Dynamic and Geometric Shifts in Wave Scattering

TL;DR

This work extends Berry's geometric-dynamic decomposition to generic wave-scattering problems by introducing the generalized Wigner-Smith operator $\hat{\bf Q}=-i\hat{S}^\dagger\partial_{\bm\rho}\hat{S}$ for a unitary $\hat{S}({\bm\rho})$. Shifts in conjugate observables (such as $\omega$, ${\bf k}$, $t$, or ${\bf r}$) decompose into a dynamic part from eigenvalue gradients and a geometric part from eigenvector gradients, via $\hat{\bf Q}=\hat{\bf Q}_d+\hat{\bf Q}_g$ with $\hat{\bf Q}_d=\hat{U}\hat{\bf Q}'\hat{U}^\dagger$ and $\hat{\bf Q}_g=\hat{U}(\hat{S}'^\dagger\hat{\bf A}\hat{S}'-\hat{\bf A})\hat{U}^\dagger$. The framework unifies and extends phenomena such as the Goos-Hänchen and Imbert-Fedorov shifts and the Pancharatnam-Berry phase within a gauge-invariant scatter­ing context, and it applies across time-varying elements, space-varying metasurfaces, beam interfaces, and 1D scattering. The results offer a principled route to engineer scattering by tailoring eigenvalues and eigenvectors, with future extensions to non-Hermitian systems and integration with inverse design or machine-learning strategies.

Abstract

Since Berry's pioneering 1984 work, the separation of geometric and dynamic contributions in the {\it phase} of an evolving wave has become fundamental in physics, underpinning diverse phenomena in quantum mechanics, optics, and condensed matter. Here we extend this geometric-dynamic decomposition from the wave-evolution phase to a distinct class of wave scattering problems, where observables (such as frequency, momentum, or position) experience shifts in their expectation values between the input and output wave states. We describe this class of problems using a unitary scattering matrix and the associated generalized Wigner-Smith operator (GWSO), which involves gradients of the scattering matrix with respect to conjugate variables (time, position, or momentum, respectively). We show that both the GWSO and the resulting expectation-values shifts admit gauge-invariant decompositions into dynamic and geometric parts, related respectively to gradients of the eigenvalues and eigenvectors of the scattering matrix. We illustrate this general theory through a series of examples, including frequency shifts in polarized-light transmission through a time-varying waveplate (linked to the Pancharatnam-Berry phase), momentum shifts at spatially varying metasurfaces, optical forces, beam shifts upon reflection at a dielectric interface, and Wigner time delays in 1D scattering. This unifying framework illuminates the interplay between geometry and dynamics in wave scattering and can be applied to a broad range of physical systems.

Figures (6)

• Figure 1: Schematic of wave scattering described by a unitary scattering matrix $\hat{S}$, which depends on a variable ${\bm\rho}$. The scattered wave experiences a shift $\Delta {\bm \pi}$ in the expectation value of the variable conjugate to ${\bm \rho}$. This shift is described by the generalized Wigner-Smith operator (GWSO). The ${\bm\rho}$-gradients of the eigenvalues and eigenvectors of $\hat{S}$ are responsible for the dynamic and geometric parts of $\Delta {\bm \pi}$ or the GWSO, respectively.
• Figure 2: Transmission of polarized light (without reflection) through an anisotropic waveplate with time-varying retardation between orthogonal polarizations, $\delta(t)$, and orientation of the optical axis, $\varphi(t)$, described by the scattering matrix \ref{['S_plate']}. The input and output polarization states are shown by elliptical arrows. The transmitted light undergoes a frequency shift $\Delta \omega$. It includes dynamic and geometric contributions \ref{['Delta_omega']} described by the corresponding parts of the GWSO, Eqs. \ref{['Qd-Qg_plate']}, and originating from the time derivatives of the parameters $\delta$ and $\varphi$, respectively.
• Figure 3: Connection between frequency shifts \ref{['Delta_omega']} and the Pancharatnam-Berry (PB) phase on the Poincaré sphere. (a) For circularly-polarized incident light ($\sigma=1$ here), the infinitesimal geometric phase shift $\Delta\omega_g dt$ is related to the infinitesimal solid angle $d\Omega$ formed by the Stokes vectors of the incident light, $\vec{\Sigma}$, transmitted light, $\vec{\Sigma}'(t)$, and $\vec{\Sigma}'(t+dt)$, via the corresponding PB phase, Eq. \ref{['PB_1']}. The dynamic shift vanishes in this case: $\Delta\omega_d=0$. (b) For linearly-polarized incident light ($\chi=1$ here), the infinitesimal total phase shift $\Delta\omega\, dt$ is related (with a minus sign) to the modified solid angle $d\tilde{\Omega}$ formed by the Stokes vectors $-\vec{\Sigma}$, $\vec{\Sigma}'(t)$, and $\vec{\Sigma}'(t+dt)$, via the modified PB phase, Eq. \ref{['PB_2']}. In both cases (a) and (b), for a cyclic evolution, when the output Stokes vector $\vec{\Sigma}'(t)$ traces a closed loop on the sphere, the total phase accumulated in the transmitted light is given by the integral PB expression \ref{['PB_cyclic']} involving the solid angle enclosed by the loop.
• Figure 4: Transmission of light through a waveplate with spatially varying parameters (a metasurface). (a) Similarly to the time-varying phase plate, Fig. \ref{['Fig_plate']}, spatial variations of the phase retardation (plate thickness), $\delta({\bf r})$, produce a dynamic wavevector shift $\Delta{\bf k}_d$. Shown is the case of linear input polarization ($\tau=1$). (b) Spatial variations of the optical-axis orientation, $\varphi({\bf r})$, generate a geometric wavevector shift $\Delta{\bf k}_g$. Shown is the case of circular input polarization ($\sigma=1$) and half-wave retardation ($\delta=\pi$). These polarization-dependent wavevector shifts are described by Eqs. \ref{['Delta_omega']} and \ref{['Qd-Qg_plate']} with the substitutions $\Delta\omega \to \Delta{\bf k}$ and $\partial/\partial t \to - \partial/\partial{\bf r}$.
• Figure 5: Total internal reflection of a Gaussian-like beam at a dielectric interface, described by the scattering matrix $\hat{S}({\bf k}_\perp)$, Eq. \ref{['S_beam']} (see explanations in the text). The $x$- and $y$-directed shifts of the reflected beam, $\Delta{\bf r}_d$ and $\Delta{\bf r}_g$, correspond to the Goos-Hänchen and Imbert-Fedorov (spin-Hall effect) shifts, respectively, and are described by the dynamical and geometric parts of the GWSO, Eqs. \ref{['Qd-Qg_beam']} and \ref{['Delta_R']}.