AUON-CLASASI7F2Il1E1D RPSTOUYCEDKHIEWSEUT EOAFILNLLT.PEHRAJNSUEAN-TClI9OO6INNAJSULCG5T4H2O4A.UTSOAAERPNTDNICGAOLSAKISD4E-8CVI-ACCES-0C2ICEO1NF9C/EG UCCEENPTETRSM 26/6 1/ ML Emhc~OhEEhEEfmhm K~hhIhhhhEI Eo momhhmhhhmhEEEI lsommmEhoEhEmEhmEhEhEhEohE EEmhEEEEmhEEEE 1-0 1*4, SC5424.AR Copy No. = STUDIES OF PHASE-CONJUGATE OPTICAL DEVICE CONCEPTS L 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 J 5 U Pochi Yeh Ian McMichael JUNE 1986 C:) L.L.J Approved for public release; distribution unlimited j- Rockwell International -- Science Center Z;:; r -.F- 'Z. 7- Rockwell International Soci%i*ence Center SC5424 .AR * " 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 C7714A/jbs Rockwell International • o Science Center % SC5424 .AR 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 NTIS CRA&I LTIC TAB 0, .' U::anno,iced Jjstif,cation 5 I. .. " " By ................... D,A: ibrtfl Ii." iv Av I hly ' .:,,-3 C7714A/jbs Dist . 7 1.0 SUMMARY 01 Rockwell International Science Center SC5424.AR 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. C77141A/jbs Rockwell International Science Center SC5424 .AR 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 .-4 -4-yJJ.. 2 C7714A/jbs - . ..L./ ~ ~ 4 .. ~... .... 01 Rockwell International Science Center SC5424 .AR 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 C7714A/jbs MCA1 O Rockwell International Science Center SC5424 AR * "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. -p *Works only partially supported by this contract. 4 C7714A/jbs . .r...c -_] *°w-w°C ) r . C ' -~ _ _ .. ' r . V L-SW vTT _ _ n - _J -, i Rockwell International Science Center SC5424.AR 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 5 | C7714A/jbs 0 Rockwell International Science Center SC5424 .AR 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. ".. L.. ' " .-. " .' '..' '.,..-. '.-. 6 C7714A/jhs "- . .' ' '. r ", - V... " " " " "'" ". " " - ".. "" ".- "" *'*" '' ' -""- "-. "- "* 0 Rockwell International Science Center SC5424 .AR 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 M1 M2 MN-1 MN 0RREFLECTOR Fig. 2-1 Phase conjugator for polarization restoration in a birefringent system. 7 C7714A/jbs O Rockwell International Science Center 4 SC5424 .AR 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 8 C7714A/jhs N* pNLk= 2, SRockwel International Science Center + (K,.C) + e( *.4)]ei(A +kz) SC5424 .AR (9) 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. 9 C7714A/jhs •-""" "--'-""""-.'" .-'"-\.--.-"-."." ., ." '", -' ' -... -'.',[-.>' •, ,'-.-,-7',,'r,,'..A ." L K! 4 & fl( T TWV"(rJ' " Y jT~~P . 1"r z* -~~ 01 Rockwell International Science Center SC5424 .AR A scheme, similar to that shown in Fig. 2-2(c), which utilizes self- - 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. SC84-26 159 * (a) PUMP BEAM PROBE BEAM ISOTROPIC NONLINEAR MEDIUM - -*z 'SE PUMP BEAM (b) PHOTOREFRACTIVE PUMP BEAM PROBE BEAM PUMP BEAM (C) Z +4 _ + 2' =,ROTATE 90 C'ABOUT Z PHOTOREFRACTIVE CRYSTAL PROBE BEAM VIED +2 =+1 * ROTATE 900 ABOUT AXIS Fig. 2- Schematic drawings of polarization-rsvigpaecnutr. 10 C771 4A /jbs @ Rockwell International Science Center SC5424.AR 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. SMIRROR OR q* REFLECTOR Uc*3-24380 I A DETECTOR B o* REFLECTOR FIBER LOOP * Fig. 2-3 Schematic of the phase-conjugate fiber-optic gyro. 11 C?? 14A/jbs *' W ciW;71kk--777 7-r' ITa WI J 01 Rockwell International Science Center SC5424.AR 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. , . , . . , ,, ,. '' ' ' ' ,' . o',' . 12 C7714A/jbs , ,, .. ._.' -:. *"..- ' -... .. ' . _' ._ , _..' .. ' " , " ,' ' ,. o Rockwell International Science Center SC5424 .AR 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 13 C7714A/jbs , ." -"...- -.-. • ,-,....v ".'.--.... . . - . J- ' - .- " . ". - - . - . -. -- ~.''~-~V~L- ~~7-~T~-'~ M.W IM W o Rockwell International Science Center SC5424 .AR 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- 14 C7714A/jbs O Rockwell International Science Center SC5424 .AR 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. I 'I P2 D 3138 BS1 P1 BS2 L1 L2 MMF MPPPC 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 15 C7714A/jhs SC85-32009 NORMAL MIRROR i Rockwell International Science Center SC5424.AR 00 900 I 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 O l Rockwell International Science Center SC5424 .AR S S 32004 B.SI M4 X/2 BS3 X/14M3 X2 aT3 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%. 17 C7714A/jbs P.-t.- '.S<'>..,> ';4..> . .. -K' -' < ' , -', >-,-- -%>;- ;, i i'-] >.>1-.i - .-, "??.?