Budden & Smith’s ‘additional memory’ and the geometric phase

Abstract

Budden & Smith considered vector waves in a z-stratified medium, driven by an arbitrary matrix A(X) depending on parameters X(z) = {X1(z),X2(z),...} which characterize the medium. They showed that in the short-wave limit there is not only the familiar optical path phase factor but an 'additional memory' factor, M, whose exponent is the line integral, along the ray, of a certain 1-form constructed from the eigenvectors of A. Their discovery anticipated the geometric phase of quantum mechanics (where z is time, and A is the hermitian hamiltonian operator). Explicit connection is made between the two formalisms. If A is symmetric, or can be made symmetric by multiplication by a constant matrix, the 1-form is integrable and M does not represent memory because it can be expressed locally, in terms of the properties of the medium at the endpoints of the ray; in quantum mechanics, symmetrizability is equivalent to the hamiltonian possessing antiunitary symmetry (e.g. time reversal). A non-integrable real M arises when light transverses a transparent medium with variable refractive index and optical activity. However, the non-integrability here is cancelled by an extra contribution from the ordinary optical path length, arising from an additional term in the constitutive relation between electric field and displacement, which is necessary in an inhomogeneous medium to ensure its transparency.