Abstract
Spatial mode (de)multiplexing of orbital angular momentum (OAM) beams is a promising solution to address future bandwidth issues, but the rapidly increasing divergence with the mode order severely limits the practically addressable number of OAM modes. Here we present a set of multivortex geometric beams (MVGBs) as highdimensional information carriers for freespace optical communication, by virtue of three independent degrees of freedom (DoFs) including central OAM, subbeam OAM, and coherentstate phase. The novel modal basis set has high divergence degeneracy, and highly consistent propagation behaviors among all spatial modes, capable of increasing the addressable spatial channels by two orders of magnitude than OAM basis as predicted. We experimentally realize the triDoF MVGB mode (de)multiplexing and data transmission by the conjugated modulation method, demonstrating lower error rates caused by center offset and coherent background noise, compared with OAM basis. Our work provides a potentially useful basis for the next generation of largescale dense data communication.
Introduction
Multiplexing of independent optical degrees of freedom (DoFs) such as polarization and wavelength have long been implemented to increase the capacity of optical communication systems^{1,2,3,4}. The exploration of spatial DoFs of optical fields has offered new possibilities that modedivisionmultiplexing (MDM) scales the capacity by a factor equal to the number of spatial modes acting as independent information channel carriers^{5,6,7,8,9}. Among all spatial modes, the use of orbital angular momentum (OAM) beams, which can accommodate theoretically infinite orthogonal modes, has generated widespread and significant interest in the last decade^{10,11,12,13,14,15,16,17,18}. However, in practice, OAM modal basis set alone cannot reach the capacity limit of a communication channel^{19}, since the beam diverges rapidly as the OAM order enlarges, which gives rise to increased power loss for a limitedsize receiver aperture. To guarantee sufficient received optical power for data recovery, the number of OAM modes that can be practically supported is severely limited under 60^{20,21}, mostly under 20^{22,23,24,25,26,27,28,29}. One can relax the limit and increase the maximum number of addressable spatial channels, by enhancing the divergence degeneracy, i.e. having more orthogonal spatial modes propagating in identical manner. To this end, incorporating both the radial and azimuthal components of Laguerre–Gaussian (LG) beams^{30,31,32,33} makes one constructive step, but the improvement is far from satisfactory. To meet the growing demand for data capacity, it is highly desirable to use a large set of spatial modes with the variation in beam quality among all modes as small as possible.
In recent years, a class of exotic structured optical fields termed raywave geometric beams has attracted much attention, whereby crafted spatial modes appear to be both wavelike and raylike^{34,35,36,37,38,39,40,41,42,43,44}. In the wave picture, the beam is a coherent laser mode, imbued with a typical OAM feature. In the ray picture, the mode is coupled with a cluster of geometric rays, unveiling new controllable DoFs that notably increase the divergence degeneracy, such as subOAM (partial vortex along each ray) and coherentstate phase (the phase to tune the ray sequence).
In this work, we demonstrate that the modal basis of raywave geometric beams outperforms the OAM and LG modal basis, in terms of approaching the capacity limit of a freespace optical communication channel. Specifically, we create a threedimensional set of orthogonal datacarrying beams, by employing three independent intrinsic DoFs of the multivortex geometric beam (MVGB), one type of raywave geometric beams, including the central OAM, subbeam OAM and coherentstate phase. We show the MVGB set is extremely densely packed in beam quality space and has a highly consistent propagation behavior that it can possess a divergence degeneracy as high as 20, a 20X increase over OAM modal basis, and a divergence variation by merely 18% among 100 independent lowest order spatially multiplexed modes, in contrast to 900% for OAM counterpart and 429% for LG counterpart. As a result, thousands of independently spatial channels in MVGB basis can be supported in a freespace optical communication system, two orders of magnitude larger than that in OAM basis. To validate the performance of the highdimensional information carriers, we analyze in detail the orthogonality of MVGB mode on different spatial indices, based on which we experimentally realize the triDoF mode (de)multiplexing and shiftkeying encoding/decoding by the conjugated modulation using digital micromirror device (DMD). The results also indicate another distinct advantage of MVGB basis in demultiplexing with much lower bit error rate (ER) caused by the center offset and the coherent background noise, and thus having lower pixel ER in freespace data transmission, compared with OAM basis. We believe the divergencedegenerate MVGB modal set provides a useful basis for boosting the capacity of future optical communication systems.
