Evolution of spatial optical solitons in biased photorefractive media under steady state conditions

__ *__ BB

a

I August 1995

OPTICS COMMUNICATIONS Optics Communications

ELSEVIER

I18 ( 1995) 569-576

Evolution of spatial optical solitons in biased photorefractive media under steady state conditions S.R. Singh, D.N. Christodoulides Depcumlent

of Electrical

Engineering

and Computer

Received 21 March

Science. Lehigh

Universiry.

1995; revised version received 10 May

Bethlehem,

PA 18015.

USA

1995

Abstract

The dynamical evolution of bright spatial solitons in biased photorefractive crystals is investigated under steady-state conditions. Our numerical study indicates that these optical solitons are stable against small perturbations whereas optical beams that significantly differ from soliton solutions tend to experience larger cycles of compression and expansion. The influence of loss in a typical photorefractive material like SBN:60 is studied in detail. The effect of the external bias field on the self-focusing behavior of optical beams is discussed and the interaction forces among optical solitons are also investigated.

1. Introduction Recently, spatial optical solitons at PW power levels were observed in photorefractive (PR) materials [ 1,2]. Since their feasibility was first proposed by Segev et al. [ 31 and Crosignani et al. [4], they have been a topic of considerable interest [I,25lo]. To date, three different kinds of PR solitons have been suggested in the literature. The first kind involves the so-called quasi-steady solitons or transient solitons [ 3-51 which arise when diffraction is exactly balanced by PR two-wave mixing phase coupling. Transient solitons require an external bias field and are typically observed experimentally over a time interval, i.e. after “the index grating” has been formed but before the external bias field is screened out [ 1 I. On the other hand, recent theoretical work predicts that two other types of PR solitons, namely screening and photovoltaic solitons. are possible under steady-state conditions [791. In particular, photovoltaic steady-state planar solitons can be supported in PR materials with appreciable photovoltaic coefficients [ 71 whereas screening 0030.4018/95/$09.50 SSDIOO30-4018(95)0031

@

solitons (like those of the quasi-steady state family) require again an external bias field [8,9]. Screening or steady-state solitons differ from their quasi-steady counterparts in terms of their properties and their dependence on light intensity. In a given physical system, the shape and width of these steady-state solitons is uniquely determined by the magnitude of the applied voltage and the ratio of their peak intensity to that of the dark irradiance [8,9]. Thus far, bright, dark and gray steady-state soliton domains have been predicted

[VI. In this Communication, we investigate numerically the dynamical evolution of bright planar screening solitons in PR media. The functional form and characteristics of bright steady-state PR solitons are discussed and their stability properties are then investigated. We show that these soliton solutions tend to be stable against small perturbations. However, beams whose wave parameters significantly depart from those of a self-trapped solution do not retain their form but instead tend to experience cycles of compression and expansion or can breakup into beam filaments. The

199.5 Elsevier Science B-V. All rights reserved l-8

S. R. Singh, D.N. Christodouiides / Optics Communications 118 (1995) 569-576

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effect of loss on the propagation of steady-state optical solitons is also investigated. In typical PR crystals with moderate optical absorption, our results indicate that loss does not significantly alter the soliton beam profile. Furthermore, we investigate how the external bias field can influence the self-focusing behavior of an optical beam. A study of optical soliton interactions is also presented. More specifically, in-phase self-trapped beams are found to attract each other whereas the out-of-phase ones are mutually repelled during propagation. For illustration purposes, pertinent examples are provided where the PR material is assumed to be strontium barium niobate (SBN:60)

[ill. 2. Theoretical formulation

- bright solitons

To study the evolution of steady-state planar spatial solitons in PR media, let us first consider an optical beam that propagates along the z-coordinate and is allowed to diffract only along the x-direction. Thus, in essence, we develop a one-dimensional (planar) diffraction theory. The PR crystal is assumed to be of the SBN type, with its optical c-axis oriented in the x-direction. The external bias electric field as well as the polarization of the incident optical beam (E) are also parallel to the c-axis. Under these conditions, the perturbed extraordinary refractive index n: (along the c-axis) is given by II,12 = n,’- nzr33Ex where r33 is the electro-optic coefficient involved [ 12)) n, is the unperturbed extraordinary index of refraction and E, = E,i is the total static field, a sum of the applied dc electric field and that induced from the space charge in this PR material. Furthermore, the absorption coefficient of this crystal is taken here to be (Y.By employing standard procedures we readily obtain the following envelope evolution equation [ 8,9] : i4, + it4

