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= STUDIES OF PHASE-CONJUGATE OPTICAL DEVICE CONCEPTS

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ANNUAL TECHNICAL REPORT FOR THE PERIOD

o

April 01, 1985 through March 31, 1986

CONTRACT NO. N00014-85-C-0219
Prepared for: Dr. Herschel Pilloff
Office of Naval Research Physics Division, Code 412
800 N. Quincy Street Arlington, VA 22217-5000

DTIC D_,LETCCTE

JUL 2 5 198

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Pochi Yeh Ian McMichael

JUNE 1986 C:)

L.L.J

Approved for public release; distribution unlimited

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TABLE OF CONTENTS

Page

1.0 SUMMARY .................................... ....................... 1

1.1 Contract Description .......................................... 1

1.2 Scientific Problem.............................

1

1.3 Progress ...................................................... I

1.4 Special Significance of Results .............................. 2

1.5 Publications and Presentations .............

3

2.U TECHNICAL DISCUSSION ...............................................

5

2.1 Phase Reversal and Doppler-Free Reflection .................... 5 2.2 Polarization Preserving Phase Conjugator ...................... 6 2.3 Phase-Conjugate Fiber Optic Gyros ............................. 11

3.0 PROGRESS ........................................................... 13

3.1 Phase-Conjugate Fiber

Gtic ro ..............................

13

3.2 Polarization Preserving Phase Conjugator ...................... 14

3.3 Phase of Phase-Conjugate Reflections .......................... 19

3.4 Photorefractive Phenomena .....................................

21

...

3.4.1 Frequency Shift of Photorefractive Resonators .......... 21 3.4.2 Frequency Shift of Self-Pumped Phase Conjugators ....... 21 3.4.3 Photorefractive Conical Diffraction .................... 22
4.0 REFERENCES ......................................................... 24

APPENDIX 5.1 Self-Pumped Phase-Conjugate Fiber-Optic Gyro .............. 26

APPENDIX 5.2 Polarization Preserving Phase Conjugator ................ 39

APPENDIX 5.3 Absolute Phase Shift of Phase Conjugators ................. 55

APPENDIX 5.4 Phase-Conjugate Fiber-Optic Gyro .......................... 63

APPENDIX 5.5 Theory of Unidirectional Photorefractive

Ring Oscillators............

...........

......

66

APPENDIX 5.6 Frequency Shift and Cavity Length in Photorefractive Resonators ................................................ 72

APPENDIX 5.7 Frequency Shift of Self-Pumped Phase Conjugator ........... 76

APPENDIX 5.8 Photorefractive Conical Diffraction in BaTi03............. 88

APPENDIX 5.9 Parallel Image Subtraction Using a Phase-Conjugate

Michelson Interferometer ...................

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

109

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LIST OF FIGURES

Figure

Page

2-1

Phase conjugator for polarization restoration in a

bi refringent system ...............................

........

7

2-2

Schematic drawings of polarization-preserving

phase conjugators .............................................. 10

2-3

Schematic drawing of the phase-conjugate fiber-optic gyro...... 11

3-1

Experiment used to demonstrate correction of polarization

scrambling in multimode fibers by polarization-preserving
phase conjugation .............................................. 15
I'.-

3-2

Correction of polarization scrambling and modal aberration

in a multimode fiber by polarization-preserving phase

conjugation .................................................... 16

3-3

Externally-pumped polarization-preserving phase

conjugate mirror ............................................... 17,.

.

3-4

Measured angle of polarization for the reflection from the

polarization-preserving phase conjugator vs the angle of polarization of the incident wave ..................... ..... 18

3-5

Reproduction of the helicity of polarized light by the

externally pumped polarization-preserving phase

,*,

conjugator ..................................................... 19

Table 1

LIST OF TABLES Page

Phase of Phase-Conjugate Reflections........................... 20

Acce.;ion For

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1.0 SUMMARY

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1.1

Contract Description

This contract studies the phase-reversal property of optical phase conjugation for navigational and other device applications. The study focuses on the development of the phase-conjugate fiber-optic gyro and the generation of new device concepts.

1.2

Scientific Problem

Although much attention is paid to the aberration correction property of phase conjugation, little attention is paid to the phase reversal property. The phase-reversal property has important applications in inertial navigation devices. The general problem for this program is to generate new device concepts using the phase-reversal property of phase conjugation.
Polarization scrdmbling is a well-known source of noise and signal fading in fiber-optic gyros. Some gyros avoid this problem by using polarization-preserving fibers and couplers to decouple the polarization modes. This program studies a new approach in which polarization-preserving phase conjugation is used to correct for polarization scrambling without the need for polarization-preserving fibers and couplers.

1.3

Progress

There are several areas of significant progress in the first year of this program that are directly related to the development of the phase-conjugate fiber-optic gyro. These include:

e First experimental observation of the phase-conjugate Sagnac phase shift.

First demonstrations of rotation sensing with a phase-conjugate gyro and with a self-pumped phase-conjugate fiber-optic gyro.

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9 First measurement of a nonreciprocal phase shift (Faraday effect) in a double phase-conjugate interferometer.

0 Development of a polarization-preserving phase-conjugate mirror that operates at milliwatt power levels.

o First demonstration of the correction of polarization scrambling in multimode fibers by polarization-preserving phase conjugation.

* First measurements of the phase of phase-conjugate reflections.

In addition to the progress mentioned above, we have also carried out other interesting scientific research and have achieved many significant results. These include:

• Frequency shifts of photorefractive resonators

* Resonator model and frequency shifts of self-pumped phase conjugate resonators

" Photorefractive conical diffraction, and

• Parallel image subtraction via phase-conjugate Michelson interferometry.

The results are published (or to be published) in the papers and conference presentations listed in Section 1.5.

1.4

Special Significance of Results

It should be noted that two of the above mentioned areas of progress are of special significance in that they are not restricted in their use to the phase-conjugate fiber-optic gyro. The polarization-preserving phase-conjugate

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mirror opens a whole new area of interferometry with multimode fibers. Measurements of the phase of the phase-conjugate reflection can be used to determine the phase shift (with respect to the intensity pattern) and the type of grating (index, absorption, gain, or mixture) involved in degenerate four-wave mixing in nonlinear media. Proper selection of nonlinear media will allow the construction of phase-conjugate interferometers that "self-quadrature" for high sensitivity and linear response. The phase shift of phase conjugators also plays an important role in the frequency shift of double phase-conjugate resonators.

"

1.5

Publications and Presentations

Publications

"Self-Pumped Phase-Conjugate Fiber-Optic Gyro," Ian McMichael and Pochi Yeh, submitted to Optics Letters, (1986).
"Polarization-Preserving Phase Conjugator," Ian McMichael, Monte Khoshnevisan and Pochi Yeh, to appear in Opt. Lett., August (1986).
"Absolute Phase Shift of Phase Conjugators," Ian McMichael, Pochi Yeh and Monte Khoshnevisan, to appear in Proc. SPIE 613, 32 (1986).
"Phase-Conjugate Fiber-Optic Gyro," Pochi Yeh, Ian McMichael and Monte Khoshnevisan, Appl. Opt. 25, 1029 (1986).
* "Theory of Unidirectional Photorefractive Ring Oscillators," Pochi Yeh, J. Opt. Soc. Am. B2, 1924 (1985).
* "Frequency Shift dnd Cavity Length in Photorefractive Resonators," M.D. Ewbank and Pochi Yeh, Opt. Lett., 10, 496-498 (1985).
* "Frequency Shift of Self-Pumped Phase Conjugator," M.D. Ewbank and Pochi Yeh, SPIE Proc. 613, 59 (1986).
* "Parallel Image Subtraction Using a Phase Conjugate Michelson Interferometer," A.E.T. Chiou and Pochi Yeh, Opt. Lett. 11, 306 (1986).

*Works only partially supported by this contract. 3
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* "Photorefractive Conical Diffraction in BaTi03," M.D. Ewbank, Pochi Yeh and J. Feinberg, to appear in Opt. Comm. (1986).

Presentations
"Self-Pumped Phase-Conjugate Fiber-Optic Gyro," Ian McMichael and Pochi Yeh, submitted to the 1986 OSA Annual Meeting in Seattle, WA.
"Measurements of the Phase of Phase-Conjugate Reflections," Ian McMichael, Pochi Yeh and Monte Khoshnevisan, presented at IQEC'86 in San Francisco, CA.
"Absolute Phase Shift of Phase Conjugators," Ian McMichael, Pochi Yeh and Monte Khoshnevisan, presented at O-E LASE'86 in Los Angeles, CA.
"Phase-Conjugate Fiber-Optic Gyro," Pochi Yeh, Ian McMichael and Monte Khoshnevisan, presented at the 1985 OSA Annual Meeting in Washington, D.C.
"Scalar Phase Conjugation Using a Barium Titanate Crystal," Ian McMichael and Monte Khoshnevisan, presented at CLEO'85 in Baltimore, MD.
* "Photorefractive Resonators," M.D. Ewbank and Pochi Yeh, paper presented at the 1985 OSA Annual Meeting in Washington, D.C. (October 14-18, 1985).
* "Frequency Shift of Self-Pumped Phase Conjugator," M.D. Ewbank and Pochi Yeh, paper presented at Conference on Nonlinear Optics and Applications, January 21-22, 1986, Los Angeles, CA.
* "Coherent Image Subtraction Using Phase Conjugate Interferometry," A.E.T. Chiou, Pochi Yeh and Monte Khoshnevisan, paper presented at Conference on Nonlinear Optics and Applications, January 21-22, 1986, Los Angeles, CA.

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2.0 TECHNICAL DISCUSSION

Optical phase conjugation has been a subject of considerable interest during the past several years. Much attention has been focused on the wavefront correction property of this process by means of degenerate four-wave mixing. 1-3 Very little attention was paid to the phase reversal property, the Doppler-free
reflection, and the phase-sensitive coupling of degenerate four-wave mixing. These properties have many interesting and important applications in inertial navigation devices. In this section, we will first briefly describe some of the nonlinear optical phenomena and then discuss the phase-conjugate fiber-optic gyro.

2.1

Phase Reversal and Doppler-Free Reflection

Phase reversal is a unique property of degenerate four-wave mixing which is not available in the conventional adaptive optics. A very interesting situation arises as a result of the phase reversal. Consider the situation when a laser beam is incident on a phase-conjugate reflector (abbreviated here as ¢* reflector). Let E exp[i(wt - kz + $)] be the incident electric field. The * reflector will generate a reflected wave of the form pE exp[i(ut + kz - 0)]. The interference pattern formed by the incident and reflected waves is of the form

I = E2 [1 + Ip12 + 21p cos (2kz - 20 + a)]

(1)

where a is the constant phase of the complex reflection coefficient p. Note that the phase 0 contains the information of the source. If the source fre-
quency fluctuates, * will be a function of time and the interference pattern
also fluctuates. This means that the phase of the interference pattern is determined by the source, not the reflector. In other words, the interference pattern is independent of the position (or motion) of the 0* reflector. This property can also be explained in terms of the Doppler-free reflection. Since there is no Doppler shift in frequency due to the motion of the 0* reflector,4

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the incident beam and the reflected beam have the same frequency which leads to a stationary interference pattern.
If the 0* reflector were replaced by an ordinary mirror, the interference pattern would have a phase which depends on the position of the mirror. Such an interference pattern would move with the mirror and does not contain any phase information about the source. Since 0 does not appear in the interference pattern, any frequency fluctuation (or phase fluctuation) of the source will not affect the pattern. In other words, the interference pattern is determined by the mirror. This property also can be explained in terms of the Doppler shift upon reflection from a moving mirror. Since the reflected wave is shifted in frequency by (2vw/c), the interference pattern is traveling at a speed equal to the speed of the mirror.
The Doppler-free reflection via four-wave mixing has been demonstrated experimentally by the author and his co-workers. 4

2.2

Polarization-Preserving Phase Conjugator

In many of the early experiments on wavefront correction,l,2 the change of polarization state upon phase-conjugate reflection had no effect on the fidelity of aberration correction because the distorting media were optically isotropic. There are many situations where the distorting media may become optically anisotropic due to external perturbations such as electric field, magnetic field, strain, etc. Under these circumstances, the polarization state of the phase-conjugated wave becomes an important issue.

Consider an optical wave of frequency w moving in the +z direction

r = IiU( )e i(ut-kz)

(2)

where X1(r) is the complex amplitude and k is the wave number. This wave satisfies the wave equation

d2

W

r,))- 0

(3)

where c is the dielectric constant and 4 is the permeability constant.

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We now consider a case where in some region of space near zo, we generate a field t2 (e.g., via degenerate four-wave mixing) which is related to the phase-conjugate of ti, and described locally by

t2 = p4I*(r+)ei((+kz)

(4)

where p is, in general, a 3 x 3 tensor. It can then be shown that the amplitude of the reflected wave t will remain p l*(+) in the region z < zo , provided r2 and 1 satisfy the same wave equation in this region. This is the basic principle of wavefront correction via optical phase-conjugation. If the dielectric function c(r), which describes the property in the region z < zo, is a tensor (i.e., has nonzero off-diagonal terms), then the wave r2 may not satisfy the wave equation (3) because the matrix multiplication is, in general, not commutative. If the phase-conjugate reflectance tensor p reduces to a scalar, then r2 also satisfies the wave equation (3), because for scalar p, pe = ep, even if E is a tensor. Thus, a scalar phase conjugator can serve to restore polarization scrambling, as well as wavefront aberration. Such a reflector is called a polarization preserving phase conjugator.
To further illustrate the polarization restoration, we consider the propagation of polarized light through a series of birefringent plates (see Fig. 2-1). At the end of the birefringent system, a phase-conjugate reflector

SC84-26142

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Fig. 2-1 Phase conjugator for polarization restoration in a birefringent system.
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retroreflects the polarized light. Let us now examine the polarization state of the light as it propagates through the system. Let XI(o) be the input polarization state and 41(L) be the output polarization state. K1 (O) and K1 (L) are related by

91(L) = MNMN_ 1 ... M3M2M1 4I(0)

(5)

where Mi (i = 1,2,..N) is the Jones matrix for the i-th plate. Upon reflection from the phase-conjugator, the polarization state becomes pI (L). When the reflected light propagates backward through the birefringent system, the final polarization state 92 (o) is given by

42(0) = M1 M2 M3 ... MNlMNPll (L)

(6)

Using Eqs. (5) and (6), this polarization state can be written

X2(0) = M1 M2 M3 ... MNMNPMNMNl ... M3 *M2 M * (0)

(7)

If p is a scalar, then Eq. (7) reduces to

X2 (0)= pA1 (0)

(8)

because all the Jones matrices are unitary (i.e., MM* = I). 5 Equation (8) indicates that Xi(0) and X2(0) have exactly the same polarization state (i.e.,
same ellipticity, handedness, helicity). If p cannot be reduced to a scalar, then Eq. (7) indicates that the polarization state K2(0) is different from Ai(0). Thus, a polarization-preserving phase-conjugator can be used to restore
the polarization state.
Consider now the tensor property of a phase-conjugator which consists
of a nonlinear isotropic medium pumped by a pair of counter-propagating beams. Let the electric fi(e(lAdts k ofr) the incident prr)obe beam and the pump beams be 4(*)ei t kz ) ei( t r) and ei(A- r), respectively. The nonlinear polarization which is responsible for the generation of the phase-conjugated wave is6

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where we assume that the material has an instantaneous polarization response (i.e., lossless) and use the relationships x1111 = 3XI122 and X1122 = X1221 =
K,-7
X1212 .7 The last two terms in the square brackets are responsible for the analogy between the degenerate four-wave mixing and holography. The first term has no holographic analog and may he the dominant term in the event of a twophoton resonance. 3 In general, all three terms contribute to the generation of the phase-conjugated wave whose amplitude is proportional to NL
According to Eq. (9), a polarization-preserving phase-conjugator can be obtained by arranging the pump beams in such a way that the holographic terms vanish. This can he achieved by making the polarization state of the pump beams orthogonal to that of the probe beam. For the case of probing incidence along the +z direction, the polarization state of the probe wave lies in the xy plane. Thus, the pump beams must be polarized along the z-direction in order to have
zero holographic terms. Such a geometry is depicted in Fig. 2-2 (a). In this scheme, the nonlinear polarization

LN=k 2xllll/*( .C)

(10)

will generate a phase-conjugated wave which preserves the polarization state. This example shows that polarization-preserving phase-conjugation exists.
Figures 2-2(b) and 2-2(c) show two other schemes which can also achieve polarization-preserving phase conjugation. In these two schemes, the phase conjugators are operated in the holographic regime (e.g., photorefractive effect) such that they respond to one linear polarization state only and have no effect on the other polarization state. By using a polarizing beam splitter (see Fig. 2-2(c)) or using two stages in cascade (Fig. 2-2(b)), it is possible to conjugate each polarization component individually and then recombine the conjugated components. By proper alignment of the crystals, it is possible to achieve polarization-preserving phase conjugation.

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A scheme, similar to that shown in Fig. 2-2(c), which utilizes self-

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pumped phase conjugation in a Michelson interferometer can also be employed to

achieve polarization-preserving phase conjugation. This approach is described

in the progress section.

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2.3

Phase-Conjugate Fiber-Optic Gyros

Polarization scrambling is a well-known noise source in fiber-optic gyros. Birefringent polarization-holding fibers can be used to decouple the two states of polarization and hence improve the sensitivity.8,9 In the phaseconjugate fiber-optic gyro, which we are studying, a polarization-preserving phase conjugator can be used to restore severely scrambled waves to their original state of polarization. This eliminates the noise due to polarization
scrambling.

Referring to Fig. 2-3, we consider a fiber-optic gyro which contains a phase-conjugate reflector (abbreviated as * reflector) at the end of the fiber loop. We now examine the phase shift of light. In the clockwise trip from the input coupling to the 4* detector, the phase shift is 01 = kL - 2iLRQ/(%c), where L is the length of the fiber, R is the radius of the loop, 9 is the rotation rate, X is the wavelength, k = 2nn/%, and c is the velocity of light. In the counterclockwise trip, the phase shift is 02 = kL + 21dRQ/(%c). Due to the phase reversal nature of the 0* reflector, the net phase shift in a round trip is AO = 02 - 01 = 41LRQ/(xc). Such a net phase change is proportional to the rotation rate and can be used for rotation sensing.

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In addition, if the * reflector can preserve the polarization state, then the polarization state will not change upon reflection. Such a polarization-preserving * reflector will produce a true time-reversed version of the incident wave and will undo all the reciprocal changes (e.g., polarization scrambling, modal aberration) when the light propagates backward from e reflector to the input coupling. Thus, the problem of polarization scrambling as well as modal aberration in multimode fibers can he solved by using polarization-preserving phase conjugation.

