1. Introduction

Encoding information needs a certain physical entity, such as photons or electrons. To increase the information-carrying capacity of photons, researchers have explored and attempted to optimize encoding in time, wavelength, polarization, phase as well as space mode [1,2]. Polarization is a typical degree of freedom of the photons. It is known that the polarization states can be used not only to transform information but also to encode it. In the optical communication system, Polarization-Division Multiplexed (PDM) transmission systems are proposed to increase the channel capacity or spectral efficiency [3–6]. Polarization encoding with orthogonal states has made success in the quantum key distribution system [7,8]. Besides, some meta-surface platforms have been proposed to use polarization as an independent channel to encode information [9]. However, these schemes use orthogonal polarization states to reduce the cross-talk problems in the decoding process. At present, the non-orthogonal polarization states have been used to minimize the polarization multiplexing angle of the PDM system [10–13]. It is noted that the non-orthogonal polarization states are still not well exploited for encoding information.

Recent studies have also shown the growing interest in encoding information using vector beams whose polarization states vary with the beam’s spatial structure [14]. High base vector beam coding and decoding can be realized using directional spatial polarization of the vector beam and has good link transmission performance. [15]. The spatial inseparability and polarization freedom of the vector beams are successfully used to encode 2 bits of information [16]. Moreover, the spatial polarization inhomogeneity of the vector modes can be used for modulus division multiplexing to improve the data transmission rate of wireless optical communication [17]. Recently, it has been shown that structured light with topological properties could potentially be used to encode and transmit information [18]. The topological optical knots formed by the structured polarization states can be generated with meta-surface holograms, such that a robust and high-capacity optical coding scheme is demonstrated [19]. These previous works show that the polarization structures contained by the vector beam have already been an interesting part of the optical information. However, when different vector beams are exploited to represent information, there must be sufficient bases prepared for encoding. Thus a re-configurable optical element, such as the spatial light modulator, is usually indispensable. A single vector beam, which contains an abundance of polarization information, has not yet been well utilized in information encoding. Besides, encoding information with the single vector beam may significantly reduce the dependence on the re-configurable optical elements.

Here, we propose a scheme to realize the non-orthogonal polarization encoding using the single vector beam. To illustrate the scheme, we use the well-known Sagnac interferometer to produce the single vector beams as the input light field. It is noted that the single vector beams can also be generated by using the interesting meta-surface platforms [9,20–22] to improve the stability of the system. Then the vector beams are projected to a specific polarization angle through a wave plate and a polarizer. The sequences of the projection angles of the polarizer implement the non-orthogonal polarization encoding. And the optical patterns projected by the polarizer are used for distinguishing those non-orthogonal polarization states. In decoding, we use the machine learning classification algorithm to identify the polarization states and then reconstruct the encoded information. We demonstrate the feasibility of the 4 bits/symbol non-orthogonal polarization coding scheme with an error rate of $0.06 \%$ for the link transmission of the 64$\times$64 pixels gray image. We further extend our scheme to 8 bits/symbol encoding scheme and realize 64x64 gray image transmission with the same vector beam. Our work requires only a single vector light field to encode and transmit information with non-orthogonal polarization states. By combining the multiple vector beams, our schemes may further improve the encoding capability.

2. Encoding/decoding theory with a single vector beam

We elaborate on the encoding process specifically through the Jones Matrix method. A vector beam can be presented as a superposition of left-hand and right-hand polarized lights with the independent complex amplitude distribution:

Due to the difference in position dependent amplitude $A_1\left (r\right )$, $A_2\left (r\right )$ and phase $\varphi _1\left (r\right )$, $\varphi _2\left (r\right )$, the transverse polarization distribution of the input vector beam will be inhomogeneous, including linear, elliptical and circular polarized states. The transformation of the incident polarized light by a linear polarizer whose fast axis is at an angle $\theta$ to the positive direction of the x-axis can be expressed as:

After going through a Quarter-Wave Plate (QWP) and a linear polarizer, the complex amplitude and Jones vector can be described as follows:

Comparing Eq. (1) and Eq. (3), we can find that two independent complex amplitudes have been projected on a certain polarization angle $\theta$. The angle $\theta$ also serves as a factor to adjust those two complex amplitudes. According to the position related $A_1\left (r\right )$, $A_2\left (r\right )$ and $\varphi _1\left (r\right )$, $\varphi _2\left (r\right )$, the output optical field produces different coherent superpositions at a different position in transverse space, which leads to different intensity patterns. The intensity of the light output from the polarizer can be described as a function associated with the spatial coordinate $r$ and projection angle $\theta$:

Whenever the projection angle of the polarizer $\theta$ changes, such as the value of $\theta$ is $5^\circ,10^\circ$ or $15^\circ$, the spatial distribution of the output light intensity $I(r,\theta )$ will change synchronously, as the example shown in the inset of Fig. 1(a). The output intensity is determined by the projection angle of the polarizer and the transverse spatial polarization distribution of the input vector beam. Thus the non-orthogonal polarization encoding and decoding scheme can be realized.

 

Fig. 1. (a) Experimental sketch of the non-orthogonal polarization encoding and decoding scheme. QWP: Quarter-Wave Plate; P: Linear Polarizer; CCD camera captures the structured optical pattern. (b) Experimental setup for generating a typical vector beam using a Sagnac interferometer. HWP: Half-Wave Plate; PBS: Polarization Beam Splitter; M: Mirror; SLM: Spatial Light Modulator. The background colors in (a) describe the orientation angles calculated by stokes parameters $S_1$ and $S_2$. The color bar beside the polarization distribution indicates that the orientation angle ranges from 0 to $\pi$.

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For experimental convenience, we use two Laguerre-Gauss modes to produce the desired complex optical field based on Eq. (1), and the output optical field can be expressed as:

In Eq. (5), $\alpha$ serves as the intensity distribution factor, which is contributed by an HWP($+\alpha ^\circ$) at the input. $LG_{p_{i}}^{l_{j}}(i,j=1,2)$ represents the complex amplitude $A_1(r)e^{i \varphi _1(r)}$ or $A_2(r)e^{i \varphi _2(r)}$ with Laguerre-Gauss mode, in which $l_j$ and $p_j$ are its azimuthal and radial indices. And based on Eq. (4), the intensity distribution after the linear polarizer in the $\theta$ direction can be described as:

It is noted that the generated vector beam described in Eq. (5) contains an abundance of polarization states which can produce different intensity patterns expressed by Eq. (6). If we choose polarization angles as the carrier of information, such as the polarization angles spaced by $5^\circ$ and start from $5^\circ$ to $80^\circ$, we can get 16 different angles available and each polarization angle corresponds to a certain coding value from 0 to 15 which can be denoted in binary as $2^4-1$. For a normal gray-scale image, each pixel of the image is denoted by the gray value from 0 to 255($2^8-1$). Since the number of the gray value is 256 which contains 8 bits ($2^8$) information and every 4 bits ($2^4$) information can be represented by a certain angle, we can use two polarization angles ($(2^4)^2$) to denote one gray value. We use "4 bits/symbol" to represent the capacity of this encoding scheme [15,23]. Suppose the polarization angles are spaced by $0.7^\circ$ and range from $0^\circ$ to $178.5^\circ$ corresponding to 256 different angles. In this condition, each polarization angle corresponds to a certain coding value from 0 to 255($2^8-1$). Then we can use one of these polarization angles($2^8$) to denote one gray value, so the capacity of the encoding scheme is improved to be "8 bits/symbol". We can use the single vector beam to implement different capacities of the encoding scheme. Thus it holds a promising application with a customized encoding scheme under the same stationary input. Then, for the decoding process, based on Eq. (6), each of the polarization angles is related to one unique intensity pattern($2^4 or 2^8$). We can perform the information reconstruction by recording and discriminating each of the patterns. Benefiting from the recent machine learning technique in classifying the different patterns, we can efficiently convert a series of light intensity patterns into the decoded gray values and thus reconstruct the encoded image. It is noted that machine learning has gradually applied to optical communications, such as the algorithms for decoding and de-multiplexing process [23–26].

