Abstract

Orbital angular momentum (OAM), a new dimension of photons, has potentials in lots of domains as high-dimensional data coding/decoding. Here we experimentally demonstrate a free-space data transmission system based on 8 bits multi-state OAM shift keying, where multiplexed optical vortices containing 8 various OAM states are employed to constitute 8 bits binary symbols. In the transmitter, the data coding of OAM shift keying is realized by switching a series of special-designed holograms. And in the receiver, the decoding is done by a single Dammann vortex grating along with image processing. We experimentally transmit data, including a gray-scale image, in free-space for 10 meters, showing zero bit-error-rate. The demonstrated results indicate a wide prospect for the future high-dimensional large data rate optical security communications.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Optical vortices are a new kind of structured beams. The complex amplitude of an optical vortex comprises the helical phase term exp(ilφ), with l an infinite integer called topological charge and φ the azimuthal angle, resulting in their carried orbital angular momentum (OAM) [1,2]. And the OAM value of each photon in optical vortices is , where ħ is the Planck constant divided by 2π. Obviously the charge l determines the OAM value of photons. Hence, parameter l also known as OAM state. Besides, optical vortices with different topological charges are orthogonal with each other. Such unique features of optical vortices contribute to their potential applications in large-capacity optical data-transmission systems, where vortices act as carriers and bring in a new mode-division multiplexing (MDM) for classical optical communications [39]. The OAM-based MDM can be done together with the common wavelength-divison multiplexing, polarization-division multiplexing and so on, where up to 160 terabits per second data transmission has been accomplished [9].

In addition to MDM, the OAMs carried by optical vortices can also be regarded as symbols to do the data coding/decoding [5,1015]. For instance, N different OAM states can act as N-ary symbols to represent N-ary numbers. By switching the OAM states continually, a time-varying symbol sequence is obtained and log2N bits per symbol data transmission is available. Such coding/decoding approach called OAM shift keying. Due to the fact that the OAM states l is theoretically infinite, each OAM symbols can carry an unlimited number of bits, making it feasible to realize high-dimensional coding/decoding for free-space optical communication through employing multiple possible OAM states of vortices. Meanwhile, another key superiority of OAM coding/decoding claimed “security”, which does not depend on mathematical or quantum-mechanical encryption methods [10].

Previous studies on OAM shift keying usually concentrate on a single OAM state [10,13]. However, such single OAM state shift keying will have limitations technically. If M bits coding/decoding is desired, 2M OAM states are essential. If (M + 1) bits coding/decoding is further desired, 2M+1 OAM states are needed, implying the inevitable additional 2M OAM states. For instance, we can employ 128 various OAM states to do the log2128 = 7 bits coding/decoding. While additional 128 OAM states must be introduced, totally 256 various OAM states, to further realize 8 bits data coding. The above statements mean that the needed OAM states will increase exponentially when more bits are coded. As for the current technology, generating optical vortices with large l is relatively difficult. Therefore, more practical schemes must be developed to achieve high-dimensional data coding/decoding with fewer OAM states. To address such issue, the OAM shift keying is proposed, where multiplexed vortices with N various OAM states are employed to accomplish N bits coding, and the number of OAM states decreases (2N-N) compared with single OAM shift keying [16,17]. However, such studies are in the stage of simulation only, and how to accomplish the data-coding/decoding practically for OAM shift keying are still the key problems to be solved.

In this paper, we experimentally demonstrate a multi-mode OAM shift keying in free space through single hologram for 10 meters. In the proposed scheme, the coding is done by switching a series of single holograms, where a time-varying multiplexed vortices sequence is generated. After free-space transmission, the decoding is done through a Dammann vortex grating, whose diffraction patterns are analyzed by the image processing. As a proof of concept, we experimentally accomplish the 8 bits multi-state OAM shift keying through 8-fold multiplexed vortices in free-space for 10 meters. A group of random 256-ary numbers, as well as a gray-scale image, is transmitted successfully with 0 bit-error-rate (BER). It is the first time, to our best knowledge, to experimentally demonstrate the practical free-space data-transmission system through one single beam multi-state OAM shift keying.

2. Data coding/decoding through multi-state OAM shift keying

Firstly, we will discuss how do N various OAM states realize N bits coding. A multiplexed vortex $|{\psi}\rangle $ consists of N different OAM states reads:

$$|\psi \rangle = \sum\limits_{k = 1}^N {{a_k}|{{l_k}} \rangle }$$
where ak is a complex coefficient. In the proposed multi-state OAM shift keying, the imaginary part of ak is forced to 0. Meanwhile, the real part of ak, which is equivalent to |ak| or ak here, is set as a binary number. In other words, ak is a real number here and can be 0 or 1. Thus all the ak contained in Eq. (1) constitute a N bits binary number sequence {ak}. When ak=1, the OAM state $|{{{l}_{k}}\rangle} $ presents. Otherwise, ak=0 and $|{{{l}_{k}}\rangle} $ is absent. Therefore, the substance of multi-state OAM shift keying is regarding the N bits binary sequence as {ak}, which determines the OAM states distribution in the multiplexed vortices. And N various OAM states can represent an N bits binary number, carrying N bits information.

The concept of multi-state OAM shift keying based data transmission is sketched in Fig. 1. The whole system consists of two main parts: the transmitter and the receiver. In the transmitter, Gaussian beams are modulated by a series of 2N-ary numbers and transformed into time-varying multiplexed vortices by switching a sequence of holograms. The holograms here are computed through pattern search assisted iterative (PSI) algorithm, which allows us to generate multiplexed vortices with selective OAM states and intensity distributions through a phase-only diffractive element [18]. The coded multiplexed vortices are then sent into free-space. After the transmission, the coded beams are captured by the receiver for decoding. In the receiver, a special-designed Dammann vortex grating (DVG) [19], which can generate N different single vortices ($|{{-}{{l}_{1}\rangle}} $, $|{{-}{{l}_{2}}\rangle} $, …, $|{{-}{{l}_{N}}\rangle} $) in N various diffraction orders when Gaussian beams are incident in, is placed. Similarly, if the coded beams $|{\psi }\rangle $ are incident, the diffraction fields still contain N different orders, but the beams in orders m ($m \in [{1, N} ]\cap {\mathbb Z}$) turn to be:

$$\exp ({ - i{l_m}\varphi } )|\psi \rangle = \sum\limits_{k = 1}^N {{a_k}|{{l_k} - {l_m}} \rangle }$$

 figure: Fig. 1.

Fig. 1. Concept of data transmission based on multi-state OAM shift keying.

