Theory of the acoustic radiation force exerted on a sphere by standing and quasistanding zero-order Bessel beam tweezers of variable half-cone angles

A rigorous theory is developed to predict the radiation force (RF) exerted on a sphere immersed in an ideal fluid by a standing or quasistanding zero-order Bessel beam of different half-cone angles. A standing or a quasistanding acoustic field is the result of counter propagating 2 equal or unequal amplitude zero-order Bessel beams, respectively, along the same axis. Each Bessel beam is characterized by its halfcone angle β_l;l = 1, 2 of its plane wave components, such that β_l = 0 represents a plane wave. Analytical expressions of RF are derived for a homogeneous viscoelastic sphere chosen as an example. RF calculations for a polyethylene sphere immersed in water are performed. Particularly, the half-cone angle dependency on the RF is analyzed for standing and quasistanding waves. Changing the half-cone angle is equivalent to changing the beamwidth. Potential applications include particle manipulation in microfluidic lab-on-chips as well as in reduced gravity environments.     

Potential-well model in acoustic tweezers--comment

The production of acoustical vortices-based potential wells for particle trapping is not only restricted to the use of a Laguerre-Gaussian beam. Other useful types of vortex beams include an r-vortex beam, a non-diffracting high-order Bessel and Bessel-Gauss beam, a fractional (diffracting) high-order Bessel beam, a non-diffracting high-order Bessel beam of fractional type α, and a hypergeometric beam to name a few. Representative types of vortex beams are chosen here, but the examples are not exhaustive and additional categories of vortex beams may be reported and investigated. Expressions for the incident acoustic pressure field of various vortex beams are provided. The results should assist in the development of a multitude of vortex-based potential-well models for particle entrapment and manipulation.     

Off-axis scattering of an ultrasound bessel beam by a sphere

In this paper, the scattering of an ultrasound zero-order Bessel beam by a rigid sphere in the off-axis configuration is studied. The beam is described through the partial wave expansion. The beam-shape coefficients which represent the amplitude of each multipole mode of the partial wave expansion are computed by numerical quadrature. Calculations are presented for both near- and far-field off-axis scattering. The far-field scattering is examined in both Rayleigh and geometrical acoustic limits. Results demonstrate that the scattered pressure in the off-axis case may significantly deviate from that in the on-axis configuration. In addition, the directive pattern of the scattered pressure is highly dependent on the relative position of the beam to the sphere.     

Magnetic tracking of acoustic radiation force-induced micro-order displacement

The dynamic behavior of a rigid magnetic sphere induced by an acoustic radiation force was investigated. The sphere was suspended in water in a simple pendulum configuration. The drag force acting on the pendulum during its motion was considered to follow a modified Stokes law for a low Reynolds number, accounting for phenomena related to its oscillatory movement. Steady forces of long (a few seconds) and short (a few milliseconds) durations were used. The movement of the magnetic sphere was tracked using a magnetoresistive sensor. From the new equilibrium position of the sphere in response to the long-duration static radiation force, the amplitude of this force was estimated. To assess the water viscosity, the relaxation movement after the acoustic force had stopped was fitted to a harmonic-motion model. Based on the results for the acoustic force and water viscosity, a theoretical profile of the sphere's micro-order displacement as a function of time caused by short-duration acoustic radiation force agreed well with experimental results.     

Comparison of stress field forming methods for vibro-acoustography

Vibro-acoustography is a method that produces images of the acoustic response of a material to a localized harmonic motion generated by ultrasound radiation force. The low-frequency, oscillatory radiation force (e.g., 10 kHz) is produced by amplitude modulating a single ultrasound beam, or by interfering two beams of slightly different frequencies. Proper beam forming for the stress field of the probing ultrasound is very important because it determines the resolution of the imaging system. Three beam-forming geometries are studied: amplitude modulation, confocal, and x-focal. The amplitude of radiation force on a unit point target is calculated from the ultrasound energy density averaged over a short period of time. The profiles of radiation stress amplitude on the focal plane and on the beam axis are derived. The theory is validated by experiments using a small sphere as a point target. A laser vibrometer is used to measure the velocity of the sphere, which is proportional to the radiation stress exerted on the target as the transducer is scanned over the focal plane or along the beam axis. The measured velocity profiles match the theory. The theory and experimental technique may be useful in future transducer design for vibro-acoustography.     

