Three-dimensional acoustic lensing with a bubbly diamond metamaterial

At a Glance

Metadata	Details
Publication Date	2021-06-25
Journal	Journal of Applied Physics
Authors	Maxime Lanoy, Fabrice Lemoult, Geoffroy Lerosey, Arnaud Tourin, Valentin Leroy
Institutions	Université Paris Sciences et Lettres, Sorbonne Université
Citations	7
Analysis	Full AI Review Included

Technical Documentation & Analysis: Bubbly Diamond Metamaterials

Executive Summary

This research demonstrates the design and numerical verification of a three-dimensional (3D) acoustic flat lens utilizing a crystalline metamaterial structure analogous to the diamond lattice. This breakthrough has significant implications for high-resolution acoustic imaging and sensing.

Core Achievement: Numerical demonstration of a 3D acoustic flat lens achieving a negative refractive index ($n = -1$) at 2929 Hz, enabling super-resolution focusing below the diffraction limit (FWHM 0.4$\lambda$₀ in the near field).
Material Design: The metamaterial consists of a periodic assembly of air bubbles (monopolar scatterers) embedded in water, arranged in a diamond crystal structure.
Key Mechanism: The bi-periodic nature of the diamond lattice introduces an optical branch with a negative slope in the dispersion relation, leading to all-angle negative refraction.
Fabrication Requirement: Practical realization relies on recent advances in high-precision 3D printed frames to maintain stable, periodic air bubble assemblies.
6CCVD Value Proposition: 6CCVD provides the ultra-hard, high-precision MPCVD diamond substrates and custom fabrication services (laser cutting, etching) necessary to create the rigid, stable 3D frames required for practical implementation of these crystalline acoustic metamaterials.
Isotropy: The diamond structure ensures highly isotropic propagation (variation in wavevectors < 0.3% at $n = -1$), critical for avoiding aberration effects in 3D lensing.

Technical Specifications

The following parameters were extracted from the numerical analysis of the bubbly diamond metamaterial:

Parameter	Value	Unit	Context
Scatterer Type	Air Bubble	N/A	Monopolar resonant scatterer in water host medium
Bubble Radius ($r$)	1	mm	Used for Minnaert Resonance calculation
Lattice Constant ($a$)	11.8	cm	Separation distance for FCC/Diamond lattice
Minnaert Resonance Frequency ($f_m$)	2890	Hz	Resonance frequency of a single 1 mm bubble
Wavelength in Water ($\lambda$₀)	52	cm	Corresponds to $f_m$ in water
Target Frequency for $n = -1$	2929	Hz	Frequency achieving $n = -1$ for standard concentration
Diluted Target Frequency for $n = -1$	2910	Hz	Frequency achieving $n = -1$ for 4x diluted concentration
Slab Thickness (Refraction Test)	1.5$\lambda$₀	N/A	Used for Gaussian beam negative refraction demonstration
Super-Resolution FWHM	0.4$\lambda$₀	N/A	Full Width at Half Maximum achieved in the near field ($n \approx -3.2$)
Refractive Index (Near-Field Focus)	$\approx -3.2$	N/A	Achieved at 2917 Hz for enhanced near-field focusing
Incident Angle (Refraction Test)	45	°	Angle of Gaussian beam relative to the interface normal

Key Methodologies

The research relied on precise structural engineering and numerical simulation to demonstrate the acoustic properties of the metamaterial.

Scatterer Selection: Utilized air bubbles in water as effective ultrasonic low-frequency monopolar resonant scatterers, exploiting the Minnaert resonance.
Lattice Design (FCC): Initial design placed 1 mm radius bubbles on the lattice sites of a Face-Centered Cubic (FCC) structure with a lattice constant $a = 11.8$ cm.
Lattice Modification (Diamond): The diamond structure was achieved by adding a second identical bubble to the primitive unit cell (at coordinates 0,0,0 and 1/4, 1/4, 1/4 in units of $a$). This bi-periodic structure is crucial for generating the negative-slope optical branch.
Numerical Modeling: The eigenvalue problem for the infinite crystal was solved using COMSOL Multiphysics with periodic boundary conditions to obtain the band structure and dispersion relations.
Lensing Simulation: Multiple scattering problems were solved for a finite slab (thickness 1.5$\lambda$₀) excited by a Gaussian beam (for refraction) or a point source (for focusing/imaging).
Damping Mitigation Strategy: Proposed solutions for practical implementation include increasing bubble concentration (to reduce attenuation) or using rigid 3D printed frames to stabilize larger bubbles (to reduce thermal and viscous damping).

