Orbital Angular Momentum Wave Generation and Multiplexing: Experiments and Analysis Using Classical and Quantum Optics
Optical fiber communication is known as the backbone of communication today for voice, video, and data transmission. Recent advancements in optical technologies are leading us to many new forms of generation of orbital angular momentum (OAM) waves and its transmission technologies. Among them are hypergeometric waveform (HG), Bessel’s waveforms, Laguerre Gaussian waveforms, and other forms of OAMs. This mode of communication is by the twisted form of electromagnetic wave which can accommodate a larger number of channels making communication simpler for wired and wireless communication. In this paper, various methods of OAM waveform generation are presented and are compared in details discussing the various needs in communication. OAM multiplexing is done by spatial division multiplexing (SDM) and wavelength division multiplexing (WDM). Methods of OAM communication, generation techniques, power analysis, and bit error analysis prove efficient transmission capability of OAM waves using low power consumption and low bit error rate in optical communication networks. OAM from fiber can be directly communicated to atmosphere which is found very useful for short distance wireless communication. Experimental validation is discussed for HG, Bessel’s waveforms, and LG waveforms. Pure Optical Vertices (OV) are generated by VCSELs (vertical cavity surface emitting lasers) and transmitted by (MMF) multimode fibers. Various atmospheric effects and challenges are also elaborated in this paper. RSoft OptSim software is used for simulation.
In traditional ways of communication generation of signals depend on frequency, time, and space, but these modes of communication are not sufficient to the present day need of high data transmission networks. In the field of optical communication, multi-input multioutput (MIMO) architecture employs a combination of OAM, spatial division multiplexing (SDM), and wavelength division multiplexing (WDM) [1–3]. The optical multiplexing system gives distinctive advantage over the other traditional multiplexing techniques. These multiplexing techniques include TDM, WDM, DWDM, OAM, SDM, and polarization division multiplexing (PDM) [1, 3, 4]. In both classical and quantum mechanics, angular momentum comprises SAM and OAM  in a combined form. OAM modes are mutually orthogonal, giving additional degree of freedom in multiplexing [1–3], increasing the data capacity of optical fibers [1, 2] in the terabit range. In WDM, one wavelength can carry multiple modes of OAM using a single multimode fiber [1, 3]. The vortex beam carries azimuthal component , where is associated with OAM mode number an integer number. The screw type phase distribution is given by an per photon [5, 6], where is topological charge and is plank’s constant. The optical beams play an important role in optical communication and quantum computation [2, 7] especially used in mode division multiplexing and optical tweezers [8–10]. These optical vortices are created by spiral phase plates [6, 8, 9, 11] and computer generated fork holograms  and in optical domain by lasers. OAM multiplexing does not carry wavelength or polarization; OAM is used in addition to WDM and PDM techniques in order to improve system capacity [2, 5, 8, 12, 13]. In demultiplexing the OAM waves, a spatial domain demultiplexer has photodetectors spatially arranged which are illuminated by each independent concentric ring; then separate individual SDM outputs are obtained. If we talk of quantum teleportation, a single photon is encoded in both spin and orbital angular momentum. The photon pairs are entangled in both degrees of freedom called hyper entanglement, which extends more degrees of freedom. These steps are towards more complex quantum system to control scalable quantum technologies . In recent research photonic structures, metasurfaces  and dendritic cell clusters  are found useful for manipulation, transmission, and absorption of light. These metasurfaces give additional degrees of freedom to the polarization of light . Some of the materials such as 2D Graphene, MoS2 molybdenum sulphide, and black phosphorous (BP) [18, 19] have properties of manipulating light in terms of polarization. These materials are proposed for OAM generation in photonic applications and also for wideband and wide-angle operation.
VCSELs have challenges as the modulation bandwidth is limited. It is of the order of several tens of GHz. So these VCSEL-based transmissions outperform electrical interconnect for long distances. VCSEL cell occupies a very small area of μm2. One of the considerations is also the electronic optical routing of optical to electrical and electrical to optical conversion which is very costly in many ways. WDM is not spectrally efficient and consumes more power, so the solution lies in switching to the spatial dimension which is only viable optical routing scheme in optical interconnects. Spatial division multiplexing (SDM) and mode division multiplexing (MDM) use single wavelength. In , is the mode no. and is the azimuthal no. in the cylindrical coordinate in MDM . Dense wavelength-division multiplexing (DWDM) proves to give a multiplicative-factor 100 while transmitting data. SDM (space division multiplexing) also plays an important role in modern communication technology. It is accomplished by using the OAM technique  and overcame the shortcomings of the communication technologies such as TDM and DWDM in terms of channel capacity.
All channels are sharing the same physical space launched in different transverse modes in a physical channel. In the oceanic communication or seaside communication, OAM wave travels at the air water interface and faces underwater scattering/turbulence below the surface. This causes power loss and modal coupling in OAM-based optical links. Further investigation in underwater communication can be a challenge . In atmosphere turbulence conditions, OAM communication leads to power fluctuation and link loss in the FSO system . The solution is the BER performance in FSO which can be increased by mode diversity. In an underwater communication, OAM multiplexing techniques could potentially enhance the data rate capacity and it can lead to larger bandwidth .
OAM waves are twisted forms of light which are not used conventionally. This mode of communication is by photon flow. Different physical dimensions of photons include modulating, multiplexing, and multicasting data information. As the demand for more number of channel accommodation and internet traffic increases, this technology is giving more opportunities for multicasting and communication in turbulence [22–27] and is found to be the most suitable. OAM has attracted interest also in a nonlinear interaction of quanta-imaging and sensing .
In this paper, OAM generation, multiplexing, and demultiplexing of the optical signals are elaborated mathematically and experimentally. The photonic circuit in the RSoft OptSim module is used. OAM is not only used for wired but also for free space communication and wireless communication using planar-phased array antennas with PCB technology for radar imaging and MIMO application . New techniques for radar imaging include quantum based measurement, which is far more precise than the classical methods. It can be carried out by OAM measurements only [2, 28, 29]. OAM is recognized as classical as well as quantum mechanics used in superresolution imaging. It has a linear momentum of electromagnetic wave at high frequency and has both spin and orbital angular momentum. Spin related to the polarization state and OAM is related to the helical phase front.
