Rigorous Ray Tracing Research Articles

A simplification of rigorous atmospheric raytracing based on judicious rectilinear paths for near-surface GNSS reflectometry

Atmospheric delays are known to cause biases in Global Navigation Satellite System Reflectometry (GNSS-R) altimetry applications, such as for sea-level monitoring. The main quantity of interest is the reflection-minus-direct or interferometric atmospheric delay. Recently, we have presented a rigorous raytracing procedure to account for linear and angular refraction in conjunction with reflection as observed from near-surface platforms. Here, we demonstrate the feasibility of simplifying the ray trajectory by imposing a rectilinear wave propagation model. Two variants were assessed, based on the apparent or refracted satellite direction on the one hand and the geometric or vacuum conditions on the other hand. The former was shown to agree with rigorous results in terms of interferometric radio length while the latter agreed in terms of the interferometric vacuum distance. Upon a judicious combination of the best aspects of the two rectilinear cases, we have defined a mixed variant with excellent agreement with rigorous raytracing in terms of interferometric atmospheric delay. We further showed that mapping functions developed for GNSS positioning cannot be reused for GNSS-R purposes without adaptations. Otherwise, the total atmospheric delay may be underestimated by up to 50% at low elevation angles. The present work facilitates the adaptation of existing atmospheric raytracing software for GNSS-R purposes.

Limitations on control performance in the Czochralski crystal growth process using bright ring measurement as a controlled variable

Abstract This paper develops a dynamical model for crystal diameter in the Czochralski process for production of monocrystalline silicon. The model combines simplified crystal growth dynamics with rigorous ray tracing to describe the camera image used for diameter control, and it is demonstrated that the resulting model captures the so-called measurement anomaly that represents a key performance limitation for crystal diameter control.

Rigorous analysis of an electric-field-driven liquid crystal lens for 3D displays

We numerically analyzed the optical performance of an electric field driven liquid crystal (ELC) lens adopted for 3-dimensional liquid crystal displays (3D-LCDs) through rigorous ray tracing. For the calculation, we first obtain the director distribution profile of the liquid crystals by using the Erickson-Leslie motional equation; then, we calculate the transmission of light through the ELC lens by using the extended Jones matrix method. The simulation was carried out for a 9view 3D-LCD with a diagonal of 17.1 inches, where the ELC lens was slanted to achieve natural stereoscopic images. The results show that each view exists separately according to the viewing position at an optimum viewing distance of 80 cm. In addition, our simulation results provide a quantitative explanation for the ghost or blurred images between views observed from a 3D-LCD with an ELC lens. The numerical simulations are also shown to be in good agreement with the experimental results. The present simulation method is expected to provide optimum design conditions for obtaining natural 3D images by rigorously analyzing the optical functionalities of an ELC lens.

고집광 태양광 발전을 위한 광학시스템 렌즈 개발

The artificial increase in the solar intensity incident on solar cells using lenses or mirrors can allow solar cells to generate equivalent power with a lower cost. There are two types of concentration optics for solar energy conversion. One is to use mirrors, and the other is to use Fresnel lenses. The gains that can be achieved with a Fresnel lens or a parabolic mirror are compared. The result showed the gains are comparable and the two configurations were developed competitively. In application areas of Fresnel lenses as solar concentrators, several variations of design were devised and tested. Some PV systems still use commercially available flat Fresnel lenses as concentrators. A convex linear Fresnel lens to improve the concentration ratio and the efficiency is devised and flat linear Fresnel lens in thermal energy collection is utilized. In this study, we designed and optimized flat Fresnel lens and the `light pipe` to develop 500X concentrated solar PV system. In the process, we compare the transmission efficiencies according to groove types. We performed rigorous ray tracing simulation of the flat Fresnel lenses. The computer aided simulation showed the `grooves in case` has the better efficiency than that of `grooves out case`. Based on the ray-trace results we designed and manufactured sample Fresnel lenses. The optical performance were measured and compared with ray-trace results. Finally, the optical efficiency was measured to be above 75%. All the design and manufacturing were performed based on that InGaP/InGaAs/Ge triple junction solar cell is used to convert the photon energy to electrical power. Field test will be made and analyzed in the near future.

