Magnitude-Phase-Decoupled Metasurfaces

Updated 23 November 2025

Magnitude-phase decoupled metasurfaces are engineered 2D structures that enable independent control over the amplitude and phase of electromagnetic waves using dual resonator or multi-layer designs.
They employ mechanisms such as coupled resonators, specialized unit-cell designs, and active reconfigurable elements to achieve near-orthogonal tuning with minimal cross-interference between amplitude and phase.
These metasurfaces facilitate advanced applications including beamforming, programmable reflectors, high-fidelity holography, and nonlinear wavefront control with enhanced efficiency.

A magnitude-phase-decoupled metasurface is an engineered two-dimensional structure whose unit cells are specifically designed to enable independent and simultaneous control over the amplitude and phase of the reflected or transmitted electromagnetic wavefront. This decoupling overcomes the intrinsic correlation observed in conventional metasurfaces, where tuning the phase generally affects the amplitude and vice versa. Recent implementations span the RF, microwave, and optical regimes and employ distributed or compact configurations of resonators, reconfigurable lumped elements, or carefully engineered unit-cell topologies.

1. Fundamental Principles and Architectures

The core function of a magnitude-phase-decoupled metasurface is to realize a reflection (or transmission) coefficient of the form

$\Gamma = |\Gamma| \, e^{j\phi}$

where $|\Gamma|$ (amplitude) and $\phi$ (phase) are independently tunable parameters for each meta-atom or pixel. Traditional single-resonator architectures, whether planar or volumetric, couple these degrees of freedom due to their single-parameter resonance—any change in loss (dissipation) or reactance (resonant frequency shift) perturbs both amplitude and phase, leading to complicated amplitude–phase loci. Magnitude-phase decoupling requires architectures offering two quasi-orthogonal control “knobs,” typically implemented using:

Coupled Resonator Approach: Parallel or concentric resonators with tunable resistive (lossy) and reactive (capacitive/inductive) elements physically or electrically separated (Ashoor et al., 2020). This enables near-orthogonal tuning: resistance $R$ (via PIN diodes, varistors) sets $|\Gamma|$ , and capacitance $C$ (via varactors) sets $\phi$ .
Multi-layer or Compound Structures: Cascaded phase-only metasurfaces (e.g., Huygens-type or locally periodic dielectric arrays) separated by a free-space or dielectric gap. Propagation between layers enables independent amplitude (via interference) and phase (via collective phase-shifting) control (Raeker et al., 2018, Raeker et al., 2021).
Specialized Unit Cell Designs: Hybrid anapole nanostructures, nonlocal BIC-based resonators, or symmetric PB-phase encodings support decoupling by virtue of multipolar cancellation or nonlinearity-selective responses (Kuznetsov et al., 2021, Sedeh et al., 8 Jul 2025). In nonlinear metasurfaces, the amplitude (harmonic efficiency) and phase (wavefront) can be controlled independently by combining resonant enhancement with spatial phase engineering.
Active and Reconfigurable Topologies: Electrically addressable MOS metasurfaces, software-defined RC/PIN diode-loaded reflectarrays, and reflective mixer-phase-shifter units combine amplitude and phase decoupling with real-time programmability (Dong et al., 4 Nov 2024, Dong et al., 16 Nov 2025, Pitilakis et al., 2022, Mayoral-Astorga et al., 12 Feb 2024).

2. Theoretical Models and Circuit Representations

Magnitude-phase decoupling in metasurfaces is analytically modeled using equivalent-circuit and boundary-condition approaches:

Equivalent Circuit Models: A coupled resonator metasurface is represented by two ladders—one dominated by $R$ (e.g., PIN-diode-loaded dipole ring), one by $C$ (varactor-loaded split ring)—both in shunt with the substrate admittance. The net impedance $Z_n(R,C)$ leads to the reflection coefficient:

$\Gamma(R,C) = \frac{Z_n(R,C) - Z_0}{Z_n(R,C) + Z_0}$

Where $Z_0$ is the free-space impedance. Sweeping $R$ at fixed $C$ varies $|\Gamma|$ with minimal phase perturbation; sweeping $C$ at fixed $R$ shifts phase over nearly the full $2\pi$ range at fixed $|\Gamma|$ (Ashoor et al., 2020).

