Diploma thesis

3.3.3 Counterpropagating signal and idler fields
Until now we only considered generating decorrelated two-photon-states with strictly
copropagating signal and idler fields. In bulk crystal on the other hand, the use of
angular phasematching is a common and widespread method.
Counterpropagating photon pairs in waveguides have already been proposed in
2002 [23]. In this scenario, the created photon pairs are separated inside the crystal
and leave the waveguide at opposing ends (see Figure 3.42).
Figure 3.42: Counterpropagation in waveguided parametric downconversion,
energy conservation and momentum conservation
Describing the impact on the spectral correlations is straightforward. The pump
envelope α(ωs, ωi) remains unaltered, whereas in the phasematching function the
momentum mismatch is altered (Equation 3.31) as follows:
∆k(ωs, ωi) = kp(ωs + ωi) − ks(ωs)+ki(ωi) − 2πm
. (3.57)
Λ
The momentum vectors of signal and idler are opposite to each other. Thus the
crystal has to take a large amount of the pump momentum, hence very small grating
periods are needed.
Accordingly, the counterpropagation directly affects the decorrelation condition
3.49 which is changed to
2
σ 2 + γL2 (k ′ p − k ′ s)(k ′ p+k ′ i) = 0. (3.58)
This simple sign change has an immense effect on the photon creation. Its impact
can be understood by considering the impacts on the formula for the phasematching
angle, which is now given by:
Θ = − arctan
� k ′ p − k ′ s
k ′ p+k ′ i
�
. (3.59)
In most materials, the denominator will be much larger than the numerator. This
results in a phasematching contour oriented along the ωs-coordinate axis, independently
from the applied material. In addition, the sign change leads to a narrow
shaped contour.
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