Geometrical aspects in optical wave-packet dynamics

Abstract

We construct a semiclassical theory for propagation of an optical wave packet in a nonconducting medium with a periodic structure of dielectric permittivity and magnetic permeability, i.e., a nonconducting photonic crystal. We employ a quantum-mechanical formalism in order to clarify its link to those of electronic systems. It involves the geometrical phase, i.e., Berry's phase, in a natural way, and describes an interplay between orbital motion and internal rotation. Based on the above theory, we discuss the geometrical aspects of the optical Hall effect. We also consider a reduction of the theory to a system without periodic structure and apply it to the transverse shift of an optical beam at an interface reflection or refraction. For a generic incident beam with an arbitrary polarization, an identical result for the transverse shift of each reflected or transmitted beam is given by the following different approaches: (i) analytic evaluation of wave-packet dynamics, (ii) total angular momentum (TAM) conservation for individual photons, and (iii) numerical simulation of wave-packet dynamics. It is consistent with a result by classical electrodynamics. This means that the TAM conservation for individual photons is already taken into account in wave optics, i.e., classical electrodynamics. Finally, we show an application of our theory to a two-dimensional photonic crystal, and propose an optimal design for the enhancement of the optical Hall effect in photonic crystals.

Representation theory for vector electromagnetic beams

“…[21] disagreed with Bliokh and Bliokh [22] on their incident beam. On the other hand, Bliokh and Bliokh [23] found that the physical properties of Onoda's incident beam [21] depend on the "incidence angle". Such a controversy concerns in fact the description of the vectorial property of a finite beam.…”

“…[21] can be described in this representation formalism as the first-order approximation of such a special beam the unit vector I of which happens to make an angle of the minus incidence angle with the propagation axis and happens to be normal to the interface. This is implied in Ref.…”

“…It is well known [2,5,13,21] that if the unit vectors p and s are defined from the wave vector k in terms of an arbitrary fixed real unit vector I as…”

“…The controversy over the incident beams of Refs. [21] and [22] is resolved. A transverse effect is also found that a beam of elliptically polarized angular spectrum is displaced from the center in the direction that is perpendicular to the plane formed by the unit vector and the propagation axis.…”

“…Recently, there appeared a controversy [21][22][23] over the physical properties of the light beams that were proposed to investigate the Imbert-Fedorov effect, a transverse displacement of a reflected [24][25][26][27][28] or a transmitted [27][28][29][30][31] beam taking place at a dielectric interface.…”