Abstract

We numerically and experimentally demonstrate an iterative method to simultaneously reconstruct two unknown interfering wavefronts. A three-dimensional interference pattern is analyzed and then Zernike polynomials and the stochastic parallel gradient descent algorithm are used to expand and calculate wavefronts.

© 2014 Optical Society of America

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References

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  1. F. Roddier, Adaptive Optics in Astronomy (Cambridge University, 2004).
  2. D. Malacara, Optical Shop Testing (Wiley, 1998).
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    [CrossRef]
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    [CrossRef]
  6. M. T. Tavassoly and A. Darudi, Opt. Commun. 175, 43 (2000).
    [CrossRef]
  7. E. A. Akhlaghi, A. Darudi, and M. T. Tavassoly, Opt. Express 19, 15976 (2011).
    [CrossRef]
  8. R. J. Noll, J. Opt. Soc. Am. 66, 207 (1976).
    [CrossRef]
  9. M. Takeda, H. Ina, and S. Kobayashi, J. Opt. Soc. Am. 72, 156 (1982).
    [CrossRef]
  10. M. Born and E. Wolf, Principles of Optics (Pergamon, 1986).
  11. R. Yazdani, M. Hajimahmoodzadeh, and H. R. Fallah, Appl. Opt. 53, 132 (2014).
    [CrossRef]
  12. M. A. Vorontsov, G. W. Carhart, and J. C. Ricklin, Opt. Lett. 22, 907 (1997).
    [CrossRef]
  13. T. Weyrauch and M. A. Vorontsov, J. Opt. Fiber. Commun. Rep. 1, 355 (2004).
    [CrossRef]
  14. M. Bass, Handbook of Optics (McGraw-Hill, 1995), Vol. 1.
  15. D. Voelz, Computational Fourier Optics: A MATLAB Tutorial (SPIE, 2011).

2014 (1)

2011 (1)

2004 (1)

T. Weyrauch and M. A. Vorontsov, J. Opt. Fiber. Commun. Rep. 1, 355 (2004).
[CrossRef]

2001 (1)

A. Darudi and M. T. Tavassoly, Opt. Lasers Eng. 35, 79 (2001).
[CrossRef]

2000 (1)

M. T. Tavassoly and A. Darudi, Opt. Commun. 175, 43 (2000).
[CrossRef]

1997 (1)

1983 (1)

M. R. Teague, J. Opt. Soc. Am. A 73, 1434 (1983).
[CrossRef]

1982 (1)

1976 (1)

1972 (1)

R. W. Gerchberg and W. O. Saxton, Optik 35, 237 (1972).

Akhlaghi, E. A.

Bass, M.

M. Bass, Handbook of Optics (McGraw-Hill, 1995), Vol. 1.

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1986).

Carhart, G. W.

Darudi, A.

E. A. Akhlaghi, A. Darudi, and M. T. Tavassoly, Opt. Express 19, 15976 (2011).
[CrossRef]

A. Darudi and M. T. Tavassoly, Opt. Lasers Eng. 35, 79 (2001).
[CrossRef]

M. T. Tavassoly and A. Darudi, Opt. Commun. 175, 43 (2000).
[CrossRef]

Fallah, H. R.

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, Optik 35, 237 (1972).

Hajimahmoodzadeh, M.

Ina, H.

Kobayashi, S.

Malacara, D.

D. Malacara, Optical Shop Testing (Wiley, 1998).

Noll, R. J.

Ricklin, J. C.

Roddier, F.

F. Roddier, Adaptive Optics in Astronomy (Cambridge University, 2004).

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, Optik 35, 237 (1972).

Takeda, M.

Tavassoly, M. T.

E. A. Akhlaghi, A. Darudi, and M. T. Tavassoly, Opt. Express 19, 15976 (2011).
[CrossRef]

A. Darudi and M. T. Tavassoly, Opt. Lasers Eng. 35, 79 (2001).
[CrossRef]

M. T. Tavassoly and A. Darudi, Opt. Commun. 175, 43 (2000).
[CrossRef]

Teague, M. R.

M. R. Teague, J. Opt. Soc. Am. A 73, 1434 (1983).
[CrossRef]

Voelz, D.

D. Voelz, Computational Fourier Optics: A MATLAB Tutorial (SPIE, 2011).

Vorontsov, M. A.

T. Weyrauch and M. A. Vorontsov, J. Opt. Fiber. Commun. Rep. 1, 355 (2004).
[CrossRef]

M. A. Vorontsov, G. W. Carhart, and J. C. Ricklin, Opt. Lett. 22, 907 (1997).
[CrossRef]

Weyrauch, T.

T. Weyrauch and M. A. Vorontsov, J. Opt. Fiber. Commun. Rep. 1, 355 (2004).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1986).

Yazdani, R.

Appl. Opt. (1)

J. Opt. Fiber. Commun. Rep. (1)

T. Weyrauch and M. A. Vorontsov, J. Opt. Fiber. Commun. Rep. 1, 355 (2004).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (1)

M. R. Teague, J. Opt. Soc. Am. A 73, 1434 (1983).
[CrossRef]

Opt. Commun. (1)

M. T. Tavassoly and A. Darudi, Opt. Commun. 175, 43 (2000).
[CrossRef]

Opt. Express (1)

Opt. Lasers Eng. (1)

A. Darudi and M. T. Tavassoly, Opt. Lasers Eng. 35, 79 (2001).
[CrossRef]

Opt. Lett. (1)

Optik (1)

R. W. Gerchberg and W. O. Saxton, Optik 35, 237 (1972).

Other (5)

F. Roddier, Adaptive Optics in Astronomy (Cambridge University, 2004).

D. Malacara, Optical Shop Testing (Wiley, 1998).

M. Born and E. Wolf, Principles of Optics (Pergamon, 1986).

M. Bass, Handbook of Optics (McGraw-Hill, 1995), Vol. 1.

D. Voelz, Computational Fourier Optics: A MATLAB Tutorial (SPIE, 2011).

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Figures (5)

Fig. 1.
Fig. 1.

(a) and (b) Simulated phase distributions and (c) interference pattern at the first observation plane.

Fig. 2.
Fig. 2.

(a) Difference between the simulated and reconstructed phase distributions of the first wave at the first observation plane. (b) Difference between the calculated PDD and the estimated one after optimization at the second observation plane.

Fig. 3.
Fig. 3.

Schematic diagram of the experimental setup.

Fig. 4.
Fig. 4.

(a) Interference pattern at the first observation plane. (b) Experimentally obtained PDD (dashed line) and the fitted quadratic curve, as expected, (solid line) along the x axis at the first observation plane.

Fig. 5.
Fig. 5.

(a) and (b) Reconstructed phase distributions.

Equations (8)

Equations on this page are rendered with MathJax. Learn more.

φ1z1(r,θ)=i=1NaiZi(r,θ),
φ2z1=φ1z1+Δφ1exp.
Δφ2cal=φ2z2φ1z2,
C=e¯=i=1Mj=1MeijM2,
C=(ee¯)2¯,
φz1z=K2n02(φz1x)2(φz1y)2,
φz2(x,y)=φz1(x,y)+Δφ(xx,yy,z2z1),
Δφ(xx,yy,z2z1)=K(xx)2+(yy)2+(z2z1)2.

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