ELECTROMAGNETIC WAVE PROPAGATION IN RANDOMWAVEGUIDES

RICARDO ALONSO⇤AND LILIANA BORCEA†

Abstract. We study long range propagation of electromagnetic waves in random waveguideswith rectangular cross-section and perfectly conducting boundaries. The waveguide is filled with anisotropic linear dielectric material, with randomly fluctuating electric permittivity. The fluctuationsare weak, but they cause significant cumulative scattering over long distances of propagation ofthe waves. We decompose the wave field in propagating and evanescent transverse electric andmagnetic modes with random amplitudes that encode the cumulative scattering e↵ects. They satisfya coupled system of stochastic di↵erential equations driven by the random fluctuations of the electricpermittivity. We analyze the solution of this system with the di↵usion approximation theorem,under the assumption that the fluctuations decorrelate rapidly in the range direction. The result isa detailed characterization of the transport of energy in the waveguide, the loss of coherence of themodes and the depolarization of the waves due to cumulative scattering.

Key words. Waveguides, electromagnetic, random media, asymptotic analysis.

AMS subject classifications. 35Q61, 35R60

1. Introduction. We study electromagnetic wave propagation in waveguides.There is extensive applied literature on this subject [18, 19, 17, 11, 5, 20] whichincludes open and closed waveguides, waveguides with losses, boundary corrugationand heterogeneous media. Here we consider the setup illustrated in Figure 1.1, for awaveguide with rectangular cross-section ⌦ = (0, L

1

)⇥ (0, L2

), filled with an isotropiclinear dielectric material. The waves are trapped by perfectly conducting boundariesand propagate in the direction z. In the applied literature this is commonly calledthe range direction. The coordinates in the plane orthogonal to z, denoted by x =(x

1

, x2

) 2 ⌦, are called cross-range. The main goal of the paper is to analyze longrange wave propagation in waveguides with imperfections. We refer to [15, 6, 9, 1, 2]and [7, Chapter 20] for rigorous mathematical studies of long range wave propagationin imperfect acoustic waveguides, and to [4, 3, 8] for their application to imagingand time reversal. We also refer to the rigorous study [14] of di↵usion of power ofa randomly perturbed Schroedinger equation with self-adjoint operator that admitsboth bound and radiation modes. It is related to the problem of wave propagation inwaveguides, and it presents a complementary probabilistic treatment to that in theworks cited above. Here we extend the theory to electromagnetic waves.

We focus attention on waveguides with imperfections due to a heterogeneousdielectric, but the ideas should extend to waveguides with corrugated boundaries.Such waveguides can be analyzed by changing coordinates to flatten the boundaryfluctuations as was done in [1] for sound waves, or by using so-called local normal modedecompositions as proposed in [19, chapter 9]. Our waveguide has straight walls andis filled with a dielectric material that has numerous inhomogeneities (imperfections).These are weak scatterers, so their e↵ect is negligible in the vicinity of the source ofthe waves. However, the inhomogeneities cause significant cumulative wave scatteringover long ranges. To quantify the cumulative scattering e↵ects we study the followingquestions: How are the modal wave components coupled by scattering? How do the

⇤Departamento de Matemática, PUC–Rio & Computational and Applied Mathematics, RiceUniversity, Houston, TX 77005. ralonso@mat.puc-rio.br

†Department of Mathematics, University of Michigan, Ann Arbor, MI 48109. borcea@umich.edu

1

x2

z

x1

L2

L1

Fig. 1.1. Schematic of the setup. The waveguide is unbounded in the range direction z and hasrectangular cross-section in the plane (x1, x2), with sides L1 and L2.

waves depolarize? How do the waves lose coherence? Can we calculate from firstprinciples the scattering mean free paths, which are the range scales over which themodal wave components lose coherence? How is energy transported at long rangesin the waveguides? Can we quantify the equipartition distance where cumulativescattering is so strong that the waves lose all information about the source? Howdoes the equipartition distance compare with the mode dependent scattering meanfree paths?

To answer these questions we model the scalar valued electric permittivity " ofthe dielectric as a random process. The random model is motivated by the fact that inapplications the imperfections can never be known in detail. They are the uncertainmicroscale of the medium, the fluctuations of "(~x) in ~x = (x, z), so we model them asrandom. The fluctuations are small, on a scale (correlation length) comparable to thewavelength. We assume that there is no dissipation in the medium, meaning that "(~x)is real, positive. Complex valued permittivities "(!,~x) which are typically required bycausality i.e., Kramers-Kronig relations, can be incorporated in the model. We do notconsider them here for simplicity, and because we are concentrating on the analysisat a single frequency !. Extensions to multi frequency analysis of wave propagationin dispersive and lossy media can be done, using techniques like in [6, 15, 1] and [7,chapter 20], but we leave them for a di↵erent publication.

The paper is organized as follows: We begin in section 2 with the setup. We stateMaxwell’s equations and the boundary conditions satisfied by the electromagneticfield. Then we follow the approach in [18] and solve for the components Ez(!,~x) andHz(!,~x) of the electric and magnetic fields in the range direction. We obtain a 4⇥ 4system of partial di↵erential equations for the components E(!,~x) and H(!,~x) ofthe fields in the cross-range plane. We analyze in section 3 its solution Eo(!,~x) andHo(!,~x) in ideal waveguides with constant permitivity "o. It is a superposition ofuncoupled transverse electric and transverse magnetic modes. The random model ofthe waveguide is introduced in section 4. Because the amplitude of the fluctuationsof "(~x) is small, of order ✏⌧ 1, the system of equations for E(!,~x) and H(!,~x) is aperturbation of that in ideal waveguides. The remainder of the paper is concerned withthe asymptotic analysis of E(!,~x) and H(!,~x) at long ranges, in the limit ✏! 0. Weconsider long ranges because the ✏! 0 limit of E(!,~x) and H(!,~x) is the same as theideal waveguide solution Eo(!,~x) and Ho(!,~x) when the waves do not propagate farfrom the source. Our analysis is based on the decomposition of E(!,~x) and H(!,~x)in transverse electric and magnetic modes, with random amplitudes that encode thecumulative scattering e↵ects, as explained in section 5. The long range scaling andthe di↵usion limit approximation for analyzing the wave field as ✏ ! 0 are stated insection 6. The main results of the paper are in section 7, where we characterize the

2

limit process. Explicitly, we describe the loss of coherence and depolarization of thewaves due to cumulative scattering, and the transport of energy. We also show thatas we let the range grow, the waves scatter so much that they eventually reach theequipartition regime, where they lose all information about the source. We end witha summary in section 8.

2. Setup. Let ~e1

, ~e2

and ~ez be the unit vectors along the coordinate axes, anduse bold letters with an arrow on top for three dimensional vectors, and bold lettersfor two dimensional vectors in the cross-range plane. Exlicitly, we write

~H = H1

~e1

+ H2

~e2

+ Hz~ez , H = (H1, H2) , (2.1)

for the magnetic field ~H(!,~x), and similarly for the electric field ~E(!,~x) and electricdisplacement ~D(!,~x). They satisfy Maxwell’s equations

~r⇥ ~H(!,~x) = ~J (!,~x) � i!~D(!,~x) , (2.2)~r⇥ ~E(!,~x) = i!µo ~H(!,~x) , (2.3)~r · ~H(!,~x) = 0 , (2.4)~r · ~D(!,~x) = ⇢(!,~x) , (2.5)

where ~J and ⇢ are the current source density and free charge density, and µo is themagnetic permeability, assumed constant. We denote by

~r = @x1

~e1

+ @x2

~e2

+ @z~ez

the three dimensional gradient and by ~r⇥ and ~r· the curl and divergence operators.The current source density

~J (!,~x) = (J (!,~x),Jz(!,~x)) = (J(!,x), Jz(!,x)) �(z) , (2.6)

models a source at the origin of range, supported in the interior of ⌦. The Fouriertransform ⇢(!,~x) of the free charge density can be obtained from the continuity ofcharge derived from (2.2-2.5)

� i!⇢(!,~x) + ~r · ~J (!,~x) = 0 . (2.7)

It vanishes at ranges z 6= 0.The electric displacement is proportional to the electric field

~D(!,~x) = "(~x)~E(!,~x) , (2.8)

with scalar valued, positive and bounded electric permittivity ". The analysis is fora single frequency, so we simplify the notation by omitting henceforth ! from thearguments of the fields.

2.1. The 4 ⇥ 4 system of equations. We study the evolution of the two di-mensional vectors E(~x) and H(~x) for z > 0. They determine the components Ez(~x)and Hz(~x) in the range direction of the electric and magnetic fields, as follows fromequations (2.2-2.3)

Hz(~x) = �i

!µor? ·E(~x) , (2.9)

Ez(~x) =i

!"(~x)

⇥

r? ·H(~x) � Jz(~x)⇤

, (2.10)

3

with r? = (�@x2

, @x1

) the perpendicular gradient in the cross-range plane. The 4⇥4system of equations for E(~x) and H(~x) is

@zE(~x) =i

!r

1

"(~x)r? ·H(~x)

�

� i!r

Jz(~x)"(~x)

�

� i!µoH?(~x) , (2.11)

@zH(~x) = �i

!µor⇥

r? ·E(~x)⇤

+ i!"(~x)E?(~x) �J ?(~x) . (2.12)

Here r = (@x1

, @x2

) is the gradient in the cross-range plane, and we let a? = (�a2

, a1

)denote the rotation of any vector a = (a

1

, a2

) by 90 degrees, counter-clockwise.Note that equations (2.9-2.12) contain all the information in the Maxwell system

(2.2-2.5). Indeed, (2.4) follows from (2.9) and (2.11)

~r · ~H(~x) = r ·H(~x) + @zHz(~x)

= r ·H(~x) � i!µo

r? · @zE(~x)

= r ·H(~x) �r? ·H?(~x)= 0 ,

because r? ·ra(~x) = 0 for any twice continuously di↵erentiable function a(~x). Sim-ilarly, (2.5) follows from (2.10) and (2.12)

~r · ~D(~x) = r ·D(~x) + @zDz(~x)

= r · ["(~x)E(~x)] + i!

⇥

r? · @zH(~x) � @zJz(~x)⇤

= r · ["(~x)E(~x)] �r? ·⇥

"(~x)E?(~x)⇤

� i!

h

r? ·J ?(~x) + @zJz(~x)i

= � i!~r · ~J (~x)

= ⇢(~x) ,

where we used (2.8) and the continuity of charge relation (2.7).

2.2. Boundary conditions. The boundary conditions at the perfectly conduct-ing boundary @⌦ are [12, Chapter 8]

~n(x) ⇥ ~E(~x) = 0 (2.13)

for ~x = (x, z) and x 2 @⌦. The outer normal ~n(x) = (n(x), 0) at @⌦ is independentof the range and is orthogonal to ~ez. Thus, equations (2.13) say that the tangentialcomponents of the electric field vanish at the boundary. Explicitly,

Ez(~x) = 0 , n?(x) ·E(~x) = 0 . (2.14)

We need more boundary conditions at @⌦ to specify uniquely the solution of (2.11-2.12), but they can be derived from Maxwell’s equations (2.2-2.3), conditions (2.14),and our assumptions on the source density (2.6), as explained in section 3.

The fields are bounded and outgoing at |z| ! ±1. We explain in section 4.1 thatthe causality of the problem in the time domain allows us to restrict the fluctuationsof "(~x) to a finite range interval, and thus justify the outgoing boundary conditions.

