECTIONS

A basic step in designing a component technology for photorefractive systems
is determining the kind of interconnections that should be used. The choice isbasically between using free-space optics or fiber optics. Multimode fiber optics was
selected because, as mentioned above, the use of connectorized fibers dispenses with
the need for alignment between modules, therefore offering enormous benefit of easeof assembly and reconfiguration. In addition, the fiber-interconnected modules derive
benefits from the substantially well-developed multimode fiber component
technology, which provides us with standardized connectors and a gamut of fiber17

terminated optical devices. What are the tradeoffs of such benefits? This brief section
clarifies the way in which the use of multimode fibers affects the modules.
One major drawback of multimode fibers as opposed to free-space, Gaussian
beams is that two-beam coupling gain is reduced, partly because of the nature of
speckled beams [Zel’dovich,'95] and partly because of polarization scrambling by thefiber. The electro-optic coefficients in barium titanate are such that the ordinary
component of the incident light polarization experiences a negligible two-beamcoupling gain, and in addition, it partially erases the grating associated with theextraordinary component. This causes a reduction of the modulation index and,
therefore, the gain. Polarization effects can be mitigated by adding small polarizers to
the modules, but in our modules we simply accept the sacrifice in gain for the sake ofsimplicity.
Multimode fibers were used as opposed to single mode, not only because it ismuch easier to couple a beam into it as compared with the latter, but mostimportantly, because a multiplicity of modes are necessary for image processing
applications, and other photorefractive signal processing applications as well, such as
the autotuning filter described in the end of this chapter. A common misconception
regarding the multimode fiber optics interconnections is that its use excludes themodules from image processing applications. This is not generally the case. The
multimode optical fiber interconnections permit one to process two-dimensional
spatial information. Naturally, two-dimensional data becomes scrambled as itpropagates down the fiber, however the information contained in the data is preserved
provided there are no severely mode-dependent losses. Many image-processing tasks18and algorithms, such as sorting or pattern recognition, are indifferent to scrambling,i.e., the presence of an unscrambled version of the processed image in the output of asystem is not essential to its function. Nevertheless, if the unscrambled output image
is desired, one can recover such image by holographically correlating the specklefield output, the scrambled output image, of a fiber-based system with theunscrambled input images. Even if the application is intolerant of scrambling, thefiber-interconnected modular approach may still allow one to explore processing
concepts en route to the design of a specialized optical circuit.
Our graded-index fiber interconnections have a 62.5µm core diameter and
125µm cladding diameter. If its numerical aperture of about 0.26 is filled, the fiber
will contain on the order of 16,000 modes if illuminated with a 514 nm laser.

2.3.

DESIGNS AND RESULTS

As shown in figure 2.1, a two-beam coupling module consists of a barium
titanate crystal located in the center of a monolithic aluminum base, and in between
two pairs of collets, one input and one output pair. An interbeam angle of about 15° istheoretically calculated to provide maximum gain for barium titanate, however, given
the collet’s diameter of 3 mm, the interbeam angle between the two collets in a pair ischosen to be 30° as a trade-off to reduce the module’s length to about 50 mm. Eachcollet lies on an independent flexured platform which can be tilted to compensate forsmall fabrication errors in the parts. This tilt is adjusted by set screws accessible from19the bottom of the base (see figure 2.2). We have tested several base versions withdifferent designs for the flexures and corresponding tilt adjustments.
A top and side view schematic of a module is given in figure 2.2. Two laser
beams enter the unit via optical fibers, are collimated by the lenses in the input
collets, and then cross in the crystal where they undergo photorefractive two-beamcoupling. After traversing the crystal, the loss and gain beams are collected by lenses
in the two output collets and focused into another pair of optical fibers. The input andoutput fibers are terminated with connectors, and the base, along with theconnectorized fibers, are assembled in a box, (as shown in figure 2.1).
This design of the modular units optimizes the photorefractive crystal
geometry. Small beam diameters between 100 µm to 820 µm allow for small crystals
and an efficient use of crystal volume: the length of the crystals typically varies
between 2.5 and 4 mm, the cross-section is about (1.5 mm)2. We get roughly 8
module crystals from a typical (5 mm)3 barium titanate piece with the appropriate caxis direction.
We have tested several modules with different combination of base designs,
lenses and crystal cuts. Below we discuss the original design followed by our
improved, latest design.

20

Gout

Lin

12.5

30°

Lout

Gin
colletBaTiO3
crystalvertical
tilt

9.25
25.4

set screws fortilt adjustment

Figure 2.2. Schematic with top and side view of two-beam coupling
module. A BaTiO3 crystal is placed at the center of the unit over asmall glass mount. Collets lie on platforms whose vertical tilt can be
adjusted by two set screws. Dimensions are in millimeters.

2.3.1. The original design

The original modules contain the collet design described below andrectangular crystals.
A cross-section of a collet is shown in figure 2.3. The collets contain a 2mmfocal-length aspheric lens (350150 GelTech) which is glued to the collet with UV
curing optical adhesive. A 62.5 µm core diameter fiber is inserted into a ferrule,attached with epoxy, and its end polished flush with the ferrule’s face to provide an
optical quality surface. The prepared ferrule is then inserted into the collet and a
specially designed tool, which moves the collet-lens assembly with respect to the21ferrule, is used to collimate a 514 nm laser beam exiting the lens. The two pieces arethen fixed with epoxy. The resulting beam diameter is approximately 820 µm.
The crystals were rectangular with a 20° or 30° cut with respect to the crystal
axis. Maximum gain is typically achieved for a 45° cut, however the benefit of suchgain maximum is associated with unwanted fanning. [Zozulya, A.,'95] Fanning
reduces the dynamic range of the two-beam-coupling module, but the effect can beameliorated by simply reducing the crystal-cut angle, which reduces the gain. The
crystal-cut must be tailored to the application at hand. For the applications in ourlaboratories, high gain is typically desired so the 30°-cut crystals were more
successful. All crystals were approximately 4 x 2 x 1.5 mm3.

aspheric lens(f=2 mm)ferrule4.0

optical fiber(62.5 µm)
2.0

8.0
d

Figure 2.3. Cross-section of a collet with an aspheric lens (Geltech,
350150). The zirconia ferrule contains a polished multimode fiber with62.5 µm core and 125 µm cladding. Distance d is adjusted to collimate
the outgoing beam. All unspecified dimensions are in millimeters.

22

2.3.1.1. The alignment procedure

The alignment procedure, as depicted in figure 2.4, begins by mounting themodule base on a rotation stage, and placing the crystal on a glass spacer at the centerof the base. Next, each of two collets is attached to a jig. The jig consists of a twodimensional translation stage with an extension arm with a beryllium-copper spring at
the end, to which the collet is attached. The collets are positioned 30° apart at the
input side of the base. They are made to completely contact the base by having the jig
exert a light pressure upon them. With this arrangement a collet has four adjustment
degrees of freedom: longitudinal and transversal translation, rotation about the axis
through the center of the base and perpendicular to it, which we call “horizontal
rotation”, and the “vertical tilt” provided by the flexured base. (The translation andhorizontal rotation stage are shown in figure 2.4, whereas the vertical tilt, in figure
2.2).
Optimization of the photorefractive gain is the main goal in aligning the input
collets. To do this 514nm Argon-Ion laser light is coupled into the input fibers andthe overlap of the two beams inside the crystal is then maximized. This isaccomplished by use of the longitudinal and transversal translation, and the verticaltilt. (Horizontal rotation is only needed when aligning the output collets.) Typicallythe gain is not uniform throughout the crystal’s volume. [MacCormack,'96]
Therefore, one can search for a higher gain by simply moving the crystal sideways, sothat the beam overlap probes different regions inside the crystal, and then select aparticularly good position. To avoid problems when coupling into the output collets,

23the beams must be parallel to the plane of the base. Once the optimal gain is obtained,
the crystal and input collets are fixed to the base with epoxy and the jigs are removed.
The same pair of jigs are now used to position the two output collets tooptimize the coupling of each beam into the respective output fiber. Finally, the two
input and two output fibers are cleaved and terminated in SC connectors and the unit
is enclosed in a 10.16 x 7.62 x 3.25 cm3 box.

translation
stage
rotation stage

power meter

Lin

G out

G in

Lout

Figure 2.4. Setup for the alignment of the input collets. The gain is
optimized by adjusting the translation stages and the set screws shownin figure 2. The rotational stage is used during the alignment of theoutput collets.

2.3.1.2.Results

Three quantities characterize these modules: gain, passive loss, and time
response.

24

The gain is measured by taking the ratio of the output intensity of the gainbeam with the loss-beam unblocked over that with the loss-beam blocked. The gain
measurements were obtained with an input intensity ratio of the loss-beam over thegain-beam of 3000. Three modules were built with 20°-cut crystals and they had
gains of 12, 13 and 19, and another three were built with 30°-cut crystals having
gains of 286, 406 and 435.
The passive losses are given by subtracting the passive signal coupling fromone. The passive signal coupling was measured by taking the output to input intensity
ratio of the gain beam while wiggling the fibers so as to impede the formation ofphotorefractive gratings in the crystal. The passive losses ranged from 72% to 84%,
with an 80% average. We also measured a significant deterioration of the passive
signal coupling due to alignment drift with a time scale of weeks. One of the modules
passive losses went from 72% to 86% in about a month, whereas two other modules
were completely misaligned after the same period.
To test the response time of the modules, we presented a step function to the
input and measured the 10 to 90% rise time in the output. The input approximation toa step function, with a rise time of about 5 ms, was generated by manuallyunblocking the input gain beam. The measurements were performed with a loss beam
power of 100 mW and a gain beam intensity of approximately 1% that of the loss
beam. The output rise time was always between 500 and 700 ms.

25

2.3.2. The next generation

This section discusses, based on the above measured results, the routes for
improving the module’s design towards a new generation. Our primary concern is to
reduce the passive losses and, most importantly, to improve the degradation with timeof the passive signal coupling due to an alignment drift. Below we list themodifications made in the new design. The alignment procedure is similar to that of
the first generation (section 2.3.1.1). The last subsection gives the results for the nextgeneration modules.

2.3.2.1. A new base design

A top view schematic of the original base is shown in figure 2.5 (a). We callattention to the pivot points of the four flexured platforms. In this original basedesign, the pivots are wider vertically than horizontally (where horizontal is the plane
parallel to the paper in the figure). We’ll therefore designate these pivots as “vertical
pivots”. In some occasions the platforms pivots broke off during the modules’alignment procedure (see section 2.3.1.1 above). This happened because the heightadjustment of the flexured platforms provided by these vertical pivots are insufficientfor correcting nonidealities of the components in the modules. For instance, thecrystal faces are not perfectly parallel, which cause the beams to be deflected. Thissuggests that the pivots could be significantly mechanically stressed even when they26

didn’t break. And most importantly, such stress is very likely the culprit for thealignment drift as the aluminum tries to relax.
To ameliorate the mechanical stress we tested several different base designmodifications. The chosen design is the one shown in figure 2.5 (b), which is similar
to the original, but instead of a vertical pivot it has a horizontal pivot (meaning thatthe pivot is wide in the horizontal plane and thin in the vertical plane). These allow
for more range of height adjustment without sacrificing the tilt adjustment.

Aluminum 7075
Anodized red

TOP VIEW

Note: Flexures arealong a vertical plane.

9.5
13.3
14.5
16.8
23.4

(a)
New flexures are wide
horizontally and thin vertically.

(b)

Figure 2.5. Details of the base design. Dimensions are in millimeters.
(a) Original design with vertical pivots. (b) New base design is similar
to the original, but with horizontal pivots which allows for greater27

adjustment range. The ellipses represent milled slots resulting on wide
but thin (about 750 µm) flexures.

2.3.2.2. Brewster-cut crystal

As explained in section 2.2 above, the ordinary (vertical) polarization partiallyerases the grating, reducing the gain. For two-beam coupling application, the greatest
component of the unwanted vertical polarized light typically comes from the pump.
So we designed a new “Brewster-cut” crystal geometry as shown in figure 2.6. The
Brewster angle for barium titanate is 67°, and is obtained by neglecting the crystal’sanisotropy and using an refractive index of 2.4. Having the pump beam enter the
crystal at Brewster’s angle increases the loss of the vertical polarization component toabout 50%, while reducing the desired horizontal polarization component loss to lessthan 2%. The average reflection loss of the signal at 37° (such that angle of pumpminus signal is 30°) is about the same as that at zero degrees, i.e. about 17%. Also,
because the pump beam comes in at a steeper angle, it is wider inside the crystal than
the signal beam: this facilitates alignment optimization for two-beam coupling. To
reduce absorption losses, the new Brewster-cut crystals are 2.5 mm long, as opposedto the previous 4 mm length of the older rectangular crystals. The optical axis of the
new crystals are oriented at 45° with respect to the input surfaces.

