= Detector pixels and optical layout
The angular dispersion at the chosen order is
$$
\frac{d\theta}{d\lambda}=\frac m{D\cos\theta},\qquad
\Delta\theta=2.182\times10^{-4}\,\mathrm{rad}.
$$
For a collecting lens with its axis directed along the selected second-order beam, the local focal-plane separation is
$$
\Delta x\simeq f\Delta\theta=\boxed{54.6\,\mathrm{\mu m}}.
$$
A basic one-pixel separation criterion is therefore $\boxed{t\lesssim55\,\mathrm{\mu m}}$. A more reliable two-pixels-per-resolution-interval choice is $\boxed{t\lesssim27\,\mathrm{\mu m}}$. Finite pixels integrate the pattern, so sampling and pixel response should not degrade the grating's optical resolution.
This <pixel sampling of a grating spectrum> depends on the layout and criterion, neither of which is fully specified. If the detector is instead described by $x=f\tan\theta$ relative to the incident-axis coordinate, the derivative contains an additional $\sec^2\theta$ and gives $65.0\,\mathrm{\mu m}$ separation, or about $32\,\mathrm{\mu m}$ for two-pixel sampling. The more approximate paraxial mapping $x\simeq fm\lambda/D$ gives $50\,\mathrm{\mu m}$. These are stated alternatives rather than one falsely precise universal pixel width.
Back to article page
