Ch. 11 – Optics Part 2
Refraction in Water
Because of the way that the direction of light changes when entering crossing a boundary where the index of refraction changes, there is a pair of phenomena that occur.
If you are “in air” and you observe an object under the water, the object appears displaced from its actual location.
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If you are “in air” and you observe an object under the water, the object appears displaced from its actual location. |
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As the angle of incidence i increases, as a consequence of Snell’s law, the angle of refraction in water reaches a maximum value of 48.6°.
The critical angle of refraction where this occurs is found from Snell’s Law:
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If this doesn’t seem strange enough, the situation when looking from underwater into the air is even stranger! Looking straight up along the normal to the surface, things look normal, but as the angle of incidence increases to 48.6°, the angle of refraction (now going into a medium of lower index of refraction) goes to 90°, and an observer looking at that angle will see objects on the surface of the water in the distance (in principle all the way to the horizon!). At larger angles of incidence, the observer will see objects reflected from under the water.
This leads to a phenomenon called the “manhole effect”. Everything above the surface is compressed into a circle of radius 48.6°.
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This is an example of total internal reflection, and is the basis for the simplest type of fiber optic cables.
However, this is the least sophisticated sort of fiber cable. Today these fibers are manufactured with an index of refraction that changes gradually from center to surface, effectively “shaping” the way the light wave propagates along the filament.
This sort of optics can be both fun and practical!
Lenses
While a prism will redirect a beam or ray of light (here we will ignore the “second-order” effect of dispersion), it doesn’t do a good job of concentrating light, even when working in tandem with other prisms.
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But by making an “infinite” number of faces (each infinitesimally small) we can concentrate the light at a point.
This is basically all a lens really does.
(Note: as we will see later, highly curved lens surfaces will produce some dispersion, however.)
The location behind the lens where parallel light rays (say from an object far away) converges is called the focal point. Its distance from the lens is the focal length of the lens. |
The lens above is an example of a double-convex lens, because both surfaces are convex outward. One can also have plano-convex lenses (one surface is a plane), double-concave, plano-concave, and meniscus lenses (one convex and one concave surface).
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Unlike the convex lens, where the focal point is on the side of the lens opposite the source of light, the focal point of a concave lens is on the same side as the light source.
The actual rays of light diverge in directions as if they came from that point. |
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Imagine an object on one side of a convex lens. If we were to follow the light coming from one part of the object, we would see that the rays converge to a point on the opposite side. If we were to place a piece of paper at that location, we could actually see a real image of the object, although it would be upside-down.
The location of the image and its actual size will depend on how far the object is from the lens. You can see the way a converging lens works by going to this Converging Lens applet. Notice that if the object is very far from the lens, the image is small and close to the focal point. As the object approaches the lens, the image gets larger and farther away. Finally, as the object reaches a distance equal to the focal length of the lens, no image is formed at all!
For concave lenses the situation is a little different. We can begin to see what happens by first looking at a mirror.
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If you look at yourself in a mirror, you certainly see something, don’t you? You and the surroundings behind you appear as if you are beyond the plane containing the mirror, but in reality you are not, of course. This is an example of a virtual image. |
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A concave lens cannot converge the light from an object to form a real image. But if you look through one at a more distant object, you will see s smaller image of it. This is another example of a virtual image. There is no real image there, but the object appears to be located there when you look at it through the lens. |
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A concave lens cannot converge the light from an object to form a real image. But if you look through one at a more distant object, you will see s smaller image of it. This is another example of a virtual image. There is no real image there, but the object appears to be located there when you look at it through the lens. The focal length of a diverging lens is a negative number.
For the way a diverging lens works, go to this Diverging Lens applet.
Thin Lenses
Lenses can be single or used in combination. By far the simplest les system is the single thin lens – one where the vertical displacement of a ray as it passes through the lens can be ignored. In this limit, the height of the image, the height of the, and their distances from the lens are closely related.
Here we see that
where hi is the height of the image, hs is the height of the subject, i is the distance from the lens where the image is formed, and s is the distance of the object from the lens. This ratio is called the magnification M of the system, for it tells how much bigger (or smaller) the image is than the subject. If we look at the angle labeled g in the figure above, we see that it is also true that
By combining these two equations and rearranging the terms, we arrive at the thin lens formula:
Note, if we were to move the object to infinity (i.e. make s infinite) then we are lefty with
This should not be too surprising, because that is how we defined the focal length to begin with! With this condition the magnification must be
Compound Lenses
It is possible to put lenses in series. If the lenses are close enough, the image that would have been formed by the first lens alone may never get a chance to do so before being affected by the second lens. Suppose the light is converging from a lens, but we place a second lens into the beam before the image can form:
The location of the new image formed by the combination of the lenses will differ from what it was originally.
Oddly enough, though, the second lens behaves as if it took the image that would have (but didn’t) form from the first lens, and used it as its subject! Such a subject is called a virtual subject.
If the lenses are close together it can be shown (see the textbook for the algebraic details) that the system acts as if it had a net focal length f of:
where f1 and f2 are the focal lengths of the first and second lenses.
With the appropriate choice of lenses, one can get magnifications that are larger or smaller than one – i.e. images that are larger or smaller than the subject.
Lens (and Mirror) Aberrations
The entire “thin lens” assumption treats a lens as if it were a lens when we wan it to be, but a pane of glass for as far as all the things real lenses do.
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Chromatic Aberration: A real lens is just a rounded prism, so it disperses the light. Each wavelength will be brought to a different focus unless “corrected by a second lens of different optical properties. In principle an “achromat” can be made with (usually) a double-convex lens plus plano-concave lens of different materials. In practice, only two wavelengths are usually brought to the same focus. More elements plus the use of novel materials, non-spherical surfaces improves the situation. Then, ray tracing is usually done by computer. |
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Spherical Aberration: Because the curvature of a lens is greater at its edge than at its center, light from the edge tends to be focused closer to the lens than light passing near its center.
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Coma: Light from a particular point on a subject will be focused at different locations, depending on which part of the lens the light passed through. |
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Astigmatism: Most lenses suffer some cylindrical curvature, so that light falling on one axis is effectively hitting a lens of different curvature than light falling along another axis. When being fitted for eyeglasses, a person always has to undergo a determination of the degree and the direction of the astigmatism of their eyes. |
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Distortion: Usually a rounded lens will produce an image that is bowed outward (“barrel distortion”) or inward (“pincushion distortion”). |
In addition to the aberrations shown above, curvature of field makes the surface on which focus occurs to be curved. This can be compensated for to some degree if the image plane is deliberately curved to the appropriate shape.
Mirrors do not suffer from chromatic aberration (since a mirror does not disperse wavelengths) but does suffer from the rest of the aberrations that lenses do. A curved mirror can, for example, form an image of a subject:
However, if the surface of the mirror is spherical in shape, it will suffer from spherical aberration:
This was the problem with the Hubble Space Telescope when first launched. It had the most precise mirror ever made, but it had been inadvertently made to precisely the wrong shape! This had to be corrected by the addition of corrective optics.
Spherical aberration in mirrors can be largely corrected for by making the mirror shaped like a parabola instead of a spherical surface. However, this introduces coma! In reality, astronomical telescope and telephoto lenses for cameras rely on a series of mirrors and lenses to minimize all the aberrations. To do so requires many precisely made surfaces, and that costs money.