Hanbury Brown–Twiss effect with electromagnetic waves

T. Hassinen; J. Tervo; T. Setälä; A. T. Friberg

doi:10.1364/OE.19.015188

1. Introduction

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 [6]. 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 γ_E (r ₁, r ₂, τ), as defined by Eq. (6) of [6]. This result is closely related to stellar intensity interferometry with a delay line. The quantity γ_E (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 [7], 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, γ_E (r ₁, r ₂, τ) and μ_E (r ₁, r ₂, ω) as given by Eq. (12) of [10], have to our knowledge not been examined. However, it is of interest to recall that unlike γ_E (r ₁, r ₂, τ), the spectral degree of electromagnetic coherence μ_E (r ₁, r ₂, ω) has been studied in the interference of two random electromagnetic beams [11]. In particular, the quantity μ_E (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 μ_E (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 [12] and demonstrate that the Hanbury Brown–Twiss effect with classical, thermal vector fields is fully specified by the spectral electromagnetic degree of coherence, 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

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) = [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 [12,13]

W (r_{1}, r_{2}, ω) = 〈 E^{*} (r_{1}, ω) E^{T} (r_{2}, ω) 〉,

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) 〉 = 〈 {| E_{x} (r, ω) |}^{2} 〉 + 〈 {| E_{y} (r, ω) |}^{2} 〉 = tr W (r, r, ω),

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]

μ_{E}^{2} (r_{1}, r_{2}, ω) = \frac{tr [W (r_{1}, r_{2}, ω) W (r_{2}, r_{1}, ω)]}{〈 I (r_{1}, ω) 〉 〈 I (r_{2}, ω) 〉} .

The degree 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]. It is real and normalized so that 0 ≤ 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.

The intensity 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

Δ I (r, ω) = I (r, ω) - 〈 I (r, ω) 〉 = I (r, ω) - tr W (r, r, ω) .

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]

〈 Δ I (r_{1}, ω) Δ I (r_{2}, ω) 〉 = {\sum_{i, j} | W_{i j} (r_{1}, r_{2}, ω) |}^{2} = tr [W (r_{1}, r_{2}, ω) W (r_{2}, r_{1}, ω)],

where W_ij are the components of W. Substitution from Eq. (3) then leads at once to the result

〈 Δ I (r_{1}, ω) Δ I (r_{2}, ω) 〉 = 〈 I (r_{1}, ω) 〉 〈 I (r_{2}, ω) 〉 μ_{E}^{2} (r_{1}, r_{2}, ω) .

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.

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]

μ_{E}^{2} (r_{1}, r_{2}, ω) = \frac{1}{2} \sum_{j = 0}^{3} {| η_{j} (r_{1}, r_{2}, ω) |}^{2},

where

η_{j} (r_{1}, r_{2}, ω) = \frac{𝒮_{j} (r_{1}, r_{2}, ω)}{{[〈 I (r_{1}, ω) 〉 〈 I (r_{2}, ω) 〉]}^{1 / 2}}, j = (0, \dots, 3),

are the so-called generalized visibility parameters and 𝒮_j(r ₁, r ₂, ω) are the two-point Stokes parameters [14–16]

𝒮_{0} (r_{1}, r_{2}, ω) = W_{x x} (r_{1}, r_{2}, ω) + W_{y y} (r_{1}, r_{2}, ω),

𝒮_{1} (r_{1}, r_{2}, ω) = W_{x x} (r_{1}, r_{2}, ω) - W_{y y} (r_{1}, r_{2}, ω),

𝒮_{2} (r_{1}, r_{2}, ω) = W_{y x} (r_{1}, r_{2}, ω) + W_{x y} (r_{1}, r_{2}, ω),

𝒮_{3} (r_{1}, r_{2}, ω) = i [W_{y x} (r_{1}, r_{2}, ω) - W_{x y} (r_{1}, r_{2}, ω)] .

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]. Moreover, the quantities |η_j(r ₁, r ₂, ω)| characterize the modulation of the Stokes parameters S_j, 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]. 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), |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.

From Eqs. (6) and (7) it readily follows that

\frac{〈 Δ I (r_{1}, ω) Δ I (r_{2}, ω) 〉}{〈 I (r_{1}, ω) 〉 〈 I (r_{2}, ω) 〉} = \frac{1}{2} \sum_{j = 0}^{3} {| η_{j} (r_{1}, r_{2}, ω) |}^{2} .

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 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.

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 form

\frac{〈 Δ I (r_{1}, ω) Δ I (r_{2}, ω) 〉}{〈 I (r_{1}, ω) 〉 〈 I (r_{2}, ω) 〉} = \frac{1}{2} [1 + 𝒫^{2} (r_{1}, r_{2}, ω)] {| η_{0} (r_{1}, r_{2}, ω) |}^{2},

where the quantity

𝒫 (r_{1}, r_{2}, ω) = \frac{\sqrt{Σ_{j = 1}^{3} {| η_{j} (r_{1}, r_{2}, ω) |}^{2}}}{| η_{0} (r_{1}, r_{2}, ω) |}

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], 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.

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.

