Electric-Magnetic Geometric Phase

Abstract

Geometric phases play an enormous role in optics and are generally associated with the evolution of light's polarization state on the Poincaré sphere, or its spin on the sphere of spin directions. Here we put forward a new kind of optical geometric phase that appears exclusively in nonparaxial light, resulting from cyclic changes to the relative amplitude and phase between the electric and magnetic fields. This phase is naturally represented on a recently introduced `electric-magnetic' sphere.

Figures (3)

• Figure 1: General geometric phase scenario. (a) The field $\vec{\boldsymbol{\Psi}}$, Eq. \ref{['bispinor']} evolves over a curve $C$ in the parameter $\bm{\rho}$ space. At the two ends of $C$ the field differs only by the phase $\Phi_\text{glo}=\Phi_\text{loc}+\Phi_\text{g}$, Eqs. \ref{['Pan_phase']}, \ref{['local_phase']}, and \ref{['geom_phase_general']}. (b) If the electric and magnetic polarization states do not vary along $C$, then $\Phi_\text{g}=-\Omega_\mathbf{W}/2$ is the EM geometric phase, proportional to the solid angle enclosed by the evolution loop on a Poincaré-like EM sphere Golat2025, Eqs. \ref{['Wvec']} and \ref{['solid_angle']}.
• Figure 2: EM geometric phase emerging in a time-modulated standing wave. (a) Diagram of scenario, designed so $\mathbf{E}$ and $\mathbf{H}$ are $\mathbf{\hat{x}}$-polarized at $z=0$, Eq. \ref{['standing_wave']}. (b) Trajectory of $\mathbf{W}$ (the star denotes $\mathbf{W}(0)=\mathbf{W}(T)$), corresponding to Eq. \ref{['solution']} with $\theta_0=\pi/6$ and $\chi_0=2\pi/3$ and enclosing solid angle $\Omega_\mathbf{W}\simeq -0.27$ sr. (c) Initial/final EM ellipse Vernon2025_1 at $z=0$. The red line is the instantaneous vector $(\real\{E_x(T)\},\real\{\eta H_x(T)\})$ of the standing wave having acquired a phase of $-\omega_0T+\Phi_\text{g}$. The grey line is that same vector for a reference standing wave that gains the phase $-\omega_0T$.
• Figure 3: EM geometric phase arising in the longitudinal components of a time-evolving focused beam carrying a first-order modal structure. (a) Schematic, where a linearly polarized beam whose mode profile $f$ is modulated over $t$, Eq. \ref{['bisp_f']}, is focused to produce longitudinal components \ref{['z_comp']} in the focal center $\mathbf{r=0}$. (b) Closed trajectory of $\mathbf{W}$ corresponding to Eqs. \ref{['z_comp']} and \ref{['ab_focused']} with $\theta_0=\pi/6$ and $\chi_0=\pi/2$. (c) Final state of the ellipse formed of the electric and magnetic z components in the beam focus, annotated similarly as in Fig. \ref{['fig2']}(c).