Section 33–1 The electric vector of light

(Linearly polarized / Elliptically polarized / Unpolarized)

 

In this section, Feynman discusses linearly polarized light, elliptically polarized light, and unpolarized light.

 

1. Linearly polarized:

“Light is linearly polarized (sometimes called plane polarized) when the electric field oscillates on a straight line; Fig. 33–1 illustrates linear polarization (Feynman et al., 1963, p. 33–2).”

 

Feynman explains that light is linearly polarized (or plane polarized) provided its electric field oscillates on a straight line. Historically, Étienne-Louis Malus (1811) defines plane of polarization as the plane that contains the direction of propagation and the magnetic field. In the Audio Recordings* [9 min: 00 sec] of this lecture, Feynman says: “historically, they did not know which field is the right one,” i.e., the term plane of polarization is confusing due to the older literature. Thus, some physicists define plane of vibration using the orientation of electric field because most materials interact more strongly with electric fields as compared to magnetic fields. The plane of vibration is not definitely constant at any point, but if it rotates uniformly, the light may be described as circularly (or helically) polarized.

 

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

 

“The motion in a straight line is a particular case corresponding to a phase difference of zero (or an integral multiple of π); motion in a circle corresponds to equal amplitudes with a phase difference of 90^o (or any odd integral multiple of π/2) (Feynman et al., 1963, p. 33–1).”

 

We can define polarized light using E = (Acos q_x)e^{i(kz - wt)} x + (Asin q_y)e^{i(kz - wt)} y. According to Feynman, linearly polarized light is a special case corresponding to a phase difference Dq = q_y − q_x = nπ, n = 0, 1, 2, 3…; circularly polarized light corresponds to equal amplitudes with a phase difference of π/2, 3π/2, or equivalent. In addition, any linearly polarized light can be formed from two different circularly polarized lights because the “left-handed rotation” component can be canceled by the “right-handed rotation” component. In the Audio Recordings [11 min: 00 sec], Feynman says: “suppose we have a left-hand circular and right-hand circular… adding two circular polarized lights in phase and equal amplitude, we get linearly polarized light.” This explanation is edited and shifted to section 33-5: “any linear polarization can be made up by superposing suitable amounts at suitable phases of right and left circular polarizations... (Feynman et al., 1963).”

 

“The real and imaginary components of a complex electric field vector are only a mathematical convenience and have no physical significance (Feynman et al., 1963, p. 33–2).”

The above sentence is potentially misleading on the real component of a complex electric field. In the Audio Recordings [3 min: 10 sec], Feynman says: “we have been using diagrams in which the real part of complex number to represent the electric fields and the imaginary part in this complex number diagram is merely a construction for the analysis of the electric fields... The diagrams on the board today are not complex number diagrams, but the real part of the electric fields in the x and y direction.” Specifically, the real component of a complex electric field does have physical significance because it can be measured. The complex electric field can also be represented by a column or row vector. In a lecture on QED, Feynman (1985) explains that a complex number is equivalent to an imaginary stopwatch hand.

 

2. Elliptically polarized:

“We have considered linearly, circularly, and elliptically polarized light, which covers everything except for the case of unpolarized light (Feynman et al., 1963, p. 33–2).”

 

Feynman distinguishes three kinds of polarized light: linearly polarized, circularly polarized, and elliptically polarized. Alternatively, Jackson (1999) defines linearly polarized light as E_x and E_y have the same phase, and q = tan^-1 E_y/E_x; if E_x and E_y have different phases, it is described as an elliptically polarized light. However, some consider linearly and circularly polarized light as special cases of elliptically polarized light. This is because these three kinds of polarized light are based on the same formula E = (Acos q_x)e^{i(kz - wt)} x + (Asin q_y)e^{i(kz - wt)} y, but some include the conditions: q_y-q_x = 0^o, 180^o means linear polarization and q = 90^o, 270^o means circular polarization. In a sense, some prefer to unify linearly, circularly, and elliptically polarized into a single category, however, it is convenient to describe how a linearly polarized light becomes circularly or elliptically polarized light through a waveplate.