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3.0 PROGRESS

During the first year of this research program, there were many areas of significant progress. These include the experimental demonstration of the phase-conjugate Sagnac effect and the polarization-preserving phase conjugator. In addition, we have proposed and demonstrated a scheme to measure the absolute phase shift of phase-conjugate reflections. This phase shift plays an important role in the detection of the Sagnac phase shift due to rotation. This progress is summarized below.

3.1

Phase-Conjugate Fiber-Optic Gyro

Our first objective was to demonstrate that the phase-conjugate fiberoptic gyro (PCFOG) described in Section 2.3 is sensitive to rotation. A proof of concept experiment was set up for this objective using an externally pumped crystal of barium titanate as the phase-conjugate mirror. Since the phaseconjugate mirror in this preliminary experiment did not preserve polarization, the fiber-optic coil was made of polarization-preserving fiber. Our report of the first demonstration of rotation-sensing is included as Appendix Section 5.4. The results of this proof of concept experiment demonstrate that as predicted, the PCFOG is sensitive to the nonreciprocal phase shift produced by the Sagnac effect and therefore it can be used to sense rotation.

In the proof of concept demonstration of the PCFOG described above, we were limited to a fiber-optic coil having an optical path length of 10 m by the coherence length of the laser. As a result, we were not able to measure low rotation rates. However, there are other configurations of the PCFOG that allow for longer lengths of fiber and hence greater sensitivity. For example, a Michelson interferometer in which both arms are terminated by the same selfpumped phase-conjugate mirror is also sensitive to nonreciprocal phase shifts. We first demonstrated this fact by measuring the nonreciprocal phase shift introduced by the Faraday effect in such an interferometer. The results of this demonstration implied that a PCFOG can he made by placing fiber-optic coils in

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the arms of this interferometer. In this configuration the two fiber-optic coils can be of any length as long as the difference between their lengths does not exceed the coherence length of the source. We recently demonstrated rotation sensing with this configuration of a PCFOG, and our report of that demonstration is included as Appendix Section 5.1. Since this configuration uses self-pumped phase conjugation, it has the obvious advantage of not having to provide external pumping waves that are coherent and form a phase-conjugate pair.

3.2

Polarization-Preserving Phase Conjugator

Our ultimate goal is the demonstration of a phase-conjugate fiber-optic gyro using multimode fiber. Such a demonstration requires the polarizationpreserving phase conjugator (PPPC) described in Section 2.2 to correct for the environmentally dependent birefringence and modal aberration of multimode fibers. With this motivation we developed the first polarization-preserving phase conjugator that operates at milliwatt power levels. The polarizationpreserving phase conjugator works by decomposing a light beam into its two polarization components, rotating one of these components with a half-wave plate, and reflecting both components from the same phase-conjugate mirror. When the two reflected components recombine they form a phase-conjugate wave that has the same polarization as the incident wave. Our report of this development is included in Appendix Section 5.2. The report presents results demonstrating that the phase-conjugate wave produced by the polarizationpreserving phase conjugator has the same ellipticity and helicity of polarization as the incident wave.

To demonstrate the ability of the polarization-preserving phase conjugator to correct for the modal and polarization scrambling of multimode fibers, we performed the experiment shown in Fig. 3-1. The highly reflective beamsplitter BS1 isolates the laser from retroreflections of its output. The polarizer P1 ensures that light entering the multimode fiber MMF is linearly polarized in the plane of the figure. Light exiting from the fiber is retro-

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reflected by either a normal mirror M, a nonpolarization-preserving phase conjugator, or by the polarization-preserving phase-conjugator PPPC. After propagating back through the fiber the light is sampled by the beamsplitter BS2, analyzed by the polarizer P2, and photographed by the camera D. The resulting photographs are shown in Fig. 3-2. The upper photographs, taken with a normal mirror at the end of the fiber, demonstrate complete polarization scrambling by the fiber. The middle photographs, taken with a phase-conjugate mirror (nonpolarization preserving) at the end of the fiber, demonstrate partial correction of the polarization scrambling. Finally, the lower photographs demonstrate complete correction of the polarization scrambling by the polarization-preserving phase conjugator.

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BS2

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Fig. 3-1 Experiment used to demonstrate correction of polarization scrambling in multimode fibers by polarization-preserving phase conjugation.
The results presented above and in Appendix Section 5.2 are for a polarization-preserving phase conjugator that utilizes self-pumped phase conjugation. A more recent experiment, with similar results, demonstrates a polarization-preserving phase conjugator that utilizes externally pumped phase conjugation. This experiment is shown in Fig. 3-3. The polarization-preserving phase conjugator consists of components BS2, M1, M2, BaTi0 3 , PBS, M3, and X/2 (not shown as a dashed line). This arrangement is a polarization-preserving
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PHASE-CONJUGATE MIRROR (NON-POLARIZATION PRESERVING)
POLARIZATIONPRESERVING PHASE-CONJUGATE MIRROR
Fig. 3-2 Correction of polarization scrambling and modal aberration in a multimode fiber by polarization-preserving phase conjugation.
phase conjugator for light incident from the left on the polarizing beamsplitter PBS. The remaining components are used to test the polarization-preserving phase conjugator. The external pumping waves for degenerate four-wave mixing in the crystal of barium titanate are provided by the reflections from mirrors M1 and M2. The components transmitted and reflected by PBS are probe waves. To test the polarization-preserving phase conjugator, either a half-wave retarder X/2 (shown as dashed line) or quarter-wave retarder X/4 is used to alter the

16 C7714A/jbs

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X/2

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Fig. 3-3 Externally pumped polarization-preserving phase-conjugate mirror.

polarization state of the light incident on the PPPC. The reflected light is sampled by the beamsplitter BS2, and analyzed by the combination of polarizer P and detector D. Since BS2 is an uncoated pellicle beamsplitter used near normal incidence (the angle of incidence is exaggerated in the figure; the actual angle of incidence is 20), the reflection coefficients for the s and p polarizations are nearly equal and the polarization measured by P and D is nearly the same as that of the reflection.
Figure 3-4 shows the measured angle of polarization for the reflection from the polarization-preserving phase conjugator, PPPC, as a function of the angle of polarization of the light incident on the PPPC, for various orientations of the half-wave retarder. Zero degrees corresponds to polarization in the plane of the previous figures. The open circles are the data (with diameters corresponding to the uncertainty), and the solid line indicates what is expected in the case of an ideal polarization-preserving phase conjugator. The measured ellipticity of the polarization for the light reflected by the PPPC (defined as the ratio of the minor polarization axis to the major polarization axis) never exceeded 5%.

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Fig. 3-4 Measured angle of polarization for the reflection from the polarization-preserving phase conjugator vs the angle of polarization of the incident wave.

The results shown in Fig. 3-4 demonstrate that the reflection from the

PPPC reproduces the angle of polarization of the incident wave. To show that it

•

reproduces the helicity of the polarization of the incident wave, the quarter-

~wave

retarder X/4 is placed between the sampling beam splitter and the polariz-

.'.%_,"i.'n'''g''m'b.e,a,mw "s'p'li.tte-r.+a,n"d ".i"s-"o-r"iente,d 'su"c"h -th'a,t"oth"e lig'ht; -i.,nc-ident on the mirror is. . .. converted from linearly polarized light to circularly polarized light. Fig-

ure 3-5 shows the measured polarization ellipses for the reflections from a nor-

mal mirror and from the polarization-preserving phase conjugator. Light reflec-

ted from the normal mirror changes helicity. After passing back through the

quarter-wave retarder, the polarization of the reflected light is orthogonal to

the incident light. This is the principle by which quarter-wave isolation works.

On the other hand, light reflected by the polarization-preserving phase conjuga-

tor has the same helicity as the incident light and returns to its original

polarization state after passing back through the quarter-wave retarder.

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POLARIZATION AFTER DOUBLE PASS THROUGH QUARTER WAVE RETARDER
Fig. 3-5 Reproduction of the helicity of polarized light by the externally pumped polarization-preserving phase conjugator.

3.3

Phase of Phase-Conjugate Reflections

The phase of the phase-conjugate reflection determines the operating point in some configurations of the phase-conjugate fiber-optic gyro. If this phase can be controlled, the PCFOG can he biased at the operating point of highest sensitivity and linear response (quadrature). With this motivation we measured the phase of phase-conjugatE reflections by determining the operating point of a phase-conjugate interferomete, and developed a theory to explain our resuilts. The detailed report of this work is included as Appendix Section 5.3 and a summary of the results is given here.

If the complex amplitude A4 of the phase-conjugate reflection of an incident wave having complex amplitude A3 is written as

A = r e AIAo2A*/IAIA21

(11)

where A1 and A2 are the complex amplitudes of the pumping waves, then 00 is

referred to as the phase of the phase-conjugate. This phase is given by,

0 :Ak + + /2

(12)

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00 06k + g + E/2t

(12)

where 06 is a term that depends on the type of grating involved in the phase conjugation (index, absorption, or gain) and g is the phase shift of the grating with respect to the light intensity pattern that produces the grating. For photorefractive media in which the index grating (oAk = 0) is shifted by T/2 radians we expect % = 0 or n radians. For thermo-optic media in which the index grating is in phase with the light intensity pattern we expect 0 = iT/2 radians.

Our experimental measurements of 0o for various nonlinear media are

given in Table 1, where e is the angle between the grating k vector and the

crystal axis. For the photorefractive materials, barium titanate and strontium

barium niobate, the measured values of % compare well with the expected values.

The small discrepancies for barium titanate at e = 00 and 1800 indicate that the

;

index grating is not shifted by exactly n/2 radians, as is often assumed. We

have verified this fact by an independent measurement of the two-wave mixing

gain as a function of the frequency detuning between the two waves. The fact

that the grating is not shifted by exactly /2 radians may be due to the exis-

tence of a photovoltaic field. Since for ruby *o = n/2 radians, the grating is

probably dominated by an index change rather than an absorption change.

Table 1 Phase of Phase-Conjugate Reflections

Material

e

00

BaTiO 3

45000
1350 1800

(19±3)0 (6±4)0
(176±3)0 (164±3)0

SBN Ruby

00 1800
-

(3±3)0 (175±5)0
(80±5)0

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These results demonstrate that the proper choice of nonlinear material for the phase-conjugate mirror can automatically bias the phase-conjugate fiberoptic gyro at the point of maximum sensitivity and linear response.

3.4

Photorefractive Phenomena

3.4.1 Frequency Shift of Photorefractive Resonators
Photorefractive resonators exhibit an extremely small frequency difference (Af/f _ 10-15) between the oscillating and pump beams. In addition, the photorefractive ring resonator seems to oscillate over a large range of cavity detuning despite the narrow gain bandwidth. A theory is developed which describes how the oscillating mode attains the round-trip phase condition. The theory predicts that the frequency difference between the oscillating and pump beams is proportional to the cavity detuning. This dependence is explained by a photorefractive phase shift due to slightly nondegenerate two-wave mixing that compensates the cavity detuning and allows the electric field to reproduce itself after each round trip. Such a theory is validated experimentally. The measured frequency and oscillating intensity agree with theory.
The details are given in the reprints of our papers which are attached as Appendix Sections 5.5 and 5.6.

3.4.2 Frequency Shift of Self-Pumped Phase Conjugators
The reflection from most photorefractive, self-pumped phase conjugators differs in frequency from the incident beam by a small amount (Aw/W 10-15). This frequency shift has been attributed to moving photorefractive gratings which Doppler shift the diff -ted light. However, the physical mechanism responsible for the moving grati is not well understood.
The frequency shift firsc manifested itself as a frequency scanning when a self-pumped BaTiO 3 was coupled to a dye laser. Since those initial observations, numerous experiments and theories involving self-pumped phase conjugators and/or photorefractive resonators have addressed, either directly or

21

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indirectly, the frequency shift issue. However, a general theory and the conclusive experiments are not available.
We have developed a theory and carried out the supporting experiments which explain such frequency shifts of most self-pumped phase conjugators. In our theory, self-pumped phase conjugation results from an internal selfoscillation. The optical resonance cavity which supports such oscillation is formed by either external mirrors or crystal surfaces. The oscillating beams provide the counterpropagating pump beams which are required in the four-wave mixing process. The frequency shift is proportional to the cavity length detuni ng.
When the self-pumping beams are spontaneously generated via photorefractive coupling in a linear resonance cavity with two external mirrors on opposite sides of a photorefractive crystal such as BaTi0 3, we observe that the frequency shift of the phase-conjugate reflection is directly proportional to cavity-length detuning. In the case where the self-pumping beams arise from internal reflections off the photorefractive crystal's surfaces, we experimentally prove that previous descriptions of the self-pumping process are inadequate and we show that a closed-looped resonance cavity forming inside the crystal is a better description.
The details are given in the reprint of a paper which is attached as Appendix Section 5.7.

3.4.3 Photorefractive Conical Diffraction
A single beam of coherent light incident on a BaTiO 3 crystal can cause a cone of light to emerge from the far face of the crystal. This cone has a polarization orthogonal to that of the incident ray and appears when the incident beam is an extraordinary ray in the crystal. There have been previous accounts of rings, fans, and other forms of photoinduced light scattering in photorefractive crystals, which have been attributed to a variety of physical
mechanisms.11-19 Recently, similar light cones in RaTiO 3 have been reported and shown to be due to stimulated two-wave mixing via the photorefractive effect. 20

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Here, we account for the phase-matching condition in BaTiO 3 for anisotropic Bragg scattering 21 by using A simple geometrical construction to predict the angular position of the light in the exit plane. We also show that precise measurements of the cone angle can be used to determine the dispersion of the birefringence, A~n = ne - no, of a BaTiO 3 sample.
The details are given in the preprint of a paper which is attached as Appendix Section 5.8.

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4.0 REFERENCES
1. B. Ya. Zel'dovich, V.I. Popovichev, V.V. Ragul'skii and F.S. Faizullov,
Sov. Phys. JETP 15, 109 (1972).
2. A. Yariv, lEE J. Quantum Electronics QE14, 65U (1978).
3. C.R. Giuliano, Physics Today 27 (April 1981).
4. P. Yeh, M. Ewbank, M. Khoshnevisan and J. Tracy, "Doppler-Free-PhaseConjugate Reflection," Opt. Lett. 9, 41-43 (1984).
5. See, for example, A. Yariv and P. Yeh, "Optical Waves in Crystals," (Wiley, NY, 1984), p. 124.
6. Reference 5, p. 553.
7. R.W. Hellwarth, "Third-Order Optical Susceptibilities of Liquids and Solids," Prog. Quant. Electr. 5, 1-68 (Pergammon Press, 1977).
8. T.G. Giallorenzi, J.A. Burcaro, A. Dandridge, G.H. Sigel, Jr., J.H. Cole, S.C. Rashleigh and R.G. Priest, "Optical Fiber Sensor Technology," IEEE, J. QE, QE-18, 626-664 (1982).
9. W.K. Burns, R.P. Moeller, C.A. Villarruel and M. Abebe, "Fiber Optic Gyroscopes with Polarization Holding Fiber," Opt. Lett. 8, 540-542 (1983).
10. P. Yeh, "Scalar Phase Conjugator for Polarization Correction," Opt. Comm. 51, 195-197 (1984).
11. W. Phillips, J.J. Amodei and D.L. Staebler, RCA Rev. 33, 94 (1972).
12. J.M. Morgan and I.P. Kaminow, Appl. Opt. 12, 1964 (1973).
13. M.R.B. Forhsaw, Appl. Opt. 13, 2 (1974). 14. R. Magnusson and T.K. Gaylord, Appl. Opt. 13, 1545 (1974).
15. S.I. Ragnarsson, Appl. Opt. 17, 116 (1978). 16. I.R. Dorosh, Yu.S. Kuzminov, N.M. Polozkov, A.M. Prokhorov, V.V. Osiko,
N.V. Tkachenko, V.V. Voronov and D.Kh. Nurligareev, Phys. Stat. Sol. (1) 65, 513 (1981). 17. E.M. Avakyan, K.G. Belabaev and S.G. Odoulov, Sov. Phys. Sol. St. 25, 1887 (1983).
18. R. Groussin, S. Mallick and S. Odoulov, Opt. Comm. 51, 342 (1984).
24 C7714A/jbs
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i Rockwell International
Science Center SC5424 .AR 19. S. Qdoulov, K. Belabaev arid 1. Kiseleva, Opt. Lett. 10, 31 (1985). 20. D.A. Tepl and C. Warde, J. Opt. Soc. Am. 1B3, 337 (1986). 21. N.V. Kukhtarev, E. Kratzig, H.C. Kulich and R.A. Rupp, Appi. Phys. B35, 17 (1984).
25 C7714A/jbs

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APPENDIX 5.1 Self-Pumped Phase-Conjugate Fiber-Optic Gyro
(Paper submitted to Optics Letters)

26 C7714A/jbs

SELF-PUMPED PHASE-CONJUGATE FIBER-OPTIC GYRO
Ian McMichael and Pochi Yeh Rockwell International Science Center
1049 Camino Dos Rios
Thousand Oaks, CA 91360 ABSTRACT
We describe a new type of phase-conjugate fiber-optic gyro that uses self-pumped phase conjugation. The self-pumped configuration is simpler than externally pumped configurations and permits the use of sensing fibers longer than the coherence length of the laser. A proof -ofprinciple demonstration of rotation sensing with the device is presented.
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Several types of phase-conjugate gyros are described in the literature,' - and we recently reported on the first demonstration of rotation sensing with a phase conjugate gyro.5 The passive phaseconjugate fiber-optic gyros described in references 3 and 5 are Michelson interferometers in which the arms contain fiber-optic coils that are terminated by externally pumped phase-conjugate mirrors. Since the phase-conjugate mirrors produce time-reversed waves, all reciprocal phase changes in the optical paths are compensated and do not effect the
output of the interferometer. However, since the phase shift produced by the Sagnac effect is nonreciprocal, the output of the interferometer is sensitive to rotation and can be used as a gyro.
Standard fiber-optic gyros 6 are Sagnac interferometers that are inherently insensitive to reciprocal phase changes and sensitive to nonreciprocal phase changes. This is true only when their operation is restricted to a single polarization mode,7 and the best fiber-optic gyros use polarization-preserving fibers and couplers.8 However, if the phase-conjugate mirrors in the phase-conjugate fiber-optic gyro preserve polarization,9 then nonpolarization preserving single-mode fibers, and even multimode fibers, can be used in the gyro.
In the externally pumped configurations described in references 3 and 5, the fiber-optic coils can be no longer than the coherence length of the laser. This limits the sensitivity of the device. It is true that longer coils can be used if a polarization-preserving fiber of equal length is used to carry the pumping waves to the phase-conjugate mirrors. However,
@1
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this defeats the above mentioned advantage in that the phase-conjugate gyro can use inexpensive multimode fibers and couplers. In this letter, we describe and demonstrate a self-pumped configuration of the phaseconjugate fiber-optic gyro that is not only simpler than the externally pumped configurations, but also allows for the use of fiber-optic coils that are longer than the coherence length of the laser.
Figure 1 shows a schematic of a self-pumped phase-conjugate fiber-optic gyro. Light from a laser is split by beamsplitter BS into two fibers Fl and F2. Fibers F1 and F2 are coiled such that light travels clockwise in Fl and counterclockwise in F2. Light waves traversing fibers Fl and F2 experience reciprocal phase shifts,

