PHYSICAL REVIEW A VOLUME 49, NUMBER 5 MAY 1994

Theory of a multimode photorefractive oscillator: Quantitative results on the frequency shift

Laurent Dambly and Hassina ZeghlacheLaboratoire de Spectroscopic Hertzienne, Universite des Sciences et Technologies de Lille 1,

59655 Villeneuve d'Ascq Cedex, France(Received 9 August 1993)

A theoretical analysis of the oscillator with a photorefractive gain is presented. The steady multimode

oscillation is considered. In such a system, the cavity light beam undergoes a frequency "pulling" due tothe nonlinear interaction between light and matter. This frequency shift is evaluated with respect to theempty-cavity frequency. We show that it can be as large as necessary depending on the experimental pa-rameters. This can be a physical interpretation for the periodic (and chaotic) oscillation of several trans-verse family modes already observed in these setups.

PACS number(s): 42.65.Hw

I. INTRODUCTION

Photorefractive materials have long been known to besuitable recording media for holography [1—4] and,hence, for data storage. For these optical processing ap-plications, many experimental and theoretical analyseshave been performed using multiwave mixing in suchcrystals [5-7]. The two-wave mixing can also be used toprovide parametric gain for oscillation in ring resonators[8—15] and when these devices operate by means of afour-wave mixing process, optical bistability has alsobeen observed [16—18]. Recently, this phenomenon wastheoretically predicted in photorefractive two-wave mix-ing [19—21].

The photorefractive oscillator is also studied from thespatiotemporal dynamics points of view [22—24]: thesedevices display rich dynamical behaviors like optical tur-bulence and defects which present a recent increase of in-terest. For these last uses of the photorefractive crystal,some theoretical support, such as full modelization, isneeded to understand the dynamics presented by the cav-ity field: for example, experimental observations [22]display dynamical oscillations (and also chaotic behavior)including up to five families of the empty-cavity trans-verse modes. These families are known to be separatedby 10 Hz. The gain line of the crystal is centered on thepump frequency and its half linewidth is about somemHz to a few Hz, depending on the nature of the crystal.These observations suppose a large enough frequencyshift to bring the modes from their passive frequency to asuitable active one very near to the pump frequency[25,26]. This phenomenon can be produced by a parame-ter variation [24] but also by the nonlinear interactionthat occurs between the beams and the medium [22,23].

In this paper, we present a general multimode steady-state theory that leads to the evaluation of the frequencyshift versus the main parameters of the two-wave mixingphotorefractive oscillator. We follow the self-consistentmodel used by Lamb to describe the laser operation [27]:the two-wave mixing in the crystal of the pump and cavi-ty beams create the material grating and the photorefrac-tive interaction between the cavity beam and the crystal

is the nonlinear source for precisely the cavity field in ac-cordance with the Maxwell formalism [28,29].

In a previous paper [30], we have evaluated the fre-quency shift that undergoes a monomode oscillation in aphotorefractively pumped cavity. To integrate the longi-tudinal dependence of the field intensities inside the crys-tal, we have used a small expansion parameter: the over-lap between the transverse profiles of the pump and theoscillating mode is shown to be a small number for allconsidered modes. In this paper, we proceed in a slightlydifFerent but equivalent way: we know from experimentalobservations that the ratio between the oscillating andthe pump intensity beams is around 10 after a singlepassage in the crystal. This makes possible a perturbativeexpansion and a multimode treatment of the problem.

This paper is organized as follows. In Sec. II wepresent a general theory that derives the frequency shiftinduced by the nonlinear interaction between the beamsand the material. Sections III and IV are then devoted tothe particular monomode and bimode cases and lead toquantitative results. Each section is, however, made upof several subsections that develop a precise point of thetheory such as the cavity model, photorefractive non-linearity, and two-wave mixing as well as the reducedmodel and its analytical and numerical results. Finally,the link with the experiments for both the monomodeand bimode sections is presented.

II. GENERAL THEORY

A. Cavity model

In this section, we extend the results of Ref. [30] to thetransverse multimode situation. The method that we useis very close to the one developed in Ref. [10], so we willoutline the main steps.

The studied system is represented in Fig. 1; it is com-posed of a ring cavity and an active medium (a pho-torefractive crystal) pumped via a two-wave mixing pro-cess. The pump E and the cavity fields E propagatealong symmetrical directions separated by 28 withrespect to the normal to the crystal input face. As we areinterested in the infiuence of the transverse nature of the

1050-2947/94/49(5)/4043(12)/$06. 00 49 4043 1994 The American Physical Society

LAURENT DAMBLY AND HASSINA ZEGHLACHE

FIG. 1. Simplified scheme of the self-induced optical ring

cavity with a photorefractive amplifier.

beams, the cavity field is expanded on the completeorthonormalized set of transverse modes of the emptycavity [31]. Each eigenvector Ed(r) and its eigenfrequen-

cy cod are characterized by the subscript d which

represents the longitudinal and the three transverse quan-

tum numbers

E(r, t)=—

where r=(x,y, z) =(ri, z) and the orthonormalizationcondition is

f E;(r) Eb(r)dV=5, bcav

and the integration is performed along the cavity volume

V„„.However, for a Cartesian (Hermite polynomials) orcylindrical description (Gauss-Laguerre functions), theeigenvector orthogonality is already realized in the trans-verse plane.

