December 1, 1994 / Vol. 19, No. 23 / OPTICS LETTERS 2015 Optimization of the field of view of a Brillouin-enhanced four-wave-mixing phase conjugator John R. Ackerman SFA, Inc., Landover, Maryland 20785 Paul S. Lebow U.S. Naval Research Laboratory, Washington, D.C. 20375-5000 Received July 11, 1994 A Brillouin-enhanced four-wave mixing phase conjugator featuring a field of view that varies by approximately an order of magnitude, i.e., from 2 to 20 mrad, is described. We control this variable field of view by utilizing a slightly noncollinear pumping geometry and by adjusting the pump-pump frequency difference to optimize the phase mismatch for the overall process. A model that describes the interplay between frequency and geometry to the Brillouin-enhanced four-wave-mixing phase mismatch is found to predict the expandable field of view. Nonlinear phase-conjugation methods such as degen- erate four-wave mixing, photorefractive wave mixing, and Brillouin-enhanced four-wave mixing (BEFWM) have typically been characterized by small numerical apertures. These narrow fields of view are an outgrowth of the precise phase matching required for establishing the appropriate interference gratings from which the phase-conjugate output is scattered. These gratings can be modified to strengthen the phase-conjugate output with adjustments to the mix- ing geometry and/or the spectral composition. ,2 BEFWM is uniquely flexible with regard to many operational parameters over which it can supply amplified phase-conjugate replicas of low- intensity signals. 3 Part of this flexibility arises from BEFWM's transient nature, i.e., high-reflectivity phase conjugation occurs only when the nonlinear medium is pumped by pulses with durations compa- rable with the medium's acoustical phonon lifetime. This condition widens somewhat the range of per- missible phase mismatch and relaxes the precise directionality of BEFWM's interacting beams, as compared with other types of wave mixing that em- ploy cw beams. Another major factor contributing to BEFWVM's wide range of operation is its ability to function over a considerable array of spectral and geometrical combinations. 4 Because the overall phase mismatch is determined from both spectral and geometrical factors, the performance of the con- jugator can be readily tailored to a given application by the appropriate selection of these parameters. An example of this performance modification result- ing in a variable field of view for BEFWM phase conjugation is described here. Critical to the high-reflectivity phase-conjugate performance of the BEFWMV mirror is the value of the phase mismatch AkL. It is the product of the interaction length L and the relevant wave-number mismatch, which is a sum of spectral and geomet- rical terms 5 ' 6 : Ak = Akfreq + Akgeom, (la) where Akfreq = ((OP 1 - COP2 + ± PC S- &SG) (n/c), Akgeom = -k~ta. (lb) Here co, are the frequencies of the four waves (two pump beams, P1 and P2, phase-conjugate beam PC, and signal beam SG), n is the refractive index, k is the average wave vector for the interacting waves, 0 is the full angle between the signal and the high- intensity pump (P1), and a is the full P1-P2 angle from collinearity [see Fig. 1(a)]. After differentiat- ing this expression with respect to 0 we see that the field of view of the conjugator, 86, depends directly on the permissible range of wave-vector mismatch and depends inversely on a: 80 = 3(Ak)/ka. (2) The primary result of our study is the verification of this relationship. A three-frequency Nd:YAG laser system, which directly supplies all three nearly degenerate in- puts to the BEFWM mirror, and the experimen- tal setup designed to measure accurately the BEFWM geometry are described in Ref. 4. The P1 and P2 pulsed pump waves are independently injection seeded, which permits user-controlled tunability of the pump-pump frequency difference. The cw signal is a 15-mWbeam (or 150 pJ integrated over the 10-ns pump pulse width), and the signal frequency is tuned to be upshifted from the P2 frequency by the medium's stimulated-Brillouin- scattering (SBS) shift. The BEFWM medium is an 8-cm path-length cell of room-temperature CS 2 (SBS shift of 3.7 GHz at 1064 nm). Each BEFWM configuration is optimized both by spectral and geometrical adjustments so as to maximize the phase-conjugate output energy. The optimized phase-conjugate output is sur- rounded by a speckle field arising from BEFWM 0146-9592/94/232015-03$6.00/0 ( 1994 Optical Society of America
Optimization of the field of view of a Brillouin-enhancedfour-wave-mixing phase conjugator
John R. Ackerman
SFA, Inc., Landover, Maryland 20785
Paul S. Lebow
U.S. Naval Research Laboratory, Washington, D.C. 20375-5000
Received July 11, 1994
A Brillouin-enhanced four-wave mixing phase conjugator featuring a field of view that varies by approximatelyan order of magnitude, i.e., from 2 to 20 mrad, is described. We control this variable field of view by utilizing aslightly noncollinear pumping geometry and by adjusting the pump-pump frequency difference to optimize thephase mismatch for the overall process. A model that describes the interplay between frequency and geometryto the Brillouin-enhanced four-wave-mixing phase mismatch is found to predict the expandable field of view.
