Photon Trajectories Research Articles

Dark matter effects in vacuum spacetime

We analyze a toy model describing an empty spacetime in which the motion of a test mass (and the trajectories of photons) evidence the presence of a continuous and homogeneous distribution of matter; however, since the energy-momentum tensor vanishes, no real matter or energy distribution is present at all. Thus, a hypothetical observer will conclude that he is immersed in some sort of dark matter, even though he has no chance to directly detect it. This suggests yet another possibility of explaining the elusive dark matter as a purely dynamical effect due to the curvature of spacetime.

DynamiX, numerical tool for design of next-generation x-ray telescopes

We present a new code aimed at the simulation of grazing-incidence x-ray telescopes subject to deformations and demonstrate its ability with two test cases: the Simbol-X and the International X-ray Observatory (IXO) missions. The code, based on Monte Carlo ray tracing, computes the full photon trajectories up to the detector plane, accounting for the x-ray interactions and for the telescope motion and deformation. The simulation produces images and spectra for any telescope configuration using Wolter I mirrors and semiconductor detectors. This numerical tool allows us to study the telescope performance in terms of angular resolution, effective area, and detector efficiency, accounting for the telescope behavior. We have implemented an image reconstruction method based on the measurement of the detector drifts by an optical sensor metrology. Using an accurate metrology, this method allows us to recover the loss of angular resolution induced by the telescope instability. In the framework of the Simbol-X mission, this code was used to study the impacts of the parameters on the telescope performance. In this paper we present detailed performance analysis of Simbol-X, taking into account the satellite motions and the image reconstruction. To illustrate the versatility of the code, we present an additional performance analysis with a particular configuration of IXO.

The effect of gravitational tidal forces on vacuum polarization: How to undress a photon

The effect of gravitational tidal forces on photon propagation in curved spacetime is investigated. It is found that the imaginary part of the local refractive index Imn(u;ω) may be negative as well as positive, corresponding to a local amplification as well as attenuation of the amplitude of the renormalized photon field. This is interpreted in terms of the effect of tidal forces on the virtual e+e− cloud surrounding the bare photon field—a positive/negative Imn(u;ω) corresponds to an increased dressing/undressing of the bare photon. Below threshold decays of the photon to e+e− pairs can occur. Photon undressing in the vicinity of a black hole singularity is described as an example. These results are shown to be consistent with unitarity and the optical theorem in curved spacetime, which is derived here both in a local form and integrated over the photon trajectory.

Statistical modeling of the intensity of light fluxes reflected by the Earth’s spherical surface

A new algorithm of the Monte Carlo method is suggested for the statistical estimation at any arbitrary point of the atmosphere of the intensity of the visible radiation caused by the reflection of solar light fluxes from the Earth’s spherical surface. The algorithm allows for the albedo of the Earth’s optically homogeneous surface to be taken into account beyond the scheme of the statistical modeling of the trajectories of photons wandering in the medium.

Extracting Rate Coefficients from Single-Molecule Photon Trajectories and FRET Efficiency Histograms for a Fast-Folding Protein

Recently developed statistical methods by Gopich and Szabo were used to extract folding and unfolding rate coefficients from single-molecule Förster resonance energy transfer (FRET) data for proteins with kinetics too fast to measure waiting time distributions. Two types of experiments and two different analyses were performed. In one experiment bursts of photons were collected from donor and acceptor fluorophores attached to a 73-residue protein, α(3)D, freely diffusing through the illuminated volume of a confocal microscope system. In the second, the protein was immobilized by linkage to a surface, and photons were collected until one of the fluorophores bleached. Folding and unfolding rate coefficients and mean FRET efficiencies for the folded and unfolded subpopulations were obtained from a photon by photon analysis of the trajectories using a maximum likelihood method. The ability of the method to describe the data in terms of a two-state model was checked by recoloring the photon trajectories with the extracted parameters and comparing the calculated FRET efficiency histograms with the measured histograms. The sum of the rate coefficients for the two-state model agreed to within 30% with the relaxation rate obtained from the decay of the donor-acceptor cross-correlation function, confirming the high accuracy of the method. Interestingly, apparently reliable rate coefficients could be extracted using the maximum likelihood method, even at low (<10%) population of the minor component where the cross-correlation function was too noisy to obtain any useful information. The rate coefficients and mean FRET efficiencies were also obtained in an approximate procedure by simply fitting the FRET efficiency histograms, calculated by binning the donor and acceptor photons, with a sum of three-Gaussian functions. The kinetics are exposed in these histograms by the growth of a FRET efficiency peak at values intermediate between the folded and unfolded peaks as the bin size increases, a phenomenon with similarities to NMR exchange broadening. When comparable populations of folded and unfolded molecules are present, this method yields rate coefficients in very good agreement with those obtained with the maximum likelihood method. As a first step toward characterizing transition paths, the Viterbi algorithm was used to locate the most probable transition points in the photon trajectories.

