The FDTD method, an introduction.
The finite-difference time-domain (FDTD) method is used to solve Maxwell's equations in the
The equations are solved numerically on a discrete grid in both space and time, and
derivatives are handled with finite differences.
It does not make any approximations or assumptions about the system and, as a result, it is highly
versatile and accurate.
Since it solves for all vector components of the electric and magnetic fields, it is
a fully-vectorial simulation method.
Because it is a time domain method, FDTD can be used to calculate broadband results from
a single simulation.
FDTD is typically used when the feature size is on the order of the wavelength.
This wavelength scale regime where diffraction, interference, coherence and other similar
effects play a critical role is called wave optics.
When the feature sizes are much larger than the wavelength, other methods such as ray
tracing are more efficient.
A simple example where we can see the difference between wave optics and ray tracing is this
triangular pattern etched in a material with a refractive index of 1.5.
This type of structure is commonly used as an efficient reflective coating because light
is always incident at 45 degrees to the normal, which is above the critical angle for total
internal reflection, and therefore we expect the device to reflect 100% of the incident
We can run an FDTD simulation to see what happens for two different wavelengths.
In the first, the incident wavelength is 400nm, or 0.4 microns, while the structure is 20
microns in pitch or period.
Since the structure is much larger than the wavelength, we typically would simulate with
As we run the FDTD simulation, we can see the result we expect from the ray optics analysis,
namely, that almost all of the light is reflected.
The second wavelength we will use is 4 microns and the structure has the same 20 micron pitch.
Now we see completely different behaviour because there is clearly a substantial transmission,
and we can even see the penetration of the evanescent field into the air.
To correctly solve the second wavelength, we need a method like FDTD.
FDTD is a general and versatile technique that can deal with many types of problems.
It can handle arbitrarily complex geometries and makes no assumptions about, for example,
the direction of light propagation.
It has no approximations other than the finite sized mesh and finite sized time step, therefore
it is very accurate.
As a time domain method, one simulation can give broadband results.
Finally, the FDTD algorithm scales well with parallelization, so it is well suited to modern,
multi-core and multi-processor computers as well as high performance computing (HPC) clusters.
Because FDTD is so versatile, it can address a wide range of applications.
These include photonic crystals, plasmonics, CMOS image sensors, nanoparticle scattering
and absorption, nano-patterned solar cells, OLEDS and LEDs, gratings, lithography, metamaterials
and integrated optics, to name just a few.
You will notice that all of these applications involve wavelength scaled structure.