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Simulation of Dispersion and Hyperbolic Wave in Metal–Dielectric Layered Metamaterial
Introduction
Optical metamaterials are artificially engineered structures composed of subwavelength building blocks that offer unique electromagnetic phenomena. Such structures exhibit anisotropic dispersion and their electrical properties (that is, permittivity, permeability, conductivity, and so on) can be controlled by changing the shape, geometry, size, orientation, and material properties of the composing unit cell. Metamaterials enable manipulation of light in unprecedented ways with extreme control over the properties of electromagnetic fields that cannot be obtained with materials available in nature.
Figure 1: Schematic drawing of an electric point dipole located in air near a metamaterial structure. The structure is composed of periodically organized metal and dielectric layers with subwavelength thickness and periodicity.
One class of extremely anisotropic metamaterials displays hyperbolic dispersion thanks to the resonance effect arising from the equivalent electrical LC circuit, due to the opposite signs of the electric or magnetic tensor components along the orthogonal optical axes. These structures are commonly referred to as hyperbolic metamaterials. That can be constructed by periodically organized (i) metal–dielectric layers and (ii) metallic nanorods embedded in a dielectric, with subwavelength periodicity and dimension. The hyperbolic waves propagating within the metamaterial structure are extremely confined and their wavelengths can be a hundred times smaller than the free-space wavelength. Such distinctive electromagnetic features of hyperbolic metamaterials are useful in novel nanophotonic applications including enhanced superlensing effect, subdiffraction imaging, sensing, negative refraction, canalization of light, energy harvesting, and quantum and thermal engineering with superior performances.
This model discusses how to simulate hyperbolic waves propagating in a hyperbolic metamaterial constructed of periodically organized metal–dielectric layers, using an electric point dipole located in the air above the structure. It also demonstrates the procedure to compute the effective relative permittivity tensor components of the metamaterial.
Model Definition
Figure 1 shows the schematic of the model. A linearly (vertically) polarized electric point dipole source is located in air near a hyperbolic metamaterial composed of periodically oriented subwavelength metal-dielectric layers. The evanescent fields radiated by the dipole couple to the structure and excite two types of waves: (i) surface plasmon polariton that propagates along the metal–air interface and (ii) hyperbolic wave that propagates within the metamaterial.
Imagine a nonmagnetic metamaterial as is shown in Figure 1. The effective permittivity of such a structure is a diagonal tensor. Within the subwavelength regime, the diagonal components can be calculated using the effective medium theory as
(1),
(2),
with Fm = tm/(tm + td) being the filling ratio of metal.
Here, εrr and εzz are the tangential and normal components of the effective relative permittivity, respectively; tm and td are thicknesses of metal and dielectric layers, respectively; and εm and εd are relative permittivities of metal and dielectric, respectively.
Equation 1 and Equation 2 show that the anisotropic dispersion of the metamaterial depends on the thickness of the metal–dielectric layers or the filling ratio. The values of εrr and εzz can be positive or negative depending on the layer thickness, and the resulting metamaterial can exhibit three different topologies: (i) dielectric (εrr > 0, εzz > 0), (ii) hyperbolic (εrr < 0, εzz > 0), and (iii) elliptic (εrr < 0, εzz < 0).
Figure 2: Real part of diagonal components of the effective relative permittivity of metamaterial as a function of filling ratio of metal. The metamaterial is composed of silver and silicon dioxide.
For further demonstration, assume a metamaterial composed of silver (metal) and silicon dioxide (dielectric). Figure 2 shows the real parts of εrr (green line) and εzz (blue line) versus the metal filling ratio, Fm, indicating the dielectric (gray box), hyperbolic (red box), and elliptic (magenta box) regimes. Here, εzz displays a resonance behavior as it depends on the electromagnetic coupling between the adjacent metal layers, whereas εrr shows a smooth variation. Within the hyperbolic regime, the metamaterial exhibits metallic (that is, inductive, εrr < 0) and dielectric (that is, capacitive, εzz > 0) response along the tangential and vertical directions, respectively. In the case of larger Fm, the value of εzz is dominated by the increased volume of metal and becomes negative, corresponding to an elliptic topology. When Fm is very small, the influence of the metal is negligible on the metamaterial properties and it behaves like an anisotropic dielectric media.
