Abstract

We study the phase retrieval (PR) technique using through-focus intensity measurements and explain the dependence of PR on the defocus distance. An optimal measurement plane in the out-of-focus region is identified where the intensity distribution on the optical axis drops to the first minimum after focus. Experimental results confirm the theoretical predictions and are in good agreement with an independent phase measurement.

© 2013 Optical Society of America

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2012

X. Liu, L. Wang, J. Wang, and H. Meng, Opt. Express 20, 15392 (2012).
[CrossRef]

A. Polo, N. van Marrewijk, S. F. Pereira, and H. P. Urbach, Proc. SPIE 8322, 832219 (2012).
[CrossRef]

A. Polo, V. Kutchoukov, F. Bociort, S. F. Pereira, and H. P. Urbach, Opt. Express 20, 237 (2012).

A. Polo, A. Haber, S. F. Pereira, M. Verhaegen, and H. P. Urbach, Opt. Express 20, 27922 (2012).
[CrossRef]

2009

P. F. Almoro and S. G. Hanson, J. Eur. Opt. Soc. Rapid Publ. 4, 09002 (2009).
[CrossRef]

2002

1999

1998

D. Paganin and K. Nugent, Phys. Rev. Lett. 80, 2586 (1998).
[CrossRef]

1996

T. F. Coleman and Y. Li, SIAM J. Optim. 6, 418 (1996).
[CrossRef]

1993

1983

R. A. Gonsalves, Proc. SPIE 0351, 56 (1983).
[CrossRef]

1982

Almoro, P. F.

P. F. Almoro and S. G. Hanson, J. Eur. Opt. Soc. Rapid Publ. 4, 09002 (2009).
[CrossRef]

Bociort, F.

A. Polo, V. Kutchoukov, F. Bociort, S. F. Pereira, and H. P. Urbach, Opt. Express 20, 237 (2012).

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge Univ., 1999).

Coleman, T. F.

T. F. Coleman and Y. Li, SIAM J. Optim. 6, 418 (1996).
[CrossRef]

Fienup, J. R.

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: the Art of Scientific Computing (Cambridge Univ., 2007).

Georges, J.

Gonsalves, R. A.

R. A. Gonsalves, Proc. SPIE 0351, 56 (1983).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (Roberts, 2005).

Haber, A.

Hanson, S. G.

P. F. Almoro and S. G. Hanson, J. Eur. Opt. Soc. Rapid Publ. 4, 09002 (2009).
[CrossRef]

Hege, E. K.

Janssen, A. J. E. M.

Jefferies, S. M.

Kutchoukov, V.

A. Polo, V. Kutchoukov, F. Bociort, S. F. Pereira, and H. P. Urbach, Opt. Express 20, 237 (2012).

Lee, D. J.

Li, Y.

T. F. Coleman and Y. Li, SIAM J. Optim. 6, 418 (1996).
[CrossRef]

Liu, X.

Lloyd-Hart, M.

Meng, H.

Nugent, K.

D. Paganin and K. Nugent, Phys. Rev. Lett. 80, 2586 (1998).
[CrossRef]

Paganin, D.

D. Paganin and K. Nugent, Phys. Rev. Lett. 80, 2586 (1998).
[CrossRef]

Pereira, S. F.

A. Polo, N. van Marrewijk, S. F. Pereira, and H. P. Urbach, Proc. SPIE 8322, 832219 (2012).
[CrossRef]

A. Polo, V. Kutchoukov, F. Bociort, S. F. Pereira, and H. P. Urbach, Opt. Express 20, 237 (2012).

A. Polo, A. Haber, S. F. Pereira, M. Verhaegen, and H. P. Urbach, Opt. Express 20, 27922 (2012).
[CrossRef]

Polo, A.

A. Polo, A. Haber, S. F. Pereira, M. Verhaegen, and H. P. Urbach, Opt. Express 20, 27922 (2012).
[CrossRef]

A. Polo, V. Kutchoukov, F. Bociort, S. F. Pereira, and H. P. Urbach, Opt. Express 20, 237 (2012).

A. Polo, N. van Marrewijk, S. F. Pereira, and H. P. Urbach, Proc. SPIE 8322, 832219 (2012).
[CrossRef]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: the Art of Scientific Computing (Cambridge Univ., 2007).

