Stacked Intelligent Metasurfaces (SIMs)

Updated 7 December 2025

Stacked Intelligent Metasurfaces (SIMs) are multilayer assemblies of reconfigurable metasurfaces with programmable meta-atoms that control electromagnetic wavefronts via phase, amplitude, and polarization tuning.
By stacking layers with subwavelength spacing and applying gradient-descent and manifold optimization techniques, SIMs approximate complex linear and nonlinear transformations at the speed of light.
SIMs offer practical advantages in communications, sensing, and wave-based computing by achieving enhanced performance, reduced hardware complexity, and near-zero processing latency.

A stacked intelligent metasurface (SIM) is a multilayer assembly of electronically reconfigurable metasurfaces, where each two-dimensional layer is densely populated with meta-atoms engineered to impart programmable electromagnetic (EM) responses—phase, amplitude, polarization—onto transmitted, reflected, or refracted waves. The vertical stacking of such metasurfaces, interleaved at subwavelength or wavelength-scale spacing, enables SIMs to perform sophisticated linear and nonlinear transformations on EM wavefronts directly in the analog wave domain, with applications spanning wave-based computing, high-dimensional MIMO communications, sensing, and signal processing at unprecedented speeds and with minimized RF hardware (Renzo, 2024, Shi et al., 23 Oct 2025, Liu et al., 2024, Jia et al., 9 Feb 2025).

1. Physical Architecture and Electromagnetic Modeling

A SIM consists of $L$ – $L\!+\!1$ contiguous planar metasurface layers, each layer comprising a $Q$ / $M$ array of meta-atoms with programmable phase (and often amplitude) response. Interlayer vertical spacing $s$ is chosen as a fraction of the operational wavelength. Each meta-atom $n$ in layer $\ell$ applies a transmission coefficient $e^{j\theta_{\ell,n}}$ (with extension to amplitude tuning possible via $a_{\ell,n}e^{j\theta_{\ell,n}}$ in active or hybrid meta-atoms), tunable by electronic biasing of varactors, PIN diodes, MMICs, or microelectromechanical systems (MEMS).

Interlayer propagation is governed by scalar or vector Rayleigh–Sommerfeld diffraction for the free-space links, encoded via coupling matrices $W_\ell$ (or $D_\ell$ in some notations) whose entries are functions of inter-meta-atom distances, local angles, and meta-atom area. For narrowband operation, the overall linear transfer matrix between input and output fields is a product of diagonal (meta-atom) matrices and interlayer propagation matrices:

$G = W_L \, \Upsilon_L \cdots W_1 \, \Upsilon_1 \, W_0$

where $\Upsilon_\ell = \mathrm{diag}(e^{j\theta_{\ell,1}}, ..., e^{j\theta_{\ell,M}})$ , $W_\ell \in \mathbb{C}^{M\times M}$ for internal layers, and $W_0$ , $W_L$ couple the SIM to external (antenna or propagation) interfaces (An et al., 2023, Renzo, 2024, Nerini et al., 2024, Liu et al., 2024).

Extensions incorporate amplitude control via $a_{\ell,n}$ , polarization control (e.g., for dual-polarized architectures), and even flexural (morphing) degrees of freedom (Magbool et al., 2 Nov 2025, Darsena et al., 2024, Zhang et al., 27 May 2025).

2. Analog Wave-Domain Computing: Successive Transformation and Matrix Approximation

Stacked architectures convert incident wavefronts through cascades of programmable layer operations, allowing SIMs to approximate, implement, or “learn” large linear (and in some cases nonlinear) transforms at light-speed. Critically:

By stacking $L$ layers, the set of achievable wave-domain transformations is dramatically enlarged compared to single-layer RIS, enabling high-rank matrix multiplication and enhanced analog computational depth (Renzo, 2024, An et al., 2023).
The achievable set of transfer matrices is, in theory, limited by the number of programmable meta-atoms per layer, the number of layers, and the physical constraints (reciprocity, energy conservation, mutual coupling) (Nerini et al., 2024, Liu et al., 2024).

