Resolution

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Page ID: 75526

Alexander Scheeline & Thomas M. Spudich
Analytical Sciences Digital Library

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Slit width/detector width and dispersion

Common slit widths are from 5 µm to 1 mm. Common pixel widths on electronic detectors are from 6 µm to 50 µm (see later discussion). Suppose one is using perfect imaging optics and a huge grating so that only the largest of the entrance slit width, exit slit width, or detector width influences resolution. The width of the slit or detector pixel is dx. Since we know the linear dispersion, the effective resolution \(\delta\)λ = d cos β \(\delta\)x/nf.

Limiting resolution for finite grating size

The number of resolution elements in a spectrum can not exceed the number of resolution elements generated by the diffraction grating. If a grating has a width W and a groove spacing of d, there are a total number of grooves G = W/d. Then the best resolution one can have is nG. (Prove to yourself that a 100 mm wide grating with 1200 grooves per mm, operating in 2^nd order has at most a resolution of 2.4 × 10⁵. For light at 300 nm, that means a resolution of no better than 300 nm/2.4 × 10⁵ or 0.00125 nm.). Remember, resolution is set by the weakest link in the chain. One computes resolution for just the grating and for the focal length, dispersion, and slit width. Whichever gives the poorest resolution (largest \(\delta\)λ) is the one to believe.

Does grating size plus slit width always limit resolution? NO!

A spectrometer is both an optical system and a dispersing system operating together. The analysis of how the grating and dispersion work ignores the imaging behavior. If the imaging is blurry, resolution will be degraded. In addition, diffraction through small apertures can limit resolution. No matter how narrow a slit is, it will appear to have a width no smaller than fλ/W. So for a spectrograph with f = 1 m, grating width W of 100 mm, and λ = 300 nm, the smallest spatial width of a focused spectral line is 3000 nm or 3 µm. Thus, the smallest number to use in the dispersion equation for \(\delta\)x is the LARGER of slit width, detector pixel width, or diffraction limited width.