Mixed orbital angular momentum amplitude shift keying through a single hologram

Shiyao Fu; Yanwang Zhai; Ci Yin; Heng Zhou; Chunqing Gao

doi:10.1364/OSAC.1.000295

1. Introduction

Optical vortices with orbital angular momentums (OAMs) are a new kind of structured light fields, whose complex amplitudes comprise the helical term exp(ilφ), with l the topological charge and φ the azimuthal angle [1]. Each photon in such vortices carries the OAM of lħ, where ħ is the Planck constant divided by 2π [1–4]. Thus such vortices are also known as OAM beams. The topological charge l is an integer and unlimited in theory, determining how much OAM does a single photon carry and hence can represent the OAM state. Moreover, OAM beams with different topological charges l are orthogonal to each other. These unique features attract more and more attention currently, and thus lead to lots of applications as optical communications [5–8], rotating detection [9–11], optical tweezers [12], producing Poincare fields [13], gravitational wave detection [14] and so on.

For the applications in optical communications, because of the orthogonality, coaxially propagated various OAM states can be separated efficiently, indicating the feasibility to employ such states to increase the capacity of communication systems through mode division multiplexing (MDM) [5–8]. Note that there is nothing special about OAM in MDM communications, due to the fact that using the complete basis is the only way to really achieve the full bandwidth [15,16]. Besides, multiple optical vortices with N different topological charges can be encoded as N different data symbols, for instance, the N-ary numbers: 0, 1, ..., (N−1) [6,17–25]. Therefore a sequence of optical vortices carrying N various OAMs sent by the transmitter represents the data information of log₂N bits. In this scenario, the encoding efficiency can be improved log₂N times compared to binary encoding. Such encoding scheme is called OAM encoding. As mentioned previously, the topological charge or OAM states can be an infinite integer. Hence it is possible to encode infinite bits per photon through OAM encoding in theory, which could achieve higher photon efficiency in quantum communications community potentially [19–21].

One of the bases of OAM encoding based data-transmission, is to create and detect OAM beams with controllable topological charges. Thanks to the development of diffractive optical elements, various holograms have been proposed to generate and detect various OAM beams [16,26–28]. Additionally, OAM beam array or complex structured lattices have also been demonstrated [29–34].

However, limitations are still present. Due to the characteristic of logarithmic function, the additional 2^B OAM states are needed if the data information per code increases from B bits to (B + 1) bits. For instance, when B = 6, additional 64 OAM states must be introduced to achieve 7 bits encoding. Meanwhile, high-speed detection of large-tuning-range OAM states is also difficult to realize relatively. Therefore, finding a practice approach to increase the higher-bit encoding but with small-tuning-range OAM states is necessary.

In this paper, we propose a high-dimensional communication scheme as mixed OAM-amplitude shift keying (OAM-ASK) through single holograms, where the OAM encoding is done simultaneously with the well-known amplitude shift keying (ASK), to further increase the encoding efficiency with fewer OAM states. We theoretically show that both the data encoding and decoding in OAM-ASK system can be realized by a single hologram. A proof-of-concept experiment is done, where the real data encoding and decoding system is given. In the experiment, a gray-scale image is transmitted in free-space through 4 bits OAM-ASK consists of octal OAM and binary amplitude, and 7 bits OAM-ASK consists of 32-ary OAM and quaternary amplitude over 10 meters, respectively. The bit-error-rate (BER) of both the two OAM-ASKs is measured as 0, indicating good performance in free-space communications.

2. Principles of OAM-ASK

2.1 Basic idea

As sketched in Fig. 1

Fig. 1 Concept of OAM-ASK. The OAM and the amplitude dimensions represent relatively independent multi-ary symbols, where the introduction of amplitude further enlarges the encoding efficiency of the common OAM encoding.

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, the proposed OAM-ASK consists of two independent dimensions. One is OAM and the other is the amplitude. The amplitude of a beam is associated with its intensity. While the OAM represents the helical wavefronts. They can be modulated separately without being interacted. Each dimension represents relatively independent multi-ary symbols. Supposing that the OAM and amplitude dimensions stand for n-ary and m-ary symbols, representing the data information of log₂n and log₂m bits, respectively. When such two dimensions are employed together, they can form mn-ary symbols. Thus the total information of each code is log₂(mn) = (log₂n + log₂m) bits, which increases log₂m bits compared to the previous common OAM encoding. From another point of view, only the OAM dimension is employed in the common OAM encoding, where m = 1, log₂m = 0. Hence, OAM-ASK can be regarded as the extension in the amplitude dimension of OAM encoding, making it possible to realize the higher-bit encoding with fewer OAM states.

