Elliptical Zone Research Articles

Off-axis reflection zone plate for quantitative soft x-ray source characterization

A compact system for high-resolution spectroscopy and quantitative photon flux and brilliance measurements of pulsed soft x-ray sources is described. The calibrated system combines a novel elliptical off-axis reflection zone plate with charge-coupled device detection for simultaneous spectral and spatial measurements. Experiments on a water-window droplet-target laser-plasma source demonstrate λ/Δλ⩾1000 spectral resolution and absolute flux and brilliance measurements. © 1997 American Institute of Physics.

Character, paleoenvironment, rate of accumulation, and evidence for seismic triggering of Holocene turbidites, Canada Abyssal Plain, Arctic Ocean

Four box cores and one piston core show that Holocene sedimentation on the southern Canada Abyssal Plain for the last 8010 ± 120 yr has consisted of a continuing rain of pelagic organic and ice-rafted clastic sediment with a net accumulation rate during the late Holocene of ⩽10 mm/1000 yr, and episodically emplaced turbidites 1–5 m thick deposited at intervals of 830 to 3450 yr (average 2000 yr). The average net accumulation rate of the mixed sequence of turbidites and thin pelagite interbeds in the cores is about 1.2 m/1000 yr. Physiography suggests that the turbidites originated on the Mackenzie Delta or its clinoform, and δ 13C values of −27 to −25%. in the turbidites are compatible with a provenance on a delta. Extant displaced neritic and lower slope to basin plain calcareous benthic foraminifers coexist in the turbidite units. Their joint occurence indicates that the turbidites originated on the modern continental shelf and entrained sediment from the slope and rise enroute to their final resting place on the Canada Abyssal Plain. The presence of Middle Pleistocene diatoms in the turbidites suggests, in addition, that the turbidites may have originated in shallow submarine slides beneath the upper slope or outer shelf. Small but consistent differences in organic carbon content and δ 13C values between the turbidite units suggest that they did not share an identical provenance, which is at least compatible with an origin in slope failures. The primary provenance of the ice-rafted component of the pelagic beds was the glaciated terrane of northwestern Canada; and the provenance of the turbidite units was Pleistocene and Holocene sedimentary deposits on the outer continental shelf and upper slope of the Mackenzie Delta. Largely local derivation of the sediment of the Canada Abyssal Plain indicates that sediment accumulation rates in the Arctic Ocean are valid only for regions with similar depositional sources and processes, and that these rates cannot be extrapolated regionally. The location of an elliptical zone of active seismicity over the inferred provenance of the turbidites suggests that they were triggered by large earthquakes. Distal turbidite sediment accumulation rates were more than two orders of magnitude greater than pelagic sediment accumulation rates on the Canada Abyssal Plain during the last 8000 years. This disparity reconciles the discrepancy between the high accumulation rates assumed by some for the Arctic Ocean because of the numerous major rivers and large ice sheets that discharge into this small mediterranean basin and the low pelagic sedimentation rates that have been reported from the Arctic Ocean.

Non-paraxial analytical solution for the generation of focal curves

Abstract A solution for the transmittance of elements forming planar and nonplanar focal curves in the non-paraxial case is proposed in this paper. The treatment is based on the scalar diffraction theory and makes use of the stationary phase method in evaluation of the diffraction integral. The equation of a stationary point enables one to establish mapping relations between the geometrical regions of the element and the corresponding points of the focal curve. These regions turn out to be described by conical curves. The mapping relations can be manipulated by a phase function imposed onto the focal curve. The amplitude factors of the derived transmittances can be omitted and the desired intensity distribution along the focal curve can then be obtained by an appropriate change in the aperture of pure phase elements. The approach is illustrated by examples of elliptical, hyperbolic, conical and circular zone plates. The focus generations for both non-paraxial and paraxial cases of these elements are compa...

Determination of the 50% probability zone for an acoustic source-locating system operating in the ground plane

A prototype source locating system (SLS) consisting of two microphone systems is modelled using a computer simulation which produces predicted locations. The simulator predicts the elliptical zone within which the SLS would have a 50% probability of finding the source.