4

2.3. Conservation of energy. The fields E(~x) and H(~x) satisfy an energyconservation relation, stated in the following proposition, and used in the analysis insection 7.

Proposition 2.1. For any z > 0, we have the conservation relation

S(z) = �Z

⌦

dx Reh

E(~x) ·H?(~x)i

= S(0+). (2.15)

where the bar denotes complex conjugate.

Note that

~S(~x) =1

2Re

h

~E(~x) ⇥ ~H(~x)i

is the time average of the Poynting vector of a time harmonic wave [12, chapter 7].Therefore,

S(z) = 2Z

⌦

dx~ez · ~S(~x)

is twice the flux of energy in the range direction, and (2.15) states that it is conservedfor all z > 0.

To derive (2.15) we obtain from (2.2-2.3) that

~r ·h

~E(~x) ⇥ ~H(~x)i

= ~H(~x) ·h

~r⇥ ~E(~x)i

� ~E(~x) ·h

~r⇥ ~H(~x)i

= i!µo�

�

�

~H(~x)�

�

�

2

� i!"(~x)�

�

�

~E(~x)�

�

�

2

� ~E(~x) · ~J (~x) ,

and from the divergence theorem that

Z

⌦

dx~r ·h

~E(~x) ⇥ ~H(~x)i

=

Z

@⌦ds(x) ~n(x) ·

n

(I � ~ez~eTz )h

~E(~x) ⇥ ~H(~x)io

+Z

⌦

dx @zn

~ez ·h

~E(~x) ⇥ ~H(~x)io

.

The boundary term vanishes because of the boundary conditions (2.14)

~n(x) · (I � ~ez~eTz )h

~E(~x) ⇥ ~H(~x)i

= Ez(~x)n(x) ·H?(~x) + Hz(~x)n?(x) ·E(~x) = 0,

and the integrand in the second term satisfies

~ez ·h

~E(~x) ⇥ ~H(~x)i

= �E(~x) ·H?(~x) .

The current source density ~J (~x) is supported at z = 0, so we conclude that

�@zZ

⌦

dxE(~x) ·H?(~x) =Z

⌦

dx

i!µo�

�

�

~H(~x)�

�

�

2

� i!"(~x)�

�

�

~E(~x)�

�

�

2

�

, z 6= 0 .

The conservation relation (2.15) follows by taking the real part in this equation.

5

3. Ideal waveguides. Maxwell’s equations are separable in ideal waveguideswith constant permitivity "o, and it is typical to solve for the longitudinal componentsEz(~x) and Hz(~x) of the electric and magnetic fields, which then define E(~x) and H(~x)[12, chapter8]. The solution is given by a superposition of waves, called modes. Theyare propagating and evanescent waves and solve Maxwell’s equations with boundaryconditions (2.14). We describe the modes in section 3.1, and then write the solutionin section 3.2.

3.1. The waveguide modes. The longitudinal components of the electric andmagnetic fields satisfy the boundary conditions

Ez(~x) = 0 and n(x) ·rHz(~x) = 0, x 2 @⌦. (3.1)

The first condition is just (2.14), and the second follows from Maxwell’s equations(2.2-2.3). Indeed, (2.3) gives

H(~x) =i

!µo

⇥

r?Ez(~x) � @zE?(~x)⇤

, (3.2)

so the normal component of H at @⌦ satisfies

n(x) ·H(~x) = i!µo

⇥

�n?(x) ·rEz(~x) + @zn?(x) ·E(~x)⇤

= 0, x 2 @⌦. (3.3)

Similarly, we obtain from equation (2.2) that

D(~x) =i

!

⇥

�r?Hz(~x) + @zH?(~x)⇤

, (3.4)

and the boundary condition (2.14) implies that

n?(x) ·D(~x) = i!

[�n(x) ·rHz(~x) + @zn(x) ·H(~x)] = 0, x 2 @⌦.

The Neumann boundary condition (3.1) on Hz follows from this equation and (3.3).The waveguide modes are solutions of Maxwell’s equations that depend on the

range z as exp(±i�z), with mode wavenumber � to be defined. We write them as

eD(x;±�)e±i�z, eH(x;±�)e±i�z, (3.5)

and similar for the longitudinal components, which satisfy

� eEz(x;±�) + (k2 � �2) eEz(x;±�) = 0, (3.6)

� eHz(x;±�) + (k2 � �2) eHz(x;±�) = 0, x 2 ⌦. (3.7)

Here � is the Laplacian in x, k = !/co is the wavenumber and co = 1/p"oµo is

the wave speed. Equations (3.7) are derived from (2.2-2.5) and direct substitution of

the electric field (eE, eEz)e±i�z and magnetic field ( eH, eHz)e±i�z, see for example [12,

Section 8.3]. In homogeneous media eD = "oeE.

3.1.1. Spectral decomposition of the Laplacian. The Laplacian operatoracting on functions with homogeneous Dirichlet conditions is symmetric negative def-inite, with countable eigenvalues ��j , where

�j =

✓

⇡j1

L1

◆

2

+

✓

⇡j2

L2

◆

2

, (3.8)

6

and eigenfunctions

eEj,z(x) = sin

✓

⇡j1

x1

L1

◆

sin

✓

⇡j2

x2

L2

◆

. (3.9)

The indexes j1

and j2

are natural numbers satisfying the constraint j21

+j22

6= 0. SinceN⇥ N is countable, we can enumerate the pairs (j

1

, j2

) by the integer j � 1, so thatthe eigenvalues are in nondecreasing order.

Similarly, the Laplacian operator acting on functions with homogeneous Neumannconditions is symmetric negative semidefinite, with the same eigenvalues as (3.8), andeigenfunctions

eHj,z(x) = cos

✓

⇡j1

x1

L1

◆

cos

✓

⇡j2

x2

L2

◆

, (3.10)

where it is understood henceforth that j1

and j2

are uniquely determined by thecounter j.

Thus, we see that the electric and magnetic fields have the same mode wavenum-bers �, which take the discrete values

p

k2 � �j . We write them as

� =

⇢

�j , j = 1, . . . , N,i�j , j > N,

for �j =q

|k2 � �j |, (3.11)

to emphasize that only the first N are real. The infinitely many modes that correspondto eigenvalues �j > k2 are evanescent. We assume that �j 6= 0 for all j, so there areno standing waves in the waveguide.

3.1.2. The transverse electric and magnetic modes. It follows from (3.2),

(3.4), (3.9) and (3.10) that eD and ( eH)? are given by superpositions of the vectorsr? eHj,z(x) and r eEj,z(x). Thus, we define the vectors

'(1)j = ↵jr?eHj,z(x) = ↵j

0

B

B

@

⇡j2

L2

cos⇣

⇡j1

x1

L1

⌘

sin⇣

⇡j2

x2

L2

⌘

�⇡j1L1

sin⇣

⇡j1

x1

L1

⌘

cos⇣

⇡j2

x2

L2

⌘

1

C

C

A

, (3.12)

and

'(2)j = ↵jr eEj,z(x) = ↵j

0

B

B

@

⇡j1

L1

cos⇣

⇡j1

x1

L1

⌘

sin⇣

⇡j2

x2

L2

⌘

⇡j2

L2

sin⇣

⇡j1

x1

L1

⌘

cos⇣

⇡j2

x2

L2

⌘

1

C

C

A

, (3.13)

normalized by

↵j =

8

>

>

<

>

>

:

2p�jL1L2

, j1

j2

6= 0,

q

2

�jL1L2, otherwise.

(3.14)

so that

k'(s)j k2 =

Z

⌦

dx�

�

�

'(s)j (x)�

�

�

2

= 1, s = 1, 2.

7

The vectors indexed by s = 1 correspond to transverse electric (TE) modes.These are waves with electric field orthogonal to the range direction. Indeed, theysatisfy

r ·'(1)(x) = 0, x 2 ⌦, (3.15)

so when we set H?(~x) = '(1)(x)ei�jz in (2.10) we get Ez(~x) = 0. Similarly, thevectors indexed by s = 2 correspond to transverse magnetic (TM) modes, which arewaves with magnetic field orthogonal to the range direction. They satisfy

r? ·'(2)(x) = 0, x 2 ⌦, (3.16)

and give Hz(~x) = 0 by equation (2.9). We refer to [12, Section 8.3] and [5, Chapter3] for more details on TE and TM modes.

The superposition of '(1)j (x) and '(2)

j (x) in the definition of the fieldseE and

( eH)? is their Helmholtz decomposition in a divergence free part and a curl free part.

3.1.3. Analogous derivation of the waveguide modes. We could have ar-rived at the same wave decomposition if we worked directly with the transverse com-ponents D and H of the fields. This observation is relevant because when the per-mittivity varies in ~x, as in the random waveguide, it is no longer possible to solveindependently for the longitudinal wave fields Ez and Dz.

We let

H(~x) = coU?(~x), (3.17)

where U is the rotated magnetic field scaled by 1/co. It is convenient to work in the Dand U variables because as we see below, they satisfy the same boundary conditionsand have the same physical units. Note from (3.12) and (3.13) that '(s)(x) areeigenfunctions of the vector Laplacian

�'(s)j (x) = rh

r ·'(s)j (x)i

+ r?h

r? ·'(s)j (x)i

= ��j'(s)j (x), (3.18)

for x 2 ⌦, with boundary conditions

n?(x) ·'(s)j (x) = 0, r ·'(s)j (x) = 0, x 2 @⌦. (3.19)

The index s satisfies 1 s Mj where Mj 2 {1, 2} accounts for the multiplicity ofthe eigenvalue �j . We can limit the multiplicity of �j by assuming that the ratio ofthe waveguide dimensions L

1

/L2

is irrational. This implies that

�j 6= �j0 , if (j1, j2) 6= (j01

, j02

) , (3.20)

where we recall that (j1

, j2

) is uniquely defined by the counter j and similarly (j01

, j02

)

is defined by j0. When either j1

or j2

is zero, Mj = 1, and only the TE modes '(1)

j (x)exist. Otherwise Mj = 2.

The eigenfunctions satisfy the orthogonality relations

D

'(s)j ,'(s0)j0

E

=

Z

⌦

dx'(s)j (x) ·'(s0)j0 (x) = �jj0�ss0 . (3.21)

andn

'(s)j

o

1sMj ,j�1is a complete set that can be used to describe an arbitrary

electromagnetic wave field in the waveguide [12, chapter8].

8

The boundary conditions (3.19) are consistent with the conditions satisfied byD(~x) and U(~x), derived from Maxwell’s equations. Indeed, equations (2.10), (2.14)and the assumption (2.6) on the source density give that

r ·U(~x) = � 1cor? ·H(~x) = 0, x 2 @⌦. (3.22)

Moreover, equation (3.3) says that

n?(x) ·U(~x) = � 1con(x) ·H(~x) = 0, x 2 @⌦. (3.23)

For the electric displacement we already know from (2.14) that

n?(x) ·D(~x) = 0, x 2 @⌦. (3.24)

The divergence condition follows from (3.4) and (3.22)

r ·D(~x) = 0, x 2 @⌦, (3.25)

and since @zDz = 0, it is consistent with the conservation of charge.

3.2. The solution in ideal waveguides. We expand D(~x) and U(~x) in the

basisn

'(s)j

o

1sMj ,j�1and associate to each '(s)j (x) a mode, which is a propagating

or evanescent wave. We rename the fields Do(~x) and Uo(~x) to remind us that we arein the ideal waveguide.

