Section 26–6 How it works

 (Photons / All possible paths / Accumulated probability)

 

In this section, Feynman discusses the nature of light as photons, explains Fermat’s principle from the perspective of all possible paths of photons, and how photons contribute to the accumulated probability of an optical image.

 

1. Photons:

“Instead the rays seem to be made up of photons, and they actually produce clicks in a photon counter, if we are using one (Feynman et al., 1963, section 26–6 How it works).”

 

Feynman provides a crude view of Fermat’s principle based on his interpretation of quantum electrodynamics. That is, the path of light, for example, from A to B, does not seem to be in the form of waves and light rays seem to be made up of photons. Note that Feynman formulated his theory in terms of paths of particles in space-time to avoid the field concept. In Wilczek’s (1999) words, “uniquely (so far as I know) among physicists of high stature, Feynman hoped to remove field-particle dualism by getting rid of the fields (p. 13).” However, Feynman was disappointed that his theory of quantum electrodynamics is mathematically equivalent to the conventional quantum theory.

 

Feynman says that light rays seem to be made up of photons instead of light rays are really photons. In addition, he explains that light rays actually produce clicks in a photon counter if we are using one. Perhaps Feynman could have clarified that the photon model adopted is just an idealization. Currently, physicists prefer the concept of quantum field instead of particles. For example, Wilczek writes, “[i]n quantum field theory, particles are not the primary reality.” On the other hand, Hobson argues using the two vacuum effects, Unruh effect and single-quantum nonlocality, to abandon the particle concept. We may also question the definition of point-particle in the sense that it cannot be found at a specific location from the viewpoint of special theory of relativity or quantum theory.

 

2. All possible paths:

“Now let us show how this implies the principle of least time for a mirror. We consider all rays, all possible paths ADB, AEB, ACB, etc., in Fig. 26–3 (Feynman et al., 1963, section 26–6 How it works).”

 

To demonstrate the principle of least time for a mirror reflection, Feynman considers all light rays and all possible paths ADB, AEB, ACB, etc., in Fig. 26–3. The path ADB makes a small contribution, but the next path, AEB, takes a quite different time, so its angle θ is quite different. However, the phrase all possible paths could be misleading because there are only thirty-two paths (instead of infinity) shown in Fig. 26-14. Of course, it is impossible to include all possible paths that are infinitely many. It is more practical to consider a minimum number of paths that is sufficient to obtain a reasonably accurate answer.

 

To have a better idea of Feynman’s sum over all paths, it can be illustrated by a thought experiment involving a point source of light, two slits, and a screen (Feynman et al., 2010). Firstly, we imagine a black plate is inserted between the light source and the two slits. Whenever we drill some holes through the plate, it creates alternative routes for the light and each of this route corresponds to a probability amplitude. Assuming it is possible to drill more holes until the plate no longer exists, what does that mean? Alternatively, we can insert more and more plates between the light source and the screen, and then drill all holes such that there is nothing left. The thought experiment shows that we can sum the amplitudes of all possible paths (or holes) from the source to the screen.

 

3. Accumulated probability:

“Almost all of that accumulated probability occurs in the region where all the arrows are in the same direction (or in the same phase) (Feynman et al., 1963, section 26–6 How it works).”

 

Feynman mentions that almost all of that accumulated probability occurs in a region where all the arrows are in the same direction. This is why the extreme parts of the mirror do not contribute much to the image formation, but it still reflects light just like the other parts of the mirror. In his lecture on QED, he uses the phrases stopwatch hand and arrow to represent quantum probability (instead of complex numbers or complex vector). Furthermore, Feynman (1985) adds that “... all the arrow to point in the same direction, and to produce a whopping final arrow - lots of light! (p. 58).” One may clarify that the meaning of arrows in the same direction means that the light rays are in phase and thus, they reinforce each other.

 

Feynman elaborates that all of the contributions from the paths which have very different times cancel themselves out because they point in different directions. In his QED lecture, he provides a clearer explanation of how the light rays from the left-hand part of the mirror cancel themselves out (with Fig. 25). In his words, “we see that some of the arrows point more or less to the right; the others point more or less to the left. If we add all the arrows together, we have a bunch of arrows going around in what is essentially a circle, getting nowhere (Feynman, 1985, p. 46).” However, a complete cancellation of arrows is not always possible because the length of the arrows is inversely proportional to the distance the light traveled (Feynman, 1985, pp 73-74).”

