Acoustic Index: Fundamentals & Applications

• Acoustic Index is a dual-use parameter that defines sound speed contrasts in structured media and gauges cardiac risk in echocardiography.
• In acoustic physics, it describes the refractive index analogue, enabling waveguiding, scattering control, and efficient design of metamaterials and gradient-index devices.
• In clinical imaging, the Acoustic Index fuses dynamic echocardiographic data and clinical metrics to accurately indicate cardiac dysfunction on a risk scale from 0 to 1.

Acoustic index is a context-dependent term with two distinct uses in the sources considered here. In acoustics, phononics, and acoustic metamaterials, it denotes the acoustic analogue of refractive index: a parameter tied to sound or elastic-wave velocity, effective wavenumber, and, in structured media, effective constitutive parameters. In that sense, slower phase velocity corresponds to higher acoustic index, and spatial or modal contrasts in acoustic index underlie guiding, scattering control, gradient-index focusing, and zero-index transport. In a separate clinical literature, “Acoustic Index” denotes a continuous AI-derived echocardiographic parameter that maps standard ultrasound sequences and clinical metadata to a risk score between 0 and 1 for cardiac dysfunction (Wang et al., 2020, Wei et al., 2020, Begiashvili et al., 17 Jul 2025).

1. Physical definitions and mathematical forms

In fluid acoustics, the acoustic refractive index is defined relative to a background medium by the sound-speed ratio

$n=\frac{c_0}{c_1},$

and, for a scatterer with density $\rho_1$ and compressibility $\beta_1$ in a background $(\rho_0,\beta_0)$, equivalently by

$$n=\sqrt{\bar{\rho}\,\bar{\beta}}, \qquad \bar{\rho}=\frac{\rho_1}{\rho_0},\;\bar{\beta}=\frac{\beta_1}{\beta_0}.$$

The associated internal wavenumber is $k_1 = k_0 n$, with $k_0=2\pi/\lambda$. High acoustic index therefore means a much lower sound speed inside the medium than in the background, or $n\gg 1$ (Wei et al., 2020).

In elastic and phononic settings, the same concept is used through wave velocity rather than a single fluid sound speed. For guided elastic waves on GaN-on-sapphire, the relevant quantity is an effective inverse acoustic velocity,

$n_{\text{ac}} \propto \frac{1}{v_{\text{ac}}},$

so a material with slower longitudinal and transverse wave speeds acts as the higher-acoustic-index region. The GaN guiding layer is slower and denser than sapphire, so GaN functions as a high-acoustic-index core and sapphire as a low-acoustic-index cladding; the index-contrast condition is expressed as $v_{t,\text{strip}}<v_{t,\text{sub}}$ and $\rho_1$0 (Wang et al., 2020).

Several literatures use closely related normalizations. In underwater gradient-index media, the normalized acoustic refractive index is written as

$\rho_1$1

or, in the notation of a transparent GRIN lens,

$\rho_1$2

In zero-index metamaterials, the effective acoustic index is defined by the wavenumber ratio

$\rho_1$3

so $\rho_1$4 corresponds to $\rho_1$5 and an effectively infinite wavelength inside the medium (Martin et al., 2015, Wang et al., 2013).

A recurrent qualification is that, in solids, the notion of a single scalar index is approximate. Acoustic waves are vector elastic waves with multiple polarizations, and the effective index is often mode-specific. This suggests that “acoustic index” is most precise when tied to a particular propagating or resonant mode rather than treated as a universal material scalar (Wang et al., 2020).

2. Index contrast, confinement, and guided-wave devices

High acoustic index contrast is the direct acoustic analogue of the high-index-core/low-index-cladding paradigm in photonics. In the GaN-on-sapphire phononic platform, this contrast enables unsuspended strip waveguides and ring resonators with sub-wavelength cross-section. The strip supports quasi-Rayleigh and quasi-Love mode families, including symmetric and anti-symmetric branches, and confinement is quantified through the mode area

$\rho_1$6

and the energy confinement ratio

$\rho_1$7

Modes with phase velocity below the substrate surface-acoustic-wave speed remain bound and decay evanescently into the substrate; as phase velocity approaches the substrate continuum, $\rho_1$8 grows and $\rho_1$9 falls (Wang et al., 2020).

