Guided TR Acoustic Modes in Fibers

• Guided torsional-radial acoustic modes are vectorial eigenmodes exhibiting coupled radial and torsional motions, fundamental for forward Brillouin scattering.
• Numerical and experimental methods solve the elastodynamic equations with stress-free boundary conditions to accurately characterize mode profiles, resonance frequencies, and Brillouin gains.
• Engineered TR modes enhance opto-acoustic coupling in fiber-optomechanical and quantum photonic devices, enabling precise RF oscillators and improved device calibration.

Guided torsional-radial acoustic modes ("TR modes," Editor's term) are vectorial eigenmodes of elastic wave propagation in cylindrical fibers and rods, exhibiting coupled radial and torsional motion. They are essential in forward Brillouin and guided acoustic wave processes in optical fibers, mediating resonant opto-acoustic interactions at MHz–GHz frequencies. These modes are characterized mathematically by solutions to the elastodynamic wave equation in isotropic media with cylindrical symmetry, subject to traction-free boundary conditions on the fiber surface. TR mode spectral features, spatial profiles, and coupling strengths determine the efficiency and selectivity of phonon-induced effects in fiber-based optomechanical and quantum photonic devices.

1. Elastodynamic Theory and Characteristic Equations

Guided TR modes arise from the vector Navier–Cauchy equation for the displacement field $u(r,\theta,z,t)$ in an isotropic, homogeneous cylinder of radius $a$, density $\rho$, and Lamé constants $\lambda$, $\mu$ (Kikuchi et al., 10 Jan 2026, Dostart et al., 2017):

$$\rho\,\frac{\partial^2 u}{\partial t^2} = (\lambda + 2\mu) \nabla\left(\nabla\cdot u\right) - \mu \nabla\times\left(\nabla\times u\right).$$

In cylindrical coordinates, harmonic TR modes are written as

$$u(r, \theta, t) = [u_r(r)\,e_r + u_\theta(r)\,e_\theta]\,e^{i(l\theta - \Omega t)},$$

where $l$ is the azimuthal order. Substitution yields coupled ODEs for $u_r(r)$ and $u_\theta(r)$.

Stress-free boundary conditions on the fiber surface $r=a$ require both the normal and shear tractions to vanish:

$\sigma_{rr}(a) = 0,\quad \sigma_{r\theta}(a) = 0.$

The modal eigenfrequencies $\Omega_{l,m}$ for TR modes are found by solving the transcendental determinant equation:

$$\det \begin{pmatrix} J_l(k_La) & J_l(k_Ta) \ k_La J'_l(k_La) & k_Ta J'_l(k_Ta) \end{pmatrix} = 0$$

where $J_l$ is the Bessel function of order $l$, $k_L = \Omega/v_L$, $k_T = \Omega/v_T$, with $v_L$ and $v_T$ the longitudinal and transverse sound velocities, respectively.

2. Mode Classification, Spatial Profiles, and Dispersion

TR mode families are indexed as $TR_{l,m}$ where $l$ determines azimuthal symmetry (number of nodal diameters) and $m$ the radial order (nodal circles) (Kikuchi et al., 10 Jan 2026):

• For $l=2$, modes couple intramodally (e.g., LP$_{01}\to$LP$_{01}$ forward Brillouin process).
• For $l=1$ or $l=3$, modes mediate intermodal coupling (e.g., LP$_{11}\to$LP$_{01}$).

A representative displacement field decomposition:

$$u_r(r) = A_{l,m}\left[J_l(k_T r) - \frac{2\mu k_T^2}{(\lambda+2\mu)k_L^2} J_l(k_L r)\right]$$

$u_\theta(r) = B_{l,m} J_l(k_T r)$

Orthonormality and $\sigma_{r\theta}(r=a)=0$ fully determine $A_{l,m}$, $B_{l,m}$.

The eigenfrequencies $\Omega_{l,m}$ are roots of the characteristic equation and yield a discrete spectrum of narrowband resonances with both shear and longitudinal character; e.g., $TR_{2,9}$ (shear) at 289 MHz, $TR_{2,10}$ (longitudinal) at 510 MHz in few-mode optical fiber (Kikuchi et al., 10 Jan 2026).

3. Numerical Modal Analysis for Acoustic Waveguides

Numerical solution of TR modes in acoustic waveguides utilizes the vector elastic wave eigenvalue problem (Dostart et al., 2017):

$$\rho(\mathbf{r})\,\partial_t^2 \mathbf{u}(\mathbf{r},t) = \nabla\cdot[\mathbb{C}(\mathbf{r}):\nabla_s \mathbf{u}(\mathbf{r},t)],$$

where $\mathbb{C}$ is the stiffness tensor. Staggered-grid finite-difference discretization assembles a sparse Hermitian operator, with boundary conditions enabling simulation of free, fixed, symmetry, and anti-symmetry surfaces. Perfectly matched layers (PMLs) facilitate modeling of leaky (radiative) modes.

