(a) Using the result of Problem 3, show that a compound lens...

(a) Using the result of Problem $3,$ show that a compound lens system can be constructed from two lenses of different indices of refraction, $n_{1 \lambda}$ and $n_{2 \lambda}$, having the property that the resultant focal lengths of the compound lens at two specific wavelengths $\lambda_{1}$ and $\lambda_{2}$, respectively, can be made equal, or $$f_{\mathrm{eff}, \lambda_{1}}=f_{\mathrm{eff}, \lambda_{2}}$$ (b) Argue qualitatively that this condition does not guarantee that the focal length will be constant for all wavelengths.

(a) Using the result of Problem $3,$ show that a compound lens system can be constructed from two lenses of different indices of refraction, $n_{1 \lambda}$ and $n_{2 \lambda}$, having the property that the resultant focal lengths of the compound lens at two specific wavelengths $\lambda_{1}$ and $\lambda_{2}$, respectively, can be made equal, or
$$f_{\mathrm{eff}, \lambda_{1}}=f_{\mathrm{eff}, \lambda_{2}}$$
(b) Argue qualitatively that this condition does not guarantee that the focal length will be constant for all wavelengths.

Key Concepts

Compound Lens System

A compound lens system consists of two or more lenses combined so that their individual optical properties interact to produce a desired overall effect. In this context, combining lenses with different dispersion properties allows one to design an optical system where the effective focal length at particular wavelengths (often chosen as the extremes of the visible spectrum) can be made equal, thereby reducing chromatic aberration for those wavelengths.

Effective Focal Length

The effective focal length of a multi-lens system is determined by both the focal lengths of the individual lenses and the spatial separation between them. For thin lenses in contact, the effective focal length can be found using the formula that involves the sum of their individual powers. In designing a compound lens system to equalize focal lengths at specific wavelengths, one typically adjusts the relative lens powers taking into account wavelength-dependent refractive indices.

Dispersion and Chromatic Aberration

Dispersion refers to the variation of a material's refractive index with wavelength, which leads to chromatic aberration—a phenomenon where different colors (wavelengths) of light do not converge to the same focal point. Even if the compound lens system is optimized so that two selected wavelengths share the same effective focal length, dispersion means that wavelengths in between (or outside) those two points may still be focused differently, resulting in residual chromatic aberration.

Achromatism

Achromatism is the property of an optical system in which two specific wavelengths are brought into focus at the same point, thereby correcting for chromatic aberration at those wavelengths. While this approach minimizes the color fringing for those two wavelengths, it does not completely eliminate chromatic aberration across the whole spectrum due to the non-linear dependence of refractive index on wavelength.

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