Descartes glare points in scattering by icicles: color photographs and a tilted dielectric cylinder model of caustic and glare-point evolution

Glare points associated with the Airy caustics of once and twice internally reflected rays are visible in the scattering by sunlit icicles. Supporting color photographs include an image of the far-field scattering. Relevant rays are analogous to the Descartes rays of primary and secondary rainbows of drops; however, the caustic conditions for the icicle are predicted to be affected by tilt of the illumination relative to the axis of the icicle. A model for the caustic evolution, given for a circular dielectric cylinder, manifests a transition in which the Airy caustic (and associated glare points) merge in the meridional plane at a critical tilt. At this critical tilt the merged glare point is predicted to be very bright. The calculations use the Bravais effective refractive index and generalized ray tracing.     

Exterior caustics produced in scattering of a diagonally incident plane wave by a circular cylinder: semiclassical scattering theory analysis

We use the semiclassical limit of electromagnetic wave scattering theory to determine the properties of the exterior caustics of a diagonally incident plane wave scattered by an infinitely long homogeneous dielectric circular cylinder in both the near zone and the far zone. The transmission caustic has an exterior/interior cusp transition as the tilt angle of the incident beam is increased, and each of the rainbow caustics has a farzone rainbow/exterior cusp transition and an exterior/interior cusp transition as the incident beam tilt angle is increased. We experimentally observe and analyze both transitions of the first-order rainbow. We also compare the predictions of the semiclassical approximation with those of ray theory and exact electromagnetic wave scattering theory.     

一种声线修正的查表方法

在利用时延测距的水声定位系统中,由于在海洋中存在声速梯度,声线传播发生弯曲,为了提高定位精度,提出了一种声线修正的查表方法。由射线声学理论可知,声传播距离和时间都是声线初始掠射角的函数,在声源深度、接收深度和声速分布已知的情况下,对每一个初始掠射角,必有唯一一个传播时间和距离与它相对应,根据声线的这一特征,可以用查表法通过声传播时间来确定相应的传播距离。通过建立传播时延与声源与接收机的水平距离的对应关系表,利用声线在水平面的投影水平距离通过圆交汇解算出目标的位置,而不是利用声线弯曲的斜距进行球面交汇解算出目标的位置。仿真和海试结果表明,在复杂水文条件下,该方法能有效提高水声定位精度。     

Generalized rainbows and unfolded glories of oblate drops: organization for multiple internal reflections and extension of cusps into Alexander's dark band

Oblate drops of water can produce caustics where, unlike a simple Airy caustic, more than two rays merge. We extend previous treatments of generalized primary rainbows based on catastrophe optics [Opt. Lett. 10, 588 (1985); Proc. R. Soc. (London) A 438, 397 (1992)] to rays having (p - 1) = 2 to 5 internal reflections. The analysis is for a horizontally illuminated ellipsoid with a vertical symmetry axis. Aspect ratios causing a vanishing of the vertical curvature at the equator for the outgoing wave front are found from generalized ray tracing. In response to infinitesimal deformation, the axial caustic of real glory rays unfolds producing cusps. Laboratory observations with laser illumination demonstrate that cusps resulting from rays with five internal reflections extend into Alexander's dark band when the drop's aspect ratio is near 1.08. The evolution of this p = 6 scattering pattern as cusps meet the quinary rainbow is suggestive of an E(6) catastrophe. For ellipsoids of varying aspect ratio and refractive index N, there is an organizing singularity associated with an exceptionally flat outgoing wave front from spheres with N = p.     

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。     

Optical Design Problems in Laser Systems

Exact mathematical expressions for the generalized rays of optics prescribed according to an artificial index is applied to obtain ray-trajectory solutions of the waveguide bound modes. Direct mapping of the bound modes via the Milne equation results in polarization-dependent zigzag rays in the film. Their continuation into the evanescent field regions are no longer straight, their curvature being an indication of the inhomogeneity of the field. This extended ray interpretation predicts inter-waveguide coupling effects, while the curvature of the rays dictates the degree of light confinement. For waveguides with a large index difference, the ray is virtually parallel to the waveguide axis at a distance of approximately a wavelength after leaving the film. The Goos-Haenchen shifts obtained from the generalized ray theory agree exactly with those obtained from the usual mode theory.     

Generalized tertiary rainbow of slightly oblate drops: observations with laser illumination

The tertiary rainbow of acoustically levitated water drops was observed in the laboratory. Nontrivial caustics were observed for relatively small values of eccentricity. The angular locations of caustics were modeled with matrix methods of generalized ray tracing. Photographs of the scattering were in general agreement with models. Possible effects on the appearance of natural tertiary bow features are discussed.     