+ &#h

- $(nzrgE,)qS

= 0,

(1)

where the optical field E = a&x, z)exp( ikz) is expressed in terms of a slowly varying envelope (b( x, z ), & = &#J/~z etc., k = (277/Ac)n, and Ao is the freespace wavelength. The intensity \c$I*of this bright optical beam is supposed to be well confined within the x-width W of the PR crystal and of course vanishes at infinity i.e. 1$12(x + *cc) = 0. In the dark regions

away from the optical beam the electric field attains a constant value, i.e. E,(x --+ fm) = Eo. In the case when the crystal is driven by a constant current source, Eo is equal to the current density divided by dark conductivity. Furthermore, Eo remains constant irrespective of the beam width. However, under a constant bias voltage, Eo can dynamically vary depending on the ratio of the spatial width of the optical beam with respect to the x-width W. Since, the beam is taken to be well confined within the crystal’s width W, we can assume that Eo remains approximately constant and is equal to VW where V is the applied bias voltage. The relevant correction factors have been worked out in Refs. [ 8,9]. Moreover, we assume as usual that No >> n, NA >> n and that ND’x NA where No is the donor concentration, NA is the acceptor trap density, Ni is the ionized donor density and n is the free-electron density. By employing the above relationships in the Vinetskii-Kukhtarev transport model, the steady-state electric field Ex is approximately given by [ 8,9,13] Id

Ex=Eo[l(x)+ld]

--

KBT e

(al/ax) [I(X)

(2) i-Id1

’

In Eq. (2), KB is Boltzmann’s constant, e is the electron charge, T is the absolute temperature and I(x) is the power density of the optical beam. Id is the socalled “dark it-radiance” which phenomenologically accounts for the rate of thermally generated electrons through the product sild, where Si is the photoexcitation cross-section. The power density Z(x) is related to the envelope 4 through the expression Z(x) = (n,/270)14~ where ~0 = (~/Eo)‘/~ is the vacuum intrinsic impedance. It is important to note that the first term in Eq. (2) arises from drift whereas the second one is associated with the process of diffusion. For large values of Eo, the drift term dominates in typical PR materials and thus E, = Eob/(l(X) + 1,) l&91. From this last expression, we see that the static electric field profile E, is lowered or screened in regions of high optical intensity, and in turn this effect justifies the term “screening soliton”. Thus by considering only the dominant drift process, the envelope evolution equation of a bright optical beam can be readily obtained by substituting the first term of Eq. (2) into Eq. ( 1). Effects arising from diffusion will be considered later on in this study. First, for simplicity the following transformations are adopted: 5 = z/( kxg), s = x/x0, r = (a/2)kxi and 4 = (2v&/n.$)“2U

S.R. Singh. D.N. Christodoulides

/ Optics Communications

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118 (1995) 569-576

where xc is an arbitrary spatial width and the beam power density has been scaled with respect to the dark irradiance Id, i.e. f(x) = (U(‘&. In this case, the normalized envelope CJ, satisfies [ 891,

where the dimensionless

P=

quantity p is given by

(koxo)*+33Eo

(4)

2

and its sign of course depends on the polarity of the externally applied electric field Ea. It can be readily shown that a transformation of the form U -+ exp( -ip[) U allows one to replace the last nonlinear termofEq. (3) withplU]*/(l+)U)*).Theresultant equation is of the nonlinear Schrodinger type with a saturable nonlinearity [ 9,141 and has also been shown to describe the so-called photovoltaic solitons in PR media [ 71. Equations of this sort have been previously considered within the context of saturable Kerr media [ 141. In this Communication, the study of steadystate PR planar solitons proceeds by numerically investigating Eq. (3) for a typical set of parameters in a biased photorefractive medium. Note, that the results presented in this study can also be applied in the case of photovoltaic PR solitons. We first begin by looking for spatial solitary optical beams in lossless PR crystals. Loss effects will be treated later in this Communication. To obtain the selftrapped solutions of Eq. (3) (when r = 0) we write U = r’/*y( s)exp( iv& where v represents ashift in the propagation constant, y(s) is a real function ranging between 0 and 1, and r is a positive scaling parameter given by r = (U,,,,,( 2 = i,ax/&. f,,, is the maximum value of the optical power density. Using Eq. (3) and the appropriate boundary conditions (dy/ds = 0 at y = I and y = 0), we readily obtain Y = -( /3/r)ln( 1-t r) as well as the first integral of Eq. (3) [8,9 1, i.e.

