Section 32–4 Independent sources

 (Incoherent sources / Partially coherent sources / Coherent sources)

 

In this section, Feynman discusses incoherent sources (distant sources), partially coherent sources (radio oscillators), and coherent sources (lasers).

 

1. Incoherent sources:

“Suppose, first, that the two sources are 7,000,000,000 wavelengths apart, not an impossible arrangement…... So if we average over regions where the phase varies very rapidly with position, we get no interference (Feynman et al., 1963, p. 32–5).”

 

Feynman’s first example involves two distant sources (7,000,000,000 wavelengths apart) that may or may not result in an observable interference pattern. The condition of a stable and clear interference pattern for Young’s double slit experiment is two coherent sources, i.e., constant phase difference between them. On the contrary, we do not expect to observe any interference pattern if two coherent sources are replaced by two thermal sources that are very close to the screen (Klauder & Sudarshan*, 2006). In general, thermal sources, such as a lightbulb or the Sun, consist of a large number of light-radiating atoms that are almost independent of each another. Some may be surprised why Feynman did not specify the nature of the two sources, but the light waves emitted could become more coherent depending on the distance traveled.

*Feynman acknowledged George Sudarshan’s contribution in 1963 stating that the “V-A theory was invented by Marshak and Sudarshan, published by Feynman and Gell-Mann, and completed by Cabibbo (Mehra, 1994, p. 477).”

 

“Then in a given direction it is true that there is a very definite value of these phase differences (Feynman et al., 1963, p. 32–5).”

 

Perhaps Feynman could have explained why there is a very definite value of the phase differences in a given direction for the two distant sources. Based on Van Cittert–Zernike theorem, wavefronts from an incoherent source will be smoothened as plane waves, i.e., any wavefront will appear more coherent after traveling long distances. For example, the light waves from the Sun are initially incoherent, but they could become effectively more coherent than laser light if the distance traveled is sufficiently far (the Sun appears as a point source). Empirically, sunlight can produce observable fringes through two pinholes. In a sense, a pinhole source can be idealized as a coherent source that allows one photon to pass through a hole and interfere with itself. One may also explain how highly correlated (or coherent state) photons pass through two holes and result in an interference pattern.

 

But, on the other hand, if we move just a hair in one direction, a few wavelengths, which is no distance at all (our eye already has a hole in it that is so large that we are averaging the effects over a range very wide compared with one wavelength) then we change the relative phase, and the cosine changes very rapidly (Feynman et al., 1963, p. 32–5).”

 

It may not be clear why there is a need to move a hair in one direction so that the relative phase would be changed very rapidly. One may clarify that this example is about two distant sources, but the light waves could become more coherent. However, light waves from the two sources after passing by two sides of the hair, would travel different distances and have varying relative phase at the photocells. In other words, there are two effective sources that are close to each other when light waves bend around the hair. According to Babinet’s principle, the interference pattern produced due to the hair is equivalent to a slit whose width is equal to the thickness of the hair. Thus, the interference is rapidly changing because the hair behaves like a moving slit (based on Babinet’s principle).

 

2. Partially coherent sources:

“Suppose that the two sources are two independent radio oscillators—not a single oscillator being fed by two wires, which guarantees that the phases are kept together, but two independent sources—and that they are not precisely tuned at the same frequency (it is very hard to make them at exactly the same frequency without actually wiring them together). Of course, since the frequencies are not exactly equal, although they started in phase, one of them begins to get a little ahead of the other, and pretty soon they are out of phase, and then it gets still further ahead, and pretty soon they are in phase again (Feynman et al., 1963, p. 32–5).”

 

Feynman’s second example involves two independent radio oscillators whose frequencies are not exactly equal. These two sources may be categorized as almost incoherent sources or partially coherent sources because the phase difference at any location due to the two sources will vary with time, but less rapidly. In essence, coherent sources emit light waves of the same frequency, incoherent sources emit light waves whose resultant have random fluctuations of amplitude and phase, whereas partially coherent sources are somewhere in between. On the other hand, we can idealize two point sources as coherent, but an extended source emits incoherent light waves. Strictly speaking, there are no completely coherent sources or completely incoherent sources, i.e., all sources are partially coherent in the real world.

