Section 27–1 Introduction

(Optics principles / Hamilton’s theory / Geometrical formula)

 

In this section, Feynman explains that geometrical optics in chapter 27 does not involves new principles, mentions the importance of Hamilton’s theory, and derives a general geometrical formula for the chapter.

 

1. Optics principle:

“One can set up the problem and make the calculation for one ray after another very easily. So the subject is really ultimately quite simple, and involves no new principles (Feynman et al., 1963, section 27–1 Introduction).”

 

Feynman initially says that geometrical optics is either very simple or very complicated. Subsequently, he adds that the subject is really ultimately quite simple and involves no new principles. This is because he has already discussed Fermat’s principle of least time that contains both the law of straight-line propagation and the law of reflection for plane mirror in section 26–3 Fermat’s principle of least time. Next, he demonstrates how Fermat’s principle can give Snell’s law of refraction within the same section. Lastly, he considers the principle of reciprocity (or principle of reversibility of light) to be an interesting consequence of the principle of least time, that is, if light can move through one direction, it can be sent through the opposite direction (section 26–4 Applications of Fermat’s principle).

 

According to Feynman, the rules of elementary or advanced optics are seldom characteristic of other fields and thus, there is no special reason to follow the subject very far except for Hamilton’s optics. However, the law of reflection is also a characteristic of other fields such as electrodynamics and surface chemistry (neutron reflection). On the other hand, the law of refraction can be found in electrodynamics and geophysics. Interestingly, a modified form of Fermat’s principle of least time is applicable to Lagrangian optics. Furthermore, Fermat’s principle can be found in wave mechanics and it is a source of the key idea in quantum mechanics.

 

2. Hamilton’s theory:

“The most advanced and abstract theory of geometrical optics was worked out by Hamilton, and it turns out that this has very important applications in mechanics (Feynman et al., 1963, section 27–1 Introduction).”

 

Feynman opines that the most advanced and abstract theory of geometrical optics was worked out by Hamilton. However, adaptive optics is also an advanced and abstract optical theory (possibly more advanced or abstract) that is useful for large optical telescopes. For example, Dyson reduced the blurring effect of atmospheric turbulence when he was involved in a project on tracking missiles and later galaxies in the early 1970s. Specifically, Dyson (1975) calculated the statistical behavior of an optical system in order to apply it to a “deformable mirror” to cancel the atmospheric distortions on the image of stars. Interestingly, Dyson also helped to synthesize Feynman’s diagrams and Schwinger’s field model.

 

Feynman mentions that Hamilton’s theory has very important applications in mechanics and it is actually even more important in mechanics than it is in optics. However, Schrödinger may not completely agree with Feynman because Hamilton’s theory is also very important in quantum mechanics. In his Nobel lecture, Schrödinger (1933) says, “… Fermat principle thus appears to be the trivial quintessence of the wave theory (p. 307).” He adds that “[i]t seemed as if Nature had realized one and the same law twice by entirely different means: first in the case of light, by means of a fairly obvious play of rays; and again in the case of the mass points, which was anything but obvious, unless somehow wave nature were to be attributed to them also (p. 308).” Then, he uses Hamilton’s theory to derive the Schrödinger’s equation.

 

3. Geometrical formula:

“In order to go on, we must have one geometrical formula, which is the following: if we have a triangle with a small altitude h and a long base d, then the diagonal s… (Feynman et al., 1963, section 27–1 Introduction).”

 

Feynman insists that we must have one geometrical formula that is in terms of a triangle with a small altitude h, a long base d, and the diagonal s. He clarifies that we need this formula to find the difference in time between two different routes (Fig. 27–1). However, it is not true that we must use the geometrical formula Δ ≈ h²/2s to derive optical formulas. For example, we can use another approximation formula such as sin q ≈ q that can be used to derive similar optical formulas instead of Δ ≈ h²/2s. Importantly, the optical formulas derived by Δ ≈ h²/2s or sin q ≈ q are only applicable to paraxial rays (close to the axis) in which the slope angles are very small to a first approximation. The triangle with a small altitude h and a long base d implies that the slope angles should be very small.

 

Feynman explains that the difference between the diagonal and the base, Δ = s – d, can be found in a number of ways. That is, we can factorize s² − d² = h² to (s − d)(s + d) = h² and use s – d = Δ and s + d ≈ 2s to obtain Δ ≈ h²/2s. However, it is easier to use sin q ≈ q that is based on an expansion of the sine of an angle. Using Maclaurin’s theorem, we have sin q = q - q³/3! + q⁵/5! - q⁷/7! + …… and we can consider the higher terms are negligible if q is close to zero. For the reason of consistency, we can also use the Maclaurin’s theorem to formulate higher orders theory of lens aberrations. For instance, we can use sin q = q - q³/3! in the third order theory of aberrations instead of deriving another geometrical formula.

 

Review Questions:

1. Is it true that the rules of elementary or advanced optics are seldom characteristic of other fields?

2. Do you agree with Feynman that the most advanced and abstract theory of geometrical optics was worked out by Hamilton?

3. Is it true that we must have the geometrical formula that is in terms of a triangle with a small altitude h, a long base d, and the diagonal s to derive optical formulas?

 

The moral of the lesson: we only need a geometrical formula such as Δ ≈ h²/2s to discuss the formation of images by curved surfaces.

 

References:

1. Dyson, F. J. (1975). Photon noise and atmospheric noise in active optical systems. JOSA, 65(5), 551-558.

2. 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.  

3. Schrödinger, E. (1933). The fundamental idea of wave mechanics. In Nobel Lectures, Physics 1922-1941. Elsevier Publishing Company, Amsterdam.