Programmable Nonreciprocal Metasurface Prism

• Programmable nonreciprocal metasurface prism is an engineered interface that maps frequency components into unique spatial angles using tailored phase and amplitude gradients.
• It integrates nonreciprocal mechanisms such as space-time modulation, unilateral amplification, and engineered gyrotropy to achieve direction-dependent wave transformations.
• The device spans optical to microwave platforms, enabling applications in isolation, beam steering, frequency conversion, and reconfigurable signal processing.

A programmable nonreciprocal metasurface prism is a thin, engineered interface that combines prism functionality—mapping frequency content or an imposed phase gradient into beam deflection—with direction-dependent scattering that violates reciprocal wave transport. In the recent literature, the term spans several closely related realizations: phase- and amplitude-gradient metaprisms with unilateral amplification, spatiotemporally modulated metasurfaces that exchange both momentum and energy with light, space-time Fresnel prisms that emulate moving refractive-index interfaces in finite size, and bidirectionally programmable nonreciprocal phase-gradient sheets (Taravati et al., 2020). Across these variants, the defining feature is that forward and backward illumination do not retrace the same transformation, so the effective scattering matrix is asymmetric, with responses such as $S_{21}\ne S_{12}$, different output angles, different frequency sidebands, different gains, or different combinations of these quantities (Guo et al., 2019).

1. Conceptual scope and relation to conventional prisms

A conventional prism decomposes incident radiation through bulk propagation and material dispersion. A metasurface prism, or metaprism, replaces this volumetric operation with a spatially varying interface response. In the active microwave formulation, it is “a thin, planar metasurface that maps temporal frequency components of an incident polychromatic electromagnetic wave into distinct spatial angles by imposing a spatially varying, frequency-dependent phase and amplitude response across an aperture,” so that each frequency component experiences a tailored phase gradient and amplitude weighting (Taravati et al., 2020). In this sense, the prism function is not restricted to beam deflection at a single carrier; it may implement an arbitrary programmed frequency-to-angle map.

The nonreciprocal extension adds a second layer of functionality. Instead of a single spatial transfer function, the surface encodes different transformations for opposite propagation directions. Depending on platform, this can mean different forward and backward phase gradients, different harmonic conversion pathways, different transmission or reflection gains, or different combinations of angle and frequency routing. In nonreciprocal phase-gradient metasurfaces, a forward incident wave experiences $\Phi_f(x)$ while a backward incident wave experiences $\Phi_b(x)\ne\Phi_f(x)$, yielding different deflection angles for opposite incidence (Lavigne et al., 2019). In space-time-modulated systems, the distinction is often even stronger: the reverse process may not recover the initial state at all, because the backward wave couples into a different Floquet channel than the one produced in the forward direction (Guo et al., 2019).

The literature also uses “prism” in a broader space-time sense. The “Space-Time Fresnel Prism” is the dynamic analogue of the conventional Fresnel prism: the transverse spatial coordinate of the usual prism is transposed to the temporal dimension $ct$, so the primary observable becomes frequency translation rather than angular diffraction (Li et al., 2023). This means that “prism” can denote either direct beam steering, frequency steering, or a compound angle–frequency operation, depending on whether the device implements a static spatial gradient, a traveling modulation, or both. A plausible implication is that the term now identifies a functional class rather than a single geometry.

2. Physical mechanisms of nonreciprocity

The most direct optical route is traveling-wave space-time phase modulation. In the Kerr metasurface demonstration, the phase profile is

$$\phi(x,t)=\phi_0+k_s x+\Delta\phi\cos(Kx-\Omega t),$$

where $k_s$ is the static phase gradient responsible for the prism function, while the traveling term provides high-speed temporal phase wobbling (Guo et al., 2019). The modulation behaves as a moving optical grating and enforces correlated transitions in momentum and energy:

$$k_{x,m}=k_{x,0}+k_s+mK,\qquad \omega_m=\omega_0+m\Omega.$$

The essential asymmetry arises because the radiative condition depends simultaneously on the sign of the transverse momentum and on the shifted frequency. For suitable $k_s$ and $K$, one branch can be pushed outside the free-space light cone in one direction, while the opposite branch remains radiative; reversing propagation does not retrace the original kinematics (Guo et al., 2019).

