IOP PUBLISHING JOURNAL OF OPTICS A: PURE AND APPLIED OPTICS

J. Opt. A: Pure Appl. Opt. 11 (2009) 085703 (8pp) doi:10.1088/1464-4258/11/8/085703

Free-space propagation of the spectraldegree of cross-polarization of stochasticelectromagnetic beamsSerkan Sahin1, Olga Korotkova1, Guowen Zhang2 and Jixiong Pu2

1 Department of Physics, University of Miami, Coral Gables, FL 33146, USA2 Department of Electronic Science and Technology, Huaqiao University, Quanzhou,Fujian 362021, People’s Republic of China

Received 5 March 2009, accepted for publication 9 April 2009Published 19 May 2009Online at stacks.iop.org/JOptA/11/085703

AbstractBased on the unified theory of coherence and polarization and the extended Huygens–Fresnelintegral the free-space evolution of the recently introduced spectral degree of cross-polarizationof a stochastic electromagnetic beam is studied. Unlike the spectral degree of coherence, thedegree of cross-polarization is unbounded and exhibits non-monotonic changes with growingpropagation distance and growing radial distance from the center of the beam, at fixedtransverse cross-sections. Our numerical results pertain to the electromagnetic GaussianSchell-model beams. We derive expressions for the cross-spectral density matrix of the EGSMbeam for arbitrary transverse and longitudinal positions. The results show that the behavior ofthe degree of cross-polarization on propagation is determined by all of the parameters of thesource radiating the beam. At sufficiently large distances from the source, the degree ofcross-polarization stabilizes for all points within the beam independently of their radialpositions.

Keywords: propagation, random beams, electromagnetic beams, polarization

(Some figures in this article are in colour only in the electronic version)

1. Introduction

Over the last several years there has been substantial interest inthe polarization changes of stochastic electromagnetic fields.It has been shown both theoretically and experimentally thatall polarization properties, namely the degree of polarization,the Stokes parameters and the parameters of the polarizationellipse of a stochastic electromagnetic beam may changeon propagation, even in free space [1–11]. All of theseproperties specify polarization content at a single spatialposition anywhere within the beam. Some publications alsoinvolved studies of polarization-related statistical propertiesdefined at a pair of spatial arguments [8]. It is worthmentioning that among all of the polarization properties ofstochastic beams, the degree of polarization is the mostimportant, providing us with the measure of how much of adeterministic portion is contained within the total beam at aparticular point in space.

In 2004, Ellis and Dogariu [12] proposed to generalizethe concept of the degree of polarization to two spatial

arguments and called the new quantity the mutual degree ofcross-polarization (MDCP) which was meant to provide themeasure of statistical similarity between the ordinary degreesof polarization at two given points. Later, Shirai and Wolf [13](see also [14]) on deriving the formula for intensity correlationsof a stochastic electromagnetic beam whose fluctuations obeyGaussian statistics obtained another quantity, which they calledthe spectral degree of cross-polarization (SDCP). It is also afunction defined at two spatial positions and reduces to theusual degree of polarization when the two points coincide.The SDCP has a form and meaning very similar to the MDCPbut involves a different normalization factor and therefore hasdifferent bounds (while MDCP takes values from −1 to 1, theSDCP is generally unbounded). Since the SDCP is a derivedquantity entering important expressions for classical fourth-order interference, while the MDCP is introduced heuristically,we will focus our attention on the former quantity.

The understanding of the behavior of the SDCP onpropagation is crucial in the analysis of ordinary polarization

1464-4258/09/085703+08$30.00 © 2009 IOP Publishing Ltd Printed in the UK1

J. Opt. A: Pure Appl. Opt. 11 (2009) 085703 S Sahin et al

evolution, as was pointed out in [15]. In fact, as was shownin [15] (see also [8]), that two sources with the same spectraldegree of coherence and the spectral degree of polarizationbut with different SDCP may generate beams which possessdifferent degrees of polarization in the far field.

In this paper we perform qualitative and quantitativeanalysis of the evolution of the SDCP of a stochasticelectromagnetic beam in free space. Numerous numericalexamples illustrate the changes in this quantity for a number ofchosen typical beams and propagation distances. In particular,we will separately consider the cases when the electric fieldcomponents are uncorrelated and partially correlated in thesource plane.

