Transverse-Mode Frequency Pulling

• Transverse-mode frequency pulling is the shift in a mode’s effective resonance due to coupling mechanisms like dipolar interactions, two-magnon scattering, and dynamic detuning.
• Models based on coupled-oscillator theory, self-energy renormalization, and Gouy-phase in optical cavities illustrate how hybridization and synchronization emerge across different platforms.
• This phenomenon is observed in magnetic vortex oscillators, ferromagnetic resonance setups, optical parametric oscillators, fiber lasers, and RF cavity systems, offering pathways to control stability and linewidth.

Searching arXiv for recent and relevant papers on transverse-mode frequency pulling and closely related phenomena. Transverse-mode frequency pulling denotes the shift of a transverse mode’s effective resonance frequency away from its uncoupled or cold-system value because of coupling, dispersive interaction, cavity geometry, dynamic detuning, or external drive. Across the literature, the phrase refers to closely related but platform-specific phenomena: gyrotropic vortex-core modes in stacked magnetic disks pulled through dipolar coupling under strain (Iurchuk et al., 2024), uniform ferromagnetic resonance pulled by hybridization with nonuniform magnons through two-magnon scattering (Peria et al., 2020), optical transverse modes made exactly or nearly degenerate in self-imaging and Gaussian resonators (Chalopin et al., 2011, Zwilich et al., 2023), transverse optical mode offsets modulated by mirror figure errors and spot motion in long suspended cavities (Zhao et al., 2015), thermally pulled propagation constants in high-power fiber systems undergoing transverse mode instability (Jauregui et al., 2020), and pump- or injection-dependent frequency shifts in cavity-QED and RF cavity oscillators (1201.01542, Arumugam, 8 May 2026). The common structure is a renormalization of mode frequencies or propagation constants that changes hybridization, synchronization, linewidth, or stability.

1. Definition and scope

In magnetic vortex oscillators, transverse-mode frequency pulling is defined operationally as the strain-induced upshift of the gyrotropic frequency of the nonmagnetostrictive Py vortex caused by dipolar coupling to the strain-tuned CoFeB vortex; the directly tuned CoFeB mode moves downward under strain, while the Py mode is indirectly pulled upward toward it (Iurchuk et al., 2024). In ferromagnetic resonance, the analogous effect is a shift of the uniform $\mathbf{q}=0$ mode produced by coupling to nonuniform magnons through two-magnon scattering; the real part of the corresponding self-energy pulls the resonance, while the imaginary part broadens it (Peria et al., 2020).

In optical systems, the phrase often refers less to direct mutual dragging of isolated eigenfrequencies and more to the way cavity geometry, residual detuning, and nonlinear interaction determine which transverse modes can share a common resonance or carrier frequency. A self-imaging optical parametric oscillator can be completely transverse-mode degenerate, so that many transverse modes resonate at the same optical frequency; slight non-degeneracy then appears as residual detuning rather than as a large intrinsic splitting (Chalopin et al., 2011). An empty Fabry–Perot conversion resonator uses the transverse-mode spacing imposed by Gouy phase to map phase-locked longitudinal sidebands onto a phase-locked set of Hermite–Gaussian modes (Zwilich et al., 2023). In long suspended cavities, by contrast, transverse-mode frequency pulling is explicitly dynamic: mirror figure errors make the transverse-mode offset frequency depend on spot position, and residual suspended-mirror motion modulates that offset in time (Zhao et al., 2015).

This diversity shows that the term is not restricted to a single microscopic mechanism. In some platforms it denotes coupled-mode avoided-crossing behavior; in others it denotes self-energy renormalization, geometry-imposed degeneracy, or drive-induced pulling toward synchronization. A frequent misconception is that all such cases involve the same kind of “mode pulling.” The cited work instead supports a broader interpretation: the invariant element is the displacement of a transverse-mode resonance from the value expected for an isolated, static, or perfectly aligned system.

