## Abstract

As a means of increasing the channel capacity in free-space optical communication systems, two types of orbital angular momentum carrying beams, Bessel–Gauss and Laguerre–Gauss, are studied. In a series of numerical simulations, we show that Bessel–Gauss beams, pseudo-nondiffracting beams, outperform Laguerre–Gauss beams of various orders in channel efficiency and bit error rates.

© 2016 Optical Society of America

## 1. INTRODUCTION

Free-space optical (FSO) communication is the transmission of information over a distance between a transmitter and a receiver using optical wavelengths, i.e., ultraviolet, visible, and infrared. FSO communication contrasts with fiber-based communication systems as it does not require a physical communication link and relies on the atmosphere (or water) as the transmission medium as opposed to an optical fiber. This is valuable when it is necessary to communicate line-of-sight between non-fixed locations or when established (fiber-based) communication systems have been destroyed by natural disasters or hostile actors. Though frequency-division-multiplexed RF communication also uses the atmosphere as its transmission medium, FSO offers several important advantages, namely, higher modulation bandwidth allowing higher information capacity [1]; smaller beam divergence, which provides larger signal intensity at the receiver [2]; and improved security to prevent eavesdropping due to directionality and non-penetration of physical obstacles [3].

Due to the complexity of the information that needs to be transmitted or the length of time allowed for transmission, it is often necessary to increase the information capacity of the data link [4]. Typically, for FSO communication, one can control the polarization, frequency, and intensity of distinct light beams and thus multiplex together different signals; additionally, spatial and temporal methods can also be considered. Another option is to control the orbital angular momentum (OAM) thus allowing beams with different mode numbers to be multiplexed together and transmitted over the same link [5,6].

OAM is a property of a coherent light beam that arises from the azimuthal components of linear momentum acting at the radius of the beam with a dependency of $\mathrm{exp}(im\theta )$. The parameter, $m\in \mathbb{Z}$, is the topological charge or mode number and indicates that there is a theoretically infinite number of modes possible; due to noise, however, this is limited [7]. This creates a twisting of light beam with a helical phase front. Bessel [8], Bessel–Gauss [9], Laguerre–Gauss [10], Hermite–Gauss [11], and Mathieu [12] are all beam types that possess OAM properties. Without the presence of turbulence, OAM beams exhibit orthogonality, which is very useful for optical FSO communication because multiplexed beams will not interfere with each other, thus allowing recovery of each mode used. However, the presence of turbulence causes mixing of information between adjacent modes, which produces channel cross talk [13,14]. This results in the degradation of the signal and a loss of information.

This study will focus on two specific OAM beams, the Laguerre–Gauss beam (LGB) and Bessel–Gauss beam (BGB). LGBs arise from solving the paraxial wave equation with cylindrically symmetric coordinates and Laguerre functions. Similarly, BGBs arise from solving the paraxial wave equation with circular cylindrical coordinates and Bessel functions. BGBs, a pseudo-nondiffracting beam, can actually heal themselves after a partial obstruction is encountered in the propagation path [15] and have very little diffraction over a fixed propagation distance [16] as compared with LGBs (order 0). These properties, and the effects atmospheric turbulence play on it, will be studied in this paper through a series of numerical simulations. We hypothesize that the BGB, due to the limited diffraction and self-healing properties, will be more resilient to atmospheric turbulence. This added ability to mitigate turbulence, compared with LGBs, will allow BGBs to have a higher channel efficiency (percent of energy recovered in the correct demultiplexing mode) and lower bit error rate (BER).

The major contribution of this paper is a detailed numerical comparison between LGBs of various orders and BGBs. We explore several understudied aspects of OAM numerical simulations, including optical transformation sorting methods, BER, and mode set spacing. These numerical simulations are conducted with varying levels of turbulence.

In Section 2, we discuss the two fundamental beams that we will study, LGB and BGB, their properties and mathematical derivations. In Section 3, we describe how we numerically simulate the propagation through a turbulent atmosphere. In Section 4, the detection of OAM mode number is described. In Section 5, we conduct several numerical simulations to describe, at various levels of turbulence, how the different beam types perform with respect to channel efficiency, optimal mode sets, and data transmission. Conclusions are given in Section 6.

