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