 Anamorphic magnification of a grating

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.