Compensation for correlated error in multilayer interferometer

Yangjin Kim; Kenichi Hibino; Naohiko Sugita; Mamoru Mitsuishi

doi:10.1364/AO.55.000171

Applied Optics
Vol. 55,
Issue 1,
pp. 171-177
(2016)
•https://doi.org/10.1364/AO.55.000171

Compensation for correlated error in multilayer interferometer

Yangjin Kim, Kenichi Hibino, Naohiko Sugita, and Mamoru Mitsuishi

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Yangjin Kim, Kenichi Hibino, Naohiko Sugita, and Mamoru Mitsuishi, "Compensation for correlated error in multilayer interferometer," Appl. Opt. 55, 171-177 (2016)

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Table of Contents Category
- Instrumentation, Measurement, and Metrology
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The topics in this list come from the Optics and Photonics Topics applied to this article.

About this Article
History
- Original Manuscript: September 17, 2015
- Revised Manuscript: November 26, 2015
- Manuscript Accepted: November 30, 2015
- Published: December 24, 2015

Abstract

Wavelength tuning interferometry is used to measure and estimate the surface shape of a sample. However, in multilayer interferometry (e.g., a lithium niobate [LNB] crystal wafer attached to a supporting plate), the correlated error between the higher harmonics and the phase-shift error causes considerable error in the calculated phase. In this study, the correlated errors calculated by various types of windowed phase-shifting algorithms are analyzed in connection with the characteristic polynomial theory and Fourier representation of the algorithms. The surface shape and optical thickness variation of the LNB wafer are measured simultaneously using the windowed phase-shifting algorithms. The results are compared in terms of the observed ripples and measurement repeatability. The experimental results show that the $4 N - 3$ algorithm is optimal and possesses the smallest repeatability error.

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Relative Frequency $(ν_{m} / ν_{1})$	Relative Amplitude	Ratio of Optical Path Difference $(D_{m} / n_{1} T_{1})$
1	0.816	$n_{1} T_{1} / n_{1} T_{1} = 1$
2	0.344	$n_{2} T_{2} / n_{1} T_{1} = 2$
3	0.783	$(n_{1} T_{1} + n_{2} T_{1}) / n_{1} T_{1} = 3$
5	1	$L / n_{1} T_{1} = 5$
6	0.439	$(L + n_{1} T_{1}) / n_{1} T_{1} = 6$
8	0.421	$(L + n_{1} T_{1} + n_{2} T_{2}) / n_{1} T_{1} = 8$

	Compensation Ability
Algorithm	Phase-Shift Error	Correlated Error
Synchronous Detection [10]	None	None
Surrel’s $2 N - 1$ [19]	$o (ε_{0})$	$o (A_{m} ε_{0})$
Hanayama’s $2 N - 1$ [21]	$o (ε_{0})$	$o (A_{m} ε_{0})$ (imperfect)
$3 N - 2$ [23]	$o (ε_{0}), o (ε_{1})$	$o (A_{m} ε_{0}), o (A_{m} ε_{1})$
$4 N - 3$ [13]	$o (ε_{0}), o (ε_{1}), o (ε_{2})$	$o (A_{m} ε_{0}), o (A_{m} ε_{1}), o (A_{m} ε_{2})$

	Repeatability Measurement Error (nm)
Algorithm Used	Optical Thickness Variation	Surface Shape
Synchronous Detection	47.231	109.345
Surrel’s $2 N - 1$	11.456	28.981
Hanayama’s $2 N - 1$	10.697	29.978
$3 N - 2$	6.078	12.069
$4 N - 3$	2.190	3.442

Relative Frequency $(ν_{m} / ν_{1})$	Relative Amplitude	Ratio of Optical Path Difference $(D_{m} / n_{1} T_{1})$
1	0.816	$n_{1} T_{1} / n_{1} T_{1} = 1$
2	0.344	$n_{2} T_{2} / n_{1} T_{1} = 2$
3	0.783	$(n_{1} T_{1} + n_{2} T_{1}) / n_{1} T_{1} = 3$
5	1	$L / n_{1} T_{1} = 5$
6	0.439	$(L + n_{1} T_{1}) / n_{1} T_{1} = 6$
8	0.421	$(L + n_{1} T_{1} + n_{2} T_{2}) / n_{1} T_{1} = 8$

	Compensation Ability
Algorithm	Phase-Shift Error	Correlated Error
Synchronous Detection [10]	None	None
Surrel’s $2 N - 1$ [19]	$o (ε_{0})$	$o (A_{m} ε_{0})$
Hanayama’s $2 N - 1$ [21]	$o (ε_{0})$	$o (A_{m} ε_{0})$ (imperfect)
$3 N - 2$ [23]	$o (ε_{0}), o (ε_{1})$	$o (A_{m} ε_{0}), o (A_{m} ε_{1})$
$4 N - 3$ [13]	$o (ε_{0}), o (ε_{1}), o (ε_{2})$	$o (A_{m} ε_{0}), o (A_{m} ε_{1}), o (A_{m} ε_{2})$

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Equations (13)

Applied Optics