Spherical–Spheroidal Mode-Mixing

• Spherical–spheroidal mode-mixing is the angular basis transformation that redistributes modal content when spherical symmetry is broken by spheroidal geometry.
• It underpins Kerr black hole analyses by relating spin-weighted spheroidal harmonics to spherical harmonics, revealing complex coupling coefficients.
• The phenomenon extends to acoustics, optics, and computational physics, where mode-mixing enables improved waveform modeling and resonance tuning.

Searching arXiv for recent and canonical papers on spherical–spheroidal mode mixing. arXiv search query: "spherical spheroidal mode mixing Kerr quasinormal harmonics" Spherical–spheroidal mode-mixing denotes the redistribution of modal content that occurs when a problem organized in a spherical basis is expressed in a spheroidal basis, or when spheroidal geometry breaks the modal separations enforced by spherical symmetry. In the literature, the phenomenon appears in several technically distinct but closely related forms: as a basis transformation between spin-weighted spherical and spin-weighted spheroidal harmonics in Kerr perturbation theory; as a finite or infinite re-expansion between spherical and spheroidal harmonics in potential theory and acoustic field representations; and as geometry-induced hybridization of resonances when a sphere is deformed into a spheroid. Across these settings, the common structure is that spherical symmetry makes certain modes pure and decoupled, while spheroidal deformation or spheroidal separation reorganizes them into mixed states with new selection rules, coupling coefficients, and physical observables (1908.10377).

1. General formulation and mode-coupling structure

In its most standard form, spherical–spheroidal mixing is a change of angular basis. For Kerr black holes, the natural angular eigenfunctions are spin-weighted spheroidal harmonics, but many waveform calculations and data products are expressed in spin-weighted spherical harmonics. The relation is written as

$${}_s S_{\ell m}(\theta,\phi; a\omega) = \sum_{\ell'=\max(|m|,|s|)}^{\ell_{\max}} C_{\ell' \ell m}(a\omega)\, {}_s Y_{\ell' m}(\theta,\phi),$$

where the coefficients $C_{\ell'\ell m}(a\omega)$ are the spherical-spheroidal mixing coefficients. When $a\omega=0$, the spheroidal harmonics reduce to spherical harmonics and the mixing is trivial,

$C_{\ell'\ell m}(0)=\delta_{\ell'\ell}.$

For nonzero $a\omega$, the angular eigenfunction spreads over multiple spherical $\ell'$ values with the same $m$ (1908.10377).

The same-$m$ selection rule is a recurrent feature. In Kerr perturbation theory, azimuthal orthogonality enforces $\delta_{m,m'}$ in the overlap integrals, so different $m$ values remain orthogonal even when $C_{\ell'\ell m}(a\omega)$0 is mixed (Berti et al., 2014). In other harmonic problems the preserved quantum number is again the azimuthal order, while the degree changes in a structured ladder. For spheroidal domains of arbitrary eccentricity, a degree-$C_{\ell'\ell m}(a\omega)$1 spheroidal harmonic expands into spherical harmonics of degrees $C_{\ell'\ell m}(a\omega)$2 with the same order $C_{\ell'\ell m}(a\omega)$3, and the inverse relation is also explicit (García-Ancona et al., 2019).

A second archetype is algebraic rather than purely geometric. In the nine-dimensional MICZ-Kepler problem, the spherical basis diagonalizes $C_{\ell'\ell m}(a\omega)$4, while the prolate spheroidal basis diagonalizes

$C_{\ell'\ell m}(a\omega)$5

The spheroidal basis is therefore a finite linear combination of spherical states, with coefficients determined by a tridiagonal recurrence in the spherical quantum number. In that setting, mode-mixing is not an approximation or numerical artifact but an exact finite-dimensional eigenvalue problem generated by changing the commuting operator that is diagonalized (Le et al., 2021).

2. Kerr black holes and gravitational-wave ringdown

The best-developed physical realization of spherical–spheroidal mode-mixing is the ringdown of a spinning Kerr black hole. Kerr perturbations separate in the Teukolsky equation using spin-weighted spheroidal harmonics $C_{\ell'\ell m}(a\omega)$6, whereas numerical-relativity waveforms and many gravitational-wave pipelines are organized in spin-weighted spherical harmonics $C_{\ell'\ell m}(a\omega)$7. The overlap between the two bases is commonly written as

$C_{\ell'\ell m}(a\omega)$8

or equivalently through coefficients $C_{\ell'\ell m}(a\omega)$9 in the expansion of a spheroidal harmonic over spherical harmonics with the same $a\omega=0$0. In the Schwarzschild limit $a\omega=0$1, the mixing matrix becomes diagonal; for spinning remnants, one spheroidal quasinormal mode contributes to several spherical multipoles (London et al., 2018).

