Stacked Intelligent Surfaces (SIS)

• Stacked Intelligent Surfaces (SIS) are multi-layered programmable metasurfaces that manipulate electromagnetic waves via cascaded phase adjustments for advanced signal processing.
• They enable robust beamforming, DOA estimation, holographic MIMO, and integrated sensing by jointly optimizing phase shifts and power through analog computation.
• SIS leverages deep neural network-like architectures and multiport EM modeling to achieve energy-efficient, high-capacity communication and sensing with practical deployment benefits.

Stacked Intelligent Surfaces (SIS) are multi-layered electromagnetic platforms composed of cascaded programmable metasurfaces, each containing dense arrays of passive or active “meta-atoms.” By tailoring the phase (and sometimes amplitude) of impinging electromagnetic waves through each layer, SIS architectures can perform advanced analog signal processing—including beamforming, spatial coding, direction-of-arrival (DOA) estimation, and over-the-air computation—in the wave domain. The stacking of metasurface layers enables functionalities that are unattainable with traditional single-layer reconfigurable surfaces, providing additional degrees of freedom, enhanced design flexibility, and multi-functional integration across communication, sensing, and computing tasks.

1. SIS Architecture and Wave-Domain Modeling

Stacked Intelligent Surfaces are physically realized by vertically integrating multiple metasurface layers, each populated by numerous programmable meta-atoms. The tunable response of each meta-atom is typically realized via electronic biasing (e.g., via FPGAs or microcontroller arrays), allowing the phase (and in some designs, amplitude) of the transmission coefficient to be dynamically adjusted: $\phi^{(l)}_n = e^{j\theta^{(l)}_n}$ for the $n$-th element in the $l$-th layer (unit-modulus phase control). Each layer is separated by a controlled distance to enable free-space propagation and mutual coupling, which is accurately modeled by complex-valued inter-layer transmission matrices (frequently based on Rayleigh–Sommerfeld diffraction theory or multiport network theory).

The overall SIS response is captured by the cascaded action of all phase shift and propagation matrices (for $L$ layers):

$$G_{\text{SIS}} = \Phi^{(L)} W^{(L)} \ldots \Phi^{(2)} W^{(2)} \Phi^{(1)}$$

where each $$\Phi^{(l)} = \operatorname{diag}(e^{j\theta^{(l)}_1},\ldots,e^{j\theta^{(l)}_N})$$ and $W^{(l)}$ encapsulates propagation physics between layers. For fully-consistent models, mutual coupling, self-impedance, and inter-element effects are incorporated using block multiport network theory (Abrardo et al., 5 Jan 2025, Nerini et al., 19 Feb 2024).

The multi-layer, stacked structure is analogous to feed-forward deep neural networks, enabling the emulation of complex, multi-stage EM transformations in a fully programmable, energy-efficient, and latency-minimizing manner (Renzo, 29 Nov 2024). SIS elements act as "neurons" with unit-modulus weights, and the cascade replicates the operation of analog neural networks in the electromagnetic domain (Zayat et al., 30 Aug 2025).

2. SIS for Beamforming, Holographic MIMO, and Parallelization

SIS architectures fundamentally extend the capabilities of reconfigurable intelligent surfaces (RIS) by achieving advanced wave-domain precoding, beamforming, and channel shaping. In multiuser downlink MISO/MIMO or holographic MIMO (HMIMO) setups, SIS can serve as both the transmit precoder and the receive combiner, enabling:

• Interference-free Parallel Channels: By jointly optimizing the phase shifts across all layers and, when present, at both transmitter and receiver SIS, the effective channel is diagonalized or closely approximates an identity matrix, enabling independent, interference-free parallel data streams (An et al., 2023, Li et al., 1 Mar 2025).
• Wave-Domain Precoding: SIS enables analog precoding, with optimization over transmission and combining matrices, eliminating the need for complex digital baseband processing and the associated RF chain count (An et al., 2023, Li et al., 1 Mar 2025).
• Near-Field Multiuser Focusing: In systems operating at large aperture or terahertz frequencies, SIS permits direct near-field spherical wavefront beamfocusing, surpassing far-field planar approximations and optimizing spatial gain and interference suppression (Jia et al., 9 Feb 2025).

