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The method of Gaussian matrices is useful for finding a paraxial solution to a given optical-design problem. This solution can then be refined by analytical methods of aberration theory or by optical-design software.
A generic way to proceed is as follows: a paraxial start layout is determined using the matrix approach. Its properties and parameter dependencies can then be analyzed in a first order of approximation with this method. The paraxial model can be implemented as a start configuration in an optical-design program. The necessary paraxial element is included in most optical-design packages. Let us say that we started with a lens in the thin-lens approximation to find and understand the first layout. Then we can make the transition to a thick spherical lens with the same focal length and extend the validity of the solution from the paraxial domain to the complete aperture while trying to keep the main properties of the layout. The control features of the software can be used for this purpose. Depending on the degrees of freedom available for the design task, we might improve the aberration control in further steps of optimization by allowing for an aspheric lens. This method also proved efficient in designing variable optical devices such as zoom lenses.
There are some approaches in the literature for extending the matrix theory that were not treated in this book: To overcome the restriction to the paraxial region, trigonometric expressions as matrix entries are used and the concept of the so-called exact matrices are introduced in Das (1991). The relation to Seidel aberrations is described by Guillemin and Sternberg (1984). And an extension of the matrix method is also proposed in Brouwer's book on instrument design (Brouwer, 1964).
In this book, the focus is on optical design, and a most natural step from the practical point of view is to use the matrix description as groundwork (and, in a way, as a framework) and then turn to aberration control for refinement of the design obtained. But the theory of Gaussian matrices also lays a solid groundwork for some in-depth theoretical studies of subjects such as Fourier optics and symplectic systems (Guillemin and Sternberg, 1984).
The matrix description is considered a good starting point to dive into the theoretical description of light propagation through optical systems by integral transformations (Guillemin and Sternberg, 1984; Hodgson and Weber, 1997).
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