Scattering-matrix analysis of periodically patterned multilayers with asymmetric unit cells and birefringent media

Fourier analysis has been successfully applied to study optical properties of photonic crystal structures, usually composed of optically isotropic media. In a commonly used formulation [D. M. Whittaker and I. S. Culshaw, Phys. Rev. B 60, 2610 (1999)], inversion symmetry of the unit cell is required. Here, we extend the treatment of Whittaker and Culshaw to structures with asymmetric unit cells that can be composed of birefringent media. As applications we consider a high-refractive-index membrane with a triangular lattice of triangular holes, where the presence of a TE-like gap at $\omega$ and of a TM-like gap at $2\omega$ is established, and a slot waveguide made of (birefringent) porous silicon, where coupling of guided modes to radiative modes is achieved through a one-dimensional periodic grating.

Figure 1

(Color online) Scheme of a periodically patterned multilayer: laboratory reference system and principal dielectric axes are indicated with continuous and dashed lines, respectively. We treat the case in which the growth direction is along the axis $Z$. The laboratory reference system ${\Sigma }_{xyz}$ is obtained from ${\Sigma }_{XYZ}$ through a clockwise rotation of an angle $\psi$ around $Z$.

Figure 2

(Color online) Scheme of a periodically patterned membrane of thickness $d$; the triangular lattice has period $a$, while the triangular hole’s side measures $L$. The Brillouin zone for the triangular lattice is also shown together with the main symmetry directions.

Figure 3

(Color online) Reflectance calculations along $\Gamma \text{−}K$ for (a) TE and (b) TM polarizations for a photonic crystal slab with a triangular pattern of triangular holes with $L∕a=0.8$, $d∕a=0.5$, and $ϵ=12.11$. The curves are slightly shifted $(\Delta R=1)$ for clarity. (c) Photonic band dispersion along $\Gamma \text{−}K$ for even (open circle) and odd (full circle) modes with respect to the symmetry ${\widehat{\sigma }}_{xy}$; photonic gaps are also indicated. The photon dispersion relation in air and in the effective core layer are given by solid and dashed lines, respectively.

Figure 4

(Color online) Reflectance calculations along $\Gamma \text{−}M$ for (a) TE and (b) TM polarizations for a photonic crystal slab with a triangular pattern of triangular holes with $L∕a=0.8$, $d∕a=0.5$, and $ϵ=12.11$. The curves are slightly shifted $(\Delta R=1)$ for clarity. (c) Photonic band dispersion along $\Gamma \text{−}M$ for even (open circle) and odd (full circle) modes with respect to the symmetry ${\widehat{\sigma }}_{xy}$; photonic gaps are also indicated. The photon dispersion relations in air and in the effective core layer are given by solid and dashed lines, respectively.

Figure 5

(a) Photonic band dispersion for even (open circle) and odd (full circle) modes along $\Gamma \text{−}K$ of a photonic crystal slab with a triangular pattern of triangular holes with $L∕a=0.8$, $d∕a=0.5$, and $ϵ=12.11$. The photon dispersion relation in air is indicated by the solid line. The dashed line represents the effective light line for an angle of incidence $\theta =20°$ using a silicon prism; its distance from the slab is $0.7a$. (b) Attenuated total reflectance spectra along $\Gamma \text{−}K$ for TE (line) and TM (dashed) polarizations at $\theta =20°$.

Figure 6

(Color online) (a) Structure scheme and parameters. (b) Main panel: reflectance calculation for TM mode at angle of incidence $\theta =0.1°$ as a function of energy. Inset: contour plot of ${\mid E_z\mid }^2$ at wavelength $\lambda =1.55\mu \mathrm{m}$ and $\theta =0.1°$ for TM polarized incident field.