Publisher Summary This chapter reviews Gaussian brackets defined on the basis of the theory of continued fractions, and summarizes the paraxial theory formulated with these Gaussian brackets for both homogeneous and inhomogeneous optical systems and also for the Gaussian beam optical system. Some examples of the application of the Gaussian brackets formulation to the analysis and synthesis of the optical system are also presented. A summary of one of the useful methods for the analysis or synthesis of an optical system in lens design is presented. The method is based on the concept named “Gaussian brackets”. Gaussian brackets are derived as the denominator of the nth convergent of a continued fraction, whose every partial numerator is equal to unity. The Generalized Gaussian Constants (GGC's) are written with the Gaussian brackets, whose elements consist of constitutional parameters of an optical system. The GGC's have a clear physical meaning, and are useful to formulate paraxial theory. The chapter explores that the Gaussian brackets' formulation can be applied not only to other types of optical systems such as a decentered optical system, but also to the aberration theory.

II Paraxial Theory in Optical Design in Terms of Gaussian Brackets

We present a new method for synthesizing and analyzing laser systems based on the use of they
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
y¯ diagram to represent these rays. This results in a representation of the beam as a single ray line on the y
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
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 a focal system and a three-element collimator.

yy¯ Diagram, a powerful optical design method for laser systems

A new nomographic diagram for design and analysis of paraxial systems has been proposed. The diagram has proved to have the following advantages: (i) Paraxial ray tracing can be performed graphically, with the result being represented by a pattern on the diagram; (ii) from the given pattern of an arbitrary ray, all the structural parameters of the system, and paths of any other rays can be obtained on the diagram; (iii) the nomographic scales devised specially for the diagram facilitate direct reading and/or plotting of the paraxial parameters, e.g., focal lengths, positions of pupils and cardinal points; and, (iv) a system with unknown structural parameters can also be treated on the diagram, provided the entrance and exit conditions are given for a pair of independent rays. It has been shown that the y-ӯ diagram can be obtained from the proposed diagram by a simple graphical transformation.

Mitsuo Takeda

Papers

New diagram for nomographic design and analysis of paraxial optical systems