Section 26–5 A more precise statement of Fermat’s principle

 (Precise statement / Philosophical principle / Single slit)

 

In this section, Feynman discusses a more precise statement of Fermat’s principle, the philosophical principle behind it, and applies the principle to a single slit.

 

1. Precise statement:

“Actually, we must make the statement of the principle of least time a little more accurately (Feynman et al., 1963, section 26–5 A more precise statement of Fermat’s principle).”

 

Feynman rephrases Fermat’s principle as “light takes a path such that there are many other paths nearby which take almost exactly the same time.” Specifically, there is no first-order change in the time and there is only a second-order change in the time for the light path taken. Alternatively, we can state Fermat’s principle as follows: “a light ray in going from point S to point P must traverse an optical path length that is stationary with respect to variations of that path (Hecht, 2002, p. 109).” In other words, the first derivative of the optical path length is zero in its Taylor series expansion. On the other hand, the second derivative of the optical path length is non-zero; it could be positive (relative minimum) or negative (relative maximum).

 

Feynman points out that Fermat’s principle is incorrectly called the principle of least time and disagrees with those who consider the light path AB in a mirror reflection to have a maximum time. However, the light path through the reflection is longer than the “shortest (direct) path without a reflection.” Although there are many curved paths that take a longer time, this reflection path can be considered as a relative maximum. Interestingly, Kaushik and Sukheeja (1984) give an example of “maximum time” in reflection on a concave spherical surface, whereas Halliday and Resnick (1980) give an example of “maximum time” in refraction (if the incidence angle is zero). Similarly, Hecht (2002) explains that the paths traveled by the light rays in an ellipsoidal mirror may be known as a relative minimum or relative maximum (p. 110).

 

2. Philosophical principle:

“… the principle of least time is a completely different philosophical principle about the way nature works (Feynman et al., 1963, section 26–5 A more precise statement of Fermat’s principle).”

 

Feynman asks whether light can smell the nearby paths, and check them against each other to choose the least-time path. His answer is yes because the wavelength tells us approximately how far away the light must “smell” the path in order to check it. One may prefer his explanation in his QED lecture for the public: “…light doesn’t really travel only in a straight line; it ‘smells’ the neighboring paths around it, and uses a small core of nearby space (Feynman, 1985, p. 54).” To show the importance of having a small core of nearby space, he adds that “if a mirror is too small for the core of neighboring paths, the light scatters in many directions, no matter where you put the mirror (p. 54).” We have to place the small mirror at a suitable region such that all of the light rays have almost the same path length in order to see the mirror image.

 

Feynman explains that the philosophical principle of Fermat does not include the idea of causality, but it is dependent on how we set up the situation. That is, light decides the path that has the shortest time or the extreme one, and then chooses that path. However, Feynman’s philosophical principle may be known as “sum over histories.” According to Dyson, Feynman mentions that “[t]he electron does anything it likes” and “[i]t just goes in any direction, at any speed, forward and backward in time, however it likes, and then you add up the amplitudes and it gives you the wave function (Dyson, 1980, p. 376).” Simply phrased, light does not need to know the path of least time in advance because it takes all paths from the starting point to ending point, but the contributions of same-time paths are greater.

 

3. Single slit:

“With a narrow slit, more radiation reaches D′ than reaches it with a wide slit! (Feynman et al., 1963, section 26–5 A more precise statement of Fermat’s principle).”

 

Feynman first discusses the paths of light through a wide slit that has rays going from S to D because it is a straight line. He explains that light will not go through the wide slit from S to D′ because the rays all correspond to different times. However, he provides a better explanation in his QED lecture with a good diagram (Fig. 33). In Feynman’s (1985) words, “[w]hen the gap between the blocks is wide enough to allow many neighboring paths to P and to Q, the arrows for the paths to P add up (because all the paths to P take nearly the same time), while the paths to Q cancel out (because those paths have a sizable difference in time) (p. 54).” In essence, the paths from the wide slit (a small core of nearby space) to D′ are different-times paths (in different phase) instead of same-time paths.

 

According to Feynman, if we prevent the light from checking the paths using a narrow slit, then it takes the only path available. As a result, more light reaches D′ through the narrow slit as compared with a wide slit. In the QED lecture, Feynman (1985) says that “[w]hen the gap is nearly closed and there are only a few neighboring paths, the arrows to Q also add up, because there is hardly any difference in time between them, either (p. 55).” In Fig. 34, he states that “[w]hen light is restricted so much that only a few paths are possible, the light that is able to get through the narrow slit goes to Q almost as much as to P, because there are not enough arrows representing the paths to Q to cancel each other out (Feynman, 1985, p. 56).” Importantly, the brightness at D′ is lesser than D because fewer rays reach D′ through same-time paths.

Review Questions:

1. How would you state Fermat’s principle of least time more precisely?

2. How would you interpret the philosophical principle of Fermat’s principle of least time?

3. What does Feynman meant by “if we close down the slit it is all right—they still go”?

 

The moral of the lesson: light takes all paths from the starting point to ending point such that a brighter image is observed provided the light rays through a small core of nearby space (e.g., wide slit or narrow slit) have same-time paths (or in phase).

 

References:

1. Dyson F. (1980). Some Strangeness in the Proportion. In H. Woolf (Ed.). A Centennial Symposium to Celebrate the Achievements of Albert Einstein. Reading, MA: Addison Wesley.

2. Feynman, R. P. (1985). QED: The strange theory of light and matter. Princeton: Princeton University Press.

3. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

4. Halliday, D., & Resnick, R. (1980). Physics (part II). New York: Wiley. 

5. Kaushik, B. K., & Sukheeja, B. D. (1984). Intricacies of Fermat’s principle. Physics Education, 19(2), 90-91.