transverse instability of energy-exchanging counterpropagating waves in photorefractive media

$: Transverse instability of energy-exchanging counterpropagating waves in photorefractive media$
Vol. 11, No. 8/August 1994/J. Opt. Soc. Am. B 1409

Transverse instability of energy-exchangingcounterpropagating waves in photorefractive media

M. Saffinan, A. A. Zozulya, and D. Z. Anderson

Department of Physics and Joint Institute for Laboratory Astrophysics, University of Colorado, Boulder, Colorado, 80309-0440

Received October 25, 1993; revised manuscript received March 2, 1994

The transverse instability of counterpropagating waves in photorefractive media that is due to the formationof reflection gratings is studied. We obtain new solutions for the instability threshold that account forthe reflection-grating-mediated transfer of energy between the pump waves. We find that the instabilitythreshold is minimized for equal-intensity pump beams inside the medium, corresponding to externallysupplied beams of unequal intensity when there is energy coupling between the pump beams. A coupling-constant-times-interaction-length product of at least 1.1 when the coupling constant is purely real, or at least6.3 when the coupling constant is purely imaginary, is sufficient for observation of transverse instabilities.Comparison with the physically available photorefractive coupling constant leads to the conclusion that avariety of commonly available materials should be suitable for the observation of transverse instabilities.In addition, new results for the instability threshold of a reflection-grating four-wave mixing oscillation areobtained that include the effects of direct coupling between the pump waves.

1. INTRODUCTION

Transverse modulational instability of two counter-propagating beams in nonlinear optical media has beenobserved in atomic vapors,1 3 liquid crystals,4 5 and pho-torefractive crystals.6 The instability manifests itself inthe far field by the appearance of conical rings, pairsof spots, or an array of spots, sometimes with hexag-onal symmetry. The transverse intensity modulationin the near field has the characteristic spatial scaleIS = 2vr/6kO, where 0, is the small angle between thegenerated satellite beams and the primary beams. Theangle 0, at which the threshold of the absolute instabilityis minimum may be found for all the above materialsby a linearized analysis of the equations of motion.7 '12

On the other hand, the nature of the nonlinear stage ofthe instability, the details of the resulting patterns, andtheir spatial and temporal stability appear to vary widelyamong different materials. In atomic vapors cones, pairsof spots, and hexagonal or other arrays of spots may all beobserved, depending on the frequency and the intensityof the primary beams. In liquid crystals the instabilityleads to a stable pattern, sometimes of hexagonal symme-try, whereas the recent observations in photorefractivemedia indicate a somewhat different behavior. Here thehexagonal pattern is well defined but temporally andspatially unstable, such that it appears as a cone in atime-averaged recording. The cones observed in atomicvapors may also be due to unstable hexagons, althoughtime-resolved data are not available at present. Thereis some reason to suspect this cause since, as was shownin Ref. 3, the cone may be collapsed into a hexagon byinjection of a weak beam at 6,. The intrinsically slowdynamics of photorefractive media simplify time-resolvedmeasurements, rendering photorefractives well suited tofurther experimental investigation of these phenomena.

A recent analysis of the modulational instability in pho-torefractive media that is due to the formation of trans-

mission gratings showed that the strong incoherent scat-tering (fanning) characteristic of photorefractive mediaconsiderably raises the otherwise quite low instabilitythreshold.12 The fanning may be effectively suppressedby the use of narrow pump beams, although this is notappropriate in the case of transmission gratings, sincethere will then be few modulation fringes across the pri-mary beams. It was shown by Honda6 that this problemmay be circumvented when the photorefractive crystal isoriented such that the instability is due to the forma-tion of reflection gratings. In this case the grating vectorkg = 2ko is parallel to the primary beams, which resultsin many modulation fringes, even with narrow beams. Ininstantaneous Kerr-type media the presence of reflectiongratings does not significantly complicate the analysis,since the resulting phase transfer between the primarybeams may be scaled out of the problem. However, theformation of reflection gratings in photorefractive mediagenerally leads to strong energy coupling between the pri-mary beams. Indeed, in the experiments of Honda thatused photorefractive KNbO3 the intensity-coupling coef-ficient between the counterpropagating pump waves wasmeasured to be 12.5 cm-. 6 The strong energy couplingand the fact that the photorefractive coupling constant iscomplex render existing treatments of this problem inap-plicable, necessitating the new analysis presented below.

The instability threshold in the geometry shown inFig. 1 depends on several characteristic quantities: thephotorefractive coupling constant y, which is complex; theangle 0 between the primary and satellite beams; the in-tensity ratio of the counterpropagating beams, given byq = IB/IF; and also the boundary conditions. The charac-teristic angle between the primary and the satellite beamsis such that the phase lag that is due to diffraction is com-pensated for by nonlinear phase transfer between the pri-mary and the satellite beams. The instability thresholdis therefore lowest when the coupling constant y is purelyreal, which maximizes the phase transfer between the

0740-3224/94/081409-09$06.00 ©1994 Optical Society of America

Saffman et al.

