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.
© 2014 Optical Society of America
Prior to 2011, observations of higher-order rainbows were limited to a handful of anecdotal reports of visual observations of the tertiary rainbow (see  for an overview of existing reports in the literature). The first photograph ever of a natural tertiary rainbow was obtained by Großmann et al. on 15 May 2011 , followed one month later by the imaging of the quaternary rainbow by Theusner . Both of these observations relied on image processing techniques that have become possible with recent technological advancements made in consumer digital cameras.
With the first four orders of natural rainbows documented photographically, the next most likely candidate rainbow to be discovered in nature was the fifth-order (quinary) rainbow. The quinary rainbow is predicted to be partly situated in Alexander’s dark band, which is the dark region between the primary and secondary bows. Alexander’s dark band is associated with minimal scattered intensity due to the absence of the zero-order glow and scattered light of the primary and secondary rainbows. The quinary rainbow is therefore much more favorably located in the sky than both the tertiary and quaternary rainbow, and for this reason Alexander’s dark band has been referred to as a “window of opportunity” . However, many years of searching for the quinary bow by rainbow researchers and enthusiasts alike have produced negative results, suggesting that the quinary bow either would be of too low color contrast to be detectable in the atmosphere or is exceedingly rare for other reasons, such as droplet flattening that can interfere with scattering, or an unfavorable droplet size distribution in rainfall.
In this paper a photograph is presented that shows the green and blue color bands of a natural quinary rainbow in Alexander’s dark band next to the secondary rainbow. The red band of the quinary rainbow is not visible in the photograph as it is obscured by the much brighter secondary rainbow. The observation of the quinary rainbow was made at near-optimal atmospheric conditions at an altitude of 3240 m above mean sea level (MSL). The photograph is compared with numerical scattering theory using the Debye series, confirming that the observed color bands of the quinary rainbow occur at the expected location within Alexander’s dark band.
2. Imaging Equipment and Image Processing
The photograph was obtained with a Nikon D700 digital single-lens reflex (DSLR) camera, which has a CMOS sensor measuring 36.0 mm by 23.9 mm in size and 14-bit analog-to-digital conversion. The camera was fitted with a Nikon AF-S Nikkor 24–70 mm 1∶2.8G ED zoom lens set at a focal length and an aperture of . The lens was fitted with a Hoya 77 mm “circular polarizer” filter, which is a linear polarizer in combination with a quarter-wave retarder (the latter located between the linear polarizer and the lens). The linear polarization filter was oriented with its axis of polarization perpendicular to the scattering plane (defined here by the sun, camera, and center of the field of view). The exposure time was 1/125 s at ISO 200 sensitivity.
Adobe Lightroom 4 was used to convert the camera raw file format (NEF) to 16-bit TIFF raster images and for image processing. Image enhancements consisted of a white balance correction followed by adjustments of contrast and brightness. For the purpose of image calibration a lens correction was applied (using a built-in profile in Adobe Lightroom specific to the lens used) to remove lens distortion effects. Both the camera and the image processing software used the sRGB color space.
From photographs of other events that occurred at accurately known times it was determined that the internal clock of the camera was ahead of actual time by at the time of the observation.
The quinary rainbow appeared in the afternoon of 8 August 2012 between 23:48 and 00:01 UTC (17:48 and 18:01 local time) at the Langmuir Laboratory for Atmospheric Research in south-central New Mexico, which is situated in the Magdalena Mountains at longitude and latitude, and at 3240 m altitude MSL. It was photographed in precipitation of a weakening thunderstorm that was a few kilometers to the east of the observatory. Light rain was also falling at the observatory. At 23:48:23 UTC, the time of the photograph shown in Fig. 1, the sun was at an azimuth of 271.9° and altitude of 26.4° above the astronomical horizon (about 27.4° above the true horizon).
