bstract

The orbital angular momentum (OAM) multiplexing holography has the advantages of large information capacity and high security, and has important application value in holographic storage, optical encryption, and optical computing. However, as the number of multiplexing channels increases, this technology suffers from deterioration in image quality, which limits its application scope. This article proposes an innovative design that introduces an optical diffractive neural network (ODNN) into OAM multiplexing holography, establishes a scientific image quality evaluation function, applies an end-to-end optimization method, and designs OAM multiplexing holograms in parallel, significantly improving the image quality of OAM holography. The design results show that compared to classical methods, the ODNN method proposed in this paper has improved diffraction efficiency and signal-to-noise ratio by 29% and 19%, respectively, and reduced mean square error and variance by 10% and 43%, respectively. Moreover, high-quality multi-channel OAM multiplexing holography has been achieved through experiments. The design method proposed in this article provides an efficient and practical way for future OAM multiplexing holographic technology to further enhance information capacity and improve security.

© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Orbital angular momentum (OAM) multiplexing holography is an emerging multiplexing holographic method. This method uses the OAM of light as an information carrier and implements it as an information channel [1]. Under the illumination of vortex beams with different topological charges, the target image of the corresponding channel is reconstructed. With the development of related research, OAM multiplexing holography has shown great potential for applications in high-security optical encryption [2–6], large capacity holographic information storage [7–10], and other fields.

Ren et al. [11] achieved OAM multiplexing holography for the first time on a phase metasurface. Subsequently, the same research group used a complex amplitude metasurface to simultaneously regulate amplitude and phase, reducing crosstalk between reconstructed images of OAM multiplexing holography [12]. Recently, Zeng et al. [13] used stacked Laguerre Gaussian beams as incident light to achieve OAM multiplexing holography. He et al. [14] proposed the idea of multiplexed coherent pixel for metasurfaces, which achieved OAM and wavelength multiplexing control based on a single-layer metasurface. Pan et al. [15] proposed the concept of metasurface vector holography based on cylindrical vector beams, whose infinite polarization order and unique polarization distribution can be used to improve information storage capacity. Meanwhile, to increase channel capacity, various improvement schemes for multi-dimensional multiplexing have been proposed, including combining OAM with other physical degrees of freedom such as wavelength and polarization [5,7,16–21], and utilizing OAM beams with other structural features [22–27]. The existing research has effectively improved the information capacity of OAM multiplexing holography. However, the design method of the above research obtains OAM multiplexing holograms by stacking multiple holograms with OAM selectivity in complex amplitude, which cannot simultaneously evaluate the image quality of multiple channels, resulting in a decrease in the image quality of holograms with an increase in the number of stacked channels, such as increased crosstalk and decreased diffraction efficiency. To achieve high-quality OAM multiplexing holography, new design methods need to be explored. Huang et al. have introduced the ODNN method into OAM multiplexing holography, achieving excellent deep multiplexing functionality [28]. Therefore, we improved the design method of OAM multiplexing holography and applied it to solve the image quality problem of single-layer OAM multiplexing holography and achieve high image quality.

This article proposes a novel design method based on ODNN [29–36] for OAM multiplexing holography. We have designed an evaluation function consisting of mean square error (MSE), variance, diffraction efficiency, and signal-to-noise ratio (SNR) to optimize the design of OAM multiplexing holograms [37–39], which can directly evaluate the image quality of reconstructed images from multiple channels simultaneously. The design results show that compared to classical methods [1], the reconstruction results of the method proposed in this paper have an average MSE reduction of 10%, an average variance reduction of 43%, an average diffraction efficiency improvement of 29%, and an average SNR improvement of 19%. This method significantly improves the image quality of reconstructed images. In addition, this article also constructed a tilted reflection optical path experimental setup using two spatial light modulators (SLM), successfully achieving holographic reconstruction of multiple target images. The experimental results are consistent with the simulation results. This work improves the imaging quality of OAM multiplexing holography and further enhances its application value in high-capacity holographic information storage and high-security optical information encryption.

