Interconnected Split-Ring Resonators

• Interconnected split-ring resonators are composite electromagnetic structures where individual SRRs are linked via shared gaps, inductive coupling, or dipole interactions to create hybrid resonant modes.
• They are studied through multiple models—full-wave hybrid mode analysis, electric dipole approximations, and coupled RLC circuits—each revealing distinct facets of resonance splitting and field localization.
• These systems enable practical applications such as band-gap engineering, topological edge and corner state formation, and enhanced magnon-photon coupling in hybrid quantum devices.

Searching arXiv for recent and foundational work on interconnected split-ring resonators and coupled SRR lattices. Interconnected split-ring resonators are composite electromagnetic structures in which individual split-ring resonators, or three-dimensional re-entrant cavities that can be interpreted as split-ring resonators by symmetry transformation, are linked by shared gaps, mutual inductive coupling, dipole-mediated interaction, or lattice-scale arrangement so that their resonances hybridize into collective modes, passbands, stopbands, localized defect states, and, in some cases, topological edge and corner states. In the literature considered here, this category includes one-dimensional lattices of 3D split-ring cavities, cage-like stereometamaterials formed from four square SRRs sharing a single gap, planar split-ring–spiral meta-resonators for photon–magnon coupling, and dimerized metasurfaces of dipole- and quadrupole-type SRRs implemented on printed circuit boards (Goryachev et al., 2014, Wang et al., 2014, Xiong et al., 2024, Rozenblit et al., 16 Jun 2026).

1. Resonator archetypes and local field structure

A foundational realization is the cylindrical re-entrant cavity. It consists of a cylindrical metal cavity of radius $R$ and height $h$, with a central post of radius $r$ and height $h-d$, leaving a narrow gap $d$ between the top of the post and the cavity ceiling. Perfect-electric-conductor boundary conditions hold on all cavity walls and on the post surface. By continuously rotating a cut 2D split-ring resonator around its axis by $2\pi$, the resulting solid is exactly this re-entrant cavity; the planar gap becomes the narrow axial gap $d$, and the ring sweeps out the cylindrical post. To first approximation, the electric field is confined almost entirely within the small gap, while the magnetic field circulates around the post and decays roughly logarithmically with radial distance. In the lumped-element picture, the dominant mode is an LC resonance with

$$C \simeq \pi \epsilon_0 r^2/d,\qquad L \simeq (\mu_0 h/2\pi)\ln(R/r),\qquad \omega_0 = 1/\sqrt{LC}.$$

This field separation is central to later lattice constructions because it suppresses inter-site capacitance while retaining strong magnetic coupling (Goryachev et al., 2014).

Planar and stereoscopic SRR realizations retain the same LC logic but redistribute the near fields differently. A cage-like stereometamaterial is formed from four identical square split-ring resonators connected by sharing a single gap in a rotational fashion with $\theta = 90^\circ$. For the specific structure studied numerically, the constituent SRR parameters are $a=300~\mathrm{nm}$, $h$0, $h$1, and $h$2, while the array period satisfies $h$3. In this geometry, hybridization among the four subresonators produces three characteristic resonances, with field patterns that can be assigned to a magnetic toroidal dipole, a magnetic dipole, and a mixed electric-dipole–magnetic-toroidal-dipole mode. The shared gap enhances the effective capacitance and makes higher-order, weakly radiating modes accessible in an array (Wang et al., 2014).

A further planar variant interconnects a double split-ring resonator and a multi-turn spiral inside the same unit cell. The DSRR uses an outer square ring of side $h$4, an inner square ring of side $h$5, strip width $h$6, and split width $h$7. The spiral has $h$8 turns, $h$9, $r$0, $r$1, and $r$2. This layout creates a bright SRR-dominated resonance and a dark spiral-dominated resonance within a single meta-resonator, with the dark mode concentrating the microwave magnetic field in a small mode volume (Xiong et al., 2024).

These examples show that “interconnection” need not mean a single mechanism. In some systems the elements are physically connected by a shared metallic gap, in others they are placed in a common enclosing cavity with the interior walls removed, and in others they are distinct resonant loops coupled through mutual inductance or dipole fields. A common feature is that the local charge accumulation at gaps and loop currents in the conductive path remain the primary degrees of freedom.

