Abstract

The use of a SLM for the three-wave lateral shearing interference is proposed, and an eight-step phase-shifting scheme is developed for extracting phase information from three-wave interferograms. The two-dimensional phase of object is reconstructed from two phase differences which are calculated from two orthogonal sheared interferograms. The flexibility of SLM can be fully utilized in the sense of dynamical controlling of the direction and amount of shear, as well as phase shift. The numerical simulation and optical experiment are carried out to demonstrate the feasibility and reliability of the proposed scheme.

© 2009 Optical Society of America

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References

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2006

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2003

2002

J. Dias and J. Leitao, "The ZπM algorithm for interferometric image reconstruction in SAR/SAS." IEEE Trans. Image Processing 11, 408-422 (2002).
[CrossRef]

2001

2000

1997

1996

1995

1988

1986

1980

Arrizón, V.

Bitou, Y.

Chung, P.

S. Zhao and P. Chung, "Digital speckle shearing interferometer using a liquid-crystal spatial light modulator," Opt. Eng. 45, 105606 (2006).
[CrossRef]

Dainty, C.

Dias, J.

J. Dias and J. Leitao, "The ZπM algorithm for interferometric image reconstruction in SAR/SAS." IEEE Trans. Image Processing 11, 408-422 (2002).
[CrossRef]

Ding, J.

Dubra, A.

Elster, C.

Flynn, T. J.

Fornaro, G.

Franceschetti, G.

Freischlad, K. R.

Galizzi, G. E.

Griffin, D. W.

Guo, C.

Itoh, M.

Jin, Z.

Kaufmann, G. H.

Kerr, D.

Koliopoulos, C. L.

Lanari, R.

Lee, H. H

Leitao, J.

J. Dias and J. Leitao, "The ZπM algorithm for interferometric image reconstruction in SAR/SAS." IEEE Trans. Image Processing 11, 408-422 (2002).
[CrossRef]

Liang, P.

Park, S. H

Paterson, C.

Ribak, E.

Sánchez-de-la-Llave, D.

Sansosti, E.

Southwell, W. H.

Takahashi, T.

Takajo, H.

Talmi, A

Tian, X.

Wang, H.

Yatagai, T.

You, J. H

Zhao, S.

S. Zhao and P. Chung, "Digital speckle shearing interferometer using a liquid-crystal spatial light modulator," Opt. Eng. 45, 105606 (2006).
[CrossRef]

Appl. Opt.

IEEE Trans. Image Processing

J. Dias and J. Leitao, "The ZπM algorithm for interferometric image reconstruction in SAR/SAS." IEEE Trans. Image Processing 11, 408-422 (2002).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Eng.

S. Zhao and P. Chung, "Digital speckle shearing interferometer using a liquid-crystal spatial light modulator," Opt. Eng. 45, 105606 (2006).
[CrossRef]

Opt. Express

Opt. Lett.

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Figures (6)

Fig. 1.
Fig. 1.

Schematic of the arrangement for generation of three-wave shearing interference. A lateral shearing interferogram is formed in the fringed area. S denotes the shear amount.

Fig. 2.
Fig. 2.

(a). Original phase used for simulation. (b). phase difference in x direction. (c). phase difference in y direction. (d). Reconstructed phase from the two phase differences after 500 iterations. (e). Relationship between reconstruction error (root mean square) and the number of iterations.

Fig. 3.
Fig. 3.

The results of simulation useing our method with 10% noise level in phase differences of shear amount s=10: (a) phase difference in x direction. (b) phase difference in y direction (c) reconstructed wave front.

Fig. 4.
Fig. 4.

(a). Optical arrangement of three-wave lateral shearing interferometer( L1, L2, imaging lenses; R-D, rotating diffuser); (b) three-wave interferogram with shear in x direction; (c) three-wave interferogram with shear in y direction

Fig. 5.
Fig. 5.

Test results of a spherical lens. (a) Phase difference in x direction. (b) Phase difference in y direction. (c) Reconstructed wave front.

Fig. 6.
Fig. 6.

Results of the optic surface testing: (a) map of the phase difference in x direction; (b) map of the phase difference in y direction; (c) the reconstructed phase distribution.

Tables (1)

Tables Icon

Table 1. Reconstruction error vs. shear amount s.

Equations (14)

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H(ξ,η)=1+βcos [2π(ξ+ξ0)/T] .
t′(x,y)=t(x,y)+(β/2)[ei2πξ0/Tt(x+λf/T,y)+ei2πξ0/Tt(xλf/T,y)]
I(x,y)=b0+b1cos[φ(x,y)φ(x+s,y)ϕ]+b2cos[φ(x,y)φ(xs,y)+ϕ]
+b3cos[φ(xs,y)φ(x+s,y)2ϕ]
ϕj=jπ4,j=0,1,,7,
Δφx(x,y)=arctan(I3+I7I1I5I0+I4I2I6)
Δφx(i,j)=φ(i+s,j)φ(is,j) i=s+1,s+2,,Ns;j=1,2,,N.
Δφy(i,j)=φ(i,j+s)φ(i,js) i=1,2,,N;j=s+1,s+2,,Ns.
U(φ)=i=s+1Nsj=1N[φ(i+s,j)φ(is,j)Δφx(i,j)]2+i=1Nj=s+1Ns[φ(i,j+s)φ(i,js)Δφy(i,j)]2
φ(i+2s,j)+φ(i2s,j)+φ(i,j+2s)+φ(i,j2s)4φ(i,j)=ρi,j
ρi,j=Δφx(i+s,j)Δφx(is,j)+Δφy(i,j+s)Δφy(i,js).
φk+1(i,j)=gi,j[φk(i+2s,j)+φk(i2s,j)+φ4(i,j+2s)+φk(i,j2s)ρi,j]
gi,j={12i=1to2sorN2stoN;j=1to2sorN2s+1toN13{i=1to2sorN2s+1toN;j=2s+1toN2sj=1to2sorN2s+1toN;j=2s+1toN2s14otherwise
φ(x,y)=2π[(x2+y22)(x2+y2+1)(x2y2)(4x2+4y25.0)]

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