28

45°

37°
gain beam
rs =.24
rp =.11

67°

67°

c-axisloss beam
rs =.50rp ≅ 0Figure 2.6. Top view of Brewster-cut crystal.

2.3.2.3. Graded-index lenses

Cheaper than the aspheric lenses, the graded-index (GRIN) lenses come inmore variety, are easier to assemble, and dispense the use of collets or other mounts
since they can be glued directly to the ferrule using optical adhesive for index
matching. Figure 2.7 below shows a cross-section of the new ferrule-lens assembly. It
is assembled by using a V-groove mount to align the center of the lens to that of theferrule while curing the optical adhesive with UV light. The external end of the GRIN
lenses were anti-reflection coated to reduce losses. We used a slightly focusing lens
pitch (Selfoc SLW1.8 - .29 - 555 from NSG America) to decrease the beam diameter
inside the crystal, resulting in beam diameters in the waist of about 100 µm. The time
response of a crystal is roughly inversely proportional to the total intensity of thebeams. [Horowitz,'91b] This means that the smaller beam diameters should result in adecrease in the response time of the modules.

29

Graded-Index lens (f=1.8)

1.8

V-groove for alignment only
5.3

Figure 2.7. Cross section of new lens-ferrule assembly. A schematicof the v–groove is shown but it is removed after alignment. The fiber
output will be imaged by the lens at a distance of about 6.3 mm fromthe output surface of the lens.

2.3.2.4. Buffered fibers

The optical fibers used in the first-generation modules were fibers having only
a 245 µm diameter coating over its 125 µm diameter cladding (62.5 µm core). The
operation of the modules is sensitive to acoustic vibrations, which disturb the fibers,change the speckle configuration, and thereby affect the two-beam coupling grating.
When the modules are tested in a ring configuration such vibrations cause the socalled “breathing” in the output, meaning that the output would slowly turn on andoff. To ameliorate this effect, we switched to fibers having a 900 µm buffer jacket
which helps isolate them from external vibrations. The buffered fiber also has theadvantage of greater mechanical resistance, are white, and therefore are easy to see ina darkened room. Not only did we use these buffered fibers in some of our new

30

modules, but also, and perhaps most importantly, we used them in all the newmodule-to-module fiber cables.

2.3.2.5. Results for the new generation

We built a total of 3 modules with the above modifications. These modules
are characterized in a similar way as described for the earlier modules (see section
2.3.1.2). The resulting small-signal gains are 526, 664 and 843; the passive losses are
68%, 73% and 72%; and the 10 to 90% rise time at the output when the input isunblocked is 120, 164 and 122 msec, respectively.
The higher gain is most likely due to a combination of factors. First, theBrewster-cut crystal geometry, which, as discussed earlier, increases the gain by
reducing the ordinary polarization component. Second, the gain characteristics of theparticular boule from which the crystals were cut. We have very little control over thematerial properties of the crystals, making it hard to discern how much each of these
two effects contributed to increasing the gain.
The passive-loss average of 71% is about 10% less (better) than that of thefirst-generation modules. This number is due to a combination of loss components.
We measured the absorption coefficient for barium titanate to be about 1.7/cm,
resulting in a absorption loss of 35% for a 2.5 mm crystal. The reflection losses areapproximately 25%: 50% reflection loss for the ordinary polarization and 0% for the31extraordinary polarization. The remaining 11% losses are consistent with typicalmeasured coupling losses.
The alignment drift was, unfortunately, still present, however it was much lesssevere than in the first-generation modules. Two months after the above loss
measurements were obtained, the measurements were redone and values had changed
to 80%, 87%, and 79% respectively, corresponding to an average loss increase of
11%. The loss increase for all three modules is less than the loss increase for theearlier modules.
As expected, due to the smaller beam diameter, these modules responded anorder of magnitude faster than the earlier ones. (Even though the pump beam
diameter is increased by the Brewster incidence angle to about 250 µm, it is still
smaller than the pump beam diameter inside the crystal in the first-generation
modules of 850 µm/ cos 15° = 880 µm.)

2.4.

DISCUSSION AND FUTURE DESIGN

The second-generation modules provided us with higher gain, lower losses
and faster response time than the previous generation. However, an alignment drift isstill present, though much less severe than before. This suggests that the componentsof the modules are performing well, but that the base is still mechanically relaxing
causing the alignment drift. We believe the cause of this relaxation is due to thespring action of the flexured platforms. Such relaxation could be enhanced by either

32temperature variations or shock. Therefore, as a future project for improving themodules we suggest that a new base be designed. Below we give some suggestions
for a new base design, but before that, for the benefit of possible future designers, wefirst estimate the errors associated with the modules’ components.
If further improvement of the base design is pursued, an assessment of errors
and tolerances within the module will become necessary. The main misalignment
causing errors are the following: the crystal’s facet perpendicularity error, typically ±.015 rad, the lens’ facet perpendicularity error, ± .006 rad, offset between the centerof the lens and the ferrule, and lens tilt with respect to the ferrule. The offset between
the center of the lens and the fiber is due to errors in the lens and ferrule radius (±5 µm for each), lens and ferrule nonconcentricity (negligible compared to diameter
error), and the height error in the V-groove mounts used to align the lens to theferrule (about ± 25 µm). The total offset between the lens and ferrule in the object
plane (±√(2.52+242) = ±25 µm) is magnified 3.7 times by the lens resulting in adisplacement of the beam in the image plane, which occurs inside the crystal, of ±93 µm. Given the distance of about 6 mm between the lens and the crystal, thisdisplacement is equivalent to an angular error of approximately ± .015 rad. Finally,
one edge of the lens can be higher then the opposite edge (by ± 25 µm), which will
result in a lens tilt with respect to the ferrule (of ± .004 rad for the 6 mm GRIN lens).
This causes an angular deviation of the out-coming beam of ± .006, calculated byusing the GRIN lens ray-tracing matrix. Adding up all the contributions we get a total
tilt error of φ=√(2.0152+2.0062)= .023 rad. The current optical length between the33

coupling lenses in the module is about 10 mm, resulting in a displacement error of
230 µm, as shown in figure 2.8.

Set screws

Ø
10mm

230 µmFigure 2.8. Representation of total displacement and tilt errors between lenses.

The main point in designing a new base is the following: Given that no
adjustments are needed after the modules are ready, all the adjustments should beexternal to the base design. The base should serve solely to support the components,
avoiding the use of internal adjustments (screws, springs, etc.), and thereby reducing
the possibility of misalignment after the module is concluded. We suggest a designfor the base where the crystal lies in the center of the base, shown in figure 2.9 (a).
The five degrees of freedom for each collet would then be externally supplied by a“jig”, a 2-D tilt stage which is held by a 3-D translation stage. The main idea is tohave the base with poles or blocks that slide up to almost meet the ferrules after they
have been aligned. (A design with wide blocks is preferable to one with thin poles, as
it is less susceptible to vibrations.) Low shrinkage epoxy is then used to fill the small
gap between the ferrules and the base by capillary action. The ferrules should not
move during the capillary process as they are held fixed by the jig. The epoxy used to34

fix the ferrule to the sliding blocks and the sliding block to the base should have low
shrinkage coefficient and low coefficient of linear thermal expansion (CTE). It should
also have a low enough viscosity (< 1000 cps) so that it can wick into the gap
between the ferrule and the pole or block. One viable epoxy is EPOTHIN from
Bhueler with a shrinkage coefficient of .0002, CTE of 62·10–6/°C, and viscosity at
25°C of 200 cps. Another possibility is the EP30 from MasterBond with a shrinkage
of .0003, which is slightly higher than that of the EPOTHIN but the CTE is lower,
35·10-6/°C, approaching that of aluminum, 24·10-6/°C, and of zirconia, 10.5·10–6/°C,
the material used in the ferrules. One could also experiment machining the base outof a material with lower thermal expansion than the currently used aluminum. There
are machinable ceramics whose CTE is close to that of the zirconia, such as
AREMCO 502-400 with a CTE of 10.8·10–6/°C. The glass-ceramic material
ZerodurTM is another possibility with CTE on the order of 10–7/°C.
Besides a new base design, below are other possible modifications to take into
consideration for a future design. Most of them involve a trade-off between gain andloss. One possible modification is to use rectangular cut crystals but with antireflection coatings on both input and output surfaces to reduce the reflection losses.
However, that would allow both polarizations to enter the crystal and, as discussed
earlier, would reduce the gain. That would also significantly increase the cost of themodule. On the other hand, one could add polarizers to increase the gain. However,
due to the speckle nature of the beams which scrambles the polarization, thepolarizers would also increase the passive losses. Naturally, one more way to35manipulate the gain/loss trade-off is via the crystal’s length, as more length implies
more gain but also more absorption losses.

Figure 2.9. Suggestion for modular base design. Ferrule-lensassembly is adjusted externally by translational and rotational stages.
Then the corresponding bases are slid upwards to meet the ferrule andglued with low shrinkage epoxy.
Finally, in a stretch of our imagination, it would be interesting if one could
find a transparent liquid/gel substance which could solidify with extremely low
shrinkage and without changing its refractive index. The ferrules could be positioned,
with the use of pre-solidified bars of the same material, inside a mold containing theliquid material. The ferrules would then be aligned and the material in the moldsolidified. Cut off the solidified bars that were originally holding the ferrules (whichshould now be sticking out), polish the six faces (assuming your mold was aparallelepiped) and you have it: a two-beam coupling module as a solid transparent
piece (like the fossils in resin that you saw as a kid) with fibers sticking out.

36

2.5.

AN APPLICATION EXAMPLE: THE AUTO-TUNING FILTER

Once the modules are in hand, building photorefractive circuits is trivial. The
two-beam coupling modules can be quickly and easily interconnected to build andoscillator rings [Yeh,'85], reflexive-coupling units [Anderson,'95], and other systems.
We tested the modules by using a ring oscillator with and without a reflexive
coupling unit. This architecture is called the auto-tuning filter, an information
processing system also known as the feature extractor [Zozulya, A. A.,'95], which we
describe in this section as an example of the versatility provided by the modules.
Given two or more temporally and spatially orthogonal signals superimposedon a common input beam, an auto-tuning filter is a self-organizing system that learns
to select the strongest input signal from within all the other input signals, in otherwords, it performs a principal component extraction. It consists of a photorefractive
ring resonator with a common input beam as the pump. The non-linear dynamics inthe crystal causes a competition of the signals for the resonator modes; the signal withthe greater optical energy wins. For simplicity, only two input signals are used in our
experiment. We built the auto-tuning filter system in the two different configurations
shown in figure 2.10: as a simple ring oscillator and as a ring oscillator with areflexive-coupling unit inside the ring.

37

Inputs S1,S2 Output
S1
S1>S2

Inputs S1, S2 Output
S1
S1>S2

(a)

(b)

Figure 2.10. Autotuning filter in two different architectures: (a)simple ring, and (b) ring with reflexive coupling. The photorefractive
crystals are represented by the circles.

