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Additional info for Design and fabrication of quasi-2D photonic crystal components based on silicon-on-insulator technology
It is apparent that, theoretically, there is no interaction between modes guided in PCS and radiation modes of the cladding. The upper boundary of the region occupied by guided modes in the ω-k plane is called light cone, and is defined by (ω ncl c ) = k . The area available to guided modes, which we call transmission window, scales roughly as ∼ ( 1 ncl ) . 9 shows an example for a triangular lattice PCS with homogeneous claddings. A linear k-scale is applied parallel to each of the directions of the outer rim of the IBZ, leading to straight lines along the Γ-M and Γ-K directions and an approximately quadratic curve along the M-K direction.
1 Guiding mechanism There are two coexisting confinement mechanisms in a PCS: in-plane confinement-given by the PBG effect- and vertical confinement-given by refractive index contrast between the PCS and its cladding. Because of its finite thickness, a PCS can support guided modes. The analysis of these modes should be three-dimensional and their properties depend on slab thickness, index contrast between the core and cladding, and vertical mirror symmetry. 7) where k is the wave-vector component in the plane of periodicity, belonging to the IBZ, G is a reciprocal lattice vector and kz is the wave-vector component perpendicular to the plane of periodicity (in the vertical direction).
PMLs are made of a nonphysical absorbing material, which strongly attenuates the incoming electromagnetic waves, while the phase velocity remains unchanged. The electromagnetic fields and the refractive index in the transverse plane (x, y) are expressed as Fourier series. In every slice, the modes and their propagation constants are computed exactly. The scattering matrix, relating the input and output fields, is computed recursively by matching the boundary conditions at the slice interfaces. This method, however, has a couple of shortcomings: (i) combining PMLs with Fourier expansions leads to more complicated eigenvalue problems (even truly guided modes will have complex propagation constants); (ii) the staircase approximation has convergence problems [Popov 2002].