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 lħ, 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 [3–9]. 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 [5,10–15]. 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, 2^M OAM states are essential. If (M + 1) bits coding/decoding is further desired, 2^M+1 OAM states are needed, implying the inevitable additional 2^M 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 (2^N-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:

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 2^N-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:

 

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

Download Full Size | PDF

For the right hand, there must be a term $|{0}\rangle $ if a_m≠0, leading to a bright spot in the beam center. Otherwise, a_m=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 {a_k}={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 {a_k}={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.

 

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 {a_k}={1,1,1,1,1,1,1,1}; (b) The case {a_k}={0,1,0,0,1,1,1,0}.

Download Full Size | PDF

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 1617 nm 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 2^N-ary numbers. At that time the decoding is finished and the transmitted information is recovered.

 

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.

Download Full Size | PDF

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 {a_k} 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 {a_k} 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.

 

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.

Download Full Size | PDF

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.

 

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.

Download Full Size | PDF

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 [22–24] 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 ({a_k}={0,1,0,0,1,0,1,0}), 99 ({a_k}={0,1,1,0,0,0,1,1}) and 255 ({a_k}={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.

 

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.

Download Full Size | PDF

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.