Section 33–4 Polarizers

 (Malus’ polarization law / Three polarizer paradox / Brewster’s angle)

 

In this section, Feynman discusses Malus’ polarization law, three polarizer paradox (or Dirac’s polarization paradox), and Brewster’s angle. It could be titled “polarization by reflection and polarizers” instead of polarizers because the previous sections are about “polarization by scattering of light” and “polarization by birefringence.”

 

1. Malus’ polarization law:

“The energy which passes through the polaroid, i.e., the intensity of the light, is proportional to the square of cos θ. Cos² θ, then, is the intensity transmitted when the light enters polarized at an angle θ to the pass direction (Feynman et al., 1963, p. 33–5).”

 

Malus’ law states that the intensity of a plane-polarized light (I) that passes through a polarizer varies with the square of the cosine of the angle (q) between the light’s initial polarization and the transmission axis of the polarizer in accordance with I = I₀cos² q, where I₀ is the initial intensity. This is an idealization because Malus could not verify the law using an accurate light intensity meter. This law was published in 1809, but Nicol prism (the first easily usable polarizer) was invented in 1828. On the other hand, Malus’ works on polarization were based on the corpuscular theory of light. Thus, Malus’ polarization law did not have a correct theoretical basis and adequate empirical support, but it involves the assumption of an ideal (or perfect) polarizer.

 

Perhaps Feynman could have included a definition of an ideal polarizer or real polarizer. For example, one may define an ideal polarizer as an optical device: (1) whose intensity is reduced in accordance with Malus’ Law; (2) which may effectively rotate linearly polarized light in a direction parallel to its transmission axis. On the contrary, a real polarizer can be defined in terms of polarization efficiency and extinction ratio. Specifically, a real polarizer may transmit about 80% of light polarized in one direction and less than 1% of light polarized in the direction at right angles to it (Born & Wolf, 1980). In the real world, we should not expect a completely linear polarized light because the needle-like molecules in the polarizer are not infinitely thin, but have a small radius.

 

Note: In the Audio Recordings [32 min: 15 sec] of this lecture*, Feynman says: “… The polaroid is not perfect, it’s not exactly zero.”

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

 

2. Three polarizer paradox:

“An interesting paradox is presented by the following situation. We know that it is not possible to send a beam of light through two polaroid sheets with their axes crossed at right angles. But if we place a third polaroid sheet between the first two, with its pass axis at 45^o to the crossed axes, some light is transmitted. We know that polaroid absorbs light, it does not create anything. Nevertheless, the addition of a third polaroid at 45^o allows more light to get through (Feynman et al., 1963, p. 33–5).”

 

The paradox presented by Feynman is sometimes known as Dirac’s polarization paradox. The crux of the paradox is related to “polaroid absorbs light, it does not create anything.” In many definitions of a polarizer, we may find words, such as “filter”, “block,” or “completely absorb.” For instance, a polarizer is an optical filter that lets light waves of a specific polarization pass through while blocking light waves of other polarizations (Source: Wikipedia). From the perspective of an ideal polarizer, one may explain that the polarizer does not only absorb light, but it effectively rotates the direction of polarization of light. The paradox may be resolved by calculating the final intensity of light using Malus’ polarization law and taking into account the order of the three polarizers tilted at different angles.

In Feynman’s words, “[w]e know that polaroid absorbs light, it does not create anything.” This statement is perhaps deliberately misleading in order to create an apparent paradox. From the perspective of a real polarizer, there is really no paradox because this polarizer can transmit light, absorb light, reflect light, and effectively rotate the plane polarization of light. (We can see the real polarizer because it does reflect light, i.e., it does not simply filter light.) More important, a real polarizer does not completely obey Malus’ polarization law and it is wavelength dependent. Strictly speaking, it is still possible to send a beam of light through two polarizers with their axes crossed at right angles.

 

Although the three polarizers paradox is attributed to Dirac, his main concern was the interaction of a photon with a polarizer (tourmaline). As he was trying to understand the experimental results from the perspective of a photon, Dirac (1947) writes, “[i]f one repeats the experiment a large number of times, one will find the photon on the back side in a fraction sin² α of the total number of times. Thus we may say that the photon has a probability sin² α of passing through the tourmaline [polarizer] and appearing on the back side polarized perpendicular to the axis and a probability cos² α of being absorbed. These values for the probabilities lead to the correct classical results for an incident beam containing a large number of photons (p. 6).” In essence, the experimental setup forces the photon to be completely into the state of perpendicular polarization or parallel polarization. It provides a quantum mechanical perspective on the paradox.

