Codex @codex  0

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.

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.

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.

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.

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.