Invisible Rainbows: Secrets of the Sky's Most Colorful Phenomena

Part I – A Few Introductions

Rainbows are certainly conspicuous. We have all seen them, we all know their colors, and many of us know about how they form when the essentially white sunlight is refracted and reflected inside raindrops—where it is dispersed into its different wavelengths that we see as colors. We all know about and experience great excitement when seeing a double rainbow, and some of us have noticed that during those occasions, it looks darker in between the primary and secondary arcs (Alexander’s dark band). We all have seen, even if not consciously noticing, that red is always in top for the primary and that the colors are inverted and much fainter and spread out for the secondary rainbow. We might have also noticed a repeated pattern of apparently pastel colors (supernumeraries) sometimes visible under the primary’s violet. As it turns out, there is so much more that we do not normally see, or that we are completely unable to see.

Photo credit: Lourdes Avilés

The primary rainbow is the brightest of the bows and the feature that is always present. A fainter and broader secondary rainbow with inverted colors is sometimes visible about eight degrees above. The darker area between the two is Alexander’s dark band. This picture was taken from the roof of the Plymouth State University science building, looking toward the school’s iconic Rounds Hall clock tower.
 

Image credit: Lourdes Avilés and Dan Bramer

A rainbow can be observed 40° above the point along the plane of the rainfall opposite to the sun, defined by the line connecting the direction of the sunlight and through the eyes of the observer. We refer to this as the antisolar point. The rainbow’s angular width is approximately two degrees, with red seen at 42° and violet at 40°.

Maybe I should first tell you why I am talking about rainbows. Before starting in my new role as a university administrator, for nearly twenty years, I was a professor of meteorology at beautiful Plymouth State University in northern New England. During that time, I taught a large number of introductory and upper level courses, and one of my favorite topics to teach whenever it fit with the course, was atmospheric optics (you know, blue skies, red sunsets, rainbows, ice halos, lunar coronas, northern lights, and so much more). Colleagues and students know about my admitted obsession and send me pictures and articles, warn me to go outside to see something interesting in the sky, join me to engage in such observations, and patiently bear with me when I give them way more information than they would have ever asked for. I have in fact been working on a textbook about atmospheric optics (a multi-year project that still has a couple more to go), where besides covering the many aspects of the physical/atmospheric science as well as how and where to observe the various optical effects, I am working on bringing in other sciences (visual system anatomy and physiology, and color perception, for example) as well as a variety of historical, social, and cultural aspects. I applied the same treatment (interdisciplinary science, history, and social aspects) to the Great New England Hurricane of 1938 in a book published by the AMS. I also wrote an early Weather Band blog post about the storm. Hurricanes are a big part of why I became an atmospheric scientist, but that is a story for another time. Rainbows were one of the first things that captured my imagination when I was a little girl pestering my parents with way too many questions, and It has been so much fun to go really broad and deep in exploring them for their corresponding chapter in my in-progress book.

Invisible Rainbows. What do I even mean with that? For our purposes here, we are talking about features that are for one reason or another not noticeable or completely impossible to see. 

Part II – Arcs, Circles, and Disks of Light

It might first be interesting to note how rainbows are not just arcs. If we were standing in a geometrically advantageous position where the entire area around the antisolar point had rain and was sunlit, we might see a full circle 40 degrees from the center. This is not easy to achieve but there are a few examples findable with a bit of internet searching. It would normally take an airplane or a drone to see this. So, rainbows are not arcs; they are more accurately circles. It is just that a large portion of those circles are hidden from us by the ground features. But as it turns out, there is more.

Image credit: Lourdes Avilés and Dan Bramer

A complete rainbow circle can be observed when the entire angular distance of 42° away from the antisolar point on the plane of the rain is in view of the observer, which can only happen from a high-enough vantage point.

Image credit: Lourdes Avilés and Dan Bramer

Because different colors refract slightly differently (they have a different refractive index), the initially white light disperses into its spectrum of colors when it enters a raindrop. Violet, the shortest of the wavelengths, refracts more and exits the droplet above the red, which refracts less, after one internal reflection off the back of the drop. An observer sees red from higher droplets and violet from lower droplets, explaining the well-known order of colors in a rainbow.

Even those who have noticed Alexander’s dark band outside the primary rainbow (above the red), might not have noticed that it is complementarily brighter inside the arc, below the violet. This is because sunlight does not come out of the raindrops back toward the observer at just one angle or small range of angles corresponding to where we see each of the colors. The so-called angles of deviation (difference between the initial and final direction of the light) for each of the colors that we see cover the entire area inside the circle. All colors together are seen as white by our eyes and what we have is a whitish disk with a smeared colorful edge. The color that bends the least, red, has the widest disk, followed by the orange one and so on. The edge occurs at the minimum angle of deviation (maximum distance from the center of the disk), where rays tend to come out together, meaning that this is also where the intensity is highest, corresponding to what we see as the bright primary rainbow. The inside of the disk, where we see all the colors together, in other words, white, is much fainter. The next time that you see a bright rainbow, or look at a picture of a rainbow, notice how the background seems lighter inside the violet than outside the red. This whiteish disk is hidden from our perception simply because we tend not to notice it.

