Section 35–2 Color depends on intensity

 (Dark-adapted vision / Purkinje effect / Peripheral vision)

 

In this section, Feynman discusses the dark-adapted vision, Purkinje effect (related to mesopic vision), and peripheral vision that depend on the intensity of light.

 

1. Dark-adapted vision:

“If the intensity of the light is very low, the things that we see have no color. It is known that this dark-adapted vision is almost entirely due to the rods, while the vision in bright light is due to the cones. As a result, there are a number of phenomena that we can easily appreciate because of this transfer of function from the cones and rods together, to just the rods (Feynman et al., 1963, p. 35–2).”

 

It is potentially misleading to say “if the intensity of the light is very low, the things that we see have no color.” In very low-light conditions, we rely on our rod cells, which are highly sensitive to low levels of light, to see in shades of gray (or grey). While rod cells do not provide detailed color information, they allow us to distinguish different levels of brightness and perceive objects in a grayscale view. On the other hand, gray is a color because it can be achieved by adding and adjusting the intensity of red, green, and blue light (see below). However, rod cells may take about 20-30 minutes for the human eye to become fully dark-adapted, that is, allowing the rod cells to reach their maximum sensitivity in the dark night.

 

Source: Butler, 2005

Dark-adapted vision primarily refers to scotopic vision, which is the vision that occurs in very low-light conditions. In his autobiography, Feynman (1997) mentions that he could see colors during an atomic explosion in the dark: “… a big ball of orange, the center that was so bright, becomes a ball of orange that starts to rise and billow a little bit and get a little black around the edges, and then you see it's a big ball of smoke with flashes on the inside, with the heat of the fire going outwards. All this took about one minute. It was a series from bright to dark, and I had seen it. I am about the only guy who actually looked at the damn thing--the first Trinity test (p. 134).” In a sense, this is another reason why it is incorrect to say that the things that we see have no color in the dark. (Interestingly, some claim to achieve beatific vision during the dark night.)

 

2. Purkinje effect:

“It turns out that the rods see better toward the blue than the cones do, and the cones can see, for example, deep red light, while the rods find that absolutely impossible to see. So red light is black so far as the rods are concerned. Thus two pieces of colored paper, say blue and red, in which the red might be even brighter than the blue in good light, will, in the dark, appear completely reversed. It is a very striking effect. If we are in the dark and can find a magazine or something that has colors and, before we know for sure what the colors are, we judge the lighter and darker areas, and if we then carry the magazine into the light, we may see this very remarkable shift between which was the brightest color and which was not. The phenomenon is called the Purkinje effect (Feynman et al., 1963, p. 35–2).”

 

Historically, Jan Evangelista Purkinje observed that his favorite flower appeared bright red in the afternoon, but it became bluish-red in the evening. We can explain Purkinje effect as follows: (1) Lighting conditions: It pertains to the shift in perceived colors under different lighting conditions, specifically during transitions from bright light (photopic vision) to low-light conditions (mesopic vision). (2) Color perception: In bright light, shorter wavelengths dominate our perception, making blue and green hues more vibrant, whereas in low-light conditions, longer wavelengths become more prominent. (3) cone cells: The effect occurs due to the different sensitivity of S-cones*, M-cones, and L-cones under different lighting conditions. In short, Purkinje effect may refer to red objects that appear darker under dim lighting conditions compared to green objects due to the reduced sensitivity of the cone cells to red wavelengths (See below).

*Humans have three types of cone cells in the retina, each sensitive to different wavelengths of light: short-wavelength cones (S-cones) are sensitive to blue light, medium-wavelength cones (M-cones) are sensitive to green light, and long-wavelength cones (L-cones) are sensitive to red light.

 

In Fig. 35–3, the dashed curve represents the sensitivity of the eye in the dark, i.e., using the rods, while the solid curve represents it in the light. We see that the peak sensitivity of the rods is in the green region and that of the cones is more in the yellow region. If there is a red-colored page (red is about 650 μm) we can see it if it is brightly lighted, but in the dark it is almost invisible (Feynman et al., 1963, p. 35–2).”

Feynman initially explains the phenomenon using two pieces of colored paper, blue and red, but elaborates the spectral sensitivity of the eye involving the cones that is more in the yellow region. However, it could be explained in terms of photopic vision and scotopic vision: (1) Photopic vision: When our eyes are exposed to bright light conditions, such as well-lit indoor environments, they are in a state of photopic vision. In photopic vision, our eyes (cones) are most sensitive to the yellow-green part of the spectrum, about 555 nanometers (nm) wavelength. (2) Scotopic vision: When our eyes are exposed to very low-light conditions, they gradually adapt to scotopic vision, which is highly sensitive to dim light. In scotopic vision, our eyes (rods) are most sensitive to about 505 nm wavelength, which corresponds to the color green on the visible spectrum.

 

3. Peripheral vision:

“Another interesting phenomenon is that the periphery of the retina is very sensitive to motion. Although we cannot see very well from the corner of our eye, if a little bug moves and we do not expect anything to be moving over there, we are immediately sensitive to it. We are all “wired up” to look for something jiggling to the side of the field (Feynman et al., 1963, p. 35–3).”

