Finite element ray tracing: a new method for ray tracing in gradient-index media

I apply the principle of finite elements, known in optics for calculating the beam propagation in waveguide structures, for calculation of meridional rays in inhomogeneous media. The plane is divided into finite elements that have a constant refractive index. The ray trajectory is calculated by a simple algorithm. Contrary to the existing methods, the model I propose in this research does not require an explicit formula for the index distribution. Only the numerical representation is sufficient, which can be a major advantage for calculation of the light propagation in real problems, such as the thermal lensing effect.     

Optical ray tracing using parallel processors

One of the instruments on the sun-synchronous Terra (EOS AM) and Aqua (EOS PM) spacecraft, the Moderate Resolution Spectroradiometer (MODIS), obtains calibration data once during every orbit. Observations of the sun permit corrections to observations of the Earth during the ensuing orbit. Although the instrument was designed to receive uniform sunlight over the entire surface of its detector, the sunlight was in fact not uniform. While this did not adversely affect the calibration, it nonetheless implied a lack of understanding of how the optical system really functioned. To learn what was wrong, NASA used an optical ray-tracing program on a DEC Alpha computer. The results correlated well with the observations made by the instrument itself, but it took nearly two weeks to complete the computer simulation, a discouragingly long time. This paper describes the algorithm and its implementation in a system with multiple digital signal processor (DSP) chips operating in parallel. Timing data show a highly linear relationship between the number of DSPs present and the speed of the computation. Administrative overhead is negligible compared to the time taken to compute ray trajectories. This implies that many more than just four DSPs could be harnessed before administrative overhead would begin to be significant.     

Fast optical ray tracing using multiple DSPs

Optical ray tracing is a computationally intensive operation that is central both to the design of optical systems and to analyzing their performance once built. The authors have previously reported on the use of parallel digital signal processors (DSPs) to reduce the time required to perform ray tracing in analyzing the performance of the moderate resolution imaging spectroradiometer (MODIS), which is presently in orbit on multiple spacecraft. The earlier work was incomplete, providing only a conservative estimate of the performance improvement that could be achieved with one to four DSPs. This paper reports on the completed project and extends the earlier work to eight DSPs. As predicted in the earlier paper, not all rays make it through the entire optical system. Many are lost along the way. This is one factor that led to reduced processing time. Another is the use of an optimizing compiler. In this paper, the authors present results showing the separate effect of each of these two independent factors on the overall processing time. The most significant finding is the extraordinarily linear relationship between the number of DSPs available and the speed of the ray tracing. By using eight DSPs, the processing time is reduced from two weeks to less than one and a half days, an improvement of nearly a whole order of magnitude. Low-cost high-speed ray tracing is now feasible using off-the-shelf plug-in processor boards.     

Improved ray tracing air mass numbers model

An improved ray tracing air mass model to calculate the air mass numbers for the entire zenith angle range is developed. The improved model uses the approach when the trajectory element of light in the atmosphere is approximated by an arc of a circle. This way the angles at the beginning and at the end of the trajectory element can be counted simultaneously. This approach gives the second-order approximation for the real light trajectory with more accurate results than the results of the approaches of Link and Neuzil (Tables of Light Trajectories in the Terrestrial Atmosphere, Hermann, 1969) and the Kasten and Young models [Appl. Opt. 28, 4735 (1989)]. The developed model allows us to avoid the calculation problems of the Link and Neuzil and Kasten models when the zenith angle is close to or equal to 90 degrees . As a result, we deliver the new air mass number table for the entire zenith angle range and provide the comparison of the developed model results with the results of the Link and Neuzil and the Kasten models.     

Complex ray tracing in uniaxial absorbing media

We theoretically analyze waves propagating between an isotropic nonabsorbing medium and a uniaxial absorbing medium in the general case where the incidence plane does not coincide with the principal section of the uniaxial crystal. Expressions for the reflection and transmission coefficients are derived by a combination of complex ray tracing and the 4 x 4 matrix method. The work presented here is useful for the design of optical systems that incorporate the use of birefringent absorbing components.     

