Section 33–7 Anomalous refraction

 (Ordinary ray / Extraordinary ray / Circularly polarized light)

 

In this section, Feynman discusses ordinary ray (o-ray), extraordinary ray (e-ray), and circularly polarized light. Simply phrased, o-ray and e-ray are symmetrical ray (spherical wavefronts) and asymmetrical ray (elliptical wavefronts), respectively, in a birefringent crystal. In a sense, o-ray and e-ray are the same light waves, but the interactions of the two perpendicularly polarized light with matter (or refractive index) are different.

 

1. Ordinary ray:

“…When this beam strikes the surface of the material, each point on the surface acts as a source of a wave which travels into the crystal with velocity v_⊥, the velocity of light in the crystal when the plane of polarization is normal to the optic axis. The wavefront is just the envelope or locus of all these little spherical waves, and this wavefront moves straight through the crystal and out the other side. This is just the ordinary behavior we would expect, and this ray is called the ordinary ray (Feynman et al., 1963, p. 33–9).”

 

Feynman’s definition of o-ray can be analyzed from the perspective of optic axis, speed of light, and wavefront. Firstly, o-ray propagates in a birefringent crystal whereby its electric fields (plane of polarization) oscillate perpendicular to the optic axis and direction of incident ray. Secondly, o-ray moves at a constant speed, v_⊥, which is dependent on the crystal’s refractive index (See Fig. 1). Thirdly, the wavefronts are spherical because of the constant speed of light and same refractive index within the crystal. However, o-ray should also be defined in terms of Snell’s law of refraction because it obeys the equation n₁sin q₁ = n₂sin q₂ in which n₁ and n₂ are the refractive index of the respective medium, whereas q₁ is the angle of incidence and q₂ is the angle of refraction.

 

Fig. 1

The concept of o-ray in optics is based on idealizations and approximations of a birefringent crystal. We idealize the crystal as free of any defects or impurities that could affect the light to deviate from its expected behavior. In the real world, all materials have some degree of imperfections that can cause the light passing through them to scatter, diffract, or absorb in unexpected ways. Essentially, the o-ray is a symmetrical ray (spherical wavefronts) oscillating perpendicular to the optic axis (approximate axis of symmetry) and experiences almost the same refractive index. If the z-axis is the direction of incident ray, we may assign the x-axis such that the plane of polarization of o-ray, xz-plane, is perpendicular to the plane that contains the incident ray and optic axis. On the contrary, we may assign the y-axis such that the plane of polarization of e-ray, yz-plane, contains the incident ray and optic axis.

 

“Anomalous refraction comes about when the optic axis, the long axis of our asymmetric molecules, is not parallel to the surface of the crystal (Feynman et al., 1963, p. 33–9).”

 

It is not generally true that anomalous refraction definitely comes about when the optic axis is not parallel to the surface of the crystal (during normal incidence). Note that the condition “perpendicular to the surface of the crystal” is a special case of “not parallel to the surface of the crystal.” That is, we cannot observe anomalous refraction when the optic axis is perpendicular to the surface of the crystal. On the other hand, anomalous refraction is also observable when the optic axis is parallel to the surface of the crystal (for incoming light) or lies in the plane of the paper (as stated in Fig. 33-7). To be precise, anomalous refraction cannot be observed when the incident ray is parallel to the optic axis of the crystal. This is because the refractive index of the crystal varies with the angle between the incident ray and optic axis.

 

2. Extraordinary ray:

“The envelope of all these elliptical waves is the wavefront which proceeds through the crystal in the direction shown. Again, at the back surface the beam will be deflected just as it was at the front surface, so that the light emerges parallel to the incident beam, but displaced from it. Clearly, this beam does not follow Snell’s law, but goes in an extraordinary direction. It is therefore called the extraordinary ray (Feynman et al., 1963, p. 33–9).”

 

Feynman’s explanation of extraordinary ray is potentially misleading because it is not generally true that v_∥ < v_⊥ in a birefringent crystal. However, we can also define e-ray ray from three perspectives, optic axis, speed of light, and wavefront: (1) optic axis: the e-ray propagates along a direction whereby its electric fields (plane of polarization) may oscillate parallel or perpendicular to the optic axis; (2) speed of light: the speed of o-ray (v_⊥) is greater than the speed of e-ray (v_⊥ ³ v ³ v_∥) in all directions except along the optic axis of a positive uniaxial crystal; in other words, v_∥ > v_⊥ in a negative uniaxial crystal (See Fig. 2); (3) wavefront: the wavefronts are ellipsoidal because the speed of e-ray and refractive index vary with angle within the crystal. Thus, the e-ray does not strictly obey Snell’s law of refraction.

