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

For thin contacting components, optical powers add. At wavelength $\lambda$ the total optical power is $\Phi (\lambda )={\rho }_1[n_1(\lambda )-1]+{\rho }_2[n_2(\lambda )-1]$. An achromatic lens has ${\Phi }_F={\Phi }_C$, henceFor normally dispersive optical glasses, both index differences are positive, so the two components have opposite curvature factors and opposite optical powers. This uses the thin Lensmaker's equation; finite thickness and separated principal planes require a more detailed design. If the denominator vanishes, use the undivided power-balance equation rather than this ratio. Two-wavelength correction alone does not eliminate the residual chromatic aberration at every intermediate wavelength.

At the $d$ line, the focal lengths obey $1/f_{id}=(n_{id}-1){\rho }_i$. ThusRearranging givesThe last equality uses the Abbe number definition. The minus sign signifies a converging and a diverging component, with the normal optical glass convention and nonzero finite focal lengths.

Set ${\Phi }_i=1/f_{id}$ and $\Phi =1/f_d$. The achromatic doublet power balance gives two linear equations:For distinct nonzero Abbe numbers, solving yields ${\Phi }_1=\Phi V_{1d}/(V_{1d}-V_{2d})$ and ${\Phi }_2=-\Phi V_{2d}/(V_{1d}-V_{2d})$, and thereforeEqual Abbe numbers force the total optical power to vanish under the achromatic condition; they cannot give a nonzero-power thin achromatic lens. Closely matched Abbe numbers require large opposing component powers and are generally unattractive because of the resulting sensitivity and other optical aberrations.

$V_d$ 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.