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A cross-disperser is a second dispersing element, often an optical prism or diffraction grating, oriented to separate optical spectra perpendicular to their principal dispersion direction.

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.

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.

Use the signed-angle convention in which the reflection grating equation is $m\lambda =d(\sin \alpha +\sin \beta )$, with positive diffraction order $m$. For fixed incidence,The second identity is the local focal-plane scale, measured about the camera axis aligned with the central diffracted ray. Since $A_{\mathrm{cam}}=W\cos \beta$, the projected beam size cancels the cosine in the grating dispersion:Writing $N=W/d$ for the number of illuminated grooves also gives $L=mN\lambda$. This connects the geometrical instrument invariant to the phase span of the illuminated diffraction grating.

Substitute the grating equation into the preceding result:Moving across the illuminated grating by one groove spacing changes the incident-plus-outgoing optical path length by $d(\sin \alpha +\sin \beta )=m\lambda$. Across length $W$, the total path difference is therefore $L$. Thus $L$ is the optical path difference between contributions from the two ends of the illuminated grating.

The number of coherent phase cycles across it is $L/\lambda =mN$, which is the intrinsic diffraction-limited spectral resolving power of the diffraction grating under the usual first-minimum criterion. In the slit-limited regime, $R\theta D=L$ also states how much sky angle can be accepted at a given resolution and aperture. Holding slit angle and resolution fixed while increasing telescope diameter requires a larger optical path span. The geometry gives $|L|\le 2W$; large incidence and diffraction angles increase resolution per unit grating length, though grazing beams become impractical.

The slit equations do not imply unlimited resolution when $\theta$ tends to zero. Once $\theta D$ is comparable to $\lambda$, finite-aperture diffraction matters and the actual resolving power is bounded by about $mN$.

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.