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Cauchy Dispersion Formula

When an electromagnetic wave is incident on an atom or a molecule, the periodic electric force of the wave sets the bound charges into vibratory motion. The frequency with which these charges are forced to vibrate is equal to the frequency of the wave. The phase of this motion as compared to the impressed electric force will depend on the impressed frequency. It will vary with the difference between the impressed frequency and the natural frequency of the charges.

Dispersion can be explained with the concept of secondary waves that are produced by the induced oscillations of the bound charges. When a beam of light propagates through a transparent medium (solid or liquid), the amount of lateral scattering is extremely small. The scattered waves travelling in a lateral direction produce destructive interference.

However, the secondary waves travelling in the same direction as the incident beam superimpose on one another. The resultant vibration will depend on the phase difference between the primary and the secondary waves. This superimposition, changes the phase of the primary waves and this is equivalent to a change in the wave velocity. Wave velocity is defined as the speed at which a condition of equal phases is propagated. Hence the variation in phase due to interference, changes the velocity of the wave through the medium. The phase of the oscillations and hence that of the secondary waves depends upon the impressed frequency. It is clear, therefore, that the velocity of light in the medium varies with the frequency of light. Also refractive index depends upon the velocity of light in the medium. Therefore the refractive index of the medium varies with the frequency (wavelength) of light.

The relation permittivity of the medium in the case of dynamic polarizability is given by

Here N is the number of electrons per unit volume, e the charge and m is the mass of the electron, ε₀ permittivity of free space, ƒ_i oscillator strengths of the substance, ω_i is the angular frequency of the electromagnetic spectrum of the substance, ω is the impressed angular frequency.

Also Σƒ_i = 1

and relative permeability in majority of substances that transmit electromagnetic waves is equal to 1.

Assuming that there is only one atomic frequency ω0 where

ω << ω₀

Using the binomial expansion,

Equation (vi) represents Cauchy’s dispersion formula. A and B are called Cauchy’s constants. The values of A and B depend on the medium. From equation (vi) it is evident that the refractive index of the medium decreases with increase in wavelength of light.

If a graph is plotted between and 1/λ² it will be a straight line. The intercept OP on the Y-axis gives value A. the slope of the line PC gives value of B.