Foci in ray pencils of general divergency

In generalized optical systems, that is, in systems which may contain thin refracting elements of asymmetric dioptric power, pencils of rays may exhibit phenomena that cannot occur in conventional optical systems. In conventional optical systems astigmatic pencils have two principal meridians that are necessarily orthogonal; in generalized systems the principal meridians can be at any angle. In fact in generalized systems a pencil may have only one principal meridian or even none at all. In contrast to the line foci in the conventional interval of Sturm line foci in generalized systems may be at any angle and there may be only one line focus or no line foci. A conventional cylindrical pencil has a single line focus at a finite distance but it can be regarded as having a second line focus at infinity. Only in generalized systems is a single line focus possible without a second at infinity or anywhere else. The purpose of this paper is to illustrate the types of pencils possible in generalized systems. Particular attention is paid to the effect of including an antisymmetric component in the divergency of the pencil.

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Ray pencils of general divergency

African Vision and Eye Health ◽

10.4102/aveh.v68i2.160 ◽

2009 ◽

Vol 68 (2) ◽

Author(s):

W. F. Harris

Keyword(s):

Homogeneous Medium ◽

Axiomatic Approach ◽

Transverse Plane ◽

Optical Systems ◽

Linear Optics ◽

Straight Line ◽

Asymmetric Power ◽

Dioptric Power ◽

Homogeneous Media ◽

Line Focus

That a thin refracting element can have a dioptric power which is asymmetric immediately raises questions at the fundamentals of linear optics. In optometry the important concept of vergence, in particular, depends on the concept of a pencil of rays which in turn depends on the existence of a focus. But systems that contain refracting elements of asymmetric power may have no focus at all. Thus the existence of thin systems with asym-metric power forces one to go back to basics and redevelop a linear optics from scratch that is sufficiently general to be able to accommodate suchsystems. This paper offers an axiomatic approach to such a generalized linear optics. The paper makes use of two axioms: (i) a ray in a homogeneous medium is a segment of a straight line, and (ii) at an interface between two homogeneous media a ray refracts according to Snell’s equation. The familiar paraxial assumption of linear optics is also made. From the axioms a pencil of rays at a transverse plane T in a homogeneous medium is defined formally (Definition 1) as an equivalence relation with no necessary association with a focus. At T the reduced inclination of a ray in a pencil is an af-fine function of its transverse position. If the pencilis centred the function is linear. The multiplying factor M, called the divergency of the pencil at T, is a real 2 2× matrix. Equations are derived for the change of divergency across thin systems and homogeneous gaps. Although divergency is un-defined at refracting surfaces and focal planes the pencil of rays is defined at every transverse plane ina system (Definition 2). The eigenstructure gives aprincipal meridional representation of divergency;and divergency can be decomposed into four natural components. Depending on its divergency a pencil in a homogeneous gap may have exactly one point focus, one line focus, two line foci or no foci.Equations are presented for the position of a focusand of its orientation in the case of a line focus. All possible cases are examined. The equations allow matrix step-along procedures for optical systems in general including those with elements that haveasymmetric power. The negative of the divergencyis the (generalized) vergence of the pencil.

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A System Matrix for Astigmatic Optical Systems: I. Introduction and Dioptric Power Relations

Optometry and Vision Science ◽

10.1097/00006324-198110000-00006 ◽

1981 ◽

Vol 58 (10) ◽

pp. 810-819 ◽

Cited By ~ 61

Author(s):

MICHAEL P. KEATING

Keyword(s):

Power Relations ◽

Optical Systems ◽

System Matrix ◽

Dioptric Power

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Inner-product spaces for quantitative analysis of eyes and other optical systems

African Vision and Eye Health ◽

10.4102/aveh.v75i1.348 ◽

2016 ◽

Vol 75 (1) ◽

Author(s):

William F. Harris ◽

Tanya Evans ◽

Radboud D. Van Gool

Keyword(s):

Quantitative Analysis ◽

Dimensional Space ◽

Three Dimensional ◽

Optical Systems ◽

Inner Product ◽

Product Spaces ◽

Inner Product Spaces ◽

Optical Character ◽

Dioptric Power ◽

Quantitative Analyses

Because dioptric power matrices of thin systems constitute a (three-dimensional) inner-product space, it is possible to define distances and angles in the space and so do quantitative analyses on dioptric power for thin systems. That includes astigmatic corneal powers and refractive errors. The purpose of this study is to generalise to thick systems. The paper begins with the ray transference of a system. Two 10-dimensional inner-product spaces are devised for the holistic quantitative analysis of the linear optical character of optical systems. One is based on the point characteristic and the other on the angle characteristic; the first has distances with the physical dimension L−1 and the second has the physical dimension L. A numerical example calculates the locations, distances from the origin and angles subtended at the origin in the 10-dimensional space for two arbitrary astigmatic eyes.

