Time Evolution of Orbital Angular Momentum Modes for Deep-Routing Multiplexing Channels

Zebin Huang; Peipei Wang; Jiafu Chen; Wenjie Xiong; Huapeng Ye; Xinxing Zhou; Ze Dong; Dianyuan Fan; Shuqing Chen

doi:10.1016/j.eng.2024.09.016

Engineering ›› 2025, Vol. 45 ›› Issue (2) :97 -104. DOI: 10.1016/j.eng.2024.09.016

Research Subwavelength Optics—Article

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Time Evolution of Orbital Angular Momentum Modes for Deep-Routing Multiplexing Channels

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Abstract

Optical orbital angular momentum (OAM) mode multiplexing has emerged as a promising technique for boosting communication capacity. However, most existing studies have concentrated on channel (de)multiplexing, overlooking the critical aspect of channel routing. This challenge involves the reallocation of multiplexed OAM modes across both spatial and temporal domains—a vital step for developing versatile communication networks. To address this gap, we introduce a novel approach based on the time evolution of OAM modes, utilizing the orthogonal conversion and diffractive modulation capabilities of unitary transformations. This approach facilitates high-dimensional orthogonal transformations of OAM mode vectors, altering both the propagation direction and the spatial location. Using Fresnel diffraction matrices as unitary operators, it manipulates the spatial locations of light beams during transmission, breaking the propagation invariance and enabling temporal evolution. As a demonstration, we have experimentally implemented the deep routing of four OAM modes within two distinct time sequences. Achieving an average diffraction efficiency above 78.31%, we have successfully deep-routed 4.69 Tbit·s⁻¹ quadrature phase-shift keying (QPSK) signals carried by four multiplexed OAM channels, with a bit error rate below 10^–6. These results underscore the efficacy of our routing strategy and its promising prospects for practical applications.

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Keywords

Orbital angular momentum / Time evolution modulation / Deep-routing technology / Unitary transformation / Mode-division communication networks

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Zebin Huang, Peipei Wang, Jiafu Chen, Wenjie Xiong, Huapeng Ye, Xinxing Zhou, Ze Dong, Dianyuan Fan, Shuqing Chen. Time Evolution of Orbital Angular Momentum Modes for Deep-Routing Multiplexing Channels. Engineering, 2025, 45 (2) : 97-104 DOI:10.1016/j.eng.2024.09.016

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1. Introduction

Vortex beams endowed with orbital angular momentum (OAM) modes hold immense promise for enhancing the capacity and efficiency of optical communications and networking [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14]. The inherent orthogonality of OAM modes introduces an additional dimension for channel multiplexing, significantly increasing communication capacity [15], [16], [17], [18], [19], [20], [21], [22]. Despite significant advances in angle-separated (de)multiplexing techniques such as fork-shaped gratings [23], [24], [25], [26], [27], [28], [29] and coordinate transformations [30], [31], the effective routing of these multiplexed channels—a crucial step for network interconnectivity—presents challenges due to the inability to selectively reallocate light paths across both spatial and temporal domains. Current methods such as cascaded modulation, which rely on the spatial alignments of helical phase plates and fork-gratings to define mode conversion relationships, offer selective routing for specific channels. However, this inadvertently affects the modes and propagation directions of non-targeted channels during processing, resulting in channel redundancy and energy consumption for optical communication [32], [33], [34], [35], [36]. Moreover, existing axial multi-focus helical modulation techniques, which enable variable-depth OAM mode conversions [37], [38], [39], are confined to the coaxial directions, limiting their utility for spatial reallocations. Given these limitations, it is imperative to develop an innovative modulation strategy that transcends the constraints of simultaneous temporal and spatial freedom modulation, thereby facilitating the selective and efficient reallocation of light paths for deep-routing purposes. This is essential for overcoming the current barriers and unlocking the full potential of mode and time domain modulation for advanced optical communication networks.