In sensing, we use nonclassical optical states such as phase, angular, and rotational variables and dispersive properties [28, 29]. Diffractive phase holograms, grating-based coupler waveguides, antenna arrays, spiral phase plate (SPP), holographic plate, and q plate are used for OAM generation.
In SAM, there are three possible polarization modes of EM beams, i.e., LHCP, RHCP, and linear polarization . Plane waves in a free space have electric and magnetic field vectors which are orthogonal to each other and to their direction of propagation. In propagation of electromagnetic waves, the Poynting vector is defined as which determines the direction of propagation of power in an electromagnetic wave. Optical vortices have fruitful quantum properties. Photon is entangled in both space and polarization state. It is a classical entanglement ; high dimensional entanglement is used in channel coding to improve high capacity optical communication . Optical vortices not only possess linear space application but also extensive uses in nonlinear communication with optical solitons . Mode multiplexing increases tremendously in OAM wave in phased array structure antennas and in radar and MIMO applications . An attractive feature is that OAM multiplexing can provide extremely high capacity with the existing modulation and multiplexing techniques. Conventional multiplexing schemes limit the communication capacity according to Shannon Cai et al. . OAM, due to the spirally distributed phase and orthogonality and mode division multiplexing features, received widespread attention . Nowadays, supervised and unsupervised learning techniques are being used to classify OAM states prorogating in a free space and a turbulent environment .
There is less coupling between adjacent modes with closet propagation mode . This is the most important issue, as in conventional method signal processing is required to reduce mode coupling. Mode conversion efficiency is very high in OAM. Its components have zero static power consumption. Experimental results obtained almost zero power consumption in transreceivers. In the future, compact silicon-based OAM transmitting  devices will be used in integrated devices and demultiplexing devices . Free-space OAM transmission link finds application in interlinks, while fiber-based OAM has application at the interrack level. So OAM finds application in optical interconnects where extremely high capacity (density) is needed for sub mm confined spaces .
Silicon photonics with integrated OAM emitters convert waveguide modes into free space OAM modes . Microsized spiral phase plates (SPPs) are fitted  in the aperture of VCSEL. OAMs are giving an unbounded number of possible states; hence, it is of great interest in optical communication, microscopy, sensing, and quantum optics. Because of low fabrication cost and high energy efficiency, this will become the future communication boon for society. Other than this, it also has high modulation bandwidth.
OAMs are achieved by an azimuthal varying phase achieved by bulk spatial light modulators (SLMs). It is an attractive device for a single photon device generation required in QKD Quantum Key distribution. OAM also contributes in optical trapping, quantum metrology, and remote detection. VCSEL aperture, each with its own integrated SPP, modulates the signal  incoming with data. OAM multiplexing, having a good modulation bandwidth, is giving several GHz of multiplexed bandwidth. Metasurfaces are also new techniques to be used for multiple OAM beam  generation.
The twisted form of an electromagnetic wave works on either spin momentum or angular momentum of light. OAM beams are orthogonal to each other, minimizing crosstalks between each other. Orthogonal beams can be made by Hermite–Gaussian beams without having an OAM pattern, but with the advantages of circular symmetry . It can be used with already available commercial components. Moreover, these are also being used in free space communication. The advantages of using OAM is that almost zero power link can be established without much modification in the available systems. A single core can carry multiple OAM modes. Using VCSELs, the systems in transmitting and receiving are minimized, which leads to reduction of cost, size, and an increase of performance [26, 36, 37].
Earlier results with experiments have been presented for Terabit communication link using 16 QAM and QPSK signals [23–25, 37]. Experimentation shows multiplexing from 1550 nm to 1570 nm wavelengths. Though the experiment was done for only four wavelengths, it can be extended to more numbers of wavelengths. As a higher number of modes diverge more rapidly  than a Gaussian beam with lower order modes, it consumes less energy. Other than this, OAM multiplexing has grown so rapidly that we are using it in multiplying free space communication and underwater communication. Optical signal in underwater communication has higher data rates . OAM modes are based on MIMO/SMM, resulting in four OAM multiplexed channels shown [23–25] for turbulent-free FSO link. In this experiment, each OAM mode carries a 100 Gbit/s quadrature phase-shift keying signal. The OAM/WDM-MIMO-based SDM and MDM  with a 1550 nm wavelength prove to give low bit error rate and turbulent-free wave propagation in FSO-optical links with QPSK input signals. Similarly, 4-level PAM  using OAM waves for 1600Gbps has been successfully implemented for FSO links. OAM with MIMO using PAM, QPSK, and QAM is coming with mitigating effects of atmospheric turbulence.
2. Mathematical Analysis of OAM Wave Generation
From a classical theory of electromagnetic wave generation, a three-dimensional Helmholtz equation is the temporal transform of the 3D wave equation. It describes the propagation of electromagnetic waves in a medium having spatially dependent refractive index .
On Fourier Transforms of this equation w.r.t to , it is defined by
In terms of propagation constant , the equation can be written as
It gives solution
is an integer. The radial equation can be defined in terms of refractive index and propagation constant using separation of variables  by
For different values of , the radial component of the em field are mutually orthogonal:
2.1. Orthogonality of OAM Waves
Continuing from earlier equations,
So again that . We go on integrating the above equation.
For cylindrical coordinates to solve the Helmholtz equation in a refractive index fluid (), in terms of radial distance,()=. Then, the equation is
In the equation, () is the complex electric field. is the field distance from the center axis of the Gaussian beam. is the center of the width of the wave, and is the topological charge of OAM. is the azimuthal angle . Such OAM beams have a helical phase structure with its wave front resembling fold corkscrew pattern . Such wave has an angular momentum along the beam axis given by around the beam, and hence, it has a spin and angular momentum both derived from Maxwell’s equations.
Wave equation in terms of propagation constant , where denotes the propagation constant denoting the direction. Information-carrying OAM beams  are given by by attaching a spiral phase mask () to the Gaussian beam . The encoding data can be described as where is the data input. is the free space wave number. The radial equation has solutions for values at a given wavelength. These solutions will represent the guided modes [6, 12] of the fiber. We can find all guided modes by starting with [6, 8, 9, 12]. These optical electromagnetic field modes are produced in the fiber by VSCEL lasers .
