Quasi-scaling of the extinction efficiency of spheres in high frequency Bessel beams (L) Philip L. Marstona) Department of Physics and Astronomy, Washington State University, Pullman, Washington 99164-2814
(Received 26 December 2013; revised 14 February 2014; accepted 3 March 2014) The extinction efficiency Qext of a sphere is defined to be the ratio of its extinction cross section to the area of its profile. For non-dissipative situations Qext reduces to the scattering efficiency. For a sphere centered on the axis of a Bessel beam Qext has a complicated dependence on dimensionless frequency ka and on the conic angle b of the beam, even in the simple case of a rigid sphere. With appropriate scaling using Babinet’s principle, however, the dependence reduces approximately to a function of ka sinðbÞ when ka is large. An example is also shown for a metal shell in water. C 2014 Acoustical Society of America. [http://dx.doi.org/10.1121/1.4868399] V PACS number(s): 43.20.Fn, 43.40.Fz, 43.25.Qp [ANN]
I. INTRODUCTION
During the 20th century analytical investigations of the scattering of sound by spheres in acoustic beams tended to be limited to low frequencies or to situations relying on high frequency approximations.1–3 Subsequently, simple modifications to the ordinary partial wave series for the scattering by a sphere in a plane wave were found for spheres placed on the axis of ordinary (zero-order) Bessel beams or on the axis of vortex (or higher-order) Bessel beams.4–6 One method of derivation uses Durnin’s representation7 (or its generalization to the vortex case6) of a Bessel beam involving a continuum superposition of tilted plane waves having a conic angle b relative to the beam’s axis. (See, e.g., Fig. 1 of Ref. 5.) There are various reasons for exploring the scattering properties of such idealized beams which include potential applications to medical ultrasonics,8,9 unusual properties of radiation forces,10–14 the synthesis of other beams by superposition of Bessel beams,5,15,16 and analogies with optically generated wave fields.7,17 The present discussion extends prior investigations of the extinction cross section rext of a sphere placed on the axis of an idealized Bessel beam.12,18,19 Extinction of sound in invariant beams such as Bessel beams18,19 has similarities with extinction from scattering in waveguides.20 Following Van de Hulst’s widely used terminology for plane waves,21 rext is normalized by the sphere’s profile area pa2 (where a is the sphere’s radius) to give the extinction efficiency Qext ¼ rext =pa2 . The limited size of sources allows Bessel wave fields in free space to be supported only over a restricted region of space,7 and the present discussion assumes the sphere is centrally located in the supported region so that the incident wave field is closely approximated by that of an unbounded beam. This paper is organized as follows. Numerical evaluation of Qext reveals a complicated dependence on frequency and the conic angle b of the beam, even for an object as simple as a rigid sphere. (The dimensionless frequency ka ¼ xa=c is used in presenting those results where x is the radian frequency and c is a)
Author to whom correspondence should be addressed. Electronic mail: [email protected]
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Pages: 1668–1671
the speed of sound in the surrounding fluid. Other terminology is reviewed in Sec. II.) Then by applying Babinet’s principle21 and a recently developed optical theorem18,19 to a partially analogous situation of a thin-rigid-circular disk placed on the beam’s axis, a scaling approximation is obtained for the situation ka 1. The associated scaling parameter ka sin b is also obtained using geometrical reasoning. In addition to a fixed rigid sphere, the example of an empty steel spherical shell is considered for a shell for which Qext had been previously evaluated22 for the special case of a plane wave incidence as a function of ka. The present application of an optical theorem to an object in a beam is unusual in that the proper modified form of the optical theorem does not appear to be well known for objects in a beam as elementary as a Gaussian beam.23 II. EXTINCTION EFFICIENCY AS A FUNCTION OF FREQUENCY
For brevity attention is restricted to a zero-order Bessel beam as the incident wave for which the complex velocity potential in cylindrical coordinates is4 wi ¼ w0 J0 ðlRÞ expðijzÞ, where l ¼ k sin b, j ¼ k cos b, and R and z denote transverse radial and axial coordinates. The time dependence is expðixtÞ and the quantity I0 ¼ ðq0 c=2Þðkw0 Þ2 characterizes the beam’s intensity where q0 denotes the density of the surrounding fluid. It is then possible to express the scattered wave in terms of a modified partial wave series (PWS) containing partial wave factors ðsn 1Þ identical in form to the case of plane wave incidence for the given sphere where here n denotes the partial wave index.4,10 The extinction power Pext ¼ Psca þ Pabs , where Psca is the total scattered power and Pabs is the power absorbed by the sphere. These quantities are related to corresponding cross sections by Pext;sca;abs ¼ I0 rext;sca;abs . Zhang and Marston12 showed by conservation of energy and direct integration, that the corresponding efficiency factors may be expressed in terms of the aforementioned coefficients sn ðkaÞ. For the purpose of the present discussion only the extinction efficiency is needed,12 Qext ¼ ðkaÞ2
1 X
2 ð2n þ 1Þ ½Pn ðcos bÞ2 Reð1 sn Þ;
(1)
n¼0
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angular spectral components of the incident beam.6,7 Notice that the scatterer need not be a sphere, and it need not be centered on the beam’s axis. When the scatterer is a sphere centered on the beam’s axis, the usual PWS for AS causes Eq. (2) to reduce immediately to Eq. (1).19 IV. EXTINCTION BY A CIRCULAR DISK ON THE BEAM AXIS
FIG. 1. Scattering efficiency as a function of the dimensionless frequency ka for a fixed rigid sphere centered in a Bessel beam for four values of the beam’s cone angle b. The dashes are decreasing in length for b of (A) 10 , (B) 20 , (C) 40 , and (D) 60 . To display the similarity between the curves a scaling of the abscissa is derived here.
