Abstract

Wavelength-tuning interferometry is commonly employed to measure the optical homogeneity of parallel plates. However, the nonlinearity of phase shifts caused by wavelength tuning errors and environmental vibration leads to a spatially uniform error in the calculated phase distribution. Herein, a wavelength-tuning interferometry method based on nonuniform fast Fourier transform (WTI-NUFFT) was developed, which solves the spectral aliasing resulting from the spatially uniform error. The characteristics of the WTI-NUFFT method were estimated through comparison with the FFT method. Both the simulated and experimental results showed that the WTI-NUFFT method can improve the accuracy of the optical homogeneity measurement of parallel plates.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
  18. L. L. Deck, “Multiple-surface phase-shifting interferometry,” Proc. SPIE 4451, 424–431 (2001).
    [Crossref]
  19. Y. Kim, K. Hibino, N. Sugita, and M. Mitsuishi, “Measurement of absolute optical thickness of mask glass by wavelength-tuning Fourier analysis,” Opt. Lett. 40(13), 3169–3172 (2015).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  21. R. H. Guo, J. X. Li, R. H. Zhu and L. Chen. “Wavelength-Tuned Phase-Shifting Calibration Based on the Fourier Transform in Time Domain,” 2009 IEEE International Conference on Service Operations, Logistics and Informatics. Chicago, U.S.A., SOLI 2009, 371–375(2009).

2018 (1)

Z. Rui, C. Lei, Z. Wen-hua, M. Shi, Z. Dong-hui, and S. Qin-yuan, “Measuring optical homogeneity of parallel plates based on simultaneous phase-shifting by lateral displacement of point sources,” Wuli Xuebao 47(1), 0112002 (2018).

2016 (2)

T. Sugimoto, “Nondestructive measurement of the refractive index distribution of a glass molded lens by two-wavelength wavefronts,” Appl. Opt. 55(28), 8116–8125 (2016).
[Crossref] [PubMed]

O. Matoušek, V. Lédl, P. Psota, and P. Vojtíšek, “Methods for refractive-index homogeneity calculation using Fourier-transform phase-shifting interferometry,” Proc. SPIE 10151, 101510Y (2016).
[Crossref]

2015 (2)

2014 (1)

J. Li, Y. R. Wang, X. F. Meng, X. L. Yang, and Q. P. Wang, “Simultaneous measurement of optical inhomogeneity and thickness variation by using dual-wavelength phase-shifting photorefractive holographic interferometry,” Opt. Laser Technol. 56(1), 241–246 (2014).
[Crossref]

2013 (3)

J J. Park, J. Jin, J. Wan Kim, and J.-A. Kim, “Measurement of thickness profile and refractive index variation of a silicon wafer using the optical comb of a femtosecond pulse laser,” Opt. Commun. 305, 170–174 (2013).
[Crossref]

K. Mantel and J. Schwider, “Interferometric homogeneity test using adaptive frequency comb illumination,” Appl. Opt. 52(9), 1897–1912 (2013).
[Crossref] [PubMed]

L. Zhu, “Inhomogeneity measurement at oblique incidence by phase measuring interferometers,” Opt. Express 21(18), 20730–20737 (2013).
[Crossref] [PubMed]

2012 (1)

2010 (1)

2005 (1)

2003 (1)

2002 (1)

K. Hibino, B. F. Oreb, and P. S. Fairman, “Improved algorithms for wavelength scanning interferometry: application to the simultaneous measurement of surface topography and optical thickness variation in a transparent parallel plate,” Proc. SPIE 4777, 212–219 (2002).
[Crossref]

2001 (1)

L. L. Deck, “Multiple-surface phase-shifting interferometry,” Proc. SPIE 4451, 424–431 (2001).
[Crossref]

2000 (1)

1996 (1)

1991 (2)

C. Ai and J. C. Wyant, “Measurement of the inhomogeneity of a window,” J. Fluids Eng. 30(9), 602–610 (1991).

D. Tentori, “Homogeneity testing of optical glass by holographic interferometry,” Appl. Opt. 30(7), 752–755 (1991).
[Crossref] [PubMed]

1969 (1)

Ai, C.

C. Ai and J. C. Wyant, “Measurement of the inhomogeneity of a window,” J. Fluids Eng. 30(9), 602–610 (1991).

Brock, N.

