## 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] 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], 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], 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, 8]).

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 [9,10]. 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]. 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 *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):

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], 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

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

and the *spectral degree of polarization 𝒫*(* ρ, z*;

*ω*) at the point

*P*(

*) is given by the expression*

**ρ**, zwhere 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], for a special class of stationary stochastic beams. A more general formulation was later given in Ref. [10]. We will briefly mention the mains results obtained in Ref. [10].

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], Ch. 7), can be shown to be given by the formula ([10], Eqs. (8) and (9))

$$\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}=\frac{1}{2}[1+\mathcal{Q}({\rho}_{1},{\rho}_{2},z;\omega )]{\mid \mu ({\rho}_{1},{\rho}_{2},z;\omega )\mid}^{2}\u3008S({\rho}_{1},z;\omega )\u3009\u3008S({\rho}_{2},z;\omega )\u3009,$$

where

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], see also, [6], Sec. 9.4.1)

where

is the Green’s function of the Helmholtz operator for paraxial propagation [see also, Ref. [5], 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], Sec. 9.4.2, Eqs. (5)–(7)]

where *σ _{i}* ≫

*δ*and

_{ij}To ensure that such a beam can be generated, these parameters must satisfy certain realizability conditions ([15]; equivalent conditions are also derived in [16]), viz.,

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

where

## 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 *A _{x}* =

*A*= 1, ${B}_{\mathrm{xy}}={B}_{\mathrm{yx}}=\frac{9}{16}$ ,

_{y}*σ*=

_{x}*σ*=

_{y}*σ*,

*δ*=

_{xx}*δ*=

_{yy}*δ*and ${\delta}_{\mathrm{xy}}={\delta}_{\mathrm{yx}}=\frac{4}{3}\delta $ . 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

where

We further assume that beam “*b*”, is characterized by the parameters *A _{x}* =

*A*= 1, ${B}_{\mathrm{xy}}={B}_{\mathrm{yx}}=\frac{9}{16}$ ,

_{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

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

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

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.

## 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

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