Singularities and perturbations for far-field caustics in the neighborhood of a crack tip

Summary Adapting the principle of least time by Fermat, the farfield caustics around a crack tip were studied as a variational problem. The singular reflected rays were probed and classified using singularity theory; furthermore the singular directions were also defined.In addition, applying perturbation theory the stress intensity factor at a crack tip is evaluated, considering higher-order terms of the Muskhelishvili complex potential.It was shown that higher approximations can be achieved with this method from caustics engendered from initial curves of a larger than usual radius convenient for problems of plasticity and elastic anisotropy.With 6 Figures     

Vision ray calibration for the quantitative geometric description of general imaging and projection optics in metrology

Exact geometric calibration of optical devices like projectors or cameras is the basis for utilizing them in quantitative metrological applications. The common state-of-the-art photogrammetric pinhole-imaging-based models with supplemental polynomial corrections fail in the presence of nonsymmetric or high-spatial-frequency distortions and in describing caustics efficiently. These problems are solved by our vision ray calibration (VRC), which is proposed in this paper. The VRC takes an optical mapping system modeled as a black box and directly delivers corresponding vision rays for each mapped pixel. The underlying model, the calibration process, and examples are visualized and reviewed, demonstrating the potential of the VRC.     

Measurement of the antiproton/proton ratio at TeV energies with the ARGO-YBJ detector

Cosmic ray antiprotons provide an important probe for the study of cosmic-ray propagation in the interstellar space and to investigate the existence of Galactic dark matter. Cosmic rays are hampered by the Moon, therefore a deficit of cosmic rays in its direction is expected (the so-called Moon shadow). The Earth–Moon system acts as a magnetic spectrometer. In fact, due to the geomagnetic field the center of the Moon shifts westward by an amount depending on the primary cosmic ray energy. Paths of primary antiprotons are therefore deflected in an opposite sense in their way to the Earth. This effect allows, in principle, the search of antiparticles in the opposite direction of the observed Moon shadow.The ARGO-YBJ experiment, in stable data taking since November 2007 with an energy threshold of a few 100s of GeV, is observing the Moon shadow with high statistical significance. Using about 1 year data, an upper limit of the flux ratio in the few-TeV energy region is set to a few percent with a confidence level of 90%.     

Extension of geometrical-optics approximation to on-axis Gaussian beam scattering. I. By a spherical particle

The geometrical-optics approximation of light scattering by a transparent or absorbing spherical particle is extended from plane wave to Gaussian beam incidence. The formulas for the calculation of the phase of each ray and the divergence factor are revised, and the interference of all the emerging rays is taken into account. The extended geometrical-optics approximation (EGOA) permits one to calculate the scattering diagram in all directions from 0 degrees to 180 degrees. The intensities of the scattered field calculated by the EGOA are compared with those calculated by the generalized Lorenz-Mie theory, and good agreement is found. The surface wave effect in Gaussian beam scattering is also qualitatively analyzed by introducing a flux ratio factor. The approach proposed is particularly important to the further extension of the geometrical-optics approximation to the scattering of large spheroidal particles.     

J-integral evaluation in elasto-plastic materials by optical shadow visualisation

The optical method of shadow visualisation is used for the experimental determination of the J-integral value. Optical relations have been obtained to describe the formation of shadow patterns (caustics) from an optical scheme making use of an imaging lens. The assumption from which the evaluation formulas are deduced is based on the nearfield stress distribution for elasto-plastic power hardening material taking into account the HRR theory. An analysis has been done on the field of application of such an approach for the interpretation of experimental data.     

A theory of caustics for mixed-mode fast running cracks

The method of caustics (shadow spot method) has proven to be a powerful optical method to measure stress intensity factors in static and dynamic fracture mechanics problems. In this paper, a theory of caustics was developed for elastodynamically propagating cracks under inplane mixed-mode conditions. Complex potentials for the general solutions of a near-tip field which have been previously derived by the authors were used in this theoretical development. Completely analytical expressions were derived for the caustic curves as well as for the initial curves for fast running cracks under inplane mixed-mode conditions. The effects of crack velocity and mixed-mode condition on the caustic pattern and the initial curve were investigated. New procedures were also proposed for the evaluation of the dynamic stress intensity factors K_I and K_II using the overall dimensions of the caustic pattern. The method of caustics developed here enables one to study quantitatively various mixed-mode dynamic fracture phenomena such as crack branching, crack curving, and crack kinking.     

The reflected caustics formed from cross-sections of twisted prismatic bars

Summary If a prismatic bar made of a homogeneous isotropic and elastic material is submitted to torsion, cross sections of the bar are twisted and warped. In this paper the shape and size of the caustics and pseudocaustics formed from reflections of light rays on these cross sections were analytically and experimentally studied. This study is concerned with cross sections of circular, elliptic, triagular, and rectangular shapes of different ratios between their characteristic dimensions. It was proved that the experimentally derived caustics are in good agreement with the theoretical curves. Then this method of reflected caustics is a sensitive device for the detection and the accurate measurement of the torsional loading of such elements of structures. With 15 Figures     

