ABSTRACT for the April 2009 Celebration of Research and Creative Activity
Creating Higher-order Mode Laser Beams with an Open Cavity Laser
Justin Tian, Marty Cohen and John Noé
Laser Teaching Center
Department of Physics & Astronomy
Stony Brook University
Laser beams can only exist in certain specific spatial patterns called modes,
which are the discrete solutions of the paraxial wave equation that describes
light moving at small angles to some axis. Most lasers are designed to
produce only the zero-order or "fundamental" mode. It has cylindrical
symmetry and a cross-sectional intensity profile I(r) described by the
Gaussian ("bell curve") function exp(-2r^2/w^2). Higher order modes with more
complex patterns are also possible, as are superpositions of higher order
modes. In the higher order modes of the Hermite-Gaussian (HG) type the
Gaussian function of the fundamental mode is multiplied by two polynomial
functions that depend on x only and y only, respectively - Hm(x) and Hn(y).
These functions are labelled by two integers m and n, which correspond to the
degree of the polynomial. A mode with order m (or n) has m (n) nodes or dark
bands along the x (y) - direction. It can easily be seen that HG modes have
rectangular symmetry. A second complete family of higher-order modes, the
Laguerre Gaussian (LG) modes labelled by l and p, is the product of functions
of the polar coordinates r and phi and has circular symmetry. Since the phase
of an LG mode varies as exp(ilphi), for l > 0 there is a singularity (dark
spot) in the center of the beam. These singular LG modes have a twisted
wavefront and are often referred to as optical vortices.
The first goal of this project is to learn how to create stable and good
quality higher-order HG modes with an open-cavity helium neon (HeNe) laser.
(In such a laser the low-pressure gas mixture is separated from the
atmosphere by a Brewster window and there is an open space between this
window and the output coupler mirror which allows objects to be placed in the
intracavity beam of the laser.) The method used so far with our laser is to
insert a human hair (diameter about 75 microns) into the exposed laser beam.
While this method does work, the results are unpredictable and unreliable. We
have recently obtained more stable higher-order modes by inserting a glass
microscope slide with fine scratches into the cavity. The slide is tilted at
Brewster's angle to suppress reflections, and the scratches (created by
wiping the glass with sandpaper) are much smaller than the width of a hair,
10 microns or less. We plan to try very fine wires next and to improve the
positioning mechanism.
The relative phase between adjacent "spots" of an HG mode pattern is
pi radians (180 degrees). Recently we have been able to demonstrate this
experimentally using a Mach-Zehnder interferometer assembled for this
purpose. One arm of the interferometer has a beam expander which creates a
reference plane wave. As the interference fringes cross the two spots of an
HG(1,0) mode they shift by one-half the fringe separation, as expected.
Currently we are setting up a "mode converter" in the interferometer arm that
doesn't contain the beam expander, to transform HG modes into LG modes. The
interference pattern for a singular LG mode (l > 0) should contain a "fork"
or spirals, depending on adjustments of the interferometer.