MICROWAVE SIGNAL PROCESSINGWITH
PHOTOREFRACTIVE DYNAMIC HOLOGRAPHY

By
Edeline Fotheringham
B.A., Diplôme d’Ingénieur,
Ecole Nationale Supérieure de Physiques de Strasbourg, 1998
M.S., Diplôme d’Etudes Approfondies,
Université Louis Pasteur de Strasbourg, 1998

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
2003

This thesis entitled:“Microwave signal processing with photorefractive dynamic holography”
written by Edeline Fotheringham
has been approved for the
Department of Electrical & Computer Engineering______________________________
Dr. Dana Z. Anderson______________________________
Dr. Leo Hollberg

Date_______________

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

iiFotheringham, Edeline (Ph.D. Electrical Engineering)
Microwave signal processing with dynamic holography
Thesis directed by Prof. Dr. Dana Z. Anderson

Abstract
Have you ever found yourself listening to the music playing from the closest stereo
rather than to the bromidic (uninspiring) person speaking to you?

Your ears receive

information from two sources but your brain listens to only one. What if your cell phone
could distinguish among signals sharing the same bandwidth too? There would be no “full”
channels to stop you from placing or receiving a call.
This thesis presents a nonlinear optical circuit capable of distinguishing uncorrelated
signals that have overlapping temporal bandwidths. This so called autotuning filter is the sizeof a U.S. quarter dollar and requires less than 3 mW of optical power to operate. It isbasically an oscillator in which the losses are compensated with dynamic holographic gain.
The combination of two photorefractive crystals in the resonator governs the filter’s winnertake-all dynamics through signal-competition for gain. This physical circuit extracts what ismathematically referred to as the largest principal component of its spatio-temporal input
space.
The circuit’s practicality is demonstrated by its incorporation in an RF-photonic system.
An unknown mixture of unknown microwave signals, received by an antenna array,
constitutes the input to the system. The output electronically returns one of the original
microwave signals. The front-end of the system down converts the 10 GHz microwave
signals and amplifies them before the signals phase modulate optical beams. The opticalcarrier is suppressed from these beams so that it may not be considered as a signal itself to the
autotuning filter.

The suppression is achieved with two-beam coupling in a singleiii

photorefractive crystal. The filter extracts the more intense of the signals present on the
carrier-suppressed input beams. The detection of the extracted signal restores the microwave
signal to an electronic form. The system, without the receiving antenna array, is packaged ina 13x18x6” briefcase. Its power consumption equals that of a regular 50 W household lightbulb. The system was shipped to different parts of the country for real-time demonstrationsof signal separation thus also validating its claim to robustness.

iv

DEDICATION

To my mother,
who lovingly offered me the keys to a blossoming life.

To my friends Gaelle and Christophe,
who opened my eyes to the world and to myself.

v

ACKNOWLEDGEMENTS

I am greatly indebted to my thesis advisor Prof. Dana Anderson. With enthusiasm andbrilliance, he taught me all the skills I had wished to learn when I started my studies
and other skills I am grateful to have aquired. I thank him for the warmth andfriendliness he showed me. I have wholeheartedly adopted his maxim:“work hard, play hard.”

I thank Leslie Czaia who I have named the fairy of the labs not only because her
knowledge of the lab is unequaled but also because she is always willing to help andgraciously assist anyone who might need it.

I thank Prof. Zoya Popovic for her guidance in the field of microwaves and for her
motherly kindness toward me through out my studies.

I thank Hans Green for the generous time he devoted to teaching me the art ofdesigning precise and reliable mechanical systems.

I thank Dr. Stefania Romisch and Paul C. Smith for the friendly and cooperative
atmosphere they maintained during trying times.

I thank Dr. Leo Hollberg who was always ready to give me advice and kindly offered
the use of his optical coating facilities.

vi

TABLE OF CONTENTS

CHAPTER 1
1. 1

INTRODUCTION ...................................................................................1

Overview............................................................................................................1

1. 1. 1The photorefractive effect..........................................................................3

1. 1. 2

The autotuning filter ..................................................................................5

1. 1. 3An optically smart antenna array ...............................................................8

1. 1. 4A photorefractive regenerative amplifier.................................................11

1. 2

Thesis outline...................................................................................................14

1. 3

Contributors to this thesis ................................................................................16

CHAPTER 2

THE AUTOTUNING FILTER ..............................................................17

2. 1

Introduction......................................................................................................17

2. 2

A heuristic explanation of the filter's dynamics...............................................20

2. 3

Experimental versions of the filter...................................................................23

2. 3. 1Fiber optic implementations of the filter .................................................23

2. 3. 2

Free space optics implementation of the filter .........................................27

2. 3. 3Results......................................................................................................31

2. 4

Theoretical analysis of the filter ......................................................................37

2. 4. 1

Varying the filter's parameters for best separation performance .............37

2. 4. 2

The filter's principal component extraction abilities................................47

2. 4. 3

Theoretical arguments to prove the filter's PC extraction capabilities.....51

CHAPTER 3

AN OPTICALLY SMART ANTENNA ARRAY.................................61

3. 1

Objective..........................................................................................................61

3. 1. 1

Blind Source Separation ..........................................................................61

3. 1. 2A portable prototype system ....................................................................62

vii

3. 2

The microwave front end .................................................................................65

3. 2. 1

The transmitters .......................................................................................65

3. 2. 2

The quasi optical antenna array ...............................................................66

3. 2. 3

The active down conversion stage ...........................................................68

3. 2. 4

Demodulation of the audio signals ..........................................................70

3. 3

The EOM and carrier suppression ...................................................................72

3. 3. 1

A two channel EO....................................................................................72

3. 3. 2

Resonant matching design .......................................................................73

3. 3. 3

EO data ....................................................................................................74

3. 3. 4The carrier suppression unit.....................................................................75

3. 4

Design of the optical stage...............................................................................80

3. 4. 1The optics.................................................................................................80

3. 4. 2The mechanics .........................................................................................83

3. 5

System performance ........................................................................................86

3. 5. 1The suitcase..............................................................................................86

3. 5. 2End-to-end characterization of the prototype...........................................88

3. 6

Remarks ...........................................................................................................93

CHAPTER 4

A PHOTOREFRACTIVE REGENERATIVE AMPLIFIER ................95

4. 1

Introduction......................................................................................................95

4. 2

A photorefrative amplifier ...............................................................................97

4. 2. 1

Design issues............................................................................................97

4. 2. 2

Manufacturing a semi-conductor laser amplifier in a research laboratory98

4. 2. 3

Immediate future work...........................................................................103

CHAPTER 5

CONCLUSIONS..................................................................................104

viii

CHAPTER 6

REFERENCES.....................................................................................108

CHAPTER 7

APPENDICES .....................................................................................115

7. 1

Making spherical crystals ..............................................................................116

7. 2

Mechanical mounts for the fiber filter ...........................................................117

7. 2. 1

Ballbearings and glue.............................................................................117

7. 2. 2

Pure mechanics ......................................................................................117

7. 3

Algorithm for studying the parameters‘ influence.........................................118

7. 4

EO mount.......................................................................................................125

7. 5

Optics baseplate .............................................................................................126

7. 6

Suitcase manual .............................................................................................127ix

LIST OF FIGURES

Figure 1.1. Schematic diagram of the autotuning filter.____________________________________ 6
Figure 1.2. Block diagram of the prototype optically smart antenna array. _____________________ 9
Figure 2.1. The filter extracts the largest principal component of its input signal space while passing
on all the other components. _______________________________________________ 18
Figure 2.2. Schematic of the autotuning filter's design. The two-beam coupling occuring in the crystal
on the left of the figure provides gain to the ring. The two-beam coupling taking place inthe crystal on the right imposes loss on the ring signals inversely proportionally to theirrelative intensities._______________________________________________________ 20
Figure 2.3. Reflexive coupling is a photorefractive selective loss element. It induces loss inversely
proportionally to the strengths of signals. _____________________________________ 21
Figure 2.4. GRIN lenses collimate (and refocus) the light coming out of (or into) a fiber held by aferrule. ________________________________________________________________ 24
Figure 2.5. BaTiO3 spherical crystals double as lenses to couple light from two fibers into two otherfibers. _________________________________________________________________ 25
Figure 2.6. a) Photographs of the two fiberized autotuning filter versions. The crystals are slabs ofspheres as it wastes less material than whole spheres. b) Design of the autotuning filter
using fibers and spherical crystals. __________________________________________ 26
Figure 2.7. Photograph of the autotuning filter using free space miniature optics. ______________ 28
Figure 2.8. The "gold bar" shaped cut of photorefractive crystals suppresses unwanted oscillations
arising between otherwize parallel faces. _____________________________________ 29
Figure 2.9. Computer generated template placed under the transparent substrate that the filter of
Figure 2.7 is built on. The miniature elements of the filter are aligned to it externally
before they are glued in place.______________________________________________ 30
Figure 2.10. Schematic of the AO test bed used to evaluate the separation performance of the
autotuning filters.________________________________________________________ 32
Figure 2.11. Separation curve taken with the filter on the top left-hand side of Figure 2.6 ________ 33xFigure 2.12: AO test bed with mixtures of two signals driving each AO. Each signal is split in two andrecombined with half of the other signal. A 180-degree phase shift is added to one half of
S2 before it is recombined.________________________________________________ 34
Figure 2.13: Separation curve taken with the filter shown on the top right-hand side of Figure 2.6 andwith the test bed of Figure 2.12 ____________________________________________ 35
Figure 2.14: Separation curve taken with the filter shown in Figure 2.7 ______________________ 36
Figure 2.15: Regime of interest in our discussion: when the signal ratio is close to unity both signals
may oscillate. Above a critical value P1/P2c only the stronger signal oscillates._______ 38
Figure 2.16: Illustration of the notations used in the equations of this section. _________________ 40
Figure 2.17. The intersection of two curves represent the pairs of (S1,S2) that yield G1=G2=1. ___ 42
Figure 2.18. The Log-Log plot of the S1/S2 vs. P1/P2 curve shows a linear region when both signals are
oscillating in the ring and a rapid increase toward infinity as the input signal ratio
approaches the boundary value where the weaker signal disappears. The parametervalues are Gr=Gg=10, m=50, and T=0.5. ___________________________________ 43
Figure 2.19. S1/S2 vs. P1/P2 curve for Gg=9, 10, and 11, left to right (T=0.5, Gr=10, m=50).______ 44
Figure 2.20. S1/S2 vs. P1/P2 curve for T=0.55, 0.50, and 0.40 left to right (Gg=Gr=10, m=25). ____ 45
Figure 2.21. Left side: S1/S2 vs. Gr for m=60, 55, 50, 45, and 40 from top to bottom with T=0.5,
Gg=10, and P1/P2=0.2 dB. Right side: same values except Gg=11.________________ 46
Figure 2.22. Simple model of our filter as a principal component extractor. The mixing on the left
hand-side accounts for the spatially finite size of the photorefractive gratings that allows
one pump beam to weakly diffract off of the other pump’s grating. ________________ 49
Figure 2.23. Separation performance of the filter. The smooth gray curve shows calculated data
points using the first principal component extraction model. The slope at the origin is 15
dB/dB. This is in contrast to the experimental data points (in black) which show a slope atthe origin of 26dB/dB. ___________________________________________________ 50
Figure 2.24. Single mode cavity model of our autotuning filter. _____________________________ 51
Figure 2.25. Gain curves for two spatially and temporally orthogonal signals._________________ 55
Figure 3.1: Diagram of the prototype optically smart antenna array. ________________________ 63xiFigure 3.2. Photograph of the battery operated transmitters. The CD player provides the audio signalthat modulates the transmitted microwaves. __________________________________ 65
Figure 3.3. Diagram of the circuit that drives the horn antennas. ___________________________ 66
Figure 3.4. Photographs of both sides of a 30-element X-band two-layer quasi-optical lens antenna
array with patch antenna elements. The one on the right also shows the ground planes of
the receiving antennas placed on the focal arc of the lens array. __________________ 67
Figure 3.5 Measured (solid line) H-plane (on the left) and E-plane (on the right) radiation patterns at
9.9 GHz have higher sidelobes than the simulated patterns for a uniformly illuminatedarray (dashed line). This is due to the non-uniform amplitude and phase associated with
the near-field spatial feed. The simulated patterns are calculated using the measured
radiation pattern of the patch element _______________________________________ 67
Figure 3.6. Down converting circuit schematics. The amplifiers imediately following the receiving
patch antennas are LNAs. ________________________________________________ 69
Figure 3.7 The demodulator uses a PLL to lock to the doubled IF frequency. The loop has afrequency divider that restores the original audio-signals that may then be sent tospeakers. _____________________________________________________________ 70
Figure 3.8. Electro-optic modulation: a channelized electro-optic modulator made from magnesium
doped lithium niobate crystal (Mg:LiNbO3) imposes spatial and temporal modulation on
the optical carrier. ______________________________________________________ 72
Figure 3.9. The two electrodes of the EO are modeled by a resistor and a capacitor. They arematched to 50 ohms by RC circuits. An LC element cancels the coupling capacitance
between the electrodes. __________________________________________________ 73
Figure 3.10. Modulation profile of the two-channel electro-optic modulator. The top graph shows therelative modulation strength of each channel along the 7 mm of the crystal (the crystal
starts at graduation 100=1 mm). The channels are driven one at a time. The bottom
graph measures in the same way (so that the graphs‘ values are comparable)but with bothchannels turned on at the same time, the power of the signal that has the beat frequency
between the two modulation signals. ________________________________________ 75xiiFigure 3.11. Two-beam coupling in a photorefractive barium titanate crystal (BaTiO3) suppresses the
optical carrier. _________________________________________________________ 76
Figure 3.12. 2D representation of single mode two-beam coupling. (a) In the case of carrier
suppression the input to the minus (“-”) port is a modulated laser beam while the input tothe plus (“+”) port is a non-modulated beam from the same laser. (b) Nonlinear coupling
between the beams causes the energy in the carrier at the minus port to be transferred tothe plus port at the output. ________________________________________________ 78
Figure 3.13: These line generators create a 30-degree fanning beam that is 0.8 mm wide. _______ 80
Figure 3.14: 3 lens subsystem. The lenses are placed in cylindrical mounts that have the exact sameouter diameter to keep the lenses centered relative to each other. The v-groove allows
only one translation. ____________________________________________________ 82
Figure 3.15: Photograph and picture of the inverted table clamps. A 1/4-20 screw with a threaded
ring attach to every subsytem on the base plate. A u-shaped piece with an integrated
screw pulls down on the ring thus immobilizing the subsystem on the base plate. _____ 83
Figure 3.16: Photograph of the 8x11 inch optical stage. __________________________________ 84
Figure 3.17: Photograph of the “suitcase project.” The two coaxial cables visible in the top rightcorner receive electrical signals directly form the patch antennas placed on the focal arcof the antenna array. The bigger gold-colored box contains the LNAs and the down
converting electronics. The output is amplified by power amplifers sitting under thevisible cooling fan (there is another one under the optics.)_______________________ 86
Figure 3.18: Configuration of the system as it was tested in the laboratory.___________________ 87
Figure 3.19. Diagram defining the context of the end-to-end system measurements. ____________ 89
Figure 3.20. Signal extraction properties of the adaptive system parameterized by the mutual angle
between the transmitters. _________________________________________________ 91
Figure 4.1 Schematic of the c-mount used to heatsink the semiconductor chip and also provide theelectrodes necessary to supply the chip with current. ___________________________ 99
Figure 4.2. PI curves of a laser diode taken a) before coating,(10 mA/div.) b) after coating one facet
and c) after coating both facets. The triangular black curves represent the current passingxiii

through the diode. The nonlinear curves represent the signal of the photodetector placedin the vacuum chamber. _________________________________________________ 101
Figure 4.3. Proposed mechanical mount to align a lens, the diode amplifier and another lens. ___ 103
Figure 5.1 Description of the method to hand-make spherical photorefractive crystals._________ 116
Figure 5.2 Mechanical mount for the first version of the autotuning filter with sphericalphotorefractive crystals (see Figure 2.6.a.) __________________________________ 117
Figure 5.3 Mechanical mount for the second version of the autotuning filter with sphericalphotorefractive crystals (see Figure 2.6.b.) __________________________________ 117
Figure 5.4. EO mount that holds the 0.3x7x30 mm crystal pressed against a piece of foam. _____ 125
Figure 5.5. Hand drawing of the 1/2 inch deep patterns that were carved out of the bottom of the
optics‘ baseplate. The light bulb shapes indicate the positions of the upside down clamps
that were shown in Figure 3.15. __________________________________________ 126

xiv

CHAPTER 1
INTRODUCTION

1. 1

Overview

This thesis aims to demonstrate the potential usefulness of photorefractive circuits in
microwave signal processing. Photorefractive crystals are nonlinear holographic materials
with a short-time memory and the ability to couple coherent optical beams at low intensities.
We take advantage, as most modern optics based communications, of the fact that a narrow
band device at optical frequencies translates into a broadband device at microwave
frequencies [1]. The photorefractive volume gratings, with which we deal in this thesis,
typically have lengths on the order of half a centimeter. This gives them a Bragg matchedbandwidth of hundreds of gigahertz, amply sufficient to cover microwave frequencies.
Prof. D.Z. Anderson’s group has a long history of information processing withphotorefractive circuits. Our group has specialized in manipulating signals with novelty filter
two-beam coupling configurations [2-5] and self-organized photorefractive resonators [6-9].
Circuits were built that performed processing tasks such as demultiplexing signals carried bydifferent frequencies [10], or present on the input at different times [11], as well as
controlling the relative intensities of signals [12]. The photorefractive community commonly
employs resonators to design information processing circuits [13-22], and less often, novelty
filters [23]. This thesis concentrates on possible uses of an oscillator-based photorefractive
1

circuit for microwave signal processing. The circuit extracts the largest principal component[24] of its input signal space. We mainly refer to it as the “autotuning filter” for reasons thatwill be explained later. The filter’s design is a variation on the feature extractor’s designpreviously reported on by Mark Saffman [25]. This thesis also shows the application of thenovelty filter configuration to the suppression of the optical carrier of phase-modulated
beams at microwave frequencies. The work on carrier suppression was pioneered by Valeria
Damiao [26] and so this report presents only the basic principles that are necessary forcontextualizing our use of it.

