the magic of the rainbow

7/30/2019 The Magic of the Rainbow

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The Magic of the Rainbow

The Optics Behind the Multi -colored Fascination

By Anthony Stauffer

Photo courtesy ofhttp://lookingtothesky.com/2011/03/double-rainbow/

For as long as humans have been thinking cognitively and wondered about the world

around us, the rainbow has always been a mystery. It has made its way into the worlds of gods,

magic, and folklore. It has become a symbol of diversity, harmony, and technological innovation.

And, to this day, is still being studied by scientists for all its optical wonder. Perhaps its most

famous usage is in the myth of leprechauns, who always hid their pot of gold at the end of the

rainbow. The search for the end of the rainbow has been an ongoing one ever since. In the 20th

century, rainbows once again appeared, but in Hollywood, with the famous song Over the

Rainbow, a part of the soundtrack of the Wizard of Oz. In this song, it is said that over the

rainbow, skies are blue. Could this be one reason why the gay and lesbian community has

adopted the rainbow as its symbol? Blue skies are peaceful, and that is what they are looking for,

and why their diverse community is standing together in the face of much hatred. Strength in

diversity is where peace may be found once you get past it

But what of the science involved? Is there really an end to a rainbow? Can we ever find

it? And why do we sometimes see two or even three rainbows at the same time? While the
http://lookingtothesky.com/2011/03/double-rainbow/http://lookingtothesky.com/2011/03/double-rainbow/http://lookingtothesky.com/2011/03/double-rainbow/http://lookingtothesky.com/2011/03/double-rainbow/


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cultural stories involving the rainbow are indeed fascinating, the science behind how rainbows

form and their properties are just as fascinating. Will somebody ever find the pot of gold at the

end of the rainbow? Read on and find out.

Ray of Light

A late afternoon summer storm has just rolled through, and, as you make your way back

outdoors, you look toward the storm and see a vibrant rainbow in the sky. Since you have

nothing to do, it being a Saturday, you decide to hop in your car and start driving towards the

rainbow in the hopes of finding the end of it. You drive towards the storm, the Sun at your back,

and you realize that the rainbow remains at the same distance from you; and it appears to dim. It

doesnt get bigger or closer. What is going on? Then, suddenly, you get close enough to the

storm that the Sun disappears behind the clouds and the rainbow vanishes. You pull off to the

side of the road and wait. In a minute the Sun reappears and the rainbow returns, the same height

as it was previously. As you watch, the rainbow brightens, and it continues to brighten as the

storm recedes.

You get on your cell phone and call your friend to tell him of your experiences. He lives

across town, about 2 miles behind you. He answers the phone and you ask him if he sees the

rainbow. The reply startles you a bit, for he says that he does see it, but its not a full arch, only

the top of it. You ask your friend if hes crazy, because youre looking right at it and it is very

bright and full. In fact, you can now see a second rainbow curving above the first. No dice, your

friend now says that he can barely make out the first. Its about to disappear. This just keeps

getting stranger. How can he not see the same thing you are? Driving back home, you sit in

silence, pondering. What you used to think about rainbows has now been seriously brought into

question. As with many things in our life, we will find out that perception is reality. From ascientific standpoint, most perception questions can be written off as a product of the human

mind. However, this time, science says that perception truly is reality. It all starts with a ray of

light.

Refraction, Reflection, and a Cone of Light

Weve all seen a glass prism, and we have all seen how it can take light, or white light in

physics terms, and turn it into all seven colors of the rainbow. But how does it do this? Lets take

one more step back. You are standing in a swimming pool a couple of hours before the storm

rolls in, and youre bored. Your mind starts to wander, and in so doing, you look down at yourfeet and notice that they seem to not be attached correctly to the rest of your body. They are

offset. This is because, as a medium for light to travel through, the air and water between your

eyes and your feet actually changes the velocity at which the light travels. By changing its

velocity, the air-water boundary causes the light to bend, or refract.

In terms of a prism, this refraction also happens, but with a little something special added

in. Because of the type of medium the prism is, glass, the angle of refraction at the boundary is


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different for all colors of light. This separation of light is known as dispersion, and it was Sir

Isaac Newton who discovered that white light is actually the combination of all colors of light.

