Orbital angular momentum of light for communications Open Access

“Structured light”—especially orbital angular momentum (OAM)—is a topic of growing interest in the optics community, not only for its inherent properties but also for its possible applications. Will it impact sensing, imaging, or micromanipulation?^1–3 This review will explore the possibilities of OAM for data communications.

Structured light generally refers to an optical beam with a tailored spatial amplitude/phase distribution and corresponding unique properties.⁴ One type of structured light involves OAM beams that are also known as vortex beams because of their donut-shaped intensity profiles with a null at the center.^5–7 In addition, such OAM beams have twisting helical phase fronts while propagating, which is often characterized by the OAM order (i.e., $ℓ$⁠). The value of $ℓ$ indicates the number of 2π phase shifts around the center of the beam's phase profile. Interestingly, OAM can be carried even by a single photon in the quantum domain.⁸ 

It is well known that Laguerre–Gaussian (LG) modes are a complete two-dimensional orthogonal modal basis set. As a subset of the LG modal basis set, different OAM modes (with different $ℓ$ values) are orthogonal with each other.^9,10 Importantly, based on such orthogonality, OAM beams could be utilized for a communication system in the following two different ways:

• (1).

  Multiplexing: The total data capacity could be potentially increased by multiplexing multiple OAM beams, each carrying an independent data channel at the transmitter side. After coaxially propagating in the free space or fiber, these OAM beams on different channels could be efficiently demultiplexed at the receiver side. Importantly, there would be little inherent channel crosstalk due to the orthogonality between different OAM beams. Consequently, the total system capacity could be increased by a factor of N, where N represents the number of multiplexed OAM beams.^11–13 

• (2).

  Encoding: Photon efficiency can be increased if a beam or single photon is encoded into one of many possible OAM values, thus making a large alphabet for possible data symbols within a discrete time window. The achievable data bits per photon increases as $log2N$⁠, where N represents the number of orthogonal encoding states.^14–17 

The field of OAM-based optical communications (i) is considered young and rich with scientific and technical challenges, (ii) holds promise for technological advances and applications, and (iii) has produced much research worldwide. In this review, a general introduction of OAM and its applications in communications will be first presented in Sec. II. Subsequently, we will describe challenges, advances, and perspectives on different aspects of the OAM-based optical communications in the following sections, including (a) OAM generation/detection and (de)multiplexing (Sec. III), (b) classical free-space optical (FSO) communication links (Sec. IV), (c) fiber-based communication links (Sec. V), (d) quantum communication links (Sec. VI), (e) OAM-based communications in different frequency ranges (Sec. VI), (f) OAM-based communications using integrated devices (Sec. VIII), and (g) novel structured beams for communications (Sec. IX).

It is well known that a light wave can carry spin angular momentum (SAM) that is associated with the polarization of the electric field.¹⁸ When the polarization of a light wave is linear, it means no SAM is carried. However, when its polarization vector rotates along the beam axis (i.e., left or right circularly polarized light), the light has an SAM of $±ℏ$ (⁠$ℏ$ is reduced Planck's constant).⁶ 

As another type of angular momentum, a light wave can also carry OAM. In general, OAM can be utilized to characterize the “twisted” helical phase front of a light beam when its wavevector spirals around the beam axis. Moreover, such a helical phase front of an OAM-carrying beam is usually represented by exp(⁠$iℓθ$⁠), where θ is the azimuthal coordinate and $ℓ$ is the number of 2π phase shifts in the phase profile of the beam.⁵ In addition, the sign of $ℓ$ (positive or negative) corresponds to the direction of the phase helices (clockwise or counterclockwise) in the phase profile. Due to the phase singularity in such a helical phase profile, an OAM beam with a nonzero $ℓ$ (i.e., OAM order) usually has a donut-shaped intensity profile, as shown in Fig. 1.

In contrast to SAM with only two different states, OAM can be theoretically quantified as an infinite number of states, which means $ℓ$ can be any integer. It should be noted that OAM beams with different $ℓ$ values are orthogonal with each other while propagating coaxially. For example, considering two OAM beams with $ℓ1$ and $ℓ2$⁠, respectively,⁷ 

where (⁠$r,θ$⁠,z) is the cylindrical coordinate and refers to the radial, azimuthal position, and propagation position. Such an orthogonality between different OAM beams can be represented by⁷ 

This is significantly important when they are utilized in communications, which will be further discussed in Sec. II B.

In general, an OAM beam could refer to any helically phased light beam, irrespective of its radial distribution. However, a complete two-dimensional (2D) modal basis can generally be characterized by two modal indices. For example, LG modes have $ℓ$ and p indices, corresponding to azimuthal and radial distribution, respectively.^9,10 The electric field of LG modes can be represented by^5,6

where the $1/e$ radius of the Gaussian term is given by $w(z)=w(0)[(z2+zR2)/zR2]1/2$ [w(0) is the beam waist], z_R is the Rayleigh range, and $(2p+|ℓ|+1) tan−1(z/zR)$ is the Gouy phase. $Lp|ℓ|$ are the generalized Laguerre polynomials and (⁠$r,θ$⁠,z) is the cylindrical coordinate.⁶ Figure 2(a) shows the intensity and phase profiles of some LG beams. For the LG beam with a non-zero $ℓ$ value, p + 1 represents the number of rings in the intensity profile, while $ℓ$ represents the number of 2π phase shifts along the azimuthal direction in the phase profile. Theoretically, LG beams with different $ℓ$ and/or p values are orthogonal with each other for a given beam waist and propagation distance, which can be represented by^9,10,20

Such orthogonality of different LG modes also ensures the potential utilization of 2D LG modal basis in communication systems, which will be further discussed in Secs. II B and IV A.

Since LG modes are a complete 2D modal basis set, a structured beam can generally be decomposed into LG modes with different $ℓ$ and p values. In general, the coefficients of LG components can be represented by a complex modal spectrum, as shown in Fig. 2(b). Moreover, it should be noted that each complex coefficient contains both amplitude and phase information, which can be tuned such that a specific structured beam would be generated by a coherent superposition of these LG components for a desired function.²¹ 

Aside from LG beams, there has recently been much interest in other novel types of spatially structured beams, such as Bessel-type beams, Airy-type beams, and pin-like beams,^4,22–29 and some of these beams have been reported and explored in various applications. Details will be further discussed in Sec. IX.

In the optical communications community, there is a growing interest in achieving high-capacity communication links. Previously, different physical properties of a light wave (e.g., wavelength, polarization, amplitude, and phase) have been explored to increase the data capacity. For example, the technique of multilevel amplitude/phase modulation formats has facilitated dramatic increases in capacity and spectral efficiency.^31–35 In addition, another typical method of increasing the data capacity is transmitting multiple independent data channels on light waves at different wavelengths or polarizations, which are wavelength-division multiplexing (WDM)^31–33 or polarization-division multiplexing (PDM),^31–35 respectively.

Recently, spatial modes have been under intense investigation in various areas.⁶ As one of the spatial orthogonal modal basis sets, OAM modes could potentially provide an infinite number of orthogonal OAM states. Therefore, OAM might be potentially utilized in communication systems, which would further increase the transmission capacity.

There are generally two ways of utilizing OAM in communications. The first approach is so-called mode-division multiplexing (MDM), which is a subset of space-division multiplexing (SDM).^12,36,37 In such a scheme, different OAM beams are utilized to carry different independent data channels. As shown in Fig. 3(a), since N independent data-carrying beams can be spatially multiplexed and simultaneously propagated over the same spatial medium (free space or fiber), the system's total data capacity would be increased by a factor of N. At the receiver side, due to the orthogonality between different OAM modes, different data-carrying beams can be efficiently demultiplexed with little inherent crosstalk. Moreover, MDM is generally compatible with the aforementioned WDM or PDM techniques. This means that in a WDM or PDM system, multiple orthogonal OAM beams that carry independent data channels could be located at each wavelength or polarization, thus further increasing the system's spectral efficiency (i.e., bit/s/Hz) and the total data capacity.^11,12,38

In addition, OAM modes can also be utilized in a data-encoding system. In this approach, N different OAM states can be encoded as N different data symbols, which represent “0.” “1,”…, “N – 1.” At the transmitter side, a sequence of OAM states are generated and sent in such a system at different times. Therefore, within each symbol period, the light beam occupies one of N possible OAM states, and the number of information bits encoded equals $log2N$⁠, as shown in Fig. 4. At the receiver, the data can be efficiently separated and decoded by checking the received OAM state with little inherent crosstalk due to the orthogonality of different OAM states. It should be noted that this approach could also be utilized in quantum communications systems in which every single photon exists in one of the N possible OAM states. As a result, this would provide a photon efficiency of up to $log2N$ bits per photon and, thus, could potentially achieve higher photon efficiency.^15,17

Recently, these two approaches have been demonstrated and employed mostly on large optical tables using bulky and expensive devices. Some examples of experimental demonstrations will be presented in Secs. IV–VII in different scenarios. However, for future OAM-based communication systems, integrated devices would seem to be significant due to their cost-effective, power-efficient, and compact features,³⁹ which will be further discussed in Sec. VIII.

On a fundamental level, these two techniques require that different modes can be efficiently combined and separated with little crosstalk, so almost any complete orthogonal modal basis set could be utilized. It should be noted that there have been some experimental demonstrations of communication links in free space or fiber based on other modal basis sets, such as LG, Hermite–Gaussian (HG), and linearly polarized (LP) modes.^{11–13,20,36,41,42} In general, an orthogonal modal basis set could be represented by two modal indices both of which can be employed in a communication link to further increase the transmission capacity.^13,43 For example, such a 2D approach was experimentally demonstrated in an FSO MDM system based on LG or HG beams,^13,20 which will be shown in detail in Sec. IV A.

In general, OAM modes could be potentially utilized in various communication systems. For example, the OAM-based MDM systems could be deployed either in free space or fiber. For different scenarios, there are different key challenges, which might induce power loss and intermodal power coupling in communication systems, as shown in Fig. 5.

For free-space links, atmospheric turbulence is one of the key issues that can cause a random phase distortion to the transverse beam profile. Such random distortion is time-variant, and thus, it could induce dynamic intermodal power coupling.⁴⁴ As a result, this issue would induce modal coupling and crosstalk in both classical and quantum communication links.^44–47 Moreover, in an OAM-based communication link, the receiver should be able to “distinguish” the transmitted spatial modes. When the receiver aperture size is limited and there is misalignment between the transmitter and the receiver, there could be relatively large power leakage in other undesired modes, and thus, the receiver could potentially fail to “distinguish” the actual transmitted modes.⁴⁸ Furthermore, the divergence of free-space higher-order beams tends to be more significant than that of lower-order OAM beams. For a receiver with a limited-size aperture, it would be harder to capture the whole higher-order OAM beams and, thus, induce power loss. Moreover, the beam truncation of a circular aperture could further induce modal power coupling to some LG modes with different p modes, leading to inter-channel crosstalk in LG-mode-based communication links.^48–50 Some details and potential mitigation methods will be discussed in Secs. IV B and IV C.

