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Correspondence High-Order Pseudo-Gaussian Scalar Acoustical Beams Farid G. Mitri, Member, IEEE Abstract—Exact solutions of the scalar Helmholtz equation describing tightly spherically-focused beams are introduced without any approximations using the complex source point method in spherical coordinates. The generalized solutions, valid for any integer degree n and order m, describe high-order pseudo-Gaussian vortex, intermediate (vortex), hollow (nonvortex), and trigonometric (non-vortex) beams having an arbitrary beam waist w0. A very useful property of these beams is the efficient and fast computational modeling of tightly focused or quasi-collimated wave-fronts depending on the dimensionless waist parameter kw0, where k is the wave number of the acoustical radiation. Examples that illustrate hollow vortex and non-vortex beams are provided, and numerical simulations for the magnitude, isosurface, and phase plots of the pressure wave field of higher-order quasi-Gaussian beams are evaluated with particular emphasis on kw0 for strongly (kw0 = 3) to weakly focused (i.e., quasi-collimated) beams (kw0 = 7). Potential applications are in beam-forming design, imaging, particle sizing and manipulation in acoustical tweezers, and phenomena related to scattering, radiation force, and torque.
Focused [1] Gaussian [2]–[4] beams are of particular importance in acoustic levitation [5], ultrasonic microscopy [6], high-intensity focused ultrasound (HIFU) [7] for the treatment of cancer, medical and nondestructive diagnostic imaging [8], and acoustical tweezers for particle manipulation [9]–[11], to name a few applications. Gaussian beams are paraxial solutions to the scalar wave equation; hence, they do not satisfy the Helmholtz equation [12]. When the wavefront becomes tightly focused (i.e., the beam waist radius is of the same order as the wavelength), it produces a beam profile that no longer remains Gaussian. Often, approximation schemes and higher-order corrections [13] are needed for beamforming design/simulation to accurately describe the propagating field. Those corrections can be computationally time-consuming and rather challenging to implement. Thus, it is of some importance to develop fast and numerically-efficient exact solutions without any approximations that can also be useful in scattering [14]–[17], which forms the basis of image formation, and the prediction of radiation forces [18]–[20] and torques [21] for particle manipulation and rotation in tweezers and acousto-fluidics [22], [23]. The objective of this correspondence is to shed light on generalized solutions of the wave equation using the complex source point (CSP) method [24]–[29], and introduce those solutions for potential research investigations in acoustics. Using the method of separation of variables in Manuscript received May 21, 2013; accepted October 1, 2013. F. G. Mitri is with Chevron-Area 52, Santa Fe, NM (e-mail: mitri@ chevron.com, [email protected]). DOI http://dx.doi.org/10.1109/TUFFC.2014.2890 0885–3010
complex spherical coordinates (R±, θ±, ϕ ) (Fig. 1), where the parameters are defined as follows; r = x 2 + y 2 + z 2, R± = x 2 + y 2 + Z ±2 , Z± = (z ± izR), zR = kw 02/2, where w0 is the beam waist and k is the wave number of the acoustic radiation, θ± = cos−1(Z±/R±) is the complex polar angle, and ϕ = tan−1(y/x) is the azimuthal angle, an exact high-order solution of positive degree n and negative or positive order m centered on a CSP, can be expressed as products of spherical Hankel functions of the first or second kind and associated Legendre functions with a complex exponential dependency on the azimuthal angle ϕ, i.e., [28, Eq. (22)]. However, such a solution is singular at the origin and may not be used to describe a physically realizable wave field. To circumvent the singularity, a sink is added to the CSP [30]. The final result of vortex nature is then expressed as a product of spherical Bessel functions of the first kind jn(·) and associated Legendre functions Pnm ( ⋅ ) with a complex exponential phase dependency on the azimuthal angle, and was given earlier in the optical context [31] for a scalar field. It is important to note here that the analysis with complex angles introduces a representation for the field’s characteristics, which allows determining the direction of the beam propagation as well as field attenuation interpretations. Moreover, the multi-valued complex distance function R± having branch point singularities on the circle defined by {x2 + y2 = z R2 , z = 0} should be transformed into a single-valued function by introducing a branch line (or cut) such that {x2 + y2 ≤ z R2 , z = 0}, for which the complex distance function is continuous at all points except the branch cut. The cost of this choice, however, is that the CSP beam solution which propagates along the ± z-direction, respectively, possesses a weak field component propagating backwards in the ∓ z-direction, respectively [32]. Therefore, the introduction of a sink along with this appropriate choice of the branch cut, makes the CSP solution free from singularities at R± = 0. The solution is termed here pseudo-Gaussian to make a clear distinction from the standard paraxial Gaussian solution that does not satisfy the wave equation. In the acoustical context, the incident time-varying pressure field of a high-order pseudo-Gaussian vortex beam of degree n and order |m| is expressed as
