This video is taken from the FDTD 100 course on Lumerical University.

## Transcript

Getting frequency domain results from FDTD.

FDTD is a time domain method, which means that we solve for the electric and magnetic

fields as a function of time.

Generally, we are interested in calculating field as a function of frequency or, equivalently,

wavelength.

We are also interested in other quantities as a function of frequency or wavelength,

such as the Poynting vector, transmission, reflection, absorption, scattering cross sections

and so on.

We calculate the frequency domain fields by Fourier transforming the time domain fields.

This Fourier transform can be calculated while the FDTD simulation is running at a predetermined

set of discrete frequency values, or it can be done after the simulation, typically by

fft, as long as the fields have been recorded over the entire simulation time.

In FDTD Solutions, frequency domain monitors perform the discrete Fourier transforms while

the simulation is running, while time domain monitors can be used to record the fields

as a function of time for subsequent analysis by fft.

In linear systems, it is most interesting to obtain the impulse response of the system

because this gives us the response to continuous wave (CW) radiation at all wavelengths in

a single simulation.

We can obtain the impulse response by normalizing the Fourier transform of the fields to the

Fourier transform of the source pulse, which we will call s, that was used to excite the

system.

Ideally, we would like s to be a delta function in the time domain and, therefore, have a

value of unity for all frequencies.

In practice, we use a very short pulse for the source pulse in the time domain, and ensure

that in the frequency domain it covers all the wavelengths of interest.

It is worth noting that with this normalization we are calculating the impulse response of

the system which is independent of the source pulse used.

This means that if we change the source pulse, for example by making it twice as long, we

will get exactly the same impulse response.

It corresponds to the CW, or monochromatic response, at each frequency or wavelength,

and this is typically the result we want for linear systems.

Please note that in nonlinear systems, the source pulse that we use will affect the response

of the system so we have to treat nonlinear systems differently.

Let's take a look practically at what this means.

We will consider a simulation of a plane wave, polarized along the z axis, in free space.

If we watch a movie of the electric field in the time domain, we can see the short source

pulse that is used in the time domain to excite the system.

We can plot both the source pulse and the electric field recorded by a point time monitor.

We see that the electric field and source pulse look nearly identical except for a time

offset because it takes some time for the pulse to arrive at the location where the

field is recorded.

We can Fourier transform these quantities and plot their magnitude as a function of

wavelength.

Since we started with a Gaussian envelope in the time domain, we have a Gaussian shape

in the frequency domain but it appears slight asymmetric because we are plotting it vs wavelength.

When we normalize the electric field to the source pulse, in the frequency domain, we

can see that the electric field is unity for all wavelengths.

This is exactly what we would expect for a plane wave of unit amplitude propagating in

free space.

While it is not plotted here, the phase of the normalized electric field is equal to

k times L where k is the wavenumber and L is the distance from the source to the monitor

that recorded the field.

We can consider a more interesting problem for FDTD which is a ring resonator.

Here we use a pulse to inject a modal field into an input waveguide.

We can see that the pulse is so short that we don't even see it overlap with itself in

the time domain simulation.

While the simulation is running, we use frequency domain monitors to perform a Fourier transform

at different wavelengths.

By doing this, we can extract the CW fields at different wavelengths and see, for example,

the fields where the ring is out of resonance and the majority of light is transmitted and

the fields when the ring is in resonance and most of the light is dropped.

Both of these results are obtained from a single time domain FDTD simulation using a

short excitation pulse.

We can calculate many other quantities from the permittivity, the electric field and the

magnetic field, known as a function of frequency or wavelength.

We can calculate reflection, transmission, scattering cross sections, absorption and

much more.

For example, the Poynting vector, often called P or S, can be calculated with E cross conjugate

of H. This allows us to calculate the power crossing a given surface, which we typically

normalize to the power of the source to obtain a quantity between -1 and 1.

This allows us to plot, in our ring resonator example, the normalized transmission through

the through and drop ports by calculating hundreds of wavelength points in a single

simulation.

Finally, we should note that the Poynting vector in the frequency domain is not the

Fourier transform of the Poynting vector in the time domain, instead it is derived from

the Fourier transforms of the electric and magnetic fields.

There are many useful properties of Fourier transforms that we can use for advanced analysis,

such as the Parseval-Plancherel theorem which tells us that the power integrated over all

time is equal to the power integrated over all frequencies.