海洋锋区的三维声线轨迹分析

鉴于射线模型明确的物理意义及海洋锋特殊的海洋环境现象，文章运用三维射线模型研究了声波在海洋锋区的传播规律，并通过仿真试验，讨论了台湾黑潮锋区的三维声线轨迹的特点及其对声呐探测带来的影响。研究的结果表明，声波在通过海洋锋区时，一般都将产生较大的垂直深度误差和水平方位误差，特别是在台湾以东的黑潮锋区，影响比较明显。     

Nonnull interferometer simulation for aspheric testing based on ray tracing

The nonnull interferometric method that employs a partial compensation system to compensate for the longitude aberration of the aspheric under test and a reverse optimization procedure to correct retrace errors is a useful technique for general aspheric testing. However, accurate system modeling and simulation are required to correct retrace errors and reconstruct fabrication error of the aspheric. Here, we propose a ray-tracing-based method to simulate the nonnull interferometer, which calculates the optical path difference by tracing rays through the reference path and the test path. To model a nonrotationally symmetric fabrication error, we mathematically represent it with a set of Zernike polynomials (i.e., Zernike deformation) and derive ray-tracing formulas for the deformed surface, which can also deal with misalignment situations (i.e., a surface with tilts and/or decenters) and thus facilitates system modeling extremely. Simulation results of systems with (relatively) large and small Zernike deformations and their comparisons with the lens design program Zemax have demonstrated the correctness and effectiveness of the method.     

Numerical determination of continuous ray tracing: the four-component method

Two interpolants are presented for Runge-Kutta ray tracing using the four-component method. Such a continuous interpolant of ray trajectory connects only two adjacent Runge-Kutta points and requires only the information needed to find the position and ray vector for interpolant of ray position and ray vector to the fourth-order error. However Runge-Kutta schemes in contrast to other schemes are not based on approximating the continuous ray path with a polynomial. For this reason an optimal interpolant is shown that gives the same accuracy for calculating ray position and ray vector as a single-step integration in the Runge-Kutta scheme.     

Accurate calculation of Narcissus signatures by using finite ray tracing

In scanning IR systems with cooled quantum detectors, the Narcissus signature usually has a limiting effect on the perceived image quality, and so it is important that it be assessed accurately before manufacture. We describe a new finite ray-tracing method in which each ray represents an equal amount of flux falling on the detector. Such methods require many rays; therefore a full error treatment is given that allows designers to estimate the necessary number of rays to obtain the required accuracy and also to calculate the standard deviation of the error in the final computed result.     

Comment on "improved ray tracing air mass numbers model"

Air mass numbers have traditionally been obtained by techniques that use height as the integration variable. This introduces an inherent singularity at the horizon, and ad hoc solutions have been invented to cope with it. A survey of the possible options including integration by height, zenith angle, and horizontal distance or path length is presented. Ray tracing by path length is shown to avoid singularities both at the horizon and in the zenith. A fourth-order Runge-Kutta numerical integration scheme is presented, which treats refraction and air mass as path integrals. The latter may optionally be split out into separate contributions of the atmosphere's constituents.     

Numerical analysis of thermoplastic composites laser welding using ray tracing method

Laser technology is a good alternative for continuous joining of thermoplastics composites structures. Presence of continuous fibers at a high fiber volume fraction (superior to 30%) does not allow using traditional development as for pure thermoplastic materials, due to the presence of fiber clusters or polymer rich areas. Those heterogeneities induce macroscopic light scattering through the structure, reducing the resulting energy level absorbed at the welding interface. The study proposed here takes into account the real microstructure of the composite in order to evaluate changes in local energy diffusion directly linked with local fiber arrangements. The objective of this work is to develop an affordable numerical simulation of the laser welding process modeled with adapted physics mechanism and taking into account the microstructure heterogeneity of the considered materials regarding optical and thermal properties. To model the optical path of the laser beam through the composite fibrous structure, a simulation tool based on geometrical optic is developed. Weldability is considered on composites with different thicknesses, showing the non linear relationship between welding energy and substrate thickness.     

A generalized two point ellipsoidal anisotropic ray tracing for converted waves

Rocks can be anisotropic due to a variety of reasons. When estimating rock velocities from seismic data, failure to introduce anisotropy into earth models could generate distortions in the final images that can have enormous economic impact. To estimate anisotropic earth velocities by tomographic methods, it is necessary to trace rays or to solve the wave equation in models where anisotropy has been properly considered. Thus, in this work we present a 3-D generalized ellipsoidal travel time formulation that allow us to trace rays in an anisotropic medium. We propose to trace rays in anisotropic media by solving a set of nonlinear optimization problems, where the group velocities for P and S wave propagation modes are 3-D ellipsoidal approximations that have been recently obtained. Moreover, we prove that this 3-D ellipsoidal anisotropic ray tracing formulation is a convex nonlinear optimization problem, and therefore any solution of the problem is a global minimum. Each optimization problem is solved by the global spectral gradient method, which requires first order information and has low computation and low storage requirements. Our approach for tracing rays in anisotropic media is a generalization in the sense that handles titled axis of symmetry and, close to the axis of symmetry, it is an accurate formulation for 2-D transversely isotropic media and 3-D orthorhombic media, depending on the input parameters. Moreover, this formulation gives the exact ray trajectories in 2-D and 3-D homogeneous isotropic media. The simplicity of the formulation and the low computational cost of the optimization method allow us to present a variety of numerical results that illustrate the behavior and computational advantages of the approach, and the difficulties when working in anisotropic media. Partially supported by Fonacit project UCV-97-003769     

