Mesoscopic sensitivity of speckles in disordered nonlinear media to changes of disordered potential.pdf

arxiv:cond-mat/9903386v1 cond-mat.mes-hall 26 mar 1999 mesoscopic sensitivity of speckles in disordered nonlinear media to changes of disordered potential b.spivak physics department, university of washington, seattle, wa 98195, usa a.zyuzin a.f.ioff e institute, 194021 st.petersburg, russia abstract we show that the sensitivity of wave speckle patterns in disordered non- linear media to changes of scattering potential increases with sample size. for large enough sample size this quantity diverges, which implies that at given coherent wave incident on a sample there are multiple solutions for the spatial distribution of the waves density. the number of solutions increases exponentially with the sample size. suggested pacs index category: 05.20-y, 82.20-w typeset using revtex 1 if a coherent wave described by a fi eld (r,) propagates in an elastically scattering medium, the spatial dependence of its ”density” n(r,) = |(r,)|2exhibits speckle: n(r,) is a random, sample specifi c function of coordinate r. here is the waves energy. in the cases of noninteracting electrons and electromagnetic waves propagating in linear media the theory of sensitivity of speckle patterns to a change in scattering potential was developed long ago 15 . it was shown that the sensitivity is very large, but fi nite. in this article we consider the same question in the case where a wave propagates in non- linear media. for the sake of concreteness we consider the situation where the propagation of the wave is described by a nonlinear schrodinger equation ( 1 2m 2 r2 + u(r) + u(r)(r,) = 0(1) here m is the wave mass, u(r) = n(r) is the eff ective nonlinear potential and u(r) is a scattering potential which is a random function of the coordinates.similar equations appear in the theory of electromagnetic waves propagating in nonlinear media 6, the theory of hydrodynamic turbulence 7, and the theory of turbulent plasma8. we will assume white noise statistics in u(r): hu(r)i = 0 , hu(r)u(r1)i = lm2(rr1). here brackets hi correspond to averaging over realizations of u(r) and l is the elastic mean free path (l k1= (2m) 1 2). let us consider the case where a coherent wave 0(r) = n 0exp(ikr) with momentum k is incident on a disordered sample of the thickness l l (see the insert in fig.1). we will show that the sensitivity of the nonlinear speckle pattern n(r) to a small change in u(r) increases with sample size l. at arbitrarily small n0and for arbitrary sign of the sensitivity become infi nite provided l is large enough. this implies that at given coherent wave incident on a sample eq.1 has many solutions. this is very diff erent from the case of uniform nonlinear media, where types of instabilities depend on the sign of . (see, for example, 6.) the r dependence of the average density hn(r,)i can be described by the diff usion equation, which is equivalent to calculation of the diagrams shown in fig.2.a. we use the usual diagram technique for averaging over realizations of random potential 9. if |n0| 2 q k lm one can neglect the nonlinear corrections to the diff usion coeffi cient d = lk 3m. we obtained this criterion by calculating the transport scattering cross section on the eff ective potential n(r). to do so we used calculated in 11,4 spatial correlation functions of the density 10. in the case of the sample geometry shown in the insert of fig.1, we have hn(r)i = n0. we can characterize the speckle pattern n(r) and its sensitivity to a small change of scat- tering potential u(r) = u(r)u(r) by correlation functions hn(r)n(r1)i and k(r,r1) = hhn(r)n(r1)ii.here n(r) = n(r) hn(r)i; n(r) = n(r,u(r) n(r,u(r), n(,r,u(r) and n(,r,u(r) are solutions of eq.1 with scattering potentials u(r) and u(r) respectively, and the brackets hhii correspond to averaging over both realizations of u(r) and realizations of u(r). we will assume that hhu(r)u(r1)ii = u2exp(|rr1| r0 ). to get the value of k(r,r1) at |r r1 | l and in the fi rst in u(r) approximation one can generalize the langevin approach for calculations of mesoscopic fl uctuations 4,11 to include nonlinear eff ects d dr j(r) = 0;j(r) = d d dr n(r) + jext(r);(2) jext(r,u(r) = jext(r,u(r) + z dr jext(r) u(r) (u(r) + u(r)(3) hji ext(r)j j ext(r1)i = 2l 3m2 hn(r)i2(r r1)ij(4) hj i ext(r) u(r) jj ext(r1) u(r1) i = 6 lk2 ij(r r1)g(r,r1)hn(r)i (hn(r1)ig(r,r) + hn(r)ig(r1,r) hn(r)ihn(r1)ig(r,r)g(r1,r)(5) u(r) = u(r) u(r) = n(r)(6) where g(r,r1) is the green function of the equation d2 d2r g(r,r1) = (r r1);(7) 3 j(r) = 1 2mim (r)d dr(r) is the current density, j(r) = j(r)hj(r)i, jext(r,u(r) is a random external current source, hjext(r)i = hjext(r) u(r) i = hji ext(r) jj ext(r ) u(r1) i = 0, and