Chromatic Aberration: The image of a white object in white light wh...
0Chromatic Aberration: The image of a white object in white light which with the help of Eqn. (4) reduces to formed by a lens is usually coloured and blurred. This defect of image is called chromatic aberration and arises due to the fact \( \frac{\omega_{1} f_{1}}{f_{1}^{2}}+\frac{\omega_{2} f_{2}}{f_{2}^{2}}=0 \) i.e., \( \frac{\omega_{1}}{f_{1}}+\frac{\omega_{2}}{f_{2}}=0 \) \( (5) \)
\( \mathrm{P} \) \( \begin{array}{ll}\text { that focal length of a lens is different for different colours. As } & \text { This condition is called condition of achromatism (for two } \\ \text { thin lenses in contact) and the lens combination which }\end{array} \)
W R.I. I of lens is maximum for violet while minimum for red, violet \( \quad \begin{array}{l}\text { thin lenses in contact) and the lens combination which } \\ \text { satisfies this condition is called achromatic lens, from this }\end{array} \) is focused nearest to the lens while red farthest from it as shown satisfiesthis condition is called achromatic lens, from this in figure. achromatic doublet:
(a) The two lenses must be of different materials.
Since, if \( \omega_{1}=\omega_{2}, \frac{1}{f_{1}}+\frac{1}{f_{2}}=0 \) i.e. \( \frac{1}{F}=0 \) or \( F=\infty \)
\( \Rightarrow-\frac{\mathrm{df}}{\mathrm{f}^{2}}=\mathrm{d} \mu\left[\frac{1}{\mathrm{R}_{1}}=\frac{1}{\mathrm{R}_{2}}\right] \)
i.e., combination plane glass plate.
(b) As \( \omega_{1} \) and \( \omega_{2} \) are positive quantities, for equation (5) to hold, \( f_{1} \) and \( f_{2} \) must be of opposite nature, i.e. if one of the lenses is converging the other must be diverging.
(c) If the achromatic combination is convergent,
Chromatic aberration of a lens can be corrected by :
Dividing Eoa. (3) by (2);
(a) providing different suitable curvatures of its two
\( -\frac{d f}{f}=\frac{d \mu}{(u-1)}=\omega\left[\omega-\frac{d u}{(u-1)}\right] \) \( = \) dispersive power surfaces.
And hence, froen Equs. (1) and (4). L. C.A. =-df \( = \) of
(b) proper polishing of its two surfaces.
L.C.A. = \( -d f=c \) ff Now, as for a single lens neither f not \( o \) can be zero, we
(c) suitably combining it with another lens.
eannot have a single lens free Condition of Achromatism:
In case of two thin lenses in contact
(d) reducing its aperture.
\[
\frac{1}{F}=\frac{1}{f_{1}}-\frac{1}{f_{2}} \text { i.c., }-\frac{\mathrm{dF}}{\mathrm{F}^{2}}=-\frac{\mathrm{df_{2 }}}{\mathrm{f}_{1}^{2}}-\frac{\mathrm{df_{2 }}}{f_{2}^{2}}
\]
The combination will be free from chromatic aberration if
\( d F=0 \)
i.e. \( \frac{d f_{1}}{f_{1}^{2}}+\frac{d f_{2}}{f_{2}^{2}}=0 \)
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