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Cloaking of a Cylindrical Scatterer with Graphene
Introduction
Electromagnetic cloaking has been an interesting research topic. Using an invisibility cloak to make an object seem transparent sounds like science fiction, but in fact it has been proven possible using different methods such as metasurfaces. Reducing the visibility and scattering of an object to practically zero is not only conceptually interesting, but also has many practical applications. For example, cloaking can be applied to aircraft to avoid detection by radar.
In this model, we introduce a cloaking method using an electrically tuned monolayer of graphene. We will show that when a cylindrical dielectric scatterer is covered in graphene, the scattering cross section is greatly reduced at the designated frequency, making it invisible. This cloak is atomically thin, offering advantages to other cloaking methods based on bulky layers.
Due to the axisymmetric nature of cylinder, the scattering simulation will be performed in 2D axisymmetry using the scattered field formulation. Compared to the 3D model, it is much more efficient. A linear polarized plane wave is used as the background excitation field. By computing the electric field distribution and the scattering cross section, we can assess the cloaking effect.
Figure 1: Linearly polarized plane wave incident on a dielectric cylinder covered by graphene. The conductivity of graphene is tuned to provide a good impedance matching such that the scattering is minimized.
Model Definition
This model demonstrates the use of the linearly polarized plane wave background field in 2D axisymmetry. Some care must be taken in the postprocessing to make sure we plot the correct results by suming over the expansion terms.
We attempt to cloak a cylindrical scatterer with permittivity ε = 3.9 at f0 = 3 THz. The diameter of the cylinder is D = λ0/5 and the length is L = 3λ0, where λ0 is the designated free space wavelength. The outer surface of the cylinder is covered by a monolayer of graphene. Cloaking at the designated frequency can be realized by electrically tuning the Fermi level of graphene to EF = 0.51 eV such that graphene provides good impedance matching. It reduces scattering by functioning similarly to an anti-reflective coating. The Fermi level can be changed to achieve cloaking effect at a different frequency.
The conductivity of graphene is calculated using the well-established Kubo formula. Both the electronic intraband transitions and interband transitions contribute to the conductivity of graphene. The intraband and interband contributions are given by
(1),
(2),
where kB is the Boltzmann constant, is the reduced Planck constant, e is the electron charge, T is the temperature, EF is the Fermi energy, τ is the relaxation time, and ω = 2πf is the angular frequency. The function H(Ω) is given by
(3).
Finally, the total 2D sheet conductivity of graphene is given by σ = σintra + σinter. In this model, we consider T = 300 K and τ = 0.5 ps. The integral in σinter can be performed using the built-in integrate() operator. Due to the atomic thickness of graphene, explicit volumetric modeling of it would be computationally expensive. We show that this can easily be avoided by using the Transition Boundary Condition, considering graphene as a 2D surface.
To investigate the performance of the graphene cloak, the scatterer is excited by a linearly polarized plane wave. The linearly polarized plane wave with arbitrary polarization and angle of incidence is not axisymmetric. Therefore, in 2D axisymmetry, the linearly polarized plane wave background field is formed using an expansion method. The key step is to expand a plane wave background field using an infinite sum of azimuthal modes with mode number m. The expansion in cylindrical coordinates yields
(4),
where Jm is the Bessel function of the first kind of order m. When choosing the linear polarized plane wave as the background field in a 2D axisymmetric model, an auxiliary sweep over the mode number will be added in the first Study. For a scatterer whose size is comparable to the wavelength, the infinite series can be truncated to only the first few terms. In this model, we set the highest mode number to 5.
Results and Discussion
The cloaking effect can be visualized in two ways. First, by plotting the electric field distribution around the cylinder without the graphene layer. Clearly, the wave front of the plane wave is strongly distorted due to scattering. By comparison, the field distribution with the graphene cloak shows minimal distortion, as if the scatterer is not present.
Figure 2: Electric field distribution around the dielectric cylinder without graphene cloak at 3 THz. The wave front is significantly distorted due to scattering.
Figure 3: Electric field distribution around the dielectric cylinder with graphene cloak at 3 THz. The wave front is minimally distorted, indicating a good cloaking effect.
