Metasurface-Based Intelligent Identification of Total Angular Momentum Spectra for Beams

Jan. 03 , 2025photonics1

Abstract

The total angular momentum (TAM), consisting of spin angular momentum (SAM) and orbital angular momentum (OAM), is a crucial indicator for characterizing the topological features of structured beams. However, current diagnostic methods have limited measurable modes, making it difficult to obtain the TAM spectrum. Here, we present a metasurface-based intelligent scheme for measuring the TAM spectrum. We designed and fabricated a metasurface to transform the TAM modes into Hermite–Gaussian-like modes for simplifying judgment and developed a deep learning network, whose core stages are several mobile inverted bottleneck convolution layers for mode decomposition, for accurate TAM spectrum identification. The favorable experimental results demonstrate that our proposal can precisely measure structured beams carrying up to 34 TAM modes. Furthermore, robustness tests of this proposal under noise, angular shift, and transverse rotation demonstrate that our model is capable of accurate performance in the presence of these adverse effects within a certain range. This work presents a new path for measuring the TAM spectrum in a miniaturized form, with high accuracy, simple operation, and wide measurable modes range, which will inspire more cutting-edge scenarios such as laser communication, high security holographic encryption, and quantum information processing.

Introduction

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The angular momentums of photons consist of two parts, the spin angular momentum (SAM) and the orbital angular momentum (OAM). The direct product of SAM and OAM under the paraxial approximation is regarded as total angular momentum (TAM), where the SAM has two eigenvalues σ = ± 1, corresponding to macroscopic left circular polarizations (LCP) and right circular polarizations (RCP), (1) while the OAM is manifested as a helical wavefront of exp(ilφ) with l the eigenvalue called topological charge and φ the azimuthal angle. (2,3) Recently, the manipulation of TAM has been applied to multiply the capacity of optical communication (4,5) and expand the dimensions of quantum information processing, (6−9) high-security holographic encryption, (10,11) and novel vectorial optical fields with structured polarization. (12,13) With the inherent orthogonality of the SAM and OAM, TAM constitutes an infinite dimensional Hilbert space, where the power distribution of each single TAM mode (14) is defined as a TAM spectrum, determining the transverse topological structure including the anisotropic wavefront and polarization distributions, which is one of the standards for the characteristic evaluation of beams. Therefore, identifying the TAM spectrum of beams accurately is of great significance. Despite that the SAM and OAM modes can be measured asynchronously, this step-by-step approach constrains their potential applications. Consequently, a single-shot measurement solution is required.

In order to measure both the SAM and the OAM components in a synchronous manner, the TAM modes are typically identified through the utilization of polarization-sensitive elements. Researchers have achieved mode sorter-based TAM sorting schemes (14−18) with Pancharatnam–Berry (PB) (19) elements, which can detect adjacent OAM modes. (14,18,20) Nevertheless, the structures of these schemes always consist of two PB elements and additional focusing lenses, restricting their application scenarios. Metasurfaces are highly efficient diffractive optical devices, ideal for miniaturization and providing superior control over the optical field using subwavelength nanostructures. (20−23) The single spin-decoupled metasurface (24−26) and the on-chip plasmonic spin-Hall nanograting structure (27) have been proposed for TAM identification with a minimal structure. However, their range of detectable TAM modes is still limited, impeding further research on vortices with multiple modes. (28−30) In essence, all of the aforementioned paths exhibit mode overlap and diffusion, which presents a significant challenge in performing a power integral when identifying multiplexing vortices that carry multiple TAM modes. This inherent limitation restricts their ability to accurately identify the TAM spectrum with precision. Consequently, the challenge persists in developing a compact structure that can accurately measure the TAM spectrum over a wide mode range.