 

One finds many books which say that two distinct light sources never interfere. This is not a statement of physics, but is merely a statement of the degree of sensitivity of the technique of the experiments at the time the book was written (Feynman et al., 1963, p. 32–5).”

 

The idea of “two distinct light sources never interfere” is related to Dirac’s (1981) view: “Each photon then interferes only with itself. Interference between different photons never occurs (p. 9).” From a quantum mechanical perspective, the wave function specifies the probability of one photon in a particular location instead of the probable number of photons in that location due to the interference of light. Dirac argues that the conservation of energy prevents two photons annihilate each other or produce four photons. Therefore, he reasons that the observed intensity is due to the interference of a single photon with itself. This is different from the statistical interpretation of quantum mechanics that relates the wave function to the probable number of photons in a location.

 

3. Coherent sources:

“The device which does this is a very complicated thing, and has to be understood in a quantum-mechanical way. It is called a laser, and it is possible to produce from a laser a source in which the time during which the phase is kept constant, is very much longer than 10⁻⁸ sec …… One can easily detect the pulsing of the beats between two laser sources (Feynman et al., 1963, p. 32–6).”

 

Feynman’s third example involves two lasers whereby one can easily detect the pulsing of the beats between these two coherent sources. To be precise, a laser beam is not really monochromatic, i.e., it is only quasi-monochromatic. The light waves emitted by a laser are dependent on the q-factor, e.g., a narrow-linewidth laser whose bandwidth is below 1 Hz. On the other hand, ultrashort pulses with few-femtosecond pulse durations can have a very large bandwidth. Essentially, the laser is idealized as a monochromatic (coherent) source, but it does have a finite bandwidth and relatively small uncontrolled fluctuations of phase and amplitude.

 

In the Audio Recordings** [3 min: 10 sec] of this lecture, Feynman says: “So first, by improving the timing of detection of precision two years ago, first time light interference was observed from two independent sources.” The precision of the equipment is dependent on the technology of the light detector, e.g., Michelson interferometer or Young interferometer. The Michelson interferometer, in essence, allows a light beam to interfere with a time-delayed version of itself. Simply phrased, the Michelson interferometer measures the temporal coherence of a light wave: the ability of the light wave to interfere with a time-delayed of light. Young interferometer measures spatial coherence: the ability of light at one point in a wave to interfere with a spatially-shifted version of itself (or the correlation between the electric fields at different locations across the light wave).

 

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

In How the Laser Happened, Townes (1999) writes: “The late Richard Feynman, a superb physicist, said once as we talked about the laser that the way to tell a great idea is that, when people hear it, they say, ‘Gee, I could have thought of that’ (p. 10).” In addition, Townes (1999) elaborates: “In his paper, Coulson presented and discussed the solutions to an equation that allowed for both the probability of microbe multiplication by division and also a probability of death. I recognized immediately that this was exactly the kind of mathematics formulation we needed to understand some aspects of the maser, in which photons are both dying (being absorbed) and being born (stimulated into existence) simultaneously, as the result of the presence of other photons (p. 84).”

 

Review Questions:

1. How would you explain Feynman’s first example involving two distant sources (7,000,000,000 l apart) that do not result in observable interference pattern?

2. How would you explain the second example involving two independent radio oscillators whose frequencies are not exactly equal?

3. How would you explain the third example involving two lasers whereby one can easily detect the pulsing of the beats between the two coherent sources?

 

The moral of the lesson: Interference pattern is stable if two independent sources are coherent (e.g., lasers), but it is also observable for two incoherent sources (e.g., thermal sources) that are sufficiently far, but all light sources are partially coherent in the real world.

