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.
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In the classical theory of coherence , 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 . A model source used in countless instances is the planar Gaussian Schell-model (GSM) source, whose CSD has the form1]. The CSD in Eq. (1) turned out to be extremely useful as a model of partially coherent circularly symmetric sources.
In a celebrated paper , 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
A basic result of the Simon-Mukunda analysis was that the twist parameter has to satisfy the following inequality:2] by imposing the non-negative definiteness property that any CSD must possess .
The source devised by Simon and Mukunda was soon realized experimentally  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 , 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 beams4] 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 .
It is interesting to note that the integral representation for W can be given the following form:Eq. (4) with the kernel given by Eq. (5) is separable.
In particular, in  the TGSM source was obtained on assuming p of the form4] 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 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 form18,19].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 form16].
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,4] used to synthesize the CSD of Eq. (2).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
The intensity distribution across the source plane is easily evaluated on letting r1 = r2 = r in Eq. (19). This gives
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 ), so that Imn is everywhere nonnegative.
As for the degree of coherence , its modulus is given by
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
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 (x1 − x2)/δ and (y1 − y2)/δ, obtained for different values of x̄ = (x1 + x2)/2.
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:
As a result, the partial transposition converts the twist phase component x1y2 − x2y1 of a higher-order TGSM CSD into the astigmatism element x2y2 − x1y1. Therefore, W̄mn 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 , 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 byEq. (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, 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  for the particular case of m = n = 0.
After some manipulations, the following beautifully simple result is obtained:Eqs. (27) and (31) we have Eq. (8)],
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  to the whole classes of higher-order twisted and astigmatic GSM sources.
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 , 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
As shown by Eq. (35), we have to Fourier transform the power of a binomial times a Gaussian function. Using Newton’s formula we write22], gives Eq. (36) we can give Sm the following more explicit form:
The latter is a polynomial of order 2m that, thanks to the following identity (see Eq. 220.127.116.11 of ):
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