Reflexive coupling [Anderson,'95] increases the competition between thesignals that oscillate inside the ring, enhancing the selectivity of the auto-tuning filter.
In simplified terms, the reflexive-coupling works as follow: A fixed fraction of theseoscillating signals is split away from the ring and folds back to cross the ring inside
the reflexive-coupling crystal. Part of these split signals are then coupled back into
the ring -the strength of this coupling is dictated by the strength of thephotorefractive grating written by each signal. These two signals compete for theavailable gain in the crystal. The weaker signal writes a weaker grating and thus
experiences greater round-trip loss, which results in an even weaker signal, and so on,while the stronger signal undergoes the opposite behavior. Ideally, this positive
feedback process continues until the weaker signal is totally inhibited by the stronger

38

one. In practice though, total inhibition of the weak signal is defeated by fanning[Zozulya, A.,'95] of the loss beam into the ring.
Figure 2.11 shows the experimental setup for the auto-tuning filter withreflexive-coupling. The laser beam is split by a polarized beam splitter; the intensity
ratio of these two input beams is adjusted by the half-wave plate preceding the beam
splitter. Each beam goes through an acousto-optic modulator (AOM) with 60%
diffraction efficiency in the first order. The AOMs perform two functions: they shift
the optical carrier frequency, resulting in a frequency separation of 280 MHz between
the two input beams, which guarantees their temporal orthogonality; and they act as
shutters to time modulate the optical carriers with the two input signals, single-sided
square pulses at 1 kHz and 625 Hz respectively. The duty cycle of these pulses is set
at 90% for low power loss. The two signal beams are then coupled into optical fibers
and mixed by a directional coupler. To have the same contribution of each signal in
the mixed input, a .50/.50 directional coupler is used (meaning that half of each
incoming signal beam is coupled to each outgoing fiber). The beam of one of theseoutgoing fibers is directed to a photodiode, and the other is photorefractively coupled
into the resonating modes of the ring by the pump unit, forming the oscillating beam.
A second directional coupler inside the ring splits the oscillating beam so that 90% is
directed into the input loss port of the reflexive-coupling, and the other 10%, into the
input gain port. Finally, a third directional coupler (.01/.99) is used to sample thering, forming the output which is monitored by a second photodiode. The intensity ofthe two photodiode signals are displayed on a spectrum analyzer (HP 356A, 100kHz).

39

The experimental setup for the simple ring oscillator is similar to that of figure 2.11
except that reflexive coupling is bypassed with an optical fiber.

Input:
S1>S2

Argon-Ion
Laser

Spectrum
Analyzer

Output:
S1>>S2Scope

λ/2 PBS

514nm

Driver 1

AOMs

Driver 2

Photo-diodes

S1

L

G

G

L

Modular
Units

Directional
Couplers

S2

Fiber
Couplers

L

G

G

L

10%
90%

Reflexive Coupling

Figure 2.11. Experimental setup for the autotuning filter withreflexive coupling. The setup for the simple ring is obtained by simply
bypassing the reflexive coupling. (PBS: polarized beam splitter, AOM:
acoustooptic modulator.)

The effectiveness of the auto-tuning filter can be given by the contrast ratio(the strongest signal intensity over the weakest signal intensity) in the output, Rout, as
compared to the contrast ratio in the input, Rin. Figure 2.12 shows the results obtainedwith an Rin of 3.17. The intensities of the input signals are shown at the top of the
figures and that of the output signals, at the bottom. Figure 2.12 (a) shows the case of
the ring with reflexive coupling. The resulting Rout is approximately 45 dB. We found40

that the system could not self-start oscillation with Rin lower than 3.17. This isbeacuse the available gain in the medium is shared by the signals, meaning that thecloser Rin is to one, the lower the gain available to the stronger signal. Eventually,
when R in < 3.17, this effective gain becomes lower than the minimum necessary forthe system to self-start.
The effect of adding the reflexive coupling to the ring can be evaluated bycomparing the result in figure 2.12 (a) with that in 2.12 (b). The absence of reflexive
coupling in the simple ring configuration decreases the competition between thesignals, resulting in a lower contrast at the output: Rout is about 38 dB, 7 dB lower
than with reflection coupling. On the other hand, the removal of reflexive coupling
also reduces the round-trip losses, which enables the system to self-start oscillation
with lower Rin. The simple ring could self-start with Rin down to 1.75.
Finally, to check for asymmetries in the system, we switched the input signalsof the ring with reflexive coupling, but keeping the ratio constant. The result is shownin figure 2.12 (c). Rout is 43 dB, 2 dB worse than the symmetric counterpart,
corresponding to a 2:3 asymmetry. We believe this asymmetry is mostly due to themode structure dependence of the directional couplers.

41(a) Ring with Reflexive Coupling:

Input: S1/S2= 3.17 ± 0.1
Output: S1/S2= 45.4dB ± 1dB (34 674)(b) Simple Ring:

Input: S1/S2= 3.17 ± 0.1
Output: S1/S2= 38.2dB ± 1dB (6607)(c) Ring with Reflexive Coupling (symmetric case):

Input: S1/S2= 3.17 ± 0.1
Output: S1/S2= 42.5dB ± 1dB (17783)

Figure 2.12. Input and output intensity spectra for the autotuning
filter. Input spectrum is shown at the top of each screen with a ratio of
3.17:1 between the two input signals (linear scale); output spectrum isshown at the bottom (dB scale). (a) Ring with reflexive coupling,
output ratio is 45 dB; (b) simple ring, output ratio is 38 dB; and (c)same as (a), except that the input signals are switched to check forasymmetries, output ratio is 43 dB. Reflexive coupling significantly
enhances the selectivity.

42

2.6.

CONCLUSIONS

On the auto-tuning filter application above, the use of the two-beam coupling
modules allowed quick verification of the properties of the system, such as the factthat adding the reflexive-coupling increases the output contrast. In this context themodules have demonstrably fulfilled its purpose of supplying a means for quick
testing of concepts and of new ideas in photorefractive systems. With a collection ofdifferent modules available, complex systems can be easily assembled and modified.
The specific modular design one chooses for a particular function, such as
two-beam coupling, is always subject to improvements. In this chapter, we haveshown the evolution from the original two-beam coupling modular design to an
improved design rendering higher gain, lower losses and less long-term alignment
drift. Suggestions for a future design to further reduce the drift is also given.
Eliminating the long-term drift would add system reproducibility – after a modular
system is disassembled it could be re-assembled exactly as before by simply using thesame modules.
When building the modules one would want to tailor it to their applications.
For example, the number of modules that can be cascaded, say, within a ring, islimited by the losses. Therefore, it might be of interest to further reduce the passive
losses of each module, but keeping in mind the typical trade-off between gain andloss, as discussed at the end of section 2.4.

43

CHAPTER 3:
BARIUM TITANATE SPHERES AND SPHERICAL DISKS

3.1.

INTRODUCTION

In the last chapter we discussed the two-beam coupling modules. In thischapter we discuss an idea with the original purpose of making two-beam coupling
units even more compact: that of having a spherical barium-titanate crystal as thegain medium. Due to its curvature and high index of refraction, the lensing and gain
effect of a photorefractive sphere can be combined to perform fiber-to-fiber twobeam coupling, dispensing the use of external lenses. The resulting two-beamcoupling units with barium-titanate spheres are described in section 3.2 below.
We also found that, due to its high photorefractive gain, it is easy to excite
whispering gallery modes (WGM) within the spherical crystal by shining a single
pump beam onto it. Several patterns were observed, but the lowest order mode, thetriangle pattern, was found to be the most robust. The WGM patterns can be better
observed and documented when excited in a spherical disk, an equatorial slab of thesphere. In section 3.3 we discuss the oscillating patterns and compare the occurrence
region of the triangle patterns with that predicted by a simple theoretical model.

44

3.2.

PREPARATION

Optical quality spheres are straightforward to produce with hand lapping andpolishing techniques. [Baxter,'50] We found that barium-titanate and lithium niobate
are not substantially more difficult to form into a sphere than optical glass. The
barium-titanate was electrically poled in its rectangular form before processing it intoa sphere. We feared that barium titanate in particular might become depoled during
processing, but such was not the case. Finished spheres of photorefractive barium
titanate in fact turn out to be more robust to shocks than their delicate rectangular-cut
counterparts, possibly due to the lack of sharp edges or corners from which cracks
could propagate.
Spherical disks were produced by first forming a sandwich of two lithiumniobate slabs surrounding a slab of barium-titanate, held together by quartz wax. The
sandwich is then lapped and polished into a sphere. Upon removing the lithiumniobate end-caps one is left with a barium-titanate disk with spherical sides, with thec-axis in the plane of the disk. Unlike the sphere, it is easy to observe through the flat
top of the disk the beams as they propagate inside the crystal. We produced sphereand disk sizes from 3 mm to 7 mm diameter.
The c-axis of a 0°-cut photorefractive crystal is normally found by placing the
crystal between two polarizers and identifying the widely-known conoscopic “ironcross”. The lensing property of the sphere morphs the iron cross into a “beach ball”
as shown in Figure 3.1, thus it is easy to find and orient the c-axis of the sphere.

45

Figure 3.1. The “beach ball” pattern of a spherical photorefractive
crystal between crossed polarizers observed directly along the c-axis.

3.3.

TWO-BEAM COUPLING WITH SPHERES

Two-beam-coupling interaction can be made compact by taking advantage ofthe geometry of the spheres and the high index of refraction of barium titanate, whichis greater than 2 throughout the visible spectrum. Due to these properties, the crystal
can directly image the light from one optical fiber to another. In a symmetricalarrangement, the collimated beam inside the sphere can be photorefractively coupled
to a second beam propagating between a second pair of fibers. Figure 3.2 shows
direct fiber-to-fiber two-beam coupling using a photorefractive sphere without otherintervening optical elements.

The sphere conveniently magnifies the interaction

region viewed by the camera. Two beams, at 514 nm wavelength, cross inside thecrystal and undergo two-beam coupling. Each beam is associated with a pair ofmultimode gradient index fibers, with a 62.5 µm core diameter. With a spherical disk46

of 4 mm diameter, the collimated size of the laser beam in the crystal is about 0.8 mm
when the fiber-to-fiber imaging is optimized. The apparatus allowed rotation of thecrystal axis continuously and permitted the angle between the beams to be chosenfrom 5° to 45° in 5° increments. Maximum gain occurred at the expected 45°
between the c-axis and input beam bisector. The optimum beam angle was observed
to be 10°±2.5° - larger than the 4°-6° typically estimated in theoretical plane-wave
calculations. [Fainman,'86] Gain was measured using a loss-to-gain beam ratio of
105:1 and a gain-beam strength of 0.5 µW. As in the previous chapter, we define gain
to be the ratio of the gain-beam intensity at the output of its fiber with the loss-beampresent to its intensity with the loss-beam blocked. Because the light from the fibers
is unpolarized, one expects an effective reduction of the coupling constant Γ by afactor of 2. We nevertheless obtained a maximum gain of 8000, which is comparable
to gains obtained with rectangular-cut crystals of similar length.1

Figure 3.2. Two-beam coupling between two pairs of fibers using a
spherical photorefractive crystal of barium titanate.

1

The two-beam coupling unit was built and tested by Edeline Fotheringham.

47

3.4.

WHISPERING-GALLERY MODE PATTERNS

We have also investigated interesting oscillation patterns that occur when aGaussian (free-space) pump is incident on the spherical disk. The oscillating patterns
are understood to be low-order whispering gallery modes which are generated bytotal internal reflection at the crystal’s boundaries.
The modes of a spherical resonator that propagate by total-internal reflection
at the surface were first analyzed by Lord Rayleigh for acoustic waves and hereferred to the analysis as “The problem of the whispering gallery”. [Lindsay,'70]
Baer demonstrated continuous-wave active excitation of whispering gallery modes
using a Nd:YAG sphere pumped with a diode laser. [Baer,'87] Demonstration ofactive oscillation of whispering gallery modes in a gamut of other materials (andstructures) followed, such as in fused-silica [Schiller,'91], polymers [KuwataGonokami,'95], semiconductors [Courtney,'98], and organic materials [Lin,'98]. In
these demonstrations oscillation arises from stimulated emission, whereas in our
photorefractive spheres oscillation arises from two-beam-coupling, a parametricinteraction between the incident pump, the loss beam, and the modes of the resonator,
the gain beams.

48

3.4.1. A family of patterns

At a wavelength of 532 nm, the critical angle for total internal reflection is
arcsin(1/2.4), about 25°, thus the lowest order whispering gallery mode of a bariumtitanate sphere or disk is a triangle. Since the maximum two-beam coupling gain forbarium titanate occurs for small loss-to-gain interbeam angles, one expects thepatterns to orient themselves such as to minimize this angle. As shown in figure 3.3
for the triangle and square, the interbeam angle is minimum when a vertex of thepattern aligns itself with either the incident pump location or with the pump’s firstreflection location. (Second and higher pump reflections are too weak to excite
patterns.) We call the former “front-side” patterns (shown with full lines in figure3.3) and the latter “back-side” patterns (shown with dashed lines). As also shown infigure 3.3, the minimum possible interbeam angle occurs for maximum displacementof the pump, giving 5° for a triangle, 20° for a square and larger for higher-orderregular polygons. Since this interbeam angle is smaller for lower order patterns, andsmaller interbeam angle translates into higher gain, we expect the triangle to be boththe strongest oscillating mode and the easiest to excite.

