Contactless Resonant Ultrasound Spectroscopy

• Contactless RUS is a non-destructive technique that optically excites and detects mechanical resonances to infer elastic constants, internal friction, and interfacial stiffness without physical contact.
• It employs pulsed laser excitation and interferometric or vibrometric detection to capture eigenfrequencies, mode shapes, and damping in both bulk and plate-based regimes.
• Applications include phase transformation studies in materials, quality control in additive manufacturing, and nanometer-sensitive thickness measurements in multilayer micro-resonators.

Contactless resonant ultrasound spectroscopy (RUS) is a non-destructive spectroscopic methodology in which mechanical resonances are excited and detected without mechanical transducers or couplant, while the underlying inference remains the classical RUS problem: measured resonance frequencies, mode shapes, and peak widths are matched to a forward elastodynamic model under free-surface boundary conditions to recover elastic constants, internal friction, geometric parameters, or interfacial stiffnesses. In current practice, the term spans at least two experimentally distinct regimes: free-body resonance measurements on mm-scale solids using pulsed laser excitation and optical vibrometry, and localized resonance spectroscopy of guided-wave zero-group-velocity (ZGV) modes in plates and multilayers using fully optical excitation and interferometric detection (Bodnárová et al., 30 Jan 2025, Janovská et al., 15 Jul 2025, Ryzy et al., 14 Oct 2025, Mezil et al., 2014).

1. Definition and relation to classical resonant ultrasound spectroscopy

Classical RUS operates on a freely vibrating solid body, often a rectangular parallelepiped, that is lightly held between piezoelectric transducers. One transducer excites vibrations across a frequency sweep, the other detects the response, and the sample’s free-vibration eigenmodes appear as peaks in the spectrum. From those resonances, together with the known geometry and density, the elastic stiffness tensor is obtained by inverse fitting. Contactless RUS preserves this logic but removes mechanical contact from excitation and detection. In the laser-based implementations described for single-crystal NiTi and cold-sprayed nickel, the sample is nominally free-standing or freely suspended in a controlled atmosphere, is driven by a pulsed infrared laser, and is read out by a scanning laser vibrometer (Bodnárová et al., 30 Jan 2025, Janovská et al., 15 Jul 2025).

A central point is that contactless operation does not alter the governing eigenproblem. For a homogeneous elastic body, the displacement field satisfies

$\rho \,\ddot u_i = C_{ijkl}\, u_{k,lj},$

with traction-free boundary conditions on the external surface, and time-harmonic solutions lead to an eigenvalue problem whose resonance frequencies depend on the elastic constants, density, and geometry. In Rayleigh–Ritz form this becomes

$T \mathbf{a} = \rho\omega^2\,E\mathbf{a},$

or, for the orthonormal Zernike–Legendre basis developed for cylindrical samples, $E=I$ and the problem reduces to a standard symmetric eigenproblem (Akins et al., 17 Feb 2025). This corrects a common misconception: contactless RUS changes the means of excitation and observation, not the spectral mechanics that determine the resonances.

The broader resonant-spectroscopy family also includes plate and multilayer implementations based on ZGV Lamb-like modes. These systems do not probe global 3D normal modes of a free bulk body, but they retain the core RUS structure: sharp resonances are measured with high precision and inverted through a forward wave model to infer mechanical parameters. The ZGV-based interfacial-stiffness method for two bonded glass plates and the frequency-domain laser ultrasound microscopy method for multilayer BAW resonators are both explicit examples of this model-based, contactless resonant inversion paradigm (Mezil et al., 2014, Ryzy et al., 14 Oct 2025).

2. Experimental architectures

The instrumentation of contactless RUS depends on the scale of the structure and the type of resonance being exploited.

