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Stacked Fluid Metasurfaces: Mutual-Coupling-Aware Modeling and Optimization

Giovanni Iacovelli, Chandan Kumar Sheemar, Symeon Chatzinotas

arXiv:2609.31591v1eess.SP

TL;DR

Compact fluid-port stacks require a model that accounts for mutual coupling as positions and interconnections vary. The paper builds a three-dimensional impedance-kernel model and analyzes the loaded stack through its induced currents; it identifies the conventional cascade as a single-pass limit and ties the stack’s efficiency cost to induced-current norm.

  • Problem

    When ports are packed below half a wavelength, moving an element changes the impedance matrix loaded by the network, so position and connectivity reshape one operator and a stack adds an axial dimension.

    Source: §I · p. 1

  • Method

    The paper models the stack with a three-dimensional impedance kernel and obtains induced currents, channel, and stack-loaded input impedance through one matrix inversion.

    Source: §VI · p. 12

  • Results

    The conventional cascade is the model’s single-pass limit, and the stack’s efficiency cost is set by the induced-current norm dissipated at the transmitter’s expense.

    Source: §VI · p. 12

  • Takeaways & Limitations

    Inter-layer connections span shorter distances than most within-panel connections in compact stacks, making layer-diagonal modeling over-restrictive.

    Source: C. Loads, interconnection patterns,… · p. 4

Abstract

from arXiv · show

Stacked intelligent metasurfaces process the transmitted field layer by layer, and fluid antennas make the position of every radiator a design variable. Combined, they pack radiators at sub-wavelength spacings within and across layers, where mutual coupling governs the physics that current models omit. This paper develops a coupling-consistent model of a fluid transmit layer illuminating a stack of passive fluid beyond-diagonal layers. The multiport impedance matrix of any port constellation is shown to be the impedance kernel sampled at the three-dimensional separations, in one closed spherical-Hankel form whose in-plane restriction is intra-layer coupling and whose axial restriction replaces the scalar Rayleigh--Sommerfeld propagator. Only the transmitter is driven: the layer currents are induced, and the familiar cascade is the single-pass limit of one matrix inversion. In the wavenumber domain the light circle still separates radiation from reaction, now with a plane-wave propagator attached, every Fourier mode sees a transmission line loaded by the layers, and an efficiency identity shows that what the stack costs in efficiency is set by its total induced-current norm, not by its layer count. Because the load is never assumed layer-diagonal, interconnections may join atoms on different sheets, and vertical networks repeated at every site are spectrally local. Alternating optimization designs precoders, positions, and loads with closed-form gradients. Numerically, fluid ports on a $\lambda/8$ grid perform within a few per cent of continuously movable ones, while a design based on the conventional cascade model can do worse than open-circuiting the stack.

I. INTRODUCTION

Compact fluid-port and stacked-layer architectures operate where mutual coupling affects radiators within and across planes, but conventional cascades omit key interactions. The paper develops a three-dimensional coupling model and uses it to formulate stack-aware analysis and design.

I. INTRODUCTION

  • Sub-wavelength spacing makes mutual coupling important within and across layers, while the standard SIM cascade omits intra-layer coupling, inter-layer reflection, transmitter feedback, polarization, and a physically meaningful power constraint.Existing fluid-element and movable-element surface models inherit this abstraction, despite mutual coupling being identified as an open challenge.Source: §I · p. 1

A. Two modeling cultures, and a third dimension

  • Prior coupling-aware stack models use fixed geometry, while flexible-position surfaces omit coupling; a differentiable, closed-form three-dimensional model combining both was missing.The field-first impedance-kernel approach had been unified with flexible positioning for a single surface, but not extended to a stack.Source: A. Two modeling cultures, and a third… · pp. 1–2

