Intensity fluctuations in random waveguides

Intensity fluctuations in random waveguides

Josselin Garnier CMAP, CNRS, Ecole polytechnique, Institut Polytechnique de Paris, 91128 Palaiseau Cedex - France (josselin.garnier@polytechnique.edu). http://www.josselin-garnier.org
Abstract

An asymptotic analysis of wave propagation in randomly perturbed waveguides is carried out in order to identify the effective Markovian dynamics of the guided mode powers. The main result consists in a quantification of the fluctuations of the mode powers and wave intensities that increase exponentially with the propagation distance. The exponential growth rate is studied in detail so as to determine its dependence with respect to the waveguide geometry, the statistics of the random perturbations, and the operating wavelength.

Keywords. Waveguide; wave propagation in random media; diffusion approximation.

AMS subject classifications. 35R60; 35L05; 60F05; 35Q60; 35Q99.

1 Introduction

We consider wave propagation in randomly perturbed waveguides. The random perturbations may affect the index of refraction within the core of the waveguide or the geometry of the core boundary. An asymptotic analysis based on a separation of scales technique can be applied when the amplitude of the random perturbation is small, its correlation length is of the same order as the operating wavelength, and the propagation distance is large so that the net effect of the perturbations is of order one. The overall result is that the scalar wavefield can be expanded on the complete basis of the modes of the unperturbed waveguide, that contains guided modes, radiating modes and evanescent modes, and the complex mode amplitudes of this decomposition follow an effective Markovian dynamics. In particular the guided mode powers form a Markovian process with a generator that describes random exchange of powers between the guided modes and power leakage (towards the radiating modes) that can be expressed as a deterministic mode-dependent dissipation. These results can be found in different forms in the physics literature [18, 7, 5] and in the mathematics literature [15, 10, 13]. In this paper we present a unified framework, we clarify the relationships between the mode-dependent dissipation coefficients and the statistics of the random perturbations, and we give a precise characterization of the mode power fluctuations, which is the main result of the paper and which can be summarized as follows.

The effective Markovian description of the guided mode powers makes it possible to analyze their first- and second-order moments (that are second- and fourth-order moments of the mode amplitudes), which in turn gives a statistical description of the intensity distribution of the wavefield. We find that the relative fluctuations of the intensity are, in general, characterized by an exponential growth with the propagation distance, whose rate can be defined as the difference of the first eigenvalues of two self-adjoint operators. When the effective dissipation is negligible, we recover the well-known equipartition result [8, 10]: The exponential growth rate is zero and the power becomes equipartitioned amongst the guided modes. When there is effective dissipation, the exponential growth rate can be positive, which means that power fluctuations may become very large, as first noticed in the physics literature by Creamer [6]. In fact we show that the exponential growth rate is positive as soon as two effective mode-dependent dissipation coefficients are different. The growth rate increases when the effective mode-dependent dissipation coefficients become more different, and it decreases when the number of guided modes increases. We analyse a special regime, the continuum approximation, in which the operating frequency is large so that the number of guided modes becomes large. Under such circumstances, we find that the exponential growth rate vanishes. The exponential growth of the intensity fluctuations can, therefore, only be observed when there is a limited number of guided modes, and we recover the standard result that, in open random medium, the wavefield behaves like a Gaussian-distributed complex field for large propagation distances and the scintillation index that measures the relative intensity fluctuations becomes equal to one.

The paper is organized as follows. In Section LABEL:sec:intro we formulate the problem and present the waveguide geometry. In Section LABEL:sec:homo we review the spectral analysis of the ideal waveguide, when the medium inside the core is homogeneous and the boundaries are straight. In Section LABEL:sec:random we explain that the wavefield in the random waveguide can be expanded on the set of eigenmodes of the ideal waveguide and we identify the set of coupled equations satisfied by the mode amplitudes. In Section LABEL:sec:effmarkov1 we present the effective Markovian dynamics for the mode amplitudes and in Section LABEL:sec:effmarkov2 we remark that the mode powers also satisfy Markovian dynamics. The long-range behavior of the mean mode powers is described in Section LABEL:sec:effmarkov3, and the fluctuation analysis in Section LABEL:sec:fluctuationanalysis reveals that the normalized variance of the intensity grows exponentially with the propagation distance.

