Stacked Intelligent Metasurfaces (SIM)

• Stacked intelligent metasurfaces are multilayer electromagnetic platforms that manipulate incident waves via cascaded amplitude and phase modulation.
• They enable analog-domain operations such as 2D Fourier transforms, multiuser beamforming, and neural computation for efficient wireless communication and sensing.
• Advanced optimization methods like alternating optimization and gradient-based learning address hardware constraints and mutual coupling challenges in SIMs.

Stacked intelligent metasurfaces (SIMs) are a multi-layer electromagnetic platform comprised of physically stacked, reconfigurable metasurface sheets which jointly manipulate incident waves entirely in the analog domain. Unlike single-layer reconfigurable intelligent surfaces (RIS), SIMs enable high-dimensional programmable transformations—including matrix-vector operations, 2D discrete Fourier transforms, multiuser beamforming, direction-of-arrival estimation, and analog neural computation—by leveraging cascaded amplitude and phase modulation across multiple metasurface layers. Controllable via global biasing networks interfaced to each meta-atom, SIMs perform wave-domain processing at the speed of light, fusing wireless communication, sensing, and analog computing into a single, energy-efficient substrate (Renzo, 29 Nov 2024).

1. Physical Architecture and Layered Structure

A stacked intelligent metasurface consists of $L$ nearly-passive metasurface layers aligned along the propagation axis, each layer comprising an $M \times M$ or $N \times N$ array of electronically tunable meta-atoms (e.g., varactor- or PIN-diode-loaded resonators). Each meta-atom in layer $\ell$ imposes a configurable amplitude $\beta_{\ell,i}$ and phase shift $\varphi_{\ell,i}$ on the local incident field. Layers are spaced at distances $d_1, d_2,\ldots,d_{L-1}$, typically on the order of a wavelength, so that the EM field exiting one layer undergoes diffraction before arriving at the next, providing a sequence of high-rank linear transformations.

The physical stack may be fabricated using planar or cavity-embedded substrates, with meta-atoms engineered for precise impedance control. Realizations span passive (lossless with |β| ≈ 1), quantized-phase programmable, or active/amplified implementations (Renzo, 29 Nov 2024, Liu et al., 4 Jul 2024). Input/output coupling is provided by feed antennas or waveguide ports, and all elements in the stack are governed by a digital controller capable of per-element dynamic programming at rates dependent on the tuning mechanism.

2. Electromagnetic and Analytical Modeling

The electromagnetic response of a SIM is captured by a multiport network formalism. At each layer $\ell$, the array implements a diagonal scattering matrix: $$S^{(\ell)} = \mathrm{diag}(\beta_{\ell,1}e^{j\varphi_{\ell,1}},\ldots,\beta_{\ell,M^2}e^{j\varphi_{\ell,M^2}}),$$ modulating the incident field vector $a^{(\ell)}$ into scattered vector $b^{(\ell)}$. Inter-layer propagation matrices $P^{(\ell,\ell+1)}$ encode full-wave diffraction between every meta-atom on adjacent layers, typically modeled by the Rayleigh–Sommerfeld formula: $$[P^{(\ell,\ell+1)}]_{ji} = h_{ji} = \frac{e^{-jk r_{ji}}}{j\lambda r_{ji}},$$ where $r_{ji}$ is the distance between meta-atoms $i, j$.

Cascading $L$ layers yields the end-to-end SIM transformation: $$b^{(L)} = S^{(L)} P^{(L-1,L)} S^{(L-1)} \ldots P^{(1,2)} S^{(1)} a^{(1)},$$ compactly denoted as the programmable linear operator $$F = \prod_{\ell=1}^L [P^{(\ell-1,\ell)} S^{(\ell)}]$$ (Renzo, 29 Nov 2024, An et al., 2023). This structure allows the SIM to be trained—by appropriate tuning of $\{\beta_{\ell,i},\varphi_{\ell,i}\}$—to directly implement analog matrix computations, including beamforming matrices, Fourier kernels, and neural network layers.

Accurate modeling for near-field and mutual coupling scenarios necessitates multiport circuit models (using S- or Z-parameter cascades), as developed in (Abrardo et al., 5 Jan 2025, Nerini et al., 19 Feb 2024). Assumptions of diagonal scattering and unilateral propagation are only justified under weak inter-element and inter-layer coupling.

