Abstract

Adding a twist phase term to the cross-spectral density (CSD) function of a partially coherent source can be done if and only if the resulting function remains nonnegative definite. Constraints on the twist term that guarantee the validity of the resulting CSD have been derived only for Twisted Gaussian Schell-model (TGSM) sources. Here, an infinite family of higher-order TGSM sources is introduced, whose CSDs are expressed as products of the CSD of a standard TGSM source times Hermite polynomials of arbitrary orders and suitable arguments. All the members present the same twist term and for all of them the twist-coherence constraint keeps obeying the form valid for a standard TGSM source. They can be used as building blocks for constructing an endless number of valid twisted CSDs, with an assigned value of the twist parameter and intensity and/or coherence features that can be very different from those of a standard TGSM source. Through partial transposition, higher-order TGSM CSDs are converted into higher-order Astigmatic Gaussian Schell-model (AGSM) CSDs. The problem of the separability of higher-order TGSM and AGSM CSDs is addressed, and conditions ensuring their separability are derived.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

In the classical theory of coherence [1], a basic role is played by the cross-spectral density (CSD), say W(r1, r2), which, for two typical points r1 and r2, represents, in a scalar approach, the field correlation function in the space-frequency domain [1]. A model source used in countless instances is the planar Gaussian Schell-model (GSM) source, whose CSD has the form

WG(r1,r2)=I0exp(r12+r224σ2)exp[(r1r2)22δ2],
where I0 is a proportionality factor, while σ2 and δ2 are the irradiance and spatial coherence variances, respectively [1]. The CSD in Eq. (1) turned out to be extremely useful as a model of partially coherent circularly symmetric sources.

In a celebrated paper [2], Simon and Mukunda noted that the most general circularly symmetric Gaussian Schell-model source could include a phase term of the form exp[iβ(r1 × r2)], with real β, the subscript ⊥ denoting the orthogonal component of r1 × r2. The presence of such a phase term would have produced a twisting of the beam generated by the source upon propagation, which could be revealed on breaking the circular symmetry with some optical device. This led Simon and Mukunda to investigate the so-called Twisted Gaussian Schell-model (TGSM) CSD, given, for two typical points r1 = (x1, y1) and r2 = (x2, y2) by

WT(r1,r2)=I0exp(r12+r224σ2)exp[(r1r2)22δ2]exp[iku(x1y2x2y1)],
where k is the wavenumber and the real constant u is the so called twist parameter.

A basic result of the Simon-Mukunda analysis was that the twist parameter has to satisfy the following inequality:

k|u|1/δ2,
which was derived in [2] by imposing the non-negative definiteness property that any CSD must possess [1].

The source devised by Simon and Mukunda was soon realized experimentally [3] and several superposition models were suggested to give intuitive views on their intriguing properties [4–7]. However, for a long time the research about partially coherent twisted beams has been essentially concentrated on the original model. Although in recent times the model choice has been extended [8–14], the need still exists for a family of genuine CSDs endowed with twist and offering an endless number of cases where a closed form expression is available. In a sense, they could be the workhorses with which to further plow the subject. To exemplify the need, it is enough to think of the twist-coherence inequality in Eq. (3). In a more general case, it is not even known whether this inequality persists with the same meaning or should be modified and/or re-interpreted.

A possible approach can be envisaged by generalizing the so-called superposition model [4], which was introduced as an integral representation of the original Simon-Mukunda formula. In this paper, we shall explore this approach and we will determine a generalized family of twisted CSDs.

2. Twisted GSM sources as superpositions of tilted Gaussian beams

As well known, a typical CSD W satisfies the non-negative definiteness condition if and only if it can be written as [16,17]

W(r1,r2)=p(ρ)H(r1,ρ)H*(r2,ρ)dρ,
where H is a suitable basic kernel, p(ρ) is a non-negative weight function, and the asterisk denotes the complex conjugate. In particular, the variable ρ can be a position vector while H can be a field specified by r and ρ. In the TGSM case, the basic kernel H can be given the form [4]
H(r,ρ)=exp[a(rρ)22πiα(sytx)],
where a > 0, α is a real parameter, and x, y (s, t) are the Cartesian components of r (ρ). It is seen that Eq. (5) describes the waist section of a TEM00 Gaussian beam centered at ρ with propagation direction inclined with respect to the z–axis. Indeed, 2παt and −2παs are the components of the mean wavevector along the x– and y–axis, respectively. Therefore, the above source can be seen as a superposition of perfectly coherent, but mutually uncorrelated, tilted Gaussian beams [15].

It is interesting to note that the integral representation for W can be given the following form:

W(r1,r2)=p(s,t)exp{a[(x1s)2+(x2s)2]+2πiαt(x1x2)}×exp{a[(y1t)2+(y2t)2]2πiαs(y1y2)}dsdt.
The exponential functions appearing here are 1D correlation functions. They are surely genuine because each of them is expressed as the product of a function evaluated in x1 (y1) and the complex conjugated of the same function evaluated in x2 (y2), thus corresponding to deterministic processes. Therefore, no matter the precise form of p, the CSD is a superposition with positive weights of products of correlation functions. This is exactly what it is meant when it is said that W(r1, r2) is separable. Therefore, any CSD derived from Eq. (4) with the kernel given by Eq. (5) is separable.

In particular, in [4] the TGSM source was obtained on assuming p of the form

p(s,t)exp[b(s2+t2)],
with positive b. For any given triplet (a, b, α), the source parameters turn out to be [4]
14σ2=ab2a+b,12δ2=a2+π2α22a+b,ku=4παa2a+b.
The very possibility of writing W in the form of Eq. (6) guarantees that such a CSD is genuine no matter how a, b, and α are chosen [16,17]. In fact, from the above expressions we can derive
k|u|δ21=2π|α|aa2+π2α21=(aπ|α|2)a2+π2α20,
showing that condition in Eq. (3) is automatically satisfied.

3. Higher-order Twisted GSM sources

In deriving Eq. (2) from Eq. (6), a key role is played by the Gaussian weight function in Eq. (7). As already noted though, it is possible to generate infinitely many CSDs by any (sensible) choice of the weight p. On the other hand, a typical form of p is useful only if the associated integral in Eq. (6) can be solved in closed form. A simple choice that ensures closed form integration is to multiply the Gaussian function by a general polynomial of the form

m=0Mn=0Ndmns2mt2n,
with positive dmn coefficients, or by a positive combination of similar polynomials. This is the case, in particular, for a sum of powers of s2 + t2 with non-negative weights, say cn, like
n=0Ncn(s2+t2)n=n=0Ncnk=0n(nk)s2kt2(nk),
which gives rise to circular symmetric weights. For example, if the cn coefficients are proportional to 1/n! a weight with flattened Gaussian profile of order N is obtained [18,19].

Using the function in Eq. (10) to multiply the Gaussian weight, the following general expression for the CSD is obtained from the superposition integral of Eq. (6):

W(r1,r2)=m=0Mn=0NdmnWmn(r1,r2),
where
Wmn(r1,r2)=s2mt2nexp[b(s2+t2)]×exp{a[(x1s)2+(x2s)2]2πiαs(y1y2)}×exp{a[(y1t)2+(y2t)2]2πiαt(x1x2)}dsdt.
Equation (12) expresses the source as the superposition of mutually uncorrelated, partially coherent sources having CSDs given by Wmn and powers proportional to dmn. The Wmn CSDs will be the basic tools of our work.

We shall begin by noticing that each of the terms Wmn can be given the factorized form

Wmn(r1,r2)=Sm(r1,r2)Tn(r1,r2),
where
Sm(r1,r2)=s2mexp(bs2)exp{a[(x1s)2+(x2s)2]2πiαs(y1y2)}ds,
and
Tn(r1,r2)=t2nexp(bt2)exp{a[(y1t)2+(y2t)2]2πiαt(x1x2)}dt.
It is worthwhile to observe that both the above integrals represent genuine CSDs by virtue of their own structure [16].

Since Sm and Tn have the same form, for their evaluation it is sufficient to consider the first of them. Before proceeding with the evaluation in the general case, however, we note that Sm (Tn) can be easily calculated when m = 0 (n = 0). In fact we have, after simple manipulations,

S0(r1,r2)=π2a+bexp[ab2a+b(x12+x22)]×exp[a22a+b(x1x2)2π2α22a+b(y1y2)2]×exp[i2πaα2a+b(x1y1x2y2x1y2+x2y1)],
showing that S0 represents the CSD of an anisotropic and astigmatic TGSM source, because the latter presents different widths and different curvatures along the orthogonal axes x and y. The same happens for T0, which corresponds to a similar TGSM source, with exactly the same twist, but where coherence widths, intensity widths, and curvature radii are interchanged between x and y. In both cases, the twisting source is obtained through a one-dimensional superposition model, which is simpler than that, two-dimensional, of [4] used to synthesize the CSD of Eq. (2).

