Simons Program Abstract
Diffraction with a Twist: Creating Optical
Vortices with a Spiral Fresnel Zone Plate
Nityan Nair, Hastings High School, Hastings, New York
John Noé, Laser Teaching Center, Stony Brook University
Optical vortices -- beams of light with helical wavefronts that resemble a
corkscrew -- have been extensively studied in recent years. Their
wavefront twist is caused by the beam's azimuthally- varying phase,
described by the complex phase factor exp(ilφ), where the topological
charge l is a property of the vortex and φ is the azimuthal angle with
respect to the beam direction. The indefinite value of φ at the center
of the beam is a phase singularity which forces the beam intensity on
axis to zero. Thus optical vortices appear as a doughnut-shaped spot of
light with a bright ring around a dark core.
Optical vortices have many current and potential applications. For
example, they are used in optical tweezers to more effectively trap
particles or to exert a torque on them. In the future they are expected to
be very useful for quantum computing on account of their ability to
represent more than two logical states.
Optical vortices are typically created using a forked diffraction grating
or a spiral phase plate, which are often implemented in a programmable
spatial light modulator (SLM). These methods can be quite expensive, and
the forked grating is relatively inefficient. In this project, an
alternative method based on diffraction by a spiral zone plate (SZP) has
been investigated. The SZP resembles the more familiar Fresnel zone plate
(FZP), which is of great importance for x-ray microscopy on account of its
ability to focus light without refraction. A FZP consists of a sequence of
annular zones which are alternately transparent and opaque, creating a
"bulls-eye" pattern. The zone radii gradually decrease in such a way that
light passing through the transparent rings is focused by constructive
interference at a sequence of distant focal spots. The SZP is similar,
except that the zone radii vary with azimuthal angle, making the zones
appear like one or more interlocked spiral arms.
Spiral zone plates corresponding to various topological charges were
effectively and inexpensively produced by photographing spiral graphs
calculated and printed out using Mathematica and Photoshop. Each 35 mm
black-and-white film negative was illuminated by a spatially-filtered
laser beam to create the corresponding optical vortex at the primary focal
point of the zone plate. A convex lens was placed near the focal point to
"freeze" the vortex (allow it to propagate unchanged). The optical
vortices produced in this manner were tested in a Mach-Zehnder
interferometer constructed for this purpose, and the characteristic
"forked" or spiral interference patterns produced confirmed the vortex
nature of the beams. Vortices produced included some with fractional
topological charge, and the properties of these mathematically interesting
objects [1] will be one focus of our future research.
We would like to thank the Simons Foundation for funding this research, and
Prof. Harold Metcalf for establishing and supporting the Laser Teaching Center.
[1] M. V. Berry, "Optical vortices evolving from helicoidal integer and
fractional phase steps," J. Optics A 6, 259 - 268 (2004).
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