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Deviation

Deviation, measured in degrees, is the angle an incident ray is turned through after passing through a prism(or other optical component).
This deviation is a minimum for a prism when the path of a light ray is symmetrical about its axis of symmetry.

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Derivation of Minimum Deviation D

In

            (i          

(D external angle of )

around points P and Q respectively,

substituting into equation (i

                       (ii          

In

                             (iii           

adding together equations (ii & (iii,

             (iv          

rearranging equation (iii,

                                   (v          

Using Snell's Law equation,

and substituting for i and r from equations (iv & (v above,

n₁ is the refractive index of air which approximates to 1. Hence the equation becomes:

Example

So for a 60^o equilateral prism made of glass(n = 1.5 approx.), the minimum deviation angle D is given by:

It follows that:

remembering that sin^-1(x) means the angle whose sine is 'x',

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Chromatic dispersion

The term Chromatic dispersion describes how refractive index changes with wavelength for a particular medium.

data courtesy of Serway & Jewett

Using refractive index theory we can formulate equations in terms of the wavelength of the light involved:

Remembering that: n_m is defined as the absolute refractive index

where,

c_o is the velocity of white light in a vacuum
c_m is the velocity of white light in the medium

It follows that: n_v is defined as the absolute refractive index for violet light

                              (i     

where, c_m(v) is the velocity of violet light in the medium

Similarly, n_r is defined as the absolute refractive index for red light

                              (ii     

where, c_m(r) is the velocity of red light in the medium

Since frequency f is constant throughout,

and

where,

    λ_o is the wavelength of white light in a vacuum
    λ_m(v) is the wavelength of violet light in the medium
    λ_m(r) is the wavelength of red light in the medium

Substituting for c_m(v) , c_m(r) and c_o into equations (i and (ii ,

cancelling f in each equation,

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Minimum angle of dispersion

This is measure of the angle of 'spread' of a spectrum when it leaves a prism. For minimum angular dispersion, the angle is derived from the difference in deviation between red and violet rays of light.

Replacing n₂ with n_r for red, and n_v for violet light in the minimum deviation equation below,

we can calculate values of deviation for each colour. Subtracting the angles gives the minimum dispersion angle for white light.

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