Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 338 / 1 / c / iv / Solution

$V_d$ is the Abbe number, a dimensionless inverse measure of optical dispersion. Large $V_d$ means weak dispersion. The Abbe diagram below spans a representative visible-light optical glass range, roughly $1.45\lesssim n_d\lesssim 2.0$ and $20\lesssim V_d\lesssim 95$. The shaded families are schematic, not exact catalogue boundaries; modern compositions overlap and include high-index crown glasses.

For a usual positive objective, choose component 1 as a higher-$V_d$ crown glass and component 2 as a lower-$V_d$ flint glass. Then $f_{1d}>0$, $f_{2d}<0$, and $|f_{2d}|/f_{1d}=V_{1d}/V_{2d}>1$: the crown glass is converging, the flint glass diverging, and their dispersive powers cancel while leaving positive net power. The Abbe numbers alone do not set their absolute focal lengths; the specified total focal length does. As a representative pair, $V_{1d}=64.17$, $V_{2d}=36.37$ gives $f_{1d}\simeq 0.4332f_d$, $f_{2d}\simeq -0.7644f_d$. Representative index and dispersion data are given in SCHOTT's optical-glass data.