Anatoli Babin

Nonlinear photonic crystals: I. Quadratic nonlinearity

Abstract We develop a consistent mathematical theory of weakly nonlinear periodic dielectric media for the dimensions one, two and three. The theory is based on the Maxwell equations with classical quadratic and cubic constitutive relations. In particular, we give a complete classification of different nonlinear interactions between Floquet–Bloch modes based on indices which measure the strength of the interactions. The indices take on a small number of prescribed values which are collected in a table. The theory rests on the asymptotic analysis of oscillatory integrals describing the nonlinear interactions.

Gaussian Wave Packets

We summarize here the basic properties of the Gaussian wave packet solution to the Schrodinger equation for the free particle.

Almost Periodic Functions and Their Time-Averages

We provide here some very basic information on real and complex-valued almost periodic functions.

Elementary Momentum Equation Derivation for NKG

We present here a direct verification of the energy and momentum conservation laws for the nonlinear Klein-Gordon equation without magnetic field.

Hamilton–Jacobi Theory

We consider in this appendix certain aspects of the Hamilton-Jacobi theory which play a very important role in the Schrodinger and de Broglie wave theories as well as in the Bohmian mechanics.

Point Charges in a Spatially Homogeneous Electric Field

We consider here an application of the Hamilton-Jacobi theory to the dynamics of a point charge in a spatially homogeneous electric field.

Clifford and Spacetime Algebras

Though we are interested primarily in Spacetime algebra (STA), it is instructive to consider it within the general framework of Clifford algebras. This chapter is a concise exposition of selected subjects from an extensive theory of Clifford algebras.

Dipole Approximation for Localized Distributed Charges

This section is devoted to dipole approximation for solutions of Maxwell equations with localized sources.

The Theory of Electromagnetic Interaction of Dressed Charges

The theory of dressed charges in many respects is similar to the theory of balanced charges developed in previous chapters. The difference is in the presence of the electromagnetic self-interaction which is absent in balanced charges. Here we show that in macroscopic regimes this difference is negligible, whereas at atomic scales balanced charges better describe Hydrogen spectrum.

Basics of Spacetime Algebra (STA)

The spacetime algebra (STA) is the Clifford algebra based on the real 4-dimensional Minkowski space. Basic concepts of the STA are introduced in this chapter.