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Abstract
Cylindrical vector beams (CVBs) exhibit great potential for multiplexing communication, owing to their mode orthogonality and compatibility with conventional wavelength multiplexing techniques. However, the practical application of CVB multiplexing communication faces challenges due to the lack of effective spatial polarization manipulation technologies for (de)multiplexing multi-dimensional physical dimensions of CVBs. Herein, we introduce a wavelength- and polarization-sensitive cascaded phase modulation strategy that utilizes multiple coaxial metasurfaces for multi-dimensional modulation of CVBs. By leveraging the spin-dependent phase modulation mechanism, these metasurfaces enable the independent transformation of the two orthogonal polarization components of CVB modes. Combined with the wavelength sensitivity of Fresnel diffraction in progressive phase modulation, this approach establishes a high-dimensional mapping relationship among CVB modes, wavelengths, spatial positions, and Gaussian fundamental modes, thereby facilitating multi-dimensional (de)multiplexing involving CVB modes and wavelengths. As a proof of concept, we theoretically demonstrate a 9-channel multi-dimensional multiplexing system, successfully achieving joint (de)multiplexing of 3 CVB modes (1, 2, and 3) and 3 wavelengths (1550 nm, 1560 nm, and 1570 nm) with a diffraction efficiency exceeding 80%. Additionally, we show the transmission of 16-QAM signals across 9 channels with the bit-error-rates below 10−5. By combining the integrability of metasurfaces with the high-dimensional wavefront manipulation capabilities of multilevel modulation, our strategy can effectively address the diverse demands of different wavelengths and CVB modes in optical communication.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Cylindrical vector beams (CVBs) with spatially helical polarization distributions offer a promising approach to enhance optical communication capacity through mode multiplexing [1–7]. Since CVB modes are independent of conventional multiplexing dimensions such as wavelength and polarization [8–10], they provide the potential for developing multi-dimensional multiplexing communication to improve spectral efficiency. Various methods have been proposed to modulate CVB modes, including Q-plates [11,12], vector Damman vortex gratings [13], and coordinate transformations [14,15]. For instance, using the relationship between left- and right-handed circularly polarized (LHCP/RHCP) components of CVB mode and diffraction order in Damman gratings, multiple off-axis CVB modes can be angularly coupled and separated simultaneously to achieve mode multiplexing and demultiplexing [8,16,17]. However, this method requires strict angular alignment and faces efficiency limitations imposed by Bragg diffraction conditions, restricting the number of supported modes. An alternative method is vectorial optical coordinate transformation, which allows efficient mode demultiplexing by applying a conjugate angular-lateral phase gradient mapping to the two orthogonal components of the CVBs [18–21]. While theoretically supporting an infinite CVB modes, challenges arise due to the irregular mode field structure, limiting its application in multiplexing scenarios. Additionally, these methods typically operate at specific wavelengths or cross broad spectral bands [9,22,23], necessitating multiple discrete optical components for wavelength multiplexing. This results in complex and bulky systems. More recently, researchers have proposed a multi-dimensional light field demultiplexing approach based on a single-layer metasurface, leveraging advances in polarization and dispersion manipulation [24–27]. This approach achieves simultaneous demultiplexing of wavelengths, polarizations, and modes by focusing multi-dimensional light fields onto distinct spatial positions in the focal plane [28]. However, it faces challenges related to the diffraction limit, requiring substantial wavelength separation to minimize crosstalk between adjacent channels, making it difficult to improve spectral efficiency. Furthermore, it suffers from low energy efficiency due to Bragg diffraction limitations in the metasurface design [29]. Therefore, the multi-dimensional (de)multiplexing of CVBs remains elusive because of the absence of effective wavelength modulation techniques and spatial polarization manipulation technologies.
Herein, we introduce a wavelength- and polarization-sensitive cascaded phase modulation strategy for achieving simultaneous (de)multiplexing of CVB modes and wavelengths using cascaded metasurfaces. These metasurfaces are composed of spatially periodic arrangements of subwavelength nanostructures, allowing them to independently modulate the orthogonal polarization components of the light field through a spin-to-orbit independent modulation mechanism [30,31]. This provides a powerful modulation platform for structure lights. By carefully configuring the size of the meta-atoms on each pixel, we can introduce independent phase shifts in the two orthogonal channels of LHCP and RHCP light, enabling the spatial transformation of CVB modes. Since the efficiency and functionality of single-layer modulation [8,16,24] are limited by grating diffraction conditions, we propose a multi-level modulation architecture [32–34]. By employing a wavefront matching strategy [35], we establish a high-dimensional mapping between the input and output mode sets and perform progressive phase modulation of the wavefront to achieve the interconversion of CVB and Gaussian modes. More critically, by introducing discrete wavelength channels in the wavefront matching process and then integrating the precisely modulated wavelength-sensitive Fresnel matrix into the transformation matrix, we establish a wavelength-spatial position mapping relationship within the CVB-Gaussian mode mapping. As a result, we can spatially sort and transform CVB modes and wavelengths, enabling the multi-dimensional (de)multiplexing of CVBs involving modes and wavelengths.
