New approach to teaching matrix optics

2020

This books deals with Matrix Method in Paraxial Optics for undergraduate student, this serves as a educational note for both teachers and students teaching or studying optics.

Journal of the Optical Society of America, 1982

The theory of canonical transforms is applied to establish the mathematical foundations of the operator algebra method, leading to useful relations between geometrical ray optics and the operator representation of wave optics.

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 ...

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.

Journal of the Optical Society of America, 1982

The canonical operator theory introduced recently for the description of lossless first-order optics is extended here to first-order systems with loss or gain, elucidating the relation between canonical operator and complex ray methods. The spread functions in the space, frequency, and hybrid domains are derived in terms of the ABCD raytransfer matrix as well as in terms of four other new matrix descriptions of geometrical optics. The fourfold correspondence among these matrices, the Hamilton characteristics, and the spread functions in the space, frequency, and hybrid domains leads to the derivation of four fundamental explicit canonical representations of the transfer operator and to their parameterization in terms of characteristic matrix elements. The relation between adjoint operators and bidirectional propagation is derived within canonical operator theory of first-order optics as well as within a more-general setting for all reciprocal (but not necessarily lossless) optical systems.

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

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.

BMJ open ophthalmology, 2022

Journal of Physics A: Mathematical and Theoretical, 2008

In geometric optics there is a natural distinction between the paraxial and aberration regimes, which contain respectively the linear and nonlinear canonical transformations of position and momentum in the phase space. In the Lie-theoretical presentation, linear inhomogeneous transformations are generated by linear and quadratic functions of the phase space, while aberrations of increasing order are generated by homogeneous functions with higher powers of these coordinates. In a way parallel but distinct from the Schrödinger quantization of continuous classical systems, we quantize the geometric optical model into discrete, finite-dimensional systems based on the Lie algebra su(2), whose wavefunctions are N-point signals, phase space is a sphere and transformations are represented by the N × N unitary matrices that form the group U(N). We factor this group into SU(2)-linear and nonlinear unitary transformations of phase space and classify all its N 2 − 4 aberrations. This offers a new parametrization of U(N) based on a chosen SU(2) subgroup.

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.

2015 ASEE Annual Conference and Exposition Proceedings, 2015

He was an invited scholar at the University of Wyoming, fall 2004, where he was recognized as an eminent engineer and inducted into tau beta pi. In 2006 he co-authored "Real-time Digital Signal Processing, from MATLAB to C with the TMS320C6x DSK" which was translated into Chinese in 2011. The second edition of this text was published in 2012.

Journal of the Optical Society of America A, 2000

The set of paraxial optical systems is the manifold of the group of symplectic matrices. The structure of this group is nontrivial: It is not simply connected and is not of an exponential type. Our analysis clarifies the origin of the metaplectic phase and the inherent limitations for optical map fractionalization. We describe, for the first time to our knowledge, an image girator and a cross girator whose geometric and wave implementations are of interest.

The Physics Teacher, 2015

Constructing ray diagrams to locate the image of an object formed by thin lenses and mirrors is a staple of many introductory physics courses at the high school and college levels, and has been the subject of some pedagogy-related articles. Our review of textbooks distributed in the United States suggests that the singular approach involves drawing principle rays to locate an object's image. We were pleasantly surprised to read an article in this journal by Suppapittayaporn et al. in which they use an alternative method to construct rays for thin lenses based on a “tilted principle axis” (TPA). In particular, we were struck by the generality of the approach (a single rule for tracing rays as compared to the typical two or three rules), and how it could help students more easily tackle challenging situations, such as multi-lens systems and occluded lenses, where image construction using principle rays may be impractical. In this paper, we provide simple “proofs” for this alternat...

Journal of the Optical Society of America A, 2006

On the basis of a matrix formalism, we analyze the paraxial optical systems composed by generalized lenses and fixed free-space intervals, suitable for orthosymplectic transformations in phase space. Flexible configurations to perform the attractive operations for optical information processing such as image rotation, separable fractional Fourier transformation, and twisting for different parameters are proposed.

Arxiv preprint arXiv:0812.0664, 2008

We propose that the height-angle ray vector in matrix optics should be complex, based on a geometric algebra analysis. We also propose that the ray's 2 × 2 matrix operators should be right-acting, so that the matrix product succession would go with light's left-to-right propagation. We express the propagation and refraction operators as a sum of a unit matrix and an imaginary matrix proportional to the Fermion creation or annihilation matrix. In this way, we reduce the products of matrix operators into sums of creation-annihilation product combinations. We classify ABCD optical systems into four: telescopic, inverse Fourier transforming, Fourier transforming, and imaging. We show that each of these systems have a corresponding Lagrange theorem expressed in partial derivatives, and that only the telescopic and imaging systems have Lagrange invariants.