Following the experimental situation, we suppose thatthe crystal is uniformly illuminated by the pump: a planewave represents the pump beam and one can write thatmore generally as

E~(r, t)=E~(r„z,t)= — po(z, t)Eo(r) .1

E

As in Ref. [30], the pd and po coefficients keep their expli-cit z dependence; this means that the power exchangealong the crystal is not neglected: experimentally, in atwo-wave mixing process, the probe beam can be multi-

plied by a factor of 20 after a single passage through acrystal length of one centimeter.

From the Maxwell equations, we get the evolution ofpd inside the cavity [1]following

COd 2 cryst~d+ —+, p~(O, t)+ f pd(z, t)dz .

Qd &, at' 'o

a'Ed(r) PN„(z,t)dV,

&s Bt cr»t

where I-«», is the crystal length. In this description, wehave included the following elements.

(i) The pd variables are constant in z outside the crys-tal.

(ii) The resonator quality factor for the d mode isdefined as Qd =cod(c/cr ). ,

(iii) Since we are interested in quantitative results, the

f o "»'pd(z, t)dz term is not neglected and in the long-timelimit it represents a constant. In Ref. [30], this term wasdropped since it has no qualitative influence on the re-sults.

(iv) In the nonlinear term (right-hand side), the integra-tion is performed only on the crystal volume since no in-teraction occurs outside the crystal.

(v) In the curly bracket (left-hand side), we have ap-proximated in the evaluation of the first term,1 (I.„—„„/I.tt ) by unity since the Rayleigh length (whichis the normalized longitudinal length of the eigenfunc-tions) is around 25 times the crystal length for the usualexperimental setups. In the second term which is relatedto the crystal integration, the z contribution of the eigen-functions is nearly constant.

At this stage, the source term PNL stands for any in-

teraction between the d mode and the matter. Equation(4) is a common cavity model containing a nonlinearmedium. The nonlinearity will be specified to the pho-torefractive interaction in the following subsection.

PN„(r,t) = sob n (r, t)E~ (r, t), (5)

where co is the vacuum dielectric constant and b, n (r, t)the grating index due to the two-wave mixing process in

the photorefractive crystal. In the stationary band-transport model of Kukhtarev et al. [32], the materialgrating index is given by the following expression:

y, p,'(z, t)p, (z, t)[E,*(r) E, (r)]An (r, t)=

Ng Ir r, t

where the total intensity IT(r, t) is expressed as

IT(r, t) =—~po(z, t) ~'1

00 oo

+—g g p,*(z,t)pd(z, t)E,"(r) Ed(r) ('7)

c =11=1

and y is the gain constant of the nonlinear process.Equations (5)—(7) give the coupling between the trans-

verse modes in the material. We transfer them in Eq. (4)to get

COg + —+Qd dt dt

pd(O, t)+ f pd(z, t)dz .0

2 y, BH„,zc,=, m Bt

B. Photorefractive coupling

The nonlinear polarization PNL that drives the resona-tor field oscillation is produced by the pump field Ez(r, t):a material index grating is created photorefractively bythe interference pattern of E~(r, t) and E(r, t). Thisphenomenon is due to the free electric charge migrationinside the nonuniformly illuminated photorefractivemedium. A space-charge field and hence an index change(via the electro-optic property of the material) settle in-

side the crystal. This nonlinear interaction is expressedusing a polarization that takes the form

49 THEORY OF A MULTIMODE PHOTOREFRACTIVE. . .

where

[Eq.EQ][EQ E, ]H&, = ppzt p, zt V.

cryst ITrt (9)

One can extract the optical frequencies from the oscilla-tion and pump fields using the following forms:

IECO lcoy f

p, (z, t) =go(z, t)e ', p~(z, t) =/b~(z, t)e (10)

The u&'s are the active mode frequencies. They must bedistinguished from the passive mode frequencies co&. Wecan then suppose that the&~(z) and/bo(z) coefficients aretime independent to be coherent with the stationary hy-pothesis of the Kukhtarev model. Replacing thedefinitions (10) into the relations (8) and (9), one can get

Icr st

(c02~ co~—)+i /iq(0)+ f pq(z)dzQ~ p

2c p I (N co~ )tg p, co,'H~, ea=1

where H,& is defined as

H~, =f dzgo(z)l p, (z)f 'drT

cryst [Eq Eo][EQ E, ]p r Iz'(r)

= fz dzlpo(z, t)l p, (z, t)Q&, (z),cryst

(12)

where the transverse integral is replaced by the functionQ,b(z) for convenient notations following the relation:

[EQ EQ][EQ Eb]Q~b(z)= drT .

CJT IT r (13)

E,(r) =E,(rz. }e (14)

Before analyzing the z dependence of the beams in thecrystal and since we are interested in an evaluation of theoscillating beam frequencies in the long-time limit, onehas to notice that the right-hand side of Eq. (11}still con-

t(a) —co~ )f

tains an e ' term that precludes definitively the ex-istence of steady states except under a stationary condi-tion.

In the multimode case, 0,b is the ctb element of a two-dimensional square matrix, ' the number of modes takeninto account gives its dimension. These functions exhibitthe normalized a component of the pump diffraction onthe index grating created by both the b mode and theputnp. The expression of IT(r) is given by the time-independent value of Eq. (7).

