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 Optical Society of America
The classic Hanbury Brown–Twiss experiment has played a major role in the development of astronomy and quantum optics [1–3]. 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, 5], the Hanbury Brown–Twiss phenomenon in the context of vector waves has also begun to attract interest.
Some years ago, in 2003, the Hanbury Brown–Twiss effect with classical vector-valued fields was briefly analyzed in the space–time domain . It was shown that for stationary electromagnetic waves obeying Gaussian statistics the normalized intensity correlation function, at points r 1 and r 2 and times separated by τ, is given by the square of the electromagnetic degree of coherence γE (r 1, r 2, τ), as defined by Eq. (6) of . This result is closely related to stellar intensity interferometry with a delay line. The quantity γE (r 1, r 2, τ) is a measure of the correlations among all the electric field components at r 1 and r 2 and it equals unity if, and only if, there is a perfect correlation between all components.
Due to physical reasons, the space–frequency representation of coherence and polarization in random electromagnetic fields has lately increasingly gained attention. The temporal and spectral degrees of coherence are related (though not so simply) in scalar theory , but for the corresponding degrees of polarization in stationary electromagnetic beams the situation is considerably more involved [8, 9]. Any analogous relationships between the electromagnetic degrees of coherence, γE (r 1, r 2, τ) and μE (r 1, r 2, ω) as given by Eq. (12) of , have to our knowledge not been examined. However, it is of interest to recall that unlike γE (r 1, r 2, τ), the spectral degree of electromagnetic coherence μE (r 1, r 2, ω) has been studied in the interference of two random electromagnetic beams . In particular, the quantity μE (r 1, r 2, ω), which accounts for the polarization and cross-polarization properties of the light at points r 1 and r 2, is in a natural way related to both the visibility of the intensity fringes and the modulation contrasts of the polarization state, as specified by the Stokes parameters in the interference pattern. Hence it can be expected that μE (r 1, r 2, ω) also has a close bearing on the spectral Hanbury Brown–Twiss effect. In this paper we show that this, indeed, is the case.
We make use of the electromagnetic theory of optical coherence in the space–frequency domain  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.
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 [12,13]10] 6]. 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 is2] Eq. (3) then leads at once to the result 2]. 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 14–16] 17]. 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 . 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.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 [18–20]. With this kind of an approach the normalized correlation of intensity fluctuations at the two points can be written in a form19], 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 ≤ θ ≤ π, demonstrating the divergence of the degree of cross-polarization at θ = π/2 when the intensity fringe visibility disappears.
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) that21]. 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.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  Eqs. (3) and (6) it then at once follows that 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 ), 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  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 [21–23].
5. Effects of the state of polarization
As the next step, we take the cross-spectral density matrix of the incident radiation to be of the formEqs. (3) and (6) it is straightforward to show that our present choice leads to the expression Eq. (20)], and Eq. (23) reduces to Eq. (18) . The other relation in Eq. (18) , on μxy(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.
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 , even though each Stokes parameter may be modulated (in the same way) on the observation screen.Eq. (13) are known. For example, if U(r 1) = U(r 2), we have 𝒫 (r 1, r 2, ω) = P(ω).
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 , is at variance with some recent works dealing with the influence of coherence and polarization on the correlations of intensity fluctuations . 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.
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
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