Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 338 / 3 / b

For thin lenses in contact, optical powers add. At the two reference wavelengths, the lensmaker's equation gives total powerAn achromatic lens has ${\Phi }_F={\Phi }_C$. Subtracting and solving for the curvature-factor ratio yieldsOpposite curvature-weighted powers compensate the two materials' optical dispersion.

At the intermediate $d$ reference wavelength, the lensmaker's equation gives $1/f_{id}={\rho }_i(n_{id}-1)$. Therefore the curvature-factor ratio from the previous part impliesand henceThe last equality uses the Abbe number definition and expresses the achromatic doublet power balance in terms of component focal lengths.

Write ${\Phi }_i=1/f_{id}$ and $\Phi =1/f_d$. The achromatic doublet power balance isSolving these two linear equations gives ${\Phi }_1=\Phi V_{1d}/(V_{1d}-V_{2d})$ and ${\Phi }_2=-\Phi V_{2d}/(V_{1d}-V_{2d})$. Taking reciprocals gives the required focal lengths:The design requires distinct Abbe numbers; identical relative dispersion cannot yield a nonzero total achromatic power from this two-element thin model.

$V_d$ is the Abbe number, a measure of inverse relative optical dispersion. Ordinary crown glasses have comparatively large $V_d$ and lower refractive index, while ordinary flint glasses have smaller $V_d$ and often higher index. The Abbe diagram below spans typical optical-glass values; the shaded family ranges are schematic, and the marked examples are catalogue data. Modern glass compositions broaden and overlap the simple crown/flint picture.

For a positive doublet with $V_{1d}>V_{2d}$, the preceding formulas require $f_{1d}>0$ and $f_{2d}<0$: a converging crown glass element and a diverging flint glass element. Their dispersion corrections cancel, while their net optical power is positive. A small difference in Abbe numbers requires large opposing component powers, which makes the design harder to correct for other optical aberrations.