2. Theoretical deduction and design method

2.1 Principle and design of ODNN based OAM multiplexing holography

The physical process of OAM multiplexing holography based on ODNN is shown in the Fig. 1. The overall structure is constructed through an input layer, a hidden layer, and an output layer. The incident Gaussian beam �� first illuminates the input layer. In the input layer, spiral phase information is encoded into the incident Gaussian beam to form a vortex beam ��⋅exp⁡(���), where � represents the imaginary unit, � represents the topological charge and � represents the azimuth angle [40]. The vortex beam propagates in free space and reaches the hidden layer. In the hidden layer, the OAM multiplexing hologram optimized by the ODNN method modulates the phase of the vortex beam, and the target image information corresponding to the topological charge � in the hologram is decoded. Subsequently, the modulated beam propagates through free space again and finally obtains the reconstructed target image corresponding to the current vortex beam in the output layer. During the iterative optimization process, multiple target images corresponding to different � are encoded into the same OAM multiplexing hologram. Therefore, the reconstructed target image of the output layer changes with the change of � in the input layer.

Fig. 1. Schematic diagram of the physical process of OAM multiplexing holography based on ODNN. The optimized OAM multiplexing hologram is used to reconstruct the corresponding target image under different vortex light irradiation.

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The physical process shown in Fig. 1 can be realized as the forward propagation process of ODNN, where the beam is represented in the form of a complex amplitude light field, and the free space propagation module is implemented using the angular spectrum method [29–32]. The neural network structure of OAM multiplexing holography based on ODNN is shown in Fig. 2. In the input layer and hidden layer, each point in the complex amplitude light field corresponds to a neuron, and each neuron independently modulates the phase of the light field. Therefore, the modulation process of the light field by the input layer and hidden layer can be expressed as:

(1)����(�,�)=���(�,�)exp⁡[��(�,�)]where ���(�,�) and ����(�,�) represents the complex amplitude of the input and output light fields, �(�,�) represents the phase delay caused by the neurons in the corresponding layers. Specifically, for the input layer and hidden layer, �(�,�) represents the spiral phase delay and OAM multiplexing hologram phase delay, respectively. According to the Huygens-Fresnel principle, every point in a complex amplitude light field modulated by neurons can be regarded as a secondary source of a wave after phase modulation. Therefore, neurons between layers are connected through secondary waves. According to the theory of angular spectrum diffraction, the free space propagation between the �th layer and the (�+1)th layer can be expressed as:

(2)����+1(�,�)=�−1{�{�����(�,�)}⋅�(�,�)}where �����(�,�) and ����+1(�,�) represents the complex amplitude of the output light field in the �th layer and the input light field in the (�+1)th layer, �(�,�) is the transfer function of angular spectrum diffraction, and � represents the Fourier transform. Finally, in the output layer, one can calculate the intensity of the complex amplitude light field and use it as the output result of the neural network. During the training phase, the evaluation function value is calculated based on the output results of the neural network and the expected target images. The neural network structure and the phase modulation values of neurons are iteratively optimized through the error backpropagation algorithm [41] to obtain the OAM multiplexing hologram.

Fig. 2. A neural network structure for OAM multiplexing holography based on ODNN. The physical process of OAM multiplexing holography is implemented as the forward propagation process of ODNN. During the training phase, the spiral phase and sampled target image are used as inputs and evaluation function labels for the neural network, respectively. Based on the output results of the network, the error backpropagation algorithm is used to iteratively optimize the neural network structure and the phase modulation values of neurons. Finally, an OAM multiplexing hologram that can reconstruct corresponding target images under different inputs is obtained at the hidden layer.

