Skew Ray Research Articles

Enhanced responsivity with skew ray excitation of reflection- and transmission-type refractometric sensors.

The responsivity of optical fibers to refractive index can be enhanced using high-order skew rays compared with using meridional rays. Skew rays can have a much higher number of reflections with increased interaction length along the core-cladding interface, which gives rise to stronger interactions with the external medium. Reflection/transmission-type refractometric sensors based on twin-coupled-core and multimode fibers showed one/two orders of magnitude increase in responsivity with skew ray excitation. The responsivity and sensitivity for the two types are ~2000%/RIU, ~1400%/RIU, and 4.9×10⁻⁵ RIU, 7.0×10⁻⁵ RIU, respectively.

Error Analysis of Ray’s Light Path for Axis-Symmetrical Optical System by Using Skew Ray Tracing

ωThis study applies a skew ray tracing approach based on a 4×4 homogeneous coordinate transformation matrix and Snell’s law to analyze the errors of a ray’s light path as it passes through a series of optical elements for axis-symmetrical optical system. The proposed error analysis methodology considers two principal sources of light path error, namely: (1) the translational errors Δxi, Δyi and Δzi and the rotational errors Δωix, Δωiy and Δωiz, which determine the deviation of the light path at each boundary surface, and (2) the differential changes induced in the incident point position and unit directional vector of the refracted / reflected ray as a result of differential changes in the position and unit directional vector of the light source. The validity of the proposed methodology is verified by analyzing the effects of optical errors in Petzval lens.

Skew ray tracing and sensitivity analysis of hyperboloid optical boundary surfaces

One of the most popular mathematical tools in fields of robotics, mechanisms and computer graphics is the 4×4 homogeneous transformation matrix. Our group's previous application of the homogeneous transformation matrix to flat and spherical optical boundaries has been extended to hyperboloid surfaces for: (1) skew ray tracing to determine the paths of reflected/refracted skew rays; (2) sensitivity analysis for direct mathematical expression of differential changes of incident points and reflected/refracted vectors with respect to changes in incident light sources and boundary geometric parameters; (3) a sensitivity analysis-based merit function derived directly from mathematical expression of catadioptric imaging system components. The presented methodology is highly suited to digital implementation and offers direct and rapid analytical statement of ray path, chief ray, marginal rays and merit functions of optical systems.

Imaging Performance Evaluation Method of Wide-View Underwater Acoustic Lens by Geometrical Skew Ray Analysis

In the image formation performance of an underwater acoustic lens, the use of geometrical skew ray analysis for usually use in optical lens design field was examined. We measured acoustic pressure fields through a single lens, from which the acoustic energy distribution was obtained by using encircled energy. On the other hand, the analyzed acoustic energy distribution was evaluated by geometrical skew ray analysis under the measurement condition including the apodization of the source and effect of water temperature. We compared the measured and analyzed convergence positions, which the diameter of 80% of total amount of acoustic energy is the minimum. Then the both convergence positions correspond. As a result, it was shown that the image performance evaluation of a wide-view acoustic lens by geometrical skew ray analysis is useful.

Skew ray tracing and sensitivity analysis of ellipsoidal optical boundary surfaces

The 4 × 4 homogeneous transformation matrix is one of the most commonly applied mathematical tools in the fields of robotics, mechanisms and computer graphics. Here we extend further this mathematical tool to geometrical optics by addressing the following two topics: (1) skew ray tracing to determine the paths of reflected/refracted skew rays crossing ellipsoidal boundary surfaces; and (2) sensitivity analysis to determine via direct mathematical analysis the differential changes of the incident point and the reflected/refracted vector with respect to changes in the incident light source. The proposed ray tracing and sensitivity analysis are projected as the nucleus of other geometrical optical computations.

The Skew Ray Ambiguity in the Analysis of Videokeratoscopic Data

Skew ray ambiguity is present in most videokeratoscopic measurements when azimuthal components of the corneal curvature are not taken into account. There have been some reported studies based on theoretical predictions and measured test surfaces suggesting that skew ray ambiguity is significant for highly deformed corneas or decentered corneal measurements. However, the effect of skew ray ambiguity in ray tracing through videokeratoscopic data has not been studied in depth. We have evaluated the significance of the skew ray ambiguity and its effect on the analyzed corneal optics. This has been achieved by devising a procedure in which we compared the corneal wavefront aberrations estimated from 3D ray tracing with those determined from 2D (meridional based) estimates of the refractive power. The latter was possible due to recently developed concept of refractive Zernike power polynomials which links the refractive power domain with that of the wavefront. Simulated corneal surfaces as well as data from a range of corneas (from two different Placido disk-based videokeratoscopes) were used to find the limit at which the difference in estimated corneal wavefronts (or the corresponding refractive powers) would have clinical significance (e.g., equivalent to 0.125 D or more). The inclusion/exclusion of the skew ray in the analyses showed some differences in the results. However, the proposed procedure showed clinically significant differences only for highly deformed corneas and only for large corneal diameters. For the overwhelming majority of surfaces, the skew ray ambiguity is not a clinically significant issue in the analysis of the videokeratoscopic data indicating that the meridional processing such as that encountered in calculation of the refractive power maps is adequate.

The Skew Ray Issue in Ocular Aberration Measurement

I investigated the "skew ray" issue with respect to ocular aberration measurement. This arises for aberrometers that can distinguish only radial components of aberrations. Using simulations and Zernike aberrations, I found that rotationally symmetric coefficients are correctly determined but that asymmetric coefficients are underestimated. This underestimation is more pronounced as radial order of the aberrations becomes smaller and as angular dependence of aberrations becomes greater. There is also "leakage" to other orders, and errors are slightly affected by the sampling pattern. Aberrometers that can distinguish only radial components of aberrations do not provide good estimates of asymmetric aberrations.

