Terms Of Geometrical Optics Research Articles

Minimum achievable height of a single-module LED low-beam projection headlamp.

Since the first full LED headlamp was introduced in 2007, technology has evolved townow provide enhanced performance, improved security, and memorable aesthetics. Further miniaturization has been continuously pursued to allow more advanced design options, but LED headlamp height has been stalled at around 40-60 mm for some years. Recently a total internal reflection-based low-beam module achieved an optical opening down to 20 mm. However, the small opening came with a housing envelope height that was larger than 40-60 mm. Conventional LED single modules are still more attractive in terms of cost and overall compactness, especially for low-cost, low-beam headlamps. A demanding question remains about the minimum achievable limit of a single-module LED low-beam projection headlamp consisting of an LED source, an elliptical reflector, a cutoff baffle, and a projection lens. Here, we answer that question using an analytical approach rather than attempt to base it on current applied design trials or parametric studies. First, we analytically investigated the baseline optical properties of an LED low-beam module in terms of geometrical optics and photometrical luminance transfer. During the analysis, we employed the ratio of the focal distances (m) of the elliptical reflectors as a design variable to efficiently consider geometrical similarities. Then we confirmed and extended the analytical results by numerical simulation, using ray-tracing software. Based on the results from the analytical and numerical analysis with low-beam regulations, we finally determined a suitable range of m (0.15-0.17), which satisfied the regulations, and found the best achievable height of a single module when m∼0.15 is 46 mm. The precise number will differ depending on the geometrical and ray-emitting properties of the LED applied, but our study provides a valid general framework for designing LED-based low-beam headlamps.

Dispersion relations in hot magnetized plasmas

In the framework of hot magnetized collisionless plasmas, dispersion relations have been extensively studied in the past [2,12,13,24,33,34,38]. This subject is still topical in plasma physics [19,27,32,36,42]. The aim of this article is to provide a rigorous derivation of the characteristic variety, based on some asymptotic analysis of the relativistic Vlasov–Maxwell system. Special emphasis is made on the modeling of Tokamaks, with spatial variations of the magnetic field and of the equilibrium distribution function. In order to take into account the inhomogeneities, the problem is formulated in terms of geometrical optics [29,31]. This allows to unify, justify and extend the preceding results. New aspects are indeed included. For instance, the dielectric tensor is defined for real frequencies through singular integrals involving the Hilbert transform.

Time-Domain Line-Integral Representations of Physical-Optics Scattered Fields

A time-domain (TD) line-integral (LI) representation of the physical-optics (PO) radiation integral is developed for the scattering from a perfectly conducting surface illuminated by an electric Hertzian dipole. The proposed LI representation, which is valid and singularity-free for all near-field observers, is derived directly in the TD by introducing dyad potentials. This TD representation can be further expressed in terms of geometric-optics and boundary-wave components of the scattering mechanisms, as in the conventional frequency-domain LI representations. Using the proposed LI representation, the computational complexity is reduced by an order of the scatterer’s linear electric size, compared with the straightforward numerical quadrature of the PO surface integral. Simple and complicated numerical examples are presented to demonstrate the efficiency and accuracy of the proposed TD-LI algorithm.

Optical behavior of surface bubbles

The observation of diamond-like light spots produced by surface bubbles obliquely illuminated is reported. The phenomenon is discussed in terms of geometrical optics, and an explanation is provided attributing the effect to the astigmatism introduced by the deformation of the liquid surface surrounding the bubble. An essential ray tracing program is outlined and used to reconstruct the observed phenomenon numerically.

Near-Horizon Celestial Phenomena, a Study in Geometric Optics

Near horizon phenomena, like blank strips in the setting sun or Fata Morganas, can be largely understood in terms of geometric optics. Here the existence of a warm layer above the observer in which light rays can be either refracted of reflected, already suffices to explain many of the phenomena; a reasoning which goes back to Alfred Wegener in the 1920’s. We review this theory, where first the atmosphere is divided into discrete, optically homogeneous layers, and where later the atmosphere is considered as a continuous, isotropic medium. Thus the theory gets embedded in differential geometry, where light rays are geodesics. A simplifying assumption is that the refraction index only depends on the elevation above the surface of the earth. Here a connection is made with the geodesic problem on a surface of revolution. In the background of this we touch the variational calculus as this runs from Bernoulli to Hamilton, all the way maintaining a completely anachronistic viewpoint.

