Geometrical Optics Limit Research Articles

GLANCE – Gravitational Lensing Authenticator using Non-modelled Cross-correlation Exploration of Gravitational Wave Signals

ABSTRACT Gravitational lensing is the phenomenon where the presence of matter (called a lens) bends the path of light-like trajectories travelling nearby. Similar to the geometric optics limit of electromagnetic waves, gravitational lensing of gravitational waves (GWs) can occur in geometric optics condition when GW wavelength is much smaller than the Schwarzschild radius of the lens, that is, $\lambda _{\mathrm{ GW}} \ll$R$^{\rm s}_{\rm lens}$. This is known as the strong lensing regime for which a multiple-image system with different magnifications and phase shifts is formed. We developed GLANCE, Gravitational Lensing Authenticator using Non-modelled Cross-correlation Exploration, a novel technique to detect strongly lensed GW signals. We demonstrate that cross-correlation between two noisy reconstruction of polarized GW signals shows a non-zero value when the signals are lensed counterparts. The relative strength between the signal cross-correlation and noise cross-correlation can quantify the significance of the event(s) being lensed. Since lensing biases the inference of source parameters, primarily the luminosity distance, a joint parameter estimation of the source and lens-induced parameters is incorporated using a Bayesian framework. We applied GLANCE to synthetic strong lensing data and showed that it can detect lensed GW signals and correctly constrain the injected source and lens parameters, even when one of the signals is below match-filtered threshold signal-to-noise ratio. This demonstrates GLANCE’s capability as a robust detection technique for strongly lensed GW signals and can distinguish between lensed and unlensed events.

Self-consistent interaction of linear gravitational and electromagnetic waves in non-magnetized plasma

This paper explores the hybridization of linear metric perturbations with linear electromagnetic (EM) perturbations in non-magnetized plasma for a general background metric. The local wave properties are derived from first principles for inhomogeneous plasma, without assuming any symmetries of the background metric. First, we derive the effective (“oscillation-center”) Hamiltonian that governs the average dynamics of plasma particles in a prescribed quasimonochromatic wave that involves metric perturbations and EM fields simultaneously. Then, using this Hamiltonian, we derive the backreaction of plasma particles on the wave itself and obtain gauge-invariant equations that describe the resulting self-consistent gravito-electromagnetic (GEM) waves in a plasma. The transverse tensor modes of gravitational waves are found to have no interaction with the plasma and the EM modes in the geometrical-optics limit. However, for longitudinal GEM modes with large values of the refraction index, the interplay between gravitational and EM interactions in plasma can have a strong effect. In particular, the dispersion relation of the Jeans mode is significantly affected by electrostatic interactions. As a spin-off, our calculation also provides an alternative resolution of the so-called Jeans swindle.

Imaging with a gravitational lens: the geometric view

ABSTRACT We investigate imaging point sources with a monopole gravitational lens, such as the Solar Gravitational Lens in the geometric optics limit. We compute the light amplification of the lens used in conjunction with a telescope featuring a circular aperture that is placed in the focal region of the lens, compared to the amount of light collected by the same telescope unaided by a gravitational lens. We recover an averaged point spread function that is in robust agreement with a wave-theoretical description of the lens, and can be used in practical calculations or simulations.

Strongly Lensed Supermassive Black Hole Binaries as Nanohertz Gravitational-wave Sources

Supermassive black hole binary systems (SMBHBs) should be the most powerful sources of gravitational waves (GWs) in the universe. Once pulsar timing arrays (PTAs) detect the stochastic GW background from their cosmic merger history, searching for individually resolvable binaries will take on new importance. Since these individual SMBHBs are expected to be rare, here we explore how strong gravitational lensing can act as a tool for increasing their detection prospects by magnifying fainter sources and bringing them into view. Unlike for electromagnetic waves, when the geometric optics limit is nearly always valid, for GWs the wave-diffraction-interference effects can become important when the wavelength of the GWs is larger than the Schwarzchild radius of the lens, i.e., . For the GW frequency range explored in this work, the geometric optics limit holds. We investigate GW signals from SMBHBs that might be detectable with current and future PTAs under the assumption that quasars serve as bright beacons that signal a recent merger. Using the black hole mass function derived from quasars and a physically motivated magnification distribution, we expect to detect a few strongly lensed binary systems out to z ≈ 2. Additionally, for a range of fixed magnifications 2 ≤ μ ≤ 100, strong lensing adds up to ∼30 more detectable binaries for PTAs. Finally, we investigate the possibility of observing both time-delayed electromagnetic signals and GW signals from these strongly lensed binary systems—that will provide us with unprecedented multi-messenger insights into their orbital evolution.

