🔊 Acoustic Lens

A Fresnel zone plate is a flat structure with concentric annular rings alternately transparent and opaque (or phase-reversing). Waves passing through the open rings arrive at the focal point with the same phase and interfere constructively. The radius of the m-th ring is r_m = √(mλf), where λ is the wavelength and f is the focal length. Zone plates can focus sound without curved geometry — useful when manufacturing a curved lens is impractical — but they are inefficient because they block (or absorb) half the incident wave. They are used in acoustic imaging, lithotripsy, and ultrasonic manipulation.

A GRIN lens has a continuously varying sound speed across its cross-section — fastest at the edges, slowest at the centre (for convergence) — rather than relying on interface refraction. Rays bend gradually rather than abruptly, eliminating interface reflections and allowing very wide aperture. GRIN acoustic lenses can be realised using structured metamaterials — periodic arrays of sub-wavelength inclusions (rods, plates) — where the effective sound speed is controlled by the fill fraction. Phononic crystal GRIN lenses have been demonstrated for ultrasonic imaging at frequencies from kHz to MHz.

For a plano-convex lens of radius of curvature R and refractive index n, the thin-lens formula gives f = R/(n – 1). The simulation estimates the focal length from the lens geometry and displays both the theoretical and observed focal distance. The observed focal length is measured by finding the position of peak pressure amplitude on the axis after the lens, which may differ from the theoretical value due to diffraction effects, finite aperture, and the fact that the thin-lens approximation breaks down when the lens diameter is comparable to f.

Focal gain is the ratio of acoustic pressure amplitude at the focal point to the incident pressure amplitude. For a circular lens of diameter D and focal length f operating at wavelength λ, the diffraction-limited gain is approximately G ≈ (πD²)/(4λf) — the "f-number" ratio determines the limit. In medical high-intensity focused ultrasound (HIFU), focused transducers achieve gains of 100–1,000, concentrating enough energy to ablate tumour tissue at the focal point while leaving surrounding tissue unharmed.

The wavelength λ = c₀/f changes with frequency, affecting diffraction. At higher frequencies the wavelength is shorter relative to the lens aperture, so geometric (ray) optics becomes a better approximation and the focal spot is sharper. At lower frequencies, diffraction spreads the beam more significantly and the focal spot broadens. Very low frequencies may not produce a visible focus at all if the wavelength exceeds the lens diameter. This frequency-dependence (chromatic aberration in acoustics) is important in wideband pulse imaging systems.

A concave lens (thinner at the centre than the edges) made of slower material causes the central wavefronts to travel through less slow medium than edge wavefronts, so the centre arrives ahead and the wavefront bows outwards — a diverging effect. Rays appear to originate from a virtual focal point on the source side of the lens. Concave acoustic lenses are used to spread sound over a wide area (e.g. in underwater sonar to achieve broad-beam coverage) or to collimate diverging waves from a point source into a planar wavefront.

A phononic crystal is a periodic arrangement of two (or more) materials with contrasting acoustic impedances, typically with periodicity on the order of the acoustic wavelength. Like photonic crystals for light, phononic crystals can create band gaps (frequency ranges that cannot propagate), produce negative effective density or bulk modulus, and achieve refractive indices not found in natural materials — including n < 0. Negative-index phononic crystals can act as "acoustic superlenses" that overcome the diffraction limit, allowing sub-wavelength imaging, an active area of research.

Medical HIFU (High-Intensity Focused Ultrasound) uses focused transducer arrays to ablate prostate, uterine fibroid, and brain tumour tissue without incisions, with FDA approvals since 2004. Lithotripsy shatters kidney stones using focused shockwaves. Acoustic tweezers use focused standing waves to trap and sort cells, bacteria, and micro-particles by size in lab-on-chip devices. Sonar arrays use phased-array beam-forming (electronic equivalent of a lens) for underwater mapping. Industrial ultrasonic testing focuses beams on welds and castings to detect internal cracks at sub-millimetre resolution.