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Switch of orbital angular momentum flux density of partially coherent vortex beams

Yongtao Zhang, Yangjian Cai, and Greg Gbur

Open Access Open Access

Abstract

We investigate the orbital angular momentum (OAM) flux density of beams which are the incoherent superposition of partially coherent vortex (PCV) beams with different topological charges and beam widths. Simulation results show that such beams can exhibit counter-rotating radial regions of the OAM flux density, and that we can “switch” the order of these regions by adjusting the topological charges and beam widths in the source plane. Furthermore, these counter-rotating regions can switch on propagation in free space without any change to the beam parameters. We discuss how these unusual OAM dynamics may find use in OAM-based applications.

© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

In recent years, there has been significant interest in the wavefield singularities present in optical fields, such as lines of zero intensity around which the phase circulates, which are known as optical vortices [1–3]. Spatially coherent beams carrying optical vortices in their core are called vortex beams, and include the familiar class of Laguerre-Gauss beams. Vortex beams carry a well-defined orbital angular momentum (OAM) as a consequence of their vortex structure [4], and the OAM of vortex beams has already been applied to a number of applications, including optical manipulation and free-space optical communication [5,6].

Wavefields with spatial or temporal fluctuations, which are called partially coherent beams, have been shown to possess many advantages in practical applications, such as free-space optical communication, particle trapping, and optical coherence tomography [7,8]. Unlike spatially coherent beams, however, partially coherent beams do not have a well-defined phase, as it fluctuates randomly in space and time, and consequently such beams typically do not possess optical vortices. Researchers have instead investigated the singularities of the complex correlation functions of partially coherent beams; these structures are called correlation singularities or coherence vortices [9–17]. Coherence vortices have been shown to be more robust as the spatial coherence of a beam is decreased, suggesting they might be useful structures for carrying information in free-space optical communication [11].

Partially coherent beams carrying coherence vortices are called partially coherent vortex (PCV) beams. The OAM of PCV beams, especially the OAM flux density in their cross-sections, was first investigated by Kim and Gbur in 2012 for first-order vortices [18]. This work used a so-called beam wander model, in which the central axis of a vortex beam is treated as a random function of position, to generate partial coherence. It was found that the OAM flux density of such beams manifest a Rankine vortex structure in their cross-sections, analogous to the Rankine structure in the topological charge of PCV beams found earlier [19]. Stahl and Gbur extended the study of the OAM flux density of beam wander model PCV beams to all azimuthal orders [20], demonstrating an entire class of PCV beams. More recently, Zhang et al. investigated the OAM flux density of the beam wander model PCV beams with any azimuthal orders and any radial orders [21]. It was found that both the azimuthal and radial orders will affect the distribution of OAM flux density of PCV beams, which provides extra freedom for controlling OAM. Furthermore, Zhang et al. showed that even more intricate OAM structures can be created with partial coherence [22]. Overall, investigations have shown that partial coherence provides great freedom in controlling the OAM state of a beam, and that this control is relatively unexplored.

In this paper, we investigate the OAM properties of more general PCV beams, considering those that are constructed as an incoherent superposition of beam wander model PCV beams. Each constituent beam is taken to have a different topological charge (azimuthal order) and beam width. It is shown that there exists a radial counter-rotating structure of the OAM flux density, and through the variation of the parameters of constituent beam wander model PCV beams, the counter-rotating regions can be “switched”. It is also found that the counter-rotating structure can spontaneously switch itself on propagation in free space. These results provide great flexibility in controlling OAM and may have potential for particle manipulation and free-space optical communication.

2. Theory

To characterize the properties of PCV beams, we work with the cross-spectral density function in the space-frequency domain, which can be defined as [23]

$$W({{{\mathbf r}_1},{{\mathbf r}_2},z} )= {\left\langle {\tilde{U}({{{\mathbf r}_1}\textrm{,}z} )U({{{\mathbf r}_2}\textrm{,}z} )} \right\rangle _\omega },$$
where ${\left\langle { \cdot{\cdot} \cdot } \right\rangle _\omega }$ represents an average over a spatial-frequency ensemble of the field $U({{\mathbf r}\textrm{,}z} )$. For convenience, we use a tilde to represent the complex conjugate throughout the paper.

