Section 30–5 Colored films; crystals

(Single surface reflection / Thin-film interference / Three-dimensional grating)

 

The three interesting concepts discussed in this section are a reflection of a light wave at a surface of a material, thin-film interference (due to front reflection and back reflection), and three-dimensional grating (reflection at atoms of crystals).

 

1. Single surface reflection:

“…when a light wave hits a surface of a material with an index n, let us say at normal incidence, some of the light is reflected. The reason for the reflection we are not in a position to understand right now; we shall discuss it later (Feynman et al., 1963, p. 30–7).”

 

Feynman says that we are not in a position to understand the reflection of a light wave right now. In Volume II, he adds that “the amplitude of a surface reflection is not a property of the material, as is the index of refraction. It is a ‘surface property,’ one that depends precisely on how the surface is made (Feynman et al., 1964, Chapter 33 Reflection from surfaces).” In short, the amount of light reflected by the surface is dependent on the smoothness of the surface or the arrangement of atoms in an object. Essentially, the free electrons of the atoms oscillate in response to the incident light waves and they may cause the reflected light waves to be either strong or weak. In other words, the reflection at the boundary between two media is a process of scattering and interference of electromagnetic waves.

 

“But there are a number of other examples, and even though we do not understand the fundamental mechanism yet, we will someday, and we can understand even now how the interference occurs (Feynman et al., 1963, p. 30–7).”

 

Feynman mentions that we do not understand the fundamental mechanism of reflection and we can understand how the interference occurs. Interestingly, physicists have argued whether two photons can be said to interfere with each other (Glauber, 1995). More important, some may prefer this explanation of reflection: “[w]hen I talk about the partial reflection of light by glass, I am going to pretend that the light is reflected by only the surface of the glass. In reality, a piece of glass is a terrible monster of complexity - huge numbers of electrons are jiggling about. When a photon comes down, it interacts with electrons throughout the glass, not just on the surface. The photon and electrons do some kind of dance, the net result of which is the same as if the photon hit only the surface (Feynman, 1985, p. 16-17).” That is, the fundamental mechanism of reflection can be explained by light waves or photons.

 

2. Thin-film interference:

“Then, if we look at the reflection of a light source in a thin film, we see the sum of two waves; if the thicknesses are small enough, these two waves will produce an interference, either constructive or destructive, depending on the signs of the phases (Feynman et al., 1963, p. 30–7).”

 

According to Feynman, if we look at the reflection of a light source in a thin film, we see an interference pattern that depends on the signs of the phases, provided the thickness of the thin film is small enough. On the other hand, there is also a strong reflection even if the “thin-film” is not small enough (e.g., single crystal x-ray diffraction). To be specific, one may clarify that the interference is observable provided the thickness of the thin film is of the order of about ¼ to 10 wavelengths of visible light. In addition, this is an interference of reflected waves at the front surface and back surface of the thin film. Thus, we may define thin-film interference as an interference of light waves that occurs when light interacts with the front and back surface of a thin film of material.

 

“So we see colors when we look at thin films and the colors change if we look at different angles, because we can appreciate that the timings are different at different angles. Feynman et al., 1963, p. 30–8).”

 

Feynman explains that we can see colors change at thin films because we can appreciate that the timings are different at different angles. Some may be surprised that his explanation is in terms of different timings instead of path difference, and he did not provide a formula. However, the principle can be based on the optical path difference due to the front reflection and back reflection of a thin film (d = nl/4 for constructive inference). In a lecture on QED, Feynman (1985) elaborates that “[t]he ‘front reflection’ arrow is drawn opposite to that of the stopwatch hand when it stops turning… The ‘back reflection’ arrow is drawn in the same direction as the stopwatch hand (pp. 28-29).” One may add that the front reflection is drawn opposite to the stopwatch hand because of a phase shift of 180 degrees when light waves move from a low refractive index medium to a high refractive index medium.

 

3. Three-dimensional grating:

“This principle is used to discover the positions of the atoms in a crystal. The only complication is that a crystal is three-dimensional; it is a repeating three-dimensional array of atoms (Feynman et al., 1963, p. 30–8).”