2

1. 1. 1 The photorefractive effectDepending on the photorefractive material, two-wave mixing may transfer phase
information between coherent beams or energy or a combination of both. The coupling
constant Γ characterizes the exchange. Γ is purely imaginary for materials in which drift
dominates the movement of free carriers generated by photoexcitation. Such materials record
holograms that couple only the phase information of coherent beams. Γ is purely real for
diffusion dominated materials that couple only the energy of interfering beams [27]. Energy
transfer results from a π/2-phase shift between the light interference pattern and the index
modulation it generates. A more in depth explanation of the phenomenon may be found in
textbooks such as [28, 29].
The optical circuits presented in this thesis use barium titanate (BaTiO3) crystals as
photorefractive media, which is a diffusion-dominated material [30]. The coupling constantsof these crystals commonly reach 20 cm-1. The value of Γ indicates the potential depth of theindex modulation and thus represents the potential diffraction efficiency of thephotorefractive grating. The effective Γ in experiments, depends on the geometry of thebeams and the orientation of these beams relative to the c-axis (or optical axis) of thebirefringent crystal.
Two-beam coupling in barium titanate is analogous to an electronic transistor.

A

transistor has two inputs and one output. The output current is an amplified copy of one of
the currents coming in. The extra current is provided by the non-amplified input. Two-beamcoupling requires coherent input beams. In the crystal, one beam undergoes amplification,
keeping all its spatio-temporal characteristics, at the expense of the other input beam’s
energy. The latter beam is often referred to as the “pump” beam. This optical transistor
outputs both the amplified and the de-amplified beams. Its small signal gain is given by3

Exp[ΓL] where L is the length of the grating in the material. For a typical 5 mm grating and apump-to-small-signal ratio of 105, gains reach 20 000 or 40 dB.
Of particular interest in this work, is two-beam coupling with temporally modulated
beams. The coupling beams may be modulated with more than one signal. If the various
signals are temporally uncorrelated, they photorefractively couple as if they were the only
signal present but with a modified coupling constant. Each signal experiences a coupling
constant diminished by the ratio of their own intensity to the total intensity in the crystal. For
example, the small signal gain for each temporal component is Exp[Isignal/Itotal⋅ΓL]. The factthat the available gain is finite means that when more signals are present the less each signal
gets amplified. This phenomenon is one of the basic building blocks for the designs of ourcompetition-for-gain driven photorefractive circuits.
The photorefractive material’s memory provides another building block for the design ofour circuits. The time constant of barium titanate is on the order of milliseconds to seconds
depending on the total intensity of the light. A photorefractive index modulation therefore,
mimics the light patterns the material is submitted to, but integrated over time. In otherwords, these holograms do not record the simple multiplication of the electrical fields
generating the interference patterns, but rather the correlation of those fields over a few
seconds. This correlation function is essential for the autotuning filter’s ability to distinguishsignals among a collection of uncorrelated signals. Milliseconds are an unnecessarily longtime to integrate radio frequency signals for correlation. The slow time constant of barium
titanate limits the adaptation speed of the autotuning filter to abruptly changing input signals.
It also limits the lowest frequency of the signals it can process to a few kilohertz. The time
constant does not, however, affect the processing of higher frequency signals.

4

1. 1. 2 The autotuning filter
The autotuning filter is designed to separate out the strongest uncorrelated signal fromits spatio-temporal input space. This space refers to multiple coherent optical beams carrying
amplitude and/or phase information. The strongest uncorrelated signal is closely related to
the largest principal component of the input space. This relationship is clarified in thetheoretical section 2. 4. 2. 1. Throughout this thesis, however, the terms “stronger signal” and“largest principal component” are used interchangeably.
The autotuning filter is fundamentally an optical ring oscillator in which gain is suppliedby photorefractive two-beam coupling [31-39]. The input beams “pump” the gain medium.
The cavity modes of the oscillator intersect these pump beams inside the photorefractive
crystal aligned to transfer the most energy to the oscillating beam. The oscillation starts with
scattered light from the pump beam that is naturally emitted in the direction of greatest gain.
This fanning phenomenon [40] is a result of local photorefractive two-beam coupling
between the scattered light and the propagating beam.
Figure 1.1 illustrates the design of the autotuning filter. The crystal on the left-hand sideof the ring is the gain medium mentioned above that couples the input beams to theoscillating beam in the ring. The additional photorefractive crystal on the right-hand side of
the ring, acts as a selective loss element. It “absorbs” a signal inversely proportionally to its
power. Since the gain medium amplifies proportionally to a signal’s relative strength, thetwo photorefractive interactions combine to facilitate the oscillation of only the largest
principal component in the ring. In steady state, the other principal components remain on
the pump beams at the output of the gain crystal. The filter effectively extracts the strongest
signal from its input. That signal may be retrieved by sampling the oscillating beam of the5resonator. Thus the filter potentially has two outputs: one output to provide the strongest
signal and the other output to provide all the other signals.

Figure 1.1. Schematic diagram of the autotuning filter.

Which (if not both) output is of interest depends on the application. For example, if the
input beams are carrying a mixture of a weak signal and a strong jammer, output #2 (refering
to Figure 1.1) delivers the desired signal while the ring extracts the unwanted jammer. If the
input mixture contains a signal with much additive noise, then output #1 provides a noisereduced signal. If all the principal components of a collection of information carrying beams
need to be extracted then “daisy-chaining” autotuning filters at the #2 outputs yields theextracted components at the #1 outputs.
Many more applications can be found in the vast field of principal component analysis(PCA) [24]. PCA is a fundamental tool in the analysis of statistical data. It is generally acomputer intensive method relegated to the post-recording processing of data. Real-time,
adaptive principal component extraction is in its infancy.

As digital signal processing

techniques improve [41, 42] and the complexity of neural networks increases [43, 44], PCA
finds applications in low frequency wireless communications. Neural network architectures
have added adaptive capabilities to principal component extractor circuits [45, 46]. Those6implementations are computer based however and are therefore also signal bandwidth
limited.
The optical filter discussed in this thesis is 5 cm2 in size and extracts the principal
components of up to 3-GHz bandwidth signals in seconds. Although the filter is based on an
optical resonator, the photorefractive dynamics give the device adapting capabilities. For
example, the circuit keeps track of slow drifts of the optical and the signal carrier frequencies.
We believe that the literature presents no other optical adaptive extractor of spatio-temporalprincipal components. Most optical systems involving PCA realize pre-determined patternrecognition [47-50], or analyze and classify optical spectra [51-54].

7

1. 1. 3 An optically smart antenna array
We chose to demonstrate the potential merit of the autotuning filter in a communications
scenario. More specifically, the filter was incorporated in the adaptive optical processor ofsignals received by a microwave antenna array.

Adaptive processing in smart antenna

systems [55, 56] can either be done at the analog front-end at the carrier frequency or after
down conversion and analog-to-digital conversion at base-band using digital signal
processing (DSP) techniques [57]. Front-end processing is fast, but expensive and complex
as the adaptation circuitry requires microwave variable phase-shifters and variable gaincontrol elements. The more common DSP approach reduces system complexity, is more
economical, and more flexible; however, it is typically power inefficient and has modest
bandwidth. Part of this thesis’s goal is to show that the use of photorefractive circuitry can
simplify adaptive antenna systems and relieve the computational burden placed on the digital
signal processing done after down conversion.
Principal component extraction on antenna array signals subscribes to the broad field of
blind source separation (BSS) [58, 59]. This field seeks to separate unknown mixtures of
unknown signals that possibly have overlapping bandwidths. The less assumptions a blindseparation method makes about the unknown signals, the more powerful it is. For principal
component separation to function as a BSS method [59], the unknown signals have to bereceived in temporally and spatially orthogonal mixtures. The temporal orthogonality or noncorrelation is guaranteed in most communications scenarios by the fact that the signals are
emitted from independent sources. The spatial orthogonality is a more restrictive condition.
However, as will be shown in Chapter 3, principal component separation as a BSS method
works only as well as the signals are spatially uncorrelated. The signals are 100% spatially8correlated only when they are emitted from the same spatial location (within the receiving
antenna’s angular resolution.)
The architecture of the processing system discussed in this report is intended to be used
in applications with a large number of antenna elements. The experimental demonstration
uses only two microwave receivers and two test sources; however, the optical processor does
not substantially change as the number of array elements increases. The prototype system is
illustrated in Figure 1.2.

Figure 1.2. Block diagram of the prototype optically smart antenna array.

The first stage of our prototype system receives two 10 GHz signals from two distinct
sources placed in the far field of the array.

Audio signals amplitude-modulate themicrowaves. The receiving front end consists of a 30-element antenna array that acts as aquasi-optical discrete lens [60] with detecting antennas placed on its focal arc. The twoelectrical signals from the detectors are then down converted to 150 MHz and amplified
before being fed to the second stage of the system. The IF signals are applied to a one
crystal, two-channel resonant electro-optic phase modulator. An optical line beam traverses
the electro-optic modulator with its top half modulated with one of the IF signals and thebottom half modulated with the other. Since this is phase modulation with low modulation9

indices, the original optical frequency is still present in the beam, necessitating a carrier
suppression element in order for the autotuning filter to work properly. If the carrier is not
suppressed it introduces artificial correlation between the signals that impairs the proper
functioning of the filter. The suppression of the carrier is achieved by two-beam coupling thesignal-bearing beam with a non signal-bearing beam. The photorefractive crystal is oriented
such that the beam carrying information transfers energy to the non-modulated beam.
Although the signals suffer a 3dB loss in the process, the carrier frequency is suppressed byover 50 dB. The final processing stage consists of the autotuning filter. It extracts theprincipal component of its two-dimensional input signal space. The oscillating beam of thefilter is sampled and photodetected. Detection without homodyning is possible because theabsence of the optical carrier turns the phase modulation into an amplitude modulation at
twice the frequency. Demodulating electronics retrieve the audio signal from the AC port ofthe photodetector and send it to a speaker.
The whole system from the down conversion stage to the electronic output is packaged
in a 13x18x6 inch briefcase. It plugs into a wall outlet consuming power equivalent to acommon household light bulb. The briefcase system has proven its robustness and portability
by demonstrating real-time principal component extraction around the country.

The

miniaturization and packaging of optics including interferometric setups in a briefcase is asignificant accomplishment in and of itself. It shows that optical table-size systems (4x6 feet)
may be considerably shrunk in size and made to work without the cumbersome vibration
isolation offered by air-cushioned tables.

10

1. 1. 4 A photorefractive regenerative amplifier
Multiple stage electronic circuits require amplifier modules within or in between stages.
The art and science of making amplifiers is a sophisticated branch of electronics. Subbranches include high power amplifiers and low noise figure amplifiers, the design skills of
which vary immensely depending on the operational frequency range. As our photorefractive
systems grow in size and complexity, laser power becomes an issue. Barium titanate crystals
generally have a high absorption constant a typical α is between 1 and 2 cm-1 (for green
wavelengths.) Taking the autotuning filter as a familiar example, this means that the throughbeam carrying all but the largest principal signal looses about half its energy passing throughthe “gain medium” of the ring (our crystals are typically around 5 mm in length.) Without anamplifier, the initial power of the beam determines how much more processing can be applied
to it.
Optical regenerative amplification is a young field compared to its electronic counterpart[61]. The field blossomed in the early 1990’s when optics were massively introduced intelecommunication systems. Optical amplifiers are, still today, developed mainly for the
purposes of the communications industry. The operating wavelengths of the vast majority ofthese devices are therefore closely matched to the 1.3 and 1.5 µm communication
wavelengths.

Industrial research also focuses on single spatial mode amplifiers; again,

because optical communication systems employ single mode fibers to propagate their beams.
Unfortunately there are no efficient photorefractive materials at communication wavelengths.
Barium titanate works best with blue and green wavelengths and rhodium doping extends its
operating range to the visible red. Other photorefractive materials function for wavelengths
up to 1.06 µm although their coupling constants Γ are not as high. In addition, the dynamics

11

of our photorefractive circuits require the spatial diversity of multi-mode beams. These facts
prompted us to develop our own amplifiers.
The basic concept of a regenerative optical amplifier is relatively simple. Stimulated
emission amplifies an optical information-carrying beam propagating through a populationinverted medium. The latter may be the same gain medium as the one used in the laser thatoriginally generated the beam:

the gain bandwidths of these materials generally cover

thousands of gigahertz. A high noise figure constitutes the main drawback of non-parametric,
i.e. simple, traveling wave amplifiers. The theoretical minimum decrease of the signal tonoise ratio is 3 dB. A high number compared to low noise electronic amplifiers that limits
the number of times a signal may be amplified before it drowns in noise.
The amplifier design presented in this thesis takes a multi-mode beam in and outputs anamplified single mode beam without any loss of temporal information. Like the autotuning
filter it is essentially a photorefractive oscillator. A photorefractive element also couples the
input to the oscillating beam. Unlike the autotuning filter, a semi-conductor laser diodeamplifier provides gain to the loop and washes out the competition for photorefractive gain.
The resonator is constrained to be single mode, which allows the use of a single mode semiconductor amplifier in the ring. The oscillating beam picks up all the temporal information of
the multi-mode input beam, since there is no competition for gain. The photorefractive
gratings have no reason to organize themselves so as to select one part of the information
over another. This oscillator thus transforms a multi-mode beam into a single mode withmore energy.
This thesis reports the first steps toward making the amplifier. The semiconductor laser
diode amplifier for red wavelengths was assembled in the laboratory, starting with uncoatedchips. Another necessary experiment for demonstrating the feasibility of our photorefractive
amplifier involved showing that a multi-mode beam could pump a resonant cavity that wasforced to be single mode. This amplifier will hopefully be the first of a series. Our toolbox
12

of photorefractive circuits would be greatly enriched by the development of high performancesemi-conductor amplifiers at red wavelengths. The privilege of that work though, is left forother graduate students. ☺

13

1. 2

Thesis outline

This thesis reports work on three separate projects. Chapter 2 presents the experimental
and theoretical progress in relation to the autotuning filter. Chapter 3 describes the making ofan opto-electronic microwave processor and evaluates its performance. Chapter 4 reports on
preliminary research for a photorefractive regenerative amplifier. The fifth and last chapter(the appendix) regroups supplementary technical information relative to various sections
from the previous chapters.
Chapter 2 is divided into three sections. The first section aims to present the reader withan intuitive understanding of the autotuning filter’s functioning. This short section offers
enough information to make sense of the experimental section and provides a background forthe discussions in the theoretical section. The second, experimental section describes thedifferent implementation trials of the autotuning filter and discusses their performances. The
third, theoretical section is divided into three parts. The first part studies the influence of thefilter’s parameters on its signal separation performance. The second part compares theexperimental filter’s data with a mathematically pure principal component extractor. The
third part demonstrates the principal component extraction ability of the filter starting withthe photorefractive differential equations embodying its design.
Chapter 3 describes in details the design, operation and measured performance of theadaptive microwave receiver. It consists of three major components: the microwave front
end, the electro-optic modulation and carrier suppression stage, and the optical processing(auto-tuning filter) stage. A separate section explains each stage and reveals the packaging
efforts that enabled to fit the system in a 13x18x6 inch briefcase. The result section reports
the signal separation performance of the suitcase from a system’s point of view. The lastsection of the chapter discusses the implementation of an extension of the 2-receiver system14to an N-receiver adaptive array, and the implications of using an optically smart quasi-opticalantenna array in radar and communications.
Chapter 4 discusses the design of a photorefractive regenerative amplifier.

It alsoreports on the experimental procedure to build a semiconductor laser diode amplifier
necessary for the implementation of the photorefractive amplifier. The procedure describedstarts out with an uncoated and unmounted InGaAlP chip. The appendices present thedetailed designs of some mechanical systems that we found particularly useful in therealization of the work presented in Chapters 2 and 3. They include practical information
relating to the manufacturing of BaTiO3 spheres and an operating manual for the suitcase
system. The appendix also provides the Mathematica codes for the numerical simulation andstudy of the autotuning filter.

