Photon Trajectories Research Articles

Precision luminosity for Z $^0$ lineshape measurements with a silicon-tungsten calorimeter

The measurement of small-angle Bhabha scattering is used to determine the luminosity at the OPAL interaction point for the LEP I data recorded between 1993 and 1995. The measurement is based on the OPAL Silicon-Tungsten Luminometer which is composed of two calorimeters encircling the LEP beam pipe, on opposite sides of the interaction point. The luminometer detects electrons from small-angle Bhabha scattering at angles between 25 and 58mrad. At LEP center-of-mass energies around the Z0, about half of all Bhabha electrons entering the detector fall within a 79nb fiducial acceptance region. The electromagnetic showers generated in the stack of 1 radiation length tungsten absorber plates are sampled by 608 silicon detectors with 38,912 radial pads of 2.5mm width.The fine segmentation of the detector, combined with the precise knowledge of its physical dimensions, allows the trajectories of incoming 45GeV electrons or photons to be determined with a total systematic error of less than 7microns. We have quantified all significant sources of systematic experimental error in the luminosity determination by direct physical measurement. All measured properites of the luminosity event sample are found to be in agreement with current theoretical expectations. The total systematic measurement uncertainty is 3.4x10^-4, significantly below the theoretical error of 5.4x10^-4 currently assigned to the QED calculation of the Bhabha acceptance, and contributes negligibly to the total uncertainty in the OPAL measurement of Gamma{invisible}/Gamma{l+l-}, a quantity of basic physical interest which depends crucially on the luminosity measurement.

Light rays in gravitating and refractive media: A comparison of the field to particle and Hamiltonian approaches

The field-to-particle method of H. P. Robertson as applied by Noonan, in order to obtain the general relativistic equations describing the trajectory of a photon in a refractive medium, is compared with Synge’s general relativistic Hamiltonian theory of waves and rays. For a photon in vacuum it is known that both approaches yield the same equation for the trajectory, i.e., a null geodesic. However for a photon in a medium, in contradistinction to the Hamiltonian theory, the field-to-particle method (a) yields equations of the photon trajectory valid only in a nondispersive medium, (b) the time component u0of the tangent to the ray remains an undetermined quantity, (c) agreement with the Hamiltonian theory is achieved by substituting into Noonan’s equations the Hamiltonian expression for u0.

Multiple light scattering in random systems: Analysis of the backscattering spot image

This study concerns the multiple scattering of light in random granular systems and the analysis of the image formed by the light scattered by a random medium. We developed an imagery method with high grey level resolution for visualization and analysis of the intensity profile in the backscattering spot image. We further present statistical models for the radiative transfer in granular media only involving the photon mean path length, the asymmetry factor and the absorption probability of photons. A renormalization of the photon trajectory in the medium gives a photon transport equation for anisotropic scattering and leads to an analytical expression for the intensity profile in the backscattering spot. We finally compare the results from imagery experiments with the predictions from numerical simulations and statistical models.

Simulation of photon acceleration in a plasma wake

The frequency shifting of an ultrashort (femtosecond) low intensity laser pulse in the presence of a plasma wave is investigated using particle simulations. One-dimensional simulations confirm the existence of photon trajectories similar to the trajectories of trapped or untrapped charged particles in a plasma wave. In the case when a plasma wake is produced by a relativistically intense laser pulse with a duration of the order of one plasma period, some full oscillations of the photon frequency take place only for the untrapped photons moving backward in the plasma wave frame before the intense laser pulse depletion due to the energy transfer to the plasma wake occurs. An analytical estimate of the probe pulse phase and frequency shift in two-dimensional (2-D) axially symmetrical plasma wake is performed. Axially symmetrical particle simulations in 2-D, with experimentally attainable parameters, display a frequency shift of the probe pulse of a few percent. In these analytical estimates and 2-D simulations special attention has been given to the modeling of the probe pulse optical collecting line that is typically used in plasma diagnostics.

