Page 290 - Organic Electronics in Sensors and Biotechnology

Organic Semiconductor Lasers as Integrated Light Sources for Optical Sensors 267

n 2
z
y
Λ n 1
x
FIGURE 7.1 One-dimensional slab waveguide structure with a periodic
variation of the refractive index.

(gratings) of the substrate. Figure 7.1 illustrates the distributed feed-
back mechanism in a slab waveguide structure with a one-dimensional,
periodic variation of the refractive index.
The lattice constant Λ of the grating transforms to the reciprocal
lattice constant G in k-space:

⎛ 0 ⎞
⎜ 2π ⎟
G = ⎜ ⎟
⎜ Λ ⎟
⎝ 0 ⎠

Lasing operation occurs in plane and can be discussed by starting
with a plane wave propagating in the slab with wave vector k:

⎛ ⎞
k
x
⎜ ⎟ 2π
k = k with | k | = n .
⎜ ⎟ eff λ
y
⎝ ⎠
0
The effective refractive index n represents the effective index of the
eff
propagating guided wave. For efficient feedback, which is essential
for laser operation, constructive interference of a scattered wave is
required. This is the case if the Bragg equation is fulfilled.
m
G
k xm = || m ∈ (7.1)
,
2
The positive integer factor m determines the scattering order. The scalar
k gives the smallest possible value of the wave vector component
x,m=1
(corresponding to largest wavelength) in the direction of the refrac-
tive index modulation for which constructive interference is possible.
In the special case of a wave traveling parallel to the reciprocal lattice
vector G, Eq. (7.1) transforms to the well-known equation
m λ
Λ=
2 n
eff