B. Contributions

  • The impedance matrix for ports on parallel planes is the impedance kernel sampled at three-dimensional separations, unifying intra-layer coupling, inter-plane transfer, and transmitter feedback.The scalar Rayleigh–Sommerfeld coefficient is its unilateral counterpart with a flat spectral weight.Source: B. Contributions · p. 2
  • Terminating the stack on one reconfigurable network yields induced currents, the multi-user channel, and stack-loaded input impedance in one inversion; the conventional cascade is a single-pass limit under explicit assumptions.The transmitter supplies the dissipation of every sheet.Source: B. Contributions · p. 2
  • In the wavenumber domain, a plane-wave propagator accompanies the radiation–reaction split, and each Fourier mode sees a loaded transmission line.Networks repeated vertically at each site are spectrally local, and connections may join atoms across sheets.Source: B. Contributions · p. 2
  • The joint design maximizes weighted sum spectral efficiency over precoders, positions, and loads with closed-form gradients; ports on a λ/8 grid perform within a few per cent of continuously movable ports.The evaluation computes coupling entries for the actual element profile.Source: B. Contributions · p. 2

A. Sheets, kernel, and complex power

  • The three-dimensional separation underlies the model, while the complex power balance distinguishes bounded resistive behavior from a reactive kernel that grows as R^-3 at short range.Dissipation on induced currents is included on each sheet, whereas radiative and reactive kernels connect sheets.Source: A. Sheets, kernel, and complex power · p. 3

B. Ports on planes and the 3-D circuit-field equivalence

The section constructs a port-level impedance model for fluid elements on parallel planes. Sampling the three-dimensional kernel captures both within-plane coupling and transfer between planes, while clarifying where the point-port approximation and scalar propagator differ.

B. Ports on planes and the 3-D circuit-field equivalence

  • As κΔ→0, mutual impedance entries on the same or different planes converge to the kernel sampled at the ports’ three-dimensional separation.The model’s port matrix matches the complex power of every excitation, with real and imaginary parts sampling the resistive and reactive kernels.Source: B. Ports on planes and the 3-D… · p. 3
  • The impedance matrix is symmetric, so transfer between two planes has the same entries in either direction.Its diagonal blocks represent in-plane coupling and its off-diagonal blocks represent inter-plane transfer.Source: B. Ports on planes and the 3-D… · p. 3
  • For compact stacks, the point-kernel limit requires port diameter much smaller than separation, so the finite-profile smoothed kernel is used when that condition may fail.For facing Gaussian-profile ports, the point-limit reactance exceeds the smoothed value by factors of 12, 5, and 1.2 at gaps of 1.25σ, 2σ, and 5σ, respectively.Source: B. Ports on planes and the 3-D… · p. 3
  • The scalar Rayleigh–Sommerfeld coefficient agrees with the kernel only near broadside and without intra-layer coupling or inter-layer reflection, not in the compact-stack regime.Its polarization-blind angular weighting overestimates E-plane transfer and underestimates H-plane transfer; in the near zone, its sign also differs from the static-dipole kernel in part of the E-plane.Source: B. Ports on planes and the 3-D… · p. 4

C. Loads, interconnection patterns, and induced currents

Only the transmit plane is driven; the passive stack is terminated by a reconfigurable network, and its induced currents and input impedance follow from the coupled multiport equations. The load model permits inter-sheet connections rather than requiring layer-diagonal structure.

C. Loads, interconnection patterns, and induced currents

  • A single network terminates all stack ports, and solving the loaded multiport equations gives induced stack currents and the transmitter’s stack-loaded input impedance.The network may have loss, represented separately from element loss and from the constraint on long connections.Source: C. Loads, interconnection patterns,… · p. 4
  • The interconnection pattern is a linear subspace of real symmetric reactance matrices and need not be block-diagonal by layer.The considered families include layer-diagonal, vertically connected, and stack-wide patterns.Source: C. Loads, interconnection patterns,… · p. 4
  • The interconnection families are nested, so their optima are ordered at fixed geometry, and the analysis requires only a passive network with a reactance matrix in the allowed linear subspace.The coupling changes both the feasible set and effective channel as positions and loads vary.Source: C. Loads, interconnection patterns,… · p. 4
  • In a compact stack, inter-layer connections span only the sheet separation, h ∈ [λ/16, λ/2], shorter than most within-panel connections, making layer-diagonal modeling over-restrictive.Each in-plane site can own a lossless L-port connecting all sheet pairs or only adjacent sheets.Source: C. Loads, interconnection patterns,… · p. 4
  • The stack-loaded input impedance accounts for dissipation on induced sheet currents, so stack loss is charged to the transmitter rather than to load absorption.The resulting input-impedance matrix depends on position and load, unlike the single-plane power matrix used without a stack.Source: C. Loads, interconnection patterns,… · p. 4

