Abstract

A non-iterative method based on principal component analysis (PCA) is presented to directly extract the phase from multiple-beam Fizeau interferograms with random phase shifts. The PCA method is the approach that decomposes the multiple-beam Fizeau interferograms into many uncorrelated quadrature signals and then applies principal component analysis algorithm to extract the measured phase without any prior guess about the phase shifts. Some factors that affect the performance of the proposed method are analyzed and discussed. Numerical simulations and experiments demonstrate that the proposed method extracts phase fast and exhibits high precision. The method can be applied in high precision interferometry.

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  24. T. E. Zander, V. Madyastha, A. Patil, P. Rastogi, and L. M. Reindl, “Phase-step estimation in interferometry via an unscented Kalman filter,” Opt. Lett. 34(9), 1396–1398 (2009).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  28. R. C. Gonzalez and R. E. Woods, Digital Image Processing, 3rd ed. (Prentice-Hall, 2007).
  29. B. Pan, Q. Kemao, L. Huang, and A. Asundi, “Phase error analysis and compensation for nonsinusoidal waveforms in phase-shifting digital fringe projection profilometry,” Opt. Lett. 34(4), 416–418 (2009).
    [CrossRef] [PubMed]

2011

2010

2009

2008

2006

2005

2004

2001

1999

B. V. Dorrío and J. L. Fernández, “Phase-evaluation methods in whole-field optical measurement techniques,” Meas. Sci. Technol. 10(3), R33–R55 (1999).
[CrossRef]

1998

P. Hariharan, “Interferometric measurements of small-scale irregularities: highly reflecting surfaces,” Opt. Eng. 37(10), 2751–2753 (1998).
[CrossRef]

1996

1995

1993

1989

G. Bonsch and H. Bohme, “Phase-determination of Fizeau interferences by phase-shifting interferometry,” Optik (Stuttg.) 82, 161–164 (1989).

1987

1983

1967

P. B. Clapham and G. D. Dew, “Surface-coated reference flats for testing fully aluminized surfaces by means of the Fizeau interferometer,” J. Sci. Instrum. 44(11), 899–902 (1967).
[CrossRef]

Ai, C.

Asundi, A.

Belenguer, T.

Blanco-García, J.

Bohme, H.

G. Bonsch and H. Bohme, “Phase-determination of Fizeau interferences by phase-shifting interferometry,” Optik (Stuttg.) 82, 161–164 (1989).

Bokor, J.

Bonsch, G.

G. Bonsch and H. Bohme, “Phase-determination of Fizeau interferences by phase-shifting interferometry,” Optik (Stuttg.) 82, 161–164 (1989).

Burow, R.

Cai, L. Z.

Chai, L.

Chen, M.

Cheng, X. C.

Clapham, P. B.

P. B. Clapham and G. D. Dew, “Surface-coated reference flats for testing fully aluminized surfaces by means of the Fizeau interferometer,” J. Sci. Instrum. 44(11), 899–902 (1967).
[CrossRef]

Dew, G. D.

P. B. Clapham and G. D. Dew, “Surface-coated reference flats for testing fully aluminized surfaces by means of the Fizeau interferometer,” J. Sci. Instrum. 44(11), 899–902 (1967).
[CrossRef]

Dong, G. Y.

Dorrío, B. V.

Doval, A. F.

Elssner, K. E.

Farrant, D. I.

Fernández, J. L.

Gao, P.

Geist, E.

Goldberg, K. A.

Grzanna, J.

Guo, H.

Han, B.

Harder, I.

Hariharan, P.

P. Hariharan, “Interferometric measurements of small-scale irregularities: highly reflecting surfaces,” Opt. Eng. 37(10), 2751–2753 (1998).
[CrossRef]

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

Hibino, K.

Huang, L.

Kemao, Q.

Lamare, M.

P. Picart, R. Mercier, and M. Lamare, “Influence of multiple-beam interferences in a phase-shifting Fizeau interferometer and error-reduced algorithms,” Pure Appl. Opt. 5(2), 167–194 (1996).
[CrossRef]

Langoju, R.

Larkin, K. G.

Li, Y.

Lindlein, N.

López, C.

Madyastha, V.

Mantel, K.

Meng, X. F.

Mercier, R.

