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 exp(imθ). The parameter, mZ, 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]:

2ut2=c22u,
where 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)eiωt, then we get the Helmholtz equation:
2U+k2U=0,
where k=ω/c=2π/λ is the wavenumber and λ is the wavelength. If we now change to cylindrical coordinates, Eq. (2) becomes
1rr(rU¯r)+2U¯z2+k2U¯=0.
Using a simplification, V(r,z)=U¯(r,z)eikz, and the paraxial assumption, 2V/z2=0, we get the paraxial wave equation:
1rr(rVr)+2ikVz=0.
By solving Eq. (4) in different coordinate systems with different symmetry assumptions, we can define several different beams that carry OAM.

If we assume the problem is cylindrically symmetric and use cylindrical coordinates (r,φ,z), where r is the radial distance from the propagation axis, φ 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

uLG(p,m)(r,θ,z)=CLG(p,m)w(z)(r2w(z))|m|Lp|m|(2r2w2(z))×exp[r2w2(z)ikr2z2(z2+zR2)]×exp[i(2p+|m|+1)ζ(z)]exp(imθ),
or when z=0,
uLG(p,m)(r,θ,z=0)=CLG(p,m)(r2w0)|m|Lp|m|(2r2)×exp(imθ).

In Eqs. (5) and (6), CLG(p,m)=2p!/(π(p+|m|)!) is a normalization constant, Lpm(·) is the generalized Laguerre–Gauss polynomial of p (radial mode) and m (angular mode), ζ(z)=tan1(z/zR) is the Gouy phase, w(z)=w01+(z/zR) is the beam radius, w0 is the beam waist, and zR=πw02/λ is the Rayleigh range. It is important to note that the Gaussian phase profile, expressed as exp(imθ), 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.

 figure: Fig. 1.

Fig. 1. (a) uLG(0,5), (b) uLG(5,5), and (c) uBG(5) with β=350 at z=0. Phase information is represented by the hue while the energy is represented by the normalized intensity. Color bar is in radians.

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Ideal Bessel beams are described as

uB(m)(r,θ,z)=CBJm(βr)exp(ikzz)exp(imθ),
where Jm is the order m Bessel function and β is the radial frequency; k=kz2+β2=2π/λ. 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:

uBG(m)(r,θ,z)=CBGw0w(z)Jm(βr1+iz/zr)×exp[i(kβ22k)zζ(z)+1w2(z)]×exp[(ik2R(z))(r2+β2zrk2)]exp(imθ),
or when z=0,
uBG(m)(r,θ,z=0)=CBGJm(βr)exp[(r/w0)2]exp(imθ),
where CBG is a constant.

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, θC, is related to the radial frequency, β, as β=ksin(θC) [9]. For a fixed θC, as z increases, the superposition of the Gaussian beams will break apart and will result in a Gaussian beam. By relating the θC to the angular spread of a Gaussian beam, Gori et al. [9] showed that for a propagation distance Z, Z=w0/θC. Using this relation, one can approximately find the radial frequency supporting a propagation distance as such: β=ksin(w0/Z). We should note here that for β=0, the special case of a Gaussian beam is revealed. An example of a BGB (β=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.

 figure: Fig. 2.

Fig. 2. Effects of multiplexing different OAM modes together. Top row is LGB (p=0) and bottom row is BGB (β=350). Left column is m={1,2}, middle column is m={1,3,4}, and right column is m={1,2,3,4}. Color bar is in radians.

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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 [13,2325], 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]:

Ψ(κ)=0.033Cn2(κ2+1/L02)11/6exp(κ2/κ2)×(1+1.082(κ/κ)0.254(κ/κ)7/6),
where κ is the spatial frequency (rad/m); L0 is the outer scale of turbulence; 0 is the inner scale of turbulence; κ=κ/(3.30); and atmospheric turbulence strength, Cn2, is the structure constant of the index refraction, a measure of the strength of the turbulence. The Fried parameter, defined as
r0=(0.423k2sec(α)PathCn2(z)dz)3/5,
where α is the zenith angle, is a measure of the quality of the transmission through the atmosphere along the defined path. Assuming that we have a constant turbulence strength over the propagation distance and α=0, we may relate Cn2 to the Fried parameter r0:
r0=(0.423k2ΔzCn2)3/5.
The phase screen P is created by
P=F1{ΨC},
where F1 is the inverse 2D Fourier transform and C is a collection of complex Gaussian random variables of the same size as the numerical propagation grid. Subharmonics are added to Eq. (13) by the methods of Lane 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.

 figure: Fig. 3.

Fig. 3. Example of two realizations of turbulence screens created by the described methods; scale is in radians.

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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, um(r,θ,Z), we cycle through the support of the mode set, un(r,θ,Z)*, where * is the conjugate, as seen in Fig. 4 forming the product um(r,θ,Z)un(r,θ,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.

 figure: Fig. 4.

Fig. 4. Conjugate mode sorting of LGB, m=5.

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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

(x,y)(ρ,θ)=(log(x2+y2),arctan(y/x)).
This transformation (implemented in an optical configuration in [34]) can be seen geometrically as mapping a ring to rectangle, as seen in Fig. 5. The mapping is translating rotation and scaling to vertical and horizontal shifts. After the mapping, a Fourier transform is applied to the new coordinate space. By measuring the intensity of the integer shifts in the Fourier plane (summed along the y dimension in this case) one can detect the relative intensity in various modes. For non-multiplexed beams taking the maximum of these Fourier intensities over the support of the mode set will result in the detected OAM mode. For multiplexed beams, a relative intensity threshold must be chosen, i.e., p% of the total Fourier intensity.

 figure: Fig. 5.

Fig. 5. Optical transformation mode sorting (log-polar) of LGB, m=5.

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In the lab, such a system could be designed, as described in [31], using two SLMs with patterns φ1(x,y) and φ2(u,v):

φ1(x,y)=2πaλf[yarctan(yx)xlog(x2+y2b)+x],
φ2(u,v)=2πabλfexp(ua)cos(va),
where a and b are scaling and translation parameters and φ1 transforms the beam and φ2 corrects the phase distortion. The authors in [35] note that the demultiplexing system is very sensitive to the placement of transforming elements, where a small displacement error can cause phase errors. We do not test them here, but improvements can be made to the log-polar sorting method by utilizing beam copy through fan-out holograms; see [35].

