The Perturbed Riemann Problem for a Geometrical Optics System

The perturbed Riemann problem for a hyperbolic system of conservation laws arising in geometrical optics with three constant initial states is solved.By studying the interactions among of the delta-shock,vacuum,and contact discontinuity,fourteen kinds of structures of Riemann solutions are obtained.The compound wave solutions consisting of delta-shocks,vacuums,and contact discontinuities are found.The single and double closed vacuum cavitations develop in solutions.Furthermore,it is shown that the solutions of the Riemann problem for the geometrical optics system are stable under certain perturbation of the initial data.Finally,the numerical results completely coinciding with theoretical analysis are presented.

Linking Diffractive and Geometrical Optics Surface Scattering at a Fundamental Level

Optical surface scattering analyses based on diffractive optics (DO) are typically applied to one surface;however, there is a need for simulating surface scattering losses for devices having many surface interactions such as light pipes. Light pipes are often simulated with geometric optics (GO) using ray tracing, where surface scattering is driven by the surface slope distribution. In the DO case, surface scattering analyses depend on the spatial frequency distribution and amplitude as well as wavelength, with the sinusoidal grating as a fundamental basis. A better understanding of the link, or transition, between DO and GO scattering domains would be helpful for efficiently incorporating scattering loss analyses into ray trace simulations. A formula for the root-mean-square (rms) scattered angle width of a sinusoidal reflection grating that depends only on the surface rms slope is derived from the nonparaxial scalar diffraction theory, thereby linking it to GO. The scatter angle’s mean and rms width are evaluated over a range of grating amplitudes and periods using scalar theory and full vector simulations from the COMSOL® wave optic module for a sinusoidal reflection grating. The conditions under which the diffraction-based solution closely approximates the GO solution, as predicted by the rms slope, are identified. Close agreement is shown between the DO and GO solutions for the same surface rms slope scattering loss due to angular filtering near the critical angle of a total internal reflection (TIR) glass-to-air interface.

Geometrical optics approximation of light scattering by large air bubbles

For large spherical bubbles in water, geometrical optics approximation is considered a better method for calculating light scattering patterns. In this paper, the basic theory of geometrical optics approximation is clarified. The change of phase for bubbles is calculated when total reflection occurs, which is different from particles with relative refractive indices larger than 1. Verification of the method was achieved by assuming a spherical particle and comparing present results to Mie scattering and Debye calculation. Agreement with the Mie theory was excellent in all directions when the dimensionless size parameter is larger than 50. Limitations of the geometrical optics approximation are also discussed.

Riemann Problem for a Geometrical Optics System

The paper solves analytically the Riemann problem for a nonstrictly hyperbolic system of conservation laws arising in geometrical optics,in which the flux contains the nonconvex function possessing an infinite number of inflection points.Firstly,the generalized Rankine–Hugoniot relations and entropy condition of delta shock waves and left（right）-contact delta shock waves are proposed and clarified.Secondly,with the help of the convex hull,seven kinds of structures of Riemann solutions are obtained.The solutions fall into three broad categories with a series of geometric structures involving simultaneously contact discontinuities,vacuums and delta shock waves.Finally,numerical experiments confirm the theoretical analysis.

Global geometrical optics method for vector-valued Schrodinger problems

We extend the theory of global geometrical optics method, proposed originally for the linear scalar high-frequency wave-like equations in [Commun. Math. Sci., 2013, 11（1）： 105-140], to the more general vector- valued Schrodinger problems in the semi-classical regime. The key ingredient in the global geometrical optics method is a moving frame technique in the phase space. The governing equation is transformed into a new equation but of the same type when expressed in any moving frame induced by the underlying Hamiltonian flow. The classical Wentzel-Kramers-Brillouin （WKB） analysis benefits from this treatment as it maintains valid for arbitrary but fixed evolutionary time. It turns out that a WKB-type function defined merely on the underlying Lagrangian submanifold can be obtained with the help of this moving frame technique, and from which a uniform first-order approximation of the wave field can be derived, even around caustics. The general theory is exemplified by two specific instances. One is the two-level SchrSdinger system and the other is the periodic SchrSdinger equation. Numerical tests validate the theoretical results.

Rendering discrete participating media using geometrical optics approximation

We consider the scattering of light in participating media composed of sparsely and randomly distributed discrete particles.The particle size is expected to range from the scale of the wavelength to several orders of magnitude greater,resulting in an appearance with distinct graininess as opposed to the smooth appearance of continuous media.One fundamental issue in the physically-based synthesis of such appearance is to determine the necessary optical properties in every local region.Since these properties vary spatially,we resort to geometrical optics approximation(GOA),a highly efficient alternative to rigorous Lorenz–Mie theory,to quantitatively represent the scattering of a single particle.This enables us to quickly compute bulk optical properties for any particle size distribution.We then use a practical Monte Carlo rendering solution to solve energy transfer in the discrete participating media.Our proposed framework is the first to simulate a wide range of discrete participating media with different levels of graininess,converging to the continuous media case as the particle concentration increases.