Nonlinear phase-conjugation methods such as degen-erate four-wave mixing, photorefractive wave mixing,and Brillouin-enhanced four-wave mixing (BEFWM)have typically been characterized by small numericalapertures. These narrow fields of view are anoutgrowth of the precise phase matching required forestablishing the appropriate interference gratingsfrom which the phase-conjugate output is scattered.These gratings can be modified to strengthen thephase-conjugate output with adjustments to the mix-ing geometry and/or the spectral composition. ,2BEFWM is uniquely flexible with regard tomany operational parameters over which it cansupply amplified phase-conjugate replicas of low-intensity signals.3 Part of this flexibility arises fromBEFWM's transient nature, i.e., high-reflectivityphase conjugation occurs only when the nonlinearmedium is pumped by pulses with durations compa-rable with the medium's acoustical phonon lifetime.This condition widens somewhat the range of per-missible phase mismatch and relaxes the precisedirectionality of BEFWM's interacting beams, ascompared with other types of wave mixing that em-ploy cw beams. Another major factor contributingto BEFWVM's wide range of operation is its abilityto function over a considerable array of spectraland geometrical combinations.4 Because the overallphase mismatch is determined from both spectraland geometrical factors, the performance of the con-jugator can be readily tailored to a given applicationby the appropriate selection of these parameters.An example of this performance modification result-ing in a variable field of view for BEFWM phaseconjugation is described here.
Critical to the high-reflectivity phase-conjugateperformance of the BEFWMV mirror is the value ofthe phase mismatch AkL. It is the product of theinteraction length L and the relevant wave-numbermismatch, which is a sum of spectral and geomet-rical terms5 ' 6:
Here co, are the frequencies of the four waves (twopump beams, P1 and P2, phase-conjugate beam PC,and signal beam SG), n is the refractive index, k isthe average wave vector for the interacting waves,0 is the full angle between the signal and the high-intensity pump (P1), and a is the full P1-P2 anglefrom collinearity [see Fig. 1(a)]. After differentiat-ing this expression with respect to 0 we see that thefield of view of the conjugator, 86, depends directly onthe permissible range of wave-vector mismatch anddepends inversely on a:
80 = 3(Ak)/ka. (2)
The primary result of our study is the verification ofthis relationship.
A three-frequency Nd:YAG laser system, whichdirectly supplies all three nearly degenerate in-puts to the BEFWM mirror, and the experimen-tal setup designed to measure accurately theBEFWM geometry are described in Ref. 4. TheP1 and P2 pulsed pump waves are independentlyinjection seeded, which permits user-controlledtunability of the pump-pump frequency difference.The cw signal is a 15-mW beam (or 150 pJ integratedover the 10-ns pump pulse width), and the signalfrequency is tuned to be upshifted from the P2frequency by the medium's stimulated-Brillouin-scattering (SBS) shift. The BEFWM medium isan 8-cm path-length cell of room-temperature CS2(SBS shift of 3.7 GHz at 1064 nm). Each BEFWMconfiguration is optimized both by spectral andgeometrical adjustments so as to maximize thephase-conjugate output energy.
