Grating dispersion ($d\beta /d\lambda =m/(d\cos \beta )$)

At fixed incidence angle, differentiating the grating equation gives $d\beta /d\lambda =m/(d\cos \beta )$. A camera with focal length $f$ therefore has local focal-plane dispersion $q=fm/(d\cos \beta )$ near its optical axis. With coordinate $X=f\tan (\beta -{\beta }_0)$, the full derivative is $q=f{\sec }^2(\beta -{\beta }_0)m/(d\cos \beta )$.