y? Diagram, a powerful optical design method for laser systems

We present a new method for synthesizing and analyzing laser systems based on the use of the y? diagram. The diagram is commonly used to represent two rays, the marginal and the chief rays, as they propagate through an optical system. Since a Gaussian beam can be represented by two rays, it is possible to use the y? diagram to represent these rays. This results in a representation of the beam as a single ray line on the y? diagram with simple graphic interpretations for the beam parameters. An equivalent representation of the Gaussian beam on the uū diagram is also presented and discussed. Complex design problems may be reduced to simple graphic problems, which often lead directly to algebraic solutions. Examples of y? diagrams are given for beam transfer through simple optical systems, including gradient optics. Diagram transformations are discussed and design examples are given of a three-element afocal system and a three-element collimator.