Exact Ray Research Articles

Exact Equations for the Back and Effective Focal Lengths of a Plano-Concave Thick Lens

In this work, we present the exact equations for the back and effective focal lengths of a plano-concave thick lens. These equations are derived through a detailed ray-tracing approach, calculating the meridional ray’s path parallel to the optical axis while taking into account lens thickness, refractive index, and surface curvature. Our analysis bridges the gap between paraxial approximations and exact ray behavior, highlighting how traditional models (e.g., thin-lens equations) can be extended to thick lenses. Additionally, we review recent advancements in negative refraction and surface phase manipulation techniques to demonstrate how our equations compare to modern numerical methods and metacoating-enhanced designs. This work provides a comprehensive and exact framework for understanding the optical properties of plano-concave thick lenses, offering new insights into their application in precision optics.

Upper limit on the coronal cosmic ray energy budget in Seyfert galaxies

Abstract The IceCube Collaboration has reported possible detections of high-energy neutrinos from nearby Seyfert galaxies. While central hot coronae are proposed as the primary neutrino production site, the exact coronal cosmic ray energy budget has been loosely constrained. In this study, we propose a new stringent upper bound on the coronal cosmic ray energy budget of Seyfert galaxies, considering both accretion dynamics and observed properties of radio-quiet Seyfert galaxies. Notably, even under the calorimetric condition where cosmic rays lose all their energy, our limit indicates that the coronal neutrino flux of NGC 1068 is about an order of magnitude fainter than the observed levels. This discrepancy suggests the need for further theoretical and observational investigations on the IceCube signals from Seyfert galaxies.

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.

Predictability of pseudophakic refraction using patient-customized paraxial eye models.

To determine whether patient-customized paraxial eye models that do not rely on exact ray tracing and do not consider aberrations can accurately predict pseudophakic refraction. Bascom Palmer Eye Institute, Miami, Florida. Prospective study. Cataract surgery patients with and without a history of refractive surgery were included. Manifest refraction, corneal biometry, and extended-depth optical coherence tomography (OCT) imaging were performed at least 1 month postoperatively. Corneal and OCT biometry were used to create paraxial eye models. The pseudophakic refraction simulated using the eye model was compared with measured refraction to calculate prediction error. 49 eyes of 33 patients were analyzed, of which 12 eyes from 9 patients had previous refractive surgery. In eyes without a history of refractive surgery, the mean prediction error was 0.08 ± 0.33 diopters (D), ranging from -0.56 to 0.79 D, and the mean absolute error was 0.27 ± 0.21 D. 31 eyes were within ±0.5 D, and 36 eyes were within ±0.75 D. In eyes with previous refractive surgery, the mean prediction error was -0.44 ± 0.58 D, ranging from -1.42 to 0.32 D, and the mean absolute error was 0.56 ± 0.46 D. 7 of 12 eyes were within ±0.5 D, 8 within ±0.75 D, and 10 within ±1 D. All eyes were within ±1.5 D. Accurate calculation of refraction in postcataract surgery patients can be performed using paraxial optics. Measurement uncertainties in ocular biometry are a primary source of residual prediction error.

KrakenOS: Python-based general exact ray tracing library

Exact ray tracing is the fundamental tool of geometric optics, it constitutes the foundation on which aberration theory and, therefore, optical design is based. We present the KrakenOS, a Python library focused on the accuracy of ray tracing and the generation of systems composed by optical surfaces with arbitrary shapes and orientations, as well as a compendium of practical examples for the use of our library. We describe a series of quality tests to compare the results obtained from ray tracing with KrakenOS with respect to those obtained with the commercial software Zemax. Among the performance computations, we include the final intersection coordinates with optical surfaces, the resulting director cosines, the calculation of surface-to-surface length of the optical path, the energy of the transmitted and reflected resulting rays, Zernike standard coefficients from wavefront fitting, and a comparison of the resulting intersection coordinates with a larger optical system. The difference between our results and those of Zemax is of the order of 9.0 × 10 − 8 mm, demonstrating that KrakenOS can be used in industrial or scientific work with high precision requirements for simple and very complex optical configurations.

