Quasi Trapped Modes in Waveguides & Photonics

• Quasi trapped modes (QTM) are near-threshold localized states that retain key features of ideal trapped modes while exhibiting weak radiative leakage due to imperfections.
• They manifest across diverse systems—including finite waveguides, photonic metasurfaces, delayed quantum networks, acoustics, nonlinear droplets, and black-hole perturbations—each with distinct trapping mechanisms.
• Understanding QTMs enables precise tuning of resonant behavior for practical applications in sensing, slow light, and optical filtering by balancing localization and leakage.

Searching arXiv for recent and foundational papers on quasi-trapped modes across waveguides, photonics, acoustics, quantum networks, droplets, and black-hole perturbations. Quasi trapped modes (QTMs) designate long-lived, weakly radiating, or near-threshold localized states that retain the spatial structure of an exactly trapped, dark, or bound mode while losing perfect confinement because of finite size, weak symmetry breaking, material loss, delayed feedback, nonlinear coupling, or an additional boundary condition. The terminology is not uniform across fields. In finite quantum waveguides, QTMs are near-threshold localized states with $\lambda \lesssim \nu_1$ and small decay rate $\kappa=\sqrt{\nu_1-\lambda}$ (Delitsyn et al., 2011). In photonics and metasurfaces, they are weakly leaky finite-size counterparts of symmetry-protected bound states in the continuum or trapped dark modes (Lü et al., 2017, Tuz et al., 2017, Evlyukhin et al., 2021). In delayed passive quantum networks, trapped modes associated with poles $p_k=-\gamma_k+i\omega_k$ are “quasi-trapped” in the practical sense because damping remains finite (Tabak et al., 2015). Acoustics, harmonically trapped quantum droplets, and non-minimally coupled black-hole perturbations use related but distinct notions of confinement, leakage, and boundary-value trapping (Dai, 2020, Zezyulin, 2023, Chiang et al., 7 Apr 2025).

1. Terminology and conceptual scope

The literature does not assign a single universal definition to QTM. Instead, the term marks a common phenomenology: a mode remains strongly localized or resonant, but the ideal trapped-state limit is weakened by a small coupling to propagating channels, by a finite-domain truncation, or by a modified boundary condition. In finite waveguides, the relevant criterion is exponential decay into long branches rather than square-integrability alone, because all eigenfunctions are already in $L^2$ (Delitsyn et al., 2011). In finite photonic structures, true bound states in the continuum do not exist, so the observable states are quasi-BICs with very large but finite $Q$ (Lü et al., 2017). In delayed networks, feedback-generated resonances are physically cavity-like yet lossy through ports, so each pole has finite linewidth and lifetime (Tabak et al., 2015). In black-hole perturbation theory, “trapped quasi-normal modes” are not standard quasi-normal modes, because the governing boundary-value problem is posed between the horizon and an unphysical pole rather than between the horizon and spatial infinity (Chiang et al., 7 Apr 2025).

Several recurring misconceptions are explicitly corrected by the cited work. A QTM is not always a symmetry-broken BIC; finite quantum waveguides realize QTMs through near-threshold exponential decay (Delitsyn et al., 2011), delayed quantum networks through finite-delay feedback poles (Tabak et al., 2015), and acoustic duct-cavity systems through total reflection of a particular combination of two guided waves (Dai, 2020). Nor is a QTM always defined by a large quality factor alone. In some settings the control parameter is the decay rate $\kappa$, in others the damping $\gamma$, the leakage rate $\Gamma$, the imaginary part of a complex resonance frequency, or the existence of a regular solution on a finite interval bounded by a pole.

This suggests that QTM functions as a family-resemblance term rather than a single formal category. The unifying idea is imperfect confinement near an exactly trapped, dark, or localized limit, while the operator, geometry, and admissible boundary conditions determine the precise mathematics.

