This thesis presents a variety of integral-equation methods that effectively tackle a range of problems in computational electromagnetism, with application to problems in electrical engineering and applied physics. In particular, the developed techniques were used to solve problems concerning waveguides in two and three dimensions and metamaterial-design problems, demonstrating the value of the integral-equation approach in this context. We conclude this thesis with a brief summary of the proposed methodologies and a discussion of potentially valuable follow-on work.
The WGF method for waveguide problems introduced in Chapters 2 and 6, enables efficient treatment of complex waveguide structures. This approach has al- ready facilitated the design of two-dimensional structures such as waveguide tapers, splitters and grating couplers such as those shown in Figures 8.1 and 8.2 (see [74] for details). This computational method provides super-algebraically accurate approxi- mations as the window sizes are increased and, as a result, the method can be orders of magnitude faster than solvers considered presently in engineering practice [74].
The proposed approach retains the attractive qualities of boundary integral equation methods, such as reduced dimensionality, efficient parallelization, and high-order accuracy for arbitrary geometries. And, while the present implementation is based on use of Nyström integral-equation solvers (which we heartily recommend), any available boundary integral method for transmission problems, such as, e.g., those based on the Method of Moments, can be easily modified to incorporate the WGF methodology.
Many problems in areas of electrical engineering require knowledge of the bound modes of three-dimensional dielectric waveguides, considered in Chapter 3, as these are used as feeds for photonic structures. In practice, a wide variety of waveguide cross sections have been used, and the evaluation of the types of open waveguide modes treated in this chapter have generally been considered challenging. As demonstrated by the numerical examples in Chapter 3, the proposed method provides an accurate and reliable tool for treatment of a problem for which other approaches have experienced difficulties.
The boundary integral equation framework for the optimization of electromag-
8.1. Future work 119 netic devices developed in Chapter 7 provides significant advantages. On the basis of its fast and highly-accurate electromagnetic solvers, as well as the new powerful adjoint techniques for gradient evaluation, the methods enable solution of challeng- ing three-dimensional simulation and optimization problems that would otherwise require prohibitively high computing costs.
The rectangular-polar method for singular integrals introduced in Chapter 4 un- derlies much of the work presented in this thesis. In view of its ability to effectively integrate the types of singular kernels that arise in the context of boundary inte- gral equations in scattering theory, the methodology was used in conjunction with suitable FFT-based acceleration methods and the GMRES linear algebra solver to produce solutions for highly-challenging three-dimensional waveguide problems and three-dimensional metamaterial structures. The rectangular-polar method is it- self demonstrated in the context of bounded obstacles, including examples for which the scattering obstacles contain open, closed, smooth and non-smooth, scattering surfaces. In all of these cases the solver produced results with high accuracy in short computing times. In particular, the rectangular-polar method is well-suited for application to general engineering configurations—where the scattering objects are provided in standard (but generally highly complex) CAD representations.
The numerical examples presented throughout this thesis demonstrate the en- abling character of the proposed approaches. We believe that the present and subsequent works, as described in the following section, will lead to significant improvements in the computational simulation and optimization capabilities in pho- tonics.
8.1 Future work
A natural continuation of the work presented throughout this thesis concerns use of the three-dimensional WGF method in Chapter 6 in conjunction with the BIE optimization framework developed in Chapter 7 to design and optimize fully three-dimensional waveguide structures. Significant advances have recently been made precisely in this direction in the two-dimensional case [74].
Figure 8.1: Illustrations of the two-dimensional mode splitter (a) and grating cou- pler (b) problems [74].
Figure 8.2: In (a) and (b), we show the intensity and real part of the fields, respec- tively, for the initial, non-optimized splitter. On the other hand, (c) and (d) present the optimized counterparts [74].
Figure 8.3: Intensity (a) and real part (b) of the fields produced by the WGF adjoint optimization algorithm for the problem of the grating coupler [74].
8.1. Future work 121 For reference in this context we present Figures 8.1 to 8.3 from [74], which demonstrate the character of the applied problems that can be treated by these methods. In particular, Figure 8.1 (a) depicts the problem of mode splitting, on which the boundaries of a waveguide junction are optimized to split the energy onto two separate waveguides. In Figure 8.1 (b), the problem of the grating coupler is depicted. In this case, a serrated structure is illuminated by a beam, and the sizes of the grating “teeth” are designed to couple as much energy as possible to the output mode. Figures 8.2 and 8.3 present the results of the optimization runs obtained for the mode splitter and the grating coupler, respectively. Forthcoming work along the lines of this effort seeks to extend and apply these methodologies to some of the most challenging and exciting three-dimensional design problems in electromagnetics in general, and photonics in particular.