O1r =f k dl, and 0 r2 f k2 d12,

(1)

respectively, where dil and d12 are elements of length along Fl and F2, and k 2 = 2rn12/X. in addition, the nonreciprocal phase shifts,

0nr, *2TrRLQ/Xc and Onr2=-2TrR 2L2Q/Xc

(2)

are due to the Sagnac effect, where R,2 and L1, 2 are the lengths and radii of the fiber loops, and 0 is the rotation rate. The net phase shifts are then, Or, + 0nrl and 0r2 * 0rr2. On reflection from the phase-conjugate mirror, the phase shifts become, -Or, - Orr, and -0r 2 - 0rr2, where we

2

have dropped the phase shift of the phase conjugator' 0." since it is common to both waves, and we are only interested in the phase difference. It should be noted that the phase shift of the phase conjugator is common to both waves, only when both waves are reflected from the same phase-conjugate mirror, or when the phase-conjugate mirrors are coupled12. In the case of self-pumped phase conjugation in barium titanate, the two incident waves interact by coherently pumping the oscillation of a resonator formed by internal ref lectons in the crystal' 3. The counterpropagating waves in the resonator provide the pumping waves for degenerate four wave mixing (DFWM) with the incident waves. Due to the resonance condition, the DFWM pumping waves may be frequency shifted' 4.15 with respect to the incident waves and result in a frequency shift or time varying phase shift for the phase conjugate reflections. But again, since the two incident waves see the same pumping waves, this phase shift is common to both and does not effect the operation of this device since it is only sensitive to the phase difference. The phase shifts for the return trip in the fiber are given by, Or, - Onrl and 0r 2 - OrW2 , noting that the sign of the reciprocal contribution is the same as before whereas the nonreciprocal contribution has opposite sign. In the round trip, the reciprocal contributions cancel and net phase shifts are given by, -20r 1 and -20rr 2. The phase difference measured by the interference at detector D,

0 = -2(0nr 2-Or-r) = 4Tr(R1L,.R 2L2)Q/Xc,

(3)

3

is proportional to the rotation rate 0, and can be used to sense rotation. This configuration has several advantages over our previously
reported configuration.5 Here, we can use self-pumped phase conjugation. with the obvious advantage of not having to provide external pump waves that are coherent and form a phase-conjugate pair. In the externally pumped configuration the pump beam(s) involved in the writing of the index grating must be coherent with the probe wave to within the response time of the phase conjugator, and the two counterpropagating pump beams must be phase conjugates of each other to obtain a high fidelity phase-conjugate reflection. In initial experiments where an entire externally pumped phase-conjugate gyro was mounted on a rotating table, due to the slow time response of phase conjugation in the barium titanate crystal used, vibrations of the mounts providing the external pumping washed out the gratings involved in the phase conjugation, and precluded the measurement of rotation. As an additional advantage of the self-pumped configuration the sensing fibers F1 and F2 can be made longer (thereby increasing the sensitivity) than the coherence length of the laser provided that they are equal in length to within the coherence length.
Figure 2 shows the experimental setup of the self-pumped phaseconjugate fiber-optic gyro. Instead of using two separate fibers as shown in Fig.l, we use the two polarization modes of a single polarization-preserving fiber coil. All experiments are done with the
4

IV

argon laser running multilongitudinal mode at 515 nm. The highly reflective beamsplitter B51 isolates the laser from retroreflections. The polarization-preserving fiber Fl couples light from the laser to the remaining part of the apparatus that is mounted on a rotating table. The output end of FI is oriented such that the polarization of light exiting from the fiber is at 450 to the plane of the figure. The component polarized in the plane of the page is transmitted by the polarizing beamsplitter PBS and travels counterclockwise in the fiber coil,

whereas the component polarized perpendicular to the page travels

clockwise in the fiber coil. The fiber coil is made of approximately 9 m

of polarization-preserving fiber coiled in a square of 0.57 m sides, and is

oriented such that the polarization of the clockwise and

counterclockwise waves are preserved. When the two waves exit from

"

the coil they are separated by a Rochon polarizer PBS2. The polarization

of the light that travels straight through PBS2 is rotated by the

polarization rotator PR such that its polarization becomes identical to

that of the light deflected by PBS2. Both beams are incident as

extraordinary waves on a barium titanate crystal such that self-pumped phase conjugation occurs' 6. The reflected waves retraverse the fiber in

an opposite sense, recombine at PBS1, and travel back toward the laser

with a phase difference 0 = 8TrRLC/Xc. These waves are sampled by the

uncoated pellicle beamsplitter 1S2. and an additional phase delay of Tr/2

radians is impressed on them when they propagate through the

quarter-wave retarder X/4. The half-wave retarder is oriented such that

5

the intensities of the interferences measured by detectors DI and D2 are proportional to sin 0 and -sin o, respectively. The signals from these detectors go to a differential amplifier and a chart recorder.
Figure 3 shows the signal from the chart recorder. For t<0, the gyro was stationary. At t=0, the gyro was rotated first clockwise, then counterclockwise in a square-wave fashion for four cycles with an amplitude of approximately 60/s. The experimentally measured phase shift is in good agreement with the predicted phase shift of 0.04 radians.

The fast rotation rate is necessary for the signal to overcome the noise

that is evident in the phase shift recorded during the time t<0. Although

*

we are not certain of the major noise source, we believe it is rapid

reciprocal phase shifts that are not corrected due to the finite response

time of the phase conjugator.

Although this experiment does not demonstrate the the correction

of polarization scrambling in multimode fibers, it does demonstrate the

measurement of the Sagnac phase shift Eq. (3). To demonstrate a

self-pumped phase-conjugate fiber-optic gyro using multimode fibers,

one must use two multimode fibers terminated by the same self-pumped

polarization-preserving phase-conjugate mirror. Simply replacing the

polarization-preserving fiber in Fig. 2 with a multimode fiber does not

work, since the polarization of light after traveling down the fiber is

scambled and when the light reaches PBS1 part of it will go to the

detectors without being reflected from the phase-conjugate mirror. In

addition to the added complication of using two fibers and associated

.

complexity of terminating them on the same self-pumped polarizationpreserving phase-conjugate mirror (four beams going into one crystal), it is necessary to insure that the light waves from the two fibers are coherent to within the response time of the phase conjugator (the change in phase shifts for the two waves due to environmental effects on the fibers must be slower than the response time of the phase conjugator). The second of the above mentioned effects can be reduced by wrapping the two fibers together so that they see nearly the same environment.
In conclusion, we have described a new type of phase-conjugate fiber-optic gyro in which self-pumped phase conjugation can be employed to allow the use of sensing fibers that are longer than the coherence length of the laser source. In previous externally pumped configurations, it is possible to use fibers longer than the coherence length of the laser, by using a fiber to carry the pumping waves. This however complicates the setup and defeats some of the advantages of using phase conjugation. We have constructed a self-pumped phase-conjugate fiber-optic gyro and demonstrated rotation sensing.
This work is supported by the Office of Naval Research contract sN0014-85-C-0219.
7

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REFERENCES
1. J.-C. Diels and 1.C. McMichael, Opt. Lett. 6. 219 (1981). 2. P. Yeh, J. Tracy, and M.Khoshnevisan, Proc. SPIE 412. 240 (1983). 3. C. J. Bord'e, in . C.IL?,CL?/n_
P. Meystre and M.0. Scully, Eds. (Plenum, New York, 1983) 269. 4. B. Fischer and S. Sternklar, Appl. Phys. Lett. A7. 1 (1985). 5. P. Yeh, 1.McMichael, and M.Khoshnevisan, App). Opt. 2M. 1029 (1986). 6. R. Bergh, H. Lefevre and H. Shaw, IEEE J. Lightwave Tech. 2. 91
(1984).
7. R. Ulrich, Opt. Lett. 5, 173 (1980). 8. W.Burns, R. Moeller, C.Villarruel. and M.Abebe, Opt. Lett. f. 540
(1983).
9. 1. McMichael, M. Khoshnevisan, and P. Yeh, Opt. Lett. J4. XXX (1986). 10. 1.McMichael, P. Yeh, and M.Khoshnevisan, Proc. SPIE 613. XXX
(1986). K1. S. Kwong, A. Yariv, M.Cronin-Golomb, and B. Fischer, J. Opt. Soc.
Am. A 3. 157 (1986). 12. M.Ewbank, P. Yeh, M.Khoshnevisan, and J. Feinberg, Opt. Lett. 10.
282 (1985). 13. M.Ewbank and P. Yeh, Proc. SPIE IL XXX (1986). 14. P. Yeh, J. Opt. Soc. Am. B 2. 1924 (1985). 15. M.Ewbank and P. Yeh, Opt. Lett. IL 496 (1985). 16. J. Feinberg, Opt. Lett. 2. 486 (1982).

8
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-' :

FIGURE CAPTIONS
Fig. I Schematic of a self-pumped phase-conjugate fiber-optic gyro. Light from a laser is split by beamsplitter BS into two fibers Fl and F2 that are coiled such that light travels clockwise in F and counterclockwise in F2. Light traversing the fibers experiences phase shifts due to thermal, mechanical, and rotational effects. The self-pumped phase-conjugate mirror PCM produces time reversed waves that compensate for the reciprocal phase changes produced by thermal and mechanical effects, but do not compensate for the nonreciprocal phase shift produced by rotation (Sagnac effect). Therefore, rotation can be sensed by measuring the interference between the recombining waves at detector D.
Fig. 2. Experimental setup of the self-pumped phase-conjugate fiberoptic gyro. Instead of the two fibers shown in Fig. 1, the experimental setup shown here uses the two polarization modes of the polarization-preserving fiber-optic coil. Light from the laser is incident on polarizing beamsplitter PBSi with its polarization at 45" to the plane of the page. The components reflected and transmitted by PBSI travel clockwise and counterclockwise respectivly, in the fiber coil. The two beams recombine at PBSI and are then split at PBS2. One of the beams has its polarization rotated by PR, and both beams are incident on a barium titanate crystal such that self-pumped phase conjugation occurs. The
9

reflected waves retraverse the fiber in an opposite sense, recombine at PBS 1, and travel back toward the laser with a phase difference 0, that is proportional to the rotation rate. These waves are sampled by the beamsplitter BS2 and an additional phase delay of Tr/2 radians is impressed on them when they propagate through the quarter-wave retarder X/4. The half-wave retarder is oriented such that the intensities of the interferences measured by detectors DI and D2 are proportional to sin 0 and -sin 0, respectively.
Fig. 3. Measurement of the Sagnac phase shift in the self-pumped phaseconjugate fiber-optic gyro This figure shows a chart recording of the output of a differential amplifier connected to detectors DI and D2 In the experimental set up of a self-pumped phase-conjugate fiber-optic gyro shown in Fig. 2. For t<O, the gyro was stationary. At t=O, the gyro was rotated first clockwise, then counterclockwise in a square-wave fashion for four cycles with an amplitude of approximately 60/s. The experimentally measured phase shift is in good agreement with the predicted phase shift of 0.04 radians.
10

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APPENDIX 5.2 Polarization-Preserving Phase Conjugator (Paper to appear in Opt. Lett., August (1986))

'"
39 C7714A/ibs

SC5424 .AR POLARIZATION-PRESERVING PHASE CONJUGATOR Ian McMichael, Monte Khoshnevisan and Pochi Yeh
Rockwell International Science Center Thousand Oaks, CA 91360 ABSTRACT
Using a single self-pumped crystal of barium titanate, we demonstrate a method for producing the phase-conjugate of an incident wave having arbitrary polarization. Our experimental results show that the phase-conjugate wave produced by this method reproduces both the ellipticity and the helicity of the polarization of the incident wave.
4.
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Most demonstrations of the correction of wavefront distortions by phase conjugation use linearly polarized light and optically isotropic wavefrontdistorting media. In some cases, however, one may also wish to correct for the changes in polarization induced by optically anisotropic media. For example, one may wish to correct for the polarization scrambling due to environmentally dependent birefringence that is a source of noise and signal fading in fiberoptic gyros. I Self-pumped phase conjugation is very efficient in barium titanate, 2 but only for extraordinary light since it utilizes the large electrooptic coefficients r42 and r51. In this letter, we demonstrate a scheme for using a single self-pumped crystal of barium titanate to produce the phaseconjugate of a wave having arihitrary polarization.
In general, the reflectivity of a phase-conjugate mirror is described a 2 x 2 tensor that determines the amplitude, phase, and polarization state of the reflected light. Only when this tensor reduces to a scalar can a phaseconjugate mirror correct for both the wavefront distortions and the changes in polarization induced by optically anisotropic wavefront-distorting media. 3 We call a phase-conjugate mirror for which the reflectivity tensor reduces to a scalar, a polarization-preserving phase conjugator (PPPC). To show that a PPPC can correct for the change in polarization state introduced by optically anisotropic media, consider the situation of a plane wave with polarization state described by a vector amplitude A,

A

A

1

is incident on an optically anisotropic medium, followed by a phaseconjugate mirror. After passing through the anisotropic medium, the amplitude becomes

A' TA,

[xx xy]

(2)

41 C7714A/jbs

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where T is a Jones matrix 4 that describes transmission through the anisotropic medium, and where txx, txy, tyx and tyy are the tensor elements. After reflection from the phase-conjugate mirror, the amplitude becomes

A" = R(TA)*,

R = ryxx ryXY

(3)

where R is a tensor that describes reflection from the phase-conjugate mirror, and where rxx, rxy, ryx and ryy are the tensor elements. After passing back through the anisotropic medium, the amplitude of the returning wave is given by

TRT*A*. (4)

Since in general this is not proportional to the phase-conjugate of the amplitude of the incident wave, the returning wave does not have the same

polarization as the incident wave. However, if the reflectivity tensor reduces to a scalar r, then the amplitude of the reflected wave becomes

Fr 0

rTT*A*, for R [Or

r

(5)

For optically anisotropic media, such as birefringent or optically active media, the Jones matrix has the property, TT* 1. In this case, the amplitude of the reflected wave becomes,

rA*, for R = r, TT* = 1.

(6)

Since this is proportional to the phase-conjugate of the incident wave, the polarization of the reflected wave is identical to that of the incident wave. Thus, we have shown that a PPPC can correct for the change in polarization state introduced hy optically anisotropic media. The PPPC cannot correct for the changes in polarization introduced by nonreciprocal phenomena such as the Faraday effect, where TT* 1.

42 C7714A/jbs

,"

V

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Several methods have been proposed for polarization-preserving phase conjugation. 3'5 ,6 Polarization-preserving phase conjugators can be made using media that have a tensor reflectivity, by decomposing the incident wave into two components having orthogonal polarizations, phase-conjugating these components, and recombining the conjugates. 5 The result is a polarization-preserving phase conjugator only if the components recombine with the correct amplitude and phase. This is insured if the losses and the amplitudes and phases of the complex phase-conjugate reflectivities are the same for both components. A schematic of our polarization-preserving phase conjugator that uses this method is shown in Fig. 1. To understand how it works, consider an incident wave having a polarization described by a complex vector amplitude A = (a,, a2 ). This wave is decomposed into two orthogonally polarized components, (a1 , 0) and (0, a2 ), by the polarizing beam splitter PBS. The component with polarization orthogonal to the plane of the figure (0, a2 ) has its polarization rotated by the half-wave retarder X/2 such that its polarization state becomes (a2, 0). Both components are incident on a barium titanate crystal at an angle such that self-pumped phase conjugation occurs. Since both components have the same polarization and are reflected from the same phase-conjugate mirror, they experience the same complex phase-conjugate reflectivity r, becoming r(al*, 0) and r(a2*, 0). The half-wave retarder restores the second of these components to its original polarization, r(O, a2*). When the two components recombine at the polarizing beam splitter, they form the phase-conjugate with the same polarization as the incident wave, r(al*, a2*) rWA*. Thus, the configuration shown in Fig. 1 acts like a polarization-preserving phase conjugator, and henceforth we refer to it as "the PPPC".
Using the self-pumped configuration in the PPPC results in a frequency shift between the components incident on the crystal and their conjugates. 8 ,9 However, since both components are incident on the same region of the crystal, they see the same pumping waves, and therefore they experience the same frequency shift and phase-conjugate reflectivity. In fact, the two beams can be thought of as two components of a single probe beam. In the experiment, the two beams are made to overlap by aligning mirror M4 while observing the scattered

........

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

43 C7714A/jbs

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light from the beams in the crystal. This procedure is sufficient to obtain the results presented here, and exact alignment is not required. The two beams interact by coherently pumping the oscillation of a resonator formed by internal reflections in the crystal. 10 The counterpropagating waves in the resonator then provide the pump waves for DFWM. Since the two beams entering the crystal see the same pump waves, they experience the same DFWM reflectivity. It should be noted that the two beams entering the crystal must be coherent to within the response time of the crystal. In the experiment, the path lengths from the polarizing beam splitter to the crystal were made equal to within the coherence length of the laser. The excellent frequency locking I1 of the conjugates when both beams are present is indicated by the fact that no beating is detected in the return from the PPPC.
Attention paid to experimental details results in improved operation of the PPPC. For example, the analysis above assumes that the polarizing beam splitter completely separates the two polarization components. Although many types of polarizing beam splitters do not do this, we now use a Rochon type that does. In our first experiments, we used a Glan-laser polarizing beam splitter that does not completely separate the two polarization components (some p polarization is reflected along with the s polarization). The p polarization that is reflected by the polarizing beam splitter is not reflected by the phaseconjugate mirror, and is lost. When the conjugated waves recombine at the polarizing beamsplitter, although the two polarization components have the correct phase relationship, they do not have the correct intensity ratio required to reproduce the polarization of the incident wave. This resulted in imperfect operation of the PPPC. For example, with a linear polarized input wave, the polarization of the reflected wave was rotated by as much as 80. The analysis also assumes no losses in the system, or at least losses that are equal for both components. Among other things, this suggests that the half-wave retarder should be antireflection coated. Our half-wave plate has a total loss of less than 0.5%. Keeping the angle between the two components incident on the crystal small helps to insure that they experience the same complex phase-conjugate reflection coefficient. In our experiment this angle is 10. It should also be
44 C7714A/jbs

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noted that this experimental configuration is an interferometer. Phase changes occurring on a time scale faster than the response time of the crystal that are not common to both components alter the polarization of the reflected wave. Therefore, the setup should be compact and rigid.

