Towards altering sound frequency at will by a linear meta-layer with time-varying and quantized properties

The study addresses how to alter acoustic wave frequency in a linear, programmable manner using materials with time-varying properties. Conventional static metamaterials shape wavefronts and wavenumbers but leave frequency unchanged. Nonlinear approaches can generate harmonics for super-resolution imaging and non-reciprocity, yet require high excitation and produce amplitude-dependent, discrete frequency conversions. Active metamaterials add tunability but often need sensors and face stability challenges. Temporal (spacetime) modulation promises linear frequency control, but practical realization of efficient frequency conversion has been limited. The research question is whether a linear, sensorless, stable mechanism can efficiently convert sound frequency to arbitrary target bands via time-varying, quantized material properties. The work proposes an acoustic meta-layer (AML) that linearly redistributes energy from a source frequency to controllable sidebands, enabling conversions such as audible-to-infrasound/ultrasound and monochromatic-to-broadband (white noise). This capability could overcome limits like Rayleigh’s criterion and enable applications in imaging, energy transport, and communication.

Prior work in spatially modulated metamaterials (photonic/sonic crystals) controls wavenumber but not frequency. Nonlinear media (optical crystals, ultrasound contrast agents) enable harmonic generation used for super-resolution imaging, rectification, and directional radiation, but require strong excitation and yield amplitude-dependent, discrete harmonics. Active metamaterials can tune effective parameters and spectral response but may need sensors and risk instability. Temporal modulation, via moving media (Doppler-like biasing) or wave/modulation of local properties (impedance), achieves linear nonreciprocity and frequency shifting, but often requires long interaction lengths (on the order of ~10 wavelengths) or significant bias. Amplitude modulation circuits historically show sum/difference frequency generation, suggesting temporal materials should enable similar linear frequency conversion. However, achieving strong, programmable, and efficient acoustic frequency conversion without nonlinearity or sensors has remained challenging. The present work builds on shunted electromechanical systems and spacetime metamaterial concepts to realize a giant, quantized impedance modulation enabling efficient linear frequency scattering.

Device architecture: The acoustic meta-layer (AML) uses a moving-coil loudspeaker diaphragm shunted by an analog RLC circuit. A MOSFET, driven by a pre-defined gate voltage sequence g(t), connects (on) or disconnects (off) the shunt, switching the diaphragm’s effective acoustic impedance between two quantized states: Z_m (mechanical only) and Z_m + ΔZ, with ΔZ = (Bl)^2/Z where Z = R + iωL + 1/(iωC). When gate voltage exceeds threshold, the MOSFET on-resistance is ~4 mΩ (shunt loaded); off-state resistance is ~4.4 kΩ (unloaded). The gating can be periodic (harmonic) at f_m or pseudo-random, band-limited within [f1, f2]. Theoretical model: The coupled electromechanical-acoustic system is modeled by lumped-parameter equations: diaphragm dynamics (mass M, damping D, stiffness K) with radiation loading 2ρ0 c0 A on both sides, and shunt circuit current I interacting via force factor Bl. Time-varying resistance R(t) = R_on + R_off[1 − g(t)] produces temporal modulation. In frequency domain, modulation introduces a convolution term Î ⊗ G(ω), where G is the Fourier transform of the square-wave sequence 2g(t)−1, coupling responses across frequencies and producing sidebands at f_s ± n f_m (dominant fundamental orders). For random g(t), G is broadband, enabling wide spectral spreading. Numerical solutions are performed in time domain. Normalization and parameters: Time is normalized by √(K/M). A normalized electric charge variable is defined to recast the system into dimensionless form with key parameters: mechanical loading d_m = (D + 2ρ0 c0 A)/M, electrical damping d_e = R/M, magnetic coupling B_m = Bl^2 k_e/K, and electric spring constant k_e = (1/√(LC))/(√(K/M)) squared. The model separates MOSFET on/off states: on-state uses the full electromechanical system; off-state reduces to purely mechanical dynamics. State transitions reset current (I→0) while charge remains constant during off-state. Time marching uses matrix exponential solutions with eigen-decomposition for each interval. Experimental setup: The AML (based on a commercial loudspeaker with DC resistance adjusted via a negative impedance converter) is clamped in a one-dimensional impedance tube with long anechoic wedges upstream and downstream (3–3.5 m each). A side-branch source injects the incident tone; microphone pairs (8 cm spacing) are used on both sides to decompose incident/reflected and transmitted/downstream-reflected waves via transfer-function methods. Static acoustic impedance measurements characterize AML in MOSFET on/off states to quantify modulation ratio. Harmonic modulation experiments use incident tones (e.g., f_s = 135 Hz, 160 Hz) and gating at f_m (e.g., 75 Hz, 141 Hz). Random modulation uses a band-limited white sequence mapped to binary gate levels (V_g = 0 or 6 V) with f_m ∈ [50, 100] Hz and f_s = 135 Hz. Spectral analysis, spectrograms, and time-domain reconstructions quantify energy redistribution. Parametric and multi-device analysis: Time-domain simulations evaluate energy scattering efficiency α_c as a function of d_m and other parameters (d_e, k_e, Bl). Predictions explore performance improvements (e.g., α_c → 0.9 for low d_m). A framework for multiple AMLs in series includes time-delayed inter-device coupling via radiation terms and echo summations at intervals of 2ΔL/c0, enabling control of sideband emphasis (e.g., low/high-pass scattering) by phase differences in modulation sequences across devices.

