In order to achieve optimal power transitivity in a Fabry-Perot interferometer, it is necessary to match the initial laser waist (from the laser source) to the natural waist of the interferometer cavity. It is, often, essential to incorporate an optics package, usually a lens or a series of lenses to refocus the laser beam to correspond with the natural waist of the cavity.

Several telescope designs consisting of two lenses were developed and experimentally tested to refocus and match an air cooled Argon ion laser beam to the natural waist size and location of a 50m Fabry-Perot interferometer. This paper will discuss these investigations as well as describe the characteristics of the laser, Fabry-Perot interferometer, the theory of mode matching, and the individual influence of these components on telescope design.

This experiment explores the theory and application of mode matching. To gain a better understanding of mode matching a fundamental knowledge of optical cavities, interferometers, lasers, and telescopes is essential. This background information will serve as a strong foundation on which a greater appreciation of optical systems can be built. This paper provides these building blocks encouraging intelligent analysis of the experimental investigations of telescope design and fabrication within this experiment.

The first is a plane parallel cavity consisting of two flat plane mirrors with the radius of curvature of infinity. Although this cavity is relatively easy to construct, it is extremely sensitive to alignment. After a few passes through the cavity, the beam "walks off " one of the mirror surfaces. Thus creating an unstable cavity. Refer to figure 1.

Stability is based on the cavity's ability to contain the beam. To ensure that a plane parallel cavity is stable the beam entering the cavity must be parallel. However, all laser beams experience some kind of divergence as they propagate - making alignment in this cavity extremely difficult.

The second cavity is a concave concave cavity comprised of two concave mirrors. Beam alignment in this cavity is much easier to expedite because this cavity has a focusing characteristic. The beam is focused during each pass. Refer to figure 2. However, concave mirrors do not necessarily guarantee a stable cavity. A stable cavity is also dependent on the dimensions of the cavity and the mirrors' radius of curvature. One condition that warrants a stable cavity is when the radius of curvature is greater than half the mirror separation.

Interferometers are optical instruments employed to study light characteristics. Exploiting laser coherence, interferometers have become a great asset in spectroscopy - particularly the Fabry-Perot interferometer.

The Fabry-Perot interferometer used in this experiment is identical to the concave concave optical cavity. The mirrors are separated by a distance of 50 meters. The radius of curvature of these mirrors is 34 meters - satisfying the above stability condition. Light inside the interferometer propagates similar to a standing wave. Refer figure 3.

Starting from the left of figure 3, the beam fills the left end mirror, pinches down to shallow point, and rises up to fill the right end mirror. The beam wavefront at the end of the cavity matches the mirrors' radius of curvature.

Figure 3. Concave Concave Optical Cavity.

The beam wavefront in the middle of cavity is planar and the beam is at its smallest size. This position is referred to as the cavity waist. The waist is the radius of the diameter enclosing 1/e² of the beam intensity. The spot size is the radius of 1/e² of the beam intensity at any other position along the beam. The cavity waist is a function of the mirrors' radius of curvature and the mirror separation. It can be expressed as:

The beam spot size is a function of z, the distance from the waist to any position on the beam. It is expressed as:

The radius of curvature at the beam wavefronts can be determined by the following equation:

The waist that occurs naturally in a cavity at a certain wavelength is the natural cavity waist. The natural waist of this Fabry-Perot cavity is 1.565 mm. By placing a planar mirror in the location of the cavity waist, a plane concave cavity is constructed. Refer to figure 4.

This type of cavity is commonly employed in laser manufacturing.

A laser is a optical oscillator consisting of an amplifying medium inside a optical cavity. Establishing a population inversion via electron excitation (pumping), spontaneous emission occurs. As the electrons drop to lower energy levels, photons are emitted. These photons travel between the cavity mirrors stimulating the emission of identical photons.

The laser used in this experiment is an air cooled Ar ion laser. The amplifying medium is singly ionized argon. The cavity of this laser is plane concave. Consequently, the radius of the beam at the plane mirror is equivalent to the laser beam waist. The laser waist is .305mm with a divergence of 1.02mrad (Omnichrome, 2). Beam divergence is the angle at which the beam transversely spreads. This is inversely related to the waist. It is expressed as:

In a stable laser cavity a set of TEM (transverse electric and magnetic) modes are naturally generated. Refer to figure 5. TEM modes are intensity patterns that can be views, if looking at the cross section of a beam. These modes are comparable to the natural modes of a standing wave. By adjusting the cavity length certain modes can be isolated.

The Ar laser lases in TEM₀₀, a cylindrically symmetric mode (Hecht,583).

Figure 5. TEM Modes

The experimental configuration consists of the Ar ion laser, two steering mirrors, a telescope, and the 50 meter Fabry-Perot interferometer. The laser, two mirrors, and telescope are located on a 4'x 8' optical table. The Fabry Perot interferometer is placed some distance from the table. Refer to figure 6. Placing the telescope between the mirrors allowed for more translation freedom of the telescope.

As with any system - mechanical, electric or optical - attaining maximum power efficiency is paramount. These various systems require different methods to ensure that optimal power is acquired. In an optical system comprising of a laser and a Fabry-Perot interferometer the technique employed is mode matching. Mode matching is the process of modifying the initial laser waist to correspond with the natural laser waist of the cavity. The instrument utilized in this procedure is a Galiean telescope.

