Pseudo-Schell model sources

: Partially coherent pseudo-Schell model sources are introduced and analyzed. They present radial symmetry and coherence characteristics depending on the diﬀerence between the radial distances of two points from the source center. As a consequence, all points belonging to circles centered on the symmetry center of the source are perfectly correlated. We show that such sources radiate ﬁelds with peculiar behaviors in paraxial propagation. In particular, when compared to beams produced by Gaussian Schell-model sources with comparable coherence parameters, the irradiance can present sharper proﬁles and higher peak values and a better beam quality parameter. Furthermore, when a pseudo-Schell model source presents a vortex, the propagated beam preserves a null of the intensity along its axis.

are presented, together with some global parameters, such as the beam quality and the beam size evolution.

Pseudo Schell-model sources
The CSD function W (ρ 1 , ρ 2 , 0) is an appropriate tool for describing in the space-frequency domain stochastic, statistically stationary up to second order, light sources [1], and expresses the correlation of the field at two arbitrary positions across the source plane. Here, we denote by ρ j (with j = 1, 2) the transverse position vectors of the two points and suppose that the source lies on the plane z = 0. The dependence on the temporal frequency is omitted for brevity. In the following, both rectangular (ξ, η) and polar (ρ, ϕ) coordinates will be used for the vector ρ.
The CSD of a physically realizable source must satisfy the non-negativeness condition [1]. While directly checking such condition for a given cross-spectral density is often not an easy task, a necessary and sufficient condition to devise genuine CSDs was established some years ago [33,34]. In the following, use will be made of such a rule.
Let us consider a light source described by a CSD function of the following form: where * denotes complex conjugation, τ (ρ) is a general complex function, and with k being the wavenumber, υ a vector with polar coordinates (υ, ψ), and H an arbitrary function of υ. In physical terms, the CSD in Eq. (1) is obtained as the incoherent superposition of a continuous set of perfectly coherent beams (parametrized by the vector υ), whose amplitudes are multiplied by the transmission function of a conical lens [47,48]. Due to the way in which W (ρ 1 , ρ 2 , 0) has been constructed, it represents a genuine CSD matrix [34,49,50]. Substitution of Eq. (2) into Eq. (1) gives where g (ρ) = 2π and J ν (·) is the Bessel function of first kind and order ν [51]. It can be observed that the irradiance of the source, I(ρ, 0) = W (ρ, ρ, 0), depends on τ (ρ), while its coherence properties are determined by the function g, since [1] whose absolute value only depends on the difference between the radial distances of the considered points from the source center. Without loss of generality, in deriving Eq. (5) we assumed g(0) = 1, obtained by suitably normalizing H(υ). Sources of this kind will be called pseudo Schell-model sources.
Note that Eqs. (3) and (4) define a whole class of sources whose coherence properties depend only on the radial distance difference of the two considered points (except for a possible phase factor). A new family of sources of this class is generated just by changing H(υ). Once a function H(υ) is chosen, different behaviors can be found depending on the shape of τ (ρ).

Pseudo Schell-model sources with Gaussian degree of coherence
To illustrate some properties of pseudo Schell-model sources, let us consider a particular choice of H(υ), i.e., where the factor kδ c /2 √ π has been introduced in order to get g(0) = 1, and δ c is a real positive parameter. The latter defines the coherence properties of the source and, as it will be shortly seen, represents a measure of the width of a ring concentric to the source center where the degree of coherence is significant. We will refer to δ c as the coherence parameter of the source. Substituting from Eq. (6) into Eqs. (4) and (5) yields [51] | µ (ρ 1 , The absolute value of the degree of coherence reaches its maximum value of 1 for ρ 1 = ρ 2 , that is, along rings concentric to the source center, and monotonically decreases on increasing | ρ 2 − ρ 1 |. Figure 1 shows the shape of the absolute value of the degree of coherence as a function of ρ 1 for different values of ρ 2 . A two-dimensional plot of | µ| is shown in Fig. 2 as a function of ρ 1 for different values of ρ 2 . It presents circular symmetry and reduces to a twodimensional Gaussian function when ρ 2 = 0. In such a case δ c coincides with the standard deviation of the Gaussian, so that its squared modulus can be taken as a measure of the coherence area of the source around that point. On moving ρ 2 away from the center, | µ| takes a donut-like shape with radius ρ 2 and width of the order of δ c . When ρ 2 is significantly greater than δ c the coherence area of the source can be taken of the order of 2πρ 2 δ c . Once the analytical expression of H(υ) is chosen, different sources are obtained on changing the modulating function τ (ρ). To get more information about properties that can be found for these families of sources, we choose τ as a Laguerre-Gaussian function [52] in which the Laguerre polynomial has order zero, that is, where A 0 is an amplitude factor, m an integer, and w 0 a positive quantity related to the source width. It reduces to a Gaussian function when m = 0 and, when m 0, presents a donut-like shape and a phase vortex of charge m. We will denote by W m (ρ 1 , ρ 2 , 0) the resulting cross spectral densities, which take the form with I 0 = | A 0 | 2 . For brevity, we will call VGPSM (for Vortex Gaussian Pseudo-Schell-Model) sources, the sources described by the CSD in Eq. (9) or, for the specific case of m = 0, GPSM (for Gaussian Pseudo-Schell-Model) sources. The results obtained for sources of this kind will be compared to the ones obtained from the corresponding (standard) Gaussian Schell-model sources with a vortex, i.e., [18,53] where the degree of coherence depends on |ρ 2 − ρ 1 |, instead of ρ 2 − ρ 1 . Such sources will be denoted by GSM (when m = 0) or VGSM (for Vortex GSM, when m 0) sources.