To verify the feasibility of the proposed strategy, we constructed a 9-channel multi-dimensional (de)multiplexer using 5-layer cascaded metasurfaces. Numerical results show that 3 CVB modes (1, 2, and 3) and 3 wavelengths (1550 nm, 1560 nm, and 1570 nm) are successfully multiplexed and demultiplexed, achieving a diffraction efficiency exceeding 80%. We further constructed a wavelength- and mode-multiplexing communication link and transmitted 16-QAM signals across 9 channels with the bit-error-rates (BERs) lower than 10−5. Moreover, we investigated the minimum wavelength interval and demonstrated that the separation and coupling efficiency of multi-dimensional CVBs can reach 80% at wavelength intervals larger than 3 nm, which is comparable to the performance of dense wavelength division multiplexing. This underscores the generalizability of the strategy. This strategy is also compatible with polarization multiplexing, as evidenced by its ability to manipulate the radial/azimuthal polarizations of CVBs. By combining the compactness of metasurfaces with the versatile wavefront manipulation capabilities of multi-level modulation, our approach offers highly integrated and powerful manipulation of high-dimensional light fields, including wavelengths, polarization, and modes. This capability can effectively meet the diverse demands for controlling light fields in multi-dimensional multiplexing communications and advance its practical applications.
2. Principles and methods
By combining the Pancharatnam-Berry (P-B) phase [36] with the propagation phase [37], the metasurfaces can achieve spin-dependent phase modulation of the circularly polarized basis vectors of light fields. This polarization-dependent phase modulation is utilized for spatial modulation of CVB, which can be understood as the coherent superposition of two orthogonal circular polarizations, as illustrated below [38]:
By harnessing the spin-dependent modulation capabilities of metasurface, we introduce a wavelength-sensitive cascaded multilevel phase modulation framework. This framework integrates two subsystems designed for the progressively processing of the RHCP and LHCP components of CVBs, aiming to achieve the coupling and separation of CVBs with different orders and wavelengths. The working principle of this multilevel phase modulation strategy is depicted in Fig. 1. With the inclusion of spatial position, this strategy allows for the construction of a multi-dimensional Hilbert space. The basis vectors within this space consist of mode, wavelength, and spatial position. A mapping matrix is used for vector transformations within this space. This matrix enables the mapping of pre-matched vectors to one another, thereby realizing spatial position translation and mode transformation for each VB. The vector mapping operation is carried out by alternately employing free space and phase modulation. The mask used for phase modulation is obtained through a wavefront matching inverse design approach.
Fig. 1. Principle of wavelength- and polarized-sensitive cascaded phase modulation strategy. M, Mask.
Wavefront matching involves calculating the phase distributions of all phase masks by ensuring alignment between the forward propagating input field (h) and the back-transmitting output field (g) on designated planes [35]. The phase distributions for each plane are updated iteratively by continuously propagating the input field forward and the output field backward numerically through the system until the evaluation function reaches convergence. Typically, phase masks designed using wavefront matching are only suitable for a specific, single wavelength [32,33]. To realize phase modulation of different wavelength channels, we construct a wavelength-sensitive Fresnel modulation matrix, which is created by introducing multiple discrete wavelength channels in the wavefront matching process. As shown in the block diagram at the bottom of Fig. 1, by selecting several specific wavelengths and concurrently monitoring the evaluation function for each wavelength channels during the iterative optimization process, the model can solve for wavelength-selective phase distributions for the phase marks. The corresponding phase distribution of the k-th layer is defined as follows:
After multiple matching iterations, the optimized masks can closely approximate the numerical solution of the mapping matrix that relates the two sets of light fields h and g within the vector space. By applying this wavefront matching algorithm to multiple wavelength channels, the optimized multi-layer phase masks for modulating the RHCP and LHCP components of CVBs can be separately determined and implemented on a set of polarization-dependent metasurfaces. This configuration forms a multi-dimensional (de)multiplexer. We employ the Particle Swarm Optimization (PSO) algorithm [39,40] to optimize the external parameters of the (de)multiplexer, which include the waists of CVB modes and Gaussian beam, the spacing of Gaussian beam, the spacing between adjacent phase masks, and the distance from Gaussian beam to the first phase mask. The evaluation function during the optimization process is the overlap integral between the output and target modes. Through iterative wavefront matching and PSO calculations, the optimal values for each parameter can be determined. More details on parameter optimization can be found in the Supplemental document.
