Grating equation ($m\lambda =d(\sin \alpha +\sin \beta )$)

For a reflection diffraction grating with both ray directions measured on the same side of its normal, adjacent grooves have path difference $d(\sin \alpha +\sin \beta )$. Constructive interference requires this to be an integer multiple $m\lambda$. Signed-angle conventions can turn the sum into a difference; the physical path difference is unchanged.

A formal grating order must have a real outgoing angle. For normal incidence, $|m|\lambda \le d$ is necessary. Oblique incidence can access orders outside that normal-incidence range; reflection Littrow geometry has $m\lambda =2d\sin \theta$. The ideal resolving power of a diffraction grating applies only to an order that exists and has sufficient signal.

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 $q=f{\sec }^2(\beta -{\beta }_0)m/(d\cos \beta )$.

At fixed wavelength, the grating equation gives $d\beta /d\alpha =-\cos \alpha /\cos \beta$. Thus an incident angular width is magnified by $\cos \alpha /\cos \beta$ in the dispersion direction. Including the ratio of camera and collimator focal lengths gives the projected slit width. The ESO B&C operating manual, Appendix A explicitly includes this anamorphic factor.

A slit of width $x$ at the focus of a collimator subtends $x/f_{\mathrm{coll}}$. Its projected width is $x{f}_{\mathrm{cam}}\cos \alpha /(f_{\mathrm{coll}}\cos \beta )$. Dividing this width by the grating dispersion gives $\Delta \lambda =xd\cos \alpha /(m{f}_{\mathrm{coll}})$. Thus the spectral resolving power is $R=m\lambda f_{\mathrm{coll}}/(xd\cos \alpha )$. Omitting the anamorphic factor gives a denominator $\cos \beta$ instead; these agree in the Littrow configuration.