There are mn different states in OAM-ASK systems. Such states must be one-to-one mapped onto mn-ary numbers:

[\begin{matrix} A_{1} | l_{1} 〉 \\ A_{2} | l_{1} 〉 \\ ⋮ \\ A_{m} | l_{1} 〉 \end{matrix} \begin{matrix} A_{1} | l_{2} 〉 \\ A_{2} | l_{2} 〉 \\ ⋮ \\ A_{m} | l_{2} 〉 \end{matrix} \begin{matrix} \dots \\ \dots \\ ⋱ \\ \dots \end{matrix} \begin{matrix} A_{1} | l_{n} 〉 \\ A_{2} | l_{n} 〉 \\ ⋮ \\ A_{m} | l_{n} 〉 \end{matrix}] \to [\begin{matrix} 0 & 1 & \dots & n - 1 \\ n & n + 1 & \dots & 2 n - 1 \\ ⋮ & ⋮ & ⋱ & ⋮ \\ (m - 1) n & (m - 1) n + 1 & \dots & m n - 1 \end{matrix}],

where

| l_{1} 〉

,

| l_{2} 〉

, …,

| l_{n} 〉

and A₁, A₂, …, A_m denote n various OAM states and m various amplitudes, respectively. In the transmitter, by switching the OAM states and intensities (intensity is amplitude square indeed, which can be used to represent amplitude), an encoded mn-ary symbol sequence is obtained. After propagating in free-space, such sequence is decoded at the receiver side, where the encoded data information is recovered.

2.2 OAM-ASK encoding

Next we will discuss how to realize the encoding / decoding in OAM-ASK technically. The fast encoding is crucial in OAM-ASK systems, which determines the information rate. Here we propose to utilize a single hologram to do the OAM-ASK encoding. Since the two dimensions in OAM-ASK are relatively independent, the hologram must meet the demands of modulating the OAM states and amplitudes simultaneously but independently. Currently, a proven technique that, designing a hologram containing the spiral phase to produce optical vortices with various topological charges, has been widely used in lots of domains [5–9]. So at present the key is to modulate the amplitude without affecting the OAM state.

To achieve such goal, a feasible way is to diffract some of the intensities of the incident beams into useless diffraction orders and then filtered out, so as to modulate the intensities in desired orders. For the designed holograms, their transmission function exp[iP(x)] can be Fourier expanded as [35]:

\exp [i P (x)] = \sum_{b = - \infty}^{+ \infty} c_{b} \exp (i b γ x),

with b the diffraction orders and γ the grating parameters that confine the grating period. c_b is the coefficient:

c_{b} = \frac{γ}{2 π} \int_{- \frac{π}{γ}}^{\frac{π}{γ}} \exp [i P (x)] \exp (- i b γ x) d x,

which define the complex amplitude in the diffraction order b:

c_{b} = | c_{b} | \exp (i ϕ_{b}) \exp (i l_{b} φ) .

In Eq. (4), |c_b|, φ_b, and l_b are the amplitude, the initial phase and topological charge of beams in diffraction order b separately. Equations (2)–(4) allow us to design holograms to obtain beams in arbitrary diffraction orders with arbitrary amplitudes, initial phases and topological charges, through forcing parameters |c_b|, φ_b, and l_b reasonably. For the OAM-ASK encoding here, a sequence of holograms are generated to produce two ( + 1st & −1st) diffraction orders in the diffraction field, where the + 1st order is our desire. We don’t care about parameters φ₊₁ and φ_-1, which are set as 0, for they are irrelevant to the encoding dimensions. The OAM states are determined directly by parameter l₊₁. While as for the modulation of amplitudes in the + 1st order, one can modify the proportion between |c₊₁| and |c_-1|. Such two parameters determine the real intensity proportion between the two diffraction orders, and thus modulate the amplitude indirectly when the intensity of the incident unencoded Gaussian beams is fixed. By now, an aperture stop is employed to filter the + 1st diffraction order, thus to obtain mn-ary OAM-ASK symbol sequence.

Figure 2

Fig. 2 Simulated encoding in OAM-ASK through a single hologram. (a)-(c), holograms to generate | + 1> with three various intensities in the + 1st diffraction order. (c) & (d), holograms to generate | + 1> and | + 3> but with identical intensities in the + 1st diffraction order. (e)-(h), corresponding diffraction patterns of (a)-(d) when Gaussian beams are incident in. (i)-(l), coaxial interference patterns of beams in (e)-(h) and Gaussian beams.

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gives some examples to show the encoding, through single hologram based on Eqs. (2)–(4), in OAM-ASK clearly. The first column [(a)–(d)] is the special designed holograms, whose simulated diffraction fields of Gaussian beams are shown in the corresponding second column [(e)–(h)]. The third column [(i)–(l)] is the coaxial interference patterns of beams in column two and Gaussian beams, to show the OAM states. Obviously beams in the + 1st diffraction order are modulated not only in OAM states, but also in amplitude, indicating the good encoding performance.

