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 $\mathrm{\Gamma }$ 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 $ϵ(\omega )$ is increased to a very positive value. For frequencies above the longitudinal optical phonon circular frequency, where $ϵ(\omega )$ 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.

A study of the band structures of a two-dimensional dispersive photonic crystal for out-of-plane propagation along with the dispersive photonic crystal iterative method are presented. In these calculations, the proposed method uses the plane wave expansion method as the foundation. The photonic crystal is a square array of circular cross-sectional dispersive MgO Lorentz single-pole rods. The band structures were calculated starting at the high symmetry points Γ , X, and M as a function of the oblique component of the propagation wave vector. The method used is discussed in detail, and a simple equation is presented for straightforward analytical calculations of points of the frequency bands in the transparency window, where the dielectric constant equals that of vacuum.

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 ${\mathit{\epsilon }}_c=n_c^2$, is treated as a periodic inhomogeneous film, with period $a_x$ and thickness $w_y$, 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 $a_x$ greater than $\lambda$ (the incident light wavelength), $w_x$ (the cylinder width), and $w_g$ (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 ${\lambda }_k=4n_c{w}_y/k$, with $k=3$, 4, and 6. The total integrated scattering in transmission decreases linearly as a function of the cylinder number.

The energies of an electron in a one-dimensional crystal are studied with both the Schrödinger and Dirac equations using the plane wave expansion method. The crystalline potential sensed by the electron in a cell was calculated by accounting for the Coulombic (electrostatic) interaction between the electron and the surrounding cores (immobile positive ions at the center of the crystal cells). The energies and wave functions of the electron were calculated as a function of four parameters: the period $a_p$ of the lattice, the dimension $n_{dim}$ of the matrix in the momentum space, the partition number $l_{pa}$ in which the unit cell is divided to calculate the potential and the number of cores $n_{co}$ that affect the electron. It was found that 8000 cores (surrounding the electron) were needed to reach our convergence criterion. An analytical equation that accurately describes the behavior of the energies in function of the cores that affect the electron was also found. As case studies, the energies for pseudo-lithium and pseudo-graphene were obtained as a first approximation for one-dimensional lattices. Subsequently, the energies of an isolated dimer nanoparticle were also calculated using the supercell method.

The electronic band structure of two-dimensional lithium is calculated using the Dirac equation. Lithium is modeled as a two-dimensional square lattice in which the two strongly bound inner electrons and the fixed nucleus are treated as a positively charged ion (+e), while the outer electron is assumed to be uniformly distributed within the cell. The electronic potential is obtained by considering Coulomb-type interactions between the charges inside the unit cell and those in the surrounding cells. A numerical method that divides the unit cell into small pieces is employed to calculate the potential and then the Fourier coefficients are obtained. The Bloch method is used to determine the energy bands, leading to an eigenvalue matrix equation (in momentum space) of infinite dimension, which is truncated and solved using standard matrix diagonalization techniques. Convergence is analyzed with respect to the key parameters influencing the calculation: the lattice period, the dimension of the eigenvalue matrix, the unit-cell partition used to compute the potential’s Fourier coefficients, and the number of neighboring cells that contribute to the electronic interaction.

In this chapter, a bandgap structure in the terahertz region is studied, considering the dependence of the dielectric constant value on the frequency of electromagnetic waves. First, the electromagnetic equation for an inhomogeneous medium is derived from Maxwell’s equations. Then, the basic theory of photonic crystals (PCs) is presented, emphasizing their geometry and how it is defined. The relevance of the dielectric constant in a PC is addressed next, showing the dispersive nature of alumina with experimental data of the dielectric constant, which exhibited values ranging from very large to very small, even showing vacuum-related values. Later, the general eigenvalue equation for any PC is obtained through the application of the plane wave expansion method. As a study case, the in-plane and out-of-plane band structures of a dispersive alumina two-dimensional PC are presented in the results section, commenting on expected but also unusual behaviors in the bands due to the contrasting values of the dielectric constant. Two absolute band gaps appear between approximately 12.6 and 17 THz. Lastly, some conclusions are given.