Metasurface Time-Reversal Focusing

• Metasurface-based time-reversal focusing is a technique that uses engineered boundaries to transform incident waves into their phase-conjugated counterparts for targeted spatial and temporal refocusing.
• It leverages both continuous GSTC-based synthesis and discrete transmitarray methods to design apertures that accurately reproduce conjugated fields in complex, reverberant media.
• Experimental implementations in hyperthermia and optical systems demonstrate effective energy concentration while addressing challenges of loss, dispersion, and modal diversity.

Metasurface-based time-reversal focusing denotes a class of wave-control methods in which an engineered boundary—typically a transmissive or reflective metasurface, or a functionally analogous programmable aperture—implements the re-emission stage of time reversal or phase conjugation so that waves refocus at a prescribed point in space and, in broadband settings, at a prescribed time. In the direct metasurface implementations represented here, the target field is obtained from the time-reversed field of a notional source placed at the focus, and the metasurface is synthesized so that a known illumination is transformed into that reversed field (Hajiahmadi et al., 2021, Rahmani et al., 17 Sep 2025). Closely related precursor work replaces direct target-side measurements by model-based Green’s-function synthesis in complex enclosures, showing that arbitrary-point focusing does not require an active source at the target if the propagation operator can be inferred or approximated from geometry (Xiao et al., 2014).

1. Time-reversal principle and its metasurface translation

The starting point is the standard time-reversal mirror principle. If a short pulse $g(t)$ is injected at the desired target location and the medium between target and transceiver has impulse response $h(t)$, then the received sona is

$s(t)=g(t)*h(t),$

and re-injection of the time-reversed sona $s(-t)$ produces

$r(t)=s(-t)*h(t)=g(-t)*h(-t)*h(t).$

The reconstruction is therefore governed by the self-correlation of the impulse response, often written through

$$r(t)=\int g(-t')f(t-t')\,dt', \qquad f(t)=h(-t)*h(t),$$

which explains the temporal focus near $t=0$ and the appearance of sidelobes from path-delay differences (Xiao et al., 2014).

In electromagnetic metasurface formulations, the same logic is expressed in the frequency domain by phase conjugation. For the recorded fields on the future metasurface plane,

$$\mathbf{E}_\text{rev}(x,y,z_\mathrm{ms})=\mathbf{E}_\text{rec}^{\ast}(x,y,z_\mathrm{ms}), \qquad \mathbf{H}_\text{rev}(x,y,z_\mathrm{ms})=-\mathbf{H}_\text{rec}^{\ast}(x,y,z_\mathrm{ms}),$$

with the negative sign reflecting that $\mathbf H$ is odd under time reversal under the $\exp(+j\omega t)$ convention. In practical terms, a source is conceptually placed at the intended focal point, the resulting field is recorded on an aperture, and the aperture is then forced to radiate the conjugated field so that propagation paths retrace and reassemble at the source point (Hajiahmadi et al., 2021).

Metasurface-based time-reversal focusing is therefore not merely “focusing by a thin lens.” The metasurface acts as a spatially distributed wave transformer that implements, exactly or approximately, the aperture field required by reciprocity and time reversal. In passive implementations this is a precomputed field transformation. In programmable interpretations it becomes a boundary-control law. The conceptual continuity with model-based synthetic sona methods is direct: instead of measuring the Green’s function from a source at the target, one estimates or computes it and engineers the boundary response that would generate the corresponding reversed field (Xiao et al., 2014, Rahmani et al., 17 Sep 2025).

2. Aperture synthesis: from conjugated fields to metasurface parameters

Two metasurface synthesis routes are explicit in the literature considered here. The first is continuous susceptibility synthesis using generalized sheet transition conditions (GSTCs). In the microwave hyperthermia design of a single transmissive metasurface forming one face of a cavity, the sheet is modeled as an axial homoanisotropic metasurface with four nonzero susceptibility components,

$h(t)$0

under the assumptions of reciprocity, nongyrotropy, and zero reflection. The synthesis equations are

$h(t)$1

$h(t)$2

with field jumps and averages defined on the metasurface plane. The known incident field is a monochromatic $h(t)$3-polarized plane wave,

$h(t)$4

and the transmitted field inside the cavity is forced to match the phase-conjugated field previously recorded from the tumor-centered dipole source (Hajiahmadi et al., 2021).

