2.1.

Working Principle of Simultaneous Sorting of VSBs

Figure 1 schematically shows the mode detection framework utilizing spin-multiplexed diffractive metasurfaces. First, let us examine the determination of mode indices in VSBs. In theoretical terms, a vectorial structured light field can be represented through the following unnormalized expression:⁵

Fig. 1

Schematic and working mechanism of VSB sorting enabled by spin-multiplexed diffractive metasurfaces. (a) The VSBs exhibit polarization DoFs and spatial DoFs with LG beams, HG beams, and BG beams (including arbitrary superpositions of them). The red line denotes the LCP component, while the blue line signifies the RCP component. (b) Schematic diagram of VSBs sorting based on spin-multiplexed diffractive metasurfaces. The input is a VSB composed of an LG beam, the hidden layer is composed of multilayer spin-multiplexed metasurfaces acting as neurons, and the output is a focused Gaussian bright spot in the planar detection area. (c) The architecture of the DNN. Phase and intensity information of the incident light is processed through several hidden layers and then optimized by an error backpropagation algorithm.

As a typical type of optical neural network, DNN has demonstrated effectiveness in manipulating complex light fields through linear transformations. By precisely adjusting the phase in the hidden layer, DNN can convert the input light field into predefined Gaussian spots across different regions, thus facilitating pattern classification. This capability presents a valuable approach for the detection and characterization of high-order VSBs.

To fully identify the mode indices of VSBs, we introduce a spin-multiplexed metasurface that independently processes two orthogonal circularly polarized waves, creating two distinct output sets, as shown in Fig. 1(b). By examining the spot position on the output plane, we can determine all mode parameters of the incident light, enabling a comprehensive classification of VSBs with a single measurement.

We set the spatial DoFs of an LG beam to be the azimuthal index $l$ and radial index $p$. The VSBs can be written in the following form:

Optimization of the hidden-layer metasurfaces enables gradual manipulation of the incident light field to yield the targeted light field. As a result, accurate identification of VSBs can be accomplished by discerning the spatial position of the focus within the detection zone. For LCP/RCP waves, the focus is assigned within the upper/lower half of the output plane. This methodology has facilitated the detection of spin angular momentum. Importantly, this framework can equally be applied to VSBs composed of BG beams and HG beams.

2.2.

Design of Spin-Multiplexed Metasurfaces

Figure 2(a) shows the schematic diagram of the spin-multiplexed metasurface. The metasurface unit consists of rectangular ${\mathrm{TiO}}_2$ nanopillars on a quartz substrate. The nanopillars have a uniform height $H$, whereas their in-plane dimensions $(D_x,D_y)$ and orientation angle ($\theta$) vary spatially. The finite-difference time-domain (FDTD) algorithm was employed to conduct full-wave simulations to scrutinize the transmission attributes of the ${\mathrm{TiO}}_2$ nanopillars. The height $H$ of the nanopillars was set to 600 nm to achieve the desired $2\pi$ phase coverage. The lattice constant was selected as 450 nm to comply with the Nyquist sampling law. Figures 2(b) and 2(c) show the simulated phase shift and transmission of $x$- and $y$-polarized lights from a nanopillar at the wavelength of ${\lambda }_0=532\mathrm{nm}$ as a function of diameter $(D_x,D_y)$. Based on the simulation results, we selected a set of 16 nanopillars to provide 16 orders of phase levels covering $2\pi$ range for ${\phi }_x$ and ${\phi }_y$. The phase and polarization conversion efficiencies (PCEs) of the nanopillars are shown in Fig. 2(d).

Fig. 2

Structure design of a spin-multiplexed metasurface. (a) Left, schematic of a metasurface composed of ${\mathrm{TiO}}_2$ nanopillars. Right, perspective view and top view of the unit cell placed on a quartz substrate. The incident wavelength is 532 nm, the nanopillar period is $U=400\mathrm{nm}$, and the height is $H=600\mathrm{nm}$. (b) and (c) Phase shifts and transmission under $x$-polarized light and $y$-polarized light, respectively. (d) Phase delay and PCE of the selected 16 nanopillars. (e) Design method for generating arbitrary spin-multiplexed metasurfaces. Given two arbitrary phase maps (${\phi }_{\mathrm{RCP}}$, ${\phi }_{\mathrm{LCP}}$), the propagation phase (${\phi }_x$, ${\phi }_y$) and the geometric phase $\theta$ of the metasurface pixels are calculated to design the in-plane sizes and rotation angles.

