On The Bloch Theorem and Orthogonality Relations

Abstract

Bloch theorem for a periodic operator is being revisited here, and we notice extra orthogonality relationships. It is shown that solutions are bi-periodic, in the sense that eigenfunctions are periodic with respect to one argument, and pseudo-periodic with respect to the other. An additional kind of symmetry between r-space and k-space exists between the envelope and eignfunctions not apparently noticed before, which allows to define new invertible modified Wannier functions. As opposed to the Wannier functions in r-space, these modified Wannier functions are defined in the k-spaces, but satisfy similar basic properties. These could result in new algorithms and other novel applications in the computational tools of periodic structures.

Coupled Mode Theory of Optomechanical Crystals

“…All other modes out of resonance will be return the fetched power from other modes soon after propagation a few lattice constants along the waveguide. The reason becomes obvious by expansion of Bloch eigenmodes [5] as E n (r; κ n ) = e −iκnx e(r; κ n ) = e −iκnx G e κn (y; G)e −iGx , (22) in which κ n is the Bloch wavevector, G = G m = 2πm are reciprocal lattice vectors, and e(r; κ n ) = e(r +x; κ n ) is the periodic envelope function of the nth harmonic. Similarly, we may use…”

“…The photo-elastic interaction [2], [3], [4] between an elastic wave with frequency Ω and electromagnetic wave at a much higher frequency ω leads to a type of nonlinear phenomenon which produces new harmonics as ω±nΩ. While both types of waves propagate in the same type of periodic medium, and observe similar Bloch-periodicity and orthogonality conditions [5], their existence are connected through the photo-elastic tensor [2], [4] which contributes through the constitutive relations for electric field and displacement vectors.…”