Goos-Hänchen Shift and Even–Odd Peak Oscillations in Edge-Reflections of Surface Polaritons in Atomically Thin Crystals

Abstract

Two-dimensional surface polaritons (2DSPs), such as graphene plasmons, exhibit various unusual properties, including electrical tunability and strong spatial confinement with high Q-factor, which can enable tunable photonic devices for deep subwavelength light manipulations. Reflection of plasmons at the graphene's edge plays a critical role in the manipulation of 2DSP and enables their direct visualization in near-field infrared microscopy. However, a quantitative understanding of the edge-reflections, including reflection phases and diffraction effects, has remained elusive. Here, we show theoretically and experimentally that edge-reflection of 2DSP exhibits unusual behaviors due to the presence of the evanescent waves, including an anomalous Goos-Hänchen phase shift as in total internal reflections and an unexpected even-odd peak amplitude oscillation from the wave diffraction at the edge. Our theory is not only valid for plasmons in graphene but also for other 2D polaritons, such as phonon polaritons in ultrathin boron nitride flakes and exciton polariton in two-dimensional semiconductors.

Scaling and Reflection Behaviors of Polaritons in Low‐Dimensional Materials

“…The distance d between a peak to sample edge has to fulfill the following constructive interference condition: where n is an integer. Quite different from the reflection phase of a plane wave (0 or π) in a 3D dielectric environment, the phase shift picked at the graphene edge is ≈(1/4)π, almost independent of the plasmon wavelength, graphene doping, or dielectric substrate . This nontrivial phase shift originates from the presence of the evanescent wave at the edge, causing electromagnetic energy storage and a phase delay known as the Goos–Hänchen phase shift .…”

“…Figure a shows a typical SNOM image of a plasmon reflection at the graphene edge. Studying the edge reflection pattern in detail, a peculiar odd–even peak oscillation has been surprisingly observed (see Figure b) . An elaborative analysis of this subtle pattern was implemented, revealing an evanescent field constrained at the graphene edge.…”

“…a) An SNOM image of plasmon reflection at the graphene edge. Reproduced with permission . Copyright 2017, American Chemical Society.…”

“…Constructive (destructive) interferences therefore occur at even‐ (odd‐) numbered antinodes, respectively. Reproduced with permission . Copyright 2017, American Chemical Society.…”

“…Quite different from the reflection phase of a plane wave (0 or π) in a 3D dielectric environment, the phase shift picked at the graphene edge is ≈(1/4)π, almost independent of the plasmon wavelength, graphene doping, or dielectric substrate . This nontrivial phase shift originates from the presence of the evanescent wave at the edge, causing electromagnetic energy storage and a phase delay known as the Goos–Hänchen phase shift . To quantitatively calculate the reflection phase, the latest cooperated research invoked the Huygens–Fresnel principle to construct a subtle model for a solution; the researchers cautiously selected a basis function for the unbounded waves in the graphene region, which made the analytical result perfectly agree with the result obtained by their simulation.…”