2 Wave propagation in waveguides

Our model consists of a two-dimensional waveguide with range axis denoted by and transverse coordinate denoted by (see Figure LABEL:fig:1). This may model a dielectric slab waveguide for instance. A point-like source at a fixed position transmits a time-harmonic signal. The wavefield satisfies the Helmholtz equation:

 [(∂2x+∂2z)+k2n2(x,z)]p(x,z)=δ(z)δ(x−xs), \hb@xt@.01(2.1)

for , where is the homogeneous wavenumber and is the index of refraction at position .

In the case of ideal (unperturbed) waveguides, the index of refraction is range-independent and equal to

 n(0)(x)={n if x∈(−d/2,d/2),1 otherwise, \hb@xt@.01(2.2)

where is the relative index of the core and is its diameter.

We are interested in randomly perturbed waveguides. In this paper we address two types of random waveguides.

Type I perturbation: in the first type, the index of refraction within the core region is randomly perturbed [2, 5, 4, 13, 15]:

 n(ε)(x,z)={n+εν(x,z) if x∈(−d/2,d/2) and z∈(0,L(ε)),1 otherwise. \hb@xt@.01(2.3)

The fluctuations are modeled by the zero-mean, bounded, stationary in random process with smooth covariance function

 \@fontswitchRI(x,x′,z′)=E[ν(x,z)ν(x′,z+z′)]. \hb@xt@.01(2.4)

It satisfies strong mixing conditions in as defined for example in [19, section 2]. The typical amplitude of the fluctuations of index of refraction is assumed to be much smaller than and it is modeled by the small and positive dimensionless parameter .

Type II perturbation: in the second type (see Figure LABEL:fig:1), the boundaries of the core are randomly perturbed [1, 3, 17, 18, 14]:

 n(ε)(x,z)={n if x∈(\@fontswitchD(ε)−(z),\@fontswitchD(ε)+(z)) and z∈(0,L(ε)),1 otherwise, \hb@xt@.01(2.5)

where

 \@fontswitchD(ε)−(z) =−d/2+εdν1(z), \hb@xt@.01(2.6) \@fontswitchD(ε)+(z) =d/2+εdν2(z). \hb@xt@.01(2.7)

The fluctuations are modeled by the zero-mean, bounded, independent and identically distributed stationary random processes and with smooth covariance function

 \@fontswitchRII(z′)=E[νq(z)νq(z+z′)],q=1,2. \hb@xt@.01(2.8)

They satisfy strong mixing conditions. The typical amplitude of the fluctuations of the boundaries is assumed to be much smaller than the core diameter and it is modeled in (LABEL:eq:Interfaces1a-LABEL:eq:Interfaces1b) by the small and positive dimensionless parameter .

We study the wavefield at , satisfying

 p(x,z)∈\@fontswitchC0((0,+∞),H2(R))∩\@fontswitchC2((0,+∞),L2(R)), \hb@xt@.01(2.9)

and to set radiation conditions, we have assumed that the random fluctuations are supported in the range interval . We will see that net scattering effect of these fluctuations becomes of order one at range distances of order , so we consider the interesting case .

3 Homogeneous waveguide

In this section, we consider an index of refraction of the form (LABEL:eq:no), which is stepwise constant. There is no fluctuation of the medium along the -axis. The analysis of the perfect waveguide is classical [16, 21], we only give the main results. The Helmholtz operator has a spectrum of the form

 (−∞,k2)∪{β2N−1,…,β20}, \hb@xt@.01(3.1)

where the modal wavenumbers are positive and . The generalized eigenfunctions , , associated to the spectral parameter in the continuous spectrum and the eigenfunctions , , associated to the discrete spectrum, are given in Appendix LABEL:app:dec. The generalized eigenfunctions are even and are odd. The eigenfunctions are even for even and odd for odd . Any function can be expanded on the complete set of the eigenfunctions of the Helmholtz operator. In particular, any solution of the Helmholtz equation in homogeneous medium can be expanded as

 p(x,z)=N−1∑j=0pj(z)ϕj(x)+∑t∈{e,o}∫k2−∞pt,γ(z)ϕt,γ(x)dγ. \hb@xt@.01(3.2)

