## Optical correlation algorithm for reconstructing phase skeleton of complex optical fields for solving the phase problem |

Optics Express, Vol. 22, Issue 5, pp. 6186-6193 (2014)

http://dx.doi.org/10.1364/OE.22.006186

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### Abstract

We propose an optical correlation algorithm illustrating a new general method for reconstructing the phase skeleton of complex optical fields from the measured two-dimensional intensity distribution. The core of the algorithm consists in locating the saddle points of the intensity distribution and connecting such points into nets by the lines of intensity gradient that are closely associated with the equi-phase lines of the field. This algorithm provides a new partial solution to the inverse problem in optics commonly referred to as the phase problem.

© 2014 Optical Society of America

## 1.Introduction

3. E. Abramochkin and V. Volostnikov, “Two-dimensional phase problem: differential approach,” Opt. Commun. **74**(3–4), 139–143 (1989). [CrossRef]

15. D. Barchiesi, “Numerical retrieval of thin aluminium layer properties from SPR experimental data,” Opt. Express **20**(8), 9064–9078 (2012). [CrossRef] [PubMed]

16. J. R. Fienup, “Phase retrieval algorithms: a personal tour [Invited],” Appl. Opt. **52**(1), 45–56 (2013). [CrossRef] [PubMed]

*i*) amplitude zeroes (also named optical vortices, or wave front dislocations, or phase singularities) are the ‘reference’, structure-forming elements, whose set constitutes a singular skeleton of a field; (

*ii*) spatially distributed amplitude zeroes obey the specific sign principle governing the characteristics (signs) of adjacent zeroes; (

*iii*) the spatial distributions of intensity and phase in complex fields are interconnected. Therefore, knowing the locations and signs of amplitude zeroes, one can predict, at least in a qualitative manner, the behavior of a field (including spatial phase distribution, with accuracy not exceeding

*viz.*homogeneously polarized coherent optical fields, is reduced to (

*i*) the development of reliable and practicable algorithms for location of amplitude of an intensity distribution with randomly located zeroes, and

*(ii*) searching for the physically most attractive algorithm for reconstruction of the spatial phase distribution (spatial phase map) of a field. Routinely, the phase problem of this kind is solved efficiently by imposing a coherent reference wave onto the field to be analyzed [19

19. N. B. Baranova, A. V. Mamaev, H. F. Pilipetsky, V. V. Shkunov, and B. Y. Zel’dovich, “Wavefront dislocations: topological limitations for adaptive systems with phase conjugation,” J. Opt. Soc. Am. **73**(5), 525–528 (1983). [CrossRef]

*viz.*assuming a homogeneously polarized field.

*i*) bicubic spline interpolation [22

22. R. Keys, “Cubic convolution interpolation for digital image processing,” IEEE Trans. Signal Processing Acoust. Speech Signal Processing **29**(6), 1153–1160 (1981). [CrossRef]

*ii*) location of the saddle points of intensity; (

*iii*) connecting the saddle points of intensity by the gradient lines.

## 2. Location of the saddle points of intensity

*dI/dx =*0 and

*dI/dy =*0. Then, if passing a stationary point one meets alternating minima and maxima (with magnitudes larger and smaller than at the specified stationary point), this point is identified as a saddle point.

## 3. Connecting saddle points by intensity gradient lines

*viz.*with probability 95%-98% as it will be argued later, see Fig. 4, the saddle points of intensity are located within the regions of rapid changing phase [18]. Therefore, the regions with small intensity gradients (smooth spatial changes of intensity) are the regions with rapid change of phase. In average, the modulo of the gradient of phase at the saddle points of intensity exceeds by

20. N. Freund, Shvartsman, and V. Freilikher, “Optical dislocation networks in highly random media,” Opt. Commun. **101**(3–4), 247–264 (1993). [CrossRef]

*the saddle points of intensity*are chosen as ‘structure-forming’, steady-state points from which

*the gradient lines of intensity*are reconstructed facilitating

*phase mapping*of complex, spatially inhomogeneous optical fields.

*i*) non-intersecting lines passing the saddle point connect phase singularities of opposite signs; (

*ii*) the form of most of the gradient lines approximately reproduces the boundaries of areas of changing phase within the intervals: 0 to π/2, π/2 to π, π to 3π/2, and 3π/2 to 2π. (Note that the choice of the mentioned intervals of changing phase is, to a certain extent, conventional. This digitization of phase is the closest one to the Rayleigh’s criterion in classical optics. In some cases, depending on the required accuracy, this criterion can be replaced by a stronger one.)

