Figure 10: Dispersion relation for photons in a material.of the matrix andω^2 are the eigenvalues of the matrix. We just need to choose numbers for them
and find the eigenvalues. Solving this over and over for different values ofkwe get a series ofωthat
solves the problem and gives us then the entire dispersion relationship.
So take some value ofk, diagonalize the matrix, find the eigenvalues (ω^2 ), take the square root of these
three values and plot them in fig. 10. Then the next step is to increase the value ofka little bit and
do the calculation again, so we have the next three points. This has to be done for all the points be-
tween zero and the brillouin zone boundary (which is half of the way to a first pointGout of the origin).
At low values forkwe get a linear dispersion relation, so the slope is the average value of the speed of
light of the material. But when we get close to the brillouin zone boundary, the dispersion relationship
bends over and then there opens a gap. So light with a wavelength close to that bragg condition will
just get reflected out again (this is called aBragg reflector).
After that the dispersion relationship goes on to the left side and then there is another gap (al-
though it is harder to see). Fig. 10 only shows the first three bands. To calculate higher bands, you
need to include more k values.
5.4 Estimate the Size of the Photonic Bandgap
Sometimes we do not need to know exactly what the dispersion relationship looks like, we just want to
get an idea how big the band gap is. Then the way to calculate is to reduce the matrix that needs to be