Labyrinthine Metasurfaces: Acoustic Wave Control

• Labyrinthine metasurfaces are planar, subwavelength acoustic structures with meandering channels that extend the effective path length to impart prescribed phase delays.
• They enable precise wavefront modulation, achieving anomalous reflection, refraction, and surface-wave conversion by engineering local phase gradients.
• Their design allows tuning of phase response, impedance, and absorption, and can be hybridized with active systems for enhanced acoustic control.

Searching arXiv for recent and foundational papers on labyrinthine metasurfaces to ground the article. arXiv search query: "labyrinthine metasurface acoustic" A labyrinthine metasurface is a planar, subwavelength-thickness acoustic structure whose constituent unit cells contain coiled or folded channels that elongate the propagation path within a compact footprint, thereby imposing a prescribed phase delay, impedance response, or resonant behavior on incident sound. In the literature summarized here, labyrinthine architectures are realized as tapered labyrinthine metamaterials, zig-zag reflective units, space-coiling quarter-wave resonators, Wunderlich-curve channels, and hybrid passive–active devices. Across these realizations, the common mechanism is the engineering of an effective acoustic path length substantially larger than the physical thickness, enabling wavefront modulation, anomalous reflection and refraction, surface-wave conversion, sound absorption, and low-frequency reflection or transmission-loss control in structures with ultralow profile (Xie et al., 2014).

1. Definition and constitutive geometry

Labyrinthine metasurfaces are built from subwavelength unit cells in which acoustic propagation occurs through meandering internal channels rather than through a straight slab. In the tapered transmissive implementation, each unit cell is a subwavelength “coiled” channel of length $L(x)$, whose cross-section $A(x)$ varies smoothly from inlet to outlet. The total physical path length exceeds the straight-line thickness $d$ by an amount $\Delta L(x)$ determined by the internal meandering. By adjusting $L(x)$, $A(x)$, and the taper profile, one engineers both the effective refractive index and the delay, yielding a nearly arbitrary phase shift $\Delta \phi(x)$ while maintaining high transmission (Xie et al., 2014).

A related reflective realization uses a 2D “zig-zag” or “labyrinth” channel of total external size $a_x \times a_y$, with internal plates on the top and bottom walls, channel width $d$, and end-caps of thickness $t$. In that geometry, the total coiled path is

$A(x)$0

In the long-wavelength limit, the channel supports only plane waves, and its effective index satisfies

$A(x)$1

This high effective index is the basis for obtaining a continuous $A(x)$2 reflection-phase range by geometric variation of the unit cell (Li et al., 2014).

Absorptive labyrinthine metasurfaces use curled perforations with one end closed and the other opening to the surface. In the 3D single-port realization, each rigid box contains one or more curled channels with cross-section $A(x)$3, total unfolded length $A(x)$4, and operating frequencies in the hundreds of hertz, so that the free-space wavelength is much larger than the cell dimensions. The same space-coiling principle appears in 3D-printed panels based on a folded quarter-wave tube derived from a first-iteration Wunderlich curve, with effective length $A(x)$5 far exceeding the panel thickness (Zhang et al., 2016).

A further geometric variant uses hierarchical Wunderlich space-filling curves. There the channel is folded along an iteration-dependent path inside a square unit cell, and the relevant geometric control parameter is the tortuous path length $A(x)$6 relative to the unit-cell pitch $A(x)$7. This is formalized by the tortuosity factor

$A(x)$8

which becomes large for higher iteration levels and shifts the relevant acoustic phenomena to low frequencies (Krushynska et al., 2017).

2. Phase engineering and generalized Snell-law synthesis

In transmissive labyrinthine metasurfaces operating in the long-wavelength, non-resonant regime, the phase imparted by one cell is

$A(x)$9

where $d$0 is the free-space wavenumber and $d$1 is the extra acoustic path length relative to a flat slab of thickness $d$2. More generally, one may retrieve an effective index $d$3 from full-wave simulations and write

$d$4

Numerical retrieval confirms that smooth variation of channel geometry can sweep $d$5 over $d$6 with low insertion loss and broad bandwidth. In the reported prototypes, $d$7–$d$8 at $d$9–$\Delta L(x)$0, and six discrete cell types cover $\Delta L(x)$1 in approximately $\Delta L(x)$2 increments around $\Delta L(x)$3 (Xie et al., 2014).

