A Novel 3D Pseudo-spectral Analysis of Photonic Crystal Slabs

摘要

We consider a double-periodic slab which is characterized by two lattice vectors a_1 and a_2 on the (x, y)-plane, the thickness h_z and a three-dimensional scalar function ε(x, y, z) specifying the dielectric constitution of the slab. Above and below the slab is free space. These assumptions imply that the z-direction is special in this problem. Therefore, following a general scheme we diagonalize the Maxwell's equations with respect to this direction. The periodicity in two directions suggests the use of spatially harmonic functions as a basis. We exploit this property; however, contrary to the traditional schemes, we propose an expansion of the fields in the form Ψ(r, z) = Σ_n f_n(z)exp(jk_n·r) allowing f_n(z) to be a fairly general function of the z-coordinate, rather than an exponential function. In this expression r is the position vector in the (x, y-) transversal plane. To guarantee maximum flexibility we discretize f in terms of finite differences. We demonstrate the superiority of our method by discussing the following properties: ⅰ) Diagonalization only involves the transversal field components, ⅱ) Diagonalization allows us easily to construct and implement various boundary conditions at the bounding surfaces z = 0 and z = h_z, ⅲ) The resulting discretized system is extraordinarily stable and robust, and facilitates fast computations; from the computational performance point of view it compares well with existing methods, while it by far applies to larger class of problems, ⅳ) It allows to use both the radian frequency ω and the wavevector K as input parameters. Therefore, the resulting discrete system can be solved at individual (ω, K)-points of interest, ⅴ) Finally, the method is applicable to both the eigenstate end the excitation problems.