Results
TriDoF MVGBs
The raywave geometric beam can be represented as the superposition of a family of eigenstates (HermiteLaguerreGaussian (HLG) modes) with subPoissonian distribution:
where N + 1 is the number of eigenmodes in the frequencydegenerate family of \({{{\mathrm{HLG}}}}_{n_0 + pK,m_0 + qK}^{\left( {\alpha ,\beta } \right)}\), p and q are ratios of transverse frequency spaces in the x and yaxis, respectively, n_{0} and m_{0} are the initial orders of transverse mode in the x and yaxis respectively, and ϕ is the coherentstate phase. In particular, when α = β = ±π/2, the HLG eigenmode degenerates to LG mode^{44}.
For the case of p = Q and q = 0, the raywave geometric beam is referred to as MVGB, having Q vortex subbeams. In a MVGB, the coherentstate phase ϕ acting as one DoF is manifested in the orientation of petallike intensity pattern, as shown in Fig. 1a that the rotation of orientation angle relative to the case of ϕ = 0 is ϕ/Q. The other two DoFs to be exploited are n_{0} and m_{0}, the values of central OAM and subbeam OAM, respectively, as demonstrated in the phase distribution in Fig. 1a. Hereinafter, we focus on the MVGB with Q = 5 as an example, the expression of which is thus abbreviated as \(\left {{{\Psi }}_{n_0,m_0}^\phi } \right\rangle ^N\) for the sake of brevity.
Before the demonstration of potential use of MVGBs in MDM communication application, it is crucial to first investigate the orthogonality of MVGBs in terms of three spatial indices, using the correlation degree as the metric. The correlation degree of 0 and 1 represent orthogonal and completely nonorthogonal conditions for two MVGBs, respectively. The correlation degree of two MVGBs with different parameters n_{0}, m_{0} and ϕ, as represented by the inner product mathematically, is given by
where sign ’∼’ meansconjugate, and \({\iint} {\widetilde {{{{\mathrm{HLG}}}}}_{n_s,m_s}^{\left( {\pi /2,\pi /2} \right)} \times {{{\mathrm{HLG}}}}_{n_r,m_r}^{\left( {\pi /2,\pi /2} \right)}{\rm{d}}x{\rm{d}}y} = \delta _{(n_s,n_r),(m_s,m_r)}\)(where \(\delta _{(n_s,n_r),(m_s,m_r)} \,\ne\, 0\) only when ns = nr and m_{s} = m_{r}).
According to Eq. (2), the theoretical results of the correlation degree analysis of MVGBs are shown in Fig. 1b. Basically, three spatial indices (n_{0}, m_{0}, and ϕ) are uncoupled and independent of each other. The orthogonalities of two MVGBs associated with each of three spatial indices are examined as follows. First, two MVGBs with different central OAM values (n_{0}) are mutually orthogonal to each other when n_{0s}−n_{0r}  ≠ ZQ (Z is an integer and \({Z}\, \leqslant \, {N}\)), as shown in Fig. 1b1 that the order spacing of the nonorthogonal mode is Q. Second, since q = 0 for MVGBs, it is natural that MVGBs with different subbeam OAM values (m_{0}) are orthogonal to each other, same as the case of general OAM beams, as illustrated in Fig. 1b2. Third, two MVGBs with the coherentstate phase (ϕ) being 0 and π respectively are orthogonal, which is analogous to the case of left and righthand circular polarization states, as depicted in Fig. 1b3. Furthermore, note that two MVGBs can be regarded as quasiorthogonal when ϕ_{s} − ϕ_{r} = π/2 (N > 5), since the correlation degree is less than 0.1, as described in Fig. 1b4. Therefore, ϕ can take up to four values (0, π/2, π, and 3π/2) when N > 5, for the realization of efficient mode (de)multiplexing. So far, we have obtained the MVGB modal basis set characterized by three spatial indices of n_{0}, m_{0}, and ϕ, enabling a combination of 4 × K_{n} × K_{m} readily available spatial modes as information carriers, where K_{n} and K_{m} are the numbers of central OAM and subbeam OAM states selected from theoretically unbounded states respectively. The detailed orthogonality analysis of general raywave geometric beam can be found in Supplementary Note 1.