To show that the PPPC reproduces the ellipticity of polarized light, we

use the experimental setup shown in Fig. 2(a). A highly reflective beam split-

ter BS1 isolates the laser from any retroreflections of its output. The polari-

zer P1 insures that the light is linearly polarized in the plane of the figure.

In all the experiments presented here, the power after P1 is - 1 mW at 514.5 nm. Either a half-wave retarder X/2 or quarter-wave retarder x/4 is used to

alter the polarization state of the light. This light is then reflected from

either a normal mirror M (multilayer dielectric high reflector), or the PPPC.

The reflected light is sampled by the beam splitter BS2, and it is analyzed by

the combination of polarizer P2 and detector D. Since BS2 is an uncoated pellicle beam splitter used near normal incidence (the angle of incidence is

exaggerated in the figure; the actual angle of iocidence is 20), the reflection

*coefficients

for the s and p polarizations are nearly equal and the polarization

-"

measured by P2 and D is nearly the same as that reflected by M.

Figure 2(b) shows the measured angle of polarization for the reflection from the polarization-preserving phase conjugator, PPPC, as a function of the measured angle of polarization for the reflection from the normal mirror M, for various orientations of the half-wave retarder. Zero degrees corresponds to polarization in the plane of the previous figures. The open circles are the data (with diameters corresponding to the uncertainty), and the solid line indicates what is expected in the case of an ideal polarization-preserving phase conjugator. The measured ellipticity of the polarization for the light reflected by the PPPC (defined as the ratio of the minor polarization axis to the major polarization axis) never exceeded 1%.

Figure 2(c) shows the measured polarization ellipse for the reflection from the polarization-preserving phase conjugator (dotted line), and the measured polarization ellipse for the reflection from the normal mirror (solid

45 C7714A/jbs

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line), for various orientations of the .quarter-wave retarder. Zero degrees corresponds to orientation of the axis of the quarter-wave retarder in the plane of the previous figures. The ellipticity of the reflected light is measured by rotating polarizer P2, and noting the minimum and maximum intensities at detector D.
The results shown in Fig. 2 demonstrate that the reflection from the PPPC reproduces the ellipticity of the polarization of the incident wave. To show that it reproduces the helicity of the polarization of the incident wave, we use the setup shown in Fig. 3(a). A quarter-wave retarder X/4 is placed between the sampling beam splitter and the mirror and is oriented such that the light incident on the mirror is converted from linearly polarized light to circularly polarized light. Figure 3(b) shows the measured polarization ellipses for the normal mirror and the polarization-preserving phase conjugator. Light reflected from the normal mirror changes helicity. After passing back through the quarter-wave retarder, the polarization of the reflected light is orthogonal to the incident light. This is the principle by which qiiarter-wave isolation works. On the other hand, light reflected by the polarization-preserving phase conjugator has the same helicity as the incident light and returns to its original polarization state after passing back through the quarter-wave retarder.
In order to quantify how well the technique works when there is an aberration in the beam, a multimode fiber is placed at the position of the quarter-wave retarder in Fig 2(a). The multimode fiber scrambles the polarization of the input wave. With a normal mirror at the end of the fiber, approximately 50% of the power returns in each of the two linear polarization states (parallel and orthogonal to the linear input polarization). With the polarization preserving phase conjugator at the end of the fiber, less than 1% of the light returns with the orthogonal polarization.
In conclusion, we have demonstrated a polarization-preserving phase conjugator that is capable of correcting for both the phase distortions and the changes in polarization induced by optically anisotropic phase distorting media. Potential applications of this device include the phase-conjugate fiber-optic gyro 12 ,13 and interferometry with multimode fibers. 14
46 C7714A/jbs

s-.

-j--

77.7

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This work is partially supported by the Office of Naval Research.
REFERENCES
1. R. Ulrich and M. Johnson, Opt. Lett. 4, 152 (1979). 2. J. Feinberg and R.W. Hellwarth, Opt. Lett. 5, 519 (1980). 3. P. Yeh, Opt. Comm. 51, 195 (1984). 4. See, for example, A. Yariv and P. Yeh, "Optical Waves in Crystals," (Wiley,
New York, 1984). 5. N.G. Basov, V.F. Efimkov, I.G. Zuberev, A.V. Kotov, S.I. Mikhailov, and
M.G. Smirnov, JETP Lett. 4, 197 (1978). 6. G. Martin, L.K. Lam, and R.W. Hellwarth, Opt. Lett. 5, 185 (1980). 7. J. Feinberg, Opt. Lett. 7, 496 (1982). 8. W.B. Whitten and J.M. Ramsey, Opt. Lett. 9, 44 (1984). 9. J. Feinberg and G.D. Bacher, Opt. Lett 9, 420 (1984). 10. M. Ewbank and P. Yeh, SPIE Proceedings 613, XXX (1986). 11. M.D. Ewbank, P. Yeh, M. Khoshnevisan, and J. Feinherg, Opt. Lett. 10, 282
(1985). 12. C.J. Borde, NATO Advanced Study Institute in Quantum Optics, "Experimental
Gravitation and Measurement Theory," P. Meystre and M.O. Scully, eds. (Plenum Press, 1983). 13. P. Yeh, I. McMichael, and M. Khoshnevisan, Appl. Opt. 25, XXX, April 1, 1986. 14. B. Fischer and S. Sternkler, Appl. Phys. Lett. 46, 113 (1985).
-

......................-

47 C7714A/jbs

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FIGURE CAPTIONS

Fig.

1 Polarization-preserving phase conjugator "PPPC". Light incident on the polarizing beam splitter PBS is decomposed into two orthogonally polarized components. One component is rotated 900 by the half-wave retarder X/2. Both components are incident on a barium titanate crystal such that self-
pumped phase conjugation occurs. When the two conjugates recombine at PBS, they form a conjugate wave that reproduces the polarization of the

incident light.

Fig. 2 Reproduction of the ellipticity of polarized light by the PPPC. (a) Experimental setup. A half- or quarter-wave retarder alters the polarization state of light incident on a normal mirror M or the PPPC. (b) Measured angle of polarization for the reflection from the PPPC vs that from M. (c) Measured polarization ellipse for the reflection from
the PPPC (dotted line) compared with that from M (solid line).

Fig. 3 Reproduction of the helicity of polarized light by the PPPC. (a) Experimental setup. The quarter-wave retarder X/4 is oriented such that the light incident on M, or the PPPC is converted from linearly polarized light to circularly polarized light. (b) Measured polarization ellipses for the reflections from M and the PPPC after retraversing X/4.

48 C7714A/jbs

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APPENDIX 5.3

SC5424 .AR

Absolute Phase Shift of Phase Conjugators

(Paper to appear in Proc. SPIE 613 (1986))

55 C7714A/jbs

*
-rature).
I*

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Absolute Phase Shift of Phase Conjugators
Ian McMichael, Pochi Yeh and Monte Khoshnevisan
Rockwell International Science Center 1049 Camino Dos Rios, Thousand Oaks, CA 91360
Abstract
We present theoretical expressions and experimental measurements of the absolute phase shifts of phase conjugators. Photorefractive media, transparent media, saturable absorbers and saturable amplifiers are considered in the analysis. Experimental measurements of the absolute phase shifts are presented for barium titanate, strontium barium niobate and ruby.
Introduction Although the reflectivity of phase conjugators has been of significant
interest recently, the phase of the phase-conjugate wave has received little attention. This phase determines the operating point of some phase-conjugate interferometers. If this phase can be controlled, the interferometers can be biased at the operating point of highest sensitivity and linear response (quad-
With this motivation, we have studied the absolute phase of phase conjugators theoretically and experimentally.
Theory We consider the usual case of degenerate four-wave mixing of two coun-
terpropagating pump waves having amplitudes Ai and A2 with a probe wave having amplitude A4 propagating in the +z direction and a phase-conjugate wave A3 propagating in the -z direction. Let the phase of the phase-conjugate reflection 03, where An : IAnlein, be written as,

56 C7714A/jbs
-.-

SC5424.AR

3 :00 + 01 + ¢2 -04

(1)

We then refer to t as the absolute phase shift of the phase conjugator. The

*

phase shift 0 can be obtained from the solution of the coupled-wave equations

describing the degenerate four-wave mixing. In general, 40 depends on the type

of grating (index, absorption, or gain) involved in the degenerate four-wave

mixing, the phase shift of this grating with respect to the light intensity

pattern that produces the grating, and the intensities of the interacting waves.

For a photorefractive medium, the complex amplitude of the phaseconjugate wave at the input to the medium (z = 0) is given by, 1

A3 (0) = A4*(O)(AI/A 2*)(eYL - 1)/(rle-YL + 1)

(2)

where L is the length of the medium, r is the pump-beam intensity ratio,

r = IA2 /AI1 2 (3)

and y is the complex coupling constant that depends on the physical process involved in the generation of the hologram. If ones makes the usual assumption that the phase grating in photorefractive media is shifted by n/2 radians with respect to the intensity pattern, then y is real and the phase shift of a photorefractive phase conjugator is

00 = 0 for y < 0

= n for y > 0

(4)

For degenerate four-wave mixing in a transparent medium with a local
response (no phase shift of the index grating with respect to the intensity
pattern), the amplitude of the phase-conjugate wave at the input of the medium is given by, 2

57 *C7714A/jbs
I-p

SC5424 .AR

A3 (0) = -iA4*(O)(I *I/K) tan I KIL where K is the complex coupling constant,

(5)

K* = (2nw/cn)XAiA 2

(6)

From Equations (5) and (6), we obtain the phase of the phase conjugator for a transparent medium with a local response,

o = -n/ 2 for x > 0

= +'R/2 for x < 0

(7)

Finally, for degenerate four-wave mixing in a saturable absorber or
amplifier with a local response, the amplitude of the phase-conjugate wave at the input of the medium is given by, 3

A3(0) = -iK*A 4*(O) [sin wL/(w cos wL + aR sin wL)J K* = iao[(l-i6)/(l + 62)] (2AIA 2/Is)/(l + 41/Is) 3/2 a = ao[(l - i6)/(1 + 62)1 (1 + 21/Is)/(I + 41/Is) 3/2 = aR " ial w = (1K12 - aR2 )11 2

(8) (9) (10) (11)

a0 is the line-center small-signal field attenuation or gain coefficient, Is is the saturation intensity for the detuning from line center 6, and I is the intensity of the pump waves (I1 = 12 = I). If the frequency of the interacting waves is on resonance with the atomic transition of the medium, then 6 = 0 and w is imaginary. From Equation (8), we obtain the phase of the phase conjugator for degenerate four-wave mixing on resonance with a saturable absorber or gain medium,

58 C7714A/bs

o0 = 0 for ao > 0 = R for ao < 0

SC5424 .AR (12)

Experiment
To measure the phase of the phase-conjugate reflection Oo, we use the experimental setup shown in Figure 1. Light from a laser is split by beamsplitter BS into two arms of an interferometer. In the reference arm of the interferometer, the light transmitted by BS passes through a photorefractive crystal XTL, and is retroreflected by mirror M1. This provides the counterpropagating pump waves for degenerate four-wave mixing in the crystal. In the signal arm of the interferometer, the light reflected by BS is then reflected by mirror M2 to provide the probe beam for degenerate four-wave mixing. The intensity measured by detector D is given by

I I + 12 - 2A7-_ cos 00

(13)

where I, and 12 are the intensities of the combining waves. This equation indicates that the operating point of this interferometer is completely determined by the absolute phase of the phase conjugator 0, and is independent of phases of the pumping and probe waves.
D

BS

M2 Fig. 1 Experimental setup used to measure the absolute phase of
phase conjugators.
59 C7714A/jbs

SC5424 .AR

To derive Equation (13), we consider a light wave of unit amplitude that is incident on the beamsplitter BS. The amplitudes of the transmitted and externally reflected waves are then t and r, respectively. The amplitudes of the pumping waves at the crystal are tei I and teic , and the amplitude of the probe wave is rei , where the phase factors D describe the accumulated phase for propagation from the beamsplitter to the nonlinear medium. These three waves interact in the nonlinear medium to produce a phase-conjugate wave with an amplitude A3,

A3 = pt~rriei(0o + + Z2 - @)

(14)

-.

where p is a real constant describing the reflectivity of the phase conjugator. When this wave returns to the beamsplitter, it combines with the wave retroreflected by mirror M1 to produce an intensity I at the detector D given by

1 : Ipt2r*ei 0o++ I" + )+ trei((1 + 42)I2

(15)

where r' is the amplitude reflection coefficient for internal reflection from the beamsplitter. Using the Stoke's relation tr* = -t*r', we can obtain Equation (13) from Equation (15).
In principle, Oo can be determined from a measurement of the intensities. However, in practice, the uncertainties in measuring the intensities do not allow for an accurate determination of Oo. A more accurate determination can be made by modulating the phase of the cosine term at a frequency W and measuring the ration R of the fundamental power to the second harmonic power at the detector. In the experiment, a piezoelectric transducer is attached to the mirror M2 to produce a phase modulation 0m sin wt. The phase modulation is faster than the response time of the nonlinear medium in which the degenerate four-wave mixing takes place so that it is not compensated by phase conjugation. The amplitude of the phase modulation is small (i << m) so that the gratings involved in the degenerate four-wave mixing are not washed out. The time varying intensity at the detector D is proportional to,

60 C7714A/jbs

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"cos(0O + 0. sin t)

(16)

From this equation, we obtain the ratio R of the fundamental power to the second harmonic power

R = [Jl(0m)/J2(0m)]tan0O

(17)

where Ji and J2 are Bessel functions. For Om << n, the absolute phase shift of the phase-conjugate reflection is given by

O0 tan-' (R$m/4 )

(18)

Results
Using the technique described above, we nave measured the phase shift of the phase-conjugate reflections from barium titanate, strontium barium niobate and ruby. Our results are given in Table 1, where 0 is the angle between the grating k vector and the crystal axis.

Table 1

Phase of Phase-Conjugate Reflections

..

Material

0

0

BaTiO 3

00

(19 ± 3)0

450

(6 ± 4)0

1350 1800

(176 ± 3)0 (164 _ 3)0

SBN

00

(3 3)0

1800

(175 ± 5)0

Ruby

-

(80 ± 5)0

61 C7714A/jbs

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For the photorefractive materials BaTi03 and SBN, the measured values of Oo

compare well with the expected values from Equation (4). The differences

between the expected values Oo = 00 and 1800, and the measured values Oo = (19 t

9-

3)0 and (164 ± 3)0 for BaTi03 at e = 0' and 1800 indicate that the index grating

is not

shifted

by

exactly

2 iT/

radians,

as

is often

assumed.

We have verified

this fact by an independent measurement of the two-wave mixing gain 1 as a func-

tion of the frequency detuning between the two waves. The fact that for ruby, Oo is close to 90', indicates that the grating involved is predominantly an

index grating rather than an absorption grating.

Summary
We have presented theoretical expressions and experimental measurements of the absolute phase shifts of phase conjugators. Photorefractive media, transparent media, saturable absorbers and saturable amplifiers were considered in the analysis. Experimental measurements of the absolute phase shifts for barium titanate, strontium barium niobate and ruby are in good agreement with the theory.

Acknowledgements This work is supported by the Office of Naval Research.

References

1. Fisher, B., Cronin-Golomb, M., White, J., and Yariv, A., "Amplified Reflection, Transmission, and Self-oscillation in Real-time Holography,"
Opt. Lett., Vol. 6, pp. 519-521, 1981.
2. Yariv, A. and Pepper, D, "Amplified Reflection, Phase Conjugation, and Oscillation in Degenerate Four-Wave Mixing," Opt. Lett., Vol. 1, pp. 16-18, 1977.
3. Abrams, R. and Lind, R., "Degenerate Four-Wave Mixing in an Absorbing Media," Opt. Lett., Vol. 2, pp. 94-96. 1978; "Errata," Opt. Lett., Vol. 3, p 205, 1978.

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APPENDIX 5.4 Phase-Conjugate Fiber-Optic Gyro (Paper appeared in Appi. Opt. 25, 1029 (1986))
63 C7714A/jbs

Reprinted from Applied Optics, Vol. 25, Page 1029, April 1, 1986 Copyright © 1986 by the Optical Society of America and reprinted by permission of the copyright owner.

Phase-conjugate fiber-optic gyro

4 rRLS!

4

Pochi Yeh, Ian McMichael, and Monte Khoshnevisan

XC

Rockwell International Science Center, 1049 Camino Dos Rios. Thousand Oaks, California 91360.
Received 23 December 1985.
0003-6935/86/071029-02$02.00!0. 9 1986 Optica Society of America.

This phase shift can be measured by using the interferente with the reference beam from the other arm. Notice that a-
a result of the phase reversal on reflection, the reciprocal phase shift kL is canceled on completion of a round trip The net phase shift left is due to anything nonreciprocal such

liteSreavteurrael.'t-y4peIsnotfhpishaLseett-ecor,nwjuegdaetescgryibroe

are proposed a new type of

in the fiber-

optic gyro that uses the phase-reversal property of polariza-

as rotation. This net phase shift is proportional to the rotation rate and
can be used for rotation sensing. In addition, if the phaseconjugate reflector is polarization-preserving,6f- it will pro-

tion-preserving phase conjugation. Although the insensitiv- duce a true time-reversed version of the incident wave and

ity of phase-conjugate gyros to reciprocal phase shifts and will undo all the reciprocal changes (e.g., polarization scram-

their sensitivity to nonreciprocal phase shifts such as the bling, modal aberration, temperature fluctuation) when the Faraday effect have been reported,. 4 to date no one has light completes the round trip in the fiber. Since the polar-

demonstrated rotation sensing. In this Letter. we report the ization scrambling and modal aberration of mult imode fiber,

*

first demonstration of rotation sensing with a phase-conju- can be corrected by polarization-preserving phase conjug.-

gate gyro.

tion. multimode fibers can replace the polarization-preser'-

Polarization scrambling is a well-known source of signal ing single-mode fiber in this new type of gyro.

fading and noise in fiber-optic gyros. Polarization-preserv-

Figure 2 shows a schematic diagram of the experimental

ing fibers and couplers must be used to decouple the two setup used to demonstrate the phase-conjugate fiber-optu states of polarization and hence improve the sensitivity.5- In gyro. Since this experiment does not use a polarization-

the phase-conjugate fiber-optic gyro, a polarization-preserv- preserving phase-conjugate mirror, it does not demonstrate

ing phase conjugator is used to restore the severely scrambled waves to their original state of polarization. 6-5 This

the correction of polarization scrambling. However, the experiment does measure the phase shift described by Eq

eliminates the signal fading and noise due to polarization (3). A highly reflective beam splitter BS1 isolates the argon

scrambling without the need for polarization-preserving fi- laser from retroreflections of its output. The light reflected

ber. Referring to Fig. 1.we consider a phase-conjugate Michel-
son interferometer 9 in which a fiber loop is inserted in the

by BS2 is focused by lens LI (60-cm focal length) into a crystal of barium titanate to provide the pumping waves for
degenerate four-wave mixing (DFWM). The light transmit-

arm that contains the phase-conjugate reflector 0*. We now ted by BS2 is split into two arms of a Michelson interferome-

examine the phase shift of light as it propagates along the ter by BS5. One arm of the interferometer contains a 10-cm

fiber. From point A to point B, the light experiences a phase radius coil of -7 m of optical fiber. Since the phase-conju-

shift of

gate mirror in this experiment is not polarization-preserving.