Case study:A simple optical inverse problem from a geometrical optics point of view

In this paper,we revisit the simple problem of reflection from a dielectric sphere for light rays and define a form of optical inverse problem in the sense of geometrical optics(GO).A general analytic formula is derived to obtain the refraction index of the sphere for any incidence light to emerge in a deflected angle.Numerical wave simulation and ray tracing are performed to verify the inverse formulae derived.

CATIA AIDED RADOME ANALYSIS USING GEOMETRIC OPTICS METHOD

In this paper, the Geometric Optics (GO) method using the approximate ray paths coupled with the Computer Aided Tri-dimensional Interface Application (CATIA) meshing modeling are implemented to analyze the performance of electric large three-dimensional dielectric radome-enclosed antenna of arbitrary contour shape. The surfaces of the radome are approximated by planar triangular patches, the influences of various number of patches on power transmission coefficient and Insertion Phase Delay (IPD) via an ogive and a conical radome are discussed by the hybrid method. The simulation results indicate that computational error from planar triangular patches can limit in one percent, meeting the engineering application requirements.

INVARIANCE AND STABILITY OF THE PROFILE EQUATIONS OF GEOMETRIC OPTICS

The profile equations of geometric optics are described in a form invariant under the natural transformations of first order systems of partial differential equations. This allows us to prove that various strategies for computing profile equations are equivalent. We prove that if L generates an evolution on L2 the same is true of the profile equations. We prove that the characteristic polynomial of the profile equations is the localization of the characteristic polynomial of the background operator at （y, dφ（y）） where φ is the background phase. We prove that the propagation cones of the profile equations are subsets of the propagation cones of the background operator.

GEOMETRIC OPTICS FOR 3D-HARTREE-TYPE EQUATION WITH COULOMB POTENTIAL

This article considers a family of 3D-Hartree-type equation with Coulomb potential ｜x｜^-1, whose initial data oscillates so that a caustic appears. In the linear geometric optics case, by using the Lagrangian integrals, a uniform description of the solution outside the caustic, and near the caustic are obtained.

Optical Image Encryption Based on Mixed Chaotic Maps and Single-Shot Digital Holography

Random phase masks play a key role in optical image encryption schemes based on double random phase technique. In this paper, a mixed chaotic method is proposed, which can efficiently solve some weaknesses that one-dimensional (1-D) single chaotic maps encounter to generate random phase masks. Based on the chaotic random phase masks, optical image encryption and decryption are realized with a single-shot digital holographic technique. In the proposed encryption scheme, the initial value and parameters of mixed chaotic maps serve as secret keys, which is convenient for the key management and transmission. Moreover, it also possesses high resistance against statistical attack, brute-force attack, noise attack and shear attack. Simulation results and security analysis verify the validity and security of the proposed encryption scheme. © 2017, Tianjin University and Springer-Verlag Berlin Heidelberg.

Wave Mechanics or Wave Statistical Mechanics

By comparison between equations of motion of geometrical optics and that of classical statistical mechanics, this paper finds that there should be an analogy between geometrical optics and classical statistical mechanics instead of geometrical mechanics and classical mechanics. Furthermore, by comparison between the classical limit of quantum mechanics and classical statistical mechanics, it finds that classical limit of quantum mechanics is classical statistical mechanics not classical mechanics, hence it demonstrates that quantum mechanics is a natural generalization of classical statistical mechanics instead of classical mechanics. Thence quantum mechanics in its true appearance is a wave statistical mechanics instead of a wave mechanics.

Non-Linear Effects in Optical Systems by Lie Algebra and Symplectic Mapping

The use of signals of different frequencies determines the geometrical deviation with respect to the optical axes of a given beam. This angle can be determined by Sympletic Map (SM), a powerful and simple mathematical tool for the characterization and construction of images in Geometrical Optics. The Sympletic Map constitutes a Lie Group, with an algebra associated: the Lie Algebra. In general, the SM can be expressed as an infinite series, where each term corresponds to different contributions produced by the optical devices that constitute the optical system (lenses, apertures, bandwidth cutoff, etc.). The level of correction to be performed on the image to recover the original object is clear and controllable by SM. This formalism can be extended easily to physical optics to describe diffraction and interference phenomena.

