2306 J. Opt. Soc. Am. B/Vol. 7, No. 12/December 1990

Mutually pumped phase conjugation in photorefractivestrontium barium niobate: theory and experiment

M. D. Ewbank, R. A. Vazquez, and R. R. Neurgaonkar

Rockwell International Science Center, 1049 Camino Dos Rios, Thousand Oaks, California 91360

Jack Feinberg

Department of Physics, University of Southern California, Los Angeles, California 90089-0484

Received December 26, 1989; accepted May 15, 1990

Two mutually incoherent laser beams can generate each other's phase-conjugate replica by sharing hologramsin a photorefractive strontium barium niobate (SBN) crystal. The path of each beam inside the SBN crystaluses three internal reflections at the crystal faces. We discuss phase-conjugate imaging, moving gratingsinduced by an applied electric field, the time response of grating formation, and the dependence of the phase-conjugate reflectivity on the incident-beam ratio. Also, we derive the amplitude coupled-wave solutions for mu-tually pumped phase conjugators with two coupled gratings, and we compare the theoretical results with ourexperimental results.

1. INTRODUCTION

A mutually pumped phase conjugator generates the phase-conjugate replica of each of two incident optical beams. 9

Each beam is converted into the phase-conjugate replica ofthe other by deflection off at least one shared hologram.The two incident beams need not be coherent with eachother, and they can even be from different lasers operat-ing at the same nominal wavelength. Mutually pumpedphase conjugation, which to date has been demonstratedonly with photorefractive crystals, may prove useful forapplications in two-way optical communication'" and ringinterferometry."

The principle of operation of a mutually pumped phaseconjugator is as follows. Consider two optical beams, Aland A2, incident upon a photorefractive crystal. Let eachbeam have a coherence length of at least a few millime-ters, but let the two beams be mutually incoherent. (Forexample, the two beams might come from two non-phase-locked semiconductor diode lasers operating at the samenominal wavelength.) Beam A, will interfere with its ownscattered light in the photorefractive crystal. The result-ing randomly oriented interference patterns will create,by the photorefractive effect, a multitude of refractive-index (fanning) gratings inside the crystal. Beam A2 willalso create its own set of fanning gratings. The particu-lar grating that scatters beam Al into the phase-conjugatereplica of beam A2 must also (by the principle of time re-versal) scatter beam A2 into the phase-conjugate replica ofbeam Al. This particular grating is reinforced by bothincident beams (whereas most of the other gratings arereinforced by only one incident beam); this shared gratingdominates the competition and eliminates the other grat-ings. In practice each beam may deflect off a series ofgratings in the crystal. The orientation of these gratingsand the particular beam path that connects them differ-entiate the various mutually pumped phase conjugators.` 7

Here we describe a new mutually pumped phaseconjugator that uses three internal reflections betweentwo distinct interaction regions inside a photorefrac-tive crystal of cerium-doped strontium barium niobate(SrO.7r BaO.25Nb 2O6, or SBN:75). Previously demonstratedmutually pumped phase conjugators used no,16- 9 one,2 ortwo3 internal reflections in addition to one 8 9 or more2

-7

gratings to connect the two incident optical beams. Thenumber of internal reflections and the number of gratingsis determined by the crystal's Pockels coefficients, dielec-tric constants, refractive indices, photorefractive chargedensity, and absorption coefficient along with the crystalcut and size.

Using this new mutually pumped phase conjugator, weshow simultaneous phase conjugation of two images, mea-sure the frequency shifts on the phase-conjugate beamsinduced by an external dc field, and determine the de-pendence of the phase-conjugate reflectivities on the rela-tive intensity of the two incident beams. Additionally, wedevelop a theory to compute the phase-conjugate reflectivi-ties of any mutually pumped phase conjugator with twointeraction regions that are interconnected through scat-tering by solving the amplitude coupled-wave equations.

2. NEW MUTUALLY PUMPED PHASECONJUGATOR: THE FROG-LEGS PHASECONJUGATOR

Figure 1(a) shows a crystal of cerium-doped SBN:75 illu-minated by two 514.5-nm beams from a cw argon-ion laser.This crystal (a x b x c = 4.9 mm x 4.7 mm 5.0 mm)was grown and poled into a single domain at RockwellInternational Science Center. The two laser beams areincident upon opposite a faces of the crystal. The laser isoperated without an 6talon, and the beam paths from thelaser to the crystal are sufficiently disparate to ensure

0740-3224/90/122306-11$02.00 C 1990 Optical Society of America

Ewbank et al.

