Analytical Ray Research Articles

High-accuracy small-angle measurement of the positioning error of a rotary table by using multiple-reflection optoelectronic methodology

To improve upon and provide a methodological alternative to published optoelectronic angle measurement systems, the paper performs high-accuracy angle measurement by use of laser diodes, dual-axis position sensing detectors, and a series of reflections between two first-surface mirrors. Measurement accuracy improves from 0.5 to 0.05 arc sec as the light ray is reflected back and forth several times between the mirrors. Analytic ray tracing is used to model the reflected light ray so as to determine the system equations implicitly in terms of the measured angles. The first-order Taylor series expansion provides a linear form of the system equations. To validate the system, a prototype is built. Calibration and stability experiments are performed. Experimental results show the resolution, accuracy, and measurement range are, respectively, 0.008, 0.05, and ±250 arc sec. Compared with traditional optical angle measurement systems, this method has many merits such as low cost, simple configuration, high sensitivity, and high linearity.

Guided self-assembly of integrated hollow Bragg waveguides

We describe the fabrication of integrated hollow waveguides through guided self-assembly of straight-sided, thin film delamination buckles within a multilayer system of chalcogenide glass and polymer. The process is based on silver photodoping, which was used to control both the stress and adhesion of the chalcogenide glass films. Straight, curved, crossing, and tapered microchannels were realized in parallel. The channels are cladded by omnidirectional dielectric reflectors designed for low-loss, air-core guiding of light in the 1550-1700 nm wavelength range. Loss as low as ~15 dB/cm was measured for channels of height ~2.5 mum, in good agreement with both an analytical ray optics model and finite difference numerical simulations. The loss is determined mainly by the reflectivity of the cladding mirrors, which is ~0.995 for the as-fabricated devices.

Uneven curing induced interfacial delamination of UV adhesive-bonded fiber array in V-groove for photonic packaging

The common approach to attaching a large number of fibers to a guided-wave device is to fabricate a linear array using V-grooves. UV-curable adhesives are used to attach the fiber in V-groove for the fabrication of fiber array. Interfacial delaminations at the adhesive fiber interfaces are common after the reliability test due to uneven curing of the UV adhesive. This paper investigated the causes of uneven curing that gave rise to delamination. An analytical ray tracing technique was used to estimate the variation in light intensity or shadowing during the UV curing of adhesive. These effects were found to be very severe when the refractive index difference between the adhesive and the cladding materials is very large. Based on this study, it can be concluded that the delamination problem can be minimized by selecting a UV-curable adhesive having the same refractive index of the cladding material. It is also recommended to light illuminate from the topside for fiber packaging

Delamination Problems of UV-Cured Adhesive Bonded Optical Fiber in V-Groove for Photonic Packaging

In this letter, we investigated the origin of the interfacial delamination due to uneven curing of ultraviolet (UV) adhesive in the fixing process of a fiber on a V-groove. The analytical ray tracing technique is used to estimate the UV intensity at different locations along the adhesive-fiber interface. Consequently, the unexposed or shaded region in the adhesive layer is determined, through which the interfacial delamination as a result of the reliability test is predicted.

Analytical and numerical ray tracing of a transient x-ray laser: Ni-like Ag laser at 139 nm

Transient soft-x-ray lasers are generated from a solid target irradiated by two intense pump laser pulses. Amplification is achieved in the plasma column thus produced. Knowledge of the beam propagation is vital for the intensity and quality of the x-ray laser output. In this paper, x-ray laser beam propagation in transient plasmas is studied both analytically and numerically. General one-dimensional formulas are developed for beams in electron density gradient media, including the exponential profile that describes the plasma created from a solid target. The gradient is predicted to limit the amplification length within the maximum gain to <2.6 mm in standard experiments. The result given by the analytical model is confirmed by numerical ray tracing of x-ray laser beams within an amplifying medium as it is defined by the full numerical simulation results of the ehybrid code.

Explicit Anisotropic P-wave Ray Velocity Functions

Sedimentary rocks encountered in seismic exploration frequently exhibit directional seismic wave velocity dependence. These anisotropic rocks often can be regarded as transversely isotropic (TI) media in which the ray velocity surface of SH-waves is elliptical, while those of P- and SV-waves are non-elliptical. Transversely isotropic phase velocity expressions exist for P-, SH- and SV-waves. Unfortunately, until now no explicit exact analytical ray velocity functions exist for P- or SV-waves propagating in TI media. In exploration seismology, explicit ray velocity functions are needed for efficient migration, velocity analysis and other seismic processing procedures.Approximate explicit analytical P-wave ray velocity functions in transversely isotropic media have now been developed. These ray velocity functions include a new dimensionless constant, A, which depends on elastic parameters C11, C33, C44, and C13 or Thomsen's parameters ε, δ, α0 and (β0. The accuracy of the functions has been tested for all the materials published by Thomsen (1986). From these ray velocity functions, the travel time between any two points in homogeneous transversely isotropic media may be calculated. Reflection travel times from a single horizontal reflector overlain by transversely isotropic media with vertical symmetrical axis (VTI) are also calculated and compared with those obtained by Tsvankin (1996). The results show that the proposed explicit ray velocity functions have high accuracy.