The second route is discrete transmitarray synthesis from a unit-cell phase-transfer curve. In the quad-layer transmissive metasurface for deep-brain hyperthermia, the conjugated aperture field is converted into local transmission-phase targets through

$h(t)$5

Because the illuminating plane wave has approximately uniform phase over the aperture, the design condition becomes

$h(t)$6

A precomputed relation $h(t)$7 is then used to map each desired phase to the nearest square-patch length $h(t)$8, producing a pixelized phase mask over the aperture (Rahmani et al., 17 Sep 2025).

This discrete implementation is explicitly passive phase engineering rather than direct replay of a measured temporal signal. The quad-layer transmitarray uses square metallic patches on dielectric substrates, periodicity $h(t)$9, substrate thickness $s(t)=g(t)*h(t),$0, air gap $s(t)=g(t)*h(t),$1, feature width $s(t)=g(t)*h(t),$2, and $s(t)=g(t)*h(t),$3. By varying $s(t)=g(t)*h(t),$4 over $s(t)=g(t)*h(t),$5 to $s(t)=g(t)*h(t),$6, the design achieves full $s(t)=g(t)*h(t),$7 transmission-phase coverage, which is the key hardware condition for reproducing the prescribed time-reversal aperture phase (Rahmani et al., 17 Sep 2025).

A broader boundary-theoretic perspective is provided by finite-time focusing theory. There the boundary is not driven by a homogeneous Green’s function as in a standard time-reversal mirror, but by the real part of a composite focusing function

$s(t)=g(t)*h(t),$8

so that

$s(t)=g(t)*h(t),$9

is reconstructed from double-sided boundary integrals. This formulation indicates that an advanced active boundary, including a sufficiently capable metasurface, would have to synthesize not only a converging phase front but also the coda required to cancel unwanted scattering through the focal plane (Meles et al., 2018).

3. Reverberation, complex environments, and effective degrees of freedom

A recurrent theme is that the environment is not a nuisance external to the focusing system; it is part of the operator being exploited. In the transmissive metasurface hyperthermia architecture, only one wall of the cavity is actively time-reversing. Because that violates the ideal complete-enclosure condition, metallic side walls and a top wall are added, and an internal scatterer is introduced to enrich the multipath environment. The stated purpose is to compensate for the incompleteness of the reversal surface by increasing scattering diversity, so that the single active wall behaves more like part of a closed time-reversal cavity (Hajiahmadi et al., 2021).

The measured improvement in the power-density figure of merit illustrates this dependence on environmental complexity. For the metasurface-only system, $s(-t)$0. With two side walls, $s(-t)$1; with three walls, $s(-t)$2; with four walls, $s(-t)$3. Adding the twice zigzag-folded rectangular PEC scatterer reduces the metric to $s(-t)$4, and optimizing the scatterer angle to $s(-t)$5 yields

$s(-t)$6

This establishes that a richer reverberant environment can partially compensate for an incomplete active aperture (Hajiahmadi et al., 2021).

A closely related mechanism appears in the Hilbert-fractal resonator work, where a planar resonant structure with subwavelength modal structure is coupled to a complex cavity. The paper defines spatial degrees of freedom $s(-t)$7, temporal degrees of freedom $s(-t)$8, and total degrees of freedom

$s(-t)$9

In the numerical multi-illumination study, 90 incident plane waves collapse to $r(t)=s(-t)*h(t)=g(-t)*h(-t)*h(t).$0 uncorrelated fields on the fractal, while $r(t)=s(-t)*h(t)=g(-t)*h(-t)*h(t).$1, yielding $r(t)=s(-t)*h(t)=g(-t)*h(-t)*h(t).$2. In the cavity-assisted one-channel experiment, the coda decay time increases from $r(t)=s(-t)*h(t)=g(-t)*h(-t)*h(t).$3 to $r(t)=s(-t)*h(t)=g(-t)*h(-t)*h(t).$4, and the reported total becomes $r(t)=s(-t)*h(t)=g(-t)*h(-t)*h(t).$5. The central interpretation is that the cavity converts spatial degrees of freedom into temporal ones, permitting high-quality single-channel time reversal on a structured planar resonator (Dupré et al., 2016).