P. Picart, R. Mercier, and M. Lamare, “Influence of multiple-beam interferences in a phase-shifting Fizeau interferometer and error-reduced algorithms,” Pure Appl. Opt. 5(2), 167–194 (1996).
[CrossRef]

Merkel, K.

Oreb, B. F.

Pan, B.

Patil, A.

Peng, H.

Pérez-Amor, M.

Picart, P.

P. Picart, R. Mercier, and M. Lamare, “Influence of multiple-beam interferences in a phase-shifting Fizeau interferometer and error-reduced algorithms,” Pure Appl. Opt. 5(2), 167–194 (1996).
[CrossRef]

Quiroga, J. A.

Rastogi, P.

Reindl, L. M.

Schwider, J.

Shen, X. X.

Soto, R.

Spolaczyk, R.

Sun, W. J.

Surrel, Y.

Vargas, J.

Wang, H.

Wang, Y. R.

Wang, Z.

Wyant, J. C.

Xu, J.

Xu, Q.

Xu, X. F.

Yao, B.

Zander, T. E.

Zhang, H.

Appl. Opt.

J. Opt. Soc. Am. A

J. Sci. Instrum.

P. B. Clapham and G. D. Dew, “Surface-coated reference flats for testing fully aluminized surfaces by means of the Fizeau interferometer,” J. Sci. Instrum. 44(11), 899–902 (1967).
[CrossRef]

Meas. Sci. Technol.

B. V. Dorrío and J. L. Fernández, “Phase-evaluation methods in whole-field optical measurement techniques,” Meas. Sci. Technol. 10(3), R33–R55 (1999).
[CrossRef]

Opt. Eng.

P. Hariharan, “Interferometric measurements of small-scale irregularities: highly reflecting surfaces,” Opt. Eng. 37(10), 2751–2753 (1998).
[CrossRef]

Opt. Express

Opt. Lett.

J. Vargas, J. A. Quiroga, and T. Belenguer, “Phase-shifting interferometry based on principal component analysis,” Opt. Lett. 36(8), 1326–1328 (2011).
[CrossRef] [PubMed]

X. F. Xu, L. Z. Cai, Y. R. Wang, X. F. Meng, W. J. Sun, H. Zhang, X. C. Cheng, G. Y. Dong, and X. X. Shen, “Simple direct extraction of unknown phase shift and wavefront reconstruction in generalized phase-shifting interferometry: algorithm and experiments,” Opt. Lett. 33(8), 776–778 (2008).
[CrossRef] [PubMed]

P. Gao, B. Yao, N. Lindlein, J. Schwider, K. Mantel, I. Harder, and E. Geist, “Phase-shift extraction for generalized phase-shifting interferometry,” Opt. Lett. 34(22), 3553–3555 (2009).
[CrossRef] [PubMed]

T. E. Zander, V. Madyastha, A. Patil, P. Rastogi, and L. M. Reindl, “Phase-step estimation in interferometry via an unscented Kalman filter,” Opt. Lett. 34(9), 1396–1398 (2009).
[CrossRef] [PubMed]

B. Pan, Q. Kemao, L. Huang, and A. Asundi, “Phase error analysis and compensation for nonsinusoidal waveforms in phase-shifting digital fringe projection profilometry,” Opt. Lett. 34(4), 416–418 (2009).
[CrossRef] [PubMed]

R. Langoju, A. Patil, and P. Rastogi, “Resolution-enhanced Fourier transform method for the estimation of multiple phases in interferometry,” Opt. Lett. 30(24), 3326–3328 (2005).
[CrossRef] [PubMed]

R. Langoju, A. Patil, and P. Rastogi, “Phase-shifting interferometry in the presence of nonlinear phase steps, harmonics, and noise,” Opt. Lett. 31(8), 1058–1060 (2006).
[CrossRef] [PubMed]

Z. Wang and B. Han, “Advanced iterative algorithm for phase extraction of randomly phase-shifted interferograms,” Opt. Lett. 29(14), 1671–1673 (2004).
[CrossRef] [PubMed]

X. F. Xu, L. Z. Cai, X. F. Meng, G. Y. Dong, and X. X. Shen, “Fast blind extraction of arbitrary unknown phase shifts by an iterative tangent approach in generalized phase-shifting interferometry,” Opt. Lett. 31(13), 1966–1968 (2006).
[CrossRef] [PubMed]

Optik (Stuttg.)

