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Fresnel Equations
Introduction
A plane electromagnetic wave propagating through free space is incident at an angle upon an infinite dielectric medium. This model computes the reflection and transmission coefficients and compares the results to the Fresnel equations.
Model Definition
A plane wave propagating through free space (n = 1) as shown in Figure 1 is incident upon an infinite dielectric medium (n = 1.5) and is partially reflected and partially transmitted. If the electric field is p-polarized — that is, if the electric field vector is in the same plane as the Poynting vector and the surface normal — then there are no reflections at an incident angle of roughly 56°, known as the Brewster angle.
Figure 1: A plane wave propagating through free space incident upon an infinite dielectric medium.
Although, by assumption, space extends to infinity in all directions, it is sufficient to model a small unit cell, as shown in Figure 1; a Floquet-periodic boundary condition applies on the top and bottom unit-cell boundaries because the solution is periodic along the interface. This model uses a 3D unit cell and applies also a periodic boundary condition in the out-of-plane direction. However, this periodic boundary condition acts like a continuity condition. The angle of incidence ranges between 0–90° for both polarizations.
To simplify the process of adding the boundary conditions, a Periodic Structure node is used. It automatically adds and configures its subnodes — the ports and the periodic conditions.
For comparison, Ref. 1 and Ref. 2 provide analytic expressions for the reflectance and transmittance1. Reflection and transmission coefficients for s-polarization and p-polarization are defined respectively as
Reflectance and transmittance are defined as
The Brewster angle at which rp = 0 is defined as
Results and Discussion
Figure 2 is a combined plot of the y component of the electric-field distribution and the power flow visualized as an arrow plot for the TE case.
Figure 2: Electric field, Ey (slice) and power flow (arrows) for TE incidence at 70° inside the unit cell.
For the TM case, Figure 3 visualizes the y component of the magnetic-field distribution instead, again in combination with the power flow.
Figure 3: Magnetic field, Hy (slice) and power flow (arrows) for TM incidence at 70° inside the unit cell.
Note that the sum of reflectance and transmittance in Figure 4 and Figure 5 equals 1, showing conservation of power. Figure 5 also shows that the reflectance around 56° — the Brewster angle in the TM case — is close to zero.
Figure 4: The reflectance and transmittance for TE incidence agree well with the analytic solutions.
Figure 5: The reflectance and transmittance for TM incidence agree well with the analytic solutions. The Brewster angle is also observed at the expected location.
References
1. J.D. Jackson, Classical Electrodynamics, 3rd Ed., Wiley, 1999.
2. B.E.A. Saleh and M.C. Teich, Fundamentals of Photonics, Wiley, 1991.
Application Library path: RF_Module/Verification_Examples/fresnel_equations
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  3D.
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.
Global Definitions
Define some parameters that are useful when setting up the geometry and the study.
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
n_air
1
1
Refractive index, air
n_slab
1.5
1.5
Refractive index, slab
lda0
1[m]
1 m
Wavelength
f0
c_const/lda0
2.9979E8 1/s
Frequency
alpha
70[deg]
1.2217 rad
Angle of incidence
beta
asin(n_air*sin(alpha)/n_slab)
0.67701 rad
Refraction angle
alpha_brewster
atan(n_slab/n_air)
0.98279 rad
Brewster angle, TM only
r_s
(n_air*cos(alpha)-n_slab*cos(beta))/(n_air*cos(alpha)+n_slab*cos(beta))
-0.54735
Reflection coefficient, TE
r_p
(n_slab*cos(alpha)-n_air*cos(beta))/(n_air*cos(beta)+n_slab*cos(alpha))
-0.20613
Reflection coefficient, TM
t_s
(2*n_air*cos(alpha))/(n_air*cos(alpha)+n_slab*cos(beta))
0.45265
Transmission coefficient, TE
t_p
(2*n_air*cos(alpha))/(n_air*cos(beta)+n_slab*cos(alpha))
0.52925
Transmission coefficient, TM
The angle of incidence is updated while running the parametric sweep. The refraction (transmitted) angle is defined by Snell’s law with the updated angle of incidence. The Brewster angle exists only for TM incidence, p-polarization (polarization is in the plane of incidence).