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For the right hand, there must be a term $|{0}\rangle $ if am≠0, leading to a bright spot in the beam center. Otherwise, am=0, and the beam center is still dark [19,20]. Figure 2 gives two examples of DVG diffractions. The incident 8-fold multiplexed vortices contain {$|{{{l}_{k}}\rangle} $}={$|{{- 10}\rangle} $, $|{{- 7}\rangle} $, $|{{- 4}\rangle} $, $|{{ - 1}\rangle} $, $|{1}\rangle $, $|{4}\rangle $, $|{7}\rangle $, $|{{10}\rangle} $}. And the DVG can produce such opposite OAM states in 8 different diffraction orders. When {ak}={1,1,1,1,1,1,1,1}, all of the above OAM states are present. One can find 8 bright center spot obviously in the diffraction patterns, as shown in Fig. 2(a). For another case, if {ak}={0,1,0,0,1,1,1,0}, only 4 OAM states are present, resulting in 4 bright center spot and 4 dark hollows in desired locations. Such phenomena imply the OAM distribution can be diagnosed by analyzing the DVG’s diffraction fields. To achieve this goal, the gray-scale algorithm is employed [21], which is an image processing method that can obtain the relative intensity distribution of various OAM states through diffraction pattern analysis.

 figure: Fig. 2.

Fig. 2. Simulated examples of multiplexed vortices ({$|{{{l}_{k}}\rangle} $}={$|{{- 10}\rangle} $, $|{{- 7}\rangle} $, $|{{- 4}\rangle} $, $|{{- 1}\rangle} $, $|{1}\rangle $, $|{4}\rangle $, $|{7} \rangle$, $|{{10}\rangle} $}) passing through a DVG. (a) The case {ak}={1,1,1,1,1,1,1,1}; (b) The case {ak}={0,1,0,0,1,1,1,0}.

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3. Experiment and results

Figure 3 displays the experimental setup of multi-state OAM shift keying based free-space data transmission for 10 meters. In the transmitter, a 1617nm laser diode is used as the source to produce fundamental Gaussian beams. After passing through a polarized beam splitter, the horizontally linear polarized Gaussian beams are coded by the first liquid-crystal spatial light modulator (SLM1) (Holoeye, PLUTO-TELCO-013-C), which is uploaded by a series of holograms computed through PSI algorithm to generate multiplexed vortices. The 4-f system (L1 & L2) along with an aperture stop filter the 1st diffraction order, in which the coded multiplexed vortices locate. By switching the holograms on SLM1, a coded time-varying multiplexed vortices sequence is obtained. The SLM2 is placed for the further OAM spectra analysis in the transmitter. When doing the data-transmission, nothing is uploaded and SLM2 acts as a reflector. After a 10 meters free-space path, the coded beams are captured by the receiver. A DVG is uploaded on SLM3, whose diffraction patterns are analyzed by gray-scale algorithm, and back-convert the coded 2N-ary numbers. At that time the decoding is finished and the transmitted information is recovered.

 figure: Fig. 3.

Fig. 3. Experimental setup. LD, laser diode; SMF, single mode fiber; Col., collimator; PBS, polarized beam splitter; SLM1∼SLM3, liquid-crystal spatial light modulator; BS, beam splitter; R1∼R4, reflector; L1∼L8, lenses; CCD1 & CCD2, infrared CCD camera.

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In the experiment, we choose 8 different OAM states: $|{{- 10}\rangle} $, $|{{- 7}\rangle} $, $|{{- 4}\rangle} $, $|{{- 1}\rangle} $, $|{1}\rangle $, $|{4}\rangle $, $|{7}\rangle $ and $|{{10}\rangle} $. By now the binary number sequence {ak} represents a 256-ary number (0, 1, 2, …, 255). Here a series of holograms that can produce multiplexed vortices with the above 8 OAM states and selective {ak} distributions must be computed. However, for computing one hologram, the PSI algorithm needs numbers of seconds, which can’t match the demand of fast shifting. Therefore we compute all 256 holograms in advance, save them in the database, then invoke and upload them on SLM1 for the data coding. Figure 4 shows some of the experimentally coded symbols (the 256-ary number 20, 102 and 249) or the generated multiplexed vortices, and their corresponding decoding patterns. The coded symbols are observed by CCD2 in the receiver, on the condition that nothing is uploaded on SLM2 and SLM3. The decoding patterns are captured after uploading the DVG shown in Fig. 2 on SLM3. One can obviously see from Fig. 4 that the center spots present at desired positions of the decoding patterns.

 figure: Fig. 4.

Fig. 4. Some of the experimentally obtained 8 bits multi-state OAM shift keying symbols (20, 102, 249) and their corresponding decoding patterns. The 8 bits binary numbers of these three symbols are also marked in the decoding patterns.

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As a proof-of-concept, we experimentally transmit 1000 256-ary random symbols, totally 8 kilobits information, through the aforementioned 8 bits multi-state OAM shift keying, in free-space for 10 meters. After decoding in the receiver, the received symbols are well recovered without seeing any error symbols, indicating 0 BER and the proposed system has good data transmission performance.

Moreover, to vividly demonstrate the data transmission performance of the proposed multi-state OAM shift keying, a 64×64 gray image (the badge of Beijing Institute of Technology) shown in Fig. 5(a) with 256 gray-scale levels is transmitted in free-space for 10 meters. Each pixel of the gray image carries information of 8 bits, implying such image is 32.768 kilobits in total. Hence such gray image can be converted to a 256-ary number sequence with 4096 256-ary numbers. The symbol length is 4096 after coding, which is reduced by 8 times compared to the classic binary symbols, and by 2.667 times compared to that of 8 single OAM states coding/decoding under the same condition (8 single-mode OAM states can represent log28 = 3 bits per symbol). By switching the corresponding holograms on SLM1, the image is coded and transformed into a series of time-varying multiplexed vortices sequence and then sent to free-space. Note that here the OAM states distributions contained by multiplexed vortices are the same with aforementioned experiments as $|{{- 10}\rangle} $, $|{{- 7}\rangle} $, $|{{- 4}\rangle} $, $|{{- 1}\rangle} $, $|{1}\rangle $, $|{4}\rangle $, $|{7}\rangle $ and $|{{10}\rangle} $. After the free-space transmission, the coded multiplexed vortices are captured by the receiver for decoding. Figure 5(b) is the received image that recovered by the receiver, which exactly recovers the transmitted one with 0 BER, indicating the successful image transmission of multi-state OAM shift keying.

 figure: Fig. 5.

Fig. 5. The 64×64 gray image transmission through the 8 bits multi-state OAM shift keying. (a) The transmitted gray image; (b) The gray image recovered by the receiver.

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The maximum refresh rate of the SLM employed here is 60 Hz, indicating that under the current condition the highest transmission rate is 8×60 = 480 bit/s. In the proof-of-concept experiment, the refresh rate is set as 1 Hz, which means 8 bits information is transmitted per second. And it takes 4096 s to transmit the 64×64 gray image. Although the refresh rate of SLM is low in this experiment and can’t meet with the demand of high-speed OAM coding. Such issue can be well addressed by using other OAM generators with high switching speed as integrated OAM devices [2224] and so on. Meanwhile, with the development of technology, we convince SLMs with much higher refresh speed will be produced. Additionally, it should be emphasized that in the proposed experiment we choose 8 various OAM states to achieve 8 bits data coding. But the alternative OAM states are not limited to this. Higher bits data coding can be accomplished if we choose more different OAM states.

