Radiative flux from a multiple-point bioluminescent or chemiluminescent source within a cylindrical reactor incident on a planar-circular coaxial detector. II. Rotationally symmetric radiation

In the previous paper [J. Opt. Soc. Am. A 28, 126 (2011)], an analytical formula was presented for calculating radiative fluxes from arbitrarily distributed and arbitrarily radiating multiple-point emitters of bioluminescent or chemiluminescent sources within cylindrical reactors, when the radiation from these point emitters propagates through two homogeneous isotropic media and reaches a planar-circular coaxial detector. This formula was based on two assumptions. The first is that radiation passes across a planar boundary interface between the two media. The second is that the surface reflections on the lateral surface and on the reactor base opposite the detector may be neglected. In this paper, the formula obtained previously was simplified for the case of uniformly distributed point emitters of bioluminescent or chemiluminescent sources emitting an identical rotationally symmetric radiation. The simplified formula is suitable for optimizing and calibrating the analyzed reactor-detector system, which is most commonly used to study the bioluminescence emitted by small biological objects and the chemiluminescence from chemical reactions. Representative data were calculated, illustrated graphically, and tabulated.     

Optical radiation flux illuminating a circular disk from an off-axis point source separated by a homogeneous plane-parallel plate

In a number of techniques that measure weak fluxes of optical radiation, it is frequently necessary to keep a detector in a medium different from that of the radiating source by separating it from the source with a planar transparent window. However, sources such as systems of light-emitting diodes, large-fiber illuminators, and microscopic living objects that emit biological luminescence may sometimes be regarded as multiple-point sources. To estimate the fluxes of optical radiation illuminating a surface from a nonuniformly distributed multiple-point source, a method for calculating fluxes from a single off-axis point source is needed. A formula is derived to estimate a flux of temporally incoherent optical radiation incident on a circular disk from a single off-axis point source separated by a plane-parallel plate (PPP). This formula is expressed by a series of single integrals of some superposed elementary functions. These functions depend on the variables that characterize the point-source-plane-parallel-plate-circular-disk geometry and on the optical properties of the media that separate the source from the PPP and the PPP from the disk. The solution was obtained for isotropic media. For illustrative purposes some examples of the use of the formula are presented. The selected results are illustrated by three-dimensional surface plots and compared with the values of the fluxes calculated for radiation incident on the disk from a point source not separated by a PPP.     

Angular distribution of an average solid angle subtended by a circular disc from multiple points uniformly distributed on a planar circular surface coaxial to the disc

Abstract

Average solid angles subtended by an external contour of a given body from several points are required for the calculation of radiation fluxes or particle beams that are incident on the body from multiple-point isotropic emitters of analogous kinds. To consider the refraction effects for fluxes of light propagating through various optical media, knowledge of the angular distributions of such average solid angles is necessary. In this paper the formula describing angular dependence of the average solid angle subtended by a circular disc from uniformly distributed points within the circular surface of a parallel and coaxial disc was derived analytically and used for the calculation of some representative results. The solution has been made in the cylindrical coordinate system. The final and some intermediate formulas were expressed as functions of the polar angle, of the radii of both discs and of the distance between their planes. These formulae were represented by superpositions of simple elementary functions, single integrals of, these superpositions and by incomplete and complete Legendre—Jacobi elliptic integrals of all three kinds. Mathematica 2.2.3 software was used to illustrate graphically the relationships between some computed data. These data indicate that the derived formula is directly applicable in any computer programs calculating the Legrendre—Jacobi elliptic integrals to estimate the fluxes of optical radiation and particle beams propagated within a non-absorbing homogeneous medium. The expressions obtained may also be used to calculate the fluxes of optical radiation propagated through various homogeneous media.     

The flux of radiation incident on a circular planar detector from a coaxial circular plane extended source

A radiometric formula was derived for calculating fluxes of radiation from a coaxial planar circular source transmitted through one homogeneous isotropic medium on a planar circular detector. This formula was obtained using fundamental laws of classical radiometry. The general solution was described by quadruple definite integrals as dependent on the source radiance, on the radii of the source and the detector, and on the source–detector distance. For a wide range of source models with various angular distributions of radiances these definite integrals can be expressed in closed form and the formula simplifies considerably. To demonstrate how this formula can be used the solution was applied to the radiation emitted by the planar circular Lambertian source model. Some calculated results were illustrated graphically and analysed.     