2. Canonical theoretical descriptions

Three descriptions recur. The first is a coupled-oscillator or coupled-mode picture. In the double-vortex system, the CoFeB gyrotropic mode $\omega_1=\omega_{\text{CoFeB}}(V_P)$ is strain-tunable, the Py mode $\omega_2=\omega_{\text{Py}}$ is approximately constant, and dipolar interaction $g$ produces hybridized eigenfrequencies

$$\omega_{\pm}=\frac{\omega_1+\omega_2}{2}\pm \sqrt{\left(\frac{\omega_1-\omega_2}{2}\right)^2+g^2}.$$

As strain lowers $\omega_1$, the two branches avoid crossing; the CoFeB-dominated branch moves downward and the Py-dominated branch is pulled upward (Iurchuk et al., 2024). In the isolated-vortex limit the relevant transverse eigenfrequency is gyrotropic,

$f_G=\frac{\kappa}{2\pi G},$

so any mechanism that changes the effective spring constant $\kappa$ changes the mode frequency (Iurchuk et al., 2024).

The second description is self-energy renormalization. In thin-film ferromagnetic resonance, the observed resonance is

$$\omega_{\text{obs}}=\omega_{FMR}+\Delta\omega_{TMS},$$

with $\Delta\omega_{TMS}$ generated by coupling to nonuniform magnons via defect-mediated two-magnon scattering. In linear-response language,

$\omega_1=\omega_{\text{CoFeB}}(V_P)$0

so $\omega_1=\omega_{\text{CoFeB}}(V_P)$1 produces frequency pulling and $\omega_1=\omega_{\text{CoFeB}}(V_P)$2 produces linewidth enhancement (Peria et al., 2020). The same real/imaginary decomposition also underlies the interpretation of linewidth broadening and frequency shift as complementary manifestations of one coupling process.

The third description is geometry-imposed transverse-mode spectra in optical cavities. In a Gaussian resonator,

$\omega_1=\omega_{\text{CoFeB}}(V_P)$3

with $\omega_1=\omega_{\text{CoFeB}}(V_P)$4 set by Gouy phase (Zwilich et al., 2023). In a self-imaging cavity, the special distances

$\omega_1=\omega_{\text{CoFeB}}(V_P)$5

make the cavity completely degenerate, so the transverse-mode-dependent frequency shift that ordinarily arises from Gouy phase is effectively cancelled (Chalopin et al., 2011). In long suspended cavities, the same transverse spacing becomes a time-dependent quantity because the effective radii of curvature $\omega_1=\omega_{\text{CoFeB}}(V_P)$6 and $\omega_1=\omega_{\text{CoFeB}}(V_P)$7 depend on spot position, so the transverse offset frequency is dynamically modulated (Zhao et al., 2015).

A quantum extension appears in the cavity-QED microlaser, where the spectrum is described by a photon-number-dependent frequency-pulling distribution $\omega_1=\omega_{\text{CoFeB}}(V_P)$8. Each photon-number sector contributes a Lorentzian centered at $\omega_1=\omega_{\text{CoFeB}}(V_P)$9, so frequency pulling is no longer a single offset but a statistical distribution associated with the photon-number distribution $\omega_2=\omega_{\text{Py}}$0 (1201.01542). This suggests that “frequency pulling” need not be a single deterministic parameter; it can itself be distributed when strong coherent light–matter coupling is resolved at the level of photon number.

3. Magnetic and spintronic realizations

A direct realization of transverse-mode frequency pulling is provided by stacked double-vortex microdisks fabricated on a PMN–PT (011) substrate. The disks have diameter $\omega_2=\omega_{\text{Py}}$1 and stack sequence Cr(3 nm) / CoFeB(25 nm) / Cu($\omega_2=\omega_{\text{Py}}$2) / Py(25 nm) / Cr(2 nm), with $\omega_2=\omega_{\text{Py}}$3, 20, or 40 nm. Both ferromagnetic layers support vortex ground states, and each layer’s low-frequency transverse mode is the gyrotropic orbit of the vortex core. CoFeB is magnetostrictive, Py is nonmagnetostrictive, and the layers are magnetostatically coupled across the Cu spacer (Iurchuk et al., 2024).

