Slow-light dispersion by transparent waveguide plasmon polaritons

We propose a classical analogue of electromagnetically induced transparency for a two-level ensemble interacting with two orthogonal optical modes. We show that a single localized plasmon resonance of a metal nanoparticle ensemble coupled to counter-propagating modes of a dielectric waveguide generates a slow transparent waveguide-plasmon polariton. Dispersion is controllable by tuning the coupling strengths of localized plasmon and waveguide modes, while maintaining extremely low loss at the system's transparency. Strong coupling in such plasmonic hybrid systems leads to large group index-bandwidth products.

Figure 1

Schematics of (a) atomic EIT and (b) classical analogues. (c) Plasmonic resonator ($w/t=50/5$ nm) array embedded in a silicone waveguide with the core thickness of 600 nm. (d) ${\left|E\right|}^2$ spectral response of probe placed 5 nm from the resonator's end facet and $\left|E\right|$${}^2$ distribution of LP mode at resonance, ${\omega }_{lp}/2\pi =189$ THz. (e) Numerically calculated dispersion relations of TM${}_0$ and TM${}_1$ bare modes for $\Lambda =275$ nm. Coupled LP-TM${}_0$-TM${}_1$ modes occur at the degeneracy point (i). When $\Lambda$ is set at 240 nm, coupled LP-TM${}_0$-TM${}_0$ modes occur at the point (ii).

Figure 2

(a) Dispersion relations of coupled LP-TM${}_0$-TM${}_1$ modes based on Eq. (1), and (b) central WPP's retrieved group index. (c) and (d) are the same for the LP-TM${}_0$-TM${}_0$ case with different detunings. In (a) and (b), $\delta$ and $\gamma =0$ for all and the coupling strengths are (i) $\hslash \left[C_0,C_1\right]=\left[10,30\right]$ meV, (ii) $\hslash \left[C_0,C_1\right]=\left[20,30\right]$ meV, and (iii) $\hslash \left[C_0,C_1\right]=\left[30,27.6\right]$ meV. In (c) and (d), $\hslash C_0=10$ meV and $\gamma =0$ for all.

Figure 3

(a) Numerically calculated dispersion relations for coupled LP-TM${}_0$-TM${}_1$ modes with resonators located 123 nm above the waveguide core's center. Numerical dispersion agrees with analytical one from Eq. (1) (dashed curves) using $\hslash \left[C_0,C_1\right]=\left[28,37\right]$ meV, $\delta /2\pi =-100$ GHz, and $2\gamma /2\pi =10.5$ THz. (b) Damping rates of the middle three modes (red, yellow and blue curves) and the retrieved group index of the central mode (black curve). Dashed curve is the group index when Eq. (2) is satisfied (resonator position at 130 nm above the core's center). (c) $\left|E\right|$${}^2$ distributions of the middle three modes at $k_z\Lambda /\pi =0.86$, labeled as (i), (ii), and (iii) in (a). (d), (e), and (f) show the LP-TM${}_0$-TM${}_0$ case with resonators located 130 nm above the core's center. The analytical dispersion (dashed curves) uses $\hslash C_0=35$ meV, $\delta /2\pi =-100$ GHz, and $2\gamma /2\pi =10.5$ THz. (f) $\left|E\right|$${}^2$ distributions of the lower three modes at $k_z\Lambda /\pi =1.0$, labeled as (i), (ii), and (iii) in (d). Note that $\left|E\right|$${}^2$ distributions of mode (ii) in (c) and (f) have been multiplied by 100 and 25, respectively, for clarity.

Figure 4

(a) A slow-light waveguide made of a large group-index region sandwiched between coupling regions. The large group-index region has 0–400 resonators located 123 nm above the core's center, while the coupling regions have 125 resonators each with the $y$ position gradually changed from 70 to 123 nm for adiabatic coupling. (b) Transmission spectra through the waveguide with a length of 70.75 $\mu$m. (c) Waveguide length dependence of transmission and averaged group index of the system.