Notable demonstrations include:

2D Discrete Fourier Transform (DFT) implementation for spatial coding and sensing, where the phase profile of each meta-atom is iteratively optimized to minimize the Frobenius norm between the SIM’s transfer matrix $G$ and the ideal 2D-DFT matrix $F$ :

$\min_{\{\xi_{\ell,m}\},\,\beta}\; \|\beta G - F\|^2_F$

solved by customized gradient descent on the phase variables (An et al., 2023, An et al., 2024).

By proper layer stacking and optimization, SIMs have been shown to approximate neural-network–like feedforward architectures, performing matrix-vector products, Fourier-type transforms, and beamforming operations directly in hardware at the speed of light (Renzo, 2024, Liu et al., 2024).

3. Optimization and Training Algorithms

Realizing a targeted analog function with a SIM is a non-convex, often highly nonlinear, optimization problem. Contemporary algorithmic strategies include:

Gradient-descent–based phase optimization: Derive layerwise or elementwise gradients of the target loss (e.g., fitting error, sum-rate, MSE), and iteratively update meta-atom phases within unit-modulus or amplitude-constrained manifolds (An et al., 2023, An et al., 2023, Renzo, 2024, Papazafeiropoulos et al., 2024).
Alternating optimization (AO): Alternately optimize metalayer responses and digital resources (precoder weights, transmit powers) (Papazafeiropoulos et al., 2024, Darsena et al., 2024, Magbool et al., 2 Nov 2025).
Projected-gradient or manifold optimization: To enforce constant modulus or amplitude-range constraints on meta-atom responses (Darsena et al., 2024, An et al., 2023).
Accelerated or learnable optimization: Recent works employ deep reinforcement learning (DRL), deep unfolding, or neural-network controllers to efficiently search the high-dimensional phase/amplitude configuration space (Liu et al., 2024, Liu et al., 2024, Renzo, 2024).
Specialized algorithms for specific transforms (e.g., layerwise gradient derivation for 2D DFT, SVD-based precoder/combiner synthesis in HMIMO, layer-by-layer water-filling for multi-stream power allocation) (An et al., 2023, Zhang et al., 27 May 2025, An et al., 2023).

Algorithmic effectiveness is subject to convergence rate, susceptibility to local minima, and the complexity of gradient or surrogate computation for hundreds to thousands of variables.

4. Applications: Communications, Sensing, and Wave-Based Signal Processing

SIMs enable several key classes of wave-domain analog processing:

a. Communications

Analog/holographic MIMO precoding and combining: SIMs implement high-rank, low-latency analog precoding/combining for multiuser MISO/MIMO and cell-free distributed massive MIMO, reducing the number of required RF chains and ADCs (An et al., 2023, Niu et al., 13 Jul 2025, Shi et al., 23 Oct 2025).
Wideband, multi-carrier OFDM processing: Stacked layers allow for simultaneous beamforming and channel diagonalization across subcarriers (Li et al., 1 Mar 2025).
Physical layer security: Multi-layer phase coding restricts energy leakage, facilitating spatial nulling and eavesdropper suppression (Shi et al., 23 Oct 2025).

b. Sensing and ISAC

Wave-domain DFTs for direction-of-arrival (DOA) estimation: Direct mapping of incoming wavefronts into spatial/angular frequency bins, with sub-microsecond latency and reduced RF hardware (An et al., 2023, An et al., 2024).
Integrated sensing and communications (ISAC): SIMs can synthesize beampatterns for simultaneous multi-user downlink and radar-like target probing, and are jointly optimized using penalties or min-max formulations to balance spectrum efficiency and sensing accuracy (Niu et al., 2024, Ranasinghe et al., 29 Apr 2025).
Bistatic and near-field beamfocusing: Layered architectures enable fine control over both far-field and near-field (spherical) wavefronts, supporting programmable focus and interference management in THz and mmWave (Jia et al., 9 Feb 2025).

c. Semantic and Hybrid Processing

Semantic encoding via SIMs as analog neural networks: Design of SIMs to encode task-relevant features (e.g., image edges, semantic cues) directly into analog wavefronts for minimal-bandwidth, real-time semantic communications (Huang et al., 14 Jun 2025, Liu et al., 2024).
Dual-polarization and flexural degrees of freedom: Dual-polarized SIMs (DPSIMs) process orthogonal polarizations independently, doubling spatial DoF. Flexible (morphable) modules enable lens-like channel shaping beyond planar architectures (Zhang et al., 27 May 2025, Magbool et al., 2 Nov 2025).

