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Achromatic aberration

Achromatic aberration :- An achromatic lens or achromat is a lens or combination of lens that is designed to limit the effects of chromatic aberration. Achromatic lenses bring two wavelengths (typically red and blue) into focus on the same plane.

Case I :- When lenses are in contact with each others

(Achromatic aberration)

Consider two lenses of focal length f₁ and f₂ are placed in contact with each other. The equivalent focal length f_eq of the combination is given by :-

$\displaystyle \frac{1}{{{f}_{eq}}}=\frac{1}{{{f}_{1}}}+\frac{1}{{{f}_{2}}}$

Differentiating above equation with respect to refractive index n, we get

$\displaystyle -\frac{1}{f_{_{eq}}^{2}}\frac{d{{f}_{eq}}}{dn}=-\frac{1}{f_{_{1}}^{2}}\frac{d{{f}_{1}}}{dn}-\frac{1}{f_{_{2}}^{2}}\frac{d{{f}_{2}}}{dn}$

$\displaystyle \Rightarrow d{{f}_{eq}}=f_{_{eq}}^{2}\left[ \frac{d{{f}_{1}}}{f_{_{1}}^{2}}+\frac{d{{f}_{2}}}{f_{2}^{2}} \right]$

For no chromatic aberration, df_eq = 0

$\displaystyle \therefore \frac{d{{f}_{1}}}{f_{_{1}}^{2}}+\frac{d{{f}_{2}}}{f_{2}^{2}}=0\text{ }..........(1)$

The above equation can also be written as,

$\displaystyle \left( \frac{d{{f}_{1}}}{{{f}_{1}}} \right)\frac{1}{{{f}_{1}}}+\left( \frac{d{{f}_{2}}}{{{f}_{2}}} \right)\frac{1}{{{f}_{2}}}=0$

Now, $\displaystyle \frac{df}{f}=\omega =\text{ Dispersive power of lens}$

$\displaystyle \frac{{{\omega }_{1}}}{{{f}_{1}}}=-\frac{{{\omega }_{2}}}{{{f}_{2}}}\text{ }..........(2)$

The negative sign shows that the nature of lenses for achromatic aberration is opposite, i.e., one is convex and the other is concave.

Case II :- When lenses are some distance apart from each others

(Achromatic aberration)

When there is a distance d between the lenses, then the equivalent focal length f_eq of the combination is given by :-

$\displaystyle \frac{1}{{{f}_{eq}}}=\frac{1}{{{f}_{1}}}+\frac{1}{{{f}_{2}}}-\frac{d}{{{f}_{1}}{{f}_{2}}}$

Differentiating above equation with respect to refractive index n, we get

$\displaystyle -\frac{1}{f_{_{eq}}^{2}}\frac{d{{f}_{eq}}}{dn}=-\frac{1}{f_{_{1}}^{2}}\frac{d{{f}_{1}}}{dn}-\frac{1}{f_{_{2}}^{2}}\frac{d{{f}_{2}}}{dn}+d\left[ \frac{1}{f_{_{1}}^{2}{{f}_{2}}}\frac{d{{f}_{1}}}{dn}+\frac{1}{f_{_{2}}^{2}{{f}_{1}}}\frac{d{{f}_{2}}}{dn} \right]$

For no chromatic aberration, $\displaystyle -\frac{1}{f_{_{eq}}^{2}}\frac{d{{f}_{eq}}}{dn}=0$

$\displaystyle \Rightarrow \frac{d{{f}_{1}}}{f_{_{1}}^{2}}+\frac{d{{f}_{2}}}{f_{2}^{2}}=\left[ \frac{d{{f}_{1}}}{{{f}_{1}}}+\frac{d{{f}_{2}}}{{{f}_{2}}} \right]\frac{d}{{{f}_{1}}{{f}_{2}}}$

$\displaystyle \Rightarrow \frac{{{\omega }_{1}}}{{{f}_{1}}}+\frac{{{\omega }_{2}}}{{{f}_{2}}}=\left( {{\omega }_{1}}+{{\omega }_{2}} \right)\frac{d}{{{f}_{1}}{{f}_{2}}}$

$\displaystyle d=\frac{{{\omega }_{1}}{{f}_{2}}+{{\omega }_{2}}{{f}_{1}}}{{{\omega }_{1}}+{{\omega }_{2}}}\text{ }..........(3)$

If ω₁ = ω₂

$\displaystyle d=\frac{{{f}_{1}}+{{f}_{2}}}{2}$

Achromatic aberration

Achromatic aberration

Case I :- When lenses are in contact with each others

(Achromatic aberration)

Case II :- When lenses are some distance apart from each others

(Achromatic aberration)

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