Spatial Characterization Of Two-Photon States - GAP-Optique

y
y
x
z
z
pump
s
i
(a)
signal
idler
(c)
1.2. Two-photon state
Figure 1.2: (a) Phase matching conditions impose certain propagation directions for
the emitted photons at frequencies ωs and ωi. (b-c) This directions define two cones,
that collapse into one in the degenerate case. (d) In the collinear configuration, the
aperture of the cones tends to zero as the direction of emission is parallel to the pump
propagation direction.
in which the signal and idler are not parallel to the pump, and collinear in
which the aperture of the cones tends to zero as the photons propagate almost
parallel to the pump, as shown in figure 1.2 (d).
Even though the phase matching condition defines the main characteristics
of the generated two-photon state, other important factors influence the measured
state of the photons, for example the detection system. The next section
describes the two-photon state mathematically, taking into account all these
factors.
1.2 Two-photon state
When an electromagnetic wave propagates inside a medium, the electric field
acts over each particle (electrons, atoms, or molecules) displacing the positive
charges in the direction of the field and the negative charges in the opposite
direction. The resulting separation between positive and negative charges of
the material generates a global dipolar moment in each unit volume known as
polarization, and defined as
(b)
(d)
P = ɛ0(χ (1) E + χ (2) EE + χ (3) EEE + · · · ) (1.2)
where ɛ0 is the vacuum electric permittivity, and χ (n) are the electric susceptibility
tensors of order n [34, 35].
For small field amplitudes, as in linear optics, the polarization is approximately
linear,
P ≈ ɛ0χ (1) E. (1.3)
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