Localized Conical Modes in Wave Physics

• Localized conical modes are wave states defined by cone geometry that enable various confinement mechanisms, including axial, angular, and spatiotemporal localization.
• They harness interference effects and spectral modifications to shape modes in systems such as dielectric cones, truncated spherical cavities, multimode fibers, and conical refraction setups.
• Practical insights include guiding high-Q resonant modes, generating quantized conical wave packets, and engineering passive longitudinal mode registers for tailored beam profiles.

Localized conical modes are wave states whose structure is governed by a cone, a conical angular spectrum, or a cone-modified angular domain, and whose confinement arises through several distinct mechanisms rather than a single universal construction. In the literature, the phrase spans at least four technically different settings: interference-localized whispering-gallery states on dielectric cones, continuous angular-spectrum cavity modes produced by conical truncation of spherical domains, spatiotemporally localized quantized conical waves in multimode fibers, and coherent or partially coherent cone-resolved fields in conical refraction. The available work therefore supports a family resemblance, not a single standardized mode class (Sumetsky, 2010, Bakr et al., 20 Dec 2025, Kibler et al., 2020, Mylnikov et al., 2022).

1. Scope and classification

The common ingredient is the role of cone geometry in reorganizing admissible propagation. What differs from one subfield to another is the meaning of “localized.” In some cases localization is axial and resonant; in others it is spatiotemporal; in others it is primarily angular-domain support or coherence-resolved cone selection. This distinction is essential because not every mode built from a conical spectrum is localized in the strict sense, and not every localized optical mode with unusual geometry is conical (Evans et al., 16 May 2026, Kumar et al., 2021).

Setting	Conical ingredient	Localization outcome
Weakly tapered dielectric cone or fiber	Real-space conical surface	Axially localized high-$Q$ resonant states
Spherical cavity with cone or wedge truncation	Cone-modified angular domain	Continuous families in $(\nu,m)$; support on truncated angular interval
Multimode optical fiber	Discrete guided analogue of conical emission	Quantized X-, O-, and Fish-wave packets
Conical refraction in biaxial crystals	Dual-cone field decomposition	Ring-, cone-, and far-field localized coherent structures
Finite-aperture Bessel-beam engineering	Conical angular spectrum	Axially addressed reconstruction, not exact localized-wave invariance
Random lasers	None	Anderson-like localization, not conical

A central conceptual divide runs between true confinement mechanisms and spectrum-engineering mechanisms. The dielectric-cone resonator is a confinement problem. Cone-truncated spherical cavities are primarily a self-adjoint-domain and angular-spectrum problem. Quantized conical waves in multimode fibers are guided spatiotemporal localized waves. Partially coherent conical refraction is a correlation-function and coherent-mode problem.

2. Interference-localized states on dielectric cones

The clearest direct realization of localized conical modes is the weakly tapered dielectric cone or optical fiber, where light forms axially localized whispering-gallery-like states even though the corresponding classical geodesic motion is not trapped on the wide side of the cone (Sumetsky, 2010). The core semiclassical representation writes the surface field as

$$\Psi(\varphi,z)\sim \sum_m S_m(\varphi,z)^{-1/2} \exp\!\left[i(\beta+i\alpha)S_m(\varphi,z)\right],$$

where $S_m$ is the geodesic length for a trajectory making $m$ turns, $\beta$ is the surface propagation constant, and $\alpha$ is attenuation. After unfolding the cone onto a plane, the relevant closed-path length is

$$S_m^0 = 2r\sin(\pi m\gamma) \sim 2\pi r m - \pi^3 r \gamma^2 m^3.$$

The cubic correction in $m$, induced by the small cone half-angle $\gamma$, is the term that enables localization.

The surprising result is that confinement does not arise from conventional geometric trapping between two classical turning points. On the narrow side of the cone, the field behaves in a turning-point-like manner. On the wide side, however, classical rays escape. Wave localization nevertheless occurs because the outgoing contributions from different winding numbers interfere destructively. In the dimensionless formulation, the localized states satisfy the quantization rule

$(\nu,m)$0

and the localization length scales as

$(\nu,m)$1

This weak scaling with taper, $(\nu,m)$2, is why extremely small fiber slopes can still support compact localized states.

