Planar metal plasmon waveguides: frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model

A numerical analysis of surface plasmon dispersion, propagation, and localization on smooth lossy films is presented. Particular attention is given to determining wavelength-dependent behavior of thin Ag slab waveguides embedded in a symmetric $\mathrm{Si}{\mathrm{O}}_2$ environment. Rather than considering Ag as a damped free electron gas, the metal is defined by the experimentally determined optical constants of Johnson and Christy and Palik. As in free electron gas models, analytic dispersion results indicate a splitting of plasmon modes—corresponding to symmetric and antisymmetric field distributions—as film thickness is decreased below $50\mathrm{nm}$. However, unlike free electron gas models, the surface plasmon wave vector remains finite at resonance with the antisymmetric-field plasmon converging to a pure photon mode for very thin films. In addition, allowed excitation modes are found to exist between the bound and radiative branches of the dispersion curve. The propagation characteristics of all modes are determined, and for thin films (depending upon electric field symmetry), propagation distances range from microns to centimeters in the near infrared. Propagation distances are correlated with both the field decay (skin depth) and energy density distribution in the metal and surrounding dielectric. While the energy density of most long-range surface plasmons exhibits a broad spatial extent with limited confinement in the waveguide, it is found that high-field confinement does not necessarily limit propagation. In fact, enhanced propagation is observed for silver films at ultraviolet wavelengths despite strong field localization in the metal. The surface plasmon characteristics described in this paper provide a numerical springboard for engineering nanoscale metal plasmon waveguides, and the results may provide a new avenue for integrated optoelectronic applications.

Figure 1

The real and imaginary components of the Ag dielectric function $({\epsilon }_1=\epsilon _1{}^{\prime }+i\epsilon _1{}^{″})$ using a free-electron model (dash-dot line type), data from Johnson and Christy (dashed), and the Palik Handbook (solid). For reference, the polynomial fit through Palik’s $\mathrm{Si}{\mathrm{O}}_2$ data is also included (dotted, plotted as $-{\epsilon }_2$). The inset shows an enlarged view of the region of anomalous dispersion for the Johnson and Christy and Palik data sets.

Figure 2

(a) Surface plasmon dispersion relation for the $\mathrm{Ag}∕\mathrm{Si}{\mathrm{O}}_2$ geometry computed using a free electron gas dispersion model. Note the existence of allowed modes (solid) for frequencies below ${\omega }_{\mathrm{SP}}$ and above ${\omega }_p$, in contrast to the forbidden (i.e., purely imaginary) modes between these frequencies (dotted). (b): Bound (SPP), radiative (RPP), and quasibound (QB) surface plasmon dispersion relation for the $\mathrm{Ag}∕\mathrm{Si}{\mathrm{O}}_2$ geometry computed using the optical constants of Johnson and Christy. Unlike the free electron dispersion of panel (a), modes are allowed throughout the entire frequency range shown. The $\mathrm{Si}{\mathrm{O}}_2$ light line (gray) is also included for reference.

Figure 3

Surface plasmon propagation length for the $\mathrm{Ag}∕\mathrm{Si}{\mathrm{O}}_2$ geometry calculated using the optical constants of Johnson and Christy (solid) and Palik (dotted). A vanishing propagation length occurs at the surface plasmon resonance, located at $\lambda =355\mathrm{nm}$ $\left(358\mathrm{nm}\right)$ for Palik (JC). The local maximum at $\lambda \sim 320\mathrm{nm}$ coincides with the transition between QB and RPP modes. Inset: Comparison of SP propagation for Ag∕Air, $\mathrm{Ag}∕\mathrm{Si}{\mathrm{O}}_2$, and $\mathrm{Ag}∕\mathrm{Si}$ geometries plotted about the SP resonance.

Figure 4

Surface plasmon electric field penetration depth into Ag and $\mathrm{Si}{\mathrm{O}}_2$, computed using the dielectric data of Johnson and Christy. The local penetration maximum at shorter wavelengths corresponds to the transition between the quasibound and radiative surface plasmon modes.

Figure 5

(Color online) Spatial distribution of the surface plasmon tangential field and corresponding electric energy density in the metal $(z>0)$ and dielectric $(z<0)$ for wavelengths characteristic of the surface plasmon-polariton modes, (a) $\lambda =500\mathrm{nm}$, (b) $\lambda =375\mathrm{nm}$, and the radiative plasmon-polariton modes, (c) $\lambda =312\mathrm{nm}$, (d) $\lambda =201\mathrm{nm}$.

Figure 6

(Color online) Characteristic electric field profile for the metal slab waveguide with thickness $z=d$. (a) The higher energy, antisymmetric field mode $(L+)$; (b) the lower energy, symmetric field mode $(L-)$. The propagation direction $x$ is indicated by arrows.

Figure 7

Dispersion for the $\mathrm{Si}{\mathrm{O}}_2∕\mathrm{Ag}∕\mathrm{Si}{\mathrm{O}}_2$ geometry for four different Ag thicknesses (12, 20, 35, and $50\mathrm{nm}$). Dispersion computed using the free electron gas dielectric function is plotted in panels (a) and (b) while dispersion computed using optical constants of Johnson and Christy is shown in (c) and (d). (a), (c) Lower energy, symmetric mode $(L-)$, (b), (d) Higher energy, antisymmetric mode $(L+)$.

Figure 8

$\mathrm{Si}{\mathrm{O}}_2∕\mathrm{Ag}∕\mathrm{Si}{\mathrm{O}}_2$ plasmon propagation lengths computed with the optical constants from Palik calculated for five different Ag thicknesses (12, 20, 35, 50, and $100\mathrm{nm}$). (a) Lower energy, symmetric field mode $(L-)$, (b) higher energy, antisymmetric field mode $(L+)$. The inset of (b) plots propagation distance for wavelengths characteristic of the regime of anamolous Ag dispersion.

Figure 9

Propagation distance as a function of in-plane wave vector for five different Ag film thicknesses (12, 20, 35, 50, and $100\mathrm{nm}$). Arrows indicate the trend of decreasing film thickness. For both the symmetric field mode [$L-$, (a)] and the antisymmetric field mode [$L+$, (b)], the optimal combination of low damping and long wave vector is achieved at ${\lambda }_{\mathrm{SP}}=210\mathrm{nm}$.

Figure 10

Skin depth of the $\mathrm{Si}{\mathrm{O}}_2∕\mathrm{Ag}∕\mathrm{Si}{\mathrm{O}}_2$ bound surface plasmon, computed using the optical constants of Johnson and Christy. (a) Lower energy, symmetric mode $(L-)$, (b) higher energy, antisymmetric mode $(L+)$ for five different film thicknesses (12, 20, 35, 50, and $100\mathrm{nm}$). The inset shows field penetration depth into the dielectric for wavelengths below the surface plasmon resonant wavelength, with the arrow indicating the trend of decreasing waveguide thickness.

Figure 11

Spatial distribution of energy density for the $\mathrm{Si}{\mathrm{O}}_2∕\mathrm{Ag}∕\mathrm{Si}{\mathrm{O}}_2$ slab geometry plotted in the metal (a),(c) and dielectric (b),(d) for both the symmetric field mode (a),(b) and the antisymmetric mode (c),(d) using the optical constants of Johnson and Christy. Ag film thicknesses shown are varied from $100\mathrm{nm}$ down to $12\mathrm{nm}$, with arrows indicating the trend of decreasing thickness. The free space wavelength is set to $\lambda =450\mathrm{nm}$, a region of maximum bound surface plasmon dispersion.