Skew Ray Tracing and Sensitivity Analysis of Paraboloidal Optical Boundary Surfaces

One of the most popular mathematical tools in fields of robotics, mechanisms and computer graphics is the 4x4 homogeneous transformation matrix. In previous work we applied this matrix to the optical domains of flat and spherical surfaces for: (1) skew ray tracing to determine the paths of skew rays being reflected/refracted; (2) sensitivity analysis to determine by direct mathematical analysis the differential change of incident point and reflected/refracted vector with respect to change in incident light source. The present work extends our previous work to include the case of parabaloidal boundary surfaces.

A New Model of Binocular Stereo Coordinate Measurement System Based on Skew Ray Tracing

Two-CCD binocular stereo vision systems using pinhole model are attracting research interest in many important fields such as 3-D coordinate measurement. However, new system modeling is difficult because of lack of fundamental imaging formation theory. In this paper, in order to measure a 3-D surface, we demonstrate how a binocular stereo vision system can be modeled by the proposed skew ray tracing method to study the geometric relations between the binocular camera images and the 3-D surface. In order to investigate the overall accuracy and resolution of this system, the influence of infinitesimal camera reading errors on measured coordinate errors is determined by sensitivity analysis. Experimental verification demonstrates that the performance of the proposed system is excellent.

9.2: Skew Ray Compensated Color Separation Prism for Projection Display Applications

AbstractProjection display systems that use polarization as a means for image recovery require phase compensation of skew rays to achieve high contrast. This paper describes a new color separation prism design that compensates skew ray phase errors using only thin film coatings. The new prism maintains high contrast for systems as fast as f/2.5.

Skew Ray Tracing and Sensitivity Analysis of Geometrical Optics

In order to improve upon the inconvenient and complicated contemporary analytic techniques employed for optical systems, this paper investigates two important optical topics: (1) the determination of light ray paths and (2) sensitivity analysis of light path parameters with respect to the light source location for occasions when light rays cross medium boundary surfaces. To this end, the traditional laws of reflection and refraction are reformulated in terms of revolution geometry. This results in a set of laws much simpler than the original, suitable for use in mathematical modeling to determine light paths and system sensitivity from location of the light source, optical component location, the equation of the optical component’s surface curve, and the refractive index. Ray tracing and sensitivity analysis of the two most popular boundary surfaces, flat and spherical, are presented as examples. In order to illustrate experimentally the integration of these boundary surfaces into optical systems, an optical measurement system for measuring surface height and orientation, containing a beam splitter and a bi-convex lens, was built. Agreement between the experimental optical system’s performance and the theoretical predictions yielded by the proposed method are excellent. [S1087-1357(00)01501-X]

Corneal topography reconstruction algorithm that avoids the skew ray ambiguity and the skew ray error.

Because corneal shape is typically measured with Placido ring targets, there is an ambiguity in determining which point on the ring corresponds to which point on the image. This ambiguity is expected to lead to errors (which we call the skew ray error) in reconstructing the corneal shape from Placido ring videokeratoscope (VKS) images. A coordinate system based on cylindrical coordinates is developed for handling the case when the corneal normal does not lie in the meridional plane. A simple solution is found for the forward problem of locating the Placido ring object given the VKS image and the corneal slope. Algorithms for solving two inverse problems are developed: (1) the inverse problem of determining the Placido image given the Placido ring object and the corneal shape, and (2) the inverse problem of reconstructing the corneal shape based on knowledge of the Placido ring object and image. The algorithm compensates for the skew ray error and does not assume the cornea to be small. Both inverse algorithms are applied to a corneal shape with eight-fold corrugations similar to what might occur in radial keratotomy (RK). The reconstruction algorithm assumes sufficient smoothness for an arc-step algorithm to be possible. We find that the reconstruction algorithm does an excellent job of obtaining the correct corneal shape without the skew ray error. The concrete reconstruction algorithm demonstrates that one can recover the correct corneal shape just from the reflection of the Placido rings. A dart board pattern is not necessary for removing the skew ray ambiguity, as long as the cornea has sufficient smoothness in the radial direction, an assumption common to all reconstruction algorithms.

Analytic Solutions of the Eikonal Equation for a GRIN-rod Lens 2. Skew Rays

Abstract Analytic solutions of the eikonal equation for a skew ray in a GRIN-rod lens are derived together with those of the ray equation by the perturbation method. It is algebraically shown that the derived solutions agree with the exact ones for a meridional ray in the sech profile and for a helical ray in the helical exact profile. The numerical accuracy is also examined by the Runge-Kutta-Gill method for the sech profile.

Skew ray results for condensing light pipes and reflective baffles

Simple criteria are presented for determining whether a general skew ray will strike an absorber at the exit of certain condensing light pipes. Several designs for ideal reflective baffles are also given and their skew ray properties discussed.

Analytic Solution of Ray Equations in Cylindrically Inhomogeneous Guiding Media Part 2: Skew Rays

Analytic solutions are found for skew ray paths in inhomogeneous cylindrical systems such as GRIN rods and gas lenses, which focus rays by virtue of a gradual radial decrease in refractive index. The solutions are obtained by employing digital computers to carry out analytic, as opposed to numerical, calculations.

Skew ray tracing through toric refracting surfaces

A simplified method for skew ray tracing through toric refracting surfaces is described and formulae are given.