Investigation of stability of four-wave mixing in photorefractive media

Volume 135, number

8,9

INVESTIGATION A.A. ZOZULYA

PHYSICS

LETTERS

A

13 March

OF STABILITY OF FOUR-WAVE MIXING IN PHOTOREFRACTIVE

1989

MEDIA

and V.T. TIKHONCHUK

P.N. Lebedev Physical Institutes Leninsky pr. 53. Moscow II 7924, USSR Received 5 October 1988; accepted Communicated by A.P. Fordy

for publication

12 January

1989

The stability of the different four-wave mixing (FWM) configurations and nonlocal response is investigated. All the configurations considered the stability region boundaries are found.

1. The investigation of four-wave mixing (FWM) in photorefractive media is of interest due to the highpotential possibilities of using the latter in optical data processing [ 11. The main attention is paid here to the consideration of stationary nonlinear states. The stability of such states has not been actually studied so far, though this problem is one of the major issues for practical applications. The stability of the nonlinear stationary FWM states has been for the first time investigated in ref. [ 21 in application to stimulated Brillouin scattering in laser-produced plasmas. Stationary solutions were shown to be unstable under some definite conditions and the onset of instability was accompanied by the excitation of the frequency satellites in the scattered radiation. The nonlinearity in photorefractive media differs substantially from the ponderomotive one, discussed in ref. [ 21. Besides both the stationary nonlinear states and their stability conditions depend strongly on the boundary conditions for the electromagnetic waves participating in FWM process. The problem of nonlinear stationary state stability for different FWM configurations in photorefractive media therefore needs special consideration. This paper deals with the investigation of the stability of FWM in photorefractive media versus amplitude perturbations for drift-diffusion nonlinearity and purely nonlocal response. We adopt the mathematical approach developed in ref. [ 2 1.

in photorefractive media for drift-diffusion nonlinearity are shown to be unstable in some range of parameters,

2. The system of dynamical equations governing the FWM in a photorefractive medium of thickness I (O
= f

0

(A,A;+A;A3)

.

Here AI-A4 are the slowly-varying amplitudes of electromagnetic waves with wavevectors k, ( k2 = -k,, k4 = -k3), u is the refractive index grating amplitude, T is the relaxation time, r-Zu ’ , where Zo= I,‘=, IA,) *, y is the nonlinear coupling coefticient; for the nonlocal response under consideration y is a real value. The parameter 0 may take two values: D= & 1. In the case r~= 1, system ( 1) describes the interaction of waves with reflective grating recording, cr= - 1 corresponds to FWM with transmissive grating recording. Consider first the FWM with reflective grating recording ((T= 1). Confining ourselves to the analysis of the stability of system ( 1) with respect to amplitude perturbations, all amplitudes A, through A4 without loss of generality may be taken real. By using the integrals of system ( 1) d,, d,, c, c2: d,(l)=A;-A:,

d,(t)=A:--A:, 447

Volume 135, number

8,9

c(t) =A,A2 -A3A4,

PHYSICS

(2)

Cl(?) =A2A‘l -A,.43 ,

system ( 1) may be reduced to a single equation the function

for

which has the form

In deriving eq. (4) it was taken into account that Z,=Qch Y, where Q=J(d, -d2)2+4c2. For d/at=0 the following stationary solution follows from (4): sh Y(‘)(x)=sh

Y(O)(O) exp(yx)

.

(5)

Relations ( 5 ) and (2) determine the fields A,(x) inside the nonlinear medium in terms of the integrals of motion. To find the latter the boundary conditions should be specified. To analyse the stability of the stationary solution ( 5 ) one should obtain from (4) the linearised equation for small deviations from the stationary state 8Y(x):

LETTERS

exp(-ift/r,)]

A2

(4

A,

(0,

[ch Y”‘(x)+l][ch YCo)(0)-l] [ch Y(‘)(x) - I] [ch Y(‘)(O) + l] ln

ch Y(‘)(x)-if ch Y(“)(O)-if

>I’

(6)

To analyze the stability of the nonlinear stationary state in a particular FWM geometry, the integrals d,, d2, c, c2 must be associated with boundary conditions and the stationary state Y(O) (x) must be determined. Then the perturbations at the nonlinear medium boundaries SY( I), &Y(O) must be expressed in terms of perturbations of integrals. Then, letting x= Z in (6 ), the dispersion equation is obtained for find448

t)

