As an ideal solution to enhance information storage and transmission capacity, multiplexing technology covers various methods such as angle multiplexing [1–3], shift multiplexing [4–6], orbital angular momentum multiplexing [7–9], and polarization multiplexing [10–21]. In the selective information storage and information transmission, this technology holds a pivotal role. By consolidating multiple signals for transmission and storage within a single physical channel, it significantly enhances the overall system efficiency. However, as the technology is applied more extensively, challenges such as multiplexing counts and image crosstalk have gradually emerged. The multiplexing count, representing the number of signals that can be concurrently transmitted or stored within the same physical channel, is directly related to the information-carrying capacity of the system. Undeniably, the augmentation in multiplexing counts widens the ability to process information for the system, a crucial aspect in the contemporary information-saturated society. However, as the multiplexing counts continue to rise, the phenomenon of image crosstalk has also become increasingly severe. Image crosstalk refers to the degradation of image quality at the receiving end due to interference among different signals during transmission, potentially resulting in unrecognizable images. This crosstalk may originate from various factors, including closely aligned signal frequencies, minor phase discrepancies, and similar polarization states. To address these challenges, the methodology leveraging the concept of orthogonality has been proposed [4,22,23].

Orthogonality refers to the mutual independence and non-interference of two or more signals on a specific dimension. In multiplexing technology, when signals exhibit orthogonality in a given dimension, they do not interfere with each other during transmission, substantially improving the multiplexing capability and reducing image crosstalk of the system. In recent years, various orthogonal multiplexing methods have been proposed. Among them, research utilizing the orthogonality of amplitude and phase to mitigate crosstalk has yielded noteworthy outcomes [4,24–28]. By precisely controlling the amplitude and phase of each signal, it can guarantee their mutual non-interference during transmission, thereby achieving high-quality image transmission. Additionally, the utilization of the orthogonal characteristics of orbital angular momentum (OAM) for image multiplexing has garnered significant research attention [9,29,30]. OAM, an inherent attribute of light waves, exhibits orthogonality in space among beams of different orders, enabling multiplexing without crosstalk. This approach holds immense potential for applications in optical data storage, 3D displays, artificial neural networks, and all-optical machine learning.

Meanwhile, polarization, as another crucial attribute of light waves, necessitates research into its utilization for multiplexing. Recently, many studies of employing non-orthogonal polarization multiplexing have been proposed to overcome the limitations of polarization multiplexing [31,32], with negligible crosstalk levels. This has opened new horizons in the field of non-orthogonal polarization multiplexing. However, challenges have been encountered in exploring orthogonality within this polarization dimension. The complexities in detecting, identifying, and recording polarization states hinder its application in image multiplexing technology. However, the introduction of polarization holography, which belongs to the category of volume holography, offers a viable solution to address the challenges of identification and recording. Additionally, the number of polarization orthogonal states is constrained currently, limiting most polarization-based multiplexing techniques to a multiplexing count of two [33]. Despite advancements in polarization multiplexing research [11], achieving a multiplexing count of four still necessitates the use of indirect methods for image separation. Consequently, the capability of polarization in multiplexing is limited to a certain extent. Furthermore, a method for achieving multi-channel vectorial holography has been proposed [34], which realizes multi-channel polarization multiplexing holography on metasufaces by precisely manipulating desired combinations of input/output polarization states. This approach increases the multiplexing count and enhances the stability and reliability of the system. When coupled with orthogonal matrices to generate arbitrary polarization orthogonal combinations, the multiplexing capability can be substantially increased. Hence, delving into polarization orthogonal matrices that encompass arbitrary polarization numbers holds immense significance.

In this paper, we demonstrate the existence of optical polarized orthogonal matrices (OPOMs), achieved by using polarization unit vectors as elements to construct the non-square matrix representation of OPOM. The implementation of OPOM can be seen as a result of superposition. By introducing polarization holography as a means of achieving superposition, we show its potential for obtaining dynamic displays and high-security applications. Leveraging the benefits of a low-cost, easily prepared, and straightforwardly recorded medium, we explore polarization holography with abilities in amplitude, phase, and polarization recording for the recording and selective reconstruction of polarization multi-channels. Our research indicates that, despite sharing the same polarization state, multiple images can be individually manipulated through orthogonal polarization combinations (PCs) within distinct polarization channels. The reconstructed images can be switched with negligible crosstalk by selecting the desired combination of input polarization states. Our demonstrated multiplexing method holds promise for advancing applications in dynamic holographic displays, switchable optical devices, data storage, and optical encryption/anti-counterfeiting, while maintaining compatibility with most optical transmission systems. We believe that the OPOM proposal holds vast potential in diverse fields that employ polarization modulation capabilities. This advancement promises to resolve intricate issues and expand perspectives in the dimension of polarization orthogonality.

