decoding for high-speed free-space optical communication

Optics Communications 452 (2019) 40–47

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High-dimensional vortex beam encoding/decoding for high-speed free-space optical communication Lei Liu, Yesheng Gao ∗, Xingzhao Liu State Key Laboratory of Advanced Optical Communication Systems and Networks, Department of Electronic Engineering, Shanghai Jiao Tong University, Shanghai 200240, China

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Keywords: Optical vortices High-dimensional High-speed Free-space optical communication Digital micromirror device

ABSTRACT The development of communication technologies has led to an ever-increasing demand for higher dimension and higher speed of optical coding technology. The orbital angular momentum (OAM) of vortex beams provides a new degree of freedom for optical coding in recent years. In this paper, we propose a novel high-dimensional vortex beam encoding/decoding scheme for high-speed free-space communication. An offaxis 4f-configuration with a digital micromirror device (DMD) and a MIP-based pattern formation algorithm are employed to generate fast switchable, accurate vortex beams (arrays) that can be used for encoding. Interference is introduced to efficiently decode data information. We experimentally demonstrate a high-quality communication link based on 64-ary vortex beam encoding/decoding and zero bit error rate is observed, which shows the validity of the proposed high-dimensional scheme. Owing to enhancement of encoding dimension and employment of high-speed DMDs, the proposed scheme enjoys a higher transfer rate exceeding other communication technologies. What is more, the data transfer rate can be further improved by introducing vortex arrays, which is demonstrated by the rapid transmission of a 24-bit true color bitmap.

1. Introduction Optical coding technologies have been a crucial component in freespace optical communication systems, which rely on physical dimensions of light beams for encoding such as wavelength/frequency, time, amplitude, phase and polarization [1,2]. Beyond these, in recent years, the orbital angular momentum (OAM) of light beam has been considered as a new degree of freedom for optical coding due to its potential ability to address the emerging transmission capacity crunch [3–5]. In 1992, Allen et al. proposed that vortex beams with helical phase fronts of exp(𝑖𝑙𝜑) (𝑙 = 0, ±1, ±2...) carry OAM, where 𝑙 is the topological charge and 𝜑 refers to the azimuthal angle [6]. Data information can be encoded into different OAM states, which provides a new method for optical coding technique. Topological charge number can take arbitrary integer number ranging from −∞ to +∞, therefore vortex beams have infinite available OAM states that can be used for encoding in theory [7]. However, the available states may be limited by specific encoding/decoding methods. In addition, generation of vortex beams with large topological charge numbers is not easy. These limitations make high-dimensional encoding/decoding with vortex beams become a challenge. To meet the demand for high dimension, Gibson et al. from University of Glasgow presented a method, in which data information was encoded into OAM beams for transmission and was recovered via a ∗

single phase hologram based on the principle of phase matching [8]. However, the method has poor system flexibility and high requirements on alignment [9]. In 2015, Wang et al. proposed a method using Bessel beams carrying OAM [10]. In this method, the decoding of vortex beams was realized by introducing the multiple inverted spiral phase patterns which can convert spiral phase fronts into planar fronts. However, all the possible inverted spiral phase patterns need to be traversed to determine the state for each vortex beam, which is unbearable for high-speed data transfer. The development of high-speed optical communication technologies has led to a growing demand for rapid generation of vortex beams [11, 12]. Many approaches have been proposed, among which the approach of utilizing Fourier transforming optical system with a liquid crystal based spatial light modulator (LC-SLM) [13–15] is more widely applied due to its high flexibility. However, low refresh rate of LC-SLM limits the speed of this approach. Digital micromirror device (DMD) [16] has attracted increasing attention owing to its high refresh rate. At present, the research on generation of vortex beams using a DMD is in progress [17–19], which is expected to increase the data transfer rate of free-space optical communication. In this paper, we propose a high-dimensional encoding/decoding scheme using Laguerre-Gaussian (LG) vortex beams in a high-speed free-space communication system. An off-axis 4f-configuration with a

Corresponding author. E-mail address: [email protected] (Y. Gao).

https://doi.org/10.1016/j.optcom.2019.06.061 Received 1 May 2019; Received in revised form 20 June 2019; Accepted 24 June 2019 Available online 26 June 2019 0030-4018/© 2019 Published by Elsevier B.V.

