Spatial modulation instability of coherent light in a weakly-relaxing Kerr medium

J. Opt. A: Pure Appl. Opt.6(2004) 461–468 PII: S1464-4258(04)73478-9

Spatial modulation instability of coherent light in a weakly-relaxing Kerr medium

H Leblond

and C Cambournac

1Laboratoire des Propri´et´es Optiques des Mat´eriaux et Applications (POMA), UMR CNRS/Universit´e d’Angers 6136, 2 boulevard Lavoisier, F-49045 Angers cedex, France

2Laboratoire d’Optique P-M Duffieux (LOPMD), UMR CNRS/Universit´e de Franche-Comt´e 6603, 16 route de Gray, F-25030 Besan¸con cedex, France E-mail: herve.leblond@univ-angers.fr and cyril.cambournac@ulb.ac.be

Received 9 December 2003, accepted for publication 4 March 2004 Published 24 March 2004

Online at stacks.iop.org/JOptA/6/461 (DOI: 10.1088/1464-4258/6/4/026) Abstract

We report on a theoretical analysis of spatial modulation instability of coherent light propagating in a nonlinear medium with a noninstantaneous Kerr response. The latter is of a relaxing Debye type and originates from diffusive molecular reorientation. We consider the examples of harmonic and pulse-like perturbations superimposed either on a monochromatic temporal background or on a quasi-monochromatic Gaussian pulse. Our analysis reveals that the finite duration of the nonlinear response is

responsible for spatiotemporal dynamic features that obviously do not exist within the framework of the usual scalar nonlinear Schr¨odinger equation, which models spatial modulation instability in an instantaneous Kerr medium.

Keywords: nonlinear optics, modulation instability, noninstantaneous Kerr effect, spatial solitons

1. Introduction

Since the advent of the laser, optics has become a major field in the study of nonlinear physics, including particularly solitonic effects and related instabilities [1–4]. Among these, the modulation instability (MI) is an ubiquitous phenomenon which refers to the self-induced break-up of a homogeneous wave into localized wavepackets due to the interplay between nonlinearity and dispersion (either temporal or spatial). The MI process is thus closely related to solitons. Either spatial or temporal, optical MI can be approximated by a degenerate four-photon parametric interaction in the frequency domain, i.e. the resonant energy transfer between interacting Fourier modes according to an appropriate phase matching between them. It is worth stressing that this phase relationship is self- matched for the spontaneous MI process, the unstable modes growing from noise. Besides the field of nonlinear optics, MI has been demonstrated in many fields such as plasma physics, the physics of fluids or recently matter-wave physics [5].

3 Present address: Service d’Optique et Acoustique, Universit´e Libre de Bruxelles, CP 194/5, 50 avenue FD Roosevelt, B-1050 Bruxelles, Belgium.

In optics most of the studies of MI have been devoted to Kerr-like nonlinear media which feature an intensity- dependent refractive index (or phase shift in the particular case of quadratic media in the so-called Kerr limit [6]).

In bulk nonsaturating Kerr media, however, MI causes random dislocation of extended beams and soliton stripes into filaments, which subsequently undergo catastrophic self- focusing due to the collapse instability [7]. Conversely, in a one-dimensional (1D) Kerr medium (e.g. a single-mode optical fibre or slab waveguide) light propagation is modelled by the nonlinear Schr¨odinger (NLS) equation which is collapse- free and thus allows stable self-phase modulation of pulses or self-focusing of beams [8]. Moreover, MI becomes controllable [9]. In the temporal domain, as applied mainly to light propagation in optical fibres, MI of continuous waves can lead to the generation of ultrahigh-repetition-rate periodic trains of soliton-like pulses [10, 11]. By equivalence, MI of quasi-plane waves can lead to the formation of arrays of soliton-like beams [12]. Recently [13], MI-induced break-up of a spatiotemporal soliton stripe (a 2D soliton along only one spatial dimension and time) in a bulk quadratic medium was

proved to evolve toward arrays of spatiotemporal filaments.

(The collapse instability was more or less prevented owing to the saturating nature of the quadratic nonlinearity.) This latter experiment paves the way for the long-awaited generation of stable light bullets, i.e. solitons in all dimensions.