 

Feynman discusses the principle for determining the positions of atoms: Based on the difference in intensity of the various images, we could find out the shape of the grating scratches, whether the grating was made of wires, sawtooth notches, and so on. Currently, some physicists prefer to use the term condition and state Bragg’s two conditions (or Bragg’s law) and Laue’s condition. Bragg’s first condition is about the regular reflection of x-rays whereby the angle of incidence equals to angle of scattering, whereas the second condition requires the path difference between two scattered waves equals to an integer number of wavelengths (2d sin q = nl). On the other hand, Laue’s condition is a relation of an incident wave and scattering wave from the crystal to the reciprocal lattice vector.

 

According to Feynman, we must use radiation of a very short wavelength, i.e., x-rays, whose wavelength is less than the space between the atoms such that there are diffraction patterns. In a sense, this is not correct because we can use electrons and neutrons instead of x-rays. Furthermore, one should elaborate that glass is an amorphous (non-crystalline) solid in which the atoms are not in regular arrangement (definite lattice pattern). More important, the symmetry (e.g., fourfold symmetry) in the diffraction pattern corresponds to the symmetrical axis or periodicity of atoms. However, the object need not be a crystal, e.g., the “cross shape” diffraction pattern indicates a helical arrangement of DNA.

As a suggestion, you may want to read his lecture on QED: “I can’t resist telling you about a grating that Nature has made: salt crystals are sodium and chlorine atoms packed in a regular pattern. Their alternating pattern, like our grooved surface, acts like a grating when light of the right color (X-rays, in this case) shines on it. By finding the specific locations where a detector picks up a lot of this special reflection (called diffraction), one can determine exactly how far apart the grooves are, and thus how far apart the atoms are (see Fig. 28). It is a beautiful way of determining the structure of all kinds of crystals as well as confirming that X-rays are the same thing as light. Such experiments were first done in 1914. It is very exciting to see, in detail, for the first time how the atoms are packed together in different substances… (Feynman, 1985, pp. 48-49).”

 

Review Questions:

1. How would you explain the reflection of light waves at the boundary between two media?

2. Would you explain that colors change at a thin film if we look at different angles because the timings are different at different angles (or path differences)?

3. How would you state the principle for determining the positions of atoms in a crystal?

 

The moral of the lesson: thin-film interference and the diffraction of light in a crystal are related to the reflection of light waves at the surfaces of a material or atoms in the crystal.

 

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. Glauber, R. J. (1995). Dirac’s Famous Dictum on Interference: One Photon or Two?. American Journal of Physics, 63(1), 12.

Section 30–4 The parabolic antenna

(Radio sources / Radio antenna / Resolving power)

 

The three interesting concepts discussed in this section are radio sources in the sky, radio antenna, and resolving power of a telescope.  

 

1. Radio sources:

“Now let us consider another problem in resolving power. This has to do with the antenna of a radio telescope, used for determining the position of radio sources in the sky, i.e., how large they are in angle (Feynman et al., 1963, p. 30–6).”

 

According to Feynman, a problem in resolving power is related to the antenna of a radio telescope that is used for determining the position of radio sources in the sky. In a sense, the phrase “radio sources in the sky” may be misleading because it takes about 23 hours and 56 minutes (a sidereal day due to the Earth’s rotation relative to the stars) for extraterrestrial radio sources to be periodically detected by the radio telescope. One should realize that radio sources can be any “warm” objects that emit radio waves. In addition, we need not criticize Feynman for not mentioning radio sources such as pulsars, quasars, active galactic nucleus, black holes, radio galaxies, or Jupiter. The term quasar was coined in May 1964 for quasi-stellar radio sources (Chiu, 1964), whereas black hole was used by Ann Ewing (a journalist) in January 1964 in an article titled Black Holes in Space.

 

“We are very interested to know whether the source is in one place or another. One way we can find out is to lay out a whole series of equally spaced dipole wires on the Australian landscape (Feynman et al., 1963, p. 30–6).”