15

1. 3

Contributors to this thesis

The design of the autotuning filter was first presented in 1991 as a frequency
demultiplexer [10]. This thesis introduces the building and testing of experimental versionsof the filter as a principal component extractor. The influences of the filter’s parameters on
the device’s signal separation performance were studied with a Mathematica program(presented in the appendices) that was greatly enhanced through discussions with
Dr. A. Zozulya. His work on the stability analysis of a photorefractive feature extractor [62]allowed the demonstration of the autotuning filter’s principal component extraction abilities
in Chapter 2.
The realization of the suitcase project (Chapter 3) was the product of teamwork among
Dr. S. Romisch and graduate student P. Smith, advised by Professor Z. Popovic, and myself.
Dr. S. Romisch and P. Smith designed and built the front-end’s electronics. I designed,
machined and packaged the optical stage of the system. Dr. S. Romisch designed and built
the demodulation electronics after the optical stage. Later, J.H. Loui helped me improve theirperformance. Dr. S. Romisch, P. Smith and I tested the different parts of the suitcase project
and

took

the

characterizing

end-to-end

measurements

of

the

system.

Professor D.Z. Anderson and Professor Z. Popovic contributed heavily to the discussion of
the system from a communications perspective.

The 4-channel implementation ofProfessor D.Z. Anderson’s multiple-channel EOM idea, was designed, built and tested by
graduate student H. Matern, advised by Professor Z. Popovic and experimentally assisted bymyself.
My experimental work on assembling a semiconductor amplifier in Chapter 4, wasguided by Dr. L. Hollberg. N. Mackie, a member of Dr. L. Hollberg’s group, supervised andassisted my anti-reflection coating trials. Graduate student S. Hugh built the single mode
cavity pumped by a multi-mode beam via a photorefractive crystal.
16

CHAPTER 2
THE AUTOTUNING FILTER

2. 1

Introduction

This chapter describes a photorefractive optical circuit that performs principal
component extraction on an ensemble of spatially and temporally modulated signals. The
circuit presented here is designed to process signals received by a microwave antenna array(see next chapter). In particular, it is designed to handle relatively large signal bandwidths asmight be present in a communications receiver.

The function of principal component

extraction is a fundamental signal and data processing task employed in a much broader
domain than simply communications [24]. It is the second order solution to the well-known
problem of blind source separation [63]. A primary aim of this work is to demonstrate that amoderately sophisticated spatio-temporal signal-processing task can be implemented to goodadvantage with photorefractive nonlinear optics. Indeed, our miniaturized optical circuit isabout the size of a U.S. quarter dollar, is capable of handling signal bandwidths in excess of
one gigahertz, and can operate with an optical input power of less than 5 mW (and no otherpower source is needed to run the processor).
In the following sections, we fully characterize the circuit’s handling of two spatiallyand temporally modulated signals. The processor is scalable to larger signal spaces without
dramatic changes to its size or design. In these characteristics—high signal bandwidth, small
17

size, low power consumption, and scalability—the photorefractive signal processor compares
favorably with a digital signal processor approach to the same function. Does the opticalsystem remain in a favorable light when one contemplates an entire system from raw signal
input to processed signal output? We believe the answer is yes. We have assembled acomplete opto-electronic system around the processor presented here: from free-space
microwave input, to the RF front-end, to the optical processor, and to the electrical output, theentire system is packaged in a standard-sized briefcase and consumes 50 W of power fromthe wall plug. This complete receiver system is described in Chapter 3. Here we focus on thedesign and characteristics of the photorefractive optical processor itself.
The photorefractive optical circuit presented in this thesis acts as a filter on the principal
components of the input signal space. The largest principal component exits one output port
while the remainder of the spatial-temporal signal space passes through the filter, as isschematically illustrated in Figure 2.1. We refer to this optical processor as an autotuning
filter, for the circuit automatically tracks and extracts the principal components without
having any a priori knowledge about the signal space.

Figure 2.1. The filter extracts the largest principal component of its input signal space while passing on all theother components.

The first section of this chapter heuristically explains how the filter functions by using
intuitive arguments of gain competition.

The second section reports on the different

18experimental implementations of autotuning filters. The performances of these filters at
separating out two signals are regrouped at the end of this experimental section. The thirdand final section of this chapter is comprised of three parts. The first part qualitatively
explores the optimization of the filter’s design parameters. The second part compares theexperimental results with those obtained with a pure principal component extractor. The
third part of this theoretical section argues that the autotuning filter does physically
implement the algebraic function of principal component extraction.

19

2. 2

A heuristic explanation of the filter's dynamics

The autotuning filter consists of a photorefractive ring oscillator [64], schematically
depicted in Figure 2.2.

The photorefractive crystal on the left-hand side of the figure

provides optical gain and the other one provides an active loss mechanism.

Figure 2.2. Schematic of the autotuning filter's design. The two-beam coupling occuring in the crystal on the leftof the figure provides gain to the ring. The two-beam coupling taking place in the crystal on the right
imposes loss on the ring signals inversely proportionally to their relative intensities.

Consider first a simple ring comprised of the gain medium alone. Also suppose that the
input “pump beam” consists of a single spatial mode beam. The optical feedback loop is
designed such that the gain is sufficient to cause oscillation in the ring. If the input beam is
temporally modulated, then the oscillation beam will also be temporally modulated withessentially the same temporal structure. Suppose now that there are two beams having
different, in particular uncorrelated, temporal structures.

Again, the gain is chosen

sufficiently large to overcome the round-trip losses so oscillation can once again take place.

20

What will be the temporal structure of the oscillating light? Will it carry the structure of one
or both input beams?
The answer is “it depends.” Ideally, the oscillating beam would have the temporal
structure of solely the stronger input beam and none of the weaker.

The second

photorefractive crystal (on the right-hand side of Figure 2.2) is designed to enhance thecircuit’s ability to approach this ideal. The interaction that takes place in this medium is a
special case of two-beam coupling referred to as reflexive coupling [12].

Figure 2.3

illustrates the effect reflexive coupling has on a multi-mode beam carrying two temporalsignals. The initial beam is split in two unequal parts. They are then coupled to each other ina photorefractive crystal that is oriented to transfer energy from the brighter beam to thedarker one. The signals in the latter get amplified proportionally to their relative intensities.
This means that the stronger signal undergoes more amplification than the weaker one. At
the output of the gain port, the intensity difference between the signals is larger than at the
input. Overall, the weaker signals suffer more loss than the strongest one.

Figure 2.3. Reflexive coupling is a photorefractive selective loss element. It induces loss inversely proportionallyto the strengths of signals.

21

Placed within the feedback loop of the gain medium, the reflexive-coupling element in
the multi-mode ring oscillator effectively enhances the intensity differences among thesignals until (ideally) only the initially strongest signal oscillates. At the same time, thisstrongest signal is largely removed from the input beam while all the other signals are passedon through the gain medium.
The oscillation in the autotuning filter starts with noisephotorefractive beam fanningin our case. The energy of the input beams alone powers the filter. If the carrier of those
input beams (optical or RF) drifts with time, the circuit's dynamics automatically adjust to thechange.

22

2. 3

Experimental versions of the filter

The first implementations of photorefractive feature extractors occupied about half thesurface of an optical table (∼2 m2). The experimental efforts in building the filters presented
in this thesis concentrated on miniaturization. There are two main reasons for doing so.
First, the cavity length of the ring oscillator determines the process bandwidth of the filter, as
it is inversely proportional to the round trip time of the light. Second, it makes the device
more robust to environmental perturbations such as air currents and vibrations.

2. 3. 1 Fiber optic implementations of the filter
Fibers were our choice candidate to help us reduce the size of our filter device. Fibers
dispense of using cumbersome adjustable mirror mounts for guiding the light. A fiberized
device also appeared as more practical for future integration in optical systems.
For the photorefractive effect to govern the dynamics of the filter, the resonant cavity ofthe oscillator may be designed to be dispersionless or highly multi-mode. Either of these
conditions ensure that the round trip phase requirement for oscillation plays no role indetermining which signal wins the competition in the ring.

2. 3. 1. 1Standard fiber technology

Photorefractive crystals are not presently integrated in fibers, so we opted to couple our
optical beams in and out of fibers using gradient-index (GRIN) lens technology. Physically,
the GRIN lenses look like small glass rods of 1.8 mm in diameter. They combine refraction at
the plane end surfaces with continuous refraction (radial gradient-index material) within therod [ref]. When cut to a specific length (pitch=0.23) they turn into a collimating lens with23

zero back focal length. Their plane end can be put into contact with the plane end of a ferrulethat holds a fiber. This theoretically leaves only two translation degrees of freedom withwhich to center the GRIN lens on the fiber tip located at the ferrule’s end. This alignment
may be adjusted using two pre-aligned v-grooves machined at the appropriate heights. Figure
2.4 shows coupling from fiber to fiber through a PR crystal using the GRIN lens and ferrule
technology.

Figure 2.4. GRIN lenses collimate (and refocus) the light coming out of (or into) a fiber held by a ferrule.

Our fiber-to-fiber coupling efficiency reached a maximum of 50% only with complex,
time-consuming alignment procedures (not counting the crystal’s absorption). Part of the
problem in our experiments was that the output beams from the GRIN lenses were never wellcollimated so that a portion of the light was emitted outside the receiving GRIN lens’s
numerical aperture.

2. 3. 1. 2

Spherical crystal technology

We greatly simplified our fiber-to-fiber coupling set up when we made sphericalphotorefractive crystals that doubled as lenses. We can couple light directly from bare fiber
tip to bare fiber tip as shown in the diagram and photograph of Figure 2.5. The fibers’
positions are constrained in v-grooves so that only one translation may be adjusted. The
24

position of the crystal sets all the remaining degrees of freedom necessary to optimize thefiber to fiber and the photorefractive coupling: x, y, z translations plus rotation of the c-axis
in the xy plane. We achieve 75% coupling efficiency from one fiber to another. The multimode fibers we used had a core size of 64.5 µm and a numerical aperture of 0.268. Due tothe birefringence of BaTiO3 deviating the beams, however, the efficiency drops to about 50%
when trying to couple 2 fibers to 2 other fibers. We still gain in ease of alignment and size ofthe filter over the previous version.

Figure 2.5. BaTiO3 spherical crystals double as lenses to couple light from two fibers into two other fibers.

Our mechanical mounts for aligning the different elements evolved over time and thetwo main systems are described in Appendix 7. 2. The first version relies on glue to holdmechanical parts together making it difficult to disassemble for future reuse of the crystals.
The second version is purely mechanically aligned and held in place. Figure 2.6.a shows
photographs of filter implementations built with each aligning system.

Another crucial

mechanical aspect that enabled the realization of these devices is the manufacturing of the25spherical crystals themselves.

Photorefractive spherical crystals are not commerciallyavailable. Appendix 7. 1 explains the process we have devised for making BaTiO3 spheres
and slabs of spheres while wasting the least material.

a.

b.
Figure 2.6. a) Photographs of the two fiberized autotuning filter versions. The crystals are slabs of spheres as it
wastes less material than whole spheres. b) Design of the autotuning filter using fibers and sphericalcrystals.

Figure 2.6.b shows the layout of the fibers in these versions of the filter. We changed
the reflexive coupling design of Figure 2.3. Instead of splitting the beam before the crystal
with a fiber directional coupler, the whole beam goes through the crystal once and is guided
back around to couple to itself. Two reasons prompted us to implement this change. One, it
26

simplifies the experimental setup while not fundamentally changing the dynamics of thereflexive coupling. Two, a multimode fiber directional coupler separates spatial modes rather
than splitting off a portion of each. This induces slightly different dynamics for different
loop paths thus breaking the degeneracy of the modes we initially designed for.

2. 3. 1. 3

Problems of multimode fibers

A problem common to all fiberized versions of the filter is the polarization scramblingproperty of multimode fiber. The photorefractive effect occurs only for polarizations that lie
in the plane of the c-axis and the direction of propagation of the beams. With every roundtrip
roughly half the light converts to the wrong polarization adding a 3dB loss factor to the loop.
In addition, the intrinsic gain in the crystal goes as ΓΙ signal
I totalso that the wrong polarized lightdecreases the available gain by half.
High roundtrip losses were the main obstacle to obtaining a device with expected signal
separation performance (see theoretical section below). Replacing multimode fibers withsingle mode fibers would require making the resonator’s cavity dispersionless. We decided
that an easier path involved abandoning fibers altogether and working on miniaturizing freespace optics.

2. 3. 2 Free space optics implementation of the filter
2. 3. 2. 1

Miniature optics

The components of the filter are two BaTiO3 crystals for the reflexive and gain units,
two planar mirrors and one spherical mirror for the ring cavity, and a beam splitter to split thebeam for the reflexive coupling. Except for the crystals, we cut out miniature versions of27

these elements from their normal size (1 inch optics) counterparts. We reduced alignment
degrees of freedom by precisely cutting their bottom surface at “exactly” 90 degrees to thereflecting surfaces. These 90 degrees were measured by comparison to a commercial 90
degree prism. This cut eliminates the need for vertical tilt adjustments on the elements.
Horizontal tilt and translation are adjusted externally before the components are glued inplace (Norland UV glue #63). The crystals are mounted so that their center height roughly
corresponds to the spherical mirror’s center height. Figure 2.7 shows the filter compared to a
U.S. quarter dollar. The two central elements are the gold bar shaped photorefractive BaTiO3
crystals. The four outer elements include the two plane mirrors, the spherical mirror (radiusof curvature: 5 cm) and the beam splitter (about 5% reflection on one side and anti-reflection
coated on the other). The substrate is a quartz disc of 25 mm diameter.

Figure 2.7. Photograph of the autotuning filter using free space miniature optics.

28

2. 3. 2. 2

Novel crystal cut

The original crystal cut we designed allows taking full advantage of the photorefractive
crystal’s available gain. In a conventional straight parallelogram cut, unwanted oscillations
started by scattered light often arise between two parallel faces of the crystal. For one, thetypical small signal gain of barium titanate is high (Γ=20/cm) and two, the index of refraction(n=2.4) induces 17% normal reflection at the crystal/air interfaces. These oscillations are in
direct competition for gain with the desired two-beam coupling interactions and so decrease
the effective available gain. Making the two opposite faces in question (the ones roughly
parallel to the propagating beam) non-parallel prevents secondary oscillations. The crystals’
geometry is shown in Figure 2.8.

The crystals proved to have excellent saturation

characteristics: a pump-beam to weak-beam ratio of one results in a gain of exactly two.
Also, to reduce the reflection losses of the signal beam, the entrance and exit faces of thecrystal are cut so that the angle of incidence is Brewster’s angle.

Figure 2.8. The "gold bar" shaped cut of photorefractive crystals suppresses unwanted oscillations arising between
otherwize parallel faces.

29

2. 3. 2. 3

Alignment

Computer generated templates placed under the transparent substrate serve as alignment
guides to position the filter’s components. In order to draw the template, the elements’shapes are carefully measured and then faithfully reproduced as computer objects. Ray
tracing on the computer then dictates the position and horizontal tilts of the elements toensure that there is a closed path. Once the elements are aligned to the template as best as
possible, just a little tweaking of the tilts of the mirrors enables the oscillation to start. Figure
2.9 is an enlarged picture of the template used to align the autotuning filter.

Figure 2.9. Computer generated template placed under the transparent substrate that the filter of Figure 2.7 is built
on. The miniature elements of the filter are aligned to it externally before they are glued in place.

30

2. 3. 3 Results
As mentioned before, our ideal filter lets only the stronger signal oscillate in the ring
while completely suppressing the weaker one. Not surprisingly, in practice this is not thecase. For example, the spatially finite grating written between one pump beam and theoscillating light in the ring induces the diffraction of a small portion of the other pump beameven if it is not exactly Bragg matched. Experimentally we measure about –30 dB (0.1%) of
the input beams diffracting off of each other’s gratings.
We evaluate how well the filter separates signals by plotting the intensity ratio of thetemporal signals inside the ring versus the input signal ratio carried by the pump beams. Theinteresting region of this plot occurs for input ratios close to one (0 dB). At exactly 0 dB thefilter cannot distinguish between the two signals since the competition for gain is completely
unbiased and therefore it also returns an output ratio of 0 dB. However, for ratios near 0 dB
the competition for gain plays a strong role. The better the filter, the greater the signal ratio
enhancement is around input ratios of 0 dB. In plots of the ring signal ratio in dB versus the
input signal ratio in dB, this translates into a higher slope at the origin.

2. 3. 3. 1

Test bed setup

We generated our pump beams by driving two acousto-optic modulators with signal
frequency f1 and f2 near 80 MHz.

The exact frequencies were 79.99960 MHz and80.00056 MHz (measured using an HP 8568B spectrum analyzer) resulting in a in frequency
difference of less than one KHz. We used 514.5nm (argon laser) and 532 nm (Verdi laser)
lasers as our light sources. The first order diffracted beams from the AOMs served as the
input signals to our filter devices. Figure 2.10 schematically illustrates the experimental
setup.

31

Figure 2.10. Schematic of the AO test bed used to evaluate the separation performance of the autotuning filters.

