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Stability Analysis of Parametric Amplification in Linear Time-Varying Electromagnetic Structures

Salehi, Mohammad Reza | 2023

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  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 55870 (05)
  4. University: Sharif University of Technology
  5. Department: Electrical Engineering
  6. Advisor(s): Memarian, Mohammad; Mehrany, Khashayar
  7. Abstract:
  8. We study the phenomenon of parametric amplification and instability in the context of time-varying electromagnetic structures, particularly dielectric slabs with periodically modulated permittivity. These structures show particular promise inasmuch as they are capable of very large amplifications when illuminated by an electromagnetic wave of half the modulation frequency. Successive studies have corroborated this finding but none have yet been able to ascertain the nature of amplification in such devices. On top of that, the problem of instability in time-periodic slabs has either been disregarded altogether or it has been subjected to loose analysis, leading to mere speculations which are wildly off the mark. Should instability occur in a time-periodic structure, it will certainly overshadow its intended application, be it parametric amplification, frequency conversion, nonreciprocity or frequency comb, among others. The ignorance concerning the phenomenon of instability—which permeates the current literature on time-periodic distributed structures—is bound to pose serious challenges to the practical realization of such devices. The problem lies in the poor understanding (or lack thereof) of the mathematical devices necessary to tackle such problems. We circumvent this problem and successfully analyze parametric amplification and instability by tapping into the rich mathematical theory of Hill's equation. In doing so, we employ the Floquet's theorem in its complete form and make use of the mathematical concept of coexistence, which to the best of our knowledge have not yet been applied in the field of time-varying optics. We also propose an equivalent circuit model for time-varying slabs which provides valuable insight into the origins of instability. This helps us work out a method to identify the unstable modes of a time-periodic slab with great accuracy. It follows from our findings that instability is much more pervasive than it was previously perceived. In fact, we demonstrate that a time-periodic slab is in most cases unstable and thus it had best be analyzed for application as a laser rather than a parametric amplifier
  9. Keywords:
  10. Instability ; Circuit Model ; Parametric Amplification ; Time-Varying Electromagnetic Structures ; Coexistence ; Hill's Equation Theory

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