2.3 OAM-ASK decoding

The aim of decoding in OAM-ASK is to diagnose the OAM states and the intensities of the received beams both accurately and rapidly, so as to recover the encoded data information. Considering the feasibility and simplicity in practice, the decoding is proposed to be done by a single hologram too, where the previously reported gray-scale algorithm [36] along with a Dammann vortex grating (DVG) [37–39] is employed. Actually, the hologram for OAM-ASK decoding is a DVG, which is designed according to the tuning-range of OAM states in the transmitter side. The so-called DVG is a kind of 0-π binary diffraction elements to produce multiple equal-intensity vortices with various topological charges or OAM states simultaneously in the form of vortices array with high efficiency. When the encoded beam (e.g. $| l 〉$ ) incident in the Dammann vortex grating, due to the back-converting, a bright center spot will emerge at the diffraction order -l, the location of which indicates the OAM states [38,39]. In addition, if we capture the far-field diffraction fields by using a CCD camera, one can understand easily that the sum of every pixel’s gray value of the bright spot received by the camera is proportional to the real intensity of the spot when the received power is lower than the camera’s threshold. Therefore, such gray sum value is suitable to stand for the relative intensity of $| l 〉$ , which is the main idea of gray-scale algorithm [36]. The demonstration of gray-scale algorithm is intended to analyse the OAM spectrum, while here is to find where the bright center spot emerge, and compute the sum of pixels’ gray value of the bright center spot. By now the OAM state and relative intensity are obtained.

Figure 3

Fig. 3 Simulated decoding in OAM-ASK through a single hologram. From left to right are the encoded vortices for decoding, DVG, the corresponding far-field diffraction patterns, and decoding results including OAM states and relative intensities, respectively. Note that the 9 circled numbers in the blue region is to show the OAM states distributions when Gaussian beams propagate through the DVG.

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shows the simulated decoding in OAM-ASK through a 3 × 3 DVG (the single hologram) along with the gray-scale algorithm, where three encoded vortices

| - 1 〉

,

| + 4 〉

, and

| + 4 〉

with intensities A², A² and A²/2 separately are presented as examples. The DVG here can produce a 3 × 3 vortices array, whose OAM state distributions from bottom left to top right are −4~ + 4 orderly, as shown in the wathet-blue part. When the three OAM states

| - 1 〉

,

| + 4 〉

, and

| + 4 〉

pass through the DVG, one can find a bright spot emerge at the locations + 1, −4 and −4 in the far-field diffractions, as circled in the third column. Meanwhile, one can also find obviously that the spot in the third line is darker than that in the second line, implying the difference of intensities. The far-field diffraction patterns are analysed through gray-scale algorithm. Then the OAM state and relative intensity are obtained in the fourth column.

Next comes the data information recover, where the anti-mapping [from right hand to left hand in Eq. (1)] is done. Here the key is to demodulate the intensity, since demodulating OAM dimension is easy to realize (just to distinguish the location of a bright center spot). Actually, as for the practical scenario, in the receiver side there is always an inevitable tiny fluctuation of beams, associated with the laser’s power stability or transmission medium. Hence the right selection of range for judgment threshold of intensities or amplitudes, is very important, where an improper judgment threshold may bring more decoding mistakes, leading to a larger BER. Here, to avoid such mistakes as much as possible, the mid-value of neighboring encoded normalized intensities are chosen as the judgment threshold. For instance, in the quaternary amplitude encoding, four normalized modulated intensities (I₁, I₂, I₃, I₄ ∈[0,1]) are presented after encoding. Then the three judgment threshold for decoding is (I₁ + I₂)/2, (I₂ + I₃)/2 and (I₃ + I₄)/2. The decoding result is I₁ if the received normalized intensity I>(I₁ + I₂)/2, I₂ if (I₁ + I₂)/2>I>(I₂ + I₃)/2, I₃ if (I₂ + I₃)/2>I>(I₃ + I₄)/2, I₄ if I<(I₃ + I₄)/2.

In fact, DVG is not the first time to be employed in OAM based data-transmissions, merely in a very different way from here, where the DVG is used to work with the gray-scale algorithm to determine the OAM states and amplitudes. For instance, Ref. [7]. is based on the mode-division multiplexing since the orthogonality of OAM modes, where two DVGs are used to combine and separate beams with various OAM states separately. Each mode in Ref. [7]. carry independent quadrature amplitude modulation signals. While in this work the proposed OAM-ASK is to enlarge further the coding efficiency of the common OAM encoding. They are two different ways to transmit data through OAM.