In addition, we experimentally analyze the OAM spectra in the transmitter and receiver. The OAM spectra in the transmitter can be measured through uploading the DVG on SLM2. And then employing gray-scale algorithm [21] to analyze the diffraction patterns captured by CCD1. Note that the reflector R1 here is to offset the mirror image of the beam splitter. Figure 6 lists the measured OAM spectra of three symbols: the 256-ary number 74 ({ak}={0,1,0,0,1,0,1,0}), 99 ({ak}={0,1,1,0,0,0,1,1}) and 255 ({ak}={1,1,1,1,1,1,1,1}), which are the 3-fold, 4-fold and 8-fold multiplexed vortices separately. One can see clearly that for the transmitter the intensity proportions of the present OAM states are not the strict identical. Such phenomena mainly result from computing the coding holograms. During the iteration of PSI algorithm, the OAM spectrum will be calculated continually, where the complex amplitude is expanded by the spiral harmonic exp(il0φ) [25], and l0 should ranging from -∞ to +∞ theoretically. But the selection of infinite range is impossible for the numerical calculation. Here we choose a finite closed interval l0∈[−15, +15], to simplify the calculation. The iterated hologram will thus deviate from the ideal identical intensity proportion for a little bit, especially when more OAM states are present.

 figure: Fig. 6.

Fig. 6. Measured OAM spectra of three coded symbols in both the transmitter and receiver. (a)–(c) are the case of 256-ary numbers 74 (01001010), 99 (01100011) and 255 (11111111), respectively.

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In the proposed experiment, the OAM states we choose is $|{{- 10}\rangle} $, $|{{- 7}\rangle} $, $|{{- 4}\rangle} $, $|{{- 1}\rangle} $, $|{1}\rangle $, $|{4}\rangle $, $|{7}\rangle $ and $|{{10}\rangle} $, where the smallest topological charge spacing is δ=3. In theory, the data-transmission performance is independent of δ, providing that the decoding algorithm is precise enough. However here the gray-scale algorithm is employed to accomplish the data decoding, which will introduce bit errors when δ is small (e.g., δ=1). The reason is that, the original intention of gray-scale algorithm is to provide an easy way to diagnose the topological charge spectra for large mode spacing δ [21]. Errors will present if δ is too small. Hence when δ is chosen very small here, for instance, δ=1, there may be decoding errors and the wrong data are recovered, thus decrease the transmission performance. However, if the more complex but more precise OAM spectra measurement approaches as the mode sorter [26,27] and so on are used here, the better performance will also be obtained if δ is small. In other words, the choice of δ determines which kind of decoding scheme is employed. And actually this is a balance between topological charge range and system complexity.

The OAM coding/decoding based data transmission has very good performance in data security. Any angular restriction, misalignment, and lateral shift will defect the unique helical wavefront and thus result in a serious spreading of OAM spectra [28]. The eavesdropper won’t cut out the whole transmitted beam, since it will be perceived by the receiver. So it can only capture some of the beams across the section. As discussed above, the OAM spectra is chaos for the stolen beams. Therefore the eavesdropper won’t recovered the transmitted data, implying that the proposed data-transmission scheme is safe.

In the 10 meters path the aberration and distortion introduced by the lenses and air will affect the vortices’ phase slightly, reflected by the measured OAM spectra in the receiver have a little difference from those in the transmitter. Although this effect has little influence on 10 meters path, it mustn’t be ignored for the practical long-distance data transmission, where the atmosphere turbulence will distort the helical phase, broaden the OAM spectrum, and finally leading to the power leakage from the desired OAM channel to the neighboring channels [29,30]. In this scenario, the decoding algorithm will recognize wrong center spots, thus increasing BER. In that case, adaptive compensation for multiplexed vortices as Shake-Hartman method [31], Gerchberg–Saxton algorithm [32], deep learning [33] and so on must be brought in.

4. Conclusions

In brief, we have demonstrated experimentally the multi-state OAM shift keying, where N various OAM states are employed to code N bits information. The coding efficiency increases N/log2N times compared with the common N single mode OAM coding. Moreover, both the coding and decoding can be accomplished by single holograms. In the proof-of-concept, we transmit data in free-space practically for 10 meters through OAM shift keying with 0 BER. Additionally, a 64×64 gray image, totally 32.768 kilobits, is also transmitted for 10 meters successfully, where the received image exactly recovers the transmitted one. This work proves the practical feasibility of multi-state OAM shift keying based data-transmission, and paves the way for the future high dimensional data coding/decoding in large capacity communication systems.

Funding

National Natural Science Foundation of China (11834001, 61905012); National Postdoctoral Program for Innovative Talents of China (BX20190036); China Postdoctoral Science Foundation (2019M650015); National Basic Research Program of China (973 Program) (2014CB340004); Beijing Institute of Technology Research Fund Program for Young Scholars.

Acknowledgments

The authors acknowledge Dr. Zheyuan Zhang of the University of Tokyo, and Miss Ci Yin of Aerospace Information Research Institute in Chinese Academy of Sciences for their helpful discussions.

Disclosures

The authors declare no conflicts of interest.