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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 (r_{1}, r_{2}, ω) = F (r_{1}, r_{2}, ω) J (r_{1}, r_{2}, ω),

where

F (r_{1}, r_{2}, ω) = 〈 a^{*} (r_{1}) a (r_{2}) 〉

and

J (r_{1}, r_{2}, ω) = 〈 {\hat{e}}^{*} (r_{1}) {\hat{e}}^{T} (r_{2}) 〉 .

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,

μ_{i j} (r_{1}, r_{2}, ω) = \frac{W_{i j} (r_{1}, r_{2}, ω)}{[〈 {| E_{i} (r_{1}, ω) |}^{2} 〉 {〈 {| E_{j} (r_{2}, ω) |}^{2} 〉}^{1 / 2}}, (i, j) = (x, y),

we may write the cross-spectral density matrix as

W (r_{1}, r_{2}, ω) = {[F (r_{1}, ω) F (r_{2}, ω)]}^{1 / 2} \times (\begin{matrix} μ_{x x} (r_{1}, r_{2}, ω) {[J_{x x} (r_{1}, ω) J_{x x} (r_{2}, ω)]}^{1 / 2} & μ_{x y} (r_{1}, r_{2}, ω) {[J_{x x} (r_{1}, ω) J_{y y} (r_{2}, ω)]}^{1 / 2} \\ μ_{y x} (r_{1}, r_{2}, ω) {[J_{y y} (r_{1}, ω) J_{x x} (r_{2}, ω)]}^{1 / 2} & μ_{y y} (r_{1}, r_{2}, ω) {[J_{y y} (r_{1}, ω) J_{y y} (r_{2}, ω)]}^{1 / 2} \end{matrix}),

where 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

μ_{x y} (r_{1}, r_{2}, ω) = μ_{x y} (r_{1}, r_{1}, ω) μ_{x x} (r_{1}, r_{2}, ω),

μ_{x x} (r_{1}, r_{2}, ω) = μ_{y y} (r_{1}, r_{2}, ω) \equiv μ (r_{1}, r_{2}, ω),

These conditions can be seen as electromagnetic extensions of the corresponding relations in the time-domain scalar theory [21]. In this context, they ensure, e.g., that the visibility of the intensity fringes in Young’s experiment equals zero if, and only if, 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.

From Eqs. (18) it readily follows that 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

W (r_{1}, r_{2}, ω) = μ (r_{1}, r_{2}, ω) {[F (r_{1}, ω) F (r_{2}, ω)]}^{1 / 2} \times (\begin{matrix} {[J_{x x} (r_{1}, ω) J_{x x} (r_{2}, ω)]}^{1 / 2} & μ_{x y} (r_{1}, r_{1}, ω) {[J_{x x} (r_{1}, ω) J_{y y} (r_{2}, ω)]}^{1 / 2} \\ μ_{y x} (r_{2}, r_{2}, ω) {[J_{y y} (r_{1}, ω) J_{x x} (r_{2}, ω)]}^{1 / 2} & {[J_{y y} (r_{1}, ω) J_{y y} (r_{2}, ω)]}^{1 / 2} \end{matrix}) .

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, 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]

P (r, ω) = {[1 - \frac{4 det J (r, ω)}{{tr}^{2} J (r, ω)}]}^{1 / 2} .

By use of Eqs. (3) and (6) it then at once follows that

\frac{〈 Δ I (r_{1}, ω) Δ I (r_{2}, ω) 〉}{〈 I (r_{1}, ω) 〉 〈 I (r_{2}, ω) 〉} = \frac{1}{2} [1 + \frac{P^{2} (r_{1}, ω) + P^{2} (r_{2}, ω)}{2}] {| μ (r_{1}, r_{2}, ω) |}^{2} .

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]), 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] 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 form

W (r_{1}, r_{2}, ω) = F (r_{1}, r_{2}, ω) U^{†} (r_{1}) J (ω) U (r_{2}),

where 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 trJ(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

\frac{〈 Δ I (r_{1}, ω) Δ I (r_{2}, ω) 〉}{〈 I (r_{1}, ω) 〉 〈 I (r_{2}, ω) 〉} = \frac{1}{2} [1 + P^{2} (ω)] {| f (r_{1}, r_{2}, ω) |}^{2},

where P(ω) is the degree of polarization [position-independent version of Eq. (20)], and

f (r_{1}, r_{2}, ω) = \frac{F (r_{1}, r_{2}, ω)}{{[F (r_{1}, r_{1}, ω) F (r_{2}, r_{2}, ω)]}^{1 / 2}}

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

\frac{〈 Δ I (r_{1}, ω) Δ I (r_{2}, ω) 〉}{〈 I (r_{1}, ω) 〉 〈 I (r_{2}, ω) 〉} = \frac{1}{2} [1 + P^{2} (ω)] {| μ (r_{1}, r_{2}, ω) |}^{2},

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.

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], 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

𝒫^{2} (r_{1}, r_{2}, ω) = [1 + P^{2} (ω)] {| \frac{f (r_{1}, r_{2}, ω)}{η_{0} (r_{1}, r_{2}, ω)} |}^{2} - 1.

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

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Hanbury Brown–Twiss effect with electromagnetic waves

Abstract

1. Introduction

2. Theory

3. Properties of the degree of cross-polarization

4. Separation of spatial correlation and the degree of polarization

5. Effects of the state of polarization

6. Conclusions

Acknowledgments

References and links

Cited By

Figures (1)

Equations (30)

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