Photonic Resonant States Overview

• Photonic resonant states are solutions of Maxwell’s equations in open structures characterized by complex eigenfrequencies and energy leakage.
• They play a critical role in applications such as ultrahigh-Q resonators, bound states in the continuum, and topologically protected light routing.
• Advanced methods like the resonant-state expansion, T-matrix, and temporal coupled-mode theory facilitate precise device design and analysis.

Photonic resonant states are solutions of Maxwell’s equations in open or structured photonic environments, characterized by complex-valued eigenfrequencies due to energy leakage (radiation) or interaction with matter. They underpin many fundamental and applied phenomena in photonic crystals, nanostructures, metasurfaces, and resonators. The unique properties of photonic resonant states—including bound states in the continuum (BICs), topologically protected modes, exceptional points, and radiative edge states—enable high spectral selectivity, robust light localization, tunable light–matter coupling, and enhanced nonlinear effects.

1. Definition and Theoretical Framework

Photonic resonant states, also termed quasi-normal modes (QNMs), are eigenmodes of Maxwell’s equations subject to outgoing-wave boundary conditions. In an isotropic, non-magnetic medium,

$$\nabla \times \nabla \times \mathbf{E}(\mathbf{r}) - \frac{\omega^2}{c^2}\varepsilon(\mathbf{r})\mathbf{E}(\mathbf{r}) = 0.$$

Unlike the standard Hermitian eigenvalue problem with real frequencies, resonant states have complex eigenfrequencies $\tilde{\omega} = \omega - i\gamma$. The real part $\omega$ represents the resonance frequency and the imaginary part $-\gamma$ encodes the decay (radiative or absorptive) rate. For open systems, their fields diverge exponentially at infinity but provide a complete and bi-orthogonal basis for expanding arbitrary solutions.

Normalization of resonant states requires both volume and surface terms: $$\int_V [\mathbf{E}_n \cdot \partial_\omega(\omega \varepsilon) \mathbf{E}_n - \mathbf{H}_n \cdot \partial_\omega (\omega \mu) \mathbf{H}_n ]\,d^3 r + \frac{i}{2\tilde{\omega}_n} \oint_{\partial V} (\mathbf{E}_n \times \mathbf{H}_n)\cdot d\mathbf{S} = 1$$ ensuring completeness and correct residue contributions for Green’s function expansions (Fischbach et al., 5 Nov 2025).

2. Formation Mechanisms and Classification

2.1 Ordinary Resonant States and Quasi-Bound Modes

Ordinary resonant states occur as solutions with finite negative $\Im\omega$ (finite lifetime) and describe leaky cavity modes, guided-resonance in photonic slabs, and Fabry–Pérot resonances in slab/waveguide geometries (Ko et al., 2022, Koshelev et al., 2022).

2.2 Bound States in the Continuum (BICs)

BICs are photonic eigenmodes with real frequencies embedded in the radiation continuum but with vanishing net radiative loss ($\Im\omega = 0$ in theory). Their defining feature is an infinite radiative $Q$-factor ($Q_{\rm rad} \to \infty$). BICs arise via several mechanisms (Koshelev et al., 2022, Ge et al., 2022, Neale et al., 2020, Ko et al., 2022):

• Symmetry-protected BICs: Symmetry precludes mode coupling to open channels. For instance, in a slab with mirror symmetry, modes with odd parity (e.g., $\rm TE_{odd}$ at normal incidence) are orthogonal to the even radiation continuum (Neale et al., 2020, Ge et al., 2022, Koshelev et al., 2022).
• Accidental or Friedrich–Wintgen BICs: Destructive interference between two or more leaky channels at specific parameter values. The Friedrich–Wintgen condition for two coupled modes with frequencies $\omega_1, \omega_2$ and radiative losses $\gamma_1, \gamma_2$ is

$$\kappa(\gamma_1 - \gamma_2) = e^{i\phi} \sqrt{\gamma_1 \gamma_2}(\omega_1 - \omega_2)$$

so that a particular superposition becomes nonradiating (Koshelev et al., 2022, Ge et al., 2022).

