Bloch-Messiah decomposition and Magnus expansion for parametric down-conversion with monochromatic pump: role of the gain

We consider the effect of different orders of Magnus expansion for the field transformation in type-I parametric down-conversion with a monochromatic pump. The exact solution, existing in this case, allows us to analyze the convergence of the Magnus expansion for the spectrum of squeezing and the angle of squeezing. We demonstrate how the convergence of the Magnus series depends on the parametric gain for various values of the phase mismatch. For each phase-mismatch angle we find the gain, which is the exact upper bound for the convergence of the Magnus series.

Effective Willis constitutive equations for periodically stratified anisotropic elastic media

A method to derive homogeneous effective constitutive equations for periodically layered elastic media is proposed. The crucial and novel idea underlying the procedure is that the coefficients of the dynamic effective medium can be associated with the matrix logarithm of the propagator over a unit period. The effective homogeneous equations are shown to have the structure of a Willis material, characterized by anisotropic inertia and coupling between momentum and strain, in addition to effective elastic constants. Expressions are presented for the Willis material parameters which are formally valid at any frequency and horizontal wavenumber as long as the matrix logarithm is well defined. The general theory is exemplified for scalar SH motion. Low frequency, long wavelength expansions of the effective material parameters are also developed using a Magnus series, and explicit estimates are derived for the rate of convergence.