Light propagation in (2+1) -dimensional electrodynamics: The case of linear constitutive laws

In this paper, we turn our attention to light propagation in three-dimensional electrodynamics. More specifically, we investigate the behavior of light rays in a continuous bi-dimensional hypothetical medium living in a three-dimensional ambient spacetime. Relying on a fully covariant approach, we assume that the medium is endowed with a local and linear response tensor which maps field strengths into excitations. In the geometric optics limit, we then obtain the corresponding Fresnel equation and, using well-known results from algebraic geometry, we derive the effective optical metric.

Gravitational Faraday effect from on-shell amplitudes

Effects of massive object’s spin on massive-massless 2 → 2 classical scattering is studied. Focus is set on the less-considered dimensionless expansion parameter λ/b, where λ is the massless particle’s wavelength and b is the impact parameter. Corrections in λ/b start to appear from mathcal{O} (G2), with leading correction terms tied to the gravitational Faraday effect, which is a special case of the Lense-Thirring effect. We compute the eikonal phase up to mathcal{O} (G2) and extract spin effect on the scattering angle and time delay up to 14th order in spin. The gravitational Faraday effect at linear order in spin [1] is reproduced by λ/b correction terms, which we compute to higher orders in spin. We find that the equivalence principle, or universality, holds up to NLO for general spinning bodies, i.e. away from geometric optics limit. Furthermore, in the black hole limit, we confirm the absence of particular spin structure observed [2–8], along with the associated shift symmetry [7], and argue that it holds to arbitrary spin order at mathcal{O} (G2) in the massless probe limit.

Designing Freeform Optical Surfaces by the Monge-Ampère Equations: A Review

Background: The equations of Monge-Ampère type which arise in geometric optics are used to design illumination lenses and mirrors. The optical design problem can be formulated as an inverse problem: determine an optical system consisting of a reflector and/or refractor that converts a given light distribution of the source into a desired target light distribution. For two decades, the development of fast and reliable numerical design algorithms for the calculation of freeform surfaces for irradiance control in the geometrical optics limit is of great interest in current research. Objective: The objective of this paper is to summarize the types, algorithms, and applications of Monge-Ampère equations. It helps scholars better grasp the research status of Monge-Ampère equations and further explore the theory of Monge-Ampère equations. Methods: This paper reviews the theory and applications of Monge–Ampère equations from four aspects. We first discuss the concept and development of Monge–Ampère equations. Then we derive two different cases of Monge–Ampère equations. We also list the numerical methods of Monge–Ampère equation in actual scenes. Finally, the paper gives a brief summary and an expectation. Results: This paper reviews the theory and applications of Monge-Ampère equations from four aspects. We first discuss the concept and development of Monge-Ampère equations. Then we derive two different cases of Monge-Ampère equations. We also list the numerical methods of the Monge-Ampère equation in actual scenes. Finally, the paper gives a brief summary and an expectation. Conclusion: Monge-Ampère equation has been widely applied in the geometric optics field since the predetermined energy distribution and the boundary condition creation can be well satisfied. Although the freeform surfaces designed by the Monge-Ampère equations is developing rapidly, there is still plenty of room for development in the design of the algorithms.

Exploiting billiard theory to calculate the mean path length of light in refractive spheroids

A technique is introduced to calculate the mean path length of light rays diffusely incident on a refractive object. It uses the phase portrait from billiard theory to determine the criteria for which chords are accessible for a given refractive index. The mean path length is given as an integral over the lengths of chords accessible via refraction, which implicitly accounts for total internal reflections. We demonstrate this method by calculating the mean path length in ellipses and spheroids. The mean path length is given by a double integral for the ellipse and a triple integral for the spheroid, which may be evaluated numerically, and also allows us to deduce simple series expansions for low eccentricity. These results give analytic expressions for the orientation averaged absorption cross sections in the geometric optics limit.

