Frequency Selective Surface, Periodic Complementary Split Ring Resonator

Frequency selective surfaces (FSS) are periodic structures with a bandpass or a bandstop frequency response. This example shows that only signals around the center frequency can pass through the periodic complementary split ring resonator layer.

Figure 1: One unit cell of the complementary split ring resonator is modeled with periodic boundary conditions to simulate an infinite 2D array. Perfectly matched layers at the top and bottom of the unit cell absorb the excited and higher order modes.

Model Definition

A split ring slot is patterned on a geometrically thin copper layer that sits on a 2 µm PTFE substrate (Figure 1). The copper layer is much thicker than the skin depth in the simulated frequency range, so it is modeled as a perfect electric conductor (PEC). The rest of the simulation domain is filled with air.

Floquet-periodic boundary conditions are used on four sides of the unit cell to simulate the infinite 2D array. Perfectly matched layers (PMLs) on the top and bottom of the unit cell absorb the excited mode from the source port and any higher order modes generated by the periodic structure. The PMLs attenuate the wave as it propagates in the direction perpendicular to the PML boundary. Since the model is solved for a range of incident angles, the wavelength in the PMLs is set to 2π/|k0cosθ|. This accounts for the angular dependence of the normal component of the wave vector inside the PMLs.

Port boundary conditions are placed on the interior boundaries of the PMLs, adjacent to the air domains. The Port boundary conditions automatically determine the reflection and transmission characteristics in terms of S-parameters. The interior port boundaries with PML backing require the slit condition. The port orientation is specified to define the inward direction for the S-parameter calculation. Since higher order diffraction modes are not of particular interest in this example, the combination of Domain-backed type slit port and PMLs is used instead of adding a Diffraction order port for each diffraction order and polarization.

The periodic boundary condition requires identical surface meshes on paired boundaries. This is accomplished in two steps: first by creating a mesh on only one of the boundaries and then using the Copy Face operation for the mesh on the other boundary. This mesh configuration is automatically set when using the physics-controlled mesh as shown in the step-by-step instructions. If you are interested in seeing more details about the mesh, build the physics-controlled mesh once and then change the mesh sequence type to the user-controlled mesh in the mesh settings. Then you can inspect the generated mesh sequence.

The periodic conditions are split by the ports, having the slit conditions. A slit condition means that the dependent variable can have different values on the two sides of the boundary. To make sure that the periodic conditions couples the correct dependent variables on the two sides of the slit condition, multiple periodic conditions are defined— both in front of the port and behind the port.

Results and Discussion

The modified multislice default plot (Figure 2) shows the electric field norm on the complimentary split ring resonator. Strong fields are observed inside the slot. The S-parameter plot in Figure 3 shows that this periodic structure functions as a bandpass filter near 4.6 THz. In Figure 4, the S-parameters appear as a function of incident angle and show that the periodic structure is penetrable at 4.6 THz over the simulated range, except for grazing angles.

The resonance frequency of this periodic structure can be quickly evaluated as 4.59 THz using an Eigenfrequency study, which is not included in this example.

Figure 2: The fields are confined in the split ring slot.

Figure 3: The S-parameter plot shows a bandpass resonance near 4.6 THz.

Figure 4: The S-parameter plot is shown as a function of incident angle.

Wave_Optics_Module/Gratings_and_Metamaterials/frequency_selective_surface_csrr

Modeling Instructions

From the menu, choose .

New

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Model Wizard

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Click

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Study 1

Step 1: Frequency Domain

In the window, under click .

In the window for , locate the section.

In the text field, type range(3.8[THz],0.1[THz],5.4[THz]).

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
theta	0[deg]	0 rad	Elevation angle

Geometry 1

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Block 1 (blk1)

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In the text field, type 15.

In the text field, type 45.

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Work Plane 1 (wp1)

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Work Plane 1 (wp1)>Plane Geometry

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Work Plane 1 (wp1)>Circle 1 (c1)

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In the text field, type 5.

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Work Plane 1 (wp1)>Circle 2 (c2)

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In the text field, type 3.5.

Work Plane 1 (wp1)>Rectangle 1 (r1)

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In the text field, type 4.

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Work Plane 1 (wp1)>Difference 1 (dif1)

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Select the object only.

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toggle button.

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Block 2 (blk2)

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In the text field, type 15.

In the text field, type 2.

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In the text field, type -1.

Block 3 (blk3)

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In the text field, type 15.

In the text field, type 80.

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Electromagnetic Waves, Frequency Domain (ewfd)

Wave Equation, Electric 1

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Perfect Electric Conductor 2

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and choose .

Select Boundary 12 only.

