Section 31–1 The index of refraction

 (Low density materials / Apparent speed c/n / Correction field)

 

The three interesting concepts in this section are low density materials (related to idealization), effective speed (related to limitation), and correction field (related to approximation).

 

1. Low density materials:

“That corresponds to a material in which the index of refraction is very close to 1, which will happen, for example, if the density of the atoms is very low (Feynman et al., 1963, p. 31–2).”

 

Feynman states an assumption of refraction index of this chapter as “very close to 1,” e.g., very low density of the atoms. In Volume II, he clarifies: “In Chapter 31 of Volume I…, we had to restrict ourselves to finding the index only for materials of low density, like gases” (Feynman et al., 1964, section 32–1 Polarization of matter). Perhaps it is more accurate to state low density materials or rarefied gases for refractive index, but physicists have developed various models based on the density of atoms. Some may emphasize the condition, homogenous medium, i.e., an optical medium which has a uniform composition throughout a material. In Volume II, Feynman suggests that we “limit ourselves to isotropic dielectrics,” which refer to isotropic media, where the electromagnetic properties are the same in all directions.

 

“We shall try to understand the effect in a very simple case. A source which we shall call “the external source” is placed a large distance away from a thin plate of transparent material, say glass (Feynman et al., 1963, p. 31–1).”

 

It is potentially confusing for some when Feynman says that we shall understand the refractive index of a thin glass plate because this is different from materials of low density, like gases. It is worth mentioning that chapter 32 of Volume II is titled “Refractive Index of Dense Materials,” whereas chapter 31 of Volume I is mainly about the refractive index of rarefied gases. The word mainly is used because section 31.1 is related to the refractive index of the thin plate. Interestingly, in chapter 32 of Volume II, Feynman mentions that “[i]n ordinary inactive materials-that are not, like lasers, light sources themselves-g is a positive number, and that makes the imaginary part of n negative.” Thus, one may include the condition, linear medium, where nonlinear optical effects are negligible provided lasers (or intense light beams) are not used.

 

2. Apparent speed c/n:

“It is approximately true that light or any electrical wave does appear to travel at the speed c/n through a material whose index of refraction is n, but the fields are still produced by the motions of all the charges — including the charges moving in the material — and with these basic contributions of the field travelling at the ultimate velocity c (Feynman et al., 1963, p. 31–1).”

 

According to Feynman, light appears to travel at the speed c/n through a material, but the basic contributions of the electric field are still travelling at the ultimate velocity c. In QED, Feynman (1985) elaborates that “the ‘slowing’ of the light is extra turning caused by the atoms in the glass (or water) scattering the light. The degree to which there is extra turning of the final arrow as light goes through a given material is called its ‘index of refraction’ (p. 109).” One may stress that light is moving as a wave at c through the electromagnetic field in any medium, but its phase is delayed due to its interaction with atoms. In short, the interactions of light with atoms in any material can be described as absorptions and re-emissions of photons. In other words, the process of scattering and re-scattering of light waves in the material causes a phase shift and the apparent speed of light as c/n.

 

“It is approximately true that light or any electrical wave does appear to travel at the speed c/n through a material whose index of refraction is n, but the fields are still produced by the motions of all the charges… (Feynman et al., 1963, p. 31–1).”

“These charges will also radiate waves back toward the source S. This backward-going field is the light we see reflected from the surfaces of transparent materials. (Feynman et al., 1963, p. 31–2).”

 

In a sense, Feynman’s descriptions of light are inconsistent because he says that light is an electrical wave and light is a backward-going field. (In Volume II, Feynman also uses the phrase “electric field of the light wave.”) Perhaps it is a distraction to include the concept of light as a field here, however, a photon is an excitation of the electromagnetic field (based on quantum theory). Alternatively, one may describe light as an electromagnetic wave that is generated by oscillating electromagnetic fields. Furthermore, a light source can produce light waves to interact with a glass plate such that an electron in the plate is influenced by the source and all other oscillating electrons. Simply put, light waves tell the electromagnetic fields how to oscillate; electromagnetic fields tell light waves how to bend and reflect.

 

“Now all oscillations in the wave must have the same frequency. (We have seen that driven oscillations have the same frequency as the driving source.) This means, also, that the wave crests for the waves on both sides of the surface must have the same spacing along the surface because they must travel together, so that a charge sitting at the boundary will feel only one frequency (Feynman et al., 1963, p. 31–2).”

