## Gerchberg-Saxton algorithm applied to a translational-variant optical setup |

Optics Express, Vol. 21, Issue 16, pp. 19128-19134 (2013)

http://dx.doi.org/10.1364/OE.21.019128

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### 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.

© 2013 OSA

## 1. Introduction

## 2. Propagation of a field through a TVOS

*x*,

*y*) on the destination-plane, to be equal to: where

*d*is the optical path of the ray joining the point source and the (

*x*,

*y*) point, and

*λ*is its wavelength. One approximation done in this step, is that the amplitude of the field on the observation-plane is constant and proportional to the amplitude of the corresponding source. After the fields due to every source are found, the superposition principle is used and the resulting field is obtained as the sum of them. This process can be performed to propagate the field in any direction through the optical system, so it is possible to propagate the field from the object plane to the image plane, and vice versa.

*(*

_{im}*x*,

*y*) is the total optical field at the destination-plane,

*A*and

_{lm}*ϕ*are the amplitude and phase of field on the source-plane at (

_{lm}*x*,

_{l}*y*), and

_{m}*θ*(

_{lm}*x*,

*y*) is the phase of the optical field on the destination-plane due to a point source located at (

*x*,

_{l}*y*) on the source-plane. When propagating forward in Fig. 1, the source-plane is the phase mask and the destination-plane is the image plane. When propagating backward, the source-plane is the image plane, and the destination-plane is the phase mask.

_{m}## 3. The implemented GS algorithm

- 0 At the beginning, a field with an amplitude given by the square root of the expected irradiance and a constant phase is taken.
- 1. The field is propagated from the image-plane to the object-plane.
- 2. The amplitude information is discarded, leaving only the phase information (for the phase mask).
- 3. The amplitude and phase of the illumination field are added to the phase information to obtain the resulting object field.
- 4. The field is propagated from the object plane to the image plane.
- 5. The resulting reconstructed image (square of the field amplitude) is compared with the expected one. By using the correlation between both images as a criterion, a decision is taken to finish the process or continue iterating.
- 6. The phase from the reconstructed image is combined with the field amplitude obtained from the expected irradiance, and the process is repeated from step 1

## 4. Details of the system used in the numerical calculations

## 5. Conclusions

12. N. Lindlein, “Simulation of micro-optical systems including microlens arrays,” Journal of Optics A: Pure and Applied Optics **4**, S1–S9 (2002) [CrossRef] .

## References

1. | R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik |

2. | J. S. Liu and M. R. Taghizadeh, “Iterative algorithm for the design of diffractive phase elements for laser beam shaping,” Opt. Lett. |

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. |

4. | C. Bay, N. Hubner, J. Freeman, and T. Wilkinson, “Maskless photolithography via holographic optical projection,” Opt. Lett. |

5. | D. C. O’Shea, A. D. Kathman, and D. W. Prather, |

6. | O. Ripoll, K. Ville, and H. P. Herzig, “Review of iterative Fourier-transform algorithms for beam shaping applications,” Opt. Eng. |

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. |

8. | Z. Zalevsky and D. Mendlovic, “Gerchberg-Saxton algorithm applied in the fractional Fourier or the Fresnel domain,” Opt. Lett. |

9. | J. W. Goodman, |

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

11. | M. Born, E. Wolf, and A. B. Bhatia, |

12. | N. Lindlein, “Simulation of micro-optical systems including microlens arrays,” Journal of Optics A: Pure and Applied Optics |

**OCIS Codes**

(080.0080) Geometric optics : Geometric optics

(140.3300) Lasers and laser optics : Laser beam shaping

**ToC Category:**

Fourier Optics and Signal Processing

**History**

Original Manuscript: April 26, 2013

Revised Manuscript: June 24, 2013

Manuscript Accepted: July 12, 2013

Published: August 5, 2013

**Virtual Issues**

Vol. 8, Iss. 9 *Virtual Journal for Biomedical Optics*

**Citation**

Ricardo Amézquita-Orozco and Yobani Mejía-Barbosa, "Gerchberg-Saxton algorithm applied to a translational-variant optical setup," Opt. Express **21**, 19128-19134 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-16-19128

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### References

- 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).
- 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]
- 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]
- C. Bay, N. Hubner, J. Freeman, and T. Wilkinson, “Maskless photolithography via holographic optical projection,” Opt. Lett.35, 2230–2232 (2010). [CrossRef] [PubMed]
- D. C. O’Shea, A. D. Kathman, and D. W. Prather, Diffractive Optics: Design, Fabrication, and Test (SPIE Publications, 2003). [CrossRef]
- 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]
- 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]
- 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]
- 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).
- N. Lindlein, “Simulation of micro-optical systems including microlens arrays,” Journal of Optics A: Pure and Applied Optics4, S1–S9 (2002). [CrossRef]

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