2. Theoretical background

We begin with a brief review of the theory of stochastic, sta-tistically stationary, at least in the wide sense, electromagneticbeam-like fields. We assume that a beam is generated in theplane z = 0 and it propagates in free space into the positivehalf-space z > 0, close to the positive z-direction. The second-order statistical properties of such a beam at angular frequencyω may be characterized by the 2 ×2 cross-spectral density ma-trix [5]

Wi j (ρ1,ρ2, ω) = ⟨E∗

i (ρ1, ω)E j (ρ2, ω)⟩, (i, j = x, y),

(1)where Ex and Ey are the mutually orthogonal components ofthe electric field transverse to the direction of propagation, ρ1

and ρ2 are two-dimensional position vectors of points in thesource plane and the angular brackets stand for the ensembleaverage in the sense of spectral coherence theory.

On free-space (paraxial) propagation the relation betweenthe cross-spectral density matrix in the source plane and that atpoints (r1, z1) and (r2, z2), z1, z2 > 0 is given by the law

Wi j(r1, z1; r2, z2; ω)

=(

k

2π

)2 1

z1z2

∫ ∫

ρ1

∫ ∫

ρ2

Wi j(ρ1,ρ2; ω)

× exp

{−ik

[(r1 − ρ1)

2

2z1− (r2 − ρ2)

2

2z2

]}d2ρ1d2ρ2, (2)

where k = ω/c is the wavenumber of the wave, and c is thespeed of light in vacuum.

The spectral degree of cross-polarization of a stochasticelectromagnetic beam-like field at two spatial positions (r1, z1)

and (r2, z2), at frequency ω, was introduced in [13] by meansof the formula

P(r1, z1; r2, z2; ω) =√√√√1 − 4Det

↔W (r1, z1; r2, z2; ω)

[Tr↔W (r1, z1; r2, z2; ω)]2

, (3)

where Det and Tr stand for the trace and the determinant.Combination of formulae (2) and (3) can now be used forevaluation of the SDCP at any two positions within thebeam.

We will now assume that the beam is generated by anelectromagnetic Gaussian Schell-model (EGSM) source [3].

EGSM sources and beams have already proved to be the mosttractable and illustrative models and are used in the majority ofanalytic and numerical calculations relating to the stochasticelectromagnetic beams. Generally, the components of thecross-spectral density matrix of the EGSM beam in the planeof the source have the form

Wi j(ρ1,ρ2, ω) = √Si (ρ1, ω)

√Sj (ρ2, ω)ηi j(ρ1,ρ2, ω),

(i, j = x, y), (4)

where Si (ρ1, ω) and Sj (ρ2, ω) are the spectral intensitydistributions and ηi j(ρ1,ρ2, ω) are the spectral correlationcoefficients defined respectively by the formulae

Sj (ρ, ω) = I j exp

(− ρ2

2σ 2

), ( j = x, y), (5)

ηi j(ρ1,ρ2, ω) = Bi j exp

(−|ρ2 − ρ1|2

2δ2i j

), (i, j = x, y).

(6)Here Bi j = |Bi j |eiϕi j . All of the parameters entering this modelmay depend on the frequency ω and have to satisfy relationsderived in [16] and [17].

On substituting from equations (4)–(6) into equation (2),after tedious integral calculation (see appendix), we obtainfor the elements of the electromagnetic cross-spectral densitymatrix of the beam propagating in free space the expressions

Wi j(r1, z1; r2, z2; ω) = Bi j√

Ii√

I j

2i j(z1, z2)

× exp

{−

[1

8σ 2+ i

z2 − z1

8kσ 2

(1

4σ 2+ 1

δ2i j

)](r1 + r2)

2

2i j(z1, z2)

}

× exp

{− (r1 − r2)

2

22i j(z1, z2)

[1

4σ 2+ 1

δ2i j

]}

× exp

{−i

k(r21 − r2

2)

2Ri j(z1, z2)

}, (7)

where

2i j(z1, z2) = 1+ z1z2

k2σ 2

(1

4σ 2+ 1

δ2i j

)+i

z2 − z1

k

(1

2σ 2+ 1

δ2i j

)

(7a)