= y[ln(l

fry*)

-y*ln(l

+r)].

(5)

The envelope y(s) of a bright soliton beams can be subsequently obtained by numerically integrating Eq. (5). It can be easily shown that this is possible only when p or EOare positive quantities. Note that Ec must be also large enough so as one can justify the neglect

fig. 1. Normalized intensity lU(s)/U(O) I2 profiles of bright spatial solitons for /3 = 35.3 and r = 0.1, I, 10 and 40 as a function of s = x/x,,. En = 40 x IO3 V/m and xo = 40pm. The system parameters are, n, = 2 33, r33 = 237 pm/V and 4) = 0.5 pm.

1 p=

35.3

,,/

Fig. 2. Intensity fwhm of bright spatial units of s = x/x* for /3 = 35.3.

PR solitons

i

versus r in

of the diffusion term in Eq. (2). As an example, the PR material is taken to be SBN:60 with the following parameters: n, = 2.33, i-33 = 237pm/V and id = 100 mW/cm* at a wavelength of A0 = 0.5 pm [ 11 I. If we let the arbitrary spatial scale to be xc = 40 ,um and look for self-trapped beams at EO = 40 x 103V/m, we find that p = 35.3. Fig. 1 depicts the normalized intensity profiles of these soliton solutions for this value of p when r= 0.1, 1, 10 and 40. Fig. 1 also illustrates that the functional form and width of these solitary states are r dependent. In Fig. 2 the variation of the full width half maximum (fwhm) of the soliton’s intensity is shown versus r, when p = 35.3. Note that the soliton spatial width can always be extrapolated from Fig. 2

S.R. Singh, D.N. Christodoulides/Optics

Fig.

3.

Dynamical

U = (0. I I ) ‘/‘exp(

evolution -s2/0.8S’)

of

an

input

Gaussian

Communications

1 I8 (1995) 569-576

beam

at p = 35.3.

for any system parameters by appropriately choosing xa such that p = 35.3. From Fig. 2 it is also apparent that the spatial soliton fwhm tends to increase for very small values of r (i.e. in the Kerr limit, when r < 0.01) as well as for large values of r (r > 100) where the nonlinearity oversaturates. Between these two regimes, the fwhm curve exhibits a plateau. It is important to remember that, for a particular system the intensity fwhm is uniquely determined by the value of the externally applied electric field Eo and r = f,,,/Zd.

3. Results and discussion The evolution of the previously found solitary states is then investigated by numerically solving Eq. (3) (with r = 0) using the so-called beam propagation method [ 15,161. As expected, our results confirm that the soliton states remain invariant with propagation distance. We then consider the stability properties of these planar solitons in lossless PR media. To do so, we follow the evolution of an optical beam whose maximum amplitude, width and functional form are slightly different from those of a PR soliton. As an example let us perturb a low-intensity soliton solution (earlier obtained for r = 0.1 and p = 35.3) by approximately matching it to an input Gaussian beam given by U = (0.11) ‘/* exp( -s*/O.85*). The evolution of this Gaussian beam is shown in Fig. 3. As we can see, the beam reshapes itself and tries to evolve into a solitary wave after a short distance. The intensity amplitude, r, from 0.11 at the input changes to 0.098 at 5 = 1. Several other cases in which the beam shape, width and amplitude were offset from that of a soliton solution

Fig. 4. Evolution of a low-intensity r = 0.1 soliton when its input amplitude is multiplied by an integer (a) N = 2 and (b) N = 3 at p = 35.3.

were studied. We have found that solitary waves are stable when perturbed in their waveforms by amounts that are much smaller than their cross-sections. For example, if the perturbation in the r = 0.1 spatial soliton discussed earlier is within lo%, the beam remains stable enough and its maximum amplitude oscillation (at 5 = 1) is found to remain below 6%. However, beams with wave parameters that deviate significantly from those of a self-trapped solution do not retain their form and experience larger magnitude cycles of compression and expansion or can breakup into filaments. Such a case arises when the input amplitude of the self-trapped solution is multiplied by an integer N > 2, i.e. UN( s, 5 = 0) = NU( s, 0). To understand the behavior of these higher-order solitons we numerically solve Eq. (3) with r = 0. Fig. 4 demonstrates the evolution of such a high-order beam when its input is a multiple of a low-intensity soliton, specifically with N = 2 and N = 3. The fundamental soliton used here is a solution previously found in this study for a SBN:60 crystal, when p = 35.3 and r = 0.1. Similar plots for a high-intensity soliton with r = 10 are shown