It is crucial to note that metasurfaces induce not only phase modulation but also amplitude crosstalk. Consequently, through parameter optimization, we make the PCE of the nanopillar basically consistent at the 532 nm wavelength, eliminating the influence of amplitude as much as possible while maintaining high efficiency ($\mathrm{PCE}>0.75$). The overall design process of the metasurface is shown in Fig. 2(e). By combining the geometric phase and the propagation phase (see Section I of Supplementary Material for detailed derivation), the in-plane size $(D_x,D_y)$ and spatial direction angle ($\theta$) of the nanopillar are determined at all the positions.

2.3.

Training of Spin-Multiplexed Diffractive Metasurfaces

To elucidate the spatial phase distribution of multilayer metasurfaces, we train a DNN. Here, we consider that each layer contains $200\times 200$ units, and thus the number of neurons reaches 40,000 on each layer, providing ample DoFs for the manipulation of the light field. The wavelength is fixed at $\lambda =532\mathrm{nm}$, and the spacing between the foremost and rearmost layers is set to be 75 times the wavelength ($75\lambda$), facilitating efficient field information transmission from the initial layer to each neuron.

Figure 3(a) shows a typical diffraction process. Throughout the training phase, the phase of each unit is treated as a learnable parameter of the network, and an error backpropagation algorithm is implemented for iterative refinement. See Supplementary Material, Section II for more details about the error backpropagation algorithm. It is evident that an increase in the number of hidden layers results in a decrease in crosstalk, which is defined as

Fig. 3

Training of diffractive metasurface for pattern detection. (a) Flowcharts of vector diffraction calculation simulation. (b) Identifying the crosstalk of mode numbers as a function of different numbers of hidden layers. (c) Scalar diffraction calculation results for incident mode identification. (d) Energy distribution matrix results of 36 modes (see the text for details on the modes).

3.1.

Simultaneous Sorting of High-Order VSBs

To assess the viability of spin-multiplexed diffractive metasurfaces for detecting VSBs, we focus on high-order vector vortex beams (HOVVBs); the results are shown in Fig. 4. The spatial DoFs for the HOVVB are represented by an LG beam with angular index $l$ and radial index $p$, as shown in Fig. 4(a). Figure 4(b) displays HOVVBs, cylindrical vector beams (VBs), and vector vortex beams (VVBs).^1,28–32 In this configuration, the angular index $l$ and radial index $p$ serve as the classification keys. Therefore, the pattern detection of HOVVBs exhibits enhanced adaptability and versatility. With our current knowledge, a stable scheme for detecting the full range of modes of HOVVB remains elusive.

Fig. 4

Characterization of spin-multiplexed diffractive metasurfaces for identifying high-order VSBs. (a) Poincaré sphere representation of HOVVBs. (b) Intensity patterns of three typical VSBs. The red line denotes the LCP component, the blue line signifies the RCP component, and the yellow line represents the linearly polarized (LP) component. (c)–(e) The polarization distributions and intensity patterns of the input light fields, and the intensities of $E_x$ and $E_y$ components. (f)–(h) Measured intensity distribution of the output plane. (i)–(k) The normalized energy ratio of 72 output channels.

Here, we employ vector diffraction to validate the functionality of the proposed neural metasurface. Figures 4(c)–4(k) show the results of the input vector light fields ${\psi }_1=(|{\mathrm{LG}}_{\mathrm{1,0}}⟩|{\sigma }_{+}⟩+|{\mathrm{LG}}_{-\mathrm{1,0}}⟩|{\sigma }_{-}⟩)/\sqrt{2}$, ${\psi }_2=(|{\mathrm{LG}}_{\mathrm{3,0}}⟩|{\sigma }_{+}⟩+|{\mathrm{LG}}_{-\mathrm{2,0}}⟩|{\sigma }_{-}⟩)/\sqrt{2}$, and ${\psi }_3=|{\mathrm{LG}}_{\mathrm{3,2}}⟩|{\sigma }_{+}⟩+|{\mathrm{LG}}_{-\mathrm{2,1}}⟩|{\sigma }_{-}⟩/\sqrt{2}$. See Supplementary Material, Section III for more detailed results of the input vector fields. For vector simulation, the phase distribution is converted into the corresponding structural parameters of the metasurface. Subsequently, the near field is obtained through FDTD simulation, followed by the extrapolation to the far field. Upon completing the calculation of the last layer, the output light intensity distribution is obtained.

