Coherent waves in finite, asymmetric, one dimensional periodic structures
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The wave properties for a one dimensional periodic structure are related to the eigenvalues and eigenvectors of two pairs of 2x2 matrices M and N, E and G. M is the complex scattering matrix of coherent waves by a single scatterer while N is constructed from M by interchanging eigenvalue and eigenvector parameters. E is the real scattering matrix for wave energy fluxes onto and away from the scatterer, while G is constructed from E by interchanging eigenvalue and eigenvector parameters. Equivalently N and G are related to M and E respectively by interchanging forward and backward scattering parameters. This matrix method allows wave solutions for any size structure, including infinite periodic structures where their “cell independent” vectors are just the eigen-vectors of M, N, E and G. It is shown that the characteristic equations (CE) for the eigenvalues of E and G can be derived from the CE for the eigenvalues of M and N. Further, all the elements of E and G can be derived from M and N CE parameters. Damped or amplified Bloch-Floquet waves (BFW) are an example of coherent periodic structure waves (PSW) where the difference of backward and forward average phase shifts is . More generally scatterers may have internal wave modes giving rise to phase sensitive wave-scatterer energy exchanges causing the difference of backward and forward average phase shifts to deviate from . Phase sensitive energy exchanges derived from M allow coherent waves to exist in asymmetric periodic structures. Energy exchanges derived from the eigenvectors of E imply incoherent wave scattering although they still depend on from coherent effects internal to a scatterer.
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