As far as i understood, in order to create a dielectric material with losses, I cannot symply propose a complex dielectric constant, but it is necessary to create a new material with a given dispersion relation that Lumerical will fit usin some FDTD-friendly function. From my intuition I would say that this is necessary because not everything defined in the frequency domain can be easily translated/used in the time domain.
But if this is true for lossy materials, why for an I can define a single value for the dielectric constant? How does Lumerical deals with that? Does it create a material that performs some kind of fitting around the dielectric constant value?

I guess this has everything to do with how FDTD simulations work. Can anyone explain me, or give me some references where to study it, how and why does it work the way it does?

This is not the case, you can create an (n,k) material with a constant real and imaginary refractive index, which will have a loss dependent on k. This method will create a fit that is compatible with the time domain solver, and has the correct values for the n and k at a single wavelength.

However, the fit may not be very good at wavelengths far from the central frequency of the simulation, so this material model should only be used for simulations where you are calculating results for a single frequency. For broadband results, you should use a fitted material model that takes into account the material dispersion.

Using (n,k) materials is discussed on this page:

This is true, the multi-coefficient model that FDTD Solutions uses creates a fit using basis functions that can easily be translated into the time domain. The material fits also must obey the Kramers-Kronig relations, so causality is not violated. There is more information on the MCM model on this page:

If you would like to learn more about the FDTD method, this page has a brief introduction and lists a few references if you would like to learn more:

These references will discuss how dispersive materials are handled in FDTD simulations.

I hope this helps. Let me know if you have any questions.

Thanks @kjohnson for the reply and all the references. These are very helpful.
There’s still one point that I’m missing though.
What about the case of NON dispersive and NON lossy media? (hence with a given value of refractive index)
Why in this case we don’t need a particular fit? As mentioned here :

For broadband simulations, n,k material is cannot be used as discussed below.
Note: For materials with n∈R, the situation is simplified as you can simply use a ‘Dielectric’ material model to avoid the following complications.

Sorry for the late reply. I am not sure why a fit is not required in this case, but I would assume that this is because the Kramers Kronig relations are satisfied if the index is purely real and constant. However, I would still recommend that you only use dielectric materials like this with simulations that have a narrow bandwidth, or when the results are only needed at single wavelength. A dispersive material would be more realistic for broadband simulations.

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