But a fundamental feature of MI in conservative Hamiltonian systems with an infinite number of degrees of freedom—such as the ones described by NLS—is its long-term periodic behaviour: the Fermi–Pasta–Ulam (FPU) recurrence [14, 15] should prevent the stabilization of soliton arrays (or trains) generated by way of MI. Indeed, such recurrence manifests itself as a periodic evolution of the MI energy spectrum rather than a thermalization, i.e. the equipartition of the initial energy among all degrees of freedom. Despite the recent demonstration of the reality of FPU recurrence of MI in temporal optics [16], no experimental work in the spatial domain [17–22] proved such behaviour yet.

It is, however, likely to be very sensitive to initial conditions, so that MI can easily evolve toward multi-soliton generation.

Because of a lack of experimental data it still remains unclear whether it is a dominant process or an exotic phenomenon that comes out from necessary incomplete theoretical models.

Among the possible departures from the ideal Kerr model (NLS), the question of the nonlocality of the nonlinear response, either spatial or temporal, is an important issue, as real systems may obviously feature such a characteristic. This explains the recent amount of attention devoted to the study of the influence of nonlocality on both the collapse instability [23]

and symmetry-breaking instabilities like, for example, the MI.

Concerning the latter, of interest for the present work, it has been proved that only some threshold level of spatiotemporal coherence is necessary for the spontaneous appearance of MI in noninstantaneous Kerr-like media, saturating [24] or not [25].

Moreover, spatial MI of coherent light has been proved to exist even in spatially nonlocal nonlinear media [26, 27].

However, the influence of temporal nonlocality on the full dynamics of the spatial MI of coherent light has, to the best of our knowledge, not been addressed to date. Our motivation stems from the experiments reported in [21].

Here, no FPU recurrence could be observed despite fulfilled initial conditions for its observation with respect to the NLS model. The authors studied the induced-MI process by favouring amplification at a given spatial frequency stronger than noise. Owing to the absence of recurrence, clear- cut arrays of quasi-one-dimensional spatial solitons could be generated. In addition, the authors observed a novel spatiotemporal dynamics: the appearance of a transversally shifted secondary array at the trailing edge of the exciting light pulse, observed as a spatial-frequency doubling on the time- integrated CCD-camera image. With the help of numerical simulations (see [21] for details) the authors ascribed this discrepancy to the finite relaxation time of the nonlinearity—

the molecular reorientation-induced optical Kerr effect—as regards the duration of the light excitation. In this paper we provide analytic support to those facts that were previously demonstrated by numerical means: the inhibition of the FPU recurrence associated to the above-mentioned spatiotemporal dynamics. Furthermore, our results enable us to bring a physical explanation to the observed results. This paper is organized as follows: in section 2 we give the evolution

equations used to model the influence of the noninstantaneous (here relaxing) nonlinearity on light propagation and reduce them to a single equation in the limit of a small relaxation time with respect to the timescale of the electromagnetic wave temporal envelope. This equation is used in sections 3 and 4 to study the spatial modulation instability of either a continuous (monochromatic) or a pulsed (quasi-monochromatic) plane wave. We finally summarize our results and conclude in section 5.

2. Relaxing Kerr nonlinearity 2.1. The model

The model considers the phenomenon of scalar MI, i.e. a self-induced instability, and only one transverse spatial dimension, thus staying consistent with the experiments. It is further restricted to a Kerr nonlinearity that originates from molecular reorientation in polar liquids, thus giving rise to a positive change of the refractive index, namely a self- focusing nonlinearity [28]. In a dispersionless and lossless focusing Kerr medium of such a type, and under the paraxial approximation, the electromagnetic field envelope obeys the following set of dimensionless equations [29, 21]:

i(∂t+v∂z)ψ+∂_x²ψ+Pψ=0, (2.1) τ∂tP+P= |ψ|², (2.2) with the initial data

ψ|z=0=ψi(x,t). (2.3) Here, v is the group velocity of the wave and τ is the relaxation time of the nonlinearity. The notation ∂t stands for the partial derivative operator∂/∂t, etc. In the context of optics, equation (2.1) rules the spatial evolution of a slowly varying electromagnetic wave envelopeψsubject to both the diffraction and the focusing molecular reorientation-induced Kerr nonlinearity which is nonlocal in time and described by the Debye diffusion equation (2.2) for nonlinear polarization density P. (Note that the spatial locality of the nonlinear response does not allow the present model to apply to molecular reorientation induced in nematic liquid crystals, which are inherently spatially nonlocal [27].)