Substituting the expansion of Do and Uo in (3.2) and (3.4) and using the identities

k2'(s)j (x) + rh

r ·'(s)j (x)i

=⇥

k2�s1 + (k2 � �j)�s2

⇤

'(s)j (x) , (3.26)

k2'(s)j (x) + r?h

r? ·'(s)j (x)i

=⇥

(k2 � �j)�s1 + k2�s2⇤

'(s)j (x) , (3.27)

we obtain

Do(~x) =NX

j=1

MjX

s=1

'(s)j (x)

s

k

�j�s1 +

r

�jk�s2

!

⇣

A±(s)j,o ei�jz + B±(s)j,o e

�i�jz⌘

+

X

j>N

MjX

s=1

'(s)j (x)

s

k

�j�s1 +

r

�jk�s2

!

E±(s)j,o e��j |z| , (3.28)

and

Uo(~x) =NX

j=1

MjX

s=1

'(s)j (x)

r

�jk�s1 +

s

k

�j�s2

!

⇣

A±(s)j,o ei�jz � B±(s)j,o e

�i�jz⌘

±

iX

j>N

MjX

s=1

'(s)j (x)

r

�jk�s1 �

s

k

�j�s2

!

E±(s)j,o e��j |z| , (3.29)

for z 6= 0. The superscripts +, � of the mode amplitudes correspond to the domainsz > 0, z < 0 respectively. The normalization coe�cients

p

k/�j andp

�j/k arenot important here, and could be absorbed in the mode amplitudes. We use them

9

for consistency with the mode expansions for the random waveguide in section 5.There the normalization is convenient because it symmetrizes the system of equationssatisfied by the mode amplitudes.

The amplitudes in (3.28-3.29) are constant on each side of the source, and aredetermined by the source density and the outgoing conditions: There are no backward

going modes to the right of the source, at positive ranges, so we can set B+(s)j,o = 0.

Similarly, we let A�(s)j,o = 0. The remaining amplitudes are obtained from the sourceconditions

Do(x, 0+) �Do(x, 0�) = �i

cokrJz(x) ,

Uo(x, 0+) �Uo(x, 0�) = �1

coJ(x) .

Substituting (3.28-3.29) in these conditions and using the orthogonality relations(3.21), we get

A+(s)j,o = �1

2co

s

k

�j�s1 +

r

�jk�s2

!

D

'(s)j ,JE

� i2cok

r

�jk�s1 +

s

k

�j�s2

!

D

rJz,'(s)jE

, (3.30)

and

B�(s)j,o = �1

2co

s

k

�j�s1 +

r

�jk�s2

!

D

'(s)j ,JE

+i

2cok

r

�jk�s1 +

s

k

�j�s2

!

D

rJz,'(s)jE

, (3.31)

for the propagating modes and

E±(s)j,o =i

2co

s

k

�j�s1 �

r

�jk�s2

!

D

'(s)j ,JE

⌥ i2cok

r

�jk�s1 +

s

k

�j�s2

!

D

rJz,'(s)jE

, (3.32)

for the evanescent modes.

3.2.1. Energy conservation. The energy conservation is obvious in this case,because the amplitudes are constant. Substituting (3.28-3.29) in the expression of theflux S(z) and using the orthogonality relations (3.21), we obtain that

S(z) = co"o

Z

⌦

Reh

Do(~x) ·Uo(~x)i

=NX

j=1

MjX

s=1

✓

�

�

�

A±(s)j,o

�

�

�

2

��

�

�

B±(s)j,o

�

�

�

2

◆

, 8z 2 R . (3.33)

The flux changes value at z = 0, where the source lies, but it is constant for z 6= 0,

S(|z|) = co"o

NX

j=1

MjX

s=1

�

�

�

A+(s)j,o

�

�

�

2

= �S(�|z|) = co"o

NX

j=1

MjX

s=1

�

�

�

B�(s)j,o

�

�

�

2

, z 6= 0 . (3.34)

The evanescent modes play no role in the transport of energy.

10

4. Statement of the problem in the random waveguide. We begin withthe model of the small fluctuations. Then we write the perturbed system of equationsfor the wave fields, which we analyze in the remainder of the paper.

4.1. Model of the fluctuations. Let us denote by n(~x) the index of refraction

n(~x) =co

c(~x)=

s

"(~x)

"o. (4.1)

It is the ratio of the electromagnetic wave speeds co and c(~x) = 1/p

"(~x)µo in thehomogeneous and heterogeneous medium, respectively. We model the electrical per-mittivity by

"(~x) = "on2(~x) , n2(~x) = 1 + ✏⌫(~x)1

(0+,zmax

)

(z) , (4.2)

where ⌫(~x) is a dimensionless random function assumed twice continuously di↵eren-tiable, with almost sure bounded derivatives. It has zero mean

E [⌫(~x)] = 0 , 8~x 2 ⌦⇥ R , (4.3)

and it is stationary and mixing in z. We refer to [16, Section 4.6.2] for a precisestatement of the mixing condition. It means in particular that the covariance

R⌫(x,x0, z) = E [⌫(x, z)⌫(x0, 0)] (4.4)

is integrable in z. The amplitude of the fluctuations in (4.2) is scaled by ✏ ⌧ 1, thesmall parameter in our asymptotic analysis.

The indicator function 1(0+,z

max

)

(z) in (4.2) limits the support of the fluctuationsto the range interval z 2 (0+, z

max

), where 0+ denotes a range that is close to zero,but strictly larger than it. The bounded support of the fluctuations is needed to statethe outgoing boundary conditions on the electromagnetic wave fields, and may bejustified in practice by the causality of the problem in the time domain. During afinite observation time t

max

, the waves are influenced by the medium up to a finiterange z

max

⇡ cotmax , so we may truncate the fluctuations beyond the range zmax.That there are no fluctuations at negative ranges may be motivated by two facts:First, the source is at z = 0 and we wish to study the waves at positive ranges.Second, we will consider a regime where the backscattered field is negligible. This letsus neglect at z > 0 the waves that come from z < 0, and truncate the fluctuationsat z = 0+. The neglect of the backscattered field is known as the forward scatteringapproximation. It is justified mathematically when the fluctuations ⌫ are smooth, asdiscussed in section 6.1.

4.2. The perturbed system of equations in the random waveguide. Wework with the electric displacement D(~x) and the scaled rotated magnetic field U(~x),defined in equation (3.17). As we explained in the previous section, this is convenientbecause the fields satisfy the same boundary conditions and have the same units.

The equations for D(~x) and U(~x) follow from (2.8), (2.11-2.12), (4.2) and (3.17).We have

@zD(~x) =i

k

�

k2n2(~x)U(~x) + r [r ·U(~x)] � n�2(~x)rn2(~x)r ·U(~x)

+

n�2(~x)@zn2(~x)D(~x) � i

cokrJz(~x) , (4.5)

11

for the electric displacement and

@zU(~x) =i

k

�

k2D(~x) + r?⇥

r? ·�

n�2(~x)D(~x)�⇤

� 1coJ (~x) , (4.6)

for the rotated magnetic field, where we used that the fluctuations are supported awayfrom the source. Morever, substituting the model (4.2) of the fluctuations, we obtain

@zD(~x) =i

k

�

k2U(~x) + r [r ·U(~x)]

� icok

rJz(~x)+

✏

⇢

@z⌫(~x)D(~x) +i

k

⇥

k2⌫(~x)U(~x) �r⌫(~x)r ·U(~x)⇤

�

+

✏2

2

�@z⌫2(~x)D(~x) +i

kr⌫2(~x)r ·U(~x)

�

+ O(✏3) , (4.7)

and

@zU(~x) =i

k

�

k2D(~x) + r?⇥

r? ·D(~x)⇤

� 1coJ (~x)�

✏i

k

�

r?⇥

⌫(~x)r? ·D(~x)⇤

+ r?⇥

D(~x) ·r?⌫(~x)⇤

+

✏2i

k

�

r?⇥

⌫2(~x)r? ·D(~x)⇤

+ r?⇥

D(~x) ·r?⌫2(~x)⇤

+ O(✏3) , (4.8)

with remainder involving powers (✏⌫)q, for q � 3. It is of order ✏3 because ⌫(~x) istwice di↵erentiable, with almost sure bounded derivatives.

The leading order terms in (4.7-4.8) involve the operators (3.26-3.27), so we havea perturbation of the problem in the ideal waveguide. The conservation of the energyflux follows from (2.15) and definitions (2.8),(4.2) and (3.17)

S(z) = co"o

Z

⌦

dx Reh

n�2(~x)D(~x) ·U(~x)i

=co"o

Z

⌦

dx⇥

1 � ✏⌫(~x) + ✏2⌫2(~x) + O(✏3)⇤

Reh

D(~x) ·U(~x)i

= S(0+) , z > 0. (4.9)

5. Mode decomposition and coupling in random waveguides. The equa-

tions in the random waveguide are no longer separable, but {'(s)j (x)}1sMj ,j�1 isan orthonormal basis, so we can still use it to decompose the wave fields for any rangez. The essential di↵erence in the decomposition is that while the mode amplitudesare constant in ideal waveguides, they vary in range in the random waveguides, dueto scattering. The range evolution of the mode amplitudes is described by a cou-pled system of infinitely many stochastic ordinary di↵erential equations. We show insection 5.3 that we can solve for the amplitudes of the evanescent modes, and thusobtain in section 5.4 a closed and finite system of equations for the amplitudes of thepropagating modes. This system is the main result of the section. We use it in section6 to obtain an explicit long range characterization of the statistical distribution of theelectromagnetic wave field.

12

5.1. Mode decomposition. We decompose the fields as

D(~x) =NX

j=1

MjX

s=1

'(s)j (x)

s

k

�j�s1 +

r

�jk�s2

!

⇣

A(s)j (z)ei�jz + B(s)j (z)e

�i�jz⌘

+

X

j>N

MjX

s=1

'(s)j (x)

s

k

�j�s1 +

r

�jk�s2

!

V (s)j (z) , (5.1)

and

U(~x) =NX

j=1

MjX

s=1

'(s)j (x)

r

�jk�s1 +

s

k

�j�s2

!

⇣

A(s)j (z)ei�jz � B(s)j (z)e

�i�jz⌘

±

iX

j>N

MjX

s=1

'(s)j (x)

r

�jk�s1 �

s

k

�j�s2

!

v(s)j (z) , (5.2)

for z 6= 0. The decomposition is similar to that in ideal waveguides, but the modeamplitudes vary in z due to scattering in the random medium. We show in section

5.2 that the forward and backward going mode amplitudes A(s)j and B(s)j and the

evanescent modes written in (5.1-5.2) as V (s)j (z) and v(s)j (z) satisfy a coupled system

of stochastic di↵erential equations driven by the random fluctuations ⌫. The termsin the first sum of (5.1-5.2) are obtained by solving the equations for the propagat-ing mode amplitudes to O(1). In ideal waveguides the evanescent modes were equal

to E(s)j,oexp(��jz), with E(s)j,o given in (3.32). The expressions of V

(s)j (z) and v

(s)j (z)

contain these terms. In addition, they contain ✏ dependent terms which do not de-cay exponentially in z and account for the coupling with the propagating modes, asexplained in section 5.3.