However, there is also some reflection that happens within the prism. As most of the light will go

through the refraction process going into and out of the prism, there will also be some amount

that gets reflected off of the outgoing surface and passes through the bottom surface.

Depending on the refraction index (a measure of a prisms ability bend a beam of light), the light

that exits the bottom surface, or other surfaces as the light continues to be reflected with in the

prism, may or may not be dispersed.

The water droplets that are necessary to form a rainbow act in much the same way as a

prism. The difference is that the air-water boundary always acts as a dispersive surface. The

reason for this issurface tension. A property of any liquid, surface tension is the force of

molecular attraction of boundary molecules to internal molecules, a force that is perpendicular to

the surface and pointing into the liquid. Therefore, while the drop of water is suspended in the

air, surface tension forms the droplet into a near perfect sphere, creating a lens. This spherical

droplet now allows for any number of refractions and internal reflections, referred to as primary,

secondary, tertiary, etc.

To form the primary rainbow, rays of light from the Sun will enter the upper hemisphere

of the raindrop, be refracted at the boundary, and then reflect off of the back of the raindrop, on

its way to a second refraction in the lower hemisphere upon exiting. The value of the angle

formed between the ray of incident sunlight and the exiting light is42 for red and 40 for blue

(the Rainbow Angle). The reason why it is best to see a rainbow during the morning or late

afternoon hours is due to theangle of the Sunin the sky behind us. Because our eyes act as a

point receptor of light, tracking all the incoming light rays will form a cone in front of us. For the

refracted light from all the raindrops to enter our eye, these rays of light must follow the conic

The photo to the left shows a wide beam of white

light being refracted and dispersed into the many

colors of the rainbow. You can also see the faint

beam going downward, which is the incident

beam, being reflected directly off of the prisms

surface. The beam going upward has been

reflected off of the outgoing surface, then it has

been refracted again as it exits the prism. Notice

with this prism that full dispersion does not occur

until the incident beam has gone through two

successive refractions.

Photo courtesy of

http://exoplanet.as.arizona.edu/~lclose/a302/lec

ture14/lecture_14.html
http://en.wikipedia.org/wiki/Surface_tensionhttp://en.wikipedia.org/wiki/Surface_tensionhttp://en.wikipedia.org/wiki/Surface_tensionhttp://www.atoptics.co.uk/rainbows/primrays.htmhttp://www.atoptics.co.uk/rainbows/primrays.htmhttp://www.atoptics.co.uk/rainbows/primrays.htmhttp://www.daviddarling.info/encyclopedia/R/rainbow.htmlhttp://www.daviddarling.info/encyclopedia/R/rainbow.htmlhttp://www.daviddarling.info/encyclopedia/R/rainbow.htmlhttp://exoplanet.as.arizona.edu/~lclose/a302/lecture14/lecture_14.htmlhttp://exoplanet.as.arizona.edu/~lclose/a302/lecture14/lecture_14.htmlhttp://exoplanet.as.arizona.edu/~lclose/a302/lecture14/lecture_14.htmlhttp://exoplanet.as.arizona.edu/~lclose/a302/lecture14/lecture_14.htmlhttp://exoplanet.as.arizona.edu/~lclose/a302/lecture14/lecture_14.htmlhttp://www.daviddarling.info/encyclopedia/R/rainbow.htmlhttp://www.atoptics.co.uk/rainbows/primrays.htmhttp://en.wikipedia.org/wiki/Surface_tension


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pattern. Therefore, the rainbow we see will be circular in shape. However, we do not see the

whole circle of the rainbow because the center of that circle is in the ground before us. Known as

the Antisolar Point, if we direct our eyes to the edge of our shadow in front of us, then our eyes

will be resting on this point and the center of the circle for the rainbow. Thus, all we see is the

arc of the rainbow that lay above the ground.

In this photograph, you can see the primary rainbow arc and its relation to the Antisolar Point

at the edge of the observers shadow. Inset A shows the refraction/reflection travel path of

light through the spherical raindrop. Notice how the red and blue light paths cross one another

after the reflection point; coupled with Inset B showing the Rainbow Ray (a.k.a. the path of

greatest intensity), this shows why the red light is at the top of the primary rainbow and the

bottom of the secondary rainbow. This will be explained further below, as will the Dark Band

and the Supernumerary bows.