The same theory identifies a slope-defined resonance regime,

$(\nu,m)$3

in which taper-induced phase accumulation dominates attenuation in setting the resonance structure. The transmission spectrum is then governed by a generalized Airy integral, producing the asymmetric oscillatory lineshapes verified experimentally on a silica fiber with $(\nu,m)$4 and $(\nu,m)$5. The fitted $(\nu,m)$6 agreed with the direct measurement to better than $(\nu,m)$7. In this setting, “localized conical mode” has its strictest meaning: a discrete, high-$(\nu,m)$8, axially confined resonant state supported by a monotonic cone through wave self-interference rather than classical boundedness.

3. Cone-modified spherical-cavity modes and continuous angular spectra

A different use of conical geometry appears in spherical electromagnetic cavities whose angular domain is altered by a cone or wedge (Bakr et al., 20 Dec 2025). Here the starting point is the separated Debye-potential ansatz

$(\nu,m)$9

with angular equation

$$\Psi(\varphi,z)\sim \sum_m S_m(\varphi,z)^{-1/2} \exp\!\left[i(\beta+i\alpha)S_m(\varphi,z)\right],$$0

The paper’s central claim is that integer $$\Psi(\varphi,z)\sim \sum_m S_m(\varphi,z)^{-1/2} \exp\!\left[i(\beta+i\alpha)S_m(\varphi,z)\right],$$1 quantization on the full sphere is not imposed by Maxwell’s equations themselves. It is enforced by two global constraints: single-valuedness under $$\Psi(\varphi,z)\sim \sum_m S_m(\varphi,z)^{-1/2} \exp\!\left[i(\beta+i\alpha)S_m(\varphi,z)\right],$$2, which forces integer $$\Psi(\varphi,z)\sim \sum_m S_m(\varphi,z)^{-1/2} \exp\!\left[i(\beta+i\alpha)S_m(\varphi,z)\right],$$3, and regularity at both poles, which restricts $$\Psi(\varphi,z)\sim \sum_m S_m(\varphi,z)^{-1/2} \exp\!\left[i(\beta+i\alpha)S_m(\varphi,z)\right],$$4 to the usual discrete values in non-sectoral families.

Once the domain is changed, the angular Sturm–Liouville operator acquires different self-adjoint realizations. A wedge replaces the azimuthal interval $$\Psi(\varphi,z)\sim \sum_m S_m(\varphi,z)^{-1/2} \exp\!\left[i(\beta+i\alpha)S_m(\varphi,z)\right],$$5 by $$\Psi(\varphi,z)\sim \sum_m S_m(\varphi,z)^{-1/2} \exp\!\left[i(\beta+i\alpha)S_m(\varphi,z)\right],$$6, yielding

$$\Psi(\varphi,z)\sim \sum_m S_m(\varphi,z)^{-1/2} \exp\!\left[i(\beta+i\alpha)S_m(\varphi,z)\right],$$7

which is generically non-integer. A conical truncation removes one pole from the domain and replaces polar regularity by a conductor boundary condition at $$\Psi(\varphi,z)\sim \sum_m S_m(\varphi,z)^{-1/2} \exp\!\left[i(\beta+i\alpha)S_m(\varphi,z)\right],$$8. For a perfectly conducting cone, the angular boundary conditions become

$$\Psi(\varphi,z)\sim \sum_m S_m(\varphi,z)^{-1/2} \exp\!\left[i(\beta+i\alpha)S_m(\varphi,z)\right],$$9

$S_m$0

For zonal modes this reduces to $S_m$1 or $S_m$2. Because $S_m$3 varies continuously, the admissible $S_m$4 also varies continuously, converting isolated full-sphere integer points into continuous branches.

The exact sectoral result

$S_m$5

is especially important. It provides a continuous dispersion curve

$S_m$6

which reproduces ordinary sectoral spherical harmonics at integer $S_m$7 but remains mathematically valid for real $S_m$8 when the azimuthal domain is modified. By contrast, tesseral and zonal families remain discrete on the full sphere because analytic continuation of the regular north-pole solution generically produces a singular south-pole component unless $S_m$9 lands on special values.

