Reflection of focused beams from opal photonic crystals.

We present a robust method for computing the reflection of arbitrarily shaped and sized beams from finite thickness photonic crystals. The method is based on dividing the incident beam into plane waves, each of which can be solved individually using Bloch periodic boundary conditions. This procedure allows us to take a full advantage of the crystal symmetry and also leads to a linear scaling of the computation time with respect to the number of plane waves needed to expand the incident beam. The algorithm for computing the reflection of an individual plane wave is also reviewed. Finally, we find an excellent agreement between the computational results and measurement data obtained from opals that are synthesized using polystyrene and poly(methyl methacrylate) microspheres.

Pseudo-spectral Analysis of Radially-Diagonalized Maxwell's Equations in Cylindrical Co-ordinates.

We present a robust and accuracy enhanced method for analyzing the propagation behavior of EM waves in z-periodic structures in (r, ö, z)-cylindrical co-ordinates. A cylindrical disk, characterized by the radius a and the periodicity length Lz, defines the fundamental cell in our problem. The permittivity of the dielectric inside this cell is characterized by an arbitrary, single-valued function å(r, ö, z) of all three spatial co-ordinates. We consider both open and closed boundary problems. Irrespective of the type of the boundary conditions on the surface r = a, our method requires the discretization of the fields in the interior of the disk only. Inside the disk volume, we expand the fields in terms of planewaves on discrete cylindrical surfaces ri = i, with being the discretization step length. The fields on the nested surfaces ri = i in the interior of the simulation domain are interrelated by the application of a simple, yet, powerful finite difference scheme. In free space outside the disk, the fields are xpanded in terms of the closed-form eigensolutions of the Maxwell's equations in cylindrical co-ordinates. In order to uniquely determine the involved unknown coefficients, the solutions in the interior- and exterior domains are matched on the disk's bounding surface. Our formulation utilizes a radially-diagonalized form of Maxwell's equations, and merely involves four (out of the six) field components. It is demonstrated that our formulation is perfectly suited, but by no means limited, to cylindrical symmetric problems.