1. Introduction

In the realm of optical communication, the quest for higher data transfer rates has led researchers to explore innovative solutions beyond conventional techniques which primarily rely on intensity, wavelength, and polarization modulation for data encoding [1]. A promising solution involves utilizing the orbital angular momentum (OAM) of light. OAM, arising from the helical phase front of the light beam, provides an extra degree of freedom, in addition to the traditional schemes. This property enables the encoding of multiple information channels onto a single beam of light via different OAM values. As a result, OAM-based communication systems have the potential to achieve unprecedented data rates, making them particularly useful for applications requiring high capacity data channels [2].

However, over significant propagation distances in free space, atmospheric turbulence induces fluctuations in the field, thereby degrading the quality of the optical beams. Specifically, turbulence effects on OAM-carrying beams result in a broadening of the OAM spectrum, consequently limiting the capacity of channels due to increased cross-talk. Addressing this challenge necessitates a comprehensive fundamental study of the effects of the fluctuations of refractive index on the light beam, particularly its OAM spectrum. However, owing to the complex nature of turbulence, researchers have encountered considerable difficulties in optimizing the interaction of light with random media. To circumvent this challenge, structured light fields with non-trivial phase, amplitude, polarization, and coherence have been investigated as a means to improve propagation characteristics.

It has been demonstrated that partially coherent beams (PCBs) exhibit higher resistance against turbulence fluctuations [3–5]. This suggests that we can mitigate the degrading effects of turbulence on the OAM spectrum by reducing the spatial coherence of beams. Transitioning from fully coherent to partially coherent can be accomplished by randomizing a characteristic feature of the light field. However, this randomization typically also affects the phase of light, leading to the redistribution of power in the OAM spectrum, which results in cross-talk between adjacent modes. Therefore, reduction of coherence has advantages and disadvantages for the reliability of data channels in propagation through turbulence. Detailed investigations are required to determine if the advantages can be leveraged to overcome the disadvantages.

Several years ago, Gbur noted that partially coherent OAM beams can be broken into three fundamental classes [6] based on how the OAM is distributed within their cross section. These classes are: (a) PCBs produced by randomizing the position of the beam axis, resulting in a Rankine vortex [7,8]; such beams are now known as Rankine model beams [9]. (b) PCBs generated by introducing a twist phase into the spatial coherence of the beam, known as twisted Gaussian-Schell model beams [10]. (c) PCBs generated with a separable vortex phase, making them fully coherent in the azimuthal direction [11]. We consider a special class of such beams that are now referred to as circularly coherent beams [12,13]. A circularly coherent vortex beam can be created via an ensemble of coherent vortex beams with varying focal distances, resulting in a field that is partially coherent only in the radial direction.

In this paper, we conduct an analytic study of the turbulence propagation of OAM-carrying beams within the aforementioned three classes of partially coherent vortex beams. We analyze the behavior of the OAM spectrum under the influence of turbulence, and consider the turbulence resistance of each class. The results provide guidelines for selecting the appropriate type of PCB to enhance the reliability of data transmission in optical communication systems, and indicate directions for future research.

2. Partial coherence and orbital angular momentum

We begin by reviewing the relevant theory relating to partially coherent beams and the corresponding OAM spectra of such beams.

Early coherence theory focused on space-time correlation functions that characterized both the spatial and temporal coherence simultaneously. However, researchers in recent years have focused more on the spatial coherence properties of light. In this case, it is natural to use the cross-spectral density (CSD) function $W(\boldsymbol {\rho }_1 , \boldsymbol {\rho }_2,\omega )$ that characterizes the spatial correlation between two points at the frequency $\omega$; it can be written as an average over an ensemble of monochromatic fields [14],

The cross-spectral density can be decomposed in a basis of azimuthal modes, often called the spiral spectrum, of the form [15]

Here, $\rho _1$, $\rho _2$ are radial coordinates and $\phi _1$ and $\phi _2$ are azimuthal coordinates. The quantity $W_{lm}(\rho _1,\rho _2)$ characterizes the spatial correlations between different azimuthal modes of order $l$ and $m$. Because a pure spiral mode of the form $\exp [il\phi ]$ is also a pure OAM state [16], we may also consider the spiral spectrum as a decomposition of the beam into different OAM modes.