The expansions (5.1-5.2) satisfy the boundary conditions (3.22-3.25) at @⌦. Theoutgoing conditions and the finite range support (0+, z

max

) of the fluctuations give

B(s)j (zmax) = 0 , (5.3)

A(s)j (0+) = A(s)j,o . (5.4)

The first equation says that there are no backward going waves coming from infinity,because there are no fluctuations beyond z = z

max

. The second equation follows fromthe source conditions

D(x, 0+) � D(x, 0�) = � icok

rJz(x) = Do(x, 0+) � Do(x, 0�) ,

U(x, 0+) � U(x, 0�) = � 1coJ(x) = Uo(x, 0+) � Uo(x, 0�) ,

and the outgoing condition A(s)j (z) = 0 at ranges z < 0, where the medium is homo-geneous. The evanescent modes satisfy

lim|z|!1

V (s)j (z) = lim|z|!1v(s)j (z) = 0 . (5.5)

13

5.2. Mode coupling. Substituting (5.1-5.2) in (4.7-4.8) and using identities(3.26-3.27) and the orthogonality relation (3.21), we obtain a system of stochasticdi↵erential equations that describes the range evolution of the mode amplitudes. Therate of change of the amplitudes of the forward going modes is given by

@zA(s)j (z) = ✏

NX

j0=1

Mj0X

s0=1

h

M (ss0)

AA,jj0(z) + ✏m(ss0)AA,jj0(z)

i

A(s0)

j0 (z) ei(�j0��j)z+

✏NX

j0=1

Mj0X

s0=1

h

M (ss0)

AB,jj0(z) + ✏m(ss0)AB,jj0(z)

i

B(s0)

j0 (z) e�i(�j0+�j)z+

✏X

j0>N

Mj0X

s0=1

h

M (ss0)

AV,jj0(z) + ✏m(ss0)AV,jj0(z)

i

V (s0)

j0 (z) e�i�jz+

✏X

j0>N

Mj0X

s0=1

h

M (ss0)

Av,jj0(z) + ✏m(ss0)Av,jj0(z)

i

v(s0)

j0 (z) e�i�jz + O(✏3) , (5.6)

for z > 0, with initial condition (5.4). The rate of change of the amplitudes of thebackward moving modes is

@zB(s)j (z) = ✏

NX

j0=1

Mj0X

s0=1

h

M (ss0)

BA,jj0(z) + ✏m(ss0)BA,jj0(z)

i

A(s0)

j0 (z) ei(�j0+�j)z+

✏NX

j0=1

Mj0X

s0=1

h

M (ss0)

BB,jj0(z) + ✏m(ss0)BB,jj0(z)

i

B(s0)

j0 (z) e�i(�j0��j)z+

✏X

j0>N

Mj0X

s0=1

h

M (ss0)

BV,jj0(z) + ✏m(ss0)BV,jj0(z)

i

V (s0)

j0 (z) ei�jz+

✏X

j0>N

Mj0X

s0=1

h

M (ss0)

Bv,jj0(z) + ✏m(ss0)Bv,jj0(z)

i

v(s0)

j0 (z) ei�jz + O(✏3) , (5.7)

for z > 0, with end condition (5.3) at z = zmax

. The evanescent components V (s)j (z)

and v(s)j (z) are described in the next section.

The coupling coe�cients in the right-hand side of equations (5.6-5.7) are sta-tionary random processes in z, defined in terms of the fluctuations ⌫. We refer toappendix A for their expression and symmetry relations. The leading order terms of

these coe�cients, denoted by the capital letter M as in M (ss0)

AA,jj0(z), are linear in ⌫,so they have zero expectation. The second order terms, denoted by the small letter

m as in m(ss0)

AA,jj0(z), are quadratic in ⌫. The coe�cients also depend on k and L1, L2,but to simplify notation we suppress these in their arguments.

14

5.3. The evanescent modes. The evanescent modes satisfy the equations

@zV(s)j (z) + �jv

(s)j (z) =✏F

(s)j (z) + ✏

X

j0>N

Mj0X

s0=1

M (ss0)

V V,jj0(z) V(s0)j0 (z) +

✏X

j0>N

Mj0X

s0=1

M (ss0)

V v,jj0(z) v(s0)j0 (z) + O(✏

2) , (5.8)

and

@zv(s)j (z) + �jV

(s)j (z) = ✏f

(s)j (z) + ✏

X

j0>N

Mj0X

s0=1

M (ss0)

vV,jj0(z) V(s0)j0 (z) + O(✏

2) , (5.9)

for z > 0, with forcing terms

F (s)j (z) =NX

j0=1

Mj0X

s0=1

h

M (ss0)

V A,jj0(z) A(s0)j0 (z) e

i�j0z + M (ss0)

V B,jj0(z) B(s0)j0 (z) e

�i�j0zi

, (5.10)

f (s)j (z) =NX

j0=1

Mj0X

s0=1

h

M (ss0)

vA,jj0(z) A(s0)j0 (z) e

i�j0z + M (ss0)

vB,jj0(z) B(s0)j0 (z) e

�i�j0zi

. (5.11)

The coupling coe�cients are described in appendix A. They are stationary processesin z that depend linearly on the fluctuations ⌫.

The system of equations (5.8-5.9) for V (s)j and v(s)j is solved in appendix B. The

solution is obtained by inverting a linear operator which is an ✏ perturbation of theidentity. The result is that we can write the evanescent modes explicitly in terms ofthe propagating mode amplitudes, as stated in the next lemma.

Lemma 5.1. The evanescent modes are given by

V (s)j (z) =E(s)j,o e

��jz +✏

2

Z 1

�1d⇣ f (s)j (z + ⇣) e

��j |⇣|+

✏

2

Z 1

0

d⇣h

F (s)j (z � ⇣) � F(s)j (z + ⇣)

i

e��j⇣ + O(✏2) , (5.12)

and

v(s)j (z) =E(s)j,o e

��jz +✏

2

Z 1

�1d⇣ F (s)j (z + ⇣) e

��j |⇣|+

✏

2

Z 1

0

d⇣h

f (s)j (z � ⇣) � f(s)j (z + ⇣)

i

e��j⇣ + O(✏2) . (5.13)

The first terms in these equations are as in ideal waveguides, with Ej,o given by(3.32). They decay exponentially with z and have a negligible contribution at longranges. The O(✏) terms capture the coupling with the propagating modes and havelong range e↵ects in equations (5.6-5.7). The remaining terms are negligible in thelimit ✏! 0.

15

5.4. Closed system for the propagating modes. The substitution of theevanescent mode equations (5.12-5.13) in (5.6-5.7) gives a closed system of ordinarydi↵erential equations for the amplitudes of the N forward and backward going modes

@zA(s)j (z) =✏

NX

j0=1

Mj0X

s0=1

h

M (ss0)

AA,jj0(z) + ✏em(ss0)AA,jj0(z)

i

A(s0)

j0 (z)ei(�j0��j)z+

✏NX

j0=1

Mj0X

s0=1

h

M (ss0)

AB,jj0(z) + ✏em(ss0)AB,jj0(z)

i

B(s0)

j0 (z)e�i(�j0+�j)z + O(✏3) , (5.14)

and

@zB(s)j (z) = ✏

NX

j0=1

Mj0X

s0=1

h

M (ss0)

BA,jj0(z) + ✏ em(ss0)BA,jj0(z)

i

A(s0)

j0 (z)ei(�j0+�j)z+

✏NX

j0=1

Mj0X

s0=1

h

M (ss0)

BB,jj0(z) + ✏ em(ss0)BB,jj0(z)

i

B(s0)

j0 (z)e�i(�j0��j)z + O(✏3) . (5.15)

Here we let

em(ss0)

AA,jj0(z) = m(ss0)AA,jj0(z) + m

(ss0)eAA,jj0(z) ,

em(ss0)

AB,jj0(z) = m(ss0)AB,jj0(z) + m

(ss0)eAB,jj0(z) ,

em(ss0)

BA,jj0(z) = m(ss0)BA,jj0(z) + m

(ss0)eBA,jj0(z) ,

em(ss0)

BB,jj0(z) = m(ss0)BB,jj0(z) + m

(ss0)eBB,jj0(z) ,

with the second terms due to the interaction via the evanescent modes. They arewritten explicitly in appendix B.1.

5.5. Energy conservation. Substituting equations (5.1-5.2) in the energy flux(4.9) and using Lemma 5.1 we obtain that

NX

j=1

MjX

s=1

�

�

�

A(s)j (z)�

�

�

2

��

�

�

B(s)j (z)�

�

�

2

�

=NX

j=1

MjX

s=1

�

�

�

A(s)j,o

�

�

�

2

��

�

�

B(s)j (0+)�

�

�

2

�

+ O(✏) . (5.16)

The evanescent modes do not contribute to leading order in the energy flux, but theyappear in the remainder O(✏). Consequently, the energy carried by the propagatingmodes is not exactly conserved for ✏ > 0. However, energy conservation holds in thelimit ✏! 0, where the remainder becomes negligible.

6. The di↵usion limit. In this section we describe the limit ✏ ! 0 of thepropagating mode amplitudes satisfying the system of equations (5.14-5.15) for z > 0,with initial conditions (5.4) at z = 0 and end conditions (5.3) at z = z

max

.

Since @zA(s)j (z) and @zB

(s)j (z) are order ✏, it is clear that the fluctuations have no

e↵ect over ranges z that are of order one, i.e., similar to the wavelength. If we let zbe of order ✏�1, the right-hand side in (5.14-5.15) becomes order one, but still thereis no net scattering e↵ect in the limit ✏ ! 0. The fluctuations average out becausethe expectation of the leading coupling coe�cients M (ss

0)

AA,jj0(z/✏), . . . , M(ss0)AA,jj0(z/✏) is

zero. See for example [13, 23] and [7, Chapter 6]. We need longer ranges, of order

16

✏�2, to see cumulative scattering e↵ects, so we let z = Z/✏2 with Z of order one, andrename the mode amplitudes in this scaling as

A(s)✏j (Z) := A(s)j

�

Z/✏2�

, B(s)✏j (Z) := B(s)j

�

Z/✏2�

, (6.1)

for j = 1, . . . , N, and 1 s Mj . Their ✏ ! 0 limit is obtained with the di↵usionapproximation theorem [22]. The result is simpler under the forward scattering ap-proximation described in section 6.1, which is valid when the covariance (4.4) of ⌫(~x)is smooth in z. The limit of the forward going mode amplitudes is described in detailin section 6.2. This is the main result of the section. We use it to analyze the longrange cumulative scattering e↵ects of the random fluctuations in section 7.

6.1. The forward scattering approximation. The di↵usion approximationtheorem applies to initial value problems, so we transform our system to such aproblem using the random propagator matrix P✏(Z). It equals the identity I atZ = 0 and relates the mode amplitudes at Z > 0 to those at Z = 0 as

✓

A✏(Z)B✏(Z)

◆

= P✏(Z)

✓

AoB✏(0)

◆

. (6.2)

Here A✏(Z) is the vector of components A(s)✏j (Z) for j = 1, . . . , N , 1 s Mj , andsimilar for B✏(Z). The backward going amplitudes are not known at Z = 0, but canbe determined from the identity

✓

A✏(Zmax

)0

◆

= P✏(Zmax

)

✓

AoB✏(0)

◆

, Zmax

= ✏2zmax

. (6.3)

The di↵usion approximation theorem [22] states that P✏(Z) converges in distri-bution as ✏! 0 to a matrix valued di↵usion process P(Z). That is to say, the entriesof P(Z) satisfy a system of stochastic di↵erential equations with initial conditionP(0) = I. We do not need to write all the details of the limit for the analysis below.Let us just note that it has the block structure

P(Z) =

✓

PAA(Z) PAB(Z)PBA(Z) PBB(Z)

◆

,

with entries determined by the z–Fourier transform bR⌫(x,x0,�) of the covariance(4.4), evaluated at various values of the wavenumber �. Explicitly, for the entries inthe block PAA(Z) that couple the j and j0 forward going amplitudes, � = �j � �j0 ,because the phases in the first sum in (5.14) are proportional to �j��j0 . Similarly, forthe entries in the blocks PAB(Z) and PBA(Z) that couple the j and j0 forward andbackward going amplitudes, � = �j + �j0 , because the phases in the second sum in(5.14) and the first sum in (5.15) are proportional to �j +�j0 . Thus, if the covarianceis smooth enough in z, so that

�

�

�

bR⌫(x,x0,�j + �j0)�

�

�

⌧ 1 , 8 j, j0 = 1, . . . , N , (6.4)

the forward and backward mode amplitudes are essentially uncoupled. Consideringthat B✏(Z) vanishes at Z

max

, we conclude that the backward going mode amplitudesare negligible, and thus justify the forward scattering approximation.