5. Performance, Hardware Complexity, and Scaling

Performance Scaling and Benchmarks

Performance metrics include sum spectral efficiency, mean squared error, NMSE to target linear transforms, beamforming gain (SNR), and energy efficiency. Multi-layer SIMs generally outperform single-layer RIS by 3–10 dB in SNR/SE, achieve 20–300% higher sum-rate or channel capacity, and substantially reduce IAI/ISI (Renzo, 2024, Li et al., 1 Mar 2025, Shi et al., 23 Oct 2025, Darsena et al., 2024).
For 2D DFT/DOA, MSE floors of $10^{-4}$ – $10^{-2}$ are achievable at moderate SNRs with grid sizes $N=4-16$ (An et al., 2023). For HMIMO, capacity can scale with the square of the total meta-atom count.

Complexity and Practical Considerations

SIMs substitute high-speed digital/analog matrix computation with passive wave propagation and reconfigurable phase coding, yielding near-zero processing latency (propagation-time limited, i.e., nanoseconds at GHz), and drastically reduced analog and digital hardware complexity.
Key system-level complexity is in electronic bias control, power/networking for thousands of meta-atoms, and rapid reconfiguration (microsecond update times remain demanding).
In practice, energy efficiency is set by the trade-off between analog losses (attenuation, loss per layer), meta-atom hardware (passive/amplified), and digital control overhead (Darsena et al., 2024, Magbool et al., 2 Nov 2025).
Multi-fiber architectures achieve equivalent DoF with fewer layers and meta-atoms, further reducing hardware count and power (Niu et al., 13 Jul 2025).

6. Challenges, Limitations, and Research Directions

EM modeling accuracy: Most studies rely on scalar diffraction or ideal propagation models; mutual coupling, substrate modes, and higher-order interactions can degrade performance, with precise modeling and EM validation required (Nerini et al., 2024, Renzo, 2024).
Amplitude-phase control: While fully passive, phase-only layers simplify hardware, combining active (amplified) and passive layers (“hybrid” SIM) enables amplitude shaping and loss compensation at increased cost (Darsena et al., 2024).
Reconfigurability and control: Updating per-atom phase/amplitude for large ( $>$ 1K) arrays at sub-millisecond timescales is a paramount challenge, spawning efforts in distributed control, DRL orchestration, and efficient joint digital-EM optimization (Liu et al., 2024, Liu et al., 2024, Renzo, 2024).
Hardware integration: Precise mechanical stacking, sub-mm alignment, and robust wiring for 3D-printed or PCB-based arrays determine achievable scaling and uniformity (Magbool et al., 2 Nov 2025, Renzo, 2024).
AI and learning-based configuration: Scalable learning or hybrid model-and-data-driven approaches for CSI-to-phase mapping, joint communication/sensing optimization, and online adaptation are emerging topics (Liu et al., 2024, Liu et al., 2024, Renzo, 2024).
Security, privacy, and new paradigms: Physical layer security, in-air analog encryption, semantic and task-driven wavefront coding, and full analog domain inference systems remain active research frontiers (Shi et al., 23 Oct 2025, Liu et al., 2024).

Stacked intelligent metasurfaces thus constitute a platform for high-dimensional, programmable, light-speed analog computing, with demonstrated and theoretical advantages over both single-layer RIS and conventional digital/hybrid mmWave/THz arrays—subject to ongoing advances in reconfigurable hardware, scalable control, and accurate EM modeling (Renzo, 2024, An et al., 2023, Shi et al., 23 Oct 2025, Li et al., 1 Mar 2025, Liu et al., 2024, Nerini et al., 2024).