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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 22 — Oct. 22, 2012
  • pp: 24910–24917
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Properties of quasi-homogeneous isotropic electromagnetic sources

Asma Al-Qasimi and Daniel F. V. James  »View Author Affiliations


Optics Express, Vol. 20, Issue 22, pp. 24910-24917 (2012)
http://dx.doi.org/10.1364/OE.20.024910


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Abstract

We study the properties of Quasi-Homogeneous Isotropic Electromagnetic (QuHIEM) Sources, a model for partially-coherent secondary light sources beyond the scalar and paraxial approximations. Our results include polarization properties in the far zone and the realizability condition. We demonstrate these results for sources with a degree of coherence described by Gaussians.

© 2012 OSA

1. Introduction

The influence of source correlations on the properties of radiated fields has been topic of considerable research interest for some time [1

1. A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. 58, 1256–1259 (1968). [CrossRef]

3

3. A. T. Friberg, ed., Selected Papers on Coherence and Radiometry, SPIE Milestone Series MS 69 (SPIE, 1993).

]. A commonly employed and intuitive mathematical model for the spatial coherence properties of a globally incoherent secondary source is the quasi-homogeneous model, in which the spatial coherence properties are assumed to be independent of the location with the source. In the scalar approximation, the cross-spectral density function at angular frequency ω of the source factorizes as follows (see ref. [4

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

], section 5.3.2, pp.90–95)
W(0)(ρ1,ρ2,ω)S(0)(ρ1+ρ22,ω)μ(0)(ρ2ρ1,ω).
(1)
Here ρi, (i = 1, 2) are the two-dimensional position vectors of two points within the source plane of a secondary source. Using this model, the radiant intensity of the field in some far-zone point with position vector r = rs (defined with respect to a suitable origin within the source) is given by:
J(rs,ω)=(2πk)2S˜(0)(0,ω)μ˜(0)(ks,ω)cos2θ,
(2)
where k = ω/c, c being the speed of light, s is the projection of the unit vector s onto the source plane (s is treated as a two-dimensional vector), and
μ˜(0)(f,ω)=1(2π)2μ(0)(ρ,ω)exp(ifρ)d2ρ
(3)
is the two-dimensional Fourier transform of the spectral degree of coherence. Similarly S̃(0)(f, ω) is the two-dimensional Fourier transform of the source power spectrum S(0) (ρ, ω).

Over the years this model has greatly contributed to the understanding of radiation sources. In particular Lambert’s Law (which states that the radiant intensity of a field is proportional to the cosine of the angle between the line of sight and the normal to the source plane) [5

5. J. H. Lambert, Photometria sive de mensure et gradibus luminis colorum et umbra (Bassel, 1760).

] is intimately linked with the spatial coherence of the source [6

6. W. H. Carter and E. Wolf, “Coherence properties of Lambertian and non-Lambertian sources,” J. Opt. Soc. Amer. 65, 1067–1071 (1975). [CrossRef]

]: within the scalar quasi-homogeneous model, a necessary and sufficient condition for a Lambertian source is that the spectral degree of coherence has the functional form μ(0) (ρ, ω) = sin()/. Another important example is the condition for spectral invariance of the radiated field: If the spectral degree of coherence obeys the scaling law, μ(0) (ρ, ω) = h(ωρ/c), then the radiated field will have the same power spectrum as the source [7

7. E. Wolf, “Invariance of the spectrum of light on propagation,” Phys. Rev. Lett. 56, 1370–1372 (1986). [CrossRef] [PubMed]

]. A source that violates the scaling law will exhibit the Wolf effect, i.e. correlation-induced spectral changes [8

8. E. Wolf, “Non-cosmological redshifts of spectral lines,” Nature 326, 363–365 (1987). [CrossRef]

, 9

9. E. Wolf and D. F. V. James, “Correlation-induced spectral changes,” Rep. Prog. Phys. 59, 771–818 (1996). [CrossRef]

].

