# Simulation of single mirror telescope optics

The optics for a single mirror telescope are defined by the , the dish
shape, and the position of the individual mirrors. The segmented
spherical mirror tiles are positioned on a dish following

The parameters describing the individual mirrors are defined through a
separate configuration file, given as input through the parameter. The
list contains the position of each mirror tile ($x_i$, $y_i$),
perpendicular to the optical axis, together with its shape, size and
individual focal length $f_i$. The $z_i$ positions of the mirror tiles
is optional and if they are omitted, the mirrors will be positioned
following variants of a shape, taking mechanical accuracies into
account, defined by the parameter . Mirror tiles can have either
individual focal lengths, defined with different $f_i$ values in the
table; all the same focal lengths, set with the parameter; or
automatically optimized focal lengths, calculated from the desired dish
shape and ($x_i$, $y_i$). The optimum focal lengths of spherical tiles
on a parabolic dish can be obtained by calculating the principal
(maximum and minimum) radii of curvature of the paraboloid at the
position of the mirror segment. Half the geometric mean, i.e.
$0.5 (r_{\text{min}} r_{\text{max}})^{1/2}$, is used for automatically
optimized focal lengths, but the simple distance of the mirror tile to
the system focus would be reasonable as well, as the two lead to almost
the same result. Mirror tiles can be of hexagonal, square, or circular
shape.

## Optical PSF

The optical PSF of the telescope is affected by

1.  the type of optics and size of mirror tiles;

2.  the micro-roughness of the mirror surface;

3.  the uncertainty on the focal length of the different mirror tiles;

4.  random misalignments and misplacements of the mirror tile on the
    dish.

Items 2 and 3 are intrinsic to each mirror tile and their effect can be
quantified in the laboratory by e.g. 2F measurements of single mirror
tile spot size. In the simulations, the small-scale roughness of the
mirror surface is taken into account with a Gaussian smearing of the
reflected photon direction according to an r.m.s. spread given by the
parameter (for a better description of the tails of the PSF, optionally
the sum of two Gaussians can be used). The uncertainty on the mirror
tile focal length is considered through random fluctuations in mirror
focal lengths and is controlled by the parameter . These two parameters
should be adapted to match the measurements in the laboratory on the
mirror tiles.

To take into account the effect of the variation of the alignment of the
mirrors with respect to the nominal alignment, two additional sets of
parameters are used to smear the horizontal and vertical directions of
the reflected photons:[^1] ($\theta_0$,$\alpha_h$, $\beta_h$,
$\gamma_h$) and ($\theta_0$,$\alpha_v$, $\beta_v$, $\gamma_v$)
respectively. These sets of parameters take into account the zenith
angle dependence of this effect into account, following any dynamical
alignment performed. The spread in the horizontal direction is given by,
$$\sigma_h = \sqrt{{\alpha_h}^{2} + {\beta_h}^{2}(\sin{\theta} - \sin{\theta_0})^2 + {\gamma_h}^{2}(\cos{\theta} - \cos{\theta_0})^{2}} \, ,$$
while the spread in the vertical direction is,
$$\sigma_v = \sqrt{{\alpha_v}^{2} + {\beta_v}^{2}(\sin{\theta} - \sin{\theta_0})^2 + {\gamma_v}^{2}(\cos{\theta} - \cos{\theta_0})^{2}} \, .$$
The zenith angle at which the PSF has been measured is given by
$\theta_0$. The parameters $\alpha_h$, $\beta_h$, $\gamma_h$,
$\alpha_v$, $\beta_v$ and $\gamma_v$ are obtained from a fit of the
experimental optical PSF of the full telescope optics, taking into
account the single-mirror tile point-spread function as described above.
The $\sigma_h$ and $\sigma_v$ are used for a Gaussian randomisation of
the mirror alignment.

[^1]: The horizontal direction refers to the azimuth or y direction in
    dish coordinates, while the vertical direction is the altitude or x
    direction in dish coordinates.