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Elementary Wave Optics
By Robert H. Webb Dover Publications, Inc.
Copyright © 1997 Robert H. Webb
All rights reserved.
ISBN: 978-0-486-14595-2
CHAPTER 1
Geometrical optics: summary
Geometrical optics deals with light (and more generally with waves) in situations where it is possible to ignore the wave character of the phenomenon. This usually means that the wavelengths involved are very much smaller than the dimensions of anything with which the waves interact. Later we will see that geometrical optics is a limiting case of wave optics. The usual value in limiting cases is their simplicity, and geometrical optics shares this asset, with reservations.
All of geometrical optics may be deduced from three simple empirical rules:
1. Light travels in straight lines in homogeneous media.
2. The angle at which light is reflected from a surface is equal to the angle at which it is incident.
3. When light passes from one medium to another, its path is described by the equation n1 sin θ1 = n2 sin θ2.
Figure 1.1 summarizes these rules and defines the various angles. A "ray" is a line along the path the light follows. We think of this as a very narrow beam of light.
We may regard rule 3, which describes refraction, as defining the relative index of refraction: n2/n1.
If we follow custom and define the index of vacuum to be nvac = 1, then n1 and n2 are the indices of each medium relative to vacuum, and are so listed in handbooks. Later we will see that this ability to characterize each medium by a single number is extremely important. Among other consequences, it will lead us to regard nk as the ratio of the speed of light in vacuum (c) to (ck), the speed of light in medium k: nk = c/ck. This in turn allows us to deduce the three rules from the more general Fermat's principle. The main asset of this principle is the esthetic one of unification. It is not essential to the conclusions of geometrical optics, although occasional simplifications are possible. But the three rules suffice.
A further statement limits the kinds of media usually considered. This is the reciprocity principle, which requires that if light can follow a certain path from A to B, then it can follow the same path from B to A. Some media do not support this principle, but they seldom occur in questions pertaining to geometrical optics.
If we apply our rules directly to plane surfaces, they describe the behavior of mirrors and prisms (1.1—1.3). Rule 3 also predicts the phenomenon of total internal reflection. When light passes from a medium with a larger index to one with a smaller index (as, for instance, from glass to air), the ray is bent toward the surface. That is, θ2 > θ1. Eventually, the ray emerging from the "denser" medium (the one with larger index) lies parallel to the surface. This occurs at the critical angle: n1 sin θc = n2 sin(90°) = n2. If θ1 is bigger than θc, no light emerges. In this case, all incident light is contained in the reflected ray so that, from inside, the surface appears to be a perfect mirror. Such mirrors are important in various optical instruments. Familiar examples are the right-angle prisms in binoculars and the light pipes which illumine hard-to-reach places.
Notice that all light incident from the outside of a totally reflecting surface will enter the surface, but it will not reach the region for which θ1 > θc. (1.4)
In applying rule 3, we find a new empirical fact: n is different for different colors of light. Later we will study the source of this dispersion in some detail, but in geometrical optics we merely use it, for example, when we separate colors with a prism. Or, we may compensate for it, as in making achromatic lenses of two kinds of glass, in which the dispersion of one compensates for that of the other. (1.5)
Rule 2 for reflection governs the behavior of instruments with curved mirrors, such as the astronomical telescope. Since the manipulation is similar to that of lenses, we can summarize the operations of the two devices together.
A converging lens is a device which brings parallel light to a single point, called the focus." Parallel light" means a beam in which any one ray is parallel to any other in the beam. Such light comes from a source so distant that the divergence of two adjacent rays is imperceptible, or (by the reciprocity principle) it comes from a source at the focal point of a converging lens. The three rules of geometrical optics tell us how to construct a real lens, but we will generally deal with existing ones, deferring consideration of their construction until needed. So, using the definition of a converging lens alone, and assuming an ideal lens, we can find how images are formed.
Without derivation, we present the thin-lens equation:
1/p + 1/q = 1/f.
This refers to a lens thin enough to have its focal points at equal distances on each side. Practically, the equation works only for rays nearly parallel to the axis of a real lens.