• Multipole cancellation BICs: Vector sum of multipoles (e.g., electric dipole, quadrupole) vanishes in every radiation direction due to group-theoretic selection rules (Koshelev et al., 2022, Ko et al., 2022).
• Parameter-tuned (Fabry–Pérot) BICs: Discrete spatial tuning (e.g., cavity spacing) yields perfect destructive interference in leakage paths (Koshelev et al., 2022).

2.3 Topological and Radiative Edge States

Resonant edge states can arise in photonic lattices with nontrivial Zak phase or nonzero Chern number per band. In open one-dimensional topological insulators (e.g., Su–Schrieffer–Heeger–like chains, bichromatic lattices with compound unit cells), edge-localized modes have complex eigenfrequencies with finite radiative decay, detectable as Lorentzian dips in reflection (Kuhl et al., 2016, Poshakinskiy et al., 2013, Slobozhanyuk et al., 2016). These states are robust against disorder and their lifetimes (radiative widths) can be calculated from the transfer-matrix formalism.

2.4 Exceptional Points and Non-Hermitian Topology

Open photonic structures are generally non-Hermitian due to radiation loss. At certain real-valued parameters, two or more resonant eigenvalues and eigenvectors can coalesce, forming exceptional points (EPs). Near second-order ($n=2$) EPs, resonance splittings and mode switching exhibit square-root singularities in parameter space: $k_{1,2} - k_* \propto \pm\sqrt{\delta - \delta_*}$ while encircling the EP induces robust mode conversion (Abdrabou et al., 2019).

3. Mathematical Formalism and Computational Approaches

Resonant states are determined by:

• Non-Hermitian Eigenvalue Problems: The Maxwell operator in open geometry yields complex eigenfrequencies. The resonant-state expansion (RSE) provides an efficient computational tool by expressing the modes of a complex photonic crystal structure as a linear expansion in the analytic resonant states of a simpler basis system (homogeneous slab, waveguide) (Neale et al., 2019, Neale et al., 2020).
• Multipole/T-matrix Formulation: Resonant states for finite assemblies of scatterers (e.g., dielectric cylinders) are obtained via T-matrix methods applied to the field-continuity equations, with outgoing Hankel-wave boundary conditions (Abdrabou et al., 2019, Andueza et al., 2018).
• Temporal Coupled-Mode Theory (TCMT): TCMT provides a reduced-order framework for BIC/absorptive resonance formation and their lineshape transitions, particularly in microcavity and layered photonic crystal systems (Krasnov et al., 2023).
• Rytov-type EMT: Effective-medium theory quantifies GMR/BIC behavior in nanophotonic lattices, predicting the location of dark (BIC) states via effective-slab transcendental equations (Ko et al., 2022).

Resonant-state theory also extends to structured photonic time crystals (PTCs), where temporal modulation introduces Floquet ladders and parametric amplification emerges from resonance conditions between Floquet sidebands and static QNMs (Valero et al., 2 Jun 2025).

4. Physical Manifestations and Experimental Signatures

Key experimental and physical phenomena associated with photonic resonant states include:

Resonant State Format	Physical Manifestation	Experimental Signature
Ordinary QNMs	Finite-$Q$ cavity/waveguide modes	Lorentzian peaks/dips in transmission/reflection
Symmetry-protected BICs	Divergent $Q$, field orthogonal to continuum	Invisible (dark) in $R/T$ at normal incidence
Accidental BICs	Parameter-tuned zero net leakage	Sharp transmission/reflection features at a resonance
Topological Edge States	Robust, spatially localized at boundaries	Exponential decay profile, Lorentzian dip in $R$
EP-induced modes	Coalescence, enhanced sensitivity	Nontrivial mode evolution upon parametric cycling
Radiative-topological	Edge states with finite leakage	Time-domain exponential tail after prompt response

