Author Archives: alexanderhaussmann
Taking photographs of tertiary and quaternary rainbows is difficult, as usually you won’t see what you are aiming at – theory predicts that the tertiary is just at the threshold of visibility , and for the quaternary the situation is even worse. I myself have been lucky only once before , (though having tried for almost a decade now), if one does not count experiments using artificial sprays . This cannot be attributed to a lack of opportunities. Over the years, I experienced several promising situations, but afterwards no higher order rainbows could be extracted from the photographs by image processing. One problem is that cloud structures in the background mess up the unsharp mask filter. But maybe also my timing was just wrong and these rainbows did not appear when I expect them to do, because I misjudged the shower and illumination geometry.
Anyway, on Aug 11th, 2020, my instincts were right. Just before I finished work in Dresden-Langebrück, a moderate shower moved in from the northeast. The sunlight was somewhat dim, which resulted in an unspectacular primary rainbow from 17:15 CEST onwards. I could see some larger drops glint in the sunlight. I went to my car on the parking lot during the next minutes, and lost sight to the east, so I do not exactly know what happened later on this side of the sky. The rain was still ongoing, but not heavy, there were people cycling around without looking too much disturbed. I’m not entirely sure if it was still the same rain shower, or another one which had meanwhile moved in. I took three photos at the parking lot (17:23-24 CEST), and then drove about 100 m to a spot with a good view towards the west. There was nothing to be seen with the naked eye, just the sun, some glinting raindrops, a cloud, and blue sky below. A major problem is that raindrops will fall on the front lens, especially when using a fisheye objective that cannot be shielded and has to be held out of the car window to make proper use of its field of view. So one needs to be fast, otherwise the images will be spoiled by artifacts from the drops. That is why I did not do any image stacks, just a threefold exposure bracket per shot for safety (to have at least one picture with a useful exposure). In between, I had to wipe the lens dry. All of the ten shots I took there from 17:26-30 showed the tertiary rainbow after some processing, and it also appeared on the earlier pictures from the parking lot, as far as the view permitted. I was rather overwhelmed to see more than the complete upper half of the tertiary against the blue sky and cloud background when I first applied an unsharp mask to the images. The quaternary rainbow (located just outside the tertiary) can also be detected beyond any doubts, especially on the left side. I found no supernumerary arcs and no traces of the seventh order  (which I had missed to pay attention to during the observation, I also did not use any polarizers, and as mentioned, I did not record image stacks).
It is well known that aerodynamic flattening of larger raindrops has an impact on rainbows through the so-called Möbius shifts, but so far the consequences for higher order rainbows have only been studied theoretically. Are there any new insights from this observation? This, of course, requires an image calibration, i.e. the assignment of scattering coordinates (scattering angle and clock angle, i.e. the sun-centered azimuth, which I will count clockwise from the rainbow’s top here) to the individual pixels. The position of the sun is easily calculated from the time the respective photograph was taken (17:27:54 CEST for the top image, checked with a radio-controlled watch) and the location (51.13° N, 13.83° E), which gives in this case an elevation of 27.6° and an azimuth of 259.5°. The pixel coordinates of the sun’s center could also be reliably determined from an image version developed as dark as possible from the RAW file in Photoshop. The projection of this specific lens I had measured eight years ago (when I newly got it), but I did also a cross-check with recent starfield pictures (abundantly available from my attempts to catch Perseid meteors the following nights). In order to determine the relevant Euler angles (elevation and azimuth of the camera’s optical axis, and the rotation of the camera sensor chip around this axis), I still needed another reference mark. Luckily there was a telecommunication tower at the horizon which I could identify and then calculate under which elevation and azimuth it is seen from the observing location.
Technical sidenote: The two coordinates of the reference mark provide me indeed with one condition more than the number of available degrees of freedom, so I can check the overall consistency of the calibration. And here it initially turned out to be not very convincing. What went wrong? When checking my hidden assumptions I found that I had pinned the piercing point of the optical axis to the precise center of the CMOS sensor (in terms of pixel coordinates). This may not be realistic, and, moreover, in a Pentax K-5 camera the sensor can move several millimeters to compensate for shaking. Even with the shaking compensation turned off there is no guarantee that it will find and stay in the optical center position (plus, there are decentering errors of the lens). From the working principle of the calibration procedure I expect the decentering error to be of quadratic order, and it may turn out to be negligible for longer focal lengths. But it matters for a fisheye lens. So I used the amounts of decentering in X and Y as further degrees of freedom and achieved consistent results for a shift of 26 pixels (0.12 mm) in the horizontal (and zero in the vertical). However, a further reference mark would be needed for a truly unique determination. Then there would still remain the assumption of a rotationally symmetric lens, but this seems to be acceptable as indicated by the recent starfield test.
From an equilateral projection in scattering coordinates it can then be deduced that the tertiary does not bend significantly over the recorded range of clock angles, which also holds for the quaternary as far as it peaks out of the noise. So seemingly they appeared as perfectly concentric circles here!
This is somewhat surprising, as theory predicts that these rainbows are also subject to Möbius shifts of various amounts along their circumference, which should become noticeable if larger (more distorted) drops are involved. Interestingly, for this sun elevation the Möbius shifts will move the tops of the 3rd and 4th order rainbows towards each other. They might even overlap for an effective drop radius of 0.5 mm . However, in nature, in most cases the drop size distribution (DSD) will cover a broad range of sizes, also including small drops. Because the shape distortion sets in (at least) quadratically with rising drop radius, it is likely to see some of the traditional concentric sphere-drop rainbows shine through in the full mixture of size dependent rainbows. This I already noted in simulations using broad Marshall-Palmer DSDs (i.e. a simple decaying exponential). As mentioned, there was no heavy rainfall going on during the observation on Aug 11th, so a dominance of the less distorted smaller sizes can be reasonably assumed – regrettably there are no direct measurements of the DSD. Lee and Laven  argue that broad DSDs tend to wipe out the tertiary’s top and leave only the sides, but their analysis was based on much lower sun elevations than occurring here.