Ultrasonic tagging of photon paths in scattering media:?parallel speckle modulation processing

Tagging of photon trajectories in scattering media is possible by application of a localized ultrasonic field to the sample and by measurement of the induced speckle modulation. Instead of using a single optical detector, which integrates the signal of many speckle grains, we propose a more efficient detection scheme that uses a CCD camera and parallel lock-in detection to record the full modulation of the speckle. The advantage of this multiplex detection is demonstrated, as well as the imaging capabilities of the process for biological tissues.

Molecular radiative transport. III. Experimental intensity decays

A critical experimental test of a previously developed theory of molecular radiative transport is described. It is concluded that the theory gives an accurate description of the effect of radiative transport on fluorescence observables. The numerical coefficients of the fluorescence decay are computed from a Monte Carlo integration procedure that mimics the photon trajectories inside a realistic sample cell, and is carried out only using known molecular and geometrical parameters. The predicted parameters are confronted with the experimental observables accessible in a typical single-photon timing experiment, rhodamine 101 in ethanol being the system studied. The theoretical predictions quantitatively describe the effects of concentration and excitation and emission wavelengths experimentally observed in optical dense nondiffusing media for the two most common geometric arrangements: front-face and right-angle detection. It is shown that radiative transport leads to spatially heterogeneous fluorescence kinetics, as a direct consequence of the existence of a spatial distribution function of electronic excitation inside the sample cell. The agreement between theory and experimental results is good, with the average decay times predicted within ≃3% accuracy for front-face detection.

A measure of photon penetration into tissue in diffusion models

In application of laser-based techniques to estimate optical parameters of tissue it is necessary to know the parts of the tissue that have been sampled by the photons. Towards this end it is desirable to define an average depth of a photon trajectory. This has been done earlier for lattice random walk models of photon trajectories where one simply counts the number of visits to a given set of sites. Here we calculate the equivalent quantity in the diffusion picture by equating the average depth of penetration to the local time of a diffusion process. We show that for a semi-infinite medium bounded by a plane the average depth probed by a photon trajectory in a continuous-wave (cw) experiment has the same functional form as that calculated from a random walk model. The local time, or average depth is shown to take a very simple form in the case of a time-gated experiment. We extend our results to the cw transillumination experiment.

Line emission from accretion discs around black holes: the analytic approach

The broad X-ray iron line observed in many active galactic nuclei spectra is thought to originate from the accretion disc surrounding the putative supermassive black hole. We show here how to perform the analytical integration of the geodesic equations that describe the photon trajectories in the general case of a rotating black hole (Kerr metric), in order to write a fast and efficient numerical code for modelling emission line profiles from accretion discs.

A numerical study of the statistics of penetration depth of photons re-emitted from irradiated media

Abstract An important question in the utilization of optical methods to estimate optical parameters non-invasively in human tissue is that of describing properties of the trajectories of laser-injected photons. It is relatively easy to calculate an expression for the average depth experienced by these photons starting from either a random-walk or a diffusion model. It is much harder to characterize more detailed descriptive parameters. Here we make use of a novel variation of the method of exact enumeration to provide essentially numerically exact results for the random-walk model of a continuous-wave experiment.

A numerical study of the statistics of penetration depth of photons re-emitted from irradiated media

An important question in the utilization of optical methods to estimate optical parameters non-invasively in human tissue is that of describing properties of the trajectories of laser-injected photons. It is relatively easy to calculate an expression for the average depth experienced by these photons starting from either a random-walk or a diffusion model. It is much harder to characterize more detailed descriptive parameters. Here we make use of a novel variation of the method of exact enumeration to provide essentially numerically exact results for the random-walk model of a continuous-wave experiment.

