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Numerical simulations of long-range plasmons

Aloyse Degiron and David R. Smith

Open Access Open Access

Abstract

We present simulations of plasmonic transmission lines consisting of planar metal strips embedded in isotropic dielectric media, with a particular emphasis on the long-range surface plasmon polariton (SPP) modes that can be supported in such structures. Our computational method is based on analyzing the eigenfrequencies corresponding to the wave equation subject to a mixture of periodic, electric and magnetic boundary conditions. We demonstrate the accuracy of our approach through comparisons with previously reported simulations based on the semi-analytical method-of-lines. We apply our method to study a variety of aspects of long-range SPPs, including tradeoffs between mode confinement and propagation distance, the modeling of bent waveguides and the effect of disorder and periodicity on the long-ranging modes.

©2006 Optical Society of America

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Figures (11)
Fig. 1.
Fig. 1. Unit cell for simulating: (a) a straight waveguide of thickness t and width w – the patterned zones represent the perfect walls placed at the boundaries of the computational domain; (b) a bent waveguide with a mean radius of curvature ρ0. The colored volume behind the boundary ρ= ρ s represents a PML. Inserts: 3D views of the two types of structures.

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Fig. 2.
Fig. 2. Real and imaginary parts of the Ag dielectric function εAg = εˊAg +iε″Ag .

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Fig. 3.
Fig. 3. Dispersion with thickness of the 4 fundamental modes supported by a straight Ag strip with rounded corners (w = 1 μm, r = 5 nm, εd = 4) at the free-space wavelength λ vac = 633 nm. For comparison, we include data points from Ref [5] for the ssb0 mode (black stars) and the sab0 mode (black crosses) supported by a rectangular Ag strip with 90 degree corners (w = 1 μm, r = 0 nm, εd = 4).

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Fig. 4.
Fig. 4. Cross-section of the Ey field component for (a) the sab0 mode when t = 200 nm; (b) the aab0 mode when t = 200 nm; (c) the sa0b mode when t = 40 nm; (d) the aab0 mode when t = 40 nm. The calculations have been made at the free-space wavelength λ vac = 633 nm. Note that the color scale has intentionally been saturated to reveal the field pattern.

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Fig. 5.
Fig. 5. Cross-section of the Ey field component for (a) the ssb0 mode when t = 200 nm; (b) the asb0 mode when t = 200 nm; (c) the ssb0 mode when t = 40 nm; (d) the asb0 mode when t = 40 nm. The calculations have been made at the free-space wavelength λ vac = 633 nm.

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Fig. 6.
Fig. 6. Effects of rounding the corners on the ssb0 mode for two Ag strips with different thicknesses t (w = 1 μm, λ vac = 633 nm). Insert: geometry of the strip.

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Fig. 7.
Fig. 7. Electric field distribution of the ssb0 mode (left panels) and the sab0 mode (right panels) supported by: (a) and (b) a rectangular strip with t = 160 nm and r = 0 nm; (c) and (d) a rounded strip with t = 160 nm and r = 80 nm; (e) and (f) a cylindrical wire with r = 80 nm. The modes have been computed at λ vac = 633 nm and the color scale is the same for all plots.

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Fig. 8.
Fig. 8. Electric field distribution for Au strips (εAu = -11.8 + i1.23 at λ vac = 633 nm) on a glass substrate. The refractive indexes of the substrate and the upper cladding are nsub = 1.52 and nup , respectively. (a) Modes supported by a rectangular strip (w = 1 μm, t = 20 nm) for increasing values of nup ; (b) Modes supported by rectangular strips of different thicknesses when nsub = 1.52 and nup = 1.55. The size of the computational domain is three to six times larger than the size of the panels.

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Fig. 9.
Fig. 9. (a) Dispersion with frequency of the ssb0 mode for a straight and a bent SPP transmission line; (b) Electric field distribution of the ssb0 mode for different radii of curvature ρ0.

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Fig. 10.
Fig. 10. Electric field pattern of the ssb0 mode in a plane perpendicular to propagation and in the upper plane of the Ag strip. The dashed zones are the horizontal electric wall and the vertical magnetic wall placed at the boundaries of the computational domain. Insert: Poynting vector above the surface.

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Fig. 11.
Fig. 11. (a) 3D and side views of a periodic structure with a square wave modulation (w = 1 μm, t=40 nm, P = 150 nm). (b) Dispersion with frequency of the ssb0 mode for three modulation heights c. Insert: zoom of the plot near the gap region. Note that the upper bands actually cross the light line, however, the mode density within the computational domain becomes so large that the solutions of interest could not be easily identified. (c) Imaginary part of frequency vs. real part of wave-vector for the lower (plain symbols) and upper (open symbols) branches of Fig. 11(b). Insert: zoom of the plot near the gap region. The same colors have been used in all plots.

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