Radiant fluxes from on- and off-axis point sources irradiating a circular disc through two different homogeneous isotropic media

The multidomain integral equation method is used to calculate fluxes of radiation from various on- and off-axis point sources passing through two different homogeneous isotropic media and striking a surface of a circular disc perpendicular to the optical axes of these sources. This method is dedicated to radiation passing through attenuating or nonattenuating media with a Fresnel interface and is applicable for arbitrary radiation patterns of point sources. The paper presents, firstly, the generalized multidomain integral equation method, expressed by double iterated integrals, for calculating radiant fluxes from arbitrary emitting point sources. This generalized method is simplified then to the form of multidomain single integral equation method applicable for rotationally symmetrical radiation patterns with optical axes perpendicular to the disc. Next, the simplified method is used for computer simulation of radiant fluxes incident on the disc from small Lambertian and Gaussian sources represented by point source models. All numerical results obtained from this simulation have shown high accuracy and efficiency of the presented method. Selected results are illustrated graphically and validated by Optical Software for Layout and Optimization (OSLO) from Lambda Research Corporation. Potential applications of the presented method include optical sensing and metrology, optical coupling, immersion microscopes, light-extraction problems and creative lighting design.     

Generalized Lorenz–Mie scattering theory for focused radiation and finite solids of revolution: case I: symmetrically polarized beams

Interaction of focused radiation with spherical and finite cylindrical homogeneous particles is considered. The aim of this investigation is to calculate the structure of the electromagnetic (EM) fields scattered by and propagated within the scattering objects. The incident EM fields are assumed to be focused fields in the image space of an aplanatic system with or without aberrations of category one. The radiation in the object space is assumed to be symmetrically polarized. The incident fields in the neighbourhood of the focus are calculated using the well-known theory of Richards and Wolf and a methodology developed by the author. At the interface between the homogeneous and the image space of the aplanatic system, the continuity conditions of the tangential components of the electric displacement and magnetic moment vectors are satisfied. The procedure results in dual discretized-Fredholm integral equations that are solved using orthogonal expansions. It is assumed that the scattered field, at large distances from the focus, is a spherical wave propagating away from the focus. Scattering by objects of various materials ranging from dielectric to perfect conductor is studied. The theory and its solution developed here allow for the scattering objects to be located anywhere along the optical axis in the image space. One of the main objectives is to calculate the energy distribution at the tip of the cylindrical homogeneous particle. Numerical calculations suggest that energy density at the tip is further enhanced if the cylindrical homogeneous particle is placed away from focus.     

Analytical solution for irradiance due to inhomogeneous Lambertian polygonal emitters

We present an analytic solution for the irradiance at a point due to a polygonal Lambertian emitter with radiant exitance that varies with position according to a polynomial of arbitrary degree. This is a basic problem that arises naturally in radiative transfer and more specifically in global illumination, a subfield of computer graphics. Our solution is closed form except for a single nonalgebraic special function known as the Clausen integral. We begin by deriving several useful formulas for high-order tensor analogs of irradiance, which are natural generalizations of the radiation pressure tensor. We apply the resulting tensor formulas to linearly varying emitters, obtaining a solution that exhibits the general structure of higher-degree cases, including the dependence on the Clausen integral. We then generalize to higher-degree polynomials with a recurrence formula that combines solutions for lower-degree polynomials; the result is a generalization of Lambert's formula for homogeneous diffuse emitters, a well-known formula with many applications in radiative transfer and computer graphics. Similar techniques have been used previously to derive closed-form solutions for the irradiance due to homogeneous polygonal emitters with directionally varying radiance. The present work extends this previous result to include inhomogeneous emitters, which proves to be significantly more challenging to solve in closed form. We verify our theoretical results with numerical approximations and briefly discuss their potential applications.     

