Experimental demonstration of multi-dimensional coding/decoding for image transfer with controllable vortex arrays

Vortex beams carrying orbital angular momentum (OAM), which featuring helical phase front, have been regarded as an alternative spatial degree of freedom for optical mode coding and multiplexing. For most reported OAM-based mode coding schemes, data information is only encoded by different OAM mode states. In this paper, we introduce a novel design technique to construct vortex array phase grating (VAPGs) for the flexible generation of vortex arrays, and employ the proposed VAPGs to realize multi-dimensional space/mode/amplitude coding/decoding. By designing VAPGs with different parameters and loading them on to a single spatial light modulator (SLM), we successfully generate vortex array with different mode states and relative power in the experiments. Moreover, a 10-bit multi-dimensional space/mode/amplitude data coding/decoding scheme for image transfer in free-space link with a zero bit-error-rate is experimentally demonstrated, which confirm the feasibility of our proposed VAPG-based coding/decoding scheme.


Introduction
Orbital angular momentum (OAM) beam carrying helical wavefront, also known as vortex beam, has been studied for decades. It was shown by Allen in 1992 that the helically phased vortex beams comprising an azimuthal phase term exp(ilφ), possess an OAM of lℏ per photon, where l is referred to topological charge and φ is azimuthal angle 1 . Due to the unique characteristics, vortex beams have seen wide applications in different areas, such as optical manipulation, optical trapping, optical sensing, and optical imaging [2][3][4][5][6][7][8] . Theoretically, vortex beams with different mode states are orthogonal with each other. In addition, there are, in principle, infinite OAM mode states, which can be employed as independent information carriers. Hence, these outstanding properties make vortex beams to have been extensively studied to increase the channel capacity for both quantum information communications and classical optical communications [9][10][11] . Generally, for vortex beam based optical communications, OAM multiplexing and OAM coding/decoding are two important ways to carry and deliver data information.
In OAM multiplexing, multiple collinear vortex beams are used to carry data information, which is known as a subset of mode division multiplexing (MDM) technique. By employing OAM-based MDM, high-capacity data transmission systems are realized in both optical fiber and free-space [12][13][14][15] . In addition to MDM, the vortex beams carrying OAM can also be regarded as symbols to realize data coding/decoding transmission, which also called OAM shift-keying. In 2004, Gibson demonstrated the transfer of information encoded as OAM modes of a light beam using computer-generated fork phase grating 16

Experimental setup
The experimental setup of the high-dimensional space/mode/amplitude coding/decoding data transfer is shown in Fig  Firstly, we test the generation of vortex array with different power distribution by employing VAPGs in experiments. Figure 3 shows the phase profiles of VAPG, corresponding simulation and experimental results for generation 5×5 vortex array (l=-12, -11, …, 0, … 11, 12) with different power distribution. The simulation results of the light field out come from the VAPG is calculated by Fresnel diffraction integral. The simulated interferograms is depicted to show the mode states of the generated OAM modes in the array. In Fig. 3(a), the power ratio of the VAPG is set to (0.8:1.5:2:1.5:0.8) in both x and y direction. In both simulation and experimental results, one can easily find that the OAM modes at the center is much brighter than the ones at the outside. The calculated power spectrum is also nearly the same, as expected. In addition, the phase distribution of the 2D VAPG in x and y directions can be set to generate OAM modes with different power ratio. In Fig.   3(b), we set the power ratio of the grating in x direction to (2:5:1:3:4), and y direction to (2:3:5:1:4). The corresponding simulation results shown in Fig. 3(b) are consistent with the preset power distribution. The captured intensity profile nearly has the same distribution with the simulation one. However, the calculated power spectrum in the experiment has a bit difference with the simulation one,  Experimental results of high-dimensional space/mode/amplitude coding/decoding By employing VAPGs, we successfully generate vortex array with different mode distribution and relative power. Here, by switching the prepared VAPGs on the SLM, we can realize multi-dimensional data coding/decoding in experiments. Figure 5 shows some examples of recorded vortex array intensity patterns for space/mode/amplitude coding/decoding. The cross dashed line marked at the center of the intensity image is used to make the position of the OAM modes clear to identify. The parameters for the generation of the vortex arrays are selected according to the coding table shown in Fig. 1. In Fig. 5(a)  In addition, to vividly demonstrate the data transmission performance of the proposed high-dimensional space/mode/amplitude data coding/decoding, an 80×80 pixels image with 256 gray-scale levels is transmitted in free-space as shown in Fig. 6(a). Each pixel of the gray image carries information of 8 bits, implying such image is 51.2 Kbit in total. Hence, the 80×80 pixels gray-scale image can be converted to 5120 10-bit symbols, which can be mapped to 5120 VAPG pattern sequence. By switching the corresponding VAPGs on the SLM, the image is coded and transformed into a series of time-varying vortex arrays. After free-space transmission, the vortex arrays are detected by the Camera for decoding. Figure   6(b) shows the received image recovered by the receiver, which exactly recovers the transmitted one with zero bit-error-rate. The obtained results indicate successful implementation of high-dimensional space/mode/amplitude data coding/decoding with favorable transmission performance. Figure 6. The transmitted and received grayscale images using the proposed multi-dimensional space/mode/amplitude data coding/decoding scheme.