Figure 1.4 defines the parameters. Notice that the way to find the image graphically is to follow two of the rays diverging from an off-axis point on the object. The one going through the center of the lens is undeflected. (This follows from rule 3, in the approximation of the thin lens, which has two close, parallel surfaces at its center.) The ray parallel to the axis must go, by definition, through the far focal point. The ray going through the near focal point becomes parallel to the axis. Two of these three rays are enough. In the situation illustrated, the rays actually come together again, and a real image (one which can be cast on a screen) is formed. Subsequently, the rays diverge from a real image just as they do from a real object. The size of the image may be inferred immediately from the geometry. The magnification M is defined by the equation:
M = Yimage/Yobject = -q/p.
(The negative sign indicates that the image is inverted.) M is called the linear magnification.
If the rays never do meet again, they will appear (by rule 1) to come from a virtual image, as shown in Figure 1.5.
The lens in Figure 1.5(b) is a diverging lens, which makes parallel light diverge as if it came from a point. We can describe all thin lenses and all situations by the same equation if we adopt a few conventions to clarify the way in which the equation is used:
1. If the incident light comes from the OBJECT, we say it is a real object, and define the distance from the lens to it as positive. Otherwise, it is virtual, and its distance is negative.
2. If the emergent light goes toward the IMAGE, we say it is a real image, and define the distance from the lens to it as positive.
3. The FOCAL LENGTH is positive for a converging lens or a concave mirror and negative for a diverging lens or a convex mirror.
The following exercise illustrates these conventions: In Figure 1.6, A is a real object for the converging lens C. B is the consequent real image, or would be if the diverging lens D did not intervene. B is also a virtual object for lens D, and E is the consequent real image. Some numerical values might be: f1 = + 5 cm; f2 = -10 cm; p1 = + 7 cm; YA = + 1 cm; and L = 12.5 cm.
First find q1, the position of B:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Its size is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Since the object distance for the lens D is the difference between q1 and L we find p2 = -5 cm, with the negative sign indicating that it is on the side of D to which the light goes. Again we apply our equation and find that
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
The object size is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Mirrors also obey the thin-lens equation, and the conventions apply in the same way. (1.6–1.10)
So far we have said nothing about the actual surfaces from which lenses and mirrors are made, nor have we provided for lenses of finite thickness. In fact, the ideal surfaces are seldom available for lenses and are often unnecessary for mirrors. The ideal surface for a lens is a very complicated curve, derivable from our rules but difficult to fabricate. That for a mirror is the simpler ellipsoid. Since a spherical surface is easiest to make, lenses (and sometimes mirrors) usually have this form.
A spherical mirror has a focal length equal to one-half the radius of curvature. Using this value, it is easy to see that only rays near the axis will be focused sharply.
For a single spherical refracting surface, the following equation holds:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],
where ninc is the index on the side from which the light is incident, nem is that on the side into which the light emerges, and R (the radius of curvature of the surface) is positive if the center of curvature is on the side toward which the light goes. For example, in Figure 1.7 an object at A yields an image at B according to
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],
or an object at B yields an image at A:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
As with lenses, we apply our equation to each surface in turn, ignoring the others. (1.11–1.12)
Keep in mind that this summary is for the simplest of applications of geometrical optics, although its assumptions are sufficient for an understanding of that subject. This explanation should enable you to use reference texts, such as Jenkins and White (see Bibliography), and help you to understand real (for example, thick) lenses, but you must expect to labor diligently to apply such knowledge. Applied geometrical optics as practiced today is one of the most complicated subjects ever derived from three simple rules.
EXERCISES
1. If a 6-ft man can just see from his feet to the top of his head in a mirror, how tall is the mirror?
2. A prism has a small peak angle, α and is made of glass with index n. Through what angle will it bend a light beam?
3. Find the size and position of the sun's image due to a lens of focal length +50 cm. (The sun subtends an angle of about 0.5 degree.)
4. Find the size and position of the moon's image due to a lens of focal length -50cm.
5. Use the equation for the image formed by a single spherical surface to find the apparent depth of a penny which is actually under 10 cm of water.
