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Electromagnetic Waves in Random Media: A Supersymmetric Approach
Roger Balian, Jean-Jacques Niez
To cite this version:
Roger Balian, Jean-Jacques Niez. Electromagnetic Waves in Random Media: A Supersymmetric Approach. Journal de Physique I, EDP Sciences, 1995, 5 (1), pp.7-69. �10.1051/jp1:1995114�. �jpa- 00247044�
Classification Physics Abstracts
41.10H 42.20 02.50
Electromagnetic Waves in Random Media: A Supersymmetric Approach
Roger Balian(~) and Jean-Jacques Niez(~)
(~) CEA, Service de Physique Théorique, CE-Saclay, 91191 Gif-sur-Yvette Cedex, France (~) DAA/Système. CEA-CESTA, 33114 Le Barp, France
(Received 13 July1994, received in final form 19 September 1994, accepted 23 September 1994)
Abstract. A general method is set up, which casts problems of electromagnetic waves in
random media into a systematic formalism akin to that of supersymmetric quantum field theory, by analogy with electrons in disordered metals. The characteristic functional of the field is related
to that of trie medium by means of a diagrammatic expansion; for a linear medium, trie vertices
are trie cumulants of trie permittivity, conductivity and permeability. Among trie auxiliary fields introduced to account for trie field equations, fermionic ones can be eliminated by discarding closed loops m the diagrams. A matrix Green function relates the expectation value of trie field to electric and magnetic monopole sources; its general structure and properties are reviewed.
The versatility of trie approach, which allows us to take advantage of perturbative and variational techniques drawn from quantum field theory or statistical mechamcs, is fllustrated by examples:
electromagnetic response of a weakly disordered medium, representation of correlations and of energy dissipation by means of four-leg diagrams. The couphng induced by disorder between
long- and short-range effects, between transverse and longitudinal parts of the wave, is accounted for. The convergence of the expansions can be improved by making them self-consistent, and
also by charactenzing the medium by the statistics of its polarizability rather than that of its
permittivity, which partly accounts for screemng
or depolanzation. The resulting expansions are
of Padé type. Bruggeman's approximation is recovered in the low-frequency, weak-disorder limit;
corrections are evaluated. An intricate structure for the electromagnetic response, involving several potes and a transition fine, anses in the high-frequency, weak-disorder limit.
1. Introduction and Outline
The propagation and attenuation of electromagnetic waves in disordered media is a subject of current interest, both for theoretical reasons and for practical or technical applications Ill. The
study of this problem requires merging two mgredients, the characterization of the geometric properties of the medium, and the solution of the wave equation. Much effort has been devoted
to the first aspect. In particular, for granular materais such as metal-insulator composites, the possibility of percolation raises many questions which are being studied extensively. We shall focus bene on the second aspect of the problem, assummg that the 8taiisVic~Li properties
© Les Editions de Physique 1995
JOURNAL DE PHYSIQUE I N°1
of the medium are explicitly given. Consider for instance a granular medium, each component of which has a linear behaviour characterized by its conductivity and susceptibilities; instead of starting from the sizes and shapes of the grains, we suppose that we have already found the
two- and many-point correlation functions for the local conductivity and susceptibilities, which
are treated as random scalar fields. From such a characterization it still remains to denve the expectation value of the electromagnetic field, as well as the correlations between the fields at varions points or at varions times, such correlations being induced by the randomness of the medium. Their whole set, for the electric and magnetic fields E and H, is embedded in the generating functional
F (jt(r, t), mt(r, t)) e In exp Î d~r dt jt(r, t) E(r, t) + mt(r, t) H(r, t)jl, (1.1)
where jt and mt are test fields, and where the average is taken over the disorder of the medium.
The expansion of F in powers of jt and mt produces to first order the required expectation values, to second order the two-point correlations and also the energy of the field, to higher
orders the cumulants of the many-point correlations. Our purpose is first to relate Fin a formai but systematic fashion to the characteristic functional for the conductivity and susceptibilities, then to evaluate it approximately by means of systematic perturbative and vanational methods.