For fiber-based links, the MDM and quantum encoding systems can also be achieved based on OAM modes. However, the fiber channel is distinct from the free space mainly in the following aspects.

• As the beam is guided in the fiber, it does not diverge.

• The fiber will often cause cross-modal power coupling within the same mode group or between mode groups due to temperature gradients, different kinds of inhomogeneities, bends, and other non-idealities. In general, the intra-modal-group power coupling is relatively stronger. Some potential methods for mitigating mode coupling will be discussed in Sec. V C.

As an orthogonal modal set, OAM modes can be manifested in many types of electromagnetic (EM) and mechanical waves and have been recently explored in acoustic, radio, millimeter, and THz waves for wireless communication links.^51–64 

For OAM-based communication links operated in different frequency ranges, there tends to be a trade-off between the effects of beam divergence and wave–matter interaction, as shown in Fig. 6:

• Divergence: OAM beams at lower frequencies have larger beam divergence. When the receiver's aperture size is limited, it becomes more difficult to capture enough of the beam for data channel recovery. Such a problem might be more serious when a higher-order OAM mode is used.

• Interaction with matter: there tends to be much more wave–matter interaction at higher frequencies. For example, optical waves suffer more atmospheric-turbulence-induced distortion and modal power coupling than radio waves in OAM-based links.

In OAM-multiplexed communication links, digital data signals can be modulated on the amplitude and phase of the optical wave temporally, as shown in Fig. 7(a). In the amplitude modulation, the bits in the data streams are mapped to multiple amplitude levels of the optical wave. For example, an on-off keying (OOK) signal has two possible amplitude levels representing the bit “0” and “1.” Since only a single photodiode is required at the receiver to recover the amplitude information, such modulation is usually considered to have a low cost and simple implementation.⁶⁵ 

In contrast, in the phase modulation, the bits are mapped to the temporal phase, and the amplitude remains constant. As an example, the quadrature phase shift keying (QPSK) has four possible phase levels, and one QPSK symbol can carry two bits. In addition, the amplitude and phase modulation can be simultaneously utilized to increase the number of bits carried by one symbol. As an example, 16-quadrature-amplitude-modulation (16-QAM) has 16 possibilities mapped on the in-phase and quadrature amplitude axes encoding four bits of information. Unlike the amplitude modulated signals, coherent detection is often used at the receiver to extract the phase information when the temporal phase is also encoded with data.⁶⁶ In a coherent receiver, the received signal is sent to a $90°$ hybrid along with the continuous-wave (CW) local oscillator (LO) laser and detected by balanced photodiodes.⁶⁶ 

The quality of the received signal can be characterized by the bit error rate (BER), which is defined as the ratio between the number of bit errors and the total number of the received bits.⁶⁷ Alternatively, error vector magnitude (EVM) can be used to determine the quality of the received signal.⁶⁷ As shown in Fig. 7(b), the received symbols are represented by the points on the constellation according to their amplitudes and temporal phases. Due to the noise and distortions during transmission, the received symbol may deviate from the transmitted symbol, resulting in an error vector $Aerror$⁠. The EVM is calculated by the ratio between the root mean square of the error vectors $Aerror$ and the reference vector $Aref$⁠,⁶⁷ 

For an OAM-based communication system, fundamental functions include OAM generation, detection, and (de)multiplexing. Generally, it is desirable to generate OAM beams with high mode purity and detect OAM beams efficiently. Moreover, in an OAM-multiplexed system, developing efficient and scalable techniques for (de)multiplexing the coaxial OAM beams is challenging. These methods could be either bulky or integrated. In this section, we will focus on the bulky methods. The integrated methods will be discussed in Sec. VIII.

Generally, there are multiple types of approaches for OAM generation and detection. The first method is the spatial phase control, which can spatially (de)modulate a single beam in the azimuthal direction θ with a factor of $exp(−iℓθ)$⁠. One straightforward way to implement this method is to use a spiral phase plate (SPP) that has a helical dielectric surface,^68,69 as shown in Fig. 8(a-1). The height gradient of the surface along the azimuthal direction is given as⁶⁸ 

where λ and n are the wavelength and refractive index of the SPP, respectively. The simplicity of the structure makes it feasible to be applied in different frequency domains.^51,70,71

However, the OAM order of the generated beam by the SPP-based approach is generally fixed. One reconfigurable approach is to use a computer-generated hologram (CGH), as shown in Fig. 8(a-2). Such a hologram could be implemented by a spatial light modulator (SLM) or a digital micromirror device. Importantly, the generated OAM beam can be modified by simply updating the hologram. However, the mode purity could be affected by the unmodulated light, which could be due to the direct reflection from the surface glass or “dead area” between the pixels.⁷² Hence, a “fork” hologram, which combines the spiral phase pattern and linear phase ramp (grating), is applied to separate the first-order modulated light and the zeroth-order unmodulated light, as shown in Fig. 8(a-3). The “fork” hologram can be characterized by⁷³ 

where $Φ$ and T are the phase distribution and the period of the linear phase ramp, respectively. It should be noted that, in most cases, the resulting beam generated by a spiral phase pattern is not a pure LG mode (e.g., $LGℓ,p=0$⁠), but rather a superposition of modes with different radial indices (⁠$∑pLGℓ,p$⁠). To generate a pure $LGℓ,p$ mode with high mode purity, one should jointly control both the phase and amplitude of the input beam.^74–77 This could be achieved by a phase-only SLM. Specifically, the phase of the beam is spatially modulated by loading a phase pattern on the SLM. In addition, the amplitude of the beam can be spatially controlled by adding a grating pattern whose diffraction efficiency is carefully designed at different positions.^74–77 

Compared with the CGH approach, which could be limited in some applications requiring a small footprint,⁷⁸ recent development of novel materials enables OAM generation using an ultrathin metasurface, which assembles a 2D array of diffractive optical elements with variant subwavelength structures along the azimuthal direction,^79–83 as shown in Fig. 8(a-4). By engineering metasurfaces, OAM beams could be generated with a high conversion efficiency over a broad bandwidth.⁸³ Moreover, different OAM orders can be achieved independently for different polarizations.^81,82

Furthermore, OAM generation could be achieved not only by manipulating a single beam but also coherently combining multiple sub-beams. The theoretical expression of OAM generation by coherently combining sub-beams with each carrying a specific phase is as follows:⁸⁴ 

where N is the number of sub-beams, u_k and $θk=2πkN$ are the electric field and azimthual location of the k-th beam, respectively, and R is the center-to-center distance between the individual beam and synthesized beam, as shown in Fig. 8(b). By tuning the relative phase between different sub-beams, a desired OAM beam could be generated. Moreover, based on high-speed phase modulation, the approach could potentially generate an OAM beam with a fast-tunable order.⁸⁵ Furthermore, similar ideas of coherent beam combinations in a circular phase array could also be applied to other frequency domains (Sec. VII) and in integrated device (Sec. VIII).

Some of the aforementioned OAM generation methods are not limited in free-space applications but could be applied to OAM generation in fiber as well. OAM generation in fiber could be achieved using various methods, including the free-space coupling of OAM beams (e.g., SPP⁷⁰ and phase hologram¹¹) or utilizing fiber-based mode converters (e.g., photonic lanterns⁸⁶ and long-period gratings⁸⁷).

In addition, most approaches of the OAM generation methods above could also be used for OAM detection.⁷ Specifically, the OAM beam could be converted back to a Gaussian-like beam by propagating through a conjugate SPP, phase hologram, or metasurface.⁷ 

Multiple approaches have been demonstrated to multiplex OAM beams with different orders. One straightforward way is to use beam splitters. Each beam splitter could coaxially multiplex two beams with a proper alignment, and N independent OAM beams could be multiplexed by N − 1 cascaded beam splitters, as shown in Fig. 9(a). Even though there is 1/N intrinsic loss theoretically, the simplicity of the beam splitter approach makes it appropriate for lab demonstrations.

A more efficient (de)multiplexing method is to use cascade interferometers, as shown in Fig. 9(b). First, the OAM beam is split into two branches of the Mach–Zehnder interferometer (MZI). For one of the branch, a pair of Dove prisms is used to perform an OAM-dependent phase shift as follows:⁸⁸ 

where $α/2$ is the relative angle between the two Dove prisms. Subsequently, the beams from the two branches with an OAM-dependent phase difference are coherently combined with constructive and destructive interference at two output ports, respectively. Hence, two different OAM modes could be efficiently separated into different output ports based on different induced phase differences between the two branches, as shown in Fig. 9(b-1). By cascading MZIs and carefully designing the phase delays of the MZI branches, multiple OAM beams with different OAM orders would be demultiplexed to different output ports of the MZIs, as shown in Fig. 9(b-2). In order to demultiplex N OAM modes, N − 1 interferometers are generally required. Furthermore, similar idea could be applied for radial modes by replacing the Dove prisms with a lens.⁸⁹ 

The functions of mode generation and multiplexing (or mode demultiplexing and detection) can be further achieved in one device simultaneously. One straightforward way is to combine multiple fork phase patterns together to form a single phase plate, that is, Dammann optical vortex grating (DOVG), as shown in Fig. 10(a). The theoretical expression of the DOVG is as follows:⁹⁰ 

where N is the total number of diffraction orders, T is the period of the grating, n is the diffraction order from $−N/2$ to $N/2$⁠, and $Δℓ$ is the interval of the OAM order. $|En|2=1/N$ is the power of the n-th order normalized with respect to the total power. From the input side, a Gaussian beam is incident from the n-th-order diffractive direction to the DOVG. At the output, the diffraction component in the same zeroth-order diffractive direction will be generated with the OAM order of $n×Δℓ$⁠. Therefore, multiple incident beams from different order diffractive directions could be used to generate and multiplex different OAM beams. Theoretically, DOVG induces $1/N$ insertion loss caused by non-zeroth-order diffraction at the output.

