Electromagnetic wave propagation in random waveguides
We study long range propagation of electromagnetic waves in random waveguides with rectangular cross-section and perfectly conducting boundaries. The waveguide is filled with an isotropic linear dielectric material, with randomly fluctuating electric permittivity. The fluctuations are weak, but they cause significant cumulative scattering over long distances of propagation of the waves. We decompose the wave field in propagating and evanescent transverse electric and magnetic modes with random amplitudes that encode the cumulative scattering effects. They satisfy a coupled system of stochastic differential equations driven by the random fluctuations of the electric permittivity. We analyze the solution of this system with the diffusion approximation theorem, under the assumption that the fluctuations decorrelate rapidly in the range direction. The result is a detailed characterization of the transport of energy in the waveguide, the loss of coherence of the modes and the depolarization of the waves due to cumulative scattering.
Key words. Waveguides, electromagnetic, random media, asymptotic analysis.
AMS subject classifications. 35Q61, 35R60
We study electromagnetic wave propagation in waveguides. There is extensive applied literature on this subject [17, 18, 16, 11, 5, 19] which includes open and closed waveguides, waveguides with losses, boundary corrugation and heterogeneous media. Here we consider the setup illustrated in Figure LABEL:fig:setup, for a waveguide with rectangular cross-section , filled with an isotropic linear dielectric material. The waves are trapped by perfectly conducting boundaries and propagate in the range direction . The cross-range coordinates are . The main goal of the paper is to analyze long range wave propagation in waveguides with imperfections. We refer to [14, 6, 9, 1, 2] and [7, Chapter 20] for rigorous mathematical studies of long range wave propagation in imperfect acoustic waveguides, and to [4, 3, 8] for their application to imaging and time reversal. Here we extend the theory to electromagnetic waves.
We focus attention on waveguides with imperfections due to a heterogeneous dielectric, but the ideas should extend to waveguides with corrugated boundaries. Such waveguides can be analyzed by changing coordinates to flatten the boundary fluctuations as was done in  for sound waves, or by using so-called local normal mode decompositions as proposed in [18, chapter 9]. Our waveguide has straight walls and is filled with a dielectric material that has numerous inhomogeneities (imperfections). These are weak scatterers, so their effect is negligible in the vicinity of the source of the waves. However, the inhomogeneities cause significant cumulative wave scattering over long ranges. To quantify the cumulative scattering effects we study the following questions: How are the modal wave components coupled by scattering? How do the waves depolarize? How do the waves lose coherence? Can we calculate from first principles the scattering mean free paths, which are the range scales over which the modal wave components lose coherence? How is energy transported at long ranges in the waveguides? Can we quantify the equipartition distance where cumulative scattering is so strong that the waves lose all information about the source? How does the equipartition distance compare with the mode dependent scattering mean free paths?
To answer these questions we model the scalar valued electric permittivity of the dielectric as a random process. The random model is motivated by the fact that in applications the imperfections can never be known in detail. They are the uncertain microscale of the medium, the fluctuations of in , so we model them as random. The fluctuations are small, on a scale (correlation length) comparable to the wavelength. We assume that there is no dissipation in the medium, meaning that is real, positive. Complex valued permittivities which are typically required by causality i.e., Kramers-Kronig relations, can be incorporated in the model. We do not consider them here for simplicity, and because we are concentrating on the analysis at a single frequency . Extensions to multi frequency analysis of wave propagation in dispersive and lossy media can be done, using techniques like in [6, 14, 1] and [7, chapter 20], but we leave them for a different publication.
