Assume a monochromatic plane electromagnetic wave incident on a transparent dielectric. It is sufficient to consider just the electric field component, represented by

, propagating in the

direction and polarized in the

direction. It is convenient to do computations with the corresponding complex forms, such as

. Assume that the polarizable electrons in the medium can be represented by Lorentz oscillators—classical charged harmonic oscillators, governed by the equation of motion

, distributed throughout the medium. Here

and

are the charge and mass of the electron,

is the fundamental frequency of the oscillator, and

is a damping constant associated with the loss of energy by radiation. Assume that

, such that, for a transparent medium, when

lies in the visible region,

is in the ultraviolet. The steady-state solution to the equation of motion reduces to

[1]. The corresponding electric current is given by

and the current density by

, where

is the number of oscillating electrons per unit volume (typically of the order of

).

To determine the radiation field produced by the collection of oscillators, we require

. The electric field of the radiation is then satisfied by a wave equation

, which pertains to the

components of

and

as functions of

and

. The subscript

indicates that

is to be evaluated at retarded times, taking account of the transmission from source points to field points at the speed of light,

. Retardation can be accounted for very simply by replacing

in the phase factor by

, where

(

will turn out to equal the index of refraction). The steady-state, far-field solution to the wave equation can be obtained by assuming that

and

both have phase factors

. This leads to an expression for the radiation field

, with the index of refraction given by

(for SI units, replace

by

). When

, the medium exhibits some absorption of the incident radiation. This can make the index of refraction a complex quantity,

, with the transmitted amplitude attenuated by a factor

. The effect is generally small and we neglect it.

[1] H. A. Lorentz,

*The Theory of Electrons and Its Applications to the Phenomena of Light and Radiant Heat*, New York: Dover Books, 2011 (original 1915).

[2] J. D. Jackson,

*Classical Electrodynamics*, 3rd ed., New York: John Wiley & Sons, 1999 pp. 246, 309–310ff.

[3] R. P. Feynman, R. B. Leighton, and M. Sands,

*The Feynman Lectures on Physics*, Vol. 2, Reading, MA: Addison–Wesley, 1964 Chap. 32.