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Abstract
Information can be encoded onto transverse spatial light modes, providing a platform that can improve the spectral efficiency of a communications link. Switching speed between these spatial modes and detection methods limit the information capacity of such links. This work demonstrates high speed capabilities of such a link. Transverse modes are created and switched at high rates by coherently coupling twisted light modes using passive optical elements in line with electro-optical modulators. Here we demonstrate the encoding and detection of two coherently coupled modes encoded with 0.5 Gbaud quadrature amplitude modulated (16- and 32-QAM) signals, for a 4X and 5X increase in spectral efficiency by exploiting both phase and amplitude of the coherently coupled modes. The receiver is able to successfully recover the signal with error rates below the forward error correction limit using passive optical techniques. The data rate of the system used is only limited by hardware, but similar devices are available that are capable of multi-Gigahertz operation.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
There is a rapidly growing interest for employing advanced multiplexing techniques to meet the ever increasing data transfer needs of modern society. Substantial data rate improvements have been made by utilizing wavelength division multiplexing (WDM), polarization division multiplexing (PDM), spatial mode division multiplexing (SDM) and the implementation of advanced modulation techniques [1,2]. Laguerre-Gaussian modes carry orbital angular momentum (OAM) and are commonly used as an SDM orthogonal mode set. This is possible due to the unique twisted wavefront and that these OAM-carrying modes possess [3] which can be described by exp(iℓϕ), where ℓ is the azimuthal mode index corresponding to the number of 2π phase rotations about the axis of propagation over one wavelength, and ϕ = (0, 2π] is the azimuthal coordinate [4].
Currently, there are two general approaches that use OAM-carrying beams of different states in optical communications applications: SDM and OAM encoding [5]. SDM uses OAM-carrying beams as carriers of different data streams, which has been widely employed in most of the recent works [6–8] for both free-space atmospheric and underwater communication. A theoretically infinite number of these orthogonal modes (ℓ = 0, ± 1, ± 2, …) can be incoherently superimposed and can be efficiently separated at a receiver [9]. SDM can be extremely useful in bandwidth limiting environments, e.g. underwater free-space links where turbid particles can cause a temporal dispersion of the signal. In these cases, utilizing OAM states in an SDM system not only increases the data rate of these bandwidth systems but also these modes tend to be more robust when propagating through extreme environments. They have several interesting properties such as improved power delivery through scattering environments [10], self-healing capabilities [11], and resistance to atmospheric turbulence in some cases [12].
The second method, OAM encoding, employs N different OAM states as N possible values of data symbols or bits [13–20]. In this case, the transmitter will switch between a discrete set of azimuthal modes that are interpreted as the symbol basis. High speed switching is demonstrated at 10 kHz and predicted out through 20 MHz [19], but at these rates this method is better suited for channel hopping applications rather than the transmission of data. One could use a discrete set of light sources, but the system runs into the issue of effective mode combination. In this work we explore a third option that allows for the rapid modulation of the spatial profile of a beam at rates more relevant for data transmission. By encoding information spatially, the spectral efficiency of a link can be significantly increased by combining this phase-shift-keying with multilevel amplitude-shift-keying, such as is used in quadrature amplitude modulation (QAM) schemes.
This third method uses the spatial encoding of information onto coherently combined OAM states, similar to the work in [17]. The coherently coupled states are conjugates of each other, each with an azimuthal mode index |ℓ| but twist in opposite directions. This produces a unique petal-like interference pattern, with 2|ℓ| distinct lobes. Similar spatial modes have been used for particle trapping [21], Doppler vorticity measurements [22], a microwave radio link [23], quantum communications [24] and free-space optical links [20,24]. The demonstration in [20] successfully utilizes a discrete set of transverse modes to transmit information despite propagation through a turbulent environment. The modes in [20] are generated and switched using a discrete set of pre-designed phase profiles encoded onto a spatial light modulator (SLM) without amplitude modulation and are recovered using advanced image processing techniques. The frame rates of SLMs and CCD cameras are typically limited to 60 Hz, which in general limits the data rate of the link. In pervious works we have explored the use of single optical elements to passively generate this coherent mode combination for underwater environments [8,9,25]. Even though the work presented in this paper merely employs OAM-carrying beams with charges ± 2, it is feasible that specific mode combinations can be spatially combined and separated in a manner similar to the SDM method mentioned above, based on the orthogonality of the conjugate mode pairs [26]. In addition, we have previously demonstrated coherent combination and detection at 532 nm for free-space underwater communications [27] at 0.5 MHz. Unfortunately, high speed optical components are not as readily available for visible wavelengths as compared to telecommunication wavelengths.