49

45°

30°

(a) φ=0°
r sinφ

25°

25°
20°

5°
65°

65°

(b) φ=90°

Figure 3.3. Diagrams of minimum interbeam angles. (a) Pump
propagates directly down the center of the crytal. (b) Pump is incident
at maximum displacement (r sinφ) from center. In every case, theminumum interbeam angle occurs when vertices are aligned with thepump incidence location (full line triangle and square) or whenvertices are aligned with pump’s first reflection location (dashed line).

Our experiments reveal that this is indeed the case. The setup used forobserving the oscillating patterns is shown in figure 3.4, and pictures of the patterns
in figure 3.5. Figure 3.5 (a) shows a front–side oscillating triangle mode in a 5.2 mm
diameter spherical disk. In the figure, the pump beam enters the crystal in thevertically downward direction. The pump beam is focused by a microscope objective
so that a beam waist occurs inside the disk and is roughly matched in position with awaist of the mode. Most of the pump energy is depleted in the first millimeter after
the pump enters the disk, indicating efficient energy transfer. Back-side oscillation50

also occurs when the crystal c-axis is appropriately rotated. In Figure 3.5 (b) gain isprovided to the triangle by the reflection of the pump off of the far-side surface. In
this instance the microscope objective is adjusted to collimate the beam in the disk.
The oscillation is weaker than the front-side case since the pump intensity is reducedby both absorption and transmission (causing the pump to exit the crystal).
Other modes are excited by changing the orientation of the c-axis, the lateralposition of the pump, as well as the pump’s angular distribution. Figures 3.5 (c) and(d) show a front-side and back-side square respectively. As explained above, they aremuch weaker than the triangle because the two-beam coupling gain falls off rapidly
with pump-signal angle. We have observed squares in isolation but they were too
weak to photograph well. Many patterns can be excited simultaneously under certain
conditions. Close inspection of Figure 3.5 (e) reveals a triangle, a square, and apentagon. For higher angular distribution of the pump, the triangles tend to getblurred, i.e., they also acquire a high angular distribution, as shown in figures 3.5 (c)and (f). Higher order modes are weaker and tend to be difficult to observe bothbecause they experience less gain as the order grows and because they are obscuredby the spherical curvature of the disk outer surface. Hexagons were the highest orderobserved and multiple patterns such as David stars (two interlaced triangles) werealso observed.

51

f=100mm

Argon-Ion Laser
514 nmλ/2
wave-plateφΨ

BaTiO3disk

lens is on a 2-D
translation stage

crystal is on arotation stage

c-axisFigure 3.4. Experimental setup for observing oscillating patterns. The
angular displacement of the pump is given by φ and the rotation of thec-axis by ψ.Note that the parameters φ and ψ are measured withrespect to the vertical line (parallel to the pump) which crosses thecenter of the crystal.

52

Figure 3.5. Oscillation patterns inside a barium titanate spherical disk.

53

3.4.2. A theory for the triangle occurrence regions

As further investigation we calculate the gain maxima for triangle excitation
using a simple model which uses two-beam coupling of the pump with one side of thetriangle. The calculations assume plane-waves for the interacting beams. Specifically,
we calculate the c-axis orientation which maximizes the gain for the triangle as afunction of the incoming pump-beam angle. Figures 3.6 (a) and (b) show thegeometrical relations used in our calculations for a front-side and a back-side triangle
respectively. For an input displacement of the pump, given by rsin(φ), the minimum
interbeam angle is given by 2θ=30°-φ′, where φ′ is the refraction angle with respect tothe surface’s normal. The two-beam coupling gain curve, given by the expression(see chapter 1, section 1.2)=

2
reff , where reff = −2 no2ne2 r42 Esc ( , ) cosn cossin(2 ),

[3.1]

has two maxima when plotted as a function of the c-axis orientation ϕ for a fixedinterbeam half-angle θ. Therefore, for each value of the interbeam angle 2θ above we
can find from the gain curve two c–axis orientations, ϕd and ϕr, corresponding to thetwo maxima. The angles ϕd and ϕr give the direction of the crystal axis with respect tothe bisector of the two interacting beams. The subscript in ϕd stands for “direct”
denoting that the c-axis direction is less than 90° apart from the bisector, whereas ϕrstands for “reverse”, and has a value between 90° and 180°. To compare our results to54

the measurements, we must reference the direct and reverse c-axis angles to thedirection of the incoming pump (the vertical in figure 3.6) thereby obtaining ψd andψr. Taking both front-side and back-side configurations in figure 3.6 into account, we
end up with four optimum c-axis orientations, yielding four theoretical curves of ψ asa function of the input pump-beam angle φ.

φ’
2θ
ϕr

φ
ψr
ψd
ϕd

ψd,r=180°−ϕd,r +φ−φ’−θψd,r =ϕd,r +φ−φ’+θ

(a)

(b)

Figure 3.6. Schematic showing the geometrical relationship between
ψ and φ for a (a) front-side and (b) back-side triangle. For a given θ,
there are two optimum values of ϕ that maximizes the two-beamcoupling gain, resulting in also two optimum values of ψ, namely ψdand ψr.

Figure 3.7 shows the occurrence map for triangles with respect to the incident
pump-beam angle (φ) and the orientation of the crystal axis (ψ). Both angles are

55

referenced to the line that crosses the center of the disk and is parallel to the external
incident pump. The four above-mentioned calculated curves (thin lines),
corresponding to the theoretical gain maxima, are superimposed over the measured
triangle regions (thick horizontal lines). The triangle structures occur over a wide
variety of beam parameters. Figure 3.7 shows a reasonable qualitative agreement
between the measured regions and the maximum gain calculations. The superposed
dotted-lined ellipses are meant to qualitatively indicate the range of parameters overwhich triangles are observed.

øΨ
c-axis rotation angle (degrees)

Figure 3.7. Triangle occurrence regions. Dotted ellipses qualitatively
show the four regions of the parameter space for which triangles areobserved (the pump beam size and divergence are fixed). The
horizontal bars show quantitative data regions of triangles. The solidlines represent the four theoretical curves of φ(ψ) that maximize gain.
(The parameter-space coordinates are defined in either figure 3.4 or
3.6.)

56

3.4.3. Unidirectional oscillation

Two qualitative observations suggest that the oscillating patterns are
unidirectional and in the forward direction defined by the pump. First, we observethat the light scattered from nonuniformities in the crystal to be in the forward
direction. Second, we never observed a steady phase-conjugate beam. In rectangularcut crystals self-pumped phase-conjugation of a single pump beam is often observed.
[Feinberg,'82] The corners of the crystals act as corner-reflectors that generates thephase-conjugated beam via four-wave mixing. Since the disks and spheres have nocorners one might wonder whether self-pumped phase-conjugation can occur. In fact,
we were never able to observe steady–state phase-conjugation of the input pumpbeam under any circumstances in any of our sphere or disk samples. We were able toobserve highly unsteady phase-conjugation of the pump beam at normal incidence tothe crystal, along the c-axis direction.1

3.5.

CONCLUSIONSThe spherical barium titanate crystal is interesting for several reasons. Most
systems built in our laboratories employ two-beam coupling as a basic function. In
many cases, the two-beam coupling unit is used to pump a ring oscillator, such as thecase of the auto-tuning filter, which we discussed in the previous chapter and will

1

Phase-conjugation observations were done by Dr. Dana Anderson and Dr. Vladimir Shkunov.

57

revisit in the next chapter. By allowing fiber-to-fiber two-beam coupling, the spheres
contribute in building compact systems. Compactness not only has the advantage ofmaking photorefractive systems portable and light weight, but also of reducing
vibrations and increasing robustness, this last due to the decreased number of
elements necessary in the system. The WGM patterns, for now, are only a scientificcuriosity. However, one can think of them as a very compact ring oscillator inside thecrystal itself. To make them useful for real systems one needs find a practical way ofcoupling some of the oscillation light out of the crystal. That could be done byfrustrating one of the reflections of the oscillating pattern [Schiller, '92] and thencoupling to an optical fiber.
Finally, the spherical disk geometry with the c-axis in the plane of the disk isconvenient for determining two-beam coupling parameters. In particular, the internalbeam angles are not restricted as they are in rectangular-cut samples by Snell’s law
and the beam geometry is practically independent of crystal orientation, though there
is some degree of optical asphericity due to crystal birefringence.

58

CHAPTER 4:
CARRIER SUPPRESSION WITH TWO-BEAM COUPLING

4.1.

INTRODUCTION

Optical carrier suppression can be used in applications where one wishes toeliminate the carrier or increase the modulation depth of an optically modulated
signal. One well known example is in photonic links, where it is necessary that thesignal be highly linear and therefore the optical carrier be very weakly modulated.
Thus, to avoid the saturation of components upon detection, and to decrease the shotnoise due to the information-free carrier, a carrier suppressor might be required toincrease the modulation depth.
Several techniques for carrier suppression have been demonstrated. A
conventional and straightforward technique is to bias a Mach-Zehnder modulator
around the minimum transmission point, such that the optical carrier destructively
interferes at the interferometer’s output, leaving only the sideband modulations.
[Montgomery, '95] Another very common technique for carrier suppression in fiberoptic links is that of using stimulated Brillouin scattering, where the carrier
propagating inside an optical fiber pumps a beam which propagates in the opposite
direction. Optical feedback of the generated beam enhances the carrier suppression
and, in fact, a record carrier suppression of 55 dB has been reported by using a59

“Brillouin-erbium fiber laser” configuration. [Loayssa, '00; Tonda-Goldstein, '00] Our
goal in this project is to build a simple photorefractive carrier suppressor, which, as
opposed to the techniques mentioned above, is adaptive due to the real-time
holographic nature of photorefractive gratings. Carrier suppression is achieved by a
two-beam coupling interaction within a photorefractive barium titanate crystal anduses the principles of an optical novelty filter [Anderson, '89; Khoury, '91]. Similar
techniques have been used for optical heterodyning [Hamel de Monchenault, '88;
Khoury, '93], the logical opposite of the carrier suppression problem.
The application of our carrier suppression system is as part of a higher levelsystem, the “optically-smart antenna array”. This system, schematically shown infigure 4.1, is composed of an RF-front end active antenna array followed by aphotorefractive optical circuit that separates the strongest incoming signal from other
weaker ones. For demonstration purposes, the current system uses only two antenna
elements and two test-sources, therefore requiring a two-channeled carrier suppressor.
The main purpose of the carrier suppressor in this system is to remove the correlation
between the modulated optical signals, which is mostly due to the optical carrier, sothat the next stage, the autotuning filter, can function optimally. The autotuning filter
is the same processor as that described in section 2.5 and its goal, as herein, is to
extract the principal component from the various signals presented at its input.
Different versions of the auto-tuning filter, some with fiber optics and spherical disks
like the ones described in the previous chapter, and others with Gaussian beam optics
and other crystal cuts have been tested. However, their use doesn’t pertain to thecarrier suppressor and will not be considered further here.

60

This chapter is divided into three further sections. In the next section we givethe theoretical background and introduce a geometrical picture to the “operator
formalism”, an original theory for describing two-beam coupling interactions. This
theory proves to be very useful and, in fact, has played a crucial role in theunderstanding and, thus, in the development of the carrier suppressor. The goal of thenext section is not to formally present the complete operator formalism, which will betreated later in the last chapter, but to give a simplified, geometrical view of thistheory. It offers an intuitive understanding of two-beam coupling, and thereby, of thecarrier suppression process. It provided us the tools necessary to appropriately set theparameters of the system, such as the modulation index or the input beam ratio of thetwo-beam coupling interaction.
After several iterations of carrier suppressors, and with the help of theoperator theory, a system was built which provided a carrier suppression of more than
70 dB, to our knowledge the best published results so far. Section 4.3 describe theexperimental setup and results of this system. We compare the results to the theory
presented in the section 4.2 by curve-fitting parameters, such as the input intensity
ratio, the gain and the input carrier intensity.
Finally, we close the chapter with a discussion of future work and the issues
involved in integrating the carrier suppressor to the above mentioned antenna array
application. We first discuss how the performance of the carrier suppressor describedin section 4.3 fulfills the performance specifications for that system. We thenintroduce a new carrier suppressor which takes advantage of a channelized
electrooptic modulator design, discuss it’s implementation, and expose an important61

limitation, that of cross-talk between the channels. Thus, further research on thechannelized design is still in progress.

