Discrete Transparent Boundary Conditions for the 2D Helmholtz Equation
We derive and implement discrete transparent boundary conditions for the 2D Helmholtz equation with smooth artificial boundary
The numerical simulation of wave propagation in integrated optics or fiber optics devices is one of the central tasks in the design of effective components. Fortunately, most of the practical structures can be modeled based on a uni- or bidirectional wave propagation. Simulation tasks of this kind can be solved by a number of different methods, most prominent here are the various types of Beam Propagation Methods (BPM). The central idea of BPM's, from its origin, has been to solve the scalar Helmholtz equation approximately by a reformulation of the boundary value problem to an initial value problem. This concept is very close to the working principle of a large number of optics components. Some structures, however, require simulation tools which are able to take into account arbitrary directions of reflections. For such type of problems it is natural, to go back to the scalar Helmholtz equation >defined on some bounded domain, and to solve it as boundary value problem. That is, one prescribes the incident field along the whole boundary and obtains the interior solution u(x,y). The central problem arising here is to construct transparent boundary conditions for general smooth boundaries and to find a suitable formulation of the interior problem, which allows for a direct implementation of these boundary conditions.
Our approach is based on a variational formulation of both the Helmholtz equation and the boundary operator. In standard fashion, we obtain the variational form of the Helmholtz equation by multiplication with a test function v and integration by parts
The function u(s) may be considered as a superposition of incoming and outgoing waves. With a boundary operator, which supplies a relation between the normal derivatives to the boundary values it follows
Here the matrices A, B and the vector r are obtained using standard finite-element test functions v. The matrix A is the conventional finite-element system matrix of the Helmholtz equation, B is the discrete version of the boundary operator and realizes the transparency of the boundary. The right hand side vector r contains the information about the incident wave. The main problem consists in the construction of a boundary operator suitable for the numerical (discrete) problem. Our approach is based on a generalization of the algebraic approach, which we have employed to construct transparent boundary conditions for Schrödinger-type equations [1,2,3].
The results of a typical simulation are given in Figure 1 and 2. The simulation concerns the reflection of the fundamental mode of a waveguide from a thin dielectric mirror. The problem is characterized by an homogeneous exterior domain. In Figure 1 the time-averaged intensity distributiopn is shown. Both reflection and diffraction of the beam becomes visuable. Figure 2 displays the corresponding amplitude distribution at a given, fixed time.
Figure 1. Reflection of the fundamental mode of a waveguide from a dielectric mirror - time averaged intensity distribution.
Figure 2. Reflection of the fundamental mode of a waveguide from a dielectric mirror - solution at a fixed time.
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