## 2. OAM IN DIFFERENT BEAMS

Mathematically, we can describe an electromagnetic wave as a field, $u(x,y,z;t)$, with spatial coordinates $(x,y,z)$ and time $t$, which follows the hyperbolic partial differential equation [17]:

where ${\nabla}^{2}$ is the Laplacian and $c$ is the speed of light. If we assume that the field variations are sinusoidal, $u(x,y,z;t)=U(x,y,z){e}^{-i\omega t}$, then we get the Helmholtz equation: where $k=\omega /c=2\pi /\lambda $ is the wavenumber and $\lambda $ is the wavelength. If we now change to cylindrical coordinates, Eq. (2) becomesIf we assume the problem is cylindrically symmetric and use cylindrical coordinates $(r,\phi ,z)$, where $r$ is the radial distance from the propagation axis, $\phi $ is the angle formed on the plane perpendicular to the propagation axis, and $z$ is the distance along the propagation axis, the LGB beam can be realized as

In Eqs. (5) and (6), ${C}_{\mathrm{LG}(p,m)}=\sqrt{2p!/(\pi (p+|m|)!)}$ is a normalization constant, ${L}_{p}^{m}(\xb7)$ is the generalized Laguerre–Gauss polynomial of $p$ (radial mode) and $m$ (angular mode), $\zeta (z)={\mathrm{tan}}^{-1}(z/{z}_{R})$ is the Gouy phase, $w(z)={w}_{0}\sqrt{1+(z/{z}_{R})}$ is the beam radius, ${w}_{0}$ is the beam waist, and ${z}_{R}=\pi {w}_{0}^{2}/\lambda $ is the Rayleigh range. It is important to note that the Gaussian phase profile, expressed as $\mathrm{exp}(im\theta )$, is what allows these beams to exhibit OAM. An example of a LGB with OAM mode $m=5$ and $p=0$ is shown in Fig. 1(a); Fig. 1(b) shows the same mode but with $p=5$.

Ideal Bessel beams are described as

where ${J}_{m}$ is the order $m$ Bessel function and $\beta $ is the radial frequency; $k=\sqrt{{k}_{z}^{2}+{\beta}^{2}}=2\pi /\lambda $. Bessel beams, in their solution to Eq. (4), exhibit distance agnostic intensity distribution and thus are considered diffraction-free beams. For a true Bessel beam to be created, it would require an infinite amount of energy to maintain the diffraction-free propagation; however, a Gaussian tapered Bessel beam, or BGB, can be created where, for a finite distance, the diffraction-free property holds (pseudo-diffraction-free beam) [16].Similar to LGBs, if we assume the problem is circularly symmetric, the BGB can be realized:

Optically, a BGB is produced by the superposition of Gaussian beams whose
axes are uniformly distributed on a cone [8]. The angular half-aperture of the cone,
${\theta}_{C}$, is related to the radial frequency,
$\beta $, as $\beta =k\text{\hspace{0.17em}}\mathrm{sin}({\theta}_{C})$ [9]. For a fixed ${\theta}_{C}$, as $z$ increases, the superposition of the
Gaussian beams will break apart and will result in a Gaussian beam. By
relating the ${\theta}_{C}$ to the angular spread of a Gaussian beam,
Gori *et al.* [9]
showed that for a propagation distance $Z$, $Z={w}_{0}/{\theta}_{C}$. Using this relation, one can
approximately find the radial frequency supporting a propagation distance
as such: $\beta =k\text{\hspace{0.17em}}\mathrm{sin}({w}_{0}/Z)$. We should note here that for
$\beta =0$, the special case of a Gaussian beam is
revealed. An example of a BGB ($\beta =350$) can be seen in Fig. 1(c).

Both BGBs and LGBs may be multiplexed together because of their orthogonality property. An example of multiplexed LGBs and BGBs can be seen in Fig. 2.

An expanded mathematical and historical comparison between LGBs and BGBs can be found in [16]. In the laboratory, OAM beams have been created by several methods, including modifying a laser beam with a computer-generated hologram [18], a spiral phase plate [19], or cylindrical lenses [20]. Furthermore, it has been shown in [5,6,21,22] that multiplexing and demultiplexing OAM beams is possible for communication links. In this report, we will rely on computer simulations, similar to [13,23–25], as we are interested in range propagation and want to control the level of turbulence in a repeatable fashion.

## 3. PROPAGATION AND TURBULENCE

To simulate the beam’s transmission through the atmosphere, we need to define how the beam will propagate from the transmitter to the receiver, as well as how the atmosphere will affect the beam’s field. The propagation of a beam can be described by the Fresnel diffraction integral with the well-known split-step Fourier method [26,27]. Turbulence was simulated by inserting random phase screens along the propagation path of the beam corresponding to the modified Kolmogorov turbulence model of Andrews [28]:

*et al.*[29] so as to closer match the theory for lower spatial frequencies. Figure 3 shows two examples of turbulence phase screens created using these methods. $N$ turbulence phase screens are created to approximate the turbulence strength over a distance $Z$. The beam at the $z=0$ plane is propagated a distance of $Z/N$. Next, a turbulence phase screen is applied to the beam’s field by multiplying the phase screen with the beams realization at the distance $z/N$. This process is continued until the beam is propagated the complete distance.