 

Some may define an elliptically polarized light as an electromagnetic wave in which the tip of its electric field vector traces an elliptical locus in space. Additionally, elliptical polarization corresponds to the case where two components of the electric field are not equal and differ in phase by an angle (e.g., q_y - q_x.¹ 0^o, 90^o, 180^o, 270^o). Elliptically polarized light can be produced by combining two linearly polarized lights having a phase difference other than 90^o or 270^o. One may distinguish an elliptically polarized light as right-hand or left-hand just like circularly polarized light. In the Audio Recordings [10 min: 05 sec], Feynman says: “This is right-hand circular, it is the way a right-hand screw would go as light comes out. One of those nonsensical, difficult matter of notations that has nothing to do with physics.”

 

3. Unpolarized light:

“If the polarization changes more rapidly than we can detect it, then we call the light unpolarized, because all the effects of the polarization average out. None of the interference effects of polarization would show up with unpolarized light. But as we see from the definition, light is unpolarized only if we are unable to find out whether the light is polarized or not (Feynman et al., 1963, p. 33–2).”

 

Feynman explains that light is unpolarized because of our inability to find out whether it is polarized or not. Thus, unpolarized light is a misnomer, but it means that the instantaneous direction of polarization can vary rapidly in time between 0 and 2π. One may prefer the term randomly polarized, but the so-called unpolarized light can be represented in terms of two arbitrary, orthogonal, linearly polarized light waves of equal amplitudes in which the relative phase difference varies rapidly and randomly. However, light is neither completely polarized nor completely randomly polarized, but it is somewhere in between. We may describe light as partially polarized because it is related to the superposition of certain amounts of completely polarized and randomly polarized light.

 

Note: In the Audio Recordings [14 min: 05 sec] of this lecture, Feynman says: “unpolarized light is only an approximate idea.”

 

“Remember that one atom emits during 10⁻⁸ sec, and if one atom emits a certain polarization, and then another atom emits light with a different polarization, the polarizations will change every 10⁻⁸ sec (Feynman et al., 1963, p. 33–2).”

Feynman could have clarified why the polarizations will change every 10⁻⁸ sec. Firstly, we do not completely understand what really occurs during the atom-transition interval of 10^-8 s. We may use the term wave packet (wave pulse) instead of polarization or infinite wave train due to the very short duration of light emission. To be more precise, each excited atom emits a polarized wave packet and then transits to a lower energy state or ground state within a duration of the order of 10^-8 s to 10^-9 s. Furthermore, a light source consists of a very large number of randomly oriented atoms that are in random motion. Thus, new randomly polarized wave packets are continuously being emitted, and the sum of the so-called polarization changes in a completely unpredictable manner (instead of changing every 10⁻⁸ sec).

 

Review Questions:

1. How would you define linearly polarized light?

2. Would you consider linearly polarized light to be a particular case of elliptically polarized light?

3. How would you explain the polarizations will change every 10⁻⁸ sec?

 

The moral of the lesson: the term linearly polarized light may be used if E_x and E_y are in phase, the term elliptically polarized light may be used if E_x and E_y are out of phase, but the term unpolarized could be replaced by randomly polarized.

 

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.

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

4. Malus, E. L. (1811). Mémoire sur de nouveaux phénomenes d’optique. Journal de physique, 72, 393-398.

Section 32–5 Scattering of light

(Rayleigh scattering / Mie scattering / Tyndall scattering)

 

In this section, Feynman discusses the mechanism of Rayleigh scattering, Mie scattering, and Tyndall scattering. Thus, the section could be titled “Rayleigh scattering, Mie scattering, and Tyndall scattering.”

 

1. Rayleigh scattering:

“The electric field of the incoming beam drives the electrons up and down, and they radiate because of their acceleration. This scattered radiation combines to give a beam in the same direction as the incoming beam, but of somewhat different phase, and this is the origin of the index of refraction (Feynman et al., 1963, p. 32–6).”