Over 100 photographs were obtained during the 13-min time interval between 23:48 and 00:01 UTC. The photograph in Fig. 1, taken at 23:48:23 UTC, was the fourth of the series. Many of the other photographs were acquired using a 16 mm full-frame fisheye lens and other wide-angle lenses without a polarization filter. The quinary bow appears with varying degrees of intensity in the region of interest in all of the contrast-enhanced photographs that were taken with a polarization filter and at different focal lengths between 24 and 70 mm, but appears best in the first seven photographs of the series (spanning about 2 min in time). Though weaker, the green band of the quinary bow is also perceptible in contrast-enhanced photographs taken without a polarization filter, including the photographs that were taken with the 16 mm fisheye lens. As the presence of the quinary rainbow was neither noticed nor expected at the time, no series of identical images was attempted for image-stacking purposes. During the 13-min time interval the primary and secondary rainbows were complete over a circular section (clock angle) of about 220°; the lower parts of both rainbows were visible well below the horizon in a deep 2 km wide canyon. However, the background of scree and vegetation prevents positive identification of the quinary bow in this area of the photographs.
Figure 1 shows the original, unprocessed photograph of 23:48:23 UTC and a contrast-enhanced version. The unprocessed photograph was only corrected for white balance, as the camera was set to a manual color temperature that was incorrect for daytime photography. The quinary bow is visible in the contrast-enhanced image as a broad green color band to the right of the secondary rainbow, bordered by a blue-violet band approximately at the center of Alexander’s dark band. Between the green band of the quinary rainbow and the onset of the red band of the secondary rainbow, hints of the quinary rainbow’s red band appear to be visible as a reddish hue of low intensity. After rigorous image processing (Sections 5–7) the first supernumerary of the quinary rainbow is also detectable. The quinary rainbow appears over the range in clock angle from the nearby mountain ridge to the top of the frame. (It is visible up to a clock angle of in another photograph, which is not reproduced here.) With scrutiny, the green band of the quinary rainbow is perceptible in the unprocessed image in Fig. 1. In the contrast-enhanced image, three supernumerary bows of the primary rainbow can be discerned and one supernumerary bow of the secondary rainbow.
4. Image Calibration
The photograph was calibrated using global positioning system (GPS) coordinates of two major landmarks, from which the azimuth and elevation angles from the observation location to both landmarks could be determined. Because one of the two landmarks was outside the field of view of the image, five additional landmarks (trees and shrubs in the image foreground for which no GPS coordinates could be obtained) were identified by measuring angular distances between all landmarks with a sextant. From these measurements the azimuth and elevation angles to the five additional landmarks could also be determined.
The azimuth and elevation angles were converted to pixel coordinates and plotted as dots in an image of the same pixel dimensions as the photograph. The conversion program assumed an ideal optical “pinhole” imaging system without lens distortions (but including geometrical distortion due to imaging a three-dimensional scene on a focal plane). For the conversion a set of input parameters was estimated for the lens focal length and the spatial resolution of the camera sensor (neither of which was accurately known), the azimuth and elevation angles corresponding to the center of the image, and rotation of the image along the optical axis of the camera. The output image was then overlaid on the digital photograph and the input parameters were adjusted until the dots overlapped with the landmarks as best as feasible. The estimated accuracy of the calibration is in scattering angle and clock angle. Relevant calibration parameters are listed in Table 1.
The image calibration yielded conversion formulae to convert coordinates between the three reference frames of azimuth and elevation, scattering angle and clock angle , and pixel coordinates in the image. Figure 2 shows the scattering coordinate frame and calibration landmarks overlaid on the contrast-enhanced image.
5. Noise Reduction
For comparison with numerical simulations, a pixel-averaging algorithm was applied to the photograph to produce a one-dimensional output image with much less image noise. This was done by averaging the RGB color values of all pixels in an averaging window that moved within a horizontal channel through the photograph. The channel was bounded by pixel rows 930 and 1169 in the photograph (of total height of 2832 rows), corresponding to an area immediately above the true horizon and below the distant cloud base with a minimum of interfering features in the sky and foreground (Fig. 2).
Although the averaging window moved horizontally, the averaging itself was done over a constant range per averaging step. The window was bounded by a horizontal top and bottom and two curves of constant scattering angle at its sides, with each pixel row of the window representing the same width and same center value , and thus had the approximate shape of a parallelogram (Fig. 2). Under these constraints the shape of the averaging window is variable while moving through the image. The step size in was 0.01° in an interval with one output pixel per step size, yielding an output image of 2001 pixels wide. To facilitate visual inspection of the output image, the single output row was stretched vertically to yield an image of 400 pixels in height.
As it moved across the image, the averaging window traversed an interval in clock angle of . The polarization filter in front of the camera was oriented such that the scattering plane along an estimated clock angle between and corresponds to polarization perpendicular to the scattering plane. The interval traversed by the averaging window therefore corresponds to clock angles within of perpendicular polarization, and within over the subinterval of that includes the region of the quinary rainbow in Alexander’s dark band.