where Re denotes the real part. For non-dissipative situations jsn j ¼ 1 and rabs ¼ 0 so that rext ¼ rsca .12 The plane wave limit is given by taking b ¼ 0 in Eq. (1). Consider now the evaluation of Eq. (1) for the simple case of a fixed rigid sphere which has4 sn ¼ hn ð2Þ ðkaÞ0 =hn ð1Þ ðkaÞ0 , where hn’s are spherical Hankel functions and primes denote differentiation with respect to the indicated argument. In this case, jsn j ¼ 1 so that Qext ¼ Qsca ¼ rsca =pa2 , the scattering efficiency. Figure 1 shows Qext for several choices for the conic angle b for the beam. Inspection of Fig. 1 reveals no obvious relationships between curves having differing values of b. In the case of a plane wave Qext is shown elsewhere and is approximately a monotonic function of ka.22
III. THE EXTENDED OPTICAL THEOREM FOR BESSEL BEAMS
For plane wave illumination the optical theorem relates rext to the forward scattering amplitude.21,24 A generalized form relates integrals involving complex scattering amplitudes to the complex amplitude in an arbitrary direction.24,25 Zhang and Marston18 extended the ordinary optical theorem to give a relation between rext and an integral involving scattering amplitudes for a broad class of invariant (or diffraction-free) beams of which a Bessel beam is an example. In the present case where the velocity potential of the incident wave is wi ¼ w0 J0 ðlRÞexpðijzÞ, the theorem reduces to18,19 "ð # 2p
rext ¼ ð2=kÞIm
AS ðb; /Þd/ ;
(2)
0
where As ðb; /Þ is the scattering amplitude (in a direction having polar and azimuthal scattering angles b and /) normalized such that the velocity potential of the far field scattered wave is wS ðr; h; /Þ ¼ w0 AS ðh; /ÞexpðikrÞ=r:
(3)
The Im in Eq. (2) indicates the imaginary part is retained. The scattering angle h is measured relative to the z axis of the beam such that h ¼ b corresponds to the dominant J. Acoust. Soc. Am., Vol. 135, No. 4, April 2014
Consider now the approximate evaluation of Eq. (2) in the case of a thin-rigid circular disk of radius a centered on the axis of the beam with its normal along the z axis. It is assumed that ka 1 and that the Kirchhoff approximation and Babinet’s principle are applicable. Consequently it is appropriate to consider the diffraction of the beam by the complementary circular aperture of radius a for which the diffracted wave [using the normalization of Eq. (3)] will be denoted by w1 ðr; h; /Þ, and the corresponding approximation for the disk scattering becomes w2 ¼ w1 . The far field limit of the Rayleigh-Summerfield integral for diffraction of a Bessel beam by a circular aperture written in polar form becomes26–28 w1 ðr; b; /Þ ¼ ik cos bIðb; /Þ expðikrÞ=2pr; ða ð 2p I ¼ dRRJ0 ðlRÞ d/0 exp½ilR cosð/0 /Þ; 0
(4) (5)
0
where l ¼ k sin b. The angular integration in Eq. (5) gives an integral representation of J0 ðlRÞ, so that29 ða I ¼ 2p RJ0 2 ðlRÞdR ¼ pa2 ½J0 2 ðlaÞ þ J1 2 ðlaÞ: (6) 0
Introducing the normalization as in Eq. (3) for the disk scattering w2 ¼ w1 gives A2 ðb; /Þ ¼ ði=2Þka2 ½J0 2 ðlaÞ þ J1 2 ðlaÞ cos b;
(7)
which does not depend on the azimuthal angle / of the field point as a consequence of the symmetry of the beam and the disk. From Eq. (2), rext becomes rext ¼ 2pa2 ½J0 2 ðlaÞ þ J1 2 ðlaÞ cos b:
(8)
In the case of the disk the half width of the shadow boundary, denoted by b ¼ acosb in Fig. 2(a), depends on the conic angle b of the Bessel beam. This is in contrast to the sphere case, Fig. 2(b), where the half width is the radius a of the sphere. When a disk is viewed tilted by an angle b, the profile becomes elliptical where the area of the ellipse is pab. It is convenient to define the extinction efficiency in the disk case as rext =pab, giving Qext ¼ 2½J0 2 ðlaÞ þ J1 2 ðlaÞ;
(9)
which depends only on la ¼ ka sin b; Qext is plotted in Fig. 3. V. COMPARISON OF SPHERES AND DISKS AND GEOMETRICAL SCALING
In the case of plane wave illumination, the similarity of the shadow boundary in the sphere and circular disk cases Philip L. Marston: Letters to the Editor
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FIG. 2. For a circular disk (a) and a sphere (b) in a Bessel beam incident from the left having a cone angle b, paths are shown to the center C of each target and to the projected or real upper shadow boundary D. Also shown in (b) is a reflection from the symmetry point E discussed in Sec. V. Both paths shown in (b) contribute to the integral in Eq. (2). (See Sec. V.) The phase of the incident zero-order beam is the same at points A and B. Length AD exceeds length BE by ½a að1 sin bÞ ¼ a sin b.