Image credit: Lourdes Avilés and Dan Bramer

Light enters a raindrop at every point on its surface, thus arriving at different angles respect to the surface of the drop, and internally reflecting off the back wall with different angles of reflection. Equally spaced rays of light (as those shown arriving to the top half of the drop) come out at different angles, with a concentration of lines at the angle outlined by the thicker line, sometimes called the rainbow ray because that is the portion of the raindrop from which an observer can see a rainbow. The rest of the rays, which are seen by the observer below the red, contribute to the whiteish disk inside primary rainbow.

Image credit: Lourdes Avilés and Dan Bramer

The rainbow is not an arc or even a circle, but a disk, or more specifically, the edge of a disk. Light coming from droplets causes the separated colors to all come together toward the eye, perceived as a whiteish brightness, and only the edge of the disk is perceivable as the rainbow. Because of the slight differences in refraction index, violet is on the inside and red on the outside of the inner white disk. The dark band and secondary rainbow surround the inner disk. An area of whiteish brightness, though less intense than the inner disk, is outside the much fainter secondary rainbow.

Part III – Higher Order Rainbows

We already know, or might suspect if we didn’t know, that the internal reflection inside the raindrops can happen twice instead of just once, and that is what gives us secondary rainbows. Because there is a lot light that comes out of the drop rather than staying inside and reflecting twice, the secondary rainbow is always fainter, sometimes so much so that we do not see it. This process can continue, and we can have much, much, fainter three, four, five reflections and so on. This leads us to the concept of higher order rainbows: third, fourth, fifth order rainbows that most of us have never seen and will most likely never see. 

Image credit: Lourdes Avilés and Dan Bramer

Multiple reflections inside a raindrop give rise to higher order rainbows, most of which are too faint to be observable except for in highly filtered images or simulations. Third and fourth order rainbows appear on the same side as the sun, obscured by its brightness. Fifth and sixth order rainbows appear in between and behind the primary and secondary rainbows but are too faint to be noticed. The zeroth order comes from no internal reflections and corresponds to an area of brightness around the sun if viewing it through a rainfall. Higher orders theoretically continue to infinity.

Under favorable conditions, the third and fourth order rainbows, which have inverted colors and bleed into each other on their respective red edges, can be seen on the same side as the sun. A really bright primary/secondary combo opposite to the sun, and dark clouds near the sun would help a lot with the possibility of seeing them. Even then, there are only a handful of third or third/forth order rainbow pictures out there. Most of us will never see these.

Image credit: Lourdes Avilés and Dan Bramer

From mathematical calculations it is possible to determine the angle for the red and violet borders of the primary and secondary rainbow, as well as of all the theoretical higher orders. The zeroth order glow is shown on the left against a blueish background. Orders 3 and 4 can be very faintly observable around the sun. All other orders are too faint to be observed in nature.

Believe it or not, it was quite exciting to calculate the angles for increasingly higher order rainbows. It’s a tricky bit of geometry but very doable once you massage the right equations so that you just need to input the desired rainbow order and the refractive index for the desired wavelength. I stopped at 12, which straddles the direction of the sun. Correspondingly, order 10 is very close to the antisolar point, much lower than the primary/secondary rainbow combo. Orders 9 and 11 are high above, near the zenith. The combo of orders 3 and 4, on the other hand, are at very similar heights than the primary/secondary, but on the other side, near the sun, as highlighted above. Beyond the fourth order, however, It is not possible to see higher order rainbows with the naked eye. They are too faint, and they can be obscured by brighter features; they are also located at unexpected heights in the sky, but this really does not matter, since they can’t be visually detected anyway. The fifth and sixth order, for example, hide within the much brighter primary rainbow. If one knows where/how to look, and if one is extremely lucky and knowledgeable about how to appropriately filter the image of an exceptionally bright rainbow, Alexander’s dark band can contain the green/blue/violet of the much, much fainter and wider 5th order rainbow, with the green closer to the secondary rainbow and the blue/violet toward the middle of the dark band. There are also a few examples of these out there on the internet. If you do find one of these examples, don’t be dazzled by the really bright primary/secondary arcs that would be needed to even have a chance to detect a fifth order rainbow; remember that you are looking for the very faint green/blue inside the dark band. The full fifth order rainbow is much, much wider (almost 8 degrees—four times wider than the primary rainbow), spilling upward into the secondary. Only the dark band allows one to have a chance to see anything, though. The 6th order is fully hidden within the whiteish brightness below the primary’s violet. Looking carefully, you can discern many of these characteristics in the accompanying diagram with the solar and antisolar red and violet limits of the various rainbow orders. A while ago, I found an article describing the detection of a seventh order rainbow (which extends from approximately 55 to 65 degrees—huge!) using astrophotography-like stacking techniques that combine the light from multiple images. In a laboratory setting, and with an extremely intense monochromatic light source (let’s say for example, a green laser), it is possible to detect a much larger number of higher order bows, reportedly up to 200 orders! Mathematically, the number of possible orders is actually infinite!