Regime	Excitation / detection	Principal observables
Free-body bulk RUS	Pulsed IR laser + scanning laser vibrometer	Eigenfrequencies, mode shapes, peak widths
Local ZGV spectroscopy in plates	Pulsed laser + interferometer	Local ZGV frequencies, Q, interface-sensitive shifts
Frequency-domain GHz spectroscopy	Modulated CW laser + Michelson interferometer + VNA	Local transfer function $U(f)$, ZGV frequency maps

In the NiTi single-crystal study, a fully contactless RUS was implemented with a pulsed Nd:YAG laser at $1.064\ \mu\text{m}$ and $8\ \text{ns}$ pulse duration for excitation, and a scanning laser Doppler vibrometer integrated in a microscope for detection. The sample, a rectangular cuboid of approximately $3.5 \times 3.2 \times 2.8\ \text{mm}^3$, was measured in a low-pressure nitrogen atmosphere of about $20\ \text{mbar}$ in a cryogenic chamber over $162$–$295\ \text{K}$, with resonances observed in the $T \mathbf{a} = \rho\omega^2\,E\mathbf{a},$0–$T \mathbf{a} = \rho\omega^2\,E\mathbf{a},$1 range (Bodnárová et al., 30 Jan 2025).

The cold-sprayed nickel study used the same contactless logic at elevated temperature. Small prismatic samples of approximately $T \mathbf{a} = \rho\omega^2\,E\mathbf{a},$2 were placed in an evacuated chamber with low-pressure $T \mathbf{a} = \rho\omega^2\,E\mathbf{a},$3 atmosphere of about $T \mathbf{a} = \rho\omega^2\,E\mathbf{a},$4, temperature control of $T \mathbf{a} = \rho\omega^2\,E\mathbf{a},$5, pulsed infrared laser excitation on the bottom face, and scanning laser vibrometry on the mirror-polished top face. FFT of the time-domain response yielded resonance spectra typically spanning $T \mathbf{a} = \rho\omega^2\,E\mathbf{a},$6–$T \mathbf{a} = \rho\omega^2\,E\mathbf{a},$7, and the same setup was used during heating and cooling up to $T \mathbf{a} = \rho\omega^2\,E\mathbf{a},$8 (Janovská et al., 15 Jul 2025).

At much higher frequency, frequency-domain laser ultrasound microscopy realizes a localized, fully optical resonant spectroscopy for multilayer micro-resonators. In that system, a $T \mathbf{a} = \rho\omega^2\,E\mathbf{a},$9 electro-absorption modulated diode laser amplified to about $E=I$0 CW is focused to a $E=I$1 spot and sinusoidally modulated up to about $E=I$2; a path-stabilized Michelson interferometer with a $E=I$3 CW Nd:YAG laser measures out-of-plane displacement, and a vector network analyzer records amplitude and phase at each drive frequency. Raster scanning with $E=I$4 lateral step size produces local spectra and resonance-frequency maps (Ryzy et al., 14 Oct 2025).

The ZGV Lamb-mode implementation for interfacial stiffness used a Q-switched Nd:YAG laser at $E=I$5, $E=I$6, and $E=I$7, with a $E=I$8 beam diameter, together with a $E=I$9 heterodyne interferometer. Two identical glass plates were held together laterally while leaving the bonded area essentially unloaded, and the long-lived local ZGV ringing was read out at the excitation spot up to $U(f)$0 (Mezil et al., 2014).

Across these architectures, the practical advantages are consistent: elimination of clamp stiffness, transducer mass loading, couplant instability, and friction; improved approximation to free boundary conditions; compatibility with cryogenic and high-temperature environments; and applicability to small samples, delicate martensitic microstructures, or multilayer devices (Bodnárová et al., 30 Jan 2025, Janovská et al., 15 Jul 2025, Ryzy et al., 14 Oct 2025).

3. Forward modeling, inversion, and dissipative observables

The inverse problem of RUS begins with a measured set of resonant frequencies $U(f)$1. A symmetry-constrained elastic tensor is chosen, predicted resonances $U(f)$2 are computed for the trial constants, and the constants are optimized to minimize the spectral mismatch. In the cold-sprayed nickel study the objective function is written explicitly as

$U(f)$3

after pairing experimental and calculated modes through measured mode shapes (Janovská et al., 15 Jul 2025). In the NiTi study, room-temperature austenite was fitted using $U(f)$4 resonant modes in the $U(f)$5–$U(f)$6 range, together with pulse-echo longitudinal velocities, to determine $U(f)$7, $U(f)$8, and $U(f)$9 with sub-GPa precision (Bodnárová et al., 30 Jan 2025).