The paper is organized as follows: We begin in section LABEL:sect:form with the setup. We state Maxwell’s equations and the boundary conditions satisfied by the electromagnetic field. Then we follow the approach in  and solve for the components and of the electric and magnetic fields in the range direction. We obtain a system of partial differential equations for the components and of the fields in the cross-range plane. We analyze in section LABEL:sect:ideal its solution and in ideal waveguides with constant permitivity . It is a superposition of uncoupled transverse electric and transverse magnetic modes. The random model of the waveguide is introduced in section LABEL:sect:asympt. Because the amplitude of the fluctuations of is small, of order , the system of equations for and is a perturbation of that in ideal waveguides. The remainder of the paper is concerned with the asymptotic analysis of and at long ranges, in the limit . We consider long ranges because the limit of and is the same as the ideal waveguide solution and when the waves do not propagate far from the source. Our analysis is based on the decomposition of and in transverse electric and magnetic modes, with random amplitudes that encode the cumulative scattering effects, as explained in section LABEL:sect:modec. The long range scaling and the diffusion limit approximation for analyzing the wave field as are stated in section LABEL:sect:diff. The main results of the paper are in section LABEL:sect:transp, where we characterize the limit process. Explicitly, we describe the loss of coherence and depolarization of the waves due to cumulative scattering, and the transport of energy. We also show that as we let the range grow, the waves scatter so much that they eventually reach the equipartition regime, where they lose all information about the source. We end with a summary in section LABEL:sect:summary.
Let , and be the unit vectors along the coordinate axes, and use bold letters with an arrow on top for three dimensional vectors, and bold letters for two dimensional vectors in the cross-range plane. Exlicitly, we write
for the magnetic field , and similarly for the electric field and electric displacement . They satisfy Maxwell’s equations
where and are the current source density and free charge density, and is the magnetic permeability, assumed constant. We denote by
the three dimensional gradient and by and the curl and divergence operators.
The current source density
models a source at the origin of range, supported in the interior of . The Fourier transform of the free charge density can be obtained from the continuity of charge derived from (LABEL:eq:M1) and (LABEL:eq:M4)
It vanishes at ranges .
The electric displacement is proportional to the electric field
with scalar valued, positive and bounded electric permittivity . The analysis is for a single frequency, so we simplify the notation by omitting henceforth from the arguments of the fields.
2.1 The system of equations
We study the evolution of the two dimensional vectors and for . They determine the components and in the range direction of the electric and magnetic fields, as follows from equations (LABEL:eq:M1)-(LABEL:eq:M2)
with the perpendicular gradient in the cross-range plane. The system of equations for and is
Here is the gradient in the cross-range plane, and we let denote the rotation of any vector by degrees, counter-clockwise.
Note that equations (LABEL:eq:f6)-(LABEL:eq:f9) contain all the information in the Maxwell system (LABEL:eq:M1)-(LABEL:eq:M4). Indeed, (LABEL:eq:M3) follows from (LABEL:eq:f6) and (LABEL:eq:f8)
because for any twice continuously differentiable function . Similarly, (LABEL:eq:M4) follows from (LABEL:eq:f7) and (LABEL:eq:f9)
where we used (LABEL:eq:f3) and the continuity of charge relation (LABEL:eq:f5).
2.2 Boundary conditions
The boundary conditions at the perfectly conducting boundary are [12, Chapter 8]
for and . The outer normal at is independent of the range and is orthogonal to . Thus, equations (LABEL:eq:f13) say that the tangential components of the electric field vanish at the boundary. Explicitly,
We need more boundary conditions at to specify uniquely the solution of (LABEL:eq:f8-LABEL:eq:f9), but they can be derived from Maxwell’s equations (LABEL:eq:M1-LABEL:eq:M2), conditions (LABEL:eq:f10), and our assumptions on the source density (LABEL:eq:f4), as explained in section LABEL:sect:ideal.
The fields are bounded and outgoing at . We explain in section LABEL:sect:formMod that the causality of the problem in the time domain allows us to restrict the fluctuations of to a finite range interval, and thus justify the outgoing boundary conditions.
2.3 Conservation of energy
The fields and satisfy an energy conservation relation, stated in the following proposition, and used in the analysis in section LABEL:sect:transp.