2. Coupling formalisms for interconnected SRRs

For arbitrary arrays of resonators, a general starting point is the energy-coupled-mode formulation derived from source-free Maxwell equations. If the coupled structure is expanded in a truncated basis of uncoupled modes,

$r$3

projection yields the matrix equations

$r$4

with

$r$5

Eliminating $r$6 gives the master eigenvalue problem

$r$7

In the common case $r$8, this reduces to $r$9. The formulation can be rewritten in a familiar “kinetic + potential” form, and for normalized modes one often identifies off-diagonal coupling coefficients $h-d$0 and diagonal induced frequency shifts $h-d$1. The resulting $h-d$2 problem provides the coupled frequencies and hybridized fields in terms of the uncoupled modes, and the paper explicitly characterizes the picture as an electromagnetic analog of molecular orbital theory (ELnaggar et al., 2013).

A complementary approximation emphasizes electric dipoles. For a single SRR, the induced dipole is written as

$h-d$3

with a dominant $h-d$4-polarized component

$h-d$5

and $h-d$6 otherwise. For two identical SRRs, the dipole field enters through the dynamic Green-function coefficients

$h-d$7

leading to a coupling coefficient

$h-d$8

In the static limit $h-d$9,

$d$0

The normal-mode frequencies obey

$d$1

The validity conditions are explicit: the particle must be small compared with the wavelength, the interaction region must remain within a “safe” electric-dipole regime, and arbitrary orientations require the full tensor structure rather than the two canonical geometries (Zeng et al., 2010).

Circuit models provide a third description that is especially useful when the interconnection is intentionally designed. In the split-ring–spiral meta-resonator, two coupled RLC tanks with mutual inductance $d$2 yield hybrid frequencies

$d$3

where $d$4. The loaded transmission response is approximated by

$d$5

This formulation makes explicit how feed coupling, internal loss, and bright–dark hybridization jointly modify $d$6 and the observable resonance splitting (Xiong et al., 2024).

Taken together, these descriptions show that interconnected SRRs can be treated as full-wave hybridized modes, as interacting electric dipoles when the geometry permits, or as coupled RLC networks when the gap capacitances and loop inductances are the dominant degrees of freedom. A common misconception is that one model subsumes all others. The literature instead treats them as regime-dependent descriptions with distinct validity conditions.

3. Re-entrant cavity lattices as interconnected 3D split-ring resonators

When $d$7 re-entrant posts are placed in a single large enclosing box and the interior walls are removed, each post-plus-gap remains a local LC oscillator. Because the electric fields stay confined to the individual gaps, nearest-neighbor posts do not share significant capacitance; the dominant interaction is magnetic coupling through overlapping $d$8-field loops. With branch fluxes $d$9 and conjugate charges $2\pi$0, the Hamiltonian is

$2\pi$1

where $2\pi$2 is the nearest-neighbor inverse mutual inductance. Defining $2\pi$3 gives the equivalent form

$2\pi$4

For a uniform infinite chain with lattice spacing $2\pi$5, Fourier transformation diagonalizes the problem and yields

$2\pi$6

with $2\pi$7 and $2\pi$8 (Goryachev et al., 2014).

The finite-$2\pi$9 eigenproblem reproduces phonon-like behavior. The lower or in-phase branch is acoustic-like, with $d$0, and its dispersion rises from $d$1 at $d$2 toward a maximum at $d$3. The out-of-phase branch is optical-like, with $d$4, and its frequency at $d$5 is $d$6, decreasing toward $d$7 at $d$8. The paper states that for finite $d$9 one finds $$C \simeq \pi \epsilon_0 r^2/d,\qquad L \simeq (\mu_0 h/2\pi)\ln(R/r),\qquad \omega_0 = 1/\sqrt{LC}.$$0 modes split into these two branches. By varying $$C \simeq \pi \epsilon_0 r^2/d,\qquad L \simeq (\mu_0 h/2\pi)\ln(R/r),\qquad \omega_0 = 1/\sqrt{LC}.$$1, through the post spacing or the post radii, the gap between the branches can be opened or closed (Goryachev et al., 2014).