In a frame moving at group velocity, i.e. using the change of variables T = t−z/vandζ = z/v, system (2.1)–(2.3) becomes

i∂ζψ+∂x²ψ+Pψ=0, (2.4) τ∂TP+P= |ψ|², (2.5) ψ|ζ=0=ψi(x,T). (2.6) Note that, despite the use of the retarded time T, the transformed initial condition (2.6) does not involve any retardation, which would have made the analysis difficult.

Besides, when considering the ideal Kerr effect, i.e. an instantaneous change of the refractive index, the well-known cubic NLS is retrieved from (2.4)–(2.6). Furthermore, it is worth stressing that the model is restricted to the influence of nonlinearity on the transverse spatial variable x only.

Indeed no group-velocity dispersion will be considered in this work and the so-called spatial MI process will then

refer to homogeneous-wave break-up along the transverse dimension x. Finally, all characteristic coherence lengths of the considered electromagnetic wavepackets are much longer than the characteristic length of the relaxing nonlinearity, i.e. we are interested in the influence of the noninstantaneous nonlinearity on coherent light.

2.2. Conserved quantities From equation (2.4) we find that

∂ζ|ψ|²=i∂x[ψ^∗∂xψ−ψ∂xψ^∗], (2.7) from which we deduce that the wave energy is conserved. More precisely,

∂_ζ

D

|ψ|²dx =0, (2.8)

where the integration runs over D=Rifψis localized, and over one period ifψ is periodic with respect to x, as in the present study of spatial modulation instability.

Making use of equation (2.5) in the conservation law (2.8), and then setting

I =∂_ζ

D

Pdx, (2.9)

we get

I=I0(x, ζ)e^−Tτ. (2.10) Thus, with regard to the time T (which is not the evolution variable of the system), the quantity I is not conserved. On the other hand, if the polarization density P is zero at the initial timet=0 for allz, thenIvanishes identically, and thus

DPdxdoes not depend onζ.

However, despite the wave energy being conserved the system is not Hamiltonian. Physically, the time evolution of the polarization densityPin the considered case of molecular reorientation strongly depends on the influence of thermal agitation. In the absence of damping, the molecular dipoles oscillate around the direction of the wave electric field. This oscillation is converted to molecular reorientation only if some damping exists. Energy dissipation due to thermal agitation yields this effect. Therefore, the phenomenon is intrinsically nonconservative. The exponential decrease (2.10) is an expression of such a characteristic. Hence, the fact that the system is not Hamiltonian seems to be a consequence of this physical feature.

2.3. Weakly-relaxing nonlinearity

Equation (2.5) can be solved explicitly, assuming some initial value of P. IfPis assumed to vanish asT tends to−∞, the system (2.4) and (2.5) reduces to a single integro-differential equation:

i∂ζψ+∂x²ψ+ψ

_T

−∞e^T−Tτ |ψ(T)|²dT

τ =0. (2.11) Equation (2.11) is a NLS equation whose nonlinearity is relaxing in time while still local in space. The relaxing nonlinearity implies that the intensity-dependent index change depends not only on local timeT but also on timeTbefore T. In practice, the leading edge of any exciting light pulse,

corresponding to the early-arriving light at a given propagation distance, also modifies the refractive index seen by the later- arriving light. This modification becomes significant for a highly-contrasted MI pattern, because it involves high enough intensities to impose a nonlinear index perturbation in the future [22]. As already mentioned, when the nonlinearity relaxes instantaneously (τ = 0) equation (2.11) reduces to the cubic NLS equation.

For small values of the relaxation time (τ 1) the integral in equation (2.11) can be evaluated with the saddle- point method [30], thus reducing the evolution equation to

i∂ζψ+∂x²ψ+ψ

N

n=0

(−τ)ⁿ∂Tⁿ|ψ(T)|²+ O(τ^N+1)=0. (2.12) Nonlocality does not appear in this limit anymore.