The optimized phase-conjugate output is sur-rounded by a speckle field arising from BEFWM
0146-9592/94/232015-03$6.00/0 ( 1994 Optical Society of America
Fig. 1. (a) Variation of the noncollinear BEFWM geom-etry by reduction of the P1-P2 angle a results in anincrease in both the SG-P1 angle 0 and the conjugator'sfield of view &O. (b) Data points are observed values ofa and 0 that result in high-reflectivity BEFWM phaseconjugation when frequency-degenerate pumps are used.The curve is a best-fit hyperbola [see Eq. (lb)] modeled tothe data. (c) Fields of view as a function of a measuredfor the BEFWM conjugator when frequency-degeneratepumps are used.
amplification of acoustic noise intrinsic to the non-linear medium.7 This speckle is most intense whenno signal of the appropriate frequency is input tothe conjugator. We can obtain an indication of thefield of view of the BEFWM mirror by measuringthe angular spread of this output speckle pattern.Video recordings of the output speckle field are madefor optimized BEFWM configurations both with andwithout the signal input to the mirror. We deter-mined the angular spread of this field from the videodata, following procedures similar to those used tomeasure 0 and a, as described in Ref. 4. We notehere that the speckle-field size does not define thepractical limits of the BEFWM field of view; some-what less-than-optimized high-performance BEFWMphase conjugation has been observed for a valuesthat lie outside the speckle observed with the videomonitor. All measured angles are adjusted for theCS2 refractive index.
As a starting point, a well-studied BEFWM con-jugator is chosen that features an equal-frequencynearly collinear pumping configuration and an anti-Stokes SBS shifted signal. After variations in thesystem's geometry are made, phase-conjugate op-eration can be retained in two basic ways. Themore customary form of variation is depicted inFig. l(a), where an adjustment in either 0 or a iscompensated by a reciprocal in the other angle whilethe frequency relationship among the interactingbeams, i.e., Akfreq, is held constant. The resultinginverse relationship between these angles, shown inFig. l(b), confirms the accuracy of Eq. (lb). The an-gular span of phase-conjugate operation, or field ofregard, for this kind of variation is -27 mrad, similarto earlier results.4 Throughout the range of theseadjustments the phase-conjugate reflectivity consis-
tently exceeds 106. Equation (2) predicts that theconjugator's field of view will be relatively large forgeometries that feature small a values. The resultspresented in Fig. l(c) confirm this trend, with thefield of view increasing smoothly from -2 to 6 mradas a decreases from 2.7 to 1.0 mrad. This trend islimited by the finite beam widths involved; as a de-creases to values below -1 mrad, 6 increases to apoint at which the signal and pump beams fail tooverlap over the entire 8-cm interaction length. Asa consequence, the phase conjugation rapidly dimin-ishes for small a values when this all-geometricalform of adjustment is used.
Another form of system variation, displayed inFig. 2, occurs when adjustments in a are comple-mented not by a reciprocal change in 0 but insteadby a change in Akfreq. Equations (1) and (2) indi-cate that an appropriate alteration in the interactingfrequencies will have an outcome on the conjuga-tor's field of view that is entirely equivalent to whenchanges are made in A kgeom only. A potential benefit[see Fig. 2(a)] of this form of frequency optimization isthat both 0 and the interaction length are held fixedduring these alterations, permitting a, and thus thefield of view, to vary over a wider range than whenonly geometrical variations are made. Figure 2(b)illustrates how frequency compensation provides forgeometries different from the degenerate pump fre-quency case. Pump geometries with a < 1 mrad arereadily achieved, and the resulting fields of view ex-pand substantially to -20 mrad at a = 0.34 mrad.The inverse relationship between field of view and abecomes prominent when this expanded data set isanalyzed; Fig. 2(b) shows that a hyperbola for 86 asa function of a fits these data quite well. Althoughthese nearly collinear pumping configurations pos-sess expanded fields of view, Fig. 2(c) reveals that
68Cc ~~~~~~~~~~SG
P2 _ PC
BEFWM cell
e _> SG
P2-TPi 0.s 7Fl
BEFV*J cell
(a)
69 (mrad)
2 3a (mrad)
BEFWMreflectivity
2 3a (mmd)
10
(b) (c)
Fig. 2. (a) Variation of the BEFWM configuration byreduction of a and by compensation with an adjustmentin Akfreq results in an increase in 80 without affecting 0.(b) Circles are fields of view measured as a function ofa when compensated by Akfreq changes. The curve is abest-fit hyperbola [see Eq. (2)] modeled to the data. Thetriangles are the data of Fig. l(c). (c) Phase-conjugatereflectivity associated with the data (circles) of (b).