Ri j (z1, z2) = √z1z2

[1 + k2σ 2

z1z2

(1

4σ 2+ 1

δ2i j

)−1]. (7b)

In particular, when in equation (7) z1 = z2 = z we obtainthe transverse cross-spectral density matrix (see also [18]where a similar expression is derived for a scalar beam)

Wi j(r1, z; r2, z; ω) = Bi j√

Ii√

I j

2i j(z)

exp

{

− (r1 + r2)2

8σ 22i j(z)

}

× exp

{

− (r1 − r2)2

22i j(z)

[1

4σ 2+ 1

δ2i j

]}

× exp

{−i

k(r21 − r2

2)

2Ri j(z)

}, (8a)

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J. Opt. A: Pure Appl. Opt. 11 (2009) 085703 S Sahin et al

Figure 1. The contour of the degree of cross-polarization P(r1 = 0, z1 = 1 m; r2 = r, z2 = z) of the EGSM beam propagating in free space.(a) δyy = 0.1 mm; (b) δyy = 0.5 mm; (c) δyy = 0.8 mm; (d) δyy = 1 mm. The values of the parameters of the beams are: λ = 632.8 nm,δxx = 0.5 mm, Ix = 3Iy .

where

2i j(z) = 1 + z2

k2σ 2

(1

4σ 2+ 1

δ2i j

)(8b)

Ri j (z) = z

[1 + k2σ 2

z2

(1

4σ 2+ 1

δ2i j

)−1]. (8c)

These expressions also agree with those given in [3].In the other special case, when in equation (7) r1 = r2 = r

we obtain the longitudinal cross-spectral density matrix

Wi j(r, z1; r, z2; ω) = Bi j√

Ii√

I j

2i j(z1, z2)

× exp

{−

[1

8σ 2+ i

z2 − z1

8kσ 2

(1

4σ 2+ 1

δ2i j

)]r2

2i j(z1, z2)

},

(9)

where 2i j(z1, z2) was given in equation (7a). Note that the

phase factor Ri j does not enter the formula in this case.

3. Numerical results

With the help of formulae (7)–(9), we will now perform nu-merical calculations which provide the quantitative informa-tion about the dependence of the SDCP on the source param-eters for a typical EGSM beam propagating in free space. Forall the numerical calculations, we fix the wavelength and therms width of the source to be λ = 632.8 nm, σ = 5 cm.

3.1. Uncorrelated electric field components

We will first discuss the cross-polarization changes in the casewhen the two components of the electric field are uncorrelated

and hence the cross-spectral density matrix (1) has no off-diagonal components. We recall that in this case the ordinarydegree of polarization usually exhibits drastic changes onpropagation in free space as compared to those for beams withpartially correlated beam components.

In figure 1 we show the contours of the degree of cross-polarization P(r1 = 0, z1 = 1 m; r2 = r, z2 = z) of the EGSMbeam propagating in free space for several different values ofsource correlation coefficients δyy, while δxx is chosen to be0.5 mm, with Ix = 3Iy . Such contour plots allow illustrationof both transverse and longitudinal components of the SDCPat the same time. When the correlation coefficients δyy and δxx

coincide (figure 1(b)) the beam is scalar-like and the contourbecomes degenerate. When the correlation coefficients δyy andδxx approach the rms width of the beam σ , the changes inthe SDCP become significant only with a slight change in δxx

(compare figures 1(c) and (d)).In order to illustrate how the initial parameters of the

EGSM beam affect its degree of cross-polarization P(r1 =0, z1 = 1 m; r2 = r, z2 = z) on propagation in free space moreexplicitly, the SDCP curves at different planes z = const. aregiven in figures 2 and 3. In particular, one can see from figure 2how the changes in the SDCP depend on the initial ordinarydegree of polarization, which in this case simply depends onthe ratio of intensities of the two components of the beam:

P(r) = Ix(r)/Iy(r) − 1

Ix(r)/Iy(r) + 1. (10)

For the four ratios Ix (r)/Iy(r) of our choice: 1, 5/3, 3and 19 the degree of polarization takes values 0 (unpolarizedbeam), 1/4, 1/2 and 9/10 (almost polarized beam). We notethat even though in the plane z = 1, where the transverseSDCP is smaller than 1 for all values of r (see figure 2(a)), in

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J. Opt. A: Pure Appl. Opt. 11 (2009) 085703 S Sahin et al

Figure 2. The degree of cross-polarization of the EGSM beam propagating in free space for different values of the initial degree ofpolarization: (a) z = 1 m; (b) z = 6 m; (c) z = 10 m; (d) z = 100 m. The other calculation parameters are: δxx = 0.5 mm, δyy = 1 mm.