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={Si}i=1M for M=15 using equations Eqs. (5) and (9) with parameter values λ=850nm and w0=5cm. For the LGB, we simulated p=0, 1, 5, 10 and for the BGB, we set β=250. The size of the numerical grid was 1024×1024 representing a 0.5 m square field.

A total propagation distance of z=1km was used with 20 turbulence screens placed equally along the propagation distance. The effective turbulence strength over the entire propagation path was chosen with Cn2={1×1016,1×1015,1×1014}, 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=1km, 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 S˜={Si}i=10M+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 p2>p1, the channel efficiency for the p2 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.

 figure: Fig. 6.

Fig. 6. LGBs with order p=0, 1, 5, and 10 and BGBs with turbulence levels CN2=1×1016 (no marker), CN2=1×1015(+) and CN2=1×1014(*). Due to symmetry we only show the positive modes. Note the y-axis is a log scale.

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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.

Tables Icon

Table 1. Collection of OAM Modes Used

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[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.

Tables Icon

Table 2. Lower Bound for the BER Found Taking the Minimum for a Threshold T[0.005,0.15] with Numerical Resolution of 0.005a

Tables Icon

Table 3. Lower Bound for the BER Found Taking the Minimum Over an Exponential Fit Thresholda

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=aexp(mb), where m is the mode number and a,b0 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.

 figure: Fig. 7.

Fig. 7. LGBs for four different mode sets and three different turbulences (Cn2=1×1016 no symbol, Cn2=1×1015+,” and Cn2=1×1014*”) encoding channel efficiencies ordered by number of bits active.

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 figure: Fig. 8.

Fig. 8. BGBs for four different mode sets and three different turbulences (Cn2=1×1016 no symbol, Cn2=1×1015+,” and Cn2=1×1014*”) encoding channel efficiencies ordered by number of bits active.

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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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27. A. Belmonte, “Feasibility study for the simulation of beam propagation: consideration of coherent lidar performance,” Appl. Opt. 39, 5426–5445 (2000). [CrossRef]  

28. L. C. Andrews, “An analytical model for the refractive index power spectrum and its application to optical scintillations in the atmosphere,” J. Mod. Opt. 39, 1849–1853 (1992). [CrossRef]  

29. R. G. Lane, A. Glindemann, and J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992). [CrossRef]  

30. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001). [CrossRef]  

31. M. P. J. Lavery, G. C. G. Berkhout, J. Courtial, and M. J. Padgett, “Measurement of the light orbital angular momentum spectrum using an optical geometric transformation,” J. Opt. 13, 064006 (2011). [CrossRef]  

32. A. Forbes, A. Dudley, and M. McLaren, “Creation and detection of optical modes with spatial light modulators,” Adv. Opt. Photonics 8, 200–227 (2016). [CrossRef]  

33. I. B. Djordjevic and M. Arabaci, “LDPC-coded orbital angular momentum (OAM) modulation for free-space optical communication,” Opt. Express 18, 24722–24728 (2010). [CrossRef]  

34. O. Bryngdahl, “Geometrical transformations in optics,” J. Opt. Soc. Am. 64, 1092–1099 (1974). [CrossRef]  

35. M. Malik, M. Mirhosseini, M. P. Lavery, J. Leach, M. J. Padgett, and R. W. Boyd, “Direct measurement of a 27-dimensional orbital-angular-momentum state vector,” Nat. Commun. 5, 3115 (2014). [CrossRef]  

36. M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1997). [CrossRef]  

37. M. P. J. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using light’s orbital angular momentum,” Science 341, 537–540 (2013). [CrossRef]  

38. J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88, 257901 (2002). [CrossRef]  

39. M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, “Communication with spatially modulated light through turbulent air across Vienna,” New J. Phys. 16, 113028 (2014). [CrossRef]  

40. D. L. Andrews and M. Babiker, eds., The Angular Momentum of Light (Cambridge University, 2012).

41. W. C. Huffman and V. Pless, Fundamentals of Error-correcting Codes (Cambridge University, 2010).

42. D. Gesbert, M. Shafi, D. Shiu, P. J. Smith, and A. Naguib, “From theory to practice: an overview of MIMO space-time coded wireless systems,” IEEE J. Sel. Areas Commun. 21, 281–302 (2003). [CrossRef]  

43. Y. Ren, Z. Wang, G. Xie, L. Li, A. J. Willner, Y. Cao, Z. Zhao, Y. Yan, N. Ahmed, N. Ashrafi, S. Ashrafi, R. Bock, M. Tur, and A. E. Willner, “Atmospheric turbulence mitigation in an OAM-based MIMO free-space optical link using spatial diversity combined with MIMO equalization,” Opt. Lett. 41, 2406–2409 (2016). [CrossRef]  

44. Y. Ren, G. Xie, H. Huang, C. Bao, Y. Yan, N. Ahmed, M. P. J. Lavery, B. I. Erkmen, S. Dolinar, M. Tur, M. A. Neifeld, M. J. Padgett, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Adaptive optics compensation of multiple orbital angular momentum beams propagating through emulated atmospheric turbulence,” Opt. Lett. 39, 2845–2848 (2014). [CrossRef]  