Optimization of a DIY parallel-optical-axes profilometer for compensation of fringe divergence.

In profilometry systems, the conventional optical configuration of fringe projection with the oblique projection and camera axis perpendicular to the field of view requires complicated calculus and exact ray trace designs to get the 3D shape under question accurately. Moreover, when the optical components of the camera and projector and their internal placement are unknown, as is the case with commercial digital video projectors, the correct alignment of these components is uncertain, needing complex methods of calibration to correct the measured shape. In this work, we re-introduce a parallel-optical-axes configuration for a DIY profilometer that simplifies the calibration by a simple trigonometric relationship obtaining the topography with a resolution and accuracy similar to conventional arrangements that require non-simple relationships among phase and height. Furthermore, an experimental method is presented to locate the divergence point within the projector, which allows optimizing the projection and observation distances. The proposed calibration procedure allows proper compensation for the apparent scale change caused by perspective. The performance of the proposed method is tested in an optimized configuration and a non-optimized one, comparing the measurements with a Coordinate-measuring machine. The optimized optical system using simple spatial phase unwrapping shows an improved RMS error of 53µm versus 203µm of the non-optimized.

Ronchi-Hartmann type null screens for testing a plano-freeform surface with a detection plane inside a caustic surface.

We have implemented an exact ray trace through a plano-freeform surface for an incident plane wavefront. We obtain two caustic surfaces and provide the critical points related to the ray tracing process. Additionally, we study the propagation of the refracted wavefronts through the plane-curved surface. Finally, by using the Ronchi-Hartmann type null screen and placing the detection plane within the caustic region, we have evaluated the shape of a plano-freeform optical surface under test, obtaining an RMS difference in sagitta value of 6.3 μm.

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.

A Ray-Tracing Algorithm Based on the Computation of (Exact) Ray Paths With Bidirectional Ray-Tracing

A novel ray-tracing algorithm to cope with the phase errors due to incorrect ray path computations in ray-launching approaches is presented. The algorithm utilizes bidirectional ray-tracing to collect information about wavefronts incident on an interaction surface and yields considerable improvements in accuracy compared to conventional unidirectional ray-tracing. The points, where exact ray paths intersect with the surface, are obtained according to the Fermat principle of least time. If the interaction surface is aligned with diffraction edges, the corresponding critical points of the second kind can also be retrieved and complicated diffraction treatments by shooting diffracted rays on Keller cones can be avoided. Thus, a substantial reduction in the number of rays can be achieved. Furthermore, a typical problem encountered in traditional ray-tracing due to the reception sphere mechanism, i.e., incorrect ray contributions, can mostly be evaded. Numerical results demonstrate the capabilities of the new algorithm and its advantages against traditional techniques.

An exact ray model for oblique incident light on planar films

During recent years, ray tracing has frequently been used to study the absorption characteristics of structured solar cells. However, wave properties such as absorption enhancement due to resonances in optically thin solar films, cannot be explained by pure classical ray models. Here we present an exact three-dimensional ray model for oblique incidence of a plane electromagnetic wave on a thin film and show that the resonant structure of the absorption cross section calculated from our ray model is identical to exact calculations by electromagnetic wave theory. Both parallel and perpendicular polarized light are described exactly by the ray model presented. We validate the resonant structure of the absorption cross section of our ray model by an experimentally realized layered film, where we obtain perfect agreement between experiment and theory. We demonstrate further that for a beam with a finite beam waist, in accordance with Beer-Lambert's law, absorption occurs along the path of the beam, while, in the case of a plane wave incident on an optically thin film, and contrary to Beer-Lambert's law, absorption occurs along the axis perpendicular to the surface of the film. • Presents a three-dimensional ray model that describes exactly the absorption properties of layered materials. • The ray model reveals the resonant structure of the absorption cross section exactly, confirmed by exact wave calculations and experiments. • The paper highlights pitfalls for explaining interference in films by rays, which modern textbooks often are not aware of. • In the case of absorption, the absorption is normal to the surface of the film and not along the path of the classical ray.

