A FLY FISHERMAN WHO SEES a fish in the water confronts the problem of where to cast the fly. The received wisdom is that the best place is just above the fish. Does the cast have to be that accurate? If it misses by a few centimeters, will the fish still see it as a fly? Robert Harmon and John Cline, who are patent attorneys in Chicago, have been looking into the optics of fly fishing. They believe the cast does have to be fairly accurate, otherwise the image seen by the fish might be too distorted by the refraction of the light rays at the surface of the water.

Light in a vacuum travels at only one speed (3 x 10⁸ meters per second). Through any transparent medium, however, its speed is lower because the light interacts with the molecules of the medium. Each interaction can be considered as a brief absorption of the light.

The easiest way to describe the net delay in the passage of the light through the medium is to say the light is moving slower. For this purpose every transparent medium is assigned an index of refraction. The effective speed of light through the medium is equal to the speed of light in a vacuum divided by the index of refraction. The index of refraction of water is approximately 1.331 and of air slightly more than 1. Hence the effective speed of light through air is almost the same as it is through a vacuum, whereas through water the speed is considerably lower.

When a light ray passes through the surface of water, it is refracted (changes its direction) because of the change in its effective speed. By convention the orientation of a ray is measured with respect to a line perpendicular to the surface crossed by the ray. Suppose the ray is incident on the water surface at an angle of 42 degrees with respect to the vertical. Part of the light is reflected from the surface at the same angle from the vertical. The rest of the light refracts into the water as a ray 30 degrees off the vertical.

Other angles of incidence yield other angles of refraction. The relation is set out in the rule named for Willebrord Snell, who proposed it in 1621. According to Snell's rule, the sine of the angle of refraction in the water is equal to a fraction of the sine of the incident angle in the air. The fraction is the ratio of the respective indexes (air: water). In every case except one the angle of refraction is smaller than the angle of incidence. The exception is when the ray is incident along the vertical line; then it goes into the water with no change in direction.

Consider a ray that comes to a fish from an object a short distance above the surface of the water. If the fish can assign a position to the origin of the ray (as a human being can), it interprets the object as lying somewhere along the ray. The object therefore appears to be at an angle in the sky higher than the true angle. The error is small if the angle of the incident ray is small and large if the angle is large.

The ray refracted the most comes from near the horizon; it is incident on the water at an angle of slightly less than 90 degrees. The angle of refraction is approximately 48.7 degrees. (The exact angle depends on the index of refraction of the water.) No ray from above the water can reach the fish at a larger angle of refraction. Hence all the rays reaching the fish from above the water fall within an imaginary cone with its apex centered on 1he fish's eye and with its sides 4X 7 degrees off the vertical.

Harmon and Cline call the intersection of this cone with the surface of the water the "window" through which the fish sees objects above the water. A ray from the horizon passes through the edge of the window and then down the side of the cone. The size of the window varies with the fish's depth in the water. When the fish is at a depth of 10 centimeters, the radius of the window is 11.3 centimeters. A greater depth gives a wider window but cannot alter the angular size of the cone. That size is set by the refraction of the rays from the horizon.

The view of the external world that arrives at the fish is anamorphic: the magnification differs in each of two perpendicular directions. Refraction warps and repositions objects in the fish's view. Perhaps a fish can interpret the anamorphic view, realizing that the objects appearing in the window lie at some distance above the surface of the water. Perhaps instead the fish regards the objects as being on the surface. In either case what does the fisherman look like to the fish?

I investigated the question by computer, calculating what the refraction would be from each of four vertical sticks at several distances from a fish. I programmed my home computer to make the calculation on the basis that each stick extends one meter above the water and 20 centimeters below it which is about right to simulate a fisherman standing in shallow water. The fish is assumed to be 1O centimeters below the surface, which is a reasonable depth for a feeding fish.