A related but more general mechanism appears in GSTC-based space-time metasurfaces. Here the surface susceptibilities or impedances are modulated as traveling waves, and the asymmetry is tied to the directional perturbation of the sheet together with the transverse wave momentum of the input wave. The 2017 GSTC formulation explicitly showed that under oblique incidence, forward and backward propagation excite different Floquet spectra, including different fundamental transmission amplitudes, even when the metasurface is phenomenologically matched (Gupta et al., 2017). The 2019 multifunctional framework further identified engineered evanescent-mode excitation as the key enabler: one illumination direction is designed to couple strongly to an evanescent resonance, while the opposite direction is phase-mismatched, producing strong nonreciprocity on the same impedance-sheet platform (Wang et al., 2019).

Nonreciprocity need not require time modulation. In transistor-based microwave implementations, unilateral active networks break time-reversal symmetry through directionally biased gain. The reflective beamsteering metasurface integrates a chain of radiating patches with unilateral, bias-controlled phase shifters, so signal flow is enforced in one direction through the chain while backward paths encounter mismatch, misphasing, and no coherent in-chain gain (Taravati et al., 2021). The 2020 metaprism similarly uses unilateral transistor amplifiers inside each super-cell, giving high forward transmission and amplification while strongly attenuating backward transmission (Taravati et al., 2020). In these architectures, the input frequency is preserved and the response is explicitly immune to undesired time harmonics (Taravati et al., 2021).

Other physical routes include self-biased gyrotropy and susceptibility programming. The passive, linear, and magnet-free microwave metasurface based on La:BaM meta-atoms exploits remanent magnetization and gyrotropic permeability to realize distinct forward and backward phase responses without external field bias (Yang et al., 2022). The susceptibility-programmable medium instead uses a structured optical pump to control ground-state populations in atoms, thereby programming the complex susceptibility tensor $\chi(\mathbf{r},t)$ in space and time; this can create direction-dependent phase and amplitude profiles and thus a nonreciprocal prism response when mapped across an interface (Zhang et al., 2024).

3. Governing laws and analytical descriptions

The static prism function is described by generalized Snell’s law. For a phase gradient $\Phi_f(x)$0 imposed on a reflective or transmissive interface,

$\Phi_f(x)$1

or equivalently

$\Phi_f(x)$2

This relation underlies both reciprocal metaprisms and the static component of nonreciprocal ones (Guo et al., 2019).

Space-time modulation extends this law to Floquet harmonics:

$\Phi_f(x)$3

with radiative condition $\Phi_f(x)$4. The asymmetry is not merely geometric. Because $\Phi_f(x)$5 depends on the shifted frequency $\Phi_f(x)$6, the forward and backward radiative conditions differ, and the scattering matrix becomes asymmetric, with $\Phi_f(x)$7 and reflected sideband content differing for opposite directions (Guo et al., 2019).

For moving-index-interface formulations, the space-time Fresnel prism adopts a different analytical language. The transmitted frequency of a space-time step from $\Phi_f(x)$8 to $\Phi_f(x)$9 moving at normalized velocity $\Phi_b(x)\ne\Phi_f(x)$0 is

$\Phi_b(x)\ne\Phi_f(x)$1

while a pure-time section gives

$\Phi_b(x)\ne\Phi_f(x)$2

The two finite-size Fresnel reductions, Prism I and Prism II, are then compared through conversion efficiencies

$\Phi_b(x)\ne\Phi_f(x)$3

with Prism II always providing higher efficiency than Prism I (Li et al., 2023). In this framework, the “deflection” is principally in frequency, although metasurface implementations can add spatial phase gradients to recover angular steering.