vortex PpG = P0e ±kz R j n(kR ±)Pnm (cos θ±)e i( m φ −ωt), (1) n, m
where the exponential exp (±kzR) is a normalization constant, P0 is a characteristic pressure amplitude, and the complex exponential exp (–iωt) denotes the time-dependence. This solution is of vortex type because the phase dependency on the azimuthal angle varies with exp (i|m|ϕ). Because (1) is an exact solution of the wave equation, an adequate superposition that remains a proper solution
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is also investigated. The high-order pseudo-Gaussian intermediate beam solution is regarded as a superposition of two high-order pseudo-Gaussian vortex beams, but with opposite order (or topological charge) and different amplitudes. Expressing the associated Legendre functions of negative order in terms of those of positive order using [33, Eq. (12.81), p. 772], the incident pressure field is expressed as 1 vortex vortex (P + PpG ), n,− m 2 pG n, m 1 (2) = P0e ±kz R j n(kR ±)Pnm (cos θ±) 2 × [e i m φ + (P1/P0)αn, m e −i m φ ]e −i ωt,
int PpG = n, m
where the real factor αn,|m| = (−1)|m|(n − |m|)!/(n + |m|)!, arises from the expression of the associated Legendre functions of negative order in terms of those of positive order, and P1 is the characteristic pressure amplitude of the second beam. It is also important to note that the weighting parameter (P1/P0)αn,|m| can be tailored to take any real (negative or positive) value so as to achieve a desired superposition with specific properties. Moreover, if an appropriate superposition of vortex beams is achieved such that the coefficient (P1/P0)αn,|m| = ±1, another solution for a beam that satisfies the Helmholtz equation can be obtained from (2) such that trig PpG = P0e ±kz R j n(kR ±)Pnm (cos θ±) n, m
mφ) e {icos( sin(mφ)}
−i ωt
, (3)
where the beam’s expression is proportional to the cosine (or the sine) function; thus, the solution is labeled a trigonometric beam. In the following, numerical simulations for the steadystate (time-independent) pressure magnitude illustrate the new solutions for two distinct values of the size parameter kw0, one corresponding to a tightly focused beam (kw0 = 3) and the other corresponding to a quasi-collimated beam (kw0 = 7). It is important to emphasize that the diffraction limit is reached for a beam waist w0 approaching half the wavelength λ/2 of the incident radiation (i.e., kw0 = π) [34]. Beams with kw0 < 3 are not directional [35] and may not be entirely physically realizable or useful. Below this limit, wave reflections at the aperture of the focused source generate exponentially decaying evanescent waves which become relatively important as kw0 → 0 [36]. An angular spectrum representation of acoustical fields yields a decomposition into homogeneous waves and evanescent waves that both satisfy the Helmholtz equation in free space [37, Section 2.9.1]. Because of its exponential distance dependence, the evanescent wave would yield infinite energy at a distance far away from its origin. Therefore, on physical grounds, the evanescent wave cannot exist in free-space and is restricted to the aperture of the acoustic source.
Fig. 1. Complex spherical coordinates system with the corresponding parameters defined in the text.
As discussed in [35], a small beam waist w0 may be achieved from a single hemispherical source, and further reduced by increasing the opening angle up to fully surround the focal (inner) region. For applications involving converging spherical waves, this may present physical restrictions because the required source amplitude becomes exponentially large as kw0 → 0. Nevertheless, hollow spherical transducers have been developed to emit/receive spherically diverging waves in the outer region for various applications [38]. In the present study, the focus is on propagating waves and disregards the evanescent contributions originating at the source. Magnitude, isosurface, and phase profile plots for the pressure wave field of higher-order quasi-Gaussian beams are evaluated using Matlab software (The MathWorks Inc., Natick, MA), with particular emphasis on the dimensionless beam waist parameter kw0 for strongly (kw0 = 3) to weakly focused beams (kw0 = 7). In the computations, the wave number is chosen to be k = 25 × 103 m−1, and the axial z and transverse (x, y) coordinates are varied by increments of δ(x, y, z) = 10−3 mm along each direction. Initially, a beam of 6th degree and zero order, i.e., (n, m) = (6, 0), is considered. This example is selected to illustrate the analysis of a high-order hollow pseudo-Gaussian beam that is not necessarily of vortex nature, because m = 0. Some properties of the fundamental (spherical) solution (0, 0) have already been investigated both in acoustics [35], [39] and optics [40], (see also the cylindrical solution counterpart [41]) and will not be reconsidered here. Figs. 2 and 3 show the axial magnitude (along z), isosurface, and cross-sectional magnitude and phase plots for kw0 = 3, and kw0 = 7, respectively. A very interesting feature of this pG6,0 beam is the hollow non-vortex nature for a tightly focused wave-front (Fig. 2); a characteristic that is
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Fig. 2. Comparison between the axial magnitude, isosurface, and crosssectional magnitude and phase plots for the pressure field of a pG6,0 beam for kw0 = 3, corresponding to a tightly focused wave-front. The units along the axes are in millimeters.
