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Robust acoustic pulling using chiral surface waves

We show that long-range and robust acoustic pulling can be achieved by using a pair of one-way chiral surface waves supported on the interface between two phononic crystals composed of spinning cylinders with equal but opposite spinning velocities embedded in water. When the chiral surface mode with a relative small Bloch wave vector is excited, the particle located in the interface waveguide will scatter the excited surface mode to another chiral surface mode with a greater Bloch wave vector, resulting in an acoustic pulling force, irrespective of the size and material of the particle. Thanks to the backscattering immunity of the chiral surface waves against local disorders, the particle can be pulled following a flexible trajectory as determined by the shape of the interface. As such, this new acoustic pulling scheme overcomes some of the limitations of the traditional acoustic pulling using structured beams, such as short pulling distances, straight-line type pulling and strong dependence on the scattering properties of the particle. Our work may also inspire the application of topological acoustics to acoustic manipulations.

The acoustic pulling force stems from the backward momentum gained by the particle from the acoustic wave during the scattering process. Due to the momentum conservation law, this requires that the scattered wave has a greater forward momentum than the incident one. To achieve pulling in free space, structured beams such as a Bessel beam with a very large cone angle [11][12][13][14][15] or multiple beams with giant obliquely incident angles [16,17], are required to minimize the forward momentum of the incident wave. The technical difficulty on realizing long-range Bessel beams and the finite beam waists of the obliquely incident beams limits the application scenarios of acoustic pulling in reality. The size, shape and material of the particle also need to be carefully customized to enhance the forward scattering and reduce the backward scattering simultaneously so that the forward momentum of the scattered wave can be large enough. Therefore, the free space acoustic pulling is not only limited to short-range in general but also dependent strongly on the scattering properties of the manipulated object.
In this paper, we propose a way to achieve long-range and robust acoustic pulling for arbitrary particles located in an interface waveguide sandwiched between two phononic crystals composed of oppositely spinning cylinders embedded in water. On the interface of the two phononic crystals, there are two topologically protected chiral surface states propagating in the same direction, the one-way propagation nature of the two surface states is independent of the shape of the interface as well as the local disorder. The channel sandwiched by the two phononic crystals can have sharp corners which do not induce the state switching of the two states, inside the channel the particle can be pulled towards the source irrespective of its size and material, following a trajectory confined by the channel. Although a similar idea of optical pulling using the photonic chiral surface waves [29] has been proposed recently, the acoustic pulling using the acoustic chiral surface waves has its unique advantage that it can be used to pull much larger particles with much stronger forces. Here, by rigorously calculating the acoustic forces using the closed-surface integral of the time-averaged radiation stress tensor [11][12][13], and quantitatively analyzing them with the response theory [30], we show that once the surface mode with the relatively small Bloch wave vector is excited, the longitudinal (along wave propagating direction) acoustic force acting on the particle located inside the channel is mostly negative. Moreover, the averaged longitudinal force for the particle moving over a lattice constant along the wave propagating direction is always negative. The negative nature of the averaged longitudinal force is guaranteed by the Bloch wave vector difference between the two surface modes. Therefore, for a particle made of any material, possessing any size, it can always be pulled backward along the interface waveguide by the surface waves due to the net pulling force.

II. Chiral surface states
The chiral surface states in classical systems have been predicted [31][32][33] and realized [34,35] in 2-dimensional magneto-optical photonic crystals by breaking time-reversal symmetry in analogy to the quantum Hall effect in electronic systems. The acoustic counterparts were proposed [36][37][38] in phononic crystals using circulating fluid coatings, where the circulating flows break the time-reversal symmetry. This acoustic chiral edge transport was observed in experiment very recently [39].
Following the idea of Ref. [36], we consider the phononic crystal consisting of spinning cylinders with the spinning angular frequency Ω arranged into a square lattice (lattice constant a ) in xy plane embedded in water with mass density 0  and sound speed 0 c . When the cylinders are static, the mass density and sound speed inside are  and c , respectively. A thin, sound permeable and unspinning shell is coated on each cylinder to avoid the direct contact between the cylinder and water, so that the spinning will not drive the water to move. Taking both the Doppler effect and Coriolis force into account, the nth order Mie coefficient of the spinning cylinder for a time-harmonic ( i t e   ) incident acoustic wave is given by [40,41] where n J and (1) n H are the Bessel function and Hankel function of the first kind, c r is the radius of   and c being the sound speed inside a static cylinder, and the auxiliary function is expressed as When 0   , Eq. (1) is reduced to the Mie coefficient of a static cylinder. The band dispersion for the lattice structure containing the spinning cylinders can be obtained by using the multiple scattering method and applying the periodic boundary conditions [42,43]. When the cylinders are static, a two-fold degeneracy at M point is enforced by the C4v and time-reversal symmetry, as shown in Fig. 1 The dispersion relations of surface states can be calculated using the multiple scattering method in conjunction with supercell calculations [44]. In calculations, the supercell contains one layer along x direction and 18 layers along y direction. In order to form a closed system with purely real eigen-frequencies, all the boundaries of the supercell are connected according to periodic boundary conditions. Cylinders in the upper 9 layers have spinning frequency  , while cylinders in the lower 9 layers have spinning frequency  . In addition, the upper 9 and lower 9 layers are further separated by a water channel of width 0.5a (therefore the dimension of the supercell is ), as shown in Fig. 2 To show that the surface states are backscattering immuned against local disorders, we did full wave simulations of the surface waves propagating on the interface and being scattered by a particle using multiple scattering technique. In simulations, 50 supercells in total are used, and an unspinning cylindrical particle is arbitrarily inside the water channel between the upper and lower phononic crystals, see Fig 2(c). To prevent the particle from entering the phononic crystals, we assume there are two hard walls (extreme thin and permeable to acoustic wave) placed at 0.575 y a   , respectively. Elastic collision occurs when the particle hits the hard walls.
Therefore, there is a 1.15 50 a a  water gap between the two phononic crystals. Two line sources with opposite initial phases are placed at 0.2 y a   , respectively, to excite the odd mode B, We can see that the surface wave can only propagate rightward indicating that the surface wave can only transport unidirectionally. Before the wave being scattered by the particle, the pressure field distribution is odd about the middle line of the water gap since only mode B is excited. After the surface wave is scattered by the particle, the odd mirror symmetry of the field distribution is broken because a portion of mode B is scattered into the even mode D. However, the field distribution on the left hand side of the particle retains the odd symmetry indicating the surface waves are not backward scattered.