In this study, we address the challenges of effective OAM mode routing by introducing a time evolution approach that enables the deep modulation of OAM modes through mode-independent modulation and the temporal evolution of wavefronts, resulting in the dynamic reallocation of light paths during transmission. Modulation is accomplished through unitary transformation, which delineates finite-dimensional orthogonal operations in inner-product spaces and facilitates the linear conversion of input and output vectors [40], [41], [42], [43], [44], [45]. By extending the modulation to encompass high-dimensional orthogonal optical fields, these transformations adeptly handle OAM mode vectors with both spatial and mode independence, due to the preservation of the matrix norm during the conversion processes [20], [46], [47], [48], [49], [50]. Furthermore, by adjusting the propagation characteristics of the wavefronts and controlling the light fields across various time sequences, deep modulation is achievable via the Fresnel diffraction matrix in forward transmission [51], [52], [53]. This technique successfully circumvents propagation invariance, fostering the temporal evolution of light fields and permitting flexible light-path reallocations across diverse spatial depths.

Although cascaded optical systems have traditionally been used for intricate light-field conversion scenarios, the advent of diffractive deep neural networks with multiple phase-modulated planes has introduced progressive wavefront modulations, integrating optical information extraction with deep learning technologies [54], [55], [56], [57], [58]. However, the heavy reliance on statistical approximations to define input-output field mappings compromises conversion and prediction accuracy due to the lack of a tangible modulation mechanism. Moreover, modulating the temporal dimension independently poses challenges without distinct modulation across varying spatial depths. Our proposed time evolution unitary transformation strategy leverages the orthogonality between the mode and space domains to create a multidimensional mapping with elevated conversion accuracy. It modulates the wavefronts of OAM modes and propagating light paths via a sequence of element-wise product operations, closely approximating the desired mathematical transformation. This process accurately manipulates both the mode and time dimensions, enabling the deep routing of light paths.

To experimentally verify the feasibility of OAM modes time evolution, we designed and verified a tri-layered phase-modulated matrix, showcasing its efficacy in the deep routing of OAM mode channels. Our experiments resulted in the successful deep routing of four OAM mode channels across two distinct time sequences, with an impressive average diffraction efficiency of over 78.31%. This enabled the deep routing of 4.69 Tbit·s⁻¹ quadrature phase-shift keying (QPSK) signals carried by these multiplexed channels, maintaining a bit error rate (BER) below 10^–6. Further investigations into the deep manipulation capabilities of time evolution were conducted by expanding the number of routing mode channels and modulating time sequences. Numerical analyses revealed that the unitary transformation accurately manages two degrees of freedom (DoFs), accommodating up to ten OAM modes and four time sequences simultaneously. These results underscore the ability of this deep manipulation strategy to leverage both time and mode DoFs, enhancing mode routing through a strategic reallocation of light paths across various time sequences. This innovative routing method holds promise for facilitating complex optical node interconnection within mode multiplexing communication networks.

2. Principle and methodology

Unitary transformations, being linear operations, maintain the matrix norms of vectors within a finite-dimensional inner product space. This property allows for mutual conversion between input and output vectors, which can be mathematically expressed as follows:

(1)

{∥v_{i}∥}_{2} = {∥v_{o}∥}_{2}

where

{∥\cdot∥}_{2}

represents the 2-norm, and the unitary transformation between input vector (

v_{i}

) and output vector

v_{o}

is described by multiplying a unitary matrix (U₀):

(2)

v_{o} = v_{i} \cdot U_{0}

where the unitary matrix will not change the matrix norms of the vectors, and

|U_{0}| = 1

. Inspired by the matrix-norm preservation characteristic of unitary transformation, Fig. 1 illustrates the mechanism of OAM mode deep routing, which involves mode-independent and time evolution modulations. The former manipulates both space and mode DoFs through unitary conversion, while the latter manipulates the space and path DoF through time evolution. OAM modes represent eigen solutions within an infinite-dimensional Hilbert space and exhibit orthogonality through invariant inner-product values. By combining OAM modes with spatial dimensions, the modulation capabilities of unitary transformations are significantly enhanced, allowing for the allocation of light paths without disrupting the completeness of the orthogonality. Therefore, the utilization of unitary transformations can efficiently convert input-output OAM mode vectors with minimal power loss and enable multidimensional mapping to facilitate mode-independent modulation. To simplify the description of modes and spatial locations, the following states are employed:

(3)

E (x, y) = |M, {SV}_{t, d}〉

where E(x, y) is the light fields of OAM modes, and M represents the OAM mode. SV_t,d represents the space vector (SV) with t time sequence and d displacement, which is further controlled by the displacement of m and the angle of

γ

. It is worth noting that the matrix norm can be preserved by the appropriate selection of M and SV satisfying the transformation principle in Eq. (2).

Following these principles, we can choose m different OAM modes with SV = 0 (coaxial center) as input vectors and generate m output vectors with varying SV while keeping M = 0 fixed. This configuration makes it possible to implement OAM mode routing, allowing for selective OAM mode conversion with a spatial shift while establishing a mode-independent linear mapping relationship. The q-dimensional unitary transformation between mode vectors can be mathematically expressed as follows:

(4)

E_{v_{o}} = [\begin{matrix} |M_{1}, {SV}_{0}〉 \\ |M_{2}, {SV}_{0}〉 \\ ⫶ \\ |M_{q}, {SV}_{0}〉 \end{matrix}] = [\begin{matrix} |M_{0}, {SV}_{t_{1}, d_{1}}〉 \\ |M_{0}, {SV}_{t_{1}, d_{2}}〉 \\ ⫶ \\ |M_{0}, {SV}_{t_{1}, d_{q}}〉 \end{matrix}] \cdot U_{1} = E_{v_{i}} \cdot U_{1}

where

U_{1}

represents the unitary matrix responsible for the conversion between input mode vector (

E_{v_{i}}

) and output mode vector (

E_{v_{o}}

), enabling multidimensional mapping during unitary transformation. To implement this transformation optically,

U_{1}

can be decomposed using singular-value decomposition and approximated through element-wise product computations involving multiple phase-modulated matrices and diffraction matrices. The diffraction matrix is derived using Fresnel diffraction theory with the paraxial approximation in the frequency domain:

(5)

H (f_{x}, f_{y}, d_{z}) = \exp (j k d_{z}) \cdot \exp [j π λ d_{z} (f_{x}^{2} + f_{y}^{2})]

where H is the diffraction matrix, (f_x, f_y) represent the spatial frequencies of (x, y),

λ

is the working wavelength, k =

2 π / λ

, and d_z is the lateral distance between layers. j represents the imaginary number and

j^{2} = - 1

. With these parameters, the mode-independent unitary transformation can be optically expressed as follows:

(6)

E_{v_{i}} \cdot U_{1} \approx E_{v_{i}} \cdot H^{1} \cdot φ^{1} \cdot H^{2} \cdot φ^{2} \cdot . . . \cdot H^{n} \cdot φ^{n}

where

φ^{n}

represents the nth phase-modulated matrix with a segment of [

0, 2 π

]. To further manipulate the temporal DoF, we employ the Fresnel diffraction matrix as a self-modulated operator to manipulate the output beam distributions, breaking the propagational invariance during propagation. This enables the deep modulation of light beams and demonstrates deep evolution. The corresponding deep evolution can be expressed as follows:

(7)

T_{2} = T_{1} \cdot H

where T₁ and T₂ are the corresponding mode vectors at different time sequences and are expressed as follows:

(8)

T_{1} = [\begin{matrix} |M_{0}, {SV}_{t_{1}, d_{1}}〉 \\ |M_{0}, {SV}_{t_{1}, d_{2}}〉 \\ ⫶ \\ |M_{0}, {SV}_{t_{1}, d_{q}}〉 \end{matrix}]

(9)