Further separating the variables,
The field solution given by where , are, respectively, the Bessel and modified Bessel function of order . If boundary conditions are imposed on as for example, the waves propagate within an the optical fiber implying
Implying that can assume only discrete values, say
In quantum mechanical language, we can write  with the total wave field given by
So that the general solution at frequency is given by the superposition .
Or in the time domain, by
Appendix A.5 for vector defining in three dimensions ; ; ; are solutions for the vector , in terms of unit vectors where defines the unit vector. Here, unit vectors satisfy also . These two-unit vectors are known as polarization vectors representing polarization properties of radiation. The solution exp(·) are orthogonal vectors satisfying the property These wave vectors are Laguerre Gauss (LG) modes defining the helical waves carrying OAM modes which are all orthogonal modes . Two waves travel perpendicular to the direction of propagation defined by Unlike other particles, the spin states of a photon are mapped to the classical aspect of light. Total angular momentum is given by [5, 41] . Laguerre Gaussian Index, , forms helical structure of beams [5, 7]. Laser field spatially modulates the electrons in a helical pattern. For a circularly polarized wave , for right and left circularly polarized light and angular moment is defined by and spin angular moment by . Azimuthal component of light’s linear momentum is also defined as per photon. Angular momentum operator and its representation in spherical polar coordinates  are defined by spin states of photons which generate quantum states. Let spin operator be defined by where its components have commutation relation where is the Levi-Civita symbol  and .
The quantum number is empirically fixed at a single half integer. States of a particle in common basis will be described as
A communication link is set by the stream of particles between two locations. Here, modulation is done in the spin state in a chosen axis. The particles also acquire an orbital angular momentum (OAM) in addition to spin. Spin component is in the direction of propagation which is compatible to the momentum operator . For there is no eigenstate. It is a solution to the field equations, for and -1; there are two spin states [8, 9]. Hence, probability of finding a particle from 0 to radius in terms of its energy is
If in the calculation of polarization parameter
Appendix A.6 in LG modes, the eigenstates of the spin operator are the circular polarizations; hence, the description of will include spin state. Spin is defined as polarization vector given by [34, 43] where the polarization states as and .
Writing for a common basis of spin and OAM,
This serves as the basis of a potential communication  scheme. In quantum communication, quantum bits are assigned to the spin states, and additional information is assigned to the OAM quantum number and associated radial number .
2.2. Scattering Matrix
Jones matrix  gives the scattering matrix. Polarization states of the scattered wave connected to that of incident wave
To generate OAM, geometric phase is used to generate of an OAM wave. The OAM scatterer is generated by a metasurface with azimuthal location . It provides a phase change of . and . Cocircularly polarized components are filtered, and a cross-circularly polarized component carries OAM. The polarization of vector fields is  controlled by a linear polarizer. If an incident wave is circularly polarized, i.e., , respectively, the output wave is having a geometric phase of .
3. OAM Modes
3.1. OAM Modes Generation by High Order Hermite–Gaussian Modes (HGMs)
Detail analysis is done for these OAM modes mathematically. Laser modes are generally Hermite–Gaussian . The electric field distributions here are mutually orthogonal . Linear harmonic oscillator using one dimensional Schrodinger wave equation is studied to understand the quantum mechanical behavior of atoms for their coherence states . The Mach–Zehnder interferometer model is used for square Hermite–Gaussian Vortex pulse beam . modes from optical tweezers are used to generate bright spot high order HGM [38, 43] in 2D arrays. In the coherent state, angular momentum is transferred to micro particles or an atom which represents the modes of an optical resonator. These modes are simulated in pumped solid-lasers . Arbitrary field distribution can be decomposed in this form . The Hermite–Gaussian functions are orthogonal sets of functions .
3.2. Mathematical Presentation of HGM Waves
The following mathematical equation represents high order HGM distribution [34, 38, 43]: is the wave number, is the beam width, is the curvature radius of phase front, is Rayleigh length, and is the order of Hermite polynomials, respectively.
Phase is given by .
and modes are superposing with phase difference which forms resultant superposed beam vortex. They are aligned in a square manner. The effect of phase difference between the two subbeams introduces a coefficient between two superposed subbeams. Resulting square vortex laser beams in the resonator are observed by changes in the phase and intensity profile [34, 43] given by
Experimental lasing is given by mode patterns. bright spots of modes are in an array manner. The and axes are defined by
Since Hermite polynomials have real values, there are cross-section points of nodal lines of mode and nodal lineof mode [34, 38]. Solution of equation is cross-section point of nodal line of mode and nodal line of mode.
High-order HGM lasing and another higher-order HGM lasing are easily achieved. Generations of helically phased beams HG01 and HG10 are combined to give optical vertices . Fundamental Gaussian mode can be converted to helically phased mode using laser. Holographic approach is used with spatial light modulators (SLMs). These SLMs are pixelated liquid crystal devices programmed through a computer. A telescope is also used to transform Hermite–Gaussian (HG) and LG modes. converters and converters, which are also called and plates , are used to transform HG to LG modes. These optical devices have many more advantages over holograms.
3.3. Different Vortex Beams and the Difference
Different types of OAM carry OAM waves in free space are L.G. beam, Hermite (HG) beams, Bessel beams, hypergeometric beam, hypergeometric-Gaussian beam, and other beams . OAM beams in free space can be mixed using different modulation schemes . Many methods have been suggested  such as photonic crystal fiber (PCF) structure with rectangular air holes  and Bessel polygon air holes. Studies found that circular air holes and noncircular air holes in photonic crystal fiber support Bessel polygon, which has very low confinement loss in order of . PCF combining different sizes of air holes are also suggested .Ring core fiber  is supposed to have good performance characteristics for OAM modes in wired communication. Several schemers for generating the OAM output are suggested . OAM with quantum entanglement properties have infinite number of Hilbert space all orthogonal basis. PCF (photonic crystal fibers) and RCF (ring core fiber) are mainly being used to generate OAM.
The oceanic turbulence Geometric Gaussian (HGG) vortex beam is also analyzed with different wavelengths . In underwater optical communication (UOC), HG are separated as Gaussian mode, modified exponential Gaussian mode, modified LG mode, and modified Bessel’s mode . In underwater communication, OAM is of more interest because it has higher multiplexing capacity. Angular momentum cross-talk is reduced in case of underwater communication when it is Bessel Gaussian beam and hypergeometric Gaussian beam [29, 47].