Random or volumetric structured media clarify the same point from another angle. In the cubic disordered bubble cloud, broadband time reversal through a multiple-scattering medium achieves focal widths governed by the structured medium’s effective wavelength rather than the host-medium wavelength. The effective-medium relation

$r(t)=s(-t)*h(t)=g(-t)*h(-t)*h(t).$6

predicts the focusing trend well, and the best reported ideal focusing reaches $r(t)=s(-t)*h(t)=g(-t)*h(-t)*h(t).$7 for perfect bubbles, while realistic viscous and thermal losses reduce this to $r(t)=s(-t)*h(t)=g(-t)*h(-t)*h(t).$8 at $r(t)=s(-t)*h(t)=g(-t)*h(-t)*h(t).$9. Although not a metasurface, this result is directly relevant because it isolates the roles of resonant phase delay, high effective wavenumber, and modal richness in far-field subwavelength time-reversal focusing (Lanoy et al., 2015).

The ray-chaotic cavity literature makes the same environmental point in electromagnetic form. In a chaotic enclosure, reverberation enlarges the effective aperture through repeated wall interactions, and focusing quality is predicted from the transmission statistics over the signal bandwidth. This suggests a transferable equivalent-aperture interpretation: reverberation and structured boundaries both increase usable wave-control degrees of freedom, but only if those degrees of freedom are accessible and accurately modeled (Xiao et al., 2014).

4. Resolution, bandwidth, and the spatiotemporal character of the focus

The spatial sharpness of a time-reversal focus is not determined by time reversal alone. In the soda-can acoustic metasurface, equally sharp focusing can be achieved without time reversal by exciting the guided mode of the locally resonant array. The focal width narrows as the frequency approaches the Helmholtz resonance from below, reaching about $$r(t)=\int g(-t')f(t-t')\,dt', \qquad f(t)=h(-t)*h(t),$$0 near $$r(t)=\int g(-t')f(t-t')\,dt', \qquad f(t)=h(-t)*h(t),$$1, but the paper emphasizes that this is not sub-diffraction when diffraction is referenced to the guided wavelength in the structured medium rather than the wavelength in air. The small spot is attributed to the short guided wavelength near resonance together with near-field concentration at the can openings (Maznev et al., 2014).

This is an important corrective for metasurface-based time-reversal focusing. The structured boundary determines the modal basis, the local density of states, and the relevant internal wavelength. Time reversal or phase conjugation then selects and synchronizes that basis. A plausible implication is that claims of “subwavelength” focusing require an explicit statement of the wavelength with respect to which the limit is being judged (Maznev et al., 2014, Lanoy et al., 2015).

Temporal fidelity matters as much as spatial fidelity. In the D-band reflecting metasurface designed for broadband focusing, the device is not a true time-reversal system, but it directly studies the conditions required for spatiotemporal refocusing: spatial phase matching across the aperture and near-linear phase-versus-frequency response at each location. The paper defines

$$r(t)=\int g(-t')f(t-t')\,dt', \qquad f(t)=h(-t)*h(t),$$2

and shows that despite highly dispersive resonators the focused pulse broadens only from $$r(t)=\int g(-t')f(t-t')\,dt', \qquad f(t)=h(-t)*h(t),$$3 to $$r(t)=\int g(-t')f(t-t')\,dt', \qquad f(t)=h(-t)*h(t),$$4, corresponding to $$r(t)=\int g(-t')f(t-t')\,dt', \qquad f(t)=h(-t)*h(t),$$5 temporal broadening over $$r(t)=\int g(-t')f(t-t')\,dt', \qquad f(t)=h(-t)*h(t),$$6–$$r(t)=\int g(-t')f(t-t')\,dt', \qquad f(t)=h(-t)*h(t),$$7. Over the reduced band $$r(t)=\int g(-t')f(t-t')\,dt', \qquad f(t)=h(-t)*h(t),$$8–$$r(t)=\int g(-t')f(t-t')\,dt', \qquad f(t)=h(-t)*h(t),$$9, the broadening is $t=0$0. The key system-level result is that strong local resonator dispersion does not automatically imply severe aperture-level pulse distortion (Hossain et al., 2023).

