Table Of ContentInstability 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. Inasmuch as Ω decreases with τ , larger
max NL
ics. Noninstantaneous nature of the nonlinearity leads p (and thus stronger nonlinearities) will be required for
to the damping of the high-frequency components in the the instability to be detectable in a medium with nonin-
spectrumoftheinitialintensityperturbation,raisingthe stantaneous nonlinearity.
1 H.M. Gibbs, “Optical Bistability: Controlling Light With 9 B. Hall, M. Lisak, D. Anderson, R. Fedele, and V.E. Se-
Light” (Academic Press, Orlando, 1985). menov,“Statisticaltheoryforincoherentlightpropagation
2 G.P.Agrawal,“NonlinearFiberOptics”(AcademicPress, in nonlinear media”, Phys.Rev.E 65, 035602 (2002).
San Diego, 1995). 10 S.E. Skipetrov and R. Maynard, “Instabilities of waves in
3 F.T. Arecchi, S. Boccaletti, P. Ramazza, “Pattern forma- nonlineardisorderedmedia”,Phys.Rev.Lett.85,736–739
tionandcompetitioninnonlinearoptics”,Phys.Rep.318, (2000) (see also cond-mat/0006136).
1–83 (1999). 11 S.E.Skipetrov,“Langevindescriptionofspeckledynamics
4 M.-F. Shih,C.-C. Jeng, F.-W.Sheu,and C.-Y.Lin, “Spa- in nonlinear disordered media”, Phys. Rev. E 67, 016601
tiotemporalopticalmodulationinstabilityofcoherentlight (2003) (see also cond-mat/0206475).
in noninstantaneous nonlinear media”, Phys. Rev. Lett. 12 A.Yu. Zyuzin and B.Z. Spivak, “Langevin description of
88, 133902 (2002). mesoscopic fluctuations in disordered media” Sov. Phys.
5 M. Soljacic, M. Segev, T. Coskun, D.N. Christodoulides, JETP 66, 560–566 (1987).
and A.Vishwanath, “Modulation instability of incoherent 13 M.C.W.vanRossumandTh.M.Nieuwenhuizen,“Multiple
beams in noninstantaneous nonlinear media”, Phys. Rev. scattering of classical waves: microscopy, mesoscopy, and
Lett. 84, 467–470 (2000). diffusion”, Rev. Mod. Phys. 71, 313–371 (1999).
6 D. Kip, M. Soljacic, M. Segev, E. Eugenieva, and D.N. 14 R. Muenster, M. Jarasch, X. Zhuang, and Y.R. Shen,
Christodoulides, “Modulation instability and pattern for- “Dye-induced enhancement of optical nonlinearity in liq-
mation in spatially incoherent light beams”, Science 290, uids and liquid crystals”, Phys. Rev. Lett. 78, 42–45
495–498 (2000). (1997).
7 J. Klinger, H. Martin, and Z. Chen, “Experiments on in- 15 M.H. Kao, K.A. Jester, A.G. Yodh, and P.J. Collings,
duced modulational instability of an incoherent optical “Observationoflightdiffusionandcorrelationtransportin
beam”, Opt.Lett. 26, 271–273 (2001). nematic liquid crystals”, Phys. Rev. Lett. 77, 2233–2236
8 Z. Chen, S.M. Sears, H. Martin, D.N. Christodoulides, (1996).
andM.Segev,“Clustering ofsolitons inweaklycorrelated
wavefronts”, Proc. Nat. Acad. Sci. 99, 5223–5227 (2002).