Discussion
In summary, we have presented a new approach to design VAPGs for the flexible generation of vortex array with high efficiency. Different vortex beams with different desired power distribution are successfully generated in the experiments.
Moreover, by loading the pre-designed VAPGs on to a single SLM, we experimentally demonstrate 10-bit (1024-ary) high-dimensional space/mode/amplitude data coding/decoding for image transfer in free-space. An 80×80 pixels gray-scale image is successfully transmitted with zero bit-error-rate.
In our proof-of-concept experiments, 16

Method
In this section, we introduce the construction process of 2D VAPG for flexible generation of vortex array, as shown in Fig. 7. Firstly, we get two fork gratings for generating single OAM mode (OAM+1 and OAM+5 in Fig. 7) in x and y direction. The fork gratings are generated with blazed grating and spiral phase distribution of OAM mode, which can only generate one OAM mode at the first diffraction order. In order to control the power of the OAM modes at each different diffraction order flexibly, one need to carefully design the phase profile of the grating.
(1) which is only determined by periodic phase ()  x of the grating. Thus, by controlling the periodic phase ()  x , one can directly control the relative output power of each diffraction order. The designing of the periodic phase distribution ()  x is a minimization problem. To solve the problem, we have proposed pattern search assisted iterative (PSI) algorithm in our previous work 25 , which is a highly effective method for phase design. In this work, the PSI algorithm is also applied to design phase gratings for vortex beam array generation. By using the method described above, we can arbitrarily manipulate the diffraction order and relative power of grating. Figure 2 shows two generated phase profiles of grating in one period and the corresponding output power distribution. The first grating generates 5 diffraction orders with equal power. The second one generates 7 diffraction orders with different relative power. Seen from Fig. 7, one can find that the generated phase grating can successfully control the diffraction order and relative power. The simulated results are nearly the same with the preset output power distribution. In addition, the calculated diffraction efficiencies of the two gratings are 92.8% and 90.3%, which shows favorable performance. After that, the linear phase profile in fork grating pattern is mapped to the corresponding desired phase profile ( Fig. 6(a)) generated by PSI algorithm. Then, we can get two complex phase profiles, which can generate 1D vortex array which desired relative power. At last, we add two phase profiles in x and y direction to generate 2D phase grating for vortex array generation as depicted in Fig. 6. The topologic charge of the OAM modes in the vortex array is determined by the origin spiral phase l and the diffraction order k. In 1D VAPG, the topological charge of the target OAM mode in the array is l×k. In 2D VAPG, the topological charge of OAM modes at each diffraction order in the array is lxkx+lyky, where lx and ly are the origin spiral phase of the x and y direction fork grating, kx and ky are the diffraction order of the target OAM mode in x and y direction, respectively. The relative power of each diffraction order in the vortex array can be expressed as ( ) 2 . Thus, we can generate vortex arrays with controlled OAM mode states and relative power by setting the corresponding parameters (diffraction order kx and ky, original spiral phase of OAM mode lx and ly, and relative power and ) of the grating in x and y directions.

Data Availability
The datasets generated or analysed during the current study are available from the corresponding author on reasonable request.