Thermodynamics of Quantum Jump Trajectories

We apply the large-deviation method to study trajectories in dissipative quantum systems. We show that in the long time limit the statistics of quantum jumps can be understood from thermodynamic arguments in terms of dynamical phases and transitions between them in trajectory space. We illustrate our approach with three simple examples: a driven 2-level system where we find a particular scale invariance point in the ensemble of trajectories of emitted photons; a blinking 3-level system, where we argue that intermittency in the photon count is related to a crossover between distinct dynamical phases; and a micromaser, where we find an actual first-order phase transition in the ensemble of trajectories.

Laser initiation of PETN containing light-scattering additives

We suggest to improve the efficiency of the laser initiation of pentaerythritol tetranitrate (PETN) by increasing the photon trajectory length in the initiated material due to multiple scattering. The proposed method has been experimentally verified. The introduction of a light-scattering additive (0.5 wt % MgO) into PETN decreases the initiation threshold by a factor of ∼3.5 for the first harmonic radiation of a neodymium laser (1060 nm).

Distinguishing between Protein Dynamics and Dye Photophysics in Single-Molecule FRET Experiments

Förster resonance energy transfer (FRET) efficiency distributions in single-molecule experiments contain both structural and dynamical information. Extraction of this information from these distributions requires a careful analysis of contributions from dye photophysics. To investigate how mechanisms other than FRET affect the distributions obtained by counting donor and acceptor photons, we have measured single-molecule fluorescence trajectories of a small α/ β protein, i.e., protein GB1, undergoing two-state, folding/unfolding transitions. Alexa 488 donor and Alexa 594 acceptor dyes were attached to cysteines at positions 10 and 57 to yield two isomers—donor 10/acceptor 57 and donor 57/acceptor 10—which could not be separated in the purification. The protein was immobilized via binding of a histidine tag added to a linker sequence at the N-terminus to cupric ions embedded in a polyethylene-glycol–coated glass surface. The distribution of FRET efficiencies assembled from the trajectories is complex with widths for the individual peaks in large excess of that caused by shot noise. Most of this complexity can be explained by two interfering photophysical effects—a photoinduced red shift of the donor dye and differences in the quantum yield of the acceptor dye for the two isomers resulting from differences in quenching rate by the cupric ion. Measurements of steady-state polarization, calculation of the donor-acceptor cross-correlation function from photon trajectories, and comparison of the single molecule and ensemble kinetics all indicate that conformational distributions and dynamics do not contribute to the complexity.