S2,...,Sn

S1
S2

Sn
S1

S1 > S2,..,Sn
Active antenna array

Carrier
Suppression

Auto-tuning filter

Figure 4.1. Schematic of the optically-smart antenna array,comprising of three parts: the RF front-end which consists of an active
antenna array, the carrier suppressor, which consists of electrooptic
modulators followed by two-beam coupling in the novelty-filter
configuration, and the auto-tuning filter, which consists of a gain
crystal pumping a ring with a reflexive-coupling interaction in its
feedback path.

62

4.2.

A GEOMETRICAL PICTURE OF TWO-BEAM COUPLING

The current section describes a geometrical picture of the operator theory1,
where the two-beam coupling interaction is viewed as a rotation operator acting on
the input fields to produce the output fields. In this simplified exposition of thetheory, we assume real coupling, i.e., only pure energy coupling takes place between
the beams. With this assumption, the fields can be represented in a two-dimensional
space. A geometrical picture of the operator’s action over the fields is thenconstructed. The goal of this section is to provide an intuitive view and understandingof the main aspects of the carrier suppression process. The complete formalism istreated later in chapter 5 and is not necessary in the context of this chapter.
Carrier suppression is schematically shown in figure 4.2. The input to the loss
port is the phase-modulated “signal” composed of the optical carrier given by ωc andthe upper and lower sidebands spaced by ωm from the carrier. The input gain port
contains the unmodulated carrier, the “pump”. (Note that the definition of signal andpump is opposite than that of previous chapters where the signal beam was the gain
port and the pump, the loss port.) In the context of this theory, we refer to the loss
port as “minus” port, and to the gain port as “plus”. The two beams cross inside thephotorefractive crystal where they experience two-beam coupling in the so-called
novelty-filter configuration, where any component of the signal that varies faster than1This theory was developed by Dr. Dana Anderson.

63the response time of the crystal (τ) will not be able to write a grating and will
therefore be transferred to the output minus-port, although some of it will passively
scatter to the plus-port due to other gratings that may be present. On the other hand,the components which are slower than τ will write a grating and thereby experience
active loss by transferring its energy to the gain port. Thus, when experiencing thetwo-beam coupling interaction the optical carrier in the signal beam will transfer mostof its power to the pump beam whereas the sidebands will scatter only some of its
power to the pump beam, therefore resulting in the suppression of the carrier at theoutput signal port. This process assumes that the modulation frequency (ωm) is greaterthan the 1/τ, typically 1-100 Hz, depending on the crystal sample. The upper boundfor the modulation frequency is given by Bragg matching considerations of themodulated beam to the holographic grating written in the crystal. Two frequencies are
considered Bragg matched if the difference in phase that they undergo when crossing
the crystal’s length l is less then :

1n

c

−

2n

c

≤ l . This results in an upper boundfrequency of a few gigahertz for a 1cm crystal. The energy-transfer process depends
on the crystal’s gain and the intensity ratio between the two input beams. The gain, inturn, depends on the crystal length and other geometrical parameters.

64

ωm
ωc

ωm
ωc

−
+

ωm

ωc

ωc

Figure 4.2. Carrier suppression via a two-beam coupling interaction.
The input to the minus port is a modulated laser beam while that of theplus port is an unmodulated beam from the same laser. The carrier inthe loss port gets transferred to the gain port resulting in a suppressed
carrier output.

In the geometrical picture of the operator theory, the different temporal fieldcomponents of the input signal are represented in a plus-minus coordinate system,
defining a “fieldvector”
 E+ 
E =  − .
E 
To obtain the different components E + , the input field at the plus port is decomposed
into an orthogonal temporal basis, for instance that of harmonic frequencies, and eachcomponent or frequency is labeled by the index ω. Similarly for the temporal fieldcomponents in the minus port, E − .

65Assuming real coupling, the fieldvector can then be represented in a plane as shownin figure 4.3. Figure 4.3 (a) shows the vectors corresponding to the fields at the inputof the system. The carrier frequency is present at both input ports, plus and minus,
and therefore is represented by a “carrier” vector Ec, which lies somewhere in the
middle of the plane; whereas, since the sidebands are only present in the minus port,
they are all aligned with the minus-axis, and can therefore be summed into one single“sideband” vector Em representing all the sidebands. As we show in the following, aprincipal eigenvector e can be defined to represent the fields. We also show that theevolution of the system as the fields propagate in the z direction through the crystal, isthen represented by a rotation of the eigenvector together with all the corresponding
fieldvectors, such that the eigenvector asymptotically approaches the plus-axis. Thisevolution can thus be represented by a single equation for (z), the angle of theprincipal eigenvector with respect to the minus-axis. The angle of the carrier vector islabeled(z).

c

The mixing between the fields in the two ports can be conveniently expressedby defining the density matrix=

1
E E† ,∑
I

where I=|Ec|2+|Em|2 is the total input intensity,[4.1]

represents the different frequency

components and the dagger represents the vector adjoint. Such representation allows
the two-beam coupling interaction to be given by a single equation for[Anderson, 2000 #77]:

66

(z)

2
where

and

∂
=( 1−
∂z

2

[4.2]

)sin(2 ),are the eigenvalues of the density matrix. The normalizedinteraction length z points towards the propagation direction in the crystal and isdefined in units of 2/ , whereis the photorefractive coupling constant, which inthis case is real. In other words, z does not just represent the physical crystal length,
but the gain experienced by the fields as they propagate through the crystal.
The solution to the above equation is given bytan (z) = e ( 1 − 2 )z tan (0) ,[4.3]

which simply means that the principal eigenvector asymptotically rotates towards theplus-axis as it propagates in z, and that this rotation is “slowed down” by the
exponential factor, where 0≤.≤1 since

is normalized. It is important tonote that, since all the fields in the crystal share the same grating, all the vectors rotate
together. One can think of the eigenvector e as “pulling” the other vectors, Ec and Em,
with it.
The main consequence of this rotation to the carrier suppression is thefollowing: As the principal eigenvector rotates towards the plus-axis, it will go
through a configuration, shown in figure 4.3 (b), where the projection of the carrier
vector on the minus-axis is zero. This means that, for a gain z=L resulting in thatparticular rotation, there is no carrier present in the output minus-port, i.e., the carrier
is perfectly suppressed. It is interesting to note that if the modulation were “off” thenEm=0 and the principal eigenvector e would therefore coincide with Ec, such that
. Thus, Ec would only asymptotically approach the plus-axis, meaning thatc67

perfect carrier suppression could not occur for a finite crystal length. However, due tothe presence of the modulation Em, the eigenvector e does not coincide with Ec. Thus,
as e approaches the plus-axis, it will go through a particular value of z where Ec
coincides with the plus-axis, meaning perfect carrier suppression at a finite crystal
length. In other words, it is the presence of the modulation which allows for perfectcarrier suppression at finite lengths. From the geometry of figure 4.3 (b), it is easy toshow that the rotation (L) corresponding to perfect carrier suppression is simply
given by[](L) = arctan e ( 1 − 2 ) L tan (0) = 90o + (0) − c (0) .

+

+

Ec

Ec

φc(0)

φ(L)
e

φ(0)

e
Es

(a)

[4.4]

Es

-

(b)

-

Figure 4.3. Carrier and sideband vectors, Ec and Es respectively, in theplus-minus coordinate system. The principal eigenvector e embodies
the dynamics of the system. (a) Initial position at z=0. (b) Position ofvectors after rotation, such that the projection of the carrier into theminus-axis is null, resulting in a carrier suppressed output at that port.

Next, we describe the carrier suppression process by using the following
experimentally convenient parameters: r is the intensity ratio between the plus and68minus input ports; and m is the modulation strength, defined as the ratio between the
power in the sidebands over the power in the carrier at the input minus port:2

Ec+ (0)I + (0)
r= −
=
2
2 ;
I (0) Ec− (0) + Em− (0)

[4.5]

2

−
I − (z) Em (z)m(z) = m−= − 2.
Ic (z) Ec (z)[4.6]

Em− (z) and Ec− (z) are the projections of the fieldvectors Em(z) and Ec(z) into the minus
axis after the corresponding rotation of the input fieldvectors by (z)- (0), where (z)is given by equation [4.3]. Similarly, Em+ (z) and Ec+ (z) are the projections into theplus axis after rotation. At z=0, the modulation strength is given by m(0)=mo. Using
equations [4.3], [4.5] and [4.6], the modulation strength, written in terms of r, m0, andthe gain z, is given bym(z) =

m0(1 − r(m +1) tan[ (z) − (0)])

2

.

[4.7]

0

We discussed above that, for a fixed value of the input modulation strength
m0, there exists a particular gain (z=L) which results in a rotation such that the carrier
is perfectly suppressed, i.e. m(L)→ . Conversely, it is also true that, for an arbitrary
fixed value of the crystal’s gain z, there exists a particular value of m 0 for which thecarrier is perfectly suppressed. Figure 4.4 shows four different curves for themodulation strength at the output, m(z), versus that at the input, m0. Each curve has apeak which goes to infinity. The peak occurs at lower input modulation strength forhigher gain and lower input intensity ratio. If the input intensity ratio increases from
r=1 to r=9, m(z) is higher everywhere except at the peak of the r=1 curve, however,69

that has the cost of reduced light intensity in the signal beam (minus beam). Also, if
the gain increases from z=6 to z=12, m(z) again becomes higher everywhere except at
the peak of the z=6 curve, but now the cost is that the peaks occur for lower m0.

Figure 4.4. Output vs. input modulation strength, for different
combinations of gain and input intensity ratio. All peaks go to infinity.
Notice that for higher gain the peak occurs for lower input modulation.
Except at the peak locations, higher gain or input intensity ratio bothresult in a general increase of the output modulation, at the cost of
decreasing the output signal strength.

In addition, we can find z=L that maximizes m(z) by using equation [4.4],
where

and

c

are written as a function of r and m0:c( r(m +1) )= arctan

[4.8]

0



2 m0r +1=arctan0 1− r + (r + 1)2 − 4r mmo+1 


0

( )

70

[4.9]

By plotting L as a function of r, as shown in figure 4.5, we find that, regardless of the
input modulation strength, perfect carrier suppression requires the least gain at equal
input intensity beams (r=1). The above theoretical results are tested in the system
discussed in the next section.

m=0.3

L
m=0.5

m=0.9

r
Figure 4.5. Gain required for suppressing the carrier as a function of
input intensity ratio for three values of the input modulation strength.
Notice that, regardless of the modulation strength, the minimum gain
occurs at r=1.

4.3.

THE CARRIER-SUPPRESSOR

The operator theory introduced in the previous section provided us with thetools necessary to optimize previous generations of carrier suppressors, and build thephotorefractive carrier suppressor described in this section.
We prepare the input signals for the carrier suppressor by modulating theminus port optical beam. For that purpose, we built phase modulators containing
lithium niobate crystals doped with magnesium oxide (MgO:LiNbO3). The

71

magnesium-doped version was selected as it is known to reduce the sensitivity oflithium niobate to optical damage. [Volk, '94] One drawback of the doping is that it
causes striations which vary boule to boule, thereby increasing the chance of
wavefront distortion as compared to undoped lithium niobate. However, for ourapplication, wavefront distortion doesn’t present any problems as the photorefractive
grating can correct for it. A MgO:LiNbO3 crystal was tested in our laboratory at
532 nm, the wavelength used in the carrier suppression experiment, and resisteddeveloping significant optical damage below 500 W/cm2, which for our applications
is considered high intensity. Two phase modulators were built, each with a 45.0 x 1.5
x 1.5 mm3 crystal with a half-wave voltage of approximately 70 V at 532 nm. (Thewavelength bandwidth for magnesium-doped lithium niobate is from 500 nm to 900nm.) The optical transmission of 66% through the modulator is rather low – due bothto the uncoated optical surfaces, and to striations inside the crystal which are evenvisible to the naked eye. Each modulator is driven through a resonant circuit mounted
directly on the modulator. The resonant circuit, shown in figure 4.6, is tuned to129.5 MHz and has a quality factor (Q) of about 15. Including the resonant circuit,the modulator has a modulation efficiency of 0.7 rad/V.

