🔬 optics

Quasinormal mode as a foundational framework for all electromagnetic Fano resonances

This paper establishes a foundational framework for all electromagnetic Fano resonances by introducing an ab initio Maxwellian approach using quasinormal modes to analytically derive the Fano asymmetry parameter, thereby replacing phenomenological fitting with a general formula validated across diverse systems.

Original authors: Mikhail Bochkarev, Nikolay Solodovchenko, Kirill Samusev, Mikhail Limonov, Tong Wu, Philippe Lalanne

Published 2026-05-01

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Mikhail Bochkarev, Nikolay Solodovchenko, Kirill Samusev, Mikhail Limonov, Tong Wu, Philippe Lalanne

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Idea: Finding the "Recipe" for a Specific Sound

Imagine you are listening to a complex piece of music. Sometimes, you hear a sound that isn't just a pure note; it's a mix of a loud, broad hum and a sharp, quiet whistle that interfere with each other. This creates a unique, lopsided sound shape called a Fano resonance.

For decades, scientists have seen these weird, lopsided shapes in light, sound, and quantum physics. They knew how to describe the shape using a number called $q$ (the asymmetry parameter), but they didn't really know why that number was what it was. Usually, they just measured the shape and guessed the number to make the math fit. It was like knowing a cake tastes sweet but not knowing how much sugar was in it.

This paper changes that. The authors, using a powerful mathematical tool called Quasinormal Modes (QNMs), have finally written down the exact "recipe" for that number $q$ . They show that you don't need to guess; you can calculate it directly from the physics of the object and the light hitting it.

The Analogy: The Swing and the Wind

To understand what the authors did, imagine a child on a swing (the resonator) and a strong wind blowing (the light).

The Old Way (Phenomenological): If you wanted to know how the swing moves, you would watch it, draw a curve on a piece of paper, and say, "Okay, the wind is pushing it this much." You are just describing what you see.
The New Way (This Paper): The authors looked at the swing, the wind, and the child's weight, and derived a formula that tells you exactly how the swing will move before you even push it.

They discovered that every single "swing" (or resonance mode) has its own unique Fano shape built into it. It's not just a random accident; it's a fundamental property of how that specific mode interacts with the world.

The "Magic Ingredient": The Overlap ( $\xi$ )

The paper introduces a new concept called $\xi$ (xi). Think of this as a measure of how well the wind fits the swing.

If the wind blows perfectly in sync with the swing's natural rhythm, you get one type of shape.
If the wind blows at a slightly different angle or timing, the shape changes completely.

The authors found a simple formula: The shape of the sound (the Fano parameter $q$ ) depends entirely on the "angle" or "phase" of this overlap ( $\xi$ ).

The Analogy: Imagine trying to push a swing. If you push exactly when the swing is coming toward you, it goes high (constructive interference). If you push when it's going away, it slows down (destructive interference). The authors found that the "Fano shape" is just a mathematical way of describing exactly when and how hard you are pushing relative to the swing's natural motion.

The Experiment: The Split-Ring

To prove their recipe works, the team built a physical experiment:

The Object: A ceramic ring with a gap in it (a split-ring resonator).
The Test: They shined microwaves (invisible light) at the ring and rotated the ring to different angles.
The Result: As they turned the ring, the "sound" of the light changed shape dramatically. Sometimes it looked like a hill, sometimes a valley, sometimes a lopsided curve.

They used their new formula to predict exactly what the shape would be for every angle. The prediction matched the real-world measurement perfectly. This proved that their "recipe" for the Fano parameter is correct and doesn't need any guessing.

Why This Matters (According to the Paper)

No More Guessing: Before this, scientists had to fit curves to data to find the Fano parameter. Now, they can calculate it directly from the design of the object.
Universal Application: The math works for almost anything: tiny particles, big rings, materials that absorb light, or materials that leak light. It's a "one-size-fits-all" solution for wave physics.
Tuning the Shape: Because the formula depends on the direction of the incoming light, you can "tune" the shape of the resonance just by changing the angle of the light beam. The paper shows you can twist the shape from a perfect hill to a perfect valley just by rotating the object.

Summary

The paper says: "We found the hidden rulebook for Fano resonances."

Instead of treating these weird, lopsided light shapes as mysterious phenomena that need to be measured and fitted, the authors showed that they are the natural result of how light waves interact with specific "modes" of an object. They provided a clear, mathematical way to predict exactly what those shapes will look like, turning a mystery into a calculable fact.

1. Problem Statement

Fano resonances, characterized by asymmetric spectral line shapes arising from the interference between a broad continuum and a narrow discrete state, are ubiquitous in wave physics. In optics and photonics, the Fano profile is traditionally described by a phenomenological formula involving an asymmetry parameter ( $q$ ) and a resonance intensity ( $\sigma_m$ ).

Limitations of Current Approaches: In optical systems, the parameter $q$ is typically treated as a fitting coefficient derived from experimental or numerical data rather than a fundamental physical quantity. Existing theoretical frameworks often rely on simplified coupled-oscillator models or specific symmetry assumptions (e.g., Mie theory for spheres) that fail to provide a general, analytical solution for arbitrary resonator shapes, materials, or background environments (such as substrates).
The Gap: There is a lack of a robust, ab initio electromagnetic foundation that analytically derives the Fano parameters ( $q_m$ and $\sigma_m$ ) directly from Maxwell's equations without relying on phenomenological fitting.