T_{2} = [\begin{matrix} |M_{0}, {SV}_{t_{2}, d_{1}}〉 \\ |M_{0}, {SV}_{t_{2}, d_{2}}〉 \\ ⫶ \\ |M_{0}, {SV}_{t_{2}, d_{q}}〉 \end{matrix}]

The right part of Fig. 1 illustrates the time evolution of a Gaussian beam using the diffraction matrix, showcasing selective beam shifts after deep evolution. Through the use of mode-independent and time evolution modulations, we can establish OAM mode deep-routing functions by appropriately designing the phase-modulated matrix and diffraction distance. These unknown parameters within the mode-independent unitary transformations can be computed using the gradient descent algorithm, which is a well-known algorithm in artificial intelligence for calculating optimal solutions [59]. For more information on the design of the mode-independent and time evolution model, interested readers can refer to Section S1 in Appendix A.

3. Results

In this work, we designed a tri-layered phase-modulated matrix with a lateral distance of 300 mm to achieve mode-independent and time evolution unitary transformation, and Fig. 2 demonstrate the schematic diagram and results of the deep routing of OAM mode multiplexed channels. Fig. 2(a) illustrates the optical implementation of OAM mode deep routing, where multiplexed OAM mode channels illuminate the center of diffractive planes and propagate until reaching the output plane. At the output plane (300 mm), the OAM modes are converted back into Gaussian modes through mode-independent unitary modulations with different light paths. Furthermore, spatial locations evolve during propagation through the use of a self-modulated diffraction matrix, enabling the time evolution of light beams for deep modulations at different times. After the second evolution plane, the Gaussian beams will continue to diverge, since there is no additional time evolution modulation to the wavefronts. It should be noted that, in order to permit better visualization of the modulation process of the unitary transformation model, we substituted transmissive phase-modulated matrices in place of the reflective architecture. The input vectors consist of coaxial light fields with different OAM modes, while the output vectors are Gaussian beams with various displacements and angles. To achieve evolution during propagation, we selected two sets of light beams with respective displacements of 0.6 and 1.2 mm, establishing a one-to-more mapping relationship during forward propagation. Moreover, the spatial locations of the output Gaussian beams are located at the center of the modulated screen, which enables higher diffraction efficiency and facilitates better convergence of the time evolution unitary transformation model. It is important to note that we intentionally set the waist radius of the output beams to be relatively smaller (0.20 mm) than that of the input beams (0.45 mm). This minimizes interlayer coupling consumption, increases the modulation accuracy, and enhances diffraction efficiency. For detailed information on the calculation strategy for finding optimal phase distributions in optical unitary transformation, interested readers may refer to Section S1.

These tri-layered unitary transformations were experimentally implemented using three reflective spatial light modulators (SLMs) with a lateral distance of 300 mm, and the experimental setups are illustrated in Section S1. The experimental results of OAM mode deep routing are shown in Figs. 2(b)–(e). We demonstrate two scenarios of OAM mode deep routings, involving light-path allocation without exchange (Figs. 2(c-i) and (c-ii)) and light-path allocation with exchange (Figs. 2(d-i) and (d-ii)), thereby showcasing the ample modulation capabilities to harness the mode and temporal DoFs using time evolution modulation. We first demonstrated the deep routing of four incident OAM modes (without exchanging light-path allocation during evolution), within two evolutional depths—namely, 300 and 600 mm—corresponding to time sequences of 1.0 and 2.0 ns, respectively. The optimal phase distributions for this scenario are shown at Section S2 in Appendix A, which were further loaded onto the SLMs to jointly facilitate selective helical conversions, spatial shifts, and beam convergences, resulting in the reallocation of light paths at different spatial locations. Consequently, we employ the metrics of diffraction efficiency to better quantify the overall performance of the time evolution unitary transformation model, which is measured by the intensity ratios between the target regions and ignores the intersection loss of the SLMs during conversion.