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References

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  1. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
    [Crossref]
  2. A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photonics 3(2), 161–204 (2011).
    [Crossref]
  3. J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
    [Crossref]
  4. 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]
  5. A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
    [Crossref]
  6. S. Yu, “Potentials and challenges of using orbital angular momentum communications in optical interconnects,” Opt. Express 23(3), 3075–3087 (2015).
    [Crossref]
  7. J. Wang, “Advances in communications using optical vortices,” Photonics Res. 4(5), B14–B28 (2016).
    [Crossref]
  8. T. Lei, M. Zhang, Y. Li, P. Jia, G. N. Liu, X. Xu, Z. Li, C. Min, J. Lin, C. Yu, H. Niu, and X. Yuan, “Massive individual orbital angular momentum channels for multiplexing enabled by Dammann gratings,” Light: Sci. Appl. 4(3), e257 (2015).
    [Crossref]
  9. L. Zhu, G. Zhu, A. Wang, L. Wang, J. Ai, S. Chen, C. Du, J. Liu, S. Yu, and J. Wang, “18  km low-crosstalk OAM + WDM transmission with 224 individual channels enabled by a ring-core fiber with large high-order mode group separation,” Opt. Lett. 43(8), 1890–1893 (2018).
    [Crossref]
  10. G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pasko, 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]
  11. A. Trichili, C. Rosales-Guzmán, A. Dudley, B. Ndagano, A. B. Salem, M. Zghal, and A. Forbes, “Optical communication beyond orbital angular momentum,” Sci. Rep. 6(1), 27674 (2016).
    [Crossref]
  12. S. Li and J. Wang, “Experimental demonstration of optical interconnects exploiting orbital angular momentum array,” Opt. Express 25(18), 21537–21547 (2017).
    [Crossref]
  13. S. Chen, S. Li, Y. Zhao, J. Liu, L. Zhu, A. Wang, J. Du, L. Shen, and J. Wang, “Demonstration of 20-Gbit/s high-speed Bessel beam encoding/decoding link with adaptive turbulence compensation,” Opt. Lett. 41(20), 4680–4683 (2016).
    [Crossref]
  14. S. Fu, Y. Zhai, C. Yin, H. Zhou, and C. Gao, “Mixed orbital angular momentum amplitude shift keying through a single hologram,” OSA Continuum 1(2), 295–308 (2018).
    [Crossref]
  15. S. Fu, Y. Zhai, H. Zhou, J. Zhang, T. Wang, C. Yin, and C. Gao, “Demonstration of free-space one-to-many multicasting link from orbital angular momentum encoding,” Opt. Lett. 44(19), 4753–4756 (2019).
    [Crossref]
  16. C. Kai, P. Huang, F. Shen, H. Zhou, and Z. Guo, “Orbital Angular Momentum Shift Keying Based Optical Communication System,” IEEE Photonics J. 9(2), 1–10 (2017).
    [Crossref]
  17. J. Li, M. Zhang, and D. Wang, “Adaptive Demodulator Using Machine Learning for Orbital Angular Momentum Shift Keying,” IEEE Photonics Technol. Lett. 29(17), 1455–1458 (2017).
    [Crossref]
  18. L. Zhu and J. Wang, “Simultaneous generation of multiple orbital angular momentum (OAM) modes using a single phase-only element,” Opt. Express 23(20), 26221–26233 (2015).
    [Crossref]
  19. N. Zhang, X. C. Yuan, and R. E. Burge, “Extending the detection range of optical vortices by Dammann vortex gratings,” Opt. Lett. 35(20), 3495–3497 (2010).
    [Crossref]
  20. S. Fu, T. Wang, S. Zhang, and C. Gao, “Integrating 5 × 5 Dammann gratings to detect orbital angular momentum states of beams with the range of −24 to + 24,” Appl. Opt. 55(7), 1514–1517 (2016).
    [Crossref]
  21. S. Fu, S. Zhang, T. Wang, and C. Gao, “Measurement of orbital angular momentum spectra of multiplexing optical vortices,” Opt. Express 24(6), 6240–6248 (2016).
    [Crossref]
  22. X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, “Integrated Compact Optical Vortex Beam Emitters,” Science 338(6105), 363–366 (2012).
    [Crossref]
  23. B. Guan, R. P. Scott, C. Qin, N. K. Fontaine, T. Su, C. Ferrari, M. Cappuzzo, F. Klemens, B. Keller, M. Earnshaw, and S. J. B. Yoo, “Free-space coherent optical communication with orbital angular, momentum multiplexing/demultiplexing using a hybrid 3D photonic integrated circuit,” Opt. Express 22(1), 145–156 (2014).
    [Crossref]
  24. J. Zhang, C. Sun, B. Xiong, J. Wang, Z. Hao, L. Wang, Y. Han, H. Li, Y. Luo, Y. Xiao, C. Yu, T. Tanemura, Y. Nakano, S. Li, X. Cai, and S. Yu, “An InP-based vortex beam emitter with monolithically integrated laser,” Nat. Commun. 9(1), 2652 (2018).
    [Crossref]
  25. Y. D. Liu, C. Gao, M. Gao, and F. Li, “Coherent-mode representation and orbital angular momentum spectrum of partially coherent beam,” Opt. Commun. 281(8), 1968–1975 (2008).
    [Crossref]
  26. G. C. Berkhout, M. P. J. 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]
  27. Y. Wen, I. Chremmos, Y. Chen, J. Zhu, Y. Zhang, and S. Yu, “Spiral Transformation for High-Resolution and Efficient Sorting of Optical Vortex Modes,” Phys. Rev. Lett. 120(19), 193904 (2018).
    [Crossref]
  28. Y. D. Liu, C. Gao, X. Qi, and H. Weber, “Orbital angular momentum (OAM) spectrum correction in free space optical communication,” Opt. Express 16(10), 7091–7101 (2008).
    [Crossref]
  29. S. Fu and C. Gao, “Influences of atmospheric turbulence effects on the orbital angular momentum spectra of vortex beams,” Photonics Res. 4(5), B1–B4 (2016).
    [Crossref]
  30. Y. Ren, H. Huang, G. Xie, N. Ahmed, Y. Yan, B. I. Erkmen, N. Chandrasekaran, M. P. J. Lavery, N. K. Steinhoff, M. Tur, S. Dolinar, M. Neifeld, M. J. Padgett, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Atmospheric turbulence effects on the performance of a free space optical link employing orbital angular momentum multiplexing,” Opt. Lett. 38(20), 4062–4065 (2013).
    [Crossref]
  31. S. Li and J. Wang, “Compensation of a distorted N-fold orbital angular momentum multicasting link using adaptive optics,” Opt. Lett. 41(7), 1482–1485 (2016).
    [Crossref]
  32. S. Fu, S. Zhang, T. Wang, and C. Gao, “Pre-turbulence compensation of orbital angular momentum beams based on a probe and the Gerchberg–Saxton algorithm,” Opt. Lett. 41(14), 3185–3188 (2016).
    [Crossref]
  33. J. Liu, P. Wang, X. Zhang, Y. He, X. Zhou, H. Ye, Y. Li, S. Xu, S. Chen, and D. Fan, “Deep learning based atmospheric turbulence compensation for orbital angular momentum beam distortion and communication,” Opt. Express 27(12), 16671–16688 (2019).
    [Crossref]

2019 (2)

2018 (4)

Y. Wen, I. Chremmos, Y. Chen, J. Zhu, Y. Zhang, and S. Yu, “Spiral Transformation for High-Resolution and Efficient Sorting of Optical Vortex Modes,” Phys. Rev. Lett. 120(19), 193904 (2018).
[Crossref]

S. Fu, Y. Zhai, C. Yin, H. Zhou, and C. Gao, “Mixed orbital angular momentum amplitude shift keying through a single hologram,” OSA Continuum 1(2), 295–308 (2018).
[Crossref]

J. Zhang, C. Sun, B. Xiong, J. Wang, Z. Hao, L. Wang, Y. Han, H. Li, Y. Luo, Y. Xiao, C. Yu, T. Tanemura, Y. Nakano, S. Li, X. Cai, and S. Yu, “An InP-based vortex beam emitter with monolithically integrated laser,” Nat. Commun. 9(1), 2652 (2018).
[Crossref]

L. Zhu, G. Zhu, A. Wang, L. Wang, J. Ai, S. Chen, C. Du, J. Liu, S. Yu, and J. Wang, “18  km low-crosstalk OAM + WDM transmission with 224 individual channels enabled by a ring-core fiber with large high-order mode group separation,” Opt. Lett. 43(8), 1890–1893 (2018).
[Crossref]

