Tilt and curvature of a slit image

When a straight slit is located parallel to the ruling of a grating, an incident light ray passing through the slit toward the center of the grating has a finite out-of-plane angle ($\gamma$) defined by the distance between the position of the ray on the slit and the center. Due to the finite out-of-plane angle, the monochromatic slit image becomes tilted and curved (Figure 5). The tilt angle on the image is given as follows (Schroeder, 1987).

$$\tan \chi = \frac{d \beta}{d \gamma} = \tan \gamma \frac{ \sin \alpha + \sin \beta}{\cos \beta} ,$$ (15)

For the Littrow configuration ( $\alpha = \beta = \theta_B$),

$$\tan \chi = 2 \tan \gamma \tan \theta_B ,$$ (16)

at the blaze wavelength. Figure 6 shows that the tilt of a slit image is sensitive to the out-of-plane angle when the blaze angle of the grating is large.

Assuming that $\gamma$ is small and integrating Eq. 15, we obtain

$$\Delta \beta \approx \left( \frac{ {\gamma}^2 }{2} \right) \lambda \frac{d \beta}{d \lambda} .$$ (17)

The slit image thus has a parabolic shape (Schroeder, 1987; Meaburn et al., 1984).

For a short slit, the slit image can be approximated by a tilted straight line. Figure 7 shows the variation of the tilt angle with the diffraction angle for an R2.0 echelle grating. It shows that the tilt angle becomes steeper with the diffraction angle when $\gamma_e$ increases.

The tilted slit images complicate the reduction of a spectrum. It can be corrected by rotating the entrance slit by the angle $\chi$. In this case, however, the spectral resolution is reduced because the entrance slit width along the direction of echelle dispersion increases by $w/\cos\chi$. In order not to reduce the spectral resolution, it is necessary to narrow the slit width by a factor of $\cos \chi$. This in turn reduces the flux by $\cos \chi$ and consequently the resolution-slit width product, i.e., the throughput, decreases by a factor of $\cos \chi$ (Schroeder & Hilliard, 1980).

Figure 5: Tilt and curvature of a slit image.

Figure 7: Variation of the tilt angle of slit images with diffraction angle for an R2.0 (blaze angle $=$ 63$.^{\circ }$5) echelle grating. The solid, long-dashed, short-dashed, and dotted lines indicate the out-of-plane angle $\gamma_e$ of 0$.^{\circ }$0, 1$.^{\circ }$0, 2$.^{\circ }$5, and 5$.^{\circ }$0, respectively