 

Elastic scattering: “Why is the sky blue?” is commonly explained using the term Rayleigh scattering by many, but it does not mean they really understand the phenomenon. On the contrary, Feynman did not mention Rayleigh, but we should acknowledge his works related to the scattering of light by particles (primarily nitrogen and oxygen) that are much smaller than the wavelength of the incoming light. Alternatively, we may use the term elastic scattering because the wavelength of scattered light is predominantly the same as the incoming light. Importantly, the word scattering means the absorption of light by a particle and the re-emission of light in almost all directions due to the oscillation of the particle. In other words, the electric field of the incoming light oscillates the electrons, and then the oscillating electrons emit scattered light.

 

“That is to say, light which is of higher frequency by, say, a factor of two, is sixteen times more intensely scattered, which is a quite sizable difference. This means that blue light, which has about twice the frequency of the reddish end of the spectrum, is scattered to a far greater extent than red light (Feynman et al., 1963, p. 32–8).”

 

Wavelength-dependent scattering: The scattering of light is frequency-dependent or wavelength-dependent, i.e., the shorter wavelengths of light (blue and violet) are strongly deflected, whereas the longer wavelengths (red and orange) are slightly deflected. By using Larmor’s formula, the power or intensity (I) of the scattered light is directly proportional to the square of the acceleration (a²) of oscillating electrons in the field of the incoming light (I µ a²). Furthermore, the acceleration of electrons is directly proportional to the square of the frequency of the incoming light (a µ -w²x). In short, one may write I µ a² µ ω⁴. Thus, the intensity of scattered blue light is 1.49⁴ (= 4.9) times more than red light if we assume the wavelength of red light and blue light are about 700 nm and 470 nm respectively (use l = c/f).

 

But if the objects are randomly located, then the total intensity in any direction is the sum of the intensities that are scattered by each atom, as we have just discussed (Feynman et al., 1963, p. 32–6).”

 

Random scattering: Strictly speaking, the objects (or scatterers) are not only randomly located and the total intensity is not simply the sum of the intensities that are scattered by each atom. Some may use the term random scattering because the air molecules are randomly oriented, in random molecular motion, and there are random microscopic fluctuations that scatter more light in one direction than another. Historically, Einstein (1910) deduced that the random thermal motion of the air results in rapid density fluctuations and causes a similar effect to Rayleigh scattering. It can be described as density fluctuation scattering because the fluctuations in the density of air would result in fluctuations in the refractive index of the medium. Based on this model, the refractive index fluctuations behave like molecular scatterers.

2. Mie scattering:

“We have just explained that every atom scatters light, and of course the water vapor will scatter light, too. The mystery is why, when the water is condensed into clouds, does it scatter such a tremendously greater amount of light? …… But if they are right next to each other, they necessarily scatter in phase, and they have a coherent interference which produces an increase in the scattering (Feynman et al., 1963, p. 32–8).”

 

Coherent scattering: Feynman discusses another mystery pertaining to the scattering of a greater amount of light by clouds. It is known as Mie scattering or coherent scattering because the light waves are coherent in the forward direction due to a lump of particles. Specifically, Mie scattering is due to aerosol particles, such as water droplets and ice crystals, but it may include dust, pollen, and smoke that are present in the atmosphere. One may define Mie scattering as the scattering of light whereby the size of a lump of “almost in-phase particles” is the same or more than the wavelength of the incoming light. Although Rayleigh scattering may be considered as the scattering of light in which the size of the particles is less than 1/10 of the light’s wavelength, it is a limiting case of Mie scattering.

 

“So as we keep increasing the size of the droplets we get more and more scattering, until such a time that a drop gets about the size of a wavelength, and then the scattering does not increase anywhere nearly as rapidly as the drop gets bigger (Feynman et al., 1963, p. 32–8).”