2. Finite waveguides and near-threshold localization

For finite quantum waveguides, the defining framework is the Dirichlet Laplacian eigenproblem in a bounded domain $D$ composed of a cavity $\Omega$ and cylindrical branches $\kappa=\sqrt{\nu_1-\lambda}$0:

$\kappa=\sqrt{\nu_1-\lambda}$1

In the infinite-waveguide setting, trapped modes are discrete eigenvalues $\kappa=\sqrt{\nu_1-\lambda}$2 below the essential spectrum $\kappa=\sqrt{\nu_1-\lambda}$3. In the finite setting there is no essential spectrum, so trapping is redefined as exponential decay of the eigenfunction into the branches, with decay controlled by the first transverse Dirichlet eigenvalue $\kappa=\sqrt{\nu_1-\lambda}$4 of the branch cross-section (Delitsyn et al., 2011).

For a rectangular branch of width $\kappa=\sqrt{\nu_1-\lambda}$5, the branch restriction takes the separated-variable form

$\kappa=\sqrt{\nu_1-\lambda}$6

If $\kappa=\sqrt{\nu_1-\lambda}$7, then $\kappa=\sqrt{\nu_1-\lambda}$8 is real and the mode decays exponentially. The cross-sectional norm satisfies

$\kappa=\sqrt{\nu_1-\lambda}$9

In general cylindrical branches, the dominant asymptotics is

$p_k=-\gamma_k+i\omega_k$0

The QTM regime is the near-threshold case $p_k=-\gamma_k+i\omega_k$1, where $p_k=-\gamma_k+i\omega_k$2 is small and the exponential decay is very slow. The mode is formally localized, but in finite branches it may be visually hard to distinguish from a nonlocalized mode if the branches are not sufficiently long (Delitsyn et al., 2011).

A central result is the reduction of the full problem on $p_k=-\gamma_k+i\omega_k$3 to a nonlinear eigenvalue problem on the cavity $p_k=-\gamma_k+i\omega_k$4:

$p_k=-\gamma_k+i\omega_k$5

with

$p_k=-\gamma_k+i\omega_k$6

Freezing $p_k=-\gamma_k+i\omega_k$7 yields a linear problem with first eigenvalue $p_k=-\gamma_k+i\omega_k$8, whose Rayleigh quotient is

$p_k=-\gamma_k+i\omega_k$9

The key criterion is necessary and sufficient: there exists a localized mode with $L^2$0 if and only if $L^2$1 (Delitsyn et al., 2011).

The constructive sufficient condition introduces

$L^2$2

and

$L^2$3

then requires

$L^2$4

For long branches this simplifies to a threshold relation of the form $L^2$5, yielding an upper bound on the minimal branch length for localization. The physical interpretation is explicit: larger $L^2$6 favors trapping, while larger $L^2$7 and $L^2$8 increase coupling to the branches and thus oppose localization (Delitsyn et al., 2011).

The examples show how QTMs emerge near geometric thresholds. In the two-dimensional L-shaped domain, the sufficient condition reduces to $L^2$9, so for equal branches $Q$0, while FEM gives $Q$1. For $Q$2, the first eigenvalue is $Q$3, with $Q$4, indicating strong decay; as $Q$5, $Q$6 becomes small and the QTM regime appears. In the bent strip, optimizing a trial family gives $Q$7 and $Q$8, while FEM yields $Q$9; near $\kappa$0 the decay rate is again small. By contrast, a truncated L-shape with triangular cavity satisfies $\kappa$1 for all $\kappa$2, so no trapped mode exists for any branch length (Delitsyn et al., 2011).

The broader significance is that finite devices require a notion of trapping different from the infinite-waveguide criterion “eigenvalue below the essential spectrum.” The finite-domain formulation exposes branch-length thresholds, switching between trapped and non-trapped regimes, and a near-threshold region in which localization is mathematically present but physically weak.

3. Photonic and metasurface realizations

In photonics, QTMs are commonly identified with quasi-BICs or trapped-mode resonances: states that would be nonradiating in an ideal infinite or perfectly symmetric structure, but become weakly leaky in finite or symmetry-broken realizations. The defining observables are near-zero group velocity, narrow Fano lineshapes, strong field concentration, and large but finite quality factor.