Another approach to calculate the anisotropic relative permittivity tensor ε of the metamaterial is to solve the constitutive relation. It can be written in terms of the electric displacement field D and the electric field E as
(3).
Equation 3 permits to calculate the radial and vertical components of ε as
(4).
For a more detailed discussion on the dispersion of hyperbolic metamaterials, see Ref. 1.
This model simulates a hyperbolic metamaterial in a 2D axisymmetric geometry using an Electromagnetic Waves, Frequency Domain interface. The metamaterial is constructed using silver and silicon dioxide thin layers with 40% metal filling ratio, with material properties taken from the built-in Optical Material Library. The Electric Point Dipole feature is used to excite the waves in the metamaterial and perfectly matched layers are employed to absorb the waves reaching the domain boundaries to minimize unexpected reflections. A Wavelength Domain study step is used to solve for the domain fields. Finally, an additional Wavelength Domain study step is performed to calculate the effective relative permittivity tensor components of the metamaterial versus wavelength. This calculation is performed following the effective medium theory, as demonstrated in Equation 1 and Equation 2, and by solving the constitutive relation as shown in Equation 4.
Results and Discussion
Figure 3 shows the electric field norm of the waves excited in the metamaterial using an electric point dipole located above it. The fields propagate within narrow channels thus resulting in a hyperbolic pattern which is different from the case in conventional materials. Fields in the surrounding air domain are not plotted here for better visualization of the hyperbolic waves.
Figure 3: Electric field norm of hyperbolic wave propagating in a hyperbolic metamaterial made of silver and silicon dioxide layers with thickness 10 nm and 15 nm, respectively. Operation wavelength is 480 nm.
Figure 4 shows the instantaneous electric field norm of the generated waves within the metamaterial and the surrounding air. It confirms the excitation of surface plasmon polariton that propagates along the metal–air interface in the tangential direction apart from the hyperbolic waves.
Figure 4: Instantaneous electric field norm of the excited wave in the complete simulation domain. Surface plasmon polariton is excited at the metal–air interface in addition to hyperbolic waves.
Figure 5: Real part of diagonal components of the metamaterial effective permittivity tensor calculated using effective medium theory (solid lines) and from simulation (markers).
Figure 5 shows the real part of the tangential (εrr) and vertical (εzz) components versus the operation wavelength. Results calculated using simulation (markers) and effective medium theory (solid lines) display excellent agreement.
Reference
1. Z. Guo, H. Jiang, and H. Chen, “Hyperbolic Metamaterials: From Dispersion Manipulation to Applications,” J. Appl. Phys., vol. 127, no. 7, p. 071101, 2016.
Application Library path: Wave_Optics_Module/Gratings_and_Metamaterials/dipole_near_hyperbolic_metamaterial
Modeling Instructions
From the File menu, choose New.
New
In the New window, click  Model Wizard.
Model Wizard
1
In the Model Wizard window, click  2D Axisymmetric.
2
In the Select Physics tree, select Optics > Wave Optics > Electromagnetic Waves, Frequency Domain (ewfd).
3
Click Add.
4
Click  Study.
5
In the Select Study tree, select Preset Studies for Selected Physics Interfaces > Wavelength Domain.
6
Click  Done.
Geometry 1
1
In the Model Builder window, under Component 1 (comp1) click Geometry 1.
2
In the Settings window for Geometry, locate the Units section.
3
From the Length unit list, choose µm.
Global Definitions
Parameters 1
1
In the Model Builder window, under Global Definitions click Parameters 1.
2
In the Settings window for Parameters, locate the Parameters section.
3
In the table, enter the following settings:
Name
Expression
Value
Description
W
2[um]
2E-6 m
Geometry width
lda0
480[nm]
4.8E-7 m
Wavelength
td
15[nm]
1.5E-8 m
Thickness, dielectric
tm
10[nm]
1E-8 m
Thickness, metal
P
td+tm
2.5E-8 m
Periodicity
tPML
250[nm]
2.5E-7 m
Thickness, PML
d
25[nm]
2.5E-8 m
Dipole-metamaterial distance
nlayer
30
30
Number of layers
hMTM
nlayer*P
7.5E-7 m
Thickness, metamaterial
Geometry 1
The geometry consists of a metamaterial composed of periodically organized metal-dielectric thin layers, and air on top of it. An electric point dipole is located in air above the metamaterial.
Top Air Domain
1
In the Geometry toolbar, click  Circle.
2
In the Settings window for Circle, type Top Air Domain in the Label text field.
3
