# Exploration into the Physics of Rainbows

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For several years, I have been contemplating this beautiful picture by photographer Brian McPhee.  I have a personal interest in the photograph because that boat is my year round home.  I also have a scientific interest in the photograph because of what it teaches me about rainbow physics.

The simplest explanation of rainbow physics is based on internal reflections in the near-spherical shape of a raindrop.

Light rays enter a raindrop from one direction (typically a straight line from the sun), reflect off the back of the raindrop, and fan out as they leave the raindrop. The light leaving the rainbow is spread over a wide angle, with a maximum intensity at the angles 40.89–42°. (Note: Between 2 and 100% of the light is reflected at each of the three surfaces encountered, depending on the angle of incidence. This diagram only shows the paths relevant to the rainbow.) — See Wikipedia Rainbow or The Theory of the Rainbow (Scientific American 1977)

Look carefully at the photo with the boat.  You will see that the sky inside the arc is much brighter than the sky outside the arc.  Some scientists claim that no such effect exists, but it’s pretty plain in the picture. The explanation is that raindrops inside the arc reflect sunlight toward me, while drops outside the arc reflect sunlight away from me.  The colors appear in the transition region where only certain colors are reflected towards my eye.

More challenging physics comes from the image of the rainbow seen on the surface of the water.   At first, I assumed that it was a reflection of the rainbow in the sky, just like reflections of blue sky and white clouds one sees on a calm day in a reflecting pool.  But then I came across Can Rainbows Cast Reflections? on the web site of noted astronomer Bob Berman.  Paraphrasing Berman. “No, they do not.  Rainbows are not 3D objects and they do not cast reflections.  In the water you see a different rainbow, not a reflection.” I spent a lot of time puzzling over that, because I didn’t understand Berman’s explanation. I also doubted its truth because I’m sure that I have seen rainbows in the rear view mirror as I drive.  It sounds like the Hollywood version of vampires that don’t make reflections in mirrors.

At first, I thought that Berman meant that the image in the water was sunlight hitting the surface and creating a rainbow effect as it was refracted back to my eye.  But no, that won’t work because water in the lake is not in the form of spherical droplets.

After much thinking, I think I’ve got it.  No vampire magic is required. The colored light you see from a rainbow is not omnidirectional, it is a unidirectional beam aimed at your eye.   By analogy, imagine a man at the far end of a hall of mirrors holding a laser pointer pointed at your eye (assume a laser suitably attenuated for safety).  The mirrors on the walls, ceiling and floor of this hall will show many images of the man, but they will not show the red dot of the laser because the laser beam doesn’t hit those mirrors.  However, if you turn your back, step to the side, and hold up a rear view mirror, you’ll see both the man and the red dot.  That is because the rear view mirror is inside the cone of light from the laser pointer.   So, to say that the red dot (or the rainbow) does reflect, and that it does not reflect are both true statements depending on which mirror it refers to.  Yet, the image of the man appears in all the mirrors.  The man is a 3D object, but the red dot is not.

So, what do we see on the surface of the lake in the picture? That image is below the horizon from the eye of the camera.  It is reflected light.  It does pass through raindrops first, but not the same raindrops as the ones creating the sky image.

This diagram illustrates, [except for the important fact that light rays from the sun are actually parallel]. The image seen in the sky comes from different raindrops than the image seen reflected in the lake.  In fact, the reflected image appears to be below the horizon (dotted line).  In that sense it is true that the image seen on the lake surface is not a reflection of the rainbow seen in the sky.  It is a reflection of a different rainbow.

Using the same logic, it is correct to say that nobody else can see the same rainbow that I do.  As I stand, a person sitting near me sees light reflected from different raindrops than I do.  And if we consider light reaching me and him from the same raindrop, I might see green light while he sees red.  That is because of the difference in angles between the drop, my eye, and his eye.  In that sense, his rainbow is not the same as mine.

I was going to end this article here, but PF mentor jedishrfu urged me to continue with other rainbow effects, so I will.  One of Wikipedia’s very best articles, Rainbow, is a rich source.

## Double, Triple, Quadruple, and Quintuple Rainbows

The primary rainbow is reflected once inside the rain drop.  It is possible to have 2, 3, 4, or 5 reflections before the beam exits the drop.   Of course, higher order rainbows are dimmer and thus harder to see.  (Picture source Tuscon News Now.)

