Numerical Ray Research Articles

High‐Latitude Off‐Great Circle Propagation Associated With the Solar Terminator

AbstractLarge‐scale ionospheric gradients associated with the solar terminator can deflect high frequency (HF) radio waves to off‐great circle paths during the morning and evening, negatively impacting technologies reliant on HF radio wave propagation. For example, geolocation algorithms used by scientific and military over‐the‐horizon radars (OTHRs) generally assume on‐great circle propagation, and thus lateral deviations from the great‐circle path can lead to positioning errors. In this study, radio wave propagation is simulated via 3D numerical ray traces though an empirical, high‐latitude model ionosphere initialized for a variety of times of the day and year to explore and quantify high‐latitude off‐great circle propagation associated with the solar terminator. Analysis of these simulations show large scale east‐west ionospheric gradients due to the solar terminator can cause lateral deviations in north‐directed propagation paths exceeding 20° at sunrise and sunset depending on radio wave frequency, though the largest portion of received signal power tends to experience maximum deflections of 5°. An exploration of the dependence of propagation direction on deflection shows that propagation paths parallel to the solar terminator tend to experience the largest deflections. Since the solar terminator at high latitudes is at an angle with respect to north in the winter and summer, propagation paths oriented west or east of north can experience larger deflections than north oriented paths at sunrise and sunset during these times of year. Impacts of these diurnal deflections on the operation of OTHR and scientific radar are discussed, as well as possible strategies for mitigating them.

Comparison of a Bottom-Up GNSS Radio Occultation Method to Measure D- and E-Region Electron Densities with Ionosondes and FIRI

High-frequency skywave propagation can be heavily impacted by D- and E-region dynamics requiring accurate global measurements to optimize performance. A standard measurement technique is to use ionosondes, but they are unable to measure below 1 MHz and are only available at a limited number of land-locked sites around the globe. In contrast, the Global Navigation Satellite System radio occultation (GNSS-RO) bottom-up method is a new approach specifically designed to generate electron density profiles in the D- and E- region ionosphere. It takes advantage of satellite constellations that currently provide over 20,000 daily measurements and global coverage. In this paper, GNSS-RO profiles were compared against ionosonde profiles at four sites covering a wide latitudinal range, and FIRI modeled profiles corresponding to the same latitude and local solar time. This comparison was completed using daytime profiles when sporadic-E (Es) was not present. The average GNSS-RO profile is found to be a few kilometers higher in altitude than the ionosonde profiles at the minimum frequency, fmin. When the ionosonde profiles are shifted so that the altitudes match at fmin, they are in good agreement up to the E-region peak altitude, hmE. Below fmin, the GNSS-RO profile is in good agreement with the FIRI profile, indicating that the profiles can measure the D- to E- transition region. The frequency of the E-region peak, foE, showed general agreement between the GNSS-RO and ionosonde measurements; however, the hmE agreement was weaker and the GNSS-RO profiles tend to have an hmE in a narrow altitude range for all profiles. Virtual heights were simulated for the GNSS-RO profiles using a numerical ray tracer for direct comparison with ionosonde observations, which showed agreement for many of the virtual heights near fmin, but also indicated a positive bias in the GNSS-RO virtual heights that may be due to low foE or elevated hmE estimates. For a quiet ionosphere, the shifted GNSS-RO electron density profiles were a good match for both measured ionosonde profiles and modeled FIRI profiles and the method is capable of providing global coverage of the D- and E-regions. Future work will require more data for seasonal and morning–afternoon comparisons as well as comparisons for the disturbed ionosphere when the sporadic-E layer is present.

Algebraic ray trace analysis of spatial heterodyne spectrometers.

Algebraic ray traces of various configurations of spatial heterodyne spectrometers are developed to derive general, approximate, formulas for resolving the power, fringe localization plane, and admissible off-axis angle for each configuration. Michelson, all-reflective, and field-widened configurations are considered separately. The derived formulas for each configuration are tested against exact numerical ray traces using optical design software and in general found to be in good agreement.

Modelling eye lengths and refractions in the periphery.