6. M. Loktev and V. Volostnikov, “Singular wavefields and phase retrieval problem,” Proc. SPIE **3487**, 141–147 (1998). [CrossRef]

8. K. G. Larkin, “Natural demodulation of two-dimensional fringe patterns. II. Stationary phase analysis of the spiral phase quadrature transforms,” J. Opt. Soc. Am. A **18**(8), 1871–1881 (2001). [CrossRef]

*per se*. This circumstance can drastically distort the processed data and the result of processing. Thus, one must search for proper procedures for data smoothing, based either on bicubic spline interpolation [22

22. R. Keys, “Cubic convolution interpolation for digital image processing,” IEEE Trans. Signal Processing Acoust. Speech Signal Processing **29**(6), 1153–1160 (1981). [CrossRef]

## 4. Conclusions

## Acknowledgments

## References and links

1. | R. H. T. Bates and M. J. McDonnell, |

2. | T. Acharya and A. K. Ray, |

3. | E. Abramochkin and V. Volostnikov, “Two-dimensional phase problem: differential approach,” Opt. Commun. |

4. | E. Abramochkin and V. Volostnikov, “Relationship between two-dimensional intensity and phase in a Fresnel diffraction zone,” Opt. Commun. |

5. | V. Volostnikov, “Phase problem in optics,” J. Sov. Laser Research |

6. | M. Loktev and V. Volostnikov, “Singular wavefields and phase retrieval problem,” Proc. SPIE |

7. | K. G. Larkin, D. J. Bone, and M. A. Oldfield, “Natural demodulation of two-dimensional fringe patterns. I. General background of the spiral phase quadrature transform,” J. Opt. Soc. Am. A |

8. | K. G. Larkin, “Natural demodulation of two-dimensional fringe patterns. II. Stationary phase analysis of the spiral phase quadrature transforms,” J. Opt. Soc. Am. A |

9. | F. Yu. Kanev, V. P. Lukin, and L. N. Lavrinova, “Correction of turbulent distortions based on the phase conjugation in the presence of phase dislocations in a reference beam,” Atmos. Oceanic Opt. |

10. | V. P. Lukin and B. V. Fortes, “Phase-correction of turbulent distortions of an optical wave propagating under conditions of strong intensity fluctuations,” Appl. Opt. |

11. | V. A. Tartakovsky, V. A. Sennikov, P. A. Konyaev, and V. P. Lukin, “Wave reversal under strong scintillation conditions and sequential phasing in adaptive optics,” Atmos. Oceanic Opt. |

12. | J. R. Fienup, “Lensless coherent imaging by phase retrieval with an illumination pattern constraint,” Opt. Express |

13. | M. Wielgus and K. Patorski, “Evaluation of amplitude encoded fringe patterns using the bidimensional empirical mode decomposition and the 2D Hilbert transform generalizations,” Appl. Opt. |

14. | M. Trusiak, K. Patorski, and M. Wielgus, “Adaptive enhancement of optical fringe patterns by selective reconstruction using FABEMD algorithm and Hilbert spiral transform,” Opt. Express |

15. | D. Barchiesi, “Numerical retrieval of thin aluminium layer properties from SPR experimental data,” Opt. Express |

16. | J. R. Fienup, “Phase retrieval algorithms: a personal tour [Invited],” Appl. Opt. |

17. | M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. |

18. | I. Mokhun, “Introduction to linear singular optics,” in |

19. | N. B. Baranova, A. V. Mamaev, H. F. Pilipetsky, V. V. Shkunov, and B. Y. Zel’dovich, “Wavefront dislocations: topological limitations for adaptive systems with phase conjugation,” J. Opt. Soc. Am. |

20. | N. Freund, Shvartsman, and V. Freilikher, “Optical dislocation networks in highly random media,” Opt. Commun. |

21. | Y. Galushko and I. Mokhun, “Characteristics of scalar random field and its vortex networks. Recovery of the optical phase,” J. Opt. A: Pure Appl. Opt. |

22. | R. Keys, “Cubic convolution interpolation for digital image processing,” IEEE Trans. Signal Processing Acoust. Speech Signal Processing |

23. | V. I. Vasil’ev and M. S. Soskin, “Analysis of dynamics of topological peculiarities of varying random vector fields,” Ukr. J. Phys. |

**OCIS Codes**

(260.2160) Physical optics : Energy transfer

(260.5430) Physical optics : Polarization

(350.4990) Other areas of optics : Particles

(350.4855) Other areas of optics : Optical tweezers or optical manipulation

**ToC Category:**

Physical Optics

**History**

Original Manuscript: January 20, 2014

Revised Manuscript: February 22, 2014

Manuscript Accepted: February 24, 2014

Published: March 7, 2014

**Virtual Issues**

Vol. 9, Iss. 5 *Virtual Journal for Biomedical Optics*

**Citation**

O. V. Angelsky, M. P. Gorsky, S. G. Hanson, V. P. Lukin, I. I. Mokhun, P. V. Polyanskii, and P. A. Ryabiy, "Optical correlation algorithm for reconstructing phase skeleton of complex optical fields for solving the phase problem," Opt. Express **22**, 6186-6193 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-5-6186

