Some last words on the SHG lineshape function.
Here is a figure showing model calculations of the line profile of a grating spectrometer for various slit widths.
These lineshape functions were calculated by performing a convolution of a box function representing the slit transmission (no diffraction) with a Gaussian function (4 pixels fwhm) representing the diffraction limited performance of the spectrometer. The choice of a Gaussian function was made based on the arguments in section 2.6.5 of Eversberg and Vollman ("Spectroscopic Instrumentation" Springer 2015).
The Gaussian function was chosen to have a fwhm of 4 pixels, consistent with measurements made using my SHG. For any given instrument, the fwhm of this limiting Gaussian is dependent on the optical speed of the collimator and camera lens systems as well as the illuminated area and groove density of the grating.
The figure is consistent with figure 12.2 in "Spectrophysics" by Thorne, Litzen and Johansson (Springer 1999). The flat topped lineshape moves to a smoother function with decreasing slit width but its fwhm is ultimately limited by diffraction, as we'd expect, and decreasing slit width below the diffraction limit will only reduce instrumental throughput without affecting spectral resolution.
The next figure shows a Gaussian fit performed on the "10 px image slit" profile in the previous figure.
You can see that a Gaussian fit to the real (modelled) lineshape function is not bad. The flat-topped feature of the real lineshape function is apparent but the fwhm and overall shape of the real function don't depart too severely from the Gaussian shape. The Gaussian shape is, thus, a pretty good approximation to the real lineshape for reasonably small slits.
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