So in order to see how much the concentricity of the tertiary and quaternary rainbows will be affected in this specific case I did some simulations for the proper sun elevation and a (guessed) DSD which contains mostly small and moderate sized drops: A Marshall-Palmer with decaying parameter (Lambda) of 4 mm-1, as previously used . There are two complementary simulation methods which I can apply: 1) GO: Geometric optic raytracing (including polarization, but neglecting interference and diffraction) for all rainbow orders up to the 7th, based on a Beard-Chuang cosine series drop shape model, with optional (2,0) quadrupole mode oscillations and Gaussian tilts of the symmetry axis from the vertical, and 2) DMK: Debye series calculations for spherical drops of various sizes, superimposed in intensity after being shifted in scattering angle by the appropriate Möbius value (depending on drop size, rainbow order, and clock angle, following Können ). These calculation include only rainbow orders up to the 5th. The Möbius shifts themselves are taken from a look-up table comprising earlier raytracing results. These were calculated from a simpler shape model (two conjoined half spheroids fitted to Beard-Chuang shapes) and do not include drop oscillations or tilts for the higher-order rainbows yet. However, this second method has the advantage of showing if supernumerary arcs can be expected under the given conditions.
I removed the most disturbing directly transmitted light (sometimes referred to as “zero order glow”) as well as the less important contributions from external reflection and the lowest two rainbow orders from the simulation, and show the resulting clear higher-order rainbows in the same sunward projection (and for the same sun elevation) as in the top image. As a reference, I also let simulations run for spherical drops with the same DSD. These, of course, turn out perfectly concentric (in scattering coordinates, not necessarily in the projected image). After having switched on drop distortions, it is reassuring to see that both methods agree in keeping the upper halves of the tertiary and quaternary well separated and still nearly concentric. However, a tendency to blur these parts can be noted, due to the contribution of larger drops. Two more pieces of information can be extracted here: Introducing moderate axis tilts and (2,0) oscillations (both their amplitude distributions set to the “standard values” used in ) does not lead to visible changes in the result (GO), and supernumeraries do not appear, neither for flattened nor spherical drops (DMK).
The latter result illustrates that in broad DSDs supernumeraries need the stabilizing “Fraser mechanism”  to become visible: If, with increasing drop size, the Möbius shifts grow in the opposite direction than the supernumeraries’ convergence towards the Descartes angle due to their shrinking angular width, there will be a certain critical drop size at which these effects compensate. Because of the resulting position stability against changes of drop radii, the supernumeraries of drops around the critical size will peak out from the unstructured background of superimposed non-aligned supernumeraries of other sizes. Traditionally, this argument is invoked for the primary rainbow (with a critical drop radius of about 0.25 mm for the first supernumerary), but it holds likewise for all other orders . If the Möbius shifts have the wrong sign (as for the tertiary and quaternary bows at the sun elevation of my observation) or are set to zero (as in the sphere reference simulations), there exists no compensation point and the averaging of all supernumeraries results in a more or less uniform intensity gradient.
The GO simulations reproduce also the 7th order rainbow, but, under the assumed conditions, do not predict any amplification effects for it caused by drop distortions or oscillations. In fact, it is not even recognizable in the simulation pictures shown here, but can be extracted by a larger intensity-to-RGB-value scaling factor (or higher gamma value).
In conclusion, the observed concentric tertiary and quaternary rainbows without supernumeraries can be consistently interpreted in the current theoretical framework of broad raindrop size distributions and drop shapes with aerodynamically plausible amounts of flattening and oscillations. Even though shape distortions have a larger influence on higher orders, they do not forbid that traditionally shaped rainbows are formed, if enough small drops are present. Of course, any observations of genuine non-spherical drop effects such as higher order twinned bows are highly welcome as they would allow for a more challenging test of the simulation models.
Twinned rainbows are rare sightings, in the sense that one may see on average only one per year in Central Europe even when paying close attention. Much rarer still, and maybe restricted to regions closer to the equator, are multi-split rainbows. Only few cases have been documented so far [1, 2, 3], though more snapshots can be found on image sharing platforms labeled as “triple rainbow” etc. It is always a very favorable situation if an archivist and analyst like myself can establish direct communication with a skilful observer, who recorded details of a rainbow display that provide some insight beyond the pretty pictures.
In April 2019 I emailed Mr. Ji Yun, who manages a Facebook group dedicated to atmospheric optical phenomena in China, asking about a spectacular photograph of a multi-split rainbow which had been shared there. He kindly relayed my request to Mr. Liu Hai-Cheng, the original observer. Mr. Liu agreed to answer a long list of questions and I also received two sets of photographs from August 12th, 2014, one from his Sony NEX-5C camera (equipped with a Nikon AF 28mm f/2.8 lens) and the other from his cell phone (Coolpad 8720L). The camera clock’s time stamps were calibrated with respect to the actual local time by comparing camera and cell phone pictures, and assuming the cell phone clock to be synchronized over the network. All time data are given here in Chinese standard time (UTC+8h).
Mr. Liu observed this rainbow rarity in the beautiful landscape of the karst mountains near the Yulong bridge (Yangshuo County, Guilin City, Guangxi province, about 400 km northwest of Hong Kong, 24.8° N, 110.4° E) during a boat trip on the Yulong river. He remembers that it was very hot that afternoon. It began to rain before he passed through the tunnel of the bridge (at about 16:50), with some heavier rain lasting for about 25 minutes. There was no lightning, thunder or strong wind.