Bending the same high-index strip into a ring produces acoustic whispering-gallery modes,

$\beta_1$0

with mode volume

$\beta_1$1

In the numerical model, only radiation into the substrate via bending is included, and the quality factor

$\beta_1$2

increases exponentially with radius. Optimized GaN-on-sapphire rings are predicted to reach $\beta_1$3 up to $\beta_1$4. The same study also reports avoided crossings and sharp $\beta_1$5 enhancement interpreted as parameter-tuned bound states in the continuum generated by destructive interference of leakage into the substrate (Wang et al., 2020).

The same platform supports directional couplers formed by parallel high-index waveguides. Their coupling is described by the coupled-mode equations

$\beta_1$6

with full transfer length

$\beta_1$7

For identical waveguides, the supermode splitting is $\beta_1$8; for dissimilar waveguides, efficient transfer requires the phase-matching condition $\beta_1$9. Interdigitated transducers deposited on the ring add direct microwave-to-phonon excitation, with impedance

$(\rho_0,\beta_0)$0

and conversion efficiency $(\rho_0,\beta_0)$1, where $(\rho_0,\beta_0)$2. By choosing the radius so that $(\rho_0,\beta_0)$3, the system reaches critical coupling, $(\rho_0,\beta_0)$4 and $(\rho_0,\beta_0)$5 (Wang et al., 2020).

A complementary airborne route to high acoustic index uses metamaterials rather than intrinsic material contrast. A periodic lattice of air-filled acoustic pipes was shown to behave as a homogeneous high-index medium with

$(\rho_0,\beta_0)$6

and a fitted low-band value $(\rho_0,\beta_0)$7. That effective medium supports acoustic analogues of dielectric slab waveguides, strip waveguides, and optical fibers, with mode profiles that are sinusoidal in the core and evanescent in air, and with open guiding rather than hard-wall confinement (Zangeneh-Nejad et al., 2018).

3. Scattering, directional sources, and acoustic angular spectra

In scattering theory, high acoustic index is valuable because it enables strong and tunable monopole and dipole resonances while higher multipoles remain comparatively small over a subwavelength size range. For a spherical acoustic scatterer, the induced monopole and dipole strengths are

$(\rho_0,\beta_0)$8

and in the deep subwavelength limit

$(\rho_0,\beta_0)$9

If both $$n=\sqrt{\bar{\rho}\,\bar{\beta}}, \qquad \bar{\rho}=\frac{\rho_1}{\rho_0},\;\bar{\beta}=\frac{\beta_1}{\beta_0}.$$0 and $$n=\sqrt{\bar{\rho}\,\bar{\beta}}, \qquad \bar{\rho}=\frac{\rho_1}{\rho_0},\;\bar{\beta}=\frac{\beta_1}{\beta_0}.$$1, the monopole and dipole are in phase and can be tuned toward equality, a condition central to directional scattering (Wei et al., 2020).

The underlying framework is the acoustic angular spectrum. For a source composed of a monopole $$n=\sqrt{\bar{\rho}\,\bar{\beta}}, \qquad \bar{\rho}=\frac{\rho_1}{\rho_0},\;\bar{\beta}=\frac{\beta_1}{\beta_0}.$$2 and dipole $$n=\sqrt{\bar{\rho}\,\bar{\beta}}, \qquad \bar{\rho}=\frac{\rho_1}{\rho_0},\;\bar{\beta}=\frac{\beta_1}{\beta_0}.$$3, the pressure field can be decomposed as

$$n=\sqrt{\bar{\rho}\,\bar{\beta}}, \qquad \bar{\rho}=\frac{\rho_1}{\rho_0},\;\bar{\beta}=\frac{\beta_1}{\beta_0}.$$4

with spectrum

$$n=\sqrt{\bar{\rho}\,\bar{\beta}}, \qquad \bar{\rho}=\frac{\rho_1}{\rho_0},\;\bar{\beta}=\frac{\beta_1}{\beta_0}.$$5

Zeros of this spectrum inside the propagating circle $$n=\sqrt{\bar{\rho}\,\bar{\beta}}, \qquad \bar{\rho}=\frac{\rho_1}{\rho_0},\;\bar{\beta}=\frac{\beta_1}{\beta_0}.$$6 yield far-field directionality, and zeros outside it yield near-field directionality toward evanescent states (Wei et al., 2020).

