Simulating the Interaction between Atom Clouds and Laguerre-Gaussian Laser Beams

Stefan Evans, John Noé, Harold Metcalf.

Laser Teaching Center,
Stony Brook University, Stony Brook, NY 11794.


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.


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.


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 = rmax, 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 ao << rmax, 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.


We thank Giovanni Milione (CCNY) for assistance with the MATLAB simulations, and RPC Photonics (Rochester) for providing the VPP-633 vortex phase plate.


[1] Basil S. Davis, L. Kaplan, and J.H. McGuire, On Exchange of Orbital 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 angular momentum of light and the transformation of Laguerre-Gaussian laser modes, Phys. Rev. A 45 (1992) 8185.

[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. pp. 11919-11924. ISSN 1094-4087 (2006).

[4] M. J. Padgett and L. Allen, The Poynting Vector in Laguerre-Gaussian laser modes, Opt. Commun. 121, 3640 (1995).

[5] V. E. Lembessis and M. Babiker, Enhanced Quadrupole Effects for Atoms in Optical Vortices, PRL 110, 083002 (2013)