• Measurement Techniques: Finite-integration and FEM eigenvalue methods, transmission/reflection spectroscopy, near-field scanning (aperture-SNOM), angle-resolved scattering, and thermal tuning via liquid-crystal-filled resonators confirm the theoretical predictions (Slobozhanyuk et al., 2016, Andueza et al., 2018, Ko et al., 2022, Krasnov et al., 2023).
• Strong Light–Matter Coupling: Tracking resonant pole splittings in the presence of dispersive media (e.g., molecular hosts, excitonic materials) unambiguously distinguishes weak from strong coupling without reliance on real-frequency spectral ambiguities (Fischbach et al., 5 Nov 2025).

5. Functional Applications and Device Implications

Photonic resonant states have driven numerous applications:

• Ultrahigh-Q Resonators: Engineering BICs (including merged/super-BICs) and topological defect modes achieves Q-factors $>10^4$, crucial for low-threshold lasing, sensing, and nonlinear optics (Ge et al., 2022).
• Nonlinear Enhancement: SHG efficiency scales as $Q_{\rm FH}^2 Q_{\rm SH}$; merged BICs in photonic crystal cavities yield order-of-magnitude enhancements for second-harmonic generation (Ge et al., 2022).
• Sensorics: High field localization and abrupt linewidth dependence, especially near BICs, enable temperature and refractive-index sensing with sub-0.01 K resolution (Krasnov et al., 2023).
• Power Limiting and Switching: Topologically protected midgap states enable reflective, high-damage-threshold limiters by field-induced loss, not index shifting (Kuhl et al., 2016).
• Light Routing and Topological Photonics: Photonic edge states in honeycomb and zigzag lattices allow controllable Rabi-like switching of valley-Hall modes and enhancement of the photonic spin Hall effect via topological protection (Zhong et al., 2019, Slobozhanyuk et al., 2016).

6. Topological, Non-Hermitian, and Floquet Extensions

Recent advances include:

• Non-Hermitian Topology: EPs, encirclement phenomena, square-root and cube-root mode splitting, and topologically protected transitions under varying system parameters (Abdrabou et al., 2019).
• Radiative Topological States: In compound-cell 1D photonic crystals, edge states survive as radiative modes with complex eigenfrequencies, each associated with the Zak or Chern invariant of the underlying band structure. These states are detectable in the time and frequency domain via exponential decay tails or phase windings in reflectivity (Poshakinskiy et al., 2013).
• Resonant Floquet States in Time-Crystals: Periodically modulated nanostructures exhibit a ladder of Floquet-modulated resonances, governed by universal quadratic shift scaling and parametric-resonance conditions between QNMs and their Floquet twins. The parametric amplification threshold $\Delta\varepsilon_{\rm thr}$ depends on the static Q-factor and mode overlap (Valero et al., 2 Jun 2025).

7. Scaling Laws, Design Principles, and Future Directions

Several design principles and scaling laws emerge:

• Scaling of $Q$-factors: Near isolated BICs, $Q \sim 1/|k - k_0|^2$; for merged BICs, $Q \sim 1/|k|^4$ (Ge et al., 2022).
• Geometric Robustness: Topological defect and midgap states (pinned by chiral or sublattice symmetry) show resilience to fabrication-induced disorder (Kuhl et al., 2016).
• Device Miniaturization: EPs and BICs have been realized in dielectric cylinder arrays and photonic molecules with footprints $<4\lambda \times 4\lambda$ (Abdrabou et al., 2019, Andueza et al., 2018).
• Functional Integration: BICs and QNMs provide the foundation for multiplexed photonic circuits, tunable metasurfaces, platform-agnostic sensors, and hybrid optomechanical and quantum photonic devices (Koshelev et al., 2022).

Emerging research explores the interplay of non-Hermiticity, topology, nonlinearity, and actively modulated environments for applications in quantum information processing, programmable optics, and space-time photonics (Valero et al., 2 Jun 2025). The systematic engineering of resonant states continues to extend the capabilities of nanophotonic technologies.