Geometrical Optics Limit Research Articles

Self-Averaging Scaling Limits for Random Parabolic Waves

We consider several types of scaling limits for the Wigner-Moyal equation of the parabolic waves in random media, the limiting cases of which include the standard radiative transfer limit, the geometrical-optics limit and the white-noise limit. We show under fairly general assumptions on the random refractive index field that sufficient amount of medium diversity (thus excluding the white-noise limit) leads to statistical stability or self-averaging in the sense that the limiting law is deterministic and is governed by one of the 6 different types of transport (Boltzmann or Fokker-Planck) equations depending on the specific scaling involved. We discuss the connection to the statistical stability of time-reversal procedure and the decoherence effect in quantum mechanics.

White-Noise and Geometrical Optics Limits of Wigner-Moyal Equation for Wave Beams in Turbulent Media

Starting with the Wigner distribution formulation for beam wave propagation in Hölder continuous non-Gaussian random refractive index fields we show that the wave beam regime naturally leads to the white-noise scaling limit and converges to a Gaussian white-noise model which is characterized by the martingale problem associated to a stochastic differential-integral equation of the Itô type. In the simultaneous geometrical optics the convergence to the Gaussian white-noise model for the Liouville equation is also established if the ultraviolet cutoff or the Fresnel number vanishes sufficiently slowly. The advantage of the Gaussian white-noise model is that its n-point correlation functions are governed by closed form equations.

Integrable Equations in Nonlinear Geometrical Optics

Geometrical optics limit of the Maxwell equations for nonlinear media with the Cole–Cole dependence of dielectric function and magnetic permeability on the frequency is considered. It is shown that for media with slow variation along one axis such a limit gives rise to the dispersionless Veselov–Novikov equation for the refractive index. It is demonstrated that the Veselov–Novikov hierarchy is amenable to the quasiclassical ‐dressing method. Under more specific requirements for the media, one gets the dispersionless Kadomtsev–Petviashvili equation. Geometrical optics interpretation of some solutions of the above equations is discussed.

Quasi-geometrical optics approximation in gravitational lensing

The gravitational lensing of gravitational waves should be treated in wave optics instead of geometrical optics when the wave lengthλ of the gravitational waves is larger than the Schwarzschild radius of the lens mass M. Wave optics is based on the diffraction integral which represents the amplification of the wave amplitude by lensing. We study the asymptotic expansion of the diffraction integral in the powers of the wave length λ. The first term, arising from the short wavelength limit λ → 0, corresponds to the geometrical optics limit. The second term, being of the order of λ/M, is the leading correction term arising from the diffraction effect. Analysing this correction term, we find that (1) the lensing magnification µ is modified to µ (1 + δ), where δ is of the order of (λ/M) 2 , and (2) if the lens has a cuspy (or singular) density profile at center ρ(r) ∝ r −α (0 <α ≤ 2), the diffracted image is formed at the lens center with magnification µ ∼ (λ/M) 3−α .

Radiative Characteristics of Opaque Spherical Particles Beds: A New Method of Prediction

We present a new method to compute the radiative characteristics of spherical particles beds. This method is applied to beds made of opaque diffusely or specularly reflecting particles with size parameters ranging in the geometric optics limit. It is based on a Monte Carlo procedure, and no hypotheses are made on dependent or independent scattering regime. Thus, the limits of validity of independent scattering hypothesis are investigated. Firstly, we present the principle and the hypothesis of the method. It is then applied to different beds and permits us to analyze the evolution of the extinction coefficient, scattering albedo, and phase function of the bed with the porosity and with the reflecting properties of the particles

Geometrical optics in nonlinear media and integrable equations

It is shown that the geometrical optics limit of the Maxwell equations for certain nonlinear media with slow variation along one axis and particular dependence of dielectric constant on the frequency and fields gives rise to the dispersionless Veselov-Novikov equation for refractive index. It is demonstrated that the last one is amenable to the quasiclassical DBAR-dressing method. A connection is noted between geometrical optics phenomena under consideration and quasiconformal mappings on the plane.

Transmission properties of a single metallic slit: from the subwavelength regime to the geometrical-optics limit.