Crepuscular rays: laboratory experiments and simulations

Model simulations of laboratory-generated and natural crepuscular rays are presented. Rays are created in the laboratory with parallel light beams that pass through artificial fogs and milk-water solutions. Light scattered by 90° in a dilute mixture of whole milk first increases in intensity with distance from the source to a maximum as a result of multiple scattering by mainly small angles before decreasing exponentially due to extinction as distance continues to increase. Crepuscular rays are simulated for three cloud configurations. In case 1, the Sun at the zenith is blocked by a cloud with an overhanging anvil. The rays appear white against blue sky and are brightest when atmospheric turbidity, β≈11. Shading by the anvil separates maximum brightness from apparent cloud edge. In case 2, a ray passes through a rectangular gap in a cloud layer. The ray is faint blue in a molecular atmosphere but turns pale yellow as β and solar zenith angle, φ(sun), increase. At φ(sun)=60° it appears most striking when the cloud is optically thick, β≈5, and the beam width Δx≈1000 m. In these cases, increasing aerosol radius, r(aer), to about 1000 nm brightens, narrows, and shortens rays. In case 3, the twilight Sun is shaded by a towering cloud or mountain. The shaded rays are deeper blue than the sunlit sky because the light originates higher in the atmosphere, where short waves have suffered less depletion from scattering. The long optical path taken by sunlight at twilight makes color and lighting contrasts of the rays greatest when the air is quite clean, i.e., for β-1?1. In all cases, the brightest rays occur when sunlight passes through an optical thickness of atmosphere, τ≈O(1).     

Semiclassical scattering of an electric dipole source inside a spherical particle.

Semiclassical scattering phenomena appearing in the far-zone scattered intensity of a point source of electromagnetic radiation inside a spherical particle are examined in the context of both ray theory and wave theory, and the evolution of the phenomena is studied as a function of source position. A number of semiclassical effects that do not occur for plane-wave scattering by the sphere appear prominently for scattering by an interior source. These include a series of scattering resonances and a new family of rainbows in regions of otherwise total internal reflection. Diffractive effects accompanying the semiclassical phenomena are also examined.     

光与影在灯具设计中的创造性运用

旨在从光与影的关系这一角度探索灯具设计的新理念和新方法。首先阐述了自然界中光与影共同存在的普遍性,并指出人类造物活动一直存在着对光与影的创造性利用,接着从"能用"、"好用"和"感性"3个层面分析灯具的发展,最后把对光与影的理解落实到设计方法上,从"光的设计"和"影的设计"2个方面概括了怎样设计"感性的灯具"。     

Scattering observations for tilted transparent fibers: evolution of airy caustics with cylinder tilt and the caustic merging transition

When a dielectric circular cylinder is obliquely illuminated, the scattering angle associated with the Airy caustics of the cylinder's primary rainbow depends on the tilt of the cylinder. We display records of the scattering pattern for a transparent poly(methyl methacrylate) fiber ranging from small values of tilt through values of tilt that are sufficiently large for the Airy caustics from both sides of the fiber to merge in a meridional plane containing the incident wave vector and the fiber's axis. The records are compared directly with the evolution of the caustic projected onto the observation plane, and certain qualitative features of the global evolution of the caustics are confirmed. Although the observations used laser illumination, they are relevant to anticipating the scattering by sunlit transparent tilted cylinders.     

Interferometry for the evaluation of the dynamical optical and mechanical properties of transparent materials

A theory is developed which leads to an interferometric method for determining the dynamic ratio, Poisson's ratio to the modulus of elasticity, and the dynamic stress-optical coefficients of transparent plates, both during the same set of tests and without attaching mechanical or electrical sensors on the surface of the specimen.

The method requires a monochromatic light source, that is, a parallel beam is directed perpendicularly to the vicinity of a crack tip which exists in the plate specimen under investigation. The rays impinging on the crack tip are transmitted through the plate and reflected by the two parallel surfaces of the plate. At the same time, because of the variation of the optical path (index of refraction and thickness of the plate) in the said area, the rays are refracted in such a way as to form two caustic surfaces, one by the transmitted and one by the reflected rays. These two caustic surfaces, when intercepted with two opaque screens, provide two strongly illuminated curves of a general epicycloid shape called caustics.

The size of the caustics is directly related to the applied load, the material properties and the distance specimen-screen. Their ratio, however, as it is taken at equally distant screens, depends only on the optical and mechanical properties of the material in question.

In addition, an interferogram consisting of fringes, as formed from rays reflected by the two surfaces of the plate, is obtained. The fringe number depends on the wavelength of the light source and the optical path. When the plate specimen is loaded, the optical path changes. This results in a change of the fringe number and consequently to a displacement of fringes with respect to a reference point as they are observed on an opaque screen.

The ratio of the two caustics, taken during a static test, and the number of displaced fringes per applied load, taken during a dynamic test, provide enough information for the determination of the aformentioned mechanical and optical parameters.

Experiments conducted according to the method developed on acrylic plates and for a tensile impulse of total time duration of 600 μsec produced a reduced by 23.5% ratio of Poisson's ratio to modulus of elasticity from the static value and a reduced by 21% stress-optical coefficient from the static value.