Vol. 7, No. 12/December 1990/J. Opt. Soc. Am. B 2307

(C)

(a)

LEG #1 LEG #2

A* (co2) A* ko,)

(b) (d)Fig. 1. Two mutually incoherent laser beams with extraordinary polarization interact by means of shared photorefractive holograms in afrog-legs phase conjugator. (a) Photograph of the beams inside the SBN:75:Ce crystal for nearly symmetric incident beams. (b) Diagramshowing the angles of incidence, the slight beam bending caused by the photorefractive gratings, and the direction of the optic axis of theSBN crystal. (The -c face was connected to the positive electrode during poling; photorefractive beam fanning is toward this face.)Phase conjugation with temporal, but no spatial, cross talk is denoted by the amplitudes Ai*(wj), where i X j. (c) Photograph of a real pairof frog legs, illustrating the resemblance to the beam paths of Fig. 1(a) and justifying the name frog-legs phase conjugator. (d) Pho-tograph of the beam interaction inside the SBN:75:Ce crystal for asymmetric incident beams.

that the two beams are not coherent with each otherwithin the crystal.'2 Each incident beam has extraordi-nary polarization inside the crystal, has a few milliwattsof power, and forms a rather extreme external angle ofincidence of -70° with the normal to the crystal face.(We could vary this angle between 45° and 80° and stillobserve phase-conjugate signals.) Figure 1(a) shows thateach beam, after traveling a millimeter or two inside thecrystal, spontaneously deviates from its incident path by

4.30 toward the -c face (which was in contact with thepositive poling electrode and is in the direction of two-wave mixing gain for this crystal). This deviation, shownschematically in Fig. 1(b), is presumably caused by deflec-tion from a self-formed refractive-index grating, as de-scribed above. The deflected beam then strikes theopposite a face of the crystal, beginning a series of threetotal internal reflections at three crystal faces, followedby a deflection from a second grating, which directs it ex-

Ewbank et al.

2308 J. Opt. Soc. Am. B/Vol. 7, No. 12/December 1990

actly opposite the other incident beam. Figure 1(c) showswhy this new configuration is called the frog-legs phaseconjugator. As shown in Fig. 1(d), the new configurationis somewhat accommodating in that it can compensate forminor changes in the position of the incident beams.

The frog-legs phase conjugator is aligned in the follow-ing manner. The two incident beams are adjusted so thattheir respective Fresnel reflections (from the far crystalfaces) cross -0.2 mm away from the -c face of the crystal.As mentioned above, the crystal is oriented so that thedirection of the fanning beams is also toward the crystal's-c face. Within a few minutes the fan of each incidentbeam will collapse into a single intense beam, with thesecollapsed fans connected by total internal reflection at the-c face of the crystal. This connection occurs eventhough initially there is no visible overlap of the two fansat the -c face. (If the incident beams contain compli-cated images, then the connecting beam will contain manyfilaments of light, but if the incident beams are simpleGaussians, then the connecting beam may contain as fewas two filaments.) As the fans collapse and connect, thephase-conjugate signals appear.

We measured the phase-conjugate reflectivity and for-mation rate of the frog-legs phase conjugator by using twofocused Gaussian incident beams with equal intensities.The intensity reflectivity was typically 25%, uncor-rected for absorption and Fresnel reflection losses. Thisreflectivity is comparable with that of the bird-wingphase conjugator2 for similar incident beams. However,the grating formation time for the frog-legs phase conju-gator was over 100 times longer than for the bird-wingphase conjugator. With two 14-mW incident beams (at514.5 nm focused to a beam diameter of -100 pum exter-nal to the SBN crystal, which corresponds to -66 W/cm2

inside the SBN crystal), the frog-legs phase-conjugate sig-nals initially required 60 sec first to appear and thenrequired an extra 10-15 sec to approach steady state,whereas the bird-wing phase-conjugate signals appearedin -0.5 sec and approached steady state -0.3 sec later

with the same 14-mW incident beams. These grating for-mation times will not necessarily become shorter if theincident beams are more tightly focused, because thephase-conjugate gratings are initiated by beam fanning(stimulated scattering), which will decrease when the spotsize is reduced. (We note that this particular cerium-doped SBN:75 crystal is also a highly efficient cat conjuga-tor,13 4 with an uncorrected phase-conjugate reflectivityas high as 60%, but only when the talon was removedfrom the argon-ion laser.)

Figure 2 shows the optical setup for phase-conjugateimaging with the frog-legs phase conjugator. Two trans-parencies T, and T2 (a U.S. Air Force resolution chart anda photograph of a moose) were illuminated by mutuallyincoherent laser beams (from one Ar' laser for the resultsdemonstrated here) and then were focused by 50-cm focal-length lenses into opposite a faces of the SBN:75:Ce crys-tal. After tens of seconds, two beams (Al* and A2*)emerged from the SBN crystal, and the image carried byeach beam was sampled by beam splitters BS, and BS 2,located between the transparencies and the lenses. Fig-ure 3 shows photographs of the phase-conjugate images.If the moose-bearing incident beam A2 was blocked be-tween transparency T2 and the pickoff beam splitter BS 2,the phase-conjugate image of the moose A2* faded slowly,while the image of the resolution chart Al* disappeared in-stantly. Similarly, blocking the resolution-chart bearingincident beam Al caused its phase-conjugate image Al*to fade slowly, while the phase-conjugate image A2* of themoose disappeared instantly. Each image was trans-formed by the crystal into the phase-conjugate replica ofthe other; this cross-readout process is an identifyingcharacteristic of a mutually pumped phase conjugator.4

Figure 3 also shows no spatial cross talk between the twoimages; i.e., no resolution chart bars appeared on themoose, and no antlers were superposed onto the resolu-tion chart.