The second and more quantitative way of evaluating the cloaking effect is to compare the scattering cross section, which is calculated as the surface integral of the dot product of the Poynting vector and the surface normal, with and without graphene. As shown in the figure below, the scattering cross section is drastically reduced at the designated frequency of 3 THz when the graphene cloak is applied.
Furthermore, even though the simulation is performed in 2D axisymmetry, we can generate full 3D distribution of the physical quantities such as the electric field by utilizing the Revolution 2D dataset. The details on how to generate the plot is discussed below in the step-by-step instructions.
Figure 4: Comparison of the scattering cross section as a function of frequency with and without graphene. When the graphene cloak is applied, the scattering cross section is drastically reduced at the designated 3 THz, indicating a good cloaking performance.
Figure 5: Comparison of the full 3D distributions of the electric field around the dielectric cylinder with and without the graphene cloak at 3 THz. With graphene cloak, the scattering cross section is reduced by over 97%.
Notes About the COMSOL Implementation
In postprocessing, we make frequent use of the withsol() operator to select results at specific mode number and frequency. The sum() operator is used to sum up the contributions from each mode. Information about those operators can be found in the COMSOL documentation under COMSOL Multiphysics > COMSOL Multiphysics Reference Manual > Global and Local Definitions > Operators, Functions, and Constants >Built-In Operators.
Reference
1. P. Chen and A. Alù, “Atomically thin surface cloak using graphene monolayers,” ACS Nano, vol. 5, no. 7, pp. 5855–5863, 2011.
Application Library path: RF_Module/Scattering_and_RCS/cylinder_graphene_cloak
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 Radio Frequency>Electromagnetic Waves, Frequency Domain (emw).
3
Click Add.
4
Click  Study.
5
In the Select Study tree, select General Studies>Frequency 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
f0
3[THz]
3E12 Hz
Designed cloaking frequency
lda0
c_const/f0
9.9931E-5 m
Designed cloaking wavelength
theta
pi/3
1.0472
Angle of incidence
D
lda0/5
1.9986E-5 m
Cylinder diameter
L
3*lda0
2.9979E-4 m
Cylinder length
eps_d
3.9
3.9
Cylinder dielectric constant
Ef
0.51[eV]
8.1711E-20 J
Fermi level of graphene
tau
0.5[ps]
5E-13 s
Scattering time
T
300[K]
300 K
Temperature
d_eff
1[nm]
1E-9 m
Effective graphene thickness
E0
1[V/m]
1 V/m
Electric field amplitude of the incident field
I0
E0^2/(2*Z0_const)
0.0013272 W/m²
Incident power
Geometry 1
Rectangle 1 (r1)
1
In the Geometry toolbar, click  Rectangle.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Width text field, type D/2.
4
In the Height text field, type L.
5
Locate the Position section. In the z text field, type -L/2.
This rectangle represents the dielectric cylinder.
Rectangle 2 (r2)
1
In the Geometry toolbar, click  Rectangle.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Width text field, type lda0.
4
In the Height text field, type L+lda0.
5
Locate the Position section. In the z text field, type -(L+lda0)/2.
6
Click to expand the Layers section. In the table, enter the following settings:
Layer name
Thickness (µm)
Layer 1
lda0/5
7
Select the Layers to the right check box.
8
Select the Layers on top check box.
This rectangle represents the simulation domain. A layer on the top, bottom, and right is added for the PML.
Rectangle 3 (r3)
1
In the Geometry toolbar, click  Rectangle.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Width text field, type D/2+lda0/5.
4
In the Height text field, type L+lda0/5.
5
Locate the Position section. In the z text field, type -(L+lda0/5)/2.
This rectangle represents a surface surrounding the scatterer. Perform the integration of the Poynting vector to calculate the scattering cross section on this surface.
6
Click  Build All Objects.
7
Click the  Zoom Extents button in the Graphics toolbar.
Definitions
Integration 1 (intop1)
1
In the Model Builder window, expand the Component 1 (comp1)>Definitions node.
2
Right-click Definitions and choose Nonlocal Couplings>Integration.
3
In the Settings window for Integration, locate the Source Selection section.