6. Numerical Simulation Using Debye Series
Numerical ray-tracing simulations for oblate-spheroidal raindrops at a solar elevation of 26° suggest that the quinary rainbow is significantly reduced in intensity (compared to its intensity from spherical raindrops) within the upper two quadrants of clock angle () as raindrops become larger and more flattened . In the photographs the quinary rainbow was discernible up to a clock angle of about , suggesting that raindrops were not significantly flattened and must therefore have been relatively small (less than diameter [6,7]). The appearance of the supernumerary of the secondary rainbow in the photograph indicates that large drops were indeed absent in the rainfall, as its appearance is indicative of a peaked rather than a broad drop size distribution .
To investigate which droplet size may have been responsible for the quinary rainbow in this observation and to compare the angular position and width of the bow with its predicted appearance, a numerical simulation of light scattering by spherical drops using the Debye series was carried out [9–12]. The Debye series expansion produces the scattered intensities for individual orders of rainbows, allowing individual rainbow orders to be studied. The order corresponds to external reflection, to the zero-order glow, to the primary rainbow, and onward. Being able to isolate and study individual orders is particularly advantageous for this observation, as the quinary rainbow () is partly overlapped by the secondary rainbow ().
Using the Debye series expansion, scattered intensities of orders through were calculated for both components of polarized light over a range of values of droplet radius and wavelength . Droplet radius ranged from 0.05 to 1.00 mm in steps of 5 μm and ranged from 380 to 700 nm in steps of 20 nm. The refractive index for air was modeled after , adjusted to an air pressure of 700 hPa at 15°C, and the complex refractive index for water as a function of at a pressure of 700 hPa and an estimated temperature of 5°C was modeled after . Scattered intensities at selected values for and were checked for accuracy by comparison with the MiePlot program by Laven [15,16].
The resulting datasets were used by another computer program to produce images in 16-bit TIFF format of rainbows of order at specified optical imaging parameters, which were identical to the photographic calibration values. A linear polarization filter was simulated with its axis oriented along . The images were produced by assuming a spectrum of sunlight as measured at ground level by Lee . The Bruton color model  was used to convert from to RGB color values, as it produces simulations that visually closely resemble the photograph, and to be consistent with MiePlot. As the spectral response of the camera is not published by the manufacturer, the analysis also relies on the color conversion to sRGB by the camera being accurate. A circular light source of 0.5° diameter was assumed to account for the angular size of the sun. The resulting images could then be directly compared to the photographs, and used for further study by extracting one-dimensional images and corresponding RGB color curves in a similar way as the noise reduction process for the photograph.
7. Comparison with Numerical Simulation
Figure 3 compares the contrast-enhanced photograph with the Debye-series simulations for the contributions of (external reflection), (primary bow), (secondary bow), and (quinary bow). All orders were calculated at , except the order , which was calculated at , being the effective radius of drops producing that rainbow as determined from the photograph (see the discussion below). As and scattering are the only significant contributions to the brightness of Alexander’s dark band, they were also added together separately in the figure for comparison.
RGB light curves were extracted from the averaged, unprocessed photograph and compared with those of simulations at various droplet sizes. Figure 4 shows three intensity light curves for , , and representing the red, green, and blue color channels. To generate the light curves, the three functions , , and from the averaged photograph were transformed to increase contrast and to eliminate a gradual decrease in intensity at increasing . The intensity gradient probably resulted from the change in sky polarization across the image in combination with the polarization filter, and affected the blue channel more than the red and green channels. The linear transformation applied to the red channel was and likewise for the green and blue channels, where and . Using this transform, the photographic light curves are intentionally somewhat displaced vertically (in intensity) to those of the simulations, facilitating visual comparison of the intensity variations due to the quinary bow.
There are a number of unknowns that may have affected the scattered intensities of the rainbows in the photograph, such as the drop size distribution, droplet flattening, Rayleigh scattering by air molecules, and an inhomogeneous image background, which the simulations do not take into account. Therefore, meaningful comparisons between the photograph and simulations can only be made by comparing the locations of maxima in the light curves; not the relative differences in color intensity between bows.
