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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.

At fixed incident and central outgoing angles, the grating equation gives $m{\lambda }_m=K=d(\sin \alpha +\sin {\beta }_0)$. Adjacent central wavelengths are separated byThis is the usual adjacent-order free spectral range. A detector width of approximately $q\Delta {\lambda }_{\mathrm{FSR}}$, where $q$ is the grating dispersion, covers one such interval.

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

In the Littrow configuration the selected diffracted ray retraces the incident ray, so $\alpha =\beta$ and the grating equation becomes $m\lambda =2d\sin \alpha$.

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.

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 $mN$.

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.

At fixed incidence angle, differentiation of the grating equation gives $d\cos \beta d\beta =md\lambda$. Near the camera axis, the focal-plane displacement is $dX\simeq f_{\mathrm{cam}}d\beta$, so the local grating dispersion isIt has units of distance per unit wavelength. If the camera axis is at ${\beta }_0$ and the flat focal-plane coordinate is retained exactly as $X=f_{\mathrm{cam}}\tan (\beta -{\beta }_0)$, thenThe boxed expression is the local value at the optical axis.

The bright peaks occur when contributions from adjacent grooves have equal phase modulo $2\pi$, so all illuminated grooves contribute by constructive interference. Use the reflection-grating convention shown below: $\alpha$ and $\beta$ are positive angles of the incident and outgoing ray lines on the same side of the normal. The incoming and outgoing path differences between neighboring grooves are $d\sin \alpha$ and $d\sin \beta$, respectively. Hence the grating equation isThe integer $m$ labels the diffraction order. With a different signed-angle convention, the same physical condition contains a difference of sines.

Let ${\beta }_0$ be the central camera direction. The grating equation fixesThe usual adjacent-order free spectral range of an echelle grating is thereforeThe constant $K$ has dimensions of length and is the path difference between adjacent grooves for light directed along the camera axis; it is also the fixed product $m{\lambda }_m$. Near the Littrow configuration, $K=2d\sin \alpha$. The detector width corresponding to this interval is approximately $qS$, with grating dispersion $q$.

There is a convention issue in reading the question literally. Adjacent order-center separation is exactly the expression above. A partition assigning each wavelength to whichever order lands closest to the vertical axis has boundaries halfway in detector displacement, not at adjacent order centers. In the small-angle detector approximation its boundaries for order $m$ are $2K/(2m+1)$ and $2K/(2m-1)$, giving width $4K/(4m^2-1)$. Both conventions give $S\sim K/m^2$ for high orders, but their exact finite-$m$ widths differ. The quoted expression is the conventional adjacent-order spacing.