For the fiberized devices the pump beams were coupled to a multimode fiber withdifferent angles thus ensuring that each beam occupied orthogonal spatial modes inside thefiber. The output of the latter was focused to pump the gain crystal of the photorefractive
oscillator. For the free space optics devices, the two pump beams (∼ 5 mm cross section
diameter) were focused with 15 cm focal length lenses and crossed inside the gain crystal
with a three to five degree angle between them.
We chose to monitor the filter's “stronger signal” output. This corresponds to sampling
the beam inside the ring. For the fiberized filters using GRIN lenses and the free space optics
filters, the sampling beam was the reflection of the oscillating beam off the front face of thegain crystal. For the filters using spherical crystals, the main loop of the oscillator was
opened just before the gain crystal and a microslide inserted to deflect some of the ring's
oscillating beam. The light was directly detected with a photodetector since AOMs generate
amplitude modulated beams. In order to use practical low frequency detectors (ease of32

alignment) and spectrum analyzer (ease of access) we lightly modulated the AOMs' driver
signals with 15 and 17 kHz sinusoids. We checked that this extra modulation had no effecton the filter's choice of signal.

2. 3. 3. 2Experiments with the fiberized filters

Figure 2.11 shows a separation curve taken with the filter on the top left-hand side of
Figure 2.6. The slope at the origin is 8 dB/dB, thus an initial signal ratio close to unity(P1/P2=1) gets amplified 6 times. For input ratios greater than 20 dB, the slope tends towards
unity (not shown on the graph); the filter does not enhance the signal contrast when theirpowers are very different to begin with.

Figure 2.11. Separation curve taken with the filter on the top left-hand side of Figure 2.6

In principle the different input signals to the filter do not need to be separately carried bydifferent spatial modes. The filter's response should be the same when the signals are
orthogonally mixed up on the spatial modes. This is further explained in the theoreticalsection 2. 4. 3. This claim was verified with the setup depicted in Figure 2.12. The AOMs'

33

driver signals are each split in two equal parts and recombined so that each AOM is driven
with some of both signals. To keep the mixing orthogonal, one of the arms has to be delayedby 180 degrees. The mixing matrix then performs a simple 45-degree rotation of the signals
in the input space.
The separation results are plotted in Figure 2.13. The data was taken with the second
version of the spherical crystal filter shown on the top right-hand side of Figure 2.6. Whentested with unmixed signals, the filter yielded the same curve as in Figure 2.11 with an
8 dB/dB slope. The slope at the origin with the mixed signals is slightly less at 6.6 dB/dB.
This decrease in performance may be explained by the fact that the weaker signal diffracts
more off of the stronger signal’s gratings. It is slightly closer to being Bragg matched tothem since they are carried by the same spatial modes. This slightly hinders the competition
for gain and decreases the slope at the origin in our plots.

Figure 2.12: AO test bed with mixtures of two signals driving each AO. Each signal is split in two and recombinedwith half of the other signal. A 180-degree phase shift is added to one half of S2 before it is recombined.

34

Figure 2.13: Separation curve taken with the filter shown on the top right-hand side of Figure 2.6 and with the test
bed of Figure 2.12

2. 3. 3. 3Free space optics version

The separation performance of the autotuning filter was greatly improved with the freespace optics implementation. The slope at the origin in Figure 2.14 is 26 dB/dB, a small
signal ratio enhancement of 400. This is nearly a two-order magnitude improvement over thefiberized versions. This performance comes at the price of a more careful alignment of the
input beams. The value of the slope is sensitive to the geometry of the pump beams. Good
performance requires a shallow angle between the pumps as well as the pumps crossing
where they intersect the ring beam. A small change in any of these requirements cannegatively affect the slope (down to 10 dB/dB). The fiber versions were not nearly sosensitive to alignment since there was no control over how the speckled pattern folded on
itself when it was focused down.

35

Figure 2.14: Separation curve taken with the filter shown in Figure 2.7

36

2. 4

Theoretical analysis of the filter

In this section we seek to gain insight about the dynamics of the autotuning filter. The
first part explores the parameter space of the filter searching to optimize signal separation
performance.

This search varies the values of the parameters of the reflexive-coupled

photorefractive ring oscillator. We analyze the steady state solutions to the differential
equations embodying the dynamics of the filter. This stability analysis consists of a fullnumerical integration of the differential equations with initial conditions set as perturbed
steady state solutions. This search does not conclude with an ideal set of parameters, but witha qualitative description of the influence of each parameter on the filter’s separation
performance.
The last part of this theoretical section demonstrates (in a somewhat non-elegant fashion,
but for the first time) that the autotuning filter separates signals by implementing themathematical function of principal component separation. Our use of the words “principal
component,” is a little off from the textbook definition.

This last part starts with aclarification of what we mean by them.

2. 4. 1 Varying the filter's parameters for best separation performance
Ideally, the desired filter perfectly suppresses anything but the stronger signal in the ring
regardless of the intensity difference between the input signals. When dealing with two
signals, a simple ring without a reflexive coupling unit theoretically achieves this. If theavailable small signal gain in the oscillator strictly equals twice the threshold gain for
oscillation, only the stronger component undergoes enough gain for oscillation.
Photorefractive gain is shared out proportionally to the relative intensities of each signal.

37

This “strictly equal” is difficult to realize experimentally, hence the need for extra degrees of
freedom supplied by the parameters of a photorefractive reflexive ring.

2. 4. 1. 1

The filter‘s adjustable parameters

The four adjustable parameters of the filter are:
- Gg, the dimensionless intensity gain supplied by the two-beam coupling gain element;- L, the multiplicative round trip losses due to absorption, mirror reflectivity (they do notinclude the splitting from the BS) and possible inserted loss elements;- m, the pump to weak beam intensity ratio at the reflexive coupling element;- Gr, the dimensionless intensity gain of the reflexive coupling element.

Figure 2.15: Regime of interest in our discussion: when the signal ratio is close to unity both signals may oscillate.
Above a critical value P1/P2c only the stronger signal oscillates.

We observe the filter’s ability to suppress the weaker signal on the same plot used toevaluate the experimental versions of the filter: the plot of the oscillating signals’ intensity inthe ring S1 and S2 versus the input signal intensity ratio P1/P2. Different points in theparameter space produce different types of plots. Figure 2.15 shows the plot type of interestin this discussion. Around a unity-input ratio (0 dB), both signals may oscillate in the ring(the on/on regime). Above some absolute critical boundary value P1/P2c, only the stronger
input signal oscillates in the ring (the on/off regime). Experimentally, this corresponds to the38stronger signal being the only one writing a grating in the crystals. As discussed earlier, theweaker signal never fully disappears from the ring. A smaller boundary value P1/P2c meansthat the filter achieves a “perfect” stronger signal extraction for input signal ratios closer tounity.
The study presented below explores how the filter’s parameters affect this boundary
value P1/P2c. In summary it shows that: a smaller gain Gg is better (too much gain washes out
the competition); a larger transmission T is better (non selective losses hinder thecompetition); a larger splitting ratio m is better (it enhances the importance of the reflexive
coupling) and given m, Gg, and T there is an optimum gain Gr.

2. 4. 1. 2

The theoretical “input signal ratio vs. output signal ratio” curve

The boundary value P1/P2c is indirectly obtained by observing the Log-Log plot of thesignal ratio in the ring S1/S2 versus the input signal or pump ratio P1/P2. When the input
signal ratio approaches P1/P2c where the weaker signal disappears from the ring, the output
signal ratio will tend to infinity. Since it is symmetric around the origin, we only observe theplot for P1 greater than P2. There are two main steps in the acquisition of the data for theseplots. First, we find the steady state solutions of the differential equations describing theevolution of the filter. Second, we determine which one makes physical sense by analyzing
the solutions’ robustness to small perturbations.
The differential equations provided by the plane-wave model of two-beam coupling for
temporally and spatially orthogonal signals yields fairly simple formulas for the steady-statevalues of the signals’ intensities. Figure 2.16 defines the notations used in the following
formulas.

39

Figure 2.16: Illustration of the notations used in the equations of this section.

The intensity of S1 just after the gain element is

S1'=

I1S1


I 
P1 Exp−Gg 1  + S1
I tot Equation 2.1

where I1=S1+P1 and Itot=I1+I2. The intensity of S1 after the reflexive coupling and the ring’s
losses L=1-T is

S1out = T

S1 '

S1 ' 
1+ mExp −Gr
S1 '+S2 ' Equation 2.2

The loop gain for S1 is given by inserting Equation 2.1 into Equation 2.2

G1(S1,S2 ) =T(S1 + P1 )
.





 (P2 +S 2 )S 2 (P1 e -Gg*I1 / Itot +S1 )I1 

P1 Exp-Gg  +S1 1 + mExp[-Gr *1/1+
 (P1 +S1 )S1 (P2 e -Gg*I2 / Itot +S 2 )
I tot Equation 2.3

40

The corresponding formula for S2 is obtained by exchanging the subscripts 1 and 2.
There are four possible steady state classes to consider: no signals oscillating in the ring,both signals oscillating, the stronger signal oscillating by itself, and the weaker signal
oscillating by itself. The off/off solution is stable when the round-trip gain is less than unity.
This case is of no interest to us and in the rest of the paper it will be assumed that the small
signal gain is always greater than the losses.
The on/off solution in which the oscillating signal corresponds to the weaker pump’s
signal is possible under special conditions of the filter’s parameters and for an input pumpratio close to unity. One of these conditions requires that the signal corresponding to theweaker pump start out as the stronger one in the ring. This regime is never observed in our
experiments since the filter starts with fanning pump light that carries the signals in the same
relative proportions as the pump beams.
The on/off regime with only the stronger signal oscillating is our desired mode ofoperation for the filter. This solution is given by finding the pairs (S1, S2=0) that make thegain of Equation 2.3 unity (assuming P1>P2). However, for input pump ratios sufficiently
close to unity, the solution is unstable: if a tiny amount of the weaker signal is introduced
into the equations, it grows until the system reaches a stable on/on regime. How close tounity the pump ratio has to be for the on/on regime to be stable depends on the otherparameters of the filter as will be described later.
Finally, searching for signals’ intensities in the on/on regime is not as straightforward as
could be expected.

For an input pump ratio close to unity, solving the equation

G1(S1,S2) = G2 (S1,S2 ) = 1 yields up to three physically allowed solutions. Figure 2.17
graphically represents three such solutions as the three intersections of the two curves

G1(S1,S2) = 1 and G2 (S1,S2 ) = 1 . A numerical integration (detailed in Appendix 7. 3) of thefull photorefractive dynamical equations for the filter, using these solutions as the initial41

conditions, shows that the two extreme solutions (left and right ones on the figure) areunstable and evolve in time to on/off regimes. The middle solution is stable: if we choose
initial conditions close to it in signal space, the ring’s intensities evolve towards this solution
which means it is robust to small perturbations. This stable solution moves to the bottom
right of the (S1, S2) space as the ratio P1/P2 is increased and eventually disappears when theon/off regime is no longer stable.

Figure 2.17. The intersection of two curves represent the pairs of (S1,S2) that yield G1=G2=1.

Figure 2.18 illustrates a typical theoretical plot of S1/S2 vs. P1/P2. In the region where
both signals oscillate at the same time the curve is mostly linear. At some critical value ofP1/P2, S2 disappears from the ring so the curve jumps to infinity. In general, the smaller thiscritical value is, the steeper the slope of the linear region. The slope directly reflects thecompetition between the signals in the auto-tuning filter.

42

Figure 2.18. The Log-Log plot of the S1/S2 vs. P1/P2 curve shows a linear region when both signals are oscillatingin the ring and a rapid increase toward infinity as the input signal ratio approaches the boundary value
where the weaker signal disappears. The parameter values are Gr=Gg=10, m=50, and T=0.5.

2. 4. 1. 3

Influence of the ring’s parameters on the signal competition

This section qualitatively describes the effect of the filter’s parameters on the criticalvalue P1/P2c of the input signal ratio above which the filter performs a theoretically perfect,
stronger signal extraction.
The gain Gg provided by the crystal coupling the input beam to the ring essentiallyserves to compensate for the losses encountered by the signals in the ring. Keeping all otherparameters constant (T=0.5, Gr=10, m=50), Figure x plots the S1/S2 vs. P1/P2 curve for Gg=9,10, and 11. Figure 2.19 shows that lowering the gain Gg reduces the on/on region andincreases the competition within it. In this particular case, Gg=9 is barely enough gain for theoscillator to self-start.

43

Figure 2.19. S1/S2 vs. P1/P2 curve for Gg=9, 10, and 11, left to right (T=0.5, Gr=10, m=50).

As mentioned before, photorefractive gain is shared among the signals in proportion totheir relative intensities. On one hand, to ensure that there is always a signal oscillating in thering the minimum available gain must be above twice the threshold gain for oscillation; when
the two signals are exactly equal, they are both allowed to oscillate. On the other hand, avery high available gain means both signals’ gains are above threshold for a wide range of
input signal ratio values both principal components may oscillate. A smaller available gain
Gg is therefore preferable as long as it stays greater than twice the threshold gain.
The passive losses of the filter are embodied by the intensity transmission coefficient T.
Keeping all other parameters constant (Gg=Gr=10, m=25), Figure 2.20 plots the S1/S2 vs.
P1/P2 curve for T=0.4, 0.5, and 0.55, illustrating that higher losses soften the filter’scompetition and increase the critical value of P1/P2. This transmission coefficient, being thesame for all signals trying to oscillate, does not help the competition between the signals.
Lowering T is not equivalent to decreasing the gain Gg from a signal competition point of

44

view; it actually hinders the competition. The critical input signal ratio decreases as thetransmission T increases.

Figure 2.20. S1/S2 vs. P1/P2 curve for T=0.55, 0.50, and 0.40 left to right (Gg=Gr=10, m=25).

The two remaining parameters are closely linked to one another. In order to study theirinfluence on the competition simultaneously, we observe slightly different plots. All thecurves have a linear region close to the origin so the gradient of the slope can be estimated bylooking at the signal separation performance for only one input signal ratio. In the following
figures we chose an input pump ratio P1/P2=0.2dB=1.047. Keeping the losses and the gain
unit’s gain constant (T=0.5, Gg=10), the left hand-side of Figure 2.21 plots S1/S2 vs. Gr fordifferent values of m. For any given value of Gr, a larger value of m provides a better signal
ratio enhancement. For a given m there is an optimum value of Gr that is higher for larger
values of m.

45

Figure 2.21. Left side: S1/S2 vs. Gr for m=60, 55, 50, 45, and 40 from top to bottom with T=0.5, Gg=10, andP1/P2=0.2 dB. Right side: same values except Gg=11.

The splitting ratio m determines the amount of power that is removed from the ring andcoupled back to it through photorefractive two-beam coupling. This reflexive coupling
process is effectively a selective loss element: the weaker signal undergoes higher losses than
the stronger one. Increasing the parameter m gives more importance to the signal competition
and therefore lowers the critical input signal ratio. Physically, Gr’s optimum value may be
explained by two facts. One, coupling back to the ring with too little gain increases theoverall losses of the filter and lets the two beam coupling at the input gain unit govern the
dynamics of the filter. Two, coupling back to the ring with too much gain dampens theselectivity of the reflexive coupling.
Overall the best signal ratio enhancement is obtained with the larger m and thecorresponding optimum Gr.

Increasing m however, imposes more difficult self-starting

conditions on the filter requiring a higher gain Gg. The right hand-side of Figure 2.21 shows
similar curves to the ones on the left hand-side with a Gg of 11 instead of 10. The signal
separation deteriorates rapidly indicating a sharp tradeoff between increasing m and46

decreasing Gg. The figure also shows that the optimal values of Gr are only slightly higher
than in the previous case. This implies that Gr depends strongly on m and weakly on theother parameters of the filter.

2. 4. 2 The filter's principal component extraction abilities
This section compares the experimental results presented earlier with theoretical dataobtained with a pure principal component extractor model. A brief review of principal
component analysis provides a context in which to describe the function of our autotuning
filter.

2. 4. 2. 1

Principal components

Consider a collection of N random time-dependent signals, s˜1 (t ), s˜2 (t ),…, s˜N (t ) , each ofwhich has a zero mean value, i.e., has no D.C. component. In general, these signals may be*correlated, meaning here that ˜si (t )s˜ j ≠i (t )T

≠ 0 , where … T indicates an average over atime T much longer than the characteristic inverse bandwidth of the signals. The originalcollection of signals can be linearly transformed to a new set of signals that are uncorrelated,
provided certain mild constraints on the signals are satisfied. One specific de-correlatingtransformation can be obtained through the correlation matrix

 s˜1 2
*1
˜C =  s˜2 s˜1P  M s˜N s˜1*s˜1 s˜2*

…

2

L
O
L

s˜2
M
s˜N s˜2*s˜1 s˜N* s˜2 ˜sN* ,M 2s˜ N 

47

Equation 2.4

or

1
s˜ s˜
C˜ ij =
P i j T

where the explicit time dependence has been dropped for compactness. The correlation
matrix is normalized by the total average power in the collection of signals,N

P = ∑ s˜i (t ) T .
2

i =1

Signal de-correlation is achieved by first diagonalizing the correlation matrix, which canbe done in the usual manner by performing a similarity transformation with a matrix V
formed from the eigenvectors and eigenvalues of the matrix.

