COMSOL 6.4 - Dispersion of Surface Plasmon Polariton in Thin Metal Embedded in Dielectric

Dispersion of Surface Plasmon Polariton in Thin Metal Embedded in Dielectric

Surface plasmon polaritons (SPPs) are a special class of surface waves that propagate along the interface between materials with negative and positive dielectric constants, and decay in the direction normal to the interface. Most of the metals at optical frequencies possess a negative dielectric constant, and are widely used materials to excite such surface waves. An SPP propagates along the metal–dielectric interface until it completely loses its energy due to the metal absorption. The thickness of the metal layer plays an important role in this context as it controls the absorption of the structure.

In the case of a metal layer with a certain thickness, embedded in a dielectric, SPPs propagating on the top and bottom interfaces couple to each other and create a supermode. Such a structure behaves similarly to an optical slab waveguide with dispersion depending on the metal thickness. SPPs supported by thin metal layers could be useful in practical applications such as short-distance signal transmission and routing as well as to design power splitters and optical couplers.

Model Definition

The schematic of the model is shown in Figure 1. A thin metal layer is covered by layers of dielectrics with certain thickness and is surrounded by air.

The expressions of the electric and magnetic fields of the TM surface waves or SPPs and the propagation constant are available in Ref. 1.

As the SPPs propagate along the surface, they lose energy due to the metal absorption. The dispersion of the structure as well as the complex SPP wavenumber kspp = kr + iki depends on the thickness of metal. Assume that the SPP wave propagates along the x direction. The electric field component of such a wave is represented by

Therefore, at a distance x, the SPP intensity has decayed by a factor of

. The propagation length, L of the SPP is defined as the distance at which the wave intensity decays by a factor 1/e, and is defined as

Figure 1: Schematic of the configuration. A thin metal layer is embedded in layers of dielectric with certain thickness. The setup is surrounded by air domains.

This model simulates SPPs propagating in a thin layer of aluminum embedded in dielectric layers, using material properties for aluminum from the built-in Optical Material Library. Numeric ports are used on the left and right boundaries of the model. The left port, with excitation turned on, launches the SPP, while the right port, with excitation turned off, absorbs the SPP without reflection. To get the mode field for both ports, two Boundary Mode Analysis study steps are added. Finally, another Boundary Mode Analysis study step with Auxiliary sweep of the wavelength is used to calculate the dispersion and the propagation length of SPP.

Results and Discussion

Figure 2 shows the electric field y-component of the SPP as it propagates from left to right. The intensity of the wave decays as it propagates due to metal absorption.

Figure 2: The y-component of the electric field for a wavelength of 290 nm (4.28 eV). The wave gets weaker, due to absorption, as it propagates from left to right.

Figure 3 shows the propagation constant of the SPP versus the photon energy and compares with the propagation constant of light in free space. As the photon energy reduces and moves away from the plasmon resonance frequency of aluminium, the SPP propagation constant converges with the propagation constant of free space light.

Figure 4 shows the propagation length versus the photon energy. The SPPs decay very fast when the photon energy is very high. This is due to the fact of strong metal absorption at high photon energy, and the loss gradually decreases as the photo energy reduces.

For more SPP simulation examples, see Ref. 1.

Figure 3: Photon energy versus the propagation constant.

Figure 4: Photon energy versus the propagation length.

Reference

1. X. Chen, “Modeling Surface Plasmon Polaritons in COMSOL®,” COMSOL Blog, 12 Oct. 2022; www.comsol.com/blogs/modeling-surface-plasmon-polaritons-in-comsol/.

Wave_Optics_Module/Waveguides/thin_metal_surface_plasmon_polariton

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Global Definitions

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In the table, enter the following settings:


Name	Expression	Value	Description
L	2[um]	2E-6 m	Width
H	3[um]	3E-6 m	Domain height
t_diel	4[nm]	4E-9 m	Dielectric layer thickness
t_metal	12[nm]	1.2E-8 m	Metal thickness
n_diel	2	2	Refractive index, dielectric
lda0	290[nm]	2.9E-7 m	Wavelength
f0	c_const/lda0	1.0338E15 1/s	Frequency