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Aberration of Lens

Chromatic Aberration (Aberration of Lens)

Aberration of Lens :- The inability of a lens to form the white image of a white object is called chromatic aberration. Image of a white object is colored and blurred because the focal length of a lens depends upon the refractive index of the material of the lens.

According to Cauchy’s formula, $\displaystyle n=A+\frac{B}{{{\lambda }^{2}}}+\frac{C}{{{\lambda }^{4}}}+...$ , a lens has different refractive Indices for different colors or wavelength of light. The refractive index is maximum for violet light and minimum for red light.

From lens makers formula, $\displaystyle \frac{1}{f}=(n-1)\left[ \frac{1}{{{R}_{1}}}-\frac{1}{{{R}_{2}}} \right]$

⇒ $\displaystyle f\propto \frac{1}{n-1}$

Hence focal length of a lens is maximum for red color and minimum for violet color.

⇒ f_Violet< f_Indigo< f_Blue< f_Green< f_Yellow< f_Orange< f_Red

Types of Chromatic Aberration (Aberration of Lens)

Chromatic aberration is of two types :-

Axial/longitudinal chromatic aberration or &
Lateral chromatic aberration

1. Axial/longitudinal chromatic aberration

Axial chromatic aberration is the spread of images along the principal axis.

This can be calculated in two cases :-

Case I :- When object lies is at infinity

Axial chromatic aberration is given by, X = f_R – f_V

Using lens maker’s formula, $\displaystyle \frac{1}{f}=(n-1)\left[ \frac{1}{{{R}_{1}}}-\frac{1}{{{R}_{2}}} \right]$

Or $\displaystyle \frac{1}{f(n-1)}=\left[ \frac{1}{{{R}_{1}}}-\frac{1}{{{R}_{2}}} \right]..........(1)$

Where f and n are respectively the focal length and refractive index of the lens for mean colour (i.e., yellow colour).

For red colour, $\displaystyle \frac{1}{{{f}_{R}}}=({{n}_{R}}-1)\left[ \frac{1}{{{R}_{1}}}-\frac{1}{{{R}_{2}}} \right]$

Using the value of $\displaystyle \left[ \frac{1}{{{R}_{1}}}-\frac{1}{{{R}_{2}}} \right]$ , from equation (1), we get

$\displaystyle \frac{1}{{{f}_{R}}}=\frac{({{n}_{R}}-1)}{f(n-1)}..........(2)$

For violet colour, $\displaystyle \frac{1}{{{f}_{V}}}=({{n}_{V}}-1)\left[ \frac{1}{{{R}_{1}}}-\frac{1}{{{R}_{2}}} \right]$

Using the value of $\displaystyle \left[ \frac{1}{{{R}_{1}}}-\frac{1}{{{R}_{2}}} \right]$ , from equation (1), we get

$\displaystyle \frac{1}{{{f}_{V}}}=\frac{({{n}_{V}}-1)}{f(n-1)}..........(3)$

Subtracting equation (2) from equation (3)

$\displaystyle \frac{1}{{{f}_{V}}}-\frac{1}{{{f}_{R}}}=\frac{1}{f(n-1)}\left[ {{n}_{V}}-1-{{n}_{R}}+1 \right]$

$\displaystyle \Rightarrow \frac{1}{{{f}_{V}}}-\frac{1}{{{f}_{R}}}=\frac{{{n}_{V}}-{{n}_{R}}}{f(n-1)}$

$\displaystyle \Rightarrow \frac{{{f}_{R}}-{{f}_{V}}}{{{f}_{V}}{{f}_{R}}}=\frac{{{n}_{V}}-{{n}_{R}}}{f(n-1)}$

Now using the relation, f_Vf_R = f²

$\displaystyle \frac{{{f}_{R}}-{{f}_{V}}}{{{f}^{2}}}=\frac{{{n}_{V}}-{{n}_{R}}}{f(n-1)}$

$\displaystyle {{f}_{R}}-{{f}_{V}}=\left( \frac{{{n}_{V}}-{{n}_{R}}}{n-1} \right)f$

Now the quantity $\displaystyle \left( \frac{{{n}_{V}}-{{n}_{R}}}{n-1} \right)$ , is dispersive power ω, so

Axial chromatic aberration, X = ωf

Case II :- When object lies at finite distance from the lens

In this case axial chromatic aberration is given by, X = v_R – v_V

Using lens formula, $\displaystyle \frac{1}{f}=\frac{1}{v}-\frac{1}{u}$

For red image, $\displaystyle \frac{1}{{{f}_{R}}}=\frac{1}{{{v}_{R}}}-\frac{1}{u}..........(4)$

And for violet image, $\displaystyle \frac{1}{{{f}_{V}}}=\frac{1}{{{v}_{V}}}-\frac{1}{u}..........(5)$

Subtracting equation (4) from equation (5), $\displaystyle \frac{1}{{{f}_{V}}}-\frac{1}{{{f}_{R}}}=\left( \frac{1}{{{v}_{V}}}-\frac{1}{u} \right)-\left( \frac{1}{{{v}_{R}}}-\frac{1}{u} \right)$

Or $\displaystyle \frac{1}{{{f}_{V}}}-\frac{1}{{{f}_{R}}}=\frac{1}{{{v}_{V}}}-\frac{1}{{{v}_{R}}}$

$\displaystyle \frac{{{f}_{R}}-{{f}_{V}}}{{{f}_{V}}{{f}_{R}}}=\frac{{{v}_{R}}-{{v}_{V}}}{{{v}_{V}}{{v}_{R}}}$

Now using the relations, f_R – f_V = ωf, f_R f_V = f² and v_V v_R = v²

We get,

$\displaystyle \frac{\omega f}{{{f}^{2}}}=\frac{{{v}_{R}}-{{v}_{V}}}{{{v}^{2}}}$

Axial chromatic aberration, $\displaystyle {{v}_{R}}-{{v}_{V}}=\frac{\omega {{v}^{2}}}{f}$

Where ω is the dispersive power, v is distance of image for mean(yellow) colour and f is mean focal length of lens.

2. Lateral chromatic aberration

Lateral chromatic aberration is the spread of images perpendicular to the principal axis.

Lateral chromatic aberration =

spread of images perpendicular to the principal axis =

Height of red image – Height of violet image

$\displaystyle \Rightarrow {{I}_{R}}-{{I}_{V}}=\frac{{{v}_{R}}O}{u}-\frac{{{v}_{V}}O}{u}$

$\displaystyle \Rightarrow {{I}_{R}}-{{I}_{V}}=({{v}_{R}}-{{v}_{V}})\frac{O}{u}$

But $\displaystyle {{v}_{R}}-{{v}_{V}}=\frac{\omega {{v}^{2}}}{f}$

∴ Lateral chromatic aberration $\displaystyle {{I}_{R}}-{{I}_{V}}=\left( \frac{\omega {{v}^{2}}}{f} \right)\frac{O}{u}$

Aberration of Lens

Aberration of Lens

Chromatic Aberration (Aberration of Lens)

Types of Chromatic Aberration (Aberration of Lens)

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