3. Results

3.1 Experimental setup

As a proof-of-concept, two experiments of 4 bits and 7 bits OAM-ASKs are done. The whole experimental setup (Fig. 4

Fig. 4 Experimental setup. DFB, distributed feedback laser; SMF, single mode fiber; Col., collimator; PBS, polarized beam splitter; SLM1 & SLM2, liquid-crystal spatial light modulators; L1~L5, plano-convex lenses; AS, aperture stop; R1~R3, reflector; CCD, infrared CCD camera.

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) consists of three parts: the transmitter, propagation in free-space, and the receiver. In the transmitter, 1.6μm Gaussian beams are produced from a distributed feedback laser (DFB) and coupled into free-space with a diameter of 3 mm. A polarized beam splitter (PBS) is placed to generate horizontally linear polarized beams so as to match the demand of phase-only modulation of the liquid-crystal spatial light modulator (SLM) (Holoeye, PLUTO-TELCO-013-C). A sequence of encoding holograms shown in Fig. 2 is addressed to a SLM1. Then a 4-f system (plano-convex lenses L1 & L2) along with an aperture stop is employed to filter the + 1st diffraction order, beams in which are the modulated symbols. After propagating in free-space, the encoded beams are captured by the receiver, where two plano-convex lenses L3 and L4 form a telescope, and a DVG is uploaded on SLM2, to produce the vortices array. Such vortices array is captured by an infrared CCD camera (Xenics, Bobcat-320-star, resolution 320 × 256) placed at the image focal plane of L5, and analysed through the gray-scale algorithm. Then the transmitted symbols are decoded, and the data are recovered.

3.2 Experimental realization of 4-bit and 7-bit OAM-ASK

Firstly the 4 bits OAM-ASK consists of octal OAM and binary amplitude is done, transmitting data information over 10 m. Eight different OAM states $(l \in {- 4, - 3, - 2, - 1, 1, 2, 3, 4})$ and two binary normalized amplitudes (1 & 0.707) are chosen, to constitute a hexadecimal symbol, and then represent a hexadecimal number. Hence each code carries information of 4 bit. The detailed mapping rules between the encoded codes and hexadecimal numbers are given in Appendix A. Figure 5(a)

Fig. 5 Encoding holograms and the experimentally captured corresponding encoded/decoded patterns of hexadecimal and 128-ary numbers in 4 bits and 7 bits OAM-ASK. (a) 4 bits OAM-ASK. (b) 7 bits OAM-ASK.

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displays three different hexadecimal numbers and their corresponding encoding hologram, encoded / decoded patterns captured in the receiver side. In this experiment, we transmit a random high-base number sequence with 5000 hexadecimal numbers, totally 2 × 10⁴ bits information, in free-space with 10 m, through switching a sequence of holograms on the SLM in the transmitter. In the receiver, a DVG that can produce 3 × 3 vortices array with topological charges −4~ + 4 are uploaded on the decoding SLM. Next the received encoded beams are diffracted, and then analysed by gray-scale algorithm to accomplish the decoding. Note that in the experiment some other factors, for instance, the resolution of SLM can’t satisfy the giant phase jump of DVG, may lead to the unequal intensity distribution among each diffraction others, which brings troubles for the amplitude dimension decoding. Hence, measuring intensity distribution of different diffraction order should be done firstly to revise the algorithm before the data transmission. The BER is measured as 0 for the 10-meter 5000 hexadecimal numbers transmission, showing favorable transmission performance of 4 bits OAM-ASK.

The 7 bits OAM-ASK is also carried out for 10 meters, where 32 different OAM states $(l \in [- 16, - 1] \cup [1, 16] \cap ℤ)$ and 4 binary normalized amplitudes (1, 0.894, 0.806, 0.707) are chosen. Then 128-ary numbers, totally 7 bits, can be represented. Their detailed mapping rules are given in Appendix B. The encoding holograms, captured encoded/decoded patterns of three different 128-ary numbers are displayed in Fig. 5(b). We transmit 2858 random 128-ary numbers, totally 20006 bits, in 10 m free-space. At this time, the SLM in the receiver is uploaded by a DVG that can produce 5 × 7 vortices array with topological charges −17~ + 17. The decoding results also show 0 BER and favourable data-transmission performance.

3.3 Image transmission through OAM-ASK

Moreover, we also transmit a gray image, the badge of Beijing Institute of Technology (BIT) shown in Fig. 6(b)

Fig. 6 50 × 50 pixel gray-image transmission through 4 bits and 7 bits OAM-ASK. (a) High-definition RGB image of BIT badge. (b) The transmitted 50 × 50 pixel gray-image of BIT badge. (c) Received gray-image through 4 bits OAM-ASK in 10 m free-space. (d) Received gray-image through 7 bits OAM-ASK in 10 m free-space.