Figure 3. The degree of cross-polarization of the EGSM beam propagating in free space for different values of δyy : (a) z = 1 m; (b) z = 6 m;(c) z = 10 m; (d) z = 100 m. The other calculation parameters are: δxx = 0.5 mm, Ix = 3Iy .

other planes it is no longer just confined to the interval between0 and 1, and will be generally larger than 1 (see figures 2(b)and (c)). It is also seen from figure 2 that both the degree ofpolarization of the source and the propagation distance affectthe degree of cross-polarization.

The influence of the source correlation coefficients andthe propagation distance on the degree of cross-polarization isshown in figure 3. In particular, in the general case, when thecorrelation coefficients are the same (dashed curves) the SDCP

remains constant for almost the whole range of r (in figure 3(b)variation in SDCP for large values of r is due to violationof non-paraxial regime). We also note that in cases whenboth correlation coefficients δyy and δxx are small comparedto the rms width of the beam σ , it is clearly seen that thechanges in the SDCP not only ‘propagate’ with distance z fromthe source but also ‘spread’ with radial distance r from theaxis of the beam (compare the solid curves in all four figuresof 3(a)–(d)).

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J. Opt. A: Pure Appl. Opt. 11 (2009) 085703 S Sahin et al

Figure 4. Contours of the transverse SDCP P(r1 = 0, r2 = r, z1 = z2 = z, ω) of the EGSM beam propagating in free space. (a) Bxy = 0; (b)Bxy = 0.1; (c) Bxy = 0.2; (d) Bxy = 0.3. The values of the parameters of the beams are: δxx = 0.15 mm, δyy = 0.225 mm,δxy = δyx = 0.25 mm, δ = π/6, Ix = Iy .

Figure 5. The transverse degree of cross-polarization of the EGSM beam propagating in free space for different values of Bxy: (a) z = 1 m;(b) z = 50 m; (c) z = 100 m; (d) z = 200 m. The other parameters are the same as in figure 4.

3.2. Correlated electric field components

Figures 4–6 illustrate the behavior of the SDCP in the moregeneral case than discussed in section 3.1, namely when the

two transverse components of the electric field are partiallycorrelated. In this situation the cross-spectral matrix (1) hasnon-zero off-diagonal elements. Since the dependence of theSDCP on initial intensities and correlation coefficients was

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J. Opt. A: Pure Appl. Opt. 11 (2009) 085703 S Sahin et al

Figure 6. The longitudinal SDCP of the EGSM beam propagating in free space for different values of Bxy: (a) z = 1 m; (b) z = 50 m;(c) z = 100; (d) z = 200 m. The other parameters are the same as in figure 4.

explored in details in section 3.1, here we will primarily focusour attention on the dependence of the SDCP on the coefficientBxy . In the EGSM beam, this coefficient alone provides themeasure of correlation of the x and y components of theelectric field in the source plane. Moreover, if in the sourceplane Ix = Iy , then Bxy solely determines the degree ofpolarization of the source:

P(r) = Bxy . (11)

In figures 4–6, four values of Bxy (or, equivalently, P(r))were chosen: 0, 0.1, 0.2 and 0.3. Higher values of Bxy

cannot be applied due to some intrinsic constraints on thechoice of parameters of the EGSM sources [17]. Also we willonly consider in this section the purely transverse or purelylongitudinal SDCP, since the general SDCP was discussed indetail above, and usually polarization properties in transverseplanes or at a fixed radial position but different distances fromthe source plane are of most interest.

Figure 4 shows the spatial distribution of the transverseSDCP for four aforementioned values of the coefficient Bxy .One can see that the change in Bxy results in the change in theSDCP especially in the central part of the beam. The same asfor the ordinary degree of polarization, in this case of correlatedfield components, the changes in the SDCP are not as dramaticas for that of the beam with uncorrelated field components.