L. Liu, Y. Gao and X. Liu

Optics Communications 452 (2019) 40–47

of 𝑧0 are illustrated in Fig. 1(i)-Fig. 1(l). Curved forklike fringes are observed, wherein the difference in the fringe number of the upper and lower portions is equal to 𝑙, and the number of the bright rings is 𝑝 + 1. It can be seen that the OAM state of corresponding vortex beam can be uniquely determined by analyzing the number of fringes in each interference pattern. In this decoding method, a camera is employed to record the interference patterns and the number of fringes in the patterns can be automatically identified by program. The specific procedure of OAM detecting is given below. Step 1: The pretreatment of interference image. Step 2: Draw a horizontal line passing through the phase singularity in the interference patterns and determine the radial mode number by calculating the number of intersection points between the horizontal line and the bright rings. The inner ring radius 𝑟1 and the outer ring radius 𝑟2 of the central bright ring can be also obtained, which will be used in Step 3. Step 3: Draw a circle with a radius of (𝑟1 + 𝑟2 )∕2 centered in the phase singularity and determine the topological charge by calculating the difference in the number of intersection points (𝑁1 and 𝑁2 ) between the upper and lower semicircles and the dark fringes.(𝑁1 − 𝑁2 = 𝑙). The proposed decoding method is easy to implement and avoids stringent alignment required by other phase distribution based detection technologies. On this basis, we propose a novel scheme of high-speed and high-dimensional encoding/decoding using LG vortex beams, and the concept and principle is illustrated in Fig. 2. At the transmitter side, base numbers of m-ary coding (0, 1, 2, ..., 𝑚−1) correspond to m different states of LG vortex beams with different topological charges (𝑙) and radial mode numbers (𝑝). A 𝑚-ary number sequence can be encoded into a LG vortex beam sequence, which can be generated by loading the corresponding patterns onto DMD. After that, the generated vortex beam sequence is interfered with a plane wave at the transmitter side. After propagation in free space, a camera is employed to directly detect the received interference pattern sequence at the receiver side. The states of the vortex beams can be determined one by one by analyzing the number of fringes of each interference pattern. Then, the number sequence can be recovered according to the identified vortex beam sequence. Fig. 2 illustrates quaternary vortex beam encoding/decoding process for free-space communication. Four different states of LG vortex beams are selected to represent quaternary base numbers respectively (𝐿𝐺02 → 0, 𝐿𝐺12 → 1, 𝐿𝐺1−2 → 2, 𝐿𝐺13 → 3). A quaternary number sequence with four numbers (1, 2, 3, 0) is encoded as a vortex beam sequence and then transmitted in free space. Four corresponding interference patterns are detected at the receiver, as shown in Fig. 2. The recovered quaternary number sequence is 1, 2, 3, 0, which is coincident with the transmitted sequence. Thus, the feasibility of the proposed scheme is evidently testified.

DMD and a novel pattern formation algorithm are employed to generate fast switchable, accurate vortex beams (arrays) that can be used for encoding. This novel generation method enjoys higher precision exceeding that of other state-of-the-art methods, such as Lee method. Interference patterns are used to decode data information with high efficiency, which avoids stringent alignment required by other phase distribution based detection technologies. Rapid transmission of a 64ary number sequence and a 24-bit true color bitmap is demonstrated respectively in the experiment. The bit error rate (BER) is evaluated and zero error is observed. Using high-speed DMDs, the scheme enjoys a higher transfer rate exceeding other communication technologies using LC-SLMs. What is more, for large-capacity communication, the scheme is able to further increase data transfer rate by introducing vortex arrays to meet the demand for high-speed communication. The rest of this paper unfolds as follows: The concept and principle of high-dimensional vortex beam encoding/decoding in a free-space communication system is given in Section 2. Rapid and accurate generation of vortex beam (array) is discussed in Section 3. Then, we give a brief introduction to the experimental setup in Section 4. Section 5 demonstrates measured results and quantitative analyses. Conclusions are drawn in Section 6. 2. Principle Optical vortex has been widely used in fields like free-space communications, quantum communications, optical micromanipulations and optical trapping [20–23]. It has a spiral phase structure with a phase singularity at its center, which is characterized by exp(𝑖𝑙𝜑), wherein 𝑙 refers to the topological charge of the phase singularity. 𝑙 will be positive if the phase increases in a right-handed sense, otherwise being negative for left-handed rotation. The phase singularity causes a dark hollow intensity distribution at the center of vortex beam. There are many types of vortex beam, among which LG vortex beam is more frequently used in optical coding owing to the superiority that it has more possible states available for encoding. The complex field [24] of a LG vortex beam that propagates along the 𝑧 direction can be written |𝑙| in Eq. (1), where 𝑝 denotes radial mode number, 𝐿𝑝 designates the Laguerre polynomial, 𝑘 is the wavenumber, 𝑓 denotes the azimuthal |𝑙| coordinate, 𝑅(𝑧) is the radius of curvature of the phase surfaces, 𝛷𝑝 (𝑧) 𝑙 refers to the Gouy phase and 𝐶𝑝 is a constant. 𝐶𝑝𝑙