2rRLQ Xc

we use single-mode polarization-preserving optical fiber. Light exiting the fiber provides the probe wave for DFWM.

where k = (21rn)/\ is the wave number and L is the length of

fiber, R is the radius of the fiber coil, 11is the rotation rate, Xis

the wavelength, and c is the velocity of light. The second

term in Eq. (1) is due to rotation. In the return trip, the

phase shift is

2TRLQ

0, -L +

,-

(2)

where we notice that the term due to rotation is reversed
because of the change in propagation direction relative to the rotation. If there were no phase conjugation, the total
round-trip phase shift due to regular mirror reflection would be 2kL. However, because of the phase reversal on phaseconjugate reflection, the round-trip phase shift becomes

The c-axis of the barium titanate crystal is parallel to the long faces of the crystal and points in the direction of beam splitter BS3. The pumping waves from mirrors M2 and M3
have powers of 18 and 3 mW, respectively, and their angle of incidence is -45*. The probe wave, exiting from the end of the fiber loop, makes a small angle (<100) with the pumping wave from mirror M2 and has a power of 0.7 mW. Under these conditions we obtain a phase-conjugate reflectivity of 50% and a response time of 0.1 s. The reference arm of the interferometer is terminated by a mirror M4 mounted on a
piezoelectric transducer so that the operating point of the interferometer can be set at quadrature. Light from both
arms combines to form complementary fringe patterns at detectors D3 and D4. Detectors DI and D2 measure the powers in the recombining waves.
The fiber coil is rotated with the rest of the setup remaining fixed at various rotation rates [first clockwise (CW), then

04

03

02

/*i

Sl

A - "02; A2- _03

Fig.~~ ILrAwhiSgaoi h hs-onuaefbrotc yo

i.2

ERimna setu th hs-Lo5jgtefbrotcg

1 Ap

/V&S1285 No. 7 PLEDOTCS 12

L" . S A0l 12A _W- D3.

'",€;-.€"€" "; ","..,".". "..""..,', ", "."-" -?'," ," '- '"." ,".," ," .'",- .'. . . ', .. ,' .. , ..

L.'

-- : .--
- ",' ."L5.

tion sensing by a phase-conjugate gyro.

This research is partially supported by the Office of Naval

02

Research.

0 1r

-

60

0

0.

60

120

ROTATION RATE ,dg,,, ,

Fig. 3. Measured phase shift as a function of applied rotation rate.

counterclockwise (CCW), etc.] in a square wave fashion for 10 cycles with an amplitude of 1200. The measured powers from all detectors are used to calculate the average phase shift between rotation in the CW and CCW directions of rotation. Figure 3 shows a plot of the measured phase shift as a function of the rotation rate. The solid line indicates the expected rotation-induced Sagnac phase shift. The large uncertainty in the data is due to rapid (faster than the response time of the DFWM) phase changes that are produced by the twisting of the fiber when the fiber loop is rotated and that act as a source of noise.
In conclusion, we have proposed a new type of fiber-optic gyro that uses polarization-preserving optical phase conjugation. and we have presented the first demonstration of rota-

Referwces

1. J.-C. Diels and I. C. McMichael, "Influence of Wave-Front-Conjugated Coupling on the Operation of a Laser Gyro." Opt. Lett. 6,

2!9 (1981). 2. P. Yeh, J. Tracy. and M. Khoshnevisan, "Phase-Conjugate Ring

Gyroscopes," Proc. Soc. Photo-Opt. Instrum. Eng. 412. 240 (1983).

3. C.J. Bode, "Phase Conjugate Optics and Applications to Interfer-

oamndetrMy eaansdurteomLeansterThGeyorroys,coPp.e.M" einysEtrxepaenrdimMen. ta0l.

GraVitation Scully, Eds

(Plenum. New York. 1983). pp. 269-291. 4. B. Fischerand Shmuel Sternklar, "New Optical Gyroscope Based
on the Ring Passive Phase Conjugator." Appl. Phys. Lett. 47, 1
(1985).
5. W.K. Burns, R.P. Moeller, C.A.Villarruel.and M.Abebe,"Fiber Optic Gyroscopes with Polarization Holding Fiber." Opt. Lett. 8, 540 (19831.
6. P. Yeh. "Scalar Phase Conjugation for Polarization Correction." Opt. Commun. 51, 195 (1984 1.
7. 1.McMichael and M. Khoshnevisan, "Scalar Phase Conjugation Using a Barium Titanate Crystal." in Technical Digest, Conference on Lasers and Eletro.Optics (Optical Society of America. Washington. D.C.. 1985), paper THN1.
8. 1. McMichael. M. Khoshnevisan. and P. Yeh. " Polarization-Preserving Phase Conjugator." submitted to Opt. Lett.
9. M. D. Ewbank. M.Khoshnevisan. and P. Yeh. "Phase-Conjugate Interferometry." Proc. Soc. Photo-Opt. Instrum. Eng. 464, 2 (1984).

I

1030 APPLIED OPTICS / Vol. 25, No. 7 1 April 1986 ",'."" --''.''"'e" ,r"."'.''" " ." ". '.,', "".',:,'' .' " '." ' -"€''" ' . '_,A '"";N"' " '

" ""

o %Rockwell International
Science Center

APPENDIX 5.5

SC5424 .AR

Theory of Unidirectional Photorefractive Ring Oscillators

66 C7714A/jbs

Reprinted from Journal of the Optical Society of America B, Vol. 2, page 1924, December 1985 Copyright C 19s5 by the Optical Society of America and reprinted by permission of the copyright owner.
Theory of unidirectional photorefractive ring oscillators

Pochi Yeh
Rvcki cliInternutional Science Center. Thousund Oukb, Culiforniu 913mu
Recoived March I1,1985:. ccepted Jul\ 11.1985 Amplification otking to holographk two-wave mixing in photorefractive crystals can be utilized toachieve unidire(tional ring oscillation. Unlike orthe conventional gain medium (e.g.. He-NeL. the gain bandwidth of photorefrative w , wave coupling is very narrok (a fewyhertz for BaTiO, . Despite this fact. the ring resonat or can still oscillate over alarge range of cavity detuning A theory is presented that describes how the oscillating mode attains the round-trip phase (ondition.

*,ne-w
*analysis.-

INTRODUCTION

The photorefractive effect in electro-optic crystals (e.g..

BaTiO.. LiNbO .. I has been widely studied for many applications. These include- real-time holography.swosptatitceal data

storagenot

cently. increasing attention has been focused on using coher-

ent

signal beam applications

amplification in two-wave mixing. These include image amplification.' vibrational

nonrecicpproocaal ttrnansmission.?4 annd laaseerr-ggyrro biisas-

ing.i The coherent signal beam amplification in two-wave

mixing can also be used to provide parametric gain for unidi-

rectional oscillation in ring resonators. Although such an

oscillation has been observed by using a BaTiO: crystal in a

ring resonator,' a general theory is not available. The pres-
ent state of the theory does not address such problems as the round-trip phase condition or even oscillation frequency.
In this paper we describe a theory of parametric ring oscillation using holographic two-wave mixing in photore-

fractive crystals. The theory shows that oscillation can occur at almost any" cavity length despite the narrow-band

nature of the coherent two-wave coupling gain, provided

that the coupling is strong enough. A similar situation also

occurs in phase-conjugate parametric oscillators.- The theory also provides explicit expressions for the oscillation fre-

quency, intensity, and threshold conditions.

FORMULATION OF THE PROBLEM
Referring to Fig. 1. we consider an optical ring resonator consisting of three partially reflecting mirrors. A photorefractive medium, which is pumped by an external laser beam, is inserted into the cavity. To investigate the properties of such oscillators, we must first treat the problem of two-wave coupling in photorefractive media. This problem has been formulated and solved by many workers.8-10 However, most work has been focused on the degenerate twowave mixing. For the purpose of developing our theory, we need to address nearly degenerate two-wave mixing.
Let us focus our attention on the region occupied by the photorefractive crystal, so that the electric field of the two waves can be written as
E)f A)(z)expji(k) •r - t)I + c.c., j = 1, 2, (1)

where z is measured along the bisector of the two beams. k and k2 are the wave vectors of the beams, and c.c. denote- a term that is the complex conjugate to the first term. In Eq.
"i. we assume for simplicity that both waves have the same

of polarization and the photorefractive medium does

exhibit tes and

optical rotation. A and

a ta

tions o

A n

2

alrye rtthe

waveeaya-mpli-

situations.

iutos In the photorefractive medium (from zero to z = 1). these

two waves generate an interference pattern (traveling it, c2l This pattern may generate and redistribute photocar-

riers. As a result, a spatial charge field (also traveling if .

- w')is created in the medium. This field induces a volume

index grating by means of the Pockels effect. In general. the

index grating will have a finite spatial phase shift relative to the interference pattern so that, following the notation of Ref. 11, we can write the fundamental component of the intensity-induced grating as

+ n= AA,"

r

n =n + I e

exp[i(K r- )I + c.c.J. (2

where

I,= I + I, - IA 2 + A,. P is real and n, is a real and positive number, K =

(3 -k

and Q.= , - W2. Here again, for the sake of simplicity. we

assume a scalar grating. The phase o indicates the degree to
which the index grating is shifted spatially with respect to the light interference pattern. According to Ref. 1. o and 0 can be written, respectively, as

0o4=o + tan - '(Q )

141

and

n2

2

An,,

(1 + '2) 1 2

where wee

7 is the decay is the sarti

time

constant of the holograph

e

of the hoordce

grating. ine

An, is the saturation value of the photoinduced index

change, and o0is a constant phase shift related to the nonlo-

cal response of the crystal under fringe illumination. Both

panadraimtsetdeirrsecAtino,nanasd

o, depend well as on

on the

the grating spacing (27,'/ material properties of the

crystal. e.g., the electro-optic coefficients. Expressions for

0740-3224/85/121924-05$02.00 C 1985 Optical Society of America

- ~7A!.&AA

/

..D~**%

Pochi Yeh

Vol. 2, No. 12/December 1985/J. Opt. Soc. Am.B 1925

d- ~rn1c0cos 0.

(1

ACos 9

The solutions for the intensities 11(z) and 12(z) are

l,(z) = 1(0) 1I ++mn-l-e-11z e

(12)

12(z) = 12(0) 1+m e

(13)

1 + me"'

7PHOTOREFRACTIVE where m is the input intensity ratio

12 MEDIUM

1,.

I

I

}

UMP

-2

1'(0)

I1M2(00)(4

(14)

.MNote
Fig. 1. Schematic drawing of a unidirectional photorefractive ring resonator.

that in the absence of absorption (c = 0), 12(z) is an increasing function of z and lj(z) is a decreasing function of z, provided that -yis positive. The sign of 1 depends on the direction of the c axis. As a result of the coupling for -"> 0 in
Fig. 1, beam 2 gains energy from beam 1. If this two-wave

mixing gain islarge enough to overcome the absorption loss,

.An,and o can be found in Refs. 10 and 12. In photorefrac-

then beam 2isamplified. oscillation.

Such an amplification is responsi-

tie media,e.g., BaTiO ,that operate by diffusion only (ii.te(.z,)

no external static field) the magnitude of ois 7r/2, with its

sigNnodwe,pebnydiunsginognetxhperedsisrieocntio(n2)offotrhen

c axis. and the

scalar-wave27

With

and 12(z) known, the phases j and ¢2 can be

integrated directly from Eqs. (9). The phase shift in tra-

versing through the photorefractive medium for beam 2 is

equation and by using the parabolic approximation (i.e.,

n°1 + ,t2(1-) 4'(0),

(15)

slowly varying amplitudes), we can derive the following cou-

dcos I

8 e'iA 2 12A1 - At, dz MO Cs 0

'

A.) dz

- CIA X10 Cos a

1 !2A2 -

A2, 2

(6)

where 0 is the half-angle between the beams and a is the absorption coefficient.
We now write

where 21rnol//\ is the phase shift in the absence of photorefractive coupling. The additional phase shift
A¢ -€2(1) - 2(0)1,16 (16)

which is due to the photorefractive two-wave coupling, can

be obtained by integrating and (13) into Eqs. (9) for I,

Eqs. and

(9). Substituting Eqs. f12) 1), respectively. we obtain

a

= 2(1) - €,,(0) =

(17)

1 + ,'

A = t exp(i,),
A 2 = <:72 exp(i4 2),

Note that this photorefractive phase shift is independent of

the absorption coefficient o. Carrying out the integration in

(7)

Eq. (17), we obtain

where j and 2 are phases of the amplitudes A, and A2 ,
respectively. Using Eqs. (7) and (3), the coupled Eqs. (6) ('an be written as

d

I 1'2

dz

11 + 12

=

-

= - log 1 +

(18)

O =

-1

+ meI

Equations (18) and (13) can now be used to investigate the

properties of the unidirectional ring oscillation.

d
dz 2

1 + 12

and

d

13 2

dz

!1+ 12

OSCILLATION FREQUENCY AND INTENSITY In a conventional ring resonator, the oscillation occurs at

those frequencies

f =

N

c .

N = integer

(19)

L

that lie within the gain curve of the laser medium (e.g., He-

" where

d dz

= 2 0

12 +

-Y - 2rn- sin o, X cos 0

(9) (10)

Ne). Here, L is the effective length of a complete loop and N

is a large integer. (19)1 are separated

For L _ 30 cm. these frequencies (Eq. by the mode spacing c/L > 1 GHz. Since

the width of the gain curve for the conventional gain medium
is typically several gigahertz, principally because of Doppler broadening, oscillation can occur at almost any cavity length

L. On the contrary, if the bandwidth of the gain curve is

+,,¢...e,.r,:.'p,'.+e,..e.+'+ + p € .+ ,-,' ,,r,+.* , . . ,+ .-_ -. ... ,, ., , . ,.* : .€ , r.-.+ . ... .. .

1926 J. Opt. Soc. Am. B/Vol. 2,No. 12/December 198

,10

3

10

109

1

M(

Pot hi Yt-h
12:41(--

102
10 g 1
10 1
10 2

1o 41

0

1

2

3

4

Fig. 2. Photorefracti\e gain 4!as a function of !- for various values of M.

1 0[ So 0

02

The gain as a function of frequency Le,('or equivalent ],, a-a
function of 0 =4e - e2l is plotted in Fig. 2 for various valueof m. Note that gain is significant only when , ,< 1
For materials such as BaTiO: and SBN, r is between I and
0.1 sec. Thus the gain bandwidth is only a few hertz. In spite of such an extremely narrow bandwidth, unidirectional oscillation can still be observed easily at any cavity length in ring resonators by using BaTiO:j crystals as the photorefrak tive medium."1 Such a phenomenon can be explained in terms of the additional phase shift [Eq. (18)] introduced 1,% the photorefractive coupling. This phase shift is a funct i,i of the oscillation frequency and is plotted in Fig. :3as a ore function of 1.. For BaTiO, crystals with 1,,/> 4-. thiphase shift can vary from -r to +r. for a frequency drift ,t
i..r= * 1. Such a phase shift is responsible for the (scilldtion of the ring resonator, which requires a round-trip phait-
shift of an integer times 27r.

0o.It

01

o S

•

10o__

__ 4

__

__

__

_

2

0

2

4

6

Fig. o3s. Photorefractive phase shift as a function of .°r for various values of m.

narrower than the mode spacing c/L. then oscillation can be
sustained, provided that the cavity loop is kept at the appropriate length.
Unlike in the conventional gain medium, the bandwidth of the photorefractive two-wave mixing is very narrow. By
using photorefractive crystals, e.g., BaTiO3 , that operate by diffusion only', the coupling constant can be written, according to Eqs. (4). (5). and (10), as

= 10/11 + (QT)2,

(20)

where 0iysothe coupling constant for the case of degenerate

two-wave mixing (i.e., 9 =w - W2 = 0) and is given by

= 4wcrAonsn5,

(21)

In deriving Eq. (20), we have used r/2 for oo in Eq. (4). The parametric two-wave mixing gain can be defined as

g 12()

1 +m e -

(22)

12(0) 1 + mC- ")

where we recall that m is the input beam ratio m = 11 (0)/12(0)
and I is the length of interaction. Note that amplification (g > 1) is possible only when -y > aand m > (1 - e-aI)/(e - aI e-'yV). Also note that g is an increasing function of m (i.e.,
*g/dm > 0 and g is an increasing function of 1,provided that
> a and

OSCILLATION CONDITIONS

lasiesrinotsecriellsatitonrgs,tothneotoestchilelaitnioitniatoifotnhiosf

the oscillation. ring resonator

Like starts

from noises that are due to physical processes such as scat -

tering and quantum fluctuation. In photorefractive crystals

the scattering dominates the noise contribution. At the

beginning, there may be a slight amount of light scattered

along the direction of the ring resonator. This slight amount of light will be amplified by the two-wave mixing

process in the photorefractive crystal, provided that the frequencies are not appreciably different. As the intensity

in the resonator builds up, the parameter m, defined by Eq. (14, decreases. The buildup of oscillation intensity leads to
a saturation of the gain (see Fig. 2; the gain decreases as rn decreases). At steady-state oscillation, the electric field must reproduce itself, both in phase and intensity, after each round trip. In other words, the oscillation conditions can be
written as

A ,+ f kds =2N~r

(241

and

gR = 1,

(25 1

where A is the additional phase shift owing to photorefrac-

tive coupling, the integration is over a round-trip beam path. the parameter R is the product of the mirror reflectivities. and g is the parametric gain of Eq. (22).

If we define a cavity-detuning parameter A" as

Ar = 2N'r - f kds.

(261

where N' is an integer chosen in such a way that Al lies

between -r and +r,then the oscillation condition IEq. (24)]

can be written as

A = AF + 2Mr,

(27)

where M is an integer. In other words, oscillation can he achieved only when the cavity detuning can be compensated for by the photorefractive phase shift.

-%

PRwhi Yeh o

. .The 0 4r
042,

7 -,1o0 8 42which

20,

20o

. ...

....

t o

o u si

for 10

4fsthraeshold

Fig. 4. Oscillation intensity and frequency as functions of cavity detuning.11F for various values of j,,1.

Equations (24) and (25) may be used to solve for the two unknown quantities m = 110)/12(0) and Q = w, - W2- If we fix the pump intensity I1(0) and the pump frequency wl, then Eqs. (24) and (25) can be solved for the oscillation
frequency u._ and the oscillation intensity 12(0). Substitut-
ing Eq. (22) forg in Eq. (25) and using Eq. (18), we obtain

A=

log(Re_,).