Mode-shape information is especially important in contactless implementations because optical vibrometry can map the displacement field over a surface. In the NiTi and cold-sprayed nickel studies, this allows safe mode pairing between experiment and numerical eigenmodes; in the GHz multilayer case, the corresponding observable is the local acoustic transfer function $1.064\ \mu\text{m}$0, measured directly in the frequency domain and fitted near a dominant ZGV peak (Bodnárová et al., 30 Jan 2025, Janovská et al., 15 Jul 2025, Ryzy et al., 14 Oct 2025).

Peak widths provide access to anelasticity. For the nickel deposits, each resonance is fitted with a Lorentzian, the full width at half maximum is extracted, and an averaged internal friction parameter is defined as

$1.064\ \mu\text{m}$1

with $1.064\ \mu\text{m}$2 for the ten tracked modes (Janovská et al., 15 Jul 2025). The NiTi study likewise tracks internal friction through the widths of the resonance peaks, revealing strongly increased damping near the martensitic transformation (Bodnárová et al., 30 Jan 2025).

For cylindrical geometries, the theoretical work on asymptotic eigenfrequency behavior replaces monomial bases by Zernike polynomials on the circular cross section and Legendre polynomials axially. This choice yields an orthonormal basis and simplifies the generalized eigenproblem. A principal asymptotic result is

$1.064\ \mu\text{m}$3

where $1.064\ \mu\text{m}$4 and the constant depends on diagonal elastic constants and sample dimensions. The same study emphasizes that very high-order modes offer diminishing new information about off-diagonal elastic constants, which is directly relevant to experimental bandwidth selection and inversion weighting in contactless RUS (Akins et al., 17 Feb 2025).

4. Resonant phenomena and parameter sensitivities

In bulk single crystals with cubic symmetry, contactless RUS directly resolves anisotropic elastic combinations. The NiTi study defines the tetragonal shear modulus

$1.064\ \mu\text{m}$5

and the Zener anisotropy factor

$1.064\ \mu\text{m}$6

At $1.064\ \mu\text{m}$7, the measured austenitic constants were $1.064\ \mu\text{m}$8, $1.064\ \mu\text{m}$9, $8\ \text{ns}$0, $8\ \text{ns}$1, and $8\ \text{ns}$2. At $8\ \text{ns}$3, for self-accommodated martensite represented in effective cubic form, the values were $8\ \text{ns}$4, $8\ \text{ns}$5, $8\ \text{ns}$6, $8\ \text{ns}$7, and $8\ \text{ns}$8. These measurements document an anisotropy switch from $8\ \text{ns}$9 in austenite to $3.5 \times 3.2 \times 2.8\ \text{mm}^3$0 in martensite (Bodnárová et al., 30 Jan 2025).

In plate and multilayer structures, the dominant resonances can be ZGV modes rather than free-body modes. A ZGV point satisfies $3.5 \times 3.2 \times 2.8\ \text{mm}^3$1 at finite $3.5 \times 3.2 \times 2.8\ \text{mm}^3$2, so the group velocity vanishes while the phase velocity remains finite. The result is lateral energy confinement and sharp local resonances. In the multilayer BAW-resonator study, a spectral collocation method including electroelastic coupling identified multiple guided branches and a pronounced first ZGV resonance around $3.5 \times 3.2 \times 2.8\ \text{mm}^3$3. For small changes in the top SiN layer, the frequency–thickness relation was calibrated as

$3.5 \times 3.2 \times 2.8\ \text{mm}^3$4

with $3.5 \times 3.2 \times 2.8\ \text{mm}^3$5 and $3.5 \times 3.2 \times 2.8\ \text{mm}^3$6. Using local spectra extracted by narrowband sweeps with $3.5 \times 3.2 \times 2.8\ \text{mm}^3$7–$3.5 \times 3.2 \times 2.8\ \text{mm}^3$8 steps and quadratic peak fitting, the method resolved nominal thickness variations of $3.5 \times 3.2 \times 2.8\ \text{mm}^3$9, $20\ \text{mbar}$0, and $20\ \text{mbar}$1, indicating sub-nanometer depth sensitivity and micrometer-scale lateral resolution (Ryzy et al., 14 Oct 2025).

