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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 16 — Aug. 2, 2010
  • pp: 17124–17129
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The influence of the degree of cross-polarization on the Hanbury Brown-Twiss effect

Asma Al-Qasimi, Mayukh Lahiri, David Kuebel, Daniel F. V. James, and Emil Wolf  »View Author Affiliations


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

The Hanbury Brown-Twiss effect [1–4

1. R. H. Brown and R. Q. Twiss, “A new type of interferometer for use in radio astronomy,” Philos. Mag. 45, 663–682 (1954).

] is generally regarded as the starting point of quantum optics. The effect is a manifestation of correlations between intensity fluctuations at two points in a cross-section of a light beam (see, for example, [5

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

], Secs. 9.9 and 14.6). The correlation between the intensity fluctuations are detected from measurements of the correlations between fluctuating current outputs of the photoelectric detectors, illuminated by the beam (see, for example, [6

6. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, Cambridge, 2007).

], Ch. 7). The effect was originally introduced in connection with attempts to measure diameters of stars but has since then found applications in high energy physics, nuclear physics, atomic physics and in neutron physics (see, for example, [7

7. G. Baym, “The Physics of Hanbury Brown-Twiss intensity interferometry: from stars to nuclear collisions,” Acta Phys. Poln. B 29, 1839–1884 (1998).

, 8

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

]).

Most of the traditional treatments of the Hanbury Brown-Twiss effect with light were carried out within the framework of the statistical theory of scalar fields. Only fairly recently, has analysis of it been made by use of the electromagnetic (vector) theory [9

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

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]

]. The analysis based on the electromagnetic theory revealed a somewhat surprising fact, namely that knowledge of the degree of coherence between the light fluctuations in the incident beams at the two detectors and of the degree of polarization of the light falling on each detector are not adequate to determine the correlation in the current output. A new statistical parameter of the incident field is needed to fully describe this effect, namely the so-called 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].

]. Whilst the degree of polarization depends on correlations between the electric field components at a particular point in space, the degree of cross-polarization depends on correlations in the field components at a pair of points (the location of the photo-detectors).

In the present paper, we provide an explicit example of this rather surprising prediction. Specifically we consider two beams generated by two sources which have the same spectral densities, the same degrees of coherence and the same degrees of polarization, but have different degrees of cross-polarization. We show that the correlations of the intensity fluctuations at two points in the far-zone are different. Thus our analysis confirms, by an explicit example, that the knowledge of the spectral density, of the degree of coherence and of the degree of polarization of the beam at the source plane are not sufficient to predict the correlation between the intensity fluctuations at a pair of points in the far-zone; it shows that is also necessary know the degree of cross-polarization. Thus the analysis clearly reveals that the knowledge of the degree of cross-polarization is needed to elucidate some physical phenomena involving the interaction of an electromagnetic field with matter.

2. Theory

We begin by recalling some basic results of the theory of stochastic electromagnetic beams. Let us consider a statistically stationary light beam generated by a planar secondary source located at the plane z = 0. Suppose that the beam propagates into the half-space z > 0 with its axis along the z direction. Let Ex(ρ, z; ω) and Ey(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

6. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, Cambridge, 2007).

], Sec. 9.1):

W(ρ1,ρ2,z;ω)[Wij(ρ1,ρ2,z;ω)]=Ei*(ρ1,z;ω)Ej(ρ2,z;ω),(i=x,y;j=x,y).
(1)

Here the asterisk denotes the complex conjugate and the angular brackets denote ensemble average. The ensemble is to be understood in the sense of coherence theory in the space-frequency domain (see, for example, [6

6. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, Cambridge, 2007).

], Secs. 4.1 and 9.1). In terms of the CSDM, the spectral density S(ρ, z; ω) at a point P(ρ, z) is given by the expression

S(ρ,z;ω)TrW(ρ,ρ,z;ω),
(2)

where Tr denotes the trace. The 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

μ(ρ1,ρ2,z;ω)TrW(ρ1,ρ2,z;ω)S(ρ1,z;ω)S(ρ2,z;ω),
(3)

and the spectral degree of polarization 𝒫(ρ, z;ω) at the point P(ρ, z) is given by the expression

𝒫(ρ,z;ω)14DetW(ρ,ρ,z;ω){TrW(ρ,ρ,z;ω)}2,
(4)

where Det denotes the determinant. However, as was mentioned earlier, these three quantities are not sufficient to determine the correlation between the intensity fluctuations at a pair of points in a cross-section of a beam. This fact was first demonstrated in Ref. [9

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]

], for a special class of stationary stochastic beams. A more general formulation was later given in Ref. [10

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]

]. We will briefly mention the mains results obtained in Ref. [10

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]

].

