SYMPLECTIC STRUCTURE ON THE SET OF GEODESICS OF A RIEMANNIAN MANIFOLD

1991 ◽  
Vol 06 (03) ◽  
pp. 431-444 ◽  
Author(s):  
JOSÉ F. CARIÑENA ◽  
CARLOS LÓPEZ
Keyword(s):  
Riemannian Manifold ◽  
Optical System ◽  
Symplectic Structure ◽  
Light Rays

A natural symplectic structure is defined on the set of oriented geodesics of a Riemannian or pseudo-Riemannian manifold. Particular examples—the set of straightlines of a plane and the set of light rays of an optical system—are developed.

Types of refractive errors and methods for their diagnosis

2020 ◽  
pp. 30-36
Author(s):  
Olga Lemzyakova
Keyword(s):  
Optical System ◽  
Contact Lenses ◽  
Vision Impairment ◽  
Refractive Errors ◽  
Vision Correction ◽  
Light Rays ◽  
Different Types ◽  
Almost All ◽  
Degrees Of Severity ◽  
The Brain

Refraction of the eye means its ability to bend (refract) light in its own optical system. In a normal state, which is called emmetropia, light rays passing through the optical system of the eye focus on the retina, from where the impulse is transmitted to the visual cortex of the brain and is analyzed there. A person sees equally well both in the distance and near in this situation. However, very often, refractive errors develop as a result of various types of influences. Myopia, or short-sightedness, occurs when the light rays are focused in front of the retina as a result of passing through the optical system of the eye. In this case, a person will clearly distinguish close objects and have difficulties in seeing distant objects. On the opposite side is development of farsightedness (hypermetropia), in which the focusing of light rays occurs behind the retina — such a person sees distant objects clearly, but outlines of closer objects are out of focus. Near vision impairment in old age is a natural process called presbyopia, it develops due to the lens thickening. Both myopia and hypermetropia can have different degrees of severity. The variant, when different refractive errors are observed in different eyes, is called anisometropia. In the same case, if different types of refraction are observed in the same eye, it is astigmatism, and most often it is a congenital pathology. Almost all of the above mentioned refractive errors require correction with spectacles or use of contact lenses. Recently, people are increasingly resorting to the methods of surgical vision correction.

scholarly journals Certain Mathematical Instruments for Graphically Indicating the Direction of Refracted and Reflected Light Rays

1906 ◽  
Vol 25 (2) ◽  
pp. 806-812
Author(s):  
J.R. Milne
Keyword(s):  
Optical System ◽  
The Other ◽  
Reflected Light ◽  
Light Rays ◽  
Image Position

The refraction equation sin i == μ sin r, though simple in itself, is apt to give rise, in problems connected with refraction, to formulæ too involved for arithmetical computation. In such cases it may be necessary to trace the course through the optical system in question of a certain number of arbitrarily chosen rays, and thence to find the course of the other rays by interpolation. Thelinkage about to be described affords a rapid and accurate means of determining the paths of the rays through any optical system.

CALCULATION OF MODULATION TRANSFER FUNCTION OF AN OPTICAL SYSTEM BY USING SKEW RAY TRACING

2009 ◽  
Vol 33 (3) ◽  
pp. 429-442 ◽  
Author(s):  
Kuo-Hwa Tseng ◽  
Chieh Kung ◽  
Te-Tan Liao ◽  
Hao-Peng Chang
Keyword(s):  
Transfer Function ◽  
Ray Tracing ◽  
Optical System ◽  
Modulation Transfer Function ◽  
Image Plane ◽  
Modulation Transfer ◽  
Light Rays ◽  
Intermediate Image ◽  
Pass Through ◽  
And Performance

The resolution and performance of an optical system can be characterized by a quantity known as the modulation transfer function (MTF), which is a measurement of an optical system’s ability to transfer contrast from the specimen to the intermediate image plane at a specific resolution. Accordingly, this study employs skew ray tracing based on a 4 × 4 homogeneous coordinate transformation matrix and Snell’s law to develop a detailed methodology for determining the spot diagram on the image plane when light rays pass through the optical system. And the authors present calculations of the MTF of an optical system by using the spot diagram on the image plane. The numerical results of the proposed methodology are demonstrated using a symmetrical optical system.