To create a simplified model of the eye by which we can specify a key optical characteristic of the crystalline lens, namely its power. Cycloplegic refraction and axial length were obtained in 60 eyes of 30 healthy subjects at eccentricities spanning 40° nasal to 40° temporal and were fitted with a three-dimensional parabolic model. Keratometric values and geometric distances to the cornea, lens and retina from 45 eyes supplied a numerical ray tracing model. Posterior lens curvature (PLC) was found by optimising the refractive data using a fixed lens equivalent refractive index ( ). Then, was found using a fixed PLC. Eccentric refractive errors were relatively hyperopic in eyes with central refractions ≤-1.44 D but relatively myopic in emmetropes and hyperopes. Posterior lens power, which cannot be measured directly, was derived from the optimised model lens. There was a weak, negative association between derived PLC and central spherical equivalent refraction. Regardless of refractive error, the posterior retinal curvature remained fixed. By combining both on- and off-axis refractions and eye length measurements, this simplified model enabled the specification of posterior lens power and captured off-axis lenticular characteristics. The broad distribution in off-axis lens power represents a notable contrast to the relative stability of retinal curvature.

Multi-input model uncertainty analysis for long-range wind farm noise predictions

One of the major sources of uncertainty in predictions of wind farm noise (WFN) reflect parametric and model structure uncertainty. The model structure uncertainty is a systematic uncertainty, which relates to uncertainty about the appropriate mathematical structure of the models. Here we quantified the model structure uncertainty in predicting WFN arising from multi-input models, including nine ground impedance and four wind speed profile models. We used a numerical ray tracing sound propagation model for predicting the noise level at different receivers. We found that variations between different ground impedance models and wind speed profile models were significant sources of uncertainty, and that these sources contributed to predicted noise level differences in excess of 10 dBA at distances greater than 3.5 km. We also found that differences between atmospheric vertical wind speed profile models were the main source of uncertainty in predicting WFN at long-range distances. When predicting WFN, it is important to acknowledge variability associated with different models as this contributes to the uncertainty of the predicted values.

Optical Quality Variation of Different Intraocular Lens Designs in a Model Eye: Lens Placed Correctly and in an Upside-Down Position

Introduction: Intraocular lenses (IOLs) may lose their optical quality if they are not correctly placed inside the capsular bag once implanted. One possible malpositioning of the IOL could be the implantation in an upside-down position. In this work, three aspheric IOLs with different spherical aberration (SA) have been designed and numerically tested to analyse the optical quality variation with the IOL flip, and misalignments, using a theoretical model eye. Methods: Using the commercial optical design software OSLO, the effect of decentration and tilt was evaluated by numerical ray tracing in two conditions: in their designed position and flipped with respect to the planned position (IOL is implanted upside down). The theoretical model eye used was the Atchison model eye. Seven IOL designs of +27.00 diopters were used: a lens with negative SA to correct the corneal SA, a lens to partially correct the corneal SA, and a lens to not add any SA to the cornea (aberration-free IOL). These lenses were designed with the aspherical surface located on the anterior and posterior IOL surface. A lens with no aspherical surfaces was also included. For the optical quality analysis, the modulation transfer function (MTF) was used, together with the Zernike wavefront aberration coefficients of defocus, astigmatism, and primary coma. Results: Off-centring and tilting the IOL reduced overall MTF values and increased wavefront aberration errors. With the IOL correctly positioned within the capsular bag, an aberration-free IOL is the best choice for maintaining optical quality. When the IOL is flipped inside the capsular bag, the optical quality changes, with the aberration-free IOL and the IOL without aspheric surfaces providing the worst results. With the lens in an upside-down position, an IOL design to partially correct corneal SA shows the best optical quality results in decentration and tilt, in terms of MTF and wavefront aberrations. Conclusion: The aberration-free IOL is the best choice when minimal postoperative errors of decentration or tilt are predicted. With IOL flip, the negative SA lens design is the best choice, regarding the root mean square wavefront aberrations. However, in a proper IOL implantation, the IOL designed to partially compensate the corneal SA including asphericity on its posterior surface is the better possible option, even in the presence of decentration or tilt.

