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.