## Hanbury Brown–Twiss effect with electromagnetic waves |

Optics Express, Vol. 19, Issue 16, pp. 15188-15195 (2011)

http://dx.doi.org/10.1364/OE.19.015188

Acrobat PDF (692 KB)

### Abstract

The classic Hanbury Brown–Twiss experiment is analyzed in the space–frequency domain by taking into account the vectorial nature of the radiation. We show that as in scalar theory, the degree of electromagnetic coherence fully characterizes the fluctuations of the photoelectron currents when a random vector field with Gaussian statistics is incident onto the detectors. Interpretation of this result in terms of the modulations of optical intensity and polarization state in two-beam interference is discussed. We demonstrate that the degree of cross-polarization may generally diverge. We also evaluate the effects of the state of polarization on the correlations of intensity fluctuations in various circumstances.

© 2011 OSA

## 1. Introduction

1. R. Hanbury Brown and R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature **177**, 27–29 (1956). [CrossRef]

4. R. Martínez-Herrero, P. M. Mejías, and G. Piquero, *Characterization of Partially Polarized Light Fields* (Springer, 2009). [CrossRef]

5. A. Luis, “An overview of coherence and polarization properties for multicomponent electromagnetic waves,” in *Advances in Information Optics and Photonics*, A. T. Friberg and R. Dändliker, eds. (SPIE Press, 2008), Chap. 9. [CrossRef]

12. J. Tervo, T. Setälä, and A. T. Friberg, “Theory of partially coherent electromagnetic fields in the space–frequency domain,” J. Opt. Soc. Am. A **21**, 2205–2215 (2004). [CrossRef]

*μ*(

_{E}**r**

_{1},

**r**

_{2},

*ω*). This result is entirely analogous to the usual scalar formulation. We also examine the influence of the spatial correlation and the state of polarization on the normalized correlations of intensity fluctuations under a variety of conditions.

## 2. Theory

*z*direction. Denoting a realization of the electric field in the space–frequency domain by a column vector

**E**(

**r**,

*ω*) = [

*E*(

_{x}**r**,

*ω*),

*E*(

_{y}**r**,

*ω*)]

^{T}, where

**r**represents position,

*ω*is angular frequency, and T denotes the transpose, the cross-spectral density matrix of the field assumes the form [12

12. J. Tervo, T. Setälä, and A. T. Friberg, “Theory of partially coherent electromagnetic fields in the space–frequency domain,” J. Opt. Soc. Am. A **21**, 2205–2215 (2004). [CrossRef]

13. M. A. Alonso and E. Wolf, “The cross-spectral density matrix of a planar, electromagnetic stochastic source as a correlation matrix,” Opt. Commun. **281**, 2393–2396 (2008). [CrossRef]

**W**(

**r**

_{1},

**r**

_{2},

*ω*) =

**W**

^{†}(

**r**

_{2},

**r**

_{1},

*ω*), where † denotes Hermitian conjugate. The mean value of the optical intensity (spectral density) at point

**r**is where tr denotes the trace. The electromagnetic (spectral) degree of coherence of the light at points

**r**

_{1}and

**r**

_{2}then is given by the expression [10

10. T. Setälä, J. Tervo, and A. T. Friberg, “Complete electromagnetic coherence in the space–frequency domain,” Opt. Lett. **29**, 328–330 (2004). [CrossRef] [PubMed]

*μ*(

_{E}**r**

_{1},

**r**

_{2},

*ω*), which is applicable to electromagnetic waves of any state of coherence and polarization, is the spectral analogue of the time-domain quantity

*γ*(

_{E}**r**

_{1},

**r**

_{2},

*τ*) [6

6. J. Tervo, T. Setälä, and A. T. Friberg, “Degree of coherence for electromagnetic waves,” Opt. Express **11**, 1137–1143 (2003). [CrossRef] [PubMed]

*μ*(

_{E}**r**

_{1},

**r**

_{2},

*ω*) ≤ 1, and it takes on the value unity only when a complete correlation exists between all field components at the two points. We note also that the formulation in Eqs. (1)–(3) holds equally well for one-, two-, and three-dimensional fields, depending on how many components the electric field

**E**(

**r**,

*ω*) has. The one-dimensional case is identical with the traditional scalar-wave formulation.