III. Acoustic force calculation
For a time-harmonic incident acoustic wave, the time-averaged acoustic force acting on a particle submerged in water can be evaluated by integrating the stress tensor T  over a surface enclosing the particle [11][12][13], where the stress tensor is expressed as with p and u being the pressure and velocity field, and I  being a 2 by 2 identity matrix. Using the idea of Lorenz-Mie theory [45][46][47][48][49], the integral can be greatly simplified when the closed surface is a circle. Then the x and y components of the acoustic force can be obtained as (see details about the derivation in the appendix)

Re
(2 ), x n n n n n n n y n n n n n n n where, 0 p is the amplitude of the line source,  Fig. 2(c)), and the result is shown in Fig. 3(a).
The rectangular region has a length a and a width 0.45a. Because the system preserves the translational symmetry, the acoustic force acting on the particle will repeat when the particle moves a lattice constant a along x direction. In Fig. 3(a) Fig. 3

(b) by the black squares.
We can see that ( ) x s F y for every s y is negative. Thus although the particle may be subject to pushing forces at some locations, as a whole it will be accelerated leftward when it moves freely inside the gap. Even if the radius and the material component of the particle are changed, ( ) x s F y are still negative for all s y , as shown by the curves of red disks and blue triangles in Fig. 3(b). In the following, we will show that so long as the incident surface wave is purely composed of mode B, ( ) x s F y is negative irrespective of the particle's size and material.

IV. Interpretation using the response theory
The response theory was proposed previously to explain the optical force in multi-port photonic systems [30]. We apply this response theory to analyze the acoustic force in our system, as optical and acoustic forces can be described by the same mathematics.
According to the response theory adapted to our system, the longitudinal acoustic force acting on the particle fulfills where 1 x and 2 x are the starting and ending points for the Fourier transform. The intensity of the surface wave component is proportional to 2 | ( , ) | g k y at any given y.
In Fig. 4, we plotted | ( , ) | g k y at different vertical coordinates y for both modes B and D. As where i g and s g are the Fourier transforms of the incident and scattered fields respectively.
The phase and amplitude of the incident field acting on the particle will determine the amplitude and phase of the scattered field. If mode B is excited, the incident field intensity obeys even symmetry about the central vertical line of the supercell, see the inset of Fig. 5(a). Thus the intensities of modes B and D of the scattered wave are also of even symmetry about the line 0 s x  , as shown in Fig. 5(a). When the particle moves a lattice constant along x direction, the phase of the incident wave will increase by ,0 B k a . The particle is the source of the scattered waves. When the particle moves a lattice constant, the scattered waves decrease a phase of direction, as shown in Fig. 5(b). According to Eq. (8), the oscillating ofB s x  will contribute no net force over a lattice constant, while the monotonous decreasing ofD s x  will lead to a negative force component. Therefore, the net optical force along x direction is always negative.
In Fig. 5 (c) and (d), we compared the acoustic forces calculated by the rigorous expressions, Eq. The slight discrepancy between the two approaches is due to the neglect of contributions from the higher order plane wave components of modes B and D.

VI. Conclusion
In summary, using a pair of chiral surface waves supported on the interface between two phononic crystals with broken time-reversal symmetry, we can achieve a new type of acoustic pulling. This acoustic pulling mechanism has advantages that are absent in traditional optical pulling schemes, such as a long pulling distance, flexible pulling trajectory and is independent of the particle's size and material. The band structures, transporting and scattering properties of the chiral surface waves have been studied using the multiple scattering technique. The acoustic forces acting on a particle changing with the particle's location inside the interface waveguide have been calculated rigorously using the Lorenz-Mie formula and analyzed according to the response theory in conjunction with the symmetry analysis. When the incident chiral surface mode has a smaller Bloch wave vector, the excitation of the other chiral surface mode due to the scattering by the particle will result in an averaged pulling force acting on the particle. The particle can be pulled towards the source irrespective of its size and material. Owing to the backscattering immunity of the chiral surface waves against local disorders, the channel between the two phononic crystals does not need to be straight line. If the defect on the interface does not break the C4v symmetry [30], such as a corner with 90 o bending angle, mode transform cannot occur before the wave is scattered by the particle. So the particle can be pulled continuously in a channel that possesses an arbitrary number of right-angled bends. As there is no restriction on the length of the channel, a long pulling distance can be easily achieved. Our work shows that topological sound waves can be used to control particles more proficiently than ordinary acoustic waves.