In this example, we will model a Faraday rotator, a device which rotates the polarization of the incident light via the magneto-optical Faraday effect. We will also show how this effect can be utilized to create an optical isolator, allowing for transmission of light in only one direction (and preventing unwanted feedback).

## Faraday Effect

The Faraday effect is a magneto-optical effect where an external magnetic field leads to circular birefringence, and the left and right circularly polarized waves propagate at different velocities. The result of this is a rotation of the plane of polarization, which is linearly proportional to the component of the magnetic field in the direction of propagation. The resulting angle of rotation β, is defined by

$$\beta = VBd$$

where V is the Verdet constant, and B is the static magnetic flux density. To model this effect in FDTD, we will use an anisotropic material combined with a grid attribute object, which allows us to apply an arbitrary unitary matrix necessary for inducing the correct rotation. Specifically, we have to rotate the reference frame such that it converts the field components from Cartesian coordinates into coordinates that represent circular polarization. To do this, we define the following unitary matrix:

$$U = \frac{1}{\sqrt{2}} \left(\begin{matrix} 1&0&i\\0 & \sqrt{2} & 0 \\ 1 & 0 & -i \end{matrix}\right)$$

and use a Matrix Transform Grid Attribute to apply this rotation around the y axis. The script faraday.lsf defines U and sets it as the transform matrix.

In faraday.fsp, we start with a plane wave polarized in the x direction, and use a linear y DFT monitor to track the change in polarization of the light as it propagates through the Faraday rotator. Here, we can define the polarization, at each point along y, by the TE fraction:

$$\frac{|E_x|^2}{|E_x|^2 + |E_z|^2}$$

The script faraday_plot.lsf plots the "TE fraction vs y" in the rotator:

We can also plot Ex and Ez as a function of y in the Visualizer to observe this change. Note that the "Duplicate" function in the Visualizer makes it very easy to plot different field components in the same figure.

## Optical Isolator

For an optical isolator (which is typically composed of a Faraday rotator and two polarizers as shown below), the polarizer in front of the Faraday rotator will not only polarize the incident light in the x direction, it will also function as a filter for the reflected light (with the opposite polarization as the incident light).

In this section, we will extend the Faraday effect described in the previous section to model the optical isolator. In the previous section, we used an incident plane wave polarized in the x direction, and found that a distance of ~11.94um will result in an angle of rotation of 45 degrees (see faraday.lsf). In isolator.fsp,a mirror is placed at the point where the angle of rotation is 45 degrees. The reflected light will continue to change in the rotator, undergoing another angle of rotation of 45 degrees when it exits the rotator at the incident surface. As a result, the reflected fields will be completely polarized in the z direction (the figure below shows the Ex and Ez components of the reflected light):

One can also use the movie monitor to observe the change in polarization as a function of time.