Compared with the DOVG approach, a more efficient and scalable method is to utilize two phase plates to achieve log-polar transformation between the spiral spatial phase of OAM beams and titled spatial phase,⁹¹ as shown in Fig. 10(b). In this scheme, the first phase plate performs a log-polar transformation that geometrically maps “concentric circles” to “parallel straight lines.” As a result, the point at the input plane (x, y) is transformed to the new point (u, v) in the output Fourier plane where $u=−a ln(x2+y2b)$ and $v=arctan(yx)$⁠. Here, a and b are the scaling factors. Then, the second phase plate is placed at the Fourier plane to correct the phase error and collimate the transformed beams. Therefore, using two phase plates could achieve the efficient sorting of different OAM beams with no inherent loss. As the transformation is reciprocal, such a log-polar-based mode sorter could also be used for OAM generation and multiplexing.

Even though the log-polar transformation is scalable to a large number of modes, it could only be applied on a 1D mode basis.⁹² One way to generate and multiplex LG modes with 2-D modal indices is to use a multi-plane light convertor (MPLC) to shape the wavefront of the input light at multiple propagating distances. A unitary transform from multiple Gaussian beams at different locations to multiple LG beams with different (⁠$ℓ,p$⁠) pairs could be accomplished as shown in Fig. 10(c).⁹² Generally, the phase patterns of MPLC could be determined by the wavefront matching method. The theoretical expression of the phase pattern at each propagation distance is as follows:⁹² 

where $uinput,i$ and $uoutput,i*$ are the i-th input field forward propagating and conjugate output field backward propagating at z = z_k, respectively. N is the number of the input and output fields. It has been experimentally demonstrated to generate 210 LG modes using a 7-plane MPLC device with an insertion loss of <–12 dB.⁹² However, this approach might require an accurate alignment of these phase plates in order to achieve a high mode purity.^92,93

In this section, various OAM generation/detection approaches are first discussed. Subsequently, different methods for (de)multiplexing multiple coaxial OAM beams are presented. In order to further improve and deploy OAM generation/detection/(de)multiplexing techniques for OAM-based communication systems, some desirable features could include (i) efficient OAM mode generation with a high mode purity and (ii) fast tunability covering a range of OAM modes.

As compared to RF links, optical links can generally provide more bandwidth and higher data capacity due to the higher carrier wave frequency. Moreover, OAM modes can be potentially used in a free-space optical communication system (MDM or encoding systems) for high data capacity. However, in free-space links, there are some critical issues that induce power loss and channel crosstalk, including divergence, misalignment, and atmospheric turbulence. In this section, we will review (i) typical experimental demonstrations of OAM-multiplexed or -encoded free-space optical communications, (ii) potential challenges in an OAM-based free-space link, (iii) reported approaches for mitigating power loss and channel crosstalk, and (iv) OAM-based communication systems for airborne platforms or underwater environments.

In general, the flexibility of the OAM-based communication enables its compatibility with (i) either direct detection or coherent detection techniques, (ii) different modulation formats, and (iii) other multiplexing techniques (e.g., WDM and PDM). In this subsection, two different ways (i.e., OAM multiplexing and OAM encoding) of utilizing OAM for communication will be discussed. Furthermore, both short-distance laboratory and long-distance field demonstrations will be shown.

Initial demonstrations of OAM-multiplexed optical communications include a free-space OOK link that multiplexed an OAM beam and a Gaussian beam.⁹⁴ Subsequently, an OAM-multiplexed link⁹⁵ with a spectral efficiency of 12.8-bit/s/Hz has been achieved using four OAM modes, each carrying a 10.7-Gbaud 16-QAM signal. Later on, the spectral efficiency is doubled by combining PDM into the OAM-multiplexed link.¹² Eight OAM channels, each carrying an independent 42.8-Gbaud 16-QAM signal, could be used to achieve a communication link with a total capacity of 1.4 Tbit/s (⁠$4×4×2×42.8$ Gbit/s), as shown in Fig. 11(a).

The following experiment further increases the link capacity to 100 Tbit/s by combining OAM multiplexing with PDM and WDM.³⁸ Figures 11(b-1)–11(b-3) show the steps used in the proof-of-concept demonstration. First, two groups of OAM beams (⁠$ℓ={+4,+10,+16}$ and ${+7,+13,+19}$⁠), are generated and multiplexed using a beam splitter. Each beam carries 42 WDM channels with the channel spacing of 100 GHz. Second, the six multiplexed beams are split into two copies. One copy is decorrelated in free space and reflected to create another six OAM beams with inverse charges. Then, these two copies are combined to prepare 12 multiplexed OAM beams. Third, the 12 multiplexed OAM beams are again split into two copies. One copy is decorrelated in free space and polarization-rotated by $90°$⁠, and then two copies are combined to generate 24 OAM beams. The observed WDM signal spectrum on one of the demultiplexed OAM beams (⁠$ℓ=+10$⁠) is shown in Fig. 11(b). Moreover, by utilizing 54.139-Gbit/s 8-QAM signals carried on 368 wavelengths, 2 polarizations, and 26 OAM beams, a free-space data link with a total data rate of 1.036 Pbit/s and a spectral efficiency of 112.6 bit/s/Hz has been demonstrated.⁹⁶ Both demonstrations indicate that OAM is compatible with existing degrees of freedom (e.g., amplitude/phase/wavelength/polarization).

For the aforementioned OAM-based communication links, the experimental demonstrations are generally conducted in the laboratory over short distances of $∼1$ m. There have been several experimental demonstrations that utilize OAM multiplexing for a long-distance free-space links in the field environment.^97,98 For example, a 120-m free-space OAM-multiplexing link with the total capacity of 400 Gbit/s has been demonstrated,⁹⁷ as shown in Figs. 12(a-1)–12(a-3). The transmitted OAM beams (⁠$ℓ=±1,±3$⁠), each carrying a 100-Gbit/s QPSK data channels, are transmitted. The transmitted OAM beams at site #1 are reflected twice by two mirrors placed 30 m away at site #2 to achieve an aggregated 120-m propagation path. The BERs of the four data channels carried by the OAM beams are achieved below the 3.8 × 10^–3 forward error correction (FEC) limit as shown in Fig. 12(a-3). In addition, a 260-m two-OAM-multiplexed communication link between two buildings has also been demonstrated. Each OAM beam carries 16-QAM signals to achieve an 80-Gbit/s aggregated data rate.⁹⁸ Potential challenges in a long-distance OAM-based free-space link will be discussed in Sec. IV B.

The OAM-encoded scheme is realized by encoding each data symbol with a unique OAM mode, i.e., representing the data stream as a sequence of different OAM beams. Based on this scheme, a classical free-space OAM-encoded link has been demonstrated.¹⁴ By utilizing the programmable SLM, one of the eight OAM states (⁠$ℓ=±16,±12,±8,±4$⁠) is transmitted at a time. To achieve a high data rate, fast switching might be required between different OAM modes. Using high-speed lithium-niobate switches, a 20-Gbit/s OAM-encoded link using four OAM modes (⁠$ℓ=±1,±3$⁠) has been demonstrated.¹⁵ The application of the OAM encoding scheme in the quantum communication link will be discussed in Sec. VI.

Aside from the efforts of achieving long-distance OAM-multiplexed links, an OAM-based encoding system over a link distance of 143 km has been achieved.¹⁶ Figure 12(b-1) shows the concept and the experimental setup for the 143-km OAM-encoded link between two islands. The transmitted beam is modulated by an SLM with specific phase patterns to generate different superpositions of OAM modes. After being magnified by a telescope, the expanded beam propagates through the free space over 143 km. At the receiver, a camera records the received mode structures. The lobed modal structure is visible for different superpositions of OAM modes $ℓ=±1, ℓ=±2$⁠, and $ℓ=±3$⁠, as shown in Fig. 12(b-2). By introducing a relative phase of π, rotation of the mode structure of the $ℓ=±3$ is clearly recorded. By distinguishing images of different OAM mode superpositions, the OAM-based encoded data are recovered with an error rate of 8.33%. Due to the effects of atmospheric turbulence, a pattern recognition algorithm based on an artificial neural network is applied. The results indicate that it is feasible to achieve the free-space OAM-based link over a kilometer-scale distance.

The aforementioned MDM-based FSO communication systems are achieved by utilizing one-dimensional modal basis. Specifically, each OAM beam carries the same p value (i.e., LG mode with p = 0 and different $ℓ$ values). In addition, LG beams with different p values and same $ℓ$ have been experimentally utilized in 200-Gbit/s MDM FSO links.²⁰ While various spatially orthogonal beams could be employed in these one-dimensional systems (varying $ℓ$ values or varying p values), one could also utilize the two-dimensional modal basis (varying both $ℓ$ and p values) to potentially support a larger set of mode-carrying channels.

As a proof of concept, a four-LG-multiplexing (⁠$LG10,LG−10,LG11,LG−11$⁠) link has been demonstrated,¹³ as shown in Fig. 13(b). Utilizing a designed hologram of SLM,⁹⁹ the amplitude and phase profiles of the input data-carrying Gaussian beam could be controlled, as shown in Fig. 13(a). As a result, a desired $LGℓp$ beam with specific radial and azimuthal indices could be generated. In such a link, each LG beam carries a 100-Gbit/s QPSK channel to achieve a total data rate of 400 Gbit/s with the BERs lower than the FEC limit, as shown in Fig. 13(c). In addition, an LG-encoded system has been demonstrated by encoding information using four LG modes.⁴³ The information of a 2-bit gray-scale image representing four possible pixel values of “00,” “01,” “10,” and “11” has been successfully encoded and decoded utilizing four $LGℓ,p$ modes (⁠$LG50,LG−50,LG21,LG−21$⁠). In order to maintain orthogonality among different LG modes, the receiver aperture would be required to fully capture the data-carrying beam. This might induce some potential challenges in LG-based free-space links, especially when the receiver has a limited-size aperture, which will be discussed in Sec. IV B.