A particularly clear example is the subwavelength ring resonator composed of dielectric nanorods. Two families are identified: below-continuum-resonance trapped modes, traced to bound modes below the light line at the first Brillouin-zone edge, and quasi-BIC trapped modes, traced to Bloch BICs above the light line in the corresponding infinite linear chain (Lü et al., 2017). The resonances obey the whispering-gallery condition

$\kappa$3

This immediately yields the existence conditions: BCR trapped modes occur only for even $\kappa$4 because near $\kappa$5 one has $\kappa$6, whereas quasi-BIC trapped modes associated with $\kappa$7 occur for any $\kappa$8 since then $\kappa$9. For $\gamma$0, the lowest BCR mode has $\gamma$1, $\gamma$2, and $\gamma$3, while the lowest quasi-BIC mode has $\gamma$4, $\gamma$5, and $\gamma$6. The fitted size scalings are

$\gamma$7

and even at $\gamma$8 the quasi-BIC mode can reach $\gamma$9 (Lü et al., 2017).

All-dielectric metasurfaces implement the same physics through controlled asymmetry and symmetry protection. A single-disk unit cell with an off-centered hole or a coaxial-sector notch excites a TE01$\Gamma$0 trapped mode that is symmetry-forbidden in the symmetric structure and appears as a sharp Fano resonance once mirror symmetry is broken (Tuz et al., 2017). The trapped-mode quality can be tuned through the asymmetry, while a short coaxial-sector “smile” notch perturbs the EH11$\Gamma$1 Mie resonance less strongly than a large off-center circular hole. The same trapped-mode family can be made polarization-insensitive by arranging four notched dielectric cylinders in a $\Gamma$2 super-cell with $\Gamma$3 or $\Gamma$4 symmetry. In the $\Gamma$5 case only two disks are active at the trapped-mode resonance; in the $\Gamma$6 case all four are active, and the response remains identical for $\Gamma$7- and $\Gamma$8-polarized normal incidence (Sayanskiy et al., 2018).

A complementary route uses weak bianisotropy rather than simple geometric asymmetry. In silicon triangular-prism metasurfaces, a two-stage strategy first matches the electric and magnetic trapped-mode conditions

$\Gamma$9

then introduces weak bianisotropy through the prism shape so that normal-incidence polarization can switch between an electric QTM and a magnetic QTM located at the same spectral position (Evlyukhin et al., 2021). With $D$0, $D$1, $D$2, base angle $D$3, and lattice period $D$4, the two resonances co-locate near $D$5–$D$6. The reported near-field enhancements are approximately $D$7 for the electric QTM and $D$8 for the magnetic QTM (Evlyukhin et al., 2021).

Anisotropic transition-metal dichalcogenide metasurfaces extend the concept to material-anisotropy-driven bianisotropy. In MoS$D$9 disk-with-hole arrays, the QTM is a quasi-BIC arising from collective magnetic-dipole ordering weakly opened to radiation by bianisotropic and nonlocal couplings and by material loss (Prokhorov et al., 2021). For $\Omega$0, $\Omega$1, $\Omega$2, and $\Omega$3, the designed telecom resonance occurs at $\Omega$4 with peak absorption $\Omega$5, $\Omega$6, and $\Omega$7; the reported sensing performance is $\Omega$8 and $\Omega$9 (Prokhorov et al., 2021). A related MoS$\kappa=\sqrt{\nu_1-\lambda}$00 metasurface with $\kappa=\sqrt{\nu_1-\lambda}$01, $\kappa=\sqrt{\nu_1-\lambda}$02, $\kappa=\sqrt{\nu_1-\lambda}$03, $\kappa=\sqrt{\nu_1-\lambda}$04, and $\kappa=\sqrt{\nu_1-\lambda}$05 supports three narrowband QTM dips under $\kappa=\sqrt{\nu_1-\lambda}$06 excitation at $\kappa=\sqrt{\nu_1-\lambda}$07, $\kappa=\sqrt{\nu_1-\lambda}$08, and $\kappa=\sqrt{\nu_1-\lambda}$09, and three polarization-switched features under $\kappa=\sqrt{\nu_1-\lambda}$10 at $\kappa=\sqrt{\nu_1-\lambda}$11, $\kappa=\sqrt{\nu_1-\lambda}$12, and $\kappa=\sqrt{\nu_1-\lambda}$13, with reported contrasts $\kappa=\sqrt{\nu_1-\lambda}$14, $\kappa=\sqrt{\nu_1-\lambda}$15, and $\kappa=\sqrt{\nu_1-\lambda}$16 (Gubin et al., 2022).