Propagation
To study the characteristics of the field upon free-space propagation we assume paraxial approximation and propagation direction along the z axis. Then, the Fresnel diffraction integral allows to obtain the CSD at any plane z = constant as [1] where r j = (x j , y j ) = (r j , θ j ) (with j = 1, 2) are two typical position vectors across the transverse plane.
For the case of the CSD in Eq. (9), the integral in Eq. (11) gives which allows to evaluate the irradiance profile and the degree of coherence of the propagated field across any transverse plane. Eq. (12) will be numerically evaluated for the cases m = 0 and m = 1 and results will be shown in Section 5.
An alternative way to characterize the properties of the propagated field is convenient. One possibility is by means of some global parameters, such as the beam width and the beam divergence, that can be calculated from the expression of CSD of the source, i.e. at z = 0. These parameters can be expressed in terms of the second order moments of the beam, that in Cartesian coordinates read [54] being P the total power of the beam, i.e., Then, the rms width of the beam across a transverse plane and in the far field can be defined as ξ 2 + η 2 = ξ 2 + η 2 and u 2 + v 2 = u 2 + v 2 , respectively, while the mixed second order moment ξu + ηv = ξu + ηv accounts for the curvature of the wavefront. These parameters allow to evaluate beam quality parameter defined as [54,55] that is invariant under propagation through rotationally symmetric ABCD optical systems and reaches the minimum value of 1/k 2 for a coherent fundamental Gaussian beam. This parameter is related to the beam propagation factor, M 2 , through the expression M 2 = k √ Q. The Q parameter is a measure of the beam quality, where a better beam quality is associated with a smaller divergence, for a fixed beam width at the source plane [54]. Due to the invariant character of Q, it can be evaluated at any transverse plane and this calculation is usually easier at the beam waist (located at z = 0 for our source), where ξu + ηv = 0.
The expressions of the second-order moments and the quality factor can be performed analytically, after substituting the CSD given by Eq. (9) into Eqs. (13). In particular, the rms beam width turns out where z R = kw 2 0 /2 is the Rayleigh length pertaining to a coherent Gaussian beam with waist size w 0 [52].
By substituting the calculated second order moments into Eq. (15), the beam quality factor is obtained as The rms beam width and the quality factor Q can be also calculated for the equivalent standard Schell-model source described by Eq. (10). Following the same procedure, it turns out that [56,57] and The obtained expressions are very similar, the only difference being the factor 2 multiplying the ratio w 2 0 /δ 2 c in the latter. This means that a Gaussian pseudo-Schell-model beam has a better quality and diverges less than the corresponding standard GSM beam. More precisely, with the same parameters, it diverges like a standard GSM beam with coherence area increased by a factor 2. Figure 3 shows the evolution of the rms beam width along the z axis and the beam quality factor Q for different values of the topological charge m (including the case m = 0). The results are compared to those of a GSM source with the same waist size w 0 and coherence parameter δ c .
Additional advantages of the sources introduced here over the equivalent GSM sources are related to the irradiance profile in free space propagation. Then, the evolution of the irradiance profile will be analyzed in Section 5. Numerical results will be shown in detail for the cases m = 0 and m = 1.