For nongyrotropic nonreciprocal phase-gradient sheets, GSTCs and sheet susceptibilities provide the most direct synthesis route. The metasurface boundary obeys

$\Phi_b(x)\ne\Phi_f(x)$4

$\Phi_b(x)\ne\Phi_f(x)$5

and nonreciprocity is obtained by violating the reciprocal bianisotropic condition through $\Phi_b(x)\ne\Phi_f(x)$6 rather than $\Phi_b(x)\ne\Phi_f(x)$7 (Lavigne et al., 2019). For prescribed forward and backward transmission phases $\Phi_b(x)\ne\Phi_f(x)$8 and $\Phi_b(x)\ne\Phi_f(x)$9, the paper provides closed-form susceptibilities, including the pure magnetoelectric “spatial gyrator” case with $ct$0 and $ct$1 (Lavigne et al., 2019).

4. Device architectures and material platforms

The architecture depends on the mechanism used to break reciprocity. Optical space-time implementations use ultrathin nonlinear resonant stacks. The Kerr metasurface consists of amorphous silicon nanobar antennas of thickness approximately $ct$2 on a $ct$3 $ct$4 spacer and $ct$5 Ag back reflector, with a supercell of three nanobars spanning $ct$6 to $ct$7 in steps of $ct$8 and a typical period $ct$9 (Guo et al., 2019). Two frequency-shifted femtosecond pumps near $$\phi(x,t)=\phi_0+k_s x+\Delta\phi\cos(Kx-\Omega t),$$0 generate the traveling interference pattern that drives the Kerr phase wobble.

At mid-infrared frequencies, the graphene platform replaces the nonlinear dielectric with a gated conductive sheet in a metal–dielectric–metal cavity. The reported device uses an optically thick metal ground plane, a $$\phi(x,t)=\phi_0+k_s x+\Delta\phi\cos(Kx-\Omega t),$$1 amorphous-Ge spacer, a $$\phi(x,t)=\phi_0+k_s x+\Delta\phi\cos(Kx-\Omega t),$$2 alumina isolation layer, and a monolayer graphene beneath six Au patch antennas per pixel. The array contains 36 one-dimensional pixels, each $$\phi(x,t)=\phi_0+k_s x+\Delta\phi\cos(Kx-\Omega t),$$3, and the three-phase RF drive implements a synthetic traveling wave with $$\phi(x,t)=\phi_0+k_s x+\Delta\phi\cos(Kx-\Omega t),$$4 (Efimov et al., 30 Aug 2025). Here the nonreciprocal prism function is realized through synthetic diffraction rather than a lithographically fixed static gradient.

Microwave active metaprisms use cascaded circuit super-cells. The 2020 programmable nonreciprocal metaprism is built from receiving and transmitting patch antennas interconnected by two unilateral transistor-based amplifiers and a phase shifter, producing a five-stage signal path whose cumulative complex transmission sets the local phase and magnitude (Taravati et al., 2020). The reflective full-duplex beamsteering metasurface uses microstrip patch radiators, unilateral amplifiers, reciprocal phase shifters, and PCB bias routing on Rogers RO4350 substrate, with chains of five supercells providing the in-chain traveling-wave behavior (Taravati et al., 2021). In both cases, bias control programs the gradient without invoking temporal modulation.

Passive self-biased gyrotropic metasurfaces form another class. The La:BaM platform uses all-dielectric Mie resonators made of self-biased hexaferrite on PTFE, with approximately $$\phi(x,t)=\phi_0+k_s x+\Delta\phi\cos(Kx-\Omega t),$$5 width, $$\phi(x,t)=\phi_0+k_s x+\Delta\phi\cos(Kx-\Omega t),$$6 height, and $$\phi(x,t)=\phi_0+k_s x+\Delta\phi\cos(Kx-\Omega t),$$7 lattice period, operating in the Ku band (Yang et al., 2022). Forward and backward phase profiles are assembled in a “LEGO-like” manner from a reduced 10-element library, since flipping the remanent magnetization swaps the forward and backward phase assignments (Yang et al., 2022).