Judging from the photos, the rainbow appeared at about 17:10 within 30 s or less. Already on the early photographs there are hints of the unusual splitting of the primary:
However, Mr. Liu’s visual impression was that the splitting became prominent only later, after the (seemingly ordinary) primary and secondary bow had appeared successively. He also noted that the visibility of the split branches changed over time, while the main primary could always be seen clearly.
Towards the end of the shower, the display reached its peak quality. The following pictures cover the full right-hand side of the rainbow and some of the left. They are presented without additional filtering to allow for a better assessment of the natural contrast conditions.
For a deeper analysis, I chose the title picture, recorded at 17:18. In the contrast-enhanced version, three primary branches are directly visible, with the most intense one in the center. The secondary rainbow, as far as it is included in the frame, does not exhibit any anomalies. This is a typical feature in (almost) all split rainbow observations known so far. My goal was now to transform the photograph into the scattering angle vs. clock angle coordinate system (in equirectangular projection), as I did on previous occasions [1, 4]. The scattering angle is the angular distance from the sun, and the clock angle the azimuth around the rainbow’s circumference, with the 0° position corresponding to its top.
The sun’s position is easily obtained from standard astronomy software (giving an elevation of 25.4°, and azimuth of 275.4°). Additionally, the precise focal length of the lens (in pixel units) and distortion characteristics need to be known, as well as the camera pointing direction in elevation and azimuth, and the angle describing the rotation of the sensor’s pixel grid with respect to the vertical.
To precisely determine these quantities, a rather extensive calibration must be carried out. Here I had to try some reasonable guessing: There is a nominal focal length in mm, the sensor data (pixel pitch) can be looked up, as well as some distortion information for this specific lens. From aerial pictures showing the river and individual mountains, the viewing direction can be estimated. The appearance of the water surface gives some clues about the camera rotation. In combination, all these estimations allow for a plausible transformation:
Assuming this reconstruction to be not too far off, it is immediately obvious that the bright central branch does indeed fit to the conventional primary rainbow locus at a constant scattering angle of about 138°. As expected, the secondary ends up at about 129°, also as a straight line. The lower branch (i.e. at higher scattering angles) can in principle be explained by aerodynamically flattened raindrops, following a long tradition in rainbow physics [5, 6, 7, 8, 4]. However, the upper branch penetrating into Alexander’s dark band requires elongated raindrops, whose existence cannot be accounted for by aerodynamics alone. Electrostatic fields  can elongate raindrops, but in the absence of any lightning activity it is speculative if any higher fields were present. Elongated shapes do also occur as transitory states during oscillations of larger drops in the appropriate (axisymmetric) modes .
The problematic element in this explanation is, however, that in the case of the rainbow we deal with a large number of contributing raindrops and a temporal average due to the finite exposure time. So we need an argument why contributions from transitory states are not simply wiped out. The resonance frequencies of the individual drops depend on their size, so no singular event such as an acoustic shock wave from thunder (if there had been any at all) can synchronize the oscillations. The only plausible idea for a formation of stable rainbow branches by drop oscillations in a stochastic ensemble might be that the two extremal states of the oscillation (flattened and elongated) are encountered with a higher probability than intermediate ones, as the momentary velocity decreases to zero at the turning points of any classical oscillation. Admittedly, this requires a rather narrow distribution of amplitudes throughout the ensemble (at least in the dominant drop size range), as otherwise the branches will be wiped out again due to the spread in extremal axis ratios. To my knowledge, there is not enough data on the statistical properties of oscillations in large ensembles of natural raindrops published yet to draw a definitive conclusion here.
Some further details of this observation are worth to be noted: The three branches of the primary bow appear each in a distinct fashion: The lowest is broad and rather diffuse, the middle one is bright and shows the features of a typical primary rainbow, the top one is narrow with a sharp uppermost outer rim. Moreover, it gives the impression of having developed a downward sub-branch in the –10°…+5° clock angle interval, resulting in a four-fold split bow there.
Rainbows certainly go on fascinating people all over the world, and rightfully so: Even in the 21st century, some outstanding displays occur from time to time that still challenge our understanding. Maybe those in hotter climates with intense rain showers have better chances of catching such rarities. In any case, we have to go out and take a look and a picture at the right time.
In my last blogpost, I described how tertiary and quaternary rainbows in the light of a halogen lamp and made by drops from a spray bottle can be photographed. The quinary rainbow I had not been able to detect back then, so I gave it another try two weeks later (on April 14th, 2018).
I chose a more conventional wide angle lens with f = 18 mm (Pentax DA 18-55 mm at a Pentax K-5 camera) instead of a fisheye this time, so that both the peak illumination intensity and the drops can be confined to a specific rainbow sector without the need to care about the rest of the rainbow circumference. Also, I hoped that a lens hood (which cannot be applied to a fisheye objective) might help somewhat against the wetting of the front lens by drifting drops. However, this did not work out, and the wetting problem did in fact worsen due to the fact that the lens has now to be pointed upward to capture the upper sections of the rainbows against the sky. This creates a much more efficient target for falling drops.
I started with a nice shot of a primary and secondary rainbow against the night sky, which might be mistaken as a lunar rainbow at first glance – but, as mentioned, both illumination and drops were purely artificial:
I then took about 40 pictures, both upwards as well as pointed horizontally to the right side against the vegetation background, without any additional polarizers. A signature of the quinary rainbow appeared in only a single frame of this whole series, recorded shortly after the one shown above. I suppose that even wiping the front less does not help to much after a while, as the lens will fog up again shortly afterwards. The diffuse background resulting from even a slightly fogged lens might be enough to mask the quinary. For the next experiment I plan to install a small battery-powered hairdryer or something of that sort to keep the lens dry. Anyway, here is the picture:
with increased contrast and saturation:
The arrow points to the green/blue stripe of the quinary rainbow inside Alexander’s dark band.