If we consider just a single term of Eq. (2) with $l=m$ and let $\boldsymbol {\rho }_1=\boldsymbol {\rho }_2=\boldsymbol {\rho }$, we get the transverse intensity of the OAM mode of order $m$, denoted by $I_m(\rho )$. By integrating this quantity over the detector aperture, we can determine the total measured intensity of the $m$th mode of orbital angular momentum,

For more details about OAM spectra in partially coherent fields, see Korotkova and Gbur [18]. In the next section, the extended Huygens-Fresnel principle is introduced as our method of propagating the CSD function through turbulence. Following this, we consider the three fundamental classes of partially coherent vortex beams and the effect of atmospheric turbulence on their mode spectra.

3. Propagation through turbulence

The propagation of PCBs through atmospheric turbulence is often calculated using the venerable extended Huygens-Fresnel principle (eHF) [19], which can be applied to coherent or partially coherent fields. If we use $\boldsymbol {\rho }_1$ and $\boldsymbol {\rho }_2$ to label the positions at the source and $\mathbf{r}_1$ and $\mathbf{r}_2$ to label the positions at a detector plane a distance $L$ from the source, the extended Huygens-Fresnel principle is of the form,

The general form of the resulting phase perturbation is quite complicated and typically can only be evaluated numerically; in order to make the calculations tractable, a quadratic approximation is employed, leading to the expression [21]

The parameter $C_n^2$ is a measure of turbulence strength and is in the range of $10^{-17} \mbox { m}^{-2/3}$ to $10^{-13}\mbox { m}^{-2/3}$ from the weaker to stronger strengths. We will consider the inner scale $l_0$ as $2\mbox { mm}$ and the outer scale $\mathcal {L}_0$ as $15\mbox { m}$ throughout.

We now turn to the three fundamental classes of partially coherent vortex beams described earlier to explore the effect of the atmosphere on the OAM spectrum of each class. In this study, we restrict ourselves to beams with OAM equivalent to a vortex of order 1. To simplify our calculations, we employed a modified version of the extended Huygens-Fresnel principle that was recently introduced [23]; this version is mathematically equivalent to the standard eHF, but can be evaluated with only two integrations.

4. Rankine model vortex beams

One direct approach for generating partially coherent beams is to create an ensemble of beams with one or more randomly varying parameters; this approach was systematized by Gori and Santarsiero [24]. In constructing partially coherent vortex beams, it is natural to look at ensembles for which every member is a deterministic vortex beam with the same vortex order. The earliest example of such a beam [25] uses the central position $\boldsymbol {\rho }_0$ of the beam as the random variable, with a probability density $P(\boldsymbol {\rho }_0)$; such beams were initially called “beam wander model” beams. The CSD of such beams may be written in the form [26]

 

Fig. 1. The Rankine model source. (a) Deviation from the central axis denoted by $\boldsymbol {\rho }_0$, (b) Combination of multiple beams with varying beam axes.

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These beams are now referred to as Rankine vortex beams, as it has been demonstrated that their normalized OAM flux density takes on the form of a Rankine vortex [8].

4.1 Rankine vortex beams at source

Let us construct our Rankine vortex beam from an ensemble of $1$st order vortex Gaussian beams in the source plane of the form,

The probability density $P(\boldsymbol {\rho }_0)$ of the beam axis position is taken to be a Gaussian distribution,

The parameter $\delta$, representing the RMS width of the Gaussian distribution of beam axis positions, may be understood as an inverse measure of spatial coherence. Specifically, $\delta =0$ signifies a fully coherent case where the beam axis does not wander; with an increase in $\delta$, the coherence decreases. By substituting from Eqs. (10) and (11) into Eq. (9) and evaluating the integral over $\boldsymbol {\rho }_0$, the CSD of the Rankine vortex beam in the source plane may be written as

As Eq. (12) is a quadratic formula in terms of the complex position variable $z_2$, we expect there will be two solutions for which $w_0=0$; these solutions will represent two distinct vortices within the transverse plane. This already suggests that the OAM spectrum of such a partially coherent beam will be significantly distorted from the coherent case.

