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

2012

2005

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

2003

2001

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

2000

1999

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

1998

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

1996

1995

1993

1992

1990

1987

1985

1984

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

1983

1975

1974

1966

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.

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

Meas. Sci. Technol.

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

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.

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

Opt. Rev.

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.)

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

Other

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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