Section 31–5 The energy carried by an electric wave

(Thin-layer approximation / Low-density approximation / Plane-wave approximation)

 

Feynman derives a formula of Poynting’s vector that is based on thin-layer approximation, low-density approximation, and plane-wave approximation.

 

1. Thin-layer approximation:

“All of our calculations have been made for a thin layer of material whose index is not too far from 1, so that E_a would always be much less than E_s (just to make the calculations easier). In keeping with our approximations, we should, therefore, leave out the term Ē_a², because it is much smaller than Ē_sĒ_a (Feynman et al., 1963, p. 31–9).”

 

In this section, Feynman only uses the term low-density approximation to derive a formula of Poynting’s vector, ϵ₀cE². However, he mentions that all of his calculations are made for a thin layer of material whose index is not too far from 1. We should recall that the angle between E_s and E_a is almost a right angle and it can be related to the approximation formula, e^{−iω(n−1)Δz/c} ≈ 1−iω(n−1)Δz/c. To be precise, this method of approximation is based on a “thin gas plate” in which Δz should be relatively thin in the formula. Otherwise, the decrease in E due to absorptions of light in a thicker plate would result in a significant loss of light energy. In short, we may include the term “thin gas plate approximation” or simply “thin-layer approximation.”

 

“For the first term we can write αĒ_s², where α is the as yet unknown constant of proportionality which relates the average value of E² to the energy being carried (Feynman et al., 1963, p. 31–9).”

 

In section 31-2, Feynman states: “If the source S (of Fig. 31–1) is far off to the left, then the field E_s will have the same phase everywhere on the plate, so we can write that in the neighborhood of the plate E_s = E₀e^iω(t−z/c).” He did not determine Ē_s² possibly because it involves calculations due to the decrease in E_s, that is, the absorption of light will end up as thermal energy in a material. Perhaps it is good to emphasize that the electric field E_s is attenuated by a factor of e^−kz in which k is the absorption index of the material. If the loss of energy is considered negligible, Feynman could have determined Ē_s² instead of writing it as αĒ_s². Note that in the next chapter, he changes the notation Ē_s² to < E_s² >.

 

2. Low-density approximation:

“One way of checking that our calculations are consistent is to see that we always keep terms which are proportional to NΔz, the area density of atoms in the material, but we leave out terms which are proportional to (NΔz)² or any higher power of NΔz. Ours is what should be called a “low-density approximation” (Feynman et al., 1963, p. 31–10).”

According to Feynman, leaving out terms that are proportional to (NΔz)² or any higher power of NΔz should be called a “low-density approximation.” However, it involves a low-density approximation due to smaller N and thin layer approximation due to shorter Δz. One may recall that η (= NΔz) is the number of charges per unit area where N is the number of atoms per unit volume of the thin plate. In Volume II, Feynman mentions: “…we had to restrict ourselves to finding the index only for materials of low density, like gases…. so we studied only the rarefied gas, where such effects are not important (Feynman et al., 1964).” It may be more accurate to state “rarefied gases approximation,” but physicists have developed various models of refractive index based on different atomic densities.

 

In section 32–3 Waves in a dielectric of Volume II, Feynman adds: “… if N is small enough so that n is close to one (as it is for a gas), then Eq. (32.27) says that n² is one plus a small number: n² = 1 + ϵ. We can then write n = √(1 + ϵ) ≈ 1 + ϵ/2, and the two expressions are equivalent (Feynman et al., 1964).” It should be worth mentioning that the refractive index of air is 1.0003. On the other hand, the term rarefied gas means that the pressure of the gas is much less than atmospheric pressure. Therefore, ϵ/2 is lesser than 0.0003, i.e., n is expected to be very close to 1, but small N may be known as a “rarefied gas approximation” or “low-pressure gas approximation.”

 

In Volume II, Feynman elaborates that “S = ϵ₀c²E×B, is called ‘Poynting’s vector,’ after its discoverer. It tells us the rate at which the field energy moves around in space. …… Believe it or not, we have already derived this result in Section 31–5 of Vol. I, when we were studying light (Feynman et al., 1964).” Perhaps Feynman could have said that he has derived Poynting’s vector ϵ₀cE² using the definition of energy density of electromagnetic wave and Poynting’s theorem. (It is also known as Umov-Poynting vector because Nikolay Umov, a Russian physicist, first proposed the concept of energy flux in a continuous medium in 1874.) In a sense, it is paradoxical that the formula ϵ₀cE² can be exactly proved using Poynting’s theorem, but Feynman’s proof is based on “low-density approximation.”

