Primary Rainbows Research Articles

Rainbows in Different Refractive Indices

The rainbow is a natural optical scattering and dispersion phenomenon that reveals the visible spectral composition of sunlight in the shape of an arc. People are instinctively attracted by its colorful appearance and curved shape. Hence, there are many serious studies about the rainbow with a long history. Recently, several simple experiments, adopting glass balls, acrylic spheres, spherical flasks, or sessile water drops, have been devised to demonstrate how the rainbow is formed. These works demonstrate the colors and shapes of the rainbow well and explain how the dispersive spectrum is produced by the refraction–reflection–refraction process. However, the influence of the refractive index is rarely illustrated. It is not difficult to see that the refractive index of raindrops and the atmosphere is closely related to the rainbow, especially the viewing angle of it. In this paper, we use spherical lenses with different materials and in different solutions to change the refractive index. Under a collimated light source, the evolution of the viewing angles of primary and secondary rainbows with respect to the refractive index is demonstrated. Experiments with refraction conditions similar to a natural rainbow are also conducted.

Influence of Regge poles on rainbow angular scattering for state-to-state chemical reactions using Heisenberg’s S matrix programme

Two powerful theories for state-to-state chemical reactions are brought together for the first time. The first theory incorporates Regge pole positions and residues into the partial wave (PW) scattering (S) matrix. The second theory is a ‘weak’ version of Heisenberg’s Scattering Matrix Programme (wHSMP). It uses four general physical principles to suggest simple parameterised forms for the S matrix. The wHSMP is particularly useful for understanding generic structures in differential cross sections (DCSs). Our initial S matrix parameterisation has no Regge poles, but it exhibits a rainbow in the DCS using a Legendre PW series. We then introduce Regge poles for three examples into the S matrix and investigate their influence on the rainbow scattering. We find that inclusion of Regge poles ‘pushes’ the rainbow angle to larger values. We also employ Nearside-Farside (NF) PW and Local Angular Momentum PW theories, including up to three resummations. The recently introduced ‘CoroGlo’ test is used to distinguish between glory and corona scattering in the forward direction. We apply full and NF asymptotic (semiclassical) rainbow theories: the uniform and transitional Airy approximations for the farside scattering. We prove that structure in the no-pole and with-pole DCSs are examples of primary and supernumerary rainbows.

Measuring the refractive index of a transparent sphere from the primary and secondary rainbows

We propose a method to obtain the refractive index of a transparent sphere from the fascinating phenomena of rainbows, which can be used as an interesting subject in the experimental courses of primary optics for undergraduate or high school students. The primary and secondary rainbows are reproduced on white walls and screens in the classroom with the sun and water drops replaced by the LED flashlight and K9 glass (one type of borosilicate glass) sphere respectively. The locations of the primary and secondary rainbows on the screen are measured, with the locations of the screen and sphere changed. By fitting the multiple measurements to the theoretical result using the least square method, the refractive index of the glass sphere, as an unknown parameter in the fitting process, is obtained. For both the primary and secondary rainbows, the locations of the red, yellow and purple bows are measured independently. Repetitive measurements give very stable and accurate output with the standard deviation ratio as low as 10−4 and the relative error all around 0.1% compared to the result obtained from special instruments in the literature. The fascinating phenomena and precise measurements enable this method to fully deepen the students’ understanding of the law of refraction and the application of the least square method.

Twisted Rainbow Light and Nature‐Inspired Generation of Vector Vortex Beams

AbstractTwisted vector light beams (optical vortices) arise from a spiral modulation of the geometric (Pancharatnam–Berry) phase converting the light spin to the orbital angular momentum. The preferred geometric‐phase elements using liquid crystals and plasmonic metasurfaces realize this conversion by structuring their building blocks, i.e., precisely orienting individual crystal molecules or plasmonic nanoantennas. Here, an analogous mechanism is discovered in the spiral phase modulation of light reflected by dielectric spheres and first demonstrated in natural phenomena, namely in the rainbow formation. The spiral geometric phase is documented by holographic imaging of full circle primary and secondary rainbows created in the laboratory. The measurement uses a wide‐angle holographic camera (field of view ≈120°) taking time‐resolved self‐correlation holograms (300 ms). The holograms allow a quantitative restoration of the spiral geometric phase of light reflected from thousands of randomly falling water drops. The capability of individual drops to generate vector vortex beams under circularly polarized illumination is proven theoretically and demonstrated in experiments using glass microspheres. The spherical reflectors are discovered as simple generators of vector vortex beams and vortex arrays, inspiring novel geometric‐phase elements.