Locate the Size and Shape section. In the Radius text field, type W/2.
4
In the Sector angle text field, type 90.
5
Click to expand the Layers section. In the table, enter the following settings:
Layer name
Thickness (µm)
Layer 1
tPML
6
Click  Build Selected.
Thin Metal Layer
1
In the Geometry toolbar, click  Rectangle.
2
In the Settings window for Rectangle, type Thin Metal Layer in the Label text field.
3
Locate the Size and Shape section. In the Width text field, type W/2.
4
In the Height text field, type tm.
5
Locate the Position section. In the z text field, type -tm.
6
Click to expand the Layers section. In the table, enter the following settings:
Layer name
Thickness (µm)
Layer 1
tPML
7
Clear the Layers on bottom checkbox.
8
Select the Layers to the right checkbox.
9
Click  Build Selected.
10
Click the  Zoom Extents button in the Graphics toolbar.
Thin Dielectric Layer
1
Right-click Thin Metal Layer and choose Duplicate.
2
In the Settings window for Rectangle, type Thin Dielectric Layer in the Label text field.
3
Locate the Size and Shape section. In the Height text field, type td.
4
Locate the Position section. In the z text field, type -tm-td.
5
Click  Build Selected.
6
Click the  Zoom Extents button in the Graphics toolbar.
Array 1 - Metal Layers
1
In the Geometry toolbar, click  Transforms and choose Array.
2
In the Settings window for Array, type Array 1 - Metal Layers in the Label text field.
3
Select the object r1 only.
4
Locate the Size section. From the Array type list, choose Linear.
5
In the Size text field, type nlayer.
6
Locate the Displacement section. In the z text field, type -P.
7
Locate the Selections of Resulting Entities section. Select the Resulting objects selection checkbox.
8
Click  Build Selected.
9
Click the  Zoom Extents button in the Graphics toolbar.
Array 2 - Dielectric Layers
1
Right-click Array 1 - Metal Layers and choose Duplicate.
2
In the Settings window for Array, type Array 2 - Dielectric Layers in the Label text field.
3
Select the object r2 only.
4
Click  Build Selected.
5
Click the  Zoom Extents button in the Graphics toolbar.
Bottom Domain
1
In the Geometry toolbar, click  Rectangle.
2
In the Settings window for Rectangle, type Bottom Domain in the Label text field.
3
Locate the Size and Shape section. In the Width text field, type W/2.
4
In the Height text field, type tPML.
5
Locate the Position section. In the z text field, type -hMTM-tPML.
6
Locate the Layers section. In the table, enter the following settings:
Layer name
Thickness (µm)
Layer 1
tPML
7
Clear the Layers on bottom checkbox.
8
Select the Layers to the right checkbox.
9
Click  Build Selected.
10
Click the  Zoom Extents button in the Graphics toolbar.
Electric Point Dipole
1
In the Geometry toolbar, click  Point.
2
In the Settings window for Point, type Electric Point Dipole in the Label text field.
3
Locate the Point section. In the z text field, type d.
4
Click  Build All Objects.
Materials
Pick silver as the metal and silicon dioxide as the dielectric from the Optical material library.
Add Material
1
In the Materials toolbar, click  Add Material to open the Add Material window.
2
Go to the Add Material window.
3
In the tree, select Built-in > Air.
4
Right-click and choose Add to Component 1 (comp1).
5
In the tree, select Optical > Inorganic Materials > Ag - Silver > Experimental data: thin film > Ag (Silver) (Ciesielski et al. 2017: Ag/SiO2; n,k 0.191-20.9 um).
6
Right-click and choose Add to Component 1 (comp1).
Materials
Ag (Silver) (Ciesielski et al. 2017: Ag/SiO2; n,k 0.191-20.9 um) (mat2)
1
In the Model Builder window, under Component 1 (comp1) > Materials click Ag (Silver) (Ciesielski et al. 2017: Ag/SiO2; n,k 0.191-20.9 um) (mat2).
2
In the Settings window for Material, locate the Geometric Entity Selection section.
3
From the Selection list, choose Array 1 - Metal Layers.
Add Material
1
Go to the Add Material window.
2
In the tree, select Optical > Inorganic Materials > O - Oxygen and oxides > Thin film > SiO2 (Silicon dioxide, Silica, Quartz) (Gao et al. 2013: Thin film; n,k 0.252-1.25 um).
3
Right-click and choose Add to Component 1 (comp1).
4
In the Materials toolbar, click  Add Material to close the Add Material window.
Materials
SiO2 (Silicon dioxide, Silica, Quartz) (Gao et al. 2013: Thin film; n,k 0.252-1.25 um) (mat3)
1
In the Settings window for Material, locate the Geometric Entity Selection section.
2
From the Selection list, choose Array 2 - Dielectric Layers.
Definitions
Material Relative Permittivity
1
In the Model Builder window, under Component 1 (comp1) right-click Definitions and choose Variables.
2
In the Settings window for Variables, type Material Relative Permittivity in the Label text field.
3
Locate the Variables section. In the table, enter the following settings:
Name
Expression
Unit
Description
epsr_air
(mat1.rfi.n11-i*mat1.rfi.ki11)^2
 
