The Digital Camera: Snell’s Window (the view from a fish’s eye)

When we look at a scene through clear air the rays from all locations arrive through straight lines. Light from parts of the scene that are directly in front of us enters our eyes at an angle of zero degrees; light from parts of the scene that are 45 degrees off to our left enters our eyes at an angle of 45 degrees.

Light rays entering the eye from various angle in clear air.

Our eyes are not equally sensitive to all of those rays: the standard field-of-view is about 180 degrees in the horizontal direction (left-right) and 110 degrees in the vertical direction (up-down). The horizontal field-of-view is 90 degrees in each direction; the vertical field-of view is about 65 degrees downward and 45 degrees upward. Some people, however, are blessed with extraordinary vision. In A Sense of Where You Are John McPhee explains that, at the peak of his college and NBA careers, former basketball great and United States Senator Bill Bradley was able to see about 98 degrees in each horizontal direction, 70 degrees downward, and an astonishing 70 degrees upward.

When our eye focuses light to form an image, it does so best when the light rays enter from a surrounding like air whose refractive index is 1.0. A fish’s eye, on the other hand, works best when the light enters from a surrounding like water that has a refractive index equal to 1.33. As long as our eye is surrounded by air, though, we can see into clear water. We see some distortion because of refraction, but, for the most part, the bottom of a clear stream or lake looks familiar when we peer through its surface. Likewise, a fish can see above the water, but, because air has a smaller refractive index than water, they experience a very different type of distortion.

When a fish looks upward they see light rays from a full 180 degree field-of-view above the water’s surface. But, because of refraction, those rays are bent and compressed into a field-of-fiew that is about 97 degrees.

The window through which a fish sees the world above water.

In our first post on Snell’s Law of refraction we used the following graph to illustrate the relationship between the angle at which a ray travels in air and the angle at which it travels after entering water:

Air-into-water refraction for all angles between 0 and 90 degrees.

Rays that are close to vertical in air — those that travel at an angle close to 0 degrees — are also close to vertical in water; however, rays that are close to horizontal in air — those that travel at an angle close to 90 degrees — are bent to an angle close to 48.6 degrees. Look at the graph, then look at the way the rays bend when traveling to the fish’s eye. The rays that are between 80 and 90 degrees in air enter the fish’s eye at roughly the same angle. Not exactly the same angle, but roughly so. Because of this phenomenon many people will say that a fish cannot see you if you keep your profile within 10 degrees of horizontal. That is not true. A fish will see you if it looks up toward the surface of the water, but, because your image will be compressed tightly with nearby parts of the scene, the fish will have trouble distinguishing you from your surroundings. But if you are blocking a bright light source like the sun, the fish will notice. Many articles about a fish’s vision get this wrong, but, as expected, Gary Borger gets it right in his discussions.

What happens if the fish moves closer to the water’s surface?

The fish's window is smaller when the fish moves closer to the surface.

When this happens, the ‘window’ through which the fish sees above the water becomes smaller. As a simple rule, you can think of the size of the ‘window’ on the water’s surface as twice as wide as the fish is deep. If the fish is 4 feet below the surface, the window will be eight feet in diameter; if the fish is 2 feet below the water, the window will be 4 feet in diameter.

When a fisherman is low, near the surface of the water, all of the light rays from the fisherman are compressed into a very small part of the fish’s window to the world. If you are the fisherman, that is good. If you are the fish, that is bad.

Compression of the light from a fisherman into a small part of the fish's vision window.

What happens, though, when the fish looks directly at the fisherman?

A fish sees the bottom when it looks directly at a fisherman that is outside of its 'window'.

In this case the surface of the water acts like a mirror, and the fish sees back toward the bottom of the lake or stream. This phenomenon is called total internal reflection and will be the topic for our next post.

© 2011 Timothy Schulz

The Digital Camera
Previous: Snell’s Law of Refraction
Next: Total Internal Reflection

Rachel ThorneDecember 11, 2014 - 7:27 am

Figure Permission request: The Digital Camera: Snell’s Window (the view from a fish’s eye) 4 August 2011,, copyright © 2011 Timothy

Pearson Education would like to request permission to use the figure listed above within our new educational publication Edexcel A Level Physics Book 1 by Miles Hudson.

Please contact me so I may forward my letter giving the details of our request along with the figure as we wish to use it.

Many thanks indeed and kind regards
Rachel Thorne
Permissions Consultant for Pearson Education Ltd
c/o 32 Earlstone Close, Cadbury Heath, Bristol BS30 8HQ, United Kingdom
Tel: +44 (0)117 9077518
Fax: +44 (0)117 9048608

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