Chromatic Aberrations in Lenses

Subject: Physics

Overview

This note provides us an information about chromatic aberrations in lenses .The inability of a lens to focus all colours of light at a single point is called chromatic aberration or axial or longitudinal chromatic aberration. It is measured by the difference in focal lengths between red and violet colours.

Chromatic Aberrations in Lenses

The inability of a lens to focus all colours of light at a single point is called chromatic aberration or axial or longitudinal chromatic aberration. It is measured by the difference in focal lengths between red and violet colours.

\begin{align*} \text {Chromatic aberration} &= f_r – f_v \\ \end{align*}

Using lens maker’s formula, for mean colour of light, we have

\begin{align*} \frac 1f &= (\mu – 1) \left ( \frac {1}{R_1} + \frac {1}{R_2} \right ) \\ \text {or,} \: \frac {1}{R_1} + \frac {1}{R_2} &= \frac {1}{f(\mu -1)} \dots (i) \\ \end{align*}

Where f is focal length of mean colour, µ is refractive index of mean colour, R1 and R2 are radii of curvature of two lens surfaces.

For violet colour, we have

\begin{align*} \frac 1f &= (\mu _v – 1) \left ( \frac {1}{R_1} + \frac {1}{R_2} \right ) \\ \text {or,} \: \frac {1}{f_v} &= (\mu _v – 1) \frac {1}{f(\mu -1)}\\ \text {or,} \: \frac {1}{f_v} &= \frac {u_v - 1}{f(\mu -1)} \dots (ii) \\ \text {where} \: \mu _v \end{align*}is refractive index of violet colour. Similarly for red colour, we have \begin{align*}\frac {1}{f_r} &=\frac {\mu_v - 1}{f(\mu -1)} \dots (iii)\\ \end{align*}

Here $\mu _r$is refractive index of red colour. Subtracting equation (iii)from equation (ii) we get \begin{align*}\frac {1}{f_v} - \frac {1}{f_r} &= \frac {\mu_v - 1}{f(\mu -1)} - \frac {\mu_r - 1}{f(\mu -1)}\\ \text {or,} \: \frac {f_r – f_v}{f_v . f_r} &= \frac { u_v – 1 - \mu _r + 1}{f(\mu -1)} \\ \text {or,} \: f_f – f_v &= \frac {(\mu _v -\mu_r) f_vf_r}{f(\mu – 1)} \dots (iv) \\ \end{align*}

Achromatic Combination of Lenses

The combination of two thin lenses in which their combination is free from chromatic aberration is called the achromatic combination of lenses.

Consider two thin lenses l and L’ of dispersive power $\omega \text {and} \omega ‘$ respectively placed in contact with each other as shown in the figure. Let $\mu, \: \mu \text {and}\: \mu _r$ are the refractive indices of L for violet, mean and red colour respectively, and fv , f and fr are the focal lengths of respective colours. Similarly $\mu, \: \mu \text {and}\: \mu _r ; f_v’, f’, f_r’$ are corresponding quantities of L’.

For lens L, focal length of mean colour is

\begin{align*} \frac 1f &= (\mu – 1) \left ( \frac {1}{R_1} + \frac {1}{R_2} \right ) \\ \text {or,} \: \frac {1}{R_1} + \frac {1}{R_2} &= \frac {1}{f(\mu -1)} \\ \end{align*}

where R1 and R2 are radii of curvature of two lens surfaces. Focal length of lens L for violet colour is

\begin{align*} \frac 1f_v &= (\mu _v – 1) \left ( \frac {1}{R_1} + \frac {1}{R_2} \right ) \\\text {or,} \: \frac {1}{f_v} &= \frac {\mu_v - 1}{f(\mu -1)} \dots (i) \\ \end{align*}Similarly, focal length of lens L’ for violet colour\begin{align*} \frac {1}{f_v’} &= \frac {\mu_v’ - 1}{f’(\mu’ -1)} \dots (ii) \\ \text {If} \: F_v \end{align*}is the combined focal length of two lenses for violet colour, then \begin{align*}\frac {1}{F_v} &= \frac {1}{f_v} + \frac {1}{f_v’} \dots (iii) \\ \frac {1}{F_v} &= \frac {\mu_v - 1}{f(\mu -1)} + \frac {\mu_v’ - 1}{f(\mu ‘-1)} \dots (iv) \\ \end{align*}In the same way for red colour,\begin{align*}\frac {1}{F_r} &= \frac {\mu_r - 1}{f(\mu -1)} + \frac {\mu_r’ - 1}{f(\mu ‘-1)} \dots (v)\\ \end{align*}

\begin{align*} \text {For achromatic combination, we have} \\ F_r &= F_v \\ \text {or,} \: \frac {1}{F_v} &= \frac {1}{F_r} \\ \text {or,} \: \frac {\mu_v - 1}{f(\mu -1)} + \frac {\mu_v’ - 1}{f(\mu ‘-1)} &= \frac {\mu_r - 1}{f(\mu -1)} + \frac {\mu_r’ - 1}{f(\mu ‘-1)} \\ \text {or,} \: \frac {\mu_v - 1}{f(\mu -1)} - \frac {\mu_r - 1}{f(\mu -1)} &= \frac {\mu_r’ - 1}{f(\mu’ -1)} - \frac {\mu_v’ - 1}{f(\mu ‘-1)} \\ \text {or,} \: \frac {\mu _v – 1 - \mu _r + 1}{f(\mu – 1)} &= \frac {\mu _v’ – 1 - \mu _r’ + 1}{f(\mu’ – 1)} \\ \text {or,} \: \frac {\mu_v - \mu_r}{f(\mu -1)} &= \frac {\mu_v’ - \mu_r’}{f(\mu ‘-1)} \\ \text {or,} \: \frac {\omega }{f} &= - \frac {\omega ‘}{f} \\ \text {where} \: \frac {\mu_v - \mu_r}{f(\mu -1)} = \omega \: \text {and} \: \frac {\mu_v’ - \mu_r’}{f(\mu ‘-1)} = \omega ‘\\ \therefore \frac {\omega }{f} + \frac {\omega ‘}{f} &= 0 \\ \end{align*}

This is the condition for achromatic combination of two lenses.

Things to remember
• The combination of two thin lenses in which their combination is free from chromatic aberration is called the achromatic combination of lenses
• \begin{align*} \text {Chromatic aberration} &= f_r – f_v \\ \end{align*}
• For lens L, focal length of mean colour is

\begin{align*} \frac 1f &= (\mu – 1) \left ( \frac {1}{R_1} + \frac {1}{R_2} \right ) \\ \text {or,} \: \frac {1}{R_1} + \frac {1}{R_2} &= \frac {1}{f(\mu -1)} \\ \end{align*}

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