Fig. 3. (a) Wave-vector mismatch and phase mismatch(Ak and AkL, respectively) calculated by use of Eq. (1)as a function of Akgeom for a variety of high-reflectivityBEFWM configurations. [The frequency-degeneratepump case occurs when Akgeom = 3.0 (arrow)]. (b) Twoexamples of the range of wave-vector and phase mismatchobtained by variation of Akgeom (by adjustment of 0)before the phase-conjugate energy falls to half its maxi-mum value.
the trade-off for this improvement is a decrease in thereflectivity available for a narrowly collimated signaldirected at the center of the speckle field to below106. The measured low-energy threshold for thesewide field-of-view configurations remains unchangedfrom the narrow field-of-view value of -10 pJ.
The well-fit hyperbola in Fig. 2(b) reflects theconstancy of wave-number mismatch, and thereforephase mismatch, for optimized BEFWM configura-tions. Figure 3(a) displays Ak and AkL computedfrom Eq. (1) for 40 distinct BEFWVV configurations(including the data shown in Figs. 1 and 2) that pos-sess a wide variety of geometries (Akgeom ranges from-0 to -6) and compensating frequency relationships(the P1-P2 pump frequency difference varies over-6 GHz, or approximately ±0.8 times the CS2 SBSshift). Average values for these measurements are0.48 ± 0.10 cm-' for Ak and 3.8 ± 0.9 for AkL, inexcellent agreement with theoretical predictions"8
(AkL 4). Similar previous measurements made inthis laboratory4 yielded phase-mismatch values thatare somewhat less (average AkL = 2.4 ± 1.8) thanobserved here, and the discrepancy is attributed totwo factors. First, refinements made in the geom-etry measuring system have improved the accuracyof the current geometrical measurements over thoseperformed in the earlier study; second, the rangeof phase mismatch that permits high reflectivityis considerable. This range is featured by the linesegments in Fig. 3(b). The endpoints of each linesegment denote the limiting phase-mismatch valuesthat generate phase-conjugate reflectivities that areone half of the maximum BEFWM reflectivity as 0is varied. Spans of several phase-mismatch units
(4.0 and 5.4) are measured. These spans are anal-ogous to the FWHM linewidth presented in Fig. 5bof Ref. 4, in which a 720-MHz FWHM linewidth, or3.7 phase-mismatch units, was obtained when the P1(and thereby PC) frequency was scanned versus thefixed P2 and SG frequencies. The similarity in thesesizable spans when either Akgeom or Akfreq is variedunderscores the equivalency of these contributions tothe overall BEFWM phase mismatch.
Other studies3,8-10 centered on BEFWM configura-tions in which the low-intensity pump is the SBSphase conjugate of the high-intensity pump. It is in-structive to extrapolate our results to this form ofBEFWM in which both a and Akgeom are constrainedto be zero. Equation (2) and the results shown inFig. 2(b) indicate that a BEFWM conjugator withcollinear pumps should have a wider field of viewthan that of the noncollinear variety, whereas the re-sults shown in Fig. 2(c) suggest that the reflectivityshould be diminished. It is our conjecture that thediminished reflectivity here is due to a reduction ingain per solid angle as the field of view expands. Thereflectivities reported in these other studies are sub-stantial, i.e., at least 106, and, to our knowledge, noother BEFWM studies have focused on the fields ofview of these phase conjugators (although the back-ground speckle shown in Fig. 7 of Ref. 10 is quitelarge). Considering the different conditions underwhich the collinear and noncollinear pumping resultswere obtained, it appears reasonable to attribute thesubstantial collinear reflectivities to other parame-ters that may differ among studies (i.e., differentBEFWM medium, total pump power, pump ratio, orsignal solid angle) and that affect BEFWM perfor-mance by means other than phase mismatch.
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