RADIANCE AND PHOTON NOISE: Imaging in geometrical optics, physical optics, quantum optics and radiology.

A fundamental way of describing a photon-limited imaging system is in terms of a Poisson random process in spatial, angular and wavelength variables. The mean of this random process is the spectral radiance. The principle of conservation of radiance then allows a full characterization of the noise in the image (conditional on viewing a specified object). To elucidate these connections, we first review the definitions and basic properties of radiance as defined in terms of geometrical optics, radiology, physical optics and quantum optics. The propagation and conservation laws for radiance in each of these domains are reviewed. Then we distinguish four categories of imaging detectors that all respond in some way to the incident radiance, including the new category of photon-processing detectors. The relation between the radiance and the statistical properties of the detector output is discussed and related to task-based measures of image quality and the information content of a single detected photon.

Exact Closed-Form Expression of the Electromagnetic Field Excited by a Uniform Current Distribution Lying on a Cartesian Quadrant

The exact analytical solution of the electromagnetic field distribution produced by uniform electric currents excited on Cartesian half-planes and quadrants is presented. The total field is expressed in terms of geometrical optics (GO) and diffracted field contributions that remain valid for arbitrary observation points and frequency. The jump discontinuity of the GO field is exactly compensated by the diffracted field whose spatial distribution is described in terms of the incomplete Hankel functions and by means of a novel special function. The expression of the diffracted field includes contributions arising from the edges and the vertex of the considered Cartesian domain, illustrating the analytical behavior of the near-field singularities and providing insight into the physical mechanisms governing the field diffractive processes. The proposed solution yields a method to determine the physical optics (PO) response of flat metallic screens excited by uniform plane waves, as well as the fields produced by pulse-shaped basis functions used in the method-of-moments (MoM) solution of electromagnetic problems, showing the relevance of the exact analytical expressions of the GO and diffracted fields arising from edges and vertices of each rectangular pulse domain. Numerical examples validate the accuracy of the proposed field representation.

Soft Polarization Diffraction Coefficient for a Conducting Cylinder-Tipped Wedge

TM, electromagnetic scattering from a perfectly conducting wedge with a cylindrical tip is formulated using a mode-matching technique. The eigenmode expansion is then written in a convenient form, where the total field is expressed as a superposition of a wedge-diffraction term based on the uniform theory of diffraction, a geometrical optics term, and a Correction Field term due to the presence of the cylindrical tip. The obtained diffraction coefficient can be easily incorporated into existing ray-tracing, high-frequency codes for the prediction of scattered fields from electrically large structures. The underlined formulation and obtained expressions are verified by comparing numerical results with the finite element method.

New Method for Evaluating Height Gain at Subscriber Station for Wireless Access Systems in Microwave Band

This paper presents a model for the variation in height of the subscriber station (SS) antenna with respect to the path loss for microwave-band wireless access systems. The propagation mechanism that causes the dependency of the height variation characteristics of the received level at an SS on the SS location and operating frequency is clarified in terms of geometrical optics (GO) using the uniform geometrical theory of diffraction (UTD). The height variation characteristics strongly depend on whether or not regular reflected waves that have a higher level than that of the diffracted wave arrive at the SS. A representation of the model is shown. The model is validated using measured data at 2.2, 5.2, and 25.15 GHz and the validity of the model is shown. This model is useful in the radio zone design of microwave-band broadband wireless access (BWA) systems operating in a non-line-of-sight environment, and in estimating the height gain at a mobile station antenna for mobile communications.