References

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  1. H. Willebrand and B. S. Ghuman, in Free Space Optics: Enabling Optical Connectivity in Today’s Networks (SAMS, 2002).
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  3. J. C. Juarez, A. Dwivedi, A. R. Hammons, S. D. Jones, V. Weerackody, and R. A. Nichols, “Free-space optical communications for next-generation military networks,” IEEE Commun. Mag. 44(11), 46–51 (2006).
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  6. G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12, 5448–5456 (2004).
    [Crossref]
  7. B. Guan, R. P. Scott, C. Qin, N. K. Fontaine, T. Su, C. Ferrari, M. Cappuzzo, F. Klemens, B. Keller, M. Earnshaw, and S. J. B. Yoo, “Free-space coherent optical communication with orbital angular, momentum multiplexing/demultiplexing using a hybrid 3D photonic integrated circuit,” Opt. Express 22, 145–156 (2014).
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  8. J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
    [Crossref]
  9. F. Gori, G. Guattari, and C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
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  10. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
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  12. J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, and S. Chávez-Cerda, “Alternative formulation for invariant optical fields: Mathieu beams,” Opt. Lett. 25, 1493–1495 (2000).
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  13. J. A. Anguita, M. A. Neifeld, and B. V. Vasic, “Turbulence-induced channel crosstalk in an orbital angular momentum-multiplexed free-space optical link,” Appl. Opt. 47, 2414–2429 (2008).
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  14. W. Nelson, J. P. Palastro, C. C. Davis, and P. Sprangle, “Propagation of Bessel and Airy beams through atmospheric turbulence,” J. Opt. Soc. Am. A 31, 603–609 (2014).
    [Crossref]
  15. M. Anguiano-Morales, M. M. Méndez-Otero, M. D. Iturbe-Castillo, and S. Chávez-Cerda, “Conical dynamics of Bessel beams,” Opt. Eng. 46, 078001 (2007).
    [Crossref]
  16. R. L. Nowack, “A tale of two beams: an elementary overview of Gaussian beams and Bessel beams,” Stud. Geophys. Geod. 56, 355–372 (2012).
    [Crossref]
  17. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 2005), Vol. 52.
  18. N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17, 221–223 (1992).
    [Crossref]
  19. M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327 (1994).
    [Crossref]
  20. M. W. Beijersbergen, L. Allen, H. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
    [Crossref]
  21. T. Su, R. P. Scott, S. S. Djordjevic, N. K. Fontaine, D. J. Geisler, X. Cai, and S. J. B. Yoo, “Demonstration of free space coherent optical communication using integrated silicon photonic orbital angular momentum devices,” Opt. Express 20, 9396–9402 (2012).
    [Crossref]
  22. H. Huang, G. Xie, Y. Yan, N. Ahmed, Y. Ren, Y. Yue, D. Rogawski, M. J. Willner, B. I. Erkmen, K. M. Birnbaum, S. J. Dolinar, M. P. J. Lavery, M. J. Padgett, M. Tur, and A. E. Willner, “100 Tbit/s free-space data link enabled by three-dimensional multiplexing of orbital angular momentum, polarization, and wavelength,” Opt. Lett. 39, 197–200 (2014).
    [Crossref]
  23. C. Chen and H. Yang, “Characterizing the radial content of orbital-angular-momentum photonic states impaired by weak-to-strong atmospheric turbulence,” Opt. Express 24, 19713–19727 (2016).
    [Crossref]
  24. L.-G. Wang and W.-W. Zheng, “The effect of atmospheric turbulence on the propagation properties of optical vortices formed by using coherent laser beam arrays,” J. Opt. A 11, 065703 (2009).
    [Crossref]
  25. J. Mendoza-Hernández, M. L. Arroyo-Carrasco, M. D. Iturbe-Castillo, and S. Chávez-Cerda, “Laguerre–Gauss beams versus Bessel beams showdown: peer comparison,” Opt. Lett. 40, 3739–3742 (2015).
    [Crossref]
  26. R. H. Hardin and F. D. Tappert, “Applications of the split-step Fourier method to the numerical solution of nonlinear and variable coefficient wave equations,” SIAM Rev. 15, 423 (1973).
  27. A. Belmonte, “Feasibility study for the simulation of beam propagation: consideration of coherent lidar performance,” Appl. Opt. 39, 5426–5445 (2000).
    [Crossref]
  28. L. C. Andrews, “An analytical model for the refractive index power spectrum and its application to optical scintillations in the atmosphere,” J. Mod. Opt. 39, 1849–1853 (1992).
    [Crossref]
  29. R. G. Lane, A. Glindemann, and J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
    [Crossref]
  30. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
    [Crossref]
  31. M. P. J. Lavery, G. C. G. Berkhout, J. Courtial, and M. J. Padgett, “Measurement of the light orbital angular momentum spectrum using an optical geometric transformation,” J. Opt. 13, 064006 (2011).
    [Crossref]
  32. A. Forbes, A. Dudley, and M. McLaren, “Creation and detection of optical modes with spatial light modulators,” Adv. Opt. Photonics 8, 200–227 (2016).
    [Crossref]
  33. I. B. Djordjevic and M. Arabaci, “LDPC-coded orbital angular momentum (OAM) modulation for free-space optical communication,” Opt. Express 18, 24722–24728 (2010).
    [Crossref]
  34. O. Bryngdahl, “Geometrical transformations in optics,” J. Opt. Soc. Am. 64, 1092–1099 (1974).
    [Crossref]
  35. M. Malik, M. Mirhosseini, M. P. Lavery, J. Leach, M. J. Padgett, and R. W. Boyd, “Direct measurement of a 27-dimensional orbital-angular-momentum state vector,” Nat. Commun. 5, 3115 (2014).
    [Crossref]
  36. M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1997).
    [Crossref]
  37. M. P. J. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using light’s orbital angular momentum,” Science 341, 537–540 (2013).
    [Crossref]
  38. J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88, 257901 (2002).
    [Crossref]
  39. M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, “Communication with spatially modulated light through turbulent air across Vienna,” New J. Phys. 16, 113028 (2014).
    [Crossref]
  40. D. L. Andrews and M. Babiker, eds., The Angular Momentum of Light (Cambridge University, 2012).
  41. W. C. Huffman and V. Pless, Fundamentals of Error-correcting Codes (Cambridge University, 2010).
  42. D. Gesbert, M. Shafi, D. Shiu, P. J. Smith, and A. Naguib, “From theory to practice: an overview of MIMO space-time coded wireless systems,” IEEE J. Sel. Areas Commun. 21, 281–302 (2003).
    [Crossref]
  43. Y. Ren, Z. Wang, G. Xie, L. Li, A. J. Willner, Y. Cao, Z. Zhao, Y. Yan, N. Ahmed, N. Ashrafi, S. Ashrafi, R. Bock, M. Tur, and A. E. Willner, “Atmospheric turbulence mitigation in an OAM-based MIMO free-space optical link using spatial diversity combined with MIMO equalization,” Opt. Lett. 41, 2406–2409 (2016).
    [Crossref]
  44. Y. Ren, G. Xie, H. Huang, C. Bao, Y. Yan, N. Ahmed, M. P. J. Lavery, B. I. Erkmen, S. Dolinar, M. Tur, M. A. Neifeld, M. J. Padgett, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Adaptive optics compensation of multiple orbital angular momentum beams propagating through emulated atmospheric turbulence,” Opt. Lett. 39, 2845–2848 (2014).
    [Crossref]

