Search my Subject Specializations: Select Users without a subscription are not able to see the full content. Rand Abstract This book bridges the gap between introductory quantum mechanics and the research front of modern optics and scientific fields that make use of light.
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More This book bridges the gap between introductory quantum mechanics and the research front of modern optics and scientific fields that make use of light. Authors Affiliations are at time of print publication. Print Save Cite Email Share. Show Summary Details. Subscriber Login Email Address. Library Card. View: no detail some detail full detail. But if we were to insert a piece of glass of just the right thickness we shall later figure out how thick it might exactly compensate the excess time that it would take the light to go at an angle!
We end up with a piece of glass that looks like Fig. This, of course, is well known to us, and we call such a device a converging lens. In the next chapter we shall actually calculate what shape the lens has to have to make a perfect focus. On any path, it goes to some mirror and comes back, and all times must be equal.
Lectures on Light: Nonlinear and Quantum Optics using the Density Matrix
Here the light always travels in air, so the time and the distance are proportional. Therefore the statement that all the times are the same is the same as the statement that the total distance is the same. An ellipse is that curve which has the property that the sum of the distances from two points is a constant for every point on the ellipse; thus we can be sure that the light from one focus will come to the other. The same principle works for gathering the light of a star.
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Imagine a star billions of miles away; we would like to cause all the light that comes in to come to a focus. Of course we cannot draw the rays that go all the way up to the star, but we still want to check whether the times are equal. Thus our curve is the locus of all points equidistant from a line and a point. Such a curve is called a parabola ; the mirror is made in the shape of a parabola.
The above examples illustrate the principle upon which such optical devices can be designed.
The exact curves can be calculated using the principle that, to focus perfectly, the travel times must be exactly equal for all light rays, as well as being less than for any other nearby path. We shall discuss these focusing optical devices further in the next chapter; let us now discuss the further development of the theory. I need a curve such that every tangent plane makes equal angles with the two rays. So is it merely a philosophical question, or one of beauty? There can be arguments on both sides. However, the importance of a powerful principle is that it predicts new things. On the other hand, according to the idea of least time, there is a definite relationship.
Another argument for the principle of least time, another prediction, is that if we measure the speed of light in water, it will be lower than in air. This is a prediction of a completely different type. It is a brilliant prediction, because all we have so far measured are angles ; here we have a theoretical prediction which is quite different from the observations from which Fermat deduced the idea of least time. It turns out, in fact, that the speed in water is slower than the speed in air, by just the proportion that is needed to get the right index!
Actually, we must make the statement of the principle of least time a little more accurately. It was not stated correctly above. It is incorrectly called the principle of least time and we have gone along with the incorrect description for convenience, but we must now see what the correct statement is. Suppose we had a mirror as in Fig. What makes the light think it has to go to the mirror? The correct statement is the following: a ray going in a certain particular path has the property that if we make a small change say a one percent shift in the ray in any manner whatever, say in the location at which it comes to the mirror, or the shape of the curve, or anything, there will be no first-order change in the time; there will be only a second -order change in the time.
In other words, the principle is that light takes a path such that there are many other paths nearby which take almost exactly the same time. The following is another difficulty with the principle of least time, and one which people who do not like this kind of a theory could never stomach. Light goes along, it sees a surface, it bends because it does something at the surface.
The idea of causality, that it goes from one point to another, and another, and so on, is easy to understand. But the principle of least time is a completely different philosophical principle about the way nature works. Instead of saying it is a causal thing, that when we do one thing, something else happens, and so on, it says this: we set up the situation, and light decides which is the shortest time, or the extreme one, and chooses that path. But what does it do, how does it find out? Does it smell the nearby paths, and check them against each other? The answer is, yes, it does, in a way.
It is hard to demonstrate this fact on a large scale with light, because the wavelengths are so terribly short. If we have a source of radiowaves, a detector, and a slit, as in Fig. One can do the same thing with light, but it is hard to demonstrate on a large scale. The effect can be seen under the following simple conditions. Find a small, bright light, say an unfrosted bulb in a street light far away or the reflection of the sun in a curved automobile bumper.
Then put two fingers in front of one eye, so as to look through the crack, and squeeze the light to zero very gently. You will see that the image of the light, which was a little dot before, becomes quite elongated, and even stretches into a long line.
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The reason is that the fingers are very close together, and the light which is supposed to come in a straight line is spread out at an angle, so that when it comes into the eye it comes in from several directions. Also you will notice, if you are very careful, side maxima, a lot of fringes along the edges too. Furthermore, the whole thing is colored.
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All of this will be explained in due time, but for the present it is a demonstration that light does not always go in straight lines, and it is one that is very easily performed. Finally, we give a very crude view of what actually happens, how the whole thing really works, from what we now believe is the correct, quantum-dynamically accurate viewpoint, but of course only qualitatively described.
Instead the rays seem to be made up of photons, and they actually produce clicks in a photon counter, if we are using one. The law for that chance is the following very strange one. The number of turns per second is the frequency of the light.
Now take another path; it has, for instance, a different time, so the vector for it is turned through a different angle—the angle being always proportional to the time. Take all the available paths and add on a little vector for each one; then the answer is that the chance of arrival of the photon is proportional to the square of the length of the final vector, from the beginning to the end!
Now let us show how this implies the principle of least time for a mirror. Eventually, we have quite a tight knot. The total probability is the distance from one end to the other, squared.
Almost all of that accumulated probability occurs in the region where all the arrows are in the same direction or in the same phase. All the contributions from the paths which have very different times as we change the path, cancel themselves out by pointing in different directions.
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If not, how come this has not been spotted before? Is an aperture implied by angle theda? It could be an error in his notes, it's possible. Michael A. Gottlieb Michael A. Gottlieb 7 7 silver badges 8 8 bronze badges. Gottlieb Apr 12 at I would wager it has been noticed, but when most people see an error in a physics book by someone like Feynman they assume the mistake is theirs.
Those who convince themselves otherwise are a small minority, only a fraction of which will bother to write about it somewhere, and not all of those reports will make it to me. Besides, I've only been collecting and correcting errata for FLP online for the past 14 years, while the book has been published for 56 years. Sign up or log in Sign up using Google. Sign up using Facebook.