Chen, L.

de Groot, P.

de Groot, P. J.

Deck, L. L.

L. L. Deck, “Fourier-transform phase-shifting interferometry,” Appl. Opt. 42(13), 2354–2365 (2003).
[Crossref] [PubMed]

L. L. Deck, “Multiple-surface phase-shifting interferometry,” Proc. SPIE 4451, 424–431 (2001).
[Crossref]

Dong-hui, Z.

Z. Rui, C. Lei, Z. Wen-hua, M. Shi, Z. Dong-hui, and S. Qin-yuan, “Measuring optical homogeneity of parallel plates based on simultaneous phase-shifting by lateral displacement of point sources,” Wuli Xuebao 47(1), 0112002 (2018).

Eom, T. B.

Fairman, P. S.

K. Hibino, B. F. Oreb, and P. S. Fairman, “Improved algorithms for wavelength scanning interferometry: application to the simultaneous measurement of surface topography and optical thickness variation in a transparent parallel plate,” Proc. SPIE 4777, 212–219 (2002).
[Crossref]

Griesmann, U.

Hayes, J.

Hibino, K.

Jin, J.

J J. Park, J. Jin, J. Wan Kim, and J.-A. Kim, “Measurement of thickness profile and refractive index variation of a silicon wafer using the optical comb of a femtosecond pulse laser,” Opt. Commun. 305, 170–174 (2013).
[Crossref]

J. Jin, J. W. Kim, C. S. Kang, J. A. Kim, and T. B. Eom, “Thickness and refractive index measurement of a silicon wafer based on an optical comb,” Opt. Express 18(17), 18339–18346 (2010).
[Crossref] [PubMed]

Kang, C. S.

Kim, J. A.

Kim, J. W.

Kim, J.-A.

J J. Park, J. Jin, J. Wan Kim, and J.-A. Kim, “Measurement of thickness profile and refractive index variation of a silicon wafer using the optical comb of a femtosecond pulse laser,” Opt. Commun. 305, 170–174 (2013).
[Crossref]

Kim, Y.

Langenbeck, P.

Lédl, V.

O. Matoušek, V. Lédl, P. Psota, and P. Vojtíšek, “Methods for refractive-index homogeneity calculation using Fourier-transform phase-shifting interferometry,” Proc. SPIE 10151, 101510Y (2016).
[Crossref]

Lei, C.

Z. Rui, C. Lei, Z. Wen-hua, M. Shi, Z. Dong-hui, and S. Qin-yuan, “Measuring optical homogeneity of parallel plates based on simultaneous phase-shifting by lateral displacement of point sources,” Wuli Xuebao 47(1), 0112002 (2018).

Li, J.

J. Li, Y. R. Wang, X. F. Meng, X. L. Yang, and Q. P. Wang, “Simultaneous measurement of optical inhomogeneity and thickness variation by using dual-wavelength phase-shifting photorefractive holographic interferometry,” Opt. Laser Technol. 56(1), 241–246 (2014).
[Crossref]

Mantel, K.

Matoušek, O.

O. Matoušek, V. Lédl, P. Psota, and P. Vojtíšek, “Methods for refractive-index homogeneity calculation using Fourier-transform phase-shifting interferometry,” Proc. SPIE 10151, 101510Y (2016).
[Crossref]

Meng, X. F.

J. Li, Y. R. Wang, X. F. Meng, X. L. Yang, and Q. P. Wang, “Simultaneous measurement of optical inhomogeneity and thickness variation by using dual-wavelength phase-shifting photorefractive holographic interferometry,” Opt. Laser Technol. 56(1), 241–246 (2014).
[Crossref]

Millerd, J.

Mitsuishi, M.

North-Morris, M.

Novak, M.

Oreb, B. F.

K. Hibino, B. F. Oreb, and P. S. Fairman, “Improved algorithms for wavelength scanning interferometry: application to the simultaneous measurement of surface topography and optical thickness variation in a transparent parallel plate,” Proc. SPIE 4777, 212–219 (2002).
[Crossref]

Park, J J.

J J. Park, J. Jin, J. Wan Kim, and J.-A. Kim, “Measurement of thickness profile and refractive index variation of a silicon wafer using the optical comb of a femtosecond pulse laser,” Opt. Commun. 305, 170–174 (2013).
[Crossref]

Park, J.

Psota, P.