2017 (3)

S. Li and J. Wang, “Experimental demonstration of optical interconnects exploiting orbital angular momentum array,” Opt. Express 25(18), 21537–21547 (2017).
[Crossref]

C. Kai, P. Huang, F. Shen, H. Zhou, and Z. Guo, “Orbital Angular Momentum Shift Keying Based Optical Communication System,” IEEE Photonics J. 9(2), 1–10 (2017).
[Crossref]

J. Li, M. Zhang, and D. Wang, “Adaptive Demodulator Using Machine Learning for Orbital Angular Momentum Shift Keying,” IEEE Photonics Technol. Lett. 29(17), 1455–1458 (2017).
[Crossref]

2016 (8)

S. Chen, S. Li, Y. Zhao, J. Liu, L. Zhu, A. Wang, J. Du, L. Shen, and J. Wang, “Demonstration of 20-Gbit/s high-speed Bessel beam encoding/decoding link with adaptive turbulence compensation,” Opt. Lett. 41(20), 4680–4683 (2016).
[Crossref]

J. Wang, “Advances in communications using optical vortices,” Photonics Res. 4(5), B14–B28 (2016).
[Crossref]

A. Trichili, C. Rosales-Guzmán, A. Dudley, B. Ndagano, A. B. Salem, M. Zghal, and A. Forbes, “Optical communication beyond orbital angular momentum,” Sci. Rep. 6(1), 27674 (2016).
[Crossref]

S. Fu, T. Wang, S. Zhang, and C. Gao, “Integrating 5 × 5 Dammann gratings to detect orbital angular momentum states of beams with the range of −24 to + 24,” Appl. Opt. 55(7), 1514–1517 (2016).
[Crossref]

S. Fu, S. Zhang, T. Wang, and C. Gao, “Measurement of orbital angular momentum spectra of multiplexing optical vortices,” Opt. Express 24(6), 6240–6248 (2016).
[Crossref]

S. Fu and C. Gao, “Influences of atmospheric turbulence effects on the orbital angular momentum spectra of vortex beams,” Photonics Res. 4(5), B1–B4 (2016).
[Crossref]

S. Li and J. Wang, “Compensation of a distorted N-fold orbital angular momentum multicasting link using adaptive optics,” Opt. Lett. 41(7), 1482–1485 (2016).
[Crossref]

S. Fu, S. Zhang, T. Wang, and C. Gao, “Pre-turbulence compensation of orbital angular momentum beams based on a probe and the Gerchberg–Saxton algorithm,” Opt. Lett. 41(14), 3185–3188 (2016).
[Crossref]

2015 (4)

T. Lei, M. Zhang, Y. Li, P. Jia, G. N. Liu, X. Xu, Z. Li, C. Min, J. Lin, C. Yu, H. Niu, and X. Yuan, “Massive individual orbital angular momentum channels for multiplexing enabled by Dammann gratings,” Light: Sci. Appl. 4(3), e257 (2015).
[Crossref]

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

S. Yu, “Potentials and challenges of using orbital angular momentum communications in optical interconnects,” Opt. Express 23(3), 3075–3087 (2015).
[Crossref]

L. Zhu and J. Wang, “Simultaneous generation of multiple orbital angular momentum (OAM) modes using a single phase-only element,” Opt. Express 23(20), 26221–26233 (2015).
[Crossref]

2014 (1)

2013 (2)

2012 (2)

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, “Integrated Compact Optical Vortex Beam Emitters,” Science 338(6105), 363–366 (2012).
[Crossref]

2011 (1)

A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photonics 3(2), 161–204 (2011).
[Crossref]

2010 (2)

N. Zhang, X. C. Yuan, and R. E. Burge, “Extending the detection range of optical vortices by Dammann vortex gratings,” Opt. Lett. 35(20), 3495–3497 (2010).
[Crossref]

G. C. Berkhout, M. P. J. 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]

2008 (2)

Y. D. Liu, C. Gao, M. Gao, and F. Li, “Coherent-mode representation and orbital angular momentum spectrum of partially coherent beam,” Opt. Commun. 281(8), 1968–1975 (2008).
[Crossref]

Y. D. Liu, C. Gao, X. Qi, and H. Weber, “Orbital angular momentum (OAM) spectrum correction in free space optical communication,” Opt. Express 16(10), 7091–7101 (2008).
[Crossref]

2004 (1)

1992 (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref]

Ahmed, N.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Y. Ren, H. Huang, G. Xie, N. Ahmed, Y. Yan, B. I. Erkmen, N. Chandrasekaran, M. P. J. Lavery, N. K. Steinhoff, M. Tur, S. Dolinar, M. Neifeld, M. J. Padgett, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Atmospheric turbulence effects on the performance of a free space optical link employing orbital angular momentum multiplexing,” Opt. Lett. 38(20), 4062–4065 (2013).
[Crossref]

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Ai, J.

Allen, L.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref]

Ashrafi, N.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Ashrafi, S.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Bao, C.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Barnett, S.

Beijersbergen, M. W.

G. C. Berkhout, M. P. J. 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]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref]

Berkhout, G. C.

G. C. Berkhout, M. P. J. 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]

Boyd, R. W.

Bozinovic, N.

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]

Burge, R. E.

Cai, X.

J. Zhang, C. Sun, B. Xiong, J. Wang, Z. Hao, L. Wang, Y. Han, H. Li, Y. Luo, Y. Xiao, C. Yu, T. Tanemura, Y. Nakano, S. Li, X. Cai, and S. Yu, “An InP-based vortex beam emitter with monolithically integrated laser,” Nat. Commun. 9(1), 2652 (2018).
[Crossref]

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, “Integrated Compact Optical Vortex Beam Emitters,” Science 338(6105), 363–366 (2012).
[Crossref]

Cao, Y.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Cappuzzo, M.

Chandrasekaran, N.

Chen, S.

Chen, Y.

Y. Wen, I. Chremmos, Y. Chen, J. Zhu, Y. Zhang, and S. Yu, “Spiral Transformation for High-Resolution and Efficient Sorting of Optical Vortex Modes,” Phys. Rev. Lett. 120(19), 193904 (2018).
[Crossref]

Chremmos, I.

Y. Wen, I. Chremmos, Y. Chen, J. Zhu, Y. Zhang, and S. Yu, “Spiral Transformation for High-Resolution and Efficient Sorting of Optical Vortex Modes,” Phys. Rev. Lett. 120(19), 193904 (2018).
[Crossref]

Courtial, J.

G. C. Berkhout, M. P. J. 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]

G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pasko, 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]

Dolinar, S.

Du, C.

Du, J.

Dudley, A.

A. Trichili, C. Rosales-Guzmán, A. Dudley, B. Ndagano, A. B. Salem, M. Zghal, and A. Forbes, “Optical communication beyond orbital angular momentum,” Sci. Rep. 6(1), 27674 (2016).
[Crossref]

Earnshaw, M.

