Spatial Characterization Of Two-Photon States - GAP-Optique
Spatial Characterization Of Two-Photon States - GAP-Optique
Spatial Characterization Of Two-Photon States - GAP-Optique
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x<br />
1.3. Approximations and other considerations<br />
pump<br />
signal<br />
beam s<br />
y z xi<br />
idler<br />
z<br />
Figure 1.3: The propagation direction of the pump beam defines the z direction of<br />
the general coordinate system. The generated pair of photons propagates with angles<br />
ϕs and ϕi with respect to the pump beam in the yz plane. Each photon coordinate<br />
system transforms to the general coordinate system through the relations in equation<br />
1.23.<br />
For the special case of a Gaussian pump beam with lp = 0 the mode function<br />
becomes<br />
τ <br />
Φ(qs, ωs, qi, ωi) ∝ dt dV<br />
0 V<br />
i<br />
y<br />
dq p exp [− w2 p<br />
4 q2 p − T 2 0<br />
4 ω2 p]<br />
× exp [ik z pzp − ik z szs − ik z i zi + iq p · r ⊥ p − iq s · r ⊥ s − iq i · r ⊥ i ]<br />
× exp [−i(ωp − ωs − ωi)t], (1.21)<br />
which assumes that the interaction time τ is longer than the spontaneous emission<br />
life time of the material.<br />
Frequency bandwidth<br />
Fields generated experimentally always contain a distribution of frequencies,<br />
and totally monochromatic fields only exist in theory. This thesis considers the<br />
frequency ω of each field as the sum of a constant central frequency ω 0 , and a<br />
small deviation from that frequency Ω, so that ω = ω 0 + Ω.<br />
As a result of integrating over the interaction time taking into account the<br />
conservation of energy, the mode function becomes<br />
<br />
Φ(qs, Ωs, qi, Ωi) ∝ dV<br />
V<br />
Coordinate transformation<br />
dq p exp [− w2 p<br />
4 q2 p − T 2 0<br />
4 (Ωs + Ωi) 2 ]<br />
× exp [ik z pzp − ik z szs − ik z i zi + iq p · r ⊥ p − iq s · r ⊥ s − iq i · r ⊥ i ].<br />
(1.22)<br />
The mode function depends on three position vectors, rp, rs, and ri. Each of<br />
them is defined in a coordinate system with the z axis parallel to the propagation<br />
direction of each field, as shown in figure 1.3. Defining all position vectors<br />
in the same coordinate system simplifies the integration of the mode function<br />
over volume.<br />
i<br />
i<br />
xs<br />
ys<br />
zs<br />
7