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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 16 — Aug. 1, 2011
  • pp: 15188–15195
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Hanbury Brown–Twiss effect with electromagnetic waves

T. Hassinen, J. Tervo, T. Setälä, and A. T. Friberg  »View Author Affiliations


Optics Express, Vol. 19, Issue 16, pp. 15188-15195 (2011)
http://dx.doi.org/10.1364/OE.19.015188


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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

The classic Hanbury Brown–Twiss experiment has played a major role in the development of astronomy and quantum optics [1

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

3

3. G. Baym, “The physics of Hanbury Brown–Twiss intensity interferometer: from stars to nuclear collisions,” Acta Phys. Polonica B 29, 1839–1884 (1998).

]. The experiment concerns the correlations between photons in two beams of light as evidenced by the correlations of the fluctuations of photoelectron currents produced by two detectors, one placed in each beam. The phenomenon has been studied in detail using both the classical and quantum theory of light. The photoelectron current is normally taken proportional to the intensity of the light falling onto the detector, and additional noise sources in the detectors and electronics are neglected. Most analyses have considered one polarization mode only which amounts to a scalar-wave treatment of the incident radiation. As vector theories for coherence and polarization in randomly fluctuating electromagnetic fields have been developed [4

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

, 5

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]

], the Hanbury Brown–Twiss phenomenon in the context of vector waves has also begun to attract interest.

We make use of the electromagnetic theory of optical coherence in the space–frequency domain [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]

] and demonstrate that the Hanbury Brown–Twiss effect with classical, thermal vector fields is fully specified by the spectral electromagnetic degree of coherence, μE (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

Consider a beam-like random optical field that propagates predominantly in the positive z direction. Denoting a realization of the electric field in the space–frequency domain by a column vector E(r, ω) = [Ex(r, ω), Ey(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

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(r1,r2,ω)=E*(r1,ω)ET(r2,ω),
(1)
where the asterisk stands for complex conjugation and the angular brackets indicate statistical average. Clearly, the cross-spectral density obeys the relation 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
I(r,ω)=|Ex(r,ω)|2+|Ey(r,ω)|2=trW(r,r,ω),
(2)
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]

]
μE2(r1,r2,ω)=tr [W(r1,r2,ω)W(r2,r1,ω)]I(r1,ω)I(r2,ω).
(3)
The degree μE (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 γE (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]

]. It is real and normalized so that 0 ≤ μE (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.

The intensity I(r, ω) = |Ex(r, ω)|2 + |Ey(r, ω)|2 of the electromagnetic field at frequency ω is a random quantity and its variation from the mean value is
ΔI(r,ω)=I(r,ω)I(r,ω)=I(r,ω)trW(r,r,ω).
(4)
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

2. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).

]
ΔI(r1,ω)ΔI(r2,ω)=i,j|Wij(r1,r2,ω)|2=tr[W(r1,r2,ω)W(r2,r1,ω)],
(5)
where Wij are the components of W. Substitution from Eq. (3) then leads at once to the result
ΔI(r1,ω)ΔI(r2,ω)=I(r1,ω)I(r2,ω)μE2(r1,r2,ω).
(6)
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, μE (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

2. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).

]. It is also directly extendable to three-dimensional fields.

In scalar theory, the degree of coherence is proportional to the visibility of the intensity fringes in two-beam interference. In the electromagnetic case, the situation is more involved due to the presence of two (or more) orthogonal components of the electric field in both beams. This brings in the added complexity of correlations of the field components in each beam and between the two beams. It has been shown that the spectral electromagnetic degree of coherence can, in general, be expressed in the form [11

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]

]
μE2(r1,r2,ω)=12j=03|ηj(r1,r2,ω)|2,
(7)
where
ηj(r1,r2,ω)=𝒮j(r1,r2,ω)[I(r1,ω)I(r2,ω)]1/2,j=(0,,3),
(8)
are the so-called generalized visibility parameters and 𝒮j(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

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]

]
𝒮0(r1,r2,ω)=Wxx(r1,r2,ω)+Wyy(r1,r2,ω),
(9a)
𝒮1(r1,r2,ω)=Wxx(r1,r2,ω)Wyy(r1,r2,ω),
(9b)
𝒮2(r1,r2,ω)=Wyx(r1,r2,ω)+Wxy(r1,r2,ω),
(9c)
𝒮3(r1,r2,ω)=i[Wyx(r1,r2,ω)Wxy(r1,r2,ω)].
(9d)
Physically, the parameter 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]

]. Moreover, the quantities |ηj(r 1, r 2, ω)| characterize the modulation of the Stokes parameters Sj, with 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]

]. As 𝒮 0(r 1, r 2, ω) is simply trW(r 1, r 2, ω), |η 0(r 1, r 2, ω)| is the usual intensity fringe visibility, while for j = (1,2,3), |ηj(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.