## 4. DETECTING OAM MODES

There are two popular methods for detecting OAM modes at the receiver: conjugate mode sorting [6,30] and optical transformation sorting [31]. For this study, we have chosen to use optical transformation sorting in our numerical simulation, but for background and to illustrate our choice, we will briefly describe both methods here.

#### A. Conjugate Mode Sorting

Conjugate mode sorting [6,30] is a method to determine the OAM mode of a detected beam based on its orthogonality properties. Given a transmitted OAM beam, ${u}_{m}(r,\theta ,Z)$, we cycle through the support of the mode set, ${u}_{n}{(r,\theta ,Z)}^{*}$, where $*$ is the conjugate, as seen in Fig. 4 forming the product ${u}_{m}(r,\theta ,Z){u}_{n}{(r,\theta ,Z)}^{*}$. If we detect intensity only at the origin, i.e., no doughnut mode, then the transmitted signal contains OAM mode $n$. This sorting method is dependent on having good alignment between the transmitter and the receiver; misalignment is shown to have comparable effects to turbulence in the correct determination of the OAM mode [6]. Due to the effect of turbulence, the normalized energy will not be concentrated exactly at the origin of the correct conjugate mode, thus we have to look at the relative energy across all the modes. For the non-multiplexing case, one can simply take the maximum value across the support of the mode set; for the multiplexing case, a threshold must be chosen so as to decide whether a mode is present or not in the signal.

In the laboratory, such a system could be designed using a single spatial light (SLM) which cycles through the various plausible conjugate modes, assuming the transmission time was sufficient to complete the range of test modes [32]. A more complex system could also be designed where the incoming beam is tested in series with multiple volume holograms with individual channel detectors [33]. Due to the limitation of testing demultiplexing modes in serial, we have chosen to simulate an optical transformation sorting method (described in the next section), which can test all demultiplexing modes at once.

#### B. Optical Transformation Sorting

In optical transformation mode sorting, a single transformation is used
to detect the presence of multiple OAM modes at once. We utilize a
method developed by Lavery *et al.* [31] called log-polar mode
sorting. In this method, the OAM beam arriving at the aperture is
first transformed from Cartesian coordinates to log-polar coordinates
according to

In the lab, such a system could be designed, as described in [31], using two SLMs with patterns ${\phi}_{1}(x,y)$ and ${\phi}_{2}(u,v)$:

#### C. Other Sorting Methods

Though not covered in this report, there are other methods for mode sorting, namely counting spiral fringes [36], measuring the Doppler effect [37], dove prism interferometers [38], and machine learning [39]. See [40] for a full discussion of these other methods.

## 5. NUMERICAL STUDIES

To test various properties of BGBs and LGBs, we developed code in MATLAB to simulate the creation, propagation through turbulence, and the detection of BGBs and LGBs of arbitrary order and mode.

For each beam type, we created a set of initial mode instances $S={\{{S}_{i}\}}_{i=1}^{M}$ for $M=15$ using equations Eqs. (5) and (9) with parameter values $\lambda =850\text{\hspace{0.17em}}\mathrm{nm}$ and ${w}_{0}=5\text{\hspace{0.17em}}\mathrm{cm}$. For the LGB, we simulated $p=0$, 1, 5, 10 and for the BGB, we set $\beta =250$. The size of the numerical grid was $1024\times 1024$ representing a 0.5 m square field.

A total propagation distance of $z=1\text{\hspace{0.17em}}\mathrm{km}$ was used with 20 turbulence screens placed equally along the propagation distance. The effective turbulence strength over the entire propagation path was chosen with ${C}_{n}^{2}=\{1\times {10}^{-16},1\times {10}^{-15},1\times {10}^{-14}\}$, representing weak to strong turbulence. We performed 1000 initializations for each beam type, mode number, and turbulence strength. These constants were chosen to match previous literature, especially [13], where the authors compared channel efficiency and independent channel BER for LGBs in free-space optical communication.

After the beams were propagated to $z=1\text{\hspace{0.17em}}\mathrm{km}$, we measured the energy contained in the OAM mode via the optical transformation sorting technique known as log-polar sorting. We looked at the finite collection of OAM states corresponding to $\tilde{S}={\{{S}_{i}\}}_{i=-10}^{M+10}$, again for $M=15$ for the possible demultiplexing modes. It is necessary to examine the OAM modes surrounding the true mode, as we expect the energy to spread from the true mode symmetrically to its neighbors.

