Transverse Laser ModesVictor Wang, Great Neck South High School, Great Neck, NY; Harold Metcalf and John Noé, Laser Teaching Center, Department of Physics and Astronomy, Stony Brook University.
It's well known that an oscillating mechanical system such as a violin string or drum head has certain natural frequencies, or modes, at which the motion tends to occur. More general motions are described as superpositions of these pure modes with specific amplitudes and phases. Laser beams also have modes, and these can be similarly described with elegant mathematical equations. The laser modes are created in the resonant cavity defined by the laser mirrors and propagate through space with the beam. The transverse modes that are the specific topic of interest can be seen as the intensity pattern in the plane perpendicular to the direction of propagation. Hermite-Gaussian (HG) modes appear as a rectangular pattern of spots, while Laguerre-Gaussian (LG) modes have a circular intensity pattern. The LG modes are of particular interest because they are related to optical vortices, which are noted not only for their intrinsic beauty but also have important applications in optical tweezers and elsewhere.
Many lasers such as the familiar He-Ne type produce only the simplest transverse mode, which has a Gaussian (bell curve) intensity distribution. In the initial phase of the project, the profile of such a Gaussian laser beam was measured as a function of the distance z past a converging lens. Two separate methods were used and gave comparable results: scanning with a 100 μm diameter pinhole and scanning with the sharp edge of a razor blade. The laser beam, initially 1.2 mm in diameter, reached a minimum diameter of 165 μm 15 cm from the lens. The measured profile versus z curve is in very good agreement with a simple one-parameter Gaussian optics calculation.
In the next phase of the project, "exotic" higher-order HG modes were created using an open cavity helium-neon laser. The open cavity design allows one mirror to be tilted and small objects such as hairs (which force a node to occur along a line in space) to be placed in the beam. Very careful tuning of these elements is needed to generate pure modes of a specific order. In the future we plan to use a stressed optical fiber to convert the HG modes into corresponding LG modes, as suggested in a paper that David McGloin recently brought to our attention. The purity of the LG modes thus obtained will be tested in a Mach-Zehnder interferometer setup.
We would like to thank the Simons Foundation for its support and Sam Goldwasser for assistance with the open cavity laser.
 David McGloin, Neil B. Simpson and Miles J. Padgett, Applied Optics 37, page 469 (1998).