Single-plane multiple speckle pattern phase retrieval using a deformable mirror

Percival F. Almoro; Jesper Glückstad; Steen G. Hanson

doi:10.1364/OE.18.019304

Optics Express
Vol. 18,
Issue 18,
pp. 19304-19313
(2010)
•https://doi.org/10.1364/OE.18.019304

Single-plane multiple speckle pattern phase retrieval using a deformable mirror

Percival F. Almoro, Jesper Glückstad, and Steen G. Hanson

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Percival F. Almoro, Jesper Glückstad, and Steen G. Hanson, "Single-plane multiple speckle pattern phase retrieval using a deformable mirror," Opt. Express 18, 19304-19313 (2010)

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Abstract

A design for a single-plane multiple speckle pattern phase retrieval technique using a deformable mirror (DM) is analyzed within the formalism of complex ABCD-matrices, facilitating its use in conjunction with dynamic wavefronts. The variable focal length DM positioned at a Fourier plane of a lens comprises the adaptive optical (AO) system that replaces the time-consuming axial displacements in the conventional free-space multiple plane setup. Compared with a spatial light modulator, a DM has a smooth continuous surface which avoids pixelation, pixel cross-talk and non-planarity issues. The calculated distances for the proposed AO-system are evaluated experimentally using the conventional free-space phase retrieval setup. Two distance ranges are investigated depending on whether the measurement planes satisfy the Nyquist detector sampling condition or not. It is shown numerically and experimentally that speckle patterns measured at the non-Nyquist range still yield good reconstructions. A DM with a surface height of 25 microns and an aperture diameter of 5.2 mm may be used to reconstruct spherical phase patterns with 50-micron fringe spacing.

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Fig. 1 (a) Setup for the conventional free-space multiple plane phase retrieval system. (b) Proposed setup based on an Adaptive Optical (AO) system using a deformable mirror.

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Fig. 2 Effects of recording the speckle field patterns at two measurement regions relative to the Nyquist sampling distance.

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Fig. 3 Numerical simulations. (a) Initial phase at object plane. (b) and (c) are the reconstructed phase maps using intensity patterns recorded at Nyquist and non-Nyquist distances, respectively.

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Fig. 4 Experimental results using conventional free-space setup. (a) and (b) are the reconstructed phase maps using speckle patterns recorded at Nyquist and non-Nyquist distances, respectively. The result in (b) demonstrates successful phase reconstruction of the test wavefront despite violations of the Nyquist condition during the speckle recording. This gives flexibility to the requirement of the surface height in the DM’s in the proposed AO-system.

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Tables (1)

Table 1 Parameters in the Green’s function

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Equations (9)

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U_{o u t, x, y} (r) = \int d r' U_{i n, x, y} (r') G (r', r),

G (r', r) = - \frac{i k}{2 π B} \exp [- \frac{i k}{2 B} (A r'^{2} - 2 r' r + D r^{2})] .

M_{f r e e s p a c e} = {\begin{matrix} 1 - \frac{2 i z}{k s_{0}^{2}} & z \\ - \frac{2 i z}{k s_{0}^{2}} & 1 \end{matrix}}

M_{A O} = (\begin{matrix} - \frac{f_{2}}{f_{1}} + \frac{- 4 f_{1} f_{2}^{2} d f + 2 i f_{1} (d f d z + f_{2}^{2}) k s_{1}^{2}}{f_{2} k^{2} s_{0}^{2} s_{1}^{2} d f} & - \frac{d z f_{1}}{f_{2}} + f_{1} f_{2} (\frac{1}{d f} - \frac{2 i}{k s_{1}^{2}}) \\ \frac{2 i f_{}}{f_{2} k s_{0}^{2}} & - \frac{f_{1}}{f_{2}} \end{matrix})

G (r', r) \propto \exp [- \frac{i k Δ f}{2 (Δ f Δ z + f_{2}^{2})} {(\frac{f_{2}}{f_{1}} r' - r)}^{2}] .

G (r', r) \propto \exp [- \frac{i k}{2 L} {(r' - r)}^{2}] .

L_{e f f} = \frac{Δ f Δ z + f_{2}^{2}}{Δ f} .

Δ L_{e f f} \equiv \frac{Δ f Δ z + f_{2}^{2}}{Δ f} - {(\frac{Δ f Δ z + f_{2}^{2}}{Δ f})}_{for Δ f \to \infty} = \frac{f_{2}^{2}}{Δ f} .

L_{e f f} = 2 h f^{2} / {(D / 2)}^{2}

Element	AO system	Free space
A/B	$\frac{f_{2}^{2}}{f_{1}^{2}} \frac{Δ f}{Δ f Δ z + f_{2}^{2}}$	$\frac{1}{z}$
1/B	$\frac{f_{2}^{}}{f_{1}^{}} \frac{Δ f}{Δ f Δ z + f_{2}^{2}}$	$\frac{1}{z}$
D/B	$\frac{Δ f}{Δ f Δ z + f_{2}^{2}}$	$\frac{1}{z}$

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Equations (9)

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