[

𝑘𝑟2 −𝑟2 −𝑖 + 𝑖𝛷𝑝|𝑙| (𝑧) − 𝑖𝑙𝜑 2 𝑤 (𝑧) 2𝑅 (𝑧) 𝑤 (𝑧) ( √ )|𝑙| ( ) 2𝑟 2𝑟2 𝐿|𝑙| × 𝑝 𝑤 (𝑧) 𝑤2 (𝑧)

𝐿𝐺𝑝𝑙 (𝑟, 𝜑, 𝑧) =

]

exp

(1)

For most of the common coding methods, 𝑝 is fixed to 0 and coding is realized only by changing the topological charge. In the proposed method of this paper, by changing the values of 𝑙 and 𝑝, more OAM states can be obtained, which makes it applicable for high-dimensional encoding/decoding. The intensity and phase distributions of LG vortex beams with different values of 𝑝 and 𝑙 are shown in Fig. 1(a)-Fig. 1(h). As shown in Fig. 1, the intensity distribution and phase distribution of vortex beams are (𝑝 + 1)-layered donut shapes and (𝑝 + 1)-layered spiral phase structure, respectively. The propagation characteristic of vortex beams is of vital importance for the optical coding technology of freespace communication. When the vortex beam propagates a distance of 𝑧0 along the 𝑧 direction in free space, its intensity distribution remains the donut shape, whose radius becomes larger with the increase of transmission distance. Equiphase lines in the phase distribution get bent after propagation, but the phase singularity remains the same. As phase distribution is difficult to acquire directly, in this paper, a plane wave is selected to interfere with the generated vortex beams at the transmitter side. And decoding is realized by detecting the interference patterns at the receiver side. We conduct a simulated experiment and the interference patterns at a propagation distance

3. Generation of vortex beam To achieve high-speed data transfer, rapid and accurate generation of vortex beams is indispensable for the proposed encoding/decoding scheme. Many approaches have been proposed to generate vortex beams, among which the approach utilizing LC-SLM is more widely applied due to its high flexibility. However, the refresh rate of commercially available LC-SLMs is commonly 60 Hz. Low refresh rate of LC-SLM limits the speed of this approach, which makes it unfit for highspeed communication applications. Thus, researchers start to capitalize on DMD to generate vortex beams due to its ultra-high refresh rate superior to LC-SLM. In this paper, a neoteric method for accurate and rapid generation of vortex beams (arrays) using a DMD is proposed. The method can generate not only LG vortex beams required by the encoding/decoding scheme, but also other modes, such as Bessel beam and perfect vortex beam. An off-axis 4f-configuration with a circular aperture is employed to perform the generation of vortex beams. It is achieved by making low pass filtering in Fourier plane of DMD. The configuration is composed 41

L. Liu, Y. Gao and X. Liu

Optics Communications 452 (2019) 40–47

Fig. 1. LG vortex beams with different values of p and l (𝐿𝐺02 , 𝐿𝐺12 , 𝐿𝐺1−2 , 𝐿𝐺13 ). (a)–(h) Intensity profiles and phase profiles of LG vortex beams at a propagation distance of 0. (i)–(l) Interference patterns of LG vortex beams at a propagation distance of 𝑧0 .

Fig. 2. Concept and principle of high-dimensional (e.g. quaternary) vortex beam encoding/decoding in a free-space optical communication system.

each value in the pattern represents the state of a micromirror. Then, illuminating DMD vertically, the collimated light beam is encoded by DMD pattern and the corresponding field 𝑓 (𝑥, 𝑦) can be written as 𝑓 (𝑥, 𝑦) =

+∞ ∑

𝛿 (𝑥 − 𝑛𝑑) 𝛿 (𝑦 − 𝑚𝑑)𝐶 (𝑥, 𝑦)

(2)

𝑛,𝑚=−∞

where each DMD pixel is simplified to a two-dimension (2-D) delta function, and the amplitude field representing DMD pattern is expressed as 𝐶(𝑥, 𝑦). 𝑥 and 𝑦 are 2-D position variables in the space domain and 𝑑 denotes pixel pitch of DMD. Next, the encoded light beam passes an off-axis 4f-configuration and finally arrives at the output plane. The two lenses are employed to perform Fourier transform and inverse Fourier transform, respectively. The reflected beam from DMD contains several diffraction orders. In our scheme, the first-order diffracted light is selected and extracted by a circular aperture to realize the of vortex beams. The circular aperture is positioned at ( generation ) 𝜆𝑓 𝜆𝑓 − 𝑠2 𝑑0 , 𝑠𝑑0 with respect to the 0th diffraction order (𝜆 is the wavelength of light, 𝑓0 denotes the focal length of the first lens, 𝑑 is the pixel pitch of DMD, and 𝑠 refers to an integer parameter). And owing to the circular aperture, the system makes low pass filtering in Fourier plane of DMD, which makes individual DMD pixels indistinguishable due to resolution declination after filtering. The two lenses are placed off-axis in order to introduce extra linear phase depending on the respective

