Surface plasmon polaritons in periodic arrays of V-shaped grooves strongly coupled to quantum emitters

We investigate the optical response of a system consisting of periodic silver V-shaped grooves interacting with quantum emitters. Two surface plasmon-polariton resonances are identified in the reflection spectrum of bare silver grooves, with the intensity of one resonance being localized near the bottom of the groove and that of the other resonance being distributed throughout the entire groove. The linear response of the hybrid silver-emitter system is thoroughly analyzed by considering the coupling between surface plasmon polaritons and emitters as the geometry of the grooves and the spatial distribution of emitters within the grooves are varied. The nonlinear response of the system is also considered by pumping the emitters with a short, high-intensity pulse. By changing the duration or the intensity of the pump, the population of emitters in the ground state at the end of the pump is varied, and it is found (upon probing with a short pulse) that an increase in the fraction of emitters in the ground state corresponds to an increase in Rabi splitting. Spatial variations in the ground-state population throughout the emitter region are shown to be a result of field retardation.

Figure 1

Schematic diagram of the system. The absorbing boundary conditions are implemented as convolutional perfectly matched layers (CPML). The periodic boundary conditions are imposed upon the left and right sides of the grid as indicated by the dashed lines. The incident wave is introduced via a total field/scattered field (TFSF) approach. The figure represents both the plane of incidence and the plane of scattering. The incident field is $p$ polarized and is normally incident on the grating as shown.

Figure 2

Linear optical response of bare V grooves at normal incidence. Panel (a) shows reflection spectrum of bare grooves with 400 nm period, ${20}^{\circ }$ groove angle, and 200 nm groove depth. Panel (b) shows time-averaged intensity in the bare groove when excited by cw plane wave at $1.2$ eV. The intensity is distributed throughout the groove. Panel (c) shows time-averaged intensity in the bare groove when excited with cw plane wave at $2.255$ eV. The intensity is localized near the bottom of the groove. The contour data in panels (b) and (c) are logarithmic and normalized to the incident intensity.

Figure 3

Optics of periodic V grooves coupled to quantum emitters. Panel (a) shows the reflection for bare grooves (dashed line) and the reflection for grooves with emitters is shown as a solid line (with emitters resonant at 1.2 eV). Panel (b) shows the upper polariton (red squares), lower polariton (blue circles), and third resonant mode (green triangles) as a function of the groove angle to sweep the SPP resonance through the emitter resonance. Panel (c) shows the same modes as in panel (b) but as functions of the groove depth to sweep the SPP resonance through the emitter resonance. The horizontal black lines in panels (b) and (c) represent the fixed emitter resonance, the curved black lines represent the SPP energy of the bare grooves for the given geometry, and the dashed lines are values predicted by the coupled-oscillator model. The density of emitters is $3\times {10}^{26}$ emitters/${\text{m}}^3$.

Figure 4

Spatially dependent coupling. The density of emitters is ${10}^{26}$ emitters/${\text{m}}^3$, the transition energy is $1.2$ eV, and the pure dephasing time is 400 fs. As the height of the molecular aggregate is increased, the coupling for the distributed resonance increases continuously whereas that for the localized resonance levels off quickly. Panel (a) shows schematics of “full” grooves versus “hollowed” grooves. Panel (b) shows the Rabi splitting as a function of the area occupied by the emitters for the more spatially distributed $1.2$ eV resonance, where blue circles indicate full grooves, red squares indicate hollowed grooves (width of emitter region is 10 nm), and green triangles indicate hollowed grooves (width of emitter region is 15 nm). Panel (c) is the same as panel (b) except the data are shown for the more localized $2.255$ eV resonance. The pure dephasing time is 600 fs.

Figure 5

Pump-probe dynamics. Panel (a) shows Rabi splitting as a function of pump amplitude. Each value is obtained by launching a short, low-intensity probe pulse immediately after the pump. Panel (b) shows ground-state population, ${\rho }_{11}$, as a function of pump amplitude. This value is averaged over the entire region occupied by emitters and obtained at the end of the pump. The density of emitters is ${10}^{26}$ emitters/${\text{m}}^3$, the pure dephasing time is 400 fs, the transition energy is $1.2$ eV, and the duration of the pump is 30 fs.

Figure 6

Spatial modulations of the molecular ground state. Each run consists of a 70 fs pump applied to the hybrid V-grooves system. The simulations shown in panels (a), (c), and (d) are run with FDTD and the Liouville–von Neumann equation whereas that in panel (b) is run by numerically integrating the Schrödinger equation for a two-level atom. For panels (a), (c), and (d) the density is ${10}^{26}$ emitters/${\mathrm{m}}^3$, the pump amplitude is $7\times {10}^8$ V/m, and the dephasing time is 400 fs. Panel (a) shows the fraction of emitters in the ground state. The characteristic length of spatial modulations is much less than the pump wavelength of 1033 nm. Panel (b) shows oscillations of the ground-state population for two emitters separated by 50 nm. Panel (c) shows that the group velocity at the transition frequency is less than $c$. The plot displays two spikes. We hypothesize that these are numerical artifacts associated with taking a finite derivative of the index of refraction in the region of anomalous dispersion, as runs performed with a smaller spatial step result in smaller spikes. Panel (d) shows the ground-state population as a function of coordinate for the system pumped on resonance ($1.2$ eV, solid line) and slightly off resonance ($1.15$ eV, dashed line).