Dislocated spots and triple splittings of natural rainbows generated by large drop distortions, oscillations, and tilts.

For an accurate modeling of natural rainbows, it is necessary to take into account the flattened shape of falling raindrops. Larger drops do also oscillate, and their axes exhibit tilt angles with respect to the vertical. In this paper, I will discuss two rare rainbow phenomena that are influenced by these effects: bright spots belonging to various rainbow orders, but appearing at remarkable angular distances from their traditional locations, as well as triple-split primary rainbows. While the former have not been observed in nature so far, the latter have been documented in a few photographs. This paper presents simulations based on natural drop size distributions using both a geometric optical model, as well as numerically calculated Möbius shifts applied to Debye series data.

Rainbows, supernumerary rainbows and interference effects in the angular scattering of chemical reactions: effect of varying the modulus of the S matrix in the context of Heisenberg’s S matrix programme

We investigate the existence of primary rainbows, supernumerary rainbows and diffraction interference effects in the product differential cross sections (DCSs) of state-to-state chemical reactions. The rainbows can be ‘pronounced’ or ‘hidden’. Our theoretical approach uses a ‘weak’ version of Heisenberg’s scattering matrix programme (wHSMP) introduced by Shan and Connor 2011 Phys. Chem. Chem. Phys. 13 8392. This wHSMP uses four general physical principles for chemical reactions to suggest simple parametrised forms for the S matrix; it does not employ a potential energy surface. We use a realistic parametrization in which the modulus of the S matrix is the sum of a smooth-step function and a gaussian function; both are functions of the total angular momentum quantum number, J. We then vary the parameters in the modulus. The phase of the S matrix is a cubic polynomial in J, which is held fixed. We demonstrate for a Legendre partial wave series (PWS) the existence of primary rainbows and supernumerary rainbows (both pronounced and hidden) as well as diffraction interference effects, in reactive DCSs. We find that reactive rainbows can be complicated in their structure. We also analyse for five examples, the angular scattering using nearside–farside (NF) PWS theory, including resummations of the PWS. In addition, we apply full and NF asymptotic (semiclassical) rainbow theories to the PWS—in particular, the uniform Airy and transitional Airy approximations for the farside scattering. This lets us prove that structures in the DCSs are indeed primary rainbows, supernumerary rainbows as well as diffraction interference effects. Our calculations complement and extend those in an earlier paper by Shan et al 2018 Phys. Chem. Chem. Phys. 20 819, in which the modulus of the S matrix is held fixed, whilst the phase is varied.

Numerical analysis of primary rainbows from a homogeneous cylinder and an optical fiber for incident low-coherent light

This work provides a numerical study of the scattering of low-coherent light by an infinite right circular cylinder and various types of optical fiber (with step- and graded-index profiles) in the vicinity of primary rainbows, caused by light that has been subjected to one internal reflection. The scattered intensity is analyzed in terms of the Fourier transform as well as in the time domain (by examining the impulse response of a fiber) with the aim to obtain a detailed information about the scattering process. The analysis reveals a wealth of information about the scattering process that is not obvious when a fiber is illuminated by a temporally coherent light source. The results also provide an idea for the characterization of the core size of step-index optical fibers.