1410 J. Opt. Soc. Am. B/Vol. 11, No. 8/August 1994

NonlinearMedium

-1 Fo

J- i F .e1'.1

0 Z

(a)

F0

Bo

NonlinearMedium

0 Z

(b)

B-1

Fo * ) ))) I I I I I I I I 4 (( -Bo

F+, (c) B+,

Fig. 1. Geometry of the optical interaction: (a) Two externallysupplied counterpropagating primary beams. (b) One externalbeam and mirror boundary conditions at z = 1. (c) Three reflec-tion gratings couple the beam pairs (Fo, Bo), [(Fo, B-1), (F+j,Bo)] and [(Fo, B+1), (F1, Bo)]. The directions of motion of themoving gratings are indicated by the short arrows.

beams. However, in the reflection geometry studied herey is almost purely imaginary. In this case the sidebandswill typically detune such that the coupling constant gov-erning the interaction of the primary beams with the side-bands acquires a larger real part and hence increasedphase transfer. The sidebands may also be detuned evenwhen y is purely real if the pump beams have unequal in-tensities. This phenomenon is well known from the the-ory of photorefractive four-wave mixing.'3

The equations of motion and the resulting dispersion re-lation giving the instability threshold for the case of real yare presented in Sections 2 and 3 below. The analyticalexpressions are derived allowing for arbitrary values of q,although numerical evaluation of the dispersion relationreveals that the threshold is lowest for q = 1, in agree-ment with the transmission-grating case.' 2 The solu-tions with real y serve as a point of comparison for thegeneral case of complex y given in Section 4. When theimaginary part of y is nonzero then q = q(z), which com-plicates the equations of motion. Analytical solutionsmay be found by choosing particular boundary condi-

tions such that q is constant even though there is strongenergy coupling between the counterpropagating beams.In Fig. 1(a) the nonlinear medium is illuminated withcounterpropagating primary beams of prescribed inten-sity. In Fig. 1(b) one end of the nonlinear medium iscoated with a perfectly reflecting surface. Both typesof boundary condition lead to a constant beam-intensityratio throughout the photorefractive medium. Numeri-cal solutions of the dispersion relations for the case ofimaginary y show that the instability threshold is in-creased several times over compared with the case ofreal y.

In Section 5 we consider the relative suitability ofparticular photorefractive media for the experimentalobservation of spatial instabilities by accounting for thedependence of the coupling constant on material param-eters. We find that the necessary coupling constant isequal to the available coupling constant within a factorof 2 in centimeter-sized samples of BaTiO3, LiNbO3 , orKNbO3. On the other hand, large variations in the cou-pling coefficient may be found in different samples of whatare nominally the same photorefractive material. Theimplication is that it may be necessary to handpick goodsamples for the observation of transverse instabilities.

2. EQUATIONS OF MOTION

Consider two strong plane waves Fo exp[i(koz - wot)] andBo exp[i(-koz - wot)] counterpropagating in a nonlinearmedium in the geometry of Fig. 1(a). We assume thatthese waves interact directly with each other throughthe formation of a reflection grating proportional toFoBO* exp(ikgz) with the grating vector kg = 2ko. Theresulting redistribution of intensity and phase betweenthe counterpropagating beams is described by the equa-tions

dFo . qdz iYPi Fo,

dBo = P 1 Bo,dz Vpl + qIo

q(z) = JBo(z)j 2

JFo(z)1

(la)

(lb)

(lc)

In writing Eqs. (la) and (lb) we have assumed that pas-sive losses that are due to absorption are negligible.The coupling constant y, is generally complex and is afunction of material parameters, the grating vector kgand a possible frequency shift fl between the interact-ing beams. The frequency-degenerate counterpropagat-ing primary beams interact through yp y(kg = 2ko2,Q = 0).

The counterpropagating beams may be absolutely un-stable to the appearance of sidebands propagating at asmall angle with respect to the main beams. Referringto Fig. 1(c), consider a weak probe wave F+j exp[i(koz +kL rL) - i(wo + r)t] with ikL <<ko, IlI w<<o. It in-teracts with the counterpropagating strong pump waveBo, resulting in a nonlinear change in the refractive in-dex of the medium proportional to F+,Bo* exp[i(2koz +kd rj) - ilt] plus its complex conjugate. Scattering ofpump wave Bo off the index grating supports F+,, and

Saffman et al.

Rage

.

FjBj 'IT-

'I0


scattering of pump wave Fo off the conjugate part sup-ports B-1. Similarly, the weak sideband F-1 exp[i(koz -k.- r±) - i(woo - fl)t] interacts with the counterpropagat-ing strong pump Bo, and scattering of the pump waves offthe refractive-index perturbation supports the sidebandsF-1 and B+1. Transmission gratings that are due to theinteraction of pump wave F0 with its sidebands F+, andF-1 and of pump wave Bo with its sidebands B+1 and B-1will in general also be formed. We assume in what fol-lows that the interaction geometry is chosen such thattransmission gratings are weak and may be neglected.This is an excellent approximation in photorefractive me-dia when the pump beams propagate along the crystal-lographic c axis. Our previous analysis' 2 considered theopposite situation, in which transmission gratings but notreflection gratings are formed.

The full set of interacting waves is written in the form

of the order of seconds). Equations (3) formally describesluggish Kerr media with the replacement

{Yo, Vp} W l 2 2 IFo(Z)12'

(1+ q) Ic~) 2 (4)

although the approximation of reflection gratings but notransmission gratings is not sensible in common Kerr-type media, where diffusion processes tend to preferen-tially wash out reflection gratings.