In this work we explore the transmission properties of a single slit in a metallic screen. We analyze the dependence of these properties on both slit width and angle of incident radiation. We study in detail the crossover between the subwavelength regime and the geometrical-optics limit. In the subwavelength regime, resonant transmission linked to the excitation of waveguide resonances is analyzed. Linewidth of these resonances and their associated electric-field intensities are controlled by just the width of the slit. More complex transmission spectra appear when the wavelength of light is comparable to the slit width. Rapid oscillations associated with the emergence of different propagating modes inside the slit are the main features appearing in this regime.

High-order pulse front tilt caused by high-order angular dispersion

We have found general expressions relating the high-order pulse front tilt and the high-order angular dispersion in an ultrashort pulse, for the first time to our knowledge. The general formulae based on Fermat's principle are applicable for any ultrashort pulse with angular dispersion in the limit of geometrical optics. By virtue of these formulae, we can calculate the high-order pulse front tilt in the sub-20-fs UV pulse generated in a novel scheme of sum-frequency mixing in a nonlinear crystal accompanied by angular dispersion. It is also demonstrated how the high-order angular dispersion can be eliminated in the calculation.

Analysis of specular resonance in dielectric bispheres using rigorous and geometrical-optics theories.

We have recently identified the resonant scattering from dielectric bispheres in the specular direction, which has long been known as the specular resonance, to be a type of rainbow (a caustic) and a general phenomenon for bispheres. We discuss the details of the specular resonance on the basis of systematic calculations. In addition to the rigorous theory, which precisely describes the scattering even in the resonance regime, the ray-tracing method, which gives the scattering in the geometrical-optics limit, is used. Specular resonance is explicitly defined as strong scattering in the direction of the specular reflection from the symmetrical axis of the bisphere whose intensity exceeds that of the scattering from noninteracting bispheres. Then the range of parameters for computing a particular specular resonance is specified. This resonance becomes prominent in a wide range of refractive indices (from 1.2 to 2.2) in a wide range of size parameters (from five to infinity) and for an arbitrarily polarized light incident within an angle of 40 degrees to the symmetrical axis. This particular scattering can stay evident even when the spheres are not in contact or the sizes of the spheres are different. Thus specular resonance is a common and robust phenomenon in dielectric bispheres. Furthermore, we demonstrate that various characteristic features in the scattering from bispheres can be explained successfully by using intuitive and simple representations. Most of the significant scatterings other than the specular resonance are also understandable as caustics in geometrical-optics theory. The specular resonance becomes striking at the smallest size parameter among these caustics because its optical trajectory is composed of only the refractions at the surfaces and has an exceptionally large intensity. However, some characteristics are not accounted for by geometrical optics. In particular, the oscillatory behaviors of their scattering intensity are well described by simple two-wave interference models.

Elastic wave propagation and scattering in solids with uniaxially aligned cracks.

In this article, elastic wave propagation and scattering in a solid medium permeated by uniaxially aligned penny-shaped microcracks are studied. The crack alignment refers to the case in which the unit normals of all cracks are randomly oriented within a plane of isotropy. The analysis is restricted to the limit of the noninteraction approximation among individual cracks. Explicit expressions for attenuations and wave speeds of the shear horizontal, quasilongitudinal, and quasishear vertical waves are obtained using stochastic wave theory in a generalized dyadic approach. The ensemble average elastic wave response is governed by the Dyson equation, which is solved in terms of the anisotropic elastic Green's dyadic. The analysis of expressions is limited to frequencies below the geometric optics limit. The resulting attenuations are investigated in terms of the directional, frequency, and damage dependence. In particular, the attenuations are simplified considerably within the low frequency Rayleigh regime. Finally, numerical results are presented and discussed in terms of the relevant dependent parameters.

Thin front propagation in steady and unsteady cellular flows

Front propagation in two-dimensional steady and unsteady cellular flows is investigated in the limit of very fast reaction and sharp front, i.e., in the geometrical optics limit. For the steady flow, a simplified model allows for an analytical prediction of the front speed vf dependence on the stirring intensity U, which is in good agreement with numerical estimates. In particular, at large U, the behavior vf∼U/log(U) is predicted. By adding small scales to the velocity field we found that their main effect is to renormalize the flow intensity. In the unsteady (time-periodic) flow, we found that the front speed locks to the flow frequency and that, despite the chaotic nature of the Lagrangian dynamics, the front evolution is chaotic only for a transient. Asymptotically the front evolves periodically and chaos manifests only in its spatially wrinkled structure.