One way to estimate the dominant droplet size in the rainfall from a photograph is by looking at the spacing between supernumeraries of the primary bow, which frequently appear together in nature. This is an unreliable method, however, because in a broad drop size distribution the supernumerary bows of the primary rainbow are selectively produced by raindrops of effective radius , at least near the top regions of the bow. This results in an underestimation of drop sizes. On the other hand, the drop size selection bias has almost no effect on the positions of the main maximum and supernumeraries of the secondary bow . Therefore, by determining the effective radius of drops producing the secondary bow, an estimate can be obtained that is representative for the actual drop size in the rainfall.
In the photograph multiple supernumeraries of the primary rainbow are visible. However, only the first of these is represented by a well-defined maximum that can be accurately located in scattering angle, and only in the red intensity curve; the other supernumeraries appear as shoulders in the curves. For this reason, instead of inferring the effective drop radius by measuring the spacing between supernumeraries , it was inferred by measuring the angular separation between the main intensity maximum of the primary bow and its first supernumerary. The supernumerary of the secondary bow (the only one that is perceptible) also shows a maximum in intensity, but also only in the red intensity curve, and therefore its corresponding drop radius could be inferred in the same way.
For the primary bow, the angular separation between supernumerary and main maximum is , which is matched by the numerical simulation at a droplet radius . Both peak intensities in the photograph have an offset in scattering angle of relative to those in the simulations, likely resulting from a small error in image calibration that lies well within the estimated calibration accuracy of 0.16°. Another possible cause for the offset is a broadened main maximum and a shift in the supernumerary bow’s position due to the drop size selection bias acting in a non-monodisperse drop size distribution.
For the secondary bow, an angular separation was found between the main maximum and that of the first supernumerary, indicating a droplet radius . The peak intensities in this area of the photograph are both offset in scattering angle by compared to the simulations. In determining the angular separation, contributions from both the and orders were taken into account, as the maximum of the red band of the secondary bow is slightly displaced into Alexander’s dark band (by about ) due to the overlap with the quinary bow.
The offsets in near the primary and secondary bows in the photograph are likely due to the inaccuracy in image calibration. Since the offsets in are opposite in sign, measurements in the region between both bows (i.e., near the quinary bow in Alexander’s dark band) are expected to be somewhat less affected by an offset. The error in due to the inaccuracy in image calibration is estimated to be in the general area of the quinary bow.
The quinary bow was accompanied by one supernumerary, and an attempt was made to measure its angular separation from the main maximum in the green channel. However, owing to its very low intensity in the photograph, the supernumerary has a noisy and broad peak that could not be very accurately located. Instead, estimates for droplet radii were obtained by measuring the scattering angles corresponding to various intensity maxima. The green band of the quinary bow is most conspicuous in the photograph, with a well-defined peak in the light curve, and gives the highest confidence for a match between photograph and simulations. Its maximum in the photograph occurs at , corresponding to , in good agreement with the value for inferred from the separation between the red intensity maxima of the secondary bow and its supernumerary.
Table 2 summarizes the various drop radii inferred from these measurements, augmented by drop radii inferred from scattering angles of the main maxima of the primary and secondary rainbows. The measured scattering angles of intensity maxima are sensitive to errors in image calibration, which negatively affect the accuracy of the estimates for the drop radii. On the other hand, the measurements of the angular separation between the main maxima and supernumeraries are insensitive to small errors in calibration, and yield the most accurate estimates for drop radii.
The value inferred from the primary bow is much smaller than the values inferred from the secondary and quinary bows. This indicates that the drop size selection bias  was effective in this case, and therefore this drop size estimate cannot be considered as a realistic estimate of the typical drop size in the rainfall. Instead, the physical meaning of the 0.275 mm value is that it represents the effective radius of the drops that produced the primary rainbow. This implies that for a realistic simulation of the rainbows, a radius for a spherical drop has to be invoked for the primary rainbow and for the secondary and quinary rainbow, as well as for the background brightness of Alexander’s dark band (). Such was done in the simulations of Fig. 3 and thereafter.
The estimates for drop radius inferred from measurements of the secondary and quinary bows are in general agreement with one another, and the radius , obtained at high accuracy from the angular separation between the secondary bow and its first supernumerary, can be regarded as a value that is representative for the typical size of the drops in the rainfall. We feel that the match in droplet radius between the secondary and the quinary bows further strengthens the case for the quinary rainbow.