Hypergeometric-Gaussian beams are to be discussed in case of wireless communication. By a computer-generated hologram, we can generate hypergeometric OAM and LG beam with different radial indices (Figure 1). Enhanced learning capabilities are introduced in the model to train a convolutional neural network. Manipulation of OAM through hologram is becoming more of an interesting topic nowadays. Hypergeometric beams have the properties of auto focusing and diffraction-free communication mode.
Another frequently used mode family is L.G. modes, which are based on polar coordinates. L.G. mode fits more in the situations where there is rotational symmetry.
OAM is susceptible to atmosphere turbulences, so there is a need to study scan diffracting beams which are found other than LG Modes. OAM techniques are analyzed in both classical and new quantum regimes. OAM waves can be generated by a hologram in both classical and quantum states. OAM states are characterized by OAM numbers . It is also described by quantum walk. In this paper, OAM waves are described by the Helmholtz equation. Hypergeometric beam is also known as Bessel’s Gaussian beam is a pseudo-non-diffracting beam family . The mathematical derivation is also explained in Section 2. The beams have autofocusing property, larger angular momentum, and self-reconstruction after an obstacle. It has great potential for wireless optical communication [28, 29]. In atmosphere turbulence , hyper-Gaussian beams [48, 49] are purposed to work strongly through moderate to strong anisotropic conditions. Anisotropy is present in the atmosphere with a layer up to 25 km or in high altitude of the stratosphere. The wireless communication using OAM will be the future study area where Bessel’s and the hypergeometric wave function will be focused for atmosphere turbulence and other under water communication.
3.4. OAM Constellation Conventional Approach and Modification/Improvement
Generation and detection are done in earlier experiments  using the Dammann optical vortex gratings (DOVGs). For beam generation, OAM beam diffraction grating is used. where is a phase function, is the grating period, is diffraction order from to , is the azimuthal angle in polar coordinator, is the interval of the topological charges, and is the power of th order normalized with respect to total power.
Micro-ring-based OAM devices are used to decode the OAM. This is also known as optical coded division multiple access (OCDMA). Here, OAM value is used as an address code of each node value. Suitable OAM transmitters and detectors enable flexible and dynamic function for multiplexing .
3.5. Hermite–Laguerre–Gaussian Modes
Hermite–Laguerre–Gaussian modes with elliptical vortices are the LG modes. These modes are defined by Appendix A.8. where is a Hermite Laguerre Polynomial. where and is Rayleigh range.
For , mode is reduced to or mode.
For or 3 the , mode is reduced to mode .
mode can be decomposed into a set of Hermite–Gaussian (HG) modes.
An earlier experiment on OAM used 28 GHz with 1 GBaud/s signals with 16-QAM modulation. 32 Gbit/s is obtained by employing 4 different OAM modes for , and on two orthogonal linear polarization states . OAM beams are generated by Gaussian beams which passes through a spiral phase plate . An inverse spiral surface in SPP can convert an OAM beam back into a Gaussian beam. These SPP are made of high-density polyethylene (HDPE) (Figure 2) which has a refractive index of 1.51 at 28 GHz. The OAM beams are generated from the interference patterns between each OAM beam and the Gaussian beam, which are combined through a beam splitter .
New wave equation consists of spherical eigenvalues with individual wave consisting of spherical waveform, equation in form of eigenvalues , where lies in waveform generator. These HG modes are plotted  in Figures 3 and 4.
These waves are represented in the form of modes, modes, and mixed wave modes (, ).
3.6. Effective Refractive Index Difference, Confinement Loss, Mode Purity, and Isolated Parameters
OV (optical vertex) has OAM per photon; is topological charge, is the reduced plank constant. OV mode for has been experimentally proposed in a four-mode fiber.
mode is composed of with phase difference. is converted to other modes. The spatial phase form of exp() () gives OAM an infinite number of orthogonal eigenstates.
In case of photonic circuit fibers used for OAM generation, it is found that circular shaped photonic crystal fiber supports several OAM modes  of 1000 nm bandwidth. Each OAM confinement loss is to db/m for this fiber. Crystal fibers with square and circular air holes  also support OAM. Confinement loss is less for these fibers. In these fibers, different modes have different propagation constant . The relationship between and can be expressed as
, is the wavenumber in the vacuum, and is the wavelength. ERI (effective refractive index)  decreases faster with the increase in a mode order. ERI increases as wavelength increases. To avoid coupling between modes, ERID should be greater than .
Good propagation features possess using PCF supports better mode quality (96%) and good isolation value about 72db that gives low crosstalk between [56, 57] guided channels. OAM covers 1520 to 1630 nanometer whole L band and C band [37, 57, 58].
is the operating wavelength, and Im() is the imaginary part of refractive index (RI). PCF properties in terms of RI are
and are Sellmeier coefficient, and is the RI that varies with wavelength. ERID is effective refractive index difference.
R is real part of RI. 38 OAM modes are supported by above mentioned PCF (wavelength 1000-2000 nm). Fused silica gives the best results in terms of low dispersion, it has lower ERID.
Light is guided between center air holes and holes in cladding. Zhang et al. proposed a scheme of C-PCF (26 OAM). Jia et al. proposed hollow core ring PCF supporting 38 OAM modes. Lower order modes show less dispersion. Higher order modes have dispersion. So lower order modes [23, 24] are stronger for stable fiber communication. Small confinement loss is necessary for robust and efficient transmission of light in fiber. A lower nonlinear coefficient is advantageous for transmitting the OAM mode into optical fibers. Reports of various PCF find circular for 100 nm and 1000 nm BW, spiral for 600 nm BW, and double for 850 nm BW, with very low confinement losses to dB/m. PCF for 76 OAM modes  is designed with 6 linear polarizations. OAM beams are used in MDM [23–25] which enhances transmission capacity. OAM has been used for optical manipulation and imaging. These designs have been proved using COMSOL multiphysics simulating software.