Ironically, I had taken this only as a fun shoot because of the twisted look of the primary, and did certainly not expect it to be the only reference image for the quinary from this series. At the location of the dark band crossing the primary, the shadow of the spray bottle was cast on the drop cloud, which suppressed part of its “rainbow response”. The remaining drops outside the shadow might have had a different size, and/or the remaining divergence of the light source did play a role. Even at a distance of 10 m from the lamp, a lateral displacement of a drop by 50 cm corresponds to a shift in the lamp position (as seen by this drop) of about 3°. So the deviation of the Minnaert cigar (which has more of an apple shape here) from an ideal cone will still have an appreciable influence. This can only be reduced by increasing the distance to the light source or by confining the drop cloud to a region closer to the camera.
As already mentioned in the last blogpost, and being also visible in the picture above, beautiful supernumeraries at both the primary and secondary rainbows can become visible for several seconds. Finally, here are several pictures that show more of their variety:
After almost 7 years since the first successful documentation of higher-order rainbows, we are now aware of at least 40 photographic observations of tertiary bows, sometimes accompanied by quaternaries. It is the more surprising that so far no one seems to have tried an outdoor experiment using artificial light and drop sources to bridge between the natural observation and single-drop scattering experiments, in which caustics are projected onto a screen.
Such an outdoor setup does not only allow to test various cameras and post-processing methods, but may also help to introduce newcomers to the challenges of observing higher order bows against the intense zero order background. Also very practical issues such as drops on the front lens or wet cameras can be directly experienced.
The setup is quite similar to what is used for diamond dust halo observations in Finland. The experiment is carried out at night in order to exploit the optimal background conditions of a dark sky. A bright searchlight lamp creates an almost parallel light beam with small opening angle, in which the camera is placed. Direct illumination of the camera is blocked by a cardboard disc placed roughly halfway between lamp and camera. This also helps in the case of photographing primary and secondary rainbows (i.e. the lens is pointing away from the light source), as stray light entering through the viewfinder on the camera backside can spoil the pictures.
I tested both Xenon (HID) and halogen lamps in the power range of 50-100 W. While Xenon lamps are brighter at the same power consumption, their non-thermal emission spectrum may lead to rainbows whose color is dominated by blue and yellow only, also the emitted light can show unwanted yellowish tinges in certain emission directions. The pictures shown here were therefore taken using the 100 W halogen lamp.
Drops were created by an ordinary spray bottle and, as judged by the appearance of the rainbows, are somewhat smaller than the ones in natural rainbows. Due to wind or movements of the bottle a spatial separation of smaller and larger drops can occur, as indicated by several well formed supernumeraries on both the primary and secondary rainbows which become visible for some moments. However, I decided not study these detail here, and tried to create a more or less spatially homogeneous spray including all available drop sizes over the exposure time of 2-5 s for each picture.
This is how the primary and secondary rainbow look like:
(camera: Pentax K-5, lens: Pentax-DA fisheye 10-17 mm at f = 10 mm, f/3.5, ISO 200, 5 s)
At that time, there was also some light natural drizzling going on, which generated only a weak primary rainbow in the lamplight. The limiting factor here is not the much lesser density of drops than in the spray (this could be helped by longer exposure times or stacking), but rather the background illumination of the sky (the pictures were taken in my garden in Hörlitz, Germany, which is a rather rural, but still pretty illuminated place, and there was also the nearly full moon behind the clouds on the evening of April 1st, 2018).
(f = 10 mm, f/3.5, ISO 200, 30 s)
When reversing the camera viewing direction, the zero order glow (=light which is transmitted through the drops without reflection) is, as expected, the dominant feature in the photographs:
(f = 10 mm, f/3.5, ISO 400, 2 s)
After strong unsharp masking, the tertiary rainbow is extracted, and can be traced almost completely around its full circumference:
As the camera was directed straight to the lamp (and rainbows thus appear as circles around the image center), it is possible to apply a radial smoothing filter to enhance the visibility further:
Here is another picture, taken with the same settings and processed similarly, which clearly shows in addition the quaternary rainbow in the upper left quadrant:
A major problem is that drops on the front lens disturb the recorded rainbows massively, as becomes apparent after unsharp masking. This problem is especially severe when using a fisheye lens (which does not have a suitable lens hood), and under windy conditions which shift the drops into unexpected directions due to swirled gusts near the ground. Periodical wiping of the front lens is therefore indispensable. Of course, the camera itself should be proof against spray water.
It is known from calculations that the contrast of the tertiary rainbow lies close to the detection limit of the human eye (see here and here), at least for purely spherical water drops. Here, no traces of it could be seen directly, even when looking through a polarizer. The main problem is that the spray lifetime is only a few seconds and the observer is constantly busy to maintain a more or less constant amount of drops in the air, which is rather distracting. A garden hose may be worth testing in the future.
So far, no unambiguous traces of the quinary rainbow (see here and here) could be extracted from Alexander’s dark band, in which its green and blue parts are expected to follow immediately the red rim of the secondary. There are several reasons which make its detection difficult here. At first, the drops are generally smaller and thus the secondary rainbow wider than in a natural setting. Next, the weak but non-zero divergence of the illumination may blur the rainbow positions further. Also, the background includes green plants in some directions which hinders the detection of green rainbow features. A more detailed study using a darker background and a narrower drop size distribution (with appreciable supernumeraries) seems necessary.
During the days of the 12th Light & Color in nature meeting (May 31st-June 3rd, 2016) in Granada, Spain, I noticed almost constantly a diffuse aureole around the sun, appearing against the background of a clear sky:
All photos were cropped to a common viewing angle of 15° x 15° and the color saturation was increased.
