Section 33–6 The intensity of reflected light

 (Basic principle / Fresnel’s sine law / Fresnel’s tangent law)

 

In this section, Feynman discusses the basic principle of deriving Fresnel’s laws of reflection and explains the derivation of Fresnel’s sine law and Fresnel’s tangent law. Some may prefer Feynman’s discussion of Fresnel’s laws of reflection in Volume II.

 

1. Basic principle:

“The principle that we must understand is as follows. The currents that are generated in the glass produce two waves. First, they produce the reflected wave. Moreover, we know that if there were no currents generated in the glass, the incident wave would continue straight into the glass (Feynman et al., 1963, p. 33–7).”

 

According to Feynman, we must understand the principle in accordance with the generated currents in the glass that produce reflected and refracted waves. In Volume II, Feynman explains that Fresnel’s laws of reflection can be obtained by boundary conditions instead of some “clever arguments” as shown in this section (Feynman et al., 1964). Specifically, the boundary conditions are based on the assumptions of linear, isotropic, and homogeneous media. In other words, this derivation of Fresnel’s equations involves certain idealizations, i.e., the materials are non-magnetic and there are no free charges or electric currents at the boundary of two media. These assumptions are made in order to idealize the effects of two refractive indices of the materials and to simplify the mathematical analysis of the electromagnetic waves.

Historically, Fresnel’s derivation of the two laws of reflection may be described as incorrect because he used an outdated model involving ether and “force vive” (Silverman, 1998). In a sense, Fresnel’s assumptions of the continuity of the tangential component of particles’ vibrations and kinetic energy are equivalent to Maxwell’s boundary conditions. In 1875, it was Lorentz who first derived Fresnel’s laws of reflection from Maxwell’s equations in his doctoral thesis (Laue, 1950). Essentially, the conditions imposed upon the fields at the boundary of two media are equivalent to stipulating the conservation of energy (or energy flow). The reflectance of the light can be related to the ratio of the number of reflected photons to the number of incident photons from the perspective of quantum mechanics.

 

2. Fresnel’s sine law:

“This is not quite true, however, because in Fig. 33–6(b) the polarization directions are not all parallel to each other, as they are in Fig. 33–6(a). It is only the component of A which is perpendicular to B, Acos (i+r), which is effective in producing B. The correct expression for the proportionality is then (33.1) b/a = B/Acos (i+r) (Feynman et al., 1963, p. 33–7).”

 

It does not seem obvious why Feynman asserts: “It is only the component of A which is perpendicular to B, Acos (i + r), which is effective in producing B.” Perhaps Feynman’s clever arguments in deriving Fresnel’s equation of reflection could be based on a trick of working backward. For example, we can deduce the relationship equation (33.8) by dividing b = −sin(i−r)/sin(i+r) by B = −tan(i−r)/tan(i+r). Thus, we get b/B = [sin(i−r)/sin(i+r)] /[tan(i−r)/tan(i+r)] = cos(i−r)/cos(i+r) where it allows one to make sense of the expression Acos (i+r) or Acos (i−r). Alternatively, we may explain that there is a proportionality relationship between refracted and reflected ray (instead of a cause-effect relationship).

 

In the Audio recording* [43 min: 45 sec], Feynman says: “in this case, I have the factors of polarization; it's only that component of this polarization which is in this direction which is effective in that case. And therefore, it isn’t just B/A, but it has an extra factor which is the cosine of the angle between here and here. That is, if I were to project this direction here, and ask for the component in this direction, which is the only component that's effective to produce capital 'B', it would be less than it would be in this case by the cosine of that angle, so you see where that comes from.” However, Feynman could have specified the component of this polarization is in the direction of B, i.e., reflected ray, and explained that the angle between refracted ray and reflected ray is i+r (See Fig 1). In addition, one may elaborate that cos (i+r) is related to how much of the two unit vectors of refracted ray and reflected ray overlap.

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

 

Fig. 1: The angle between the refracted ray and the "reflected ray" is (i+r).

“This field is not observed, and therefore the currents generated in the glass must produce a field of amplitude −1, which moves along the dotted line. Using this fact, we will calculate the amplitude of the refracted waves, a and A (Feynman et al., 1963, p. 33–7).”

