Source: https://jmrplens.github.io/phonometry/materials/diffusers/metadiffusers/ # Metadiffusers A Schroeder diffuser earns its diffusion with depth: the [design equations](https://jmrplens.github.io/phonometry/materials/diffusers/diffusers/) turn a phase sequence into wells whose maximum depth approaches half the design wavelength, so a diffuser for the low hundreds of hertz is tens of centimetres thick, and that thickness is what keeps phase-grating diffusers out of small rooms, studios and vehicles. A **metadiffuser** (Jiménez, Cox, Romero-García and Groby, 2017) reaches the same reflection phases from a panel one to two orders of magnitude thinner: each well becomes a thin closed slit loaded by Helmholtz resonators, and the resonators slow the sound inside the slit so drastically that a 2 cm panel behaves like wells 27 cm deep. This guide covers the slow-sound mechanism, the published quadratic-residue metadiffuser evaluated end to end, and the design recipe that maps any Schroeder sequence onto a deep-subwavelength panel. ## 1. Slow sound in a loaded slit The unit cell is the same one the [slow-sound absorber](https://jmrplens.github.io/phonometry/materials/absorbers/metamaterial-absorbers/) is built from: a slit of height $h_n$ runs the depth $L$ of the panel and is loaded, along its upper wall, by an array of Helmholtz resonators on a lattice step $a$. The slit chain is evaluated with the transfer-matrix method, $$ T = M_{\Delta l}\,\prod_m \left(M_\mathrm{s}\,M_\mathrm{HR}^{(m)}\,M_\mathrm{s}\right), $$ half-lattice slit steps $M_\mathrm{s}$ around each resonator, the resonators as shunt point scatterers $[[1,0],[1/Z_\mathrm{HR},1]]$, and a slit-radiation end correction $M_{\Delta l}$, with visco-thermal effective parameters in the slit and in the square necks and cavities (Stinson 1991). Below the resonance of the loading resonators the shunt compliance adds to the compliance of the slit itself, and the effective phase velocity of the slit mode collapses: the retrieved effective wavenumber $k_\mathrm{eff}$ of the chain corresponds to a phase speed of a few tens of metres per second. The slit therefore reaches its quarter-wavelength condition, and any intermediate reflection phase, at a fraction of the depth an air-filled well would need. Two consequences make the mechanism ideal for diffusers rather than just absorbers: - **The reflection phase is tunable per slit.** The rigidly backed slit reflects with $R_n = (T_{11}\cos\theta - Z_0 T_{21})/(T_{11}\cos\theta + Z_0 T_{21})$, and the resonator geometry tunes the phase of $R_n$ through the full $2\pi$ range at the working frequency while $|R_n|$ stays close to one. A phase profile that a classical grating writes with depth, the metadiffuser writes with resonator tuning. - **Critical coupling adds a third state.** Driving one slit to [critical coupling](https://jmrplens.github.io/phonometry/materials/absorbers/metamaterial-absorbers/) balances its visco-thermal loss against its leakage and produces a reflection zero: $R_n \approx 0$. Ternary sequences (reflection values $+1$, $-1$, $0$) need exactly that perfectly absorbing state, which no rigid well can provide. `metadiffuser_reflection` runs that chain once per well (two-dimensional resonators, visco-thermal losses and end corrections included) and returns the per-well complex reflection $R_n(f)$; `metadiffuser_polar_response` and `metadiffuser_diffusion_spectrum` reduce that spatial profile through the same Fraunhofer far field and ISO 17497-2 coefficient used for the [classical designs](https://jmrplens.github.io/phonometry/materials/diffusers/diffusers/). A `None` entry in the well sequence stands for a