In Eq. (11), we have partially taken account (for thepz's) of the steady-state condition and separated the lon-gitudinal and transverse integrations in the crystal by us-ing the relative smallness of the crystal compared to theRayleigh length: this implies neglecting the z dependenceof the resonator eigen vector inside the crystal.Mathematically, it can be expressed as

We generalize the multimode oscillation the system ofordinary differential equations that describes the energyexchange along the crystal [15,30]. We use the waveequations for the two beams, express the photorefractivenonlinearity, integrate the transverse expansion [Eqs. (1)and (3)], and finally keep the slowly varying amplitudeapproximation with respect to time and z (the& function)of the fields. In the long-time limit, we easily get:

dp~2ik& +i co&oibo/l~ =2cpo&ocod lao(z) I'

dz

X g y~, (z)Q~, (z),a=1

d p2lkp + lcopcTibpko=2c)MQEoco+Q(z)

dz

X g y~,'(z) g pb(z)Q, b .a=1 b=1

The functions Q,b(z) are still defined by Eq. (13).For consistency, we use the notations of Ref. [30], and

take A. =A,&=A, which corresponds to fixed fringe pat-terns. When A, WA.„,the medium response time is soslow that this difference can be neglected as shown inRef. [15]. We also define a as the linear absorptioncoefficient of the photorefractive material by a=co.pp,replace z/cos(8} by z (a new longitudinal variabledefinition) and introduce

co 2' +lgk =np —= np and y, =l, e (17)

C. The stationary condition

The evaluation of the frequency shift in a multimodecase needs to consider the long-time limit (or the steady-state situation). The dynamical creation of the indexgrating is not explained and no dynamical characteristictimes of the system can be involved in our formulation.The existence of steady solutions in Eq. (11) imposes thatall the active frequencies are degenerate, while the pas-sive frequencies are fixed by the geometry of the resona-tor. We know from experiments that a slight differencebetween the pump and the cavity frequencies is usuallymeasured. This dynamical frequency mismatch is of theorder of the inverse of the photorefractive characteristictime (a few Hertz). Our work will demonstrate that thephotorefractive optical activity is able to move the oscil-lating beam frequencies (in the general multimode case)from their passive values to the same final frequency: thisfrequency difference (shift) can be larger than 20 MHz(the empty cavity frequency spacing) depending on theexperimental setup. The dynamical multimode behaviorof the oscillating multimode beam can then lead to aslight temporal extension of this model.

Then the following stationary condition has to be add-ed to Eq. (11):

co,' =co' for a =1, . . . , 00 . (15)

The next subsection will be devoted to specify the zdependence ofpo(z) and+, (z) [defined in Eq. (10)] insidethe crystal through the analysis of the two-wave mixingoperation.

D. Two-wave mixing

LAURENT DAMBLY AND HASSINA ZEGHLACHE 49

+—p„=—iQ, (z) ~' g p.ea=1

(18a)

where I, is the gain factor and y, is the phase mismatchbetween the light interference grating and the materialindex grating related to the a mode,

T

The smallness of the crystal length, compared to the Ray-leigh length, is introduced, so one can neglect the zdependence of the resonator eigenvector inside the crys-tal following Eq. (14).

Using the following polar expansion:

d a a 'Pa+ Po i/0(Z) g e / g/b~ bdz 2 ~ 12 b =1

(18b)

+ i%'0 +i+dgo= +Ioe ' and pd =+Id e (19)

one finally deals with two real equations for the intensi-ties,

+a ID=ID g I, g QI, Ib ImIe 'e ' g bI,a=1 b=1(20a)

+a Id=Io g I', QI, Id ImIe 'e ' "gd, j,

a=1 (20b)

and two equations for the mode phases

d%0 - I.dZ

12 b

number Ad is defined as' 1/2

cryst Id (z)

Id(0)i [%„(z)—4~(O}]

(24)

d%d - I. iy i(% —4d )

Io g —QI, /Id ReIe 'eZ

12

(21a) to make a possible separation between real and imaginaryparts in Eq. (23). Then the real part gives the formula forthe frequency shift from the passive to the undeterminedactive frequency of the 2' mode

(21b)

The energy transfer along the crystal from the pump tothe transverse modes takes a quite different expression in

the monomode and multimode situations because of thepresence of additive coupling terms. However, one cannote that, even in the multimode case, the total intensityis conserved when the linear absorption is negligible or,more generally,

+a Io(z)+ g Id(z) =0 .d=1

(22)

E. Frequency shift

We transfer the results of this last section in the cavitymodel [Eqs. (11) and (12)]. The following equation is val-

id outside the crystal for consistency:I

(cod co')+i —I +Id(0)e

Qd

Formally, Eqs. (20) and (21) fully describe the z behaviorof the transverse modes along the crystal, and the com-plete stationary cavity model treatment can be pursued inthe same way.

2c EOCdd CO

— Co %dId (0)

(25)

1 ~d 2ccO

td Qd s +Id(0)(26)

%d and 8d are defined by the following real and imagi-

nary parts:

e—iW (0)

%„=Re 1+—i% (0)d

1+JVd

l, e 'Hd,a=1

g I', e 'Hd,a=1

(27)

(28)

CO CO

dd

td yd(29)

The difference (cod —co' ) is much smaller than the opticalfrequencies, so we can approximate the last relation by

Equation (26) introduces the decay time of the photondensity in the 2' mode without interaction (t„=10 s).This time is much less than the characteristic time ofphotorefractive medium (r= 1 s).

