## Generalized Stokes parameters of three-dimensional stochastic electromagnetic beams |

Optics Express, Vol. 18, Issue 22, pp. 22826-22832 (2010)

http://dx.doi.org/10.1364/OE.18.022826

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### Abstract

The generalized Stokes parameters of 2D stochastic electromagnetic beams are developed to the 3D case, which can be addressed as certain linear combinations of the 3 × 3 cross-spectral density matrix in terms of the nine Gell-Mann matrices. Using the electromagnetic Gaussian Shell-model source as an example, we investigate their precise propagation laws of coherence properties and polarization properties with the help of the 3D generalized Stokes parameters. Some numerical examples and detailed comparisons of the obtained results with the 2D case are made. It is shown that 3D generalized Stokes parameters are required for the exact description of stochastic electromagnetic beams.

© 2010 OSA

## 1. Introduction

1. F. Zernike, “The concept of degree of coherence and its application to optical problems,” Physica (Utrecht) **5**(8), 785–795 (1938). [CrossRef]

2. L. Mandel and E. Wolf, “Spectral coherence and concept of cross-spectral purity,” J. Opt. Soc. Am. **66**(6), 529 (1976). [CrossRef]

4. A. Luis, “Degree of coherence for vectorial electromagnetic fields as the distance between correlation matrices,” J. Opt. Soc. Am. A **24**(4), 1063–1068 (2007). [CrossRef]

6. E. Wolf, “Correlation-induced changes in the degree of polarization, the degree of coherence, and the spectrum of random electromagnetic beams on propagation,” Opt. Lett. **28**(13), 1078–1080 (2003). [CrossRef] [PubMed]

7. O. Korotkova and E. Wolf, “Generalized stokes parameters of random electromagnetic beams,” Opt. Lett. **30**(2), 198–200 (2005). [CrossRef] [PubMed]

8. T. Setälä, M. Kaivola, and A. T. Friberg, “Degree of polarization in near fields of thermal sources: effects of surface waves,” Phys. Rev. Lett. **88**(12), 123902 (2002). [CrossRef] [PubMed]

10. T. Setälä, K. Lindfors, and A. T. Friberg, “Degree of polarization in 3D optical fields generated from a partially polarized plane wave,” Opt. Lett. **34**(21), 3394–3396 (2009). [CrossRef] [PubMed]

## 2. Theoretical analyses

8. T. Setälä, M. Kaivola, and A. T. Friberg, “Degree of polarization in near fields of thermal sources: effects of surface waves,” Phys. Rev. Lett. **88**(12), 123902 (2002). [CrossRef] [PubMed]

*λ*(

_{j}*j*= 0,1,…,8) [12

12. X. Du, D. Zhao, and O. Korotkova, “Changes in the statistical properties of stochastic anisotropic electromagnetic beams on propagation in the turbulent atmosphere,” Opt. Express **15**(25), 16909–16915 (2007). [CrossRef] [PubMed]

*S*

_{0,3D}(

**r**,

*ω*)being equal to the spectral density of the field, the spectral degree of coherence of the field at a pair of point being given by the formula

**r**and frequency

*ω*,

11. A. Luis, “Polarization distribution and degree of polarization for three-dimensional quantum light fields,” Phys. Rev. A **71**(6), 063815 (2005). [CrossRef]

12. X. Du, D. Zhao, and O. Korotkova, “Changes in the statistical properties of stochastic anisotropic electromagnetic beams on propagation in the turbulent atmosphere,” Opt. Express **15**(25), 16909–16915 (2007). [CrossRef] [PubMed]

13. K. Duan and B. Lü, “Partially coherent vectorial nonparaxial beams,” J. Opt. Soc. Am. A **21**(10), 1924–1932 (2004). [CrossRef]

## 3. Numerical calculation results and comparative analyses

*λ*and Rayleigh distance specified by

*z*=

_{R}*πσ*

^{2}/

*λ*to normalize the corresponding transverse and longitudinal distances. Figure 1 is the spectral density of a 3D stochastic electromagnetic beam in the plane