 

References:

1. Dirac, P. A. M. (1981). The principles of quantum mechanics. Oxford 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. Klauder, J. R., & Sudarshan, E. C. G. (2006). Fundamentals of quantum optics. New York: Dover.

4. Mehra, J. (1994). The beat of a different drum: The life and science of Richard Feynman. Oxford: Clarendon Press.

5. Townes, C. H. (2002). How the laser happened: Adventures of a scientist. New York: Oxford University Press.

Section 32–3 Radiation damping

 (Oscillator’s Q-factor / Classical electron radius / Atoms’ resonance width)

 

In this section, Feynman discusses an oscillator’s Q-factor, classical electron radius, and atoms’ resonance width that are related to radiation damping or radiation resistance of the oscillator.

 

1. Oscillator’s Q:

“What is the Q of such an oscillator, caused by the electromagnetic effects, the so-called radiation resistance or radiation damping of the oscillator? The Q of any oscillating system is the total energy content of the oscillator at any time divided by the energy loss per radian: Q = −W/(dW/dϕ) (Feynman et al., 1963, p. 31–4).”

 

In the Audio Recordings* [16 min: 10 sec] of this lecture, Feynman actually asks: “What is the Q of such an oscillator, [not caused by the resistance of the motion of the charge in goo, but merely] caused by the electromagnetic effects, the so-called radiation resistance or radiation damping of the oscillator?” (goo is a viscid substance.) In a sense, we can define the Q-factor of an antenna as the power stored in the antenna divided by its radiated power. The term per radian in Feynman’s definition of Q-factor means that there is a cycle (or 2p) in the definition because an oscillatory motion can be viewed as the projection of uniform circular motion onto one axis. That is, we may also define Q as 2p(max stored energy per cycle)/(energy lost per cycle), or Q = (total stored energy)/(energy lost per radian) in which 2p is hidden in per radian. However, Feynman’s explanation of Q in terms of 2p [16 min: 35 sec]* is omitted in the edited lecture.

 

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

 

“Or (another way to write it), since dW/dϕ = (dW/dt)/(dϕ/dt) = (dW/dt)/[ω(dW/dt)/ω], (32.8) Q = −ωW/(dW/dt). If for a given Q this tells us how the energy of the oscillation dies out, dW/dt =−(ω/Q)W, which has the solution W=W₀e^−ωt/Q if W₀ is the initial energy (at t = 0). (Feynman et al., 1963, p. 31–4).”

 

In the Audio Recordings [16 min: 55 sec]* of this lecture, Feynman says: “no trick, it is just the same expression as here,” but it may involve some non-trivial tricks. There are possibly three tricks or hidden concepts: (1) Decay law or exponential function, (2) Chain rule, and (3) working backward that is related to a damping oscillator. Firstly, dW/dt = −(ω/Q)W is similar to the radioactive decay law dN/dt = −lN in which l is the decay constant. Secondly, dW/dϕ = (dW/dt)/(dϕ/dt) is based on a chain rule to introduce ω = dϕ/dt. Thirdly, (dW/dt)/(dϕ/dt) = (dW/dt)/[ω(dW/dt)/ω] is a “working backward” trick because we have the end result in mind and it justifies the use of ω or 2p in the final formula. Some may need to revise the damping oscillation equation md²x/dt²+ γmdx/dt + mω₀²x = 0 that contains γ as a dissipative constant (See chapter 23).

 

2. Classical electron radius:

“It has been given a name, the classical electron radius, because the early atomic models, which were invented to explain the radiation resistance on the basis of the force of one part of the electron acting on the other parts, all needed to have an electron whose dimensions were of this general order of magnitude (Feynman et al., 1963, p. 31–4).”