For cylindrical rods:

1. Circular grid, isotropic elastic constants within rod.
2. Free boundary enforced ($\sigma_{nn}=0, \sigma_{nt}=0$).
3. Eigenvalue solver finds $\omega^2$ and modes.
4. Validation against analytic solutions yields agreement to $<1\%$ for mode shapes and dispersion.

This modal solver framework supports mode identification in complex waveguide geometries and allows direct computation of $u_r$, $u_\theta$, $u_z$ in physical space.

4. Experimental Observation and Spectral Characterization

Forward TR modes in optical fibers are interrogated via spontaneous Brillouin scattering using heterodyne detection (Kikuchi et al., 10 Jan 2026). The setup:

• Narrow-linewidth laser (e.g., 1064 nm) is split into pump and local-oscillator arms.
• Pump is launched in the desired fiber spatial mode (LP$_{01}$, LP$_{11}$) using SLMs.
• Forward-scattered light is mode-selected via polarization or spatial filtering.
• Scattered Stokes and anti-Stokes components heterodyne with the LO on a fast photodiode, producing symmetric peaks at $\pm\Omega_{l,m}$ in the electrical spectrum.

Measured and theoretical values for prominent modes:

Acoustic Mode	Theory $g_B$ (W$^{-1}$km$^{-1}$)	Measured $g_B$	Theory $\Omega/2\pi$ (MHz)	Measured $\Omega/2\pi$	$\Gamma/2\pi$ (MHz)
TR$_{2,9}$ (S)	0.94	2.0 ± 0.5	289.0	289	1.1
TR$_{2,10}$ (L)	0.25	0.28 ± 0.08	510.1	511	2.7
TR$_{1,14}$ (S)	0.27	0.46 ± 0.14	423.1	423	1.2
TR$_{1,14}$ (L)	1.66	0.52 ± 0.13	677.2	677	2.6

Resonances exhibit line-widths in the 1–3 MHz range and fiber Brillouin gain coefficients up to 2 W$^{-1}$km$^{-1}$.

5. Opto-Acoustic Coupling and Nonlinear Gain Mechanisms

The Brillouin gain coefficient $G_B$ for a TR acoustic mode $u(r)$ is quantified by the spatial overlap of optical pump and scattered fields through the photoelastic perturbation $\Delta\epsilon$ (Kikuchi et al., 10 Jan 2026):

$$G_B = \frac{k Q^2}{8 n^2 c \rho \Gamma \Omega_{l,m}}$$

where $Q = \int u\cdot f\,dA$ and $f(r)$ is the electrostriction force density.

In hybrid opto-electro-mechanical oscillator implementations, a CW optical pump excites acoustic TR modes via electrostriction. The generated strain field modulates the probe phase via the photoelastic effect, which is read out interferometrically and fed back to drive sustained RF oscillations. In single-mode fibers, axisymmetric radial modes dominate, but the same formalism applies to TR$_{l,m}$ with nonzero $u_\theta$ and off-diagonal stress.

6. Applications and Relevance in Fiber Optomechanics

Guided TR acoustic modes underpin advanced forward Brillouin and guided acoustic wave Brillouin scattering (GAWBS) processes:

• Deterministic RF oscillators exploiting single-mode resonance (e.g., R$_{0,7}$ mode at 319 MHz, Q ≈ 66, side-mode suppression >40 dB) (London et al., 2018).
• Spectrally selective phonon interactions for quantum photonic applications.
• Fundamental references for calibrating optomechanical gain and spatial mode overlap in few-mode and multi-core fiber systems.

A plausible implication is that engineered spatial overlap and stress-tuning enable controlled selection of TR resonance frequencies and gain for specific optomechanical tasks.

7. Methodological Advances and Simulation Frameworks

Recent numerical elastodynamic solvers provide a robust framework for simulating TR modes in arbitrary cross-section waveguides (Dostart et al., 2017):

• Sparse Hermitian (or complex symmetric) matrix assembly with staggered-grid FDTD analogs.
• Efficient implementation of symmetry, anti-symmetry, and traction-free boundary conditions.
• PML incorporation for leaky mode analysis.

Convergence, accuracy, and cross-validation against analytical and FEM benchmarks demonstrate correct recovery of TR, radial, shear, and longitudinal mode spectra and spatial profiles. The solver paradigm directly parallels electromagnetic eigenmode techniques, streamlining integration into optomechanical device design workflows.

This comprehensive overview delineates the theoretical foundations, numerical formulation, experimental validation, and opto-acoustic significance of guided torsional-radial acoustic modes in fiber-based structures, with reference to state-of-the-art preprints and implemented methods (London et al., 2018, Kikuchi et al., 10 Jan 2026, Dostart et al., 2017).