We investigate a 4×8 OPOM scheme for achieving eight-channel multiplexing using polarization-sensitive birefringent media, as illustrated in Fig. 1. By leveraging polarization-sensitive holographic technology, we demonstrate the spatial multiplexing of eight distinct pieces of information at a single position using the 4×8 OPOM. Based on the polarization-sensitive holographic properties, PCs with OPOM characteristics are utilized as a modulation mechanism to sequentially record eight different images into the media, thus preserving the information within it. During information retrieval, PCs with OPOM characteristics served as a key to illuminate the media. This PC interacts with the holograms, and after interference cancellation, only one hologram is reconstructed. As shown in Fig. 1(a), selective reconstruction of eight holograms is achieved at a single position in the media by using input PC reference waves with different PCdek=PCenk(k=1to8). Each input PC reference wave originates from a column vector of the 4×8 OPOM. Under the eight orthogonal states of PCenk(k=1to8), crosstalk among images can be effectively reduced through interference cancellation. We select the Arabic numerals 1 through 8 as the reconstructed images and successfully reconstructed eight independent images with high resolution and fidelity under different polarization orthogonal channel illuminations [Fig. 1(b)]. Furthermore, the reconstructed images exhibit the same polarization state but can be reconstructed separately by orthogonal incident light waves, demonstrating the orthogonal selectivity of the information stored in the media.

For achieving multi-channel holography, we have to explore the polarization matrix with mutual orthogonality and construct it.

It is well known that any polarization state can be represented by the Jones vector; the typical Jones vectors of vertically polarized, s, and horizontally polarized, p, lights are represented by Eq. (1): s=[01],p=[10].(1)

To characterize the OPOM of minimum order, we utilize a 2×2 square matrix. Unlike traditional orthogonal matrices, the elements in this matrix are vectors representing polarization states, not scalars. This construction results in a matrix that possesses polarization orthogonality characteristics. We assume the expression as Eq. (2): O2×2=[O11O12O21O22],(2)where Oij (the subscripts, i and j, take values of either “1” or “2”) represents Jones vector of unknown polarized lights. To find a solution that satisfies the conditions for an orthogonal matrix, O2×2 must adhere to Eq. (3): O2×2O2×2T=nE2×2,(3)where the superscript, T, denotes the transpose symbol, n is a positive coefficient, and E2×2 represents the identity matrix of size 2×2. By substituting Eq. (3) into Eq. (2), we derive Eq. (4): [O112+O122O11O21+O12O22O11O21+O12O22O212+O222]=n[1001].(4)

For Eq. (4) to hold, the following conditions in Eqs. (5) and (6) must be satisfied: O112+O122=O212+O222≠0,(5)O11O21+O12O22=0.(6)

In the basic Poincaré spheres, a pair of orthogonal polarized lights, such as p- and s-polarized lights, can be identified. The p-polarized light is defined as the electric field oscillating in the x-z plane and being perpendicular to the direction of wave propagation, while the s-polarized light is defined as the electric field oscillating in the y-z plane. Moreover, the following characteristics in Eq. (7) can be observed: $$\begin{gathered} s·p=0,p·s=0,s·(−p)=0,p·(−s)=0, \\ s·s=1,p·p=1,s·(−s)=−1,p·(−p)=−1, \end{gathered}$$(7)where s and −s represent linearly polarized light with polarization angles of 90° and 270°, respectively. Similarly, p and −p signify linearly polarized light with polarization angles of 0° and 180°, respectively.

Clearly, Eq. (5) is always satisfied. Therefore, it is only necessary to analyze Eq. (6) and divide it into three scenarios to discuss, namely, Eqs. (8) and (9): O11O21=−O12O22=0,(8)O11O21=−O12O22=±1.(9)

Among them, to prevent the elements of the column vector from having the same polarization state, the scenario described by Eq. (9) is not considered.