Transmission-Line and Surface-Impedance Models: For doubly-loaded metasurfaces (RC or RL in each unit), the reflection coefficient is given by

$\Gamma = \frac{Z_s - Z_0}{Z_s + Z_0}, \qquad Z_s = R_s + j X_s$

Where amplitude and phase can be solved from $R_s$ , $X_s$ for target $|\Gamma|$ , arg $\Gamma$ (Pitilakis et al., 2022).

Boundary-Condition and Propagation Models: In multi-layer structures, each metasurface applies a local phase jump; amplitude shaping derives from propagation and interference. Analytical expressions involve Fresnel-Kirchhoff integrals or transfer-matrix descriptions, yielding algorithms (e.g. modified Gerchberg–Saxton) to achieve arbitrary amplitude and phase at the target plane (Raeker et al., 2021, Raeker et al., 2018).
Multipolar and Symmetry Analysis: In hybrid anapole or QTM metasurfaces, distributed multipolar moments (electric, toroidal, magnetic) combine to suppress scattering in undesired channels, thereby enforcing high transmission at arbitrary phase (Kuznetsov et al., 2021, Sedeh et al., 8 Jul 2025). In nonlinear scenarios, phase modulation can be made frequency- or polarization-selective via resonant field confinement and geometric phase.

3. Mechanisms for Amplitude–Phase Decoupling

The essential physical and electronic mechanisms underlying magnitude-phase decoupling include:

Orthogonal Degrees of Freedom: Physical separation of dissipative and reactive tuning elements ensures that $|\Gamma|$ and $\phi$ map to different physical processes—loss vs. resonance shift.
Multi-resonator Coupling: Superposition of independent scattering channels enables amplitude and phase to be modulated independently, in contrast to single-resonator devices where control variables interact nontrivially.
Compound Metaoptics: By distributing phase-only metasurfaces across a propagation gap, amplitude and phase manipulation become independent optimization variables, with the intermediate field profile acting as an algorithmic design parameter (Raeker et al., 2021).
Multipolar Interference: Complete cancellation of radiating and toroidal moments in hybrid-anapole meta-atoms yields transmission $|t|\approx 1$ with phase control solely by geometry, regardless of neighbor proximity (Kuznetsov et al., 2021).
Nonlinear Resonant Enhancement: Selectively enhanced QTM or BIC resonances for nonlinear harmonic generation allow the amplitude of higher-harmonic processes (e.g., third harmonic) to be set by resonance strength and the phase by local rotation (Pancharatnam–Berry phase), decoupling via mode- and frequency-selective response (Sedeh et al., 8 Jul 2025).

4. Representative Implementations and Experimental Performance

Magnitude-phase-decoupled metasurfaces have been demonstrated in various architectural embodiments:

Implementation	Regime	Decoupling Mechanism
DRR + SRR with PIN diodes/varactors	Microwave	Tunable R and C with concentric dual-resonator
Two-layer phase-only metaoptics	Optical	Fresnel interference via spaced phase layers
Hybrid-anapole nanocylinder arrays	Optical	Multipolar cancellation, neighbor-insensitive
MOS-capacitor loaded nanoantennas	Near-IR	Carrier refraction in ITO for constant
PIN-diode patch + switch reflectors	Microwave	Mixer-phase shifter with separate modulation
QTM all-dielectric structures	NLO	Frequency- and geometry-selective phase/amplitude
RC-loaded patch metasurfaces	RF	Surface impedance engineering with lumped loads

Specific metrics and results include:

Continuous tuning of $|\Gamma|$ from near 0 to 1 and $\phi$ over $0\ldots360^\circ$ with insertion loss $<0.3$ dB (Ashoor et al., 2020).
Optical analogs show $|T| > 0.93$ across $\sim$ 80% phase range; multi-beam formers and complex holograms with 70–80% efficiency (Raeker et al., 2021).
Near-perfect transmission ( $|t|\sim0.99$ ) with full-range phase control in hybrid-anapole metasurfaces, robust under substantial disorder (Kuznetsov et al., 2021).
Electrically tunable MOS metasurfaces demonstrate $\sim30^\circ$ $\phi$ shift at constant $|r| \sim0.5$ , with design extensions promising 360 $^\circ$ (Mayoral-Astorga et al., 12 Feb 2024).
Nonlinear QTM metasurfaces exhibit $>10^3\times$ THG enhancement, and phase mapping (PB-phase) decoupled from amplitude, frequency-selectively (Sedeh et al., 8 Jul 2025).