4.2 Rankine vortex beams in free space

In this paper the term “free space” means “in the absence of turbulence effects.” The propagation of the CSD function in free space may be derived from the traditional Huygens-Fresnel diffraction formula,

In the above expression, the quantities $w_L$ and $\sigma _s^\prime$ refer to the beam sizes associated with the initial coherent beam and the partially coherent Rankine model beam during free space propagation to distance $L$, respectively. Additionally, $\sigma _g^\prime$ and $R_L^\prime$ represent the coherence width and the average radius of curvature of the Rankine model beam, respectively, after propagation to distance $L$ in free space. These quantities are derived as

In the above expressions, $L_0=kw_0^2/2$ is the Rayleigh range of the coherent beam. The parameter $\beta _L$ and the weight factors $F_1$ and $F_2$ are introduced as

It is to be noted that the beam has retained its overall mathematical structure on free-space propagation, as can be seen by comparing Eqs. (12) and (19).

4.3 Rankine vortex beams in turbulence

The propagation of the Rankine vortex beam in turbulence can be found by evaluating Eq. (5) for the source given in Eq. (12); after some effort, the CSD function through turbulence takes the form

The weight factors $T_i$ with $i=0,1,2$ are obtained as

The CSD in turbulence given in Eq. (27) recovers the CSD in free space expressed in Eq. (19) in the limit of $Q(L)\rightarrow 0$, which results in $\alpha =1$. It is to be noted that the cross-spectral density in turbulence again has a very similar form to the propagated cross-spectral density in free space.

4.4 OAM spectrum of Rankine vortex beams

Having the cross-spectral density characterized in both free space and turbulence, we can now calculate the OAM spectrum for each case and evaluate the turbulence resistance.

Starting with the free space case, we substitute the CSD of Eq. (12) into the angular integrals of Eq. (3); the mode intensity $I_m(\rho )$ can be expressed in the form (see [27], Chap. 6)

To determine the overall OAM spectrum in the beam, it is necessary to integrate across the entire detector aperture, as in Eq. (4). In this manuscript, this radial integral will be evaluated numerically.

Given that the propagation in free space does not alter the distribution of energy among different OAM modes, we can affirm that the OAM spectrum of the source, as derived from Eq. (38), can be treated as the OAM spectrum detected in free space as well, provided we consider an infinite aperture, i.e. $R_d\rightarrow \infty$.

Following similar calculation steps, the OAM mode distribution at a radial distance $r$ on propagation through turbulence takes the form,

We consider the case of an infinite aperture in Eq. (4), so that $R_d\rightarrow \infty$.

Figure 2 displays the resulting OAM spectrum in free space and turbulence for illustrative coherence widths of the Rankine vortex source. The beam is propagated $1$ km through a turbulent media with the strength $C_n^2=10^{-14}$ m$^{-2/3}$. It can be seen that, as spatial coherence decreases, the two spectra become increasingly similar, suggesting that the beam OAM spectrum is less affected by turbulence. However, the mode spectrum for $\sigma _g=0.5\mbox { cm}$ is overall much wider than the more coherent $\sigma _g=2\mbox { cm}$ case, indicating that this resistance is a pyrrhic victory. This seemed like the most likely outcome at the beginning of these studies, considering the mode spectrum of a partially coherent Rankine vortex is much wider than that of a coherent vortex. There is still the possibility that a Rankin vortex beam has its OAM spectrum spread more slowly than a coherent beam, and we will see that this is the case in Section 7..

 

Fig. 2. OAM spectrum of Rankine vortex beam at the source and in $L=1000$ m propagation through turbulence with the strength $C_n^2=10^{-14}$ m$^{-2/3}$ for various coherence widths $\sigma _g = 0.5 , 1 , 2$ cm measured on an infinitely sized detector with $R_d\rightarrow \infty$. The beam size at the source is $\sigma _s=1$ cm with the wavelength $\lambda =632$ nm.

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5. Twisted Gaussian-Schell model beams

Twisted Gaussian-Schell model (tGSM) beams, first introduced by Simon and Mukunda in 1993 [28], are modifications of the standard Gaussian-Schell model beams in which a phase twist is introduced into the state of coherence. In this type of partially coherent beam, the twist phase gives a handedness to the beam, which in turn results in a non-zero OAM for the beam. A tGSM source is characterized by the CSD,

5.1 Twisted GSM beams in free space

Propagation of tGSM beams in free space is obtained by evaluating Eq. (18) when the source is given by Eq. (40). The resulting CSD at the detector plane, located at distance $L$, is determined as

The parameter $A$ is defined as

As in the Rankine vortex case, we note that the propagated CSD has the same mathematical structure as the source CSD.

