This example shows how to excite a TE (s-polarized) surface plasmon on a graphene sheet placed on top of a photonic crystal structure. The surface plasmon is excited by a near-infrared (1.31 μm) plane-wave source inside the glass substrate (Kretschmann configuration). As demonstrated in this example, the coupling between the source and the TE mode is strongest when the projection of the source's wave vector onto the y direction matches the corresponding component of the surface plasmon's wave vector.
FDE simulation setup
A simple 1D FDE simulation can be used to demonstrate that the shown structure supports a TE surface plasmon mode. The file graphene_exciting_TE_sp.lms sets up the 1D FDE simulation shown below. The graphene material type (with a scattering rate of 0.11meV, a chemical potential of 0.5eV and a temperature of 300K) together with the 2D rectangle geometry described in Modeling methodology are used to model the graphene layer following the surface conductivity approach.
At 1.31 μm, the graphene layer has a conductivity of
$$\sigma = (15.98 - i27.69) \mathrm{\mu S}$$
FDE results
The 1D FDE simulation generates a surface plasmon mode with two magnetic field components (Hx and Hz) and one electric field component (Ey) as shown below.
The effective index of the surface plasmon mode is
$$n_{eff} = 1.116 + i0.002$$
Observe that the fields extend all the way to the left PML boundary, so the mode exhibits some radiation.
FDTD simulation setup
As mentioned in the introduction, the excitation is strongest when the projection of the source's wave vector along the y direction matches the corresponding projection of the surface plasmon's wave vector. In this case, this occurs when the angle of incidence of the source is
$$\theta_{SP} = \mathrm{arcsin}\left(\mathrm{Re}(n_{eff})/n_{glass} \right) \approx 50.5^{\circ}$$
This can be verified by using the 2D FDTD simulation set up by the project file graphene_exciting_TE_sp.fsp, which is described below. PML boundary conditions are used along the x direction and Bloch periodic boundaries along the y direction. As in the FDE simulation, the graphene material type and the 2D rectangle geometry are used to model the graphene layer. Two monitors are used to collect the transmitted and reflected power.
FDTD results
The script graphene_exciting_TE_sp.lsf collects the reflected and transmitted power for source angle values between 43 and 51 degrees. The collected results are compared with analytic results in plot below. These results are essentially the same as those presented in Fig. 2 of the paper referenced at the beginning of this page. The only difference is that the results presented there were obtained using a slightly different surface conductivity model where different scattering rates are employed for the interband and intraband contributions.
Observe that, for all source angles
$$\theta\gt\theta_{TIR}$$
the fields in the photonic crystal region are purely evanescent. In addition, observe that a dip in the reflected power occurs for the source angle given earlier in (3). This dip is due to the excitation of the surface plasmon mode. To confirm this, the following image generated by the script graphene_exciting_TE_sp.lsf shows how the electric field intensity along the x direction changes for source angle values between 43 and 51 degrees.
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NOTE: Plot resolution In order to reduce the execution time for the angle sweep, the original angle resolution set in graphene_exciting_TE_sp.lsf is coarser than the one used in the FDTD results shown above. The angle resolution can be adjusted by modifying the angle values in the arrays src_theta_TIR, src_theta_rest and src_theta_dip at the start of the script; an angle step of 0.05 degrees was used here. |
Related publication
- I. Degli-Eredi, J. E. Sipe, and N. Vermeulen, “TE-polarized graphene modes sustained by photonic crystal structures”, Opt. Lett. Vol. 40, No. 9, pp. 2076-2079 (2015).