For any , we have the conservation relation
where the bar denotes complex conjugate.
is the time average of the Poynting vector of a time harmonic wave [12, chapter 7]. Therefore,
is twice the flux of energy in the range direction, and (LABEL:eq:ENC) states that it is conserved for all .
To derive (LABEL:eq:ENC) we obtain from (LABEL:eq:M1)-(LABEL:eq:M2) that
and from the divergence theorem that
The boundary term vanishes because of the boundary conditions (LABEL:eq:f10)
and the integrand in the second term satisfies
The current source density is supported at , so we conclude that
The conservation relation (LABEL:eq:ENC) follows by taking the real part in this equation.
3 Ideal waveguides
Maxwell’s equations are separable in ideal waveguides with constant permitivity , and it is typical to solve for the longitudinal components and of the electric and magnetic fields, which then define and [12, chapter8]. The solution is given by a superposition of waves, called modes. They are propagating and evanescent waves and solve Maxwell’s equations with boundary conditions (LABEL:eq:f10). We describe the modes in section LABEL:sect:ModeDec, and then write the solution in section LABEL:sect:amplit.
3.1 The waveguide modes
The longitudinal components of the electric and magnetic fields satisfy the boundary conditions
The first condition is just (LABEL:eq:f10), and the second follows from Maxwell’s equations (LABEL:eq:M1-LABEL:eq:M2). Indeed, (LABEL:eq:M2) gives
so the normal component of at satisfies
Similarly, we obtain from equation (LABEL:eq:M1) that
and the boundary condition (LABEL:eq:f10) implies that
The Neumann boundary condition (LABEL:eq:BCHz) on follows from this equation and (LABEL:eq:f11).
The waveguide modes are solutions of Maxwell’s equations that depend on the range as , with mode wavenumber to be defined. We write them as
and similar for the longitudinal components, which satisfy
Here is the Laplacian in , is the wavenumber and is the wave speed.
3.1.1 Spectral decomposition of the Laplacian
The Laplacian operator acting on functions with homogeneous Dirichlet conditions is symmetric negative definite, with countable eigenvalues
The indexes and are natural numbers satisfying the constraint . We associate the pair to the index because is countable, and enumerate the eigenvalues in increasing order.
Similarly, the Laplacian operator acting on functions with homogeneous Neumann conditions is symmetric negative semidefinite, with the same eigenvalues as (LABEL:eq:P10), and eigenfunctions
Thus, we see that the electric and magnetic fields have the same mode wavenumbers , which take the discrete values . We write them as
to emphasize that only the first are real. The infinitely many modes that correspond to eigenvalues are evanescent. We assume that , so there are no standing waves in the waveguide.
3.1.2 The transverse electric and magnetic modes
It follows immediately from (LABEL:eq:bH), (LABEL:eq:bD), (LABEL:eq:P10Ez) and (LABEL:eq:P10Hz) that and are given by superpositions of the vectors and . Thus, we define the vectors
The vectors indexed by correspond to transverse electric (TE) modes. Indeed, they satisfy
so when we set in (LABEL:eq:f7) we get . Similarly, the vectors indexed by correspond to transverse magnetic (TM) modes. They satisfy
and give by equation (LABEL:eq:f6).
The superposition of and in the definition of the fields and 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 arrived at the same wave decomposition if we worked directly with the transverse components and of the fields. This observation is relevant because when the permittivity varies in , as in the random waveguide, it is no longer possible to solve independently for the longitudinal wave fields and .
where is the rotated magnetic field scaled by . It is convenient to work in the and variables because as we see below, they satisfy the same boundary conditions and have the same physical units. Note from (LABEL:eq:TE) and (LABEL:eq:TM) that are eigenfunctions of the vector Laplacian
for , with boundary conditions
The index corresponds to the multiplicity of the eigenvalues. We can limit the multiplicity of by assuming that the waveguide dimensions satisfy . This implies that
When and either or are zero, , and only the TE modes exist. Otherwise .