G. Bonsch and H. Bohme, “Phase-determination of Fizeau interferences by phase-shifting interferometry,” Optik (Stuttg.) 82, 161–164 (1989).

Pure Appl. Opt.

P. Picart, R. Mercier, and M. Lamare, “Influence of multiple-beam interferences in a phase-shifting Fizeau interferometer and error-reduced algorithms,” Pure Appl. Opt. 5(2), 167–194 (1996).
[CrossRef]

Other

Y. Surrel, “Fringe analysis,” in Photomechanics, P. K. Rastogi, ed., Vol. 77 of Topics in Applied Physics (Springer, 2000), pp. 55–102.

K. Creath, “Temporal phase measurement method, ” in Interferogram Analysis, D. W. Robinson and G. T. Reid, eds. (Institute of Physics, Bristol, UK, 1993), pp. 94–140.

D. Malacara, M. Servín, and Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, New York, 1998).

R. C. Gonzalez and R. E. Woods, Digital Image Processing, 3rd ed. (Prentice-Hall, 2007).

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Figures (3)

Fig. 1
Fig. 1

Simulation results: (a) the multiple-beam Fizeau interferogram, (b) the wrapped phase, (c) the unwrapped phase and (d) the phase error of the proposed algorithm.

Fig. 2
Fig. 2

The relations between the phase error of the proposed algorithm and (a) reflectivity coefficient (r 2),(b) signal-to-noise ratio (SNR) and (d) the number of frames used (N).

Fig. 3
Fig. 3

Experimental results. (a) multiple-beam Fizeau interferogram,(b) the phase by the proposed PCA algorithm,(c)the phase by least-squares based iterative algorithm, (d) the reference phase, (e) the phase error of (b), and (f) the phase error of (c).

Tables (1)

Tables Icon

Table 1 PV and RMS values of the retrieved phases and the phase errors

Equations (15)

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I n = A [ 1 B 1 C cos ( φ + θ n ) ] + η n
B = ( 1 r 1 2 ) ( 1 r 2 2 ) 1 + r 1 2 r 2 2 , C = 2 r 1 r 2 1 + r 1 2 r 2 2
I n = A [ a 0 2 + j = 1 a j cos j ( φ + θ n ) + η n
a 0 = 2 ( r 1 2 + r 2 2 2 r 1 2 r 2 2 ) 1 r 1 2 r 2 2 , a j = 2 ( 1 r 1 2 ) ( 1 r 2 2 ) r 1 2 r 2 2 1 ( r 1 r 2 ) j , j = 1 , 2 , 3 ,
I n = A a 0 2 + a j j = 1 [ cos ( j θ n ) cos ( j φ ) sin ( j θ n ) sin ( j φ ) ] + η n = A a 0 2 + a j j = 1 [ b j cos ( j φ ) + c j sin ( j φ ) ] + η n
x = 1 M x y = 1 M y { cos [ j φ ( x , y ) ] x = 1 M x y = 1 M y cos [ j φ ( x , y ) ] } { sin [ k φ ( x , y ) ] x = 1 M x y = 1 M y sin [ k φ ( x , y ) ] } = x = 1 M x y = 1 M y { cos [ j φ ( x , y ) ] sin [ k φ ( x , y ) ] } 0
x = 1 M x y = 1 M y { cos [ j φ ( x , y ) ] cos [ k φ ( x , y ) ] } 0 f o r j k x = 1 M x y = 1 M y { sin [ j φ ( x , y ) ] sin [ k φ ( x , y ) ] } 0 f o r j k
I = [ I 1 , I 2 , , I n , , I N ] T
C I = [ I μ I ] [ I μ I ) ] T
C I ϕ i = λ i ϕ i , i = 1 , , N
Φ = [ ϕ 1 , ϕ 2 , , ϕ n , , ϕ N ] T
C I = Φ Λ Φ 1
Y = { y 1 y 2 y N } = Φ ( I μ I ) = { ϕ 1 ϕ 2 ϕ N } ( I μ I )
( I μ I ) = Φ T Y
φ = arctan ( y 2 / y 1 )   or   φ = 0.5 arctan ( y 4 / y 3 )

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