Acoustic Beamforming Using Different Types of Lenses

Abstract

Acoustic beamforming is a fundamental technique for controlling the direction and shape of sound waves, enabling targeted delivery and precise sound manipulation. Traditionally achieved through arrays of sound sources, beamforming can also be realized using acoustic lenses that manipulate the wavefront through refraction, diffraction, or metamaterial properties. This white paper explores various types of acoustic lenses, including refractive, axicon, phased, metamaterial-based, and gradient-index (GRIN) lenses, highlighting their principles, applications, and specific advantages. Additionally, we discuss dynamic focusing techniques using variable thickness lenses, sectioned lenses with tunable fluid properties, and frequency-dependent phase manipulation. Through these advanced methods, acoustic lenses can achieve flexible and precise beam steering, offering significant potential for applications ranging from ultrasound imaging to particle manipulation and spatial audio systems. The integration of dynamic focusing mechanisms, such as the VARI-SOUND lens and sectioned acoustic lenses, marks a pivotal advancement, offering the ability to adjust focal points in real time, which is crucial for adaptive sound control in various technological contexts.

Acoustic Beamforming using Acoustic Lenses

Acoustic beamforming is a technique employed to control both the direction and shape of sound waves as they propagate through space. This is accomplished either by manipulating multiple sound sources-such as speaker arrays or transducers-or by utilizing specially designed lenses or structures that focus or steer sound energy toward a specific location. In essence, beamforming creates a “beam” of sound: instead of broadcasting sound in all directions, it allows for the targeted delivery of sound, much like a flashlight focuses light into a beam.

Applications of Acoustic Beamforming

Examples of applications include, but are not limited to:

Ultrasound imaging: Focusing sound deep inside the body for clear internal scans.
Acoustic communication: Directing signals along specific paths, such as in underwater sonar or wireless acoustic telemetry.
3D audio systems: Simulating directional sound perception in headsets or speakers.
Particle manipulation: Using focused acoustic waves to trap and move small objects (acoustic tweezers).
Noise control: Steering sound away from sensitive areas.

When beamforming is achieved by manipulating sound sources (like speaker arrays and transducers), no lens is necessary. However, lenses provide the opportunity to control the direction and shape of a sound wave without altering the source, allowing for beamforming and steering with only a single source. Acoustic lenses are widely used in the ultrasonic regime but are also applied in the audio range (20 Hz–20 kHz). In the audio regime, lenses can be designed to focus or steer sound waves for specific applications, though they present unique challenges compared to the ultrasound regime.

Principles and Types of Acoustic Lenses

Acoustic lenses generally operate based on the principle of refraction: sound travels at different speeds through different materials. When a sound wave passes from one medium to another (for example, from air into plastic), its speed changes, causing the wave to bend-this phenomenon is known as refraction. Acoustic lenses exploit this by using curved surfaces or inhomogeneous materials to focus sound at a desired location.

Acoustic lenses typically fall into one of three categories:

a. Refractive Lenses (Convex or Concave)

Utilize curved shapes and uniform materials.
The shape causes wavefronts to bend and either converge (focus) or diverge.

Operation Principle

These lenses function based on the refraction of sound waves at the interface between two materials. When a sound wave passes through a curved interface between two media of different acoustic impedance and sound speed, it bends (refracts). This behavior is governed by the acoustic form of Snell’s Law: Commonly used in water or air, functioning similarly to optical lenses.

$\frac{sin \theta_1}{c_1} = \frac{sin \theta_2}{c_2}$

Where:

$\theta_1$ : Incident angle in the lens
$\theta_2$ : Refracted angle in the surrounding medium (e.g., water or tissue)
$c_1$ : Speed of sound in the lens material
$c_2$ : Speed of sound in the surrounding medium

Depending on the curvature and the speed of sound in the two media, the lens can cause the sound field to diverge or converge. For example, a convex lens with $c_1 < c_2$ will cause the wave to bend away from the normal, resulting in beam spreading (a diverging field). Such diverging acoustic lenses are useful in 3D imaging, as they expand the field of view by spreading acoustic energy over a wider angle. Audoin’s¹ work demonstrates a convex diverging acoustic lens, depicted in blue in Figure 1, positioned atop the transducer. The red line represents the sound beam emitted from one of the transducer elements, showing a wider spread compared to the scenario without the lens, indicated by the black dashed line. The resulting field of view (FOV) of the transducer, shown in beige, is determined by the lens’s radius of curvature and the ratio between the speed of sound in the lens material $v_1$ and the imaging medium $v_2$ ¹.

