Reduced Multimode Bilayer Theory

• The paper introduces a reduced multimode bilayer theory that compresses the full Maxwell–LLG problem into a self-energy formulation, retaining odd standing-spin-wave families.
• It employs transfer-matrix and effective mode matrix methods to elucidate how geometry, symmetry, and exchange interactions control the collective magnon-polariton spectrum.
• Design insights include leveraging antinode-compatible film placements for enhanced coupling and using slight asymmetries to activate dark-derived spectral features.

Reduced multimode bilayer theory is a family-resolved effective description of bilayer planar-cavity magnonics introduced for the exchange regime $J\neq 0$, in which two magnetic films embedded inside the same microwave cavity are represented not by the full seven-region Maxwell–Landau–Lifshitz–Gilbert boundary-value problem, but by a compact self-energy formulation that retains the odd standing-spin-wave families and their bright/dark bilayer reorganization (Solihin et al., 13 Apr 2026). In that setting, the theory extends an exact two-film scattering treatment in the macrospin limit $J=0$ and isolates how geometry, symmetry, and exchange-driven mode hierarchy control the collective magnon-polariton spectrum (Solihin et al., 13 Apr 2026).

1. Physical setting and benchmark structure

The underlying system is a one-dimensional planar microwave cavity of length $L$, bounded by two partially transmitting walls at $x=0$ and $x=L$, with a static bias field applied transversely so that the cavity microwave field drives magnetization dynamics. Two magnetic films are embedded in the cavity, with thicknesses $d_1$ and $d_2$, separated by a nonmagnetic spacer of length $s$. The remaining cavity length is shared symmetrically between outer spacers of length

$\ell=\frac{L-d_1-d_2-s}{2}.$

For identical films, $d_1=d_2=d$, the film centers are

$J=0$0

These coordinates determine whether the films sit near antinodes or nodes of a given cavity standing-wave mode and therefore determine the effective coupling strength (Solihin et al., 13 Apr 2026).

Electromagnetically, the cavity is treated as one-dimensional with harmonic time dependence $J=0$1. In nonmagnetic spacers $J=0$2, while inside magnetic films with real permittivity $J=0$3, $J=0$4. The walls are described by an opacity parameter $J=0$5, with transmission and reflection amplitudes

$J=0$6

The seven-region geometry is then amenable to an exact transfer-matrix treatment in the macrospin limit.

A central benchmark of the bilayer scattering theory is the zero-gap half-thickness limit. If a single magnetic slab of thickness $J=0$7 is split into two slabs of thickness $J=0$8 with zero gap,

$J=0$9

the bilayer transmission satisfies the exact identity

$L$0

This identity is the formal check that the bilayer construction reduces to the known one-film planar result when the two films are fused into a single slab.

2. Macrospin bilayer scattering and the need for reduction

In the macrospin limit $L$1, each film behaves as a homogeneous FMR-like scatterer with internal effective wave number

$L$2

where

$L$3

and

$L$4

For a magnetic slab, the dielectric mismatch is

$L$5

and the single-film cavity transmission reduces to the established one-film planar expression used as the benchmark for the bilayer theory.

The bilayer case is not equivalent to simply increasing the amount of magnetic material. The effective transmission and reflection amplitudes of two slabs in series depend on their individual scattering amplitudes and on the intermediate spacer phase $L$6, so the overall coupling is a coherent sum over the cavity field evaluated at the two film positions. This yields the first important physical conclusion: antinode-compatible placements enhance effective coupling, whereas node-compatible placements suppress it (Solihin et al., 13 Apr 2026).

That point resolves a common misconception. The bilayer does not merely strengthen the magnon-photon interaction by adding magnetic volume. Rather, it enables position-dependent control of the collective bright channel. In the symmetric limit, with cavity mode profile proportional to $L$7, the effective coupling of film $L$8 scales as $L$9. When both films lie at antinode-compatible positions, the bright channel acquires the standard $x=0$0 enhancement familiar from Tavis–Cummings scaling; when the fields at the two films are small or opposite in sign, the collective coupling is partially or almost completely canceled.