X-ray focusing optics using flat or bent crystals with two-dimensionally modulated surfaces

Calculations of focus characteristics for x-ray focusing systems using either flat or bent crystals with two-dimensional modulated surfaces are presented. Based on the wave-optics approach, the analytical formulas are derived for elliptical Fresnel zone structures (flat crystals) and linear Bragg zone structures (toroidally bent crystals). It is shown that for a given Si(111) reflection and Ti Kα radiation, a micron-size focal spot can be achieved by using only one flat crystal with a modulated surface. The peak intensity is then nearly 100 times higher than that of the flat crystal without modulations. However, for a toroidally bent crystal with both Fresnel and Bragg diffraction amplitudes, the latter becomes dominant, since, due to the bending, the rapid oscillations of the Fresnel diffraction amplitude vanish. It is also shown that a focal spot of 0.3 μm for a bent crystal with linear Bragg zone structures is approximately four times narrower than that of an unmodulated surface. From the analysis, the modulated surfaces for cylindrically and spherically bent crystals are deduced to be a combination of Bragg and Fresnel zones.

Imaging of extraaxial points by zone plates: stigmatism at a particular order

We present exact analytical expressions for the boundaries of individual zones of a zone plate that yields stigmatic imaging between two extraaxial conjugate points. Our treatment incorporates imaging at any order. It is found that the boundaries of the zones are elliptical and they are not concentric. We introduce dimensionless variables to arrive at convenient expressions for eccentricity and shift at the centre of the elliptical zone boundaries. Our approach enables us to obtain simple formulae for the two parameters. Approximations involved in the same are valid for most practical applications.

Correction of Myopia and Astigmatism Using an Ablatable Mask

Most excimer laser refractive procedures use a computer driven mechanical diaphragm to shape the laser beam. Studies are currently underway using an ablatable polymethylmethacrylate (PMMA) mask to transfer a new spherical or toric curve to the cornea for the correction of myopia and astigmatism; it may leave a smoother corneal surface than diaphragm procedures. As part of a Phase IIb FDA clinical study, 25 eyes of 25 patients underwent excimer laser photorefractive keratectomy using a hand held ablatable mask. Fifteen eyes had attempted spherical corrections of up to 6.00 diopters (D) and 10 had toric corrections of up to 6.00 D of sphere and 2.75 D of astigmatism. Seventy-four percent of all eyes achieved uncorrected visual acuity of 20/40 or better--86% in the spherical group and 63% in the astigmatism group. Sixty-nine percent of eyes were within +/- 1 D of the attempted correction. In eyes treated for astigmatism, mean astigmatism decreased from 1.48 D preoperatively to 0.86 D postoperatively. Approximately one half of the eyes treated for astigmatism had a decrease in cylinder of more than 0.5 D. One eye lost 2 Snellen lines of best spherical corrected visual acuity. Video keratography showed toric ablations to result in an elliptical optical zone. Analysis of centration of the procedure showed 66% of ablations centered within 1.0 mm of the center of the pupil aperture. The ablatable mask represents a promising modality for the treatment of eyes with both myopia and myopic astigmatism.

Generalized Zone Plates Focusing Light into Arbitrary Line Segments

Abstract A new computer-generated generalized zone plate is proposed, which is able to concentrate the incident beam into a line segment of various length and arbitrary inclination to the optical axis, as well as unrestricted distribution of the longitudinal intensity. The approach is based on the energy conservation principle and equations of the paraxial ray tracing, both considered in the cartesian coordinate system. Hyberbolic, elliptic, linear, and conical zone plates are shown to be limiting cases of the proposed element. Other specific cases, e.g. a new element focusing into a segment of the optical axis, are also derived.

Evaluation of the E-Test for susceptibility testing of pneumococci

The E-Test (AB Biodisk, Sweden) is an antibiotic gradient strip that is applied to an inoculated agar plate and results in an elliptical zone of inhibition that intercepts the graded strip, producing a quantitative (microgram per milliliter) result. Pneumococci (100) were used in this study, 38 penicillin-susceptible [minimal inhibitory concentrations (MICs), ≤0.12 μg/ml], 42 intermediately resistant (MICs, 0.12–1.0 μg/ml), and 20 resistant (MICs, >1 μg/ml). E-Test strips were evaluated on Mueller-Hinton agar plates with 5% sheep blood. Agar dilution MICs were determined by the National Committee for Clinical Laboratory Standards (NCCLS) method. Penicillin MICs for the E-Test tended to be slightly lower (one log 2 dilution) than reference MICs due to the continuous scale from which E-Test MICs were read. All but two penicillin-susceptible isolates were correctly categorized by the E-Test method. Of the 62 penicillin-resistant strains, 59 had E-Test MICs of ≥0.12 μg/ml, with 88% of these strains having E-Test MICs within one doubling dilution of the reference MICs. However, using the current NCCLS breakpoint MICs, many of the penicillin-resistant strains with reference MICs of 2 μg/ml were categorized as intermediate by the E-Test, with MICs of 0.38–1 μg/ml. For chloramphenicol, erythromycin, and tetracycline, correlation of the two methods was excellent. E-Test chloramphenicol MICs provided clearer separation of susceptible and resistant strains than did the reference method. We conclude that the E-Test is a reliable method for determination of MICs of the antibiotics evaluated for pneumococci.