Optimization frameworks typically maximize the downlink sum spectral efficiency under transmit power and unit-modulus constraints. Techniques include alternating optimization (jointly over phase shifts and power allocation), projected gradient descent (with Armijo step size), block coordinate descent with penalty convex-concave procedures, deep reinforcement learning (DRL)-based actor-critic schemes, and complex-valued neural network modeling using GPU-accelerated autograd (Papazafeiropoulos et al., 29 May 2024, Liu et al., 14 Feb 2024, Zayat et al., 30 Aug 2025).

3. Integrated Sensing and Communications (ISAC) via SIS

SIS platforms have demonstrated outstanding potential for Integrated Sensing and Communications (ISAC) systems, where a single hardware entity jointly performs high-throughput communications and target/environmental sensing:

• Joint Beam Pattern Synthesis: SIS can generate customized beampatterns for simultaneous downlink data transmission and high-gain target illumination, optimizing both user sum rate and beampattern power in desired sensing (radar) directions. Constraints on the beampattern are handled as hard inequalities or via penalty terms in the spectral efficiency objective (Niu et al., 19 Aug 2024, Li et al., 5 Sep 2024).
• Dual-Objective Optimization: Multi-objective algorithms, such as dual-normalized differential gradient descent (D³), balance gradients from communication and sensing tasks to ensure a tunable trade-off between communication sum-rate and sensing beampattern error (Li et al., 5 Sep 2024).
• Cramér–Rao Bound Minimization: For extended target models, the estimation error on the target response matrix is minimized, typically under SINR and power constraints for communication users. Solutions generally involve alternating optimization and semidefinite relaxation (SDR) (Wang et al., 2 May 2024).
• Bistatic and Doubly-Dispersive Channels: In bistatic settings, SIS optimization is formulated as a min-max problem (maximizing weakest channel path), and compressed sensing-based probabilistic data association (PDA) algorithms are applied for radar parameter estimation under OTFS, OFDM, or AFDM waveforms (Ranasinghe et al., 29 Apr 2025).

The experimental validation confirms that multi-layer SIS consistently improves both communication throughput and sensing resolution, with performance scaling favorably with the number of layers and overall aperture size (Wang et al., 2 May 2024). System-level simulations demonstrate rapid convergence of proposed algorithms (typically within tens of iterations) and substantial gains in both spectral efficiency and sensing accuracy (Niu et al., 19 Aug 2024, Li et al., 5 Sep 2024).

4. Computational and Physical Modeling Frameworks

Accurate modeling and optimization of SIS require a physically consistent, multiport EM network framework:

• Multiport Network & Scattering/Impedance Formalism: The SIS is represented as an electromagnetic collaborative object with ports at each meta-atom location. The system EM response is described by comprehensive Z-parameter or S-parameter matrices. The optimal transfer function or channel matrix depends not only on phase configuration but also on inter-element mutual impedances and reflection coefficients (Abrardo et al., 5 Jan 2025, Nerini et al., 19 Feb 2024).
• Assumptions and Limitations: Conventional models that ignore electromagnetic coupling and assume unilateral propagation yield tractable, low-complexity optimization (O(PMK²)), but may deviate significantly from actual performance for dense, multi-layer SIS. Physically consistent models, using block diagonals and inversion of full coupling matrices, restore accuracy at the cost of higher computational complexity (Abrardo et al., 5 Jan 2025).
• Connection to Backpropagation: Efficient blockwise or layerwise algorithms (using Neumann series expansion and banded matrix structures) allow scalable numerical solution, analogous to backpropagation in deep neural networks.
• Meta-Fiber Integration: To address energy attenuation and phase shift optimization complexity in deep stacks, the use of engineered meta-fibers allows for meta-atom interconnections that reduce the required number of layers (e.g., enabling a two-layer SIS to match or outperform seven-layer conventional designs), cut cumulative losses, and allow closed-form AO-based phase updates with substantial capacity and hardware advantages (Niu et al., 13 Jul 2025).