Across these implementations, the physical mechanisms vary—band-edge slow light, Bloch BIC inheritance, weak symmetry breaking, and bianisotropic magnetoelectric coupling—but the operational picture is stable. A QTM is a weakly leaky descendant of a dark or perfectly bound mode, and its observables are controlled by how strongly the protected state is coupled back to the radiative continuum.

4. Open-system realizations: delayed quantum networks and acoustic duct–cavity systems

In passive linear quantum stochastic networks with finite propagation delays, trapped modes arise as poles of the transfer function generated by delayed feedback (Tabak et al., 2015). The basic passive linear QSDE model is

$\kappa=\sqrt{\nu_1-\lambda}$17

with transfer function

$\kappa=\sqrt{\nu_1-\lambda}$18

For a delay network,

$\kappa=\sqrt{\nu_1-\lambda}$19

and the poles satisfy

$\kappa=\sqrt{\nu_1-\lambda}$20

In the elementary SISO cavity,

$\kappa=\sqrt{\nu_1-\lambda}$21

These poles have the standard parametrization

$\kappa=\sqrt{\nu_1-\lambda}$22

so the modes are cavity-like but damped. The paper therefore treats them as “trapped modes” in the passive formalism while also noting that, in practical terms, they are quasi-trapped because $\kappa=\sqrt{\nu_1-\lambda}$23 or coupling to external ports gives finite damping (Tabak et al., 2015).

The identification method is based on the Blaschke–Potapov factorization

$\kappa=\sqrt{\nu_1-\lambda}$24

where $\kappa=\sqrt{\nu_1-\lambda}$25 is the Blaschke–Potapov product carrying the zero–pole pairs and $\kappa=\sqrt{\nu_1-\lambda}$26 is a singular inner term associated with feedforward-only delays. Each modal factor isolates a single resonance,

$\kappa=\sqrt{\nu_1-\lambda}$27

and can be translated back into a finite-dimensional passive QSDE block. This yields a physically realizable reduced-order model that preserves the resonant poles and zeros over a chosen frequency band (Tabak et al., 2015).

Open acoustic waveguides supply a different mechanism. In a two-dimensional cavity segment open to two semi-infinite ducts, the pressure satisfies

$\kappa=\sqrt{\nu_1-\lambda}$28

with modal solutions

$\kappa=\sqrt{\nu_1-\lambda}$29

The central result is that an embedded trapped mode is not, in general, the interference of two closed-cavity eigenmodes. Rather, it is built from two coexisting standing waves generated by total reflection of a specific combination of two propagative guided waves at each cavity–duct interface (Dai, 2020). The feedback-loop closure principle encodes this through

$\kappa=\sqrt{\nu_1-\lambda}$30

and a trapped mode occurs when

$\kappa=\sqrt{\nu_1-\lambda}$31

For a trapped mode, the phase changes of both cavity guided waves around the feedback loop must be integral multiples of $\kappa=\sqrt{\nu_1-\lambda}$32 (Dai, 2020).

The quasi-trapped regime appears when the cavity length is detuned away from the trapped-mode value. The resulting resonance frequency becomes complex, with small but nonzero imaginary part. The examples reported are $\kappa=\sqrt{\nu_1-\lambda}$33 for $\kappa=\sqrt{\nu_1-\lambda}$34, $\kappa=\sqrt{\nu_1-\lambda}$35, and $\kappa=\sqrt{\nu_1-\lambda}$36 for $\kappa=\sqrt{\nu_1-\lambda}$37, $\kappa=\sqrt{\nu_1-\lambda}$38. Near such QTMs, transmission is governed by two acoustic channels and exhibits Fano-type scattering; the exact $\kappa=\sqrt{\nu_1-\lambda}$39 is accurately reproduced by retaining only the two cavity channels associated with the two guided waves (Dai, 2020).

These two open-system literatures sharpen the meaning of QTM in non-Hermitian settings. A QTM is not merely “high-$\kappa=\sqrt{\nu_1-\lambda}$40”; it is a resonance identified by the analytic structure of a transfer function or by a closure condition in a multimode scattering loop, with finite loss or leakage explicitly encoded in the pole structure.