The phase-conjugate nature of the images generated bythe frog-legs phase conjugator was demonstrated by plac-

+2* BS2 L2 PD 2

Fig. 2. Typical optical setup for performing phase-conjugate imaging by using a mutually pumped phase conjugator. M's, mirrors; BS's,beam splitters; L's, lenes; BE's, beam expanders; FI's, Faraday isolators; T's, transparencies; PD's, phase distorters; MPPC, mutuallypumped phase conjugator; (P's, phase-conjugate images. Two expanded laser beams (either from two separate lasers or split by BSo froma single laser) illuminate two different transparencies T1 and T2 and are focused into the mutually pumped phase conjugator. The twophase-conjugate images /0* and 02* are observed by means of pick-off beam splitters BS, and BS2.

Ewbank et al.

Vol. 7, No. 12/December 1990/J. Opt. Soc. Am. B 2309

LEG 1 LEG 2

NORMAL MIRROR(NO DISTORTER)

FROG-LEGSPHASE CONJUGATOR(NO DISTORTER)

NORMAL MIRROR(WITH DISTORTER)

FROG-LEGSPHASE CONJUGATOR(WITH DISTORTER)

Fig. 3. Simultaneous phase-conjugate images generated by the frog-legs phase conjugator for incident beams, bearing images of aU.S. Air Force resolution chart and a moose. The phase-conjugate images are compared with those produced by reflection from normalmirrors, both without and with double passing through phase distorters.

ing severe phase distorters PD1 and PD2 (smeared Ducocement on glass slides) in the respective incident beampaths. Figure 3 shows that the moose-bearing image wasrestored, almost free from aberration. (The residual dis-tortions are probably due to the conjugator's limited nu-merical aperture.) The crystal transformed the incident

distorted image of the resolution chart into an image that,after passing through the second distorter, emerged bear-ing the undistorted image of a moose.

The resolution of the phase-conjugate images in Fig. 3 is-7 lines/mm. This resolution is less than that previ-ously demonstrated with other phase conjugators. 2"3

Ewbank et al.

2310 J. Opt. Soc. Am. B/Vol. 7, No. 12/December 1990

Fig. 4. Simplified diagram of the frog-legs phase conjugatorwith two interaction regions, as used in the four-wave mixingcoupled-wave analysis. The left-hand interaction region extendsfrom 1l z 12. The four beams are 4, left-hand incidentbeam; 1, fanned beam propagating toward the right interactionregion; 2, fanned beam coming from the right-hand interactionregion; 3, phase conjugate of beam 4. The right-hand interac-tion region extends from 1l' < z' < 12'. The four beams are4', right-hand incident beam; 1', fanned beam propagating to-ward the left-hand interaction region; 2', fanned beam comingfrom the left-hand interaction region; 3', phase-conjugate ofbeam 4'.

However, we note that the fidelity of the images observedwith the frog-legs phase conjugator did not degrade if bothof the incident beams were simple TEMoo modes, in con-trast with that of images from the double phase-conjugatemirror."8 ,9

3. THEORY OF THE MUTUALLY PUMPEDPHASE CONJUGATOR WITH TWOINTERACTION REGIONS

Here we present a coupled-plane-wave theory for anymutually pumped phase conjugator that has two inter-action regions (i.e., two separate gratings) interconnectedby only scattered (i.e., diffracted) light. While in prin-ciple this theory can be extended to the more realisticcase of many interaction regions, we solve only the two-interaction-region case. This theory predicts the effi-ciency of mutually pumped phase conjugation for planewaves only; we do not attempt the more difficult problemof image-bearing beams.

Cronin-Golomb et al.'" developed a coupled-plane-wavetheory to describe a variety of single-beam conjugatorswith a single interaction region. MacDonald and Fein-berg' 6 extended this theory to the case of two interactionregions to describe another single-beam conjugator (thecat conjugatorl3 ). By modifying the boundary and conti-nuity conditions of the latter theory, we describe thepresent case of a mutually pumped phase conjugator withtwo interaction regions. (In fact, the equations presentedhere for the mutually pumped phase conjugator prove to besimpler than those for the single-beam cat conjugator.)Recently the same type of mutually pumped phase conju-gator with two interaction regions interconnected by onlyscattered light was analyzed by using an intensity coupled-mode theory'7 and also by extending the theory for thesingle interaction region to two coupled junctions.' Thistype of two-interaction-region mutually pumped phaseconjugator, with each interaction region pumped by onlyone incident beam, should be distinguished from those in

which both interaction regions are pumped by both inci-dent beams.' 9

Preserving the notation of Refs. 15 and 16, we considertwo beams, denoted 4 and 4', that are incident upon a pho-torefractive crystal, as shown in Fig. 4. The two beamsare interconnected by deflection from two separate grat-ings, G and G'. Beam 4 is deflected by grating G intobeam 1, which then propagates through the crystal (and ispossibly routed by reflections at the crystal faces) to be-come beam 2', which is incident upon grating G'. Grat-ing G' deflects beam 2' into beam 3', which exits thecrystal as the phase-conjugate replica of beam 4'. Simi-larly, beam 4' is deflected by grating G' into beam 1',propagates to become beam 2, and is deflected by gratingG into beam 3, which is phase conjugate to beam 4.