*I*(

**r**,

*ω*) = |

*E*(

_{x}**r**,

*ω*)|

^{2}+ |

*E*(

_{y}**r**,

*ω*)|

^{2}of the electromagnetic field at frequency

*ω*is a random quantity and its variation from the mean value is We now assume that the incident electromagnetic beam is thermal in nature. It then follows, by use of the moment theorem for complex Gaussian random processes, that the correlation of the intensity fluctuations at two positions may be expressed as [2] where

*W*are the components of

_{ij}**W**. Substitution from Eq. (3) then leads at once to the result Hence, the correlation between intensity fluctuations, at a pair of points

**r**

_{1}and

**r**

_{2}, depends on the mean intensities, 〈

*I*(

**r**

_{1},

*ω*)〉 and 〈

*I*(

**r**

_{2},

*ω*)〉, and on the degree of electromagnetic coherence,

*μ*(

_{E}**r**

_{1},

**r**

_{2},

*ω*). The normalized correlation of intensity fluctuations is equal to the square of the degree of coherence for electromagnetic fields. This is one of the main results of this paper. We emphasize that it is fully analogous with the corresponding relation for the intensity fluctuations and the degree of coherence in the scalar theory [2]. It is also directly extendable to three-dimensional fields.

11. T. Setälä, J. Tervo, and A. T. Friberg, “Contrasts of Stokes parameters in Young’s interference experiment and electromagnetic degree of coherence,” Opt. Lett. **31**, 2669–2671 (2006). [CrossRef] [PubMed]

*(*

_{j}**r**

_{1},

**r**

_{2},

*ω*) are the two-point Stokes parameters [14

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

16. T. Setälä, J. Tervo, and A. T. Friberg, “Stokes parameters and polarization contrasts in Young’s interference experiment,” Opt. Lett. **31**, 2208–2210 (2006). [CrossRef] [PubMed]

*𝒮*

_{0}(

**r**

_{1},

**r**

_{2},

*ω*) represents the sum of the correlations of the

*x*and

*y*components of the field at points

**r**

_{1}and

**r**

_{2}, whereas the other parameters with

*j*= (1, 2, 3) can be interpreted as the differences of the correlations of the orthogonal field components in different coordinate systems obtained from the Cartesian one via unitary transformations [17

17. J. Tervo, T. Setälä, A. Roueff, Ph. Réfrégier, and A. T. Friberg, “Two-point Stokes parameters: interpretation and properties,” Opt. Lett. **34**, 3074–3076 (2009). [CrossRef] [PubMed]

*η*(

_{j}**r**

_{1},

**r**

_{2},

*ω*)| characterize the modulation of the Stokes parameters

*S*, with

_{j}*j*= (0,..., 3), on the observation screen in a Young’s interference experiment with an electromagnetic field of arbitrary state of coherence and polarization incident on the pinholes located at

**r**

_{1}and

**r**

_{2}[16

16. T. Setälä, J. Tervo, and A. T. Friberg, “Stokes parameters and polarization contrasts in Young’s interference experiment,” Opt. Lett. **31**, 2208–2210 (2006). [CrossRef] [PubMed]

*𝒮*

_{0}(

**r**

_{1},

**r**

_{2},

*ω*) is simply tr

**W**(

**r**

_{1},

**r**

_{2},

*ω*), |

*η*

_{0}(

**r**

_{1},

**r**

_{2},

*ω*)| is the usual intensity fringe visibility, while for

*j*= (1,2,3), |

*η*(

_{j}**r**

_{1},

**r**

_{2},

*ω*)| are the modulation contrasts of the corresponding polarization Stokes parameters. Hence, a complete representation of the electromagnetic degree of coherence includes the modulations of both the optical intensity and the polarization state. Equation (7) can obviously be regarded as a natural generalization of the classic two-pinhole scalar result.

*μ*(

_{E}**r**

_{1},

**r**

_{2},

*ω*). Conversely, determination of the modulation contrasts of both the optical intensity and the polarization state in a two-beam interference experiment results, by Eq. (10), in the normalized intensity (and photoelectron current) fluctuations. This is a consequence of the correlations between the electric field components among the two beams since, as we have noted, electromagnetic coherence (full or partial) can manifest itself not just in the formation of intensity fringes but also, or sometimes only, in the modulation of the polarization properties on interference.

