Ray Release Based on a Plane Electromagnetic Wave

Ray Release Based on a
Plane Electromagnetic Wave

This tutorial shows how to set up a ray release based on the incident electric field at a boundary. First the Electromagnetic Waves, Frequency Domain interface is used to solve for the electric field of a plane wave. Then rays are released with initial intensity and polarization matching that of the electric field at the releasing boundary.

Model Definition

This model offers some guidelines for multiscale modeling of electromagnetic wave propagation. The main idea is to use the Electromagnetic Waves, Frequency Domain interface to compute the electric field over a region that is similar in size to the wavelength, then use the Geometrical Optics interface to model propagation over longer distances.

Motivation

The Electromagnetic Waves, Frequency Domain interface can be used to obtain an accurate full-wave solution to Maxwell’s Equations using the finite element method (FEM). However, the finite element mesh must be fine enough to resolve the individual oscillations of the electric field. Following the Nyquist criterion, the mesh should have at least 10 linear or 5 second-order mesh elements per wavelength. For example, if the wavelength is 500 nm and the domain is a cube that is 10 μm on each side, then the simulation domain is about 20 wavelengths in each direction. For second-order shape functions, a swept mesh of this domain would include one million elements. This 10 μm cube would be a very difficult problem to solve on a desktop computer; a room several meters in width is simply infeasible.

In contrast, the Geometrical Optics interface can be used to model electromagnetic wave propagation over very large distances because it does not require the mesh to be fine enough to resolve individual wavelengths. However, a ray tracing approach treats each ray as a wavefront that is locally plane, and therefore effects like diffraction are not considered.

One compromise is to solve Maxwell’s Equations using the Electromagnetic Waves, Frequency Domain interface in the immediate vicinity of any object similar in size to the wavelength, and then trace rays over longer distances that lack such fine geometric details. As the rays are released, information about their initial intensity, polarization, and phase can be obtained from the electric field in the adjacent domain.

Although the rays in this example are traced over a distance comparable to the waveguide, this is mainly for visualization and validation purposes. The rays could easily propagate over any arbitrarily large number of wavelengths.

Ray Intensity Initialization

In the Geometrical Optics interface, ray intensity can be initialized from an electric field by converting the electric field amplitude to a set of Stokes parameters,

where the ray is assumed to propagate in the z-direction so that the transverse electric field components are Ex and Ey (SI unit: V/m). If the light is not monochromatic, then time-averaging of the electric field components is required. The Stokes parameters themselves have units of intensity (W/m2). In addition,

•

c = 299,792,458 m/s is the speed of light in a vacuum,

•

ε0 = 8.854187817 × 10-12 F/m is the permittivity of vacuum, and

•

n (dimensionless) is the refractive index.

The medium is assumed to be nonmagnetic (μr = 1). The middle column shows how the Stokes parameters can be defined phenomenologically. Here I(θ,δ) (SI unit: W/m2) is the intensity that would be transmitted by a polarizer that only accepts light that is polarized at angle θ, with a phase delay of δ between the x- and y- components. With this definition, the second, third, and fourth Stokes parameters each describe the difference in intensity between one polarization state and an orthogonal state.

•

s0 is the total ray intensity,

•

s1 is the preference for horizontal polarization over vertical polarization,

•

s2 is the preference for polarization in the direction y = x, over polarization in the direction y = -x, and

•

s3 is the preference for right-handed circular polarization over left-handed circular polarization.

Altogether, the four Stokes parameters are capable of characterizing any polarization state of a quasi-monochromatic wave, including linearly polarized, circularly polarized, elliptically polarized, unpolarized, and partially polarized. If the light is monochromatic, or otherwise fully polarized, then

Model Setup

The electric field of a plane wave is first computed using the Electromagnetic Waves, Frequency Domain interface. Depending on the application, sometimes the Electromagnetic Waves, Beam Envelopes interface can be used instead.

The model geometry is a block that is several wavelengths long. A plane wave propagates in the positive z-direction. To excite the wave, the boundary condition is used at one face parallel to the xy-plane. The is used to excite the wave instead of the because the allows the input power to be specified more easily.

The is used at the other end of the domain, to absorb the outgoing wave and prevent any energy from being reflected back toward the . For best results, the should be used to absorb waves at normal incidence. If waves are likely to be incident at very large angles to the surface normal, a better approach is to use a .