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2.2 Evaluation and optimization of ODNN based OAM multiplexing holography

To improve the comprehensive image quality of OAM multiplexing holography, we defined an evaluation function (i.e., the loss function in deep learning) to assess the output results of the neural network during forward propagation. The smaller the evaluation function value (i.e. loss value), the better the current network optimization output result, that is, the higher the image quality. We use four indicators: mean square error (MSE), variance, diffraction efficiency, and signal-to-noise ratio (SNR), to construct the evaluation function [37–39]. Mean square error measures the similarity between the output image and the target image, with smaller values indicating higher similarity. Variance measures the uniformity of the output image, with smaller values indicating better uniformity. Diffraction efficiency measures the energy utilization efficiency of the output image, with higher values indicating higher utilization efficiency. Signal-to-noise ratio measures the effective signal strength in the output image, with higher values indicating a higher proportion of effective signals. Taking into account all the four indicators, the complete evaluation function (�) can be expressed as:

(3)�(�,�)=�⋅���(�,�)+�⋅���2(�)−�⋅log⁡[�(�)]−�⋅���(�),

(4)���(�,�)=1����∑�∈����[�(�)−�(�)]2,

(5)���2(�)=1���∑�∈���[�(�)−���(�)]2,

(6)���(�)=1���∑�∈����(�),���(�)=1���∑�∈����(�),

(7)�(�)=∑�∈����(�)∑�∈�����(�),

(8)���(�)=���(�)���(�)

Among them, � and � represent the normalized light intensity distribution images of ODNN output and target, respectively. �(�) and �(�) represent the normalized light intensity values of the output image and target image at pixel point �, respectively. Due to the requirement of OAM multiplexing holography to perform discrete sampling of the target image at certain intervals, the actual target image consists of two parts: pixels containing target information (���) and background pixels without target information (���). Therefore, ����, ��� and ��� represents the set of all pixels, pixels containing target information, and background pixels in the image, ����, ��� and ��� represents the number of pixels in the corresponding set, with ����=���∪���, ���∩���=∅, ����=���+���.

In Eq. (3), ���(�,�) represents the mean square error between the output image and the target image, as shown in Eq. (4). ���2(�) represents the variance of pixels containing target information in the output image, as shown in Eq. (5). ���(�) and ���(�) respectively represent the mean values of pixels containing target information and background pixels in the output image, as shown in Eq. (6). �(�) represents the diffraction efficiency of the output image, as shown in Eq. (7). ���(�) represents the signal-to-noise ratio of the output image, as shown in Eq. (8). The coefficients �,�,�,� respectively represent the weight factors of the four indicators, with a value range of [0,1] and satisfying �+�+�+�=1. By adjusting the weight factors, it is possible to flexibly adapt to different inputs and target images, while controlling the emphasis of the optimization process on different indicators. In addition, �(�) and ���(�) take negative logarithms and opposite numbers, respectively, to make the direction of evaluation function value decrease consistent with the optimization direction. Based on this evaluation function definition, the optimization problem of improving the imaging quality of output images converts to the problem of minimizing the evaluation function value.

To train the required OAM multiplexing holograms, we used the error back-propagation algorithm and the gradient descent algorithm. The gradient of the evaluation function with respect to the OAM multiplexing hologram needs to be calculated, which is then used to update the OAM multiplexing hologram during each iteration process. The specific process can be expressed as:

(9)��+1=��−�⋅��(��)���where �� represents the current OAM multiplexing hologram, which is the trainable weight of the neural network, ��+1 represents the updated OAM multiplexing hologram, � represents the learning rate constant. The learning rate is used as a hyper-parameter, which takes an initial value of 1.00 throughout this work, during the training phase to control the magnitude of a single weight update. We also apply a primitive adaptive learning rate strategy, which will be stated in detail later.

The complete optimization process of OAM multiplexing holography based on ODNN is shown in Fig. 3. Firstly, initialize the OAM multiplexing hologram � as the trainable weight of the neural network. Then, in each iteration, select an input spiral phase map and its corresponding target image, calculate the forward propagation and evaluation function, and optimize the trainable weights. According to the gradient descent algorithm, �, as the independent variable of the evaluation function, continuously moves towards the negative gradient direction during iteration, resulting in a continuous decrease in the evaluation function value and achieving the optimization goal. Iterative optimization is repeated on the training dataset, and the average evaluation function value of each epoch is used as the basis for judging whether the algorithm converges. To further balance the convergence speed and the accuracy, we have introduced an adaptive strategy of the learning rate, which is, if the evaluation function value has not decreased during the last 4 epochs, the learning rate � should drop to 1/10 of the previous value. Finally, when the evaluation function value no longer decreases during the last 8 epochs, the training ends, and the optimal OAM multiplexing hologram is obtained.