Fig. 3. System structure of generation of vortex beams (arrays).

of a DMD, two lenses, a circular aperture and a camera, as shown in Fig. 3. DMD is a spatial light modulator composed of multiple micromirrors with the capability of spatial binary amplitude control of light [25]. The binary pattern is written on DMD at first, wherein 42

L. Liu, Y. Gao and X. Liu

Optics Communications 452 (2019) 40–47

Fig. 4. (a)(b) System structure and experimental setup of high-speed and high-dimensional encoding/decoding using vortex beams (arrays).

position in the output plane. The output field in the output plane is illustrated in Eq. (3), where the spectrum of ℎ(𝑥, 𝑦) refers to the circular 2𝐽 (𝜌) function 𝑐𝑖𝑟𝑐(𝑤𝑥 , 𝑤𝑦 ), ∗ denotes convolution operator, 𝑗𝑖𝑛𝑐 (𝜌) = 1𝜌 is known √ as the jinc function. 𝐽1 (𝜌) is the one order Bessel function, 2𝑟𝑑 𝑥2 + 𝑦2 . 𝜌= 𝜆𝑓

𝐸𝑡 arg 𝑒𝑡 (𝑖, 𝑗) =

𝐶𝑝𝑙

2

2

(𝑖,𝑗) exp[ −𝑟𝜔2(𝑖,𝑗) − 𝑖 𝑘𝑟2𝑅(𝑧) + 𝑖𝛷𝑝𝑙 (𝑧) + 𝑖𝑙𝜑(𝑖, 𝑗)] (𝑧) (√ ) ( ) 2 2𝑟(𝑖,𝑗) 𝐿𝑙𝑝 2𝑟𝜔2(𝑖,𝑗) × 𝜔(𝑧) (𝑧) 𝜔(𝑧)

(6)

0

( 𝑟 (𝑥, 𝑦) =

𝑓 (𝑥, 𝑦) 𝑒 [

+∞ ∑

=

𝑥 −𝑗 2𝜋 𝑦 𝑗 2𝜋 𝑠2 𝑑 𝑒 𝑠𝑑

) ∗ ℎ (𝑥, 𝑦)

𝐶 (𝑥, 𝑦) 𝛿 (𝑥 − 𝑛𝑑) 𝛿 (𝑦 − 𝑚𝑑)𝑒

𝑚,𝑛=−∞

𝑗 2𝜋 𝑥 −𝑗 2𝜋 𝑦 𝑠2 𝑑 𝑒 𝑠𝑑

𝛥𝐸 (𝑖, 𝑗) = 𝐸𝑜𝑢𝑡 (𝑖, 𝑗) − 𝐸𝑡 arg 𝑒𝑡 (𝑖, 𝑗)

] ∗

𝐽1 (𝜌) 𝜌

−𝜀 ≤ Re (𝛥𝐸 (𝑖, 𝑗)) ≤ 𝜀 −𝜀 ≤ Im (𝛥𝐸 (𝑖, 𝑗)) ≤ 𝜀

(3) It can be deduced from Eq. (3) that the response of a single DMD pixel is a jinc function with corresponding extra phase in the corresponding position of the output plane. The main lobe width of the jinc function is determined by the size of the circular aperture, which will affect the resolution of the system. The size of the aperture is variable. In this encoding/decoding scheme, the diameter of the circular aperture 𝜆𝑓 is set to 2𝑑𝑠0 and 𝑠 is designated to 4, thus the system resolution is 8 × 8 pixels per diffraction limited spot. And the output field is the sum of all single DMD pixel responses. Based on this, the pattern written on the DMD can be obtained and the corresponding vortex beams can be shown in the output plane. The algorithm of obtaining DMD pattern is essential to the proposed method. As the single pixel response is a spatially unbounded jinc function, the value of each pixel in the output field is related to all the pixels of DMD pattern, as illustrated in Eq. (4). Where the size of DMD pattern is 𝑁𝑝 × 𝑁𝑝 pixels. (𝑖, 𝑗) is the 2-D coordinate of the output plane (Row 𝑖, Column 𝑗), (𝑛, 𝑚) is the 2-D coordinate of DMD (Row 𝑛, Column 𝑛𝑚 denotes the response on the output plane evoked by the single 𝑚). 𝐸𝑜𝑢𝑡 DMD pixel positioned at (𝑛, 𝑚) of DMD, wherein the response positioned 𝑛𝑚 (𝑖, 𝑗). C is a binary matrix composed at (𝑖, 𝑗) on the output plane is 𝐸𝑜𝑢𝑡 of 𝑐 𝑛𝑚 , which represents DMD pattern. 𝐸𝑜𝑢𝑡 denotes the output field of the output plane. 𝑁