Study 1 (TE)
1
In the Model Builder window, click Study 1.
2
In the Settings window for Study, type Study 1 (TE) in the Label text field.
Step 1: Frequency Domain
1
In the Model Builder window, under Study 1 (TE) click Step 1: Frequency Domain.
2
In the Settings window for Frequency Domain, locate the Study Settings section.
3
In the Frequencies text field, type f0.
Geometry 1
First, create a block composed of two domains. Use layers to split the block.
Block 1 (blk1)
1
In the Geometry toolbar, click  Block.
2
In the Settings window for Block, locate the Size and Shape section.
3
In the Width text field, type 0.2.
4
In the Depth text field, type 0.2.
5
In the Height text field, type 0.8.
6
Click to expand the Layers section. In the table, enter the following settings:
Layer name
Thickness (m)
Layer 1
0.4
7
Click  Build All Objects.
8
Click the  Zoom Extents button in the Graphics toolbar.
Choose wireframe rendering to get a better view of each boundary.
9
Click the  Wireframe Rendering button in the Graphics toolbar.
Electromagnetic Waves, Frequency Domain (TE)
Set up the physics based on the direction of propagation and the E-field polarization. First, assume a TE-polarized wave which is equivalent to s-polarization or perpendicular polarization. Ex and Ez are zero while Ey is dominant.
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, type Electromagnetic Waves, Frequency Domain (TE) in the Label text field.
Periodic Structure 1
1
In the Physics toolbar, click  Domains and choose Periodic Structure.
2
In the Settings window for Periodic Structure, locate the Port Mode Settings section.
3
In the α1 text field, type alpha.
The Periodic Structure node automatically adds and configures its port and periodic condition subnodes.
Wave Equation, Electric 1
1
In the Model Builder window, expand the Periodic Structure 1 node, then click Wave Equation, Electric 1.
2
In the Settings window for Wave Equation, Electric, locate the Electric Displacement Field section.
3
From the Electric displacement field model list, choose Refractive index.
Materials
Now set up the material properties based on refractive index. The top half is filled with air.
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
Select Domain 2 only.
4
Locate the Material Contents 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
n_air
1
Refractive index
Refractive index, imaginary part
ki_iso ; kiii = ki_iso, kiij = 0
0
1
Refractive index
Glass
The bottom half is glass.
1
Right-click Materials and choose Blank Material.
2
In the Settings window for Material, type Glass in the Label text field.
3
Select Domain 1 only.
4
Locate the Material Contents 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
n_slab
1
Refractive index
Refractive index, imaginary part
ki_iso ; kiii = ki_iso, kiij = 0
0
1
Refractive index
Mesh 1
In the Model Builder window, under Component 1 (comp1) right-click Mesh 1 and choose Build All.
Study 1 (TE)
Parametric Sweep
1
In the Study toolbar, click  Parametric Sweep.
2
In the Settings window for Parametric Sweep, locate the Study Settings section.
3
Click  Add.
4
In the table, enter the following settings:
Parameter name
Parameter value list
Parameter unit
alpha (Angle of incidence)
range(0,2[deg],88[deg])
deg
5
In the Study toolbar, click  Compute.
Results
Electric Field (emw, TE)
1
In the Settings window for 3D Plot Group, type Electric Field (emw, TE) in the Label text field.
The default plot is the E-field norm for the last solution, which corresponds to almost tangential incidence. Replace the expression with Ey, add an arrow plot of the power flow (Poynting vector), and choose a more interesting angle of incidence for the plot.
Multislice 1
1
In the Model Builder window, expand the Electric Field (emw, TE) node, then click Multislice 1.
