## Abstract

The fluctuation characteristics of the orbital angular momentum (OAM) are studied numerically in the computer experiment simulating the propagation of an OAM-carrying laser beam through a turbulent atmosphere and OAM registering by a finite-size receiver aperture. The wave-optics numerical simulation employs the conventional split-operator method (with phase screens) and the Monte Carlo averaging technique. OAM statistical characteristics in the turbulent medium are compared for two employed definitions of the specific (normalized) orbital angular momentum. For the non-vortex Gaussian beam, the correlation between OAM fluctuations and intercepted power is found to be negligibly small at arbitrary size of the receiver aperture. For the vortex Laguerre—Gaussian beam, the correlation between OAM fluctuations and intercepted power is significant (the correlation coefficient can achieve 0.7). In this case, the values of OAM variance for these definitions differ almost twice. It is shown that the OAM value averaged over realizations of the turbulent medium for the aperture size intercepting almost the entire beam power becomes equal to that in the homogeneous medium. As the aperture size decreases, the average OAM value decreases, and when the aperture size tends to zero, it tends to zero as well. The results obtained in the paper can be used in development and improvement of methods for optical transmission of information by optical beams with OAM, as well as in development of optical means for manipulating micron-sized particles.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

In our recent paper [1], we have performed the numerical experiment on the propagation of a vortex laser beam in the turbulent atmosphere and studied statistical characteristics of the orbital angular momentum of a Laguerre—Gaussian beam as a function of the intensity of atmospheric turbulence and the value of initial OAM. The statements that the law of OAM conservation is true in a turbulent medium for the OAM value averaged over realizations of the medium and that OAM fluctuations in the case of weak turbulence and an initially circularly symmetric beam are much lower than intensity fluctuations of a plane wave or a collimated beam, which were derived analytically in [2–4], have been confirmed. According to the classical definition of OAM [5–7], it was assumed in [1] that the received OAM is calculated through integration of the OAM density [8] over the “infinite-size” aperture. However, it is often hard to obtain the conditions in the turbulent atmosphere, under which the receiver aperture could be considered as infinite-size (complete interception of the beam).

The experimental implementation of complete interception of the beam is a complicated problem. Moreover, such a communication system will be expensive. This circumstance prompts researchers to look for the possibility of implementing communication systems without complete interception of the beam. This issue is now considered in the growing number of papers.

It is known [9–11] that even in a homogeneous medium the vortex beam having OAM diverges faster than the fundamental Gaussian beam, leading to the larger power loss at the finite-size receiver aperture. Thus, the effective width of the Laguerre—Gaussian laser beam LG* _{pm}* with the radial and azimuthal indices

*p*= 0 and

*m*, respectively, $\sqrt {|m |+ 1}$ times exceeds that of the fundamental Gaussian beam LG

_{00}. Since a beam broadens and wanders randomly in a turbulent medium due to the effect of inhomogeneities of the refractive index, the effective width of the beam increases even more. The complete interception of the beam by the receiver aperture becomes more technically complicated with an increase of the azimuthal index of the Laguerre—Gaussian mode. Overcoming the effect of the finite-size of a receiver aperture in comparison with the effective width of OAM beams is one of the challenges in development of OAM-multiplexing communication systems [11–13].

It also seems important to estimate the effect of the finite size of the receiver aperture on the characteristics of the received signal and, in particular, on statistical characteristics of the orbital angular momentum measured by the finite-size receiver aperture. This estimate was recently obtained in [14] by the analytical methods used earlier by us in [3,4] for the case of an optical source generating a spherical wave. In this paper, we apply the approaches used in [1] to study the average value and the variance of fluctuations of OAM of the laser beam LG_{01} intercepted by a finite-size receiver aperture. In this case, two definitions of the normalized OAM can be introduced. In [14], as in other similar papers, to derive analytical equations, the authors study ОАМ statistical moments for the beam after propagation through a finite-size aperture with OAM normalized to the average power of radiation inside the aperture. In this paper, we also consider fluctuations of OAM normalized to the instantaneous value of radiation power inside the aperture. We compare the results of simulation of statistical moments of OAM normalized by these two methods.

As far as we know, OAM statistical moments determined by these two methods have not been ever compared by other authors.