If the jth beam has an optical electric field E. = Aexp(ikj x - it) + c.c. (where c.c. denotes complex con-jugate), then the two interaction regions couple the opticalfields as follows (neglecting absorption):

dA= +,ygdz Io

dA' _ +Yg1 dz' - 1' A4'

dA4* -ygA *dz Io l

dA4'* Yg'dz' I lA-'*,

dA3 _ -ygdz 10

dA3' = -_YAdz' Io, 2,

dA2* +ygdz Io A

dA 2 '* +Y'9 '*

dz' I-0' *'

(la)

(lb)

(lc)

(ld)

(le)

(if)

(lg)

(lh)

where y and y' are the amplitude coupling coefficients forthe two interaction regions, Io and Io' are the total intensi-ties in each interaction region, i.e.,

Io = A j2 + 1A212 + A312 + A412, (2a)

Io' = A,'I 2 + 1A2'12 + 1A3'I2 + JA4'I2,

and the quantities g and g' are defined as

g AlA4 * + A2 *A3,

g' Al'A4 '* + A2'*A3'.

(2b)

(3a)

(3b)

Note that the equations that describe the grating G' (whichhas boundaries 1' s z' < 12') are simply the primed ver-sion of the corresponding equations for grating G (whichhas boundaries 1l < z c 12).

The boundary conditions at the edges of the two inter-action regions are

A1(l) = 0,

Ai'(11') = 0,

(4a)

(4b)

A3(12) = 0, (4c)

Ewbank et al.

Vol. 7, No. 12/December 19901J. Opt. Soc. Am. B 2311

A3A(12 ') = 0. (4d)

The continuity equations connecting the two interactionregions are

A2(12) = AlVARe'o,

AA'12') = Al,)Re'o,

(5a)

(5b)

where R is the amplitude Fresnel-reflection coefficientat the crystal face(s) (which is included for generalityeven though it is unity for the total internal reflectionsthat occur in the frog-legs phase conjugator) and isthe accumulated phase shift from those reflections andfrom propagation between the two interaction regions.

* From energy conservation the following quantities areconstant15,16:

Tq determine the conserved quantity c = c', we mustuse the complete analytical solutions to the coupled-waveequations [Eqs. l(a)-(h)]. They are'5"16

Al(z) _ +2c tanh[A(z - 1)]

A2 *(z) -A tanh[4(z - 11)] + r

A'(z') +2c' tanh[1'(z' - 1')]

A2'*(Z') A' tanh[1'(z' - 1')] + r'

A3 (z) _ -2c tanh[,u(z - 12)]

A4 *(z) A tanh[A(z - 12)] + r

A3 '(z') - 2c' tanh[1Z(z' - 12')]

A4'*(Z') A' tanh[u'(z' - 12')] + r'

(lOa)

(lOb)

(lOc)

(lOd)

where

d = A 12 + 1A412 = I + 4 = 4(11),

di'= IA,'I 2 + IA4 ' 2 = I + I4' = 14'(11),

d2 = IA 212 + IA3!2 = 2 + I3 = I2(12),

d2 ' = IA2 '12 + 1A3 '12 = 12' + 13' = I2'(12').

The following quantity is also conserved:

c = AiA2 + A3A4 = A,'A 2 ' + A3A4 ' = c',

(6a)

(6b)

(6c)

(6d)

e+x - e Xtanh x

e+x + ex

A 3 d2 - d,

A 3 d2 ' - d'

r (A2+ 41c12)/2,

r' 3 (A'2 + 4I 2)/2,(6e)

as can be shown by manipulating the coupled-wave equa-tions [Eqs. (la)-(lh)]. The equality c = c' in Eq. (6e) isproved by application of the boundary and continuity con-ditions [Eqs. (4c), (4d), (5a), and (5b)]. (Note that c • c' inRef. 16 because of the different boundary conditions forthat problem).