Photo courtesy ofhttp://cliffmass.blogspot.com/2012/11/triple-rainbow.html.
http://cliffmass.blogspot.com/2012/11/triple-rainbow.htmlhttp://cliffmass.blogspot.com/2012/11/triple-rainbow.htmlhttp://cliffmass.blogspot.com/2012/11/triple-rainbow.htmlhttp://cliffmass.blogspot.com/2012/11/triple-rainbow.html


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All Your Colors in a Row

For the primary bow, since an angle of 42 between the incident white light and the

exiting red is the angle of greatest intensity, and since the angle of greatest intensity decreases as

the color progresses towards blue light, the outer edge of our light cone will be red. Moving

inward from the red light takes us through orange, yellow, green, and then blue, with violet andindigo (or purple) finding its way in there too, though its intensity is very low because of its

actual color. Below the 40 refraction angle of blue light, the area under the primary bow is quite

bright as all the remaining light gets scattered by additional raindrops not aiding in the formation

of the rainbow, but whose light is still within our light cone and directed towards our eyes.

Outside the primary rainbow the sky looks a whole lot darker. This is for the very opposite

reason just stated above for the bright region below the rainbow; at refraction angles above 42,

the light is scattered outside of our light cone and appears much darker, and is named, aptly, the

Dark Band.

Above the Dark Band, if conditions permit, we may see a secondary, less intenserainbow. These form due to double reflections of incident white light into properly positioned

raindrops, with the lack of intensity simply due to the double reflection and the loss of overall

intensity as a percentage of the light refracts out of the raindrop at each reflection point. This

concept can be seen in the photo of the prism above. The refraction angles of the individual

colors do not change, but because of the two internal reflections, the placement these secondary

rainbows have is about 8 higher in the sky than the primary. And, as the physics would have it,

from two reflections the red light now has the lower valued rainbow angle than the blue light;

hence, the red light is on the bottom of the secondary bow. From the picture below, you can also

see how, to get proper orientation for the observer to see the secondary rainbow, the incident

light must enter the raindrops in the lower hemisphere and exit from the upper hemisphere.

The orientation angles for primary and

secondary rainbows prove the reason for

the bright region and dark band below and

above the primary rainbow, respectively.

Note: the dark band is also known as

Alexanders dark band after Alexander of

Aphrodisias, who first took note of it in 200

AD.

Picture courtesy of

http://www.daviddarling.info/encyclopedia/

R/rainbow.html.
http://www.daviddarling.info/encyclopedia/R/rainbow.htmlhttp://www.daviddarling.info/encyclopedia/R/rainbow.htmlhttp://www.daviddarling.info/encyclopedia/R/rainbow.htmlhttp://www.daviddarling.info/encyclopedia/R/rainbow.htmlhttp://www.daviddarling.info/encyclopedia/R/rainbow.html


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As you guessed it, there can also be higher orders of rainbows; but, with each successive

internal reflection, higher orders of rainbows have much less intensity and tend to be much

broader than their primary and secondary cousins. Tertiary and quaternary rainbows are even

harder to see as they are seen only when looking sunward, with the Sun itself as the center of

these bows. Third and fourth order rainbows form sunward because the third and fourth

reflections of the light within the raindrops cause the exiting light to be in the same direction as

the incident sunlight. Fifth and sixth order rainbows indeed form opposite the Sun; however, due

to their very low intensity and very broad arrangement, they would be hard-pressed to be seen.

Yet, since the outer red edge of the fifth order bow has an angle of about 53, it may be possible

to see the green and blue sections of these bows within Alexanders dark band.

On Rainbow Pond

Returning, now, to the point in time when you saw the double rainbow; lets change the

geography for a moment and put you on the beach (lucky you!) with your back to the water and

the Sun. As you gaze at the double rainbow, the Suns intensity increases behind you as the

remainder of the storm clouds move to the east; suddenly, there appears in front of you a third

rainbow. However, this particular rainbow appears in between the first two, but at a skewed

angle. Where did this strange, new rainbow come from? Well, if you think about your change in

geography, the answer should be intuitive. The third rainbow is skewed because the center of thisarc is now above the horizon line before you; and it is the result of the Suns reflection off of the

surface of the water behind you. This reflection now forms an analogous reference point to the

Antisolar Point discussed earlier, which is known as theAnthelic Point.