 

Feynman explains that all oscillations in a wave must have the same frequency because driven oscillations have the same frequency as the driving source and a charge at the boundary of two media will feel only one frequency. However, Franken, Hill, Peters, and Weinreich (1961) show that the color of a laser’s light in a medium could be changed, which led to uses such as LASIK eye surgery. This also results in the development of nonlinear optics that studies the interaction of light with matter in which the response of materials to the applied electromagnetic field is not linear. Specifically, the frequency of illuminated light could be doubled or tripled in the materials if light intensities are relatively high. Perhaps Feynman was not familiar with this phenomenon when the lecture was delivered on 27 Feb 62.

 

3. Correction field E_a:

“To see where we are going, let us first find out what the ‘correction field’ E_a would have to be if the total field at P is going to look like radiation from the source that is slowed down while passing through the thin plate… But if it appears to travel at the speed c/n then it should take the longer time nΔz/c or the additional time Δt = (n−1)Δz/c (Feynman et al., 1963, p. 31–3).”

 

It may be unclear to some why Feynman states the additional time needed to pass through a thin plate as Δt = (n−1)Δz/c. In a sense, the refractive index n of a medium is a number that tells us how many more wavelengths can be squeezed within the medium as compared to vacuum. In general, the additional time needed is Δt = (n₂−n₁)Δz/c in which n₁ and n₂ are the refractive index of vacuum and thin plate respectively. If we assume the thin plate mainly consists of rarefied gases, then n₂ and n₁ are equal to 1.0003 and 1 respectively, but n₂−n₁ becomes 0.0003. (Although the refractive index of air is equal to 1.0003, it is sometimes approximated as 1.) However, this chapter is not completely about rarefied gases because this section is partly related to the refractive index of a thin glass plate.

 

“The delay due to slowing down in the plate would delay the phase of this number, that is, it would rotate E_s through a negative angle. But this is equivalent to adding the small vector E_a at roughly right angles to E_s But that is just what the factor –i means in the second term of Eq. (31.8). It says that if E_s is real, then E_a is negative imaginary or that, in general, E_s and E_a make a right angle (Feynman et al., 1963, p. 31–3).”

 

According to Feynman, if E_s is real, then the induced field E_a is negative imaginary or that, in general, E_s and E_a make a right angle. This implies that the resultant of E_s and E_a would be longer than E_s and he should clarify whether this violates the law of conservation of energy. However, one may explain that the angle between E_s and E_a is a right angle based on the approximation formula: e^{−iω(n−1)Δz/c} ≈ 1−iω(n−1)Δz/c. To be more accurate, E_a would be rotated by an angle that is close to 90 degrees if we consider the contributions of the remaining terms of the exponential function. Importantly, the resultant of E_s and E_a should be slightly shorter than E_s (see Fig 31-3) such that the law of conservation of energy is not violated.

 

Review Questions:

1. Should Feynman state low density materials (rarefied gases) or very low density atoms as an idealization for the refractive index in this chapter?

2. Why do light waves appear to travel at the speed c/n in a medium? Do you agree with Feynman that all oscillations in the wave must have the same frequency or provided we limit ourselves to low intensity light?

3. Is it correct for Feynman to say that E_s and E_a make a right angle (Approximation)?

 

The moral of the lesson: light waves tell the electromagnetic fields how to oscillate, whereas electromagnetic fields tell light waves how to bend and reflect; this is due to the process of scattering and re-scattering of light waves in the material that causes a phase shift and the apparent speed of light as c/n.

 

References:

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

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

4. Franken, P.A., Hill, A.E., Peters, C.W., & Weinreich, G. (1961). Generation of Optical Harmonics. Physical Review Letters, 7, 118-119.

Section 30–7 The field of a plane of oscillating charges

 (Radiation field / Infinite plane / Finite charges)

 

In this section, the three interesting concepts involved in a problem are radiation field, an infinite plane of in-phase oscillating charges, and a finite amount of charges on the plane. This section is about an idealization of a plane of in-phase oscillators, but the title could be known as “The far field of an infinite plane of oscillating charges.” This section not only aptly ends with the concept of diffraction that is due to an infinite number of sources, but a formula is derived to explain the origin of refractive index for the next chapter.

 

1. Radiation field:

Suppose that we have a plane full of sources, all oscillating together, with their motion in the plane and all having the same amplitude and phase. What is the field at a finite, but very large, distance away from the plane? … (We cannot get very close, of course, because we do not have the right formulas for the field close to the sources.) (Feynman et al., 1963, p. 30–10).”