The phase-conjugate intensity reflectivities Rp. andRo.' of the two incident beams at the appropriate bounda-ries (z = 1 and z' = 1,') can be expressed in terms of theconserved quantities as

A 3l,) 2 C 2

R =| A(11) = | (7a)

A3V(l')2 2 ,Re =- (7b)

A4 '*(l1') di'

The transmissivities T and T' of the two incident beamsthrough the conjugator are readily obtained from thephase-conjugate reflectivities "':

T _ lA31V 1, 2 _ R' _I C 2

A4(11) q did,'

= T' = A,3(1 ) qR2 , (8)

where q is the incident-beam ratio:

A 4(l,) 2 dq A |'( 1 = d= * (9)

The equality in Eq. (9) is derived from Eqs. (6a) and (6b)evaluated at the outer boundaries of the two interactionregions. Note that the two transmissivities in Eq. (8) areequal, since c = c' [see Eq. (6e)]. The only unknownquantity in Eqs. (7) and (8) is the conserved quantityc = c'.

and

yr/I=2Io'

y'r'

2Io'

(llf)

(llg)

Equations (lOa), (lOb), and (lOd) are identical to the cor-responding equations in Ref. 16; only Eq. (lOc) is dif-ferent, because of the difference in the boundary andcontinuity relations between the frog-legs and the catphase conjugators.

Evaluating Eqs. (lOc) and (lOd) at the boundaries z = 11and z' = 1' [see Eqs. (7a) and (7b)] yields

Y(2 - l)r] r[ 2Io J Io

[Y'( 2 ' - l')r' 1 r'ta h L 2 I' Io10'

(12a)

(12b)

These two transcendental equations can be rewritten bysubstituting for r/Io and r'/Io' in Eqs. (12a) and (12b):

t ~l[(d - d )2 + 4C12]12)

tanh=2(d2 + di) J_[(d 2 - d 2 + 4 I2I1/ 2

(d2 + di)

h y I [(d,' - d) 2 + 4 2] 1/2

2(d2' + d,') J[(d2 - d )

2+ 41II 2 P12

(d2' + d,')

I (13a)

(13b)

where I 12 - 1 and 1' 12' - 1'. By multiplying onecontinuity equation [Eq. (5b)] by the complex conjugate ofthe other [Eq. (5a)] and using the conserved quantities

(lla)

(llb)

(lic)

(lid)

(lie)

Ewbank et al.

2312 J. Opt. Soc. Am. B/Vol. 7, No. 12/December 1990

[Eqs. (6c)-(6e)], we obtain

2 d2d2.JR= 1 I2. (14)

The three equations, (13a), (13b), and (14), contain threeunknowns (ICd 2, d2, and d2') and can be solved numerically(specifically for JCl 2) in terms of the independent variables(the incident intensities d and di' and the couplingstrengths yl and 'l'). Again, R 2 is a loss parameterincluded to account for reflection losses at the crystal sur-faces [see Eqs. (5a) and (5b)]; it can also approximate ab-sorption and scattering losses.

Figure 5 shows the calculated phase-conjugate reflec-tivities of both input beams as a function of couplingstrength for incident-beam ratios of q _ I4/I4' of 1, 2, or 4.For all these plots we assume no loss (RI 2

= 1) except forone lossy case (R1 2

= 0.7). We find that the phase-conju-gate reflectivities are zero below a threshold photore-fractive coupling strength, which is yl = 2.493 for q = 1.This threshold increases as q departs from unity. Thetwo phase-conjugate reflectivities are necessarily equalwhen the incident intensities are the same (q = 1). Bothphase-conjugate reflectivities increase when the couplingstrength is increased; the reflectivity of the right-handincident beam asymptotically approaches the incident-beam ratio q, while the reflectivity of the left-hand inci-dent beam asymptotically approaches q'. For extremelylarge coupling strengths, all the light that is incident uponone side of the mutually pumped phase conjugator is effi-ciently channeled to the other side. In Fig. 6 the phase-conjugate reflectivities and transmissivities are plotted asa function of incident-beam intensity ratio q for three dif-ferent coupling strengths. As predicted by Eq. (8), all thetransmissivities and phase-conjugate reflectivities areequal for q = 1. As the beam ratio q increases, the two(equal) transmissivities and the phase-conjugate reflectiv-ity of the more intense incident beam decrease monotoni-cally, but the phase-conjugate reflectivity of the lessintense incident beam increases to a maximum and then

2.5

'U-JU-'U

'UI-

CD

0

'UWi

a-

1.0

0.0 2.5 3.0 3.5

AMPLITUDE COUPLING STRENGTH = y/Fig. 5. Calculated phase-conjugate reflectivities versus ampli-tude coupling strength yi for various incident beam ratios q -I./I4': q = 1 (dotted curve, R0, = RV'), q = 2 (solid curves, forRo. and R-'), q = 4 (dashed curves, for Rpt and R.'), all with noloss (R12 = 1); q = 1 (dashed-dotted curve, Rg = R.'), with aloss of RI2

= 0.7.

52I-20 >.

'U

-

oa.

LUu)

I

1 = .

