Glory Rays Research Articles

Analysis of scattering by spheres having a negative acoustical refractive index

Electromagnetic waves having oppositely directed phase and group velocities propagate in metamaterials having a negative permeability and negative permittivity [J. B. Pendry and D. R. Smith, Phys. Today 57(6), 37–44 (2004)]. Such materials are predicted to have unusual electromagnetic scattering properties [R. Ruppin, Solid State Commun. 116, 411–415 (2000)]. If it is possible to fabricate acoustical materials having a simultaneously negative effective elastic modulus and density (in a dynamical sense), the mechanical energy flux will have the opposite direction as the wave-vector associated with phase evolution. Rays descriptive of the energy flux refracted by such hypothetical materials at interfaces with ordinary fluids would be characterized by a negative acoustical refractive index. Partial-wave-series calculations of high frequency scattering by fluid spheres having an acoustical refractive index at (or close to) 1 reveal backscattering enhancements associated with glory rays which, unlike ordinary spheres [P. L. Marston and D. S. Langley, J. Acoust. Soc. Am. 73, 1464–1475 (1983)], require only a single internal chord. Generalized Lamb waves on elastic shells having opposite phase and group velocities also cause enhanced backscattering associated with unusual rays [G. Kaduchak, D. H. Hughes, and P. L. Marston, J. Acoust. Soc. Am. 96, 3704–3714 (1994)].

A time-resolved glimpse of the terahertz glory

If you stand above a cloud, you will see bright coloured rings around your shadow. These ‘glory’ rings are most commonly seen around the shadow of an aeroplane. What causes this strong backscattering of light by droplets of water? It has been hard to isolate the glory rays from other scattered components — but this has now been done using ultra-short laser pulses and timing the arrival of components that take different paths thought the drop. This experiment confirms the idea that glory is caused by interference between surface waves on the water droplets.

Edge contribution to forward scattering by spheres

Edge functions T1 and T2, which describe the polarization-dependent edge contribution to forward scattering by spheres, are derived from the exact Mie solution. All the relative refractive indices and the 64 < x < 2048 size parameter range are considered. The edge functions significantly improve the approximation methods that can be used to calculate forward-scattering patterns. For m close to 1, an asymptotic approximation is used. Otherwise, the familiar geometrical optics approximation and the similar physical optics approximation for glory rays are used. Both geometrical and physical optics equations can be deduced from the above-mentioned asymptotic approximation.

Backscattering due to ray reverberations between nonconcentric surfaces of a thick shell: Ray stability, transitions in wavefront shapes, and observed scattering enhancements due to focusing.

The paraxial ray stability analysis previously developed for laser resonators is applied to high‐frequency elastic wave reverberations across the thickness of a shell having nonconcentric surfaces. A previous summation of backscattered amplitudes from rays reverberant within concentric surfaces [S. G. Kargl and P. L. Marston, J. Acoust. Soc. Am. 88, 1114–1122 (1990)] is found to lie on the boundary between stable and unstable resonators. Shifting the center of the inner surface of the shell away from the direction of the source produces a stable resonator. Rays reverberant between such nonconcentric surfaces produce a far‐field focusing that is periodic in the number of internal bounces. Exact ray tracing, used to identify scattered wavefront shapes near focal conditions, reveals the presence of glory rays (off‐axis rays which are exactly backscattered). Shifting the inner surface of the shell toward the source produces an unstable resonator and no focused rays. The existence of backscattering enhancement...

Generalized rainbows and unfolded glories of oblate drops: organization for multiple internal reflections and extension of cusps into Alexander's dark band

Oblate drops of water can produce caustics where, unlike a simple Airy caustic, more than two rays merge. We extend previous treatments of generalized primary rainbows based on catastrophe optics [Opt. Lett. 10, 588 (1985); Proc. R. Soc. (London) A 438, 397 (1992)] to rays having (p - 1) = 2 to 5 internal reflections. The analysis is for a horizontally illuminated ellipsoid with a vertical symmetry axis. Aspect ratios causing a vanishing of the vertical curvature at the equator for the outgoing wave front are found from generalized ray tracing. In response to infinitesimal deformation, the axial caustic of real glory rays unfolds producing cusps. Laboratory observations with laser illumination demonstrate that cusps resulting from rays with five internal reflections extend into Alexander's dark band when the drop's aspect ratio is near 1.08. The evolution of this p = 6 scattering pattern as cusps meet the quinary rainbow is suggestive of an E(6) catastrophe. For ellipsoids of varying aspect ratio and refractive index N, there is an organizing singularity associated with an exceptionally flat outgoing wave front from spheres with N = p.