Fig. 3. The same as in Fig. 2, but kw0 = 7, corresponding to a quasicollimated beam. It is noted that the null in magnitude at the center of the beam (Fig. 2) is transformed into a maximum when kw0 increases. The units along the axes are in millimeters.
not observed for high-order Bessel vortex beams [42]. Typically, Bessel hollow beams have an axial null with vortex (i.e., helicoidal) nature, and this distinguishes them from the hollow non-vortex pseudo-Gaussian beam (6, 0). Comparison of the plots in Figs. 2 and 3 shows that the hollow pG6,0 beam is transformed into a quasi-collimated continuous beam with a maximum in magnitude at its center, when the parameter kw0 increases. This is a characteristic of the quasi-collimated beam. Plotting the spatial phase diagrams in the cross-sectional plane (x, y) is important to show the effects of the beam’s degree n and order m for which phase singularities/discontinuities can occur. To further explore the characteristics of high-order pseudo Gaussian beams of vortex type (i.e., m > 0), additional computations are performed for (n, m) = (1, 1) for a pG1,1 beam. Fig. 4 shows the axial magnitude, isosurface, and crosssectional magnitude and phase plots for a beam having kw0 = 3. The first and second columns correspond to a hollow vortex beam, computed using (1), the third and fourth columns correspond to an intermediate vortex beam computed using (2), and the fifth and sixth columns correspond to a trigonometric (non-vortex) beam computed using (3). The trigonometric beam for which α1,1 = −1 is realized by superposing two vortex beams with unit-
magnitude topological charge but with opposite handedtrig vortex ness and different amplitudes, i.e., PpG11 − , = 1/2(PpG 1,1 vortex PpG 1,−1 ). Although the phase profiles of both the vortex and intermediate beams show similarity of helicoidal nature, the spatial distributions of the pressure magnitudes are quite distinct. Moreover, the vortex beam clearly exhibits a doughnut shape as demonstrated by the isosurface plot (first row, second column), whereas the trigonometric beam clearly shows a null along the direction x = 0, which may be attributed to a geometric phase [41], [44]. When the beam waist parameter increases (Fig. 5), the beam becomes directional (in the broad sense) such that the intermediate pG1,1 beam still manifests itself as a vortex. In summary, high-order pseudo Gaussian beam solutions that exactly satisfy the Helmholtz equation are introduced and some of their intrinsic characteristics delineated for tightly focused and quasi-collimated acoustical wave-fronts. It is anticipated that the present analysis is of potential use in beam-forming design and the fast numerical simulation of tightly focused or quasi-collimated (vortex or non-vortex) beams, in contrast to the rather timely-consuming simulation programs (such as Abersim [45], Field II [46], FOCUS [47] or k-Wave [48]). Fast and
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Fig. 4. Comparison between the axial magnitude, isosurface, and cross-sectional magnitude and phase plots for the pressure field of a pG1,1 beam for kw0 = 3, corresponding to a tightly focused wave-front. The first and second columns correspond to a hollow vortex beam as computed using (1), the third and fourth columns correspond to an intermediate vortex beam computed using (2), and the fifth and sixth columns correspond to a trigonometric (non-vortex) beam computed using (3). The units along the axes are in millimeters.
effective numerical predictions/simulations allow appropriate design of experimental tools and development of novel devices operating with such beams including acoustical tweezers, HIFU, and nondestructive and medical imaging applications. Moreover, the results can be utilized to compute the beam-shape coefficients (BSCs) (See the procedure described by [17, Eqs. (20)–(28)]), which are the crux in the evaluation of the arbitrary acoustical scattering phenomenon that plays a major role in imaging applications, the acoustic radiation forces and torques for
Fig. 5. The same as in Fig. 4, but the size parameter kw0 = 7.
particle confinement and manipulation in 3-D, and possibly other areas in physical and biomedical acoustics using tightly focused vortex or non-vortex quasi-Gaussian beams. In addition, linear wave propagation has been assumed throughout this analysis. In some cases, however, acoustical beams often exhibit nonlinear wave-propagation effects [49] not accounted for in the present analysis. Nevertheless, it is important to note that the second-order (nonlinear) pressure depends on quadratic quantities that partly involve the linear (first-order) pressure field (see