The expression of Wmn in the general case, obtained through the evaluation of the integrals in Eqs. (15) and (16), is reported in the Appendix. The final result is

Wmn(r1,r2)=4π(1)m+n[4(2a+b)]n+m+1exp[ab2a+b(r12+r22)]×exp[a2+π2α22a+b(r1r2)2+i4πaα2a+b(x1y2x2y1)]×H2m[πα(y1y2)+ia(x1+x2)2a+b]H2n[πα(x1x2)ia(y1+y2)2a+b],
Hn being the Hermite polynomial of order n [20], which we use to define the CSD of a higher-order Twisted GSM source.

A clearer insight into Eq. (18) is obtained if the natural parameters of a TGSM source, namely, σ, δ and u, are used instead of a, b, and α. The relations linking these two sets of parameters are given in Eq. (8). Then, Wmn takes the form

Wmn(r1,r2)=Amn(1)m+nexp(r12+r224σ2)exp[(r1r2)22δ2iku(x1y2x2y1)]×H2m[c(y1y2)+ic+(x1+x2)2δ]H2n[c(x1x2)ic+(y1+y2)2δ],
where Amn is a positive proportionality factor and
c+=cosϕ;c=sinϕ,
with
ϕ=arctan(παa).

The intensity distribution across the source plane is easily evaluated on letting r1 = r2 = r in Eq. (19). This gives

Imn(r)=Amn(1)m+nexp(r22σ2)H2m(i2c+xδ)H2n(i2c+yδ).

In deriving the latter equation it has been taken into account that the involved Hermite polynomials have even order, so that they don’t change on swapping the sign of their argument. The presence of imaginary arguments in the expression of the intensity might be surprising, but the function (−1)nH2n (it), with real t, is a polynomial of order n in t2 with positive coefficients (see Eq. (5.5.4) of [20]), so that Imn is everywhere nonnegative.

As for the degree of coherence [1], its modulus is given by

|μmn(r1,r2)|=exp[(r1,r2)22δ2]|H2m[c(y1y2)+ic+(x1+x2)2δ]||H2m(i2c+x1δ)H2m(i2c+x2δ)|×|H2n[c(x1x2)+ic+(y1+y2)2δ]||H2n(i2c+y1δ)H2n(i2c+y2δ)|.

It is apparent that in the case n = m = 0 Eq. (19) gives rise to the standard TGSM CSD of Eq. (2), but very different behaviors can be obtained on varying the indices m and n. It should be stressed that all the sources of this class have the same twist parameter and that for each of them the non-negativeness condition remains the one given in Eq. (3). Of course, an infinite number of twisted sources is obtained on combining CSDs of the above kind with positive weights. Note that the procedure we followed in deriving the expression of higher-order FGSM CSDs starts from the superposition integral in Eq. (6) and is no way ascribable to the insertion of an amplitude filter altering the transverse profile of a standard TGSM source.

To give a simple example of what can be changed by non-zero values of m and/or n let us evaluate W10. Equation (19) gives

W10(r1,r2)=2A10exp(r12+r224σ2)exp[(r1r2)22δ2iku(x1y2x2y1)]×{12[c(y1y2)+ic+(x1+x2)2δ]2},
to which the following intensity:
I10(r)=2A10exp(r22σ2)(1+4c+2x2δ2),
and degree of coherence:
|μ10(r1,r2)|=exp[(r1,r2)22δ2]|12[c(y1y2)δ+ic+(x1+x2)δ]2|(1+4c+2x12δ2)(1+4c+2x22δ2)
correspond.

A 3D plot of Eq. (25) for a = b = α is shown in Fig. 1 as a function of (x/δ, y/δ), in arbitrary units.

 figure: Fig. 1

Fig. 1 Plot of the intensity in Eq. (25) as a function of x/δ and y/δ, with a = b = α.

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While the modulus of the degree of coherence is purely Gaussian in the CSD of Eq. (2), it can be considerably different in the present case. Just to give an example, contour plots of |μ10| for a = b = α are shown in Fig. 2 as a function of (x1x2)/δ and (y1y2)/δ, obtained for different values of = (x1 + x2)/2.

 figure: Fig. 2

Fig. 2 Contour plot of |μ10|, obtained from Eq. (26), as a function of (x1x2)/δ and (y1y2)/δ for a = b = α and different values of /δ: 0 (a); 0.5 (b); 1.0 (c); 1.5 (d).

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The difference with respect to the purely Gaussian law will be more and more considerable passing to higher values of m (and/or n) where a complex polynomial of degree 2m (or 2n) will be involved, and considering superpositions of TGSM sources with different orders.

4. Higher-order Astigmatic GSM sources

As we noted above, all CSDs obtained from Eq. (6) are separable. As a consequence, they remain valid under the partial transposition operation, which consists in interchanging, e. g., x1 and x2 while leaving y1 and y2 unchanged (or vice-versa). As it can be seen from Eq. (19), the partial transposition converts higher-order twisted CSDs into the following ones:

W¯mn(r1,r2)=Amn(1)m+nexp(r12+r224σ2)exp[(r1r2)22δ2iku(x2y2x1y1)]×H2m[c(y1y2)+ic+(x1+x2)2δ]H2n[c(x1x2)+ic+(y1+y2)2δ],
where the parity of H2n has been taken into account.

As a result, the partial transposition converts the twist phase component x1y2x2y1 of a higher-order TGSM CSD into the astigmatism element x2y2x1y1. Therefore, mn is the CSD of what we can call a higher-order Astigmatic GSM (AGSM) source, and is surely genuine and separable, due to the way it has been derived. Like for the family of the higher-order TGSM CSDs, these functions form a set of genuine CSDs endowed with the same astigmatic phase.

The parameter u has now to be interpreted as an astigmatism parameter, whose value, however, is limited by the inequality in Eq. (3). We may wonder whether the CSD of a higher-order AGSM source in which k|u| violates such inequality can be, notwithstanding, a genuine CSD. While in the case m = n = 0 a positive answer is reached by simple inspection [21], it is not so if m and/or n differ from zero.

We now show that a change of the basic kernel H can solve the problem. To this end, we introduce the kernel H′, given by

H(r,ρ)=exp[a(r,ρ)2+2πiα(xs)(yt)].
Differently from Eq. (5), where an inclination factor depending on ρ appears, H′ depends on ρ only for a translation. On inserting this kernel into Eq. (4) we find, for the new CSD,
Wmn(r1,r2)=s2mt2neb(s2+t2)exp[a(r1ρ)2+2πiα(x1s)(y1t)]×exp[a(r2ρ)22πiα(x2s)(y2t)]dsdt,
which is a superposition of mutually uncorrelated astigmatic Gaussian beams with parallel axes [15]. We know that the expression in Eq. (29) represents a genuine CSD, but its form does not imply its separability, so that we still don’t know if it is separable as well, or under which conditions it is so. The same problem has been recently addressed in [21] for the particular case of m = n = 0.

The closed-form expression of W′mn can be directly obtained on comparing Eq. (29) to the analogous expression for mn, derived from Eq. (13) through partial transposition, that is,

W¯mn(r1,r2)=s2mt2neb(s2+t2)exp{a[(x1s)2+(x2s)2]2πiαt(x1x2)}×exp{a[(y1t)2+(y2t)2]2πiαs(y1y2)}dsdt.

After some manipulations, the following beautifully simple result is obtained:

Wmn(r1,r2)=W¯mn(r1,r2)exp[2πiα(x1y1x2y2)],
so that from Eqs. (27) and (31) we have
Wmn(r1,r2)=Amn(1)m+nexp(r12+r224σ2)exp[(r1r2)22δ2ikv(x2y2x1y1)]×H2m[c(y1y2)+ic+(x1+x2)2δ]H2n[c(x1x2)+ic+(y1+y2)2δ],
where [see Eq. (8)],
kv=4παa2a+b+2πα=2πbα2a+b.

Equation (32) coincides with Eq. (27), with the only difference that u has been replaced by v. But no upper bound limits the modulus of v. In fact we have

kvδ2=πbαa2+π2α2,
which apparently can take any assigned value on suitably choosing the parameters a, b and α.

The above analysis allows us to answer to the question about the separability of W′mn. The latter, in fact, is a valid CSD because of the way it has been built [see Eq. (29)], but it turns out to be separable if and only if its partial transposed is valid too. Now, the partial transposed of W′mn is the CSD of a higher-order TGSM source with twist parameter v, and is valid only if k|v| does not exceed 1/δ2. Beyond such limit W′mn is still a genuine CSD, but entangled (not separable). The present result extends the one obtained in [21] to the whole classes of higher-order twisted and astigmatic GSM sources.