Multiple light scattering and optomechanical forces

When off-resonant light travels through a transparent medium, light scattering is the primary optical process to occur. Multiple-particle events are relatively rare in optically dilute systems: scattering generally takes place at individual atomic or molecular centers. Several well-known phenomena result from such single-center interactions, including Rayleigh and Raman scattering, and the optomechanical forces responsible for optical tweezers. Other, less familiar effects may arise in circumstances where throughput radiation is able to simultaneously engage with two or more scattering sites in close, nanoscale, proximity. Exhibiting the distinctive near-field electromagnetic character, inter-particle interactions such as optical binding and a variety of inelastic bimolecular processes can then occur. Although the theory for each two-center process is well established, the connectivity of their mechanisms has not received sufficient attention. To address this deficiency, and to consider the issues that ensue, it is expedient to represent the various forms of multi-particle light scattering in terms of transitions between different radiation states. The corresponding quantum amplitudes, registering the evolution of photon trajectories through the material system, can be calculated using the tools of quantum electrodynamics. Each of the potential outcomes for multi-particle scattering generates a set of amplitudes corresponding to different orderings of the constituent photon-matter interactions. Performing the necessary sums over quantum pathways between radiation states is expedited by a state-sequence development, this formalism also enabling the identification of intermediate states held in common by different paths. The results reveal the origin and consequences of linear momentum conservation, and they also offer new insights into the behavior of light between closely neighboring scattering events.

Understanding interference experiments with polarized light through photon trajectories

Bohmian mechanics allows to visualize and understand the quantum-mechanical behavior of massive particles in terms of trajectories. As shown by Bialynicki-Birula, Electromagnetism also admits a hydrodynamical formulation when the existence of a wave function for photons (properly defined) is assumed. This formulation thus provides an alternative interpretation of optical phenomena in terms of photon trajectories, whose flow yields a pictorial view of the evolution of the electromagnetic energy density in configuration space. This trajectory-based theoretical framework is considered here to study and analyze the outcome from Young-type diffraction experiments within the context of the Arago-Fresnel laws. More specifically, photon trajectories in the region behind the two slits are obtained in the case where the slits are illuminated by a polarized monochromatic plane wave. Expressions to determine electromagnetic energy flow lines and photon trajectories within this scenario are provided, as well as a procedure to compute them in the particular case of gratings totally transparent inside the slits and completely absorbing outside them. As is shown, the electromagnetic energy flow lines obtained allow to monitor at each point of space the behavior of the electromagnetic energy flow and, therefore, to evaluate the effects caused on it by the presence (right behind each slit) of polarizers with the same or different polarization axes. This leads to a trajectory-based picture of the Arago-Fresnel laws for the interference of polarized light.

Emission Line Profiles and Images of Geometrically Thin and Optically Thick Accretion Disks

Under the assumption that the accretion disk around a Kerr (spinning) black hole is geometrically thin and optically thick, the trajectories of photons in Kerr metric are calculated by using the photon tracing method. And by numerical calculations, we have made a study on the relativistic iron line profiles and images of thin accretion disks. The result shows that viewing at large inclination angles, because of the contribution of the photons from the lower surface of the accretion disk, the line profile becomes double-peaked and the flux image is also significantly modified.

Photon-By-Photon Analysis of Single Molecule Fluorescence Trajectories of a Fast Folding Protein

Förster resonance energy transfer (FRET) experiments of single protein molecules can be used to measure the structural and dynamical properties of sub-populations, as well as the kinetics of transitions between sub-populations. However, the timescale for study of both the dynamics and kinetics has been limited. To obtain folding and unfolding rate coefficients from FRET trajectories, for example, the bin size must be sufficiently long to calculate an accurate FRET efficiency, but also much shorter than the waiting times in the folded and unfolded states. In cases where the photon count rate is too low for these conditions to hold, an alternative approach is to analyze the photon-by-photon trajectories using maximum likelihood methods. Unlike ensemble methods, where only the sum of the rate coefficients can be measured for a two state system, maximum likelihood methods allow the determination of the individual rate coefficients, and therefore also the equilibrium constant. Here we apply a maximum likelihood method recently developed by Gopich and Szabo (J.Phys.Chem.113pp10965-10973(2009)) to the photon-by-photon trajectories of a two-state designed protein, α3D, that folds in ∼ 1ms, too fast to measure accurately from waiting time distributions in FRET trajectories. Experiments were carried out on both freely-diffusing and immobilized molecules. The FRET efficiency distributions in the free diffusion experiment are broadened by the folding/unfolding transitions occurring within the time bin, a phenomenon similar to line broadening in NMR experiments. Mean FRET efficiencies and rate coefficients extracted using the Gopich-Szabo method were found to be reliable by comparing the sum of the rates with the relaxation rates obtained from the donor-acceptor cross correlation function. Finally, photon trajectories can be divided into folded and unfolded segments at a single photon level using the hidden Markov model (Viterbi algorithm) with extracted parameters.