2016 (3)

2015 (1)

2014 (6)

H. Huang, G. Xie, Y. Yan, N. Ahmed, Y. Ren, Y. Yue, D. Rogawski, M. J. Willner, B. I. Erkmen, K. M. Birnbaum, S. J. Dolinar, M. P. J. Lavery, M. J. Padgett, M. Tur, and A. E. Willner, “100 Tbit/s free-space data link enabled by three-dimensional multiplexing of orbital angular momentum, polarization, and wavelength,” Opt. Lett. 39, 197–200 (2014).
[Crossref]

M. Malik, M. Mirhosseini, M. P. Lavery, J. Leach, M. J. Padgett, and R. W. Boyd, “Direct measurement of a 27-dimensional orbital-angular-momentum state vector,” Nat. Commun. 5, 3115 (2014).
[Crossref]

B. Guan, R. P. Scott, C. Qin, N. K. Fontaine, T. Su, C. Ferrari, M. Cappuzzo, F. Klemens, B. Keller, M. Earnshaw, and S. J. B. Yoo, “Free-space coherent optical communication with orbital angular, momentum multiplexing/demultiplexing using a hybrid 3D photonic integrated circuit,” Opt. Express 22, 145–156 (2014).
[Crossref]

W. Nelson, J. P. Palastro, C. C. Davis, and P. Sprangle, “Propagation of Bessel and Airy beams through atmospheric turbulence,” J. Opt. Soc. Am. A 31, 603–609 (2014).
[Crossref]

Y. Ren, G. Xie, H. Huang, C. Bao, Y. Yan, N. Ahmed, M. P. J. Lavery, B. I. Erkmen, S. Dolinar, M. Tur, M. A. Neifeld, M. J. Padgett, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Adaptive optics compensation of multiple orbital angular momentum beams propagating through emulated atmospheric turbulence,” Opt. Lett. 39, 2845–2848 (2014).
[Crossref]

M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, “Communication with spatially modulated light through turbulent air across Vienna,” New J. Phys. 16, 113028 (2014).
[Crossref]

2013 (1)

M. P. J. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using light’s orbital angular momentum,” Science 341, 537–540 (2013).
[Crossref]

2012 (3)

T. Su, R. P. Scott, S. S. Djordjevic, N. K. Fontaine, D. J. Geisler, X. Cai, and S. J. B. Yoo, “Demonstration of free space coherent optical communication using integrated silicon photonic orbital angular momentum devices,” Opt. Express 20, 9396–9402 (2012).
[Crossref]

R. L. Nowack, “A tale of two beams: an elementary overview of Gaussian beams and Bessel beams,” Stud. Geophys. Geod. 56, 355–372 (2012).
[Crossref]

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

2011 (1)

M. P. J. Lavery, G. C. G. Berkhout, J. Courtial, and M. J. Padgett, “Measurement of the light orbital angular momentum spectrum using an optical geometric transformation,” J. Opt. 13, 064006 (2011).
[Crossref]

2010 (1)

2009 (1)

L.-G. Wang and W.-W. Zheng, “The effect of atmospheric turbulence on the propagation properties of optical vortices formed by using coherent laser beam arrays,” J. Opt. A 11, 065703 (2009).
[Crossref]

2008 (1)

2007 (1)

M. Anguiano-Morales, M. M. Méndez-Otero, M. D. Iturbe-Castillo, and S. Chávez-Cerda, “Conical dynamics of Bessel beams,” Opt. Eng. 46, 078001 (2007).
[Crossref]

2006 (1)

J. C. Juarez, A. Dwivedi, A. R. Hammons, S. D. Jones, V. Weerackody, and R. A. Nichols, “Free-space optical communications for next-generation military networks,” IEEE Commun. Mag. 44(11), 46–51 (2006).
[Crossref]

2005 (1)

2004 (1)

2003 (1)

D. Gesbert, M. Shafi, D. Shiu, P. J. Smith, and A. Naguib, “From theory to practice: an overview of MIMO space-time coded wireless systems,” IEEE J. Sel. Areas Commun. 21, 281–302 (2003).
[Crossref]

2002 (1)

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88, 257901 (2002).
[Crossref]

2001 (1)

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
[Crossref]

2000 (2)

1997 (1)

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1997).
[Crossref]

1994 (1)

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327 (1994).
[Crossref]

1993 (1)

M. W. Beijersbergen, L. Allen, H. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[Crossref]

1992 (4)

N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17, 221–223 (1992).
[Crossref]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref]

L. C. Andrews, “An analytical model for the refractive index power spectrum and its application to optical scintillations in the atmosphere,” J. Mod. Opt. 39, 1849–1853 (1992).
[Crossref]

R. G. Lane, A. Glindemann, and J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
[Crossref]

1987 (2)

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref]

F. Gori, G. Guattari, and C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[Crossref]

1974 (1)

1973 (2)

R. H. Hardin and F. D. Tappert, “Applications of the split-step Fourier method to the numerical solution of nonlinear and variable coefficient wave equations,” SIAM Rev. 15, 423 (1973).

A. Siegman, “Hermite–Gaussian functions of complex argument as optical-beam eigenfunctions,” J. Opt. Soc. Am. 63, 1093–1094 (1973).
[Crossref]

Ahmed, N.

Allen, L.

M. W. Beijersbergen, L. Allen, H. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[Crossref]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref]

Andrews, L. C.

L. C. Andrews, “An analytical model for the refractive index power spectrum and its application to optical scintillations in the atmosphere,” J. Mod. Opt. 39, 1849–1853 (1992).
[Crossref]

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 2005), Vol. 52.

Anguiano-Morales, M.

M. Anguiano-Morales, M. M. Méndez-Otero, M. D. Iturbe-Castillo, and S. Chávez-Cerda, “Conical dynamics of Bessel beams,” Opt. Eng. 46, 078001 (2007).
[Crossref]

Anguita, J. A.

