Ultrasmall Volume Plasmons, yet with Complete Retardation Effects

Nanoparticle plasmons are attributed to quasistatic oscillations with no wave propagation due to their subwavelength size. However, when located within a band-gap medium (even in air if the particle is small enough), the particle interfaces act as wave mirrors, incurring small negative retardation. The latter, when compensated by a respective (short) propagation within the particle, generates a constructive interference based resonator. The unusual wave interference in the subwavelength regime (modal volume $<0.001{\lambda }^3$) significantly enhances the $Q$ factor, e.g., 50 vs 5.5 of the quasistatic limit.

Figure 1

The complex (normal incidence) reflection coefficient ($\rho$) showing the negative refection phase compared to $\pi$ in dashed brown line [for the case of inset (b)] and to 0 in dashed blue line [for the case of inset (c)]. The continuous brown and blue lines are the reflection coefficients for perfect electric conductor and low refractive index dielectric mirrors, respectively. The insets are (a) Fresnel equation for plane-waves reflection coefficient; (b) metal-air-metal based plane-waves cavity, showing the reflection phase progress relatively to PEC mirror, due to the plasmonic mirrors (less than $\pi$ phase); (c) The 3D nanoparticle cavity and the related reflection phase progress (less than 0 phase).

Figure 2

A three-dimensional illustration, of the particle-plasmon shielded cavity. The structure consists of an inner nanocylindrical particle and longer (coaxial) outer shielding layer, both made of plasmonic metal (gold).

Figure 3

Cavity characteristics as a function of the nanoparticle radius (${\lambda }_0=1.5\mu \mathrm{m}$). (a) The various characteristic length scales: blue (or dark gray)—shortest cavity length, red (or gray)—effective modal dimension and black—half the plasmonic wavelength in the resonator segment; (b) reflection phase (${\phi }_{\rho }$). Dashed red line—Fresnel-like approximation using the effective index of the lowest order mode (${\mathrm{TM}}_0$) in all segments for $b=2a$. Inset: illustration of the phase accumulation cycle.

Figure 4

Cavity merits as a function of the nanoparticle radius (${\lambda }_0=1.5\mu \mathrm{m}$). (a) $Q$ factor values. Black dashed line: quasistatic value ($\sim 5$); (b) corresponding $Q/V$ figure of merit. Inset: the effective modal $\mathrm{\text{dimension}}=V^{1/3}$.

Figure 5

Field intensity distribution and its cross section for a typical gold and air cavity mode: $a=30\mathrm{nm}$, $b=90\mathrm{nm}$, $L=60\mathrm{nm}$, $Q=30$ ($Q_{\mathrm{statics}}=5.5$), $V=(75\mathrm{nm}{)}^3$, ${\lambda }_0=1.5\mu \mathrm{m}$.

Figure 6

Field intensity of (2D) Au (${\epsilon }_M=-86.1-i8.16$) dimer excited by a $y$-polarized plane wave. Diameter is 20 nm, and intercenter distance is 23 nm. $Q$ factor: quasistatic calculations is $\sim 6$ and $Q$ factor full wave calculations is $\sim 11$.