Strain is generated electrically in the PMN–PT substrate with an electric field $\omega_2=\omega_{\text{Py}}$4 per volt of $\omega_2=\omega_{\text{Py}}$5. In CoFeB, the corresponding magnetoelastic energy

$\omega_2=\omega_{\text{Py}}$6

softens the restoring potential, so $\omega_2=\omega_{\text{Py}}$7 and the CoFeB gyrotropic frequency decreases with increasing $\omega_2=\omega_{\text{Py}}$8 (Iurchuk et al., 2024). Py has negligible intrinsic magnetoelastic contribution, so any shift of its gyrotropic resonance is coupling-mediated rather than directly strain-induced.

The spacer thickness controls the coupling regime. For $\omega_2=\omega_{\text{Py}}$9 nm, coupling is strong: the CoFeB frequency starts to drop and the Py frequency starts to rise at about $g$0 V, and for $g$1 V the two vortices gyrate at almost the same frequency. For $g$2 nm, the CoFeB downshift begins around 6 V, Py pulling remains visible but weaker, and synchronization occurs for $g$3 V. For $g$4 nm, the CoFeB downshift begins near 8 V, Py shows essentially no measurable pulling, and the synchronized regime is not reached within the available $g$5 range (Iurchuk et al., 2024). In this system, transverse-mode frequency pulling is therefore not merely a frequency shift but a pathway to synchronized double-vortex oscillation.

Ferromagnetic resonance provides a second magnetic realization with a different mechanism. In Fe$g$6Ga$g$7 thin films of thickness 17, 26, and 33 nm, two-magnon scattering couples the uniform FMR mode to nonuniform magnons when the magnetization is in-plane. The effect is absent or strongly suppressed for perpendicular-to-plane magnetization because the relevant degeneracies in the magnon dispersion disappear (Peria et al., 2020). If frequency pulling is neglected, in-plane and perpendicular fits yield inconsistent $g$8 factors and $g$9. When the real part of the same self-energy used to fit linewidths is included, consistency is restored. In these films the pulling is a red shift that approaches nearly 1 GHz at the lowest measured frequencies, and its magnitude correlates with the two-magnon linewidth, as predicted by the shared self-energy description (Peria et al., 2020).

These magnetic examples establish two distinct but compatible meanings of transverse-mode frequency pulling. In one case, a tunable mode hybridizes with a neighboring transverse mode and pulls it toward synchronization. In the other, a primary mode is renormalized by a continuum of transverse side modes, and frequency pulling must be included to recover the correct bare material parameters.

4. Optical cavities, degeneracy engineering, and spatial-mode locking

In multimode quantum optics, transverse-mode frequency pulling is often approached through deliberate cavity degeneracy engineering. A self-imaging OPO formed by a plane mirror, a converging lens of focal length $$\omega_{\pm}=\frac{\omega_1+\omega_2}{2}\pm \sqrt{\left(\frac{\omega_1-\omega_2}{2}\right)^2+g^2}.$$0, a spherical mirror of radius $$\omega_{\pm}=\frac{\omega_1+\omega_2}{2}\pm \sqrt{\left(\frac{\omega_1-\omega_2}{2}\right)^2+g^2}.$$1, and a 10 mm PPKTP crystal becomes completely transverse-mode degenerate when the cavity distances satisfy $$\omega_{\pm}=\frac{\omega_1+\omega_2}{2}\pm \sqrt{\left(\frac{\omega_1-\omega_2}{2}\right)^2+g^2}.$$2 and $$\omega_{\pm}=\frac{\omega_1+\omega_2}{2}\pm \sqrt{\left(\frac{\omega_1-\omega_2}{2}\right)^2+g^2}.$$3. In the reported implementation, $$\omega_{\pm}=\frac{\omega_1+\omega_2}{2}\pm \sqrt{\left(\frac{\omega_1-\omega_2}{2}\right)^2+g^2}.$$4 mm, $$\omega_{\pm}=\frac{\omega_1+\omega_2}{2}\pm \sqrt{\left(\frac{\omega_1-\omega_2}{2}\right)^2+g^2}.$$5 mm, total cavity length is 128 mm, finesse is $$\omega_{\pm}=\frac{\omega_1+\omega_2}{2}\pm \sqrt{\left(\frac{\omega_1-\omega_2}{2}\right)^2+g^2}.$$6, bandwidth is approximately 4.7 MHz, and escape efficiency is approximately 60% (Chalopin et al., 2011). Because the cavity is completely degenerate, many transverse modes resonate at the same optical frequency; the natural basis for the quantum description is the eigenbasis of the parametric down-conversion coupling matrix, which in many practical cases is very close to a set of Hermite–Gauss modes (Chalopin et al., 2011).

