## Reconstructing the phase distribution of two Interfering wavefronts by analysis of their nonlocalized fringes with an iterative method |

Optics Express, Vol. 19, Issue 17, pp. 15976-15981 (2011)

http://dx.doi.org/10.1364/OE.19.015976

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### Abstract

This paper presents a technique for reconstructing two interfering wavefronts by analyzing their 3D interference field pattern. The method is based on the numerical inverse problem and will present a robust algorithm for reconstructing of wavefronts. Several simulations are done to validate the proposed method.

© 2011 OSA

## 1. Introduction

6. M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. A **73**(11), 1434–1441 (1983). [CrossRef]

7. N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. **49**(1), 6–10 (1984). [CrossRef]

8. J. Ragazzoni, “Pupil plane wavefront sensing with an oscillating prism,” J. Mod. Opt. **43**(2), 289–293 (1996). [CrossRef]

10. Z. Zalevsky, D. Mendlovic, and R. G. Dorsch, “Gerchberg-Saxton algorithm applied in the fractional Fourier or the Fresnel domain,” Opt. Lett. **21**(12), 842–844 (1996). [CrossRef] [PubMed]

11. K. R. Freischlad, “Absolute interferometric testing based on reconstruction of rotational shear,” Appl. Opt. **40**(10), 1637–1648 (2001). [CrossRef] [PubMed]

12. R. Kowarschik, L. Wenke, T. Baade, M. Esselbach, A. Kiessling, G. Notni, and K. Uhlendorf, “Optical measurements with phase-conjugate mirrors,” Appl. Phys. B **69**(5-6), 435–443 (1999). [CrossRef]

13. C. Ai and J. C. Wyant, “Absolute testing of flats by using even and odd functions,” Appl. Opt. **32**(25), 4698–4705 (1993). [CrossRef] [PubMed]

14. U. Griesmann, “Three-flat test solutions based on simple mirror symmetry,” Appl. Opt. **45**(23), 5856–5865 (2006). [CrossRef] [PubMed]

15. R. de la Fuente and E. López Lago, “Mach-Zehnder diffracted beam interferometer,” Opt. Express **15**(7), 3876–3887 (2007). [CrossRef] [PubMed]

16. M. T. Tavassoly and A. Darudi, “Reconstruction of interfering wavefronts by analyzing their interference pattern in three dimensions,” Opt. Commun. **175**(1-3), 43–50 (2000). [CrossRef]

*z*. B) The phase difference distributions (PDDs)

16. M. T. Tavassoly and A. Darudi, “Reconstruction of interfering wavefronts by analyzing their interference pattern in three dimensions,” Opt. Commun. **175**(1-3), 43–50 (2000). [CrossRef]

## 2. Two wavefronts reconstruction

### 2.1. Theory and algorithm

### 2.2. Wavefront propagator

21. D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. **164**(4-6), 233–245 (1999). [CrossRef]

### 2.3. COST unction

*E*) which measures the degree of phase matching. For this purpose we compare

*e*is a 2D array and represents difference between measured and calculated phase distribution at the coordinate

## 3. Simulation results

*z*axis (mean propagation direction) and its size is 5mm×5mm with 500×500 resolution. In order to lowering the calculation time, the resolution of phase distribution is decreased to 50×50 pixels by averaging over 10×10 window. Simulations were performed on a PC with 2.6 GHz clock rate and triple cores processor (AMD Phenom(tm) II X3 710). The optimization time and convergence behavior of the COST function strongly depends on complexity of the wavefronts, grid size of the reconstructed waves and optimization algorithm. Normally with simple optimization algorithm the optimization loop is trapped on local minima and results has less accuracy. We have used the simulated annealing for optimization algorithm which reduce the computation time and also improve the accuracy of the reconstruction.

### 3.1 Interference of spherical and cylindrical waves

### 3.2 Interference of two cylindrical waves

## 4. Conclusion

16. M. T. Tavassoly and A. Darudi, “Reconstruction of interfering wavefronts by analyzing their interference pattern in three dimensions,” Opt. Commun. **175**(1-3), 43–50 (2000). [CrossRef]

17. A. Darudi and M. T. Tavassoly, “Interferometric specification of lens (single and doublet) parameters,” Opt. Lasers Eng. **35**(2), 79–90 (2001). [CrossRef]

## References and links

1. | D. Malacara, M. Servin, and Z. Malacara, |

2. | D. Malacara, |

3. | R. Tyson, |

4. | J. M. Geary, |

5. | R. V. Shack and B. C. Platt, “Production and use of a lenticular Hartmann screen,” J. Opt. Soc. Am. |

6. | M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. A |

7. | N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. |

8. | J. Ragazzoni, “Pupil plane wavefront sensing with an oscillating prism,” J. Mod. Opt. |

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

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

11. | K. R. Freischlad, “Absolute interferometric testing based on reconstruction of rotational shear,” Appl. Opt. |

