Abstract

In phase shifting interferometry, the fringe contrast is preferred to be at a maximum when there is no phase shift error. In the measurement of highly-reflective surfaces, the signal contrast is relatively low and the measurement would be aborted when the contrast falls below a threshold value. The fringe contrast depends on the design of the phase shifting algorithm. The condition for achieving the fringe contrast maximum is derived as a set of linear equations of the sampling amplitudes. The minimum number of samples necessary for constructing an error-compensating algorithm that is insensitive to the jth harmonic component and to the phase shift error is discussed. As examples, two new algorithms (15-sample and (3N2)-sample) were derived that are useful for the measurement for highly-reflective surfaces.

© 2014 Optical Society of America

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  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. Creath, “Phase measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (North-Holland, 1988).
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    [CrossRef]
  4. J. C. Wyant, “Use of an ac heterodyne lateral shear interferometer with real-time wavefront correction systems,” Appl. Opt. 14(11), 2622–2626 (1975).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  15. K. Hibino, “Error-compensating phase measuring algorithms in a Fizeau interferometer,” Opt. Rev. 6(6), 529–538 (1999).
    [CrossRef]
  16. P. de Groot, “Measurement of transparent plates with wavelength-tuned phase-shifting interferometry,” Appl. Opt. 39(16), 2658–2663 (2000).
    [CrossRef] [PubMed]
  17. 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]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  23. K. Patorski, Z. Sienicki, and A. Styk, “Phase-shifting method contrast calculations in time-averaged interferometry: error analysis,” Opt. Eng. 44(6), 065601 (2005).
    [CrossRef]
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    [CrossRef]
  25. R. Juarez-Salazar, C. Robledo-Sanchez, F. Guerrero-Sanchez, and A. Rangel-Huerta, “Generalized phase-shifting algorithm for inhomogeneous phase shift and spatio-temporal fringe visibility variation,” Opt. Express 22(4), 4738–4750 (2014).
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    [CrossRef] [PubMed]

2014 (1)

2012 (1)

2005 (1)

K. Patorski, Z. Sienicki, and A. Styk, “Phase-shifting method contrast calculations in time-averaged interferometry: error analysis,” Opt. Eng. 44(6), 065601 (2005).
[CrossRef]

2004 (2)

2003 (1)

2001 (1)

C. S. Vikram, “Phase error effect on contrast measurement in Schwider-Hariharan phase-shifting algorithm,” Optik (Stuttg.) 112(3), 140–141 (2001).
[CrossRef]

2000 (1)

1999 (1)

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

1998 (1)

A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Self-calibrating algorithm for three-sample phase-shift interferometry by contrast levelling,” Meas. Sci. Technol. 9(5), 744–750 (1998).
[CrossRef]

1997 (1)

1996 (1)

1995 (3)

1993 (1)

1992 (1)

1990 (1)

1987 (1)

1985 (1)

1984 (1)

J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23(4), 350–352 (1984).
[CrossRef]

1983 (1)

1975 (1)

1974 (1)

1966 (1)

P. Carre, “Installation et utilisation du comparateur photoelectrique et interferential du Bureau International des Poids et Mesures,” Metrologia 2(1), 13–23 (1966).
[CrossRef]

Apostol, D.

A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Self-calibrating algorithm for three-sample phase-shift interferometry by contrast levelling,” Meas. Sci. Technol. 9(5), 744–750 (1998).
[CrossRef]

Brangaccio, D. J.

Brohinsky, W. R.

Bruning, J. H.

Burke, J.

Burow, R.

Carre, P.

P. Carre, “Installation et utilisation du comparateur photoelectrique et interferential du Bureau International des Poids et Mesures,” Metrologia 2(1), 13–23 (1966).
[CrossRef]

Creath, K.

Damian, V.

A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Self-calibrating algorithm for three-sample phase-shift interferometry by contrast levelling,” Meas. Sci. Technol. 9(5), 744–750 (1998).
[CrossRef]

de Groot, P.

Dobroiu, A.

A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Self-calibrating algorithm for three-sample phase-shift interferometry by contrast levelling,” Meas. Sci. Technol. 9(5), 744–750 (1998).
[CrossRef]

Eiju, T.

Elssner, K. E.

Fairman, P. S.

Farrant, D. I.

Freischlad, K.

Gallagher, J. E.

García-Márquez, J.

Greivenkamp, J. E.

