vik dhillon: phy217 - imagers

example problems

An 8 m, f/8 telescope is equipped with a re-imager and a detector of 2000 x 2000 pixels, where each pixel is 24 μm in size. What magnification must the re-imager have to give a field of view of 8' x 8'?

The focal length of the telescope is F = 8 x 8 = 64 m. Hence the plate scale is 206265 / 64000 = 3.2 "/mm. The field of view without the re-imager is therefore 2000 x 0.024 x 3.2 = 154". Hence the required re-imager magnification is 154"/480" = 0.32x.

In the above example, what focal ratio must the camera have?

The focal length of the telescope plus re-imager, F_sys, is given by:

F_sys = F_tel M.

Since M = F_cam / F_coll, we can write:

F_sys = F_tel F_cam / F_coll.

Now, F_tel = f_tel D_tel, F_cam = f_cam D_cam and F_coll = f_coll D_coll. Hence:

F_sys = f_tel D_tel f_cam D_cam / f_coll D_coll.

In order to collimate the beam from the telescope, the focal ratio of the collimator must be equal to the focal ratio of the telescope, i.e. f_coll = f_tel. Hence:

F_sys = D_tel f_cam D_cam / D_coll.

Also, since both are in the same collimated beam, the diameter of the camera must be at least as big as the diameter of the collimator, but there is no point in making it any larger, i.e. D_cam = D_coll. So,

F_sys = D_tel f_cam.

Hence the focal ratio of the camera must be:

f_cam = F_sys / D_tel = F_tel M / D_tel = f_tel M = 8 x 0.32 = f/2.6.

A 4 m, f/10 telescope is equipped with a re-imager with a collimator of focal length 300 mm and a camera of focal length 100 mm. If the seeing is 0.8", what detector pixel size in μm is required to sample the image optimally?

The Nyquist sampling theorem tells us that we require at least 2 pixels across the seeing disc in order to sample the image optimally, i.e. the pixel size should be 0.4".

To convert this to μm, we need to know the plate scale of the telescope plus re-imager, p_sys:

p_sys = 206265 / F_sys,

where F_sys is the focal length of the telescope plus re-imager. In problem 2 above, we derived a relation between F_sys and the focal lengths of the telescope, collimator and camera:

F_sys = F_tel F_cam / F_coll.

Hence

p_sys = 206265 F_coll / F_tel F_cam = (206265 x 300) / (40000 x 100) = 15.5 "/mm.

Therefore, each pixel must be of size 0.4 x 1000 / 15.5 = 26 μm.