6CCVD Solutions & Capabilities

The practical realization of complex 3D acoustic metamaterials, such as the bubbly diamond structure, requires ultra-precise, rigid, and chemically inert structural components. 6CCVD’s MPCVD diamond materials are ideally suited to serve as the high-fidelity frames or substrates necessary to maintain the exact periodic arrangement of scatterers.

Applicable Materials

To replicate or extend this research into a practical device, 6CCVD recommends the following materials for the structural frame/substrate:

Material Recommendation	Rationale & Application	6CCVD Capability
Optical Grade SCD	Required for ultimate precision in small unit cells or high-frequency applications where structural tolerances are tight (Ra < 1 nm). Provides maximum rigidity and thermal stability.	SCD thickness from 0.1 µm up to 500 µm. Polishing to Ra < 1 nm.
Large Area PCD Substrates	Ideal for scaling the metamaterial into large-area flat lenses (e.g., 112$\lambda$₀ width used in the simulation). PCD offers excellent mechanical stability and chemical inertness in water environments.	Plates/wafers up to 125 mm in diameter. Thickness up to 500 µm (or 10 mm for substrates). Polishing to Ra < 5 nm.
Boron-Doped Diamond (BDD)	If the application requires integrated electrodes or active tuning mechanisms within the frame structure (e.g., for electro-acoustic control), BDD provides a conductive, robust platform.	Custom doping levels available for specific resistivity requirements.

Customization Potential for Metamaterial Fabrication

The creation of the precise, periodic diamond lattice structure requires advanced micro-fabrication capabilities that 6CCVD specializes in:

Custom Dimensions: The simulated slab width (112$\lambda$₀) is substantial. 6CCVD can supply large-format PCD wafers up to 125 mm, allowing researchers to build large-scale acoustic lenses.
Precision Etching/Laser Cutting: To create the intricate 3D frame structure necessary to hold the bubbles in the exact diamond lattice positions (0,0,0 and 1/4, 1/4, 1/4), 6CCVD offers high-resolution laser cutting and etching services. This ensures the structural periodicity ($a = 11.8$ cm) is maintained with micron-level accuracy.
Custom Thickness: The slab thickness (1.5$\lambda$₀) is critical for focusing performance. We provide SCD and PCD materials in custom thicknesses from 0.1 µm to 10 mm to match specific acoustic impedance and wavelength requirements.
Metalization Services: While this paper focuses on acoustic waves, future integration with electronic control or sensing elements (e.g., piezoelectric transducers) may require metal contacts. 6CCVD offers in-house metalization using Au, Pt, Pd, Ti, W, and Cu.

Engineering Support

The successful transition from numerical simulation to a functional 3D acoustic metalens hinges on overcoming the damping issue and achieving high structural fidelity.

Acoustic Metamaterial Design: 6CCVD’s in-house PhD engineering team specializes in the material science of CVD diamond and can assist researchers in selecting the optimal diamond grade (SCD vs. PCD) and thickness for their specific acoustic impedance matching requirements.
Structural Stability: We provide consultation on designing ultra-rigid diamond frames that minimize mechanical vibration and maintain the precise lattice constant ($a$) required for stable negative refraction and super-resolution focusing projects.

For custom specifications or material consultation, visit 6ccvd.com or contact our engineering team directly.

View Original Abstract

A sound wave travelling in water is scattered by a periodic assembly of air bubbles. The local structure matters even in the low frequency regime. If the bubbles are arranged in a face-centered cubic (fcc) lattice, a total bandgap opens near the Minnaert resonance frequency. If they are arranged in the diamond structure, which one obtains by simply adding a second bubble to the unit cell, one finds an additional branch with a negative slope (optical branch). For a single specific frequency, the medium behaves as if its refractive index (relative to water) is exactly n=−1. We show that a slab of this material can be used to design a three-dimensional flat lens. We also report super-resolution focusing in the near field of the slab and illustrate its potential for imaging in three dimensions.

Tech Support

Original Source

DOI: https://doi.org/10.1063/5.0046090

References

1967 - Electrodynamics of substances with simultaneously negative values of ɛ and μ [Crossref]
2001 - Experimental verification of a negative index of refraction [Crossref]
2000 - Negative refraction makes a perfect lens [Crossref]
1998 - Low frequency plasmons in thin-wire structures [Crossref]
1996 - Extremely low frequency plasmons in metallic mesostructures [Crossref]
1999 - Magnetism from conductors and enhanced nonlinear phenomena [Crossref]
2000 - Locally resonant sonic materials [Crossref]
2006 - Ultrasonic metamaterials with negative modulus [Crossref]
2004 - Double-negative acoustic metamaterial [Crossref]
2015 - Soft 3D acoustic metamaterial with negative index [Crossref]