Inclination and Curvature of Spectral Images

Each order spectrum taken with a cross-dispersed echelle spectrograph has an inclination and curvature with respect to the dispersion direction of the echelle grating (Figure 3). The inclination is caused by the cross-disperser whose dispersion direction is perpendicular to that of the echelle grating in general. The inclination of the spectrum at a given wavelength is

$$\tan \psi = \frac{d \beta_c}{d \beta_e} = \frac{\sigma_e m_c}{\sigma_c m_e} \frac{ \cos \gamma_e}{\cos \gamma_c} \frac{\cos \beta_e}{\cos \beta_c}.$$ (5)

This equation shows that the inclination increases with decreasing order number $m_e$, or increasing wavelength.

The curvature is caused mainly because the out-of-plane angle ($\gamma_c$) against the cross-disperser varies with wavelength in the beam which has already been dispersed by the echelle grating. Figure 4 shows the effect of out-of-plane angles on a spectrum. The out-of-plane angle against cross-disperser is given by

$$\gamma_c = (\beta_e - \beta_{x_o,e}) / M_p + \gamma_{c0} ,$$ (6)

where $\beta_e$ is the diffraction angle of the echelle grating, $\beta_{x_o,e}$ is the diffraction angle toward the optical axis (center of the detector) against the echelle grating, $\gamma_{c0}$ is the out-of-plane angle toward the optical axis (center of the detector) against the cross-disperser, and $M_p$ is a pupil magnification between the echelle grating and cross-disperser. The dispersion angle $\beta_c$ can be written as follows,

$$\beta_c ( \lambda ) = \beta_c ( \lambda ) \vert _{\gamma_c ( \lambda ) = \gamma_{c0}} + \Delta \beta_c ( \gamma_c ( \lambda )) .$$ (7)

The dependence of $\beta_c$ on $\gamma_c$ is given as follows,

$$\frac{d \beta_c}{d \gamma_c} = \tan \gamma_c \lambda \frac{d \beta_c}{d \lambda} .$$ (8)

When $\gamma_c$ changes from 0 to $\gamma_{\chi}$ ( $\gamma_{\chi} \ll 1$), $\beta_c$ changes roughly by the following amount,

$$\Delta \beta_c \approx \left( \frac{ {\gamma_{\chi}}^2 }{2} \right) \lambda A ,$$ (9)

where $A$ is the angular dispersion of the cross-disperser. Since $\gamma_c$ is a function of $\beta_e - \beta_{x_o}$, the absolute value of $\gamma_c$ takes its largest values at the both ends of the spectrum at each order. Thus, the spectrum is curved toward the longer wavelength side ($+$Y) as seen in Figure 4. When $\gamma_{c0}$ is non-zero, the whole spectrum shifts to the longer wavelength ($+$Y) with its inclination changing. When $\gamma_{c0}$ has a positive value, the spectrum will be steeper, because the absolute value of $\gamma_c$ increases linearly with $\beta_e$. On the other hand, when $\gamma_{c0}$ has a negative value, the absolute value of $\gamma_c$ increases when $\beta_e < \beta_{X_o}$ and decreases when $\beta_e > \beta_{x_o}$. As a result, the inclination becomes gentler and eventually reverses with decreasing $\gamma_{c0}$.

Figure 4: Variation of inclinations and curvatures with the out-of-plane angle of a cross-disperser. The diffraction angle increases with increasing X-coordinate. This graph is calculated with an echelle grating which has the groove number of 24.35 lines/mm and blaze angle of 70$^\circ$ (R2.75) and a cross-disperser which has the groove number of 150 lines/mm and blaze angle of 8$.^{\circ }$63. We assumed $\beta_{B,e} =\beta_{x_o,e}$.