Abstract

In optical metrology synchronous phase-stepping algorithms (PSAs) estimate the measured phase of temporal linear-carrier fringes with respect to a linear-reference. Linear-carrier fringes are normally obtained using closed-loop, feedback, optical phase-stepped devices. On the other hand, open-loop phase-stepping devices usually give fringe patterns with nonlinear phase steps. The Fourier spectrum of linear-carrier fringes is composed of Dirac deltas only. In contrast, nonlinear phase-shifted fringes are wideband, spread-spectrum signals. It is well known that using linear-phase reference PSA to demodulate nonlinear phase stepped fringes, one obtains a spurious-piston. The problem with this spurious-piston is that it may wrongly be interpreted as a real thickness in any absolute phase measurement. Here we mathematically find the origin of this spurious-piston and design nonlinear phase-stepping PSAs to cope with nonlinear phase-stepping interferometric fringes. We give a general theory to tailor nonlinear phase-stepping PSAs to synchronously demodulate nonlinear phase-stepped wideband fringes.

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

Full Article  |  PDF Article
OSA Recommended Articles
Synchronous phase-demodulation and harmonic rejection of 9-step pixelated dynamic interferograms

J. M. Padilla, M. Servin, and J. C. Estrada
Opt. Express 20(11) 11734-11739 (2012)

Design of nonlinearly spaced phase-shifting algorithms using their frequency transfer function

Manuel Servin, Moises Padilla, Guillermo Garnica, and Gonzalo Paez
Appl. Opt. 58(4) 1134-1138 (2019)

References

  • View by:
  • |
  • |
  • |

  1. 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]
  2. K. Freischlad and C. L. Koliopoulos, “Fourier description of digital phase-measuring interferometry,” J. Opt. Soc. Am. A 7(4), 542–551 (1990).
    [Crossref]
  3. M. Servin, A. Quiroga, and J. M. Padilla, Fringe Pattern Analysis for Optical Metrology, Theory Algorithms and Applications (Wiley-VCH, 2014), Chap. 2.
  4. Y. Y. Cheng and J. C. Wyant, “Phase shifter calibration in phase-shifting interferometry,” Appl. Opt. 24(18), 3049–3052 (1985).
    [Crossref] [PubMed]
  5. C. Ai and J. C. Wyant, “Effect of piezoelectric transducer nonlinearity on phase shift interferometry,” Appl. Opt. 26(6), 1112–1116 (1987).
    [Crossref] [PubMed]
  6. 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]
  7. Y. Surrel, “Phase-shifting algorithms for nonlinear and spatially nonuniform phase shifts: comment,” J. Opt. Soc. Am. 15(5), 1227–1233 (1998).
    [Crossref]
  8. K. Hibino, “Error compensating phase measuring algorithms in a Fizeau interferometer,” Opt. Rev. 6(6), 529–538 (1999).
    [Crossref]
  9. 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]
  10. 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]
  11. Y. Kim, K. Hibino, N. Sugita, and M. Mitsuishi, “Absolute optical thickness measurement of transparent plate using excess fraction method and wavelength-tuning Fizeau interferometer,” Opt. Express 23(4), 4065–4073 (2015).
    [Crossref] [PubMed]
  12. K. Hibino, “Phase-shifting algorithm inside an optical cavity for absolute length measurement,” Appl. Opt. 55(5), 1101–1106 (2016).
    [Crossref] [PubMed]
  13. Y. Kim, K. Hibino, and M. Mitsuishi, “Interferometric profile measurement of optical-thickness by wavelength tuning with suppression of spatially uniform error,” Opt. Express 26(8), 10870–10878 (2018).
    [Crossref] [PubMed]
  14. J. R. Klauder, A. C. Price, S. Darlington, and W. J. Albersheim, “The theory and design of chirp radars,” Bell Svs. Tech. J. 39(4), 745–808 (1960).
    [Crossref]
  15. R. Millet, “A matched-filter pulse-compression system using a nonlinear FM waveform,” IEEE Trans. Aerosp. Electron. Syst. 6(1), 73–78 (1970).
    [Crossref]
  16. B. P. Lathi and Z. Ding, Modern Digital and Analog Communication Systems, 4th ed.(Oxford University Press, 2009).
  17. F. Träger, Springer Handbook of Lasers and Optics, 2nd ed., (Springer-Verlag, 2012).
  18. F. J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transforms,” Proc. IEEE 66(1), 51–83 (1978).
    [Crossref]
  19. 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]

2018 (1)

2016 (1)

2015 (1)