Furthermore, the influence of relaxation is better enlightened from a physical point of view: in equation (2.12) appear nonlinear terms of different order in the derivative of the wave temporal envelope with respect to the transverse dimension T. Whereas the zero-order term is the usual self-phase modulation one an important remark concerns the first- order term. The same one indeed arises when considering the intrapulse Raman scattering in optical fibres. This scattering process is responsible for the self-frequency shift of ultrashort pulses and the decay of higher-order solitons into its constituents [31]. Its effect is to induce a continuous redshift of an ultrashort-pulse carrier frequency because of a reciprocally wide enough temporal spectrum. Like Raman scattering, a molecular reorientation-induced Kerr effect is inherently noninstantaneous, which explains the observed redshifts of temporal spectra of pulsed spatial solitons generated in CS2

planar waveguides [32]. As the nonlinear terms are weighted by powers ofτ, it is likely that the influence of the first-order term be predominant compared with higher-order ones. Its influence on the spatial modulation instability studied here is shown in sections 3 and 4 below.

3. Spatial MI of a monochromatic plane wave

3.1. Linearized problem

Equation (2.11) admits the plane wave solutionψ0=Ae^i|A|2ζ. For the sake of simplicity, we assume Areal. We further considerAas being temporally constant in order to describe the case of a monochromatic excitation (CW). We are interested in the spatial modulation instability of Benjamin–Feir type [33]

of this solution. By setting ψ = ψ0 + εψ1, ε being a small quantity, equation (2.11) is linearized around ψ0 and the solutions are sought under the form

ψ1=[u1(T)e^ikx +u^∗₂(T)e^−ikx]e^iA2ζ. (3.1) We get the following equations foru1andu2:

i∂ζu1−k²u1+A²

_∞

0

[u1(T−T) +u2(T−T)]e^−T/τdT

τ =0, (3.2)

−i∂ζu2−k²u2+A²

_∞

0

[u1(T−T) +u2(T−T)]e^−T/τdT

τ =0. (3.3)

3.2. Approximate solution

An approximate solution of interest can be found for smallτ, that is by use of the reduction (2.12). Equations (3.2) and (3.3) then reduce to

(iλ−k²)uˆ1+A²(uˆ1+uˆ2)−A²τ∂T(uˆ1+uˆ2)+· · · =0, (3.4) (−iλ−k²)uˆ2+A²(uˆ1+uˆ2)−A²τ∂T(uˆ1+uˆ2)+· · · =0, (3.5) where we setuj= ˆuje^λ(T)ζ for j=1,2. At leading order in τ, the solutions of equations (3.4) and (3.5) are

ˆ

u₁=(k²+ iλ)a and uˆ₂=(k²−iλ)a, (3.6) with

λ=λ0=k

2A²−k². (3.7)

Here,λ0is the usual instability gain in the limit ofτtending to zero [9]. We seek after a solution of equations (3.4) and (3.5) of the form (3.6) witha =a(T)andλ=λ0+τλ1+· · ·. At first order inτ, we computeλ1and get

λ=λ0− τk A²

√2A²−k²

∂lna

∂T . (3.8)

Thereforeλ > λ0whena(T)is decreasing andλ < λ0when a(T)is increasing. A pulse-like perturbation will thus undergo a faster growth on its rear and a slower on its front. It seems then to move towards increasing values ofT, i.e. to the rear side of the pulse-like perturbation.

3.3. Exact solution

Equations (3.2) and (3.3) can be solved exactly, assuming that u1(T)andu2(T)are given atζ =0. We setu1+u2=Uand u1−u2 = V. Summing up and subtracting equations (3.2) and (3.3) yield

i∂_ζV−k²U+ 2A²

_∞

0

U(T−T)e^−T/τdT

τ =0, (3.9) i∂_ζU−k²V =0. (3.10) By use of the Fourier transform

U = +∞

−∞

U˜(ω)e^iωTdω, (3.11) we get

(∂_ζ²+k⁴)U˜(ω)= 2k²A²

1 + iωτU˜(ω). (3.12) The general solution of (3.12) isU˜ =Be^λζ+Ce^−λζwhere BandCare constants and

λ²= 2A²k²

1 + iωτ −k⁴. (3.13) The constants B and C can be computed explicitly from

˜

u1(ζ =0)andu˜2(ζ =0). Thenu˜1andu˜2are easily deduced and u₁and u₂are found by inverse Fourier transform. It is easily checked that the above solution coincides with (3.8) at first order inτ.