Gradient-index rods as flux concentrators with applications to laser fiber optic surgery

Gradient-index rod lenses have been developed and ana- lyzed as imaging optical elements, but their capability as flux concentra- tors has not been explored. Both analytic methods and computer ray- trace simulations are used for a comprehensive evaluation of the concentration and efficiency of these rods. They appear to be well suited as concentrators for the distal end of laser fiber optic surgical units, toward improving surgical efficacy and reducing expensive laser power. © 2001 Society of Photo-Optical Instrumentation Engineers. (DOI: 10.1117/1.1347998) Subject terms: concentrators; biomedical optics; nonimaging optics; gradient in- dex; optical design; geometric optics.

Determination of P-Wave Velocity Structures, Earthquake Hypocenters, and Focal Mechanisms in the Morgan Hill Region of Central California

P-wave velocity structure, earthquake hypocenters, and focal mechanisms are investigated by applying an analytic ray tracing technique to 2437 P-wave arrivals from 154 locally recorded earthquakes from the Morgan Hill region of central California. Taking as our reference the results of a previous investigation by Michael (1988) on the effects of laterally varying structure on these parameters of hypocenters and focal mechanisms, we found that to the first order the principal structures are preserved, but that features associated with the fault zone tend to he better focused and of higher contrast when the analytic ray tracer is used. In particular, the low velocities associated with the Madrone Springs and Calaveras faults are confined to a narrow zone of about 5 km width. Contrasts in velocities associated with this zone are 2-3% larger at shallow depth. These contrasts gradually decrease with depth, but are evident to depths of at least 12 km. Hypocenters relocated in this structure shift systematically to the east by 1-2 km and their depths generally increase by as much as 5 km. The combined effect of the epicenter and depth shift is to decrease the apparent dip of the seismic zone by 4° and to move the surface intercept of the seismic zone to the east by 0.8 km. Compared with those determined by approximate ray tracing, initial ray directions calculated with the analytic ray tracer are within+5° in azimuth but can be as much as 30° different in take-off angle. The danger of attempting focal mechanism analysis with only a few arrivals is therefore evident. The effects of these differences on the deduced focal mechanisms themselves appear to be on the order of 10° in the strike and dip of the probable fault plane.

Multiple quasi-parabolic presentation of the IRI profile

The paper describes the representation of any given density profile with multiple quasi-parabolic (QP) segments and the technique is applied to IRI profiles. An optimization method is introduced that automatically finds the segment boundaries and minimizes the number of segments for a specified fitting accuracy. This representation of the profiles makes it possible to use analytical ray tracing techniques for ionospheric radio wave propagation studies. The Lowell Digisondes now provide the multiple QP profiles in real time as an additional output.

An evaluation of a new two‐dimensional analytic ionospheric ray‐tracing technique: Segmented method for analytic ray tracing (SMART)

Analytic ray tracing is considerably less time consuming than numerical ray tracing. However, despite the advantage of speed, analytic ray tracing has a major limitation: the difficulty of including the effects on the ray path of horizontal gradients in electron density. In the past this difficulty has been addressed using ionospheric models whereby an effective tilt is introduced by displacing the center of the ionosphere from the center of the Earth. In this paper, ray equations for the ground range, phase path, group path, and divergent power loss are developed for a new two‐dimensional analytic ray tracing technique, known as the segmented method for analytic ray tracing (SMART), which is able to deal with horizontal gradients in electron density. Ray tracing using two tilting methods are compared with ray tracing using SMART. Further, these results are compared with a numerical ray tracing program known as homing‐in ray tracing (HIRT).

