Abstract

The standard Gerchberg–Saxton (GS) algorithm is normally used to find the phase (measured on two different parallel planes) of a propagating optical field (usually far-field propagation), given that the irradiance information on those planes is known. This is mostly used to calculate the modulation function of a phase mask so that when illuminated by a plane wave, it produces a known far-field irradiance distribution, or the equivalent, to calculate the phase mask to be used in a Fourier optical system so the desired pattern is obtained on the image plane. There are some extensions of the GS algorithm that can be used when the transformations that describe the optical setup are non-unitary, for example the Yang-Gu algorithm, but these are usually demonstrated using nonunitary translational-invariant optical systems. In this work a practical approach to use the GS algorithm is presented, where raytracing together with the Huygens-Fresnel principle are used to obtain the transformations that describe the optical system, so the calculation can be made when the field is propagated through a translational-variant optical system (TVOS) of arbitrary complexity. Some numerical results are shown for a system where a microscope objective composed by 5 lenses is used.

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References

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  1. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik35, 227–246 (1972).
  2. J. S. Liu and M. R. Taghizadeh, “Iterative algorithm for the design of diffractive phase elements for laser beam shaping,” Opt. Lett.27, 1463–1465 (2002).
    [CrossRef]
  3. Q. Li, H. Gao, Y. Dong, Z. Shen, and Q. Wang, “Investigation of diffractive optical element for shaping a Gaussian beam into a ring-shaped pattern,” Opt. Laser Technol.30, 511–514 (1998).
    [CrossRef]
  4. C. Bay, N. Hubner, J. Freeman, and T. Wilkinson, “Maskless photolithography via holographic optical projection,” Opt. Lett.35, 2230–2232 (2010).
    [CrossRef] [PubMed]
  5. D. C. O’Shea, A. D. Kathman, and D. W. Prather, Diffractive Optics: Design, Fabrication, and Test (SPIE Publications, 2003).
    [CrossRef]
  6. O. Ripoll, K. Ville, and H. P. Herzig, “Review of iterative Fourier-transform algorithms for beam shaping applications,” Opt. Eng.43, 2549–2556 (2004).
    [CrossRef]
  7. G. Yang, B. Gu, J. Zhuang, and O. K. Ersoy, “Gerchberg-Saxton and Yang-Gu algorithms for phase retrieval in a nonunitary transform system: a comparison,” Appl. Opt.33, 209–218 (1994).
    [CrossRef] [PubMed]
  8. Z. Zalevsky and D. Mendlovic, “Gerchberg-Saxton algorithm applied in the fractional Fourier or the Fresnel domain,” Opt. Lett.21, 842–844 (1996).
    [CrossRef] [PubMed]
  9. J. W. Goodman, Introduction to Fourier Optics (Roberts & Co, 2005).
  10. A. Shoemaker, 40X Microscope objective, US Patent 3893751 (1975).
  11. M. Born, E. Wolf, and A. B. Bhatia, Principles of Optics (Pergamon Press, 1975).
  12. N. Lindlein, “Simulation of micro-optical systems including microlens arrays,” Journal of Optics A: Pure and Applied Optics4, S1–S9 (2002).
    [CrossRef]

2010 (1)

2004 (1)

O. Ripoll, K. Ville, and H. P. Herzig, “Review of iterative Fourier-transform algorithms for beam shaping applications,” Opt. Eng.43, 2549–2556 (2004).
[CrossRef]

2002 (2)

J. S. Liu and M. R. Taghizadeh, “Iterative algorithm for the design of diffractive phase elements for laser beam shaping,” Opt. Lett.27, 1463–1465 (2002).
[CrossRef]

N. Lindlein, “Simulation of micro-optical systems including microlens arrays,” Journal of Optics A: Pure and Applied Optics4, S1–S9 (2002).
[CrossRef]

1998 (1)

Q. Li, H. Gao, Y. Dong, Z. Shen, and Q. Wang, “Investigation of diffractive optical element for shaping a Gaussian beam into a ring-shaped pattern,” Opt. Laser Technol.30, 511–514 (1998).
[CrossRef]

1996 (1)

1994 (1)

1972 (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik35, 227–246 (1972).

Bay, C.

Bhatia, A. B.

M. Born, E. Wolf, and A. B. Bhatia, Principles of Optics (Pergamon Press, 1975).

Born, M.

M. Born, E. Wolf, and A. B. Bhatia, Principles of Optics (Pergamon Press, 1975).

Dong, Y.

Q. Li, H. Gao, Y. Dong, Z. Shen, and Q. Wang, “Investigation of diffractive optical element for shaping a Gaussian beam into a ring-shaped pattern,” Opt. Laser Technol.30, 511–514 (1998).
[CrossRef]

Ersoy, O. K.

Freeman, J.

Gao, H.

Q. Li, H. Gao, Y. Dong, Z. Shen, and Q. Wang, “Investigation of diffractive optical element for shaping a Gaussian beam into a ring-shaped pattern,” Opt. Laser Technol.30, 511–514 (1998).
[CrossRef]

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik35, 227–246 (1972).