In cavity-based electromagnetic time reversal, temporal metrics are explicit. The synthetic-sona paper quantifies reconstruction quality by peak-to-peak voltage $t=0$1, focus ratio, and transfer ratio. The reconstruction peak at the focal time is governed by the transmission spectrum through

$t=0$2

so at $t=0$3 the dominant control quantity is the in-band average of $t=0$4. The paper defines

$t=0$5

and argues that $t=0$6 scales approximately linearly with $t=0$7. This gives a practical bandwidth-averaged transfer-function figure of merit that transfers naturally to metasurface apertures (Xiao et al., 2014).

Loss introduces a persistent amplitude–contrast tradeoff. In the superconducting cut-circle cavity, the amplitude decay times are $t=0$8 and $t=0$9. The superconducting reconstruction has higher $$\mathbf{E}_\text{rev}(x,y,z_\mathrm{ms})=\mathbf{E}_\text{rec}^{\ast}(x,y,z_\mathrm{ms}), \qquad \mathbf{H}_\text{rev}(x,y,z_\mathrm{ms})=-\mathbf{H}_\text{rec}^{\ast}(x,y,z_\mathrm{ms}),$$0, but the focus ratio drops from $$\mathbf{E}_\text{rev}(x,y,z_\mathrm{ms})=\mathbf{E}_\text{rec}^{\ast}(x,y,z_\mathrm{ms}), \qquad \mathbf{H}_\text{rev}(x,y,z_\mathrm{ms})=-\mathbf{H}_\text{rec}^{\ast}(x,y,z_\mathrm{ms}),$$1 in the normal state to $$\mathbf{E}_\text{rev}(x,y,z_\mathrm{ms})=\mathbf{E}_\text{rec}^{\ast}(x,y,z_\mathrm{ms}), \qquad \mathbf{H}_\text{rev}(x,y,z_\mathrm{ms})=-\mathbf{H}_\text{rec}^{\ast}(x,y,z_\mathrm{ms}),$$2 in the superconducting state because long reverberation builds stronger sidelobes. In the bubbly-medium work, the same tradeoff appears spectrally: large $$\mathbf{E}_\text{rev}(x,y,z_\mathrm{ms})=\mathbf{E}_\text{rec}^{\ast}(x,y,z_\mathrm{ms}), \qquad \mathbf{H}_\text{rev}(x,y,z_\mathrm{ms})=-\mathbf{H}_\text{rec}^{\ast}(x,y,z_\mathrm{ms}),$$3 near resonance is not sufficient when dissipation broadens modal linewidths and destroys mode separability, causing the near-resonant BW1 focus to fail in the realistic lossy case while BW2 retains $$\mathbf{E}_\text{rev}(x,y,z_\mathrm{ms})=\mathbf{E}_\text{rec}^{\ast}(x,y,z_\mathrm{ms}), \qquad \mathbf{H}_\text{rev}(x,y,z_\mathrm{ms})=-\mathbf{H}_\text{rec}^{\ast}(x,y,z_\mathrm{ms}),$$4 focusing (Xiao et al., 2014, Lanoy et al., 2015).