 

The phenomenon described by Feynman is related to the distribution of rod cells in the human retina and how it affects our peripheral vision. We may define peripheral vision (or side vision) as follows: 1. Field of vision: It covers a wider field of view compared to central vision. 2. Sensitivity to motion: It is highly sensitive to motion and is crucial for detecting movement in the surrounding environment due to a higher density of rod cells. 3. Lower acuity and color perception: It has lower visual acuity and reduced color perception compared to central vision due to a lower density of cone cells. Thus, the periphery of the retina is better suited for detecting motion and objects in dim lighting conditions.

 

“Another interesting effect of the fact that the number of cones decreases as we go farther to the side of the field of view is that even in a bright light color disappears as the object goes far to one side. The way to test that is to look in some particular fixed direction, let a friend walk in from one side with colored cards, and try to decide what color they are before they are right in front of you. One finds that he can see that the cards are there long before he can determine the color. When doing this, it is advisable to come in from the side opposite the blind spot, because it is otherwise rather confusing to almost see the color, then not see anything, then to see the color again (Feynman et al., 1963, p. 35–3).”

 

It is unclear if one can almost see the color of moving cards, then not see anything, and see the color again while we are looking in a fixed direction. Firstly, the phenomenon related to the blind spot typically occurs when observing a stationary object, not a moving one. When the eyes are fixated on a specific point, the brain fills in the missing information from the blind spot, creating the illusion that the object is still visible, even though it is not being detected by the photoreceptor cells in the blind spot area. This occurs seamlessly and almost instantaneously, so one does not notice the gap in the visual field due to the blind spot. On the other hand, the decrease in the number of cones and the dominance of rod cells in the peripheral vision result in reduced color perception and decreased visual acuity in the outer edges of the visual field. This effect can make colors less noticeable as the object moves toward the side of our visual field.

 

Feynman initially mentions that another interesting effect is: even in a bright light, color disappears (instead of almost disappear) as the object goes far to one side. This is related to the near absence of cones and abundance of rods in the outer part of retina that is the reason for our not getting color or detailed information from our peripheral vision. However, a more interesting phenomenon is Troxler’s fading, which occurs when the eyes fixate on a particular point. As a result, details of objects located in the periphery of the fixated point can fade away or become less visible. One common example of Troxler's fading is staring at a small, high-contrast object (like a dot) in the center of a large, uniform background as shown below.

Source: Pegoraro, 2016

 

Review Questions:

1. How would you explain dark-adapted vision and whether one may see colors under very low light conditions?

2. How would you explain the Purkinje effect?

3. How would you define peripheral vision under normal light conditions?

 

The moral of the lesson: human vision depends on the intensity of light because it is based on rods (sensitive to dim light) and three types of cone cells: S-cones (sensitive to blue light), M-cones (sensitive to green light), and L-cones (sensitive to red light).

 

References:

1. Butler, Y. (Ed.). (2005). The Advanced Digital Photographer's Workbook: Professionals Creating and Outputting World-class Images. Taylor & Francis.

2. Feynman, R. P. (1997). Surely You’re Joking, Mr. Feynman! : Adventures of a Curious Character. New York: Norton.

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. Pegoraro, V. (2016). Handbook of Digital Image Synthesis: Scientific Foundations of Rendering. CRC Press.

Section 35–1 The human eye

 (Functions of eye / Structure of retina / Eye-brain system)

 

In this section, Feynman discusses functions of human eyes, the structure of human retina, and the eye-brain system.

 

1. Functions of eye:

“Light enters the eye through the cornea; we have already discussed how it is bent and is imaged on a layer called the retina in the back of the eye, so that different parts of the retina receive light from different parts of the visual field outside (Feynman et al., 1963, p. 35–1).”

 

Perhaps Feynman could have explained the human eye, a biological organ, using more physics-oriented explanations, physical principles, or physical analogies. For example, the human eye functions in a manner similar to a camera, both collecting, detecting, and processing light from the surrounding environment. In addition, light enters the eye through the cornea, the transparent front part of the eye. Similarly, light also enters the camera through the camera lens, which is analogous to the cornea in the human eye. On the other hand, the retina contains two types of photosensitive cells or photoreceptors: rods (responsible for night vision) and cones (responsible for color vision). It is analogous to the camera’s sensor that acts like tiny photosensitive receptors, detecting the intensity of light that falls on it.

 

In his autobiography, Feynman shared how he viewed the explosion of atomic bomb using the truck windshield. In his words, “They gave out dark glasses that you could watch it with. Dark glasses! Twenty miles away, you couldn't see a damn thing through dark glasses. So I figured the only thing that could really hurt your eyes (bright light can never hurt your eyes) is ultraviolet light. I got behind a truck windshield, because the ultraviolet can't go through glass, so that would be safe, and so I could see the damn thing (Feynman, 1997, p. 134).” It is worthwhile to have some knowledge of human eye functions and how to prevent damage in the retina. For example, the lens of the eye can absorb some ultraviolet (UV) radiation and provide a certain degree of protection to the retina, but this natural UV filtering capacity is limited.

