In this page we compare the Analytic results from STACK with 1D FDTD simulation for a 4 layer dielectric.
We use the plane wave source technique to calculate the transmission and reflection from an n = 1:1.5:2.5:1.5 dielectric stack as a function of angle. Results for s-polarization (TM) are shown here. The simulation can be easily modified to calculate the p-polarization.
The above figure shows the simulation file plane_4layer.fsp. The n = 1:1.5:2.5:1.5 dielectric stack is visible, along with the plane wave source (white line, 1 um single wavelength) and profile monitors (yellow line). A plane wave source with Bloch boundary conditions is the typical setup for this type of simulation, which makes the simulation volume is basically 1D. Each simulation will provide R and T at a single angle. A series of simulations must be run to calculate R and T vs theta. After opening the fsp file, run the parameter sweep followed by the plane_4layer.lsf script file to reproduce the following results.
- This technique can easily be extended to periodic gratings
- Simulation volume is very small, so each simulation is fast
- One simulation is required for each angle
An alternate simulation setup is described on the Dipole technique page.
The parameter sweep is set up to run 30 simulations at angles from 0 to 60 degrees. Once the parameter sweep is complete, the script plots the results from the parameter sweep with the theory. Below you can see the reflection and transmission for an s-polarized (TM) plane wave.
The results shown in the above figure are good, but can be improved upon. The fsp file and script are setup to use a 10 nm mesh, which gives good results while keeping the simulation time reasonably short. Re-running the calculation with a 5nm mesh will give much better agreement. To use a 5nm mesh, change the following settings.
In the simulation file, go to the objects tree and edit the following objects
- FDTD region: Set the "x span" to 5nm. Set "dx","dy" to 5nm.
In the simulation file, go to the optimization and sweeps window, then
- Edit the sweep, and set the number of points to 60. As a result the reflection and transmission plots will contain more points so they will look smoother.
The simulation region boundary conditions in the y-direction use SCPML with the "steep angle" profile to prevent any reflections from light incident at steep angles. More information on the PML profile settings can be found at PML boundary conditions .
Detailed comparison with analytic results
The above results were generated with a 10nm mesh. Therefore, the thicknesses of each layer was resolved to within 10 nm. It is interesting to see how sensitive the analytic solution is to variations of 10 nm in the layer thicknesses.
The figure above shows the same FDTD reflection result as in the previous figure. The other two lines show the minimum and maximum reflections that will occur by changing the thickness of the two middle layers by +5nm or -5nm, as calculated with the analytic formula. The results are clearly within the error of the mesh.