We will specifically consider beams which are constructed as the incoherent superposition of a finite number of beam wander model PCV beams with different topological charges and beam widths, for which the total cross-spectral density function may be expressed as [24]

$$W({{{\mathbf r}_1}\textrm{,}{{\mathbf r}_2},z} )= \sum\limits_{\alpha = 1}^N {\int {{{\tilde{U}}_\alpha }({{{\mathbf r}_1}{\mathbf - }{{\mathbf r}_0},z} ){U_\alpha }({{{\mathbf r}_2}{\mathbf - }{{\mathbf r}_0},z} )f({{{\mathbf r}_0}} ){\textrm{d}^2}{{\mathbf r}_0},} }$$
where $f({{{\mathbf r}_0}} )$ is the probability density for the position of the axis. We take the probability density to be of Gaussian form,
$$f({{{\mathbf r}_0}} )= \frac{1}{{\pi {\delta ^2}}}\exp \left( { - \frac{{r_0^2}}{{{\delta^2}}}} \right),$$
where ${r_0} = \sqrt {x_0^2 + y_0^2} $ is the transverse position of the axis and $\delta $ is the width of the probability function. In the limit $\delta \to 0$, the position of the beam axis is fixed and the field is spatially coherent; an increase in $\delta $ consequently results in the decrease of spatial coherence. Each individual term of Eq. (2) represents the cross-spectral density of a beam with a wandering central axis.

The field $U({{\mathbf r}\textrm{,}z} )$ is taken to be a Laguerre-Gauss beam of radial order $n = 0$ and arbitrary azimuthal order m; at any propagation distance z, we have the well-known expression

$${U_\alpha }({{\mathbf r}\textrm{,}z} )\textrm{ = }{C_\alpha }(z ){r^{|{{m_\alpha }} |}}\exp \left[ { - \frac{{{r^2}}}{{\sigma_\alpha^2}}} \right]\exp ({i{m_\alpha }\varphi } )\exp [{ - i{\Phi _\alpha }(z )({|{{m_\alpha }} |+ 1} )} ],$$
with
$${C_\alpha }(z )= \sqrt {\frac{2}{{\pi w_\alpha ^2(z )|{{m_\alpha }} |!}}} {\left( {\frac{{\sqrt 2 }}{{{w_\alpha }(z )}}} \right)^{|{{m_\alpha }} |}},{w_\alpha }(z )= {w_\alpha }\sqrt {1 + {{{z^2}} / {z_\alpha ^2}}} ,$$
$${\sigma _\alpha } = {1 / {\sqrt {\frac{1}{{w_\alpha ^2(z )}} + \frac{{ik}}{{2{R_\alpha }(z )}}} }},\,{R_\alpha }(z )= z + {{z_\alpha ^2} / z},$$
$${\Phi _\alpha }(z )= \arctan ({{z / {{z_\alpha }}}} ),\,{z_\alpha } = {{\pi w_\alpha ^2} / \lambda },$$
where ${w_\alpha }(z )$ and ${w_\alpha }$ represent the beam widths of the beam at propagation distance z and z = 0, respectively. ${\sigma _\alpha }$ represents a complex propagation parameter, and ${R_\alpha }(z )$ represents the wavefront curvature of the beam. Furthermore, ${\Phi _\alpha }(z )$ is the Gouy phase shift, ${z_\alpha }$ is the Rayleigh range, and $\varphi $ is the azimuthal angle. On substitution of Eqs. (3) and (4) into Eq. (2), the cross-spectral density $W({{{\mathbf r}_1}\textrm{,}{{\mathbf r}_2}\textrm{,}z} )$ can be evaluated analytically, and it takes on the form
$$\begin{array}{l} W({{{\mathbf r}_1},{{\mathbf r}_2},z} )= \sum\limits_{\alpha = 1}^N {\frac{{{{|{{C_\alpha }} |}^2}}}{{{\delta ^2}}}\exp \left[ { - \frac{{r_1^2}}{{{A_\alpha }{{\tilde{\sigma }}_\alpha }^2{\delta^2}}}} \right]\exp \left[ { - \frac{{r_2^2}}{{{A_\alpha }{\sigma_\alpha }^2{\delta^2}}}} \right]\exp \left[ { - \frac{{{{({{{\mathbf r}_1} - {{\mathbf r}_2}} )}^2}}}{{{A_\alpha }{{|{{\sigma_\alpha }} |}^4}}}} \right]} {\sum\limits_{{l_\alpha } = 0}^{{m_\alpha }} {\left( {\begin{array}{{c}} {{m_\alpha }}\\ {{l_\alpha }} \end{array}} \right)} ^2}\frac{{\Gamma ({{l_\alpha } + 1} )}}{{{A_\alpha }^{2{m_\alpha } - {l_\alpha } + 1}}}\\ \times {\left[ {\left( {\frac{1}{{{\sigma_\alpha }^\textrm{2}}} + \frac{1}{{{\delta^2}}}} \right)({{x_1} \mp i{y_1}} )- \frac{1}{{{\sigma_\alpha }^\textrm{2}}}({{x_2} \mp i{y_2}} )} \right]^{{m_\alpha } - {l_\alpha }}}{\left[ {\left( {\frac{1}{{{{\tilde{\sigma }}_\alpha }^2}} + \frac{1}{{{\delta^2}}}} \right)({{x_2} \pm i{y_2}} )- \frac{1}{{{{\tilde{\sigma }}_\alpha }^2}}({{x_1} \pm i{y_1}} )} \right]^{{m_\alpha } - {l_\alpha }}}, \end{array}$$
with
$${A_\alpha } = \frac{2}{{w_\alpha ^2(z )}} + \frac{1}{{{\delta ^2}}},$$