Effects of multi-scattering on the performance of a single-beam acoustic manipulation device

The effects of multiple scattering on acoustic manipulation of spherical particles using helicoidal Bessel-beams are discussed. A closed-form analytical solution is developed to calculate the acoustic radiation force resulting from a Besselbeam on an acoustically reflective sphere, in the presence of an adjacent spherical particle, immersed in an unbounded fluid medium. The solution is based on the standard Fourier decomposition method and the effect of multi-scattering is taken into account using the addition theorem for spherical coordinates. Of particular interest here is the investigation of the effects of multiple scattering on the emergence of negative axial forces. To investigate the effects, the radiation force applied on the target particle resulting from a helicoidal Bessel-beam of different azimuthal indexes (m = 1 to 4), at different conical angles, is computed. Results are presented for soft and rigid spheres of various sizes, separated by a finite distance. Results have shown that the emergence of negative force regions is very sensitive to the level of cross-scattering between the particles. It has also been shown that in multiple scattering media, the negative axial force may occur at much smaller conical angles than previously reported for single particles, and that acoustic manipulation of soft spheres in such media may also become possible.     

Assessment of shear modulus of tissue using ultrasound radiation force acting on a spherical acoustic inhomogeneity

An ultrasound-based method to locally assess the shear modulus of a medium is reported. The proposed approach is based on the application of an impulse acoustic radiation force to an inhomogeneity in the medium and subsequent monitoring of the spatio-temporal response. In our experimental studies, a short pulse produced by a 1.5-MHz highly focused ultrasound transducer was used to initiate the motion of a rigid sphere embedded into an elastic medium. Another 25 MHz focused ultrasound transducer operating in pulse-echo mode was used to track the displacement of the sphere. The experiments were performed in gel phantoms with varying shear modulus to demonstrate the relationship between the displacement of the sphere and shear modulus of the surrounding medium. Because the magnitude of acoustic force applied to sphere depends on the acoustic material properties and, therefore, cannot be used to assess the absolute value of shear modulus, the temporal behavior of the displacement of the sphere was analyzed. The results of this study indicate that there is a strong correlation between the shear modulus of a medium and spatio-temporal characteristics of the motion of the rigid sphere embedded in this medium.     

Microparticle trapping in an ultrasonic Bessel beam

This paper describes an acoustic trap consisting of a multi-foci Fresnel lens on 127?μm thick lead zirconate titanate sheet. The multi-foci Fresnel lens was designed to have similar working mechanism to an Axicon lens and generates an acoustic Bessel beam, and has negative axial radiation force capable of trapping one or more microparticle(s). The fabricated acoustic tweezers trapped lipid particles ranging in diameter from 50 to 200?μm and microspheres ranging in diameter from 70 to 90?μm at a distance of 2 to 5?mm from the tweezers without any contact between the transducer and microparticles.     

Matrix method for acoustic levitation simulation

A matrix method is presented for simulating acoustic levitators. A typical acoustic levitator consists of an ultrasonic transducer and a reflector. The matrix method is used to determine the potential for acoustic radiation force that acts on a small sphere in the standing wave field produced by the levitator. The method is based on the Rayleigh integral and it takes into account the multiple reflections that occur between the transducer and the reflector. The potential for acoustic radiation force obtained by the matrix method is validated by comparing the matrix method results with those obtained by the finite element method when using an axisymmetric model of a single-axis acoustic levitator. After validation, the method is applied in the simulation of a noncontact manipulation system consisting of two 37.9-kHz Langevin-type transducers and a plane reflector. The manipulation system allows control of the horizontal position of a small levitated sphere from -6 mm to 6 mm, which is done by changing the phase difference between the two transducers. The horizontal position of the sphere predicted by the matrix method agrees with the horizontal positions measured experimentally with a charge-coupled device camera. The main advantage of the matrix method is that it allows simulation of non-symmetric acoustic levitators without requiring much computational effort.     

Mathieu Function Solutions for the Photoacoustic Effect in Two- and Three-Dimensional Structures and Optical Traps

The wave equation for the photoacoustic effect in a three-dimensional spherically symmetric, or two-dimensional structure where the compressibility or density varies sinusoidally in space reduces to an inhomogeneous Mathieu equation. As such, exact solutions for the photoacoustic pressure can be found in terms of either Mathieu functions, integer order Mathieu functions, or fractional order Mathieu functions, the last of these being of importance for problems pertaining to structures of finite dimensions. Here, frequency domain solutions are given for a spherical structure with material properties varying radially, and a two-dimensional structure with material variations in one direction. Solutions for the acoustic pressure are found that give closed form expressions for the resonance frequencies. It is also shown that Mathieu functions give solutions for the motion of an optically levitated sphere trapped in an intensity modulated, Gaussian laser beam. By determining the frequencies at which the motions of the sphere are largest, that is, where the Mathieu functions become unstable, it is shown that the trap can act to determine the radiation force relative to the gravitational force on the sphere.     