2. Methodology

The authors propose a rigorous ab initio Maxwellian approach utilizing Quasinormal Mode (QNM) theory.

QNM Framework: They treat electromagnetic resonances as source-free solutions to Maxwell's equations with complex eigenfrequencies ( $\tilde{\omega}_m = \omega_m - i\gamma_m/2$ ), where the imaginary part accounts for decay into the continuum (via radiation, absorption, or leakage).
Field Expansion: The scattered field is expanded as a sum of QNMs. To ensure mathematical completeness (especially in the presence of substrates or branch cuts), the authors employ a QNM-regularization formalism using Perfectly Matched Layers (PMLs). This allows for a unique expression of the excitation coefficients ( $\alpha_m$ ) derived via the Lorentz reciprocity theorem.
Derivation: By substituting the QNM expansion into the expression for the extinction cross-section, the authors perform algebraic manipulations to isolate the contribution of each individual QNM. They demonstrate that the extinction contribution of a single mode naturally takes the form of a Fano profile.
Approximation: To resolve the frequency dependence of the overlap integrals, they employ the "Resonant Frequency Approximation" (RFA), calculating the key overlap term at the real part of the eigenfrequency ( $\omega_m$ ). This approximation is shown to be highly accurate even for low-quality factor (Q) resonances.

3. Key Contributions

The paper provides a closed-form analytical solution for the Fano parameters, fundamentally linking them to the overlap between the incident field and the QNM fields.

Analytical Formula for the Fano Parameter ( $q_m$ ):
The authors derive that the asymmetry parameter is determined by the argument of a complex overlap integral $\xi_m$ :
$q_m = -\cot\left(\frac{\arg(\xi_m)}{2}\right)$
where $\xi_m$ is the product of two overlap integrals involving the dielectric contrast ( $\Delta\varepsilon$ ), the incident background field ( $\mathbf{E}_b$ ), and the normalized QNM field ( $\tilde{\mathbf{E}}_m$ ).
Analytical Formula for Resonance Intensity ( $\sigma_m$ ):
$\sigma_m = \frac{1}{P_0} \frac{\omega^2}{2\gamma_m} |\xi_m|$
Physical Insight:
- Intrinsic Fano Nature: The study reveals that every individual QNM inherently exhibits a Fano profile. The interaction between the localized resonance and the continuum is naturally embedded in the QNM formalism, eliminating the need to artificially assume a "background" continuum.
- Tunability: Unlike traditional models where $q$ is an intrinsic property of the resonance, this theory shows that $q_m$ explicitly depends on the incident field direction and polarization. By rotating the incident wave, $q_m$ can be tuned continuously from $0$ to $\pm\infty$ .
- Reciprocity: The derived parameters are invariant under the reversal of the incident wave direction ( $\mathbf{k}_b \to -\mathbf{k}_b$ ), consistent with the principle of reciprocity in linear systems.

4. Results

The theoretical framework was validated through both experimental measurements and numerical simulations.

Experimental Validation:
- Setup: A split-ring resonator made of a high-index ceramic was placed in an anechoic chamber. Extinction spectra were measured for various incident angles ( $\phi$ ) relative to the resonator.
- Findings: The measured spectra for different modes ( $m=30$ and $m=31$ ) were fitted using the classical Fano formula. The extracted parameters ( $q_m, \sigma_m$ ) showed strong quantitative agreement with the values predicted by the new QNM theory (Eqs. 7 and 8) across a wide range of angles.
- Observation: The experiment confirmed the theoretical prediction that $q_m$ transitions from $+\infty$ to $-\infty$ at specific angles where the phase of the overlap integral $\xi_m$ crosses zero.
Numerical Validation:
- Simulations were performed on diverse systems, including dielectric and plasmonic particles in free space and on substrates.
- The theory successfully described complex scenarios, including low-Q resonators, spectrally overlapping resonances, and systems with exceptional points (where modes coalesce).
- The results demonstrated that the QNM-based formula outperforms previous theoretical attempts (e.g., those based on symmetry assumptions) which failed to capture the correct behavior or lacked invariance under directional reversal.

5. Significance

This work fundamentally shifts the understanding of Fano resonances in electromagnetism:

Unification: It establishes QNM theory as the unifying framework for Fano effects, integrating discrete and continuum states into a single formalism applicable to arbitrary geometries and materials.
Design and Optimization: The explicit, closed-form expressions for $q_m$ and $\sigma_m$ allow for the rapid optimization of spectral line shapes. Engineers can now tune resonator geometries and illumination conditions to achieve desired asymmetry without iterative fitting.
Future Applications: The framework provides a solid foundation for advanced applications, including:
- Optimizing optical sensors (enhancing detection limits by tuning $q$ ).
- Inverse design of photonic structures.
- Integration with computational bounds and deep learning approaches for spectral response engineering.

In summary, the paper moves the description of Fano resonances from a phenomenological curve-fitting exercise to a rigorous, predictive, and tunable electromagnetic theory based on the fundamental properties of Quasinormal Modes.