Spin-Split Dispersion of Leaky Surface plasmons in Inversion- Symmetric System

TL;DR

The paper addresses realizing spin-split dispersion in inversion-symmetric metasurfaces, where conventional spin-orbit effects typically rely on symmetry breaking. It employs Fourier-domain leakage-radiation measurements with full polarization Mueller matrix analysis and a two-step Mueller–Jones model of focusing and grating anisotropy, represented as $M(d_f,\delta_f,\phi)$ and $M(d_p,\delta_p,0)$. The key finding is a spin-momentum-locked dispersion of leaky surface plasmons on a 1D gold grating, with spin-dependent shifts in $k_{x,out}$ and an experimentally extracted geometric transverse momentum $k_g^{x} \approx 1.2\ \mathrm{rad}/\mu\mathrm{m}$; the dispersion follows $k_{x,out}=k_{spp}-\frac{2\pi}{d} \pm k_g^{x}$, driven by a geometric phase gradient $\nabla \Phi_g$. This work establishes a geometric LB-LD mechanism for spin control in simple centrosymmetric metasurfaces, offering a new pathway for spin-based dispersion engineering in nano-optics.

Abstract

Spin-dependent dispersion and Rashba effect are manifestations of universal spin orbit interaction associated with the breaking of the spatial inversion symmetry in condensed matter and in optical systems. In sharp contrast to this, we report a spin-split dispersion effect of leaky surface plasmons in an inversion-symmetric one dimensional plasmonic grating system. In our system, the signature of spin-momentum locking and the resulting spin-polarization dependent splitting of dispersion of the surface plasmons are observed through the leakage radiation detected in a Fourier (momentum) domain optical arrangement. The setup enables single-shot recording of the full polarization-resolved dispersion (frequency vs transverse momentum (k)) of the leaky surface plasmons. Momentum domain polarization analysis identified a transverse momentum (k) dependent linear birefringence-linear dichroism effect (referred to as the geometric LB-LD effect) responsible for the observed spin-split dispersion. This unconventional SOI effect is reminiscent of the recently reported LB-LD effect resulting in giant chirality in centrosymmetric crystal, albeit with geometric origin. It is demonstrated that the interplay of the geometrical polarization transformation in focused polarized light and subsequent interaction of the structured field polarization with the plasmonic grating leads to the evolution of strong geometrical phase gradient or spin(circular polarization)-dependent transverse momentum of light resulting in spin-split dispersion. Our study offers a new paradigm of spin-based dispersion engineering and spin-enabled nano-optical functionalities in simple symmetric metasurfaces using geometric LB-LD effect.

Figures (4)

• Figure 1: Experimental embodiment for recording SOI of light in the momentum ($k_x,k_y$) domain and for the detection of the polarization-resolved dispersion ($E/\lambda$ vs $k_x$) of the surface plasmon modes of the plasmonic grating through the leakage radiation or scattered light in the far-field Fourier domain:(a) Schematic of the experimental setup, PSG, DFC (dark field condenser), PSA, L1 and L2 (lenses), M (mirror), BS (beam splitter), and SP (spectrograph). (b) Scanning electron microscope (SEM) image of the 1D plasmonic grating with a period of $500 nm$. (c) Simulated and (d) the experimentally recorded polarization-blind momentum domain intensity arc segments of the diffraction pattern formed by the leaky surface plasmons. (e) Dispersion characteristics ($k_x$ vs. $E/\lambda$) is recorded from the region marked by the black rectangle in (d). The bottom axis shows the energy ($E=\hbar \omega$) and the corresponding wavelength ($\lambda$) is shown in the top axis. (f) Spectral response obtained at different transverse momentum values ($k_x$) along the dispersion curve, showing the plasmon resonances.
• Figure 2: Manifestation of spin-split dispersion of the leaky surface plasmons and Spin momentum locking in the recorded polarization-resolved dispersion features and in the momentum domain Mueller matrix images: (a) Spin-dependent azimuthal (($\phi$)) intensity lobes in the recorded polarization-resolved momentum domain intensity arc segments of the leakage radiation for the spectral band $\lambda = 610$–$635 \ \text{nm}$. Left: Differential circular polarization response under polarization-blind or unpolarized illumination (corresponding to $M_{41}$ Mueller matrix element). Right: Intensity difference between RCP and LCP (corresponding to $M_{14}$ element). (b) The corresponding spin-dependent dispersion across $\lambda = 450$–$700 \ \text{nm}$ are shown as circular polarization response differences (left) and the splitting of RCP and LCP branches (right). (c) Schematic illustration of the mechanism of the underlying geometric LB–LD effect. The two-step polarization evolution of tightly focused light and its propagation through the anisotropic plasmonic grating are illustrated as two sequential linear diattenuating retarder polarization transformations with different relative orientation of anisotropy axes. The azimuthal ($\phi$) orientation of the anisotropy axis of focusing yields $\phi$-dependent relative orientation with respect to the fixed anisotropy axis of the plasmonic grating, leading to the geometric LB-LD effect. (d) Polarization-resolved dispersion plots ($k_{x}$ vs. $E/\lambda$) in the form of Mueller matrix. $M_{11}$ shows the polarization-blind dispersion. Linear diattenuation and linear (circular) retardance descriptor elements are highlighted with blue solid and magenta solid (dashed) boxes. Circular anisotropy elements $M_{14}$ (orange dashed box) and $M_{41}$ (green dashed box) reveal spin-dependent dispersion for LCP ($\sigma = -1$) and RCP ($\sigma = +1$) states.
• Figure 3: Unraveling the geometric origin of the LB-LD effect from the momentum ($\mathbf{k}$)- resolved spectral Mueller matrix of the plasmonic grating: Spectral Mueller matrices are shown for the $k_{x,out}$-integrated response (cyan-blue) and $k_{x,out}$-resolved $k_x = \pm 5 \ rad/ \mu m$ (blue for positive and red for negative $k_{x,out}$). The off-diagonal elements $M_{13/31},\ M_{14/41},\ M_{23/32},\ \text{and}\ M_{24/42}$ reveal the geometric signature of the LB–LD effect through polarity reversal between $+k_{x,out}$ and $-k_{x,out}$. In contrast, the $k_{x,out}$-integrated response eliminates these signatures and subsequently yields near zero magnitudes of the $M_{14/41}$ elements.
• Figure 4: Extraction of the spin-dependent transverse momentum ($k_g$) due to geometric LB-LD effect from the experimentally recorded spin-split dispersion Mueller matrix elements $M_{14} (E,k_{x,out})$: a) Schematic illustration of the construction of the circular overlays corresponding to the Bragg diffraction orders ($-1$ and $+1$). Experimentally recorded arc segments from polarization-blind momentum-domain images are analyzed for two spectral windows: $510–550$ nm (green circle) and $610–635$ nm (red circle). The red circles indicate zeroth ($n=0$) and first-order ($n=\pm1$) diffraction rings for 610–635 nm, while the black dotted circle marks the numerical aperture of the imaging system. (b) Clockwise and counter-clockwise rotation of RCP/ LCP polarized intensity lobes in the momentum domain $M_{14}(k_{x}, k_{y})$ element (left panel). The corresponding dispersion of the Mueller matrix element $M_{14}$ as a function of transverse momentum $k_x$ and photon energy $E$ (or wavelength $\lambda$), showing spin-dependent splitting.