Out-of-plane band structure of a two-dimensional dispersive photonic crystal.

We calculated the out-of-plane band structure of a two-dimensional dispersive photonic crystal (PC). To achieve this goal, the plane wave expansion method was implemented in conjunction with a numerical algorithm, the dispersive photonic crystal iterative method. The PC is an array of circular cross-sectional dispersive MgO Lorentz single-pole rods in a square lattice. The frequency bands are calculated starting at Γ as a function of the oblique component of the wave vector. For the lowest frequencies, it was found that the modes bend drastically to the horizon as the dielectric constant ϵ(ω) is increased to a very positive value. For frequencies above the longitudinal optical phonon circular frequency, where ϵ(ω) has very low positive values, the expected degeneration occurs in the transparency window, and a line of modes behaves close to the line of light.

Scattering of a Gaussian beam from a row of cylinders with rectangular cross section.

The transverse magnetic Gaussian beam diffraction from a finite number, equally spaced and rectangular cross section dielectric cylinder row is studied. The infinitely long cylinders' axes are perpendicular to the beam's direction of propagation. The cylinder row, with dielectric constant εc=nc2, is treated as a periodic inhomogeneous film, with period ax and thickness wy, bounded by two semi-infinite homogeneous media. With this restriction, the method is valid only for square or rectangular cross section cylinders. The supercell and the plane wave expansion methods are used to calculate the eigenfrequencies and eigenvectors supported for a one-dimensional photonic crystal. Then, these eigenfrequencies and eigenvectors are used to expand the field in the inhomogeneous film. Numerical results are presented for ax greater than λ (the incident light wavelength), wx (the cylinder width), and wg (Gaussian beam waist). Two cases are studied. In the first (second) case, the unit cell contains one cylinder (a cylinder row), which simulates the scattering from a single cylinder (an inhomogeneous thin film). The total integrated scattering in transmission (reflection) shows three well-defined minima (maxima), which are due to interference effects. Its positions can be approximately obtained with the formula λk=4ncwy/k, with k=3, 4, and 6. The total integrated scattering in transmission decreases linearly as a function of the cylinder number.