COMSOL 6.4 - Time-Domain Modeling of Dispersive Drude

Time-Domain Modeling of Dispersive Drude–Lorentz Media

Plasmonic hole arrays have attracted a lot of scientific interest, since the discovery of extraordinary transmission through sub-wavelength hole arrays (compare with Ref. 1). The classical Bethe theory predicts that transmittance through a sub-wavelength circular hole of diameter d in a PEC screen scales as (d/λ)4, where λ is the wavelength. Yet, transmission through holes in realistic metallic films can exceed 50% and even approach 100%. This phenomenon was attributed to surface plasmon polaritons that can tunnel electromagnetic energy through the hole even if it is very much smaller than the wavelength.

This particular model is intended as a tutorial that shows how to model the full time-dependent wave equation in dispersive media, such as plasmas and semiconductors (and any linear medium describable by a sum of Drude–Lorentz resonant terms). The dispersion of the medium in the frequency domain is assumed to be of the form

(1)

where the constant ε∞ > 1 absorbs contributions from high-frequency contributions that are not modeled explicitly, ωp is the plasma frequency, Γi is a damping coefficient, and ωi is a resonance frequency, The particular case when the resonance frequency ωi is zero is known as plasma (or Drude medium), and it covers most metals in the optical frequency range, from mid-IR to visible. For lossless plasmas, when the damping coefficient also is zero (ωi = Γi = 0), modeling simplifies significantly since then the polarization density is linearly related to the magnetic vector potential.

In this model, the wave equation for the magnetic vector potential

(2)

where the electric displacement field is defined by

(3)

is solved together with an ordinary differential equation for the polarization field, obtained by a Fourier transformation of Equation 1,

(4)

Here f is an oscillator strength (normally set to 1).

Notice that this model is not primarily intended to demonstrate the anomalously high transmission through hole arrays, but rather to demonstrate temporal dispersion modeling.

Model Definition

The geometry consists of a single dispersive slab of thickness 1 μm with a slit of width 0.5 μm in it. The wavelength used is 1 μm. Periodic boundary conditions are applied to make the structure physically appear as an array of slits. The source of electromagnetic radiation is a plane wave pulse with flat front and Gaussian temporal shape.

Results and Discussion

Figure 1 shows the probe plot of the y-component of the electric field at the input boundary. The left part of the curve represents the incoming wave, whereas the right part shows the reflected wave returning to the input boundary.

Figure 1: The y-component of the electric field at the input boundary. The left part shows the incident pulse and the right part shows the reflected pulse.

Figure 2 shows the probe plot of the y-component of the polarization at a point in the entrance of the slit. Notice the propagation delay between the incident field, shown in Figure 1, and the onset of the polarization oscillations at this point.

Figure 2: The y-component of the polarization at a point at the entrance of the slit.

Figure 3 shows the probe plot of the y-component of the polarization field at a point at the rear end of the slit.

Figure 3: The y-component of the polarization at a point at the exit of the slit.

Figure 4 shows a field plot of the y-component of the polarization field after the last time step (100 fs).

Figure 4: The y-component of the polarization field after 100 fs.

Finally, the out-of-plane component of the magnetic field and, as an overlaid contour plot, the y-component of the polarization field are shown in Figure 5, after 100 fs.

Figure 5: The out-of-plane component of the magnetic field and the y-component of the polarization field (contours) after 100 fs.

Reference

1. T.W. Ebbesen H.J. Lezec, H.F. Ghaemi, T. Thio, and P.A. Wolff, “Extraordinary Optical Transmission Through Sub-wavelength Hole Arrays,” Nature, vol. 391, pp. 667–669, 1998.

Wave_Optics_Module/Gratings_and_Metamaterials/drude_lorentz_media

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select > > .

Click .

Click

In the tree, select > .

Click

Global Definitions

Parameters 1

Add some parameters that will define the geometry and the properties of the incident field.

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
lda0	1[um]	1E-6 m	Wavelength
E0	1[V/m]	1 V/m	Electric field amplitude
k0	2*pi/lda0	6.2832E6 1/m	Wave number in vacuum
t0	25[fs]	2.5E-14 s	Time delay
dt	10[fs]	1E-14 s	Pulse duration

Definitions

Variables 1

In the window, under right-click and choose .

Now add some variables that defines the incident field and the material properties.

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Unit	Description
omega0	2pi[rad]c_const/lda0	rad/s	Angular frequency
E_bnd	E0cos(omega0t-k0*x)	V/m	Plane-wave factor for electric field
E_pulse	exp(-(t-t0)^2/dt^2)		Temporal factor for electric field
omega_p	1.5*omega0	rad/s	Plasma frequency
omega_1	0.5*omega_p	rad/s	Resonance frequency
gamma_1	0.1*omega_1	rad/s	Damping coefficient