Imaging with four spherical mirrors

We investigate unobstructed, plane-symmetric imaging systems of four spherical mirrors. Fifteen parameters are necessary to specify the configuration of such a system. Constraints are determined that ensure that any resultant system possesses a given set of first-order properties. These constraints remove four parameters as available degrees of freedom. To illustrate the efficacy of this design approach, we present two example studies: one for a class of systems with the object at infinity and another for finite-conjugate projection systems. For each study a global optimizer is used as the primary search tool. Example systems from these studies are presented.     

Imaging with three spherical mirrors

Unobstructed, plane-symmetric systems of three spherical mirrors are investigated. Twelve parameters are necessary to specify the configuration of such a system. Constraints are determined to eliminate four of these parameters as independent degrees of freedom. These constraints ensure appropriate first-order behavior and are used to aid in two example design studies-one for a class of systems with the object at infinity and another for a class of finite conjugate projection systems. For the first study, a portion of the associated merit-function space is systematically evaluated and plotted, and the results are compared with those obtained when a global optimizer is used. For the second study, a global optimizer is employed as the primary search tool. Example systems from both studies are presented.     

Bessel-Gauss resonator with spherical output mirror: geometrical- and wave-optics analysis

A detailed study of the axicon-based Bessel-Gauss resonator with concave output coupler is presented. We employ a technique to convert the Huygens-Fresnel integral self-consistency equation into a matrix equation and then find the eigenvalues and the eigenfields of the resonator at one time. A paraxial ray analysis is performed to find the self-consistency condition to have stable periodic ray trajectories after one or two round trips. The fast-Fourier-transform-based Fox and Li algorithm is applied to describe the three-dimensional intracavity field distribution. Special attention was directed to the dependence of the output transverse profiles, the losses, and the modal-frequency changes on the curvature of the output coupler and the cavity length. The propagation of the output beam is discussed.     

Deformation mechanism in α-Cu-Al single crystals with a spherical indentor

The subject of sliding friction is of great technological importance and has been studied extensively on an engineering basis, but many of the underlying microscopic factors that control friction or wear remain obscure. The present study is part of an effort to relate sliding deformation to the crystal structure of the contacting surfaces. The method was to scratch a steel ball in various crystal directions on an accurately finished surface of Cu-Al single crystals. For a scratch on the (1 1 1) face, the scratched track width and the width of the slipped area were found to be wider in the [¯2 1 1] direction than in the [2 ¯1 ¯1]. The mechanism of the active slip produced by the scratch can be explained in terms of stress patterns and interactions among the dislocations produced in the subsurface region. The distributions of dislocation density on the cross-section perpendicular to the (1 1 1) face were elucidated.     

Theory of spherical holographic gratings recorded by use of a multimode deformable mirror

We present the theory of spherical holographic gratings recorded by use of a deformable plane mirror and consider its application to the optimized Rowland Mounting. We illustrate the efficiency of such a mounting by computing two high-resolution gratings (3800 grooves/mm) with f/24 and f/10 apertures.     

A finite line crack in a pressurized spherical shell

The deformation of a thin sheet having initial spherical curvature is shown to be associated with that of an initially flat plate resting upon an elastic foundation. Using an integral formulation the coupled Reissner equations for a shell with a crack of length 2c are solved for the in-plane and Kirchhoff bending stresses, and, among other things, it is found that the explicit nature of the stresses near the crack point depends upon the inverse half power of the non-dimensional distance from the point e. The character of the combined extension-bending stress field near the crack tip is investigated in detail for the special case of a radial crack in a spherical cap which is subjected to a uniform internal pressure q_o and is clamped at the boundary \(\overline {\text{R}} = \overline {\text{R}} _{\text{o}}\). Pending a complete study of the solution, approximate results for the combined surface stresses near the crack tip normal and along the line of crack prolongation are respectively of the form

$$\sigma _y (\varepsilon ,0)|_{\nu = {1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}} \approx 0.45\sqrt {{1 \mathord{\left/ {\vphantom {1 \varepsilon }} \right. \kern-\nulldelimiterspace} \varepsilon }} \tfrac{{q_O R}}{h} + ...$$     