There are some critical issues in an OAM-based FSO system, including channel power loss, inter-modal power coupling, and inter-channel crosstalk. As shown in Fig. 14, there are many causes of these issues for OAM-based FSO links:

• Divergence: The divergence of free-space higher-order OAM beams tends to be more significant than that of OAM beams with lower mode orders. When the receiver aperture size is limited, it is harder to capture the whole OAM beam with a higher mode order and, thus, induces power loss. Moreover, the beam truncation in the radial direction could further induce modal power coupling to some LG modes with different p values.^9,48–50

• Misalignment: In an OAM link, the receiver should be able to “distinguish” the transmitted modes. When the transmitter and receiver are misaligned (i.e., not coaxial), the limited-size receiver could potentially fail to fully capture the phase change in the azimuthal direction. At the receiver side, there could be relatively higher power leakage in other undesired modes, making it difficult for the receiver to “distinguish” the actual transmitted modes..^48,100

• Turbulence: When an optical beam propagates through a turbulent medium, a random phase distortion is induced on the transverse beam profile. Such phase distortion breaks the orthogonality between different modes. Thus, the time-variant random phase distribution could induce dynamic intermodal power coupling. This might be in the order of milliseconds.^44–47 

In an FSO link, the system is sometimes limited by the size of optical components (e.g., limited-size receiver aperture), and thus the beam divergence could increase the difficulty of fully capturing the beam profile. This could induce signal power loss, which lowers the signal-to-noise ratio (SNR) and, thus, affects the BER performance of the recovered signal in a communication system. In general, various system parameters are related to beam divergence, including wavelength, propagation distance, and transmitted beam size.⁴⁸ Specifically, the beam divergence of an MDM system is also dependent on the mode indices. The beam width is a function of propagation distance z and is defined as:^9,101

where $σℓ,p(z)$ is the square root of the second moment of the beam intensity profile.^9,101 It was investigated that, for a given $LGℓ,p$ beam, the beam width $Wℓ,p(z)$ is related to the beam waist w(z), as shown in Eq. (14),^9,101

where $M=2p+|ℓ|+1$⁠,^9,101,102 and ℓ and p are the LG mode indices for azimuthal and radial directions, respectively. Equation (14) indicates that the divergence effects become more significant with increasing mode order^9,49,102,103 and, thus, make it harder to fully capture higher-order beams with a limited-size receiver aperture. This would result in power loss of the transmitted channel, as shown in Fig. 15. Moreover, for a communication link using LG modes with various p values, the divergence effects could potentially induce both power loss and inter-channel crosstalk. This is due to the fact that the truncation effect of a circular limited-size aperture at the receiver side can also lead to a modal leakage from the desired mode to other LG modes with various p values.^48,49,104 Therefore, the divergence-related parameters should be carefully chosen to mitigate channel power loss and inter-channel crosstalk in an LG-based communication link.⁴⁸ 

In an ideal FSO link, the receiver and the optical axes would be aligned perfectly. However, the jitter and vibration of the transmitter/receiver platform could lead to transmitter–receiver misalignment, including relative lateral displacement and receiver angular error.

In a communication link using Gaussian beams, the transmitter–receiver misalignment could lead to power loss, as the limited-size receiver aperture might fail to fully capture the beam profile. However, in an OAM-based communication link, such misalignment could induce not only power loss but also inter-channel crosstalk. This is due to the fact that the receiver with a limited-size aperture might fail to fully recover azimuthal phase change and, thus, induces inter-modal power coupling. Figures 16(a-1)–16(a-3) show the scheme of system alignment and the received phase profiles when the transmitter and receiver are perfectly aligned, aligned with lateral displacement, and aligned with receiver angular error, respectively.

The lateral displacement and angular error induced modal coupling have been theoretically investigated.¹⁰⁰ It has been shown that both types of misalignment would cause spreading of the OAM spectrum and induce power coupling to other modes. The normalized power distribution of different modes when the transmitter and receiver are aligned with lateral displacement or receiver angular error is shown in Figs. 16(b) and 16(c), respectively.⁴⁸ As the lateral displacement increases, the normalized power on the other modes increases, while the normalized power on the transmitted mode decreases. This is due to a larger mismatch between the received beams and the receiver aperture. Such modal coupling could potentially induce inter-channel crosstalk in an OAM-based link. Furthermore, power coupling from OAM +3 to OAM +1 and OAM +5 is lower than that of neighboring OAM modes (i.e., OAM +2, +4). Therefore, an OAM-based link with larger spacing of OAM order tends to be more tolerant to lateral displacement. Moreover, the simulation results indicate that there is a similar trend regarding receiver angular error. When the receiver angular error increases, the normalized power on the undesired mode increases, and the power on the transmitted mode decreases. Therefore, various misalignment issues, including lateral displacement and receiver angular error, should be considered in an OAM-based communication link.

Turbulence effects may be manifest in atmospheric and underwater environments. In this subsection, we discuss atmospheric turbulence effects. Specifically, temperature fluctuations and air currents in a turbulent atmosphere cause the appearance of eddies. These eddies lead to fluctuations of the local index of refraction at any position in the atmosphere.⁴⁴ Typically, the average size of the turbulent eddies could be characterized by two parameters. One is the inner scale l₀, and the other is outer scale L₀, typically on the order of millimeters and meters, respectively.⁴⁴ When a light beam passes through a turbulent atmosphere, dynamic phase shifts at different cross-sectional locations will be induced. Such differential phase shifts can cause a phase front distortion to the transmitted beam.¹⁰⁵ Under typical conditions, the dynamic phase front distortion could vary on a millisecond timescale.^44,106

The turbulence strength and turbulence fluctuation⁴⁴ could be quantified by the refractive index structure constant $Cn2$ and the Rytov variance $σR2$⁠, respectively. $Cn2$ is also known as atmospheric refractive index inhomogeneities. The values of $Cn2$ in the atmosphere are typically in the range from $10−17 m−2/3$ (relatively weaker turbulence strength) to $10−13 m−2/3$ (relatively stronger turbulence strength).⁴⁴ $σR2$ represents the intensity fluctuation of the optical beam after propagating through a turbulent medium. The value of $σR2$ is typically related to the refractive index inhomogeneities $Cn2$⁠, optical wavelength λ, and the propagating distance L.⁴⁴ Generally, a higher value of $σR2$ represents stronger turbulence fluctuation.

Various models have been reported to simulate beam propagation through a turbulent atmosphere.^44,107,108 One of the simplified turbulence models is the Kolmogorov turbulence model.^44,108 The power spectral density of the refractive index fluctuations given by the Kolmogorov model is⁴⁴ 

where κ is the angular spatial frequency vector. According to the Kolmogorov spectrum model, the relationship between $Cn2$ and $σR2$ can be represented as⁴⁴ 

where $k=2π/λ$ is the wavenumber and L is the propagation distance. As an alternative to $Cn2$⁠, the Fried parameter, r₀, is the radius representing atmospheric coherence length.^109,110r₀ defines an circular area outside of which atmospheric turbulence becomes uncorrelated.¹⁰⁹ In the Kolmogorov turbulence model, r₀ of a plane wave is given by^109,110

There have been several methods to investigate the atmospheric turbulence effect on an OAM-based link.^45–47,110 Figure 17(a) shows a rotating phase plate used as one of the turbulence effect emulation methods. The modal power leakage is characterized by measuring the modal power ratio of the distorted beam, as shown in Fig. 17(b). Under the relatively weaker turbulence (⁠$Cn2=2.5×10−16m−2/3$⁠), most power of the desired OAM mode (i.e., OAM +3) could be received. However, under relatively stronger turbulence (⁠$Cn2=2.2×10−14m−2/3$⁠), the power ratio of the undesired modes increases, which leads to relatively large signal power loss and inter-modal crosstalk.

The challenges in an OAM-based free-space communication system discussed in Sec. IV B may cause power loss and modal crosstalk among the data channels. In order to achieve a good performance, some mitigation methods have been reported to potentially alleviate the deleterious effects caused by these challenges, as shown but not limited to the approaches in Fig. 18. In this section, we will discuss the mitigation methods for the three main challenges of the OAM-based free-space communication systems in terms of the benefits and limitations.

Due to the divergence of the OAM beam, the receiver aperture size may be required to be larger to capture a sufficient potion of the beam when the transmission distance becomes longer. However, the requirement of a larger aperture size might increase the cost of the receiver. Therefore, it might be desirable to reduce the OAM beam size at the receiver aperture. This can be achieved by placing lenses at the transmitter to focus the transmitted OAM beams.¹¹¹ The concept of using the transmitter lenses is shown in Figs. 19(a) and 19(b). A pair of lenses with focus lengths of f₁ and f₂ is placed at the transmitter. The multiplexed OAM beams pass through the lenses before transmitting out of the transmitter aperture. The lens pair has an equivalent focus length of f₀, satisfying¹¹¹ 

where d is the spacing between the two lenses. By tuning the spacing d, the beam size of the focused OAM beams at receiver can be changed.

Figure 19(c) shows the simulated power loss as functions of transmission distances with both a transmitter and receiver diameter of 10 cm. The simulation results show that power loss reduction using a lens pair with f₀ of 1 km could be achieved within a distance up to 2 km. However, a transmitter lens pair with a given f₀ could be a potential solution up to a specific propagation distance. For a longer distance, more accurate adjustment of d might be required.¹¹¹ In addition, it is shown that the transmitter lenses could help one to enhance the system robustness under angular errors. However, this method would have a higher power penalty under lateral displacement compared to the link without transmitter lenses.¹¹¹ 

Due to the divergence of OAM beams, the size of the receiver aperture is sometimes limited with the consideration of cost, such that it can only partially receive the transmitted beam. Such a receiver is referred to as “partial receiver.”^113,114 As the OAM beam has a ring-like intensity profile, a partial receiver at the center of the beam may cause a relatively higher power loss. If the partial receiver is placed on the intensity maxima of the ring (off the beam center), the received power can be increased, as shown in Fig. 20(c). However, such displacement of the receiver might cause modal coupling, as shown in Fig. 20(d).

Moreover, placing two partial apertures symmetrically at the beam annulus might help one to further reduce the power loss and modal coupling,¹¹² the concept of which is shown in Fig. 20(a). As an example, to receive the OAM +2 beam, the captured portion of the intensity and phase profiles of the two apertures are shown in Fig. 20(b). Figures 20(c) and 20(d) show the improvement of the received power and crosstalk using double partial receiver apertures. Compared to the case of placing a single aperture on the ring, the received power of double apertures has an increment of $∼6$ dB. As shown in Fig. 20(d), with double apertures, power coupling from OAM +2 to the modes that have an odd number of charge difference to OAM +2 (e.g., OAM –1, OAM +1, OAM +3) is reduced by $∼10$ dB compared to the case of a single aperture. Therefore, a specific set of OAM modes might be selected in order to achieve a low crosstalk. Furthermore, using a partial receiver with more apertures might potentially further reduce the power loss and modal coupling effects. However, such a partial receiver cannot fully capture the azimuthal phase change, which induces modal coupling to other undesired modes. As a result, the number of OAM modes that could be used in a partial-receiver-based system might be limited. It should be noted that the similar concept could also be applied in millimeter-wave OAM communications.