The eigenvectors of thecorrelation matrix are commonly referred to as the principal components of the signal space
while its eigenvalues are referred to as the principal values. For the present we assume thatthe correlation matrix is non-singular, and furthermore that its eigenvalues {λ 1, λ2 ,…, λ N }are non-degenerate so that they can be ordered, λ 1 > λ 2 > … > λ N .

The eigenvector

corresponding to the largest eigenvalue is loosely referred to as the “largest principal
component” and similarly for the eigenvector corresponding to the second largest eigenvalue,and so forth.
Corresponding to the matrix V is a unique unitary matrix U that can be used to transformthe original, correlated signals, to a new set of signals {sˆ1 (t ), sˆ2 (t ),…, sˆN (t )} that areuncorrelated and furthermore have the property thatsˆi

2

= λ i . In the remainder of thischapter we twist the conventional jargon a bit by referring to the transformed signals48

themselves as the principal components, e.g., to sˆ1 (t ) as the largest principal component,sˆ2 (t ) as the second largest, and so forth.

2. 4. 2. 2Comparison of experimental results with pure PC extraction

Figure 2.22. Simple model of our filter as a principal component extractor. The mixing on the left hand-side
accounts for the spatially finite size of the photorefractive gratings that allows one pump beam to
weakly diffract off of the other pump’s grating.

Figure 2.22 illustrates how we obtained the theoretical data we are comparing to ourexperimental data set. The mixing box on the left hand-side of the Figure 2.22 allows forsome cross talk between the two input beams to account for the finite size of the gratings
written in the photorefractive crystals. It is represented mathematically by a mixing matrix
mij. If we call Esource the vector whose components are the temporally uncorrelated signals’
field values E1 and E2, the input signal vector Ein to the principal component extractor is m E + m12 E 2 
Ein = M • Esource =  11 1
.
 m21 E1 + m22 E 2 

Equation 2.5

We assume the mixing matrix M is symmetric with m11=m22=0.9995 andm12=m21=0.0316 (these numbers preserve the intensity of each source signal and make the49intensity cross talk 0.1%). The right hand-side box of Figure 2.22 extracts the largest
principal component of Ein.
The smooth curve in Figure 2.23 represents the calculated signal intensity ratio in thelargest principal component versus the input pump intensity ratio E12/E22. The experimental
data points are the crosses in Figure 2.23.

The size of the “x” marks covers themeasurements’ uncertainty of ± 0.5dB.

Figure 2.23. Separation performance of the filter. The smooth gray curve shows calculated data points using thefirst principal component extraction model. The slope at the origin is 15 dB/dB. This is in contrast to
the experimental data points (in black) which show a slope at the origin of 26dB/dB.

The model provides a remarkably good fit considering that no photorefractive related
equations are utilized. We are comparing the performance of the filter to the purely algebraic
technique of principal component extraction.

Only the known cross talk between thechannels is physically accounted for. The slope at the origin of the theoretical curve is50

15 dB/dB corresponding to a small signal ratio enhancement of 32. For the experimental dataof Figure 2.23, this ratio is 26 dB/dB. The strict principal component extraction model“underestimates” the role of the competition for gain when input ratios are close to unity.

2. 4. 3 Theoretical arguments to prove the filter's PC extraction capabilities
2. 4. 3. 1

Model and equations

The auto-tuning filter’s complex dynamics led us to find a simpler model for its
theoretical analysis. We model our filter by a single-mode dispersionless cavity comprising
only the photorefractive gain crystal. The single-mode resonator has the same effect as ourreflexive coupling on the real cavity both allow only one spatial mode to oscillate. The
dispersionless quality replaces the highly multimode cavity.

Figure 2.24. Single mode cavity model of our autotuning filter.

The modeled filter depicted in Figure 2.24 has a two-dimensional spatial input as well asa two-dimensional temporal input. The optical pump beams are spatially orthogonal channels
that carry mixtures of two temporal signals. The ring brings one other spatial channel to the51system, orthogonal to the input channels. This section answers the question: What is the
temporal content of the ring mode given the temporal contents of the input channels?
The answer is that only the energy corresponding to the largest principal component of
the input space pumps the ring while the rest of the energy passes through it.

Wedemonstrate this with an input beam containing two spatial channels carrying two temporalsignals.

The photorefractive equations
As simple as our filter’s model looks, there are still 8-coupled variables to account for.
They all depend on both space and time. The numbers 1 and 2 refer to input spatial channel
information and the Greek letters α and β refer to temporal information. The variables are:- 2 gratings G1 and G2 in the photorefractive crystal: the two spatial pumps each write agrating with the ring’s spatial mode (we assume they do not write a grating with one another)- 4 variables to describe the pump beams’ temporal content: two variables Piα and Piβ(i=1,2) for each spatial pump-

2 variables Rα and Rβ to describe how much of each temporal signal is in the ring.

First we study the simplest of all input scenarios: temporally uncorrelated signals α andβ (they cannot write a grating together) each carried by a different pump. The classicalcoupled equations describing the photorefractive dynamics relative to our 6 field and 2
grating variables are:

52

 ∂

τ + 1G1 (z,t) = ∂t 
Γ**Rα (z,t)P1α (z,t) + Rβ (z,t)P1β (z,t)
2ΙΤ
 ∂

τ + 1G2 (z,t) = ∂t 
ΓRα (z,t)P2α * (z,t) + Rβ (z,t)P2 β * (z,t)
2ΙΤ

(

)

(

)

∂
*
P1α (z,t) = −G1 (z,t)Rα (z,t)∂z∂*P1β (z,t) = −G1 (z,t)Rβ (z,t)∂z∂*P2α (z,t) = −G2 (z,t)Rα (z,t)∂z∂*P2 β (z,t) = −G2 (z,t)Rβ (z,t)∂z∂R (z,t) = G1 (z,t)P1α (z,t) + G2 (z,t)P2α (z,t)∂z α∂R (z,t) = G1(z,t)P1β (z,t) + G2 (z,t)P2 β (z,t)∂z βEquation 2.6

where τ is the photorefractive time constant, Γ is the coupling constant of the crystal and IT isthe total intensity present in the medium.
After a short study of this case we will show how any input to the filter can be related tothis one. It is then straightforward to find the temporal content of the ring given any temporalcontent of the input channels.

2. 4. 3. 2

The simple case

The above equations are not, as far as we know, analytically solvable. To find thesolutions we consider, as in section 2. 4. 1, all possible steady states in the ring and analyze
them one by one. The options are an off/off state where no signal oscillates in the ring; twoon/off states where only one of the temporal features is present in the ring and finally an
53

on/on state where both temporal features oscillate in the ring. The latter may have more than
one solution if the ring can support different combinations of the signals.
The only parameters that are not input dependent are the available gain Γl (l is the lengthof the gratings) of the photorefractive crystal and the round trip losses. Clearly, the off/off
state occurs when the small signal photorefractive gain Exp[Γl] is smaller than the round trip
losses. This case is not of interest to us. The on/off states of the model filter were thoroughly
studied in the past with temporally and spatially orthogonal signals [62].

That paper

demonstrates that the only stable on/off state is the oscillation of the temporal signal
corresponding to the overall strongest pump.
We now claim that no on/on steady state is possible with our simple input scenario. The
on/on state exists if (but not only if) both signals can reach a unity round trip gain in the loop.
The gain of each signal in the ring, αGain and βGain, depends on the following parameters
and input variables: P1α(z=0) and P2β(z=0) (P2α(z=0)=P1β(z=0)=0 here), the intensity of the ring
signals Rα and Rβ, the value of Γl and the absorption losses.
For different pairs of P1α(z=0) and P2β(z=0), we numerically calculated the pairs of (Rα ,Rβ)yielding αGain=1 and those yielding βGain=1. Smooth gain curves like those of Figure 2.25
are obtained. However, no combination of values of P1α(z=0) and P2β(z=0) (the best casescenario being when they have very similar values), Γl and loss make the curves intersect.
This means that no non-zero linear combination of (Rα,Rβ) results in unity gain for both
temporal signals in the ring at once. The on/on state is not a steady state.

54

Figure 2.25. Gain curves for two spatially and temporally orthogonal signals.

Summarizing this paragraph, when the filter’s input consists of temporally uncorrelated
signals carried by different pumps, the circuit always allows only the signal corresponding to
the more intense pump to oscillate. The following paragraphs generalize this statement to all
spatially orthogonal mixtures of the uncorrelated signals on the pump beams.

2. 4. 3. 3

Spatially orthogonal mixtures

The photorefractive gain crystal in our model sees three spatial beams: one bears thering signal and the other two carry the input pump signals. These may be represented as threespatial ports and their contents written as the components of a vector. Different vectors areused for different temporal signals. With the notation from the photorefractive-coupledequations, we have two vectors Vα and Vβ:

55

R 
R 
α
β



Vα =  P1α  and Vβ =  P1β .


P 
P 
 2α 
 2β 

Equation 2.7

So far, we have studied input vectors of the shape:

R  Rα β  and V =  0 ,
Vα = P1α

β 
 


P
0
 
 2β 

Equation 2.8

and concluded that when we require a unity gain on the first port, either Rα or Rβ goes to zero
whether P1α or P2β is weaker respectively.
Now let us rotate the axes of the pump subspace around the ring mode axis (the first axishere). In the new basis, the sub-vectors’ expressions (P1α,0) and (0,P2β) of Vα and Vβ, willchange to:

 R



Rα
β




Vα =  cos θ ⋅ P1α  and Vβ =  sin θ ⋅ P2 β ,
−sin θ ⋅ P 
cos θ ⋅ P2 β 

1α 



Equation 2.9

where θ is the angle of rotation of the subspace around the first axis. By construction, thisrotation leaves any vector that lies in the plane perpendicular to the rotation axis, in that sameplane. Any vector that did not previously have a component along the rotation axis will still
not have a component along it, in our new basis. In other words, it still holds true in this56

basis that our steady-state vector will show only one temporal feature along the ring modeaxis.

It is straightforward to reverse the reasoning that if we start with a spatially orthogonal
mixture of the uncorrelated temporal signals on the pump beams. We can just rotate thespatial pump subspace so that the two temporal signals at the input lie along separate axes
(this is the eigen subspace of the pumps). In this space our “simple case” study holds and we
can once again conclude that the ring mode’s temporal features will correspond to only thoseof the stronger signal.

2. 4. 3. 4

Non spatially and temporally orthogonal mixtures

What are the temporal features of the ring beam if the two channels carry a linear
mixture of some signals Sa and Sb that are somewhat correlated?
The previous paragraphs determined that the filter extracts what it sees as the stronger of
two spatially and temporally orthogonal signals. This paragraph establishes that those signals
are none other than the principal components of the input space.
First we establish that they are indeed spatially and temporally orthogonal. Let us
represent our input space with the vector P. The rows describe the temporal contents of input
channels 1 and 2 and the symbols Sia and Sib represent how much of each temporal signal ison each channel:

 Ch   S + S1b 
P =  1  =  1a.
Ch2   S2a + S2b 

57

Equation 2.10

By definition, the principal components are the eigenvectors of the following correlation
matrix ρ: Ch1 ** ⊗ Ch1 Ch2
Ch2 
 Ch 2Ch1Ch2* 
1
t
t

=
.
2
 Ch2Ch1*

Ch
2



(

ρ =

)

t

Equation 2.11t

The eigenvectors are orthogonal by construction. In our case this ensures that they arespatially orthogonal. The particular shape of the correlation matrix also ensures that they aretemporally orthogonal. In general, the correlation matrix for complex variables may be
written as:

 m2
ρ =  * nm

mn * 
2 ,n 

Equation 2.12

where m and n in our case are the temporal signals on the two input channels.

The

2

2

eigenvalues of ρ are always 0 and 1 (provided the matrix has been normalized by m + n )and the eigenvectors written in the channel basis are: n− m 


 1 

and

 m
 n .
 
1

58

Equation 2.13

The temporal correlation of the PCs is then:

 n
 m

PC1 ⋅ PC2 t = − m + n  m + n 
 m
 n
 t
=−

n m
n
m
.
m m −
mn + n m + nn t
m n
m
n
t
t
t

Equation 2.14

=0

Are the PCs the only possible spatially and temporally orthogonal signals obtained froma linear combination of the original input signals? Let us suppose there is another spatiotemporal orthogonal pair (L1, L2) that is also a linear combination of the input signals. The
spatial orthogonality constraint means that this new set can be written as the rotation of thePCs plus a possibly different scaling factor, G and H, on each vector:

L1 = G(cos θ ⋅ PC1 + sin θ ⋅ PC2 )L2 = H(−sin θ ⋅ PC1 + cos θ ⋅ PC2 ).

.

Equation 2.15

Are these combinations still temporally orthogonal? Using Equation 2.14 we calculate thecorrelation product:
L1 ,L2 t = G(cos θ ⋅ PC1 + sin θ ⋅ PC2 ),H(−sin θ ⋅ PC1 + cos θ ⋅ PC2 ) t .

(= GH cos θ sin θ PC12t

− PC22t).

Equation 2.16

This product is zero only if θ is zero (no rotation) or if the PCs have the same energy. The
latter is a degenerate case we have also observed in our experiments. When the two input
signals have equal intensity the auto-tuning filter is unable to choose one over the other.

59

2. 4. 3. 5

Conclusion

To summarize the last few sections, the temporal signal that oscillates in the filter’s ringin steady state has the same features as the temporal content of the largest principal
component of the input signal space. If the eigenvalues happen to be degenerate the temporalcontent of the ring may be any linear combination of the principal components. In the nondegenerate case, the largest principal component may be expressed as a linear combination of
the original signals Sa and Sb by finding the coordinates of the principal components in thechannel basis. The channels’ temporal contents are then themselves a linear combination of
the signals.

60

CHAPTER 3
AN OPTICALLY SMART ANTENNA ARRAY

3. 1

Objective

3. 1. 1 Blind Source Separation
When you are in a room full of people your brain has no problem hearing many
conversations at once but tuning in to only one. Without much effort, your brain solves what
is known as the “cocktail party problem.” The common perception of television or cell phonechannels is that more of them require more bandwidth: they each need their own separate binof frequencies. What if all the channels shared the same frequency bin and we possessed a
black box that could hear them all at once while listening to any one of them on command?
This black box performs what is called blind source separation (BSS). It is capable ofdistinguishing among a multitude of completely unknown incoming signals that may overlap
in frequencies; just as our brain does with audio signals at a cocktail party.
There are various algorithms that solve the problem of blind source separation to somedegree. In the last decade a method called independent component analysis (ICA) has
emerged as a powerful method for blindly separating signals [65]. It is a computer-intensive
method that relies on the statistics of the signals. Its real-time application has mostly been
limited to the blind separation of speech type signals; simultaneous processing of multiple

61high-bandwidth signals is exceedingly challenging for digital signal processing approaches[66]. Our group is currently working on an opto-electronic implementation of one particular
ICA technique. We see no obstacles to operate our system with megahertz bandwidth signals
in real time. The first part of the ICA algorithm involves separating the principal components
of a signal mixture. Chapter 2 described in length and demonstrated that we have an opticaldevice capable of doing principal component extraction.

The primary objective that

motivates the work presented in this chapter is the assessment of the design issues andtradeoffs associated with the incorporation of holographic optical signal processing in a
microwave receiver. If the incorporation of the autotuning filter in such a system can bedemonstrated as practical, then it seems reasonable to expect that the technology will alsoenable the practical realization of the optical circuitry required for the higher-order signal
processing associated with ICA.

3. 1. 2 A portable prototype system
In the following pages, we demonstrate the technology to make a miniature (but not
integrated) optical system robust and portable. This chapter describes the making of, andevaluates the performance of a smart antenna array with optical adaptive processing. The
array operates at a 10 GHz RF carrier and a 140 MHz IF signal. The input wave is an
unknown superposition of a number of uncorrelated signals. These signals are incident fromdifferent directions and possibly have changing spatial and temporal characteristics. The
output electrical signals are adaptively separated and ordered according to their respective
signal strengths.

This function is accomplished with the autotuning filter, which iscompletely internal to the system. The processor and associated electronics are packaged in astandard-size aluminum briefcase with a single external power plug, and consumes less than
50 W of CW power.
62

Overview of the system
The first stage of our prototype system, schematically shown in Figure 3.1, receives two
10 GHz signals from two distinct sources placed in its far field. Audio signals frequencymodulate the microwaves. The receiving front end consists of a 30-element antenna arraythat acts as a quasi-optical discrete lens with the detecting antennas placed on its focal arc.
The two electrical signals from the detectors are then down converted to 140 MHz andamplified before feeding the second stage of the system.

Source 2

Source 1

QO
Array

Optical carrier
suppression

Receivers andEO modulator
Pre-processing
at RF carrier

Signal 2

Signal 1
Adaptive
Opticalprocessor

Figure 3.1: Diagram of the prototype optically smart antenna array.

