In this work the paraxial optical imaging is generally described by means of three square matrices: one unitary system matrix and two operation matrices with determinants equal the Lagrange-Helmholtz invariant. Elements of system matrix are functions of design parameters while elements of the operation matrix depend on input and output coordinates of characteristic rays. Each matrix has only three independent elements. Internal system parameters are determined from equations created of system matrix elements, which values are dependent on the operation matrix. Matrix approach enables the solution of only three non-linear equations with respect to system parameters. Matrix approach has also another advantage. It enables the determination of number of degrees of freedom. We have a superiority of parameters over the number of equations when the number of components is bigger than 2. The more complex is model the higher degree of freedom it has. There are special ways of reducing the number of degree of freedom: by selection of spaces between component, introduction of additional requirements and criteria of distribution of optical powers. Significant help is in defining all the spaces between components, what means full control of the components position and their dimensions. In such a case the only thing left is the determination of optical powers, while the number of degree of freedom is equal k-1 (k is the number of components). In this work the computer program realizing described algorithms has been developed. This program was tested with specially selected examples. Results of calculation for two interesting applications are also given.
Original matrix formulas obtained by differentiation of the system matrix in respect to movements of components are derived. Components kinematics for the three zoom systems realized by means of interactive graphical software is presented. An optical system may be structurally designed by successive steps and its parameters determined to fulfil requirements, such as optical conjugation, focal lengths or magnifications. Improved software developed in this work serves both determination of optical powers and separations and movements of components.Developed methodology covers different types of fixed and zoom systems, the latter type with electronic or optical compensation. One may consider any optical system, such as the reproduction lens, objective lens or telescope system, because matrix optics distinguishes them remarkably easy. Kinematics pertaining to a full tract of the zoom system is determined at a discrete number of positions.Movements of so-called basic variable components are determined in a full cycle of work by means of iterative methods, while movements of supplementary components may be inserted by means of exponential-parabolic functions also including their linear form. Any component of the zoom system may act as a variable, supplementary or fixed component, but it is mainly dependent on the structural design.Parameters of characteristics are computed as elements of a certain matrix. Designing is that to set these elements on required values by means of system parameters or movements of components. In this way, one may create complex multi-group systems with characteristics and movements which we accept. Properties of these systems are presented by numerical and graphical forms.Advantages of these systems are their more compact construction, more smooth kinematics, and better possibilities of optimization, what is particularly valuable for zoom systems with a high zooming ratio.
New matrix formulas for structural optical design have been obtained from analysis of derivative of the system matrix in respect to construction parameters and movements of components. Functional parameters of the optical system become elements of the matrix, presenting working conditions of the optical system. Developed methodology of structural design multi-group zoom systems with unlimited number of components and with mechanical-electronic compensation is presented. Any optical system, such as the objective lens, reproduction system, or telescopic system, can be analyzed with this methodology. Kinematics of components pertaining to a full tract of the zoom system is determined for a discrete number of positions. Three examples of the structural design of complex zoom systems with five-components and high zooming ratio are provided.
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