I first considered a stick two meters from the fish horizontally. A ray from the submerged part of the stick is not refracted and is perceived (if the fish can see that far) in its proper place. A ray from just above the waterline on the stick passes through the edge of the window and travels along the side of the imaginary cone that marks the limit of the rays reaching the fish from above the water. The fish might interpret this ray as originating somewhere back along a line making the same angle with the vertical. If it does, the waterline of the stick would seem to lie along a line 48.7 degrees from the vertical.

A ray from the top of the stick passes slightly closer to the center of the window. Its angle of refraction is about 42 degrees. The fish might see the ray as originating along a line that is a rearward extrapolation of the refracted ray. If the fish does, the top of the stick would seem to lie on a line 42 degrees off the vertical. Hence if the fish has depth perception, the stick would seem to lie somewhere in the air between 42 and 48.7 degrees off the vertical.

The situation is represented in Figure 5. The image of the stick curves between those angles. In order to leave room for the other components of the illustration the image is shown as being separated from the window by about as much as the stick actually is.

Do not take the drawing literally. I do not know if the fish can mentally extrapolate light rays. I also do not know if it can even recognize a stick for what it is. Surely a fish cannot conclude that the seemingly warped object is a vertical, rigid stick. Much of a human being's ability to assign depth and shape to objects comes from experience with those objects.

With my computer I calculated angular positions for three other sticks. In all four cases the fish sees two images of the stick. The part above the surface of the water is seen through the window. The submerged part is seen in its true position and is well separated from the image of the part above the surface. As I move a stick closer to the window the images of the two parts get closer to each other, finally merging when the stick reaches the edge of the window.

The illustration in Figure 1 offers a flat view of the sticks as they are seen through the refraction of the window. A fish without depth perception or any understanding of what it is seeing probably depends on such a flat picture of the external world. To keep the sticks from overlapping in the illustration I have repositioned them so that they lie in a circle around the fish. The sizes of the sticks and the distances from them to the fish are the same as before. The submerged parts are not shown because they are too far away to fit into the illustration. Marks on the sticks indicate several heights above the waterline.

In the illustration the bottom of the part of a stick above the surface appears at the edge of the window and the top appears along a radial line and closer to the center of the window. A stick two meters from the fish is compressed into a small area. The bottom of the part of the stick above the surface is compressed more than the top because of the strong refraction of the rays from the bottom. The image of the stick takes up less than 2.5 centimeters along a radius of the window. Since many other objects around a body of water would show up along the edge of the window, the stick might be lost in the clutter.

Less compression is apparent in the sticks 1.5 meters and one meter from the fish. Since they extend more toward the center of the window, however, they are noticeably tapered. The stick 50 centimeters from the fish is even more tapered and distorted. The full image of the part above the surface takes up about 70 percent of a radial line in the window and therefore must be quite noticeable to a fish.

My stick is equivalent to a short fisherman. Such a fisherman two meters or more from the fish is compressed into a miniature that occupies only a small part of the window and may be lost in the clutter at the edge. As the fisherman moves closer to the fish he takes up more angle in the fish's field of view and occupies more of the window. The submerged part of the fisherman also gets larger in the fish's field of view.

At some point the motion of one of these images warns the fish of possible danger. The motion of the part of the fisherman above the water shows up as an image that starts at the edge of the window and grows radially toward the center. Perhaps the fish watches for motion that looks as though it might cast a full image from the edge to the center.

Similar optics applies to the appearance of a fly cast near a fish. Some possibilities are represented on the left side of Figure 1. For the sake of convenience I have considered a narrow rectangular fly extending 2.5 centimeters above the surface and .2 centimeter below it. (The height is about the same as that of a size 4 dry fly. The width of the fly along the surface is not important.) Although a rectangular fly is not likely to be inviting to a fish, it serves to demonstrate the distortion caused by refraction.

I programmed my computer to find the image the fly makes in the window. If the waterline of the fly is five centimeters from the center of the window, the part of the fly above the surface of the water lies across only 1.3 centimeters of a radial line in the window. The part below the surface, which is compressed, merges into the image of the part above the surface.

