Matrix decompositions for nonsymmetrical optical systems

Journal of the Optical Society of America, 1983

In the paraxial approximation a symmetrical optical system may be represented by a 2 X 2 matrix. It has been the custom to describe each optical element by a transfer matrix representing propagation between the principal planes or through an interface for thin elements. If the focal-plane representation is used instead, any focusing element or combination of elements is represented by the same antidiagonal matrix whose nonzero elements are the focal lengths: The matrix represents propagation between the focal planes. For propagation between any two arbitrary planes, the system transfer matrix can be decomposed into the product of two upper triangular matrices and an antidiagonal matrix. This decomposition yields the above-mentioned focal-plane matrix, and the two upper triangular matrices represent propagation between the input and the output planes and the focal planes. Because the matrix decomposition directly yields the parameters of interest, the analysis and the synthesis of optical systems are simpler to carry out. Examples are given for lenses, diopters, mirrors, periodic sequences, resonators, lenslike media, and phase-conjugate mirror systems.

SPIE Proceedings, 2000

The optimization of an optical system benefits greatly from a study of its aberrations and an identification of each of its elements' contribution to the overall aberration figures. The matrix formalism developed by one of the authors was the object of a previous paper and allows the expression of image-space coordinates as high-order polynomials of object-space coordinates. In this paper we approach the question of aberrations, both through the evaluation of the wavefront evolution along the system and its departure from the ideal spherical shape and the use of ray density plots. Using seventh-order matrix modeling, we can calculate the optical path between any two points of a ray as it travels along the optical system and we define the wavefront as the locus of the points with any given optical path; the results are presented on the form of traces of the wavefront on the tangential plane, although the formalism would also permit sagital plane plots. Ray density plots are obtained by actual derivation of the seventh-order polynomials.

Applied Optics, 2008

The properties of first-order optical systems are described paraxially by a ray transfer matrix, also called the ABCD matrix. Here we consider the inverse problem: an ABCD matrix is given, and we look for the minimal optical system that consists of only lenses and pieces of free-space propagation. Similar decompositions have been studied before but without the restriction to these two element types or without an attempt at minimalization. As the main results of this paper, we found that general lossless onedimensional optical systems can be synthesized with a maximum of four elements and two-dimensional optical systems can be synthesized with six elements at most.

Optik - International Journal for Light and Electron Optics, 2005

The various non-linear transformations incurred by the rays in an optical system can be modelled by matrix products up to any desired order of approximation. Mathematica software has been used to find the appropriate matrix coefficients for the straight path transformation and for the transformations induced by conical surfaces, both direction change and position offset. The same software package was programmed to model optical systems in seventh-order. A Petzval lens was used to exemplify the modelling power of the program.

Journal of the Optical Society of America A, 2007

Based on the eigenvalues of the ray transformation matrix, a classification of ABCD systems is proposed and some nuclei (i.e., elementary members) in each class are described. In the one-dimensional case, possible nuclei are the magnifier, the lens, and the fractional Fourier transformer. In the two-dimensional case we have-in addition to the obvious concatenations of one-dimensional nuclei-the four combinations of a magnifier or a lens with a rotator or a shearing operator, where the rotator and the shearer are obviously inherently twodimensional. Any ABCD system belongs to one of the classes described in this paper and is similar (in the sense of matrix similarity of the ray transformation matrices) to the corresponding nucleus. Knowledge of a nucleus may be helpful in finding eigenfunctions of the corresponding class of first-order optical systems: one only has to find eigenfunctions of the nucleus and to determine how these functions propagate through a firstorder optical system.

Ophthalmic & physiological optics : the journal of the British College of Ophthalmic Opticians (Optometrists), 2017

To show that 14-dimensional spaces of augmented point P and angle Q characteristics, matrices obtained from the ray transference, are suitable for quantitative analysis although only the latter define an inner-product space and only on it can one define distances and angles. The paper examines the nature of the spaces and their relationships to other spaces including symmetric dioptric power space. The paper makes use of linear optics, a three-dimensional generalization of Gaussian optics. Symmetric 2 × 2 dioptric power matrices F define a three-dimensional inner-product space which provides a sound basis for quantitative analysis (calculation of changes, arithmetic means, etc.) of refractive errors and thin systems. For general systems the optical character is defined by the dimensionally-heterogeneous 4 × 4 symplectic matrix S, the transference, or if explicit allowance is made for heterocentricity, the 5 × 5 augmented symplectic matrix T. Ordinary quantitative analysis cannot be ...

Education in Optics, 1991

The classical 2x2 matrix approach to geometrical optics is very limited for practical use, because fundamental decompositions of system matrices do not yield matrices with physically significant properties. The use of permuted matrices proposed by us previously (J. Opt. Soc. Am. 73, 1350-1359) yields a much more powerful representation that is more useful for teaching from beginning undergraduate to advanced graduate levels. The matrices of the previous theory are still valid, but when they are treated in what might be called the focal plane representation, the matrices obtained by the LDU decomposition have a simple and direct physical meaning. The relationship between the older matrix theory and this one is analogous to the relationship between the Descartes and the Newton formalisms of geometrical optics: matrix components are simplified by measuring all distances from the foci. This facilitates synthesis problems, for which the standard approach is not well adapted. In addition to simple applications like lenses, mirrors and diopters, the theory can be applied to more complex cases like lenslike media, resonators, Fourier transform systems and phase-conjugate mirrors. This theory can be directly generalized for nonsymmetrical systems using a 4x4 matrix formalism. The other theory, where distances are measured from the principal planes, cannot be generalized for nonsymmetrical systems having no principal planes.

DESCRIPTION Lecture notes (17 pages) : The matrix methods in geometrical optics can be developed in terms of three basic operators (matrices) for reflection, refraction and translation and can be applied with ease even for an optical system consisting of a large number of optical elements. An elementary but thorough discussion of this method is presented here.

BMJ open ophthalmology, 2022

Journal of the Optical Society of America A, 1997

Systematic procedures are presented for determining the optical components needed to produce an arbitrary transformation of a Gaussian light beams's spot size, radius of curvature, displacement, and direction of propagation. As an example, an optical system is considered that spatially separates the two coincident Gaussian beams produced by a high-diffraction-loss resonator that uses a Gaussian variable-reflectivity output coupler. In addition, an ABCDGH reverse matrix theorem and an ABCDGH Sylvester theorem are also derived. These matrix theorems may be used to satisfy special constraints inherent in the design of multipass and periodic optical systems. © 1997 Optical Society of America [S0740-3232 882