Quantum primary rainbows in transmission of positrons through very short carbon nanotubes

This paper is devoted to a quantum mechanical consideration of the transmission of positrons of a kinetic energy of 1MeV through very short (11, 9) single-wall chiral carbon nanotubes. The nanotube lengths are between 50 and 320nm. The transmission process is determined by the rainbow effects. The interaction potential of a positron and the nanotube is deduced from the Molire’s interaction potential of the positron and a nanotube atom using the continuum approximation. We solve numerically the time-dependent Schrödinger equation, and calculate the spatial and angular distributions of transmitted positrons. The initial positron beam is assumed to be an ensemble of non-interacting Gaussian wave packets. We generate the spatial and angular distributions using the computer simulation method. The examination is focused on the spatial and angular primary rainbows. It begins with an analysis of the corresponding classical rainbows, and continues with a detailed investigation of the amplitudes and phases of the wave functions of transmitted positrons. These analyses enable one to identify the principal and supernumerary primary rainbows appearing in the spatial and angular distributions. They also result in a detailed explanation of the way of their generation, which includes the effects of wrinkling of each wave packet during its deflection from the nanotube wall, and of its concentration just before a virtual barrier lying close to the corresponding classical rainbow. The wrinkling of the wave packets occurs due to their internal focusing. In addition, the wave packets wrinkle in a mutually coordinated way. This explanation may induce new theoretical and experimental investigations of quantum rainbows occurring in various atomic collision processes.

Polarization-resolved simulations of multiple-order rainbows using realistic raindrop shapes

This paper presents selected results of a simulation study of the first five (primary–quinary) rainbow orders based on a realistic, size-dependent shape model for falling raindrops, taking into account that the drops’ bottom part is flattened to higher degree than the dome-like top part. Moreover, broad drop size distributions are included in the simulations, as it is one goal of this paper to analyze, whether the predicted amplification and attenuation patterns for higher-order rainbows, as derived from previous simulations with monodisperse drop sizes, will still be pronounced under the conditions of natural rainfall. Secondly, deviations of the multiple rainbow orders’ polarization state from the reference case of spherical drops are discussed. It is shown that each rainbow order may contain a small amount of circularly polarized light due to total internal reflections. Thirdly, it is investigated, how the conditions that generate twinned primary rainbows will affect the higher orders. For the simulations, geometric-optic ray tracing of the full Stokes vector as well as an approximate approach using appropriately shifted Debye series data is applied.

Twin primary rainbows scattered by micro-channel with circular cross-section

基于几何光学理论及Debye理论，研究圆形截面微通道这一双层圆柱产生的双一阶彩虹现象。由于全反射现象，双一阶彩虹并非始终存在，故通过数值模拟，得出内外径比的临界值来判断双一阶彩虹的存在性。当双一阶彩虹存在时，双一阶彩虹中的彩虹总是可被观察到；彩虹在某些情况下会因光强较弱而被淹没在以二阶彩虹为主的其他散射结构中。研究发现彩虹位于二阶彩虹主峰右侧时，其可被观测到。为此分析彩虹与二阶彩虹两者主峰散射角重叠这一临界情况，进而提出判断彩虹能否被被观测到的方法。最后，构建实验系统，以充满去离子水的高硼硅玻璃毛细管为对象进行实验研究，通过CCD相机拍摄彩虹图像。结果表明，毛细管内径为550 m、外径为600 m时，其内外径比大于上述临界值0.7485，故双一阶彩虹存在。但是由上述判断方法得出彩虹无法被观测到，这一结论与仅能从拍摄图像中获得彩虹结构的现象相符。

双层圆柱的双一阶彩虹频谱结构

双层圆柱的双一阶彩虹频谱结构

Application of vector ray tracing to the computation of Möbius shifts for the primary and secondary rainbows.

The Möbius approximation for the primary rainbow and the Können approximation for the secondary rainbow have been modified to yield consistent predictions of the Möbius shift of the top and bottom rainbows, respectively. The applicability ranges of the Möbius and Können approximations are investigated by comparison to vector ray tracing (VRT) simulations. For the primary rainbow, these results indicate that the Möbius approximation is valid for spheroidal water droplets (m=1.333) in the range of aspect ratios 0.98≤a/c≤1.02. For the secondary rainbow, the Können approximation predicts the Möbius shift well for spheroidal water droplets within the range 0.99≤a/c≤1.01. For a spheroidal droplet with side-on incidence, the difference between the approximations and VRT simulations are discussed. Furthermore, the dependence of Möbius shifts on the relative refractive index of droplet is discussed.

