Floquet-mode analysis of pillar-type photonic crystal waveguide using spectral-domain approach

Photonic crystal is a periodic structure consisting of highly contrast dielectrics, in which the electromagnetic wave cannot transmit in a specific wavelength range. It is therefore known that, if localized defects are introduced in the photonic crystal, the electromagnetic fields are strongly confined around the defects. For example, point defects in the photonic crystal work as resonance cavities and line defects work as waveguides. This paper presents a Floquet-mode analysis of the photonic crystal waveguide (PCW) using the spectral-domain approach. For the straight waveguides, the structure maintains the periodicity in the propagation direction, and the Floquet theorem asserts that the electromagnetic fields in the structure can be expressed by superposition of the Floquet-modes [1]. The Floquet-modes of the PCW are obtained by the eigenvalue analysis of the transfer matrix for one periodicity cell in the propagation direction. The periodicity cell that makes up the PCW has imperfect periodicity in the direction perpendicular to wave propagation. Therefore the fields in the structure have continuous spectra. The present analysis uses the pseudo-periodic Fourier transform (PPFT) [2] to consider the discretization scheme in the wavenumber space. The PPFT and its inverse are formally given by f(x; ξ) = Σ
^∞
m = −∞
f (x − md)e
^imdξ
(1) f(x) = 1/k
d
∫
^kd/2
−kd/2
f(x; ξ)d ξ (2) where d is a positive value usually chosen to be equal with the structural period, ξ is a transform parameter, and k
d
= 2π/d is the inverse lattice constant.