3. Experimental setup

Our crucial part of the encoding and decoding scheme is illustrated in Fig. 1(a). An arbitrary vector beam is used as the input of the system. The polarizer is rotated by an electric motor controlled by the computer to generate the corresponding polarization code. The CCD camera is used to capture the decoded optical patterns. The recognition of those intensity patterns is automatically performed by the computer-based Machine Learning (ML) technique. The ML technique works best for the obvious structural differences of the light patterns, and the capacity of the encoding scheme depends on the polarization angles’ step. The small polarization angle difference increases the difficulty of identifying intensity patterns while increasing the encoding capacity. In our experimental test, we build a two-fold Sagnac interferometer [27–30] with an embedded Spatial Light Modulator (SLM), which has been widely used to produce the vector beams, as shown in Fig. 1(b). A 532nm laser beam passes through a polarizer and a half-wave plate to generate an adjustable light intensity ratio between the horizontal and vertical components. Horizontally polarized light in the Sagnac interferometer can directly pass through PBS, while vertically polarized light will be reflected by PBS. For both polarization components to be modulated by SLM, the vertically polarized light needs to be transformed into a horizontally polarized one through an HWP at $45^\circ$. Horizontally polarized and vertically polarized light is reflected in two different regions of the SLM. Thus the two components carry different independent complex amplitudes. After these two components pass through a QWP with $45^\circ$, the desired vector beam is generated.

4. Experimental results

Firstly, to verify the performance of the setup for generating a desired vector beam, we choose $l_1=2,p_1=0,l_2=-3,p_2=0$ based on Eq. (5) as an example and measure all of the Stokes components using the methods in Ref. [31]. Then we can reconstruct the polarization distribution of the output light field from the Sagnac interferometer shown in Fig. 1(b). It can be seen that our experimentally reconstructed polarization structure distribution shown in Fig. 2(b) is consistent with the simulation results shown in Fig. 2(a). Thus it can clearly show that we have produced the desired vector beam with good performance.

Then, to demonstrate our proposed encoding and decoding scheme, we use the generated vector beam to perform two typical coding schemes: 4 bits/symbol and 8 bits/symbol. We choose the polarization angles from $5^\circ$ to $80^\circ$ and take an intensity image every $5^\circ$. The theoretical simulation of each intensity pattern is shown in Fig. 2(c), while the experimental results are shown in Fig. 2(d). Although the intensity modulation imposed on SLM is not ideally linear, we can also trust the generated vector beam for that each interference pattern of the experiment well matches the simulated one. The different polarization angles correspond to different intensity patterns. So we can use the 16 non-orthogonal polarization states to represent the hexadecimal numbers and then perform the encoding process. For the decoding part of the transmission system, we only need to distinguish the transmitted 16 intensity patterns. Here, we choose an effective method–Convolutional Neural Network (CNN) with a 50-layer network structure to decode these intensity patterns. Since the convolution layers in CNN can gradually extract features from input intensity patterns and change the dimension for the output classification, it can be conveniently applied to subtle differential image identification. However, it is noted that various other Neural Network methods that can classify the images are also suitable for usage here. We record 16 videos under 16 different polarization angles, and each video under the same polarization angle contains 100 frames for the robustness against environmental turbulence. These 100 frames under the same polarization angle provide enough features of a certain intensity pattern under environmental turbulence. A total of 1600 images obtained from these videos provide enough training data for the 50-layer CNN to decode the 16 intensity patterns. And $80\%$ of these images are used as training sets, $10\%$ are used as test sets, and the last $10\%$ are used as verification sets to avoid CNN over-fitting. The training results show that 16 intensity patterns can be identified with $100\%$ accuracy.

 

Fig. 2. Comparison between theory and experiment. (a) Polarization structure of the theoretical vector beam. (b) Polarization reconstruction of the experimentally generated vector beam. (c) Encoding basis in theory. (d) Encoding basis measured in the experiment. (e) Part of the theoretical intensity patterns from $70^\circ$ to $80.5^\circ$. (f) Part of the experimentally measured intensity patterns from $70^\circ$ to $80.5^\circ$.