The results shown in Fig. 2(c-i) illustrate that, at a propagation distance of 300 mm (corresponding to a time sequence of 1.0 ns), the four incident OAM modes were routed in different directions with a spatial shift of 0.6 mm. The average diffraction efficiencies of the four beams at 300 mm were 87.22% ± 1.77% (marked as “w/o ex” in Fig. 2(e-i)). Subsequently, self-modulated phase distributions were applied for deep evolution after the unitary transformation, which was evident from the simulated complex-value distributions. This enabled the time evolution to selectively control the spatial location of the beams for deep routing. For better visualizing the time evolution procedure of OAM mode deep routing, the intensity distributions of different times are shown in Section S2. The results shown in Fig. 2(c-ii) indicate that the output beams were deeply routed, with a displacement of 1.2 mm, and the average diffraction efficiencies at 2.0 ns were 80.25% ± 1.89% (marked as “w/o ex” in Fig. 2(e-ii)). This represents a decrease of approximately 6.96% compared with the previous time sequence. The lower diffraction efficiency at 2.0 ns can be attributed to the fact that deep routing at 1.0 ns was directly realized by the unitary transformation, which has abundant modulation abilities, while routing at 2.0 ns primarily relied on a diffraction unitary operator. However, this reduction in diffraction efficiencies during deep routing does not significantly affect the overall quality and provides additional DoFs for manipulating the light field. We also verify the scalability of this OAM mode deep-routing technology and experimentally demonstrate OAM mode deep routing with larger OAM modes and different type of structural light beams (Section S3 in Appendix A). Since the tri-layered phase modulation architecture is experimentally demonstrated using multiple-plane light conversion system comprising three reflective SLMs, the performance of time evolution unitary transformation is highly affected by the spatial alignment of optical components. The discussion of robustness assessments of tri-layered phase modulation architecture are shown at Section S4 in Appendix A.

To showcase the light-path reallocation capabilities of deep routing, we reconfigured the allocations to enable the exchange of propagation directions during evolution at different time sequences. In this scenario, the light paths initially share the same direction as in the previous section, at an evolution depth of 300 mm (corresponding to a 1.0 ns time sequence). Consequently, they exchange spatial locations during evolution, while the output Gaussian modes are preserved. The experimental results are displayed in Figs. 2(d-i) and (d-ii). We successfully achieved the deep routing of four OAM modes, with average diffraction efficacies of 69.40% ± 1.44% at 1.0 ns and 65.06% ± 2.42% at 2.0 ns (see the section labeled “w ex” in Figs. 2(e-i) and (e-ii)).

It is worth noting that a lower overall diffraction efficiency was observed compared with the previous scenario, which is primarily due to the limitations in the modulation capabilities of the proposed time evolution mechanism. More specifically, using only Fresnel diffraction with a fixed propagation distance does not provide sufficient light-field processing capability to achieve high-accuracy OAM mode and spatial location conversion. In addition, increasing the shifting distance required for evolution between time sequences from 0.6 to 1.8 mm for light-path exchange introduces greater modulation difficulties and further deteriorates the performance. However, the complete transformed Gaussian beam structure clearly demonstrates the significant light-path reallocation capabilities of this unitary transformation for deep-routing applications. While the energy utilization is reduced relative to the previous scenario by approximately 17%, it still markedly outperforms conventional OAM mode routing technologies using fork-grating or cascaded helical modulations, which can be further implemented in the networking of mode optical communication networks. Moreover, the mode deep-routing technique provides additional DoFs for information encoding, enabling the arbitrary allocation of optical terminals within depth planes. This capability enhances routing functionalities and increases the number of terminals to meet the complex requirements of optical communication systems. Furthermore, this deep-routing strategy can reduce receiver congestion by distributing signals across multiple output depth planes, thereby enhancing the capability of optical equipment to handle more OAM mode channels. The separation of OAM modes can also help to reduce inter-channel crosstalk, further improving the communication quality.