Index profile of multilayer fibers of an elliptical cross-section

A mathematical formula is derived from the multiple-beam Fizeau fringers crossing an elliptical multilayer fiber. A formula for the two-beam interference fringes crossing this fiber is also given. A model that divides the elliptical fiber cross-section into elliptical zones of constant refractive index is assumed. The index profile of the poly(aryl-ether-ether-ketone) fiber is calculated by using the mathematical formulas. Microinterferograms of both the multiple-beam Fizeau fringes and the totally duplicated image of the fiber from a two-beam intereference microscope are used for the determination of the refractiveindex profile of the fiber. The cross-sectional shape is determined from the diffraction pattern of a He-Ne laser beam. Microinterferograms are given for illustration.

Astigmatism correction in x-ray scanning photoemission microscope with use of elliptical zone plate

We report the impact of an elliptical, high resolution zone plate on the performance of an initially astigmatic soft x-ray scanning photoemission microscope. A zone plate with carefully calibrated eccentricity has been used to eliminate astigmatism arising from transport optics, and an improvement of about a factor of 3 in spatial resolution was achieved. The resolution is still dominated by the source size and chromatic aberrations rather than by diffraction and coma, and a further gain of about a factor of 2 in resolution is possible. Sub 100 nm photoemission microscopy with primary photoelectrons is now within reach.

Microstructure and properties of supersaturated ZA27 alloy during natural ageing

The decomposition of the supersaturated solid solution of ZA27 alloy at room temperature has been investigated by X-ray diffraction, TEM and mechanical properties testing. Based on the results obtained, both continuous precipitation and cellular reaction occur during the decomposition process. The continuous precipitation follows the sequence: α′ → α1′ + spherical GP zones → α2′ + elliptical GP zones → α3′ + R → α + η. The cellular reaction can be written: α → α + η + e. The properties of the alloy depend on the microstructure. After 1 month of ageing, a series of changes of microstructure have taken place. The properties of the alloy are: σb=500 MPa, δ=13%,Hv=148.

Accuracy of Derived Ground Reaction Force Curves for a Rigid Link Human Body Model

This study investigated whether force data could be derived accurately using segment inertia data determined by the elliptical zone method (Jensen, 1976), automatic digitizing from high-speed video using a Motion Analysis VP110 system, and for an activity that does not require flexion of the thorax. The criterion fonctions were the force-time records of the jumps recorded at 500 Hz by a Kistler 9281B force platform. A second-order Butterworth digital filter was used to smooth the derived data, with frequency cutoffs being selected on the basis of root mean square error of the smoothed function with respect to the criterion force function. In a second procedure, the criterion function was the directly measured force-time record after filtering with a second-order Butterworth digital filter at 5 Hz to remove the high frequency part of the force signal. The closeness of fit of the derived data to the low frequency part of the criterion force was then assessed. It was concluded that, using the techniques described, the low frequency components of the ground reaction forces of drop jumps could be derived accurately.

Surface deformation and energy release rates for constant stress drop slip zones in an elastic half‐space

Free surface displacements, stress intensity factors, and energy release rates are calculated for planar slip zones in an elastic half‐space subjected to a prescribed shear stress drop. Although the method can treat arbitrarily shaped planar zones and distributed stress drops, for simplicity, results are presented only for circular and elliptic zones and uniform stress drops. Calculations of the stress intensity factors and energy release rates for various geometries indicate that solutions for the half‐space differ by less than 10% from those in the full space if the distance from the slip zone center to the free surface is greater than the downdip width of the slip zone. In addition, the influence of the free surface is greater for decreasing dip angle. For slip zones that are near the free surface and, especially, those that break the surface there is a coupling between slip and normal relative displacement. That is, for a prescribed shear stress drop and zero normal stress change, slip induces relative normal displacement. As example applications, these solutions are used to reexamine the coseismic geodetic data from three earthquakes: 1966 Parkfield, 1983 Borah Peak, and 1987 Whittier Narrows. The geometries, moments, and stress drops are similar to those inferred in previous studies using dislocation methods. However, the stress drop inferred here may be more reliable because stress drop is one of the parameters adjusted to fit the observed surface deformations. In addition, the method makes it possible to estimate the critical energy release rate at the termination of rupture. Values for the Parkfield, Borah Peak, and Whittier Narrows earthquakes are 1.5.×106 J/m2, 1.2×106 J/m2, and 2×108 J/m2, respectively.