2SD0925 Theory of single-molecule photon trajectories and FRET efficiency distributions(2SD Bridging Single Molecule Biophysics and System Biology:New Experimental and Theoretical Challenges,The 48th Annual Meeting of the Biophysical Society of Japan)

2SD0925 Theory of single-molecule photon trajectories and FRET efficiency distributions(2SD Bridging Single Molecule Biophysics and System Biology:New Experimental and Theoretical Challenges,The 48th Annual Meeting of the Biophysical Society of Japan)

Analysis of Wireless Optical Communications Feasibility in Presence of Clouds Using Markov Chains

Free Space Optical (FSO) communications is a practical solution for creating a three dimensional global broadband communications grid, offering bandwidths far beyond possible in Radio Frequency (RF) range. However, attributes of atmospheric turbulence (scintillation) and obscurants such as clouds impose perennial limitations on availability and reliability of optical links. To design and evaluate optimum transmission techniques that operate under realistic atmospheric conditions, a good understanding of the channel behavior is necessary. In most prior works, Monte-Carlo Ray Tracing (MCRT) algorithm has been used to analyze the channel behavior. This task is quite numerically intensive. The focus of this paper is on investigating the possibility of simplifying this task by a direct extraction of state transition matrices associated with standard Markov modeling from the MCRT computer simulations programs. We show that by tracing a photon's trajectory in space via a Markov chain model, the angular distribution can be calculated by simple matrix multiplications. We also demonstrate that the new approach produces results that are close to those obtained by MCRT and other known methods. Furthermore, considering the fact that angular, spatial, and temporal distributions of energy are inter-related, mixing time of Monte-Carlo Markov Chain (MCMC) for different types of aerosols is calculated based on eigen-analysis of the state transition matrix and possibility of communications in scattering media is investigated.

An improved photon path length probability density function–based radiative transfer model for space‐based observation of greenhouse gases

We present an improved model to describe the photon path length probability density function (PPDF) that effectively accounts for both aerosol and thin cloud effects for rapid retrieval of greenhouse gas data from space‐based high spectral resolution measurements. The reasonably simple PPDF and effective transmittance parameterization permit vertical inhomogeneity of gas absorption and three plane‐parallel arbitrarily located layers to account for light‐scattering effects due to aerosol and clouds. The basic assumption to construct the PPDF model refers mainly to the presentation of PPDF in terms of weakly correlated aerosol and cloud path length components. The model was validated using Monte Carlo simulations of photon trajectories and tested for a representative set of atmospheric conditions in which both aerosol and clouds modified the path length significantly. This study also focused on the connection of PPDF parameters to atmospheric optical characteristics commonly utilized in the solution of radiative transfer equations. These characteristics were converted into PPDF parameters by PPDF retrieval from a very limited spectral range of simulated radiance including at least one gas absorption line. The results demonstrate that this method can be used effectively for rapid radiative transfer spectral calculations over a rather wide spectral range of radiance at given atmospheric optical characteristics. Another important application of such conversion is to account for a priori knowledge of atmospheric optical characteristics when retrieving amounts of gases within PPDF‐based or other methods, such as full physics algorithms that utilize the solutions of radiative transfer equations.

