Question

The refractive indices of a material for a lens for violet, yellow and red colors of light are respectively $1.66$, $1.44$, and $1.62$. The mean focal length of the lens is 10cm. Determine the chromatics aberration of the lens between the violet and the red colors.
A. $0.225cm$
B. $0.324cm$
C. $0.576cm$
D. $0.625cm$

Answer 1

Hint: First calculate the dispersive power of the lens using the refractive indices of the material and then apply its value in the formula of chromatic aberration. Dispersive power of a lens shows how much a particular color or fragment of a light is getting scattered and interfered by other colors of the light.

Complete step by step answer:
Chromatic aberration, also referred to as “color fringing” or “purple fringing”, has been a common optical problem that happens when a lens is either unable to bring all the wavelengths of colors to the same focal plane, or when wavelengths of color are focused at different positions within the focal plane. It is also possible that both the cases occur simultaneously. Chromatic aberration is caused by lens dispersion, with different colors of white light travelling at different speeds while passing through a lens. As a result, the image can look blurred or colored edges can be seen around objects, especially in high-contrast situations. A perfect lens would focus all wavelengths into one focus, where the simplest focus with the “circle of least confusion” is found. The index of refraction for every wavelength is different in lenses, which causes two sorts of aberration – Longitudinal aberration and Lateral aberration. Now let \[\omega \]be the dispersive power of the lens and \[{n_r},{n_y},{n_v}\] be the refractive indices of red, yellow and violet respectively. The frequencies of the colors of red, violet and yellow are \[{f_r},{f_v},{f_y}\] respectively. The chromatic aberration of lens between the colors violet and red are as follows:
\[{f_r} - {f_v} = \omega \times {f_y}\]
Here \[\omega \] is the dispersive power of the lens and in terms of refractive indices, it given as follows:
\[\omega = \dfrac{{{n_v} - {n_r}}}{{{n_y} - 1}}\]

Thus the chromatic aberration will given by:
\[
{f_r} - {f_v} = \dfrac{{{n_v} - {n_r}}}{{{n_y} - 1}} \times {f_y} \\
\Rightarrow {f_r} - {f_v} = \dfrac{{(1.66 - 1.62)}}{{(1.64 - 1)}} \times 10 \\
\therefore {f_r} - {f_v} = 0.625cm \\
\]
Thus option D is correct.

Note: Here, chromatic aberration can also be calculated through wavelength of light, as chromatic aberration means the shifting of wavelength of light from its original wavelength. Although we don’t use wavelength as it is mathematically simpler to calculate it through frequency.