ABSTRACT
This chapter, presented in a step-by-step computational example (CE) and case study (CS) solution format, is intended as a mathematical and computer program algorithm supplement to the rigorous coupled wave analysis (RCWA) method (based on state variable (SV) theory) discussed in Chapter 8. The basic mathematical algorithms and equations presented in Chapter 8, many of which are only summarized, require for their detailed derivation complex variable integration in the complex plane (Cauchy’s integral and higher-order integral formulas, i.e., residue theorem, for example), the detailed derivation of complicated bipolar coordinate trigonometric equations and relations, detailed derivations of the EM želds as calculated in the bipolar coordinate geometry, and many other computations as well. For reproducing the numerical results presented in Chapter 8 and producing new numerical results for cases not studied herein, many small computer programs and small computer subprograms need to be constructed in order that an overall system program can be constructed. The computer programs that need to be constructed carry out tasks such as (1) the cross-checking of the correctness of complicated trigonometric and hyperbolic trigonometric bipolar coordinate formulas; (2) the cross-checking of the accuracy and correctness of complex variable integrals; (3) the calculation of eigenvalues and eigenfunctions associated with solving Maxwell’s equations in bipolar coordinates (and plots of these eigenfunctions in order to check if sensible results have been obtained); (4) the checking, testing, and studying the orthogonality of the eigenfunctions found from solving Maxwell’s equations in bipolar coordinates; (5) the calculation of cascade-ladder transfer matrices used to formulate žnal matrix equations from which the overall unknowns of the scattering system may be determined; (6) the calculation and display of the geometry of the scattering system in bipolar coordinates; (7) the calculation and display of the electric and magnetic želds once an EM želd solution has been obtained; (8) the calculation of the bistatic line widths (i.e., also called scattering cross section); and (9) the calculation and cross-checking of many other quantities associated with the bipolar RCWA algorithm as well.