2
In the Settings window for Multislice, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 1 (comp1) > Electromagnetic Waves, Frequency Domain (TE) > Electric > Electric field - V/m > emw.Ey - Electric field, y-component.
3
Locate the Multiplane Data section. Find the X-planes subsection. In the Planes text field, type 0.
4
Find the Z-planes subsection. In the Planes text field, type 0.
5
Locate the Coloring and Style section. From the Color table list, choose WaveLight.
Arrow Volume 1
1
In the Model Builder window, right-click Electric Field (emw, TE) and choose Arrow Volume.
2
In the Settings window for Arrow Volume, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 1 (comp1) > Electromagnetic Waves, Frequency Domain (TE) > Energy and power > emw.Poavx,...,emw.Poavz - Power flow, time average.
3
Locate the Arrow Positioning section. Find the Y grid points subsection. In the Points text field, type 1.
4
Locate the Coloring and Style section. From the Color list, choose Green.
Electric Field (emw, TE)
1
In the Model Builder window, click Electric Field (emw, TE).
2
In the Settings window for 3D Plot Group, locate the Data section.
3
From the Parameter value (alpha (deg)) list, choose 70.
4
In the Electric Field (emw, TE) toolbar, click  Plot.
5
Click the  Zoom Extents button in the Graphics toolbar.
The plot should look like that in Figure 2.
S-Parameter (emw, TE)
A 1D plot with the reflection and transmission versus the angle of incidence is added by default.
1
In the Model Builder window, under Results click S-Parameter (emw).
2
In the Settings window for 1D Plot Group, type S-Parameter (emw, TE) in the Label text field.
3
Click to expand the Title section. From the Title type list, choose None.
4
Locate the Plot Settings section. Select the x-axis label checkbox.
Replace the dB-scale variables with the linear-scale S-parameter variables.
5
In the y-axis label text field, type S-parameter.
6
Locate the Legend section. From the Position list, choose Middle left.
Global 1
1
In the Model Builder window, expand the S-Parameter (emw, TE) node, then click Global 1.
2
In the Settings window for Global, locate the y-Axis Data section.
3
In the table, enter the following settings:
Expression
Unit
Description
abs(emw.S11)^2
1
Reflectance
abs(emw.S21)^2
1
Transmittance
4
In the S-Parameter (emw, TE) toolbar, click  Plot.
5
Click to expand the Coloring and Style section. Find the Line style subsection. From the Line list, choose None.
6
Find the Line markers subsection. From the Marker list, choose Cycle.
7
From the Positioning list, choose Interpolated.
Global 2
1
In the Model Builder window, right-click S-Parameter (emw, TE) 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
abs(r_s)^2
 
Reflectance, analytic
n_slab*cos(beta)/(n_air*cos(alpha))*abs(t_s)^2
 
Transmittance, analytic
4
Locate the x-Axis Data section. From the Unit list, choose °.
5
Click to expand the Legends section. From the Legends list, choose Manual.
6
In the table, enter the following settings:
Legends
Reflectance, analytic
Transmittance, analytic
7
In the S-Parameter (emw, TE) toolbar, click  Plot.
Compare the resulting plots with Figure 4.
Smith Plot (emw, TE)
1
In the Model Builder window, under Results click Smith Plot (emw).
2
In the Settings window for Smith Plot Group, type Smith Plot (emw, TE) in the Label text field.
3
In the Model Builder window, expand the Smith Plot (emw, TE) node.
Color Expression 1
1
In the Model Builder window, expand the Results > Smith Plot (emw, TE) > Reflection Graph 1 node, then click Color Expression 1.
2
In the Settings window for Color Expression, locate the Expression section.
3
In the Expression text field, type alpha.
4
From the Unit list, choose °.
Smith Plot (emw, TE)
1
In the Model Builder window, under Results click Smith Plot (emw, TE).
2
In the Settings window for Smith Plot Group, click to expand the Title section.