This basis mismatch is the leading explanation for the anomalous post-merger behavior of some spherical-harmonic modes in numerical relativity. The canonical example is the $a\omega=0$2 mode, whose amplitude bumps and instantaneous-frequency oscillations were shown to arise, to leading order, from leakage of the dominant $a\omega=0$3 Kerr quasinormal mode into the spherical $a\omega=0$4 channel. In that interpretation, the observed “mode-mixing” is largely the shadow of a spheroidal Kerr ringdown projected onto a spherical extraction basis (1212.5553). A later numerical study of perturbed Kerr black holes quantified these overlaps directly and emphasized that the coefficients are complex-valued functions of the Kerr spin $a\omega=0$5, with off-diagonal mixing increasing as the spheroidal deformation parameter $a\omega=0$6 grows (Berti et al., 2014).

The practical implication is that a spherical multipole $a\omega=0$7 is not, in general, a single quasinormal mode. Instead, it is a linear superposition of spheroidal modes with the same $a\omega=0$8 and different spheroidal $a\omega=0$9, together with mirror-mode contributions when those are included in the model (Dhani et al., 2021). This matters most for sub-leading modes within a fixed $C_{\ell'\ell m}(0)=\delta_{\ell'\ell}.$0-sector. In nonspinning binary-black-hole merger simulations with $C_{\ell'\ell m}(0)=\delta_{\ell'\ell}.$1, the modes $C_{\ell'\ell m}(0)=\delta_{\ell'\ell}.$2, $C_{\ell'\ell m}(0)=\delta_{\ell'\ell}.$3, $C_{\ell'\ell m}(0)=\delta_{\ell'\ell}.$4, $C_{\ell'\ell m}(0)=\delta_{\ell'\ell}.$5, and $C_{\ell'\ell m}(0)=\delta_{\ell'\ell}.$6 were found to be strongly affected, while the leading mode of each $C_{\ell'\ell m}(0)=\delta_{\ell'\ell}.$7-sector was only minimally affected because its self-mixing coefficient is close to unity and contamination from other $C_{\ell'\ell m}(0)=\delta_{\ell'\ell}.$8 values is at most a few percent (Dhani et al., 2021).

This structure has been incorporated directly into waveform models. For nonspinning binaries, the $C_{\ell'\ell m}(0)=\delta_{\ell'\ell}.$9 and $a\omega$0 spherical modes were modeled by first approximately “unmixing” them into spheroidal-like ringdown modes, fitting the smoother unmixed quantities, and then reapplying the mixing coefficients to reconstruct the observable spherical modes. The resulting analytical waveform family includes explicit mode mixing and achieves faithfulness $a\omega$1 (Mehta et al., 2019). In this literature, mode-mixing is therefore not merely a correction to waveform aesthetics; it is a required component of accurate ringdown spectroscopy and higher-multipole inference.

3. Computation, fitting, completeness, and phase conventions

The numerical treatment of spherical–spheroidal mixing has matured into a distinct computational subfield. The qnm Python package computes Kerr quasinormal-mode frequencies, separation constants, and spherical-spheroidal mixing coefficients using a hybrid strategy: the radial sector is solved with Leaver’s continued-fraction method, while the angular sector is treated by the Cook–Zalutskiy spectral method. In that formulation, the angular Teukolsky equation becomes a matrix eigenvalue problem in a spin-weighted spherical-harmonic basis, so the mixing coefficients appear directly as components of the angular eigenvector. They are therefore obtained “for free” once the angular eigenproblem has been solved (1908.10377).

For repeated evaluations over black-hole spin, interpolation and caching are essential. The same package stores previously computed modes at selected spin values, interpolates from the cache to generate initial guesses, and then refines the result by root polishing. This exploits the smooth spin dependence of the quasinormal frequencies and mixing coefficients and makes practical computations faster and more robust (1908.10377).

A complementary development is phenomenological modeling across the full spin range $a\omega$2. A study of Kerr black-hole modeling introduced greedy-multivariate-polynomial regression for quasinormal frequencies and greedy-multivariate-rational regression for spherical-spheroidal mixing coefficients. For the latter, the spin variable was remapped using

$a\omega$3

which makes the spin dependence much smoother near extremality. GMVR was then used to construct explicit rational fits for dominant multipoles up to $a\omega$4, extending previous fits and connecting prograde and retrograde branches in a single spin-continuous model (London et al., 2018).

