Book contents
- Frontmatter
- Contents
- Preface
- 1 Introduction
- 2 Optical models
- 3 Material model I: Semiconductor band structures
- 4 Material model II: Optical gain
- 5 Carrier transport and thermal diffusion models
- 6 Solution techniques for optical equations
- 7 Solution techniques for material gain equations
- 8 Solution techniques for carrier transport and thermal diffusion equations
- 9 Numerical analysis of device performance
- 10 Design and modeling examples of semiconductor laser diodes
- 11 Design and modeling examples of other solitary optoelectronic devices
- 12 Design and modeling examples of integrated optoelectronic devices
- Appendices
- Index
6 - Solution techniques for optical equations
Published online by Cambridge University Press: 09 October 2009
- Frontmatter
- Contents
- Preface
- 1 Introduction
- 2 Optical models
- 3 Material model I: Semiconductor band structures
- 4 Material model II: Optical gain
- 5 Carrier transport and thermal diffusion models
- 6 Solution techniques for optical equations
- 7 Solution techniques for material gain equations
- 8 Solution techniques for carrier transport and thermal diffusion equations
- 9 Numerical analysis of device performance
- 10 Design and modeling examples of semiconductor laser diodes
- 11 Design and modeling examples of other solitary optoelectronic devices
- 12 Design and modeling examples of integrated optoelectronic devices
- Appendices
- Index
Summary
The optical mode in the cross-sectional area
There are numerous optical mode solvers that deal with optical eigenvalue problems in a scalar form as shown in equation (2.34), or in a more comprehensive vectorial version, in order to obtain the optical field distribution (i.e., the optical mode) in the cross-sectional area. In contrast to dealing with a 1D slab waveguide problem where an analytical approach (such as a transfer matrix method) exists, a general 2D eigenvalue problem has to be treated numerically. Among many different numerical approaches, the finite difference method (FD) seems to be a popular one for its balance between implementation complexity, computational efficiency and accuracy.
In the 2D domain, equation (2.34) is posed as a boundary value problem of a PDE of elliptical type. Hence, discretization of equation (2.34) under the FD scheme is fairly straightforward since stability is not a concern. For example, we can always stay with the center discretization scheme to gain a second order accuracy. The boundary treatment, however, is crucial especially for those 2D structures with piecewise uniformity, which is most commonly seen in semiconductor optoelectronic devices.
At physical boundaries inside the computation domain, the refractive index is discontinuous, whereas as the solution to equation (2.34), the scalar optical mode must be continuous. Therefore, special treatments are necessary at such boundary points. For example, in many numerical solvers we do not select any mesh grid point at the boundary points to avoid refractive index assignment at these boundaries.
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- Optoelectronic DevicesDesign, Modeling, and Simulation, pp. 172 - 199Publisher: Cambridge University PressPrint publication year: 2009
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