for

"Improvement of Laser Efficiency using Mode Selection Techniques"

A Full 1999 Summer Stipend

at

Eastern Oregon University

by

Anthony A. Tovar, Ph. D.,

Assistant Professor, Physics and Engineering Programs

**Work Completed:**

The basic proposal for the full summer stipend consisted of a laser resonator design that could be used to increase its efficiency. The increase in efficiency was to be achieved by designing a laser resonator to produce an intracavity beam mode which yielded a higher mode volume than the conventional Gaussian beam. The expected product of the work was a manuscript to be submitted to a major optics/lasers journal and a local presentation.

A manuscript related to this subject has been written
and submitted to the Journal of the Optical Society of America A [1]. However,
it appears that an additional manuscript will have to be completed to finish
the proposed project. Since the time of the proposal submission, Dr. Tovar
has been invited to be on the program committee for Photonics West 2000,
an international conference hosted by the SPIE. The conference will take
place in January of 2000 in San Jose, California. Dr. Tovar has also be
giving an invited speaker presentation at that conference. The title of
his presentation will be "Multi-Gaussian Beams - a Super-Gaussian Alternative,"
and will summarize the work done this summer during the period of the stipend.

[1] | A. A. Tovar, ,
Journal of the Optical Society of America A, , vol. 18, pp. 1897 - 1904
(2001).
Propagation of flat-topped multi-Gaussian beamsThe abstract is web published at http://physics.eou.edu/~atovar/research/pub17.html |

**Technical Summary:**

A top-hat laser beam profile is highly sought in
a wide variety of applications. An undesirable feature of the top-hat profile,
represented by a circ function, however is that it destabilizes as it propagates,
leading to undesirable nulls in the beam profile. An oft-used alternative
is the Gaussian-profiled beam which has the advantage of retaining its
smooth shape as it propagates through free space. The disadvantage of the
Gaussian shape is that the slow intensity drop off at the sides of the
beam often does not make it a good approximation to the desired top-hat
shape. For example, in laser resonator design the ability of a laser resonator
to extract energy from its intracavity amplifier is limited by the shape
of the beam profile. In the typical Gaussian mode resonator, the center
of the Gaussian beam saturates the amplifier and the gain at the sides
of the beam is wasted. If the beam mode had a circ shape, it would increase
the energy extraction efficiency for a short length, but would quickly
degenerate into a beam much worse than the Gaussian at extracting energy
from the amplifier.

A popular alternative to the Gaussian and the circ
profiles is the super-Gaussian profile. However, the evaluation of the
free space propagation of the field with this profile cannot be performed
in a closed form, and numerical techniques are required. This difficulty
would be alleviated if one could write the beam profile as a sum of Gaussian
beams, as the propagation characteristics of Gaussian beams are well known.
With this and acoustics applications in mind, Wen and Breazeale proposed
a beam profile that consists of a sum of superimposed complex Gaussian.
Their proposed beams have the following form:

They obtained the complex A_{k} and B_{k} coefficients
by a computer optimization. This superposition was also used as a model
for an aperture function to obtain the propagation characteristics of Bessel,
Gaussian, and similar beams through apertures.

Another alternate to super-Gaussian functions involves writing the
profile as a sum of polynomial-Gaussian functions of the form

These "flattened Gaussian beams" can be rewritten as a superposition of Laguerre-Gaussian beams, whose propagation characteristics are also well known.

In this study, a new class of beams, known as multi-Gaussian beams, is proposed which have the advantages of the super-Gaussian, while having diffraction characteristics which are analytically solvable. The multi-Gaussian beams consist of a small sum of finite-width Gaussian beams side-by-side each of which represents an intuitive component of the entire beam. Unlike the Wen and Breazeale beams, all of the Gaussians have the same width, phase curvature, and absolute phase. Unlike the flattened Gaussian beams, each of the multi-Gaussian beam components can be traced individually without resort to further series expansion.

The general formula for the shape of the proposed multi-Gaussian functions is

From the following figure, it can be easily seen how a pulse function
can be written in the above sum:

The above figure corresponds to a 2nd order beam. Higher order
beams are shown in the next figure:

Again, it can be easily seen that higher order multi-Gaussian beams approach a top-hat shape.

**Results:**

The propagation characteristics of multi-Gaussian beams through optical
systems represented by real or complex ABCD systems have been obtained.
The theory represents an important alternative to super-Gaussian beams
because of its analytical solvability. Because of the use of transfer
matrix techniques in the solutions, previous laser resonator theory may
be applied to the design of a laser resonator whose cavity mode is multi-Gaussian.
A laser with such a resonator would have increased energy extraction efficiency.
This efficiency is especially important in high power lasers, as extra
energy is converted to heat which destabilizes and under a variety of circumstances
destroys the laser.