Erkmen, B. I.

Fan, D.

Fazal, I. M.

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Ferrari, C.

Fontaine, N. K.

Forbes, A.

A. Trichili, C. Rosales-Guzmán, A. Dudley, B. Ndagano, A. B. Salem, M. Zghal, and A. Forbes, “Optical communication beyond orbital angular momentum,” Sci. Rep. 6(1), 27674 (2016).
[Crossref]

Franke-Arnold, S.

Fu, S.

Gao, C.

S. Fu, Y. Zhai, H. Zhou, J. Zhang, T. Wang, C. Yin, and C. Gao, “Demonstration of free-space one-to-many multicasting link from orbital angular momentum encoding,” Opt. Lett. 44(19), 4753–4756 (2019).
[Crossref]

S. Fu, Y. Zhai, C. Yin, H. Zhou, and C. Gao, “Mixed orbital angular momentum amplitude shift keying through a single hologram,” OSA Continuum 1(2), 295–308 (2018).
[Crossref]

S. Fu, T. Wang, S. Zhang, and C. Gao, “Integrating 5 × 5 Dammann gratings to detect orbital angular momentum states of beams with the range of −24 to + 24,” Appl. Opt. 55(7), 1514–1517 (2016).
[Crossref]

S. Fu, S. Zhang, T. Wang, and C. Gao, “Measurement of orbital angular momentum spectra of multiplexing optical vortices,” Opt. Express 24(6), 6240–6248 (2016).
[Crossref]

S. Fu and C. Gao, “Influences of atmospheric turbulence effects on the orbital angular momentum spectra of vortex beams,” Photonics Res. 4(5), B1–B4 (2016).
[Crossref]

S. Fu, S. Zhang, T. Wang, and C. Gao, “Pre-turbulence compensation of orbital angular momentum beams based on a probe and the Gerchberg–Saxton algorithm,” Opt. Lett. 41(14), 3185–3188 (2016).
[Crossref]

Y. D. Liu, C. Gao, X. Qi, and H. Weber, “Orbital angular momentum (OAM) spectrum correction in free space optical communication,” Opt. Express 16(10), 7091–7101 (2008).
[Crossref]

Y. D. Liu, C. Gao, M. Gao, and F. Li, “Coherent-mode representation and orbital angular momentum spectrum of partially coherent beam,” Opt. Commun. 281(8), 1968–1975 (2008).
[Crossref]

Gao, M.

Y. D. Liu, C. Gao, M. Gao, and F. Li, “Coherent-mode representation and orbital angular momentum spectrum of partially coherent beam,” Opt. Commun. 281(8), 1968–1975 (2008).
[Crossref]

Gibson, G.

Guan, B.

Guo, Z.

C. Kai, P. Huang, F. Shen, H. Zhou, and Z. Guo, “Orbital Angular Momentum Shift Keying Based Optical Communication System,” IEEE Photonics J. 9(2), 1–10 (2017).
[Crossref]

Han, Y.

J. Zhang, C. Sun, B. Xiong, J. Wang, Z. Hao, L. Wang, Y. Han, H. Li, Y. Luo, Y. Xiao, C. Yu, T. Tanemura, Y. Nakano, S. Li, X. Cai, and S. Yu, “An InP-based vortex beam emitter with monolithically integrated laser,” Nat. Commun. 9(1), 2652 (2018).
[Crossref]

Hao, Z.

J. Zhang, C. Sun, B. Xiong, J. Wang, Z. Hao, L. Wang, Y. Han, H. Li, Y. Luo, Y. Xiao, C. Yu, T. Tanemura, Y. Nakano, S. Li, X. Cai, and S. Yu, “An InP-based vortex beam emitter with monolithically integrated laser,” Nat. Commun. 9(1), 2652 (2018).
[Crossref]

He, Y.

Huang, H.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

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]

Y. Ren, H. Huang, G. Xie, N. Ahmed, Y. Yan, B. I. Erkmen, N. Chandrasekaran, M. P. J. Lavery, N. K. Steinhoff, M. Tur, S. Dolinar, M. Neifeld, M. J. Padgett, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Atmospheric turbulence effects on the performance of a free space optical link employing orbital angular momentum multiplexing,” Opt. Lett. 38(20), 4062–4065 (2013).
[Crossref]

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Huang, P.

C. Kai, P. Huang, F. Shen, H. Zhou, and Z. Guo, “Orbital Angular Momentum Shift Keying Based Optical Communication System,” IEEE Photonics J. 9(2), 1–10 (2017).
[Crossref]

Jia, P.

T. Lei, M. Zhang, Y. Li, P. Jia, G. N. Liu, X. Xu, Z. Li, C. Min, J. Lin, C. Yu, H. Niu, and X. Yuan, “Massive individual orbital angular momentum channels for multiplexing enabled by Dammann gratings,” Light: Sci. Appl. 4(3), e257 (2015).
[Crossref]

Johnson-Morris, B.

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, “Integrated Compact Optical Vortex Beam Emitters,” Science 338(6105), 363–366 (2012).
[Crossref]

Kai, C.

C. Kai, P. Huang, F. Shen, H. Zhou, and Z. Guo, “Orbital Angular Momentum Shift Keying Based Optical Communication System,” IEEE Photonics J. 9(2), 1–10 (2017).
[Crossref]

Keller, B.

Klemens, F.

Kristensen, P.

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]

Lavery, M. P. J.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
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Y. Ren, H. Huang, G. Xie, N. Ahmed, Y. Yan, B. I. Erkmen, N. Chandrasekaran, M. P. J. Lavery, N. K. Steinhoff, M. Tur, S. Dolinar, M. Neifeld, M. J. Padgett, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Atmospheric turbulence effects on the performance of a free space optical link employing orbital angular momentum multiplexing,” Opt. Lett. 38(20), 4062–4065 (2013).
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G. C. Berkhout, M. P. J. 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).
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T. Lei, M. Zhang, Y. Li, P. Jia, G. N. Liu, X. Xu, Z. Li, C. Min, J. Lin, C. Yu, H. Niu, and X. Yuan, “Massive individual orbital angular momentum channels for multiplexing enabled by Dammann gratings,” Light: Sci. Appl. 4(3), e257 (2015).
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Y. D. Liu, C. Gao, M. Gao, and F. Li, “Coherent-mode representation and orbital angular momentum spectrum of partially coherent beam,” Opt. Commun. 281(8), 1968–1975 (2008).
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J. Zhang, C. Sun, B. Xiong, J. Wang, Z. Hao, L. Wang, Y. Han, H. Li, Y. Luo, Y. Xiao, C. Yu, T. Tanemura, Y. Nakano, S. Li, X. Cai, and S. Yu, “An InP-based vortex beam emitter with monolithically integrated laser,” Nat. Commun. 9(1), 2652 (2018).
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J. Li, M. Zhang, and D. Wang, “Adaptive Demodulator Using Machine Learning for Orbital Angular Momentum Shift Keying,” IEEE Photonics Technol. Lett. 29(17), 1455–1458 (2017).
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A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
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Li, Y.