 

Review Questions:

1. Would you assume light rays are made up of photons in your explanation of Fermat’s principle?

2. Did Feynman consider all possible paths ADB, AEB, ACB… in Fig. 26-14?

3. How does the ultimate picture of photons relate to the accumulation of arrows?

 

The moral of the lesson: the ultimate picture (or image formation) of photons is dependent on the probability of arrival of photons or an accumulation of arrows based on the principle of least time.

 

References:

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

2. Feynman, R. P., Hibbs, A. R., & Styer, D. F. (2010). Quantum mechanics and path integrals (Emended ed.). New York: Dover.

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. Wilczek, F. (1999). The persistence of ether. Physics Today, 52(1), 11-13.

5. Wilczek, F. (2001). Fermi and the Elucidation of Matter. arXiv preprint physics/0112077.

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.

Section 26–4 Applications of Fermat’s principle

(Least-time path / Same time paths / Relative refractive index)

 

In this section, Feynman applies Fermat’s principle from the perspective of least-time path and same-time paths, and apply the principle to deduce the relationship between relative refractive index and absolute refractive index.

 

1. Least-time path:

“… instead of the light deciding to come in the straightforward way, it also has a least-time path by which it goes into the region where it goes faster for awhile, in order to save time (Feynman et al., 1963, section 26–4 Applications of Fermat’s principle).”

 

Feynman mentions that when the sun appears to go below the horizon, it is actually already well below the horizon. He offers a one-line explanation: light travels more slowly in air than in a vacuum, and so the sunlight can reach point S beyond the horizon more quickly if it avoids the dense regions where it goes slowly by getting through them at a steeper tilt. One may add that a greater refraction occurs for light rays from the lower parts of the sun that travels through the denser air in the lower atmosphere. Thus, the sun appears to be flattened vertically. Perhaps Feynman should explain that the position of the sun can be deduced from the moon’s eclipse. The fact that we can see the sun and partial eclipse of the moon (well above the horizon) at the same time implies that the sun should be well below the horizon.

 

According to Feynman, the mirage we see on the road is a reflection of light from the sky. Essentially, light goes faster in the hotter air near the ground than the cooler air that is higher because thinner air has a lower refractive index. Because the temperature change in the air and the change in its refractive index are gradual, the rays later slowly curve upward (to achieve the least-time path). Furthermore, we can see some objects in the original positions, with an inverted image below it, as if there is a horizontal mirror. Another point is: the hot road due to the sunlight also warms the air and creates turbulence, thereby resulting in a blurry mirage (or shimmering effect).

 

2. Same-time paths:

“Therefore the problem of making a focusing system is merely to arrange a device so that it takes the same time for the light to go on all the different paths! (Feynman et al., 1963, section 26–4 Applications of Fermat’s principle).”

 

Feynman says that the sum of the two distances r₁ and r₂ traveled by the light (in the air) within the ellipsoidal mirror must be a constant. Then, he defines an ellipse as a  curve which has the property that the sum of the two distances from the two foci is a constant for every point on the curve. In a lost lecture, Feynman defines the ellipse as “the curve that can be made, by taking one string and two tacks and putting a pencil here and going around (Feynman et al., 1997, p. 149).” He also explains the reflection property of the ellipse and says that “… light is reflected as though the surface here were a plane tangent to the actual curve (Feynman et al., 1997, p. 150).” However, the paths traveled by the light rays within the ellipsoidal mirror may be considered as a relative maximum (Hecht, 2002, p. 110).

 

Feynman explains that the mirror is made in the shape of a parabola for gathering the light of a star. In addition, he defines a parabola as a curve which has the property that the sum of the distances XX′+X′P′ is a constant, no matter where X is chosen. One may elaborate that a parabola is a limiting case of an ellipse and thus, it has the same reflection property as the ellipse. On the other hand, the usefulness of the parabolic mirror to focus parallel lights is related to the focal property of the parabola. The focal property can be stated as “The tangent and normal at any point P of a parabola bisect the angles formed by the line joining P with the focus and the diameter through P (Korn & Korn, 2013, p. 53)” and can be proved without words (Ayoub, 1991).