The frequency shift is formally given by

g I,e 'co'Hd, , (23)s'(1+JVd )

I = 1 dSd N-

2td d(30)

where Hd, is given by Eq. (12) and contains the crystalinhuence obtained in the previous section. The complex

To go further in our calculations one needs to specify theproblem. As in Sec. III, we shall be concerned with some

49 THEORY OF A MULTIMODE PHOTOREFRACTIVE. . .

restrictive cases of this general theory and their applica-tions, it is interesting to make some links with experience.In the next subsection, we shall analyze the parametersand their relative weight to derive possible perturbativeexpansions. As a general remark, we note that for allconsidered situations, one has to solve first the single pas-sage equations [(20) and (21)] and then determine the fre-quency shift following Eq. (30).

1,b( )=

P,'(z)pf (z)X 5ab g g Pab efc=if=i Vo(z)~

where 9',b,f now has a precise definition,

(31}

p b f f drT[E "(r).Eb(r)] [E, (r).Ef(r)] . (32)

F. Experimental reality, theory, and pertnrbative expansion

In the formal previous expressions for the frequencyshift an underlying complex problem exists essentially ifone plans to develop some analytical study. The matrixH of numbers (H,b) appears through an integral [Eq.(12)] over the crystal volume and the functions Q,b(z)contain the z dependence of the fields in the total intensi-ty term present in the denominator. Thus a completeseparation of the longitudinal and transverse coordinatesseems impossible to realize. In the preceding paper [30],we have treated a monomode oscillating situation: thiscase allows us to approximate the functions Q,b(z} with aratio (obtained by a numerical fit operation) and replacecompletely the transverse integration in a single parame-ter fz. Such a fit operation depends on a too large num-ber of undefined parameters (f, )in the multim"ode situa-tion. For this reason, we chose in this paper to solvedifferently the problem according to existing experiences.To go towards more realistic and quantitative results, oneneeds to define and estimate the parameters and intro-duce them in the model.

For a centimeter crystal length, the following data areavailable [24): At the material entrance, the ratioP=Io(0)/gd", Id(0) is in between 10 and 10 whereasthe ratio 6, defined as Iz(L,~„)/gd", Id(0), is estimat-ed to be 10. The transfered intensity between the pumpand probe beams is exactly given by the relation

g Id(L„„„)4=1

Io(~cryst )

—aL cryst6'

For a =0.3 cm ' and Ldrys,= 1 cm, this ratio is of the or-

der 10 -10 . These measurements have been carriedout in the configuration in Fig. 1. This results in the fol-lowing:

(1) The transfer along the crystal axis is drastically re-lated to the material length (a developed discussion willbe presented in the monomode section), and

(2} the relatively high losses inside the optical cavity(such as the reflection of the entrance face of the crystalbut also the presence of the cavity mirrors) reduce the in-trinsic gain obtained after a single passage in the materi-al. Mathematically, these two combined effects lead tothe presence of a small parameter: the pump intensity isalways much larger than the cavity intensity.

For these reasons we shall approximate the functionsQ,b(z) by their first-order expansion versus the intensityratio:

Equation (32) is a transverse overlap integral reduced tofour modes because of the perturbation expansion. Ituses the functions E, (r) . (and not the vectors) becauseonly the s polarization of the empty cavity transversemodes may oscillate. We shall evaluate P,b,f for thedifferent considered cases and for several empty cavitymodes.

The frequency shift is obtained with the following ex-panded form for Hd,

Hd, =f dz+I, (z)e

QIb (z)I, (z )X 5.~ —y y P~., bc

b=ic=i O

I [%,(z)—% b(z)]Xe (33)

We note that the inliuence on mode Z of the other trans-verse modes appears only via the nonlinear term which isproportional to Pd, b, .

III. THE MONOMODE PHOTOREFRACTIVEOSCILLATOR

A. Theory

I. Two-muue mixing

Let us introduce the last approximations in the formalresults of Sec. II and consider the monomode case. Thesystem of Eqs. (20) can be reduced to

Id(z)+a Id(z)=I dId(z) ~ 1 Fd sin(—yd),o z

(34a}

Id (z)+a Io(z) = —I dId(z) ~ 1 Fd si—n(tpd ) .dz Ioz

(34b)

%e give in Table I the values of Fd corresponding toseveral d empty cavity modes when it oscillates alone. InEq. (32), these values correspond to a =e =f =d while bstands for the plane-wave characteristics since the pumpbeam shines uniformly on the crystal. One can comparethese equations to those of Ref. [30] and conclude a simi-lar role for the Fd parameters even if the way to derivethem is quite different. The two methods that we havepresented to simplify the G function are equivalent in thesense that the fitting definition of the fd parameter leads

LAURENT DAMBLY AND HASSINA ZEGHLACHE 49

TABLE I. Values of the Fd parameter in the monomode caseand for different cavity transverse modes.

%d(z) —Vd(0)= —,' —cot((pd)[az+ln[X(z)]] . (37b)

1=01=11=21=31=41=51=61=71=8

p=00.07960.05970.04480.03730.03260.02940.02690.0250.0234

p=10.03970.03730.02980.02560.02290.02080.01920.01800.0169

0.02740.02800.02320.02030.01840.01690.01570.01480.0140

p =3

0.02110.02270.01920.01710.01560.01440.01350.01270.0121

The field phases have globally the same behavior as inRef. [30].

2. Frequency shift

The frequency shift of the d mode is given by Eq. (30).The functions %d and 8d are still defined by Eqs. (27)and (28). If one introduces the following definitions:

y t d Z ~+d[z] +d[p)]I (z)

Id(0)

to such small fd parameters that it allows one to performthe perturbative expansion versus fd (or its multiplicativecomponent, the intensity ratio) in a second step. Onethen finds the nonlinearities obtained with a direct expan-sion versus the ratio of intensities at first order. But nowthe weight parameter fd is replaced by Fd which has aclearer mathematical formulation.

