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Abstract: Mie theory offers an exact solution to the problem of scattering of sunlight by spherical drops of water. Until recently, most applications of Mie theory to scattering of light were restricted to a single wavelength. Mie theory can now be used on modern personal computers to produce full-color simulations of atmospheric optical effects, such as rainbows, coronas, and glories. Comparison of such simulations with observations of natural glories and cloudbows is encouraging.
Abstract: The scattering of light from homogeneous spheres might be considered to be a trivial problem because rigorous solutions, such as Mie theory, were developed almost 100 years ago. Nevertheless, full-colour simulations of atmospheric optical effects, such as rainbows, coronas and glories, reveal several intriguing issues. Calculations using the Debye series can help us to understand the scattering mechanisms causing specific effects: for example, the atmospheric glory seems to be caused by light rays that have suffered one internal reflection within water drops.
Abstract: Atmospheric optical effects can teach much about physics and especially optics. Coronae—coloured rings around the sun or moon—are large-scale consequences of diffraction, which is often thought of as only a small effect confined to the laboratory. We describe coronae, how they are formed and experiments that can be conducted on ones in the sky. Recognizing that this is not always convenient, we show how students can also learn about coronae and thus diffraction from experiments with accurate full-colour computer simulations and laboratory demonstrations.
Abstract: Mie theory can be used to provide full-color simulations of atmospheric glories. Comparison of such simulations with images of real glories suggests that most glories are caused by spherical water droplets with radii between 4 and 25 µm. This paper also examines the appearance of glories taking into account the size of the droplets and the width of the droplet size distributions. Simulations of glories viewed through a linear polarizer compare well with the few available pictures, but they show some features that need corroboration by more observations.
Abstract: Mie theory can be used to generate full-color simulations of atmospheric glories, but it offers no explanation for the formation of glories. Simulations using the Debye series indicate that glories are caused by rays that have suffered one internal reflection within spherical droplets of water. In 1947, van de Hulst suggested that backscattering (i.e., scattering angle θ = 180°) could be caused by surface waves, which would generate a toroidal wavefront due to spherical symmetry. Furthermore, he postulated that the glory is the interference pattern corresponding to this toroidal wavefront. Although van de Hulst’s explanation for the glory has been widely accepted, the author offers a slightly different explanation. Noting that surface waves shed radiation continuously around the droplet (not just at θ = 180°), scattering in a specific direction θ = 180° - δ can be considered as the vector sum of two surface waves: one deflecting the incident light by 180° - δ and the other by 180° + δ. The author suggests that the glory is the result of two-ray interference between these two surface waves. Simple calculations indicate that this model produces more accurate results than van de Hulst’s model.
Abstract: A ray-theoretic account of the passage of light through a radially inhomogeneous transparent sphere has been used to establish the existence of multiple primary rainbows for some refractive index profiles. The existence of such additional bows is a consequence of a sufficiently attractive potential in the interior of the drop, i.e., the refractive index gradient should be sufficiently negative there. The profiles for which this gradient is monotonically increasing do not result in this phenomenon, but nonmonotone profiles can do so, depending on the form of n. Sufficiently oscillatory profiles can lead to apparently singular behavior in the deviation angle (within the geometrical optics approximation) as well as multiple rainbows. These results also apply to systems with circular cylindrical cross sections, and may be of value in the field of rainbow refractometry.
Abstract: The atmospheric glory caused by backscattering of sunlight from clouds usually has circular colored rings. However, glories with noncircular rings are frequently observed, especially along the edges of clouds. Noting that the angular radius of the rings of glories is a sensitive indicator of the size of the water droplets in clouds, several images of glories have been examined in an attempt to explain the formation of noncircular glories.
Abstract: Atmospheric glories are caused by backscattering of sunlight from spherical droplets of water (e.g., from fog or clouds). But what would glories look like if they were caused by scattering from more exotic substances, such as clouds of ethane as found on Titan? Examining backscattering as a function of the refractive index n of spherical droplets leads to the surprising conclusion that a glory's appearance is almost independent of n (at least for 1.03 < n <1.7) ‐ unlike the colors of rainbows, which are critically dependent on the variation of n across the visible spectrum.