Suppose that a statistically stationary electromagnetic beam is incident on two detectors, placed at the points 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

6. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, Cambridge, 2007).

], Ch. 7), can be shown to be given by the formula ([10

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]

], Eqs. (8) and (9))

C(ρ1,ρ2,z;ω)ΔI(ρ1,z;ω)ΔI(ρ2,z;ω)
=12[1+𝒬(ρ1,ρ2,z;ω)]μ(ρ1,ρ2,z;ω)2S(ρ1,z;ω)S(ρ2,z;ω),
(5)

where

𝒬(ρ1,ρ2,z;ω)=2Tr[W(ρ1,ρ2,z,ω)·W(ρ1,ρ2,z;ω)]TrW(ρ1,ρ2,z;ω)21
(6)

is called the 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.

Suppose that a stationary stochastic electromagnetic beam propagates a distance 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]

], see also, [6

6. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, Cambridge, 2007).

], Sec. 9.4.1)

W(ρ1,ρ2,z;ω)=W(ρ1,ρ2,0;ω)G*(ρ1ρ1,z;ω)G(ρ2ρ2,z;ω)d2ρ1d2ρ2,
(7)

where

G(ρρ,z;ω)=ik2πzexp[ik(ρρ)22z]
(8)

is the Green’s function of the Helmholtz operator for paraxial propagation [see also, Ref. [5

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

], Eq.(5.6–17)].

Suppose that the beam is of Gaussian Schell-model type. The elements of the CSDM at the source plane z = 0 are then given by the expressions [see, for example, Ref. [6

6. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, Cambridge, 2007).

], Sec. 9.4.2, Eqs. (5)–(7)]

Wij(ρ1,ρ2,0;ω)=AiAjBijexp[(ρ124σi2+ρ224σj2)]exp[(ρ2ρ1)22δij2],i=x,y;j=x,y,
(9)

where σiδij and

Bij=1wheni=j,
(10a)
Bij<1whenij,
(10b)
Bij=Bij*,
(10c)
δij=δji.
(10d)

To ensure that such a beam can be generated, these parameters must satisfy certain realizability conditions ([15

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

]; equivalent conditions are also derived in [16

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]

]), viz.,

max{δxx,δyy}δxymin{δxxBxy,δyyBxy}.
(11)

Suppose now that the beam has propagated some distance 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

6. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, Cambridge, 2007).

], Sec. 9.4.2, Eq. (10), (11))

Wij(ρ1,ρ2,z;ω)=AiAjBijΔij2(z)exp[(ρ1+ρ2)28σ2Δij2(z)]exp[(ρ2ρ1)22ΩijΔij2(z)]exp[ik(ρ22ρ12)2Rij(z)],
(12)

where

Rij=z[1+(kσΩijz)2],
(13a)
1Ωij2=14σ2+1δij2,
(13b)
Δij(z)=1+(zkσΩij)2.
(13c)

3. Example

We will now return to our main problem namely to show that it is possible to have two planar sources with the same spectral densities, the same spectral degrees of coherence and the same spectral degrees of polarization, which may generate beams with different correlations between the intensity fluctuations at a pair of points. To demonstrate this result, we consider two Gaussian Schell-model beams, “a” and “b”, produced by two different planar secondary sources. We assume that the beam “a” is characterized by parameters Ax = Ay = 1, Bxy=Byx=916 , σx = σy = σ, δxx = δyy = δ and δxy=δyx=43δ . It can readily be shown that these parameters obey the realizability conditions [Eq. (11)]. From Eq. (9), it readily follows that the CSDM W⃡(a) of beam a, at the source plane z = 0 has the form