30

/I = 2.7- 1 2 3 4 5 6 7 8

INCIDENT-BEAM RATIO q

Fig. 6. Theoretical dependence of the transmissivities (dottedcurves) and phase-conjugate reflectivities (solid curves for oneincident beam and dashed curves for the other) on the incident-beam ratio q _ I4/I4' for amplitude coupling strengths yl of 2.7,3.0, and 3.4, all with no loss (R12 = 1).

0.7 ,

0.6-

F ~ ~~~~~ /A 0.5

_uJ 0

a 0.3

ZZ

aZC.0.2- A

o AA0 .1

A

0.00 1 2 3 4 5 6 7 8 9 10 11

INCIDENT-BEAM RATIO q

Fig. 7. Experimental dependence of the transmissivities (filledtriangles for one beam and open triangles for the other) andphase-conjugate reflectivities (filled circles for one beam andopen circles for the other) on the incident-beam ratio q _ I4/I4'.The theoretical fit has a coupling strength y = 4.3 and a lossparameter R12 = 0.28.

decreases for larger q. Therefore, for sufficiently largecoupling strength and beam ratio, the phase-conjugatereflectivity of the less intense beam can exceed unity.Under these conditions the frog-legs phase conjugator canbe used with an ordinary mirror to form a self-oscillatingresonator cavity.

We measured the phase-conjugate reflectivities andtransmissivities as a function of incident-beam ratio forthe frog-legs phase conjugator, and our results are plottedin Fig. 7. The data agree only qualitatively with the theo-retical predictions of Fig. 6. A best fit to the data wasobtained with an amplitude coupling strength yl = 4.3and a loss parameter RI2 = 0.28. The discrepancies be-tween the experimental data and our fit are possibly dueto our simplifying assumption that there are only two in-teraction regions. In reality there are many interactionregions, as can be seen by photographs of the actual beam

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

-...... .......... _ .................

. , //~~~~~~~~~~~~~.......

_ ///~~~~~~~............

-

- - - - - -

Ewbank et al.

2.0

1.5

0.51

Vol. 7, No. 12/December 1990/J. Opt. Soc. Am. B 2313

paths inside the frog-legs phase conjugator, which reveal agentle curvature of the beams rather than two simplesharp deflections and which imply that the beam path iscomposed of a series of short segments.

4. FREQUENCY SHIFTS AND APPLIEDFIELDS IN THE FROG-LEGS PHASECONJUGATORWe performed experiments to measure any frequency shiftimparted to a beam transmitted through the frog-legsphase conjugator. Figure 8 shows a pair of Mach-Zehnderinterferometers constructed to cause each incident beamto interfere with its corresponding transmitted beam,with care taken to match the optical path lengths, sincethe multilongitudinal-mode laser had a short (-3 cm) co-herence length. Any constant frequency shift would berevealed by scanning the fringes at the output of the inter-ferometer. Such frequency shifts can be caused by reflec-tion from a moving photorefractive grating in the crystal.

With no external electric field applied to the SBN crys-tal, the fringes were stationary. Applying a dc electricfield along the c axis of the crystal (with the positive elec-trode attached to the crystal's -c face) caused the fringesto scan in a direction that was consistent with gratingsthat move in the same direction as the applied electricfield, i.e., toward the crystal's +c face. For example, aconstant beat frequency of -0.1 Hz was detected at theoutput of both interferometers when 250 V of electricitywas applied across the 5-mm-thick crystal for incidentbeams of equal intensity (again, 14-mW beams focused to-66 W/cm2 inside the SBN crystal). Figure 9 shows the

Q> BS2 IX ...-4

+

II

>MPPC I

ArefA2

A7 A; N 8S4

Fig. 8. Optical setup, incorporating dual Mach-Zehnder interfer-ometers to detect scanning fringes caused by moving photorefrac-tive gratings (M's, mirrors; BS's, beam splitters; D's, detectors).Two mutually incoherent Are-laser beams Al and A2, incidentupon the mutually pumped phase-conjugating crystal (MPPC),produce phase-conjugate reflections A1* and A2*. One interfer-ometer (short-dashed lines for its nonoverlapping beam paths)combines A2* with reference beam Al'f at BS3 to form interfer-ence fringes at Di. Similarly, the other interferometer (long-dashed lines for its nonoverlapping beam paths) combines Al*with reference beam A2ref at BS4 to form interference fringes atD2. Note that the optical path lengths of both arms in each in-terferometer must be equal to within the coherence length of thelaser in order to achieve high-contrast fringes.

