Abstract

This paper presents the derivation of a 17-sample phase-shifting algorithm that can compensate the miscalibration and first-order nonlinearity of phase shift error, coupling error, and bias modulation of the intensity and satisfy the fringe contrast maximum condition. The phase error of measurements performed using the 17-sample algorithm is discussed and compared with those of measurements obtained using other algorithms. Finally, the optical thickness variation of a BK7 optically transparent plate obtained using a wavelength tuning Fizeau interferometer and the 17-sample algorithm are presented. The experimental results indicate that the optical thickness variation measurement accuracy for the BK7 plate was 3 nm.

© 2015 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Interferometric measurement of surface shape by wavelength tuning suppressing random intensity error

Yangjin Kim, Kenichi Hibino, Naohiko Sugita, and Mamoru Mitsuishi
Appl. Opt. 55(23) 6464-6470 (2016)

Simultaneous measurement of surface shape and optical thickness using wavelength tuning and a polynomial window function

Yangjin Kim, Kenichi Hibino, Naohiko Sugita, and Mamoru Mitsuishi
Opt. Express 23(25) 32869-32880 (2015)

Interferometric profile measurement of optical-thickness by wavelength tuning with suppression of spatially uniform error

Yangjin Kim, Kenichi Hibino, and Mamoru Mitsuishi
Opt. Express 26(8) 10870-10878 (2018)

References

  • View by:
  • |
  • |
  • |

  1. D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing, Vol. 61 of Optical Engineering Series (Marcel Dekker, 1998), pp. 169–245.
  2. J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13(11), 2693–2703 (1974).
    [Crossref] [PubMed]
  3. K. Creath, “Phase measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (North-Holland, 1988).
  4. P. Hariharan, “Digital phase-stepping interferometry: effects of multiply reflected beams,” Appl. Opt. 26(13), 2506–2507 (1987).
    [Crossref] [PubMed]
  5. Y. Kim, K. Hibino, N. Sugita, and M. Mitsuishi, “Design of phase shifting algorithms: fringe contrast maximum,” Opt. Express 22(15), 18203–18213 (2014).
    [Crossref] [PubMed]
  6. J. Schwider, R. Burow, K. E. Elssner, J. Grzanna, R. Spolaczyk, and K. Merkel, “Digital wave-front measuring interferometry: some systematic error sources,” Appl. Opt. 22(21), 3421–3432 (1983).
    [Crossref] [PubMed]
  7. P. Hariharan, B. F. Oreb, and T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26(13), 2504–2506 (1987).
    [Crossref] [PubMed]
  8. K. G. Larkin and B. F. Oreb, “Design and assessment of symmetrical phase-shifting algorithms,” J. Opt. Soc. Am. A 9(10), 1740–1748 (1992).
    [Crossref]
  9. Y. Surrel, “Phase stepping: a new self-calibrating algorithm,” Appl. Opt. 32(19), 3598–3600 (1993).
    [Crossref] [PubMed]
  10. K. Hibino, B. F. Oreb, D. I. Farrant, and K. G. Larkin, “Phase shifting for nonsinusoidal waveforms with phase-shift errors,” J. Opt. Soc. Am. A 12(4), 761–768 (1995).
    [Crossref]
  11. J. Schmit and K. Creath, “Extended averaging technique for derivation of error-compensating algorithms in phase-shifting interferometry,” Appl. Opt. 34(19), 3610–3619 (1995).
    [Crossref] [PubMed]
  12. P. Groot, “Derivation of algorithms for phase-shifting interferometry using the concept of a data-sampling window,” Appl. Opt. 34(22), 4723–4730 (1995).
    [Crossref] [PubMed]
  13. Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. 35(1), 51–60 (1996).
    [Crossref] [PubMed]
  14. R. Onodera and Y. Ishii, “Phase-extraction analysis of laser-diode phase-shifting interferometry that is insensitive to changes in laser power,” J. Opt. Soc. Am. A 13(1), 139–146 (1996).
    [Crossref]
  15. Y. Surrel, “Additive noise effect in digital phase detection,” Appl. Opt. 36(1), 271–276 (1997).
    [Crossref] [PubMed]
  16. Y. Surrel, “Design of phase-detection algorithms insensitive to bias modulation,” Appl. Opt. 36(4), 805–807 (1997).
    [Crossref] [PubMed]
  17. K. Hibino, B. F. Oreb, D. I. Farrant, and K. G. Larkin, “Phase-shifting algorithms for nonlinear and spatially nonuniform phase shifts,” J. Opt. Soc. Am. A 14(4), 918–930 (1997).
    [Crossref]
  18. P. de Groot, “Measurement of transparent plates with wavelength-tuned phase-shifting interferometry,” Appl. Opt. 39(16), 2658–2663 (2000).
    [Crossref] [PubMed]
  19. K. Hibino, B. F. Oreb, and P. S. Fairman, “Wavelength-scanning interferometry of a transparent parallel plate with refractive-index dispersion,” Appl. Opt. 42(19), 3888–3895 (2003).
    [Crossref] [PubMed]
  20. K. Hibino, B. F. Oreb, P. S. Fairman, and J. Burke, “Simultaneous measurement of surface shape and variation in optical thickness of a transparent parallel plate in wavelength-scanning Fizeau interferometer,” Appl. Opt. 43(6), 1241–1249 (2004).
    [Crossref] [PubMed]
  21. R. Hanayama, K. Hibino, S. Warisawa, and M. Mitsuishi, “Phase measurement algorithm in wavelength scanned Fizeau interferometer,” Opt. Rev. 11(5), 337–343 (2004).
    [Crossref]
  22. Y. Kim, K. Hibino, R. Hanayama, N. Sugita, and M. Mitsuishi, “Multiple-surface interferometry of highly reflective wafer by wavelength tuning,” Opt. Express 22(18), 21145–21156 (2014).
    [Crossref] [PubMed]
  23. K. Freischlad and C. Koliopoulos, “Fourier description of digital phase-measuring interferometry,” J. Opt. Soc. Am. A 7(4), 542–551 (1990).
    [Crossref]
  24. P. J. de Groot, “Correlated errors in phase-shifting laser Fizeau interferometry,” Appl. Opt. 53(19), 4334–4342 (2014).
    [Crossref] [PubMed]
  25. Y. Kim, K. Hibino, N. Sugita, and M. Mitsuishi, “Surface profile measurement of a highly reflective silicon wafer by phase-shifting interferometry,” Appl. Opt. 54(13), 4207–4213 (2015).
    [Crossref]
  26. P. Groot, “Phase-shift calibration errors in interferometers with spherical Fizeau cavities,” Appl. Opt. 34(16), 2856–2863 (1995).
    [Crossref] [PubMed]
  27. K. Creath and P. Hariharan, “Phase-shifting errors in interferometric tests with high-numerical-aperture reference surfaces,” Appl. Opt. 33(1), 24–25 (1994).
    [Crossref] [PubMed]
  28. K. Liu and M. G. Littman, “Novel geometry for single-mode scanning of tunable lasers,” Opt. Lett. 6(3), 117–118 (1981).
    [Crossref] [PubMed]