- SRockwell International Science Center SC5424 .AR 900 uj z 0 cc * Z0DIN- 450 0 SC6 32007 o L9 0 , u) 00( 0 0 454 PAOe POLARIZATION ANGLE 90 0 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. .w . ' 18 , C7714A/jbs X ° x lI - 2 / VI Rockwell International Science Center SC5424. AR 9SOI 32004 00 Y 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) 19 C7714A/jbs -~ . . .*. . * , .. .*. , . O Rockwell International Science Center SC5424 .AR 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 20 C7714A/jbs I-- -- -- - .. . . -. g - .. . |O l% Rockwell International Science Center SC5424.AR 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 C7714A/jbs i .:1 ' Rockwell International Science Center SC5424 .AR 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 22 C7714A/jbs 0 Rockwell International Science Center SC5424 .AR 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. ," bI 23 C7714A/jbs ..........'.......................-..,',...,'.-... ..,............-.------ o Rockwell International Science Center SC5424.AR 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 'p% 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 oil Rockwell International Science Center - SC5424 .AR 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. V4 .. ,. , .. . - .V .. . . .... '~P P - ~ ~P - " 'P -~ oPilR* V 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 .1; ,' .. %-.,-.%" -.'..-.... ".."....."...".".."-................"."..".,".'.........",...........".........".............,......."...".....,..........."....."........,......"..."., 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 ,.-.- ,. . -. . a... - . . . .. . 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 I'~.,.-:.'".',<,:,- .-.'.:>-i''?.'-". .. '. - -"L -'. L . - :.- -;i-': ;." _ -..-.--:.:- '.': ...-, .. ,-",-., -' : 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. ow>0 :0 Oi% Rockwell International Science Center APPENDIX 5.3 SC5424 .AR Absolute Phase Shift of Phase Conjugators (Paper to appear in Proc. SPIE 613 (1986)) 55 C7714A/jbs * -rature). I* SC5424.AR 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 SC5424 .AR "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 SC5424 .AR 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. ~ - °-.,,,. - * -S ""*. "'''''' 62 C7714A/jbs -" , . ._-"..•. ".",".•"-"•"..-* -. - *.,.-. *"-•."- -" ' ..-.. . " . -." ,.. - o. "' .., . "."." ° ,•-"-.-" "-" i Rockwell International o % Science Center SC5424 .AR 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 ")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 ( -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) -0.00 --0.04 -0.02 0 0.02 0.04 0.061 deroiedth romintthrfeeogromtisn a D1 D2andD3, espctiely Figure 4 -U -2 - 0 1 2 3 I , I 0 20 P'iT VOLTAGE (VI I , I * I * I 60 120 A0 1SO 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. ARON~ SON LASER 9(614. num) M , F RDAY IS-o- - l -1000 urnn ,J X.]o~t - s %\V _]1 MI I L3S -.]02 M -MIRROR U KAM SPLTER L - LENS D - DETECTOR 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. A --l0.9 iH, ,9A ... 4..... 2090 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 ............................................... " ......................" " " r, * 4' . J*. ' , " , ''. * '- ' . '' ' 1.' ,. '' . '" ' " ." . "" . '' ., ,,, ' ' . '' , ' . ,., " .. " ... ,. ' -~~~~~~- - --- . -Mr-- - Z. IR . wv 4-WAVE MIXING Figure 7 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 .4 / . (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. -' . . ... *..., . . .. . . . . . . ... . .. . . ... . - -*: 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). -4 . .*.*.'- fJ YyW U YWh ~~-- ~ W J~~ ~ b Vl " '1 --'" T,-V'1. T7-.-VWV-.--Iv7 -v - WT - V.wlrr-l~ w- .w- ylw 1w-v 0 %Rockwell International Science Center SC5424 .AR 4'4' APPENDIX 5.8 Photorefractive Conical Diffraction in BaTiO 3 '88 C774Ajb 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. .4 J7612TC/jbs 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 2 J7612TC/jbs .hr '...¢'..,gt, .... " .. .',.. ,". .. '......'......-'*.--*,...-."."--.-"**...."...*...-"..."..'.-.,... :."- ",,* ,- - - - " " - beams interfere with the incident beam and create photorefractive index gratings with wavevectors given by g=f "~ • (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 I", 3 J76121C/jbs .. . RD-Nil M STUDIES OF PNSE-CONJU4TE OPTICAL DEVICE CONCEPTS(U) 2'2 ROCKWELL INTERNATIONAL THOUSAND OAKS CA SCIENCE CENTER P YEN ET AL. JUN 86 SC5424.AR NN*14-S-C-S219 UNCLRSSIFIED F/G 20/6 ML Ill IIIIIEEEElIII 1-0- - -l