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.