It is tempting to consider eliminating the direct inter-action of the pump waves by arranging for them to bemutually temporally incoherent or to have orthogonal po-larization states. Putting y, = 0 in Eqs. (3) leads to twodecoupled sets of single-sideband four-wave mixing equa-tions:

F(r±, z, t) = Fo(z){1 + F+1 exp[i(kL r, - fit)]

+ F-1 exp[i(-k±. r, + fAt)]}

X exp[i(koz - oot)],

B(r±, z, t) = Bo(z){1 + B+, exp[i(k,. r - t)]

+ B-1 exp[i(-k 1 . rL + fAt)]}

( d + ikd)F+ = i 1q X+o(F+l + B1*),

(2a)

d + ikd)B-l* = i X + _ *),

X exp[i(-koz - wot)].

Linearizing the equations of motion in the amplitudes ofthe sidebands results in the set

d + ikd)F+1 = i q [(X+yo - yp)F+l + pB+

+ X+yoB-,*], (3a)

d - ikd )F-* = -i q [(x- yo* - * )F-*

+X- Yo*B+1 + y B- ],

d ikd) B+11 *

=1 + *F+1 + X-Yo Fy-

+ (X-o* - yp*)B+,],

(d+ ikd)B-* = 1+[X+y.F+l + YPF_1*

+ (X + o - yp)B-*]. (3d)

The characteristic diffraction wave vector is given bykd = k /2ko; the frequency-shifted sidebands interactwith the primary beams through y(2ko ± k, ±fl) =x±y(2ko ± k±,O), where X± = 1/(1 iflr); and, since1k1I1 << k, y(2ko ± k, 0) has been approximatedby y(2ko, 0) yo. We have allowed for differentpump-pump (yp) and pump-sideband (yo) coupling con-stants in Eqs. (3). These may arise if the pump beamshave different polarization states or have imperfect tem-poral coherence, which would result in lyp < Vol. Wehave also assumed that the medium is sluggish and re-sponds only at frequencies much lower than the opticalfrequency coo and that the characteristic propagation time1/c (I is the length of the medium, and c is the speed oflight) is small compared with the characteristic relax-ation time T (for most photorefractive media the latter is

(d - ikd F_,* = -i q X *Y *(F-l* + B+'),( dz 1+ q +1

-- ikd) = -i 1_y(*Y*F +l+B+).( dz d}+1q

(6a)

(6b)

Despite the fact that kd explicitly appears in Eqs. (5)and (6), they are completely decoupled, which means thatthey do not describe the appearance of a dual-sidebandtransverse instability with an explicit dependence on kd.Analysis shows that this is true even when the mir-ror boundary conditions of Fig. 1(b) are used, which docouple the two sets of sidebands. This explains the ex-perimental observation reported in Ref. 6 that the trans-verse instability disappears when the pump beams aremade mutually temporally incoherent. In this respectthe physics of the reflection-grating-mediated transverseinstability is quite different from the transmission-gratingcase, where the instability occurs in the absence ofany direct pump-pump interaction. It may be notedthat Eqs. (5) and (6) are equivalent to those given inRef. 14, which describe the on-axis stimulated-Brillouin-scattering instability of counterpropagating beams. Thedifference between Eqs. (5) and (6) and those describingthe stimulated-Brillouin-scattering case is that here thenonlinearity is proportional to the modulation depth ofthe field interference as opposed to being proportional tothe optical intensity.

With these results in mind we therefore set y = yo,which leads to the equations of motion:

(d + ikd F+= i q yo[(X+ -1)F+lkdz dJ B + q

+ B + X+B-,*], (7a)

(5a)

(2b)

(5b)

Saffman et al.


(d .q *d- ikd)F-l* = - i Yo -

+ X-B+i +

( d -ikd) B+l = -i y*[F + X-F-1

(d + ikd)Bl* = 1+qyo[F + F_

+ (X+-l)B-1*] .

kd = kd + 2

(7b) q -

(9c)

(9d)

These results and those that follow below were derivedwith r assumed to be real such that X-* = X+. This is not

(7c) necessary, but it simplifies the resulting formulas. Theboundary conditions F+1(0) = F_1*(0) = 0, F+1(l) = B+1(1),FP1*(1) = B_1*(1), corresponding to an absolute instabilityin the geometry of Fig. 1(b), yield the dispersion relation

(7d) kd (q 1 )[X+2 - (1 - 2 x+)q ] + (1- +)2q2

Analysis of the solutions of these equations constitutesthe remainder of this paper.