Photographic observation of a natural fifth-order rainbow.

A photograph has been obtained of a natural fifth-order (quinary) rainbow. The photograph was acquired on 8 August 2012 with a digital camera and a polarization filter to maximize contrast of the rainbows with the background. The quinary rainbow, together with its first supernumerary, appears in a contrast-enhanced version of the photograph as broad green and blue-violet color bands within Alexander's dark band between the primary and secondary rainbows. The red band of the quinary rainbow is obscured by the much brighter secondary rainbow. A comparison with a numerical simulation using the Debye series confirms that the color bands of the quinary rainbow appear at the expected location. The numerical simulation produces a good match with the photograph for a droplet radius of 0.46 mm. The green band of the quinary rainbow is even faintly discernible in the unprocessed photograph, suggesting that under exceptional viewing conditions the green band of the quinary rainbow may be observed visually with the aid of a polarization filter.

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.

Primary and secondary rainbows in the late evening at West Kirby on the Wirral peninsula (Merseyside)

WeatherVolume 68, Issue 9 p. E1-E1 Inside cover photograph Primary and secondary rainbows in the late evening at West Kirby on the Wirral peninsula (Merseyside) First published: 27 August 2013 https://doi.org/10.1002/wea.2153Read the full textAboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat No abstract is available for this article. Volume68, Issue9September 2013Pages E1-E1 RelatedInformation

Primary and secondary rainbows at the old Davidstow Airfield on Bodmin Moor (Cornwall)

WeatherVolume 68, Issue 9 p. E2-E2 Inside Cover Photograph Primary and secondary rainbows at the old Davidstow Airfield on Bodmin Moor (Cornwall) First published: 27 August 2013 https://doi.org/10.1002/wea.2231Read the full textAboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat No abstract is available for this article. Volume68, Issue9September 2013Pages E2-E2 RelatedInformation

Origin of ringlike angular distributions observed in rainbow channeling in ultrathin crystals

Using the theory of crystal rainbows, we prove that the well-known doughnut patterns observed in ion channeling in thin crystal membranes are manifestations of the rainbow effect. This is done by a detailed morphological study of the high-resolution experimental angular distributions of 2 MeV protons channeled in a 55-nm-thick (001) silicon crystal tilted away from the [001] direction. The inner side of the doughnut is the dark side of the rainbow, analogous to the Alexander's dark band, occurring between the primary and secondary meteorological rainbows.

Photographic evidence for the third-order rainbow

The first likely photographic observation of the tertiary rainbow caused by sunlight in the open air is reported and analyzed. Whereas primary and secondary rainbows are rather common and easily seen phenomena in atmospheric optics, the tertiary rainbow appears in the sunward side of the sky and is thus largely masked by forward scattered light. Up to now, only a few visual reports and no reliable photographs of the tertiary rainbow are known. Evidence of a third-order rainbow has been obtained by using image processing techniques on a digital photograph that contains no obvious indication of such a rainbow. To rule out any misinterpretation of artifacts, we carefully calibrated the image in order to compare the observed bow's angular position and dispersion with those predicted by theory.

Debye series for light scattering by a coated nonspherical particle

By using the extended boundary condition method, the Debye series is developed for light scattered by a coated nonspherical particle in order to interpret the angular dependence of the scattered intensity in terms of various physical processes. Numerical calculations are performed to study the influence of the coating thickness and the ellipticity of a coated spheroid on the angular position of the {alpha} and {beta} primary rainbows, which are produced by partial waves experiencing one internal reflection. The hyperbolic umbilic focal section is demonstrated and is analyzed for both the {alpha} and the {beta} rainbows.

Rainbows from inhomogeneous transparent spheres: a ray-theoretic approach

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.