As the fly moves closer to the edge its image stretches. For example, when the fly's waterline is 10 centimeters from the center of the window, the image of the part of the fly above the surface takes up three centimeters along a radial line. That is more than the true height of the fly. The image of the part below the surface, still attached to that of the part above the surface, is also stretched slightly, which should make the fly more noticeable to the fish.

When the fly moves past the edge of the window, the image of the part above the surface begins to contract and that of the part below the surface separates from it. The illustration shows the situation when the fly is 15 centimeters from the center of the window. The top of the part below the surface is seen at its proper distance from the center. The bottom of the part above the surface appears at the edge. The top of the fly, which is actually 2.5 centimeters above the waterline, shows up only 1.9 centimeters from the edge of the window. The fly is no longer easy to see.

When the fly is moved to 20 centimeters from the center of the window, the apparent contraction of the part above the surface is greater. The bottom of that part still lies at the edge of the window and the top now appears at about .8 centimeter from the edge. This contraction of the image of the part above the surface gives the fish a highly distorted view of the fly. Moreover, the image of the part above the surface may be lost in the clutter at the edge of the window. Recognizing the fly is now more difficult. In addition the image of the part below the surface is well removed from the image in the window. Even if both images are still perceptible, a fish is likely to see two objects, both of them small.

Harmon and Cline say that if you are fishing with a fly and can see the fish, cast the fly as close to it as you can. If you can put the fly within the fish's window, it may be recognizable as a fly. At least the images of the part of the fly above the water and of the part below the water are merged. If the fly lies inside the window near the edge, the image of the part above the water is magnified in the sense that its length along a radial line of the window is larger than the true height of the fly.

If your cast is off by a few centimeters, the fly may be outside the fish's window. The separation of the images of the part below the surface and of the part above makes the fly look less like a fly. The compression of the image of the part above the surface may even make that part so small that it is lost in the clutter at the edge of the window.

The problem is. particularly difficult if the fisherman is in the same direction from the fish as the fly is; his image adds to the clutter. In this situation his only chance of attracting the fish is with the image of the part of the fly below the surface, which the fish will see without distortion by refraction. Harmon and Cline suggest it would be well if that part of the fly were brightly colored.

So far I have assumed that the index of refraction of water has a single value. In reality it differs at different wavelengths of light. Red light, at the long-wavelength end of the visible range, has an index of about 1.331. Blue light, at the short-wavelength end, has an index of about 1.343. Suppose a ray of white light, consisting of all the colors, passes into water. Refraction spreads the colors through a small range of angles. The ray with the smallest angle of refraction is blue; the one with the largest angle of refraction is red. The colors at intermediate wavelengths have intermediate angles of refraction. This separation of colors is called dispersion.

Harmon and Cline point out that dispersion plays a minor role in the image a fish sees in its window. To investigate dispersion I considered the rays of white light extending from the top of my imaginary rectangular fly. One ray refracts at the water surface to send a red ray to the fish. Another ray refracts slightly closer to the center of the window to send a blue ray. The fish sees a colored image where the rays cross through the window. Although the blue image is slightly closer to the center Of the window, the dispersion of the colored image is weak unless the fly is well outside the window. Even then the spread amounts to no more than about a millimeter in the window.

What the fish sees on the surface of the water outside the window is largely a reflection of rays that have scattered off the bottom. Although any refraction of light through the surface and into the air must obey Snell's rule, for some rays refraction is impossible. Whether or not a ray refracts depends on the angle of incidence. If the angle is less than 48.7 degrees, part of the light refracts through the surface and the rest reflects downward. According to Snell's rule, the angle of refraction (now in the air) must be larger than the angle of incidence. The angle of refraction can be as much as 90 degrees, however, which it is when the refracted ray barely skims over the surface of the water.

