Multidimensional multiplexing holography based on optical orbital angular momentum lattice multiplexing

Abstract. The use of orbital angular momentum (OAM) as an independent dimension for information encryption has garnered considerable attention. However, the multiplexing capacity of OAM is limited, and there is a need for additional dimensions to enhance storage capabilities. We propose and implement orbital angular momentum lattice (OAML) multiplexed holography. The vortex lattice (VL) beam comprises three adjustable parameters: the rotation angle of the VL, the angle between the wave normal and the z axis, which determines the VL’s dimensions, and the topological charge. Both the rotation angle and the VL’s dimensions serve as supplementary encrypted dimensions, contributing azimuthally and radially, respectively. We investigate the mode selectivity of OAML and focus on the aforementioned parameters. Through experimental validation, we demonstrate the practical feasibility of OAML multiplexed holography across multiple dimensions. This groundbreaking development reveals new possibilities for the advancement of practical information encryption systems.


Introduction
][7][8][9][10][11] This approach is constrained by limited spatial channel availability and significant cross talk.3][14] This technique retains the OAM property and enables selective image reconstruction based on the encoded OAM.][21][22][23][24][25][26][27] Despite these advancements, the capacity for encoded information in OAM multiplexed holography remains limited.Moreover, conventional OAM multiplexed holography lacks an additional degree of freedom to enhance both information security and capacity.A newly proposed approach, MHC-OAM multiplexed holography, introduces new degrees of freedom by multiplexing different angular momentum modes of light beams, enabling the simultaneous transmission of multiple information channels. 28Overall, the use of OAM in holography has the potential to revolutionize information transmission and storage, and ongoing research seeks to further optimize this technology.
In this study, we introduce and demonstrate the effectiveness of orbital angular momentum lattice (OAML) multiplexed holography, an innovative technique that utilizes a vortex lattice (VL) beam that is independently adjustable within three crucial parameters: the rotation angle of the VL, the angle between the wave normal and the z axis (which determines the VL's size), and the topological charge.Notably, the rotation angle and the size of the VL introduce additional encrypted dimensions, specifically in the azimuthal and radial directions.We conduct an extensive investigation into the selectivity of OAML modes based on these parameters, enabling us to achieve multidimensional multiplexed holography.Our experimental results affirm the practicality of OAML multiplexed holography across various dimensions, holding significant promise in the field of information encryption.This methodology unlocks new possibilities for robust optical encryption and a wide range of classical or quantum information applications.

Materials and Methods
The proposed VL beam significantly enhances the encryption capacity of OAM holography.Figure 1 shows a comparison between OAML holography and conventional OAM holography.In the case of the conventional OAM beam [top of Fig. 1(a)], an OAM-preserved hologram is designed to maintain the OAM property of incident OAM beams in each pixel of reconstructed holographic images, allowing OAM to serve as an independent information carrier.Conversely, for the VL beam, the rotation angle of the VL, the angle between the wave normal and the z axis, and the topological charges are configured as independent information carriers.In the upper part of Fig. 1(b), a helical phase plate is overlaid with an OAM-preserved hologram to create an OAM-selective hologram for the conventional OAM beam.This hologram contains spatial-frequency components that carry a helical wavefront.Due to OAM conservation, only a specific incident OAM beam with an inverse helical mode index can be transformed into a Gaussian mode, selectively reconstructing a holographic image encoded in an OAM image channel.Similarly, for the VL beam, when a VL phase is added to an OAM-preserved hologram, an OAM-selective hologram is achieved.It is worth noting that for the OAML-preserved hologram, different hologram images correspond to different VLs with varying rotation angles and sizes in the Fourier plane.The different VLs with different rotation angles correspond to different points on the same circle, while the different VLs with different sizes correspond to various points on concentric rings with different radii.As these different points on the concentric ring or the single circle are orthogonal, the aforementioned VLs are also orthogonal (see Supplementary Note 1 in the Supplementary Material).This hologram likewise contains spatial-frequency components.Due to the conservation of OAM, only a specific incident VL beam, with its topological charge reversed compared to the topological charge at the back focal plane of the lens for the OAM-selective hologram, can be transformed into a Gaussian mode in the Fourier transform plane.The VL phase φ generating the VL beam can be calculated as where k ¼ 2π∕λ, λ is the wavelength set as 532 nm in this paper, ðx; yÞ represents the rectangular coordinate, and α, β, l, and θ present the rotation angle of the VL, the angle between the wave normal and the z axis, topological charge, and azimuth angle, respectively.As the chosen example is the VL, which is the quadruple rotational symmetry, α ranges from 0 to π∕2.β is too large to make the four holographic images overlap, which identifies the encrypted image and cannot achieve the image encryption.β should have a small value and determines the size of the VL.To simplify matters, this paper chooses the square VL as an example, even though the VL can have an arbitrary shape.
The spatial-frequency distribution of the VL beams based on the Fourier integral theorem is expressed as where circðr∕RÞ is the circle function used to describe the circular aperture stop, r and R represent the radial coordinate and the normalization factor of the radial coordinate, respectively, and ðρ; ϕÞ indicate the polar coordinates in the image plane.
In computer-generated holography, a Fourier pair is established between the electric field in the image plane and the holographic plane.As a result, the electric field of the reconstructed image is where E h and E OAM are the complex amplitudes of the hologram and the VL beam, respectively.The operators I and * are the Fourier transform and convolution, respectively.If the sampling array of the target image is correlated with the spatial frequency of the VL-OAM beam, the OAM properties will be preserved in the reconstructed image.

