5 Photonic Crystals
5.1 Intruduction: Plane Waves in Crystals
A crystal lattice is a periodic arrangement of lattice points. If a plane wave hits the crystal, it scat-
ters from every point in it. The scattered waves typically travel as spherical waves away from the
points. There are certain angles where all of these reflections add constructively, hence they generate
a diffraction peak in this direction. In fig. 6 one point is defined as the origin. An incoming radio
Figure 6: Scattering amplitude.wave scatters from the origin and goes out into a direction where a detector is positioned. Another
ray scatters in some general positionrand then also travels out towards the detector. The beam that
goes through the origin has by definition phase 0. The beam that goes through the pointrtravels an
extra length(rcosθ+rcosθ′).
The phase of the beam scattered atris (^2 λπ(rcosθ+rcosθ′)) = (k−k′)r. So the phase factor
isexp(i(k−k′)r). If the phase difference is 2 πor 4 π, then the phase factor is 1 and the rays add
coherently. If the phase isπor 3 π, then the phase factor is− 1 and the rays add destructively.
Now consider a scattering amplitudeF.
F=∫
dV n(r) exp(
i(k−k′)r)
=∫
dV n(r) exp (−i∆kr) (35)The amplitude of the scattered wave atris assumed to be proportional to the electron densityn(r)
in this point. A high concentration of electrons inrleads to more scattering from there, if there are
no electrons inrthen there is no scattering from that point. In eq. (35) the 3rd term is rewritten
because of∆k=k′−k→k−k′=−∆k.