𝐸𝑜𝑢𝑡 (𝑖, 𝑗) =

𝑐 𝑛𝑚 = 0, 1

1 ≤ 𝑖, 𝑗, 𝑛, 𝑚 ≤ 𝑁𝑝

𝑛𝑚 𝐸𝑜𝑢𝑡 (𝑖, 𝑗) ⋅ 𝑐 𝑛𝑚

(4)

𝑛=1 𝑚=1

Thus, to generate accurate vortex beams, the correlations among all pixels must be considered in getting DMD pattern, which is hard to realize due to the increased computation complexity. To solve this, we put forward a DMD pattern formation algorithm based on mixed integer programming (MIP) [26]. The algorithm converts the DMD pattern formation problem into a MIP optimization problem that the objective function equals to 1 and the constraints are shown as ,𝑚=𝑗+ 𝑁−1 𝑛=𝑖+ 𝑁−1 2 2

𝐸𝑜𝑢𝑡 (𝑖, 𝑗) =

∑

𝑛𝑚 𝐸𝑜𝑢𝑡 (𝑖, 𝑗) ⋅ 𝑐 𝑛𝑚

(8)

(9)

(10)

where 𝐸𝑡𝑎𝑟𝑔𝑒𝑡 and 𝐸𝑜𝑢𝑡 denote the target field and the output field in the output plane, respectively, and 𝛥𝐸 is the difference between the target field and the output field. As the energy of the jinc function is mainly concentrated near its main lobe, a truncate jinc function with 𝑁 × 𝑁 pixels is employed in this optimization model to replace the intact jinc function, as shown in Eq. (5), which is able to reduce the computation complexity of the model while maintaining the high precision of the generation result by choosing the appropriate value of 𝑁. The value of 𝑁 is typically chosen such that the main lobe of jinc function is included in the range of 𝑁 × 𝑁 pixels. 𝜀 is the precision parameter, and if it is infinitely small, the output field will be incredibly close to the desired target field, which indicates that the proposed method is able to generate vortex beams at high precision. The model belongs to a MIP problem and can be solved by many optimization solvers to obtain the DMD patterns. The proposed method can generate the desired single vortex beam on the output plane by loading the obtained pattern on the DMD. Moreover, vortex arrays can also be easily generated by the proposed method. To generate vortex arrays, the pattern loaded onto DMD should be composed of multiple parts and each part represents the DMD pattern corresponding to a vortex beam of the array and can be obtained by the proposed pattern formation algorithm. The proposed generation method has a superior performance in precision compared with other conventional methods, such as Lee holography [17,18,27], which is guaranteed by considering the correlations among all pixels when obtaining DMD pattern. And fast switching among vortex beams can be achieved by the proposed method, whose speed far exceeds that achieved by LC-SLMs. High switching speed of the method makes it applicable to high-speed communication.

𝑁

𝑝 𝑝 ∑ ∑

(7)

(5)

𝑛=𝑖− 𝑁−1 ,𝑚=𝑗− 𝑁−1 2 2

43

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Optics Communications 452 (2019) 40–47

Fig. 5. The generation of two LG vortex beams 𝐿𝐺13 and 𝐿𝐺1−2 : (1) Simulated results under the proposed method and Lee holography: (a)–(d) DMD patterns loaded on the DMD, (e)–(l) intensity distributions and phase distributions, Last two columns: error analysis results based on RMSE and fidelity. (2) Measured results under the proposed method: (m)–(p) measured intensity distributions and interference patterns.

4. Experimental setup

circular aperture. The DMD patterns are obtained by MIP- based pattern formation algorithm, in which parameter N is chosen to be 33 here.