i,j are the coordinates indices. eqs.2,3,7 require the usual diff usion boundary conditions: g(r,r) = 0, = n0at x = 0 and n rg(r,r ) = n d rhn(r)i = n jext(r) = 0 at the closed samples boundaries. here n is a the unit vector normal to the boundary. eqs.2-7 are a closed system which diff er from 4,11 by the term in eq.3 proportional to . they are equivalent to the summation of diagrams shown in fig.2b-g. diagrams, shown in fig.2h, are responcible for the small nongaussian part of the distribution function of jext(r) u(r) . they are proportional to a small parameter 1 k2ll 1 in the three dimentional case (d=3) and can be neglected. all diagrams responcible for localisation eff ects can be neglected as well. let us fi rst consider the linear case = 0, u(r) = 0. index (0) will indicate quantities calculated at = 0. solving eqs.2-5,7 at |r r1| l in d = 3 case we get 4 k(0)(r,r1) = hhn(0)(r)n(0)(r1)ii (d f )2 n2 0 k2l|r r1|( d f )2(8) where d= l2 d , and 1 f = r0u l characterizes the change in scattering potential. this can also be obtained by calculating the diagrams shown in fig.2b,c. the characteristic time (0) f l2 d corresponds to a complete change in the speckle pattern due to the change of the scattering potential u(r). one can get the same estimate from the requirement that an additional phase (0) r l2 d(0) f , which the traveling wave aquires due to the change in the potential u(r), is of order . if impurities have a cross-section of order 1 k2 and they are shifted from their initial position by distances of order 1 k then in the d = 3 case one has to change the positions of the n(0)= llk2impurities to in order change the speckle pattern signifi cantly 2. characteristic changes of energy (0) = d l2 and of the angle of incidence (0)= 1 kl (see the insert in fig.1), which change the speckle pattern signifi cantly, can be obtained in a similar way 1,4. let us now turn to the case 6= 0. expanding eqs.2-6 up to second order in , and performing the average over realizations of u(r) and u(r) in the d=3 case we get the 4 correction to k(0)(r,r1). k(1)(r,r1) k(0)(r,r1)(9) where = (3 2 n0 )2(l l )3(10) the index (1) indicates quantities proportional to 2. it can also be obtained by calculating the diagrams shown in fig.2d-g or by estimating the additional phase which the wave trav- eling along a typical diff usion path will pick up due to the change in the eff ective potential u(1)(r) = n(0)(r) hh(1)2ii = (k 2 )2hh z dsds1n(0)(r(s)n(0)(r(s1)ii hh(0)2ii(11) here integration is taken along typical diff usion paths of length l2 l . eqs.9,11 imply that hh(1)2ii hh(0)2ii and that eq.1 has many solutions. let us estimate the number of the solutions in the d=3 case. it is convenient to expand u(r) = d l x m m 1 3 umnm(r) (12) over a complete set of eigenstates nm (r) of diff usion equation eq.7 d d2 d2r nm(r) = emnm(r)(13) where em 1 d m 2 3are eigenvalues of eq.13 and m = 1,2. labels the eigenmodes. let us fi rst regard umas independent variables. the solution of eq.1 can be written as n(r) = n(r, u1, uk.). than using the selfconsistency equation u = n we get 1 2m 2 3 um= fm( u1, uk.) (14) where fm( u1,.) = kl1n1 0 m 1 3l 1 2 r drn(r, u1.)nm (r) are random sample specifi c functions. the problem of the investigation of properties of fm( u1,.) as a function of unis equiv- alent to the linear problem considered in 15. to characterize the dimensionless functions 5 fmwe calculate the following correlation functions with the help of eqs.2-5,7: (a)mesoscopic fl uctuations of modes with m 6= n are uncorrelated hfmfni = 0, where fm= fmhfmi; (b) h(fm)2i 1; and (c) hfm( u1, un+ un,) fm( u1 un,)2i h(fm)2i ( un)2.(15) eq.15 means that the characteristic period of random oscillations of fmas a function of un is of order unity, un 1. in anticipation of these results we have introduced the factor m 1 3in eq.12. it is important that eqs.14 with large enough m m = 3 4have unique solutions um m 2 3 1, provided the values um1 = 0 in the fi rst of eqs.14 for m = 1 and get the equation 1 2 u1= f1( u1,0,0,) (16) it is equivalent to substitution in eq.1 u(r) d l u1n1(r). (then expanding eq.1 with respect to powers of one can reproduce the values of the diagrams fig.2b-g with the precision of the factor of order of unity.) in fig.1 we show a qualitative ”graphical” solution of eq.16 which corresponds to intersection of two functions:f1( u1,0,0) and 1 2 u1. it follows from fig.1 that at 1 both eq.16 with a typical realization of the potential u(r), and consequently eq.1, have many solutions. in this case the sensitivity defi ned as ( f d) 2k(r,r) diverges. the main contribution to this divergency comes from realizations of u(r), when f1( u1,0,0) and 1 2 u1are tangent to each other. the criterion 1 is equivalent to the inequality h( l3