5 Limitations

The PD algorithm is based on several assumptions which must be well verified to obtain a good accuracy on the results. A list of possible error sources is given below. Quantitative results are essentially given on experimental data and with additional simulation data when necessary. The global procedure of data reduction can be found in Sect. 6. The error sources can be decomposed in three parts: the errors due to a non-perfect knowledge of the system (calibration errors or uncertainties), the errors due to the image acquisition and pre-processing (noise, residual background, etc.) and the errors due to limitation of the algorithm (spectral bandwidth, amplitude of the estimated aberrations, etc.).

5.1 System limitations

Let us first focus on the errors due to the imperfect system knowledge.

   
5.1.1 Defocus distance

The major assumption of the PD principle is the addition of a known distortion (defocus in our case) between two images. An error on the defocus induces an error on the coefficients of radially symmetric aberrations, with a main part on the estimated defocus itself (see Fig. 4). The difference between the maximal and the minimal estimated value of the defocus coefficient is 55 nm. For the spherical aberration, it is 10 nm and for the other ones, it is less than 5 nm.

Figure 4: Influence of the error on defocus distance on estimated aberration by phase diversity. The CONICA camera C50S and the narrow band filter ${\rm Br_\gamma }$ are used. The focused and defocused images are obtained using the 0-4 pinhole pair (that is a theoretical defocus of 4 mm between the two images). For the same couple of images several defocus distances (from 3.8 to 4.2 mm) serve as input parameters.

Figure 5: Influence of the error on defocus distance on the estimation of the defocus. Experimental data have been used. The results are given for the CONICA camera C50S and two narrow band filters: FeII (1.527 $\mu$m) and ${\rm Br_\gamma }$ (2.166 $\mu$m). The pinhole pair 0-2 (that is a theoretically defocus of 2 mm) and 0-4 (that is a theoretically defocus of 4 mm) are respectively used for FeII and ${\rm Br_\gamma }$ filters. For each couple of images several defocus distances (respectively from 1.5 to 2.5 mm and from 3.5 to 4.5 mm for FeII and ${\rm Br_\gamma }$ filters) serve as input parameters.

In addition one can show in Fig. 5 that for reasonable errors on the known defocus, the propagation error coefficient is equal to one. Hence, the uncertainty on the known defocus distance yields directly the uncertainty on the estimated defocus aberration.

This uncertainty on the defocus distance can be due to:

• an uncertainty on the physical position of the CONICA pinholes in the entrance focal plane. This uncertainty is estimated to $\pm 0.15$ mm corresponding to $\Delta a_4=\pm 24$ nm RMS;
• a systematic uncertainty on the F/D ratio (estimated to be less than two percent);
• the precision of a focus adjustment with the NAOS deformable mirror for the dichroic aberration estimation. This error is estimated to less than a few percent.

All these items lead to a precision of the estimated defocus roughly equal to a $\pm 30$ nm. It will be shown in the following that the error on the defocus distance is by far the dominant error for our application.

5.1.2 Camera pixel scale

The camera pixel scale is needed to calculate the oversampling factor. An error on this factor induces an error on the coefficients of all radially symmetric aberrations (defocus, spherical aberration ...) as shown in Fig. 6.

Figure 6: Influence of the pixel scale error on estimated aberration by phase diversity. Experimental data have been used. The CONICA camera C50S and the narrow band filter ${\rm Br_\gamma }$ are used. The focused and defocused images are obtained using the 0-4 pinhole pair. For the same couple of images several pixel scales (from 13.05 to 13.45 mas) serve as input parameter for the PD algorithm.

Figure 7: Influence of the pixel scale error. The reference value is set to 13.25 mas. For this value it is assumed that the error on the defocus coefficient estimation is zero (the experimental conditions are the same as in Fig. 6).

An error of the pixel scale is essentially propagated to the defocus aberration estimation. The difference between the maximal and the minimal estimated value of the defocus coefficient is 17 nm. A slight error of 6 nm can be seen on spherical aberration but remains negligible in comparison to the one on the defocus. In Fig. 7 the evolution of the estimation error of the defocus coefficient is plotted as a function of a pixel scale measurement error. It is assumed that the true value is 13.25 mas measured during the firs on-sky tests of the AO system. Since the accuracy on the pixel scale measurement is better than 0.2 mas, one can estimate the wavefront error (WFE) due to this uncertainty to be less than a few nanometers and therefore to remain negligible.

5.1.3 Pupil shape

An exact knowledge of the pupil shape (diameter, central obstruction, global shape) is required. In particular, few percent of mis-alignment of the pupil leads to an error of a few tens nanometers on the phase estimation.