2. Quasi-homogeneous isotropic electromagnetic sources

In this note, we introduce and study the quasi-homogeneous, isotropic electro-magnetic (QuHIEM) model source, in which the coherence properties of the source, in addition to obeying the spatial invariance inherent in the quasi-homogeneous model, is also isotropic, in the sense that the coherence properties are independent of the orientation of the axes in the source plane. For such a source we have the following cross-spectral density matrix (see ref. [4

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

], Section 9.1):
Wα,β(0)(ρ,ρ2,ω)=S(ρ1+ρ22,ω)μα,β(ρ2ρ1,ω),(α,β=x,y)
(4)
where the isotropic spectral degree of coherence is given by (see ref. [16

16. G. K. Batchelor, The Theory of Homogeneous Turbulence (Cambridge Science Classics, 1990).

]):
μ(α,β)(ρ,ω)=δα,βA(kρ)+ραρβρ2B(kρ).
(5)
Thus the assumption of isotropy means that the correlation properties of such sources are specified entirely by the two real-valued scalar functions A() and B() (rather than the three real and three complex functions needed for more general sources). We have assumed the spectral degree of coherence obeys the scaling law [7

7. E. Wolf, “Invariance of the spectrum of light on propagation,” Phys. Rev. Lett. 56, 1370–1372 (1986). [CrossRef] [PubMed]

], since such sources seems to be very prevalent in nature, and further, this makes the mathematical description more compact. Relaxing this criterion is relatively straightforward. Note that the degree of polarization in the source is everywhere zero.

Implicit in the use of the quasi-homogeneous model is the assumption that the source is globally incoherent, i.e. the coherence area is very small compared to the overall size of the source. Different models, such as the Schell model [17

17. A. C. Schell, “A technique for determination of the radiation pattern of a partially coherent aperture,” IEEE Trans. Antennas Propag. AP15187–188 (1967). [CrossRef]

], have been introduced to describe sources with more spatial coherence in the scalar approximation; the extension of such models to the electromagnetic case remains an open problem.

3. Radiation, degree of polarization, and the realizability condition for the QuHIEM sources

In order to calculate the radiation field generated by the QuHIEM sources one needs to employ vector diffraction theory [18

18. G. Toraldo di Francia, Electromagnetic Waves (Interscience, 1955), pp. 218–221.

20

20. M. A. Alonso, O Korotkova, and E. Wolf, “Propagation of the electric correlation matrix and the van CittertZernike theorem for random electromagnetic fields,” J. Mod. Opt. 53, 969–978 (2006). [CrossRef]

]. If E(0)(ρ,ω)={Ex(0)(ρ,ω),Ey(0)(ρ,ω)} is the two-dimensional projection of the electric field in the source plane (the far-zone diffracted field does not depend on the field component perpendicular to the source plane), let us define the following two-dimensional vector field in the radiation zone:
(rs,ω)=ikexp(ikr)2πrE(0)(ρ,ω)exp(iksρ.)d2ρ.
(6)
This is the vector radiation field one would predict by simply applying scalar diffraction theory (without inclination factors) to the two components separately; however the electric field in the radiation zone is a three-component vector perpendicular to s, so ℰ(rs, ω) is not the electric field. However, the electric field is given in terms of ℰ(rs, ω) as follows:
E()(rs,ω)={szx(rs,ω),szy(rs,ω),sxx(rs,ω)syyx(rs,ω)}.
(7)
It can be seen that s ·E(∞)(rs, ω) = 0. Note that since sz = cosθ, the inclination factors appear as a direct consequence of the vector diffraction formulation.

If we write s in terms of standard angular variables, viz.,
se^r={sinθcosϕ,sinθsinϕ,cosθ},
(8)
then it will be useful to write E(∞)(rs, ω) in terms of the the following two vectors:
e^θ={cosθcosϕ,cosθsinϕ,sinθ},
(9)
e^ϕ={sinϕ,cosϕ,0}.
(10)
In that case, eq.(7) may be re-written as follows:
[Eθ()(rs,ω)Eϕ()(rs,ω)]=P(θ)R(ϕ)[x(rs,ω)y(rs,ω)],
(11)
where the 2 × 2 matrices P(θ) and R(ϕ) are given by
P(θ)=[100cosθ]
(12)
R(ϕ)=[cosϕsinϕsinϕcosϕ]
(13)

Fig. 1 Geometry for diffraction of Electromagnetic sources. The source lies in the xy plane, containing the origin O; a typical position vector of a point in the source is ρ = {x, y} (not shown). The field is observed at a far-zone point P, with position vector rs and spherical polar coordinates {r, θ, ϕ}. The field at P is conveniently described using the unit vectors {êr, êθ, êϕ}, given by eqs.(8)(10).