The IF signals are applied to a one crystal, two-channel resonant electro-optic phasemodulator. An optical line beam comes out of the electro-optic modulator with its top half
modulated with one of the IF signals and the bottom half modulated with the other. Since
this is phase modulation the original optical frequency is still present in the beam,necessitating a carrier suppression element in order for the third stage of the system to work
properly. The suppression of the optical carrier is achieved by two-beam coupling the signalbearing beam with a non-signal bearing beam in a photorefractive crystal where the former

63

beam is the pump (loss port). Although the signals suffer a 3dB loss, the carrier frequency issuppressed by over 50 dB.
The final processing stage consists of the autotuning filter, a 5 cm2 photorefractive ringoscillator. Two barium titanate crystals in the oscillator combine to extract the principal
component of the input signal space, in this case the stronger signal, while passing on theother components.

The oscillating beam is sampled and photodetected.

Demodulating

electronics retrieve the audio signal (originally imposed in the 10 GHz microwave
transmitter) from the AC port of the photodetector, which is then sent to a speaker.

64

3. 2

The microwave front end

3. 2. 1 The transmitters
Although we process 140 MHz RF, we transmit 10 GHz microwaves across space forthe practical reasons that it makes the far field of the source much closer in space, and greatly
reduces the dimensions of the receiving antenna arrays well as the space needed todemonstrate the system! Figure 3.2 is a photograph of the battery-operated transmitters. The
horn antennas as well as the transmitters’ circuitry were designed and built in the Active
Antenna Lab (Prof. Z. Popovic’s lab).

Figure 3.2. Photograph of the battery operated transmitters. The CD player provides the audio signal thatmodulates the transmitted microwaves.

Modulation scheme
Portable CD players generate our audio signals (we use only the right or left ear
channel). The relatively slow changing voltage of each audio signal is amplified and drives a~100 MHz Voltage Controlled Oscillator. A mixer imposes this frequency modulated signal
as sidebands on a ~10 GHz high frequency signal from a Dielectric Resonator Oscillator.
65

The output of the mixer is amplified up to 200 mW before being emitted by the hornantennas. Figure 3.3 illustrates the transmitter's circuit diagram and the spectrum that they
emit.

Figure 3.3. Diagram of the circuit that drives the horn antennas.

3. 2. 2 The quasi optical antenna array
3. 2. 2. 1

Principle of operation

The lens antenna array shown schematically in Figure 3.4 is a quasi-optical array
analogous to a Rotman lens [67, 68]. It is a passive discrete lens with 30 patch antenna
elements, in which the lensing is accomplished with varying delay lines across the array
between input and output antenna elements. The left photograph shows the patch antenna
elements fed at the non-radiating edges and the microstrip delay lines, which are connectedwith via holes to the orthogonally polarized patches on the other side of the two-layer lensarray.

66

Figure 3.4. Photographs of both sides of a 30-element X-band two-layer quasi-optical lens antenna array withpatch antenna elements. The one on the right also shows the ground planes of the receiving antennas placed on
the focal arc of the lens array.

Figure 3.5 Measured (solid line) H-plane (on the left) and E-plane (on the right) radiation patterns at 9.9 GHz
have higher sidelobes than the simulated patterns for a uniformly illuminated array (dashed line). This is due to the
non-uniform amplitude and phase associated with the near-field spatial feed. The simulated patterns are calculatedusing the measured radiation pattern of the patch element673. 2. 2. 2

Antenna data

The antenna is designed for 10 GHz, with a few percent bandwidth (hundreds of MHz).
The F-number of the lens is 0.6, corresponding to a focal length of 7.5cm. The array lattice istriangular, with a period of 0.65λ0, chosen to minimize grating lobes when the beam isscanned. The measured radiation patterns of the lens fed at the focal point with an open Xband waveguide feed are shown in Figure 3.5.
The lens has a number of imperfect focal points, and in our project we use two of thesepoints along the H-plane focal arc to place receivers, each preferentially receiving one spatial
beam. The photograph on the right of Figure 3.4 shows the back of two receiving patch
antennas placed at those focal points. The two transmitters are placed in the far field of thelens array symmetrically on either side of the array's normal. Each receiver preferentially
receives from one of the sources, but some signal from the other source is also present, andreferred to as the “cross-talk” signal. The relative amount of the cross talk depends on thespatial separation between sources as well as the radiation pattern of the lens.

3. 2. 3 The active down conversion stage
There are two reasons we need a down conversion stage in our prototype system. One
the autotuning filter it is built around processes up to 3 GHz signals, not 10 GHz. Second, themicrowave signals are transferred to optical beams by phase modulation. We found nocommercial electro-optic modulator (EOM) for green light small enough to fit our size
criteria. Since phase modulation was not the focus of our project we built one in the lab but
for lower frequencies to simplify the RF circuitry.

68

Figure 3.6. Down converting circuit schematics. The amplifiers imediately following the receiving patch
antennas are LNAs.

Figure 3.6 presents the down conversion box's schematics. The receiving patch antenna
feeds are directly connected by coaxial cable to low noise amplifiers (LNA) at the input ofthe down conversion box. The LNAs are designed for optimal noise figure (CHA2063
United Monolithic Semiconductors, with a 7-13 GHz bandwidth, 16 dB gain and 2 dB noise
figure). The output power for the LNAs at 1 dB-compression point is 10 dBm, with low DC
power consumption (40 mA at 5 V). A mixer down converter follows the LNA. The IF
amplifiers are CLC522 National Semiconductors wideband variable gain amplifiers and they
provide more than 40 dB gain control through a single high impedance voltage input. The
ability to control the levels of the signals at the input of the optical processor is important for
proper optical processing, as discussed at the end of this chapter. The outputs of the IF
amplifiers drive power amplifiers, the 0 to 30 dBm outputs of which drive the electro-optic
modulator.

69

3. 2. 4 Demodulation of the audio signals
The down converted IF signals phase modulate an optical beam so that they may be
optically processed. The proper functioning of the autotuning filter requires that the opticalcarrier be suppressed from the beams. This brings one advantage and one drawback to theprocess of retrieving the original signals.

With no optical carrier the two microwave

sidebands combine to produce an amplitude-modulated beam at twice the IF frequency. This
modulation is easily detectable with a fast photodetector. There is no need for a heterodyne
detection.

For demonstration purposes, the original microwave signals were frequency

modulated with audio signals. The frequency-doubled audio at the output of the filter cannot
be sent to speakers without hurting the audience’s ears. Figure 3.7 shows the diagram of thecircuit that was used to retrieve the original audio signal by performing a square root
operation on the detected signal. The doubled IF frequency is first divided by four with aprescaler to give a signal that is half the IF frequency. It is then compared with the halved
output frequency of a voltage-controlled oscillator that locks onto the original IF carrier
frequency via a phase lock loop. The filtered output of the comparator, which drives the
VCO in the PLL, restores the original audio signals.

Figure 3.7 The demodulator uses a PLL to lock to the doubled IF frequency. The loop has a frequency divider
that restores the original audio signals that may then be sent to speakers.

70With only one transmitted signal, the circuit performed well with a high pass filter (>200MHz) and an automatic amplitude gain control module inserted after the photodetector.
Unfortunately the PLL did not do well at extracting both transmitted audio signals when the
power-contrast between the two IF carriers at the output of the detector was less than 30 dB.
This was the case when the antenna array received the transmitted signals with similar
powers and the system was most convincingly useful.
The audio modulation in the system was added to the demonstration to enhance theaudience’s understanding of the system’s separation performance. Unclear audio signals at
the crucial moment of the demonstration defeated the purpose. Instead, we chose to show theaudience the spectrum of the IF signals at the output of the photodetector. To allow the realtime demonstration of the system we filmed the screen of a hand held spectrum analyzer(Bantham model) with a camcorder and projected its output through a laptop.

71

3. 3

The EOM and carrier suppression

3. 3. 1 A two channel EO
The 100 MHz IF signals are encoded as phase-modulation sidebands on an opticalcarrier using a single electro-optic crystal modulator as shown in Figure 3.8. The optical line
shaped beam passes through a magnesium doped lithium niobate crystal with a pair ofhorizontal microstrip electrodes. The other side of the crystal has a corresponding pair ofgrounded microstrips. The top and bottom halves of the beam are spatially modulated withthe temporal signals of the two electronic channels.

Figure 3.8. Electro-optic modulation: a channelized electro-optic modulator made from magnesium doped
lithium niobate crystal (Mg:LiNbO3) imposes spatial and temporal modulation on the optical carrier.

The electro-optic crystal is a thin slab (0.3 by 7 by 35 mm) where the optical facets havebeen AR coated in order to minimize back-reflections. The two signal electrodes are two
strips, approximately 3 mm wide and separated by about 1 mm. Our first idea to apply theseelectrodes was to coat these electrodes directly onto the crystal. We evaporated 50 nm ofchromium and then gold by electroplating for increased thickness. Something in the processrendered our crystal 90 % opaque. After a couple of trials we abandoned the idea of coating.
The electrodes were finally etched out as a pattern on copper covered plastic sheets and72

simply mechanically pressed against the crystal. This mount that also houses the electronic
matching circuit is detailed in Appendix 7. 4.

3. 3. 2 Resonant matching design
At the operating IF frequency of 140 MHz, the modulator is a lumped-element circuit;the microstrip transmission lines are only a few electrical degrees long. The electro-optic
crystal and electrodes are modeled as two capacitively coupled capacitors. The measured
capacitance of the 3 mm-wide electrodes is 53 nF. The resonating circuit that matches theelectrodes to 50 ohms and cancels the coupling capacitance is drawn in Figure 3.9.

Figure 3.9. The two electrodes of the EO are modeled by a resistor and a capacitor. They are matched to 50 ohms
by RC circuits. An LC element cancels the coupling capacitance between the electrodes.

Independently of the suitcase project, a four-channel EO was built using a crystal withthe same dimensions. The complexity of such a device lies in the design of a matchingcircuit capable of canceling coupling capacitances between adjacent and non-adjacentchannels. The built device had –10 dB cross talk between the electrodes.

733. 3. 3 EO data
3. 3. 3. 1

Half-wave voltage Vπ

The modulation efficiency has been measured to be about 0.2 rad/V for both electrodes,
fed separately with a narrowband matching circuit. This gives a Vπ of 15.7 Volts. The value
is close to the theoretical Vπ calculated from the material constants of LiNbO3 and thegeometry of the cut. The formula is

Vπ =

d λo
L r n3where the thickness d equals 0.3 mm, the length L equals 30 mm, the wavelength λo equals
532 nm, the index n equals 2.2 and the electro-optic coefficient r equals 30.8⋅10-12 m/V.
These values yield a theoretical Vπ of 16.2 Volts.

3. 3. 3. 2

Cross talk

The cross talk between the two electrodes, due to RF leakage, measured with a HP8510
Network Analyzer, is –24 dB. A more pertinent way to measure this cross talk is to measureeach channel’s spatial distribution of modulation.
suppression unit with a photodetector.

We scan the beam after the carrier

Direct detection is possible because the phasemodulation without the carrier becomes an amplitude modulation. The scanning results areshown in Figure 3.10. The two curves on the top plot show the spatial distribution ofmodulation for each electrode turned on one after the other. The bottom plot shows theintensity of the beat frequency along the beam when both channels are turned on. The level74

of this beat frequency stays low compared to the levels of modulation of the desired signals.
This indicates that the cross talk between the channels has been efficiently canceled.

Figure 3.10. Modulation profile at the output of the two-channel electro-optic modulator along the optical beam.
The top graph shows the relative modulation strength of each channel along the 7 mm of the line shaped
beam that starts at graduation 100=1 mm and ends at 800=8 mm. The channels are driven one at a time.
The bottom graph measures the power of the signal that has the beat frequency between the twomodulation signals in the same way (so that the graphs‘ values are comparable) but with both channels
turned on at the same time,.

3. 3. 4 The carrier suppression unit
The autotuning filter distinguishes various signals by correlation. The optical carrierthat remains after phase modulation is common to all the signal channels and therefore
introduces an unwanted false correlation.

We implement an optical carrier suppression

75system for this reason. Carrier suppression is accomplished through two-beam coupling
within a barium titanate photorefractive crystal and uses the principles of an optical novelty
filter [2]. The beam coupling interaction takes place between two beams coming from thesame laser, one of which has been modulated with the IF signals by the EOM as
schematically illustrated in Figure 3.11. As a result of this interaction the modulated beam istransmitted from the crystal with the optical carrier largely suppressed, while the other beam
has increased carrier power and also a portion of the IF signal power.

Figure 3.11. Two-beam coupling in a photorefractive barium titanate crystal (BaTiO3) suppresses the opticalcarrier.

Two-beam coupling coherent beams that have the same frequency, transfers the energyof one beam to the other through a holographic grating that these same beams induce in the
crystal. We refer to the beam gaining energy as the "plus" port and to the beam losing energy
as the "minus" port. The amount of energy transferred depends on two parameters: the gainof the crystal, G, and the power ratio between the two input beams. Each of these twoparameters, in turn, depends on certain geometrical parameters, such as the angle between the
beams at which they are incident upon the crystal.

76

When one beam consists of multiple frequencies, as in the carrier suppression case, theprocess becomes more complex. However, it is this process of energy transfer that allows us
to suppress the carrier from the modulated beam.

A complete treatment of the carrier

suppression problem has been published in [69]. Here we present a short and intuitive
version of the theory relevant to our system. We simplify it further by assuming we have asinusoidal modulation.

Principle of photorefractive carrier suppression
The dynamics of two beam coupling with a purely real coupling constant occurring
between two single mode beams may be pictured in an abstract two dimensional field space
as illustrated in Figure 3.12. The horizontal axis represents the fields in the minus port, andthe vertical axis represents those in the plus port. Each vector in this space holds all theinformation about all the fields at the same frequency. In the case of the carrier suppression
scenario we have two vectors: the one that describes the optical carrier frequency Eωc, hascomponents along the minus as well as the plus ports. The vector related to the modulation
frequency Eωm has a non-zero coordinate only along the minus axis.

Ignoring bulk

absorption, the energy transfer from the minus port to the plus port is a simple rotation. The
remarkable fact is that the value of the rotation angle is the same for all the vectors (under thecondition that the frequencies are within the photorefractive grating’s bandwidth). The angle
depends only on the input vectors in a highly nonlinear way. If λ1 and λ2 are the eigenvaluesof the density matrix of the input vectors defined by

ρ=

(

)

(

)

Iω c
Iω
+
+
Eω c ⊗ Eω c + m Eω m ⊗ Eω m .
Itotal
Itotal77

The rotation angle may be expressed as

θ = Arc tan[e( λ − λ )Γl tan θ o ],12where θo is the input angle of ρ’s eigenvector associated with the largest eigenvalue λ1, Γ isthe coupling constant and l is the length of the grating.

This formula describes how

photorefractive two-beam coupling transfers energy such that the eigenvector associated withthe largest eigenvalue tends toward the plus axis. Only infinite gain perfectly aligns the two.

Figure 3.12. 2D representation of single mode two-beam coupling. (a) In the case of carrier suppression the input
to the minus (“-”) port is a modulated laser beam while the input to the plus (“+”) port is a nonmodulated beam from the same laser. (b) Nonlinear coupling between the beams causes the energy inthe carrier at the minus port to be transferred to the plus port at the output.

In our carrier suppression picture our largest eigenvector lies somewhere between Eωcand Eωm. With enough gain (i.e., a long enough grating) the rotation representing the energy
transfer will align Eωc along the plus axis as illustrated in Figure 3.12.b. At that point, Eωchas no component along the minus axis; it is “perfectly” suppressed from the initially78

modulated beam. The picture also shows that this suppression comes at the cost of losing
some of Eωm's energy from that beam. The quantitative version of what we just describedshows that the gain to achieve this carrier suppression is minimum when the intensity at theplus port initially equals the intensity at the minus port no matter what the modulation depthof the modulated beam is. The value of this minimum gain decreases as the modulation depth
increases. This fact makes the experimental implementation of this photorefractive carrier
suppression very convenient:

the intensity ratio between the non-modulated and themodulated beam only needs to be adjusted once. The other convenient fact about this way of
suppressing the carrier is that photorefractive crystals adapt to slow drifts of the laser beam's
frequency. This is a major advantage over Mach-Zender configurations.
Very careful experiments resulted in 70 dB of carrier suppression. This means that thecarrier on the modulated beam was decreased by seven orders of magnitude resulting in thebest carrier suppression reported in the literature to date. Not so careful alignment of thephotorefractive carrier suppression still yields 50 dB suppression. This was enough, in ourcase, to dwarf the carrier compared to the modulated signals at the input of the autotuning
filter. Figure 3.12.b shows that rotating all the vectors by the 45 degrees, required for best
carrier suppression, adds 3 dB of loss on the modulated signals. This is physically explainedby the fact that even though the signals cannot write a stable grating with the carrier, they canread a grating written by the two carriers (a 0.5 cm finite length grating has hundreds of GHz
bandwidth.) Some of the signals’ energies are therefore lost to the plus port.

