Stacked Intelligent Metasurfaces (SIM) in the Nonlinear Regime: A Multiport Network Model Approach

Abstract: We present a physically consistent multiport framework for stacked intelligent metasurfaces (SIMs) with linear and explicit nonlinear terminations. The model provides closed-form input--output relations in the linear case and fixed-point forward evaluation in the nonlinear case, with adjoint-based gradients for optimization in both settings. Under stage-isolated SIM structure, complexity remains $\mathcal{O}(QK^3)$. In a 28 GHz near-field localization case study, nonlinear terminations improve transfer-function matching and reduce mean localization error, close to the ideal benchmark.

• The paper presents a fixed-point formulation for nonlinear SIM terminations coupled with an adjoint-based gradient method for efficient parameter optimization.
• The multiport network model accurately captures internal electromagnetic interactions, reducing localization error in 28 GHz near-field scenarios.
• The methodology bridges nonlinear EM device physics with system-level signal processing, enabling advanced wave-domain manipulation for beyond-5G and 6G applications.

Stacked Intelligent Metasurfaces with Nonlinear Terminations: Multiport Network Modeling

Introduction and Motivation

Stacked intelligent metasurfaces (SIMs) represent a substantial advancement in wave-domain control for communications, sensing, and electromagnetic localization in beyond-5G and 6G wireless architectures. By orchestrating multiple tunable transmissive layers—each implemented as reconfigurable intelligent surfaces (T-RISs)—SIMs unlock richer electromagnetic manipulation than legacy single-layer RIS designs. Traditional system models, often relying on cascading channel abstractions, are insufficient for capturing the complex internal EM interactions, boundary conditions, and inter-port couplings characteristic of tightly stacked metasurfaces. Addressing this gap, multiport network modeling based on S-parameters offers a physically consistent description linking device-level EM behaviors with end-to-end signal processing objectives.

Recent advances in nonlinear hardware components for metasurfaces further motivate the extension of multiport SIM models beyond linear terminations, enabling explicit nonlinear wave manipulation anchored in rigorous network theory. The paper introduces:

• A tractable fixed-point formulation for nonlinear SIM terminations.
• Adjoint-based gradient analytics for scalable parameter optimization.
• Preservation of complete multiport Tx-SIM-Rx structure, supporting both modeling fidelity and algorithmic practicality.

Linear SIM Multiport Modeling

The SIM system comprises $Q$ stages, each consisting of two facing T-RIS layers with $K$ ports per layer, yielding $N = 2QK$ internal ports. The Tx-SIM-Rx stack forms an $N_t = L+N+M$ port network, whose S-parameter matrix $S \in \mathbb{C}^{N_t \times N_t}$ encapsulates all internal and external EM interactions. Ports are partitioned into Tx, SIM internal, and Rx groups, with boundary conditions enforcing zero reflection at Tx and Rx.

SIM terminations are implemented via a sparse, block-diagonal matrix $T(n)$ parametrized by control vector $n$, with each cell modeled as a tunable 2-port. The overall transfer function is given in closed form:

$y = (S_{RT} + S_{RE}T(n)S_{ET})a_s,$

where $a_s$ is the input excitation. End-to-end optimization relies on an adjoint-state gradient computation, with complexity scaling as $O(QK^3)$ under stage-isolated assumptions.

Nonlinear SIM Termination Framework

Extending to nonlinear terminations, each T-RIS cell is described by an explicit nonlinear mapping $K$0. This model generalizes ideal phase-shifters to accommodate nonlinear amplitude- and phase-dependent behaviors, such as those arising from embedded anti-parallel diodes or other nonlinear circuit elements.

Forward evaluation transitions from closed-form expressions to numerical fixed-point iterations:

• For each excitation, solve $K$1 iteratively.
• Structured sparsity in $K$2 and cell-wise separability in $K$3 ensure computational efficiency, with per-iteration complexity linear in $K$4.

Gradient evaluation adapts the adjoint method to the nonlinear regime by exploiting the Jacobian of the nonlinear mapping, with the adjoint equation maintaining identical structure to the linear case. The gradient computation remains cell-local and utilizes scalar chain-rule derivatives, preserving the block-recursive solver efficiency and $K$5 scaling.

Case Study: Anti-Parallel Diode Limiter Modeling

A relevant nonlinear cell model is constructed by embedding an anti-parallel diode pair as a shunt across the transmission line. Under the fundamental component approximation, diode-induced amplitude compression is characterized with negligible phase distortion. The soft-limiter AM/AM nonlinearity is accurately modeled using the Rapp function:

$K$6

Parameters are fit to numerically computed diode responses. The nonlinear two-port law incorporates both amplitude compression and programmable phase control, facilitating enhanced manipulation of incident electromagnetic signals within the SIM stack.

Numerical Results: Near-Field Localization

A 28 GHz near-field localization scenario with five stacked layers is investigated. The SIM is optimized to implement a polar-domain mapping, partitioning the region of interest into angle-range bins for user position estimation. The nonlinear SIM model achieves improved transfer-function matching (error reduction from 2% linear to 1% nonlinear) and reduced mean localization error (from 5.78 cm linear to 4.75 cm nonlinear, approaching the ideal benchmark of 4.17 cm at SNR = 10 dB). The introduction of nonlinear terminations yields consistent quantitative gains in both signal processing fidelity and functional accuracy, with no increase in computational complexity.

Implications and Future Directions

The formal advancement of physically consistent SIM modeling to include explicit nonlinear termination laws provides a unified analytical pipeline for EM-aware optimization and signal processing. Practically, this enables the design of SIM architectures that leverage nonlinear wave manipulation for superior performance in near-field localization, communications, and sensing. The framework’s cell-local computational structure, preserved scaling, and direct applicability to large-scale stacked metasurfaces position it as a viable tool for next-generation SIM deployment in dynamic wireless environments.

Theoretically, the methodology bridges the gap between nonlinear EM device physics and system-level objectives, facilitating the exploration of richer nonlinear metasurface functionalities. Future directions may include integration with time-varying or actively reconfigurable nonlinear elements, extension to non-diagonal terminations with arbitrary multiport coupling, and joint modeling of hardware impairments and environmental variability.

Conclusion

The paper presents a unified multiport network model for stacked intelligent metasurfaces under both linear and explicit nonlinear cell-terminals, enabling physically consistent evaluation and adjoint-based optimization while maintaining tractable complexity. Numerical evidence demonstrates improved performance in near-field localization tasks, substantiating the practical value of nonlinear SIMs in advanced wave-domain processing applications (2605.23713).