Diploma thesis

Figure 3.5: 1D square-wave periodical poling period
We have to treat this varying nonlinear index by introducing spatial variations in
the previously constant χ (2) . The new nonlinear index is modelled as a square wave
function:
χ (2) (z) = χ (2) � � ��
2πz
sgn sin . (3.28)
Λ
The physical meaning of this function can be understood if we transform Formula
3.28 into a Fourier series:
χ (2) (z) = χ (2) �
∞�
�
1 2πm π
x+i
Re ei Λ 2 m = ±1, ±3, ±5 . . . (3.29)
m
m=−∞
.As in the previous discussion, we neglect the phase factor and the PDC Hamiltionian
is altered in respect to Eq. 3.23:
|ψs,i〉 = A
∞�
m=−∞
1
m
� ∞ � ∞
0
0
dωs dωie − (ωs+ωi−ωp)2 2σ2 e −γ
�
(kp−ks−ki − 2πm
Λ )L
2
� 2
â † s (ωs) â †
i (ωi) |0〉 .
(3.30)
Hence the grating period introduces an additional term in the momentum mismatch,
named quasiphasematching vector kQP M = 2πm
Λ . The momentum conservation
is modified:
∆k = kp − ks − ki − kQP M
(3.31)
The influence of a grating period is visualized in Figure 3.6. The phasematching
function gets shifted in the frequency plane. That way, periodic poling is a very
powerful tool to generate photon pairs at, in principle, arbitrary frequencies. This
has been applied, in the laboratory, to produce degenerate downconverted photon
pairs around 800 nm, where efficient detectors are available, or at the telecommunication
wavelength of 1550 nm. At this frequency, the transmission loss through
optical fibers is minimal.
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