3. DISPERSION RELATIONS FORREAL COUPLING CONSTANTSSolutions of Eqs. (7) yield dispersion relations for thethreshold of the absolute instability that are functions ofq(z). We wish to find the value of q(z) that minimizes theinstability threshold. When the wave interaction is dueto the formation of transmission gratings, the equationscorresponding to Eqs. (7) may be solved for arbitrary val-ues of q(z) and yo.12 Equations (7) do not admit of suchgeneral solutions, and we consider first the case of real yo.Solutions for complex yo are given in Section 4 below.When yo is real (o = e), q(z) = constant, and Eqs. (7)are readily integrated in closed form. The boundary con-ditions F+1(0) = F 1*(0) = B+1(l) = B*(l) = 0, corre-sponding to the geometry of Fig. 1(a), yield the followingdispersion relation for the threshold of the absolute in-stability:

kd 2[x+2 + (2 - 4X+ + x+2)q2 ] + O2 (1 - x+)2 q2 (q2- 1)

4

+ (2 - )[-d2X+2 + (1 - X+)2q2] cos(rl)cos(sl)

+ (1q-2) d4x 2 + kd-y-(1- X+2 )[X+2 + (1- )q2]

q2(1 - X+)3[1 + x+(l-22)]} sin(rl)sin(sl)

(8)

where

r = kd2- -[1 - - 2q(1 - X+)]

4

+ Yo'Ikd2[x+2 + q2(1 - 2+)]

+ 2 - X+) 2q2(q2 - 1)It

S2 kd2 _ ° [1 - +2 - 2q2(1 - X+)]

4

- o kd2[x+2 +1 q2 (1 - 2+)]

+ Y21 - x+)2q2(q2-

x [ d 2 - q] + {-kd 2X+2 + (1 _X+)q

X [-kd+2 + y0.2]jcos(rl)cos(sl)

+ kdlkd 2 O[x+2(1 + X+) - (+ 2 + 2 x+ - 1)q2]

-kd X+(1 - X+)2 q2- 8 (I - X+)2 q2[l + X+(1 - 2q2)]

4 8

-kd 3 X+2 sin(rl)sin(sl) =o. (10)

The beam-intensity ratio is in this case given by q = R.R being the mirror reflectivity.

When the counterpropagating beams have equal inten-sities, we have q = 0, and the dispersion relations sim-plify to

1 + cos(rl)cos(sl) + [2kd2 - (r2 + 52)]

sin(rl)sin(sl)x = 0rs

(11)

for external-beam boundary conditions [Fig. 1(a)] and

cos(rl)cos(sl) + kd(kd - TgX+) sin(rl)sin(sl) 0rs

(12)

for mirror boundary conditions [Fig. 1(b)]. In Eqs. (11)and (12), r = (kd + yo'x+/2)2 - To 2/4 and S2 (kd -yo'X+/2)2 - d2 /4.

The dispersion relations have an infinite number ofsolution branches. Consider first the external-beam caseof Fig. 1(a). In the limit kd >> lYrl the two sidebandgratings decouple, and Eq. (8) reduces to

X 2

+ q2

(x+2

-4 X+ + 2) + (1 - q)Y+

X cos{yg[x+ 2 + q2(1 - 2X+)i" 2 l} = 0. (13)

(9a) Numerical solution of Eq. (13) shows that the minimum(9a) threshold occurs for Q = 0, in which case Eq. (13) further

reduces to

1 + cos[2Ty'/ /(q + 1)] = 0. (14)

The threshold is given by yo'l (N + 1/2)(q + 1)1r/>q,with N an arbitrary integer, which for q = 1 reduces

(9b) to the well-known result for reflection-grating four-wavemixing oscillation in photorefractive media with a local

Saffman et al.


response.' 3 When q 1, Eq. (13) extends the resultsof Ref. 13 for the oscillation threshold by including theeffect of a direct interaction between the counterpropa-gating pump beams. The dependence of the thresholdon q as given by Eq. (14) is shown in Fig. 2. To com-pare the present results with those of Ref. 13, wehave used y = yo'/(l - fln) in the threshold conditioncosh[-iyl/2 + ln(q)/2] = 0 [Eq. (2.15) of Ref. 13]. Wesee that the direct interaction of the counterpropagatingpump beams tends to raise the oscillation threshold asmeasured by lyo'l. There is also a qualitative differencewhen the effect of the pump-pump grating is included.When the pump-pump grating is absent, four-wave mix-ing oscillation in the presence of unequal-intensity pumpbeams is possible only with detuned sidebands. Whenthe pump-pump grating is included, there is no fre-quency shift of the sidebands even with unequal pumpbeams. The coincidence of the threshold condition forq = 1 calculated here with that calculated in Ref. 13 issomewhat fortuitous and is related to the fact that forq = 1 there is reflection symmetry about the line z = 1/2.

We have plotted the lowest-lying solution branches ofEq. (8) in Fig. 3 by finding the values of y'l and flr thatminimize Iydo'1 while satisfying the dispersion relation foreach value of kdl. The minimum threshold of l' 1ll 2.1occurs for q = 1, in which case the focusing (( > 0)and defocusing (o < 0) branches are equivalent: chang-ing the sign of T' corresponds to relabeling the positive2 direction. The scalloped form of the threshold curvesarises because the dispersion relation factors into a prod-uct of terms,1 01 2 leading to several solutions for each valueof kdl. For q = 1, factorization of Eq. (11) gives

[tan(rl/2)tan(sl/2) + aj][tan(rl/2)tan(sl/2) + a2]

X [tan(rl/2)tan(sl/2) + /al]

X [tan(rl/2)tan(sl/2) + 1/a2] = 0,

(15a)

al,2 = [(ed ± yo'/2)2 - 4 x+2]12. (15b)

Only the lowest of the solutions is shown in Fig. 3, whilethe sharp cusps correspond to points where the solutionbranches cross. For q 0 1 and kdl 2 the lowest thresh-old occurs for fl = 0, in which case Eq. (8) factors into theform