 

3. Plane wave approximation:

“We now go back to Eq. (30.19), which tells us that for large z E_a = NΔzq_ev(ret by z/c)/2ϵ₀c (31.26) (Feynman et al., 1963, p. 31–10).”

 

Eq. (30.19), as mentioned by Feynman, is based on the assumption of an infinite plane of constant density of oscillating charges. More importantly, the phrase “for large z” means that a spherical wave may appear like a plane wave because the curvature becomes negligible when it is significantly farther away from the source. Some may prefer the term monochromatic plane wave because this is a single-frequency spherical wave from a faraway source that would look like a plane wave after traveling a very large distance. For example, the wavefronts of light from stars are effectively parallel. It may be known as a plane wave approximation as the direction of electric fields is perpendicular to the motion of the plane wave.

 

In section 31.2, Feynman explains that in the neighborhood of the plate E_s = E₀e^iω(t−z/c) if the source S (Fig. 31–1) is far off to the left (Feynman et al., 1963, p. 31–4). That is, the source S is idealized to be infinitely far and thus the electric fields of oscillating electrons are in the same direction and have the same amplitude E₀. Specifically, the field E_s is weaker in the z-direction and it is sometimes called an inhomogeneous plane wave (Jackson, 1999). In short, the homogeneous plane wave becomes an inhomogeneous plane wave when it enters a medium. Some may prefer to describe this method as “inhomogeneous plane-wave approximation.”

 

Review Questions:

1. How would describe Feynman’s method of calculations that is based on a thin layer of material whose refractive index is close to 1?

2. Is the energy carried by an electric wave calculated by only using low density approximation?

3. Would you describe a wave of the electric field after traveling a large z as a “monochromatic plane wave” (Griffiths, 2005) or “inhomogeneous plane wave” (Jackson, 1999)?

 

The moral of the lesson: A simplified formula of Poynting’s vector ϵ₀cE² for a chromatic (or inhomogeneous) plane wave can be derived using thin-layer approximation, low-density approximation, and plane-wave approximation.

 

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.

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

3. Griffiths, D. J. (2005). Introduction to Electrodynamics (3rd ed.). New Jersey: Pearson Education.

4. Jackson, J. D. (1999). Classical Electrodynamics (3^rd ed.). New York: John Wiley & Sons.

Section 31–4 Absorption

 (Absorption index / Absorption of light / Absorption spectrum)

 

In this section, the three interesting concepts are absorption index, absorption of light, and absorption spectrum.

 

1. Absorption index:

“We can see what such a complex index means by going back to Eq. (31.6), which is the equation of the wave after it goes through a plate of material with an index n (Feynman et al., 1963, p. 31–8).”

 

Feynman suggests that we can write n = n′−in′′ in which n′ and n″ are real numbers in order that n″ will turn out to be a positive number. In Volume II, he modifies the symbol slightly: “Let’s say that we write n as the sum of a real and an imaginary part: n = n_R−in_I, (32.35) where n_R and n_I are real functions of ω (Feynman et al., 1964).” However, one may clarify that this is a matter of convention, e.g., an alternative is using n = n′ + in′′ (instead of n = n′ - in′′), but n′′ will be a negative number that corresponds to a loss of energy. In addition, the amplitude of light wave in the material is gradually decreased and thus n′′ may be known as the absorption index. In essence, the use of complex index corresponds to the light wave that is represented by a complex function.

 

“We see that the imaginary part n″ of a complex index of refraction represents an absorption (or “attenuation”) of the wave. In fact, n″ is sometimes referred to as the ‘absorption index’ (Feynman et al., 1963, p. 31–8).”

 

It is good that Feynman relates the absorption index to the absorption of the light wave. Perhaps some may elaborate that there is a profound connection between the real index of refraction and the absorption index. Specifically, the real index (n′) and absorption index (n″) are related through the Kramers–Kronig relations. This was deduced independently by Ralph Kronig in 1926 and by Hans Kramers in 1927. In a sense, the relation between n′ and n″ should be expected because the Lorentz Oscillator model assumes the existence of “friction force.” Simply put, the friction force affects the absorption of light in the medium (including the absorption index) and the dispersion of light with respect to different frequency of light (as shown in the refractive index curve).

 

2. Absorption of light:

“As the wave goes through the material, it is weakened. The material is ‘absorbing’ part of the wave. The wave comes out the other side with less energy. We should not be surprised at this, because the damping we put in for the oscillators is indeed a friction force and must be expected to cause a loss of energy (Feynman et al., 1963, p. 31–8).”