RFsignal80 - 90 nH

15 - 60 pF

EOM

1 - 10 KΩ

Figure 4.6. Resonant circuit of the electrooptic modulator used in the
carrier suppressor.

72

The carrier suppressor is built with 1” optical height standard, resulting in acompact system of about 38 x 38 cm2, which reduces the system sensitivity tovibrations and acoustic noise. A picture of the carrier suppressor is shown in figure
4.7 (a). Figure 4.7 (b) shows the setup schematic of the test system. The laser is adoubled Nd:YAG at 532 nm wavelength. A telescope brings the 2.2 mm diameterbeam down to about 500 µm. A half-wave plate, followed by a polarizing beam
splitter, allows for a variable splitting of the beam. We adjust the splitting intensity
ratio so that r=1 at the input of the photorefractive crystal, which, as mentioned in theprevious section, minimizes the gain necessary for optimum carrier suppression. Oneof the beams enters the electrooptic modulator and is modulated by a 129 MHz
signal. Since optimum photorefractive interaction occurs with pure-extraordinary
input polarization, a polarizer with a 10-5 extinction ratio is placed at the output of themodulator to reduce the ordinary polarization component from the signal beam. This
beam then crosses the unmodulated beam inside a barium-titanate crystal. A
heterodyne detection scheme1 enabled the direct measurement of the output carrier
and modulation sideband strengths. The heterodyne beam is shifted by 80 MHz withrespect to the optical carrier via an acoustooptic modulator.

1

Suggested by Bruce Tiemann.

73

BS1

λ/2

L2

L1

λ/2
L3

BS2

EOM

Pol.

PBS

BaTiO3(a)

Nd:Yag532 nm
L1 L2
BS1

λ/2

λ/2

AOM

L3
L4

BS2

Pol.
PBS

BaTiO3Spectrum
Analyzer(b)

Figure 4.7. (a)Picture and (b)Setup of the carrier suppressor. (At thetime the picture was taken the acoustooptic modulator had been
temporarily removed. BS: beam-splitter, PBS: polarizing beamsplitter, L1 and L3: 20 mm focal-length lenses, L2 and L4: 6.15 mm
focal-length lenses, EOM: electrooptic modulator, AOM: acoustooptic
modulator, λ/2: half-wave plate, Pol.: 10-5 extinction-ratio polarizer.
All unspecified optics are mirrors.)74

To characterize this system we varied the input modulation strength imposedby the modulator and observed the resulting carrier output as displayed by thespectrum analyzer. From a theoretical standpoint, we could have also varied the gain,
but this is not as easy to do experimentally in a well-controlled manner. The
measured intensity of the carrier at the minus output port ( Ic− ) is plotted in figure 4.8
as a function of the modulation index ( ).1 A theoretical curve is also shown. We
notice two dips of the output carrier strength. The dip at β=2.4 corresponds to acarrier nulling due to the phase modulation process. This zero is not of interest as,
first, it is not adaptive and tends to suffer drifts, and second, at this modulation index,
the modulation of the signal component is highly nonlinear and has negative slope.
Our interest is focused in the first dip occurring at a modulation index of β≅0.11,
which is due to the photorefractive carrier suppression. The parameters to thetheoretical curve are found by least-squares curve-fitting the data to the theory
presented in the previous section. Both the theoretical curve and the data arereferenced to the value of the output carrier power when the suppressor is “off”, i.e.,
except for passive losses, the output beam is a replica of the input beam. However,since this reference could not be reliably measured, we included it as a parameter inthe fit, along with the input intensity ratio r, and the normalized length z. We also
assumed a noise floor to avoid numerical instabilities caused by the infinite dips in
the fitting function. The noise floor was set at the measured noise floor of the system,–100 dBm as given by the spectrum analyzer. This value was due to a combination of1

This particular set of data was taken by Amy Sullivan.

75noise in the detection process and leakage of the carrier through the acoustooptic
modulator in the heterodyne detection beam. The parameters r and z were found to be
1.22 and 6.47, with standard errors of 0.16 and 0.04, respectively.
This first dip shows an output carrier suppression of –71.4 dB ± 1.3 dB. The
theoretical curve, the data and the noise floor at –76.8 dB is showed with respect tothe fitted reference, which is normalized to 0dB. Notice that at the higher data values
the fitted function is higher than the data due to systematic errors. Otherwise, there isgood qualitative agreement between the theoretical and experimental curves. The
average error between the data and fitted function is 3.7 dB.

Modulation Index

Figure 4.8. Measured output carrier power as a function of modulationindex. The output carrier is normalized to the value obtained when the
carrier suppression is “off”. The first dip of –71.4 dB is due to thephotorefractive interaction, whereas the second dip is due to a null of
the carrier at the phase modulation process (when β=2.4).

76

From the curve in figure 4.8 we obtain the curve for the output vs. input
modulation strength, which is shown in figure 4.9 (compare to the theoretical plots
shown in figure 4.4). This curve results in a peak output modulation strength of aboutm(z)=44 dB.

Input Modulation Strength (dB)Figure 4.9. Output vs. input modulation strength obtained from fitted
carrier suppression data.

As discussed earlier, the original application for the carrier suppressor is aspart of the optically-smart antenna array system. The next section discusses how thiscarrier suppression fulfills, on a per-channel basis, the performance-requirements forthe antenna array system. However, the antenna array system has space-requirements,
demanding a re-design of the carrier suppressor to process several channels in acompact space. In that section, we introduce a design that can potentially fulfill theserequirements, and discuss its current limitations.

77

4.4.

FUTURE WORKAs described in the introduction to this chapter, the application for our carrier
suppressor is as an intermediate stage of the optically-smart antenna array (see figure4.1), where its function is to provide uncorrelated signals to the following stage, theautotuning filter, by suppressing the carrier. The goal of the work described in thischapter is to demonstrate carrier suppression performance necessary for integration
with the antenna array.
A carrier suppression of more than 70 dB was obtained which is above the
30 dB estimated to be necessary at the input of the autotuning filter. Additionally, thesignal-to-pump ratio of 1:1 makes the minimum input power for the autotuning filterof 10 mW easily attainable.
After integrating the carrier suppressor with the autotuning filter, the next stepis the integration of the optical processor, consisting of both the suppressor and thefilter, to the RF front-end. In order to maintain optimum carrier suppression, for aparticular gain of the photorefractive crystal, it is necessary that the input modulation
strength m0 be maintained at a specific value which results in a peak of themodulation depth, as discussed in section 4.2. For a variable input RF signal, it
becomes necessary to have a mechanism that automatically adjusts the modulation
strength to the optimum level. We suggest the use of an automatic gain control
amplifier in front of the modulator’s driver.

78

Adequate carrier suppression performance is demonstrated in this chapter,
however the incorporation of the carrier suppressor in the antenna array system
requires a design that can process several channels in a compact space. The antenna
array system’s first iteration contains only two channels, but the design should be
expandable to more channels for future iterations. In the following, we introduce a
double-channeled carrier suppressor design that fulfills these requirements. Suchdesign is still under development as the first implementation attempt ran against somecross-talk problems, as discussed below.
The main difference between this new carrier suppressor and the previous one
resides in a new electrooptic modulator design, shown in figure 4.10. This modulator
design integrates two modulation channels in the same crystal. The modulator’s input
laser beam is shaped into a vertical line and the upper and lower portions of the beam
are modulated by different signals. The modulated beam then crosses a similarly
shaped but unmodulated beam inside the photorefractive crystal, suppressing thecarrier from the output beam and thereby revealing the modulation structure of thebeam. The channelized modulator design can be expanded to integrate more than two
channels without significantly changing the size of the optical system. In fact, using
this compact modulator design, we built a 41 x 20 cm2 double-channel carrier
suppressor, more compact than the previous single-channel one, even though theyboth utilize 1”-optics. Figure 4.11 shows a picture of this system with its
corresponding setup.

79

Figure 4.10. Channelized electrooptic modulation and carrier
suppression. The modulator imposes spatial and temporal modulationon the optical carrier. The optical carrier is then suppressed by aphotorefractive two-beam coupling interaction, revealing the spatial
modulation.

The channelized electrooptic modulator contains a thin 0.7 x 5.2 x 45.5 mm3
slab of magnesium-doped lithium niobate crystal, where the optical facets have been
cut at Brewster’s angle, about 65°, in order to minimize reflections. The electrodesconsist of two copper strips, approximately 2 mm wide and separated by about
1.5 mm. Each electrode is separately fed with a narrow-band matching circuit similar
to that used in the modulators in the previous section. The center frequency is set to
105 MHz giving a modulation efficiency of about =0.4 rad/V.1
Other than the channelized electrooptic modulator, there are very few
differences between the setup of this system and that of the previous one. To couple
light into the electrooptic modulator, a 30° laser line generator lens (from Edmund
Scientific Inc.) is used. The heterodyne beam for the output measurement is split
directly out of the pump beam. The transmission through the modulator in this setup
was about 85%.

1

Dr. Stefania Romisch collaborated with the modulator’s assembly and characterization.

80

(a)

Nd:Yag532 nmCL2
Pol
BaTiO3

PBS

λ/2

LG

CL6
CL3

CL5

CL1

λ/2

L1

CL4

Spectrum
Analyzer(b)

Figure 4.11. (a) Picture and (b) corresponding setup of carrier
suppressor with the channelized electrooptic modulator. (Component
abbreviations are the same as those in figure 4.9. Additionally, LG:
30° line generator, CL: cylindrical lenses (CL1: fy=8 mm, CL2:
fx=30 mm)).

81

However, when testing this system, we found a severe limitation - that ofcross-talk between the two channels.

The cross-talk was first revealed by aqualitative, visual method: by using the photorefractive crystal to suppress the carrier,we drive each channel, one at a time, and visually check how the signal beams
overlap at the output. We found a significant overlap. To quantify this overlap, we
measured the beam directly at the output of the modulator. The data in figure 4.12
shows the intensity of the signal component, the carrier, and second order sideband,as seen by the spectrum analyzer. The measurements were made by moving thephotodiode detector along the beam line and optimizing the alignment of theheterodyne detection at each point. The modulator was driven at 105 MHz with
25 dBm. In channel 2 the modulation index achieved higher values (going past the
dip at =2.4) than that in channel 1. We find a significant cross-talk between the twochannels with the signal from channel 1 extending significantly over the otherchannel’s side, resulting in a cross-talk of about –10 dB. By changing the couplingalignment into the modulator crystal, we could vary which channel displayed theworst coupling into the other channel’s spatial domain, and even adjust for asymmetric coupling, however, the cross-talk remains.

82Intensity of modulation components inchannel 1 vs. position on line
dBm

0

-10
-20

J1
J0

-30

J2

-40
-50
-60
0

2

4

6
mm

(a)

Intensity of modulation components inchannel 2 vs. position on line
dBm 0
-10
-20
J1
-30

J0
J2

-40
-50
-60
0

2

4

6

mm

(b)

Figure 4.12. Output intensity of the carrier (J0), signal (J 1) and second
order component (J2) on (a) channel 1 and (b) channel 2. The signal on
channel 1 is more weakly modulated and extends into the spatial
domain of channel 2.

The origin of the cross-talk is currently a matter of debate. The most likely
cause is the RF coupling between the resonant circuits that drive the electrooptic

83

modulator. To attack the cross-talk issue, research in this new system design is still
under progress.
Alternatively, one could develop a new design which can potentially process
several channels. For instance, we suggest using a grating to generate separate beams
which can be made parallel by an adjustable prism system. The beams would thenenter separate modulators, which however are mounted on a single block. RF
coupling could be reduced by a metal shielding between the driving circuits, while
optical coupling would be eliminated by the air space between the modulating
crystals.

84

CHAPTER 5:
AN OPERATOR APPROACH TO TWO-BEAM COUPLING:
THE 2-BY-2 COMPLEX-COUPLING CASE

5.1.

INTRODUCTION

In this chapter we present an operator theory for describing the process of
two-beam coupling. This photorefractive interaction is represented by a black box
which operates over the input optical fields to produce the output fields. The input
and output fields are represented by vectors, the “fieldvectors”, and the black box
transformation by the coupling operator T:

Eout = T Ein .