2014 (1)

2003 (1)

1999 (1)

K. Hibino, “Error compensating phase measuring algorithms in a Fizeau interferometer,” Opt. Rev. 6(6), 529–538 (1999).
[Crossref]

1998 (1)

Y. Surrel, “Phase-shifting algorithms for nonlinear and spatially nonuniform phase shifts: comment,” J. Opt. Soc. Am. 15(5), 1227–1233 (1998).
[Crossref]

1997 (1)

1995 (1)

1990 (1)

1987 (1)

1985 (1)

1978 (1)

F. J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transforms,” Proc. IEEE 66(1), 51–83 (1978).
[Crossref]

1974 (1)

1970 (1)

R. Millet, “A matched-filter pulse-compression system using a nonlinear FM waveform,” IEEE Trans. Aerosp. Electron. Syst. 6(1), 73–78 (1970).
[Crossref]

1960 (1)

J. R. Klauder, A. C. Price, S. Darlington, and W. J. Albersheim, “The theory and design of chirp radars,” Bell Svs. Tech. J. 39(4), 745–808 (1960).
[Crossref]

Ai, C.

Albersheim, W. J.

J. R. Klauder, A. C. Price, S. Darlington, and W. J. Albersheim, “The theory and design of chirp radars,” Bell Svs. Tech. J. 39(4), 745–808 (1960).
[Crossref]

Brangaccio, D. J.

Bruning, J. H.

Cheng, Y. Y.

Darlington, S.

J. R. Klauder, A. C. Price, S. Darlington, and W. J. Albersheim, “The theory and design of chirp radars,” Bell Svs. Tech. J. 39(4), 745–808 (1960).
[Crossref]

Fairman, P. S.

Farrant, D. I.

Freischlad, K.

Gallagher, J. E.

Groot, P.

Hanayama, R.

Harris, F. J.

F. J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transforms,” Proc. IEEE 66(1), 51–83 (1978).
[Crossref]

Herriott, D. R.

Hibino, K.

Kim, Y.

Klauder, J. R.

J. R. Klauder, A. C. Price, S. Darlington, and W. J. Albersheim, “The theory and design of chirp radars,” Bell Svs. Tech. J. 39(4), 745–808 (1960).
[Crossref]

Koliopoulos, C. L.

Larkin, K. G.

Millet, R.

R. Millet, “A matched-filter pulse-compression system using a nonlinear FM waveform,” IEEE Trans. Aerosp. Electron. Syst. 6(1), 73–78 (1970).
[Crossref]

Mitsuishi, M.

Oreb, B. F.

Price, A. C.

J. R. Klauder, A. C. Price, S. Darlington, and W. J. Albersheim, “The theory and design of chirp radars,” Bell Svs. Tech. J. 39(4), 745–808 (1960).
[Crossref]

Rosenfeld, D. P.

Sugita, N.

Surrel, Y.

Y. Surrel, “Phase-shifting algorithms for nonlinear and spatially nonuniform phase shifts: comment,” J. Opt. Soc. Am. 15(5), 1227–1233 (1998).
[Crossref]

White, A. D.

Wyant, J. C.

Appl. Opt. (6)

Bell Svs. Tech. J. (1)

J. R. Klauder, A. C. Price, S. Darlington, and W. J. Albersheim, “The theory and design of chirp radars,” Bell Svs. Tech. J. 39(4), 745–808 (1960).
[Crossref]

IEEE Trans. Aerosp. Electron. Syst. (1)

R. Millet, “A matched-filter pulse-compression system using a nonlinear FM waveform,” IEEE Trans. Aerosp. Electron. Syst. 6(1), 73–78 (1970).
[Crossref]

J. Opt. Soc. Am. (1)

Y. Surrel, “Phase-shifting algorithms for nonlinear and spatially nonuniform phase shifts: comment,” J. Opt. Soc. Am. 15(5), 1227–1233 (1998).
[Crossref]

J. Opt. Soc. Am. A (2)

Opt. Express (3)

Opt. Rev. (1)

K. Hibino, “Error compensating phase measuring algorithms in a Fizeau interferometer,” Opt. Rev. 6(6), 529–538 (1999).
[Crossref]

Proc. IEEE (1)

F. J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transforms,” Proc. IEEE 66(1), 51–83 (1978).
[Crossref]

Other (3)

B. P. Lathi and Z. Ding, Modern Digital and Analog Communication Systems, 4th ed.(Oxford University Press, 2009).

F. Träger, Springer Handbook of Lasers and Optics, 2nd ed., (Springer-Verlag, 2012).

M. Servin, A. Quiroga, and J. M. Padilla, Fringe Pattern Analysis for Optical Metrology, Theory Algorithms and Applications (Wiley-VCH, 2014), Chap. 2.

