# Poor fitting of imaginary refractive index

#1

Hi,

I’m having trouble getting a good model fit for Si in the 7-13 um wavelength band.

As can be seen in the following picture:

The FDTD model tracks the experimental data acceptably for the real part of the refractive index, but not at all for the imaginary part. As can be seen, the fit chosen by FDTD has only two coefficients whereas I would prefer a large number of coefficients to get a tighter fit.

I also tried changing the weighting of the imaginary part over a range of values (1e1 - 1e5). This resulted in either a poor compromise where the fit was unacceptable for both parts, or a good fit for the imaginary component and a poor fit for the real one.

• Why are the two functions coupled in the fitting process? Seems like fitting each function separately (or at least having the option to) would have been a better approach here.

• Is it possible for me to input a functional model myself? i.e. find the fit I would like to work with in a separate program (e.g. Matlab) and input the model directly?

Would appreciate any input on this subject,
Cheers

#2

Fitting the material properties can indeed be very tricky! In the frequency domain, the relation between the electric field and displacement field is pretty simple:

But FDTD is a time domain technique. In the time domain, this relation becomes a convolution product:

This implies limitations in the form of the permittivity:

• It must be able to solve the convolution product in the time domain.
• It must be causal (the material cannot respond to fields in the future)

The second point links the real part and imaginary part (they have to satisfy the Kramers-Kronig relationships).

This explains why the 2 functions are coupled and why it is not possible to input any custom model.

In a lot of cases, one question is how good do we know the actual material properties. If you look at the imaginary part of Si index beyond the 7-13um range, it is 0 except for 2 points. So are these 2 points relevant, or are they within the measurement uncertainty?