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Explore the mathematical foundations of electromagnetic waveguides in this comprehensive lecture that introduces key concepts for understanding wave propagation in guiding structures. Begin with general second-order wave propagation models and learn about closed waveguides with heterogeneous cross sections, focusing on guided modes characterized by frequency ω and axial wave number β. Examine two fundamental approaches to finding guided modes: searching for possible frequencies given a real wave number, and finding possible wave numbers for a given real frequency, both formulated as eigenvalue problems in the waveguide's cross section. Discover how the first approach yields a self-adjoint eigenvalue problem with necessarily real frequencies, while the second presents a quadratic eigenvalue problem where solutions may be complex. Study the application of these concepts to scalar wave equations before delving into the more complex electromagnetic case. Master essential material for Maxwell equations, including differential operators and related functional spaces, and understand how finding electromagnetic modes reduces to determining the spectrum of a self-adjoint operator with compact resolvent. Learn about the sophisticated mathematical techniques required for this analysis, including Helmholtz decomposition of 2D vector fields and associated compactness results. Investigate the special case of constant electromagnetic wave speed c in the cross section using transverse field formulation, where mode-finding becomes equivalent to finding eigenvalues of a positive self-adjoint transverse operator A with compact resolvent. Conclude by exploring how eigenmodes can be classified into TE (transverse electric), TM (transverse magnetic), and potentially TEM modes depending on the waveguide's cross-sectional connectivity.