Because of the dry and often cloudless summer weather we had back then, it seems unlikely that any kind of water drops did cause the phenomenon. On the other hand, the angular radius was way too small for Bishop’s ring, which at first seemed to be a plausible option as we had observed some haze towards Africa shortly before our plane landed in Malaga on May 30th.
No pronounced color pattern was visible to the naked eye, nor through a gray filter, but the saturation increase in the image processing revealed a typical corona structure with alternating colors. Thinking of pollen as possible scattering particles, the large amount of olive trees (olea europaea) in Andalusia immediately comes to mind. Furthermore, we witnessed ourselves that the olive trees were blooming these days when we visited a grove at Monachil in the vicinity of Granada – some of the visitors’ shirts or backpacks got covered with green dust after coming too close to the trees.
In order to check this hypothesis I looked up the shape and size of olive pollen: They are almost spherical with a mean polar diameter of 20.1 µm and mean equatorial diameter of 21.5 µm. For most of the observations, the sun elevation was high enough to simply approximate the pollen as spheres of 21.5 µm in size. I calculated the resulting corona from the solar spectrum using simple diffraction theory (which at this particle sizes is justified):
Both the photograph and the simulation (right hand side) were cropped to a field of view of 10° x 10°. For the simulation, I assumed a relative spread in the pollen size (standard deviation of a Gaussian distribution divided by the mean diameter) of 15%, convoluted the result with the sun’s disk and added a gray background. It matches the photograph quite well, though the contrast of the natural corona remains lower than that of the simulation. Maybe there were other scattering particles with a broader size distribution present, which added another, rather colorless aureole “layer” on top of the pollen corona, thereby diminishing its contrast. Surprisingly, I could not find any previous reports about “olive pollen coronae”, though the phenomenon should be quite prominent during the right season in the olive-growing regions.
In 2014, Harald Edens reported ten cases of photographically detected natural quinary rainbows, recorded during 2009-2013 in New Mexico, USA, at altitudes of 1.8-3.2 km. These and some newer observations can also be found on his website.
So far, no reports from other locations have been published. In the German observers’ network, we analyzed many candidate photographs showing bright primary and secondary rainbows, but from most of them no reliable traces of quinary rainbows could be extracted. Such analyses are not easy, as the quinary signal is weak compared to the neighboring secondary rainbow, and processing methods such as unsharp masking can cause a leakage of colors into Alexander’s dark band. Furthermore, the processing operator will experience disturbing afterimage issues from the intense renditions of the primary and secondary on the screen after a couple of minutes.
Despite these difficulties, we now believe that we have identified three cases of genuine quinary rainbows. In cases 1 and 3, the quinary could be extracted from several photographs. Nonetheless, in order to keep this blogpost brief, we restricted ourselves to show only one image (or the results from one polarization series in case 1) per observation. We chose a straightforward processing method (= only increasing contrast and saturation, no local filtering such as unsharp masks) similar to the one applied by Harald Edens to allow for an easier comparison with his results. Alternative processing routes will be presented at a later stage.
1) April 22nd, 2012, near Göttingen, Germany (51° 31’ N, 9° 58’ E, altitude 250 m), 19:16 CEST, sun elevation 10.2°, photographed by Frank Killich after a moderate shower
The original intention of Frank Killich was to use the primary and secondary rainbows as test objects for a home-built photopolarimetric setup made from a Canon 20D camera and a linear polarizer precisely rotatable by a stepper motor. By recording four successive images at polarizer positions of 0°, 45°, 90° and 135° with respect to the vertical, it is possible to reconstruct the first three components of the Stokes vector for each viewing direction (pixel coordinates) and color channel (red, green, blue) individually. These images can be numerically combined to reconstruct the unpolarized intensity (= the ordinary photographic result without a polarizer) and, moreover, the linearly polarized portion of the recorded light distribution (= the total intensity with the unpolarized background removed for each pixel). In the case of rainbows, this corresponds effectively to a subtraction of the radial (weak) component from the azimuthal (strong) polarization component equally all along the visible part of the circumference. As known from theory, also the quinary will be easier to detect in such a polarization contrast image.
Unpolarized intensity as calculated from the original images, f = 22 mm:
Unpolarized intensity, increased saturation and contrast:
Linearly polarized portion as calculated from the original images:
Linearly polarized portion, increased saturation and contrast:
The expected broad bands of green and blue are clearly visible in the processed linearly polarized portion picture, and might be slightly visible also in the unpolarized intensity.
The other two photographic observations were carried out without any polarizers, i.e. only the unpolarized intensity information is available in these cases.
2) March 20th, 2013, near Pforzheim, Germany (48° 56’ N, 8° 36’ E, altitude 312 m), 16:21 CET, sun elevation 21.1°, photographed by Michael Großmann after an intense shower
Original (Canon EOS 450D, f = 22 mm):
Increased saturation and contrast:
A slight green/blue hue is visible inside the secondary at and slightly above the horizon.
3) May 15th, 2016, Mt. Zschirnstein, Germany (50° 51’ N, 14° 11’ E, altitude 560 m), 19:57 CEST, sun elevation 6.2°, photographed by Alexander Haußmann after a moderate shower
Original (Pentax K-5, f= 17 mm, cropped):
Increased saturation and contrast:
Again, a slight green/blue hue appears close to the horizon.
At this point it is of course not possible to draw any statistical conclusions about the frequency of detectable quinary rainbows. However, it seems worthwile that every rainbow observer re-examines his photographical treasure trove for previously overlooked rarities, even if no polarizer enhancement was involved during photographing.