Figure 1. Convex diverging acoustic lens (blue) placed on a transducer, spreading the emitted sound beam (red) compared to the no-lens case (black dashed line), increasing the field of view (beige).

Axicon Lens

Another form of acoustic refracting lens is the axicon. An acoustic axicon is specifically engineered to transform a spherical or planar wavefront into a conical wavefront, thereby generating a nondiffracting beam-commonly known as a Bessel-like beam-that remains stable over a designated distance.

Unlike conventional focusing lenses, which concentrate acoustic energy at a single focal point, an axicon produces a line focus. This creates a narrow beam that retains a consistent width over a significant range. This unique property makes axicons particularly advantageous for applications such as:

Deep tissue imaging
High aspect-ratio particle trapping
Therapeutic ultrasound, where maintaining beam uniformity over depth is crucial

An acoustic axicon typically features a conical geometry with the following characteristics:

Base: Flat and directly coupled to the transducer face
Tip: Aligned along the axis of sound propagation

The inclined surface of the axicon causes wavefronts to refract at different angles according to their radial position, with outer rays bending more steeply than inner ones.

To explain the axicon’s operation, we refer back to Snell’s law as previously discussed. However, unlike the lens depicted in Figure 1, the inverted conical shape of the axicon shown in Figure 2 leads to the formation of a nondiffracting (Bessel-like) beam.

Due to this conical configuration, each portion of the wavefront refracts at a distinct radial angle, resulting in constructive interference of the refracted wavefronts along the central axis, which produces a narrow, core beam. An example of an axicon can be observed in Figure 2 ².

Figure 2. An example of an axicon²

b. Phased or Metamaterial-Based Lenses

The invention of Fresnel lenses in optics demonstrated that stepped-profile lenses can replicate the focusing effect of conventional lenses. This concept also holds true in acoustic lenses. For instance, within the beams produced by axicons, it is feasible to create stepped profiles that replicate the refraction characteristics of axicons through phase modulation. Figure 3 shows a fraxicon mounted on a transducer, which is a stepped-profile lens designed to mimic the function of an axicon. fraxicons ³ have shown potential in applications such as acoustic manipulation and tweezers and can even be employed as acoustic bullets.

Stepped, phased, and metamaterial-based acoustic lenses represent advanced designs for controlling sound propagation. Stepped lenses are composed of discrete segments or layers, each engineered with specific curvature or thickness to modulate the wavefront’s phase and direction. Phase shifts are precisely controlled, enabling dynamic beam shaping and steering for adaptive focusing. Metamaterial-based lenses exploit artificially engineered materials to achieve unconventional acoustic behaviors, including negative refraction and superlensing, facilitating precise and innovative sound field manipulation.

Figure 3. A fraxicon mounted on an ultrasound transducer³

VARI-SOUND ⁴

One example of an acoustic metamaterial is the VARI-SOUND lens, as shown in Figure 4.It is a varifocal acoustic lens created using 3D-printed acoustic metamaterials. It is designed to mimic the function of optical devices for sound, enabling compact, adjustable focusing of audio much like a zoom lens does for light. This technology allows a standard speaker to be transformed into a directional sound source and enables the creation of devices such as acoustic “spotlights” and “magnifying glasses” for sound. The VARI-SOUND lens can dynamically adjust its focus, opening up new possibilities for immersive and personalized audio experiences without the need for bulky speaker arrays or headphones.

