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,ϕ,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 e(r,ϕ,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 =iΔ, with Δ being the discretization step length. The fields on the nested surfaces ri =iΔ 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 expanded 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.