5 Image quality versus estimated aberrations

5.1 Strehl ratios by PD and focal plane image

The PD calibration data can be used to investigate the available image quality in different ways. First, the knowledge of the wavefront allows us to calculate a SR. After the reduction of the calibration data the wavefront is described by a set of Zernike coefficients. Furthermore, we can just refer to the in-focus image and calculate a SR with the measured point spread function. In the following we give a more detailed explanation of how these SRs are obtained.

Strehl by PD

For small wavefront deviations the SR can be determined via the coherent energy referring to the wavefront variance $\sigma^2$ in radian. The PD estimation yield the wavefront expanded in terms of Zernike coefficients a_i. For small $\sigma^2$

$${\rm SR} \simeq {\rm e}^{-\sigma^2} \simeq 1- \sigma^2 \simeq 1 - \sum_{i=4}^m a_i^2$$ (5)

allows us to calculate the SR directly by the output of PD estimation. In principal the sum runs to infinity ( $m = \infty$) but for our purpuse we stop at m = 15. We can compare these SR numbers to the ones that are directly determined by the in-focus images.

Strehl on image

A straight-forward way to calculate a SR on the focus image (PSF) is to construct a theoretical diffraction-limited image ${\rm PSF_{diff}}$ taking into account the wavelength, the pixel scale, the aperture and the central obscuration. Having normalized the total intensity of the PSF and ${\rm PSF_{diff}}$ to 1, the fraction of these values yields the SR (see Eq. (6)).

In particular, in the case of the PSF sampling being close to the Nyquist criterion this approach has the disadvantage of being sensitive to the exact position of the PSF peak with respect to the pixel center. Furthermore, since the total intensity has to be determined by the integrated signal over a wider region around the PSF, the reliability of the SR value depends on a precise background correction. If the background is overestimated, then the SR will be overestimated, too, and vice versa. The reliability of the SR values can be enhanced when we switch from the image space to the Fourier space by

$${\rm SR} = \frac{{\rm PSF}(\alpha = 0)}{{\rm PSF}_{{\rm diff... ...= \frac{\int{{\rm OTF}(f)}}{\int{{\rm OTF}_{{\rm diff}}(f) }},$$ (6)

where OTF is the optical transfer function. Since in Fourier space only spatial frequencies are considered, a shift of the PSF is of no importance anymore. Aside from that, an elegant and reliable background correction can be performed using the zero spatial frequency. We calculate the SRs by the following procedure:

• The image is corrected by its corresponding background;
• The OTF is calculated. It is given by the real part of the Fourier transform of the image;
• The residual background is corrected by the zero frequency. A fit of the very first spatial frequencies is used to extrapolate the true zero frequency value. The difference of the measured and the extrapolated value for the zero frequency yields the residual background;
• The noise level is subtracted using the high frequencies beyond the diffraction limit;
• A theoretical telescope OTF is constructed and multiplied by a Bessel function to account for the spatial spread due to the object size;
• The detector response is taken into account. This is done by a further multiplication of the theoretical telescope OTF with the Fourier transform of the detector response;
• The SR is obtained by the division of the normalized integrals from the measured and the theoretical OTF. All points with spatial frequencies higher than the diffraction limit are excluded.

Figure 7 gives an example of how the described OTFs look for a PSF taken through the filter H2(1_0)S(7). To allow for a one dimensional representation, the circular mean of the two dimensional OTF is calculated. The raw, untreated OTF of the image is labeled "OTF image''. "OTF corr'' displays the image OTF which is corrected for the residual background at zero frequency and for the noise level. The noise level of the uncorrected image OTF can be seen as plateau beyond the cutoff frequency $D/\lambda$ and averages 1% of the maximum value. The theoretical diffraction-limited telescope OTF includes the reduction due to the object size and the correction for the detector modulation transfer function (MTF). It is labeled "MTF Det'' and is located at the top of the plot. The detector response is constructed by the assumption that roughly 3% of the total intensity is contained at each of the adjacent pixels and 1.5% in the corner pixels. Here we refer to Finger et al. (2000). In this paper a measurement of the response of a comparable infrared array is described. Due to the lack of a precise knowledge of the detector response it is constructed by a quadratic scaling in relation to the different pixel size. A quadratic scaling is implied by a linear behaviour of the diffusion of the minority carriers in the detector material in one dimension for small distance variations.

