Instability of speckle patterns in random media with noninstantaneous Kerr nonlinearity S.E. Skipetrov Laboratoire de Physique et Mod´elisation des Milieux Condens´es CNRS, 38042 Grenoble, France 3 Onset of the instability of a multiple-scattering speckle pattern in a random medium with 0 Kerr nonlinearity is significantly affected by the noninstantaneous character of the nonlinear 0 medium response. The fundamental time scale of the spontaneous speckle dynamics beyond the 2 instability threshold is set by the largest of times T and τ , where T is the time required for D NL D n the multiple-scattered waves to propagate through the random sample and τ is the relaxation NL a time of the nonlinearity. Inertial nature of the nonlinearity should complicate the experimental J observation of theinstability phenomenon. 5 1 Accepted for publication in Optics Letters (cid:13)c 2003 Optical Society of America, Inc. ] n n Instabilities, self-oscillations and chaos are widely en- dom sample (D =cℓ/3 is the diffusion constant and c is - s countered in nonlinear optical systems1,2,3. In particu- the speed of light in the average medium). We derive i lar, the so-called modulation instability (MI), consist- explicit expressions for the characteristic time scale τ of d . ing in the growth of small perturbations of the wave spontaneous intensity fluctuations beyond the instabil- t a amplitude and phase, resulting from the combined ef- itythresholdandthemaximalLyapunovexponentΛmax, m fect of nonlinearity and diffraction, can be observed for characterizing the predictability of the system behavior. - both coherent2,4 and incoherent5,6,7,8,9 light. In the lat- Finally, we discuss the experimental implications of our d ter case, propagation of a powerful, spatially incoherent results. n quasimonochromaticlightbeamthroughanopticallyho- Weconsideraplanemonochromaticwaveoffrequency o mogeneous nonlinear crystal results in ordered or disor- ω incident on a random sample of typical size L ≫ ℓ. c 0 [ deredspatialpatterns5,6,7,8,9. Recently,instabilitieshave The realdielectric constantofthe sample canbe written 1 also been predicted10,11 for diffuse light in random me- asε(r,t)=ε0+δε(r)+∆εNL(r,t),whereε0andδε(r)are dia with Kerr nonlinearity (sample size L ≫ mean free theaveragevalueandthefluctuatingpartofthelineardi- v path ℓ ≫ wavelength λ). The strength of the nonlin- electricconstant,respectively,and∆ε (r,t)denotesthe 2 NL 6 earity has to exceed a threshold for the instability to nonlinearpartofε. Withoutlossofgenerality,weassume 2 develop, similarly to the case of incoherentMI (coherent ε0 =1andhδε(r)δε(r1)i=4π/(k04ℓ)δ(r−r1). Undercon- 1 MI has no threshold), but in contrast to the latter, (a) ditions of weak scattering k ℓ ≫ 1 (where k = ω /c), 0 0 0 0 initially coherent wave loses its spatial coherence inside the average intensity hI(r)i of the scattered wave inside 3 the medium, due to the multiple scattering on hetero- the sample can be described by a diffusion equation, the 0 geneities of the refractive index, (b) temporal coherence short-range correlation function of intensity fluctuations / t ofthe incidentlightisassumedtobe perfect(inthe case δI(r,t) = I(r,t)−hI(r)i decreases to zero at distances a m ofincoherentMI,thecoherencetimeoftheincidentbeam of order λ, and the long-range correlation of δI(r,t) can is much shorter than the response time of the medium), be found from the Langevin equation12 (see Ref. 13 for - d (c) incident light beam is completely destroyed by the a review of wave diffusion in random media): n scattering after a distance ∼ ℓ and light propagation is o diffusive in the bulk of the sample, and (d) scattering ∂ δI(r,t)−D∇2δI(r,t)=−∇·j (r,t). (1) c providesadistributedfeedbackmechanism,absentinthe ∂t ext : v case of MI. i In a nonlinear medium, the random external Langevin X Since the noninstantaneous nature of nonlinearity currents jext(r,t) obey a dynamic equation11 r proved to be important for the development