S. R. Singh. D.N. Chrisrodoulides / Optics Communications 118 (I 995) 569-576

Fig. S. Evolution multiplied

of a r = IO soliton when its input amplitude

by an integer (a)N

= 2 and (b)N

is

= 3 at p = 35.3.

in Fig. 5. It is clearly seen that these optical beams do not retain their shape but instead alternate between expansion and compression. This process could be used advantageously to achieve spatial compression. Note however, that there is a difference between the initial behavior of a low and a high intensity beam. Typically, we have found that as long as N2r x I+ the wave experiences compression during the initial part of the cycle. In the case where N2r > 10, however the beam will initially diffract and will experience compression later on in the cycle. This behavior can be intuitively explained from Fig. 2. More specifically, in the lowintensity limit. solitons with higher amplitudes have narrower widths. and as a result experience compression during the initial stages of propagation. On the other hand, in the high-intensity range, solitons with higher peak values exhibit larger widths, and thus UN initially diffracts and undergoes compression during the later part of the cycle. We will now investigate the influence of linear loss on the propagation of optical solitons in photorefractive media. The effect of loss on the dynamical evolution of optical beams can be readily accounted for by assuming a finite f in Eq. (3). The beams at the input

513

Fig. 6. Effect of loss on (a) a low-intensity r = 0.1 soliton and (b) a high-intensity r = 40 soliton when I‘ = I .2.

are assumed to be the soliton solutions of ELq. (3) in the absence of loss. Two solitary waves, found earlier in this study for a SBN:60 crystal with x0 = 40pm, ho = 0.5 pm, /3 = 35.3 at r = 0.1 and I = 40 are propagated when the normalized constant r is 1.2, which corresponds to an actual absorption coefficient of cr = 0.51 cm-‘. The dynamics of these two waves is depicted in Fig. 6. The figure shows that the beams retain their shape as long as 5 < 0.2 , which corresponds to an actual distance of less than one centimeter. This is more evident in Fig. 7 where the normalized beam intensity profiles at 5 = 0 and 5 = 0.2 are plotted against each other. Thus, we find that loss plays a relatively small role in the overall dynamics of optical solitons provided that absorption effects are moderate. Having conserved their shapes till 5 zz 0.2, the low-intensity beam (r = 0.1) then gradually starts to widen while the high-intensity one (r = 40) begins to compress. The initial expansion of a low intensity beam ( r < 1) and the compression of a high intensity ( r > 10) one can be explained from Fig. 2. In particular, as the wave propagates, its peak amplitude decreases and in essence it moves adiabatically on the curve of Fig. 2, that is towards lower r’s. For low T’S, solitons with

514

S.R. Singh, D.N. Christodouliaks /Optics Communications 118 (1995) 569-576

-2

-1.5

-1

-0.5

0

S

05

1

1.5

2

Fig. 8. Normalized intensity output profiles of a r = 0.1 sol&on at EO = 40 x IO3 V/m and /3 = 35.3 for different applied electric field strengths, i.e. EO = 0, -40 x 103,40 x ld and 150 x lo3 V/m. The output beams at .$ = 0.2 ate compand with that at the input.

S

Fig. 7. Normalized intensity profiles at the input (8 = 0) and at 5=0.2 (z z I cm) when f = 1.2 of (a) a low-intensity r = 0.1 soliton and (b) a high intensity r = 40 soliton.

lower peak amplitudes exhibit larger widths, hence the input beam with r = 0.1 tends to expand. On the other hand, for high r values, a reduced r implies a smaller soliton width. Thus optical beams with high r values at the input will initially experience compression. However, as r further decreases (because of loss) the beam eventually diffracts. Though loss is eventually expected to destroy these solitons, in cases of practical interest where the product aL is relatively low, we find that it does not significantly affect their beam profiIe. Hence for simplicity, from this point on we will ignore any loss effects in this study. As previously noted, the solitary wave solutions of Eq. (3) (with r = 0) can always be determined provided that the set of parameters /I and I is known. The normalized quantity /3 on the other hand, depends on the crystal parameters n, and rs3, on the choice of the scaling factor xc, the wavelength Aa, and on the

Fig. 9. Evolution of a r = 0.1 soliton at & = 40 x ld V/m and fi = 35.3 when the applied electric field strengthis 150x ld V/m.

