# Digital Holographic Positioning Sensor for a Small Deployable Space Telescope
## 4 Data Processing Algorithm and Sensitivity  
The most obvious way to process a 2D periodic pattern, such as the interferogram is to convert it into the Fourier domain. So we compute fast Fourier transforms for each of the interferograms and further operate with their modules \(I=|FFT\{j\}|\). Figure 7 demonstrates the main stages of the data processing. We consider three alignment variables: axial misplacements of the vertical primary segment \(\Delta z_{L1}\), misplacements of one of the two lower segments \(\Delta z_{L2}\), and of the secondary mirror \(\Delta z_{M}\). Figure 7A show the Fourier spectrum for a perfectly aligned system, which is common for all the variables. Figure 7B show the Fourier spectrum evolution with the largest deviation of each variable and the features of the image, which we extract for the further analysis. Figure 7C show a 2D intensity distribution for the selected feature and definition of the variables used to construct the alignment metrics. The red solid curves there correspond to the 0 deviation, i.e., the case from Figure 7A, while the rest of the colors and dashes indicate some deviations and the key values extracted from each plot are marked for the maximum deviation as in Figure 7B. As one can see from the first column in Figure 7, displacement of the vertical segment leads to a symmetrical blurring of the top and bottom orders in the Fourier spectrum. Then we take a vertical section crossing the orders maxima \(I_{max}\) and construct the following metric  
\[m_{1}(\Delta z_{L1})=\frac{I_{max}}{I_{max}0}, \tag{11}\]  
Figure 7: The interferogram post-processing scheme: (**A**) reference Fourier spectrum with 0 deviations; (**B**) distorted Fourier spectra generated with misplaced optical elements: (**C**) 1D section of the spectrum used for the metrics computations. where \(I_{max0}\) is the maximum intensity in the ideal case. Similarly, in the case of a displacement of the lower segment, the top and bottom orders of the spectrum are asymmetrically distorted. Therefore, we take an average over seven horizontal rows containing the two top orders and compute the difference between the two maxima:  
\[m_{2}(\Delta z_{L2})=|\frac{I_{1 max}-I_{2 max}}{max(I_{1 max}-I_{2 max})}|. \tag{12}\]  
This anisotropy in the interferogram decoding is one of the reasons to consider the geometry with three segments, which does not have a double-plane symmetry. Finally, the secondary mirror axial displacement creates a rotationally symmetrical blurring of all the orders in the spectrum. In order to quantify it, we extract a box containing one of the lateral orders, perform averaging along the 10 rows, and then compute the FWHM for the horizontal section:  
\[m_{3}(\Delta z_{M})=\sum\frac{I}{I_{max}}>0.5. \tag{13}\]  
Note that the limits of secondary mirror displacement are narrower. The higher sensitivity to this parameter is expected, since it doubles the optical path change. Using the proposed simple metrics we can estimate the system sensitivity of each of the individual misalignments. The results of iterative computations of the metrics are presented in Figure 8.  
The plots show that the metrics behave as non-linear functions of the parameters to be measured. At very low values of the argument they become relatively noisy, mainly due to the use of discrete values. This effect is more notable for the primary mirror. However, there are no clear zones of insensitivity or saturation on the curves. In order to quantify the sensitivity of proposed metrics we perform polynomial fitting for the computed datasets (see the dashed curves in the plots). We make the following conservative assumptions: the sensor digital output has 256 levels of the intensity and the widening of the Fourier spectrum orders may be measured with the precision of 1 element, which corresponds to a \(128~{}\times~{}128~{}\upmu\)m (\(8\times 8\) pixels) element in the space domain. To compute the sensitivity we find a derivative for the fitting polynomial.