A Galiean telescope is implemented into the experimental configuration to modify and match the initial Ar waist to the natural cavity waist. A Galiean telescope consists of a concave lens and a convex lens. This typical design is a varifocal system, where the effective focal length of the telescope can be varied by adjusting the distance between the lenses (Levi, 489). The focal lengths and the location of the telescope are restricted by several experimental parameters - the lens size, the available lens selection, the location of the cavity waist, and the optical table dimensions. Several methods can be employed to determine the telescope design. In this experiment, ray matrix, computer ray tracing and basic optical formulas were applied.

Ray matrix is the description of ray propagation through various medium and optical elements via matrices. These calculations begin with two equations that describe ray travel . These equations

relate the output ray characteristic with the ray input. These equations describe the position and the slope of the ray, respectively. They can be expressed by the following 2x2 matrix:

The general expression of these equations are

The general ABCD matrix can be derived to describe ray propagation through individual components (Verdeyen, 36). Refer to figure 7.

Figure 7. Examples of various ray matrices (Yariv, 37).

A ray matrix is required for each medium and optical element the ray travels through. Application of the thin lens matrix is required in addition to the length matrix. The lenses utilized were thin lenses of 1' diameter. This particular experiment requires five ray matrices - three medium and two lens matrices. Once the appropriate series of ray matrices has been selected, it is necessary to combine these individual matrices into one matrix, the transmission matrix. The transmission matrix provides a complete description of the total ray propagation. By applying the ABCD law it's possible to calculate the system unknowns. Due to the limited lens selection, the focal lengths of the lenses and the distance of the cavity waist from the second lens were designated as the knowns. The lens separation, d, and the distance from the laser source to the first lens, d1 were the variables. These calculations were manipulated by a program generated on Mathematica. Mathematica was able to calculate all solutions of a specified lens combinations -real and complex. The feasible solutions were extracted and tested experimentally. However due to ,what is believed to be, a program glitch, certain combinations result in enormous values for d1. But through extrapolation, these values could be determined. The solution set of each combination was extremely small and any deviation from this set resulted in a pole.

Computer ray tracing is the standard ray tracing performed by a computer program. Through the application of two computer programs, Paraxia and Beam Four solution sets were mathematically determined and tested. Paraxia and Beam Four enables the user to stimulate a specified experimental set-up. Paraxia, an user friendly program based on guassian optics, animates beam propagation accounting for the waist, wavelength, and beam divergence associated with the wavelength. Even though Paraxia's computer animation provided excellent mathematical results, the reproducibility of these solutions in the lab were difficult. Beam Four requires comprehension of the program syntax to stimulate ray propagation. Beam Four was applied late in the telescope design development to understand the Paraxia inconsistencies. Although Beam Four generated reproducible results, an experimental time constraint prevented the collection of substantial data on its performance.

Thin lens formulas were also applied to determine telescope parameters. These fundamental formulas are expressed as :

They can be derived from the basic thin lens combination formulas. Due to the beam divergence negligence, these equations served as solution filters - sifting out unattainable solutions. The following table displays the calculated solutions from all techniques and the experimental results.

Although the spot size produced experimentally was not the desired result, many objectives were met. Impractical solutions were filtered out theoretically and experimentally. The relationship between the project limitations and the results were analyzed. Through this analysis, better adapted telescope designs can be fabricated to be perform more efficiently within these limitations.

Several non -experimental factors also influenced the experimental outcome. Due to the lack of x and y translation of the optical elements the alignment process was crude and at time inefficient. The spot size measurement was done by eye. The eye becomes saturated and visualizes a spot size bigger than 1/e². Air scattering, beam divergence due to temperature gradients and particles in the air, had a small influence.

If these factors were controlled, the probability of producing the desired waist would be high. With the x and y translation mounts incorporated into the system beam alignment is more precise, resulting in a good symmetric beam spot. Measuring the spot size with a power meter would result in a more accurate assessment of the beam spot size. In a controlled environment, a temperature regulated clean room, air scattering can be reduced to a minimum. With these system upgrades the desired results are attainable.

First I would like to extend my gratitude to Dr. Davenport, Diane Ingram, Elliot M^cCroy and the rest of the SIST committee. I greatly appreciate this opportunity you have given me.

To my supervisor, Frank Nezrick I extend the deepest and most heartfelt thank you. I could not have asked for a better supervisor. I greatly appreciated the patience, hospitality, and knowledge you extended to me. You are great resource. Finally I would like to give a special thanks to the Nezrick crew.

Hecht, Eugene. Optics. Volume I. California : Addison Wesley Publishing, 589. 1987.

Levi, Leo. Applied Optics Aguide to Optical System Design. Vol. I. New York : John Wiley & Son, 489. 1968.

Omnichrome Instruction Manual. California : Omnichrome, 2. 1988.

Verdeyen, Joesph. Laser Electronics. New Jersey : Prentice Hall, 36. 1995.

Yariv, A. Optical Electronics. New York : Oxford University Press, 37. 1991.