The Equation of Echellogram

The echellogram of an echelle spectrograph with a perfect optics can be described by a simple combination of the dispersion equations of an echelle grating and a cross-disperser. The dispersion equation of the cross-disperser depends on its type. The equation of an echellogram, in addition, depends on the configuration of the two dispersing elements. In this document, I will consider a grating equation for the case when a cross-disperser is placed after an echelle grating, i.e., the post-disperser configuration, in order to apply the results to IRCS which uses a low-dispersion grating as a cross-disperser.

We use the grating equation given by Schroeder (1987). The following equations represent the echellogram with x-axis being the direction of the echelle dispersion and y-axis being the direction of the cross-disperser dispersion.

$$x = f_{cam,e} \tan \left\{ \sin^{-1} \left( \frac{m_e \lambda}{\sigma_e \cos \gamma_e} - \sin \alpha_e \right) \right\} - x_o ,$$ (3)

$$y = f_{cam,c} \tan \left\{ \sin^{-1} \left( \frac{m_c \lambda }{ \sigma_c \cos \gamma_c} - \sin \alpha_c \right) \right\} - y_o,$$ (4)

where the subscript $e$ and $c$ mean echelle and cross-disperser, respectively, the position ($x_o$, $y_o$) is the detector center, $\gamma$ is the out-of-plane angle, $m$ is the order number, $\sigma$ is the groove spacing, and $\alpha$ is the incident angle of a ray. Figure 2 shows the angles($\alpha$,$\beta$,$\gamma$) against a reflection grating. FThe parameters $f_{cam,e}$ and $f_{cam,c}$ are the effective focal lengths of the optics following the echelle and cross-disperser, respectively. For IRCS, $f_{cam,e}$ is equal to $f_{cam,c}$. They are not the same when there is optics that has pupil magnification between the echelle and cross-disperser gratings.

Figure 3: An echellogram of IRCS. The image is a $L$-band frame.