5.2 Twisted GSM beams in turbulence

Substituting the correlation function of the tGSM source given by Eq. (40) into the extended Huygens-Fresnel integral of Eq. (5) gives the CSD function of the tGSM beam at the detector plane in the form

5.3 OAM spectrum of tGSM beam

To analytically evaluate the OAM spectrum of the tGSM beam, we represent the twist phase term by a Bessel series representation (see [29], Sec.16.3),

By applying the above relation in the expressions of tGSM beams at the source and in turbulence, we can characterize the intensity of each OAM mode on concentric circles in the beam cross-section through analytical calculations.

By substituting from Eq. (40) into Eq. (3) and evaluating the azimuthal integrals, the intensity of the OAM mode of order $m$ at the radial distance $\rho$ from the beam axis in the source plane is given as

Again, since the OAM spectrum is preserved in free space propagation, the above result can be utilized to describe the OAM spectrum of the beam in free space propagation as well.

The distribution of intensity among OAM modes due to propagation through turbulence can be found by substituting from Eq. (48) into Eq. (3). The resulting expression is of the form

Figure 3 illustrates the OAM spectrum on $1\mbox { km}$ propagation through free space and turbulence for selected values of source coherence width $\sigma _g$. The twist magnitude was taken in each case to be the maximum allowed by Eq. (41), i.e. $u=1/k\sigma _g^2$. In free space, we can see that the twist of the beam is represented by a mode spectrum skewed towards the positive modes. Analogous to the Rankine vortex case, we can see that reducing the spatial coherence results in a greater resemblance between the free space and the turbulence mode spectrum, suggesting that the mode spectrum in this case is more resistant to turbulence. But as in the Rankine case, this comes at the cost of a significantly broadened mode spectrum to begin with, which indicates that tGSM beams will not necessarily alleviate cross-talk in OAM-based communications systems.

 

Fig. 3. OAM spectrum of twisted Gaussian-Schell model beam with maximum possible twist magnitude $u=1/k\sigma _g^2$ at the source and in propagation through turbulence (with the strength $C_n^2=10^{-14}$ m$^{-2/3}$) for various coherence widths $\sigma _g = 0.5 , 1 , 2$ cm. The beam size at source is $\sigma _s=1$ cm, and the wavelength is $\lambda =632$ nm. The propagation distance is $L=1000$ m to an infinitely sized detector with $R_d\rightarrow \infty$.

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It is to be noted, however, that tGSM beams with low coherence do maintain their overall OAM spectra, as is evidenced by the skewness of the mode distribution being maintained. If a method can be found to discriminate tGSM beams by their twist parameter $u$, this twist parameter could potentially be a distinct method of encoding information in OAM-based communications. Currently, however, we are not aware of any method for doing such twist discrimination.

6. Partially coherent beams with circular coherence

Circular coherence, introduced in 2017, describes a category of partially coherent sources that display perfect coherence along any ring concentric with the beam axis within the beam cross-section [12]. There is partial coherence between any two points on concentric rings with different radii, with coherence decreasing as the radial difference increases. The CSD function characterizing a circularly coherent beam (CCB) may be expressed as

Here, the average amplitude and phase is characterized by $U(\boldsymbol {\rho })$, while the degree of coherence is characterized by the function $g$, satisfying the condition $g(0)=1$. A general partially coherent source can be defined by the method of Gori and Santarsiero [24],

Let us take the kernel $H(\boldsymbol {\rho },\nu )$ to be a coherent field $U(\boldsymbol {\rho })$ with a finite radius of curvature [13],

In this study, we have followed Santarsiero et al. [12] and taken the probability density to be of the form

In this case, if $\nu _0$ is positive, the members of the ensemble will typically be focused, while negative $\nu _0$ means the members will typically be defocused.

