SPATIAL . OPTICAL ACTIVITY.

PROBLEM Let’s consider a medium, where the dielectric depends on the vector of light k

(0) ε푙푚 = 휀푙푚 + 푖훾푙푚푛푘푛,

(0) where 휀푙푚 is the part of the dielectric permittivity, which does not depend on the wave vector of light and 훾푙푚푛 is a phenomenological , which describes the dependence of the dielectric permittivity on various projections of the wave vector. Assume that light-matter interaction is a non-dissipative process. a) Does 훾푙푚푛 contribute to the symmetric or antisymmetric part of ε푙푚? b) Will 훾푙푚푛 result in circular or linear birefringence? c) Can we experimentally distinguish between birefringence induced by 훾푙푚푛 and by (magnetic field)? d) What are the point groups that allow observing the effect of wave vector dependence of the dielectric permittivity?

Note about spatial dispersion In the linear optical approximation, we have assumed that the electric displacement 퐃(푡, 퐫) at time point 푡 and space point 퐫 is defined by the electric field in the very same space point 퐫, where 퐫 is the vector from the centre of the coordinate system to the point.

퐃(푡, 퐫) = ∫ 휀(푡′)퐄(푡 − 푡′, 퐫)푑푡′.

This integration is necessary, if the in the frequency domain the dielectric permittivity 휀 depends on the frequency of light. The fact of such a dependence is called time dispersion.

Strictly speaking, light-matter interaction requires a more general approach which account for non- locality of the response in space

퐃(푡, 퐫) = ∬ 휀(퐫, 퐫′, 푡′)퐄(푡 − 푡′, 퐫′)푑푡′ 푑푉′, where 푉′ is volume and states for integration over the space of vectors 퐫′.

For simplicity we limit our consideration here to interaction of matter with monochromatic light at a frequency ω. Hence there is no need to account for time dispersion and the equation which describes light-matter interaction states

퐃(퐫) = ∫ 휀(ω, 퐫, 퐫′)퐄(퐫′) 푑푉′

For anisotropic media the very same equation is slightly different

′ ′ 퐷푙(퐫) = ∫ 휀푙푚(ω, 퐫, 퐫 )퐸푗(퐫′) 푑푉 Such a non-locality of optical response is called spatial dispersion. Assume that the spatial dispersion is weak and the vector of electric displacement can be represented by two terms of the Taylor series

(0) 휕퐸푙 퐷푙 = 휀푙푚 퐸푚 + 훾푙푚푛 , 휕휁푛 where 휁 is one of the coordinates (푥, 푦 or 푧). If 퐸푙 is the electric field of a monochromatic wave (퐸(푥, 푡) = 퐸(푚푎푥)exp (푖(휔푡 − 퐤퐫)) + 푐. 푐, the equation can be written in terms of effective local response

(푒푓푓) 퐷푙 = 휀푙푚 퐸푚 (푒푓푓) (0) 휀푙푚 = 휀푙푚 + 푖훾푙푚푛푘푛