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,
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
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
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
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
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
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.