5. Conclusions

In this paper, starting from the generalized form of a superposition model first devised for a TGSM source, we introduced a whole family a CDSs endowed with twist. These generalized CSDs turn out to be expressed as products of the original Simon-Mukunda CSD times Hermite polynomials of arbitrary orders, whose arguments are suitable combinations of the observation point coordinates. Accordingly, such CSDs have been termed higher-order TGSM. The coherence features of these CSDs can be very sophisticated, especially when Hermite polynomials of high orders are involved. Yet, the twist-coherence constraint keeps obeying the original Simon-Mukunda form for any order. Since any combination with positive coefficients of higher-order TGSM CSDs is a valid CSD, they represent the building blocks to devise very general twisted CSDs. The extension of such result to the case of electromagnetic sources, where the twist phase can affect the source polarization properties in a nontrivial way [24], is a subject worth examining.

Through partial transposition of higher-order TGSM CSDs, another family of CSDs is obtained, the so called higher-order Astigmatic GSM CSDs, where the twist phase is replaced by an analogous element, responsible for an astigmatism of the source. For those sources no limitation exists for the astigmatism parameter of the source.

The concept of separability plays a fundamental role in quantum contexts but, how it has been quite recently realized, this property may be relevant even in classical physics, in which case the term “classical entanglement” is often used. In particular, significant recent results concern the role of the entanglement in classical, either scalar and vectorial, light fields [25–32]. The issue of the separability of the CSDs of the sources introduced here has been addressed. While the higher-order TGSM CSDs turn out to be alway separable (whenever valid), the corresponding astigmatic CSDs are separable only for those values of the astigmatism parameter fulfilling the constraints required for a TGSM source.

Appendix: Evaluation of the functions Wmn

We begin by evaluating the integrals in Eqs. (15) and (16). Expanding binomial squares and putting s–independent terms out of the integral in Eq. (15) we obtain, letting s = ζ + x12,

Sm(r1,r2)=exp[ab2a+b(x12+x22)]exp[g(x1x2)22πiγ12x12]×(ζ+x12)2me(2a+b)ζ2e2πiγ12ζdζ,
where
g=a22a+b,x12=a(x1+x2)2a+b,γ12=α(y1y2).

As shown by Eq. (35), we have to Fourier transform the power of a binomial times a Gaussian function. Using Newton’s formula we write

Sm(r1,r2)=exp[ab2a+b(x12+x22)]exp[g(x1x2)22πiγ12x12]×k=02m(2mk)x12kζ2mke(2a+b)ζ2e2πiγ12ζdζ,
which, through repeated applications of the derivative theorem for Fourier transforms [22], gives
Sm(r1,r2)=π2a+bexp[ab2a+b(x12+x22)]exp[g(x1x2)22πiγ12x12]×k=02m(2mk)x12k(2πi)2mkd2mkdγ122mk[eπ2γ122/(2a+b)].

The derivative in the last raw of Eq. (38) can be written in a closed form because [see Eq. (5.5.3) of [20]]

dndxnex2=(1)nex2Hn(x),
so that
Sm(r1,r2)=π2a+bexp[ab2a+b(x12+x22)]×exp[g(x1x2)22πiγ12x12π2γ1222a+b]×k=02m(2mk)x12k(2i2a+b)2mkH2mk(πγ122a+b).
By recalling the definitions in Eq. (36) we can give Sm the following more explicit form:
Sm(r1,r2)=π2a+bexp[ab2a+b(x12+x22)]×exp[a2(x1x2)2+π2α2(y1y2)22a+b]×exp[i2πaα2a+b(x1y1x2y2x1y2+x2y1)]Fm(x1+x2,y1y2),
where the function
Fm(ξ,η)=k=02m(2mk)(aξ2a+b)k(i22a+b)2mkH2mk(παη2a+b)
has been introduced.

The latter is a polynomial of order 2m that, thanks to the following identity (see Eq. 4.5.1.7 of [23]):

k=0n(nk)(2y)nkHk(x)=Hn(x+y),
can be written as
Fm(ξ,η)=[14(2a+b)]mH2m(παη+iaξ2a+b).
We can similarly find Tn, obtaining
Tn(r1,r2)=π2a+bexp[ab2a+b(x12+x22)]×exp[a2(y1y2)2+π2α2(x1x2)22a+b]×exp[i2πaα2a+b(y1x1y2x2y1x2+y2x1)]Fn*(y1+y2,x1x2).

Finally, on taking Eq. (44) into account, the product of the quantities in Eqs. (41) and (45) gives

Wmn(r1,r2)=π(1)m+n4m+n(2a+b)m+n+1exp[ab2a+b(r12+r22)]×exp[a2+π2α22a+b(r1r2)2+i4πaα2a+b(x1y2x2y1)]×H2m[πα(y1y2)+ia(x1+x2)2a+b]H2n[πα(x1x2)ia(y1+y2)2a+b].

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24. S. A. Ponomarenko and G. P. Agrawal, “Asymmetric incoherent vector solitons,” Phys. Rev. E 69, 036604 (2004). [CrossRef]  

25. R. J. C. Spreeuw, “A classical analogy of entanglement,” Found. Phys. 28, 361–374 (1998). [CrossRef]  

26. B. N. Simon, S. Simon, N. Mukunda, F. Gori, M. Santarsiero, R. Borghi, and R. Simon, “A complete characterization of pre-Mueller and Mueller matrices in polarization optics,” J. Opt. Soc. Am. A 27, 188–199 (2010). [CrossRef]  

27. B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010). [CrossRef]   [PubMed]  

28. X. Qian and J. H. Eberly, “Entanglement and classical polarization states,” Opt. Lett. 36, 4110–4112 (2011). [CrossRef]   [PubMed]  

29. P. Chowdhury, A. S. Majumdar, and G. S. Agarwal, “Nonlocal continuous-variable correlations and violation of Bell’s inequality for light beams with topological singularities,” Phys. Rev. A 888, 195 (2013).

30. P. Ghose and A. Mukherjee, “Entanglement in classical optics,” Rev. Theor. Sci. 2, 274–288 (2014). [CrossRef]  

31. A. Aiello, F. Töppel, C. Marquardt, E. Giacobino, and G. Leuchs, “Quantum-like nonseparable structures in optical beams,” New J. Phys. 17, 043024 (2015). [CrossRef]  

32. X. F. Qian, A. N. Vamivakas, and J. H. Eberly, “Entanglement limits duality and vice versa,” Optica 5, 942–947 (2018). [CrossRef]  

References

  • View by:

  1. L. Mandel L. and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
    [Crossref]
  2. R. Simon and N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10, 95–109 (1993).
    [Crossref]
  3. A. T. Friberg, E. Tervonen, and J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11, 1818–1826 (1994).
    [Crossref]
  4. D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
    [Crossref]
  5. F. Gori, M. Santarsiero, R. Borghi, and S. Vicalvi, “Partially coherent sources with helicoidal modes,” J. Mod. Optics 45, 539–554 (1998).
    [Crossref]
  6. S. A. Ponomarenko, “Twisted Gaussian Schell-model solitons,” Phys. Rev. E 64, 036618 (2001).
    [Crossref]
  7. F. Gori and M. Santarsiero, “Twisted Gaussian Schell-model beams as series of partially coherent modified Bessel-Gauss beams,” Opt. Lett. 40, 1587–1590 (2015).
    [Crossref] [PubMed]
  8. R. Borghi, F. Gori, G. Guattari, and M. Santarsiero, “Twisted Schell-model beams with axial symmetry,” Opt. Lett. 40, 4504–4507 (2015).
    [Crossref] [PubMed]
  9. Z. Mei and O. Korotkova, “Random sources for rotating spectral densities,” Opt. Lett. 42, 255–258 (2017).
    [Crossref] [PubMed]
  10. W. Fu and H. Zhang, “Evolution properties of a radially polarized partially coherent twisted beam propagating in a uniaxial crystal,” Eur. Phys. J. D 71, 181 (2017).
    [Crossref]
  11. F. Gori and M. Santarsiero, “Devising genuine twisted cross-spectral densities,” Opt. Lett. 43, 595–598 (2018).
    [Crossref] [PubMed]
  12. R. Borghi, “Twisting partially coherent light,” Opt. Lett. 43, 1627–1630 (2018).
    [Crossref] [PubMed]
  13. C. S. D. Stahl and G. Gbur, “Twisted vortex Gaussian Schell-model beams,” J. Opt. Soc. Am. A 35, 1899–1906 (2018).
    [Crossref]
  14. J. Wang, H. Huang, Y. Chen, H. Wang, S. Zhu, Z. Li, and Y. Cai, “Twisted partially coherent array sources and their transmission in anisotropic turbulence,” Opt. Express 26, 25974–25988 (2018).
    [Crossref] [PubMed]
  15. A. E. Siegman, Lasers (University Science Books, 1986).
  16. F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32, 3531–3533 (2007).
    [Crossref] [PubMed]
  17. R. Martinez-Herrero, P. M. Mejías, and F. Gori, “Genuine cross-spectral densities and pseudo-modal expansions,” Opt. Lett. 34, 1399–1401 (2009).
    [Crossref] [PubMed]
  18. F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
    [Crossref]
  19. V. Bagini, R. Borghi, F. Gori, A. M. Pacileo, M. Santarsiero, D. Ambrosini, and G. Schirripa Spagnolo, “Propagation of axially symmetric flattened Gaussian beams,” J. Opt. Soc. Am. A 13, 1385–1394 (1996).
    [Crossref]
  20. G. Szegö, Orthogonal Polynomials, 4th edition (American Mathematical Society, 1975).
  21. Arvind, S. Chaturvedi, and N. Mukunda, “Entanglement and complete positivity: Relevance and manifestations in classical scalar wave optics,” Fortschr. Phys. 66, 1700077 (2017).
    [Crossref]
  22. J. W. Goodman, Introduction to Fourier Optics, 4th edition (W. H. Freeman, 2017).
  23. A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integral and Series (Gordon and Breach, 1986, Vol. 2).
  24. S. A. Ponomarenko and G. P. Agrawal, “Asymmetric incoherent vector solitons,” Phys. Rev. E 69, 036604 (2004).
    [Crossref]
  25. R. J. C. Spreeuw, “A classical analogy of entanglement,” Found. Phys. 28, 361–374 (1998).
    [Crossref]
  26. B. N. Simon, S. Simon, N. Mukunda, F. Gori, M. Santarsiero, R. Borghi, and R. Simon, “A complete characterization of pre-Mueller and Mueller matrices in polarization optics,” J. Opt. Soc. Am. A 27, 188–199 (2010).
    [Crossref]
  27. B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
    [Crossref] [PubMed]
  28. X. Qian and J. H. Eberly, “Entanglement and classical polarization states,” Opt. Lett. 36, 4110–4112 (2011).
    [Crossref] [PubMed]
  29. P. Chowdhury, A. S. Majumdar, and G. S. Agarwal, “Nonlocal continuous-variable correlations and violation of Bell’s inequality for light beams with topological singularities,” Phys. Rev. A 888, 195 (2013).
  30. P. Ghose and A. Mukherjee, “Entanglement in classical optics,” Rev. Theor. Sci. 2, 274–288 (2014).
    [Crossref]
  31. A. Aiello, F. Töppel, C. Marquardt, E. Giacobino, and G. Leuchs, “Quantum-like nonseparable structures in optical beams,” New J. Phys. 17, 043024 (2015).
    [Crossref]
  32. X. F. Qian, A. N. Vamivakas, and J. H. Eberly, “Entanglement limits duality and vice versa,” Optica 5, 942–947 (2018).
    [Crossref]

2018 (5)

F. Gori and M. Santarsiero, “Devising genuine twisted cross-spectral densities,” Opt. Lett. 43, 595–598 (2018).
[Crossref] [PubMed]

R. Borghi, “Twisting partially coherent light,” Opt. Lett. 43, 1627–1630 (2018).
[Crossref] [PubMed]

C. S. D. Stahl and G. Gbur, “Twisted vortex Gaussian Schell-model beams,” J. Opt. Soc. Am. A 35, 1899–1906 (2018).
[Crossref]

J. Wang, H. Huang, Y. Chen, H. Wang, S. Zhu, Z. Li, and Y. Cai, “Twisted partially coherent array sources and their transmission in anisotropic turbulence,” Opt. Express 26, 25974–25988 (2018).
[Crossref] [PubMed]

X. F. Qian, A. N. Vamivakas, and J. H. Eberly, “Entanglement limits duality and vice versa,” Optica 5, 942–947 (2018).
[Crossref]

2017 (3)

Arvind, S. Chaturvedi, and N. Mukunda, “Entanglement and complete positivity: Relevance and manifestations in classical scalar wave optics,” Fortschr. Phys. 66, 1700077 (2017).
[Crossref]

Z. Mei and O. Korotkova, “Random sources for rotating spectral densities,” Opt. Lett. 42, 255–258 (2017).
[Crossref] [PubMed]

W. Fu and H. Zhang, “Evolution properties of a radially polarized partially coherent twisted beam propagating in a uniaxial crystal,” Eur. Phys. J. D 71, 181 (2017).
[Crossref]

2015 (3)

F. Gori and M. Santarsiero, “Twisted Gaussian Schell-model beams as series of partially coherent modified Bessel-Gauss beams,” Opt. Lett. 40, 1587–1590 (2015).
[Crossref] [PubMed]

R. Borghi, F. Gori, G. Guattari, and M. Santarsiero, “Twisted Schell-model beams with axial symmetry,” Opt. Lett. 40, 4504–4507 (2015).
[Crossref] [PubMed]

A. Aiello, F. Töppel, C. Marquardt, E. Giacobino, and G. Leuchs, “Quantum-like nonseparable structures in optical beams,” New J. Phys. 17, 043024 (2015).
[Crossref]

2014 (1)

P. Ghose and A. Mukherjee, “Entanglement in classical optics,” Rev. Theor. Sci. 2, 274–288 (2014).
[Crossref]

2013 (1)

P. Chowdhury, A. S. Majumdar, and G. S. Agarwal, “Nonlocal continuous-variable correlations and violation of Bell’s inequality for light beams with topological singularities,” Phys. Rev. A 888, 195 (2013).

2011 (1)

X. Qian and J. H. Eberly, “Entanglement and classical polarization states,” Opt. Lett. 36, 4110–4112 (2011).
[Crossref] [PubMed]

2010 (2)

B. N. Simon, S. Simon, N. Mukunda, F. Gori, M. Santarsiero, R. Borghi, and R. Simon, “A complete characterization of pre-Mueller and Mueller matrices in polarization optics,” J. Opt. Soc. Am. A 27, 188–199 (2010).
[Crossref]

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
[Crossref] [PubMed]

2009 (1)

R. Martinez-Herrero, P. M. Mejías, and F. Gori, “Genuine cross-spectral densities and pseudo-modal expansions,” Opt. Lett. 34, 1399–1401 (2009).
[Crossref] [PubMed]

2007 (1)

F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32, 3531–3533 (2007).
[Crossref] [PubMed]

2004 (1)

S. A. Ponomarenko and G. P. Agrawal, “Asymmetric incoherent vector solitons,” Phys. Rev. E 69, 036604 (2004).
[Crossref]

2001 (1)

S. A. Ponomarenko, “Twisted Gaussian Schell-model solitons,” Phys. Rev. E 64, 036618 (2001).
[Crossref]

1998 (2)

F. Gori, M. Santarsiero, R. Borghi, and S. Vicalvi, “Partially coherent sources with helicoidal modes,” J. Mod. Optics 45, 539–554 (1998).
[Crossref]

R. J. C. Spreeuw, “A classical analogy of entanglement,” Found. Phys. 28, 361–374 (1998).
[Crossref]

1996 (1)

V. Bagini, R. Borghi, F. Gori, A. M. Pacileo, M. Santarsiero, D. Ambrosini, and G. Schirripa Spagnolo, “Propagation of axially symmetric flattened Gaussian beams,” J. Opt. Soc. Am. A 13, 1385–1394 (1996).
[Crossref]

1994 (3)

F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
[Crossref]

A. T. Friberg, E. Tervonen, and J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11, 1818–1826 (1994).
[Crossref]

D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
[Crossref]

1993 (1)

R. Simon and N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10, 95–109 (1993).
[Crossref]

Agarwal, G. S.

P. Chowdhury, A. S. Majumdar, and G. S. Agarwal, “Nonlocal continuous-variable correlations and violation of Bell’s inequality for light beams with topological singularities,” Phys. Rev. A 888, 195 (2013).

Agrawal, G. P.

S. A. Ponomarenko and G. P. Agrawal, “Asymmetric incoherent vector solitons,” Phys. Rev. E 69, 036604 (2004).
[Crossref]

Aiello, A.

A. Aiello, F. Töppel, C. Marquardt, E. Giacobino, and G. Leuchs, “Quantum-like nonseparable structures in optical beams,” New J. Phys. 17, 043024 (2015).
[Crossref]

Ambrosini, D.