 

3. Relative refractive index:

“So if we measure the speed of light in all materials, and from this get a single number for each material, namely its index relative to vacuum, called n_i…, then our formula is easy... (Feynman et al., 1963, section 26–4 Applications of Fermat’s principle).”

 

Feynman says that we can predict the index for a new pair of materials from the indexes of the individual materials that are both against air or against vacuum. It could be clearer to some students if he uses the two terms absolute refractive index and relative refractive index. Firstly, the absolute refractive index of a medium n₁ is a measured value that determines the angle of refraction when a ray of light passed from a vacuum into the medium 1. Mathematically, it is the ratio of the speed of light in vacuum to its speed in the medium 1. Secondly, the relative refractive index n₂₁ of a second medium relative to the first medium is the ratio of the speeds of light, v₁ and v₂, in the first and second medium respectively. In other words, we can predict the relationship between absolute refractive index and relative refractive index using Fermat’s principle.

 

Feynman elaborates that it is easy to get our formula (refractive index) if we can measure the speed of light in all materials and get its index relative to vacuum. In the next chapter, he explains the concept of chromatic aberration and then clarifies that “[t]he principle of least time is only an approximation, and it is interesting to know how much error can be allowed...” Strictly speaking, the definition of refractive index assumes the condition of no dispersion because different colors of light have different speeds in different refraction indices. It is related to the phenomenon of chromatic aberration in which the image of a white spot will have colors such that when we focus the red color, the violet color is out of focus, and vice versa. However, it is not easy as it may seem to measure the speed of light (different colors) in all materials.

 

Review Questions:

1. How would you explain the sun is actually well below the horizon during sunset?

2. How would you explain the reflection property of an ellipsoidal mirror?

3. Is it easy to measure the speed of light in different materials in order to determine the relative refractive index?

 

The moral of the lesson: we can apply Fermat’s principle to understand the position of the sun during sunset, the nature of mirage, and the principle of ellipsoidal mirror and parabolical mirror.

 

References:

1. Ayoub, A. B. (1991). Proof without Words: The reflection property of the parabola. Mathematics Magazine, 64(3), 175-175.

2. Feynman, R. P., Goodstein, D. L., & Goodstein, J. R. (1997). Feynman’s lost lecture: the motion of planets around the sun. London: Vintage.

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. Hecht, E (2002). Optics (4th edition). San Francisco: Addison Wesley.

5. Korn, G. A. & Korn, T. M. (2013). Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review. New York: Dover.

Section 26–3 Fermat’s principle of least time

 (Law of reflection / Law of refraction / Infinitesimal change)

 

In this section, Feynman discusses the law of reflection and Snell’s law of refraction from the perspective of Fermat’s principle of least time in which the first order infinitesimal change is zero.

 

1. Law of reflection:

“…the angle of incidence equals the angle of reflection is equivalent to the statement that the light goes to the mirror in such a way that it comes back to the point B in the least possible time (Feynman et al., 1963, section 26–3 Fermat’s principle of least time).”

 

According to Feynman, Fermat states the principle of least time in about 1650 as “out of all possible paths that it might take to get from one point to another, light takes the path which requires the shortest time.” Specifically, in a 1657 letter to Cureau de la Chambre, Fermat states that the law of refraction might be deduced from a minimum principle. Currently, the principle can be stated as: a ray of light going from one point to another takes the path that has the shortest optical path length. The optical path length (nd) is the distance moved by a light ray in a vacuum that is equivalent to the distance moved (d) by the light ray in a medium of index n in the time taken and the same number of wavelengths. In other words, the path of least time corresponds to the path of minimum number of wavelengths (or optical path length).

 

Feynman initially says that the first way of thinking that made the law about the behavior of light evident was discovered by Fermat. One may argue that this is not correct because Hero of Alexandria is the first to suggest that the light travels by the path of shortest distance in a plane mirror reflection in about 100AD. In a sense, Damianus could be the first to suggest the path of least time in about 200AD. In Mach’s (1953) words, “Damianus says ‘… the nature of our ray of vision did not permit of aimless wandering, the ray would be bent (reflected) at equal angles.’ As already mentioned, it is not quite clear whether the teleological conception refers to a minimum of time, or space, or perhaps both (p. 28).”