4
From the Geometric entity level list, choose Boundary.
5
Select Boundaries 6, 12, and 17 only.
The integration is performed on the boundaries of Rectangle 3.
Analytic 1 (an1)
1
In the Definitions toolbar, click  Analytic.
2
In the Settings window for Analytic, type H in the Function name text field.
3
Locate the Definition section. In the Expression text field, type sinh(hbar_const*x/(k_B_const*T))/(cosh(hbar_const*x/(k_B_const*T))+cosh(Ef/(k_B_const*T))).
4
Locate the Units section. In the table, enter the following settings:
Argument
Unit
x
rad/s
5
Locate the Plot Parameters section. In the table, enter the following settings:
Argument
Lower limit
Upper limit
Unit
x
0
1e16
rad/s
Variables 1
1
Right-click Definitions and choose Variables.
2
In the Settings window for Variables, locate the Variables section.
3
In the table, enter the following settings:
Name
Expression
Unit
Description
Omega
1[rad/s]
rad/s
Variable of integral
integral
integrate((H(Omega)-H(emw.omega/2))/(emw.omega^2-4*Omega^2),Omega,0[rad/s],1e16[rad/s])
s
Integral in the interband conductivity equation
sigma_intra
((2*k_B_const*T*e_const^2)/(pi*hbar_const^2))*(log(2*cosh(Ef/(2*k_B_const*T)))*(-j/(emw.omega-j/tau)))
S
Intraband conductivity
sigma_inter
(e_const^2/(4*hbar_const))*(H(emw.omega/2)-(j*4*emw.omega/pi)*integral)
S
Interband conductivity
sigma
sigma_intra+sigma_inter
S
Total graphene conductivity
C_scatt
intop1(nr*emw.relPoavr+nz*emw.relPoavz)/I0
 
Scattering cross section
Next, set up the calculation for computing the graphene conductivity and the scattering cross section.
Perfectly Matched Layer 1 (pml1)
1
In the Definitions toolbar, click  Perfectly Matched Layer.
2
Select Domains 1 and 5–8 only.
3
In the Settings window for Perfectly Matched Layer, locate the Geometry section.
4
From the Type list, choose Cylindrical.
The PML is used for absorbing outgoing radiation.
Materials
Air
1
In the Model Builder window, under Component 1 (comp1) right-click Materials and choose Blank Material.
2
In the Settings window for Material, type Air in the Label text field.
3
Locate the Material Contents section. In the table, enter the following settings:
Property
Variable
Value
Unit
Property group
Relative permittivity
epsilonr_iso ; epsilonrii = epsilonr_iso, epsilonrij = 0
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
Dielectrics
1
Right-click Materials and choose Blank Material.
2
In the Settings window for Material, type Dielectrics in the Label text field.
3
Select Domain 4 only.
4
Locate the Material Contents section. In the table, enter the following settings:
Property
Variable
Value
Unit
Property group
Relative permittivity
epsilonr_iso ; epsilonrii = epsilonr_iso, epsilonrij = 0
eps_d
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
Electromagnetic Waves, Frequency Domain (emw)
1
In the Model Builder window, under Component 1 (comp1) click Electromagnetic Waves, Frequency Domain (emw).
2
In the Settings window for Electromagnetic Waves, Frequency Domain, locate the Formulation section.
3
From the list, choose Scattered field.
4
From the Background wave type list, choose Linearly polarized plane wave.
5
In the θ text field, type theta.
6
In the Highest mode number text field, type 5. This is sufficient for a scatterer whose size is comparable to the wavelength, where only a few terms in the expansion are needed to obtain an accurate result.
7
In the E0 text field, type E0.
8
Click Set up Sweep. This generates two parameters: modeNum and highestMode. It also adds an auxiliary sweep in Study 1 that sweeps modeNum from -highestMode to highestMode.
Transition Boundary Condition 1
1
In the Physics toolbar, click  Boundaries and choose Transition Boundary Condition.
2
In the Settings window for Transition Boundary Condition, locate the Transition Boundary Condition section.
3
From the Electric displacement field model list, choose Relative permittivity.
4
Select Boundaries 8, 10, and 16 only.