Most of you have probably seen a double rainbow in real life.  People living close to the arctic circle have probably seen triple rainbows.

If you see higher order rainbows, they are always outside and concentric from the primary rainbow.  If the order of colors in the odd-numbered rainbows is ROYGBIV, then the order in the even numbered rainbows is VIBGYOR.

## Twinned Rainbow

Let me make an assertion.   To see a rainbow, many or most of the raindrops must be of uniform size and shape.  My argument for that is that drop size and shape are related.   As the drops fall, air resistance flattens small drops more than large drops. Flattened drops reflect at a different angle.  Flattened drops might also oscillate or tumble. If the statistics of the raindrops are diverse, then light will be reflected in many directions other than toward your eye, so you won’t see a rainbow.  If you do see a rainbow, that is direct evidence of the uniformity uniformity or correlation of the drop statistics.

A twinned rainbow happens when there are two predominant drop sizes.  The small flattened drops reflect at a different angle than the larger drops.

## Circular Rainbows

A rainbow is usually seen as less than half a circle.  In the right circumstances, full-circle rainbows can be seen above you in the sky, or below when you view from high up.  However, the physics is the
same.

Contrary to my prior beliefs, I learned that full circle rainbows are not the same physics as a glory or a 22° halo.

Wikipedia says that the glory is believed to happen due to classical wave tunneling,  when light nearby a droplet tunnels through air inside the droplet and is emitted backwards due to resonance effects. (Picture Brocken Inaglory)

The 22° halo is due to reflections in ice crystals that are hexagonal, not spherical. (Picture Gladson Machado)

Sun dogs are closely related.  The image below shows very bright sun dogs in Fargo, North Dakota. Also visible are parts of the 22° halo (the arcs passing through each sundog), a sun pillar (the vertical line) and the parhelic circle (the horizontal line).

## Supernumerary Rainbow

A supernumerary rainbow—also known as a stacker rainbow—is an infrequent phenomenon, consisting of several faint rainbows on the inner side of the primary rainbow, and very rarely also outside the secondary rainbow. Supernumerary rainbows are slightly detached and have pastel color bands that do not fit the usual pattern.

It is not possible to explain their existence using classical geometric optics. The alternating faint rainbows are caused by interference between rays of light following slightly different paths with slightly varying lengths within the raindrops.

The very existence of supernumerary rainbows was historically a first indication of the wave nature of light, and the first explanation was provided by Thomas Young in 1804 — Wikipedia Rainbow

A supernumerary rainbow picture by Andrew Dunn

## Reflected and Reflection Rainbows

The picture of the boat shows a reflected rainbow.  We already discussed the physics of that.

A reflection rainbow is seen when the sunlight is reflected from the water surface first, then up to raindrops in the sky, then internally reflected by raindrops back toward your eye.

## Monochrome  Rainbows

When the sunlight is red, such as at dawn or dusk, then rainbows can still be formed but they will be monochrome.   (Image by  www.rodjonesphotography.co.uk)

A monochrome rainbow in a red sky

When rainbows are formed by moonlight, they appear to be the

same white color as The Moon.   Fogbows also appear to be mostly monochromatic.  Fogbows are formed by the same physics as rainbows, but they are made with much smaller drops closer to the horizon.

Someday (probably well beyond my lifetime) we could have a close passage by a large comet with a brilliant tail.  If that happens, rainbows could be formed with comet light.  They would show us the spectrum of the comet’s light source, monochromatic or not.

Could we see a rainbow formed by the light from an aurora in the stratosphere?  That sounds difficult (because the aurora is not a point source) but not quite impossible.  That is a fun speculation.  Maybe someone could finance a year for me in Iceland so that I could research that.

## Mystery Rainbow

This remarkable picture, by dabrandner, appears to show a
third rainbow sandwiched between the primary and secondary double rainbows.  I call it a mystery because I’m not sure if the third rainbow is a twinned rainbow or a reflection rainbow, or some other kind. [Reader comments are welcome.]

## Conclusions

I confess that when I started this article with just a picture of my boat, I had no idea how far my research would take me into interesting physics phenomena. Thank you jedishrfu for egging me on.