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Advanced 3D roughness analysis for the characterization of ceramic surfaces

Additional Conferences (Device Packaging HiTEC HiTEN & CICMT) ◽

10.4071/cicmt-2011-wp12 ◽

2011 ◽

Vol 2011 (CICMT) ◽

pp. 000207-000214

Author(s):

Karl-Heinz Strass

Keyword(s):

Surface Roughness ◽

Surface Properties ◽

Surface Preparation ◽

Surface Characteristics ◽

Optical Systems ◽

Single Line ◽

Past Century ◽

Roughness Parameters ◽

Roughness Analysis ◽

3D Metrology

Advanced processes often hinge on the ability to reproduce specific surface characteristics. While throughout the past century, the mean roughness value, also expressed as Ra has often been used as the main parameter to classify a surface's ability to retain oil and provide a functional bearing surface, the materials, processes and required functionalities of modern surfaces go far beyond that application. Superficial interfaces now often determine the ability of a consecutive layer to adhere to the substrate or, amongst others, enhance a surface's ability to exchange electrons or other elements. In many applications the surface itself enables the process the device is intended to deliver. The paper discusses various approaches to measure surface roughness, and takes a closer look at more advanced surface roughness parameters that allow the user to more accurately and precisely determine a surface's ability to function within its desired boundary conditions. We will review various technologies widely used to measure surface topographies, from the traditional stylus profilometry, to white light interferometry, confocal microscopy to optical systems that combine several of the advantages into one technology. Almost none of the surfaces functions today are based on surface properties of a single line, hence the paper will review the correlation between single line scan metrology and real 3D metrology. Acquiring 3-dimensional data provides the user with significantly more information. Any directionality of surface properties can now be evaluated by taking advantage of more advanced and complex roughness parameters that indicate for instance, anisotropies of surface preparation processes.

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Tandem scanning reflected light microscopy: How it works and applications

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100142104 ◽

1986 ◽

Vol 44 ◽

pp. 84-87

Author(s):

Alan Boyde ◽

Milan Hadravský ◽

Mojmír Petran ◽

Timothy F. Watson ◽

Sheila J. Jones ◽

...

Keyword(s):

Light Microscopy ◽

Focal Plane ◽

Central Symmetry ◽

Single Line ◽

Scanning Microscope ◽

Reflected Light ◽

Illuminating Light ◽

Illuminating Beam

The principles of tandem scanning reflected light microscopy and the design of recent instruments are fully described elsewhere and here only briefly. The illuminating light is intercepted by a rotating aperture disc which lies in the intermediate focal plane of a standard LM objective. This device provides an array of separate scanning beams which light up corresponding patches in the plane of focus more intensely than out of focus layers. Reflected light from these patches is imaged on to a matching array of apertures on the opposite side of the same aperture disc and which are scanning in the focal plane of the eyepiece. An arrangement of mirrors converts the central symmetry of the disc into congruency, so that the array of apertures which chop the illuminating beam is identical with the array on the observation side. Thus both illumination and “detection” are scanned in tandem, giving rise to the name Tandem Scanning Microscope (TSM). The apertures are arranged on Archimedean spirals: each opposed pair scans a single line in the image.

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Atomic Resolution in Crystal Aperture Stem

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100179932 ◽

1990 ◽

Vol 48 (1) ◽

pp. 236-237

Author(s):

J T Fourie

Keyword(s):

Optical System ◽

Spherical Aberration ◽

Three Dimensional ◽

Transmission Function ◽

Optical Systems ◽

Bragg Angle ◽

Electron Optical System ◽

Intimate Contact ◽

Diffraction Effects ◽

Design Aspect

The attempts at improvement of electron optical systems to date, have largely been directed towards the design aspect of magnetic lenses and towards the establishment of ideal lens combinations. In the present work the emphasis has been placed on the utilization of a unique three-dimensional crystal objective aperture within a standard electron optical system with the aim to reduce the spherical aberration without introducing diffraction effects. A brief summary of this work together with a description of results obtained recently, will be given.The concept of utilizing a crystal as aperture in an electron optical system was introduced by Fourie who employed a {111} crystal foil as a collector aperture, by mounting the sample directly on top of the foil and in intimate contact with the foil. In the present work the sample was mounted on the bottom of the foil so that the crystal would function as an objective or probe forming aperture. The transmission function of such a crystal aperture depends on the thickness, t, and the orientation of the foil. The expression for calculating the transmission function was derived by Hashimoto, Howie and Whelan on the basis of the electron equivalent of the Borrmann anomalous absorption effect in crystals. In Fig. 1 the functions for a g220 diffraction vector and t = 0.53 and 1.0 μm are shown. Here n= Θ‒ΘB, where Θ is the angle between the incident ray and the (hkl) planes, and ΘB is the Bragg angle.

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