5
From the εr list, choose User defined. From the μr list, choose User defined. From the σ list, choose User defined. In the associated text field, type sigma/d_eff.
6
In the d text field, type d_eff.
The Transition Boundary Condition is added on the boundary of the dielectric cylinder. The conductivity and effective thickness are set to that of graphene.
Mesh 1
1
In the Model Builder window, under Component 1 (comp1) click Mesh 1.
2
In the Settings window for Mesh, locate the Physics-Controlled Mesh section.
3
From the Element size list, choose Extremely fine.
Since the 2D axisymmetric model is very efficient to solve, you can use a very fine mesh to ensure accuracy.
Study 1
Step 1: Frequency Domain
1
In the Model Builder window, under Study 1 click Step 1: Frequency Domain.
2
In the Settings window for Frequency Domain, locate the Study Settings section.
3
From the Frequency unit list, choose THz.
4
In the Frequencies text field, type range(1,0.2,5).
Next, add a second study to simulate the case without graphene cloak. In this study, the Transition Boundary Condition is disabled. An auxiliary sweep that sweeps modeNum from -highestMode to highestMode is manually added.
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 General Studies>Frequency Domain.
4
Click Add Study in the window toolbar.
5
In the Home toolbar, click  Add Study to close the Add Study window.
Study 2
Step 1: Frequency Domain
1
In the Settings window for Frequency Domain, locate the Physics and Variables Selection section.
2
Select the Modify model configuration for study step check box.
3
In the tree, select Component 1 (comp1)>Electromagnetic Waves, Frequency Domain (emw)>Transition Boundary Condition 1.
4
Right-click and choose Disable.
5
Locate the Study Settings section. From the Frequency unit list, choose THz.
6
In the Frequencies text field, type range(1, 0.2, 5).
7
Locate the Study Extensions section. Select the Auxiliary sweep check box.
8
Click  Add.
9
In the table, enter the following settings:
Parameter name
Parameter value list
Parameter unit
modeNum (Azimuthal mode number)
range(-highestMode,1,highestMode)
 
Now, run the two studies.
Study 1
In the Home toolbar, click  Compute.
Study 2
Click  Compute.
Results
Add two Mirror 2D datasets for postprocessing the field distribution.
Mirror 2D 1
1
In the Results toolbar, click  More Datasets and choose Mirror 2D.
2
In the Settings window for Mirror 2D, click to expand the Advanced section.
3
Select the Define variables check box.
Mirror 2D 2
1
In the Results toolbar, click  More Datasets and choose Mirror 2D.
2
In the Settings window for Mirror 2D, locate the Data section.
3
From the Dataset list, choose Study 2/Solution 2 (sol2).
4
Locate the Advanced section. Select the Define variables check box.
Total Field (With Graphene Cloak)
1
In the Results toolbar, click  2D Plot Group.
2
In the Settings window for 2D Plot Group, type Total Field (With Graphene Cloak) in the Label text field.
3
Locate the Data section. From the Dataset list, choose Mirror 2D 1.
4
From the Parameter value (freq (THz)) list, choose 3.
5
Click to expand the Title section. From the Title type list, choose Manual.
6
In the Title text area, type Total field, z-component (with graphene cloak).
7
Clear the Parameter indicator text field.
Surface 1
1
Right-click Total Field (With Graphene Cloak) and choose Surface.
2
In the Settings window for Surface, locate the Expression section.
3
In the Expression text field, type sum(withsol('sol1',emw.Ez,setval(freq,freq),setind(modeNum,index)),index,1,2*highestMode+1).
4
Click to expand the Range section. Select the Manual color range check box.
5
In the Minimum text field, type -1.
6
In the Maximum text field, type 1.
Filter 1
1
Right-click Surface 1 and choose Filter.
2
In the Settings window for Filter, locate the Element Selection section.
3
In the Logical expression for inclusion text field, type mir1x>0.
Surface 2
1
In the Model Builder window, right-click Total Field (With Graphene Cloak) and choose Surface.
2
In the Settings window for Surface, locate the Expression section.