have motivated comparisons of the near-forward scattering properties when ka is large. See, e.g., Morse and Feshbach.30 In the present case of illumination by a Bessel beam, the comparison is also of interest because of the combined dependence on ka sin b in Eq. (9) and the apparent complexity of the extinction for a rigid sphere shown in Fig. 1. Figure 4 shows that when Qext for the sphere is plotted as a function of ka sin b: (i) provided ka sin b > 3, the curves for the sphere become somewhat similar in shape and magnitude; and (ii) these also display a similarity with the disk result from Eq. (9), provided ka sin b > 3. Inspection of Fig. 4 shows also, however, some superposed oscillatory structure in the sphere case not present in the disk case. Those oscillations are relatively noticeable in the curves for b ¼ 40 and 60 . The geometrical construction shown in Fig. 2(b) suggests a mechanism for interference between ray contributions to the scattering for which the relative phases depend on the scaling parameter ka sin b. Notice that the part of the scattering amplitude AS which contributes to rext in Eq. (2) is the part scattered at an angle b relative to the z axis. Figure 2(b) shows that in addition to diffraction near the shadow boundary, point D, there is also a specular reflection contribution from point E. Considering only the relative phases associated with propagation, construction of the path lengths shows that the path length for reflection at E is less than that for diffraction at D. The phase difference is
FIG. 3. Scattering efficiency as a function of the scaling parameter ka sin b for a circular disk according to the approximate result given in Eq. (9). 1670
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FIG. 4. Scattering efficiency of a fixed rigid sphere as in Fig. 1 but now plotted as a function of the scaling parameter ka sin b (dashed curves). The b values are as in Fig. 1. The solid curve is from Eq. (9).
d ¼ 2ka½1 ð1 sin bÞ ¼ 2ka sin b:
(10)
The associated oscillation period for which the increment in d is 2p predicts an increment in ka sin b of p, which is close to the period of oscillation visible in Fig. 4. Notice also that the interference mechanism diagramed in Fig. 2(b) should not be present in the disk case, in agreement with the absence of significant corresponding oscillations for the solid curves in Figs. 3 and 4. The dependence of the path difference in the sphere case on the combined parameter ka sin b provides an alternate motivation for considering the scaling shown in Fig. 4 in addition to the disk analysis given in Eq. (9). VI. SPHERICAL SHELL CASE, DISCUSSION, AND CONCLUSIONS
The aforementioned considerations indicating a scaling dependence of Qext for spheres on ka sin b are not limited in applicability to the rigid sphere example shown in Fig. 4. Analogous scaling has been confirmed by replacing the sn in Eq. (1) by the predicted values for empty spherical shells.5 Two cases were investigated giving generally similar scaling. Both cases were stainless steel shells in water having walls that were sufficiently thick to have negative buoyancy. One of the cases was chosen because the extinction cross section was previously evaluated for plane wave illumination, and some forward scattering properties were measured in experiments.22 That was the case of an empty shell made of 440c stainless steel having a thickness to radius ratio h/a ¼ 0.162. The other was an ordinary stainless steel shell having h/a ¼ 0.05, selected because similar shells are sometimes used in ocean acoustics experiments.5 While both examples display similarities with Fig. 4 when ka sin b > 5, only the results for the thicker shell are shown here to facilitate ease of comparison with the plane-wave case previously considered. Figure 5 shows Qext for this shell where the identity of the dashed curves correspond to the same b values shown in Fig. 4. (The material and water properties are listed in Ref. 22.) In the case of plane wave illumination, Fig. 1 of Ref. 22, Qext displays substantial elastic enhancements below ka of 10. The associated elastic response significantly alters the appearance of the dashed curves in Fig. 5 in the Philip L. Marston: Letters to the Editor
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2
FIG. 5. Scattering efficiency plotted as in Fig. 4 as a function of the scaling parameter ka sinb but now shown with empty spherical shell coefficients in Eq. (1) for the shell in water studied in Ref. 22. The b values are as in Figs. 1 and 4 (dashed curves). The solid curve is from Eq. (9).
region ka sin b < 5. Above that value the curves are relatively similar to the solid curve for the disk and the corresponding curves for the rigid sphere case, except for the addition of some narrow features also known to be present in the plane wave case. The superposed narrow features tend to be more pronounced, and are sometimes more difficult to resolve in the case of the shell having h/a ¼ 0.05. In conclusion, the apparently irregular behavior of the extinction efficiency when plotted as a function of ka for widely differing values of the Bessel beam cone angle b can produce quasi-similar behavior when plotted as a function of ka sin b as shown by comparing Figs. 1 and 4 for the case of a rigid sphere. In addition application of Babinet’s principle to the case of a circular disk produces a curve which is generally similar, provided ka sin b > 3; however, that curve lacks interference features derived from consideration of the reflection of sound by the sphere. The intercept of Qext ¼ 2 predicted by Eq. (9) in the limit of vanishing ka sin b, is in agreement with elementary considerations: vanishing b corresponds to the plane wave-limit of a Bessel beam and large values of ka in the case of plane wave illumination of a disk at normal incidence are known to give rext ¼ 2pa2 , a limiting case commonly referred to as the “extinction paradox” (p. 107 of Ref. 21). The intercept with the vertical axis in Fig. 3 is consistent with that limit. Computations for empty stainless steel spherical shells in water display a generally similar scaling except there are superposed narrow features associated with the specific elastic response of each shell. The geometrical considerations shown in Fig. 2 are also applicable to spheres centered in Bessel vortex beams and predict ka sin b quasi-scaling for Qext . To extend Eqs. (5)–(9) in the simplest case of a first-order vortex involves the introduction of special functions not considered here. ACKNOWLEDGMENT
