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Calibration Access and Data Handbook

Procedure

The grating mis-alignment is parameterized with a set of Gaussian functions, whose parameters are stored in the CCF. The corresponding probability density is according to

$$p(\beta) = \sum_{i=1}^{N} \left [ a_i\ e^{-\frac{1}{2} \left... ...a-f \dot (1+C)\mu_i}{f\dot (1+C)\sigma_i} \right)^2} \right ]$$

The parameters $a_i$, $\mu_i$, $\sigma_i$ are stored in the CCF. $N$ is the number of parameters available in the CCF. $C$ is the chromatic magnification that scales the width of the grating mis-alignment distribution as a function of angle of incidence and diffraction angle. It is defined as

$$C = \frac{\sin\alpha}{\sin\beta_{\rm m}}$$

and is an input argument.

Additionally to the chromatic magnification, the alignment distribution is homogeniously scaled by a factor $f$, which is stored in the CCF.

The average center of all Gaussians is calculated by

$$a_i' = a_i\ \sigma_i\ \sqrt{2\pi}$$

$$\mu_0 = \frac{\sum a_i'\cdot \mu_i}{\sum a_i'}\ .$$

The function returns the probability distribution $P(j)$ which calculated from $p(\beta)$ by integration per channel $j$

$$P(j) = \sum_{i=1}^N a_i'\ \int_{\beta_j}^{\beta_{j+1}} e^{-\... ...)+\mu_0]} {(1+f\dot C)\sigma_i} \right)^2} \ {\rm d}\beta \ .$$

The distribution will be calculated such that the peak is at the center of the bins.