Theoretical principle of APF-WFS
The APF-WFS method relies on the analysis of the Fourier properties of an AO-corrected image acquired after an asymmetric hard-stop mask has been placed in the pupil. The SCExAO instrument is equipped with two of these masks with the asymmetric feature at distinct position angles, so that every part of the instrumental pupil can be accounted for. A rotation wheel, located in a plane conjugated with the pupil of the instrument (see Fig. 3 of Jovanovic et al. 2015b) makes it possible to move the masks in and out of the beam as required by the observer.
 | Fig. 1 Images of the pupil (left) and the focal plane (right) acquired by the SCExAO internal science camera. In addition to the four Subaru telescope spiders, the thick arm visible on the left hand side of the pupil introduces the asymmetry required for the wavefront sensing technique. The thick dot visible in the bottom pupil quadrant is induced by a dead actuator on the DM. In the focal plane, the presence of this asymmetry results in an additional set of diffraction spikes along a direction that is perpendicular to that of the arm and a lumpier first diffraction ring. |
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Figure 1 shows an image of one of the asymmetric masks in the pupil and the point spread function (PSF) it produces. Combined with the use of the 2000-element deformable mirror (DM), this simple alteration of the pupil is a powerful tool used to control the low-order aberrations of the instrument PSF. All images featured in this paper were acquired using an H-band filter, centered on wavelength 1.65 μm and with an effective bandwidth of 0.3 μm. The pixel scale of the internal science camera is 12.1 mas per pixel which, for this wavelength, provides a sampling better than Nyquist.
It was shown that, in the low-aberration regime, typical of what is left over after a first layer of AO correction is applied, the phase Φmeasured in the Fourier transform of an image I and the instrumental pupil phase ϕ are linearly related. On the internal calibration source, unaffected by atmospheric turbulence, the Strehl of images used in this study (such as during the calibration) is typically of the order of 80%. On-sky, since the results featured here were acquired before the XAO loop is closed, the Strehl is significantly lower, of the order of 50%, which is a sufficiently good starting point for approximation to be valid.
The target phase information, associated with the spatial structures of the observed object Φ0, is also present in the Fourier plane and simply adds to this instrumental Fourier phase. When wavefront aberrations are low (below ~1 radian), the classical image-object convolution relation 
(1)can therefore be reformulated, if one works with the phase part of the Fourier transform of this image as follows: 
(2)where A is an operator that describes the way the pupil phase ϕ propagates into the Fourier-plane.
When observing a point source, for which Φ0 = 0 (or if the object is known), this relation can be inverted if one introduces an asymmetry in the pupil (Martinache 2013). A direct focal plane image, with only a small amount of additional diffraction generated by the pupil asymmetry (see Fig. 1), can therefore serve as a wavefront sensor.
To determine the structure of the operator A, a discrete representation of the instrument pupil needs to be built – including the asymmetric mask – following a regular grid with a step such that the sampling density is reasonably representative of the original pupil. Then the way this discrete model projects into equivalent interferometric baselines in the Fourier plane needs to be studied. The model currently used on SCExAO is provided in Fig. 2. It reduces the masked pupil to a 292-component vector that projects onto a 675-element vector in the Fourier domain. The phase transfer matrix A that establishes the mapping between the two spaces (Φ = A × ϕ) is calculated using the PYSCOsoftware, which is used for wavefront sensing, as well as for kernel-phase data analysis of diffraction limited images.
 | Fig. 2 Discrete model of the asymmetric pupil mask used for the calibration of the non-common path error in SCExAO. The pupil is discretized into a 292-element vector that projects onto a set of 675 equivalent interferometric baselines (or UV points) in the Fourier domain. The linear transformation that relates the wavefront to the phases measured in the Fourier transform of an image is entirely determined from this model. The presence of the asymmetry in the pupil ensures that an inverse relation for this phase-transfer matrix exists. |
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The presence of the asymmetry in the pupil ensures that an inverse relation for this phase transfer matrix exists, and can be used to infer the pupil-phase vector ϕ from the Fourier phase Φ, using the relation 
(3)where A+ is a Moore-Penrose pseudoinverse of the phase-transfer matrix A, computed after rejecting modes associated with low singular values. The geometry of the asymmetric feature of the mask used for this work is not the result of an optimization and simply follows the shape used in the concept paper of Martinache (2013). A smaller asymmetric feature would be expected to result in a lower sensitivy but a systematic study of the sensitivity impact of the geometry of the asymmetry has yet to be done. In the mean time, the curious reader can check the experimental work of Pope et al. (2014), which shows that, in the case of a segmented aperture, the technique remains effective, even with a minimum of asymmetry (a single segment of the aperture) and suddenly breaks, if no asymmetry is present at all, validating the mathematical model this approach relies on.
Integrating a real system
The case featured in Martinache (2013) was somewhat idealized. Working on monochromatic images, and with a perfect DM, it was able to exactly generate the wavefront correction determined by the analysis. To deploy this method on an actual closed-loop system is not as direct and requires us to take into account the actual properties of the DM, such as the response curve of the actuators, their influence functions, as well as a careful mapping of these DM actuators on the instrument pupil (Blain 2013). While possible, this type of model is very prone to errors and its maintenance is demanding because of small changes in the internal instrument alignment that are induced by temperature drifts or after a telescope slew.
AO systems in operation usually choose to rely on a more pragmatic approach that encapsulates this kind of model in a transparent manner: individual DM actuators or groups of actuators (pre-defined modes) are sequentially excited and the system response is recorded and assembled in a matrix. Filtering of the noisy modes before inversion (using SVD or similar procedures) leads to the obtention of a control matrix that can be used to directly multiply an input vector made of the wavefront sensor raw input data.
This pragmatic approach is the one that was retained for this implementation of APF-WFS. The control software was designed to sense and control eight low-order Zernike modes (Zernike 1934) that correspond to classical optical aberrations: focus, two terms of coma, astigmatism and trefoil, and spherical aberration. Figure 3 shows how these modes map on the discrete pupil model used to describe the instrument.