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From the slit profile, the spectral resolving power obeys $R=\lambda q/p$. Substitute into the preceding optical-invariant relation:The slit-image width cancels. Although $q$ is a detector length per unit wavelength, $\lambda q$ is a detector length, so this expression for $L$ has units of length, as does $R\theta D$.

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 $mN$.

In the simple slit-image approximation, the projected slit width is $w=x{f}_{\mathrm{cam}}/f_{\mathrm{coll}}$. Equating this width to the separation $q\Delta \lambda$ of barely resolved features, using the grating dispersion, givesThis recovers the stated spectral resolving power under the assumption that the grating has unit anamorphic magnification.

For arbitrary distinct $\alpha$ and $\beta$, the anamorphic magnification of a grating must be included. At fixed wavelength, the grating equation gives $|d\beta /d\alpha |=\cos \alpha /\cos \beta$, so the slit image instead has width $w=x{f}_{\mathrm{cam}}\cos \alpha /(f_{\mathrm{coll}}\cos \beta )$. Thus the general slit-limited resolving power of a grating isThe two expressions agree in the Littrow configuration, $\alpha =\beta$. Without that condition or the unit-magnification approximation, the quoted $\cos \beta$ expression is not the general slit-limited result. Finite grating size, detector sampling, and optical aberrations can lower the actual resolution further.

For field of view, fiber-fed spectrographs commonly cover wider sky areas: fibers can pick targets across a broad focal plane while feeding a compact spectrograph with a fixed output slit. A multi-slit spectrograph must image its field through the spectrograph optics, and spectra must fit on the detector without overlap, which restricts both field and target layout.

For spectral resolution, fibers can feed an optimized, stable high-dispersion instrument, including an echelle grating. The fiber image acts as its entrance width. In a multi-slit spectrograph, slit width and dispersion similarly determine the spectral resolving power. Neither feed type alone imposes a universal resolution ranking: narrower fibers or slits improve resolution at the cost of losing source light, and both can be designed for high or low resolution.

For faintness limit, slit masks often have an advantage for individual faint objects because they avoid fiber coupling and transmission losses, permit a slit width matched to astronomical seeing, and sample local sky along the slit. Fibers can admit more sky through a fixed circular aperture and require sky subtraction from separate locations; focal-ratio degradation can also reduce throughput. The actual limit depends on throughput, aperture size, background stability, and detector noise through the signal-to-noise ratio in photon counting. Well-designed fiber instruments can nevertheless be very efficient for wide-field surveys, so the comparison is conditional on the optical design and observing conditions.

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.

A spectrograph records the optical spectrum dispersed by an instrument. Its spectral resolving power describes which nearby wavelengths it separates.