Lorentz Oscillator Model for Refractive Index

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.

References

[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.