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Appendix A: The projection operators – A history of POAM in adaptive optics
Fugate et al. (1991) created AO, and in doing so, demonstrated a means to compensate for atmospheric turbulence through the use of a wavefront sensor and a deformable mirror. The wavefront sensor measures phase and, through processing, returns the deformable mirror commands that compensate for the phase disturbance. Since the measurement contains noise, in order to ensure optimal performance, the measurements must be filtered; in AO, this process is called reconstruction of the phase, and the matrix that does the reconstruction is called the reconstructor.
In retrospect, the AO reconstructor is the keystone that allowed for creating the projection operator onto ℋ_{Rot}. Sections A.1–A.5 illuminate how this is so.
Appendix A.1: Minimization of noise – The first reconstructor
Following Fugate (2001), assume the sensor is a ShackHartmann wavefront sensor; this sensor discretizes the field and returns phase gradients. Let each discretely measured element be called a subaperture. For each subaperture, we wish to estimate the deformable mirror command required to correct the phase aberration in that subaperture. In this light, let a be the deformable mirror actuator commands for all subapertures (i.e. the optical path difference (OPD) required to remove phase in each subaperture), and let g be the phase gradients for all subapertures.
Adaptive optics fundamentally assumes that the atmosphere is a smooth disturbance that exists in the pupil of the telescope, and further that the AO system is optically conjugate to the turbulence. Under these assumptions (and no noise), g is continuous and (A.1)where Γ is a matrix that converts actuator commands (i.e. phase) into phase gradients. We need to solve for the deformable mirror commands, a, given the measurements g, and this must be done in the presence of noise. (Note that under the stated assumptions, the discontinuous part of g is considered noise.)
Letting â be the actuator commands in the presence of noise, the minimum error solution in the least mean square sense is given by the minimization of g − Γâ. The solution is given by (A.2)where Γ^{†} is the pseudo inverse. The left hand side of Eq. (A.2) is the AO compensation which returns the optimal phase for imaging.
Equation (A.2) is the reconstruction process and Γ^{†} is the AO reconstructor.
Appendix A.2: Branch points in AO
Fried (1998) when studying the branch point problem in AO, considered both the gradients associated with â, as well as, the remainder of the gradients. In doing so, he illustrated that the total phase, ϕ is uniquely comprised of two phases, ϕ_{LMS} and ϕ_{Rot}, with ϕ_{Rot} caused by branch points, and further that the gradients of these phases, ∇ϕ_{LMS} and ∇ϕ_{Rot}, are orthogonal i.e. ϕ = ϕ_{LMS} + ϕ_{Rot} with In the terms of the previous subsection, ; also, the gradients ∇ϕ_{Rot} are considered noise by the reconstructor, and hence, are rejected. This fact led Fried (1998) in his seminal paper to call ϕ_{Rot} the “hidden phase”.
In the evolution of theory, the phase which was originally viewed as a least mean square solution to a noise problem, ϕ_{LMS}, now also defines another phase, ϕ_{Rot}, which is due to branch points. At this point in conventional wisdom, since branch points were seen to randomly appear and disappear in all experimental data, they were still considered a noise problem.
Appendix A.3: Branch points as enduring features of the traveling wave
Sanchez & Oesch (2009) and Oesch et al. (2010) later demonstrated that a component of ϕ_{Rot} was not noise at all, but an enduring feature of the traveling wave. That branch points seemingly randomly appeared and disappeared in experimental data was shown to be a measurement problem. Note for clarity, in experimental data, both ϕ_{LMS} and ϕ_{Rot} contain noise, and while noise must be dealt with when analyzing data, it is not our concern here.
At this point in the evolution of theory, both ϕ_{LMS} and ϕ_{Rot} are physical features of the traveling wave, caused directly and indirectly by turbulence, respectively.
Appendix A.4: The two orthogonal Hilbert spaces and the projection operators
Consider the gradients g_{â} such that (A.5)The operator ΓΓ^{†} is a projection of the total gradients, g, onto g_{â}. Denote the remainder of the gradients . Then, (A.6)where is the identity operator. is a projection operator of g onto . Brennan (2007 priv. comm. TR1648) demonstrated that the gradients g_{â} and define two orthogonal Hilbert spaces, ℋ_{LMS} and ℋ_{Rot}, such that ℋ = ℋ_{LMS} ⊕ ℋ_{Rot}, and he derived a sparse basis for ℋ_{Rot}. In this light, ΓΓ^{†} is a projection operator (called earlier in the text) that selects the gradients in ℋ_{LMS}. Similarly, is a projection operator onto ℋ_{Rot}. Note, since SH WFSs discretize the phase, and are matrices.
In the notation of Fried (1998) and writing the phase in units of OPD, the gradients due to branch points define a Hilbert space ℋ_{Rot}. Since the gradients are random variables, knowing that the gradients lie in Hilbert spaces, gives a ready means, through the norm of the Hilbert space, to estimate their size.
At this point in the evolution of theory, a means was in hand to estimate the relative quantity of the stochastically generated nonzero elements in ℋ_{LMS} and ℋ_{Rot} (this was presented as the ratio η earlier in the text).
Appendix A.5: POAM in the traveling wave
To complete the history, Sanchez & Oesch (2011a,b) later established that implies that the traveling wave is carrying nonzero POAM, and realized a short time later, it would be possible to show that a good fraction, perhaps most, of the photons in the universe carry nonzero POAM.
Appendix B: Plots of the on sky data
Fig. B.1
49 Ceti. Left column: [[H]]. Right column: η. 

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Fig. B.2
HR 1529. Left column: [[H]]. Right column: η. 

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Fig. B.3
HR 1577. Left column: [[H]]. Right column: η. 

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Fig. B.4
HR 1784. Left column: [[H]]. Right column: η. 

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Fig. B.5
HR 1895. Left column: [[H]]. Right column: η. 

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© ESO, 2013