3
In the Expression text field, type sum(withsol('sol1',emw.Ez*exp(-j*modeNum*pi),setval(freq,freq),setind(modeNum,index)),index,1,2*highestMode+1).
4
Locate the Coloring and Style section. Clear the Color legend check box.
5
Click to expand the Inherit Style section. From the Plot list, choose Surface 1.
Filter 1
1
Right-click Surface 2 and choose Filter.
2
In the Settings window for Filter, locate the Element Selection section.
3
In the Logical expression for inclusion text field, type mir1x<0.
4
In the Total Field (With Graphene Cloak) toolbar, click  Plot.
The default plots show the z-components of the scattered field and the background field. First, visualize the z-component of the total field in the case with graphene cloak. The wave front is only minimally distorted thanks to the graphene cloak.
Total Field (Without Graphene Cloak)
1
In the Home toolbar, click  Add Plot Group and choose 2D Plot Group.
2
In the Settings window for 2D Plot Group, type Total Field (Without Graphene Cloak) in the Label text field.
3
Locate the Data section. From the Dataset list, choose Mirror 2D 2.
4
From the Parameter value (freq (THz)) list, choose 3.
5
Locate the Title section. From the Title type list, choose Manual.
6
In the Title text area, type Total field, z-component (without graphene cloak).
7
Clear the Parameter indicator text field.
Surface 1
1
Right-click Total Field (Without Graphene Cloak) and choose Surface.
2
In the Settings window for Surface, locate the Expression section.
3
In the Expression text field, type sum(withsol('sol2',emw.Ez,setval(freq,freq),setind(modeNum,index)),index,1,2*highestMode+1).
4
Locate the Range section. Select the Manual color range check box.
5
In the Minimum text field, type -1.
6
In the Maximum text field, type 1.
Filter 1
1
Right-click Surface 1 and choose Filter.
2
In the Settings window for Filter, locate the Element Selection section.
3
In the Logical expression for inclusion text field, type mir2x>0.
Surface 2
1
In the Model Builder window, right-click Total Field (Without Graphene Cloak) and choose Surface.
2
In the Settings window for Surface, locate the Expression section.
3
In the Expression text field, type sum(withsol('sol2',emw.Ez*exp(-j*modeNum*pi),setval(freq,freq),setind(modeNum,index)),index,1,2*highestMode+1).
4
Locate the Coloring and Style section. Clear the Color legend check box.
5
Locate the Inherit Style section. From the Plot list, choose Surface 1.
Filter 1
1
Right-click Surface 2 and choose Filter.
2
In the Settings window for Filter, locate the Element Selection section.
3
In the Logical expression for inclusion text field, type mir2x<0.
4
In the Total Field (Without Graphene Cloak) toolbar, click  Plot.
Similarly, visualize the z-component of the total field for the case without graphene cloak as a comparison. The field is noticeably distorted due to scattering.
Scattering Cross Section
1
In the Home toolbar, click  Add Plot Group and choose 1D Plot Group.
2
In the Settings window for 1D Plot Group, type Scattering Cross Section in the Label text field.
3
Locate the Data section. From the Parameter selection (modeNum) list, choose First.
4
Locate the Plot Settings section.
5
Select the x-axis label check box. In the associated text field, type Frequency (THz).
6
Select the y-axis label check box. In the associated text field, type Scattering cross section (m^2).
7
Click to expand the Title section. From the Title type list, choose Manual.
8
In the Title text area, type Scattering cross section.
Global 1
1
Right-click Scattering Cross Section 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
sum(withsol('sol1',C_scatt,setval(freq,freq),setind(modeNum,index)),index,1,2*highestMode+1)
1
With graphene cloak
sum(withsol('sol2',C_scatt,setval(freq,freq),setind(modeNum,index)),index,1,2*highestMode+1)
1
Without graphene cloak
4
Locate the x-Axis Data section. From the Axis source data list, choose freq.
5
Click to expand the Legends section. From the Legends list, choose Manual.
6
In the table, enter the following settings:
Legends
With graphene cloak
Without graphene cloak
7
In the Scattering Cross Section toolbar, click  Plot.
8
Click the  y-Axis Log Scale button in the Graphics toolbar.