Diffraction of elastic sphericalP waves by a cylindrical cavity

Summary This paper deals with the three dimensional diffraction of impulsive compressional waves by a cylindrical cavity. The cavity is embedded in an unbounded, isotropic, homogeneous elastic medium. The compressional point source, generating the incident pulse, is supposed to be situated outside the cavity. The formal solution to the problem is obtained in integral form. The integrals are evaluated asymptotically for the early time motion by the residue-Cagniard method.With 6 Figures     

The point-characteristic function, wavefronts, and caustic of a spherical wave refracted by an arbitrary smooth surface

The aim of this paper is to obtain expressions for the k-function, the wavefront train, and the caustic associated with the light rays refracted by an arbitrary smooth surface after being emitted by a point light source located at an arbitrary position in a three-dimensional homogeneous optical medium. The general results are applied to a parabolic refracting surface. For this case, we find that when the point light source is off the optical axis, the caustic locally has singularities of the hyperbolic umbilic type, while the refracted wavefront, at the caustic region, locally has singularities of the cusp ridge and swallowtail types.     

A marching-on-in-time method for 2-D transient electromagnetic scattering from homogeneous, lossy dielectric cylinders using boundary elements

A method for determining the fields scattered by arbitrarily shaped cylindrical, lossy dielectric structures with a transient incident wave is described. The transient scattering problem is reduced to the solution of a time-domain integral equation which is solved directly in the time domain by a time-stepping method. As the scatterer is homogeneous, the solution can be obtained by means of a boundary integral formulation and the problem-independent free-space Green's function. The approximate electromagnetic impulse response for a number of cylindrical targets is calculated using this method.<>     

Investigation of the Mechanism of Radiation Propagation at Side Illumination of a Light Guide

A design of a fiberoptic distributedtype detector of radiation sources arbitrarily located in the technological control zone is proposed. We have elaborated a method for recording scattered and directed radiation flows supplied to the side surface of the optical fiber based on measurement of the luminous flux diffusely scattered by the technological microdefects of the optical fiber. It has been established that the scattered flux is concentrated mainly in the optical fiber cladding. The detector sensitivity to the radiation from a gas flame, a laser diode, and background sources has been investigated.     

Energy distribution and radiation loading of a cylindrical source suspended within a nonconcentric fluid cylinder

Acoustic radiation from a cylindrical source undergoing angularly periodic and axially-dependent harmonic surface vibrations while eccentrically suspended within a fluid cylinder, which is submerged in an infinite fluid medium, is analyzed in an exact manner. This configuration, which is a realistic idealization of a cylindrical liquid-filled acoustic lens with focal point inside the lens when used as a sound projector, is of practical importance with applications in underwater and biomedical ultrasonics. The formulation utilizes the appropriate wave-field expansions along with the pertinent translational addition theorem to develop an infinite series solution. The analytical results are illustrated with a numerical example in which the basic acoustic quantities, such as the modal acoustic radiation impedance load on the source, on-axis pressure magnitude, radiated far-field pressure directivity patterns, and the radiation intensity distribution are evaluated for representative values of the parameters characterizing the system. Numerical results clearly illustrate that, in addition to source frequency and surface velocity distribution, its position (eccentricity) can be of great significance in sound radiation.     

Numerical investigation of the characteristics of radiant heat transfer in a turbine grid

The Monte Carlo method is used to investigate the distribution of radiant heat fluxes in the interblade channel of a turbine grid and directionality diagram of the emitted radiation.     

A new synthetic formalism for true coincidence summing calculations

Simultaneous detection in one sensing crystal of photons emitted in cascade from the same radioactive atom gives rise to true coincidence summing effects (excess and deficit counting). Presented in this paper is a general formula to correct for true coincidence summing. This analytical formula enables calculation of the correction factor and its associated uncertainty for any gamma-ray placed in a decay scheme of any radionuclide (all decay modes treated) in any measuring configuration. It takes account of any cascade of photons either through the nature of the involved photons (γ-rays, X-rays, and annihilation photons) or the unlimited number of photons emitted in cascade. The requirement for this calculation is twofold. From the radionuclide part, theoretical knowledge of the decay scheme is necessary along with its characteristics (gamma-ray intensities and internal conversion coefficients for instances). Some atomic quantities such as fluorescence yields have also to be known. From the detector part, calibrations for full-energy peak efficiency and total efficiency in the measuring configuration have to be available. The demonstration of the formula is done with restrictive conditions (point source, no angular correlation, and no metastable state between the initial and final energy states) but adaptation can be implemented to partially or totally lift these limitations or approximations. Implementation of the formula in a computer code named Coincal is mentioned. Its successful application to point sources of ¹⁵²Eu placed 10 cm and 15 mm away from an n-type coaxial HPGe detector with 60% relative efficiency is also described.     

