While acoustic vortex beams have many potential applications, the full implication of the phase information available in scattering experiments has not been developed. The present paper concerns observables in measured near-backward scattering from a sphere in water raster scanned through a first-order acoustic vortex beam. Symmetrically placed transducer elements were operated in a transmit-receive mode. Helicity-dependent projections of the spatial evolution of the scattering were used to display magnitude and phase information. The resulting phase swirl patterns were projection dependent and especially sensitive to the transverse position of the sphere. The magnitude also depended on the sphere's position relative to the beam's axial null.

An early investigation of acoustic vortex beam propagation measured aspects of the beam's amplitude and phase spatial evolution and suggested applications to scattering, alignment, and to orbital angular momentum and the associated acoustic radiation torque.1 While there have been several investigations of the radiation torque of such beams,2–4 there has been less attention given to measurements associated with alignment and closely related phase aspects of measured scattering by such beams.5,6 The present paper concerns measured aspects of the backscattering by a sphere raster scanned through a first-order acoustic vortex beam in water. By recording the scattering with an array of receivers, the dependence of the scattering on the sphere's location is displayed using helicity projections described in Ref. 6 and extended here for the purposes of the present investigation. While some of the results have been previously presented,7,8 the present discussion should help make the results more widely available.

Nye and Berry9 described what they called a helicoidal wave solution of the Helmholtz equation in cylindrical polar coordinates (r, ϕ, z) of the following form:

p(r,ϕ,z)=(Ar|m|+Br|m|)exp(ikz+imϕ),
(1)

where r and z are radial and axial coordinates and ϕ is the angular coordinate. The sign of the integer m determines what Nye and Berry termed the “handedness” of the wave, though in the present context the term “helicity” is useful; m is now often called the order or topological charge.1 In the present context, p becomes a complex pressure for which the corresponding acoustic pressure is Re[p(r, ϕ, z)exp(−iωt)], where Re denotes the real part, t is time, ω = 2πf, f is the frequency, k = ω/c, and c is the speed of sound. The coefficients A and B may be complex, though in the present context B = 0 to avoid an axial singularity. To suppress the growth in magnitude with increasing r, Eq. (1) is only applied close to the z-axis. Vortex beam solutions of the Helmholtz equation applicable over a larger range of r include Leguerre-Gauss beams1 and Bessel beams.10 The beam properties most relevant to the present investigation are the dependence of phase on ϕ and the vanishing axial magnitude.

Figure 1 diagrams the experiment used to investigate the scattering by a sphere illuminated in water by a vortex beam. Like Ref. 1, the beam is generated by appropriately phased excitation of a symmetric four-element array of sources. Unlike Ref. 1, the source elements are isosceles triangles with the base of each corresponding to the sides of a square, each base having a length of 10.16 cm. Each element was excited by an f = 125 kHz Tukey-windowed three-cycle tone burst. (Measurements similar to the ones displayed here were also taken using longer bursts.) The elements consisted of piezo-composite material manufactured and assembled as an array by Materials Systems, Inc. (now known as MSI Transducers Corporation). The dissipation of the piezo-material was such that the source elements had sufficient bandwidth and similar phase and amplitude properties to facilitate the present investigation and experiments on acoustic data transmission using modulated helicity.6,8,11 An acrylic (PMMA) sphere having a radius a = 12.7 mm was placed in a plane nearly perpendicular to the z-axis of the generated beam at a range z = 1.67 m. The sphere was raster scanned in its (x, y) plane, where the coordinates x and y correspond to the center of the sphere with (x, y) = (0, 0) offset slightly from the beam's z-axis. A procedure was developed for aligning the plane of the source sufficiently parallel to that of the raster scan for the purpose of this investigation.8 The source and the sphere were positioned in a 3.6 m diameter 2.2 m deep water tank in such a way that echoes from the tank's boundaries were separated in time from the sphere's echo. A braided polymer 0.43 mm diameter thread having a negligible echo tethered the sphere. With the sound speed c = 1484 m/s the wavelength λ = 11.9 mm and ka = 6.7. This ka was displaced from those of the sphere's resonances.12 The triangular shape of the four-elements in the source and the distance to the plane of the raster scan were selected in such a way that phase anomalies offset from the beam's axis visible in Fig. 4 of Ref. 1 were suppressed in the region of the raster scan. Since it was not feasible to scan a hydrophone with accurate position registration, the wave-field was verified by numerical evaluation of the Kirchhoff approximation of the wave-field, Eq. (2) of Ref. 13. The equiphase swirls of the beam at the range of the sphere are predicted to be approximately Fermat spirals near the axis of the form (ϕϕ0) ≈ kr2/(2z), where ϕ0 is a constant. It was predicted in Ref. 1 from symmetry considerations that symmetric objects such as spheres on the axis of vortex beams have vanishing backscattering amplitudes; this was computationally verified for various spheres placed in the axis of vortex Bessel beams.10 