The Southern African neutron monitor program: A regional network to study global cosmic ray modulation

Neutron monitor (NM) observations provide a unique, long-term record of galactic cosmic ray intensities, from which many valuable insights have been drawn as to the transport and modulation of these particles. In this paper, the African Neutron Monitor Network, which provides NM observations for broad range of cutoff rigidities, and spanning an exceptional period of time, is introduced. Aspects of these NM observations from this network are discussed, followed by a demonstration of time-dependent numerical cosmic ray modulation modelling done at the North-West University in South Africa, which hosts this network.

The concept of a stigmatic flat-field X-ray spectrograph based on conical diffraction

The optical scheme of a flat-field stigmatic X-ray spectrograph, which includes a grazing-incidence focusing mirror and a concave fan-type diffraction grating crossed with respect to the mirror and mounted in a conical diffraction scheme, is calculated analytically. Spectral images of a point monochromatic source are obtained by numerical ray tracing, confirming the high quality of spectral images at a level of 1 × 4 μm. It is assumed that the use of small grazing incidence angles in combination with the application of multilayer reflective coatings, including aperiodic ones, will make it possible to extend the working spectral range of the stigmatic spectrograph to the region of the ‘tender’ (ℏω ≈ 1.5 – 6 keV) X-ray range.

Effect of decentration, tilt and rotation on the optical quality of various toric intraocular lens designs: a numerical and experimental study.

Toric intraocular lenses (T-IOLs) may lose their optical quality if they are not correctly positioned inside the capsular bag once implanted. In this work, T-IOLs with cylinder powers of +1.50, +4.50 and +7.50 D and differing degrees of spherical aberration have been designed, manufactured and tested in vitro using a commercial optical bench that complies with the requirements of standard ISO 11979-2. Moreover, the effect of tilt and rotation on optical quality was assessed by means of numerical ray tracing on an astigmatic eye model, while the effect of decentration was evaluated numerically and experimentally.

A detailed comparison of techniques used to model drift in numerical cosmic ray modulation models

Drifts have long been known to play a significant role in the transport and modulation of galactic cosmic rays (GCRs), occurring due to gradients in, as well as the curvature of, the heliospheric magnetic field (HMF). GCRs also drift along the wavy current sheet, a structure separating regions of opposite HMF polarity. Furthermore, the influence of drift effects is solar cycle dependent, as can, for example, be seen in the 22-year cycle observed in neutron monitor observations, corresponding to the 22-year HMF polarity cycle. As such, over the previous decades several techniques have been proposed to incorporate these effects into numerical cosmic ray modulation models. The present study compares the effects of employing these techniques, using a 3D, stochastic, ab initio GCR modulation model with a simulated solar cycle dependence. It is found that, for solar minimum conditions, all methods considered yield similar GCR intensities, but that the choice of method becomes significant when heliospheric conditions approaching solar maximum are to be modelled.

Design of a high-resolution VLS monochromator for synchrotron radiation

A high-resolution monochromator with a broad spectral range of 125-4200 Angstrem is designed for a measuring beamline of the projected synchrotron radiation source "SKIF" (Novosibirsk). The optical configuration of the monochromator comprises a grazing-incidence concave mirror, a plane VLS grating, and an exit slit. It is planned to use two replaceable VLS gratings with central groove frequencies of 600 and 150 mm-1 intended for subranges of 125-1000 Angstrem and 900-4200 Angstrem, respectively. Wavelength tuning in each of the two subranges is carried out by solely the rotation of the VLS-gratings. Due to the proper choice of p1 VLS-grating coefficients, the focal distance varies only slightly over the entire spectral range, and the p2 VLS-grating coefficients are used to suppress the aberrations of the mirror and the gratings. The resolving power of the configuration obtained by numerical ray tracing exceeds 1000 in the 125-1000 Angstrem range and 2000 in the 900-4200 Angstrem range. Keywords: VUV, spectroscopy, VLS-grating, synchrotron radiation.

A Continuity Flow Based Tomographic Reconstruction Algorithm for 4D Multi-Beam High Temporal-Low Angular Sampling.