Alternating resonator parameters within a unit cell creates explicit band-gap engineering. If successive posts differ in $$C \simeq \pi \epsilon_0 r^2/d,\qquad L \simeq (\mu_0 h/2\pi)\ln(R/r),\qquad \omega_0 = 1/\sqrt{LC}.$$2 or $$C \simeq \pi \epsilon_0 r^2/d,\qquad L \simeq (\mu_0 h/2\pi)\ln(R/r),\qquad \omega_0 = 1/\sqrt{LC}.$$3, for example through alternating post radii $$C \simeq \pi \epsilon_0 r^2/d,\qquad L \simeq (\mu_0 h/2\pi)\ln(R/r),\qquad \omega_0 = 1/\sqrt{LC}.$$4 or gaps $$C \simeq \pi \epsilon_0 r^2/d,\qquad L \simeq (\mu_0 h/2\pi)\ln(R/r),\qquad \omega_0 = 1/\sqrt{LC}.$$5, the reduced Brillouin zone becomes $$C \simeq \pi \epsilon_0 r^2/d,\qquad L \simeq (\mu_0 h/2\pi)\ln(R/r),\qquad \omega_0 = 1/\sqrt{LC}.$$6 and a gap opens at the zone boundary. Potential-well lattices are produced by gradually increasing inter-post spacing away from the center so that $$C \simeq \pi \epsilon_0 r^2/d,\qquad L \simeq (\mu_0 h/2\pi)\ln(R/r),\qquad \omega_0 = 1/\sqrt{LC}.$$7 falls quadratically with $$C \simeq \pi \epsilon_0 r^2/d,\qquad L \simeq (\mu_0 h/2\pi)\ln(R/r),\qquad \omega_0 = 1/\sqrt{LC}.$$8; in the continuum limit the Hamiltonian density acquires position-dependent $$C \simeq \pi \epsilon_0 r^2/d,\qquad L \simeq (\mu_0 h/2\pi)\ln(R/r),\qquad \omega_0 = 1/\sqrt{LC}.$$9 and $\theta = 90^\circ$0, and the modes localize near $\theta = 90^\circ$1. The same platform also supports impurity physics: an interstitial defect can split the chain into two weakly coupled subchains, while a substitutional defect can behave either as a wall or as a weak link, depending on the defect radius. With a 1D Fibonacci sequence of bond lengths or post types, the spectrum fragments into a Cantor-like set of bands and gaps, directly emulating a 1D phononic quasicrystal (Goryachev et al., 2014).

The significance of this architecture lies in the combination of a 3D high-$\theta = 90^\circ$2 cavity environment and a discretized lattice Hamiltonian. The paper also states that the system is easily scalable to simulate 2D and 3D lattices, and it outlines extensions in which Josephson junctions add a $\theta = 90^\circ$3 term, mechanically compliant gaps create multimode optomechanical networks, and spin ensembles or magnons are placed in high-$\theta = 90^\circ$4 field regions between posts. This suggests that interconnected 3D split-ring cavities are not limited to passive dispersion engineering but can serve as microwave analog simulators for nonlinear and hybrid quantum systems.

4. Shared-gap stereometamaterials and multipolar hybridization

In the cage-like SRR stereometamaterial, four identical square SRRs are connected by sharing a single gap in a rotational fashion. Each constituent SRR is modeled as a lumped series LC circuit, and the four-fold connection enhances the effective capacitance to

$\theta = 90^\circ$5

while mutual inductive coupling $\theta = 90^\circ$6 modifies the mode-dependent inductance according to

$\theta = 90^\circ$7

The resonance frequencies are therefore

$\theta = 90^\circ$8

with the explicit forms

$\theta = 90^\circ$9

The shared gap lowers the fundamental frequency and enables hybridization that gives rise to dark high-order modes (Wang et al., 2014).