#### A. Channel Efficiency

Channel efficiency can be characterized by examining the energy that is demultiplexed into each mode. The plot in Fig. 6 shows the results for BGBs and LGBs (of various orders $m$) with respect to different turbulence levels. We first notice, as expected, increased turbulence strength causes a decrease in channel efficiency. Due to the spreading of the beam for higher mode ($m$) numbers, there is a greater effect on these high modes. For our simulations, we also note that the BGB performed better than the LGBs of all orders ($p$) tested across all turbulence levels. This indicates, when possible due to propagation distances, that one should favor a BGB for communication tasks as it maximizes the received energy for each mode. We also note there is an improvement in channel efficiency when increasing the order of the LGB. Across the different turbulence levels and modes we notice that for LGBs with order ${p}_{2}>{p}_{1}$, the channel efficiency for the ${p}_{2}$ beam will be greater. From the results, it is also clear there is a convergence of the channel efficiency for the LGB; note the difference between the $p=0$ and $p=5$ compared with the $p=5$ and $p=10$ beams.

#### B. Multiplexing Channel Efficiency

Here we simulate the effects of multiplexing beams. We look at four sets of OAM modes, which are defined in Table 1. At random, we encode 4-bit numbers a total of 5000 times with multiplexed BGB and LGB OAM beams. For each bit-encoding, the OAM modes in each mode set are used to represent the binary digits. As in the channel efficiency experiment, multiplexed modes are propagated through various levels of turbulence a distance of 1 km. At the receiver, we examine the normalized energy found for the modes in the mode set under consideration. A threshold $T$ is applied to the normalized energies. For each mode, if its normalized energy is above $T$ then that mode is declared to be present and its bit representation is on; otherwise the mode is declared not present and its bit representation is off.

To measure the accuracy of the mode set at a particular turbulence we use the BER. The BER is defined as the ratio of incorrect bits to total bits; we average over the 5000 samples.

We begin by optimizing a constant threshold, $T\in [0.005,0.15]$, which produces the results shown in Table 2. We notice that the BGB outperforms the LGB in the multiplexing experiment as it did in the single channel efficiency experiment. It is also evident, for most cases, that increasing the separation between OAM modes, decreases the BER. This makes sense as turbulence causes OAM modes to spread energy into adjacent modes; we also noticed this in the channel efficiency experiments. It should be noted for the results in this paper concerning BER we are not stating that such a system could be realized without additional technologies, such as forward error correcting codes [41], multiple-input multiple-output (MIMO) [42], spatial diversity [43], adaptive optics [44], low-density parity-check (LPDC) codes [33], etc. For this paper, we simply want to compare the BGBs and LGBs, and multiplexed communication is one such means to do this.

Recall that the channel efficiency results presented in Fig. 6 indicated an exponential decrease in channel efficiency as the mode number increases. This indicates that an exponential threshold should give better results as large mode numbers have a lower channel efficiency. In Table 3 we optimize an exponential threshold, $T=a\text{\hspace{0.17em}}\mathrm{exp}({m}^{-b})$, where $m$ is the mode number and $a,b\ge 0$ are constants. We see an improvement (where BER was not zero in the linear case) in BER for all modes and turbulence levels. The largest improvements can be seen from moving from single mode spacing to a spacing of two; after this, continual improvement is seen in some cases but the effects are less. Due to the spreading of the beams at higher mode numbers, one cannot simply increase the mode spacing and hope for continual improvements as less energy tends to fall on the detector due to diffraction.

Similar to Section 5.A, we can also look at the channel efficiencies, which here we will call encoding efficiencies. We measure the ratio of received energy in the bits that are on to the total energy received for the encoding mode set; these results are shown in Figs. 7 and 8. These results match closely with what is presented in the BER tables. Comparing the BGB to LGB, we noticed that the spread in the efficiencies for increasing the mode spacing is more pronounced in the BGB indicating it is more of an advantage. This is especially noticeable in the higher turbulence cases, which can be explained by the BGB pseudo-nondiffracting window. The results presented in Figs. 7 and 8 do not make any assumption on the number of modes present in a received signal unlike in Fig. 6 where only one mode was assumed to be present; thus only a general comparison should be made, i.e., the trend of lower efficiencies with higher turbulence values.

## 6. CONCLUSIONS

In this study, we examined how the presence of turbulence affected the transmission of diffracting and pseudo-nondiffracting OAM beams. We have shown that both LGBs and BGBs are acceptable for use in free-space optical communication with a preference for BGBs or higher-order LGBs. For individual channel efficiency, we showed a large improvement can be made in a communication system by utilizing BGBs over LGBs, especially at high levels of turbulence. We also verified earlier results which indicated a drop in channel efficiency with increasing mode numbers and a reduction in coding errors when increasing the mode spacing.

We see future work in expanding upon the types of beams tested, e.g., Airy, Mathieu, Hermite–Gauss, as well as the different propagation distances.

## Funding

U.S. Naval Research Laboratory (NRL) NRL 6.1 Base Program.

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