In terms of em field (,), an effective area is given by
The following is the numerical aperture:
Dual guided photonic crystal fiber supports large capacity transmission  in high speed optics communication with LP modes. Twisted fibers, SPP, cylindrical lenses, helical lattices, twisted hollow core fibers, and step index fibers have been used for OAM transmission.
Dispersion in fibers is given by
In this equation, is the real value of the effective refractive index, and is velocity of light.
Nonlinear property of PCF is expressed as  , where is the nonlinear coefficient of fused silica .
4. Effect of Atmospheric Turbulence on Optical Signal Propagation
OAM modes suffer loss of information in atmospheric turbulence due to intermodal dispersion in fibers and higher order modes. 850 nm VCSEL has been used for OAM generation . It is used with multimode fibers (MMF). Optical beam-carrying multiplexed OAM is directly transmitted to an atmospheric channel (Figure 5) from the fiber termination point. Atmospheric turbulence affects the signal at the receiver side optical beam is directly focused to the second MMF core . Hence, this setup is used for short distance communication. Hybrid FSO communication links do not require costly electric to optical and optical to electrical signal converters, and no amplification stages are needed before transmitting and receiving through the atmospheric channel.
10 GHz Ethernet is applied using this technology for LAN , and it can be used for metronetwork and enterprise connectivity broadband heterogeneous wireless services. It is also being used as an extension of radio over fiber link . Phase distortions are caused due to turbulence which is the reason for intermodal cross-talk . It can be reduced if the phase front of an OAM beam is partially adjusted. An experiment reports  adaptive optics techniques to mitigate the effect of atmospheric turbulence.
In a random medium such as the atmosphere, ocean, and tissue, optical beam intensity fluctuates in the propagation paths . Scintillation is caused by temperature variations in the random medium, which results in index-of-refraction fluctuations. This is also known as optical turbulence.  OAM can be used high-definition video broadcasts, video conferences, distance education, and webcasts, which demand increased network capacity. Laser beams are expected to be the next generation communication tool, having large-capacity for satellite, ground, and unmanned aerial vehicles (UAV) using spatial propagation. This has also advantages of reduced size and weight of optical terminals.
In oceanic communication spatial correlations, Kolmogorov power spectrum is a method used to analyze the atmospheric effect . Backward scattering method is also used for OAM analysis in turbulent atmosphere . Scintillation index reduces with the increase of the topological charge it makes OAM beam, propagating longer distances , but mode scattering increases for higher order modes. Angular momenta is conserved for the free-space propagation  but not for an inhomogeneous medium. Scintillation severely impacts the operation of target designation  experiments done on the model simulations of discontinuity of statistical variations of atmospheric conditions. For geometrical optics, Monte Carlo simulations are done for the statistical  models. A multibeam reduces the effect of the atmospheric turbulence. The multibeam is obtained by  a mirror and a beam splitter using laser and telescope. For turbulent-free space communication, OAM with 4 channels  has been proved experimentally using WDM/SMM technique at 1550 nm wavelength.
High-order Bessel beams also have an advantage of OAM in multiplexing FSO communication. Wide-spectrum lasers reduce Scintillation effects. Supercontinuum decreases atmospheric turbulence. For this, wide-spectrum OAM beams are used. In weak turbulence channels, the performance is measured by SI, PF, and BER . Fog, dispersion, absorption, rain, and snow , these are the range-limiting factors in atmospheric communication which is called atmospheric turbulences. In wired and space communication, lasers of 1550 nm wavelength have lower optical losses than 830 nm wavelength. Atmospheric turbulence is governed by the Navier–Stokes equations; the statistical approaches are developed to solve this. OAM beam patterns do not change during the propagation in atmospheric turbulence, and it evolves into a Gaussian beam as propagation distance increases. After propagating 1000 m, beam array transforms into a fan structure in moderate turbulence; then, it turns to a single vortex beam after propagating 5000 m. Also, optical vortex beams are able to propagate longer distances in weak turbulent atmosphere. When distance exceeds 500 km, the output beam finally loses the OAM property and gradually becomes a Gaussian-shaped beam. Scintillation affects more to the LG vortex beams.
Figure 6 shows the experimental setup of VCSEL converting bit stream into an OAM wave. A spatial analyzer is used to measure the OAM wave.
These OAM waves carry the random data which we use to multiplex them. Similarly, at the receiving end, the demultiplexed data are obtained. Figures 5 and 7 show the transmission link for OAM waves. In our photonic circuit, VCSELs are used for generating twisted light beams (Laguerre–Gaussian laser beams). VCSEL (vertical surface emitting laser) is a multimode transmitter; it has inbuilt spatial light modulator or multimode generator by changing the parameters of the LG mode expression, both SAM and OAM of a photon beam  can be visualized. Different modes with different azimuth and different radial component are generated using VCSEL which are shown in Figure 8. The wave has a corkscrew pattern [1, 2, 5] of the scalar field. Optically dense and white lines are observed. Modes are generated for intensity profile of . Doubled parabolic light energy curves are shown representing different radial modes
At , it shows a Gaussian beam . For experiments, the wavelength of light is taken as 1550 nm.
5.1. Case 1: OAM Wave Generation by VCSELs of Same Wavelength (Appendix A.8)
Using light of single wavelength 1550 nm in VCSEL, the following modes are observed by varying spin quantum numbers and azimuthal quantum numbers: , (Figure 8(a)); , (Figure 8(b)); , (Figure 8(c)); (d) , (Figure 8(d)); , (Figure 8(e)); and , (Figure 8(f)).
These OAM modes are being generated by individual lasers separately. So we can use each OAM to send information through the fiber. Fibers carrying these are multimode fibers. Linearly polarized light is converted to spherically polarized light by VCSEL, and multiple modes of OAM are generated. To check the modes again, spatial analyzers are used.
5.2. Case 2: OAM Wave Generation by VCSELs of Different Wavelengths
With a single VCSEL, we generate OAM waves of different wavelengths as shown in Figure 9.Wavelengths from 1550 nm to 1560 nm are used at a wavelength difference of 5 nm. Each input wavelength is varied, which is at a difference of 5 nm to the laser. At 1550 nm, outputs in the form of intensity profile are shown in Figure 9(a) (, ), at 1555 nm output by Figure 9(b) (, ), at 1560 nm Figure 9(c) (, ), and at 1570 nm (, ) Figure 9(d). Figures 9(a)–9(d) show modes for , ; , ; , ; and , , respectively.