Applying this formulation to the case of random fields, let us define
𝒲α,β(r1s1,r2s2,ω)α*(r1s1,ω)β(r2s2,ω)=k2exp[ik(r2r1)](2π)2r1r2Wα,β(0)(ρ1,ρ2,ω)exp[ik(s1ρ1s2ρ2)]d2ρ1d2ρ2.
(14)
The coherency matrix Ji,j(rs, ω), expressed in the {êθ, êϕ} coordinate system, is then given by a matrix transformation of the 2 × 2 matrix 𝒲α,β(rs, rs, ω), viz.,
J=[Jθ,θJθ,ϕJϕ,θJϕ,ϕ][Eθ()*Eθ()Eθ()*Eϕ()Eϕ()*Eθ()Eϕ()*Eϕ()]=𝒫(θ)(ϕ)[𝒲x,x𝒲x,y𝒲y,x𝒲y,y](ϕ)T𝒫(θ)T,
(15)
where we have suppressed the arguments rs and ω for clarity. That the properties of the far-zone field might be reduced to a single 2 × 2 tensor, rather than a 3 × 3 tensor (as considered in [20

20. M. A. Alonso, O Korotkova, and E. Wolf, “Propagation of the electric correlation matrix and the van CittertZernike theorem for random electromagnetic fields,” J. Mod. Opt. 53, 969–978 (2006). [CrossRef]

], for example) is due to the suppression of the longitudinal component of the field in the far zone. The field is, locally, a plane wave propagating along the direction s. For plane waves, Gauss’s law, ∇·E(rs, ω) = 0, implies that the field component parallel to s is zero. Thus the use of spherical polar coordinates (rather than the cartesian coordinates used in [20

20. M. A. Alonso, O Korotkova, and E. Wolf, “Propagation of the electric correlation matrix and the van CittertZernike theorem for random electromagnetic fields,” J. Mod. Opt. 53, 969–978 (2006). [CrossRef]

]) thus allows the expression for the field to be made more compact.

To proceed, we must now substitute our model source, defined in eqs.(4)(5) into eq.(14). We find
𝒲α,β(rs,rs,ω)=(2πk)2r2S˜(0)(0,ω)μ˜α,β(0)(ks,ω).
(16)
Substituting from eq.(5), the two-dimensional Fourier transforms reduce to one-dimensional Hankel transforms, and after some algebra we obtain the following expression:
μ˜α,β(0)(ks,ω)=1k2{δα,β[I1(θ)+12I2(θ)]sαsβsin2θI2(θ)},(α,β=x,y)
(17)
where
I1(θ)=12π0[A(ξ)+12B(ξ)]J0(ξsinθ)ξdξI2(θ)=12π0B(ξ)J2(ξsinθ)ξdξ,
(18)
=12π0[2ξsin2θB(ξ)B(ξ)]J0(ξsinθ)ξdξ
(19)
Jn(u) being the n-th order Bessel function of argument u. Substituting from eq.(17) into eq.(15) and performing the requisite matrix multiplications, we find that the coherency matrix in the {êθ, êϕ} basis is given by:
J(rs,ω)=(2π)2r2S˜(0)(0,ω)[I1(θ)12I2(θ)00{I1(θ)+12I2(θ)}cos2θ].
(20)
The Poynting vector of the field in the radiation zone is therefore given by
S()(rs,ω)=sε0c2Tr{J(rs,ω)}=s(2π2ε0cr2)S˜(0)(0,ω){(1+cos2θ)I1(θ)12sin2θI2(θ)}.
(21)
The degree of polarization of the radiated field is
P()(θ)=14Det{J(rs,ω)}Tr{J(rs,ω)}2=|sin2θI1(θ)12(1+cos2θ)I2(θ)|(1+cos2θ)I1(θ)12sin2θI2(θ).
(22)