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### References

- R. H. T. Bates and M. J. McDonnell, Image Restoration and Reconstruction (Caledon Oxford, 1986).
- T. Acharya and A. K. Ray, Image Processing – Principles and Applications (Wiley InterScience, 2006).
- E. Abramochkin, V. Volostnikov, “Two-dimensional phase problem: differential approach,” Opt. Commun. 74(3–4), 139–143 (1989). [CrossRef]
- E. Abramochkin, V. Volostnikov, “Relationship between two-dimensional intensity and phase in a Fresnel diffraction zone,” Opt. Commun. 74(3–4), 144–148 (1989). [CrossRef]
- V. Volostnikov, “Phase problem in optics,” J. Sov. Laser Research 11(6), 601–626 (1990). [CrossRef]
- M. Loktev, V. Volostnikov, “Singular wavefields and phase retrieval problem,” Proc. SPIE 3487, 141–147 (1998). [CrossRef]
- K. G. Larkin, D. J. Bone, M. A. Oldfield, “Natural demodulation of two-dimensional fringe patterns. I. General background of the spiral phase quadrature transform,” J. Opt. Soc. Am. A 18(8), 1862–1870 (2001). [CrossRef] [PubMed]
- K. G. Larkin, “Natural demodulation of two-dimensional fringe patterns. II. Stationary phase analysis of the spiral phase quadrature transforms,” J. Opt. Soc. Am. A 18(8), 1871–1881 (2001). [CrossRef]
- F. Yu. Kanev, V. P. Lukin, L. N. Lavrinova, “Correction of turbulent distortions based on the phase conjugation in the presence of phase dislocations in a reference beam,” Atmos. Oceanic Opt. 14, 1132–1169 (2001).
- V. P. Lukin, B. V. Fortes, “Phase-correction of turbulent distortions of an optical wave propagating under conditions of strong intensity fluctuations,” Appl. Opt. 41(27), 5616–5624 (2002). [CrossRef] [PubMed]
- V. A. Tartakovsky, V. A. Sennikov, P. A. Konyaev, V. P. Lukin, “Wave reversal under strong scintillation conditions and sequential phasing in adaptive optics,” Atmos. Oceanic Opt. 15, 1104–1113 (2002).
- J. R. Fienup, “Lensless coherent imaging by phase retrieval with an illumination pattern constraint,” Opt. Express 14(2), 498–508 (2006). [CrossRef] [PubMed]
- M. Wielgus, K. Patorski, “Evaluation of amplitude encoded fringe patterns using the bidimensional empirical mode decomposition and the 2D Hilbert transform generalizations,” Appl. Opt. 50(28), 5513–5523 (2011). [CrossRef] [PubMed]
- M. Trusiak, K. Patorski, M. Wielgus, “Adaptive enhancement of optical fringe patterns by selective reconstruction using FABEMD algorithm and Hilbert spiral transform,” Opt. Express 20(21), 23463–23479 (2012). [CrossRef] [PubMed]
- D. Barchiesi, “Numerical retrieval of thin aluminium layer properties from SPR experimental data,” Opt. Express 20(8), 9064–9078 (2012). [CrossRef] [PubMed]
- J. R. Fienup, “Phase retrieval algorithms: a personal tour [Invited],” Appl. Opt. 52(1), 45–56 (2013). [CrossRef] [PubMed]
- M. S. Soskin, M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001).
- I. Mokhun, “Introduction to linear singular optics,” in Optical Correlation Techniques and Applications, Ed. O. V. Angelsky, (2007), Chap. 1, TA 1630.A6, 1–133.
- N. B. Baranova, A. V. Mamaev, H. F. Pilipetsky, V. V. Shkunov, B. Y. Zel’dovich, “Wavefront dislocations: topological limitations for adaptive systems with phase conjugation,” J. Opt. Soc. Am. 73(5), 525–528 (1983). [CrossRef]
- N. Freund, Shvartsman, V. Freilikher, “Optical dislocation networks in highly random media,” Opt. Commun. 101(3–4), 247–264 (1993). [CrossRef]
- Y. Galushko, I. Mokhun, “Characteristics of scalar random field and its vortex networks. Recovery of the optical phase,” J. Opt. A: Pure Appl. Opt. 11094017 (2009).
- R. Keys, “Cubic convolution interpolation for digital image processing,” IEEE Trans. Signal Processing Acoust. Speech Signal Processing 29(6), 1153–1160 (1981). [CrossRef]
- V. I. Vasil’ev, M. S. Soskin, “Analysis of dynamics of topological peculiarities of varying random vector fields,” Ukr. J. Phys. 52, 1123–1129 (2007).

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