In this article, a periodic gradient metasurface-based intelligent scheme is proposed to identify the TAM spectra of multiplexing vortices. Different vortices are transformed into Hermite–Gaussian-like (HGL) modes on a diffraction plane through the periodic gradient structure (PGS); these HGL modes have different intensity distributions (see evaluating details in the Supporting Information, Figure S1–S3). It has been confirmed that OAMs with opposite eigenvalues can be distinguished by a single-shot measurement. (31) Furthermore, the HGL modes, transformed from OAMs in a broad mode space, do not suffer serious divergence, thereby enabling the measurement of power spectra across a wide range of modes. Moreover, a spin-controlled PGS is proposed for simultaneously converting OAM modes and decoupling SAM modes, which is materialized via a single polarization-sensitive (spin-decoupled) metasurface (32−35) fabricated by chemical vapor deposition (CVD) and electron beam lithography (EBL). Given that the interference of HGL modes results in the diffracted patterns becoming more complex when vortices carry a large number of TAM modes, we design a spin-coupled adjusted depth TAM identification net (SADT-Net) for the purpose of measuring the power distribution of the TAM modes through the decomposition of the HGL modes from the complex interferograms. By employing the PGS metasurface for manipulation and utilizing SADT-Net for the identification of diffracted patterns, a broad range of TAM spectrum identification is accomplished with fast response and high precision.

A series of experiments has been carried out to verify the feasibility of this proposal. The favorable experimental results demonstrate that our scheme supports 34 single TAM channels and accurately identifies the TAM spectra with a mean squared error (MSE) lower than 10^–6, a sufficiently wide measuring range, and high diagnosis accuracy. In order to assess the universality of the proposed scheme, a series of experimental robustness analysis are conducted. The results demonstrate that the scheme is capable of functioning effectively even in adverse conditions, such as the presence of significant noise and spatial dislocations. This work opens a new insight for complicated TAM mode recognition in a compact size, high precision, and quite intelligence and is expected to pave the way for novel laser communications and quantum information processing.

Results

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Concept of TAM Spectral Identification

The concept of the TAM spectral identification scheme is depicted in Figure 1. The PGS metasurface is designed to decouple the SAM modes and affect a linear transformation of the OAM modes concurrently. When a multiplexing vortex is focused on the metasurface, it is converted into two diffracted beams by the phase modulation of the PGS, reducing the difficulty of feature extraction. The orthogonal SAM components of the incident vortex are modulated by opposite linear gradient phases, resulting in the spatial separation of SAM. These two diffracted patterns are captured by a camera, and the patterns of SAM modes |σ = −1⟩ and |σ = +1⟩ are distributed on the left and right sides, respectively. The aforementioned patterns are derived from the respective OAM components, whose intensity distributions are a consequence of interference of the transformed HGL modes. Next, the proposed SADT-Net is applied for obtaining the TAM spectrum. The captured image is cropped and split into two channels, corresponding to the decoupled SAM components. And, the size of the transformed image is W × H, where W is the width and H is the height of the image. Subsequently, the preprocessed image is decomposed into TAM intrinsic modes via the network, whose core components are a series of mobile inverted bottleneck convolution layers (MBConvs) (36) utilized for feature decomposition. Ultimately, the TAM spectrum is accurately identified from the decomposed features by a global average pooling layer, a fully connected layer, and a calibration.

Figure 1. Schematic of the identification principle. A multiplexing vortex is focused onto the PGS metasurface, resulting in the generation of two diffracted beams captured by a camera. Subsequently, the TAM spectrum of the vortex contained in the captured image is identified via the proposed SADT-Net.

PGS Metasurface Design

In order to achieve the desired TAM mode decoupling at a wavelength of 1550 nm, an array of unit cells with a stationary periodicity of 650 nm is employed to construct the designed phase distribution. As illustrated in Figure 2a, each unit consists of amorphous silicon cuboids fabricated on a silicon dioxide substrate, the design of which is based on the principles of the PB phase.

Figure 2. Design, fabrication, and evaluation of the TAM feature extraction metasurface. (a) Unit structure: an amorphous silicon cuboid on a SiO₂ substrate. (b) and (c) are the phase modulation (Φ) and transmission efficiency (T), respectively, versus heights and rotation degrees (θ) of a silicon cuboid unit. (d) Phase difference between the y and x polarizations (ΔΦ) of the designed metasurface versus widths and lengths of a silicon cuboid unit. (e) and (f) are the phase modulation (Φ) and transmission efficiency (T) of the designed metasurface, respectively, versus incident wavelengths and rotation degrees (θ) of a silicon cuboid unit. (g) Slow-axis orientation angle (θ) distribution of the designed unit cell array and the separation of the orthogonal SAM components through PB phase modulation, where the phase distribution is shown in the lower left panel. (h) SEM image of the fabricated metasurface. (i) Experimentally measured transmission efficiency of different SAM ratios and different OAM modes.

