## The influence of the degree of cross-polarization on the Hanbury Brown-Twiss effect |

Optics Express, Vol. 18, Issue 16, pp. 17124-17129 (2010)

http://dx.doi.org/10.1364/OE.18.017124

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### Abstract

We show, by an example, that the knowledge of the degree of coherence and of the degree of polarization of a light beam incident on two photo detectors is not adequate to predict correlations in the fluctuations of the currents generated in the detectors (the Hanbury Brown-Twiss effect). The knowledge of the so-called degree of cross-polarization, introduced not long ago, is also needed.

© 2010 Optical Society of America

## 1. Introduction

8. D. Kleppner, “Hanbury Brown’s steamroller,” Physics Today **61**, 8–9 (2008). [CrossRef]

9. T. Shirai and E. Wolf, “Correlations between intensity fluctuations in stochastic electromagnetic beams of any state of coherence and polarization,” Opt. Commun. **272**, 289–292 (2007). [CrossRef]

10. S. N. Volkov, D. F. V. James, T. Shirai, and E. Wolf, “Intensity fluctuations and the degree of cross-polarization in stochastic electromagnetic beams,” J. Opt. A, Pure Appl. Opt. **10**, 055001 (2008). [CrossRef]

*degree of cross-polarization*[11

11. In Refs. [9] and [10] the degree of cross-polarization was defined in the space-frequency domain. A definition of the degree of the degree of cross-polarization in the space-time domain was introduced in Ref. [12]. Another two-point generalization of the degree of polarization, called complex degree of *mutual polarization* was introduced in Ref. [13].

## 2. Theory

*z*= 0. Suppose that the beam propagates into the half-space

*z*> 0 with its axis along the

*z*direction. Let

*E*(

_{x}**,**

*ρ**z*;

*ω*) and

*E*(

_{y}**,**

*ρ**z*;

*ω*) be the Cartesian components at frequency

*ω*, of the members of the statistical ensemble of the fluctuating electric field, in two mutually orthogonal

*x*and

*y*directions, perpendicular to the beam axis, at a point

*P*(

**,**

*ρ**z*). The second-order correlation properties of the beam at a pair of points

*P*

_{1}(

*ρ*_{1},

*z*),

*P*

_{2}(

*ρ*_{2},

*z*) in any cross-sectional plane

*z*= constant > 0 may be characterized by the so-called

*cross-spectral density matrix*(to be abbreviated by CSDM), whose elements are given by ([6], Sec. 9.1):

*spectral density S*(

**,**

*ρ**z*;

*ω*) at a point

*P*(

**,**

*ρ**z*) is given by the expression

*spectral degree of coherence μ*(

*ρ*_{1},

*ρ*_{2},

*z*;

*ω*) at a pair of points

*P*

_{1}(

*ρ*_{1},

*z*) and

*P*

_{2}(

**ρ**

_{2},

*z*) is defined by the formula

*spectral degree of polarization 𝒫*(

*;*

**ρ**, z*ω*) at the point

*P*(

*) is given by the expression*

**ρ**, z9. T. Shirai and E. Wolf, “Correlations between intensity fluctuations in stochastic electromagnetic beams of any state of coherence and polarization,” Opt. Commun. **272**, 289–292 (2007). [CrossRef]

10. S. N. Volkov, D. F. V. James, T. Shirai, and E. Wolf, “Intensity fluctuations and the degree of cross-polarization in stochastic electromagnetic beams,” J. Opt. A, Pure Appl. Opt. **10**, 055001 (2008). [CrossRef]

10. S. N. Volkov, D. F. V. James, T. Shirai, and E. Wolf, “Intensity fluctuations and the degree of cross-polarization in stochastic electromagnetic beams,” J. Opt. A, Pure Appl. Opt. **10**, 055001 (2008). [CrossRef]

*P*

_{1}(

**ρ**

_{1},

*z*) and

*P*

_{2}(

*ρ*_{2},

*z*), in a cross-sectional plane

*z*= constant > 0 of the beam. The correlation

*C*(

*ρ*_{1},

*ρ*_{2},

*z*;

*ω*) between the intensity fluctuations at these two points, which is proportional to the correlation between the current fluctuations in the two detectors ([6], Ch. 7), can be shown to be given by the formula ([10

**10**, 055001 (2008). [CrossRef]

*degree of cross-polarization*. In Eq. (6) the dot symbolizes ordinary matrix multiplication. Equations (5) show that the correlation between intensity fluctuations, at a pair of points, does not depend only on the spectral density S and on the spectral degree of coherence of the incident beams

*μ*, but depends also on the degree of cross-polarization

*𝓠*. The expressions [Eqs. (5) and (6)] have been derived with the assumption that the random fluctuations of the electric field in the beam obey Gaussian statistics.