Use Floquet-periodic conditions on all side boundaries.

Periodic Condition 1

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Select Boundaries 4, 7, 10, and 24–26 only.

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Periodic Condition 2

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Periodic Condition 3

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Periodic Condition 4

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and choose .

Select Boundaries 2, 14, 17, and 21 only.

In the window for , locate the section.

From the list, choose .

The wave is excited from the port on the top.

Port 1

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From the list, choose .

For the first port, wave excitation is by default.

Select the check box.

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Click .

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Specify the H0 vector as


0	x
1	y
0	z

In the α1 text field, type theta.

The maximum frequency in the setting window will be used only when button is clicked to generate Diffraction Order features handling higher order mode individually. In this model, PML absorbs all higher order modes, so this setting is ineffective.

Port 2

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Select Boundary 6 only.

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Select the check box.

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Click .

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Specify the H0 vector as


0	x
1	y
0	z

Scattering Boundary Condition 1

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and choose .

Select Boundary 3 only.

Definitions

Perfectly Matched Layer 1 (pml1)

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Select Domain 5 only.

In the window for , locate the section.

From the list, choose .

In the text field, type 2*pi/abs(ewfd.k0*cos(theta)).

Perfectly Matched Layer 2 (pml2)

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Select Domain 1 only.

In the window for , locate the section.

From the list, choose .

In the text field, type 2*pi/abs(ewfd.k0*cos(theta)).

Add Material

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Materials

Material 2 (mat2)

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Select Domain 3 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	2.1	1	Basic
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

Mesh 1

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Definitions

Hide for Physics 1

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Select Boundaries 10, 11, 13, and 14 only.

Mesh 1

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Study 1

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Results

Electric Field (ewfd)

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Multislice 1

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Find the subsection. In the text field, type 0.

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In the text field, type -1.

In the toolbar, click

Click the

button in the toolbar.

Click the

button in the toolbar.

This reproduces Figure 2.

Reflectance, Transmittance, and Absorptance (ewfd)

Identify the resonant frequency of the periodic structure from the S-parameter plot Figure 3.

Next, evaluate the reflectivity and transmittivity performance of the model with different incident angles.

Add Study

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to open the window.

Go to the window.

Study 1

Solver Configurations

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to close the window.

Study 2

Parametric Sweep

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Click

In the table, enter the following settings:


Parameter name	Parameter value list	Parameter unit
theta (Elevation angle)	range(0[deg],5[deg],85[deg])	rad

Step 1: Frequency Domain

In the window, click .

In the window for , locate the section.

In the text field, type 4.6[THz].

In the toolbar, click

Results

Multislice 1

In the window, expand the node, then click .

In the window for , locate the section.

Find the subsection. In the text field, type 0.

Find the subsection. From the list, choose .

In the text field, type -1.

In the toolbar, click

Global 1

In the window, expand the node, then click .

In the window for , locate the section.

From the list, choose .

In the toolbar, click

This is the S-parameter plot as a function of incident angle shown in Figure 4.

Analyze the same model with a much finer frequency resolution using based on asymptotic waveform evaluation (AWE). When a device presents a slowly varying frequency response, the AWE provides a faster solution time when running the simulation on many frequency points. The following example with the AWE can be computed 50 times faster than regular Frequency Domain sweeps with a same finer frequency resolution.

Electromagnetic Waves, Frequency Domain (ewfd)

Port 1

In the window, under click .

In the window for , locate the section.

Click

In the dialog box, type Port 1 in the text field.

Click .

Port 2

In the window, click .

In the window for , locate the section.

Click

In the dialog box, type Port 2 in the text field.

Click .

Add Study

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to open the window.

Go to the window.

Find the subsection. In the tree, select .

Click in the window toolbar.

In the toolbar, click

to close the window.

Study 3

Step 1: Adaptive Frequency Sweep

In the window for , locate the section.

In the text field, type range(3.8[THz],0.01[THz],5.4[THz]).

Use a ten times finer frequency resolution.

A slowly varying scalar value curve works well for AWE expressions. When is set to in the study settings, abs(comp1.ewfd.S21) is used automatically for two-port devices.

Because such a fine frequency step generates a memory-intensive solution, the model file size will increase tremendously when it is saved. When only the frequency response of port related variables are of interest, it is not necessary to store all of the field solutions. By selecting the check box in the section, we can control the part of the model on which the computed solution is saved. We only add the selection containing these boundaries where the port variables are calculated. The port size is relatively small compared to the entire modeling domain, and the saved file size with the fine frequency step is more or less that of the regular discrete frequency sweep model when only the solutions on the port boundaries are stored.