## Correlations of polarization in random electro-magnetic fields |

Optics Express, Vol. 19, Issue 17, pp. 15711-15719 (2011)

http://dx.doi.org/10.1364/OE.19.015711

Acrobat PDF (967 KB)

### Abstract

Random electromagnetic fields have a number of distinctive statistical properties that may depend on their origin. We show here that when two mutually coherent fields are overlapped, the individual characteristics are not completely lost. In particular, we demonstrate that if assumptions can be made regarding the coherence properties of one of the fields, both the relative average strength and the field correlation length of the second one can be retrieved using higher-order polarization properties of the combined field.

© 2011 OSA

## 1. Introduction

5. I. Freund and R. Berkovits, “Surface reflections and optical transport through random media: Coherent backscattering, optical memory effect, frequency, and dynamical correlations,” Phys. Rev. B Condens. Matter **41**(1), 496–503 (1990). [CrossRef] [PubMed]

7. I. Freund, M. Rosenbluh, and S. Feng, “Memory effects in propagation of optical waves through disordered media,” Phys. Rev. Lett. **61**(20), 2328–2331 (1988). [CrossRef] [PubMed]

9. F. C. MacKintosh, J. X. Zhu, D. J. Pine, and D. A. Weitz, “Polarization memory of multiply scattered light,” Phys. Rev. B Condens. Matter **40**(13), 9342–9345 (1989). [CrossRef] [PubMed]

11. A. Dogariu and E. Wolf, “Coherence theory of pairs of correlated wave fields,” J. Mod. Opt. **50**(11), 1791–1796 (2003). [CrossRef]

13. D. Burckel, S. H. Zaidi, A. Frauenglass, M. Lang, and S. R. J. Brueck, “Subfeature speckle interferometry,” Opt. Lett. **20**(3), 315–317 (1995). [CrossRef] [PubMed]

## 2. Coherent superpostion of random electromagnetic fields

**r**is a position vector, and

*x,*where

*β*continues to increase,

*x*and

*y*field components. The unpolarized field can be caused by any number of strongly scattering media but, for the purpose of this paper, it is assumed that the unpolarized field is examined near its source and, therefore, the field correlation length is of the order of a wavelength [17

17. B. Shapiro, “Large intensity fluctuations for wave propagation in random media,” Phys. Rev. Lett. **57**(17), 2168–2171 (1986). [CrossRef] [PubMed]

18. M. J. Stephen and G. Cwilich, “Intensity correlation functions and fluctuations in light scattered from a random medium,” Phys. Rev. Lett. **59**(3), 285–287 (1987). [CrossRef] [PubMed]

*x*. This can result from the scattering from a rough surface, ballistic scattering, and other types of scattering that conserve the state of polarization [1,19

19. H. Fuji, T. Asakura, and Y. Shindo, “Measurement of surface roughness properties by means of laser speckle techniques,” Opt. Commun. **16**(1), 68–72 (1976). [CrossRef]

*β*, these two correlation lengths directly influence the length scales of the resulting REF.

*β*,

*f*factors in Eqs. (5) and (6) are all global properties, evaluated as ensemble averages. While

20. J. Broky and A. Dogariu, “Complex degree of mutual polarization in randomly scattered fields,” Opt. Express **18**(19), 20105–20113 (2010). [CrossRef] [PubMed]

21. J. Ellis and A. Dogariu, “Complex degree of mutual polarization,” Opt. Lett. **29**(6), 536–538 (2004). [CrossRef] [PubMed]

*r*and a chosen reference polarization state:

12. B. Ruth, “Superposition of Two Dynamic Speckle Patterns–An application to Non-contact Blood Flow Measurements,” J. Mod. Opt. **34**(2), 257–273 (1987). [CrossRef]

*a*

_{1}and

*a*

_{2}denote the individual, normalized

*x*field components

*β*and

*β*and are also characterized by their, possibly different, coherence lengths

*β*but is not influenced at all by

## 3. Numerical simulations of overlapping REF

*r*, with a coherence length

*r*to produce different values that are larger than

*β*). We have also fitted the power spectral densities to the formulation in Eqs. (9) and (10) using the magnitudes

*I*

_{1}and

*I*

_{2}and the three Gaussian widths as fitting parameters. The results are included with continuous lines in Fig. 2. The first two terms correspond to the power spectral densities of the individual fields P and U. Since the CDMP maps for the individual fields do not change with