More abstract but broadly applicable platforms model the device as an impedance or admittance sheet on a grounded dielectric substrate. This description is central to multifunctional space-time metasurfaces and to the Space-Time Fresnel prism, where the actual implementation may use varactors, RF switches, CMOS drivers, electro-optic modulators, graphene, or other time-programmed unit cells to realize an effective $$\phi(x,t)=\phi_0+k_s x+\Delta\phi\cos(Kx-\Omega t),$$8 or a moving space-time index step (Wang et al., 2019). A plausible implication is that “metasurface prism” now denotes an overview methodology as much as a specific material stack.

5. Programmability and design strategies

Programmability enters through a small set of physically transparent control variables. In the Kerr nonreciprocal prism, the static steering angle is set by $$\phi(x,t)=\phi_0+k_s x+\Delta\phi\cos(Kx-\Omega t),$$9 through the supercell period $k_s$0; the transverse momentum transfer per sideband is set by $k_s$1 through the pump-beam crossing angle; the sideband spacing is set by $k_s$2 through the pump frequency splitting; and the conversion efficiency is controlled by $k_s$3 through $k_s$4 (Guo et al., 2019). With $k_s$5 and $k_s$6, the reported example gives $k_s$7 at normal incidence; with $k_s$8, the $k_s$9 sideband exits at $$k_{x,m}=k_{x,0}+k_s+mK,\qquad \omega_m=\omega_0+m\Omega.$$0 while the $$k_{x,m}=k_{x,0}+k_s+mK,\qquad \omega_m=\omega_0+m\Omega.$$1 sideband is evanescent (Guo et al., 2019).

The Space-Time Fresnel prism provides a finite-size design methodology for continuous-wave operation. Prism II is selected for broadband behavior because its period,

$$k_{x,m}=k_{x,0}+k_s+mK,\qquad \omega_m=\omega_0+m\Omega.$$2

is independent of $$k_{x,m}=k_{x,0}+k_s+mK,\qquad \omega_m=\omega_0+m\Omega.$$3 when the interluminal condition $$k_{x,m}=k_{x,0}+k_s+mK,\qquad \omega_m=\omega_0+m\Omega.$$4 is satisfied (Li et al., 2023). Delay-line interconnection then aligns silence intervals through analytically prescribed switch positions and switching times, and the reported FDTD validation shows that interconnected Prism I and II outputs match theory within $$k_{x,m}=k_{x,0}+k_s+mK,\qquad \omega_m=\omega_0+m\Omega.$$5 in frequency and $$k_{x,m}=k_{x,0}+k_s+mK,\qquad \omega_m=\omega_0+m\Omega.$$6 in amplitude, while a $$k_{x,m}=k_{x,0}+k_s+mK,\qquad \omega_m=\omega_0+m\Omega.$$7 modulation-period jitter yields $$k_{x,m}=k_{x,0}+k_s+mK,\qquad \omega_m=\omega_0+m\Omega.$$8 amplitude error (Li et al., 2023). This architecture is programmable through synchronized switching rather than through a single continuous traveling modulation.

Active microwave metaprisms are programmed through bias. In the 2020 platform, generalized Snell’s law is used to compute the required phase gradients from the target map $$k_{x,m}=k_{x,0}+k_s+mK,\qquad \omega_m=\omega_0+m\Omega.$$9, after which the desired phase and magnitude at each super-cell are implemented by amplifier bias voltages and phase-shifter settings (Taravati et al., 2020). The proof-of-concept was manually controlled, but the device is explicitly described as compatible with FPGA-based digital control of amplifier biases and varactor voltages (Taravati et al., 2020). In the reflective beamsteering architecture, the phase shifters’ electrical length is tuned by DC bias, changing the aperture gradient $k_s$0 and thus the reflection angle according to

$k_s$1

Because the device is linear time-invariant for fixed bias, the reconfiguration is rapid and does not generate new spectral lines (Taravati et al., 2021).

Digital space-time coding offers a different route to programmability. The metamaterial antenna implementation toggles the unit-cell phase constant between two states via varactors and uses an FPGA to apply a spatiotemporal coding matrix with circular shifts across time slots, thereby synthesizing a traveling modulation with programmable $k_s$2 and $k_s$3 (Vosoughitabar et al., 2023). In a transmissive or reflective prism interpretation, the same mechanism allows the static gradient and the moving modulation to be digitally co-designed.