The experiment demonstrated simultaneous squeezing of three transverse modes at the same wavelength: 1.2 dB in TEM$$\omega_{\pm}=\frac{\omega_1+\omega_2}{2}\pm \sqrt{\left(\frac{\omega_1-\omega_2}{2}\right)^2+g^2}.$$7, 0.6 dB in TEM$$\omega_{\pm}=\frac{\omega_1+\omega_2}{2}\pm \sqrt{\left(\frac{\omega_1-\omega_2}{2}\right)^2+g^2}.$$8, and 0.3 dB in TEM$$\omega_{\pm}=\frac{\omega_1+\omega_2}{2}\pm \sqrt{\left(\frac{\omega_1-\omega_2}{2}\right)^2+g^2}.$$9, measured at a 3 MHz sideband on a 1064 nm carrier. The pump waist was approximately $\omega_1$0, much larger than the crystal coherence length $\omega_1$1, and the theoretical number of significantly squeezed modes was approximately 7 (Chalopin et al., 2011). Strictly speaking, exact degeneracy removes the need for pulling among split transverse frequencies; the relevant effect appears when the cavity is not completely degenerate and slight detuning is present on higher-order modes. Then nonlinear gain allows several modes with small residual detunings to remain squeezed at the same carrier frequency, which is a near-degenerate form of effective locking rather than large-amplitude mutual dragging (Chalopin et al., 2011).

A complementary optical example is the empty two-mirror conversion resonator used to transform longitudinal mode locking into transverse mode locking. The cavity consists of a plane mirror and a concave mirror with $\omega_1$2, cavity length $\omega_1$3, longitudinal free spectral range $\omega_1$4, and measured transverse spacing $\omega_1$5. By choosing the modulation frequency to satisfy $\omega_1$6, carrier and sidebands are simultaneously resonant with successive $\omega_1$7 modes, producing phase-locked superpositions of up to five transverse modes (Zwilich et al., 2023). In this setting, cavity geometry fixes the transverse frequency ladder, and cavity-length tuning selects which transverse order $\omega_1$8 is resonant with the carrier. The resulting transmitted field is a transverse mode-locked state whose intensity distribution moves across the transverse plane at $\omega_1$9 (Zwilich et al., 2023).