The design of beam steering follows the generalized Snell’s law for a transmitted wave through a surface with spatial phase profile $\Delta L(x)$4:

$\Delta L(x)$5

For a periodic array of period $\Delta L(x)$6, the local gradient is approximated by $\Delta L(x)$7. At normal incidence,

$\Delta L(x)$8

Two gradients were experimentally tested, $\Delta L(x)$9 and $L(x)$0, and the measured $L(x)$1 at $L(x)$2 agrees with $L(x)$3 to within experimental error. The spatial implementation uses a super-cell of six labyrinthine elements, each imparting an approximately $L(x)$4 phase step, repeated to form a one-dimensional metasurface (Xie et al., 2014).

For reflected-wave control, the same synthesis principle is applied to the reflection phase. The reflected angle satisfies

$L(x)$5

A constant gradient yields anomalous reflection, whereas a designed nonlinear profile yields focusing or caustic trajectories. For an ultrathin planar lens focusing to $L(x)$6, the prescribed phase is

$L(x)$7

up to a global reference. For self-accelerating beams along $L(x)$8, the phase gradient is set by the local trajectory angle through

$L(x)$9

In the reported realization, eight discrete values of the end-cap thickness $A(x)$0 provide the required phase coverage for arbitrary reflection control (Li et al., 2014).

This suggests that labyrinthine metasurfaces should be understood less as a single structure type than as a general phase-synthesis platform in which the meander geometry functions as the control variable for local acoustic momentum transfer.

3. Reflection, refraction, diffraction, and surface-wave conversion

The earliest transmissive demonstrations established that a tapered labyrinthine metasurface can steer an acoustic beam as expected from generalized Snell’s law, but the same device also exhibits surface-wave conversion, extraordinary beam-steering, and apparent negative refraction through higher-order diffraction. These effects arise when the imposed phase gradient approaches or exceeds the radiative momentum range of propagating transmitted waves (Xie et al., 2014).

Surface-wave conversion occurs when the transmitted plane wave becomes evanescent and bound to the interface. The in-plane wavenumber is

$A(x)$1

and the condition for surface-wave conversion is

$A(x)$2

Under normal dispersion, the critical incident angle is

$A(x)$3

Beyond $A(x)$4, the normal component

$A(x)$5

becomes imaginary, and energy is carried along the surface. Measured surface-wave fields and their spatial Fourier spectra confirm $A(x)$6 and evanescent decay away from the interface (Xie et al., 2014).

When the super-cell period $A(x)$7 is not deeply subwavelength, the metasurface also acts as a grating. The generalized Snell law then acquires a reciprocal-lattice term,

$A(x)$8

with integer diffraction order $A(x)$9. An equivalent form is the grating equation

$\Delta \phi(x)$0

In the non-radiating regime, the diffraction order can jump to satisfy momentum matching. The reported consequences are an extreme dependence $\Delta \phi(x)$1, described as extraordinary beam steering, and apparent negative refraction when the refracted beam emerges on the same side of the normal as the incident beam. Both were observed at $\Delta \phi(x)$2 when $\Delta \phi(x)$3 and the phase gradient was $\Delta \phi(x)$4 (Xie et al., 2014).

Reflective labyrinthine metasurfaces provide a complementary route to wavefront control. Analytical design and experiment demonstrated anomalous reflection, an ultrathin planar lens with adjustable focal point, and non-paraxial or paraxial acoustic self-accelerating beams with arbitrary trajectories. In one realization, the target reflected beam was centered at $\Delta \phi(x)$5 with more than $\Delta \phi(x)$6 of the energy; in another, a half-circle trajectory with radius $\Delta \phi(x)$7 was produced in close agreement with analytical prediction (Li et al., 2014).

A common misconception is that labyrinthine metasurfaces operate only as local phase plates. The diffraction and surface-wave results show that periodicity, reciprocal-lattice momentum, and evanescent conversion can be equally decisive when the super-cell period is not negligible relative to the wavelength.

4. Resonant absorption and bandwidth engineering

A major branch of labyrinthine metasurface research concerns sound absorption rather than wavefront deflection. In the 3D single-port labyrinthine acoustic metamaterial, a closed–open channel of length $\Delta \phi(x)$8 and cross-section $\Delta \phi(x)$9 has input impedance

$a_x \times a_y$0

where $a_x \times a_y$1 and $a_x \times a_y$2 accounts for internal losses. Coupling to the background through an aperture of area $a_x \times a_y$3 gives an effective metasurface impedance $a_x \times a_y$4, from which the reflection coefficient

$a_x \times a_y$5

and absorption

$a_x \times a_y$6

follow directly. Perfect absorption requires impedance matching, $a_x \times a_y$7, and the corresponding resonances occur at

$a_x \times a_y$8

Critical coupling is obtained when the absorptive and radiative quality factors are balanced, $a_x \times a_y$9 (Zhang et al., 2016).