D. Effective channel and the cascade limit

The effective channel is obtained from a coupled impedance formulation in one inversion, while the familiar cascade emerges only under restrictive coupling and load assumptions. The same formulation exposes limits on single-reactance layer control and relates stack efficiency to induced currents.

D. Effective channel and the cascade limit

  • One Ns × Ns inversion gives the position-dependent channel while retaining multiple reflections, intra-layer coupling, and feedback among all planes.The channel is the impedance-form counterpart of the loaded-scatterer model.Source: D. Effective channel and the cascade… · p. 5
  • The cascade follows from the self-consistent layer equations only when loads are layer-diagonal and field coupling is nearest-neighbour forward, with direct illumination of deeper sheets and backward coupling discarded.Keeping only backward-block truncation is the weaker alternative; vertical inter-sheet connections are excluded by the layer-diagonal assumption.Source: D. Effective channel and the cascade… · p. 5
  • With diagonal, lossless loads and neglected intra-layer coupling, one real reactance per atom traces a restricted response circle, so amplitude and phase cannot be assigned independently.The cascade's unit-modulus phase mask is not realizable with one lossless reactance per element; lifting the lock requires another degree of freedom.Source: D. Effective channel and the cascade… · p. 5
  • In compact stacks, discarded backward blocks retain near-field strength, while the omitted direct transmitter-to-deeper-sheet blocks do not fade as the stack is stretched.For two resonant facing atoms, strong mutual impedance can reduce the true currents even as the cascade predicts large currents.Source: D. Effective channel and the cascade… · p. 5
  • At fixed radiated power, the stack's efficiency cost is set by the total induced-current norm rather than the number of layers.Induced currents dissipate in their own conductors, and stronger control and stronger dissipation coincide near element resonance.Source: D. Effective channel and the cascade… · p. 5

E. What a single sheet can do

A lossless single sheet cannot choose transmission phase and power independently, while the wavenumber-domain model extends the radiation–reaction split across planes and resolves inter-plane coupling mode by mode.

E. What a single sheet can do

  • A lossless sheet links transmission phase and power, so changing the transmitted field also sends an equal-amplitude wave backward.The transmitted power is |t|^2 = cos^2(ψ/2); a phase of ±π/4 costs 3 dB, while ±π/2 returns all incident power.Source: E. What a single sheet can do · p. 6

F. Multi-user signal model and constraints

  • The multi-user formulation constrains transmitter power, amplifier voltage, and induced layer currents through quadratic forms of the precoders.The power constraint includes radiation and dissipation in sheets and loads; the induced-current constraint limits current through tunable components.Source: F. Multi-user signal model and… · p. 6

III. WAVENUMBER-DOMAIN REPRESENTATION ACROSS PLANES

  • The wavenumber-domain representation preserves the light-circle split across planes, diagonalizes inter-plane impedance blocks, and represents a fluid port on any plane as a codeword.Each Fourier mode sees a transmission line loaded by the stack; its reaction includes both Fresnel-phase and evanescent parts.Source: §III · p. 6, A. The light circle with a propagator · p. 6

A. The light circle with a propagator

  • Across planes, radiation remains confined to the visible disc, while reaction includes Fresnel-phase contributions on the disc and evanescent contributions that decay off it.The propagator adds the factor e^(jk_z h), and its spectral weight includes an obliquity-and-polarization term absent from the scalar coefficient.Source: A. The light circle with a propagator · p. 6
  • At separations h ≲ 0.17λ, evanescent modes dominate the axial reactive coupling, contributing 93% at 0.1λ, 67% at 0.15λ, and 21% at 0.2λ.For a Gaussian element profile of standard deviation λ/20, the facing reactance at h = λ/16 is 1.9Rr rather than 23Rr.Source: A. The light circle with a propagator · p. 6

B. Fourier modes across planes

Fourier modes asymptotically decouple the inter-plane power bookkeeping, turning each mode into a loaded transmission line. The resulting structure clarifies when layer networks remain spectrally local and computationally efficient.