Liquid-filled spherical reflectors: Analysis of glory ray amplitudes

Thin spherical shells filled with a liquid having a low speed of sound ci are known to have enhanced high-frequency backscattering. It is generally recognized that when the acoustic refractive index N=c/ci is between √2 and 2, the enhancement is associated with a two-chord off-axis backscattered ray. A geometrical analysis of the amplitude diverges since the Gaussian curvature of the outgoing wave front vanishes. In the present research, a physical-optics analysis of glory ray scattering [P. L. Marston and D. S. Langley, J. Acoust. Soc. Am. 73, 1464–1475 (1983)] was applied to the ray of interest. Attention was limited to neutrally buoyant spheres and shell effects were neglected in both the analysis and the partial-wave series (PWS) calculations. The form function contribution is proportional to (ka)1/2, where a is the sphere radius and k=ω/c. For large ka, the analysis gives the major contribution to the PWS calculations except for N near √2 and 2. In this range for N the contribution has a concise analytical approximation that depends on the ray impact parameter b = a(N/2)√4−N2. [Work supported by ONR.] a)Present address: EXP Group, Inc., 44063 Fremont Blvd., Fremont, CA 94538.

Optical glory of small freely rising gas bubbles in water: observed and computed cross-polarized backscattering patterns

Light scattered from spherical bubbles in water manifests an enhancement in the backward direction analogous to the well-known optical glory of a drop. Unlike the glory for water drops, in which the rays travel partially on the drop’s surface, the glory for bubbles is due to rays that are refracted and multiply reflected within the scatterer and is an example of a transmitted wave glory. Photographs of glory scattering for freely rising air bubbles in water are displayed for bubbles having diameters of less than 300 μm. A physical-optics model for backscattering is developed for spherical bubbles. Computed glory patterns from both Mie-series calculations and the physical-optics model agree with the observed patterns. The patterns of freely rising air bubbles having a diameter of ≲300 μm are essentially those predicted for a spherical scatterer. The interference of different classes of glory rays is more clearly seen for bubbles in water than for the previously studied case of bubbles in oil.

Geometrical Optics Backscattered Field from Dielectric Spheroids with Evaluation of the Ring Caustic Field

Abstract A hybrid high-frequency formulation for determining the scattering characteristics of dielectric bodies has been developed using the principles of geometrical optics and Huygens equivalent sources. An application of the method to the problem of determining the near backscattered field due to once internally-reflected rays from dielectric spheroids is presented here, with particular consideration given to the dominating rear-axial and glory rays. The field distribution as predicted by geometrical optics is found to occupy a finite circular aperture and exhibit a “caustic ring” along the outer perimeter of the aperture where the field becomes singular. In order to specify a finite value for the field at the caustic and provide a continuous transition from lit to shadow regions, the Huygens-Fresnel diffraction principle is utilized.

Analysis of internally reflected and diffracted fields in electromagnetic backscattering by dielectric spheroids

For a prolate spheroid of arbitrary size various components of the scattering mechanism responsible for the backscattered energy (which is of interest to monostatic radar systems) are analyzed. In the case of a spherical structure, due to the symmetry, the conditions for the existence of glory rays and internal surface waves are straightforward. It is shown that for the prolate spheroid with arbitrary direction of incidence, these conditions are more complex, and it is advantageous to resort to a numerical method of ray tracing. The solution yielded by this numerical method, along with some geometrical arguments, shows that for most of the cases of arbitrary scatterers, the single bounce glory ray will not exist and therefore will not contribute to the backscattered energy. For the axial case of incidence, conditions for the existence of internal surface waves have also been developed. Finally, some results for the approximate backscattered signal incorporating the above components, are presented.

Glory- and rainbow-enhanced acoustic backscattering from fluid spheres: Models for diffracted axial focusing

Transmitted-wave contributions to backscattering from large spheres are calculated. Amplitudes due to glory rays are enhanced, relative to those of axial reflections, because of focusing. The glory pressure amplitudes are proportional to a(ka)1/2, where a is the sphere’s radius and k is the wavenumber. The model, though similar in form to one for optical scattering from air bubbles [D. S. Langley and P. L. Marston, Phys. Rev. Lett. 47, 913–916 (1981)], contains new features to facilitate the summing of amplitudes in a range of angles. An additional enhancement due to rainbow focusing is modeled for certain sound velocity ratios M; for these M the backscattered amplitude is proportional to a(ka)2/3. Major features of the exact scattering are reproduced by these models when ka=1000 and (with defects) ka=100. The enhancements are not intrinsically due to resonance. Applications to the design of passive sonar targets are noted.

Focusing and diffraction of backscattering from fluid spheres: Comparison of the partial‐wave sum with a model

Consider the scattering of a plane wave from a sphere of radius a imbedded in a fluid with a sound speed c0 = Mci. Backscattered rays with impact parameters b ≠ 0 are known as glory rays; their contributions fn to the total normalized form function f are enhanced due to axial focusing. Naive application of ray acoustics predicts that |fn|→∞ as γ→0 where γ is the angle relative to the backward axis. We derive an approximation which removes this divergence and describes the effect of diffraction on axial focusing. It is similar to our model for the optical glory of bobbles [D. S. Langley and P. L. Marston, Phys. Rev. Lett. 47, 213–916 (1981)] and it gives |fn| = (ka)1/2Gn|J0(kb sinγ)|, where k is the wavenumber and Gn is a function of the number of internal chords n and of fluid properties. The modeled f is obtained by interfering several fn with axial reflections. It was compared with an exact partial‐wave sum for f with ka = 100 and 1000 and various M < 1; our model correctly describes the main features. ...

