The non-linear constitutive relation in solids at optical frequencies: II

This is the authors' abstract. We don't add key points for this paper.

This paper forms a natural extension of previous work of the authors published in 1963. An alternative formula to that previously obtained is derived for the rth-order conductivity tensor σμα1...αr (r)(ω1,..., ωr) of a solid by calculating the time development operator of the charged particles in the solid. This leads to a general explicit expression for σ(r) in terms of the energy eigenstates of the solid, which provides a generalization to arbitrary order of the Kramers-Heisenberg dispersion formula given essentially by σ(1). This alternative formula can also be cast into a form which reveals the so-called overall permutation symmetry property of the tensors. This says that the related susceptibility tensor χμα1...αr (r)(ω1,..., ωr) = iσμα1...αr (r)(ω1,..., ωr)/(ω1 + ω2 +... + ωr) is invariant under all (r + 1)! permutations of the pairs μ(- ω1 - ω2 -... - ωr), α1ω1, α2ω2,..., αrωr for frequencies lying in any part of the complex frequency plane. The significance of this symmetry relation is discussed for various special cases. In particular, it is valid at real frequencies when all singularities of the tensor are avoided so that no energy transfer can take place between the solid and the radiation field. It is then shown, among other things, to allow a derivation of the Manley-Rowe relations for a dispersive medium. The consequences of the invariance of the unperturbed Hamiltonian of the solid under time reversal are discussed. It is shown that the rth-order conductivity tensor changes sign when all frequencies are negated.