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, through 4 bits and 7 bits OAM-ASK respectively, in free-space for 10 m. Each pixel in the gray image has 256 gray values (0~255), thus carries the information of 8 bits. Here the badge image is chosen as 50 × 50 pixel, 2500 pixels in total. To achieve this goal, 2500 × 8 = 20000 bits data should be transmitted. For 4 bits and 7 bits OAM-ASK, the encoding symbol length is 5000 and 2858, which is reduced by 4 and 6.998 times separately compared to the original binary data. Figures 6(c) and 6(d) show the received gray images after the decoding of 4 bits and 7 bits OAM-ASK in the receiver. One can clearly see that, the received image exactly the same as the transmitted one, confirming the successful image transmission of OAM-ASK. The BERs of both the two OAM-ASK systems are also evaluated as 0, among the 20000 bits data transmission in this experiment. Note that transmitting a clearer image (much more total bits) through the proposed 4 bits and 7 bits OAM-ASK is feasible in theory. What we should do is to increase the encoding symbol length.

4. Discussions

In this present work, the OAM-ASK encoding is done through the hologram designed based on the Fourier expansion of transmission function, which is not the only way here. We can also reduce the phase depth for each OAM mode from 2π to 2xπ. For example, if x = 0 then no light goes into this OAM mode, with the amplitude fraction increasing with parameter x until you have 100% of the light [40]. The OAM-ASK decoding is done here by the gray-scale algorithm. Such process can also be realized by the OAM mode sorter, which can separate photons with different OAM states in different positions of the image plane [41–44].

It should also be noticed that, the 0 BER performance obtained here is on the indoor condition of the using of few modes and transmit data for 10 meters. For the long-distance outdoor OAM-ASK based communications, effects of atmosphere turbulence won’t be ignored, since it will distort the helical wavefront of OAM beams and broaden the OAM spectra, contributing to the increasing of interchannel crosstalk and BER values [45,46]. Besides, long-distance transmission need larger transmitting and receiving diameters, which may bring errors. In these scenarios, adaptive compensation schemes like Shack–Hartmann method [47,48], Gerchberg–Saxton algorithm [49,50] or Zernike polynomial-based stochastic parallel gradient descent algorithm [51] and so on must be introduced.

There may be one doubt about relationships between the crosstalk between different encoding modes and the BER performance in the experiment. The crosstalk between different encoding modes or the mode purity here may not be zero. There may be slightly power leakage from the desired mode to neighboring modes in the experiment. Usually such crosstalk is not very serious with no turbulence, and it will have little influence on the BER performance. The reason is, when decoding through the gray-scale algorithm, we look for the diffraction orders with the strongest center spot. So we can determine the OAM states accurately. Meanwhile, we also set a judgment threshold for the amplitude. If the measured intensity is located in the proper section, the discretized amplitude will be determined correctly. In another case, if the crosstalk is very serious (usually results from the turbulence and other factors), the decoding system can’t determine the OAM states and the amplitudes accurately, thus the BER value will increase. And adaptive compensation schemes discussed above must be introduced.

In the practice applications, one thing should be concerned about is the data transmission speed, associated with the encoding and decoding speed. In our scheme, such speed is limited by the frame rate of the SLM and response time of the CCD camera. With the development of hardware, such issues will be overcome in the future. Furthermore, high-speed OAM state encoding can also be realized by on-chip optoelectronic devices [52,53], where the switching rate of OAM states has been reduced to about 20 μs through integrated photonic circuits with thermal tuning [53].

5. Conclusions

In summary, we have demonstrated a single hologram based mixed OAM-ASK to improve further the encoding efficiency considerably. Schemes of how to accomplish the encoding and decoding in OAM-ASK through a single hologram are proposed. Two proof-of-concept experiments of 4 bits and 7 bits OAM-ASK in 10 m free-space are done separately, showing 0 BER performance. A gray image, totally 20 kbits, is also transmitted in free-space, where the received image exactly recovers the transmitted one. Such favorable results indicate the successful implement of OAM-ASK, which opens a new insight for mixed data encoding / decoding systems.

Appendix A Mapping rules for 4 bits OAM-ASK

The one-to-one mapping rules between the encoded codes and hexadecimal numbers for 4 bits OAM-ASK are given in appendix Table 1

Table 1. Mapping Rules for 4 Bits OAM-ASK

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.

Appendix B Mapping rules for 7 bits OAM-ASK

The one-to-one mapping rules between the encoded codes and 128-ary numbers for 7 bits OAM-ASK are given in appendix Table 2

Table 2. Mapping Rules for 7 Bits OAM-ASK

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.