 

In his Nobel lecture titled Radio Telescopes of Large Resolving Power, Ryle (1975) explains: “…the forerunners for this type of instrument were realized in the early days when observations in both Australia and England with aerial elements having a range of separations were used to determine the distribution of radio brightness across the solar disc.” Interestingly, Feynman says that one way to locate radio sources is to lay out equally spaced dipole wires on the Australian landscape (instead of England landscape). One may clarify that radio telescopes cover a region of the sky ±45^o from zenith, that is, mostly the southern sky if they are in Australia (northern sky if they are in England). Historically, Hanbury Brown’s research proposal was not accepted by the referees, thus he left England. Subsequently, Hanbury managed to find support and build an observatory in Australia.

 

2. Radio antenna:

“Some radio antennas are made in a different way… we may arrange them not in a line but in a curve, and put the receiver at a certain point where it can detect the scattered waves … This is an example of what is called a reciprocity principle.  (Feynman et al., 1963, p. 30–7).”

 

Feynman explains a reciprocity principle of radio antenna as “the receiving pattern of an antenna is exactly the same as the intensity distribution we would get if we turned the receiver around and made it into a transmitter.” He adds that this principle is generally true for any arrangement of antennas, angles, and so on. However, one may elaborate that the essence of reciprocity principle is similar to action and reaction are equivalent, but Newton’s third law of motion does not always hold. Better still, this principle should include a condition of validity because it relates two possible solutions in a linear system (or linear medium) where the radio sources and radio receivers are interchanged. It is worth mentioning that Rayleigh formulates the principle of reciprocity in acoustic and electromagnetism.

 

“The arranging of the antennas on a parabolic curve is not an essential point. It is only a convenient way to get all the signals to the same point with no relative delay and without feed wires (Feynman et al., 1963, p. 30–7).”

 

Feynman clarifies that we may arrange radio antennas on a parabolic curve, but this is not an essential point. However, extraterrestrial radio signals are extremely weak because the wavelengths could be 100 kilometers and longer (or billions of times weaker than the signals used by communication systems). We can apply the principles of Hanbury-Brown-Twiss effect by connecting two radio antennas to analyze the correlation between the fluctuations of radio signal intensities. In his book titled QED: The Strange Theory of Light and Matter, Feynman (1985) writes: “[t]his phenomenon, called the Hanbury-Brown-Twiss effect, has been used to distinguish between a single source and a double source of radio waves in deep space, even when the two sources are extremely close together (p. 75).” This effect has helped to develop quantum optics and it is related to Dirac’s incorrect dictum on interference: “Interference between two different photons can never occur.”

 

Many physicists including Feynman had difficulty in accepting the Hanbury-Brown-Twiss effect. In Radhakrishnan’s (2002) words: “I was present at a Caltech colloquium at which Hanbury talked about it, and Richard Feynman jumped up and said, ‘It can’t work!’ In his inimitable style, Hanbury responded, ‘Yes, I know. We were told so. But we built it anyway, and it did work.’ Late that night, Feynman phoned and woke Hanbury up to say ‘you are right.’ He also wrote a letter in which he magnanimously admitted his mistake and acknowledged the importance of this phenomenon that, at first sight, appears counterintuitive, even to quantum theorists (2002, p. 76).”

 

3. Resolving power:

“Now we are describing a telescope mirror, of course. We have found the resolving power of a telescope! Sometimes the resolving power is written θ = 1.22λ/L, where L is the diameter of the telescope (Feynman et al., 1963, p. 30–7).”

 

It may seem strange that the discussion of radio antenna is changed to the resolving power of a circular telescope within a paragraph. However, a side-view of the circular telescope is parabolic in shape, but there could be a new paragraph to explain the resolving power formula θ = 1.22λ/L (what if you substitute the wavelength of radio signals l = 100 km into the formula?). To be specific, the magnified image of a star seen through the telescope is not the star’s physical body, but it is a diffraction pattern (or “moving” diffraction pattern due to the Earth’s rotation). The resolution of image seen depends on the sky conditions as well as the diameter of the eyepiece and the size of the pupil. One may explain that the resolving power or resolution is based on at least three mathematical concepts: “Abbe’s diffraction limit,” “Airy disk diameter,” and “Rayleigh’s criterion.”

 

“… thus we can appreciate that the effective diameter is a little shorter than the true diameter, and that is what the 1.22 factor tells us. In any case, it seems a little pedantic to put such precision into the resolving power formula (p. 30–7).”