The functions A() and B() have a number of constraints. First, the definition eq.(5) requires that B(0) = B′(0) = 0. Additionally, Gori et al. [21

21. F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Realizability condition for electromagnetic Schell-model sources,” J. Opt. Soc. Am A 25, 1016 (2008). [CrossRef]

] have demonstrated the realizability condition, based on the requirement that the cross-spectral tensor is a positive-definite function. The condition entails the following:
|μ˜xy(f,ω)|μ˜xx(f,ω)μ˜yy(f,ω),
(23)
For our model, this implies:
(I1(θ)±I2(θ)2)0.
(24)
Furthermore, we require that J(rŝ, ω) is zero when θ = π/2, i.e. there is no radiation parallel to the source plane. This requirement is born of the fact that the solid angle subtended by a planar source is zero when θ = π/2 and hence the radiated intensity should be zero as well. (This seems the usual circumstance in radiometry, although we are not aware of a rigorous proof.) Mathematically, this requirement implies that
(I1(π/2)I2(π/2)2)=0.
(25)

Some general results follow immediately. First at θ = 0, i.e. when viewing the source along the normal, we find I2(0) = 0 from the properties of J2(x); and hence
J(re^z,ω)=(2π)2r2S˜(0)(0,θ)I1(0)[1001].
(26)
In other words, the field on-axis from an isotropic source must be completely unpolarized. This is a reasonable result, simply because there is nothing to break the isotropic symmetry. Off-axis, however, is symmetry broken, and it is indeed plausible that the êθ component of the field might be greater than the êϕ component, or vice-versa.

4. Examples

A special example of a QuHIEM source has already been analyzed in ref. [15

15. D. F. V. James, “Polarization of light radiated by black-body sources,” Opt. Commun. 109, 209 (1994). [CrossRef]

], which investigated the radiation from a thermal source. In that case, the functions A() and B() were given by:
A(kρ)=j0(kρ)j1(kρ)/kρ
(27)
B(kρ)=j2(kρ).
(28)
It was found that, in this case, the radiated field obeys Lambert’s law, i.e. the normalized amplitude of the Poynting vector 𝒮(∞)(θ) = S(∞)(rs, ω)/S(∞)(rêz, ω) is given by
𝒮()(θ)=cosθ.
(29)
Further, the degree of polarization P(∞)(θ) = 0 for all values of θ.

As an example to illustrate the general formulation presented here, suppose that the functions A(ρ) and B(ρ) in eq.(5), are given by two Gaussians, defined as follows:
A(kρ)=12exp(k2ρ2/2a2),
(30)
B(kρ)=B0k2ρ2exp(k2ρ2/2b2),
(31)
where k = ω/c and a and b are dimensionless quantities. In this case, the integrals, I1 and I2, are given by:
I1(θ)=14πk2[a2exp(12a2sin2θ)+B0b4(2b2sin2θ)exp(12b2sin2θ)]
(32)
I2(θ)=12πk2B0b6sin2θexp(12b2sin2θ).
(33)
If we invoke the criterion eq.(25), which ensures that 𝒮(∞)(π/2) = 0, we find that B0 = a2 exp(−a2/2)/2b4(b2 − 1)exp(−b2/2), and hence we obtain the following expressions for the radiation pattern and degree of polarization:
𝒮()(θ)=(1b2)(1+cos2θ)exp(a2cos2θ/2)(1+cos2θb2sin2θ)exp(b2cos2θ/2)2[(1b2)exp(a2/2)exp(b2/2)]
(34)
P()(θ)=|(1b2)sin2θ[exp(a2cos2θ/2)exp(b2cos2θ/2)]|(1b2)(1+cos2θ)exp(a2cos2θ/2)(1+cos2θb2sin2θ)exp(b2cos2θ/2)
(35)
These somewhat complex expressions nevertheless reveal some useful results. In particular, from eq.(35) we find the conditions under which P(∞)(θ) = 0. First, the formula suggests the criterion b = 1, implying that the coherence length of the second part of the cross-coherency tensor is equal to the wavelength; however the condition eq.(25) cannot be satisfied if b = 1, and hence we discard this possibility. The second criterion is a = b, implying that the two parts of the cross-coherency tensor have the same coherence lengths; in this case, the realizability condition requires that a > 1.

