Supernumerary Bows Research Articles

Airy theory revisited with the method combining vectorial complex ray model and physical optics.

Airy published his theory in the 1830s to remedy the problem of infinite intensity in the rainbow angles of a spherical droplet predicted by geometrical optics. This theory has been studied by mathematicians and physicists since then from different points of view. In what concerns the scattering diagram around the rainbow angles, Airy theory has been improved by researchers in order to predict correctly the intensity of the supernumerary bows. However, it is known that the positions and the intensities of the supernumerary bows predicted by Airy theory differ from those of rigorous Debye theory with increasing order p and the scattering angles from the rainbow angle. In the present Letter, we will show that this discrepancy is caused by the approximations in Airy theory and can be revised by combining the vectorial complex ray model and physical optics (PO). The former permits us to calculate rigorously the amplitudes and phases of all rays and predicts precisely the scattering pattern except for the main bows. The combination with PO predicts very precisely all the supernumerary bows for both perpendicular and parallel polarization. This method can be applied directly to the light scattering of non-spherical particles with smooth surfaces.

Supernumerary bows: interference theory with the zero wavefront as a basic element.

This study relates to the prediction of the angular positions of supernumerary screenbows and rainbows, in the case of a refractive sphere illuminated by a point source placed at a distance of h from its center; for h→∞, the incident light beam becomes parallel. The screenbow appears on a spherical screen whose center is that of the sphere and which intercepts the tangential caustic surface. The rainbow, specific to the water drop, but here generalized to any refractive sphere, corresponds to a screenbow produced on a "screen" placed at an infinite distance. This paper uses exact graphical representations of the wavefronts associated with rainbows resulting from k internal reflections to illustrate how the angular positions of the supernumerary rainbows and the positions of the corresponding supernumerary bows on screens are to be calculated. All considerations are made within the framework of geometrical optics being, on the one hand, the limit of the electromagnetic theory as the wavelength goes to 0, and, on the other hand, complemented by the Gouy phase shift theory.

Supernumerary bows: caustics of a refractive sphere and analysis of the relative overall Gouy phase shift of supernumerary rays.

The modified Young's theory of interference related to supernumerary rainbows is based on a difference of 90° in the Gouy phase shifts for the parallel rays producing these bows. An observation screen placed at a given distance from a refractive sphere illuminated by a point source of light should also show supernumerary screen bows. An extensive description and analysis of the caustics involved are given. For any k order, k being the number of reflections inside the sphere, a procedure is established to determine the number of Gouy phase shifts encountered by any ray along its path from the source to the screen. Special consideration is given to the order k=0. For any k supernumerary bow, on any spherical screen whose center is that of the sphere, the difference in the Gouy phase shifts for the two rays producing a bow always amounts to 90°. An indirect proof of this characteristic is given. All considerations are made within the framework of geometrical optics being, on the one hand, the limit of the electromagnetic theory as the wavelength goes to 0, and being, on the other hand, complemented by the Gouy phase shift theory.

Photodetachment and photoionization rainbows and glories

Quantum scattering has many similarities with the physics of the atmospheric rainbow. Diffraction effects, including rainbows and glories, have long been introduced in the physics of scattering, and particularly in nuclear, atomic and molecular physics. In this paper we describe the striking similarity between the optics of the primary rainbow and supernumerary bows and photodetachment microscopy, with the latter term referring to the photodetachment of a structureless anion in the presence of a static electric field. Further, we extend the aforementioned analogy to the more complex and fertile case of photoionization microscopy. Despite the fact that in the latter situation the analogy is only approximate, we demonstrate the emergence of additional features that are also found in classical optics, like higher-order bows and glories. Finally, based on the conclusions drawn from the above analyses, we discuss the significant contribution of glories in threshold photoelectron spectroscopy.

Markov chain formalism for generalized radiative transfer in a plane-parallel medium, accounting for polarization

A Markov chain formalism is developed for computing the transport of polarized radiation according to Generalized Radiative Transfer (GRT) theory, which was developed recently to account for unresolved random fluctuations of scattering particle density and can also be applied to unresolved spectral variability of gaseous absorption as an improvement over the standard correlated-k method. Using Gamma distribution to describe the probability density function of the extinction or absorption coefficient, a shape parameter a that quantifies the variability is introduced, defined as the mean extinction or absorption coefficient squared divided by its variance. It controls the decay rate of a power-law transmission that replaces the usual exponential Beer-Lambert-Bouguer law. Exponential transmission, hence classic RT, is recovered when a→∞. The new approach is verified to high accuracy against numerical benchmark results obtained with a custom Monte Carlo method. For a<∞, angular reciprocity is violated to a degree that increases with the spatial variability, as observed for finite portions of real-world cloudy scenes. While the degree of linear polarization in liquid water cloudbows, supernumerary bows, and glories is affected by spatial heterogeneity, the positions in scattering angle of these features are relatively unchanged. As a result, a single-scattering model based on the assumption of subpixel homogeneity can still be used to derive droplet size distributions from polarimetric measurements of extended stratocumulus clouds.