*z*> 0 from the source plane

*z*= 0. It can readily be shown that the cross-spectral density matrix at a pair of points

*P*

_{1}(

*ρ*_{1},

*z*) and

*P*

_{2}(

*ρ*_{2},

*z*), at a cross-sectional plane

*z*= constant > 0 is given by ([14

14. 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**, 1078–1080 (2003). [CrossRef] [PubMed]

*z*= 0 are then given by the expressions [see, for example, Ref. [6], Sec. 9.4.2, Eqs. (5)–(7)]

*σ*≫

_{i}*δ*and

_{ij}15. H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. **249**, 379–385 (2005). [CrossRef]

16. F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Realizability condition for electromagnetic Schell-model sources,” J. Opt. Soc. Am. A **25**, 1016–1021 (2008). [CrossRef]

*z*> 0. Using the propagation law [Eq. (7), it may be shown that the elements of the CSDM, at a pair of points in that plane, are given by ([6], Sec. 9.4.2, Eq. (10), (11))

## 3. Example

*a*” and “

*b*”, produced by two different planar secondary sources. We assume that the beam “

*a*” is characterized by parameters

*A*=

_{x}*A*= 1,

_{y}*σ*=

_{x}*σ*=

_{y}*σ*,

*δ*=

_{xx}*δ*=

_{yy}*δ*and

^{(a)}of beam

*a*, at the source plane

*z*= 0 has the form

*b*”, is characterized by the parameters

*A*=

_{x}*A*= 1,

_{y}*σ*=

_{x}*σ*=

_{y}*σ*,

*δ*=

_{xx}*δ*=

_{yy}*δ*=

_{xy}*δ*=

_{yx}*δ*. Using Eq. (9) again, one readily finds that the CSDM of the beam “

*b*”, at the source plane has the form

*𝒜*is the same quantity as in Eq. (15a) and

*ρ*with the choice

*δ*= 0.001m and

*σ*= 0.01m.

*z*=

*z*

_{0}> 0, the correlation in the intensity fluctuations associated with each of them may become significantly different. To see this, we choose the parameters

*δ*= 0.001m and

*σ*= 0.01m. Recalling the definition Eq. (5) and using formula (12), one can calculate the correlation between the intensity fluctuations at a pair of points in any cross-section of the beam. Figure 2 shows the variations of this correlation function with the half-separation distance

*ρ*of two diametrically opposite points in a beam cross-section, at a distance

*z*= 10km from the source. Figures 1 and 2 clearly show that degree of cross-polarization of a field affects, in general, the correlations in the intensity fluctuations of an electromagnetic beam.

## Acknowledgement

## References and links

1. | R. H. Brown and R. Q. Twiss, “A new type of interferometer for use in radio astronomy,” Philos. Mag. |

2. | R. H. Brown and R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature (London) |

3. | R. H. Brown and R. Q. Twiss, “Interferometry of the intensity fluctuations in light, I: basic theory: the correlation between photons in coherent beams of radiation,” Proc. Roy. Soc. (London) Sec. A |

4. | R. H. Brown and R. Q. Twiss, “Interferometry of the intensity fluctuations in light, II: an experimental test of the theory for partially coherent light,” Proc. Roy. Soc. (London) Sec. A , |

5. | L. Mandel and E. Wolf, |

6. | E. Wolf, |

7. | G. Baym, “The Physics of Hanbury Brown-Twiss intensity interferometry: from stars to nuclear collisions,” Acta Phys. Poln. B |

8. | D. Kleppner, “Hanbury Brown’s steamroller,” Physics Today |

9. | T. Shirai and E. Wolf, “Correlations between intensity fluctuations in stochastic electromagnetic beams of any state of coherence and polarization,” Opt. Commun. |