Axisymmetric analysis of a tube-type acoustic levitator by a finite element method

A finite element approach was taken for the study of the sound field and positioning force in a tube-type acoustic levitator. An axisymmetric model, where a rigid sphere is suspended on the tube axis, was introduced to model a cylindrical chamber of a levitation tube furnace. Distributions of velocity potential, magnitudes of positioning force, and resonance frequency shifts of the chamber due to the presence of the sphere were numerically estimated in relation to the sphere's position and diameter. Experiments were additionally made to compare with the simulation. The finite element method proved to be a useful tool for analyzing and designing the tube-type levitator.     

Radiation force on a nonlinear microsphere by a tightly focused Gaussian beam

We determine the characteristics of the radiation force that is exerted on a nonresonant nonlinear (Kerr-effect) rigid microsphere by a strongly focused Gaussian beam when diffraction and interference effects are significant (sphere radius a < or = illumination wavelength lambda). The average force is calculated from the surface integral of the energy-momentum tensor consisting of incident, scattered, and internal electromagnetic field vectors, which are expressed as multipole spherical-wave expansions. The refractive index of a Kerr microsphere is proportional to the internal field intensity, which is computed iteratively by the Rytov approximation (residual error of solution, 10(-30). The expansion coefficients for the field vectors are calculated from the approximated index value. Compared with that obtained in a dielectric (linear) microsphere in the same illumination conditions, we find that the force magnitude on the Kerr microsphere is larger and increases more rapidly with both a and the numerical aperture of the focusing objective. It also increases nonlinearly with the beam power unlike that of a linear sphere. The Kerr nonlinearity also leads to possible reversals of the force direction. The proposed technique is applicable to other types of weak optical nonlinearity.     

Production of high-order Bessel beams with a Mach-Zehnder interferometer

A new experimental setup is demonstrated to produce high-order Bessel beams. It is based on the field decomposition of the Bessel beam into its even and odd field components. The implementation is performed over the spectral components with a Mach-Zehnder interferometer that synthesizes the components into the desired Bessel beam. The main advantage of our setup is that the required annular transmittances have only discrete phase changes of pi radians instead of a continuous change of phase.     

Single Gaussian beam interaction with a Kerr microsphere: characteristics of the radiation force

We analyze the characteristics of the radiation force that is generated when a highly focused unpolarized Gaussian beam interacts with a nonabsorbing microsphere whose refractive index exhibits a first-order dependence on the beam intensity. The behavior of the force exerted on the sphere is analyzed as a function of beam power, axial distance, sphere radius, refractive-index difference between the sphere and the surrounding liquid, and wavelength. The force characteristics are compared with those of the radiation force that is generated when the electro-optic Kerr effect is absent. Our results show that a reversal in the net force direction is introduced when the Kerr effect becomes significant, which occurs at sufficiently high beam intensities.     

Liquid lens using acoustic radiation force

A liquid lens is proposed that uses acoustic radiation force with no mechanical moving parts. It consists of a cylindrical acrylic cell filled with two immiscible liquids (degassed water and silicone oil) and a concave ultrasound transducer. The focal point of the transducer is located on the oil-water interface, which functions as a lens. The acoustic radiation force is generated when there is a difference in the acoustic energy densities of different media. An acoustic standing wave was generated in the axial direction of the lens and the variation of the shape of the oil-water interface was observed by optical coherence tomography (OCT). The lens profile can be rapidly changed by varying the acoustic radiation force from the transducer. The kinematic viscosity of silicone oil was optimized to minimize the response times of the lens. Response times of 40 and 80 ms when switching ultrasonic radiation on and off were obtained with a kinematic viscosity of 200 cSt. The path of a laser beam transmitted through the lens was calculated by ray-tracing simulations based on the experimental results obtained by OCT. The transmitted laser beam could be focused by applying an input voltage. The liquid lens could be operated as a variable-focus lens by varying the input voltage.     

Propagation of high-order Bessel–Gaussian beams through a hard-aperture misaligned optical system

In this paper, we consider the propagation of high-order Bessel–Gaussian beams (HBGBs) passing through a hard-aperture misaligned optical system. By expanding the hard-aperture function into a finite sum of complex Gaussian functions, a general propagating formula of HBGBs is derived in terms of the generalized diffraction integrals. Based on the derived formula, the diffraction properties of HBGBs propagating through a simple misaligned lens system are numerically illustrated. This method provides a convenient tool for studying the propagation and transformation properties of a high-order Bessel–Gaussian beam through an apertured misaligned optical system.     