These equations can be reduced to a single ordinarydifFerential equation if one uses the following variables:Id 0(z)e '=Id 0(z) and the relation (24). ThenId(z)+ID(z) =Id(0)+ID(0) =Id(0}+ID(0)=K(const).

The final intensity equation for the probe beam is givenby

Id (z)Id(z) = I dId(z) ~ 1 Fd — sin(q&d ) .

dz Ir: Id z— (35)

This equation is easily integrated and leads to implicitsolutions. If one defines new variables as the adimension-al intensities X(z)=Id(z)/Id(0), Y(z) =Io(z)/Io(0) andthe parameter g =I d sinyd, the solutions verify

X(z)[1+P e'(Fd+1)X—(z)] =(P Fd ) "e'"—X(z)+PF(z) =(P+1)e (36)

In comparison with the results of Ref. [30], these vari-ables describe the same behavior versus the crystallength. They show an intensity transfer from the pumpbeam to the probe beam all along the crystal axis. Thistransfer reaches a maximum value then decreases to zerobecause of the losses (a). Qualitatively, the infiuence ofthe Fd parameter is also similar in spite of the differentorder of the expansions. As soon as Fd decreases (themodes have globally a more complicated transverse struc-ture with respect to the single spot of the TEMOO), thetransfer along the crystal is activated and the nonlinearcoupling is increased. However in real experimental con-ditions, the relevant part of the transfer intensity is limit-ed by the crystal length to the beginning of the operationsince no saturation occurs along the full length of thecrystal. Consequently and at this stage, the inhuence ofthe transverse nature of the modes is negligible.

The phase of the transverse mode can be exactly solvedif the intensity is known,

=N'"'(z)+iN "(z),cryst ([%'&(z)—4'd(0)+q&d ] d

(38)

= c "(z)+ic"'(z), (39)

the frequency shift takes the following easy formal ex-pression,

1 C'"'(1+N'"')+N "C"2r C"(1+N'"')—N"C'"' (40)

„1.0

0.5

0.0—0.5 0.0I

0.5(~ rad)

10 15

If JVd has a small modulus, one can retrieve the ratioC'"'/C' of our previous work [30]. However, this is notalways the case: depending on the experimental situa-tions, the ratio Id(L„„„)/Ie(0)can take large values.The denominator of Eq. (40} still vanishes in this formu-lation and Eq. (40} presents poles as defined in Ref. [30],corresponding to "infinite" values of the frequency shift.The presence of Ad just shifts the poles in the parameterspace.

We have evaluated numerically the frequency shift intwo cases presented in Figs. 2 and 3. The first calculationconcerns zero order in Fd (Fd « 1), and Fig. 2 shows thepresence of large frequency shifts (poles) for variousphase mismatch between the light interference and thematerial index gratings yd and crystal lengths L„„„.It

') 5

q'd(z) —0'd(0) = ——,' cot(yd ) ln[X(z)e ']

which can be also written as

(37a) FIG. 2. In the monomode case, zero-order calculation of the

frequency shift in the parameter plane (y, L,~„)with a=0.46cm ' and I =5 cm

49 THEORY OF A MULTIMODE PHOTOREFRACTIVE. . .

9.0

8.0

6.0

5.0Q~4.0

L

3.0

2.0

1.0

0.0 0.O O. 5(~ rad)

cess starts again with the mode I =I „.We can suggesta physical interpretation for this order based on the non-linear interaction between the cavity field and the crystal.

In the first column of Table I, we give the values of thetransverse overlap integral F& in the monomode case forthe modes p =O. One can remark that Fd decreases withincreasing I. This implies that the energy exchange in thetwo-wave mixing process is increased, and then the opti-cal activity is more efFective. This phenomenon is ob-served in Figs. 4(a) and 4(b) where we represent the fre-quency shift for several transverse modes: the more com-plex the mode, the more important the frequency pullingthat may bring it in the material gain line and allow itsoscillation.

0-

FIG. 3. Frequency shift with the complete model, for themode {0,0) alone, in the parameter plane (y, I ) with the follow-ing values of parameters: L„„„=1cm, a=0.46 cm ', and

P i0+4

exists at least two poles for small crystal sizes. As thelength becomes larger, six regions in the (q&, L) plane mayexist where the frequency shift tends to "infinity. " Wenote that the limit Fd=0 is all the Inore valid since itconcerns modes which present very complicated trans-verse patterns (very large p and I).

We have also considered the "exact" calculation (orfirst order in Fz or intensity ratio) in two cases: for thecouple p =0 and I =0 (TEMOO) (Fig. 3) but also p =3 andI =7 which corresponds to a transverse mode with 4X14spots distributed on four rings. The two cases presentqualitatively the same features, and Fig. 3 shows two re-gions in the parameter space (y, I ) for a crystal length ofone centimeter, where the frequency shift can reach verylarge values. We also conclude with a weak inhuence ofthe transverse pattern, which is due to the small variationof the transverse parameter Fd when the mode becomesmore and more complex.

B. Application

-8-

—12-

—160.83

i00—

50-

M hV

I

0.84(vr rad)

0.85

Arecchi's group [22] have observed the so-called"periodic itinerancy" that corresponds to a finite se-quence of modes, each of them appearing in a periodicorder. Some arguments based on the influence of thetransverse shape of the modes can justify the passage or-der in these sequences.