Fourier modes across planes

  • For separated planes, modal Gram blocks become diagonal in the large-aperture limit, and the reactive Gram converges even for evanescent modes.Visible-mode off-diagonal leakage is bounded by a term that vanishes with aperture size; at zero separation, the truncated-mode reactive Gram instead diverges logarithmically.Source: B. Fourier modes across planes · p. 7
  • Each Fourier mode sees the layers as shunt loads on a transmission line, with the transmitter entering at the first sheet after the standoff delay.For unconnected sheets, the modal system is tridiagonal and costs O(L) operations.Source: B. Fourier modes across planes · p. 7
  • A spectrally local network preserves independent modal systems; a repeated vertical L-port is one such network, with nearest-neighbour ladders costing O(L) per mode.A dense vertical L-port costs O(L^3) per mode, still below the O(N_l^2) cost of one fully connected layer.Source: B. Fourier modes across planes · p. 7
  • Element-varying diagonal reactances generally mix Fourier modes, whereas a shift-invariant beyond-diagonal layer is spectrally local.For a network that is not spectrally local, the modal image is full and the system is best formed in the port domain.Source: B. Fourier modes across planes · pp. 7–8

C. Fluid ports on every plane

The paper represents fluid-port positions on each plane through mode-domain codewords, extending the position-to-codeword mapping to the full stack. In the diagonal limit, complex power also separates by mode.

C. Fluid ports on every plane

  • The profile-smoothed user channel on a plane is obtained from the transposed codeword map, as a superposition of codeword evaluations.The coefficients collect the Fourier components of the user functional on that plane.Source: C. Fluid ports on every plane · p. 8
  • In the diagonal limit, complex power splits over modes, and collecting the port indices recovers the port-domain quadratic form 1/2 ι^H Z ι.The resulting impedance blocks are formed from the plane codeword maps and the corresponding modal blocks.Source: C. Fluid ports on every plane · p. 8
  • A fluid port’s position on any plane is represented by a constant-modulus codeword, extending the single-surface codeword maps plane by plane.The stack’s mode-domain dictionary treats positioning as modulation, layering as propagation, and interconnection as a load.Source: C. Fluid ports on every plane · p. 8

IV. COUPLING-AWARE JOINT DESIGN

The design jointly optimizes precoders, port positions, and loads under power, voltage, current, interconnection, and spacing constraints. The resulting problem is non-convex and is solved by alternating over three blocks with closed-form ingredients.

IV. COUPLING-AWARE JOINT DESIGN

  • The objective is weighted sum spectral efficiency in bit/s/Hz, with lossless loads assumed for concreteness.The model is monochromatic and does not certify a bandwidth for converting this objective to a rate in bit/s.Source: A. Problem · p. 8
  • The joint design problem is non-convex because coupling makes both the feasible set and effective channel vary with positions and loads.The position constraints keep ports within their apertures and at least d_min apart on each plane.Source: A. Problem · p. 8
  • The authors alternate over three design blocks, whose ingredients are all closed-form.The master problem jointly covers precoders, loads, and positions under the constraints collected from the earlier formulation.Source: A. Problem · p. 8

B. Precoder block

The precoder, load, and position blocks optimize the weighted sum spectral efficiency in turn, using whitening and WMMSE for precoding and closed-form gradients for loads and positions.