Figure 7: Visualization of the theoretical and measured OTF at the example of filter FeII1644.

  
5.2 Comparison of Strehl ratios

The resulting SRs for the narrow-band filters in J, H, and K are presented in Figs. 8 and 9. For each filter two SRs are given: the SR by PD and the SR on the image.

A number of error sources contribute to the error of the SR values on image. Beside of small error contributions due to uncertainties of the pixel scale and the flatfield, the remaining uncertainty of the background correction and the detector response lead us to estimate an absolute error of $\pm 4$%. The expected wavelength dependency of the MTF error is minor with respect to the remaining background error. Therefore it is neglected and we use the constant value given above derived from experience in reducing the experimental data.

Recall that the SR by PD has a maximal wavelength-dependent error of $\pm 5$% at 1 $\mu$m and $\pm 1$% at 2 $\mu$m taking into account an error of $\pm 35$ nm RMS for the focus estimation (i=4). The main contributor to this error is a systematic error in the precision of the pinhole positions in the Zernike tool (see Sect. 4.1).

In general the PD SRs exceed the other SR values. This reflects the fact that the wavefront is expanded by a limited number of Zernike coefficients and the higher order aberrations are cut off. Note that it is not astonishing that in the case of very low SR values (worse than 50%) the PD SR value may lie below the image SRs (Fig. 9). Such strong wavefront errors violate the condition under which Eq. (5) is valid. Thus, we expect Eq. (5) to yield underestimated values.

The comparison of the SR values determined by the different methods turns out to be consistent. The longer the wavelength, the more the image and PD SR values approach each other. This shows that the influence of aberrations scales with the wavelength. In other words, the fact that we cut off at a certain Zernike number (i=15) has a greater impact at short wavelengths.

Figure 8: Comparison of SR versus wavelength in J and H band calculated directly and derived from PD results.

Figure 9: Comparison of SR versus wavelength in K band calculated directly and derived from PD results.

  
5.3 Focus adjustment

Having in mind the small estimated wavefront errors that we presented in the previous sections we become conscious of the required precision of the most trivial aberration we regard: the focus. It is striking that even in the focus determination we depend on the precision of PD calibration. This becomes evident when we look at the conventional procedure of focus tuning and regard the loss of SR caused by the detected aberrations.

To tune the focus of CONICA, the in-focus pinhole of the Zernike tool is imaged on the detector. Now, a focus curve is obtained by taking images at different axial position of the detector stage (see Sect. 3.1). The maximum of the SRs indicate the proper focus position of the stage. The maximum of the obtained focus curves for the different cameras can be located with an accuracy of about 50 nm rms. For this wavefront error, Eq. (5) yields a loss of SR of 2.5% at a wavelength of 2 $\mu$m and almost 10% at 1 $\mu$m. Thus, in particular in the J- and H-band, the inaccuracy of determining the focus only by moving the detector stage gives reason for a significant loss of SR. Furthermore the whole effort of fine-tuning for the remaining static wavefront aberrations becomes irrelevant when the remaining focus error is in the regime of the highest higher-level aberrations (Zernike number $i \ge 5$). Compare the aberrations for focus with astigmatism in Figs. 4 and 5. The only way to achieve a significant improvement of the wavefront error, and therefore of the SR after closed loop compensation, is to ensure that the residual focus deviation is corrected properly, too. This is guaranteed by following the procedure:

• Determination of the rough nominal focus position of the CONICA detector for each camera with one reference filter. The in-focus pinhole of the Zernike tool serves as a reference;
• Determination of the nominal focus for the whole instrument. The calibration point source in the NAOS entrance focal plane serves as a reference. The data points for the focus curve are obtained by moving the field selector in closed loop. This has to be done for every NAOS dichroic;
• The corresponding data base entries are updated by the nominal focus positions (CONICA internal and NAOS). The nominal focus deviations are included in the data base. They are compensated for by moving the field selector in the case of switching the NAOS dichroics and by moving the CONICA detector stage in the case of switching the camera objectives;

• Then the PD estimation reveals the residual focus error for each configuration, in particular for each filter. They are entered into the data base together with the higher order aberrations. For a certain instrument configuration the corresponding values are fetched automatically and delivered to the AO system. The DM corrects for the residual focus deviations.