=Q2,

A,(&

=rA,(L

1989

t)

t)>

=o

4(L

>

t> =r&(O,

t>

,

(7)

where r is the feedback loop transmission coefficient. Introducing the nonlinear reflection coefficient R(t) =A, (I, t) /a, and dimensionalizing all integrals with respect to the signal wave intensity a:, one obtains by using boundary conditions (7 ), d, =r2D(3-4D), c=rJD<1-D,,

dz=D, c2=rJ5(2D-1))

(8)

where D(t)=t[l+Jl-R2(f)/r2]. The substitution of (8) into relation determines the function Y, yields sh Y(0)=2r2

SY(O)=

(3), which

J1-0(20-1)

Q

'

J1-0 (d+2r’D),

y(f)=2

&DQ2 [2-D(

(9)

1+2D)

i-r2D(2D-3)(40-3)1&D, sY(f)=SY(O)-

if

2(l+f*)

1 + -_( lff2

t)

,

where f=f’ +if” is the complex frequency, t,=r’“‘(x)Zo(x)Q-l (note that T in eq. (4) depends on coordinate x through Z,(x) and 7, does not). The solution of this equation yields:

Xln

13 March

ing the complex frequency f=f’ + if”. The stationary state is unstable, if f"> 0, the instability threshold (the stationary state bifurcation) corresponds to f n=O. Below we shall study, as an example, the stability of some FWM configurations. In the case of the passive ring mirror (fig. 1) the boundary conditions are as follows:

sh

Y(x, t)=YCo)(x)+Re[6Y(x)

A

l

DJiZ

SD.

(10)

From (9), (5) one finds the stationary value of the function D(O) and, thus, the value of the nonlinear reflection coefficient R (‘) of a passive ring mirror in the stationary state [ 41:

Fig. 1. FWM in the passive ring mirror grating recording. M: mirrors.

geometry

with reflective

Volume 135, number

1 +2r2D”’

PHYSICS

8.9

=exp(yl)

r2(2D”‘-1)

.

dc+-r2,

&-+I,

c-0,

and from (6) for x=1, keeping in mind ( lo), one obtains the dispersion equation whose solution yields f= - 2i [ yl- (~l)~~] / (~l)~~. Thus, near the threshold, at which the nonlinear stationary state arises, the latter turns out to be aperiodically stable. The instability of a stationary state arises at considerable threshold excesses, when 1 -R2/r2=r4(

1 +r*) -* exp( -2~1) << 1 .

In this case the dispersion form 1 -r2

- -

--

2r2

=exp

rl

equation

(6) takes the

’

f

kif=

ln[(l-r2)/2r21,

x( 2N+ 1) In[ (I-r2)/2r2]

’

A,(O,

t)=O,

A,(I,l)=a,,

A,([,

Introducing the reflectivity for a signal wave R(t)=A:(/, t>/a: and the parameters r=a:/a:, q=ai/ (a: +a$) characterizing the relative intensity of pump and signal waves and making all the system integrals dimensionless with respect to a:, one obtains from (2) with due regard for the above boundary conditions: d2(t)=r-q(l+r)R(t),

c(t)=,/&,

d,(t)=l-qr-‘(l+r)(&-signyfl),

A,([,

t)=signy[q(

1+r)R(t)/r]“2Az(l,

t) .

(14)

x{(1+r.)(l+q)[exp(2yl)+r+q+rqJ -r[exp(yl)-112}-‘ . The study of the stability of this stationary state is similar to that in the previous case. By using relations (3), (14) one finds SY(l), S!P(O) and then by means of (6) one determines the complex frequency f as a function of r, q, yl. Consider first the case of a weak signal wave q-0. In this case

(13)

where N is an integer. The minimum bifurcation threshold ( yl)b,f,m,n- 2n takes place for r2 - exp ( - rc). As the second example we shall consider the FWM configuration with two external pump waves (fig.2) meeting the conditions A,(O,t)=a,,

pump waves (re-

R’O’=r[exp(yl)-l]*

(12)

According to ( 12 ), the instability of a stationary state (bifurcation) arises at r* < f . At the bifurcation threshold f’ =O one obtains the following estimations for the coupling coefficient and for the frequency: (y0,ir=(l+f&)

with two external

The stationary solution of (5) by using ( 14) allows one to obtain the following expression for R (O) as a function of r and q [ I]:

l+f21n2

>I

1 ln 3-if 1 +f’ ( l-if

Fig. 2. FWM configuration flective grating).