Infrared and submillimeter spectroscopy of grooved silicon structures

Transmission of grooved silicon structures with 4-and 7-μm periods is studied by polarization-sensitive IR and submillimeter spectroscopy in a wide spectral range. The experimental results obtained in the range 1–10-μm are explained in terms of geometrical optics taking into account the scattering of radiation. In the far-IR (20–2000 μm) range, the structures exhibit a strong birefringence, which can be described in terms of the effective medium model in the electrostatic approximation. An influence of photoexcitation on the optical transmission and its anisotropy is observed; this effect can be explained in terms of the effective medium model taking into account the interaction of radiation with free carriers.

A Local Level Set Method for Paraxial Geometrical Optics

We propose a local level set method for constructing the geometrical optics term in the paraxial formulation for the high frequency asymptotics of two-dimensional (2-D) acoustic wave equations. The geometrical optics term consists of two multivalued functions: a travel-time function satisfying the eikonal equation locally and an amplitude function solving a transport equation locally. The multivalued travel-times are obtained by solving a level set equation and a travel-time equation with a forcing term. The multivalued amplitudes are computed by a newEulerian formula based on the gradients of travel-times and takeoff angles. As a byproduct the method is also able to capture the caustic locations. The proposed Eulerian method has complexity of $O(N^2{\rm Log }N)$, rather than $O(N^4)$ as typically seen in the Lagrangian ray tracing method. Several examples including the well-known Marmousi synthetic model illustrate the accuracy and efficiency of the Eulerian method.

An optical shadowgraph microscope for a semiconductor wafer bump height measurement

We describe an optical micro shadowgraph for measuring the height of a micro solderball on a semiconductor wafer. The proposed optical system consists of a white-light microscopic inclination illumination and a telecentric objective. The solderball shadow image resulting from a slant illumination onto the protruded bump/solderball on the wafer is clearly captured. In terms of geometrical optics, a formula is derived to resolve the bump height from the length of the resulting shadow on the wafer surface. Experimental results demonstrate that the bump height measurement can be readily implemented.

Geometrical optics of moving media

The propagation of a monochromatic light wave in a medium moving at a spatially nonuniform velocity is described in terms of geometrical optics. An eikonal equation is derived to the first order in the parameter v/c (where v is the velocity of the medium and c is the velocity of light). It is shown that the nonuniformity of the motion of the medium leads to a shift and bending of the light rays and to a rotation of the plane of polarization. The estimates obtained demonstrate the feasibility of observing the revealed effects in real experiments. These effects can be used to analyze the distributions of the velocity of motion of fluids and, possibly, gases.

Interference in a Double Rainbow

Atmospheric optical phenomena, especially rainbows, are fascinating. Various texts and the scientific literature 1 1. See, for example, J. D. Walker, Am. J. Phys. 44, 421 (1976) https://doi.org/10.1119/1.10172. have done well in explaining the rainbow in terms of geometric optics. The phenomenon of the double rainbow is shown to be one collective optical pattern not always seen or recognized as such. In particular, knowledgeable observers have aptly described the dark zone between the primary and the secondary rainbows as an optical interferometric condition. The effect of the interference is clearly visible in the photograph. However, the destructive interference in the dark zone is not complete; witness the mountain image in the background.BRIDIE SACCOCIOPPT|High resolutionREFERENCESection:ChooseTop of pageREFERENCE <<1. See, for example, J. D. Walker, Am. J. Phys. 44, 421 (1976) https://doi.org/10.1119/1.10172. Google ScholarCrossref, ISI© 2003 American Institute of Physics.