The mathematical status of the spheroidal basis has also been clarified. Because physical Kerr spheroidal harmonics are mode-dependent through $a\omega$5, they do not form a conventional orthogonal family under one common operator. A bi-orthogonal framework was therefore introduced using adjoint-spheroidal harmonics $a\omega$6, with completeness expressed as

$a\omega$7

Within fixed-overtone subsets, especially $a\omega$8, the physical spheroidal harmonics were shown to be minimal and complete, and the adjoint basis suppresses the spherical-style leakage terms that otherwise contaminate mode estimation (London, 2020).

Phase choice is an additional technical issue. Spin-weighted spheroidal functions have an overall phase freedom, and the complex mixing coefficients depend on how that phase is fixed. A recent study defined and compared Cook–Zalutskiy and spherical-limit phase-fixing schemes and proposed the spherical-limit continuous convention $a\omega$9 as the default. The recommended prescription fixes the function, or if necessary its derivative, to be real at the equator and matches the sign to the spherical limit. The stated motivation is that poor phase conventions can introduce artificial discontinuities in tabulated sequences and hinder comparisons between spherical and spheroidal mode data (Cook et al., 23 Mar 2026).

4. Exact harmonic and algebraic relations beyond Kerr

Outside black-hole perturbation theory, spherical–spheroidal mode-mixing often appears as an exact re-expansion between separable solutions. For spheroidal domains of arbitrary eccentricity, the relation

$\ell'$0

shows explicitly that a spheroidal harmonic of degree $\ell'$1 is a finite linear combination of spherical harmonics of degrees $\ell'$2 with the same order $\ell'$3. The coefficients are algebraic and independent of the eccentricity parameter $\ell'$4; the entire $\ell'$5-dependence appears through the powers $\ell'$6. The inverse relation is also explicit, and analogous formulas hold for the Garabedian harmonics and for monogenic polynomials (García-Ancona et al., 2019).

The same paper extends the picture from sphere-to-spheroid conversion to conversion between two different spheroidal eccentricities. For Garabedian harmonics, the basis at eccentricity $\ell'$7 can be expanded in the basis at eccentricity $\ell'$8 with coefficients containing the Gaussian hypergeometric function

$\ell'$9

This demonstrates that “mode-mixing” is not limited to the spherical versus spheroidal dichotomy; changing the spheroidal shape itself induces a structured mixing across degrees while preserving azimuthal order (García-Ancona et al., 2019).

Potential theory near a sphere provides another exact setting in which spheroidal re-expansions are advantageous. For Laplace’s equation with a point source outside a sphere, irregular spherical harmonics can be re-expanded in irregular offset spheroidal harmonics $m$0, whose intrinsic singularity matches the line singularity of the analytic continuation of the solution. For the corresponding internal-potential problem, ordinary regular spheroidal harmonics do not match the correct singular support, but a Kelvin-transformed, radially inverted spheroidal coordinate system does. The resulting basis $m$1 was shown to converge much faster because it fits the semi-infinite image-line singularity exactly (Majić et al., 2017).

The nine-dimensional MICZ-Kepler problem gives an algebraic counterpart to these analytic transformations. There, the spherical-to-spheroidal coefficients $m$2 satisfy a tridiagonal recurrence,

$m$3

which can be written as a finite matrix eigenvalue problem. The direct overlap integrals involving confluent Heun, Laguerre, and Jacobi functions are thereby replaced by an algebraic solution (Le et al., 2021). This makes clear that spherical–spheroidal mode-mixing can be exact, finite, and operator-theoretic, not only asymptotic or perturbative.

5. Resonance engineering, cavities, and acoustic field representations

In wave-scattering and resonator problems, spheroidal deformation changes mode content in a more physical, less purely representational sense. For high-index dielectric nanoparticles, a sphere can satisfy the first Kerker condition $m$4 only on the weak-scattering tail of its resonances, where both dipole amplitudes are relatively small. An oblate spheroid, by contrast, shifts the electric and magnetic dipole resonances so that they can overlap at the resonance maximum, with $m$5. The result is near-zero backscattering together with much stronger forward scattering. In the numerical examples reported, for $m$6 a spheroid with $m$7 gives about $m$8 higher forward scattering than the best sphere, and for $m$9 an oblate spheroid with $m$0 can reach $m$1 with nearly zero backscattering, whereas a sphere gives only $m$2 (Luk`yanchuk et al., 2014). In this optics setting, mode-mixing refers to shape-induced hybridization of the electric and magnetic dipolar resonances.