By identifying the solutions that satisfy Eq. (8) based on the given conditions in Eq. (7), the two column vectors of O2×2 can be solved, yielding four respective solutions as shown in Eq. (10): $$\begin{gathered} O1=[O11O21],O2=[O12O22], \\ O1=O2=[sp]or[ps]or[s−p]or[p−s]. \end{gathered}$$(10)

Moreover, any orthogonal matrix must have full rank, which is a necessary condition for satisfying Eq. (11): R(O2×2)=2.(11)

As a result, O2×2 can be determined with six solutions as shown in Eq. (12): O2×2=[O1O2]=[spps]or[ssp−p]or[spp−s]or[pss−p]or[sp−p−s]or[pps−s].(12)

The six solutions of O2×2 are summarized, and it can be observed that their solutions arise from the pairwise combination of four column vectors. These four column vectors are then consolidated into a matrix named OPOM2×4 as shown in Eq. (13): OPOM2×4=[spspps−p−s].(13)

This result represents the complete form of the minimum-order OPOM, with an orthogonal combinations number of four. The s,−s,p, and −p in Eq. (13) can also be substituted with other orthographically linearly polarized lights. For instance, by employing s+p,−s−p,s−p, and −s+p, an alternative form of OPOM can be expressed as Eq. (14): OPOM2×4′=[s+ps−ps+ps−ps−ps+p−s+p−s−p].(14)

After deriving the minimum-order OPOM, the solution of any higher-order OPOM can be obtained by combining it with the Hadamard matrices, leveraging their orthogonality. Utilizing the recursive Kronecker product, the expression for OPOM2m×4m with an orthogonal combinations number of 4m is as Eq. (15): OPOM2m×4m=OPOM2×4⊗Hm×m,(15)where Hm×m denotes a Hadamard matrix with element size m×m, which is written as Eq. (16): $$\begin{gathered} Hm×m=H2×2⊗Hm2×m2, \\ H2×2=[111−1]. \end{gathered}$$(16)

As depicted in Fig. 2, within the basic OPOM2×4 extension layer, using OPOM2×4 as the minimum unit involves replicating a rectangle with a side length of m (where m is an even number) to generate an extension layer. The Hadamard layer is formed by arranging the elements of the Hadamard matrix. The element modules in the Hadamard layer are mapped one-to-one with the elements in the basic OPOM2×4 extension layer, and their respective positions are multiplied to generate OPOM. In Fig. 2, an example is provided wherein OPOM4×8 is derived by multiplying H2×2 with the respective positions of the basic OPOM2×4 extension layer. Only the portion mapped to the Hadamard matrix with an element value of “−1” experiences a 180° increase in polarization angle, leaving the rest unchanged. The expression is articulated as the following Eq. (17): OPOM4×8=[spspspspps−p−sps−p−sspsp−s−p−s−pps−p−s−p−sps].(17)

The proposal of OPOM breaks through the limitation of the polarization orthogonal dimension and lays the groundwork for enhancing polarization modulation capabilities, especially in the context of high-dimensional polarization multiplexing. This paper provides an example of using OPOM4×8 to achieve single-point eight-channel information multiplexing in polarization holography. As shown in Fig. 3, OPOM4×8 was divided into eight PCs based on column vectors, with each PC consisting of four polarization components.

In Table 1, A and B represent the intensity and polarization response coefficients of the material, respectively; a and b represent any real numbers; Sig., Ref., and Rec. represent the signal, reference, and reconstructed wave, respectively. In polarization holography based on tensor theory [35], with specific values assigned to the signal wave and interference angle, orthogonally polarized light served as a reference wave for recording and reading, resulting in null reconstruction characteristics [36] achieved under the condition of A+B=0, while faithful reconstruction characteristics [36] can be achieved using the same polarization as the reference wave for both recording and reading under identical conditions, ensuring the faithful reconstruction of recorded information. The condition of A+B=0 implies that the factor of exposure time needs to be considered during recording [37]. However, by setting b=0 and utilizing the polarization orthogonal bases s and p as both the reference waves for recording and reading, we can overcome the limitations imposed by A and B, enabling a dual-channel system without exposure constraints, as demonstrated in Table 2.Table 1.