Based on the third method, this work demonstrates a coherent OAM free-space optical communication link at 1550 nm over a propagation distance of 0.5 m using 500 MHz signals, far outstripping the rates achievable by SLM switching. While only propagating a short distance [20], shows success of a 3 km link. The method presented in this paper is primarily limited by the size of the fabricated optical elements, the divergence of the beam, and signal levels. The overall concept is illustrated in Fig. 1 and Fig. 2. Both the amplitude and phase of the coherent OAM beam are modulated separately for signal encoding, creating a spatialequivalent to QAM where combination of transverse mode and optical power is used to represent each symbol. In order to verify this approach, a 32–QAM encoding technique is introduced using the interference of two OAM beams based on optical multiplexing and optical demultiplexing. This is the first demonstration of data transmission using the controlled generation and recovery of coherently combined OAM modes through spatial amplitude and phase modulation. Bit error rate (BER), symbol error rate (SER), and system limitations are furthermore discussed. In addition, the multiplexing applications of these beams provides a basis for a further increase of spectral efficiency in a system by opening up the ability for multiplexing capabilities and multidimensional constellation mapping. It is important to note that it may be possible to achieve multidimensional constellation mapping using other coherent orthogonal mode combinations but the previous study of similar vector beams [17,24] provides an excellent basis for signal recovery and constellation mapping of coherent OAM combinations while still allowing for the additional benefit of polarization division multiplexing. This unique method of encoding information onto optical signals paves a path for a variety of encoding schemes that exploit the phase and amplitude of coherent optical modes which provides the means for a multidimensional constellation space, where the symbols are mapped to the full sphere shown in Fig. 1(a).
Fig. 1 (a) Modulation concept illustrating coherent coupling of twisted light modes, note that the optical path lengths must ensure temporal and spatial alignment. (b) Simulated output images for an 8-symbol system compared to (c) corresponding experimental images, which can then be mapped to (d) constellation space on the equator of a Bloch Sphere, with symbol locations indicated. See Visualization 1 for a demonstration of symbol mapping.
Fig. 2 Three-channel signal recovery concept illustrating received signals for amplitude and phase modulation. The input beam is modulated in space and amplitude, then three detections are performed, the first monitoring the total input power, and the second and third perform the optical phase matching necessary for mode recovery using binary phase plates. Corresponding ideal output waveforms are simulated for two amplitudes with the phase linearly changing from 0 to 2π twice. See Visualization 2 for experimental demonstration of beam profile, I(t), and Q(t) signals.
2. Dynamic mode control and signal encoding
The general analytic expression of an OAM beam after passing through a spiral vortex phase plate can be expressed as [28]
where B(t) gives total amplitude information; A(ρ) is a radial amplitude function that describes the beam profile; ℓ is the charge number mentioned above; and ϕ is the azimuthal angle. In this case, for two coherent beams with equal amplitude and opposite charge number Gℓ(ρ,ϕ) = B(t)A(ρ)exp(-iℓϕ) and G-ℓ(ρ,ϕ) = B(t)A(ρ)exp(iℓϕ), the interference is the sum of two complex fields. In addition, if the phase of G-ℓ(ρ,ϕ) is changed by a time-changing phase, θ(t), then the complex field can be expressed aswhere ϕ is the azimuthal plane, and θ is the relative phase delay between the two modes and represents the rotation in the azimuthal plane. The modes used in this work are mapped to a Bloch Sphere as shown in [29,30] for equal amplitudes of the two modes, and will therefore be mapped to the equator as in Fig. 1(d).As B(t) and θ(t) change in time, the spatial profile and amplitude of the E-field, U(ρ,ϕ,t), can be temporally controlled as is illustrated in Figs. 1(a)-1(c).
