Plasmonic Wire Grating

A plane electromagnetic wave is incident on a wire grating on a dielectric substrate. The model computes transmission and reflection coefficients for the refraction, specular reflection, and first order diffraction.

Model Definition

Figure 1 shows the considered grating, with a gold wire on a dielectric material with refractive index nβ. The grating constant, or the distance between the wires, is d. A plane-polarized wave traveling through a medium with refractive index nα is incident on the grating, at an angle α in a plane perpendicular to the grating.

Figure 1: The modeled grating. The model considers a unit cell of a slice through this geometry. The grating is assumed to consist of an infinite number of infinitely long wires.

If the wavelengths involved in the model are sufficiently shorter than the grating constant, one or several diffraction orders can be present. The diagram in Figure 2 shows two transmissive paths taken by light incident on adjacent cells of the grating, exactly one grating constant apart.

Figure 2: The geometric path lengths of two transmitted parallel beams. The optical path length is the geometric path length multiplied by the local refractive index.

The criterion for positive interference is that the difference in optical path length along the two paths equals an integer number of vacuum wavelengths, or:

(1)

with m = 0, ±1,± 2,..., λ0 the vacuum wavelength, and βm the transmitted diffracted beam of order m. For m = 0, this reduces to refraction, as described by Snell’s law:

Because the sine functions can only vary between −1 and 1, the existence of higher diffraction order requires that

The model instructions cover only first order diffraction, and are hence only valid under the condition

(2)

Note that for the special cases of perpendicular and grazing incidence, the right-hand side of the inequality evaluates to dnβ and d(nα + nβ), respectively.

Figure 3 shows the corresponding paths of the reflected light.

Figure 3: The geometric path lengths of two parallel reflected beams.

For positive interference we get

(3)

where αm is the reflected beam of diffraction order m. Setting m = 0 in this equation renders

or specular reflection. The condition for no reflected diffracted beams of order 2 or greater being present is

(4)

The model uses nα = 1 for air and nβ = 1.2 for the dielectric substrate. Allowing for arbitrary angles of incidence and with a grating constant d = 400 nm, Equation 2 sets the validity limit to vacuum wavelengths greater than 440 nm. The model uses λ0 = 441 nm. For the wire, a complex-valued permittivity of −1.75 − 5.4i approximates that of gold at the corresponding frequency.

The performance of the grating depends on the polarization of the incident wave. Therefore both a transverse electric (TE) and a transverse magnetic (TM) case are considered. The TE wave has the electric field component in the z direction, out of the modeling xy-plane. For the TM wave, the electric field vector is pointing in the xy-plane and perpendicular to the direction of propagation, whereas the magnetic field has only a component in the z direction. The angle of incidence is for both cases swept from 0 to π/2, with a pitch of π/40.

Results and Discussion

As an example of the output from the model, Figure 4 and Figure 5 show the electric field norm for an angle of incidence equal to π/5, for the TE and TM case respectively.

Figure 4: Electric field norm for TE incidence at π/5.

Figure 5: Electric field norm for TM incidence at π/5.

All the computed transmission and reflection coefficients for TE incidence are plotted in Figure 6. R0, the coefficient for specular reflection, increases rather steadily with the angle of incidence. This is both because of reflection in the material interface and because the wave “sees” the wire as increasingly wider at greater angles — the same effect as achieved by a Venetian blind. T0, the refracted but not diffracted transmission, decreases accordingly. For the considered wavelength to period length ratio, the transmitted diffracted beam T−1 is propagating only for nearly perpendicular incidence. The reflected diffraction order R1 would need a shorter wavelength or a larger grating period to show up. Instead, the most prominent diffraction orders are R−1 and T1.

The sum of all coefficients is consistently less than 1 due to the dielectric losses in the wire. This is even more apparent for TM incidence, as Figure 7 shows. Here, approximately half of the wave is absorbed in the wire. Another important feature of the TM case is that there is very little specular reflection (R0) around 60 degrees.

Figure 6: Transmission and reflection coefficients for TE incidence.

Figure 7: Transmission and reflection coefficients for TM incidence.

Notes About the COMSOL Implementation

The model is set up for one unit cell of the grating, flanked by Floquet boundary conditions describing the periodicity. As applied, this condition states that the solution on one side of the unit equals the solution on the other side multiplied by a complex-valued phase factor. The phase shift between the boundaries is evaluated from the perpendicular component of the wave vector. Because the periodicity boundaries are parallel with the y-axis, only the x-component is required. Due to the continuity of the field, the phase factor for the refracted and reflected beams is the same as for the incident wave.

Port conditions are used for specifying the incident wave and also for letting the resulting solution leave the model without any nonphysical reflections. In order to achieve perfect transmission through the port boundaries, one port for each mode (m = 0, m = −1, m = 1) in each direction must be present. This gives a total of 6 ports.

