Spatial Characterization Of Two-Photon States - GAP-Optique

1. General description of two-photon states
virtual state
pump
ground state
signal
idler
Figure 1.1: In spontaneous parametric down conversion the interaction of a molecule
with the pump results in the annihilation of the pump photon and the creation of
two new photons.
1.1 Spontaneous parametric down-conversion
Spontaneous parametric down-conversion is one of the possible resulting processes
of the interaction of a pump photon and a molecule, in the presence of
the vacuum field. In spdc, the pump-molecule interaction leads to the generation
of a single physical system composed of two photons: signal and idler, as
figure 1.1 shows.
The name of the process reveals some of its characteristics. spdc is a
parametric process since the incident energy totally transfers to the generated
photons, and not to the molecule. And, it is a down-conversion process since
each of the generated photons has a lower energy than the incident photon.
The conservation of energy and momentum establish the relations between the
frequencies (ωp,s,i) and wave vectors (kp,s,i) of the photons,
ωp = ωs + ωi
kp = ks + ki. (1.1)
where the subscripts p, s, i stand for pump, signal and idler. Macroscopically,
spdc results from the interaction of a field with a nonlinear medium. Commonly
used mediums include uniaxial birefringent crystals. The conditions in
equation 1.1 can be achieved for this kind of crystals, since their refractive index
changes with the frequency and the polarization of an incident field [34, 35].
This thesis considers type-i configurations in uniaxial birefringent crystals.
In such configuration the pump beam polarization is extraordinary since it is
contained in the principal plane, defined by the crystal axis and the wave vector
of the incident field. The polarization of the generated photons is ordinary,
which means that it is orthogonal to the principal plane.
For a given material, the conditions imposed by equation 1.1 determine the
characteristics of the generated pair. The second line in equation 1.1 defines the
spdc geometrical configuration, and it is known as a phase matching condition.
The generated photons are emitted in two cones coaxial to the pump, as shown
in figure 1.2 (b). The apertures of the cone are given by the emission angles ϕs,i
associated to the frequencies ωs,i. Once a single frequency or emission angle is
selected, only one of all possible cones is considered. In degenerate spdc, both
photons are emitted at the same angle ϕs = ϕi and the cones overlap, as in
figure 1.2 (c). There are two kinds of degenerate spdc processes: noncollinear
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