Spatial Characterization Of Two-Photon States - GAP-Optique

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