From Eqs. (6) and (7) it readily follows that
ΔI(r1,ω)ΔI(r2,ω)I(r1,ω)I(r2,ω)=12j=03|ηj(r1,r2,ω)|2.
(10)
Measurement of the normalized correlation of the intensity fluctuations in a spectral Hanbury Brown–Twiss experiment with random vector fields then yields, in view of Eq. (6), the electromagnetic degree of coherence μE (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.

3. Properties of the degree of cross-polarization

In some recent papers, Eq. (10) has been re-expressed in a form where η 0(r 1, r 2, ω) has been separated out on the right-hand side of the expression [18

18. 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]

20

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]

]. With this kind of an approach the normalized correlation of intensity fluctuations at the two points can be written in a form
ΔI(r1,ω)ΔI(r2,ω)I(r1,ω)I(r2,ω)=12[1+𝒫2(r1,r2,ω)]|η0(r1,r2,ω)|2,
(11)
where the quantity
𝒫(r1,r2,ω)=Σj=13|ηj(r1,r2,ω)|2|η0(r1,r2,ω)|
(12)
has been termed the degree of cross-polarization of the electromagnetic beam. We note that 𝒫(r 1, r 2, ω) is generally not normalized but may assume arbitrarily large values [19

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. 10, 055001 (2008). [CrossRef]

], although it reduces to the usual degree of polarization when r 1 = r 2. Since the normalized correlation of the intensity fluctuations remains finite, Eq. (11) leads to ambiguities in cases when the fringe visibility η 0(r 1, r 2, ω) goes to zero and the fields at r 1 and r 2 are at least partially correlated. To compensate for the vanishing of η 0(r 1, r 2, ω) the degree of cross-polarization 𝒫(r 1, r 2, ω) consequently has to approach infinity.

As an illustration we consider a simple experiment where the fields at r 1 and r 2 are linearly polarized and completely correlated, implying that the left-hand side of Eq. (11) equals one. The polarization at r 1 is taken horizontal and the plane of vibration at r 2, oriented at angle θ, is rotated from horizontal to vertical. Then, as a function of θ, simply |η 0(r 1, r 2, ω)|2 = cos2 θ and 𝒫 2(r 1, r 2, ω) = 2sec2 θ – 1. The behavior of these quantities is shown in Fig. 1 for the range 0 ≤ θπ, demonstrating the divergence of the degree of cross-polarization at θ = π/2 when the intensity fringe visibility disappears.

Fig. 1 Square of the intensity fringe visibility |η 0(r 1, r 2, ω)|2 (dash, blue), square of the degree of cross-polarization 𝒫 2(r 1, r 2, ω) (solid, green), and the normalized correlation of intensity fluctuations [left-hand side of Eq. (11)] (dash-dot, red), as a function of the polarization-plane rotation angle θ on a logarithmic scale.

4. Separation of spatial correlation and the degree of polarization

To gain further insight, let us examine the influences of the spatial correlations and the state of polarization of the field on the correlations of the intensity fluctuations. To that end, we assume first that the spectral electric field is of the form 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
W(r1,r2,ω)=F(r1,r2,ω)J(r1,r2,ω),
(13)
where
F(r1,r2,ω)=a*(r1)a(r2)
(14)
and
J(r1,r2,ω)=e^*(r1)e^T(r2).
(15)
The normalization implies that trJ(r 1, r 1, ω) = trJ(r 2, r 2, ω) = 1. On introducing the (spectral) correlation coefficients of the electric field components,
μij(r1,r2,ω)=Wij(r1,r2,ω)[|Ei(r1,ω)|2|Ej(r2,ω)|21/2,(i,j)=(x,y),
(16)
we may write the cross-spectral density matrix as
W(r1,r2,ω)=[F(r1,ω)F(r2,ω)]1/2×(μxx(r1,r2,ω)[Jxx(r1,ω)Jxx(r2,ω)]1/2μxy(r1,r2,ω)[Jxx(r1,ω)Jyy(r2,ω)]1/2μyx(r1,r2,ω)[Jyy(r1,ω)Jxx(r2,ω)]1/2μyy(r1,r2,ω)[Jyy(r1,ω)Jyy(r2,ω)]1/2),
(17)
where F(r, ω) = F(r, r, ω) and Jij(r, ω) = Jij(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
μxy(r1,r2,ω)=μxy(r1,r1,ω)μxx(r1,r2,ω),
(18a)
μxx(r1,r2,ω)=μyy(r1,r2,ω)μ(r1,r2,ω),
(18b)
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]

]. In this context, they ensure, e.g., that the visibility of the intensity fringes in Young’s experiment equals zero if, and only if, μij(r 1, r 2, ω) = 0 for all i and j, where r 1 and r 2 are the positions of the pinholes.