Gaussian pseudo Schell-model sources
In this case, the irradiance at the source plane corresponds to that of a Gaussian with spot size w 0 . It is well known that the irradiance profile of a standard GSM beam remains invariant, except for a scale factor, in free space propagation [28,29]. However, the irradiance profile of the generated beam by a GPSM source changes upon free propagation in a way that depends on the selected coherence parameter δ c in Eq. (7).
As an example, Fig. 4 shows the evolution of the irradiance at several propagation distances for δ c = 0.2w 0 . A somewhat unexpected behavior is found: in a certain range of propagation  distances, the irradiance profile becomes narrower and presents a maximum value higher than that at the beam waist (note that the vertical scale in each part of Fig. 4 is different).
In order to analyze in a deeper detail such a behavior, the transverse shape of the irradiance (normalized to its maximum value) as a function of the radial coordinate ρ is shown in Fig. 5(a). In general, it is Gaussian only at the source plane but it evolves during propagation in a way depending on the coherence parameter δ c . In particular, for coherence parameters smaller than the waist size and short propagation distances, it presents a shoulder. But the most significant aspect is that during propagation the profile becomes sharper than that at the source plane. For the chosen values of the parameters, the maximum sharpness is reached at a distance of the order of z R /5. Figure 5(b) reports the transverse full width at half maximum (FWHM) as a function of the propagation distance for different values of the coherence parameter δ c . It clearly shows the sharpening of the transverse irradiance occurring during propagation when δ c is lower than w 0 , thus confirming what we already noticed in Fig. 5(a). This minimum FWHM is smaller and nearer to the source plane for lower and lower values of the coherence width across the source.
It should be stressed that such a result does not contradict the fact that the width of the beam, defined in terms of the second-order moment of the intensity profile, presents its minimum value at the source plane and then increases monotonically with z, as shown in Sec. 4 (see Eq. (16)). Actually, the smaller the coherence parameter, the faster the increasing of the rms width of the beam profile. The variance of the profile, indeed, also depends on the irradiance at points very far away from the center and, in some cases does not carry the same information as the transverse FWHM plotted in Fig. 5(b). For example, the curves in Fig. 5(a) corresponding to z = 0 and to z = 0.7 z R seem to correspond to quite similar values of the FWHM. For the latter, however, the tail is higher, and this gives rise to a greater value of the variance. Figure 6 shows the behavior of the maximum value of the transverse irradiance profile (which, in this case, is found at the beam center) as a function of the propagation distance, for several  values of the coherence parameter δ c . It can be noted that for large values of δ c the maximum irradiance reduces monotonically with propagation, in a way quite similar to that occurring for a GSM beam. On the contrary, for lower values of the coherence parameter (of the order or smaller than w 0 ), the axial irradiance presents a peak, at distances of the order of z R /2 or less from the source, where its value is greater than that at the source plane. The peak value of the maximum irradiance increases and the distance at which its maximum is reached decreases on decreasing the coherence parameter δ c .
Similar sharpening of the beam profile have been already described for some partially coherent sources with non shift invariant coherence characteristics [6,9,44].

Vortex Gaussian pseudo Schell-model sources with m=1
In the next example we will consider a Gaussian vortex with topological charge m = 1, i.e., the τ (ρ) function in Eq. (8) evaluated for m = 1.
The irradiance profile at the source plane in this case corresponds to the shape of a usual coherent Laguerre-Gaussian beam of zero order and with topological charge m = 1 as it can be seen in the upper left part of Fig. 7. However, due to the coherence characteristics imposed to the source, the irradiance profile of the propagated beam may present some interesting feature.
First of all, the irradiance is zero everywhere along the beam axis, as it can be seen from Eq. (12), because the Bessel functions J m in the integral vanish for r 1 = 0 and r 2 = 0 (if m 0), so that the donut-like shape is preserved during propagation. This is not the case for usual partially coherent hollow beams of the Schell-model type [58][59][60]. Furthermore, as we shall see, the irradiance profile of the propagated beam at certain planes may be much narrower than that across the source. As a consequence, the maximum irradiance value of the propagated beam may be much higher than that at the source plane. The evolution of the transverse profile during propagation is quite complex and is shown in Fig. 7. Close to the plane z = 0, the profile is very similar to that of the source but after a certain propagation distance a second, smaller donut appear around the beam axis, which gradually becomes predominant. This behavior is made evident in Fig. 8, where the transverse profile is shown as a function of the radial coordinate for different values of δ c . It appears that the transverse size of the second donut depends on the coherence parameter of the source. A consequence of such a behavior is that the FWHM of the beam (defined as the largest value of the radius at which the irradiance reaches the half of its maximum) is not monotonic with the propagation distance (see Fig. 9) but presents an abrupt reduction at a propagation distance that depends on the coherence parameter. This effect becomes more and more evident on reducing the value of the coherence parameter and could be used in particle trapping, due to the small confinement region and the large gradient of the irradiance in the region where the inner donut appears.
The behavior of the maximum irradiance of the beam profile during propagation is shown in Fig. 10. It can be traced back to the one of the beam width, being related to the peculiar way in which the profile of the irradiance evolves for relatively low values of the coherence parameter δ c , as it was shown in Figs. 7 and 10.