Abstract: Craig Bohren has offered a million-dollar prize to anyone who can devise a detector that accepts scattered light but rejects diffracted light. This challenge was examined from a theoretical perspective by considering the scattering of red light by a spherical droplet of water with diameter 20 μm. Illumination of the droplet by short pulses (e.g. a duration of 5 fs) could allow a detector to distinguish between light scattered by various mechanisms, such as diffraction, transmission, reflections and surface waves. Although such techniques would not satisfy the precise terms of the challenge, the time domain approach can deliver remarkable insights into the details of the scattering processes.
Abstract: We computed the Debye series p = 1 and p = 2 terms of the Mie scattered intensity as a function of scattering angle and delay time for a linearly polarized plane wave pulse incident on a spherical dielectric particle and physically interpreted the resulting numerical data. Radiation shed by electromagnetic surface waves plays a prominent role in the scattered intensity. We determined the surface wave phase and damping rate and studied the structure of the p = 1, 2 surface wave glory in the time domain.
Abstract: The p = 0 term of the Mie–Debye scattering amplitude contains the effects of external reflection and diffraction. We computed the reflected intensity in the time domain as a function of the scattering angle and delay time for a short electromagnetic pulse incident on a spherical particle and compared it to the predicted behavior in the forward-focusing region, the specular reflection region, and the glory region. We examined the physical consequences of three different approaches to the exact diffraction amplitude, and determined the signature of diffraction in the time domain. The external reflection surface wave amplitude gradually replaces the diffraction amplitude in the angular transition region between forward-focusing and the region of specular reflection. The details of this replacement were studied in the time domain.
Abstract: This is a feature issue devoted to optical phenomena that can be observed in nature, primarily with the naked eye. Many of the papers published in this feature issue are based on presentations given at the “Light & Color in Nature” conference held in June 2010 at St. Mary’s College of Maryland.
Abstract: Rainbows, coronas and glories are caused by the scattering of sunlight from water droplets in the atmosphere. Although these optical phenomena are seen fairly frequently, even scientifically minded people sometimes struggle to provide explanations for their formation. This paper offers explanations of these phenomena based on numerical computations the scattering of a 5 fs pulse of red light by a spherical droplet of water. The results reveal the intricate details of the various scattering mechanisms, some of which are essentially undetectable except in the time domain.
Abstract: Naturally occurring tertiary rainbows are extraordinarily rare and only a handful of reliable sightings and photographs have been published. Indeed, tertiaries are sometimes assumed to be inherently invisible because of sun glare and strong forward scattering by raindrops. To analyze the natural tertiary’s visibility, we use Lorenz–Mie theory, the Debye series, and a modified geometrical optics model (including both interference and nonspherical drops) to calculate the tertiary’s (1) chromaticity gamuts, (2) luminance contrasts, and (3) color contrasts as seen against dark cloud backgrounds. Results from each model show that natural tertiaries are just visible for some unusual combinations of lighting conditions and raindrop size distributions.
Abstract: In this article, we derive a physically-based model for simulating rainbows. Previous techniques for simulating rainbows have used either geometric optics (ray tracing) or Lorenz-Mie theory. Lorenz-Mie theory is by far the most accurate technique as it takes into account optical effects such as dispersion, polarization, interference, and diffraction. These effects are critical for simulating rainbows accurately. However, as Lorenz-Mie theory is restricted to scattering by spherical particles, it cannot be applied to real raindrops which are nonspherical, especially for larger raindrops. We present the first comprehensive technique for simulating the interaction of a wavefront of light with a physically-based water drop shape. Our technique is based on ray tracing extended to account for dispersion, polarization, interference, and diffraction. Our model matches Lorenz-Mie theory for spherical particles, but it also enables the accurate simulation of nonspherical particles. It can simulate many different rainbow phenomena including double rainbows and supernumerary bows. We show how the nonspherical raindrops influence the shape of the rainbows, and we provide a simulation of the rare twinned rainbow, which is believed to be caused by nonspherical water drops.
Abstract: Although scattering of light by a coated sphere is much more complicated than scattering by a homogeneous sphere, each of the partial wave amplitudes for scattering of a plane wave by a coated sphere can be expanded in a Debye series. The Debye series can then be rearranged in terms of the various reflections that each partial wave undergoes inside the coated sphere. For a given number of internal reflections, it is found that many different Debye terms produce the same scattered intensity as a function of scattering angle. This is called path degeneracy. In addition, some of the ray trajectories are repeats of those occurring for a smaller number of internal reflections in the sense that they produce identical time delays as a function of scattering angle. These repeated paths, however, have a different intensity as a function of scattering angle than their predecessors. The degenerate paths and repeated paths considerably simplify the interpretation of scattering within the coated sphere, thus making it possible to catalog the contributions of the various paths.