Figure 1.Compared gain curves of spatial modulation instability of a continuous plane wave in the presence of a relaxing nonlinearity:

unperturbed CW (full curve); sinusoidally modulated CW at ωτ=0.2 (broken curve) andωτ=7 (dotted curve). (Parameter:

|A|²=1.)

3.3.1. Harmonic perturbation. First we consider the simple case of harmonic perturbation of a continuous wave. In such a case,a(T) ≡e^iωT and the instability gain is exactly given by equation (3.13). Figure 1 illustrates the gain behaviour in this interesting case of practical importance. Without any temporal perturbation (full curve of figure 1 corresponding to ω =0), the relaxation has obviously no effect on spatial MI and the usual steady-state gain curve is obtained. Conversely, for a sinusoidally modulated CW, the gain curve now appears to be open: there is no cut-off frequency any more (broken and dotted curves of figure 1). This means that the FPU recurrence may be inhibited in such a dynamic system. Indeed, energy partition among spatial-frequency modes is not strongly limited to low-frequency harmonics, i.e. thermalization in such a dynamic system has time to establish [34, 35]. In the case of a high enough temporal modulation frequency, the gain curve does not have any more maximum. In such a case, the MI process itself can be inhibited because no spatial frequency can be favoured in the amplification process. These effects are, however, transient, as they occur on the timescale of the nonlinear response only. In practice, such behaviour can arise from a time-varying coherent beam, either coming from a temporally dynamic noise or from an induced temporal modulation. This has recently been experimentally observed in photorefractive media owing to their slow and tunable relaxing nonlinearity [36].

3.3.2. Pulse-like perturbation. Using fast Fourier transforms the solution can be quickly computed from this procedure. For the sake of simplicity, we assume that the growing component only is present (i.e.C =0) and

ˆ

u1= iλ+k²

2k² Be^λζ. (3.14)

We assume that the amplitudeBhas a Gaussian shape B=2k²B0e^−ω2/s2. (3.15) The evolution is obviously dominated by an exponential growth, in which we are not especially interested. In order to make more apparent the other features of the evolution, we multiply|u1|by a conveniently chosen decreasing exponential.

Figure 2.Evolution of a pulse-like perturbation to a continuous plane wave, corrected from the dominant exponential growth

|u1|exp(−0.55ζ ). (Parameters:k=0.5,τ=0.5,s=5,B0=1, A=1.)

As shown in figure 2, during the propagation the pulse-like perturbation becomes wider and goes towards the positive values ofT. For larger values ofkandτ, striations can appear, as shown in figure 3. However, this effect is rather limited.

4. Quasi-monochromatic excitation: Gaussian temporal envelope

In the previous section we studied the spatial modulation instability of the constant solution to equation (2.11) and described the evolution in the presence of harmonic and pulse- like temporal perturbations. This study has shown that the noninstantaneous character of the nonlinear response shifts a bell-shaped perturbation toward the rear. In this section we show that this effect can be responsible for part of the experimentally observed results. Experimentally, however, not only the perturbation but also the wave itself has a finite duration. Therefore we look for a solution of equation (2.11) which is uniform but not constant, in the form

ψ0= A(T)e^iω(T)ζ. (4.1) It is easily seen that such a solution exists for any function

A(T), the phase factor being ω(T)=

_∞

0

|A(T−T)|²e^−T/τdT

τ . (4.2)

In the following, we study the modulation instability about this solution.

4.1. Correction to the growth rate

Still considering a modulation instability of Benjamin–Feir type [33], we look for solutions of equation (2.11) of the form ψ =ψ0+εg(ζ,x,T)e^iω(T)ζ, (4.3) where

g(ζ,x,T)=u1(ζ,T)e^ikx +u^∗₂(ζ,T)e^−ikx, (4.4)

Figure 3.The same as figure 2, except thatk=1.3 andτ=1.2.