We shall rely on several techniques issued from statistical mechanics and quantum field
theory. In the former case the randomness of the field is due to thermal fluctuations, in the latter case to quantum fluctuations; here this randomness follows from that of the medium.
The electromagnetic field is supposed to be generated by some externat currents and charges;
these play the same rôle as the sources which are used in quantum field theory to generate by
denvation the Green functions, but the sources have here a physical interpretation. We regard
them as given once and for ail while we are averaging over the probabilistic realizations of the medium.
There exist two systematic field theoretic approaches to the statistical mechanics of disor- dered media. The first one, the so-called rephca trick [2j, is based on the introduction of n
identical mental copies of the system under study. Averaging over disorder generates an effec- tive interaction between them. The physical quantities are obtained by taking a limit n - o,
as m the theory of polymers [31. This method has recently been adapted to the scattering of
electromagnetic waves in composite media [4, Si. However its use requires an analytic contin- uation from positive integer values of n towards n - o. The validity of this extrapolation
may be questioned [6]. Moreover the possibility of breaking the replica symmetry should be controlled.
We therefore resort to the second general method, that of supersymmetric fields. The basic idea consists in introducing, beside the physical random electromagnetic field, other fluctuating auxiliary fields, which account for the varions equations, and which allow us to average over
the randomness [7, 8]. Some of these fields are ordinary, "bosonic" ones, some are "fermionic",
I.e., are elements of an anticommuting Grassmann algebra, and there is a supersymmetry of
the theory under transformations mixing them. The outcome will be a representation of the
generating functional F as a functional integral over the above fields. The formai analogy
of such an expression with a partition function or with the generating functional for Green functions in field theory will then enable us to make use of standard but powerful, perturbative
or variational, techniques of quantum field theory.
Such a method has already been applied successfully to the propagation of quantum elec-
tronic waves in random media [9]. Its extension to the propagation of electromagnetic waves
raises, however, a number of questions, some technical and some physical. (i) On the one hand, there is a formai analogy between the Maxwell equation in a random dielectric medium
with permittivity e(r) at a given frequency uJ/2~ and the Schrôdinger equation for an elec- tron in a random potential V(r) at a given energy E, within the replacement of uJ~/Lce(r) by 2mh~~ [E- V(r)]. Nevertheless, as discussed at length in reference [4], physical phenomena such
as localization are expected to differ significantly. Indeed, it is only at high frequencies that we may expect similar behaviours: low frequencies for the field correspond to low-energy electrons moving in a weak random potential; a dielectric constant everywhere positive corresponds to
an energy E lying above the highest value of the potential. The nature of electromagnetic transport and the very existence of localization thus cannot be inferred from similar properties for electrons, and refinements of the techniques used in electronic systems are required. (ii) On
the other hand, due to gauge invariance, the vector nature of the electromagnetic field is more subtle thon trie mere occurrence of three components. While the propagation of transverse
waves in a homogeneous medium resembles the motion of electrons in a metal, the longitudi-
nal field describes long-range effects which have no equivalent for electrons. Randomness of the medium entangles the transverse and longitudinal parts, and thus couples the short- and
long-range properties in a way that we must control. (iii) In addition, we deal with classical
electromagnetic waves ~&.hereas electrons m random media are described by their quantum wave function. The latter is not a directly measurable quantity, since wave functions in quantum mechanics are nothing but tools for evaluating expectation values of physical observables; such expectation values are always bilinear (or quadratic) in the wave functions. In contrast, the Green function for electromagnetic waves is in itself the value at some point of space of the measurable field produced by a source lying at some other point. (iv) Moreover, while quantum
wave functions always obey linear equations, the phenomenological equations for electromag-
netic waves in matter can be non-linear. Although we shall non dwell on such types of media, the formalism described below paves the way to their study. (v) Finally screening or depo-
larization effects are long-range features specific to electromagnetic waves. We shall be able to take them partly into account for composite media by means of a change of field variables, performed before averagmg over disorder. As explained in the second half of this introduction and as discussed with more details in Sections 7 and 8, the technique uses is a new extension of the supersymmetry scheme. Here agam the approach has no counterpart in the theory of
electrons.