Figures 8 and 10 show intensity profile of beam diffracted from SLM (modulator) emitted by VCSEL integrated SPPs by , up to so on. is the helical wave front of OAM beam. Higher topological charge gives larger diameter of annular far field. It gives scope for free space quantum communication.
Figure 11(a) shows OAM waves carrying 0.3 to 3 m V of the OAM generated by different wavelengths. Power used by the four OAM waves in multiplexing is shown in Figure 11(b). Experiments show that OAM waves consume almost zero power in multiplexing. Power of different baseband signals being modulated by these OAM waves is shown in Figure 12(a).
Multiple waves of OAM are multiplexed as shown in Figure 7. These waves are multiplexed using 1550–1570 nm laser waves at a difference of 5 nm (1550 nm, 1555, 1560, and 1565 nm). OAM number in spin and azimuth can be calculated by their rings and centers in pair marked in red color as shown in Figure 9 showing (spin quantum number) and (azimuth quantum number). Maximum power shown is 0.25 milliwatt. Four signals are multiplexed with the spatial mode using four input ports. Keeping the radial component asand azimuth component as 0,1,2,3, a multimode fiber carries spatial modes without much dispersion [2, 5, 12]. Figures 10(a) and 10(b) show the multiplexing  of these modes with power spectrum. LP modes were multiplexed [12, 13] and sent through the photonic circuit. Laguerre Gaussian (LG) beam is the most common optical vortex beam [8, 42]. For multiplexing of information carrying OAM, beams are given by
The resultant field is given by
Positive SAM state can be obtained by combining the OAM waves. Figure 10(c) shows the wavelength division multiplexing all wavelengths of equal power, and the wavelengths are generated at a nanometers. The waves are modulated by using four baseband signals of low frequency input signals. Power used is almost zero as shown in Figure 10(b). Signal magnitudes of multiplexed wavelengths are of 0.0025 W. After multiplexing, the total power is also shown by the simulation in Figure 11(b). Signal eye diagram presents signal variation between 0 and 0.2 m V in 10 GHz range. These baseband signals obtained are of very low power as shown in dBm Figure 12(a). Signal output shown in the eye diagram has maximum output of 1.5 V. Figure 12 is the complete multiplexed OAM signal and the corresponding output power in dBm . OAM multiplexing enhances the performance of the multiuser system of these channels [31, 32]. Output after demultiplexing power spectrum of different baseband signals was achieved as shown in Figure 12(a). Figure 11(a) shows amplitude of multiplexed wavelengths. Figure 10 shows that if we increase the OAM waves, the output power consumed decreases. Hence, in optical networks, higher numbers of OAMs give better results as almost OAM waves carrying information at zero power is consumed.
If we compare with other techniques, there is no power analysis so far has been seen in case of OAM generated by planar antennas using PCBs and also holographic techniques. Although we have a portable OAM devices using PCBs, it needs extra hardware for test. PCB technology lags in analysis of power driven by the system. The eye diagram shown in Figure 13 represents that noise, inter symbol interference, and jitter are almost nil in the communication channel using multiplexing with OAM waves.
The bit error rate is nil (Figure 13(b)) and shows good signal to noise ratio. Small dispersion can be seen in the output of multiplexed signal (Figure 14). The distortion is less; there is good ratio at the sampling points (Figure 13(b)). The open eye area Figure 13(a) shows the best ratio achieved. Earlier done experiments were 80/160 Tbit/s capacity achieved using 1600 nm wavelength with 16 QAM data channel with 80 wavelengths and two polarizations .
Experimentally pure OAM waves are generated using VCSEL at low cost with high efficiency SLM. The results were obtained for superposition of waves OAM for efficient topological charge. Relative phase of OAM can be turned from 0 to , and the corresponding fringe patterns are achieved. These experiments are simulated using RSoft OptSim tool.
5.4. Demultiplexed Signal
Detected power  at the demultiplexer is represented mathematically as where is constant offset. are the coefficient of the Fourier series of azimuthally varying intensity patterns, and and are arbitrary constants. When an inverse operation is to be performed, spiral phase mask can be used to remove the azimuthal phase dependence of an OAM beam.
are the and azimuthal number. Only one OAM beam is converted back to beam with the phase removed. All other waves remain the same. In practical applications, this beam is passed through a spherical lens to get back original beam.
Figure 15 shows demultiplexed signals having ring-shaped intensity profiles. Intensity is low at the center and radii proportional to the azimuthal index. This output shows OAM beams of 4 OAM channel after demultiplexing. The ring-shaped beam is converted into bright light intensity spots. These are coupled to optical fibers for detection .
6. Comparison with the Earlier Experiments
Earlier experiments done for 6 OAM modes for capacity of , 6 Tb/s over 119 km . It gives high capacity [31, 32] in free space. Fiber-based OAM is also proven for 1.1 km high capacity 1 Tb/s . Orbital angular momentum multiplexing/demultiplexing  studies have been done. OAM beams are employed to encode information onto a laser beam for transmitting not only in fiber but also in free space communication [23, 46]. This also increases security in the link [23, 46]. The next-generation FSO will be achieved by OAM. This research focuses on the quantum and classical aspects. This research brings overall methodologies and discussions on OAM using mathematical derivations of all the OAM waves and its experimental validation for LG waves using VCSELs with the help of simulations. Some of the derivations will be analyzed using quantum mechanics in the near future.
The two OOK signal modulations of 1.5 Gb/s were directly modulated with 445 nm laser diode. These channels were carried by two OAM modes used for underwater optical communication . The results with lowest cross-talk have been proved with optical  vortices of Si waveguides. Several methods have been shown by earlier work. We can generate OAM by diffraction grating. Beams and incident coaxial optical vertices beam carrying massive OAM and demultiplexing using single Dammann optical vortex gratings (DOVGs), by  topological charge equal to zero. From one DOVG, several OAM can be generated and similarly be detected using sufficient diffraction angle interval to arrange appropriate filters and detectors.
