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.