The pupil

The pupil of the spectrograph is an image of the primary mirror formed by the secondary and the collimator. The pupil diameter $D_{pupil}$ is simply set by the expansion of the beam as it reaches the collimator, as shown in Figure 2, and its location is set by the focal length of the collimator.

$$D_{pupil} = f_{coll} / N_{tel} = \frac{D_{tel} f_{coll}}{f_{tel}}$$

$D_{pupil}$ gives the minimum size of a dispersing element in the collimated beam. This is a critical number since it sets the minimum size of a grating, grism, or other optical device that the instrument requires.

Off-axis beams, which are displaced by $D_{field}/2$ in the focal plane, pass through the pupil at an angle

$$\theta_{offaxis} = \frac{D_{field}}{2f_{coll}}.$$

All the light from on- and off-axis beams passes through a ``waist'' at the pupil. If we introduced a screen into the light path at the pupil, we would see an in-focus donut-shaped image of the primary mirror. If the screen were placed ahead of or behind the pupil, the image of the primary would be not clearly focused.

The pupil size interacts with the field of view indirectly. Once the collimator focal length $f_{coll}$ is chosen, increasing the field of view does not increase the pupil size, since all the off-axis beams pass through the ``waist'' of the pupil. However, if we choose a long $f_{coll}$, then off-axis light enters the collimator at a less extreme angle, so the collimator lens is slower and easier to design. But long $f_{coll}$ requires a larger pupil. Equivalently, from the equations from $N_{coll}$ above, we see that increasing $D_{pupil}$ relative to $D_{field}$ makes the collimator slower. So although field of view does not depend directly on pupil size, in real optical designs it is difficult to image a large field through a small pupil.