In the case of a weak Fresnel number, the experiments[22,23] show that the chosen modes oscillating in the cav-ity are those with p =0 in regular notations (correspond-ing to the absence of a full ring shape) and proportionalto sin(ly), where I is the azimuthal number. The experi-mental observations [22] conclude that the first modewhich appears is the I =I mode according to theFresnel number which is fixed by the losses in the resona-tor. This mode collapses to be replaced by the modesI =I,„—1. This continues regularly, until the model=0 disappears. Then the beam switches off. The pro-

—50-

—|000.83

I

0.84(vr rad)

0.85

FIG. 4. Complete calculation of the frequency shift in themonomode case for different transverse modes in the neighbor-hood of the creation of a double pole. (a) This presents thebehavior before (I =5.2 cm ') and (b) presents the behaviorafter the creation of the double pole (I =5.3 cm '). Except Iall the parameters have the same values in both figures:L,~„=1cm; a=0.46 cm ', and m =10 . In (b) the stars, tri-angles, and circles stand, respectively, for the modes (0,0), (0,1),and (0,2).

4050 LAURENT DAMBLY AND HASSINA ZEGHLACHE

IV. THE BIMODE PHOTOREFRACTIVKOSCILLATOR

A. Theory

—2I z sin(q&z+ q1z —qi, )

I1I2XF1z cos(%'z —%'1),

Io(41a)

In the bimode case, the system of Eqs. (22) and (23) canbe reduced to two ordinary differential equations for theoscillating modes still using the 'conservative" relation(26) to deduce the pump intensity:

d F11I1+F,2I2+a I, =I 1sin(y, )I, 1—

z Io

d8 I1=(vi) —viz) —

I v11F11 F—1z(rl1+zlz) Idz Io

I2+ I 92F22 Flz( 91 92) IIo

I1+F1z [ 711 cos(8)+ zl'1 sin(8) IIo

I2+F12 I '9z cos(8)+zlz sin(8) I

Io(43c}

where g,. =I, cosy, and g,' =I,. sing, To close

the system one has to keep in mind that Io(z)=

I Io(0)+I1(0)+ Iz (0) I e "—I, (z) —Iz (z).For realistic values of the parameters, the numerical

integration of the system of equations (43) is presented in

Figs. 5(a) for the intensities, and S(b} for the phases. One

d +a Iz —I z sin(lpz}Iz 1—dz

F12I1 +F22I2

Io

—2I1 sin(1p1+ 4z —q11)

XF1z cos(%'z —4', ) .0

(41b)

The equations for the phases become

d+1 I F11I1+F12I2cos(p1) 1—

0

I2+ I' cos(hz+0'z —0'1)F1z cos(%'z —%', ), (42a)Io

1 I

1

z (crn)

lirT}.it of validit. y

0.00

diaz 1,dz 2

cos(yz) 1—F12I1+F22I2

Io (b)

I1+&, cos(y1 —+z+q1])F cos(q'z —F11) . (42b)

Io

One can see that, as in any usual bimode situation, thephases arrange themselves such that only their difference4=%2—4, is a relevant variable. Using slightly differentnotations one can reduce the problem to solving a systemof three ordinary difFerential equations (ODE's} for I

„Iz,

and 0=2%:

dI1 i i F11I1 +E12I2=(n1 —a)I

1.5

I1I2{zlz+zlz cos(8)+gz sin(8) I, (43a)

Io

I

1z (crn)

dI2 F11I1+F12I2=(viz —a)Iz —

viz zdz 0

I1I2—F1z Ig1+ q1 cos(8)+zl1 sin(8) I, (43b)o

FIG. 5. Behavior of two-wave mixing in a bimode situation.

(a) This presents the energy exchange along the crystal for the

following parameters: I 1=4 cm ', I 2=3 cm ', y&=y2=m/2,a=0.46 cm ', L„„„=1 cm, and P, =Pz=10+ . (b) This shows

the z dependence of the phases of the mode for the same modes

and the same value of the parameter except for y&=yz =~.

49 THEORY OF A MULTIMODE PHOTOREFRACTIVE. . . 4051

can observe that the energy exchange (via the intensities)still occurs between the pump beam and the two com-ponents of the probe beam. We remark that qualitativelythe variable behavior along the crystal length remains thesame as compared to the monomode case [30]. In spite ofthe limitations introduced by the perturbative expansion(the horizontal line gives a ratio of intensities equal to10 ), we have pursued the numerical integration for un-

realistic crystal length to follow the variables evolutionalong the z axis and to confirm the assumption that theperturbative expansion version the intensity ratio is

equivalent to the transverse parameter expansion (Fd).Moreover, for a crystal length of a centimeter and sinceI" is around 5 cm, the validity of the expansion is al-

ways fulfilled. Note that the maximum exchange in in-

tensity occurs for y(=(pi=ad/2, while in phase it hap-pens for q)( =(p2 =0 (mod n }.