Called a reflection bow, they can form anytime there is a large enough body of water

behind you that can reflect the Suns light into the trailing edge of the storm. And, just as with

The third-order (tertiary) rainbow (left),

accompanied by the fourth-order

(quaternary) rainbow (right). They

appear on the sunward side of the sky,

at approximately 40 and 45,

respectively, from the Sun. This is the

first picture ever of a quaternary

rainbow in nature and the second

picture ever of a tertiary rainbow. Credit:

Michael Theusner/Applied Optics

Photo courtesy of

http://www.science20.com/news_article

s/triple_rainbows_confirmed_exist_and_

apparently_quadruple-83301
http://dictionary.reference.com/browse/anthelion?s=thttp://dictionary.reference.com/browse/anthelion?s=thttp://dictionary.reference.com/browse/anthelion?s=thttp://www.science20.com/news_articles/triple_rainbows_confirmed_exist_and_apparently_quadruple-83301http://www.science20.com/news_articles/triple_rainbows_confirmed_exist_and_apparently_quadruple-83301http://www.science20.com/news_articles/triple_rainbows_confirmed_exist_and_apparently_quadruple-83301http://www.science20.com/news_articles/triple_rainbows_confirmed_exist_and_apparently_quadruple-83301http://www.science20.com/news_articles/triple_rainbows_confirmed_exist_and_apparently_quadruple-83301http://www.science20.com/news_articles/triple_rainbows_confirmed_exist_and_apparently_quadruple-83301http://www.science20.com/news_articles/triple_rainbows_confirmed_exist_and_apparently_quadruple-83301http://dictionary.reference.com/browse/anthelion?s=t


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your run-of-the-mill rainbow, reflection bows can also produce secondaries, tertiaries, and

quaternaries. Unfortunately, primary reflection bows are typically not brighter than secondary

rainbows from the fact that they are produced from a reflection themselves.

The Eye of the Beholder

Remember how we discussed that the reason for a rainbow being an arc was the result of

our eyes acting as a point receptor of light, thereby forming a cone? This is also the reason why,

when you saw the rainbow reappear after ending your chase to find the rainbows end, your

rainbow, and the secondary one with it, was not the same as seen by your friend who was

standing two miles behind you. The rainbows you see are those associated with your own

personal light cone. Standing in close proximity to another person can bring your light cones

close enough together that you could semantically say that you are seeing the same rainbow, but,

in truth, the rainbow you see is your very own.

So, why did your friend only see so little of his rainbow, to the point of watching it

disappear? The answer is the distance he was from the rear of the storm. As the storm proceeded

away from him, the curvature of the Earth came into play. The horizon for an average human

being on flat earth is three miles. Therefore, as the storm moved on, the rain wall that

provided the mechanism for rainbow formation was also receding. When the storm passed the

horizon line, the line-of-sight on the rain wall started to be affected, and the light cone began to

get cut off. All that could be seen was the upper portion of the arch, until the storm got far

enough away that your friends rainbow disappeared.

Photo on the right is provided by Anna Jensen-Clem, taken around 3:15 PM Saturday,

November 24th

, 2012, in Kenmore, WA, just south of the intersection of 68th Ave NE and Bothell

Way. The picture on the left is a schematic representation of the geometry involved in

reflection bow formation, with the same rainbow angles for both types of bows. Photo and

picture published onhttp://cliffmass.blogspot.com/2012/11/triple-rainbow.html.
http://cliffmass.blogspot.com/2012/11/triple-rainbow.htmlhttp://cliffmass.blogspot.com/2012/11/triple-rainbow.htmlhttp://cliffmass.blogspot.com/2012/11/triple-rainbow.htmlhttp://cliffmass.blogspot.com/2012/11/triple-rainbow.html


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Depending on how small the raindrops were that formed your primary rainbow, you may

have seen another phenomenon within the arc towards the top. A non-classical phenomenon of

geometric optics, what you may have seen wassupernumerary rainbows. They are a product of

interference patterns developed by dispersed rays of light following slightly different paths

within the raindrops. Through this, they form bands of very bright arcs that appear almost pastel

in color (though they do indeed have all the colors within them) due to constructive interference

(waves reinforcing each other) with dark bands in between due to destructive interference.

