Free spectral range of an echelle grating

At fixed incident and central outgoing angles, the grating equation gives $m{\lambda }_m=K=d(\sin \alpha +\sin {\beta }_0)$. Adjacent central wavelengths are separated byThis is the usual adjacent-order free spectral range. A detector width of approximately $q\Delta {\lambda }_{\mathrm{FSR}}$, where $q$ is the grating dispersion, covers one such interval.

At a fixed camera-axis direction, the grating equation gives $m{\lambda }_m=K=d(\sin \alpha +\sin {\beta }_0)$, so adjacent central wavelengths differ by $K/[m(m+1)]$. This differs from the exact wavelength cell where an order is nearest the optical camera axis. For ${\beta }_0=0$ and $x=f\tan \beta$, equal distances of orders $m$ and $m+1$ mean opposite diffraction angles; adding their diffraction grating equations gives $\lambda =K/(m+1/2)$. The full order-$m$ cell is bounded by $K/(m+1/2)$ and $K/(m-1/2)$, hence has width $K/(m^2-1/4)$ for $m\ge 2$ and accessible neighbouring orders. Both widths approach $K/m^2$ at high order.