The Ansys Lumerical FDTD is used to numerically simulate the optical characteristics of the array (see Methods for simulation details). By rotating the silicon cuboid, the phase shift (Φ) and transmission efficiency (T) at the working wavelength are shown in Figure 2b and Figure 2c, respectively, with the parameter h_u varying from 400 to 1050 nm. Since the ideal unit can fulfill phase modulation from −π to π with high transmittance during its rotation, h_u = 1020 nm is finally selected with the further consideration of fabrication difficulty, where the simulated phase modulation and transmittance performances of the metasurface meet the requirement. Given the PB phase profiles, each unit cell should work as a miniature half-wave plate, wherein the optimal phase difference between the y and x polarizations (ΔΦ) is represented by the value of π. With the scanning of the length (l_u) and width (w_u) to accomplish the requirement, ΔΦ at the l_u varying from 450 to 550 nm and w_u varying from 200 to 300 nm are given in Figure 2d. The optimized length and width are 240 and 500 nm, respectively, where ΔΦ is 1.01π. Then, the phase shift (Φ) and transmission efficiency (T) at the wavelength of 1350 to 1650 nm are shown in Figure 2e and Figure 2f, respectively, with the rotation of the silicon cuboid, indicating that the phase coverage achieves −π ∼ π and the transmission efficiency approaches 1.

As the metasurface works as a spin-controlled PGS, the slow-axis orientation angle distribution θ(x,y) is denoted as

θ (x, y) = \frac{π x}{a - b y} + \frac{π (a - c) x}{a c}

$θ (x, y) = \frac{π x}{a ? b y} + \frac{π (a ? c) x}{a c}$

(1)

where (x, y) is the reference plane shown in Figure 2g and a, b, and c are the transformation parameters (see effects of parameters in the Supporting Information, Figures S4–S6). The first term represents the PGS phase, and the second term corresponds to the linear gradient phase resulting in the spatial separation of orthogonal SAM components (see the parameter setting in Methods). The structure of the unit cell array and the separation of the orthogonal SAM components based on the PB phase are sketched in Figure 2g. Moreover, the separation angle between the two diffracted beams of light is 2tan^–1 (λ/c).

Under the parameters and the orientation angle distribution above, the metasurface was fabricated with the use of an e-beam lift-off method (see details in Methods) and the scanning electron microscopy (SEM) image of the metasurface is shown in Figure 2h. Since the metasurface has a tiny effective area, a convex lens with a focal length of 100 mm was employed to focus the incident vortex on the metasurface, which ensures the complete phase modulation.

Ideally, each TAM component should be extracted without any power loss or with a uniform loss. However, it is almost impossible to achieve due to the fabricating error, even though we keep it within 5%. To evaluate the practical performance of the fabricated metasurface, it was illuminated by TAM-carried beams for evaluation (see evaluating details in the Supporting Information, Figure S7), and it is of paramount importance that the incident beam is suggested being focused into the diameter of 100–265 μm at the central of the metasurface. Figure 2i shows the SAM intensity response (transmission) at different SAM ratios. It indicates that the metasurface can convert the orthogonal SAM components completely from each other. As for the OAM components, Figure 2i also gives the transmission of two diffraction patterns when the incident beams carry two equivalent SAM components with different OAM modes, illustrating that the transmissions of different OAM components are not ideal, which leads to a necessary calibration.

Intelligent TAM Spectral Identification

When a TAM carried beam focused on the metasurface, the two diffraction orders, displayed as Figure 3a1 and Figure 3a2, were captured by a camera. The eigenvalues of the single OAM modes can be obtained by calculating the number of stripe gaps in the captured images. Then, the TAM spectrum was obtained by grayscale value integration, but there is no difference between the intensity of two orthogonal polarization modes. The metasurface introduces a nonlinear transformation, which serves as the kernel function, transforming the vortex mode into an HGL mode (see the Supporting Information, Figures S1–S3 for detail). This process enables identification of the single mode. However, the diffraction patterns become complex when the incident beam is composed of multiple TAM modes. As the ratio of each single TAM mode in different vortices is distinct, the interference among the HGL modes gives rise to different patterns. Thus, it is feasible to decompose the TAM modes of mutual interference by utilizing the designed SADT-Net, a feature extraction algorithm.