This research was supported by ONR. 1
G. C. Gaunaurd and H. Uberall, “Acoustics of finite beams,” J. Acoust. Soc. Am. 63, 5–16 (1978).
J. Acoust. Soc. Am., Vol. 135, No. 4, April 2014
J.-R. Wu and G.-H. Du, “Acoustic radiation force on a small compressible sphere in a focused beam,” J. Acoust. Soc. Am. 87, 997–1003 (1990). 3 X. C. Chen and R. E. Apfel, “Radiation force on a spherical object in an axisymmetric wave field and its application to the calibration of highfrequency transducers,” J. Acoust. Soc. Am. 99, 713–724 (1996). 4 P. L. Marston, “Scattering of a Bessel beam by a sphere,” J. Acoust. Soc. Am. 121, 753–758 (2007). 5 P. L. Marston, “Acoustic beam scattering and excitation of sphere resonance: Bessel beam example,” J. Acoust. Soc. Am. 122, 247–252 (2007); 125, 4092(E) (2009). 6 P. L. Marston, “Scattering of a Bessel beam by a sphere: II. Helicoidal case and spherical shell example,” J. Acoust. Soc. Am. 124, 2905–2910 (2008). 7 J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987). 8 J.-Y. Lu and J. F. Greenleaf, “Ultrasonic nondiffracting transducer for medical imaging,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 37, 438–447 (1990). 9 G. T. Silva, “Off-axis scattering of an ultrasound Bessel beam by a sphere,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 58, 298–304 (2011). 10 P. L. Marston, “Axial radiation force of a Bessel beam on a sphere and direction reversal of the force,” J. Acoust. Soc. Am. 120, 3518–3524 (2006). 11 F. G. Mitri, “Negative axial radiation force on a fluid and elastic spheres illuminated by a high-order Bessel beam of progressive waves,” J. Phys. A: Math. Theor. 42, 245202 (2009). 12 L. K. Zhang and P. L. Marston, “Geometrical interpretation of negative radiation forces of acoustical Bessel beams on spheres,” Phys. Rev. E 84, 035601R (2011). 13 A. Novitsky, C.-W. Qiu, and H. F. Wang, “Single gradientless light beam drags particles as tractor beams,” Phys. Rev. Lett. 107, 203601 (2011). 14 D. Baresch, J.-L. Thomas, and R. Marchiano, “Three-dimensional acoustic radiation force on an arbitrarily located elastic sphere,” J. Acoust. Soc. Am. 133, 25–36 (2013). 15 J. Lekner, “Acoustic beam invariants,” Phys. Rev. E 75, 036610 (2007). 16 P. L. Marston, “Quasi-Gaussian beam analytical basis and comparison with an alternative approach,” J. Acoust. Soc. Am. 130, 1091–1094 (2011). 17 J. Chen, J. Ng, Z. F. Lin, and C. T. Chan, “Optical pulling force,” Nat. Photon. 5, 531–534 (2011). 18 L. K. Zhang and P. L. Marston, “Axial radiation force exerted by general non-diffracting beams,” J. Acoust. Soc. Am. 131, EL329–EL335 (2012). Four lines above Eq. (B1), replace ps with ps. 19 L. K. Zhang and P. L. Marston, “Optical theorem for acoustic nondiffracting beams and application to radiation force and torque,” Bio. Opt. Express 4, 1610–1617 (2013); (E) 4, 2988 (2013). 20 P. Ratilal and N. C. Makris, “Extinction theorem for object scattering in a stratified medium,” J. Acoust. Soc. Am. 110, 2924–2945 (2001). 21 H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), pp. 105–107, 124–130. 22 S. G. Kargl and P. L. Marston, “Ray synthesis of Lamb wave contributions to the total scattering cross section for an elastic spherical shell,” J. Acoust. Soc. Am. 88, 1103–1113 (1990). 23 J. A. Lock, J. T. Hodges, and G. Gouesbet, “Failure of the optical theorem for Gaussian-beam scattering by a spherical particle,” J. Opt. Soc. Am. A 12, 2708–2715 (1995). 24 R. G. Newton, “Optical theorem and beyond,” Am. J. Phys. 44, 639–642 (1976). 25 P. L. Marston, “Generalized optical theorem for scatterers having inversion symmetry: applications to acoustic backscattering,” J. Acoust. Soc. Am. 109, 1291–1295 (2001). 26 P. L. Marston, “Geometrical and catastrophe optics methods in scattering,” Phys. Acoust. 21, 1–234 (1992). See Sec. 2.1, pp. 6–9. 27 J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1999), pp. 481 and 140. 28 B. Lu, W.-L. Huang, and B. Zhang, “Fraunhofer diffraction of a Bessel beam focused by an aperture lens,” Opt. Commun. 119, 6–12 (1995). 29 M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), pp. 360 and 484. 30 P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGrawHill, New York, 1953), Vol. 2, pp. 1551–1555.
