Identifying the Topological Charge of Optical Vortices Through Diffraction

This project studied the properties of optical vortices diffracted by hard apertures and how diffraction can be used to measure the topological charge of the vortex. Optical vortices are a type of transverse (spatial) laser mode that contains a phase singularity. Hermite-Gauss (HG) modes have rectangular symmetry while Laguerre-Gauss (LG) modes have cylindrical symmetry. It is well known [1] that cylindrical lenses can be used to convert HG laser modes to LG laser modes, and vice versa. The key feature of optical vortices, the singularity at the center of the beam, is a result of its azimuthal phase variation. In part because optical vortices carry orbital angular momentum (OAM), the study of their properties has become a rich field in physics in the last two decades with many applications, such as in optical tweezing and quantum computation.

In 2006 J.M Hickmann and collaborators at the Universidade Federal de Alagoas in Brazil [2] showed that if an optical vortex is incident on a triangular aperture, the resulting far field diffraction pattern is a triangular lattice of points that gives information on both the sign and absolute value of the topological charge in a very straightforward way. The topological charge of a vortex is an integer that says how many 2 π phase windings there are in one wavelength.

We started by recreating the work of Ref. [2] and continued by studying the diffraction of vortices by other types of apertures. The triangular and rectangular apertures were made in the Advanced Technology Lab (ATL) in the department by the EDM (Electrical Discharge Machining) method. In EDM a voltage is applied across two electrodes submersed in oil and the resulting discharge creates an aperture which matches the previosuly fabricated active electrode. The circular aperture was a commercial 1000 micron dia. pinhole. We worked with an open cavity HeNe laser and by introducing a student hair into the cavity were able to produce a wide variety of higher order HG modes. Using a standard two-cylinder-lens mode converter [1] we converted the HG modes into LG modes and tested these to determine how different topological charges affected the diffraction pattern for different apertures. Working with the triangular aperture and the set up previously described [2] we were able to reproduce the results of the Hickmann et al. experiment. We also attempted to see how other apertures (rectangular and circular), could be used to measure the topological charge of the vortex. With the rectangular aperture we saw a pattern that depended on the absolute value of the topological charge. Using a circular aperture an interesting pattern was seen that gave information on both the sign and absolute value of the charge. The exact nature of this pattern is still being studied, but we believe it has applications in easily measuring the topological charge of optical vortices and understanding their diffractive properties. Finally, we studied how higher topological charges affected the double slit interference pattern [3]. Although the pattern did depend on the value of the charge, this method's main usefulness is determining the sign of the charge.

This work was supported by the National Science Foundation (PHY-0851594). We would also like to thank Pete Davis (ATL) for his invaluable help in creating the apertures.

[1] M.W. Beijersbergen et al., Optics Communications 96 , 123-132 (1993)

[2] J.M. Hickmann et al., Phys. Rev. Lett. 105, 053904 (2010)

[3] H.I. Sztul et al., Optics Letters 31, Issue 7, 999-1001 (2006)