 

Feynman feels that it is pedantic to put such precision into the resolving power formula. However, one may argue that it is not strictly pedantic because we can theoretically compare Rayleigh’s criterion with Houston’s criterion, Abbe’s criterion, and Sparrow’s criterion. From a practical perspective, we can compare the resolving power of different telescopes such as a refractor telescope and reflector telescope, or other optical systems. On the other hand, the diffraction limit of the eye can be calculated using Rayleigh’s criterion where D is the diameter of the eye’s pupil. If you are wondering about the factor “1.22,” it is based on the Bessel function (of the first kind) of order one, J₁(x).

 

Review Questions:

1. How would you describe the radio sources in the sky?  

2. How would you explain a reciprocity principle of radio antenna?

3. Does the factor 1.22 seem pedantic to be included in the resolving power formula?

 

The moral of the lesson: The diffraction patterns of radio sources are so weak that we cannot simply rely on the Rayleigh’s criterion, but it is important to apply the principles of Hanbury-Brown-Twiss effect.

 

References:

1. Chiu, H. Y. (1964). Gravitational collapse. Physics Today, 17, 21–34.

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. Radhakrishnan, V. (2002). Obituary: Robert Hanbury Brown. Physics Today, 55(7), 75–76.

5. Ryle, M. (1975). Radio Telescopes of Large Resolving Power. Reviews of Modern Physics, 47, 557–566.

Section 30–3 Resolving power of a grating

 (Rayleigh’s criterion / Grating’s resolving power / Reciprocal time difference)

 

In this section, Feynman discusses Rayleigh’s criterion of resolution, derives the resolving power of a diffraction grating, and relates it to the reciprocal time difference between extreme paths of light.

 

1. Rayleigh’s criterion:

“In order to be able to just make out the double bump, the following simple criterion, called Rayleigh’s criterion, is usually used. It is that the first minimum from one bump should sit at the maximum of the other (Feynman et al., 1963, p. 30–6).”

 

According to the Rayleigh criterion, two images are just resolved when the maximum intensity of one Airy disk (smallest diffraction-limited spot size) coincides with the first minimum of the other disk. While this provides a useful rule of thumb, it remains an arbitrary convention rather than a fundamental physical limit. Importantly, Rayleigh’s intention was to use this criterion to compare the resolution of different optical instruments. In Investigation in Optics, Rayleigh (1880) writes “[a]ccording to the principles of common optics, there is no limit to resolving-power, nor any reason why an object, sufficiently well lighted, should be better seen with a large telescope than with a small one.” However, his effort to quantify resolving power in practical terms, imply the choice of a 26.3% intensity drop relative to the maximum is a matter of convenience rather than an inherent property of light. The Rayleigh criterion has historical significance, but modern imaging and computational methods surpass its limitations.

In Rayleigh’s (1880) words, “[w]e conclude that a double line cannot be fairly resolved unless its components subtend an angle exceeding that subtended by the wavelength of light at a distance equal to the horizontal aperture.” In essence, Rayleigh initially suggested a criterion of resolution whereby the subtended angle due to a wavelength of light ensure the double line to be “fairly resolved” (or well resolved). Subsequently, Rayleigh (1896) built upon Airy’s work and proposed the following: “[i]t appeared that two neighbors, whether constituting a single pair of points or forming part of an extended series of equidistant points, could not be properly distinguished if the distance were less than half the wavelength of the light employed.” In short, this criterion requires a minimum of half wavelength of the light such that the double line can be “just resolved.”

 

2. Grating’s resolving power:

“That is, we want Δ to be exactly one wavelength λ more than mnλ. That is, Δ=mnλ+λ = mnλ′. Thus if λ′=λ+Δλ, we find (30.9) Δλ/λ=1/mn (Feynman et al., 1963, p. 30–6).”

 

Feynman’s derivation of resolving power of a grating could be “confusing” to some students (e.g., why Δ=mnλ+λ = mnλ′?). Below is an alternative derivation:

The maximum intensity for λ (order m) corresponds to nf/2 = mnp.