5. Nonlinear and many-body trapped-mode families

Effectively one-dimensional quantum droplets in a symmetric Bose–Bose mixture confined by a parabolic trap provide a nonlinear trapped-mode setting in which the paper does not use the phrase “quasi trapped modes,” but the trapped-state families fit the same qualitative idea of long-lived, localized, parameter-sensitive modes (Zezyulin, 2023). The dimensionless governing equation is

$\kappa=\sqrt{\nu_1-\lambda}$41

with the default symmetric-mixture choice $\kappa=\sqrt{\nu_1-\lambda}$42. Stationary states $\kappa=\sqrt{\nu_1-\lambda}$43 satisfy

$\kappa=\sqrt{\nu_1-\lambda}$44

The nonlinear families bifurcate from the linear harmonic-oscillator eigenstates

$\kappa=\sqrt{\nu_1-\lambda}$45

and the leading nonlinear shift is

$\kappa=\sqrt{\nu_1-\lambda}$46

Thus $\kappa=\sqrt{\nu_1-\lambda}$47 initially decreases as the particle number increases from zero (Zezyulin, 2023).

The principal structural result is that the families $\kappa=\sqrt{\nu_1-\lambda}$48 are nonmonotonic for $\kappa=\sqrt{\nu_1-\lambda}$49 and possess minima $\kappa=\sqrt{\nu_1-\lambda}$50 at finite $\kappa=\sqrt{\nu_1-\lambda}$51. For the strong trap $\kappa=\sqrt{\nu_1-\lambda}$52, the reported minima occur at $\kappa=\sqrt{\nu_1-\lambda}$53, $\kappa=\sqrt{\nu_1-\lambda}$54, $\kappa=\sqrt{\nu_1-\lambda}$55, and $\kappa=\sqrt{\nu_1-\lambda}$56, while the drop $\kappa=\sqrt{\nu_1-\lambda}$57 is nearly universal, $\kappa=\sqrt{\nu_1-\lambda}$58, close to the free-droplet bound $\kappa=\sqrt{\nu_1-\lambda}$59 (Zezyulin, 2023). Because of the minimum in $\kappa=\sqrt{\nu_1-\lambda}$60, there are bistability regions in which two stationary states of the same family coexist at the same chemical potential but with different particle numbers and widths.

The stability structure is mode-dependent. The ground family $\kappa=\sqrt{\nu_1-\lambda}$61 is stable for all $\kappa=\sqrt{\nu_1-\lambda}$62, and the single-node family $\kappa=\sqrt{\nu_1-\lambda}$63 is also stable near the linear limit and remains stable throughout. By contrast, the $\kappa=\sqrt{\nu_1-\lambda}$64 and $\kappa=\sqrt{\nu_1-\lambda}$65 families are unstable near the linear limit, then stabilize above family-dependent thresholds, and may show additional finite instability windows before stabilizing again at sufficiently large $\kappa=\sqrt{\nu_1-\lambda}$66 (Zezyulin, 2023). The spectral problem is formulated through

$\kappa=\sqrt{\nu_1-\lambda}$67

with the self-adjoint operators

$\kappa=\sqrt{\nu_1-\lambda}$68

The harmonic trap also yields exact eigenvalues $\kappa=\sqrt{\nu_1-\lambda}$69 and $\kappa=\sqrt{\nu_1-\lambda}$70, corresponding to phase invariance and center-of-mass oscillation (Zezyulin, 2023).

At large density, the paper derives a modified Thomas–Fermi profile

$\kappa=\sqrt{\nu_1-\lambda}$71

with particle number

$\kappa=\sqrt{\nu_1-\lambda}$72

Under a slow ramp of the trap strength to zero, the ground-state solution evolves into the free-space solitonlike droplet, while excited trapped states break into several moving quantum droplets: two for $\kappa=\sqrt{\nu_1-\lambda}$73, three for $\kappa=\sqrt{\nu_1-\lambda}$74, and four for $\kappa=\sqrt{\nu_1-\lambda}$75 (Zezyulin, 2023). This makes the “quasi” aspect especially clear. Confinement is genuine in the trap, yet parameter variation can convert an apparently robust trapped family into a breakup process determined by its nodal structure.