 

2. Structure of retina:

“The retina is not absolutely uniform: there is a place, a spot, in the center of our field of view which we use when we are trying to see things very carefully, and at which we have the greatest acuity of vision; it is called the fovea or macula (Feynman et al., 1963, p. 35–1).”

 

In the Audio Recordings* [5 min: 25 sec] of this lecture, Feynman says: “it is called the fovea and that’s here” instead of “it is called the fovea or macula.” The macula is the central portion of the retina with a high concentration of cones, contributing to central vision and color perception. The fovea is a specialized pit structure within the macula that contains only cones and provides the greatest acuity of vision, making it responsible for the highest detailed and sharp visual perception. We may rephrase the above-statement (highlighted in yellow) as: “The retina is not uniform, and there is a specialized region in the center of our field of view known as the fovea, which is part of the larger macula. The fovea is the area where we have the greatest acuity of vision and use for tasks that require precise and detailed visual perception.”

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

 

The retina can be distinguished into at least 3 regions: fovea, macula, and periphery (see figure below). In the fovea (0.35 mm in diameter), the density of cones is highest, while the rods are completely absent. Moving away from the fovea, the density of cones is lower in the macula, but it is relatively high compared to the peripheral retina. This allows for good visual detail and color perception in the central part of our vision. As we move towards the periphery, the density of cones decreases, but the density of rods increases. The periphery of the retina is more sensitive to low intensity light and is better suited for detecting motion and objects in the dark. Specifically, rods are more numerous in the peripheral retina, enhancing our ability to perceive movement and objects in our side vision.

Source: Why don't we get color or detail information from our peripheral vision? | Socratic

3. Eye-brain system:

“Now the interesting thing is that in the retina each of the cells which is sensitive to light is not connected by a fiber directly to the optic nerve, but is connected to many other cells, which are themselves connected to each other. There are several kinds of cells: there are cells that carry the information toward the optic nerve, but there are others that are mainly interconnected “horizontally.” There are essentially four kinds of cells... (Feynman et al., 1963, p. 35–2).”

 

Feynman mentions that there are essentially four kinds of cells and there are cells that are interconnected “horizontally.” Perhaps he could have explained that the horizontal cells are interconnected laterally instead of horizontally. The term “horizontal cells” is derived from their horizontal (side-to-side) connections with the neighboring retinal cells, emphasizing their function in integrating and regulating signals across the retina. Instead of saying four kinds of cells, some may emphasize that there are six kinds of retinal cells, namely, horizontal, bipolar, amacrine, interplexiform, ganglion, and photoreceptor cell (Dowling, 2012). For example, amacrine cells are involved in fine-tuning the signals transmitted between bipolar cells and ganglion cells, i.e., they play a role in modulating contrast and sensitivity to different light intensities.

 

Source: Human eye - Retina, Optic Nerve, Vision | Britannica

“…the information from the various cells does not immediately go to the brain, spot for spot, but in the retina a certain amount of the information has already been digested, by a combining of the information from several visual receptors. It is important to understand that some brain-function phenomena occur in the eye itself (Feynman et al., 1963, p. 35–2).”

 

Although the human retina is physically located within the eye, its role extends beyond simple light detection. The retina’s functions and connections are closely tied to the brain, which processes the visual information collected by the retina. In essence, the retina is an integral part of the eye-brain system, where the eye captures light and initial visual information, whereas the brain processes and translates this information into the visual experiences that humans perceive. The processing and interpretation of visual information occurs in the brain, making the retina and the brain tightly interconnected. In a sense, photoreceptor cells can be compared to the pixels on a camera sensor. Just as pixels on a camera sensor detect light and convert it into electrical signals, photoreceptor cells in the retina (rods and cones) detect light and convert it into electrical signals, initiating the process of visual information.

 

Note: In Chapter 36, Feynman explains: “As a matter of fact, people who study anatomy and the development of the eye have shown that the retina is, in fact, the brain: in the development of the embryo, a piece of the brain comes out in front, and long fibers grow back, connecting the eyes to the brain. The retina is organized in just the way the brain is organized and, as someone has beautifully put it, ‘The brain has developed a way to look out upon the world.” The eye is a piece of brain that is touching light, so to speak, on the outside. So it is not at all unlikely that some analysis of the color has already been made in the retina’ (Feynman et al., 1963, p. 36–2).”

 

Review Questions:

1. How would you explain the functions of human eyes?

2. How would you explain the structure of the retina (fovea, macula, and periphery)?

3. How would you explain whether the eye is a part of the brain?

 

The moral of the lesson: the eye-brain system contains two types of photoreceptors: rods (for night vision) and cones (for color vision) that are related to six kinds of retinal cells (horizontal, bipolar, amacrine, interplexiform, ganglion, and photoreceptor).

 

References:

1. Dowling, J. E. (2012). The retina: An approachable part of the brain. Cambridge, MA: Harvard University Press.

2. Feynman, R. P. (1997). Surely You’re Joking, Mr. Feynman! : Adventures of a Curious Character. New York: Norton.

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.

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.