A closely related parameter sensitivity appears in bonded plate systems. For two identical glass plates coupled by a thin layer modeled as normal and tangential springs, the symmetric characteristic function depends only on the normal stiffness $20\ \text{mbar}$2, while the antisymmetric characteristic function depends only on the tangential stiffness $20\ \text{mbar}$3. This decoupling allows symmetric ZGV frequencies to determine $20\ \text{mbar}$4 and antisymmetric ZGV frequencies to determine $20\ \text{mbar}$5. For the tested bonding layers, the inferred stiffnesses were $20\ \text{mbar}$6, $20\ \text{mbar}$7, and $20\ \text{mbar}$8 in units of $20\ \text{mbar}$9 for water, oil, and salol, respectively; $162$0 was below $162$1 for water and oil and $162$2 for salol (Mezil et al., 2014).

These examples show that contactless RUS is not tied to a single observable. Depending on the geometry and forward model, the spectroscopic quantities may be full elastic tensors, effective symmetry classes, internal friction, layer thickness, or interface-specific normal and shear stiffnesses (Bodnárová et al., 30 Jan 2025, Ryzy et al., 14 Oct 2025, Mezil et al., 2014).

5. Representative materials studies

The NiTi single-crystal work is a canonical demonstration of contactless RUS for phase-transforming materials. The sample underwent the full B2 $162$3 B19$162$4 martensitic transformation between $162$5 and $162$6. Optical microscopy, EBSD, and EDS showed fine self-accommodated martensitic microstructures with TiC inclusions and Ti$162$7Ni precipitates around carbides. A key result was that temperature-induced B19$162$8 martensite, though monoclinic at the single-variant level, exhibits effectively cubic macroscopic elastic behavior when formed as a fine mixture of variants. The contactless spectra further showed pronounced non-linear softening of $162$9 before the transition, nearly complete vanishing of elastic anisotropy prior to transformation, strong damping near the transformation, and a low-temperature anisotropy that is consistent with mechanical instability of the B19$295\ \text{K}$0 lattice with respect to $295\ \text{K}$1 shear. To connect experiment and theory, the study used DFT-based single-variant elastic constants from Wagner & Windl and from Haskins & Lawson, transformed them, constructed a six-laminate self-accommodated microstructure, and obtained effective cubic constants by Hill homogenization that agreed well with the extrapolated experimental $295\ \text{K}$2 values (Bodnárová et al., 30 Jan 2025).

The cold-sprayed polycrystalline nickel study establishes contactless RUS as a probe of magneto-elastic softening, recrystallization, and residual stress in additively manufactured ferromagnetic materials. Two nearly isotropic deposits were analyzed by tracing ten resonant modes through the Curie point and extracting $295\ \text{K}$3, $295\ \text{K}$4, and $295\ \text{K}$5. The first ten modes were noted to be strongly dominated by the shear modulus, with frequency sensitivities to shear versus bulk modulus ranging from $295\ \text{K}$6 to $295\ \text{K}$7. Below $295\ \text{K}$8, magnetostriction-induced softening reduced the modulus when domain rotation was permitted; residual stress suppressed that effect; annealing relaxed stress and restored a large $295\ \text{K}$9 effect. In the annealed CS1 deposit, the relative drop in shear modulus between $T \mathbf{a} = \rho\omega^2\,E\mathbf{a},$00 and the minimum of $T \mathbf{a} = \rho\omega^2\,E\mathbf{a},$01 during cooling was about $T \mathbf{a} = \rho\omega^2\,E\mathbf{a},$02. The same spectra captured broad recrystallization-related damping peaks and showed that specimens cut from different depths in CS2 had essentially identical $T \mathbf{a} = \rho\omega^2\,E\mathbf{a},$03 and $T \mathbf{a} = \rho\omega^2\,E\mathbf{a},$04, indicating homogeneous residual stress across thickness (Janovská et al., 15 Jul 2025).