Figure 5 illustrates the dependence of the transverseSDCP calculated for a pair of points in the fixed plane zon the radial distance r. We see that with the increasingpropagation distance the SDCP becomes more uniform acrossthe transverse cross-section, saturating at certain non-zerovalues, depending on the coefficient Bxy . This result isin striking contrast with the behavior of the spectral degreeof coherence which always tends to 1 at sufficiently largedistances on free-space propagation.

Figure 6 illustrates the dependence of the longitudinalSDCP at a pair of points on the axis of the beam at fourdistances z1 from the source while distance z2 varies, forseveral chosen values of coefficient Bxy . We note that thechange is pronounced only for relatively small separationdistances |z1 − z2|; for larger separations the SDCP saturatesat certain fixed values, not necessarily 0. It is also seen thatthe longitudinal SDCP changes its shape with distance z1.Parameter Bxy , on the other hand, influences the magnituderather than the shape of the SDCP.

4. Summary

To summarize, we have derived the expressions for theelements of the 2×2 cross-spectral density matrix of an EGSMbeam propagating in free space for two arbitrary points inthe beam (not necessarily belonging to the same transversecross-section) and then studied the changes in the SDCP. Bya number of numerical examples we have demonstrated thedependence of the SDCP of the propagating beam on all ofthe parameters of the source. We considered separately thecases of uncorrelated and correlated components of the electricfield in the source plane and also purely transverse and purelylongitudinal SDCPs. In particular we found that the behaviorof the SDCP of an EGSM beam depends strongly on the(constant) spectral degree of polarization in the source plane.Unlike the spectral degree of coherence and the spectral degreeof polarization the SDCP may take on any non-negative value.Unlike the spectral degree of coherence, the transverse SDCPdoes not tend to 1 for sufficiently large distances from thesource plane. Among the other not too obvious results isthat the longitudinal SDCP saturates at certain (generally non-zero) values after the beam propagates over a sufficiently longdistance in free space.

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J. Opt. A: Pure Appl. Opt. 11 (2009) 085703 S Sahin et al

On finishing we would like to note that atmosphericpropagation of the degree of cross-polarization has alsorecently been studied by the authors and belongs to a separatepublication [19].

Acknowledgments

OK acknowledges the AFOSR support via grant FA 9550-08-1-0102. The research of JP is supported by KeyProject of Science and Technology of Fujian Province (GrantNo. 2007H0027).

Appendix

Under paraxial approximation the EGSM source

W (0)i j (ρ1,ρ2, ω) = √

Ii I j Bi j exp

{−ρ2

1 + ρ22

4σ 2

}

× exp

{− (ρ2 − ρ1)

2

2δ2i j

}(A.1)

can radiate an EGSM beam, whose cross-spectral density inthe z plane (z > 0) can be given by the expression

Wi j(r1, z1; r2, z2; ω) =(

k

2π

)2 1

z1z2

×∫ ∫ ∫ ∫ ∞

−∞W (0)

i j (ρ1,ρ2, ω)

× exp

{−ik

[(r1 − ρ1)

2

2z1− (r2 − ρ2)

2

2z2

]}d2ρ1 d2ρ2.

(A.2)

On substituting from (A.1) into (A.2) we obtain theformula

Wi j(r1, z1; r2, z2; ω)

= √Ii I j Bi j

(k

2π

)2 1

z1z2exp

{− ik

2

(r2

1

z1− r2

2

z2

)}

×∫ ∫ ∫ ∫ ∞

−∞exp

{−ρ2

1 + ρ22

4σ 2

}

× exp

{− (ρ2 − ρ1)

2

2δ2i j

}

× exp

{−ik

[ρ2

1

2z1− ρ2

2

2z2− r1ρ1

z1+ r2ρ2

z2

]}d2ρ1 d2ρ2,

(A.3)

which can be rewritten as

Wi j(r1, z1; r2, z2; ω)

= √Ii I j Bi j

(k

2π

)2 1

z1z2exp

{− ik

2

(r2

1

z1− r2

2

z2

)}

×∫ ∫

exp

{−

(1

4σ 2+ 1

2δ2i j

+ ik

2z1

)ρ2

1

}

× exp

{−i2π

(iρ2

2πδ2i j

− r1

λz1

)}d2ρ1

×∫ ∫

exp

{−

(1

4σ 2+ 1

2δ2i j

− ik

2z2

)ρ2

2 − ikr2ρ2

z2

}d2ρ2.