In the spatially incoherent limit, i.e. a, b → 0, we would expect the results to be independent of the functional form of A(ξ) and B(ξ). In this case the two expressions become:
𝒮()(θ)=cos2θμ(5+cos2θ)46μ4
(36)
P()(θ)=μsin2θμ(5+cos2θ)4,
(37)
where
μ=a2b2a2,
(38)
and the realizability condition requires that μ ≥ 1. In other words, for highly spatially incoherent sources it appears that the degree of polarization will always be directionally dependent in the far zone.

As a specific example, we have plotted the radiation pattern and the degree of polarization for such sources for a variety of values of a and b (see Fig. 2).

Fig. 2 Normalized Radiation Patterns and Degrees of Polarization versus θ. This plot is based on the Gaussian model QuHIEMS defined by eq.(31). The blue solid line shows the normalized radiation pattern 𝒮(∞)(θ), defined by eq.(34); the red dashed line is the degree of polarization P(∞)(θ) defined by eq.(35). The parameters used were as follows: (i) incoherent limit (a → 0) with μ = (a2b2)/a2 = 1; (ii) incoherent limit with μ = 2; (iii) incoherent limit with μ → ∞; (iv) a = 1, b = 0.0; (v) a = 1, b = 1.35; (vi) a = 1, b = 1.9318 (the largest value consistent with the realizability condition); (vii) a = 5, b = 0.0; (viii) a = 5, b = 1.3; (ix) a = 5, b = 5.1486.

5. Conclusions

In conclusion, the isotropic quasi-homogeneous model source provides a compact, mathematically tractable, and physically plausible description of globally spatially incoherent electromagnetic sources. The pertinent properties of the far-zone field, namely the radiation pattern and the degree of polarization, can be predicted by a few relatively straightforward integrals involving the functions describing the local correlation properties in the source. Our results demonstrate the prevalence of coherence-induced changes in polarization: in the limit of complete spatial incoherence, it seems that this effect will always occur.

Acknowledgments

This work was supported by NSERC.

References and links

1.

A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. 58, 1256–1259 (1968). [CrossRef]

2.

E. Wolf, “Coherence and radiometry,” J. Opt. Soc. Am. 68, 6–17 (1978). [CrossRef]

3.

A. T. Friberg, ed., Selected Papers on Coherence and Radiometry, SPIE Milestone Series MS 69 (SPIE, 1993).

4.

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

5.

J. H. Lambert, Photometria sive de mensure et gradibus luminis colorum et umbra (Bassel, 1760).

6.

W. H. Carter and E. Wolf, “Coherence properties of Lambertian and non-Lambertian sources,” J. Opt. Soc. Amer. 65, 1067–1071 (1975). [CrossRef]

7.

E. Wolf, “Invariance of the spectrum of light on propagation,” Phys. Rev. Lett. 56, 1370–1372 (1986). [CrossRef] [PubMed]

8.

E. Wolf, “Non-cosmological redshifts of spectral lines,” Nature 326, 363–365 (1987). [CrossRef]

9.

E. Wolf and D. F. V. James, “Correlation-induced spectral changes,” Rep. Prog. Phys. 59, 771–818 (1996). [CrossRef]

10.

D. F. V. James, “Change of polarization of light-beams on propagation in free-space,” J. Opt. Soc. Am. A 111641–1643 (1994). [CrossRef]

11.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, and G. Guattari, “Beam coherence-polarization matrix,” Pure and Applied Optics 7941–951 (1998). [CrossRef]

12.

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312, 263–267 (2003). [CrossRef]

13.

J. Tervo, T. Setälä, and A. T. Friberg, “Theory of partially coherent electromagnetic fields in the space-frequency domain,” J. Opt. Soc. Am. A 212205–2215 (2004). [CrossRef]

14.

O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. 246, 35–43 (2005). [CrossRef]

15.

D. F. V. James, “Polarization of light radiated by black-body sources,” Opt. Commun. 109, 209 (1994). [CrossRef]

16.

