Diploma thesis

By inserting the corresponding terms into the two-photon-state we obtain:
� ∞ � ∞ � b � b
|Ψs,i〉 = A dωs dωi dx dyEp(x, y)Es(x, y)Ei(x, y)
0
0
0
0
e − (ωs+ωi−ωp)2 2σ2 ∆kL
−γ(
e 2 )2
â † s (ωs) â †
i (ωi) |0〉 . (3.33)
The waveguiding structures in our laboratory are tooth shaped as shown in Fig.
3.16. To calculate the modal overlap we approximate this waveguide as a twodimensional
rectangular structure of width and height b, with perfect conducting
edges, as depicted in Fig. 3.17.
Figure 3.16: Real waveguide (50x) Figure 3.17: Assumed 2D waveguide
The transversal modes for this scenario are:
�
nπ
Eµ(x, y) = Aµsin
b x
� �
mπ
sin
b y
�
n, m ∈ N\{0}. (3.34)
Equation 3.33 is separable, i.e. E(x, y) = E(x)E(y), and leads to two independent
integrals.
� b � b
dx dyEp(x, y)Es(x, y)Ei(x, y)
0
0
� L
� L
= dxEp(x)Es(x)Ei(x) dyEp(y)Es(y)Ei(y) (3.35)
0
We solve this one dimensional integration as follows:
� b �
nπ
dx sin
0 b x
� �
mπ
sin
b x
� �
oπ
sin
b x
�
= b
4π (
1
1
cos(π(n + m + o)) +
cos(π(−n + m − o))
n + m + o −n + m − o
1
1
+ cos(π(n − m − o)) +
cos(π(−n − m + o))) (3.36)
n − m − o −n − m + o
16
0