COMSOL 6.4 - Embedded Scatterer on a Substrate

A plane transverse electric (TE)–polarized electromagnetic wave is incident on a metallic sphere embedded on a substrate. In this electromagnetic scattering problem, the far-field variables are computed for a few different elevation angles of incidence.

Model Definition

Figure 1 shows the schematic of the geometry. One half of the spherical metallic scatterer is embedded within the dielectric substrate. The substrate is considered to occupy the entire z<0 half space, while the z>0 half space is air. The interface between the air and the substrate is planar and located in the xy–plane at z=0. A plane electromagnetic wave, with frequency 2.3 GHz, is incident at an elevation angle θ. The wave is plane–polarized with the electric field vector tangential to the interface between the air and the substrate.

Figure 1: The modeled geometry. The gray domain represents the dielectric substrate, and the gold spherical domain represents the metallic scatterer. The incident wave propagates at an elevation angle θ.

The model uses na = 1 for air and nb = 1.7 for the dielectric substrate. The metallic sphere is modeled as a perfect electric conductor. The model calculates the S–parameters without the scatterer and the far–field variables in the presence of the scatterer.

Results and Discussion

As is explained in Notes About the COMSOL Implementation, the model first computes a background field from the plane wave incident on the substrate, and then uses that to arrive at the total field with the scatterer present.

Figure 2 shows the instantaneous electric field norm of the total field due to the superposition of the incident plane wave and the field reflected from the substrate, for θ = π/4. This total electric field is not affected by the scatterer. In the air, the total field is a superposition of the incident and the reflected plane waves. In the substrate, only the transmitted plane wave exists.

Figure 2: Instantaneous electric field norm of the total field without the scatterer for θ = π/4.

The total electric field solved for without the scatterer is used as the background field to solve for the total electric field in the presence of the scatterer. Figure 3 shows the instantaneous norm of the total electric field for θ = π/4, after it has been influenced both by the material interface and by the scatterer. Figure 4 shows the instantaneous norm of the background electric field for the same elevation angle of incidence, which is similar to Figure 2.

Figure 3: Instantaneous norm of the total electric field in the presence of the scatterer for θ = π/4.

Figure 4: Instantaneous norm of the background electric field for θ = π/4.

Figure 5 shows a polar plot of the far–field norm radiation pattern. Radiation patterns are plotted in plane of incidence, xz–plane, for different elevation angles of incidence, θ = 0, θ = π/6, and θ = π/4.

Figure 5: Radiation pattern of the far-field norm in the xz-plane.

Notes About the COMSOL Implementation

The Electromagnetic Waves, Frequency Domain interface provides as an option to solve for the scattered field, a perturbation to the total field caused by a local scatterer embedded on a substrate. If the scatterer is suspended in free space, the incident field that is launched — for instance, a Gaussian or plane wave — is simply entered as the background electric field. With the scatterer embedded on a substrate, the analytical expression for the background field becomes more complicated. It needs to be the correct superposition of an incident field, the field reflected from the substrate, and the transmitted field in the substrate.

A simple and general way to avoid having to derive and enter the analytical background field is to use a full field solution of the problem without the scatterer. To achieve this full field solution, the simulation is set up with two Port and two Periodic Condition nodes. One Port node defines the incident plane wave and allows for specular reflection. The other one absorbs the transmitted plane wave. The Periodic Conditions nodes allow the solution of one side of the geometry to be equal to the solution on the other side multiplied by a complex-valued phase factor. This effectively turns the model into a periodic cell of a geometry that extends indefinitely.

A second Electromagnetic Waves, Frequency Domain interface introduces the scatterer and a Perfect Electric Conductor node is used to define it. The total field solution from the first interface is used as the background electric field. This solves for the total field, including the field scattered by the scatterer, and allows calculating the far-field variables using a Far-Field Domain, Inhomogeneous node.

RF_Module/Scattering_and_RCS/embedded_scatterer_on_substrate

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

Electromagnetic Waves, Frequency Domain (emw)

Update the Electric displacement field model settings to Refractive index. This enables you to specify the materials using a Refractive index material model.

Wave Equation, Electric 1

In the window, under > click .

In the window for , locate the section.

From the list, choose .

Electromagnetic Waves, Frequency Domain 2 (emw2)

In the window, under > click .

In the window for , locate the section.

From the list, choose .

Global Definitions

Define some parameters that are useful for setting up the geometry and the study.

Parameters 1

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
f0	2.3[GHz]	2.3E9 Hz	Frequency
lda0	c_const/f0	0.13034 m	Free space wavelength
w	1.5*lda0	0.19552 m	Width of physical geometry
h_air	lda0*0.8	0.10428 m	Air domain height
h_subs	lda0/2	0.065172 m	Substrate domain height
t_pml	lda0/3	0.043448 m	PML thickness
r_scatterer	3[cm]	0.03 m	Scatterer radius
na	1	1	Refractive index, air
nb	1.7	1.7	Refractive index, substrate
theta	0	0	Elevation angle of incidence in air