Abstract

A random phase-shifting interferometry insensitive to environmental noises is proposed. The relationship between intensity and phase in each pixel is obtained from a large amount of phase-shifting interferograms. In the phase-solving algorithm, the phase shift step length is not taken as a parameter, but the temporal intensity maximum and minimum in each pixel are needed. For finding the extreme values, random passive phase shifts caused by environmental noises are adopted to make the intensity ergodic. Supplementary active phase shifts, which are not accurately controlled or calibrated, are performed to shorten the measurement cycle. Finally, averaging statistically uncorrelated data over a long enough period of time can effectively reduce most random errors. A minitype Fizeau interferometer applying this random phase-shifting method demonstrated the feasibility of it.

© 2009 Optical Society of America

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References

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2004

D. Wu, R. Zhu, L. Chen, and J. Li, Optik (Jena) 115, 343 (2004).

2002

J. Hayes, Laser Focus World 38, 109 (2002).

2001

1995

Brock, N.

Chen, L.

D. Wu, R. Zhu, L. Chen, and J. Li, Optik (Jena) 115, 343 (2004).

DeGroot, P.

Frejlich, J.

Freschi, A. A.

Hao, Q.

Q. Hao, Q. Zhu, Y. Hu, and L. Tang, Chinese patent application 200810188355.2 (filed Dec. 25, 2008).

Hayes, J.

Hu, Y.

Q. Hao, Q. Zhu, Y. Hu, and L. Tang, Chinese patent application 200810188355.2 (filed Dec. 25, 2008).

Li, J.

D. Wu, R. Zhu, L. Chen, and J. Li, Optik (Jena) 115, 343 (2004).

Meneses-Fabian, C.

Millerd, J.

Ngoi, B. K. A.

North-Morris, M.

Novak, M.

Robledo-Sanchez, C.

Rodriguez-Zurita, G.

Sivakumar, N. R.

Tang, L.

Q. Hao, Q. Zhu, Y. Hu, and L. Tang, Chinese patent application 200810188355.2 (filed Dec. 25, 2008).

Toto-Arellano, N.

Vazquez-Castillo, J. F.

Venkatakrishnan, K.

Wu, D.

D. Wu, R. Zhu, L. Chen, and J. Li, Optik (Jena) 115, 343 (2004).

Wyant, J.

Zhu, Q.

Q. Hao, Q. Zhu, Y. Hu, and L. Tang, Chinese patent application 200810188355.2 (filed Dec. 25, 2008).

Zhu, R.

D. Wu, R. Zhu, L. Chen, and J. Li, Optik (Jena) 115, 343 (2004).

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Figures (3)

Fig. 1
Fig. 1

Intensity curve against the frame index.

Fig. 2
Fig. 2

Configuration of a minitype interferometer adopting the random phase-shifting method. BE, beam expander; BS, beam splitter; CL, collimating lens; SP, standard plane; WP, workpiece; PZT, piezoelectric transducer.

Fig. 3
Fig. 3

Wavefront measurement result of a plane. (a) Typical interferograms distorted. (b) Gray scale curve against the frame index of an arbitrary pixel. (c) Reconstructed wavefront of the plane.

Equations (7)

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I ( x , y , i ) = [ I b ( x , y ) + I a ( x , y ) ] { cos [ Φ ( x , y ) + δ ( x , y , i ) ] } ,
{ I max ( x , y ) = I b ( x , y ) + I a ( x , y ) I min ( x , y ) = I b ( x , y ) I a ( x , y ) } .
{ I b ( x , y ) = [ I max ( x , y ) + I min ( x , y ) ] 2 I a ( x , y ) = [ I max ( x , y ) I min ( x , y ) ] 2 } .
φ ( x , y , i ) = arccos [ I ( x , y , i ) I b ( x , y ) I a ( x , y ) ] .
φ ( x , y ) = 1 N i = 1 N W [ φ ( x , y , i ) ] ,
φ ( x , y , i ) = Φ ( x , y ) + [ δ a ( i ) + δ p 1 ( i ) ] + δ p 2 ( x , y , i ) ,
φ ( x , y ) = W [ Φ ( x , y ) ] + 1 N i = 1 N [ δ a ( i ) + δ p 1 ( i ) ] + 1 N i = 1 N δ p 2 ( x , y , i ) .

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