17

6.2. The coupled mode di↵usion process. Equations (5.14) simplify as

@ZA✏(Z) ⇡ 1

✏G

A✏(Z), ⌫

✓

·, Z✏2

◆

,Z

✏2

�

+ g

A✏(Z), ⌫

✓

·, Z✏2

◆

,Z

✏2

�

, Z > 0 , (6.5)

with initial conditions A✏(0) = Ao, and right-hand side

G

A✏(Z), ⌫

✓

·, Z✏2

◆

,Z

✏2

�

= M

⌫

✓

·, Z✏2

◆

,Z

✏2

�

A✏(Z) , (6.6)

g

A✏(Z), ⌫

✓

·, Z✏2

◆

,Z

✏2

�

= em

⌫

✓

·, Z✏2

◆

,Z

✏2

�

A✏(Z) . (6.7)

Here we let M be the matrix with entries M (ss0)

AA,jj0(Z/✏2)ei(�j��j0 )Z/✏

2

, and emphasize

in the notation that it depends on Z/✏2 via the fluctuations ⌫ and the phase. A similarnotation applies to matrix em. The approximation sign in (6.5) reminds us that wemade the forward scattering approximation and neglected the O(✏) remainder thatplays no role in the limit ✏! 0.

To apply the di↵usion approximation theorem stated and proved in [22] to (6.5),we rewrite the system in real form, for the concatenated vector (A✏R,A

✏I) of real and

imaginary values of A✏. As pointed out in [7, Section 20.3.1], the generator of thelimit process A = AR + iAI has a simpler form when written in terms of A and itscomplex conjugate A. Therefore, we recall from complex di↵erentiation that

rAR = rA + rA, rAI = i�

rA �rA�

,

and if we let GR and GI be the real and imaginary parts of G, we can write

(GR,GI) · (rAR ,rAI ) = G ·rA + G ·rA .

With these observations we state in the next lemma the limit ✏ ! 0 given by thedi↵usion approximation theorem.

Lemma 6.1. The mode amplitudes {A(s)✏j (Z)}1jN,1sMj converge in distri-bution as ✏ ! 0 to a di↵usion Markov process denoted by {A(s)j (Z)}1jN,1sMj ,with generator G. It is defined on smooth enough, scalar valued test functions '(A,A)as follows

G'(A,A) = limT!1

Z T

0

d⌧

T

Z 1

0

dz E�⇥

G [A, ⌫(·, 0), ⌧ ] ·rA + G [A, ⌫(·, 0), ⌧ ] ·rA⇤

⇥⇥

G [A, ⌫(·, z), ⌧ + z] ·rA + G [A, ⌫(·, z), ⌧ + z] ·rA⇤

'(A,A)+

limT!1

Z T

0

d⌧

TE�

[g [A, ⌫(·, 0), ⌧ ] ·rA + g [A, ⌫(·, 0), ⌧ ]] ·rA

'(A,A).

6.3. Conservation of energy. Recall the conservation relation (5.16), andrewrite it using the forward scattering approximation as

NX

j=1

MjX

s=1

�

�

�

A(s)✏j (Z)�

�

�

2

=NX

j=1

MjX

s=1

|Aj,o|2 + R(✏) , (6.8)

18

with negligible remainder R(✏) as ✏! 0. The di↵usion limit gives that

NX

j=1

MjX

s=1

�

�

�

A(s)✏j (Z)�

�

�

2 ✏!0�!NX

j=1

MjX

s=1

�

�

�

A(s)j (Z)�

�

�

2

=NX

j=1

MjX

s=1

|Aj,o|2 , (6.9)

where the convergence is in probability, because the limit is deterministic.

7. Cumulative scattering e↵ects. We use the limit stated in Lemma 6.1 toderive the main result of the paper: a detailed characterization of cumulative scatter-ing e↵ects of the random fluctuations of the electric permeability.

We begin in sections 7.1 and 7.2 with the calculation of the first and secondmoments of the mode amplitudes. They determine the coherent part of the wavesand the intensity of their fluctuations. Then, we start from the energy conservationrelation (6.9) and derive in section 7.3 an important matrix identity, needed in sections7.4 and 7.5 to describe the loss of coherence of the waves and the energy exchangebetween the modes. We also prove in section 7.5 that as the range grows, the wavesscatter so much that they enter the equipartition regime, where they forget all theinformation about the source. We illustrate the results of the analysis with numericalsimulations.

7.1. The mean mode amplitudes. Let us denote by

hAi(Z) = E {A(Z)} , (7.1)

the expectation of the mode amplitudes with respect to their limit distribution. Usingthe generator G in Lemma 6.1 and Kolmogorov’s equation [21, chapter 8], we obtain

@Z hAji(Z) = Qj hAji(Z) , Z > 0 , (7.2)

with initial condition given by (3.30)

hAji(0) = Aj,o . (7.3)

The system (7.2) is block diagonal, for vectors Aj of components A(s)j . There are N

blocks Qj 2 CMj⇥Mj . Each one of them is constant, with entries given by

Q(ss0)

j =NX

l=1

MlX

q=1

Z 1

0

dz En

M (sq)AA,jl(z)M(qs0)AA,lj(0)

i

ei(�l��j)z + En

m(ss0)

j (0)o

, (7.4)

where we introduced the simplified notation

mj(z) := emAA,jj(z). (7.5)

We give a few details of the calculation of Qj in appendix C, and use the result inthe numerical simulations of sections 7.4 and 7.5. Here it su�ces to point out thatthe last term in (7.4) is purely imaginary, so we can write it as

E {mj(0)} = ij , (7.6)

with real matrix j 2 RMj⇥Mj . This is the only term that is a↵ected by the couplingof the propagating modes with the evanescent ones.

19

The mean amplitudes are decoupled for di↵erent indexes j of the modes. However,for each j we have Mj coupled transverse electric and magnetic mode amplitudes, asdescribed by the matrix exponential in

hAji(Z) = eQjZAj,o , j = 1, . . . , N. (7.7)

We expect from physical arguments that the right-hand side in (7.7) decays with Z, on

some mode dependent range scales S(s)j , the scattering mean free paths. The coherentpart of the amplitudes, the entries in hAji(Z), become negligible beyond these scales,and all the energy lies in their random fluctuations.

It is di�cult to see the loss of coherence directly from (7.4). The expression ofQj in appendix C is useful for numerical calculations, but it is too complicated toprove that the spectrum of Qj lies in the left half of the complex plane. However, theresult follows from the energy conservation relation (6.9), as explained in section 7.4.

7.2. The mean powers. We denote the mean power matrices of the amplitudesof the modes with wavenumber �j by

Pj(Z) =⇣

P ss0

j (Z)⌘

1s,s0Mj:= E

�

Aj(Z) ⌦Aj(Z)

. (7.8)

They are Hermitian, positive definite matrices, satisfying a coupled system of di↵eren-tial equations derived from the generator in Lemma 6.1 and Kolmogorov’s equation.Explicitly, we have

@ZPj(Z) =QjPj(Z) + Pj(Z)Q?j+

NX

l=1

Z 1

�1dzE

�

MAA,jl(z)Pl(Z)M?AA,jl(0)

ei(�l��j)z , (7.9)

for Z > 0, with initial condition

Pj(0) = Aj,o ⌦Aj,o. (7.10)

The matrix Qj is defined in (7.4), and the star superscript denotes complex conjugateand transpose.

Equations (7.9) describe the exchange of energy between the modes and the lossof polarization of the waves. Say for example that the source emits a single transverseelectric mode indexed by j

Pl(0) = �lj

✓

|A(1)j,o |2 00 0

◆

, 8 l = 1, . . . , N .

Cumulative scattering distributes the energy to all propagating modes for Z > 0, asgiven by (7.9), and the wave loses its initial polarization.

7.3. Conservation of energy identity. The conservation of energy relation(6.9) states that the mean power matrices satisfy

NX

j=1

trace[Pj(Z)] =NX

j=1

trace[Pj(0)] =NX

j=1

MjX

s=1

�

�

�

A(s)j,o

�

�

�

2

. (7.11)

20

Therefore, equations (7.9) and the properties of the trace operator imply that

NX

j=1

trace⇥�

Qj + Q?j + Cj

�

Pj(Z)⇤

= 0 , 8 Z � 0 , (7.12)

with Hermitian matrix Cj defined by

Cj =NX

l=1

Z 1

�1dz E

�

M?AA,lj(z)MAA,lj(0)

ei(�l��j)z . (7.13)

The terms in this sum are the power spectral densities of the stationary, matrix valuedprocesses MAA,jl(z), evaluated at the wavenumber di↵erence �j � �l. This impliesthat Cj is a positive definite matrix, as shown in appendix D.

The following lemma gives a matrix identity used in the next sections to provethe loss of coherence of the waves and the equipartition regime as Z ! 1.

Lemma 7.1. The matrices Qj and Cj defined by (7.4) and (7.13) satisfy

Qj + Q?j + Cj = 0 , 8 j = 1, . . . , N. (7.14)

Proof. The result is a consequence of the fact that (7.12) holds for any correlationmatrices Pj(Z) and all Z � 0. Indeed, let Xj be the M2j dimensional vector space ofMj ⇥Mj Hermitian matrices with inner product

(U, V )Xj = trace[UV?] , 8 U, V 2 Xj .

Let also X = X1

⇥ X1

. . . ⇥ XN be the vector space defined by the product of thespaces Xj , with inner product

(U,V)X =NX

j=1

(Uj , Vj)Xj , 8 U = (U1, . . . , UN ), V = (V1, . . . , VN ), Uj , Vj 2 Xj .

Equation (7.12) evaluated at Z = 0 becomes

(Q + Q? + C,Po)X = 0 , 8 P(0) = Po 2 X.

We can take in particular the initial conditions

Po = (0, . . . ,0,Pj,o,0, . . . ,0) , 8 Pj,o = Aj,o ⌦A?j,o 2 Xj ,

and conclude that�

Qj + Q?j + Cj ,Pj,o

�

Xj= 0.

The statement of the lemma follows from this equation and the observation thatmatrices like Pj,o span Xj . For example,

✓

10

◆

(1, 0) =

✓

1 00 0

◆

,

✓

01

◆

(0, 1) =

✓

0 00 1

◆

,

✓

11

◆

(1, 1) =

✓

1 11 1

◆

,

✓

i1

◆

(�i, 1) =✓

1 i�i 1

◆

,

is a basis of Xj . ⇤21

0 10 20 30 40 50 6010−3

10−2

10−1

100

Propagating mode j

1/µ

j,q

1/µj,2SjEquip. dist.

0 10 20 30 40 50 60 70 8010−3

10−2

10−1

100

Propagating mode j

1/µ

j,q

1/µj,2SjEquip. dist.