## 2. Mathematical problem statement and basic definitions

The propagation of a laser beam in the turbulent medium is considered in the approximation of a parabolic equation for the complex amplitude of the beam optical field *E*

In this paper, we study the propagation of the Laguerre—Gaussian beam LG_{0m}, whose initial field distribution has the form

*m*is the value of the topological charge and, at the same time, the azimuthal index of the mode LG

_{0m}, ${P_0} = \int\limits_{ - \infty }^\infty {\int\limits_{ - \infty }^\infty {I({{\mathbf r},0} )} } \textrm{d}{\mathbf r}$ is the radiation power, $I(\mathbf{r}, z)$ is the intensity of the field $E({{\mathbf r},z} ).$ $I({\mathbf r},z) = {|{E({{\mathbf r},z} )} |^2}.$ Equation (2) describes the Gaussian beam if

*m*= 0 and the circular mode LG

_{0m}of the Laguerre—Gaussian beam if

*m*is a positive or negative integer number. We solved Eq. (1) with the split-operator method [15–18]. Atmospheric turbulence was represented by a series of phase screens [17]. The simulation algorithms were organized identically to the algorithms in [1,19]. We used the modified Andrews spectrum of fluctuations of the refractive index [20], which has the following form

*κ*

_{0}= 2

*π*/

*M*

_{0},

*κ*= 3.3

_{a}*m*

_{0},

*m*

_{0}and

*M*

_{0}are the inner and outer scales of turbulence, $C_n^2$ is the structure characteristic of the refractive index. In our calculations, we assumed that

*M*

_{0}= 20

*a*and

*m*

_{0}= 0.08

*a.*The turbulence was considered as homogeneous and isotropic. The turbulent conditions of propagation were specified by the Rytov parameter $\beta _0^2 = 1.23C_n^2{k_0}^{{7 / 6}}{z^{{{11} / 6}}},$ which depends on $C_n^2,$ the path length

*z*, and the wave number of laser radiation [20].

For this paper, the Rytov parameter was fixed as 1.2, which corresponds to average turbulent conditions. We consider laser beams with initial azimuthal index of the beam (topological charge of the optical vortex) *m *= 0 (Gaussian) and *m *= 1 (Vortex) propagating along the path length (in Rayleigh diffraction lengths *z*_{d}* *= *k*_{0}*a*^{2}) *z *= 0.1, *z *= 1.0 and *z *= 3.0.

To find the value of the orbital angular momentum, it is first necessary to calculate the transversal component of the Pointing vector ${{\mathbf P}_ \bot }({\mathbf r},z)$ (energy flux density vector) [21], which can be written in the following form in the paraxial approximation

**n**

*is the unit vector in the direction of the radiation propagation axis.*

_{z}One can calculate the specific angular momentum of the laser beam at the distance *z*, correspondingly, by the following equation:

In our previous publication [1] and in this paper, we use Eq. (7) for construction of schemes for numerical simulation. From the theoretical point of view, Eqs. (6) and (7) are identical. However, our comparative analysis [1] has shown that numerical algorithms based on Eq. (7) are characterized by higher stability.

Then we assume that the radiation is received by a circular aperture with the finite radius ${a_t}.$ We are interested in the specific OAM ${L_z}({z;{a_t}} )$ [7, 23] measured within this aperture

where${M_z}({{a_t}} )$ is OAM of the beam after passage through the finite-size aperture, $P({{a_t}} )$ is the power of radiation within the aperture. In Eqs. (9) and (10), $\rho ,\theta ,z$ are cylindrical coordinates, ${\mathbf r}({\rho ,\theta } )$ is the radius vector of the point in the transverse plane.

Definition (8) is the basis for quantitative measurements of the OAM of optical beams with the integer and fractional topological charge [23–26]. These measurements can be carried out with registering of the beam intensity moments upon propagation through cylindrical lenses or with pre-decomposition of the beam into azimuthal modal components [27,28] or with the Shack–Hartmann wavefront sensor [29].

