Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 338 / 1 / a / ii / Solution

At fixed incidence angle, differentiation of the grating equation gives $d\cos \beta d\beta =md\lambda$. Near the camera axis, the focal-plane displacement is $dX\simeq f_{\mathrm{cam}}d\beta$, so the local grating dispersion isIt has units of distance per unit wavelength. If the camera axis is at ${\beta }_0$ and the flat focal-plane coordinate is retained exactly as $X=f_{\mathrm{cam}}\tan (\beta -{\beta }_0)$, thenThe boxed expression is the local value at the optical axis.