V. Bagini, R. Borghi, F. Gori, A. M. Pacileo, M. Santarsiero, D. Ambrosini, and G. Schirripa Spagnolo, “Propagation of axially symmetric flattened Gaussian beams,” J. Opt. Soc. Am. A 13, 1385–1394 (1996).
[Crossref]

D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
[Crossref]

Arvind,

Arvind, S. Chaturvedi, and N. Mukunda, “Entanglement and complete positivity: Relevance and manifestations in classical scalar wave optics,” Fortschr. Phys. 66, 1700077 (2017).
[Crossref]

Bagini, V.

V. Bagini, R. Borghi, F. Gori, A. M. Pacileo, M. Santarsiero, D. Ambrosini, and G. Schirripa Spagnolo, “Propagation of axially symmetric flattened Gaussian beams,” J. Opt. Soc. Am. A 13, 1385–1394 (1996).
[Crossref]

D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
[Crossref]

Borghi, R.

R. Borghi, “Twisting partially coherent light,” Opt. Lett. 43, 1627–1630 (2018).
[Crossref] [PubMed]

R. Borghi, F. Gori, G. Guattari, and M. Santarsiero, “Twisted Schell-model beams with axial symmetry,” Opt. Lett. 40, 4504–4507 (2015).
[Crossref] [PubMed]

B. N. Simon, S. Simon, N. Mukunda, F. Gori, M. Santarsiero, R. Borghi, and R. Simon, “A complete characterization of pre-Mueller and Mueller matrices in polarization optics,” J. Opt. Soc. Am. A 27, 188–199 (2010).
[Crossref]

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
[Crossref] [PubMed]

F. Gori, M. Santarsiero, R. Borghi, and S. Vicalvi, “Partially coherent sources with helicoidal modes,” J. Mod. Optics 45, 539–554 (1998).
[Crossref]

V. Bagini, R. Borghi, F. Gori, A. M. Pacileo, M. Santarsiero, D. Ambrosini, and G. Schirripa Spagnolo, “Propagation of axially symmetric flattened Gaussian beams,” J. Opt. Soc. Am. A 13, 1385–1394 (1996).
[Crossref]

Brychkov, Y. A.

A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integral and Series (Gordon and Breach, 1986, Vol. 2).

Cai, Y.

J. Wang, H. Huang, Y. Chen, H. Wang, S. Zhu, Z. Li, and Y. Cai, “Twisted partially coherent array sources and their transmission in anisotropic turbulence,” Opt. Express 26, 25974–25988 (2018).
[Crossref] [PubMed]

Chaturvedi, S.

Arvind, S. Chaturvedi, and N. Mukunda, “Entanglement and complete positivity: Relevance and manifestations in classical scalar wave optics,” Fortschr. Phys. 66, 1700077 (2017).
[Crossref]

Chen, Y.

J. Wang, H. Huang, Y. Chen, H. Wang, S. Zhu, Z. Li, and Y. Cai, “Twisted partially coherent array sources and their transmission in anisotropic turbulence,” Opt. Express 26, 25974–25988 (2018).
[Crossref] [PubMed]

Chowdhury, P.

P. Chowdhury, A. S. Majumdar, and G. S. Agarwal, “Nonlocal continuous-variable correlations and violation of Bell’s inequality for light beams with topological singularities,” Phys. Rev. A 888, 195 (2013).

Eberly, J. H.

X. F. Qian, A. N. Vamivakas, and J. H. Eberly, “Entanglement limits duality and vice versa,” Optica 5, 942–947 (2018).
[Crossref]

X. Qian and J. H. Eberly, “Entanglement and classical polarization states,” Opt. Lett. 36, 4110–4112 (2011).
[Crossref] [PubMed]

Friberg, A. T.

A. T. Friberg, E. Tervonen, and J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11, 1818–1826 (1994).
[Crossref]

Fu, W.

W. Fu and H. Zhang, “Evolution properties of a radially polarized partially coherent twisted beam propagating in a uniaxial crystal,” Eur. Phys. J. D 71, 181 (2017).
[Crossref]

Gbur, G.

C. S. D. Stahl and G. Gbur, “Twisted vortex Gaussian Schell-model beams,” J. Opt. Soc. Am. A 35, 1899–1906 (2018).
[Crossref]

Ghose, P.

P. Ghose and A. Mukherjee, “Entanglement in classical optics,” Rev. Theor. Sci. 2, 274–288 (2014).
[Crossref]

Giacobino, E.

A. Aiello, F. Töppel, C. Marquardt, E. Giacobino, and G. Leuchs, “Quantum-like nonseparable structures in optical beams,” New J. Phys. 17, 043024 (2015).
[Crossref]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 4th edition (W. H. Freeman, 2017).

Gori, F.

F. Gori and M. Santarsiero, “Devising genuine twisted cross-spectral densities,” Opt. Lett. 43, 595–598 (2018).
[Crossref] [PubMed]

R. Borghi, F. Gori, G. Guattari, and M. Santarsiero, “Twisted Schell-model beams with axial symmetry,” Opt. Lett. 40, 4504–4507 (2015).
[Crossref] [PubMed]

F. Gori and M. Santarsiero, “Twisted Gaussian Schell-model beams as series of partially coherent modified Bessel-Gauss beams,” Opt. Lett. 40, 1587–1590 (2015).
[Crossref] [PubMed]

B. N. Simon, S. Simon, N. Mukunda, F. Gori, M. Santarsiero, R. Borghi, and R. Simon, “A complete characterization of pre-Mueller and Mueller matrices in polarization optics,” J. Opt. Soc. Am. A 27, 188–199 (2010).
[Crossref]

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
[Crossref] [PubMed]

R. Martinez-Herrero, P. M. Mejías, and F. Gori, “Genuine cross-spectral densities and pseudo-modal expansions,” Opt. Lett. 34, 1399–1401 (2009).
[Crossref] [PubMed]

F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32, 3531–3533 (2007).
[Crossref] [PubMed]

F. Gori, M. Santarsiero, R. Borghi, and S. Vicalvi, “Partially coherent sources with helicoidal modes,” J. Mod. Optics 45, 539–554 (1998).
[Crossref]

V. Bagini, R. Borghi, F. Gori, A. M. Pacileo, M. Santarsiero, D. Ambrosini, and G. Schirripa Spagnolo, “Propagation of axially symmetric flattened Gaussian beams,” J. Opt. Soc. Am. A 13, 1385–1394 (1996).
[Crossref]

F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
[Crossref]

D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
[Crossref]

Guattari, G.

R. Borghi, F. Gori, G. Guattari, and M. Santarsiero, “Twisted Schell-model beams with axial symmetry,” Opt. Lett. 40, 4504–4507 (2015).
[Crossref] [PubMed]

Huang, H.

J. Wang, H. Huang, Y. Chen, H. Wang, S. Zhu, Z. Li, and Y. Cai, “Twisted partially coherent array sources and their transmission in anisotropic turbulence,” Opt. Express 26, 25974–25988 (2018).
[Crossref] [PubMed]

Korotkova, O.

Z. Mei and O. Korotkova, “Random sources for rotating spectral densities,” Opt. Lett. 42, 255–258 (2017).
[Crossref] [PubMed]

Leuchs, G.

A. Aiello, F. Töppel, C. Marquardt, E. Giacobino, and G. Leuchs, “Quantum-like nonseparable structures in optical beams,” New J. Phys. 17, 043024 (2015).
[Crossref]

Li, Z.

J. Wang, H. Huang, Y. Chen, H. Wang, S. Zhu, Z. Li, and Y. Cai, “Twisted partially coherent array sources and their transmission in anisotropic turbulence,” Opt. Express 26, 25974–25988 (2018).
[Crossref] [PubMed]

Majumdar, A. S.

P. Chowdhury, A. S. Majumdar, and G. S. Agarwal, “Nonlocal continuous-variable correlations and violation of Bell’s inequality for light beams with topological singularities,” Phys. Rev. A 888, 195 (2013).

Mandel L., L.

L. Mandel L. and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
[Crossref]

Marichev, O. I.

A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integral and Series (Gordon and Breach, 1986, Vol. 2).

Marquardt, C.

A. Aiello, F. Töppel, C. Marquardt, E. Giacobino, and G. Leuchs, “Quantum-like nonseparable structures in optical beams,” New J. Phys. 17, 043024 (2015).
[Crossref]

Martinez-Herrero, R.

R. Martinez-Herrero, P. M. Mejías, and F. Gori, “Genuine cross-spectral densities and pseudo-modal expansions,” Opt. Lett. 34, 1399–1401 (2009).
[Crossref] [PubMed]

Mei, Z.

