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0:00
- [Child] Look at that! So pretty! - The other day my son asked me why rainbows are curved. And I could have given him a simple explanation but instead I made this video.
0:14
With beautiful demonstrations I've never seen before. This is the perspective of a rainbow from a single raindrop. And the best animations ever created on the subject. Because I promise you, almost every explanation out there, double rainbow, (intense music) is an oversimplification.
0:32
For example, if raindrops spread white light into colors like a prism, then why do you never see a rainbow when looking in the direction of the sun? Or why is it darker above a rainbow than under it? How can you make a rainbow disappear with sunglasses? What's going on here? Why is this rainbow so much smaller than usual? And how did this phenomenon directly lead to a Nobel Prize? Oh yeah, we're going deep on this one because the full explanation is so much more satisfying
1:04
than anything you've seen before.
1:07
(rain splattering) To make a rainbow, you need three things, raindrops, the sun and you, an observer. Okay, this is one of those experiments that seems so simple, but I've never seen anyone do it before. I have a glass sphere and that represents my raindrop because as we've learned in previous videos, raindrops, of course, are essentially spherically shaped. (air whooshing) Just preparing the particulate so that I can see the laser beams. Over here I have a laser.
1:41
And that laser beam represents a ray from the sun.
1:46
Rays of light from the sun reach a raindrop essentially parallel to each other because the sun is so far away.
1:53
When light strikes the sphere, some of it reflects back off the front surface and some is transmitted into the sphere. Then at the back, again, some of the light reflects off the back surface and some is transmitted.
2:06
Every time light goes from one medium into another, some of it will be reflected and some transmitted. And exactly how much depends on the angle of the light, its polarization and the nature of the two media. This is actually helpful here because I can use the reflections to make sure the laser is lined up properly. I think I've got this red laser lined up, so it's hitting the middle of the sphere and it's reflecting here, but some of the light goes through
2:31
and some of the light reflects at the back surface, and then some of the light goes through for the wall. Now I'm going to keep the laser horizontal and move it up the sphere. So let's call the distance from the central axis, the impact parameter. (rhythmic music) As I move the laser up, the reflection off the front surface goes up. It's just bouncing off that curve surface. And I can tell you, this is a simple, boring reflection. Nothing interesting happens with it.
2:59
It's not involved in rainbows, so it's there, but we're basically going to ignore it for the rest of the video. What's much more interesting is the reflection off the back surface.
3:10
Now here comes that ray here, this spot is on the table.
3:15
As the laser moves up, it goes down.
3:20
In fact, the whole beam inside the sphere bends down. That's because as the light enters the sphere, it slows down, and so it refracts.
3:31
But why does light slow down when it enters a dense medium like glass? Well, I think a lot of people can tell you that light is an electromagnetic wave without really thinking about what that means. You know the electric field around a charged balloon that pulls on your hair or makes it stick to a wall and the magnetic field around a bar magnet that makes iron filings line up? Well, light is what happens if you could rip the electric field off the charges,
3:58
the magnetic field off the magnet, smoosh them together, and send them out traveling through space, sort of. I mean in practice, electromagnetic waves are made by accelerating charges like by wiggling them up and down. Then the changing electric and magnetic fields, they create, team up as light and head off on their own.
4:19
The clearest explanation I know of for how light is slowed down in a medium comes from Grant over at 3Blue1Brown. I asked him if I could summarize his explanation for this video and he graciously agreed. So when those electromagnetic waves encounter charges in a medium like those in the first layer of our sphere, the light pushes them back and forth. You can think of each charge as a little mass on a spring and the changing electric and magnetic fields cause it
4:46
to vibrate at the same frequency as the light.
4:49
But now you have wiggling that is accelerating charges. So they too must create their own electromagnetic waves. And the net electromagnetic field is just a sum of the incident wave plus this new wave. The result is almost exactly like the original wave, except it has shifted back slightly. It receives a phase kick and each layer of the material adds another phase kick. So the net effect of all this is the wavelength of the radiation decreases in the new medium.
5:23
And since the frequency stays the same, a shorter wavelength decreases the speed of light through the material. The speed of light in a vacuum divided by the speed of light in a medium is called the refractive index. It's around 1.5 for glass and 1.33 for water. So when light enters a new medium at an angle, the part of the wave crest that enter the new medium first, slow down first. And this changes the angle of all the wave crests. And since the direction of a beam of light is perpendicular
5:55
to the wave crests, this means the light changes direction.
6:00
So to recap, light causes charges to wiggle, so they create their own electromagnetic wave, which gives the light a phase kick shortening its wavelength which slows it down and so it bends.