flat rigid strip ($R = 1$), the $+1$ state of ternary designs. ## 2. The published quadratic-residue metadiffuser The published design packs a whole $N = 5$ quadratic-residue diffuser into a 35 cm x 2 cm panel: five slits, each loaded by two resonators, on a 7 cm pitch. Its first slit reaches critical coupling (the reflection zero the ternary `0` state is built from), and at the 2 kHz evaluation frequency the panel scatters like the 27.4 cm deep QRD it mimics. As in the published far-field comparison, the polar response repeats the five-well sequence six times (`periods=6`, a 2.1 m panel): ```python import numpy as np from phonometry import ( HelmholtzResonator, MetadiffuserWell, metadiffuser_polar_response, metadiffuser_reflection, ) # The published quadratic-residue metadiffuser: five slits with two # resonators each in a 35 cm x 2 cm panel (7 cm pitch), tuned to mimic a # QRD designed for 500 Hz whose wells would run up to 27.4 cm deep. mm = 1e-3 rows = [ # slit h, neck l_n, cavity l_c, neck w_n, cavity w_c [mm] (14.7, 13.0, 16.4, 6.2, 9.0), (30.9, 9.1, 4.3, 3.5, 9.0), (30.9, 9.1, 4.3, 3.5, 9.0), (15.7, 13.3, 17.0, 6.3, 9.0), (20.3, 18.0, 20.7, 3.2, 9.0), ] wells = [ MetadiffuserWell( h * mm, 2 * (HelmholtzResonator(ln * mm, wn * mm, lc * mm, wc * mm),), ) for h, ln, lc, wn, wc in rows ] f = np.arange(1800.0, 2601.0, 5.0) panel = metadiffuser_reflection(f, wells, depth=0.02, period=0.07) alpha1 = panel.well_absorption[0] print(round(float(alpha1.max()), 2), int(f[alpha1.argmax()])) # 0.99 2305 # Far-field comparison: the five-well sequence repeated six times (2.1 m). polar = metadiffuser_polar_response(2000.0, wells, depth=0.02, period=0.07, periods=6) print(round(polar.coefficient, 2)) # 0.32 ```

Show the code for this figure

```python import numpy as np from phonometry import ( HelmholtzResonator, MetadiffuserWell, materials, metadiffuser_polar_response, ) mm = 1e-3 rows = [ (14.7, 13.0, 16.4, 6.2, 9.0), (30.9, 9.1, 4.3, 3.5, 9.0), (30.9, 9.1, 4.3, 3.5, 9.0), (15.7, 13.3, 17.0, 6.3, 9.0), (20.3, 18.0, 20.7, 3.2, 9.0), ] wells = [ MetadiffuserWell( h * mm, 2 * (HelmholtzResonator(ln * mm, wn * mm, lc * mm, wc * mm),), ) for h, ln, lc, wn, wc in rows ] # The metadiffuser panel and the QRD it was tuned to at 2 kHz, both with # six repetitions of the period (2.1 m panels). meta = metadiffuser_polar_response(2000.0, wells, depth=0.02, period=0.07, periods=6) sequence = np.roll(materials.quadratic_residue_sequence(5), -1) depths = sequence * (343.0 / 500.0) / (2 * 5) qrd = materials.predict_diffuser_polar_response( 0.07, 2000.0, depths=depths, periods=6, include_obliquity=False, ) ax = meta.plot(marker="", linewidth=2.2, label="Metadiffuser, panel 2 cm") qrd.plot(ax=ax, marker="", linewidth=1.6, linestyle="--", label="QRD, wells up to 27.4 cm") ax.legend(loc="lower center") ```

The far-field overlay above is the frequency-domain summary; the FDTD animation below meshes both panels for real (the metadiffuser at 0.25 mm, slits, necks and cavities included) and lets the same 2 kHz wavefront hit them: the 27 cm QRD and the 2 cm panel throw out nearly the same scattered fan. [Watch the high-resolution video (WebM)](https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/anim_fdtd_metadiffuser.webm) ## 3. The design recipe: phases first, geometry second Designing a metadiffuser is a two-step translation. First pick the Schroeder sequence exactly as for a [classical design](https://jmrplens.github.io/phonometry/materials/diffusers/diffusers/): the quadratic residues $s_n = n^2 \bmod N$ give the target well depths $d_n = s_n \lambda_0/(2N)$ and hence the target reflection phases $\varphi_n = -2 k d_n$ at the evaluation