Full-wave simulations were carried out in COMSOL Multiphysics (RF Module) using tetrahedral meshes with minimum element size $a=300~\mathrm{nm}$0 near metal edges and gaps, periodic boundary conditions on the $a=300~\mathrm{nm}$1-$a=300~\mathrm{nm}$2 faces, and perfectly matched layers in $a=300~\mathrm{nm}$3. Gold was modeled with $a=300~\mathrm{nm}$4 from Johnson–Christy (1972), the background was air, and the excitation was a normally incident plane wave with electric field polarized along the SRR gap. For $a=300~\mathrm{nm}$5, the transmission spectrum exhibits three resonances at approximately

$a=300~\mathrm{nm}$6

with estimated quality factors $a=300~\mathrm{nm}$7, $a=300~\mathrm{nm}$8, and $a=300~\mathrm{nm}$9 (Wang et al., 2014).

The radiative content of these resonances was analyzed through multipole decomposition of the induced current density $h$00, using the electric dipole,

$h$01

the magnetic dipole,

$h$02

the toroidal dipole,

$h$03

and the electric quadrupole tensor

$h$04

Mode 1 is a magnetic toroidal dipole at $h$05: four in-phase current loops form a head-to-tail arrangement of magnetic dipoles in the $h$06-$h$07 plane, giving a pronounced $h$08. Mode 2 is a magnetic dipole at $h$09: second-order currents on each SRR cancel the toroidal loops but leave a net $h$10, with weaker electric quadrupole content. Mode 3 is a hybrid electric–magnetic toroidal dipole at $h$11, dominated by $h$12 and a small electric dipole from corner charges, and it is weakly radiating in an isolated C-SRR because of phase cancellation (Wang et al., 2014).

A key point is the contrast between an isolated cage and an infinite array. A single isolated C-SRR shows two strong peaks at approximately $h$13 and $h$14, with only a faint shoulder near $h$15. In the infinite 2D array, transmission dips are clearly resolved at $h$16, $h$17, and $h$18. The symmetry argument given in the paper is that periodic boundary conditions break the exact local cancellation condition for mode 3 and allow constructive coupling of mode-3 currents across cells. This is a direct example of a recurrent theme in interconnected SRRs: a mode that is dark or nearly dark in an isolated meta-atom may become spectrally prominent once lattice coherence is introduced (Wang et al., 2014).

The same resonances serve as multiband stop-bands near $h$19, $h$20, and $h$21. For $h$22, the reported filter characteristics are $h$23 with $h$24, $h$25 with $h$26, and $h$27 with $h$28. The paper states that tuning $h$29, $h$30, and $h$31 shifts the three bands independently.

5. Hybrid split-ring–spiral resonators and magnon–photon coupling

A distinct class of interconnected SRR structure combines a planar DSRR with an Archimedean-type spiral resonator in the same unit cell. The substrate is Rogers TMM laminate with $h$32, $h$33, substrate thickness $h$34, and copper thickness $h$35. The interconnection is primarily inductive: the split-ring and spiral are modeled as two coupled RLC tanks with mutual inductance $h$36. The constituent parameters are written as

$h$37

$h$38

$h$39

and

$h$40

summed over $h$41 gaps. The mutual inductance is written as $h$42, where $h$43 depends on the spacing $h$44 and planar overlap (Xiong et al., 2024).

The coupling reorganizes the bare resonances into bright and dark hybrid modes. The loaded resonator imparts a shunt impedance $h$45 to the feedline, and the two-port response near resonance is approximated by a Lorentzian $h$46. Bandwidth broadening and peak splitting follow from the hybridization; the resonance separation is given approximately by

$h$47

while each peak’s loaded quality factor includes an additional bright–dark loss channel. The design guidelines state, for example, that Rogers TMM yielded $h$48, that $h$49 was chosen for $h$50 and $h$51, and that $h$52 gives $h$53. A spiral–SRR spacing $h$54 gave $h$55, while reducing the spacing to $h$56 raises $h$57 but increases parasitic capacitance (Xiong et al., 2024).

The same resonator becomes a hybrid magnonic platform when a resonator mode of frequency $h$58 couples to the uniform magnon Kittel mode $h$59. The minimal Hamiltonian is

$h$60

with collective coupling

$h$61

where $h$62, $h$63, $h$64 is the sample volume, and $h$65 is the resonator–magnon overlap factor. At magnetic bias satisfying $h$66, the transmittance peaks anti-cross, and the splitting at resonance gives $h$67 (Xiong et al., 2024).