High divergence degeneracy
The beam propagation dynamics in free space is vital for freespace optical communication^{45} and governed by the beam quality factor M^{2} entirely. The dynamic transmission characteristics of the beam include the beam size and divergence angle, where the divergence angle of the beam determines the transverse spatial frequency of the beam. For a LG_{pl} mode and a MVGB mode as a superposition of multiple higherorder eigenmodes (\({{{\mathrm{HLG}}}}_{n,m}^{(\alpha ,\beta )}\)), the beam quality factors are respectively expressed as^{46}
where \(c_{nm} = \frac{1}{{2^{N/2}}}\left( {\begin{array}{*{20}{c}} N \\ K \end{array}} \right)^{1/2}\) are normalized amplitudes for the eigenmodes \({{{\mathrm{HLG}}}}_{n_0 + QK,m_0}^{\left( {\alpha ,\beta } \right)}\) with n = n_{0} + QK and m = m_{0} in Eq. (1), and the total power \(\mathop {\sum}\nolimits_{m = 0}^\infty {\mathop {\sum}\nolimits_{n = 0}^\infty {\left {c_{nm}} \right^2} } = \frac{1}{{2^{N/2}}}\mathop {\sum}\nolimits_{K = 0}^N {\left( {\begin{array}{*{20}{c}} N \\ K \end{array}} \right)^{1/2}} = 1\)
It can be seen from Eq. (4) that the beam quality factor of the MVGB depends entirely on the family of eigenmodes it contains and the corresponding normalized weighting factor. Notably, the coherentstate phase parameter does not affect the superposition components and weighting factors, thus the additional DoF of ϕ can scale the beam quality degeneracy by a factor of 4 (equal to the number of employed values of ϕ), compared with twodimensional LG modal basis. For instance, as Fig. S3 shows, among the 100 lowest orders of MVGB modes, by combinations of m_{0} = {0,1,2,3,4}, n_{0} = {0,1,2,3,4}, and ϕ = {0,π/2,π,3π/2}, the maximum beam quality degeneracy reaches as high as 20, that up to 20 modes share the same beam quality factor of M^{2} = 17.5. This leads to only a total of 9 beam quality factors from all the 100 modes: M^{2} = {13.5,14.5,15.5,16.5,17.5,18.5,19.5,20.5,21.5}. In contrast, the 100 lowest orders of LG_{pl} modes, by combinations of p and l both taking 10 integer values from 0 to 9, have 28 integer values of M^{2} from 1 to 28.
Similarly, the divergence and beam waist diameter of MVGBs have high degeneracies. The beam size can be calculated by the second moment of intensity^{47}, thus we have the beam waist diameter D_{0,m} and divergence θ_{m} of morder LG modes and MVGBs expressed as:
where I_{0}(r, ϕ) is the intensity distribution of beam waist crosssection.
Figure 2a compares the divergences of OAM modes, LG modes and MVGBs in respective 100 lowest orders, all normalized to the divergence angle of fundamental Gaussian beam (θ_{0}). Note that the maximum divergence degeneracy is 20, that 20 modes share the same divergence of θ_{m} = 4.18 θ_{0}, and the divergence varies by merely 18% among 100 independent lowest order spatially multiplexed modes of MVGBs, in contrast to 900% for OAM counterpart and 429% for LG counterpart. This brilliant feature of MVGB basis results in a highly consistent propagation behavior of data channels, which is beneficial for the beam tracking, and alignment control of receiver optics and adaptive optics^{6}. The corresponding variations in beam waist diameter for LG modes and MVGBs are compared in Fig. S4.