This formalism supports two canonical source classes. A Huygens acoustic source satisfies the acoustic Kerker condition

$$n=\sqrt{\bar{\rho}\,\bar{\beta}}, \qquad \bar{\rho}=\frac{\rho_1}{\rho_0},\;\bar{\beta}=\frac{\beta_1}{\beta_0}.$$7

which suppresses backward scattering. A Janus acoustic source uses a quadrature relation between monopole and dipole, implemented for a realistic scatterer as

$$n=\sqrt{\bar{\rho}\,\bar{\beta}}, \qquad \bar{\rho}=\frac{\rho_1}{\rho_0},\;\bar{\beta}=\frac{\beta_1}{\beta_0}.$$8

and achieves face-selective coupling to nearby guided modes. In the abstract source model, one Janus condition is

$$n=\sqrt{\bar{\rho}\,\bar{\beta}}, \qquad \bar{\rho}=\frac{\rho_1}{\rho_0},\;\bar{\beta}=\frac{\beta_1}{\beta_0}.$$9

These constructions depend on high acoustic index because the required amplitude and phase relations between $k_1 = k_0 n$0 and $k_1 = k_0 n$1 are difficult to reach with low-contrast scatterers (Wei et al., 2020).

4. Gradient-, zero-, and equivalent-index media

Spatially varying acoustic index profiles underpin gradient-index devices. A classical example is the acoustic Luneburg lens with radial profile

$k_1 = k_0 n$2

implemented experimentally as a two-dimensional cylinder composed of 701 aluminum columns with various radii. That device focuses incoming sound on the opposite edge over 1,000–3,200 Hz, with amplitude enhancement by 3–4 times, corresponding to 10–15 dB, and was proposed as a sonar element (Kim, 2014).

Later work introduced a reduced aberration acoustic Luneburg lens using

$k_1 = k_0 n$3

where $k_1 = k_0 n$4 is optimized through the aberration error

$k_1 = k_0 n$5

With $k_1 = k_0 n$6, the resulting 2D and 3D additively manufactured lenses were shown theoretically, numerically, and experimentally to provide broadband and omnidirectional collimation and focusing over 20–40 kHz, while reducing aberrations relative to a modified Luneburg profile (Zhao et al., 2020).

A distinct underwater GRIN program used a transparent lens in which the effective index is lower than the water background, so propagation control occurs by phase advance rather than phase delay. There the normalized index is

$k_1 = k_0 n$7

with $k_1 = k_0 n$8 corresponding to $k_1 = k_0 n$9. The index gradient is generated by varying the filling fraction of hollow, air-filled aluminum tubes in a sonic-crystal lattice that is impedance matched to water over a broad bandwidth. Focusing was observed in the homogenization limit with intensity magnifications in excess of 7 dB and reflected pressure levels $k_0=2\pi/\lambda$0 dB (Martin et al., 2015).

At the opposite limit, zero-index metamaterials realize $k_0=2\pi/\lambda$1, $k_0=2\pi/\lambda$2, and almost uniform phase inside the medium. In a two-dimensional waveguide containing a zero-index slab and a cylindrical solid defect, the transmission coefficient is

$k_0=2\pi/\lambda$3

Total transmission occurs when

$k_0=2\pi/\lambda$4

whereas total reflection occurs when

$k_0=2\pi/\lambda$5

and $k_0=2\pi/\lambda$6. These conditions show how longitudinal and transverse elastic speeds in the defect govern wave transport through the zero-index region (Wang et al., 2013).

A further generalization appears in homogenization by perforation. For many small soft obstacles in $k_0=2\pi/\lambda$7, with $k_0=2\pi/\lambda$8 and minimum distance $k_0=2\pi/\lambda$9 for $n\gg 1$0, the far field converges to that of a penetrable medium obeying

$n\gg 1$1

with effective squared index

$n\gg 1$2

Here $n\gg 1$3 is the capacitance-derived geometry factor and $n\gg 1$4 the local obstacle density, so the equivalent index factorizes into geometric and distributional contributions (Ahmad et al., 2014).

5. Design automation and parameter retrieval

Practical use of acoustic index concepts depends on two technical capabilities: realizing prescribed index landscapes and retrieving effective parameters from measurements. The GRadient Index Pick-and-Place algorithm, GRIPP, addresses the first problem. It takes as input an analytic refractive-index distribution $n\gg 1$5 and a pre-defined library of unit cells with known effective refractive indices and impedances, and outputs a 3D model that is ready to be 3D printed. The demonstration uses a Gaussian-type profile

$n\gg 1$6

with $n\gg 1$7 mm, $n\gg 1$8 mm, and $n\gg 1$9 mm. Its 3D-cross unit-cell library spans effective indices from $n_{\text{ac}} \propto \frac{1}{v_{\text{ac}}},$0 to $n_{\text{ac}} \propto \frac{1}{v_{\text{ac}}},$1, with corresponding normalized impedances from $n_{\text{ac}} \propto \frac{1}{v_{\text{ac}}},$2 to $n_{\text{ac}} \propto \frac{1}{v_{\text{ac}}},$3, and the method exports the assembled geometry as an STL file (Xie et al., 2018).