Fig. 7.1. We plot in green Sj

, the reciprocal of the minimum eigenvalue of C

j

, and in blue

the reciprocal of the maximum eigenvalue. The equipartition distance is shown in red. We show

results for two waveguides (from left to right) (1) L1 = 3.03 and L2 = 5.84 giving N = 64, and (2)L1 = 4.08 and L2 = 5.77 giving N = 84. The abscissa is the mode index j and the ordinate is inunits of the wavelength �.

7.4. The loss of coherence. Lemma 7.1 and equation (7.2) give that

@Zk hAji(Z)k2 = �hAji?(Z)Cj hAji(Z), Z > 0, k hAji(0)k2 = kAj,ok2,

where k ·k is the Euclidian norm, and we recall that Cj is Hermitian, positive definite.The result stated in the next theorem follows from Gronwall’s lemma:

Theorem 7.2. Let µj,q > 0 be the eigenvalues of Cj in increasing order, for allj = 1, . . . , N and 1 q Mj. We have that

e�µj,2ZkAj,ok2 k hAji(Z)k2 e�µj,1ZkAj,ok2, if Mj = 2, (7.15)

and

k hAji(Z)k2 = e�µj,1ZkAj,ok2, if Mj = 1. (7.16)

Thus, the mean amplitudes decay exponentially with Z, on mode dependent rangescales (scaled scattering mean free paths)

Sj = 1/µj,1 . (7.17)

Discussion and numerical illustration. The decay of the mean mode amplitudesis a manifestation of the loss of coherence of the modes. This is a gradual process,with the last indexed modes losing coherence faster than the first ones, as illustratedby the numerical results displayed in Figure 7.1. We plot Sj = 1/µj,1 in green and1/µj,Mj in blue. Note that Sj are the scaled scattering mean free paths, as followsfrom (6.1). The actual scattering mean free paths are given by S✏j = Sj/✏2, and aremuch larger than the wavelength �. The matrix Cj is computed as in (7.13), using thecoe�cients defined in appendix A, for an isotropic random medium that is stationaryin x

1

, x2

and z, with covariance

E[⌫(~x)⌫(~x0)] = exp✓

� |~x� ~x0|2

2`2

◆

, ` = �.

The left plot is for a waveguide with dimensions L1

= 3.03� and L2

= 5.84�, so thatN = 64. In the right plot L

1

= 4.08� and L2

= 5.77�, so that N = 84.

22

Note that for any j the eigenvalues µj,s are almost the same for 1 ⌧ s ⌧ Mj ,indicating that the equality in (7.15) holds independent of the multiplicity Mj . More-over, Sj decreases with j, and the rate of decrease acellerates for j close to N . Thatthe curves in Figure 7.1 are not monotone is due to the counting of the amplitudes i.e.,of the pairs (j

1

, j2

) that define the eigenvalues. In the analysis we kept the multiplicityof the eigenvalues to at most 2 by asking that L

1

/L2

be irrational. In the simulationsthe ratio L

1

/L2

is a rational number, and the high indexes where the curves oscillatein the figure correspond to eigenvalues with multiplicity larger than 2.

The scale shown with red in Figure 7.1 is the equipartition distance, up to the ✏�2

factor. This is the range where cumulative scattering by the medium distributes theenergy of the waves uniformly over the modes, independent of their initial state. Wegive more details in the next section, but it is important to note that the equipartitiondistance is larger, by a factor of ten, than all the scattering mean free paths. Thisis very similar to the result obtained for sound waves in random waveguides withstraight boundaries [3, Figure 4.2].

To interpret the results, let us note that the modes '(s)j exp(i�jz), for 1 s Mj ,are superpositions (component-wise) of the plane waves exp(i~Kj ·~x), with wave vectors

~Kj = (±⇡j1/L1,±⇡j2/L2,�j) .

The plus and minus signs are due to the reflections of the waves at the walls ofthe waveguide. Recall that �j =

p

k2 � �j , with eigenvalues �j defined by (3.8) andenumerated in increasing order. When j is small, the wave vector ~Kj is almost alignedwith the range axis, and the waves propagate with large (group) range velocity

1/�0j(!) = co

q

1 � �j/k2 ⇡ co.

They arrive quickly to range Z because they travel along shorter paths, with a smallnumber of reflections at the walls, and are least a↵ected by the random medium. Forthe high index modes �j ⇡ k2, and the wave vectors ~Kj are almost orthogonal tothe range axis. The waves propagate very slowly along range because they strike thewaveguide walls many times. The interaction with the random medium accumulatesover the long travel paths of these modes, and the waves lose coherence over shorterrange scales, as modeled by the small scattering mean free paths.

For any given j the modes '(s)j exp(i�jz) are the superposition of the same planewaves for s = 1 and 2, so their interaction with the medium is the same. This is whythe eigenvalues µj,1 and µj,2 are almost equal.

7.5. The equipartition regime. The transport of energy in the waveguides ismodeled by the evolution equations (7.9). Our goal in this section is to describe theirsolution in the limit Z ! 1.

We begin by writing equations (7.9) as

@ZP(Z) = ⌥�

P�

(Z) = ⌥+�

P�

(Z) �⌥��

P�

(Z), Z > 0, (7.18)

with initial condition P(0) = Po. Here ⌥,⌥± : X ! X are linear operators acting onthe vector space X defined in section 7.3, with values in X. We have ⌥ = ⌥+ � ⌥�and

⌥+�

P�

j(Z) =

NX

l=1

⌥+jl�

Pl�

(Z) , ⌥��

P�

j(Z) =

NX

l=1

⌥�jl�

Pj�

(Z), (7.19)

23

with operators ⌥±jl : Xl ! Xj acting on the spaces Xl of Hermitian matrices definedin section 7.3, and given by

⌥+jl�

U�

=

Z 1

�1dzE

�

MAA,jl(z) U M?AA,jl(0)

ei(�l��j)z ,

⌥�jl�

U�

= �⇣

Ql U + U Q?l

⌘

�jl . (7.20)

We may think of ⌥+jl and ⌥�jl as modeling the inflow/outflow of energy of the j ⌧ l

modes, because

⌥+jl(U) � 0 and trace�

⌥�jl(U)�

� 0 , (7.21)

for 1 l, j N, and all U 2 Cl, the cone of positive semidefinite matrices in Xl.The limit of P(Z) as Z ! 1 depends on the spectrum of the operator ⌥, and inparticular its kernel, described in the next theorem.

Theorem 7.3. The operator ⌥ has the following spectral properties:(i) The eigenvalues of ⌥ lie in (�1, 0].(ii) The Kernel

�

⌥�

is not trivial, it has an eigenbase, and it intersects the cone

C = C1

⇥ C2

⇥ . . .CN ⇢ X.(iii) The Kernel

�

⌥�

is one dimensional under the additional assumption that

⌥+jl�

U�

> 0 , 8 0 6= U 2 Cl and 1 j, l N. (7.22)

For any initial condition Po 2 C we have P(Z) 2 C for all Z, as shown in appendixE. This is why we are interested in the cone C of the space X. The assumption(7.22) says that there is positive flux of energy for all the waveguide modes. Itis the generalization of the condition stated in [7, Section 20.3.3] which gives theequipartition regime for sound waves. The statement there is that the power spectraldensity of the fluctuations of the wave speed does not vanish when evaluated at thedi↵erences of the wavenumbers of the modes. Our condition (7.22) is similar, butwith Mj ⇥Mj matrices. The following corollary follows immediately from parts (i)and (iii) of Theorem 7.3.

Corollary 7.4. Suppose that condition (7.22) holds, and let Uo be the uniquevector that spans Kernel

�

⌥�

\ C, normalized by kUok = kPok. We have�

�P(Z) �Uo�

� C (1 + Z)m⌥ e�⌥ Z ,

where �⌥

is the smallest (in magnitude) nonzero eigenvalue of ⌥, and m⌥

is its

multiplicity.

We display in Figure 7.2 the element Uo 2 Kernel�

⌥�

for the same two randomwaveguides considered in Figure 7.1. We normalize Uo so that its maximum entry isequal to one. Because Uo is a concatenation of N matrices of size Mj⇥Mj , we embedit in a square matrix for display purposes. The entries of interest in the square matricesdisplayed in Figure 7.2 are the Mj ⇥Mj blocks along the diagonal. We note from thefigure that the result is almost the matrix identity. Therefore, Corollary 7.4 says thatin the limit Z ! 1, the energy is distributed uniformly over the modes, independentof the initial mode power distribution Po. This is the equipartition regime, and it isreached when the waves travel beyond the equipartition distance

Leq

= 1/|�⌥

|. (7.23)24

Fig. 7.2. The element (matrix) Uo

2 Kernel�⌥�for the same two waveguides considered in

Figure 7.1. The top plots are for the waveguide with N = 64 modes and the bottom plots for thewaveguide with N = 84 modes. We display U

o

from the lateral side (left) to show the magnitude of

its entries (left), and from above (right) to show that it is almost diagonal.

Proof of Theorem 7.3:Item (i). Let 0 6= � 2 C be an eigenvalue of ⌥ and U 2 X an associated

eigenvector. Therefore, ⌥�

U�

= �U, or componentwise,

�Uj = ⌥�

U�

j= ⌥

�

U�?

j=�

�Uj�?

= �Uj , 1 j N,

where we used definitions (7.19-7.20) to obtain the second equality. Consequently,the eigenvalues of ⌥ are real valued. To see that they cannot be positive, we use thatP(Z) 2 C for all Z as shown in appendix E, and the conservation of energy

NX

j=1

trace�

Pj(Z)�

=NX

j=1

trace�

Pj,o�

, Z � 0.

Since Pj(Z) 2 Cj , we have 0 P ssj (Z) trace�

Pj(Z)�

, for all Z � 0, 1 j N and1 s Mj . Moreover

�

�P 12j (Z)�

� q

P 11j (Z)P22

j (Z) 12

trace�

Pj(Z)�

, Z � 0,

when Mj = 2, where we used the Cauchy-Schwarz inequality. Therefore

supZ�0

�

�P(Z)�

� NX

j=1

trace(Po), 8Po 2 C.

25

Now consider an arbitrary initial state in X, not necessarily in C. We denote such astate by ePo to distinguish it from a physical initial mode power state that is necessarilyin C, and the corresponding solution of (7.18) by eP (Z). Since any ePo can be writen asa linear combination of elements in C, and (7.18) is linear, eP (Z) is a linear combinationof solutions with initial states in C, which are bounded as shown above. We concludethat all trajectories are bounded, independent of the initial state.

Now, let U be an eigenvector of ⌥ for an eigenvalue �, and set eP0

= U. Thesolution of (7.18) is eP(Z) = U e�Z and it is uniformly bounded if and only if � 0.

Item (ii). To characterize Kernel(⌥) we rewrite the conservation of energy as�

⌥(U),1�

X=�

U,⌥?(1)�

X= 0 , 8U 2 X, (7.24)

where 1 is the vector of concatenated Mj ⇥Mj identity matrices, for j = 1, . . . , N ,and ⌥? is the adjoint of ⌥. We conclude that 1 2 Kernel

�

⌥⇤�

, and therefore that thekernel of ⌥ is not trivial.

We also infer from the boundedness of the solutions of (7.18) that Kernel�

⌥�

hasan eigenbase. Otherwise the solutions could grow polynomially in Z.