εbeing a small quantity. The equations foru1andu2are now i∂ζu1−k²u1

+A(T)

_∞

0

[A^∗u1+Au2](T−T)e^−T/τdT

τ =0, (4.5)

−i∂ζu2−k²u2

+A(T)

_∞

0

[A^∗u1+Au2](T−T)e^−T/τdT

τ =0. (4.6) For small values ofτwe can use the formula (2.12). Keeping only two terms in the expansion, equations (4.5) and (4.6) then yield

i∂ζu1−k²u1+A(A^∗u1+Au2)−τA∂T(A^∗u1+Au2)=0, (4.7)

−i∂_ζu2−k²u2+A(A^∗u1+Au2)−τA∂T(A^∗u1+Au2)=0.

(4.8) For the sake of simplicity, we restrict to the case where A= A(T)is real and look for exponentially growing solutions uj= ˆuje^λζ, forj=1,2. Equations (4.7) and (4.8) thus reduce to

(iλ−k²)uˆ1+A²(uˆ1+uˆ2)−Aτ∂TA(uˆ1+uˆ2)=0, (4.9) (−iλ−k²)uˆ2+A²(uˆ1+uˆ2)−Aτ∂TA(uˆ1+uˆ2)=0. (4.10) At orderτ⁰, the solutions of equations (4.9) and (4.10) are

ˆ

u1=(k²+ iλ)a(T) and uˆ2=(k²−iλ)a(T), (4.11) with

λ=λ0=k

2A²−k². (4.12) It does not differ from the solution found at this order for A constant in section 3. We seek for a solution of equations (4.7) and (4.8) of the form (4.11) withλ=λ0+τλ1+· · ·. At first order inτ, we computeλ1and get

λ= |k|

2A²−k²− τ|k|A²

√2A²−k²

∂ln(Aa)

∂T . (4.13) The absolute value |k| expresses the fact that, while k can take both signs, we are interested in the exponentially growing solution only, i.e. we requireλ >0. The result obtained for a constant Ahas thus been generalized. However, the exact

Figure 4.One spatial period of the analytic solution after introduction of the retardation induced by the noninstantaneous character of the nonlinearity. (Parameters:κ=π/(2√

2), T/C=0.8.)

resolution of equations (4.5) and (4.6) by means of a Fourier transform for a constant A cannot be generalized. Indeed, although the differential equation is linearized it has variable coefficients in the present case. Thus a convolution arises and prevents us from achieving the computation.

According to (4.4) and (4.11), the perturbation amplitudeg(ζ,x,T)expresses as

g(ζ,x,T)=2(k²+ iλ)a(T)e^λζcoskx. (4.14) We write it as

g=G(T)e^iϕ(T)e^λζcoskx, (4.15) withG(T)= |2(k²+ iλ)a(T)| =2√

2A(T)ka(T)+ O(τ)and ϕ(T)=arg[(k²+ iλ)a(T)].

The correction to the growth rate is thus λ1= − |k|A²

√2A²−k²

∂lnG

∂T . (4.16)

4.2. Time shift

We assume that the initial wave and perturbation amplitudes have the same Gaussian shape:

A= A0e^−T2/s2 and G=G0e^−T2/s2. (4.17) At a given propagation distanceζ, we compute the maximal value of the perturbation amplitude|g|. The positionTmof the maximum of the pulse is a solution of d(Ge^τλ1ζ)/dT =0. We solve this equation using a power series expansion inτ, which yields at leading order

Tm 2A²₀ζτ|k|

2A²₀−k²

. (4.18)

The approximation obtained in this way is checked by considering higher-order terms. It is valid only for small values of the distanceζ. Since the following term in the expansion

Figure 5.One spatial period of the analytic solution, which accounts for the FPU recurrence for instantaneous nonlinearity.

(Parameter:κ=π/(2√ 2).)

is proportional to 1/s², the approximation is better for longer pulses. This corresponds to the experimental situation where the relaxation timeτ is about 5% that of the pulse duration (FWHM) [21].

