Number Of Internal Reflections Research Articles

Geometric optics of a refringent sphere illuminated by a point source: caustics, wavefronts, and zero phase-fronts for every rainbow "k" order.

This study relates to a refringent sphere illuminated by a point source placed at a distance h from its center; for h→∞ the light beam becomes parallel. A selection of variables, principally angular with the center of the sphere as a common point, allows a global, straightforward, and geometrically transparent way to the rays, caustics, and wavefronts, internal as well as external, for every k order, k being the number of internal reflections. One obtains compact formulas for generating the rays and the wavefronts.

High-order rainbows of a spherical particle produced by near-grazing incident light.

This study is concerned with the formation of high-order rainbows by near-grazing light incident on a spherical particle. As the number of internal reflections involved increases, the incident Descartes ray strikes the sphere surface increasingly closer to its edge, where the predictions of ray theory and Airy theory become invalid. The deflection angle of the confluence of the stationary points of the phase of the partial wave scattering amplitudes is studied as a function of rainbow order and sphere radius. It is found that as the rainbow order increases, the angular interval over which the upper supernumerary ray stationary point occurs shrinks to zero. In addition, for deflection angles beyond the confluence of the upper supernumerary ray with the tunneling ray, intensity oscillations are due to interference of the field of the lower supernumerary ray with that of the edge region Fock transition, rather than interference between the upper and lower supernumerary rays.

Use of a Geometry Optimized Fiber-Optic Surface-Enhanced Raman Scattering Sensor in Trace Detection

A novel SERS (surface-enhanced Raman scattering) sensor has been recently developed; its peculiar geometry is able to increase considerably both the SERS active surface and the number of internal reflections at the interface between silica and silver, thus allowing an increase of the signal intensity. The aim of this work is to demonstrate that this sensor could be efficiently used to detect some molecules such as illegally used veterinary medicine (crystal violet and malachite green) below the ppb detection limit. The advantages of this sensor with respect to other detection techniques are not only the higher sensitivity, but also the fast response and the possibility of coupling with a portable Raman spectrometer for "on-field" measurements. The ability of the sensor to work under real environmental conditions in the presence of many cationic and anionic species has been tested both in solutions containing sodium and chlorine ions and in water coming from the aqueduct of Milan and from the (normally polluted) river Serio.

Geometrical-optics code for computing the optical properties of large dielectric spheres.

Absorption of electromagnetic radiation by absorptive dielectric spheres such as snow grains in the near-infrared part of the solar spectrum cannot be neglected when radiative properties of snow are computed. Thus a new, to our knowledge, geometrical-optics code is developed to compute scattering and absorption cross sections of large dielectric particles of arbitrary complex refractive index. The number of internal reflections and transmissions are truncated on the basis of the ratio of the irradiance incident at the nth interface to the irradiance incident at the first interface for a specific optical ray. Thus the truncation number is a function of the angle of incidence. Phase functions for both near- and far-field absorption and scattering of electromagnetic radiation are calculated directly at any desired scattering angle by using a hybrid algorithm based on the bisection and Newton-Raphson methods. With these methods a large sphere's absorption and scattering properties of light can be calculated for any wavelength from the ultraviolet to the microwave regions. Assuming that large snow meltclusters (1-cm order), observed ubiquitously in the snow cover during summer, can be characterized as spheres, one may compute absorption and scattering efficiencies and the scattering phase function on the basis of this geometrical-optics method. A geometrical-optics method for sphere (GOMsphere) code is developed and tested against Wiscombe's Mie scattering code (MIE0) and a Monte Carlo code for a range of size parameters. GOMsphere can be combined with MIE0 to calculate the single-scattering properties of dielectric spheres of any size.

Theory of transient spontaneous emission by an atom in a planar microcavity

We study the transient regime of spontaneous emission by an atom excited in a planar high-Q microcavity. The variation of the spatial distribution of the radiation outside the cavity is determined as a function of the increasing number of internal reflections for transition dipole moments parallel and perpendicular to the mirrors and for mirror separations equal to $\ensuremath{\lambda}/2$ and $\ensuremath{\lambda}/4.$ Particular attention is given to the parallel dipole in the $\ensuremath{\lambda}/2$ cavity, where the number of reflections is closely proportional to the elapsed time. It is shown that the transient regime extends over times of the order of the cavity decay time, in contrast to the effectively one-dimensional cavities previously studied, where the steady state is achieved after the cavity round-trip time. The field distribution inside the cavity tends to a steady-state form whose dimension parallel to the mirrors is of the order of the transverse coherence length. Despite this transverse localization of the field excitation, it is shown that the conditions for the achievement of strong atom-field coupling and the observation of Rabi oscillations cannot be met in the planar microcavity.

