Time Refraction in Magnetoplasmon-Polariton

• Time refraction of magnetoplasmon-polaritons is a phenomenon where time-varying material parameters modulate the propagation, frequency, and polarization of hybrid plasmonic modes.
• It employs dynamic modulation of dielectric functions and magnetic fields to induce effects like time circular birefringence, Berry curvature corrections, and mode splitting.
• These insights enable ultrafast switching and non-reciprocal control in photonic circuits, with applications in pulse shaping, frequency conversion, and topologically protected edge states.

Time refraction of magnetoplasmon-polaritons refers to the temporal modification of propagation, frequency, and field distribution of hybrid electromagnetic–plasmonic modes under time-dependent changes in system parameters, especially in the presence of magnetic effects or nontrivial dielectric dispersion. Central to this phenomenon are mechanisms that break time-reversal symmetry, induce ultrafast birefringence, or modulate Berry curvatures in the frequency domain, all of which enable dynamic control over plasmonic excitations and facilitate novel light manipulation protocols at subwavelength and ultrafast scales.

1. Fundamental Principles and Definitions

Magnetoplasmon-polaritons are hybrid excitations that arise from the coupling of electromagnetic waves with charge-density oscillations in the presence of a magnetic field or magnetic order, typically within metal–dielectric interfaces, quantum Hall systems, or plasmonic resonators. Time refraction denotes the change in frequency content, group velocity, or modal coupling resulting from temporal variation of material parameters—analogous to spatial refraction at an interface. In plasmonic systems, this often involves temporal modulation of the dielectric function, external magnetic field, magnetoelectric tensor, or carrier density.

Recent work demonstrates that in dispersive systems (with explicit frequency dependence in ε(ω)), Maxwell's equations transform into a non-standard eigenvalue problem, where propagation eigenstates acquire Berry curvature corrections in frequency/momentum space (Deng et al., 18 Aug 2025). These corrections mediate anomalous dynamics in the presence of time refraction and can induce swinging or deflection of the propagation trajectory.

2. Magnetoplasmon-Polariton Dispersion and Magnetic Control

The propagation of magnetoplasmon-polaritons in Voigt geometry (magnetization along z, propagation along x) is governed by anisotropic or gyrotropic dielectric tensors:

$$\varepsilon = \begin{pmatrix} \varepsilon_r & \varepsilon_{xy} & 0 \ -\varepsilon_{xy} & \varepsilon_r & 0 \ 0 & 0 & \varepsilon_r \end{pmatrix}$$

where $\varepsilon_{xy} = gM_z$ introduces magnetization-dependent off-diagonal terms (Nikolova et al., 2013). Dispersion relations, for example at a single metal–magnetic dielectric interface, are derived from boundary conditions on exponentially decaying field profiles:

$$\varepsilon_m \varepsilon_r \kappa_{yd} - i \varepsilon_m \varepsilon_{xy} k_x + (\varepsilon_r^2 + \varepsilon_{xy}^2)\kappa_{ym} = 0$$

This symmetry breaking leads to different propagation constants for left- and right-propagating modes ($k_+, k_-$), rendering the mode structure sensitive to magnetization.

In metal-insulator-metal (MIM) waveguide cavities, the magnetization can strongly skew mode profiles between upper and lower interfaces, allowing for control over the coupling between emitters and guided modes. Numerical simulations confirm that a point dipole can switch its emission into the propagating mode by modulating core magnetization (Nikolova et al., 2013). These mechanisms enable magnetic field control over emission, coupling, and far-field radiation distributions, providing a channel for ultrafast or dynamic photonic circuit operation.

3. Temporal Modulation: Time Circular Birefringence and Time Refraction

Time-dependent axion-type magnetoelectric (ME) media exhibit time circular birefringence (TCB), wherein a temporally varying pseudoscalar field $\Theta(t)$ induces two degenerate circularly polarized modes for each wave vector $k$:

$$\frac{d^2 T_{\pm}}{dt^2} + \frac{d\ln\varepsilon}{dt}\frac{dT_{\pm}}{dt} + v^2 k[k \pm \mu \frac{d\Theta}{dt}] T_{\pm} = 0, \quad v^2 = \frac{1}{\varepsilon \mu}$$

(Zhang et al., 2015)

At a temporal boundary ("time interface"), a linearly polarized wave splits into two modes with distinct frequencies, the so-called "time refracted" and "time reflected" waves. The superposition of these modes produces "time Faraday rotation": the polarization axis rotates coherently in the time domain, with an angular velocity set by the frequency splitting. Experimentally, time-dependent magnetoelectric modulation can be achieved via molecular fluids subject to pulsed parallel electric and magnetic fields; the refractive-index difference between TCB branches under realistic conditions is estimated at $10^{-17}$.

In the context of magnetoplasmons, this framework predicts splitting and mixing of edge/polariton modes under timed external field variation, resulting in modulated energy flow, polarization rotation, and frequency conversion.