2015 (1)

2014 (3)

2004 (2)

2003 (1)

2000 (1)

1997 (3)

1996 (2)

1995 (4)

1994 (1)

1993 (1)

1992 (1)

1990 (1)

1987 (2)

1983 (1)

1981 (1)

1974 (1)

Brangaccio, D. J.

Bruning, J. H.

Burke, J.

Burow, R.

Creath, K.

de Groot, P.

de Groot, P. J.

Eiju, T.

Elssner, K. E.

Fairman, P. S.

Farrant, D. I.

Freischlad, K.

Gallagher, J. E.

Groot, P.

Grzanna, J.

Hanayama, R.

Y. Kim, K. Hibino, R. Hanayama, N. Sugita, and M. Mitsuishi, “Multiple-surface interferometry of highly reflective wafer by wavelength tuning,” Opt. Express 22(18), 21145–21156 (2014).
[Crossref] [PubMed]

R. Hanayama, K. Hibino, S. Warisawa, and M. Mitsuishi, “Phase measurement algorithm in wavelength scanned Fizeau interferometer,” Opt. Rev. 11(5), 337–343 (2004).
[Crossref]

Hariharan, P.

Herriott, D. R.

Hibino, K.

Y. Kim, K. Hibino, N. Sugita, and M. Mitsuishi, “Surface profile measurement of a highly reflective silicon wafer by phase-shifting interferometry,” Appl. Opt. 54(13), 4207–4213 (2015).
[Crossref]

Y. Kim, K. Hibino, R. Hanayama, N. Sugita, and M. Mitsuishi, “Multiple-surface interferometry of highly reflective wafer by wavelength tuning,” Opt. Express 22(18), 21145–21156 (2014).
[Crossref] [PubMed]

Y. Kim, K. Hibino, N. Sugita, and M. Mitsuishi, “Design of phase shifting algorithms: fringe contrast maximum,” Opt. Express 22(15), 18203–18213 (2014).
[Crossref] [PubMed]

K. Hibino, B. F. Oreb, P. S. Fairman, and J. Burke, “Simultaneous measurement of surface shape and variation in optical thickness of a transparent parallel plate in wavelength-scanning Fizeau interferometer,” Appl. Opt. 43(6), 1241–1249 (2004).
[Crossref] [PubMed]

R. Hanayama, K. Hibino, S. Warisawa, and M. Mitsuishi, “Phase measurement algorithm in wavelength scanned Fizeau interferometer,” Opt. Rev. 11(5), 337–343 (2004).
[Crossref]