[tan(rl/2)tan(sl/2) + r/s][tan(rl/2)tan(sl/2) + sr] = 0,(16a)

r = kd2 + kdYo'(1 - q2)12 ,

s2= kd2

- kdY(1- q-

(16b)

(16c)

Thus, when the lowest threshold occurs for fl = 0, thedispersion curves with q 1 are identical to those withq = 1 but with yo' scaled by 1/(1 - q2)12. For q 1 andsmall kdl the focusing branches have il # 0, whereas thedefocusing branches always have fl = 0. Additional cal-culations, not plotted in Fig. 3, show that as q increasesthe value of kdl at which the focusing branch becomes

0

detuned increases. This is complementary to the trans-mission grating case,' 2 for which the value of kdl at whichthe focusing branch detunes decreases with increasing q.

For mirror boundary conditions [Fig. 1(b)] the limit ofkd >> IYo'1 gives

qX+2 - q2(1 - 2x+) - X+2(1 + q)

X cos{yo'[x+2 + 42(1 - 2X+)]/21} = 0 (17)

for the dispersion relation. When the pump beams are ofequal intensity, Eq. (17) reduces to cos(ydx+l) = 0. Theminimum threshold is thus yo'l = w/2 and l = 0, whichis half the minimum threshold without mirror boundaryconditions. This result may be attributed to the mirror'sfeeding energy from the sidebands back into the inter-action region. The instability thresholds calculated fromEq. (10) for q equal to 1, 1/5, and 1/25 are shown inFig. 4. The absolute minimum threshold of yo'lI 1.1is found on the focusing side with q = 1 at kdl - 1.4 andis roughly a factor of 2 lower than the minimum thresholdwithout a mirror. The presence of the mirror breaks thereflection symmetry about z = 1/2, so even for q = 1 the

12

0 20 40 q 60 80 100

Fig. 2. Four-wave mixing oscillation threshold as a function ofthe pump-beam intensity ratio q. Curve i is calculated fromEq. (14) and curves ii from the theory of Ref. 13. Solid curves,lyo'll; dashed curve, flT.

0

.4

8 12 16 20kdl

Fig. 3. Instability thresholds for real yo = yo' and externallysupplied beams. The curves are labeled with the value of q,and fl- is identically zero for all branches except for the q = 25focusing branch. The focusing (yo' > 0) and defocusing (yo' < 0)branches for a given value of q coincide, except where fT 0 0.Solid curves Ilo'Zl; long-dashed curves, fTr.

0 -

.r 25-L - -_ 1,5, 25 -

I

Saffman et al.

14


10

7.5

2.5

04 8 kd 1 1 2 16

Fig. 4. Instability thresholds for real yo = yo' and mirrorboundary conditions. The curves are labeled with the valueof q, and flr is identically zero for all branches except for theq = 1/25 focusing branch. Solid curves, focusing branches;dotted curves, defocusing branches; long-dashed curves, Mr.

focusing and the defocusing branches are not equivalent.As the pump-beam intensities become more unequal, theintertwined focusing and defocusing branches move apartfrom each other, with the defocusing branches having thelowest threshold. Referring to Eqs. (1), we see that thiscorresponds physically to the case in which the phases ofFo and Bo are minimized at the mirror.

4. DISPERSION RELATIONS FORCOMPLEX COUPLING CONSTANTS

When the coupling constant has a finite imaginary part,the counterpropagating pump beams exchange energy,and q(z) becomes position dependent, complicating theanalysis of Eqs. (7). The exact conditions used in theexperimental observation of hexagons in photorefractivemedia6 are therefore not readily analyzed. However,even when the counterpropagating beams exchange en-ergy it is possible to obtain q(z) = 1 by correct choice ofthe boundary conditions. The case q(z) = 1 is also of in-terest because it probably represents the minimum insta-bility threshold.s This is certainly the case when yo ispurely real, as was shown in Section 3 above, and also forthe transmission grating case with an arbitrary complexcoupling constant 2 or for combined transmission and re-flection gratings with a real coupling constant.' 0 Notealso that the solutions of Eqs. (7) remain unchanged if wemake the replacement q - 1/q; since the physical situa-tion is identical, we have merely renamed the forward andthe backward propagating waves. It follows that q(z) = 1is an extremum of the threshold condition with respectto variations in q. We argue that this extremum is infact the minimum both by analogy with the known re-sults cited above and physically, because the strength ofthe induced change in the refractive index of the mediumis proportional to j/ /(1 + q), which has a maximum atq = 1.

To obtain q(z) = 1, note that all solutions of Eqs. (lb)satisfy IFo(z)12 - Bo(z)12 = constant. The constant iszero, giving q(z) = 1, for lFo(0)12/IBo(l)12 = exp(y 'l),where the complex coupling coefficient has been writ-ten as yo = yo + iyo". Alternatively, using the geometryof Fig. 1(b), the constant is identically zero independent

of the coupling constant for a perfectly reflecting sur-face at z = 1. It should be emphasized that even thoughq(z) = 1 inside the photorefractive medium there is largevariation in the pump-beam intensities along z owing tothe strong coupling of energy between the counterpropa-gating pump beams, and the externally supplied beamshave unequal intensities at the two ends of the medium.

