Subaperture translation estimation accuracy in transverse translation diversity phase retrieval

Dustin B. Moore; James R. Fienup

doi:10.1364/AO.55.002526

Applied Optics
Vol. 55,
Issue 10,
pp. 2526-2536
(2016)
•https://doi.org/10.1364/AO.55.002526

Subaperture translation estimation accuracy in transverse translation diversity phase retrieval

Dustin B. Moore and James R. Fienup

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Dustin B. Moore and James R. Fienup, "Subaperture translation estimation accuracy in transverse translation diversity phase retrieval," Appl. Opt. 55, 2526-2536 (2016)

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Table of Contents Category
- Instrumentation, Measurement, and Metrology
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About this Article
History
- Original Manuscript: October 12, 2015
- Revised Manuscript: January 17, 2016
- Manuscript Accepted: February 9, 2016
- Published: March 22, 2016

Abstract

For optical metrology by transverse translation diversity phase retrieval (or ptychography), information theoretic limits on the ability to estimate subaperture translation, essential for accurate metrology, are assessed as a function of the optical aberrations of the system being measured. Special attention is given to the case that an unknown linear phase aberration, or equivalent detector or target motion, is present that varies with each point spread function in the measured data.

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Equations (34)

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$j$	$n$	$Z_{j} (u, v)$	Aberration
1	0	1	Piston
2	1	$2 u$	$u$ linear
3	1	$2 v$	$v$ linear
4	2	$\sqrt{3} (2 u^{2} + 2 v^{2} - 1)$	Defocus
5	2	$2 \sqrt{6} u v$	Astig. 45°
6	2	$\sqrt{6} (u^{2} - v^{2})$	Astig. 0°
7	3	$\sqrt{8} (3 u^{2} v + 3 v^{3} - 2)$	$v$ coma
8	3	$\sqrt{8} (3 u^{3} + 3 u v^{2} - 2)$	$u$ coma
9	3	$\sqrt{8} (3 u^{2} v - v^{3})$	$v$ trefoil
10	3	$\sqrt{8} (u^{3} - 3 u v^{2})$	$u$ trefoil
11	4	$\sqrt{5} [6 {(u^{2} + v^{2})}^{2} - 6 (u^{2} + v^{2}) + 1]$	Spherical

$j^{'}$	$j = 2$	3	4	5	6	7	8
4	$2 \sqrt{3} q$	$2 \sqrt{3} r$
5	$\sqrt{6} r$	$\sqrt{6} q$
6	$\sqrt{6} q$	$- \sqrt{6} r$
7	$6 \sqrt{2} q r$	$3 \sqrt{2} q^{2} + 9 \sqrt{2} r^{2}$	$2 \sqrt{6} r$	$2 \sqrt{3} q$	$- 2 \sqrt{3} r$
8	$9 \sqrt{2} q^{2} + 3 \sqrt{2} r^{2}$	$6 \sqrt{2} q r$	$2 \sqrt{6} q$	$2 \sqrt{3} r$	$2 \sqrt{3} q$
9	$6 \sqrt{2} q r$	$3 \sqrt{2} (q^{2} - r^{2})$		$2 \sqrt{3} q$	$2 \sqrt{3} r$
10	$3 \sqrt{2} (q^{2} - r^{2})$	$- 6 \sqrt{2} q r$		$- 2 \sqrt{3} r$	$2 \sqrt{3} q$
11	$2 \sqrt{5} q + 12 \sqrt{5} (q^{3} + q r^{2})$	$2 \sqrt{5} r + 12 \sqrt{5} (q^{2} r + r^{3})$	$4 \sqrt{15} (q^{2} + r^{2})$	$4 \sqrt{30} q r$	$2 \sqrt{30} (q^{2} - r^{2})$	$2 \sqrt{10} r$	$2 \sqrt{10} q$

$j^{'}$	$j = 2$	3	4	5	6	8
4	$2 \sqrt{3}$
5		$\sqrt{6}$
6	$\sqrt{6}$
7				$2 \sqrt{3}$
8			$2 \sqrt{6}$		$2 \sqrt{3}$
9				$2 \sqrt{3}$
10					$2 \sqrt{3}$
11	$2 \sqrt{5}$					$2 \sqrt{10}$

$j^{'}$	$j = 2$	3	4	5	6	7
4		$2 \sqrt{3}$
5	$\sqrt{6}$
6		$- \sqrt{6}$
7			$2 \sqrt{6}$		$- 2 \sqrt{3}$
8				$2 \sqrt{3}$
9					$2 \sqrt{3}$
10				$- 2 \sqrt{3}$
11		$2 \sqrt{5}$				$2 \sqrt{10}$

Term	Mean Value ( ${rad}^{2}$ )
$σ_{2}^{2}$	$9.44 \times 10^{- 5}$
$σ_{3}^{2}$	$9.49 \times 10^{- 5}$
$σ_{4}^{2}$	$9.15 \times 10^{- 5}$
$σ_{5}^{2}$	$1.58 \times 10^{- 4}$
$σ_{6}^{2}$	$1.55 \times 10^{- 4}$
$σ_{7}^{2}$	$1.84 \times 10^{- 5}$
$σ_{8}^{2}$	$1.83 \times 10^{- 5}$

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Applied Optics