[5.1]

The operator theory is based on the same underlying physics as the standard
approach for two-beam coupling. [Kukhtarev, '79] However, the nature of operators
confers to this theory its main advantage over the standard approach: a mathematicalnotation that can deal with complicated spatial modes. This lends the power ofrepresenting the information as is the most convenient for the information processing
application in hand. For example, if the input optical signals are faces, in the standard
approach the faces are decomposed into a superposition of plane-waves, and a85

differential equation would result for each pair of plane waves, resulting in acomplicated problem. With the operator theory the spatial basis may be chosen to bea set of orthogonal facial features, bridging the gap between the mathematicalnotation and the information processing task at hand.
Because the operator theory is based on the same physical model as thestandard approach, it shares the same limitations and approximations. In particular,
this theory assumes steady-state gratings, in other words, the spatial and temporalvariations of the fields must be such that the gratings reach a stationary state. Thismeans that the temporal components should vary fast enough such that a component
in one beam only writes a grating with the corresponding component in the otherbeam, and with no other. At the other end, the temporal variations should not vary as
fast as to cause the Bragg degeneracy between different optical frequencies to bebroken. As mentioned earlier in this thesis, for barium titanate, these conditions limit
the frequency components to lie between about hundreds of hertz and a few gigahertz
for a 1cm crystal. This theory also assumes that the different spatial components are
Bragg distinct, in which case we can think of each spatial component as having its
own independent grating. Finally, the operator theory assumes that the coupling
between the beams in each port are all described by the same coupling constant Γ.
This coupling constant is in general complex, meaning that both energy or phaseexchange may occur between the coupled beams. [Yeh, '93]

Γ = Γ ei .

86

[5.2]

The fieldvectors in equation [5.1] has two components, E+ and E-,
corresponding respectively to the gain and loss port of the two-beam couplinginteraction. In their turn E+ and E- can be decomposed into their various spatial
modes. In general, the number of orthogonal spatial modes at the gain port, p, may bedifferent of that in the loss port, m.

 E+ ,1  M  E+   E+ , p 
E=  =E  E− 
 − ,1  M E  −, m [5.3]

The operator theory for the general case of p+m spatial modes can be solved
numerically and are treated in [Anderson, '00; Anderson, '99], and its details will not
be covered in this chapter. Our interest is in the simple case of p=m=1, meaning that
there is a single spatial component in each port, resulting in a 2x2 dimensionality forthe coupling matrix T. For this “2-by-2 case”, there exists closed-form solutions forthe two-beam coupling evolution and the coupling matrix T.
This chapter follows with three sections. Section 5.2 introduces the general
operator formalism.1 Section 5.3 treats the 2-by-2 case with complex coupling
culminating with the solution of T for this case. We also present the special cases ofpurely real and purely imaginary coupling.

1

The general theory was developed by Dr. Dana Anderson.

87

Back in chapter 4, section 4.4, we presented a simplified geometrical pictureof the operator approach for the 2-by-2 case, assuming purely real coupling and real
input fields. Although the current chapter stands alone, the simplified picture
presented in chapter 4 may help lend a preliminary intuitive understanding of theoperator algebra. Other approaches using an operator or matrix formulation for twobeam coupling are given in[Liu, '93; Ringhofer, '00; Stojkov, '92].

5.2.

THE OPERATOR FORMULATION

Given spatially and temporally varying beams as the input to the two-beamcoupling, the fields can be decomposed into a spatial basis, as shown in equation[5.3], as well as into a temporal basis. We will label the different temporalcomponents of the input beams by the subscript, but it is understood that thesecomponents are not necessarily harmonic components, but any orthogonal temporalbasis. The evolution of each temporal component as it propagates through the crystal
is given by equation [5.1], rewritten here as

E (z) = T(z) E (0) ,[5.4]

where z, the interaction length in the direction of propagation, is measured in units of
1/|Γ|, where |Γ| is the coupling constant modulus, so the only part of the coupling
constant that appears in our equations is it’s phase factor
88

as defined in equation

[5.2]. The field values at any position z within the crystal is determined by solving
for the coupling operator T. As we will see later, T corresponds to a rotation operator
which conserves the length of the fieldvectors (meaning that the transformation islossless) and is therefore Unitary, i.e. T † = T −1 . The formulation involved in solving
for T follows.
In an analogy to quantum mechanics, we introduce a “Hamiltonian” H, atransformation that determines the evolution of the fieldvector such thatE′(z) = −iH(z) E(z),[5.5]where the prime denotes derivation with respect to z. The Hamiltonian operator
represents the effect of the grating on the evolution of the fields in each location z. It
will later be defined such as to embody the two-beam coupling physics in the context
of our operator formulation, but because the grating itself is formed by the fields, we
can anticipate that H(z) will depend on the fieldvectors. However, this fact does notaffect the validity of equation [5.5].
The black-box view, given by the coupling operator T, and the above
fieldvector evolution, given by H, can be connected by simply substituting equation[5.4] in the above equation, giving

T ′(z)E (0) = −iH(z)T (z)E (0) ,

89

[5.6]

which, by eliminating E (0), gives us a differential equation for the coupling
operator:

T ′ = −iH T ,[5.7]

where the explicit z-dependence is dropped.
Our problem involves the mixing between two input beams or ports, whichtranslates into a mixing of the components of the fieldvector. Thus a “density
operator”, , which represents the mixing between the components is introduced:=

1
E × E† ,
∑
I

[5.8]

where E † is the adjoint (complex-transpose) of E , and I is the total intensity:

I = ∑ I = ∑ E† E .

In the 2-by-2 case, the density operator in the gain and loss port space is given by thedensity matrix:

2
1  E ,+= ∑ *
I  E , +E , −90

E , + E * ,− 
2 E ,− 

[5.9]

Characterizing the system by the density operator is equivalent to characterizing it by
its fieldvectors, with the advantage that it simplifies and suits the problem. [CohenTannoudji, '77b] The density operator is Hermitian, and, because it is normalized, it
has a unity trace:

Tr ( ) = ∑ ii = 1.
iJust as we had represented the fieldvector in the black-box view of equation [5.4], we
can equivalently represent the density matrix at some location z as a function of the
input density matrix by simply substituting equation [5.4] in equation [5.8], obtaining= T (0 ) T † .

[5.10]

And, again, equivalently to equation [5.5] for the fieldvectors, we can write adifferential equation for the density operator, which is obtained by combining theabove equation [5.10] with equation [5.7], giving′ = −iHT (0 )T † + i T ( 0)T † H = −i[ H,],[5.11]

where [ A, B] = AB − BA is the “commutator” of A and B.
The equations above hold independently of the physics of the problem. As
mentioned above the physics is included by defining an appropriate Hamiltonian,91

which satisfies the assumptions and approximations of the physical model describedin the introduction of this chapter. It is given by

H=

where

i i 3
e [ 3,
4],[5.12]

is the same density operator defined above and σ3 is an operator which

embodies the asymmetry of the coupling mechanism, and, in the 2-by-2
representation, is given by the Pauli spin matrix

3

1 0 =.
 0 −1

The factor of i in the Hamiltonian makes it Hermitian. This two-beam coupling
Hamiltonian is not derived here from first principles, but, upon substitution inequation [5.11], the resulting equations for the 2-by-2 case can be shown to reduce tothat of the standard approach. [Anderson, '00]

5.3.

A CLOSED-FORM SOLUTION TO THE 2-BY-2 CASE.

The operators defined in the previous section are valid in the general case,when a superposition of spatial modes are present in each beam (the n-by-n case). We
now focus on the 2-by-2 case where only one spatial mode is present in each beam.

92

For this simple case, closed-form solutions exist for both the density operator

andthe coupling operator T. From this point forward, we will cease to refer to them as
operators but instead refer to their 2-by-2 matrix representation.
Any 2-by-2 matrix can be represented by a superposition of the unit matrix
and the Pauli spin matrices. As we progress on the problem of deriving the coupling
matrix, we will see that such a representation proves to be very well suited for thedensity matrix [Cohen-Tannoudji, '77a]:

=

1 3
+ ∑s
2 i =1 ii,[5.13]

where1

 0 1=,
 1 0

2

0
=
i

−i 
1 0 
, a n d 3 = 0 0 −1

and

si =

1
Tr(
2

i

).

[5.14]The Pauli spin matrices have several interesting properties. First, they are idempotent,
meaning that2i

= 1. Thus e ia i = cosa + i i sina , where a is a scalar. Second, they

are cyclic upon multiplication, i.e.,ij= i ijk93

kwhere i j k andijk

is 1 for an even

permutation of i,j,k and –1 for an odd permutation. Third, they anti-commute,meaning that { i ,

j

}=

i

j

+

j

i

= 0 where i j.

The coefficients of the spin matrices can be conveniently expressed inspherical coordinates:

s1 = Λ sin(2 ) cos(2 )s2 = Λ sin( 2 ) sin(2 )[5.15]

s3 = Λ cos(2 )where Λ = s12 + s22 + s23 = (λ1−λ2)/2 where λ1 and λ2 are the eigenvalues of , with
λ1>λ2. The -matrix in the spherical coordinates is simply given by: 21 + Λcos(2 ) Λsin(2 )e − i2 =.
1 Λ sin(2 )ei2−
Λcos(2)
2[5.16]

We prefer to write the solution of the density matrix and coupling matrix interms of the spin matrices: 0// = 1 cos( 2 ) + 2 sin ( 2 ) =  i 2
ee −i 2 

0 

 0
⊥ =
2 cos( 2 ) − 1 sin ( 2 ) = i  i2
e−e−i 2 
0 

94[5.17]

As can be deduced from the expressions above, are orthogonal and are obtained byrotating
into the

1

and

1

2

around the3“axis”, such that//is aligned with the projection of-plane, as shown in figure 5.1.

2

σ3
ρ
2θ
σ⊥

σ2
2φ
σ//

σ1

Figure 5.1. Representation of

in

1

2

-coordinate space. The

3“vectors” // and ⊥ are the parallel and orthogonal projection of
on the 1 2-plane and they rotate in the plane as rotates in space.

Using these, we can rewrite the density matrix as:

=

1
+Λ
2

where

95

[5.18]

=

ei 2

⊥

3

=

//

sin( 2 ) +3cos(2 ) ,points in the direction of .
The expression for the Hamiltonian is obtained by substituting the aboveinequation [5.12], giving:

H= −Substituting the above H andforΛ
sin(2 )(
2

⊥

cos +

//

sin ) .

[5.19]

in equation [5.11], provides the differential equations

and :

′=−

Λ
cos sin(2 )2Λ′ = sin cos(2 )2The solution for[5.20]is straightforward:= arctan( e−Λ z cos tan 0 ),96

[5.21]

where= (0) and the same convention will be used from this point forward with all

0variables. Solving

by using ′ =∂∂=′, we get:

 sin(2 0 ) 
tan
ln
+ 0
2
 sin(2 ) [5.22]

where

sin(2 ) =sin( 2 0 )e Λz cos cos2 0 + e−Λ z cos sin 2 0[5.23]

cos(2 ) =The solution for

e

±i 2

 sin(2 0) = sin(2 ) Λz cos−Λ z cose
cos 0 − esinΛz cos2−Λ z cose
cos 0 + e
sin 2
2

2

0
0

is more usefully casted as:± i tane ±i 2 0 = (e Λz cos cos2 0 + e −Λz cos sin 2 0 )With the solution ofandat hand,shows the shape for the evolution ofin the±i tane± i2 0[5.24]

is now solved. The plot in figure 5.2

12-space (as shown in figure 5.1)

3

for =1.5, i.e., 0< <π/2. Figure 5.2 (a) shows a view of the south pole, whereas figure
5.2 (b) shows the north pole, which points in the direction of97. Given the initial3

conditions

and,will point to a particular point in the curve and then, as itpropagates through the crystal,
north pole. The value of 2

will follow the curve asymptotically towards theis an expression of the intensity ratio between the beams.