Figure 3. Performance of SADT-Net for TAM spectrum identification. The experimental diffraction patterns, where (a1) is the left one and (a2) is the right one of a TAM carried beam ψ₁ modulated by the metasurface. (b) Structure of the SADT-Net. CBS denotes a convolutional block, consisting of a convolution layer, a batch normalization layer, and a swish activation function. MBConv1 and MBConv6 are mobile inverted bottleneck convolution layers (MBConv blocks) for TAM mode decomposition, the number (M) of which is repeatable. PF consists of an average pooling layer and a fully connected layer for mapping to the TAM spectrum from the extracted features. The calibration is used for correcting the nonideal transmission of the metasurface modulation. (c) MSE of the different numbers of layers in training and validation. (d1) Experimental output TAM spectrum of ψ₁ via the SADT-Net and calibration. (d2) Intensity distribution of the input beam ψ₁. (e) Performance of the spectral identification with vortices carrying different numbers of TAM modes. (f) Performance of the spectral identification versus different single TAM modes.

Before the input of the SADT-Net, the images need to be preprocessed. First, the intensity distributions captured are divided into I_–1 and I₊₁, corresponding to |σ = +1⟩ and |σ = −1⟩_, respectively. Then, I_–1 and I₊₁ are preprocessed by subtracting the camera background noise for increasing the identification accuracy. Then, the size of the images are reshaped into the required size 256 × 256 by cropping and bilinear interpolation. Lastly, the two reshaped images are stacked into 256 × 256 × 2 in the channel dimension for the required size of the SADT-Net.

The structure of the SADT-Net is given in Figure 3b. A convolutional block, consisting of a convolution layer, a batch normalization layer, and a swish activation function, is employed for preliminary feature extraction. The core stages mainly consist of several layers of mobile inverted bottleneck convolution layers (MBConv) for orthogonal mode decomposition, which are repeatable, leading to accuracy improvement. Then, the extracted high-dimensional features are mapped to the TAM spectrum through an average pooling layer and a fully connected layer. Ultimately, the output spectrum is calibrated by the transmission curve given in Figure 2i.

Since the MBConv is repeatable, the repetition of the MBconv blocks needs to be determined. We have trained a series of models with different numbers of layers, which are demonstrated in Figure 3c. The mean square error, denoted as MSE = ∥I_t – I_e∥₂²/N, is used for error evaluation, where I_t denotes the practical TAM spectrum, I_e is the calculated TAM spectrum, and N is the number of the validation data. The number of layers (M) is determined with the least MSE, where the M of the network is 4. However, it is indicated that M = 4 might be the local optimal, where the MSE might be lower than M = 4. With the consideration of balancing the accuracy and training difficulty, we choose M = 4 for this task (see structure, training details, additional experimental validation examples, and model interpretation of the SADT-Net in the Supporting Information, Table S1 and Figure S9; data set and implementation are given in Methods).

The TAM spectrum identified by the SADT-Net of vortex ψ₁ is given in Figure 3d1. Compared with the ground truth, the MSE is 7.608 × 10^–6, which has a high accuracy. To evaluate the performance of the concept, we have measured the TAM spectra on 2006 sets of beams (see Methods for optical characterization; the experimental setup is given in the Supporting Information, Figure S12). Figure 3d1 exhibits the result of ψ₁, whose TAM components are shown by the cyan bars. To quantitatively analyze the error of the TAM spectrum, MSE = ∥C_t – C_e∥₂²/N is used, where C_t denotes the theoretical TAM spectrum (ground truth), C_e is the experimental TAM spectrum, and N is the number of the TAM channels. The TAM spectrum MSE of the vortex ψ₁ is 7.608 × 10^–6, and the average MSE of the whole sets is 1.616 × 10^–5; the standard deviation is 2.127 × 10^–8, which illustrates that the experimental result is consistent with the theory.