As mentioned above, an OAM beam has an intensity null at the beam center, which induces power loss when a limited-size receiver aperture is placed at the center. For an LG beam with a non-zero-radial-index (i.e., p > 0 LG beam), the size of the inner-most ring can be smaller than that of the ring of the zero-radial-index (i.e., p = 0 LG beam) LG beam with the same $ℓ$ for a limited transmission distance, as shown in Fig. 21(a). Thus, utilizing LG beams with a p > 0 radial index might increase the received power with a limited receiver aperture size.¹⁹ 

Figures 21(b-1) and 21(b-2) show the simulated power loss of the LG beams with various receiver aperture diameters when the transmitted beam diameter is 8 cm. As shown in Fig. 21(b1), when the transmission distance is 500 m, LG beams with p > 0 radial indices have less power loss than LG beams with p = 0 radial indices and the same $ℓ$ value for relatively small receiver aperture sizes. As the receiver aperture size increases, the power loss of the p > 0 LG beams would become higher than that of the p = 0 LG beams. This might be due to the fact that the outer ring of the p > 0 LG beam is larger than the ring of the p = 0 LG beam. It can also be seen that when the transmission distance increases from 500 m to 1 km, the power loss reduction tends to decrease, as shown in Fig. 21(b2). This might be because that the p > 0 LG beam diverges faster than the p = 0 LG beam when both the $ℓ$ value and the transmitted beam waist are the same. Furthermore, this method may also induce additional channel crosstalk in an MDM system based on the multiplexing of LG modes with different radial indices. This is due to the fact that the limited-size aperture may affect the orthogonality among these modes, which has been discussed in Sec. IV B.

In general, each independent data channel is transmitted on an optical beam using a single LG mode in an LG-based MDM system. However, the performance of such an MDM FSO system could be affected by a limited-size receiver. This impairment would result in deleterious signal power loss for the transmitted LG mode and power coupling from the transmitted LG mode to other LG modes with different p values, that is, modal crosstalk. One potential approach is to transmit each data channel on a single beam composed of multiple LG modes with complex coefficients to achieve smaller data-carrying beams at the Rx to mitigate the effects of the limited-size receiver, as shown in Fig. 22. By carefully designing the coefficients of each LG component, these data-carrying multi-mode beams could be spatially tailored and remain orthogonal to each other at the receiver.

The modal coupling caused by the limited aperture between the LG modal set can be described by a transmission matrix H. This approach is based on the singular value decomposition (SVD) of H, as follows:¹¹⁵ 

For each channel, the transmitted beam is specifically designed and generated such that it comprises a set of LG modes. The complex coefficients of these LG components are obtained by column vectors of the V. Due to the orthogonality between the column vectors of the V, the transmitted beams on different channels are orthogonal with each other. After propagating over a certain distance and experiencing a limited-size receiver aperture, the independent beams still comprise such a set of LG modes. It should be noted that the complex weights of LG components for these resulting beams are the row vectors in the U matrix multiplied by the singular values in the $Σ$ matrix. Consequently, these beams are still orthogonal with each other, which enables efficient demultiplexing at the receiver with little interchannel crosstalk.

Figure 23(a) presents measured power loss and crosstalk with different receiver aperture sizes for beams composed of single mode or multiple modes. When using single beam composed of one LG mode for each channel, the power loss and crosstalk increase as the aperture size decreases. However, when using a single beam comprising multiple LG modes, the power loss is relatively low when the aperture radius is $1.0−1.6$ mm. In addition, the crosstalk for both channels also remains at a low level in this range due to the orthogonality preservation between the two beams after experiencing the limited-size aperture. It should be noted that this method can also be used to mitigate the power loss and modal crosstalk caused by the receiver aperture misalignment, which will be discussed later. To use this method, the transmission matrix H need to be measured. Since dynamic link impairments might cause a dynamic change in H, the measurement needs to be updated. Therefore, a feedback from the receiver to the transmitter might be required.

A beam tracking system is important for the free-space OAM-based communication systems, especially when the link is dynamic that there may be residual misalignment between the transmitter and receiver. The beam tracking with a beacon beam at a different wavelength than the signal beam is shown in Fig. 24(a). A beacon beam at the wavelength λ₂ with the fundamental Gaussian mode is transmitted co-axially with the OAM signal beams at the wavelength λ₁. At the receiver, a tracking system is implemented to detect and compensate for the displacement between the transmitter and the receiver. In this system, a position-sensitive detector (PSD) is first utilized to detect the position information of the beacon Gaussian beam. Such a PSD has four cell quadrants (Q1 to Q4), which can detect the beam position by measuring the received optical power on them, as shown in Fig. 24(b). The outputs of the PSD are voltages V_x and V_y, which can be evaluated by¹¹⁹ 

Subsequently, V_x and V_y are sent to the controller as a feedback signal such that the angle of the FSM can be dynamically tuned to keep the transmitted OAM beams close to the PSD center. As an example, Figs. 24(c) and 24(d) show the result of the tracking for the displacement in one direction of the transmitted OAM beam. The beam centroids without tracking and with tracking using the beacon Gaussian beam are shown. The results indicate that the misalignment in the ±5-mm displacement range can be compensated for using the tracking system. However, such a tracking system requires an additional Gaussian beam as a beacon beam and an accurate alignment between the beacon beam and the transmitted OAM beam.

In an OAM-based communication link, the transmitted OAM beam itself can be used for the tracking without using an additional Gaussian beacon beam. In general, the gradient of the OAM beam's intensity profile has (1) a direction pointing to the beam center along the radial direction and (2) varying strengths at different distances to the beam center. Due to the unique characteristics, the intensity gradient of the OAM beam can be potentially used to determine the position of the transmitted OAM beam. As shown in Fig. 25(a), the received OAM beam can be first split into two branches by a beam splitter. One of the two branches can be sent to a three-pixel detector to measure the gradient information by vector summation of two orthogonal components in horizontal and vertical directions. Subsequently, the measured intensity gradient can be used to determine the position of the beam and used as the correction signal to a tracking system. As shown by the intensity distributions in Figs. 25(b-1) and 25(b-2), the OAM beam may have a higher intensity gradient than the Gaussian beam. However, it should be noted that a higher order OAM beam could provide higher tracking accuracy but may operate over a smaller range of misalignment.¹¹⁸ 

Without a tracking system, the aforementioned SVD-based beam orthogonalization using LG modes for divergence mitigation can also help one to reduce the power loss and crosstalk induced by the misalignment, as shown in Fig. 22. Figures 26(a) and 26(b) show the experimental results for the power loss and crosstalk mitigation of the misalignment effects using the SVD-based methods. Figure 26(a) shows that for the case of beams comprising one LG mode with different p values (LG₁₀, LG₁₁), as the horizontal displacement increases, both the power loss and crosstalk increases. However, for the case of beams comprising multiple LG modes, the power loss for both channels becomes lower when the displacement varies from 0.4 to 0.7 mm. In addition, the crosstalk for the beams comprising multiple LG modes on both channels remains at a relatively low level (⁠$<−27$ dB). The case for the beams comprising multiple LG modes with different $ℓ$ values (LG₁₀, LG₋₁₀) is also investigated, as shown in Fig. 26(b). It is observed that using such an approach, the power loss can be reduced when the displacement varies from 0.8 to 1.1 mm. Moreover, the crosstalk for both channels also remains at a relatively low level (⁠$<−17$ dB).

In common radio systems, multiple–input–multiple–output (MIMO) electronic channel-equalization digital-signal-processing (DSP) algorithms are widely used to “unwind” channel crosstalk when multiple independent transmitting antennas are communicating with multiple receiving antennas.¹²¹ In an OAM-based communication link, multiple OAM beams can be considered as originating from multiple transmitting antennas. Therefore, similar approaches can be adopted to free-space OAM-based communication links to mitigate the crosstalk induced by the atmospheric turbulence. Figure 27(a) shows the concept of using a 4 × 4 MIMO equalization to recover four OAM channels after propagating through atmospheric turbulence. Four coherent receivers are used to detect the signals $xi$ (⁠$i=1,2,3,4$⁠) carried by the four OAM beams as the input of the MIMO equalization. The output of the MIMO equalization can be expressed as¹²² 

where $wij$(⁠$i,j=1,2,3,4$⁠) is the coefficient vector. The coefficient vectors could be adaptively estimated to mitigate the channel crosstalk induced by the turbulence. Figure 27(b) shows the measured signal constellations of a four OAM-multiplexed link with turbulence effect with and without the MIMO equalization.¹²⁰ Four OAM beams each carrying a 20-Gbit/s QPSK signal are multiplexed and propagated co-axially through the emulated turbulence. Without the MIMO equalization, the received signals suffer from turbulence-induced crosstalk, resulting in EVMs of 24%, 46%, 33%, and 46%. With the MIMO equalization, the EVMs are reduced to 14%, 14%, 15%, and 21% for the four channels, respectively. The results indicate that the MIMO equalization method could mitigate the channel crosstalk among the transmitted OAM channels. However, the performance of this method could be degraded if most of the OAM beam power is coupled to the modes that are not detected and compared by the algorithm.¹²³ In addition, the MIMO equalization requires the reception of multiple data channels and may increase the electrical DSP complexity when increasing the number of OAM-carrying data channels.

Without increasing the electrical DSP complexity, one optical method to compensate the turbulence effect is adaptive optics (AO). The concept of AO for mitigating turbulence effect in an OAM-based communication link is shown in Fig. 28. A Gaussian probe beam is transmitted co-axially with the OAM beams at different polarizations. An AO compensation system is built at the receiver, in which the turbulence-distorted Gaussian beam is separated from the distorted OAM beams to serve as the probe for wavefront distortion estimations and required correction-pattern retrieval with a wavefront sensor (WFS). A feedback controller is used to update the two wavefront correctors (e.g., DMD or SLM) to compensate the phase front of the Gaussian probe and the distorted OAM beams.¹²⁴ The far-field intensity profiles of the Gaussian beam and OAM beams, as shown in Figs. 28(b) and 28(c), indicate that the turbulence-induced distortion can be compensated by the AO system. It is noted that the probe beam could also be prepared at a separate wavelength instead of a different polarization.¹²⁵ In addition, the wavefront correction pattern can also be used for the pre-compensation of the OAM beams transmitted in the other direction for a bi-directional free-space OAM-based communication link.¹²⁶ However, the probe beam and the feedback loop used in the AO system could increase the complexity of the optical system. In addition, in order to accurately probe the distorted wavefront of beams, accurate alignment between the probe beam and OAM beams might be required especially for a long-distance link.

