Ultrasonic super-oscillation wave-packets with an acoustic meta-lens

The Schrödinger equation is a fundamental equation to describe the wave function of a quantum-mechanical system. The similar forms between the Schrödinger equation and the paraxial wave equation allow a paradigm shift from the quantum mechanics to classical fields, opening up a plethora of interesting phenomena including the optical super-oscillatory behavior. Here, we propose an ultrasonic meta-lens for generating super-oscillation acoustic wave-packets with different spatial momenta and then superimposing them to a diffraction-limit-broken spot, visually represented by the ring-shaped trapping of tiny particles. Moreover, based on the focused super-oscillation packets, we experimentally verify proof-of-concept super-resolution ultrasound imaging, opening up the arena of super-oscillation ultrasonics for advanced acoustic imaging, biomedical applications, and versatile far-field ultrasound control.


Supplementary Note 1 | Convolution and deconvolution
In mathematics, the convolution between the functions ( , ) f x y and ( , ) h x y is defined by , where * denotes the convolution operator, ( , ) f x y is the source function before the convolution process, ( , ) h x y is the convolutional interaction function and ( , ) g x y is the solution function after convolution. In experimental measurements, the functions h x y and ( , ) g x y are discretized into matrices. Then the Eq. (S1) can be expressed into where (

Supplementary Note 2 | Local wavenumber near the super-oscillation region
For a single-belt Fresnel zone plate, the intensity distribution at the focal region is Here we define n as the unit normal vector pointing away from the surface S. Then, we can obtain the continuity equation by applying the Gauss's theorem The second governing equation is the famous Navier-Stokes equation. It is a motion equation for the Eulerian velocity field that is directly related to the conservation of momentum density  v . In analogy with the mass conservation, the rate of momentum change is given by On the other hand, the momentum P can change both by the advection and by the action of forces obeying the Newton's second law. The forces can be divided into two parts which are the body force that acts on the interior of volume Ω (gravity) and the contact force that acts on the surface S of volume Ω (pressure and viscosity force). Thus, the change rate of momentum can also be written as body ( ) where body f describes an external body force density in the entire fluid body. The tensor ( )  v v denotes the advection of momentum  v into the volume Ω. p and   are the pressure and the viscosity stress tensor, respectively. Analyses show that the stress tensor   must be symmetric for small velocity gradients in microfluidics. The stress tensor satisfies the following equation where the first term relates to the dynamic shear viscosity  of incompressible fluid, and the second term appears when the compressibility-induced dilatational viscosity cannot be neglected.  is the ratio of the shear viscosity to the dilatational viscosity.
Then we can obtain the Navier-Stokes equation for Newtonian fluids by applying the In the end, we deduce the energy conservation equation in the studied system. It is derived by using an approach similar to the one employed to obtain the continuity equation and the Navier-Stokes equation. We consider the change rate of energy inside the volume Ω. The energy density is given by the sum of kinetic energy density 2 / 2 v  and the internal energy density  . On the other hand, the total energy E can change via the energy advection through the surface S of volume Ω, by friction forces acting on the surface as well as the thermal gradients at the surface.
Therefore, we obtain where T and th  represent the temperature and the thermal conductivity, respectively.

Supplementary Note 4 | The calculation of acoustic radiation force
In this work, we analyze the acoustic radiation force acting on a compressible, spherical, micrometer-sized particle of radius a suspended in a thermo-viscous fluid, where the wavelength of ultrasound  is much larger than the particle radius a, that is, a   . In this case, the micro-particle is treated as a weak point scatterer, which can be handled by the first-order scattering theory. An expression for the acoustic radiation force where n is the unitary normal vector pointing out of the particle surface S0(t) and out of the static enclosure surface S1. Finally, given that the time-averaged change rate of momentum is zero for the propagating ultrasound field, we thus obtain By taking the second-order perturbation and using the expansions v v , and 0 1 2        , the Eq. (S13) becomes where we assumed that the time average of the first-order component of the harmonic wave is zero. In the regions sufficiently far from the boundary layers, ultrasound waves propagate like free-space mechanical waves with little damping. Therefore, the viscous and thermal effects are negligible. In this case, Eq. (S14) can be solved by placing an arbitrary static surface enclosing the particle S1 in the far field, which is also valid in the near field by considering the thermo-viscous effect. In the far field, the governing equations in fluid are where the first-order perturbation equations are derived as follows After taking the time derivative t  of Eq. (S16a) and substituting it with Eq. (S16b), we can obtain the first-order wave equation of The time-averaged second-order perturbation of the governing equation takes the form  that is proportional to the square of particle volume (a 6 ) and is therefore negligible, (iii) in sc   that is proportional to particle volume (a 3 ) and dominantly contributes to rad F . By keeping only the mixed term in sc   and using the index notation, the i-th component of Eq.