B. Precoder block

  • Whitening reduces the precoder block to a weighted-sum spectral-efficiency problem solved by WMMSE, with second-order cone constraints enforced by penalties and projected-subgradient updates.Each precoder pass is monotone in the objective.Source: B. Precoder block · p. 8

C. Load block: projected gradient on the interconnection pattern

  • Projected gradients update symmetric loads on any permitted interconnection pattern, using a mask projection and derivatives assembled after one inversion.For diagonal layers, the update becomes a per-element reactance step rather than a phase update.Source: C. Load block: projected gradient on… · pp. 8–9

D. Position block: closed-form 3-D gradients

  • Closed-form 3-D kernel gradients propagate position changes through the stack, with sparse derivatives supporting projected-gradient updates on every plane.A moving transmit port affects its corresponding coupling entries; a stack port affects its row and column in the stack impedance matrix.Source: D. Position block: closed-form 3-D… · p. 9

E. Certificates, initialization, algorithm, and complexity

A holographic upper bound relaxes transmit-plane codewords while retaining the stack, and the resulting configuration initializes an alternating design whose blocks converge to a partial stationary point.

E. Certificates, initialization, algorithm, and complexity

  • The holographic upper bound relaxes the transmit-plane codeword set but retains the stack couplings, and bounds the design objective at fixed stack geometry and loads.The fully connected layer pattern upper-bounds sparser patterns at fixed geometry.Source: E. Certificates, initialization,… · p. 9
  • The initialization projects the bound solution onto the transmit codeword manifold, starts layers on a λ/2 grid, and selects among uniform-reactance load starts.The alternating algorithm then updates precoders, loads, and positions using WMMSE and projected-gradient steps.Source: E. Certificates, initialization,… · pp. 9–10
  • Monotone block updates and a bounded objective yield convergence of the objective sequence and a partial stationary point of the design problem.The implementation uses one inversion per outer iteration and requires no full-wave solver or nested cascade.Source: E. Certificates, initialization,… · p. 10
  • For spectrally local layers, the matrix inversion is replaced by NI tridiagonal solves of size L, with total cost O(NI L) independent of layer port count.The general implementation costs O(N_s^3) for the stack inversion.Source: E. Certificates, initialization,… · p. 10

V. NUMERICAL RESULTS

Numerical tests compare coupling-aware designs against simplified models and vary wiring, depth, and port mobility. A λ/8 fluid-port grid stays close to continuous movement, whereas cascade-designed loads can perform worse than an open-circuited stack.

A. Scenario, conventions, and schemes

  • The default scenario uses two 5 × 5 atom sheets on a λ/2 grid, three users, and eight paired drops; designs are scored under the exact model against an open-circuited direct link.The users lie 8λ to 14λ beyond the last sheet, within the aperture’s radiating near field.Source: §V · p. 10, A. Scenario, conventions, and schemes · p. 10

B. Convergence

  • The projection initialization reaches 13.15 bit/s/Hz, within 0.04 of the λ/2-square initialization, and reaches 99% of its limit after seven outer iterations.A random constellation ends 0.9 bit/s/Hz lower.Source: B. Convergence · p. 10, Figure 1 · p. 11

C. What the cascade costs

  • At h = λ/2, cascade-based designs lose up to 1.71 bit/s/Hz under (A2), 1.74 under (A1)+(A2), 1.66 under the mask, and 0.66 under (A2−).At h = λ/16, induced-current relative errors are 3.1 for (A2−), 4.7 for (A2), 6.3 for (A1)+(A2), and 4.1 for the mask.Source: C. What the cascade costs · pp. 10–11, Figure 2 · p. 11

D. How to wire a stack

  • Connecting gains up to 0.43 bit/s/Hz between 0.15λ and 0.35λ, while moving gains 0.62 at λ/16, 0.29 at λ/10, and 0.43 at λ/2; together they retain 81% to 89% of their summed gains.The movable connected design stays within 1.7% of 13.94 bit/s/Hz over a plateau beginning at h = 0.15λ.Source: D. How to wire a stack · p. 11
  • At comparable interconnection dimension, an in-plane chain beats the vertical ladder at both spacings, while stackwide wiring reaches 12.50 and 15.59 bit/s/Hz at h = 0.125λ and 0.375λ.Stackwide wiring uses 5 to 29 times the reactances of column patterns.Source: D. How to wire a stack · p. 11, Figure 3 · p. 12