 

Feynman says that the damping we put in for the oscillators is a friction force, but it causes a loss of light energy. However, the term friction force may be considered as a metaphor or an assumption of the Lorentz oscillator model, and it is not needed to explain the absorption of light. From the perspective of quantum physics, the energy of a light wave is absorbed by an electron provided the frequency of the light wave through a medium is equal to a resonant frequency of an electron in an atom. After the absorption of light, the electron interacts quantum mechanically with nearby atoms and converts its vibrational energy into thermal energy. In short, a photon is absorbed when an electron in an atom transits from one energy level to another.

 

“We may also point out that an imaginary part to the index n corresponds to bending the arrow E_a in Fig. 31–3 toward the origin. It is clear why the transmitted field is then decreased (Feynman et al., 1963, p. 31–8).”

 

It may not be clear to some why Feynman points out that an imaginary part to the index n corresponds to bending the arrow E_a toward the origin (See Fig. 31–3). One may explain that the arrow should be almost perpendicular in order that the resultant arrow E_a is decreased slightly. Perhaps some may prefer Feynman’s (1985) explanation in his public lecture on QED: “For substances that absorb light, the minor arrows are at less than right angles to the main arrow (Fig. 69b). This causes the final arrow to be shorter than the main arrow, indicating that the probability of a photon going through partially opaque glass is smaller than through transparent glass… (p. 109).” In other words, the almost perpendicular arrow E_a corresponds to the absorption of light in a medium such as transparent glass.

 

3. Absorption spectrum:

“It is just this effect that gives the dark lines in the spectrum of light which we receive from the sun. The light from the solar surface has passed through the sun’s atmosphere (as well as the earth’s), and the light has been strongly absorbed at the resonant frequencies of the atoms in the solar atmosphere (Feynman et al., 1963, p. 31–9).”

 

Feynman explains that the light from the solar surface has been strongly absorbed by the atoms in the solar atmosphere depending on their resonant frequencies. However, one may clarify that the absorption of light occurs in the cooler atmosphere that is further away from the solar surface (Dwivedi & Phillips, 2001). To be specific, the sun’s corona (the outer solar atmosphere) is hundreds of times hotter than the solar surface. (It is analogous to feeling warmer when you are walking farther away from a fireplace.) Furthermore, the atoms in the cooler (and farthest) atmosphere will re-emit absorbed photons in random directions subsequently. Thus, the dark lines of the absorption spectrum are not really black lines, but dimmer lines because much lesser photons are re-emitted in the same direction as the original photons.

 

“The observation of such spectral lines in the sunlight allows us to tell the resonant frequencies of the atoms and hence the chemical composition of the sun’s atmosphere. The same kind of observations tell us about the materials in the stars. From such measurements we know that the chemical elements in the sun and in the stars are the same as those we find on the earth (Feynman et al., 1963, p. 31–9).”

 

Feynman claims that the chemical elements in the sun and in the stars are the same as those we find on the earth. However, some may ask whether all atoms are exactly the same because these atoms could be compressed to a smaller size due to the very high pressure in the sun and their life span may appear to be different because of stronger gravitational forces. It is worth mentioning that some physicists study spectral lines from distant quasars to investigate whether physical laws remain constant. For example, the study of the fine structure constant in distant quasars suggests a modification of electromagnetic force. Interestingly, the study of the quasar spectrum also indicates that the absorption of light depends on the density of hydrogen in the Universe, as predicted by Gunn and Peterson (1965).

 

Review Questions:

1. Would you represent the complex index using n = n′ + in′′ or n = n′ - in′′?

2. Would you explain the absorption of light in a medium using the concept of frictional force?

3. Do you agree with Feynman that the chemical elements in the sun and in the stars are exactly the same as those we find on the earth?

 

The moral of the lesson: An absorption index is a complex number because it corresponds to the absorption of light that is represented by a complex function, whereas the absorption spectrum is due to the absorption of light that occurs in the cooler atmosphere further away from the solar surface.

 

References:

1. Dwivedi B. N., & Phillips K. J. (2001). The paradox of the sun's hot corona. Scientific American, 284(6), 40-7.

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. Gunn, J. E. & Peterson, B. A. (1965). On the density of neutral hydrogen in intergalactic space. Astrophysical Journal, 142, 1633‒1636.

Section 31–3 Dispersion

 (Normal dispersion / Anomalous dispersion / Refining dispersion equation)

 

The three interesting concepts in this section are normal dispersion, anomalous dispersion, and the refinements for the dispersion equation.

 

1. Normal dispersion:

“The phenomenon that the index depends upon the frequency is called the phenomenon of dispersion, because it is the basis of the fact that light is “dispersed” by a prism into a spectrum (Feynman et al., 1963, p. 31–6).”