The same work makes explicit an analytic trade-off between relative absorption bandwidth and thickness. For the $d$0th resonance of a one-channel metasurface,

$d$1

and in terms of minimum slab thickness $d$2 and resonant wavelength $d$3,

$d$4

The stated implication is that for deep subwavelength thickness, $d$5 is inevitably small; increasing $d$6 or the loss parameter $d$7 broadens the band but also requires a thicker slab to maintain critical coupling. Experimentally, a narrowband prototype achieved unity absorption at $d$8 with $d$9 and $t$0, corresponding to $t$1. A broadband design using six parallel channels merged six peaks into a continuous band from $t$2 to $t$3 with $t$4 and $t$5, corresponding to $t$6 (Zhang et al., 2016).

A distinct broadband strategy is the “acoustic rainbow” panel, in which multiple space-coiling quarter-wave cells of different dimensions are arranged in a quasi-periodic lattice so that their narrowband absorption peaks superimpose into a broader response. For a single unit cell, the lowest resonance satisfies

$t$7

with thermo-viscous corrections encapsulated in effective parameters. In the reported experiments, the resonance frequency scales approximately linearly with thickness and lateral size in the considered range, with

$t$8

Using two principal cell sizes and multiple thicknesses, a $t$9 panel of total thickness up to $A(x)$00 was designed for $A(x)$01–$A(x)$02. Reverberation-room measurements showed that $A(x)$03 rises from near zero below $A(x)$04 to $A(x)$05 between $A(x)$06 and $A(x)$07, peaking at $A(x)$08 around $A(x)$09 (Nistri et al., 2021).

These absorptive results distinguish two regimes. One is the critically coupled, single-port resonator regime, where perfect or near-perfect absorption is achieved at designed quarter-wave resonances. The other is the multi-resonant “rainbow” regime, where geometrically detuned cells broaden the overall response by area-weighted superposition. A plausible implication is that labyrinthine metasurfaces can trade phase control for loss engineering without abandoning the same underlying space-coiling geometry.

5. Tortuosity, band gaps, and low-frequency reflection

Direct modeling of wave propagation in folded channels shows that labyrinthine channels allowing propagation opposite to the incident-wave direction do not behave like straight slits of equivalent length. The difference is attributed to activated tortuosity effects. In the Wunderlich-curve framework, the relevant descriptor is again

$A(x)$10

where $A(x)$11 is the channel length at iteration level $A(x)$12 and $A(x)$13 is the periodic cell pitch. Parameterizing the channel centerline by arc length $A(x)$14 and setting $A(x)$15 transforms the one-dimensional Helmholtz equation into

$A(x)$16

which makes explicit that the folded channel behaves as a medium with enhanced effective propagation constant (Krushynska et al., 2017).

For Bloch propagation, the dispersion relation is

$A(x)$17

Band gaps open when

$A(x)$18

corresponding to

$A(x)$19

Since $A(x)$20, this recovers Fabry–Pérot frequencies

$A(x)$21

but the crucial observation is not simply resonant retardation. The reported effect is $A(x)$22 wave reflection at band-gap frequencies, and this total reflection is found to be insensitive to thermo-viscous dissipation in air (Krushynska et al., 2017).

For a single monolayer backed by a rigid substrate, the input impedance is written as

$A(x)$23

and the reflection coefficient is

$A(x)$24

At the band-gap condition, $A(x)$25, which yields $A(x)$26 and $A(x)$27. The reported interpretation is destructive interference in the tortuous channel. Example values given for $A(x)$28 and $A(x)$29 place the lowest band-gap near $A(x)$30 for $A(x)$31, $A(x)$32 for $A(x)$33, and $A(x)$34 for $A(x)$35 (Krushynska et al., 2017).

This body of work corrects a second common simplification: labyrinthine structures are not fully characterized by effective path length alone. Tortuosity, folded-path topology, and the possibility of local propagation against the incident direction materially affect dispersion and reflection behavior.

6. Implementations, performance envelopes, and hybridization

The published implementations span transmission, reflection, absorption, and hybrid noise control. In the tapered transmissive metasurface, the reported performance metrics include full $A(x)$36 phase coverage in a single transmissive layer, transmission efficiency greater than $A(x)$37 in the pass-band, and roughly equal phase spacing from $A(x)$38 to $A(x)$39, corresponding to approximately $A(x)$40 relative bandwidth. The stated design trade-offs are that smaller thickness $A(x)$41 reduces the phase range per cell, larger cross-section $A(x)$42 lowers viscous loss but reduces phase tunability, and excessively large period $A(x)$43 increases diffraction orders and potential sidelobes (Xie et al., 2014).