For “${\pm} $”and “${\mp} $” in Eq. (8), the top symbols are for the beams carrying positive topological charges, and the bottom symbols are for the beams having negative topological charges.

The OAM flux density of a partially coherent beam can be shown to be related to the cross-spectral density $W({{{\mathbf r}_1}\textrm{,}{{\mathbf r}_2}\textrm{,}z} )$ by the expression [1]

$${L_{orb}}({{\mathbf r}\textrm{,}z} )={-} \frac{{{\varepsilon _0}}}{k}{\mathop{\rm Im}\nolimits} {[{{y_1}{\partial_{{x_2}}}W({{{\mathbf r}_1},{{\mathbf r}_2},z} )- {x_1}{\partial_{{y_2}}}W({{{\mathbf r}_1},{{\mathbf r}_2},z} )} ]_{{{\mathbf r}_1} = {{\mathbf r}_2} = {\mathbf r}}},$$

On substitution from Eq. (8) into Eq. (10), we can get the analytical expression of the OAM flux density along the $z$ axis

$$\scalebox{0.85}{$\displaystyle{L_{orb}}({{\mathbf r}\textrm{,}z} )= \sum\limits_{\alpha = 1}^N { \pm \frac{{{\varepsilon _0}{{|{{C_\alpha }} |}^2}}}{{k{\delta ^2}}}\exp \left[ { - \frac{{2{r^2}}}{{{\delta^2}{A_\alpha }{w_\alpha }^2(z )}}} \right]{{\sum\limits_{{l_\alpha } = 0}^{{m_\alpha }} {\left( {\begin{array}{{c}} {{m_\alpha }}\\ {{l_\alpha }} \end{array}} \right)} }^2}\frac{{\Gamma ({{l_\alpha } + 1} )({{m_\alpha } - {l_\alpha }} )}}{{{A_\alpha }^{{l_\alpha } + 1}}}{r^{2({{m_\alpha } - {l_\alpha }} )}}{{({{A_\alpha }{\delta^2}} )}^{ - 2({{m_\alpha } - {l_\alpha }} )+ 1}}} ,$}$$

In this expression, “+” is for the beam constituents carrying positive topological charges, and “-” is for the constituents having negative topological charges.

The OAM flux density gives the spatial distribution of OAM within the cross section of the beam. The magnitude of this OAM flux density arises, however, from two distinct effects: the strength of circulation of the phase and the intensity of the beam, i.e., the density of photons. To better understand the physics of OAM distribution, we define the normalized OAM flux density

$${l_{orb}} = \frac{{\hbar \omega {L_{orb}}({{\mathbf r}\textrm{,}z} )}}{{S({{\mathbf r}\textrm{,}z} )}},$$
where $S({{\mathbf r}\textrm{,}z} )$ is the z component of the Poynting vector, which is of the form
$$S({{\mathbf r}\textrm{,}z} )= \frac{k}{{{\mu _0}\omega }}W({{\mathbf r}\textrm{,}{\mathbf r}\textrm{,}z} ),$$

The total average OAM per photon can be given by the ratio of the integrated ${L_{orb}}({{\mathbf r}\textrm{,}z} )$ and $S({{\mathbf r}\textrm{,}z} )$,

$${l_t} = \frac{{\hbar \omega \int {{L_{orb}}({{\mathbf r}\textrm{,}z} ){d^2}r} }}{{\int {S({{\mathbf r}\textrm{,}z} ){d^2}r} }},$$

The total OAM is always conserved on propagation; the normalized OAM flux density ${l_{orb}}$, which shows the spatial distribution of OAM, will in general exhibit changes. We will show in the next part that we can control these changes, by a careful choice of the beam width, topological charge and spatial coherence. The evolution behavior of the OAM flux density on propagation will also be discussed.