4. Ultrafast Manifestations and Nonlinear Time Dynamics

Magnetoplasmonic crystals display ultrafast evolution of the magneto-optic Kerr effect (TMOKE) within femtosecond pulses, specifically in iron-based structures under resonant SPP excitation (Shcherbakov et al., 2014). The SPP dispersion takes the form:

$$k_{SPP}(M) = \frac{\omega}{c} \sqrt{\frac{\varepsilon}{\varepsilon+1}(1+\alpha g(M))}$$

where $g(M)$ encodes magnetization dependence and $\alpha$ derives from the dielectric permittivity.

When excited by a femtosecond pulse, the reflected envelope $I(M,t)$ features a time-varying magneto-optic response:

$\delta(M, t) = \frac{I(M, t) - I(-M, t)}{I(0, t)}$

and $\delta(M,t)$ exhibits opposite time derivatives on either side of the SPP resonance. This arises due to the magnetization-driven shift and asymmetric resonance shape: different parts of the pulse experience distinct phase shifts, directly mirroring time refraction. The ability to tailor time-varying optical response through magnetic control enables active pulse shaping and ultrafast switching in plasmonic circuits.

Negative time refraction, as demonstrated in strongly coupled plasmonic antenna–ENZ systems, relies on time-dependent nonlinear polarization oscillating at doubled frequency (induced by optical pumping) for efficient generation of phase-conjugate and negative refracted beams. The conversion efficiency is boosted by Rabi splitting and near-field energy density enhancement, yielding frequency mixing and new directions in ultrafast light-matter interactions (Bruno et al., 2019).

5. Topological and Edge-State Effects under Time Refraction

2D magnetoplasmons with intrinsic particle-hole symmetry host topologically protected edge states (chiral, one-way propagation), which are analogous to chiral Majorana states in p+ip superconductors (Jin et al., 2016). The bulk spectrum is gapped by the cyclotron frequency, with zero modes localized at edges or domain walls.

Parameter modulation in time (such as instantaneous change in cyclotron frequency $\omega_c$ or electron density) can realize time-refraction events that switch or convert edge modes, with persistence guaranteed by their topological character. Time-resolved experiments on quantum anomalous Hall (QAH) insulators demonstrate non-reciprocal chiral edge magnetoplasmon (EMP) propagation, where collective plasmon velocities exceed equilibrium drift velocities and the plasmon frequency evolves in response to temporal environmental changes (Martinez et al., 2023). Heating or microwave power modulates bulk dissipation channels, which directly alter EMP time-of-flight and decay: these features provide clear signatures of time refraction.

6. Frequency Domain Berry Curvature and Anomalous Wave-Packet Dynamics

When the dielectric function is dispersive (e.g., $\varepsilon(\omega) = 1-\omega_p^2/\omega^2$), Maxwell's equations become a non-standard eigenvalue problem with frequency inside the operator. Following an off-shell variational framework, the Berry curvature in the frequency domain emerges as a correction to wave-packet dynamics (Deng et al., 18 Aug 2025). The equations of motion for the magnetoplasmon-polariton center involve group velocity and anomalous velocity terms:

$$\dot{x} = v_g - [...] \Omega_{\omega r} - [...] \Omega_{\omega k}$$

$\dot{k} = F + [...] \Omega_{\omega r} + [...]$

Here, $\Omega_{\omega r}$ and $\Omega_{\omega k}$ are Berry curvatures involving frequency derivatives. During adiabatic modulation (e.g., sweeping $\omega_p$), these induce lateral swinging of the wave packet, so the ray trajectory deviates transversely from the propagation direction, governed by the geometric phase accumulated through instantaneous changes. Such geometric wave phenomena are essential for controlled light manipulation in dispersive, topologically nontrivial media.

7. Experimental Realizations and Applications

Systems engineered for time refraction of magnetoplasmon-polaritons include:

• Magnetic slot waveguides or MIM cavities with dynamically tunable magnetization, allowing selective switching and propagation control (Nikolova et al., 2013).
• Magnetoplasmonic crystals subjected to ultrafast optical pulses, where time-varying Kerr effects manipulate pulse envelopes (Shcherbakov et al., 2014).
• Time-dependent ME media realized via fluidic platforms with parallel electric and magnetic fields, achieving measurable time circular birefringence and Faraday rotation (Zhang et al., 2015).
• Strongly coupled plasmonic antenna–ENZ metasurfaces, leveraging nonlinear dynamics for frequency mixing, phase-conjugation, and negative time refraction (Bruno et al., 2019).
• QAH insulators or 2DEG samples probed by microwave pulses, exhibiting time domain switching between edge and bulk transport, with direct applications in non-reciprocal devices and integrated signal routing (Martinez et al., 2023).

Potential applications span ultrafast switches, temporal cloaking, robust frequency converters, time-dependent polaritonic devices, and topological photonic circuits.

8. Theoretical and Practical Significance

Time refraction in magnetoplasmon-polariton systems exemplifies the interplay between ultrafast optoelectronic modulation, topological protection, and geometric phase phenomena. Insights from analytical models (Zhang et al., 2015, Deng et al., 18 Aug 2025), numerical simulation (Nikolova et al., 2013), and experimental protocol (Shcherbakov et al., 2014, Martinez et al., 2023) confirm the feasibility and utility of dynamic control over light–matter interactions. Berry curvature effects in the frequency domain imply new routes for geometric manipulation of optical and plasmonic excitations, offering improved robustness and control in future quantum and classical photonic technologies.