Centrifugal instability and the rib vortices in the cylinder wake

The physical mechanism for generation of streamwise vortices (orrib vortices) in the cylinder wake is numerically investigated with a finite-differencescheme. Rayleigh’s theory of centrifugal instability for inviscid axisymmetric flow isextended to analyze the 2-D primary flows. Accordingly, an analytical dimensionlessgroup Ray=-(r/vθ)■vθ/■r-1 is derived, where vθ represents the velocity of a fluidelement relative to the oncoming flow, r is the local curvature radius of the elementpathline. Centrifugal instability occurs when Ray>0. Stability analyses are carriedout with this discriminant for primary flows at different time levels in a half sheddingperiod of the von Karman (or vK) vortices. Unstable areas are identified and thelocations of rib vortices are coincident well with the unstable areas within the firstwavelength of vK vortices behind the cylinder. The numerical results also show thatrib vortices experience amplification in this region. It is apparent that centrifugalinstability plays an important role in the generation of rib vortices in the cylinderwake.

The effect of large‐scale ionospheric gradients on backscatter ionograms

This paper presents the results of the synthesis of a range of backscatter ionograms using ray tracing through model ionospheres. The backscatter ionograms were obtained by the Jindalee over‐the‐horizon radar facility at Alice Springs in northern Australia. Sample ionograms obtained during 1990 were used, and the study concentrated on reproducing effects due to sunrise‐sunset gradients and the equatorial anomaly. Backscatter ionograms were synthesized using both analytical and numerical ray tracing through ionospheric models based on FAIM (fully analytic ionospheric model). To make the synthesis realistic, signal strength was calculated taking account of ray divergence, ionospheric absorption, and antenna patterns. Analytical ray tracing produced quite realistic results when horizontal gradients were small but did not reproduce prominent features observed during sunrise‐sunset or when propagation occurred through the equatorial anomaly region. Since the analytical ray tracing was restricted to a single vertical profile which could be tilted, this result shows that gradients in ionospheric electron density, rather than simple tilts, are most significant in determining propagation characteristics. Numerical ray tracing through ionospheric models based on the FAIM model reproduced dominant features of backscatter ionograms for those situations when analytical ray tracing proved inadequate. Major seasonal variations were also reproduced. Overall, the results of this initial study show that many premier features on backscatter ionograms, including the power variation of the backscattered signals, can be realistically modeled using ray tracing and ionospheric models. Further work is required before all the detailed structure of backscatter ionogram traces can be synthesized and accurately interpreted in terms of ionospheric structure.

Analytic ray parameters for the quasi‐cubic segment model of the ionosphere

Models of the ionosphere giving analytic ray solutions are very useful for the study of ionospheric propagation. Probably the most useful is the quasi‐parabolic model which gives analytic solutions for a spherical ionosphere when the effects of the Earth's magnetic field are ignored. Furthermore, its use in analytic ray tracing has been extended recently by developments in which real ionospheres are described by fitting quasi‐parabolic segments (QPS) to measured vertical profiles. While numerical ray tracing is required to take account of the range of horizontal gradients that occur in the ionosphere, in some real‐time applications, analytic results are still preferred because of the shorter computation time. Even though the QPS approach has greatly extended the utility of analytic ray tracing, it produces model ionospheres which are continuous in only the first derivative of the refractive index. The lack of continuity of the second derivative can lead to relatively abrupt changes in ray quantities with, for example, elevation angle. This drawback has been addressed by developing a model based on quasi‐cubic segments (QCS) which also provide analytic ray solutions. In this paper the QCS model is described, and analytic solutions are derived for range, group, and phase paths. An example of an application is presented in which the QCS model was used to check the performance of a numerical ray tracing code.

The equation and properties of a ray path in a linearly varying velocity field

The equation of ray path in a linearly varying velocity field in one-dimensional (1-D) case corresponds to an equation of circle whose center is defined by the initial take-off angle and the constant parameters of the velocity field. The physical ray path can exist only on the half circle, where the velocity is positive. The initial take-off angle of a ray passing through the two specified points is derived, the expression of which assures an analytical two-point ray tracing. The linearly varying velocity field in two-dimensional (2-D) or three-dimensional (3-D) can be transformed to that in 1-D. This transformation reduces the equation of ray path in a linearly varying velocity field in 2-D or 3-D to one in 1-D in the transformed coordinates. The results can be implemented for two-point ray-tracing calculation as well as analytical ray path and travel time calculation, both in forward and inverse problems.

A two‐dimensional analytic ray tracing technique accommodating horizontal gradients

A new two‐dimensional analytic ray tracing method, known as segmented method for analytic ray tracing (SMART), which automatically segments the ionosphere along the ray path is described. Unlike other techniques, SMART is able to accurately ray trace through horizontal gradients which vary with altitude along and in the direction of the ray path. Computer run times are approximately 10 times faster than those of numerical ray tracing packages, and ground range errors are typically less than 2.5%.