Three quarters of a double rainbow, plus an accidental snapshot of a tertiary, Mt. Zschirnstein, Germany, May 15th, 2016
Over the past two decades it has become a tradition among my friends to carry out a bicycle tour to the Elbe Sandstone Mountains (“Saxon Switzerland“) at the Pentecost weekend. We then often pay a visit to a table hill named “Großer Zschirnstein“ (561 m), which features a remarkable cliff of 70 m in height at its south-eastern edge.
Almost 15 years ago, on the evening of June 3rd, 2001, we had the opportunity to observe from there a rainbow extending well below the horizon almost down towards its bottom. Unfortunately, we only had a compact camera without a fisheye lens at hand back then, so the old photos show only some sections of the whole phenomenon.
This year, on May 15th, we were finally granted the proverbial second chance. I already anticipated some rainbow potential in the “Icelandic” weather that day. In the early afternoon, there had already been a rain shower while the sun was shining, but as we had not yet ascended the mountain and the sun was still high in the sky, there was no chance for a rainbow observation.
Some minutes after reaching the plateau in the evening, we had to retreat to the shelter when a rather strong shower of hail and rain set in. To the west a stripe of clear sky widened, and sunshine seemed at hand soon. It took longer than expected, as the clouds were moving rather slow. On the left side, a small rainbow fragment suddenly appeared at the horizon, resulting from sunlit drops a few kilometers off. It was a rather unusual observation to see this rainbow streak vanish and reappear again, as its sight was repeatedly obstructed by scudding (and non-illuminated) mist around the Zschirnstein massif:
(19:42 CEST, f = 88 mm, Pentax K-5)
Finally the great moment came: Sunshine was reaching the Zschirnstein while the shower, now mostly composed of rain instead of hail, still continued. Within a few minutes we could enjoy this marvelous view:
(19:56 CEST, f = 10 mm / fisheye)
Unfortunately there was no safe way to access a viewpoint which would have allowed to study the missing quarter, as this would have required some careful climbing around the sandstone rocks for which I already felt too excited at that moment. The fisheye picture can hardly express how huge both rainbows looked like, and how beautiful the raindrop clusters glittered as they drifted around the cliff some 10 m further down. These are certainly the moments that make you understand that famous “double rainbow enthusiasm”, thought not everyone is as outgoing as other people on the internet. Maybe we also stayed a bit calmer because the strong and cold wind added a rather painful component to the taking of photographs and videos.
Later the right part of the primary close to the horizon became especially bright:
(19:59 CEST, f = 80 mm)
This photo has been processed in a way that no color channel reaches saturation, which is a necessary prerequisite for analyzing possible kinks in the rainbow. In this case, the red rim looks as if would bend inside a bit below the horizon, but this might only be an illusion due to the intensity gradient.
The primary’s right foot above the horizon remained still visible for a rather long time, as the shower withdrew in this direction:
(20:19 MESZ, f = 50 mm)
But the story does not end here. When going through the pictures later at home, I suddenly realized that I had missed to look for higher order rainbows, or to deliberately take some pictures in the appropriate directions. I was a bit disappointed about my inattentiveness, since this had been my best rainbow display in years and, moreover, I had not been hindered by the limited field of view from a window in a city building. I am often forced to decide between the sunward or antisolar hemisphere when observing rainbows from there.
Luckily I had taken two pictures (an exposure bracket) towards the sun just at the moment when the three-quarter rainbows started to evolve. The reason for this was only the lighting atmosphere – it was the moment when the sun rays had first reached the Zschirnstein plateau. As I deduced later from the movement direction of the shower, there had been rather good conditions for the formation of tertiary and quaternary rainbows when the picture pair was taken. So I decided to apply the strong filtering procedures which are needed to extract higher-order rainbows from photographs. The shorter exposure just gave noise in the interesting region. However, in the longer exposed version something interesting popped up.
(19:54 MESZ, f = 17 mm / fisheye)
Slightly to the right above the stone pillar, a red-green stripe in the color ordering of the tertiary rainbow can be discerned. For an unambiguous identification it would, however, be necessary to calibrate the picture in order to assign scattering coordinates to the photo’s pixel matrix. Though I had previously calibrated the projection of the lens for the used focal length (the upper end of the zoom range), I would need two reference marks with known elevation and azimuth which are included in this specific photograph to complete the analysis. On the horizon, no distinct remote references could be found. This means that I would have to reconstruct my precise position on the plateau to minimize parallax errors, and then to record a starfield image from there at night, enabling me finally to use the stone pillar or nearby trees as references. Unfortunately, it would take an inconvenient amount of time to access the spot again and the effort for such a trip would be a bit over-the-top for the sole purpose of calibrating a photograph.
But there was still a piece of hope: From the shorter exposed version (-2 EV), I could estimate the position of the sun quite accurately, as there is only a small overexposed area around it. This allowed me at least to draw lines of constant angular distance from the sun into the photograph in order to decide if the colored stripe appeared at the correct position or not. Using the previously measured spectral sensor response of my camera, and estimating the temperature of the water drops to be around 5°C, I derived the following values for the Descartes angles of the tertiary and quaternary rainbows: 41.7° / 43.7° (red, 620 nm), 40.6° / 45.1° (green, 530 nm), and 39.3° / 46.8° (blue, 460 nm). In the following animation, these angular distances from the estimated position of the sun have been marked by their respective colors:
The colored stripe seems to fit reasonably well to the Descartes angles of the tertiary rainbow, especially when taking into account that the positions of maximal intensity are shifted a bit inward from the Descartes angles for the tertiary (and outward for the quaternary) due to wave-optical effects. This shift was also noted in the analysis of the very first photograph of a tertiary rainbow. Further contributions form distorted drop shapes are of minor importance here, as the sun elevation is small and we are looking at the rainbow’s sides. Therefore the effective cross section of the drops should remain nearly circular, even if they are squeezed in the vertical. I leave it to the readers to decide if also traces of the quaternary might be visible among the color noise slightly to the left above the stone pillar.