O. Matoušek, V. Lédl, P. Psota, and P. Vojtíšek, “Methods for refractive-index homogeneity calculation using Fourier-transform phase-shifting interferometry,” Proc. SPIE 10151, 101510Y (2016).
[Crossref]

Qin-yuan, S.

Z. Rui, C. Lei, Z. Wen-hua, M. Shi, Z. Dong-hui, and S. Qin-yuan, “Measuring optical homogeneity of parallel plates based on simultaneous phase-shifting by lateral displacement of point sources,” Wuli Xuebao 47(1), 0112002 (2018).

Roberts, F. E.

Rui, Z.

Z. Rui, C. Lei, Z. Wen-hua, M. Shi, Z. Dong-hui, and S. Qin-yuan, “Measuring optical homogeneity of parallel plates based on simultaneous phase-shifting by lateral displacement of point sources,” Wuli Xuebao 47(1), 0112002 (2018).

Schwider, J.

Shi, M.

Z. Rui, C. Lei, Z. Wen-hua, M. Shi, Z. Dong-hui, and S. Qin-yuan, “Measuring optical homogeneity of parallel plates based on simultaneous phase-shifting by lateral displacement of point sources,” Wuli Xuebao 47(1), 0112002 (2018).

Sugimoto, T.

Sugita, N.

Tentori, D.

Vojtíšek, P.

O. Matoušek, V. Lédl, P. Psota, and P. Vojtíšek, “Methods for refractive-index homogeneity calculation using Fourier-transform phase-shifting interferometry,” Proc. SPIE 10151, 101510Y (2016).
[Crossref]

Wan Kim, J.

J J. Park, J. Jin, J. Wan Kim, and J.-A. Kim, “Measurement of thickness profile and refractive index variation of a silicon wafer using the optical comb of a femtosecond pulse laser,” Opt. Commun. 305, 170–174 (2013).
[Crossref]

Wang, Q.

Wang, Q. P.

J. Li, Y. R. Wang, X. F. Meng, X. L. Yang, and Q. P. Wang, “Simultaneous measurement of optical inhomogeneity and thickness variation by using dual-wavelength phase-shifting photorefractive holographic interferometry,” Opt. Laser Technol. 56(1), 241–246 (2014).
[Crossref]

Wang, Y. R.

J. Li, Y. R. Wang, X. F. Meng, X. L. Yang, and Q. P. Wang, “Simultaneous measurement of optical inhomogeneity and thickness variation by using dual-wavelength phase-shifting photorefractive holographic interferometry,” Opt. Laser Technol. 56(1), 241–246 (2014).
[Crossref]

Wen-hua, Z.

Z. Rui, C. Lei, Z. Wen-hua, M. Shi, Z. Dong-hui, and S. Qin-yuan, “Measuring optical homogeneity of parallel plates based on simultaneous phase-shifting by lateral displacement of point sources,” Wuli Xuebao 47(1), 0112002 (2018).

Wyant, J.

Wyant, J. C.

C. Ai and J. C. Wyant, “Measurement of the inhomogeneity of a window,” J. Fluids Eng. 30(9), 602–610 (1991).

Yang, X. L.

J. Li, Y. R. Wang, X. F. Meng, X. L. Yang, and Q. P. Wang, “Simultaneous measurement of optical inhomogeneity and thickness variation by using dual-wavelength phase-shifting photorefractive holographic interferometry,” Opt. Laser Technol. 56(1), 241–246 (2014).
[Crossref]

Zhu, L.

Appl. Opt. (7)

J. Fluids Eng. (1)

C. Ai and J. C. Wyant, “Measurement of the inhomogeneity of a window,” J. Fluids Eng. 30(9), 602–610 (1991).

Opt. Commun. (1)

J J. Park, J. Jin, J. Wan Kim, and J.-A. Kim, “Measurement of thickness profile and refractive index variation of a silicon wafer using the optical comb of a femtosecond pulse laser,” Opt. Commun. 305, 170–174 (2013).
[Crossref]

Opt. Express (4)

Opt. Laser Technol. (1)

J. Li, Y. R. Wang, X. F. Meng, X. L. Yang, and Q. P. Wang, “Simultaneous measurement of optical inhomogeneity and thickness variation by using dual-wavelength phase-shifting photorefractive holographic interferometry,” Opt. Laser Technol. 56(1), 241–246 (2014).
[Crossref]

