Exact closed forms for the transmittance of electromagnetic waves in one-dimensional anisotropic periodic media

Abstract

In this work, we obtain closed expressions for the transfer matrix and the transmittance of electromagnetic waves propagating in finite 1D anisotropic periodic stratified media with an arbitrary number of cells. By invoking the Cayley-Hamilton theorem on the transfer matrix for the electromagnetic field in a periodic stratified media formed by N cells, we obtain a fourth-degree recursive relation for the matrix coefficients that defines the so-called Tetranacci Polynomials. In the symmetric case, corresponding to a unit-cell transfer matrix with a characteristic polynomial where the coefficients of the linear and cubic terms are equal, closed expressions for the solutions to the recursive relation, known as symmetric Tetranacci Polynomials, have recently been derived, allowing us to write the transfer matrix and transmittance in a closed form. We show as sufficient conditions that the $4\times4$ differential propagation matrix of each layer in the binary unit cell, $Δ$, a) has eigenvalues of the form $\pm p_1$, $\pm p_2$, with $p_1\ne p_2$, and b) its off-diagonal $2\times2$ block matrices possess the same symmetric structure in both layers. Otherwise, the recursive relations are still solvable for any $4\times4$-matrix and provide an algorithm to compute the N-th power of the transfer matrix without carrying out explicitly the matrix multiplication of N matrices. We obtain analytical expressions for the dispersion relation and transmittance, in closed form, for two finite periodic systems: the first one consists of two birefringent uniaxial media with their optical axis perpendicular to the z-axis, and the second consists of two isotropic media subject to an external magnetic field oriented along the z-axis and exhibiting the Faraday effect. Our formalism applies also to lossy media, magnetic anisotropy or optical activity.

Figures (6)

• Figure 1: Dispersion relation for a unit cell composed by two birefringent uniaxial media with their optical axis perpendicular to the $z$-axis, with ordinary refraction indexes, $n_o^{(1)}= 1.6$ and $n_o^{(2)}= 1.1$, and extraordinary refraction indexes $n_e^{(1)}= 1.9$ and $n_e^{(2)}= 1.4$. The corresponding filling fractions are $f_1 = 0.4$ and $f_2 = 0.6$, respectively. We have assumed both the isotropic ambient and the substrate to have the same refraction index, $n_0=n_2=1$ and for simplicity, normal incidence is assumed, $\theta_0=0$. We have set the angles of the optical axes with respect to $x$-axis, $\psi_1$ and $\psi_2$, of the first and second layer, $\psi_1=0$ and $\psi_2=\pi/4$, respectively, to allow the so-called mode coupling.
• Figure 2: Dispersion relation for a unit cell composed by two isotropic media that exhibit the Faraday effect when an external magnetic field is applied along the $z$-axis, with Faraday rotation parameters, $\gamma_1/\epsilon_v=0.36$ and $\gamma_2/\epsilon_v=0.001$, respectively. The corresponding refraction indexes of the first and second layer are, $n_o^{(1)}= 1.47$ and $n_o^{(2)}= 1.7$, with filling fractions $f_1 = 0.4$ and $f_2 = 0.6$, respectively. Again, we have assumed both the isotropic ambient and the substrate to have the same refraction index, $n_0=n_2=1$, and for simplicity, normal incidence is assumed, $\theta_0=0$.
• Figure 3: Transmittance for normally incident $p$-polarized light on the finite periodic optical system with unit cell described in Figure 1 and consisting of $N=16$ cells. The results obtained with both methods, $A$ and $B$, are shown. They are indistinguishable.
• Figure 4: (Color online) Transmittance of the same system of Figure 1 as function of the number of cells and for normally incident $s$-polarized light.
• Figure 5: Transmittance of the same system of Figure 1, as function of the number of cells for normally incident $p$-polarized light.