Fig. 1.

Idealized experimental configuration in which a first-order vortex beam illuminates a sphere that is scanned in the xy-plane. Ideally the source plane (with the associated coordinates xsys, not shown) is parallel to the scanned plane. The actual scanned plane is slightly tilted and offset. The drawing is not to scale: the length of each side of the square boundary of the source is 10.16 cm, the distance to the xy-plane is 1.67 m, and the sphere diameter is 25.4 mm.

Fig. 1.

Idealized experimental configuration in which a first-order vortex beam illuminates a sphere that is scanned in the xy-plane. Ideally the source plane (with the associated coordinates xsys, not shown) is parallel to the scanned plane. The actual scanned plane is slightly tilted and offset. The drawing is not to scale: the length of each side of the square boundary of the source is 10.16 cm, the distance to the xy-plane is 1.67 m, and the sphere diameter is 25.4 mm.

Close modal

Ideally the backscattering on the z-axis could be measured by placing a small hydrophone at the apex of the four triangular transducer elements, referred to here as panels. That was impractical and by itself would not provide the data needed here. Each panel was connected to a transmit-receive (T-R) circuit that enabled each to be used both as a receiver and as a source. The received signal from each was separately amplified and digitized. The corresponding voltage was interpreted as proportional to the spatially averaged acoustic pressure scattered by the sphere. To describe the signal processing each panel is indexed by n = 1–4 as shown in Fig. 2 with the recorded voltages designated as Vn(t). The simplest processing option (referred to here as the helicity-neutral projection) is to sum the recorded voltages VHN(t) = V1(t) + V2(t) + V3(t) + V4(t). For scattering by a small sphere, VHN(t) would be approximately proportional to the signal from a small axial hydrophone. To motivate the other processing projections, consider the ϕ-dependence of scattering by a sphere placed exactly on the axis of a first-order vortex beam, Eq. (9) of Ref. 10. The resulting angular phase evolution of the Vnwould ideally be as indicated in the upper section of Fig. 2. Consequently, if each Vn had the same magnitude, VHN = 0 as expected from the symmetry considerations discussed in Ref. 1. Now consider co-helical processing shown on the left of Fig. 2. Shift the phase of Vn by θCOn = −(π/2)(n − 1). The processed signal for the ideally aligned case has no phase evolution as shown in lower left of Fig. 2. The shift introduced by θCOn cancels the vortex-beam-associated evolution. The phase shift introduced for cross-helical processing is shown on the right of Fig. 2: θCRn = (π/2)(n − 1). The processed signal for the ideally aligned case has an angular phase shift rate twice as large as the neutral processing case and thus becomes analogous to the phase evolution for scattering by a beam having |m| = 2. The cross- and co-terminology here correspond to those used in the earlier discussion of forward scattering, where the phase shifts were introduced by longitudinal displacements of individual receivers in a four-element array.6 In the present experiment the non-vanishing θCOn and θCRn were introduced either by shifting a time index of a digitized signal or (in the case of |θCOn| or |θCRn| = π) by sign reversal. The digitization rate and f = 125 kHz were selected such that the desired phase shifts corresponded to an integer time-index shift.