Furthermore, the AO-based method has also been demonstrated to simultaneously mitigate turbulence effects and demultiplex OAM channels using a wavefront-shaping-and-diffusing (WSD) approach^128,129 or a single MPLC device.^92,130 However, these approaches might require additional optical devices (e.g., an optical diffuser) or more complicated mitigation algorithms, which would increase the system complexity.

In general, using beam forming at the transmitter to mitigate channel crosstalk is a well-understood approach in radio.¹³² Similarly, free-space optical beam forming (or beam shaping) can be accomplished to mitigate turbulence effects.^127,131 For example, this method could be achieved by transmitting each channel on a single beam comprised of multiple modes, such that each composite mode can have a complex coefficient in amplitude and phase, as shown in Fig. 29(a).¹²⁷ In order to obtain the complex coefficient of each composite mode, the complex transmission matrix is first retrieved using simple in-fiber power measurements without using WFS.¹²⁷ Subsequently, the complex coefficients are calculated based on the inverse of the measured transmission matrix. By transmitting each channel on a single beam composed of two OAM modes with the designed complex coefficients, the turbulence-induced channel crosstalk could be mitigated. This is because that, for each channel, the turbulence-induced modal coupling to the other undesired mode would experience a destructive interference with the corresponding modal component of the transmitted beam.

As an example, the measured normalized transmission intensity matrices without turbulence effects, without and with turbulence compensation, are shown in Figs. 29(b-1)–29(b-3), respectively. With the turbulence effects, the crosstalk of the two channels increases to −8.7 and −5.5 dB in the absence of turbulence compensation. With turbulence compensation, the crosstalk decreases to −22.1 and −17.8 dB for the two channels. However, the number of required measurements could increase with the increasing size of the transmission matrix, thus resulting in a higher requirement of the measurement and feedback speed. In addition, this method might induce some power loss, as only a limited number of OAM modes are utilized.

In order to manipulate a larger number of modes, one phase conjugation approach to individually shape the amplitude and phase of each transmitted beam has been implemented in a 340-m outdoor free-space link,¹³¹ as shown in Fig. 30(a). In such an approach, the amplitude and phase profiles of each turbulence-distorted OAM mode could be first retrieved from a back-propagating probe beam with the same OAM order using off-axis digital holography analysis.¹³¹ Subsequently, each data-carrying beam is shaped based on the phase conjugation of the back-propagating probe beam. After the transmitted phase-conjugated beam propagates through the turbulence, an OAM beam with a relatively high mode purity and a relatively low modal coupling could be retrieved at the receiver, as shown in Fig. 30(b). However, in order to obtain the turbulence distortion information at the transmitter, back-propagating probe beams and other relatively complicated devices^133,134 might be needed, which would increase the system complexity.

With the developing demand for the data communications of airborne platforms over recent years, there has been an increasing need for high-capacity FSO links between these airborne platforms and the corresponding ground stations.^135–138 

Figure 31 illustrates various types of airborne and satellite FSO links. Each of these links has particular characteristics and corresponding challenges.

(1) Satellite-to-satellite links could have an ultra-long-distance transmission distance of >1000 km. The beam divergence is a challenge due to the long propagation distance.¹³⁹ (2) Satellite-to-ground-station links require a long transmission distance through the atmosphere. In such a link, a laser beam generally propagates through the Earth's atmosphere. Therefore, both the beam divergence and the accumulated turbulence are the main challenges.¹³⁹ (3) Airplane-to-ground-station links have a length of ∼1–100 km. As the airplane is usually moving fast, the misalignment could degrade the performance. In addition, the atmospheric turbulence is also an important issue in such links. (4) Unmanned-aerial-vehicles (UAV)-to-ground-station communications generally have relatively slow-moving UAVs hovering around the ground station. The link between the UAV and the ground station has a relatively shorter transmission distance of < 1 km. Such links may also be affected by atmospheric turbulence.^140–143 

UAVs such as flying drones have gained interest recently because of their potential applications.^140–143 In order to achieve high-speed communications, the OAM multiplexing technique might be potentially utilized in such a UAV platform. As shown in Fig. 32(a), an example of the OAM-multiplexed UAV communication link has been demonstrated.¹⁴⁰ Two OAM beams, each carrying 40-Gbit/s QPSK signals, are multiplexed at the ground station. The beams are transmitted to the UAV, which is flown up to $∼50$ m away from the transmitter. Subsequently, the beams are reflected by a retro-reflector on the UAV to the receiver, which is located at the same place as the transmitter. A tracking system using a probe Gaussian beam is also employed to mitigate misalignment issues.

In such platforms, atmospheric turbulence might be one challenge, which could potentially affect the system performance. As discussed before, the transmitted OAM beam can be distorted by the turbulence. As a result, part of the transmitted power on one OAM mode can be coupled into other OAM modes, resulting in inter-channel crosstalk. The atmospheric turbulence-induced channel crosstalk could be mitigated using the aforementioned MIMO equalization algorithm. It has been shown that a 2 × 2 MIMO equalization is utilized in an OAM-multiplexed communication link between a hovering UAV and the ground station with a 100-m link length. Figure 32(b-1) illustrates the constellations of the received QPSK signals carried by the two OAM beams. The EVMs could be reduced with the MIMO equalization. It is observed that the MIMO equalization can reduce the EVM from 32% and 54% to 26% and 27%, respectively, for OAM $ℓ$ = +1 and $ℓ$ = –3 beams. Figure 32(b-2) shows the received BERs with various the transmitted power. For both OAM channels, the BER is reduced by the MIMO equalization, which indicates that the turbulence-induced channel crosstalk could be mitigated. In addition to MIMO equalization, some other approaches for mitigating turbulence effects might also be helpful, such as the AO system or beam shaping.

To enable the OAM-based airborne communication links in the future, the corresponding challenges for different scenarios need to be investigated and mitigated. For satellite-to-satellite links, the beam diverges after the long-distance propagation. At the receiver, the received power can be reduced as the aperture size is often limited. Therefore, a highly sensitive detector at the receiver can benefit such communication links. In addition, mitigation methods such as transmitter lenses and partial receivers can also be potentially used to mitigate the loss and crosstalk in such ultralong links. For satellite-to-ground-station and airplane-to-ground-station links, both beam divergence and atmospheric turbulence need to be considered. Therefore, besides the methods for beam divergence, some aforementioned mitigation methods for turbulence effects might also help to improve the system performance.

Moreover, there are some common features favored for these types of FSO links. For example, (1) low size, weight, and power (SWaP) are desired, which might be achieved with photonic integrated circuits and will be discussed in Sec. VIII),¹⁴⁵ and (2) accurate pointing, acquisition, and tracking (PAT) systems are favored, which might mitigate the power coupling among the OAM modes.⁴⁸ 

In addition, the data capacity demand of underwater communication links has also been increasing because of the various applications.¹⁴⁶ Although acoustic waves can be utilized in such underwater links, the data rate transmitted could be quite limited due to the small bandwidth resources at a low carrier frequency.¹⁴⁷ In contrast, optical light waves can provide a higher bandwidth, allowing a higher data rate being transmitted. Specifically, the blue-green visible light region has a low attenuation in the underwater link, enabling high-capacity optical underwater communications.¹⁴⁸ In these communication systems, the OAM multiplexing technique could potentially further enhance the data capacity.

There are several reports of OAM beam propagation through underwater environments with water currents, particle scattering, and turbulence.^149,151 Figure 33(a) illustrates the schematic of an underwater OAM-multiplexing optical communication link. Data-carrying OAM beams are multiplexed, transmitted through the underwater environment, and de-multiplexed at the receiver. Such links can be affected by various water conditions, as shown by the profiles of a Gaussian beam, OAM + 1 and + 3 beams in Fig. 33(b). Among these effects, (i) the tap water is not likely to distort the donut-shaped intensity profile of the OAM beams. (ii) The water current can slightly change the intensity profiles. (iii) The scattering effect is emulated by adding Maalox solution to water. It causes a dynamic change to the profiles over time as the particles move in the water. (iv) The turbulence effect in the water is emulated by mixing cold and hot water to create a thermal gradient. It tends to cause a large distortion to the beam profiles.¹⁴⁹ 

Besides the transmission of a single OAM beam, another work explored using combinations of concentric optical vortex beams for underwater links with scattering effects.¹⁵⁰ Figure 33(c) shows the intensity profiles of the “petal” pattern generated by the periodic constructive and destructive interference of the concentric vortices. The profiles are little affected by the clean water and slightly more distorted by the turbid water because of the underwater scattering effects.¹⁵⁰ 

Blue-green light underwater links with non-OAM beams have been reported over a $∼100$-m transmission distance.¹⁴⁸ Moreover, to further boost the data capacity, OAM-multiplexing has been demonstrated in underwater links with a transmission distance of a few meters^149,151

Generally, the signal could be modulated on the blue-green light by the internal modulation of the light source. For instance, an 1.5-Gbit/s OOK signal is directly modulated on the driving current of a 445-nm laser diode.¹⁵¹ By multiplexing two independent OOK data channels carried by two OAM modes, an underwater communication link with a 3-Gbit/s total capacity is achieved over a 2.96-m transmission distance.¹⁵¹ 

However, the direct modulation on commercial laser diodes might have a limited bandwidth. Therefore, the data rate of direct modulated data channels could be limited. To achieve higher data rates, a high-speed modulated blue-green light can be generated based on the second harmonic generation. For example, a 10-Gbit/s OOK signal can be first generated by using a commercial high-speed 1064-nm lithium niobite modulator. Subsequently, a module for a second harmonic generation is used such that the carrier wavelength of the signal is converted to 532 nm.¹⁴⁹ Figure 33(d) presents the BERs as a function of the received power for the two multiplexed OAM channels, each carrying a 10-Gbit/s OOK signal. As the link is affected by the turbulence in water, the BER is degraded to above the FEC limit. With the constant modulus algorithm (CMA)-based equalization, the BER can be reduced to below the FEC limit.¹⁴⁹ 

In addition to OAM-based links in underwater environments, there are applications where the OAM beams originate above the water and need to be in contact with a station below the water. In this scenario, the OAM beams may propagate through several “layers” at the air-water interface, including maritime turbulence, non-uniform aerosols above water, time-varying water curvatures, and underwater scattering/turbulence below the surface,^152,153 as shown in Fig. 34. The combination of these issues could potentially cause power loss and modal coupling in an OAM-based optical link, which might be alleviated by using aforementioned mitigation approaches. Further investigations in this field could be an interesting challenge.