 

Feynman explains that dispersion is a phenomenon in which the refractive index depends upon the frequency and it is based on the fact that light is dispersed by a prism into a spectrum. However, one may define dispersion as a spreading of light waves into their full spectrum of wavelengths in a dispersive medium. Furthermore, we may state the condition of dispersive medium whereby the speed of a light wave is dependent on its frequency, i.e., all material media are dispersive except vacuum. Specifically, normal dispersion refers to the increase in refractive index provided the frequency (w) increases toward a resonant frequency (w₀) and (w₀² - w²) decreases in accordance with the equation (31.19). In other words, the slope dn/dw is positive as long as w is not too close to one of the resonant frequencies of the dispersive medium.

 

Setting ω₀ = 0 in our dispersion equation yields the correct formula for the index of refraction for radiowaves in the stratosphere, where N is now to represent the density of free electrons (number per unit volume) in the stratosphere. But let us look again at the equation, if we beam x-rays on matter, or radiowaves (or any electric waves) on free electrons the term (ω₀²−ω²) becomes negative, and we obtain the result that n is less than one (Feynman et al., 1963, p. 31–6).”

 

According to Feynman, if we beam x-rays or radiowaves (or any electric waves) on free electrons in the stratosphere, the term (ω₀²−ω²) becomes negative and it means n is less than one. This is not quite correct because the term (ω₀²−ω²) would be positive for frequencies of radiowaves that are lesser than ω₀. In addition, there are several resonant frequencies (ω_0i) depending on various materials as shown in the equation (31.20). For example, the stratosphere also contains ozone molecules, which protect life on earth by reducing the harmful ultraviolet radiation because ozone molecules have stronger absorption in the ultraviolet region. Note that British geophysicists, Farman, Gardiner, and Shanklin (1985) reported a significant decrease in stratospheric ozone levels over the Antarctic stations.

 

In spite of the fact that it is said that you cannot send signals any faster than the speed of light, it is nevertheless true that the index of refraction of materials at a particular frequency can be either greater or less than 1. This just means that the phase shift which is produced by the scattered light can be either positive or negative… It is this advance in phase which is meant when we say that the ‘phase velocity’ or velocity of the nodes is greater than c (Feynman et al., 1963, p. 31–6-7).”

 

According to Feynman, the refractive index can be greater or less than 1 means that the phase shift which is due to the scattered light can be either positive or negative, and specifically, advance in phase means the “phase velocity” is greater than c. Importantly, the phase velocity is the velocity of any point within the wave, i.e., it is not necessarily the nodes. Perhaps Feynman could have proved this fact by using kx - wt = constant, which may refer to any point (or constant phase) in the wave. In short, d(kx - wt)/dt = 0 Þ dx/dt = w/k = c, which is independent of w and k for a sinusoidal wave. On the other hand, it is not strictly correct to say that the refractive index can be greater or less than 1 because it may be a complex number or less than zero. (In 1968, Veselago hypothesized the possibility of a negative refractive index.)

 

2. Anomalous dispersion:

“Very near the resonant frequencies, however, there is a small range of ω’s for which the slope is negative. Such a negative slope is often referred to as ‘anomalous’ (meaning abnormal) dispersion, because it seemed unusual when it was first observed, long before anyone even knew there were such things as electrons. From our point of view both slopes are quite ‘normal’! (Feynman et al., 1963, p. 31–8).”

 

Feynman explains that it seemed unusual to have a negative slope (the rate of change of refractive index with respect to frequency) when it was first observed without the knowledge of electrons. In short, the dispersion is normal if dn/dω is greater than zero (dn/dω > 0 or dn/dl < 0), i.e., the dispersion is anomalous if dn/dω is less than zero. In other words, anomalous dispersion of light refers to the refraction spectra in which the normal order of the separation of component waves is reversed in the region near the resonant frequency. In this frequency range, the material is almost opaque and it may be described as “regions of resonant absorption” (Jackson, 1999, p. 130). The anomalous dispersion regions may not be observed because the medium strongly absorbs energy from the light waves within the absorption band.

 

During Weiner’s interview on March 5, 1966, Feynman says: “They asked me which color was on top of a rainbow. I said I didn’t know, but I could figure it out. They said, all right, figure it out, and I drew a drop of water and said, let’s see, now, the red rays are bent more than the blue — And the professor said, ‘Would you draw a curve of index refraction against wavelength?’ And I thought: aha, I got it wrong. So when I drew the index versus wavelength I turned it around so that the index was higher for the blue end than for the red end — the opposite of what I just said. When I looked at my curve I just drew, I said, ‘Oh yes, excuse me, it’s the other way around.’ But I drew the curve because I knew from the question it was backward, and I thought, I’m fooling him. Of course, he’d just given me a hint that I was wrong on the other.”