Electromagnetic-Energy Flow in Anisotropic Metamaterials: The Proper Choice of Poynting’s Vector

We study the controversy about the proper determination of the electromagnetic energy-flux field in anisotropic materials, which has been revived due to the relatively recent experiments on negative refraction in metamaterials. Rather than analyzing energy-balance arguments, we use a pragmatic approach inspired by geometrical optics, and compare the predictions on angles of refraction at a flat interface of two possible choices on the energy flux: and . We carry out this comparison for a monochromatic Gaussian beam propagating in an anisotropic non-dissipative anisotropic metamaterial, in which the spatial localization of the electromagnetic field allows a more natural assignment of directions, in contrast to the usual study of plane waves. We compare our approach with the formalism of geometrical optics, which we generalize and analyze numerically the consequences of either choice.

Anti-detection technology of cat eye target based on decentered field lens

Optoelectronic imaging equipment is easy to expose to active laser detection devices because of ＂cat eye＂ effect. In this paper, we propose a new structure of optical system to reduce the retroreflector effect of a cat eye target. Decentered field lens structure is adopted in the design without sacrificing imaging quality and clear aperture. An imaging system with±30° field of view is taken for example. The detailed design and simulation results are presented. The results indicate that this kind of optical system can reduce the retroreflection signal substantially and maintain acceptable imaging performance.

Oscillation Modes of Weld Pool in Stationary GTA Welding Using Structure Laser Method

Researchers have recently attempted to monitor pool oscillations using the three-dimensional laser vision method.However,the deficiency of simulation software will result in significant capital expenditure.Both simulations and experiments are performed in this study,and the Bessel equation is used to analyze the oscillation mode of a weld pool.The laser dot matrix images of(0,1),(1,1),(2,1),and(0,2)oscillation modes at different times are obtained via structured laser optical measurement simulation.The oscillation mode of a stationary gas tungsten arc weld pool is analyzed based on laser dot matrix images obtained from a structure laser experiment.Results show that the simulated laser dot matrix images are consistent with the experiment results.The oscillation mode of the weld pool can be recognized based on the laser dot matrix image.This study not only provides conditions for assessing the penetrating state of a weld pool,but also enable a further understanding of the oscillation mode of a weld pool and the development of more effective observation methods and measurement tools to effectively control and improve welding quality.

First-Principles Calculation on Geometric,Electronic and Optical Properties of Fully Fluorinated Stanene:a Large-Gap Quantum Spin Hall Insulator

The searches for large-gap quantum spin Hall insulators are important for both practical and fundamental inter- ests. In this work, we present a theoretical observation of the two-dimensional fully fluorinated stanene （SnF） by means of density functional theory. Remarkably, a significant spin-orbit coupling is observed for the SnF monolayer in the valence band at the F point, with a considerable indirect band gap of 278 meV. The direct gap of the SnF monolayer is at the F point, which is slightly larger by as much as 381 meV. In addition, the elastic modulus of the SnF monolayer is about 20J/m^2, which is comparable with the in-plane stiffness of black phos- phorus monolayer along the x-direction （~28.94 J/m^2）. Finally, the optical properties of stanene, SnF monolayer and stanene/SnF bilayer are calculated, in which the stanene/SnF bilayer is supposed to be an attractive sunlight absorber.

Spatial-Variant Geometric Phase of Hybrid-Polarized Vector Optical Fields

We investigate a novel spatial geometric phase of hybrid-polarized vector fields consisting of linear, elliptical and circular polarizations by Young＇s two-slit interferometer instead of the widely used Mach-Zehnder interferometer. This spatial geometric phase can be manipulated by engineering the spatial configuration of hybrid polarizations, and is directly related to the topological charge, the local states of polarization and the rotational symmetry of hybrid-polarized vector optical fields. The unique feature of geometric phase has implications in quantum information science as well as other physical systems such as electron vortex beams.

Geometric optics interpretation for rainbow scattering of a chiral sphere

Research on light scattering from a large chiral sphere shows that the rainbow phenomenon is different from that of an isotropic sphere. A chiral sphere with certain chirality generates three first-order rainbows. In this Letter,we present a geometric optics interpretation for the phenomenon and make a calculation of the rainbow angles.The ray traces inside the sphere are determined by the reflection and refraction laws of light at the achiral–chiral interface and the chiral–achiral interface. The calculated rainbow angles achieve good agreements with those obtained by the analytical solutions. The effects of chirality and the refractive index of the sphere on rainbow angles are analyzed.

Analysis of eccentric photorefraction by Fourier optics

Eccentric photorefraction usually is used as early eyesight diagnostic test of infants and small children. Unlike currently approved geometrical optical model of eccentric photorefractometer, the crescent formation and the light-intensity distribution in the pupil image of a myopic eye are analyzed by Fourier optics with the assumption of an isotropic scattering retina. In the case of little circular light source and rectangular slit, the simulation results of different myopic diopters are obtained by geometrical optical theory and Fourier optics respectively. It is found that the simulation results by Fourier optics are similar as those obtained by geometrical optics, and all simulations are almost corresponding to the experimental result. The result demonstrates that the new method presented here is feasible.