The analytical dependence on z of the intensities andphases is not available, however, one can express formallythe frequency shift. We define N~(z(z), N", 2(z) as the realand imaginary parts of the JV, 2 functions, following Eq.(38). We also use the notations:

I',"'=I) z I) z 1 —F)) —F)2 cos I (p, +%'((z) —0'((0) ]

—2IF„ I, I o0,—0, o ~+0, —% 00 0

(44a)

(~

) cryst I, I2I~ =I

~z I] z 1 F~& F]2 sin g)+0 ) z +& 0

—2I2F,2 z I, z I cos%'2 —0, sin qr2+%2 z —0&

00 0

(44b)

crystI,"=r, I, 1 —F„—F;, o q, +e, —e, o

2I &F]2 z I2 z I cos 42 4] cos p&+%'& z —%'2 00 0

(44c)

cryst I) I2I&' =I z dz+I&(z) 1 —F,2

—Fi2 sin[ad)&+0'z(z) —0'2(o)]

—2I ]F&2 z Ig z cos %2 0&

sin p&+%'& z —0'2 00 Io

(44d)

1

2td

cled

or, equivalently,

i %22t i ( cij i cij2 )

cFi t2

(45)

We must make a comment on the information that can beobtained from Eqs. (45) and (46). The ratio Ad /8d givesdirectly the frequency shift that undergoes the d modedue to the nonlinear interaction with the crystal andwhich brings it from the passive frequency cod to the ac-tive one co', very close to co& to allow oscillation. While inEq. (46), the difFerence co, —co& is directly related to thetransverse frequency spacing hvT. This physical parame-

such that the frequency shift of the Z mode (d =1,2} isgiven by

I'"'(1+N'"')+N "I"t d((J'(1+N("'}—N(')I(")

ter is fixed by the cavity geometry and is given bybvz =Ic/mL„„]cos 'I+1 ~/2L~„] where ~ is thecurvature of the mirror M' (Fig. 1). Then, following con-ventional notations, one can write co, —co2=2mhvThq,where q =2p +I is an integer and denotes the transverse

family of modes characterized by the same frequency[31]. For example, if one is concerned by two immediateneighbor families, then hq =1. For the experimental set-

up described in Ref. [24], hvz is about 70 MHz and,thus, for a mode photon cavity lifetime (t, ) equal to6.25 X 10 s, one easily finds that the difference2ti(co) ra&)=(A/8') is 5.40—rad. By plotting the ratioR /I versus the parameters, that we shall define, one canobtain for a chosen couple of modes the parameter spacefor which these modes are moved in frequency until theyreach the same final value (the stationary condition); thenthey coexist in the material gain line. The frequency shiftof each of them can be zero, infinity, or any value; whatappears is the difFerence between the two frequency shiftsof the modes that realize the stationary condition.

The formal expressions (45} and (46) will be treated nu-

LAURENT DAMBLY AND HASSINA ZEGHLACHE

ection. However, to give athe roblem parameters an o

hll t tb th 1tions that one can expect, we s a s arcase corresponding to I"; =

B. Numerical simulabons an ppd a lication

1. Zero orderin I',6

two-wave mixing [Eqs. (41) and (42 ] issimp i e an1'fi d d the exact solutions are t e o o

t

I

I (z) =Id(0), [I d sin(qd )—a]z,d Z d

Id4 (z) —Vd(0}= cos(yd )z .d Z d

(47a)

(47b}

)

Io o

l ~

/

(4&)

' t can be calculated analytically. EvenThe frequency shift can e caletel the information on e

'

fi d th ib'lit f li-es wecan

find

epos' ' '

f hing the stationary conditio .' 'n. The ratio o

(35) can be written as

X~( 1 +Sd ) + Yd Td

Yd(1+Sd ) —Xd Td

l 0(-

se zero-order calculation of the fre-FIG. 6. In the bimode case, zero-or er chift in the parameter plane (y~,quency s i t in e

' and t, =t, .r =r =7cm ', a=0.46cm-, an

where Xd ~d ~d and Td

X = e[I'd sin(yd )—a]z/2

0

X cos cos(pd )z +gdId

dz,2

Y = e[I d sin(yd) —a]z/2'd0

X sin cos(yd)z+yz dz,2

e[I d sin(yd )—a)z/2

0

IdX os cos( /id )z dzC

(49a)

(49b)

(49c)

then for b,y=0 (mod2m), thehf1 their respective frequencymodes are identica,

I (in t units vanis es.equal and the ratio Rionarit condition is rea izeother cases the stat ri y

h the same featuresthe arameters (p„pz

I' /I =0 ( d, / =0)'t. Fi ures7and8s ow e

we just notee that the relation'

n (only mode 2) whilerepresents t e mhe monomode situation on yth symmetrical bimodeprres onds to t e sym

situation. eTh crystal length is a ways acentimeter.

"e [I d sin(q d }—a]z/2d

X sin cos(y&)zId

dz .2

(49d)

d solve the problem(46) at order zero and soOne can use Eq.

d d '"' "der '" thdth i fl ofth

we have stu ie z

b ode case the lar eand we discussed t e in ua L . Intis imo

h ~ of. 1eters makes necessary enumber of parame eh d fi es some of them.arameters between the modes an x

A' '/8 ' valuesel the parameters (y„hy=yzp

w en a arameter is studied, for exam-atch, the ot er parapie the phase misma c

etrical and refer-kee at least a symmeto be identical to eephe absence of relativearacterized by t e a sen

(%/el=0). In Fig.nce situation, c

optical activity

}

(o

1 0

e case, zero-order calculation of the fre-G. . In the bunode case, zequency s i t inh f '

the parameter plane (y&,=0.46 cm ', Ay=0, and t, =t2.cm, a=

49 THEORY OF A MULTIMODE PHOTOREFRACTIVE. . . 4053

O

0

FIG. 8. In the bimode case, zero-order calculation of the fre-quency shift in the parameter plane (y&, t& It&) with I &=I 2=7cm ', a=0.46 cm ', and hq=0.

