VISUALIZING THE GOUY PHASE OF A LASER BEAM

Thomas E. Videbaek, Marty Cohen and John No� 
Laser Teaching Center, Department of Physics & Astronomy, Stony Brook University

The propagation of laser beams is described by a precise theory that specifies
the amplitude, curvature and phase of the evolving wavefronts as a function of
radius r and distance z.  The simplest such beams have a radially-symmetric
Gaussian intensity profile I(r) which remains Gaussian-shaped as the beam
propagates through space or radially-symmetric optical elements. The changing
beam radius w(z) follows a hyperbola, and the point of minimum radius w0 at a
"focus" is called the waist.

An interesting and easily over-looked feature of the theory is the Gouy phase
[1], a small correction to the on-axis wavefront phase compared to a reference
plane wave; it varies from -pi/2 to +pi/2 as a beam moves through a waist. The
Gouy phase can be visualized using a Mach-Zehnder interferometer setup by
placing a suitable lens in one arm. Peatross and Pack [2] have described an
alternative method that utilizes the "ghost beam" created by internal
reflections in an uncoated plano-convex (PC) lens. When the PC lens is used to
collimate diverging laser light the weak ghost beam forms a compact waist
within the broad main beam and the resulting ring-shaped interference patterns
can be viewed with a camera. The intensity pattern inverts as the camera is
moved through the waist as a result of the changing Gouy phase.

In this project we investigated the ghost-beam method and compared it to the
classic interferometer setup.  We found that the former method has numerous 
advantages, including several not mentioned in Ref. [2]. We also found that
the secondary lens used [2] to control the divergence of the beam incident on
the primary PC lens is unnecessary. It is sufficient to pick a primary lens
that matches the intrinsic divergence of the laser beam, which is easily
determined by beam profile measurements at one or more distances from the
laser. Our recorded interference patterns are in generally good agreement
with a model of the interference process that we created in Mathematica.
Finally, we were able to derive some interesting mathematical relationships
relevant to our simplified ghost-beam method.

This work was supported by the National Science Foundation (Phy-0851594).

[1] R.W. Boyd, J. Opt. Soc. Am. 70, 877-880 (1980).

[2] J. Peatross and M.V. Pack, Am. J. Phys. 69, 1169-117 (2001).