(28)

This equation can now be used to solve for the oscillation frequency QT. For the case of pure diffusion, using Eq. (4)
for o with o = 7r/2 and Eqs. (10) and (11), we obtain from
Eq. (28)

= -2A# _ -21F + 2M)

(29)

al - log R al - log R

where A" is the cavity detuning and is given by Eq. (26).
Substituting Eq. (22) forg in Eq. (25), we can solve for m and obtain

I1(0) I - Re - ' r. 12(0) .._ Re-"__t_ -"(0 -~(30)

Since . must e positive, we obtain from Eq. (30) the threshold condition for oscillation

"lI> 1tl a! - log R,

(31)

%%here", is the threshold parametric gain constant. Since -Y

is a function of frequency it, Eq. (31) dictates that the para-

metric gain is above threshold only in a finite spectral re-

gime. When Eq. (20) is used for -y,Eq. (31) becomes

k /
i'Q <

\i2P)
-log R- 1

(32)

where we recall that -Y)is the parametric gain at 1= - W2 = 0. Inequality (32) defines the spectral regime where the parametric gain - is above threshold (i.e., -y > -y,).
We have thus far obtained expressions for the oscillation frequency [Eq. 129)1 and the spectral regime where the gain
is ahove threshold. The ring resonator will oscillate only when the oscillation frequency falls within this spectral region. The oscillation frequency W2, = W1 - S1is determined by Eq. (29). with A being the cavity detuning JEq. (26)j.

Vol, 2. No. 12/December t985/J. Opt. Soc. Am. B 1927

same oscillation frequency must also satisfy expression (32). Thus we obtain the following oscillation condi-

tion:

2IA,

I

1.2

-0 _ 1],

(33)

10

<I
al - log R [a - log R

can also be written as

"YoI > ")tl+ ± (2A )2 - Gil,

(34)

where-, is the threshold parametric gain of Eq. (31) for the
case when Aik = 0 and G, may be considered the threshold
gain for the case when A * 0. According to Eq. (34), the
gain increases as a function of the cavity detuning
A. The cavity detuning Al' not only determines the oscillation frequency [Eq. (29)] but also determines the threshold gain G.
The AF in Eq. (26) is the cavity detuning and is defined between -ir and 7r. However, the photorefractive phase shift [Eq. (18)] can be greater than 7r. When this happens, the unidirectional ring resonator may oscillate at more than one frequency. These frequencies are given by Eq. (29), with M = 0, ±-1, ±-2. etc., and with their corresponding threshold gain given by

G il = -, I + 1

+ 2M7r)12

(35)

In other words, for each cavity detuning AF, the ring resonator can support multimode oscillation, provided that the
coupling constant "0 is large enough. Figure 4 shows the
oscillation intensity as well as the oscillation frequency as functions of cavity detuning A. Note that for larger "01 the resonator can oscillate at almost any cavity detuning AF, whereas for small 101 oscillation occurs only when the cavity detuning is limited to some small region around Ai = 0.

CONCLUSION AND DISCUSSION
In conclusion, we have derived a theory of unidirectional ring oscillators using parametric photorefractive two-wave
mtaiinxinang. exBpryesussiionng ftohrethseimpphloetocroeufrpalcetdi-vmeopdheastheesohriyft,. weSuocbh-
a photorefractive phase shift can compensate for cavity detuning and thus can allow the oscillation to occur. According to this theory, the oscillation frequency will be slightly detuned from the pump frequency. Such a frequency offset is necessary to produce the photorefractive phase shift to compensate for the cavity detuning. The photorefractive phase shift is proportional to the coupling constant. Thus, when materials with a large coupling constant (e.g.. BaTiO d are used. oscillat ion can occur at almost any cavity detuning. Such a theory has been validated by the author and his coworker.'" The same theory can also be applied to linear oscillators and thus can be employed to explain the frequency shift of self-pumped phase conjugatorsi6.' :
ACKNOWLEDGMENTS
The author acknowledges helpful discussions with M. Khoshnevisan and M. Ewbank (Rockwell Science Center) and J. Feinberg (University of Southern California).

192S J. Opt. Soc. Am. B/Vol. 2, No. 12/December 1985

Pochi Yeli

REFERENCES

9 V 1. Vinetskii. N. V. Kukhtarev. S. G. Odulov. and NI S. Sriskin. "lDvnamic self-diffraction of coherent light beanic-

1. J. P. Huignard and A. Marrackchi, "Coherent signal beam am plification in two-wave mixing experiments with photorefrac-
tive BSO crystals.- Opt. Commun. 38. 249 (1981). 2 ,J.P. Huignard and A. Marrackchi. "Two-wave mixing and ener
g' transfer in B12,SiO.,0 crvstals: amplification and vibration analysis.- Opt. Let. 6. 622 (19811. 3. P. Yeh. "Contradirectional two-wave mixing in photorefractive media." Opt. Common. 45, 323 (19S:0.

So% Phys. sp 22. 742 (1979j. 10. N V. Kukhtares. V. B Marko%. S.G. Odulov. M. S.Soskin. and
V L. Vinetskii. 'Holographic storage in elect ro-opt ic.s cry'stalbeamn coupling dnd light amplification." Ferroelectrics 22, 4f i (1491
11 B. FischCr. MI.Cronin Golomb. .J.0. White. and A Yark. "Art. plified reflection. transmission, and self-oscillation in real-tina holographY.' Opt Lett 6.519(198!1

4. P. Yeh. "Electromagnetic propagation in a photorefractive ]ay.

12. ,J.Feinberg, D. Heiman. A. H. '1anguay. and H. Hellw&arth .1

ered medium." J. Opt. Soc. Am. 73. 1268 (19831. 5. P. Yeh. "Photorefractive coupling in ring resonators." Appl.
Opt. 23, 2974 119M). 6. .1 0. White. M. ('ronin-Golomb. B. Fischer. and A. Yariv. "Co-
herent oscillation b,, self-induced gratings, in photorefractive

AppI Phvs. 51, 1297 (198W. 13. J. Feinberg and G. D. Bacher. "Self-scanning of a continuii
wave dye laser having a phase-conjugating resonator (a%t, Opt Lett. 9,420 c19841 14. M. 1). Ewbank and 1'. Yeh. "Frequency shift and ca\ it' dtncu

crystals." AppI. Phi-s. Lett. 40, 45(1 (1). P. Y'eh. "TheorY of phase-conjugate oscillators,-* . Opt Soc Am.A 2. 727-730 (19s5.

ing in photorefractive resimnators," Opt Lett 10, 41. 4 oI] 9h.5 I

8. D. L. Staehler and .1. .1. Amodei. "Coupled wave analysis oif

J

~holographic storage in LiNi()1 1 ppl. l'h\-s 34, 10142 u19721.

4

k

01 Rockwell International
Science Center
SC5424 .AR
APPENDIX 5.6 Frequency Shift and Cavity Length in Photorefractive Resonators

72 C7714A/jbs

Reprinted from Optics Letters, Vol. 10, page 496, October, 1985
FrequCoepynrigchyt ©s1h985ifbty and cavity the Optical Society of lAemnerigcatahnd irnepripntehdobytpoerrmeissfiornaofcttheivcoepyrrighetsoownnera. tors

M. D. Ewbank and P. Yeh
Rockwell International Science Center. Thousand Oaks. California 91360
Received April 22. 1985, accepted July 22, 1985 Photorefractive resonators exhibit an extremely small frequency difference (Aw/w - 10- 1s) between the oscillating and pumping beams. The observed frequency difference is proportional to cavity-length detuning. This dependence is explained by a photorefractive phase shift that is due to slightly nondegenerate two-wave mixing that compensates for cavity detuning and satisfies the round-trip phase condition for steady-state oscillation. The measured onset or threshold of oscillation as a function of photorefractive gain and intensity agrees with theory.

Despite the attention that self-pumped phase conju-
gators and optical resonators utilizing photorefractive BaTi0 3 have received recently,' -3 two dilemmas remain unresolved. First, self-pumped phase conjugation in BaTi0 3 exhibits a slight frequency shift (-1-Hz), 4- 7 attributed to a Doppler shift from moving photorefractive phase gratings.4 Second, resonators using photorefractive gain media apparently oscillate at any optical cavity length.4.8 In this Letter we show that these two dilemmas are interrelated and reveal the or-
igin of the moving gratings and frequency shifts. For simplicity, consider a unidirectional ring oscil-
]ator with photorefractive two-wave mixing providing
the gain. The optical arrangement (Fig. 1) is chosen because only two-wave mixing occurs. The frequency
difference between the unidirectional oscillation beam - and the pump can be controlled by small changes (<X) in ring-cavity length. In fact, both the sign and the
magnitudeof the frequency shift exhibit a one-to-one
correspondenceto the cavity-length detuning. The above observations are predicted by a theory for
unidirectional photorefractive ring resonators.9 Oscillation occurs when the two-wave mixing gain dominates cavity losses and the round-trip optical phase reproduces itself (to within an integer multiple of 2ir). The condition on phase is unique because of a signifi-
cant opticalphase shift owing to nondegeneratetwowave mixing. This condition is satisfied at any cavity length if the oscillation frequency is detuned from the
pump frequency, since the photorefractive phase shift depends on the detuning. The frequency difference Q. between the pumping and oscillating beams is9

= - [2(Ar + 2m7)/TA],

(1)

where AF is the cavity-length detuning with respect to

an integer multiple of optical pump waves in the cavity,m is an integer, - is the photorefractive time response,

and A represents the total cavity loss. There are

threshold conditions for oscillation involving cavity loss and gain (taking m to be zero):

Ji1 (1/r)(y-L/A - 1)1/2,

(2a)

IAfl < (A/2)(-yL/A - 1)1/2,

(2b)

where -y is the degenerate two-wave mixing coupling coefficient, where L is the interaction length, and A =

-ln(RT Tp) [with R the product of the reflectivities of the cavity mirrors and output coupler; T, the transmission through the BaTiO3 sample accounting for the absorption, Fresnel reflections, and scattering (or beam fanning); and Tp the effective transmission through the
pinhole aperture]. This theory predicts that the unidirectional ring
resonator will oscillate at a frequency different from the
pump frequency by an amount directly proportional to the cavity-length detuning. Furthermore, in a photo-
refractive material with moderate slow r, the theory postulates a threshold where oscillation will cease if the cavity detuning (frequency difference) becomes too
large. The experiments performed to examine the above
theory will now be discussed in detail. Referring to Fig. 1, a single-mode argon-ion laser (514.5 nm) serves multiple purposes: It pumps the BaTiO 3 crystal with the beam reflected from BS, and focused by lens L1 (F = 700 mm) to a beam diameter of 0.5 mm at the crystal, it is a reference beam to determine interferometrically
the relative frequency of the ring-cavity oscillation at BS5, and it provides an accurate method to align the ring-cavity components (unblocking the beam reflected from BS2 ) by monitoring the Fabry-Perot peaks when

W.. "

A to

L °'

,,,
-
-.,ocS s,A

A
L , -,

"-
0,

Fig. 1. Optical setup for the photorefractive unidirectional ring resonator with variable cavity length. The beat frequency between the self-oscillation and pump beams is derived from the motion of the interferograms at D2 or D3.

0146-9592/85/100496-03$2.00/0 (C1985, Optical Society of America

October 1985 / Vol. 10, No. 10 / OPTICS LETTERS 497

z

-,

0_-

- 1bi 04-

0-

I OT4U,4N

0

4

04~- ture

-' 3-

- ~liIlIIlliliIII,,I

1

_-

il

40
20°C I

a,

12

,

-

away from this origin, the fringe motion becomes faster and the frequency difference increases. Figure 2(b)
shows the linear dependence of the frequency difference on cavity detuning with the ramping period equal to 20,000 sec for improved resolution. This frequency differ[esnimceilairs teostFimig.a2te(ad)]frboymmtehaesubreiantg-frtheequtiemnceyinstiegrnvaa-l between intensity maxima.
moTvhese tfhrreoquugehnctyhedoifrfiegrienn. ceThcheafnrgeqesuesnigcny aosf oMsc, islllaotwiolyn
is upshifted with respect to the frequency of the cpourmrepsipnogndbineagmtowahdenecrtehaesepoinsirtionng-coafvMity, liesnngethgaftriovme, that where no frequency shift occurs. The observed

|_sign is consistent with the sign of the phase shift be-

o

,

-2

4

6

,ZTDRIVEVVEOLLATTIAVEGE(VI

Fig. 2. Characteristics of the unidirectional self-oscillation adsetaufnuinncgt,iownheorferi1n0g0-%cavimitypllieensgathde(it.eu.n, iPnZgTofvoonlteagfeulol rocpatviciatyl

wave): (a) ring-cavity intensity (right) and beat-frequency signature (left); (b) frequency difference between the selfoscillation and the pumping beam.

rapidly scanning the piezo mirror PZT-Mi. Two-wave

mixing in BaTiO 3 couples light into the unidirectional ring cavity, formed by two planar mirrors (MI and M 2) and a planar beam splitter (BS 3), even when the alignment beam is blocked. The ring-cavity beam is sam-

pled through the output coupler BS 3, its intensity being detected at D, while the beat frequency between it and
the pumping beam is determined using complementary

fringe patterns formed at detectors D2 and D3. Without a ring-cavity pinhole aperture, unidirectional oscillation

is observed at any cavity length. However, dynamically unstable multiple spatial modes are evident 4"0 in the fringe patterns at D2 and D3 . To obtain a single mode (inantdheclreinang fcraivngitey.patterns), a 2 0 0 -Am pinhole is placed

The basic premises of the theory9 are verified by

slowly ramping the PZT voltage and observing the beat

frequency, along with the ring-cavity oscillation in-tensity. Typical results are shown in Fig. 2(a) for an

80-mW pump beam incident at 400 from the c axis of
BaTi0 3 and at 200 from the oscillating beam (both angles are external in air). A triangle waveform (ampli-

tude 250 V and period 10,000 sec) drives the PZT. The

slow PZT scanning rate approximates steady-state two-wave mixing in BaTiO3 while simultaneously permitting a controlled variation of the cavity length.

The unidirectional ring-cavity intensity versus cavity

length [Fig. 2(a)] indicates threshold gain conditions

[expressions (2a) and (2b)]. The beat frequency be-

tween oscillating and pumping beams, as observed in

the time variation of the 2(a)], clearly corresponds

fraitnihtgoeep-posaitttieornn

intensity (Fig. of the PZT-M

This beat frequency is not simply due to the Doppler

shift caused by the moving PZT-M, since this Doppler

shift is 3 orders of magnitude smaller than the observed

beat frequency. When M, is exactly at the correct po-

sition (chosen as the origin), the fringe pattern is stationary, i.e., there is no frequency shift. As M, moves

tween the light intensity pattern and index modulation that determines the direction of energy exchange in
two-wave mixing.
periodic function of PZT mirror position. The
Fabry-Perot peak spacing detected at D, during alignment of the passive ring cavity calibrates the PZT motion (-50-V/free-spectral range for 514.5 nm). The observed beat-frequency signature reproduces itself with a M, displacement of every - X/2, as expected (i.e.,
a cavity length detuning periodicity of X). Experimentally, the frequency threshold for oscilla-
tion is approximately a linear function of the pumping-beam intensity, as shown in Fig. 3(a). According to Eq. 2(a), this frequency threshold is inversely proportional to r, but T can be approximately proportional to the inverse of the pump intensity (assuming that the cavity intensity is negligible by comparison) when the photoconductivities dominate." Therefore, the observed dependence [Fig. 3(a)] agrees with theory.
The oscillation conditions for the unidirectional ring resonator are dependent on the two-wave mixing gain (-yL) in the photorefractive medium. yL is varied by

22"

-

i

°
9

C . T POW,.......
'0 30 20 .0
-
3 -

03

25
,
0S,
0 1

ANGLBEETWEEPNUMP

30

3S 4O.6

so

3
I

0 0

0 so to 0

0 ,,.o.....

(a)

S

6 •..

.o

Ub)

Fig. 3. Oscillation threshold behavior for the unidirectional n pgurmespoinngat-obre:am(ao) rmrianxgi-mcauvmitybepaotwferreqaluoenngcywiaths aa fliunnecatriofint (solid line); (b) maximum beat frequency (left) and cavity
detuning (right) as a function of two-wave mixing gain, "L, where -yL is related to the external angle that the pumping beam makes the crystal's c axis as shown (top scale). Note:
the two solid curves in (b) correspond to the evaluation of expressions (2a) and (2b) as described in text.

77.

-

0

498 OPTICS LETTERS / Vol. 10. No. 10 / October 1985 4stead

'ternal
gal

PINHOLE
T s2 MY,

Fig. 4. Self-pumped phase conjugator using external reflectors to generate the self-oscillation with a frequency shift
6 and the phase-conjugate reflection with a frequency shift
25, where 6 is proportional to the linear cavity length.

rotating the BaTiO:j crystal with respect to the pumping and oscillating beams.i 2 When the gain is too small, no unidirectional oscillation is observed, regardless of ring-

cavity length. For "yLjust above threshold, two pro-

nounced differences are evident, contrasting with IL

large. First, the amount of cavity detuning that is ac-

commodated before oscillation ceases is greatly reduced.

Second, the maximum frequency difference between the pumping and oscillating beams is much less. The quantitative trends of these two effects are given in Fig.

3(b) for a pump power of 80 mW.

The threshold oscillation conditions given in ex-

pressions (2a) and (2b) agree with the data [Fig. 3(b)].

The solid curve associated with the left-hand scale of Fig. 3(b) is generated from expression (2a) for A = 5.1

and r = 0.53 sec. This cavity-loss factor, A, is estimated

independently from R = 0.99 X 0.91 X 0.81 (for M1 , M 2 and BS3 , respectively), T, = 0.52 and T, = 0.016 (for a cavity length of 50 cm). Accumulating these contri-

butions gives A = 5.2, in excellent agreement with the

observed 5.1. The right-hand scale of Fig. 3(b) shows

the dependence of threshold cavity detuning (i.e., the

maximum detuning that will still support self-oscilla-

tion) on -yL, along with the prediction from Eq. 2(b),

where Alr is normalized by 27r. Remarkable agreement

is obtained using A = 5.1 from Fig. 3(a) and no adjust-

able parameters.