*β*and

*β*is constant, all the magnitudes remain unchanged and only the width of the third component changes as

*β*is varied. This corresponds to a gradual progression of different polarization regimes. Again, the most interesting features lie in the high spatial frequencies. When fitting the results of the simulation, only the magnitudes of the Gaussians are altered since now the underlying field correlations of the different components are unchanged. As a result, the curves are almost parallel to each other in the high spatial frequency range, as can be clearly seen in the inset. One can also note that, at low

*β*, the influence of the correlation length of field P is minimal. This is because, when the average strength of the uniformly polarized component increases, the overall content of high spatial frequencies decreases due to a decrease in the magnitude of the second term in Eq. (9). The values of this magnitude are 3.1, 3.0, and 2.6 for the PSDs labeled A, B, and C, respectively. The behavior seen in Fig. 3 demonstrates that, if the correlations of the underlying fields do not vary during the transition from polarized to globally unpolarized regimes, the shape of the PSD remains relatively unchanged.

## 4. Conclusion

## Acknowledgments

## References and Links

1. | J. W. Goodman, |

2. | P. Sebbah, O. Legrand, and A. Z. Genack, “Fluctuations in photon local delay time and their relation to phase spectra in random media,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics |

3. | M. J. Stephen and G. Cwilich, “Intensity correlation functions and fluctuations in light scattered from a random medium,” Phys. Rev. Lett. |

4. | P. A. Lee and A. D. Stone, “Universal conductance fluctuations in metals,” Phys. Rev. Lett. |

5. | I. Freund and R. Berkovits, “Surface reflections and optical transport through random media: Coherent backscattering, optical memory effect, frequency, and dynamical correlations,” Phys. Rev. B Condens. Matter |

6. | A. H. Gandjbakhche and G. H. Weiss, “Random walk and diffusion-like model of photon migration in turbid media,” in |

7. | I. Freund, M. Rosenbluh, and S. Feng, “Memory effects in propagation of optical waves through disordered media,” Phys. Rev. Lett. |

8. | I. Freund, M. Kaveh, R. Berkovits, and M. Rosenbluh, “Universal polarization correlations and microstatistics of optical waves in random media,” Phys. Rev. B Condens. Matter |

9. | F. C. MacKintosh, J. X. Zhu, D. J. Pine, and D. A. Weitz, “Polarization memory of multiply scattered light,” Phys. Rev. B Condens. Matter |

10. | T. S. McKechnie, “Speckle reduction,” in |

11. | A. Dogariu and E. Wolf, “Coherence theory of pairs of correlated wave fields,” J. Mod. Opt. |

12. | B. Ruth, “Superposition of Two Dynamic Speckle Patterns–An application to Non-contact Blood Flow Measurements,” J. Mod. Opt. |

13. | D. Burckel, S. H. Zaidi, A. Frauenglass, M. Lang, and S. R. J. Brueck, “Subfeature speckle interferometry,” Opt. Lett. |

14. | G. G. Stokes, Trans. Cambridge Philos Soc. 9 (1852) 399, in |

15. | E. Wolf, “Can a light beam be considered to be the sum of a completely polarized and a completely unpolarized beam?” Opt. Lett. |

16. | E. Collett, |

17. | B. Shapiro, “Large intensity fluctuations for wave propagation in random media,” Phys. Rev. Lett. |

18. | M. J. Stephen and G. Cwilich, “Intensity correlation functions and fluctuations in light scattered from a random medium,” Phys. Rev. Lett. |

19. | H. Fuji, T. Asakura, and Y. Shindo, “Measurement of surface roughness properties by means of laser speckle techniques,” Opt. Commun. |

20. | J. Broky and A. Dogariu, “Complex degree of mutual polarization in randomly scattered fields,” Opt. Express |

21. | J. Ellis and A. Dogariu, “Complex degree of mutual polarization,” Opt. Lett. |

**OCIS Codes**

(260.5430) Physical optics : Polarization

(290.5855) Scattering : Scattering, polarization

**ToC Category:**

Physical Optics

**History**

Original Manuscript: June 17, 2011

Manuscript Accepted: July 7, 2011

Published: August 1, 2011

**Citation**

J. Broky and A. Dogariu, "Correlations of polarization in random electro-magnetic fields," Opt. Express **19**, 15711-15719 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-17-15711

