Have you ever sat on the bottom of a swimming pool and pondered your watery ceiling? Most of the surface is a sheet of light blue, and you can’t see through it, even though the water is clear. But right above you, there’s a round window of transparency.
And here’s the awesome thing: Through this ring you get a fish-eye view that shows you not just the sky, but stuff around the side of the pool, like trees or people sipping mai tais on the pool deck. This cool effect is caused by the optical properties of water, and it has a name: Snell’s window.
You can see this even if you don’t spend much time underwater. Perhaps, like me, you prefer to watch spearfishing videos on YouTube. Here is a beautiful example of Snell’s window from the channel YBS Youngbloods (the link takes you right to the 15-second segment of interest).
One curious thing to notice there: As the diver (Brodie) and the camera person descend, the window seems to stay the same size. So what, you ask? Well, think about it: If you filmed a window in your home as you backed away from it, it would appear to get smaller.
In fact, Snell’s window is getting bigger—see how the diver on the surface fills less and less of it? But unlike a window or anything else on dry land, its angular size, as perceived by your eye, stays the same as the distance increases.
Mysteries of the deep! There’s some beautiful physics behind all this, so let’s investigate, shall we?
Refraction and Snell’s Law
Since light is an electromagnetic wave, it doesn’t need a medium to “wave in” (unlike sound). That means it can travel through empty space—as sunlight does, luckily for us. Since light travels at a speed of 3 x 108 meters per second, this trip from the sun to Earth takes about eight minutes.
But something happens when the light enters a transparent medium like our atmosphere: It slows down. Air slows it by just 0.029 percent, but when light enters water it loses around 25 percent of its speed. It’s just like how you slow down when you run from the beach into the ocean, because water is denser than air.
This speed differential varies for different media, and it is described by its index of refraction (n), which is the ratio of the speed of light in a vacuum to the speed in a particular material. The higher the index of refraction, the slower light travels in that medium. In air, n = 1.00027. In water, n = 1.333. In glass, n = 1.5
But here’s the thing: Changing speed also causes the direction of the light to change. That’s actually what we mean by “refraction.” You see it when you look at a straw in a glass of water: The part of the straw underwater doesn’t match up with the part above. Why? The bending of light off the underwater portion causes you to see it somewhere that it’s not.
Maybe this classic analogy will help: Imagine you have a pair of wheels on an axle rolling down your driveway. When the wheels run off the concrete into grass, they slow down. But what if the axle hits the boundary between lawn and driveway at an angle? In that case, one wheel slows down while the other continues at the original speed, causing the path of the axle to change as it enters the grass.
You can see this same effect with waves at the beach. When waves are coming into shore, the speed of the wave decreases when the water becomes shallower. For waves coming in at an angle, you can see the bending of the wave front just like with light going from air into water.
We can calculate the amount that a light wave bends using Snell’s law. For a given light source in the air, we measure the “incident angle” with respect to a vertical line perpendicular to the surface (see the picture below). We use the Greek letter theta for the angle, so θ1 is the angle of the light in the air. Then the refracted light in the water will be traveling at an angle θ2.
Here n1 and n2 are the indexes of refraction for air and water respectively. Oh, and “sin” is not a moral lapse, it’s the sine function. You can see that the final direction of the light wave in water depends on the initial direction of the light in air. As θ1 increases, θ2 must also increase. You can play around with this using this awesome simulator from PhET.
Of course Snell’s window has something to do with Snell’s law. If you start with a light beam pointing straight down at the water from a source overhead, the incident angle θ1 is 0 degrees; then θ2 must also be zero, and there is no refraction.
But as the light source is rotated toward the horizon, resulting in larger angles of incidence, the light bends as it enters the water. Because the two materials have different indexes of refraction, the angles will be different. Here’s a plot of the incident angle (θ1) versus the refracted angle (θ2 ).
You can see that as θ1 increases, θ2 also increases, but not as much, In fact, the refracted angle has a maximum value. Even when the incident light enters the water at 89.999 degrees, the refracted angle never exceeds 49 degrees.
Seeing Stuff
Now for the window part of Snell’s window. If you are underwater and you want to see something above the water, light from that object has to travel in such a way that it enters your eye. Remember that our eyes are basically light detectors—they don’t shoot “vision” out of them (unless you are Superman).
Suppose a bird flies overhead and sunlight reflects off it to hit the water at an angle of 60 degrees. The graph above tells us that this light will refract to a 40-degree angle in the water. If you point to where you see the bird through Snell’s window, you’ll actually be pointing at empty sky.
What happens as the bird moves toward the horizon? The light hits at a larger and larger angle of incidence (θ1 approaches 90 degrees), but the angle of refraction hardly changes anymore. So light from the bird will reach your eye, but it’s going to be squished together with other light with high angles of incidence, so everything gets jumbled and distorted at the edges of the window.
In fact, if you are underwater, the largest angle of light your eyes can detect is 48.6 degrees from the vertical, creating a cone of light with an angular diameter of 97.2 degrees. This limiting angle is Snell’s window. But check it out. The “size” of this window depends only on an angle. That means that as you move deeper, the apparent size (the angular size) of the window is constant.
Total Internal Reflection
Remember in our pool example at the very beginning, we noted that most of the water surface (i.e., apart from Snell’s window), when seen from below is a light blue sheet through which nothing is visible? This is caused by a related phenomenon called total internal reflection.
Imagine that we have light that starts in water and travels up to the boundary between water and air. (Maybe it’s an underwater laser pointer.) Using Snell’s law, we can get an expression for the angle in air (θ1) based on the angle in water (θ2).
Remember that the index of refraction for water is 1.33 and air is approximately 1. This means that n2 /n1 is going to be a number larger than 1. This is important because the value of sin(θ1) can only be between –1 and +1 (because it deals with ratio of sides for a right triangle—but you know that because you paid attention in your trig class).
This gives us a problem. The left-hand side has to be less than 1, but part of the right hand side is greater than 1. Consider the case where θ1 is 90 degrees. Solving for θ2 we get 48.6 degrees. Oh wow—that’s just like Snell’s window. But this says that if the angle of the light in water is at an angle greater than 48.6 degrees, the light doesn’t pass into the air at all.
This seems crazy, so let’s test it out just to make sure. Here’s a clear box of water. If I shine a laser so that it goes through the water at a large angle, the light doesn’t pass into air but instead reflects back into the water.
Now back to Snell’s window. If you are underwater there is this circular area that has light from above the water coming down to your eyes. But what about the rest of the surface? Is it light blue because the sky is blue, or the water is blue? Nope, it’s because your backyard swimming pool is likely painted blue. You’re seeing the reflection of light off the bottom of the pool.
Here’s a picture of me in the ocean. You can see the blue sky and clouds through Snell’s window, but the green color around it in this case is light reflecting off the sand on the seafloor. It’s light that comes from the sun, enters the water, and then bounces between the bottom and the water surface to come back to the camera. Pretty cool.
Also, it’s possible that you are using total internal reflection right now. One common application of this phenomenon is in fiber optics. These are thin filaments of glass. Remember we said glass has a high index of refraction? Because of total internal reflection, you can send a light beam into the fiber and all the light will be transmitted even if there is a bend in the fiber. If you look at a fiber-optic cable, the ends look bright because light can enter the sides but only exit through the ends.
This is quite useful for sending data from one place to another. It turns out that by using light, you can transmit data at a much higher rate than by electrical signals, so it’s great for streaming hi-def videos of spearfishing. But don’t worry, if you don’t have fiber-optic internet at home, you can just go outside and jump in the swimming pool. And be sure to look for Snell’s window!