5. Advanced Functionalities: DOA Estimation, Analog Computing, and Dual Polarization

SIS architectures have been leveraged for diverse analog computing and sensing functions:

• Wave-Domain Fast Fourier Transform: By optimizing the layerwise phase configuration, SIS can approximate high-dimensional 2D discrete Fourier transforms (DFT) in the EM domain, enabling ultra-fast, energy-efficient DOA estimation with sub-millidegree accuracy and mean-squared-error on the order of $10^{-4}$ (An et al., 2023, An et al., 13 Feb 2024).
• Analog Neural Networks: The cascade of programmable layers in SIS naturally enables the physical realization of deep neural network operations (e.g., matrix multiplications or convolution) directly on the propagating wavefront, achieving computational throughput at optical speed with minimal hardware cost (Renzo, 29 Nov 2024, Liu et al., 4 Jul 2024).
• Dual-Polarization SIS: The dual-polarized SIS (DPSIM) architecture employs dual-polarized meta-atoms, independently controlling two orthogonal polarizations, thereby doubling spatial degrees of freedom, suppressing polarization cross-interference and inter-stream interference, and supporting more parallel, ISI-free data streams. Optimization using layerwise gradient descent with water-filling achieves SE and EE levels approaching theoretical upper bounds, even under polarization imperfections (Zhang et al., 27 May 2025).

6. Algorithmic Optimization and Learning-Based Configuration

The high-dimensional, non-convex optimization inherent to SIS design is addressed with several methodologies:

• Alternating Optimization (AO): Decomposition into power allocation and phase-shift update subproblems, typically involving iterative water-filling for power and gradient ascent for phases (or projected gradient descent with Armijo backtracking), with convergence in tens of iterations for moderate system sizes (An et al., 2023, Papazafeiropoulos et al., 29 May 2024).
• Convex-Concave Procedures and Closed-Form Updates: Block coordinate descent and penalty-based relaxation enable low-complexity minimization of MSE or fitting error between desired and actual channel matrices, with closed-form phase shift update expressions providing rapid and monotonic convergence (Li et al., 1 Mar 2025, Niu et al., 13 Jul 2025).
• Deep Reinforcement Learning (DRL): Continuous-action actor-critic DRL methods (e.g., DDPG) enable near-real-time, data-driven configuration of SIS phase profiles and power allocation under imperfect knowledge and environmental uncertainty. Whitening processes and action randomization are used for enhanced exploration and robustness (Liu et al., 14 Feb 2024, Liu et al., 9 Aug 2024).
• End-to-End Differentiable Neural Optimization: Modeling the SIS as a deep, fully differentiable, complex-valued neural network allows for direct gradient-based optimization of all phase parameters, with the flexibility to optimize for throughput, detection error, symbol error rate, or arbitrary multi-objective metrics. GPU-accelerated auto-differentiation enables near-online adaptation and robust performance under dynamic channels (Zayat et al., 30 Aug 2025).

7. Practical Deployment, Physical Limitations, and Future Directions

SIS presents a transformative, multifunctional platform for next-generation wireless systems, but several challenges and design trade-offs must be addressed:

• Hardware Non-Idealities: Manufacturing imperfections, amplitude-phase coupling in real meta-atoms, inter-layer and inter-element coupling, and finite phase quantization may degrade practical performance (Liu et al., 4 Jul 2024).
• Physical Layer Modeling: Accurate EM modeling and calibration are critical. Simplified scalar and unit-modulus phase shift models may be inadequate for dense, high-frequency, or near-field regimes (Abrardo et al., 5 Jan 2025, Nerini et al., 19 Feb 2024).
• Complexity and Energy Efficiency: The introduction of meta-fibers, optimized dual-polarization structures, and efficient algorithmic schemes (closed-form AO, DRL, differentiable CV-NN) can significantly lower complexity and processing energy, making large-aperture, high-capacity SIS feasible for practical deployment (Niu et al., 13 Jul 2025, Zhang et al., 27 May 2025).
• Integrated Sensing, Computing, and Communications: The natural fusion of communication, sensing, and wave-domain computing tasks in SIS positions it as a foundational technology for future 6G networks, digital twins, UAV/drone platforms, and smart radio environments (Renzo, 29 Nov 2024, Liu et al., 4 Jul 2024).

Continued research is required in electromagnetic-compliant optimization, integration of hybrid analog–digital schemes, robust and adaptive learning-based reconfiguration, and experimental system validation under realistic deployment constraints. The insights gained from algorithmic developments, modeling advances, and dual-functional hardware offer a comprehensive roadmap for future SIS-enabled architectures.