Fig. 4 shows the experimental setup for high-speed and highdimensional encoding/decoding using vortex beams in a free-space optical communication system. It is composed of a 532-nm laser source, a TI DLP6500 DMD, two Fourier lenses with focal length of 30 cm, two beam splitters with splitting ratio of 50:50, a circular aperture with a diameter of 1.84 mm, three mirrors, an adjustable attenuator and a Hamamatsu C13440-20CU CMOS camera. The laser source contains a power-adjustable laser with up to 50 mW output power, a pinhole and a collimated lens. The DMD has pixel pitch of 10.8 μm, high resolution of 1920 × 1080 pixels and high refresh rate of 9500 Hz. The C13440-20CU CMOS camera has pixel pitch of 6.5 μm, high resolution of 2048 × 2048 pixels and high output bit depth of 16 bit. The frame rate of this camera at full resolution is 100 frames/s. By shrinking the acquisition region, the frame rate can reach 25655 frames/s. The laser source is used to generate a collimated laser beam. An offaxis 4f-configuration with a DMD is employed to generate the desired vortex beam by writing the corresponding pattern on the DMD. The 4f-configuration is composed of a DMD, two Fourier lenses and a

By altering the patterns loaded on the DMD, the time-varying vortex beam sequence carrying information can be obtained on output plane I of the transmitter side. Then the vortex beam sequence will propagate in free space and finally arrive at the receiver side. Two reflective mirrors (M2 and M3) are just used to change the propagation direction of vortex beam in free space. At the transmitter side, the interference is performed by two beam splitters and a mirror. Beam splitter 1 is used for splitting the laser source into two beams, and one of the two passes by reflective mirror 1 and then interferes with the other beam used for generation of vortex beam at beam splitter 2. At the receiver side, the camera on output plane II is used for capturing the received interference pattern sequence. Data sequence can be recovered by analyzing the forklike fringes of the patterns. In this system, the distance of free-space communication is from output plane I to output plane II. 44

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Optics Communications 452 (2019) 40–47

5. Experimental results and discussions 5.1. Generation of LG vortex beams and its quantitative analyses We first demonstrate the generation of LG vortex beams using the proposed method. Here, 𝐿𝐺13 and 𝐿𝐺1−2 modes are used to demonstrate the performance. DMD patterns of three LG modes obtained by MIPbased pattern formation algorithm are shown in Fig. 5(a)-Fig. 5(b). These patterns are written on the DMD and the three reconstructed fields are obtained on output plane I, whose intensity profiles are shown in Fig. 5(e)-Fig. 5(f). The figures show the multiple-layered donut shapes symbolizing LG vortex beams, and the number of bright rings in each figure is 𝑝 + 1, which accords with the theoretical value. The corresponding phase profiles are illustrated in Fig. 5(i)-Fig. 5(j), which present typical vortex phase structures. The simulation results show the feasibility and effectiveness of the proposed method. A measured experiment is performed to verify the simulation results. The measured intensity distributions captured by CMOS camera are shown in Fig. 5(m)-Fig. 5(n), which coincide with the simulation results. And the generated vortex beam is interfered with a plane wave to verify its phase property. The measured interference patterns are displayed in Fig. 5(o)-Fig. 5(p). We observe forklike fringe patterns, and the difference in the number of fringes of the upper and the lower portions is equal to the topological charge value, which testifies the presence of optical vortex. The simulated and measured results indicate evidently that LG vortex beams can be well generated by the proposed method, and the results match well with the ideal vortex beams shown in Fig. 2. Experimental results on the generated three LG modes using Lee holography are also given in Fig. 5. Lee holography [27] is the most common approach to generate vortex beams with a DMD, which is realized by introducing the binary amplitude holograms [17,18]. In Lee holography, the size of the spatial filter remains the same and the spatial carrier frequency is set to 2𝜋 𝑑 −1 to obtain maximum 12 precision. As seen in Fig. 5, it is clear that the intensity profile under Lee holography appears obviously less smooth than that of the proposed method, and the phase profile is a little distorted. To analyze the modulation precision of the two methods more accurately, quantitative error analysis is introduced and is given by RMSE [28] and fidelity [29]. RMSE represents the overall differences between the numerical values of target field and output field. Fidelity describes the similarity between target field and output field. The expressions of two measures are shown in Eq. (11) and Eq. (12), wherein the size of the target field is 𝑁𝑥 × 𝑁𝑦 pixels. The analysis results are also illustrated in Fig. 5. It is found that the RMSEs of the proposed method are less than 0.0192 and the corresponding fidelities are more than 0.9917. Compared with Lee holography, the proposed method can reduce RMSE by 86.4% and improve fidelity by 0.2%. The quantitative results evidently testify that the proposed method enables accurate generation of vortex beams and enjoys higher modulation precision exceeding that of other state-of-the-art methods, such as Lee holography. √ √ ∑𝑖=𝑁 ,𝑗=𝑁 √ |2 𝑦 | √ 𝑖,𝑗=1𝑥 |𝐸𝑜𝑢𝑡 (𝑖, 𝑗) − 𝐸𝑡 arg 𝑒𝑡 (𝑖, 𝑗)| √ | | 𝑅𝑀𝑆𝐸 = (11) 𝑁𝑥 𝑁𝑦 | |2 |𝐸𝑜𝑢𝑡 𝐸𝑡∗arg 𝑒𝑡 | | | 𝐹 𝑖𝑑𝑒𝑙𝑖𝑡𝑦 = |𝐸𝑜𝑢𝑡 | ||𝐸𝑡 arg 𝑒𝑡 || | || |