Optical data storage system with a planoellipsoidal solid immersion mirror illuminated directly by a point light source

A new solid immersion mirror called the planoellipsoidal (PE) solid immersion mirror (SIM) for the near-field optical storage is proposed and developed. The PE SIM has a small aperture on the apex of the ellipsoidal surface. The intensity distribution of the transmitted field is calculated by using the vector diffraction theory. Compared with a conventional solid immersion lens (SIL), the proposed PE SIM has the following features. A PE SIM replaces three optical elements of the collimator, objective, and SIL in a conventional SIL optical storage system, so that the optical system equipped with the PE SIM is not only simple in its assembly but is also effective in making an optical head unit. The PE SIM obtains light from a point light source and focuses it directly on the recording layer, which may be useful for a compact optical data storage system. The convex ellipsoidal surface of the PE SIM can reduce the risk of the SIM touching the surface of the recording medium. In addition, the spreading of the spot size with the increase of distance is very small in the PE SIM.     

High-aspect-ratio line focus for an x-ray laser by a deformable mirror

A high-aspect-ratio line focus is required on a plane target in x-ray laser experiments for obtaining a high gain-length product. Inherent wave-front aberrations in line-focusing optics, which consist of a cylindrical lens and a spherical lens, are discussed with respect to beam diameter. The nonuniformity of the linewidth that is due to the aberrations is also calculated by the ABCD matrix method. A deformable mirror of a continuous plate type with a diameter of 185 mm provides an adequate wave-front distribution for compensating for the wave-front aberration. The wave-front control by the deformable mirror realizes a fine linewidth of 25 mum and 18.2 mm long, corresponding to the aspect ratio of 728. The linewidth is three times the diffraction limit. The intensity distribution along the line focus is also improved.     

Geometric theory of wavefront aberrations in an off-axis spherical mirror

We present, analyze, and evaluate expressions for the wavefront aberrations in an off-axis spherical mirror. These formulas are derived from the optical path difference between an ellipsoid and a sphere, assuming a relatively small pupil and a small angle of incidence, as will be described in detail. Some well-known and also some useful new aberration expressions are obtained. They can be used to design and analyze cavities, spectrographs, and retinal adaptive optics imaging systems.     

Image of a point force in a spherical container and its connection to the Lorentz reflection formula

The image system for a velocity field of the Oseen tensor in a fluid region bounded by a rigid spherical container is derived. The Green's function and image system due to a nearby boundary constitute two themes explored in the pioneering (1896) paper by Lorentz. The special structure of our image system facilitates its incorporation as kernels for integral representations of velocity fields (another theme in the Lorentz paper) for a domain bounded by a spherical wall. The reflection formula for a plane wall is derived as a limiting case of the new solution.Correspondence concerning this article should be sent to Sangtae Kim.     

Thermal averages in a quantum point contact with a single coherent wave packet

A novel formal equivalence between thermal averages of coherent properties (e.g., conductance) and time averages of a single wave packet arises for Fermi gases and certain geometries. In the case of one open channel in a quantum point contact (QPC), only one wave packet history, with the wave packet width equal to the thermal length, completely determines the thermally averaged conductance. The formal equivalence moreover allows very simple physical interpretations of interference features surviving under thermal averaging. Simply put, pieces of the thermal wave packet returning to the QPC along independent paths must arrive at the same time in order to interfere. Remarkably, one immediate result of this approach is that higher temperature leads to narrower wave packets and therefore better resolution of events in the time domain. In effect, experiments at 4.2 K are performing time-gated experiments at better than a gigahertz. Experiments involving thermally averaged ballistic conductance in 2DEGS are presented as an application of this picture.     

Local-field resonance in light scattering by a single water droplet with spherical dielectric inclusions

Light scattering by an evaporating water droplet several micrometers in size with spherical dielectric inclusions was investigated. The evolution of the droplet radius and the effective refractive index was determined. A deviation from predictions by standard effective-medium theories in the form of a resonance was encountered. Simple analysis of the phenomenon was conducted, and a qualitative explanation was proposed.     