frequency. Second, tune each slit (its height, and the neck and cavity of its resonators) until the chain reproduces that phase from the fixed panel depth $L$. The per-well reflection returned by `metadiffuser_reflection` is the quantity to match — the inverse problem is not automated, so the matching is done by evaluating candidate geometries; for the published design at 2 kHz the five slits land within a few degrees of the QRD targets while keeping $|R_n|$ near one: ```python import numpy as np from phonometry import materials, metadiffuser_reflection # wells: the five published slits from the example above. panel = metadiffuser_reflection(np.array([2000.0]), wells, depth=0.02, period=0.07) phases = np.degrees(np.angle(panel.reflection[:, 0])) print(np.round(np.abs(panel.reflection[:, 0]), 2)) # [0.97 1. 1. 0.97 0.98] print(np.round(phases)) # [ 75. -71. -71. 74. 2.] # The targets: the 500 Hz N = 5 QRD it mimics, evaluated at 2 kHz. sequence = np.roll(materials.quadratic_residue_sequence(5), -1) depths = sequence * (343.0 / 500.0) / (2 * 5) # up to 27.4 cm deep k = 2 * np.pi * 2000.0 / 343.0 targets = np.degrees(np.angle(np.exp(-2j * k * depths))) print(np.round(targets)) # [ 72. -72. -72. 72. 0.] ``` A 2 cm panel hitting the phases of 27.4 cm wells to within a few degrees is the whole trick, and the limits of the trick are worth stating plainly: - **Losses are not optional.** The same visco-thermal boundary layers that make critical coupling possible shave a few percent off $|R_n|$ everywhere, so a metadiffuser always absorbs a little where a rigid grating would not (the published designs stay in the percent range away from the critically coupled slit). The per-well and face-averaged absorption are exposed as `well_absorption` and `absorption` for exactly this check. - **The phase match is dispersive.** A rigid well's phase grows linearly with frequency; a resonator-loaded slit's does not. The sequence is exact at the frequency it was tuned for and degrades away from it, so a broadband design is a compromise across the working band, checked with `metadiffuser_diffusion_spectrum` band by band rather than assumed from the design frequency alone. - **Below resonance is the working side.** Above the resonance of the loading resonators the slow-sound branch closes and the slit stops tracking the target phase; designs place the resonances above the working band. The reduction to a graded coefficient is then identical to the classical workflow: `metadiffuser_polar_response` for one frequency, `metadiffuser_diffusion_spectrum` for the band-by-band normalised coefficient, both against the same-footprint flat reference of ISO 17497-2 Formula (7). The model is locally reacting — wells do not couple internally — and the reduction inherits the Fraunhofer limits of the [classical page](https://jmrplens.github.io/phonometry/materials/diffusers/diffusers/): accuracy falls at low frequency, at grazing angles and where edge diffraction matters, so the chain grades candidate designs rather than replacing an ISO 17497-2 measurement. ## See also - [Diffusers and Their Coefficients](https://jmrplens.github.io/phonometry/materials/diffusers/diffusers/): the ISO 17497-1/-2 coefficients this panel is graded by, and the classical Schroeder designs whose phase sequences it borrows. - [Metamaterial Absorbers](https://jmrplens.github.io/phonometry/materials/absorbers/metamaterial-absorbers/): the same slit and resonator cell tuned for perfect absorption instead of controlled phase, with the critical-coupling theory behind the ternary `0` state. - [Porous and Multilayer Absorbers](https://jmrplens.github.io/phonometry/materials/absorbers/porous-absorbers/): the