Recovery of the amplitude and phase information B(t) and θ(t) requires that Eq. (2) be decomposed into two orthogonal expressions, which can then be used to correlate with the received signal. If we denote these two orthogonal expressions as UI and UQ as shown in Fig. 1(d), the maximum correlation will occur at θ = 0 and θ = π/2 respectively. For θ = 0, Eq. (2) becomes
and for θ = π/2, Eq. (2) becomesIn order to demultiplex the information from the received signal, an optical correlation must be performed on these orthogonal projections which represent the I and Q coordinates. In order to realize a phase-only filter representation of these respective basis functions, the phase arguments of UI and UQ are taken which yields a function that alternates between 0 and π for UI, and a function that alternates between –π/4 and 3π/4 for UQ [20]. Therefore, both functions have the same phase profile, where one is rotated π/(4ℓ) radians relative to the other as shown in Eq. (4). The corresponding phase functions for the I and Q components are
where the argument of a real sinusoidal function alternates between 0 and π where the sign of the function is positive and negative respectively. These two phase expressions are encoded into a diffractive element, which is then used to optically correlate with the signal function to determine the components for the I and Q detection scheme presented in the next section.For the case in this work where ℓ = 2, the phase profiles of I and Q are described by
where δI = 0 and δQ = π/(4ℓ) = π/8 are the rotation angles of the phase plates in radians according to the value of ϕ that produces a maximum correlation of Eq. (7) with Eqs. (5) and (6) respectively, and are illustrated in Figs. 2 and 3(a-b).Fig. 3 Simulated power plot of (I, Q) before shifting with arbitrary units shown with the corresponding orientation of the interference pattern as θ(t) is varied from -π to π. PB and PC correspond to the power collected after propagation through BPP optics (a) and (b) respectively, optics comprised of alternating optical path lengths 0 and π radians for the 1550 nm source.
In this work, both the amplitude modulation B(t) and phase modulation θ(t) are used for information encoding, properly recovering the two modulation signals from the detected |U(ρ,ϕ,t)|2 will determine how the detection methods are designed. Figures 1(b) and 1(c) compares the analytical and experimental beam profiles for a random bit sequence with amplitude and phase modulation.
3. Amplitude and phase recovery
Optical correlators are commonly used in many SDM applications [6,8,31]. In general, the phase-match element is designed to cancel a matched phase profile of the incident beam, which forms a correlation peak in the Fourier plane of a lens placed after the element.
In order to compute this correlation spot, the electric-field from Eq. (2) can be simplified down to
Equation (8) demonstrates that as the relative phase between the two OAM modes θ(t) changes, the cosine function results in a shift in the ϕ direction, resulting in the apparent rotation of the beam and the total power will ideally remain unchanged. Therefore, the total amplitude, B(t), can be easily detected by monitoring the total signal power. On the other hand, θ(t) must be recovered by mapping the spatial profile to the two orthogonal expressions in Eqs. (3) and (4). This is done using a two-channel optical correlator that uses two binary phase plates (BPP) designed to function as phase-match elements according to Eq. (5) and (6), and illustrated in Figs. 3(a) and 3(b). The BPPs used in this work are custom fabricated in our cleanroom facilities using a standard photolithographic patterning process and etched from the photoresist to the fused silica wafer. The BPPs were etched to a depth of 1.755 µm corresponding to a phase delay of π radians for fused silica with a refractive index of 1.44 for the designed wavelength of 1550 nm.The optical correlators function by exploiting the Fourier transforming property of lenses. The incident beam propagates through the BPPs which produces an inner-product of the two. The inner-product then passes through a lens; the Fourier-plane of this lens therefore contains the correlation information of the incident beam and the BPP. The on-axis power in the Fourier-plane indicates how closely the phase of the incident mode matches the phase of the BPP. Using the Fourier-transform based optical propagation method, the on-axis correlation of Eq. (2) and (7) can be expressed as
where A'(ρ') is a radial amplitude function that comes from the far-field of A(ρ).The power of the correlation spot can be written as follows by applying the half-angle formula to the intensity ,
This is the resulting power of the on-axis information contained in the diffraction limited spot. is proportional to the power in the correlation spot, which is integrated over a radius ρo, determined by the diffraction limit of the system.Substituting ℓ = 2 into Eq. (10) yields the power collected by the two detectors: and for δI = 0 and δQ = π/8 respectively, as illustrated in Fig. 3. From these equations, the orthogonal components of θ(t) can be measured by subtracting PRX(t), which can easily be performed using AC coupling of the photodetector resulting in the following expressions
where I(t) and Q(t) map a full circle centered at the origin representing θ(t) varying from -π to + π radians.Changing the total power of signal will result in phase circles with different radii. This is how B(t) is incorporated into this QAM equivalent encoding scheme. Figure 4 shows a block- diagram of the receiver side overview to help better understanding the I and Q equivalent 2-channel correlation detection, where ℓ = 2 and Const is a fixed constant ratio that is related to system losses, approximately 1.4 after signal amplification. Signals were collected on a scope and processed in MATLAB for specific sample points at t = Tb.