Analysis of the gradual transition from the near to the far field in single-slit diffraction

In Optics it is common to split up the formal analysis of diffraction according to two convenient approximations, in the near and far fields (also known as the Fresnel and Fraunhofer regimes, respectively). Within this scenario, geometrical optics, the optics describing the light phenomena observable in our everyday life, is introduced as the short-wavelength limit of near-field phenomena, assuming that the typical size of the aperture (or obstacle) that light is incident on is much larger than the light wavelength. With the purpose to provide an alternative view on how geometrical optics fits within the context of the diffraction theory, particularly how it emerges, the transition from the near to the far field is revisited here both analytically and numerically. Accordingly, first this transition is investigated in the case of Gaussian beam diffraction, since its full analyticity paves the way for a better understanding of the paradigmatic (and typical) case of diffraction by sharp-edged single slits. This latter case is then tackled both analytically, by means of some insightful approximations and guesses, and numerically. As it is shown, this analysis makes explicit the influence of the various parameters involved in diffraction processes, such as the typical size of the input (diffracted) wave or its wavelength, or the distance between the input and output planes. Moreover, analytical expressions have been determined for the critical turnover value of the slit width that separates typical Fraunhofer diffraction regimes from the behaviors eventually leading to the geometrical optics limit, finding a good agreement with both numerically simulated results and experimental data extracted from the literature.

Dynamics of irregularly shaped cometary particles subjected to outflowing gas and solar radiative forces and torques

ABSTRACTThe dynamics of irregularly shaped particles subjected to the combined effect of gas drag and radiative forces and torques in a cometary environment is investigated. The equations of motion are integrated over distances from the nucleus surface up to distances where the gas drag is negligible. The aerodynamic forces and torques are computed assuming a spherically symmetric expanding gas. The calculations are limited to particle sizes in the geometric optics limit, which is the range of validity of our radiative torque calculations. The dynamical behaviour of irregular particles is quite different to those exhibited by non-spherical but symmetric particles such as spheroids. An application of the dynamical model to comet 67P/Churyumov–Gerasimenko, the target of the Rosetta mission, is made. We found that, for particle sizes larger than ∼10 μm, the radiative torques are negligible in comparison with the gas-driven torques up to a distance of ∼100 km from the nucleus surface. The rotation frequencies of the particles depend on their size, shape, and the heliocentric distance, while the terminal velocities, being also dependent on size and heliocentric distance, show only a very weak dependence on particle shape. The ratio of the sum of the particles projected areas in the sun-to-comet direction to that of the sum of the particles projected areas in any direction perpendicular to it is nearly unity, indicating that the interpretation of the observed u-shaped scattering phase function by Rosetta/OSIRIS on comet 67P coma cannot be linked to mechanical alignment of the particles.

Eikonal quasinormal modes of black holes beyond general relativity. III. Scalar Gauss-Bonnet gravity

In a recent series of papers, we have shown how the eikonal/geometrical optics approximation can be used to calculate analytically the fundamental quasinormal mode frequencies associated with coupled systems of wave equations, which arise, for instance, in the study of perturbations of black holes in gravity theories beyond General Relativity. As a continuation to this series, we focus here on the quasinormal modes of nonrotating black holes in scalar Gauss-Bonnet gravity assuming a small-coupling expansion. We show that the axial perturbations are purely tensorial and are described by a modified Regge-Wheeler equation, while the polar perturbations are of mixed scalar-tensor character and are described by a system of two coupled wave equations. When applied to these equations, the eikonal machinery leads to axial quasinormal modes that deviate from the general relativistic results at quadratic order in the Gauss-Bonnet coupling constant. We show that this result is in agreement with an analysis of unstable circular null orbits around black holes in this theory, allowing us to establish the geometrical optics--null geodesic correspondence for the axial quasinormal modes. For the polar quasinormal modes, the small-coupling approximation forces us to consider the ordering between eikonal and small-coupling perturbative parameters, one of which we show, by explicit comparison against numerical data, yields the correct identification of the quasinormal modes of the scalar-tensor coupled system of wave equations. These corrections lift the general relativistic degeneracy between scalar and tensorial eikonal quasinormal modes at quadratic order in Gauss-Bonnet coupling in a way reminiscent of the Zeeman effect. In general, our analytic, eikonal quasinormal mode frequencies (normalized to the General Relativity ones) agree with numerical results with an error of $\mathcal{O}(10%)$ in the regime of small coupling constant. Finally, we find that the analytical expressions for the quasinormal modes are common to a broad class of scalar-Gauss-Bonnet theories to leading eikonal order, showing a degeneracy between the quasinormal modes of nonrotating black holes in particular scalar-Gauss-Bonnet theories in the geometrical optics limit.