The scattering cross section as a function of frequency is plotted for the cases with and without graphene. When the graphene cloak is applied, the scattering cross section is drastically reduced at the designated frequency of 3 THz as expected.
Next, visualize the field distribution in 3D by utilizing the Revolution 2D dataset.
Revolution 2D
1
In the Model Builder window, under Results>Datasets click Revolution 2D.
2
In the Settings window for Revolution 2D, click to expand the Revolution Layers section.
3
In the Revolution angle text field, type 360.
4
Click to expand the Advanced section. Select the Define variables check box.
Revolution 2D 1
1
In the Model Builder window, click Revolution 2D 1.
2
In the Settings window for Revolution 2D, locate the Revolution Layers section.
3
In the Revolution angle text field, type 360.
4
Locate the Advanced section. Select the Define variables check box.
Comparison of the Total Field in 3D
1
In the Home toolbar, click  Add Plot Group and choose 3D Plot Group.
2
In the Settings window for 3D Plot Group, type Comparison of the Total Field in 3D in the Label text field.
3
Locate the Data section. From the Parameter value (freq (THz)) list, choose 3.
4
Click to expand the Title section. From the Title type list, choose None.
5
Locate the Plot Settings section. Clear the Plot dataset edges check box.
6
Locate the Color Legend section. Clear the Show legends check box.
Slice 1
1
Right-click Comparison of the Total Field in 3D and choose Slice.
2
In the Settings window for Slice, locate the Expression section.
3
In the Expression text field, type sum(withsol('sol1',emw.Ez*exp(-j*modeNum*rev1phi),setval(freq,freq),setind(modeNum,index)),index,1,2*highestMode+1).
4
Locate the Plane Data section. From the Plane list, choose XY-planes.
5
In the Planes text field, type 2.
6
Click to expand the Range section. Select the Manual color range check box.
7
In the Minimum text field, type -1.
8
In the Maximum text field, type 1.
Slice 2
1
Right-click Slice 1 and choose Duplicate.
2
In the Settings window for Slice, locate the Plane Data section.
3
From the Plane list, choose ZX-planes.
4
In the Planes text field, type 1.
5
Click to expand the Inherit Style section. From the Plot list, choose Slice 1.
Surface 1
1
In the Model Builder window, right-click Comparison of the Total Field in 3D and choose Surface.
2
In the Settings window for Surface, locate the Expression section.
3
In the Expression text field, type sum(withsol('sol1',emw.Ez*exp(-j*modeNum*rev1phi),setval(freq,freq),setind(modeNum,index)),index,1,2*highestMode+1).
4
Click to expand the Range section. Locate the Coloring and Style section. Click  Change Color Table.
5
In the Color Table dialog box, select Thermal>Thermal in the tree.
6
Click OK.
Filter 1
1
Right-click Surface 1 and choose Filter.
2
In the Settings window for Filter, locate the Element Selection section.
3
In the Logical expression for inclusion text field, type sqrt(rev1x^2+rev1y^2)<D/2 && abs(rev1z)<L/2.
Comparison of the Total Field in 3D
1
In the Model Builder window, under Results click Comparison of the Total Field in 3D.
2
In the Settings window for 3D Plot Group, locate the Data section.
3
From the Parameter value (freq (THz)) list, choose 3.
Slice 3
1
Right-click Comparison of the Total Field in 3D and choose Slice.
2
In the Settings window for Slice, locate the Data section.
3
From the Dataset list, choose Revolution 2D 1.
4
From the Parameter value (freq (THz)) list, choose 3.
5
Locate the Expression section. In the Expression text field, type sum(withsol('sol2',emw.Ez*exp(-j*modeNum*rev2phi),setval(freq,freq),setind(modeNum,index)),index,1,2*highestMode+1).
6
Locate the Plane Data section. From the Plane list, choose XY-planes.
7
In the Planes text field, type 2.
8
Locate the Inherit Style section. From the Plot list, choose Slice 1.
Translation 1
1
Right-click Slice 3 and choose Translation.
2
In the Settings window for Translation, locate the Translation section.
3
In the x text field, type 200.
4
In the y text field, type 100.
