Abstract

In phase measurement systems that use phase shifting techniques, phase errors that are due to nonsinusoidal waveforms can be minimized by applying synchronous phase shifting algorithms with more than four samples. However, when the phase shift calibration is inaccurate, these algorithms cannot eliminate the effects of nonsinusoidal characteristics. It is shown that, when a number of samples beyond one period of a waveform such as a fringe pattern are taken, phase errors that are due to the harmonic components of the waveform can be eliminated, even when there exists a constant error in the phase shift interval. A general procedure for constructing phase shifting algorithms that eliminate these errors is derived. It is shown that 2j + 3 samples are necessary for the elimination of the effects of higher harmonic components up to the jth order. As examples, three algorithms are derived, in which the effects of harmonic components of low orders can be eliminated in the presence of a constant error in the phase shift interval.

© 1995 Optical Society of America

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References

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    [CrossRef]
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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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  10. K. G. Larkin, B. F. Oreb, “Design and assessment of symmetrical phase-shifting algorithms,” J. Opt. Soc. Am. A 9, 1740–1748 (1992).
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  11. K. Freischlad, C. L. Koliopoulos, “Fourier description of digital phase-measuring interferometry,” J. Opt. Soc. Am. A 7, 542–551 (1990).
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  12. P. Hariharan, B. F. Oreb, T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26, 2504–2506 (1987).
    [CrossRef] [PubMed]

1993

1992

1990

1987

1985

1984

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

1983

1982

1974

1966

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

Brangaccio, D. J.

Brohinsky, W. R.

Bruning, J. H.

Burow, R.

Carre, P.

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

Creath, K.

K. Creath, “Phase measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1988), Vol. 26, pp. 349–393.
[CrossRef]

Eiju, T.

Elssner, K. E.

Freischlad, K.

Gallagher, J. E.

Greivenkamp, J. E.

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

Grzanna, J.

Hariharan, P.

Herriott, D. R.

Koliopoulos, C. L.

Larkin, K. G.

Merkel, K.

Morgan, C. J.

Oreb, B. F.

Rosenfeld, D. P.

Schwider, J.

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

J. Schwider, “Advanced evaluation techniques in interferometry,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1990), Vol. 28, pp. 271–359.
[CrossRef]

Spolaczyk, R.

Stetson, K. A.

Surrel, Y.

White, A. D.

Appl. Opt.

J. Opt. Soc. Am. A

Metrologia

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

Opt. Eng.

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

Opt. Lett.

Other

J. Schwider, “Advanced evaluation techniques in interferometry,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1990), Vol. 28, pp. 271–359.
[CrossRef]

K. Creath, “Phase measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1988), Vol. 26, pp. 349–393.
[CrossRef]

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Figures (2)

Fig. 1
Fig. 1

Fourier transform of the sampling amplitudes bi [F1(ν)] and ai [F2(ν)] for the seven sample algorithm.

Fig. 2
Fig. 2

Fourier transforms of the sampling amplitudes bi [F1(ν)] and ai [F2(ν)] for the 11-sample algorithm.

Tables (1)

Tables Icon

Table 1 Minimum Number of Samples and Typical Phase-Shift Intervals Necessary to Eliminate the Effects of Harmonic Components up to the jth Order in the Presence of a Constant Phase-Shift Errora

Equations (49)