Using the sky as a scientific instrument, we can observe:

• How light behaves inside the raindrops.
• The reflection behavior of unidirectional versus omnidirectional light beams.
• The statistical distribution of drop size and shape.
• Some quantum effects (Superluminary)
• Evanescent wave coupling, an exotic field of optics I never heard of before.
• A spectrograph of the light source.
• Science history from 1804.

I’m sure you will agree, that is very cool.

Thanks to jedishrfu and to Greg Bernhardt for their assistance and encouragement.

88 replies
1. anorlunda says:

[QUOTE=”OmCheeto, post: 5335695, member: 103343″]Being that the rainbow in the “What caused the Quadruple Rainbow?” thread looks suspiciously like your “mystery” rainbow, I’m going to guess that yours was a “reflected” rainbow.[/QUOTE]

I think you mean “reflection” rainbow, not “reflected” in the way that I used those terms in the article. The reflected rainbow is seen on the lake surface. The reflection rainbow is seen in the sky.

2. anorlunda says:

[QUOTE=”spareine, post: 5338523, member: 351563″]Why denying virtual objects?[/QUOTE]
I think you missed the point, as astronomer Berman said, “Rainbows are not 3D objects.” I too struggled with what that means.

Refer back to the diagram in the article that shows my eye, two drops (one high one low), and a sky image plus a lake image. Now suppose that the rain just began so that the high drop is there but there is no low drop. Then I will see the sky image but no lake image at all. Conversely, if the rain abruptly ended, the low drop could be there but no high drop, so I would see the lake image with no sky image. 3D objects, parallax, vanishing points, and virtual images are optical phenomena that propagate at the speed of light, so that if I saw the sky image I should simultaneously see the lake image.

The same applies if you stand beside me. Your eyes see rainbow light from different raindrops than my eyes see. Because those other raindrops may be missing, or have other properties, your eye sees a different rainbow than my eye. In most, [U]but not all[/U] circumstances, they look alike. That’s why the conventional optics rules for 3D images don’t apply.

The point may be clearer if we refer back to the article’s analogy with a man pointing a perfectly collimated laser at my eye. My eye sees the man and a red dot. My other eye (or you standing beside me) sees the man but no red dot. Conventional optics, including vanishing point, apply to the image of the man, but not to the red dot. The rainbow you see is more analogous to the red dot than to the image of the man.

3. S
sophiecentaur says:

One way to think of a rainbow is that it is just a highly distorted virtual image of the Sun. There is a large amount of Chromatic aberration and there is also a lot of spatial distortion – so much so that we see the image as a ring. Because of the apparent direction of arrival of the sections of the bow, there is no parallax so the image appears at infinity. All very confusing and not like other things we see in the sky.
[QUOTE=”spareine, post: 5338523, member: 351563″]Maybe it is useful to add that your statement that two persons cannot see the same rainbow is a denial of the concept of virtual objects.[/QUOTE]
I am sure that, if rainbows didn’t look so gorgeous, they wouldn’t have got into folk lore and they wouldn’t have that extra magical quality that seems to make people treat them differently from other optical phenomena.
I am not being Mr Grumpy about this. I am just suggesting that our intuition is not the best way to attempt descriptions and explanations of rainbows. People are really bad witnesses when asked to describe what they actually see; they seems to believe that the bow actually goes into the ground, for instance. Fact is that, very often, you can actually see a hint of rainbow in front of the ground. (Paradoxical if you want to place the rainbow ‘somewhere’ but no more so than looking at an image in {behind} a mirror).

[QUOTE=”anorlunda, post: 5338681, member: 455902″]The point may be clearer if we refer back to the article’s analogy with a man pointing a perfectly collimated laser at my eye.[/QUOTE]
I don’t wee why it has to be a laser beam. The same thing would apply to any object that happens to be obscured from one viewer and visible to the other. One observer is aware of the object and the other is not.
And, for an object / image to be “3D’ there has to be a spread of distances from the observer to different parts of the object and parallax effects should also be seen. It’s all at infinity so that hardly applies.

4. anorlunda says:

[QUOTE=”sophiecentaur, post: 5338727, member: 199289″]I don’t wee why it has to be a laser beam.[/QUOTE]
Because light reflected from an ordinary object goes in many directions. It is (at least partially) omnidirectional light.

