To better appreciate the impact of the phase-transfer model, one can look at the effect of the projection in the Fourier plane and then back in the pupil plane using the linear relations of Eqs. (2) and (3). The original phase ϕ becomes 
(4)Depending on the number of modes kept in the determination of the pseudo-inverse A + of the phase-tranfer matrix, the reconstruction goes from perfect (if all modes are preserved) to very partial (if few modes are preserved). The modes discarded correspond to low singular values which, for a given level of signal-to-noise in actual data, would result in amplified noise. On SCExAO, for the control of these eight low-order modes, 150 out of the 291 available modes are maintained in the computation of the pseudo inverse. Under these conditions (see Fig. 4), the reconstruction appears satisfactory, and confirms that the technique can indeed be used to control low-order modes, assuming that the linear model holds (and that aberrations are small).
| Fig. 4 The eight Zernike modes reconstructed by the linear model when 150 out of the 291 modes are kept in the computation of the pseudo-inverse of the phase transfer matrix A+. |
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Calibration
The calibration procedure for this implementation of the APF-WFS follows the linear control framework. After the asymmetric mask has been inserted, one image labeled as reference is acquired, followed by a sequence of images acquired after a Zernike mode of appropriate amplitude has been applied to the DM. Figure 4 features one of these calibration data-set, which was acquired with the focal camera of SCExAO on its internal calibration source (super-continuum laser) using a standard H -band filter for a 30 nm RMS deformation of the DM. We note that this displacement actually translates into a 60 nm wavefront amplitude modulation (the DM being a reflective system).
| Fig. 5 Calibration data for the APF-WFS acquired by the SCExAO science camera. Top left: the reference PSF, acquired with the system in its starting state. From left to right and top to bottom: the PSF after the corresponding Zernike mode has been applied. A non-linear scale is used to better show the impact of a 30 nm RMS DM modulation. |
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Each image is recentered and windowed by a super-gaussian function that filters out high-spatial frequencies, and reduces the impact of detector readout noise. It is then Fourier-transformed and the Fourier-phase is extracted according to the sampling model featured in the right hand side of Fig. 2 to populate a vector Ψ.
To each Zernike mode, a Fourier-phase vector Φ i can be associated after subtracting the phase Ψref measured in the initial or reference state: 
(5)The wavefront associated with this Fourier-phase signature can be recovered using the pseudo-inverse A + previously computed and applying Eq. (3). This wavefront in radians can, in turn, be converted into a DM displacement map (in microns) after being multiplied by the proper λ / 4 π scaling factor; where λ is the wavelength expressed in microns and 4 π contains the ×2 factor caused by the reflection. Figure6 features an example of experimentally recovered modes. One will observe that the reconstruction from the Fourier analysis of actual images appears visually satisfactory. Differences in the reconstruction with the modes plotted in Fig. 4 can be attributed to imperfections of the pupil discretization model (see Fig. 4 of Martinache 2013), combined with practical subtleties like the fact that the dynamical range on the camera is limited and that classical noises, such as photon and readout, do apply.
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| Fig. 6 Experimentally recovered Zernike modes. Save for the spherical aberration, it can be seen that the modes extracted from the analysis of the images of Fig. 5 reproduce the features expected after looking at the theoretically reconstructed modes presented in Fig. 4. |
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| Fig. 7 Comparative quality of the modal reconstruction: for each mode, the local value of the DM displacement (in nm) for the theoretical Zernike mode of predefined amplitude (here 30 nm, labeled as the excitation) is plotted against its experimental measurement. |
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To complete this description of the general aspect of the modes with a quantitative estimate of the quality of the reconstruction, Fig. 7directly compares the experimental reconstruction E to the theoretical Zernike modulation T, by plotting the local deduced DM displacement against its predicted value. One can confirm that all modes are not equally reproduced by the analysis and that, for this experimental setup, the sensor is most sensitive to astigmatism (for which the correlation coefficient is the strongest) and not particularly suited to the sensing of spherical aberration. Table 1 accompanies this figure and provides the value of the Pearson product-moment correlation coefficient for all modes: 
(6)
Table 1
Pearson product-moment correlation coefficient of the experimentally reconstructed mode E with their theoretical counterpart T.
Table 1 also shows that, with a correlation coefficient ~0.5, spherical aberration is significantly less well reconstructed than the other modes that exhibit a correlation coefficient >0.8. The specificity of the response to spherical aberration is, however, not an intrinsic limit of the sensing approach and can, in fact, simply be explained by the 2D geometry of this aberration and how it fits within the footprint of the Subaru Telescope and its large (~30%) central obstruction. By looking for instance at Fig. 3, we can see that the donut shape of the spherical aberration results in a pretty uniform distribution of the phase, which only varies near the inner and outer edges of the pupil. The basis of Zernike polynomials is defined for a complete circular aperture: quoted amplitudes correspond to a given wavefront RMS over the entire circular aperture. For the spherical aberration, the presence of the central obstruction naturally filters out a lot of the effect of the spherical aberration. This is further confirmed after a close examination of scatter plots of Fig. 7: whereas all Zernike stimuli (along the horizontal axis of the plots) have the same amplitude, we can see that the resulting range of local DM displacement (corresponding to the horizontal spread of the data points) is appreciably shorter for the spherical aberration than it is for the other modes.