The exchange regime $x=0$1 invalidates the macrospin reduction because each film then supports a ladder of standing spin waves. The exact seven-region scattering solver is no longer used there; instead, the theory switches to a reduced multimode formulation designed to retain the exchange hierarchy while compressing the full spatial matching problem into a small set of effective mode families.

3. Core formulation for $x=0$2

With finite exchange, the effective magnetic field includes

$x=0$3

and a single film with pinned boundary conditions supports standing spin waves with quantized wave vector $x=0$4 and resonance frequencies

$x=0$5

Only odd indices have substantial coupling to the cavity mode profile considered in the paper, so the retained family set is

$x=0$6

For a single film in cavity mode $x=0$7, the reduced multimode transmission is

$x=0$8

For a bilayer, each film $x=0$9 carries its own standing-spin-wave ladder,

$x=L$0

with

$x=L$1

A naive bilayer self-energy would simply sum the two film ladders without inter-film mode mixing, but that does not expose the bright/dark bilayer combinations within each odd family.

The reduced multimode bilayer theory introduces, for each odd family $x=L$2, an effective $x=L$3 magnetic mode matrix

$x=L$4

a damping matrix

$x=L$5

and a cavity-coupling vector

$x=L$6

The family-resolved self-energy is then

$x=L$7

and the total reduced bilayer transmission becomes

$x=L$8

This is the core formula of the reduced multimode bilayer theory (Solihin et al., 13 Apr 2026).

The reduction is controlled but not exact. Only odd standing-spin-wave families are retained; the family sum is truncated at a finite $x=L$9 in numerics; inter-film mixing is included only within a fixed family $d_1$0, not between different $d_1$1; and the detailed seven-region Maxwell–LLG matching is compressed into the effective parameters $d_1$2, $d_1$3, $d_1$4, and $d_1$5. The paper explicitly emphasizes that this is not the full exact $d_1$6 scattering solver.

4. Family-resolved bright and dark channels

In the symmetric bilayer limit for a fixed odd family $d_1$7,

$d_1$8

the natural basis is

$d_1$9

In this basis, the cavity-coupling vector becomes proportional to $d_2$0, so only the bright combination couples to the cavity, with enhanced coupling

$d_2$1

while the dark combination is decoupled to leading order (Solihin et al., 13 Apr 2026).

This family-by-family bright/dark decomposition is the essential conceptual content of the theory. The statement that “odd standing-spin-wave families reorganize into family-resolved bright and dark bilayer channels” means that each odd $d_2$2 produces a pair of collective bilayer modes: one that inherits the cavity weight and one that is dark in the symmetric limit.

Weak symmetry breaking makes the dark channel visible without immediately erasing the main bright anticrossing. In the macrospin sector the paper implements symmetry breaking by detuning the two films’ FMRs through different bias fields,

$d_2$3

or equivalently

$d_2$4

In the exchange regime the same logic applies more generally when

$d_2$5

Then the bright/dark basis no longer diagonalizes the magnetic sector and the dark combination acquires finite cavity overlap. In self-energy language, the dark-derived pole acquires finite residue, so each odd family can contribute an extra weak spectral feature.

A second misconception is therefore addressed. The emergence of a dark-derived spectroscopic branch under weak asymmetry does not immediately destroy the main avoided crossing. The theory predicts, and the paper highlights, a regime in which the additional branch becomes visible while a substantial bright splitting remains.

5. Spectral signatures, mode hierarchy, and design logic

The reduced transmission formula is used to compute magnon-polariton spectra for several thicknesses and asymmetries. For $d_2$6, the spectra are dominated mainly by the $d_2$7 family, and the symmetric bilayer shows a larger anticrossing than the single-film case. For $d_2$8, higher standing-spin-wave families become visible and generate a richer pattern of avoided crossings; the symmetric bilayer again strengthens the bright-family anticrossings (Solihin et al., 13 Apr 2026).

