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Analysis of Wave Propagation Eigenproblem in Periodic Structures

Faghihifar, Ehsan | 2020

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  1. Type of Document: Ph.D. Dissertation
  2. Language: Farsi
  3. Document No: 53427 (05)
  4. University: Sharif University of Technology
  5. Department: Electrical Engineering
  6. Advisor(s): Akbari, Mahmood
  7. Abstract:
  8. The Fourier modal method is one of the most important methods in the analysis of flat periodic structures (gratings). Using this method, the problem of wave propagation in the periodic medium leads to an eigenproblem, in which eigenvalues represent the propagation constants and eigenvector or eigenfunctions determine the filed distribution of the modes. On the other side, considering all the generalizations and modifications reported so far, the Fourier modal method still faces two fundamental problems. First, for problems involving large dielectric constants or high contrasts, the matrix form of the eigenproblem (the modal matrix) can be large, dense, and require a high computational cost. Second, due to the discrete structure of the gratings, the emergence of the Gibbs phenomenon in the field distribution of the modes and the consequent loss of the exponential convergence rate is inevitable. This dissertation deals with theoretical studies on these two fundamental problems and the less explored aspects of the wave propagation eigenproblem. In this regard, by asymptotic analysis of the modal matrix and defining the eigenvalue pattern, new methods for estimating the propagation constants of gratings have been proposed, without the need to completely solve the eigenvalue equation. We have specifically shown that the eigenvalue pattern of a grating can be expressed as a weighted sum of the corresponding patterns of uniform slabs comprised of the same dielectrics that appear in the grating. We have also proved that for conventional optical gratings, nearly every propagation constant appears on the main diagonal of the modal matrix. On the other side, with a systematic study of the spectral methods for the resolution of the Gibbs phenomenon and their evaluation in the reconstruction of modal fields, an attempt has been made to open a new door to photonic problems. To this end, by introducing a new criterion for the definition of test functions, we have shown that except for the direct reprojection method with Gegenbauer polynomials, almost all other spectral reconstruction techniques for the resolution of the Gibbs phenomenon, are inefficient in a framework of practical problems such as recovering the grating modes. At the same time, by scrutinizing existing methods, especially the Gegenbauer method, we have proposed ideas for their optimization, improving their performance, overcoming their inherent limitations, and expanding their applications
  9. Keywords:
  10. Periodic Structure ; Eigenvalue Estimation ; Fourier Modal Analysis ; Gibbs Phenomenon ; Gegenbauer Polynomials ; Wave Propagation ; Spectral Method

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