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

the type of optics and size of mirror tiles;
the micro-roughness of the mirror surface;
the uncertainty on the focal length of the different mirror tiles;
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.