Plasmonic Rainbow Trapping Structures for Light Localization and Spectrum Splitting

“Rainbow trapping” has been proposed as a scheme for localized storage of broadband electromagnetic radiation in metamaterials and plasmonic heterostructures. Here, we articulate the dispersion and power flow characteristics of rainbow trapping structures, and show that tapered waveguide structures composed of dielectric core and metal cladding are best suited for light trapping. A metal-insulator-metal taper acts as a cascade of optical cavities with different resonant frequencies, exhibiting a large quality factor and small effective volume comparable to conventional plasmonic resonators.

Figure 1

Schematic descriptions of (a)–(c) mode conversion mechanism, (d)–(f) $n_{\mathrm{eff}}$, and (g)–(i) $v_E$ of IMI (${\epsilon }_{\mathrm{I}}=-8.5$, ${\epsilon }_{\mathrm{II}}=10$) ${\mathrm{TM}}_0$, INI (${\epsilon }_{\mathrm{I}}={\mu }_{\mathrm{I}}=-3$, ${\epsilon }_{\mathrm{II}}={\mu }_{\mathrm{II}}=1$) ${\mathrm{TM}}_2$, and MIM (${\epsilon }_{\mathrm{I}}=10$, ${\epsilon }_{\mathrm{II}}=-1$) ${\mathrm{TM}}_2$ modes versus $\alpha k_0$. In (d)–(f), the real part and imaginary part of $n_{\mathrm{eff}}$ are represented as solid and dashed curves, respectively. Lossless and lossy ($\mathrm{Im}\{\epsilon \}/\mathrm{Re}\{\epsilon \}=0.03$ for metal and $\mathrm{Im}\{\epsilon \}/\mathrm{Re}\{\epsilon \}=\mathrm{Im}\{\mu \}/\mathrm{Re}\{\mu \}=0.03$ for negative index metamaterial) cases are plotted as thin blue and thick red curves, respectively, in (d)–(i). Dotted vertical lines indicate the degeneracy point ${\alpha }_d$, radiation point ${\alpha }_r$, and the mode cutoff ${\alpha }_c$.

Figure 2

Energy density distribution $u(x,z)$ of (a) IMI (${\epsilon }_{\mathrm{I}}=-8.5$, ${\epsilon }_{\mathrm{II}}=10+0.01i$) ${\mathrm{TM}}_0$ and (b) MIM (${\epsilon }_{\mathrm{I}}=10$, ${\epsilon }_{\mathrm{II}}=-1+0.001i$) ${\mathrm{TM}}_2$ modes. Boundaries between core and claddings are indicated by white solid lines. (c),(d) Mode amplitudes of $|f+⟩$ (red solid), $|f-⟩$ (blue dashed), $|b+⟩$ (orange dotted), and $|b-⟩$ (purple dash-dotted) modes as functions of core thickness in the (c) IMI and (d) MIM structures. Dotted vertical lines indicate ${\alpha }_d$ and ${\alpha }_c$.

Figure 3

${\alpha }_d$ (blue dashed), ${\alpha }_c–{\alpha }_d$ [blue dotted, 100 times magnified in (b)] and $n_d$, the effective phase index of the mode at $\alpha ={\alpha }_d$ (red solid) of MIM (${\epsilon }_{\mathrm{I}}=10$, $\theta =2°$) (a) ${\mathrm{TM}}_1$ and (b) ${\mathrm{TM}}_2$ modes versus $\omega /{\omega }_p$. ${\alpha }_d$ and ${\alpha }_c$ are normalized by $k_p^{-1}=c/{\omega }_p$. The inset in (a) shows the schematic of a MIM rainbow trapping structure. $Q$ versus $\omega /{\omega }_p$ of (c) ${\mathrm{TM}}_1$ modes for $\Gamma /{\omega }_p=\{0(\mathrm{\text{red circles}}),0.001(\mathrm{\text{blue squares}}),0.01(\mathrm{\text{orange triangles}})\}$ and (d) ${\mathrm{TM}}_2$ modes for $\Gamma /{\omega }_p=\{0(\mathrm{\text{red circles}}),0.01(\mathrm{\text{orange triangles}}),0.1(\mathrm{\text{purple diamonds}})\}$.

Figure 4

$Q$ and $Q/V_{\mathrm{eff}}$ versus ${\theta }^{-1}$ of (a),(c) MIM (${\epsilon }_{\mathrm{I}}=10$) ${\mathrm{TM}}_1$ modes at $\omega /{\omega }_p=0.277$ for $\Gamma /{\omega }_p=\{0(\mathrm{\text{red circles}}),0.001(\mathrm{\text{blue squares}}),0.01(\mathrm{\text{orange triangles}})\}$ and (b),(d) ${\mathrm{TM}}_2$ modes at $\omega /{\omega }_p=0.6$ for $\Gamma /{\omega }_p=\{0(\mathrm{\text{red circles}}),0.01(\mathrm{\text{orange triangles}}),0.1(\mathrm{\text{purple diamonds}})\}$. The insets in (a) and (c), respectively, plot $Q$ and $Q/V_{\mathrm{eff}}$ for $\Gamma =0$ in full range. $Q/V_{\mathrm{eff}}$ is normalized by $({\lambda }_0/n_{\mathrm{I}}{)}^{-3}$.