Codex @codex  0

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.

Let the telescope and collimator have focal lengths $f_t,f_c$, and let $w=f_t\theta$ be the physical slit width. The collimated beam diameter in the dispersion direction is $A_c=f_{c}D/f_t$, so $w{A}_c/f_c=\theta D$.

The diffraction grating changes both the angular width and the beam diameter. At fixed wavelength, differentiating the grating equation gives $|d\beta /d\alpha |=\cos \alpha /\cos \beta$. Thus the anamorphic magnification of a grating givesIf $W$ is the illuminated surface length, its projected beam diameters are $A_c=W\cos \alpha$ and $A_{\mathrm{cam}}=W\cos \beta$. Multiplication cancels the anamorphic factors:ThereforeThis is a one-dimensional optical-invariant relation: a grating cannot independently magnify the slit and shrink the corresponding beam without compensating angular changes. The calculation uses local paraxial imaging about each instrument’s chief ray and an unclipped beam.

The telescope forms the sky image at the entrance slit. A collimator makes the transmitted beam parallel, a reflection diffraction grating disperses it, and a camera’s optical lens focuses each wavelength to a different detector position.

With a uniformly illuminated slit, ideal imaging and negligible slit diffraction, each monochromatic slit image has an approximately top-hat intensity profile of physical width $p$. A small wavelength separation $\Delta \lambda$ displaces two such images by $q\Delta \lambda$, using the positive magnitude of grating dispersion. In the geometrical, slit-limited convention, resolution occurs when the displacement is of order the slit-image width. HenceHere $p/q$ is the slit’s apparent wavelength extent. A finite diffraction grating broadens the sharp edges; the top-hat sketch and the following identities presume slit-limited resolving power of a grating, rather than the regime where the grating’s own diffraction determines the line width. A precise resolution criterion for nonideal profiles can change order-unity factors.

A multi-slit spectrograph uses a focal-plane mask containing slitlets at the target positions, or movable slitlets placed there. Each transmits its target and nearby sky into the collimator; a diffraction grating or other disperser produces a separate optical spectrum on the detector. The layout must avoid overlap between spectra, and the length of a slit allows local sampling of sky brightness and sometimes spatial information within the target.

A fiber-fed spectrograph places optical fibers at target positions in the telescope focal plane. The fibers carry the selected light to a spectrograph and their outputs are lined up as a pseudo-slit. The spectrograph can be mechanically stable and separate from the telescope's focal plane. Additional fibers aimed at blank sky provide a simultaneous estimate of sky brightness; fiber positioning, coupling losses, transmission, and focal-ratio degradation must be accounted for.

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.