As a proof of concept, we introduce a prototype optical communication system designed for the deep routing of OAM mode multiplexed channels. Detailed information and the construction principles concerning the optical communication system can be found in Appendix A Section S5. In our demonstration, we illustrate the feasibility of time evolution unitary transformation and the achievement of deep routing for four OAM modes at two evolutionary time sequences. To better showcase the deep routing of OAM modes at varying evolution depths, we captured a period of the signal in each mode channel before routing and routing at different time sequences. The selected waveform diagrams in Figs. 3(a-i)–(b-ii)) depict the transmission of four signals during the time evolution in two different scenarios (without and with light-path exchange), where the digital signals were successfully transferred with high similarity. We also assessed the BER curves of the four OAM modes at these two evolution depths under two scenarios, as shown in Fig. 4. All channels achieved a BER below 10⁻⁶ as the received power increased to −20.5 dBm. Furthermore, the BER remained below the hard-decision forward error correction (FEC) of 3.8 × 10⁻³ at −26.0 dBm. The communication sensitivity was only 2 dB lower than in a back-to-back (B2B) scenario, approaching the maximum performance of this system.

The consistent BER curves at different evolution depths indicate that deep routing provides additional modulation DoFs for optical communication while maintaining quality. We also evaluated the error vector magnitude of the received signals and depicted the constellation diagrams of these two scenarios at the received power of −20.5 dBm (Figs. 4(b-i)–(b-iv) and (d-i)–(d-iv), respectively). The results indicated that all channels were successfully recovered with good convergence, and the average error vector magnitude values were 17.95% ± 0.58% in the without-exchange scenario and 17.51% ± 0.27% in the with-exchange scenario.

Furthermore, we verified the compatibility of the OAM mode deep-routing technology with wavelength-division multiplexing technology. We selected and multiplexed 24 wavelengths ranging from 1541.32 to 1559.78 nm with a spacing of 0.8 nm by means of a wavelength (de)multiplexer; the spectrum measured before and after the free space optical communication system is shown and BER curves are measured in Section S4. The BER curve shows that ten selected wavelengths and OAM mode channels drop below 10⁻⁶ as the power increases to −19.0 dBm. Compared with the communication quality without utilizing wavelength multiplexing, the communication sensitivity is only 1.5 dB worse, indicating that the deep-routing technique is highly compatible with wavelength-division multiplexing.

4. Discussion

To demonstrate the advantages of time evolution unitary transformation for routing OAM mode multiplexed channels, we undertook a review of OAM mode multiplexed channel routing technologies (Table 1 [24], [33], [60], [61]). Our review assessed the performance of related work in terms of modulation capability, DoFs, diffraction efficiency, and routing channels. In the realm of mode routing techniques, one method—initially proposed in Ref. [25]—employs cascaded and sub-regional helical phases, which utilize phase plates with multiple functions to establish a linear mapping relationship for mode conversions. However, its modulation capability and DoFs are constrained by its coaxial conversion mechanism, requiring a cumbersome optical system for separating the mode channels to reallocate light paths. Inspired by coupling and separating technologies in mode (de)multiplexing, other approaches—such as angle-separated vortex gratings and coordinate transformation principles—have been proposed for routing mode multiplexed channels with appropriate channel selection and reallocation. For example, Yan et al. [21] employed a Dammann vortex grating to diffract multiplexed mode channels and route them to different diffraction orders based on their incident angle and mode status. While this approach offers abundant modulation functions, it sacrifices diffraction efficiency to 1/N as the number of multiplexed channels (N) increases. A detailed comparison of time evolution unitary transformation with a Dammann vortex can be found in Appendix A Section S6.

Energy-consumption issues are further addressed by the coordinate transformation proposed in Ref. [51], which achieves channel routing by converting OAM modes to lateral lines. Moreover, through the utilization of polarization-matching modulation in meta-atoms, Wang et al. [52] designed a polarization-independent gradient metasurface, enabling the routing of x-axis and y-axis OAM modes. It is worth noting that these existing approaches rely heavily on prior knowledge and physical descriptions to design routing functions, which makes them less suitable for flexible light-path reallocation—a crucial requirement for node information interconnection in mode channel routing.