6:14
There's a simple mathematical expression that relates the angles of incidence and refraction to the indexes of refraction of the two media. It's known as Snell's law, even though it was independently discovered by a handful of people, some well before Snell. (intense music) And this is what we're seeing in the sphere. The higher the laser hits the sphere, the larger the angle of incidence and hence the more it bends down due to refraction. Most of this light exits out the back of the sphere
6:42
but some of it is reflected. And it's this reflected ray that we can see coming out the front of the sphere below the incident beam. Now I wanna graph the angle of this reflected ray as I move the beam up the sphere. When the laser is dead center, the reflected beam comes straight back at the source. So let's call that 0 degrees. Then as I move the laser up, this angle increases. So the light's coming back at 5 degrees, then 10 degrees, and it keeps increasing the higher I go.
7:15
But now we come to the critical point. Watch this spot on the table. As I move the incident beam up, this dot is moving in. The reflected ray is coming closer and closer and closer.
7:29
But there's a certain point right there where it stops coming closer.
7:35
Look at that. And even as I keep moving this beam higher and higher, it doesn't get any closer. (intense music) And then it goes back the other way.
7:49
So what we're seeing there is a maximum angle this reflected ray reaches before it turns around and goes back the other way.
7:58
(intense music) And this is really important. It means that over a range of impact parameters, a range of heights of the laser, the reflected beam comes out at essentially the same angle, which means the light is becoming concentrated at that angle. And a concentration of light rays is called a caustic.
8:21
Curved surfaces all tend to create caustics, from coffee mugs to glasses or even just rippling water. Caustics create the light patterns we see by concentrating light rays. In the case of red light through a sphere of water, the maximum scattering angle, and hence this caustic always occurs at 42 degrees below the horizontal. Since my sphere is made of glass rather than water, well, the angle is different, but the principle is the same.
8:50
Now you might ask, why does this reflected ray reach a maximum angle and then turn around? Well, the answer is just geometry. As I move the laser up the sphere, although the ray refracts down, the point on the back of the sphere where it reflects continues to move up until you get to this special point, which is about 7/8 the radius of the sphere. And here, the angle of incidence is so steep that the refracted ray stops hitting the back higher and starts hitting it lower.
9:21
That is why the reflection turns around and therefore we get a maximum scattering angle and the concentration of light rays at that angle. But the precise maximum scattering angle depends on the color of light.
9:36
To see why, let's go back to the idea of the charges in the sphere as masses on springs. They have a natural frequency, a frequency at which they would oscillate if not driven at any particular frequency. And in most materials, this natural frequency is pretty high, much higher than the frequencies of visible light. Now, when light pushes a charge back and forth, the amplitude of the resulting vibration depends on the difference between the frequency of light
10:02
and the natural frequency of the charge. The closer the two frequencies are, the greater the amplitude of the resulting vibration, which makes sense if you've ever pushed someone on a swing. The closer your pushing frequency is to the natural frequency of the swing, the higher they'll go. This means that higher frequency light like blue light will cause the charges to wiggle with greater amplitude. And because of this, the charges produce higher amplitude electromagnetic radiation,
10:30
which creates a bigger phase kick, which shortens the wavelength proportionally more, making higher frequency light travel slower and bend more than lower frequency light. All right, I'm gonna change lasers.
10:45
So when I repeated the experiment with green light, it refracted more than red light and therefore the green dot turned around sooner than the red dot. Whoa oh
10:56
there's the minimum deflection for green and it's significantly different than for red. In other words, its maximum scattering angle was smaller than for red light. If the sphere were water, it would occur at around 41 degrees below the horizontal. Similarly, for blue light, the maximum scattering angle approaches 40 degrees. I do have a specialty laser, which is very bright blue, but it's very dangerous so we're gonna use it very carefully. how are we gonna use it very carefully, Derek?
11:28
(tool thuds) That is a good question. (tape ripping) Maybe we're just gonna tape it on here.
11:36
Yeah. (playful music)
11:46
So the blue only makes it to there, green here, red to here. So it is a pretty serious spread here. I think my experiment may be a little wonky honestly, I think this one wasn't perfectly horizontal.
12:00
To really see the importance of the caustic, imagine we illuminate the sphere uniformly with red light. Well, more light is going to hit the sphere at higher impact parameters because the further out you go, the more area there is. So I've adjusted these sections so that they all have the same area. Then using our graph of the scattering angle, you can see where all of this red light will end up after reflecting off the back surface. Most of it ends up at the maximum scattering angle.