4. Architecture of the communication system
The coherent OAM communication system shown in Fig. 5 includes two main parts: the transmitter, which generates and controls the dynamic OAM modes with signal encoding; and the receiver, which detects the transmitted OAMs modes and decodes the spatial information.
The transmitter—a fiber-to-free space Mach-Zehnder interferometer—consists of an Agilent 8164A 1550 nm seed laser source, a Nufern NUA-1550-PB-0010-B3 Erbium doped fiber amplifier (EDFA), a Thorlabs LN81S-FC fiber-coupled optical amplitude modulator (AM) for optical pulse shaping and amplitude modulation, a Thorlabs LN65S-FC fiber-coupled electro optic phase modulator (PM) and a 50:50 fiber splitter and the corresponding free-space optics. All fiber components are polarization maintained. The two fiber outputs are collimated and passed through spiral phase plates with OAM charge ℓ = ± 2 and then combined with a polarization independent 50:50 cube beam splitter. The output of the system is the coherent combination of OAM beams with ℓ = ± 2, which produces a four-petal intensity profile. A two-channel Tektronix arbitrary waveform generator (AWG 7052) provides modulation signals to the AM and PM for amplitude, B(t), and phase modulation, θ(t), which will project onto the intensity and rotation of the output as described in section 2. The signal sent to the PM was measured to be 0.72 Vpp and the signal sent to the AM was measured to be 0.25 Vpp before amplification through a Tektronix PSPL5865 amplifier. These signals were chosen such that after amplification, the voltage signal to the PM is mapped to eight equally spaced phase values from 0 to 2π, for a half-wave voltage of 6.1 V, and the voltage signal to the AM is mapped to four amplitude values according to a half-wave voltage of 6.0 V.
The output of the transmitter is then propagated 0.5 m through free space to the receiver. The receiver consists of three detection measurements. First, 10% of the incident beam is sampled by a 90:10 beam splitter and focused to a high speed Thorlabs photo-detector (PD, DET08CL) by a lens (f = 20 mm) to monitor the total power of the signal. The PDs used have a responsivity of approximately 1 A/W at 1550 nm. The remaining 90% of the incident beam is 50:50 split into two legs for the I and Q measurements. Legs B and C are optical correlators each containing a BPP at a specific orientation, an AR-coated lens (f = 150 mm), a 100 μm pinhole and a PD (DET08CL) at the Fourier plane of the lens. The received electrical signals pass throu10.gh a DC block and a preamplifier (Fairview Microwave, SLNA-030-32-30-SMA) and then are analyzed by a real-time TDS7404B Tektronix oscilloscope.
Two BPPs oriented at angles δB = 0° ± 1° and δC = 22.5° ± 1° = π/(4ℓ) relative to the x-axis, are used to perform an optical correlation with the incident beam at high speed [26,30–32]. The specific profile of the BPPs are determined using Eq. (3) such that the leg B and C are mapped onto UI and UQ. More detail of the de-modulation will be provided in section 4. The amplitude and phase value mapping follows the concept of Poincare-sphere equivalent OAM sphere introduced in [31]. Instead of mapping to the whole sphere, only the equator-plane is mapped by keeping the ratio of the two OAM beams to be equal.
The power loss from the 50:50 beam splitter was measured to be −0.61 dB total. The BPPs are not AR coated and therefore introduce losses of the −0.22 dB and −0.28 dB for legs B and C respectively. The mean loss of the phase-modulated signal through the pinholes is −6.11 dB and −5.83 dB respectively.