3
From the Title type list, choose Manual.
4
In the Title text area, type Reflection Graph: S-parameter, Color: Angle of incidence (deg).
5
In the Smith Plot (emw, TE) toolbar, click  Plot.
Polarization Plot (emw, TE)
1
In the Model Builder window, under Results click Polarization Plot (emw).
2
In the Settings window for 1D Plot Group, type Polarization Plot (emw, TE) in the Label text field.
3
In the Polarization Plot (emw, TE) toolbar, click  Plot.
The polarization plot shows that the field is linearly polarized, with a pure out-of-plane polarization. The out-of-plane component of the Jones vector represents the amplitude of a wave polarized in the y direction, whereas the in-plane component represents the amplitude of a wave with a polarization that is orthogonal to both the out-of-plane direction and the propagation direction, resulting in a polarization in the x − z plane.
Electromagnetic Waves, Frequency Domain (TE) (emw)
The following instructions are for the case of a TM-polarized wave (p-polarization or parallel polarization). In this case, Ey is zero while Ex and Ez characterize the wave. In other words, Hy is dominant while Hx and Hz are zero.
In the Model Builder window, under Component 1 (comp1) right-click Electromagnetic Waves, Frequency Domain (TE) (emw) and choose Copy.
Electromagnetic Waves, Frequency Domain (TM)
1
In the Model Builder window, right-click Component 1 (comp1) and choose Paste Electromagnetic Waves, Frequency Domain.
2
In the Messages from Paste dialog, click OK.
3
In the Settings window for Electromagnetic Waves, Frequency Domain, type Electromagnetic Waves, Frequency Domain (TM) in the Label text field.
Study 1 (TE)
Step 1: Frequency Domain
1
In the Model Builder window, under Study 1 (TE) click Step 1: Frequency Domain.
2
In the Settings window for Frequency Domain, locate the Physics and Variables Selection section.
3
In the Solve for column of the table, under Component 1 (comp1), clear the checkbox for Electromagnetic Waves, Frequency Domain (TM) (emw2).
Electromagnetic Waves, Frequency Domain (TM) (emw2)
Periodic Structure 1
1
In the Model Builder window, expand the Component 1 (comp1) > Electromagnetic Waves, Frequency Domain (TM) (emw2) node, then click Periodic Structure 1.
2
In the Settings window for Periodic Structure, locate the Port Mode Settings section.
3
From the list, choose Transverse magnetic (TM). This will change the port mode fields to be polarized parallel to the plane of incidence.
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 the Add Study button 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 Study Settings section.
2
In the Frequencies text field, type f0.
3
Locate the Physics and Variables Selection section. In the Solve for column of the table, under Component 1 (comp1), clear the checkbox for Electromagnetic Waves, Frequency Domain (TE) (emw).
4
In the Solve for column of the table, under Component 1 (comp1), select the checkbox for Electromagnetic Waves, Frequency Domain (TM) (emw2).
5
In the Model Builder window, click Study 2.
6
In the Settings window for Study, type Study 2 (TM) in the Label text field.
Parametric Sweep
1
In the Study toolbar, click  Parametric Sweep.
2
In the Settings window for Parametric Sweep, locate the Study Settings section.
3
Click  Add.
4
In the table, enter the following settings:
Parameter name
Parameter value list
Parameter unit
alpha (Angle of incidence)
range(0,2[deg],88[deg])
deg
5
In the Study toolbar, click  Compute.
Results
Magnetic Field (emw2, TM)
In the Settings window for 3D Plot Group, type Magnetic Field (emw2, TM) in the Label text field.
Multislice 1
1
In the Model Builder window, expand the Magnetic Field (emw2, TM) node, then click Multislice 1.
2
In the Settings window for Multislice, locate the Expression section.
3
In the Expression text field, type emw2.Hy.
4
Locate the Multiplane Data section. Find the X-planes subsection. In the Planes text field, type 0.