Relative permittivity, air
epsr_m
(mat2.rfi.nr(ewfd.lambda0)-i*mat2.rfi.ni(ewfd.lambda0))^2
 
Relative permittivity, metal
epsr_d
(mat3.rfi.nr(ewfd.lambda0)-i*mat3.rfi.ni(ewfd.lambda0))^2
 
Relative permittivity, dielectric
epsr_ave
(epsr_air+epsr_d)/2
 
Average relative permittivity
The corresponding materials are dispersive and their relative permittivities are calculated as a function of wavelength.
Metamaterial Domains
1
In the Definitions toolbar, click  Explicit.
2
In the Settings window for Explicit, type Metamaterial Domains in the Label text field.
3
Locate the Input Entities section. Click  Paste Selection.
4
In the Paste Selection dialog, type 2-61 in the Selection text field.
5
Click OK.
This selection will be used to define a user-controlled mesh in the metamaterial domain.
Perfectly Matched Layer 1 (pml1)
1
In the Definitions toolbar, click  Perfectly Matched Layer.
2
Select Domain 63 only.
Artificial Domains
Perfectly Matched Layer 2 (pml2)
1
In the Definitions toolbar, click  Perfectly Matched Layer.
2
In the Settings window for Perfectly Matched Layer, locate the Domain Selection section.
3
Click  Paste Selection.
4
In the Paste Selection dialog, type 1,64-124 in the Selection text field.
5
Click OK.
6
In the Settings window for Perfectly Matched Layer, locate the Geometry section.
7
From the Type list, choose Cylindrical.
8
Locate the Scaling section. From the Typical wavelength from list, choose User defined.
9
In the Typical wavelength text field, type lda0.
Electromagnetic Waves, Frequency Domain (ewfd)
1
In the Model Builder window, under Component 1 (comp1) click Electromagnetic Waves, Frequency Domain (ewfd).
2
In the Settings window for Electromagnetic Waves, Frequency Domain, locate the Components section.
3
From the Electric field components solved for list, choose In-plane vector, as only the in-plane polarization will be included in the simulation.
Electric Point Dipole 1
1
In the Physics toolbar, click  Points and choose Electric Point Dipole.
2
Select Point 63 only.
3
In the Settings window for Electric Point Dipole, locate the Dipole Parameters section.
4
In the p text field, type 1.
The electric point dipole is vertically polarized.
Mesh 1
Waves propagating inside the metamaterials possess very large wavenumber. Therefore, User-controlled mesh with reduced maximum element size is used to properly resolve the waves.
1
In the Model Builder window, under Component 1 (comp1) click Mesh 1.
2
In the Settings window for Mesh, locate the Sequence Type section.
3
From the list, choose User-controlled mesh.
Size
1
In the Model Builder window, under Component 1 (comp1) > Mesh 1 click Size.
2
In the Settings window for Size, locate the Element Size Parameters section.
3
In the Maximum element size text field, type 0.06.
Size 1
1
In the Model Builder window, click Size 1.
2
In the Settings window for Size, locate the Geometric Entity Selection section.
3
From the Selection list, choose Metamaterial Domains.
4
Locate the Element Size Parameters section. In the Maximum element size text field, type 0.01.
Adequate choice of Maximum element size is important to properly resolve the electromagnetic waves inside the metamaterial. A value of 10 nm is sufficient for this example model.
5
Click  Build All.
Study 1
Step 1: Wavelength Domain
1
In the Model Builder window, under Study 1 click Step 1: Wavelength Domain.
2
In the Settings window for Wavelength Domain, locate the Study Settings section.
3
In the Wavelengths text field, type lda0.
4
In the Study toolbar, click  Compute.
Results
Study 1/Solution 1 (sol1)
In the Model Builder window, expand the Results > Datasets node, then click Study 1/Solution 1 (sol1).
Selection
1
In the Results toolbar, click  Attributes and choose Selection.
2
In the Settings window for Selection, locate the Geometric Entity Selection section.
3
From the Geometric entity level list, choose Domain.
4
Click  Paste Selection.
5
In the Paste Selection dialog, type 2-62 in the Selection text field, to exclude the PML domains.
6
Click OK.
Revolution 2D 1
1
In the Model Builder window, under Results > Datasets click Revolution 2D 1.
2
In the Settings window for Revolution 2D, click to expand the Revolution Layers section.
3
In the Start angle text field, type 0.
4
In the Revolution angle text field, type 180.
5
Click  Plot.
Cut Plane 1
1
In the Results toolbar, click  Cut Plane.
