Integrated plasmonic waveguides: A mode solver based on density of states formulation

We express the density of states (DOS) near guided resonances of plasmonic waveguides by using multiple-scattering theory. In direct analogy with the case of localized electronic defect states in condensed matter, we demonstrate that optical DOS variations follow a lorentzian profile near guided modes resonances. The lorentzian shape gives quantitative information on the guided modes (effective index, propagation length, and polarization state). We numerically investigate both leaky and bound (lossy) modes supported by dielectric-loaded surface-plasmon-polariton waveguides.

Figure 1

(Color online) (a) DLSPPW configuration. A dielectric ridge [thickness $t$, width $w$, optical index $n_{diel}={\left({ϵ}_{diel}\right)}^{1/2}$] is deposited on a metal film of thickness $d$ and optical index $n_{\text{metal}}={\left({ϵ}_{\text{metal}}\right)}^{1/2}$. The substrate index is $n_{sub}={\left({ϵ}_{sub}\right)}^{1/2}$. (b) Reference system: substrate/metal/air slab. See text for details.

Figure 2

(Color online) (a) DOS variations calculated for a DLSPPW. The white lines indicate the fundamental mode propagation constant at telecom wavelength $\lambda =1.55\mu \text{m}$. The dispersion curve slope gives a group velocity $v_g/c=0.6$ at this wavelength. This mode is extensively studied in the following. Panels (b), (c), and (d) show the $x$-, $y$-, and $z\text{-DOS}$ contributions to the total DOS variation, respectively (note the different scales). The structure is made of a $600\text{nm}\times 600\text{nm}$ PMMA strip deposited on a 100 nm gold film. Dielectric constants are taken from Ref. 24.

Figure 3

(Color online) Cross section of dispersion relation (Fig. 2) at given frequency (corresponding to $\lambda =1.55\mu \text{m}$). Fit curve is a lorentzian fit [Eq. (8)] with parameters indicated on the figure.

Figure 4

(Color online) Single waveguide characterization. (a) Partial DOSs calculated as a function of $k_y/k_0$. $x\text{-DOS}$ is magnified $50\times$ to be visible. (b) (Time-averaged) electric intensity calculated 50 nm above the dielectric ridge when the system is excited by a (2D-) Gaussian beam by total internal reflection from the substrate centered at incident angle $\text{arcsin}\left(n_{eff}/n_{sub}\right)=53.8°$. The blue curve is an exponential fit with decay length $L_{\text{SPP}}^{2\text{D}}=44.4\mu \text{m}$. (c) Mode intensity pattern ${\mathbf{E}}_{2\text{D}}\left({\mathbf{r}}_{\parallel }\right)$. (d) Electric field intensity computed at a certain time, far from the incident excitation spot [see Fig. 4b], revealing the mode propagation. The white arrows indicate the electric field orientation. “$--$” and “$++$” represent the charges density signs at the metal surface, deduced from the field polarization. The dielectric ridge $\left(n_{diel}=1.535\right)$ is $t=600\text{nm}$ thick and $w=600\text{nm}$ width. The gold film and glass substrate optical indices are $n_{\text{metal}}=0.55+i11.5$ and $n_{sub}=1.6$, respectively. Incident vacuum wavelength is $\lambda =1.55\mu \text{m}$. Electric intensity is normalized with respect to the incident intensity at the focal point $\left(y=0,z=0\right)$.

Figure 5

(Color online) Coupled DLSPPWs. (a) DOS difference computed for two guides separated by $d=500\text{nm}$ (edge to edge). (b) Electric intensity calculated 50 nm above the DLSPPWs when the top waveguide is excited with a (spatially truncated) 2D-Gaussian beam at incident angle $53.8°$. (c) Coupled waveguides DOS as a function of separation distance. (d) Coupling length for several separation distance deduced from DOS variations (“DOS”) or effective index model (“EIM”). The dotted line is an exponential fit with a slope $k_x=1/290{\text{nm}}^{-1}$.

Figure 6

(Color online) (a) DOS variation for a high index $\left(n=2.437\right)$ strip above a 50 nm gold film. (b) Mode intensity profile in the $\left(XZ\right)$ plane.

Figure 7

(Color online) Mode intensity profile in the $\left(YZ\right)$ plane for (a,b) PMMA $\left(t=w=600\text{nm}\right)$ and (c,d) chalcogenide $\left(t=w=300\text{nm}\right)$ DLSPPW. The gold film thickness is (a,c) 50 nm or (b,d) 10 nm. The vacuum wavelength is $\lambda =1.55\mu \text{m}$. In (b), $k_{\text{substrate}}$ represents the leaky mode wave vector into the substrate. The mode effective indices and propagation length obtained from DOS calculation are indicated on the figures.

Figure 8

(Color online) Comparison of the (a) effective indices and (b) propagation length obtained with either effective index model, differential method, or DOS calculation. The chalcogenide strip thickness is 300 nm and its width varies from 0 to $1\mu \text{m}$. In (a), the horizontal line at the substrate index separates bound and leaky modes.