Modeling a Negative Refractive Index

It is possible to engineer the structure of materials such that both the permittivity and permeability are negative. Such materials are realized by engineering a periodic structure with features comparable in scale to the wavelength. It is possible to model both the individual unit cells of such a material, as well as to model to properties of a bulk negative index material. This example demonstrates the correct way to model a metamaterial domain with bulk negative permittivity and permeability.

Figure 1: A plane wave of light incident upon an infinite half-space of material with negative permittivity and permeability.

Model Definition

A plane wave (λ = 1 m) traveling through vacuum (εr = 1, μr = 1) is incident at an angle of 30? upon an infinite half-space of metamaterial with bulk negative permittivity and permeability (εr = −1, μr = −1) as shown in Figure 1. The objective of the analysis is to observe the fields in the metamaterial.

Since the model space is infinite and invariant along the interface, it is possible to model a finite-sized unit cell around the interface, and use Floquet-periodic boundary conditions. The incident wave is modeled using a Port boundary condition, which both launches the incident wave and absorbs the reflected wave.

Two different ways are demonstrated for modeling boundary to the infinite metamaterial domain. In the first case, a Port boundary condition is used. This boundary condition requires computing the wave vector in the metamaterial, and manually adjusting the propagation constant at the boundary to account for the negative index. In the other case, a Perfectly Matched Layer (PML) is used to truncate the domain. The PML acts as an absorbing medium for all energy incident upon it, but must also be adjusted to account for the negative index. It is simpler to use, but increases the model size.

The transition between the two materials requires some special care. The natural boundary condition between domains of different material properties does not account for the change in direction of the flux. An additional degree of freedom has to be added to the model. This can be done by using the Transition Boundary Condition (TBC), which allows for a change in flux across the boundary. The TBC takes as input the material properties on one side of the domain, it does not matter which side. The thickness of the TBC should be approximately 1/1000 of the wavelength. If it is too small, it can introduce numerical difficulties. If it is too big, it alters the results significantly. The TBC only needs to be used in this way if the effective refractive index is similar in magnitude, but opposite in sign.

Results and Discussion

The electric field is plotted in Figure 2. The fields within the PML are not plotted, since these have no physical meaning. The wave is observed to switch direction at the interface between vacuum and the metamaterial.

Figure 2: The electric fields in vacuum and in the metamaterial. The waves are observed to switch direction.

Wave_Optics_Module/Gratings_and_Metamaterials/negative_refractive_index

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

Load parameters from a file that are useful when setting up the physics and the materials.

Parameters 1

In the window, under click .

In the window for , locate the section.

Click

Browse to the model’s Application Libraries folder and double-click the file negative_refractive_index_parameters.txt.

Here, c_const is a predefined COMSOL constant for the speed of light in vacuum.

Geometry 1

Two port model

In the toolbar, click

In the window for , type Two port model in the text field.

Locate the section. In the text field, type 3.

Click to expand the section. In the table, enter the following settings:


Layer name	Thickness (m)
Layer 1	1.5

The first rectangle consists of two rectangular domains representing the vacuum and metamaterial, respectively.

PML model

In the toolbar, click

In the window for , type PML model in the text field.

Locate the section. In the text field, type 4.

Locate the section. In the text field, type 1.05.

In the text field, type -1.

Locate the section. In the table, enter the following settings:


Layer name	Thickness (m)
Layer 1	1
Layer 2	1.5

The second rectangle consists of three rectangular domains representing the vacuum, metamaterial, and PML.

Click

Click the

button in the toolbar.

Electromagnetic Waves, Frequency Domain (ewfd)

Next, set up the physics. Use two different ways to model the infinite metamaterial domain.

In the window, under click .

In the window for , locate the section.

From the list, choose .

Wave Equation, Electric 1

In the window, under click .

In the window for , locate the section.

From the list, choose .

Port 1

In the toolbar, click

and choose .

Select Boundary 5 only.

For the first port, wave excitation is by default.

In the window for , locate the section.

From the list, choose .

Locate the section. Specify the E0 vector as


0	x
0	y
1	z

In the α text field, type alpha.

Locate the section. In the n text field, type n_a.

Port 2

In the toolbar, click

and choose .

Select Boundary 2 only.

In the window for , locate the section.

From the list, choose .

Locate the section. Specify the E0 vector as


0	x
0	y
1	z

Locate the section. In the n text field, type n_b.

The first method uses two ports.

Periodic Condition 1

In the toolbar, click

and choose .

Select Boundaries 1, 3, 6, and 7 only.

In the window for , locate the section.

From the list, choose .

Port 3

In the toolbar, click

and choose .

Select Boundary 14 only.

In the window for , locate the section.

From the list, choose .

Locate the section. Specify the E0 vector as


0	x
0	y
1	z

In the α text field, type alpha.

Locate the section. In the n text field, type n_a.

Periodic Condition 2

In the toolbar, click

and choose .

Select Boundaries 8, 10, 12, and 15–17 only.

In the window for , locate the section.

From the list, choose .

Definitions

Perfectly Matched Layer 1 (pml1)

In the toolbar, click

Select Domain 3 only.

In the window for , locate the section.

In the text field, type -1.

The second method uses one port and the PML.

Electromagnetic Waves, Frequency Domain (ewfd)

Transition Boundary Condition 1

In the toolbar, click

and choose .

Select Boundaries 4 and 13 only.

In the window for , locate the section.

From the list, choose .

From the μr list, choose . In the associated text field, type mu_a.

From the εr list, choose . In the associated text field, type e_a.

From the σ list, choose . Leave the default value 0.

In the d text field, type lda0/1000.

Materials

Material 1 (mat1)

In the window, under right-click and choose .

Select Domains 2 and 5 only.

In the window for , locate the section.

In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Relative permittivity	epsilonr_iso ; epsilonrii = epsilonr_iso, epsilonrij = 0	e_a	1	Basic
Relative permeability	mur_iso ; murii = mur_iso, murij = 0	mu_a	1	Basic
Electrical conductivity	sigma_iso ; sigmaii = sigma_iso, sigmaij = 0	0	S/m	Basic

Material 2 (mat2)

Right-click and choose .

Select Domains 1, 3, and 4 only.

In the window for , locate the section.

In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Relative permittivity	epsilonr_iso ; epsilonrii = epsilonr_iso, epsilonrij = 0	e_b	1	Basic
Relative permeability	mur_iso ; murii = mur_iso, murij = 0	mu_b	1	Basic
Electrical conductivity	sigma_iso ; sigmaii = sigma_iso, sigmaij = 0	0	S/m	Basic