  
5.1.4 Differential object structures

The algorithm is based on the assumption that the same object is used to obtain the focused and the defocused image. If two different objects are used (case of CONICA pinholes), errors could be induced if they are not completely identical.

Figure 8: Comparison of the estimated aberrations of various pinhole pairs. Camera C50S with FeII narrow band filter (1.644 $\mu$m).

Figure 8 shows for the narrow band filter FeII (1.64 $\mu$m) and the objective C50S the estimated aberration for different pinhole pairs. It shows a good agreement between all the pinholes, except for the pinhole pair 0-1. The discrepancy in the estimated aberrations can probably be attributed to the small defocus distance. Disregarding the pair 0-1, the good agreement of all the other pairs leads us to assume the pinholes close to be identical. Indeed, the main part of the WFE is due to the defocus coefficient and highlights the uncertainties on pinhole positions as already mentioned in Sect. 5.1.1.

  
5.1.5 Translation in the entrance focal plane

The maximal detector translation along the optical axis is not enough to introduce significant diversity between focused and defocused images. For the calibration of CONICA aberrations, we introduce defocus by translating "the object'' in the entrance focal plan. As we said before (Sect. 3.1), this implementation does not create exactly a pure defocus but also in first order, some spherical aberration. We have quantified the deviation from a pure defocus by using the optical design software ZEMAX and shown that it can be neglected (a translation of 4 mm in the entrance focal plan induces defocus and a negligible spherical aberration a₁₁=0.14 nm).

  
5.1.6 Field aberrations

The images obtained by the pinhole 1 and 2 are not located at the same position on the detector than those obtained by the pinhole 0 and 4 (separation of 100 pixels in y). It induces that some focused and defocused images (for example pair 0-1 and 0-2) do not see exactly the same aberrations. The evolution of the field aberrations has been evaluated by optical calculations using ZEMAX. This study has shown that the main influence concerns the astigmatism a₅ but in a negligible way (its variation is less than 5 nm for a separation of 100 pixels in x and 100 pixels in y).

5.2 Image limitations

5.2.1 Signal to noise ratio

The accuracy of the PD algorithm is directly linked to the signal to noise ratio in the images (the definition of $S\!NR$ is given in Sect. 4).

Figure 9: Evolution of the wavefront error as function of $S\!NR$ on focal plane image. Simulation presented in Sect. 4 have been used. A 1/$S\!NR$theoretical behavior is plotted for comparison.

We present in Fig. 9 the WFE error evolution as a function of image focal plane $S\!NR$ This figure, obtained on simulated data, shows the perfect agreement between simulation and expected (theoritical) 1/$S\!NR$ behavior (Meynadier et al. 1999). For the NAOS-CONICA aberration estimation it has been checked that the $S\!NR$ is high enough to make this error source negligible in comparison to the others since the typical values of $S\!NR$ are greater than a few hundred. That leads to a WFE of a few nanometers due to signal to noise ratio. This has been experimentally checked on one set of data. Four pairs of focused-defocused images have been recorded sequentially in the same CONICA configuration (FeII filter and camera C50S). The $S\!NR$ on these images is 400. Aberrations are then estimated for each couple and the WFE fluctuation on this four set of coefficients is equal to 2.2 nm RMS, which is in perfect agreement with Fig. 9 (obtained on simulation).

  
5.2.2 Residual background features

The principle of the PD method is the minimization of a criterion (Eq. (10)) which is based on a convolution image model (Eq. (1)). Thus the image should be perfectly corrected for all instrumental features (background, dead pixels, etc.) in order to match the model. In practice, residual features are still present. In particular, in the case of CONICA images, a background fluctuation due to pick-up noise can induce residual features on the images even after a proper background calibration (see Sect. 6.2). These features are interpreted as signal by the phase-diversity algorithm. Therefore they induce bias on the aberration estimation. The effect of such fluctuations is highlighted in Fig. 10 on experimental data. The difference of the PD results obtained with and without residual background features yield the WFE which is plotted as a function of the image size. This can be understood as a function of the residual background influence, too, because it obviously depends on the image size: the smaller the images, the less important the residual background in comparison to the signal. Nevertheless, the image size should be large enough to contain the whole signal. Furthermore, the modelisation of the pupil shape (see Sect. 5.3.3) must also be taken into account to choose the right image size. In Sect. 6.2 we describe a pre-processing algorithm which allows to remove these residual background features.