6.1. Comparison of Methods with Previous Known Methods
PCBs and printed circuit boards are used . Other techniques are antenna arrays, holographic plates, and inhomogeneous birefringent devices for generating OAM beam. One experiment  for multiplexing OAM was proposed designing a nonpolarizing beam splitter where 4 OAMs are generated by 4 SPPs and using nonpolarizing beam splitter; multiplexing was done. Similarly, antenna array is used to generate intermodulated OAM modes. Other low-cost OAM generation is by using different holes on a planar antenna substrate in PCB technology [34, 54]. Metasurfaces generate dual modes with polarization-dependent transmission and reflection properties. These muxed waves are the key to QKD Quantum Key Distribution protocols when we superimpose , - quantum optics  two-dimensional QKD for higher dimension quantum optics.
Spatial high modulation is limited to a millisecond response time. Greater conversion efficiency is achieved through SPP based spiral phase plates. SLM has no possibility of miniaturization and integrated components. The innovation research on OAM by Willner et al.  shows exactly the given solution of the research and proves the results obtained in Figures 8–Figure 13 discussed mathematically in this paper.
Li et al. demonstrated a 64 Gbps  FSO link using PAM-4 modulations experimented using VCSEL for 88 Gbps data transmission over 555 m free-space length. It uses WDM and PAM-4 techniques . SPP (spiral phase plates) used to give OAM waves giving an angle of phase shift over the full circle when incident EM wave falls upon SPP wave.
Hardware  is used for the OAM generation. The intensity profiles we get are using RSoft analyzers. In addition to this, RSoft-simulated work validates the multiplexing of signals. These signals are good especially for short-reach data interconnection and QKD. In this experiment, Gaussian beam is sent to laser and we get OAM waves at the outputs. We use spiral phase plates to generate different intensity profiles of OAM as shown in Figure 16.
Experiments also show the superimposed beams of OAM . It also shows diffraction in the multiplexed output beams. Figure 17(e) shows the superimposition of waves , , and , which shows the intensity profile of the OAM waves. Figure 17(e) shows the diffraction in the multiplexed beams. OAM beams are generated by cylindrical lenses, spherical plates, and diffractive optics. The number of channels in OAM is an important property. Some of the techniques are photonic crystal fiber, with spiral PCF; hence, circular PCF has been designed at  1000 nm wavelength.
Inverse phase hologram is used to recover data on one of the OAM beams .
Another approach that gained much attention is to transmit independent data, each in a different core using multicore fibers (MCF) [11, 67]. By increasing the number of spatial modes, fiber capacity and the spectral efficiency of the channel increases.
In this research, classical and quantum behaviors of OAM waves were analyzed. It proves a successful implementation of wired photonic circuit using VCSEL for a single wavelength and multiple wavelength OAM generation and multiplexing/demultiplexing using OAM. It is observed that OAM waves are not distorted when demultiplexed. It proves that OAM requires very low power in milliwatt range in multiplexing and demultiplexing.
Multiplexing in OAM modes and higher order LP modes are composed  of , two eigenmodes . OAM is combined by two fiber eigenmodes. They have the same propagation constant as compared to LP mode components of different propagation constants; OAM mode components of same propagation constants will not suffer from intrinsic mode walk off. So OAM gives a stable propagation through a certain length of fiber. Single-ring fiber  of high index maintains radially fundamental modes and reduces cross-talk during mode excitement. To increase the number of eigenmode coupling through the fiber, either the core radius is increased  or the RI difference between core and cladding is increased.
Single fiber gives 1-D MDM. By using multiple ring fiber interring cross-talk  can be minimized, OAM modes with different phases into high-index ring regions are multiplexed in spatial dimensions.
OAM beams are used for free space and fiber-based communication links. Multiple OAM beams each having a unique twisting value carries an independent data stream which are multiplexed at the transmitter. At the receiver side, these OAM modes are demultiplexed and the resultant Gaussian beam is propagated through a single mode fiber for data recovery. 16 QAM data capacity of 1.4Terabit/sec additionally 100 Tb/s free-space data link was reported by simultaneously combining/multiplexing OAM. QPSK signal of 1008 channels was successfully tested  with a capacity 100.8Tbot/s.
7.1. Further Advancement and Connected Technologies
OAM can be categorized as OAM–SK and OAM–DM (shift keying and division multiplexing). It has increased the data rate from terabit to petabit . 100 Tbit/s data capacity achieved through multiplexing of 42 wavelengths, 120 AM beams, and two polarization states.
For demultiplexing and detection mechanism, optical vortex grating is used for the multiplexing of OAM channels with individual modulation schemes and simultaneous detection mechanisms . Four-level PAM for FSO link of  16000Gbps input streams is successfully implemented with OAM having low BER value with clear eye diagrams. QPSK inputs with SMM/SMD in MIMO lusing 1550 nm wavelength OAM modes are proves to give turbulent-free FSO. OAM-based MIMO systems are the next coming technology which has greater channel capacity  than conventional MIMO systems.
8. Applications of OAM
OAM finds applications in wired and wireless communication technologies. For wired technologies, encoding with OAM is done. Assuming a network of users transmitting photonic qubits, data encoding is done with three degree of freedom-path, polarization, and OAM. Photon degrees of freedom is used in terms of polarization, time, frequency, phase, and OAM. This gives different quantum communication protocols. If OAM is used for multiplexing under atmospheric turbulences, it provides higher security in terms of total secrecy. OAM can be controlled by controlling Q factor of cavity. This opens a door towards the next-generation molecular sciences and applications.
Optical vortices have advanced applications as optical tweezers. Quantum mechanics is used in high order quantum entanglement, nonlinear optics, and classical models of optical vortices.
In free space communication (FSO), the FSO receiver link [8, 38] is designed based on the orthogonality of the Laguerre–Gaussian (LG) beams with different azimuthal mode numbers. Transmitters and receiver links are designed. This enables ultra-high-speed transmission to end-users, allowing interoperability of various RF and optical technologies. It increases the BER performance of the modulation and reduces the complexity of receivers. FSO is being used in UWB communication, in satellite communication, deep space communication, satellite to aircraft communication, and aircraft to satellite communication. In all the cases, atmospheric turbulence does not affect the communication. This technology will meet the high demand of increasing bandwidth and channel capacities in today’s world.