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T. Lei, M. Zhang, Y. Li, P. Jia, G. N. Liu, X. Xu, Z. Li, C. Min, J. Lin, C. Yu, H. Niu, and X. Yuan, “Massive individual orbital angular momentum channels for multiplexing enabled by Dammann gratings,” Light: Sci. Appl. 4(3), e257 (2015).
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T. Lei, M. Zhang, Y. Li, P. Jia, G. N. Liu, X. Xu, Z. Li, C. Min, J. Lin, C. Yu, H. Niu, and X. Yuan, “Massive individual orbital angular momentum channels for multiplexing enabled by Dammann gratings,” Light: Sci. Appl. 4(3), e257 (2015).
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T. Lei, M. Zhang, Y. Li, P. Jia, G. N. Liu, X. Xu, Z. Li, C. Min, J. Lin, C. Yu, H. Niu, and X. Yuan, “Massive individual orbital angular momentum channels for multiplexing enabled by Dammann gratings,” Light: Sci. Appl. 4(3), e257 (2015).
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Liu, Y. D.

Y. D. Liu, C. Gao, X. Qi, and H. Weber, “Orbital angular momentum (OAM) spectrum correction in free space optical communication,” Opt. Express 16(10), 7091–7101 (2008).
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J. Zhang, C. Sun, B. Xiong, J. Wang, Z. Hao, L. Wang, Y. Han, H. Li, Y. Luo, Y. Xiao, C. Yu, T. Tanemura, Y. Nakano, S. Li, X. Cai, and S. Yu, “An InP-based vortex beam emitter with monolithically integrated laser,” Nat. Commun. 9(1), 2652 (2018).
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T. Lei, M. Zhang, Y. Li, P. Jia, G. N. Liu, X. Xu, Z. Li, C. Min, J. Lin, C. Yu, H. Niu, and X. Yuan, “Massive individual orbital angular momentum channels for multiplexing enabled by Dammann gratings,” Light: Sci. Appl. 4(3), e257 (2015).
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A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
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J. Zhang, C. Sun, B. Xiong, J. Wang, Z. Hao, L. Wang, Y. Han, H. Li, Y. Luo, Y. Xiao, C. Yu, T. Tanemura, Y. Nakano, S. Li, X. Cai, and S. Yu, “An InP-based vortex beam emitter with monolithically integrated laser,” Nat. Commun. 9(1), 2652 (2018).
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A. Trichili, C. Rosales-Guzmán, A. Dudley, B. Ndagano, A. B. Salem, M. Zghal, and A. Forbes, “Optical communication beyond orbital angular momentum,” Sci. Rep. 6(1), 27674 (2016).
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Niu, H.

T. Lei, M. Zhang, Y. Li, P. Jia, G. N. Liu, X. Xu, Z. Li, C. Min, J. Lin, C. Yu, H. Niu, and X. Yuan, “Massive individual orbital angular momentum channels for multiplexing enabled by Dammann gratings,” Light: Sci. Appl. 4(3), e257 (2015).
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Padgett, M. J.

Y. Ren, H. Huang, G. Xie, N. Ahmed, Y. Yan, B. I. Erkmen, N. Chandrasekaran, M. P. J. Lavery, N. K. Steinhoff, M. Tur, S. Dolinar, M. Neifeld, M. J. Padgett, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Atmospheric turbulence effects on the performance of a free space optical link employing orbital angular momentum multiplexing,” Opt. Lett. 38(20), 4062–4065 (2013).
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A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photonics 3(2), 161–204 (2011).
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G. C. Berkhout, M. P. J. 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).
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Qi, X.

Qin, C.

Ramachandran, S.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

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).
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A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
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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).
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Y. Ren, H. Huang, G. Xie, N. Ahmed, Y. Yan, B. I. Erkmen, N. Chandrasekaran, M. P. J. Lavery, N. K. Steinhoff, M. Tur, S. Dolinar, M. Neifeld, M. J. Padgett, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Atmospheric turbulence effects on the performance of a free space optical link employing orbital angular momentum multiplexing,” Opt. Lett. 38(20), 4062–4065 (2013).
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J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
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A. Trichili, C. Rosales-Guzmán, A. Dudley, B. Ndagano, A. B. Salem, M. Zghal, and A. Forbes, “Optical communication beyond orbital angular momentum,” Sci. Rep. 6(1), 27674 (2016).
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L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
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Strain, M. J.

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A. Trichili, C. Rosales-Guzmán, A. Dudley, B. Ndagano, A. B. Salem, M. Zghal, and A. Forbes, “Optical communication beyond orbital angular momentum,” Sci. Rep. 6(1), 27674 (2016).
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A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
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J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
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Wang, A.

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L. Zhu, G. Zhu, A. Wang, L. Wang, J. Ai, S. Chen, C. Du, J. Liu, S. Yu, and J. Wang, “18  km low-crosstalk OAM + WDM transmission with 224 individual channels enabled by a ring-core fiber with large high-order mode group separation,” Opt. Lett. 43(8), 1890–1893 (2018).
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J. Zhang, C. Sun, B. Xiong, J. Wang, Z. Hao, L. Wang, Y. Han, H. Li, Y. Luo, Y. Xiao, C. Yu, T. Tanemura, Y. Nakano, S. Li, X. Cai, and S. Yu, “An InP-based vortex beam emitter with monolithically integrated laser,” Nat. Commun. 9(1), 2652 (2018).
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S. Li and J. Wang, “Compensation of a distorted N-fold orbital angular momentum multicasting link using adaptive optics,” Opt. Lett. 41(7), 1482–1485 (2016).
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J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
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L. Zhu, G. Zhu, A. Wang, L. Wang, J. Ai, S. Chen, C. Du, J. Liu, S. Yu, and J. Wang, “18  km low-crosstalk OAM + WDM transmission with 224 individual channels enabled by a ring-core fiber with large high-order mode group separation,” Opt. Lett. 43(8), 1890–1893 (2018).
[Crossref]

J. Zhang, C. Sun, B. Xiong, J. Wang, Z. Hao, L. Wang, Y. Han, H. Li, Y. Luo, Y. Xiao, C. Yu, T. Tanemura, Y. Nakano, S. Li, X. Cai, and S. Yu, “An InP-based vortex beam emitter with monolithically integrated laser,” Nat. Commun. 9(1), 2652 (2018).
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Wang, T.

Weber, H.