G. K. Batchelor, The Theory of Homogeneous Turbulence (Cambridge Science Classics, 1990).

17.

A. C. Schell, “A technique for determination of the radiation pattern of a partially coherent aperture,” IEEE Trans. Antennas Propag. AP15187–188 (1967). [CrossRef]

18.

G. Toraldo di Francia, Electromagnetic Waves (Interscience, 1955), pp. 218–221.

19.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, 1975), p. 437.

20.

M. A. Alonso, O Korotkova, and E. Wolf, “Propagation of the electric correlation matrix and the van CittertZernike theorem for random electromagnetic fields,” J. Mod. Opt. 53, 969–978 (2006). [CrossRef]

21.

F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Realizability condition for electromagnetic Schell-model sources,” J. Opt. Soc. Am A 25, 1016 (2008). [CrossRef]

OCIS Codes
(030.1670) Coherence and statistical optics : Coherent optical effects
(050.1960) Diffraction and gratings : Diffraction theory

ToC Category:
Coherence and Statistical Optics

History
Original Manuscript: September 18, 2012
Revised Manuscript: October 11, 2012
Manuscript Accepted: October 11, 2012
Published: October 16, 2012

Citation
Asma Al-Qasimi and Daniel F. V. James, "Properties of quasi-homogeneous isotropic electromagnetic sources," Opt. Express 20, 24910-24917 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-22-24910


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References

  1. A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am.58, 1256–1259 (1968). [CrossRef]
  2. E. Wolf, “Coherence and radiometry,” J. Opt. Soc. Am.68, 6–17 (1978). [CrossRef]
  3. A. T. Friberg, ed., Selected Papers on Coherence and Radiometry, SPIE Milestone Series MS 69 (SPIE, 1993).
  4. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, 2007).
  5. J. H. Lambert, Photometria sive de mensure et gradibus luminis colorum et umbra (Bassel, 1760).
  6. W. H. Carter and E. Wolf, “Coherence properties of Lambertian and non-Lambertian sources,” J. Opt. Soc. Amer.65, 1067–1071 (1975). [CrossRef]
  7. E. Wolf, “Invariance of the spectrum of light on propagation,” Phys. Rev. Lett.56, 1370–1372 (1986). [CrossRef] [PubMed]
  8. E. Wolf, “Non-cosmological redshifts of spectral lines,” Nature326, 363–365 (1987). [CrossRef]
  9. E. Wolf and D. F. V. James, “Correlation-induced spectral changes,” Rep. Prog. Phys.59, 771–818 (1996). [CrossRef]
  10. D. F. V. James, “Change of polarization of light-beams on propagation in free-space,” J. Opt. Soc. Am. A111641–1643 (1994). [CrossRef]
  11. F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, and G. Guattari, “Beam coherence-polarization matrix,” Pure and Applied Optics7941–951 (1998). [CrossRef]
  12. E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A312, 263–267 (2003). [CrossRef]
  13. J. Tervo, T. Setälä, and A. T. Friberg, “Theory of partially coherent electromagnetic fields in the space-frequency domain,” J. Opt. Soc. Am. A212205–2215 (2004). [CrossRef]
  14. O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun.246, 35–43 (2005). [CrossRef]
  15. D. F. V. James, “Polarization of light radiated by black-body sources,” Opt. Commun.109, 209 (1994). [CrossRef]
  16. G. K. Batchelor, The Theory of Homogeneous Turbulence (Cambridge Science Classics, 1990).
  17. A. C. Schell, “A technique for determination of the radiation pattern of a partially coherent aperture,” IEEE Trans. Antennas Propag.AP15187–188 (1967). [CrossRef]
  18. G. Toraldo di Francia, Electromagnetic Waves (Interscience, 1955), pp. 218–221.
  19. J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, 1975), p. 437.
  20. M. A. Alonso, O Korotkova, and E. Wolf, “Propagation of the electric correlation matrix and the van CittertZernike theorem for random electromagnetic fields,” J. Mod. Opt.53, 969–978 (2006). [CrossRef]
  21. F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Realizability condition for electromagnetic Schell-model sources,” J. Opt. Soc. Am A25, 1016 (2008). [CrossRef]

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