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(Received 26 December 2013; revised 14 February 2014; accepted 3 March 2014) The extinction efficiency Qext of a sphere is defined to be the ratio of its extinction cross section to the area of its profile. For non-dissipative situations Qext reduces to the scattering efficiency. For a sphere centered on the axis of a Bessel beam Qext has a complicated dependence on dimensionless frequency ka and on the conic angle b of the beam, even in the simple case of a rigid sphere. With appropriate scaling using Babinet’s principle, however, the dependence reduces approximately to a function of ka sinðbÞ when ka is large. An example is also shown for a metal shell in water. C 2014 Acoustical Society of America. [http://dx.doi.org/10.1121/1.4868399] V PACS number(s): 43.20.Fn, 43.40.Fz, 43.25.Qp [ANN]
I. INTRODUCTION
During the 20th century analytical investigations of the scattering of sound by spheres in acoustic beams tended to be limited to low frequencies or to situations relying on high frequency approximations.1–3 Subsequently, simple modifications to the ordinary partial wave series for the scattering by a sphere in a plane wave were found for spheres placed on the axis of ordinary (zero-order) Bessel beams or on the axis of vortex (or higher-order) Bessel beams.4–6 One method of derivation uses Durnin’s representation7 (or its generalization to the vortex case6) of a Bessel beam involving a continuum superposition of tilted plane waves having a conic angle b relative to the beam’s axis. (See, e.g., Fig. 1 of Ref. 5.) There are various reasons for exploring the scattering properties of such idealized beams which include potential applications to medical ultrasonics,8,9 unusual properties of radiation forces,10–14 the synthesis of other beams by superposition of Bessel beams,5,15,16 and analogies with optically generated wave fields.7,17 The present discussion extends prior investigations of the extinction cross section rext of a sphere placed on the axis of an idealized Bessel beam.12,18,19 Extinction of sound in invariant beams such as Bessel beams18,19 has similarities with extinction from scattering in waveguides.20 Following Van de Hulst’s widely used terminology for plane waves,21 rext is normalized by the sphere’s profile area pa2 (where a is the sphere’s radius) to give the extinction efficiency Qext ¼ rext =pa2 . The limited size of sources allows Bessel wave fields in free space to be supported only over a restricted region of space,7 and the present discussion assumes the sphere is centrally located in the supported region so that the incident wave field is closely approximated by that of an unbounded beam. This paper is organized as follows. Numerical evaluation of Qext reveals a complicated dependence on frequency and the conic angle b of the beam, even for an object as simple as a rigid sphere. (The dimensionless frequency ka ¼ xa=c is used in presenting those results where x is the radian frequency and c is a)
Author to whom correspondence should be addressed. Electronic mail: [email protected]
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J. Acoust. Soc. Am. 135 (4), April 2014
Pages: 1668–1671
the speed of sound in the surrounding fluid. Other terminology is reviewed in Sec. II.) Then by applying Babinet’s principle21 and a recently developed optical theorem18,19 to a partially analogous situation of a thin-rigid-circular disk placed on the beam’s axis, a scaling approximation is obtained for the situation ka 1. The associated scaling parameter ka sin b is also obtained using geometrical reasoning. In addition to a fixed rigid sphere, the example of an empty steel spherical shell is considered for a shell for which Qext had been previously evaluated22 for the special case of a plane wave incidence as a function of ka. The present application of an optical theorem to an object in a beam is unusual in that the proper modified form of the optical theorem does not appear to be well known for objects in a beam as elementary as a Gaussian beam.23 II. EXTINCTION EFFICIENCY AS A FUNCTION OF FREQUENCY
For brevity attention is restricted to a zero-order Bessel beam as the incident wave for which the complex velocity potential in cylindrical coordinates is4 wi ¼ w0 J0 ðlRÞ expðijzÞ, where l ¼ k sin b, j ¼ k cos b, and R and z denote transverse radial and axial coordinates. The time dependence is expðixtÞ and the quantity I0 ¼ ðq0 c=2Þðkw0 Þ2 characterizes the beam’s intensity where q0 denotes the density of the surrounding fluid. It is then possible to express the scattered wave in terms of a modified partial wave series (PWS) containing partial wave factors ðsn 1Þ identical in form to the case of plane wave incidence for the given sphere where here n denotes the partial wave index.4,10 The extinction power Pext ¼ Psca þ Pabs , where Psca is the total scattered power and Pabs is the power absorbed by the sphere. These quantities are related to corresponding cross sections by Pext;sca;abs ¼ I0 rext;sca;abs . Zhang and Marston12 showed by conservation of energy and direct integration, that the corresponding efficiency factors may be expressed in terms of the aforementioned coefficients sn ðkaÞ. For the purpose of the present discussion only the extinction efficiency is needed,12 Qext ¼ ðkaÞ2
1 X
2 ð2n þ 1Þ ½Pn ðcos bÞ2 Reð1 sn Þ;
(1)
n¼0
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angular spectral components of the incident beam.6,7 Notice that the scatterer need not be a sphere, and it need not be centered on the beam’s axis. When the scatterer is a sphere centered on the beam’s axis, the usual PWS for AS causes Eq. (2) to reduce immediately to Eq. (1).19 IV. EXTINCTION BY A CIRCULAR DISK ON THE BEAM AXIS
FIG. 1. Scattering efficiency as a function of the dimensionless frequency ka for a fixed rigid sphere centered in a Bessel beam for four values of the beam’s cone angle b. The dashes are decreasing in length for b of (A) 10 , (B) 20 , (C) 40 , and (D) 60 . To display the similarity between the curves a scaling of the abscissa is derived here.