Geometric Aberration Theory of Offner Imaging Spectrometers.

A third-order aberration theory has been developed for the Offner imaging spectrometer comprising an extended source; two concave mirrors; a convex diffraction grating; and an image plane. Analytic formulas of the spot diagram are derived for tracing rays through the system based on Fermat’s principle. The proposed theory can be used to discuss in detail individual aberrations of the system such as coma, spherical aberration and astigmatism, and distortion together with the focal conditions. It has been critically evaluated as well in a comparison with exact ray tracing constructed using the commercial software ZEMAX. In regard to the analytic formulas, the results show a high degree of practicality.

The Mirascope: an explanation on a conceptual level

The main objective of this paper is to provide an explanation on a conceptual level of how the Mirascope and the occurring image formation work. The physics behind a Mirascope—a well-known device consisting of two concave mirrors—is very complex. In literature, only either high level or very simplified and deficient explanations can be found. These explanations often lack a basic but detailed description on a conceptual level of the phenomena observable with the human eye. Therefore, this paper aims to give a physical explanation on a conceptual level to support learning processes in optics. Phenomena such as the diverse direction of the projected real image and the formation of the virtual image are clarified. As the focus lies on providing a basis to foster a conceptual understanding, there is no claim for a full and detailed explanation on the level of an exact ray analysis. In addition to a clarification of the content on a conceptual and educational level, suggestions for further investigations of the Mirascope with everyday items are given. The Mirascope is seen as an appropriate device for creating a learning environment following an inquiry-based approach in physics (teacher) education. This paper focuses on describing and explaining aspects of the Mirascope on a conceptual level.

Slowness‐domain kinematical characteristics for horizontally layered orthorhombic media. Part II: Pre‐critical slowness match

ABSTRACTPart II of this paper is a direct continuation of Part I, where we consider the same types of orthorhombic layered media and the same types of pure‐mode and converted waves. Like in Part I, the approximations for the slowness‐domain kinematical characteristics are obtained by combining power series coefficients in the vicinity of both the normal‐incidence ray and an additional wide‐angle ray. In Part I, the wide‐angle ray was set to be the critical ray (‘critical slowness match’), whereas in Part II we consider a finite long offset associated with a given pre‐critical ray (‘pre‐critical slowness match’). Unlike the critical slowness match, the approximations in the pre‐critical slowness match are valid only within the bounded slowness range; however, the accuracy within the defined range is higher. Moreover, for the pre‐critical slowness match, there is no need to distinguish between the high‐velocity layer and the other, low‐velocity layers. The form of the approximations in both critical and pre‐critical slowness matches is the same, where only the wide‐angle power series coefficients are different. Comparing the approximated kinematical characteristics with those obtained by exact numerical ray tracing, we demonstrate high accuracy. Furthermore, we show that for all wave types, the accuracy of the pre‐critical slowness match is essentially higher than that of the critical slowness match, even for matching slowness values close to the critical slowness. Both approaches can be valuable for implementation, depending on the target offset range and the nature of the subsurface model. The pre‐critical slowness match is more accurate for simulating reflection data with conventional offsets. The critical slowness match can be attractive for models with a dominant high‐velocity layer, for simulating, for example, refraction events with very long offsets.