Figure 10: Evolution of the wavefront error as a function of the image size. Experimental data have been used.

5.2.3 Number of estimated Zernike polynomials

As presented in Sect. 2.3, the phase regularization in our algorithm is provided by a truncation of the solution space through the use of a finite (and small) number of unknowns (typically the first twenty Zernike coefficients). Figure 11 shows, on experimental data, the influence of the number of estimated Zernike on the reconstruction quality. Note that, in the case of measurements of CONICA stand-alone aberrations, the pupil is unobscured and thus the Zernike polynomials are strictly orthogonal.

Figure 11: Evolution of the aberration estimation as a function of the number of Zernike in PD algorithm. Experimental data have been used.

There exists a limit to the number of Zernike polynomials that can be estimated with a reasonable accuracy. Of course this limit depends on the signal to noise ratio on the images. In the present case, this number is equal to 36. Note that if a more sophisticated regularization term is introduced in the PD algorithm both on the object and the aberrations this limitation should be overcome. Nevertheless such a regularization is not needed here since the aberration amplitudes are negligible (less than a few nanometers) for Z_iabove i=11 (i.e. the spherical aberration). The WFE between estimated Zernike coefficients 4-15 and estimated Zernike coefficients 4-36 is about 1.3 nm RMS. This WFE is very small and thus shows that the aliasing of the Zernike polynomials above 15 on the estimated coefficients 4-15 is negligible.

5.3 Algorithm limitations

5.3.1 Spectral bandwidth

The phase diversity concept proposed here is a monochromatic wave-front sensor (theoretically the concept can be applied on polychromatic images but it induces an important modification of the algorithm to model the data (Seldin & Paxman 2000)). Nevertheless it has been shown (Meynadier et al. 1999) that the use of broadband filters does not significantly degrade the accuracy as long as $\frac{\Delta \lambda}{\lambda}$ is lower than a few tens of percents (typically $\frac{\Delta \lambda}{\lambda} \le 0.15$).

5.3.2 Image centering

As mentioned above, the PD algorithm can not estimate a tip-tilt between the two images larger than $2\pi$. Therefore, a fine centering between focused and defocused images must be done before the aberration estimation (see Sect. 6.2).

  
5.3.3 Pupil model

Since we consider here experimental data (see Sect. 6), the modelisation of the pupil shape in the algorithm is critical, in particular the pixelisation effects. Indeed, in PD algorithm the pupil definition depends on the image size and on the oversampling factor. For example, images of camera C50S in K band oversample with a factor of 2 and a $32\times32$ image will lead to a pupil diameter of 8 pixels (see Fig. 12). In this case, the pixelisation effects on the shape of the pupil will induce aberration estimation error. These effects are illustrated in Fig. 13.

Figure 12: Pupil shape in the PD algorithm for three numbers of pixel used in the pupil sampling [8, 32, 64 and 256]. The oversampling factor of 2 in K band leads to corresponding image sizes equal to [32, 128, 256 and 1024].

Therefore, large images are recommended to well model the pupil and to obtain accurate results. Nevertheless, two problems may occur with the processing of large images:

• a residual background problem (see Sect. 5.2.2). The larger the image, the more important the effects of residual background unless the images are tapered outside their central region;
• a computation time problem. Because of the iterative resolution of criterion defined in Eq. (7), the increase of images size will lead to an increase of computation time.

This highlights the choice of a good compromise between pupil model in the PD algorithm to avoid phase reconstruction error and a reasonable image size.

The evolution of the reconstruction error as a function of pupil sampling is presented in Fig. 13. To minimize the residual background effects, all the background pixels (that is pixels with no PSF signal) has been put to zero in the images.

Figure 13: Evolution of the wavefront reconstruction error as function of pupil sampling in PD algorithm. The x axis gives the pupil diameter in pixel in the PD algorithm.

Considering the results shown in Figs. 13 and 10 along with the computation load lead us to choose an image size equal to $128\times128$ pixels for the K band and $64\times64$ pixel for the J band.

5.4 Conclusion

In this part, we have analyzed and quantified, on experimental and simulated data, the possible sources of errors in the static aberration estimation for NAOS-CONICA. It is shown that the main source of errors is due to an imperfect knowledge of the system (that is calibration errors). In particular a precise knowledge of the defocus distance between focused and defocused planes is essential.

If very high precisions are required on the estimation and on the correction of static aberrations (for instance in the case of a future very high SR AO for exo-planet detection), the PD must be taken into account in the early design of the system in order to optimize with respect to the constraints and error sources listed above.