Synthesis and Photophysics of Core‐Substituted Naphthalene Diimides: Fluorophores for Single Molecule Applications

The synthesis and photophysics of two new aminopropenyl naphthalene diimide (SANDI) dyes are reported. A general and convenient method for the synthesis of the precursor mono-, di-, and tetrabrominated 1,4,5,8-naphthalene tetracarboxylic dianhydrides is described. The two core-substituted SANDIs exhibit many of the photophysical properties required for fluorescence labeling applications including high photostability and high fluorescence quantum yields (>0.5) in the visible region of the spectrum. The emission wavelength is sensitive to the number of substituents on the NDI core, and the fluorescence decay times are in the range of approximately 8-12 ns for both compounds in the solvents investigated. Preliminary fluorescence emission data from single molecules of the compounds embedded in poly(methyl methacrylate) films are also reported and show that single molecules have very low yields of photobleaching, particularly the di-substituted system. Furthermore, only a small proportion (<10 %) of the single molecules studied display fluorescence intermittencies or "blinks" in their photon trajectory. The compounds appear to be excellent candidates for applications at the single molecule level, for example, as FRET labels.

Hidden markov analysis of short single molecule intensity trajectories.

Photon trajectories from single molecule experiments can report on biomolecule structural changes and motions. Hidden Markov models (HMM) facilitate extraction of the sequence of hidden states from noisy data through construction of probabilistic models. Typically, the true number of states is determined by the Bayesian information criteria (BIC); however, constraints resulting from short data sets and Poisson-distributed photons in radiative processes like fluorescence can limit successful application of goodness-of-fit statistics. For single molecule intensity trajectories, additional information criteria such as peak localization error (LE) and chi-square probabilities can incorporate theoretical constraints on experimental data while modifying normal HMM. Chi-square minimization also serves as a stopping point of the iteration in which the system parameters are trained. Peak LE enables exclusion of overfitted and overlapped states. These constraints and criteria are tested against BIC on simulated single molecule trajectories to best identify the true number of emissive levels in any sequence.

Decoding the Pattern of Photon Colors in Single-Molecule FRET

Conformational dynamics of a single molecule can be studied using Forster resonance energy transfer (FRET) by recording a sequence of photons emitted by a donor and an acceptor dye attached to the molecule. We describe a simple and robust method to estimate the rates of transitions between different conformational states and the FRET efficiencies associated with these states. For a photon trajectory with measured interphoton times, the pattern of colors is decoded by maximizing the appropriate likelihood function. This approach can be used to analyze bursts of photons from diffusing molecules as well as photon trajectories generated by immobilized molecules. The procedure is illustrated using simulated photon trajectories corresponding to two-state and three-state molecules. The method works even when the photon colors appear to be scrambled because of high background noise, the photophysical properties of the conformers are similar, or the conformational and photon count rates are comparable. The consistency of the model with the data can be checked by recoloring the photon trajectories and comparing the predicted and observed FRET efficiency histograms.

STRONG FIELD EFFECTS ON PULSAR ARRIVAL TIMES: CIRCULAR ORBITS AND EQUATORIAL BEAMS

If a pulsar orbits a supermassive black hole, the timing of pulses that pass close to the hole will show a variety of strong field effects. To compute the intensity and timing of pulses that have passed close to a nonrotating black hole we introduce here a simple formalism based on two "universal functions," one for the bending of photon trajectories and the other for the photon travel time on these trajectories. We apply this simple formalism to the case of a pulsar in circular orbit that beams its pulses into the orbital plane. In addition to the "primary" pulses that reach the receiver by a more-or-less direct path, we find that there are secondary and higher order pulses. These are usually much dimmer than the primary pulses, but they can be of comparable or even greater intensity if they are emitted when pulsar is on the side of the hole furthest from the receiver. We show that there is a phase relationship of the primary and secondary pulses that is a probe of the strongly curved spacetime geometry. Analogs of these phenomena are expected in more general configurations, in which a pulsar in orbit around a hole emits pulses that are not confined to the orbital plane.

A FAST NEW PUBLIC CODE FOR COMPUTING PHOTON ORBITS IN A KERR SPACETIME

Relativistic radiative transfer problems require the calculation of photon trajectories in curved spacetime. We present a novel technique for rapid and accurate calculation of null geodesics in the Kerr metric. The equations of motion from the Hamilton-Jacobi equation are reduced directly to Carlson's elliptic integrals, simplifying algebraic manipulations and allowing all coordinates to be computed semi-analytically for the first time. We discuss the method, its implementation in a freely available FORTRAN code, and its application to toy problems from the literature.