Exact ray theory for the calculation of the optical generation rate in optically thin solar cells

There is a profound duality between rays and waves. In fact, 70 years ago, in the context of quantum mechanics, Feynman showed that rays, properly equipped with phases and correctly summed, provide exact solutions of the quantum mechanical wave equation. In this paper, constructing explicit, exact ray solutions of the one-dimensional Helmholtz equation as a model for optically thin solar cells, we show that the ray-wave duality is also exact in the context of the electromagnetic wave equations. We introduce a complex index of refraction in order to include absorption. This have so far not been treated in the quantum ray-splitting literature. We show that inclusion of exact phases is mandatory and that a ray theory without phases may result in amplitude errors of up to 60%. We also show that in the case of multi-layered solar cells the correct summation order of rays is important. Providing support for the notion that rays provide the “skeleton” of electromagnetic waves, we perform a Fourier transform of the (experimentally measurable) solar cell reflection amplitude, which reveals the rays as peaks in the optical path length spectrum. An application of our exact ray theory to a silicon solar cell is also provided. Treating the one-dimensional case exactly, our paper lays the foundation for constructing exact ray theories for application to solar cell absorption cross section in two and three dimensions.

Ray-tracing Analysis of the Corneal Power From Scheimpflug Data.

To describe a method by which mapped corneal elevation data can be used for ray-tracing analysis of the effective corneal power. Mapped elevation data of the front and back surface of the cornea exported by a clinical Scheimpflug camera was triangulated into a polygonal format and imported into a commercial optical engineering software. The focal length of the cornea was determined by exact ray tracing analysis of the distance giving the sharpest point spread function (PSF) at the selected image plane. The effective corneal power could then be determined as the reciprocal of the observed focal length "reduced to air." The corneal power determined by the ray-tracing procedure was checked for reproducibility and effect of pupil size and finally compared with standard keratometry methods. Twenty random cases referred for cataract or refractive lens surgery were investigated. The ray-traced corneal power was found to be highly reproducible with a maximum error of 0.023 diopters (D) between repeated ray-tracing experiments. The mean ray-traced corneal power of 42.34 D (assuming a 3-mm pupil) was found to be 1.02 D lower than the standard keratometry reading assuming a keratometric index of 1.3375 (P < .001). The ray-traced corneal power was found to be higher than the true net power (P < .01) but not significantly different from the total corneal refractive power reported by the Scheimpflug device (P > .05). The ray-traced corneal power increased 0.31 D when the pupil size increased from 3 to 5 mm, which was attributed to spherical aberration. Exact ray tracing can be used on mapped tomography data to analyze for the effective corneal power. This technique was found to be highly reproducible and may be a promising tool in the analysis of the true power of the cornea of any shape. [J Refract Surg. 2018;34(1):45-50.].

Fourth-order normal moveout velocity in elastic layered orthorhombic media — Part 1: Slowness-azimuth domain

Considering all types of pure-mode and converted waves, we derive the azimuthally dependent, fourth-order normal moveout (NMO) velocity functions, and hence the corresponding effective anellipticity functions, for horizontally layered orthorhombic media. We emphasize that this paper does not suggest a new nonhyperbolic traveltime approximation; rather, it provides exact expressions of the NMO series coefficients, computed for normal-incidence rays, which can then be further used within known azimuthally dependent traveltime approximations for short to moderate offsets. We do not assume weak anisotropy or acoustic approximation for P-waves. At each layer, the elastic parameters, thickness, and azimuth of the orthorhombic vertical symmetry planes are considered to be different. We distinguish between two different azimuths: slowness azimuth (part 1 of this paper) and offset azimuth (part 2 of this paper). In part 1, the slowness-azimuth domain NMO is approximated as a series of either infinitesimal horizontal slowness (slowness-azimuth/slowness domain) or infinitesimal offsets (slowness-azimuth/offset domain). Similarly, in part 2, we distinguish between two offset-azimuth domains: offset-azimuth/slowness and offset-azimuth/offset. Note that the azimuthally dependent NMO velocity functions of each of the four cases are different. The validity of the method is tested by introducing our derived azimuthally dependent, fourth-order effective anellipticity, into the well-known azimuthally dependent, asymptotic nonhyperbolic traveltime approximation, in which we compare the traveltime approximation versus exact numerical ray tracing for short to moderate offsets. It is clearly shown that for these types of azimuthally anisotropic layered models, the fourth-order terms are essential even for relatively small horizontal-slowness values or short offsets.