To address these challenges, we propose a time evolution unitary transformation for deep OAM mode channel routing. This approach enables the reallocation of light beams, allowing manipulation of both mode and spatial locations during transmission. It offers a wide range of modulation functions by utilizing time, mode, and space DoFs, enabling the routing of 4 × 2 multiplexed channels with an average diffraction efficiency exceeding 78.31%. Furthermore, this approach allows for more flexible routing directions, facilitating the three-dimensional (3D) deep routing of light paths. This proposed time evolution unitary transformation exhibits distinct advantages and is expected to provide a versatile platform for light-path reallocation in mode routing within optical networks.

The deep modulation capabilities of the self-modulated diffraction operator in time evolution unitary transformation are worth exploring. This modulation capability can be leveraged to enhance the functionalities by utilizing time and mode DoFs. Consequently, we increased the number of modulated time sequences while maintaining a fixed OAM mode range (l = ±1 and l = ±2, where l is the topological charge of OAM mode); the numerical results are presented in Section S6. In this setup, the deep evolution interval is set at 150 mm for each time sequence, with a 0.6 mm spatial shift per spatial depth. The numerical results reveal that, in the scenario of a single time sequence, mode routing relies solely on the unitary transformation, achieving maximal diffraction efficiency. However, as the number of time sequences increases, the self-modulated diffraction operator becomes a crucial modulation operator for mode routing. Notably, the diffraction speckles become more prominent as the time sequence reaches four, indicating a trade-off between modulation function and diffraction efficiency.

We also examined the modulation capabilities of OAM mode routing with a fixed time sequence of two using tri-layered unitary transformation. We selected ten OAM modes ranging from −5 to +5 as the input, with corresponding Gaussian beams at 36° intervals as output (Section S7 in Appendix A). Compared with previous results, the output Gaussian beam exhibited more speckles in the first time sequence (150 mm) due to the increased number of modes required within the same modulation capability. However, the intensity distributions in the second time sequence (300 mm) were primarily controlled by the diffraction matrix, resulting in minimal impact on spot quality and diffraction efficiency. This phenomenon can be attributed to the limited modulation function in tri-layered time evolution unitary transformation, which could potentially be improved by increasing the number of phase layers (refer to Section S7).

5. Conclusions

In summary, we have introduced a time evolution unitary transformation strategy for reallocating light paths and demonstrated its application in the deep routing of OAM mode multiplexed channels. This innovative approach harnesses both mode and time DoFs for selective light-path reallocation and deep evolution during transmission, overcoming the limitations of traditional mode multiplexed channel routing technologies. Our experimental results confirm the effectiveness of this approach, as we successfully deep-routed four multiplexed OAM modes across two spatial depths using a tri-layered modulation matrix, achieving a remarkable diffraction efficiency of 78.31%. We further demonstrated the practical application of deep routing by transmitting 4.69 Tbit·s⁻¹ QPSK signals. In this experiment, all channels were successfully routed with a BER below 10⁻⁶. In addition, we discussed the versatility of time evolution unitary transformation in terms of deep routing, highlighting its capabilities in modulating time-sequence and mode processing. We anticipate that this proposed OAM mode deep evolution strategy will provide a new approach for deep routing, enabling complicated interaction among optical nodes and facilitating the development of communication networks.

Acknowledgments

This study was funded by the National Natural Science Foundation of China (62271322), the Guangdong Basic and Applied Basic Research Foundation (2022A1515011003 and 2023A1515030152), and the Shenzhen Science and Technology Program (JCYJ20210324095610027 and JCYJ20210324095611030).

Compliance with ethics guidelines

Zebin Huang, Peipei Wang, Jiafu Chen, Wenjie Xiong, Huapeng Ye, Xinxing Zhou, Ze Dong, Dianyuan Fan, and Shuqing Chen declare that they have no conflict of interest or financial conflicts to disclose.

Appendix A. Supplementary material

Supplementary data to this article can be found online at https://doi.org/10.1016/j.eng.2024.09.016.

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