Funding

National Basic Research Program of China (973 Program) (2014CB340002, 2014CB340004); Graduate Technological Innovation Project of Beijing Institute of Technology (2017CX10003, 2018CX10020)

Acknowledgment

We acknowledge Miss Tonglu Wang for her helpful discussions.

References

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

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

3. M. J. Padgett, “Orbital angular momentum 25 years on [Invited],” Opt. Express 25(10), 11265–11274 (2017). [CrossRef] [PubMed]

4. H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19(1), 013001 (2017). [CrossRef]

5. 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]

6. 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]

7. 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]

8. 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]

9. M. P. J. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using light’s orbital angular momentum,” Science 341(6145), 537–540 (2013). [CrossRef] [PubMed]

10. S. Fu, T. Wang, Z. Zhang, Y. Zhai, and C. Gao, “Non-diffractive Bessel-Gauss beams for the detection of rotating object free of obstructions,” Opt. Express 25(17), 20098–20108 (2017). [CrossRef] [PubMed]

11. L. Fang, M. J. Padgett, and J. Wang, “Sharing a common origin between the rotational and linear Doppler effects,” Laser Photonics Rev. 11(6), 1700183 (2017). [CrossRef]

12. M. J. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5(6), 343–348 (2011). [CrossRef]

13. S. Fu, Y. Zhai, T. Wang, C. Yin, and C. Gao, “Tailoring arbitrary hybrid Poincare beams through a single hologram,” Appl. Phys. Lett. 111(21), 211101 (2017). [CrossRef]

14. M. Granata, C. Buy, R. Ward, and M. Barsuglia, “Higher-order Laguerre-Gauss mode generation and interferometry for gravitational wave detectors,” Phys. Rev. Lett. 105(23), 231102 (2010). [CrossRef] [PubMed]

15. N. Zhao, X. Li, G. Li, and J. M. Kahn, “Capacity limits of spatially multiplexed free-space communication,” Nat. Photonics 9(12), 822–826 (2015). [CrossRef]

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

17. 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]

18. A. J. Willner, Y. Ren, G. Xie, Z. Zhao, Y. Cao, L. Li, N. Ahmed, Z. Wang, Y. Yan, P. Liao, C. Liu, M. Mirhosseini, R. W. Boyd, M. Tur, and A. E. Willner, “Experimental demonstration of 20 Gbit/s data encoding and 2 ns channel hopping using orbital angular momentum modes,” Opt. Lett. 40(24), 5810–5813 (2015). [CrossRef] [PubMed]

19. 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]

20. Y. Ren, C. Liu, K. Pang, J. Zhao, Y. Cao, G. Xie, L. Li, P. Liao, Z. Zhao, M. Tur, R. W. Boyd, and A. E. Willner, “Spatially multiplexed orbital-angular-momentum-encoded single photon and classical channels in a free-space optical communication link,” Opt. Lett. 42(23), 4881–4884 (2017). [CrossRef] [PubMed]

21. M. Mirhosseini, O. S. Magaña-Loaiza, M. N. O’Sullivan, B. Rodenburg, M. Malik, M. P. J. Lavery, M. J. Padgett, D. J. Gauthier, and R. W. Boyd, “High-dimensional quantum cryptography with twisted light,” New J. Phys. 17(3), 033033 (2015). [CrossRef]

22. 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]

23. M. Krenn, J. Handsteiner, M. Fink, R. Fickler, R. Ursin, M. Malik, and A. Zeilinger, “Twisted light transmission over 143 km,” Proc. Natl. Acad. Sci. U.S.A. 113(48), 13648–13653 (2016). [CrossRef] [PubMed]

24. M. P. J. Lavery, C. Peuntinger, K. Günthner, P. Banzer, D. Elser, R. W. Boyd, M. J. Padgett, C. Marquardt, and G. Leuchs, “Free-space propagation of high-dimensional structured optical fields in an urban environment,” Sci. Adv. 3(10), e1700552 (2017). [CrossRef] [PubMed]

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

26. 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]

27. S. Fu and C. Gao, “Progress of detecting orbital angular momentum states of optical vortices through diffraction gratings,” Wuli Xuebao 67, 034201 (2018).