3. Results

With the help of Eqs. (8), (11), (12) and (13), we calculated the normalized OAM flux density ${l_{orb}}$ for different beam widths ${w_\alpha }$, topological charges ${m_\alpha }$, spatial coherence parameters $\delta$ and propagation distances z. We take $N = 2$ and $\lambda = 632.8\textrm{nm}$ for the remainder of the paper.

 figure: Fig. 1.

Fig. 1. The distributions of normalized OAM flux density ${l_{orb}}/\hbar$ at z = 0 for different beam widths w1 and w2 with ${m_1} ={+} 1$ , ${m_2} ={-} 1$, and $\delta = 1\textrm{mm}$.

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 figure: Fig. 2.

Fig. 2. The distributions of normalized OAM flux density ${l_{orb}}/\hbar$ at z = 0 for different azimuthal orders m1 and m2 with ${w_1} = 1\textrm{mm}$, ${w_2} = 5\textrm{mm}$, and $\delta = 1\textrm{mm}$.

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 figure: Fig. 3.

Fig. 3. The distributions of normalized OAM flux density ${l_{orb}}/\hbar$ at z = 0 for different states of coherence with ${w_1} = 1\textrm{mm}$, ${m_1} ={+} 1$, and ${m_2} ={-} 1$.

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 figure: Fig. 4.

Fig. 4. The distributions of normalized OAM flux density ${l_{orb}}/\hbar$ for different values of propagation distance z with ${w_1} = 1\textrm{mm}$ ${w_2} = 5\textrm{mm}$, ${m_1} ={+} 1$, ${m_2} ={-} 1$ $\delta = 1\textrm{mm}$ , and .

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 figure: Fig. 5.

Fig. 5. The distributions of normalized OAM flux density ${l_{orb}}/\hbar$ for several values of propagation distance z with ${w_1} = 1\textrm{mm}$ ${w_2} = 5\textrm{mm}$, ${m_1} ={+} 1$, ${m_2} ={-} 1$, and $\delta = 1\textrm{mm}$.

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Figure 1 illustrates the distributions of normalized OAM flux density by adjusting the relative widths of a pair of beam wander model PCV beams. In Fig. 1, using Eqs. (8), (11), (13) and (14), the total average OAM per photon is calculated to be zero. However, the normalized OAM flux density is shown to have a positive OAM flux density in the core and a negative OAM flux density in the outskirts. In other words, the beam has counter-rotating regions in its cross-section, something that is not typically seen in coherent vortex beams; this can be explained as follows. The OAM flux density of beam wander PCV beams manifests a Rankine vortex structure, with a rigid body rotational behavior near the core and a fluid rotational behavior outside of this, a behavior that does not exist for spatially coherent beams. The incoherent superposition of two Rankine vortices with different signs and widths will generally result in a radially varying imbalance of OAM flux, allowing for counter-rotating regions.

We choose ${m_1} ={+} 1$, ${m_2} ={-} 1$ and ${w_1} < {w_2}$ in Fig. 1, which means the smaller constituent beam carries positive topological charge, and the larger constituent beam carries negative topological charge. If we switch the signs of the topological charges of the smaller and larger beams, we will get a negative OAM flux density core and a positive OAM flux density outskirts. Here, we can see it is possible to create a counter-rotating structure of OAM flux density in the beam with an appropriate choice of beam widths and topological charges, and the radial locations of these regions can be “switched” by changing the relationship between beam widths and topological charges.

Furthermore, by changing the topological charges of the constituent beams, the size and shape of the core region is also readily adjusted, as is shown in Fig. 2.