(A.4)

On recalling that the two-dimensional Fourier transformof a Gaussian function, g(ρ) = exp(−aρ2), is

F{g(ρ)} =∫ +∞

−∞exp(−ax2) exp(−i2πwx x) dx

×∫ +∞

−∞exp(−ay2) exp(−i2πwy y)dy

= π

aexp

(−π2w2

a

), (A.5)

we now perform integration with respect to variable ρ1 inequation (A.4) to obtain the expression∫ ∫

exp

{−

(1

4σ 2+ 1

2δ2i j

+ ik

2z1

)ρ2

1

}

× exp

{−i2π

(iρ2

2πδ2i j

− r1

λz1

)}d2ρ1 = π

14σ 2 + 1

2δ2i j

+ ik2z1

× exp

{− π2r2

1(1

4σ 2 + 12δ2

i j+ ik

2z1

)λ2z2

1

}

× exp

{ρ2

2(1

4σ 2 + 12δ2

i j+ ik

2z1

)4δ4

i j

+ iπr1ρ2(1

4σ 2 + 12δ2

i j+ ik

2z1

)λz1δ

2i j

}. (A.6)

On substituting from (A.6) into (A.4) we get theexpression

Wi j(r1, z1; r2, z2; ω) = √Ii I j Bi j

(k

2π

)2 1

z1z2

× exp

{− ik

2

(r2

1

z1− r2

2

z2

)}π

14σ 2 + 1

2δ2i j

+ ik2z1

× exp

{− π2r2

1(1

4σ 2 + 12δ2

i j+ ik

2z1

)λ2z2

1

}

×∫ ∫

exp

{−

[1

4σ 2+ 1

2δ2i j

− ik

2z2

− 1(

14σ 2 + 1

2δ2i j

+ ik2z1

)4δ4

i j

]ρ2

2

}

× exp

{−i2π

[r2

λz2− r1

2(

14σ 2 + 1

2δ2i j

+ ik2z1

)λz1δ

2i j

]ρ2

}

− ikd2ρ2. (A.7)

Similarly, the integration with respect to variable ρ2 canalso be evaluated and we have

Wi j(r1, z1; r2, z2; ω) = √Ii I j Bi j

(k

2π

)2 1

z1z2

× π1

4σ 2 + 12δ2

i j− ik

2z2− 1(

14σ2 + 1

2δ2i j

+ ik2z1

)4δ4

i j

× π1

4σ 2 + 12δ2

i j+ ik

2z1

exp

{− π2r2

1(1

4σ 2 + 12δ2

i j+ ik

2z1

)λ2z2

1

}

7

J. Opt. A: Pure Appl. Opt. 11 (2009) 085703 S Sahin et al

× exp

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

−π2

[r2λz2

− r1

2(

14σ2 + 1

2δ2i j

+ ik2z1

)λz1δ

2i j

]2

14σ 2 + 1

2δ2i j

− ik2z2

− 1(1

4σ2 + 12δ2

i j+ ik

2z1

)4δ4

i j

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

× exp

{− ik

2

(r2

1

z1− r2

2

z2

)}. (A.8)

Finally, after performing the simplification of equa-tion (A.8) we obtain the formula

Wi j(r1, z1; r2, z2; ω) = Bi j√

Ii I j

2(z1, z2)

× exp

{−

[1

8σ 2+ i

z2 − z1

8kσ 2

(1

4σ 2+ 1

δ2i j

)](r1 + r2)

2

2(z1, z2)

}

× exp

{− (r1 − r2)

2

22(z1, z2)

[1

4σ 2+ 1

δ2i j

]}

× exp

{−i

k(r21 − r2

2)

2R(z1, z2)

}, (A.9)

where

2(z1, z2) = 1 + z1z2

k2σ 2

(1

4σ 2+ 1

δ2i j

)

+ i

[z2 − z1

k

(1

2σ 2+ 1

δ2i j

)],

R(z1, z2) = √z1z2

[1 + k2σ 2

z1z2

(1

4σ 2+ 1

δ2i j

)−1].

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