Z. Mei and O. Korotkova, “Random sources for rotating spectral densities,” Opt. Lett. 42, 255–258 (2017).
[Crossref] [PubMed]

Mejías, P. M.

R. Martinez-Herrero, P. M. Mejías, and F. Gori, “Genuine cross-spectral densities and pseudo-modal expansions,” Opt. Lett. 34, 1399–1401 (2009).
[Crossref] [PubMed]

Mukherjee, A.

P. Ghose and A. Mukherjee, “Entanglement in classical optics,” Rev. Theor. Sci. 2, 274–288 (2014).
[Crossref]

Mukunda, N.

Arvind, S. Chaturvedi, and N. Mukunda, “Entanglement and complete positivity: Relevance and manifestations in classical scalar wave optics,” Fortschr. Phys. 66, 1700077 (2017).
[Crossref]

B. N. Simon, S. Simon, N. Mukunda, F. Gori, M. Santarsiero, R. Borghi, and R. Simon, “A complete characterization of pre-Mueller and Mueller matrices in polarization optics,” J. Opt. Soc. Am. A 27, 188–199 (2010).
[Crossref]

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
[Crossref] [PubMed]

R. Simon and N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10, 95–109 (1993).
[Crossref]

Pacileo, A. M.

V. Bagini, R. Borghi, F. Gori, A. M. Pacileo, M. Santarsiero, D. Ambrosini, and G. Schirripa Spagnolo, “Propagation of axially symmetric flattened Gaussian beams,” J. Opt. Soc. Am. A 13, 1385–1394 (1996).
[Crossref]

Ponomarenko, S. A.

S. A. Ponomarenko and G. P. Agrawal, “Asymmetric incoherent vector solitons,” Phys. Rev. E 69, 036604 (2004).
[Crossref]

S. A. Ponomarenko, “Twisted Gaussian Schell-model solitons,” Phys. Rev. E 64, 036618 (2001).
[Crossref]

Prudnikov, A. P.

A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integral and Series (Gordon and Breach, 1986, Vol. 2).

Qian, X.

X. Qian and J. H. Eberly, “Entanglement and classical polarization states,” Opt. Lett. 36, 4110–4112 (2011).
[Crossref] [PubMed]

Qian, X. F.

X. F. Qian, A. N. Vamivakas, and J. H. Eberly, “Entanglement limits duality and vice versa,” Optica 5, 942–947 (2018).
[Crossref]

Santarsiero, M.

F. Gori and M. Santarsiero, “Devising genuine twisted cross-spectral densities,” Opt. Lett. 43, 595–598 (2018).
[Crossref] [PubMed]

R. Borghi, F. Gori, G. Guattari, and M. Santarsiero, “Twisted Schell-model beams with axial symmetry,” Opt. Lett. 40, 4504–4507 (2015).
[Crossref] [PubMed]

F. Gori and M. Santarsiero, “Twisted Gaussian Schell-model beams as series of partially coherent modified Bessel-Gauss beams,” Opt. Lett. 40, 1587–1590 (2015).
[Crossref] [PubMed]

B. N. Simon, S. Simon, N. Mukunda, F. Gori, M. Santarsiero, R. Borghi, and R. Simon, “A complete characterization of pre-Mueller and Mueller matrices in polarization optics,” J. Opt. Soc. Am. A 27, 188–199 (2010).
[Crossref]

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
[Crossref] [PubMed]

F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32, 3531–3533 (2007).
[Crossref] [PubMed]

F. Gori, M. Santarsiero, R. Borghi, and S. Vicalvi, “Partially coherent sources with helicoidal modes,” J. Mod. Optics 45, 539–554 (1998).
[Crossref]

V. Bagini, R. Borghi, F. Gori, A. M. Pacileo, M. Santarsiero, D. Ambrosini, and G. Schirripa Spagnolo, “Propagation of axially symmetric flattened Gaussian beams,” J. Opt. Soc. Am. A 13, 1385–1394 (1996).
[Crossref]

D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
[Crossref]

Schirripa Spagnolo, G.

V. Bagini, R. Borghi, F. Gori, A. M. Pacileo, M. Santarsiero, D. Ambrosini, and G. Schirripa Spagnolo, “Propagation of axially symmetric flattened Gaussian beams,” J. Opt. Soc. Am. A 13, 1385–1394 (1996).
[Crossref]

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, 1986).

Simon, B. N.

B. N. Simon, S. Simon, N. Mukunda, F. Gori, M. Santarsiero, R. Borghi, and R. Simon, “A complete characterization of pre-Mueller and Mueller matrices in polarization optics,” J. Opt. Soc. Am. A 27, 188–199 (2010).
[Crossref]

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
[Crossref] [PubMed]

Simon, R.

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
[Crossref] [PubMed]

B. N. Simon, S. Simon, N. Mukunda, F. Gori, M. Santarsiero, R. Borghi, and R. Simon, “A complete characterization of pre-Mueller and Mueller matrices in polarization optics,” J. Opt. Soc. Am. A 27, 188–199 (2010).
[Crossref]

R. Simon and N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10, 95–109 (1993).
[Crossref]

Simon, S.

B. N. Simon, S. Simon, N. Mukunda, F. Gori, M. Santarsiero, R. Borghi, and R. Simon, “A complete characterization of pre-Mueller and Mueller matrices in polarization optics,” J. Opt. Soc. Am. A 27, 188–199 (2010).
[Crossref]

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
[Crossref] [PubMed]

Spreeuw, R. J. C.

R. J. C. Spreeuw, “A classical analogy of entanglement,” Found. Phys. 28, 361–374 (1998).
[Crossref]

Stahl, C. S. D.

C. S. D. Stahl and G. Gbur, “Twisted vortex Gaussian Schell-model beams,” J. Opt. Soc. Am. A 35, 1899–1906 (2018).
[Crossref]

Szegö, G.

G. Szegö, Orthogonal Polynomials, 4th edition (American Mathematical Society, 1975).

Tervonen, E.

A. T. Friberg, E. Tervonen, and J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11, 1818–1826 (1994).
[Crossref]

Töppel, F.

A. Aiello, F. Töppel, C. Marquardt, E. Giacobino, and G. Leuchs, “Quantum-like nonseparable structures in optical beams,” New J. Phys. 17, 043024 (2015).
[Crossref]

Turunen, J.

A. T. Friberg, E. Tervonen, and J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11, 1818–1826 (1994).
[Crossref]

Vamivakas, A. N.

X. F. Qian, A. N. Vamivakas, and J. H. Eberly, “Entanglement limits duality and vice versa,” Optica 5, 942–947 (2018).
[Crossref]

Vicalvi, S.

F. Gori, M. Santarsiero, R. Borghi, and S. Vicalvi, “Partially coherent sources with helicoidal modes,” J. Mod. Optics 45, 539–554 (1998).
[Crossref]

Wang, H.

J. Wang, H. Huang, Y. Chen, H. Wang, S. Zhu, Z. Li, and Y. Cai, “Twisted partially coherent array sources and their transmission in anisotropic turbulence,” Opt. Express 26, 25974–25988 (2018).
[Crossref] [PubMed]

Wang, J.

J. Wang, H. Huang, Y. Chen, H. Wang, S. Zhu, Z. Li, and Y. Cai, “Twisted partially coherent array sources and their transmission in anisotropic turbulence,” Opt. Express 26, 25974–25988 (2018).
[Crossref] [PubMed]

Wolf, E.

L. Mandel L. and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
[Crossref]

Zhang, H.

W. Fu and H. Zhang, “Evolution properties of a radially polarized partially coherent twisted beam propagating in a uniaxial crystal,” Eur. Phys. J. D 71, 181 (2017).
[Crossref]

Zhu, S.