6.1 Circularly coherent vortex beams at source

We may introduce a circularly coherent beam (CCB) carrying OAM simply by choosing the coherent field $U(\boldsymbol {\rho })$ to be a vortex beam; curiously, this possibility has not yet been explored in detail [30]. Let us take the case of a first-order vortex,

6.2 Circularly coherent vortex beams in free space

By inserting the CCB source into the Huygens-Fresnel integral given in Eq. (18), the corresponding CSD function in free space is given as

6.3 Circularly coherent vortex beams in turbulence

Using the extended Huygens-Fresnel principle, the correlation function of CCB between the positions $\mathbf{r}_1$ and $\mathbf{r}_2$ at the detector plane after propagation through turbulence can be written as

As is evident, turbulence effects give rise to a CSD function characterized by the superposition of Gaussian-Schell model vortex beams with different spot sizes $\Sigma _s(\nu )$, coherence widths $\Sigma _g(\nu )$ and radii of curvature $R_L(\nu )$, all defined as a function of probability variable $\nu$ with the following expressions,

The relative weight factors $C_i$ with $i=0,1,2$ are introduced as

The CSD function in turbulence, as expressed in Eq. (65), produces the free space result, given in Eq. (61), in the limit $Q(L)\rightarrow 0$, which indicates $\alpha =1$.

6.4 OAM spectrum of circularly coherent vortex beam

From Eqs. (60) and (61), it is evident that the entire intensity of the circularly coherent vortex beam is concentrated in the OAM mode of order 1 at the source and in free space. The important question we will attempt to answer is whether a circularly coherent vortex beam performs better in turbulence – maintains a narrower mode spectrum – than a fully coherent vortex beam of the same order.

Due to the interaction with turbulent media, the power of the beam is distributed among higher-order modes. By substituting from Eq. (65) into Eq. (3) and evaluating the angular integrals, we get

To explore the impact of transitioning from perfect coherence to partially circular coherence on resistance against turbulence effects, we compare the OAM spectrum of a fully coherent vortex beam with a flat phase front to that of a circularly coherent vortex beam; the results are in Fig. 4. The CCB is modeled by combining $N=5$ beams with varying radii of curvature with values of $\nu$ equally spaced within the range of $(\nu _0-\Delta \nu /2, \nu _0+\Delta \nu /2)$, with $\Delta \nu =0.001/\lambda L$. This combination is determined by the probability distribution function given in Eq. (58). As is evident from Figs. 4(a) and 4(c), the defocused beams with negative curvature display unfavorable effects, with a broader spectrum compared to the coherent case. Conversely, in Figs. 4(b) and 4(d), positive curvature provides advantages over the fully coherent beam.

 

Fig. 4. OAM spectrum of fully coherent and circularly coherent vortex beams for different central values of radius of curvature of the superimposed beams at source in $L=1000$ m propagation through turbulence (with the strength $C_n^2=10^{-14}$ m$^{-2/3}$). The beam size at source is $\sigma _s=1$ cm, and the wavelength is $\lambda =632$ nm. The number of superimposed beams at source is $N=5$.

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Upon closer examination of Figs. 4(b) and 4(d), it is demonstrated that the radius of curvature $R_0=L$, equivalent to $\nu _0=1/2\lambda L$, results in a spectrum profile with less spread and higher central peak, suggesting greater resistance against turbulence compared to the beam with a curvature $R_0=2L$, equivalent to $\nu _0=1/4\lambda L$.

From the results of Fig. 4, it can be concluded that transitioning to partially circular coherence does not always provide better resistance of the OAM spectrum on propagation through turbulence. Selecting a judicious value for the radius of curvature at the source is essential to optimize the generated beam for an effective turbulence propagation. We have discussed the circular coherence case in more detail in an upcoming publication [31], in which it is demonstrated that the wavefront curvature, determined by $\nu _0$, plays a greater role in turbulence resistance of the CCB than its spatial coherence, affected by $\Delta \nu$.

In the upcoming section, we will compare the behavior of all three types of PCBs in interaction with atmospheric turbulence. This comparison will be facilitated by examining the standard deviation of their OAM spectrum profiles.

7. Comparing the three PCB classes

After introducing the three classes of OAM-carrying partially coherent beams and delving into the propagation of their cross spectral density function through turbulence, as well as analyzing the behavior of their respective OAM spectra under turbulence disturbances individually, we now compare the resilience of the three beam types against the effects of turbulence.

We have noted that both the Rankine model beam and the CCB are produced by manipulating the parameters of a fully coherent Gaussian vortex source. In the Rankine model, the position of the coherent beam axis is randomized, whereas in the CCB, coherent beams with various radii of curvature are superimposed. In a tGSM beam, the initial source is a zero-order Gaussian beam without any OAM, and one can show that the OAM is induced by using an ensemble of beams that are all tilted to produce a net handedness [32].