Roggemann, M. C.

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: the Art of Scientific Computing (Cambridge Univ., 2007).

Urbach, H. P.

A. Polo, V. Kutchoukov, F. Bociort, S. F. Pereira, and H. P. Urbach, Opt. Express 20, 237 (2012).

A. Polo, N. van Marrewijk, S. F. Pereira, and H. P. Urbach, Proc. SPIE 8322, 832219 (2012).
[CrossRef]

A. Polo, A. Haber, S. F. Pereira, M. Verhaegen, and H. P. Urbach, Opt. Express 20, 27922 (2012).
[CrossRef]

van Marrewijk, N.

A. Polo, N. van Marrewijk, S. F. Pereira, and H. P. Urbach, Proc. SPIE 8322, 832219 (2012).
[CrossRef]

Verhaegen, M.

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: the Art of Scientific Computing (Cambridge Univ., 2007).

Wang, J.

Wang, L.

Welsh, B. M.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge Univ., 1999).

Appl. Opt.

J. Eur. Opt. Soc. Rapid Publ.

P. F. Almoro and S. G. Hanson, J. Eur. Opt. Soc. Rapid Publ. 4, 09002 (2009).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Express

Phys. Rev. Lett.

D. Paganin and K. Nugent, Phys. Rev. Lett. 80, 2586 (1998).
[CrossRef]

Proc. SPIE

A. Polo, N. van Marrewijk, S. F. Pereira, and H. P. Urbach, Proc. SPIE 8322, 832219 (2012).
[CrossRef]

R. A. Gonsalves, Proc. SPIE 0351, 56 (1983).
[CrossRef]

SIAM J. Optim.

T. F. Coleman and Y. Li, SIAM J. Optim. 6, 418 (1996).
[CrossRef]

Other

J. W. Goodman, Introduction to Fourier Optics (Roberts, 2005).

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: the Art of Scientific Computing (Cambridge Univ., 2007).

M. Born and E. Wolf, Principles of Optics (Cambridge Univ., 1999).

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Figures (3)

Fig. 1.
Fig. 1.

(a) Example of an arbitrary phase distribution, (b) performance of the PR as a function of u, with the intensity distribution along the axis (red curve) and the reconstructed RMS wavefront deviation (marked curves).

Fig. 2.
Fig. 2.

Intensity distribution along the optical axis for a system free of aberrations (red dashed curve); integral of the minimum interference term achievable along the optical axis (blue solid curve). The maximum of this term is located in the region close to u=4π.

Fig. 3.
Fig. 3.

Through-focus intensity measurements. From top to bottom, u=0 (focal plane), u=2π, u=4.2π, and u=6π. From left to right, astigmatism, coma and second-order astigmatism.

Tables (1)

Tables Icon

Table 1. PR RMS Wavefront Deviations for Different Aberrations and Different out-of-Focus Measurement Planes

Equations (8)

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

P(ξ⃗;αnm)=A(ξ⃗)exp(iϕ(ξ⃗;αnm)).
ϕ(ξ⃗;αnm)=n8m=0nαnmZnm(ξ⃗).
U(r,f)=exp[ik2fr]iλfF[P(ξ⃗;αnm)].
Uz(r)=F1{F[U(r,f)]expikzz},
Uz(r)=Gz[P(ξ⃗;αnm)].
Uz(r)Gz[A(ξ⃗)]+iGz[A(ξ⃗)ϕ(ξ⃗;αnm)].
|Uz(r)|2|Gz[A(ξ⃗)]|2+|Gz[A(ξ⃗)ϕ(ξ⃗;αnm)]|2+2Re{iGz[A(ξ⃗)]Gz[A(ξ⃗)ϕ(ξ⃗;αnm)]*}.
argminαnm|2Re{iGz[A(ξ⃗)]Gz[A(ξ⃗)ϕ(ξ⃗;αnm)]*}|dr(|Gz[A(ξ⃗)]|dr)12(|iGz[A(ξ⃗)ϕ(ξ⃗;αnm)]*dr)12,

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