The spatial coherence parameter $\delta$ can also play a significant role in adjusting the distribution of normalized OAM flux density. Figure 3 gives the evolution of OAM flux density for different values of the coherence parameter $\delta$. It is shown in Fig. 3(a) that for $\delta = 0.01{mm}$, i.e., the case of high spatial coherence, the core of the OAM flux density acts much like a fluid rotator (OAM flux density independent of radial distance $r$). As the coherence parameter $\delta$ increases, the spatial coherence decreases, the OAM flux density behaves like a rigid rotator in the core, with a quadratic radial dependence.

What is particularly surprising is that the counter-rotating regions in this class of PCV beams can switch locations during free-space propagation. The evolution behavior of the OAM flux density on propagation is given in Figs. 4 and 5. It is to be noted that we do not change any parameters of the beam during the propagation. As seen in Fig. 4, there still exists a positive OAM flux density core and a negative OAM flux density outskirts over short propagation distances; however, as the propagation distance increases, the radial locations of the counter-rotating regions can switch, resulting in a negative OAM flux density core and a positive OAM flux density outskirts. This switch of orbital angular momentum flux density is reminiscent of the correlation-induced spectral shifts discovered by Wolf [25,26]. In the spectral shifts example, the changes arise due to diffraction effects of different frequencies associated with the spatial coherence length of the beam.

We can also find from Fig. 4 that there exists a special distance, after which the counter-rotating regions will switch. At the special distance, the beam widths of the constituent beams are equal, then the OAM flux density will be zero, and there is no circulation. With the help of Eq. (5), the special distance in Fig. 4 is calculated to be 24.823 m. It is to be noted that the position of this switch depends on the spatial coherence of the beams, and can be changed by changing the spatial coherence of one or both beams.

To highlight the OAM switch, we plot the OAM flux density for several values of propagation distance z in Fig. 5. This phenomenon may be explained by the expression of ${w_\alpha }(z )$ in Eq. (5), where ${w_\alpha }(z )$ depends on both the initial beam width ${w_\alpha }$ and the propagation distance z. So with the increase of propagation z, the relationship between ${w_1}(z )$ and will ${w_2}(z )$ change, while the topological charges ${m_1}$ and ${m_2}$ do not change, which will change the radial locations of the counter-rotating regions. We also calculated in Fig. 6 the normalized spectral density for the superposed PCV beams, both in the source plane and after propagation. It can be seen from Fig. 6 that the distributions of special density are significantly different than those of OAM flux density.

 figure: Fig. 6.

Fig. 6. The distributions of normalized spectral density for several values of propagation distance z with ${w_1} = 1\textrm{mm}$, ${w_2} = 5\textrm{mm}$, ${m_1} ={+} 1$, ${m_2} ={-} 1$, and $\delta = 1\textrm{mm}$.

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4. Summary

We have studied the spatial distribution of normalized OAM flux density, for the beam which is the incoherent superposition of beam wander model PCV beams with different topological charges and beam widths. The results show that though the total OAM is constant, in the source plane we can get a counter-rotating structure, and by adjusting the beam widths ${w_\alpha }$ and topological charges ${m_\alpha }$, the counter-rotating structure can be “switched”. The more interesting finding in this paper is that after propagation, the counter-rotating structure can be maintained over short propagation distances, while the radial locations of the counter-rotating regions will switch itself with the increase of propagation distances. It is also found that by changing the spatial coherence parameter $\delta$, there can exist a fluid rotator, a rigid rotator, and a hybrid between them, i.e., a Rankine vortex.

These results show that spatial coherence effects can be used to create distributions of OAM flux density in PCVs that are apparently not achievable in their fully coherent counterparts. The robustness of the OAM flux density distributions on propagation suggests that they might serve as an alternative way to convey information in optical communications systems – provided that a method of easily measuring them can be determined. The most straightforward possibility to measure OAM flux density might be to measure the motion of microscopic particles in the path of the beam, deducing the strength and handedness of the rotation by the direction the particle moves. The freedom in controlling OAM in the transverse plane will be most useful in trapping and micromanipulation applications, and the counterrotating regions could potentially be used to drive light-driven micromachines [27,28] and optical tweezers [29]. Furthermore, PCV beams are robust to environment noise such as the turbulence of natural media, which make them have unique advantages in communication and sensing systems [30–34].