Slice 4
1
In the Model Builder window, right-click Comparison of the Total Field in 3D and choose Slice.
2
In the Settings window for Slice, locate the Data section.
3
From the Dataset list, choose Revolution 2D 1.
4
From the Parameter value (freq (THz)) list, choose 3.
5
Locate the Expression section. In the Expression text field, type sum(withsol('sol2',emw.Ez*exp(-j*modeNum*rev2phi),setval(freq,freq),setind(modeNum,index)),index,1,2*highestMode+1).
6
Locate the Plane Data section. From the Plane list, choose ZX-planes.
7
In the Planes text field, type 1.
8
Locate the Inherit Style section. From the Plot list, choose Slice 1.
Translation 1
1
Right-click Slice 4 and choose Translation.
2
In the Settings window for Translation, locate the Translation section.
3
In the x text field, type 200.
4
In the y text field, type 100.
Surface 2
1
In the Model Builder window, right-click Comparison of the Total Field in 3D and choose Surface.
2
In the Settings window for Surface, locate the Data section.
3
From the Dataset list, choose Revolution 2D 1.
4
From the Parameter value (freq (THz)) list, choose 3.
5
Locate the Expression section. In the Expression text field, type sum(withsol('sol2',emw.Ez*exp(-j*modeNum*rev2phi),setval(freq,freq),setind(modeNum,index)),index,1,2*highestMode+1).
6
Click to expand the Inherit Style section. From the Plot list, choose Surface 1.
Translation 1
1
Right-click Surface 2 and choose Translation.
2
In the Settings window for Translation, locate the Translation section.
3
In the x text field, type 200.
4
In the y text field, type 100.
Filter 1
1
In the Model Builder window, right-click Surface 2 and choose Filter.
2
In the Settings window for Filter, locate the Element Selection section.
3
In the Logical expression for inclusion text field, type sqrt(rev2x^2+rev2y^2)<D/2 && abs(rev2z)<L/2.
Annotation 1
1
In the Model Builder window, right-click Comparison of the Total Field in 3D and choose Annotation.
2
In the Settings window for Annotation, locate the Annotation section.
3
In the Text text field, type With graphene cloak.
4
Locate the Position section. In the X text field, type -100.
5
In the Z text field, type 230.
6
Locate the Coloring and Style section. Clear the Show point check box.
Annotation 2
1
Right-click Annotation 1 and choose Duplicate.
2
In the Settings window for Annotation, locate the Annotation section.
3
In the Text text field, type Without graphene cloak.
4
Locate the Position section. In the X text field, type 150.
5
In the Z text field, type 260.
Annotation 3
1
Right-click Annotation 2 and choose Duplicate.
2
In the Settings window for Annotation, locate the Annotation section.
3
In the Text text field, type C_scatt = eval(sum(withsol('sol1',C_scatt,setval(freq,freq),setind(modeNum,index)),index,1,2*highestMode+1), um^2) um^2.
4
Locate the Position section. In the X text field, type 0.
5
In the Z text field, type 0.
Annotation 4
Right-click Annotation 3 and choose Duplicate.
Translation 1
1
In the Model Builder window, right-click Annotation 3 and choose Translation.
2
In the Settings window for Translation, locate the Translation section.
3
In the x text field, type -130.
4
In the y text field, type -20.
5
In the z text field, type -220.
Annotation 4
1
In the Model Builder window, under Results>Comparison of the Total Field in 3D click Annotation 4.
2
In the Settings window for Annotation, locate the Annotation section.
3
In the Text text field, type C_scatt = eval(sum(withsol('sol2',C_scatt,setval(freq,3[THz]),setind(modeNum,index)),index,1,2*highestMode+1), um^2) um^2.
Translation 1
1
Right-click Annotation 4 and choose Translation.
2
In the Settings window for Translation, locate the Translation section.
3
In the x text field, type 140.
4
In the y text field, type 40.
5
In the z text field, type -200.
6
Click the  Show Grid button in the Graphics toolbar.
7
Click the  Show Axis Orientation button in the Graphics toolbar.
At 3 THz, the scattering cross section is reduced by over 97% when the graphene cloak is applied.