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To further show the real information encoding and decoding with those non-orthogonal polarization states, we transmit a 64x64 gray-scale image. The picture is converted into 8192 (64x64x2) symbols, and every two symbols encode one gray value of the pixel in the initial gray image. For example, when the gray value is 93 (a decimal number), it can be represented by the hexadecimal numbers 5 and 13 ($5*16+13=93$). And these two hexadecimal numbers 5 and 13 are mapped to the corresponding polarization angles $30^\circ$ and $70^\circ$ respectively. So the capacity of the encoding scheme is 4 bits/symbol. Then the related intensity patterns at the receiver can be distinguished by the CNN and decoded into the gray value of pixels. The whole process can be seen in the top line of the transmission diagram in Fig. 3(a). By comparing the sending and the receiving polarization angles, the cross-talk matrix of the system is obtained in the left of Fig. 3(b). The transmission accuracy of the image is $99.94\%$ and the bit error rate is only $0.06\%$, which show the excellent performance of the system. To expand the information capacity of the system, we also try to carry out an 8 bits/symbol encoding scheme. We take the polarization projection angles spaced by $0.7^\circ$, and a total of 256 angles associated with 256 intensity patterns are generated. Then, followed by a similar principle, a total of 264,760 experimental images obtained from the 256 videos for the 256 angles are taken as the CNN data set. It is noted that more than 1000 frames are collected for each polarization angle, and these frames are sampled from the corresponding video at equal intervals. The ’8:1:1’ data proportion of training set, test set, and verification set gives us limited accuracy. Considering the inherent danger of over-fitting, we tried other proportions between training data set and verification data set to improve the generalization ability of our net [32]. We find the proportion ’14:1:5’ shows better identification results in our scheme. It is noted that the proportion used here is suitable for experimental transmission analysis but is not necessarily the best one for real-world scenarios. Typically, the size of the validation data set is around 1/3 of the training data set because an insufficient training data set leads to the under-fitting problem, while an insufficient verification set may risk over-fitting [32]. Due to the increase in encoding capacity, the total number of the transmitted symbols is reduced by half, as shown in the bottom line of the transmission diagram in Fig. 3(a). The gray values (such as $27, 93, 211$) of the different pixels can be presented by a single polarization angle. According to the reconstructed gray image and cross-talk matrix in Fig. 3(c), we find that the image transmission accuracy is about $65.58\%$ and the bit error rate is $34.42\%$ under 256 intensity patterns.

 

Fig. 3. (a) Encoding and decoding process of 4bits/symbol(top) and 8bits/symbol(bottom) schemes for a 64x64 gray image. (b) Cross matrix between Sent and Received patterns under the basis of 16 intensity patterns. (c) Cross matrix between Sent and Received patterns under the basis of 256 intensity patterns. (d) Local accuracy of different areas of the transmitted image.

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For further analyzing the 8 bits/symbol scheme, we divide the transmitted image into 16 different areas and calculate the accuracy separately, as shown in Fig. 3(d). Since there is a similar line in the left part of the Lena image, which corresponds to slight variations in gray values, one can see that the transmitted symbols in the left column tend to be more easily mistaken. Compared with the 4 bits/symbol scheme, which uses two patterns to encode one pixel’s gray value, the 8 bits/symbol scheme only uses one pattern to encode. So the error increased seriously due to the reduced identification accuracy of the similar pixel’s gray value. Besides, since the pixels of the initial gray image are encoded column by column, the other parts of the errors may be caused by the laser power fluctuation or the environmental interference experimentally. If the input power is controlled within a certain range by the optical isolator and the free space interferometer is replaced by an integrated polarization element, such as the meta-surface platforms [20–22], we expect that our 8 bits/symbol coding scheme will achieve higher accuracy.

5. Conclusion

In summary, we have exploited a single vector beam to realize the information encoding and decoding with the non-orthogonal polarization states. The experimental results show that our non-orthogonal polarization coding scheme is reliable for optical communication. Besides, one can achieve different capacities of encoding bit by changing the step of the polarization angle with the same single vector beam. Thus it holds a promising application with a customized encoding scheme under the same stationary input. If the input vector beams are generated by an integrated polarization element, such as the meta-surface platforms [9,20–22], we expect that our scheme may be more robust for environmental influence and can extend to high capacity with highly accurate information transmission system and even make the system more integrated. If considering the singularity of the vector beams, our schemes may further improve the encoding capability seriously [33]. Besides, it is also noticed that ML is not the only approach to decode the information reliably. Many other techniques can also be used for decoding. For example, recent studies have shown that polarization can be classified at different diffraction angles assisted by meta-surface [34,35]. Our decoding process efficiency can be further improved by using the combined way. Overall, our work shown here requires only a single vector beam to encode and transmit the massive information, and considering the scalability of the system, the scheme holds a promising application in optical communication systems.