Polarization Holography with Arbitrary Orthogonal Reference Waves under Fixed Interference Angle of 90°

The experimental setup was designed as shown in Fig. 4(a). The light source was a fundamental TEM00 532 nm laser with a waist radius of approximately 0.75 mm. The beam after collimating and expanding was divided into s-polarized and p-polarized lights through a polarization beam splitter (PBS1). The s-polarized light served as the incident light on the reference wave path, while the p-polarized light served as the signal wave. In the reference wave path, PBS2 was employed to separate the beam into s- and p-polarized lights. Subsequently, half wave plates (HWP1 and HWP2) were employed to modulate the polarization state individually. A beam splitter (BS1) was then used to generate a parallel beam with two distinct polarization states by non-overlapping. The purpose of the HWP6 before the PBS2 was to adjust the components of the s- and p-polarization after the PBS2. Similarly, BS2 was employed for beam splitting, while HWP3 and HWP4 were employed to modulate the polarization states. Another BS3 was employed for non-overlapping beam combining, resulting in parallel beams with four distinct polarization states. By rotating the fast axis positions of HWP1–HWP4, four different polarization states can be designed.

In the signal wave path, an amplitude-based spatial light modulator (A-SLM) was employed to modulate the signal images. A 4f imaging system was used to improve image quality. The signal wave, along with the modulated reference wave, was incident into the phenanthrenequinone-doped polymethyl methacrylate (PQ/PMMA) photopolymers material with a 90° interference angle to create the polarization hologram. The optical power of the signal wave was 850 μW, whereas the optical power of the reference wave was 3.328 mW. Additionally, the recording duration for the series of eight holograms ranged from approximately 10 to 15 s. After eight recording cycles, using only the reference wave to illuminate the material, a diffracted beam can be received by a charge-coupled device (CCD detector) in the reconstruction wave path. Notably, in the recording process, the used reference wave was replaced with PCen1−PCen8, adhering to the OPOM4×8 rule, to record the Arabic numerals 1–8 respectively, thus completing the preparation of the polarization hologram. The reference wave was divided into multiple modules with different polarizations, corresponding to the elements in a single orthogonal vector unit under OPOM. Multiple exposures were performed using such class of reference waves. Exposing the grating to a matched incident wave facilitated vector superposition, allowing for the independent reconstruction of various polarizing holographic images in the direction of 90° propagation.

To generate the PC reference wave for the OPOM4×8 column vector, one can either combine multiple polarization components or adopt a spatially separated approach. The latter method is used in this paper. HWP1–HWP4 are utilized to individually modulate the polarization state of each polarization component, ensuring that altering the fast axis positions of the HWPs does not impact the optical path difference. By rotating the fast axis positions of HWP1–HWP4, four different polarization states can be designed. As illustrated in Fig. 4(b), different fast axis positions of HWP1–HWP4 allowed for the realization of different polarization combinations on various column vectors of OPOM4×8. To fully demonstrate the characteristics akin to the inner product of orthogonal matrices in polarization holography, it is imperative to overlap the propagation direction of each element of the reconstructed wave, thereby enabling vector superposition of the resultant reconstructed wave. Consequently, the polarization components within the PC reference wave must be positioned side-by-side.

Figure 5(a) illustrates the experimental results of polarization hologram multi-channel multiplexing. Each individual information image can be selectively reconstructed using an input PC reference wave with PC of PCdek=PCenk(k=1−8), respectively. Different input PC reference waves will reconstruct Arabic numerals 1–8 at a 90° angle, using PQ/PMMA photoanisotropic materials (see Section 5) under a 532 nm laser.

From the results of the eight-channel multiplexing implemented by OPOM4×8, the small circle around the eight Arabic numerals can be used to calculate crosstalk between information. When different PCdek were used, the information reconstruction results on the PCenk channel were compared, as depicted in Fig. 5(b); the crosstalk between messages can be ignored, indicating the robustness and efficiency of this device.

In addition to high-dimensional polarization multiplexing, information retrieval from OPOM-multiplexed polarizing holograms can be achieved by decoding the corresponding information using the PCde. This establishes an optical information transmission network in the field of optical communication. The Arabic numerals 1 to 8 obtained by illuminating holograms with PCde are also expected to be used in designing a 3-bit ciphertext for optical encryption.