Deformed square resonator lasers illustrate yet another optical role for transverse-mode pulling: not the creation of degeneracy, but the control of transverse spacing and the mitigation of unwanted mutual pulling by geometry. In circular-side square resonators, the deformation amplitude $f_G=\frac{\kappa}{2\pi G},$0 adjusts both $f_G=\frac{\kappa}{2\pi G},$1 factors and transverse mode intervals. For $f_G=\frac{\kappa}{2\pi G},$2, simulated $f_G=\frac{\kappa}{2\pi G},$3 factors rise from approximately $f_G=\frac{\kappa}{2\pi G},$4, $f_G=\frac{\kappa}{2\pi G},$5, and $f_G=\frac{\kappa}{2\pi G},$6 for 0th-, 1st-, and 2nd-order modes at $f_G=\frac{\kappa}{2\pi G},$7 to approximately $f_G=\frac{\kappa}{2\pi G},$8, $f_G=\frac{\kappa}{2\pi G},$9, and $\kappa$0 at $\kappa$1, and to approximately $\kappa$2, $\kappa$3, and $\kappa$4 at $\kappa$5 (Weng et al., 2015). Experimentally, the transverse spacing sensitivity between $\kappa$6 is $\kappa$7, $\kappa$8, and $\kappa$9, and stable dual-mode lasing was observed with separations 5.98, 3.43, 2.46, and 1.26 nm for $$\omega_{\text{obs}}=\omega_{FMR}+\Delta\omega_{TMS},$$0, 1.1, 1.3, and 1.5 $$\omega_{\text{obs}}=\omega_{FMR}+\Delta\omega_{TMS},$$1, respectively (Weng et al., 2015). The stationary condition for dual-mode emission is attributed to the fact that the high-$$\omega_{\text{obs}}=\omega_{FMR}+\Delta\omega_{TMS},$$2 confined modes have totally different mode numbers, reducing cross saturation and therefore reducing mutual frequency dragging.

5. Dynamic modulation, instability, and suppression

Long optical cavities show that transverse-mode frequency pulling can be intrinsically time dependent rather than quasistatic. In the context of three-mode parametric instability, earlier modelling assumed that transverse optical modes were stable in frequency except for slow thermal drifts. The 74 m Gingin cavity experiments and associated modelling showed instead that mirror figure errors make the transverse-mode offset frequency depend on beam spot position, and the low-frequency residual motion of suspended mirrors modulates that offset (Zhao et al., 2015). In an Advanced LIGO arm cavity, a curvature change of 1 m, corresponding to a sagitta change of about 0.3 nm over the beam diameter, changes the TEM$$\omega_{\text{obs}}=\omega_{FMR}+\Delta\omega_{TMS},$$3 spacing by approximately 13 Hz. In the Gingin cavity, TEM$$\omega_{\text{obs}}=\omega_{FMR}+\Delta\omega_{TMS},$$4–TEM$$\omega_{\text{obs}}=\omega_{FMR}+\Delta\omega_{TMS},$$5 spacing fluctuations with peak-to-peak amplitudes of a few kHz were observed and correlated with spot displacement on the input test mass (Zhao et al., 2015).

This modulation enters the three-mode detuning as

$$\omega_{\text{obs}}=\omega_{FMR}+\Delta\omega_{TMS},$$6

with normalized amplitude $$\omega_{\text{obs}}=\omega_{FMR}+\Delta\omega_{TMS},$$7. The effective parametric gain becomes

$$\omega_{\text{obs}}=\omega_{FMR}+\Delta\omega_{TMS},$$8

so transverse-mode frequency modulation suppresses instability whenever the modulation depth is comparable to or larger than the transverse-mode linewidth (Zhao et al., 2015). The paper concludes that gain suppression factors of 10–20 could be achieved for individual modes by exploiting this mechanism, and that suppression can be enhanced by spot dithering or fast thermal modulation (Zhao et al., 2015). A common misconception addressed by this work is that mirror imperfections are only detrimental. The reported result instead shows that imperfect figure combined with unavoidable residual motion can act as a passive suppression mechanism for parametric instability.

High-power fiber lasers provide the waveguide analogue. There the relevant transverse eigenvalues are propagation constants,

$$\omega_{\text{obs}}=\omega_{FMR}+\Delta\omega_{TMS},$$9

and thermally induced changes in $\Delta\omega_{TMS}$0 pull the modal phase velocities and beat lengths. The beat length between two modes is

$\Delta\omega_{TMS}$1

so thermal lensing, gain saturation, and photodarkening shift the spatial frequency of the modal interference pattern, which in turn imprints a thermally induced long-period grating (Jauregui et al., 2020). The experimental phenomenology is a threshold-like onset of rapid beam fluctuations: below a system-specific $\Delta\omega_{TMS}$2 the output beam is stable and nearly Gaussian; at threshold there is sudden degradation, nearly complete energy transfer between the fundamental mode and higher-order modes on the millisecond time scale, and RF signatures in the 100 Hz–kHz range whose scaling with core size is compatible with thermal diffusion times (Jauregui et al., 2020).