Microwave backscatter from non-gaussian seas

Rough-surface scattering theory is applied to study microwave backscattering from seas characterized by a non-Gaussian wave-height distribution. The relationship of the geometrical optics limit of rough-surface scattering theory to the probability density of surface slopes is used to relate the coefficients of Gram-Charlier expansions, describing measured slope statistics, to the wavenumber spectra of non-Gaussian surface components. Functional forms for the spectra consistent with measured slope statistics are assumed, and the backscatter predicted by rough-surface scattering theory is compared with measured cross sections. The predicted upwind-downwind asymmetry of scattering cross sections is comparable to that observed, and a measurable dependence of cross sections on atmospheric stability is predicted.

Retro‐MACHOs: π in the Sky?

Shine a flashlight on a black hole, and one is greeted with the return of a series of concentric rings of light. For a point source of light, and for perfect alignment of the lens, source, and observer, the rings are of infinite brightness (in the limit of geometric optics). In this manner, distant black holes can be revealed through their reflection of light from the Sun. Such retro-MACHO events involve photons leaving the Sun, making a π rotation about the black hole, and then returning to be detected at the Earth. Our calculations show that although the light return is quite small, it may nonetheless be detectable for stellar-mass black holes at the edge of our solar system. For example, all (unobscured) black holes of mass M or greater will be observable to a limiting magnitude , at a distance given by 0.02 pc × [10(M/10 M☉)2]1/3. Retro-MACHOs offer a way to directly image the presence of black holes and would be a stunning confirmation of strong-field general relativity.

NON-MINIMAL COUPLING, VARIABLE SPEED OF LIGHT AND COSMOLOGY

Presently, there exists some renewed interest in time varying speed of light theories as possible solutions of the major cosmological problems1. It is often believed that the local Lorentzian invariance is broken if the speed of light in a vacuum is not a constant. We point out that this belief is not necessarily founded and that a variable speed of light is perfectly consistent with general relativity under the assumption of non-minimal coupling between electromagnetism and curvature. Two kinds of arguments may be invoked in favour of such an assumption. First, a theorem due to Horndeski2shows that in a four-dimensional space-time the Einstein-Maxwell field equations are not the only second-order vector potential field equations which stem from a Lagrangian scalar density, are consistent with the charge conservation and reduce to Maxwell's equations in a flat space-time (see also3). Second, according to QED4,5, vacuum polarization induces tidal gravitational effects which imply that photons propagating in a curved space-time have velocities exceeding the value of the "Lorentzian structural constant" c. The modified electromagnetic field equations given by Horndeski2are studied here in the geometrical optics limit. Considering the case of Friedmann-Robertson-Walker cosmological models, we find the value of the speed of light as a function of the energetic content of the universe. We deduce from this result a new equation of state for a photon gas and we discuss the consequences of this equation on the evolution of the scale factor during the radiation-dominated era.

Front speed enhancement in cellular flows.

The problem of front propagation in a stirred medium is addressed in the case of cellular flows in three different regimes: slow reaction, fast reaction and geometrical optics limit. It is well known that a consequence of stirring is the enhancement of front speed with respect to the nonstirred case. By means of numerical simulations and theoretical arguments we describe the behavior of front speed as a function of the stirring intensity, U. For slow reaction, the front propagates with a speed proportional to U(1/4), conversely for fast reaction the front speed is proportional to U(3/4). In the geometrical optics limit, the front speed asymptotically behaves as U/ln U. (c) 2002 American Institute of Physics.

A localized beam representation of high‐frequency wave fields using a Wilson basis

A rigorous orthogonal basis is described for the representation of high‐frequency wave fields using beam superpositions parameterized by both spatial position and spatial direction of each beam. This representation has many desirable features not found in conventional phase‐space‐based expansions such as the discrete or continuous Gabor schemes. The representation is highly efficient in the geometrical optics limit, that is, when it is strongly localized in the phase‐space.