The aluminum dislocation-density study used a lightly contacting rather than fully contactless RUS configuration, but its theoretical content is directly transferable to contactless implementations because the spectral inversion and the dislocation–wave coupling do not depend on transducer contact. The work generalized Granato–Lücke theory and obtained

$T \mathbf{a} = \rho\omega^2\,E\mathbf{a},$05

linking changes in shear-wave speed to changes in the dimensionless dislocation parameter $T \mathbf{a} = \rho\omega^2\,E\mathbf{a},$06. Measured RUS-derived dislocation trends compared favorably with modified Williamson–Hall X-ray diffraction, and the study emphasized that the same framework can be transplanted directly into fully contactless RUS using laser or air-coupled excitation and optical detection (Mujica et al., 2012).

At the micro- and nanoscale, the fully optical frequency-domain ZGV approach extends contactless resonant spectroscopy to thin films and MEMS. The demonstration on solidly mounted BAW resonators with sub-$T \mathbf{a} = \rho\omega^2\,E\mathbf{a},$07 SiN height patterns showed that a single local ZGV resonance can serve as a quantitative probe of thickness. The paper also states that the broader spectrum contains multiple ZGV and other guided resonances and that a multi-parameter inversion could, in principle, recover several layer thicknesses, elastic constants, densities, and internal stresses, making the method a true localized GHz-frequency contactless RUS for layered devices (Ryzy et al., 14 Oct 2025).

6. Limitations, misconceptions, and methodological direction

A recurrent misconception is that contactless operation automatically guarantees easier interpretation. In practice, removal of transducer contact simplifies boundary conditions but shifts the burden to optical access, signal quality, and forward modeling. The laser-based bulk implementations require reflective or prepared surfaces for reliable vibrometry, and the excitation must remain non-ablative. High spectral quality is necessary not only to locate many modes but also to fit overlapping Lorentzians when damping becomes large, as occurred after recrystallization in nickel and near phase transformation in NiTi (Janovská et al., 15 Jul 2025, Bodnárová et al., 30 Jan 2025).

Another common simplification is to treat all contactless RUS as global free-body spectroscopy. ZGV-based methods contradict that reduction. Their resonances are local, laterally confined, and guided by layered dispersion relations rather than by the 3D normal modes of a finite body. This does not place them outside the resonant-ultrasonic framework; it means that the forward model changes from free-body eigenmodes to plate or multilayer dispersion with ZGV extrema. The interfacial-stiffness and FreDomLUS studies make this distinction explicit while preserving the core logic of resonant inversion (Mezil et al., 2014, Ryzy et al., 14 Oct 2025).

Theoretical limitations are equally important. The asymptotic cylinder analysis assumes a homogeneous single-phase material, linear elasticity, uniform density, negligible damping, and perfectly traction-free surfaces. It also shows that high-index modes become dominated by asymptotic spectral behavior controlled by diagonal elastic constants and geometry, so indiscriminate extension to ever higher frequencies does not necessarily improve identifiability of the full tensor (Akins et al., 17 Feb 2025). In multilayer ZGV imaging, the calibration factor $T \mathbf{a} = \rho\omega^2\,E\mathbf{a},$08 is mode- and structure-specific; different layers or larger thickness changes require recomputation, and simultaneous variation of several layers makes a single-mode inversion underdetermined (Ryzy et al., 14 Oct 2025).

Field environments introduce a different constraint. In the nickel magneto-elastic study, contactless RUS could not be used under applied magnetic field because magnetic forces would move the free-standing sample; the authors therefore reverted to conventional clamped RUS for $T \mathbf{a} = \rho\omega^2\,E\mathbf{a},$09 at room temperature (Janovská et al., 15 Jul 2025). This is a reminder that “contactless” is not intrinsically superior in every environment.

The methodological direction indicated by the current literature is therefore specific rather than generic: tighter integration of contactless measurement with full-wave forward solvers, symmetry analysis, DFT and homogenization where phase microstructures demand it, and multi-mode inversion where localized GHz resonances are available. In that sense, contactless RUS is best understood not as a single instrument class but as a family of resonance-based, model-driven, non-contact elastic spectroscopies spanning bulk crystals, ferromagnetic deposits, bonded interfaces, and multilayer micro-resonators (Bodnárová et al., 30 Jan 2025, Janovská et al., 15 Jul 2025, Ryzy et al., 14 Oct 2025, Akins et al., 17 Feb 2025).