Section 34–9 The momentum of light

 (Radiation pressure / Linear momentum / Angular spin momentum)

 

In this section, Feynman discusses radiation pressure, linear momentum, and angular spin momentum of light. Alternatively, this section could be titled as “radiation pressure,” however, the chapter ends with the angular momentum of photons that connects to the later chapter on Quantum Behavior. The term radiation pressure is also used in the context of solar sail spacecraft and optical trapping. Historically, James Clerk Maxwell deduced the magnitude of radiation pressure in 1873 using his theory of electrodynamics. However, John Henry Poynting could be acknowledged for his contribution on radiation pressure and angular momentum of light.

 

1. Radiation pressure:

“Let us determine how strong the radiation pressure is. Evidently it is F = qvB or, since everything is oscillating, it is the time average of this, ⟨F⟩. From (34.2) the strength of the magnetic field is the same as the strength of the electric field divided by cc, so we need to find the average of the electric field, times the velocity, times the charge, times 1/c: ⟨F⟩=q⟨vE⟩/c. But the charge q times the field E is the electric force on a charge, and the force on the charge times the velocity is the work dW/dt being done on the charge! Therefore the force, the “pushing momentum,” that is delivered per second by the light, is equal to 1/c times the energy absorbed from the light per second! (Feynman et al., 1963, p. 34–11).”

 

Some physicists may feel confused with Feynman’s derivation of equation 34.24 because it refers to “radiation force” instead of radiation pressure. Furthermore, the equation, E = cB, is not a fundamental equation of electromagnetism, but it is a simplified relation that describes the electric field (E) and magnetic field (B) in a specific context: electromagnetic waves in vacuum. Some may expect the radiation pressure formula to be P_rad = Intensity of Light (I)/ Speed of Light (c) and prefer it to be derived from the Poynting vector (S), which represents the energy flow per unit area per unit time. However, it is worthwhile for students to show that the radiation pressure exerted on a perfectly absorbing surface is equal to the energy density (or field energy per unit volume) of the wave (Jackson, 1999). In his PhD thesis, de Broglie (1925) showed that radiation pressure equals to one third of the energy density in a cavity filled with black body radiations at temperature T by assuming an isotropic distribution of velocities.

 

“Therefore, when light is shining on a charge and it is oscillating in response to that light, there is a driving force in the direction of the light beam. This is called radiation pressure or light pressure (Feynman et al., 1963, p. 34–11).”

 

According to Feynman, radiation pressure is a driving force in the direction of the light beam due to oscillating charges in response to the light shining on them. Alternatively, we can define radiation pressure as follows: (1) Linear momentum: The cause of radiation pressure is a transfer of linear momentum (or field momentum) of light. (2) Radiation Pressure formula P_rad = I/c: The radiation pressure (P_rad) is equal to the intensity of the electromagnetic radiation (I) divided by the speed of light (c). (3) Nature of the surface: Radiation pressure is influenced by the surface it interacts with, whether it reflects, absorbs, transmits, or scatters light. In essence, the surface of an object determines how photons from the incident light interact with the object, leading to the momentum transfer (or energy transfer) and resulting in radiation pressure.

 

2. Linear momentum:

Equation (34.27) can be written more elegantly as p_μ= ℏk_μ, a relativistic equation, for a particle associated with a wave. Although we have discussed this only for photons, for which k (the magnitude of k) equals ω/c and p = W/c, the relation is much more general (Feynman et al., 1963, p. 34–11).”

 

We can distinguish the linear momentum of light in the context of electrodynamics, special relativity, and quantum physics. (1) Electrodynamics: By dividing ⟨F⟩ = (dW/dt)/c (34.24) by area, we get P_rad = I/c, i.e., radiation pressure is due to the transfer of field momentum from the light wave to an object’ surface, which is directly proportional to the intensity of the electromagnetic field. (2) Special relativity: Einstein’s relationship between energy (E) and linear momentum (p) for a particle can be stated as E² = (m₀c²)² + (pc)²; E = pc for a massless photon. (3) Quantum physics: de Broglie suggested that the wavelength associated with a particle is related to the de Broglie’s momentum, p =  h/l. In summary, the three linear momentums of light require different conception of lights (field momentum, photon’s momentum, and de Broglie’s momentum), but it shows these three perspectives are interrelated in a consistent framework.