The modes for are guided, the modes for are radiating, the modes for are evanescent. Indeed, the complex mode amplitudes satisfy

 ∂2zpj+β2jpj =0,j=0,…,N−1, \hb@xt@.01(3.3) ∂2zpt,γ+γpt,γ =0,γ∈(−∞,k2), \hb@xt@.01(3.4)

for any . Therefore, if the source is of the form (LABEL:eq:pressure0), we have for :

 p(x,z)= N−1∑j=0aj,s√βjeiβjzϕj(x)+∑t∈{e,o}∫k20at,γ,sγ1/4ei√γzϕt,γ(x)dγ \hb@xt@.01(3.5)

where the mode amplitudes are constant and determined by the source:

 aj,s= √βj2ϕj(xs),j=0,…,N−1, \hb@xt@.01(3.6) at,γ,s= |γ|1/42ϕt,γ(xs),γ∈(−∞,k2),t∈{e,o}. \hb@xt@.01(3.7)

4 Random waveguide

We consider the two types of random perturbations described in Section LABEL:sec:intro. In both cases we can write

where the perturbation is of the form

 V(ε)(x,z)=εν(x,z) \hb@xt@.01(4.1)

for type I perturbations, and

 V(ε)(x,z)= (n2−1)[−1(−d/2,−d/2+εdν1(z))(x)1(0,+∞)(ν1(z)) +1(−d/2+εdν1(z),−d/2)(x)1(−∞,0)(ν1(z))] +(n2−1)[1(d/2,d/2+εdν2(z))(x)1(0,+∞)(ν2(z)) \hb@xt@.01(4.2)

for type II perturbations.

The solution of the perturbed Helmholtz equation (LABEL:eq:pressure0) can be expanded as (LABEL:eq:modalexpansion) and the complex mode amplitudes satisfy the coupled equations for :

 ∂2zpj+β2jpj =−k2N−1∑l=0C(ε)j,l(z)pl−k2∑t′∈{e,o}∫k2−∞C(ε)j,t′,γ′(z)pt′,γ′dγ′, \hb@xt@.01(4.3)

for ,

 ∂2zpt,γ+γpt,γ =−k2N−1∑l=0C(ε)t,γ,l(z)pl−k2∑t′∈{e,o}∫k2−∞C(ε)t,γ,t′,γ′(z)pt′,γ′dγ′, \hb@xt@.01(4.4)

for and , with

 C(ε)j,l(z)= (ϕj,ϕlV(ε)(⋅,z))L2, \hb@xt@.01(4.5) C(ε)j,t′,γ′(z)= (ϕj,ϕt′,γ′V(ε)(⋅,z))L2, \hb@xt@.01(4.6) C(ε)t,γ,l(z)= (ϕt,γ,ϕlV(ε)(⋅,z))L2, \hb@xt@.01(4.7) C(ε)t,γ,t′,γ′(z)= (ϕt,γ,ϕt′,γ′V(ε)(⋅,z))L2, \hb@xt@.01(4.8)

and stands for the standard scalar product in (see (LABEL:scalarproduct)). These equations are obtained by substituting the ansatz (LABEL:eq:modalexpansion) into (LABEL:eq:pressure0) and by projecting onto the eigenmodes.

From the definitions (LABEL:eq:newindrefI) or (LABEL:eq:newindref) of and the Taylor expansions of the eigenfunctions and around , we obtain power series (in ) expressions of the coefficients :

 C(ε)j,l(z)= εCj,l(z)+ε2cj,l(z)+o(ε2), \hb@xt@.01(4.9) Cj,l(z)= \hb@xt@.01(4.10) cj,l(z)= {0 type I(n2−1)d22{−ν21(z)∂x[ϕjϕl](−d2)+ν22(z)∂x[ϕjϕl](d2)} type II, \hb@xt@.01(4.11)

and similarly for , , and .

