Abstract

This Letter presents a simple radial shearing interferometery with circular gratings to measure the radial first-order derivative of arbitrary wavefront phase. Two spatial phase-shifted radial shearing interferograms can be simultaneously obtained by the system, and a two-step phase-shifting algorithm with arbitrary phase-shift value is proposed for phase retrieval. The measurement for spherical wave has shown the feasibility and validity, and the optical system is used to measure the radial first-order derivative of projection wavefront phase of a propane flame with plane incident wave.

© 2013 Optical Society of America

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References

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2012 (1)

2011 (1)

Y.-y. Chen, Y. Song, Z.-h. Li, and A.-z. He, Opt. Commun. 284, 2648 (2011).
[CrossRef]

2009 (2)

1996 (1)

M. D. Pritt, IEEE Trans. Geosci. Remote Sens. 34, 728 (1996).
[CrossRef]

1989 (2)

Q.-S. Ru, N. Ohyama, T. Honda, and J. Tsujiuchi, Appl. Opt. 28, 3350 (1989).
[CrossRef]

Q.-S. Ru, N. Ohyama, and T. Honda, Opt. Commun. 69, 189 (1989).
[CrossRef]

1988 (1)

1981 (1)

1972 (2)

A. W. Lohmann and D. E. Silva, Opt. Commun. 4, 326 (1972).
[CrossRef]

D. E. Silva, Appl. Opt. 11, 2613 (1972).
[CrossRef]

1971 (1)

1970 (1)

Bar-Ziv, E.

Bryngdahl, O.

Buzug, T. M.

T. M. Buzug, Computed Tomography (Springer, 2008), p. 303.

Cai, L. Z.

Chen, Y.

Chen, Y.-y.

Y.-y. Chen, Y. Song, Z.-h. Li, and A.-z. He, Opt. Commun. 284, 2648 (2011).
[CrossRef]

Glatt, I.

Guo, J. P.

He, A.

He, A.-z.

J. Wang, Y. Song, Z.-h. Li, N. Sun, and A.-z. He, J. Opt. Soc. Am. A 29, 1686 (2012).
[CrossRef]

Y.-y. Chen, Y. Song, Z.-h. Li, and A.-z. He, Opt. Commun. 284, 2648 (2011).
[CrossRef]

Honda, T.

Q.-S. Ru, N. Ohyama, and T. Honda, Opt. Commun. 69, 189 (1989).
[CrossRef]

Q.-S. Ru, N. Ohyama, T. Honda, and J. Tsujiuchi, Appl. Opt. 28, 3350 (1989).
[CrossRef]

Kafri, O.

Keren, E.

Li, A. M.

Li, Z.-h.

J. Wang, Y. Song, Z.-h. Li, N. Sun, and A.-z. He, J. Opt. Soc. Am. A 29, 1686 (2012).
[CrossRef]

Y.-y. Chen, Y. Song, Z.-h. Li, and A.-z. He, Opt. Commun. 284, 2648 (2011).
[CrossRef]

Lohmann, A. W.

A. W. Lohmann and D. E. Silva, Opt. Commun. 4, 326 (1972).
[CrossRef]

Meng, X. F.

Ohyama, N.

Q.-S. Ru, N. Ohyama, T. Honda, and J. Tsujiuchi, Appl. Opt. 28, 3350 (1989).
[CrossRef]

Q.-S. Ru, N. Ohyama, and T. Honda, Opt. Commun. 69, 189 (1989).
[CrossRef]

Peng, X.

Pritt, M. D.

M. D. Pritt, IEEE Trans. Geosci. Remote Sens. 34, 728 (1996).
[CrossRef]

Ru, Q.-S.

Q.-S. Ru, N. Ohyama, and T. Honda, Opt. Commun. 69, 189 (1989).
[CrossRef]

Q.-S. Ru, N. Ohyama, T. Honda, and J. Tsujiuchi, Appl. Opt. 28, 3350 (1989).
[CrossRef]

Silva, D. E.

A. W. Lohmann and D. E. Silva, Opt. Commun. 4, 326 (1972).
[CrossRef]

D. E. Silva, Appl. Opt. 11, 2613 (1972).
[CrossRef]

Song, Y.

Sun, N.

Tsujiuchi, J.

Wang, J.

Wang, Y. R.

Zhao, Z.

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Figures (7)

Fig. 1.
Fig. 1.

Optical schematic diagram of radial shearing interferometery.

Fig. 2.
Fig. 2.

(a) Filter 1 and (b) Filter 2 (transmitting within pink dotted area).

Fig. 3.
Fig. 3.

Interferograms of spherical wave (a) original interferograms, (b) noise removed by Gaussian filter, and (c) background removed by low-pass filter.

Fig. 4.
Fig. 4.

Radial first-order derivative of spherical wave phase (a) extracted by two-step phase-shifting method and (b) computed by Eq. (6).

Fig. 5.
Fig. 5.

Comparison of experimental result and simulation result (Y=160).

Fig. 6.
Fig. 6.

Interferograms with propane flame (a) original interferograms, (b) noise removed by Gaussian filter, and (c) background removed by low-pass filter.

Fig. 7.
Fig. 7.

(a) Self-light image of propane flame and (b) radial first-order derivative of wavefront phase in tested zone.

Equations (6)

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u1(r,φ)exp[ikΦ(r,φ)],
I1(r,φ)=M{1+cos[2πΦ(r,φ)rΔaπλΔa2]}I+1(r,φ)=M{1+cos[2πΦ(r,φ)rΔa+πλΔa2]},
tan[2πΔaΦ(r,φ)r]=I1(r,φ)I+1(r,φ)I1(r,φ)+I+1(r,φ)cotπλΔa2,
W[2πΦ(r,φ)rΔa]=tan1[I1(r,φ)I+1(r,φ)I1(r,φ)+I+1(r,φ)cotπλΔa2].
Φ(r,φ)r=a2πΔ{W[2πΦ(r,φ)rΔa]±2Kπ}(K=0,1,2,).
D2+r2r=rD2+r2.

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