3. Results and analyses
To validate the feasibility of the proposed scheme, we constructed a joint (de)multiplexer consisting of 5-layer cascaded metasurfaces, as shown in Fig. 2(a). Each metasurface contains 256 × 256 meta-atoms. The optimized design parameters include: the spatial distance between adjacent metasurface of 253 µm, the distance from the Gaussian beam array to the first metasurface of 77.7 µm, and the mode field diameter of CVBs of 19.3 µm. The Gaussian beam array (x-polarization) features a mode-field diameter of 12 µm and a pitch of 15 µm. We designed the (de)multiplexer for three wavelengths (1550 nm, 1560 nm, and 1570 nm) and three CVB modes (l = 1, 2, 3). The optimized phase distributions of phase marks for the LHCP and RHCP channels are shown in Fig. 2(b). The performances of multiplexing and demultiplexing multi-dimensional CVBs is demonstrated in Fig. 3. In Fig. 3(a), the two rows show the inputs and outputs at the multiplexing stage, where the inputs are a 3 × 3 array of Gaussian beams with positions corresponding to the wavelengths and modes of the CVBs after multiplexing. It should be noted that the metasurface has different initial phases for the gradient phase modulation of the RHCP and LHCP. This phase difference leads to an effect similar to that of a half wavelength plate, which deflects the polarization of the output CVB. Because this polarization deflection is fixed, it does not affect the performance of the (de)multiplexer. Figure 3(b) shows the inputs and outputs at demultiplexing stage, where the inputs consist of coaxial CVBs carrying different wavelengths and different modes, and the output Gaussian beams are positioned according to the wavelength and modes of the corresponding input CVBs. Figure 3(c) presents the calculated diffraction efficiencies of the RHCP and LHCP channels after multiplexing, which exceed 80% for all channels. Figure 3(d) shows the efficiency of the output Gaussian beam in the target and non-target regions after demultiplexing, which were obtained by normalizing the light intensity in the target and non-target regions for different channels. The results shows that the energy percentage of the target area is above 90%. This is consistent with the spatial separation of CVBs with different wavelengths and modes. Although the intensities of the output multiplexed CVBs exhibit non-uniformities and losses limited by the currently optimization algorithm, the overall results agree well with theory.
Fig. 2. (a) Functional schematic of the (de)multiplexer. FA, Fiber Array; MS, Metasurface; S, The spacing of the metasurface. (b) Phase of the mask corresponding to RHCP and LHCP, respectively.
Fig. 3. Results of multiplexing and demultiplexing. (a-b) Input and output corresponding multiplexing and demultiplexing. (c) Diffraction efficiency of LHCP and RHCP channels when Gaussian beams multiplexed. (d) Energy percentage of the target area and the non-target area after demultiplexing.
To validate the communication performance of the designed multi-dimensional CVB (de)multiplexer, we constructed a 9-channel multi-dimensional multiplexing link consisting of 3 CVB modes and 3 wavelengths. Each channel carries 16-QAM signals. Specifically, nine x-polarized Gaussian beams are grouped into three, with signals at 1550 nm, 1560 nm, and 1570 nm, respectively, forming a 3 × 3 Gaussian beam array. The array is incident on the first metasurface of the (de)multiplexer and converted into coaxial CVBs with different modes after passing through the first to the fifth metasurface sequentially. After propagating a certain distance, the coaxial CVBs are reverse-incident on the fifth metasurface, passing through the fifth to the first metasurface, eventually being demultiplexed back to a Gaussian beam array. In Fig. 4(a), we can observe the BERs of the nine CVB channels over different SNRs. When the SNR exceeds 19 dB, the BERs of all channels are below the forward error correction (FEC) threshold of 3.8 × 10−4. At 24 dB SNR, the BERs reduce to below 10−4 for all channels. Notably, the three channels with different wavelengths (1550 nm, 1560 nm, and 1570 nm) and the same mode (l = 2) have significantly higher BERs than the other six because their demultiplexed Gaussian beams are all located in the middle and have higher crosstalk. These results are also reflected in the error vector magnitude (EVM) curves shown in Fig. 4(b). Additionally, in Fig. 4(c), the constellation diagrams of two example channels (wavelengths and modes are [1550 nm, 1] and [1560 nm, 2], respectively) at SNR of 20 dB and 24 dB are depicted. It can be seen that the constellation points of channels with different wavelengths and modes are separated from each other and converge with increasing SNR, which is in line with theoretical expectations. The constellation diagrams of the two channels are very similar, indicating that channels with different wavelengths and modes receive similar crosstalk. These results show that the designed (de)multiplexer is feasible and reliable in multi-dimensional CVB multiplexing communications.