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[50, after (III.4b)]). Therefore, the linear pressure field of a high-order pseudo-Gaussian acoustical beam can be used to advantage in evaluating the second-order pressure providing that an appropriate extension of the theory to include nonlinear effects is performed. Finally, the model equations presented here do not take into account dissipative effects and their contributions to the wave propagation phenomenon. Thermoviscous and streaming effects will affect the wave propagation and beam-forming unless the (fluid) medium of wave propagation has a negligible viscosity. A full theory including nonlinear wave propagation and the generation of harmonics as well as thermoviscous effects and acoustic streaming is beyond the scope of the present work and will be the subject of a future investigation. References [1] H. T. O’Neil, “Theory of focusing radiators,” J. Acoust. Soc. Am., vol. 21, no. 5, pp. 516–526, 1949. [2] J. J. Wen and M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am., vol. 83, no. 5, pp. 1752–1756, 1988. [3] D. Ding, Y. Shui, J. Lin, and D. Zhang, “A simple calculation approach for the second harmonic sound field generated by an arbitrary axial-symmetric source,” J. Acoust. Soc. Am., vol. 100, no. 2, pp. 727–733, 1996. [4] G. Du and M. A. Breazeale, “Theoretical description of a focused Gaussian ultrasonic beam in a nonlinear medium,” J. Acoust. Soc. Am., vol. 81, no. 1, pp. 51–57, 1987. [5] J. Wu and G. Du, “Acoustic radiation force on a small compressible sphere in a focused beam,” J. Acoust. Soc. Am., vol. 87, no. 3, pp. 997–1003, 1990. [6] R. A. Lemons and C. F. Quate, “Acoustic microscope—Scanning version,” Appl. Phys. Lett., vol. 24, no. 4, pp. 163–165, 1974. [7] J. E. Soneson and M. R. Myers, “Gaussian representation of highintensity focused ultrasound beams,” J. Acoust. Soc. Am., vol. 122, no. 5, pp. 2526–2531, 2007. [8] L. Schmerr and J.-S. Song, Ultrasonic Nondestructive Evaluation Systems: Models and Measurements. New York, NY: Springer, 2007. [9] J. Wu, “Acoustical tweezers,” J. Acoust. Soc. Am., vol. 89, no. 5, pp. 2140–2143, 1991. [10] J. Shi, D. Ahmed, X. Mao, S.-C. S. Lin, A. Lawit, and T. J. Huang, “Acoustic tweezers: Patterning cells and microparticles using standing surface acoustic waves (SSAW),” Lab Chip, vol. 9, no. 20, pp. 2890–2895, 2009. [11] K. H. Lam, H.-S. Hsu, Y. Li, C. Lee, A. Lin, Q. Zhou, E. S. Kim, and K. K. Shung, “Ultrahigh frequency lensless ultrasonic transducers for acoustic tweezers application,” Biotechnol. Bioeng., vol. 110, no. 3, pp. 881–886, 2013. [12] S. Nemoto, “Nonparaxial Gaussian beams,” Appl. Opt., vol. 29, no. 13, pp. 1940–1946, 1990. [13] J. P. Barton and D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys., vol. 66, no. 7, pp. 2800–2802, 1989. [14] F. G. Mitri and G. T. Silva, “Off-axial acoustic scattering of a highorder Bessel vortex beam by a rigid sphere,” Wave Motion, vol. 48, no. 5, pp. 392–400, Jul. 2011. [15] G. T. Silva, “Off-axis scattering of an ultrasound Bessel beam by a sphere,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control, vol. 58, no. 2, pp. 298–304, 2011. [16] F. G. Mitri, “Generalized theory of resonance excitation by sound scattering from an elastic spherical shell in a nonviscous fluid,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control, vol. 59, no. 8, pp. 1781–1790, 2012. [17] F. G. Mitri, “Arbitrary scattering of an acoustical high-order Bessel trigonometric (non-vortex) beam by a compressible soft fluid sphere,” Ultrasonics, vol. 53, no. 5, pp. 956–961, 2013.
195 [18] G. T. Silva, “An expression for the radiation force exerted by an acoustic beam with arbitrary wavefront (L),” J. Acoust. Soc. Am., vol. 130, no. 6, pp. 3541–3544, 2011. [19] G. T. Silva, J. H. Lopes, and F. G. Mitri, “Off-axial acoustic radiation force of repulsor and tractor Bessel beams on a sphere,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control, vol. 60, no. 6, pp. 1207–1212, 2013. [20] M. Azarpeyvand and M. Azarpeyvand, “Acoustic radiation force on a rigid cylinder in a focused Gaussian beam,” J. Sound Vibrat., vol. 332, no. 9, pp. 2338–2349, 2013. [21] G. T. Silva, T. P. Lobo, and F. G. Mitri, “Radiation torque produced by an arbitrary acoustic wave,” Europhys. Lett., vol. 97, no. 5, art. no. 54003, Mar. 2012. [22] H. Bruus, J. Dual, J. Hawkes, M. Hill, T. Laurell, J. Nilsson, S. Radel, S. Sadhal, and M. Wiklund, “Forthcoming Lab on a Chip tutorial series on acoustofluidics: Acoustofluidics-exploiting ultrasonic standing wave forces and acoustic streaming in microfluidic systems for cell and particle manipulation,” Lab Chip, vol. 11, no. 21, pp. 3579–3580, 2011. [23] J. Friend and L. Y. Yeo, “Microscale acoustofluidics: Microfluidics driven via acoustics and ultrasonics,” Rev. Mod. Phys., vol. 83, no. 2, pp. 647–704, 2011. [24] B. D. Seckler and J. B. Keller, “Geometrical theory of diffraction in inhomogeneous media,” J. Acoust. Soc. Am., vol. 31, no. 2, pp. 192–205, 1959. [25] G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett., vol. 7, no. 23, pp. 684–685, 1971. [26] J. B. Keller and W. Streifer, “Complex rays with an application to Gaussian beams,” J. Opt. Soc. Am., vol. 61, no. 1, pp. 40–43, 1971. [27] A. L. Cullen and P. K. Yu, “Complex source-point theory of the electromagnetic open resonator,” Proc. R. Soc. Lond. A, vol. 366, no. 1725, pp. 155–171, 1979. [28] M. Couture and P. A. Belanger, “From Gaussian beam to complexsource-point spherical wave,” Phys. Rev. A, vol. 24, no. 1, pp. 355– 359, 1981. [29] L. B. Felsen, “Geometrical theory of diffraction, evanescent waves, complex rays and Gaussian beams,” Geophys. J. R. Astron. Soc., vol. 79, no. 1, pp. 77–88, 1984. [30] M. V. Berry, “Evanescent and real waves in quantum billiards and Gaussian beams,” J. Phys. Math. Gen., vol. 27, no. 11, art. no. L391, 1994. [31] Z. Ulanowski and I. K. Ludlow, “Scalar field of nonparaxial Gaussian beams,” Opt. Lett., vol. 25, no. 24, pp. 1792–1794, 2000. [32] E. Heyman, B. Z. Steinberg, and L. B. Felsen, “Spectral analysis of focus wave modes,” J. Opt. Soc. Am., vol. 4, no. 11, pp. 2081–2091, 1987 [33] G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 6th ed., San Diego, CA: Academic, 2005. [34] A. E. Siegman, Lasers. Mill Valley, CA: University Science Books, 1986. [35] O. A. Sapozhnikov, “An exact solution to the Helmholtz equation for a quasi-Gaussian beam in the form of a superposition of two sources and sinks with complex coordinates,” Acoust. Phys., vol. 58, no. 1, pp. 41–47, 2012. [36] P. L. Marston, “Quasi-Gaussian beam analytical basis and comparison with an alternative approach (L),” J. Acoust. Soc. Am., vol. 130, no. 3, pp. 1091–1094, 2011. [37] E. G. Williams, “Plane waves,” in Fourier Acoustics, London, UK: Academic Press, 1999, ch. 2. [38] S. Alkoy, A. Dogan, A. C. Hladky, P. Langlet, J. K. Cochran, and N. E. Newnham, “Miniature piezoelectric hollow sphere transducers (BBs),” IEEE Trans. Ultrason. Ferroelectr. Freq. Control, vol. 44, no. 5, pp. 1067–1076, 1997. [39] F. G. Mitri, “Interaction of an acoustical Quasi-Gaussian beam with a rigid sphere: Linear axial scattering, instantaneous force, and time-averaged radiation force,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control, vol. 59, no. 10, pp. 2347–2351, 2012. [40] F. G. Mitri, “Quasi-Gaussian electromagnetic beams,” Phys. Rev. A, vol. 87, no. 3, art. no. 035804, 2013. [41] F. G. Mitri, “Cylindrical quasi-Gaussian beams,” Opt. Lett., vol. 38, no. 22, pp. 4727–4730, 2013. [42] F. G. Mitri, “Acoustic scattering of a high-order Bessel beam by an elastic sphere,” Ann. Phys., vol. 323, no. 11, pp. 2840–2850, Nov. 2008. [41] M. Berry, “Geometric phase memories,” Nat. Phys., vol. 6, no. 3, pp. 148–150, 2010.
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Correspondence High-Order Pseudo-Gaussian Scalar Acoustical Beams Farid G. Mitri, Member, IEEE Abstract—Exact solutions of the scalar Helmholtz equation describing tightly spherically-focused beams are introduced without any approximations using the complex source point method in spherical coordinates. The generalized solutions, valid for any integer degree n and order m, describe high-order pseudo-Gaussian vortex, intermediate (vortex), hollow (nonvortex), and trigonometric (non-vortex) beams having an arbitrary beam waist w0. A very useful property of these beams is the efficient and fast computational modeling of tightly focused or quasi-collimated wave-fronts depending on the dimensionless waist parameter kw0, where k is the wave number of the acoustical radiation. Examples that illustrate hollow vortex and non-vortex beams are provided, and numerical simulations for the magnitude, isosurface, and phase plots of the pressure wave field of higher-order quasi-Gaussian beams are evaluated with particular emphasis on kw0 for strongly (kw0 = 3) to weakly focused (i.e., quasi-collimated) beams (kw0 = 7). Potential applications are in beam-forming design, imaging, particle sizing and manipulation in acoustical tweezers, and phenomena related to scattering, radiation force, and torque.