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 (11)

Fig. 1
Fig. 1 Panel (a) shows the standard case of using a linear-carrier reference r1(t) for phase demodulation of a chirp (quadratic phase) data I(t). Panel (b) shows the herein proposed strategy, which consist on synchronously following the nonlinear phase variations of the fringe data I(t) by a chirped reference r2(t).
Fig. 2
Fig. 2 Panel (a) shows in blue linear phase-shifting, and in red nonlinear phase-shifting. Panel (b) shows linear carrier fringes. Panel (c) shows nonlinear carrier fringes.
Fig. 3
Fig. 3 Panel (a) shows the Fourier spectra of linear-carrier fringes. In panel (b), the red rectangle schematically/abstractly represents wideband spectral lobes.
Fig. 4
Fig. 4 FTF spectra of two nonlinear-reference PSAs. Panel (a) shows (in green) the FTF of a square-window, nonlinear-reference PSA. Panel (b) shows the FTF of a nonlinear-reference PSA with Gaussian window [18,19]. The red rectangles schematically represent the wideband spectrum of nonlinear-carrier fringes. The FTF in panel (b) is a smooth approximation of a Hilbert quadrature filter (for more details see chapter 5, page 229 in [3]).
Fig. 5
Fig. 5 Fundamental (in green) and harmonic (in red) FTF response for the nonlinear-reference PSA centered at π/2. The surviving distorting harmonics are {...,-7,-3,5,9,...} [3].
Fig. 6
Fig. 6 Panel (a) shows the linear (in blue) and a strong chirp-phase (in red). Panel (b) shows the chirp-carrier fringes sampled at a constant sampling rate. Note that the time interval [0,T] has more than three temporal fringes and has been sampled 13 times .
Fig. 7
Fig. 7 Panel (a) shows the real component of the complex-valued chirp impulse response. Panel (b) shows the corresponding FTF which smoothly approximate a Hilbert quadrature filter [3].
Fig. 8
Fig. 8 Schematic of nonlinear-carrier fringe spectrum (in red), and the FTF of the Gaussian-window PSA (in green) having negligible response at the negative frequencies.
Fig. 9
Fig. 9 Phase error given by Eq. (30). Note the vertical scale is within [-0.03,0.03] radians. This phase error has about the same magnitude as the one obtained in [12] using a linear reference PSA.
Fig. 10
Fig. 10 Spectral response (FTF) for the square-window, nonlinear-phase reference PSA. This square window (wn = 1.0) has large response at the left side of the fringe spectrum. This FTF is a bad approximation of a one-sided Hilbert quadrature filter [3].
Fig. 11
Fig. 11 The blue trace shows the phase-error for the 13-step square-window nonlinear-reference PSA. For comparison, the red-trace is the phase error corresponding to the Gaussian window PSA seen previously in Fig. 9. Note that the vertical scale is now [-0.1,0.1] radians.

Equations (36)

