Calibration Access and Data Handbook

Procedure

This description is a copy of in [3]. The following definitions are used:

$k = 2\pi/\lambda$, where $\lambda$ is the wavelength at which the LSF is to be evaluated

$\sigma_s$ is the r.m.s. surface roughness for two scatter models; $s$ is either small or large for small angle scatter, and large angle scatter, respectively. This is available from CCF LineSpreadFunc. Note that the surface roughness for the large angle scatter is parameterized as a function of wavelength $\lambda$ and its value is obtained with
CAL_rgsGetLAScatterRoughness (section 3.4.18)

$l_s$ is the correlation length of the scatter components; $s$ is either small or large for small angle scatter, and large angle scatter, respectively. This is available from CCF LineSpreadFunc.

$\beta_{\rm m}$ is the center angle of the unscattered light distribution

The fraction of scattered light for small/large angle scatter is

$$F_{\rm small,large} = 1 - e^{-[k\sigma_{\rm small,large} (\sin\alpha + \sin\beta_{\rm m})]^2}\ ,$$ (13)

with a correction factor of the LSF probability density $P(\beta)$ for large angle scattering of

$$R = \frac{a_0+a_1x+a_2x^2+a_3x^3+a_4x^4} {b_0+b_1x+b_2x^2+b_3x^3+b_4x^4}$$ (14)

where $a_i$ & $b_i$ are stored in CCF LineSpreadFunc, and $x$ is defined by

$$x = \frac{1}{k\,l_{\rm large} \sin^2\beta_{\rm m}}\ .$$

The power spectral densities are defined with respect to $p$, with

$$p = k (\cos\beta_{\rm m} - \cos\beta)\ ,$$

and finally the density functions for small and large angle scatter are

$$W_{\rm small}(p) = \frac{1}{\sqrt{\pi}}\,l_{\rm small}\,\sigma_{\rm small}^2 \,e^{-l_{\rm small}^2\,p^2}$$ (15)

$$W_{\rm large}(p) = \frac{1}{\pi}\,l_{\rm large}\,\sigma_{\rm large}^2 \,\frac{1}{1 + l_{\rm large}^2\,p^2}\ .$$ (16)

The intensity of the LSF for one bin in $\beta$ that has a range from $\beta-\Delta$ to $\beta+\Delta$ is then given by

$$LSF(\beta) = F_{\rm large}\, \frac{R\cdot \sin\beta_{\rm m... ...}{(\sin\alpha+\sin\beta_{\rm m})^2}\, W_{\rm large}\ d\beta +$$ (17)

$$(1- F_{\rm large}) F_{\rm small} \frac{\sin\beta_{\rm m}\,... ...}{(\sin\alpha+\sin\beta_{\rm m})^2}\, W_{\rm small}\ d\beta +$$ (18)

$$(1- F_{\rm large}) (1-F_{\rm small}) \int_{\beta-\Delta}^{\beta+\Delta} \delta(\beta-\beta_{\rm m})\ d\beta \ ,$$ (19)

where the total fraction of non-scattered light is given by $(1- F_{\rm large}) (1-F_{\rm small})$. The function that is implemented in the CAL does not include the last term (the $\delta$-function), as is only the unscattered light and its contribution is taken care of by the calling task (rgsrmfgen).

In order to obtain the LSF, this needs to be convolved with the PSF of the mirror, which acts as a perturbation of $\alpha$. This is not performed by this function.