Boundary conditions for interfaces of electromagnetic crystals and the generalized Ewald-Oseen extinction principle

The problem of plane-wave diffraction on semi-infinite orthorhombic electromagnetic (photonic) crystals of a general kind is considered. Boundary conditions are obtained in the form of infinite system of equations relating amplitudes of incident wave, eigenmodes excited in the crystal, and scattered spatial harmonics. The generalized Ewald-Oseen extinction principle is formulated on the base of deduced boundary conditions. The knowledge of properties of infinite crystal’s eigenmodes provides an option to solve the diffraction problem for the corresponding semi-infinite crystal numerically. In the case when the crystal is formed by small inclusions which can be treated as point dipolar scatterers with fixed direction the problem admits complete rigorous analytical solution. The amplitudes of excited modes and scattered spatial harmonics are expressed in terms of the wave vectors of the infinite crystal by closed-form analytical formulas. The result is applied for the study of reflection properties of metamaterial formed by cubic lattice of split-ring resonators.

Figure 1

Geometry of an infinite electromagnetic crystal.

Figure 2

Illustration for the replacement of the dielectric permittivity by the local polarizability.

Figure 3

Geometry of an semi-infinite electromagnetic crystal.

Figure 4

Geometries of scatterers which can be modeled as dipoles with fixed orientation: (a) split-ring resonator, (b) inductively loaded wire.

Figure 5

Geometry of an infinite electromagnetic crystal formed by uniaxial dipolar scatterers.

Figure 6

Geometry of a semi-infinite electromagnetic crystal formed by uniaxial dipolar scatterers.

Figure 7

Dependence of $\Pi$ vs normalized frequency $ka∕\left(2\pi \right)$.

Figure 8

(Color online) Dependencies of the normalized wave vectors $q_{x}a∕\pi$ of excited eigenmodes (imaginary and real parts) and the reflection coefficient $R$ calculated using (55) on normalized frequency $ka$ for a semi-infinite cubic lattice of split-ring resonators with $A=0.1{\mu }_0{a}^3$ and ${\omega }_0=1∕\left(a\sqrt{{\epsilon }_0{\mu }_0}\right)$.