J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23(4), 350–352 (1984).
[CrossRef]

Groot, P.

Grzanna, J.

Guerrero-Sanchez, F.

Hanayama, R.

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.

Juarez-Salazar, R.

Koliopoulos, C. L.

Larkin, K. G.

Malacara-Doblado, D.

Merkel, K.

Mitsuishi, M.

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]

Nascov, V.

A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Self-calibrating algorithm for three-sample phase-shift interferometry by contrast levelling,” Meas. Sci. Technol. 9(5), 744–750 (1998).
[CrossRef]

Oreb, B. F.

Patorski, K.

K. Patorski, Z. Sienicki, and A. Styk, “Phase-shifting method contrast calculations in time-averaged interferometry: error analysis,” Opt. Eng. 44(6), 065601 (2005).
[CrossRef]

Rangel-Huerta, A.

Robledo-Sanchez, C.

Rosenfeld, D. P.

Schmit, J.

Schwider, J.

Sienicki, Z.

K. Patorski, Z. Sienicki, and A. Styk, “Phase-shifting method contrast calculations in time-averaged interferometry: error analysis,” Opt. Eng. 44(6), 065601 (2005).
[CrossRef]

Spolaczyk, R.

Stetson, K. A.

Styk, A.

K. Patorski, Z. Sienicki, and A. Styk, “Phase-shifting method contrast calculations in time-averaged interferometry: error analysis,” Opt. Eng. 44(6), 065601 (2005).
[CrossRef]

Surrel, Y.

Téllez-Quiñones, A.

Vikram, C. S.

C. S. Vikram, “Phase error effect on contrast measurement in Schwider-Hariharan phase-shifting algorithm,” Optik (Stuttg.) 112(3), 140–141 (2001).
[CrossRef]

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.

Wyant, J. C.

Appl. Opt. (12)

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]

J. C. Wyant, “Use of an ac heterodyne lateral shear interferometer with real-time wavefront correction systems,” Appl. Opt. 14(11), 2622–2626 (1975).
[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]

K. A. Stetson and W. R. Brohinsky, “Electro-optic holography and its application to hologram interferometry,” Appl. Opt. 24(21), 3631–3637 (1985).
[CrossRef] [PubMed]

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

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

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]

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

Meas. Sci. Technol. (1)

A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Self-calibrating algorithm for three-sample phase-shift interferometry by contrast levelling,” Meas. Sci. Technol. 9(5), 744–750 (1998).
[CrossRef]

Metrologia (1)

P. Carre, “Installation et utilisation du comparateur photoelectrique et interferential du Bureau International des Poids et Mesures,” Metrologia 2(1), 13–23 (1966).
[CrossRef]

Opt. Eng. (2)

J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23(4), 350–352 (1984).
[CrossRef]

K. Patorski, Z. Sienicki, and A. Styk, “Phase-shifting method contrast calculations in time-averaged interferometry: error analysis,” Opt. Eng. 44(6), 065601 (2005).
[CrossRef]

Opt. Express (1)

Opt. Rev. (2)

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

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]

Optik (Stuttg.) (1)

C. S. Vikram, “Phase error effect on contrast measurement in Schwider-Hariharan phase-shifting algorithm,” Optik (Stuttg.) 112(3), 140–141 (2001).
[CrossRef]

Other (1)

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

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Figures (4)

Fig. 1
Fig. 1

Interferogram of (a) mirror (R = 60% at wavelength 632.8 nm) and (b) fused silica plate (R = 4% at wavelength 632.8 nm).

Fig. 2
Fig. 2

Sampling functions of (a) synchronous detection (4-sample), (b) Scwider-Hariharan 5-sample, (c) Larkin-Oreb N + 1 (N = 6), (d) Surrel 2N1 (N = 6), (e) Hibino 11-sample, (f) de Groot 13-sample algorithm.

Fig. 3
Fig. 3

Fringe contrast of Schwider-Hariharan 5-sample algorithm.

Fig. 4
Fig. 4

Fourier transforms of the sampling amplitudes for (a) 15-sample algorithm, (b) 3N2 algorithm (N = 15).

