Formation of high-order acoustic Bessel beams by spiral diffraction gratings

The formation of high-order Bessel beams by a passive acoustic device consisting of an Archimedes' spiral diffraction grating is theoretically, numerically, and experimentally reported in this paper. These beams are propagation-invariant solutions of the Helmholtz equation and are characterized by an azimuthal variation of the phase along its annular spectrum producing an acoustic vortex in the near field. In our system, the scattering of plane acoustic waves by the spiral grating leads to the formation of the acoustic vortex with zero pressure on axis and the angular phase dislocations characterized by the spiral geometry. The order of the generated Bessel beam and, as a consequence, the size of the generated vortex can be fixed by the number of arms in the spiral diffraction grating. The obtained results allow for obtaining Bessel beams with controllable vorticity by a passive device, which has potential applications in low-cost acoustic tweezers and acoustic radiation force devices.

Figure 1

Scheme of the first-order Bessel beam formation by the Archimedes' spiral grating. The incident plane wave is scattered into a converging and diverging conical wave front in which the phase of the wave ($\varphi$) is a linear function of the angle ($\theta$). A corkscrew dislocation in the wave front is produced, leading to an acoustic vortex and a zero pressure on-axis $r=0$. The minimum along the symmetry axis results from the destructive interference of first-order diffraction generated at opposite sides of the axis.

Figure 2

Examples of Archimedes' spirals with multiple arms (a) $n=1$, (b) $n=3$, (c) $n=7$, and (d) $n=20$.

Figure 3

(a) Longitudinal pressure distribution along the first lobe of the Bessel beam obtained from Eq. (1) for (blue) $M=40$, (gray) $M=1000$, (red) $k$-space simulation considering elastic scatters, and (black dashed) analytic Eq. (17). (b) Pressure map distribution obtained by numerical evaluation of the Rayleigh-Sommerfeld diffraction integral Eq. (1) for a spiral of $M=40$. (c) Transversal pressure distribution at $z/\lambda =25$ obtained from Eq. (1) for (blue) $M=40$, (red) $k$-space simulation considering elastic scatters, and (black dashed) analytic Eq. (17). (d) Far field showing the characteristic ring of the Bessel beam, where $k_r=2\pi /a$.

Figure 4

$n\mathrm{th}$ order Bessel beams formed by spiral gratings of $n$ arms. (a)–(d) Transversal pressure distribution at $z=z_F$ and $y=0$, (continuous line) Eq. (17), and (markers) $k$-space numerical solution of the wave equation assuming a steel grating embedded in water. Pressure magnitude obtained at (e)–(h) $z=F$ and (i)–(l) $x=0$. Color bars in normalized units $p/p_{max}$. (m)–(p) The phase of the field is calculated at $z=z_F$; color bars are in normalized units of $\varphi /\pi$.

Figure 5

(a) Spiral grating used in experiments. The inset shows a zoom were the spiral pattern can clearly be seen. (b) Pressure map distribution obtained by numerical integration of the Rayleigh-Sommerfeld integral. (c) Axial pressure distribution at $x=0.45\lambda$ obtained by the Rayleigh integral for the finite aperture grating (continuous), theoretical for infinite aperture grating (dashed), and experiments (markers). (d), (i), and (n) Experimental field magnitude and (e), (j), and (o) theoretical cross sections at $z_1=8.8\lambda$ (6 mm), $z_2=17.6\lambda$ (12 mm), and $z_1=26.5\lambda$ (18 mm), respectively; (f), (k), and (p) and (g), (l), and (q) corresponding phases. Color bars in normalized units ($p/p_0$ for the field magnitude and $\varphi /\pi$ for the phase). (h), (m), and (r) Traversal beam cross sections at different axial distances obtained by the Rayleigh integral for the finite aperture grating (continuous), analytic expression of the first-order Bessel beam (dashed), and experiments (markers). The vertical dotted line represents the position where the field is null.