OAM has tremendous ability to increase the capacity of channels in terms of data in the terabit transmission. Optical communication having larger bandwidth increases the data spectral efficiency by using polarization, amplitude, and phase of the optical signal.
For this, SDM, WDM, MDM are to be used, which gives scope for optical communication in terms of low attenuation of signals, better bit error rate, low interchannel interference, and less cross-talk, leaving behind copper communication. The system performance can be improved by the Laguerre–Gaussian vortex, hypergeometric beams, and combined Gaussian-vortex beams. Thus, OAM beams in today’s network provide an essential aid for free space communication, underwater optical communication, satellite communication, and sensing systems. The OAM waves received after scattering from target give information of azimuth hence, finds tremendous scope in target detection. In future, only optical communication has the ability to provide large capacity high-speed communication which can meet the demand of network broadband services and scientific communities. It will be further expanded in fiber communication using quantum optics for QKD and higher security channel designs.
9.1. Future Scope
These advantages with OAM waves assure that in the near future, quantum technologies will dominate the traditional communication techniques because of very low power consumption and ease of generating these modes. This also opens an area for quantum computing and quantum cryptography. Future work lies on metasurfaces, where by controlling the polarization, we can achieve these optical transmitters and receivers in compact designs.
EM wave has angular momentum component is in the direction. A component of linear momentum is in the , plane. Linear momentum density is given by and angular momentum density
is the dielectric permittivity of the fiber; and are electric and magnetic fields.
Here, the photon is in the eigenstate along the axis defined by angular momentum operates in classical theory as
In terms of components, , , and are, respectively, the three Cartesian components of the angular momentum operator in quantum mechanics. They also represent the generators of rotations around the three axes in the representation (). In a representation of the rotation group where the squared total angular momentum operator has eigenvalue , and have eigenvalues . In particular, if the representation is , , , and have eigenvalues of 1, 0, and -1. Polarized light can be either left circularly polarized, right circularly polarized, or a linear combination of these two states. Thus, polarized light has eigenvalues +1 or -1 for any component of the angular momentum. The 0 eigenvalue does not occur. We thus say that the photon is a spin one particle. If we wish to describe waves produced by other particles having, say, spin , then we must use the other representations of the rotation group, i.e., for , in the case of Boson’s and half integers the case of fermions.
LP modes in (,) defines the cylindrical waveguide, and classical wave is represented in this form. It can also be represented as a quantum wave in azimuthal and spin angular momentum operator where is the azimuthal angle.
In the quantum theory of electromagnetic fields, moment of vector potential which satisfies the homogeneous wave equations, where its spatial solutions are given by . Specifying a volume of a cavity length  with electromagnetic wave travelling inside the cavity assuming a vector potential  . Here, is a unit vector, and is specifying product of temporal and spatial part.
Electric field and magnetic field in the cavity in terms of scalar wave potential are
From curl of vector ,
The vector potential:
This is the solution of spatial part.
Temporal part solution is
From Maxwell’s equations k, with . Assuming
OAM waves can also be represented by  vector potential as where
is defined as probability of finding the particle at point from 0 to which is inversely proportional to its classical momentum given by  and
The energy of particle in OAM is
is a basis defined in
Azimuthal phase dependence is .
Vector potential is finally expressed as
where using quantum mechanics
is the photon state.
Increasing or decreasing the radial number by one unit, with unchanged creates (anhiliate) one positive quantum and one negative quantum number. “” is the quantum number  because of degeneracy in .
The following shows the transmission of LP modes in optical fiber:
The spatial modes travel in the form of linearly polarized modes in the fiber.
Optical fiber wave equation is given by
The vector quantities (p=px, py, pz)
Expression gives the LG modes. defined in terms of give solution to the LP modes. The electrons are distributed in a spiral along the beam axis, which gives a helical density modulation. Bunching factor is given by integrating over the where the normalized distribution of the trapped electron is, is wave number and is azimuthal mode index.
The combined form of OAM wave is given by are two rectilinear polarizations with respect to propagation wave vector. OAM modes  generated in the fibers or the cylindrical waveguide are represented as a complete basis of LG mode represented by and
It is defined as Gouy phase.
is the Rayleigh range of beam,  is the beam waist, and are the associated Laguerre Polynomials with is the azimuthal index giving of . per photons, and is number of radial nodes in the intensity distribution. For LG modes comprise concentric rings with zero on axis intensity.
Laguerre polynomial is given by .
is defined as azimuthal index of OAM given by per photon, and is the number of radial nodes for .
In defining ,
Probability distribution shows dark concentric rings.
It is a radial number operator of the Laguerre Gaussian mode.
|MIMO:||Multiple Input and Multiple Output|
|OAM:||Orbital angular momentum|
|SDM:||Spatial domain multiplexing|
|VCSELs:||Vertical cavity surface emitting lasers|
|WDM:||Wavelength division multiplexing|
|FSO:||Free space communication|
|OAMBS:||OAM Beam Splitter|
|SAM:||Spin angular momentum|
|RSOFT:||RSOFT photonic device tool OptSim used for simulation|
|SLMs:||Spatial light modulators|
|SLM:||Spatial light modulators|
|UOC:||Underwater optical communication|
|OAM–SK:||OAM shift keying|
|OAM–DM:||OAM division multiplexing|
|PCF:||Photonic crystal fibers|
|UAV:||Unmanned aerial vehicles RCF (ring core fiber)|
|LHCP:||Left hand circularly polarized|
|RHCP:||Right hand circularly polarized|
|QAM:||Quadrature amplitude modulation|
|DOVG:||Dammann optical vortex gratings|
|ERI:||Effective refractive index|
|SMM/SMD:||Spatial multimode multiplexing/spatial mode multiplexing|
|FSO:||Free Space Optics|
|PAM:||Pulse amplitude modulation|
|QPSK:||Quadrature phase shift keying QKD -Quantum Key Distribution|
All the data presented has been presented in the paper, and other data required are available upon request.
This manuscript is not based on data of human tissues or studies on the data of animals.
Conflicts of Interest
There is no conflict of interest. This study was completed as part of employment in the Department of Electronics and Communication Engineering Netaji Subhash University of Technology East Campus Delhi.
The authors are thankful to the Department of Electronics and Communication NSUT East Campus for extending help in optical communication laboratories to complete the research.
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