Solving Eqs. (7) with q = 1 and external-beam bound-ary conditions yields

cosh(y'x+l) + cos(rl)cos(sl)

+ [2kd2 _ 1(r2 + S2)] sin(rl)sin(sl) = 0 (18)

for the dispersion relation. Solving Eqs. (7) with q = 1and mirror boundary conditions yields

cos(rl)cos(sl) + Yod cos(rl)sin(sl) + si(rl) cos(sl)]c o s ~ r l c o s ~ s l 2+ -S r

+ [ kd(kd - YOX+) + yo 2] sin(rl)sin(sl) = 0 (19)

for the dispersion relation. In Eqs. (18) and (19) r2

(kd + yX+/2)2 - IyoI2/4 and 2 = (d - y+/2)2 -

IyoI2 /4. Equation (18) is independent of the sign of yd",whereas the dispersion relation (19) for the case of mirrorboundary conditions does depend on the sign of yd'. ForYT" > 0 the optical intensity reaches its minimum at themirror, and for yo' < 0 the optical intensity is a maximumat the mirror; these are physically distinct configurations.Note also that when Yo' = 0 then Eq. (19) has no depen-dence on X+ and in fact no solutions. Mirror boundaryconditions are therefore of limited utility when yo is al-most purely imaginary.

Equations (18) and (19) have an infinite number ofsolution branches. In the limiting case of kd >> 1Yo0Eq. (18) reduces to

cos(yc(X+l) + cosh(y'x+l) = 0. (20)

The lowest branch has a minimum threshold yo = r andQ = 0, which was determined in Section 3. The depen-dence of the four-wave mixing threshold on arg(yo) isshown in Fig. 5. The theory of Ref. 13 when evaluatedfor q = 1 yields the same threshold coupling and detuningas shown in Fig. 5 except that fl < 0 when arg(yo) is inthe first or the third quadrants. The dispersion relationfor mirror boundary conditions, Eq. (19), reduces to

cos(yo'X+l) = 0 (21)

with yd' arbitrary. The large kd threshold with mirrorboundary conditions is thus independent of arg(yo) pro-vided that yc( # 0.

To evaluate the instability threshold with varying de-grees of energy coupling, we numerically solve the disper-sion relations with arg(yo) fixed while allowing IyoI and Qto vary. The lowest-lying branches with external-beamboundary conditions are shown in Fig. 6(a) for I yol = IyoI,that is, arg(yo) = vr/4 and arg(yo) = 3X/4, and in Fig. 6(b)for arg(yo) = IT/2. The threshold curves for arg(yo) =5ir/ 4, 7ir/4 are not shown, since they coincide with thecurves for arg(yo) = 3ir/ 4 , Ir/4 , respectively. This is a

15::~~~~~~/

1/2 1,15,12

Saffman et al.

I

.0.4a= 0200


IC

20 component of the electro-optic tensor, and Esc(kg) is thephotoinduced space-charge field along kg. The phase ofYpr is determined by the space-charge field, which is givenby

1 6115

10 a 5

5

0

arg(y o )

Fig. 5. Four-wave mixing oscillation threshold as a function ofarg(yo) for q = 1. Solid curve, yoll; dotted curve, fQr.

consequence of Eq. (18), which is independent of the signof yd'. For the worst case of arg(yo) = r/2 the mini-mum threshold is increased to 1yoll 6.3 as comparedwith lyoll 2.1 for arg(yo) = 0. Corresponding resultsfor mirror boundary conditions are shown in Fig. 7 forarg(yo) = -/4, 3/4, 5/4, 77r/4. With mirror bound-ary conditions some of the solutions with energy coupling[arg(yo) = 7/4] actually have lower minimum thresholdsthan solutions without energy coupling [arg(yo) = ir], al-though the absolute minimum occurs for no energy cou-pling. Note also that all the curves shown in Fig. 7 haveflT = 0 and are thus qualitatively different than theexternal-beam thresholds.

When the real part of yo is nonzero, mirror boundaryconditions generally give the lowest threshold, whereasfor a purely imaginary yo instability is possible onlywith external-beam boundary conditions. In the photo-refractive media to be discussed in the following sectionthe reflection-grating coupling constant is almost purelyimaginary. To determine which type of boundary condi-tions are preferable in this situation, we plot the mini-mum value of Iyol as a function of arg(yo) in Fig. 8. Thecrossover point between external-beam and mirror bound-ary conditions occurs at arg(yo) - 7r/ 2 - 0.08. Sincemost photorefractive media in the reflection-grating ge-ometry have arg(yo) closer to 7r/ 2 than this, external-beam boundary conditions will typically be preferable.

5. PHOTOREFRACTIVE MEDIA

We turn now to a comparison of several characteristictypes of photorefractive medium. The coupling coeffi-cient describing the interaction of counterpropagating op-tical waves traveling along the symmetry axis of a pho-torefractive crystal is

ypr(kg, fl) = ( wonr )Es(kg)\.2c )Eck)

3

0

2

1

0

20

16

12

8

4

0

0 4 8 dI 1 2kdl

(a)

U 4 kdl It 10 M

(b)Fig. 6. Instability thresholds with external beams and q = 1:(a) arg(yo) = O. r/4, 31r/4; (b) arg(yo) = 1/2. In (a) the curvesare labeled with the value of arg(yo). Solid curves, yoll; dashedcurves, fQr.

