4 Simulation results

In order to validate the algorithm and to quantify its precision, we first consider simulated images. The conditions for this simulation are given by a point-like object, an imaging wavelength of 2.166 $\mu$m and a pure defocus equal to $\lambda$ (peak to valley) between the two images, corresponding to a defocus coefficient a₄^d of 641.5 nm RMS. We degrade the PSF by a wavefront deformation described by its Zernike coefficients. The coefficients are arbitrary but chosen to have comparable values as observed in the calibration procedure (see Paper II). The phase is generated with the first 15 Zernike polynomials, note that the estimated phase will be expanded on the same polynomials. We add white noise to each image in order to obtain a $S\!NR$ of 200 which corresponds roughly to the $S\!NR$ of the CONICA data (this is the typical average $S\!NR$ corresponding to the time exposure used, which has been estimated on several measurements). The $S\!NR$ is defined as the ratio of the maximum flux in the focus image over the RMS noise. The same noise statistics is applied to the defocused image. That results in a lower $S\!NR$ on this image since the defocus spreads the PSF and reduces its maximum value. The focused and defocused images are presented in Fig. 2. In this example only two possible limiting parameters (see Sect. 5) have been taken into account: noise and image re-centering. The system is assumed to be perfectly adjusted and the images to be perfectly pre-processed without having any residual background features.

Figure 2: Simulated focused and defocused images (logarithmic scale). Noise is added to obtain a $S\!NR$ of 200 in the focused image.

Figure 3 shows a comparison between the input images of the simulation and the PSFs being reconstructed by the estimated aberrations and visualizes the quality of the aberration estimation. This visualization is helpful to judge real calibration data when only in-focus and out-of-focus images are available. Table 2 quantifies the performance of wavefront estimation by comparing the true and the estimated Zernike coefficients. On each coefficient a good accuracy is obtained: the errors are less than a few nanometers. The maximum error is for the two astigmatisms. The influence of $S\!NR$ on estimation results is analyzed in Sect. 5.

The slight tip-tilt introduced between the two input images ( $\vec{a}_2=70$ nm rms and $\vec{a}_3=-103$ nm RMS) are estimated with a high precision, too. The error amounts to less than 1% (the tip-tilt values are not shown in Table 2). In the next section, we focus on the possible sources for losses of estimation accuracy.

Figure 3: Comparison between images (left) and reconstructed PSFs (right) from estimated aberrations. On top the focused images, at the bottom the defocused images (logarithmic scale and zoom$\times$2 are considered for each image).

 

 

Table 2: Comparison between true and estimated Zernike coefficients applying phase diversity to simulated data with an image $S\!NR$ equal to 200. The absolute value of the error is given for each coefficient. The total error is equal to 9.3 nm RMS.

Zernike number	true (nm)	estimated (nm)	error (nm)
4	60.5	61.5	1
5	-39.3	-46.7	7.4
6	58.1	61.7	3.6
7	-16.2	-15.9	0.3
8	-14.1	-10.9	3.1
9	-2.5	-3.6	1.1
10	13.7	12.9	0.7
11	-24.3	-26.	1.7
12	0.5	1.2	0.7
13	-3.2	-4.9	1.7
14	2.8	3.3	0.4
15	-2.4	-2.2	0.2