Transmodal Fabry–Pérot Resonance

• Transmodal Fabry–Pérot resonance is the efficient energy conversion between distinct wave modes achieved by engineered mode coupling and specific phase-matching conditions.
• It is realized through material anisotropy and geometric perturbations in elastic, photonic, and plasmonic systems, leading to hybridized modes and tunable transmission spectra.
• This phenomenon has practical applications in ultrasound imaging, nondestructive testing, and photonic devices by enabling precise wave manipulation and selective mode conversion.

A transmodal Fabry–Pérot resonance refers to the phenomenon where maximal energy conversion occurs between two distinct wave modes within a resonant cavity or structure, in contrast to conventional unimodal Fabry–Pérot resonances where energy is transmitted within the same mode. The essential feature is strong coupling or hybridization between different polarization, symmetry, or spatial modes—such as longitudinal and shear elastic waves, anapole and cavity modes, or bulk and surface plasmonic states—often enabled by anisotropy, material structuring, or geometric perturbation. Transmodal resonances are governed by specific phase-matching conditions and result in nontrivial field distributions, avoided crossings, and tunable transmission spectra with practical implications for wave manipulation, device design, and sensing.

1. Physical Principles and Definition

Transmodal Fabry–Pérot resonance (TFPR) arises when two (or more) otherwise distinct wave modes are coherently coupled within a structured medium or resonant cavity, allowing energy initially associated with one mode to be efficiently transmitted or converted into another. In the context of elastic waves, a paradigmatic example is the maximal transmission of a longitudinal mode to a shear mode (or vice versa) through an anisotropic, mode-coupled elastic layer, achieved when the phase difference accumulated by the two eigenmodes matches specific odd multiples of $\pi$ (Kweun et al., 2016).

The general phenomenon is underpinned by the following:

• Coexistence of multiple eigenmodes (polarizations, transverse modes, internal resonances) within the structure.
• Engineered coupling that breaks the orthogonality of these modes, allowing energy exchange.
• Phase-matching or resonance condition arising from the difference in propagation constants (wavenumbers) over the layer/cavity thickness.

Mathematically, the transmodal resonance condition is of the form

$$\Delta \phi = (k_2 - k_1) d = (2m+1)\pi, \qquad m \in \mathbb{Z}$$

where $k_{1,2}$ are modal wavenumbers, $d$ is the thickness/cavity length, and $\Delta\phi$ is the phase difference traversed by the modes (Kweun et al., 2016, Luo et al., 18 Sep 2025).

2. Theoretical Formulations Across Physical Platforms

2.1 Elastic Metamaterials

For an elastic layer with coupled longitudinal (L) and shear (S) wave modes, the Christoffel equation,

$\det [k^2 \Gamma - \rho\omega^2 I] = 0$

with Christoffel matrix $$\Gamma = \begin{bmatrix} C_{11} & C_{16} \ C_{16} & C_{66} \end{bmatrix}$$, yields two eigenmodes (quasi-longitudinal, QL, and quasi-shear, QS). TFPR occurs under the condition

$\Delta\phi = (k_{QS} - k_{QL}) d = (2m+1)\pi$

maximizing the cross-mode power transmission $T_s \equiv |t_s|^2$ from L to S (Kweun et al., 2016).

2.2 Photonic and Plasmonic Cavities

In layered photonic or plasmonic systems, similar behavior is observed:

• In metal–dielectric–Bragg mirror structures, resonances evolve continuously from two Fabry–Pérot standing-wave modes to a single localized Tamm plasmon as the dielectric contrast increases (Durach et al., 2012). The hybrid (“transmodal” [editor’s term]) regime is characterized by mode hybridization and spectral coalescence.
• In thin-film Fabry–Pérot/Surface-Plasmon microcavities, transmission maps reveal high-transmission branches that interpolate between bulk FP resonance ($k_{in} d + \phi_T = m\pi$) and coupled surface-plasmon resonance ($k_{In} d - \ln|T_{LMH}| = 0$), with the transition point interpreted as a transmodal FP resonance (Arosa et al., 28 Oct 2024).

2.3 Cavity Mode Hybridization

In Fabry–Pérot microcavities with imperfect mirrors, transverse-mode coupling induced by spatial inhomogeneities or finite size leads to avoided crossings and mixing between Hermite–Gaussian modes. The resulting eigenmodes are admixtures, and the resonance splitting $\Delta\omega = 2|\kappa_{nm}|$ at degeneracy is observed experimentally (Benedikter et al., 2015).