28. C. Rosales-Guzmán, N. Bhebhe, N. Mahonisi, and A. Forbes, “Multiplexing 200 spatial modes with a single hologram,” J. Opt. 19(11), 113501 (2017). [CrossRef]

29. S. Fu, T. Wang, and C. Gao, “Perfect optical vortex array with controllable diffraction order and topological charge,” J. Opt. Soc. Am. A 33(9), 1836–1842 (2016). [CrossRef] [PubMed]

30. J. Yu, C. Zhou, Y. Lu, J. Wu, L. Zhu, and W. Jia, “Square lattices of quasi-perfect optical vortices generated by two-dimensional encoding continuous-phase gratings,” Opt. Lett. 40(11), 2513–2516 (2015). [CrossRef] [PubMed]

31. S. Fu, S. Zhang, T. Wang, and C. Gao, “Rectilinear lattices of polarization vortices with various spatial polarization distributions,” Opt. Express 24(16), 18486–18491 (2016). [CrossRef] [PubMed]

32. S. Fu, C. Gao, T. Wang, S. Zhang, and Y. Zhai, “Simultaneous generation of multiple perfect polarization vortices with selective spatial states in various diffraction orders,” Opt. Lett. 41(23), 5454–5457 (2016). [CrossRef] [PubMed]

33. S. Fu, T. Wang, Z. Zhang, Y. Zhai, and C. Gao, “Selective acquisition of multiple states on hybrid Poincare sphere,” Appl. Phys. Lett. 110(19), 191102 (2017). [CrossRef]

34. C. Rosales-Guzmán, N. Bhebhe, and A. Forbes, “Simultaneous generation of multiple vector beams on a single SLM,” Opt. Express 25(21), 25697–25706 (2017). [CrossRef] [PubMed]

35. L. A. Romero and F. M. Dickey, “Theory of optimal beam splitting by phase gratings. I. one-dimensional gratings,” J. Opt. Soc. Am. A 24(8), 2280–2295 (2007). [CrossRef] [PubMed]

36. 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] [PubMed]

37. J. N. Mait, “Design of Dammann gratings for two-dimensional, nonseparable, noncentrosymmetric responses,” Opt. Lett. 14(4), 196–198 (1989). [CrossRef] [PubMed]

38. 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] [PubMed]

39. S. Fu, T. Wang, S. Zhang, and C. Gao, “Integrating 5 Ч 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] [PubMed]

40. C. Rosales-Guzmán and A. Forbes, How to Shape Light with Spatial Light Modulators (SPIE SpotLight, 2017).

41. 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] [PubMed]

42. M. P. J. Lavery, D. J. Robertson, G. C. G. Berkhout, G. D. Love, M. J. Padgett, and J. Courtial, “Refractive elements for the measurement of the orbital angular momentum of a single photon,” Opt. Express 20(3), 2110–2115 (2012). [CrossRef] [PubMed]

43. M. Mirhosseini, M. Malik, Z. Shi, and R. W. Boyd, “Efficient separation of the orbital angular momentum eigenstates of light,” Nat. Commun. 4(1), 2781 (2013). [CrossRef] [PubMed]

44. 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] [PubMed]

45. 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] [PubMed]

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

47. Y. Ren, G. Xie, H. Huang, C. Bao, Y. Yan, N. Ahmed, M. P. J. Lavery, B. I. Erkmen, S. Dolinar, M. Tur, M. A. Neifeld, M. J. Padgett, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Adaptive optics compensation of multiple orbital angular momentum beams propagating through emulated atmospheric turbulence,” Opt. Lett. 39(10), 2845–2848 (2014). [CrossRef] [PubMed]

48. 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] [PubMed]

49. 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] [PubMed]

50. S. Fu, T. Wang, S. Zhang, Z. Zhang, Y. Zhai, and C. Gao, “Non-probe compensation of optical vortices carrying orbital angular momentum,” Photon. Res. 5(3), 251–255 (2017). [CrossRef]

51. G. Xie, Y. Ren, H. Huang, M. P. J. Lavery, N. Ahmed, Y. Yan, C. Bao, L. Li, Z. Zhao, Y. Cao, M. Willner, M. Tur, S. J. Dolinar, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Phase correction for a distorted orbital angular momentum beam using a Zernike polynomials-based stochastic-parallel-gradient-descent algorithm,” Opt. Lett. 40(7), 1197–1200 (2015). [CrossRef] [PubMed]

52. 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] [PubMed]

53. M. J. Strain, X. Cai, J. Wang, J. Zhu, D. B. Phillips, L. Chen, M. Lopez-Garcia, J. L. O’Brien, M. G. Thompson, M. Sorel, and S. Yu, “Fast electrical switching of orbital angular momentum modes using ultra-compact integrated vortex emitters,” Nat. Commun. 5(1), 4856 (2014). [CrossRef] [PubMed]

Mixed orbital angular momentum amplitude shift keying through a single hologram

Abstract

1. Introduction

2. Principles of OAM-ASK

2.1 Basic idea

2.2 OAM-ASK encoding

2.3 OAM-ASK decoding

3. Results

3.1 Experimental setup

3.2 Experimental realization of 4-bit and 7-bit OAM-ASK

3.3 Image transmission through OAM-ASK

4. Discussions

5. Conclusions

Appendix A Mapping rules for 4 bits OAM-ASK

Appendix B Mapping rules for 7 bits OAM-ASK

Funding

Acknowledgment

References

Cited By

Figures (6)

