4 "Cross-cleaning''

There is a solution for taking out cross-photons. It consists in selecting among the null-sum triplets only those having at least one $\Delta$-coordinate not shared with any other null-sum triplet. As each $\Delta$-coordinate of a cross-photon is shared, by definition, with a real photon, all cross-photons will be removed. The draw-back is that in some cases, valid photons may be removed.

Let (a₁,b₁,c₁) the $\Delta$-coordinates of a real photon. If the frame contains a large number of photons, there may be three cross-photons with $\Delta$-coordinates: (a₁,...,...), (...,b₁,...), and (...,...,c₁). Hence, this photon would be removed, causing a loss in overall quantum efficiency. With the simulator, we measured this attenuation factor after cross-cleaning. Measurements were made for different values of $\bar N$ and $\varepsilon$ (Fig. 3a). Poisson's law was used to draw the value of N for each frame. To take into account a non-uniformly illuminated field, more subject to a quantum efficiency loss by cross-cleaning than a flat-field, we also used input photons from stellar speckle data acquired with a Ranicon camera at a 2 meter diameter telescope (the V-Cygni star observation). The field illuminated during speckle observations is illustrated by Fig. 4. Poisson's law was used to draw photons sequentially from a speckle interferometry data file. Figure 3b shows that a reduced field of illumination does not strongly affect the quantum efficiency: it remains fair at $\varepsilon\leq 4$ and $\bar N\leq 5$.

  

Figure 3: Cross-cleaning removes the cross-photons but reduces the quantum efficiency. This figure shows the percentage of transmitted photons vs. mean number of photons per frame ($\bar N$) for different tolerances $\varepsilon$ and in case of: a) flat-field imaging, b) speckle imaging (unresolved star, 2$^{\prime\prime}$ speckle pattern and 4$^{\prime\prime}$ field)

Cross-photons are not noticeable in the integrated images, but they cause an artifact in the integrated autocorrelations. This would cause problems for second order moment methods such as speckle-interferometry. As expected, the artifact disappears when cross-cleaning is applied to the photon list. Figures 5a and 5b shows the autocorrelations of a simulated flat-field respectively without and with cross-cleaning.

  

Figure 4: 10 ms exposure of an unresolved star in photon counting mode showing the size of the speckle pattern compared with the field of the DELTA camera (the field boundaries are white lines). The original data used for this simulation were acquired with a Ranicon camera

The star-shaped pattern and the central peak in the autocorrelation of images without cross-cleaning are due to the cross-photons. The loss of quantum efficiency due to cross-cleaning is acceptable up to $\bar N=4$ photons per frame, with the null-sum test tolerance set to $\varepsilon=2$.

  

Figure 5: Cross-photons generate an artifact in the autocorrelation. This artifact disappears when these are removed form the photon list. Autocorrelations of flat-fields obtained: a) without cross-cleaning, b) with cross-cleaning