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Achromatism in Prism

Derivation without dispersion: The dispersive powers of the materials of different prisms are different. The dispersive power if a flint glass prism is higher than the dispersive power of a crown glass prism, i.e. the angular separation of the spectral colours in a flint glass prism is greater than that for a crown glass prism of the same angle. Further, when a composite beam of light is passed through a prism, the beam is dispersed as well as deviated. In fig. C and F are two prisms of crown and flint glass of refracting angles and respectively. Let and be the refractive indices of the two materials for blue, yellow and red rays of light. Then

The angular separation for the blue and red rays of light is

For no dispersion

(δ_b – δ_r) + (δ’_b – δ_r) = 0

The negative sign shows that the refracting angles of the two prisms are in opposite directions. If is the refracting angle of the crown glass prism, the refracting angle of the flint glass prism can be calculated from equation (ii). The combination of two such prisms which satisfies equation (ii) is achromatic and a beam of composite light refracted through the two prisms will only be deviated and not dispersed. The angle δ and δ’ are given by

Substituting these values of and in equation (i)

∴ ωδ + ω’δ’ = 0 (iii)

where ω and ω’ are the dispersive powers of the materials of the crown and flint glass prisms respectively.

As for achromatism, the refracting angles of the prisms are different, the deviations produced by the two prisms are different and are in opposite directions. The derivation of the mean ray is

D = δ + δ’

But, from equation (i)

Substituting this value of in equation (iv)