Now let us write the solutions of (7.18) as

P(Z) = Uo +nX

j=1

mjX

q=0

cjq Ujq (Z + 1)qe�jZ , (7.25)

where Uo 2 Kernel�

⌥�

, and n is the number of distinct eigenvalues �j < 0 of ⌥,with multiplicity mj . The finite sequences {Ujq}

mjq=1 are linear independent sets of

generalized eigenvectors corresponding to the eigenvalue �j . Equation (7.25) implies�

�P(Z) �Uo�

� C (1 + Z)m⌥ e�⌥ Z , (7.26)

where we denote by �⌥

the eigenvalue with smallest magnitude and m⌥

its multiplicity.Now, take 0 6= Po 2 C, and therefore, P(Z) 2 C for all Z � 0. Letting Z ! 1 in(7.26) we conclude that Uo 2 C i.e., Uo 2 Kernel

�

⌥�

\ C. Moreover, Uo 6= 0 by theconservation of energy.

Item (iii). We prove that condition (7.22) guarantees a one dimensional kernel ofthe adjoint operator ⌥?, and therefore of ⌥. We consider first a large family of linearmappings defined in terms of ⌥⇤ and restricted to the subspace D ⇢ X of vectors ofdiagonal matrices. We show that this family has the one dimensional kernel span{1}in D. Then we show that Kernel(⌥?) = span{1} in X.

To define the family of mappings, we recall that the elements U = (U1

, . . . , UN ) ofX are vectors of Hermitian matrices which are diagonalized by similarity transforma-tions with orthogonal matrices of their eigenvectors. For any vector V = (V

1

, . . . , VN )of orthogonal matrices we define the transformation 'V : X ! X by

'V�

U�

= V? UV :=�

V ?1

U1

V1

, · · · , V ?NUNVN�

,

with dimensions of Uj and Vj assumed to match for each 1 j N . We also definethe family of operators ⌥?V : X ! X by ⌥?V = '

�1V �⌥? � 'V, or more explicitly,

⌥?V�

U�

= V⌥?�

V? UV�

V? , U 2 X. (7.27)

Its restriction to the subspace D ⇢ X of vectors of diagonal matrices is denoted by

⌥?V�

�D

�

D�

= ⌥?V�

D�

for any D 2 D,

26

and we wish to prove that

Kernel�

⌥?V�

�D

�

= span{1}, 8V. (7.28)

Note that ⌥?V�

�D(1) = 0 by the definition (7.27) and ⌥?(1) = 0.

The statement of the theorem is implied by (7.28). Indeed, take an arbitraryU 2 Kernel

�

⌥?�

. Since U is a vector of Hermitian matrices, there exists a vector V oforthogonal matrices and a vector D of diagonal matrices such that U = V⇤ DV. ThenD 2 Kernel

�

⌥?V�

�D

�

and (7.28) implies that D = ↵1 for some ↵ 2 R. Consequently,

U = V?↵1V = ↵1,

which means that Kernel(⌥?) = span{1}.The proof of (7.28) is based on the Perron–Frobenius theorem for irreducible ma-

trices. We start by computing the matrix representation of the operator ⌥?V using theproperties of the adjoint operators ⌥± ?jl : Xj ! Xl that define ⌥?. Then, we extracta suitable irreducible matrix ⇤V from it to apply the Perron–Frobenius theorem.

The matrix representation of ⌥?V consists of N2 blocks, where the lj–block is

the matrix representation of the operator ⌥+?Vlj �⌥�?Vlj , with ⌥

±?Vlj : Xj ! Xl defined

naturally, in light of (7.27), as

⌥±?Vlj(U) = Vl⌥± ?jl

�

V ?j U Vj�

V ?l , U 2 Xj .

Recall that the dimension of Xj is M2j , so the lj–block has dimension M2

l ⇥M2j . Tobe more precise, consider the case Mj = Ml = 2, and write explicitly

⌥±?Vlj =⇣

�± ss0

lj

⌘

, 1 s M2l , 1 s0 M2j ,

In terms of the canonical basis for Xj

⇢

⇣ 1 00 0

⌘

,⇣ 0 0

0 1

⌘

, 1p2

⇣ 0 11 0

⌘

, 1p2

⇣ 0 i�i 0

⌘

�

=:�

E1

, E2

, E3

, E4

,

we have

�+ ss0

lj =�

Es,⌥+?Vlj(Es0)

�

Xl

=�

Es, Vl⌥+ ?jl

�

V ?j Es0 Vj�

V ?l�

Xl

=�

V ?l Es Vl,⌥+ ?jl

�

V ?j Es0 Vj��

Xl. (7.29)

Note that E1

and E2

are positive semidefinite matrices, and so are V ?l E1 Vl andV ?j E2 Vj . It follows from the explicit expression of ⌥

+ ?jl computed from (7.20) as

⌥+ ?jl (U) =

Z 1

�1dz E

�

M?AA,jl(z)UMAA,jl(0)

e�i(�l��j)z, (7.30)

and equation (7.29), that

�+ ss0

lj � 0 , 1 s, s0 2. (7.31)

27

The remaining entries of ⌥+?Vlj do not have a definite sign, in general. The entries of

⌥�?Vlj are computed similarly, and take the form

�� ss0

lj =�

Es,⌥�?Vlj(Es0)

�

Xl

=�

Es, Vl⌥� ?jl

�

V ?j Es0 Vj�

V ?l�

Xl

=�

Es,�

Vl Cl V?l

�

Es�

Xl�ss0 �jl � 0 , 1 s, s0 2. (7.32)

We used that EsEs0 = Es�ss0 for s, s0 2 {1, 2} for the third equality, and propertiesof positive definite matrices in the last inequality. In summary, the lj–block is of theform

⇣

�+ ss0

lj

⌘

�⇣

�� ss0

lj

⌘

=

�+lj ·· ·

!

�

��ll ·· ·

!

�lj , (7.33)

with 2 ⇥ 2 matrices �±lj with nonnegative entries, and nonnegative diagonal matrices��ll . Cases where Mj or Ml or both are equal to one are handled similarly.

Now, define the M⇥M square matrix

⇤V = Block�

�+lj�

� Block�

��lj�

, (7.34)

where M =NX

j=1

Mj , and observe that the dimension of its kernel satisfies

1 Dim�

Kernel�

⌥?V�

�D

�

Dim�

Kernel�

⇤V�

. (7.35)

This is because for any D 2 Kernel�

⌥?V�

�D

�

, by construction of ⇤V, the M vector

formed by the diagonal entries of D lies in Kernel�

⇤V�

. In particular,�

1, 1, · · · , 1�

2Kernel

�

⇤V�

so the entries in each row of ⇤V sum to zero. Then, Gershgorin’s circletheorem and the special structure of ⇤V give that any eigenvalue �V of ⇤V satisfies

�V v+ssjj � v�jj + v

+ss0

jj +X

l 6=j

MlX

s00=1

v+ss00

jl = 0,

for all 1 s, s0 Mj and s 6= s0. This shows that the largest eigenvalue of �V is zero.Consider now the matrix

⇤V + ↵I,

with ↵ a su�ciently large positive real number such that the diagonal is positive.The spectrum of this matrix is a translation by ↵ of the spectrum of ⇤V, therefore,by the previous discussion ↵ must be its largest positive real eigenvalue. Assumingthat Block

�

�+lj�

is irreducible, ⇤V + ↵I is a Perron–Frobenius matrix implying thatits largest positive real eigenvalue, in this case ↵, is simple, and therefore, zero is asimple eigenvalue of ⇤V. Thus, the kernel of ⇤V is one dimensional and so is thekernel of ⌥?

V�

�D, by (7.35). This is precisely (7.28).

It remains to show that the condition (7.22) ensures that Block�

�+lj�

is irreducible,regardless of the transformation 'V. We already know that this is a matrix with

28

nonnegative entries, but we need to show that each entry is positive. From (7.29) weobserve that this means that1

�+,ss0

lj = trace⇣

Es Vl⌥+ ?jl

�

V ?j Es0 Vj�

V ?l

⌘

> 0 , 1 s Ml, 1 s0 Mj ,

which is equivalent to saying that the diagonal entries of the matrix

Vl⌥+ ?jl

�

V ?j Es0 Vj�

V ?l , 1 s0 Mj ,

are positive. To see that this is true, observe that 0 6= V ?j Es0 Vj 2 Cj for 1 s0 Mj ,and that condition (7.22) and definition (7.30) imply

⌥+ ?jl�

V ?j Es0 Vj�

> 0 , 1 s0 Mj .

The result follows immediately from this. ⇤8. Summary. We presented a rigorous analysis of electromagnetic wave prop-

agation in waveguides with rectangular cross-section. The dielectric materials thatfill the waveguides are lossless isotropic, and contain numerous weak inhomogeneities(imperfections). Consequently, their electric permittivity "(~x) has small fluctuationsin ~x that are uncertain in applications, which is why we model them with a randomprocess. The main result of the paper is a detailed characterization of long rangecumulative scattering e↵ects in random waveguides.

Our method of analysis decomposes the electromagnetic wave field in transverseelectric and magnetic modes, which are propagating and evanescent waves. The modesare coupled by scattering in the random medium, so their amplitudes are randomprocesses. They satisfy a stochastic system of equations driven by the random fluctu-ations of the permittivity "(~x), and can be analyzed at long range using the di↵usionapproximation theorem. The result is a detailed characterization of the loss of co-herence of the modes, the depolarization of the waves due to scattering, and thetransport of energy by the modes. Loss of coherence means that the expectation ofthe mode amplitudes (the coherent part) is overwhelmed by their random fluctuations(the incoherent part) once the waves travel beyond distances called scattering meanfree paths. These are range scales that depend on the modes, the wavelength andthe covariance of the fluctuations of "(~x). Our analysis of long range transport ofenergy shows how scattering in the random medium redistributes the energy amongthe waveguide modes. In particular, it identifies a range scale, called the equiparti-tion distance, beyond which the energy is uniformly distributed among the waveguidemodes, independent of the initial conditions.

Our results have applications in long range communications and imaging in waveg-uides. See for example the imaging and time reversal studies [4, 3, 8] that arebased on the theory of sound wave propagation in random waveguides developedin [15, 6, 9, 1, 2]. Here we extended the theory to electromagnetic wave propagationin random waveguides.

Acknowledgements. The work of R. Alonso was partially supported by theAFOSR Grant FA9550-12-1-0117 and the ONR Grant N00014-12-1-0256. The workof L. Borcea was partially supported by the AFOSR Grant FA9550-12-1-0117, theONR Grant N00014-12-1-0256 and by the NSF Grants DMS-0907746, DMS-0934594.

Appendix A. The coupling coe�cients. Let us denote by

O✏DD =h

@z⌫(~x) �✏

2@z⌫

2(~x)i

I , (A.1)

1For dimension one, it is understood that the basis is given by {E1} = {1}.

29

and

O✏DU = ik⌫(~x)I�i

kr⌫(~x)r · + i✏

2kr⌫2(~x)r· , (A.2)

the perturbation operators in (4.7) acting on D and U, where I is the identity. Sim-ilarly, we let O✏UD be the perturbation operator in (4.8) acting on U. The couplingcoe�cients in equations (5.6-5.7) are linear combinations of

↵(ss0)✏

jj0 (z) =

r

�jk�s1 +

s

k

�j�s2

!

D

'(s)j ,O✏DD'(s0)j0

E

s

k

�j0�s01 +

r

�j0

k�s02

!

,

�(ss0)✏

jj0 (z) =

r

�jk�s1 +

s

k

�j�s2

!

D

'(s)j ,O✏DU'(s0)j0

E

r

�j0

k�s01 +

s

k

�j0�s02

!