The north pole corresponds to all the energy being present at the gain port, and thesouth pole, to it all being present at the loss port. Thus, naturally, the equator
corresponds to equal intensities between the two beams. On the other hand, 2
corresponds to the phase-difference between the two beams and the spiraling
indicates a phase coupling between the beams. As indicated by the cos(2 ) factor inequation [5.20] for , the spiraling in the northern hemisphere is in the opposite
direction as that in the southern hemisphere. At the equator, no spiraling can occur as
the curve is perpendicular to the equator line. This has a simple physicalinterpretation. Unlike energy transfer, two-beam coupling phase transfer issymmetric, and thus, when the beams have equal intensity they both change theirphases at equal rates, resulting in no net change in the phase difference between thetwo beams.

98

(a)

(b)

Figure 5.2. Plot of the evolution of the density matrix as a functionof propagation z through the medium, for =1.5. The initial conditionsare =1.5, close to the south pole (a), where all the energy is in theloss port, and =0. As the density matrix evolves, it spirals up andswitches direction at the equator, where ’ is maximum ‘=0, and thenasymptotically spirals up to the north pole (b).

So, in summary, for the general case of complex coupling, where 0< <π/2, theenergy coupling causes

to approach the north pole whereas the phase couplingcauses it to rotate around the polar axis

, resulting in the spiral shape shown in3

figure 5.2. We know from photorefractive physics that a real coupling constant,which occurs for

0 (see equation [5.2]), corresponds to pure energy coupling andthus, for this special “real case” we expect the curve to go directly from the south to99

the north pole without spiraling. That is indeed the case and is shown in figure 5.3(a), plotted for

0. Conversely, an imaginary coupling constant, which occurs for

π/2, corresponds to pure phase coupling and is represented by a pure rotation
around. We call this the “imaginary case” and it is shown in figure 5.3 (b).

3Naturally, the longitude of the curve in the real case is given by the constantand similarly, the latitude of the curve in the imaginary case, by. In the real caseis simply given by substituting

0 in equation [5.21]. In theimaginary case, the complex-case expression for

is not valid as it becomesthe equation for

undetermined at that limit. Notice that when −> π/2 is plugged in equation [5.22],
since tan( )−> ∞and ln( 1) −> 0,is undetermined. However, since in this caseis constant, the expression for

directly follows from its differential equation [5.20],

giving (z) = Λ2z cos(2 0 ) + 0 .

100

(a)

(b)

Figure 5.3. Evolution of the density matrix for the special cases of (a)real and (b) imaginary coupling. In the real case of pure energy
coupling, follows a longitude line towards the north pole, thelongitude being determined by. In the imaginary case, rotatesaround the pole with constant speed, the latitude being determined by.

As described above, the solution to the density matrix provides us with animportant tool for understanding the behavior of the system, but our ultimate goal is
to solve for the coupling matrix which represents the two-beam coupling black-box.
If the chain rule is applied to the differential equation for the coupling matrix,
equation [5.7], and the expressions for ’ ’, and H (equations [5.20] and [5.19]
respectively) are substituted, we get:

101

∂T
∂T
∂T
Λ
′+
′+
= i sin (2 )(
∂
∂
∂z
2

⊥

cos +

//

sin

)T

[5.25]

which has the following solution for T:

T =T T T =e (
−i

− 0)

⊥

e (
−i

− 0) 3i Λz sine2

The special limit cases are straightforward by substituting
case, and

TRe = e (−i −rotation of.

[5.26]

0 for the realand

π/2for the imaginary case:andThe action of0

onaround0) ⊥and TIm = e (−i − 0) 3

iΛz

e2

0

=e

− i Λzcos ( 2 0 ) 3
2

i Λz

e2

0

.

is simple: each factor of the type T = eiA i corresponds to aiby an amount A. So the complex solution in equation [5.26]

represents a general rotation of the density matrix decomposed into the following
rotations: the first factor to act on , T , rotatesaround itself, a degenerate rotation

which does not affect ; the second factor T then rotates
finally the last term T rotatesaround⊥byaround

3

by

; and

, which is a rotation directly up

towards the north pole (see figure 5.1).
The interpretation of the factors T and T

are clear, the first represent therelative phase difference between the beams picked up by the phase transfer, whereas

102the latter represents the energy coupling. However the interpretation of T issomewhat subtle. It is interesting to note that if this factor is removed from thesolution, the resulting expression still satisfies the relation= T (0 ) T † however itwill not satisfy the differential equation T ′ = −iH T . We believe this is because
doesn’t have any information about the evolution of the absolute phases of the fields,
but only the relative phase between them, so it is indifferent to T . However, the twobeam coupling equations provide information about how much phase is beingtransferred from each beam to each other beam, and not only the difference. This
information is therefore contained in T by this degenerate-rotation of . For example,
we mentioned above that ’ is always zero at the equator, however that does notmean that the phase transfer stops at that location, it just means that the phase transfer
is symmetrical, and this information is contained in T by T .

103CONCLUSIONSIn this chapter we discuss applications and future work regarding the different
aspects of the research presented in this thesis. This thesis revolves around one
fundamental photorefractive interaction, the two-beam coupling effect. The
contributions described in this thesis are divided into three aspects of photorefractive
systems: The first is that of developing a component technology, which includes thedemonstration of the two-beam coupling modules described in chapter 2 and by thestudy of the oscillating patterns occurring inside a barium-titanate spherical disk, inparticular the triangle pattern, described in chapter 3. The second is that of building
and optimizing a specific photorefractive application, the carrier suppressor,
described in chapter 4. Finally, the third is that of developing an analytical tool thatcan simplify the modeling and prediction of photorefractive systems. In this case, thecontribution is the presentation of a general solution for the operator theory in thecase of a single spatial mode in each beam, described in chapter 5. These different
aspects sometimes overlap. For example, the operator theory provided a model on
which to cast the carrier suppression effect, resulting in an understanding of thesystem which proved to be crucial for its optimization. Below, we give concluding
remarks and related applications on the topics covered in this thesis.
The fiber-interconnected two-beam coupling modules are pre-aligned building
blocks for photorefractive system design. They provide a way to quickly test an idea
without having to invest the time in critical optical alignment which is necessary104when building a conventional system. However, it is important to note that, because
the geometry within a module is fixed, they are not well suited for optimization ofsystems. The utility of the two-beam coupling modules as a component technology
was illustrated in the example given in section 2.5, where the inclusion of a reflexivecoupling unit in the autotuning filter’s ring provided an increase of the output contrast
between two signals which are being separated. It took only a few minutes to include
the reflexive-coupling element and test the performance. Future work on the modules
should seek to improve their long term stability. In section 2.4 some suggestions for
future designs were given, but the main issue to keep in mind in a new design is thatthe adjustments be external to the final module, thereby reducing the susceptibility tolong term alignment drifts. Among the useful applications of the modules, we suggestthat of testing different ring configurations. Because the geometry of the modules are
fixed, the differences in behavior can then be attributed to the different architecturesand not to the geometrical factors of the two-beam coupling configurations. For
example, one can compare the behavior of the autotuning filter architecture shown infigure 2.10 (b) with one where the pump crystal is instead located in the reflexive
loop (directly in front of the plus input port of the reflexive coupling crystal).
Understanding the different rings could result in new ideas for future oscillator-based
photorefractive processors.
We believe that our research group is the first to observe the oscillating
patterns occurring within a barium-titanate spherical disk. For now, the oscillating
patterns observed inside the spherical disks stand mostly as a scientific curiosity,
however, in order to devise possible applications for this phenomenon, it is necessary105

to first understand it. In fact, the model presented in section 3.4 suggests that the disk
patterns are unidirectional miniature oscillators pumped mostly by a single two-beamcoupling interaction, and therefore hold the promise for a future integrated ringprocessor such as the autotuning filter [Anderson, '92].
As part of a system oriented group, it is natural that within the contributions isthat of building and optimizing a photorefractive system for a specific processingtask, in this case, that of carrier suppression. The carrier suppressor was developed
for a specific application as part of a higher level system, where its function is toremove the correlation between temporal signals which are to be discriminated by thenext stage, the autotuning filter. Another application for the carrier suppressor
demonstrated in chapter 4 would be in laser tuning. In Ye et al [Ye, '96] a laser beam,
whose frequency is to be tuned, is shifted by an acoustooptic modulator and thenlocked to a reference cavity. In this case, the reference cavity is locked to the 87Rb D2line at 780 nm, however the same concept could be used with the I2 hyperfine
resonances at 532 nm [Jungner, '95]. The carrier suppressor used on such a system
would allow the use of electrooptic modulators, therefore providing higher
modulation bandwidth and, thus, shorter reference cavities which are typically easier
to stabilize. The carrier suppressor described in chapter 4 was the evolved result of
several other carrier suppressor predecessors. The earlier systems were not coveredby this thesis because, first, they didn’t accomplish the specifications of ourapplication, and second, all the important aspects of the carrier suppression physics iscovered in the description of the last, optimized system. This system rendered a
suppression of more than 70 dB, the highest value reported in the literature [Loayssa,

106

'00; Tonda-Goldstein, '00]. Despite this successful result, there is room for
improvement in other aspects of the system. Section 4.4 provides preliminary resultsin the implementation of a future carrier suppressor which utilizes a channelized
electrooptic modulator design. Although the integration provided by this design is
attractive for two or three channels, as the number of channels increases, we believe
this design will prove impractical due to crystal size and cross-talk issues. In thatcase, a possible alternative design is that of using a grating to direct the several beams
to independent modulators which are, however, placed next to each other on the same
mount.
Finally, in chapter 5, we present a closed-form solution for two-beamcoupling for the case of only one spatial component on each port. This solution is castin terms of an operator algebra which was originally developed by Anderson[Anderson, '00; Anderson, '99] and is summarized in section 5.2. We believe that thesolution presented is the most general available in the literature as it includes bothcomplex coupling and any number of temporal components. The work of Saxena et al[Saxena, '90] solves the problem of N mutually incohenrent pairs of temporal beams.
In this case, they show that complete energy transfer between the two write beams
may occur in a finite interaction length, as opposed to the infinite thickness requiredin the standard case of only one temporal component on each beam. This is the sameconclusion we arrived at, for example, in the case of the carrier suppression, where
the presence of a sideband allowed for perfect suppression at a finite interaction
length. However, their solution includes only the case for real coupling, this is, pureenergy transfer. The work from Ringhofer et al [Ringhofer, '00] provides the general107complex coupling solution for a phase modulated beam in the gain port, with
arbitrary phase modulation strength. However, their solution describes only two
temporal components, the carrier and the modulation sidebands, as opposed to an
unlimited number of temporal modes as in the solution provided in chapter 5.3. In
their paper they also provide an application example, where phase modulation is usedin the active stabilization of two-beam coupling by means of an electronicallyintroduced phase feedback. For the case of a single spatial mode on each beam,
besides the signal processing applications in our laboratories, such as thedemultiplexer [Saffman, '91] or the autotuning filter (previously known as the feature
extractor) [Anderson, '92], other applications include RF filtering [Hong, '93], laser
bandwidth narrowing [Chomsky, '91] or signal amplification [Hamel deMonchenault, '87]. An application that has currently been brought into the spotlight isthat of using heterodyne detection with two-beam coupling for laser ultrasonics [Ing,'91; Puoet, '96; Scruby, '90]. Heterodyne detection with two-beam coupling offers
several advantages over the conventional optical heterodyne scheme [Hamel deMonchenault, '88]: First, the signal beam may consist of an arbitrary wave-front, suchas speckles for instance, while, in the conventional heterodyne, the signal is limited toa spatial mode which is collinear with the local oscillator plane wave. Second, since
the grating adapts to the incident signal wave front, the heterodyne detection is not
susceptible to time varying phase distortions which are slow compared to thephotorefractive response time.

The photorefractive heterodyne detection is well

suited for laser ultrasonics, a noncontact method for performing ultrasonicnondestructive evaluation measurements. Unlike the conventional piezoelectric108contact transducers, the noncontact nature of laser ultrasonics allows inspection in
high-temperature or otherwise hostile environments. The method typically consists ofgeneration of ultrasound with a high-power pulsed laser followed by detection of theultrasonic motion of the scattering surface by a photorefractive heterodyne detection
setup (see, for example, [Murray, '00]).
In conclusion, this thesis covers important aspects in the development ofphotorefractive system that utilizes two-beam coupling. It covers, first, thedevelopment of component technologies which allow for easy testing of concepts,
second, the development of a specific processing system, the carrier suppressor, andfinally, it provides a solution for the general case of a single spatial mode two-beamcoupling cast in terms of an operator formalism.

109

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