J. Wang, H. Huang, Y. Chen, H. Wang, S. Zhu, Z. Li, and Y. Cai, “Twisted partially coherent array sources and their transmission in anisotropic turbulence,” Opt. Express 26, 25974–25988 (2018).
[Crossref] [PubMed]

Eur. Phys. J. D (1)

W. Fu and H. Zhang, “Evolution properties of a radially polarized partially coherent twisted beam propagating in a uniaxial crystal,” Eur. Phys. J. D 71, 181 (2017).
[Crossref]

Fortschr. Phys. (1)

Arvind, S. Chaturvedi, and N. Mukunda, “Entanglement and complete positivity: Relevance and manifestations in classical scalar wave optics,” Fortschr. Phys. 66, 1700077 (2017).
[Crossref]

Found. Phys. (1)

R. J. C. Spreeuw, “A classical analogy of entanglement,” Found. Phys. 28, 361–374 (1998).
[Crossref]

J. Mod. Opt. (1)

D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
[Crossref]

J. Mod. Optics (1)

F. Gori, M. Santarsiero, R. Borghi, and S. Vicalvi, “Partially coherent sources with helicoidal modes,” J. Mod. Optics 45, 539–554 (1998).
[Crossref]

J. Opt. Soc. Am. A (5)

R. Simon and N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10, 95–109 (1993).
[Crossref]

A. T. Friberg, E. Tervonen, and J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11, 1818–1826 (1994).
[Crossref]

C. S. D. Stahl and G. Gbur, “Twisted vortex Gaussian Schell-model beams,” J. Opt. Soc. Am. A 35, 1899–1906 (2018).
[Crossref]

V. Bagini, R. Borghi, F. Gori, A. M. Pacileo, M. Santarsiero, D. Ambrosini, and G. Schirripa Spagnolo, “Propagation of axially symmetric flattened Gaussian beams,” J. Opt. Soc. Am. A 13, 1385–1394 (1996).
[Crossref]

B. N. Simon, S. Simon, N. Mukunda, F. Gori, M. Santarsiero, R. Borghi, and R. Simon, “A complete characterization of pre-Mueller and Mueller matrices in polarization optics,” J. Opt. Soc. Am. A 27, 188–199 (2010).
[Crossref]

New J. Phys. (1)

A. Aiello, F. Töppel, C. Marquardt, E. Giacobino, and G. Leuchs, “Quantum-like nonseparable structures in optical beams,” New J. Phys. 17, 043024 (2015).
[Crossref]

Opt. Commun. (1)

F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
[Crossref]

Opt. Express (1)

J. Wang, H. Huang, Y. Chen, H. Wang, S. Zhu, Z. Li, and Y. Cai, “Twisted partially coherent array sources and their transmission in anisotropic turbulence,” Opt. Express 26, 25974–25988 (2018).
[Crossref] [PubMed]

Opt. Lett. (8)

F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32, 3531–3533 (2007).
[Crossref] [PubMed]

R. Martinez-Herrero, P. M. Mejías, and F. Gori, “Genuine cross-spectral densities and pseudo-modal expansions,” Opt. Lett. 34, 1399–1401 (2009).
[Crossref] [PubMed]

F. Gori and M. Santarsiero, “Devising genuine twisted cross-spectral densities,” Opt. Lett. 43, 595–598 (2018).
[Crossref] [PubMed]

R. Borghi, “Twisting partially coherent light,” Opt. Lett. 43, 1627–1630 (2018).
[Crossref] [PubMed]

F. Gori and M. Santarsiero, “Twisted Gaussian Schell-model beams as series of partially coherent modified Bessel-Gauss beams,” Opt. Lett. 40, 1587–1590 (2015).
[Crossref] [PubMed]

R. Borghi, F. Gori, G. Guattari, and M. Santarsiero, “Twisted Schell-model beams with axial symmetry,” Opt. Lett. 40, 4504–4507 (2015).
[Crossref] [PubMed]

Z. Mei and O. Korotkova, “Random sources for rotating spectral densities,” Opt. Lett. 42, 255–258 (2017).
[Crossref] [PubMed]

X. Qian and J. H. Eberly, “Entanglement and classical polarization states,” Opt. Lett. 36, 4110–4112 (2011).
[Crossref] [PubMed]

Optica (1)

X. F. Qian, A. N. Vamivakas, and J. H. Eberly, “Entanglement limits duality and vice versa,” Optica 5, 942–947 (2018).
[Crossref]

Phys. Rev. A (1)

P. Chowdhury, A. S. Majumdar, and G. S. Agarwal, “Nonlocal continuous-variable correlations and violation of Bell’s inequality for light beams with topological singularities,” Phys. Rev. A 888, 195 (2013).

Phys. Rev. E (2)

S. A. Ponomarenko and G. P. Agrawal, “Asymmetric incoherent vector solitons,” Phys. Rev. E 69, 036604 (2004).
[Crossref]

S. A. Ponomarenko, “Twisted Gaussian Schell-model solitons,” Phys. Rev. E 64, 036618 (2001).
[Crossref]

Phys. Rev. Lett. (1)

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
[Crossref] [PubMed]

Rev. Theor. Sci. (1)

P. Ghose and A. Mukherjee, “Entanglement in classical optics,” Rev. Theor. Sci. 2, 274–288 (2014).
[Crossref]

Other (5)

J. W. Goodman, Introduction to Fourier Optics, 4th edition (W. H. Freeman, 2017).

A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integral and Series (Gordon and Breach, 1986, Vol. 2).

A. E. Siegman, Lasers (University Science Books, 1986).

L. Mandel L. and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
[Crossref]

G. Szegö, Orthogonal Polynomials, 4th edition (American Mathematical Society, 1975).

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Figures (2)

Fig. 1
Fig. 1 Plot of the intensity in Eq. (25) as a function of x/δ and y/δ, with a = b = α.
Fig. 2
Fig. 2 Contour plot of |μ10|, obtained from Eq. (26), as a function of (x1x2)/δ and (y1y2)/δ for a = b = α and different values of /δ: 0 (a); 0.5 (b); 1.0 (c); 1.5 (d).

Equations (46)

