Fresnel Equations

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.

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.

RF_Module/Verification_Examples/fresnel_equations

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select .

Click .

Click

In the tree, select .

Click

Global Definitions

Define some parameters that are useful when setting up the geometry and the study.

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
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_aircos(alpha)-n_slabcos(beta))/(n_aircos(alpha)+n_slabcos(beta))	-0.54735	Reflection coefficient, TE
r_p	(n_slabcos(alpha)-n_aircos(beta))/(n_aircos(beta)+n_slabcos(alpha))	-0.20613	Reflection coefficient, TM
t_s	(2n_aircos(alpha))/(n_aircos(alpha)+n_slabcos(beta))	0.45265	Transmission coefficient, TE
t_p	(2n_aircos(alpha))/(n_aircos(beta)+n_slabcos(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)

In the window, click .

In the window for , type Study 1 (TE) in the text field.

Step 1: Frequency Domain

In the window, under click .

In the window for , locate the section.

In the text field, type f0.

Geometry 1

First, create a block composed of two domains. Use layers to split the block.

Block 1 (blk1)

In the toolbar, click

In the window for , locate the section.

In the text field, type 0.2.

In the text field, type 0.8.

Click to expand the section. In the table, enter the following settings:


Layer name	Thickness (m)
Layer 1	0.4

Click

Click the

button in the toolbar.

Choose wireframe rendering to get a better view of each boundary.

Click the

button in the toolbar.

Electromagnetic Waves, Frequency Domain (emw)

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.

Wave Equation, Electric 1

In the window, under click .

In the window for , locate the section.

From the list, choose .

In the window, click .

In the window for , type Electromagnetic Waves, Frequency Domain (TE) in the text field.

Port 1

In the toolbar, click

and choose .

Select Boundary 7 only.

In the window for , locate the section.

From the list, choose .

For the first port, wave excitation is by default.

Locate the section. Specify the E0 vector as


0	x
1	y
0	z

Locate the section. In the n text field, type n_air.

Locate the section. In the α1 text field, type alpha.

The maximum frequency in the setting window will be used only when button is clicked to generate features representing higher order modes. In this model, no diffraction is expected from the given geometry, so this setting is ineffective.

Port 2

In the toolbar, click

and choose .

Select Boundary 3 only.

In the window for , locate the section.

From the list, choose .

Locate the section. Specify the E0 vector as


0	x
1	y
0	z

Locate the section. In the n text field, type n_slab.

The bottom surface is an observation port. The S21-parameter from Port 1 and Port 2 provides the transmission characteristics.

Periodic Condition 1

In the toolbar, click

and choose .

Select Boundaries 1, 4, 10, and 11 only.

In the window for , locate the section.

From the list, choose .

Periodic Condition 2

In the toolbar, click

and choose .

Select Boundaries 2, 5, 8, and 9 only.

In the window for , locate the section.

From the list, choose .

Materials

Now set up the material properties based on refractive index. The top half is filled with air.

Air

In the window, under right-click and choose .

In the window for , type Air in the text field.

Select Domain 2 only.

Locate the 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.

Right-click and choose .

In the window for , type Glass in the text field.

Select Domain 1 only.

Locate the 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 window, under right-click and choose .

Study 1 (TE)

Parametric Sweep

In the toolbar, click

In the window for , locate the section.

Click

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

In the toolbar, click

Results

Electric Field (emw, TE)

In the window for , type Electric Field (emw, TE) in the 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

In the window, expand the node, then click .

In the window for , click in the upper-right corner of the section. From the menu, choose .

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

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

Locate the section. Click

In the dialog box, select in the tree.

Click .

Arrow Volume 1

In the window, right-click and choose .

In the window for , click in the upper-right corner of the section. From the menu, choose .

Locate the section. Find the subsection. In the text field, type 1.

Locate the section. From the list, choose .

Electric Field (emw, TE)

In the window, click .

In the window for , locate the section.

From the list, choose .

In the toolbar, click

Click the

button in the 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.

In the window, under click .

In the window for , type S-Parameter (emw, TE) in the text field.

Click to expand the section. From the list, choose .

Locate the section. Select the check box.

Replace the dB-scale variables with the linear-scale S-parameter variables.

In the text field, type S-parameter.

Locate the section. From the list, choose .

Global 1

In the window, expand the node, then click .

In the window for , locate the section.

In the table, enter the following settings:


Expression	Unit	Description
abs(emw.S11)^2	1	Reflectance
abs(emw.S21)^2	1	Transmittance

In the toolbar, click

Click to expand the section. Find the subsection. From the list, choose .

Find the subsection. From the list, choose .

From the list, choose .

Global 2

In the window, right-click and choose .

In the window for , locate the section.

In the table, enter the following settings:


Expression	Unit	Description
abs(r_s)^2		Reflectance, analytic
n_slabcos(beta)/(n_aircos(alpha))*abs(t_s)^2		Transmittance, analytic

Locate the section. From the list, choose .

In the toolbar, click

Compare the resulting plots with Figure 4.

Smith Plot (emw, TE)

In the window, under click .

In the window for , type Smith Plot (emw, TE) in the text field.

In the window, expand the node.

Color Expression 1

In the window, expand the node, then click .

In the window for , locate the section.

In the text field, type alpha.

From the list, choose .

Smith Plot (emw, TE)

In the window, under click .

In the window for , click to expand the section.

From the list, choose .

In the text area, type Reflection Graph: S-parameter, Color: Angle of incidence (deg).

In the toolbar, click

Polarization Plot (emw, TE)

In the window, under click .

In the window for , type Polarization Plot (emw, TE) in the text field.

In the toolbar, click

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 have no effect. Thus, the H-field is perpendicular to the plane of incidence and it is convenient to specify the port mode fields as the H-field.

In the window, under right-click and choose .

Electromagnetic Waves, Frequency Domain (TM)

In the window, right-click and choose .

In the dialog box, click .

In the window for , type Electromagnetic Waves, Frequency Domain (TM) in the text field.

Study 1 (TE)

Step 1: Frequency Domain

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Physics interface	Solve for	Equation form
Electromagnetic Waves, Frequency Domain (TE) (emw)	√	Automatic (Frequency domain)
Electromagnetic Waves, Frequency Domain (TM) (emw2)		Automatic (Frequency domain)

Electromagnetic Waves, Frequency Domain (TM) (emw2)

Port 1

In the window, expand the node, then click .

In the window for , locate the section.

From the list, choose .

Specify the H0 vector as


0	x
1	y
0	z

Port 2

In the window, click .

In the window for , locate the section.

From the list, choose .

Specify the H0 vector as


0	x
1	y
0	z

Add Study

In the toolbar, click

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 2

Step 1: Frequency Domain

In the window for , locate the section.

In the text field, type f0.

Locate the section. In the table, enter the following settings:


Physics interface	Solve for	Equation form
Electromagnetic Waves, Frequency Domain (TE) (emw)		Automatic (Frequency domain)
Electromagnetic Waves, Frequency Domain (TM) (emw2)	√	Automatic (Frequency domain)