Helmholtz Equation In Three - Layer Planar Waveguides Based On The Characteristics Of

Slab waveguides are commonly used in photonic integrated circuits. So their accurate modeling is essential for the development of new, higher performance of optical components required by high-bandwidth communications system.Most numerical calculations of normal modes of slab optical waveguides are carried out within a finite computational domain, which leads to a difficulty when the investigated mode extends outside the domain. Thus it is important to use accurate transverse boundary conditions or absorbing layers to simulate the open nature of the cross-section beyond the domain boundary.In this thesis, we use the Perfectly Matched Layer (PML) as a material absorbing boundary condition (ABC) to solve the Helmholtz equation. The perfectly matched layer (Berenger, 1994) is an artificial interface between two half spaces, which has the virtue that surrounding the computational domain it can theoretically absorb without refection any kind of traveling waves towards boundaries. The loss of lossy half space is in the direction normal to the interface. As such, the PML medium has been commonly used as the best absorbing boundary condition for numerical solution of partial differential equations.It has been demonstrated that PML in Cartesian coordinates is equivalent to coordinate stretching in the complex space, through a change of variables (Chew and Weedon, 1994). In brief, by reformulating the wave equation, the problem is deducted to a complex eigenvalue problem. Here, the formalism of complex coordinate stretching allows us to reuse all the eigenmode formulas derived for the non-PML case without modification, simply by allowing the cladding thickness to assume complex values (Bienstman et al., 2001) (Derudder et al., 2001).Then, using the alternative numerical discretization by the Finite Difference Method, the Helmholtz equation leads to an algebraic eigenvalue equation. Since the...