5
Find the Z-planes subsection. In the Planes text field, type 0.
6
Locate the Coloring and Style section. From the Color table list, choose WaveLight.
Arrow Volume 1
1
In the Model Builder window, right-click Magnetic Field (emw2, TM) and choose Arrow Volume.
2
In the Settings window for Arrow Volume, locate the Expression section.
3
In the X-component text field, type emw2.Poavx.
4
In the Y-component text field, type emw2.Poavy.
5
In the Z-component text field, type emw2.Poavz.
6
Locate the Arrow Positioning section. Find the Y grid points subsection. In the Points text field, type 1.
7
Locate the Coloring and Style section. From the Color list, choose Green.
8
In the Magnetic Field (emw2, TM) toolbar, click  Plot.
This reproduces Figure 3.
S-Parameter (emw2, TM)
1
In the Model Builder window, under Results click S-Parameter (emw2).
2
In the Settings window for 1D Plot Group, type S-Parameter (emw2, TM) in the Label text field.
3
Locate the Title section. From the Title type list, choose None.
4
Locate the Plot Settings section. Select the x-axis label checkbox.
5
In the y-axis label text field, type S-parameter.
6
Locate the Legend section. From the Position list, choose Middle left.
Global 1
1
In the Model Builder window, expand the S-Parameter (emw2, TM) node, then click Global 1.
2
In the Settings window for Global, locate the y-Axis Data section.
3
In the table, enter the following settings:
Expression
Unit
Description
abs(emw2.S11)^2
1
Reflectance
abs(emw2.S21)^2
1
Transmittance
4
Locate 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 Cycle.
6
From the Positioning list, choose Interpolated.
Global 2
1
In the Model Builder window, right-click S-Parameter (emw2, TM) 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
abs(r_p)^2
 
Reflectance, analytic
n_slab*cos(beta)/(n_air*cos(alpha))*abs(t_p)^2
 
Transmittance, analytic
4
Click to expand the Legends section. From the Legends list, choose Manual.
5
In the table, enter the following settings:
Legends
Reflectance, analytic
Transmittance, analytic
6
Locate the x-Axis Data section. From the Unit list, choose °.
7
In the S-Parameter (emw2, TM) toolbar, click  Plot.
The plot should look like Figure 5. The Brewster angle is observed around 56 degrees, which is close to the analytic value.
Smith Plot (emw2, TM)
1
In the Model Builder window, under Results click Smith Plot (emw2).
2
In the Settings window for Smith Plot Group, type Smith Plot (emw2, TM) in the Label text field.
3
In the Model Builder window, expand the Smith Plot (emw2, TM) node.
Color Expression 1
1
In the Model Builder window, expand the Results > Smith Plot (emw2, TM) > Reflection Graph 1 node, then click Color Expression 1.
2
In the Settings window for Color Expression, locate the Expression section.
3
In the Expression text field, type alpha.
4
From the Unit list, choose °.
Smith Plot (emw2, TM)
1
In the Model Builder window, under Results click Smith Plot (emw2, TM).
2
In the Settings window for Smith Plot Group, locate the Title section.
3
From the Title type list, choose Manual.
4
In the Title text area, type Reflection Graph: S-parameter, Color: Angle of incidence (deg).
5
In the Smith Plot (emw2, TM) toolbar, click  Plot.
Polarization Plot (emw2, TM)
1
In the Model Builder window, under Results click Polarization Plot (emw2).
2
In the Settings window for 1D Plot Group, type Polarization Plot (emw2, TM) in the Label text field.
3
In the Polarization Plot (emw2, TM) toolbar, click  Plot.
Also in this case, the polarization plot shows that the field is linearly polarized, in this case with a pure in-plane polarization.
1
Note that depending on the sign convention for what is defined as a positive polarization for the reflected wave with in-plane (p-polarization), the sign for the reflection coefficient rp can differ for different authors. However, the total field solution will always be the same.