c2(t)=&&--d2),

if

[ I-if--

1989

(11)

This state exists at values of the coupling coefficient yl greater than the threshold value (yl),,= In [ ( 1 + 2r2)/r’]. As the coupling coefficient increases, the nonlinear reflection coefficient R(O) monotonously grows tending to the maximum value R (‘Lr. Near the threshold, when yl- (y&-c (~2),~, D(O)-+ 1,

13 March

LETTERS A

sY(O)=-Jql(l+r)r-‘[exp(yl)+r]ZiR, SY(l)=

-r6!P(O)

,

and the dispersion equation reduces to the form - r= exp [ yl/ ( 1 - if) 1. This equation possesses unstable solutions at r> 1 for yl> 0 and at r-=z1 for yl< 0, at the bifurcation threshold (yl)bif=(l+f&)lnr,

f~,f=Ir(2N+1)/Inr.

(15)

l)=a,.

The solutions

of ( 15) are known and correspond

to 449

Volume 135. number

8.9

PHYSICS

the self-excitation threshold in a nonlinear medium being in the field of two antiparallel pump waves [ 5 1. The analysis of the dispersion equation in the opposite limit q+m yields stable solutions only. Thus, the instability of a stationary state takes place within the finite range of the signal wave intensities 0 < q < q* only. The value of q* is found from the equation 6Y( I) = - 6 Y( 0 ) for 1yf ( -co, whose solution yields: q*= Jl-%-

1-2r

2(l+r)

=-

r-l rtl

’

’

y/c0

(r< 1))

yl>O

(r> 1).

dz(r)=A:+A:,

c(t)=A,A&A3A4,

cz(t)=A*A‘$-A,A3)

system (1) can be reduced the function !?(x, t)=2

(16)

to a single equation

I3 March

where P = yQ/ I(,. The equation for small perturbations Y(x, t)=YY’O)(x)+Re[6Y(x)

1989

&Y(x):

exp( -i.ft/r)]

.

with respect to the stationary state ( 19) is obtained from ( 18 ) similarly to the case considered above. Its solution can be expressed as follows:

for

sin

X

“I’-“I

y/CO)(x)

sin y

(20)

>

The integral in ( 20) is expressed in terms of a hypergeometric function [ 61. Relations ( 19), (20) are used, similarly to ( 5 ), (6), for analysing the stability of any FWM configuration with transmissive grating recording. Below, the stability of FWM schemes in the passive ring mirror (fig. 3) and semi-linear mirror (fig. 4) geometries are considered. The problem of the stability of a ring mirror on a transmissive grating is quite similar, from the mathematical point of view, to the problem of the stability of the SBS process in laser-produced plasma, which has been considered in ref. [ 2 1. The boundary conditions for the configuration of fig. 3 are as follows:

+arctg(c2_~~+d:)

arctg($$)

’

(17) This equation

A

+&($+j

3. Now we shall pass to the analysis of the stability of FWM configurations with transmissive grating recording ( CJ= - 1 in eqs. ( 1) ). Restricting ourselves, as before, to the analysis of the stability of system ( 1) with respect to amplitude perturbations, by using integrals d,, d,, c, c2: d,(t)=A:+A$,

LETTERS

is (cf. ref. [ 2 ] )

(18) .42 (I,

where Q=J(d, -d2)2+4c2, I,=d,+d,. The stationary solution Y(O) follows from ( 18 ) for a/at=0 and has the form tg[ YCo)(x)/2]

=tg[Y”)(0)/2]

exp(-p)

,

(19)

A,(/,

I)

t),

=o

>

AI(O, t)=rA3(0,

t) .

(21)

Making the integrals of the system dimensionless with respect to as and introducing the nonlinear reflection coefficient R(t) =A, (1, t) /az, one obtains from (21):

x

‘1

=a,,

.4,(0, t)=rAz(O,

(16),

M

t)

3

I --

4

e

<,

M

Fig. 3. FWM in the passive ring mirror geometry sive grating recording.

450

with transmis-

Fig. 4. FWM in the semi-linear grating).

mirror

geometry

(transmissive

Volume

135, number

d, =r2,

d,=l,

8,9

PHYSICS

c=R(t),

cr==dm.