 

One may define radiation field as an electromagnetic field that provides the energy radiated by accelerated charges and it is inversely proportional to the distance from the charges. However, the term radiation field could be distinguished into two regions, namely near field and far field, that can be expressed by two different mathematical formulas. To be specific, we can define far field using the equation (28.6) in which the field produced by an accelerating charge is moving non-relativistically and it is located at a very large distance r. This problem also includes the far field of an infinite number of sources that are located at infinity. On the other hand, we do not need the right formulas here for the near field that is close to the sources.

Note: It may be confusing to some why Feynman says “… we do not have the right formulas for the field close to the sources,” but we may use near field to explain the maximum limit of refractive index of a medium (Andreoli et al., 2021).

 

“We know that the radiation field is proportional to the acceleration of the charge, which is −ω²x₀e^iωt…… Using this value for the acceleration as seen from P in our formula for the electric field at large distances from a radiating charge, we get (30.11) (Electric field at P from charge at Q) ≈ q/4πϵ₀c²ω²x₀e^iω(t−r/c)/r (Feynman et al., 1963, p. 30–10).”

 

Feynman clarifies that the formula (30.11) is not quite right because it should not only include the acceleration of the charge but its component perpendicular to the line QP. Theoretically, the component perpendicular to QP of the far field due to the charges would not be canceled out because the oscillations are all in the same direction despite having cylindrical symmetry. From a practical point of view, it is the far field that allows electromagnetic waves to propagate far distances with weak signal strength, but it can still be picked up by receivers or antennas. However, some may prefer saying the acceleration field (instead of radiation field or far field) produced by an accelerating charge varies as 1/r. This is different from the velocity field (Coulomb field) of the charge that is independent of a, but its field varies as 1/r².

 

2. Infinite plane:

“When ρ = 0, we have r = z, so the limits of r are z to infinity… Now e^−i∞ is a mysterious quantity. Its real part, for example, is cos (−∞), which, mathematically speaking, is completely indefinite (although we would expect it to be somewhere—or everywhere (?)—between +1 and −1!). (Feynman et al., 1963, p. 30–11).”

 

Feynman says that e^−i∞ is a mysterious quantity and its real part, cos(−∞), is indefinite. On the contrary, mathematicians prefer to write lim e^−iz and include the condition z ® ∞, but they consider e^−i∞ to be illegal. Furthermore, they would investigate whether a function is holomorphic using Cauchy-Riemann equations u_x = v_y and u_y = –v_x. In general, physicists may define terms involving infinity without rigor or quote a statement that is attributed to Einstein: “Two things are infinite, the universe and human stupidity, and I am not yet completely sure about the universe.” More importantly, the problem based on the infinite plane of constant density of charge is unrealistic and an infinite amount of charges on a “circular ring element” (2pr)Dr divided by the distance r that is infinity could be considered as an undefined quantity.

 

In Fig. 30–11 we have drawn the first five pieces of the sum. Each segment of the curve has the length Δr and is placed at the angle Δθ = −ωΔr/c with respect to the preceding piece (Feynman et al., 1963, p. 30–11).

 

The first term of e^iθ = e^−iωz/c has the most contribution to the total field because z is the shortest distance from the plane to P, but ωz/c is relatively small compared to z and thus the angle of the first arrow (θ) should be small. Similarly, the remaining each segment of the curve has the length Δr and it is placed at the angle Δθ = −ωΔr/c with respect to the preceding piece. One should realize that the real part (x component) starts to decrease in Fig. 30–11 when we add the fourth and fifth term, but this may not really correspond to the physical situation. In other words, not only e^−i∞ is a mysterious quantity, but it is remarkable that e^−iωz/c starts to oscillate right at the beginning. This suggests that the idealized model based on e^−iωz/c has a serious limitation and thus needs tweaking.

 

In Surely You’re Joking, Mr. Feynman!, Feynman (1997) challenged Paul Olum to give him an integral that most people could evaluate with only contour integral, but he could use other methods: “One time I boasted, ‘I can do by other methods any integral anybody else needs contour integration to do.’ So Paul puts up this tremendous damn integral he had obtained by starting out with a complex function that he knew the answer to, taking out the real part of it and leaving only the complex part. He had unwrapped it so it was only possible by contour integration! He was always deflating me like that. He was a very smart fellow (pp. 195-196).” Perhaps Feynman could have explained how the physical situations result in another Cornu spiral or contour integral based on Fermat’s principle of extremum path.

 

3. Finite charges:

“In any real situation the plane of charges cannot be infinite in extent, but must sometime stop... If, however, we let the number of charges in the plane gradually taper off at some large distance from the center (or else stop suddenly but in an irregular shape so for larger ρ the entire ring of width dρ no longer contributes), then the coefficient η in the exact integral would decrease toward zero. (Feynman et al., 1963, p. 30–11).”

 

Feynman suggests tweaking the problem by decreasing the number of charges in the plane that are farther from the center, i.e., the coefficient η in the exact integral would decrease toward zero. We can also include the projection of the acceleration on the plane perpendicular to the line PQ, but we would feel unlucky to even approximate the integral involving an additional 1/r. More importantly, some may argue whether it is mathematically legal to let e^−i∞ equal to zero due to various physical situations. For instance, the coefficient η could be specified as a function that varies with 1/r or e^−r because there is no infinite amount of charges in the real world. Alternatively, Feynman could have discussed whether including the projection of the acceleration on the plane perpendicular to the line PQ would definitely improve the formula, however, he claims that the formula (30.18) or (30.19) is correct at any distance z.

 

It is interesting to note that (iωx₀e^iωt) is just equal to the velocity of the charges, so that we can also write the equation for the field as Total field at P=−ηq²ϵ₀c[velocity of charges] at t−z/c (Feynman et al., 1963, p. 30–11).”

 

Feynman concludes the chapter by saying the formula derived is fortunately rather simple and it is valid for distances far from the plane of oscillating charges… If he assumes that η is a function that varies with 1/r or e^−r, then the integration method has to be changed and it would result in a different formula. The so-called simple formula is a result of idealizations and approximations because we have avoided realistic models such as including the projection of the acceleration on the plane perpendicular to the line PQ (cosine factor). In a sense, this problem beautifully shows the daily life of a physicist in cheating (idealizing models) and tweaking (approximating formulas). However, Feynman could end this chapter by explaining the importance of the formula derived because it relates the delay in phase of the field to refractive index.

Note: It may be worth mentioning this section was delivered at the beginning of the lecture of chapter 31 on refractive index.

 

Review Questions:

1. How would you define the radiation field at a distance far from a plane of oscillators (Idealization)?

2. Should we assume the problem to be based on an infinite plane of constant density of oscillating charges (Limitation)?

3. Is it mathematically legal to let e^−i∞ equal to zero such that we can obtain a reasonable approximate answer (Approximation)?

 

The moral of the lesson: We can understand diffraction further by idealizing (cheating) a plane of in-phase oscillating charges, approximating (tweaking) the density of charges on the plane, and confessing the limitation of the model. In a sense, this is an unfortunate problem, but it comes out—fortunately a rather simple formula by “cheating” and “tweaking” (creating luck).

 

References:

1. Andreoli, F., Gullans, M. J., High, A. A., Browaeys, A., & Chang, D. E. (2021). Maximum Refractive Index of an Atomic Medium. Physical Review X, 11, 011026.

2. Feynman, R. P. (1997). Surely You’re Joking, Mr. Feynman! : Adventures of a Curious Character. New York: Norton.

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.

 

Section 30–6 Diffraction by opaque screens

(Effective light sources / Cornu’s spiral / Geometrical shadow edge)

 

The three interesting concepts discussed in this section are effective light sources, Cornu’s spiral, and geometrical shadow edge.

 

1. Effective light sources:

“Of course, actually there are no sources at the holes, in fact that is the only place that there are certainly no sources… If we use the theorem that we have not yet proved, then we can replace the actual problem by a set of effective sources uniformly distributed over the open space beyond the object (Feynman et al., 1963, p. 30–8).”

 

Some may enjoy how Feynman explained the diffraction of light waves at an opaque sheet with holes in it: We have assumed that there are sources distributed with uniform density across the open holes, but there are actually no sources at the holes. However, there is a gap in his explanation because he has not proved the theorem that justifies why we can replace the problem with a set of effective light sources uniformly distributed over the open space. For example, one may state the theorem as Huygens-Fresnel Principle and then explain that every point on a wavefront is the source of spherical wavelets... In short, the idealized light sources on the wavefront do not exist as physical objects, but they are useful mathematical objects. Perhaps Feynman did not like this principle partly because he wrote that “[a]ctually Huygens’ principle is not correct in optics… (Feynman, 1942, p. 91)” in his PhD thesis.

 

Nevertheless, we get the correct diffraction patterns by considering the holes to be the only places that there are sources; that is a rather peculiar fact. We shall explain later why this is true, but for now let us just suppose that it is (Feynman et al., 1963, p. 30–8).

 

Feynman says that we can get the correct diffraction patterns by considering the holes to be the only places where there are sources, but he has planned to explain later why this is true. Note that he does not provide the explanation for the diffraction patterns in this chapter. In section 31–6 Diffraction of light by a screen, he mentions: “[w]e have the result that the field at P when there are holes in a screen (case b) is the same (except for sign) as the field that is produced by that part of a complete opaque wall which is located where the holes are!... (Feynman et al., 1963).” Furthermore, he remarks that this theory of diffraction is only approximate, and it is valid only if the holes are not too small. In the real world, there are real sources, for example, a beam of radiation incident on an atom causes the electrons in the atom to oscillate and thus the electrons can radiate in various directions (Feynman et al, 1963, Chapter 32).

 

2. Cornu’s spiral:

“To construct that curve involves slightly advanced mathematics, but we can always construct it by actually drawing the arrows and measuring the angles. In any case, we get the marvelous curve (called Cornu’s spiral) shown in Fig. 30–8. (Feynman et al., 1963, p. 30–9).”

 

According to Feynman, Fig. 30–8 shows a marvelous curve that is called Cornu’s spiral. Strictly speaking, the figure does not really show a curve but a series of arrows representing the addition of amplitudes for many in-phase oscillators or antennas. Simply phrased, the arrows having the same length means that the idealized antennas (instead of a continuous line source) have the same electric field strength and are equally spaced. The slightly advanced mathematics is essentially Fresnel integrals that is also known as Euler’s identity: ò₀^u e^i(p/2)u² du = ò₀^u cos (p/2)u² du + iò₀^u sin (p/2)u² du. In other words, Cornu’s spiral is a continuous curve in the complex plane of the points Z = C(z) + iS(z) in which C(z) and S(z) are Fresnel Integrals.

 

A property of Cornu’s spiral is its curvature at any point is linearly proportional to its arc length (distance along the spiral) from the origin. The curve spirals towards a point relatively quickly because the diffraction pattern is mainly due to a small region of effective sources. In essence, the intensity of diffraction pattern is mainly contributed by light rays of the shortest and shorter paths. We can find similar spirals for phenomena including reflection and refraction. Similarly, in his lecture on QED, Feynman (1985) wrote: “[b]elow the graph is the direction of each arrow, and at the bottom is the result of adding all the arrows. It is evident that the major contribution to the final arrow’s length is made by arrows E though I, whose directions are nearly the same because the time of their paths is nearly the same (p. 43).”

 

3. Geometrical shadow edge:

“The intensity near the edge of a shadow. The geometrical shadow edge is at x₀ (Feynman et al., 1963, p. 30–9).”

 

Perhaps Feynman could have clarified the meaning of geometrical shadow edge or geometrical shadow. For instance, the geometrical shadow may be explained as the idealized shadow that would have been seen, assuming there are no diffraction effects. That is, the diffraction of light due to a semi-infinite opaque screen causes the edge of geometrical shadow to be fuzzy and thus physicists define the geometrical shadow edge at x₀. Better still, the geometrical shadow edge could be defined as the boundary between the illuminated region and shadow region. Additionally, the intensity at x₀ is ¼I₀ whereby I₀ is the unobstructed intensity if there is no semi-infinite opaque screen. Perhaps Fig. 30–9 could be modified as shown below. 

Some may ponder whether it is appropriate to say “the intensity is ¼ of the incident light.” This is related to the explanation “actually there are no sources at the holes, in fact that is the only place that there are certainly no sources,” but it is good to stress that I₀ is not the maximum intensity. Theoretically, the intensity at the geometrical shadow edge (x₀) is ¼I₀ implies that I₀ would be the intensity of a plane wave that is parallel to the semi-infinite opaque screen. In other words, the intensity of the incident light is I₀ at all points on the same wavefront and the intensity at the screen or anywhere would remain I₀ if there is no obstacle. Alternatively, we can deduce the intensity at x₀ to be ¼I₀ without using Cornu’s spiral by simply explaining the amplitude is halved because of the semi-infinite opaque screen. However, the plane wave does not exist in the real world and “actually there are no sources at the holes.” (It should also be a plane wave because "we have light coming in from infinity" as mentioned by Feynman.) 

 

Review Questions:

1. Would you explain that there are effective sources or no sources at the holes?

2. Would you say that Fig. 30–8 shows a marvelous curve that is called Cornu’s spiral?

3. How would you define the geometrical shadow edge?

 

The moral of the lesson: The intensity of diffraction pattern is dependent mainly on light rays that traveled by the shortest path to the screen followed by those traveled by slightly shorter paths.

 

References:

1. Feynman, R. P. (1942/2005). Feynman’s thesis: A New Approach to Quantum Theory. Singapore: World Scientific.

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.