Spatial Characterization Of Two-Photon States - GAP-Optique

1. General description of two-photon states
The semiclassical approximation
As a consequence of the low efficiency of the nonlinear process, the incident
field is several orders of magnitude stronger than the generated fields. In that
case, it is possible to consider the pump as a classical field, while the signal
and idler are considered as quantum fields, this is known as the semiclassical
approximation.
By defining the pump as a classical field, with a spatial amplitude distribution
Ep(q p) and a spectral distribution Fp(ωp):
Ep(rp, t) ∝ dqp dωpEp(qp)Fp(ωp) exp [iqp · r ⊥ p + ik z pzp − iωpt], (1.16)
the mode function reduces to
τ
Φ(qs, ωs, qi, ωi) ∝ dt dV dqp dωpEp(qp)Fp(ωp) Gaussian beam approximation
0 V
× exp [ik z pzp − ik z szs − ik z i zi + iqp · r ⊥ p − iqs · r ⊥ s − iqi · r ⊥ i ]
× exp [−i(ωp − ωs − ωi)t]. (1.17)
According to equation 1.17, the mode function depends on the spatial and
temporal profiles of the pump beam. To write these profiles explicitly we
assume a pump beam with a Gaussian temporal distribution,
Fp(ωp) = exp − T 2 0
4 ω2
p
(1.18)
where T0 is the pulse duration (standard deviation), which tends to infinity for
continuous wave beams. Also, we assume that the pump beam is an optical
vortex with orbital angular momentum lp per photon. As will be described
in section 3.1, under these conditions the pump spatial profile is given by the
Laguerre-Gaussian polynomials
1
2
wp
Ep(qp) =
2π|lp|!
|lp|
−iwp
√ qp exp −
2 w2 p
4 q2
p + ilpθp
(1.19)
that are functions of the pump beam waist wp, and the transversal vector
magnitude qp and phase θp, given by
6
qp =
θp = tan −1
(q x p ) 2 + (q y p) 2
q y p
q x p
.
(1.20)