The reported enhancement is specific. A $h$68 diameter YIG sphere on a DSRR at $h$69 yields $h$70, whereas the same sphere on the s-DSRR spiral mode at approximately $h$71 yields $h$72, described as a $h$73 enhancement. Similar $h$74 gain is reported for a YIG film and for the higher s-DSRR mode at $h$75. The design guidance attributes this to the dark mode’s high local magnetic field and reduced mode volume, and it recommends placing the magnetic sample at the spiral center where $h$76 peaks and can be $h$77 stronger than near the SRR arm (Xiong et al., 2024).

This case illustrates that interconnected SRRs need not be used only to generate spectral multiplicity. They can also be arranged so that one resonance serves primarily as a field concentrator for matter coupling, while another remains more directly addressable through the feedline.

6. Topological and localization phenomena in SRR networks

Interconnected SRRs also support deliberately engineered localization beyond ordinary defect physics. A recent example is a higher-order topological metasurface composed of two types of SRRs: a Type-1 single-gap copper ring whose fundamental eigenmode at approximately $h$78 carries an out-of-plane electric dipole ($h$79) moment, and a Type-2 two-gap copper ring, each gap bridged by a lumped capacitor $h$80, whose lowest antisymmetric mode at approximately $h$81 carries a quadrupolar ($h$82) charge distribution. These resonators are patterned on standard PCB using copper on F4BM255 of thickness $h$83, then mounted on a low-$h$84 foam spacer into a dimerized square lattice. A unit cell of side $h$85 contains two parallel Type-1 SRRs and two perpendicular Type-2 SRRs; adjacent cells are separated by $h$86, with $h$87 corresponding to weak coupling $h$88 and $h$89 to strong coupling $h$90 (Rozenblit et al., 16 Jun 2026).

The effective Hamiltonian is written as

$h$91

where $h$92 alternates in sign and magnitude to realize a net $h$93-flux per plaquette. In a $h$94-space basis $h$95, the model becomes

$h$96

with eigenvalues

$h$97

The paper states that the ordering of $h$98 character flips as $h$99 passes through unity, i.e. a band inversion occurs (Rozenblit et al., 16 Jun 2026).

The topological invariant is formulated through nested Wilson loops. One first computes

$r$00

with Berry connection $r$01. The phases of $r$02 define branches, and a nested Wilson loop along $r$03 yields a quantized quadrupole moment $r$04. The bulk polarization vanishes but the second moment is fractional, so boundary charge accumulates at corners (Rozenblit et al., 16 Jun 2026).

The finite-size manifestation is a hierarchy of bulk, edge, and corner localization. In an $r$05 lattice with open boundaries, the spectrum shows two bulk bands, one-dimensional gapped edge bands in the range $r$06, and zero-dimensional corner modes pinned near mid-gap at $r$07. The localization is quantified by the inverse participation ratio: bulk states scale as $r$08, edge states as $r$09, and corner states as $r$10. Experimental validation used a $r$11 PCB array with $r$12, a corner-placed electric-dipole transmitter, and a scanning $r$13-field probe $r$14 above the SRR plane; maps at $r$15, $r$16, and $r$17 were reported to agree well with numerics despite an approximately $r$18 spread in capacitor values (Rozenblit et al., 16 Jun 2026).

Non-topological localization appears in earlier interconnected-SRR systems as well. In the 3D re-entrant cavity lattice, gradual variation of inter-post spacing creates a potential well that localizes modes near the center, while interstitial and substitutional defects reshape the spectrum into weakly coupled subchains or weak links; Fibonacci ordering produces a Cantor-like fragmentation of bands and gaps (Goryachev et al., 2014). The contrast is instructive: one class of localization is produced by spatially varying couplings and defects, whereas the higher-order topological metasurface uses sign-alternating couplings and a synthetic $r$19-flux to realize corner and edge states protected by chiral and inversion symmetries.

A common misconception is that strongly localized SRR modes are necessarily defect-induced or fabrication-accidental. The PCB metasurface demonstrates a deliberate route to edge and corner localization, while the re-entrant lattice shows an equally deliberate but non-topological route through coupling profiles, impurities, and quasiperiodic order.