In asymmetric bilayers, extra weak branches appear between the principal bright anticrossings. The paper interprets these as dark-derived standing-spin-wave branches. Line cuts for $d_2$9 and $s$0 at increasing $s$1 show a middle peak emerging between the two main bright peaks. The $s$2 family is more sensitive than $s$3: its dark-derived peak becomes visible at smaller $s$4, which the paper attributes to a smaller intrinsic splitting and thus more fragile dark-state protection.

The resulting exchange-driven mode hierarchy is explicit. The $s$5 family has the strongest coupling, the largest splitting, and the most robust bright character. Higher odd families such as $s$6 have smaller bright splittings and are more easily activated by symmetry breaking. This produces a family-dependent hierarchy of spectroscopic visibility.

The theory also yields direct design logic. Geometry fixes the native bright-channel strength through the film placements $s$7 and $s$8. Symmetry controls whether an ideal dark partner exists and how easily it is activated. Exchange produces the ladder of odd standing-spin-wave families and therefore the hierarchy of bright and dark subchannels. The paper accordingly identifies three practical tendencies: antinode-compatible placements maximize $s$9-type bright enhancement; controlled asymmetry such as a small $\ell=\frac{L-d_1-d_2-s}{2}.$0 reveals additional dark-derived branches; and tuning thickness $\ell=\frac{L-d_1-d_2-s}{2}.$1, exchange $\ell=\frac{L-d_1-d_2-s}{2}.$2, and cavity frequency selects which odd families dominate the observed multimode spectrum (Solihin et al., 13 Apr 2026).

6. Broader methodological context

Within cavity magnonics, the reduced multimode bilayer theory is presented as conceptually related to Tavis–Cummings models and to single-film planar cavity magnonics. Its self-energy expressions are Tavis–Cummings-like, but the “spins” are exchange standing-spin-wave modes, and the bilayer reorganizes those modes into bright and dark channels family by family rather than merely duplicating a single collective oscillator (Solihin et al., 13 Apr 2026).

Related reduced bilayer constructions also occur in other areas of physics, although they address different microscopic problems. In the $\ell=\frac{L-d_1-d_2-s}{2}.$3 quantum Hall bilayer, a reduced multimode description retains the symmetric and antisymmetric gauge modes and identifies the dominant angular-momentum pairing channels within a modified-RPA Eliashberg framework (Lotrič et al., 2023). In hole-doped reduced bilayer nickelate La$\ell=\frac{L-d_1-d_2-s}{2}.$4Ni$\ell=\frac{L-d_1-d_2-s}{2}.$5O$\ell=\frac{L-d_1-d_2-s}{2}.$6, an orbital-space bilayer model compresses a five-orbital problem into upper and lower effective modes separated by a large $\ell=\frac{L-d_1-d_2-s}{2}.$7, with superconductivity optimized in an incipient-band regime (Kamiyama et al., 12 Mar 2026). In cavity polariton theory for multilayered materials, microscopic light-matter Hamiltonians are reduced to finite matrix models whose bright excitonic channels are selected by layer position and cavity mode structure (Mandal et al., 2023). In elasticity, reduced bilayer theories similarly replace full three-dimensional descriptions by high-order effective interface or plate models, either for buckling of film/substrate systems (Wang et al., 2022) or for isometry-constrained bilayer plates derived through $\ell=\frac{L-d_1-d_2-s}{2}.$8-convergent reduction (Bartels et al., 2015).

This broader usage suggests a common methodological pattern rather than a single universal formalism. In each case, a bilayer system with many microscopic or spatial degrees of freedom is projected onto a smaller set of collective channels—bright and dark bilayer magnons, symmetric and antisymmetric gauge modes, orbital-space upper and lower bands, or effective curvature modes—chosen so that the reduced model preserves the dominant spectral, geometric, or pairing structure. In the specific magnon-polariton setting, that pattern is realized most explicitly through the family-resolved self-energy

$\ell=\frac{L-d_1-d_2-s}{2}.$9

which makes the reduced multimode bilayer theory both a compact computational framework and a classification scheme for the collective bright and dark channels of bilayer planar cavities (Solihin et al., 13 Apr 2026).