*z*= 15

*z*for different parameters

_{R}*f*and

_{j}*f*value. For the convenience of comparison the corresponding 2D results are also given in the figures, which are represented by the dashed lines. One can see from Fig. 1 that when the value of parameters

_{αβ}*f*and

_{j}*f*is very small, the spectral density of 3D beams coincide with the 2D results quite well, so that for this case the 2D results hold true and the

_{αβ}*z*component is very small and can be negligible. However, for the large values of parameters

*f*and

_{j}*f*, the difference between the 3D results and 2D results become obvious, the longitudinal component also become large and cannot be neglected. Figure 2 is the contour graphs of the spectral density

_{αβ}*S*

_{0,2D}and

*S*

_{0,3D}of a stochastic electromagnetic beam in the plane

*z*= 15

*z*when the value of parameters

_{R}*f*and

_{j}*f*is larger. From Fig. 2(a), we can see when the results were considered 2D case, the spectral density distribution

_{αβ}*S*

_{0,2D}is axially rotational symmetry. However, when we consider 3D field distribution, shown as Fig. 2(b), the spectral density distribution

*S*

_{0,3D}loses their circular symmetry and become inclined elliptical symmetry due to the existence of the longitudinal component.

*z*-axis direction of a stochastic electromagnetic Gaussian Schell-model beam when the pair of field points are located symmetrically with respect to the

*z*-axis, i. e.

**ρ**= -

_{2}**ρ**. The 3D results of spectral degree of coherence coincide with the 2D results quite well for the small values of parameters

_{1}*f*and

_{j}*f*, so that for this case the 2D results hold true. Whereas for the large values of parameters

_{αβ}*f*and

_{j}*f*, the initial spectral degrees of coherence exist obvious difference, but both of the 2D and 3D results tend to 1 with increasing propagation distance.

_{αβ}*P*

_{2D}and

*P*

_{3D}, three-dimensional distributions of spectral degree of polarization and corresponding contour graphs of 2D and 3D stochastic electromagnetic beams in the plane

*z*= 15

*z*are plotted in Figs. 5 and 6 . It can be seen from Figs. 5 and 6 that the difference between

_{R}*P*

_{2D}and

*P*

_{3D}is very obvious. The 2D spectral degree of polarization

*P*

_{2D}is Gaussian distribution and axially rotational symmetry, but the 3D spectral degree of polarization

*P*

_{3D}is not Gaussian distribution and loses their circular symmetry. The

*P*

_{3D}first decreases to a minimum and then rises with increasing transverse coordinate. This is due to the existence of the longitudinal component of the 3D light field.

*f*and

_{j}*f*are key parameters for determine the propagation behavior of 3D random electromagnetic beams.

_{αβ}*f*is proportional to the ratio of the wavelength to the beam width, and

_{j}*f*is proportional to the ratio of the wavelength to the correlation length. The large values of parameters

_{αβ}*f*mean the beam width that is comparable with or less than

_{j}*λ*results in the beam nonparaxiality. The large values of parameters

*f*mean weak coherence that leads to the large divergence angle. Here, the longitudinal component must be taken into consideration.

_{αβ}## 4. Conclusions

## Acknowledgments

## References and links

1. | F. Zernike, “The concept of degree of coherence and its application to optical problems,” Physica (Utrecht) |

2. | L. Mandel and E. Wolf, “Spectral coherence and concept of cross-spectral purity,” J. Opt. Soc. Am. |

3. | J. Tervo, T. Setälä, and A. T. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express |

4. | A. Luis, “Degree of coherence for vectorial electromagnetic fields as the distance between correlation matrices,” J. Opt. Soc. Am. A |

5. | C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, New York, 1998). |

6. | E. Wolf, “Correlation-induced changes in the degree of polarization, the degree of coherence, and the spectrum of random electromagnetic beams on propagation,” Opt. Lett. |