of optical a ∞ instabilities1,2,3,4,5,6,7,8,9, it is natural to consider its ef- ∂ j (r,t) = d3r′ d∆tq(r,r′,∆t) fect on the instability of diffuse waves in random non- ∂t ext Z Z V 0 linear media. Up to now, the latter phenomenon has ∂ ′ × ∆ε (r ,t−∆t), (2) only been studied assuming the instantaneous medium NL ∂t response10,11. In the present letter, we show that while the absolute instability threshold does not depend on wheretheintegrationisoverthevolumeV oftherandom the nonlinearity response time τ , the dynamics of the medium and q(r,r′,∆t) is a random, sample-specific re- NL speckle pattern above the threshold is extremely sensi- sponsefunctionwithzeroaverageandacorrelationfunc- tivetotherelationbetweenτ andthetimeT =L2/D tion given by the diagrams of Fig. 1. NL D thattakesthediffusewavetopropagatethroughtheran- Supplemented by a Debye-relaxation relation for 2 FIG. 1: Diagrams contributing to the correlation function FIG. 2: Instability threshold as a function of the excitation q(i)(r,r′,t−t′)q(j)∗(r ,r′,t −t′) oftherandomresponse frequency Ω (in units of T−1) for several ratios of the non- 1 1 1 1 D D E linearity relaxation time τ and the time T required for functions q entering into Eq. (2). Solid lines represent the NL D light todiffusethroughtherandom sample. Dashed linecor- retarded and advanced Green’s functions of the linear wave responds to τ /T =0. Inset: Lyapunov exponent Λ vs. Ω equation,dashedlinesdenotescatteringofthetwoconnected (both in unitsNLof TD−1) slightly above the threshold (p=1.1) wave fields on the same heterogeneity. Wavy lines are k2 D 0 for the same values of τ /T as in themain plot. vertices. NL D ∆ε (r,t): NL For givenΩ andp, Eq.(4) determines the correspond- ∂ ing value of the Lyapunov exponent Λ. Λ > 0 signifies τ ∆ε (r,t)=−∆ε (r,t)+2n I(r,t), (3) NL NL NL 2 the instability of the speckle pattern with respect to the ∂t excitations at frequency Ω. If p < 1, then Λ < 0 for where n2 is the nonlinear coefficient, Eqs. (1) and (2) all Ω. As p exceeds 1, Λ becomes positive inside some can now be used to perform a linear stability analysis frequency band (0,Ω ) (see the inset of Fig. 2). We max of the multiple-scattering speckle pattern in a random show the frequency-dependent threshold following from medium with noninstantaneous nonlinearity. Assuming Eq. (4) in Fig. 2 for different values of τ /T . The NL D δI(r,t)=δI(r,α)exp(αt) with α=iΩ+Λ (Ω>0), and threshold value of p at Ω&τ−1 deviates from the result NL similarly for jext(r,t) and ∆εNL(r,t), we use Eqs. (2) obtained for the instantaneous nonlinearity (dashed line and (3) to express the correlation function of Langevin inFig.2),whiletheabsolute thresholdisp=1atΩ=0, currents je(xi)t(r,α)je(xj)t∗(r1,α) in terms of the intensity independent of τNL/TD. correlatioDn function hδI(r,α)δEI∗(r ,α)i and Eq. (1) to To study the onset of the instability, we consider the 1 limit of 0 < p −1 ≪ 1 in more detail. In this limit, express the latter correlation function in terms of the a series expansion of the function F in Eq. (4) can be former. For large enough sample size L/ℓ ≫ k ℓ and 0 found: F(x,y) ≃ 1+ C x2 +y(C − C x2), where we moderate frequency ΩT ≪ [L/(k ℓ2)]2 we can neglect 1 2 3 D 0 keep only the terms quadratic in x and linear in y, and the short-range correlation of intensity fluctuations and the numerical constants are C ≃ 0.031, C ≃ 0.59, write the condition of consistency of the two obtained 1 2 C ≃0.045. At a given frequency Ω, the threshold value equations[andhencetheconditionofconsistencyofEqs. 