Simulating the Interaction between Atom Clouds and LaguerreGaussian Laser Beams
Stefan Evans, John Noé, Harold Metcalf.
Laser Teaching Center, Stony Brook University, Stony Brook, NY 11794.
Abstract
Among certain atomic transitions is a class called electric quadrupole
transitions, which require properties of the light field that conflict with
those of ordinary laser beams. However, certain kinds of beams, including
Laguerre Gaussian (LG) beams possessing orbital angular momentum
(OAM), satisfy the needed conditions [1,5]. We simulate and experiment with
obstructions, aiming to understand how these beams interact with clusters
of atoms.
Introduction
OAM comes about from the unique helical wave front of these LG beams
[3]. The energy flow of the light (Poynting vector S) circulates around the
beam axis similar to the way water rotates in a whirlpool and thus has no
intensity in the center. The OAM is quantified by ℓ, the "topological charge":
the number of times the phase varies from 0 to 2π per wavelength.
Simulation
We use the MATLAB program to simulate the interation between LG beams and
large cluster of atoms with simple obstructions to see
the effect on angular momentum.
An LG beam electric field profile is written as
We plot this function onto a meshgrid where w is a portion of
total pixels and make a distribution of points where
E = 0, centered at specific r about the beam axis. In
this case the standard deviation is w = 5 and r = r_{max},
the LG beam's maximum intensity at
A proportionality between the Skew angle and OAM
state has been confirmed [3]. Here we Fourier Transform
the intensity profile and distribution, examining the
Guoy phase shift of the Swew angle, the angle between
the LG beam's Poyning vectors and the axis of propagation,
written as
where we consider only the φ component for a collimated
beam [4].
We now plot a distribution of zero intensity points and examine them
in the far field, corresponding to a Fourier Transform function in MATLAB.
Figure 1: Near and far fields for point distributions in
different locations. The π/2 Guoy phase acts on the Skew angle,
rotating the dim and bright areas of the beam.

When we cover up the majority of the beam profile, we
have a similar result, where the rotation in the clockwise
direction is determined by the local k vectors. Therefore
a small obstruction on one side effects the profile in the
same manner, depending how much intensity was blocked
out. The next step in this simulation is to analyze the
loss of OAM. Since the beams have "refilled" the obstructed
areas to some extent, we have a dimmer LG beam superimposed
with a mode of different or zero OAM, so the
two must be decomposed.
In the case of interaction with atoms, we know that
a_{o} << r_{max}, but also that the region of the beam that
interacts with the atom, as seen in the case of scattering,
is far lager than the size of the atom. Simulating an atom
will require a new approach.
Laguerre Gaussian Beam Generation
We generate an LG beam using a vortex phase plate
(VPP), where the thickness of the plate varies to give a
633 nm TEM 00 Gaussian beam a helical wavefront.
Figure 2: The plates range in charge from
ℓ = 1  8, where each square is a centimeter wide.

We can also remove the OAM by sending the beam
back through the VPP. The VPP imposes a flipped
charge from the opposite side. The purpose is to observe
the change in OAM of the final output when we introduce
obstructions into the LG beams's path, as demonstrated
in the simulations. We expect to have an Airy pattern
in the far field after removing helical phase from an LG
beam, due to the null intensity in the center.
Figure 3: Setup to make an ℓ = +4 and bring it back to zero.
We take advantage of the plates on the borders of the VPP,
which are all equal in charge, as shown in Figure 2.

Figure 4: Top: ℓ = +4 beam in the far field.
Diffraction caused
by steps in the VPP has been cleaned. Bottom: Far field of
the beam after second pass through the VPP, in agreement
with a Matlab simulation of an ℓ = +4 beam with helical
phase removed, transformed into the far field.

Further Investigation
We would like to look more closely into the patter we observe after
the second through the VPP. The ring gains a bright spot in its center when
we look in the far field, which tells us that we no longer have the same OAM
value, and that it is likely zero. Another interesting way to test the
effectiveness of the VPP, and to confirm that sending a beam in through opposite
directions yields opposite topological charges is to send it though a plate of
a different charge!
For example, sending the beam through an ℓ = +4 and back through an ℓ
= 3 plate should yield an ℓ = 1 LG beam, null in the center.
Now that we have imposed and removed helical phase
from the beam, the next step is to introduce obstructions,
where a rotation of the intensity profile in the far field
indicates a nonzero OAM.
Acknowledgements
We thank Giovanni Milione (CCNY) for assistance with the MATLAB simulations,
and RPC Photonics (Rochester) for providing the VPP633 vortex phase plate.
References
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Angular Momentum Between Twisted
Photons and Atomic Electrons, J. Opt. 15 (2013) 035403
[2] L. Allen, M.W. Beijersbergen, R.J.C. Spreeuw, J.P. Woerdman, Orbital
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[3] Leach, J., Keen, S., Padgett, M.J., Saunter, C., and Love,
G.D., Direct measurement of the skew angle of the Poynting vector in a
helically phased beam, Optics Express, 14.
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[4] M. J. Padgett and L. Allen, The Poynting Vector in
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[5] V. E. Lembessis and M. Babiker, Enhanced Quadrupole
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(2013)