 

Multiple scattering: Feynman explains that there is more scattering of light if the size of the water droplets is increased till the wavelength of the incoming light. Instead of saying more and more scattering, we may use the term multiple scattering which depends on the density of particles, size of scatterers (air molecules or aerosol particles), and path of light. For example, the paths of sunlight near the horizon (sunrise or sunset) are longer than the path through the zenith (noon), i.e., we expect more scattering through longer paths. The color of the sky is not simply blue, but it continues to vary depending on the Sun’s position, atmospheric conditions, and locations of the observer (direction of viewing). Multiple scattering of light by water droplets may result in the appearance of a white cloud or dark cloud depending on the density and height of the clouds.

 

3. Tyndall scattering:

“We use a solution of sodium thiosulfate (hypo) with sulfuric acid, which precipitates very fine grains of sulfur. As the sulfur precipitates, the grains first start very small, and the scattering is a little bluish (Feynman et al., 1963, p. 32–9).”

 

Feynman ends the lecture using a demonstration to show the scattering of light by colloidal particles of sulfur. This may be described as Tyndall scattering experiment (See Fig. 1) that uses a glass tube to simulate the sky and a light source to represent the Sun. However, Tyndall scattering experiment does not completely explain how the blue sky varies due to different meteorological or humidity conditions. In his paper titled On the blue colour of the sky, and on the polarization of light, Tyndall (1869) writes: “[f]rom the illuminated bluish cloud, therefore, polarized light was discharged, the direction of maximum polarization being at right angles to the illuminating beam… (p. 224).” To acknowledge his findings, Tyndall scattering may be defined as the lateral scattering of unpolarized light by colloidal particles whereby maximum polarized light is observable at 90^o to the incoming light.

 

Fig. 1

So if the incoming light has an electric field which changes and oscillates in any direction, which we call unpolarized light, then the light which is coming out at 90^o to the beam vibrates in only one direction! (Feynman et al., 1963, p. 32–9).”

 

Lateral scattering: We may adopt the term lateral scattering because the scattered light is perpendicular to the incoming light, however, it is misleading to say that the light vibrates in only one direction. Perhaps Feynman could have emphasized that the unpolarized light vibrates in all planes that are perpendicular to the direction of light propagation and thus, the scattered light does not vibrate in the same direction as the incoming light. If the scattered light is moving upward in the vertical direction, it should be horizontally polarized (See Fig. 2). Additionally, we observe vertically polarized light if the scattered light emerges in the horizontal direction. You should try to connect the direction of lateral scattering of blue light to Brewster’s angle or Heaviside-Feynman’s formula for the electric field of an accelerated charge.

 

Fig. 2

According to Rayleigh scattering, the color of the sky should be violet instead of blue, but it could be clarified from the perspective of a light source and observer. Firstly, the sunlight is not an equal mix of all colors, but there is more blue than violet light. This is related to Planck’s radiation law that is formulated to explain the spectral-energy distribution of radiation emitted by a blackbody. Secondly, our eyes are more sensitive to blue than violet light. Based on Brown and Wald's (1964) experiments, there are three types of color cones in the retina of the human eye that are relatively more sensitive to red, green, and blue light. To summarize, the sky is blue partly because of more blue than violet light emitted by the Sun and our eyes are more sensitive to blue light than violet light.

 

Review Questions:

1. How would you explain the amount of scattering of light is inversely proportional to the fourth power of its wavelength?

2. How would you explain the scattering of light by clouds results in a greater amount of light?

3. How would you explain the scattered light is perpendicular to the incoming light?

 

The moral of the lesson: The phenomenon of blue sky is related to Rayleigh scattering that involves (1) elastic scattering, (2) wavelength-dependent scattering, (3) random scattering, (4) multiple scattering, (5) lateral scattering, (6) Sun emits more blue than violet light, and (7) eye-sensitivity to blue.

 

References:

1. Brown, P. K., & Wald, G. (1964). Visual pigments in single rods and cones of the human retina. Science, 144(3614), 45-52.

2. Einstein, A. (1910). The Theory of the Opalescence of Homogeneous Fluids and Liquid Mixtures near the Critical State. Annalen der Physik, 33, 1275-1298.

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. Tyndall, J. (1869). On the blue colour of the sky, and on the polarization of light. Phil. M., (4), 37, 384-394.