Practical realization of a sub-λ/2 acoustic jet

Studies in optics and acoustics have employed metamaterial lenses to achieve sub-wavelength localization, e.g. a recently introduced concept called ‘acoustojet’ which in simulations localizes acoustic energy to a spot smaller than λ/2. However previous experimental results on the acoustojet have barely reached λ/2-wide localization. Here we show, by simulations and experiments, that a sub-λ/2 wide localization can be achieved by translating the concept of a photonic jet into the acoustic realm. We performed nano- to macroscale molecular dynamics (MD) and finite element method (FEM) simulations as well as macroscale experiments. We demonstrated that by choosing a suitable size cylindrical lens, and by selecting the speed-of-sound ratio between the lens material(s) and the surrounding medium, an acoustic jet (‘acoustic sheet’) is formed with a full width at half maximum (FWHM) less than λ/2. The results show, that the acoustojet approach can be experimentally realized with easy-to-manufacture acoustic lenses at the macroscale. MD simulations demonstrate that the concept can be extended to coherent phonons at nanoscale. Finally, our FEM simulations identify some micrometer size structures that could be realized in practice. Our results may contribute to starting a new era of super resolution acoustic imaging: We foresee that jet generating constructs can be readily manufactured, since suitable material combinations can be found from nanoscale to macroscale. Tight focusing of mechanical energy is highly desirable in e.g. electronics, materials science, medicine, biosciences, and energy harvesting.


FEM SIMULATIONS
The FEM simulations were modelled in 2D geometry, the actual geometries used in Fig. E1 are shown in Figs. S1 and S2. Two coupled physics interfaces were used in COMSOL Multiphysics® (v 5.2): the pressure acoustics module for liquids and the solid mechanics module for solid parts. The pressure pulse used in the experiments was approximated to be continuous in the simulations to permit using a frequency domain study instead of a transient study. The frequency domain approach was chosen since it is computationally more efficient than a full transient simulation, especially since we needed to run parametric sweeps across frequencies and geometries. The frequency domain simulations give the same result as the steady state solution for a sufficiently long transient simulation. Because our experiments were done with a pulsed acoustic signal in a finite geometric domain, the frequency domain simulation takes into account some echoes from the structure that are absent in the experimental results. The effect of these echoes was small enough that the frequency domain simulations give a sufficiently accurate representation of the experimentally measured intensity.
The basic structure for the models is shown in Fig. S1. Here the geometry is the one used for ethanol in the centimeter scale, same as Fig. S2a. The geometry consist of a solid, liquid filled tube ( Fig. S2a-b) or a solid cylinder structure (Fig. S2c-d), a surrounding medium that is either liquid (Fig. S2a- (Table S2), generic values 22,23 were used for the polystyrene cylinder (Table S1). The materials used in the micro scale simulations ( Fig. 1c-d, Fig. E1c-d) in metals were from COMSOL's material library (Table S1). A pressure boundary condition was used at the bottom edge of the geometry to produce the acoustic wave. In Fig

MOLECULAR DYNAMICS (MD) SIMULATIONS
The simulation system was built by creating a diamond slab of size of 210×4×210 (x×y×z) unit cells of 0.54 nm. The total number of atoms was 1.4×10 6 . A cylindrical metamaterial lens 28 nm in diameter featuring germanium atoms was located at the center of the system (Fig. S4). The rest of the system featured silicon atoms. The simulation system was prepared by minimizing the potential energy using the conjugate gradient method. All acoustic simulations were done using this system in which the number of atoms, the volume and the energy remained constant. Periodic boundary conditions were applied along the x and y directions and open boundary conditions along the z direction.
The interaction between the atoms was described by the Stillinger-Weber potential 24 . The parameter σ of the germanium part of the potential was set to the same value as for silicon to minimize the interfacial stresses. Simulations were performed using the LAMMPS 25 MD code. Speed of sound for silicon and germanium used in this work were determined from LAMMPS simulations to be 8328 m/s and 5092 m/s, respectively.
Data analysis and visualization was performed using Ovito 26 . The average hydrostatic pressure squared presented in Fig. 1c in this manuscript was calculated as the time average over 30 ps after the pulse initiation. This time was long enough for the entire pulse to pass through the Ge cylinder but short enough that the wave reflection from the open bottom surface did not affect the results.
Supplementary Figure 4 -Structure of the molecular dynamics simulation system. Atoms representing the medium (Si) are dark blue whereas atoms constituting the metamaterial lens (Ge) are cyan.
Instantaneous hydrostatic pressure is visualized in Fig. S5. The snap shot is taken 23 ps after wave initiation. A focusing of acoustic energy is seen.
Supplementary Figure 5 -MD simulated stress field. Hydrostatic pressure (not intensity) as obtained from MD simulations (in MPa) at 23 ps after the wave packet initiation. The germanium cylindrical lens is presented as a purple circle. Excitation frequency was 1 THz.

EXPERIMENTS
We measured the speed of sound (Fig. S6) and density (Fig. S7) of the materials at room temperature. A Panametrics 5072PR pulser launched an acoustic pulse through a Karl Deutsch S 24 HB 0,3-1,3piezoelectric transducer. The transducer was coupled to a beaker containing the liquid. We varied the distance that the acoustic wave traveled in the liquid by changing the volume of the fluid.
The density measurement was conducted by placing a beaker on top of a precision scale (Precisa 410AM-FR) and varying the volume of fluid in the beaker.
Supplementary Figure 6 -Speed of sound measurement results. The speed of sound is the slope of the linear fit (Table S1). The error bars correspond to twice the thickness of the beaker marks used in the speed of sound measurements. There were no repeats of this experiment.
Supplementary Figure 7 -Density measurement results. We measured the mass of different volumes of fluid with a precision scale. The fluid density is the slope of the linear fit (Table S2). The error bars correspond to the volume uncertainty described by the beaker manufacturer. There were no repeats of this experiment.

DATA ANALYSIS
From a single A-line (Fig. S11), we examined a time window of two periods. We did this to minimize the contribution of noise (Figs. S11 and S12). The time window was positioned in the middle of the five-cycle ballistic wave. The square of the voltage was integrated across the time window. We calculated this value for each point in x-z-plane to construct an intensity map (Figs. E2 and E3).
The time when the acoustic burst arrives to the hydrophone changes as the hydrophone moves further away from the cylinder. We compensated for this by adding = to the examined time window. Here is the speed of sound in the medium of study, and is the distance from the cylinder along the z-axis. While moving along the x-axis, the varying distance was not compensated for since the shift in time of the ballistic wave was barely noticeable.
Supplementary Figure 11 -A-line in ethanol experiment at x = 0 mm and z = 2 mm. The ballistic wave and first echo arising from the ethanol-air boundary are indicated. A two-cycle time window (orange line) from the center of the ballistic wave was examined to minimize the contribution of noise (Fig. S12).