Moreover, the average MSE of the method when beams carrying different numbers of TAM modes are measured is given in Figure 3e. With the increase of the number of TAM channels, the measurement accuracy is basically equivalent, the result of which is related to the data set distribution. The average MSE versus different single TAM modes is shown in Figure 3f, giving the MSE results of different single TAM channels, indicating that the MSE is relatively consistent within the range of the eigenvalue of the OAM component from −5 to +5 except for Gaussian mode. The experimental multi-TAM mode generation is based on the phase modulation of a spatial light modulator (SLM). Moreover, the holograms encoded on the SLM are generated by calculating the angle resulting from the sum of the helix phases exp(ilφ) where l = −5 to +5, which inevitably produces mode broadening, (37) leading to the experimental beams carrying OAM modes ranging from −8 to +8 except for the Gaussian mode. Since the overflow modes ranging from −8 to −6 and +6 to +8 have much lower intensities than those of −5 to +5, the network struggles to learn features of the overflow OAM modes, resulting in excessive errors.

Robustness Representation against Several Potential Perturbation

In realistic scenarios, the incident vortex is susceptible to distortion due to numerous uncontrollable potential perturbations, leading to significant TAM mode crosstalk. This is detrimental to the functionality of practical communication links and quantum computing. Thus, the robustness of a TAM spectrum identification system is crucial. Here, we employ a quantitative analysis to examine the performance of the SADT-Net in the presence of noise and spatial dislocations, namely, angular shift (AS) and transverse rotation (TR). These effects are illustrated in Figure 4b1–d1, which shows the MSE of the spectral identification as a function of the specified conditions. The scheme is considered to be robust under these conditions if it can accurately output the TAM spectra that correspond to the distorted ground truths, and the evaluation of the model was carried out using a randomly selected beam ψ₂.

Figure 4. Robustness analysis of SADT-Net against adverse conditions. (a1) Schematic of the optical transformation part with beam ψ₂ to measure. (a2) Identified TAM spectrum of ψ₂ via the SADT-Net without any adverse conditions. Error of the spectral identification versus noise strength (b1), angular shift (c1), and transverse rotation (d1), where the left panel is the schematic diagrams of the generation of the adverse conditions. The identified TAM spectrum of ψ₂ via the SADT-Net under the noise with standard deviation of 0.273 (b2), under the angular shift of 0.16 mrad (c2), and under the transverse rotation of 0.188 rad (d2). The corresponding patterns captured by the camera are also given in the upper left panel.

Additional Gaussian noise is always added to the captured images when a camera without a cooling module is used for a long time. The random Gaussian noise is added to the diffraction pattern of ψ₂ with a mean of 0 and a standard deviation of 0.0001 to 0.3. The error of the measurement results is presented in Figure 4b1, representing that the sliding average MSE increases steadily with the increase of the noise intensity, which is changing from 1.627 × 10^–5 to 1.642 × 10^–4. Moreover, the TAM spectrum under a standard deviation of 0.273 is given in Figure 4b2, whose MSE is 1.371 × 10^–4, illustrating that there are some intensity changes in certain modes, but there is no significant change in the overall distribution. In general, SADT-Net exhibits strong robustness against Gaussian noise.

Another crucial factor is the misalignment between the generation and detection modules, which is usually a consequence of the AS and is illustrated in the top left panels of Figure 4c1. It is an unavoidable obstacle, as it is an inevitable aspect of realistic configurations. As shown in Figure 4c1, the averaged MSE barely changes from 1.946 × 10^–5 to 1.661 × 10^–4 with the angular shift angle varying from −0.2 to 0.2 mrad. Moreover, the TAM spectrum under the angular shift angle of 0.16 mrad is given in Figure 4c2, whose MSE is 1.086 × 10^–4. It is noticed that the curve is not symmetric, which might be caused by the error of metasurface modulation, leading to the captured pattern distortion.