By the virtue of high divergence degeneracy, we claim that the MVGB basis can achieve capacity beyond OAM and LG counterparts. To confirm this, we count the number of MVGB modes that fit into a lineofsight freespace communication system with a space–bandwidth product (SBP) of 2R_{0} × 2NA/λ, where R_{0} and NA are the aperture radius and numerical aperture of both circular apertures of transmitter and receiver, and λ is the wavelength. Following the procedure of ref. ^{19}, we define a dimensionless parameter S, which is π/4 times the SBP, and then estimate the lower and upper bound on the number of independently addressable spatial subchannels Q, counting all MVGB modes that satisfy \(M_{{{{\mathrm{MVGB}}}}}^2 \, \leqslant \, S\), as given by
where ⌊⌋ represents the floor integration.
The estimated subchannel numbers of MVGB multiplexing in an SBPlimited freespace optical communication system based on Eq. 6 and Eq. 7 are compared with those of OAM multiplexing and LG mode multiplexing in Fig. 2b. It is noteworthy that although the lowest beam quality of MVGB is higher than OAM beams and LG modes, corresponding to a larger intercept in the xaxis, the lower and upper bound curves of MVGB have far steeper slopes versus S (equivalently the SBP) than OAM and LG modal basis, thereby accomodating far more data channels. For example, at S = 30 (SBP = 38), Q^{OAM} is about 60, Q^{LG} lies between 460 and 1200, and Q^{MVGB} lies between 1200 and 5000, which is about two orders of magnitude larger than OAM basis. The superiority of MVGB modal basis would become even more distinct, in terms of addressable independent spatial subchannels, when the communication system has higher SBP and larger scales, as indicated by the trend in Fig. 2b. The freespace propagating performance of MVGBs in the presence of atmospheric turbulence is investigated in Supplementary Note 2.
Low ER in MVGB demultiplexing
Despite the widely applied sorting approaches developed for OAM beams and LG beams, such as Dammann vortex grating^{23,48,49}, Gouy phase radial mode sorter^{50}, logpolar based azimuthal mode sorters^{51,52}, interference and diffraction method^{53,54,55}, and deep learning^{56}, identification and sorting of raywave geometric beams such as MVGBs is still at its infancy, due to the intrinsic complex structure and rich controlling parameters. Inspired by the recent work of digital cavityfree tailoring^{44}, here we demonstrate the sorting and demultiplexing of superposed MVGBs in the following experiments, using the demultiplexed conjugated holographic masks that are designed by diffracting each beam component into different location^{57,58}, as detailed in “Materials and methods”. Figure 3 shows the experimental results of MVGB demultiplexing associated with each of three spatial indices. For the DoF of ϕ, subfigures a_{1} to a_{4} demonstrate the intensity profiles of demultiplexed beam components separated along the x direction. The corresponding input of collinearly superposed MVGBs containing one to four beam components are \({\sum} {_{\phi = 0}\left {{{\Psi }}_{5,10}^\phi } \right\rangle ^5}\), \({\sum} {_{\phi = 0,\pi }\left {{{\Psi }}_{5,10}^\phi } \right\rangle ^5}\), \({\sum} {_{\phi = 0,\pi /2,3\pi /2}\left {{{\Psi }}_{5,10}^\phi } \right\rangle ^5}\)and \({\sum} {_{\phi = 0,\pi /2,\pi ,3\pi /2}\left {{{\Psi }}_{5,10}^\phi } \right\rangle ^5}\), respectively. Corresponding demultiplexed conjugated holographic masks are designed as \(T(x,y) = \frac{1}{2} + \frac{1}{2}{{{\mathrm{sign}}}}\left[ {\cos \left( \Phi \right) + \cos \left( {{{{\mathrm{arcsin}}}}A} \right)} \right]\), where A and Φ are respectively the amplitude and phase of noncollinearly superposed conjugated optical field of MVGBs \({\sum} {_{\phi = 0,\pi /2,\pi ,3\pi /2}\left {\widetilde {{\Psi }}_{5,10}^\phi } \right\rangle ^5\exp \left( {{{{\mathrm{i}}}}2\pi u_\phi x} \right)}\) (see details in “Materials and methods”). The four dotted circles in each subfigure of a_{1–4} indicate the target diffracting locations of all four beam components by the holographic mask design, among which the yellow ones are signal locations, corresponding to those beam components that are practically multiplexed in the input beams, and the blue ones are nonsignal locations. The same method is applied to OAM demultiplexing for the comparison in the next subsection, as shown in Fig. 4. Moreover, an 8bit and 16bit hybrid shiftkeying encoding/decoding scheme with triDoF MVGBs are demonstrated with zero bit ER, as detailed in Supplementary Note 3. Another advantage of MVGB multiplexing is manifested in the low bit ER in the demultiplexing process of conjugated modulation.