The interdependence of the optical cavity length and

the beat frequency between the oscillating and pumping

beams is a general property of photorefractive resona-

tors. These results are not unique to the optical setup

shown in Fig. 1. Similar behavior is observed with other ccororynssftiiaglluiirnnatiFigoign..s11. baceanFaenmnirtbsseetr,salttheereotdhriseonattbchaetuaitottnhheofpuuhthmepBianagTaiOnnd3t onresopcilslaleacltefi-dnpgubmbyepcairmnygsstoaeclnscteuorrfst.hs4teroaSneftacicuoemnbdu,bttahirneiusBmuacThniiaOowb3aacytaent1h3b.a'e4t

(nominally undoped and cerium doped). Third, a lin-

ear resonator (Fig. 4) acts as a self-pumped phase conjugator.1 The observed frequency shift of the

phase-conjugate beam is exactly twice that of the self-

oscillation, which is necessary to satisfy energy con-

servation for slightly nondegenerate four-wave mixing. is

In all three variations, the measured frequency differ-

ences correlate with cavity length detuning; results

eqFuiivnaallelyn,t

to those consider

shown in Fig. 2 are obtained. another self-pumped phase conju-

FR.

gator, in which the four-wave mixing pumping beams

arise from internal reflections at the crystal faces in-

.W .. -. -. V,.

r,T-7 _W1_V

of external mirrors.3-7 A resonance cavity containing the self-pumping beams is formed from the in-

length

reflections cannot be

off the crystal surfaces systematically varied

(but the cavity and the spatial

modes cannot be restricted with an aperture). The

frequencies of the beams can again ferometrically. One self-pumping

be compared interbeam is monitored

using the uncollimated light escaping the crystal's corner during reflection. Preliminary experiments indicate that the phase-conjugate beam sometimes exhibits a frequency shift exactly double that of a self-pumping

beam, just as in Fig. 4. Even though four-wave mixing

is occurring in all cases of self-pumped phase conjugators, two-wave mixing might be the dominating process in determining the oscillation conditions necessary for generating the self-pumping beams.
In conclusion, the frequency difference between the self-oscillating and pumping beams in the unidirectional ring resonator experimentally depends on the optical cavity length. This dependence supports a theory9 that uses a photorefractive phase shift associated with slightly nondegenerate two-wave mixing to satisfy the round-trip phase-oscillation condition for the resonating beam. Similarly, the observed frequency shifts in other photorefractive resonators, including selfpumped phase conjugators, may also be explained by
the same mechanism.

The authors acknowledge discussions with M. Khoshnevisan and A. Chiou of Rockwell International

Science Center and J. Feinberg of the University of Southern California.

References
1. J. 0. White, M. Cronin-Golomb, B. Fischer, and A.Yariv, Appl. Phys. Lett. 40, 450 (1982).
2. R. McFarlane and D. Steel, Opt. Lett. 8, 208 (1983). 3. J. Feinberg, Opt. Lett. 8, 480 (1983). 4. J. Feinberg and G. D. Bacher, Opt. Lett. 9, 420 (1984).
5. K. R. MacDonald and J. Feinberg, J. Opt. Soc. Am. A 1,
1213 (A) 1984. 6. J. F. Lam, J. Opt. Soc. Am. A 1, 1223 (A) 1984. 7. W. Whitten and J. Ramsey, Opt. Lett. 9, 44 (1984). 8. H. Rajbenbach and J. P. Huignard, Opt. Lett. 10, 137
(1985). 9. P. Yeh, J. Opt. Soc. Am. B 2 (to be published, November
1985). 10. G. Valley and G. Dunning, Opt. Lett. 9, 513 (1984). 11. P. Gunter, Phys. Rep. 93, 199 (1982).
12. As the orientation of the two-wave mixing fringe pattern
with respect to the crystal axes changes, the effective electro-optic coefficient (and coupling efficiency of the index grating) is modified. -yL is independently measured by removing ring-cavity mirror M2 and using an attenuated probe beam from BS 2 with an external angle of 20 between probe and pump beams. 13. Provided by R. Neurganokar, Rockwell International. Thousand Oaks, Calif. 14. B. Fischer, M. Cronin-Goulomb, J. 0. White, A. Yariv, and
Neurgaonkar, Appl. Phys. Lett. 40, 863 (1982). 15. P. Yeh, M. D. Ewbank, M. Khoshnevisan, and J. NI.
Tracy, Opt. Lett. 9, 41 (1984).

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Rockwell International

Science Center

SC5424 .AR

APPENDIX 5.7 Frequency Shift of Self-Pumped Phase Conjugator

76 C7714A/jbs

*cavity * 4When
*we

Frequency shifts of self-pumped phase conjugators
M.D. Ewbank and P. Yeh
Rockwell International Science Center 1049 Camino Dos Rios, Thousand Oaks, California 91360
Abstract
The reflection from most photorefractive, self-pumped phase conjugators differs in frequency from the incident beam by a small amount (Lw/w - 10-15). A theory and the supporting experiments which explain such frequency shifts are presented. In our theory, four-wave mixing is responsible for the generation of the conjugated wave where a selfoscillation arising from photorefractive coupling provides the customary pumping beams. The frequency of these pumping beams is determined by a resonance cavity geometry and may be slightly different from that of the incident beam. Nondegenerate four-wave mixing using these self-oscillating pumping beams give rise to the frequency shift of the phaseconjugate reflection. Experimental results are in good agreement with theory.
Introduction
Self-pumped phase-con3ugate reflectors 1 - 3 using photorefractive BaTiO3 have recently received considerable attention because 30-50% reflectivities are relatively easy to achieve even with low-power lasers. In self-pumped (or passive) phase conjugators, the counterpropagating pumping beams needed in the four-wave mixing process are automatically generated by light photorefractively diffracted out of the incident beam. However, the phaseconjugate reflection is generally shifted slightly in frequency when compared to the incident beam (on the order of 1 Hz depending on intensity). This frequency shift has been attributed to moving photorefractive gratings which Doppler shift the diffracted light.' The physical mechanism, which is responsible for the moving gratings, is, however, not well understood.
The frequency shift first manifested itself as a frequency scanning when self-pumped BaTiO3 was coupled to a dye laser.4- 6 Since those initial observations, numerous experiments and theories involving self-pumped phase conjugators and/or photorefractive resonators have addressed, either directly or indirectly, the frequency shift issue. 7 - 16 However, a general theory and the conclusive experiments are not available.
In this paper we present a theory and the supporting experiments which explain such frequency shifts of most self-pumped phase conjugators. In our theory, self-pumped phase conjugation results from an internal self-oscillation. The optical resonance cavity which supports such oscillation is formed by either external mirrors or crystal surfaces. The oscillating beams provide the counterpropagating pump beams which are required in the fourwave mixing process. The theory shows that despite the narrow gain bandwidth of the photorefractive two-wave coupling, internal oscillation can still occur over a large range of cavity length detuning. The frequency shift is proportional to the cavity length detuning. Such a dependence is explained by a photorefractive phase shift that is due to slightly nondegenerate two-wave mixing. The additional photorefractive phase shift compensates for
length detuning and satisfies the round-trip phase condition for steady-state oscillation.
the self-pumping beams are spontaneously generated via photorefractive coupling in a linear resonance cavity with two external mirrors on opposite sides of a photorefractive crystal such as BaTiO3, we observe that the frequency shift of the phase-conjugate reflection is directly proportional to cavity-length detuning. In the case where the selfpumping beams arise from internal reflections off the photorefractive crystal's surfaces,
experimentally prove that a previous description 2 of the self-pumping process is inadequate and we show that a closed-looped resonance cavity forming inside the crystal is a better description.
Theory
The theory is an extension of our earlier theory on unidirectional photorefractive ring oscillators. 1 6 In this theory, amplification owing to holographic two-wave mixing in the photorefractive crystal is responsible for the self-oscillation. When the configuration of the resonance cavity relative to the crystal supports bi-directional oscillation, a phaseconjugate beam is generated via the four-wave mixing process. The parametric two-wave mixing gain, defined as the output to input intensity ratio, is given by 16

.

.

.

.

.

.

.

.

--

1 M m + me-

-at

-at

g

where m is the ratio of the pump beam intensity to oscillating beam intensity, a is the
bulk absorption coefficient, I is the length of interaction y is the intensity coupling
constant, and go is the gain where there is no loss. For crystals such as BaTiO3, this gain can be several thousand per pass. Thus oscillation can be sustained even in cavities with high scattering, diffraction, and/or mirror losses.

isThe phase shift in traversing through the photorefractive medium for the oscillating beam

2(2n-Io+ A4 '

(2)

where 2tn 1I/X is the phase shift in the absence of photorefractive coupling. The additional phase shift, Aq,, is due to the photorefractive two-wave coupling and is given by16

*A =_!log(

+

)=-log l (g )

(3)

1.+ me~ 1 ~

where 0 is the phase-coupling constant. 16 Note that this phase shift, Ad,, is independent

of absorption losses. According to a nonlinear model of the photorefractive two-wave mixing, these coupling
constants (i.e., 0 and y) are given by

2"in 1 sin4 (4)
X cose

w

0 = X cos8 cosX ,

(5)

where e is the half-angle between the pump beam and the oscillating beam. The parameters
and n, can be written, respectively,

t= 'o + tan- 1 (Q)

(6)

and

n =

n s

(7)

/i + 02 It2

where r is the decay time constant of the holograph grating, Ans is the saturation value of the photoinduced index change for degenerate two-wave mixing, $ is a constant phase shift related to the nonlocal response of the crystal under fringe illumination, and Q is the frequency shift between the oscillating and pumping beams. According to Equation (1), amplification is possible (i.e., g > i) only when y > a. Note that gain is significant only when OT < 1. For materials such as BaTiO3 and SBN, T is typically between 0.1 s and 1.0 s for nominal laser intensities (1-10 W/cm 2 ). Thus, the gain bandwidth is only a few
Iertz. In spite of such an extremely narrow bandwidth, self-oscillation can still be observed easily at "any" cavity length using BaTiO3 crystals as the photorefractive medium.
Such a phenomenon can be explained in terms of the additional phase shift (Equation (3)) introduced by the photorefractive coupling. This phase shift is a function of the oscilla-
tion frequency. For BaTiO3 crystals with y7 > 4n, this phase shift can vary from -n to +% .for a frequency drift of AQ% - * 1. Such a phase shift contributes to the round-trip phase shift so that the latter can be an integer times 2 %, a condition required for oscillation.

Concerning the initiation of the oscillation in the photorefractive resonator, like laser oscillators, the oscillation starts from noises that are due to physical processes such as
scattering and quantum fluctuation. In photorefractive crystals, the scattering dominates the noise contribution. At the beginning, a slight amount of light may be scattered along the direction of the resonator. This slight amount of light will be amplified by the two-
wave mixing process in the photorefractive crystal, provided that the frequencies are not
appreciably different. As the intensity in the resonator builds up, the beam ratio parameter, m, decreases. The buildup of oscillation intensity leads to a saturation of the gain (see Equation (1) where the gain decreases as m decreases). At steady-state oscillation,

4

. . - .. .

..

.

.- ;

-"

the electric field must reproduce itself, both in phase and intensity, after each round trip. In other words, the oscillation conditions can be written as

64, + f kds = 2Nn

(8)

and

gR=

,

(9)

where 64, is the additional phase shift owing to photorefractive coupling, the integration is over a round-trip beam path, N is an integer, the parameter R represents the cavity losses (e.g., the product of the mirror reflectivities), and g is the parametric gain of Equation (1).

If we define a cavity-detuning parameter tr as

Ar - 2N'n - f kds

(10)

where N' is an integer chosen in such a way that Ar lies between - : and + %, then the oscillation condition (Equation (8)) can be written as

4, = Ar + 2Mn ,

(11

where M is an integer. In other words, oscillation can be achieved only when the cavity detuning can be compensated by the photorefractive phase shift.

Equations (8) and (9) may be used to solve for the two unknown quantities m and 0. If we fix the pump intensity and the pump frequency, then Equations (8) and (9) can be solved for the oscillation frequency and the oscillation intensity. Substituting Equation (1) for g in Equation (9) and using Equation (3), we obtain

bA - Y log (Re-

(12)

This equation can now be used to solve for the frequency shift Q. For the case of pure diffusion, using Equation (6) for 0 with *o = %/2 and Equations (4) and (5), we obtain from
Equation (12)

-2 A_ - log R

-2(Ar + 2Mn) at - log R

(13)

where eAr is the cavity detuning and is given by Equation (10). Substituting Equation (1) for g in Equation (9), we can solve for m and obtain

m = ReR-*aeI-e-- Y1

(14)

Since m must be positive, we obtain from Equation (14) the threshold condition for

oscillation

yX > -ftl - a - log R

(15)

where Yt is the threshold parametric gain constant. Since y is a function of frequency Q, Equation (15) dictates that the parametric gain is above threshold only in a finite spectral regime. Using Equation (1) for y, Equation (15) becomes

< [JcxtYlo-log RR -1 1/2

(16)

where yo is the parametric gain at 0 = 0. Equation (16) defines the spectral regime where the parametric gain y is above threshold (i.e., y > yt).
We have obtained expressions for the oscillation frequency (Equation (13)) and the spectral regime where the gain is above threshold. The self-oscillation will be sustained only when the parametric gain is greater than the round-trip cavity loss and the oscillation frequency falls within this spectral region. The frequency shift is determined by Equation (13), with Ar being the cavity detuning given by Equation (10). According to Equations

(15) and (16), the resonator can be made to fall below threshold by decreasing the reflectivity, R. When this happens, oscillations ceases.

The Ar in Equation (10) is the cavity detuning and is defined between -n and %. However,
the photorefractive phase shift (3) can be greater than n. When this happens, the internal oscillation may occur at more than one frequency. These frequencies are given by Equation (13), with M = 0, *1, *2,... etc. In other words, for each cavity detuning ar, the resonator can support multimode oscillation, provided the coupling constant yo is large enough. Note that when yol is large compared to the natural logarithm of the cavity losses, the resonator can oscillate at almost any cavity detuning Ar; whereas when yOl is small, oscillation occurs only when the cavity detuning is limited to some small region around Ar = 0.

The interdependence of the cavity length detuning and the frequency shift between the

oscillating and pumping beams is a general property of photorefractive resonators. Such

frequency shifts have been conjugators using external

measured mirrors.

1

5in

unidirectional ring resonators and self-pumped phase In these two types of resonators, the cavity length,

cavity losses and two-wave mixing gain (via crystal orientation) can be varied in a con-

trolled fashion. The results, described below, show that the frequency shift of the self-

oscillation is directly proportional to cavity length detuning for both of these resonators

in excellent agreement with the above theory.

In self-pumped phase conjugator using internal reflections, bi-directional internal oscillations must simultaneously be present. These two counterpropagating beams act as the customary pumping beams of the four-wave mixing process. If each of these pumping beams is
frequency shifted by Q, then the generated phase conjugate beam has a frequency shift of 2Q as required by conservation of energy.

Unidirectional Ring Oscillator

Experiments

Before describing the frequency shift experiments with self-pumped phase conjugators, let's first review similar experiments performed on a unidirectional ring oscillator 1 5 with
only two-wave mixing photorefractive gain. The optical arrangement is shown in Figure 1 where two-wave mixing in BaTiO3 couples light from an argon ion laser into the unidirec-
tional ring cavity, formed by two planar mirrors (MI and M2 with the former being piezoelectrically (PZT) driven) and a planar beamsplitter (BS3). The unidirectional oscillation in the ring cavity (confined to a sing.e mode by the pinhole aperture1 7 ) is sampled via the output coupler BS3 , its intensity being measured at detector D1 and the frequency shift between the self-oscillating and pumping beams being determined interferometrically using
complementary fringe patterns at detectors D2 and D3.

LA EML

PZ-,

0S - AMSPLrMTR
U *MIRROR
0 *DETECTOR LENS

M300

PINHOLE APERTURE
U2

-A V IIMM P

]0

:ILOCK

a

as,

Figure 1 Optical setup for a photorefractive unidirectional ring resonator with variable cavity length. Two-wave mixing coupling in BaTiO3 provides the gain for selfoscillation. The beat frequency between the pumping and self-oscillating beams is oerived from the motion of the interferograms at D2 and D3.
Figure 2a bhows the beat frequency, along with the ring cavity oscillation intensity, as a function of PZT mirror position or cavity detuning. A slow ramping rate of the PZT mirror is used to mimic steady-state for the two-wave mixing process in the slow photorefractive BaTiO3 while, at the same time, permitting a controlled variation of the ring cavity
d

i

CAVITY DETUNING (%)

-12

-8

-4

0

4

r.

1b)

.

!

.

'

'

".-U0.4

004 UJ

-0.4

040

g- .8 0
_J U-

'SE4C0
~-SE-CS'-E0C

'

20:

Z 3-

-2
LU

Was ,", 12
.
1,0ci-

.00

-6

-4

-2

0

2

4

6

RELATIVE PZT DRIVE VOLTAGE (VI

Figure 2 Characteristics of the unidirectional self-oscillation as a function of ring cavity length (i.e., PZT voltage or cavity detuning where 100% implies a detuning of one full optical wave): (a) ring cavity intensity (right) and beat-frequency signature (left); (b) frequency difference between the selfoscillation and pumping beam (solid line is a linear least squares fit to data).

length. The frequency difference between the pump beam and the unidirectional oscillation, as observed in the time-variation of the fringe pattern intensity shown in Figure 2a, is clearly related to position of the PZT mirror. When the mirror is exactly at the correct position (here, arbitrarily assigned to be the "origin"), the fringe pattern is stationary, i.e., no frequency shift. The farther the PZT moves away from this "origin," the faster the fringe motion and, hence, the larger the frequency difference, until the selfoscillation ceases (recall Equation 16 above). Figure 2b shows the linear dependence of the frequency difference on cavity detuning, determined from the time intervals between the maxima in the beat-frequency signature. This dependence agrees with Equation (13). Minor deviations from the linear behavior could be due to air currents or the nonlinear response in the PZT. Note that the observed beat-frequency signature vs cavity length also reproduces itself periodically as the PZT mirror moves every half optical wavelength (see Equation (11)).
In the above experiment, unidirectional oscillation is observed only when the ring cavity length is "tuned" to an appropriate length by the PZT mirror and the beat frequency between the unidirectional oscillation and pumping beam is directly proportional to the cavity length detuning. These observations are explained by the theory presented above. Specifically , self-oscillation occurs only when the two-wave mixing gain, which is a function of the frequency shift, is sufficient to overcome the cavity losses and when the ring cavity roundtrip optical phase reproduces itself (to within an integer multiple of 21).
Self-Pumped Phase Conjugator with External Linear Cavity
The frequency shifts in the self-pumped phase conjugator with two external mirrors forming a linear cavity are observed to be very similar to those described above for the unidirectional ring resonator. Specifically, both the sign and magnitude of the frequency shift can be controlled by the linear cavity length detuning. "he optical setup is shown in Figure 3 where light from a single incident beam is photor. -actively coupled by the BaTiO3 crystal into the linear cavity formed between two highly retf -ting beam splitters (BS8 and BS9). This self-oscillation serves as the counterpropagating k mping beams in a traditional four-wave mixing geometry to phase conjugate the incident beam.1

ARGON ION LASER
1514.5 nml

M - MIRROR SIS - BEAM SPLrrTER

3 T

]DARDA0

;
LI I-

_~

e T10

I

Figure 3

Optical setup for a self-pumped phase conjugator with external linear cavity. Photorefractive coupling in BaTiO3 generates the self-oscillations in the resonant cavity (with variable length) formed by beam splitters BS8 and BS9. The beat frequencies for the phase-conjugate reflection and both of the two counter-propagating self-oscillations (relative to the incident beam) are derived from the motion of the interferograms at Dl, D2 arnd D3, respectively.