Besides, TR usually occurs when the detection module is independently rotated. Thus, it is necessary to determine whether the SADT-Net is rotation-invariant. As displayed in Figure 4d1, the averaged MSE changes from 1.914 × 10^–5 to 5.643 × 10^–4 when the rotation angle varies from −3π/50 rad to 3π/50 rad, and the symmetric curves indicate identical performances of the two rotation directions. The TAM spectrum under a maximum rotation angle of 3π/50 rad is given in Figure 4d2, where the method performs poorly on the mode with high intensity. The SADT-Net is robustness within the TR angle varying from −2π/50 rad to 2π/50 rad.

Conclusions

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In summary, we introduce a novel scheme to identify the TAM spectra of beams. The designed PGS-based metasurface is applied to separate the orthogonal SAM modes of the incident beam and maps the OAM spectra of the respective components to the intensity distribution of two diffraction orders. Then, we propose an intelligent algorithm SADT-Net to decompose the diffraction patterns into the orthogonal HGL modes, aiming to calculate the TAM spectrum. The favorable experimental results illustrate that the proposed scheme supports TAM spectral measurement for beams carrying up to 34 single TAM modes, a sufficiently wide measuring range. The mean MSE of the TAM spectral measurement is 1.616 × 10^–5 with a standard deviation of 2.127 × 10^–8, which is fairly precise. Moreover, the accuracy of this scheme under noise, angular shift, and transverse rotation illustrates good robustness of the proposal and performs uniformly on each TAM channel. This work provides a fast, accurate, and compact scheme for TAM spectral measurement and contributes to many advanced scenarios like novel laser communications and quantum information processing.

Methods

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Optical Element Simulation and Fabrication

We simulate the silicon cuboid by using the commercial simulation program Lumerical FDTD Solutions. The cuboid structure is placed on the substrate (+z), and the plane wave source with the propagation direction of +z is placed below the SiO₂ substrate. The transmitted fields are recorded from a field monitor that is placed above the structures. For the silicon cell simulation, periodic boundary conditions (BCs) are used along the x axis and y axis, and perfectly matched layers (PMLs) of BC are used along the z axis. We scan the parameters of the cell, including its length l_u, width w_u, height h_u, and wavelength λ of the incident beam. Then, we get the best size of the silicon cuboid at l_u = 500 nm, w_u = 240 nm, and h_u = 1020 nm.

The TAM extraction metasurface in our experiment is fabricated by electron beam etching on a silica substrate with the diameter of 2 in. coated with a silicon film where the parameters a = 27.2 μm, b = 0.02, and c = 4 μm of ?(x, y).

The 1020 nm-thick silicon epitaxial layer is grown on a silica substrate using chemical vapor deposition (CVD) equipment. Then, silicon cuboids are patterned by electron beam lithography according to the slow-axis orientation angle distribution that we designed. It has an effective area 500 μm × 500 μm, etch depth 1020 nm, minimum line spacing 650 nm, and minimum line width 240 nm. The processing error is less than 5%.

Data Set and the Implementation of the Network

For the data set generation of the network, the incident beams with different TAM spectra are transformed into the diffraction patterns captured by two identical infrared cameras with the modulation of the metasurface. The two captured patterns of each beam form a data pair with their corresponding practical ionization of OAM spectra as labels, respectively.

In this work, we experimentally collect 28,000 pairs of training set, 5000 pairs of validation set, and 2025 pairs of test set, which consist of OAM modes ranging from −8 to +8.

The training and testing of the SADT-Net are implemented using the PyTorch framework version 2.0.1 running on a Linux server (NVIDIA A6000 GPU, Intel Xeon E5-2699 v4 CPU with 88 cores, 192 GB of RAM, and the Ubuntu 20.04.6 LTS operating system).

Optical Characterization

The TAM mode preparation detail is given in the Supporting Information, Figure S7, mainly employing a 1550 nm laser diode (Thorlabs, FPL1009P), a series of polarization elements, and two liquid-crystal SLMs (Holoeye, PLUTO-TELCO-013-C). The experimental setup of the optical extraction stage was basically consistent with the theoretical set. A convex lens with a focal length of 100 mm was involved to focus the TAM beam on our metasurface mounted on the rotation mirror mount (MM4P-2T, JCOPTiX) and a pitch yaw rotary platform (PRY70, LBTEK). An Infrared CCD camera (CRED-3, LUSTER) is used for capturing diffracted patterns. The cameras are placed 150 mm behind the metasurface.