strength and polarity of the applied bias field Ea. The effect of the external bias on the self-focusing or defocusing behavior of steady-state optical beams in PR media will now be investigated. To illustrate this, let us consider first a low-intensity soliton solution as an input, i.e. at z = 0. This input solitary beam was earlier obtained for r = 0.1, no = 40pm and I& = 40 x lo3 V/m (p = 35.3). Its intensity fwhm is approximately 40,~m. The dynamics of this optical beam for different values of Eo are then studied by numerically solving Eq. (3). Moreover, their output beamwidths at a distance of 5 = 0.2 (or z % 1 cm) are compared against each other. In the absence of any bias ( EO = 0) the wave undergoes diffraction as expected. The intensity fwhm increases from 40 pm to 5 1 pm at f = 0.2. Under the influence of a negative electric

S.R. Singh, D.N. Christodoulides

/ Optics Communications

field, say Ec = -40 x lo3 V/m, the beam experiences self-defocusing on top of diffraction and it expands to 62 pm. On the other hand, an increased positive bias, leads to an increase in the self-focusing effect. This becomes evident in Fig. 8 where the normalized waveforms obtained at .$ = 0.2 are presented as a function of the bias field. It is clear that at EO = 40 x lo3 V/m, the beam remains unchanged. However, for higher electric fields, the PR self-focusing effect overcomes diffraction and the beam initially undergoes compression; that is at EO = 60 x lo3 V/m and EO = 150 x lo3 V/m the width decreases by 15% and 66% respectively at 5 = 0.2. Fig. 9 shows the evolution of a r = 0.1 solitary beam when Eo = 150 x lo3 V/m. At such high field strengths, the beamwidth initially reaches a minimum, and thereafter it follows a cyclic behavior. This initial compression can be explained by noting that soliton solutions of the same peak amplitude have narrower widths at higher electric fields. Furthermore, we have found that if the field is quite high, e.g. Eo = 300 x lo3 V/m, the beam can breakup into filaments after a particular distance. However, we find that for fields less than EO = 150 x IO3 V/m, this wave maintains a smooth single beam profile along the length of the crystal. The interaction forces among multiple planar solitons can also be investigated using Eq. (3). Such interactions may be potentially useful in all-optical switching applications. In this study we will briefly discuss how a pair of self-trapped beams will evolve in a biased PR crystal. As input, we use the solitary wave solutions of Eq. (3). In general, we have found that the interaction scenario is quite similar to that of a nonlinear Schriidinger equation [ 161. As expected, the interaction forces were found to depend on a number of variables such as the soliton separation (As), the soliton relative amplitude and phases. In particular two self-trapped beams of the same phase are attracted to each other when their distance of separation is comparable to their spatial widths. After having merged with each other, the resultant beam is found to go through cycles of expansion and compression. On the other hand, anti-phase solitary waves tend to repel each other. In Fig. 10, we show the evolution of a pair of equal amplitude low-intensity r=O.l solitons, obtained earlier at p = 35.3, x0 = 40,um and A0 = .5 ,cLm. The two beams at the input are 40 pm wide and 70,~m apart which corresponds to a nor-

s

4

118 (1995) 569-576

3

2

,

0

515

-,

-2

-3

-4

-5

s

Wg. 10. Interaction between a pair of r = 0.1 solitons at j3 = 35.3 with a separation distance of As = 1.75 when they are (a) in-phase and (b) anti-phase.

Fig. 11. Self-deflection of a r = 10 soliton at fl = 35.3 when the diffusion parameter y = 0.57.

malized separation As = 1.75. The development of the in-phase pair is shown in Fig. 10a and that of the anti-phase one in Fig. lob. The separation between the solitons is found to change slightly at 5 = 0.2 (z M 1 cm), specifically reducing to 65 pm and increasing to 75 pm for the in-phase and anti-phase couple respectively. Nevertheless, such interactions are expected to be more pronounced in longer PR crystals. Finally, we investigate the effects of diffusion on the dynamics of planar bright solitons in photorefractive

516

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/ Optics Communications

media. To do so, we retain the second term of Eq. (2)) i.e. (&T/e) (af/&r) [i(x) + Zd] -I, which is known to dominate diffusion related effects in the case of relatively broad optical beams [ 91. Substituting Eq. (2) into Eq. ( 1), we obtain

NJ

I

iU,f+ $J.y.s-

1

+

,u,2

+

e=, 1+1(/l*

’