In order to assess and compare the robustness of the OAM spectra for these beams, we consider the standard deviation of the OAM spectrum associated with each beam type; the standard deviation quantifies the dispersion of power among different OAM modes. A lower standard deviation indicates a narrower OAM spectrum, suggesting better mode purity in the face of turbulence and a beam that is more useful for OAM-based communications.

In Figure 5, a comparison is presented of the standard deviations associated with a fully coherent vortex beam, Rankine model vortex beam, twisted Gaussian-Schell model beam and partially coherent vortex beam with circular coherence. As is evident, for propagation distances less than about $250\mbox { m}$, the Rankine an tGSM beams have broader OAM spectra compared to the fully coherent case. In contrast, the CCB demonstrates a significantly narrower spectrum compared to a fully coherent beam.

 

Fig. 5. Standard deviations of the OAM spectra versus propagation distance through turbulence (with the strength $C_n^2=10^{-14}$ m$^{-2/3}$) corresponding to a fully coherent Gaussian vortex beam, Rankine model vortex beam, twisted Gaussian-Schell model beam and circularly coherent beam. The beam size at source is $\sigma _s=1$ cm, and the wavelength is $\lambda =632$ nm. Coherence width of the Rankine and tGSM model beam is $\sigma _g=0.5$ cm. The CCB is comprised of $N=5$ Gaussian vortex beams with various radii of curvature in the range $(\nu _0-\Delta \nu /2 , \nu _0+\Delta \nu /2)$ with $\Delta \nu =0.001/\lambda L$ and $\nu _0=3/4\lambda L$, equivalent to $R_0=2L/3$. The radius of the detector is $R_d=1$ m.

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As the distance surpasses $700\mbox { m}$, both the coherent beam and CCB exhibit a notable rise in standard deviation. Meanwhile, the standard deviation profiles of the Rankine and tGSM beams exhibit a gentle slope, indicating consistent stability and greater resistance. Our initial hypothesis, mentioned at the end of Section 4, was that randomized partially coherent vortex beams might have a slower spread of their OAM spectra in turbulence, and this appears to be the case.

It should be noted that the radius of the detector in Fig. 5 is $R_d=1$ m, which is much larger than the effective size of the beams on the receiver plane, since the largest spot size of the beams at the receiver, achieved at $z=3500$ m, falls within the range of $9\leq \Sigma _s\leq 18$ cm.

8. Conclusion

We have conducted an analytic investigation into how reducing the coherence of an OAM-carrying beam affects its resistance to atmospheric turbulence. “Resistance,” in this case, refers to how well the beam maintains its OAM purity on propagation through turbulence. We considered three physically distinct models of partially coherent OAM beams: Rankine model, twisted Gaussian-Schell model and circularly coherent beams. By examining an OAM beam corresponding to each class and deriving its mode spectrum after propagation through turbulence, we analyzed the advantages and disadvantages of coherence reduction associated with each class.

We use the standard deviation of the OAM spectrum to quantify the purity of the OAM state of the beam. The standard deviation is viewed as a basic metric for assessing the reliability of data transmission using OAM modes. A larger standard deviation suggests that data channels just be given wider OAM spacing in order to avoid crosstalk, which reduces the number of channels available.

The comparison of the standard deviations of the three types of PCBs and the coherent beam has indicated that for short propagation distances, the coherent beam and an optimized CCB have better performance of the four. However, over longer propagation distances, the OAM spectrum of coherent beam and CCB expand dramatically, leaving the Rankine and tGSM beams with narrower spectra. The advantages offered by the Rankine and tGSM beams become increasingly prominent over longer propagation distances; however, it should be noted that these beams have a significant standard deviation to begin with that may limit their usefulness.

Considering the results presented, we can infer that OAM beams with lower coherence at the source could potentially serve as an optimal selection for long propagation distances or when encountering strong turbulence. However, due to their tendency to spread more rapidly in propagation, over short distances, reduction in coherence at the source does not provide any significant advantage over coherent beams in the OAM-based applications. We expect that the OAM spectrum of other types of PCBs will exhibit similar behavior in propagation through turbulent atmosphere.

Our results provide guidelines for further investigations into the propagation of partially coherent vortex beams in turbulence. It is possible to encode OAM in partially coherent beams in more sophisticated ways beyond the classes considered here, for instance by considering twisted vortex GSM beams [33] or beams with polarization singularities encoded [34]; these will provide avenues for future study.