$\star$ Subtraction of the interference rings

The images taken in the filters close to infra-red (I and Z) present sometimes fringes. In this field wavelength, CCDs are sufficiently thin to allow the interference of the light of the emission lines of molecules OH of the atmosphere, powerful beyond 800 Nm.

The figures of fringes are particular characteristics of each CCD: they schematically characterize the thickness of the CCD according to the position and are stable during the life of the CCD.

The construction of the images of fringes is done starting from the two types of charts of quantum effectiveness built for the quantum correction of effectiveness. As we saw, the light of the twilight has a primarily solar spectrum (a black body mainly) whose contributions are mainly with short wavelengths, on the other hand, the light of the night sky to strong contributions to larger wavelength. The first thus present figures of fringes much less significant.

The difference in these two images thus gives us the chart of fringes.

The second stage consists in evaluating the intensity of these fringes on the image to be corrected. Indeed, even if the figures of interferences are reproducible, their intensity very strongly depends on the quantity of light coming from the sky. Two methods are used:

• From a chart of preset fringe, with the positions of maximum and of the minima of intensity of the fringes, one calculates the intensity like the difference peak with peak of the amplitudes.

• One seeks the intensity ($k$) fringes which minimizes the dispersion of the défrangée image, in order to reduce the noise which determines our limit of weak target detection. We thus minimize least square:

  $$\chi^2 = \sum_{i,j}(I_{i,j} - k F_{i,j})^2$$ (7.14)

  
  

  where $F_{i, j}$ the chart of fringe. The minimum is reached for:

  $$k = \frac{\sum_{i,j}(F_{i,j}I_{i,j})}{\sum_{i,j} F_{ij}^2} = \frac{Cov(I,F)}{Var{F}}$$ (7.15)

  
  

One cannot guarantee that the illumination of the bottom of sky is completely uniform, in particular because of the moon. To take into account these possible variations, one adjusts simultaneously $P_n$bottom polynômial:

$$\chi^2 = \sum_{i,j}(I_{i,j} - k F_{i,j} +P_n(x,y)))^2$$ (7.16)

The coefficients of the polynomial are linear in the adjustment, the solution is thus analytical and relatively rapid. In practice, one uses a polynomial of degree 3.

The result of this procedure is presented in figure 7.9 . One does not distinguish any more the fringes on the corrected image. The amplitude of the residual fringes which was about $0.2\%$.

Figure 7.9: In top: Comparison of the images before and after subtraction of the fringes. To note, the bottom is withdrawn on the défrangée image. In bottom, comparison of the distribution of the basic pixels of sky before and after subtraction of the fringes. It is noticed that the standard deviation of the bottom of sky on the défrangée image is weaker than on the image with the fringes. Moreover, the form of the distribution for the image défrangée perfectly compatible with the Gaussian one is awaited for the distribution of the bottom of sky. In both cases, excess with dimensions positive is due to the astrophysical sources.