where Re denotes the real part. For non-dissipative situations jsn j ¼ 1 and rabs ¼ 0 so that rext ¼ rsca .12 The plane wave limit is given by taking b ¼ 0 in Eq. (1). Consider now the evaluation of Eq. (1) for the simple case of a fixed rigid sphere which has4 sn ¼ hn ð2Þ ðkaÞ0 =hn ð1Þ ðkaÞ0 , where hn’s are spherical Hankel functions and primes denote differentiation with respect to the indicated argument. In this case, jsn j ¼ 1 so that Qext ¼ Qsca ¼ rsca =pa2 , the scattering efficiency. Figure 1 shows Qext for several choices for the conic angle b for the beam. Inspection of Fig. 1 reveals no obvious relationships between curves having differing values of b. In the case of a plane wave Qext is shown elsewhere and is approximately a monotonic function of ka.22
III. THE EXTENDED OPTICAL THEOREM FOR BESSEL BEAMS
For plane wave illumination the optical theorem relates rext to the forward scattering amplitude.21,24 A generalized form relates integrals involving complex scattering amplitudes to the complex amplitude in an arbitrary direction.24,25 Zhang and Marston18 extended the ordinary optical theorem to give a relation between rext and an integral involving scattering amplitudes for a broad class of invariant (or diffraction-free) beams of which a Bessel beam is an example. In the present case where the velocity potential of the incident wave is wi ¼ w0 J0 ðlRÞexpðijzÞ, the theorem reduces to18,19 "ð # 2p
rext ¼ ð2=kÞIm
AS ðb; /Þd/ ;
(2)
0
where As ðb; /Þ is the scattering amplitude (in a direction having polar and azimuthal scattering angles b and /) normalized such that the velocity potential of the far field scattered wave is wS ðr; h; /Þ ¼ w0 AS ðh; /ÞexpðikrÞ=r:
(3)
The Im in Eq. (2) indicates the imaginary part is retained. The scattering angle h is measured relative to the z axis of the beam such that h ¼ b corresponds to the dominant J. Acoust. Soc. Am., Vol. 135, No. 4, April 2014
Consider now the approximate evaluation of Eq. (2) in the case of a thin-rigid circular disk of radius a centered on the axis of the beam with its normal along the z axis. It is assumed that ka 1 and that the Kirchhoff approximation and Babinet’s principle are applicable. Consequently it is appropriate to consider the diffraction of the beam by the complementary circular aperture of radius a for which the diffracted wave [using the normalization of Eq. (3)] will be denoted by w1 ðr; h; /Þ, and the corresponding approximation for the disk scattering becomes w2 ¼ w1 . The far field limit of the Rayleigh-Summerfield integral for diffraction of a Bessel beam by a circular aperture written in polar form becomes26–28 w1 ðr; b; /Þ ¼ ik cos bIðb; /Þ expðikrÞ=2pr; ða ð 2p I ¼ dRRJ0 ðlRÞ d/0 exp½ilR cosð/0 /Þ; 0
(4) (5)
0
where l ¼ k sin b. The angular integration in Eq. (5) gives an integral representation of J0 ðlRÞ, so that29 ða I ¼ 2p RJ0 2 ðlRÞdR ¼ pa2 ½J0 2 ðlaÞ þ J1 2 ðlaÞ: (6) 0
Introducing the normalization as in Eq. (3) for the disk scattering w2 ¼ w1 gives A2 ðb; /Þ ¼ ði=2Þka2 ½J0 2 ðlaÞ þ J1 2 ðlaÞ cos b;
(7)
which does not depend on the azimuthal angle / of the field point as a consequence of the symmetry of the beam and the disk. From Eq. (2), rext becomes rext ¼ 2pa2 ½J0 2 ðlaÞ þ J1 2 ðlaÞ cos b:
(8)
In the case of the disk the half width of the shadow boundary, denoted by b ¼ acosb in Fig. 2(a), depends on the conic angle b of the Bessel beam. This is in contrast to the sphere case, Fig. 2(b), where the half width is the radius a of the sphere. When a disk is viewed tilted by an angle b, the profile becomes elliptical where the area of the ellipse is pab. It is convenient to define the extinction efficiency in the disk case as rext =pab, giving Qext ¼ 2½J0 2 ðlaÞ þ J1 2 ðlaÞ;
(9)
which depends only on la ¼ ka sin b; Qext is plotted in Fig. 3. V. COMPARISON OF SPHERES AND DISKS AND GEOMETRICAL SCALING
In the case of plane wave illumination, the similarity of the shadow boundary in the sphere and circular disk cases Philip L. Marston: Letters to the Editor
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FIG. 2. For a circular disk (a) and a sphere (b) in a Bessel beam incident from the left having a cone angle b, paths are shown to the center C of each target and to the projected or real upper shadow boundary D. Also shown in (b) is a reflection from the symmetry point E discussed in Sec. V. Both paths shown in (b) contribute to the integral in Eq. (2). (See Sec. V.) The phase of the incident zero-order beam is the same at points A and B. Length AD exceeds length BE by ½a að1 sin bÞ ¼ a sin b.