Keeping in mind these varions points, we start from scratch and resume for completeness m
Section 2 the known technique of auxiliary supersymmetric fields. We illustrate its successive steps by specializing it to the simple example of propagation in a linear medium. However,
we make use of completely general notations so as to better exhibit the versatility of the approach. Indeed, although we shall focus m the subsequent sections on applications to a
medium which behaves linearly and which is macroscopically homogeneous, isotropic and non-
chiral, the formalism of Section 2 would apply without change to any more general situation.
We build up in Section 3 the diagrammatic perturbative expansion for the characteristic functional (1.1), which describes the statistics of waves m a random linear medium. To this aim, we rely on the formol analogy with quantum field theory. It tums ont that introducing
fermionic auxiliary fields was just an algebraic means for eliminating closed loops. We thus find
a twofold simplification, since we can both disregard these fermionic fields and retain only trie diagrams with the following structure: a set of open solid fines refernng to the electromagnetic field, connected to themselves and to one another by dotted fines representing the statistics of the medium (Fig. l). This feature, well known for electrons, is completely general, and applies to ail, linear or non-linear, random media, as shown in the Appendix. The diagrams
thus obtained have the same topological features as the ones obtained through more standard calculations Iii. However the present approach is more systematic: expectation values are taken belote any perturbation expansion, which gives us a complete freedom to use ail available
10 JOURNAL DE PHYSIQUE I N°1
r' r~,,----,, r~ r
. > > .
Fig. I. A diagram contributing to the expectation value of the electromagnetic field. The end dot
corresponds to the point r where the field is evaluated, the origin r' describes the sources, the solid hnes represent unperturbed propagators, and the dotted fines represent cumulants of the dielectric
constant or susceptibility (here, the two-point correlation between ri and r2).
approximation techniques.
The expectation value of the electric and magnetic fields is related to the corresponding
sources (electric and magnetic monopole currents) by means of a matrix Green function (Fig.
l). This object constitutes a basic tool in diagrammatic perturbation theory. It also describes
m itself the physical properties of the average fields, this average being taken over disorder. We give in Section 4 a systematic account of its general properties for an arbitrary, macroscopically
homogeneous medium, namely symmetry and time-reversai properties, mequahties expressing the positivity of dissipation, and short-wavelength behaviour. In particular, in the latter limit, the Fourier transforms of (Dl and (El are not simply proportional; their transverse parts have
a ratio (El and their longitudinal parts a ratio (e~~)~~, which generally differ as soon as the dielectric constant is not uniform.
In Section 5 we apply this formalism to the evaluation of the average electromagnetic re-
sponse of a weakly disordered, macroscopically homogeneous and isotropic medium. Our main purpose is to illustrate the flexibility of the method, which allows us as in quantum field theory
to combine perturbative and variational techniques. We thus evaluate the Green function by starting from an unperturbed form which satisfies at least in part the expected general prop- erties, and which depends on parameters to be determined self-consistently. We shall see that
self-consistency con be implemented in varions ways, fitted to the question of interest.
Section 6 illustrates another type of outcome of this approach. By expanding (1.1) in powers of the test fields we con obtain correlations between fields at different points of space-time or
at the same point. These are quantities which, for a random medium, cannot be deduced from the sole knowledge of the response function, that is, of the Green function, since averaging
over randomness should be performed in the end. We evaluate in particular the dissipation,
a quantity which, being quadratic in the fields, involves such correlations. While the global
energy balance has the same form as for a homogeneous medium, corrections are found for the local balance. We even exhibit an example of situation where dissipation occurs in the medium although the average magnetic field vanishes and the average electric fields does not propagate. This shows clearly that the knowledge of the average electromagnetic fields in a random medium is not suflicient to characterize the dissipation. Likewise, for a disordered metal, the resistivity does not depend only on the single-electron Green's function. Moreover,
m contrast to the localization of the electromagnetic field itself, the study of the localization of its energy reqmres the evaluation of the four-leg diagrams involved in the transport of energy.