In order to demonstrate the feasibility of this communication link, a 16- and 32-star QAM map is used to transmit a data signal. MATLAB is used to create a uniform pseudorandom bit sequence using a Mersenne Twister pseudorandom number generator with a seed of zero. The signals are encoded using grey symbol mapping in order to minimize symbol errors. The electrical amplitude signals were 2- and 4-level pulse signals, filtered with a 30%-width Gaussian filter; the phase signal for both schemes was a 30%-width Gaussian-filtered 8-level return-to-zero (RZ) code. Sample digital waveforms used for the generation of a 32-QAM signal are shown below in Fig. 6.
Fig. 6 Sample of normalized and filtered waveforms generated in MATLAB which are scaled on the AWG and applied to the amplitude and phase modulators for 32-QAM signals with ideal sample locations indicated.
The symbols were transmitted at a rate of 0.5 GHz. First, a 16-star QAM signal was transmitted by using a map with 2-amplitude and 8-phase levels. 50,000 symbols were transmitted and recovered using photodiodes that convert optical power to an electrical current with approximately a 1:1 ratio as noted in section 3. The electrical current is then converted to a voltage with a 50 Ω load and displayed on the scope. Therefore, the symbol maps recovered by the scope, shown in Fig. 7, are amplified by a factor of approximately 50. The 16-QAM transmission, shown in Fig. 7(a), had a SER of 1.18 × 10−3, with a corresponding BER of 2.36 × 10−4, well below the acceptable forward error correction limit of 3.8 × 10−3 [7]. Next a 50,000 32-QAM symbols with 4 amplitude levels and 8 phase levels was transmitted and is shown in Fig. 7(b) with a measured SER of 1.75 × 10−2, corresponding to a BER of 3.51 × 10−3. The number of bits that each symbol can represent is given by log2(N), where N is the number of symbols in the system. This result is a 4X and 5X increase in the number of bits transmitted per symbol over that of a system with binary encoding for the 16- and 32-QAM symbol maps respectively using purely spatial amplitude and phase modulation.
Fig. 7 Density plot of constellation diagram of an equivalent spatial (a) 16-QAM and (b) 32-QAM signals with corresponding ideal grey-encoded symbol locations. Color map indicates bin counts out of 50,000 total transmitted symbols.
5. Discussion and conclusion
Noise in the system results in an azimuthal and radial spreading of the signal as a result of the mapping technique used. Assuming a 0.72 Vpp signal with Gaussian-distributed noise, the azimuthal spread will be 26.07°. This is the best-case scenario for the AWG with an amplitude accuracy of ± (3% of the amplitude + 2 mV) and a 5.75 dB noise figure from an 19x electrical amplifier, assuming a 6 V half-wave voltage. In the azimuthal direction, 99% of azimuthal signal is contained within 28.9°. Assuming a 0.312 Vpp signal with Gaussian-distributed noise, the radial spread will be 20.9% of the total amplitude-modulated signal. This is the best-case scenario for the amplitude signal with the noise and amplification specifications mentioned previously. The average radial spread is measured to be approximately 21.5% of the received amplitude, assuming 99% of Gaussian-distributed noise. This noise results in a measured optical signal-to-noise ratio of approximately 15.3 dB for 16-QAM and 14.6 dB for 32-QAM.
Noise in the both the azimuthal and radial direction will result from electrical intersymbol interference of the waveforms, which was minimized using a Gaussian filter and amplitude pulsing as mentioned above. Additionally, temporal misalignment of the amplitude and phase signals, sampling time, and optical alignment will increase the noise of the system. Temporal alignment resolution is limited to 200 ps, and therefore minimally affects the 500 MHz signals with a full-width half-maximum of 600 ps. In addition, misalignment of optical setup introduces amplitude fluctuations detected on the power monitor which can be seen as an increase in noise along one axis of the constellation plot. This is a result of a slight misalignment of the transmitted beam such that the phase modulation signal can be seen as on top of the amplitude monitor, PRX. This increased the noise on the amplitude signal by approximately 3.4%, primarily along the axis that the signal is shifted, PA = PB. The additional noise from the power monitor will primarily show up along the axis which the signals are shifted back to center, but minimally affects the other symbol locations.
There are several methods that could be employed to improve the system. Temporal and spatial alignment is key and could be greatly improved with more precise control of the electrical waveforms and higher precision of the detection phase plate alignment. The symbol rate of the system in this work is limited to 500 MBd due to current hardware limitations in the present setup for waveform generation and sampling. Amplitude and phase modulators are available with operation in excess of 40 GHz. In addition, it is important to take into account phase sensitivity of a Mach-Zehnder interferometry setup. Temperature fluctuations and air currents result in a phase drift of the system. Modulation at high speeds minimizes the impact on the system, but a correction sequence was necessary to compensate. This can also be mitigated using a Differential Phase Shift Keying approach to eliminate the phase wanders.