3 of p then becomes p ≃ 1 + C +(τ /T )2 (ΩT )2, (1–3)] as 1 NL D D while at a given p the maxim(cid:2)um excited freq(cid:3)uency is p≃F(ΩTD,ΛTD)H(ΩτNL,ΛτNL), (4) Ωmax = TD−1 C1+(τNL/TD)2 −1/2(p − 1)1/2 and the shortest typica(cid:2)l time of spont(cid:3)aneous intensity fluctua- where a numerical factor of order unity has been omit- tions is τ =1/Ωmax. ted, p =h∆n i2(L/ℓ)3 is the effective nonlinearity pa- As follows from the above analysis, a continuous low- NL rameter, h∆n i = n hI(r)i is the average value of the frequency spectrum offrequencies (0,Ω ) is excitedat NL 2 max nonlinear part of refractive index, and hI(r)i is assumed p > 1, and the Lyapunov exponent Λ decreases mono- to be approximately constant inside the sample. The tonically with Ω (see the inset of Fig. 2). This allows function F in Eq. (4) is the same as in the case of in- us to hypothesize that at p = 1 the speckle pattern un- stantaneousnonlinearity11,whileH(x,y)=x2+(1+y)2 dergoes a transition from a stationary to chaotic state. originates from the noninstantaneous nature of the non- Sucha behavior shouldbe contrastedfromthe “routeto linear response. In the limit of τ → 0 (instantaneous chaos” through a sequence of bifurcations, characteris- NL nonlinearity), H ≡ 1 and Eq. (4) reduces to Eq. (7) of tic of many nonlinear systems1,2,3. Given initial condi- Ref. 11. tions at t = 0 and deterministic dynamic equations, the 3 evolution of a chaotic system can only be predicted for instability threshold for Ω & τ−1, as we show in Fig. 2. NL t < 1/Λ . We find the maximal Lyapunov exponent Thisphenomenonissimilarinitsorigintotheincreaseof max Λ =T−1(p−1)/(C +2τ /T ). Itisnoweasytosee thespeedofcoherentMIpatternswithdecreasingτ in max D 2 NL D NL thatthefundamentaltimescaleofthespecklepatterndy- ahomogeneous nonlinearmedium4. Similarlytothecase namics at p>1 is set by the largestof the times T and of coherent MI4, the instability is arrested in the limit D τ . Indeed, in the case of fast nonlinearity (τ ≪T ) of τ → ∞, in contrast to the case of incoherent MI, NL NL D NL we have Ω ∝ T−1(p−1)1/2, τ ∝ T (p−1)−1/2, and where τ →∞ (more precisely, τ ≫ than the coher- max D D NL NL Λ ∝T−1(p−1). In the opposite limit of slow nonlin- encetimeoftheincidentbeam)isanexplicitassumption max D ear response (τ ≫T ) we find Ω ∝ τ−1(p−1)1/2, of the theoretical models5,6,9. NL D max NL τ ∝τ (p−1)−1/2,andΛ ∝τ−1(p−1),respectively. NL max NL The origin of the instability of the speckle pattern To conclude, we discuss the experimental implications I(r,t) in a nonlinear random medium and its sensitivity of our results. ∆n up to 10−3 can be realized in in- NL to τ can be qualitatively understood by considering I organicphotorefractivecrystals6,7,8. Ifscatteringcenters NL asaresultofinterferenceofmanypartialwavestraveling (opticaldefects)areintroducedinthe crystal,L/ℓ∼102 alongvariousdiffusionpaths. An infinitely smallpertur- will suffice to reach the instability threshold p ≃ 1. Al- bationofI atsomepointr attimet diffusesthroughout ternatively, in nematic liquid crystals both nonlinear- 0 0 the medium [according to Eq. (1)] and, after some time ity (∆n up to 0.1, see, e.g., Ref. 14) and scattering NL ∆t,modifiesthespecklepatterninsideasphereofradius (L/ℓ > 10 has been reported in Ref. 15) can be suffi- R∼(D∆t)1/2 around r . Due to the nonlinearity of the cientlystrongtoobtainp≃1. Thenonlinearityisrather 0 medium, this leads to a change of the nonlinear part of slow in the above cases, τ exceeds T (T ∼10÷100 NL D D thedielectricconstant∆ε insidethesamesphereafter ns for ℓ ∼ 0.1÷1 mm and L ∼ 1 cm), and the results NL atime∼τ [accordingtoEq.(3)]and,consequently,to obtained in the present letter are, therefore, of partic- NL a modification of the phases of the partial diffuse waves ular relevance. Finally, although the noninstantaneous that interfere to produce a new speckle pattern I(r,t). nature of nonlinearity does not affect the absolute insta- If the strength of the nonlinearity exceeds the threshold bility threshold, it should complicate the experimental given by Eq. (4), I(r ,t)−I(r ,t ) can be larger than observation of the instability phenomenon. Indeed, in a 0 0 0 the initial perturbation. The latter is hence amplified real experiment only sufficiently fast fluctuations can be and the speckle pattern develops spontaneous dynam- detected. 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