In this literature, the conical effect is primarily spectral rather than defect-bound. The modes are “localized” mainly by support on a truncated angular interval or, in the large-$m$0 sectoral case, by equatorial concentration through

$m$1

The paper is explicit that this is not cone-tip localization. Numerical validation for a cavity of radius $m$2 showed that as $m$3 varied from $m$4 to $m$5, the fundamental zonal TM branch moved continuously from $m$6 to $m$7, with theory and HFSS frequencies agreeing within $m$8. The limiting point $m$9 is also singular in a physically important way: the Debye potential remains non-trivial, but every electromagnetic field component vanishes identically, so the endpoint is not a physical EM mode.

4. Quantized conical waves in multimode optical fibers

In multimode optical fibers, localized conical modes appear as guided spatiotemporal wave packets in which the continuous conical spectrum of bulk media is replaced by a discrete modal spectrum (Kibler et al., 2020). The field is expanded as

$\beta$0

and each modal spectral component evolves with propagation constant $\beta$1. The generalized conical-wave family condition is

$\beta$2

This selects modal-frequency points $\beta$3 that share a common affine residual phase relation. The corresponding group velocity is

$\beta$4

Because the waveguide supports only a finite discrete set of modes, the ideal conical-wave spectrum is not a continuous curve in $\beta$5, as in bulk X-wave theory, but a discrete set

$\beta$6

The resulting multimode superposition is propagation invariant in intensity in the ideal limit. In the realistic finite-bandwidth case,

$\beta$7

different modes acquire different group delays, so the structure becomes quasi-invariant rather than exactly invariant. This is the sense in which the paper distinguishes ideal quantized conical waves from spontaneously generated nonlinear realizations.

The nonlinear simulations interpret these states as multimode resonant radiation emitted by a moving localized nonlinear structure such as a shock front or split pulse. In a $\beta$8 step-index MMF with pure silica core, $\beta$9 diameter, NA $\alpha$0, pumped at $\alpha$1 with a $\alpha$2, $\alpha$3 Gaussian pulse in LP$\alpha$4, the output modal-frequency spectrum formed a discrete X pattern. The broadening saturated after about $\alpha$5, following pulse splitting and trailing-edge steepening. The inferred phase-matching parameters were $\alpha$6 and $\alpha$7. At $\alpha$8 and $\alpha$9, the same fiber produced a Fish-wave pattern, combining a high-frequency X-wave tail and a low-frequency O-wave structure. A $$S_m^0 = 2r\sin(\pi m\gamma) \sim 2\pi r m - \pi^3 r \gamma^2 m^3.$$0 GRIN fiber with $$S_m^0 = 2r\sin(\pi m\gamma) \sim 2\pi r m - \pi^3 r \gamma^2 m^3.$$1 diameter and NA $$S_m^0 = 2r\sin(\pi m\gamma) \sim 2\pi r m - \pi^3 r \gamma^2 m^3.$$2, pumped at $$S_m^0 = 2r\sin(\pi m\gamma) \sim 2\pi r m - \pi^3 r \gamma^2 m^3.$$3 with $$S_m^0 = 2r\sin(\pi m\gamma) \sim 2\pi r m - \pi^3 r \gamma^2 m^3.$$4, $$S_m^0 = 2r\sin(\pi m\gamma) \sim 2\pi r m - \pi^3 r \gamma^2 m^3.$$5, also yielded an X-shaped modal spectrum, now with branch curvature determined by the GRIN dispersion landscape.

This framework is among the closest guided-wave analogues of strict localized conical modes. The localization is genuinely spatiotemporal, the conical morphology survives in the $$S_m^0 = 2r\sin(\pi m\gamma) \sim 2\pi r m - \pi^3 r \gamma^2 m^3.$$6 and $$S_m^0 = 2r\sin(\pi m\gamma) \sim 2\pi r m - \pi^3 r \gamma^2 m^3.$$7 representations, and the quantization is produced by the discrete guided-mode set rather than by quantum mechanics.