,

⌘(ss0)✏

jj0 (z) =

s

k

�j�s1 +

r

�jk�s2

!

D

'(s)j ,O✏UD'(s0)j0

E

s

k

�j0�s01 +

r

�j0

k�s02

!

,

where h·, ·i denotes inner product in L2(⌦). These expressions become after integra-tion by parts

↵(ss0)✏

jj0 (z) =

r

�jk�s1 +

s

k

�j�s2

!

s

k

�j0�s01 +

r

�j0

k�s02

!

⇥h

@z (ss0)jj0 (z) �

✏

2@z

(ss0)jj0 (z)

i

, (A.3)

and

�(ss0)✏

jj0 (z) =i

k

r

�jk�s1 +

s

k

�j�s2

!

r

�j0

k�s01 +

s

k

�j0�s02

!

⇥n

(k2 � �j0�s02) (ss0)

jj0 (z) +⇥(ss0)jj0 (z) +

✏

2

h

�j0�s02 (ss0)jj0 (z) � ✓

(ss0)jj0 (z)

io

, (A.4)

and

⌘(ss0)✏

jj0 (z) =i�j�s1p

k�j

s

k

�j0�s01 +

r

�j0

k�s02

!

h

(ss0)

jj0 (z) � ✏ (ss0)jj0 (z)

i

, (A.5)

where we introduced the notation

(ss0)

jj0 (z) =

Z

⌦

dx ⌫(~x)'(s)j (x) ·'(s0)j0 (x) ,

(ss0)

jj0 (z) =

Z

⌦

dx ⌫2(~x)'(s)j (x) ·'(s0)j0 (x) , (A.6)

and

⇥(ss0)

jj0 (z) =

Z

⌦

dx ⌫(~x)r ·'(s)j (x)r ·'(s0)j0 (x) ,

✓(ss0)

jj0 (z) =

Z

⌦

dx ⌫2(~x)r ·'(s)j (x)r ·'(s0)j0 (x) . (A.7)

30

Note that since r · �(1)j = 0, we have

⇥(ss0)

jj0 (z) = �s2�s02⇥(22)

jj0 (z) , (A.8)

and similar for ✓(ss0)

jj0 (z).The coupling coe�cients in (5.6) are given by

M (ss0)

AA,jj0(z) + ✏m(ss0)AA,jj0(z) =

1

2

h

↵(ss0)✏

jj0 (z) + �(ss0)✏jj0 (z) + ⌘

(ss0)✏jj0 (z)

i

, (A.9)

M (ss0)

AB,jj0(z) + ✏m(ss0)AB,jj0(z) =

1

2

h

↵(ss0)✏

jj0 (z) � �(ss0)✏jj0 (z) + ⌘

(ss0)✏jj0 (z)

i

, (A.10)

M (ss0)

AV,jj0(z) + ✏m(ss0)AV,jj0(z) =

1

2

h

↵(ss0)✏

jj0 (z) + ⌘(ss0)✏jj0 (z)

i

, (A.11)

M (ss0)

Av,jj0(z) + ✏m(ss0)Av,jj0(z) = �

i

2(�1)s

0�(ss

0)✏

jj0 (z) , (A.12)

and those in (5.7) are

M (ss0)

BA,jj0(z) + ✏m(ss0)BA,jj0(z) =

1

2

h

↵(ss0)✏

jj0 (z) + �(ss0)✏jj0 (z) � ⌘

(ss0)✏jj0 (z)

i

, (A.13)

M (ss0)

BB,jj0(z) + ✏m(ss0)BB,jj0(z) =

1

2

h

↵(ss0)✏

jj0 (z) � �(ss0)✏jj0 (z) � ⌘

(ss0)✏jj0 (z)

i

, (A.14)

M (ss0)

BV,jj0(z) + ✏m(ss0)BV,jj0(z) =

1

2

h

↵(ss0)✏

jj0 (z) � ⌘(ss0)✏jj0 (z)

i

, (A.15)

M (ss0)

Bv,jj0(z) + ✏m(ss0)Bv,jj0(z) = �

i

2(�1)s

0�(ss

0)✏

jj0 (z) . (A.16)

Because �(ss0)✏

jj0 (z) and ⌘(ss0)✏jj0 (z) are imaginary, we obtain the relations

M (ss0)

BB,jj0(z) = M(ss0)AA,jj0(z) , m

(ss0)BB,jj0(z) = m

(ss0)AA,jj0(z) , (A.17)

M (ss0)

BA,jj0(z) = M(ss0)AB,jj0(z) , m

(ss0)BA,jj0(z) = m

(ss0)AB,jj0(z) , (A.18)

M (ss0)

BV,jj0(z) = M(ss0)AV,jj0(z) , m

(ss0)BV,jj0(z) = m

(ss0)AV,jj0(z) , (A.19)

M (ss0)

Bv,jj0(z) = M(ss0)Av,jj0(z) , m

(ss0)Bv,jj0(z) = m

(ss0)Av,jj0(z) . (A.20)

Similarly, the coupling coe�cients in equations (5.8-5.9) satisfied by the evanes-cent modes are given by

M (ss0)

V A,jj0(z) + ✏m(ss0)V A,jj0(z) = ↵

(ss0)✏jj0 (z) + �

(ss0)✏jj0 (z) , (A.21)

M (ss0)

vA,jj0(z) + ✏m(ss0)vA,jj0(z) = i(�1)

s⌘(ss0)✏

jj0 (z) , (A.22)

M (ss0)

V V,jj0(z) + ✏m(ss0)V V,jj0(z) = ↵

(ss0)✏jj0 (z) , (A.23)

M (ss0)

V v,jj0(z) + ✏m(ss0)V v,jj0(z) = �i(�1)

s0�(ss0)✏

jj0 (z) , (A.24)

M (ss0)

vV,jj0(z) + ✏m(ss0)V v,jj0(z) = i(�1)

s⌘(ss0)✏

jj0 (z) , (A.25)

and by

M (ss0)

V B,jj0(z) = M(ss0)V A,jj0(z) , m

(ss0)V B,jj0(z) = m

(ss0)V A,jj0(z) , (A.26)

M (ss0)

vB,jj0(z) = M(ss0)vA,jj0(z) , m

(ss0)vB,jj0(z) = m

(ss0)vA,jj0(z) . (A.27)

31

In equations (5.8-5.9) we use only the leading part of these coe�cients denoted by

the capital letter M , as in M (ss0)

V A,jj0(z).

Appendix B. Analysis of the evanescent modes. Consider the 2⇥2 system

@zV(s)j (z) + �j✓

0 11 0

◆

V(s)j (z) = ✏F(s)j (z,V) , (B.1)

for vectors

V(s)j (z) =

V (s)j (z)

v(s)j (z)

!

, (B.2)

which we string together in the infinite vector V(z). The system (5.8-5.9) is of thisform, with right-hand side

F (s)j (z,V) =

F (s)j (z)

f (s)j (z)

!

+X

j0>N

Mj0X

s0=1

M (ss0)

V V,jj0(z) M(ss0)V v,jj0(z)

M (ss0)

vV,jj0(z) 0

!

V(s0)

j0 (z) .

(B.3)We neglect the O(✏2) remainder because it plays no role in our setup.

We diagonalize (B.1) by writing V(s)j in the orthonormal basis {u+, u�} of R2,

where u± = 1p2

✓

1±1

◆

are the eigenvectors of matrix

✓

0 11 0

◆

for eigenvalues

±1. Explicitly, we write

V(s)j (z) = ✓(s)+j (z)u

+ + ✓(s)�j (z)u� , (B.4)

where ✓(s)±j (z) are decaying and growing evanescent waves, satisfying

[@z ± �j ] ✓(s)±j (z) = ✏u± ·F (s)j (z,V) , (B.5)

the initial condition

✓(s)+j (0+) = u+ · V(s)j (0+) =

p2E(s)j,o , (B.6)

and the end condition

✓(s)�j (zmax) = 0. (B.7)

We obtain after integrating equations (B.5) and using (B.4) that

V (s)j (z) =E(s)j,oe

��jz +✏p2

Z z

0

d⇣u+ ·F (s)j (⇣,V)e��j(z�⇣)�

✏p2

Z zmax

zd⇣ u� ·F (s)j (⇣,V)e�j(z�⇣) , (B.8)

and

v(s)j (z) =E(s)j,oe

��jz +✏p2

Z z

0

d⇣u+ ·F (s)j (⇣,V)e��j(z�⇣)+

✏p2

Z zmax

zd⇣ u� ·F (s)j (⇣,V)e�j(z�⇣) , (B.9)

32

for j > N and 1 s Mj . We only take the term E(s)j,oe��jz because of the conditions(B.6) and (B.7).

Equations (B.8-B.9) form an infinite system of integral equations for the vectorV(z), which we write in compact form as

[I� ✏Q]V(z) = F(z) . (B.10)Here F(z) is the vector given by the concatenation of

F(s)j (z) =

✓

11

◆

h

E(s)j,oe��jz +

✏

2

Z z

0

d⇣ u+ ·

F (s)j (⇣)

f (s)j (⇣)

!

e��j(z�⇣)i

+

✓

�11

◆

✏

2

Z zmax

zd⇣ u� ·

F (s)j (⇣)

f (s)j (⇣)

!

e�j(z�⇣),

for j > N and 1 s Mj . The operator in the left-hand side is a perturbation ofthe identity I, with Q the linear integral operator that takes the infinite vector V(z)and returns the infinite vector obtained by concatenating the entries

1p2

Z z

0

d⇣X

j0>N

Mj0X

s0=1

u+ ·

M (ss0)

V V,jj0(⇣) M(ss0)V v,jj0(⇣)

M (ss0)

vV,jj0(⇣) 0

!

V(s0)

j0 (⇣)e��j(z�⇣) ⌥

1p2

Z zmax

zd⇣

X

j0>N

Mj0X

s0=1

u� ·

M (ss0)

V V,jj0(⇣) M(ss0)V v,jj0(⇣)

M (ss0)

vV,jj0(⇣) 0

!

V(s0)

j0 (⇣)e�j(z�⇣) ,

for j > N and 1 s Mj . We show in Lemma B.1 in Appendix B.2 that Q is abounded linear operator, so we can solve (B.10) using Neumann series

V(z) = [I + ✏Q + . . .]F(z) ,to obtain

V (s)j (z) =E(s)j,o e

��jz +✏

2

Z z

0

d⇣h

F (s)j (⇣) + f(s)j (⇣)

i

e��j(z�⇣)�

✏

2

Z zmax

zd⇣

h

F (s)j (⇣) � f(s)j (⇣)

i

e�j(z�⇣) + O(✏2) , (B.11)

and

v(s)j (z) =E(s)j,o e

��jz +✏

2

Z z

0

d⇣h

F (s)j (⇣) + f(s)j (⇣)

i

e��j(z�⇣)+

✏

2

Z zmax

zd⇣

h

F (s)j (⇣) � f(s)j (⇣)

i

e�j(z�⇣) + O(✏2) , (B.12)

for j > N and 1 s Mj . The result in Lemma 5.1 follows from (B.11-B.12) andthe approximations

Z z

0

d⇣ (⇣)e��j(z�⇣) =

Z

0

�zdt (z + t)e��t ⇡

Z

0

�1dt (z + t)e��t ,

andZ z

max

zd⇣ (⇣)e�j(z�⇣) =

Z zmax

�z

0

dt (z + t)e��j |t| ⇡Z 1

0

dt (z + t)e��j |t| ,

for an arbitrary bounded function (z). The error in these approximations is similarto e��jz, and we can negle