Diploma thesis

For the volume integration, we restrict ourselves to a strictly collinear propagation
of the pump, signal and idler beams through the crystal (see Chapter 3.1.3 for
details). The integration over the crystal length yields the formula:
1
V
�
V
dV e i∆� k(ωp,ωs,ωi)�r → 1
L
� L
dze i∆k(ωp,ωs,ωi)z
�
∆kL
= sinc
2
0
Finally, we obtain the following description of the two-photon-state:
|Ψs,i〉 = 2πA ′ � ∞ � ∞
L dωs dωie − (ωs+ωi−ωp)2 2σ2 �
∆kL
sinc
2
0
0
�
∆kL
i
e 2 (3.20)
�
∆kz
i
e 2 â † s (ωs) â †
i (ωi) |0〉
(3.21)
For further discussion, we focus on the spectral properties of PDC. The phase contributions
are not considered in this thesis:
� ∞ � ∞
|ψs,i〉 = A dωs dωie − (ωs+ωi−ωp)2 2σ2 � �
∆kL
sinc â
2
† s (ωs) â †
i (ωi) |0〉 . (3.22)
0
0
Furthermore, we approximate the sinc contribution with a Gaussian distribution
(sinc(x) ≈ exp � −γx2� with γ = 0.193 . . .).
� ∞ � ∞
|ψs,i〉
y,z = A
0
0
dωs dωie − (ωs+ωi −ωp)2
2σ2 ∆kL
−γ(
e 2 )2
â † s (ωs) â †
i (ωi) |0〉 (3.23)
As one can readily verify the spectral contributions of the two-photon state are given
by two Gaussian functions. The first one,
α(ωs + ωi) = e − (ωs+ω i −ωp)2
2σ (3.24)
is named the pump envelope. It contains the pump parameters, pump central frequency
ωp and pump width σ. It implies the energy conservation condition. The
second Gaussian distribution,
� �
∆kL
φ(ωs, ωi) = sinc
2
∆kL
−γ(
≈ e 2 )2
(3.25)
is referred to as the phasematching function and ensures momentum conservation.
In SFG and PDC the same formula occurs (3.3 and 3.18) i.e., both processes share
the same phasematching.
Both pump envelope and phasematching function constitute the joint spectral
amplitude (JSA) of the PDC-created two-photon state:
f(ωs, ωi) = e − (ωs+ωi−ωp)2 ∆kL
−γ( 2σ e 2 )2
Finally, we write the two-photon state in a compact notation:
� ∞ � ∞
|ψs,i〉 = A
10
0
0
(3.26)
dωs dωif(ωs, ωi)â † s(ωs)â †
i (ωi) |0〉 . (3.27)