The fiber review does not use the phrase “frequency pulling,” but its thermo-optic coupled-mode description is structurally one of pulled transverse eigenvalues. As average power rises, the thermal lens increases the separation of mode-dependent effective indices, thereby compressing or stretching the interference pattern and changing the dynamical beat frequency that drives the instability (Jauregui et al., 2020). This suggests that transverse-mode frequency pulling and transverse mode instability are not separate subjects; the former is one of the dynamical ingredients of the latter.

6. Quantum, driven, and application-oriented extensions

The cavity-QED microlaser generalizes frequency pulling beyond classical mode competition. In the off-resonant microlaser, dispersive atom–cavity interaction produces a photon-number-dependent pulling distribution $\Delta\omega_{TMS}$3, so the spectrum becomes a sum of Lorentzians centered at $\Delta\omega_{TMS}$4, each weighted by the steady-state photon-number distribution. Unlike conventional lasers, the mean pulling varies periodically with pump parameter because of strong atom–cavity coupling and Rabi dynamics (1201.01542). The reported result is important conceptually because it shows that frequency pulling can be pump dependent even above threshold and can contribute spectral broadening through the width of the pulling distribution itself, rather than through a single static shift (1201.01542).

An RF implementation appears in a self-sustained oscillator built from an RF feedback loop around an open TEM cavity containing a cesium Rydberg vapor cell. Injection of a microwave signal pulls the oscillator frequency according to Adler-type phase dynamics,

$\Delta\omega_{TMS}$5

with effective coupling $\Delta\omega_{TMS}$6 (Arumugam, 8 May 2026). The pulling is read out optically as an intermediate-frequency signal in a Rydberg superheterodyne architecture, converting microwave input power into a measurable frequency shift. The reported peak responsivity is $\Delta\omega_{TMS}$7 for initial detuning $\Delta\omega_{TMS}$8, with a lower peak responsivity of about $\Delta\omega_{TMS}$9 for $\omega_1=\omega_{\text{CoFeB}}(V_P)$00, and sensitivity is enhanced near synchronization (Arumugam, 8 May 2026). Although the cavity supports a dominant TEM mode rather than a rich family of spatial modes, the work belongs to the same broader class of pulling phenomena in which an external signal drags an oscillator toward a common frequency.

Across applications, transverse-mode frequency pulling is not uniformly beneficial or harmful. In spintronics it provides an additional degree of freedom for manipulation of dynamical regimes and synchronization conditions in double-vortex oscillators, controlled by voltage and tunable by strain, with relevance to neuromorphic computing, spectrum analysis, and wireless communication (Iurchuk et al., 2024). In quantum optics it enables simultaneous squeezing of several transverse modes at one carrier frequency and phase-locked superpositions whose spatial structure is directly controllable (Chalopin et al., 2011, Zwilich et al., 2023). In semiconductor microlasers it can be minimized or predictably controlled by designing large intrinsic transverse spacing and weak cross saturation, which is important for stationary dual-mode emission and fiber-based comb generation (Weng et al., 2015). In magnetic resonance, ignoring pulling leads to incorrect inference of bare anisotropy parameters, and in gravitational-wave interferometry dynamic pulling must be included to model and suppress parametric instability correctly (Peria et al., 2020, Zhao et al., 2015).

Taken together, these results support a precise but broad definition: transverse-mode frequency pulling is the renormalization, locking, or modulation of transverse-mode resonance conditions by coupling, dispersion, geometry, or drive. What changes across platforms is the microscopic agent—magnetoelasticity, dipolar hybridization, two-magnon scattering, Gouy-phase engineering, mirror figure error, thermo-optic feedback, strong atom–cavity coupling, or injection dynamics—not the underlying fact that transverse modes acquire effective frequencies different from those of the uncoupled idealized system.