Study of sporadic-E clouds by backscatter radar

Abstract. It is shown that swept-frequency backscatter ionograms covering a range of azimuths can be used to study the dynamics of sporadic-E clouds. A simple technique based on analytic ray tracing can be used to simulate the observed narrow traces associated with Es patches. This enables the location and extent of the sporadic-E clouds to be determined. The motion of clouds can then be determined from a time sequence of records. In order to demonstrate the method, results are presented from an initial study of 5 days of backscatter ionograms from the Jindalee Stage B data base obtained during March-April 1990. Usually 2–3 clouds were observed each day, mainly during the evening and up to midnight. The clouds lasted from 1–4 h and extended between 30°–80° in azimuth and 150-800 km in range. The clouds were mostly stationary or drifted generally westward with velocities of up to 80 m s–1. Only one cloud was observed moving eastward.

Propagation of sound above an impedance plane in a downward refracting atmosphere

An asymptotic analysis has been carried out to study the propagation of sound at long ranges in an arbitrary downward refracting sound‐speed profile. It has been shown that the solution can be expressed in a form of the Weyl–Van der Pol formula and the present theory offers an extension for the established theory to the case of a vertically stratified medium. The analysis starts from the full wave equation and the derivation assumes that the ambient properties only depend on the height above an impedance ground and that there is no wind or turbulence in the medium. In addition, the theory also assumes that the sound speed of the atmosphere increases monotonically with height. For such an atmospheric condition, the medium forms a waveguide that enhances the sound levels at long ranges along the ground surface. It is demonstrated that the acoustical path length derived by the asymptotic method is identical to that derived by the analytical ray trace model. Furthermore, the asymptotic method also allows a proper inclusion of the surface wave term rigorously.

The effect of realistic ground impedance on the accuracy of ray tracing

The heuristic model of sound propagation in the atmosphere has been developed by L’Espérance et al. [Appl. Acoust. 37, 111–139 (1992)] to provide rapid calculation of sound-pressure levels. This model uses an analytic ray trace method to calculate ray paths from the source to the receiver. The effects of finite ground impedance are treated by calculating the spherical wave reflection factor as if the atmosphere was homogeneous. The actual reflection angles and times of flight are used in this formulation. Since the speed of sound varies linearly with height, no focusing factors are calculated. The heuristic model has been compared to full wave calculations and is found to agree well at long ranges. In this paper the agreement between the heuristic model and the fast field program is investigated in detail. Criteria for the accuracy of ray tracing are discussed. It is demonstrated that the heuristic model produces realistic predictions even though certain criteria are violated. The finite ground impedance is shown to remove those rays for which the ray tracing criteria do not hold.

Finding eigenrays in environments prone to numerical instability and chaos by directly optimizing the travel time integral

In classical ray tracing, eigenrays are determined by a ‘‘shooting’’ approach whereby the launch angles of rays are varied until the rays intersect the receiver. In nonseparable range-dependent environments, the ray paths computed by conventional methods are sometimes chaotic, thereby putting a fundamental limit on the accuracy of ray tracing. Previous researchers [M. D. Collins and W. A. Kuperman, ‘‘Overcoming ray chaos,’’ J. Acoust. Soc. Am. 95, 3167–3170 (1994)] have suggested that ray chaos can be overcome by recasting the problem in terms of Fermat’s principle of minimum propagation time. The problem then becomes amenable to so-called ‘‘direct methods’’ of optimization theory. For the specialized duct investigated by Collins and Kuperman we have easily found eigenrays using a simple Rayleigh–Ritz method for directly minimizing the travel time integral. The structure of the ray equations for the duct suggests, however, that the numerical instability may actually be due to ‘‘stiffness’’ rather than chaos. To investigate chaos independently of numerical issues such as stiffness, we consider several nonseparable, range-dependent ducts with known piecewise analytic ray solutions. Results obtained from a shooting method and from Fermat’s principle for several such ducts are presented and discussed. [Work supported by ONR.]

Analytic calculation of the ordinary (O) and extraordinary (X) mode nose frequencies on oblique ionograms

On oblique ionograms, the maximum frequencies of the ordinary and extraordinary modes are referred to as nose frequencies. The difference in the nose frequencies depends on parameters such as the length of the propagation circuit and the direction of propagation. In this paper, the difference in nose frequencies is studied using the frequency scaling technique of Bennett et at. [(1991) Appl. Comput. Electromagn. Soc. Jl 6, 192]. For long paths, an explicit formula is obtained which gives the difference approximately as a function of the local magnetic dip and azimuth of propagation at the ray mid point. For shorter paths, it is shown how analytic ray tracing can be used to determine the difference.