have motivated comparisons of the near-forward scattering properties when ka is large. See, e.g., Morse and Feshbach.30 In the present case of illumination by a Bessel beam, the comparison is also of interest because of the combined dependence on ka sin b in Eq. (9) and the apparent complexity of the extinction for a rigid sphere shown in Fig. 1. Figure 4 shows that when Qext for the sphere is plotted as a function of ka sin b: (i) provided ka sin b > 3, the curves for the sphere become somewhat similar in shape and magnitude; and (ii) these also display a similarity with the disk result from Eq. (9), provided ka sin b > 3. Inspection of Fig. 4 shows also, however, some superposed oscillatory structure in the sphere case not present in the disk case. Those oscillations are relatively noticeable in the curves for b ¼ 40 and 60 . The geometrical construction shown in Fig. 2(b) suggests a mechanism for interference between ray contributions to the scattering for which the relative phases depend on the scaling parameter ka sin b. Notice that the part of the scattering amplitude AS which contributes to rext in Eq. (2) is the part scattered at an angle b relative to the z axis. Figure 2(b) shows that in addition to diffraction near the shadow boundary, point D, there is also a specular reflection contribution from point E. Considering only the relative phases associated with propagation, construction of the path lengths shows that the path length for reflection at E is less than that for diffraction at D. The phase difference is
FIG. 3. Scattering efficiency as a function of the scaling parameter ka sin b for a circular disk according to the approximate result given in Eq. (9). 1670
J. Acoust. Soc. Am., Vol. 135, No. 4, April 2014
FIG. 4. Scattering efficiency of a fixed rigid sphere as in Fig. 1 but now plotted as a function of the scaling parameter ka sin b (dashed curves). The b values are as in Fig. 1. The solid curve is from Eq. (9).
d ¼ 2ka½1 ð1 sin bÞ ¼ 2ka sin b:
(10)
The associated oscillation period for which the increment in d is 2p predicts an increment in ka sin b of p, which is close to the period of oscillation visible in Fig. 4. Notice also that the interference mechanism diagramed in Fig. 2(b) should not be present in the disk case, in agreement with the absence of significant corresponding oscillations for the solid curves in Figs. 3 and 4. The dependence of the path difference in the sphere case on the combined parameter ka sin b provides an alternate motivation for considering the scaling shown in Fig. 4 in addition to the disk analysis given in Eq. (9). VI. SPHERICAL SHELL CASE, DISCUSSION, AND CONCLUSIONS
The aforementioned considerations indicating a scaling dependence of Qext for spheres on ka sin b are not limited in applicability to the rigid sphere example shown in Fig. 4. Analogous scaling has been confirmed by replacing the sn in Eq. (1) by the predicted values for empty spherical shells.5 Two cases were investigated giving generally similar scaling. Both cases were stainless steel shells in water having walls that were sufficiently thick to have negative buoyancy. One of the cases was chosen because the extinction cross section was previously evaluated for plane wave illumination, and some forward scattering properties were measured in experiments.22 That was the case of an empty shell made of 440c stainless steel having a thickness to radius ratio h/a ¼ 0.162. The other was an ordinary stainless steel shell having h/a ¼ 0.05, selected because similar shells are sometimes used in ocean acoustics experiments.5 While both examples display similarities with Fig. 4 when ka sin b > 5, only the results for the thicker shell are shown here to facilitate ease of comparison with the plane-wave case previously considered. Figure 5 shows Qext for this shell where the identity of the dashed curves correspond to the same b values shown in Fig. 4. (The material and water properties are listed in Ref. 22.) In the case of plane wave illumination, Fig. 1 of Ref. 22, Qext displays substantial elastic enhancements below ka of 10. The associated elastic response significantly alters the appearance of the dashed curves in Fig. 5 in the Philip L. Marston: Letters to the Editor
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2
FIG. 5. Scattering efficiency plotted as in Fig. 4 as a function of the scaling parameter ka sinb but now shown with empty spherical shell coefficients in Eq. (1) for the shell in water studied in Ref. 22. The b values are as in Figs. 1 and 4 (dashed curves). The solid curve is from Eq. (9).
region ka sin b < 5. Above that value the curves are relatively similar to the solid curve for the disk and the corresponding curves for the rigid sphere case, except for the addition of some narrow features also known to be present in the plane wave case. The superposed narrow features tend to be more pronounced, and are sometimes more difficult to resolve in the case of the shell having h/a ¼ 0.05. In conclusion, the apparently irregular behavior of the extinction efficiency when plotted as a function of ka for widely differing values of the Bessel beam cone angle b can produce quasi-similar behavior when plotted as a function of ka sin b as shown by comparing Figs. 1 and 4 for the case of a rigid sphere. In addition application of Babinet’s principle to the case of a circular disk produces a curve which is generally similar, provided ka sin b > 3; however, that curve lacks interference features derived from consideration of the reflection of sound by the sphere. The intercept of Qext ¼ 2 predicted by Eq. (9) in the limit of vanishing ka sin b, is in agreement with elementary considerations: vanishing b corresponds to the plane wave-limit of a Bessel beam and large values of ka in the case of plane wave illumination of a disk at normal incidence are known to give rext ¼ 2pa2 , a limiting case commonly referred to as the “extinction paradox” (p. 107 of Ref. 21). The intercept with the vertical axis in Fig. 3 is consistent with that limit. Computations for empty stainless steel spherical shells in water display a generally similar scaling except there are superposed narrow features associated with the specific elastic response of each shell. The geometrical considerations shown in Fig. 2 are also applicable to spheres centered in Bessel vortex beams and predict ka sin b quasi-scaling for Qext . To extend Eqs. (5)–(9) in the simplest case of a first-order vortex involves the introduction of special functions not considered here. ACKNOWLEDGMENT