79

3. 4

Design of the optical stage

Building a miniature optical system involved resolving two main types of problems.
One was designing the optical layout with space constraints. This is what I called opticalmode matching: obtaining a specific beam size and divergence in a specific place. The
second type of problem was purely mechanical. Miniature systems are not common enoughthat optics companies carry a wide assortment of miniature mounts and subsystems. Many ofthe mechanical parts, that are usually pulled out of a drawer or ordered from a catalog, had tobe replaced with “homemade” versions. Most of these mechanical designs are described in
the appendices.

3. 4. 1 The optics
3. 4. 1. 1

A line shaped beam

A natural choice for the system’s optical set up would be to mirror the electronics layoutby splitting the laser beam in two and propagating these two channels all the way to the
autotuning filter. We chose a single vertical line-shaped beam instead. This facilitates
coupling in and out of the electro-optic crystal and simplifies suppressing the carrier fromtwo beams at once. The geometry is shown in Figure 3.11 above. This arrangement also
avoids having to use two sets of lenses any time the beam needs shaping. A 30-degree line
generator (Edmund Scientific), shown in Figure 3.13, creates our beam.

Figure 3.13: These line generators create a 30-degree fanning beam that is 0.8 mm wide.

80The line generator is followed by a horizontal cylindrical lens that collimates the beam’s
height through out the system to 6 mm in order to fit in the LiNiO3 EO crystal. The
photorefractive crystal configured for carrier suppression (45-degree cut based on the group’s
past experience with novelty filters) is cut to accommodate this dimension. It also has
Brewster windows that are oriented to reduce reflections for the modulated beam.

3. 4. 1. 2

Mode matching with 3-lens subsystems

Certain elements in our system require very specific beam sizes. For example, the line
generator needs a 0.8 mm focused beam, and the EO crystal requires a beam less than 0.3 mm
wide over a distance of several centimeters. A gaussian laser beam is fully characterized, at
any point in space, by its waist W and radius of curvature R. These two parameters areclassically embodied by the complex parameter q defined byq(z) =

1
1+.
R(z) iW(z)This is particularly convenient because propagating q through a simple lens system mayeasily be done with the ABCD matrix formulation. The formula relating the input and output

q parameters is

qout =

Aqin + B
C qin + D

81

Equation 3.1

Transforming the input q into the desired output q using three lenses gives theoretically
7 adjustable parameters.

Three parameters determine the focal lengths.

Four otherparameters define the positions of the lenses and the location at which the beam has thedesired input and output q. Our constraints are the total length between the locations of the
input and output q, and two equations provided by the real and imaginary parts ofEquation 3.1. It may seem at first that we do not need 7 adjustable parameters when we have
only 3 constraints. However the three focal lengths are only partially adjustable since wewished to use common lenses available through catalogs. The choice of three lenses was
simply a good compromise between the number of elements and the time spent optimizing
their positions. Figure 3.14 shows a picture of a three-lens subsystem. The lenses are placedin cylindrical mounts that have the exact same outer diameter. This enables a single v-groove
to keep them all centered, while allowing one degree of translation for position adjustment.

Figure 3.14: 3 lens subsystem. The lenses are placed in cylindrical mounts that have the exact same outer diameterto keep the lenses centered relative to each other. The v-groove allows only one translation.

82

3. 4. 2 The mechanics
The base plate and the clamping mechanism
For our prototype system to be robust and portable enough to travel, we had to ensure
the good alignment of the optics was solidly fixed and that the system functioned properly
when not on an optical cushioned table. The first problem was solved with an original
clamping system from underneath. The mechanism resembles an upside down table clamp.
The left of Figure 3.15 shows a photograph of the clamps. Each subsystem has a 1/4-20
screw attached to it that goes through the base plate. The clamps pull on a ring that threads
onto this screw as illustrated on the right of the figure. Designing a simplified miniature
optical table solved the second problem. It decouples the optical stage from any relevant andreasonably expectable vibrations the suitcase could be submitted to. Vibration modes of theplate itself, which is a one big piece of steel, are avoided by carving out a girder structure outof the bottom and all the holes for the screws mentioned above help too. Outside vibrations
are cut off by isolating the plate from the rest of the suitcase with foam feet. The plate is heldin place with c-shaped metal ribbons (see appendix 7. 5 for details).

Figure 3.15: Photograph and picture of the inverted table clamps. A 1/4-20 screw with a threaded ring attach to
every subsytem on the base plate. A u-shaped piece with an integrated screw pulls down on the ring
thus immobilizing the sub sytem on the base plate.

83Layout of the optics
Figure 3.16 shows the 8x11 inch optical stage. The laser is a frequency doubled solidstate laser emitting 150 mW of continuous power at 532 nm (Compass 315M-150 from
Coherent).

Figure 3.16: Photograph of the 8x11 inch optical stage.

The first three spherical lenses after the laser mode-match the laser beam for the line
generator. Three more cylindrical lenses shape the beam for the EO crystal housed in the box
connected to the two coaxial cables bringing in the IF signals. A beam splitter divides thelaser beam in equal parts before one half enters the EOM. The two beams are then coupled ina photorefractive crystal so that the carrier is suppressed from the modulated beam. The last
set of lenses and the half wave plate prepare the beam for the autotuning filter’s input. The
84

filter sits sideways under the orange roof that absorbs reflected or scattered green light.
Finally, a 90-degree prism sends the output of the filter to a fast 300 MHz photodetector
placed under the base plate. The laser’s control electronics and power supply also hide
beneath the plate.

85

3. 5

System performance

3. 5. 1 The suitcase
Figure 3.17 is a photograph of the adaptive antenna processing system. The system ispackaged in a 13x18x6 inch aluminum briefcase with a single power plug. The system(mostly the laser) consumes a maximum of 90 W of power at start up.

The power

consumption drops to 50 W after 4 minutes.

Figure 3.17: Photograph of the “suitcase project.” The two coaxial cables visible in the top right corner receive
electrical signals directly form the patch antennas placed on the focal arc of the antenna array. The
bigger gold-colored box contains the LNAs and the down converting electronics. The output is
amplified by power amplifers sitting under the visible cooling fan (there is another one under the86

optics). The two electronic channels then drive the EOM in the optical stage to the left of the picture.
The suitcase measures 13x18x6 inches when it is closed and plugs into a regular wall outlet,
consuming less than 50 W of power.

The electronic portions of the system sit on the right side of the case, while the optical
portion sits to the left of center. The laser power supply is visible at the far left, while thepower for the electronics is distributed from the gold box with stripe-shaped vents in its
cover. A fan under the optics combined with a fan on top of the power amplifiers create
enough of an air flow to keep the suitcase’s elements cool.
Figure 3.18 shows the system as it was tested end-to-end. The processor sits towards theleft of the photograph, with the antenna lens array sitting just to its right. On the right are thetwo transmitters with their horn antennas. The two transmitters are placed symmetrically
about the antenna array boresite.

Figure 3.18: Configuration of the system as it was tested in the laboratory.

87

3. 5. 2 End-to-end characterization of the prototype
The outputs of our two-channel system do not, in general, recover the original source
signals in the face of multi-path and other interference effects. We built this prototype
around the autotuning filter and not around the full circuit performing independent
component extraction for reasons mentioned in the introduction of this chapter. The output of
our adaptive processor always provides the principal components of the signal space. The
principal components are proportional to the two transmitted signals if and only if three
conditions are satisfied: 1) the sources are independent, 2) they produce different received
powers and (3) the signals incident on the optical processor correspond to spatiallyorthogonal signals. Condition 1 is always satisfied in practice unless some spurious effectcauses an apparent correlation (such as the presence of the optical carrier, in our case). The
limits imposed on our system by condition 2 are already revealed by the signal extraction
curve of the auto-tuning filter, which shows a finite slope through the origin, discussed in theprevious chapter. This requirement reflects a fundamental limitation of principal component
analysis. Condition 2 can be relaxed with a more sophisticated optical circuit capable ofhigher-order (than correlation) signal processing. The fact, though, is that this requirement isusually satisfied in practice. Condition 3 is another matter. It reflects a particular limitationof our holographic circuit that needs a more complicated optical circuit geometry to be
overcome. As it stands, condition 2 is the most confining one and is the focus of the end-toend characterization of the system.

We assessed the performance of the system by taking signal extraction curves
reminiscent of the enhancement of the auto-tuning filter (see previous chapter) for threedifferent angles between the two transmitters. Figure 3.19 provides a context in which todescribe the measurements. In general, each of the two detector outputs will be composed of88

a linear superposition of the transmitter signals.

Since the transmitted signals areuncorrelated, one can write:

R1 = R11 + R12
R2 = R21 + R22

where Rjk is the received signal power from transmitter Tk after down conversion to the IF.

Figure 3.19. Diagram defining the context of the end-to-end system measurements.

With a large angle between the transmitters, the received power R1 at the IF on one channelof the receiver is primarily from transmitter T1 while the received power R2 at the otherchannel is primarily from transmitter T2. In other words, for equal transmitted powers R12
and R21 are small compared to R11 and R22. We measure a cross talk between receiving
channels by setting the transmitter so that the received powers R11 and R22 are equal and thendetermine C j = R jk / R jj ( i ≠ k ). From the symmetry of our experimental arrangement, we
expect the two values of cross talk to be the same, C1 ≈ C2 ≡ C . Table 1 shows this crosstalk C for three different angles. At the largest angle of 48° between the transmitters the
cross talk is –15dB. At the smallest angle of 20° it is 0 dB.

89

In this experiment, as in the previous chapter, we optically sample and detect only thepower in the filter’s ring, which provides the higher power of the two principal components.
We define the input signal power ratio as R11 / R22 . We define the output power ratio

S1 / S2 , the ratio of detected power corresponding to the temporal signals of the twotransmitters T1 and T2. Ideally this output ratio would be zero or infinity, depending one
which signal was larger, that received from T1 or that received from T2 but otherwise
independent of the input ratio.
Qualitatively, we can expect that the principal components will become more and more asuperposition of the two source signals as the transmitters are moved closer together inangular space. When they are at the same angular position, there is nothing available to theoptical processor to distinguish one source from two sources. Figure 3.20 plots the measured
output signal ratio versus the input ratio for three different angles between the transmitters.
The two curves for the larger angles have a characteristic “S” shape. When the received
signals have identical powers they are indistinguishable by the auto-tuning filter (this isbecause the two eigenvalues of the signal correlation matrix are degenerate). However, evena small power difference can provide the needed distinction. Of particular interest therefore
is the slope near the origin of the curve. For the largest angle of incidence it is 76 dB/dB.
This means that if the signals differ by 0.04 dB at the input to the auto-tuning filter, their ratio
is enhanced to about 3 dB at the output. Once the output ratio is above 10 dB or so, theenhancement begins to level off. With the smaller angle between the transmitters of 35° theslope at the origin of the “S” curve becomes much shallower. The value of this slope is 6.1
dB/dB. For an angle of 20° the “enhancement” is smaller with a value of about 1.3 dB/dB.
At this latter angle we note that the cross talk is 0 dB, meaning that all four IF powers Rjk areabout the same size for equal transmitted powers. It would seem for this smallest angle thatthe two transmitters are unresolved by the antenna and therefore their two signals should be90

Figure 3.20. Signal extraction properties of the adaptive system parameterized by the mutual angle between thetransmitters.

Angle between

ReceiverMaximumtransmitters (α)

cross talkslope enhancement[degrees]

[dB]

(dB)

48

-15

76

35

-10

6.1

20

0

1.3

Table 1. Cross talk and enhancement slope of the antenna system for three different angles between thetransmitters in the far field.

91

indistinguishable from one. However, the analysis in terms of power omits the important roleof the signal phase and this is the reason there is still some observed signal contrast
enhancement, however small. Overall the performance of the system at separating out thetwo source signals degrades gracefully as our mixture of signals becomes less orthogonal.

92

3. 6

Remarks

The role of the antenna lens array in our system design is subtler than might first be
apparent. It is true that the array performs something akin to a spatial Fourier transform on
the signal space; roughly speaking, it converts an angle of arrival to a spatial region in the
focal surface. In this sense it does some preprocessing of the incoming signals. If theincoming signals are known to have well-defined and distinguishable directions, the adaptive
processor is less relevant. The focal-surface detectors already contain fairly well separated
signals. That does not help in cases of multipath and other forms of interference. In principle
it makes no difference to the auto-tuning filter whether or not this “Fourier” transformation of
the signals takes place. The actual practical benefit of the antenna lens array enters throughdynamic range considerations: in a typical communications scenario one can expect thepower incident on the antenna lens array to be more or less uniform over the array, while thepower at the focal surface varies by a larger amount. Consider two angularly well-separated
sources for example, one giving rise to a received power of 1 W, and the second giving rise toa received power of 0.1 W. At the array, each antenna element receives the same power of
1.1/N watts, where N is the number of array elements. At the focal surface, one detector
receives 1.0 W while the second receives 0.1 W. One can clearly amplify the signal from thesecond detector so that the output powers from the front-end channels are more similar forboth signals. This could not be done were the signals taken directly at the antenna array.
Such dynamic range considerations remain valid when the signals are subject to a multipath
environment.
Our prototype demonstration is designed for processing two signals. Among the major
benefits of the optical processing are its scaling characteristics. A larger input signal space,say of size N, can be accommodated merely by increasing the number of electrodes on theelectrooptic modulator to N —the autotuning filter remains largely unchanged. Increasing
93

the number of output channels from 2 to M, requires a total of (M-1) autotuning filters. Thusin general the adaptive processor scales linearly with the size of the desired signal space,rather than quadratically, as is often the case with adaptive electronic processors.
Our system will optimally process multipath communication signals provided arrival
time differences are negligible compared to the inverse bandwidth of the signal. Different
paths are received by different focal surface receivers. The autotuning filter combines thereceived signals coherently while the noise from different receivers adds incoherently. Whenthe multipath delay times are much greater than the inverse signal bandwidth, the autotuning
filter treats each as a separate source and extracts the strongest one.
The prototype demonstrated here processes signals with bandwidths in the 100 MHz
range. None of the components are however limited to this bandwidth. Broadband antennas,
such as second resonant slots with up to 20% bandwidth can be used instead of the patches at
a higher carrier frequency, allowing for several gigahertz IF frequencies. A possible issue inthis case may be the angle-of-arrival detection quality due to higher grating lobes at the lower
band edge, which is a design parameter. The electro-optic modulator presented here iseffectively a lumped-element circuit, but its architecture accommodates a set of N coupled
traveling wave transmission lines, and such an EO modulator can cover several gigahertz of
bandwidth with good modulation efficiency.

94

CHAPTER 4
A PHOTOREFRACTIVE REGENERATIVEAMPLIFIER

4. 1

Introduction

The autotuning filter of Chapter 2 extracts one principal component at a time.
Extracting all the components requires cascading as many filters as there are signals at the
input, minus one. The filter runs on a minimum of 3 mW of optical power supplied by the
input signal-carrying beams. After the filter, the through beam is weaker since the ringextracts some of its energy and the gain crystal also absorbs nearly half of it. Current smalllaser sources—that easily fit in a briefcase for example—output up to a couple hundred
milliwatts. This limits how many autotuning filters, or any other photorefractive circuit, may
be daisy-chained before there is simply not enough optical power left to add a new one.
Adding more laser sources is a complicated solution as our circuits rely on coherent beams:
the added sources would have to be injection-locked. Amplification appears as a simpler
solution in the long run.
This short chapter presents the first steps toward making photorefractive regenerative
amplifiers. For the first amplifier, in hopefully a longer series, we chose to work on an
amplifier that takes a multimode beam in and outputs a single mode beam with the same95

temporal characteristics as the input beam. This amplifier could, for example, be added at theoutputs of cascaded autotuning filters (the outputs that locally extract the largest principal
component of their input space.) The same amplifier could not be inserted in between
cascaded filters, as the spatial diversity needs to be preserved for the input of the following
filter.

96

4. 2

A photorefrative amplifier

4. 2. 1 Design issues
We chose to use a photorefractive ring oscillator to convert a multimode beam into asingle mode beam.

As mentioned in section 2. 2, a ring oscillator pumped withphotorefractive two-beam coupling transfers all of the temporal features of the input to theoscillating beam. This is true weather both or either of the input and ring beams are single
mode or multimode. In the case of a single mode cavity, the photorefractive gratings adapt sothat the diffracted input signal is always on resonance with the length of the loop [64]. Someof the photorefractive gain is sacrificed to induce the necessary phase shift if the signal is off
resonance. In any case, this simple ring transforms a multimode beam into a single modeone. It does not overall provide gain: sampling the ring produces the output beam so thefinite gain limits how much light can be taken out. To provide real gain we have two choices.
One consists of adding a semiconductor laser amplifier in series with the single mode output
beam. The second choice involves incorporating the semiconductor amplifier in the ring. It
is a more elegant but also more complex solution. The challenge of inserting an amplifier inan adaptive resonator is that it will naturally tend to make the cavity lase at the frequency thatyields the highest energy in the ring. This is due to the fact that the spontaneous emission of
a semiconductor amplifier is equivalent to an optical broadband noise source. The design ofa regenerative amplifier with a semiconductor laser diode in a photorefractive ring must
therefore prevent the lasing and ensure that all the available gains serve to amplify a given
signal.
We started our photorefractive amplifier experimental investigations with work
necessary for both of the amplifier solutions mentioned above. We first learned how to pumpa photorefractive ring resonator that was forced to be single mode. This constraint was97

achieved by obliging the oscillating beam to couple in and out of a single mode fiber inserted
in the loop. Second, we learned how to make a semiconductor laser diode amplifier since
they are not commercially available. The following section provides the technical details
about how to make the device.