Coherent light sources are widely exploited in MDM communication, since the mode coding can be done by a modulation device and does not require the spatial coupling. However, coherence may bring difficulty to signal decoding and thus increase the bit ER, which is defined as the proportion of bit values that are incorrectly identified according to the demultiplexing results. On the one hand, the intensity peak of a coherently superposed beam may deviate from the copropagating optical axis, leading to the center offset, lateral displacement of intensity peak relative to the target location, of decoding spot and thus possibly introducing a bit error that the bit value of 1 is incorrectly identified as 0, e.g. the marked peak in Fig. 4(d_{4}) for the case of OAM beams. The bit ER caused by center offset depends on the radius of the discrimination region (RD) that the bit error induced by center offset is valid only when the offset is larger than the RD. On the other hand, the background noise embedded in different beam components may be coherently superposed near the target diffracting position, resulting in a noise peak beyond the discrimination threshold (DT) that may introduce a bit error that bit value of 0 is incorrectly identified as 1, e.g. the marked peaks in subfigures e_{1} and e_{2} of Fig. 4. The bit ER caused by background noise depends on DT that a bit error is valid when the intensity peak of background noise is higher than DT in the discrimination region.
We emphasize that MVGBs have remarkably stronger capability, in contrast to OAM beams, in suppressing the bit ER caused by center offset and nonnegligible background noise in the demultiplexing process of coherent light, thereby efficiently improving the signaltonoise ratio and reducing the intermodal crosstalk. First, due to the inherently more complicated intensity and phase distribution of MVGB than OAM beam, the intensity profile of superposed MVGBs tends to maintain the centrosymmetric feature especially on the DoF of ϕ, which is vital in mitigating the center offset. Second, for the input beam components that do not match with the demultiplexed mask, the complex phase structure of MVGB enables the responses at the nonsignal locations as weak dispersing speckles (see Fig. 3), rather than concentrated spot patterns which are common in the case of OAM beams (see Fig. 4).
To further verify the superiority, we compare the demultiplexing performance of MVGBs and OAM beams in terms of the measured bit ERs caused by center offset and background noise, as shown in Fig. 5a, b, respectively. The average offsets of the MVGBs on three DoFs (ϕ, n_{0}, m_{0}) are 2.69 µm, 4.47 µm, and 3.95 µm, respectively, while those for OAM beams (mode spacing of ∆l = 1, 2, and 3) are 6.62 µm, 5.31 µm, and 5.25 µm, respectively. As a result, the bit ER averaging among all values of RD (from 4 µm to 11 µm) are 0.16, 0.37, and 0.31 for MVGBs (ϕ, n_{0}, m_{0}), and are 0.59, 0.48, and 0.46 for OAM beams (∆l = 1, 2, and 3), respectively, as shown in Fig. 5a. It is of particular interest that MVGBs on the DoFs of ϕ and n_{0} have zero bit ER with RD above 5 µm and 8 µm, respectively. Figure 5b shows the bit ER results merely induced by the background noise, in which a large RD as 30 µm is used to prevent the influence of center offset on the bit ER. It is impressive that MVGB has zero bit ER for demultiplexing on all three DoFs at \({0.4}\,\leqslant\,{\mathrm{DT}}\,\leqslant\,{0.6}\), in which the lower bound of zero bit ER for the case of ϕ can reach as low as 0.2 (not shown in the plot). In contrast, demultiplexing of OAM beams yields a high bit ER by background noise, which increase from 0.071 with DT = 0.6 to 0.196 with DT = 0.4, for the cases of OAM mode spacing of 1. When the OAM mode spacing increases to 3, the bit ER reduces to zero but at a narrow DT range \(({0.4}\,\leqslant\,{\mathrm{ST}}\,\leqslant\,{0.45})\). Note that the bit ER increases rapidly for all the cases with DT higher than 0.6, which is not caused by an extremely high noise level beyond DT, but by the uneven intensity responses among different beam components of signal, causing that certain beam component has a smaller signal intensity than DT and the bit value at corresponding channel is incorrectly identified as 0. All these experimental results show that the triDoF MVGBs outperform the general OAM beams in terms of lowbit ER, indicating that triDoF MVGBs are advantageous as potential highdimensional information carriers.