Identifying type II strongly lensed gravitational-wave images in third-generation gravitational-wave detectors

Strong gravitational lensing is a gravitational wave (GW) propagation effect that influences the inferred GW source parameters and the cosmological environment. Identifying strongly-lensed GW images is challenging as waveform amplitude magnification is degenerate with a shift in the source intrinsic mass and redshift. However, even in the geometric-optics limit, type II strongly-lensed images cannot be fully matched by type I (or unlensed) waveform templates, especially with large binary mass ratios and orbital inclination angles. We propose to use this mismatch to distinguish individual type II images. Using planned noise spectra of Cosmic Explorer, Einstein Telescope and LIGO Voyager, we show that a significant fraction of type II images can be distinguished from unlensed sources, given sufficient SNR ($\sim 30$). Incorporating models on GW source population and lens population, we predict that the yearly detection rate of lensed GW sources with detectable type II images is 172.2, 118.2 and 27.4 for CE, ET and LIGO Voyager, respectively. Among these detectable events, 33.1%, 7.3% and 0.22% will be distinguishable via their type II images with a log Bayes factor larger than 10. We conclude that such distinguishable events are likely to appear in the third-generation detector catalog; our strategy will significantly supplement existing strong lensing search strategies.

Analytic expressions for geometric cross-sections of fractal dust aggregates

ABSTRACT In protoplanetary discs and planetary atmospheres, dust grains coagulate to form fractal dust aggregates. The geometric cross-section of these aggregates is a crucial parameter characterizing aerodynamical friction, collision rates, and opacities. However, numerical measurements of the cross-section are often time consuming as aggregates exhibit complex shapes. In this study, we derive a novel analytic expression for geometric cross-sections of fractal aggregates. If an aggregate consists of N monomers of radius R0, its geometric cross-section G is expressed as $G=AN\pi R_0^2/[1+(N-1)\tilde{\sigma }]$, where $\tilde{\sigma }$ is an overlapping efficiency, and A is a numerical factor connecting the analytic expression to the small non-fractal cluster limit. The overlapping efficiency depends on the fractal dimension, fractal prefactor, and N of the aggregate, and its analytic expression is derived as well. The analytic expressions successfully reproduce numerically measured cross-sections of aggregates. We also find that our expressions are compatible with the mean-field light scattering theory of aggregates in the geometrical optics limit. The analytic expressions greatly simplify an otherwise tedious calculation and will be useful in model calculations of fractal grain growth in protoplanetary discs and planetary atmospheres.

Numerical design of dual-scale foams to enhance radiation absorption

<abstract> Unlike silicon carbide ceramic, the alumina ceramic and zirconia ceramic are by its very nature weakly absorbing in visible and near infrared wavebands. Therefore, open-cell foams made of such ceramics are rarely used in applications requiring strong absorption of radiation energy, such as solar absorbers. In order to improve the potential of such foams, open-cell foams with dual-scale pores were digitally designed in the limit of geometric optics. The first-order pores are obtained by Voronoi tessellation technique and then the second-order pores are realized by the design of 'porous strut'. A Monte Carlo Ray-tracing method is applied in the dual-scale foam structure for radiative transfer calculation. The parameterized study is conducted on the radiation absorption of alumina foam sheets and zirconia foam sheets. The results show that for the present cases, the designed dual-scale alumina foams with strut porosity <italic>p</italic>_s = 0.3 can increase the normal absorptance within the wavebands from 0.4 μm to 7.0 μm by 62.7% than the foams with single-scale pores. For the zirconia foams, this increase is 115.9% within the wavebands from 1.2 μm to 7.6 μm. The findings can provide strong confidence on dual-scale foams to enhance the absorption of radiation energy in potential applications. </abstract>

A class of conserved currents for linearized gravity in the Kerr spacetime

We construct a class of conserved currents for linearized gravity on a Kerr background. Our procedure, motivated by the current for scalar fields discovered by Carter (1977), is given by taking the symplectic product of solutions to the linearized Einstein equations that are defined by symmetry operators. We consider symmetry operators that are associated with separation of variables in the Teukolsky equation, as well as those arising due the self-adjoint nature of the Einstein equations. In the geometric optics limit, the charges associated with these currents reduce to sums over gravitons of positive powers of their Carter constants, much like the conserved current for scalar fields. We furthermore compute the fluxes of these conserved currents through null infinity and the horizon and identify which are finite.