The physics of the VARI-SOUND lens is based on acoustic metamaterials-engineered structures made from conventional materials (like plastic) but with carefully designed internal microstructures that manipulate sound waves in novel ways. These metamaterials are assembled into thin metasurfaces, each composed of sub-wavelength unit cells that locally control the phase of incoming sound waves. By arranging these unit cells with varying geometries, the lens can shape and focus sound much like an optical lens bends and focuses light, but at a fraction of the wavelength’s thickness. The design leverages principles from optics, such as the thin-lens equation, to achieve tunable focusing. By mechanically adjusting the configuration of the metasurfaces, the focal length of the lens can be changed in real time, allowing for dynamic control over where sound is concentrated or directed. This approach overcomes traditional limitations of acoustic lenses-namely, their bulkiness and static nature-making them practical for real-world applications in spatial audio, VR, and beyond.

Figure 4. An adjustable focus acoustic lens, i.e., the equivalent of a zooming objective for light⁴

c. GRIN (Gradient-Index) Lenses

GRIN (Gradient-Index) lenses operate based on a gradual change in the speed of sound within the lens material, which results in a smooth bending of the wave path. These lenses are typically constructed using engineered metamaterials or materials with graded porosity, allowing for precise control over the acoustic refractive index. For instance, a GRIN lens may be designed so that the refractive index is higher (corresponding to a slower sound speed) at the center and lower (faster sound speed) at the edges. This configuration causes incoming sound waves to bend toward the focal point, effectively focusing the acoustic energy.

Applications

GRIN lenses allow for compact, flat, and highly efficient focusing or beam-shaping of sound, with uses in ultrasound imaging, non-destructive testing, and directional audio delivery. A notable example is the 3D acoustic Luneburg lens ⁵, made from a lattice of 3D-cross-shaped metamaterial unit cells with a refractive index gradient from 1 at the edge to 1.5 at the center. This design enables wide field-of-view acoustic imaging with minimal aberration.

d. Dynamic Focusing in Acoustic Lenses

While most of the lenses discussed in the previous sections have fixed focal points, the VARI-SOUND lens represents a dynamic focusing lens. Dynamic focusing can be achieved by leveraging the structure and underlying principles of the lens. Below are a few additional examples that demonstrate this concept.

1. Variable Thickness Lenses

Adjusting the thickness profile of a lens in real time (e.g., using liquid-filled lenses where internal pressure alters curvature) enables dynamic focusing by changing phase delays across the lens. An acoustic lens with dynamically adjustable thickness modulates the time waves spend traversing the lens material. For example, a curved lens-with greater thickness at the center-delays peripheral rays more than central rays, especially if the material propagates sound faster than the surrounding medium. Modifying the lens curvature in real time alters the thickness profile and hence the phase delays, enabling dynamic focusing.

Figure 5. Example of a variable thickness acoustic lens, where curvature changes the phase delay imparted on sound rays to focus them at different depths

Dynamic curvature control has been experimentally demonstrated, notably through liquid-filled lenses where internal pressure alters the curvature profile ⁶

2. Sectioned Lenses⁷

Sectioned lenses consist of channels filled with materials or fluids of different sound speeds. By dynamically altering properties such as salinity in each channel, the phase profile across the lens can be tuned in real time, allowing for dynamic focus control. For example, a concentric channel lens filled with saltwater solutions at varying concentrations can modulate the phase profile and achieve constructive interference at the desired focal point.

Figure 6 shows a side view of a concentric channel lens with uniform depth of but filled with different internal materials or fluids, each exhibiting a distinct sound speed. As sound waves travel through these sections, they undergo varying phase shifts . If water is used in each section, for instance, adjusting the salinity levels can change the speed of sound, thereby modifying the phase shift. By dynamically altering these fluid properties, the phase contribution from each section can be precisely tuned in real time.

Rostami and Mobley^7,8 demonstrated a concentric channel lens filled with saltwater solutions at varying concentrations. These variations modulate the phase profile across the lens aperture, allowing for dynamic focus control.

Figure 6. Illustration of a sectioned acoustic lens where sound speed is varied across channels by modulating fluid properties such as salinity, resulting in real-time adjustable focus. By appropriately adjusting the salt concentration-and thus the sound speed-in

3. Frequency-Dependent Phase Manipulation

Phase modulation can also be achieved through frequency control. Binary acoustic lenses consist of alternating structures, such as a lens made with channels where the walls are composed of Polylactic Acid (PLA) and the channels are filled with water. These materials exhibit different acoustic propagation speeds, creating a phase shift profile across the lens. Since the phase shift introduced by each section is frequency-dependent, altering the input frequency changes the effective phase delay imparted by each element. This, in turn, shifts the location of constructive interference, thereby moving the focal point.