The input to each periodic port is an electric or magnetic field amplitude vector and an angle of incidence. The angle of incidence is defined as

where k is the propagation vector of the incident wave, n is the normalized normal vector, k is the wave number, α is the angle of incidence, and z is the unit vector in the z direction. Note that this definition means that the angle of incidence on the opposite sides have opposite signs. To automatically create ports for the diffraction orders, you also provide the refractive index at the port boundary and the maximum frequency (which in this model is the single frequency that is used).

The table below lists the parameters names used in the model. “Internal” means that the variable is not provided as an input parameter.

Table 1: parameter names.
model description	model	description
nα	na	Refractive index, air
nβ	nb	Refractive index, dielectric
α	alpha	Angle of incidence
α1	Internal	Reflected diffraction angle, order 1
α-1	Internal	Reflected diffraction angle, order -1
β0	beta	Refraction angle
β1	Internal	Refracted diffraction angle, order 1
β-1	Internal	Refracted diffraction angle, order -1

RF_Module/Tutorials/plasmonic_wire_grating

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

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
na	1	1	Refractive index, air
nb	1.2	1.2	Refractive index, dielectric
d	400[nm]	4E-7 m	Grating constant
lam0	441[nm]	4.41E-7 m	Vacuum wavelength
f0	c_const/lam0	6.798E14 1/s	Frequency
alpha	0[deg]	0 rad	Angle of incidence
beta	asin(na*sin(alpha)/nb)	0 rad	Refraction angle

Although the angle of incidence will not remain constant at 0, it needs to be specified as a parameter to be accessible to the parametric solver.

Study 1

Step 1: Frequency Domain

In the window, under click .

In the window for , locate the section.

In the text field, type f0.

Geometry 1

Create the geometry entirely in terms of the grating constant, for easy scalability.

Rectangle 1 (r1)

In the toolbar, click

In the window for , locate the section.

In the text field, type d.

In the text field, type 3*d.

Click

Rectangle 2 (r2)

In the toolbar, click

In the window for , locate the section.

In the text field, type d.

In the text field, type 3*d.

Locate the section. In the text field, type -3*d.

Click

Circle 1 (c1)

In the toolbar, click

In the window for , locate the section.

In the text field, type d/5.

Locate the section. In the text field, type d/2.

Click

Click the

button in the toolbar.

The geometry now consists of two rectangular domains for the air and the dielectric, and a circle centered on their intersection. You can remove the line through the circle if you first create a union of the objects.

Union 1 (uni1)

In the toolbar, click

and choose .

Click in the window and then press Ctrl+A to select all objects.

In the window for , click

Delete Entities 1 (del1)

In the window, right-click and choose .

On the object , select Boundary 6 only. This is the horizontal diameter of the circle in the center of the geometry.

In the window for , click

Form Union (fin)

In the window, click .

In the window for , click

Electromagnetic Waves, Frequency Domain (emw)

Before setting up the materials, define which constitutive relations you want to use in the Electromagnetic Waves interface.

Wave Equation, Electric 1

In the window, under click .

In the window for , locate the section.

From the list, choose .

Wave Equation, Electric 2

In the toolbar, click

and choose .

Select Domain 3 only.

It might be easier to select the correct domain by using the window. To open this window, in the toolbar click and choose . (If you are running the cross-platform desktop, you find in the main menu.)

In the window for , locate the section.

From the list, choose .

Materials

Air

In the window, under right-click and choose .

In the window for , type Air in the text field.

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


Property	Variable	Value	Unit	Property group
Refractive index, real part	n_iso ; nii = n_iso, nij = 0	na	1	Refractive index

Dielectric

Right-click and choose .

In the window for , type Dielectric in the text field.

Select Domain 1 only.

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


Property	Variable	Value	Unit	Property group
Refractive index, real part	n_iso ; nii = n_iso, nij = 0	nb	1	Refractive index

Gold

Right-click and choose .

In the window for , type Gold in the text field.

Select Domain 3 only.

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


Property	Variable	Value	Unit	Property group
Relative permittivity (imaginary part)	epsilonBis_iso ; epsilonBisii = epsilonBis_iso, epsilonBisij = 0	5.4	1	Dielectric losses
Relative permittivity (real part)	epsilonPrim_iso ; epsilonPrimii = epsilonPrim_iso, epsilonPrimij = 0	-1.75	1	Dielectric losses
Relative permeability	mur_iso ; murii = mur_iso, murij = 0	1	1	Basic
Electrical conductivity	sigma_iso ; sigmaii = sigma_iso, sigmaij = 0	0	S/m	Basic

Electromagnetic Waves, Frequency Domain (emw)

In the first version of this model, you will assume a TE-polarized wave. This means that Ex and Ey will be zero throughout the geometry, and that you consequently only need to solve for Ez.

In the window, under click .

In the window for , locate the section.

From the list, choose .

Port 1

Now define the excitation port. A periodic port assumes that the structure is periodic and simplifies the setup of ports for the diffraction orders.

In the toolbar, click

and choose .

Select Boundary 5 only.

It might be easier to select the correct boundary by using the window. To open this window, in the toolbar click and choose . (If you are running the cross-platform desktop, you find in the main menu.)

In the window for , locate the section.