As mentioned above, atmospheric turbulence could lead to power fluctuation and link loss in a single-channel FSO system.⁴⁴ One way to mitigate the turbulence effects is to utilize AO, which generally consists of a wavefront sensor, a wavefront corrector, and a feedback control unit.^155,156 Without using a feedback control loop for an AO system, diversity schemes could be employed to potentially provide system redundancy and thus reduce the outage probability of the FSO link.^44,156,157 In a diversity scheme, multiple copies of the same data streams are simultaneously carried by multiple beams.^44,158,159 Each beam experiences a different channel condition, which reduces the variance of the overall received power, and thus the outage probability of the link could be lower than that of the link without diversity.^44,158,159 Specifically, spatial diversity can be realized by transmitting the same data stream with multiple apertures at the transmitter and/or receiver (i.e., aperture diversity).^44,158,159 Instead of using multiple apertures that are physically separated, another form of spatial diversity, namely, mode diversity, can be employed utilizing coaxial spatial modes from different or the same spatial mode groups at the same Tx/Rx aperture.^{154,160–162}

Recently, a single-channel FSO link using mode diversity from different spatial mode basis sets has been experimentally demonstrated.¹⁵⁴ Specifically, two beams (i.e., $HGm=2,n=2$ and $LGℓ=2,p=1$⁠) carrying the same data stream are combined and transmitted at a single aperture as shown in Fig. 35(a). Beams with the same mode order (⁠$m+n=|ℓ|+2p$⁠) have similar beam divergence after propagation, and thus they tend to have a similar field size at the receiver. In this case, the $HGm=2,n=2$ and $LGℓ=2,p=1$ beams are selected due to their mutual orthogonality. As these two orthogonal beams with different spatial profiles could potentially interact differently with the turbulent medium, the data channel carried by each data-carrying beam tends to have an independent outage probability. By receiving the combined beams at the receiver, the overall outage probability in the mode diversity scheme tends to be smaller compared to that in the single-mode scheme. As proof of concept, the BER performance of a 1-kbit/s FSO link under varying turbulence strength (r₀) are measured as shown in Fig. 35(b). It is shown that employing mode diversity could help reduce the BER under different turbulence strength.

Furthermore, OAM-based mode diversity has also been experimentally implemented using the modes in the same mode group (e.g., multiple OAM modes).^161,162 In addition, the mode diversity could also be potentially used in an OAM-multiplexing system to increase the system robustness.¹⁶⁰ 

In this section, some experimental demonstrations of FSO OAM multiplexing and encoding links are first presented. Subsequently, different potential challenges (beam divergence, misalignment, and turbulence) and the corresponding mitigation methods in OAM-based FSO communication systems are also discussed. Table I illustrates various experimental demonstrations mentioned in this section of OAM-based optical communication links in free space, in an underwater environment, and between a ground station and a UAV. Major parameters, including number of channels, total data rate, and transmission distance, are summarized. To further explore and develop the mitigation techniques for OAM-based communication links, one may need to consider several issues, including but not limited to: (1) scalability: for scaling the compensation scheme to the system that multiplexes a larger number of modes; (2) processing speed: for real-time mitigation of various environmental effects; and (3) cost and complexity: for enabling the deploy of a compensation scheme in a real OAM-based link.

Optical fibers have been widely deployed in wired communications, providing a high data capacity. In order to further boost the data rate and spectral efficiency of fiber communication systems, OAM modes can be potentially facilitated in optical fibers that support multiple OAM modes.^11,163 In this section, we will discuss several different types of the fibers that can allow the propagation of OAM beams. The random mode coupling is a key issue of the OAM-based fiber communication systems, and several mitigation methods will be introduced.

In general, a conventional multi-mode fiber (MMF) has a relatively large core diameter that can support multiple modes to be propagated. Theoretically, the optical beam in an MMF can be described with the Helmholtz equation,¹⁶⁴ 

where E is the electrical field of the optical beam, $k0=ω/c=2π/λ$⁠, and n is the refractive index as a function of the location in the fiber cross section. Different types of MMFs can be characterized by different n distributions,^165,166 such as step-index fibers, as shown in Fig. 36.¹⁶⁵ 

Equation (22) can be solved with the method of variable separation.¹⁶⁴ The exact solutions lead to the vector eigenmodes of the fiber, including the HE_νm and EH_νm modes. When the mode index ν = 0, they are also referred to as the transverse-electric (TE) and transverse-magnetic (TM) modes. Applying the weakly guiding approximation,¹⁶⁷ the vector eigenmodes can be expressed as¹⁶⁸ 

where $Rℓ,m(r)$ is the radial field distribution (⁠$ℓ=0,1,2,…$⁠; $m=1,2,…$ are two mode indices). In addition, $(r,θ,z)$ is the cylindrical coordinate and refers to the radial, azimuthal position in a fiber cross section, and propagation position along the fiber. $x̂$ and $ŷ$ represent the unit vector of the X and Y polarizations, respectively. e and o denote the even and odd modes, respectively. Each mode solution has a propagation constant β, and the effective index $neff$ is defined as $neff=λβ/2π$⁠. The modes that have a similar propagation constant are generally classified into the same “mode group.”

Under the weakly guiding approximation, another set of mode solutions with linear polarizations can be obtained, which are referred to as the LP modes.¹⁶⁴ The LP_ℓ,m mode solution along one polarization (e.g., x polarization) is given as¹⁶⁸ 

We note that the LP modes can be constructed by a linear superposition of vector eigenmodes,¹⁶⁸ 

It is observed that, based on Eq. (24), OAM can be represented by a combination of the LP modes or vector eigenmodes, as shown in Fig. 37. For example, a combination of LP modes¹⁶⁹ 

can represent an OAM mode of order $ℓ$ with a linear polarization. In order to increase the data capacity in a fiber-based MDM link, OAM modes can be potentially used as a modal basis in the MMFs.¹⁷⁰ 

Although the fiber medium is quite different than free space, OAM-based fiber communication systems also have important mode-related challenges. For instance, while the beams could be excited into orthogonal modes (e.g., vector modes or LP modes) when launching into an MMF, they will often be coupled to other modes as a result of the temperature variation, different kinds of inhomogeneities, bends, and other non-idealities.¹⁷¹ 

In general, different modes in a fiber might have different β values. When the difference between the β values for two modes is smaller, intermodal power coupling gets higher. If the two modes have the same β, the power coupling becomes the highest.¹⁶⁴ Thus, the modes from the same mode group tend to have a relatively strong coupling (i.e., intra-modal-group power coupling) to each other during fiber transmission. In contrast, modes from different mode groups have large propagation constant differences such that they have a relatively weak mode coupling (i.e., inter-modal-group power coupling).¹⁷² It has been shown that an effective index difference of $Δneff>10−4$ (⁠$neff=λβ/2π$⁠) can potentially result in a low mode coupling between the modes and stable propagation for a certain transmission length.¹⁷³ As mentioned above, an OAM mode could propagate in conventional central-core fibers as a combination of the vector eigenmodes or LP modes within the same group. Moreover, these mode components (vector eigenmodes or LP modes) that have similar β (or $neff$⁠) tend to randomly couple power into each other. Therefore, the OAM modes that are combination of these modes could be distorted, thus inducing mode coupling to other OAM modes.¹⁶⁸ 

Generally, the mode coupling effect in the fibers could induce channel crosstalk for an OAM-multiplexed fiber communication system.¹⁶³ Therefore, mitigation methods or novel designed fibers for the issue of the random mode coupling are often required to enable the OAM multiplexing.

The concept of MIMO equalization mentioned previously to compensate for turbulence effects in free-space links can be used to compensate the OAM channel crosstalk caused by the random mode coupling in central-core fibers without adding optical complexities. An example of the receiver configuration is shown in Fig. 38(a). In this scheme, two PDM QPSK signals are carried by OAM +1 and −1. The two OAM beams are transmitted through a 5-km few-mode fiber (FMF) and demultiplexed to two single-mode fibers (SMF). A local oscillator is split to four branches to be mixed with the four data signals. Subsequently, the four data channels are detected and sampled simultaneously. Offline DSP is used to perform a 4 × 4 MIMO equalization.¹⁷⁰ Figure 38(b) shows the received constellations of the 20-Gbit/s QPSK signals carried by the four multiplexed OAM beams, which have the corresponding EVMs below 25%. It should be noted that this approach might also be potentially used for simultaneous mitigation of both inter- and intra-group mode couplings.^122,174 However, the MIMO equalization could have high electrical complexity when the number of modes increases.

In some applications, it might be desired to use an optical approach to compensate for the modal crosstalk without increasing the electrical DSP complexity. The AO system introduced previously for the turbulence mitigation can be used to compensate the mode crosstalk in MDM fiber communication systems.

The schematic of intra-group mode coupling mitigation in an FMF utilizing AO is shown in Fig. 39(a). The mode coupling of the OAM +1 and -1 modes in an FMF can be described with a complex 2 × 2 transmission matrix H. The amplitude and phase components indicate the power coupling and phase change between the two OAM modes, respectively. An inverse matrix S can be applied to the received beams with mode coupling¹⁷⁵ 

where $s1T=[s11 s12]$ and $s2T=[s21 s22]$ are the row vectors of matrix S. A phase pattern can be generated as a superposition of the demultiplexing phase patterns of the two OAM modes. The complex weights are determined by the row vectors (⁠$s1T$ and $s2T$⁠) for the OAM +1 and −1 channels, respectively. By applying such a superposed phase pattern at the receiver, the crosstalk between the two OAM channels can be mitigated. Figure 39(b) shows the measured crosstalk as a function of time for the $ℓ=+1$ channel. The crosstalk is reduced to a relatively low level (⁠$<−11$ dB) when the AO is applied. The AO is applied again when the crosstalk increases to a threshold. In addition, a similar “optical equalization” approach, called digital optical phase conjugation, has also been used for the mitigation of the inter-group mode coupling.¹⁷⁶ However, these optical approaches might require a feedback loop, which would increase the complexity of the optical system.

Without increasing system complexity, one can consider to designing novel fibers to support the transmission of OAM modes.^11,178–180 In order to reduce the OAM mode coupling, the design of such fibers generally has two goals: (1) large effective index difference between the vector eigenmodes and (2) a fiber profile that is compatible with the donut-shaped OAM beams.¹⁸¹ One potential candidate might be the ring-core fiber of which the typical refractive index distributions are shown in Fig. 40(a).