 

“In quantum mechanics all particles, not only photons, exhibit wavelike properties, but the frequency and wave number of the waves is related to the energy and momentum of particles by (34.27) (called the de Broglie relations) even when p is not equal to W/c (Feynman et al., 1963, p. 34–11).”

 

According to Feynman, the angular frequency and wave number of the waves is related to the energy and momentum of particles by E = ħw and p = ħk even when p is not equal to W/c, i.e., the rest mass of the particles is non-zero. However, in de Broglie’s (1925) words: “One may imagine that, by cause of a meta law of Nature, to each portion of energy with a proper mass m₀, one may associate a periodic phenomenon of frequency ν₀, such that one finds: (1.1.5) hν₀ = m₀c². The frequency ν₀ is to be measured, of course, in the rest frame of the energy packet. This hypothesis is the basis of our theory: it is worth as much, like all hypotheses, as can be deduced from its consequences.” Some may deduce that de Broglie’s wavelength is a result of the marriage of Einstein’s equation E = m₀c² and Planck’s equation E = hν₀, which may be simplified as m₀c = hν₀ /c = h/l. In his Nobel lecture, de Broglie (1929) showed that the momentum p = h/l by using W = hν and the relationship v_phase ´ v_group = c² (as shown below).

 

de Broglie's (1925) derivation

Perhaps Feynman could have clarified why a photon has linear momentum, but its mass is zero. In Volume II, Feynman explains: “A photon of frequency ω₀ has the energy E₀=ℏω₀. Since the energy E₀ has the relativistic mass E₀/c² the photon has a mass (not rest mass) ℏω₀/c² and is ‘attracted’ by the earth (Feynman et al., 1964, section 42–6).” Thus, it suggests that light has linear momentum and the effective mass E₀/c² that is equivalent to its energy. However, some may argue that the relativistic mass of the photon is not zero, but it is a matter of definition or semantic problem.

 

3. Angular momentum:

“In the quantum picture, a beam of circularly polarized light is regarded as a stream of photons, each carrying an angular momentum ±ℏ, along the direction of propagation. That is what becomes of polarization in the corpuscular point of view—the photons carry angular momentum like spinning rifle bullets. But this ‘bullet’ picture is really as incomplete as the ‘wave’ picture, and we shall have to discuss these ideas more fully in a later chapter on Quantum Behavior (Feynman et al., 1963, p. 34–11).”

 

Perhaps Feynman could have used the term spin angular momentum that is different from orbital angular momentum, total angular momentum, and fractional angular momentum. Spin angular momentum is an intrinsic form of angular momentum associated with particles, such as electrons and photons. Orbital angular momentum of light refers to the angular momentum associated with a helical or twisted wavefront (Allen et al., 1992). Furthermore, the total angular momentum of light is an unequal mixture of spin and orbital contributions (Ballantine, Donegan, & Eastham, 2016). Recently, the light’s fractional angular momentum is demonstrated experimentally by shining a laser beam through a biaxial crystal (Ballantine, Donegan, & Eastham, 2016).

 

Feynman explains that the angular momentum of a photon ±ℏ using the ‘spinning bullet’ analogy and adds that it is as incomplete as the ‘wave’ picture. Note that ℏ, the Dirac constant (Feynman simply called it Planck constant), is defined as h/(2π) and can be traced back to the efforts of Paul Dirac in formulating a relativistic quantum theory for electrons. Currently, the spin angular momentum of light is regarded as a circulating flow of energy and it is related to the Poynting’s theorem. In other words, the concept of spin for light is due to the fields of a circularly polarized light wave. Perhaps the section could be concluded by saying the radiation pressure, linear momentum, and spin angular momentum of light are all related to the Poynting’s theorem and Quantum physics, however, the angular momentum of light was not covered during Feynman’s lecture because of limited time.

 

Historically, Pauli was uncomfortable with the concept of spin as a form of angular momentum and famously described the electron’s spin as “classically indescribable two-valueness.” In Volume II, Feynman explains: “a circulating momentum means that there is angular momentum. So there is angular momentum in the field. Do you remember the paradox we described in Section 17–4 about a solenoid and some charges mounted on a disc? It seemed that when the current turned off, the whole disc should start to turnWhere did the angular momentum come from? The answer is that if you have a magnetic field and some charges, there will be some angular momentum in the field. It must have been put there when the field was built up. When the field is turned off, the angular momentum is given back. So the disc in the paradox would start rotating. This mystic circulating flow of energy, which at first seemed so ridiculous, is absolutely necessary. There is really a momentum flow. It is needed to maintain the conservation of angular momentum in the whole world (Feynman et al., 1964, section 27–6 Field momentum).”

