Spectrometers -- Optical Design
Plane grating, Crossed with prism or grating, Multiple detector (continued)

A somewhat simplified situation can be described by modeling the echelle as operating in Littrow mode, so
a = b = 70° at the center of every order and nl = 2d sin 70°. Suppose one wants to have 325 nm right in the center of an order. Then n × 325 nm = 2 d × 0.93969. d = n × 172.93 nm. One equation, two variables, what can one do? Choose the desired dispersion. For 25 µm pixels, one may wish 0.003 nm resolution (10 pm is approximately the width of atomic emission lines, and this way we have 3 pixels across each line). That works out to 0.003 nm/0.025 mm or 0.12 nm/mm = d cos b/nf. d/nf = (0.12 nm/mm)/cos 70° or d/nf = 0.3509 nm/mm. d/n = 172.93 nm, so 172.93/f = 0.3509, f = 606.74 mm. This is not a standard focal length for a mirror. 600 mm is a sensible value. But we still have no way to separate n from d. One might check to see what d spacings are in grating catalogs. The Richardson Grating Laboratory, now part of Newport Corp., has a catalog of possible gratings. Groove densities for gratings with 70° blaze include 27, 44.41, 158, and 316 grooves/mm. Let's tabulate the behavior with which these correspond, both at 325 nm and at 200 nm (we'll need to know how many orders span this range momentarily).

We thus can have anywhere from (348-214) = 134 to (30-18) = 12 orders to cover the portion of the UV spectrum in which a majority of atomic emission lines are found. The cross-dispersion has to be sufficient so that adjacent orders don't overlap, confounding identification of spectral lines. If we want to cover half the CCD with 200 to 325 nm, we need to know how far apart adjacent orders will be and thus how many pixels must separate each order. While the free spectral range, the range of wavelengths in a given order, varies from order to order (higher orders have shorter free spectral range), we can get an initial guestimate by ignoring the change. Taking order difference divided by wavelength range, we see that average free spectral range varies from 125 nm/134 orders to 125 nm/12 orders. That's anywhere from 1 to 10 nm per order. But wait! We know the dispersion near 325 nm! It's 0.003 nm/pixel, and the typical CCD has 1000 pixels across! That means we can't have more than 3 nm across a single order at 325 nm or we'll lose information off the edge of the CCD. That means we need an average of less than 3 nm per order. If we pluck 2.5 nm/order out of thin air, we're looking for 125 nm/2.5 nm/order = 50 orders. NONE of our choices match this! Fortunately, now that we have a better idea of what free spectral range and number of orders we desire, we can go back to the grating catalog and choose a blaze NEAR (but not equal to) 70°, and something between 45 and 158 grooves/mm to get something that will work. This iterative approach (choose a parameter, work out its implications, refine the constraints, and repeat until the system is workable) is typical of how one chooses instrumental parameters.
Grooves/mm
27
44.41
158
316
d (nm)
37037
22518
5329
3164.6
n (325 nm, rounded)
214
130
31
18
n (200 nm, rounded)
348
212
50
30
 Atomic Emission Spectroscopy