COMSOL 6.4 - 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 nb. The grating constant, or the distance between the wires, is d. A plane-polarized wave traveling through a medium with refractive index na 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 dnb and d(na + nb), 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 na = 1 for air and nb = 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) (also called s-polarization) and a transverse magnetic (TM) (or p-polarization) 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 T1 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 T-1.

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. 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 unphysical 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.

To add the port and periodic conditions, a Periodic Structure node is added. It automatically adds and configures its subnodes — the ports and the periodic conditions.

The table below lists the parameters names used in the model.

Table 1: parameter names.
model description	model	description
na	na	Refractive index, air
nb	nb	Refractive index, dielectric
α	alpha	Angle of incidence

Wave_Optics_Module/Gratings_and_Metamaterials/plasmonic_wire_grating

Modeling Instructions

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Global Definitions

Parameters 1

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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