PROBLEMS
These problems are intended to supplement, as well as illustrate, the material covered above. Relevant sections of the text are followed by problem numbers in parentheses. Solutions are given for selected problems at the end of the book.
1.1 Three mirrors are set at right angles to each other to form a "corner reflector." If a light is shone into this system, where does the beam go?
1.2 Where does the light ray in the figure hit the screen? Suppose that at each interface one-half the light intensity goes into the reflected ray and one-half into the refracted (transmitted) ray. Find the second brightest spot on the screen.
1.3 A fish watcher looks at an aquarium from a point on a line diagonally through it. For a fish on this line of sight, 5 cm from the corner, how many images does the watcher see, and what are their locations. Does the fish appear different in the different images? What does the fish see?
1.4 A skin diver shines his flashlight at the surface on the water so that the beam makes an angle of 60 degrees with the vertical.
(a) Where does the beam go? Assume that there is no reflected beam if there is a transmitted one.
(b) Oil of index 1.2 is now spread on the water. Where does the beam go?
(c) Many layers of oil are spread on the water, as shown. Sketch the path.
(d) The air over a blacktop road is hottest near the road surface. The index of air far from the surface is 1.0003. An observer sees the road surface only if he looks down at an angle of 89 degrees or less. What is the index of air at the surface?
1.5 The index of the glass used to make a prism is 1.55 for red light and 1.65 for blue. Design an arrangement whereby only red light emerges from the prism. Take the index of air to be 1.00. Hint: Let the beam enter perpendicularly to a surface.
1.6 Find and describe the image in the following lens combinations:
(a) Simple magnifier:
Yobj = 1 mm, p = 5 cm, f = 6 cm.
(b) Compound microscope:
Yobj = 0.01 mm, p1 = 1.1 cm,
f1 = 1 cm, f2 = 10 cm, L = 18 cm.
Does the image appear bigger or smaller? Hint: Keep at least the second term in q1 before approximating. (See Appendix A for help with approximation.)
(d) Opera glass (Gallilean telescope):
Yobj = 2m,
p1, = 50 m,
f1 = 0.10 m,
f2 = -0.05 m,
L = 0.05 m.
Does the image appear bigger or smaller?
1.7 Find and describe the image in the following mirror combinations:
(a) Simple magnifier:
Yobj = 1 mm,
p = 5 cm,
f = 6 cm.
(b) Compound microscope:
Yobj = 0.01 mm,
p1 = 1.1 cm,
f1 = 1 cm,
f2 = 10 cm,
L = 18 cm.
Comment on possible restrictions on the first mirror.
(c) Astronomical telescope: Y
obj = 2.5 thousand miles, p1 = 0.25 million miles,
f1 = 10x miles,
f2 = x miles,
10x < 0.25 million.
Does the image appear bigger or smaller? Hint: Keep at least the second term in q1 before approximating. (See Appendix A for help with approximation.)
1.8 An ideal camera lens of focal length f and diameter d is used to photograph an object at a distance p in front of the lens. A real image is formed on the film at a distance q behind the lens. The object is a square of area A, each point of which radiates light isotropically. The object radiates a total power P. The intensity at its surface is I = P/A.
(a) Find the power P' delivered to the film.
(b) Find the intensity I' in the image formed on the film. Express your answer as a function of I, d, f, and p.
(c) Show that for 10f ≤ p = ≤ ∞, one makes an error of less than 25 percent by regarding I' as a function only of I and the ratio f/d (the f number of the lens).
1.9 Three children line up for their Christmas photo, as shown. Only the one at distance p is in sharp focus. A point on one of the other children forms not a point but a small "blur" circle of diameter δ on the film. If we keep this circle no bigger than a typical silver grain (say, 1 µm, or 10—6 m), then the picture appears correctly focused.
(a) Find d in terms of Δ, f, p, and the lens diameter d.
(b) If Δ/p < 1, express δ in terms of Δ, d, and the object and image sizes.
(c) For best depth of focus on a given subject, what must we do? What does this imply for a "pinhole" camera?
(Continues...)
Excerpted from Elementary Wave Optics by Robert H. Webb. Copyright © 1997 Robert H. Webb. Excerpted by permission of Dover Publications, Inc..
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