 

Review Questions:

1. How would you define radiation pressure?

2. How would you derive the linear momentum of light using radiation pressure?

3. How would you explain a photon has an angular (spin or orbital?) momentum ±ℏ along the direction of propagation?

 

The moral of the lesson: The radiation pressure applied by an electromagnetic wave (due to momentum transfer) on a perfectly absorbing surface is equal to intensity of light (I)/speed of light (c), or simply the field energy density of the wave.

 

References:

1. Allen, L., Beijersbergen, M. W., Spreeuw, R. J. C., & Woerdman, J. P. (1992). Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes. Physical review A, 45(11), 8185.

2. Ballantine, K. E., Donegan, J. F., & Eastham, P. R. (2016). There are many ways to spin a photon: Half-quantization of a total optical angular momentum. Science Advances, 2(4), e1501748.

3. de Broglie, L. (1925). Research on the theory of quanta. In Annales de Physique (Vol. 10, No. 3, pp. 22-128). In Foundation of Louis de Broglie (English translation by A.F. Kracklauer, 2004. ed.)

4. de Broglie, L. (1929). The wave nature of the electron. Nobel lecture, 12, 244-256.

5. 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.

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

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

Section 34–8 Aberration

(Definition / Derivation / Application)

 

In this section, Feynman’s discussion of stellar aberration could be analyzed from the perspective of definition, derivation, and application. This section may be titled as “stellar aberration” or “aberration of starlight” instead of “aberration” that is vague.

 

1. Definition:

“This effect, that a telescope has to be tilted, is called aberration, and it has been observed (Feynman et al., 1963, p. 34–10).”

 

Perhaps Feynman could have used the term stellar aberration instead of aberration that has other definitions in Optics. We may define stellar aberration from the perspective of angular shift, relative motion, and reference frame: (1) Angular shift: It is an astronomical phenomenon that manifests as an angular shift (typically on the order of seconds of arc) with respect to the apparent position of a star or celestial object when observed at two different times of a year. (2) Relative motion: It depends on the velocity of light and velocity of the Earth, along with the application of the Lorentz transformation. (3) Reference frame: The change of the observer’s reference frame, whether it is based on Earth’s based telescope or satellite-based telescope, determines the apparent angular shift. In short, stellar aberration can be distinguished as annual aberration (orbital motion of the Earth about the Sun), diurnal aberration (Earth’s rotation), and secular aberration (motion of the solar system).

 

There could be an analogy using the apparent direction of falling rain (See figure below) to explain stellar aberration. In Schwinger’s (2002) words: “One favorite analogy for this effect of motion imagines someone standing in vertically descending rain, using a vertically held umbrella for maximum shielding. When that person walks, the umbrella must be inclined forward to maintain the shielding, the more so the faster the pace. Turn the rain into starlight, the umbrella into a telescope, and the umbrella carrier into the Earth in its orbit, and you have Bradley's explanation of the aberration of starlight (p. 27).” However, a limitation of the analogy is that the speed of “rain” is dependent on the velocity of the observer. One may emphasize that the stellar aberration is due to the orbital velocity of the Earth instead of its position (commonly known as stellar parallax).

Source: 5. One for Martin Gardner | Ehrenfest Paradox (wordpress.com)

 

2. Derivation:

“To find out, we will have to write down the four components of k_μ and apply the Lorentz transformation. The answer, however, can be found by the following argument: we have to point our telescope at an angle to see the light. Why? Because light is coming down at the speed c, and we are moving sidewise at the speed v, so the telescope has to be tilted forward so that as the light comes down it goes “straight” down the tube. It is very easy to see that the horizontal distance is vt when the vertical distance is ct, and therefore, if θ′ is the angle of tilt, tan θ′=v/c. How nice! How nice, indeed—except for one little thing: θ′ is not the angle at which we would have to set the telescope relative to the earth, because we made our analysis from the point of view of a “fixed” observer (Feynman et al., 1963, p. 34–10).”