4. 2. 2 Manufacturing a semi-conductor laser amplifier in a research laboratory
4. 2. 2. 1

The method

Conceptually, a traveling wave optical amplifier is simple. It is a laser diode gain
medium through which an optical beam passes once.

The first step to making asemiconductor amplifier is the acquisition of a semiconductor laser diode chip. In this worldof cheap red laser pointers, it is surprisingly difficult to get small quantities of bare laser
diode chips.

Eventually the company Semiconductor Laser International sold us someuncoated, double heterostructured InGaAlP diodes, initially designed to emit at 655nm. The
second manufacturing step consists of attaching the chips to a mount that allows us to deliver
200 mA of current to the chip and that acts as a heatsink. The third step is anti-reflection
coating both facets of the chips.

4. 2. 2. 2

The techniqueMounting the chips
We used one of the industry’s standard diode laser submounts called c-mounts withsome slight modification. Figure 4.1 shows the modified c-mount. The top part of the mountis thinned down to be roughly the same thickness as the diode chip’s length (about
500 microns.)

This is so the full 30-degree cone of acceptance of the chips stays

unobstructed on both sides. Our chips are designed to be single spatial mode laser sources98which leads to a smaller, 1x3 µm-aperture than commercial multi-spatial modes power
amplifiers which have 1x100 or 200 µm-apertures.
We glued the chips down on the mounts with very highly electrically and thermallyconductive glue from Epoxy Technology (2 part epoxy, model # EPO-TEX H20E). We
manipulate the chips with a Vaccum Parts Handling System from Techni-Tool. It is a penshaped tube, connected to a small vacuum pump on one end and fitted with a metal syringe
tip on the other (plastic syringe tips turned out to pick up static electricity too well.).
Miniature wood tweezers (carved out of a wood Q-tip stick) proved to be another useful tool
to manipulate the chips. Once glued in place, the top metal sides of the chips are wedgebonded to the ribbon-like electrode illustrated in Figure 4.1.

Figure 4.1 Schematic of the c-mount used to heatsink the semiconductor chip and also provide the electrodes
necessary to supply the chip with current.

99Coating the chips
InGaAlP has an index of refraction of 3.5 so that the facets of the chip produce natural
30% reflecting mirrors. These offer a resonating cavity to the spontaneously emitted light,
imposing a lasing current threshold on the semiconductor amplifier. In addition, the presenceof a cavity induces ripples on the wavelength dependent gain-curve that distort theamplification of a broad bandwidth signal. The better the anti-reflection coating and thesmaller those ripples. Optical amplifiers became commercially viable when the reflections of
their input and output facets were reduced to less than 0.01%.
The diodes’ coatings were optimized by monitoring the laser output power versus theinjection current during the coating process. We chose to deposit a single quarter-wavelengthlayer of HfO2 by electron-beam evaporation. The diode’s output-power-versus-current (PI)curve was continuously observed during the process: the diode was subject to a periodic
ramp of current that was displayed on an oscilloscope along with the optical beam’s power.
When the lasing current threshold was crossed, the slope of the PI curve exhibited a suddenchange. As the coating approached the optimal thickness, the lasing disappeared altogether.
We then closely observed the peak power of the spontaneous emission decrease.

Thedeposition process was stopped when the decrease-rate slowed down, which was sign that thelight emission was at a minimum and about to increase again. Figure 4.2 shows the PI curvesof a diode measured before coating and then after each facet has been coated. Reflection
curves taken with an optical spectrum analyzer and microscope revealed the center
wavelength and bandwidth of the coatings. We achieved less than 0.01% reflection over tensof nanometers.

100

a

b

c
Figure 4.2. PI curves of a laser diode. The triangular black curves represent the current passing through the diode.
Their vertical scale is 100 mV for 10 mA. The nonlinear blue curves represent the laser ouptut powermonitored by a photodetector placed in the vacuum chamber, taken a) before coating,(Ith = 22 mA), b)
after coating one facet (Ith=33 mA), and c) after coating both facets (lasing Ith not apparent below 65 mA).

101

The thickness of Hf02 deposited was monitored through the resonant frequency of apiezo-electric crystal being coated alongside the chips. Unfortunately our calibration of the
thickness deposited was not repeatable enough. The center frequency of the anti-reflection
coating changed (over 100 nm) with nearly every run. The yield of usable diodes was low so
we sacrificed performance for speed: we sent our diodes to be coated in bulk to a coatingcompany (Spectrum Thin Films, NY.) The diodes were packaged so that the c-mounts wereattached to a single metal “window frame” (both sides of the diodes had to be open forcoating) and each had its electrodes shorted. The diodes survived the trip and came back witha good uniform coating for 655 nm.

4. 2. 2. 3

Results

The uncoated diode chips purchased from SLI were originally intended to become
35 mW laser diodes with 80 mA of current. The anti-reflection coated diode chips, pumped
with 100 mA of current, amplified a 0.2 mW, 655 nm beam up to 20 mW. This is anunsaturated gain of 20 dB.

102

4. 2. 3 Immediate future work
The immediate next step will consist of integrating the c-mounted diode in a pre-aligned
lens-diode-lens unit. Figure 4.3 shows the proposed design for the mechanical alignment of
the unit with 5 degrees of freedom (the unnecessary 6th is the rotation around the diode’s
waveguide axis). The pre-aligned unit could then be inserted into any circuit where a diodeamplifier needs to be tried out.

Figure 4.3. Proposed mechanical mount to align a lens, the diode amplifier and another lens.

103

CHAPTER 5
CONCLUSIONSThis thesis summarized the development of three projects. The first project concerned
the building and the analysis of a photorefractive oscillator. We named it an autotuning filter
for its ability to automatically tune in to the strongest principal component of its spatiotemporal input space.

This filter was the object of Chapter 2. The following chapter

described the insertion of that optical circuit in a sophisticated RF-photonics system, which
implemented principal component extraction on mixtures of two microwave signals received
by a 10 GHz quasi-optical lens antenna array.

Finally, Chapter 4 presented the firstdevelopment stages of an optical amplifier adapted to photorefractive information processing.
All three projects achieved progress toward engineering photorefractive circuits forcommercial applications.
The autotuning filter’s ancestors are circuits such as the feature extractor and thefrequency demultiplexer. Their experimental implementations occupied about half a standard
optical table (about 2 m2) and required hundreds of milliwatts to operate. The signals
processed by those circuits were within audio frequency range. The free space optics versionof the autotuning filter, described in Chapter 2, processes signals with up to 3 GHz of
bandwidth. The circuit fits on a quartz plate that is 3 cm in diameter and requires only 3 mWof optical power to operate. The earlier, fiber versions of the autotuning filter required about
10 mW of optical power and could process signals with bandwidths of hundreds of MHz.
104

Their signal separation performance was not as good as the performance of the free space
optics version of the filter but the results were very repeatable. This is not the case with thefree space optics version whose performance varies greatly with the alignment of the input
pump beams to the oscillator. A non-optimized alignment yields a slope at the origin of
8 dB/dB, which is as good as the fiberized versions of the filter. An optimized alignment
yields a quasi bistable device for input signals with powers within 10% of each other. Future
work on this device should include a way to make the alignment of the input beams
repeatable so that the device’s performance may also be uniform from one use to the next.
One solution would be to fiberize the input with single mode fibers whose alignment can be
optimized once and then fixed in place.
Chapter 3 demonstrated that the use of dynamic holographic optics in microwave
systems could be implemented with reasonable robustness, power consumption and size, tobe convenient in practical environments. The prototype system we presented performed
principal component extraction on an ensemble of signals incident on a microwave antenna
array. At the end of Chapter 3 we discussed widening the processing bandwidth of oursystem as well as increasing the number of signals it could handle. Another direction for
future work lies in implementing a more powerful method for blind source separation (BSS.)
Principal component analysis (PCA) separates signals by forming a set of uncorrelated
spatio-temporal signals. The characterization section of Chapter 3 showed how the PCA
method fails to recognize the original source signals when the received mixtures are notspatially orthogonal. There is a BSS method called independent component analysis (ICA)
that does not require spatially orthogonal mixtures. ICA yields the original source signals if
the temporal behaviors of the source signals are not correlated to any degree and that thereceived mixtures are linear combination of these signals. ICA may be implemented with
optics and electronics in two stages. The first stage performs PCA on the received mixtures.
The second stage consists of a winner-takes-all opto-electronic loop that extracts one of the105

independent components from the principal components at its input. A prototype of such asystem is currently being developed in our laboratories.
Finally, Chapter 4 presented the making of an optical semiconductor amplifier for red
wavelengths. To render this device practical, the mounted chip must be incorporated in a unitthat facilitates the handling and the coupling of light into and out of the amplifier. This
convenient single mode amplifier unit could be inserted in a single mode ring pumped by amulti-mode beam. It was explained in Chapter 4 how that oscillator would transform a multimode beam carrying a temporal signal to a single mode beam carrying essentially the same
information. Last and most difficult would be the realization of a fully spatially multi-mode
amplifier.
In conclusion, this thesis described the miniaturization of a photorefractive oscillator
capable of extracting the strongest principal component of its input signal space.

The

analysis and simulation of the circuit provided insight into the circuit’s dynamics. This
autotuning filter physically performs a task that is otherwise computationally intensive forsuch high bandwidth signals. Chapter 3 showed that the insertion of this photorefractive
circuit in a microwave processor was reasonably practical and robust to transportation. The
work on optical amplifiers adapted to our way of handling information opened the gate fordesigning more complex and smarter photorefractive circuits.

106

107

CHAPTER 6
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67. Hollung, S., A.E. Cox, and Z.B. Popovic, A bi-directional quasi-optical lens amplifier.
Ieee Transactions on Microwave Theory and Techniques, 1997. 45(12): p. 2352-2357.
68. Popovic, Z. and A. Mortazawi, Quasi-optical transmit/receive front ends. Ieee
Transactions on Microwave Theory and Techniques, 1998. 46(11): p. 1964-1975.
69. Anderson, D.Z., et al., -70 dB optical carrier suppression by two-beam coupling inphotorefractive media. Journal of Applied Physics, 2000 (?). invited: p. 743-748.

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CHAPTER 7
APPENDICES

These appendices regroup technical information relative to the work presented in theprevious chapters. The first appendix reveals the manufacturing process of photorefractive
spherical crystals with tools commonly available in an experimental laboratory. The second
appendix details the mechanical mounts that were designed for the autotuning filter versions
using slabs of spherical crystals.

The third appendix is transcript of the code to the

Mathematica program written to study the influence of the filter’s parameters on its signal
separation performance. The fourth appendix details the mount that was designed to housethe fragile electro-optic crystal and the electronic circuit that impedance matches theelectrodes to 50 ohms. The fifth appendix presents a mechanical method to avoid vibration
modes in a large piece of metal, the suitcase’s optics base plate in our case. The sixth
appendix describes the procedure for demonstrating the suitcase’s signal separation abilities
in front of an audience.

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7. 1

Making spherical crystals

Figure 7.1 Description of the method to hand-make spherical photorefractive crystals.

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7. 2

Mechanical mounts for the fiber filter

7. 2. 1 Ballbearings and glue

Figure 7.2 Mechanical mount for the first version of the autotuning filter with spherical photorefractive crystals(see Figure 2.6.a.)7. 2. 2 Pure mechanics

Figure 7.3 Mechanical mount for the second version of the autotuning filter with spherical photorefractive
crystals (see Figure 2.6.b.)

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7. 3

Algorithm for studying the parameters‘ influence

Following is the Mathematica code that numerically simulates the autotuning filter.
Comments were added throughout the program to enable the reader to follow its structure.

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119

Algorithm for the time loop:

Gg 1,2
TT
t
T

S 1,2
P 1,2

S
x1 P
1

0

Lg

gg

BS

Gr1,2
T
T
t
0 T

Lr

xr

Update the 2 gratings Ggi in the gain crystal with the previous loop’s intensities.
Initialize the intensities of the beams just before the gain crystal:
Signal beams @ xg = 0 = T x beams @ xr= Lr
Propagate the beams through the gain crystal.
Update the 2 gratings Ggr in the reflexive crystal with the previous loop’s intensities.
Initialize the intensities of the beams just before the gain crystal.
Signal beams @ xr = 0 = BS ratio x beams @ xg= Lg
Propagate the beams through the reflexive crystal.

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121

122

123

124

7. 4

EO mount

The thin (0.3 mm) Mg:LiNbO3 electro-optic crystal is sandwiched between a piece ofgrounded copper that was lapped flat and a piece of foam. A spring-loaded piece of metal
pushes on the foam and holds the crystal in place. The copper electrodes (etched out from acopper covered plastic sheet) are inserted between the crystal and the piece of foam (plastic
away from the crystal.)
The square-inch, resonant, matching circuit screws in sideways with its ground plane flat
against the square wall visible in the center of the mount.

Figure 7.4. EO mount that holds the 0.3x7x30 mm crystal pressed against a piece of foam.

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7. 5

Optics baseplate

The optics base plate is made out of a 3/4 inch thick piece of aluminum. To cut out
vibration modes of the plate, 1/2 inch deep patterns were carved out of the bottom. These
patterns are shown as striped patterns in the figure below. The light bulb shapes indicate thelocations of the upside down table clamps.

Figure 7.5. Hand drawing of the 1/2 inch deep patterns that were carved out of the bottom of the optics‘ baseplate.
The light bulb shapes indicate the positions of the upside down clamps that were shown in Figure 3.15.

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7. 6

Suitcase manual

PART LIST
The suitcase•

Suitcase + power cord (white)•

10dB directional coupler with BNC to SMA adapters•

2 BNC cables (one labeled CH2 and the other labeled PD)•

TOOLS:•

Flat-head screwdriver•

Wrench (SMA connector size)

Allen key (to fit fixing down srews)

The transmitters•

2 transmitter boxes with their respective horn antennas•

AA and NiMH 6V batteries (or 2 PC power supplies and an extension cord)•

TOOLS: cross and flat-head screwdrivers, voltmeter

Handheld spectrum analyzer (Bantam)
To get the backlit display, press the contrast button and select backlight option ON thenthe PRIOR MENU button to return to the normal screen.

To get a preset screen environment (=freq. range and ref. level) press the SAVE/RCL
button and then the SETUP MEM option. Select the desired setup by highlighting it andthen press the option RECALL.

When fully charged, the spectrum analyzer has a two-hour autonomy.
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Receiving antennas•

Lens antenna•

2 detecting patch antennas on Plexiglas poles and their cables.

•

A Plexiglas antenna holder•

TOOLS: 5 min.-epoxy glue!

Well before the demonstration:
Make sure the Plexiglas antenna holders survived the trip. If not glue the parts that cameundone.

For battery operated transmitters:•

Put in the two AA batteries in first.

•

Put in the four 6V batteries.

•

With a voltmeter make sure that the batteries are properly in place by checking thevoltages at the connecting pins.

INSTRUCTIONS FOR SETTING UP THE PROJECT
Suitcase•

Plug the white power cord into the suitcase and to the wall.

•

Loosen the optical table (3 screws).

•

Insert the 10dB directional coupler between the channel 2 output of the down
conversion box (the big gold box) and the channel 2 input to the power amplifier(black box with a fan on top).

•

Connect the BNC cable labeled PD to the output of the photodiode.

•

Turn the power on.
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Receiving antennas
Put the antenna holder in place and connect the detecting antennas’ cables to thecorresponding cables coming out of the suitcase.

Transmitters
Place the transmitters roughly in the right place indicated by the strings attached to thebase of the detecting antennas’ poles.

For wall supplied transmitters:•

Plug the brown power cord extension into a wall outlet.

•

Set the two power supplies to 13V.

•

Connect the supplies to the transmitters and plug them into the brown extension
cord.

Just before the demonstration•

Switch on the power supply for transmitter #1 (the green LED should come on).

•Plug the directional coupler BNC cable into the little spectrum analyzer and selectscreen environment #2 (freq. range around 141MHz)•

Iteratively adjust the DRO and VCO adjustment screws so that the peaks that appear
on the screen at first collapse together, centered on MARKER 1 (freq=142.5MHz)•

Switch on the power supply for transmitter #2.

•

Adjust the new peaks to collapse together, centered on MARKER 2(freq=140.4MHz).

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•

Unplug the directional coupler BNC cable and plug the photodiode BNC cable into
the S.A.

•

Select the screen environment #1 (freq. range around 282MHz).

DEMONSTRATING THE PROJECT
Three peaks appear on the screen environment #1: the middle one is due to someintermodulation product of the signals. The extreme peak to the left is the signal fromtransmitter 2 and the extreme peak to the right is the signal from transmitter 1. The extreme
peaks are roughly the same height because the transmitted powers are equal and there is very
little competition between the signals at that point.
By putting a hand right in front of the horn antennas you attenuate one of the transmitted
signal by a couple of dBs. You should see the corresponding peak go down and the otherpeak go up.

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