Third-Order Theory of Spectacle Lenses Applied to Correction of Peripheral Refractive Errors

To demonstrate that relatively simple third-order theory can provide a framework which shows how peripheral refraction can be manipulated by altering the forms of spectacle lenses. Third-order equations were used to yield lens forms that correct peripheral power errors, either for the lenses alone or in combination with typical peripheral refractions of myopic eyes. These results were compared with those of finite raytracing. The approximate forms of spherical and conicoidal lenses provided by third-order theory were flatter over a moderate myopic range than the forms obtained by rigorous raytracing. Lenses designed to correct peripheral refractive errors produced large errors when used with foveal vision and a rotating eye. Correcting astigmatism tended to give large errors in mean oblique error and vice versa. When only spherical lens forms are used, correction of the relative hypermetropic peripheral refractions of myopic eyes that are observed experimentally, or the provision of relative myopic peripheral refractions in such eyes, appears impossible in the majority of cases. The third-order spectacle lens design approach can readily be used to show trends in peripheral refraction.

Analysis of thin-film silicon solar cells with plasma textured front surface and multi-layer porous silicon back reflector

The optical performance of thin crystalline silicon solar cells with plasma textured front surface and with porous silicon stack back reflector is analysed. A rigorous analytical ray tracing model, that uses an accurately estimated porous silicon refractive index, is elaborated to predict the optical absorption, carrier generation, external quantum efficiency, and total photogenerated current in the cell. Calculated results best fitted to those measured for an experimental demonstration cell show that surface recombination is a major loss factor in the demonstration cell. The study also concludes that the plasma textured front surface has poor light diffusion properties, and predicts that the maximum gain in the photocurrent from back reflection at the porous silicon stack equals 10% compared to 20% if the back reflector is ideal. However, the proposed cell is realistic and promises excellent performance when state of the art front surface texturing, passivation and front metallization techniques are implemented.

The acoustical equivalence principle for ray tracing

While the acoustical wave functions for various acoustical problems can be mapped into each other by a coordinate transformation (the acoustical equivalence principle), the corresponding ray tracings cannot be so mapped. Applying the acoustical equivalence principle [A. E. Faraggi and M. Matone, Int. J. Mod. Phys. A 15, 1869–2017 (2000) for the analogous quantum equivalence principle] to the underlying acoustical Hamilton–Jacobi equation for ray tracing renders a modified acoustical Hamilton–Jacobi equation which does obey the quantum equivalence principle. Solutions of the modified acoustical Hamilton-Jacobi equation, which are either acoustical Hamilton’s principal function or the acoustical reduced action (acoustical Hamilton’s characteristic function), are the generators of motion for rigorous ray tracing.

First symbiosis between rigorous ray tracing and the parabolic approximation

Rigorous ray tracing originates from the trajectory representation of quantum mechanics while the (Leontovich–Fock) parabolic equation was originally applied to electromagnetic propagation. Rigorous ray tracing (and quantum trajectory representation) have had theoretical success while the parabolic approximation has been a computational success. The parabolic approximation inspires a similar equivalent approximation to the generalized Hamilton–Jacobi equation for rigorous ray tracing. For separable coordinates, the resulting approximations render the same approximate wavefunction. In return, the rigorous ray leads to a parabolic approximation incorporating reflection. The trajectory representation has rendered a synthesized single wavefunction that includes an incident wave of coefficient α and a reflected wave of coefficient β. This synthesized wavefunction can generate a new alternative parabolic equation that incorporates reflection, which for small reflection (β≪α) becomes Fzz+i2k[1−(2β/α)cos(2kx)]exp[i(2β/α)sin(2kx/B)]Fx+(κ2−k2)F=0.

Analysis of radiation characteristics of a polyhedron‐approximate reflector antenna using the aperture integration method and the current distribution method

AbstractIn this paper, the radiation characteristics and its analysis method are discussed for a polyhedron‐approximated reflector antenna in which the reflector is made of several planar facets as in a large deployable antenna. Based on the current distribution method considering the law of reflection and the effect of diffraction, the aperture integration method using the ray tracing containing an approximation in the law of reflection is used for the electric field distribution of a polyhedron‐approximated antenna. It is found that the electric field of the present antenna becomes maximal at the joint between the facets. It is shown that the far‐field characteristics of the antenna derived by the aperture integration method based on the approximate ray tracing and by the current distribution method are identical. the relationship was found between the equivalent aperture plane where the field distributions derived by the two methods are identical. Further, by means of the rigorous ray tracing method, the three dimensional ray behavior is shown. It is found that the aperture integration method based on the rigorous ray tracing method is inappropriate for the calculation of the radiation characteristics of the present antenna.