5. Conical refraction, coherent modes, and partial coherence

Conical refraction provides a different but closely related arena in which localized conical structures are naturally decomposed into mode families (Mylnikov et al., 2022). In the paraxial Belsky–Khapalyuk–Berry formulation, the coherent field is built from CR amplitudes

$$S_m^0 = 2r\sin(\pi m\gamma) \sim 2\pi r m - \pi^3 r \gamma^2 m^3.$$8

with

$$S_m^0 = 2r\sin(\pi m\gamma) \sim 2\pi r m - \pi^3 r \gamma^2 m^3.$$9

The index $m$0 labels the orbital channels carrying extra OAM. For partially coherent input, the relevant second-order quantities are the orbital correlation functions

$m$1

The dual-cone decomposition,

$m$2

is particularly important. It isolates the converging and diverging CR cones and permits a direct definition of their mutual coherence,

$m$3

In the high-coherence regime the paper derives

$m$4

where $m$5 is the source degree of coherence evaluated at a characteristic CR separation $m$6. In the low-coherence regime it obtains the universal scaling

$m$7

with $m$8. This reduction of cone-cone coherence explains the disappearance of the dark Poggendorff ring in the Lloyd plane: partial coherence suppresses interference between the two conical branches while leaving the conical basis itself intact.

For a Gaussian Schell-model source, the paper also gives a Mercer decomposition into Laguerre–Gaussian coherent modes. After propagation through the biaxial crystal, partially coherent conical refraction becomes an incoherent superposition of standard coherent CR modes rather than an amorphous random field. In the low-coherence focal-plane limit, the intensity of each cone tends to

$m$9

so the CR ring narrows instead of broadening. In the far field, the low-coherence result

$\gamma$0

implies a propagation-invariant radial profile, while the Raman-spot maximum shifts to

$\gamma$1

Thus conical refraction supports localized conical optical structures in both coherent and partially coherent regimes, but the localization is best understood through correlation functions, dual-cone coherence, and coherent-mode decompositions rather than through a discrete eigenvalue problem of the conical medium alone.

6. Passive conical-spectrum engineering and boundaries of the term

A broader contemporary usage attaches “conical” to fields built from Bessel-beam angular spectra, but not all such work constitutes localized conical modes in the strict sense (Evans et al., 16 May 2026, Kumar et al., 2021). In a finite-aperture Bessel-beam architecture, the defining statement is that “An ideal Bessel beam is formed by the interference of plane waves distributed on a cone of fixed half-angle $\gamma$2,” with

$\gamma$3

The paper exploits the one-to-one mapping between radial SLM coordinate and axial reconstruction distance,

$\gamma$4

together with the annulus-thickness relation

$\gamma$5

to create a passive longitudinal-mode register. Different annuli of a single static phase mask reconstruct different transverse fields at different $\gamma$6-planes. In the demonstrated sequence, the beam passes through a zero-order Bessel-Gaussian at $\gamma$7, a Bessel vortex with $\gamma$8 at $\gamma$9, a Hermite-Gaussian-Bessel mode at $(\nu,m)$00, and an Airy caustic mode at $(\nu,m)$01, with measured intensity fidelities $(\nu,m)$02, $(\nu,m)$03, $(\nu,m)$04, and $(\nu,m)$05, and crosstalk $(\nu,m)$06 for each region. This is conical-wave engineering, but the paper itself distinguishes it from exact localized-wave physics: the beam is monochromatic, finite-aperture, segmented, and only piecewise reconstructed near selected axial slabs.

This boundary matters for terminology. The spherical-cavity study with cones and wedges likewise does not primarily establish defect-bound localization near a cone surface; it establishes continuous families of admissible angular parameters on a cone-modified domain. By contrast, the dielectric-cone resonator and the quantized multimode-fiber X/O/Fish-wave constructions do deliver strong claims of localization, respectively axial-resonant and spatiotemporal. A plausible implication is that “localized conical modes” is most precise when reserved for cases in which the cone geometry participates directly in the confinement mechanism, and less precise when it serves mainly as a spectral scaffold.

The contrast with localized random lasers makes the boundary even sharper. In the quasi-1D DCM-doped PMMA random laser with 125 air grooves, the modes are spatially localized and approximately exponential, with localization lengths estimated numerically as $(\nu,m)$07 and experimentally as $(\nu,m)$08, and selective pumping suppresses gain competition, cross-saturation, and spatial hole burning. But the work explicitly contains no conical wavevector surface, no Bessel-like conical state, no ring-shaped near field, and no conical-mode terminology. It is therefore a study of localized optical modes, not of localized conical modes.

Across these literatures, the phrase denotes a technically heterogeneous class unified by one principle: cone geometry can alter the balance among diffraction, dispersion, boundary conditions, and interference so profoundly that new localized or quasi-localized wave structures emerge. What remains non-universal is the localization mechanism itself.