Goodman, J. W.

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

Gu, B.

Herzig, H. P.

O. Ripoll, K. Ville, and H. P. Herzig, “Review of iterative Fourier-transform algorithms for beam shaping applications,” Opt. Eng.43, 2549–2556 (2004).
[CrossRef]

Hubner, N.

Kathman, A. D.

D. C. O’Shea, A. D. Kathman, and D. W. Prather, Diffractive Optics: Design, Fabrication, and Test (SPIE Publications, 2003).
[CrossRef]

Li, Q.

Q. Li, H. Gao, Y. Dong, Z. Shen, and Q. Wang, “Investigation of diffractive optical element for shaping a Gaussian beam into a ring-shaped pattern,” Opt. Laser Technol.30, 511–514 (1998).
[CrossRef]

Lindlein, N.

N. Lindlein, “Simulation of micro-optical systems including microlens arrays,” Journal of Optics A: Pure and Applied Optics4, S1–S9 (2002).
[CrossRef]

Liu, J. S.

Mendlovic, D.

O’Shea, D. C.

D. C. O’Shea, A. D. Kathman, and D. W. Prather, Diffractive Optics: Design, Fabrication, and Test (SPIE Publications, 2003).
[CrossRef]

Prather, D. W.

D. C. O’Shea, A. D. Kathman, and D. W. Prather, Diffractive Optics: Design, Fabrication, and Test (SPIE Publications, 2003).
[CrossRef]

Ripoll, O.

O. Ripoll, K. Ville, and H. P. Herzig, “Review of iterative Fourier-transform algorithms for beam shaping applications,” Opt. Eng.43, 2549–2556 (2004).
[CrossRef]

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik35, 227–246 (1972).

Shen, Z.

Q. Li, H. Gao, Y. Dong, Z. Shen, and Q. Wang, “Investigation of diffractive optical element for shaping a Gaussian beam into a ring-shaped pattern,” Opt. Laser Technol.30, 511–514 (1998).
[CrossRef]

Shoemaker, A.

A. Shoemaker, 40X Microscope objective, US Patent 3893751 (1975).

Taghizadeh, M. R.

Ville, K.

O. Ripoll, K. Ville, and H. P. Herzig, “Review of iterative Fourier-transform algorithms for beam shaping applications,” Opt. Eng.43, 2549–2556 (2004).
[CrossRef]

Wang, Q.

Q. Li, H. Gao, Y. Dong, Z. Shen, and Q. Wang, “Investigation of diffractive optical element for shaping a Gaussian beam into a ring-shaped pattern,” Opt. Laser Technol.30, 511–514 (1998).
[CrossRef]

Wilkinson, T.

Wolf, E.

M. Born, E. Wolf, and A. B. Bhatia, Principles of Optics (Pergamon Press, 1975).

Yang, G.

Zalevsky, Z.

Zhuang, J.

Appl. Opt. (1)

Journal of Optics A: Pure and Applied Optics (1)

N. Lindlein, “Simulation of micro-optical systems including microlens arrays,” Journal of Optics A: Pure and Applied Optics4, S1–S9 (2002).
[CrossRef]

Opt. Eng. (1)

O. Ripoll, K. Ville, and H. P. Herzig, “Review of iterative Fourier-transform algorithms for beam shaping applications,” Opt. Eng.43, 2549–2556 (2004).
[CrossRef]

Opt. Laser Technol. (1)

Q. Li, H. Gao, Y. Dong, Z. Shen, and Q. Wang, “Investigation of diffractive optical element for shaping a Gaussian beam into a ring-shaped pattern,” Opt. Laser Technol.30, 511–514 (1998).
[CrossRef]

Opt. Lett. (3)

Optik (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik35, 227–246 (1972).

Other (4)

D. C. O’Shea, A. D. Kathman, and D. W. Prather, Diffractive Optics: Design, Fabrication, and Test (SPIE Publications, 2003).
[CrossRef]

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

A. Shoemaker, 40X Microscope objective, US Patent 3893751 (1975).

M. Born, E. Wolf, and A. B. Bhatia, Principles of Optics (Pergamon Press, 1975).

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Figures (8)

Fig. 1
Fig. 1

Diagram of a typical Fourier system.

Fig. 2
Fig. 2

Block diagram of the algorithm used.

Fig. 3
Fig. 3

Optical setup simulated to obtain the numerical results.

Fig. 4
Fig. 4

Irradiance distributions observed at the image plane: (a) without and (b) with aberration correction

Fig. 5
Fig. 5

Spot radius (RMS) vs displacement.

Fig. 6
Fig. 6

(a) Target image and (b) reconstructed image when the standard GS algorithm is used.

Fig. 7
Fig. 7

Modified GS algorithm evolution.

Fig. 8
Fig. 8

Correlation between the reconstructed image and the target image, as a function of the number of iterations.

Equations (2)

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θ = 2 π d λ
Θ im ( x , y ) = l m A l m exp ( ι ( θ l m ( x , y ) + ϕ l m ) )

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