W(a)(ρ1,ρ2,0;ω)=exp[ρ12+ρ224σ2](𝒜𝓑𝓑𝒜),
(14)

where

𝒜=exp[(ρ2ρ1)22δ2],
(15a)
𝓑=916exp[9(ρ2ρ1)232δ2].
(15b)

We further assume that beam “b”, is characterized by the parameters Ax = Ay = 1, Bxy=Byx=916 , σx = σy = σ, δxx = δyy = δxy = δyx = δ. Using Eq. (9) again, one readily finds that the CSDM of the beam “b”, at the source plane has the form

W(b)(ρ1,ρ2,0;ω)=exp[ρ12+ρ224σ2](𝒜𝒞𝒞𝒜),
(16)

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

𝒞=916exp[(ρ2ρ1)22δ2].
(17)

On using Eqs. (2)–(4), (14) and (16), one can readily verify that both beams have the same distributions of spectral densities, of spectral degrees of coherence and of spectral degrees of polarization at the sources, namely

S(a)(ρ,0;ω)=S(b)(ρ,0;ω)=2exp[ρ22σ2],
(18a)
μ(a)(ρ1,ρ2,0;ω)=μ(b)(ρ1,ρ2,0;ω)=exp[(ρ2ρ1)2δ22],
(18b)
𝒫(a)(ρ,0;ω)=𝒫(b)(ρ,0;ω)=916.
(18c)

However, as can be shown by using Eq. (6), the two beams have different distribution of the degree of cross-polarization at the source plane.

In Fig. 1 the variation of degree of cross-polarization, at a pair of diametrically opposite points at the source plane, has been plotted with half-separation distance ρ with the choice δ = 0.001m and σ = 0.01m.

As the beams propagate some distance 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.

Fig. 1. Variation of the degree of cross-polarization 𝒬(ρ,−ρ, 0;ω) with ρ, at a pair of diametrically opposite points in two Gaussian Schell-model sources separated by distance 2ρ, for δ = 0.001m and σ = 0.01m. The labels (a) and (b) refer to the beams which are labeled in this way in the text.
Fig. 2. Variation of the correlation between the intensity fluctuations C(ρ,−ρ, z 0;ω) with ρ, at two diametrically opposite points separated by distance 2ρ, for the two beams, in a plane at distance z 0 = 10km from the sources. The labels (a) and (b) have the same meaning as in Fig. 1.

Acknowledgement

The research was supported by the US Air Force Office of Scientific Research under grant No. FA9550-08-1-0417, by the Air Force Research Laboratory (ARFL) under contract number 9451-04-C-0296, and by NSERC (Canada).

References and links

1.

R. H. Brown and R. Q. Twiss, “A new type of interferometer for use in radio astronomy,” Philos. Mag. 45, 663–682 (1954).

2.

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

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 242, 300–324 (1957). [CrossRef]

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 , 243, 291–319 (1957). [CrossRef]

5.

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

6.

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, Cambridge, 2007).

7.

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

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]

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].

12.

D. Kuebel, “Properties of the degree of cross-polarization in the spacetime domain,” Opt. Commun. 282, 3397–3401 (2009). [CrossRef]

13.

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

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]

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]

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

  1. R. H. Brown, and R. Q. Twiss, “A new type of interferometer for use in radio astronomy,” Philos. Mag. 45, 663–682 (1954).
  2. R. H. Brown, and R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature 177, 27–29 (1956). [CrossRef]
  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 242, 300–324 (1957). [CrossRef]
  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 243, 291–319 (1957). [CrossRef]
  5. L. Mandel, and E. Wolf, Optical Coherence and Quantum Optics (Cambridge, Cambridge University Press, 1995).
  6. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, Cambridge, 2007).
  7. G. Baym, “The Physics of Hanbury Brown-Twiss intensity interferometry: from stars to nuclear collisions,” Acta Phys. Poln. B 29, 1839–1884 (1998).
  8. D. Kleppner, “Hanbury Brown’s steamroller,” Phys. 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]
  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].
  12. D. Kuebel, “Properties of the degree of cross-polarization in the space-time domain,” Opt. Commun. 282, 3397–3401 (2009). [CrossRef]
  13. J. Ellis, and A. Dogariu, “Complex degree of mutual polarization,” Opt. Lett. 29, 536–538 (2004). [CrossRef] [PubMed]
  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]
  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]

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