Fluctuations of OAM of a laser beam registered by finite-size receiver aperture after propagation through a turbulent atmosphere were analyzed based on the mean value ${\left\langle {{L_z}({z;{a_t}} )} \right\rangle _N}$ and the variance of fluctuations of the specific OAM

*N*denote the statistical averaging over the ensemble of

*N*= 4000 realizations of the turbulent medium. If the receiver aperture has finite size, then both the numerator and denominator in Eq. (8) are generally fluctuating variables. To obtain analytical estimates for the variance of fluctuations of OAM, $P({{a_t}} )$ in Eq. (8) is replaced with a non-random parameter, such as the statistical average $\left\langle {P({{a_t}} )} \right\rangle$ of $P({{a_t}} )$ in an inhomogeneous medium [14]. To estimate the effect from this replacement, we study not only the OAM value given by Eq. (8), but also the following parameter

## 3. Results and discussions

#### 3.1 Principle of conservation of OAM for the finite-size receiver aperture

As was shown in our paper [2], the condition

*a*→ ∞, it is also true that It follows from conditions (14) and (15) that

_{t}It can be shown that in the homogeneous medium the principle of conservation of the specific OAM is also true for the finite-size receiver aperture. For the beams with the axisymmetric intensity distribution (for example, Laguerre—Gaussian beams), Eq. (9) transforms to the following form:

*φ*= 2

*πm*. Consequently, we obtain It follows from here that that is, the specific OAM for the receiver aperture of any size takes the value equal to the initial value of OAM [7,23].

However, this condition violates as the beam propagates in the turbulent atmosphere. The results shown in Fig. 1 demonstrate that the average value of the specific OAM *L _{z}*(z;

*a*) varies from zero to

_{t}*m*which was taken

*m*= 1 in the considered calculations, as the aperture size changes from zero to infinity. This behavior of OAM values for the finite-size aperture can be explained by the fact that in the turbulent atmosphere the average circular phase progression (average circulation of the phase gradient) of the radius $\rho$ no longer takes values multiple of 2

*π*[30, 31]. Actually, neglecting the correlation between the intensity and the phase in the equation for $\left\langle {{M_z}({a_t})} \right\rangle$ obtained from Eq. (17), we have

*a*. With an increase of the circle radius, this parameter behaves similarly to the behavior of average OAM in Fig. 1. As was shown numerically and analytically [30, 31], it increases from zero up to 2

_{t}*πm*.

In addition, the results shown in Fig. 1 indicate that for the limited aperture the OAM definitions given by Eqs. (8) and (12) are not identical and lead to different values. It can be seen that for small apertures these values differ more than twice. It follows from the figure that the average OAM becomes equal to its initial value (14) at the receiver aperture size sufficient to intercept almost the entire beam power.

It is obvious that the value of normalized OAM determined by Eq. (8) is significantly affected by the degree of correlation between the numerator and denominator, whereas the value of normalized OAM determined by Eq. (12) is independent of the degree of correlation of OAM and instantaneous power received by the finite-size aperture. Consider how correlated are fluctuations of the numerator and denominator in Eq. (8). To estimate the degree of correlation of two random parameters *X* and *Y*, we calculate the Pearson correlation coefficient [32]

*X*= ${M_z}({{a_t}} )$,

*Y*= $P({{a_t}} ).$ The calculated correlation coefficient is shown in Fig. 2. The calculations are given for three propagation distances and moderate turbulence. In addition, this figure depicts the dependence of statistical characteristics of received power fluctuations

It can be seen that for the vortex beam (Figs. 2(a)–(c)), power fluctuations of the intercepted signal become maximal when the aperture intercepts approximately half of the beam power. The correlation coefficient of ${M_z}({a_t})$ and $P({a_t})$ is maximal for approximately the same aperture size for distances comparable with and longer than the diffraction length *z _{d}*

_{ }=

*k*

_{0}

*a*

^{2}. It is obvious that as the size of the receiver aperture increases, the averaging impact of the receiver aperture on intensity fluctuations comes into effect [20]. As a result, the received power no longer fluctuates, and its variance tends to zero.

It follows from Fig. 2 that for the vortex beam, the position of the maxima in the correlation coefficient and the standard deviation of fluctuations of the received power depends significantly on the propagation path length. Their shift from the center increases as the beam broadens along the path. To be noted is the significant difference in the values of the Pearson coefficient for the vortex Laguerre—Gaussian beam (Figs. 2(a)–(c)) and the Gaussian beam (Figs. 2(d)–(e)). It is obvious that the correlation is high for the numerator and denominator in Eq. (8) for the vortex beam and nearly zero for the numerator and denominator for the Gaussian (non-vortex) beam.