DECAY OF SOLUTIONS OF THE SYSTEM OF THERMOELASTICITY OF TYPE III

This paper is devoted to analyzing the long time behavior of solutions of the system of thermoelasticity of type III in a bounded domain of ℝn (n = 1,2,3) and in the whole space ℝn. For the first case, we introduce a decoupled system that allows to reduce the problem of the asymptotic behavior for the original system to a suitable observability inequality for the Lamé system. In this way most of the existing results for the classical system of thermoelasticity are shown to hold for this system too. In particular, we show that: (1) For most domains the energy of the system does not decay uniformly; (2) Under suitable conditions on the domain that may be described in terms of Geometric Optics, the energy of the system decays exponentially; and (3) For most domains in two space dimensions, the energy of smooth solutions decays polynomially. For the problem in the whole space ℝn, first, based on Fourier analysis and Lyapunov's second method, we show that the energy of longitudinal and thermal waves of the system decays as that of the classical heat equation (while that of the transversal wave component is conservative). Then, by means of a careful spectral analysis, we give a sharp description on the decay rate of the high frequency longitudinal and thermal waves of the system.

Paraxial geometrical optics for quasi-P waves: theories and numerical methods

The quasi-P wave in anisotropic solids is of practical importance in obtaining maximal imaging resolution in seismic exploration. The geometrical optics term in the asymptotic expansion for the wave characterizes the high frequency part of the quasi-P wave by using two functions: a phase (traveltime) function satisfying an eikonal equation and an amplitude function satisfying a transport equation. Based on a paraxial eikonal equation satisfied by the traveltime corresponding to the quasi-P downgoing waves, two new advection equations for take-off angles provide essential ingredients for computing amplitude functions on uniform Cartesian grids. However, the radiation problem of the eikonal equation has an upwind singularity at the point source which renders all finite-difference eikonal solvers to be first-order accurate near the source. Extending an isotropic adaptive eikonal solver to the paraxial quasi-P eikonal equations can treat this singularity efficiently and yield highly accurate traveltimes and amplitudes. Numerical experiments for quasi-P traveltimes and amplitudes in transversely isotropic media with vertical symmetry axes verify that the numerical methods are efficient and accurate.

Geometrical optics Terms and concepts

Geometrical optics Terms and concepts

Geometric Optics and Long Range Scattering¶for One-Dimensional Nonlinear Schrödinger Equations

With the methods of geometric optics used in [2], we provide a new proof of some results of [10], to construct modified wave operators for the one-dimensional cubic Schrodinger equation. We improve the rate of convergence of the nonlinear solution towards the simplified evolution, and get better control of the loss of regularity in Sobolev spaces. In particular, using the results of [9], we deduce the existence of a modified scattering operator with small data in some Sobolev spaces. We show that in terms of geometric optics, this gives rise to a “random phase shift” at a caustic.

High-frequency surface currents induced on a spherical reflector by the edge-diffraction of obliquely incident waves

An explicit expression of the high-frequency surface current excited on a perfectly conducting spherical cap by the edge-diffraction of an obliquely incident wave is derived. The result is given in the GTD terminology and shows the influence of the incidence angles on the transfer functions. To this end the real (physical) space is embedded into a twofold extended abstract space in which an equivalent canonical problem is established. This latter yields a system of dual integral equations whose kernels were not previously treated. By inverting these new kernels, the system of dual equations is reduced to two independent Wiener-Hopf problems. From the asymptotic solutions of these problems, one derives the expression of the edge-excited current along with the known geometrical optics term. The resutlt can be used directly in practical applications. If the incident ray becomes normal to the edge, then the resulting expressions are reduced to the already known expressions. The basic idea and general formulas used here can also be used to solve other diffraction problems with similar geometry.

High‐frequency asymptotic acceleration of the fast multipole method

The plane wave translation operator of the fast multipole method (FMM) is evaluated asymptotically in the high‐frequency limit. This operator relates the plane wave components of a source region to a plane wave expansion for the radiated fields over a region away from the source. The asymptotic evaluation exploits the source‐to‐receiver directivity of the operator, leading to terms analogous to the geometrical optics (GO) and diffracted fields in the uniform theory of diffraction. Most importantly, the GO term identifies a “lit” region which may be used to define a windowing function for filtering out weak plane wave translations in the “shadow” region. The reduction in plane waves lowers the operational count of the FMM from O(N3/2) to O(N4/3) without increasing the complexity of the implementation.