5. Implementations and experimental demonstrations

The most direct application represented here is microwave hyperthermia for deep brain tumors. In the first metasurface-only study, the head is approximated as a two-layer phantom with dimensions $$\mathbf{E}_\text{rev}(x,y,z_\mathrm{ms})=\mathbf{E}_\text{rec}^{\ast}(x,y,z_\mathrm{ms}), \qquad \mathbf{H}_\text{rev}(x,y,z_\mathrm{ms})=-\mathbf{H}_\text{rec}^{\ast}(x,y,z_\mathrm{ms}),$$5 cm, $$\mathbf{E}_\text{rev}(x,y,z_\mathrm{ms})=\mathbf{E}_\text{rec}^{\ast}(x,y,z_\mathrm{ms}), \qquad \mathbf{H}_\text{rev}(x,y,z_\mathrm{ms})=-\mathbf{H}_\text{rec}^{\ast}(x,y,z_\mathrm{ms}),$$6 cm, $$\mathbf{E}_\text{rev}(x,y,z_\mathrm{ms})=\mathbf{E}_\text{rec}^{\ast}(x,y,z_\mathrm{ms}), \qquad \mathbf{H}_\text{rev}(x,y,z_\mathrm{ms})=-\mathbf{H}_\text{rec}^{\ast}(x,y,z_\mathrm{ms}),$$7 cm, shell parameters $$\mathbf{E}_\text{rev}(x,y,z_\mathrm{ms})=\mathbf{E}_\text{rec}^{\ast}(x,y,z_\mathrm{ms}), \qquad \mathbf{H}_\text{rev}(x,y,z_\mathrm{ms})=-\mathbf{H}_\text{rec}^{\ast}(x,y,z_\mathrm{ms}),$$8, $$\mathbf{E}_\text{rev}(x,y,z_\mathrm{ms})=\mathbf{E}_\text{rec}^{\ast}(x,y,z_\mathrm{ms}), \qquad \mathbf{H}_\text{rev}(x,y,z_\mathrm{ms})=-\mathbf{H}_\text{rec}^{\ast}(x,y,z_\mathrm{ms}),$$9, brain parameters $\mathbf H$0, $\mathbf H$1, and a spherical tumor of radius $\mathbf H$2 mm centered at

$\mathbf H$3

At 2 GHz, with a $\mathbf H$4 metasurface placed at $\mathbf H$5, the optimized design with four metallic walls and a properly oriented internal scatterer yields thermal predictions after 20 minutes of exposure of about $\mathbf H$6 in the tumor, while the rest of the brain stays below $\mathbf H$7 and close to $\mathbf H$8 almost everywhere (Hajiahmadi et al., 2021).

The later transmitarray study extends this program to fabrication and phantom validation. Numerically, the realistic-head study uses a nominal frequency of $\mathbf H$9, a cubic time-reversal cavity of side length $\exp(+j\omega t)$0, a spherical tumor of radius $\exp(+j\omega t)$1 at

$\exp(+j\omega t)$2

and the same brain and shell dielectric properties $\exp(+j\omega t)$3 and $\exp(+j\omega t)$4. Experimentally, the operating frequency is shifted to $\exp(+j\omega t)$5, the head is replaced by a spherical phantom of diameter $\exp(+j\omega t)$6, and the target is placed at

$\exp(+j\omega t)$7

The abstract reports a normalized loss-density figure of merit

$\exp(+j\omega t)$8

while Table III reports $\exp(+j\omega t)$9 for the realized metasurface and $h(t)$00 for the ideal time-reversal phase sheet in one realistic-head PEC-wall comparison, indicating sensitivity to configuration and implementation error (Rahmani et al., 17 Sep 2025).

The experimental hardware consists of a quad-layer transmissive metasurface, FR-4 substrate of thickness $h(t)$01, PEC walls made from $h(t)$02 aluminum foil, and a spherical electrothermal head phantom synthesized to approximate

$h(t)$03

A horn antenna of model ETS 3160-03, operational over $h(t)$04–$h(t)$05, is placed $h(t)$06 from the metasurface. After 20 min irradiation, the tumor-center temperature increases by about $h(t)$07 to $h(t)$08, surrounding tissue remains about $h(t)$09 cooler than the tumor region, and the measured hot-spot size is $h(t)$10 along $h(t)$11 and $h(t)$12 along $h(t)$13 (Rahmani et al., 17 Sep 2025).

