Page 291 - Organic Electronics in Sensors and Biotechnology

268 Chapter Seven

In general the constructive scattering of an incident wave with the
vector k occurs if the Laue condition is fulfilled:
i
k ⎛ xm ⎞
,
⋅
2kG =|| 2 with k = ⎜ k ⎟ (7.2)
G
i i ⎜ y ⎟
⎝ 0 ⎠

This formulation of the feedback condition states that Bragg scatter-
ing can occur if the incident photon k vector is located on a Bragg
plane, being the perpendicular bisector of a line connecting the origin
with the reciprocal lattice point G. A wave vector k which is located
i
on a Bragg plane is scattered to k and vice versa. The wave vector k
d i
of the incident and k of the scattered wave correlate in the form
d
k = k + G (7.3)
d i
Energy conservation results in the additional relation

|k | = |k | = const
i d
Figure 7.2a shows first-order scattering (m = 1) of a wave vector k. If
i
the incident wave vector k complies with the condition in Eq. (7.2), it
i
is scattered as stated in Eq. (7.3). Second order scattering (m = 2) is
depicted in Fig. 7.2b. In this case there are two possibilities for the scat-
tering process. The scattering condition (Bragg) determines only the
component k of the wave vector. Therefore a nonzero z component of
x
the wave vector results since energy conservation has to be met. This
means the incident wave is scattered out of the waveguide.
Second-order laser resonators are advantageous for applications
with free space optics due to their good light extraction efficiencies.

k d
G
G
k
k d i
k i k z
k y
π π 2π π π 2π k
– Λ Λ – Λ – Λ Λ Λ x

(a) First-order scattering (b) Second-order scattering
FIGURE 7.2 Feedback through scattering processes in a one-dimensional
Bragg slab waveguide.