We considered the versatile switching capabilities of polarization orthogonal matrices in image display, and successfully generated holograms utilizing OPOM4×8 to record eight Tetris pieces with varying shapes. Input PCde beams can selectively reconstruct different holographic images from the polarization hologram. As a result, the eight individual segments of the Tetris image were reconstructed in the experiment by using input PCde beams corresponding to PCen1−PCen8, respectively [Figs. 6(a)–6(h)]. This opens up the possibility of using the beam of PCs as an optical switch for dynamic displays. By linearly superimposing these eight PCen, a mixed image can be obtained. For instance, by superimposing PCen1 to PCen8 as the PCde, a complete Tetris image was experimentally obtained [Fig. 6(i)]. Similarly, using the PCde superimposed by PCen1 to PCen4, the mixed image of Tetris I–IV was obtained [Fig. 6(j)], and using the PCde superimposed by PCen5 to PCen8, the mixed image of Tetris V–VIII was obtained [Fig. 6(k)]. Without the correct PCde, it is impossible to discern the specific shape of a single Tetris, making it suitable for image encryption. Moreover, it is anticipated that OPOM will play a role in the design of logical optical switches, and distinct logical optical switches can be devised based on different orders of OPOM. The introduction of OPOM may address the bottleneck issue in preparing polarization holography devices for optical logic gates using multiple PCs.

We have proposed a novel construction method for OPOM and successfully validated its high-dimensional multiplexing capabilities in polarization holography by experiment. Our demonstrated OPOM can enable the expansion of multiplexing technology in the polarization dimension, paving the way for multiple information retrieval and dynamic holographic displays. The receiver can solely access precise transmitted information (such as digital display scenarios) by utilizing the appropriate polarization PC key. By escalating the complexity of the image (such as using some Tetris blocks to form a rectangular pattern) and modulating the spatial position of the reconstructed images, higher flexibility and dynamic switching can be achieved. This polarization-sensitive birefringent material boasts advantages such as cost-efficiency, straightforward preparation process, scalability in size, and enduring storage life [38], making it advantageous in information storage, image display, and communication transmission.

The constructed OPOM exhibits the capacity to extend high-order polarization orthogonality, rendering it suitable for any scenario requiring polarization modulation. Media akin to metasurfaces, which possess comparable capabilities to polarization holography, can modulate the amplitude, phase, and polarization of light waves individually and simultaneously [39,40]. The introduction of OPOM is anticipated to promote the modulation prowess of metasurfaces in polarization dimension. In addition, the OPOM can also promote the application of polarization holography in anti-counterfeiting and encryption, as the desired image can only appear individually when the incident light is a specific polarization-combined light that adheres strictly to the OPOM mapping rules.

Our proposal of OPOM introduces innovative concepts and opens up possibilities for exciting applications in multi-channel broadcasting in optical communication, multidimensional optical storage, optical logic devices, anti-counterfeiting, and ultra-high-security optical encryption, among others. This work represents a significant contribution to the field of orthogonal matrices and has the potential to drive further research and development in these areas.

The polarization multiplexing holograms designed in the experiment were obtained through interferometric recording within the polarization-sensitive PQ/PMMA material. Therefore, the fabrication of this material [41–43] is of paramount importance. The preparation procedure consisted of two steps. In the first step, phenanthrenequinone (PQ) and 2, 2-azobisisobutyronitrile (AIBN) were dissolved in a glass bottle filled with methyl methacrylate (MMA). PQ served as a photosensitizer, AIBN acted as a thermo-initiator, and MMA functioned as a liquid monomer. The concentrations of PQ and AIBN were both 1% in mass fraction. The solution was thoroughly mixed using an ultrasonic water bath, and impurities were removed through filtration using a mesh filter. Subsequently, the glass bottle was placed on a magnetic stirrer and maintained at a constant temperature until the solution reached a uniformly viscous state. In the second step, the resulting syrup was poured into a glass mold, which was then heated to 45°C for 15 h followed by an additional 8 h at 60°C to solidify the mixture. Finally, the solidified material was removed from the mold and set aside.