Focused [1] Gaussian [2]–[4] beams are of particular importance in acoustic levitation [5], ultrasonic microscopy [6], high-intensity focused ultrasound (HIFU) [7] for the treatment of cancer, medical and nondestructive diagnostic imaging [8], and acoustical tweezers for particle manipulation [9]–[11], to name a few applications. Gaussian beams are paraxial solutions to the scalar wave equation; hence, they do not satisfy the Helmholtz equation [12]. When the wavefront becomes tightly focused (i.e., the beam waist radius is of the same order as the wavelength), it produces a beam profile that no longer remains Gaussian. Often, approximation schemes and higher-order corrections [13] are needed for beamforming design/simulation to accurately describe the propagating field. Those corrections can be computationally time-consuming and rather challenging to implement. Thus, it is of some importance to develop fast and numerically-efficient exact solutions without any approximations that can also be useful in scattering [14]–[17], which forms the basis of image formation, and the prediction of radiation forces [18]–[20] and torques [21] for particle manipulation and rotation in tweezers and acousto-fluidics [22], [23]. The objective of this correspondence is to shed light on generalized solutions of the wave equation using the complex source point (CSP) method [24]–[29], and introduce those solutions for potential research investigations in acoustics. Using the method of separation of variables in Manuscript received May 21, 2013; accepted October 1, 2013. F. G. Mitri is with Chevron-Area 52, Santa Fe, NM (e-mail: mitri@ chevron.com, [email protected]). DOI http://dx.doi.org/10.1109/TUFFC.2014.2890 0885–3010
complex spherical coordinates (R±, θ±, ϕ ) (Fig. 1), where the parameters are defined as follows; r = x 2 + y 2 + z 2, R± = x 2 + y 2 + Z ±2 , Z± = (z ± izR), zR = kw 02/2, where w0 is the beam waist and k is the wave number of the acoustic radiation, θ± = cos−1(Z±/R±) is the complex polar angle, and ϕ = tan−1(y/x) is the azimuthal angle, an exact high-order solution of positive degree n and negative or positive order m centered on a CSP, can be expressed as products of spherical Hankel functions of the first or second kind and associated Legendre functions with a complex exponential dependency on the azimuthal angle ϕ, i.e., [28, Eq. (22)]. However, such a solution is singular at the origin and may not be used to describe a physically realizable wave field. To circumvent the singularity, a sink is added to the CSP [30]. The final result of vortex nature is then expressed as a product of spherical Bessel functions of the first kind jn(·) and associated Legendre functions Pnm ( ⋅ ) with a complex exponential phase dependency on the azimuthal angle, and was given earlier in the optical context [31] for a scalar field. It is important to note here that the analysis with complex angles introduces a representation for the field’s characteristics, which allows determining the direction of the beam propagation as well as field attenuation interpretations. Moreover, the multi-valued complex distance function R± having branch point singularities on the circle defined by {x2 + y2 = z R2 , z = 0} should be transformed into a single-valued function by introducing a branch line (or cut) such that {x2 + y2 ≤ z R2 , z = 0}, for which the complex distance function is continuous at all points except the branch cut. The cost of this choice, however, is that the CSP beam solution which propagates along the ± z-direction, respectively, possesses a weak field component propagating backwards in the ∓ z-direction, respectively [32]. Therefore, the introduction of a sink along with this appropriate choice of the branch cut, makes the CSP solution free from singularities at R± = 0. The solution is termed here pseudo-Gaussian to make a clear distinction from the standard paraxial Gaussian solution that does not satisfy the wave equation. In the acoustical context, the incident time-varying pressure field of a high-order pseudo-Gaussian vortex beam of degree n and order |m| is expressed as
vortex PpG = P0e ±kz R j n(kR ±)Pnm (cos θ±)e i( m φ −ωt), (1) n, m
where the exponential exp (±kzR) is a normalization constant, P0 is a characteristic pressure amplitude, and the complex exponential exp (–iωt) denotes the time-dependence. This solution is of vortex type because the phase dependency on the azimuthal angle varies with exp (i|m|ϕ). Because (1) is an exact solution of the wave equation, an adequate superposition that remains a proper solution
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is also investigated. The high-order pseudo-Gaussian intermediate beam solution is regarded as a superposition of two high-order pseudo-Gaussian vortex beams, but with opposite order (or topological charge) and different amplitudes. Expressing the associated Legendre functions of negative order in terms of those of positive order using [33, Eq. (12.81), p. 772], the incident pressure field is expressed as 1 vortex vortex (P + PpG ), n,− m 2 pG n, m 1 (2) = P0e ±kz R j n(kR ±)Pnm (cos θ±) 2 × [e i m φ + (P1/P0)αn, m e −i m φ ]e −i ωt,
int PpG = n, m
where the real factor αn,|m| = (−1)|m|(n − |m|)!/(n + |m|)!, arises from the expression of the associated Legendre functions of negative order in terms of those of positive order, and P1 is the characteristic pressure amplitude of the second beam. It is also important to note that the weighting parameter (P1/P0)αn,|m| can be tailored to take any real (negative or positive) value so as to achieve a desired superposition with specific properties. Moreover, if an appropriate superposition of vortex beams is achieved such that the coefficient (P1/P0)αn,|m| = ±1, another solution for a beam that satisfies the Helmholtz equation can be obtained from (2) such that trig PpG = P0e ±kz R j n(kR ±)Pnm (cos θ±) n, m
mφ) e {icos( sin(mφ)}
−i ωt
, (3)
where the beam’s expression is proportional to the cosine (or the sine) function; thus, the solution is labeled a trigonometric beam. In the following, numerical simulations for the steadystate (time-independent) pressure magnitude illustrate the new solutions for two distinct values of the size parameter kw0, one corresponding to a tightly focused beam (kw0 = 3) and the other corresponding to a quasi-collimated beam (kw0 = 7). It is important to emphasize that the diffraction limit is reached for a beam waist w0 approaching half the wavelength λ/2 of the incident radiation (i.e., kw0 = π) [34]. Beams with kw0 < 3 are not directional [35] and may not be entirely physically realizable or useful. Below this limit, wave reflections at the aperture of the focused source generate exponentially decaying evanescent waves which become relatively important as kw0 → 0 [36]. An angular spectrum representation of acoustical fields yields a decomposition into homogeneous waves and evanescent waves that both satisfy the Helmholtz equation in free space [37, Section 2.9.1]. Because of its exponential distance dependence, the evanescent wave would yield infinite energy at a distance far away from its origin. Therefore, on physical grounds, the evanescent wave cannot exist in free-space and is restricted to the aperture of the acoustic source.