Two-photon pattern in a second-order interference

An interferometric arrangement utilizing a pair of twin photons, generated by parametric down-conversion, is devised to show second-order interferences revealing de Broglie's wavelength of the photon pair as a whole and the Pancharatnam phase depending on the number of photons. This proposed experiment utilizes the phenomenon ``induced coherence without stimulated emission'' to generate a coherent superposition of states distinct from the usual superposition with beam splitters, where the photons are independently scattered. In this experiment interference occurs due to recombination of photon trajectories on a beam splitter and also through frustration or enhancement of the down-conversion processes.

The optical-mechanical analogy for stationary metrics in general relativity

The optical-mechanical analogy allows for a common description of the motion of particles in mechanics and for light in geometrical optics. In a recent series of articles in this journal, it has been shown that the optical-mechanical analogy can be extended to general relativity for the case of static metrics expressible in isotropic coordinates. In this paper, we extend the optical-mechanical analogy in general relativity to the case of stationary metrics. A variational principle for the trajectories of both photons and particles is derived which takes the form of Fermat’s principle or the principle of Maupertuis. When the (suitably defined) spatial portion of the metric is written as or restricted to an isotropic form, exact equations of motion for both massless and massive particles are obtained in the form of Newtonian mechanics, describing objects moving in a medium with a spatially varying index of refraction. Such restrictions of the metric commonly occur, for example, when orbital motion is considered in a plane perpendicular to the axis of rotation of a rotating black hole or spacetime. The Newtonian form of the equations of geodesic motion are illustrated by applications to a uniformly rotating reference system and a rotating black hole (the Kerr metric).

Photon propagation in Einstein and higher derivative gravity

We derive the wave equation obeyed by electromagnetic fields in curved space-time. We find that there are Riemann and Ricci curvature coupling terms to the photon polarisation which result in a polarisation-dependent deviation of the photon trajectories from null geodesics. Photons are found to have an effective mass in an external gravitational field and their velocity in an inertial frame is, in general, less than c. A physically relevant consequence of the analysis is that the curvature corrections to the propagation of electromagnetic radiation (in an homogenous and isotropic space-time) keep the velocities subluminal provided the strong energy condition is satisfied. We further show that the claims of superluminal velocities in higher derivative gravity theories are erroneous and arise due to the neglect of Riemann and Ricci coupling terms in the wave equation of Einstein gravity.

Practical model for the calculation of multiply scattered lidar returns

An equation to predict the intensity of the multiply scattered lidar return is presented. Both the scattering cross section and the scattering phase function can be specified as a function of range. This equation applies when the cloud particles are larger than the lidar wavelength. This approximation considers photon trajectories with multiple small-angle forward-scattering events and one large-angle scattering that directs the photon back toward the receiver. Comparisons with Monte Carlo simulations, exact double-scatter calculations, and lidar data demonstrate that this model provides accurate results.

Pulsar X‐Ray and Gamma‐Ray Pulse Profiles: Constraint on Obliquity and Observer Angles

We model the thermal X-ray profiles of Geminga, Vela, and PSR 0656+14, which have also been detected as γ-ray pulsars, to constrain the phase space of obliquity and observer angles required to reproduce the observed X-ray pulsed fractions and pulse widths. These geometrical constraints derived from the X-ray light curves are explored for various assumptions about surface temperature distribution and flux anisotropy caused by the magnetized atmosphere. We include curved spacetime effects on photon trajectories and the magnetic field. The observed γ-ray pulse profiles are double peaked with phase separations of 0.4-0.5 between the peaks. Assuming that the γ-ray profiles are due to emission in a hollow cone centered on the magnetic pole, we derive the constraints on the phase space of obliquity and observer angles, for different γ-ray beam sizes, required to produce the observed γ-ray peak phase separations. We compare the constraints from the X-ray emission to those derived from the observed γ-ray pulse profiles and find that the overlapping phase space requires both obliquity and observer angles to be smaller than 20°-30°, implying γ-ray beam opening angles of at most 30°-35°.