This research was supported by ONR. 1
G. C. Gaunaurd and H. Uberall, “Acoustics of finite beams,” J. Acoust. Soc. Am. 63, 5–16 (1978).
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J.-R. Wu and G.-H. Du, “Acoustic radiation force on a small compressible sphere in a focused beam,” J. Acoust. Soc. Am. 87, 997–1003 (1990). 3 X. C. Chen and R. E. Apfel, “Radiation force on a spherical object in an axisymmetric wave field and its application to the calibration of highfrequency transducers,” J. Acoust. Soc. Am. 99, 713–724 (1996). 4 P. L. Marston, “Scattering of a Bessel beam by a sphere,” J. Acoust. Soc. Am. 121, 753–758 (2007). 5 P. L. Marston, “Acoustic beam scattering and excitation of sphere resonance: Bessel beam example,” J. Acoust. Soc. Am. 122, 247–252 (2007); 125, 4092(E) (2009). 6 P. L. Marston, “Scattering of a Bessel beam by a sphere: II. Helicoidal case and spherical shell example,” J. Acoust. Soc. Am. 124, 2905–2910 (2008). 7 J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987). 8 J.-Y. Lu and J. F. Greenleaf, “Ultrasonic nondiffracting transducer for medical imaging,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 37, 438–447 (1990). 9 G. T. Silva, “Off-axis scattering of an ultrasound Bessel beam by a sphere,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 58, 298–304 (2011). 10 P. L. Marston, “Axial radiation force of a Bessel beam on a sphere and direction reversal of the force,” J. Acoust. Soc. Am. 120, 3518–3524 (2006). 11 F. G. Mitri, “Negative axial radiation force on a fluid and elastic spheres illuminated by a high-order Bessel beam of progressive waves,” J. Phys. A: Math. Theor. 42, 245202 (2009). 12 L. K. Zhang and P. L. Marston, “Geometrical interpretation of negative radiation forces of acoustical Bessel beams on spheres,” Phys. Rev. E 84, 035601R (2011). 13 A. Novitsky, C.-W. Qiu, and H. F. Wang, “Single gradientless light beam drags particles as tractor beams,” Phys. Rev. Lett. 107, 203601 (2011). 14 D. Baresch, J.-L. Thomas, and R. Marchiano, “Three-dimensional acoustic radiation force on an arbitrarily located elastic sphere,” J. Acoust. Soc. Am. 133, 25–36 (2013). 15 J. Lekner, “Acoustic beam invariants,” Phys. Rev. E 75, 036610 (2007). 16 P. L. Marston, “Quasi-Gaussian beam analytical basis and comparison with an alternative approach,” J. Acoust. Soc. Am. 130, 1091–1094 (2011). 17 J. Chen, J. Ng, Z. F. Lin, and C. T. Chan, “Optical pulling force,” Nat. Photon. 5, 531–534 (2011). 18 L. K. Zhang and P. L. Marston, “Axial radiation force exerted by general non-diffracting beams,” J. Acoust. Soc. Am. 131, EL329–EL335 (2012). Four lines above Eq. (B1), replace ps with ps. 19 L. K. Zhang and P. L. Marston, “Optical theorem for acoustic nondiffracting beams and application to radiation force and torque,” Bio. Opt. Express 4, 1610–1617 (2013); (E) 4, 2988 (2013). 20 P. Ratilal and N. C. Makris, “Extinction theorem for object scattering in a stratified medium,” J. Acoust. Soc. Am. 110, 2924–2945 (2001). 21 H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), pp. 105–107, 124–130. 22 S. G. Kargl and P. L. Marston, “Ray synthesis of Lamb wave contributions to the total scattering cross section for an elastic spherical shell,” J. Acoust. Soc. Am. 88, 1103–1113 (1990). 23 J. A. Lock, J. T. Hodges, and G. Gouesbet, “Failure of the optical theorem for Gaussian-beam scattering by a spherical particle,” J. Opt. Soc. Am. A 12, 2708–2715 (1995). 24 R. G. Newton, “Optical theorem and beyond,” Am. J. Phys. 44, 639–642 (1976). 25 P. L. Marston, “Generalized optical theorem for scatterers having inversion symmetry: applications to acoustic backscattering,” J. Acoust. Soc. Am. 109, 1291–1295 (2001). 26 P. L. Marston, “Geometrical and catastrophe optics methods in scattering,” Phys. Acoust. 21, 1–234 (1992). See Sec. 2.1, pp. 6–9. 27 J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1999), pp. 481 and 140. 28 B. Lu, W.-L. Huang, and B. Zhang, “Fraunhofer diffraction of a Bessel beam focused by an aperture lens,” Opt. Commun. 119, 6–12 (1995). 29 M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), pp. 360 and 484. 30 P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGrawHill, New York, 1953), Vol. 2, pp. 1551–1555.
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