2. The "eract" calculation

The full model is now considered. Qualitatively, theresults are the same as above and for the zero-order ex-pansion [30]. The stationary condition for the bimodecase can be realized for the sensible parameter values.The numerical simulations at first order of the expansionin the intensity ratio are presented in Fig. 9 for the modes(p =0, 1=3) corresponding to six spots, and (p =0,I =4) with eight spots. In that case, the transverse (over-lap integrals) parameters are the following:F~& =0.037 302 F&2 =0.02176 F22 =0.3264. Thesemodes belong to successive families, (co, —co&)=70 MHz,for L„„„=1 cm, L „=150cm (the losses are a=0.46cm '). We have taken an asymmetrical situation since1,=5 cm ' and I 2/I', =1.3, t, =6.25X10 s and

t, /t2 =0.7, giving a relative optical activity for the van-ishing phase mismatch. The intensity ratio of each modeon the pump is 10 and the initial phases of the modesare zero.

In Fig. 10, we choose to display the stationary condi-tion realization in a degenerate case with a bimode beamwhose components belong to the same family q =2:(p = 1, I =0) and (p =0, I =2). In that case, this relationco&

—co&=0 (hq =0) prevails: the modes are shifted sepa-rately to the same frequency co' for the plotted values ofthe phase mismatch, so they may oscillate together.

From an experimental point of view, the monomodeanalysis has given a theoretical justification to the or-dered sequence of modes in "periodic itinerancy. " Thepresent section results, and more precisely the summaryin Fig. 9, exhibit the simultaneous presence of two pas-sively nondegenerated transverse modes in a narrow ma-terial gain curve.

The results displayed in Fig. 10 and corresponding tothe mixing of two degenerated modes, (p =0, l =2) and(p =1, 1=0), and their simultaneous oscillation, have

been also observed with the same modes in the experi-ment of the Lille group [24].

V. CONCLUSION

We have presented a theoretical analysis of a pho-torefractive oscillator that explains some experimentalobservations on spatiotemporal dynamics [23—25]. Theself-consistent model we have used leads, in the steady-state case, to the frequency shift that undergoes the pas-sive modes of the resonator. Depending on whether wetreat the monomode or the multimode oscillation, theformulation of the problem is different. In the first case,the notion of the frequency shift is enough to explain thejumps in the oscillation of transverse modes belonging to

2.0

1.5

I1.0

0.5

()Q

P

/

( /

(

r~/I'~g) ~.O

loo——0.5 0.0

FIG. 9. The frequency shift with the complete model in a bi-mode situation with a nondegenerate mode. We consider themodes (0,3) and (0,4): F» =0.03730, F» =0.02176, andF2q =0.03264.

FIG. 10. The frequency shift with the complete model in abimode situation with two degenerated modes, (1,0) and (0,2),for the same parameters as in Fig. 9 except F»=0.04676,Fl2 =0.03979, and F~2 =0.01989.

4054 LAURENT DAMBLY AND HASSINA ZEGHLACHE 49

different families and then separated by large frequencyspacing, the stream of modes being controlled by a pa-rameter variation or not. In the multimode case, the ad-ditional condition imposed by stationarity poses the prob-lem differently: the imposed frequency locking of thesteady-state solutions allows the modes to be presentsimultaneously in the material gain curve, and thenmakes possible their oscillation, all parameters beingfixed.

The main result of our analysis is that, for realistic ex-perimental parameters (cavity and crystal lengths, reso-nator losses, etc.), this locking condition can always befulfilled for suitable material gains, or photon lifetimes, orphase mismatch parameters. Zero-order calculationsgive a restriction: in a symmetrical bimode situation,only degenerate modes fulfill the stationary condition.But at first order, the addition of the transverse parame-ters suppresses the symmetrical case. Then this restric-tion disappears.

%e have presented the bimode results in terms of rela-tive parameters: an all-plane parametric description. Wemean that for any "fixed" mode parameter, the secondmode has several possibilities to "adjust" itself with itscorresponding parameter: for example, the stationarycondition is realized for q&=0. 5 rad, and F2=0.5, 0.5,1.0, and 1.S rad. This is due to the presence of multiplepoles and is responsible for the richness of the experimen-

tal observations. The reduction to a plane-parameter rep-resentation is probably due to the perturbative expansionthat allows the bimode oscillation to be close to two addi-tive monomode situations. %e note that even when wehave taken some of the parameters as close as possible toexperimental reality, we have performed our analysis fol-lowing some generality to make it fitting with a max-imum number of setups (crystals and resonators). Thefull multimode problem can also be numerically treatedbecause we give the formal expression of the frequencyshift: one can then obtain the parameter configurationthat can lead to the full transverse multimode oscillationwhich is directly related to the turbulence.

The generality of our analysis and results, with respectto the parameter space, allows us to conclude the pres-ence of a generic phenomenon produced by the nonlinearinteraction between the cavity modes and the crystal inthe resonator environment.

ACKNOWLEDGMENTS

The authors wish to thank Professor P. Glorieux, Pro-fessor D. Dangoisse, and Dr. D. Hennequin for veryfruitful discussions and their encouragement. The La-boratoire de Spectroscopic Hertzienne is associe auCNRS (URA No. 249).

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