Opt. Lett. (2)

Proc. SPIE (3)

O. Matoušek, V. Lédl, P. Psota, and P. Vojtíšek, “Methods for refractive-index homogeneity calculation using Fourier-transform phase-shifting interferometry,” Proc. SPIE 10151, 101510Y (2016).
[Crossref]

K. Hibino, B. F. Oreb, and P. S. Fairman, “Improved algorithms for wavelength scanning interferometry: application to the simultaneous measurement of surface topography and optical thickness variation in a transparent parallel plate,” Proc. SPIE 4777, 212–219 (2002).
[Crossref]

L. L. Deck, “Multiple-surface phase-shifting interferometry,” Proc. SPIE 4451, 424–431 (2001).
[Crossref]

Wuli Xuebao (1)

Z. Rui, C. Lei, Z. Wen-hua, M. Shi, Z. Dong-hui, and S. Qin-yuan, “Measuring optical homogeneity of parallel plates based on simultaneous phase-shifting by lateral displacement of point sources,” Wuli Xuebao 47(1), 0112002 (2018).

Other (1)

R. H. Guo, J. X. Li, R. H. Zhu and L. Chen. “Wavelength-Tuned Phase-Shifting Calibration Based on the Fourier Transform in Time Domain,” 2009 IEEE International Conference on Service Operations, Logistics and Informatics. Chicago, U.S.A., SOLI 2009, 371–375(2009).

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Figures (13)

Fig. 1
Fig. 1 Measurement procedure. (a) Cavity of the Fizeau interferometer employed for testing a parallel plate. (b) The empty cavity.
Fig. 2
Fig. 2 Sampling process by equispaced grid points.
Fig. 3
Fig. 3 Flow chart of the NUFFT-WTI process.
Fig. 4
Fig. 4 Phase shift errors with increasing sampling times.
Fig. 5
Fig. 5 Spectrograms obtained using the FFT and NUFFT methods.
Fig. 6
Fig. 6 Simulated and calculated phases of the optical homogeneity of a parallel plate. (a) Simulated optical homogeneity. (b) Optical homogeneity calculated using FFT. (c) Optical homogeneity calculated using NUFFT.
Fig. 7
Fig. 7 Residual phases of optical homogeneity with the simulated system. (a) Residual phase calculated using FFT. (b) Residual phase calculated using NUFFT.
Fig. 8
Fig. 8 PV values of the residual phase with different linearity values in different methods.
Fig. 9
Fig. 9 RMS values of the residual phase with different linearity values in different methods.
Fig. 10
Fig. 10 Photographic image of a square brick in the wavelength-tuning Fizeau interferometer. (a) Hardware configuration. (b) The experimental system.
Fig. 11
Fig. 11 Four consecutive sampled interferograms.
Fig. 12
Fig. 12 Experimentally obtained spectrograms calculated using the FFT and NUFFT methods.
Fig. 13
Fig. 13 Optical homogeneities of the square brick calculated using different methods. (a) Optical homogeneity obtained using the flipping method and a Zygo interferometer. (b) Optical homogeneity calculated using FFT. (c) Optical homogeneity calculated using NUFFT.

Tables (3)

Tables Icon

Table 1 Optical path distances and cavity lengths of each interference fringe

Tables Icon

Table 2 Simulation results

Tables Icon

Table 3 Results obtained through calculations using the three different methods

Equations (13)

Equations on this page are rendered with MathJax. Learn more.

Δn= W 2 n 0 ( W 7 W 6 W 1 ) 2t
I i = a i + b i cos( ϕ i + δ i )
ϕ i =4π h i / λ 0 =2π W i / λ 0
δ i 4π h i tΔλ/ λ 0 2 = f i t
I(x)= i=1 6 I i
g τ (x)= e x 2 /4τ
I τ (x)=I(x) g τ (x)= π π I(y) g τ (xy)dy
I τ (mΔx)= j=0 N1 I( x j ). e (mΔx x j ) 2 /4τ ,m=0,1,, M r 1
B τ (k) 1 M r m=0 M r 1 I τ (mΔx) e ikmΔx ,k=0,1,, M r 1
B(k)= B τ (k)/G(k),k=0,1,, M r 1
G(k)= 4τπ e k 2 τ
ϕ i = F 1 ( F l ),l=1,2,6
W l = ϕ l λ 0 /2π

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