Rigorous ray tracing for frequency-dependent, stratified media

The equations of motion for rigorous ray tracing for a frequency-dependent, horizontally stratified media are the same as those for simply a horizontally stratified media. In the Hamilton-Jacobi representation, both time and range are still cyclic variables. The behavior of the motion of cyclic variables can be determined by Routh's procedure, and so the trajectory of the ray path in the vertical plane is described by the equations of motion of rigorous ray tracing. The concept of “group velocity” may be shown in a Hamilton-Jacobi representation to be an application of Routh's procedure where the velocity is explicitly dependent upon the momentum conjugate to time (i.e., frequency). As a by-product, since the effective velocity of rigorous ray tracing is dependent upon the momentum conjugate to horizontal range, the relationship between rigorous ray tracing and classical ray tracing is analogous to the relationship between “group velocity” and “phase velocity.”

The manifestation of transmission and reflection in Hamilton's principal function for rigorous ray tracing

Hamilton's principal function for rigorous ray tracing is shown to manifest reflection and transmission at a step in a piecewise-constant index of refraction. The first and second derivatives of Hamilton's principal function are continuous across the step. This specifies the generator of the motion, Hamilton's principal function, which is sufficient to describe propagation including the bifurcation of the incoming wave function into the reflected and transmitted wave functions at the step in the index of refraction. This alternative description is consistent with the contemporary description determined by matching the wave function and its first derivative at the step in the index of refraction.

Closed-form solutions for rigorous ray tracing

Closed-form solutions and expressions for rigorous ray tracing are presented for the effective sound velocity profile, the generating function, the equations of motion, and the initial conditions. These closed-form solutions may be expressed in terms of the closed-form solutions of the corresponding Helmholtz equation. Even if the closed-form solutions for the corresponding Helmholtz equation are unknown, we may still process rigorous ray tracing in terms of the symbolic solutions of the Helmholtz equation. Comparisons are made to both normal modes and classical ray tracing. An Ermakov invariant for rigorous ray tracing is developed and evaluated with closed-form solutions. A Green’s function that is couched in terms of closed-form rigorous-ray-tracing solutions is developed for acoustic sources.

The existence of caustics and cusps in a rigorous ray tracing representation

With rigorous ray tracing, caustics and their associated cusps in addition to those of classical ray tracing are encountered. These additional caustics are associated with classical (WKB) vertex points. Rigorous ray tracing also confirms those caustics that exist under a classical ray tracing representation, albeit both displaced from the classical locations into the classical shadow zone and perhaps deformed with additional cusps.

Foundations of rigorous ray tracing

Rigorous ray tracing may be derived by inserting an ansatz into the Helmholtz equation to develop an alternate Hamilton‐Jacobi equation for rigorous ray tracing expressed by (∂W∂z2) + Cm−2 − C−2 = −12ω2 ∂3W/∂z3∂W/∂z + 34ω2 (∂2W/∂z2∂W/∂z)2, where W is Hamilton's characteristic function (the generator of motion), z is depth, C is the sound velocity profile, Cm is the constant of motion (vertex velocity), and ω is the radial frequency. The left side of the above equation manifests the Hamilton‐Jacobi equation for classical ray tracing. The terms on the right side of the above equation, which contain the common factor ω−2, compensate for finite wavelength. This alternate Hamilton‐Jacobi equation does not have any associated auxiliary equation and may be related to the original Hamilton‐Jacobi equation for rigorous ray tracing [E. R. Floyd, J. Acoust. Soc. Am. 75, 803–808 (1984)]. Equations of motion for both ray paths and wave normals follow from the generator of motion W.

The existence of caustics and cusps in a rigorous ray tracing representation: Floyd, E.R., 1986 J. acoust. Soc. Am., 79(6):1741–1747. Arctic Submarine Lab., NOSC, San Diego, CA 92152–5000, USA

The existence of caustics and cusps in a rigorous ray tracing representation: Floyd, E.R., 1986 J. acoust. Soc. Am., 79(6):1741–1747. Arctic Submarine Lab., NOSC, San Diego, CA 92152–5000, USA

Ray directions, wave normals, and rigorous ray tracing

Herein some exact effective sound velocity profiles for rigorous ray tracing are developed numerically. The equations of motion for rigorous ray tracing are developed and it is shown that in general a ray is not normal to the wave surfaces of constant phase.

Modified phase integral approximation for a more rigorous ray tracing technique

The phase integral approximation is modified to describe wave propagation in three dimensions without any singularities at classical turning points. For stratified media a rigorous ray tracing scheme is derived. Stratified media are shown in general to induce dispersion. The connections to both the normal mode and filter theory methods are discussed. The effects of oceanic surface thermoclines upon rigorous ray tracing are shown to be frequency dependent. Subject Classification: [43]30.20; [43]20.15.