## 4. Separation of spatial correlation and the degree of polarization

**E**(

**r**,

*ω*) =

*a*(

**r**,

*ω*)

**ê**(

**r**,

*ω*), where

*a*(

**r**,

*ω*) and

**ê**(

**r**,

*ω*) are random functions of position and

**ê**(

**r**,

*ω*) is normalized, i.e., for each realization |

**ê**(

**r**,

*ω*)| = 1. If, further,

*a*(

**r**,

*ω*) and

**ê**(

**r**,

*ω*) are independent, we find from Eq. (1) that where and The normalization implies that tr

**J**(

**r**

_{1},

**r**

_{1},

*ω*) = tr

**J**(

**r**

_{2},

**r**

_{2},

*ω*) = 1. On introducing the (spectral) correlation coefficients of the electric field components, we may write the cross-spectral density matrix as

*F*(

**r**,

*ω*) =

*F*(

**r**,

**r**,

*ω*) and

*J*(

_{ij}**r**,

*ω*) =

*J*(

_{ij}**r**,

**r**,

*ω*) with (

*i, j*) = (

*x,y*). Since the assumption of Gaussian statistics does not pose restrictions on the form of second-order correlation functions, we may consider vector fields that obey These conditions can be seen as electromagnetic extensions of the corresponding relations in the time-domain scalar theory [21

21. L. Mandel and E. Wolf, “Coherence properties of optical fields,” Rev. Mod. Phys. **37**, 231–287 (1965). [CrossRef]

*μ*(

_{ij}**r**

_{1},

**r**

_{2},

*ω*) = 0 for all

*i*and

*j*, where

**r**

_{1}and

**r**

_{2}are the positions of the pinholes.

*μ*(

_{yx}**r**

_{1},

**r**

_{2},

*ω*) =

*μ*(

_{yx}**r**

_{2},

**r**

_{2},

*ω*)

*μ*(

_{xx}**r**

_{1},

**r**

_{2},

*ω*). Hence, in these circumstances, we may re-express Eq. (17) as

**r**

_{1}and

**r**

_{2}. Therefore, we choose to rotate the coordinate axes at

**r**

_{1}and

**r**

_{2}such that the intensities of the

*x*and

*y*components of the electric field become equal. It is always possible to do so (the operation may be different at the two points). At both points

**r**

*,*

_{s}*s*= (1, 2), we then have, in the local coordinate system,

*J*(

_{xx}**r**

*,*

_{s}*ω*) =

*J*(

_{yy}**r**

*,*

_{s}*ω*) = 1/2, and moreover, in these circumstances the quantities |

*μ*(

_{xy}**r**

_{1},

**r**

_{1},

*ω*)| and |

*μ*(

_{yx}**r**

_{2},

**r**

_{2},

*ω*)| simply are the degrees of polarization

*P*(

**r**

*,*

_{s}*ω*) at those points, given by [2] By use of Eqs. (3) and (6) it then at once follows that This result shows that when the conditions in Eq. (18) hold, the effects of spatial correlations and the degree of polarization of the field on the normalized correlations of the intensity fluctuations at a pair of points separate for random fields of the form of Eq. (13), even when the degree of polarization varies with position.

21. L. Mandel and E. Wolf, “Coherence properties of optical fields,” Rev. Mod. Phys. **37**, 231–287 (1965). [CrossRef]

21. L. Mandel and E. Wolf, “Coherence properties of optical fields,” Rev. Mod. Phys. **37**, 231–287 (1965). [CrossRef]

**37**, 231–287 (1965). [CrossRef]

23. T. Hassinen, J. Tervo, and A. T. Friberg, “Cross-spectral purity of electromagnetic fields,” Opt. Lett. **34**, 3866–3868 (2009). [CrossRef] [PubMed]

## 5. Effects of the state of polarization

*F*(

**r**

_{1},

**r**

_{2},

*ω*) is a scalar correlation function,

**U**(

**r**

_{1}) and

**U**(

**r**

_{2}) are arbitrary unitary matrices, and

**J**(

*ω*) is a polarization matrix normalized so that tr

**J**(

*ω*) = 1. In other words, we consider a field whose degree of polarization is constant, but the state of polarization may change. The variation of the polarization state is effected pointwise through the deterministic unitary operator

**U**(

**r**). For instance,

**U**(

**r**) may correspond to rotation of the electric field or any other non-intensity-changing polarization modulation. Using Eqs. (3) and (6) it is straightforward to show that our present choice leads to the expression where

*P*(

*ω*) is the degree of polarization [position-independent version of Eq. (20)], and is a normalized form of the scalar correlation function. If we assume, further, that

**U**(

**r**) =

**U**, then not just the degree but also the state of polarization remains invariant. Under this circumstance Eq. (23) reduces to where

*μ*(

**r**

_{1},

**r**

_{2},

*ω*) is as specified in Eq. (18) . The other relation in Eq. (18) , on

*μ*(

_{xy}**r**

_{1},

**r**

_{2},

*ω*), is likewise satisfied and clearly the result (25) is consistent with Eq. (21) when the degree of polarization is constant. The difference between Eqs. (23) and (25) is that the function

*f*(

**r**

_{1},

**r**

_{2},

*ω*) equals the degree of coherence only if the field is uniformly polarized and the degree of polarization

*P*(

*ω*) = 1. Otherwise

*f*(

**r**

_{1},

**r**

_{2},

*ω*) represents correlation but it is not a degree of correlation of a fixed Cartesian component.