Still another kind of flexibility of the method of Section 2 will be featured in Section 7. As
a matter of fact, each auxiliary variable appears as conjugate to one of the equations which
determine the field. For instance, in order to account for the Maxwell equation curl H e(r)j~
= j~
,
(1.2)
where the permittivity e(r) is random, we introduced in Section 3 an auxiliary field Ec con- jugate to E, and the effective action thus constructed contains a term f d~r dl [Ec curl H
-e(r)Ec ôE lût], where E, Ec and H behave as quantum fields. The averaging over e(r) con
then easily be performed since such an action is linear in e, and this leads to a new effective action which is no longer hnear in
ç~ e Ec ôE lût, that is, quadratic in the fields; the resulting
field theory involves vertices in ç~~ (contributing for instance to Fig, l), in ç~~,... However, replacing the field equation (1.2) by another equivalent equation con completely change the final perturbation expansion constructed from the general formalism of Section 2. To fix ideas, let us first imagine that (1.2) is replaced by
iJ(r)curl H j~ = ~(r)j~
,
(1.3)
where q(r) à [e(r)]~~. We thus characterize the medium by the statistics of q(r) rather thon by the statistics of e(r), which is as easily feasible. We now introduce an auxihary field Ed which multiplies (1.3) this leads to a new effective action containing a term f d~r dt [q(r)Ed curl H -Ed ôE/ôt]. Since this effective action is linear in q, we con readily perform the averaging
over q. We thus obtain a field theory for E, Ed and H, involving vertices m x~,x~,..., where x e Ed curl H. Obviously the very nature of the fields entering the theory as well as the structure of the final effective action depend in a crucial fashion on the way the statistical properties of the medium are implemented. A non-linear change of variables such as going from
e to e~~ deeply affects the whole formalism. Although the final exact results are obviously
the same in both cases, the second field theory involving Ed is not related through any simple change of variables to the first theory involving Ec, smce the ratio e(r) = Ed/Ec of these fields has beforehand been integrated over. Changing the original equation (1.2) into some other one such as (1.3), and accordingly implementing the statistics of the medium through q(r) instead of e(r), can thus provide at little cost a rather eflicient means for improving the convergence of the perturbative expansions, since this amounts to completely transform bath the unperturbed
terms and the interaction terms of the effective action. In the above example, mstead of
expanding the physical quantities in powers of the cumulants of e (and possibly performing
associated partial summations through self-consistency), we find as expansion parameters the cumulants of e~~. Going directly from one theory ta the other, for instance by means of a diagrammatic partial summation, would be eut of reach.
This idea underlies the systematic approach developed in Section 7, which accounts for
screening or depolarization effects. Dur starting point is the same as for Bruggeman's approx-
imation. Consider a polarizable ellipsoid, embedded in a homogeneous medium with permit- tivity eext. If a uniform static field em is applied, the interior electric field E;nt and induction D;nt are uniform and satisfy
~eÎÀ D"nt + (1 À)E>nt " OEm
,
(IA)
where A is a positive tensor with trace 1, which depends on the excentricity of the ellipsoid, and which reduces to 1/3 for a sphencal inclusion [la]. The field
OE(r) e eQ(A (D(r) + eext (A~~ 1) E(r)) (1.5)
equals em both inside the inclusion and at large distances from it. It is thus expected to vary in space less rapidly than E(r) itself. If we regard a random material as a set of inclusions
embedded in an average medium, we are led to introduce hkewise a field OE(r) = L~~jD(r) +1 E(r)j
,
(1.6)
similar to (1.5). Since OE(r) behaves as an effective field applied to each piece of the material, it
is expected to have relative statistical fluctuations weaker than E(r). The values of L and are