Uw

I-

cas

2

2U-

I-

2 C-4.4

-

_

a1. I

* 4

0 10 20 30 40 50TIME (sec)

Fig. 9. Oscilloscope photograph of the time-dependent intensi-ties at the outputs of the two Mach-Zehnder interferometers,caused by scanning fringe patterns, for an applied dc voltage of250 V across the crystal.

intensity variations caused by the motion of the fringespast the aperture of each detector. For a given appliedvoltage, the fringes moved at the same rate for both detec-tors because both beams deflected off the same movinggratings. Figure 10 shows the dependence of the mea-sured frequency downshift on the applied electric field.Note that the fringes do not move when the appliedvoltage is below the threshold voltage of 150 V; a nearlylinear dependence is observed for larger voltages. In ad-dition, the phase-conjugate reflectivities monotonicallydecreased from approximately 24% to 6% as the externallyapplied voltage was increased from 150 to 500 V

According to theory, 20 when an external dc electric fieldE. is applied to a photorefractive crystal, the amplitudecoupling coefficients y and y' for the two interaction re-gions become functions of that field and any frequencyshifts 8 and 8':

y(E., ) 1 Y2 OiT (ED + Eq)(Eo + iED),

°' 1 + i ED[EO + i(ED + E)]

y'(E., 8') = Y' X (ED' + Eq') (EO + iED')

1 + iT'' ED'[EO + i(ED' + Eq')]

(15a)

(15b)

where [or '] is the photorefractive time response. Thecharacteristic fields are ED = k(kBTle) [or ED' =

kg'(kB Tie)] for diffusion and E = eNeff/(ekg) [or E' =eNeff/(E'kg')] for saturation, where kg [or kg'] is the gratingwave number, [or e'] is the effective dielectric constantdictated by the grating orientation for each interaction re-gion, and Neff is the effective photorefractive charge den-sity. Equations (12a) and (12b) [or (13a) and (13b)] can besolved only if the coupling coefficients y(E., 8) and y'(E.,8') are real. Setting the imaginary parts of Eqs. (15a)and (15b) to zero constrains the frequency shifts 8 and 8'to depend on the external field20 :

(-1fr)EqEoE0

2 + ED(ED + E)

at (-1/r')Eq`Eo

E. 2 + ED'(ED' + Eq')

(16a)

(16b)

B------

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2314 J. Opt. Soc. Am. B/Vol. 7, No. 12/December 1990

0.4

0.3

N

2" 0.20

0).0U.-

0.1

0.00 100 200 300 400 500

Applied Voltage (V)Fig. 10. Measured frequency downshift of the phase-conjugate beams as a function of external voltage applied across the SBN: 75: Cecrystal.

The cumulative frequency shift for a two-interaction-region mutually pumped phase conjugator, such as thefrog-legs phase conjugator, will be the sum of the individualfrequency shifts ( + 8'). Equations (16a) and (16b) pre-dict a linear dependence of the frequency shift on the ex-ternal field E., in agreement with our experiment, butonly if E 2 << ED(ED + E). The frog-legs phase conjuga-tor exhibits a beam-crossing angle of 4.3° and a gratingorientation of 26° off the c axis (see Section 5 below); also,we previously determined Neff = 0.9 x 1016 cm-3 andTi = 0.1 cm2 /(W-sec) (see crystal G in Table II of Ref. 21).Using these values and assuming identical interaction re-gions for the symmetric frog-legs phase conjugator, weestimate that ED = ED' = 540 V/cm and Eq = Eq' =270 V/cm. These parameters give a calculated maximumfrequency shift 18 + 8' = 3.2 Hz at E, = 650 V/cm, whichis at least an order of magnitude larger than the fre-quency shifts observed experimentally. Furthermore, forE, as large as 1000 V/cm with the above parameters, thefunctional dependence of 1 + 8'I on E, should extendbeyond the linear regime, reaching a maximum and thenbeginning to decrease. These disagreements betweentheory and experiment can be explained by noting thatthe interaction regions of the frog-legs phase conjugatorare illuminated by tightly focused beams and that thissame nonuniform illumination can cause a substantialreduction in the size of E, because of photoconductive

22screening. Using the observed frequency shift, weestimate that the localized E is reduced by approxima-tely an order of magnitude from the externally appliedfield because of screening, which ensures that E 2 <<ED(ED + Eq) and thereby maintains the linear dependenceof frequency shift on applied voltage, as seen in Fig. 10.The observed threshold voltage of 150 V may also be dueto the screening of the applied dc field by charges in thecrystal. In a final comparison between theory and thefrequency-shift measurements, we note that Eqs. (16a)and (16b) predict a negative detuning, in agreement withthe observed frequency downshift of both phase-conjugatereflections.

5. DISCUSSION

A mutually pumped phase conjugator is a clever (althoughlimited) optical computer. Given a pair of input beams,the conjugator self-generates a series of holograms thatconnects the two beams with the maximum possible effi-ciency. The optimum orientation and spacing of each ofthese self-generated holographic gratings is determinedby various crystal parameters, including the Pockels coef-ficients, the dc dielectric constants, the refractive indices,and the photorefractive charge density.

If a mutually pumped phase conjugator has only one in-teraction region (such as in the double phase-conjugatemirror'), the orientation and spacing of the single self-generated grating are fixed by the directions of the twoincident beams. However, when a conjugator uses morethan one interaction region, it gains the freedom to chooseboth the orientation and the spacing of its self-generatedgratings. In general, the conjugator will maximize thetwo-beam-coupling gain coefficient and the beam-overlaplength in each interaction region (increasing the net two-beam-coupling gain) while minimizing the optical-pathdistance between adjacent interaction regions (decreasingthe absorptive loss). If necessary, the conjugator can alsouse total internal reflection at its crystal faces to connectadjacent interaction regions and thereby minimize reflec-tion losses.

