Vectoral Beam Propagation for Anisotropic Waveguides
Introduction
To rigorously evaluate the propagation characteristics of an inhomogeneous and/or anisotropic waveguide, a Vectorial wave analysis is necessary, with at least two field components. These formulations are fundamentally more accurate than scalar forms, since they can represent true hybrid modes in a general dielectric waveguide. A semi-vectorial Beam Propagation Method (BPM) can identify polarization dependence; however,…
Mathematical Formulations
We consider a longitudinal varying 3-D optical waveguide, as shown in Figure 1, and the transverse cross-section of the optical waveguide surrounded by PML regions I, II, and III with thickness d, as shown in Figure 2, where x and y are the transverse directions and z is the propagation direction. Figure 1: Longitudinally varying 3-D optical…
Continuous Wave Equations for Anisotropic Media
Consider an anisotropic medium described by relative electric permittivity Therefore, the propagation of the electromagnetic field is described by the curl-curl equation where Φ , p , and q are shown in Table 6. Φ p q E 1 ε ˜ E ε–1 ˜ 1 Table 6: Definitions of Φ , p , and…
Full E-vector Formulation
E-Formulation: The vector wave equation for the electric fields On the transverse plane Equation (19) becomes: The solution of the wave equation Equation 31 can be separated as a slowly varying envelope and a fast-oscillating phase term: Using Gauss’ Law, we get: If the refractive index varies slowly along the propagation direction , which is valid…