 

The classical electron radius r₀ is defined by equating the electrostatic potential energy of a sphere of charge (U = e²/4pe₀r_e) with the mass-energy of the electron, E = m_ec². Perhaps Feynman could have explained why he did not consider another electron’s radius such as Bohr radius (a_B ≈ 5.3 × 10⁻¹¹ m) or Compton wavelength of the electron λ_c. (Note: r_e = αλ_c = α²a_B; α ≈ 1/137 is the fine structure constant.) In section 32.1, Feynman mentions that an electron is not a “little ball,” i.e., the size of an electron is dependent on the nature of the experiment or theoretical model. We may expect a different effective electron size in an antenna due to the radiation damping depending on the antenna frequency.

 

In Chapter 36, Feynman adds: “the electron, for example, was thought to behave like a particle, and then it was found that in many respects it behaved like a wave. So it really behaves like neither (Feynman et al., 1963, p. 36-1).” For example, Bender et al. (1984) found from electron scattering experiments at 29 GeV that the upper limit of the size of the electron is about 10^-18 m. Thus, theoretical physicists consider an electron to be a point particle because there is no measurable structure down to 10^-18 m. However, Wilczek (2013) explains: “[t]here is tension between these two observations, that the electron is a simple point-particle, and that it contains the world (p. 31).” In other words, the electron may appear to spread over a region of space that is limited by the size of the universe.

3. Atoms’ resonance width:

“The effective resistance term γ in the resistance law for the oscillator can be found from the relation 1/Q = γ/ω₀, and we remember that the size of γ determines how wide the resonance curve is (Fig. 23–2). Thus we have just computed the widths of spectral lines for freely radiating atoms! (Feynman et al., 1963, p. 31–5).”

 

Instead of writing the relation 1/Q = γ/ω₀, one may prefer to use the symbol Q₀ and state another definition of Q-factor. That is, Q₀ is a ratio of the resonant frequency ω₀ to the full width at half-maximum bandwidth Δω of the resonance curve, i.e., Q₀ = ω₀/Δω. In section 23-2, Feynman elaborates: “we ask for the width Δω of the curve, at one half the maximum height, the full width at half the maximum height of the curve is Δω = γ, supposing that γ is small (Feynman et al., 1963).” It is worthwhile mentioning that the “energy loss” definition Q = −W/(dW/dϕ) is applicable to any ω, whereas the bandwidth definition Q₀ = ω₀/Δω is dependent on ω₀. In essence, these two definitions give different numerical results, but they are approximately equivalent for larger Q values (i.e., lower γ or damping).

 

“This is valid only for atoms which are in empty space, not being disturbed in any way. If the electron is in a solid and it has to hit other atoms or other electrons, then there are additional resistances and different damping (Feynman et al., 1963, p. 31–5).”

Feynman’s computation of the widths of spectral lines for freely radiating atoms in empty space is based on the classical electron radius. Historically, the early atomic models were invented to explain the radiation damping and all that are needed to have an electron whose dimensions were of this general order of magnitude. Currently, the classical electron radius does not have any physical significance in the sense that whether the electron really has such a radius. We may conceptualize the electron to have a different size depending on the solid (e.g., antenna) involved. Perhaps Feynman could have discussed the Q of an antenna empirically and deduced the effective size of the electron in the antenna.

 

Review Questions:

1. Would you define the Q of an oscillator in terms of 2p, e.g., Q = 2p(max stored energy per cycle)/(energy lost per cycle)?

2. How would you explain the physical significance of classical electron radius?

3. Would you compute the widths of spectral lines using classical electron radius that does not have any physical significance?

 

The moral of the lesson: The widths of spectral lines for freely radiating atoms based on Feynman’s computation are valid only for atoms that are in empty space and are not being disturbed in any way (except radiation damping).

 

References:

1. Bender, D., Derrick, M., Fernandez, E., Gieraltowski, G., Hyman, L., Jaeger, K., ... & Va'Vra, J. (1984). Tests of QED at 29 GeV center-of-mass energy. Physical Review D, 30(3), 515-527.

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. Wilczek, F. (2013). The enigmatic electron. Nature, 498(7452), 31-32.