The Spectral Directional Emissivity of Photovoltaic Surfaces

Photovoltaic solar cells are used for the direct conversion of solar radiation to electric power. To evaluate the efficiency of this energy conversion process, all in- and outgoing fluxes in the thermodynamic balance equations for energy and entropy must be known. The spatial and spectral distribution of radiation energy intensities must be known to calculate the radiation energy fluxes. To calculate the entropy fluxes, additional information on the coherence properties of the radiation field is essential. This information is expressed by the degree of polarization. First results of measurements of the optical properties of a solar cell are presented. The calculation procedure to obtain the outgoing energy and entropy fluxes is described. The experimental apparatus introduced in this paper yields the spectral directional emissivity by comparing the sample radiation with the radiation from an isothermal cavity. The degree of polarization of the emitted radiation is determined by a retarder/polarizer set within the apparatus. Both quantities are measured in the infrared region for wavelengths between 4.0 and 20.0 m.     

Shaping of the end of a multimode optical fiber for efficient coupling light from a laser diode

An entrance surface of a multimode optical fiber with the highest efficiency of coupling radiation from a laser diode is considered. The geometry of the surface is designed with the purpose of keeping a refracted ray inside the fiber with simultaneous minimization of the reflection coefficient in each point of the surface. As a good approximation to the ideal surface, the composite entrance surface of a cylinder-wedge shape is proposed. It has clearly improved the coupling efficiency as compared with that of the purely cylindrical surface. Analysis of the entrance surfaces is based on their transmission coefficients, depending on the incident ray direction.     

Integral equation formulation for reflection by a mirror

When light is incident on a mirror, it induces a current density on its surface. This surface current density emits radiation, which is the observed reflected field. We consider a monochromatic incident field with an arbitrary spatial dependence, and we derive an integral equation for the Fourier-transformed surface current density. This equation contains the incident electric field at the surface as an inhomogeneous term. The incident field, emitted by a source current density in front of the mirror, is then represented by an angular spectrum, and this leads to a solution of the integral equation. From this result we derive a relation between the surface current density and the current density of the source. It is shown with examples that this approach provides a simple method for obtaining the surface current density. It is also shown that with the solution of the integral equation, an image source can be constructed for any current source, and as illustration we construct the images of electric and magnetic dipoles and the mirror image of an electric quadrupole. By applying the general solution for the surface current density, we derive an expression for the reflected field as an integral over the source current distribution, and this may serve as an alternative to the method of images.     

Resonances of pumping and higher stokes components in fiber Brillouin lasers and a method of setting them

An algorithm is presented for setting a fiber Brillouin laser that allows simultaneous provision of the resonance of pumping and higher Stokes components. The proposed radiation source combines a low noise level and low lasing threshold characteristic of double-resonance lasers with the advantages of multifrequency optical emitters.     

Transmitting boundaries for time-harmonic elastodynamics on infinite domains

A finite element method with transmitting boundary is developed for time-harmonic elastodynamics on infinite domains. Only a finite inner domain needs to be discretized into conventional finite elements, while the effect of the exterior domain is simulated by introducing the transmitting boundary stiffness matrix on the degrees-of-freedom of the transmitting boundary nodes. In order to incorporate the Sommerfeld radiation condition, the reciprocity relation, on the transmitting boundary for the radiating elastodynamic state under consideration and the one which is induced by a finite number of point source is used to calculate the transmitting boundary stiffness matrix. The effectiveness of the transmitting boundary technique has been demonstrated by solving the stress concentration problems of a homogeneous isotropic full space with a circular cylindrical cavity and of the one with a spherical cavity, which are subjected to plane incident waves, and the results are compared with the ones of the conventional FEM and the exact data.