3. Wave-Domain Communication, Beamforming, and Sensing

SIMs can serve as real-time, analog-domain precoders for a wide range of wireless functions:

• Holographic MIMO (HMIMO): At the transmitter, a SIM replaces or supplements the digital precoder, enabling spatial multiplexing or SVD-based diagonalization of the MIMO channel entirely in the EM domain. At the receiver, a stacked SIM can implement analog combining (Renzo, 29 Nov 2024, An et al., 2023, Li et al., 1 Mar 2025).
• Multiuser Beamforming: By stacking layers, the SIM achieves control over multiple angular lobes, enabling simultaneous interference-suppression and user-specific beam directionality. The effective beamforming gain is

$G_k = |a_R(\theta_k)^{H} F a_T(\psi_k)|^2,$

with the phase profiles across layers optimized for the instantaneous channel state (An et al., 2023, An et al., 2023).

• Integrated Sensing and Communications (ISAC): Layer parameters are jointly optimized to balance spectral efficiency for communication users and beampattern sharpness for radar targets, typically using alternating or penalized optimization: $$\underset{\{\beta,\varphi\}}{\max} \; \alpha R_{\mathrm{comm}} + (1-\alpha) \mathrm{SNR}_{\mathrm{radar}}$$ (Renzo, 29 Nov 2024, Niu et al., 19 Aug 2024, Ranasinghe et al., 29 Apr 2025).
• Wave-Domain 2D DFT and Direction-of-Arrival (DOA) Estimation: SIMs trained via gradient descent can directly implement the 2D DFT, mapping spatial samples onto angular spectrum peaks and enabling superfast, low-power DOA estimation at the speed of light (An et al., 13 Feb 2024, An et al., 2023).
• Semantic Encoding & On-Device Computing: SIMs can act as electromagnetic neural networks (EMNNs), combining source and semantic encoding (such as image classification) within the wave domain and leveraging mini-batch-trained amplitude/phase profiles. This supports paradigm-shifting physical-layer semantic communications and imaging (Huang et al., 21 Jul 2024, Huang et al., 14 Jun 2025).

4. Optimization and Control Methodologies

The high-dimensional, nonconvex nature of SIM control (with $LM^2$ programmable variables for $L$ layers of $M \times M$ meta-atoms) motivates a range of algorithms:

• Alternating Optimization (AO): Decomposes into power allocation (e.g., water-filling) for a fixed SIM, and phase/amplitude tuning (e.g., gradient ascent or projected gradient for discrete/continuous constraints) (An et al., 2023, An et al., 2023, Papazafeiropoulos et al., 29 May 2024).
• Gradient-Based Learning: For analog computing or DFT tasks, the loss (e.g., fitting error, cross-entropy on energy patterns) is iteratively minimized by back-propagating through each cascaded layer. Closed-form gradients exploit the multiplicative structure (An et al., 13 Feb 2024, An et al., 2023, Huang et al., 21 Jul 2024, Huang et al., 14 Jun 2025).
• Deep Reinforcement Learning (DRL): Model-free approaches (e.g., DDPG) map observed environment state to optimized SIM phase configurations and power levels, particularly effective under CSI uncertainty or mobility (Liu et al., 9 Aug 2024, Liu et al., 14 Feb 2024).
• Multiport Network Optimization: For rigorous physical models including non-diagonal and coupled scattering, matrix-derivative calculus is applied to Z/S-parameter cascades, with computational shortcuts leveraged under diagonal/unilateral assumptions (Abrardo et al., 5 Jan 2025, Nerini et al., 19 Feb 2024).

The configuration rate is limited by the controller speed and underlying meta-atom technology, but full reconfiguration at sub-nanosecond scales has been reported (An et al., 2023).