 

2. Law of refraction:

“…let us demonstrate that the principle of least time will give Snell’s law of refraction (Feynman et al., 1963, section 26–3 Fermat’s principle of least time).”

 

Applying the principle of least time to give Snell’s law, Feynman (1985) imagines a beautiful girl has fallen out of a boat and she is screaming for help in the water. In his lecture on QED for the public, Feynman rephrases the problem as “suppose you’re the lifeguard, sitting at S, and the beautiful girl is drowning at D (p. 51).” Some may prefer Feynman’s explanation in this lecture that distinguishes four kinds of path: (1) path of least time, (2) path of least water, (3) path of least distance, and (4) path of certainly not least time. Furthermore, he adds that “light seems to go slower in water than it does in air, which makes the distance through water more ‘costly’, so to speak, than the distance through air (p. 51).” The word costly is appropriate because Fermat names his law as the “principle of natural economy.”

Historically, Maupertuis proposes the principle of least action because it did not make sense to him how light can go by the shortest distance or by the least time. Interestingly, Feynman’s Ph.D. thesis is titled “the principle of least action in quantum mechanics.” More important, there is a type of laser beam (spacetime wave packets) that does not follow the law of refraction. Bhaduri et al. (2020) explain that “[s]pace-time refraction defies our expectations derived from Fermat’s principle…” In essence, the spacetime wave packets follow (or light beam) different rules when they refract through different materials, but it should be more appropriately explained by the principle of least action.

 

3. Infinitesimal change:

“Of course, there is an infinitesimal change of a second order; we ought to have a positive increase for displacements in either direction from C. (Feynman et al., 1963, section 26–3 Fermat’s principle of least time).”

Feynman explains that if we compare the time needed at points near C, there is essentially no change in time (in the first approximation) because the slope is zero at the bottom of the curve (Fig. 26–5). In other words, we can find optical laws by comparing the time at various points that are very close together (infinitesimally short distance). To have a better understanding of first approximation, we may use Taylor’s theorem that can be expressed as f(x) = f(a) + (x−a)f′(a)/1! + (x–a)²f′′(a)/2! + … (x – a)ⁿfⁿ(a)/n! + … Specifically, we can obtain the first order approximation by using the second term (x−a)f′(a)/1! that is a first-order polynomial. That is, we are using a straight-line graph to achieve the first approximation. The first term f(a) is also known as the zeroth order approximation.

Feynman remarks that there is an infinitesimal change of a second-order and in this example, there is a positive increase for displacements in either direction from C (Fig. 26–5). Perhaps some may not understand the meaning of infinitesimal change of second order. We can obtain the second-order approximation by using the term (x–a)²f′′(a)/2! that is a second order polynomial. As a result, we can use a quadratic curve to achieve the second approximation. In volume II, Feynman elaborates that “one of the properties of the minimum is that if we go away from the minimum in the first order, the deviation of the function from its minimum value is only second order… (Feynman et al., 1964).” One may clarify that this second order is related to the second derivative of a function.

 

Review Questions:

1. Do you agree with Feynman that “the first way of thinking that made the law about the behavior of light evident was discovered by Fermat”?

2. How would you explain the law of refraction using Fermat’s principle of least time?

3. What does the infinitesimal change of a first-order mean?

 

The moral of the lesson: we can derive optical laws (e.g., law of reflection and law of refraction) by finding the location whereby there is essentially no change in time in the first approximation.

 

References:

1. Bhaduri, B., Yessenov, M., & Abouraddy, A. F. (2020). Anomalous refraction of optical spacetime wave packets. Nature Photonics, 14, 416-421.

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. Feynman, R. P., Leighton, R. B., & Sands, M. (1964). The Feynman Lectures on Physics, Vol II: Mainly electromagnetism and matter. Reading, MA: Addison-Wesley.

5. Mach, E. (1953). The Principles of Physical Optics: An Historical and Philosophical Treatment. New York: Dover.