On the two sides of the domain where

, the default boundary condition is used. On the two other sides, where

, the condition is applied.

After solving for the electric field using the study (the study is also a valid choice), the Geometrical Optics interface is used to continue tracing rays outside of the domain. The node is used to release rays directly from the boundary.

In the settings for the study, it is crucial that the correct study is chosen in the section (typically a or study step is chosen), or else the rays might be released with zero intensity.

Results and Discussion

A plot of the x-component of the electric field is shown in Figure 1. Then the x-component of the electric field for both the FEM calculation and the ray tracing solution are shown side-by-side in Figure 2. It is clear that the amplitude, direction, and phase of the electric field along the rays matches that of the FEM solution in the adjacent domain. This can also be shown by plotting the electric field along one of the edges of the domain, then plotting the electric field on rays as a function of average position with a plot, as in Figure 3. The sum of the power of all rays can also be evaluated and compared to the input power in the settings for the feature. The two results are similar but are not exactly the same due to discretization error in the FEM solution.

Figure 1: Electric field of a monochromatic plane wave, x-component.

Figure 2: Electric field of a monochromatic plane wave, x-component, computed using FEM within the domain and by releasing rays from its surface.

Figure 3: Electric field x-component as a function of the z-coordinate. This illustrates how the released rays are assigned both the amplitude and phase of the incident electric field.

Ray_Optics_Module/Tutorials/plane_em_wave_to_ray_release

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

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
W	1[um]	1E-6 m	Waveguide width
L	5[um]	5E-6 m	Waveguide length
lam0	1[um]	1E-6 m	Wavelength

Geometry 1

First create the waveguide geometry, which is a simple block.

In the window, under click .

In the window for , locate the section.

From the list, choose .

Block 1 (blk1)

In the toolbar, click

In the window for , locate the section.

In the text field, type W.

In the text field, type L.

Click

Electromagnetic Waves, Frequency Domain (ewfd)

Add the boundary conditions. This model uses a to excite the plane wave, a boundary condition to absorb it, and pairs of and conditions on the opposite walls. The node is already present by default.

Port 1

In the window, under right-click and choose .

Select Boundary 3 only.

In the window for , locate the section.

Specify the E0 vector as


1	x
0	y
0	z

Locate the section. In the Pin text field, type 1e-9[W].

Locate the section. In the β text field, type ewfd.k0.

Scattering Boundary Condition 1

In the toolbar, click

and choose .

Select Boundary 4 only.

Perfect Magnetic Conductor 1

In the toolbar, click

and choose .

Select Boundaries 2 and 5 only.

Add Material

In the toolbar, click

to open the window.

Go to the window.

In the tree, select .

Click in the window toolbar.

In the toolbar, click

to close the window.

Study 1

Step 1: Wavelength Domain

In the window, under click .

In the window for , locate the section.

In the text field, type lam0.

In the toolbar, click

Results

The default plot shows the electric field norm. This appears quite noisy because the electric field norm should ideally be uniform in this domain. Instead, plot the real part of the electric field amplitude.

Multislice 1

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 .

In the toolbar, click

. Compare the resulting plot to Figure 1.

Add Physics

Now set up the Geometrical Optics physics interface.

In the toolbar, click

to open the window.

Go to the window.

In the tree, select .

Find the subsection. In the table, clear the check box for .

Click in the window toolbar.

In the toolbar, click

to close the window.

Add Study

In the toolbar, click

to open the window.

Go to the window.

Find the subsection. In the table, clear the check box for .

Find the subsection. In the tree, select .

Click in the window toolbar.

In the toolbar, click

to close the window.

Geometrical Optics (gop)

In the window, under click .

In the window for , locate the section.

In the text field, type 0.

Locate the section. From the list, choose .

Select the check box.

Release from Electric Field 1

In the toolbar, click

and choose .

Select Boundary 4 only.

In the window for , locate the section.

In the N text field, type 9.

Locate the section. From the E list, choose .

Select the check box. With this check box selected, the wavelength input in the node will be ignored. Instead, the vacuum wavelength will be taken directly from the previous study.

Study 2

Step 1: Ray Tracing