The FDTD simulation region allows you to specify the background index, but this feature is limited to real part of the index only. The following steps demonstrate a simple trick that allows you to use arbitrary material or complex index as a background for your simulation.
Add a rectangle to your simulation and set its position and size in a way that it covers the entire simulation region
Set the material of the rectangle to the material that you want to use as a background index:
If the default list of materials does not include the material of your interest, you can import arbitrary material data or use one of the many supported material modes as it is shown in this post. Note that this includes a simple n,k material model that allows you to quickly specify a complex index.
Set the mesh order of the rectangle to lower priority(higher number) in order to make sure that any objects inserted in the simulation region will have higher mesh order priority. The default mesh order for any inserted object is 2, so it is a good idea to set the mes order to 3 or larger:
You can read more about mesh order settings here.
At this point, the background index is set to your desired value, but the rectangle is obstructing the view of your simulation region. This can be easily fixed by setting the render type of the rectangle to “wireframe” in the rectangle properties/Graphical rendering tab:
Here is the final unobstructed view: