Section 27–2 The focal length of a spherical surface

 (Spherical surface / Plane surface / Spherical mirror)

 

In this section, Feynman discusses the formation of an image through a single spherical surface, plane surface, and spherical mirror.

 

1. Spherical surface:

“The farther ones may deviate if they want to, unfortunately, because the ideal surface is complicated, and we use instead a spherical surface with the right curvature at the axis. (Feynman et al., 1963, section 27–2 The focal length of a spherical surface).”

 

Feynman says that the simplest situation is a single refracting surface and we desire to arrange the curved surface such that every ray from O which hits the surface, at any point P, will be bent to reach the focal point O′. That is, the time it takes for the light rays to go from O to P plus n×O′P is equal to a constant independent of the point P. However, the curved surface is a fourth-degree curve sometimes known as the Cartesian ovals (or oval of Descartes) that may be used for lens design. We may derive the curved surface using simple analytic geometry based on the equation OP + n×O′P = Ö[(x – o)² + y²] + nÖ[(x – o′)² + y²] = c in which o and o′ are the x co-ordinate of O and O′ respectively. The fourth-degree curve can be obtained after expanding the equation Ö[(x – o)² + y²] + nÖ[(x – o′)² + y²] = c.  

 

According to Feynman, the law that connects the distances s and s′, and the radius of curvature R of a single spherical surface is (1/s) + (n/s′) = (n−1)/R. The main point of the equation is: the excess time (due to extra path) along route OP (h²/2s) and the other route (nh²/2s′) is balanced by the excess time (due to extra refractive index) needed for (n−1)VQ = (n−1)(h²/2s). The equation derived is known as the Gaussian formula for a single spherical surface. We can also derive the formula (27.3) by assuming the angles of incident light rays are so small such that the sines of angles can be replaced by the angles (in radians). In section 27-3, he adds that most of the lenses have two surfaces, but this is not a common definition of a lens.

 

2. Plane surface:

“… if we look into a plane surface at an object that is at a certain distance inside the dense medium, it will appear as though the light is coming from not as far back (Feynman et al., 1963, section 27–2 The focal length of a spherical surface).”

 

In the case of a plane surface, Feynman let R equal to infinity and obtain (1/s) + (n/s′) = 0. It can be simplified as s′ = −ns, which means that if we can look from a dense medium into a rare medium and see a point in the rare medium, the image appears to be deeper by a factor n. When we look at the bottom of a pool from above, it is not exactly correct to say that the apparent depth of the pool is ¾ of the real depth just because the reciprocal of the refraction index of water is ¾. Specifically, the apparent depth would vary with the angle of incidence or any point in the pool with respect to the location of an observer’s eye. The factor of ¾ is approximately correct for paraxial rays that are small enough if the cosine of the incident angle is almost equal to one or the sine of the incident angle is almost equal to the same incident angle (in radians).

 

To illustrate the meaning of negative s′ (or negative image), Feynman uses a ray diagram (Fig. 27–3) whereby the light rays which are diverging from O, will be bent at the refracting surface, and they will not come to a focus. That is, O is too close to the refracting surface such that the image is formed “beyond infinity.” Feynman explains that the image is a virtual image if the rays appear to come from a fictitious point that is different from the original location. Although the virtual image cannot be directly observed on a screen, it is still possible for the light rays to converge and form an observable image if a suitable lens is chosen. On the other hand, a real image is not only observable by using a screen, its aerial image through the lens axis (or optical axis) can be seen directly by naked eyes.

 

3. Spherical mirror:

“… we leave it to the student to work out the formula for the spherical mirror, but we mention that it is well to adopt certain conventions concerning the distances involved s (Feynman et al., 1963, section 27–2 The focal length of a spherical surface).”

 

According to Feynman, the object distance s is positive if the point O is to the left of the surface, whereas the image distance s′ is positive if the point O′ is to the right of the surface. In addition, the radius of curvature of the surface is considered positive if the center is to the right of the surface. However, there should be clarifications on the freedom of adopting certain conventions in optics. For example, we have the freedom to let the object distance to be positive or negative if the object O is to the left of the spherical mirror. On the other hand, we also have the freedom to let the image distance to be positive or negative if the image O′ is to the right of the spherical mirror. Thus, we have at least four conventions based on the object distance and image distance.

 

Feynman mentions that if we had used a concave surface, our formula (27.3) would still give the correct result if we merely make R a negative quantity. The convention is commonly known as “Real is Positive” convention, but some students may not be able to determine the location of an image whether it is at the left or right of the spherical mirror. To have a better understanding of the nature of the image, we can use ray diagrams to show whether the image is upright or inverted. One may explain that the concave mirror behaves like a convex lens in the sense that the light rays refracted to the right, are instead reflected to the left. This is possible provided the concave mirror is small compared with its radius of curvature such that parallel rays are focused to a common point (or focal point).

 

Review Questions:

1. How would you explain the derivation of the formula of a single spherical surface, (1/s) + (n/s′) = (n−1)/R?

2. How would you explain the apparent depth of a swimming pool does not look as deep as it really is?

3. What are the pros and cons of “real is positive” convention for a spherical mirror?

 

The moral of the lesson: the equation (1/s) + (n/s′) = (n−1)/R can be explained by how the excess time of two diagonal rays (due to extra path) is balanced by the excess time of a horizontal ray (due to extra refractive index).

 

References:

1. 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.   

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.