MERITS OF CROSS-DISPERSED
ECHELLE SPECTROGRAPHS

A useful figure of merit for a grating spectrograph is its throughput, which is the spectral resolving power multiplied by its entrance slit width (Vogt, 1987). It is given by

$$R\phi = \frac{Wm\lambda}{\sigma D} = \frac{d_{col}{\rm cos}\gamma({\rm sin}\alpha + {\rm sin}\beta)}{D \cdot {\rm cos}\alpha} ,$$ (1)

where $R$ is the resolving power, $\phi$ is the entrance slit width projected on the sky (radian), $W$ is the projected length of the collimated beam on the grating, $\lambda$ is the wavelength of order number $m$, $\sigma$ is the groove spacing of the grating, $D$ is the diameter of the telescope primary, $d_{col}$ is the collimated beam diameter, $\alpha$ and $\beta$ are the angles of incidence and diffraction, respectively, and $\gamma$ is the out-of-plane angle (Figure 1 and 2). For the Littrow case ( ${\alpha} = {\beta} = {\theta_B}, {\gamma} = 0$)

$$R\phi = \frac{2d_{col}{\rm tan}{\theta_B}}{D} ,$$ (2)

where $\theta _B$ is the blaze angle of the grating.

In order to achieve a larger throughput, a larger beam size and blaze angle are required. The beam size becomes the largest when the focal ratio of the collimator equals the effective focal ratio of the telescope, i.e., $F_{col}=F_{tel}$.

An echelle grating has a large blaze angle and has high throughput compared to the other conventional diffraction gratings of the same size. Moreover, it is possible to utilize two-dimensional detector array space more efficiently by using a cross-disperser for order separation. The cross-disperser disperses the spectrum into the direction perpendicular to the echelle dispersion, allowing us to cover wide spectral ranges with a single exposure.

Figure 1: Schematic drawing of a slit spectrograph with a reflection grating.

Figure 2: Schematic showing angles of incident angle $\alpha$, diffraction angle $\beta$, and out-of-plane angle $\gamma$.