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This paper introduces a distributed circuit model (DCM) to accurately predict the quality factor (Q-factor) of magnetic polariton (MP) resonances in various metal-insulator-metal (MIM) structures. The proposed DCM accounts for radiation loss by incorporating a radiation resistance, overcoming limitations of existing lumped-parameter RLC models. The model provides a unified analytical expression for Q-factors of arbitrary-order MPs and demonstrates accurate predictions when validated against rigorous coupled-wave analysis (RCWA).

Existing RLC circuit models lead to inaccurate Q-factor predictions for magnetic polariton (MP) resonances under imperfect absorption conditions due to the omission of radiation loss.
The lumped-parameter nature of RLC models limits their applicability for predicting higher-order MP modes, requiring separate circuits for each mode.
Previous models, like the one by Sakurai et al., are only accurate near perfect absorption and show substantial deviations when significant transmission/reflection occurs, as they neglect radiation losses.

A distributed circuit model (DCM) is proposed for predicting the Q-factor of MP resonances, overcoming limitations of existing models.
A radiation resistance is introduced into the DCM to characterize radiation loss.
A unified analytical expression for the Q-factor of arbitrary-order MPs is derived by establishing a mapping between distributed and lumped circuit parameters.
The model is validated against rigorous coupled-wave analysis (RCWA), demonstrating its accuracy for various MIM structures and MP orders.

The paper proposes a Distributed Circuit Model (DCM) to predict the Q-factors of magnetic polariton (MP) resonances.

Radiation Loss Incorporation: A radiation resistance, Rrad, is introduced into the DCM to account for radiative losses. This is achieved by adding Rrad in series with the kinetic resistance Rk in an equivalent RLC circuit.

Parameter Mapping: A mapping is established between the distributed circuit parameters (distributed kinetic resistance Rk0, distributed inductance L0, and distributed capacitance C0) and the corresponding lumped circuit parameters. This allows for the prediction of Q-factors of arbitrary-order MPs within a unified framework.

Unified Analytical Expression: By incorporating radiative losses and establishing the parameter mapping, a unified analytical expression for the Q-factors of arbitrary-order MPs is derived. This expression relates the Q-factor to resonance frequency, inductance, capacitance, and the sum of kinetic and radiation resistances.

Correction Factor: A correction coefficient, c1, is introduced to account for non-uniform loss distributions among current loops, particularly in cases where the assumption of uniformity is not met (e.g., small slit widths or substrate-induced asymmetry).

Validation: The theoretical predictions from the DCM are compared with results obtained from rigorous coupled-wave analysis (RCWA) for various MIM structures and MP orders to validate the model's accuracy.

Architecture: The model extends a previously proposed DCM by incorporating a radiation resistance (Rrad) and establishing a mapping between distributed and lumped circuit parameters. For higher-order MP resonances (MPm), m current loops are considered, each representable by an equivalent RLC circuit.
Key Equations:
- The Q-factor for a series RLC circuit with radiative losses is given by: Q = (ωrL) / (Rrad + Rk) = (1 / √LC) / (Rrad + Rk)
- The ratio between radiation resistance and kinetic resistance is related to peak absorptance (αr): Rrad / Rk = (1 - αr) / αr
- The resonance condition for the mth-order MP mode in a substrate-free vertical MIM structure is: ωMPm = (mπ) / √(hLo(ωMPm)Co(ωMPm))
- The Q-factor for arbitrary-order MP resonances is given by: QMPm = (ωMPmL(@MPm)) / (Rrad(@MPm) + Rk(@MPm))
Loss/Objective: The objective is to accurately predict the Q-factor of MP resonances by accounting for both Ohmic losses (represented by kinetic resistance Rk) and radiative losses (represented by radiation resistance Rrad).
Optimization: A correction coefficient c1 is used to improve accuracy, especially when loss distribution is non-uniform.
Data Handling: The model's predictions are validated against results obtained from rigorous coupled-wave analysis (RCWA).
Implementation Details: The Drude model is used to describe the optical response of Ag, with parameters: plasma frequency ωp = 1.39×10¹⁶ rad/s, scattering rate Γ = 2.70×10¹³ rad/s, and high-frequency constant ɛ∞ = 3.4. ZnSe is treated as a nondispersive dielectric with n = 2.4. RCWA calculations employed 181 Fourier orders.

Datasets: The model is validated on substrate-supported horizontal MIM structures and substrate-free vertical MIM structures. Specific geometric parameters like slit width (b), period (Λ), metal strip thickness (w), and slit length (h) are varied.
Metrics: The primary metric is the Q-factor. Mean Relative Error (MRE) is used to quantify prediction accuracy.
Baselines: Predictions are compared against results from RCWA and a conventional RLC model without radiation loss.
Numeric Results:
- The RLC model incorporating radiation resistance Rrad significantly improves prediction accuracy, reducing relative errors from 348.29% to 11.76% and from 752.13% to 7.65% for specific cases.
- For vertical MIM structures, the incorporation of radiation loss reduces the mean relative error (MRE) from 504.85% to 9.14%.
- For horizontal MIM structures, MRE decreases from 192.77% to 3.25% when radiation loss is included for a specific configuration.
- For vertical MIM structures, MRE decreases from 254.18% to 7.33% when radiation loss is included for a specific configuration.
Runtime/Training Cost: Not stated in the paper.

The paper mentions that further developments are required to describe strong radiative-Ohmic coupling and MP–SPP hybridization.
A correction coefficient c1 is needed to account for non-uniform loss distributions, indicating that the assumption of uniform loss distribution among current loops is not always valid, especially at small slit widths or with substrate-induced asymmetry.
A relatively larger discrepancy is observed for MP5 in one case due to its proximity to the SPP mode.

Method evidence: "By introducing a radiation resistance Rrad, to characterize radiative losses and connect it in series with the kinetic resistance Rk in the original RLC model, as illustrated for the right slit in Fig. 1(d)." (section: Page 4)
Result evidence: "The incorporation of radiation loss reduces the mean relative error (MRE) from 504.85% to 9.14%." (section: Page 6)
Limitation evidence: "Although further developments are required to describe strong radiative-Ohmic coupling and MP–SPP hybridization, the proposed model provides a simple yet powerful framework for designing high-performance metamaterial emitters and absorbers with tailored spectral selectivity." (section: Page 11)

Distributed Circuit Model for Predicting the Quality Factor of Magnetic Polariton Resonance

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