Within the broader QTM vocabulary, these results extend the idea of quasi-trapping from linear wave leakage to nonlinear localization and stability windows. The relevant imperfection is not radiative decay but conditional dynamical robustness.

6. Trapped quasi-normal modes and pole-induced confinement in black-hole perturbations

In Einstein–Proca theory with non-minimal couplings, matter perturbations on a Schwarzschild background can develop unphysical poles in their linearized equations. Physical solutions are then confined to the region between the event horizon and a pole, and the appropriate eigenvalue problem is no longer the standard quasi-normal-mode problem on $\kappa=\sqrt{\nu_1-\lambda}$76 (Chiang et al., 7 Apr 2025). The action is

$\kappa=\sqrt{\nu_1-\lambda}$77

and on Ricci-flat Schwarzschild the linearized field equation reduces to

$\kappa=\sqrt{\nu_1-\lambda}$78

The non-minimal coupling introduces kinetic factors $\kappa=\sqrt{\nu_1-\lambda}$79 whose zeros generate poles outside the horizon for suitable $\kappa=\sqrt{\nu_1-\lambda}$80 (Chiang et al., 7 Apr 2025).

For the monopole, the master equation is

$\kappa=\sqrt{\nu_1-\lambda}$81

with

$\kappa=\sqrt{\nu_1-\lambda}$82

and the pole lies outside the horizon when $\kappa=\sqrt{\nu_1-\lambda}$83. For axial modes,

$\kappa=\sqrt{\nu_1-\lambda}$84

and the pole lies outside when $\kappa=\sqrt{\nu_1-\lambda}$85 (Chiang et al., 7 Apr 2025). The physical boundary conditions are ingoing at the horizon and regular at the pole. For the monopole near $\kappa=\sqrt{\nu_1-\lambda}$86,

$\kappa=\sqrt{\nu_1-\lambda}$87

and regularity requires $\kappa=\sqrt{\nu_1-\lambda}$88. For the axial mode near $\kappa=\sqrt{\nu_1-\lambda}$89,

$\kappa=\sqrt{\nu_1-\lambda}$90

and again regularity requires $\kappa=\sqrt{\nu_1-\lambda}$91 (Chiang et al., 7 Apr 2025).

The interior problem on $\kappa=\sqrt{\nu_1-\lambda}$92 defines the trapped quasi-normal modes. Their spectra differ qualitatively from standard black-hole QNMs. Axial-vector modes are unstable, and the growth rate increases with multipole number. The large-$\kappa=\sqrt{\nu_1-\lambda}$93 estimate for the fundamental mode is

$\kappa=\sqrt{\nu_1-\lambda}$94

The monopole sector reveals a separate instability mechanism. The near-pole effective potential forms a negative well, and the fundamental interior QTM becomes unstable when $\kappa=\sqrt{\nu_1-\lambda}$95; higher overtones destabilize at larger $\kappa=\sqrt{\nu_1-\lambda}$96 (Chiang et al., 7 Apr 2025). Representative spectral values are $\kappa=\sqrt{\nu_1-\lambda}$97 for the monopole with $\kappa=\sqrt{\nu_1-\lambda}$98, $\kappa=\sqrt{\nu_1-\lambda}$99, and $p_k=-\gamma_k+i\omega_k$00 for the axial mode with $p_k=-\gamma_k+i\omega_k$01, $p_k=-\gamma_k+i\omega_k$02, $p_k=-\gamma_k+i\omega_k$03 or $p_k=-\gamma_k+i\omega_k$04.

The exterior problem on $p_k=-\gamma_k+i\omega_k$05 remains stable and obeys standard-type outgoing or quasi-bound conditions, so the pole splits the spectrum into an interior trapped sector and an exterior radiative sector. This is a precise sense in which the black-hole QTM is “trapped”: the pole behaves as an additional boundary that prevents continuation of regular solutions through the full exterior region (Chiang et al., 7 Apr 2025).

Taken together with the earlier sections, this gravitational example shows how far the QTM concept has traveled from its waveguide and photonic origins. The trapping mechanism can be geometric, symmetry-protected, feedback-induced, nonlinear, or pole-generated. What remains common is the presence of a mode that is not fully extended, not fully closed, and whose observable behavior is controlled by a small but decisive departure from an ideal trapped limit.