Beyond hyperthermia, operator-based selective focusing in optics provides a complementary model of what a programmable metasurface could do once the scattering operator is calibrated. In the DORT method, one measures the optical backscattering matrix $h(t)$14, decomposes the time-reversal operator through the singular value decomposition

$h(t)$15

and uses the input singular vectors $h(t)$16 as optimal incident wavefronts for selective focusing. Experimentally, the method yields intensity enhancement $h(t)$17 at $h(t)$18 gold nanobead positions and works even through an aberrating $h(t)$19 PDMS layer. Although implemented with an SLM rather than a metasurface, this is a direct operator-theoretic template for matrix-calibrated metasurface time reversal (Popoff et al., 2011).

6. Limitations, misconceptions, and current technical frontiers

A first misconception is that time reversal alone explains extraordinary focal sharpness on structured surfaces. The soda-can study argues explicitly that time reversal “certainly has an advantage” for arbitrary field shaping and off-center focusing, but that it is not what makes the focal maximum so narrow; the guided mode of the resonant metasurface and the localized field at the resonator openings set the achievable spot size (Maznev et al., 2014). A second misconception is that a favorable effective wavelength or a strong resonance automatically guarantees useful focusing. The bubbly-medium results show that dissipation can preserve high nominal $h(t)$20 while destroying modal separability, which is fatal for broadband time reversal (Lanoy et al., 2015).

Loss and heterogeneity are a central unresolved difficulty for realistic biological applications. In absorbing media, strict time-reversal symmetry is broken. The heterogeneous instantaneous time mirror work addresses this by making the temporal disruption itself spatially nonuniform, so that stronger reversal amplitude is generated where the forward wave suffered greater attenuation. The paper models the temporal jump by

$h(t)$21

and uses local permittivity changes to control the amplitude of the reversed wave. In the deep-seated bladder target example, homogeneous ITM gives a focus accuracy error of $h(t)$22, whereas a heterogeneous ITM using the full three-step selection-and-weighting procedure achieves $h(t)$23 accuracy, $h(t)$24 time persistence, and $h(t)$25 50%-contour area (Wu et al., 2021). This indicates that path-dependent amplitude compensation, not phase conjugation alone, is required in lossy heterogeneous environments.

Model fidelity is equally decisive. Synthetic-sona focusing in chaotic cavities succeeds when the synthetic duration $h(t)$26 is comparable to the decay time $h(t)$27, but deteriorates when geometric uncertainty accumulates and long orbits become inaccurate. In the bowtie cavity with inserts, the best synthetic duration falls to about $h(t)$28, quoted as $h(t)$29, because later parts of the synthetic sona contribute more to sidelobes than to the focal peak (Xiao et al., 2014). The same issue reappears in patient-specific metasurface hyperthermia: accurate tumor localization is assumed, the head models are simplified, and static fabricated metasurfaces must be redesigned when the target or anatomy changes (Hajiahmadi et al., 2021, Rahmani et al., 17 Sep 2025).

A further frontier is boundary control beyond classical time reversal. Finite-time focusing theory shows that if two open boundaries are available, one can design a boundary wavefield whose in- and output are finite in time and whose interaction with the layer embedding the focal point is reduced relative to standard TRM. In the realistic head model, the normalized $h(t)$30-norm in the whole brain changes by $h(t)$31 for standard double-sided Marchenko focusing and $h(t)$32 for finite-time focusing relative to TRM, while in the “red cones” the reductions are $h(t)$33 and $h(t)$34, respectively (Meles et al., 2018). This suggests that future metasurface-based time-reversal platforms may need to implement not just conjugated aperture fields but full spatiotemporal focusing functions.

Taken together, these results define the present technical picture. Metasurface-based time-reversal focusing is most mature as a passive or quasi-passive implementation of monochromatic phase-conjugated aperture fields for patient-specific microwave hyperthermia (Hajiahmadi et al., 2021, Rahmani et al., 17 Sep 2025). Its deeper theoretical structure, however, is broader: model-based Green’s-function synthesis can replace target-side training (Xiao et al., 2014); modal richness and reverberation determine whether subwavelength focusing is physically available (Dupré et al., 2016, Lanoy et al., 2015); operator decompositions enable selective focusing and aberration correction (Popoff et al., 2011); and loss-aware, finite-time, reduced-exposure focusing requires spatially and temporally tailored boundary control beyond uniform time reversal (Wu et al., 2021, Meles et al., 2018).