Fig. 3. The overall design flowchart of OAM multiplexing holography based on ODNN, where the physical process in Fig. 1 is implemented as the forward propagation process of the neural network. In one iteration, a spiral phase map and a target image are input into the ODNN, and the OAM multiplexing hologram is updated through forward propagation, evaluation function calculation, and gradient descent. A training epoch consists of multiple iterations, and each spiral phase map and target image in the training dataset is input into ODNN to iteratively optimize the OAM multiplexing hologram. The complete training process consists of multiple epochs, and the convergence of the algorithm is determined by monitoring the average evaluation function value of each epoch, that is, whether the imaging quality no longer improves.

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Compared to the classical OAM multiplexing holography with our work based on ODNN, one can figure out that the ODNN method can achieve better performance in imaging quality. The optimization process of the classical method is divided into two steps [1], while the optimization process of our new ODNN method is simplified to end-to-end, requiring only one step [29]. In the classical method, multiple target images are first subjected to discrete sampling, iterative optimization, and spiral phase superposition operations to obtain the corresponding OAM selective hologram for each target image, which can be represented as ��(�=1,2,…,�). Afterward, all OAM selective holograms are overlaid with complex amplitudes to obtain a single OAM multiplexing hologram, which can be represented as ����=arg⁡[∑�=1�exp⁡(���)]. Therefore, the classical method cannot evaluate and guide the optimization process on real imaging results during the optimization process. In the ODNN method, the entire optimization process is end-to-end, with a single OAM multiplexing hologram iteratively optimized on all input and target images, obtaining phased and realistic imaging results in each iteration step and conducting imaging quality evaluation and hologram optimization.

3. Simulation results and analysis

3.1 Simulation of ODNN based OAM multiplexing holography

We first designed OAM multiplexing holography based on ODNN for 36 target images and conducted numerical simulations. Then, taking different numbers of target images as examples, we applied ODNN and classical methods to optimize the design of OAM multiplexing holograms, and analyzed and discussed the results.

In the simulation, the target image consists of 10 digits and 26 uppercase letters. Each target image corresponds to an input vortex beam, and the topological charge � of the vortex beam starts from 3 and increases sequentially at intervals of 5, alternating positively and negatively (i.e. �=3,−3,8,−8,…,88,−88). The wavelength of the incident Gaussian beam is 632.8 nm; The matrix of the complex amplitude light field is of size 724 × 724; Pixel size is 8 × 8 �m; The diffraction distance from OAM multiplexing hologram to the image plane is 20 cm. After the optimization process shown in Fig. 3, the target image and its corresponding vortex beam information are obtained and then encoded into the OAM multiplexing hologram. Subsequently, we fix the OAM multiplexing hologram and perform forward propagation under different input vortex beams to obtain reconstructed target images, which are the numerical simulation results.

The normalized intensity images of 36 targets reconstructed from OAM multiplexing hologram based on ODNN under 36 different vortex beams are shown in Fig. 4(a). The target image can only be reconstructed under the illumination of its corresponding vortex beam. The enlarged normalized intensity image of the reconstructed number “0” under the illumination of a vortex beam with a topological charge of �=3 is shown in Fig. 4(b). The simulation results show that the OAM multiplexing holography based on ODNN can reconstruct each target image in 36 multiplexing channels.

Fig. 4. Simulation results of OAM multiplexing holography based on ODNN. (a) Normalized light intensity images of 36 targets reconstructed by holography under 36 different vortex beams, with the number in the lower right corner representing the corresponding topological charge � of the input vortex beam. (b) The normalized light intensity image of the digit “0” in holographic reconstruction, with the corresponding input spiral phase map in the lower right corner, and the inset at the bottom shows an enlarged image of a 29 × 29 pixel area around a single light spot in the reconstructed image, along with its intensity profile.