We finally introduce the generalized forward-going and backward-going mode amplitudes:

 {aj(z),bj(z), j=0,…,N−1}  and  {at,γ(z),bt,γ(z), γ∈(0,k2)}, \hb@xt@.01(4.12)

for , which are defined such that

 pj(z)= 1√βj(aj(z)eiβjz+bj(z)e−iβjz), ∂zpj(z)= i√βj(aj(z)eiβjz−bj(z)e−iβjz),j=0,…,N−1, \hb@xt@.01(4.13)

and

 pt,γ(z)= 1γ1/4(at,γ(z)ei√γz+bt,γ(z)e−i√γz), ∂zpt,γ(z)= iγ1/4(at,γ(z)ei√γz−bt,γ(z)e−i√γz),γ∈(0,k2),t∈{e,o}. \hb@xt@.01(4.14)

We can substitute (LABEL:eq:guidedFBLABEL:eq:radFB) into (LABEL:eq:cma2aLABEL:eq:cma2b) in order to obtain the first-order system of coupled random differential equations satisfied by the mode amplitudes (LABEL:eq:amplitudes):

 ∂zaj(z)= ik22N−1∑l′=0C(ε)j,l′(z)√βl′βj[al′(z)ei(βl′−βj)z+bl′(z)ei(−βl′−βj)z] +ik22∑t′∈{e,o}∫k20C(ε)j,t′,γ′(z)4√γ′√βj[at′,γ′(z)ei(√γ′−βj)z+bt′,γ′(z)ei(−√γ′−βj)z]dγ′ +ik22∑t′∈{e,o}∫0−∞C(ε)j,t′,γ′(z)√βjpt′,γ′(z)e−iβjzdγ′, \hb@xt@.01(4.15)
 ∂zat,γ(z)= ik22N−1∑l′=0C(ε)t,γ,l′(z)4√γ√βl′[al′(z)ei(βl′−√γ)z+bl′(z)ei(−βl′−√γ)z] +ik22∑t′∈{e,o}∫k20C(ε)t,γ,t′,γ′(z)4√γ′γ[at′,γ′(z)ei(√γ′−√γ)z+bt′,γ′(z)ei(−√γ′−√γ)z]dγ′ +ik22∑t′∈{e,o}∫0−∞C(ε)t,γ,t′,γ′(z)4√γpt′,γ′(z)e−i√γzdγ′, \hb@xt@.01(4.16)

with similar equations for and . This system is complemented with the boundary conditions at and :

 aj(0)=aj,s,bj(L(ε))=0,at,γ(0)=at,γ,s,bt,γ(L(ε))=0,

where and are defined by (LABEL:eq:defajs-LABEL:eq:defagammas). The evanescent mode amplitudes , , , satisfy (LABEL:eq:cma2b).

5 The effective Markovian dynamics for the mode amplitudes

We rename the complex mode amplitudes in the long-range scaling as

 aεj(z)=aj(zε2),bεj(z)=bj(zε2),j=0,…,N−1, \hb@xt@.01(5.1) aεt,γ(z)=at,γ(zε2),  bεt,γ(z)=bt,γ(zε2),γ∈(0,k2),t∈{e,o}. \hb@xt@.01(5.2)

We can follow the lines of [13] to get the following results.

1) In the regime the evanescent mode amplitudes, that satisfy (LABEL:eq:cma2b), can be expressed to leading order in closed forms as functions of the guided and radiating mode amplitudes (LABEL:eq:rescAmplitudesa-LABEL:eq:rescAmplitudesb). Indeed it is possible to invert the operator in (LABEL:eq:cma2b) for by using the Green’s function that satisfies the radiation condition and to obtain:

 +∫k20Ct,γ,t′,γ′(z′)4√γ′[aεt′,γ′(z)ei√γ′z′+bεt′,γ′(z)e−i√γ′z′]dγ′}e−√|γ||zε2−z′|dz′ +O(ε2), \hb@xt@.01(5.3)

for and . Here we recognize that is the Green’s function of the equation for .

2) Under the assumption that the power spectral density for type-I perturbations (or for type-II perturbations) has compact support or fast decay, the forward-scattering approximation can be proved, i.e. the coupling between forward-going and backward-going mode amplitudes is negligible, so that we have

 bεj(z)≈0,j=0,…,N−1,bεt,γ(z)≈0,  γ∈(0,k2),  t∈{e,o}.

3) The forward-going guided mode amplitudes and radiating mode amplitudes then satisfy a closed linear system of the form

 d\itbfaεdz=1εF(zε2)\itbfaε+G(zε2)\itbfaε+o(1),

with initial conditions for at . Here , resp. , is an operator with zero mean, resp. non-zero mean, and ergodic properties inherited from those of the processes .