K. Hibino, B. F. Oreb, and P. S. Fairman, “Wavelength-scanning interferometry of a transparent parallel plate with refractive-index dispersion,” Appl. Opt. 42(19), 3888–3895 (2003).
[Crossref] [PubMed]

K. Hibino, B. F. Oreb, D. I. Farrant, and K. G. Larkin, “Phase-shifting algorithms for nonlinear and spatially nonuniform phase shifts,” J. Opt. Soc. Am. A 14(4), 918–930 (1997).
[Crossref]

K. Hibino, B. F. Oreb, D. I. Farrant, and K. G. Larkin, “Phase shifting for nonsinusoidal waveforms with phase-shift errors,” J. Opt. Soc. Am. A 12(4), 761–768 (1995).
[Crossref]

Ishii, Y.

Kim, Y.

Koliopoulos, C.

Larkin, K. G.

Littman, M. G.

Liu, K.

Merkel, K.

Mitsuishi, M.

Onodera, R.

Oreb, B. F.

Rosenfeld, D. P.

Schmit, J.

Schwider, J.

Spolaczyk, R.

Sugita, N.

Surrel, Y.

Warisawa, S.

R. Hanayama, K. Hibino, S. Warisawa, and M. Mitsuishi, “Phase measurement algorithm in wavelength scanned Fizeau interferometer,” Opt. Rev. 11(5), 337–343 (2004).
[Crossref]

White, A. D.

Appl. Opt. (17)

P. Hariharan, “Digital phase-stepping interferometry: effects of multiply reflected beams,” Appl. Opt. 26(13), 2506–2507 (1987).
[Crossref] [PubMed]

J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13(11), 2693–2703 (1974).
[Crossref] [PubMed]

Y. Surrel, “Phase stepping: a new self-calibrating algorithm,” Appl. Opt. 32(19), 3598–3600 (1993).
[Crossref] [PubMed]

P. Hariharan, B. F. Oreb, and T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26(13), 2504–2506 (1987).
[Crossref] [PubMed]

Y. Surrel, “Additive noise effect in digital phase detection,” Appl. Opt. 36(1), 271–276 (1997).
[Crossref] [PubMed]

P. Groot, “Derivation of algorithms for phase-shifting interferometry using the concept of a data-sampling window,” Appl. Opt. 34(22), 4723–4730 (1995).
[Crossref] [PubMed]

Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. 35(1), 51–60 (1996).
[Crossref] [PubMed]

K. Hibino, B. F. Oreb, and P. S. Fairman, “Wavelength-scanning interferometry of a transparent parallel plate with refractive-index dispersion,” Appl. Opt. 42(19), 3888–3895 (2003).
[Crossref] [PubMed]

K. Hibino, B. F. Oreb, P. S. Fairman, and J. Burke, “Simultaneous measurement of surface shape and variation in optical thickness of a transparent parallel plate in wavelength-scanning Fizeau interferometer,” Appl. Opt. 43(6), 1241–1249 (2004).
[Crossref] [PubMed]

K. Creath and P. Hariharan, “Phase-shifting errors in interferometric tests with high-numerical-aperture reference surfaces,” Appl. Opt. 33(1), 24–25 (1994).
[Crossref] [PubMed]

Y. Surrel, “Design of phase-detection algorithms insensitive to bias modulation,” Appl. Opt. 36(4), 805–807 (1997).
[Crossref] [PubMed]

P. J. de Groot, “Correlated errors in phase-shifting laser Fizeau interferometry,” Appl. Opt. 53(19), 4334–4342 (2014).
[Crossref] [PubMed]

J. Schwider, R. Burow, K. E. Elssner, J. Grzanna, R. Spolaczyk, and K. Merkel, “Digital wave-front measuring interferometry: some systematic error sources,” Appl. Opt. 22(21), 3421–3432 (1983).
[Crossref] [PubMed]

Y. Kim, K. Hibino, N. Sugita, and M. Mitsuishi, “Surface profile measurement of a highly reflective silicon wafer by phase-shifting interferometry,” Appl. Opt. 54(13), 4207–4213 (2015).
[Crossref]

P. Groot, “Phase-shift calibration errors in interferometers with spherical Fizeau cavities,” Appl. Opt. 34(16), 2856–2863 (1995).
[Crossref] [PubMed]

J. Schmit and K. Creath, “Extended averaging technique for derivation of error-compensating algorithms in phase-shifting interferometry,” Appl. Opt. 34(19), 3610–3619 (1995).
[Crossref] [PubMed]

P. de Groot, “Measurement of transparent plates with wavelength-tuned phase-shifting interferometry,” Appl. Opt. 39(16), 2658–2663 (2000).
[Crossref] [PubMed]

J. Opt. Soc. Am. A (5)

Opt. Express (2)

Opt. Lett. (1)

Opt. Rev. (1)

R. Hanayama, K. Hibino, S. Warisawa, and M. Mitsuishi, “Phase measurement algorithm in wavelength scanned Fizeau interferometer,” Opt. Rev. 11(5), 337–343 (2004).
[Crossref]

Other (2)

D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing, Vol. 61 of Optical Engineering Series (Marcel Dekker, 1998), pp. 169–245.