Abstract: Numerical computations were made of scattering of an incident electromagnetic pulse by a coated sphere that is large compared to the dominant wavelength of the incident light. The scattered intensity was plotted as a function of the scattering angle and delay time of the scattered pulse. For fixed core and coating radii, the Debye series terms that most strongly contribute to the scattered intensity in different regions of scattering angle-delay time space were identified and analyzed. For a fixed overall radius and an increasing core radius, the first-order rainbow was observed to evolve into three separate components. The original component faded away, while the two new components eventually merged together. The behavior of surface waves generated by grazing incidence at the core/coating and coating/exterior interfaces was also examined and discussed.
Abstract: Rainbows, coronas and glories are examples of atmospheric optical phenomena caused by the scattering of sunlight from spherical drops of water. It is surprising that the apparently simple process of scattering of light by spherical drops of water can result in this wide range of colourful effects. However, the scattering mechanisms are very complicated. Eminent scientists (such as Descartes, Newton, Young, Airy and many others) offered various explanations for the formation of rainbows—thus making major contributions to our understanding of the nature of light. The basic features of rainbows can be explained by geometrical optics but, in the early 1800s, supernumerary arcs on rainbows provided crucial supporting evidence for the wave theory of light. In 1908, Mie provided a rigorous (but very complicated) solution to the problem of scattering of light by spherical particles. More than 100 years later, Mie’s solution can now be used to produce excellent full-colour simulations. Examples of such simulations show how the appearance of these phenomena vary with the size of the water drops, as well as describing the scattering mechanisms that are responsible for their formation.
Abstract: We present a new analysis of Robert Grosseteste’s account of color in his treatise De iride (On the Rainbow), dating from the early 13th century. The work explores color within the 3D framework set out in Grosseteste’s De colore [see J. Opt. Soc. Am. A 29, A346 (2012)], but now links the axes of variation to observable properties of rainbows. We combine a modern understanding of the physics of rainbows and of human color perception to resolve the linguistic ambiguities of the medieval text and to interpret Grosseteste’s key terms.
Abstract: This is a feature issue devoted to optical phenomena in nature. Many of the papers published in this feature issue are based on presentations given at the “Light & Color in Nature” conference held in August 2013 at the University of Alaska—Fairbanks.
Abstract: The atmospheric corona is a well-known diffraction phenomenon, typically seen as colored rings surrounding the Sun or Moon. In many respects, Fraunhofer diffraction provides a good explanation of the corona. As the angular sizes of the corona’s rings are inversely proportional to the radius, r, of the spherical particles causing the corona, it should be easy to estimate the particle size from observations and photographs. Noting that some of the techniques commonly used for particle sizing based on diffraction theory can give misleading results for coronas caused by the scattering of sunlight, this paper uses Mie theory simulations to demonstrate that the inner 3 red rings of the corona have angular radii of θ ≈ 16/r, 31/r, and 47/r, when θ is measured in degrees and r is measured in μm.
Abstract: We calculated scattering of an electromagnetic plane wave by both a radially-inhomogeneous particle and bubble, the square of whose refractive index profile is parabolic as a function of radius. Depending on the value of the two adjustable parameters of the parabola, the particle or bubble can have either a refractive index discontinuity at its surface, or the refractive index can smoothly merge into that of the exterior medium. Scattering was analyzed in ray theory, and various novel features of the scattering, including the details of the curved ray paths, transmission rainbows, and near-critical-angle scattering were apparent and were contrasted with their behavior for scattering by a homogeneous sphere.
Abstract: We calculated scattering of an electromagnetic plane wave by a radially inhomogeneous particle and a radially inhomogeneous bubble when the square of the refractive index profile is parabolic as a function of radius. Such a particle or bubble is called a generalized Luneburg lens. A wide variety of scattering phenomena can occur, depending on the value of the two adjustable parameters of the parabola. These phenomena, including transmission rainbows, the weak caustic for near-critical-angle scattering by a bubble, surface orbiting, the interior orbiting paths of morphology-dependent resonances, and the separation of diffraction are studied here using wave theory and time domain scattering. These phenomena are also compared with their appearance or absence for scattering by a homogeneous sphere.