 

Historically, Leroux discovered the phenomenon of anomalous dispersion involving iodine vapor whose absorption bands fall within the visible region. In Leroux’s (1862) words, “Iodine vapor disperses light in different direction to any substance yet studied; that is, a prism full of iodine vapor refracts red rays to a greater extent than blue rays (p. 246).” That is, a prism formed using iodine vapor will deviate the red rays more than the violet, giving a different spectrum as compared to a substance having normal dispersion. When it was later discovered that transparent substances (e.g., glass) also possess absorption regions, such as ultraviolet and infrared radiation, the term anomalous is no longer appropriate. Although Leroux adopted the term abnormal dispersion, it could be better to replace it with resonance dispersion.

 

3. Refining dispersion equation:

“Let us now look again at our dispersion equation... To be completely accurate we must add some refinements. First, we should expect that our model of the atomic oscillator should have some damping force (otherwise once started it would oscillate forever, and we do not expect that to happen). We have worked out before (Eq. 23.8) the motion of a damped oscillator and the result is that the denominator in Eq. (31.16), and therefore in (31.19), is changed from (ω₀²−ω²) to (ω₀²−ω²+iγω), where γ is the damping coefficient (Feynman et al., 1963, p. 31–7).”

 

Feynman says that our model of the atomic oscillator should have some damping force so that it would not oscillate forever. The presence of damping force means that the amplitude of oscillations would not approach infinity when the driving frequency of light waves is equal to the resonant frequency of electrons. Some may explain that it is difficult to observe the region of anomalous dispersion because the electrons strongly absorb energy from the light waves whose frequencies are close to their resonant frequency. For example, metals are opaque because the electrons in the metals absorb the light waves and re-emit them during their passage through a metal. However, Lorentz oscillator model has already included damping force and thus another refinement could be including free and bound electrons (Hecht, 2002).

 

“We need a second modification to take into account the fact that there are several resonant frequencies for a particular kind of atom. It is easy to fix up our dispersion equation by imagining that there are several different kinds of oscillators, but that each oscillator acts separately, and so we simply add the contributions of all the oscillators. Let us say that there are N_k electrons per unit of volume, whose natural frequency is ω_k and whose damping factor is γ_k. We would then have for our dispersion equation n = 1 + (q_e²/2ϵ₀m)[∑kN_k/(ω²k−ω²+iγ_kω)] (Feynman et al., 1963, p. 31–7-8).”

 

In an endnote of chapter 31, Feynman adds that the dispersion equation (31.20) n = 1 + (q_e²/2ϵ₀m)[∑kN_k/(ω_k²−ω²+iγ_kω)] is still valid in quantum mechanics, but its interpretation is somewhat different. To be more accurate, an atom with one electron, like hydrogen, has several resonant frequencies. However, in the summary of the lecture (as shown on the blackboard), Feynman has also included the equation n – 1 = (Nq_e²/2ϵ₀m)[∑f_i/(ω_k²−ω²+iγ_kω)] in which f_i = fractional strength of oscillator i. Currently, we may explain that the f_i terms, which satisfy the requirement that å_if_i = 1, are weighting factors known as oscillator strengths (Hecht, 2002). Alternatively, the f_i terms are known as transition probabilities, and perhaps Feynman could have cited the Thomas-Reiche-Kuhn sum rule.

 

Review Questions:

1. Do you agree with Feynman that the refractive index n is definitely less than one if we beam radiowaves (or any electric waves) on free electrons?

2. Would you adopt the term anomalous dispersion?

3. How would you refine and interpret the new dispersion equation?

 

The moral of the lesson: Anomalous dispersion (or resonance dispersion) occurs in the region near a resonant frequency, whereas normal dispersion occurs below the resonant frequency; these two phenomena repeat in other resonant frequencies as shown below.

 

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.

2. Farman, J. C., Gardner, B. G. & Shanklin, J. D. (1985). Large losses of total ozone in Antarctica reveal seasonal Cl0x/NOx interaction. Nature, 315, 207–210.

3. Hecht, E. (2002). Optics (4th edition). San Francisco: Addison Wesley.

4. Jackson, J. D. (1999). Classical Electrodynamics (3rd ed.). John Wiley & Sons, New York.

5. Leroux, M. F.-P. (1862). Researches on the refractive indices of bodies which only assume the gaseous condition at high temperatures. Abnormal dispersion of iodine vapour. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 24(160), 245-247.