Spatial Characterization Of Two-Photon States - GAP-Optique

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