scholarly journals An energy conservative hp-scheme for light propagation using Liouville’s equation for geometrical optics

2020 ◽  
Vol 238 ◽  
pp. 02005
Author(s):  
Robert A.M. van Gestel ◽  
Martijn J.H. Anthonissen ◽  
Jan H.M. ten Thije Boonkkamp ◽  
Wilbert L. IJzerman
Keyword(s):  
Optical System ◽  
Light Propagation ◽  
Energy Propagation ◽  
Conservation Of Energy ◽  
Adaptive Scheme ◽  
Propagation Of Light ◽  
Light Rays ◽  
Liouville’S Equation ◽  
Full Domain ◽  
Space Description

In this contribution an alternative method to standard forward ray-tracing is briefly outlined. The method is based on a phase-space description of light propagating through an optical system. The propagation of light rays are governed by Hamilton’s equations. Conservation of energy and étendue for a beam of light, allow us to derive a Liouville’s equation for the energy propagation through an optical system. Liouville’s equation is solved numerically using an hp-adaptive scheme, which for a smooth refractive index field is energy conservative. A proper treatment of optical interfaces ensures that the scheme is energy conservative over the full domain.

scholarly journals STABILITY OF ANOSOV HAMILTONIAN STRUCTURES

2010 ◽  
Vol 02 (04) ◽  
pp. 419-451 ◽  
Author(s):  
WILL J. MERRY ◽  
GABRIEL P. PATERNAIN
Keyword(s):  
Riemannian Manifold ◽  
Symplectic Form ◽  
Negative Curvature ◽  
Symplectic Structure ◽  
Hamiltonian Structure ◽  
Riemannian Metric ◽  
Hamiltonian Structures ◽  
Negatively Curved ◽  
Closed Riemannian Manifold

Let (Mn, g) denote a closed Riemannian manifold (n ≥ 3) which admits a metric of negative curvature (not necessarily equal to g). Let ω1 := ω0 + π*σ denote a twisted symplectic form on TM, where σ ∈ Ω2(M) is a closed 2-form and ω0 is the symplectic structure on TM obtained by pulling back the canonical symplectic form dx ∧ dp on T*M via the Riemannian metric. Let Σk be the hypersurface [Formula: see text]. We prove that if n is odd and the Hamiltonian structure (Σk, ω1) is Anosov with C1 weak bundles, then (Σk, ω1) is stable if and only if it is contact. If n is even and in addition the Hamiltonian structure is 1/2-pinched, then the same conclusion holds. As a corollary, we deduce that if g is negatively curved, strictly 1/4-pinched and σ is not exact then the Hamiltonian structure (Σk, ω1) is never stable for all sufficiently large k.

scholarly journals On a class of even-dimensional manifolds structured by an affine connection

2002 ◽  
Vol 29 (11) ◽  
pp. 681-686
Author(s):  
I. Mihai ◽  
A. Oiagă ◽  
R. Rosca
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Conformal Transformation ◽  
Symplectic Structure ◽  
Affine Connection ◽  
Parallel Connection

We deal with a2m-dimensional Riemannian manifold(M,g)structured by an affine connection and a vector field𝒯, defining a𝒯-parallel connection. It is proved that𝒯is both a torse forming vector field and an exterior concurrent vector field. Properties of the curvature2-forms are established. It is shown thatMis endowed with a conformal symplectic structureΩand𝒯defines a relative conformal transformation ofΩ.