An example of such ring-core fibers for OAM mode propagation is shown in Fig. 40(b). The measured (dots) and numerically calculated (lines) effective index differences of the first-order modes with respect to the LP₀₁ are shown in Fig. 40(c).¹¹ The result shows separation among the effective index of the OAM mode, the TE₀₁ mode, and the TM₀₁ mode, which indicates that this OAM mode would be preserved within a certain transmission distance. Moreover, 1.6-Tbit/s data capacity with two OAM beams on ten wavelengths have been achieved through such a fiber over 1.1 km.¹¹ 

Furthermore, to increase the number of supported modes, a high refractive index difference between the ring-shaped core and the inner core is preferred.¹⁸² For example, the air–core fiber has been proposed with the air as the inner-core medium.^177,183 As shown in Fig. 41(a), the simulated effective indices of a specially designed air–core fiber have a relatively large difference between most of the OAM modes. As a result, MIMO-free data transmission of 12 OAM modes over 1.2-km air-core fiber is achieved with 60 WDM channels, each carrying 10-Gbaud QPSK signals.¹⁷⁷ The received BERs of all the OAM modes on three wavelengths are shown in Fig. 41(b). However, such novel fibers have different structures compared to the current deployed fibers. In order to broadly deploy them in the actual links, more resolve might be needed.

In addition to random mode coupling, the power loss caused by fiber attenuation might be another challenge for OAM-based fiber communication systems, which may limit the transmission distance. In order to achieve long-distance OAM-based fiber transmission links, the inline OAM fiber amplifiers might help us to compensate for the power loss in the fiber link. For an OAM-based fiber communication system, a desired OAM fiber amplifier might have features, including high gain and small gain difference on different OAM modes and different wavelengths.

As similar to the case of conventional single-mode fiber amplifiers, there are generally two types of OAM fiber amplifiers, including the OAM erbium-doped fiber amplifiers (EDFA)^184,186 and OAM Raman amplifiers,¹⁸⁵ as shown in Fig. 42. The OAM EDFA uses the specially designed erbium-doped fiber (EDF) as the gain medium, which has a central air hole and a ring-shaped erbium-doped core in its cross section,¹⁸⁴ as shown in Fig. 42(a-1). The OAM beams can be amplified by launching them to the EDF along with a 976-nm pump laser. The measured gain spectrum has a gain peak of $∼15.7$ dB located at $∼1565$ nm, as shown in Fig. 42(a-2). Figure 42(a-3) shows the interferograms of the amplified OAM + 1 and –1 beams at the output of the EDFA.¹⁸⁴ This OAM EDFA shows a gain of >10 dB with a uneven curve in the $∼20$-nm wavelength range. Recently, this type of OAM amplifier has been utilized in a two-OAM-multiplexed communication link with each OAM carrying 10-Gbaud 16-QAM signals on four wavelengths.¹⁸⁶ 

The OAM Raman amplifier uses the normal ring-core fiber for both transmitting and amplifying the OAM modes, as shown in Fig. 42(b-1). In this example, to amplify the OAM beams, two 1455-nm pump lasers are launched along with them in the forward Raman pumping configuration. Another two 1455-nm pump lasers and a 1360-nm pump laser are launched in the other direction in the backward pumping configuration.¹⁸⁵ The measured gain spectra for OAM + 4 and + 5 modes are shown in Fig. 42(b-2). The intensity profiles of the output OAM modes after the Raman amplifier are shown in Fig. 42(b-3). In contrast to the EDFA example, this OAM Raman amplifier achieves a $∼3$-dB gain with a relatively even curve in the $∼30$-nm wavelength range.

In this section, we first discuss the propagation of OAM modes and random mode coupling in fibers. Subsequently, different approaches and novel fibers for mitigating mode-coupling-induced channel crosstalk in OAM-based links are presented. In addition, different types of inline OAM fiber amplifiers are discussed to compensate for the power loss caused by the fiber attenuation. Table II illustrates the experimental demonstrations mentioned in this section of the OAM-based fiber communication systems. Major parameters of each experiment, including the number of channels, total data rate, fiber type, and fiber length, are summarized. In order to further deploy OAM modes in fiber-based communication links, novel fibers with more complex designs, more advanced OAM fiber amplifiers, or other techniques might be needed to improve system performance in fibers over a long transmission distance.¹⁶⁸ 

Quantum optical communication links have attracted much interest since they could potentially enhance the system security.^187–189 In order to increase the photon efficiency (i.e., bits/photon), OAM modes can be potentially used in quantum communication links.^190–193 In this section, we will first illustrate the basic concept of the OAM-based quantum encoding system. Subsequently, the quantum key distribution (QKD) systems based on OAM modes are introduced. In addition, experimental demonstrations of OAM-based QKD systems in different scenarios are presented. Some practical challenges and corresponding mitigation methods are discussed.

Conventionally, data could be encoded on two orthogonal quantum states of the photon, for example, polarizations.¹⁹⁴ In such a quantum scheme, only one bit of information is encoded on each photon. In order to improve photon efficiency, a larger number of orthogonal states might be utilized.^190–193 As mentioned above, OAM is another property of photons that can also be potentially used in communication links. The experimental accumulated intensity structure of single photons with OAM +1 is shown in Fig. 43.¹⁹⁵ For single photons, the intensity profiles gives the probability of detecting a photon at a certain point. If a triggered single-photon camera is used to detect many heralded OAM photons, the donut-shaped intensity profile of the OAM mode emerges.¹⁹⁵ Generally, the OAM occurs in discrete steps of $ℓℏ$⁠, where $ℓ$ is an integer.⁵ Therefore, if multiple orthogonal OAM states are used for encoding, such a high-dimensional quantum link could potentially improve the photon efficiency. Figure 43 presents the concept of a quantum link by encoding data on OAM modes. In each symbol period, a single Gaussian photon is converted to and occupies one of the M OAM states, inducing a photon efficiency of up to $log2M$ bit/photon.¹⁹⁶ 

Quantum cryptography, such as the QKD, implements a cryptographic protocol and, thus, enables secure communication links between many parties.¹⁸⁷ Typically, QKD systems utilize polarization states of photons for data encoding.^197–199 For example, in the BB84 protocol,¹⁸⁷ two pairs of orthogonal polarization states are usually used, which are two mutually unbiased bases (MUBs). The usual polarization state pairs used are the rectilinear basis [vertical (0°) and horizontal (90°)] and the diagonal basis (45° and 135°). The sender (Alice) creates a random bit (0 or 1) and randomly chooses one of the two bases.²⁰⁰ Therefore, she randomly prepares a photon based on the bit and chosen basis. Subsequently, she transmits the photon to the receiver (Bob) using the quantum channel. Since Bob does not know which basis the photon is encoded in, he randomly chooses a basis for the photon measurement.²⁰⁰ After Bob has measured all the photons, he communicates with Alice about the selected bases over a public classical channel. In this way, this QKD protocol establishes a secure key between Alice and Bob. This is due to the fact that, if an eavesdropper selects the incorrect basis, he or she might obtain no meaningful information but instead introduce a disturbance in the system, which results in its detection by Alice and Bob.¹⁸⁷ 

Although MUBs based on polarization states offer security against eavesdropping, only a maximum of one bit of information per photon can be transmitted, resulting in a limited key generation rate. As another degree of freedom, the OAM state could be potentially utilized for achieving higher-dimensional QKD schemes.^17,202,203 Figure 44 presents the experimental setup and the two MUBs for a seven-dimensional OAM-based QKD system. In this scheme, Alice randomly chooses photons from two MUBs for information encoding. One base is a set of OAM modes, whose phase profiles can be characterized by $ψOAMℓ=eiℓθ$⁠, where $ℓ∈$ ${−3:+3}$⁠. Moreover, the other basis is constructed by a linear combination of these OAM modes based on the following formula:¹⁷ 

where dimension M = 7 in the experiment. These modes are referred to as the ANG modes, as shown in Fig. 44(b). The ANG modes form a MUB with respect to the OAM basis,¹⁷ 

At Alice's side, photons with OAM and ANG states are generated by shining the attenuated pulse on a digital micro-mirror device. At Bob's side, single photons are efficiently separated by a mode sorter based on their OAM/ANG states and subsequently collected by an array of fibers. As similar to the case of the polarization-based MUBs, an eavesdropper who measures a photon in the OAM basis might not obtain useful information about its ANG state and vice versa. Based on such a seven-dimensional alphabet, a channel capacity of 2.05 bits per sifted photon is achieved with an average symbol error rate (SER) of 10.5%.

In general, OAM-based QKD links in the laboratory have been experimentally demonstrated over only several meters. However, when OAM photons are used in a long-distance quantum link outside the laboratory, the system performance might be degraded due to several potential issues, such as beam divergence, system misalignment, and atmospheric turbulence effects.

For example, a high-dimensional FSO OAM-based QKD link has been demonstrated between two buildings over 300 m, as shown in Fig. 45.²⁰¹ Four-dimensional quantum states were utilized based on two OAM modes and two vector modes. Under moderate turbulence without active wavefront correction, a BB84 protocol was performed, and a quantum bit error rate of 11% was achieved with a secret key rate of 0.65 bits per sifted photon.

As mentioned before, atmospheric turbulence distorts the phase front of light in the classical domain, thus inducing both power loss and crosstalk. Similarly, in the quantum domain, atmospheric turbulence is also a key challenge for OAM-based quantum FSO links. Since it would distort the phase front of photons, intermodal crosstalk might be induced.

One potential method for mitigating the turbulence effect is performing an AO system in an OAM-based quantum communication link. This approach has been experimentally demonstrated in an OAM-based encoding link with emulated turbulence effects, as shown in Fig. 46.²⁰⁴ A classical Gaussian probe beam (λ₂, Pol. 2) is transmitted coaxially with the quantum channel (λ₁, Pol. 1). A wavefront sensor and wavefront corrector are used for phase detection and correction, respectively. Subsequently, a PBS, wavelength filter, and OAM converter are used to efficiently separate the classical and quantum channels due to their different polarizations, wavelengths, and OAM orders.

Figure 47 shows the channel transfer matrices (upper line) between OAM modes ${ℓ=−3,…,+3}$⁠, and the photon count ratio (lower line) on the received OAM modes when sending only OAM $ℓ=+1$ photons in different cases. The results show that, under atmospheric turbulence distortion, the mode purity of photons carrying OAM $ℓ=+1$ is reduced from ∼57% to ∼31%. With AO compensation, the mode purity of photons is improved from ∼31% to ∼52%. Consequently, quantum symbol error rates (QSERs) are reduced by ∼76% in such a 10-Mbit/s OAM-based quantum encoding link. In addition, registered photon rates are also improved by ∼64%. Since OAM-based encoding might be performed in the QKD protocols, the AO approach could be potentially used for mitigating turbulence effects in an OAM-based QKD system.²⁰⁶