A grating can send one color in several directions
Imagine a narrow beam falling on a shiny surface covered with evenly spaced grooves. Light reflected from one groove travels a slightly different distance from light reflected from its neighbor. At certain outgoing angles, those differences are whole numbers of wavelengths. The reflected waves then reinforce each other and form bright beams.
Each whole number labels a diffraction order. The first order has one extra wavelength between neighboring grooves, the second has two, and so on. One color can therefore leave the grating in several bright directions. Change the color a little and those directions shift.
At a high order, a small change of wavelength produces a larger angular shift. A camera lens turns that shift into a larger distance between bright spots. This is useful when two spectral lines lie very close together, as they often do in light from a hot gas or a star.
Sources: [3]
High orders bring a second color into the same direction
The difficulty appears when two different whole numbers work for two different colors. A wavelength of 505 nanometres in order 100 gives the same product as 500 nanometres in order 101. At a fixed angle of incidence, both satisfy the same outgoing-angle condition. A camera placed there receives both colors on top of one another. The grating equation states that coincidence compactly. The groove spacing and the two beam angles fix the product of order and wavelength. Knowing the outgoing direction alone leaves more than one possible color.
An echelle uses relatively widely spaced grooves with steep, polished facets. Those facets direct much of the useful light toward high orders, where nearby wavelengths separate strongly. The name refers to the staircase-like groove profile. Those facets concentrate useful light into high orders, whose overlapping colors still need a second direction of separation.
Sources: [3]
A perpendicular disperser turns the spectrum into rows
Place another dispersing element after the echelle and turn its direction of spreading through a right angle. Two colors that previously traveled together now bend by different amounts in this second direction. The camera records them at different heights. Along each stripe, the echelle still supplies the fine separation between nearby wavelengths.
The result resembles several short rainbows laid one above another. Each stripe belongs to one high order over the camera's wavelength range. Taken together, the stripes cover more wavelengths than a single long, highly spread rainbow would fit on the detector.
The UVES manual illustrates this principle with a reflective echelle, a cross disperser and camera optics. Its actual folded, out-of-plane layout manages the beams and pupils of a large instrument. The useful idea fits a simpler picture: one optical element separates fine differences along a stripe, and another separates the stripes across the camera.
Sources: [1]
Known lamp lines put wavelengths on the stripes
A photograph supplies pixel positions and brightness, so the stripes still need a wavelength scale. A smooth lamp exposure first reveals where each stripe runs, including any tilt or curvature. Light from known narrow lines then places recognizable marks along it. The measured centers of those marks connect pixel columns to wavelengths.
A line kept out of that fit makes a useful check. Its predicted wavelength should agree with its known value. Dark exposures measure the camera's background, and uniformly illuminated flat exposures reveal differences between pixel responses. These observations let an instrument turn a striped photograph into a usable spectrum.
The stripes need enough space between them. A wide slit or too little perpendicular dispersion can blur neighboring orders together. Once their light overlaps strongly on the same pixels, extracting two separate spectra becomes ambiguous. The second direction solves the order overlap only when the camera actually resolves the stripes.
Sources: [2]