Retrieval becomes more involved when a metamaterial sample and an impedance tube have different sizes. In that case, the sample and surrounding air gap are modeled as a bilayer inside the duct. From measured complex transmission and reflection coefficients, the effective properties are obtained by solving a set of eight linear equations once the dimensions are known. The final retrieval formulas use reconstructed averaged pressures and volume velocities at the sample faces to determine the effective impedance

$n_{\text{ac}} \propto \frac{1}{v_{\text{ac}}},$4

and the effective refractive index

$n_{\text{ac}} \propto \frac{1}{v_{\text{ac}}},$5

Numerical evaluation showed retrieval errors below 1% for effective refractive index and impedance (Khodaei et al., 2021).

Taken together, these methods indicate that acoustic index is not only a theoretical descriptor but also an engineering variable that can be specified, discretized, fabricated, and experimentally recovered. A plausible implication is that the maturity of acoustic-index workflows increasingly resembles that of optical effective-index design, especially in metamaterial and integrated-phononics contexts (Xie et al., 2018, Khodaei et al., 2021).

6. Acoustic Index as an echocardiographic risk parameter

In echocardiography, “Acoustic Index” denotes a different concept altogether. It is introduced as a single, continuous, AI-derived echocardiographic parameter that quantifies global cardiac dysfunction on a scale from 0 (low risk) to 1 (high risk). Formally, it is defined as a nonlinear fusion of dynamical risk, clinical risk, and cross-domain interaction, passed through a calibrated sigmoid,

$n_{\text{ac}} \propto \frac{1}{v_{\text{ac}}},$6

where $n_{\text{ac}} \propto \frac{1}{v_{\text{ac}}},$7 are Koopman eigenfunctions and eigenvalues from Extended Dynamic Mode Decomposition, $n_{\text{ac}} \propto \frac{1}{v_{\text{ac}}},$8 are attention weights, and $n_{\text{ac}} \propto \frac{1}{v_{\text{ac}}},$9 is a vector of normalized clinical parameters such as EF, age, and dimensions (Begiashvili et al., 17 Jul 2025).

The computational pipeline treats the heart as a nonlinear dynamical system and approximates the Koopman operator through EDMD. For video frames $v_{t,\text{strip}}<v_{t,\text{sub}}$0, the method constructs

$v_{t,\text{strip}}<v_{t,\text{sub}}$1

with eigendecomposition providing modes and eigenvalues whose real parts encode growth or decay and whose spatial structure localizes motion patterns. Each mode contributes dynamical risk through

$v_{t,\text{strip}}<v_{t,\text{sub}}$2

where $v_{t,\text{strip}}<v_{t,\text{sub}}$3. Clinical metadata are embedded through a Siamese network into a latent manifold, with

$v_{t,\text{strip}}<v_{t,\text{sub}}$4

and a geodesic cross-domain interaction term $v_{t,\text{strip}}<v_{t,\text{sub}}$5. The model uses five standard echo views—PLAX, PSAX-MP, PSAX-AV, A4C, and A2C—with 2D/Doppler sequences at 30–60 Hz from Philips EPIQ and GE Vivid T8 systems (Begiashvili et al., 17 Jul 2025).

The reported study is a prospective observational cohort of 736 patients, split into 624 for training and 112 for held-out testing, with disease-positive and disease-negative labels derived from standard clinical and echocardiographic evaluation and validated by two observers. On the independent test set, the Acoustic Index achieved an AUC of approximately 0.89. Across five folds, sensitivity and specificity both exceeded 0.8 for thresholds around 0.45, and test-set classification accuracy ranged from 77% to 81%. The intended motivation is to provide a reproducible, vendor-agnostic, operator-independent parameter that uses video dynamics rather than single-frame or speckle-tracking summaries and that can support early detection, triage, and longitudinal monitoring (Begiashvili et al., 17 Jul 2025).

The paper also emphasizes limitations. The cohort size is modest, external multi-center validation is absent, labels remain expert-derived rather than outcome-based, the output is a global risk score rather than a disease-specific diagnosis, and some architectural details are proprietary. This suggests that, unlike the physically grounded acoustic-index literature in wave mechanics, the clinical Acoustic Index is presently best understood as a specialized biomarker name rather than as a general acoustic property (Begiashvili et al., 17 Jul 2025).