The purpose of this post is to find an equation for the resolving power of an spectrograph with a circular slit.
The resolving power of an spectrograph working in Littrow configuration for a rectangular slit is given by:
where f is the focal distance of the collimator, w the slit width and θ the Littrow angle (the ray incidence angle and diffracted angle with respect to the grating normal are the same). This equation is valid for spectrographs including free aberration optics. However, in a more realistic situation, the image profile of the slit on the CCD is not a perfect rectangular function. Due to aberrations and specially for narrow slits, the profile approaches a gaussian function. In this case, the slit width w is represented by the FWHM of the gaussian function. The same reasoning applies to a circular slit but a different FWHM.
In order to find the resolving power for a circular slit, we need to find the variation of the FWHM between a rectangular and circular slits. In Figure 1 we have superposed the two profiles. For simplicity we consider only the half of a circle. The result is the same for the whole circle.
Figure 1. Rectangular and circular slits. Only the half is shown for simplicity in the calculations
For the rectangular slit, the FWHM is simply the width of the slit. In Figure 1 is 2R. As for the circle, we have to find the profile of the PSF. During the data reduction, the columns of pixels perpendicular to the dispersion are binned to find the PSF. Assuming an evenly illumination of the circle, each pixel will contain the same number of electrons No. If Yi is the number of pixels for a given pixel position Xi, the flux of that column will be YiNo. We can see that the profile of the circle is the circle function y = (R^2 – x^2)^1/2
By doing y = R/2, we find x = R√3/2, therefore the FWHM = R√3
By replacing the FWHM of the circle and slit in equation 1, the resolving power by a circle is
Rcircle = √3 /2 Rslit ~ 1.15 Rslit
Figure 2 shows the spectrum of the sodium doublet obtained with FLECHAS spectrograph using a fibre and a rectangular slit. Note the elliptical shape of the image of the fibre due to the anamorphic magnification of the spectrograph configuration. In order to find the FWHM at different pixel samplings, the fibre and slit were wide opened (400 um) to cover a large number of pixels.
Figure 2. Sodium doublet with a 400 um fibre and with a 400 um slit
The FWHM were