 Pixel sampling of a grating spectrum

A normal-incidence diffraction grating of pitch $D$ has order $m$ at $D\sin \theta =m\lambda$ and angular separation $\Delta \theta \simeq m\Delta \lambda /(D\cos \theta )$. A lens whose axis follows the selected order maps a small angular interval to $\Delta x\simeq f\Delta \theta$. Detector pixels substantially smaller than this separation are needed; two pixels per resolution interval correspond to $t\lesssim \Delta x/2$. If instead detector coordinates obey $x=f\tan \theta$ relative to the original axis, the scale is $\Delta x\simeq f{\sec }^2\theta \Delta \theta$. A numerical pixel limit therefore requires an optical layout and a sampling criterion as well as the resolving power of a diffraction grating.