Section 31–6 Diffraction of light by a screen

 (Opaque screen / Complementary screens / “Small hole” screen)

 

This section is about the diffraction of light by an opaque screen, complementary screens, and “small hole” screen that are related to Babinet’s principle of diffraction. Another possible title for this section is “Babinet’s principle of diffraction.”

 

1. Opaque screen:

“‘What is an opaque screen?’ Suppose we have a completely opaque screen between a source S and an observer at P, as in Fig. 31–6(a). If the screen is ‘opaque’ there is no field at P (Feynman et al., 1963, p. 31–10).”

 

According to Feynman, an opaque screen would have charges moving within the screen due to an electric field E_s and it would generate a new field to exactly cancel the field E_s on the back side of the screen. In addition, the opaque screen has a large and imaginary index such that the electric field is absorbed exponentially as it goes deeper into the material of the screen based on the theory of refractive index. Alternatively, opaque screen can be defined as a screen that is sufficiently large with a thickness and consists of uniformly distributed electron-oscillators. It should be good to clarify that this section is related to Babinet’s principle of diffraction. This principle shows the equivalence of a hole (or opening) to an obstacle with respect to the diffraction patterns.

 

“So if we make the screen thick enough, there is no residual field, because there is enough opportunity to finally get the thing quieted down… You know, of course, that a thin enough sheet of the most opaque material, even gold, is transparent (Feynman et al., 1963, p. 31–10).”

 

Idealization: Perhaps Feynman would state Babinet’s principle in terms of a “thick screen” because he explains that there is no residual electric field if the screen is thick enough and the electric wave is absorbed exponentially (as it goes through). However, we can idealize Babinet’s principle by conceptualizing an infinitesimally thin and perfectly conducting plane screen. In Jackson’s (1998) words, “[a] rigorous statement of Babinet’s principle for electromagnetic fields can be made for a thin, perfectly conducting plane screen and its complement (p. 489).” That is, the very thin and perfectly conducting (metallic) screen would absorb the electric wave such that there is no residual electric field for all frequencies. Although Feynman adds that gold is transparent provided it is thin enough, one may clarify that it can reflect red and yellow lights and allow greenish-blue light to pass through it (Hecht, 2002).

 

2. Complementary screens:

“We 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, p. 31–11).”

 

We may define complementary screens as two screens that are “complementary” in the sense that the first screen that has transparent regions can be replaced by the second screen whereby the transparent regions become opaque regions, or vice versa. In short, “Babinet’s principle states that the diffracted fields from complementary screens are the negative of each other (Carcione & Gangi, 1999, p. 1485).” In other words, complementary screen means that we can replace a hole (or opening) with an obstacle or replace an obstacle with a hole and still have the same diffraction pattern. Intuitively, it may not be clear to many (e.g., Poisson) why the Arago spot can be observed in the shadow of an obstacle. A beauty of Babinet’s principle is that the Arago spot can be simply explained by the negative electric field, E₁ = -E_2.

 

“The field at P is certainly zero in case (c), but it is also equal to the field from the source plus the field due to all the motions of the atoms in the walls and in the plugs. We can write the following equations: Case (b): E_{at P} = E_s+E_wall, Case (c): E′_{at P} = 0 = E_s+E′_wall+E′_plug (Feynman et al., 1963, p. 31–11).”

 

In Landau’s (1971) words, “Let us consider the Fraunhofer diffraction from two screens which are ‘complementary’: the first screen has holes where the second is opaque and conversely (p. 155).” That is, a limitation of Babinet’s principle is that the diffraction patterns of the two complementary screens are identical provided the condition of Fraunhofer diffraction is met. (In the Audio Recordings* [50 min: 20 sec] of this lecture, Feynman says: “The point is all the fields from the infinity impinges the opaque screen……,” but this is missing in the edited Feynman’s Lectures.) Although the Fraunhofer diffraction requirement means the source should be located at infinity, we may expect a similar diffraction pattern if the source is relatively far compared to the width of the hole. The assumption of infinite source distance allows the incident wave to appear like a plane wave such that all “fields” (or secondary sources) on the same wavefront near the hole are in phase.

 

*The Feynman Lectures Audio Collection: https://www.feynmanlectures.caltech.edu/flptapes.html

 

3. “Small hole” screen:

“Now if the holes are not too small (say many wavelengths across), we would not expect the presence of the plugs to change the fields which arrive at the walls except possibly for a little bit around the edges of the holes. Neglecting this small effect, we can set E_wall = E^′_wall and obtain that E_{at P} = −E′_plug (Feynman et al., 1963, p. 31–11).”

 

Feynman briefly mentions that the theory of diffraction is approximately correct provided the holes are not very small. He explains that the E′_plug term will be small and then the difference between E′_wall and E_wall (it is taken to be zero) may be comparable to or larger than the small E′_plug term, and our approximation will no longer be valid. One may clarify that the approximation is based on the assumption that the interactions between the electron oscillators around the edges of the holes are negligible. In the real world, the edge effect is minimal if the screen, hole, or obstacle is sufficiently thin. This is why physicists idealize an infinitesimally thin and perfectly conducting plane screen. In short, size matters (i.e., the diffraction pattern is dependent on the width and thickness of the obstacles or openings).

 

“We remark again that this theory of diffraction is only approximate, and will be good only if the holes are not too small (Feynman et al., 1963, p. 31–11).”

 

In general, one may apply Keller’s (1962) geometrical theory of diffraction to deduce the edge effects of very small holes. From the perspective of experiment, it is a challenge to investigate the darker diffraction pattern of a very small hole because only very little light passes through the hole. However, it could be apt to end the chapter with the beauty of Arago spot (Poisson’s spot) or Babinet’s principle. The diffraction pattern of a hole is almost identical to the diffraction pattern of an obstacle that is sufficiently thin. For practical situations, Babinet’s principle is also approximately valid and applicable to curved screens whose radii of curvature are large compared to the size of the hole (Jackson, 1998).

 

Review Questions:

1. How would you idealize an opaque screen (infinitely thin or perfectly conducting)?

2. How would you state Babinet’s principle of diffraction (e.g., by including the limitation)?

3. Is Babinet’s principle of diffraction applicable to very small holes?

 

The moral of the lesson: Babinet’s principle of diffraction states the equivalence of a hole (opening) to an obstacle from the perspective of diffraction pattern, but it is approximately valid depending on the size (or thickness) of the hole and obstacle.

 

References:

1. Carcione, J. M., & Gangi, A. F. (1999). Babinet’s principle for elastic waves: A numerical test. The Journal of the Acoustical Society of America, 105(3), 1485-1492.

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

4. Jackson, J. D. (1998). Classical Electrodynamics (3rd ed.). John Wiley & Sons, New York.

5. Keller, J. B. (1962). Geometrical theory of diffraction. Journal of the Optical Society of America, 52(2), 116-130.

6. Landau, L. D., & Lifshitz, E. M. (1971). The classical theory of fields (Vol. 2). Oxford: Pergamon.

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.