5. Applications and Limitations

Magnitude-phase-decoupled metasurfaces support advanced functionalities unattainable by conventional phase-only or amplitude-only architectures:

Beamforming and Gain Shaping: Dynamic control of array factor amplitude and phase enables beam steering with variable, pattern-shaped gain, low sidelobe synthesis (Chebyshev/binomial tapers), and multi-beam generation (Ashoor et al., 2020, Pitilakis et al., 2022).
Programmable and Adaptive Reflectors: Real-time reconfiguration for smart wireless environments, including adaptive reflectarrays for wireless communications and backscatter transmitters (Dong et al., 4 Nov 2024, Dong et al., 16 Nov 2025).
Holography and 3D Wavefront Engineering: Simultaneous arbitrary amplitude and phase synthesis facilitates high-fidelity, efficiency-optimized holograms and complex optical field manipulations (Raeker et al., 2021, Raeker et al., 2018).
Nonlinear Wavefront Control: Third-harmonic beam steering and polarization-sensitive conversion in silicon photonic devices, leveraging decoupled enhancement and PB-phase imposition (Sedeh et al., 8 Jul 2025).
Ultra-Compact Metaoptics: Subwavelength, disordered, or tightly packed arrays made feasible by negligible inter-cell coupling (hybrid-anapole designs) (Kuznetsov et al., 2021).

Identified limitations include:

Routing and integration complexity scales unfavorably with independent R and C (or equivalent control) networks per pixel in large arrays.
Bandwidth is generally limited by resonator $Q$ -factor and loss-dispersion trade-offs.
Continuous, real-time decoupling requires either two actively controlled elements per cell, or multiple physical layers, increasing device fabrication complexity.
Quantization and switching limitations constrain the achievable resolution in amplitude–phase space for digital implementations (Dong et al., 4 Nov 2024, Dong et al., 16 Nov 2025).
In active architectures, nonlinearities and device drift (as in PIN and MOS elements) can impact long-term stability and speed dependent on device selection (Mayoral-Astorga et al., 12 Feb 2024).

6. Design Algorithms and Guidelines

Robust design flows have been established:

For coupled-resonator or RC-type reflective metasurfaces, sweep $R$ to target $|\Gamma|$ , $C$ for target $\phi$ at each frequency point; utilize analytical formulas for $Z_s$ to back-calculate required values (Ashoor et al., 2020, Pitilakis et al., 2022).
In compound metaoptics, employ a two-step iterative algorithm: (1) retrieve a complex intermediate field so amplitude matches target post-propagation, (2) correct for phase at the final layer; typically solved using modified Gerchberg–Saxton or direct inversion techniques (Raeker et al., 2021, Raeker et al., 2018).
In multipolar architectures, exploit precomputed abacus (lookup table) relationships—for example, geometry (Si nanocylinder radius) to transmission phase, ensuring negligible $|t|$ variance (Kuznetsov et al., 2021).
For programmable platforms, calibrate 2D LUTs (lookup tables) mapping control voltages to the desired $|\Gamma|$ , $\phi$ for real-time driving via DAC/GPIO (Dong et al., 16 Nov 2025).
Nonlinear metasurfaces require controlled symmetry breaking (e.g., local notches, hole offset) to enable frequency- or process-selective geometric phase decoupling (Sedeh et al., 8 Jul 2025).

7. Future Directions

Research is advancing toward metasurfaces with physically and electronically reconfigurable unit cells at nanosecond timescales, multi-functional and multi-harmonic control, and integrated systems for communications, lidar, and advanced photonics. Key areas include:

Increasing the dynamic range and speed of decoupled active elements, especially in THz and optical regimes.
Integration with CMOS and on-chip control networks to realize fully addressable software-defined metasurfaces (Pitilakis et al., 2022).
Exploring topological and nonlocal mechanisms for robust, disorder-immune decoupling (Kuznetsov et al., 2021, Sedeh et al., 8 Jul 2025).
Extending decoupling concepts to polarization, frequency, and space-time domains, including arbitrary baseband modulation combined with precision RF beamforming (Dong et al., 4 Nov 2024, Dong et al., 16 Nov 2025).
Developing scalable nanofabrication approaches enabling subwavelength sampling without mutual coupling artifacts or bandwidth limitations.

Magnitude-phase-decoupled metasurfaces thus represent a key technology for next-generation wavefront engineering, wireless infrastructure, programmable optics, and nonlinear photonics.