Figure 3 shows the dependence of statistical characteristics of the OAM

*B*,

_{M}*B*, and ${\left\langle {P({{a_t}} )} \right\rangle _N}$ reach their saturation levels, the correlation between ${M_z}({{a_t}} )$ and $P({{a_t}} )$ vanishes. It is noticeable that the saturation level of OAM variance for the vortex beam (Figs. 3(a)–(c)) appears to be lower than that for the non-vortex beam (Figs. 3(d)–(e)). The similar result was already demonstrated in [1].

_{P}#### 3.2 Fluctuations of the OAM in the case of the finite-size receiver aperture

Figure 4 shows the calculated variance of fluctuations of the specific angular momentum (received OAM normalized to the received power) as a function of the size of the receiver aperture.

The calculations were performed for the conditions of moderate turbulence at an atmospheric path. The variances of OAM fluctuations calculated by Eqs. (11) and (13) for the vortex beam (Figs. 4(a)–(c)) vary from minimal values at the small size of the receiver aperture to the maximal one at the increased aperture size and then tend to the residual level. At the same time, the maximum of OAM fluctuations is achieved at smaller sizes of the receiver aperture than the maximum of fluctuations of the received power. The definition of specific OAM (12) proposed by Charnotskii [14] gives higher OAM variance for the vortex beam. It should be also noted that the maximal differences in values of the OAM variance determined according to Eqs. (8) and (12) are observed in the region of high values of the correlation coefficient for fluctuations of the numerator and denominator in Eq. (8).

However, for the non-vortex Gaussian beam (mode LG_{00}) the results of calculation by Eqs. (8) and (12) coincide. In this case, as follows from the results depicted in Fig. 2, there is practically no correlation between fluctuations of the numerator and denominator in Eq. (8). Thus, in the absence of correlation between fluctuations of OAM and received power, both Eqs. (8) and (12) are applicable to calculations for laser beams initially free of optical vortex. It should be noted that the maximum in the dependence of the variance of OAM fluctuations on the aperture size was observed at a path of any length used in the calculations for the vortex beam, but only at the short path (*z* = 0.1) for the non-vortex beam.

## 4. Conclusions

In the numerical experiment by the Monte Carlo technique, we have studied the statistical characteristics of the orbital angular momentum of the Laguerre—Gaussian beam propagating in the turbulent atmosphere. It has been found that the finite size of the receiver aperture affects the average value and the variance of fluctuations of OAM normalized to the received power of the beam.

It is shown that the OAM value averaged over realizations of the turbulent medium for the aperture intercepting almost the entire beam power becomes equal to that in a homogeneous medium. The average OAM value decreases as the aperture size decreases and tends to zero as the aperture size tends to zero.

From the calculation of the Pearson correlation coefficients, we have determined the sizes of receiver apertures, at which fluctuations of received OAM and power can be considered as uncorrelated. The maximum of power fluctuations of the intercepted signal is observed for the aperture intercepting approximately half of the beam power. For distances comparable with and longer than the diffraction length *z _{d}*

_{ }=

*k*

_{0}

*a*

^{2}, the maximal value of the correlation coefficient of OAM and instantaneous power is observed for approximately the same aperture size. It has been shown that for the Gaussian beam, OAM fluctuations and fluctuations of the received power can be considered as uncorrelated at an arbitrary size of the receiver aperture.

The variances of fluctuations of laser beam OAM and power received by finite-size receiver aperture tend to their saturation level, as the aperture size increases. The saturation level of the OAM variance of the vortex beam appears to be lower than that for the non-vortex beam. The OAM statistical characteristics calculated at two different definitions of the specific OAM in the turbulent medium have been compared. In the first version, the random value of the OAM is normalized to the random values of the received power. In the second version, the random value of OAM is normalized to the value of the received power averaged over realizations of the medium. It has been shown that the use of the second definition gives higher values of the OAM variance. The maximal difference between the OAM values determined in the different ways is observed for high values of the coefficient of correlation between fluctuations of OAM and the received power. For the non-vortex beam, the variances of OAM fluctuations obtained with the different normalization methods coincide.

The results obtained in the paper can be used to assess the potentials of wireless information transmission systems employing wave beams carrying OAM, as well as in development of optical means for manipulating micron-sized particles.

## Funding

Russian Foundation for Basic Research (18-29-20115\18); Ministry of Science and Higher Education of the Russian Federation (budget funds for IAO SB RAS).

## 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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