12:30
To make this more obvious, we can add more light and we can do the same thing for orange and yellow and all of the other colors. And this is what gives us the rainbow. (soft choral music) It's not enough to say that a raindrop spreads white light into its component colors because all of the light that hits closer to the middle is spread too. But since the reflections all overlap as they come out, the colors mix and produce white again. It's only the difference in maximum scattering angles
13:03
and the caustics they produce that gives us the rainbow. (soft music)
13:11
So now we know what happens along a single radius of the sphere. So what happens if we uniformly illuminate the whole sphere with white light? (bright music) Well, I blacked out my window and cut a hole just big enough for afternoon sunlight to cover the sphere.
13:30
You can see there's a circle of white light coming from the reflections off the back of the sphere, and then around it, there's a ring of rainbow colors.
13:42
Come on.
13:44
One raindrop creates a cone of light. The inside is all white, and the ring around the outside is colored.
13:54
This is the perspective of a rainbow from a single raindrop.
13:58
All these different light rays coming in at different places reflect back off the front surface and the back surface, that reflection off the back surface reaches a maximum angle. And for blue, green, yellow, and red, the maximum angle is different. So the red maximum angle is the furthest. That's why it's on the outside here. So good, so good. (light music) What I really wanted to see is if I could observe the cone of different colored caustics that produced that ring. Ho ho ho ho ho ho ho
14:30
ho ho ho ho. You've got to see this color cone. You have got to see this.
14:37
My eye is right in the color cone here, I can see the color cone. That is so cool.
14:42
(intense music) So this is the crazy focus on the back of the sphere.
14:51
If you stick your finger in there, it gets burnt very quick. Ow! That thing is a focal.
14:59
(intense music) Now this really looks like a rainbow, but remember, this is just the light coming away from a single droplet. When you see a rainbow, there are billions of raindrops, and each one is projecting a rainbow cone back toward the sun. So how does all of this create a single unified rainbow? Well, for your eye to see a color, let's pick red in a certain part of the sky, then the red caustic from a raindrop there must go directly into your eye.
15:30
And this only happens when the angle from the sun to the raindrop to your eye is 42 degrees. And this explains why rainbows take the form of an arch with a 42 degree angle. (intense music) Now, the violet light from these same raindrops passes above or beside your eye, so you can't possibly see it. But there are raindrops below and inside the arc of those red giving raindrops whose violet caustics do intersect your eye. They form a shallower angle of 40 degrees between the sun and your eye.
16:06
And of course, there are raindrops at all intermediate angles that send you all the other colors of the rainbow.
16:14
So a rainbow really is the ultimate optical illusion from billions of droplets each projecting a rainbow cone. You see a single static arch of color, but the droplets sending you those colors are constantly changing. A single drop as it falls might send to your eye first red, then orange, yellow, green, blue, indigo, and violet. (soft music) And because a rainbow must form an angle of 40 to 42 degrees with your eye, the center of the arch must be on a line
16:47
that passes from the sun through the back of your head. So your shadow is the center of your rainbow. This means no two people can ever see the exact same rainbow. In fact, your left and right eyes don't even see the same rainbow. A rainbow is an optical illusion made unique for each perspective. This also explains why in most parts of the world, you can only see a rainbow in the early morning or late afternoon, not in the middle of the day.
17:18
The higher the sun is, the lower the top of the rainbow is. And when the sun is more than 42 degrees above the horizon, no rainbow is visible from the ground. (water splattering) But even when rainbows are visible, you can turn them invisible using sunglasses that is as long as they are polarized.
17:41
Light from the sun is unpolarized, which means the electric fields of the light are all randomly oriented, oscillating back and forth equally in all directions. But it just so happens that when the light in the rainbow ray reflects off the back of the droplet, it does so very close to a special angle known as Brewster's angle. At this angle, all light with its electric field oriented parallel to the plane of reflection is transmitted. So it passes out the back of the droplet,
18:12
and therefore the only light that is reflected has its electric field perpendicular to the plane of reflection. This is the light that creates the rainbow.
18:23
This means rainbow light is polarized along the direction of the rainbow, so horizontal at the top and closer to vertical on the sides. This is why you can use a polarizing filter to make a rainbow disappear or to make it brighter if you orient the filter to allow that polarized light to pass through.