Addendum: A short video clip from the observation can be found here.
Have you ever wondered how many photos of outstanding atmospheric phenomena may exist “out there” without us knowing about them, just because they are not posted on our regular websites, blogs or forums? From time to time, I do Google image search queries on atmospheric optics related subjects to see if something interesting and yet unknown might show up. Some weeks ago, I encountered this way a true rainbow rarity on a Japanese website. The picture had already been publicly accessible for over two years, but went unnoticed by the European or US atmospheric optics community so far. Using the automatic translation function I identified the photographer and contacted him to learn more about his (as of now) unique observation.
Kunihiro Tashima noticed an approaching rain shower on the evening of August 5th, 2012, in the town of Yobuko, Saga prefecture, Kyushu island, Japan (33.54° N, 129.90° E). According to his experience, these showers appear quite regularly after sunny days in the Japanese summer. At 18:24 JST he took the first photographs of a marvellous rainbow display made up from a triple-split primary and an undisturbed secondary (photograph 1, unsharp masked; photograph 2, unsharp masked) from a parking lot. Kunihiro used a Nikon D7000 camera equipped with either a AF-S DX NIKKOR 18-55 mm or a Tokina AT-X 116 PRO DX II 11-16 mm lens at 18 mm and 11 mm focal length, respectively. The sun was located at 9.7° in elevation and 283.8° in azimuth when these pictures were taken.
Within the next minute the shower intensified at his position, so he had to withdraw into his car. Photos taken at 18:25 through the windscreen give the impression that the middle branch had by then already merged with the uppermost one, resulting in a rather broad “traditional” twinned rainbow (photograph 3, unsharp masked). Around 18:32, only an ordinary single primary and a weak secondary were left in front of receding clouds and the blue sky (photograph 4, unsharp masked). At this time, the sun’s position was 8.1° in elevation and 284.9° in azimuth.
Twinned rainbows are nowadays a well-documented phenomenon  and several promising steps have been taken to explain their formation [2, 3]. In one of my earliest reports on simulations of rainbows generated by flattened drops with broad size distributions, I pointed out the idea that also split rainbows with three or four branches might occur at very rare occasions [4, p. 117]. However, up to now, no photographs or clear observation records of such highly exotic rainbow displays have been known to the community. Some old reports of multiple rainbows do exist , but these are difficult to evaluate due to the lack of further details. Hence Kunihiro’s photos provide to my knowledge the first reliable evidence that multi-split (>2) rainbows exist.
A reflection rainbow generated by mirrored sunlight from a horizontal water surface can be excluded as an explanation here, since the angular deviation from the original bow would have to be larger at this solar elevation. Furthermore, the secondary bow remained unaffected by any anomalies, which is a familiar feature seen in many split rainbow displays.
For further analyses it is necessary to assign scattering coordinates (scattering angle and clock angle) to the individual pixels of the photographs. Unfortunately, no starfield calibration photos or position data for reference objects in the photos are available. Nonetheless I tried to estimate the three orientation angles for one of the images (2nd photo from 18:24) using azimuthal positions of roof-edges etc. as calculated from Google Maps aerial pictures and additional constraints such as the vertical orientation of lampposts and the approximately constant scattering angle of the secondary bow. The lens distortions (deviations from the ideal rectilinear projection) were corrected with predefined, lens-specific data in the RAW converter software UFRaw. Though this estimation procedure is only an error-prone stopgap solution (compared to a true calibration with a starfield image) the results are quite convincing. This can be seen best when the rainbow photos are morphed into an equirectangular projection in scattering coordinates (0° in clock angle = rainbow vertex).
I calculated such projections for the 1st and 2nd photo from 18:24, as well as for the last photo from 18:32. The orientation angles I only estimated once (for the 2nd picture from 18:24), whereas I pursued a “dead reckoning” approach using some reference objects to transfer the initial orientation calibration (including its errors) to the other two photos. This allows for a consistency check of the method by evaluating the last picture which shows an ordinary rainbow display. The non-split primary appears, according to the expectation, as an almost straight line with only a slight curvature towards the antisolar point around its vertex.
With the orientation being now somewhat trustable, I took a closer look at the finer details in the triple-split bow. The uppermost branch of the primary is shifted by approximately 1° for clock angles > –60° into Alexander’s dark band, i.e. towards the secondary, when compared to its left foot at around –70° in clock angle. Such a behaviour cannot be explained by the current theory for rainbows generated by flattened drops, since it predicts an inward shift of the primary at its vertex, i.e. away from the secondary, for this elevation of the sun. Elongated rather than flattened drops will yield a shift towards the secondary, but such shapes far from the equilibrium are not stable and will occur only temporarily during drop oscillations. Since these oscillations have periodicities in the range of milliseconds for common raindrop sizes, it is doubtful that a well-defined rainbow, required to be stable over the typical exposure time of a camera (or the human eye), can be generated by oscillating drops with considerable amplitudes. Obviously, such oscillation blurring will be reduced for smaller amplitudes as the oscillations damp out over time, but simultaneously the drop shapes will converge towards their flattened equilibrium states.
Summing all up this means that Kunihiro’s pictures do not only represent the first photographic proof for multi-split bows, but will also give the rainbow theorists something to think about. It might be that we have to take into account additional influences such as electrostatic fields, refractive index variations, or anomalous wind drag.
On Sept 25th and 27th, 2014, I was traveling by plane from Dresden to Brussels and back, with stops at Frankfurt and Munich, respectively. As usual, I booked window seats to study sky phenomena. The sunward side was not very interesting, since these short-distance flights are carried out at heights below the cirrus clouds and therefore no sub-horizon halos can be observed (at least in autumn). On Sept 25th only a single 22° halo appeared in the cirrus clouds above the plane, whereas on Sept 27th ice crystal clouds seemed to be fully absent.