Fig. 1. Complex spherical coordinates system with the corresponding parameters defined in the text.
As discussed in [35], a small beam waist w0 may be achieved from a single hemispherical source, and further reduced by increasing the opening angle up to fully surround the focal (inner) region. For applications involving converging spherical waves, this may present physical restrictions because the required source amplitude becomes exponentially large as kw0 → 0. Nevertheless, hollow spherical transducers have been developed to emit/receive spherically diverging waves in the outer region for various applications [38]. In the present study, the focus is on propagating waves and disregards the evanescent contributions originating at the source. Magnitude, isosurface, and phase profile plots for the pressure wave field of higher-order quasi-Gaussian beams are evaluated using Matlab software (The MathWorks Inc., Natick, MA), with particular emphasis on the dimensionless beam waist parameter kw0 for strongly (kw0 = 3) to weakly focused beams (kw0 = 7). In the computations, the wave number is chosen to be k = 25 × 103 m−1, and the axial z and transverse (x, y) coordinates are varied by increments of δ(x, y, z) = 10−3 mm along each direction. Initially, a beam of 6th degree and zero order, i.e., (n, m) = (6, 0), is considered. This example is selected to illustrate the analysis of a high-order hollow pseudo-Gaussian beam that is not necessarily of vortex nature, because m = 0. Some properties of the fundamental (spherical) solution (0, 0) have already been investigated both in acoustics [35], [39] and optics [40], (see also the cylindrical solution counterpart [41]) and will not be reconsidered here. Figs. 2 and 3 show the axial magnitude (along z), isosurface, and cross-sectional magnitude and phase plots for kw0 = 3, and kw0 = 7, respectively. A very interesting feature of this pG6,0 beam is the hollow non-vortex nature for a tightly focused wave-front (Fig. 2); a characteristic that is
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Fig. 2. Comparison between the axial magnitude, isosurface, and crosssectional magnitude and phase plots for the pressure field of a pG6,0 beam for kw0 = 3, corresponding to a tightly focused wave-front. The units along the axes are in millimeters.
Fig. 3. The same as in Fig. 2, but kw0 = 7, corresponding to a quasicollimated beam. It is noted that the null in magnitude at the center of the beam (Fig. 2) is transformed into a maximum when kw0 increases. The units along the axes are in millimeters.
not observed for high-order Bessel vortex beams [42]. Typically, Bessel hollow beams have an axial null with vortex (i.e., helicoidal) nature, and this distinguishes them from the hollow non-vortex pseudo-Gaussian beam (6, 0). Comparison of the plots in Figs. 2 and 3 shows that the hollow pG6,0 beam is transformed into a quasi-collimated continuous beam with a maximum in magnitude at its center, when the parameter kw0 increases. This is a characteristic of the quasi-collimated beam. Plotting the spatial phase diagrams in the cross-sectional plane (x, y) is important to show the effects of the beam’s degree n and order m for which phase singularities/discontinuities can occur. To further explore the characteristics of high-order pseudo Gaussian beams of vortex type (i.e., m > 0), additional computations are performed for (n, m) = (1, 1) for a pG1,1 beam. Fig. 4 shows the axial magnitude, isosurface, and crosssectional magnitude and phase plots for a beam having kw0 = 3. The first and second columns correspond to a hollow vortex beam, computed using (1), the third and fourth columns correspond to an intermediate vortex beam computed using (2), and the fifth and sixth columns correspond to a trigonometric (non-vortex) beam computed using (3). The trigonometric beam for which α1,1 = −1 is realized by superposing two vortex beams with unit-
magnitude topological charge but with opposite handedtrig vortex ness and different amplitudes, i.e., PpG11 − , = 1/2(PpG 1,1 vortex PpG 1,−1 ). Although the phase profiles of both the vortex and intermediate beams show similarity of helicoidal nature, the spatial distributions of the pressure magnitudes are quite distinct. Moreover, the vortex beam clearly exhibits a doughnut shape as demonstrated by the isosurface plot (first row, second column), whereas the trigonometric beam clearly shows a null along the direction x = 0, which may be attributed to a geometric phase [41], [44]. When the beam waist parameter increases (Fig. 5), the beam becomes directional (in the broad sense) such that the intermediate pG1,1 beam still manifests itself as a vortex. In summary, high-order pseudo Gaussian beam solutions that exactly satisfy the Helmholtz equation are introduced and some of their intrinsic characteristics delineated for tightly focused and quasi-collimated acoustical wave-fronts. It is anticipated that the present analysis is of potential use in beam-forming design and the fast numerical simulation of tightly focused or quasi-collimated (vortex or non-vortex) beams, in contrast to the rather timely-consuming simulation programs (such as Abersim [45], Field II [46], FOCUS [47] or k-Wave [48]). Fast and
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Fig. 4. Comparison between the axial magnitude, isosurface, and cross-sectional magnitude and phase plots for the pressure field of a pG1,1 beam for kw0 = 3, corresponding to a tightly focused wave-front. The first and second columns correspond to a hollow vortex beam as computed using (1), the third and fourth columns correspond to an intermediate vortex beam computed using (2), and the fifth and sixth columns correspond to a trigonometric (non-vortex) beam computed using (3). The units along the axes are in millimeters.
