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A cross-dispersed echelle spectrograph uses an echelle grating for strong dispersion and a cross-disperser in the perpendicular direction to separate overlapping diffraction orders. It records many orders simultaneously in an echellogram.

An echelle grating works in high diffraction orders. A separate cross-disperser separates their overlapping optical spectra on a two-dimensional detector.

Built by André Baranne, Michel Mayor, Didier Queloz and colleagues, the ELODIE fiber-fed spectrograph carried light from Haute-Provence's 1.93-metre telescope through optical fibers. Its echelle grating dispersed light in high diffraction orders; an optical prism and grism cross-disperser separated them perpendicularly on a charge-coupled device. This cross-dispersed echelle spectrograph achieved spectral resolving power $R=42000$. A simultaneous thorium reference tracked instrumental drift. Wavelength calibration mapped detector positions to wavelengths; spectral cross-correlation combined many spectral lines into a fitted profile. A barycentric velocity correction removed the observer's motion.^[ref]

Why can a spectrograph measure a Doppler shift much smaller than one spectral resolution element? Resolution concerns separating neighbouring features, whereas locating the centre of a known profile uses information across that profile and across many spectral lines. For a chosen example, at $\lambda =500\mathrm{nm}$ a recession velocity of $60\mathrm{m}{\mathrm{s}}^{-1}$ givesAt $R=42000$, one spectral resolution element corresponds to $c/R\simeq 7.14\mathrm{km}{\mathrm{s}}^{-1}$, about $119$ times larger. A well measured centre can move by a small fraction of that width. More photons and more useful spectral lines improve the estimate, but do not remove wavelength calibration errors or stellar activity.

A simple mathematical picture of spectral cross-correlation makes the readout concrete. Define $x=c\log (\lambda /{\lambda }_0)$, with fixed reference wavelength ${\lambda }_0$, and let $d(x)$ be continuum-normalized line absorption. For small velocities, a common Doppler shift approximately translates $d(x)$. A positive absorption mask $m(x)$ then givesIf $d(x)=m(x-v_r)$, the maximum occurs at $u=v_r$; a convention correlating the received observed radiative flux instead produces a dip. A fitted correlation centre is an effective collective line position, not a claim that each individual spectral line has been resolved into a narrower feature. The lamp tracks instrumental drift, while the barycentric velocity correction removes the observer's changing line-of-sight velocity; neither removes changes in the star itself.

Choose photodetector $x$ along increasing main grating dispersion, and $y$ along increasing wavelength for the cross-disperser. The echellogram shows two order traces and their intersections with the $y$ axis, which passes through the optical camera optical axis.

For fixed incidence, $d\beta /d\lambda =m/(d\cos \beta )>0$, so increasing wavelength goes to the right along each order. The cross-disperser also sends increasing wavelength upward in the chosen orientation; the traces therefore slope upward. At $x=0$, ${\lambda }_m=K/m>{\lambda }_{m+1}=K/(m+1)$, so the trace labeled $m$ lies above the trace labeled $m+1$. Reflecting either plotting axis would reverse its arrow but leave the optical content unchanged. The cross-disperser separates orders that would overlap in the main dispersion direction.

The echellogram contains nearly parallel traces, one for each diffraction order. Choose the main echelle grating dispersion to increase wavelength toward the right and the cross-disperser to increase wavelength upward. At a fixed horizontal coordinate, order $m+1$ has a shorter wavelength than order $m$, so it lies below order $m$ in this convention.