Designation of lenses with a single freeform surface for multiple point sources

Optical lenses with a freeform surface can be designed for diverse illumination profiles with uniformity. However, most of the previous studies formulate the problem for a single point source, and the lens topology has freeform top and spherical bottom surfaces. In this study, the formulation is extended for multiple point sources, and a flat surface is included in the lens bottom topology for ease of prototyping and manufacturing. The extended formulation for multiple point sources requires only a freeform surface to design. The formulation of overdetermined coupling equations is solved by applying the weighted least-square method. The weightings are correlated with the emitting intensities of sources in terms of an inverse gamma function. The weighting scheme gives a parameter space for designation of illumination profile fit and uniformity. The adequacy of the extended formulation is demonstrated by simulation. Examples of circular and rectangular illumination for single and multiple point sources are studied. The simulation results show that unbalanced luminance distribution can be induced by an offset source and collimated by a lens, which is designated by taking the offset into account. For multiple point sources, illumination profile fit and uniformity are designated in trade off based on the parameter design.     

Simple yet accurate noncontact device for measuring the radius of curvature of a spherical mirror

An easily reproducible device is demonstrated to be capable of measuring the radii of curvature of spherical mirrors, both convex and concave, without resorting to high-end interferometric or tactile devices. The former are too elaborate for our purposes, and the latter cannot be used due to the delicate nature of the coatings applied to mirrors used in high-power CO(2) laser applications. The proposed apparatus is accurate enough to be useful to anyone using curved optics and needing a quick way to assess the values of the radii of curvature, be it for entrance quality control or trouble shooting an apparently malfunctioning optical system. Specifically, the apparatus was designed for checking 50 mm diameter resonator (typically flat or tens of meters concave) and telescope (typically some meters convex and concave) mirrors for a high-power CO(2) laser, but it can easily be adapted to any other type of spherical mirror by a straightforward resizing.     

Recording high-dispersion spherical holographic gratings in a modified Rowland mounting by use of a multimode deformable mirror

To reduce the uncorrected higher-order aberrations for holographic gratings requiring an extreme dispersion, we have modified the Rowland mounting by moving the recording laser sources away from the grating. Then, with a multimode deformable plane mirror to record the grating, the correction of all the aberrations up to the fourth order inclusive is found sufficient to obtain a high-quality image. Applied to the FUSE-LYMAN grating, with a groove density of as much as 5740 grooves/mm, for which a resolution of 30,000 was required, this new recording device produces a resolution from 139,000 to 222,000 over the spectral range.     

Imaging the charge distribution within a single molecule

Scanning tunnelling microscopy and atomic force microscopy can be used to study the electronic and structural properties of surfaces, as well as molecules and nanostructures adsorbed on surfaces, with atomic precision, but they cannot directly probe the distribution of charge in these systems. However, another form of scanning probe microscopy, Kelvin probe force microscopy, can be used to measure the local contact potential difference between the scanning probe tip and the surface, a quantity that is closely related to the charge distribution on the surface. Here, we use a combination of scanning tunnelling microscopy, atomic force microscopy and Kelvin probe force microscopy to examine naphthalocyanine molecules (which have been used as molecular switches) on a thin insulating layer of NaCl on Cu(111). We show that Kelvin probe force microscopy can map the local contact potential difference of this system with submolecular resolution, and we use density functional theory calculations to verify that these maps reflect the intramolecular distribution of charge. This approach could help to provide fundamental insights into single-molecule switching and bond formation, processes that are usually accompanied by the redistribution of charge within or between molecules.     

Simulating gas-discharge processes at a single cold microscopic point

The development of ionization avalanches in nitrogen at atmospheric pressure near a single cold microscopic point on a cathode surface has been simulated under the conditions of E/P ≫ 1 kV/(cm Torr), where E is the electric field strength and P is the gas pressure. It is established that a layer of dense gas-discharge plasma with a density of ∼10¹⁶ cm⁻³ is formed within a period of ∼1 ps as a result of the gas ionization by electrons emitted from the cathode. The current of fast electrons, which appears due to gas ionization is more than ten times greater than the field emission current and can reach I ∼ 1 A for one microscopic point.