5. Performance Metrics, Scaling, and Trade-Offs

Key metrics for SIM design include:

• Beamforming Gain: $G = |h_e^H F s|^2$, optimized over all programmable phase/amplitude coefficients (Renzo, 29 Nov 2024).
• Spatial/Azimuthal Resolution: $\Delta\theta \approx \lambda/(M d)$, scaling with aperture and layer count (Renzo, 29 Nov 2024, An et al., 13 Feb 2024).
• Spectral and Computing Throughput: $O(M^2 L f_{\text{rep}})$ operations per second, with f_{\text{rep}} determined by meta-atom reconfiguration rate (Renzo, 29 Nov 2024).
• Energy Efficiency: Dramatically improved over digital architectures for large M, since each meta-atom draws microwatts; total power is $$P_{\text{tot}} \simeq P_{\text{bias}} + P_{\text{ctrl}}$$ (Renzo, 29 Nov 2024, An et al., 2023).
• Capacity Scaling: For HMIMO, capacity scales quadratically with meta-atom count in the large system regime (An et al., 2023); in wideband, SIM enables over 300% increase in channel capacity compared to single-layer beamformers (Li et al., 1 Mar 2025).

Trade-offs critical to deployment include insertion loss (which accumulates over $L$ layers), mutual coupling (necessitating full S-parameter modeling), finite phase quantization, and joint optimization complexity.

6. Implementation Issues, Hardware Prototypes, and Modeling Limitations

Realistic SIM deployment faces several challenges:

• Hardware Constraints: Loss per metasurface layer, limited phase quantization (often 1–4 bits), finite tuning bandwidth, and cross-talk between meta-atoms all impact realized performance (Renzo, 29 Nov 2024, An et al., 2023, Liu et al., 4 Jul 2024).
• Modeling Fidelity: Ideal diagonal scattering and unidirectional propagation approximations can fail under strong coupling or small interlayer spacings; full multiport network modeling is required for high-fidelity performance and in optimization for tasks like 2D DFT (Abrardo et al., 5 Jan 2025, Nerini et al., 19 Feb 2024).
• Calibration and Control: Real-time channel estimation and calibration become increasingly difficult as $L, M$ grow. Scalable control architectures (e.g., FPGA/ASIC with hierarchical control) and calibration methods are active research areas (Renzo, 29 Nov 2024, Liu et al., 4 Jul 2024).

Hybrid digital–wave architectures are also under exploration, leveraging coarse, high-DSP-throughput SIMs for analog precoding and digital fine-tuning for robust adaptation under channel variation (Renzo, 29 Nov 2024).

7. Open Research Directions and Future Applications

Current and emerging research frontiers include:

• Beyond-DFT Analog Computing: Extending SIMs to implement convolutional transforms and physically realizable neural networks for tasks beyond classical communication (Renzo, 29 Nov 2024).
• Ultra-Low-Latency Multi-Modal SemCom: Integration with generative models and semantic-oriented encoding for drastically reduced bandwidth image and scene communication (Huang et al., 14 Jun 2025, Huang et al., 21 Jul 2024).
• Robust Optimization under Hardware Impairments: Accounting for meta-atom non-idealities, fabrication tolerances, and channel uncertainties. Incorporating robust and data-driven control in high-dimensional phase spaces (An et al., 2023, Liu et al., 9 Aug 2024).
• Wideband and Near-Field Applications: Expanding the frequency range of operation (mmWave/THz), addressing dispersion, and leveraging near-field beamfocusing for enhanced spatial degrees of freedom (Li et al., 1 Mar 2025, Jia et al., 9 Feb 2025).
• Reduced-Complexity Topologies: Meta-fiber architectures that compress deep, multi-layer SIMs into two-layer designs without sacrificing DoF, offering significant gains in both capacity and hardware efficiency (Niu et al., 13 Jul 2025).
• Cooperative and Cell-Free Deployments: Distributed SIM panels for dense, ultra-massive connectivity scenarios (Darsena et al., 27 Oct 2025).
• Integrated Sensing and Communications (ISAC): Jointly optimizing for communication and radar beampatterns, pushing the boundaries of physically co-designed platforms (Niu et al., 19 Aug 2024, Ranasinghe et al., 29 Apr 2025).

In summary, stacked intelligent metasurfaces fuse the underlying physics of programmable electromagnetics with advanced analog-domain computation, creating a highly scalable, energy-efficient, and ultra-fast platform for next-generation wireless communication, joint sensing, and beyond-digital computation in the wave domain (Renzo, 29 Nov 2024).