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W G ( r 1 , r 2 ) = I 0 exp ( r 1 2 + r 2 2 4 σ 2 ) exp [ ( r 1 r 2 ) 2 2 δ 2 ] ,
W T ( r 1 , r 2 ) = I 0 exp ( r 1 2 + r 2 2 4 σ 2 ) exp [ ( r 1 r 2 ) 2 2 δ 2 ] exp [ i k u ( x 1 y 2 x 2 y 1 ) ] ,
k | u | 1 / δ 2 ,
W ( r 1 , r 2 ) = p ( ρ ) H ( r 1 , ρ ) H * ( r 2 , ρ ) d ρ ,
H ( r , ρ ) = exp [ a ( r ρ ) 2 2 π i α ( s y t x ) ] ,
W ( r 1 , r 2 ) = p ( s , t ) exp { a [ ( x 1 s ) 2 + ( x 2 s ) 2 ] + 2 π i α t ( x 1 x 2 ) } × exp { a [ ( y 1 t ) 2 + ( y 2 t ) 2 ] 2 π i α s ( y 1 y 2 ) } d s d t .
p ( s , t ) exp [ b ( s 2 + t 2 ) ] ,
1 4 σ 2 = a b 2 a + b , 1 2 δ 2 = a 2 + π 2 α 2 2 a + b , k u = 4 π α a 2 a + b .
k | u | δ 2 1 = 2 π | α | a a 2 + π 2 α 2 1 = ( a π | α | 2 ) a 2 + π 2 α 2 0 ,
m = 0 M n = 0 N d m n s 2 m t 2 n ,
n = 0 N c n ( s 2 + t 2 ) n = n = 0 N c n k = 0 n ( n k ) s 2 k t 2 ( n k ) ,
W ( r 1 , r 2 ) = m = 0 M n = 0 N d m n W m n ( r 1 , r 2 ) ,
W mn ( r 1 , r 2 ) = s 2 m t 2 n exp [ b ( s 2 + t 2 ) ] × exp { a [ ( x 1 s ) 2 + ( x 2 s ) 2 ] 2 π i α s ( y 1 y 2 ) } × exp { a [ ( y 1 t ) 2 + ( y 2 t ) 2 ] 2 π i α t ( x 1 x 2 ) } d s d t .
W m n ( r 1 , r 2 ) = S m ( r 1 , r 2 ) T n ( r 1 , r 2 ) ,
S m ( r 1 , r 2 ) = s 2 m exp ( b s 2 ) exp { a [ ( x 1 s ) 2 + ( x 2 s ) 2 ] 2 π i α s ( y 1 y 2 ) } d s ,
T n ( r 1 , r 2 ) = t 2 n exp ( b t 2 ) exp { a [ ( y 1 t ) 2 + ( y 2 t ) 2 ] 2 π i α t ( x 1 x 2 ) } d t .
S 0 ( r 1 , r 2 ) = π 2 a + b exp [ a b 2 a + b ( x 1 2 + x 2 2 ) ] × exp [ a 2 2 a + b ( x 1 x 2 ) 2 π 2 α 2 2 a + b ( y 1 y 2 ) 2 ] × exp [ i 2 π a α 2 a + b ( x 1 y 1 x 2 y 2 x 1 y 2 + x 2 y 1 ) ] ,
W m n ( r 1 , r 2 ) = 4 π ( 1 ) m + n [ 4 ( 2 a + b ) ] n + m + 1 exp [ a b 2 a + b ( r 1 2 + r 2 2 ) ] × exp [ a 2 + π 2 α 2 2 a + b ( r 1 r 2 ) 2 + i 4 π a α 2 a + b ( x 1 y 2 x 2 y 1 ) ] × H 2 m [ π α ( y 1 y 2 ) + i a ( x 1 + x 2 ) 2 a + b ] H 2 n [ π α ( x 1 x 2 ) i a ( y 1 + y 2 ) 2 a + b ] ,
W m n ( r 1 , r 2 ) = A m n ( 1 ) m + n exp ( r 1 2 + r 2 2 4 σ 2 ) exp [ ( r 1 r 2 ) 2 2 δ 2 i k u ( x 1 y 2 x 2 y 1 ) ] × H 2 m [ c ( y 1 y 2 ) + i c + ( x 1 + x 2 ) 2 δ ] H 2 n [ c ( x 1 x 2 ) i c + ( y 1 + y 2 ) 2 δ ] ,
c + = cos ϕ ; c = sin ϕ ,
ϕ = arctan ( π α a ) .
I m n ( r ) = A m n ( 1 ) m + n exp ( r 2 2 σ 2 ) H 2 m ( i 2 c + x δ ) H 2 n ( i 2 c + y δ ) .
| μ m n ( r 1 , r 2 ) | = exp [ ( r 1 , r 2 ) 2 2 δ 2 ] | H 2 m [ c ( y 1 y 2 ) + i c + ( x 1 + x 2 ) 2 δ ] | | H 2 m ( i 2 c + x 1 δ ) H 2 m ( i 2 c + x 2 δ ) | × | H 2 n [ c ( x 1 x 2 ) + i c + ( y 1 + y 2 ) 2 δ ] | | H 2 n ( i 2 c + y 1 δ ) H 2 n ( i 2 c + y 2 δ ) | .
W 10 ( r 1 , r 2 ) = 2 A 10 exp ( r 1 2 + r 2 2 4 σ 2 ) exp [ ( r 1 r 2 ) 2 2 δ 2 i k u ( x 1 y 2 x 2 y 1 ) ] × { 1 2 [ c ( y 1 y 2 ) + i c + ( x 1 + x 2 ) 2 δ ] 2 } ,
I 10 ( r ) = 2 A 10 exp ( r 2 2 σ 2 ) ( 1 + 4 c + 2 x 2 δ 2 ) ,
| μ 10 ( r 1 , r 2 ) | = exp [ ( r 1 , r 2 ) 2 2 δ 2 ] | 1 2 [ c ( y 1 y 2 ) δ + i c + ( x 1 + x 2 ) δ ] 2 | ( 1 + 4 c + 2 x 1 2 δ 2 ) ( 1 + 4 c + 2 x 2 2 δ 2 )
W ¯ m n ( r 1 , r 2 ) = A m n ( 1 ) m + n exp ( r 1 2 + r 2 2 4 σ 2 ) exp [ ( r 1 r 2 ) 2 2 δ 2 i k u ( x 2 y 2 x 1 y 1 ) ] × H 2 m [ c ( y 1 y 2 ) + i c + ( x 1 + x 2 ) 2 δ ] H 2 n [ c ( x 1 x 2 ) + i c + ( y 1 + y 2 ) 2 δ ] ,
H ( r , ρ ) = exp [ a ( r , ρ ) 2 + 2 π i α ( x s ) ( y t ) ] .
W m n ( r 1 , r 2 ) = s 2 m t 2 n e b ( s 2 + t 2 ) exp [ a ( r 1 ρ ) 2 + 2 π i α ( x 1 s ) ( y 1 t ) ] × exp [ a ( r 2 ρ ) 2 2 π i α ( x 2 s ) ( y 2 t ) ] d s d t ,
W ¯ m n ( r 1 , r 2 ) = s 2 m t 2 n e b ( s 2 + t 2 ) exp { a [ ( x 1 s ) 2 + ( x 2 s ) 2 ] 2 π i α t ( x 1 x 2 ) } × exp { a [ ( y 1 t ) 2 + ( y 2 t ) 2 ] 2 π i α s ( y 1 y 2 ) } d s d t .
W m n ( r 1 , r 2 ) = W ¯ m n ( r 1 , r 2 ) exp [ 2 π i α ( x 1 y 1 x 2 y 2 ) ] ,
W m n ( r 1 , r 2 ) = A m n ( 1 ) m + n exp ( r 1 2 + r 2 2 4 σ 2 ) exp [ ( r 1 r 2 ) 2 2 δ 2 i k v ( x 2 y 2 x 1 y 1 ) ] × H 2 m [ c ( y 1 y 2 ) + i c + ( x 1 + x 2 ) 2 δ ] H 2 n [ c ( x 1 x 2 ) + i c + ( y 1 + y 2 ) 2 δ ] ,
k v = 4 π α a 2 a + b + 2 π α = 2 π b α 2 a + b .
k v δ 2 = π b α a 2 + π 2 α 2 ,
S m ( r 1 , r 2 ) = exp [ a b 2 a + b ( x 1 2 + x 2 2 ) ] exp [ g ( x 1 x 2 ) 2 2 π i γ 12 x 12 ] × ( ζ + x 12 ) 2 m e ( 2 a + b ) ζ 2 e 2 π i γ 12 ζ d ζ ,
g = a 2 2 a + b , x 12 = a ( x 1 + x 2 ) 2 a + b , γ 12 = α ( y 1 y 2 ) .
S m ( r 1 , r 2 ) = exp [ a b 2 a + b ( x 1 2 + x 2 2 ) ] exp [ g ( x 1 x 2 ) 2 2 π i γ 12 x 12 ] × k = 0 2 m ( 2 m k ) x 12 k ζ 2 m k e ( 2 a + b ) ζ 2 e 2 π i γ 12 ζ d ζ ,
S m ( r 1 , r 2 ) = π 2 a + b exp [ a b 2 a + b ( x 1 2 + x 2 2 ) ] exp [ g ( x 1 x 2 ) 2 2 π i γ 12 x 12 ] × k = 0 2 m ( 2 m k ) x 12 k ( 2 π i ) 2 m k d 2 m k d γ 12 2 m k [ e π 2 γ 12 2 / ( 2 a + b ) ] .
d n d x n e x 2 = ( 1 ) n e x 2 H n ( x ) ,
S m ( r 1 , r 2 ) = π 2 a + b exp [ a b 2 a + b ( x 1 2 + x 2 2 ) ] × exp [ g ( x 1 x 2 ) 2 2 π i γ 12 x 12 π 2 γ 12 2 2 a + b ] × k = 0 2 m ( 2 m k ) x 12 k ( 2 i 2 a + b ) 2 m k H 2 m k ( π γ 12 2 a + b ) .
S m ( r 1 , r 2 ) = π 2 a + b exp [ a b 2 a + b ( x 1 2 + x 2 2 ) ] × exp [ a 2 ( x 1 x 2 ) 2 + π 2 α 2 ( y 1 y 2 ) 2 2 a + b ] × exp [ i 2 π a α 2 a + b ( x 1 y 1 x 2 y 2 x 1 y 2 + x 2 y 1 ) ] F m ( x 1 + x 2 , y 1 y 2 ) ,
F m ( ξ , η ) = k = 0 2 m ( 2 m k ) ( a ξ 2 a + b ) k ( i 2 2 a + b ) 2 m k H 2 m k ( π α η 2 a + b )
k = 0 n ( n k ) ( 2 y ) n k H k ( x ) = H n ( x + y ) ,
F m ( ξ , η ) = [ 1 4 ( 2 a + b ) ] m H 2 m ( π α η + i a ξ 2 a + b ) .
T n ( r 1 , r 2 ) = π 2 a + b exp [ a b 2 a + b ( x 1 2 + x 2 2 ) ] × exp [ a 2 ( y 1 y 2 ) 2 + π 2 α 2 ( x 1 x 2 ) 2 2 a + b ] × exp [ i 2 π a α 2 a + b ( y 1 x 1 y 2 x 2 y 1 x 2 + y 2 x 1 ) ] F n * ( y 1 + y 2 , x 1 x 2 ) .
W m n ( r 1 , r 2 ) = π ( 1 ) m + n 4 m + n ( 2 a + b ) m + n + 1 exp [ a b 2 a + b ( r 1 2 + r 2 2 ) ] × exp [ a 2 + π 2 α 2 2 a + b ( r 1 r 2 ) 2 + i 4 π a α 2 a + b ( x 1 y 2 x 2 y 1 ) ] × H 2 m [ π α ( y 1 y 2 ) + i a ( x 1 + x 2 ) 2 a + b ] H 2 n [ π α ( x 1 x 2 ) i a ( y 1 + y 2 ) 2 a + b ] .

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