12. | R. Kowarschik, L. Wenke, T. Baade, M. Esselbach, A. Kiessling, G. Notni, and K. Uhlendorf, “Optical measurements with phase-conjugate mirrors,” Appl. Phys. B |

13. | C. Ai and J. C. Wyant, “Absolute testing of flats by using even and odd functions,” Appl. Opt. |

14. | U. Griesmann, “Three-flat test solutions based on simple mirror symmetry,” Appl. Opt. |

15. | R. de la Fuente and E. López Lago, “Mach-Zehnder diffracted beam interferometer,” Opt. Express |

16. | M. T. Tavassoly and A. Darudi, “Reconstruction of interfering wavefronts by analyzing their interference pattern in three dimensions,” Opt. Commun. |

17. | A. Darudi and M. T. Tavassoly, “Interferometric specification of lens (single and doublet) parameters,” Opt. Lasers Eng. |

18. | A. Darudi, “Interferometry without reference wave,” (PhD Thesis, Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan, Iran, 2001). |

19. | M. Born, and E. Wolf, |

20. | F. Shen and A. Wang, “Fast-Fourier-transform based numerical integration method for the Rayleigh-Sommerfeld diffraction formula,” Appl. Opt. |

21. | D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. |

22. | M. Bass, |

23. |

**OCIS Codes**

(120.3180) Instrumentation, measurement, and metrology : Interferometry

(120.5050) Instrumentation, measurement, and metrology : Phase measurement

(070.7345) Fourier optics and signal processing : Wave propagation

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: March 9, 2011

Revised Manuscript: April 19, 2011

Manuscript Accepted: April 25, 2011

Published: August 5, 2011

**Citation**

Ehsan A. Akhlaghi, Ahmad Darudi, and M. Taghi Tavassoly, "Reconstructing the phase distribution of two Interfering wavefronts by analysis of their nonlocalized fringes with an iterative method," Opt. Express **19**, 15976-15981 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-17-15976

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### References

- D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, 1998).
- D. Malacara, Optical Shop Testing (John Wiley & Sons, 1998).
- R. Tyson, Principles of Adaptive Optics (CRC Press, 2010).
- J. M. Geary, Introduction to Wavefront Sensors (SPIE Press, 1995).
- R. V. Shack and B. C. Platt, “Production and use of a lenticular Hartmann screen,” J. Opt. Soc. Am. 61, 656 (1971).
- M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. A 73(11), 1434–1441 (1983). [CrossRef]
- N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49(1), 6–10 (1984). [CrossRef]
- J. Ragazzoni, “Pupil plane wavefront sensing with an oscillating prism,” J. Mod. Opt. 43(2), 289–293 (1996). [CrossRef]
- R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).
- Z. Zalevsky, D. Mendlovic, and R. G. Dorsch, “Gerchberg-Saxton algorithm applied in the fractional Fourier or the Fresnel domain,” Opt. Lett. 21(12), 842–844 (1996). [CrossRef] [PubMed]
- K. R. Freischlad, “Absolute interferometric testing based on reconstruction of rotational shear,” Appl. Opt. 40(10), 1637–1648 (2001). [CrossRef] [PubMed]
- R. Kowarschik, L. Wenke, T. Baade, M. Esselbach, A. Kiessling, G. Notni, and K. Uhlendorf, “Optical measurements with phase-conjugate mirrors,” Appl. Phys. B 69(5-6), 435–443 (1999). [CrossRef]
- C. Ai and J. C. Wyant, “Absolute testing of flats by using even and odd functions,” Appl. Opt. 32(25), 4698–4705 (1993). [CrossRef] [PubMed]
- U. Griesmann, “Three-flat test solutions based on simple mirror symmetry,” Appl. Opt. 45(23), 5856–5865 (2006). [CrossRef] [PubMed]
- R. de la Fuente and E. López Lago, “Mach-Zehnder diffracted beam interferometer,” Opt. Express 15(7), 3876–3887 (2007). [CrossRef] [PubMed]
- M. T. Tavassoly and A. Darudi, “Reconstruction of interfering wavefronts by analyzing their interference pattern in three dimensions,” Opt. Commun. 175(1-3), 43–50 (2000). [CrossRef]
- A. Darudi and M. T. Tavassoly, “Interferometric specification of lens (single and doublet) parameters,” Opt. Lasers Eng. 35(2), 79–90 (2001). [CrossRef]
- A. Darudi, “Interferometry without reference wave,” (PhD Thesis, Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan, Iran, 2001).
- M. Born, and E. Wolf, Principles of Optics (Cambridge University Press, 2003).
- F. Shen and A. Wang, “Fast-Fourier-transform based numerical integration method for the Rayleigh-Sommerfeld diffraction formula,” Appl. Opt. 45(6), 1102–1110 (2006). [CrossRef] [PubMed]
- D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164(4-6), 233–245 (1999). [CrossRef]
- M. Bass, Handbook of Optics (McGraw-Hill, 1995), Vol. I.
- http://www.zemax.com

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