Results and Discussion
The experimental setup for OAML holography is presented in Fig. 3.We employ a laser with a wavelength of 532 nm as the light source.To ensure optimal beam characteristics, we expand and collimate the laser beam using a spatial filter comprising an objective lens, a pinhole, and an additional lens to adjust the incident beam size, aligning it with the phase-only spatial light modulator (SLM, UPOLabs-HDSLM80R Pro, 1920 pixels × 1200 pixels, pixel pitch of 8 μm).Since the SLM is sensitive solely to the horizontally polarized component of the incident beam, we insert a polarizer between the spatial filter and the SLM, generating a horizontally polarized beam.The modulated beam, reflected by the beam splitter, is directed through a CMOS camera (FLIR, GS3-U3-123S6C-C, 4096 pixels × 3000 pixels, pixel pitch of 3.45 μm) responsible for capturing the reconstructed holographic image.In this specific experiment, the system is simplified, as only one SLM is utilized.Rather than directly illuminating the hologram pattern with a VL beam, we superimpose the decoded VL phase distribution onto the hologram.The hologram itself is then illuminated by a planar beam, as shown in Fig. 3(a).During the decryption process, the hologram can be represented as the superposition of the decoded VL phase and the OAML hologram.Therefore, the mathematical representation of the phase-only hologram can be described as where Φ i represents the phase information of each image channel, ψ i−de represents the decoded VL phase distribution, and N represents the number of multiplexing channels.The design principle of the hologram loaded into the SLM is shown in Fig. 3(b).We've thoroughly analyzed the impact of adding another SLM, factoring in misalignment (see Supplementary Note 2 in the Supplementary Material).
Figure 4 provides a visual representation of the schematic diagram that elucidates the concept of l-encrypted OAML multiplexed holography.The experimental setup involves encoding four distinct target images labeled as Arabic numerals "1," "2," "3," and "4" into separate holograms while preserving crucial OAM information.This encoding process utilizes the GS algorithm to achieve optimal results.To accomplish this, we employ four VL phase modes, each characterized by specific parameters, denoted as (l ¼ 1, 11, 21, 31, α ¼ 0.2π∕2, 0.2π∕2, 0.2π∕2, 0.2π∕2, β ¼ 0.001, 0.001, 0.001, 0.001).These parameters enable the creation of corresponding OAM-selective holograms.These individual holograms are then combined to produce a unified OAM multiplexed hologram, as shown in Fig. 4(a).To assess the practical feasibility of l-multiplexed holography, we conducted both numerical simulations and physical experiments, as meticulously presented in Figs.4(b)-4(e).During the experimental phase, the OAML multiplexed hologram associated with a specific "l" key was subjected to various incident VL beams.These beams were characterized by parameters such as (l fαg Ã ¼ −0.2π∕2, fβg Ã ¼ −0.001).Intriguingly, this arrangement resulted in the reconstruction of four distinct images: "1," "2," "3," and "4" at the lens's focal plane for each corresponding case.Consequently, these results conclusively demonstrate the effectiveness of utilizing specific "l" values associated with incident VL beams to achieve the encryption of four discrete images from a single multiplexed hologram.Figures 4(f)-4(i) illustrate the capture intensity distributions of the aforementioned VL beams, respectively.Upon illumination of the multiplexed OAML-preserved hologram by a planar beam, four images manifest simultaneously, appearing indistinguishable from each other, as shown in Fig. 4(j).Let us consider the case of OAM holography, which encodes singular OAM information to produce a single image.In this scenario, we employ a blazed grating to modulate the phase of the holography, allowing us to selectively extract the first-order diffraction through the SLM.Within the experimental framework, the efficiency of both generating and decoding the holography is quantified by the ratio between the energy content of the reconstructed image and that of the background.While the measured efficiency currently stands at 12.95%, it is possible to optimize the experimental setup further to bring it closer to its numerical simulation value of 71.84%.

Conclusion
In this study, we introduced OAML multiplexed holography as an innovative approach to enhance holographic multiplexing capabilities.Our method involves modulating key parameters, including the rotation angle of the square lattice, the angle between the wave normal and the z axis (which determines the size of the square lattice), and the topological charge.By manipulating these parameters, we achieved multidimensional multiplexing within holography.It is worth noting that these parameters are mutually orthogonal, allowing for independent control.Furthermore, we conducted a comprehensive investigation into the selectivity of the OAML mode based on the aforementioned parameters, enabling efficient multiplexing in holography.Experimental verification unequivocally demonstrated the feasibility of OAML multiplexed holography and its potential applications in enhancing information encryption.This significant advancement in multiplexed holography holds great promise for various fields, including optical communication, optical encryption, and 3D display.

Fig. 1
Fig. 1 Schematic diagrams of two types of holograms: (a) an OAM-preserved hologram and (b) an OAM-selective hologram.These holograms are designed to transfer the OAM property from an incident OAM beam to a holographic image and to reconstruct specific OAM channels, respectively.Top: the conventional OAM beam.Bottom: the proposed VL beam.fa 1 g Ã and fβ 1 g Ã are the equivalent values of α 2 and β 1 for the conjugate phase of the encoded phase, respectively.

Fig. 2
Fig. 2 OAML mode selectivity.(a) Design concept for an OAM-preserved hologram and an OAMselective hologram.(b) Mode selectivity of the constant l.(c) Mode selectivity of α.(d) Mode selectivity of β.

Fig. 3
Fig. 3 (a) Schematic diagram of the experimental setup of OAML hologram.(b) The hologram loaded into the SLM consists of two components: the decoded phase and the OAM hologram.