7. | O. Korotkova and E. Wolf, “Generalized stokes parameters of random electromagnetic beams,” Opt. Lett. |

8. | T. Setälä, M. Kaivola, and A. T. Friberg, “Degree of polarization in near fields of thermal sources: effects of surface waves,” Phys. Rev. Lett. |

9. | A. Luis, “Degree of polarization for three-dimensional fields as a distance between correlation matrices,” Opt. Commun. |

10. | T. Setälä, K. Lindfors, and A. T. Friberg, “Degree of polarization in 3D optical fields generated from a partially polarized plane wave,” Opt. Lett. |

11. | A. Luis, “Polarization distribution and degree of polarization for three-dimensional quantum light fields,” Phys. Rev. A |

12. | X. Du, D. Zhao, and O. Korotkova, “Changes in the statistical properties of stochastic anisotropic electromagnetic beams on propagation in the turbulent atmosphere,” Opt. Express |

13. | K. Duan and B. Lü, “Partially coherent vectorial nonparaxial beams,” J. Opt. Soc. Am. A |

**OCIS Codes**

(030.1640) Coherence and statistical optics : Coherence

(260.0260) Physical optics : Physical optics

(260.5430) Physical optics : Polarization

**ToC Category:**

Coherence and Statistical Optics

**History**

Original Manuscript: August 24, 2010

Revised Manuscript: October 7, 2010

Manuscript Accepted: October 8, 2010

Published: October 13, 2010

**Citation**

Zhangrong Mei, "Generalized Stokes parameters of three-dimensional stochastic electromagnetic beams," Opt. Express **18**, 22826-22832 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-22-22826

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### References

- F. Zernike, “The concept of degree of coherence and its application to optical problems,” Physica (Utrecht) 5(8), 785–795 (1938). [CrossRef]
- L. Mandel and E. Wolf, “Spectral coherence and concept of cross-spectral purity,” J. Opt. Soc. Am. 66(6), 529 (1976). [CrossRef]
- J. Tervo, T. Setälä, and A. T. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express 11(10), 1137–1143 (2003). [CrossRef] [PubMed]
- A. Luis, “Degree of coherence for vectorial electromagnetic fields as the distance between correlation matrices,” J. Opt. Soc. Am. A 24(4), 1063–1068 (2007). [CrossRef]
- C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, New York, 1998).
- E. Wolf, “Correlation-induced changes in the degree of polarization, the degree of coherence, and the spectrum of random electromagnetic beams on propagation,” Opt. Lett. 28(13), 1078–1080 (2003). [CrossRef] [PubMed]
- O. Korotkova and E. Wolf, “Generalized stokes parameters of random electromagnetic beams,” Opt. Lett. 30(2), 198–200 (2005). [CrossRef] [PubMed]
- T. Setälä, M. Kaivola, and A. T. Friberg, “Degree of polarization in near fields of thermal sources: effects of surface waves,” Phys. Rev. Lett. 88(12), 123902 (2002). [CrossRef] [PubMed]
- A. Luis, “Degree of polarization for three-dimensional fields as a distance between correlation matrices,” Opt. Commun. 253(1-3), 10–14 (2005). [CrossRef]
- T. Setälä, K. Lindfors, and A. T. Friberg, “Degree of polarization in 3D optical fields generated from a partially polarized plane wave,” Opt. Lett. 34(21), 3394–3396 (2009). [CrossRef] [PubMed]
- A. Luis, “Polarization distribution and degree of polarization for three-dimensional quantum light fields,” Phys. Rev. A 71(6), 063815 (2005). [CrossRef]
- X. Du, D. Zhao, and O. Korotkova, “Changes in the statistical properties of stochastic anisotropic electromagnetic beams on propagation in the turbulent atmosphere,” Opt. Express 15(25), 16909–16915 (2007). [CrossRef] [PubMed]
- K. Duan and B. Lü, “Partially coherent vectorial nonparaxial beams,” J. Opt. Soc. Am. A 21(10), 1924–1932 (2004). [CrossRef]

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