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These experimentally obtained pupil-phase modes ϕi are stored in a 8 × 291 matrix Z, referred to as the response matrix. In practice, unless the DM registration were to change in a dramatic manner, the calibration is quite robust: a response matrix acquired using the internal calibration source can be very well used during on-sky observations if the filter remains unchanged, and if the change of exposure time does not result in a saturated PSF core (see the discussion in Sect. 4).
On SCExAO, the acquisition of this response matrix only takes a few seconds, so it can easily be repeated if neessary after acquisition of a new target. In practice, it seems a response matrix that is acquired using the stable internal calibration source provides the best results.
Closed-loop operation
Just as during the calibration, focal-plane images acquired on-sky with the asymmetric mask are dark-subtracted, recentered, and windowed by a super-Gaussian function before being Fourier-transformed. After extraction of the Fourier phase, a wavefront is produced and directly projected onto the basis of modes (without subtracting the reference), to find the coefficients associated with all eight Zernike components. If the current wavefront sensor signal is ϕ, the instant Zernike coefficients (α) are the solution of Z α = ϕ. The least square solution 
of this system is the solution to 
(7)The solution 
to this well-behaved system of equations is used as an input for a control loop algorithm. The loop in operation on SCExAO implements a simple proportional controller, with a gain common to all Zernike modes with value contained between 0.05 and 0.3, depending on the overall stability of the wavefront provided by the upstream AO. When looking at the internal calibration source, we can reliably use the highest gain. Since, for now, it is only used for a very short time (typically ~15 s), at the time of target acquisition to flatten the static component of the wavefront, the current implementation of the algorithm proves satisfactory. Once the non-common path error is accounted for, the asymmetric mask is taken out of the optical path and the system is ready for observation using the full pupil of the telescope.
Performance
On-sky demonstration
| Fig. 8 Illustration of the impact of the APF-WFS. Left: 0.5 ms PSF acquired by SCExAOs internal science camera after the upstream AO loop has been closed. Right: identical exposure acquired 30 s after the APF-WFS loop has been closed. Despite residual imperfections owing to dynamic changes, the PSF quality is obviously improved. |
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The technique was successfully deployed and proved to be effective at reducing the non-common path error during on-sky observations behind Subaru Telescopes facility AO system AO188 (Minowa et al. 2010). Figure 8 illustrates the impact of the approach, with two 500 μ s exposures of the target (Altair) acquired by SCExAOs internal science camera on UT 2015-10-30.
The first image shows the PSF after the AO188 loop has been closed on the target: although it features a well-defined diffraction core, the PSF clearly exhibits some static aberrations that can be attributed to the non-common path error between AO188 and SCExAOs focal plane. The second image shows the PSF about 30 s after the APF-WFS loop has been closed. The gain in Strehl is low (of the order of 5%), but the PSF is improved at where it matters most for high contrast imaging and no longer features any obvious low-order aberration signature. Residual inhomogeneity of the first diffraction ring can be attributed to a combination of instantaneous AO residuals combined with the effect of the asymmetric arm.
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SCExAOs internal science detector is a fast but low-sensitivity detector that can acquire images at up to 170 Hz full-frame rate whose specifications are given by Jovanovic et al. (2015b). APF-WFS seems to exhibit sufficient sensitivity to be used in a fast closed-loop that could very well track low-order aberrations with frequencies up to a tenth of the camera frame rate.
At the moment, the goal of the loop is to calibrate the quasi-static non-common path error. The control software keeps a rolling average of the 20 last wavefront estimations, and corrects for the average of these estimations at each iteration, thus filtering vibration-induced fast varying component. Combined with the acquisition, the (non-optimized) computation of the wavefront makes the loop run at a frequency of~8 Hz.