Arabaci, M.

Arroyo-Carrasco, M. L.

Ashrafi, N.

Ashrafi, S.

Bao, C.

Barnett, S. M.

M. P. J. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using light’s orbital angular momentum,” Science 341, 537–540 (2013).
[Crossref]

G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12, 5448–5456 (2004).
[Crossref]

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88, 257901 (2002).
[Crossref]

Beijersbergen, M. W.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327 (1994).
[Crossref]

M. W. Beijersbergen, L. Allen, H. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[Crossref]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref]

Belmonte, A.

Berkhout, G. C. G.

M. P. J. Lavery, G. C. G. Berkhout, J. Courtial, and M. J. Padgett, “Measurement of the light orbital angular momentum spectrum using an optical geometric transformation,” J. Opt. 13, 064006 (2011).
[Crossref]

Birnbaum, K. M.

Bock, R.

Boyd, R. W.

Bryngdahl, O.

Cai, X.

Cao, Y.

Cappuzzo, M.

Chávez-Cerda, S.

Chen, C.

Coerwinkel, R. P. C.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327 (1994).
[Crossref]

Courtial, J.

M. P. J. Lavery, G. C. G. Berkhout, J. Courtial, and M. J. Padgett, “Measurement of the light orbital angular momentum spectrum using an optical geometric transformation,” J. Opt. 13, 064006 (2011).
[Crossref]

G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12, 5448–5456 (2004).
[Crossref]

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88, 257901 (2002).
[Crossref]

Dainty, J. C.

R. G. Lane, A. Glindemann, and J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
[Crossref]

Davis, C. C.

Djordjevic, I. B.

Djordjevic, S. S.

Dolinar, S.

Y. Ren, G. Xie, H. Huang, C. Bao, Y. Yan, N. Ahmed, M. P. J. Lavery, B. I. Erkmen, S. Dolinar, M. Tur, M. A. Neifeld, M. J. Padgett, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Adaptive optics compensation of multiple orbital angular momentum beams propagating through emulated atmospheric turbulence,” Opt. Lett. 39, 2845–2848 (2014).
[Crossref]

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Dolinar, S. J.

Dudley, A.

A. Forbes, A. Dudley, and M. McLaren, “Creation and detection of optical modes with spatial light modulators,” Adv. Opt. Photonics 8, 200–227 (2016).
[Crossref]

Durnin, J.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref]

Dwivedi, A.

J. C. Juarez, A. Dwivedi, A. R. Hammons, S. D. Jones, V. Weerackody, and R. A. Nichols, “Free-space optical communications for next-generation military networks,” IEEE Commun. Mag. 44(11), 46–51 (2006).
[Crossref]

Earnshaw, M.

Eberly, J. H.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref]

Erkmen, B. I.

Fazal, I. M.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Ferrari, C.

Fickler, R.

M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, “Communication with spatially modulated light through turbulent air across Vienna,” New J. Phys. 16, 113028 (2014).
[Crossref]

Fink, M.

M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, “Communication with spatially modulated light through turbulent air across Vienna,” New J. Phys. 16, 113028 (2014).
[Crossref]

Fontaine, N. K.

Forbes, A.

A. Forbes, A. Dudley, and M. McLaren, “Creation and detection of optical modes with spatial light modulators,” Adv. Opt. Photonics 8, 200–227 (2016).
[Crossref]

Franke-Arnold, S.

G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12, 5448–5456 (2004).
[Crossref]

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88, 257901 (2002).
[Crossref]

Geisler, D. J.

Gesbert, D.

D. Gesbert, M. Shafi, D. Shiu, P. J. Smith, and A. Naguib, “From theory to practice: an overview of MIMO space-time coded wireless systems,” IEEE J. Sel. Areas Commun. 21, 281–302 (2003).
[Crossref]

Ghuman, B. S.

H. Willebrand and B. S. Ghuman, in Free Space Optics: Enabling Optical Connectivity in Today’s Networks (SAMS, 2002).

Gibson, G.

Glindemann, A.

R. G. Lane, A. Glindemann, and J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
[Crossref]

Gori, F.

F. Gori, G. Guattari, and C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[Crossref]

Gorshkov, V. N.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1997).
[Crossref]

Guan, B.

Guattari, G.

F. Gori, G. Guattari, and C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[Crossref]

Gutiérrez-Vega, J. C.

Hammons, A. R.

J. C. Juarez, A. Dwivedi, A. R. Hammons, S. D. Jones, V. Weerackody, and R. A. Nichols, “Free-space optical communications for next-generation military networks,” IEEE Commun. Mag. 44(11), 46–51 (2006).
[Crossref]

Handsteiner, J.

M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, “Communication with spatially modulated light through turbulent air across Vienna,” New J. Phys. 16, 113028 (2014).
[Crossref]

Hardin, R. H.

R. H. Hardin and F. D. Tappert, “Applications of the split-step Fourier method to the numerical solution of nonlinear and variable coefficient wave equations,” SIAM Rev. 15, 423 (1973).

Heckenberg, N. R.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1997).
[Crossref]

N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17, 221–223 (1992).
[Crossref]

Huang, H.

Huffman, W. C.

W. C. Huffman and V. Pless, Fundamentals of Error-correcting Codes (Cambridge University, 2010).

Iturbe-Castillo, M. D.

Jones, S. D.

J. C. Juarez, A. Dwivedi, A. R. Hammons, S. D. Jones, V. Weerackody, and R. A. Nichols, “Free-space optical communications for next-generation military networks,” IEEE Commun. Mag. 44(11), 46–51 (2006).
[Crossref]

Juarez, J. C.

J. C. Juarez, A. Dwivedi, A. R. Hammons, S. D. Jones, V. Weerackody, and R. A. Nichols, “Free-space optical communications for next-generation military networks,” IEEE Commun. Mag. 44(11), 46–51 (2006).
[Crossref]

Keller, B.

Klemens, F.

Krenn, M.

M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, “Communication with spatially modulated light through turbulent air across Vienna,” New J. Phys. 16, 113028 (2014).
[Crossref]

Kristensen, M.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327 (1994).
[Crossref]

Lane, R. G.

