We have calculated the optical force on a particle in the previous examples. In this example, we will look at the optical force on a metal wall as well as a dielectric wall and compare the results to the easily calculated theoretical predictions to further confirm the validity of this technique.

## Simulation setup

A planewave source with wavelength 0.5 um is injected at normal incidence towards a slab which represents the wall. Two cases will be simulated: a metal slab and a dielectric slab. The material of the metal wall is set as a perfect electric conductor (PEC). The background index is 1.

Two 2D power monitors, one in front and one behind the slab are used to define the closed surface S around the slab. Since periodic boundary conditions are used at the sides, all of the light is assumed to eventually either reflect back to transmit through the slab.

The analysis script located in the optical_force_slab analysis object integrates the Maxwell's stress tensor to provide the total force on the particle. There are two analysis groups contained in the optical_force_slab analysis group (one for each of the two z power monitors) which are used to calculate the stress tensor.

## Analysis and results

When the simulation is finished, run force_wall.lsf to run the analysis group script and calculate the optical force on the wall at wavelength 0.5 um. Note that the power monitor above the conductor wall will have zero reading because no light is supposed to be transmitted through the perfect conductor.

The force_wall.lsf script also calculates the theoretical prediction of the same force. For a perfect metal slab, the reflection coefficient is 1.

In every second, the source emits N photons each with energy E and momentum p, for a total power of P.

$$\begin{array}{l}{N=\frac{P}{E}=\frac{P}{h v}} \\ {P=\frac{h}{\lambda}=\frac{n h}{\lambda_{0}}}\end{array}$$

NOTE: This momentum is the Minkowski or canonical momentum which is appropriate for the calculation of optical force, for details see http://rsta.royalsocietypublishing.org/content/368/1914/927.full. |

The force on the wall is equal to the change in the momentum of all the photons. Since the photons hit the wall and return with the same speed, the change in their momentum is 2\(p|):

$$F=2 N_{p}^\rho=\left(\frac{P \lambda_{0}}{h c}\right) \times 2 \times\left(\frac{n h}{\lambda}\right)=\frac{2 n P}{c}$$

Once the script is run, both the theoretical and FDTD analysis method results are calculated. The values are in agreement as expected.

The same procedure can be carried out for a dielectric wall. The only difference in this case is that not all the photons are reflected back, and so both of the power monitors above and below the wall will have non-zero readings.

The reflection coefficient of the dielectric is extracted from the Fresnel equations. The fraction of the intensity of the reflected field to that of the incident field can then easily be calculated:

$$R=\left|\frac{n_{\text {air}}-n_{\text {wall}}}{n_{\text {air}}+n_{\text {wall}}}\right|^2$$

$$\frac{I_{r}}{I_{i}}=\frac{4 R \sin ^{2}(\frac {2 \pi d_{\text {wall}} n_{\text {wall}}}{\lambda})}{(1-R)^{2}+4 R \sin ^{2}(\frac{2 \pi d_{\text {wall}} n_{\text {wall}}}{\lambda})}$$

Then, the force is :

$$F=\frac{2 P}{c} \cdot \frac{I_{r}}{I_{i}}$$

Edit the rectangle object so that the material is a user defined dielectric with index 1.5. In the script, the variable PEC to 0 on line 9 and rerun the script. Again the analysis group results and the theoretical results are in agreement as expected.