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# Lorentz Oscillator Model for Refractive Index

The index of refraction is often described as an apparent decrease in the speed of light from to as it passes through a dielectric medium. In fact, light photons do not actually slow down, but the effect is simulated by a retarding phase shift in the emerging electromagnetic waves. This is caused by superposition of the incident wave with a retarded wave produced by radiation from the electrons in the medium.
This Demonstration describes a highly idealized and simplified classical atomic model for the refractive index. Using the sliders, you can vary the wavelength of the incident light, , over the visible region 400–700 nm. You can also vary the natural frequency of the electron oscillators, , which generally lies in the ultraviolet, and , the number of oscillating electrons per unit volume. The lower half of the graphic shows the phase retardation of the amplitude for the transmitted radiation compared to the incident radiation. The parameters approximate those of glass, with .

### DETAILS

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.

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