Collimated light is unidirectional. A collimated beam can be aimed at your left eye and 0% of its light reaches your right eye.

So a man holding a paper with a red dot printed on it, is very different from a man holding a red laser pointer.

5. S
sophiecentaur says:

[QUOTE=”anorlunda, post: 5338738, member: 455902″]Because light reflected from an ordinary object goes in many directions. It is (at least partially) omnidirectional light.

Collimated light is unidirectional. A collimated beam can be aimed at your left eye and 0% of its light reaches your right eye.

So a man holding a paper with a red dot printed on it, is very different from a man holding a red laser pointer.[/QUOTE]
I can hold my hand in front of me so that I can see an object with one eye but not the other. Is there any (relevant) difference. Colimation is a way over the top requirement for this explanation. Speaking as one who did all the basic Physics learning in the absence of handy laser pointers, I often find that people reach for a virtual laser to prove points when simple shadows can do just as well. What did the 19th century opticians do when they wanted to explain things?

6. anorlunda says:

[QUOTE=”sophiecentaur, post: 5338745, member: 199289″]I can hold my hand in front of me so that I can see an object with one eye but not the other. Is there any (relevant) difference[/QUOTE]
A big difference. Zero photons from the laser reach the other eye, with our without your hand.

7. Orodruin says:

[QUOTE=”anorlunda, post: 5338750, member: 455902″]A big difference. Zero photons from the laser reach the other eye, with our without your hand.[/QUOTE]
But in the case of the rainbow you are making a virtual image of the sun at infinity. Even if the different rays from this virtual image passes through different rain drops, it does not make them part of a different virtual image. When you construct virtual images of objects using lenses, the rays take different paths as well, passing through different parts of the lens. The important thing is that a virtual image is constructed. It [I]is[/I] this virtual image which is reflected. Fine, the rays did not pass through the same rain drops, but the virtual image is not where the raindrops are.

[QUOTE=”anorlunda, post: 5338681, member: 455902″]Now suppose that the rain just began so that the high drop is there but there is no low drop[/QUOTE]
You could likely get similar effects with the virtual images created by lenses by placing a mirror beyond the lens, clearly the virtual image is not going to be reflected.

8. S
spareine says:

[QUOTE=”anorlunda, post: 5338681, member: 455902″]I think you missed the point, as astronomer Berman said, “Rainbows are not 3D objects.” I too struggled with what that means.[/QUOTE]

I agree that rainbows are not 3D objects. Our difference is simply that I prefer to extrapolate the light rays back to infinity. For me raindrops are merely mirror particles at a finite distance, they are certainly not the location of the rainbow. Extrapolate the light rays back to infinity to find the virtual object. The celestial sky is the location of the rainbow. Everybody sees the rainbow at the same location in the celestial sky.

9. S
sophiecentaur says:

[QUOTE=”Orodruin, post: 5338868, member: 510075″]When you construct virtual images of objects using lenses, the rays take different paths as well, passing through different parts of the lens.[/QUOTE]
The imaging forming structure in a rainbow is different from what happens in a lens – it’s more like a multiplicity of lenses, with each lens contributing within a narrow angle. It’s a bit like what happens with a lenticular screen or a fresnel lens. I think it’s a bit pointless to try to make the rainbow fit in with the more straightforward images that we see. Of course the image is not ‘real’ because the light behaves as if it comes from way behind the image forming structure. There is no parallax against distant objects so it can be classed as infinitely far away.
[QUOTE=”spareine, post: 5338924, member: 351563″]Everybody sees the rainbow at the same location in the celestial sky.[/QUOTE]
I don’t think they do, exactly. The distances are so large that it would be difficult to spot but when you move to the left, the bow moves to the left, with you. So it would be moving across the sky relative to the distant stars. The centre of the bow is in line with the Sun and the back of your head. But a rainbow at night? Weird idea! Perhaps it’s an experiment you could do with the Moon – if you could arrange the rain to come at the right time of the day and month. But you would need to travel quite a distance sideways to see the effect against the moonscape as a background. (many km to observe a recognisable movement of a fuzzy thing like a rainbow.

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