Abstract
The physical objective of solving for eigen-modes of a 1D quasiperiodic structure in photonics has been achieved. This was achieved thru considering this structure as a 1D projection or cut of a 2D periodic structure. And the problem is solved in a manner similar to 2D periodic photonic structures. A mechanical analogy (quasiperiodic orbits) helps to bring conceptual clarity.
Highlights
The idea is to use the construction of a 1D quasi-periodic structure
It was verified above that uk ( x) is periodic with period a. It is much clearer how the Bloch Theorem for the quasi-periodic problem related to the Bloch Theorem of the periodic problem; through the diagonal sectioning transformation (3.36)
It is much clearer where the periodicity pops up in a quasi-periodic problem; the physical field H ( x) of the quasi-periodic problem inherits the periodicity from the physical field H ( x, y) from the parent periodic lattice
Summary
The construction uses the diagonal sectioning of a (parent) 2D periodic tiling, the diagonal having an irrational slope (see for example [1]). This construction is used to come up with a (Fourier) series expansion. This series expansion may later be used to solve the EVP (Eigen-Value Problem) Master Equation of Photonics (see [2]). It is found that instead of solving for a single frequency/eigenvalue, we need to solve for two—one for each dimension in the associated 2D periodic tiling problem. It is found that to find the two frequencies, we need to solve the 2D periodic problem as well as the 1D quasi-periodic problem. The method of solution and the solution itself closely resembles that of periodic problems to the extent that concepts like reciprocal lattice, periodicity and a Bloch’s Theorem still apply
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