In summary, it can be concluded that the quinary bow’s intensity was mainly contributed to by droplets of radius in a peaked drop size distribution. The variations between the light curves of the photograph and those of the simulations can be attributed to Rayleigh scattering, brightness variations in the image background, accuracy of the image calibration, accuracy of color conversion, residual image noise, and the droplet size distribution in the rainfall.
Table 3 lists the measured and predicted scattering angles at intensity maxima of the secondary and quinary bows for a droplet radius of 0.46 mm. Figure 5 shows an image composite of the simulated bows overlaid on the photograph in the same coordinate frame. As in Figs. 3 and 4, a droplet radius was assumed in the simulation for the primary bow, and for the secondary and quinary bows as well as for scattering. A background hue was added to the simulation image to more closely resemble the photograph. Visually, the match between simulations and photograph is good, reproducing the quinary bow with approximately the same appearance in the same location as seen in the photograph.
8. Summary and Conclusion
In comparing the numerical Debye-series simulations with the photograph, it can be concluded that the green and blue-violet color hues in Alexander’s dark band in the photograph are part of the quinary rainbow. To our knowledge, Fig. 1 represents the first time that this rainbow has been recognized in nature. The rainbow is surprisingly conspicuous in a single photograph after only basic contrast enhancements.
The presence of a supernumerary of the secondary bow indicates a peaked drop size distribution, peaking at an effective drop radius of 0.46 mm. This value also yields a good match between the photograph and simulations for the observed angular position of the quinary bow and its first supernumerary. Droplets of 0.46 mm radius are flattened by about 1% . The peaked drop size distribution in a regime where droplets are small and close to spherical in shape, which becomes more important for the visibility of higher-order rainbows, likely was a key factor that caused the quinary bow to appear.
The green band of the quinary bow is also faintly discernible in the original photograph when viewed on a high-quality computer display. This suggests that under optimal viewing conditions the quinary bow’s green band may be visible with the naked eye when looking through a polarization filter. However, the contribution from external reflection in Alexander’s dark band is polarized, as is the quinary bow, and the use of a polarization filter does not help as much in enhancing the contrast of the quinary bow with its background as it does for the tertiary and quaternary bows . It seems therefore not surprising that the quinary bow’s green band is also visible in other contrast-enhanced photographs of 8 August 2012 that were taken without a polarization filter. This suggests that to an astute observer the green band of the quinary bow may even become visible with the naked eye without the use of a polarization filter.
One important question is what atmospheric circumstances were conducive to the visibility of the quinary bow in this observation, in particular since it is relatively conspicuous in the contrast-enhanced photograph. In an attempt to answer this question, the author undertook a thorough search for other appearances of the quinary bow in 2013, yielding four additional photographic observations, some of which also included the tertiary and quaternary bows. In addition, five earlier appearances of the quinary bow were discovered in the author’s collection of digital rainbow photographs that spans the years 2008–2013. Two of these observations are from 2009 and three from 2010. All 10 observations were made in New Mexico: seven at Langmuir Laboratory, and three on the desert plains between 1.8 and 2.0 km altitude MSL (Table 4). The solar elevation in these observations ranges from 1° to 34°. Out of the 10 observations, the observation of 8 August 2012 discussed here shows the quinary rainbow most clearly and at the highest intensity, with a conspicuous blue-violet color band. For illustration, Fig. 6 shows the second-oldest observation found in the author’s digital collection.
Photographs from all 10 cases are reproduced in . The quinary rainbow may not be that rare, at least not in this region, and relatively easy to image with a DSLR camera of high dynamic range. An explanation for this is that in the desert region of the southwestern United States, small, high-based thunderstorms often form by orographic convection, producing rainfall in which rainbows can be very bright. Moreover, the atmosphere in this region tends to be exceptionally clear, containing low amounts of aerosols. In addition, the atmosphere is of lower density at higher altitude. Clean air of low density causes less scattering by aerosols and the air itself; such scattering reduces the already low color contrast of faint phenomena such as the quinary rainbow and may render it undetectable.
The author is indebted to Gunther Können, Les Cowley, Philip Laven, and Raymond Lee for their helpful comments and guidance. Major support for Langmuir Laboratory comes from the Irving and Marion Langmuir family, the Geophysical Research Center of the New Mexico Institute of Mining and Technology, and the United States Forest Service.
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