Abstract

The objective of this paper is to describe a novel method for phase extraction from multiple-beam Fizeau interferograms with arbitrary phase shifts. The approach begins with applying FFT method to estimate the phase shifts and then utilizes least-squares iterative algorithm to extract phase and phase shifts simultaneously. If the spatial carrier frequency of the fringes is high enough to separate the phase of the first-order maximum in the Fourier domain, the proposed method requires only two iterative cycles to accurately extract phase information from seven multiple-beam Fizeau interferograms with arbitrary phase shifts. Numerical simulations and experiments demonstrate the effectiveness of the proposed algorithm. A comprehensive analysis of the influences of systematic errors (spatial carrier frequency, reflectivity coefficient, and random noise) on the evaluation of phase shifts and phase is presented. The method has applications in high precision interferometry.

© 2008 Optical Society of America

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References

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    [CrossRef]
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2006

2005

2004

2001

2000

Y. Surrell, "Fringe Analysis," Top. Appl. Phys. 77, 55-102 (2000).

1999

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

1998

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

1997

1996

1995

1993

1989

G. Bonsch and H. Bohme, ‘‘Phase-determination of Fizeau interferences by phase-shifting interferometry,’’ Optik 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, 899-902 (1967).
[CrossRef]

Ai, C.

Blanco-Garcia, J.

Bohme, H.

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

Bokor, J.

Bonsch, G.

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

Burow, R.

Cai, L. Z.

Chen, M.

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, 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, 899-902 (1967).
[CrossRef]

Dong, G. Y.

Dorrio, B. V.

Dorrío, B. V.

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

Doval, A. F.

Elssner, K.E.

Farrant, D. I.

Fernandez, J. L.

Fernández, J. L.

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

Goldberg, K. A.

Grzanna, J.

Guo, H.

Han, B.

Hariharan, P.

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

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

Hibino, K.

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, 167-194 (1996)
[CrossRef]

Langoju, R.

Larkin, K. G.

Lopez, C.

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, 167-194 (1996)
[CrossRef]

Merkel, K.

Oreb, B. F.

Patil, A.

Perez-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, 167-194 (1996)
[CrossRef]

Rastogi, P.

Schwider, J.

Shen, X. X.

Soto, R.

Spolaczyk, R.

Surrel, Y.

Surrell, Y.

Y. Surrell, "Fringe Analysis," Top. Appl. Phys. 77, 55-102 (2000).

Wang, Z.

Wyant, J. C.

Xu, X. F.

Zhao, B.

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, 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, R33-R55 (1999).
[CrossRef]

Opt. Eng.

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

Opt. Express

Opt. Lett.

Optik

G. Bonsch and H. Bohme, ‘‘Phase-determination of Fizeau interferences by phase-shifting interferometry,’’ Optik 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, 167-194 (1996)
[CrossRef]

Top. Appl. Phys.

Y. Surrell, "Fringe Analysis," Top. Appl. Phys. 77, 55-102 (2000).

Other

K. Creath, ‘‘Temporal phase measurement method,’’ in Interferogram Analysis, D.W. Robinson and G. T. Reid, eds., (Inst. of Phys., Bristol, UK, 1993) pp. 94-140.

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

D. C.  Ghiglia and M. D.  Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms and Software (John Wiley and Sons Inc., New York, 1998).

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Figures (7)

Fig. 1.
Fig. 1.

Schematic drawing of frequency spectrum of multiple-beam Fizeau interferogram with spatial carrier frequency k 0.(a) r 1=0.2, r 2=0.7 and (b) r 1=0.7, r 2=0.9

Fig. 2.
Fig. 2.

(a). multiple-beam Fizeau interferogram, (b). phase obtained by the proposed iterative algorithm, and (c) and (d) the phase errors for no iteration and two iterative cycles respectively.

Fig. 3.
Fig. 3.

(a). phase-shift error and (b) phase error for no iteration and two iterative cycles at different spatial carrier frequencies.

Fig. 4.
Fig. 4.

(a). phase-shift error and (b) phase error for no iteration and two iterative cycles at different reflectivity coefficients of test surface.

Fig. 5.
Fig. 5.

(a). phase-shift error and (b) phase error for no iteration and two iterative cycles at different noise levels.

Fig. 6.
Fig. 6.

(a). phase-shift error and (b) phase error for no iteration and two iterative cycles at different quantization bits.

Fig. 7.
Fig. 7.

Experimental results. (a) multiple-beam Fizeau interferogram, (b) the intensity gray at the middle row of Fig. 7(a), (c) phase by the proposed algorithm and (d) phase by Zygo’s standard interferometer with traditional algorithm.

Tables (1)

Tables Icon

Table 1. Phase shifts for FFT-estimation (no iteration) and two iterative cycles. Unit: radian.

Equations (27)

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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 1 2 1 ( r 1 r 2 ) j , j = 1 , 2 , 3 ,
φ ( r ) = Φ 0 ( r ) + k 0 r
I n ( r ) = A ( r ) { a 0 2 + j = 1 a j exp ( ij θ n ) exp [ ij ( Φ 0 ( r ) + k 0 r ) ] +
j = 1 a j exp ( ij θ n ) exp [ ij ( Φ 0 ( r ) + k 0 r ) ] } + η n ( r )
G n ( k ) = A 0 ( k ) + j = 1 exp ( ij θ n ) C j ( k j k 0 ) + j = 1 exp ( ij θ n ) C j * ( k + j k 0 ) + ζ n ( k )
G n ( k 0 ) = A 0 ( k 0 ) + exp ( i θ n ) C 1 ( 0 ) + j = 2 exp ( ij θ n ) C j ( ( 1 j ) k 0 ) +
j = 1 exp ( ij θ n ) C j * ( ( 1 + j ) k 0 ) + ζ n ( k 0 ) exp ( i θ n ) C 1 ( 0 )
θ n tan 1 [ Im ( G n ( k 0 ) ) Re ( G n ( k 0 ) ) ] θ 0
I n = b 0 + j = 1 3 b j cos [ j ( φ + θ n ) ] + j = 4 P b j cos [ j ( φ + θ n ) ] + η n
I n g n = i = 0 6 X j S j ( n ) + η n
g n = j = 4 P b j cos [ j ( φ + θ n ) ]
E ( X 0 , X 1 , X 6 ) = n = 1 N [ I n g n i = 0 6 X i S i ( n ) ] 2
AX = Y
A ik = n = 1 N S i ( n ) S k ( n ) i , k = 0 , 1 , 2 , 6
Y i = n = 1 N ( I n g n ) S i ( n ) i = 0 , 1 , 2 , 6
φ = tan 1 ( X 2 X 1 )
b j = X 2 j 1 2 + X 2 j 2 j = 1 , 2 , 3
b j = b 2 j 1 b 1 j 1 j > 3
E ( S 0 ( n ) , S 1 ( n ) S 6 ( n ) ) = m = 1 M [ I n ( m ) g n ( m ) i = 0 6 X i ( m ) S i ( n ) ] 2
B S ' ( n ) = Z ( n )
B ik = m = 1 M X i ' ( m ) X k ' ( m ) i , k = 0 , 1 , 2 , 3 6
Z k ( n ) = m = 1 M ( I n ( m ) g n ( m ) X k ( m ) k = 0 , 1 , 2 6 ; n = 1 , 2 N
θ n = tan 1 ( S 2 ( n ) S 1 ( n ) )
max ( θ n q θ 1 q ) ( θ n q 1 θ 1 q 1 ) < ε

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