Tables (2)

Equations (4)

OSA Continuum

Hexadecimal number	0	1	2	3	4	5	6	7
OAM state l	−4	−3	−2	−1	+ 1	+ 2	+ 3	+ 4
Normalized amplitude	1	1	1	1	1	1	1	1
Hexadecimal number	8	9	10	11	12	13	14	15
OAM state l	−4	−3	−2	−1	+ 1	+ 2	+ 3	+ 4
Normalized amplitude	0.707	0.707	0.707	0.707	0.707	0.707	0.707	0.707

128-ary number	0	1	2	3	4	5	6	7
OAM state l	−16	−15	−14	−13	−12	−11	−10	−9
Normalized amplitude	1	1	1	1	1	1	1	1
128-ary number	8	9	10	11	12	13	14	15
OAM state l	−8	−7	−6	−5	−4	−3	−2	−1
Normalized amplitude	1	1	1	1	1	1	1	1
128-ary number	16	17	18	19	20	21	22	23
OAM state l	+ 1	+ 2	+ 3	+ 4	+ 5	+ 6	+ 7	+ 8
Normalized amplitude	1	1	1	1	1	1	1	1
128-ary number	24	25	26	27	28	29	30	31
OAM state l	+ 9	+ 10	+ 11	+ 12	+ 13	+ 14	+ 15	+ 16
Normalized amplitude	1	1	1	1	1	1	1	1
128-ary number	32	33	34	35	36	37	38	39
OAM state l	−16	−15	−14	−13	−12	−11	−10	−9
Normalized amplitude	0.894	0.894	0.894	0.894	0.894	0.894	0.894	0.894
128-ary number	40	41	42	43	44	45	46	47
OAM state l	−8	−7	−6	−5	−4	−3	−2	−1
Normalized amplitude	0.894	0.894	0.894	0.894	0.894	0.894	0.894	0.894
128-ary number	48	49	50	51	52	53	44	55
OAM state l	+ 1	+ 2	+ 3	+ 4	+ 5	+ 6	+ 7	+ 8
Normalized amplitude	0.894	0.894	0.894	0.894	0.894	0.894	0.894	0.894
128-ary number	56	57	58	59	60	61	62	63
OAM state l	+ 9	+ 10	+ 11	+ 12	+ 13	+ 14	+ 15	+ 16
Normalized amplitude	0.894	0.894	0.894	0.894	0.894	0.894	0.894	0.894
128-ary number	64	65	66	67	68	69	70	71
OAM state l	−16	−15	−14	−13	−12	−11	−10	−9
Normalized amplitude	0.806	0.806	0.806	0.806	0.806	0.806	0.806	0.806
128-ary number	72	73	74	75	76	77	78	79
OAM state l	−8	−7	−6	−5	−4	−3	−2	−1
Normalized amplitude	0.806	0.806	0.806	0.806	0.806	0.806	0.806	0.806
128-ary number	80	81	82	83	84	85	86	87
OAM state l	+ 1	+ 2	+ 3	+ 4	+ 5	+ 6	+ 7	+ 8
Normalized amplitude	0.806	0.806	0.806	0.806	0.806	0.806	0.806	0.806
128-ary number	88	89	90	91	92	93	94	95
OAM state l	+ 9	+ 10	+ 11	+ 12	+ 13	+ 14	+ 15	+ 16
Normalized amplitude	0.806	0.806	0.806	0.806	0.806	0.806	0.806	0.806
128-ary number	96	97	98	99	100	101	102	103
OAM state l	−16	−15	−14	−13	−12	−11	−10	−9
Normalized amplitude	0.707	0.707	0.707	0.707	0.707	0.707	0.707	0.707
128-ary number	104	105	106	107	108	109	110	111
OAM state l	−8	−7	−6	−5	−4	−3	−2	−1
Normalized amplitude	0.707	0.707	0.707	0.707	0.707	0.707	0.707	0.707
128-ary number	112	113	114	115	116	117	118	119
OAM state l	+ 1	+ 2	+ 3	+ 4	+ 5	+ 6	+ 7	+ 8
Normalized amplitude	0.707	0.707	0.707	0.707	0.707	0.707	0.707	0.707
128-ary number	120	121	122	123	124	125	126	127
OAM state l	+ 9	+ 10	+ 11	+ 12	+ 13	+ 14	+ 15	+ 16
Normalized amplitude	0.707	0.707	0.707	0.707	0.707	0.707	0.707	0.707