In conclusion, we have demonstrated the successful spatial and amplitude modulation of coherent coupled OAM beams employing the equivalent of a 16- and 32- QAM modulation scheme, with error rates below the FEC limit. This is equivalent to a multiplication factor of 4 and 5 in spectral efficiency using only three detection measurements. In addition, this increase in spectral efficiency is only limited by the source coherence length, hardware stability and electrical timing. To further increase the data capacity of these spatial links, we will explore more complex spatial mapping techniques that have the potential to further increase the spectral efficiency of a link through the dynamic control of relative amplitudes and alternative mode combinations.
Funding
Office of Naval Research (ONR) (N00014-161-3090, N00014-17-1-2779).
References and links
1. H. Gnauck, P. J. Winzer, S. Chandrasekhar, X. Liu, B. Zhu, and D. W. Peckham, “Spectrally efficient long-haul WDM transmission using 224-Gb / s polarization-multiplexed 16-QAM,” J. Lightwave Technol. 29(4), 373–377 (2011). [CrossRef]
2. A. Sano, H. Masuda, T. Kobayashi, M. Fujiwara, K. Horikoshi, E. Yoshida, Y. Miyamoto, M. Matsui, M. Mizoguchi, H. Yamazaki, Y. Sakamaki, and H. Ishii, “Ultra-high capacity WDM transmission using spectrally-efficient PDM 16-QAM modulation and C- and extended L-band wideband optical amplification,” J. Lightwave Technol. 29(4), 578–586 (2011). [CrossRef]
3. L. Allen, M. W. Beijersbergen, R. J. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992). [CrossRef] [PubMed]
4. M. P. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using light’s orbital angular momentum,” Science 341(6145), 537–540 (2013). [CrossRef] [PubMed]
5. S. Li and J. Wang, “Experimental demonstration of optical interconnects exploiting orbital angular momentum array,” Opt. Express 25(18), 21537–21547 (2017). [CrossRef] [PubMed]
6. J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012). [CrossRef]
7. N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340(6140), 1545–1548 (2013). [CrossRef] [PubMed]
8. J. Baghdady, K. Miller, K. Morgan, M. Byrd, S. Osler, R. Ragusa, W. Li, B. M. Cochenour, and E. G. Johnson, “Multi-gigabit/s underwater optical communication link using orbital angular momentum multiplexing,” Opt. Express 24(9), 9794–9805 (2016). [CrossRef] [PubMed]
9. G. C. Berkhout, M. P. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, “Efficient sorting of orbital angular momentum states of light,” Phys. Rev. Lett. 105(15), 153601 (2010). [CrossRef] [PubMed]
10. B. Cochenour, K. Morgan, K. Miller, E. Johnson, K. Dunn, and L. Mullen, “Propagation of modulated optical beams carrying orbital angular momentum in turbid water,” Appl. Opt. 55(31), C34–C38 (2016). [CrossRef] [PubMed]
11. M. McLaren, T. Mhlanga, M. J. Padgett, F. S. Roux, and A. Forbes, “Self-healing of quantum entanglement after an obstruction,” Nat. Commun. 5, 3248 (2014). [CrossRef] [PubMed]
12. C. Gopaul and R. Andrews, “The effect of atmospheric turbulence on entangled orbital angular momentum states,” New J. Phys. 9(4), 94 (2007). [CrossRef]
13. G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas’ko, S. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12(22), 5448–5456 (2004). [CrossRef] [PubMed]
14. M. Malik, M. O’Sullivan, B. Rodenburg, M. Mirhosseini, J. Leach, M. P. J. Lavery, M. J. Padgett, and R. W. Boyd, “Influence of atmospheric turbulence on optical communications using orbital angular momentum for encoding,” Opt. Express 20(12), 13195–13200 (2012). [CrossRef] [PubMed]
15. D. Zhang, X. Feng, and Y. Huang, “Encoding and decoding of orbital angular momentum for wireless optical interconnects on chip,” Opt. Express 20(24), 26986–26995 (2012). [CrossRef] [PubMed]