10. | S. N. Volkov, D. F. V. James, T. Shirai, and E. Wolf, “Intensity fluctuations and the degree of cross-polarization in stochastic electromagnetic beams,” J. Opt. A, Pure Appl. Opt. |

11. | In Refs. [9] and [10] the degree of cross-polarization was defined in the space-frequency domain. A definition of the degree of the degree of cross-polarization in the space-time domain was introduced in Ref. [12]. Another two-point generalization of the degree of polarization, called complex degree of |

12. | D. Kuebel, “Properties of the degree of cross-polarization in the spacetime domain,” Opt. Commun. |

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

14. | 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. |

15. | H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. |

16. | F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Realizability condition for electromagnetic Schell-model sources,” J. Opt. Soc. Am. A |

**OCIS Codes**

(030.1640) Coherence and statistical optics : Coherence

(260.3160) Physical optics : Interference

(260.5430) Physical optics : Polarization

**ToC Category:**

Coherence and Statistical Optics

**History**

Original Manuscript: May 27, 2010

Revised Manuscript: July 3, 2010

Manuscript Accepted: July 19, 2010

Published: July 28, 2010

**Citation**

Asma Al-Qasimi, Mayukh Lahiri, David Kuebel, Daniel F. V. James, and Emil Wolf, "The influence of the degree of cross-polarization on the Hanbury Brown-Twiss effect," Opt. Express **18**, 17124-17129 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-16-17124

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### References

- R. H. Brown, and R. Q. Twiss, “A new type of interferometer for use in radio astronomy,” Philos. Mag. 45, 663–682 (1954).
- R. H. Brown, and R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature 177, 27–29 (1956). [CrossRef]
- R. H. Brown, and R. Q. Twiss, ““Interferometry of the intensity fluctuations in light, I: basic theory: the correlation between photons in coherent beams of radiation,” Proc. Roy. Soc. (London) Sec. A 242, 300–324 (1957). [CrossRef]
- R. H. Brown, and R. Q. Twiss, ““Interferometry of the intensity fluctuations in light, II: an experimental test of the theory for partially coherent light,” Proc. Roy. Soc. (London) Sec. A 243, 291–319 (1957). [CrossRef]
- L. Mandel, and E. Wolf, Optical Coherence and Quantum Optics (Cambridge, Cambridge University Press, 1995).
- E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, Cambridge, 2007).
- G. Baym, “The Physics of Hanbury Brown-Twiss intensity interferometry: from stars to nuclear collisions,” Acta Phys. Poln. B 29, 1839–1884 (1998).
- D. Kleppner, “Hanbury Brown’s steamroller,” Phys. Today 61, 8–9 (2008). [CrossRef]
- T. Shirai, and E. Wolf, “Correlations between intensity fluctuations in stochastic electromagnetic beams of any state of coherence and polarization,” Opt. Commun. 272, 289–292 (2007). [CrossRef]
- S. N. Volkov, D. F. V. James, T. Shirai, and E. Wolf, “Intensity fluctuations and the degree of cross-polarization in stochastic electromagnetic beams,” J. Opt. A, Pure Appl. Opt. 10, 055001 (2008). [CrossRef]
- In Refs. [9] and [10] the degree of cross-polarization was defined in the space-frequency domain. A definition of the degree of the degree of cross-polarization in the space-time domain was introduced in Ref. [12]. Another two point generalization of the degree of polarization, called complex degree of mutual polarization was introduced in Ref. [13].
- D. Kuebel, “Properties of the degree of cross-polarization in the space-time domain,” Opt. Commun. 282, 3397–3401 (2009). [CrossRef]
- J. Ellis, and A. Dogariu, “Complex degree of mutual polarization,” Opt. Lett. 29, 536–538 (2004). [CrossRef] [PubMed]
- 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, 1078–1080 (2003). [CrossRef] [PubMed]
- H. Roychowdhury, and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249, 379–385 (2005). [CrossRef]
- F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Realizability condition for electromagnetic Schell model sources,” J. Opt. Soc. Am. A 25, 1016–1021 (2008). [CrossRef]

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