6. Demonstrations and reported performance

The clearest optical demonstration of a space-time nonreciprocal prism is the ultrathin Kerr metasurface operating near $k_s$4, corresponding to $k_s$5. The static anomalous reflection efficiency is approximately $k_s$6 at $k_s$7, and the interaction length is only about $k_s$8 (Guo et al., 2019). With $k_s$9, $K$0, $K$1, and $K$2, the moving-phase velocity is approximately $K$3, or about $K$4 (Guo et al., 2019). Under normal incidence, the forward wave exhibits a downward-converted sideband at $K$5 and $K$6, while the upward sideband is evanescent; in the reverse test, incidence at $K$7 with $K$8 yields further downward conversion to $K$9 exiting at normal direction, so the back-reflected wave does not return to the original forward state (Guo et al., 2019). The measured nonreciprocal conversion spans a $\chi(\mathbf{r},t)$0 3-dB bandwidth around $\chi(\mathbf{r},t)$1–$\chi(\mathbf{r},t)$2, with first-order conversion efficiency of approximately $\chi(\mathbf{r},t)$3.

The active microwave metaprism reports a different operating point and a different figure of merit. The fabricated $\chi(\mathbf{r},t)$4 programmable nonreciprocal metasurface prism operates from $\chi(\mathbf{r},t)$5 to $\chi(\mathbf{r},t)$6 with thickness $\chi(\mathbf{r},t)$7 at $\chi(\mathbf{r},t)$8 (Taravati et al., 2020). In near-field horn measurements under normal incidence, the forward transmission gain exceeds $\chi(\mathbf{r},t)$9 across the operating band, the backward transmission loss exceeds $\Phi_f(x)$00, and the isolation exceeds $\Phi_f(x)$01 (Taravati et al., 2020). In far-field decomposition of a three-tone incident wave, the device routes $\Phi_f(x)$02 to $\Phi_f(x)$03 with gain $\Phi_f(x)$04 and isolation $\Phi_f(x)$05, $\Phi_f(x)$06 to $\Phi_f(x)$07 with gain $\Phi_f(x)$08 and isolation $\Phi_f(x)$09, and $\Phi_f(x)$10 to $\Phi_f(x)$11 with gain $\Phi_f(x)$12 and isolation $\Phi_f(x)$13 (Taravati et al., 2020).

The reflective full-duplex beamsteering metasurface emphasizes angle-dependent gain asymmetry rather than frequency splitting. At $\Phi_f(x)$14–$\Phi_f(x)$15, the forward reflection gains exceed $\Phi_f(x)$16 in most measured cases, while the reverse direction shows much lower gain and different output angles (Taravati et al., 2021). Reported examples include $\Phi_f(x)$17 with gain $\Phi_f(x)$18 and isolation $\Phi_f(x)$19, $\Phi_f(x)$20 with gain $\Phi_f(x)$21 and isolation $\Phi_f(x)$22, and $\Phi_f(x)$23 with gain $\Phi_f(x)$24 and isolation $\Phi_f(x)$25 (Taravati et al., 2021).

In the mid-IR graphene platform, the prism functionality is synthetic and explicitly frequency nonreciprocal. Using a QCL near $\Phi_f(x)$26, a $\Phi_f(x)$27 three-phase RF modulation, and Littrow geometry with synthetic period $\Phi_f(x)$28, the $\Phi_f(x)$29 synthetic orders occur at $\Phi_f(x)$30 for $\Phi_f(x)$31 (Efimov et al., 30 Aug 2025). With phase sequence $\Phi_f(x)$32, the forward reflection undergoes $\Phi_f(x)$33 while the reverse test yields $\Phi_f(x)$34, with no return to $\Phi_f(x)$35; reversing the phase sequence switches the operation to $\Phi_f(x)$36 forward and $\Phi_f(x)$37 backward (Efimov et al., 30 Aug 2025). The same study connects such directional sideband asymmetry to a spectral-directional breakdown of Kirchhoff’s law.