While only two-wave mixing occurs in the unidirectional ring oscillator previously des-

cribed, both two- and four-wave mixing are occurring simultaneously in this self-pumped

phase conjugator with external mirrors forming a linear cavity. The photorefractive cou-

*

pling process is considerably more complicated in this latter situation.

As illustrated in Figure 3, the frequency shifts (relative to the incident beam) appearing

*

on the phase-conjugate reflection and the two counterpropagating self-oscillations are si-

multaneously measured at detectors, Dj, D2 and D3, using the same interferometric techniques

* described in the unidirectional ring oscillator experiment. The resulting beat-frequency

signatures, as a function of linear cavity length detuning, are shown in Figure 4 for the

phase-conjugate reflection and one self-oscillating beam. Note that the other counter-

propagating self-oscillation beat-frequency signature appears identical to the one shown.

FRACTIONAL CHANGE IN CAVITY LENGTH INORMAIiZED TO OPTICAL WAVELENGTH)

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0

0.02

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Self-pumped phase conjugator wTIiMtEhISEI

linear caveixttye:rnaDlependence of

the beat frequencies (relative to the incident beam) for the phase-conjugate

reflection (top) and one self-oscillation (bottom) on cavity length detuning.

Regarding the beat-frequency signatures shown in Figure 4, two important features are evident. First, the frequency shift of the phase-conjugate reflection is exactly a factor of two larger than that of either self-oscillation. In fact, this is simply the conserva-
tion of energy constraint for slightly nondegenerate four-wave mixing when the two counterpropagating pump beams are the same frequency. 1 8 Second, the signs and magnitudes of the beat frequencies of the phase-conjugate reflection and the counterpropagating selfoscillations depend on the external linear cavity length detuning, similar to the unidirec-
tional ring oscillator. That is, the beat frequencies are directly proportional to the detuning (see Equation (13)), becoming faster and faster as the cavity length detuning increases or decreases away from the length that gives no frequency shift.

Two additional observations concerning the beat-frequency signatures are not shown in Figure 4. First, if the cavity length detuning is increased or decreased far enough, the selfoscillation ceases because the frequency shift required by the phase oscillation condition could not be supported by the slow response time of the photorefractive BaTiO3 (i.e., the threshold condition for oscillation as described by Equation (16)). Second, the beat-frequency signatures are periodic in cavity length detuning with the entire patterns repeating
for every half wavelength change in PZT beam splitter position (see Equation (11)). Note that both of these effects are also present in the unidirectional ring resonator.

Self-Pumped Phase Conjugator with Internal Reflections

The self-pumped phase conjugator, where the "pumping beams" for the four-wave mixing

process arise entirely from internal reflections at the crystal faces instead of external

|.

mirrors, is the simplest self-pumped phase conjugator configuration 2 since it does not

require any additional optical components. This self-pumped phase conjugator also exhibits

a frequency shift in its phase-conjugate reflection.4

We have speculated that the frequency shift observed in this self-pumped phase conjugator using internal reflections is due to the "oscillation conditions" (see Equations (8) and (9)) involving an optical resonance cavity, 1 5' 16 just as in the unidirectional ring oscilla-
tor and in the self-pumped phase conjugator with an external linear cavity described above. A number of experiments investigating the self-pumped phase conjugator with internal reflections provide conclusive evidence that the aforementioned speculation is indeed the case.

of Inthethephafsires-tcoenxjpuegraitmeenrtefwlheecrteiontheisopctoimcpaalredsettuop firesqusehnowcny ishnifFtisguroen 5th,e thientferrneaqluensceylf-shift
pumping beams. The beat frequency of the phase-conjugate reflection is determined in the usual way (i.e., interferometrically) at detector, D1 . The frequency shifts on the internal self-pumping beams are inferred by observing the beat frequency for the scattered light that emanates from the primary self-pumping corner of the crystal. Upon interfering with a portion of the incident beam as shown in Figure 5, this scattered light forms a discernable fringe pattern at detector, D2, only after it is spatially filtered to some degree by an aperture.

The results of this frequency shift comparison are shown in Figure 6. During the period where the frequency shift of the phase-conjugate reflection is constant, the beat-frequency
signature of the scattered light is also reasonably consistent considering the poor quality of the fringe pattern used to make the determination. After taking the average of the time intervals between maxima in the beat-frequency signatures, we note that the frequency shift of the phase-conjugate reflection is approximately twice that of the scattered light. Just as with the self-pumped phase conjugator with an external linear cavity, this factor of two
results from conservation of energy for slightly nondegenerate four-wave mixing (assuming,
of course, that the frequency shift of the scattered light is the same as the frequency
shift of the internal pumping beams). The deviation from two may be due to a multimode oscillation inside the crystal.

Unlike the unidirectional ring resonator and the self-pumped phase conjugator with an external linear cavity, any resonant cavity length in the self-pumped phase conjugator using internal reflections cannot be varied by simply moving a PZT mirror as was done
previously. Any resonant cavity in the self-pumped phase conjugator with internal reflections is completely contained inside the photorefractive crystal. In a second experiment, attempts have been made to systematically vary the internal cavity length via thermal expansion by controlling the temperature. 1 9 The results of this investigation are currently inconclusive because small changes in temperature (< 10C) induce instabilities in the frequency shift and intensity of the phase-conjugate reflection. We speculate that these instabilities are due to competition between the multiple spatial resonant cavity modes supported by a variety of internal reflections from the crystal surfaces. In the unidirectional ring oscillator and the self-pumped phase conjugator with an external linear cavity, the spatial modes of the resonance cavities were well-defined by the pinhole aper-
ture. In this experiment with the self-pumped phase conjugator using internal reflections, it is impossible to place an aperture inside the crystal for mode selection.

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Figure 5

Optical setup for a self-pumped phase conjugator using internal reflections. Photorefractive coupling and internal reflections from the BaTiO3 crystal surface automatically generate the self-pumping beams inside the crystal. The beat frequencies for the phase-conjugate reflection and the light scattered from the primary self-pumping corner (relative to the incident beam) are derived from the motion of the interferograms at Di and D2, respectively.

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120

TIME ISEC)

Figure 6

Self-pumped phase conjugator using internal reflections off crystal faces: correlation of the beat frequencies (relative to the incident beam) for the phase-conjugate reflection (top) and the light scattered from the primary self-pumping corner (bottom).

In a third experiment, we conclusively show that the old model 2 , 1 2 ,2 0 for the self-pumped pohsacsiellactoinojnugactoonrdituisionng aisnstoecrinaatled rweiftlhectaiocnlsoseidn-lBaoToipO3resiosnainncceorrceacvtityandisthaaptplitcheablpehasteo the
self-pumped phase conjugator using internal reflections, as well as the previously described photorefractive resonators.1 5 The picture of the old model 2' 1 2 ,2 for the selfpumped phase conjugator using internal reflections in BaTiO3 is schematically illustrated in Figure 7. Simply stated, this model assumed a pair of four-wave mixing interaction regions where two counterpropagating (and mutually phase-conjugated) self-pumping beams

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Schematic diagram for the self-pumped phase conjugator using internal
reflections off crystal faces, showing the pair of four-wave mixing regions where the incident beams interact with the self-pumping beams to generate the phase-conjugate reflection.

underwent total internal reflection in one corner (the primary self-pumping corner) of the BaTiO3 crystal. Using this model, the incident beam and its phase-conjugate reflection can also serve as the four-wave mixing "pumping beams" to produce the double-phase-conjugate oscillation8 '9 between the two interaction regions which make up the self-pumping beams.
The accuracy of this old model 2 '1 2 '2 0 can be ascertained by examining the pictures shown in Figure 8. An actual micrograph of the interacting beams in self-pumped BaTiO3 is shown in Figure 8a and tends to support the old model. However, by increasing the exposure time by a factor of ten in the same micrograph, it becomes evident that more than the two selfpumping beams are present, as shown in Figure 8b.

A

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/

.

(a)

Wb

W e

Figure 8

Microscope photographs of self-pumping process using internal reflections
in a crystal of BaTiO3, (a) 12 s exposure showing only the primary selfpumping beams along with the incident beam, (b) 120 s exposure showing primary and secondary self-pumping beams along with the incident beam and (c) 120 s exposure after painting lower-left crystal face black showing the
incident beam and the broad fan of photorefractively scattered light. Note that the phase-conjugate reflection and all self-pumping beams vanished after painting even though primary self-pumping corner (right corner) was not
painted or changed in any way.

Z

Finally, we have proven that these secondary beams which are apparent in Figure 8b are absolutely crucial to the operation of the self-pumped phase conjugator using internal

reflections. It has been suggested that the surface reflectivity in photorefractive BaTiO3 can be modified by painting the crystal faces. 1 1 ' 2 1 We attempted to reduce the reflectivity of the lower-left surface of the crystal, as indicated in Figure 8c, by covering its entire width (all the way to the corners) with Krylon ultra-flat black paint, thereby

attempting to eliminate the secondary beams shown in Figure 6b. When this painting was

carried out "in-situ" (i.e., without disturbing the optical alignments used to obtain the

self-pumping beam pictures shown in Figures 8a and 8b), not only did the secondary beams
disappear, but the phase-conjugate reflections and primary self-pumping beams also vanish (as can be seen in Figure 8c) even though primary self-pumping corner of the BaTiO3 crystal

remained undisturbed. Only a broad fan of photorefractively scattered light, along with

the incident beam, remains visible in Figure 8c. This observation agrees with our theory.

According to Equation (16), the threshold oscillation condition depends on the roundtrip mirror reflectivity R. By decreasing the refectivity R, the internal cavity falls below

threshold and thus oscillation dies. Furthermore, after painting, the crystal would not

self-pump in any orientation (i.e., at any angle or position of the incident beam). This

conclusively from just one

shows that corner and

self-pumping that the old

in BaTiO model 2 '

3 1

2

involves ' 2 0 for

more than the process

the is

internal reflections not correct. Also,

the resonator model for the self-pumped phase conjugator using internal reflections which we proposed is consistent with the series of pictures shown in Figure 8.

Summary

In conclusion, we have presented a general theory and the supporting experiments which
explain the frequency shifts of self-pumped phase conjugators. The cause of the slight frequency shifts (- 1 Hz) observed in both the photorefractive unidirectional ring resonator and the self-pumped phase conjugator with an external linear cavity is unequivocally established 1 5 and is well understood. In addition the previous description of the selfpumped phase conjugator using internal reflections , 12,20 is proven inadequate. We view all photorefractive, self-pumped phase conjugators which exhibit a frequency shift in the phase-conjugate reflection as being almost equivalent. Note that only two known photorefractive, self-pumped phase conjugators do not show the - 1 Hz frequency shifts: the ring conjugator 2 2 '2 3 and the stimulated-backscattering (2k-grating) conjugator. 2 4 The self-pumped phase conjugators which do exhibit a frequency shift all employ some sort of resonant cavity (using only internal reflections from crystal surfaces or using only external reflections from ordinary mirrors or using a combination of both) to automatically generate the self-pumping beams. Because a closed-loop resonance cavity forms, the frequency shift on the phase-conjugate reflection is dictated by the phase oscillation condition for this resonance cavity.

Acknowledgments
The authors acknowledge helpful discussions with M. Khoshnevisan (Rockwell Science Center), J. Feinberg (University of Southern California), S.K. Kwong (Caltech) and M. Cronin-Golomb (Ortel). This research is supported, in part, by the Office of Naval Research.
References
1. J.O. White, M. Cronin-Golomb, B. Fischer and A. Yariv, Appl. Phys. Lett. 40, 450 (1982).
2. J. Feinberg, Opt. Lett. 7, 486 (1982); J. Feinberg, Opt. Lett. 8, 480 (1983). 3. R.A. McFarlane and D.G. Steel, Opt. Lett. 8, 208 (1983). 4. J. Feinberg and G.D. Bacher, Opt. Lett. 9,7420 (1984). 5. W.B. Whitten abd J.M. Ramsey, Opt. Lett. 9, 44 (1984). 6. F.C. Jahoda, P.G. Weber and J. Feinberg, Opt. Lett. 9, 362 (1984). 7. H. Rajbenbach and J.P. Huignard, Opt. Lett. 10, 137 (1985). 8. M.D. Ewbank, P. Yeh, M. Khoshnevisan and J. Feinberg, Opt. Lett. 10, 282 (1985). 9. M. Cronin-Golomb, B. Fischer, S-K. Kwong, J.O. White and A. YarivF-Opt. Lett. 10, 353 (1985). 10. J.M. Ramsey and W.B. Whitten, Opt. Lett. 10, 362 (1985). 11. P. Gunter, E. Voit, M.Z. Zha and J. Albers, Opt. Comm. 55, 210 (1985). 12. K.R. MacDonald and J. Feinberg, Phys. Rev. Lett. 55, 82--(1985). 13. A. Yariv and S-K. Kwong, Opt. Lett. 10, 454 (1985T. 14. S-K. Kwong, A. Yariv, M. Cronin-Golomib and I. Ury, Appl. Phys. Lett. 47, 460 (1985). 15. M.D. Ewbank and P. Yeh, Opt. Lett. 10, 496 (1985). 16. P. Yeh, J. Opt. Soc. Am. B2, 1924 (T985). 17. G.C. Valley and G.D. DunnTf-ng, Opt. Lett. 9, 513 (1984). 18. P. Yeh, M.D. Ewbank, M. Khoshnevisan and 7.M. Tracy, Opt. Lett. 9, 41 (1984). 19. M. Khoshnevisan, Rockwell International Science Center, Thousand-Oaks, CA, private communication.

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20. K.R. MacDonald and J. Feinberg, J. Opt. Soc. Am. 73, 548 (1983). 21. S-K. Kwong, California Institute of Technology, Pasadena, CA, private communication. 22. M. Cronin-Golomb, B. Fischer, J.0. White and A. Yariv, Appl. Phys. Lett. 42, 919 (1983). 23. M. Cronin-Golomb, J. Paslaski and A. Yariv, Appl. Phys. Lett. 47, 1131 (1985). 24. T.Y. Chang and R.W. Hellwarth, Opt. Lett. 10, 408 (1985).
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APPENDIX 5.8

Photorefractive Conical Diffraction in BaTiO 3

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PHOTOREFRACTIVE CONICAL DIFFRACTION IN BaTiO 3 M.D. Ewbank and Pochi Yeh
Rockwell International Science Center Thousand Oaks, California 91360 and Jack Feinberg Department of Physics
University of Southern California Los Angeles, California 90089-0484
ABSTRACT A laser beam incident on BaTiO 3 can cause a cone of light to exit the crystal. If the incident beam is polarized as an extraordinary ray, tle cone of light is formed by ordinary rays. The cone angle is fixed by a phase-matching condition for the incident and cone beams. Measurement of this cone angle as a function of the incident angle is a simple and sensitive method for determining the birefringence of a BaTiO 3 crystal over the entire range of wavelengths where the sample is photorefractive.
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A single beam of coherent light incident on a BaTi03 crystal can cause a cone of light to emerge from the far face of the crystal. This cone has a polarization orthogonal to that of the incident ray and appears when the
incident beam is an extraordinary ray in the crystal. There have been previous accounts of rings, fans, and other forms of photoinduced light scattering in photorefractive crystals, which have been attributed to a variety of physical mechanisms. 1 9 Recently, similar light cones in BaTi0 3 have been reported and shown to be due to stimulated two-wave mixing via the photorefractive effect. 1 0 Here, we account for the phase-matching condition in BaTiO 3 for anisotropic Bragg scattering 11 by using a simple geometrical construction to predict the angular position of the light in the exit plane. We also show that precise measurenents of the cone angle can be used to determine the dispersion of the birefringence, An = ne - no, of a BaTi0 3 sample.
Figure I shows the experimental setup, with a laser beam incident on one of the a-faces of a BaTiO 3 crystal. The incident beam makes an angle e in air with the face normal and is polarized to be an extraordinary ray, with its electric-field vector in the plane of incidence defined by the beam direction and the c-axis of the crystal. A broad fan 12 of extraordinary light is observed on the +c-axis side of the transmitted beam, as shown in Fig. 2a. Simultaneously, a single ring of light with ordinary polarization appears on the negative c-axis side of the transmitted beam (see the multiple exposure photograph in Fig. 2b). For an incident beam intensity of - IW/cm 2, the fan and the ring appear within a few seconds. As shown in Fig. 2b, the shape of the ring varies with the angle of incidence. The ring is visible for both positive and negative
angles of incidence e (with positive 0 defined in Fig. 1), although for negative
angles the ring intensity is diminished because self-pumped phase conjugation 13 depletes the incident beam intensity.
The rings observed in Fig. 2b for BaTiO 3 result from anisotropic Bragg scattering1 1 of the incident beam off photorefractive gratings formed during beam fanning.1 0 The incident beam, with wavevector i scatters from defects or impurities into a broad fan having a range of wavevectors f These scattered
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beams interfere with the incident beam and create photorefractive index gratings with wavevectors given by

g=f

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(1)

The incident beam then Bragg-scatters off these gratings and either reinforces or depletes the fanning beams by two-wave mixing 14 depending on the sign of the projection of onto the positive c-axis direction. The collection of all amplified scattered beams is a broad fan of light directed towards the positive c-axis side of the crystal.
The photorefractive grating wavevectors formed during bean fanning can also deflect the incident beam into a cone of light. As illustrated in Fig. 3a, some of these photorefractive gratings will have wavevectors of exactly the right length and direction - to deflect the extraordinary incident wavevector i into an ordinary ring beam
r

r

1

(2)

Eliminating from Eqs. (1) and (2) gives the phase-matching condition:

= 2i " f

(3)

Equation (3) selects a cone of wavevectors r as can be seen in the

following simple geometric interpretation.
lr

The locus of all possible

r

(ordinary ring beams) is a sphere of radius 2no/oX. The locus of all possible

f (extraordinary fanned beams) is an ellipsoid of revolution with semi-minor

,

and semi-major axes of lengths 2 -ne/, and 2no/X, respectively. Displace the

centerl1of the ellipsoid from the center of the sphere by an amount 2 i. Then the intersection of the ellipsoid and the sphere selects a cone of phase-matched wavevectors r" Figure 3b shows this geometric construction in the x-z plane,

while Fig. 3c extends it to three dimensions. As can be seen in Fig. 3c, the

intersection of the two normal surfaces is a ring centered around the direction

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