Fig. 2.

The transducer panels are indexed by n in the lower-middle diagram. The upper middle diagram represents the idealized evolution of the phase of the scattering detected by the nth panel for an axially aligned sphere. For neutral projection processing, the receiver signals are added. On the lower left and right are shown co-helical and cross-helical projection processing.

Fig. 2.

The transducer panels are indexed by n in the lower-middle diagram. The upper middle diagram represents the idealized evolution of the phase of the scattering detected by the nth panel for an axially aligned sphere. For neutral projection processing, the receiver signals are added. On the lower left and right are shown co-helical and cross-helical projection processing.

Close modal

The raster scan of the sphere position was accomplished using a two-dimensional stepper motor scanner, where following each position increment, signals were not acquired until the sphere was stationary. The step increment (Δx, Δy) = (0.3 mm, 0.3 mm) was selected to allow a 28 cm × 28 cm region to be scanned in a practical duration. With the sphere at the midpoint of each of the boundaries of the scanned region, the backscattering angle for an observer at the center of the transducer becomes γ = tan−1(14 cm/167 cm) = 4.8°. Spheres as small as ka = 6.7 are known to exhibit specular-like reflection, and the early-time response (associated with specular reflection) was separated from the subsequent elastic response. Given the small |γ| the specular reflection coefficient is expected to have a negligible dependence on (x, y) for the scanned range. Consequently, for helicity-neutral processing the amplitude is dominated by the beam's properties in the plane where the sphere is scanned. For each signal projection the phase and magnitude were extracted using standard Hilbert processing operations.8 

Figure 3 shows the processed measurements. Consider first the spatial evolution of the measured phase displayed in the right-hand column. The helicity neutral case (d) displays the expected |m| = 1 phase spiral of first-order vortex beam associated with the local illumination of the sphere. The cross-helical case (e) displays the phase spiral analogous to a |m| = 2 beam in that the phase evolves by 4π along a circular trajectory centered on the symmetry point. This is consistent with doubled rate of phase evolution shown on the right of Fig. 2. In the co-helical case (f) the spiral is removed by the processing (as on the left of Fig. 2) such that all that remains is a slow radial phase evolution associated with the combined spreading of waves due to propagation from the source to the sphere and back to the receivers. The magnitude of the helicity neutral data (a) shows a sharp dip when the sphere is at the symmetry point of the spiral in (d). This is from the suppression of near backward scattering previously mentioned.1,10 The magnitude evolution was also plotted on a linear scale along a slice through (or very near) the center of the neutral-case phase spiral for the three processing cases. In each case the minima are offset slightly from the center of the scanned region since it was not practical to determine in advance the precise axial location of the acoustic beam. The neutral-case magnitude is approximately linear in the shift of the sphere (from the apparent axis) for shifts smaller than 5 cm. For the cross and co-helical cases, it is approximately quadratic.8 

Fig. 3.

(Color online) Projections recorded for a raster scan of the sphere in the xy-plane: helicity neutral (a, d), cross-helical (b, e), and co-helical (c, f). Upper row: the magnitude in dB (upper right scale) relative to an undetermined level. Lower row: the phase relative to an undetermined constant. The phase-spiral center in (d) is at (x, y) = (1.5, −0.6) cm.

Fig. 3.

(Color online) Projections recorded for a raster scan of the sphere in the xy-plane: helicity neutral (a, d), cross-helical (b, e), and co-helical (c, f). Upper row: the magnitude in dB (upper right scale) relative to an undetermined level. Lower row: the phase relative to an undetermined constant. The phase-spiral center in (d) is at (x, y) = (1.5, −0.6) cm.