Cross-talk
Zernike polynomials (Zernike 1934) form a convenient basis to describe a wavefront within a circular aperture: designed to form an orthonormal basis, the first terms of the series happen to correspond to classical monochromatic optical aberrations like focus, astigmatism or coma. As previously seen, the presence of a central obstruction in the pupil makes this basis no longer perferctly orthogonal. Substitutes have been proposed (Mahajan 1981) to accomodate for the presence of this rather common feature of telescopes, but the asymmetric arm required for the wavefront sensing (see Sect. 1) would also require an adaptation.
| Fig. 9 Control modes inner product matrices characterizing the orthogonality properties of the implementation of the APF-WFS described in this paper on SCExAO. From left to right: 1. the mostly perfectly diagonal case of the Zernike polynomials basis; 2. the seemingly identical inner product matrix for the theoretical modes after reconstrucion by the linear model and filtering by the SVD; and 3. the inner product matrix for the experimentally reconstructed modes. The overall diagonal allure of the latter characterizes the sensor as suited to the control of the low-order Zernike modes. Numerical values for the experimental inner products are provided in Table 2. |
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Instead of trying to specify a new orthogonal basis perfectly adapted to our case, we have judged it more appropriate to stick to the conventional Zernike basis and verify a posteriori how orthogonal the different modes actually are. Figure 9 does this by plotting the 8 × 8matrix of inner products between the eight control modes (Z T Z) for three cases: the input Zernike polynomials (given in Fig. 3), the theoretical reconstruction of the linear model with 150 out of the 291 eigen modes kept in the phase transfer matrix inversion (given in Fig.4), and the experimentally acquired modes (given in Fig. 6). An orthogonal basis will result in a perfectly diagonal inner-product matrix, whereas non-orthogonality would become manifest with strong non-diagonal components.
One can therefore observe, looking at the left hand side panel of Fig. 9, that the Zernike modes form a satisfactory, nearly orthonormal basis, with a mostly uniform diagonal and with a limited amount of cross terms standing out (except in the case of the 16% cross-correlation between focus (Z4) and spherical (Z11). For the sake of consistency with the rest of the data presented in the paper, Fig. 9 also shows in its central panel, that the modes reconstructed by the linear model reproduce most of these features, although one can observe~20% degradation of the relative strength of the focus signal. What we observe here is the effect of the filtering of low singular values in the construction of the pseudo-inverse A + as used in Eq. (4). With 150 out of the possible 291 modes kept in the construction of the pseudo-inverse, the inner product matrix for the experimentally recovered modes is also mostly diagonal.
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Table 2 provides the numerical values for the experimental inner products, also graphically represented in the right hand side panel of Fig.9. Although some of the cross terms are non-neglibible, the terms along the diagonal still dominate, indicating that a control loop that relies on this calibration dataset will reliably converge toward a state that will cancel the non-common path aberration.
Table 2
Numerical values for the experimental modes inner-product.
Range of linear response
As mentioned in Sect. 2.1, the APF-WFS relies on the assumption that an upstream AO correction is provided. The system is expected to deal with with small residual wavefront errors, and the calibration procedure described above, typically employs DM modulation amplitudes of ~30−50 nm. To determine the amount of aberration the technique is able to deal with, we performed a systematic exploration of the response of the sensor to stimuli of variable amplitude. The instantaneous response of the sensor is projected onto the basis of modes following the procedure outlined in Sect. 2.5. Figure 10 summarizes the results of this systematic exploration of the response of the sensor, over a 150 nm range of DM modulation amplitude.
| Fig. 10 Experimental response of the APF-WFS obtained on the SCExAO internal (super-continuum) source in the H -band. Each plot features (on the vertical axis) the reponse of the sensor to a Zernike mode of RMS amplitude that varies over a 150 nm range (units for both axis are in nm). While nearly linear over the entire range for most modes, the sensor only exhibits a significant non-linear behavior for the coma 2, and the two trefoil modes when the DM Zernike amplitude is larger than 80−100 nm. We note that this limit is on the DM surface, which must be doubled if refering to aberrations on the wavefront. |
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Although not perfectly linear, for Z4 (focus), Z5, Z6 (astigmatism), Z7 (coma 1), and Z11 (spherical), the response remains monotonic over the entire 150 nm range. For Z8 (coma 2), Z9 and Z10 (trefoil 1 and 2), the response is only monotonic over the 80 nm modulation range beyond which the sensor cannot be used reliably. The drastic difference of response between Z7 and Z8, which corresponds with the same type of aberration (i.e. coma), can be explained by the azimuth of the asymmetric arm in the pupil stop, oriented such that it almost entirely masks out one of the two antisymmetric bumps that are characteristic of this aberration (see for instance the top left panel of Fig. 3).
A strong non-linearity of the response is experienced when the pupil-phase peak-to-valley (P2V) wavefront becomes larger than 2 π (which results in a phase wrap). The presence of the asymmetric stop at its current azimuth essentially divides the P2V by a factor of two in the case of Z7 (i.e. coma 1), hence making the sensor able to handle twice as much coma along the horizontal axis than along the vertical axis. We note that the same effect (to a lesser extent) can also be observed when comparing Z9 and Z10. We can nevertheless conclude that, in the H -band under normal operating conditions, the sensor is able to operate linearly as long as the RMS error on either mode is less than 200 nm on the wavefront.