2
In the Settings window for Cut Plane, locate the Plane Data section.
3
From the Plane list, choose XZ-planes.
4
Click  Plot.
The fields will be plotted over this plane.
Electric Field (ewfd)
1
In the Model Builder window, under Results click Electric Field (ewfd).
2
In the Settings window for 2D Plot Group, locate the Data section.
3
From the Dataset list, choose Cut Plane 1.
Filter 1
1
In the Model Builder window, expand the Electric Field (ewfd) node.
2
Right-click Surface 1 and choose Filter.
3
In the Settings window for Filter, locate the Element Selection section.
4
In the Logical expression for inclusion text field, type z<=0. This enables to visualize the fields inside the metamaterial only.
Surface 1
1
In the Model Builder window, click Surface 1.
2
In the Settings window for Surface, locate the Coloring and Style section.
3
From the Color table list, choose GrayBody.
4
From the Color table transformation list, choose Nonlinear.
5
In the Color calibration parameter text field, type -1, for better visualization of the fields.
The fields propagating inside the metamaterial exhibit a hyperbolic shape. This is because of the inductive-capacitive (LC) resonance arising from the metallic and dielectric response along the orthogonal optical axes.
Instantaneous Electric Field norm (ewfd)
1
In the Model Builder window, right-click Electric Field (ewfd) and choose Duplicate.
2
In the Settings window for 2D Plot Group, type Instantaneous Electric Field norm (ewfd) in the Label text field.
Surface 1
1
In the Model Builder window, expand the Instantaneous Electric Field norm (ewfd) node, then click Surface 1.
2
In the Settings window for Surface, locate the Expression section.
3
In the Expression text field, type ewfd.normEi.
Filter 1
1
In the Model Builder window, expand the Surface 1 node.
2
Right-click Filter 1 and choose Delete, to visualize the fields in both air and metamaterial.
3
In the Instantaneous Electric Field norm (ewfd) toolbar, click  Plot.
Instantaneous Electric Field norm (ewfd)
Now, the radial and vertical components of the electromagnetic waves will be plotted.
Electric Field Components (ewfd)
1
In the Model Builder window, right-click Instantaneous Electric Field norm (ewfd) and choose Duplicate.
2
In the Settings window for 2D Plot Group, type Electric Field Components (ewfd) in the Label text field.
3
Click to expand the Plot Array section. From the Array type list, choose Linear.
Surface 1
1
In the Model Builder window, expand the Electric Field Components (ewfd) node, then click Surface 1.
2
In the Settings window for Surface, locate the Expression section.
3
In the Expression text field, type ewfd.Er, to plot the radial component.
4
Locate the Coloring and Style section. From the Color table list, choose ThermalWaveDark.
5
From the Scale list, choose Linear symmetric.
6
Click to expand the Range section. Select the Manual color range checkbox.
7
In the Minimum text field, type -1.1E16.
8
In the Maximum text field, type 1.1E16.
It helps to properly visualize the field profile.
Surface 2
1
In the Model Builder window, right-click Surface 1 and choose Duplicate.
2
In the Settings window for Surface, locate the Expression section.
3
In the Expression text field, type ewfd.Ez, to plot the vertical component.
4
Click to expand the Inherit Style section. From the Plot list, choose Surface 1.
Electric Field Components (ewfd)
1
In the Model Builder window, click Electric Field Components (ewfd).
2
In the Settings window for 2D Plot Group, locate the Color Legend section.
3
From the Position list, choose Bottom.
4
In the Electric Field Components (ewfd) toolbar, click  Plot.
5
Click the  Zoom Extents button in the Graphics toolbar.
Definitions
This part of the step-by-step instructions demonstrates how to calculate the effective relative permittivity tensor components of the metamaterial from the simulated data, and compares with the ones computed using the effective medium theory.
Relative Permittivity, Effective Medium Theory
1
In the Model Builder window, under Component 1 (comp1) right-click Definitions and choose Variables.
2
In the Settings window for Variables, type Relative Permittivity, Effective Medium Theory in the Label text field.
3
Locate the Variables section. In the table, enter the following settings:
Name
Expression
Unit
Description
f
tm/(tm+td)
 