From Eqs. (18) it readily follows that μyx(r 1, r 2, ω) = μyx(r 2, r 2, ω)μxx(r 1, r 2, ω). Hence, in these circumstances, we may re-express Eq. (17) as
W(r1,r2,ω)=μ(r1,r2,ω)[F(r1,ω)F(r2,ω)]1/2×([Jxx(r1,ω)Jxx(r2,ω)]1/2μxy(r1,r1,ω)[Jxx(r1,ω)Jyy(r2,ω)]1/2μyx(r2,r2,ω)[Jyy(r1,ω)Jxx(r2,ω)]1/2[Jyy(r1,ω)Jyy(r2,ω)]1/2).
(19)
The normalized correlation of the intensity fluctuations, or the square of the electromagnetic degree of coherence in view of Eq. (6), is invariant under arbitrary unitary transformations at points 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, Jxx(r s, ω) = Jyy(r s, ω) = 1/2, and moreover, in these circumstances the quantities |μxy(r 1, r 1, ω)| and |μyx(r 2, r 2, ω)| simply are the degrees of polarization P(r s, ω) at those points, given by [2

2. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).

]
P(r,ω)=[14detJ(r,ω)tr2J(r,ω)]1/2.
(20)
By use of Eqs. (3) and (6) it then at once follows that
ΔI(r1,ω)ΔI(r2,ω)I(r1,ω)I(r2,ω)=12[1+P2(r1,ω)+P2(r2,ω)2]|μ(r1,r2,ω)|2.
(21)
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.

It is of interest to note that Eq. (21) is analogous to a classic result on thermal-beam intensity correlations (Eq. (6.26) of [21

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

]), derived differently in time domain and assuming certain cross-spectral purity properties for the polarization components. Indeed, our relations in Eq. (18) , which deal with normalized correlation functions at two spatial points in frequency domain, are mathematically formally identical with the purity conditions employed in [21

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

] for normalized correlation functions at a single point but for two instants of time. Relatively little is known, however, of the true physical meanings of these electromagnetic purity conditions in either domain [21

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

23

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

We emphasize that Eq. (22) with constant 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]

], even though each Stokes parameter may be modulated (in the same way) on the observation screen.

Finally, comparing Eqs. (11) and (23), we obtain at once that
𝒫2(r1,r2,ω)=[1+P2(ω)]|f(r1,r2,ω)η0(r1,r2,ω)|21.
(26)
This relation simplifies further if the exact forms of the unitary transformations in Eq. (13) are known. For example, if U(r 1) = U(r 2), we have 𝒫 (r 1, r 2, ω) = P(ω).

6. Conclusions

Making use of the space–frequency representation of random electromagnetic fields, we have analyzed the classic Hanbury Brown–Twiss phenomenon in terms of vector waves that obey Gaussian statistics. As the primary result we showed that the normalized correlations of the intensity fluctuations, which are proportional to the fluctuations of the photoelectron currents produced by the two detectors, are fully determined by the spectral electromagnetic degree of coherence as defined in Eq. (3). This conclusion, which is consistent with the traditional scalar-wave relation and whose space–time domain counterpart was briefly mentioned in [6

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

], is at variance with some recent works dealing with the influence of coherence and polarization on the correlations of intensity fluctuations [20

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]

]. The differences in the physical conclusions are a consequence of the different definitions of the electromagnetic degree of coherence.

We also examined the degree of cross-polarization, demonstrating that it diverges in certain conditions. Further, we studied the role of polarization of the incident radiation on the correlations of intensity fluctuations. We showed that the effects of spatial correlations and of the degree and state of polarization can in certain conditions be separately identified. In particular, a space-frequency domain analogue of a classic result on temporal intensity fluctuations of thermal light beams was derived under the circumstance when the random vector field exhibits certain purity conditions of its polarization components. If the polarization state is constant, the field’s polarization properties then can be regarded as pure in a well-defined sense.

Acknowledgments

This work was supported by the Academy of Finland (grants 118329, 118951, and 128331) and by the Ministry of Education of Finland (Nanophotonics Research and Development Project).

References and links

1.

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

2.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).

3.

G. Baym, “The physics of Hanbury Brown–Twiss intensity interferometer: from stars to nuclear collisions,” Acta Phys. Polonica B 29, 1839–1884 (1998).

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]

6.

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

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. 20, 623–625 (1995). [CrossRef] [PubMed]

8.

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]

9.

M. Lahiri, “Polarization properties of stochastic light beams in the space–time and space–frequency domains,” Opt. Lett. 34, 2936–2938 (2009). [CrossRef] [PubMed]

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]

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]

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]

14.

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

15.

O. Korotokova and E. Wolf, “Generalized Stokes parameters of random electromagnetic beams,” Opt. Lett. 30, 198–200 (2005). [CrossRef]

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]

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]

18.

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]

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. 10, 055001 (2008). [CrossRef]

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]

21.

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

22.

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

23.

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

24.

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

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


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References

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