Carnegie Mellon
Department of Mathematical 

Guy Bouchitte, Universite du Toulon

"Homogenization of wire photonic crystals leading to Left-Handed Medi"


We consider a domain of $ \mathbb{R}^3$ filled periodically (period $ \eta$) by parallel rods of radius $ r$ and high permittivity $ \epsilon$. We are looking for the asymptotics of the harmonic diffraction problem (electromagnetic waves with a exp $ (-i\omega t)$ dependence) as the small parameters $ \eta,r,\epsilon^{-1}$ tend to zero. Two different asymptotic behaviors are presented. The first one corresponds to the capacitary case (i.e. the capacity of the fibers remains constant) and in the polarized case we are led to a possibly negative effective permittivity $ \epsilon_{\rm eff}(\omega)$. The second one is new and quite surprising: it may produce a negative effective permeability.

Case I The filling ratio of the fibers vanishes $ ( r << \eta$) and the permittivity $ \epsilon$ has a large imaginary part (high conductivity). Assuming that all the fibers are vertical, we find that the vertical component $ E_3$ of the limit electric field induces a volumic current $ j$ density (solution of a 1D- propagation equation in which $ E_3$ acts as a source term). The resulting diffraction problem involves $ (E,H)$ and $ j$. It is non local in $ (E,H)$. However, in the case of a $ e_3$- polarized electric field, we recover an homogenized medium characterized by an effective permittivity $ \epsilon_{\rm eff}(\omega)$. This $ \epsilon_{\rm eff}(\omega)$ which depends explicitly of the wave number becomes negative below a breakdown frequency.

Case II The filing ratio of the fibers is positive $ (r \sim \eta)$ but now the permittivity $ \epsilon$ is real and scaled like $ \frac{1}{\eta^2}$. Assuming a $ e_3$-polarized magnetic field, we find an homogenized medium with a magnetic activity characterized by a permeability $ \mu_{\rm eff}(\omega)$. For certain ranges of frequencies $ \mu_{\rm eff}(\omega),\ \epsilon_{\rm eff}(\omega)$ can be negative.

Combining the two cases, we are led to propose a mathematical foundation for the so-called left-handed media for which the experimental $ \mu_{\rm eff}(\omega),\ \epsilon_{\rm eff}(\omega)$ are negative.

THURSDAY, April 22, 2004
Time: 4:30 P.M.
Location: PPB 300