A local weakening of the brittle-ductile transition can explain some intraplate seismic zones

A decrease in the strength of a localized area of the lower crust would decrease the depth to the brittle-ductile transition and concentrate stress in the stronger elastic crust around and above the zone of decreased strength. Deformation within the zone of weakened lower crust would occur through viscous or dislocation creep in response to regional plate stress as well as by elastic failure in small earthquakes. Two-dimensional finite-element models of zones of weakness subjected to a regional plate stress predict stress amplification of 10–100% surrounding the local decrease in strength. A finite-element model for an irregular zone of weakness in the lower crust can generate stress directions and relative magnitudes that satisfy the focal mechanisms and seismicity of southeastern Tennessee. An analysis of the displacements and stresses in the central area of decreased crustal strength suggests that strike-slip faulting should dominate. Above this zone the strike-slip faulting should exhibit a strong thrust component. The compression and extension of the crust surrounding the weakened zone at the level of the brittle-ductile transition predicts that the dominant strike-slip faulting should exhibit components of normal faulting on the edges of the weak zone which are near a line through the weak zone parallel to the regional stress and reverse faulting on the edges which are near a line transverse to the regional stress. The seismicity of southeastern Tennessee is diffused over a narrow elliptical zone trending NE. The greatest concentration of activity is near the center. The central zone is characterized by deep-focus strike-slip events with predominantly N or E striking nodal planes. These events are responding directly to compression in the direction of the regional compressive stress and extension perpendicular to the regional compressive stress within the weak zone. The area surrounding the central zone is characterized by focal mechanisms with larger components of reverse or normal fault movements. Events with normal components are dominant on the edges in line with the regional stress as predicted by the stress model for a weak central zone under stress. The agreement between observed earthquake focal mechanisms in southeastern Tennessee and models of crustal stress surrounding a zone of weakness suggests that these events may be caused by a zone of weakness in the lower crust.

Fabrication of high-resolution multilayer reflection zone plate microlense for the soft x-ray range

The fabrication of linear, circular, and elliptical reflective zone plate lenses by a multistep process using microfabrication technologies is reported. The diffractive elements were generated by electron-beam lithography on a Mo/C multilayer mirror obtained by means of the triode sputtering technique. The patterns were transferred anisotropically into the multilayer mirror by reactive ion etching in a fluorinated plasma. An intermediate metallic mask made by the lift-off process was used for the transfer process. The groove depth could be monitored by following the reflectivity ofthe structure with a helium-neon laser during etching. Linear, circular (with 0.4-μm outer zones), and elliptical Fresnel zone plates (with 0.8-μ.m outer zones) were produced. Test patterns with a 50-nm top and 90-nm bottom resolution were produced. The groove profile and dimensional control were also investigated.

Fabrication of holographic microlenses using a deep UV lithographed zone plate

Holographic microlens arrays and elliptical zone plates are fabricated on holographic plates and photoresists by interference between the first- and second-order diffracted waves from an ion etched Fresnel zone plate, which is made by electron-beam scanning and deep UV lithography. This method of fabrication is simple, and a holographic lens with a large numerical aperture and short focal length is obtained. Experimental results of focusing and imaging by a lens array and an elliptical zone plate are given. Two methods of fabricating holographic zone plates for an IR or UV wavelength are also proposed.

Analysis of the thermally stressed state of shells of arbitrary shape

Geometrically nonlinear relationships are derived for thin elastic shells subjected to nonuniform heating. A discrete system describing the thermally stressed state of a shell subject to significant shape variation is obtained using the concept of finite elements. The strain state is determined by the iteration method, based on computation of the first and second variations of the system's energy. Examples are cited for the calculation of a plate with an elliptical heating zone and a shell of canonical shape under nonstationary heating.

One-dimensional image and orthogonal Fourier transform with an elliptical zone plate

An elliptical zone plate is used for imaging along one direction the coherent light distribution and simultaneously obtaining the orthogonal 1-D Fourier transform. Experimental results are shown.

Display of phase-space functions by elliptical zone plates

A coherent processor to display the phase-space functions for 1-D signals is proposed. An elliptical zone plate is used as a single optical element. It is shown that, for a given 2-D input, the output plane shows the 1-D Fourier transform in one direction and the image in the other one. This lensless coherent processor can be used to display the Ambiguity Function, the Wigner Distribution Function and the Local Spectrum for some 1-D functions. Experimental results are presented.