Error compensation in a pointing system based on Risley prisms.

Risley prisms are widely used for beam pointing in several optical systems. The exact solution for the inverse problem does not exist, except using numerical methods. However, the errors introduced by misalignment are usually greater than the approximation errors. We present a novel method to compensate alignment errors in pointing systems based on Risley prisms. The prism model that we used is based on paraxial approximation with an additional vector to compensate typical alignment errors. Simulation and experimental results show that the improvement in pointing accuracy is achievable even in comparison with exact ray tracing methods.

Design of Hartmann type null screens for testing a plano-convex aspheric lens with a CCD sensor inside the caustic.

A new method to design Hartmann type null screens to test either qualitatively or quantitatively fast plano-convex aspherical lenses is presented. We design both radial and square null screens that produce arrays of circular spots uniformly distributed at predefined planes, considering that the CCD sensor is solely placed inside the caustic region. The designs of these null screens are based on knowledge of the caustic by refraction and on exact ray tracing. The null screens also serve to improve the alignment in optical systems.

Empirical assessment of the validity limits of the surface wave full ray theory using realistic 3-D Earth models

The surface wave full ray theory (FRT) is an efficient tool to calculate synthetic waveforms of surface waves. It combines the concept of local modes with exact ray tracing as a function of frequency, providing a more complete description of surface wave propagation than the widely used great circle approximation (GCA). The purpose of this study is to evaluate the ability of the FRT approach to model teleseismic long-period surface waveforms (T ∼ 45–150 s) in the context of current 3-D Earth models to empirically assess its validity domain and its scope for future studies in seismic tomography. To achieve this goal, we compute vertical and horizontal component fundamental mode synthetic Rayleigh waveforms using the FRT, which are compared with calculations using the highly accurate spectral element method. We use 13 global earth models including 3-D crustal and mantle structure, which are derived by successively varying the strength and lengthscale of heterogeneity in current tomographic models. For completeness, GCA waveforms are also compared with the spectral element method. We find that the FRT accurately predicts the phase and amplitude of long-period Rayleigh waves (T ∼ 45–150 s) for almost all the models considered, with errors in the modelling of the phase (amplitude) of Rayleigh waves being smaller than 5 per cent (10 per cent) in most cases. The largest errors in phase and amplitude are observed for T ∼ 45 s and for the three roughest earth models considered that exhibit shear wave anomalies of up to ∼20 per cent, which is much larger than in current global tomographic models. In addition, we find that overall the GCA does not predict Rayleigh wave amplitudes well, except for the longest wave periods (T ∼ 150 s) and the smoothest models considered. Although the GCA accurately predicts Rayleigh wave phase for current earth models such as S20RTS and S40RTS, FRT's phase errors are smaller, notably for the shortest wave periods considered (T ∼ 45 s and T ∼ 60 s). This suggests that the FRT approach is a useful means to build the next generation of elastic and anelastic surface wave tomography models. Finally, we observe a clear correlation between the FRT amplitude and phase errors and the roughness of the models. This allows us to quantify the limits of validity of the FRT in terms of model roughness thresholds, which can serve as useful guides in future seismic tomographic studies.

Measuring conic constant and vertex radius of fast convex conic surfaces from a set of Hartmann patterns

A method for measuring conic constant and vertex radius of fast conic surfaces from Hartmann patterns obtained for different axial positions of the surface is proposed. For each pattern the evaluation of the surface is carried out through an exact ray tracing algorithm, including the vector form of the Law of Reflection. Every reconstructed surface is fitted to an even radial polynomial; analyzing the behavior of the coefficients for the second, fourth, and sixth order terms of the polynomial in terms of the axial position, the right values of the conic constant and the vertex radius are found. Experimental results for a sphere with radius 7.78mm and a paraboloid with vertex radius 6.50mm are shown.