This approach has been experimentally demonstrated by Hu et al ⁹. Binary acoustic lenses hold significant potential for dynamic focusing on applications like medical ultrasound, non-invasive brain stimulation, and targeted acoustic delivery systems.

Conclusion

Acoustic lenses represent a powerful approach to beamforming, allowing for focused and directional sound without the need for complex multi-source arrays. By leveraging principles such as refraction and phase modulation, lenses can effectively control sound wave propagation, leading to applications in medical imaging and audio engineering. Among the explored lens types, refractive lenses like convex and concave designs offer fundamental control over wave convergence and divergence, while axicons generate nondiffracting beams for precise long lines of focus beams. Advanced lenses, including phased, metamaterial-based, and GRIN designs, enable more nuanced acoustic manipulation, particularly in complex environments. The development of dynamically adjustable lenses, such as the VARI-SOUND and sectioned lenses, further expands the capability of acoustic beamforming by introducing real-time focus adaptation. These innovations provide practical solutions for both ultrasonic and audio frequency ranges, offering new possibilities in adaptive sound systems, medical technologies, and spatial audio applications. As research progresses, continued optimization of lens materials, geometries, and dynamic control methods will enhance the versatility and precision of acoustic beamforming, promoting its adoption across diverse scientific and industrial domains.

References

M. Audoin, A. Salari, B. G. Tomov, J. Arendt Jensen, and E. V. Thomsen, “Novel Diverging Acoustic Lens Geometries for Row-Column Array Transducers,” in 2023 IEEE International Ultrasonics Symposium (IUS) (IEEE, 2023) pp. 1–4. doi:10.1109/IUS51837.2023.10306503
R. Lirette, and J. Mobley, “Broadband wave packet dynamics of minimally diffractive ultrasonic fields from axicon and stepped fraxicon lenses,” J Acoust Soc Am 146, 103–108 (2019). doi:10.1121/1.5116011
R. Lirette, and J. Mobley, “Ultrasonic near-field based acoustic tweezers for the extraction and manipulation of hydrocarbon droplets,” , doi: 10.1063/5.0122269. doi:10.1063/5.0122269
G. Memoli, L. Chisari, J. P. Eccles, M. Caleap, B. W. Drinkwater, and S. Subramanian, “VARI-SOUND,” in Proceedings of the 2019 CHI Conference on Human Factors in Computing Systems (New York, NY, USA, ACM, 2019) pp. 1–14. doi:10.1145/3290605.3300713
Y. Xie, Y. Fu, Z. Jia, J. Li, C. Shen, Y. Xu, H. Chen, et al., “Acoustic Imaging with Metamaterial Luneburg Lenses,” Sci Rep 8, 16188 (2018). doi:10.1038/s41598-018-34581-7
Z. Li, R. Guo, C. Fei, D. Li, D. Chen, C. Zheng, R. Wu, et al., “Liquid lens with adjustable focus for ultrasonic imaging,” , doi: 10.1016/j.apacoust.2020.107787. doi:10.1016/j.apacoust.2020.107787
Sina Rostami, The characterization of radiation forces using tunable acoustic lenses, (Oxford, University of Mississippi , 2025).
Sina Rostami, and Joel Mobley, “Liquid Channel Tunable Lenses for Phased-based Ultrasonic Beamforming.,”
Z. Hu, Y. Yang, L. Xu, Y. Hao, and H. Chen, “Binary acoustic metasurfaces for dynamic focusing of transcranial ultrasound,” , doi: 10.3389/fnins.2022.984953. doi:10.3389/fnins.2022.984953

Complete Communications Engineering

Acoustic Beamforming Using Different Types of Lenses

Abstract

Acoustic Beamforming using Acoustic Lenses

Applications of Acoustic Beamforming