A mathematical framework and accompanying numerical algorithm exploiting the continuity equation for 4D reconstruction of spatiotemporal attenuation fields from multi-angle full-field transmission measurements is presented. The algorithm is geared towards rotation-free dynamic multi-beam X-ray tomography measurements, for which angular information is sparse but the temporal information is rich. 3D attenuation maps are recovered by propagating an initial discretized density volume in time according to the advection equations using the Finite Volumes method with a total variation diminishing monotonic upstream-centered scheme (TVDMUSCL). The benefits and limitations of the algorithm are explored using dynamic granular system phantoms modelled via discrete elements and projected by an analytical ray model independent from the numerical ray model used in the reconstruction scheme. Three phantom scenarios of increasing complexity are presented and it is found that projections from only a few (unknowns:equations > 10) angles can be sufficient for characterisation of the 3D attenuation field evolution in time. It is shown that the artificial velocity field produced by the algorithm sub-iteration, which is used to propagate the attenuation field, can to some extent approximate the true kinematics of the system. Furthermore, it is found that the selection of a temporal interpolation scheme for projection data can have a significant impact on error build up in the reconstructed attenuation field.

Bubble optics: Leonardo's cross revisited-Part 3, nonparaxial analytical methods.

In about 1508, Leonardo da Vinci first noted and sketched a cross-shaped caustic produced by a plane wave diagonally incident on the circularly symmetric meniscus surrounding a bubble on the surface of a water-filled container. This caustic was analyzed in two companion papers using a numerical ray theory spot diagram approach, and paraxial analytical methods. In this study we employ nonparaxial analytical methods, which are necessary when the outer bubble meniscus rises high above the ambient water surface. We also use a simple exactly soluble approximation to the outer meniscus shape, and obtain a distorted astroid caustic. We show that one of the refraction astroid cusp points is blocked by an aperture effect for steeply diagonally incident light, and the missing cusp is approximately replaced by a new cusp due to light transmitted through the bubble into the water, and reflecting from the underside of the outer meniscus.

Bubble optics: Leonardo's cross revisited-Part 2, paraxial analytical methods.

Around 1508, Leonardo da Vinci first noted and sketched a cross-shaped caustic produced by a plane wave diagonally incident on the meniscus surrounding a bubble on the surface of a water-filled container. This caustic was analyzed in a companion paper using a numerical ray theory spot diagram approach and is analyzed here using analytic ray theory approaches appropriate to the paraxial regime. We demonstrate that the caustic assumes the standard astroid form when the maximum meniscus slope is small, and the shape starts to distort as the maximum meniscus slope increases. We show that competition between generalized coma and generalized astigmatism, with generalized coma dominating, is responsible for the cusp points of the astroid in the plane of the incident light being reversed with respect to their images in the exit plane of the optical system. We also show that Leonardo's cross is the result of the twofold symmetry of the astigmatic focusing of the diagonally incident plane wave by the circularly symmetric meniscus.

Improved Analytical Theory of Ophthalmic Lens Design

A revisited form of the classic third-order ophthalmic lens design theory that provides a more precise and meaningful use of aspheric surfaces and a generalization of the standard oblique errors is presented. The classical third-order theory follows from the application of the Coddington equations to a ray trace through the lens and the expansion of the incidence angles and the surface sagittas appearing on them up to order two of the radial coordinate. In this work we show that the approximations for surface sagittas and angles can be decoupled, and the lens oblique powers predicted by the proposed theory provides a better fit to the numerical results obtained by exact raytracing and multi-parametric optimization than the classical third-order theory does. Modern ophthalmic lens design uses numerical optimization and exact ray tracing, but the methods presented in this paper provide a deeper understanding of the problem and its limitations. This knowledge and the more general merit functions that are also presented may help guide the numerical approaches.