The SBN:75:Ce crystal prefers the frog-legs geometry(with its three internal reflections) over the simpler bird-wing geometry2 (which has only one reflection). As indi-cated above, incident optical beams generate a set ofgratings that (1) provides the largest two-beam-couplinggain and (2) connects the two incident beams. In SBNthe optimum grating for two-beam coupling has its k vec-tor parallel to the crystal's c axis, since r33 is the largestelectro-optic coefficient. However, when we use an inci-dent beam at an extremely steep input angle, we rule outthis grating orientation. In fact, for the particular SBNcrystal used here (crystal G in Ref. 21) and with a beamincident at an exterior angle of 700 to the normal to the a

Ewbank et al.

Vol. 7, No. 12/December 1990/J. Opt. Soc. Am. B 2315

face, we calculate that the highest two-beam-coupling gainis obtained with the fanned beam inclined from the inci-dent beam by -4° toward the -c direction, so that thegrating k vector is rotated from the c axis by 260. [In-spection of Fig. 1(a) shows that this fanned beam actuallysprings up at a measured angle of 4.3°, which is in goodagreement with our calculated value.] For this particu-lar SBN:75:Ce crystal, the coupling strength decreasesmore rapidly for minor deviations in beam-crossing angle(or grating spacing) compared with deviations in gratingorientation.

The geometric constraints imposed by the crystal sizealso play a role in determining which beam paths achievethe best coupling. This 40 beam-crossing angle betweeneach incident beam and its corresponding fanned beam istoo small to permit the two fanned beams to connect bymeans of a single reflection at the -c face of the crystal,as occurs in the bird-wing phase conjugator. (If our crys-tal were considerably wider, so that its aspect ratio wereapproximately 4: 1, then the beams could connect by meansof a single reflection at the bottom of the crystal.) Instead,the crystal chooses to preserve the small beam-crossingangle by using additional reflections at the crystal faces toconnect the two fanned beams. We attempted to test thishypothesis by translating the crystal so that the incidentbeams entered nearer the -c face of the crystal, therebypossibly permitting the two deflected beams, even withtheir small 4.3° bending angle, to connect by means of asingle reflection at the crystal's -c face. However, we didnot observe any phase-conjugate signal with this geome-try, perhaps because, by forcing the crystal to use only onereflection, we had effectively decreased by half the avail-able interaction length for each interaction region in thecrystal. Inspection of Fig. 1(a) shows that the bendingof the incident beams by the grating occurs over a mostthe entire length of the crystal and that, by reducing theavailable interaction length for each region to 1/2, we haddecreased the two-wave-coupling gain below the thresholdamplitude coupling strength yl > 2.5 required for mutu-ally pumped phase conjugation.

The frog-legs phase conjugator will operate only if theinput beams are incident at large enough angles that thedeflected beams strike the far face of the crystal at anangle exceeding the critical angle for total internal reflec-tion. For a refractive index of ne = 2.34 the critical angleinside the crystal is 25.3°. Subtracting 40 (which is theoptimum angle for two-beam coupling in our sample ofSBN in this geometry) from this critical angle and usingSnell's law, we compute that the angle of incidence outsidethe crystal should exceed -58° for optimum operation.In practice the frog-legs phase conjugator still operated atincident-beam angles as small as 450, probably by choosinga fanned beam with a slightly larger than optimum de-flection angle, thereby trading some two-beam-couplinggain in order to achieve total internal reflection at the farcrystal face.

6. CONCLUSIONSWe have presented and analyzed a new configuration formutually pumped phase conjugation in photorefractivecrystals-the frog-legs geometry, which uses three total

internal reflections at the crystal faces and at least twointeraction regions. The frog-legs phase conjugator typi-cally has phase-conjugate reflectivities of approximately20-30% but formation times that are relatively slow (min-utes for nominal cw laser intensities). This device demon-strates the remarkable ability of photorefractive crystalsto invent new geometries that optimize the gain of aphase-conjugate beam. With two incident beams, andafter an initial period of beam fanning, the crystal even-tually channels its scattered light along that path with thelargest two-beam-coupling gain that still permits thebeams to link up through total internal reflections atthe crystal faces. The final beam path is determined bythe crystal's aspect ratio and its photorefractive chargedensity as well as by the relative sizes of its Pockels coeffi-cients and dc dielectric constants. The two-interaction-region theory, based on coupled-wave amplitudes anddescribed above, accounts for the major features of thefrog-legs phase conjugator but is too simple to predict ac-curately the complex behavior exhibited by this device.

ACKNOWLEDGMENTSWe gratefully acknowledge helpful discussions withR. Saxena and R Vachss from the Rockwell InternationalScience Center.

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