Funding

National Key Research and Development Program of China (2019YFA0705000, 2022YFA1404800); National Natural Science Foundation of China (12174173, 12192254, 92250304, 11974218); Natural Science Foundation of Fujian Province (2022J02047); Air Force Office of Scientific Research [United States] (FA9550-21-1-0171); Office of Naval Research, MURI (N00014-20-1-2558).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (6)
Fig. 1.
Fig. 1. The distributions of normalized OAM flux density ${l_{orb}}/\hbar$ at z = 0 for different beam widths w1 and w2 with ${m_1} ={+} 1$ , ${m_2} ={-} 1$, and $\delta = 1\textrm{mm}$.

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Fig. 2.
Fig. 2. The distributions of normalized OAM flux density ${l_{orb}}/\hbar$ at z = 0 for different azimuthal orders m1 and m2 with ${w_1} = 1\textrm{mm}$, ${w_2} = 5\textrm{mm}$, and $\delta = 1\textrm{mm}$.

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Fig. 3.
Fig. 3. The distributions of normalized OAM flux density ${l_{orb}}/\hbar$ at z = 0 for different states of coherence with ${w_1} = 1\textrm{mm}$, ${m_1} ={+} 1$, and ${m_2} ={-} 1$.

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Fig. 4.
Fig. 4. The distributions of normalized OAM flux density ${l_{orb}}/\hbar$ for different values of propagation distance z with ${w_1} = 1\textrm{mm}$ ${w_2} = 5\textrm{mm}$, ${m_1} ={+} 1$, ${m_2} ={-} 1$ $\delta = 1\textrm{mm}$ , and .

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Fig. 5.
Fig. 5. The distributions of normalized OAM flux density ${l_{orb}}/\hbar$ for several values of propagation distance z with ${w_1} = 1\textrm{mm}$ ${w_2} = 5\textrm{mm}$, ${m_1} ={+} 1$, ${m_2} ={-} 1$, and $\delta = 1\textrm{mm}$.

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Fig. 6.
Fig. 6. The distributions of normalized spectral density for several values of propagation distance z with ${w_1} = 1\textrm{mm}$, ${w_2} = 5\textrm{mm}$, ${m_1} ={+} 1$, ${m_2} ={-} 1$, and $\delta = 1\textrm{mm}$.

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Equations (14)