Accordingly, the viewing direction towards the antisolar point proved to be much more interesting. As most of the Atmospheric Optics enthusiasts I had seen glories and cloudbows before (especially when traveling to the Light&Color meetings in the US) but this time the conditions seemed to be especially favorable. I could observe an an almost textbook-like development of both phenomena right after piercing through an Altocumulus layer after the take off from Dresden (Sept 25th, 11:13 CEST):
From Debye series simulations (intensity sum of the p = 0 to p = 11 terms in order to prevent artifacts from the small-scale inter-p-interferences as present in the Mie results) a mean drop radius of about 8 µm with 0.5 µm standard deviation can be estimated (assuming a Gaussian drop size distribution):
This simulation was calculated for the original lens projection with added ad-hoc gray background. It is also available as a fisheye view centered on the antisolar point without background , together with the corresponding simulation for monodisperse drops (no spread in size) of 8 µm in radius .
Unsharp masking and saturation increase processing of the photograph reveals that the sequence of supernumeraries can be traced until they merge with the glory rings:
Over the next minute I mounted the fisheye lens to my camera in order to record a broader view. Unfortunately, some of the outer glory rings and inner supernumeraries had already vanished, indicating an increase in the drop size spread:
Note the smaller angular size of the plane’s shadow as the distance to the Ac layer had further increased. A well fitting simulation to this photo can be calculated by assuming again a mean drop radius of 8 µm and setting the standard deviation now to 1 µm:
For comparison, the fisheye simulation centered on the antisolar point was calculated for the 1 µm drop size spread as well . Furthermore, I recorded a video sequence showing the movement of both glory and cloudbow across the uniform Ac layer (11:15, ). When later the edge of the Ac field was reached, the glory showed an appreciable degree of distortion (11:18 CEST , processed version ).
On Sept 27th, not a uniform but a fractured Ac layer was present after the take off from Brussels. Nonetheless the glory appeared circular (12:34 CEST , processed version , video at 12:37 CEST ), with the exception of occasional larger disturbances in the layer (12:34 CEST ). The cloudbow was not as prominent as two days earlier. During the later part of the flight only occasional Cumulus clouds were present, which did not allow for further glory observations until the plane started descending when approaching Munich. At this point the angular size of the clouds became large enough again to act as suitable canvas for the glory (13:14 CEST  ). During the final passage through a Cu cloud I recorded a further video (13:15 CEST ). Remarkably, the angular size of the plane’s shadow varies rapidly (indicating the distance to the drops) whereas the the angular size of the glory remains rather stable (indicating the drop radius).
Photos and videos were taken with a Pentax K-5 camera equipped with either a Pentax 10-17 mm fisheye or Pentax-DA 18-55 mm standard zoom lens. A gallery view of my photos can be seen here .
Chasing the circumhorizontal arc (CHA) has become a quite popular activity among the German halo observers. Depending on the latitude, there is only a 1-2 h time slot at noon for a few weeks around the summer solstice. Even the highest elevation the sun can reach is still a few degrees lower than the optimal value for CHA formation. This might only be beaten by the moon in a suitable position with respect to the ecliptic.
I was keen on observing the CHA this year as well, and had not had any luck so far. On Saturday, June 28th, there had been a single 22° ring before noon at my home in Hörlitz (51° 32’ N, 13° 57’ E). At 12:45 CEST I got on my bicycle for a visit in the neighbouring village. Already after 500 m I had to stop: The 22° ring intensified, and although there was still nothing else visible with the naked eye, I decided to take a fisheye picture at 12:51 for a later analysis. As seen in the unsharp masked version, the complete circumscribed halo and parhelic circle were already accompanying the 22° halo. With an ordinary wide-angle lens I took a “blindfold” picture deep in the south a minute later, and after unsharp masking both the CHA and the infralateral arc could be distinguished.
Of course this was unknown to me during the observation, but I felt some kind of suspicion that there might be more in the sky than I just saw (even by looking through a grey filter or using a black watch glass mirror). Around 12.53 I noticed the parhelic circle high in the sky, which had a diameter only slightly larger that of the 22° ring (~29°). Within the next few minutes the circumscribed halo became bright enough to appear clearly separated from the 22° ring at the sides. There were no traces of plate halos such as the 120° parhelia which I took as a bad sign for the CHA. There were now also cumulus clouds gathering in the south.
I moved on a bit, but stopped again after a 1 km: The sight of this huge “wedding ring”-like pattern in the sky was just too fascinating. I also scrutinized the south from time to time: Wasn’t there any colourful band appearing in the gaps between the Cu clouds? From time to time I thought that that I could see a part of the CHA, and the photos later proved that it was actually there, but I was not sure if I were just imagining something after staring too long into the sky. Consequently, I do not count this as a successful visual CHA observation. After reaching my destination at about 13.25, the Cu clouds were obstructing larger and larger parts of the sky as the halos were fading away in the gaps. I really had the luck to observe a parhelic circle at almost the highest possible solar elevation at my place (61.7° at 13.07)! Only 0.2° were missing to the ultimate maximum a week before the observation.
When going through the pictures again, I also found the upper part of the Parry arc in the filtered versions. Remarkably, the part below the parhelic circle is missing, and I do not have an explanation for this at hand at the moment. Nonetheless, the presence of the Parry arc allows to discard plates at all: The CHA may as well be generated by Parry crystals, as seen in this HaloSim simulation. However, when the portion of Parry crystals is increased to the point at which the CHA is rendered at a reasonable intensity, the Parry arc appears too bright.
A representative selection of images from this observation is available here.