Possible Tests of Curvature Effects in Weak Gravitational Fields

We analyze the possibility of detecting, with optical methods, particle and photon trajectories predicted by general relativity for a weak spherically-symmetric gravitational field. We discuss the required sensitivities and the possibility of performing specific experiments on the Earth.

Bending of Light by Gravity Waves

We describe the statistical properties of light rays propagating though a random sea of gravity waves and compare with the case for scalar metric perturbations from density inhomogeneities. For scalar fluctuations the deflection angle grows as the square-root of the path length $D$ in the manner of a random walk, and the rms displacement of a ray from the unperturbed trajectory grows as $D^{3/2}$. For gravity waves the situation is very different. The mean square deflection angle remains finite and is dominated by the effect of the metric fluctuations at the ends of the ray, and the mean square displacement grows only as the logarithm of the path length. In terms of power spectra, the displacement for scalar perturbations has $P(k) \propto 1/ k^4$ while for gravity waves the trajectories of photons have $P(k) \propto 1/k$ which is a scale-invariant or `flicker-noise' process, and departures from rectilinear motion are suppressed, relative to the scalar case, by a factor $\sim (\lambda / D)^{3/2}$ where $\lambda$ is the characteristic scale of the metric fluctuations and $D$ is the path length. This result casts doubt on the viability of some recent proposals for detecting or constraining the gravity wave background by astronomical measurements.

Effects of Surface Properties on Radiative Transfer in a Cylindrical Tube With a Nonparticipating Medium

Radiative heat transfer inside a cylindrical tube is modeled using a statistical method called the discrete probability function (DPF) method. The DPF method involves solution of the equation of radiative heat transfer using Lagrangian simulations of representative photon trajectories on a discrete spatial grid. The DPF method is different from the Markov Chain method in terms of associating a probability with each state of the photon rather than a transition from one state to another. The advantages and disadvantages of the DPF method in comparison to the Markov Chain method are demonstrated in this paper using two practical applications of the cylindrical tube radiative heat transfer problem. The cylindrical tube has a hot source at one end and a detector at the other end. The cylindrical wall absorbs and reflects (both diffusely and specularly) the radiation incident on it. The present calculations have applications in: (1) intrusive pyrometry with collimating light guides, and (2) measurement of the spectral absorption and reflection coefficients of coatings using two, coated cylindrical tubes as specimen. The results show that: (1) the effect of light guide surface properties on errors in pyrometry must be carefully assessed, and (2) the method can be used for a convenient evaluation of radiative properties of coatings.

The optical–mechanical analogy in general relativity: New methods for the paths of light and of the planets

The optical–mechanical analogy involves the expression of geometrical optics and particle mechanics in the same mathematical language. In this paper, an especially simple version of the optical–mechanical analogy is extended to general relativity. A variational principle for the trajectories of photons and particles is obtained which is applicable to a broad class of metrics. This permits us to cast the exact equations of motion for both massive and massless particles into the form of Newtonian mechanics. The new equations of motion are illustrated by applications to the Schwarzschild metric.

General Relativistic Radiative Transfer in Hot Astrophysical Plasmas: A Characteristic Approach

In this paper we present a characteristic method for solving the transfer equation in differentially moving media in a curved spacetime. The method is completely general, but its capabilities are exploited at best in presence of symmetries, when the existence of conserved quantities allows to derive analytical expressions for the photon trajectories in phase space. In spherically--symmetric, stationary configurations the solution of the transfer problem is reduced to the integration of a single ordinary differential equation along the bi--parametric family of characteristic rays. Accurate expressions for the radiative processes relevant to continuum transfer in a hot astrophysical plasma have been used in evaluating the source term, including relativistic e--p, e--e bremsstrahlung and Compton scattering. A numerical code for the solution of the transfer problem in moving media in a Schwarzschild spacetime has been developed and tested. Some applications, concerning ``hot'' and ``cold'' accretion onto non--rotating black holes as well as static atmospheres around neutron stars, are presented and discussed.