Abstract

We show a practical way for building wideband phase-shifting algorithms for interferometry. The idea presented combines first- and second-order quadrature filters to obtain wideband phase-shifting algorithms. These first- and second-order quadrature filters are analogous to the first- and second-order filters commonly used in communications engineering, named building blocks. We present a systematic way to develop phase-shifting algorithms with large detuning robustness or large bandwidth. In general, the approach presented here gives a powerful frequency analysis and design tool for phase-shifting algorithms robust to detuning for interferometry.

© 2009 Optical Society of America

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References

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1990 (1)

1987 (3)

1985 (1)

1984 (1)

J. E. Greivenkamp, Opt. Eng. (Bellingham) 23, 350 (1984).

1983 (1)

1982 (1)

1975 (1)

1974 (1)

Ai, C.

Brangaccio, D. J.

Bruning, J. H.

Burow, R.

Cheng, Y.-Y.

Eiju, T.

Elssner, K.-E.

Freishlad, K.

Gallagher, J. E.

Greivenkamp, J. E.

J. E. Greivenkamp, Opt. Eng. (Bellingham) 23, 350 (1984).

Grzanna, J.

Hariharan, P.

Herriott, D. R.

Koliopoulos, C. L.

Malacara, D.

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

Malacara, Z.

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

Merkel, K.

Morgan, C. J.

Oreb, B. F.

Rosenfeld, D. P.

Schwider, J.

Servin, M.

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

Spolaczyk, R.

White, A. D.

Wyant, J. C.

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Figures (4)

Fig. 1
Fig. 1

(a) shows the graphic of Eq. (4), and (b) shows the graphic of Eq. (5). (c) and (d) show the graphics of Eqs. (4, 5), shifted (tuned) at ω = ω 0 , respectively.

Fig. 2
Fig. 2

(a) shows the first- and second-order building block filters, and (b) shows the product of these building blocks. We can see in (b) that the quadrature filter obtained removes the dc term at ω = 0 and the frequency component at ω = ω 0 .

Fig. 3
Fig. 3

(a) shows the frequency locations of the basic building blocks used to construct the wideband quadrature filter shown in (b). The obtained quadrature filter is the product of these building blocks.

Fig. 4
Fig. 4

(a) is the ground true phase, (b) is the generated signal, and (c) is the absolute phase error of estimated phase using the wideband phase-shifting algorithm of Eq. (12). Note: The presentation of graphic (b) suffers from aliasing owing to the limited resolution of the software used to generate the graphics.

Equations (12)

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s ( t ) = a x , y + b x , y cos ( ϕ x , y + ω 0 t ) , t I ,
h 1 ( t ) = i 2 [ δ ( t 1 ) δ ( t + 1 ) ] ,
h 2 ( t ) = 2 δ ( t ) δ ( t 1 ) δ ( t + 1 ) ,
H 1 ( ω ) = sin ( ω ) ,
H 2 ( ω ) = 2 2 cos ( ω ) .
H ( ω ) = H 1 ( ω ) H 2 ( ω ω 0 ) = sin ( ω ) [ 2 2 cos ( ω ω 0 ) ] .
h ( t ) = [ 2 δ ( t ) δ ( t 2 ) δ ( t + 2 ) ] sin ( ω 0 ) 2 + i [ 2 δ ( t 1 ) 2 δ ( t + 1 ) ] 2 i [ δ ( t 2 ) δ ( t + 2 ) ] cos ( ω 0 ) 2 .
ϕ x , y = tan 1 [ 2 s ( 1 ) 2 s ( 1 ) [ s ( 2 ) s ( 2 ) ] cos ( ω 0 ) [ 2 s ( 0 ) s ( 2 ) s ( 2 ) ] sin ( ω 0 ) ] ,
ϕ x , y = tan 1 [ 2 s ( 1 ) 2 s ( 1 ) 2 s ( 0 ) s ( 2 ) s ( 2 ) ] ,
H ( ω ) = H 1 ( ω ) H 2 ( ω ω 0 ) H 2 ( ω ω 1 ) H 2 ( ω ω 2 ) ,
h ( t ) = 1 2 δ ( t 4 ) ( 2 2 + 3 ) δ ( t 2 ) + ( 4 2 + 5 ) δ ( t ) ( 2 2 + 3 ) δ ( t + 2 ) + 1 2 δ ( t + 4 ) i [ ( 2 + 1 ) δ ( t 3 ) ( 3 2 + 5 ) δ ( t 1 ) + ( 3 2 + 5 ) δ ( t + 1 ) ( 2 + 1 ) δ ( t + 3 ) ] .
ϕ x , y = tan 1 [ ( 2 + 1 ) s ( 3 ) ( 3 2 + 5 ) s ( 1 ) + ( 3 2 + 5 ) s ( 1 ) ( 2 + 1 ) s ( 3 ) 1 2 s ( 4 ) ( 2 2 + 3 ) s ( 2 ) + ( 4 2 + 5 ) s ( 0 ) ( 2 2 + 3 ) s ( 2 ) + 1 2 s ( 4 ) ] .

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