Fabry–Pérot microcavities with transverse mirror misalignment provide a related resonator example. There the fundamental cavity mode couples to higher-order Hermite–Gauss modes. For finite-diameter spherical cap mirrors, the mode-mixing method agrees with clipping loss to within about an order of magnitude in most regions, but also reveals a protective effect near transverse degeneracies: hybridization can move power away from the mirror edge, make the footprint more compact, and reduce diffraction loss. For Gaussian mirrors, whose curvature varies across the profile, the same mechanism generally produces stronger distortion, resonance splitting, and high-loss bands (Hughes et al., 2023). The authors explicitly relate the spherical cap case to the classic spherical–spheroidal mode-mixing problem in open resonators.

Spatial audio supplies a further basis-conversion instance. Spheroidal ambisonics expands sound fields in prolate spheroidal wave functions and then analytically transcodes them into the spherical ambisonics format used by the existing ecosystem. The key conversion is an infinite mode-mixing relation between spheroidal and spherical Helmholtz solutions, with parity restriction

$m$3

in the degree coupling. The final transcoding formula maps the real spheroidal coefficient pairs $m$4 into complex spherical HOA coefficients $m$5 through the spheroidal expansion coefficients $m$6. Numerical experiments reported that the sound field reconstructed from the transcoded coefficients has a zone of accurate reconstruction prolonged toward the long axis of a prolate spheroidal microphone array (Kaneko, 2021).

These examples share a common structural point. In one class, the spheroidal basis is chosen because it is better adapted to geometry or boundary conditions, and the spherical coefficients are recovered by an explicit mixing transform. In the other, spheroidal deformation alters the resonant spectrum itself, so the “mixed” modes are physically different eigenresponses rather than merely a different representation of the same field.

6. Interpretation, misconceptions, and related geometrical reorganizations

A recurring misconception is that spherical–spheroidal mode-mixing is merely a numerical nuisance. The literature does not support that interpretation. In Kerr ringdown, the effect is a geometric consequence of using a spherical basis for radiation whose natural angular eigenfunctions are spheroidal (Berti et al., 2014). In harmonic analysis, it is an exact change of basis with explicit coefficients (García-Ancona et al., 2019). In nanoparticle optics and resonator design, it can be deliberately exploited to overlap resonances or reduce loss (Luk`yanchuk et al., 2014).

A second misconception is that off-diagonal mixing must be large to matter. Ringdown analyses show the opposite: even when off-diagonal coefficients are small, a strongly excited contaminating mode can dominate a sub-leading spherical multipole. This is why the leading mode in a fixed $m$7-sector is usually little affected, whereas the next mode can display pronounced amplitude and instantaneous-frequency modulations (Dhani et al., 2021). The practical consequence is that same-$m$8 sectors should be modeled simultaneously rather than as independent spherical channels.

A third misconception is that all “mode-mixing” in spheroidal geometries is the same phenomenon. The surveyed work instead suggests a family resemblance rather than strict identity. In Kerr perturbation theory the central object is a complex overlap coefficient between angular bases. In the MICZ-Kepler problem the mixing is generated by the action of $m$9 in the spherical basis. In optics the decisive effect is the geometry-induced coincidence of electric and magnetic dipole resonances. A related but distinct reorganization appears in low-Reynolds-number hydrodynamics of spheroidal squirmers, where the spheroidal geometry separates odd and even slip modes into translation and stresslet channels rather than preserving the familiar spherical two-mode structure (Poehnl et al., 2019). This suggests that “spherical–spheroidal mode-mixing” is best understood as a broader symmetry-breaking pattern: once spherical symmetry is relaxed or a spheroidal eigenbasis is adopted, the original spherical mode labels cease to define isolated physical channels.

The topic therefore sits at the intersection of special-function theory, spectral methods, perturbation theory, and wave physics. Its technical importance comes from the fact that mode labels are often treated as if they were invariant physical attributes, while the cited work shows that they can be basis-dependent, geometry-dependent, and phase-convention-dependent. Correctly accounting for spherical–spheroidal mode-mixing is thus essential whenever spheroidal geometry or Kerr rotation is strong enough that the spherical description no longer tracks the true eigenstructure of the problem.