Method of wavefront phase retrieval from wavefront curvature sensing using membrane modes

Xuan Xie; Bocheng Wang; Bocheng Wang; Hairen Wang; Hairen Wang

doi:10.1364/AO.472772

Applied Optics
Vol. 61,
Issue 33,
pp. 10043-10048
(2022)
•https://doi.org/10.1364/AO.472772

Method of wavefront phase retrieval from wavefront curvature sensing using membrane modes

Xuan Xie, Bocheng Wang, and Hairen Wang

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Xuan Xie, Bocheng Wang, and Hairen Wang, "Method of wavefront phase retrieval from wavefront curvature sensing using membrane modes," Appl. Opt. 61, 10043-10048 (2022)

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Abstract

Wavefront phase retrieval is one of the most critical problems in adaptive optics. Here, phase retrieval by solving the transport of intensity equation using membrane vibration modes is proposed. Our study shows that the wavefront curvature sensing signal on the pupil can be expanded as a set of corresponding membrane vibration modes. The analytic expressions of the reconstructed phase are given. The coefficients of the functions are obtained by the integral over the pupil and boundary. Several representative Zernike circular and annular polynomials are respectively fitted by eigenfunctions and membrane modes in the absence of noise. In addition, wavefront recovery from noisy curvature data of the simulated atmospheric turbulence phase based on Zernike modes and Kolmogorov spectrum is demonstrated to verify the accuracy and robustness of the proposed method.

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Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Equations (28)

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$Z_{n}$	4	5	8	9	11	12
Eigenfunctions	2.30	1.15	4.98	9.80	8.82	7.40
Membrane modes	0	0	0.01	0	0.08	0.04

$Z_{n}^{*}$	4	6	7	9
Eigenfunctions	1.43	0.49	2.32	5.06
Membrane modes	0	0	0.00148	0.0185

$Z_{2}$	$r \cos θ$	$ξ_{2}$	−0.3504
$Z_{3}$	$r \sin θ$	$ξ_{3}$	0.0832
$Z_{4}$	$2 r^{2} - 1$	$ξ_{4}$	−0.0710141
$Z_{5}$	$r^{2} \sin 2 θ$	$ξ_{5}$	0.271403
$Z_{6}$	$r^{2} \cos 2 θ$	$ξ_{6}$	0.0205757
$Z_{7}$	$(3 r^{3} - 2 r) \sin θ$	$ξ_{7}$	−0.0565685
$Z_{8}$	$(3 r^{3} - 2 r) \cos θ$	$ξ_{8}$	0.124168
$Z_{9}$	$r^{3} \sin 3 θ$	$ξ_{9}$	−0.133502
$Z_{10}$	$r^{3} \cos 3 θ$	$ξ_{10}$	−0.0039598
$Z_{11}$	$6 r^{4} - 6 r^{2} + 1$	$ξ_{11}$	0.0225843
$Z_{12}$	$(4 r^{4} - 3 r^{2}) \cos 2 θ$	$ξ_{12}$	−0.000632456
$Z_{13}$	$(4 r^{4} - 3 r^{2}) \sin 2 θ$	$ξ_{13}$	−0.18025
$Z_{14}$	$r^{4} \cos 4 θ$	$ξ_{14}$	−0.188472
$Z_{15}$	$r^{4} \sin 4 θ$	$ξ_{15}$	0.0739973

$Z_{4}^{*}$	( $2 r^{2} - 1 - ε^{2}) / (1 - ε^{2})$
$Z_{6}^{*}$	$r^{2} \sin 2 θ / (1 + ε^{2} + ε^{4})^{1 / 2}$
$Z_{7}^{*}$	$[3 (1 + ε^{2}) r^{3} - 2 (1 + ε^{2} + ε^{4}) r] \cos θ / {(1 - ε^{2}) [(1 + ε^{2}) (1 + 4 ε^{2} + ε^{4})]^{1 / 2}}$
$Z_{9}^{*}$	$[6 r^{4} - 6 (1 + ε^{2}) r^{2} + 1 + 4 ε^{2} + ε^{4}] / (1 - ε^{2})^{2}$

$Z_{n}$	4	5	8	9	11	12
Eigenfunctions	2.30	1.15	4.98	9.80	8.82	7.40
Membrane modes	0	0	0.01	0	0.08	0.04

Abstract

Supplementary Material (1)

Data availability

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Equations (28)

Applied Optics