6 Practical example

In this section we give a detailed description on a practical example of the global procedure used to estimate NAOS-CONICA static aberrations. This procedure is quite general but the particularities of the CONICA and NAOS dichroic aberration measurements are underlined. All the illustrations are obtained for the following configuration of CONICA: objective C50S (pixel size equal to 13.25 mas) and ${\rm Br_\gamma }$ filter.

  
6.1 Input data

The input data are a focused and a defocused image ( $\vec{i}_{1,2}$) with their associated backgrounds ( $\vec{b}_{1,2}$) and a known defocus distance expressed in mm in the entrance focal plane of CONICA. This distance is given by the pinhole choice in the case of CONICA measurements (see Table 1) or by the defocus introduced by the DM in the case of the NAOS dichroic aberration measurements. In the example we consider the first approach and introduce the defocus by the pinhole choice.

  
6.2 Pre-processing

The pre-processing of the images is required before the wavefront can be estimated. We split the pre-processing in several steps:

• Conventional background subtraction in order to remove the main part of detector defects (bad pixels, background level, possible background features, etc.):
  

  $$\vec{i}^{\rm corr}_{1,2} = \vec{i}_{1,2} - \vec{b}_{1,2} .$$ (14)

  
  
  The division by a flat-field pattern is recommended to increase the accuracy of the results even if it is not done in this example.

• The focused and defocused images are re-centered by a correlation procedure. For each filter we computed the relative shifts of the PSFs against each other. The median of these shifts was determined and serves to re-center the defocused images. This procedure ensures that re-centering is accurate enough to obtain a relative tip-tilt between the two images lower than 2$\pi$.

• Removing of the residual background feature: the most important feature is a residual sine function in vertical direction due to pick-up noise. Its amplitude is greater than the noise level. An estimation of the residual background features is necessary and performed directly on the images using a median filter in horizontal direction applied to each image column. A comparison of the images before and after this residual feature removal is presented in Fig. 14.

  
  

  Figure 14: Comparison of focused and defocused images before (on top) and after (at the bottom) the application of an algorithm which removes residual background features (log scale). The estimated $S\!NR$ is equal to 400.

  
  

• Image windowing: the image size is a trade-off between a good numerical pupil modeling and a reasonable computation time. We achieve reliable results by using 128$\times$128 frames.

Note that we have assumed that all the bad pixels have been removed by the background subtraction. If some of them are still present in the pre-processed images, they must be removed by hand to ensure that they do not induce reconstruction errors in the PD algorithm.

  
6.3 CONICA aberration estimation

When both focused and defocused images have been pre-processed as described above, the PD algorithm can be applied. The inputs of the PD algorithm are:

• pixel scale: 13.25 mas (camera C50S);
• central obstruction given by the fraction of the pupil diameter: 0 (full pupil);
• wavelength (in $\mu$m): 2.166 ( ${\rm Br_\gamma }$ filter);
• highest estimated Zernike number: 15 (estimation of Zernike polynomials from 4 up to 15);
• defocus coefficient a^d₄ (in nm RMS): in the present example, d = 4 mm which leads to a^d₄ = -641.5 nm;
• the focused and defocused images obtained after pre-processing.

 

 

Table 3: Measured aberrations (in nm RMS) for the CONICA camera C50S and the ${\rm Br_\gamma }$ filter. The defocus distance between the two images is 4 mm and the estimated $S\!NR$ is 400. Only the 12 first Zernike 4-15 are given. The raw values are obtained without residual background subtraction. The corrected ones are obtained after subtraction of the residual background features.

Zernike	4	5	6	7	8	9
aberration (nm)
raw	91	-27	48	3	-9	-4
aberration (nm)
corrected	112	-24	47	1	-5	-1
Zernike	10	11	12	13	14	15
aberration (nm)
raw	19	-20	-5	-1	1	-2
aberration (nm)
corrected	17	-19	-3	-2	-3	-3

The results obtained are summarized in Table 3. A bad background correction leads to an important error on the defocus ( $\backsimeq$20 nm). A comparison between focused and defocused images and reconstructed PSFs from the estimated Zernike is proposed in Fig. 15.

Figure 15: Comparison between images (left) and reconstructed PSF from estimated aberrations (Right). (up) focused image, (down) defocused image (log scale are considered for each image).

The estimated SR on the 12 estimated Zernike is equal to 87%. It compares nicely to the SR directly computed on the focal plane image, which is equal to 85%.