classical equivalent-fluid and resonant layers the metamaterial cells are built from. - [2D FDTD wave simulation](https://jmrplens.github.io/phonometry/simulation/fdtd-simulation/): the wave solver behind the animation above, meshing the real slit and resonator geometry. - API reference: [`materials.diffusers.metadiffuser`](https://jmrplens.github.io/phonometry/reference/api/materials/metadiffuser/) and [`materials.diffusers.design`](https://jmrplens.github.io/phonometry/reference/api/materials/design/). ## References - Jiménez, N., Cox, T. J., Romero-García, V., & Groby, J.-P. (2017). Metadiffusers: Deep-subwavelength sound diffusers. *Scientific Reports*, 7, 5389. [doi:10.1038/s41598-017-05710-5](https://doi.org/10.1038/s41598-017-05710-5). The metadiffuser model implemented here: slits loaded by Helmholtz resonators reproduce Schroeder phase profiles and ternary sequences from panels 1/46 to 1/20 of the design wavelength thick, including the quadratic-residue design of section 2. - Jiménez, N., Cox, T. J., Groby, J.-P., & Romero-García, V. (2019). Beyond phase grating diffusers using locally-resonant metamaterials. *Proceedings of the 23rd International Congress on Acoustics (ICA 2019)*, Aachen. [Proceedings PDF](https://pub.dega-akustik.de/ICA2019/data/articles/000706.pdf). The congress companion to the 2017 paper: the transfer-matrix chain, the slow-sound dispersion picture and the far-field comparisons that section 3 condenses. - Cox, T. J., & D'Antonio, P. (2017). *Acoustic absorbers and diffusers: Theory, design and application* (3rd ed.). CRC Press. ISBN 978-1-4987-4099-9. [doi:10.1201/9781315369211](https://doi.org/10.1201/9781315369211). The Schroeder sequences, the Fraunhofer far-field model and the diffusion-coefficient machinery the metadiffuser is evaluated with. - Jiménez, N., Groby, J.-P., Pagneux, V., & Romero-García, V. (2017). Iridescent perfect absorption in critically-coupled acoustic metamaterials using the transfer matrix method. *Applied Sciences*, 7(6), 618. [doi:10.3390/app7060618](https://doi.org/10.3390/app7060618). The slit + Helmholtz-resonator transfer-matrix chain each well is evaluated with, and the critical-coupling condition behind the ternary `0` state. - Jiménez, N., Huang, W., Romero-García, V., Pagneux, V., & Groby, J.-P. (2016). Ultra-thin metamaterial for perfect and quasi-omnidirectional sound absorption. *Applied Physics Letters*, 109(12), 121902. [doi:10.1063/1.4962328](https://doi.org/10.1063/1.4962328). The resonator impedance (Eq. A23) and its radiation end corrections (Eqs. A24-A27) used in every loaded slit. - Jiménez, N., Umnova, O., & Groby, J.-P. (Eds.). (2021). *Acoustic waves in periodic structures, metamaterials, and porous media* (Topics in Applied Physics, Vol. 143). Springer. [doi:10.1007/978-3-030-84300-7](https://doi.org/10.1007/978-3-030-84300-7). The book-length treatment of resonant metamaterial absorbers and diffusers; the modern companion to Cox & D'Antonio for this whole family. ## Standards No standard governs the metadiffuser model itself; it is a journal method (Jiménez et al. 2017) implemented clean-room from the cited sources. Its output is graded by ISO 17497-2:2012 (diffusion coefficient), covered in [Diffusers and Their Coefficients](https://jmrplens.github.io/phonometry/materials/diffusers/diffusers/), and its unit cell is the slit and Helmholtz-resonator chain of [Metamaterial Absorbers](https://jmrplens.github.io/phonometry/materials/absorbers/metamaterial-absorbers/). Numerical conformance against the published designs and closed forms is tracked in [CONFORMANCE.md](https://jmrplens.github.io/phonometry/reference/conformance/).