From equation (4.18) we see that the deviation of the pulse is proportional to|k|. This statement can explain part of the experimental observations. When the noninstantaneous character of the nonlinearity is neglected, the field evolution can be described by means of an analytic solution of the NLS equation which features the FPU recurrence [37, 38]. With the notations of the present paper, this solution is

ψ=

1−

√2κ³

δ coshδζ+ i√

2κsinhδζ

√2κ

δ coshδζ−cosκx

e^{−i arccos(1−κ2)+iζ}, (4.19) with δ = κ√

2−κ². It satisfies the standard NLS, which follows from equation (2.11) withτ =0:

i∂ζψ+∂x²ψ+ψ|ψ|²=0. (4.20) Let us callψ(ζ,˜ k,T)the Fourier transform ofψ relative to the spatial variable x. According to the previous analysis, the effect of the perturbation−τψ^∂|ψ|_∂T² on solution (4.19) of equation (4.20) is to shift each Fourier componentψ(ζ,˜ k,T) towards the front for a time proportional toζ|k|. In a first approximation, ψ˜d(ζ,k,T) = ˜ψ(k,T −Cζ|k|) withC = 2A²₀τ/(√

2A0). Since the analytic solution (4.19) is a function of(x, ζ), we rather useψ˜d(ζ,k,T)= ˜ψ(k, ζ−T/(C|k|)). In figure 4 is presented the result obtained for one spatial period, and forT/C = 0.8. It can be compared to solution (4.19) drawn in figure 5. Increasing the shift parameterT/Cto 1.5, we get the contour plot of figure 6. Intermediate maxima of intensity arise, as observed in experiments. It is worth stressing that, in accordance with solution (4.19) and [21], the case highlighted here corresponds to the induced-MI experimental situation in that only the unstable modes of maximal gain are initially present. Nevertheless, the agreement between these computations and the experimental results allows us to conclude that the mentioned time shift is responsible for the observations.

Figure 6.One spatial period of the analytic solution after introduction of the retardation induced by the noninstantaneous character of the nonlinearity. (Parameters:κ=π/(2√

2), T/C=1.5.)

5. Conclusion

In this paper we provided new insights into the phenomenon of modulation instability (MI) in nonlinear Kerr media. Indeed, we showed how the addition of the temporal transverse dimension due to a weakly-noninstantaneous nonlinearity of the relaxing (Debye) type alters the behaviour of spatial modulation instability of a coherent optical wavepacket in a cavityless Kerr-like medium, i.e. in the travelling-wave propagation regime. First, we considered both harmonic and pulsed perturbations superimposed on a monochromatic light excitation (continuous temporal envelope). As previously shown [36], spatial MI can be overcome due to a high- frequency temporal modulation of the electromagnetic-field envelope: this situation arises because of an absence of any favoured spatial frequency within the unstable-frequency range. However, this suppression of instability occurs on the timescale of the nonlinear response only. Nevertheless, at low-frequency temporal modulation, MI remains whereas an inhibition of the Fermi–Pasta–Ulam recurrence is revealed, thus allowing the stabilization of MI-generated arrays of soliton beams, that is a multi-soliton generation. The latter situation arises because a spatial frequency of maximal gain still exists together with an unstable-frequency range of infinite extent. Second, we considered the above-mentioned perturbations superimposed on a quasi-monochromatic light excitation (here a Gaussian temporal envelope). The analysis showed that the FPU recurrence inhibition is associated with a spatiotemporal dynamics, causing the appearance of a secondary array. The relaxation of the nonlinearity indeed introduces a temporal shift between the different Fourier components that exchange energy during the MI process. These conclusions are in fairly good agreement with recent experiments on spatial MI in a planar relaxing Kerr waveguide [21, 22]. In view of applications to all-optical signal handling, such conclusions reveal that stable periodic arrays of self-induced graded-index waveguides can be generated by way of modulation instability in a relaxing Kerr medium.

Acknowledgments

CC acknowledges Professor Eric Lantz, Dr Mathieu Chauvet, and Dr Herv´e Maillotte for their continuous support during his PhD thesis. He is greatly indebted to them for the work and discussions they shared together during this period.

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