The stationary state is governed by the following dependence of reflection coefficient R (‘) on the coupling coefficient ~2: 1+r’_Q’o’+2rJJZ--R’O’Z

This stationary solution takes place at negative ues of the coupling coefficient

As yl decrease beginning from (~l)~,,, R(O), according to (22), monotonously grows from 0 tending to the maximal possible value, R(O)+r. The analysis of the dispersion equation, obtained from (20) for x=/, shows that it is stable near the stationary state threshold. As in the case of the reflective grating ring mirror considered above, the stationary state instability arises high above the threshold and takes place at r2< f. At the instability (bifurcation) threshold the formulae for ( Y/),,,~,f&r coincide with those obtained for the reflective grating mirror. In the semi-linear mirror geometry (fig. 4) the boundary conditions have the form A,(& l)=O,

A,(/, t)=O,

A2(1, t)=rA,(/,

2).

(23)

Introducing R(t) =A,(O, t)/a, and making integrals dimensionless with respect to a:, one obtains from (16), (I?‘), (23):

c2=

d,=rR(t),

c=R(t)

,

R(t)[r-R(t)].

(24)

In the stationary state the dependence vi is as follows [ 11: l+rR(“)-Q(o)

l+rRCo)+QCo)

Solutions

=exp(yl

1+“IrE:oi)

.

13 March

1989

(2+r’)2

2r( 1 +r2)3’2

2JiG

val-

y/c($),,=-(l+r’)(l-r2)ln[(l+r’)/2r2].

A‘l(0, t)=a4,

A

lution happens to be double-valued, namely, two reflectivity values R’O) Rth correspond to a single value of yl. Near the stationary state threshold 1yf- (yl),,( << lyllth the solution of the dispersion equation (20) yields:

f= -it

1 +r2+Q’0’+2r~~

d, = 1,

LETTERS

of R(O) on

(25)

of (25) exist for

where t=R (‘) -R,,,. It follows from (26) that the upper branch of the stationary solution R :“’ > R,, (e> 0) is stable, whereas the lower branch R ?) < Rrh (t ~0) is unstable. The analysis of the solution of the dispersion equation conserves this statement throughout the region of existence of a stationary solution. 4. In conclusion, one should make some remarks. The relations obtained above , which determine the region of existence of stable stationary solutions, may turn out to be “optimistic” for several reasons. First, the question of stability of stationary solutions with respect to phase perturbations has not been considered. Second, the loss of the system’s stability may be due to distortion of a transversal structure of the field of electromagnetic radiation beams. This question can be considered, strictly speaking, only within the framework of a three-dimensional system of equations; however, some results may be obtained in the plane wave model as well, if one supposes that the perturbations may propagate along the directions not coinciding with the propagation directions of the waves A,-& (see ref. [ 71 for example). Third, as the numerical calculations show [ 8 1, when the system possesses several stationary states within the given range of parameters, then it can be stable with respect to infinitely small perturbations. However, finite (though very small) perturbations will result in the system’s breaking-away from the given stationary state.

References and correspond to the hard stationary state excitation threshold: for rl= (yl),,, R’“)=R,,,=r/(2+rZ). Throughout the region of existence the stationary so-

[ 1 ] M. Cronin-Golomb, B. Fisher, J.O. White and A. Yariv, IEEE J. Quantum Electron. 20 ( 1984) 12.

451

Volume 135, number

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PHYSICS

[ 21 V.P. Silin. V.T. Tikhonchuk and M.V. Chegotov, Fiz. Plazmy 12 (1986) 350. [ 31 N.V. Kukhtarev, V.B. Markov, S.G. Odulov, M.S. Soskin and V.L. Vinetskii, Ferroelectrics 22 (1979) 949. [4] A.A. Zozulya and V.T. Tikhonchuk, Kwantovaya Elektron. 15 ( 1988) , to be published. [S] A. Yariv and D.M. Pepper, Opt. Lett. 1 (1977) 16.

452

LETTERS

A

[ 61 I.S. Gradstein

13 March

1989

and I.M. Ryzhik, Tables of integrals, sums, series and products (Moscow, 197 1) [in Russian]. [7] V.P. Silin and M.V. Chegotov, Preprint FIAN No. 97. Moscow (1986). [8] N.E. Andreev, A.A. Zozulya, A.V. Kuprin, V.P. Silin, V.T. Tikhonchuk and M.V. Chegotov, Fiz. Plazmy 13 ( 1987 ) 37 1.