In this example, we will determine the optimal shape of an SOI taper using MODE 2.5D variational FDTD solver.
[[Note:]] that this taper can be simulated using the eigenmode expansion (EME) solver as well.
We will start by parameterizing the design of this taper as follows:
$$w(x) = \alpha(Lx)^m + w_2$$
$$w(0) = w_1$$
$$w(L) = w_2$$
$$\alpha = (w_1  w_2)/L^m$$
The taper design in this case will be proportional to \(x\) to the power of exponent \(m\). At the two ends of the taper, we are constrained to the waveguide width of \(w_1\) and \(w_2\).
\(m=0.5\) corresponds to a square root shaped taper. 
\(m=2\) corresponds to a parabolic shaped taper. 
The file taper_design.lms contains a 2.5D propagator simulation region with a slab gaussian beam as the source. The slab gaussian beam is set up to focus at a distance of 25 um away from the source position.


A parameter sweep project has been set up to track the transmission into the output waveguide as a function of the exponent m. You can use the animate function to see how the shape of the taper changes as a function of m.
The result of the parameter sweep for the exponent m from 0.1 to 4 is shown below. One can see that the transmission changes quite significantly as m changes from 0.1 to 4. The peak value is close to 1 (corresponding to a linear taper), but if we run the parameter sweep project again, which sweeps m over a much narrower range from 0.8 to 1.7, we find that the optimal value is at around 1.15.


We can also look at the propagation of light in this taper (form=1.15) with the movie monitor.
It is important to note that this simulation (~30um by 30um by 2um) would have taken a very long time to run with 3D FDTD. The 2.5D Propagator is ideal here in that it allows us to find the optimal shape of this SOI taper very quickly. On the next page, we will calculate the transmission into individual modes of the output waveguide and show that the results are very close to the 3D FDTD results.
Comparing results with 3D FDTD
In the previous section, we found the optimal shape of the SOI taper by parameterizing it an exponential function and using a parameter sweep to find the optimal parameter values. Now we will demonstrate how to calculate the transmission from the input waveguide into individual modes of the output waveguide, and compare the results with 3D FDTD.
We will use the same taper with the optimal value m = 1.15 found in the previous section. Here, we will inject the mode source from the left waveguide "w1", and measure the transmission into the TE modes of the right waveguide "w2". The figure below shows the simulation model and resultant field profile.
To measure the transmission into the modes of the output waveguide, we added a transmission monitor and a Mode Expansion Monitor at w2. The mode expansion monitor allows one to select an arbitrary number of modal fields with the "user select" mode selection option. One can then shiftselect the desired modes from the mode list. For this example, we will select the first 5 even TE modes for the expansion, which are modes #2, 6, 10, 14, 18.
Once the simulation finishes running, rightclick on the expansion monitor to visualize the results. The Visualizer screen shot below shows the forward transmission into the first 5 even TE modes of the output waveguide.
Because collapsing the vertical structure into an effective slab works very well in a wide region like that of the taper, no approximations are made for light propagation in the waveguide slab and one can get results very close to 3D FDTD with this 2.5D FDTD treatment. The figure below compares the results between 2.5D FDTD and 3D FDTD (using taper.fsp), and you can see that the results are almost identical to 3D FDTD.