R. G. Lane, A. Glindemann, and J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
[Crossref]

Lavery, M. P.

M. Malik, M. Mirhosseini, M. P. Lavery, J. Leach, M. J. Padgett, and R. W. Boyd, “Direct measurement of a 27-dimensional orbital-angular-momentum state vector,” Nat. Commun. 5, 3115 (2014).
[Crossref]

Lavery, M. P. J.

Leach, J.

M. Malik, M. Mirhosseini, M. P. Lavery, J. Leach, M. J. Padgett, and R. W. Boyd, “Direct measurement of a 27-dimensional orbital-angular-momentum state vector,” Nat. Commun. 5, 3115 (2014).
[Crossref]

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88, 257901 (2002).
[Crossref]

Li, L.

Mair, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
[Crossref]

Majumdar, A. K.

A. K. Majumdar and J. C. Ricklin, Free-Space Laser Communications: Principles and Advances (Springer, 2010), Vol. 2.

Malik, M.

M. Malik, M. Mirhosseini, M. P. Lavery, J. Leach, M. J. Padgett, and R. W. Boyd, “Direct measurement of a 27-dimensional orbital-angular-momentum state vector,” Nat. Commun. 5, 3115 (2014).
[Crossref]

M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, “Communication with spatially modulated light through turbulent air across Vienna,” New J. Phys. 16, 113028 (2014).
[Crossref]

Malos, J. T.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1997).
[Crossref]

McDuff, R.

McLaren, M.

A. Forbes, A. Dudley, and M. McLaren, “Creation and detection of optical modes with spatial light modulators,” Adv. Opt. Photonics 8, 200–227 (2016).
[Crossref]

Méndez-Otero, M. M.

M. Anguiano-Morales, M. M. Méndez-Otero, M. D. Iturbe-Castillo, and S. Chávez-Cerda, “Conical dynamics of Bessel beams,” Opt. Eng. 46, 078001 (2007).
[Crossref]

Mendoza-Hernández, J.

Miceli, J. J.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref]

Mirhosseini, M.

M. Malik, M. Mirhosseini, M. P. Lavery, J. Leach, M. J. Padgett, and R. W. Boyd, “Direct measurement of a 27-dimensional orbital-angular-momentum state vector,” Nat. Commun. 5, 3115 (2014).
[Crossref]

Naguib, A.

D. Gesbert, M. Shafi, D. Shiu, P. J. Smith, and A. Naguib, “From theory to practice: an overview of MIMO space-time coded wireless systems,” IEEE J. Sel. Areas Commun. 21, 281–302 (2003).
[Crossref]

Neifeld, M. A.

Nelson, W.

Nichols, R. A.

J. C. Juarez, A. Dwivedi, A. R. Hammons, S. D. Jones, V. Weerackody, and R. A. Nichols, “Free-space optical communications for next-generation military networks,” IEEE Commun. Mag. 44(11), 46–51 (2006).
[Crossref]

Nowack, R. L.

R. L. Nowack, “A tale of two beams: an elementary overview of Gaussian beams and Bessel beams,” Stud. Geophys. Geod. 56, 355–372 (2012).
[Crossref]

Padgett, M. J.

M. Malik, M. Mirhosseini, M. P. Lavery, J. Leach, M. J. Padgett, and R. W. Boyd, “Direct measurement of a 27-dimensional orbital-angular-momentum state vector,” Nat. Commun. 5, 3115 (2014).
[Crossref]

H. Huang, G. Xie, Y. Yan, N. Ahmed, Y. Ren, Y. Yue, D. Rogawski, M. J. Willner, B. I. Erkmen, K. M. Birnbaum, S. J. Dolinar, M. P. J. Lavery, M. J. Padgett, M. Tur, and A. E. Willner, “100 Tbit/s free-space data link enabled by three-dimensional multiplexing of orbital angular momentum, polarization, and wavelength,” Opt. Lett. 39, 197–200 (2014).
[Crossref]

Y. Ren, G. Xie, H. Huang, C. Bao, Y. Yan, N. Ahmed, M. P. J. Lavery, B. I. Erkmen, S. Dolinar, M. Tur, M. A. Neifeld, M. J. Padgett, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Adaptive optics compensation of multiple orbital angular momentum beams propagating through emulated atmospheric turbulence,” Opt. Lett. 39, 2845–2848 (2014).
[Crossref]

M. P. J. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using light’s orbital angular momentum,” Science 341, 537–540 (2013).
[Crossref]

M. P. J. Lavery, G. C. G. Berkhout, J. Courtial, and M. J. Padgett, “Measurement of the light orbital angular momentum spectrum using an optical geometric transformation,” J. Opt. 13, 064006 (2011).
[Crossref]

G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12, 5448–5456 (2004).
[Crossref]

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88, 257901 (2002).
[Crossref]

Padovani, C.

F. Gori, G. Guattari, and C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[Crossref]

Palastro, J. P.

Pas’ko, V.

Phillips, R. L.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 2005), Vol. 52.

Pless, V.

W. C. Huffman and V. Pless, Fundamentals of Error-correcting Codes (Cambridge University, 2010).

Qin, C.

Ren, Y.

Ricklin, J. C.

A. K. Majumdar and J. C. Ricklin, Free-Space Laser Communications: Principles and Advances (Springer, 2010), Vol. 2.

Rogawski, D.

Scheidl, T.

M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, “Communication with spatially modulated light through turbulent air across Vienna,” New J. Phys. 16, 113028 (2014).
[Crossref]

Scott, R. P.

Shafi, M.

D. Gesbert, M. Shafi, D. Shiu, P. J. Smith, and A. Naguib, “From theory to practice: an overview of MIMO space-time coded wireless systems,” IEEE J. Sel. Areas Commun. 21, 281–302 (2003).
[Crossref]

Shapiro, J. H.

Shiu, D.

D. Gesbert, M. Shafi, D. Shiu, P. J. Smith, and A. Naguib, “From theory to practice: an overview of MIMO space-time coded wireless systems,” IEEE J. Sel. Areas Commun. 21, 281–302 (2003).
[Crossref]

Siegman, A.

Smith, C. P.

