Duality relation for the Maxwell system
F. Zolla
1
and S. Guenneau
2,
*
1
Institut Fresnel, UMR 6133, Faculte ´ de Saint Je ´ro ˆme, Case 162, 13397 Marseille Cedex 20, France
2
Department of Mathematical Sciences, University of Liverpool, Liverpool L69 3BX, United Kingdom
Received 26 June 2002; published 24 February 2003
This paper is intended to establish a link between the vector Maxwell system for three-dimensional 3D and
2D finite photonic crystals in the low-frequency limit. For this, we generalize the classical results of Keller and
Dykhne chessboard problem to periodic media described by piecewise continuous permittivity profiles: our
theorem enlights the result of Mendelson polycrystalline and multiphase media in the framework of homog-
enization theory of elliptic operators. In fine, we give illustrative examples by using both integral equation and
variational approaches via the so-called method of fictitious charges and finite-element method.
DOI: 10.1103/PhysRevE.67.026610 PACS numbers: 41.20.-q, 42.25.-p, 78.20.Bh, 41.20.Cv
I. INTRODUCTION
Some mathematical works devoted to the theory of com-
posites go back to Wiener 1 by interchanging the roles of
the background and matrix in the Lorentz-Lorenz formula. A
classical theorem is that of Keller 2, which found a relation
between the transverse effective conductivity of an array of
cylinders and the conductivity when the phases are inter-
changed. The essence of Keller’s theorem is that if in a two-
dimensional 2D potential problem for Laplace’s equation u
is a solution for a problem with one type of inclusion, then
its Cauchy-Riemann partner v is the solution for the problem
with the dielectric constant inverted and the electric field
rotated by 90°. In 1970, Dykhne 3, using the fact that a
divergence-free field when rotated locally at each point by
90° produces a curl-free field and vice versa could generalize
Keller’s result to isotropic multiphase and polycrystalline
media. He noted that the duality relations implied exact for-
mulas for the conductivity of phase-interchange-invariant
two-phase media such as checkerboards and for polycrys-
tals constructed from a single crystal. In Sec. III, we gener-
alize the results of Keller 2, Nevard and Keller 4, and
Dykhne 3: we express the homogenized permittivity of a
two-dimensional electrostatic problem in terms of the ho-
mogenized inverse permittivity up to a rotation of 90°. In the
particular case of a checkerboard, i.e., a planar body of
square symmetry characterized by a piecewise constant per-
mittivity which takes the values
1
and
2
, it can be then
deduced the well-known formula 3
eff
=
1
1
.
This formula has been generalized by Kozlov in the random
case and also the asymptotic behavior of
eff
when
1
is
small in term of percolation thresholds 5. This formula was
independently derived by Golden and Papanicolaou 6.
The pioneering works of Keller and Dykhne stimulated a
large body of research: Berdichevski 7 derived an exact
formula for the effective shear modulus of a checkerboard of
two incompressible phases. The duality and phase inter-
change relations of Berdichevski were extended to aniso-
tropic composite materials by Helsing, Milton, and Movchan
8 together with numerical results of high accuracy and
Nemat-Nasser and Ni 9 obtained duality transformations
for three-dimensional anisotropic bodies with stress and
strain fields independent of the x
3
coordinate. Milton and
Movchan 10 found an equivalence between planar elastic-
ity problems and antiplane elasticity problems in inhomoge-
neous bodies. These results and additional duality relations
were then discussed in detailed by Helsing, Milton, and
Movchan 8. A plethora of three-dimensional exact relations
from pyroelectricity to thermopiezoelectric composites have
recently been given by Grabovsky, Milton, and Sage 11.
Among other things, when no explicit formulas are avail-
able, it is an important matter to have a priori estimates on
the effective matrix in term of the statistical properties of
each component of the composite. This is the so-called
theory of bounds which motivated a lot of contributions in
several fields of physics, mechanics, and mathematics. The
pioneering work was done by Hashin and Shtrikman in 12
where a complete description of all possible effective tensors
was derived. This result was proved later by Tartar 13 who
extended the result to the anisotropic case. Then many papers
appeared in the literature among them, of which we may
quote Benveniste 14 who obtained bound results in piezo-
electricity and Francfort 15 who made a correspondence
between the equations of incompressible elasticity and the
duality relations for conductivity.
To conclude, the rigorous mathematical theory of the ho-
mogenization of elliptic operators with random coefficients
proposed by Jikov, Kozlov, and Oleinik 16 is closely re-
lated to the Bergman-Milton theory of bounds 17–21 who
obtained some bounds for the effective modulli of periodic
and random structures.
II. LOW-FREQUENCY LIMIT OF THE VECTOR
MAXWELL SYSTEM
The homogenization of 3D dielectric photonic crystals
has been independently performed by Ke-Da et al. with the
Rayleigh method two-phase media with spherical inclu-
sions22 and by Guenneau and Zolla with the multiple-
*FAX: +44 0 151 794 4061
Electronic address: guenneau@liverpool.ac.uk
PHYSICAL REVIEW E 67, 026610 2003
1063-651X/2003/672/0266108/$20.00 ©2003 The American Physical Society 67 026610-1