K. Creath, “Phase measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (North-Holland, 1988).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1
Fig. 1 Design procedure for 17-sample phase shifting algorithm. Insensitivity to (a) mth harmonics, (b) phase-shift error, (c) coupling error, and (d) bias modulation, and (e) satisfaction of fringe contrast maximum requirement.
Fig. 2
Fig. 2 Sampling functions iF1 and F2 of 17-sample algorithm.
Fig. 3
Fig. 3 RMS errors of phase-shifting algorithms listed in Table 1 as functions of phase-shift miscalibration when reference surface reflectivity is 4% and sample reflectivity is (a) 4% and (b) 30%.
Fig. 4
Fig. 4 Characteristic diagram of (a) new 17 algorithm (N = 8), (b) synchronous detection (N = 18), (c) Larkin-Oreb N + 1 algorithm (N = 18), (d) Surrel 12 algorithm (N = 6) and (e) Hanayama 2N – 1 algorithm (N = 8).
Fig. 5
Fig. 5 Wavelength tuning Fizeau interferometer used to measure optical thickness variation of BK7 transparent plate. PBS denotes polarization beam splitter; QWP is quarter-wave plate; HWP is half-wave plate.
Fig. 6
Fig. 6 (a) Laboratory photo of BK7 plate in wavelength tuning Fizeau interferometer; (b) raw interferogram at wavelength of 632.8 nm.
Fig. 7
Fig. 7 Optical thickness variation of BK7 plate at 633 nm wavelength.

Tables (2)

Tables Icon

Table 1 Representative Phase-Shifting Algorithm

Tables Icon

Table 2 Measurement Repeatability Error

Equations (20)

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

I( α r )= m=1 A m cos( φ m m α r ) = I 0 [ 1+ m=1 γ m cos( φ m m α r ) ] = I 0 + I 0 γ 1 cos( φ 1 α r )+ I 0 γ 2 cos( φ 2 2 α r )+,
φ * =arctan r=1 M b r I( α r ) r=1 M a r I( α r ) ,
α r = α 0r [ 1+ε( α 0r ) ]= α 0r [ 1+ ε 0 + ε 1 α 0r π + ε 2 ( α 0r π ) 2 ++ ε p ( α 0r π ) p ],
Δφ= φ * φ 1 =ο( A k )+ο( ε q )+ο( A k ε q ),
P( x )= r=1 M ( a r +i b r ) x r1 ,
P( x )= (x1) 2 ( x ζ 2 ) 2 ( x ζ 3 ) 3 ( x+1 ) 2 ( x ζ 1 ) 3 ( x ζ 2 ) 2 ( x ζ 3 ) 2 ,
a r =( 1, 2 ,0,3 2 ,8,5 2 ,7 2 ,14,7 2 ,5 2 ,8,3 2 ,0, 2 ,1 ),
b r =( 0, 2 ,4,3 2 ,0,5 2 ,12,7 2 ,0,7 2 ,12,5 2 ,3 2 ,4, 2 ,0 ).
F 1 ( ν )= r=1 M b r exp( i α r ν ) ,
F 2 ( ν )= r=1 M a r exp( i α r ν ) ,
i F 1 ( ν )= F 2 ( ν )=0( ν=0,2,,6 ).
di F 1 ( ν ) dν | ν=1 = d F 2 ( ν ) dν | ν=1 ,
d 2 i F 1 ( ν ) d ν 2 | ν=1 = d 2 F 2 ( ν ) d ν 2 | ν=1 .
d 2 i F 1 ( ν ) d ν 2 = d 2 F 2 ( ν ) d ν 2 =0( ν=2,3,6 ).
di F 1 ( ν ) dν | ν=0 = d F 2 ( ν ) dν | ν=0 =0.
di F 1 ( ν ) dν | ν=1 = d F 2 ( ν ) dν | ν=1 =0.
σ mis = 1 2 2 | i F 1 ( ν ) F 2 ( ν ) 1 |.
σ cou = 1 2 m=2 γ m γ 1 [ i F 1 ( mν ) i F 1 ( ν ) ] 2 + [ F 2 ( mν ) F 2 ( ν ) ] 2 ,
σ= σ mis 2 + σ cou 2 ,
δλ= λ 2 4πnT δφ0.0503nm.

Metrics