Simplified models for ultrasonic cleaning: wave incidence angle effects

We have previously studied simple axisymmetric finite element models for ultrasonic cleaning. In these models, we sought the magnitude of the particle removal force for a 1 GHz acoustic plane wave incident on a rigid 1 micron diameter sphere lying on a rigid plane. Now we wish to extend this work and determine the effect of the wave incidence angle on the horizontal and vertical sphere forces.By employing a Fourier decomposition of the ambient plane wave we reduce the three-dimensional problem to two two-dimensional problems. Calculation of the sphere forces for various wave incidence angles reveals that the magnitude of the vertical force is not significantly affected by scattering for angles below about 60 degrees. However, the horizontal force is a complicated function of the scattered and ambient wave solutions. In addition, we find that the horizontal force magnitude at 30 degrees incidence is almost as large as that at 60 degrees.     

Magnetic Interaction Force and a Couple on a Superconducting Sphere in an Arbitrary Dipole Field

The calculation of the magnetostatic potential and levitation force due to a point magnetic dipole placed in front of a superconducting sphere in the Meissner state is readdressed. Closed-form analytical expression for the scalar potential function that yields the image system for an arbitrarily oriented magnetic dipole located in the vicinity of a superconducting sphere is given. Analytic expression for the lifting or levitation force acting on the sphere is extracted from the solution for a general dipole. A special case of our expression where the initial magnetic dipole makes an angle with the z-axis is derived. Our expression for the force in this particular case shows that a recently obtained result (J. Supercond. Nov. Magn. 21:93–96, 2008) for an arbitrary dipole is incorrect. A brief discussion of another erroneous result (J. Supercond. Nov. Magn. 15:257–262, 2002) for a transverse/tangential dipole–sphere configuration, corrected elsewhere recently, is reproduced. Correct expressions for the interaction energy with some limiting cases are also provided. The result derived here demonstrates that the value of the levitation force for a dipole that makes an angle with z-axis lies between the values for a radial dipole–sphere and transverse dipole–sphere configurations providing upper and lower bounds. It is found that for a magnetic dipole making an angle with z-axis, there exits a second force component along the negative y-direction, which influences a couple acting on the superconducting sphere. It is also shown that the couple is proportional to the second force component and that both the couple and second force components vanish for a radial dipole–sphere and transverse dipole–sphere configurations, respectively. These results appear to be new and have not had received due attention in the context of superconductivity.     

Generalized theory of resonance excitation by sound scattering from an elastic spherical shell in a nonviscous fluid

This work presents the general theory of resonance scattering (GTRS) by an elastic spherical shell immersed in a nonviscous fluid and placed arbitrarily in an acoustic beam. The GTRS formulation is valid for a spherical shell of any size and material regardless of its location relative to the incident beam. It is shown here that the scattering coefficients derived for a spherical shell immersed in water and placed in an arbitrary beam equal those obtained for plane wave incidence. Numerical examples for an elastic shell placed in the field of acoustical Bessel beams of different types, namely, a zero-order Bessel beam and first-order Bessel vortex and trigonometric (nonvortex) beams are provided. The scattered pressure is expressed using a generalized partial-wave series expansion involving the beam-shape coefficients (BSCs), the scattering coefficients of the spherical shell, and the half-cone angle of the beam. The BSCs are evaluated using the numerical discrete spherical harmonics transform (DSHT). The far-field acoustic resonance scattering directivity diagrams are calculated for an albuminoidal shell immersed in water and filled with perfluoropropane gas, by subtracting an appropriate background from the total far-field form function. The properties related to the arbitrary scattering are analyzed and discussed. The results are of particular importance in acoustical scattering applications involving imaging and beam-forming for transducer design. Moreover, the GTRS method can be applied to investigate the scattering of any beam of arbitrary shape that satisfies the source-free Helmholtz equation, and the method can be readily adapted to viscoelastic spherical shells or spheres.     

Drop Tower Experiments and Numerical Modeling on the Combustion-Induced Secondary Flow in Standing Acoustic Fields

From the microgravity experiments on single fuel droplet combustion and on spark-ignited premixed flame in acoustic field, that have been done in this decade, secondary flow has been confirmed. The flow occurs only when combustion occurs and always in the direction to the neighboring node of velocity oscillation in standing acoustic field. This flow is thought to be driven by a kind of acoustic radiation force that arises from density non-uniformity due to combustion. According to the hypothesis, the driving force is expected to be very similar to buoyancy, being a volumetric force proportional to density difference. Since the mechanism tells that only the density difference due to combustion is the requirements for the occurrence of the secondary flow, a simpler system is employed for the hypothesis validation. A hot wire in a standing acoustic field gives a local temperature rise and the secondary flow is successfully reproduced. To avoid buoyancy that covers the radiation force, microgravity conditions are used. Through direct- and modeled numerical simulations and with drop tower experiments, the validity of the hypothesis is checked out. As a result, the supposed radiation force is found to be able to explain well the characteristics of the secondary flow. It is found that adding an acoustic radiation force term determined from acoustic parameters into conventional CFD calculation, one can sufficiently reproduce and predicts the behavior of the secondary flow numerically.