Twin primary rainbows scattering by a liquid-filled capillary

The twin primary rainbows scattered by a liquid-filled capillary are investigated with Debye theory and geometric optics. From the numerical simulations, a critical radius ratio of the core to the coating of the coated cylinder is proposed to judge the existence of the α and β supernumerary bows. When the ratio is less than or around the critical value, the α and β supernumerary bows disappear. On the premise that the α and β supernumerary bows exist, the β rainbow can always be detected. However, the α supernumerary bows sometimes submerge in the other scattering structure so that the α rainbow cannot be detected under this condition. Four frequency peaks in the angular frequency spectrum of the twin primary rainbows are investigated numerically. Furthermore the relationship between F3 (the third frequency peak which is similar to the ripple frequency of the scattering by a homogeneous cylinder) and the external radius of the capillary is obtained. Experiments are carried out to verify the numerical works with capillary filled with deionized water.

Physically-based simulation of rainbows

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.

Imaging polarimetry of the fogbow: polarization characteristics of white rainbows measured in the high Arctic

The knowledge on the optics of fogbows is scarce, and their polarization characteristics have never been measured to our knowledge. To fill this gap we measured the polarization features of 16 fogbows during the Beringia 2005 Arctic polar research expedition by imaging polarimetry in the red, green and blue spectral ranges. We present here the first polarization patterns of the fogbow. In the patterns of the degree of linear polarization p, fogbows and their supernumerary bows are best visible in the red spectral range due to the least dilution of fogbow light by light scattered in air. In the patterns of the angle of polarization α fogbows are practically not discernible because their α-pattern is the same as that of the sky: the direction of polarization is perpendicular to the plane of scattering and is parallel to the arc of the bow, independently of the wavelength. Fogbows and their supernumeraries were best seen in the patterns of the polarized radiance. In these patterns the angular distance δ between the peaks of the primary and the first supernumerary and the angular width σ of the primary bow were determined along different radii from the center of the bow. δ ranged between 6.08° and 13.41°, while σ changed from 5.25° to 19.47°. Certain fogbows were relatively homogeneous, meaning small variations of δ and σ along their bows. Other fogbows were heterogeneous, possessing quite variable δ- and σ-values along their bows. This variability could be a consequence of the characteristics of the high Arctic with open waters within the ice shield resulting in the spatiotemporal change of the droplet size within the fog.

Fountain rainbows

We present the first measurements of radiance spectra of rainbows. The bows on two sunny days (3 and 6 June 2008) were produced by the fountain in the Parque de las Ciencias, Granada, Spain, that consists of a rectangular perimeter of 40 spray nozzles. Optical thickness of the spray from each nozzle was approximately 0.5. Spectral purity of the primary bow was highest for orange and blue, reaching values of 23% and 7%, respectively, while skylight 90 degrees from the Sun had a color purity of 34% (on 6 June). The secondary bow had much lower color purity with red absent because the regions around the bows and in Alexander's dark band were pale blue. The narrow sickle shape of the chromaticity curves for the primary bows and the absence of supernumerary bows indicated that the drop radius was between 0.2 and 0.4 mm.

Why can the supernumerary bows be seen in a rain shower?

Although the spectra of drop radii in rain showers are broad, the supernumerary bows are caused by only those drops with radii of about 0.25 mm. The angle of minimum deviation, the rainbow angle, is a function of drop size, being large for big drops, owing to drop distortion, and large for small drops, owing to interference. Between these extremes, there is a minimum rainbow angle. The drops that cause it give rise to the supernumerary bows.

Angular scattering and rainbow formation in pendant drops

Angular scattering measurements obtained with a polar nephelometer employing a linearly polarized laser source are used to examine the general scattering behavior and rainbow generation of pendant water drops, a type of near-spherical particle that has certain similarities to the shape of distorted raindrops. Comparison of the experimental data with theoretical predictions of spherical drop scattering reveals that in many respects the near-spherical particles behave like spheres when the measurements are performed in the horizontal scattering plane, the plane in which the drops display circular cross sections. Furthermore, the angular positions of the rainbow intensity maxima corresponding to the main rainbow peak and supernumerary bows are shown to be predicted accurately by the approximate Airy theory for both the primary and secondary rainbows. Pendant drops whose shapes are significantly elongated in the vertical direction are indicated to generate anomalously strong rainbows from three or more internal reflections. The implications of these findings to rainbow formation in the atmosphere are discussed.