Synthesis
The CSD function of any source can be expressed by means of the following Mercer expansion: where λ n are positive coefficients (the eigenvalues) and the functions Ψ n (ρ) (the modes) form a set of orthonormal functions [1,61]. Sometimes, expansions of the form (20) are also possible with positive coefficients and nonorthogonal functions. In such cases, the latter are called pseudo-modes [34,62]. In both cases, the physical meaning of the above expansion is that a partially coherent source can be thought of as the superposition of a set of mutually uncorrelated, perfectly coherent, and suitably weighted fields. Of course, this also represents a way to physically synthesize partially coherent source [19,22,23,26,40,63], especially when they are not of the Schell-model type [6,44,45]. The superposition usually involves an infinite number of coherent fields but for practical applications a finite number is often enough for a good representation of the CSD function [27,44]. Using a known result obtained for the case of a 1D GSM source [27,64], the following expansion can be derived for the CSD in Eq. (9): where and being H n (·) the Hermite polynomial of order n [51], and In general, the functions ψ m,n are not orthogonal in 2D, and they neither are normalized. But we can define a set of pseudo-modes by introducing the normalized functions where β m,n is such that and 2 F 1 is the generalized hypergeometric function [65]. Therefore, a pseudo-modal expansion for W m can be written in the form (see Eqs. (21) and (25)) where the coefficients λ m,n are λ m,n = γ n 2 n n! β 2 m,n .
For the case of GPSM source, the behavior of the coefficients λ 0,n for n = 0 to n = 10 is shown in Fig. 11 as a function of the ratio of coherence parameter δ c to the beam width at its waist. It can be observed that when this ratio is large, for example δ c /w 0 > 4, the source is highly coherent and then the contribution of only two modes could be enough to represent the source. However, when this ratio decreases so the source becomes more and more incoherent, it is necessary to take into account the contribution of more and more pseudo-modes. For example, in the inset can be observed that for a ratio δ c /w 0 < 1/2 more than ten modes are needed to accurately represent the source.
For the case of VGPSM, the coefficients λ m,n present a similar behavior. However, for a given value of the δ c /w 0 , a higher number of modes is necessary to accurately represent the source, Fig. 11. Coefficients of the pseudo-modal expansion for the GPSM source as a function of the coherence parameter δ c .  specially for lower and lower values of such a ratio. This can be deduced from observing Fig. 12 where it can be observed that, for example, the coefficient λ 1,50 shows a greater value for δ c /w 0 around 0.05 than the lower order modes. Figure 13 shows the set of the first 21 (41) coefficients for GPSM (VGPSM) sources with different values of δ c /w 0 ratio. It can be noted that for highly coherent GPSM and VGPSM sources, the most significant contribution is that of the zero-th order mode. On the other hand, for lower and lower coherence, the most important contribution to the source comes from higher and higher order modes.

Conclusion
In the present work, a type of partially coherent sources is proposed and analyzed. Sources of this kind present a degree of coherence that is radially shift invariant, so that points at the same distance from the source center are perfectly coherent, but the coherence generally decreases for points on circles of different radii. This radial shift-invariance makes the coherence area growing with the distance from the source center. Pseudo-Schell model sources with Laguerre-Gaussian irradiance profile and Gaussian degree of coherence have been studied in detail. Beams radiated from sources of this kind show better beam quality parameters than equivalent Schell-model beams with comparing parameters. An interesting property of the propagated beams is that, at certain distances, the FWHM of their irradiance profile can be significantly smaller than that at the beam waist. Furthermore, when the irradiance profile presents a null at the source center, the irradiance of the radiated beam remains equal to zero at the beam axis for any propagation distance, giving rise to a dark hollow partially coherent beam. The above properties could be exploited for particle trapping, due to the great irradiance gradient obtained in the radial direction. Finally, a pseudo-modal expansion of the CSD has been derived, which allows to build up the source with an adequate superposition of pseudo modes.