"I * ° kdl - I 1 U

Fig. 7. Instability thresholds with mirror boundary conditionsand q = 1. The curves are labeled with the value of arg(yo),and flT = 0 for all the curves.

02

(22)

where kg = 2ko2. The coupling coefficient is due to pho-toinduced charge redistribution that results in a space-charge field that modifies the index of refraction by thePockels effect. Here c is the speed of light in vacuum,no is the ordinary index of refraction, r 3 is the relevant

20

16

12

0

8

4

0

Esc(kg, fl) = Esc°(kg) 1 - i7(kg)

0 IU2

(23)

_ _ - o16 20

6

I1 ., , .. . I . o

OL

Saffman et al.

I o . I: o


10

external

4 -mirror - _

boundary -

2

0 0 0.1 0.2 0.3 0.4 0.5

arg(yo4fI

Fig. 8. Absolute minimum instability thresholds as a functionof arg(yo) for external-beam and mirror boundary conditions.

with

E8,C(kg) = Em.(kg) Eo(kg) + iEdiff(kg) (24a)EmaE(kg) + Ediff(kg) - iEo

T(kg) t E,(kg) + Ediff(kg) - iEo (24b)tEmax(kg) + Ediff (kg) - iEo

Eo = Eapp + Epv (24c)

The space-charge field is expressed in terms of a num-ber of characteristic fields: Emax = eN/e3kg, the limit-ing space-charge field (e is the electronic charge, N is thedensity of traps, and 3 is the third component of the dcdielectric tensor); Ediff = kBTkg/e, the diffusion field (kgis Boltzmann's constant, and T is the absolute tempera-ture); E, = yDN/,ukg, the drift field (D is the recom-bination rate of charge carriers, and ,u is the mobility);and Eo, which is due to the combined effect of the photo-voltaic field EPV and any external field Eapp applied alongthe c axis. We use a simple model for the photovoltaiceffect such that Epv is the local part of the photovoltaicfield directed along the c axis1 7 and is assumed to haveno dependence on kg. The time constant T scales withthe characteristic material response time to, which is in-versely proportional to the optical intensity. The valuesof all these parameters may vary considerably betweendifferent samples of any particular material. The valuesgiven in Table 1 are meant to be representative; see e.g.,Ref. 18 for references to the original literature.

The coupling and time constants for BaTiO3, LiNbO3,and KNbO 3 are shown in the last three lines of Table 1.When the interaction is due to the formation of trans-mission gratings with correspondingly small values forkg, BaTiO3 and LiNbO3 have quite different properties;BaTiO3 is diffusion dominated, leading to arg(yo) - 7r/2,whereas LiNbO3 is drift dominated, leading to arg(yo) - 0.However, in the reflection grating case, for which kg islarge compared with the Debye screening wave vector, thephotorefractive materials considered here are all diffusiondominated and hence have similar characteristics. Themost significant difference between BaTiO3 , LiNbO3, andKNbO3 in this configuration is simply the magnitude ofyo. Equal trap densities in all three materials were as-sumed in Table 1 for the sake of comparison, which leadsto the conclusion that KNbO3 is the preferred material.In practice trap densities vary considerably betweensamples, and any of the three materials examined herecould be suitable for observation of instabilities.

In BaTiO3 and KNbO3 arg(yo) is close to v/2, soexternal-beam boundary conditions give the lowestthreshold. For LiNbO3 the value of arg(yo) is very closeto the crossover point in Fig. 8, so either type of boundarycondition should give similar thresholds. Note, however,that the value of arg(yo) in LiNbO3 is strongly dependenton the value of the photovoltaic field, which may varyby 4 orders of magnitude in LiNbO 3. In samples withlarge photovoltaic fields arg(yo) will decrease, and mirrorboundary conditions are to be preferred. On the otherhand, samples of LiNbO3 in which the photovoltaic fieldis strong are known to exhibit strong fanning and may,on those grounds alone, be unsuitable for the observationof transverse instabilities. Using the calculated couplingconstant values from Table 1 for external-beam bound-ary conditions implies that a nonlinear medium length of1-2 cm should be sufficient for experimental observationof the instability.

It is possible to make a rough comparison with theexperimental observations of Ref. 6, which were basedon illumination of a 5-mm-long sample of KNbO3 withunequal-intensity beams. The experimental conditionsdid not correspond exactly to any of the calculations re-ported here, since the externally supplied beams werenot of the correct intensity to make the beam-intensityratio independent of position. Apart from the internalvariation of the beam-intensity ratio, the calculations ofFig. 6(b) correspond fairly closely to the experimental

Table L Physical Parameters of Some Photorefractive Crystalsa

Parameter Symbol BaTiO3 LiNbO3 KNbO3

Refractive index no 2.46 2.34 2.23Dc dielectric constant E3 106 29 55Electro-optic coefficient (pm/V) r13 19.5 8.6 28Photovoltaic field (V/cm) EPV 10. 1. x 104 1. X 103Trap density (m- 3) N 4. X 1022 4. X 1022 4. X 1022

Recombination rate (m3/s) YD 5. x 10-14 5. x 10-'4 5. x 10-16Mobility (m2/V s) A 5. x 10-5 8. x 10-5 5. x 1o-

5

Magnitude of coupling constant (cm-l) h/oI 1.9 2.4 3.9Argument of coupling constant arg(yo) 1.57 1.45 1.56Argument of time constant arg(T) 1.5 x 10-4 8.0 x 1o-4 -1.0 X 10-3

aThe last three rows are calculated for 514-nm laser light and reflection gratings with kg along the z axis.