Lastly, we demonstrate the data transmission by shiftkeying method using the triDoF MVGBs. The data packet used is a 4bit 16level grayscale image composed of 64 × 64 pixels with an equalized graylevel histogram, as shown in Fig. 6a, e. We use four modes by shiftkeying coding that the 4bit grayscale information of each pixel is fully encoded into one pulse. For instance, the grayscale ^{′}0101^{′} is encoded in a collinearly superposed MVGBs \({\sum} {_{\phi = 0,\pi } {{{\Psi }}_{5,10}^\phi } \rangle ^5}\) on the DoF of ϕ (see details in “Materials and methods”). We decode the received information with RD = 11 µm and DT = 0.5. The reconstructed images using MVGB carriers in three DoFs (ϕ, n_{0}, m_{0}) as shown in Fig. 6b–d, are compared with that using OAM beam carriers with different mode spacings (∆l = 1, 2, and 3) as shown in Fig. 6f–h, in which the pixels that receive incorrect graylevel information are marked in green. It can be seen in Fig. 6 that the triDoF MVGB carrier reliably transmits the image, outperforming the general OAM beams in terms of pixel ER. In addition, the pixel ER measured for the MVGB case matches the bit ERs results for DT = 0.5 in Fig. 5b.
Discussion
The emergence of structured light offers a possible solution to meet future demands for communication capacity, by utilizing its spatial DoFs from highdimensional orthogonality. In this work, we introduce a novel modal basis of MVGBs with three DoFs including the central OAM, subbeam OAM and coherentstate phase. At the heart of our work is the exploitation of modes in raywave duality state, which allows us access to higher divergence degeneracy and more consistent propagation behavior among all modes, dramatically increasing the addressable number of independent spatial channels. We validate the potentials of spatially multiplexed MVGB as highdimensional information carriers, by proofofconcept experiments of the triDoF mode (de)multiplexing and shiftkeying encoding/decoding. Notably, we make the challenging demultiplexing task of triDoF modes possible, by proposing a novel approach based on conjugated modulation that can fully resolve the raywave duality state, removing the longstanding obstacle in the identification and sorting of raywave geometric beam that has prohibited its progress. The decoding results show that MVGB modal basis has significant strengths in suppressing bit ERs induced by center offset and coherent background noise, leading to successful datapacket transmission with much lower pixel ER than OAM beams.
The MVGB multiplexing is compatible with and can combine with other techniques, such as wavelength and polarization division multiplexing, and may also work within fibers, to further increase capacity and facilitate the implementation of nextgeneration highcapacity freespace communication network. Our technique could be extended to other types of raywave geometric beams, to explore even more spatial DoFs and higher divergence degeneracy. The concept of triDoF modal basis can also be applied to encoding and decoding in the quantum data channels. In the future, we will exploit the nonseparability among multiple DoFs to further explore its value in realizing highdimensional multipartite entanglement.