A Field Emission System For Transmission Electron Microscopy

1973 ◽  
Vol 31 ◽  
pp. 298-299
Author(s):  
Michel Troyonal ◽  
Huei Pei Kuoal ◽  
Benjamin M. Siegelal
Keyword(s):  
Electron Microscope ◽  
Field Emission ◽  
Optical System ◽  
High Vacuum ◽  
Ultra High Vacuum ◽  
Objective Lens ◽  
Electron Optical System ◽  
Cross Sectional ◽  
Transmission Electron ◽  
Emission System

A field emission system for our experimental ultra high vacuum electron microscope has been designed, constructed and tested. The electron optical system is based on the prototype whose performance has already been reported. A cross-sectional schematic illustrating the field emission source, preaccelerator lens and accelerator is given in Fig. 1. This field emission system is designed to be used with an electron microscope operated at 100-150kV in the conventional transmission mode. The electron optical system used to control the imaging of the field emission beam on the specimen consists of a weak condenser lens and the pre-field of a strong objective lens. The pre-accelerator lens is an einzel lens and is operated together with the accelerator in the constant angular magnification mode (CAM).

Information and estimation in image processing

1987 ◽  
Vol 45 ◽  
pp. 14-17
Author(s):  
B. Roy Frieden
Keyword(s):  
Image Processing ◽  
Optical System ◽  
Large Amplitude ◽  
Communication Theory ◽  
Image Data ◽  
Direct Approach ◽  
Ill Posed

Despite the skill and determination of electro-optical system designers, the images acquired using their best designs often suffer from blur and noise. The aim of an “image enhancer” such as myself is to improve these poor images, usually by digital means, such that they better resemble the true, “optical object,” input to the system. This problem is notoriously “ill-posed,” i.e. any direct approach at inversion of the image data suffers strongly from the presence of even a small amount of noise in the data. In fact, the fluctuations engendered in neighboring output values tend to be strongly negative-correlated, so that the output spatially oscillates up and down, with large amplitude, about the true object. What can be done about this situation? As we shall see, various concepts taken from statistical communication theory have proven to be of real use in attacking this problem. We offer below a brief summary of these concepts.

Atomic Resolution in Crystal Aperture Stem

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.

STEM and ELS Observation of Early Oxide Formation on the Surface of Cr Thin Films

1980 ◽  
Vol 38 ◽  
pp. 412-413
Author(s):  
Fumio Watari ◽  
J. M. Cowley
Keyword(s):  
Thin Films ◽  
Optical System ◽  
Room Temperature ◽  
Oxide Formation ◽  
Initial Nucleation ◽  
Diffraction Patterns ◽  
Relationship Of ◽  
Multiple Twinning ◽  
Moderately High Temperature

STEM coupled with the optical system was used for the investigation of the early oxidation on the surface of Cr. Cr thin films (30 – 1000Å) were prepared by evaporation onto the polished or air-cleaved NaCl substrates at room temperature and 45°C in a vacuum of 10−6 Torr with an evaporation speed 0.3Å/sec. Rather thick specimens (200 – 1000Å) with various preferred orientations were used for the investigation of the oxidation at moderately high temperature (600 − 1100°C). Selected area diffraction patterns in these specimens are usually very much complicated by the existence of the different kinds of oxides and their multiple twinning. The determination of the epitaxial orientation relationship of the oxides formed on the Cr surface was made possible by intensive use of the optical system and microdiffraction techniques. Prior to the formation of the known rhombohedral Cr2O3, a thin spinel oxide, probably analogous to γ -Al203 or γ -Fe203, was formed. Fig. 1a shows the distinct epitaxial growth of the spinel (001) as well as the rhombohedral (125) on the well-oriented Cr(001) surface. In the case of the Cr specimen with the (001) preferred orientation (Fig. 1b), the rings explainable by spinel structure appeared as well as the well defined epitaxial spots of the spinel (001). The microdif fraction from 20A areas (Fig. 2a) clearly shows the same pattern as Fig. Ia with the weaker oxide spots among the more intense Cr spots, indicating that the thickness of the oxide is much less than that of Cr. The rhombohedral Cr2O3 was nucleated preferably at the Cr(011) sites provided by the polycrystalline nature of the present specimens with the relation Cr2O3 (001)//Cr(011), and by further oxidation it grew into full coverage of the rest of the Cr surface with the orientation determined by the initial nucleation.

Sign in / Sign up

Export Citation Format

Share Document