In electromagnetic scattering of an incident beam by a single particle possessing a reasonably high degree of symmetry, the Debye series decomposes the partial wave scattering and interior amplitudes into the contributions of a number of intuitive physical processes. We describe the Debye series for scattering by a sphere, a coated sphere, a multi-layer sphere, a tilted cylinder, and a prolate spheroid. We also comment on the meaning of the various Debye series terms, and briefly recount the method by which the formulas of ray scattering can be derived from them. We also consider time-domain scattering of a short pulse by a single particle and describe the way in which the time-domain scattering signature naturally separates out the various Debye series terms. Lastly, we show how time-domain scattering further separates a number of cooperating sub-processes present in individual Debye series terms.
Abstract: While making airborne measurements of cloud particles, a bright glory was observed on a thin layer cloud. By deliberately flying through this glory-producing cloud on several occasions, cloud particle size distributions were obtained. We found that warm liquid clouds with narrow cloud droplet size distributions are responsible for producing the observed glory. This paper presents these results and compares the results of Mie theory simulations with an image of the glory.
Abstract: We consider transmission scattering of a plane wave by a radially inhomogeneous sphere containing a localized region of refractive index decrease. In ray theory, the boundary conditions on the deflection angle at axial and grazing incidence determine that transmission scattering gives rise to an even number of bows, half of them being relative maximum bows and half being relative minimum bows. For a model refractive index profile, we determine the conditions under which different numbers of bows occur, and we suggest physical mechanisms responsible for producing them. We also verify that these bows occur in wave scattering in the short wavelength limit, both in the frequency domain and time domain.
Abstract: Near-forward scattering of sunlight generates coronas and iridescence on clouds. Coronas are caused by diffraction, whereas iridescence is less easily explained. Iridescence often appears as bands of color aligned with the edges of clouds or as apparently random patches of color on clouds. This paper suggests that iridescence is due to interference between light that has been diffracted by a spherical droplet of water and light that has been transmitted through the same droplet.
Abstract: Supernumerary arcs on rainbows are historically important because in the early 1800s they provided evidence in favor of the wave theory of light. The success of Airy’s rainbow integral has overshadowed the earlier contribution from Young, who proposed that supernumerary arcs were caused by interference between two geometrical rays that emerge from the raindrop at the same scattering angle. Airy dismissed Young’s idea as “the imperfect theory of interference” because it predicted supernumerary arcs at the wrong angles. Young was unaware that a light ray encountering a focal line can suffer a phase shift of 90°. If these phase shifts are taken into account, the theory of interference becomes surprisingly accurate.
Abstract: In static and dynamic light scattering, it has frequently been claimed that cross-polarized scattering cannot occur for single-scattering by a homogeneous spherical particle. Although this is true for both plane wave and on-axis Gaussian beam incidence, we show that cross-polarized scattering does occur when the beam is translated off-axis incidence perpendicular to the scattering plane. We find that the existence of cross-polarized scattering is a direct result of the breaking of the circular symmetry of the beam with respect to the center of the particle when the beam is translated off-axis in wave theory, and to the constraint of the incident beam being a solution of Maxwell’s equations.
Abstract: In static and dynamic light scattering, it has frequently been claimed that cross-polarized scattering cannot occur for single-scattering by a homogeneous spherical particle. Although this is true for both plane wave and on-axis Gaussian beam incidence, it does occur when the beam is translated off-axis incidence perpendicular to the scattering plane. An approximation to the co-polarized and cross-polarized scattering amplitudes is developed for which the sums over azimuthal modes can be evaluated analytically. This approximation provides a close fit to the exact generalized Lorenz-Mie polarization-resolved intensity for a number of off-axis locations of the incident beam.
Abstract: Diffraction in the near-forward direction is examined for the localized model of an off-axis focused Gaussian beam scattered by a sphere when the beam is translated off-axis either in the scattering plane or perpendicular to it. The results are analyzed using the Fraunhofer diffraction approximation. We also obtain various Debye series contributions to the exact GLMT co-polarized VV and HH scattering amplitudes and cross-polarized VH and HV scattering amplitudes, and discuss their interpretation. Time-domain scattering is also considered for both the exact GLMT scattering amplitudes and our approximation to them, for which the sum over azimuthal modes is evaluated analytically. It is found that although the approximation accurately reproduces the overall magnitude of the scattered intensity, it is not as accurate in reproducing the phase of high-frequency oscillations in the intensity.
Page updated on 12 January 2019
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