18:44
But why is it brighter under a rainbow than above it? Well, this is because the raindrops beneath the rainbow are reflecting all colors of light at you off their back surfaces. This is what created the white disk in my glass sphere experiment. In contrast, the raindrops above the top of the rainbow are not reflecting any light to you off their back surfaces. Your eye is now outside the maximum deflection angle of all of the colors. But if you look up even further,
19:15
sometimes you see a second fainter rainbow with its colors inverted. Double rainbow. So where does this come from? Well, it comes from an additional reflection inside the raindrops. Now, instead of reflecting once off the inside of the sphere, light reflects twice. These reflections also create colored caustics, though much fainter because light is lost with each reflection. If you look at deflection angles, this light starts going out the back of the raindrop. So at an angle of 180 degrees,
19:47
but the further out light hits from the center, the smaller the angle light reflects back at until it reaches a minimum of around 50 degrees for red light. Then it turns around and goes back the other way. So between 42 and 50 degrees, it is dark because no light reflected once or twice inside a raindrop comes out at this angle. This is known as Alexander's Dark Band. Now, there is photographic evidence of third and fourth order rainbows formed after three or four internal reflections,
20:23
but this light comes out the back of a raindrop. So they are the only kinds of rainbows that you could expect to see when looking in the direction of the sun. But they're so faint that conditions would have to be perfect. Under lab conditions, up to 200th order rainbows have been detected,
20:41
but that is not what is going on here. This is known as a supernumerary rainbow. Multiple rainbow like bands show up under a primary rainbow, but this only occurs when the raindrops are all really small, just 10ths of a millimeter in diameter. Now, the light rays that passed just above and below the primary rainbow ray end up coming out at similar angles under 40 degrees, but they travel slightly different distances on the order of a wavelength.
21:12
And because of this, those light rays can interfere constructively and destructively, producing a series of light and dark bands inside the main rainbow. Different colors in these supernumerary rainbows overlap more than in the main rainbow, so they can produce strange colors like magenta, a combination of blue and red. Supernumeraries also offer a clue to how these small rainbows work. Whenever I saw images like this or even observed this sort of thing from an airplane,
21:44
I wondered how a rainbow could be so small. (soft choral music) These are known as glories or Brocken bows. Instead of the usual 42 degrees, these circles of color are only around two to four degrees wide. Well, the key is that just like in supernumeraries, glories are due to interference. So they too require tiny water droplets just 10ths of a millimeter in diameter. These are the sorts of droplets you'd find in fog or clouds.
22:15
Light that strikes the edge of the drop can go around the back and come straight back at the source. You can see that with the laser on the glass sphere, but effectively in the presence of parallel light rays, tiny little droplets become a ring source of light. But for these tiny droplets, the distance from one point to all edges of the sphere can vary on the order of a wavelength. So take for example, the point right out in front of the drop. Well, now the distance to all edges is the same.
22:50
So the light interferes constructively here and produces a bright spot. But a little bit off to one side and now half of the light, on average, has traveled an extra half a wavelength. And so we get a dark spot here. If you go a little further, well now the light has traveled an extra whole wavelength. So now the light is arriving in phase again, and we get a bright spot here. So we can rotate this around 360 degrees and extend it out in all directions.
23:17
And what we get is a fuzzy bullseye pattern. And of course, all the different colors of light have different wavelengths. And so these bullseye patterns aren't completely overlapping. So when they're superimposed, what we see is rings of color.
23:33
Now, this is just the pattern from a single droplet, but just like with a rainbow, if you have millions or billions of these droplets, they all contribute to produce this same pattern with your shadow at the center.
23:46
(soft music) And it was just such a pattern that inspired a Nobel Prize winning discovery. In September of 1894, a scientist named CTR Wilson was visiting an observatory in the Scottish Hills. It was then that he observed "the coloured rings surrounding the shadow cast on mist or cloud."
24:08
He recalls that these glories, "greatly excited my interest and made me wish to imitate them in the laboratory." So Wilson invented the cloud chamber for the explicit purpose of observing glories. Of course, once he discovered the cloud chamber, made the tracks of energetic particles visible, he abandoned his original aim and was later awarded the Nobel Prize. But it all started with the mystery of rings of color in the fog.
24:41
- [Child] Wow. Wow, looks so nice. - How did you see that? So now I hope you know my son, why rainbows are curved, and why they're polarized and why they exist at all. And even more than that, I hope you know why I find such enjoyment in learning about our world, why it is worth figuring things out. For millennia, rainbows have been this blatant challenge held up to us by nature. But can you figure this out? And it's satisfying to say we have.
25:15
- [Child] I'm looking at a different rainbow than you. - It's true. (celestial music)
25:23
When I started researching this video, I thought I already knew about rainbows. I mean, I'd learned all of the colors of the rainbow in school and that it was caused by light refracting and reflecting, but I realized now that I'd only really just memorize things about rainbows and not really understood how they work. Learning should be about mastering a subject, not memorizing a list of facts. And that's why we asked Brilliant to sponsor this video.
25:49
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26:16
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26:42
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27:08
and I wanna thank you for watching.
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