Smith, P. J.

D. Gesbert, M. Shafi, D. Shiu, P. J. Smith, and A. Naguib, “From theory to practice: an overview of MIMO space-time coded wireless systems,” IEEE J. Sel. Areas Commun. 21, 281–302 (2003).
[Crossref]

Soskin, M. S.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1997).
[Crossref]

Speirits, F. C.

M. P. J. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using light’s orbital angular momentum,” Science 341, 537–540 (2013).
[Crossref]

Sprangle, P.

Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref]

Su, T.

Tappert, F. D.

R. H. Hardin and F. D. Tappert, “Applications of the split-step Fourier method to the numerical solution of nonlinear and variable coefficient wave equations,” SIAM Rev. 15, 423 (1973).

Toyoshima, M.

Tur, M.

Ursin, R.

M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, “Communication with spatially modulated light through turbulent air across Vienna,” New J. Phys. 16, 113028 (2014).
[Crossref]

van der Veen, H.

M. W. Beijersbergen, L. Allen, H. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[Crossref]

Vasic, B. V.

Vasnetsov, M.

Vasnetsov, M. V.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1997).
[Crossref]

Vaziri, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
[Crossref]

Wang, J.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Wang, L.-G.

L.-G. Wang and W.-W. Zheng, “The effect of atmospheric turbulence on the propagation properties of optical vortices formed by using coherent laser beam arrays,” J. Opt. A 11, 065703 (2009).
[Crossref]

Wang, Z.

Weerackody, V.

J. C. Juarez, A. Dwivedi, A. R. Hammons, S. D. Jones, V. Weerackody, and R. A. Nichols, “Free-space optical communications for next-generation military networks,” IEEE Commun. Mag. 44(11), 46–51 (2006).
[Crossref]

Weihs, G.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
[Crossref]

White, A. G.

Willebrand, H.

H. Willebrand and B. S. Ghuman, in Free Space Optics: Enabling Optical Connectivity in Today’s Networks (SAMS, 2002).

Willner, A. E.

Willner, A. J.

Willner, M. J.

Woerdman, J. P.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327 (1994).
[Crossref]

M. W. Beijersbergen, L. Allen, H. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[Crossref]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref]

Xie, G.

Yan, Y.

Yang, H.

Yang, J.-Y.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Yoo, S. J. B.

Yue, Y.

Zeilinger, A.

M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, “Communication with spatially modulated light through turbulent air across Vienna,” New J. Phys. 16, 113028 (2014).
[Crossref]

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
[Crossref]

Zhao, Z.

Zheng, W.-W.

L.-G. Wang and W.-W. Zheng, “The effect of atmospheric turbulence on the propagation properties of optical vortices formed by using coherent laser beam arrays,” J. Opt. A 11, 065703 (2009).
[Crossref]

Adv. Opt. Photonics (1)

A. Forbes, A. Dudley, and M. McLaren, “Creation and detection of optical modes with spatial light modulators,” Adv. Opt. Photonics 8, 200–227 (2016).
[Crossref]

Appl. Opt. (2)

IEEE Commun. Mag. (1)

J. C. Juarez, A. Dwivedi, A. R. Hammons, S. D. Jones, V. Weerackody, and R. A. Nichols, “Free-space optical communications for next-generation military networks,” IEEE Commun. Mag. 44(11), 46–51 (2006).
[Crossref]

IEEE J. Sel. Areas Commun. (1)

D. Gesbert, M. Shafi, D. Shiu, P. J. Smith, and A. Naguib, “From theory to practice: an overview of MIMO space-time coded wireless systems,” IEEE J. Sel. Areas Commun. 21, 281–302 (2003).
[Crossref]

J. Mod. Opt. (1)

L. C. Andrews, “An analytical model for the refractive index power spectrum and its application to optical scintillations in the atmosphere,” J. Mod. Opt. 39, 1849–1853 (1992).
[Crossref]

J. Opt. (1)

M. P. J. Lavery, G. C. G. Berkhout, J. Courtial, and M. J. Padgett, “Measurement of the light orbital angular momentum spectrum using an optical geometric transformation,” J. Opt. 13, 064006 (2011).
[Crossref]

J. Opt. A (1)

L.-G. Wang and W.-W. Zheng, “The effect of atmospheric turbulence on the propagation properties of optical vortices formed by using coherent laser beam arrays,” J. Opt. A 11, 065703 (2009).
[Crossref]

J. Opt. Netw. (1)

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (1)

Nat. Commun. (1)

M. Malik, M. Mirhosseini, M. P. Lavery, J. Leach, M. J. Padgett, and R. W. Boyd, “Direct measurement of a 27-dimensional orbital-angular-momentum state vector,” Nat. Commun. 5, 3115 (2014).
[Crossref]

Nat. Photonics (1)

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Nature (1)

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
[Crossref]

New J. Phys. (1)

M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, “Communication with spatially modulated light through turbulent air across Vienna,” New J. Phys. 16, 113028 (2014).
[Crossref]

Opt. Commun. (3)

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327 (1994).
[Crossref]

M. W. Beijersbergen, L. Allen, H. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[Crossref]

F. Gori, G. Guattari, and C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[Crossref]

Opt. Eng. (1)

M. Anguiano-Morales, M. M. Méndez-Otero, M. D. Iturbe-Castillo, and S. Chávez-Cerda, “Conical dynamics of Bessel beams,” Opt. Eng. 46, 078001 (2007).
[Crossref]

Opt. Express (5)

Opt. Lett. (6)

Y. Ren, Z. Wang, G. Xie, L. Li, A. J. Willner, Y. Cao, Z. Zhao, Y. Yan, N. Ahmed, N. Ashrafi, S. Ashrafi, R. Bock, M. Tur, and A. E. Willner, “Atmospheric turbulence mitigation in an OAM-based MIMO free-space optical link using spatial diversity combined with MIMO equalization,” Opt. Lett. 41, 2406–2409 (2016).
[Crossref]

Y. Ren, G. Xie, H. Huang, C. Bao, Y. Yan, N. Ahmed, M. P. J. Lavery, B. I. Erkmen, S. Dolinar, M. Tur, M. A. Neifeld, M. J. Padgett, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Adaptive optics compensation of multiple orbital angular momentum beams propagating through emulated atmospheric turbulence,” Opt. Lett. 39, 2845–2848 (2014).
[Crossref]