The passive self-biased gyrotropic platform shows that nonreciprocal prism behavior can be achieved without time modulation or active gain. The reported metasurface reaches transmittance up to $\Phi_f(x)$38 and operation angle up to $\Phi_f(x)$39 (Yang et al., 2022). For the nonreciprocal deflector near $\Phi_f(x)$40, the backward first-order efficiency is approximately $\Phi_f(x)$41 in simulation and $\Phi_f(x)$42 in measurement, with deflection around $\Phi_f(x)$43 simulated and $\Phi_f(x)$44 measured, while the forward zeroth-order transmission is approximately $\Phi_f(x)$45 simulated and $\Phi_f(x)$46 measured (Yang et al., 2022).

7. Applications, limitations, and research directions

The application space is broad because a programmable nonreciprocal metasurface prism can steer, isolate, convert frequency, amplify, or selectively absorb depending on platform. Reported use cases include optical isolation and one-way beam steering, frequency-agile routing and signal processing, lidar and sensing with nonreciprocal glare suppression, photonic integrated circuits, full-duplex reflective beamsteering, holography, and wireless telecommunications (Guo et al., 2019). In the mid-IR regime, the same space-time nonreciprocity has been connected to energy conversion, radiative cooling, sensing, and imaging through unequal absorptivity and emissivity channels (Efimov et al., 30 Aug 2025).

Several misconceptions recur in this area. Nonreciprocity does not require magnetic bias: it can be achieved through traveling-wave space-time modulation, unilateral active circuitry, self-biased gyrotropic media, or susceptibility programming (Guo et al., 2019). Conversely, not every “space-time prism” steers by angle; in the 1+1D Space-Time Fresnel prism, the main observable is frequency translation rather than apex diffraction, and angle control is explicitly outside the scope of the base formulation (Li et al., 2023). It is also incorrect to equate complete asymmetry with high efficiency. The Kerr demonstration shows completely asymmetric reflections but only approximately $\Phi_f(x)$47 first-order conversion efficiency, limited by pump–probe temporal mismatch, spatial overlap, and thermal constraints at $\Phi_f(x)$48 repetition rate (Guo et al., 2019).

The main limitations are mechanism-specific. Optical space-time modulation offers ultrafast multi-THz operation and subwavelength interaction length, but presently with modest conversion efficiency and pump-power requirements (Guo et al., 2019). The Space-Time Fresnel prism avoids infinite device length but requires accurate timing, delay-line management, and synchronization (Li et al., 2023). Active transistor platforms achieve gain and strong isolation without time harmonics, but introduce DC power consumption, amplifier noise, and stability constraints (Taravati et al., 2021). Graphene-based mid-IR devices require careful RF grounding, heating management, and high-quality transfer, while cavity $\Phi_f(x)$49 and modulation depth jointly determine the efficiency–bandwidth trade-off (Efimov et al., 30 Aug 2025). Passive gyrotropic metasurfaces avoid modulation and gain but presently rely on absorption-prone ferrites and, in the reported implementation, circular polarization (Yang et al., 2022).

The forward-looking directions stated in the literature are correspondingly diverse. For optical space-time metaprisms, further increases in conversion efficiency are expected through improved pump–probe overlap, lower repetition-rate pumping, ENZ materials, and cavity-assisted metasurface designs (Guo et al., 2019). The Space-Time Fresnel concept points toward slab, gradient, and accelerated profiles, including space-time chirping and time-lens operations (Li et al., 2023). Integrated photonic work on susceptibility-programmable media suggests reconfigurable, strong, and fastly switchable isolation based on a programmable susceptibility tensor and structured optical pumps, which in turn suggests a route to nonreciprocal prism behavior without relying on weak Kerr effects (Zhang et al., 2024). Taken together, these developments indicate that the programmable nonreciprocal metasurface prism is evolving from a single beam-steering element into a general platform for direction-dependent control of angle, frequency, gain, and thermal channel occupancy.