Inhomogeneities in the Color and Intensity of the Rainbow

It has long been recognized that when the sun is low in the sky, the supernumerary bows disappear in the vertical portion of the rainbow. It is noted here that the red part of the rainbow also varies in intensity near the vertical portion of the bow. Both of these phenomena are ascribed to the nonspherical shape of large raindrops.

Infrared Rainbow

Radition in the near-infrared spectral region should produce a rainbow that is not visible to the human eye. An infrared photograph is shown which displays the primary bow, the secondary bow, and two supernumerary bows in side the primary bow.

Scattering of Visible Light by Large Water Spheres

The intensity and the degree of polarization of the radiation scattered by a large sphere were computed using the Mie theory at sufficiently small interval of the scattering angle for obtaining a complete picture of all the characteristics of the field of the scattered radiation. The results are presented for four different sizes of the water sphere (radius = 6.25 micro, 12.5 ;micro, 25.0 micro, and 50.0 micro) assumed to be illuminated by an unpolarized beam of monochromatic radiation with wavelength 0.4 micro. A detailed comparison is then made between the results obtained using the exact Mie theory and those obtained using an approximate approach based on the application of the other laws of the geometrical and physical optics. The angular positions of the primary and secondary rainbows, as well as those of their supernumerary bows as obtained using the approximate method, agree with those obtained from the Mie theory only if the size parameter of the sphere is of the order of 800. Besides the phenomenon of glory which is not amenable to explanation in terms of the geometrical and physical optics, the Mie computations bring out several distinct maxima and minima whose occurrence cannot be explained in likewise manner.

On Cusped Waves of Light and the Theory of the Rainbow

Diagrams were shown of the forms assumed by a plane wave of light falling on a spherical raindrop and twice reflected from the interior of the drop, as well as the waves emerging from the drop. The waves in general have cuspidal edges which run along the caustic surfaces. This relation between the caustic and the cusps on the waves was pointed out by Wood in connection with the similar waves produced by reflection at a spherical surface. It had been noticed earlier by Potter, Jamin and Mace de Lepinay. The phase over a wave of this type is not constant, the two portions on opposite sides of a cusp differing in general by a quarter period. Attention was drawn to the advantage of regarding the distribution of light in the rainbow as the consequence of the interference of the cusped waves which run down to the observer's eye along the direction of minimum deviation. This way of looking at the matter is shown to be equivalent to Mascart's approximate method of explanation of the formation of the supernumerary bows by interference of disturbances coming from the two poles on the special wave-form used by Airy.

The Fog Bow

THE complete theory of the rainbow, as developed by Sir George Airy (Camb. Phil. Trans., vi. p. 379, 1836), besides explaining the supernumerary bows, shows that the main bow has a radius somewhat smaller than that calculated on the ordinary geometrical theory. The smaller the drops the greater is the discrepancy. With the tiny drops composing a fog, the discrepancy is so marked that the bow receives a new name—the fog-bow, or “arc-en-ciel blanc.” Mr. Mohn's (NATURE, February 23, p. 391) nearly simultaneous measurements of the fog-bow and Ulloa's rings afford a capital opportunity of putting the theory to the test, for from the latter phenomenon we can readily calculate the average size of the particles.

A Remarkable Rainbow

THE phenomenon of supernumerary bows noticed by “L. C.” on September 24, has been repeatedly observed and described. Various explanations have been suggested; and “L. C.” will probably find what he wants in Archdeacon Pratt's paper in Phil. Mag., 4th series, vol. v. pp. 78–86 (1853).

Letters to Editor

IN NATURE, vol. x. p. 438, Mr. Schuster complains that in text-books no mention is made of supernumerary rainbows, and that the theory of them is to be sought in original memoirs, not generally accessible. Allow me to mention that in Sir John Herschel's Meteorology (a little work published by Black, price three and sixpence, and originally an article in the Encycl. Britann.), a complete explanation of the rainbow, and of the supernumerary bows as well, on the principle of interference, is to be found.

XV. The supernumerary bows in the rainbow arise from interference

XV. The supernumerary bows in the rainbow arise from interference