The spin-multiplexed diffractive metasurfaces are trained based on four hidden layers and achieve convergence after 500 iterations. Figures 4(c)–4(e) show the intensity and polarization distribution of the input light field, along with the corresponding electric field components of $E_x$ and $E_y$. Figures 4(f)–4(h) showcase the light-intensity distribution within the detection area, revealing two distinct Gaussian bright spots on the output plane, from which we can accurately identify the spatial pattern of the incident light field. The complete diffraction efficiency of the four-layer metasurface stands at 15.3%, which can be attributed to the inherent loss of the metasurface and the energy dissipation throughout the diffraction process. Further enhancement of efficiency can be attained through the optimization of the metasurface units. Therefore, with a single measurement, such a metadevice can fully resolve all modes of HOVVBs, encompassing spin angular momentum ($s$), angular quantum number ($l$), and radial quantum number ($p$).

Finally, we calculated the normalized energy ratio of the output channels to evaluate the crosstalk, with the results presented in Figs. 4(i)–4(k). The inset depicts the intensity and phase distribution of the LCP and RCP components of the incident light. It is evident that the energy peak accurately aligns with the corresponding mode position, and its proportion exceeds 70% even with the consideration of the radial index $p$.

3.2.

Simultaneous Sorting of Superimposed VSBs

For further demonstration, we then investigate the recognition of arbitrary VSBs, as shown in Fig. 5. We use three typical structural beam modes to compose very complex DoFs. Among them, the mode indices of the LG beam are $l=-2$, $-1$, 1, 2 and $p=1$, 2, 3. The mode indices of the HG beams are $m=1$, 2, 3, 4 and $n=1$, 2, 3. The topological charge of the BG beam of $l=1,2,\dots ,12$ is selected. Employing all 36 modes as the input, the output of spin-multiplexed diffractive metasurfaces consistently remained a Gaussian light spot at the predetermined position. This approach uniquely enables the identification of arbitrary vectorial structured beams in a single detection, which was unachievable with prior methods.

Fig. 5

Characterization of spin-multiplexed diffractive metasurfaces for identifying arbitrary VSBs. (a)–(c) The polarization distributions and intensity profiles of the input light fields, and the intensity results of the $E_x$ and $E_y$ components. (d)–(f) Measured intensity distribution of the output plane. (g)–(i) The normalized energy ratio of 72 output channels.

We then evaluated the performance of the components through vector components. To demonstrate its capabilities, we select three vector modes of ${\psi }_1=(|{\mathrm{LG}}_{\mathrm{2,2}}⟩|{\sigma }_{+}⟩+|{\mathrm{LG}}_{-\mathrm{1,3}}⟩|{\sigma }_{-}⟩)/\sqrt{2}$, ${\psi }_2=(|{\mathrm{HG}}_{\mathrm{3,2}}⟩|{\sigma }_{+}⟩+|{\mathrm{HG}}_{\mathrm{2,4}}⟩|{\sigma }_{-}⟩)/\sqrt{2}$, and ${\psi }_3=(|{\mathrm{BG}}_2⟩|{\sigma }_{+}⟩+|{\mathrm{BG}}_6⟩|{\sigma }_{-}⟩)/\sqrt{2}$. See Supplementary Material, Section IV for more details. Figures 5(a)–5(c) show the patterns of input VSBs, characterized by distinct polarization distributions and field intensity patterns. These beams propagate through multilayer neural metasurfaces and generate varied light spots on the output plane, as shown in Figs. 5(d)–5(f). The figures show the light-intensity distribution in the detection area as two distinct Gaussian spots on the output plane. Furthermore, the normalized energy ratio of the output channels is quantified to assess the crosstalk, as shown in Figs. 5(i)–5(k). These outcomes confirm the precise recognition of the input light fields. Simultaneously, the method demonstrates the classification and identification of multiple groups of nonorthogonal vector beams, a task previously deemed unattainable by existing methodologies.

In addition, we consider the experimental feasibility of the proposed four-layer multiplexed diffractive metasurface. The ${\mathrm{TiO}}_2$ metasurfaces can be fabricated using standard nanofabrication processes with a combined process of electron-beam lithography and reactive ion etching. Further, to align cascaded metasurfaces accurately, a high-precision six-dimensional displacement stage and a high-resolution microscope can be employed for optimal alignment.²³ In addition, photoresist can serve as a spacer layer for preparing multilayer integrated cascade metasurfaces. This approach achieves layer-to-layer alignment during the fabrication process; thus the monolithic diffractive metasurface eliminates the need for subsequent manual alignment.