N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17, 221–223 (1992).
[Crossref]

J. Mendoza-Hernández, M. L. Arroyo-Carrasco, M. D. Iturbe-Castillo, and S. Chávez-Cerda, “Laguerre–Gauss beams versus Bessel beams showdown: peer comparison,” Opt. Lett. 40, 3739–3742 (2015).
[Crossref]

H. Huang, G. Xie, Y. Yan, N. Ahmed, Y. Ren, Y. Yue, D. Rogawski, M. J. Willner, B. I. Erkmen, K. M. Birnbaum, S. J. Dolinar, M. P. J. Lavery, M. J. Padgett, M. Tur, and A. E. Willner, “100 Tbit/s free-space data link enabled by three-dimensional multiplexing of orbital angular momentum, polarization, and wavelength,” Opt. Lett. 39, 197–200 (2014).
[Crossref]

J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, and S. Chávez-Cerda, “Alternative formulation for invariant optical fields: Mathieu beams,” Opt. Lett. 25, 1493–1495 (2000).
[Crossref]

Phys. Rev. A (2)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref]

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1997).
[Crossref]

Phys. Rev. Lett. (2)

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref]

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88, 257901 (2002).
[Crossref]

Science (1)

M. P. J. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using light’s orbital angular momentum,” Science 341, 537–540 (2013).
[Crossref]

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[Crossref]

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Figures (8)

Fig. 1.
Fig. 1. (a)  u LG ( 0 , 5 ) , (b)  u LG ( 5 , 5 ) , and (c)  u BG ( 5 ) with β = 350 at z = 0 . Phase information is represented by the hue while the energy is represented by the normalized intensity. Color bar is in radians.
Fig. 2.
Fig. 2. Effects of multiplexing different OAM modes together. Top row is LGB ( p = 0 ) and bottom row is BGB ( β = 350 ). Left column is m = { 1 , 2 } , middle column is m = { 1 , 3 , 4 } , and right column is m = { 1 , 2 , 3 , 4 } . Color bar is in radians.
Fig. 3.
Fig. 3. Example of two realizations of turbulence screens created by the described methods; scale is in radians.
Fig. 4.
Fig. 4. Conjugate mode sorting of LGB, m = 5 .
Fig. 5.
Fig. 5. Optical transformation mode sorting (log-polar) of LGB, m = 5 .
Fig. 6.
Fig. 6. LGBs with order p = 0 , 1, 5, and 10 and BGBs with turbulence levels C N 2 = 1 × 10 16 (no marker), C N 2 = 1 × 10 15 ( + ) and C N 2 = 1 × 10 14 ( * ) . Due to symmetry we only show the positive modes. Note the y -axis is a log scale.
Fig. 7.
Fig. 7. LGBs for four different mode sets and three different turbulences ( C n 2 = 1 × 10 16 no symbol, C n 2 = 1 × 10 15 + ,” and C n 2 = 1 × 10 14 * ”) encoding channel efficiencies ordered by number of bits active.
Fig. 8.
Fig. 8. BGBs for four different mode sets and three different turbulences ( C n 2 = 1 × 10 16 no symbol, C n 2 = 1 × 10 15 + ,” and C n 2 = 1 × 10 14 * ”) encoding channel efficiencies ordered by number of bits active.

Tables (3)

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Table 1. Collection of OAM Modes Used

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Table 2. Lower Bound for the BER Found Taking the Minimum for a Threshold T [ 0.005 , 0.15 ] with Numerical Resolution of 0.005 a

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Table 3. Lower Bound for the BER Found Taking the Minimum Over an Exponential Fit Threshold a

Equations (16)

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2 u t 2 = c 2 2 u ,
2 U + k 2 U = 0 ,
1 r r ( r U ¯ r ) + 2 U ¯ z 2 + k 2 U ¯ = 0 .
1 r r ( r V r ) + 2 i k V z = 0 .
u LG ( p , m ) ( r , θ , z ) = C LG ( p , m ) w ( z ) ( r 2 w ( z ) ) | m | L p | m | ( 2 r 2 w 2 ( z ) ) × exp [ r 2 w 2 ( z ) i k r 2 z 2 ( z 2 + z R 2 ) ] × exp [ i ( 2 p + | m | + 1 ) ζ ( z ) ] exp ( i m θ ) ,
u LG ( p , m ) ( r , θ , z = 0 ) = C LG ( p , m ) ( r 2 w 0 ) | m | L p | m | ( 2 r 2 ) × exp ( i m θ ) .
u B ( m ) ( r , θ , z ) = C B J m ( β r ) exp ( i k z z ) exp ( i m θ ) ,
u BG ( m ) ( r , θ , z ) = C BG w 0 w ( z ) J m ( β r 1 + i z / z r ) × exp [ i ( k β 2 2 k ) z ζ ( z ) + 1 w 2 ( z ) ] × exp [ ( i k 2 R ( z ) ) ( r 2 + β 2 z r k 2 ) ] exp ( i m θ ) ,
u BG ( m ) ( r , θ , z = 0 ) = C BG J m ( β r ) exp [ ( r / w 0 ) 2 ] exp ( i m θ ) ,
Ψ ( κ ) = 0.033 C n 2 ( κ 2 + 1 / L 0 2 ) 11 / 6 exp ( κ 2 / κ 2 ) × ( 1 + 1.082 ( κ / κ ) 0.254 ( κ / κ ) 7 / 6 ) ,
r 0 = ( 0.423 k 2 sec ( α ) Path C n 2 ( z ) d z ) 3 / 5 ,
r 0 = ( 0.423 k 2 Δ z C n 2 ) 3 / 5 .
P = F 1 { Ψ C } ,
( x , y ) ( ρ , θ ) = ( log ( x 2 + y 2 ) , arctan ( y / x ) ) .
φ 1 ( x , y ) = 2 π a λ f [ y arc tan ( y x ) x log ( x 2 + y 2 b ) + x ] ,
φ 2 ( u , v ) = 2 π a b λ f exp ( u a ) cos ( v a ) ,

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