LENSINGGW: a PYTHON package for lensing of gravitational waves

Advanced LIGO and Advanced Virgo might be able to observe the first lensed gravitational waves in the coming years. With the addition of the KAGRA and LIGO India detectors to the detector network and with the future construction of theEinsteinTelescope we might be able to observe hundreds of lensed events. Ground-based gravitational-wave detectors can resolve arrival-time differences on the order of the inverse of the observed frequencies. The LIGO and Virgo frequency band spans from a few Hz to a few kHz, therefore the typical time resolution of current interferometers is on the order of milliseconds. When microlenses are embedded in galaxies or galaxy clusters, lensing can become more prominent and result in observable time delays at LIGO and Virgo frequencies. Therefore, gravitational waves might offer an exciting alternative probe of microlensing. However, only a few lensing configurations have currently been worked out in the context of gravitational-wave lensing. In this paper, we presentLENSINGGW, a PYTHONpackage designed to handle both strong lensing and microlensing of compact binaries and the related gravitational-wave signals in the geometrical optics limit. This synergy paves the way for systematic parameter space investigations and for the detection of arbitrary lens configurations and compact sources. Here we focus on the LIGO and Virgo frequencies. We demonstrate the working mechanism ofLENSINGGW and its use in studying microlenses that are embedded in galaxies.

Conserved currents for electromagnetic fields in the Kerr spacetime

We construct a variety of conserved currents for test electromagnetic fields on a Kerr background. Our procedure, which involves the symplectic product for electromagnetism and symmetry operators, generates the conserved currents given by Andersson et al [2015 Surveys in Differential Geometry (Hong Kong: International)], as well as a new conserved current. These currents reduce to the sum of (positive powers of) the Carter constants of photons in the geometric optics limit, and generalize the current for scalar fields discovered by Carter [1977 Phys. Rev. D 16 3395–3414] involving the Killing tensor. We furthermore show that the fluxes of our new current through null infinity and the horizon are finite.

Wave effects in the microlensing of pulsars and FRBs by point masses

ABSTRACT Wave effects are often neglected in microlensing studies; however, for coherent point-like sources, such as pulsars and fast radio bursts (FRBs), wave effects will become important in their gravitational lensing. In this paper, we describe the wave-optics formalism, its various limits, and the conditions for which these limits hold. Using the simple point lens as an example, we show that the frequency dependence of wave effects breaks degeneracies that are present in the usual geometric optics limit, and constructive interference results in larger magnifications further from the lens. This latter fact leads to a generic increase in cross-section for microlensing events in the wave-optics regime compared to the geometric optics regime. For realistic per cent-level spectral sensitivities, this leads to a relative boost in lensing cross-section of more than an order of magnitude. We apply the point-lens model to the lensing of FRBs and pulsars and find that these radio sources will be lensed in the full wave-optics regime by isolated masses in the range of $0.1\!-\!100\,{\rm M}_\oplus$, which includes free-floating planets (FFPs), whose Einstein radius is smaller than the Fresnel scale. More generally, the interference pattern allows an instantaneous determination of lens masses, unlike traditional microlensing techniques that only yield a mass inference from the event time-scale.

A Stochastic Green’s Function for Solution of Wave Propagation in Wave-Chaotic Environments

We present a statistical, mathematical, and computational model for prediction and analysis of wave propagation through complex, wave-chaotic environments. These are generally enclosed environments that are many wavelengths in extent and are such that, in the geometric optics limit, ray trajectories diverge from each other exponentially with distance traveled. The wave equation solution is expressed in terms of a novel stochastic Green's function that includes both coherent coupling due to direct path propagation and incoherent coupling due to propagation through multiple paths of the scattering environment. The statistically fluctuating portion of the Green's function is characterized by random wave model and random matrix theory. Built upon the stochastic Green's function, we have derived a stochastic integral equation method, and a hybrid formulation to incorporate the component-specific attributes. The proposed model is evaluated and validated through representative experiments.

Evolution of the electric field along null rays for arbitrary observers and spacetimes

We derive, in a manifestly covariant and electromagnetic gauge independent way, the evolution law of the electric field $E^\alpha = Ee^\alpha\ (e^\alpha e_\alpha=1)$, relative to an arbitrary set of instantaneous observers along a null geodesic ray, for an arbitrary Lorentzian spacetime, in the geometrical optics limit of Maxwell's equations in vacuum. We show that, in general, neither the magnitude $E$ nor the direction $e^\alpha$ of the electric field (here interpreted as the observed polarization of light) are parallel transported along the ray. For an extended reference frame around the given light ray, we express the evolution of the direction $e^\alpha$ in terms of the frame's kinematics, proving thereby that its expansion never spoils parallel transport, which bears on the unbiased inference of intrinsic properties of cosmological sources, such as, for instance, the polarization field of the cosmic microwave background (CMB). As an application of the newly derived laws, we consider a simple setup for a gravitational wave (GW) interferometer, showing that, despite the (kinematic) shear induced by the GW, the change in the final interference pattern is negligible since it turns out to be of the order of the ratio of the GW and laser frequencies.