In reflective devices, fabrication was carried out in ABS-like thermoplastic with 3D printing at $A(x)$44 precision, and measurements were performed in a 2D waveguide formed by two parallel Plexiglas plates separated by $A(x)$45. The operating band was around $A(x)$46, corresponding to wavelength approximately $A(x)$47, and the metasurface strip had length $A(x)$48. The reported limitations include bandwidth set by the labyrinthine resonance, approximately $A(x)$49 around $A(x)$50, angular range limited by maximum attainable phase gradient, increasing viscous and thermal loss for smaller channels, and phase-step accuracy limited by fabrication precision (Li et al., 2014).

A more recent extension combines a labyrinthine metasurface with multichannel feedforward active noise control. In that hybrid device, alternating open and meandering channels induce a $A(x)$51 phase shift at the design frequency by extending the channel length by $A(x)$52 relative to the open channels. The metasurface comprises $A(x)$53 bricks, arranged as $A(x)$54 open and $A(x)$55 labyrinthine elements, with design frequency $A(x)$56 and total width $A(x)$57. The passive metasurface exhibits a pronounced transmission-loss peak of approximately $A(x)$58 centered at approximately $A(x)$59 with a $A(x)$60 bandwidth of about $A(x)$61 (Hernandez et al., 27 Sep 2025).

The active component uses $A(x)$62 miniature loudspeakers and an overdetermined microphone array, with cost function

$A(x)$63

and optimal control

$A(x)$64

Two hybrid placements were evaluated. With control sources behind open cells, the transmission loss is approximately $A(x)$65 below $A(x)$66, approximately $A(x)$67 at $A(x)$68, and approximately $A(x)$69 above $A(x)$70. With control sources behind labyrinth cells, the transmission loss is approximately $A(x)$71 below $A(x)$72, approximately $A(x)$73 at $A(x)$74, and rises to approximately $A(x)$75 at $A(x)$76. Spectroscopic analysis attributes the narrowband peak at $A(x)$77 to the labyrinthine metasurface and identifies weak-coupling Fano-type behavior in one placement and strong-coupling splitting into two polariton-like resonances in the other (Hernandez et al., 27 Sep 2025).

The hybrid results show that labyrinthine metasurfaces need not remain purely passive. A plausible implication is that the narrowband, geometry-imposed resonance of a labyrinthine element can be used as a structured spectral feature that active control then broadens, shifts, or deepens.

7. Applications, scope, and interpretive boundaries

The applications explicitly identified for labyrinthine metasurfaces include beam-steering, surface-wave manipulation, high efficiency sound absorption, acoustic imaging, ultrasound lens design, acoustic switching, surface-wave sensing and coupling, sound absorption and manipulation, portable flat beam-steering devices, subwavelength planar lenses for ultrasound imaging, and low-frequency reflection or filtering (Xie et al., 2014).

Several application classes emerge from the reported demonstrations. One class concerns wavefront synthesis: anomalous transmission, anomalous reflection, focusing, and self-accelerating beams all follow from prescribed local phase gradients or phase masks. A second class concerns spectral control: quarter-wave resonance, critical coupling, and multi-resonant superposition support narrowband perfect absorption or broadband absorption bands from $A(x)$78–$A(x)$79 or $A(x)$80–$A(x)$81, depending on architecture (Zhang et al., 2016). A third class concerns reflection-dominated control at low frequencies, where hierarchical tortuosity creates broad band gaps with total reflection in a metamaterial monolayer (Krushynska et al., 2017). A fourth class concerns hybrid passive–active treatments, in which a labyrinthine resonance is coupled to feedforward control to increase transmission loss over the $A(x)$82–$A(x)$83 range (Hernandez et al., 27 Sep 2025).

The literature also places clear boundaries on interpretation. Not every labyrinthine metasurface is broadband: one reflective implementation explicitly reports bandwidth of approximately $A(x)$84 around $A(x)$85, while transmissive phase spacing remains robust over $A(x)$86–$A(x)$87 and rainbow absorption requires deliberate multi-cell detuning (Li et al., 2014). Not every labyrinthine response is local and effective-medium-like: higher-order diffraction, reciprocal-lattice momentum transfer, and tortuosity-driven band gaps show that periodicity and channel topology can dominate. Not every low-frequency absorber is ultrathin without compromise: the 3D single-port work makes explicit the trade-off between relative absorption bandwidth and thickness (Zhang et al., 2016).

Taken together, the reported studies define the labyrinthine metasurface as a family of acoustic metastructures in which space-coiling geometry is used as the primary design degree of freedom. Depending on whether the objective is phase delay, impedance matching, destructive interference, or hybrid active coupling, the same underlying geometric principle can be specialized into transmissive wavefront modulators, reflective phase surfaces, critically coupled absorbers, band-gap reflectors, broadband rainbow panels, or active–passive transmission-loss treatments.