Materials and methods
Experimental setup
The experimental setup is shown in Fig. 7. The beam from a solid laser source (CNI laser, MGLIII532nm) is expanded to a nearplane wave by passing through a telescope (F_{1}, focal length of 25 mm; F_{2}, focal length of 300 mm) with the magnification of 1:12, and illuminates DMD #1 loaded with a hologram of the target light field. Then the first order of the beam is selected and reflected by a high reflective (HR) mirror, image relayed by a 4f system with F_{3} and F_{4}, both with the focal length of 150 mm, and transmitted to DMD#2 loaded with holograms of corresponding conjugated optical field. The modulated beam is focused into a spot by a convex lens (F_{5}, focal length of 150 mm), the intensity profile of which is captured by a CCD camera located at the focal plane of F_{5}.
The discrimination and measurement of the correlation degree of raywave geometric beams can be realized by mode projective measurement. The hologram for producing the input beams is loaded into DMD #1, then we load a series of the corresponding conjugated holographic mask into DMD #2 sequentially and capture the corresponding focal spots using CCD camera, which performs a modal decomposition on every DoF. The measured results of correlation degree are shown in Fig. S2. Meanwhile, in the demultiplexing experiments for MVGBs and OAM beams, DMD #1 is loaded with the hologram of the collinearly superposed light field for generating the multiplexed beam, and DMD #2 is loaded with the demultiplexed hologram for separating and identifying beam components in the multiplexed beam.
Data transmission by shiftkeying coding
Experiments of shiftkeying data transmission rely on the demultiplexing results of MVGBs and OAM beams in Fig. 3 and Fig. 4. Taking the DoF of ϕ in Fig. 3 as an example, the states of 1.5π, π, 0.5π, and 0 from left to right correspond to the 4th, 3rd, 2nd, and 1st bit of a 4bit binary signal, respectively. In mode shiftkeying coding, N bits binary sequence corresponds to the collinear superposition coefficients of the beam components in the multiplexed MVGBs, that is, the corresponding MVGB component is presented when the signal bit value is 1, and absent when it is 0. Therefore, the results in Fig. 3(a_{2}) correspond to ^{′}0101^{′} in the binary system and ^{′}5^{′} in the decimal system. The rest of the mode shiftkeying process follows the same principle.
In the experiment, DMD #1 and #2 shown in Fig. 7 serve as the encoder and decoder respectively. First, a sequence of timevarying multiplexed MVGBs or OAM beams is obtained by the holograms loaded on DMD #1 according to the encoded signal. Then the encoded multiplexed beams propagate in the free space and illuminate DMD #2 loaded with a constant demultiplexed conjugated holographic mask, getting separated into 4 diffraction positions, with the focal spots recorded by CCD. Finally, we obtain the results in Fig. 6 by recovering the signal.
Demultiplexing design of MVGBs
The DMD transmission function of the hologram of conjugated optical field modulation is given as:
where\(M(\alpha ,\beta ) = \frac{1}{2} + \frac{1}{2}{{{\mathrm{sign}}}}\left[ {\cos \left( \beta \right) + \cos \left( {{{{\mathrm{arcsin}}}}\alpha } \right)} \right]\) (see details in Supplementary Note 2). The target diffracting position of a beam component is determined by the linear grating period (u_{0}, v_{0}). The demultiplexed conjugated holographic mask is calculated by a noncollinearly superposed conjugated optical field with different periods of linear grating, which means the diffraction direction of each beam component is separated, as shown in Fig. 8a. The noncollinearly superposed conjugated optical field of different linear grating to the corresponding conjugate optical field of MVGBs is:
where \(\widetilde {SU}^n\) are a set of orthogonal MVGB components. According to Eq. 8, different linear grating is added to each conjugate optical field, as described in Fig. 8b. The demultiplexed conjugated holographic mask can be obtained as:
where A^{D} and Φ^{D} are normalized amplitude and phase of CSU, respectively. The u_{n} and v_{n} are the reciprocal of the period of the linear grating in the x and y direction of the nth multiplexed mode, respectively.
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This work was funded by the National Natural Science Foundation of China (61975087).
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Wan, Z., Shen, Y., Wang, Z. et al. Divergencedegenerate spatial multiplexing towards future ultrahigh capacity, low errorrate optical communications. Light Sci Appl 11, 144 (2022). https://doi.org/10.1038/s41377022008344
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DOI: https://doi.org/10.1038/s41377022008344
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