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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3. <strong>Spatial</strong> correlations and OAM transfer<br />
-1<br />
0 1 2<br />
OAM content<br />
in ħ units<br />
Phase front<br />
Intensity<br />
profile<br />
Figure 3.1: The Laguerre-Gaussian modes are characterized by their phase front<br />
distribution and intensity profile. The figure shows the phase fronts and the intensity<br />
profiles for the modes with oam contents from −1 to 2.<br />
3.1 Laguerre-Gaussian modes and OAM content<br />
In an analogous way to the decomposition of an electromagnetic field as a<br />
series of planes waves, it is possible to decompose the field in other bases. For<br />
instance, paraxial fields can be decomposed as a sum of Laguerre-Gaussian (lg)<br />
modes. This basis is especially convenient since the lg modes are eigenstates of<br />
the orbital angular momentum (oam) operator. That is, the state of a photon<br />
in a lg mode has a well defined oam value [51]. This section describes the<br />
properties of the lg modes, and shows how to calculate the weight of each<br />
mode in a given decomposition.<br />
<strong>Photon</strong>s carrying oam different from zero have at least one phase singularity<br />
(or vortex) in the electromagnetic field, a region in the wavefront where the<br />
intensity vanishes. This is precisely one of the most distinctive characteristics<br />
of Lagurre-Gaussian modes as figure 3.1 shows. Each lg mode is defined by<br />
the number p of non-axial vortices, and the number l of 2π-phase shifts along<br />
a close path around the beam center. The index l also describes the helical<br />
structure of the phase front around the singularity, and more important here,<br />
l determines the orbital angular momentum carried by the photon in units.<br />
The state of a single photon in a lg mode is<br />
<br />
|lp〉 = dqLGlp(q)â † (q)|0〉, (3.1)<br />
where the mode function lglp(q) is given by Laguerre-Gaussian polynomials<br />
<br />
1<br />
2 wp!<br />
LGlp(q) =<br />
2π(|l| + p)!<br />
|l|<br />
wq<br />
√2 L |l|<br />
2 2 w q<br />
p<br />
2<br />
<br />
× exp − w2q2 <br />
exp ilθ + iπ(p −<br />
4<br />
|l|<br />
2 )<br />
<br />
(3.2)<br />
as a function of the beam waist w, the modulus q and the phase θ of the<br />
transversal vector, and the associated Laguerre polynomials L |l|<br />
p defined as<br />
30<br />
L |l|<br />
p [x] =<br />
p<br />
i=0<br />
i<br />
l + p (−x)<br />
. (3.3)<br />
p − i i!