effective numerical predictions/simulations allow appropriate design of experimental tools and development of novel devices operating with such beams including acoustical tweezers, HIFU, and nondestructive and medical imaging applications. Moreover, the results can be utilized to compute the beam-shape coefficients (BSCs) (See the procedure described by [17, Eqs. (20)–(28)]), which are the crux in the evaluation of the arbitrary acoustical scattering phenomenon that plays a major role in imaging applications, the acoustic radiation forces and torques for
Fig. 5. The same as in Fig. 4, but the size parameter kw0 = 7.
particle confinement and manipulation in 3-D, and possibly other areas in physical and biomedical acoustics using tightly focused vortex or non-vortex quasi-Gaussian beams. In addition, linear wave propagation has been assumed throughout this analysis. In some cases, however, acoustical beams often exhibit nonlinear wave-propagation effects [49] not accounted for in the present analysis. Nevertheless, it is important to note that the second-order (nonlinear) pressure depends on quadratic quantities that partly involve the linear (first-order) pressure field (see
mitri: high-order pseudo-gaussian scalar acoustical beams
[50, after (III.4b)]). Therefore, the linear pressure field of a high-order pseudo-Gaussian acoustical beam can be used to advantage in evaluating the second-order pressure providing that an appropriate extension of the theory to include nonlinear effects is performed. Finally, the model equations presented here do not take into account dissipative effects and their contributions to the wave propagation phenomenon. Thermoviscous and streaming effects will affect the wave propagation and beam-forming unless the (fluid) medium of wave propagation has a negligible viscosity. A full theory including nonlinear wave propagation and the generation of harmonics as well as thermoviscous effects and acoustic streaming is beyond the scope of the present work and will be the subject of a future investigation. References [1] H. T. O’Neil, “Theory of focusing radiators,” J. Acoust. Soc. Am., vol. 21, no. 5, pp. 516–526, 1949. [2] J. J. Wen and M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am., vol. 83, no. 5, pp. 1752–1756, 1988. [3] D. Ding, Y. Shui, J. Lin, and D. Zhang, “A simple calculation approach for the second harmonic sound field generated by an arbitrary axial-symmetric source,” J. Acoust. Soc. Am., vol. 100, no. 2, pp. 727–733, 1996. [4] G. Du and M. A. Breazeale, “Theoretical description of a focused Gaussian ultrasonic beam in a nonlinear medium,” J. Acoust. Soc. Am., vol. 81, no. 1, pp. 51–57, 1987. [5] J. Wu and G. Du, “Acoustic radiation force on a small compressible sphere in a focused beam,” J. Acoust. Soc. Am., vol. 87, no. 3, pp. 997–1003, 1990. [6] R. A. Lemons and C. F. Quate, “Acoustic microscope—Scanning version,” Appl. Phys. Lett., vol. 24, no. 4, pp. 163–165, 1974. [7] J. E. Soneson and M. R. Myers, “Gaussian representation of highintensity focused ultrasound beams,” J. Acoust. Soc. Am., vol. 122, no. 5, pp. 2526–2531, 2007. [8] L. Schmerr and J.-S. Song, Ultrasonic Nondestructive Evaluation Systems: Models and Measurements. New York, NY: Springer, 2007. [9] J. Wu, “Acoustical tweezers,” J. Acoust. Soc. Am., vol. 89, no. 5, pp. 2140–2143, 1991. [10] J. Shi, D. Ahmed, X. Mao, S.-C. S. Lin, A. Lawit, and T. J. Huang, “Acoustic tweezers: Patterning cells and microparticles using standing surface acoustic waves (SSAW),” Lab Chip, vol. 9, no. 20, pp. 2890–2895, 2009. [11] K. H. Lam, H.-S. Hsu, Y. Li, C. Lee, A. Lin, Q. Zhou, E. S. Kim, and K. K. Shung, “Ultrahigh frequency lensless ultrasonic transducers for acoustic tweezers application,” Biotechnol. Bioeng., vol. 110, no. 3, pp. 881–886, 2013. [12] S. Nemoto, “Nonparaxial Gaussian beams,” Appl. Opt., vol. 29, no. 13, pp. 1940–1946, 1990. [13] J. P. Barton and D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys., vol. 66, no. 7, pp. 2800–2802, 1989. [14] F. G. Mitri and G. T. Silva, “Off-axial acoustic scattering of a highorder Bessel vortex beam by a rigid sphere,” Wave Motion, vol. 48, no. 5, pp. 392–400, Jul. 2011. [15] G. T. Silva, “Off-axis scattering of an ultrasound Bessel beam by a sphere,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control, vol. 58, no. 2, pp. 298–304, 2011. [16] F. G. Mitri, “Generalized theory of resonance excitation by sound scattering from an elastic spherical shell in a nonviscous fluid,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control, vol. 59, no. 8, pp. 1781–1790, 2012. [17] F. G. Mitri, “Arbitrary scattering of an acoustical high-order Bessel trigonometric (non-vortex) beam by a compressible soft fluid sphere,” Ultrasonics, vol. 53, no. 5, pp. 956–961, 2013.
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