Fig. 4. Simulation results of multi-dimensional CVB (de)multiplexing communication. (a) BER as a function of SNR for multiplexed 9 CVB channels. FEC, forward error correction. (b) EVM of 9 CVB channels. (c) Constellations diagrams of two CVB channels (wavelengths and modes are [1550 nm, 1] and [1560 nm, 2], respectively).
4. Discussion
Coupling and separating CVBs carrying different wavelengths and modes are crucial procedures in multi-dimensional CVB multiplexing communication. Based on the Jones matrix, CVBs can be generated using a pair of orthogonally circularly polarized lights with opposite helical phases. This characteristic allows for CVB modulation through a polarization- dependent phase modulation method. In this study, we propose a cascaded phase modulation strategy that is sensitive to both wavelength and polarization, which enables the coupling and separation of multi-dimensional CVBs. To achieve this, we obtained two sets of wavelength-sensitive phase masks for the LHCP and RHCP channel mode mapping. These phase masks are obtained using a wavelength-variant wavefront matching algorithm. Subsequently, these phase masks are implemented onto a set of polarization-sensitive metasurfaces, which act as (de)multiplexer. These metasurfaces independently phase modulate the two orthogonal circular polarizations, allowing for the modulation of multi-dimensional CVBs, including wavelengths and modes. It is important to note that although only a few wavelengths are included in the algorithm, the number of modes and wavelengths that can be (de)multiplexed can be increased by designing the phase mask in a rational manner.
To further demonstrate the universality of the proposed strategy in the wavelength dimension, we trained (de)multiplexers for different wavelength intervals. In Fig. 5(a), we compare the performance of (de)multiplexers trained at different wavelength intervals, taking channel 1 (1550 nm, l = 1) as a reference. The first and second rows of Fig. 5(a) show the light intensity of channel 1 when multiplexed and demultiplexed using optimized (de)multiplexers trained with different wavelength intervals. When a Gaussian light is input, these multiplexers with different wavelength intervals can generate relatively satisfactory CVBs, and their diffraction efficiency increases with the widening of wavelength intervals. For intervals greater than 3 nm, the diffraction efficiencies exceed 85%. When the input is a CVB, the output Gaussian beams maintain over 80% energy in the target region for intervals greater than 3 nm, as shown in Figs. 5(b-c). These results indicate that the proposed multi-dimensional CVB (de)multiplexing strategy is not limited to a few wavelengths, but can flexibly select wavelengths for separation and coupling. Furthermore, the approach is also compatible with the polarization dimension, which is demonstrated by the radial and azimuthal distribution of CVB spatial polarization profiles [41]. We have shown the (de)multiplexing of radial CVBs, and the results of (de)multiplexing of azimuthal CVBs can be found in the Supplemental document. By incorporating a polarization beam splitter, the designed multiplexer enables three-dimensional (de)multiplexing of polarized-wavelength-CVB modes.
Fig. 5. (a) Multiplexing and Demultiplexing results of channel 1 (wavelength is 1550 nm and CVB mode is 1) when the training wavelength intervals are different. (b) Diffraction efficiency of multiplexing. (c) Energy percentage of the target area and the non-target area after demultiplexing.
5. Conclusion
In conclusion, we have proposed a wavelength- and polarization-sensitive cascaded phase modulation strategy for multi-dimensional CVB (de)multiplexing. By utilizing the spin-dependent phase modulation of metasurfaces and the wavelength selectivity of multi-level modulation, we can achieve mutual conversion between Gaussian beams at different off-axis positions and wavelengths, resulting in coaxial CVBs with different modes and enabling the (de)multiplexing of multi-dimensional CVBs. We have successfully demonstrated the (de)multiplexing of 9 CVB channels with 3 wavelengths (1550 nm, 1560 nm, and 1570 nm) and 3 modes (l = 1, 2, and 3) using 5-layer metasurfaces. The mode conversion efficiency exceeds 80%. Additionally, we have numerically constructed a 9-channel multi-dimensional communication link and successfully transmitted 16-QAM signals with the BERs below 10−5, validating the performance of the designed (de)multiplexer. Our demonstrations provide a universal approach for joint (de)multiplexing wavelength and CVB modes, and the combination with polarization modulation technology enables flexible manipulation of CVBs. This work offers new insights into multi-dimensional spatial optical field modulation techniques and opens up possibilities for advanced multi-dimensional communication systems.
Funding
National Natural Science Foundation of China (62271322, 62275162); Guangdong Basic and Applied Basic Research Foundation (2021A1515011762, 2022A1515011003, 2023A1515030152); Shenzhen Science and Technology Program (JCYJ20200109144001800, JCYJ20210324095610027, JCYJ20210324095611030, SZWD2021013); Natural Science Foundation of Top Talent of SZTU (GDRC202204).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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