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3.2 Comparative analysis with classical OAM multiplexing holography

To compare the imaging quality of the ODNN method and the classical method, we replicated the classical method [1] and trained OAM multiplexing holography based on ODNN and classical OAM multiplexing holography under different target image quantities, and calculated the imaging quality indicators. All simulations use the same parameters, inputs, and target images. Based on the simulation results, four imaging quality indicators MSE, ���2, � and SNR were calculated. The comparison results are shown in Fig. 5, where the line graph represents the indicator values of the ODNN method and the classical method, and the bar graph represents the performance improvement percentage of the ODNN method compared to the classical method.

Fig. 5. Comparison of imaging quality between the ODNN method and the classical method. (a-d) represent the comparison results of MSE, ���2, � and SNR. Each indicator was simulated and calculated for 2-36 target quantities. The red and blue lines represent the indicator values of the classical method and the ODNN method, respectively. The yellow bar chart represents the percentage of performance improvement of the ODNN method compared to the classical method. The yellow dashed line represents the average performance improvement percentage across all target quantities.

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According to the information shown in Fig. 5, the ODNN method shows better results than the classical method on each metric, with MSE and ���2 decreased by an average of 10% and 43%, respectively, while � and SNR increased by an average of 29% and 19%, respectively. These indicators indicate that the holographic reconstruction results have higher similarity with the target image, more uniform energy distribution, higher energy utilization efficiency, and clearer information expression. It is worth mentioning that, as shown in the bar graph, for the MSE, �, and SNR metrics, as the number of target images increases, the imaging quality improvement of the ODNN method shows an upward trend compared to the classical method, indicating that ODNN method has obvious advantages in the field of multiplexing holography.

4. Experimental results and analysis

Taking 6 digital target images as examples, we designed an OAM multiplexing holographic system based on ODNN and constructed an experimental setup using two reflective phase SLMs, as shown in Fig. 6. The light source uses a helium-neon laser (Thorlabs HNLS008L-EC) with a wavelength of 632.8 nm. The Gaussian beam emitted by the laser is amplified into a high-quality linearly polarized Gaussian beam after passing through an attenuator, polarizer, lenses, pinhole filter, and aperture. After two SLMs (Holoeye PLUTO-2.1-VIS-130 and CAS MICROSTAR FSLM-2K70-P02), the final imaging result is received by an industrial array camera (HIKROBOT MV-CH120-10UC). The resolutions of two SLMs are both 1920 × 1080 with pixel size of 8 × 8 �m. The resolution of the industrial camera is 2592 × 2048 with pixel size of 4.8 × 4.8 �m. The distance between two SLMs, as well as the distance between the second SLM and the industrial camera, is 20 cm.

Fig. 6. Schematic diagram of the experimental setup for OAM multiplexing holography based on ODNN. The spiral phase map and OAM multiplexing hologram are loaded into SLM1 and SLM2, respectively. After filtering and beam expansion, the coherent light is modulated into a vortex beam by SLM1, which is irradiated onto SLM2 to decode the corresponding target information in the OAM multiplexing hologram. Finally, the target image is reconstructed at the camera.

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Considering the alignment error of the experimental optical path, we quantitatively analyzed the influence of the position of the vortex beam incident on the second SLM on the diffraction efficiency. In the optical path system of OAM multiplexing holography based on ODNN, the translation misalignment of the �-axis and �-axis between two SLMs, as well as the translation misalignment of the �-axis between the second SLM and the industrial camera, can have a significant impact on the imaging quality. When the offset reaches 1 mm along any coordinate axis, the diffraction efficiency of holographic reconstruction decreases to below 50% of its original value. Therefore, we used a 6-axis adjustment bracket (including 3-axis translation and 3-axis rotation) to fix these SLMs in the experiment, to achieve comprehensive micrometer-level control capability. In addition, by independently loading phase plates capable of holographic reconstruction of cross patterns on two SLMs and observing the positional relationship between the cross patterns, we were able to control the error within 8 pixels, that is, within 64 �m, demonstrating the excellent diffraction efficiency stability of our experimental system.