It is related to I = I₀sin²(nf/2)/sin²(f/2) = I₀sin²(npdsin q/l)/sin²(pdsin q/l).

Similarly, the first minimum for λ′ (order m) corresponds to nf/2 = mnp-p.

(Longer wavelength λ′=λ+Δλ Þ more deviation and lesser phase mnp-p needed.)

Thus, we have npdsin q/(l+Dl)= mnp-p ---(1) and npdsin q/l = mnp ---(2).

Equation (2)/Equation (1) gives (l+Dl)/l = mnp/(mnp-p) = mn/(mn-1)

Dl/l = (mn – mn + 1)/(mn – 1) = 1/(mn – 1) or approximately 1/mn.

 

Rayleigh is the first person to derive the formula of the resolving power of a grating. In an article titled On the manufacture and theory of diffraction gratings, Rayleigh (1874) provides a derivation by drawing and explaining a diagram: “[s]uppose now that l + dl is the wavelength for which BQ gives the principal maximum, then (mn + 1)l = mn(l + dl); whence dl/l = 1/mn which shows that the resolving power varies directly as m and n.” Rayleigh’s derivation of the formula is essentially the same as Feynman’s derivation, but it is based on his earlier criterion that corresponds to one wavelength. Current textbook authors may prefer to use the equation df = (2pd cos q)dq/l and let the phase difference between a maximum and the first adjacent minimum Df to be equal to 2p/n.

 

3. Reciprocal time difference:

“…this formula is equivalent to the formula that the error in frequency is equal to the reciprocal time difference between extreme paths that are allowed to interfere: Δν = 1/T (Feynman et al., 1963, p. 30–6).”

 

According to Feynman, the formula of resolving power λ/Δλ is equivalent to the formula Δν = 1/T. To prove the equivalence of the two formulas, we can use the hint provided in the footnote “In our case T = Δ/c = mnλ/c, where c is the speed of light. The frequency ν = c/λ, so Δν = cΔλ/λ²”. However, the notation T used in the footnote could be confusing. Let’s recall chapter 27 where Feynman mentions that if the distance of separation of two points is D and if the opening angle of the lens is θ, then the inequality t₂ − t₁ > 1/ν is exactly equivalent to D > λ/nsin θ. As a suggestion, we could replace T = Δ/c = mnλ/c by t₂ − t₁ = mnλ/c. Therefore, Δν = cΔλ/λ² = (c/λ)(Δλ/λ) = (mn/[t₂ − t₁])(1/mn) = 1/(t₂ − t₁).

 

Feynman suggests that we should remember the general formula Δν = 1/T because it works not only for gratings, but for any other instrument, while the formula dl/l = 1/mn is applicable only to gratings. Interestingly, in Investigations in Optics, Rayleigh (1880) concludes that “[i]t is not easy to decide whether the highest resolving-power is more likely to be obtained by gratings or by prisms.” To resolve a double line, he stipulates that the aggregate thickness of the prisms (t) should exceed the value given by the equation t = l/dm in which dm is the change in refractive index. He adds that the resolving-power of a prismatic spectroscope of a dispersive material is proportional to the total thickness used, but it is independent of the number, angles, or setting of the prisms, is perhaps the most important proposition in this subject.

 

Review Questions:

1. How would you state Rayleigh’s criterion of resolution for an optical system?

2. Would you use Rayleigh’s method (or Feynman’s method) that is based on his earlier criterion?

3. How would you show that the formula of resolving power λ/Δλ is equivalent to the formula Δν = 1/T?

 

The moral of the lesson: we may derive the resolving power of a diffraction grating as 1/(mn – 1) » 1/mn by taking the phase difference Df between a maximum and the first adjacent minimum to be equal to 2p/n.

 

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. Strutt, J. W. (Lord Rayleigh) (1874). On the manufacture and theory of diffraction gratings. Philosophical Magazine, 47(310), 81-93.

3. Strutt, J. W. (Lord Rayleigh) (1880). Investigations in optics, with special reference to the spectroscope. Philosophical Magazine, 8(49), 261-274.

4. Strutt, J. W. (Lord Rayleigh) (1896). On the theory of optical images, with special reference to the microscope. Philosophical Magazine, 42(255), 167-195.