3. Origin and Interpretation of Spectral Features

The occurrence of “wiggly” or non-monotonic transmission spectra is a direct signature of multi-mode interference and structural instability:

• In elastic TFPR, the envelope of maximal L→S conversion is periodically modulated by oscillatory terms from internal reflections, quantified by a structural instability factor $F_S = (C_{11} C_{66} - C_{16}^2) / (C_{11} C_{66})$; pronounced spectral ripples correspond to strong mode-coupling (Kweun et al., 2016).
• In optical hybrid structures, the transfer matrix formalism reveals the analytic evolution of mode profiles, from pure standing waves to localized interface-bound states as parameters (e.g., Bragg contrast, cavity length) are tuned (Durach et al., 2012, Arosa et al., 28 Oct 2024).
• In microcavities, crossing points between unperturbed mode frequencies manifest as avoided crossings in spectra, with splitting and additional loss channels governed by overlap integrals of the mixed modes and the mirror perturbation profile (Benedikter et al., 2015).

4. Experimental Realization and Modulation

TFPR effects have been demonstrated in multiple platforms using both intrinsic material properties and engineered metamaterials:

• Elastic Metamaterials: Anisotropic layers are realized by laser-cutting arrays of oblique micro-slits in aluminum plates, where geometrical tuning of slit length $\ell$, thickness $t$, and angle $\theta$ allows continuous control of the mode-coupling coefficient $C_{16}$ and resultant TFPR frequency. Weakly and strongly coupled regimes were predicted (e.g., $(C_{11}, C_{66}, C_{16}) = (45.2, 17.9, 10.4)$ GPa vs. $(20.5, 12.9, 17.2)$ GPa), and experimentally verified via ultrasonic Lamb wave transmission and magnetostrictive transducer measurements (Kweun et al., 2016).
• Hybrid Cavity-Resonant Metasurfaces: Anapole metasurfaces embedded in THz FP cavities exploit length tuning to select between strong (ultrastrong polaritonic splitting with $2\Omega_R \gtrsim 30\%$ of the anapole frequency) and weak (linewidth compression and LDOS enhancement) coupling regimes, with on-demand switching via the FP resonance condition (Luo et al., 18 Sep 2025).
• Optical Microcavities: Mode mixing is observed and controlled by the curvature, finite size, and surface profile of fiber-based planar-curved Fabry–Pérot mirrors, allowing precise mapping of resonance splitting, loss, and field distributions (Benedikter et al., 2015).

5. Applications and Technological Implications

Transmodal Fabry–Pérot resonance empowers new strategies in wave control and device engineering:

• Ultrasound Imaging and Nondestructive Testing: Efficient L↔S conversion at TFPR frequencies in elastic metamaterials enables selective generation and detection of shear modes, critical for improved resolution and contrast in medical and materials diagnostics (Kweun et al., 2016).
• Photonics and Polaritonic Devices: Modal hybridization allows for tunable switching between narrowband (high-Q) and broadband (polaritonic) states, enhanced field localization, and mode volume engineering in applications such as sensing, nonlinear optics, and quantum information processing (Luo et al., 18 Sep 2025, Durach et al., 2012, Arosa et al., 28 Oct 2024).
• Mode-Selective Cavities: In microcavities, intentional or avoided transmodal resonances can be leveraged for engineered diffraction loss, spatial mode selection, and mitigation of performance degradation due to mode mixing (Benedikter et al., 2015).

6. Future Directions and Research Outlook

Continued research in transmodal Fabry–Pérot phenomena targets several frontiers:

• Extension to multilayer, higher-dimensional, and nonreciprocal structures—facilitating spatiotemporal wavefront shaping, isolation, and chirped-grating functionalities.
• Integration with quantum materials and emitters, exploiting strong coupling, polariton splitting, and field enhancement for quantum optics and light-matter interaction studies (Luo et al., 18 Sep 2025, Durach et al., 2012).
• Tunable and programmable metamaterial platforms that expand the range of accessible TFPR conditions by geometric or external (e.g., time-modulated) control (Kweun et al., 2016).

Transmodal Fabry–Pérot resonance thus establishes a unifying conceptual framework for phase-matched modal conversion in resonant systems, enabling precise spectral and spatial control across acoustics, mechanics, photonics, and beyond (Kweun et al., 2016, Durach et al., 2012, Arosa et al., 28 Oct 2024, Luo et al., 18 Sep 2025, Benedikter et al., 2015).