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).”

 

Feynman connects Rayleigh’s criterion to a question of angular resolution: “how different in angle must it be in order for us to be able to distinguish two light sources?” In addition, he simplifies the criterion as “the first minimum from one bump should sit at the maximum of the other.” It should be worthwhile to have a refined criterion: “two images are just resolved when the maximum intensity of an Airy disk (minimum spot size) is directly over the first minimum of the other disk.” On the other hand, it is incorrect to describe the criterion as Rayleigh’s curse because his intention was to use it 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.”

 

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.

Section 30–2 The diffraction grating

(Diffraction grating / Reflection grating / Transmission grating)

 

In this section, Feynman discusses the principle of diffraction grating and specifically, reflection grating as well as transmission grating.

 

1. Diffraction grating:

“In one of its forms, a diffraction grating consists of nothing but a plane glass sheet, transparent and colorless, with scratches on it. (Feynman et al., 1963, p. 30–4).”

 

Feynman says that a diffraction grating may consist of a plane glass sheet that is transparent and colorless, with scratches (reduce light transmission) on it. There are often several hundred scratches within a millimeter that are carefully arranged so as to be equally spaced. However, we can define a diffraction grating as an optical device consisting of a system of narrow slits or grooves, which by “diffracting light” to result in an “interference pattern” (due to interfering light beams). This device is able to split electromagnetic radiation into its constituent wavelengths and it is preferred over a prism because it does not absorb much ultraviolet or infrared radiation. Currently, we may use holographic diffraction gratings that are generated by recording an interference pattern in a photo-resist coated substrate.

 

According to Feynman, if the wire spacing is greater than the wavelength, we will get a strong intensity of scattering in the normal direction, and in certain other directions given by dsin θ = mλ (30.6). In short, the principle of diffraction grating is based on path difference dsin θ that may be a multiple of wavelength. However, the principle of diffraction grating involves the following: 1. Idealization: we assume the slits in a diffraction grating are all in phase, as if they are connected to a plane wave (from a light source that is “practically at infinity”) and parallel to the plane of the slits. 2. Approximation: The grating equation dsin θ = mλ is approximately correct because the slits are all very narrow and the screen is very far away from the grating such that the angles of diffracting waves (or interference fringes) are almost the same.

 

2. Reflection grating:

“So not only do we get a beam in the same direction as the incoming beam but also one in another direction, such that the angle of incidence is equal to the angle of scattering. This we call the reflected beam (Feynman et al., 1963, p. 30–5).”

 

Feynman mentions that a grating is often made with little “sawtooth” cuts instead of little symmetrical notches. Then, he adds the basic principle of reflection grating: the incoming light wave generates motions of the atoms in the reflector, and the reflector then regenerates a new light wave. However, a reflection grating may be defined as a system of equally spaced ridges or grooves on a reflective screen (or made from a mirror). One may elaborate that the surface of the grating should be periodic such that the diffracted light waves can constructively interfere in certain special directions. If Feynman were alive today, he might discuss why we see a rainbow of colors that is apparently “reflected through” a compact disc.

 

A simple reflection grating is a mirror that has hundreds or thousands of narrow, parallel grooves on its surface. In QED, Feynman (1986) explains that “[s]uch a mirror is called a diffraction grating, and it works like a charm. Isn’t it wonderful – you can take a piece of mirror where you didn’t expect any reflection, scrape away part of it, and it reflects (pp. 46-47).” In essence, we can modify a mirror so that it behaves like a reflection grating, but a reflection grating does behave like a mirror. Importantly, we should distinguish the “reflected beam” of a reflection grating and a mirror.

 

Reflected beam: Feynman says that there is a reflected beam because we get another beam in the same direction as the incoming beam such that the angle of incidence is equal to the angle of scattering. Mathematically, sin θ_out = sin θ_in may mean that θ_out is the supplement of θ_in because the light comes out in the same direction as the light which was exciting the grating. However, the term reflected beam used by Feynman is potentially misleading. The so-called reflected beam is an effect of diffracting waves and interfering waves that occurs after reflection (or scattering). Instead of saying reflected beam, it may be more appropriately replaced by “interference after scattering” or “diffraction and interference after reflection.”

 

3. Transmission grating:

“Next, we discuss the special case when d→0. That is, we have just a solid piece of material, so to speak, but of finite length. In addition, we want the phase shift from one scatterer to the next to go to zero (Feynman et al., 1963, p. 30–5).”

 

d→0: Feynman explains that when we put more antennas between the other ones, so that each of the phase difference approaches zero, but the number of antennas is increasing such that the total phase difference, between one end of the line and the other, is constant. In addition, if we recognize n²I₀ as I_m, the maximum intensity at the center of the beam, we get (30.8) I = 4I_msin² ½ Φ/Φ² and this limiting case is shown in Fig. 30–2. However, the antennas are analogous to the slits, and the effect of an infinite number of very narrow slits is equivalent to one wide slit. Perhaps Feynman should elaborate on Fig 30.1: it would appear to be a circular arc instead of a polygon. This is because as the number of sides of a polygon increases to infinity, the length of each side and the phase difference between adjacent antennas approach zero.

 

d<λ: Note that m = 0 is the only solution if d is less than λ because sin θ_out = sin θ_in, which means that θ_out is the supplement of θ_in so the light comes out in the same direction as the light which was exciting the grating. (It means the atoms of the grating absorb the light, but new light was exiting the grating.) That is, the light “does not really go right through” the grating because new light is generated by scattering at the grating. Essentially, the wavelength of light is too long (λ > d) to allow constructive inference to occur at a larger angle in accordance to the formula sin θ = λ/d (there is no θ whereby sin θ = λ/d > 1). It should be worthwhile mentioning that the phrase “distance D” used by Feynman does not refer to the path difference dsin θ, but it is Lsin θ as shown in Fig. 30–3.

 

Review Questions:

1. How would you explain the principle of diffraction grating?

2. Would you use the term reflected beam that was suggested by Feynman?

3. How would you define the equatorial plane in Fig. 30–5?

 

The moral of the lesson: the principle of diffraction grating is based on the grating equation dsin θ = mλ, but it also involves diffraction and interference as well as idealization (source at infinity) and approximation (screen at infinity).

 

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.