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### References

- J. W. Goodman, Speckle Phenomena in Optics, 1st ed. (Roberts & Co., 2007).
- P. Sebbah, O. Legrand, and A. Z. Genack, “Fluctuations in photon local delay time and their relation to phase spectra in random media,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 59(2), 2406–2411 (1999). [CrossRef]
- M. J. Stephen and G. Cwilich, “Intensity correlation functions and fluctuations in light scattered from a random medium,” Phys. Rev. Lett. 59(3), 285–287 (1987). [CrossRef] [PubMed]
- P. A. Lee and A. D. Stone, “Universal conductance fluctuations in metals,” Phys. Rev. Lett. 55(15), 1622–1625 (1985). [CrossRef] [PubMed]
- I. Freund and R. Berkovits, “Surface reflections and optical transport through random media: Coherent backscattering, optical memory effect, frequency, and dynamical correlations,” Phys. Rev. B Condens. Matter 41(1), 496–503 (1990). [CrossRef] [PubMed]
- A. H. Gandjbakhche and G. H. Weiss, “Random walk and diffusion-like model of photon migration in turbid media,” in Progress in Optics, E. Wolf, ed. (Elsevier Science, 1995), pp. 333–402.
- I. Freund, M. Rosenbluh, and S. Feng, “Memory effects in propagation of optical waves through disordered media,” Phys. Rev. Lett. 61(20), 2328–2331 (1988). [CrossRef] [PubMed]
- I. Freund, M. Kaveh, R. Berkovits, and M. Rosenbluh, “Universal polarization correlations and microstatistics of optical waves in random media,” Phys. Rev. B Condens. Matter 42(4), 2613–2616 (1990). [CrossRef] [PubMed]
- F. C. MacKintosh, J. X. Zhu, D. J. Pine, and D. A. Weitz, “Polarization memory of multiply scattered light,” Phys. Rev. B Condens. Matter 40(13), 9342–9345 (1989). [CrossRef] [PubMed]
- T. S. McKechnie, “Speckle reduction,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, 1984).
- A. Dogariu and E. Wolf, “Coherence theory of pairs of correlated wave fields,” J. Mod. Opt. 50(11), 1791–1796 (2003). [CrossRef]
- B. Ruth, “Superposition of Two Dynamic Speckle Patterns–An application to Non-contact Blood Flow Measurements,” J. Mod. Opt. 34(2), 257–273 (1987). [CrossRef]
- D. Burckel, S. H. Zaidi, A. Frauenglass, M. Lang, and S. R. J. Brueck, “Subfeature speckle interferometry,” Opt. Lett. 20(3), 315–317 (1995). [CrossRef] [PubMed]
- G. G. Stokes, Trans. Cambridge Philos Soc. 9 (1852) 399, in Polarized Light, W. Swindell, ed., (Dowden, Hutchinson, and Ross, Inc., 1975).
- E. Wolf, “Can a light beam be considered to be the sum of a completely polarized and a completely unpolarized beam?” Opt. Lett. 33(7), 642–644 (2008). [CrossRef] [PubMed]
- E. Collett, Polarized Light: Fundamentals and Applications (Marcel Dekker, 1993).
- B. Shapiro, “Large intensity fluctuations for wave propagation in random media,” Phys. Rev. Lett. 57(17), 2168–2171 (1986). [CrossRef] [PubMed]
- M. J. Stephen and G. Cwilich, “Intensity correlation functions and fluctuations in light scattered from a random medium,” Phys. Rev. Lett. 59(3), 285–287 (1987). [CrossRef] [PubMed]
- H. Fuji, T. Asakura, and Y. Shindo, “Measurement of surface roughness properties by means of laser speckle techniques,” Opt. Commun. 16(1), 68–72 (1976). [CrossRef]
- J. Broky and A. Dogariu, “Complex degree of mutual polarization in randomly scattered fields,” Opt. Express 18(19), 20105–20113 (2010). [CrossRef] [PubMed]
- J. Ellis and A. Dogariu, “Complex degree of mutual polarization,” Opt. Lett. 29(6), 536–538 (2004). [CrossRef] [PubMed]

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