**U**separates the spatial dependence of the cross-spectral density matrix from the state of polarization and so it can be viewed to correspond to polarization purity (or non-entanglement). In particular, such fields possess ‘pure’ polarization in the sense that the polarization state does not change on two-beam interference [24

24. F. Gori, J. Tervo, and J. Turunen, “Correlation matrices for completely unpolarized beams,” Opt. Lett. **34**, 1447–1449 (2009). [CrossRef] [PubMed]

## 6. Conclusions

6. J. Tervo, T. Setälä, and A. T. Friberg, “Degree of coherence for electromagnetic waves,” Opt. Express **11**, 1137–1143 (2003). [CrossRef] [PubMed]

20. A. Al-Quasimi, M. Lahiri, D. Kuebel, D. F. V. James, and E. Wolf, “The influence of the degree of cross-polarization on the Hanbury Brown–Twiss effect,” Opt. Express **18**, 17124–17129 (2010). [CrossRef]

## Acknowledgments

## References and links

1. | R. Hanbury Brown and R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature |

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

3. | G. Baym, “The physics of Hanbury Brown–Twiss intensity interferometer: from stars to nuclear collisions,” Acta Phys. Polonica B |

4. | R. Martínez-Herrero, P. M. Mejías, and G. Piquero, |

5. | A. Luis, “An overview of coherence and polarization properties for multicomponent electromagnetic waves,” in |

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

7. | A. T. Friberg and E. Wolf, “Relationships between the complex degrees of coherence in the space–time and in the space–frequency domains,” Opt. Lett. |

8. | T. Setälä, F. Nunziata, and A. T. Friberg, “Differences between partial polarizations in the space–time and space–frequency domains,” Opt. Lett. |

9. | M. Lahiri, “Polarization properties of stochastic light beams in the space–time and space–frequency domains,” Opt. Lett. |

10. | T. Setälä, J. Tervo, and A. T. Friberg, “Complete electromagnetic coherence in the space–frequency domain,” Opt. Lett. |

11. | T. Setälä, J. Tervo, and A. T. Friberg, “Contrasts of Stokes parameters in Young’s interference experiment and electromagnetic degree of coherence,” Opt. Lett. |

12. | J. Tervo, T. Setälä, and A. T. Friberg, “Theory of partially coherent electromagnetic fields in the space–frequency domain,” J. Opt. Soc. Am. A |

13. | M. A. Alonso and E. Wolf, “The cross-spectral density matrix of a planar, electromagnetic stochastic source as a correlation matrix,” Opt. Commun. |

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

15. | O. Korotokova and E. Wolf, “Generalized Stokes parameters of random electromagnetic beams,” Opt. Lett. |

16. | T. Setälä, J. Tervo, and A. T. Friberg, “Stokes parameters and polarization contrasts in Young’s interference experiment,” Opt. Lett. |

17. | J. Tervo, T. Setälä, A. Roueff, Ph. Réfrégier, and A. T. Friberg, “Two-point Stokes parameters: interpretation and properties,” Opt. Lett. |

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

19. | 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. |

20. | A. Al-Quasimi, M. Lahiri, D. Kuebel, D. F. V. James, and E. Wolf, “The influence of the degree of cross-polarization on the Hanbury Brown–Twiss effect,” Opt. Express |

21. | L. Mandel and E. Wolf, “Coherence properties of optical fields,” Rev. Mod. Phys. |

22. | C. Brosseau, |

23. | T. Hassinen, J. Tervo, and A. T. Friberg, “Cross-spectral purity of electromagnetic fields,” Opt. Lett. |

24. | F. Gori, J. Tervo, and J. Turunen, “Correlation matrices for completely unpolarized beams,” Opt. Lett. |

**OCIS Codes**

(030.1640) Coherence and statistical optics : Coherence

(260.2110) Physical optics : Electromagnetic optics

(260.3160) Physical optics : Interference

(260.5430) Physical optics : Polarization

**ToC Category:**

Coherence and Statistical Optics

**History**

Original Manuscript: May 26, 2011

Revised Manuscript: July 1, 2011

Manuscript Accepted: July 1, 2011

Published: July 22, 2011

**Citation**

T. Hassinen, J. Tervo, T. Setälä, and A. T. Friberg, "Hanbury Brown–Twiss effect with electromagnetic waves," Opt. Express **19**, 15188-15195 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-16-15188