E. How deep, and how many ports

  • FAS on a λ/8 grid trails continuously movable MA by at most 0.36 bit/s/Hz (3%) through N0 = 4, while CAS falls below DIRECT from N0 = 4 onward.CAS losses reach 1.45 bit/s/Hz at N0 = 1 and 9.52 bit/s/Hz at N0 = 8.Source: Figure 4 · p. 12, E. How deep, and how many ports · p. 12
  • Six sheets double the rate with four streams, from 7.56 to 15.48 bit/s/Hz for spread users and from 5.05 to 10.38 for clustered users; 88% of the gain is reached by L = 3.With four streams, each sheet beyond L = 3 adds at most 0.44 bit/s/Hz.Source: E. How deep, and how many ports · p. 11, Figure 4 · p. 12

VI. CONCLUSION

  • The conclusion reports that the λ/8 fluid-port grid performs within a few per cent of continuous movement, while cascade-designed stacks can fall below the direct link.It identifies bandwidth accounting, coupling-aware acquisition of per-plane spectra, dielectric spacers, and electrically significant vertical connections as future work.Source: §VI · p. 12

APPENDIX A PROOF OF THEOREM 1

The appendices establish the impedance-kernel construction and derive its power, scattering, and wavenumber-domain consequences. They also evaluate the facing-port kernel by separating propagating and evanescent contributions.

APPENDIX A PROOF OF THEOREM 1

  • The plane-current integrals reduce to a correlation form, and expanding the smooth kernel at distinct ports yields the three-dimensional Hankel-form coupling and reciprocal symmetry.The expansion requires both Δ ≪ R and κΔ ≪ 1.Source: Appendix A · p. 14

APPENDIX B PROOF OF LEMMA 1

  • Eliminating the induced stack currents gives a symmetric Schur-complement input impedance and a power balance that includes sheet dissipation, load absorption, and radiation.The stack’s real-power terms include its radiation, conduction loss, and interference with the transmitter’s field.Source: Appendix B · p. 14

APPENDIX C PROOF OF LEMMA 2

  • A tangential-current sheet scatters symmetrically mode by mode, so transmission equals one plus reflection; two connected sheets can realize an arbitrary lossless two-port per mode.For a lossless sheet, Re{r} = −|r|^2; the two-sheet case can realize r = 0 with freely chosen transmission phase.Source: Appendix C · p. 14

APPENDIX D PROOF OF THEOREM 2

  • The Weyl expansion gives the planar spectrum a propagating-disc and evanescent-region split: the former contributes radiation, while the latter is purely imaginary and decays with separation.The spectral factor includes the plane-wave propagation term for h > 0.Source: Appendix D · p. 14

APPENDIX E PROOF OF COROLLARY 2

  • For facing ports, radial integration separates the disc’s radiative contribution from its reactive contribution and the evanescent ring’s reactive contribution.The radial weight includes 1−ϱ^2/(2κ^2); the two reactive terms sum to χ(0, h).Source: Appendix E · pp. 14–15

APPENDIX F PROOF OF LEMMA 3

The appendices establish convergence conditions for the wavenumber representation, identify a transmission-line structure across layers, and prove a modal upper bound for the original design.

APPENDIX F PROOF OF LEMMA 3

  • The spectral approximation converges under the stated smoothness and integrability conditions, with the evanescent diagonal converging for positive layer separation.Outside the disc, the weight decays exponentially for h > 0, allowing the dominated-convergence argument over the plane.Source: Appendix F · p. 15

APPENDIX G PROOF OF COROLLARY 4

  • Across L sheets, the modal coupling has a semiseparable transmission-line form whose inverse is tridiagonal, and the layer-diagonal solve costs O(L).The inverse fails only if a gap has sin µ_l = 0, which cannot occur for an evanescent mode.Source: Appendix G · p. 15

APPENDIX H PROOF OF PROPOSITION 3

  • The modal mapping preserves the channel and power of feasible transmit designs, while optimizing over the larger modal feasible set gives an upper bound.The modal problem drops constraints C2 and C3, so enlarging the feasible set cannot lower the optimum.Source: Appendix H · p. 15

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