Close modal

The simplest model for the near-axis scaling is to invoke reciprocity and to replace each of the triangular transducer elements by point-like elements offset from the z-axis. The spherical scatterer is also taken to be point-like. The sources are taken to be excited with phase offsets corresponding to those of the actual sources such that magnitude and phase evolution in a distant plane parallel to the source plane becomes proportional to ψs ≈ r exp(iϕ) close to the z axis. From reciprocity the processing introduces an additional factor proportional to ψj ≈ r|j|exp(i), where the integer j depends on the processing case. From Fig. 2 the neutral case has j = 0, and the cross and co-helical cases have j = 1 and −1, respectively. The combined factors give ψsψj ≈ r(1+|j|) exp[i(1 + j)ϕ] as suggested by the data; notice that for j = ±1, |ψsψj| ≈ r2. This scaling is also found near the axis for direct superposition for point-like sources and a point-like target.8 

To explain the scaling of the processed scattering away from the axis the following numerical approach was used. The scatterer is again taken to be point-like but now the vortex beam is modeled using the Kirchhoff approximation and the phase-weighted average for each receiver element is evaluated. The dependence of the signal on the target coordinates (x, y) becomes proportional to the function,

ΨKj(x,y)=dxsdysdxrdyrexp[ik(Rs+Rr)+iβ]/(Rs+Rr),
(2)

where the integration is over the source and receiver coordinates (xs, ys) and (xr, yr) and Rs and Rr are the corresponding distances to the target at (x, y) in the plane at z. The phase function β(xs, ys xr, yr) accounts for the combined effects of the excitation phase for the nth element and processing phase shift introduced is shown in Fig. 2; the contribution to β depends on the source and receiver element. The resulting spatial dependence of the magnitude and phase of ψKj shown in Fig. 4 has the important features observed in Fig. 3. When the |ψKj| are plotted on a linear scale the aforementioned features are evident near the z-axis.

Fig. 4.

(Color online) Modeled projections identified as in Fig. 3 but computed for a monopole point-like scatterer of arbitrary strength (see text). The transducer and target planes are taken to be perfectly aligned with the z-axis.

Fig. 4.

(Color online) Modeled projections identified as in Fig. 3 but computed for a monopole point-like scatterer of arbitrary strength (see text). The transducer and target planes are taken to be perfectly aligned with the z-axis.

Close modal

The signal projections use measurements from a symmetrically placed array of receivers. The features evident in the observations depend on the projection-class of the processed data. While Fig. 3 displays projections for the specular part of the echo, the patterns are similar in form when evaluated for the subsequent elastic contributions, though the angular offsets of the phase swirls can differ. The spatial evolution of the projections in Fig. 3 depend on phase information sometimes not retained in modeling efforts,14 though for scattering by an on-axis sphere the angular phase structure has long been known.10 (See also Ref. 15 where the phase of the off-axis scattering can be computed in the Bessel beam illumination case.) The double swirl in the cross-helical projection resembles the phase behavior of a second-order beam. This was previously reported for the corresponding projection of forward scattering.6 The 3 mm step position increment for the sphere was smaller than the 11.9 mm wavelength λ. The resolution of the phase swirl appears to suffice for sub-wavelength positioning of spheres that is noteworthy given the range (1.67 m) and compact size of the source (10.16 cm square). Considering potential radiation force, medical, and communications applications of vortex beams,4,6,15–21 the signal projections demonstrated here might be useful for more than sonar purposes noted in Ref. 1. The measurements in Fig. 3 concern a well-aligned situation. When the scanned plane is tilted relative to the transducer plane by as small as 1°, the resulting projections are found to be less symmetric.8 

The demonstrated helicity projection processing of scattering could be extended to higher-order vortex beams using transducers having finer angular spacing with appropriate source excitation and processing of received signals. This was impractical to explore with the available apparatus.

This material is based upon research supported by the U. S. Office of Naval Research under Award No. N000141912039 and prior Award Nos. N000141512603 and N000141010093. T. M. Marston contributed to the testing and modeling of the source in 2009 and D. S. Plotnick and D. J. Zartman made helpful suggestions concerning signal processing in 2012–2014. Recently, A. R. Fortuner also provided assistance.

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