We can finally apply a diffusion approximation theorem to establish the following result (see [13] for the full statement or [15] for a first version in which the contributions of the evanescent modes is neglected, which means that the operator is missing in the expression of the generator ).

Proposition 5.1

The random process

 ((aεj(z))N−1j=0,(aεt,γ(z))γ∈(0,k2),t∈{e,o})

converges in distribution in , the space of continuous functions from to , to the Markov process

 ((aj(z))N−1j=0,(at,γ(z))γ∈(0,k2),t∈{e,o})

with infinitesimal generator . Here is equipped with the weak topology and the infinitesimal generator has the form where , , are the differential operators:

 \@fontswitchL1= 12N−1∑j,l=0Γjl(aj¯¯¯¯¯aj∂al∂¯¯¯¯al+al¯¯¯¯al∂aj∂¯¯¯¯aj−ajal∂aj∂al−¯¯¯¯¯aj¯¯¯¯al∂¯¯¯¯aj∂¯¯¯¯al)1j≠l +12N−1∑j,l=0Γ1jl(aj¯¯¯¯al∂aj∂¯¯¯¯al+¯¯¯¯¯ajal∂¯¯¯¯aj∂al−ajal∂aj∂al−¯¯¯¯¯aj¯¯¯¯al∂¯¯¯¯aj∂¯¯¯¯al) +12N−1∑j=0(Γjj−Γ1jj)(aj∂aj+¯¯¯¯¯aj∂¯¯¯¯aj)+i2N−1∑j=0Γsjj(aj∂aj−¯¯¯¯¯aj∂¯¯¯¯aj), \hb@xt@.01(5.4) \@fontswitchL2= −12N−1∑j=0(Λj+iΛsj)aj∂aj+(Λj−iΛsj)¯¯¯¯¯aj∂¯¯¯¯aj, \hb@xt@.01(5.5) \@fontswitchL3= iN−1∑j=0κj(aj∂aj−¯¯¯¯¯aj∂¯¯¯¯aj). \hb@xt@.01(5.6)

In these definitions we use the classical complex derivative: if , then and , and the coefficients of the operators (LABEL:eq:defL1-LABEL:eq:defL3) are defined for , as follows:

- For all , and are given by

 Γjl= k42βjβl∫∞0\@fontswitchRjl(z)cos((βl−βj)z)dz, \hb@xt@.01(5.7) Γsjl= k42βjβl∫∞0\@fontswitchRjl(z)sin((βl−βj)z)dz, \hb@xt@.01(5.8)

with defined by

 \@fontswitchRjl(z) :=E[Cj,l(0)Cj,l(z)], \hb@xt@.01(5.9) E[Cj,l(0)Cj′,l′(z)] =⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩∫R∫Rϕjϕl(x)\@fontswitchRI(x,x′,z)ϕj′ϕl′(x′)dxdx′ type I(n2−1)2d2[ϕjϕlϕj′ϕl′(−d2)+ϕjϕlϕj′ϕl′(d2)]\@fontswitchRII(z) type II \hb@xt@.01(5.10)

- For all :

 Γ1jl= k42βjβl∫∞0E[Cj,j(0)Cl,l(z)]dz.

- For all , is defined by

 Λj= \hb@xt@.01(5.11)

and

 Γjj= −N−1∑l=0,l≠jΓjl,Γsjj=−N−1∑l=0,l≠jΓsjl, Λsj= ∑t∈{e,o}∫k20k42√γβj∫∞0\@fontswitchRj,t,γ(z)sin[(√γ−βj)z]dzdγ, κj= ∑t∈{e,o}∫0−∞k42√|γ|βj∫∞0\@fontswitchRj,t,γ(z)cos(βjz)e−√|γ|zdzdγ+k22βjE[cj,j(0)],

where is defined as in (LABEL:def:calRjl) upon substitution for and

 E[cj,j(0)]={0 type I(n2−1)d2\@fontswitchRII(0)∂x[ϕ2j](d2) type II

We give some remarks before focusing our attention on the mode powers.

1) The convergence result holds in the weak topology. This means that we can only compute quantities of the form for any test functions and . These quantities are the limits of