<p>This work presents a compact theoretical framework for classifying magnetic multipole regimes in coupled split-ring resonator (SRR) metamolecules. Each SRR is modeled as an effective resonant magnetic element, while the collective response is described through the coupling ratio <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\chi =|M|/L\)</EquationSource> </InlineEquation>, the normalized modal splitting <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(S_{\omega }\)</EquationSource> </InlineEquation>, the dipolar cancellation factor <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\Gamma _D\)</EquationSource> </InlineEquation>, the reduced quadrupolar indicator <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\Gamma _Q\)</EquationSource> </InlineEquation>, and the toroidal-like circulation factor <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\Gamma _T\)</EquationSource> </InlineEquation>. A reference four-SRR metamolecule is used to distinguish independent SRRs, bright collective dipoles, dark dipole-suppressed states, quadrupolar-like configurations, and toroidal-like magnetic arrangements. The results show that modal splitting is controlled by <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(\chi \)</EquationSource> </InlineEquation>, while multipolar behavior appears when the net magnetic dipole is suppressed. The mutual inductance is calibrated from the SRR geometry, the imposed sign patterns are obtained from a coupled-mode eigenanalysis of the metamolecule, and the reduced indicators are cross-checked against a field-based Cartesian multipole decomposition, showing that the toroidal-like circulation factor coincides with the exact discrete toroidal dipole magnitude. The proposed regime map does not replace full-wave electromagnetic simulation, but provides a physically transparent pre-design tool for magnetic multipole engineering in SRR-based metamaterials.</p>

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Theoretical regime map of magnetic multipole modes in coupled split-ring resonator metamolecules

  • Álvaro Garay

摘要

This work presents a compact theoretical framework for classifying magnetic multipole regimes in coupled split-ring resonator (SRR) metamolecules. Each SRR is modeled as an effective resonant magnetic element, while the collective response is described through the coupling ratio \(\chi =|M|/L\) , the normalized modal splitting \(S_{\omega }\) , the dipolar cancellation factor \(\Gamma _D\) , the reduced quadrupolar indicator \(\Gamma _Q\) , and the toroidal-like circulation factor \(\Gamma _T\) . A reference four-SRR metamolecule is used to distinguish independent SRRs, bright collective dipoles, dark dipole-suppressed states, quadrupolar-like configurations, and toroidal-like magnetic arrangements. The results show that modal splitting is controlled by \(\chi \) , while multipolar behavior appears when the net magnetic dipole is suppressed. The mutual inductance is calibrated from the SRR geometry, the imposed sign patterns are obtained from a coupled-mode eigenanalysis of the metamolecule, and the reduced indicators are cross-checked against a field-based Cartesian multipole decomposition, showing that the toroidal-like circulation factor coincides with the exact discrete toroidal dipole magnitude. The proposed regime map does not replace full-wave electromagnetic simulation, but provides a physically transparent pre-design tool for magnetic multipole engineering in SRR-based metamaterials.