Properties of seismic fault zone waves and their utility for imaging low‐velocity structures

A two‐dimensional solution for the scalar wave equation in a model of two vertical layers between two quarter spaces is used to study properties of seismic waves in a laterally heterogeneous low‐velocity structure. The waves, referred to as seismic fault zone waves, include head waves, internal fault zone reflections, and trapped waves. The analysis aims to clarify the dependency of the waves on media velocities, media attenuation coefficients, layer widths, and source‐receiver geometry. Additional calculations with extreme low‐velocity layers provide examples that may be relevant for volcanic and geothermal domains. The interference patterns controlling seismic fault zone waves change with the number of internal reflections in the low‐velocity structure. This number increases with propagation distance along the structure, decreases with fault zone width, and increases (for given length scales) with the velocity contrast. The relative lateral position of the source within the low‐velocity layer modifies die length scales associated with internal reflections and influences the resulting interference pattern. Low values of Q affect considerably the dominant period and overall duration of the waves. Thus there are significant tradeoffs between propagation distance along the structure, fault zone width, velocity contrast, source location within the fault zone, and Q. The lateral and depth receiver coordinates determine the particular point where the interference pattern is sampled and observed motion is a strong function of both coordinates. The zone connecting sources generating fault zone waves and observation points with appreciable wave amplitude can be over an order of magnitude larger than the fault zone width. Calculations for cases with layer P wave velocity of ∼200 m s−1, modeling a vertical dike or crack with fluid and gas, show conspicuous persisting oscillations. The results resemble aspects of seismic data in volcanic domains. If these waves exist in observed records, their explicit recognition and modeling will help to separate source and structural effects and aid in the interpretation of volcano‐seismology signals. Although the tradeoffs in parameters governing seismic fault zone waves are significant, each variable has its own signature, and the parameters may be constrained by additional geophysical data. Simultaneous modeling of many waveforms with an appropriate solution can resolve the various parameters and provide a high‐resolution structural image.

Efficient Source-To-Fiber Coupling Method Using a Diamond Rod: Theory and Application to Multimode Evanescent-Wave IR Absorption Spectroscopy

The use of a diamond rod to couple a blackbody source optically to a chalcogenide glass fiber is described and shown to be useful for evanescent-wave IR absorbance measurements. One end of a short 0.5-mm-diameter diamond rod abuts an optical fiber of the same diameter; the other end is in direct contact with a glowing silicon carbide IR source. The close match of the refractive indices of SiC, diamond, and AsSeTe fiber reduces optical losses at the interfaces, allowing a high overall throughput as well as transmission into the fiber of a wider range of transverse optical modes than is available with the use of through-air focusing methods. Use of the higher-order modes for evanescent-wave spectroscopy leads to increases in sensitivity, due both to increased evanescent pathlength and to a larger number of internal reflections per unit length.

Solar Radiative Fluxes for Broken Cloud Fields above Reflecting Surfaces

Abstract A statistical bidirectional method for including the effects of underlying reflecting surfaces in Monte Carlo simulations of atmospheric photon transport is presented. It is illustrated for the idealized Lambertian surface and a general particulate surface. It is shown that the simple Lambertian surface should be an adequate approximation (to mildly anisotropic surfaces) for calculation of overall fluxes for broken cloud fields. The problem of incorporating multiple internal reflections of solar radiation between cloud and surface is reviewed, and cloud field reflectances to upwelling radiation rk as a function of the number of internal reflections k are calculated. Unlike plane-parallel and regular arrays of cloud, all rk are generally not equal for more realistic scaling cloud fields. Strictly speaking, this prohibits use of the familiar geometric sum formulas for flux calculation in a multiple reflecting system. Fortunately, though fortuitously, the geometric sum formulas can be used in such c...

Rainbows: Mie computations and the Airy approximation

Efficient and accurate computation of the scattered intensity pattern by the Mie formulas is now feasible for size parameters up to x = 50,000 at least, which in visual light means spherical drops with diameters up to 6 mm. We present a method for evaluating the Mie coefficients from the ratios between Riccati-Bessel and Neumann functions of successive order. We probe the applicability of the Airy approximation, which we generalize to rainbows of arbitrary p (number of internal reflections = p - 1), by comparing the Mie and Airy intensity patterns. Millimeter size water drops show a match in all details, including the position and intensity of the supernumerary maxima and the polarization. A fairly good match is still seen for drops of 0.1 mm. A small spread in sizes helps to smooth out irrelevant detail. The dark band between the rainbows is used to test more subtle features. We conclude that this band contains not only externally reflected light (p = 0) but also a sizable contribution f rom the p = 6 and p = 7 rainbows, which shift rapidly with wavelength. The higher the refractive index, the closer both theories agree on the first primary rainbow (p = 2) peak for drop diameters as small as 0.02 mm. This may be useful in supporting experimental work.