Filling ratio, metal
epsr_rr_EMT
f*epsr_m+(1-f)*epsr_d
 
Effective relative permittivity, rr-component
epsr_zz_EMT
((f/epsr_m)+((1-f)/epsr_d))^-1
 
Effective relative permittivity, zz-component
Effective medium theory is valid when the thickness of metal/dielectric layers, and periodicity of the unit cell are within the subwavelength regime.
Metal Thickness Variable
1
In the Model Builder window, right-click Definitions and choose Variables.
2
In the Settings window for Variables, type Metal Thickness Variable in the Label text field.
3
Locate the Geometric Entity Selection section. From the Geometric entity level list, choose Domain.
4
From the Selection list, choose Array 1 - Metal Layers.
5
Locate the Variables section. In the table, enter the following settings:
Name
Expression
Unit
Description
t
tm
m
Metal thickness variable
This variable will be used in the expression to calculate the effective relative permittivity.
Dielectric Thickness Variable
1
Right-click Metal Thickness Variable and choose Duplicate.
2
In the Settings window for Variables, type Dielectric Thickness Variable in the Label text field.
3
Locate the Geometric Entity Selection section. From the Selection list, choose Array 2 - Dielectric Layers.
4
Locate the Variables section. In the table, enter the following settings:
Name
Expression
Unit
Description
t
td
m
Dielectric thickness variable
This variable will also be used in the expression to calculate the effective relative permittivity.
Metal-dielectric Interior Boundaries
1
In the Definitions toolbar, click  Adjacent.
2
In the Settings window for Adjacent, type Metal-dielectric Interior Boundaries in the Label text field.
3
Locate the Input Entities section. Under Input selections, click  Add.
4
In the Add dialog, select Metamaterial Domains in the Input selections list.
5
Click OK.
6
In the Settings window for Adjacent, locate the Output Entities section.
7
From the Exterior boundaries list, choose None.
8
Select the Interior boundaries checkbox.
Average 1 (aveop1)
1
In the Definitions toolbar, click  Nonlocal Couplings and choose Average.
2
In the Settings window for Average, locate the Source Selection section.
3
From the Geometric entity level list, choose Boundary.
4
From the Selection list, choose Metal-dielectric Interior Boundaries.
Calculated Effective Relative Permittivity
1
In the Model Builder window, right-click Definitions and choose Variables.
2
In the Settings window for Variables, type Calculated Effective Relative Permittivity in the Label text field.
3
Locate the Variables section. In the table, enter the following settings:
Name
Expression
Unit
Description
meanDr
(up(ewfd.Dr)*up(t)+down(ewfd.Dr)*down(t))/(tm+td)
C/m²
Average electric displacement field, r-component
meanEr
(up(ewfd.Er)*up(t)+down(ewfd.Er)*down(t))/(tm+td)
V/m
Average electric field, r-component
meanDz
(up(ewfd.Dz)*up(t)+down(ewfd.Dz)*down(t))/(tm+td)
C/m²
Average electric displacement field, z-component
meanEz
(up(ewfd.Ez)*up(t)+down(ewfd.Ez)*down(t))/(tm+td)
V/m
Average electric field, z-component
epsr_rr_cal
aveop1(meanDr/(epsilon0_const*meanEr))
 