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$$W({{{\mathbf r}_1},{{\mathbf r}_2},z} )= {\left\langle {\tilde{U}({{{\mathbf r}_1}\textrm{,}z} )U({{{\mathbf r}_2}\textrm{,}z} )} \right\rangle _\omega },$$
$$W({{{\mathbf r}_1}\textrm{,}{{\mathbf r}_2},z} )= \sum\limits_{\alpha = 1}^N {\int {{{\tilde{U}}_\alpha }({{{\mathbf r}_1}{\mathbf - }{{\mathbf r}_0},z} ){U_\alpha }({{{\mathbf r}_2}{\mathbf - }{{\mathbf r}_0},z} )f({{{\mathbf r}_0}} ){\textrm{d}^2}{{\mathbf r}_0},} }$$
$$f({{{\mathbf r}_0}} )= \frac{1}{{\pi {\delta ^2}}}\exp \left( { - \frac{{r_0^2}}{{{\delta^2}}}} \right),$$
$${U_\alpha }({{\mathbf r}\textrm{,}z} )\textrm{ = }{C_\alpha }(z ){r^{|{{m_\alpha }} |}}\exp \left[ { - \frac{{{r^2}}}{{\sigma_\alpha^2}}} \right]\exp ({i{m_\alpha }\varphi } )\exp [{ - i{\Phi _\alpha }(z )({|{{m_\alpha }} |+ 1} )} ],$$
$${C_\alpha }(z )= \sqrt {\frac{2}{{\pi w_\alpha ^2(z )|{{m_\alpha }} |!}}} {\left( {\frac{{\sqrt 2 }}{{{w_\alpha }(z )}}} \right)^{|{{m_\alpha }} |}},{w_\alpha }(z )= {w_\alpha }\sqrt {1 + {{{z^2}} / {z_\alpha ^2}}} ,$$
$${\sigma _\alpha } = {1 / {\sqrt {\frac{1}{{w_\alpha ^2(z )}} + \frac{{ik}}{{2{R_\alpha }(z )}}} }},\,{R_\alpha }(z )= z + {{z_\alpha ^2} / z},$$
$${\Phi _\alpha }(z )= \arctan ({{z / {{z_\alpha }}}} ),\,{z_\alpha } = {{\pi w_\alpha ^2} / \lambda },$$
$$\begin{array}{l} W({{{\mathbf r}_1},{{\mathbf r}_2},z} )= \sum\limits_{\alpha = 1}^N {\frac{{{{|{{C_\alpha }} |}^2}}}{{{\delta ^2}}}\exp \left[ { - \frac{{r_1^2}}{{{A_\alpha }{{\tilde{\sigma }}_\alpha }^2{\delta^2}}}} \right]\exp \left[ { - \frac{{r_2^2}}{{{A_\alpha }{\sigma_\alpha }^2{\delta^2}}}} \right]\exp \left[ { - \frac{{{{({{{\mathbf r}_1} - {{\mathbf r}_2}} )}^2}}}{{{A_\alpha }{{|{{\sigma_\alpha }} |}^4}}}} \right]} {\sum\limits_{{l_\alpha } = 0}^{{m_\alpha }} {\left( {\begin{array}{{c}} {{m_\alpha }}\\ {{l_\alpha }} \end{array}} \right)} ^2}\frac{{\Gamma ({{l_\alpha } + 1} )}}{{{A_\alpha }^{2{m_\alpha } - {l_\alpha } + 1}}}\\ \times {\left[ {\left( {\frac{1}{{{\sigma_\alpha }^\textrm{2}}} + \frac{1}{{{\delta^2}}}} \right)({{x_1} \mp i{y_1}} )- \frac{1}{{{\sigma_\alpha }^\textrm{2}}}({{x_2} \mp i{y_2}} )} \right]^{{m_\alpha } - {l_\alpha }}}{\left[ {\left( {\frac{1}{{{{\tilde{\sigma }}_\alpha }^2}} + \frac{1}{{{\delta^2}}}} \right)({{x_2} \pm i{y_2}} )- \frac{1}{{{{\tilde{\sigma }}_\alpha }^2}}({{x_1} \pm i{y_1}} )} \right]^{{m_\alpha } - {l_\alpha }}}, \end{array}$$
$${A_\alpha } = \frac{2}{{w_\alpha ^2(z )}} + \frac{1}{{{\delta ^2}}},$$
$${L_{orb}}({{\mathbf r}\textrm{,}z} )={-} \frac{{{\varepsilon _0}}}{k}{\mathop{\rm Im}\nolimits} {[{{y_1}{\partial_{{x_2}}}W({{{\mathbf r}_1},{{\mathbf r}_2},z} )- {x_1}{\partial_{{y_2}}}W({{{\mathbf r}_1},{{\mathbf r}_2},z} )} ]_{{{\mathbf r}_1} = {{\mathbf r}_2} = {\mathbf r}}},$$
$$\scalebox{0.85}{$\displaystyle{L_{orb}}({{\mathbf r}\textrm{,}z} )= \sum\limits_{\alpha = 1}^N { \pm \frac{{{\varepsilon _0}{{|{{C_\alpha }} |}^2}}}{{k{\delta ^2}}}\exp \left[ { - \frac{{2{r^2}}}{{{\delta^2}{A_\alpha }{w_\alpha }^2(z )}}} \right]{{\sum\limits_{{l_\alpha } = 0}^{{m_\alpha }} {\left( {\begin{array}{{c}} {{m_\alpha }}\\ {{l_\alpha }} \end{array}} \right)} }^2}\frac{{\Gamma ({{l_\alpha } + 1} )({{m_\alpha } - {l_\alpha }} )}}{{{A_\alpha }^{{l_\alpha } + 1}}}{r^{2({{m_\alpha } - {l_\alpha }} )}}{{({{A_\alpha }{\delta^2}} )}^{ - 2({{m_\alpha } - {l_\alpha }} )+ 1}}} ,$}$$
$${l_{orb}} = \frac{{\hbar \omega {L_{orb}}({{\mathbf r}\textrm{,}z} )}}{{S({{\mathbf r}\textrm{,}z} )}},$$
$$S({{\mathbf r}\textrm{,}z} )= \frac{k}{{{\mu _0}\omega }}W({{\mathbf r}\textrm{,}{\mathbf r}\textrm{,}z} ),$$
$${l_t} = \frac{{\hbar \omega \int {{L_{orb}}({{\mathbf r}\textrm{,}z} ){d^2}r} }}{{\int {S({{\mathbf r}\textrm{,}z} ){d^2}r} }},$$
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