16. J. Du and J. Wang, “High-dimensional structured light coding/decoding for free-space optical communications free of obstructions,” Opt. Lett. 40(21), 4827–4830 (2015). [CrossRef] [PubMed]
17. Y. Zhao and J. Wang, “High-base vector beam encoding/decoding for visible-light communications,” Opt. Lett. 40(21), 4843–4846 (2015). [CrossRef] [PubMed]
18. R. D. Niederriter, M. E. Siemens, and J. T. Gopinath, “Continuously tunable orbital angular momentum generation using a polarization-maintaining fiber,” Opt. Lett. 41(14), 3213–3216 (2016). [CrossRef] [PubMed]
19. N. Radwell, D. Brickus, T. W. Clark, and S. Franke-Arnold, “High speed switching between arbitrary spatial light profiles,” Opt. Express 22(11), 12845–12852 (2014). [CrossRef] [PubMed]
20. M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, “Communication with spatially modulated light through turbulent air across Vienna,” New J. Phys. 16(11), 113028 (2014). [CrossRef]
21. S. Franke-Arnold, J. Leach, M. J. Padgett, V. E. Lembessis, D. Ellinas, A. J. Wright, J. M. Girkin, P. Öhberg, and A. S. Arnold, “Optical ferris wheel for ultracold atoms,” Opt. Express 15(14), 8619–8625 (2007). [CrossRef] [PubMed]
22. M. P. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using light’s orbital angular momentum,” Science 341(6145), 537–540 (2013). [CrossRef] [PubMed]
23. F. Spinello, C. G. Someda, R. A. Ravanelli, E. Mari, G. Parisi, F. Tamburini, F. Romanato, P. Coassini, and M. Oldoni, “Radio channel multiplexing with superpositions of opposite-sign OAM modes,” Int. J. Electron. Commun. 70(8), 990–997 (2016). [CrossRef]
24. B. Ndagano, I. Nape, M. A. Cox, C. Rosales-Guzman, and A. Forbes, “Creation and detection of vector vortex modes for classical and quantum communication,” J. Lightwave Technol. 36(2), 292–301 (2018). [CrossRef]
25. K. Morgan, J. K. Miller, W. Li, Y. Li, and E. G. Johnson, “Dynamic free-space propagation of interfering composite vortices.” In Propagation Through and Characterization of Atmospheric and Oceanic Phenomena Conference pp. in Imaging and Applied Optics (Optical Society of America, 2017) paper PTh2D–4. [CrossRef]
26. K. Morgan, W. Li, J. K. Miller, I. R. Srimathi, and E. G. Johnson, “Diffractive optics for the generation and detection of dynamic composite optical vortices,” In CLEO: Science and Innovations (Optical Society of America, 2017) Paper JW2A–29.
27. K. Morgan, J. K. Miller, Y. Li, W. Li, and E. G. Johnson, “Higher order Bessel beams integrated in time (HOBBIT) for free space underwater sensing and communication,” In OCEANS (Marine Technical Society/Institute of Electrical and Electronics engineers, 2017) pp 1–4.
28. E. G. Johnson, J. Stack, and C. Koehler, “Light coupling by a vortex lens into graded index fiber,” J. Lightwave Technol. 19(5), 753–758 (2001). [CrossRef]
29. B. Jack, A. M. Yao, J. Leach, J. Romero, S. Franke-Arnold, D. G. Ireland, S. M. Barnett, and M. J. Padgett, “Entanglement of arbitrary superpositions of modes within two-dimensional orbital angular momentum state spaces,” Phys. Rev. A 81(4), 043844 (2010). [CrossRef]
30. M. J. Padgett and J. Courtial, “Poincaré-sphere equivalent for light beams containing orbital angular momentum,” Opt. Lett. 24(7), 430–432 (1999). [CrossRef] [PubMed]
31. A. Forbes, A. Dudley, and M. McLaren, “Creation and detection of optical modes with spatial light modulators,” Adv. Opt. Photonics 8(2), 200–227 (2016). [CrossRef]
32. G. C. G. Berkhout, M. P. J. Lavery, M. J. Padgett, and M. W. Beijersbergen, “Measuring orbital angular momentum superpositions of light by mode transformation,” Opt. Lett. 36(10), 1863–1865 (2011). [CrossRef] [PubMed]