 

3. Brewster’s angle:

“It was discovered empirically by Brewster that light reflected from a surface is completely polarized if the reflected beam and the beam refracted into the material form a right angle. (Feynman et al., 1963, p. 33–5).”

 

We may define Brewster’s angle (or polarizing angle) in terms of three perspectives: (1) s-polarized light: when the incident light is polarized normal to the plane of incidence, the reflected light is maximally polarized at the polarizing angle; it means that light can be polarized by reflection; (2) p-polarized light: when the incident light is polarized parallel to the plane of incidence, there is “no reflection” of p-polarized light; it indicates that light is a transverse wave; (3) unpolarized light: the reflected light and refracted light form a right angle; it leads to the equation tan q_B = n (specifically, tan q_B = n₂/n₁) in which n is the refractive index of the medium. Perhaps Feynman could have used the phrase maximally polarized (instead of completely polarized) because of multiple interactions of light with the reflection surface.

 

“From Fig. 33–4 it is clear that only oscillations normal to the paper can radiate in the direction of reflection, and consequently the reflected beam will be polarized normal to the plane of incidence. If the incident beam is polarized in the plane of incidence, there will be no reflected light (Feynman et al., 1963, p. 33–6).”

 

In a sense, the phrase “it is clear” may be misleading because some explanations are needed to understand why the reflected light will be polarized normal to the plane. Firstly, some may need an explanation of how oscillating charges produce a donut-shaped radiation whereby there is no radiation along the axis of oscillation. Secondly, the main contribution of reflected light comes from a small critical region at the interface of the reflection surface where light can travel shortest distances to reach an observer (Fermat’s principle of least action). Thirdly, there is no reflected light at Brewster’s angle unless light is a longitudinal wave. That is, charges oscillating at the interface of two media that produce reflected light and refracted light “should” have the same polarization as shown below. However, the surface of reflection is not perfectly smooth and we may describe the reflected light at Brewster’s angle as not exactly zero or it is minimally polarized.

 

In his seminal paper on the blue sky, Tyndall (1869) writes: “the polarization of the beam by the incipient cloud has thus far proved to be absolutely independent of the polarizing-angle. The law of Brewster does not apply to matter in this condition, and it rests with the undulatory theory to explain why. Whenever the precipitated particles are sufficiently fine, no matter what the substance forming the particles may be, the direction of maximum polarization is at right angles to the illuminating beam…” Rayleigh cited Tyndall’s words and responded, “I venture to think that the difficulty is entirely imaginary, and is caused mainly by misuse of the word reflection (Strutt, 1871, p.107).” Interestingly, Feynman provides a good explanation of the word reflection: “this so-called reflected light is not simply that the incident beam is reflected; our deeper understanding of this phenomenon tells us that the incident beam drives an oscillation of the charges in the material, which in turn generates the reflected beam.” However, Brewster’s angle could be related to the maximally polarized light scattered at 90^o which is briefly discussed in Section 33.2.

 

Review Questions:

1. How would you define an ideal polarizer or real polarizer?

2. Do you agree with Feynman that “it is not possible to send a beam of light through two polaroid sheets with their axes crossed at right angles”?

3. (a) How could one tell the absolute direction of polarization from a piece of polaroid? (Source: Page 212 of Surely You’re Joking, Mr. Feynman!)

(b) How would you explain Brewster’s angle pertaining to the reflection on a grain of salt?

 

The moral of the lesson: The concept of Malus’ polarization law, Dirac’s polarization paradox, and Brewster’s angle may help us to take a better picture using a polarizer filter as shown below.

 

References:

1. Born, M., & Wolf, E. (1980). Principles of optics: electromagnetic theory of propagation, interference, and diffraction of light (6^th ed.). New York: Pergamon.

2. Dirac, P. A. M. (1981). The principles of quantum mechanics. London: Oxford university press.

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

4. 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.

5. Strutt, J. W. (1871). XV. On the light from the sky, its polarization and colour. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 41(271), 107-120. 

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