Fig. 6. Lookup table for mapping 64 states of LG vortex beams to 64-ary base numbers.

64-ary base numbers respectively, as illustrated in Fig. 6. The figure consists of 64 subfigures containing vortex beams, each representing a number varying from 0 to 64. Thereout, the number sequence to transmit can be encoded as a LG vortex beam sequence. Next, at the transmitter side, we generate the desired LG vortex beam sequence by switching the corresponding patterns loaded onto the DMD. MIP-based pattern formation algorithm is employed to obtain the DMD patterns. After encoding, the vortex beam sequence propagates for a distance of 0.8 m and finally arrives at the receiver side. For decoding, the interference between the vortex beam and a plane wave is performed at the transmitter side at first, and then at the receiver side, the interference patterns after propagation are captured by CMOS camera. For example, the received interference patterns of number sequence 5, 6, 4, 7, 3, 8, 2, 9, 1, 10, 45, 46, 44, 47, 43, 48, 40, 51, 29, 30, 27, 32, 59, 60 are shown in Fig. 7. It is clear to observe the desired curved forklike fringe patterns. By analyzing the number of fringes in each pattern, the state of its corresponding vortex beam can be identified and its corresponding 64-ary number can be found based on the lookup table. The recovered number sequence exactly coincides with the transmitted one, which confirms the successful implementation of free-space communication based on the proposed encoding/decoding scheme. In this communication link experiment, we transmit a number sequence with 1000 64-ary numbers and investigate the BER performance. BER denotes the ratio of the number of received error symbols to the number of total symbols received, where each N-ary number corresponding to each vortex beam is considered to be a symbol. In this experiment, zero BER is achieved, showing the favorable transmission performance. Moreover, the scheme is able to perform the encoding/decoding with higher dimension by utilizing more OAM states. Compared with conventional methods, the proposed high-dimensional scheme has more alternative code and higher data transfer rate. For example, for conventional hexadecimal coding methods, it takes three vortex beams to complete the transmission of 12 bits of information, whereas only two vortex beams are required for the proposed scheme, which shrinks the transmission time. In addition to high dimension, the proposed scheme offers a large improvement in data transfer rate also due to its ability of rapid generation of vortex beams.

(12)

5.2. High-dimensional encoding/decoding experiment using vortex beam We complete a free-space visible-light communication link using high-dimensional vortex beam encoding/decoding. A random 64-ary number sequence is transmitted through the communication link. We first select 64 different states of LG vortex beams to represent the

5.3. High-speed encoding/decoding experiment using vortex arrays To further improve the data transfer speed, optical coding technique based on vortex arrays is developed for free-space communication. By 45

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Optics Communications 452 (2019) 40–47

Fig. 7. The received interference patterns of a 64-ary number sequence, and the corresponding LG vortex beams and recovered number sequence obtained by analyzing interference patterns.

Fig. 8. 24-bit Lena color bitmap free-space transmission based on vortex array encoding/decoding.

extending the encoding dimension to dimension of space, we can easily implement simultaneous transmission of several numbers each time, greatly increasing communication speed. The 24-bit true color bitmap attracts much attention due to its rich colors and realistic effects. 24-bit bitmap is composed of abundant pixels and each pixel contains 3 bytes (24 bits) of color information, which means that its data amount is quite large. Various algorithms have been proposed for image transmission but few of them is suitable for the 24-bit color bitmap. Huge data amount makes the transmission of the 24-bit bitmap at high speed become a challenge. We choose a 128 × 128 pixel 24-bit Lena color bitmap in the experiment, as shown in Fig. 8. For the color image, we first extract the color values of three channels (R, G, B) of each pixel from the image, which are 8 bits each channel. Then the color value of each pixel is converted to four 64-ary numbers. Hence, the bitmap with 128 × 128 pixels can be converted to a number sequence with 65536 64-ary numbers. It will take a long time to transmit this color bitmap in the conventional way of transmitting one number each time. Here, four numbers are transmitted simultaneously each time, which will dramatically shrink the transmission time and increase the data transmission rate. A vortex array composed of four LG vortex beams

