 
Summary: Analysis of the Radiation Properties of a Planar
Antenna on a Photonic Crystal Substrate
Habib Ammari , Natacha Bereux , and Eric Bonnetier
Abstract
This paper is concerned with the rigorous investigation of the radiation properties
of a planar patch antenna on a photonic crystal substrate. Under the assumptions that
the driving frequency of the antenna lies within the band gap of the photonic crystal
substrate and that the crystal satises a symmetry condition, we prove that the power
radiated into the substrate decays exponentially. To do this, we reduce the radiation
problem to the study of the wellposedness of a weakly singular integral equation on
the patch antenna, and to the study of the asymptotic behavior of the corresponding
Green's function. We also provide a mathematical justication of the use of a photonic
crystal substrate as a perfect mirror at any incidence angle.
Key words. photonic crystals, patch antennas, dielectric re
ectors, perfect mirrors, inte
gral equations, Green's functions
AMS subject classications. 35Q60, 78A40
1 Introduction
A photonic crystal is a periodic dielectric structure that possesses the feature that electro
magnetic waves of certain frequencies cannot propagate inside. The range of the prohibited
frequencies is called the complete band gap. The propagation of electromagnetic waves in