(6)

where the parameter y associated with the diffusion term is given by y = (KaT/e) (k$~un~rss/2). As previous studies have indicated, the last term of Eq. (6) is responsible for beam self-deflection effects [ 5,17191. Numerical simulations of Eq. (6) lead also to similar results. Fig. 11 depicts the evolution of a planar spatial soliton obtained at r = 10, p = 35.3 and x0 = 40pm. For this case, y in Eq. (6) is equal to 0.57. The shift in the beam at 5 = 0.2 (z z 1 cm) is found to be 7,~m. The figure clearly shows that the spatial soliton bends with distance without experiencing significant change. The spatial shift of the soliton beam follows approximately a parabolic trajectory whereas the shift in its angular spectrum is linear with distance. A detailed theory of the soliton self-bending effect will appear elsewhere. In all cases previously discussed, we have found that the inclusion of diffusion effects in Eq. (3) does not appreciably alter the dynamics of single beams apart from introducing self-bending. However, its role in multiple-beam interaction is more complicated and is currently under investigation.

The effect of the external bias field on the selffocusing behavior of optical beams was discussed and the interaction forces among spatial solitons were also investigated. Beam self-bending effects arising from the process of diffusion were also considered.

References [l] G.C. Duree, J.L. Shultz, G.J. Salamo, M. Segev, A. Yariv, B. Crosignani, PD. Porto, E.J. Sharp and R.R. Neurgaonkar, Phys. Rev. Lett. 71 (1993) 533. [2] G. Duree, G. Salamo, M. Segev, A. Yariv, B. Crosignani, PD. Port0 and E. Sharp, Optics Lea. 19 (1994) 1195. [3] M. Segev, B. Crosignani, A. Yariv and B. Fischer, Phys. Rev. Lett. 68 (1992) 923. 141 B. Crosignani, M. Segev, D. Engin, PD. Porto, A. Yariv and G. Salamo, J. Opt. Sot. Am. B 10 (1993) 446. [ 51 D.N. Christodoulides and M.I. Carvalho, Optics Lett. 19 (1994) 1714. [6] M.D. Castillo, PA. Aguilar, J.J. Mondragon, S. Stepanov and V. Vysloukh, Appl. Phys. Lett. 64 (1994) 408. [7] G.C. Valley, M. Segev, B. Crosignani, A. Yariv, M.M. Fejer and M.C. Bashaw, Phys. Rev. A 50 (1994) R4457. 181 M. Segev, G.C. Valley, B. Crosignani, P DiPotto and A. Yariv, Phys. Rev. Lett. 73 ( 1994) 3211. 191 D.N. Christodoulides and M.I. Carvalho, Bright, dark and gray spatial soliton states in photorefractive media, unpublished. [IO] A.A. Zozulya and D.Z. Anderson, Phys. Rev. A 51 ( 1995) 1520. [ 111R.A. Vazquez, R.R. Neurgaonkar and M.D. Ewbank, J. Opt. Sot. Am. B 9 (1992)

1416.

[ 121 P Gunter and J.P Huignard, [ 131

4. Conclusion In conclusion, we have investigated the evolution of bright planar solitons in biased photorefractive media under steady-state conditions. In our numerical studies we have found that these optical solitons are stable against small perturbations. Optical beams on the other hand, whose wave profiles depart significantly from that of a self-trapped solution tend to experience cycles of compression and expansion. We have also demonstrated that in typical PR crystals with moderate absorption losses, the shape of a planar soliton does not change appreciably.

118 (1995) 569-576

[ 141 [ 151 [ 161

Photorefractive Materials and Their Applications 1 and 11 (Springer, Berlin, 1988). V.L. Vinetskii and N. Kukhtarev, Sov. Phys. 16 (1975) 2414; N. Kukhtarev, V.B. Markov. S.G. Odulov, M.S. Soskin and V.L. Vinetskii, Ferroelectrics 22 (1979) 949. S. Gatz and J. Henmann, J. Opt. Sot. Am. B. 8 ( 1991) 2296. D. Yevick and B. Hermansson, Optics Comm. 47 (1983) 101. G.P. Agrawal, Nonlinear Fiber Optics (Academic, Boston,

1989). 1171J. Feinberg, J. Opt. Sot. Am. 72 (1982) 46. 1181M. Segev, Y. Ophir and B. Fischer, Optics Comm. 77 ( 1990) 265. and V.V. Shkunov, Sov. J. Quantum I191 O.V. Lyubomudrov Electron. 22 (1992) 1121.