 

Some may derive the stellar aberration formula using Lorentz transformation between two frames of reference or specifically, the relativistic velocity addition formulas. In the Earth’s frame of reference, we can use c_x and c_y to represent the horizontal and vertical velocity of light respectively and thus the light beam is tilted at an angle θ such that tan θ = c_y/c_x. In the “fixed star’s frame,” if we assume the orbital velocity of the Earth is v (x-direction relative to the Sun), then the light beam is tilted at an angle θ¢ such that tan θ¢ = c_y¢/c_x¢ = c_y/g(c_x+v) by using the relativistic velocity addition formulas. Alternatively, the formula can be derived using the wave four-vector, cos [k(xcos θ + ysin θ) + wt], but it may appear to be complicated. However, it should be instructive for students to derive the result using at least two different methods involving Lorentz transformation.

 

tan⁡(�)=��/����′=��When we said the horizontal distance is vt, the man on the earth would have found a different distance, since he measured with a “squashed” ruler. It turns out that, because of that contraction effect, tan θ = (v/c)/√(1−v²/c²)−(34.22) which is equivalent to sin θ = v/c. (34.23) It will be instructive for the student to derive this result, using the Lorentz transformation (Feynman et al., 1963, p. 34–10).”

 

Feynman suggests students deriving the stellar aberration formula using the Lorentz transformation. For stars close to zenith, we can derive the formula using the second postulate of special relativity and Pythagorean theorem. Pictorially, the hypotenuse of the triangle for Earth frame and the vertical line in the fixed-star frame both represent the velocity of light that is constant in all inertial frames (see Figure below). In addition, the horizontal component of the triangle is v and the vertical component is Ö(c² – v²) by using the Pythagorean theorem. Thus, tan Dθ = v/Ö(c² – v²) = (v/c)/Ö[1 – (v/c)²)]. Similarly, we have sin Dθ = v/c by simply using “opposite over hypotenuse”.

 

3. Applications:

“How can we observe it? Who can say where a given star should be? Suppose we do have to look in the wrong direction to see a star; how do we know it is the wrong direction? Because the earth goes around the sun. Today we have to point the telescope one way; six months later we have to tilt the telescope the other way. That is how we can tell that there is such an effect (Feynman et al., 1963, p. 34–10).”

 

It should be worth mentioning that observing stellar aberration is not an easy task due to several factors, including the relatively small magnitude of the angular shift and instrumental challenges. The maximum stellar aberration of stars is about 20.5 arc seconds, which is roughly the diameter of the full Moon (e.g., Fortson, 2022).  Specifically, the Earth’s atmosphere can cause the light from stars to be refracted, scattered, or distorted as it passes through different layers of air. On the other hand, observing stars at the right times and in the right directions is crucial for maximizing the effect and minimizing errors. Alternatively, we may use satellite-based telescope because it offers a stable observational environment whereby there are no atmospheric turbulence, weather-related changes, and light pollution.

 

Perhaps Feynman could have highlighted seven implications (or applications) of stellar aberration experiment: (1) Confirmation of Earth’s motion around the Sun: It provided evidence supporting the heliocentric model of the solar system. (2) Determination of the Speed of Light: It helped to verify the speed of light with reasonable accuracy. (3) Ether drag hypothesis: Airy's modified aberration experiment using a water-filled telescope provided evidence against the idea of ether drag. (4) Einstein’s light postulate: The realization that the apparent shifts in the positions of all stars in a specific direction have the same value and are independent of the brightnesses or distances of the stars. (5) Determination of Earth’s orbital velocity: Stellar aberration has been utilized to measure the Earth’s orbital velocity around the Sun. (6) Determination of Earth’s nutation:  Bradley’s discovery of the nutation of the Earth’s axis helped to understand the net gravitational forces between the Earth, the Moon, and the Sun. (7) Investigate the motion of our solar system: The combined effect of stellar aberration from numerous stars helps in determining the motion of our solar system around the center of the galaxy, contributing to our understanding of galactic dynamics.

Review Questions:

1. How would you define the term stellar aberration?

2. How would you derive the stellar aberration formula? 

3. What are the implications or applications of stellar aberration? 

 

The moral of the lesson: The observer's changing reference frame, due to Earth's orbital motion, leads to the apparent deviation of starlight, providing astronomers with important insights into the dynamics of celestial objects and the fundamental nature of light, however, some of stars may not exist now.

 

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. Fortson, N. (2022). Discovering the nature of light: The Science and the Story. Singapore: World Scientific.

3. Schwinger, J. S. (2002). Einstein's legacy: the unity of space and time. New York: Dover.