- Giant impedance modulation: Static measurements show an acoustic impedance modulation ratio α_m = |Z_m + ΔZ|/|Z_m| up to 45 at 135 Hz, with elevated ratios across 30–500 Hz. This is at least two orders of magnitude higher than prior space-time modulation or optical biasing techniques (typical ~0.14–0.21 or 10^−4–10^−3, respectively). - Passband-to-stopband switching: With MOSFET on, the AML switches from acoustically soft to rigid in 92–184 Hz, reducing transmittance by ~5× compared to off-state, behaving like an instant phase-change material. - Harmonic modulation (f_s = 135 Hz, f_m = 75 Hz): Transmitted wave is well reconstructed by three components at f_s, f_− = 60 Hz, and f_+ = 210 Hz with normalized amplitudes 0.279, 0.184, 0.186, respectively. Energy at difference and sum bands relative to source are 43.6% and 44.4%; overall energy scattering efficiency α_c ≈ 1 − 1/(1+0.436+0.444) = 46.8%. Numerical predictions agree reasonably with measured spectra and time traces. - Parameter dependence: Measured point (α_c ≈ 0.468 at d_m ≈ 2.63) aligns with predictions. Simulations show α_c increases markedly as d_m → 0, reaching ~0.9 at d_m = 0.14. Increasing electrical damping d_e reduces α_c; reducing d_e slightly improves it. Changing electric spring constant (e.g., k_e = 2) can lower α_c relative to the calibrated value (~1.07). - Audible-to-infrasound conversion: Using f_s = 160 Hz and f_m = 141 Hz generates a strong infrasound component at f_− = 19 Hz downstream; the f_+ = 301 Hz component is largely absorbed by wedges. Decomposed upstream/downstream waves confirm dominance of 19 Hz in transmission and reflection characteristics. - Broadband applicability: With the same circuit parameters, the AML is effective for incidence from 40 to 640 Hz (four octaves), suggesting potential to convert broadband low-frequency noise into imperceptible infrasound (e.g., 19 Hz). - Predicted audible-to-ultrasound conversion: Time-domain modeling indicates f_s = 5 kHz with f_m = 25 kHz can produce ultrasounds at 20 kHz and 30 kHz, suggesting routes to super-resolution and depth-penetration trade-off mitigation in ultrasound imaging. - Random modulation (f_s = 135 Hz; f_m ∈ [50,100] Hz): Binary, band-limited random gating converts a monochromatic input to band-limited white noise at f_− ∈ [35,85] Hz and f_+ ∈ [185,235] Hz, with randomized component amplitude ~1.19 Pa versus residual tone 1.36 Pa (incident 4.56 Pa), corresponding to ~43.4% of transmitted energy in the randomized band and ~56.6% residual at f_s. Spectrograms show clear emergence of two broadband bands during modulation-on. - Leakage emission: With modulation on but no incident sound, small emissions (~28 dB) occur due to MOSFET gate leakage and grounding ripples, otherwise the AML does not self-radiate.

The AML demonstrates linear, sensorless, and programmable frequency conversion by temporally quantized impedance switching, directly addressing the need for efficient, amplitude-independent control of wave frequency. The giant modulation ratio (~45) enables substantial scattering of source energy into sidebands using a single thin layer, outperforming nonlinearity-based devices that require high excitation and yield discrete harmonics. Experiments verify both deterministic (harmonic) and stochastic (random) modulation modes: harmonic modulation produces strong sum/difference sidebands with nearly half the transmitted energy shifted, while randomized modulation converts a monochromatic tone into band-limited white noise, offering encryption and masking capabilities. The platform’s tunability across four octaves and capacity for near-audible to infrasound conversion demonstrate practical routes for inaudible, long-range energy transport and noise management. Modeling indicates performance can be further enhanced by reducing mechanical and electrical damping and tuning the electric spring constant, and by cascading multiple AMLs with controlled modulation phases to emphasize desired sidebands (low- or high-pass behavior) through interference. Extension to higher frequencies suggests potential to overcome trade-offs in medical ultrasound by frequency upconversion, enabling deep penetration with fine resolution through staged conversion. Overall, the results validate temporal-spacetime material concepts for frequency control and broaden the toolkit for linear wave manipulation.

The work introduces a linear acoustic meta-layer that uses MOSFET-controlled, time-quantized impedance to scatter energy from a source frequency into programmable sidebands with giant modulation efficiency. Key contributions include: (1) experimental realization of efficient linear frequency conversion with modulation ratios up to 45 and energy scattering approaching 50% for a single layer; (2) demonstration of audible-to-infrasound conversion and predicted audible-to-ultrasound conversion; (3) randomized modulation that transforms a monochromatic tone into band-limited white noise, enabling encryption and potential noise control; and (4) a modeling framework identifying parameter regimes (low mechanical/electrical damping, tuned electric spring) and multi-layer phase control for enhanced or selective scattering. Future work should focus on device scaling (e.g., MEMS implementations) to reach ultrasonic regimes without modal complications, cascading architectures for spectral shaping, integration as parametric gain media, and applications in super-resolution/holographic imaging, acoustic diodes, underwater communication, and Doppler deception.

- The ultrasonic conversion was not experimentally realized; it is supported by modeling and requires scaled-down devices (e.g., MEMS) to avoid diaphragm modal effects. - Anechoic wedges effectively absorb audible frequencies but are inadequate for infrasound, leading to residual reflections and measurement distortions at ~19 Hz. - Numerical models do not fully capture frequency-dependent diaphragm damping, resistor behavior, or finite anechoic terminations, contributing to discrepancies with measurements. - The tested diaphragm appears mechanically overloaded for optimal scattering; improved performance is predicted with lower mechanical/electrical damping and better parameter tuning. - Small unintended emissions (~28 dB) occur during modulation-on with no incident sound due to MOSFET gate leakage and grounding ripples; while minor, they indicate practical circuit considerations. - Only single-layer experiments were performed; multi-layer interference control is supported by theory/supplementary simulations but not yet experimentally validated.