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Y. Wen, I. Chremmos, Y. Chen, J. Zhu, Y. Zhang, and S. Yu, “Spiral Transformation for High-Resolution and Efficient Sorting of Optical Vortex Modes,” Phys. Rev. Lett. 120(19), 193904 (2018).
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A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

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]

Y. Ren, H. Huang, G. Xie, N. Ahmed, Y. Yan, B. I. Erkmen, N. Chandrasekaran, M. P. J. Lavery, N. K. Steinhoff, M. Tur, S. Dolinar, M. Neifeld, M. J. Padgett, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Atmospheric turbulence effects on the performance of a free space optical link employing orbital angular momentum multiplexing,” Opt. Lett. 38(20), 4062–4065 (2013).
[Crossref]

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

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L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
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J. Zhang, C. Sun, B. Xiong, J. Wang, Z. Hao, L. Wang, Y. Han, H. Li, Y. Luo, Y. Xiao, C. Yu, T. Tanemura, Y. Nakano, S. Li, X. Cai, and S. Yu, “An InP-based vortex beam emitter with monolithically integrated laser,” Nat. Commun. 9(1), 2652 (2018).
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A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
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Y. Ren, H. Huang, G. Xie, N. Ahmed, Y. Yan, B. I. Erkmen, N. Chandrasekaran, M. P. J. Lavery, N. K. Steinhoff, M. Tur, S. Dolinar, M. Neifeld, M. J. Padgett, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Atmospheric turbulence effects on the performance of a free space optical link employing orbital angular momentum multiplexing,” Opt. Lett. 38(20), 4062–4065 (2013).
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J. Zhang, C. Sun, B. Xiong, J. Wang, Z. Hao, L. Wang, Y. Han, H. Li, Y. Luo, Y. Xiao, C. Yu, T. Tanemura, Y. Nakano, S. Li, X. Cai, and S. Yu, “An InP-based vortex beam emitter with monolithically integrated laser,” Nat. Commun. 9(1), 2652 (2018).
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Xu, X.

T. Lei, M. Zhang, Y. Li, P. Jia, G. N. Liu, X. Xu, Z. Li, C. Min, J. Lin, C. Yu, H. Niu, and X. Yuan, “Massive individual orbital angular momentum channels for multiplexing enabled by Dammann gratings,” Light: Sci. Appl. 4(3), e257 (2015).
[Crossref]

Yan, Y.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Y. Ren, H. Huang, G. Xie, N. Ahmed, Y. Yan, B. I. Erkmen, N. Chandrasekaran, M. P. J. Lavery, N. K. Steinhoff, M. Tur, S. Dolinar, M. Neifeld, M. J. Padgett, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Atmospheric turbulence effects on the performance of a free space optical link employing orbital angular momentum multiplexing,” Opt. Lett. 38(20), 4062–4065 (2013).
[Crossref]

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Yang, J. Y.

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

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A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photonics 3(2), 161–204 (2011).
[Crossref]

Ye, H.

Yin, C.

Yoo, S. J. B.

Yu, C.

J. Zhang, C. Sun, B. Xiong, J. Wang, Z. Hao, L. Wang, Y. Han, H. Li, Y. Luo, Y. Xiao, C. Yu, T. Tanemura, Y. Nakano, S. Li, X. Cai, and S. Yu, “An InP-based vortex beam emitter with monolithically integrated laser,” Nat. Commun. 9(1), 2652 (2018).
[Crossref]

T. Lei, M. Zhang, Y. Li, P. Jia, G. N. Liu, X. Xu, Z. Li, C. Min, J. Lin, C. Yu, H. Niu, and X. Yuan, “Massive individual orbital angular momentum channels for multiplexing enabled by Dammann gratings,” Light: Sci. Appl. 4(3), e257 (2015).
[Crossref]

Yu, S.

L. Zhu, G. Zhu, A. Wang, L. Wang, J. Ai, S. Chen, C. Du, J. Liu, S. Yu, and J. Wang, “18  km low-crosstalk OAM + WDM transmission with 224 individual channels enabled by a ring-core fiber with large high-order mode group separation,” Opt. Lett. 43(8), 1890–1893 (2018).
[Crossref]

J. Zhang, C. Sun, B. Xiong, J. Wang, Z. Hao, L. Wang, Y. Han, H. Li, Y. Luo, Y. Xiao, C. Yu, T. Tanemura, Y. Nakano, S. Li, X. Cai, and S. Yu, “An InP-based vortex beam emitter with monolithically integrated laser,” Nat. Commun. 9(1), 2652 (2018).
[Crossref]

Y. Wen, I. Chremmos, Y. Chen, J. Zhu, Y. Zhang, and S. Yu, “Spiral Transformation for High-Resolution and Efficient Sorting of Optical Vortex Modes,” Phys. Rev. Lett. 120(19), 193904 (2018).
[Crossref]

S. Yu, “Potentials and challenges of using orbital angular momentum communications in optical interconnects,” Opt. Express 23(3), 3075–3087 (2015).
[Crossref]

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, “Integrated Compact Optical Vortex Beam Emitters,” Science 338(6105), 363–366 (2012).
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Sci. Rep. (1)

A. Trichili, C. Rosales-Guzmán, A. Dudley, B. Ndagano, A. B. Salem, M. Zghal, and A. Forbes, “Optical communication beyond orbital angular momentum,” Sci. Rep. 6(1), 27674 (2016).
[Crossref]

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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]

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, “Integrated Compact Optical Vortex Beam Emitters,” Science 338(6105), 363–366 (2012).
[Crossref]

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Figures (6)

Fig. 1.
Fig. 1. Concept of data transmission based on multi-state OAM shift keying.
Fig. 2.
Fig. 2. Simulated examples of multiplexed vortices ({$|{{{l}_{k}}\rangle} $}={$|{{- 10}\rangle} $, $|{{- 7}\rangle} $, $|{{- 4}\rangle} $, $|{{- 1}\rangle} $, $|{1}\rangle $, $|{4}\rangle $, $|{7} \rangle$, $|{{10}\rangle} $}) passing through a DVG. (a) The case {ak}={1,1,1,1,1,1,1,1}; (b) The case {ak}={0,1,0,0,1,1,1,0}.
Fig. 3.
Fig. 3. Experimental setup. LD, laser diode; SMF, single mode fiber; Col., collimator; PBS, polarized beam splitter; SLM1∼SLM3, liquid-crystal spatial light modulator; BS, beam splitter; R1∼R4, reflector; L1∼L8, lenses; CCD1 & CCD2, infrared CCD camera.
Fig. 4.
Fig. 4. Some of the experimentally obtained 8 bits multi-state OAM shift keying symbols (20, 102, 249) and their corresponding decoding patterns. The 8 bits binary numbers of these three symbols are also marked in the decoding patterns.
Fig. 5.
Fig. 5. The 64×64 gray image transmission through the 8 bits multi-state OAM shift keying. (a) The transmitted gray image; (b) The gray image recovered by the receiver.
Fig. 6.
Fig. 6. Measured OAM spectra of three coded symbols in both the transmitter and receiver. (a)–(c) are the case of 256-ary numbers 74 (01001010), 99 (01100011) and 255 (11111111), respectively.

Equations (2)

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| ψ = k = 1 N a k | l k
exp ( i l m φ ) | ψ = k = 1 N a k | l k l m

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