An extinction theorem for optical scattering

From a many-body theory of optical scattering we derive a relation for the collective behaviour of incoherently scattered light emerging from a molecular fluid. The relation involves a surface integral and has precisely the mathematical form of the Ewald-Oseen extinction theorem; it is exact in the sense of that theorem. We prove from the relation that scattered radiation is refracted and reflected at the surface of the scattering medium according to rules inferred from macroscopic optics. The relation provides a construction of contributions to the scattering with any number of internal reflections; it applies also to scattering restricted by stops. Explicit expressions for the surface dependent scattering in terms of a Fresnel transmissivity correct a previously reported theory of bulk scattering.

A Kirchhoff Solution to the Diffraction by an Elastic Wedge

The scattering of an incident plane wave from an elastic wedge has been calculated using the Kirchhoff approximation. In this method reflected and transmitted waves are first calculated using Snell's law. The boundary values obtained in this way for the pressure and its normal derivative on the surface of the wedge are then substituted into the Helmholtz integral. The radiation pattern of the diffracted wave has been calculated for infinite steel wedges in water and compared with rigid wedge solutions. Close agreement is found except in certain directions corresponding to transmitted plane waves not present in the rigid case. The effect of penetration on the diffracted wave becomes more important as the number of internal reflections within the wedge increases. The method also introduces some spurious infinites in a finite number of directions.

Some New Direct Vision Dispersion Prism Systems

Direct vision prisms are useful for quick examination of any spectrum. Most of the commercial direct vision instruments utilize the cemented type Amici prism consisting of both crown and flint prisms. In this paper a type of direct vision prism made of single glass is discussed. It utilizes two refractions for dispersive purposes and an even number of internal reflections. In addition some other types of direct vision prisms are described differing from the usual Amici type.

On Talbot's Bands and the Theory of the Lummer-Gehrcke Interferometer

New Simple Method of Observing Talbot's Bands.-If a thin strip of cover glass is immersed in a dispersive medium, such as a suitable mixture of CS 2 and benzene, with a refractive index such that the group velocity in it is less than in the plate, and if a vertical slit illuminated with white light is observed through the glass strip held in a vertical plane very oblique to the plane of the incident light, two out of the four spectra formed by the edges of the plate exhibit Talbot's bands. The theory of the formation of the bands in this way is closely similar to that of the Lummer-Gehrcke plate, and as in the latter, the maxima are narrow and the minima crossed by secondary maxima when the number of internal reflections is small. The appearance, too, is not affected by opening the slit wide. Interference Bands Analogous to Those given by a Fabry-Perot etalon were also observed with the same simple arrangement. The appearance and behavior of these bands are described and explained.

A Graphic Treatment of Cusped Wave-fronts and of the Rainbow

A method of drawing the cusped wave-fronts produced by refraction and reflection at a spherical surface and a graphic treatment of the elementary theory of the rainbow are described in the Paper. The method is based upon the properties of the centre of junction. In the case of reflection at a spherical surface in which aplanatic points are not available the position of the junction-centre is obtained by elementary geometry. By the use of junction-centres the distribution of the successive foci obtained on refraction and reflection at a spherical surface is readily plotted. If in the case of a rainbow it is assumed that the emergent portion of the effective pencil, as well as the incident portion, is one of parallel rays, then the distribution of foci with regard to the drop is a symmetrical one. The caustics on refraction and reflection are also readily drawn as loci of points and envelopes of rays. Hence the wave-surfaces can be obtained. These in some regions are cusped and the caustics are the loci of the cusps. In the case of pencils that are effective in producing rainbows, the lengths of the incident portion of the chief ray intercepted by the spherical drop of radius, r, and a concentric sphere of radius, r. μ, are in the ratio 1 to n + 1, where n is the number of internal reflections. This leads to a geometrical construction for finding the points of incidence of the chief rays of the effective pencils. These rays are at minimum deviation, and the same chief ray remains at minimum deviation although the associated pencil may be converging or diverging.