 

Feynman deduces Fresnel’s laws of reflection by assuming the electric field in the glass must produce oscillations of the charges, which generate a field of amplitude −1. However, one may suggest using the ratios of electric fields of the incident ray, reflected ray, and refracted ray respectively (E_i : aE_i: bE_i & E_i : AE_i: BE_i) for both polarizations; the two ratios can be simplified as 1: a: b and 1: A: B. Instead of saying “an extra factor which is the cosine of the angle between here and here,” one may explain that Acos (i+r) is the component of a (refracted ray) that is in the direction of the reflected ray. On the other hand, Acos (i − r) is the component of A that is in the direction of the incident ray. It does not seem necessary to use the concept of negative amplitude along the dotted line in the derivation of Fresnel’s laws.

 

3. Fresnel’s tangent law:

“But we see from part (b) of the figure that only the component of A that is normal to the dashed line has the right polarization to produce this field, whereas in Fig. 33–6(a) the full amplitude a is effective, since the polarization of wave a is parallel to the polarization of the wave of amplitude −1. Therefore we can write (33.2) Acos(i − r) / a = −1 (Feynman et al., 1963, p. 33–7).”

 

Feynman’s explanation in deriving Acos(i − r)/a = −1 seems unnatural by relating it to the polarization of wave a that is parallel to the polarization of the wave of amplitude −1 (In the audio recordings [61 min: 35 sec], Feynman was asked to re-explain the derivation because a student said that he did not get anything……). The derivation is more complicated for polarization perpendicular to the plane of incidence (or paper) than polarization parallel to the plane of incidence. According to Feynman, the component of A that is normal to the dashed line has the right polarization to produce this field is Acos (i − r) is shown below in Fig 2. Perhaps Feynman could have specified this component of A is in the direction of the incident ray. Furthermore, cos (i − r) is related to how much of the two unit vectors of refracted ray and incident ray overlap.

 

Fig. 2 The angle between the refracted ray and the incident ray is (i − r).

“It is possible to go on with arguments of this nature and deduce that b is real. To prove this, one must consider a case where light is coming from both sides of the glass surface at the same time, a situation not easy to arrange experimentally, but fun to analyze theoretically (Feynman et al., 1963, p. 33–8).”

 

Feynman proves that b is a real number by considering a case where light is coming from both sides of the glass surface at the same time. However, this is really an idealization because the refractive index could be defined as a complex number instead of a real number. That is, the real part of the refractive index is related to the effective speed of light in a material, while the imaginary part is related to the absorption (or attenuation) of the light as it passes through the material. In other words, we can introduce complex refractive index that has an impact on the reflectance and transmittance of the light at the boundary of two media. For example, when the imaginary part of the refractive index is large, the light may be strongly attenuated and only a little of it may be transmitted through the material.

 

“One can show by these arguments that (33.8) b = −sin(i−r)/sin(i+r), B = −tan(i−r)/tan(i+r). These expressions for the reflection coefficients as a function of the angles of incidence and refraction are called Fresnel’s reflection formulas (Feynman et al., 1963, p. 33–8).”

 

Note that the minus sign in Feynman’s derivation of Fresnel’s tangent law is different from B = +tan(i−r)/tan(i+r) which can be found in many textbooks (e.g., Hecht, 2002). Firstly, the minus sign implies that the electric field assumed by him should be in the opposite direction. Secondly, Feynman could have explained that the reflected wave of s-polarized light (normal to the plane of incidence) experiences a phase shift of 180 degrees during the reflection at the boundary of the two media. Thirdly, we could replace Feynman’s symbols a, b, A, and B using r_^ , t_^, r_∥, and t_∥ respectively. It allows us to have a better picture of reflection coefficients and transmission coefficients (refracted ray) by writing the following equations t_^ + (−r_^) = 1 (holds for all angle of incidence) and t_∥ +  r_∥ = 1 that is true only at normal incidence (Hecht, 2002).

Review Questions:

1. Should the basic principle of Fresnel’s laws of reflection be based on the generated currents in the glass that produce reflected and refracted waves?

2. How would you explain the derivation of Fresnel’s sine law?

3. How would you explain the derivation of Fresnel’s tangent law?

 

The moral of the lesson: Fresnel’s sine law and Fresnel’s tangent law are related to Acos (i + r) and Acos (i − r) which are the components of A in the direction of reflected ray and incident ray respectively.

In other words, Fresnel’s sine law and Fresnel’s tangent law are dependent on how the unit vector of refracted ray overlaps with the unit vector of the reflected ray and incident ray respectively.

 

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

3. Hecht, E. (2002). Optics (4th edition). San Francisco: Addison Wesley.

4. Laue, M. V. (1950). History of physics. New York: Academic Press.

5. Silverman, M. P. (2018). Waves and grains: reflections on light and learning. Princeton: Princeton University Press.