Saffinan et al.


situation. Translating the results to the experimentalconditions, Fig. 6(b) predicts a minimum-intensity cou-pling constant of 2yoll 12.6 and an external angle be-tween the primary beam and the satellites of 0.9°. Thevalues reported in Ref. 6 were 2yoll 6.3 and an ex-ternal angle of 0.9-1.1°. While the predicted angle isin good agreement with the experiment, the value of thecoupling constant is not. The source of the discrepancyis unknown.

To recapitulate, the transverse instability of counter-propagating beams in photorefractive media that is due tothe formation of reflection gratings has been studied. Weobtained analytic solutions including the effect of strongenergy coupling between the primary waves by fixing theboundary conditions such that the ratio of the counter-propagating intensities inside the medium remains con-stant. This implies that when the coupling constant iscomplex the externally supplied beams should have un-equal intensities. The calculated minimum thresholdscorrespond, using commonly available photorefractive me-dia, to a nonlinear medium thickness of the order of 1 cm,and the generated satellite beams are predicted to be de-tuned from the pump beams unless mirror boundary con-ditions are used.

ACKNOWLEDGMENTS

This work was supported by National Science Foundation(NSF) grant PHY 90-12244, U.S. Air Force Office of Scien-tific Research (AFOSR) grant AFOSR-90-0198, U.S. Officeof Naval Research grant N00014-88-K-0083, and the Op-toelectronic Computing Systems Center, a NSF Engineer-ing Research Center. M. Saffman thanks the AFOSR fora laboratory graduate fellowship.

REFERENCES AND NOTES

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2. A. Petrossian, M. Pinard, A. Maitre, J.-Y. Courtois, andG. Grynberg, "Transverse-pattern formation for counter-propagating beams in rubidium vapor," Europhys. Lett. 18,689-695 (1992).

3. J. Pender and L. Hesselink, "Conical emissions and phaseconjugation in atomic sodium vapor," IEEE J. QuantumElectron. 25, 395-402 (1989); "Degenerate conical emissionin atomic-sodium vapor," J. Opt. Soc. Am. B 7, 1361-1373(1990).

4. R. Macdonald and H. J. Eichler, "Spontaneous optical pat-tern formation in a nematic liquid crystal with feedback mir-ror," Opt. Commun. 89, 289-295 (1992).

5. M. Tamburrini, M. Bonavita, S. Wabnitz, and E. Santamato,"Hexagonally patterned beam filamentation in a thin liquid-crystal film with a single feedback mirror," Opt. Lett. 18,855-857 (1993).

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9. G. Grynberg and J. Paye, "Spatial instability for a standingwave in a nonlinear medium," Europhys. Lett. 8, 29-33(1989).

10. W. J. Firth, A. Fitzgerald, and C. Par6, "Transverse instabili-ties due to counterpropagation in Kerr media," J. Opt. Soc.Am. B 7, 1087-1097 (1990).

11. G. G. Luther and C. J. McKinstrie, Transverse modula-tional instability of collinear waves," J. Opt. Soc. Am. B7, 1125-1141 (1990); "Transverse modulational instabilityof counterpropagating light waves," J. Opt. Soc. Am. B 9,1047-1060 (1992).

12. M. Saffman, D. Montgomery, A. A. Zozulya, K. Kuroda,and D. Z. Anderson, "Transverse instability of counterpropa-gating waves in photorefractive media," Phys. Rev. A 48,3209-3215 (1993).

13. M. Cronin-Golomb, B. Fischer, J. 0. White, and A. Yariv,"Theory and applications of four-wave mixing in photorefrac-tive media," IEEE J. Quantum Electron. 20, 12-30 (1984).

14. B. Ya. Zel'dovich and V. V. Shkunov, Characteristics ofstimulated scattering in opposite pump beams," KvantovayaElektron. 9, 393-395 (1982) [Sov. J. Quantum Electron. 12,223-225 (1982)].

15. Note, however, that in the stimulated Brillouin scatteringof counterpropagating beams discussed in Ref. 14 the oscil-lation threshold is a minimum when the pump beams haveunequal intensities.

16. N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin,and V. L. Vinetskii, "Holographic storage in electrooptic crys-tals. I. Steady state," Ferroelectrics 22, 949-960 (1979).

17. B. Sturman, "The photogalvanic effect-a new mechanismof nonlinear wave interaction in electrooptic crystals," Kvan-tovaya Elektron. 7, 483-485 (1980) [Sov. J. Quantum Elec-tron. 10, 276-278 (1980)].

18. P. Giinter and J.-P. Huignard, eds., Photorefractive Materi-als and Their Applications I (Springer-Verlag, Berlin, 1988);M. P. Petrov, S. I. Stepanov, and A. V. Khomenko, Pho-torefractive Crystals in Coherent Optical Systems (Springer-Verlag, Berlin, 1991).

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transverse instability of energy-exchanging counterpropagating waves in photorefractive media

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