Leaky and bound modes of surface plasmon waveguides

While theoretical studies of finite width plasmonic waveguides have focused on the bound modal solutions of metallic structures in homogeneous or near-homogeneous dielectric matrices, experimental studies have mainly probed the leaky surface plasmon modes along the air-exposed surfaces of metallic stripes on glass substrates. Combining a full-vectorial, magnetic field finite-difference method with complex coordinate stretching perfectly matched layer boundary conditions, we have solved for both the leaky and bound modes of metallic slab and stripe waveguides. Solutions for the leaky modes excited via attenuated total reflection in the Kretschmann configuration provide added insight into the results of recent near and far-field experimental studies. For these cases, an analytical approximation for the number of allowed modes as a function of waveguide width is derived based upon the guidance condition of dielectric waveguide theory. Then, consistent with a ray optics interpretation for surface plasmon-polariton propagation, a direct comparison is made between our simulations and published results with regard to field profiles and propagation lengths.

Figure 1

(a) Schematic view of momentum conservation in Kretschmann Configuration ATR which allows far-field radiation modes in the glass to couple to SPPs at air-metal interface. (b) Dispersion relation for this system which demonstrates that SPPs excited by ATR are leaky modes which can reradiate in the glass.

Figure 2

Transverse magnetic field intensity for the two-dimensional leaky surface plasmon mode as calculated by present FVH-FDM with CCS-PMLs and the RPM.

Figure 3

(Color online) Simulated field profiles for two leaky quasi-TM surface plasmon modes of a $3.5\mu \mathrm{m}$ wide, $55\mathrm{nm}$ tall Au stripe waveguide on glass substrate $(\lambda =800\mathrm{nm})$: (a) lowest order leaky mode and (b) second order leaky mode. Note that these leaky modes are localized at top air-metal interface and radiate through the glass substrate into the PML region $(y<0)$. (All fields have been normalized such that $\max \mid H_x\mid =1\mathrm{A}∕\mathrm{m}$, and the simulated refractive index profile has been provided alongside each mode for reference.)

Figure 4

(Color online) Simulated field profiles for two of the four bound quasi-TM surface plasmon modes of a $3.5\mu \mathrm{m}$ wide, $55\mathrm{nm}$ tall Au stripe waveguide on glass substrate $(\lambda =800\mathrm{nm})$: (a) lowest order bound mode and (b) fourth order bound mode. Note that these bound modes are localized at bottom glass-metal interface, and were calculated without the use of a PML region. (All fields have been normalized such that $\max \mid H_x\mid =1\mathrm{A}∕\mathrm{m}$, and the simulated refractive index profile has been provided alongside each mode for reference.)

Figure 5

Complex propagation constants for the eight lowest order leaky, quasi-TM surface plasmon modes of varying width Au stripe waveguides ($t=55\mathrm{nm}$, $\lambda =800\mathrm{nm}$). Inset shows the simulated geometry.

Figure 6

Complex propagation constants for the nine lowest order bound, quasi-TM surface plasmon modes of varying width Au stripe waveguides ($t=55\mathrm{nm}$, $\lambda =800\mathrm{nm}$).

Figure 7

Comparison with FVH-FDM simulation results for analytical approximation of the number of (a) leaky and (b) bound quasi-TM surface plasmon modes supported as a function of waveguide width.

Figure 8

(Color online) Transverse magnetic field profiles and electric field intensities for leaky, quasi-TM surface plasmon modes of varying width Au stripe waveguides ($t=55\mathrm{nm}$, $\lambda =800\mathrm{nm}$); (a) sole, lowest-order mode for $W=1.5\mu \mathrm{m}$; (b) sole, lowest-order mode for $W=2.5\mu \mathrm{m}$; (c) lowest-order mode for $W=3.5\mu \mathrm{m}$; and (d) second-order mode for $W=3.5\mu \mathrm{m}$. (Note that the electric field intensity is plotted for a height of $105\mathrm{nm}$ above the substrate surface, and dashed line denotes location of Au stripe.)

Figure 9

Comparison of experimental results in Ref. 9 to simulated propagation lengths for lowest-order leaky, quasi-TM surface plasmon modes of varying width Ag stripes $({\epsilon }_{\mathrm{Ag}}=-18.3136+0.47941i)$ (see Ref. 33) on glass substrate with and without $50\mathrm{nm}$ $\text{silica}∕50\mathrm{nm}$ Al screening layer $({\epsilon }_{\mathrm{Al}}=-51.314+19.43i)$ (see Ref. 26).