Ground-Based Infrared Remote Sensing of Cloud Properties over the Antarctic Plateau. Part II: Cloud Optical Depths and Particle Sizes

Abstract One full year of twice-daily longwave atmospheric emission spectra measured from the surface at 1-cm−1 resolution are used to infer optical thicknesses and ice crystal sizes in tropospheric clouds over the Antarctic Plateau. The method makes use of the cloud's emissivity at 10- and 11-μm wavelength and the cloud's transmittance of stratospheric ozone emission in the 9.6-μm band. Knowledge of the cloud-base temperature and the vertical distributions of ozone and temperature is required; these are available at South Pole Station from radiosondes and ozonesondes. The difference in emissivity between 10 and 11 μm is sensitive to ice particle size because the absorption coefficient of ice varies greatly between these two wavelengths. The retrieval of optical depth (expressed as its value in the geometric-optics limit τg) is limited to τg < 5, and the effective particle radii reff are distinguished only for reff < 25 μm, but 80% of the clouds observed have τg and reff in the retrievable range. These cl...

A new approximate phase function for isolated particles and polydispersions

An accurate and expeditious modelling of the radiative heat transfer is essential for the prediction of combustion equipment performance. For anisotropically scattering media the knowledge of the scattering phase function is of utmost importance. In the present paper, a new phase function modelling, based on the geometrical optics limit, is presented. The new model is easily applicable to polydispersions and does not require the knowledge of any other parameter than the asymmetric factor and efficiency factors. Also, a curve-fitting approach is presented herein to calculate the above-mentioned required parameters. Moreover, this model allows for the analytical and expeditious determination of the radiative energy fraction that is scattered into a finite solid angle, feature that is particularly useful when using solution methods for the radiative heat transfer equation that discretise the space into a finite number of solid angles. When compared with the Mie theory results, the proposed model has proven to be very accurate, provided that the particles can be considered opaque.

Tachyons, Lamb shifts and superluminal chaos

An elementary account on the origins of cosmic chaos in an open and multiply connected universe is given; there is a finite region in the open 3-space in which the world-lines of galaxies are chaotic, and the mixing taking place in this chaotic nucleus of the universe provides a mechanism to create equidistribution. The galaxy background defines a distinguished frame of reference and a unique cosmic time order; in this context superluminal signal transfer is studied. Tachyons are described by a real Proca field with negative mass square, coupled to a current of subluminal matter. Estimates on tachyon mixing in the geometric optics limit are derived. The potential of a static point source in this field theory is a damped periodic function. We treat this tachyon potential as a perturbation of the Coulomb potential, and study its effects on energy levels in hydrogenic systems. By comparing the induced level shifts to high-precision Lamb shift measurements and QED calculations, we suggest a tachyon mass of 2.1 keV/c2 and estimate the tachyonic coupling strength to subluminal matter. The impact of the tachyon field on ground state hyperfine transitions in hydrogen and muonium is investigated. Bounds on atomic transition rates effected by tachyon radiation as well as estimates on the spectral energy density of a possible cosmic tachyon background radiation are derived.

Retrieval of Ice Cloud Parameters Using a Microwave Imaging Radiometer

Abstract Based on the radiative transfer theory, the microwave radiance emanating from ice clouds at arbitrary viewing angles is expressed as an analytic function of the cloud ice water path (IWP), the particle effective diameter (De), and the particle bulk density (ρi). Thus, for a given particle density, the earth-viewing measurements at two frequencies (e.g., 340 and 89 GHz) can provide an estimate of De and IWP for submillimeter-size particles. This physical retrieval is tested using data from the Millimeter-wave Imaging Radiometer (MIR). A comparison among MIR, radar, and infrared sensor measurements shows that the MIR frequencies are affected primarily by thick ice clouds such as cirrus anvil and convection. Over highly convective areas, the measurements from 89 to 220 GHz are nearly identical since the scattering by large ice particles aloft approaches the geometric optics limit, which is independent of wavelength. Under these conditions, only the lower MIR frequencies (89 and 150 GHz) are used to ...