Analysis of Vase Shaped Pumping Cavity for Solar-Pumped Laser

Although sunlight is a promising renewable energy source, the light is incoherent and difficult to use directly. Therefore, a solar-pumped laser, which directly converts sunlight into coherent laser light of, is a promising technology. A solar-pumped laser collects sunlight into the laser medium to realize laser oscillation. In order to realize an efficient solar-pumped laser system, it is necessary to design a pumping cavity that absorbs maximal sunlight into the laser medium with minimal thermal shock. In this research, the pumping cavity shape was studied using a numerical ray tracing simulation. As a result, it was found that a cone shaped pumping cavity can be expected to improve the absorption rate by approximately 30% over a cylindrically shaped pumping cavity. Furthermore, the absorption power density distribution can be flattened by a vase shaped pumping cavity, while maintaining the same absorption efficiency. The vase shaped pumping cavity has almost half the dispersion of the absorbed power density in the laser medium when compared with the cone shaped pumping cavity.

Схема VLS-монохроматора высокого разрешения для синхротронного излучения

A high-resolution monochromator with a broad spectral range of 125 – 4200 Angstrem is designed for the measuring beamline for the projected synchrotron radiation source "SKIF" (Novosibirsk). The optical configuration of the monochromator comprises a grazing-incidence concave mirror, a plane VLS grating, and an exit slit. It is planned to use two replaceable VLS gratings with central line frequencies of 600 mm-1 and 150 mm-1 intended for subranges of 125 – 1000 Angstrem and 900 – 4200 Angstrem , respectively. Wavelength tuning in each of the two subranges is effected by only the rotation of the VLS-gratings. Due to the proper choice of p_1 VLS-grating coefficients, the focal distance varies only slightly over the entire spectral range, and the p_2 VLS-grating coefficients are used to suppress the aberrations of the mirror and the gratings. The resolving power of the configuration obtained by numerical ray tracing exceeds 1000 in the 125 – 1000 Angstrem range and 2000 in the 900 – 4200 Angstrem range.

FANM: A software for focus and aberrations of nuclear microprobe

Focus and Aberrations of Nuclear Microprobe (FANM) is a new beam optics package to achieve fast and accurate design of a nuclear microprobe. FANM achieves a balance between speed of focusing and accuracy of high order aberrations. A combined method proposed in FANM is to achieve focusing conditions using a matrix method and to calculate aberration coefficients using a numerical ray tracing method. FANM has two optional optimization algorithms (derivative-free method and particle swarm optimization), offering a powerful choice for multi-variable optimization. Numeric variables in FANM are stored as 64-bit (8-byte) double-precision floating-point values to reduce calculation errors. Results obtained with FANM are compared with six existing computational tools (PRAM, TRANSPORT-PSI, TRANSPORT-PBO, WinTRAX, GEANT4-nanobeam and Zgoubi). The reasons for discrepancies between the different packages are discussed. Calculated currents of magnetic quadrupole lenses obtained with FANM are compared with experimental values of a nuclear microprobe in the National University of Singapore.

Implementation of the toroidal absorption cell with multi-layer patterns by a single ring surface.

We developed a type of toroidal multi-pass cell with multi-layer patterns based on the off-axis model. The effective path length of the original toroidal multi-pass cell is extended several roundtrips in comparison with the single-layer pattern, since the inner surface of the toroidal multi-pass cell is more efficiently utilized. The light pattern has been achieved by using the simple ring surface, which is easy to fabricate. The exact analytical equations for the design of the toroidal multi-pass cell were derived based on analytical vector calculations. A series of numerical ray tracing simulations is presented, and the maximum theoretical optical path length that can be reached is 30 m with a setup of 5 cm column radius. Furthermore, two practical spot patterns are demonstrated with a path length of 8.3 m for a two-layer pattern and 10 m for a three-layer pattern, with respective effective volumes of 63 mL and 94 mL. Furthermore, the fringe effect is substantially reduced to less than 0.5% by the usage of our designed mask.

Oblique and vertical incidence ionogram simulations with the presence of Es layer

The methodology making available an analysis of high frequency (HF) wave propagation in ionosphere with the presence of sporadic E layer (Es) is presented. A numerical ray tracing code is exploited to estimate the ray-paths through ionosphere comprising E, Es, F1 and F2 layers. A number of vertical and oblique sounding ionograms were emulated in the frame of NIM-RT propagation model. Fairly good resemblance between measured and simulated ionograms with the presence of Es layer was achieved. Numerical modelling provides an assessment of a number of parameters defining Es layer features including virtual height of the layer and its transparency.