The eigenfunctions satisfy the orthogonality relations
and is a complete set that can be used to describe an arbitrary electromagnetic wave field in the waveguide [12, chapter8].
The boundary conditions (LABEL:eq:P9) are consistent with the conditions satisfied by and , derived from Maxwell’s equations. Indeed, equations (LABEL:eq:f7), (LABEL:eq:f10) and the assumption (LABEL:eq:f4) on the source density give that
Moreover, equation (LABEL:eq:f11) says that
For the electric displacement we already know from (LABEL:eq:f10) that
The divergence condition follows from (LABEL:eq:bD) and (LABEL:eq:CU1)
and since , it is consistent with the conservation of charge.
3.2 The solution in ideal waveguides
We expand and in the basis and associate to each a mode, which is a propagating or evanescent wave. We rename the fields and to remind us that we are in the ideal waveguide.
Using the identities
we obtain that
for . The normalization coefficients and are not important here, and could be absorbed in the mode amplitudes. We use them for consistency with the mode expansions for the random waveguide in section LABEL:sect:modec. There the normalization symmetrizes the system of equations satisfied by the mode amplitudes.
The amplitudes in (LABEL:eq:ID1-LABEL:eq:ID2) are constant on each side of the source, and are determined by the source density and the outgoing boundary conditions. There are no backward going modes to the right of the source, at positive ranges, so we can set . Similarly, we let . The remaining amplitudes are obtained from the source conditions
Substituting (LABEL:eq:ID1-LABEL:eq:ID2) in these conditions and using the orthogonality relations (LABEL:eq:orthog), we get
for the propagating modes and
for the evanescent modes.
3.2.1 Energy conservation
The energy conservation is obvious in this case, because the amplitudes are constant. Substituting (LABEL:eq:ID1-LABEL:eq:ID2) in the expression of the flux and using the orthogonality relations (LABEL:eq:orthog), we obtain that
The flux changes value at , where the source lies, but it is constant for ,
The evanescent modes play no role in the transport of energy.
4 Statement of the problem in the random waveguide
We begin with the model of the small fluctuations. Then we write the perturbed system of equations for the wave fields, which we analyze in the remainder of the paper.
4.1 Model of the fluctuations
Let us denote by the index of refraction
It is the ratio of the electromagnetic wave speeds and in the homogeneous and heterogeneous medium, respectively. We model the electrical permittivity by
where is a dimensionless random function assumed twice continuously differentiable, with almost sure bounded derivatives. It has zero mean
and it is stationary and mixing in . We refer to [15, Section 4.6.2] for a precise statement of the mixing condition. It means in particular that the covariance
is integrable in . The amplitude of the fluctuations in (LABEL:eq:f16) is scaled by , the small parameter in our asymptotic analysis.
The indicator function in (LABEL:eq:f16) limits the support of the fluctuations to the range interval , where denotes a range that is close to zero, but strictly larger than it. The bounded support of the fluctuations is needed to state the outgoing boundary conditions on the electromagnetic wave fields, and may be justified in practice by the causality of the problem in the time domain. During a finite observation time , the waves are influenced by the medium up to a finite range so we may truncate the fluctuations beyond the range . That there are no fluctuations at negative ranges may be motivated by two facts: First, the source is at and we wish to study the waves at positive ranges. Second, we will consider a regime where the backscattered field is negligible. Thus, we may neglect at the waves that come from , and truncate the fluctuations at .
4.2 The perturbed system of equations in the random waveguide
We work with the electric displacement and the scaled rotated magnetic field , defined in equation (LABEL:eq:defU). As we explained in the previous section, this is convenient because the fields satisfy the same boundary conditions and have the same units.
The equations for and follow from (LABEL:eq:f3), (LABEL:eq:f8-LABEL:eq:f9), (LABEL:eq:f16) and (LABEL:eq:defU). We have
for the electric displacement and
for the rotated magnetic field, where we used that the fluctuations are supported away from the source. Morever, substituting the model (LABEL:eq:f16) of the fluctuations, we obtain