Sort: Year | Journal | Reset

### References

- R. Hanbury Brown and R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature 177, 27–29 (1956). [CrossRef]
- L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).
- G. Baym, “The physics of Hanbury Brown–Twiss intensity interferometer: from stars to nuclear collisions,” Acta Phys. Polonica B 29, 1839–1884 (1998).
- R. Martínez-Herrero, P. M. Mejías, and G. Piquero, Characterization of Partially Polarized Light Fields (Springer, 2009). [CrossRef]
- A. Luis, “An overview of coherence and polarization properties for multicomponent electromagnetic waves,” in Advances in Information Optics and Photonics , A. T. Friberg and R. Dändliker, eds. (SPIE Press, 2008), Chap. 9. [CrossRef]
- J. Tervo, T. Setälä, and A. T. Friberg, “Degree of coherence for electromagnetic waves,” Opt. Express 11, 1137–1143 (2003). [CrossRef] [PubMed]
- A. T. Friberg and E. Wolf, “Relationships between the complex degrees of coherence in the space–time and in the space–frequency domains,” Opt. Lett. 20, 623–625 (1995). [CrossRef] [PubMed]
- T. Setälä, F. Nunziata, and A. T. Friberg, “Differences between partial polarizations in the space–time and space–frequency domains,” Opt. Lett. 34, 2924–2926 (2009). [CrossRef] [PubMed]
- M. Lahiri, “Polarization properties of stochastic light beams in the space–time and space–frequency domains,” Opt. Lett. 34, 2936–2938 (2009). [CrossRef] [PubMed]
- T. Setälä, J. Tervo, and A. T. Friberg, “Complete electromagnetic coherence in the space–frequency domain,” Opt. Lett. 29, 328–330 (2004). [CrossRef] [PubMed]
- T. Setälä, J. Tervo, and A. T. Friberg, “Contrasts of Stokes parameters in Young’s interference experiment and electromagnetic degree of coherence,” Opt. Lett. 31, 2669–2671 (2006). [CrossRef] [PubMed]
- J. Tervo, T. Setälä, and A. T. Friberg, “Theory of partially coherent electromagnetic fields in the space–frequency domain,” J. Opt. Soc. Am. A 21, 2205–2215 (2004). [CrossRef]
- M. A. Alonso and E. Wolf, “The cross-spectral density matrix of a planar, electromagnetic stochastic source as a correlation matrix,” Opt. Commun. 281, 2393–2396 (2008). [CrossRef]
- J. Ellis and A. Dogariu, “Complex degree of mutual polarization,” Opt. Lett. 29, 536–538 (2004). [CrossRef] [PubMed]
- O. Korotokova and E. Wolf, “Generalized Stokes parameters of random electromagnetic beams,” Opt. Lett. 30, 198–200 (2005). [CrossRef]
- T. Setälä, J. Tervo, and A. T. Friberg, “Stokes parameters and polarization contrasts in Young’s interference experiment,” Opt. Lett. 31, 2208–2210 (2006). [CrossRef] [PubMed]
- J. Tervo, T. Setälä, A. Roueff, Ph. Réfrégier, and A. T. Friberg, “Two-point Stokes parameters: interpretation and properties,” Opt. Lett. 34, 3074–3076 (2009). [CrossRef] [PubMed]
- 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]
- A. Al-Quasimi, M. Lahiri, D. Kuebel, D. F. V. James, and E. Wolf, “The influence of the degree of cross-polarization on the Hanbury Brown–Twiss effect,” Opt. Express 18, 17124–17129 (2010). [CrossRef]
- L. Mandel and E. Wolf, “Coherence properties of optical fields,” Rev. Mod. Phys. 37, 231–287 (1965). [CrossRef]
- C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, 1998).
- T. Hassinen, J. Tervo, and A. T. Friberg, “Cross-spectral purity of electromagnetic fields,” Opt. Lett. 34, 3866–3868 (2009). [CrossRef] [PubMed]
- F. Gori, J. Tervo, and J. Turunen, “Correlation matrices for completely unpolarized beams,” Opt. Lett. 34, 1447–1449 (2009). [CrossRef] [PubMed]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

### Figures

Fig. 1 |

« Previous Article | Next Article »

OSA is a member of CrossRef.