Generalized phase-shifting interferometry with arbitrary unknown phase steps for diffraction objects

L. Z. Cai; Q. Liu; X. L. Yang

doi:10.1364/OL.29.000183

Optics Letters
Vol. 29,
Issue 2,
pp. 183-185
(2004)
•https://doi.org/10.1364/OL.29.000183

Generalized phase-shifting interferometry with arbitrary unknown phase steps for diffraction objects

L. Z. Cai, Q. Liu, and X. L. Yang

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L. Z. Cai, Q. Liu, and X. L. Yang, "Generalized phase-shifting interferometry with arbitrary unknown phase steps for diffraction objects," Opt. Lett. 29, 183-185 (2004)

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Abstract

A general method of extracting the arbitrary unknown and unequal phase steps in phase-shift interferometry from interferograms recorded on the diffraction field of an object and then reconstructing the object wave front digitally with our derived formulas is proposed. The phase steps are first calculated based on the statistical nature of the diffraction field and are further improved by an iterative approach. This method is simple, highly accurate, and usable for any frame number $N$ ( $N \geq 3$ ) and for both smooth and diffusing objects, as is verified by a series of computer simulations.

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Equations (15)

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	Smooth Surface				Diffusing Surface
	$α_{1}$	$α_{2}$	$α_{3}$		$α_{1}$	$α_{2}$	$α_{3}$
Assumed value	1.3963	1.8326	1.6581	Assumed value	1.3963	1.8326	1.6581
Value extracted by Eqs. (9) and (13)	1.3976	1.8354	1.6625	Value extracted by Eqs. (9) and (13)	1.3978	1.8315	1.6606
Iteration				Iteration
4	1.3936	—	$\times$	1	1.3968	—	$\times$
7	1.3966	1.8324	$\times$	2	—	1.8325	$\times$
9	1.3964	—	$\times$	7	1.3963	—	$\times$
10	—	1.8325	$\times$	8	—	1.8326	$\times$
11	1.3963	—	$\times$	11	—	—	1.6576
12	—	1.8326	$\times$	16	—	—	1.6581
16	—	—	1.6585
20	—	—	1.6581

	Smooth Surface				Diffusing Surface
	$α_{1}$	$α_{2}$	$α_{3}$		$α_{1}$	$α_{2}$	$α_{3}$
Assumed value	1.3963	1.8326	1.6581	Assumed value	1.3963	1.8326	1.6581
Value extracted by Eqs. (9) and (13)	1.4202	1.8334	1.6817	Value extracted by Eqs. (9) and (13)	1.4012	1.8354	1.6629
Iteration				Iteration
2	1.4002	—	$\times$	1	1.3912	—	$\times$
7	1.3952	—	$\times$	3	1.3932	—	$\times$
8	1.3962	1.8324	$\times$	6	1.3962	1.8324	$\times$
10	1.3964	—	$\times$	8	—	1.8326	$\times$
11	1.3963	1.8325	$\times$	9	1.3963	—	$\times$
12	—	1.8326	$\times$	14	—	—	1.6579
14	—	—	1.6617	16	—	—	1.6581
18	—	—	1.6577
22	—	—	1.6581

	Smooth Surface				Diffusing Surface
	$α_{1}$	$α_{2}$	$α_{3}$		$α_{1}$	$α_{2}$	$α_{3}$
Assumed value	1.3963	1.8326	1.6581	Assumed value	1.3963	1.8326	1.6581
Value extracted by Eqs. (9) and (13)	1.3976	1.8354	1.6625	Value extracted by Eqs. (9) and (13)	1.3978	1.8315	1.6606
Iteration				Iteration
4	1.3936	—	$\times$	1	1.3968	—	$\times$
7	1.3966	1.8324	$\times$	2	—	1.8325	$\times$
9	1.3964	—	$\times$	7	1.3963	—	$\times$
10	—	1.8325	$\times$	8	—	1.8326	$\times$
11	1.3963	—	$\times$	11	—	—	1.6576
12	—	1.8326	$\times$	16	—	—	1.6581
16	—	—	1.6585
20	—	—	1.6581

	Smooth Surface				Diffusing Surface
	$α_{1}$	$α_{2}$	$α_{3}$		$α_{1}$	$α_{2}$	$α_{3}$
Assumed value	1.3963	1.8326	1.6581	Assumed value	1.3963	1.8326	1.6581
Value extracted by Eqs. (9) and (13)	1.4202	1.8334	1.6817	Value extracted by Eqs. (9) and (13)	1.4012	1.8354	1.6629
Iteration				Iteration
2	1.4002	—	$\times$	1	1.3912	—	$\times$
7	1.3952	—	$\times$	3	1.3932	—	$\times$
8	1.3962	1.8324	$\times$	6	1.3962	1.8324	$\times$
10	1.3964	—	$\times$	8	—	1.8326	$\times$
11	1.3963	1.8325	$\times$	9	1.3963	—	$\times$
12	—	1.8326	$\times$	14	—	—	1.6579
14	—	—	1.6617	16	—	—	1.6581
18	—	—	1.6577
22	—	—	1.6581

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