Suppose that maybe, due to a storm updraft, the raindrops forming your rainbow arent

completely spherical; perhaps, they are shaped likeoblate spheroids, like the Earth. Or, maybe

there are two different sizes of water droplets in the air at the same time. Whatever the

mechanism, its possible that you may see a twinned bow, where the arc of the bow appears tosplit into two arcs with the same color organization. Rarely seen, these phenomena have been

photographed, as shown below.

A twinned primary rainbow produced

through computer simulation. The

rainbow is split because of the

interaction of light with two types of

water drops: some smaller, spherical

ones, and some larger water drops

that become non-spherical.

Photo courtesy of

http://www.huffingtonpost.com/201

2/08/10/twinned-rainbows-

formation_n_1764331.html.

Supernumerary rainbows captured by

Lisa Gonelli over the town of

Pilesgrove, NJ, on October 27th, 2010.

You can clearly see three distinct

bands inside the main arc.

Courtesy of

http://epod.usra.edu/blog/2011/01/s

upernumerary-bows.html
http://www.atoptics.co.uk/rainbows/supers.htmhttp://www.atoptics.co.uk/rainbows/supers.htmhttp://www.atoptics.co.uk/rainbows/supers.htmhttp://en.wikipedia.org/wiki/Oblate_spheroidhttp://en.wikipedia.org/wiki/Oblate_spheroidhttp://en.wikipedia.org/wiki/Oblate_spheroidhttp://www.huffingtonpost.com/2012/08/10/twinned-rainbows-formation_n_1764331.htmlhttp://www.huffingtonpost.com/2012/08/10/twinned-rainbows-formation_n_1764331.htmlhttp://www.huffingtonpost.com/2012/08/10/twinned-rainbows-formation_n_1764331.htmlhttp://www.huffingtonpost.com/2012/08/10/twinned-rainbows-formation_n_1764331.htmlhttp://epod.usra.edu/blog/2011/01/supernumerary-bows.htmlhttp://epod.usra.edu/blog/2011/01/supernumerary-bows.htmlhttp://epod.usra.edu/blog/2011/01/supernumerary-bows.htmlhttp://epod.usra.edu/blog/2011/01/supernumerary-bows.htmlhttp://epod.usra.edu/blog/2011/01/supernumerary-bows.htmlhttp://www.huffingtonpost.com/2012/08/10/twinned-rainbows-formation_n_1764331.htmlhttp://www.huffingtonpost.com/2012/08/10/twinned-rainbows-formation_n_1764331.htmlhttp://www.huffingtonpost.com/2012/08/10/twinned-rainbows-formation_n_1764331.htmlhttp://en.wikipedia.org/wiki/Oblate_spheroidhttp://www.atoptics.co.uk/rainbows/supers.htm


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Pot o Gold?

Sadly, now that we know what we do about rainbows, we must accept the fact that

nobody will ever find the pot of gold, or the leprechaun to steal it from, at the end of the

rainbow. But, as a consolation prize, there is a treasure to be had from exploring the world of

atmospheric optics. Aside from the wonders we have discussed above, there are numerous otherways to see refraction and reflection at work.Anythingfrom Moonbows to Dewbows , Ice Halos

to Cloud Bows, the different ways that light can be refracted by water and ice in the atmosphere

provides for a cornucopia of interesting photo opportunities. And you can also take heart in the

fact that, whatever you see, is mathematically for your eyes only. Somebody may see the same

phenomenon as you, in close enough proximity to see something identical; but geometry proves

that nobody will see exactly what you see.
http://www.atoptics.co.uk/bows.htmhttp://www.atoptics.co.uk/bows.htmhttp://www.atoptics.co.uk/bows.htmhttp://www.atoptics.co.uk/bows.htm

the magic of the rainbow

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