is employed to represent the value of a pixel, wherein each vortex beam corresponds to a 64-ary number respectively. Thus, the bitmap can be mapped to a vortex array sequence of 16384 vortex arrays. At the transmitter side, the generation of the desired LG vortex array sequence is achieved by switching the corresponding patterns loaded onto the DMD. Then a plane wave is introduced to interfere with the vortex array sequence. After propagation in free space, the interference patterns are detected at the receiver side. Each pattern contains four forklike fringe patterns, representing four vortex beams of a vortex array respectively. Hence, the vortex arrays can be identified one by one by analyzing the forklike fringe patterns. When the received 64ary number sequence is recovered, it is further converted to a 24-bit true color bitmap. For example, suppose that R value, G value, B value of a pixel is 20,33,5 in decimal, it can be converted to four 64-ary numbers (5)64 , (2)64 , (4)64 , (5)64 . These four numbers are mapped onto the vortex beams of 𝐿𝐺01 , 𝐿𝐺04 , 𝐿𝐺02 , 𝐿𝐺01 , respectively, which will further constitute a vortex array. And another pixel has R value of 112, G value of 81, B value of 28 and its corresponding numbers are (28)64 , (5)64 , (4)64 , (28)64 . Four vortex beams of 𝐿𝐺22 , 𝐿𝐺01 , 𝐿𝐺02 , 𝐿𝐺22 constitute another vortex array. The received interference patterns of two pixels are shown in Fig. 8. By analyzing the patterns, two vortex arrays are 46

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References

identified and the corresponding 64-ary numbers are determined. Then four 64-ary numbers representing each pixel are converted to three 8 bit values for recovering the values of R, G, B channels of each pixel. Experimental results show that the values of two pixels are accurately recovered. And the received image exactly recovers the transmitted one, which confirms the successful image transmission based on vortex array encoding/decoding. The BER of the image transmission is zero, showing favorable transmission performance. Remarkably, the proposed scheme has a superiority of high data transfer rate. Most conventional methods use phase-only LC-SLMs with refresh rate of 60 Hz to generate vortex beams for encoding. Generally speaking, these methods can just implement hexadecimal encoding, and only a few of them enables 32-ary coding. When the Lena 24-bit color map is transmitted using these conventional methods, the color value of each pixel is converted to six hexadecimal numbers and the image is converted into a number sequence with 98304 hexadecimal numbers. Employing the proposed high-dimensional scheme, the color value of each pixel is converted to four 64-ary numbers and the image is converted into a number sequence with 65536 64-ary numbers. By transmitting only one number each time, the data transfer rate under conventional methods is 240 bit/s and the image transmission time is 1638.4 s. DMD with refresh rate of 9500 Hz is used in the proposed scheme and the data transfer rate is 57 kb/s. By contrast, the image transmission time is greatly shrunken to 6.9 s. By employing vortex arrays to achieve simultaneous transmission of four numbers each time, transmitting speed of this 24-bit true color bitmap can reach 228 kb/s. Note that it is simple to generate larger-scale vortex arrays using the proposed scheme by changing the patterns loaded on DMD. And simultaneous transmission of 48–72 bits of information each time can be supported in this communication system with the help of largerscale vortex arrays, which makes the proposed scheme applicable to high-speed data transmission for large-capacity communication.

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6. Conclusion An innovative high-dimensional vortex beam (array) encoding/decoding scheme for free-space optical communication has been proposed in this paper. An off-axis 4f-configuration with a DMD and a MIP-based pattern formation algorithm are employed to generate accurate, fast switchable vortex beams (arrays) that can be used for encoding. It is experimentally demonstrated that the generated vortex beams have a high fidelity of 0.9917 and a low RMSE of 0.0192. A plane wave is selected to interfere with vortex beams (arrays) at the transmitter side, and transmitted interference patterns are detected at the receiver side and then analyzed to recover the data information. We experimentally demonstrate a high-quality communication link based on 64-ary vortex beam encoding/decoding with zero BER. Owing to augment of dimension and employment of high-speed DMDs, the proposed scheme enjoys a higher transfer rate exceeding other communication technologies. What is more, data transfer rate can be further improved by introducing vortex arrays. The rapid transmission of a 24-bit true color bitmap is successfully implemented at a speed of 228 kb/s by employing vortex arrays. The proposed scheme has a promising application prospect in the field of high-speed data transfer for large-capacity communication. Acknowledgments The authors want to express their gratitude to editors and anonymous reviewers who gave their valuable comments and suggestions to this paper. Funding This work was supported by the National Natural Science Foundation of China under Grant 61601285. 47