In the experiment, the spiral phase map was loaded onto SLM1 to modulate the incident Gaussian beam into a vortex beam. The OAM multiplexing hologram obtained through training optimization is loaded onto SLM2, and under the illumination of a vortex beam, the corresponding target image information is decoded. By switching the spiral phase maps with different topological charges on SLM1, different reconstructed target images can be received at the industrial camera, achieving multiplexing holography of six target images. The experimental results are shown in Fig. 7, where the first row represents the spiral phase distribution of OAM generation when input topological charges �=3,−3,8,−8,13,−13, respectively. The second and third rows represent the normalized intensity images of the simulated and experimental results relative to the OAM mode in the first row, respectively. One can observe perfect imaging of the decoded numbers as displayed. These results indicate that OAM multiplexing holography based on ODNN can accurately decode the corresponding target images under different input vortex beams, and there is a clear distinction between pixels containing target information and background pixels. In terms of specific imaging quality indicators, the average values of � and SNR in the simulation results are 14.8% and 301, respectively, while the average values of � and SNR in the experimental results are 11.5% and 267, respectively. The imaging quality indicators of the experimental and simulation results are consistent, indicating that designing an evaluation function to improve imaging quality and using the ODNN method for end-to-end optimization can significantly improve the imaging quality of OAM multiplexing holography. Compared to simulation, the decrease in image quality of experimental results may be caused by alignment errors and other error sources during the experimental process. Although we were unable to conduct experiments on the classical method due to limitations in experimental conditions, we believe that the performance improvement demonstrated by the ODNN method in the simulation of various indicators (see Fig. 5) and the experimental verification of the ODNN method are sufficient to demonstrate its advantages and value.

Fig. 7. Experimental results of OAM multiplexing holography based on ODNN. The first row represents the spiral phase map of input �=3,−3,8,−8,13,−13, the second row represents the normalized light intensity image of the corresponding simulated reconstructed target, and the third row represents the normalized light intensity image of the reconstructed target obtained in the experiment. The values in the upper left and lower right of the simulation and experimental results represent � (%) and SNR (in number), respectively.

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5. Conclusion

This article proposes a design method for OAM multiplexing holography based on ODNN, which significantly improves the imaging quality of OAM multiplexing holography. This article focuses on the physical process of OAM multiplexing imaging, constructs a new ODNN model, designs an evaluation function consisting of four image quality indicators, and optimizes the OAM multiplexing hologram end-to-end for different incident OAM modes and their corresponding target images, obtaining the best OAM multiplexing imaging results. The simulation results show that compared with the classical method, MSE and variance of this method have been reduced by an average of 10% and 43%, respectively, while the diffraction efficiency and SNR have been improved by an average of 29% and 19%, respectively. We built an experimental setup and successfully achieved OAM multiplexing holography of digital target images based on ODNN. The experimental results were consistent with the simulation results, proving the feasibility of this method. This method is expected to be combined with technologies such as wavelength OAM multiplexing and polarization OAM multiplexing to further enhance the information capacity and imaging quality of OAM multiplexing. Our work implies great value in promoting the development of OAM multiplexing holography in the fields of high-security optical information encryption and high-capacity holographic data storage.

Funding

National Natural Science Foundation of China (61275167); Shenzhen Higher Institution Stability Support Plan (20200812103045003); LingChuang Research Project of China National Nuclear Corporation; Shenzhen Fundamental Research (JCYJ20170817101827765, JCYJ20180305125430954); Basic and Applied Basic Research Foundation of Guangdong Province (2023A1515010168).

Disclosures

The authors declare no conflicts of interest.

Data availability

The data that support the findings of this study are available from the authors on reasonable request, see author contributions for specific data sets.

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