Angular Dispersion and Deviation without Dispersion

Angular Dispersion

When a beam of white light passes through a prism, light of different colours are deviated by different amounts. The mean deviation of light is measured by the deviation of yellow light as this deviation is nearly the average of all deviation. As shown in the figure, the red light is least deviated and the violet light is the most deviated. The angular dispersion is defined as angle between the two extreme colours of light (i.e. violet and red colours)

\begin{align*} \text {angular dispersion} \: &= \delta _v - \delta _r \dots (i) \\ \end{align*}Since the deviation is small prism \begin{align*}\delta = A(\mu -1), \: \text {then deviation for violet light,} \: \delta _v &= A (\mu _v – 1) \\ \text {and deviation for red light,} \: \delta _r &= A (\mu _v – 1) \\ \text {where, A is angle of prism, and} \mu _v \: \text {and} \: \mu _r \: \end{align*}are refractive indices of violet and red light respectively. Substituting the values of \begin{align*}\delta _v \: \text {and} \: \delta _r \: \text {and equation} \: (i), \text {we get} \\ \text {angular dispersion} &= \delta _v - \delta _r \\ &= A (\mu _v – 1) - A (\mu _r – 1) \\ &= A (\mu _v - \mu _r) \\ \text {or,} \: \delta _v - \delta _r &= A (\mu _v - \mu _r) \\ \end{align*}

Dispersive power, \(\omega \)

Dispersive power of a prism is defined as the ratio of angular dispersion to the deviation of mean colour.

\begin{align*} \text {Dispersive power,} \omega &= \frac {\delta _v - \delta _r}{\delta } \\ &= \frac { A (\mu _v – 1) - A (\mu _r – 1)}{ A (\mu – 1)} \\ \end{align*}$$Here,\delta = A(\mu -1) $$is angle of deviation for mean color.\begin{align*}\text {or,} \: \omega &= \frac { A (\mu _v - \mu _r)} { A (\mu – 1)} \\ \therefore \omega &= \frac { \mu _v - \mu _r} { \mu – 1} \\ \end{align*}

Deviation without Dispersion

The two glass prisms are arranged as shown in the figure. The first prism is crown glass prism while the second prism in flint glass prism. The glass prism has reflecting angle A and the flint glass has A’. Let µ_v, µ, and µ_r be the refractive index of crown glass for violet, mean and red light while µ ’_v, µ’ and µ’_r be the corresponding value for flint glass. When white light is incident on the crown glass prism it is splitted into seven colours. Then the splitted colours of white light are again passed through the flint glass prism. Deviation produced by flint glass prism for a colour is equal and opposite which produced by crown glass prism. As a result, when the seven colours of light come out through the flint glass prism, they combine to give a white light. Hence, there is no dispersion of light.

\begin{align*} \text {Angular dispersion produced by crown glass prism is given by} \\ \delta _v - \delta _r &= A (\mu _v - \mu _r) \dots (i)\\ \end{align*} Angular dispersion produced by flint glass prism is given by \begin{align*}\delta _v’ - \delta _r’ &= A’ (\mu _v’ - \mu _r’) \dots (i)\\ \end{align*}{For the dispersion of the combined prism to be zero, we should have net dispersion = 0 \begin{align*}\text {or,} \: \delta _v - \delta _r &= \delta _v’ - \delta _r’ \\ A(\mu _v - \mu _r) &= A’ (\mu _v’ - \mu _r’) \dots (iii) \\ \end{align*}This is the condition for the combined prism to have no dispersion.

\begin{align*}\text {Now,} \\ \text {For crown glass prism,} \delta &= A (\mu -1) \\ \text {For flint glass prism,} \delta ‘ &= A’ (\mu ‘ -1) \\ \therefore A = \frac {\delta }{\mu – 1} \: \text {and} \: A’ = \frac {\delta ‘}{\mu ‘– 1} \dots (iv)\\ \text {Now, from equation} \: (iii) \text {and} \: (iv) \\ \left (\frac {\delta }{\mu – 1}\right )(\mu _v - \mu _r) &= \left (\frac {\delta ‘}{\mu ‘– 1} \right )(\mu _v’ - \mu _r’) \\ \text {or,} \: \delta \left (\frac {\mu _v - \mu _r }{\mu – 1}\right ) &= \delta ‘ \left (\frac {\mu _v’ - \mu _r’ }{\mu ‘ – 1}\right ) \\ \text {or,} \: \delta \omega &= \delta ‘ \omega ‘ \\ \text {or,} \: \frac {\delta }{\delta ‘} &= \frac {\omega ‘}{\omega } \dots (v) \\ \end{align*}

Hence, for dispersion, the two prism should be selected that their dispersive power are in the inverse ratio of the deviations by them for the mean light.

Net Deviation

Net deviation suffered by the light due to combination of two prisms is given by

\begin{align*} \text {Net deviation} &= \delta - \delta ‘ \\ &= \delta \left ( 1 - \frac {\delta ‘}{\delta } \right ) \\ \text {Net deviation} &= \delta \left ( 1 - \frac {\omega}{\omega ‘ } \right ) \end{align*}

Conclusion

1. It may be pointed out that the values of refractive index and dispersive power of flint glass are greater than that of crown glass, \(\omega > \omega \) and \(\mu ‘ > \mu \) . so,
   $$ \frac {\omega }{\omega ‘} < 1$$
   The factor \( 1 - \omega / \omega \), is positive which justifies that the combination will produce deviation in the same direction as produced by crown glass prism since \(\delta > \delta ‘ \)
2. From equation (i), we have
   \begin{align*} A (\mu _v - \mu _r ) &= A’ (\mu _v’ - \mu _r’ )\\ \text {or,} \: A(\mu – 1) \frac {\mu _v - \mu _r}{\mu -1} &= A’(\mu ‘– 1) \frac {\mu _v’ - \mu _r’}{\mu ‘ -1} \\ \text {or,} \: A(\mu – 1) \omega &= A’(\mu ‘– 1)\omega ‘ \\ \text {Since} \: \omega ‘ >\omega , A(\mu -1) >A’(\mu -1). \\ \text {Again,} \: \mu ‘> \mu \: \text {so} \: A(\mu -1) > A’(\mu ‘-1). \text {Hence} \: A > A’ \\ \end{align*}It means angle of prism of crown glass is greater than that of flint glass.
3. Of two prisms are of same material. So \(\mu = \mu ‘\) and \( \omega = \omega ‘\). Then equation (iii) becomes
   $$\text {Net deviation} = \delta \left ( 1 - \frac {\omega }{\omega ‘} \right ) = 0 $$
   it means when two prisms are of same materials, there is no dispersion as well as no deviation.