Simulation of X-ray beamlines with the new ray tracing tool XTrace

A novel X-ray tracing simulation tool XTrace has been developed recently in order to calculate the beam properties from the X-ray source (bending magnet, undulator, wiggler, etc.) through all optical elements (slit, filter, window, mirror, crystal monochromator, multilayer, etc.) and the sample to the detector. In XTrace, the diffraction by a perfect crystal is described by the dynamical theory taking into account refraction and absorption effects. The code has been used to simulate the X-ray beamline of the Synchrotron Laboratory for Environmental Studies (Synchrotron Umwelt-Labor (SUL)) at ANKA. XTrace has been successfully used to simulate precisely the beam parameters such as position, size, divergence, photon flux, transmission, heat load, etc. at any distance from the source. For a better insight it is also possible to visualize the beam profile, energy spectrum, transmission spectrum, intensity distribution, power density, heat load, foot print, DuMond diagram, ray propagation diagram, etc. An excellent agreement between measured and simulated flux intensities over the whole energy range at the sample position for the X-ray beamline SUL has been found. XTrace has been proven to be a reliable, powerful, fast and easy to use tool for describing existing and designing and optimizing new X-ray beamlines in the future.     

Reconstruction of optical characteristics of waveguide lenses by the use of ray tracing

A method that uses the data of ray tracing for optical waveguide lens diagnostics is described. This method permits a direct reconstruction of the optical characteristics of a waveguide without the optical or the physical thickness being measured. Conditions are determined for the mathematical problem of diagnostics by ray tracing to have a unique solution, and a technique to obtain a numerical solution from noisy experimental data is described.     

Modeling of micro cat's eye retroreflectors using a matrix-based three-dimensional ray tracing technique

In this paper we develop a three-dimensional (3D) ray tracing tool based on the ABCD ray transfer matrices. With symmetric optical components and under paraxial approximation, two sets of 2×2 ABCD matrices, each for a two-dimensional subspace, can be used to describe the 3D ray propagation completely. Compared to commercial ray-tracing software packages, our tool requires no tedious drawing, and the results for various conditions, such as different device dimensions and incident angles, can be easily obtained by simply changing the parameter values used for the calculation. We have employed this matrix-based 3D ray tracing tool to model cat's eye retroreflectors. The cat's eye performance, including the retroreflection efficiency, acceptance angle (i.e., field of view), and beam divergence and deviation, is fully studied. The application of this 3D ray tracing technique can be further extended to other optical components.     

Analysis and design of optical systems by use of sensitivity analysis of skew ray tracing

Optical systems are conventionally evaluated by ray-tracing techniques that extract performance quantities such as aberration and spot size. Current optical analysis software does not provide satisfactory analytical evaluation functions for the sensitivity of an optical system. Furthermore, when functions oscillate strongly, the results are of low accuracy. Thus this work extends our earlier research on an advanced treatment of reflected or refracted rays, referred to as sensitivity analysis, in which differential changes of reflected or refracted rays are expressed in terms of differential changes of incident rays. The proposed sensitivity analysis methodology for skew ray tracing of reflected or refracted rays that cross spherical or flat boundaries is demonstrated and validated by the application of a cat's eye retroreflector to the design and by the image orientation of a system with noncoplanar optical axes. The proposed sensitivity analysis is projected as the nucleus of other geometrical optical computations.     

Use of Rayleigh imaging and ray tracing to correct for beam-steering effects in turbulent flames

Laser Rayleigh imaging has been applied in a number of flow and flame studies to measure concentration or temperature distributions. Rayleigh cross sections are dependent on the index of refraction of the scattering medium. The same index of refraction changes that provide contrast in Rayleigh images can also deflect the illuminating laser sheet. By applying a ray-tracing algorithm to the detected image, it is possible to correct for some of these beam-steering effects and thereby improve the accuracy of the measured field. Additionally, the quantification of the degree of beam steering through the flow provides information on the degradation of spatial resolution in the measurement. Application of the technique in a well-studied laboratory flame is presented, along with analysis of the effects of image noise and spatial resolution on the effectiveness of the algorithm.     

Full angular profile of the coherent polarization opposition effect

We use the rigorous vector theory of weak photon localization for a semi-infinite medium composed of nonabsorbing Rayleigh scatterers to compute the full angular profile of the polarization opposition effect. The effect is caused by coherent backscattering of unpolarized incident light and accompanies the well-known backscattering intensity peak.