Echelle grating

An echelle grating is a diffraction grating with few, widely spaced grooves, typically 30 to 300 lines per millimeter, whose facets are cut at a steep blaze angle, commonly 63.4° or 76.0° (blaze tangents of 2 and 4, labeled R2 and R4). It is used near Littrow at high angles of incidence, so the light diffracts into orders of tens to more than a hundred. A 79 line/mm R2 echelle (groove spacing 12.66 µm) works in order 45 near 500 nm and in order 14 or 15 near 1550 nm. The high order gives high angular dispersion and resolving power from a small grating; the cost is a short free spectral range, so neighboring orders overlap and must be separated by a second dispersing element.

Resolving power and dispersion

The grating equation, $m\lambda = d(\sin\alpha + \sin\beta)$, and the resolving power $R = mN$ are covered in the diffraction-grating entry. Substituting $m$ from the grating equation gives a form that does not contain the groove density:

$$R = \frac{W(\sin\alpha + \sin\beta)}{\lambda},$$

where $W = Nd$ is the illuminated width of the grating. In Littrow, with $\alpha = \beta = \theta_B$, this becomes $R = 2W\sin\theta_B/\lambda$, and the angular dispersion is

$$\frac{d\beta}{d\lambda} = \frac{2\tan\theta_B}{\lambda}.$$

Resolving power is therefore set by the width of the grating and how steeply it is used. For the R2 echelle above, 100 mm wide at 500 nm, $R = 2(0.1\ \mathrm{m})(0.894)/(500\ \mathrm{nm}) \approx 358{,}000$, a resolvable interval of 1.4 pm. A 1200 line/mm grating of the same width used in first order gives $R = mN = 120{,}000$. The echelle's angular dispersion, 0.46°/nm, is about six times that of the first-order grating at the same wavelength, 0.072°/nm.

Overlapping orders and cross-dispersion

The free spectral range in order $m$ is about $\lambda/m$: 11.1 nm in order 45 at 500 nm. Order 46 at 492.3 nm leaves the grating in the same direction as order 45 at 503.2 nm, and only a short span of wavelengths around the blaze direction falls in each order with good efficiency. An echelle spectrometer therefore places a low-dispersion element, a prism or a coarse first-order grating, so that it disperses perpendicular to the echelle. Each order is displaced sideways by a different amount, and the spectrum is laid out on a two-dimensional detector as a stack of short, nearly parallel strips, the echellogram. One exposure then covers a broad band, often the whole visible, at full resolution.

Reducing an echellogram means tracing each order on the detector, extracting it, applying a wavelength solution from a calibration lamp, and correcting the blaze function, the efficiency variation along each order that makes a flat spectrum look scalloped. Because order spacing in the cross-dispersed direction changes with wavelength, the strips crowd together at one end of the detector, which limits slit height and sets how much background light from neighboring orders is mixed in.

Where echelles are used

Echelle spectrographs are standard in astronomy for stellar spectroscopy and radial-velocity work, in atomic emission spectrometry for elemental analysis, and in compact high-resolution spectrometers and laser wavelength analyzers. The same principle appears in integrated photonics as the etched or echelle grating demultiplexer: a curved, deep-etched grating facet in a slab waveguide on silicon-on-insulator or InP, used in a moderately high order to separate channels for wavelength-division multiplexing. It competes with the arrayed waveguide grating; the echelle is usually smaller, while its performance depends strongly on the verticality and roughness of the etched facets.

Pitfalls

• Order overlap: without a cross-disperser or a narrow order-sorting filter, a single detector pixel receives light from several wavelengths, one per order.
• Blaze efficiency falls off away from the center of each order, so throughput at the order edges can be well under half the peak.
• High-order operation magnifies small errors in groove position: ghosts and scattered light are more prominent than in a first-order grating, and the grating's temperature must be stable because thermal expansion shifts every order.
• The resolving power formula assumes the full width is coherently illuminated; in practice the slit width and pixel size usually set the achieved resolution.

Common questions

What does "echelle" mean?

The word is French for ladder. It refers to the staircase profile of the grooves, whose long, steep facets do the reflecting at high angles of incidence.

How is an echelle different from an ordinary blazed grating?

Both are blazed; an echelle has much coarser grooves and a much steeper blaze, and it is used in high orders instead of first or second order. Its resolving power for a given width is higher, and its free spectral range is far shorter.

Why are R2 and R4 echelles common?

The labels give the tangent of the blaze angle, 2 or 4. Since the Littrow dispersion is $2\tan\theta_B/\lambda$, an R4 echelle has twice the angular dispersion of an R2 at the same wavelength, at the cost of a larger grating length for the same beam width.

References: E. G. Loewen and E. Popov, Diffraction Gratings and Applications (Marcel Dekker, 1997); D. J. Schroeder, Astronomical Optics, 2nd ed. (Academic Press, 2000); E. Hecht, Optics, 5th ed. (Pearson, 2017).