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I ( x , y , α ) = k = 0 s k ( x , y ) cos [ k α ϕ k ( x , y ) ] ,
ϕ = arctan i = 1 m b i I i i = 1 m a i I i ,
I i = I ( α i ) ,
l = { m / 2 for even m ( m + 1 ) / 2 for odd m .
i = 1 m a i I i = s 1 cos ϕ 1 ,
i = 1 m b i I i = s 1 sin ϕ 1 .
i = 1 m a i sin ( k α i ) = 0 ,
i = 1 m a i cos ( k α i ) = δ ( k , 1 ) ,
i = 1 m b i sin ( k α i ) = δ ( k , 1 ) ,
i = 1 m b i cos ( k α i ) = 0 .
n j + 2 .
i = 1 m a i cos ( n 1 ) α i = i = 1 m a i cos α i .
a i = [ 2 / ( j + 2 ) ] cos [ 2 π ( i l ) / ( j + 2 ) ] , i = 1 , . . . , j + 2 ,
b i = [ 2 / ( j + 2 ) ] sin [ 2 π ( i l ) / ( j + 2 ) ] , i = 1 , . . . , j + 2 .
I i = I [ ( 1 + ) α i ] ,
ϕ = arctan { k = 0 i = 1 m b i s k [ cos ( k α i ϕ k ) k α i sin ( k α i ϕ k ) + o ( 2 ) ] k = 0 i = 1 m a i s k [ cos ( k α i ϕ k ) k α i sin ( k α i ϕ k ) + o ( 2 ) ] } ,
ϕ = arctan [ s 1 sin ϕ 1 k = 1 j i = 1 m b i α i k s k sin ( k α i ϕ k ) s 1 cos ϕ 1 k = 1 j i = 1 m a i α i k s k sin ( k α i ϕ k ) ] = ϕ 1 + 2 k = 1 j i = 1 m k s k α i × [ a i sin ( k α i ϕ k ) cos ϕ 1 b i sin ( k α i ϕ k ) sin ϕ 1 ] sin ( 2 ϕ 1 ) + o ( 2 ) ,
i = 1 m b i α i sin ( k α i ϕ k ) sin ϕ 1 = i = 1 m a i α i sin ( k α i ϕ k ) cos ϕ 1
i = 1 m b i ( i l ) ( cos ϕ 1 ) [ sin ( k α i ) cos ϕ k cos ( k α i ) sin ϕ k ] = i = 1 m a i ( i l ) ( sin ϕ 1 ) [ sin ( k α i ) cos ϕ k cos ( k α i ) sin ϕ k ] .
i = 1 m a i ( i l ) sin ( k α i ) = 0 , k = 2 , . . . , j ,
i = 1 m a i ( i l ) cos ( k α i ) = 0 , k = 1 , . . . , j ,
i = 1 m b i ( i l ) sin ( k α i ) = 0 , k = 1 , . . . , j ,
i = 1 m b i ( i l ) cos ( k α i ) = 0 , k = 2 , . . . , j ,
i = 1 m a i ( i l ) sin α i = i = 1 m b i ( i l ) cos α i ,
ϕ = arctan [ 2 ( I 4 I 2 ) I 1 + 2 I 3 I 5 ] ,
( a 1 , a 2 , a 3 , a 4 , a 5 , a 6 , a 7 ) = ( 0 , ¼ , 0 , ½ , 0 , ¼ , 0 ) ,
( b 1 , b 2 , b 3 , b 4 , b 5 , b 6 b 7 ) = ( , 0 , , 0 , , 0 , ) .
ϕ = arctan [ I 1 3 I 3 + 3 I 5 I 7 2 ( I 2 + 2 I 4 I 6 ] ,
I ( α ) = s 0 + s 1 cos ( α ϕ 1 ) + s 2 cos ( 2 α ϕ 2 ) .
Δ ϕ = ϕ ϕ 1 = ( π ) 2 [ s 2 s 1 sin ϕ 1 cos ϕ 2 1 16 sin ( 2 ϕ 1 ) ] + o ( 3 ) ,
ϕ = arctan [ 3 ( I 2 4 I 3 7 I 4 6 I 5 + 6 I 7 + 7 I 8 + 4 I 9 + I 10 ) 2 I 1 5 I 2 6 I 3 I 4 + 8 I 5 + 12 I 6 + 8 I 7 I 8 6 I 9 5 I 10 2 I 11 ] ,
I ( α ) = s 0 + k = 1 4 s k cos ( k α ϕ k ) ,
Δ ϕ = ϕ ϕ 1 = ( π ) 2 9 [ 2 3 sin ( 2 ϕ 1 ) + s 2 s 1 sin ( ϕ 1 + ϕ 2 ) + 3 s 3 s 1 sin ϕ 1 cos ϕ 3 + 4 s 4 s 1 sin ( ϕ 1 ϕ 4 ) ] + o ( 3 ) .
f 1 ( α ) = i = 1 m b i δ ( α α i ) ,
f 2 ( α ) = i = 1 m a i δ ( α α i ) ,
ϕ = arctan [ f 1 ( α ) I ( α ) d α f 2 ( α ) I ( α ) d α ] = arctan [ F 1 ( ν ) J ( ν ) d ν F 2 ( ν ) J ( ν ) d υ ] ,
F 1 ( ν ) = ( 1 / 4 ) [ 3 sin ( π ν / 2 ) sin ( 3 π ν / 2 ) ] ,
F 2 ( ν ) = ½ [ ( 1 cos ( π ν ) ] .
i = 1 m a 1 sin ( k α i ) = 0 , k = 1 , . . . , j ,
i = 1 m a 1 cos ( k α i ) = δ ( k , 1 ) , k = 0 , 1 , . . . , j ,
i = 1 m b i sin ( k α i ) = δ ( k , 1 ) , k = 1 , . . . , j ,
i = 1 m b i cos ( k α i ) = 0 , k = 0 , 1 , . . . , j ,
i = 1 m a i ( i l ) sin ( k α i ) = 0 , k = 2 , . . . , j ,
i = 1 m a i ( i l ) cos ( k α i ) = 0 , k = 1 , . . . , j ,
i = 1 m b i ( i l ) sin ( k α i ) = 0 , k = 1 , . . . , j ,
i = 1 m b i ( i l ) cos ( k α i ) = 0 , k = 2 , . . . , j ,
i = 1 m a i ( i l ) sin α i = i = 1 m b i ( i l ) cos α i .
i = 1 m a i cos [ ( j p ) α i ] = i = 1 m a i cos [ ( p + 2 ) α i ] ,
i = 1 m a i sin [ ( j p ) α i ] = i = 1 m a i sin [ ( p + 2 ) α i ] ,

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