If the incident angle is larger than 48.7 degrees, refraction is impossible. The light can only reflect, a situation that is called total internal reflection since the light is unable to escape from the water. Any light that reflects to the fish from the underside of the window must have an angle of incidence smaller than 48.7 degrees. There part of the light also refracts into the air. A ray that reflects just at the window's edge has an angle of incidence of 48.7 degrees, sending a refracted component along the surface of the water. Any light that reflects to the fish from the rest of the surface must have an angle of incidence larger than 48.7 degrees. All this light is internally reflected. The reflections from the window region are likely to be lost in the glare of light from the sky, but the reflections elsewhere might be- bright enough to give the fish a mirror-like picture of the bottom.

The optics I have been discussing applies to a situation in which a fish looks out through the sides of an aquarium. Here, of course, the window is in a vertical plane. The anamorphic distortion resulting from refraction would change the geometry of objects outside the aquarium. For example, an object that is in fact square would have the shape of a pincushion.

The human eye open in water does not see any of these optical distortions because it is adapted for vision in air. About two-thirds of the refraction necessary for focusing normally takes place at the surface of the eye. Since the eye has almost the same index of refraction as water, a submerged eye loses that refraction. It cannot focus on objects imaged in the window. You can regain focus if you wear a face mask to trap air next to your eye. Is there a window then? There is none if the plane of the mask is parallel to the surface of the water. When the rays pass from the water into the air in the mask, the refraction reinstates their original directions of travel. The cone limiting the rays is eliminated and therefore so is the window. You might want to investigate other orientations of the face mask.

I have briefly considered another refraction problem common to fishing. Can you see a fish in its true location? The problem is crucial if you fish, as a few people do, with a bow and arrow. Should you aim the arrow directly at the fish as you see it? The answer is no. Unless the fish is just below the surface, you should aim lower in your field of view. The rays reaching you from the fish refract according to Snell's rule, ending up with larger angles with respect to the vertical than they had initially. When you receive one of the rays, you mentally extrapolate back along it to find the source, being misled into thinking that the fish is in that direction.

Lawrence E. Kinsler analyzed similar problems about the refraction of rays from a submerged object. He pointed out that the depth of an object is misjudged even when your view is from directly above it. Much of your decision about the distance to the object derives from the angle through which each eye must turn so that the eyes together can converge their lines of sight on the object. Since the rays of light are refracted before they reach the eyes, the point of convergence lies above the object, leaving you with the illusion that the object is not as deep as it actually is.

Observations from other angles also involve such an error in the assignment of depth. Kinsler's results (for a fish) are summarized in the illustration on the left. One ray is included to represent the light that travels from the fish to the observer. Actually each eye receives a ray from a slightly different direction. The observer believes the fish lies along a rearward extrapolation of the rays. In the illustration the extrapolation is indicated for the single representative ray. The convergence of the lines of sight from the eyes determines where along the extrapolation the fish appears to be. The result is that the fish seems to be higher on a vertical line running through its true location.

Such is the illusion for a normal view of a fish. Suppose the observer lies on a dock with his eyes directed downward in a vertical plane. As before the fish seems to be on a rearward extrapolation of the rays reaching the eyes. This time they seem to come from a place higher and closer to the observer.

You can check these illusions with a simple demonstration. Fill a tub with water. Look at a coin on the bottom. When your line of sight is well off the vertical, the apparent depth of the coin is obviously inconsistent with the depth of the tub. When you then move your head so that your eyes are in a vertical plane, the apparent position of the coin immediately shifts so that the coin seems to be higher and closer to you.

Bibliography

IMAGING OF UNDERWATER OBJECTS. Lawrence E. Kinsler in American Journal of Physics, Vol. 13, No. 4, pages 255-257; August, 1945.

AT THE EDGE OF THE WINDOW. Robert Harmon and John Cline in Rod and Reel, No. 7, pages 41-45; March/April, 1980.

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