Effective relative permittivity, rr-component (calculated)
epsr_zz_cal
aveop1(meanDz/(epsilon0_const*meanEz))
 
Effective relative permittivity, zz-component (calculated)
The expressions used to calculate the average electric displacement field and electric field components use up and down operators. These operators are performed on the metal-dielectric interfaces within the metamaterial, and helps to evaluate fields with discontinuity on each side of the interfaces. Then, the average operator is used to perform integration of the constitutive relation D=epsilon0_const*E on the boundaries to calculate the metamaterial relative permittivity.
Add Study
1
In the Home toolbar, click  Add Study to open the Add Study window.
2
Go to the Add Study window.
3
Find the Studies subsection. In the Select Study tree, select Preset Studies for Selected Physics Interfaces > Wavelength Domain.
4
Right-click and choose Add Study.
5
In the Home toolbar, click  Add Study to close the Add Study window.
Study 2
Step 1: Wavelength Domain
1
In the Settings window for Wavelength Domain, locate the Study Settings section.
2
In the Wavelengths text field, type range(450[nm],25[nm],900[nm]).
3
In the Model Builder window, click Study 2.
4
In the Settings window for Study, locate the Study Settings section.
5
Clear the Generate default plots checkbox.
6
In the Study toolbar, click  Compute.
Results
Effective Relative Permittivity
1
In the Results toolbar, click  1D Plot Group.
2
In the Settings window for 1D Plot Group, type Effective Relative Permittivity in the Label text field.
3
Locate the Data section. From the Dataset list, choose Study 2/Solution 2 (sol2).
Global 1
1
Right-click Effective Relative Permittivity and choose Global.
2
In the Settings window for Global, locate the y-Axis Data section.
3
In the table, enter the following settings:
Expression
Unit
Description
real(epsr_rr_EMT)
1
 
real(epsr_zz_EMT)
1
 
imag(epsr_rr_EMT)
1
 
imag(epsr_zz_EMT)
1
 
4
Locate the x-Axis Data section. From the Parameter list, choose Expression.
5
In the Expression text field, type ewfd.lambda0.
6
Click to expand the Legends section. From the Legends list, choose Manual.
7
In the table, enter the following settings:
Legends
Re(epsr_rr_EMT)
Re(epsr_zz_EMT)
Im(epsr_rr_EMT)
Im(epsr_zz_EMT)
Global 2
1
Right-click Global 1 and choose Duplicate.
2
In the Settings window for Global, locate the y-Axis Data section.
3
In the table, enter the following settings:
Expression
Unit
Description
real(epsr_rr_cal)
1
 
real(epsr_zz_cal)
1
 
imag(epsr_rr_cal)
1
 
imag(epsr_zz_cal)
1
 
4
Click to expand the Coloring and Style section. Find the Line style subsection. From the Line list, choose None.
5
Find the Line markers subsection. From the Marker list, choose Circle.
6
Locate the Legends section. In the table, enter the following settings:
Legends
Re(epsr_rr_cal)
Re(epsr_zz_cal)
Im(epsr_rr_cal)
Im(epsr_zz_cal)
Effective Relative Permittivity
1
In the Model Builder window, click Effective Relative Permittivity.
2
In the Settings window for 1D Plot Group, click to expand the Title section.
3
From the Title type list, choose None.
4
Locate the Plot Settings section.
5
Select the y-axis label checkbox. In the associated text field, type Relative permittivity.
6
Locate the Legend section. From the Layout list, choose Outside graph axis area.
7
From the Position list, choose Top.
8
In the Number of rows text field, type 2.
9
In the Effective Relative Permittivity toolbar, click  Plot.
Metamaterial relative permittivity tensor components calculated from the simulation and effective medium theory show excellent agreement.