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

J(t)=a+bcos(φ+ ω 0 t);t[0,T].
I(t)=a+bcos[φ+ ω 0 t+Δ(t)];t[0,T].
[ ω 0 + dΔ(t) dt ](0,π);t[0,T].
F{ a+ b 2 e i[φ+ ω 0 t] + b 2 e i[φ+ ω 0 t] }=aδ(ω)+ b 2 e iφ δ(ω ω 0 )+ b 2 e iφ δ(ω+ ω 0 ).
F{ a+ b 2 e iφ e i[ ω 0 t+Δ(t)] + b 2 e iφ e i[ ω 0 t+Δ(t)] }=aδ(ω)+ b 2 e iφ C(ω)+ b 2 e iφ C * (ω),
C (ω)=F{ e i[ ω 0 t+Δ(t)] };C(ω)=F{ e i[ ω 0 t+Δ(t)] }.
A e iφ = n=0 N1 [ c n e in θ 0 ] LinearReference [ a+bcos(φ+n θ 0 ) ] LinearCarrierFringes ; θ 0 = ω 0 T/N,( c n ).
A 1 e i(φ+Piston) = n=0 N1 [ d n e in θ 0 ] LinearReference [ a+bcos( φ+n θ 0 + Δ n ) ] NonlinearCarrierFringes ;( d n ).
A 2 e iφ = n=0 N1 [ w n e i(n θ 0 + Δ n ) ] NonlinearReference [ a+bcos( φ+n θ 0 + Δ n ) ] NonlinearCarrierFringes ;( w n ).
A 1 e i[φ+Piston] = n=0 N1 d n e in θ 0 I n = n=0 N1 d n e in θ 0 { a+ b 2 e iφ e i(n θ 0 + Δ n ) + b 2 e iφ e i(n θ 0 + Δ n ) }
A 1 e i[φ+Piston] =a[ n=0 N1 d n e in θ 0 ]+ b 2 e iφ [ n=0 N1 d n e i Δ n ]+ b 2 e iφ [ n=0 N1 d n e i(2n θ 0 + Δ n ) ]
[ n=0 N1 d n e in θ 0 ]=0,and[ n=0 N1 d n e i(2n θ 0 + Δ n ) ]=0.
A 1 e i(φ+Piston) = b 2 e iφ [ n=0 N1 d n e i Δ n ];Piston=arg{ n=0 N1 d n e i Δ n }.
A 2 e iφ = n=0 N1 w n e i(n θ 0 + Δ n ) I n = n=0 N1 w n e i(n θ 0 + Δ n ) [ a+ b 2 e iφ e i(n θ 0 + Δ n ) + b 2 e iφ e i(n θ 0 + Δ n ) ]
A 2 e iφ =a[ n=0 N1 w n e i(n θ 0 + Δ n ) ]+ b 2 e iφ [ n=0 N1 w n ]+ b 2 e iφ [ n=0 N1 w n e i(2n θ 0 + Δ n ) ]
[ n=0 N1 w n e i(n θ 0 + Δ n ) ]=0;and[ n=0 N1 w n e i(2n θ 0 + Δ n ) ]=0.
A 2 e iφ = b 2 e iφ n=0 N1 w n ;( w n ).
h(t)= n=0 N1 w n e i[ ω 0 t+Δ(t)] δ(tn T s ) ;( w n ).
H(ω)= n=0 N1 w n e i[ ω 0 n T s +Δ(n T s )] e inω = n=0 N1 w n e i(n θ 0 + Δ n ) e inω .
H(ω)0forω[π,0] H(ω)0forω(0,π) .
I(t)= n=0 N1 { a+bcos[φ+ ω 0 t+Δ(t)]+η(t) }δ(tn T s ) .
S(ω)=F[ R ηη (τ) ]=F[ E{ η(t)η(t+τ) } ]= η 0 2 ;ω[π,π].
SNR 1 = SignalEnergy NoiseEnergy = ( b 2 ) 2 | n=0 N1 d n e i Δ n | 2 ( η 0 2 ) n=0 N1 | d n | 2 .
SNR 2 = SignalEnergy NoiseEnergy = ( b 2 ) 2 | n=0 N1 w n | 2 ( η 0 2 ) n=0 N1 | w n | 2 .
| n=0 N1 w n | 2 | n=0 N1 d n e i Δ n | 2 ,
SNR 2 SNR 1 .
I n =[ 1+cos(φ+n θ 0 + ε 2 n 2 T s 2 ) ];n{0,1,...,12},( θ 0 =0.35π, ε 2 =0.055 rad sec 2 ).
A 2 e iφ = n=0 12 w n e i( θ 0 n+ ε 2 n 2 T s 2 ) I n ; w n = e 0.1 [ n6 ] 2 .
H(ω)=F[ h(t) ]=F[ n=0 12 w n e i( ω 0 t+ ε 2 t 2 ) δ(tn T s ) ]= n=0 12 w n e i[ θ 0 n+ ε 2 n 2 T s 2 ] e inω .
φ Error =φarg[ n=0 12 w n e i( n θ 0 + ε 2 n 2 T s 2 ) I n ];φ[0,2π].
A 2 e i φ Square = n=0 12 e i[ n θ 0 + ε 2 n 2 T s 2 ] { 1+cos[ φ+n θ 0 + ε 2 n 2 T s 2 ] } .
I(t)=a+bcos[φ+ ω 0 t+Δ(t)];t[0,T].
[ ω 0 + dΔ(t) dt ](0,π);t[0,T].
A 2 e iφ = n=0 N1 w n e i[ n θ 0 + Δ n ] I n ;( w n ).
H(ω)=F[ h(t) ]=F{ n=0 N1 w n e i[ ω 0 n+ Δ n ] δ(tn T s ) }= n=0 N1 w n e i[ ω 0 n+ Δ n ] e inω .
H(ω)0forω[π,0], H(ω)0forω(0,π).

Metrics