Observation of the acoustic glory: Scattering from an elastic sphere in near backward directions

We have measured the scattering for small angles γ (relative to the backward axis) from a fused silica sphere of radius a≈52 mm. Tone bursts in water corresponding to ka≈450 were incident on the sphere; their short duration permitted glory and axial returns to be separated in time. The γ for the probe hydrophone was scanned to test a model [P. L. Marston and L. Flax, J. Acoust. Soc. Am. Suppl. I 68, S81 (1980)] of diffractive effects on backward axial focusing. Observations tend to support the model as adapted to fused silica: (1) from the arrival time, the strongest echo is evidently due to the 4‐chord shear glory ray; (2) its amplitude is ∝J0(kb sin γ) where b is the calculated glory circle radius; (3) its amplitude at γ = 0, though slightly smaller than predicted, exceeds that of the first axial reflection; and (4) the times, amplitudes, and γ dependences of other echoes are correlated to predictions. The first null of the strongest echo occurs at γ≃1°. Consequently, we demonstrate for the first time the diffraction limited backward focusing of echoes from a sphere. [Work supported by ONR. Marston is an Alfred P. Slosh Research Fellow.]

Backscattering of light from a water droplet: The glory effect

We discuss the theoretical attempts at explaining the glory effect and describe a simple setup that permits visualization of the glory ray.

Glory in backscattering: Mie and model predictions for bubbles and conditions on refractive index in drops

Mie calculations illustrate the dependences of scattered intensity on polarization, angle, and size for bubbles with radii ≃0.3 mm in a siloxane liquid. Computations give evidence of the axial focusing observed in recent experiments and predicted by a model. Conditions for glory rays in drops and bubbles are derived, and errors in the literature are noted.

Glory contribution to backscatter from large elastic spheres

Partial wave theory for backscatter by large aluminum spheres and cylinders in water give a form function for spheres several times greater than that for cylinders when each is normalized to the geometric scattering of a rigid reflector of the same size and shape [L. Flax, J. Acoust. Soc. Am. 62, 1502 (1977)]. The physical‐optics approximation is used in the present work to show that twice‐reflected, twice‐refracted glory rays are a significant contribution to the largeness of the backscatter by elastic spheres. These rays lead to a back‐facing toroidal wavefront which appears to originate at a virtual ring‐like source of radius b < the sphere radius a. The modulus of the glory contribution to the normalized form function obtained by computing the farfield diffraction of this wavefront is |fa| = (ka)1/2G|J0 (kb sin γ)|, where k is the wavenumber in water, G is a function of material properties, γ = 180° − θ, and θ is the scattering angle. When all glory rays in aluminum are S waves, b/a ≃ 0.27 and |fa ≃ 2...

Calculation of the glory scattering with anisotropic molecule-atom interactions in sudden approximation

For specified rotational states glory phenomena are investigated in sudden approximation under influence of angle dependent terms in the intermolecular potential. The distance at which the intermolecular potential is probed by the anisotropic glory undulation is R ≈ 0.96 R m, for LJ 12-6 potentials, i.e. the well defined probing distance nearly coincides with the classical turning point of the glory ray. For a specific choice of the ratio of anisotropy parameters, q 2,12/ q 2,6, ≈ 1.5, the anisotropic glory changes sign. Quenching of the state averaged glories practically disappears for q 2,12/q 2,6 ≈ 2 (e.g. for NO-Ar). The method of calculation is tested against a full partial wave treatment.

Geometrical optics of inhomogeneous particles: glory ray and the rainbow revisited

The glory ray and the rainbow are analyzed by considering the scattering of light from inhomogeneous particles. It is shown that melting ice crystals may be strong contributors to the glory ray. Geometrical optics is used to investigate and catalog a wide variety of particle inhomogeneities which support rainbow and glory rays.

Near-field impulse response examination of backscattering from dielectric spheres

The backscattered component of the normalized Poynting vector impulse response is evaluated at selected times on a plane normal to the direction of pulse propagation and in the near field of a dielectric sphere. The computations for spheres with refractive indices of 1.60 + 0.01i and 2.25 + 0.01i are consistent with the physical interpretations of the high frequency analogs of axial, stationary, and glory rays and internal surface waves. Flux maxima associated with glory and stationary rays appear first on the axis normal to the plane of polarization and change to the axis of polarization as their analogs change polarity. Latter stages of the glory ray appear to propagate in a mode similar to internal surface waves.

Far-field impulse response verification of selected high-frequency optics backscattering analogs

Impulse response approximations are presented for selected refractive indices to illustrate the variety of pulse returns for dielectric spheres. High frequency analogs for the rear axial and glory rays are verified by comparison with isolated returns in appropriate impulse responses.