This is not a bad time to point out that that disk that is not just a circle that is not just an arc, is also not just a disk. It is indeed a cone of light through which we are looking. This cone starts at the observer and spreads outward at the rainbow angle. This is not something we can ever notice from our vantage point.

Image credit: Lourdes Avilés and Dan Bramer

Different order rainbow angles placed based on calculations exist either above or below the plane connecting the observer to the solar and antisolar point (A), but they occur along the entire circle, therefore creating a cone (B). Each rainbow order thus corresponds to the edge of a cone of light through which the observer would be looking. This explains why some of the rainbow orders have inverse colors from other orders (C).

Part IV – Ultraviolet and Infrared Rainbows

There are also truly invisible portions of the rainbow that we might never think about. The spectrum of solar radiation includes mostly visible light (which we have defined as such because it is the portion of the electromagnetic spectrum that our human visual system is able to detect and interpret as visible light), a little bit of ultraviolet (with wavelengths shorter than violet), and some infrared (with wavelengths longer than red—specifically, what we call “near infrared”). We consider visible light as that with wavelengths between 400 and 700 nanometers (with a nanometer being 0.000000001 meters, which equal approximately 0.00000004 inches). Given the intensity of sunlight at the various wavelengths and atmospheric absorption of some of these wavelengths via a variety of processes, the range received at the surface of the Earth with enough intensity is between 300 and 1800 nanometers). Using the formulas mentioned above, it is straightforward to calculate (using a rainbow order of one, and the appropriate refractive index for the edge wavelengths) that the ultraviolet portion (300–400 nm) would extend approximately 2 degrees below the violet, and the infrared portion approximately four degrees above the red in the primary rainbow, filling a good portion of Alexander’s dark band, about half of it. The other half is correspondingly filled with near IR coming from the secondary arc. The dark band is therefore not so dark in reality, at least in the near infrared; we just can’t see it. Using the right type of photographic equipment, however, one can detect at least a portion of these invisible extensions of the rainbow. As a fun aside, bees can detect ultraviolet light (why some flowers that are visually bland, are quite showy in the ultraviolet), and it is fun to imagine that a bee would see a rainbow differently than we would—and let us not worry about the fact that bees would not interpret what they might see as a rainbow. The same is true for most animals, they would see rainbows differently if they could notice them: some see a shifted spectrum (like the bees), some see less colors (dogs, for example) or more colors (birds can detect more color gradations, and some additionally have a shifted spectrum) than humans. 

https://twitter.com/statto/status/1297538557033287680/photo/3

There is, as one might expect, so much more to rainbows into which we did not go here: for example, how they come out of much more complicated mathematical simulations using Mie Scattering Theory (which can also produce a host of other effects for smaller droplets: fogbows, glories, coronas, cloud iridescence). And how about light polarization? We don’t have the space to go into the details of how it works here, but rainbow light is polarized in the direction along the arc. With a polarizer sheet—it is easy to buy small squares online—or with polarized sunglasses, one can make portions of a rainbow disappear! I have had exactly ONE chance to play with that when I was carrying a polarizer in my pocket when a rainbow formed right outside the window of our weather center. Polarized sunglasses are vertically polarized (to block out added glare from reflections off horizontal surfaces). If you can fit a good portion of the rainbow arc in your view, the top of the arc will disappear. If you turn the sunglasses 90 degrees, the vertical legs of the arc will then disappear. Try it sometime!

Photo credit: Lourdes Avilés, Image composite credit: Dan Bramer

Sunlight is not polarized, and most of the light that we see in the sky is not polarized, but rainbows are. The polarization direction is parallel to the rainbow arc. A linearly polarized filter like one of the square sheets shown in the picture (or polarized sunglasses) can be used to make the portions of the rainbow polarized 90 degrees from the filter’s polarization disappear by being completely blocked. There is still enough light to see the background provided by the light that the polarizer lets through. The third image shows the rainbow that was visible without a polarized filter. These photos were taken out the window of the Plymouth State Meteorology Weather Center.

It is indeed amazing that the seemingly simple rainbow can provide so much hidden “under the hood” wonder even to those of us that are more knowledgeable about how it all works. We do not need to go there, however, as even with its hidden features, the simple fully visible primary rainbow has provided inspiration and wonder since humankind has had eyes to see it appear in the sky.