Equations (34)

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I( x,y, α r )= I 0 ( x,y ){ 1+γcos[ α r φ( x,y ) ] },
φ*=arctan r=1 M b r I( α r ) r=1 M a r I( α r ) ,
γ= 1 A [ r=1 M a r I( α r ) ] 2 + [ r=1 M b r I( α r ) ] 2 ,
A= 1 M r=1 M I( α r ) .
α r = α 0r [ 1+ ε 1 + ε 2 α 0r π + ε 3 ( α 0r π ) 2 ++ ε p ( α 0r π ) p1 ],
δγ γ = δA A + δ{ [ r=1 M a r I( α r ) ] 2 + [ r=1 M b r I( α r ) ] 2 } 2 [ r=1 M a r I( α r ) ] 2 + [ r=1 M b r I( α r ) ] 2 ,
dγ d ε 1 | ε 1 =0 =0.
r=1 M a r cos( φ α 0r ) r=1 M α 0r a r sin( φ α 0r ) + r=1 M b r cos( φ α 0r ) r=1 M α 0r b r sin( φ α 0r ) = 1 2 ( r=1 M α 0r a r sin α 0r r=1 M α 0r b r cos α 0r ) + 1 2 cos2φ( r=1 M α 0r a r sin α 0r + r=1 M α 0r b r cos α 0r )=0,
r=1 M a r sin( m α 0r ) =0form=1,2,,j,
r=1 M a r cos( m α 0r ) =δ( m,1 )form=0,1,,j,
r=1 M b r sin( m α 0r ) =δ( m,1 )form=1,2,,j,
r=1 M b r cos( m α 0r ) =0form=0,1,,j.
r=1 M α 0r a r sin α 0r =0.
r=1 M α 0r b r cos α 0r =0.
r=1 M α 0r ( a r cos α 0r + b r sin α 0r ) =0,
r=1 M α 0r ( a r cos α 0r b r sin α 0r ) =0,
r=1 M α 0r ( a r sin α 0r + b r cos α 0r ) =0.
r=1 M α 0r a r sin( m α 0r ) =0,
r=1 M α 0r a r cos( m α 0r ) =0,
r=1 M α 0r b r sin( m α 0r ) =0,
r=1 M α 0r b r cos( m α 0r ) =0,
a r = a M+1r , b r = b M+1r , α r = α M+1r ,
M=( j+2 )+2+( j1 ) =2j+3 =2N1,
f 1 ( α )= r=1 M b r δ( α α r ) ,
f 2 ( α )= r=1 M a r δ( α α r ) ,
F 1 ( ν )= r=1 M b r exp( i α r ν ) =i r=1 M b r sin( α r ν ) ,
F 2 ( ν )= r=1 M a r exp( i α r ν ) = r=1 M a r cos( α r ν ) ,
di F 1 dν | ν=1 = r=1 M α 0r b r cos α 0r ,
d F 2 dν | ν=1 = r=1 M α 0r a r sin α 0r .
a r =( a 1 , a 2 , a 3 , a 4 , a 5 , a 6 , a 7 , a 8 , a 7 , a 6 , a 5 , a 4 , a 3 , a 2 , a 1 ), b r =( b 1 , b 2 , b 3 , b 4 , b 5 , b 6 , b 7 , b 8 , b 7 , b 6 , b 5 , b 4 , b 3 , b 2 , b 1 ),
a 1 = 2 64 , a 2 =0, a 3 = 3 2 64 , a 4 = 1 8 , a 5 = 5 2 64 , a 6 =0, a 7 = 7 2 64 , a 8 = 1 4 , b 1 = 2 64 , b 2 = 1 16 , b 3 = 3 2 64 , b 4 =0, b 5 = 5 2 64 , b 6 = 3 16 , b 7 = 7 2 64 , b 8 =0.
tanφ*= ( I 1 I 15 )+2 2 ( I 2 I 14 )+3( I 3 I 13 )5( I 5 I 11 )6 2 ( I 6 I 10 )7( I 7 I 9 ) ( I 1 + I 15 )3( I 3 + I 13 )4 2 ( I 4 + I 12 )5( I 5 + I 11 )+7( I 7 + I 9 )+8 2 I 8 ,
a r = 2 N w r cos 2π N ( r 3N1 2 ), b r = 2 N w r sin 2π N ( r 3N1 2 ),
w r = 1 N 2 [ 1 2 r( r+1 ) ]( 1rN ), w r = 1 N 2 [ 1 2 N( N+1 )+( rN )( 2Nr1 ) ]( N+1r2N2 ), w r = 1 N 2 [ 1 2 ( 3Nr1 )( 3Nr ) ]( 2N1r3N2 ).

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