Fig. 2

 

“Clearly, this beam does not follow Snell’s law, but goes in an extraordinary direction (Feynman et al., 1963, p. 33–9).”

 

There were some laughters when Feynman explained that e-ray does not follow Snell’s law, but it goes in an extraordinary direction. One may add that the e-ray does not obey Snell’s law even during normal incidence (q = 0). Furthermore, the e-ray feels a different refractive index due to the crystal’s anisotropic nature. Some may suggest that Snell’s law still holds for both the o-ray and e-ray, i.e., the difference being that the refractive index is no longer a constant for the e-ray. We may justify an extension Snell’s law for the e-ray by taking into account the anisotropic properties of the crystal. The modified Snell’s Law can have an effective refractive index that is dependent upon the incident ray, its polarization, and the relationship with the optical axis (e.g., Lekner, 1991; Wu & Zhang, 2013).

 

There could be a table to compare an o-ray and e-ray as shown in the table below. Firstly, the o-ray obeys Snell’s law, but e-ray does not obey Snell’s law even at normal incidence. Secondly, the electric fields of o-ray are perpendicular to the plane containing the incident ray and optic axis, whereas the electric fields of e-ray are parallel to the plane containing the incident ray and optic axis. Thirdly, the wavefronts of o-ray are spherical and the wavefronts of e-ray are ellipsoidal. Fourthly, the refractive index for o-ray is constant n_o, but the refractive index for e-ray varies from n_o to n_e depending on the angle of incidence with respect to the optic axis. Fifthly, the speed of o-ray in all directions remains constant as v_⊥ (c/n_o), but the speed of e-ray varies from v_⊥ to v_∥ (c/n_e). The nature of o-ray and e-ray are the same except the optic axis is oriented differently with respect to the electric fields of the two perpendicularly polarized light.

	o-ray	e-ray
Snell’s law	obeys Snell’s law	does not obey Snell’s law
Electric fields	perpendicular to the plane containing the incident ray and optic axis	parallel to the plane containing the incident ray and optic axis
Wavefronts	spherical	ellipsoidal
Refractive index	constant no	varies from no to ne
Speed of light	constant v⊥ (c/no)	varies from v⊥ to v∥ (c/ne)

 

3. Circularly polarized light:

“We see therefore that a beam of right circularly polarized light containing a total energy E carries an angular momentum (with vector directed along the direction of propagation) E/ω. For when this beam is absorbed that angular momentum is delivered to the absorber. Left-hand circular light carries angular momentum of the opposite sign, −E/ω (Feynman et al., 1963, p. 33–10).”

 

Feynman ended this lecture by discussing the o-ray and e-ray, but the inclusion of the concept of angular momentum of circularly polarized light may seem to be a misfit. However, he did not clarify that the spin angular momentum of a photon may be either +ħ or -ħ in which the signs mean left or right-handedness respectively (using E = ħω and the convention for ±signs in Optics). Interestingly, photons can have half-integer values of reduced Planck’s constant (ħ) when they are confined to two dimensions due to their interactions with one another in certain materials. This is related to the term anyons for excitations that satisfy any statistics interpolating bosons and fermions (Wilczek, 1982). The light’s fractional angular momentum can now be demonstrated experimentally by shining a laser beam through a biaxial crystal (Ballantine, Donegan, & Eastham, 2016).

 

Review Questions:

1. How would you define an ordinary ray?

2. Would you consider an extraordinary ray to be a misnomer?

3. Would you explain that a beam of circularly polarized light containing a total energy E carries an angular momentum E/ω that is equal to the reduced Planck’s constant?

 

The moral of the lesson: an ordinary ray is a symmetrical (spherical wavefronts) light ray and an extraordinary ray is an asymmetrical (ellipsoidal wavefronts) light ray because the optic axis is oriented differently with respect to the electric fields of the two perpendicularly polarized light.

 

References:

1. 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.

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. Lekner, J. (1991). Reflection and refraction by uniaxial crystals. Journal of Physics: Condensed Matter, 3(32), 6121.

4. Wilczek, F. (1982). Quantum mechanics of fractional-spin particles. Physical Review Letters, 49(14), 957.

5. Wu, J. F., & Zhang, Y. T. (2013). The relation between the propagation of extraordinary ray and the optical axis in the uniaxial crystal. Optik - International Journal for Light and Electron Optics, 124(17), 2667-2669.

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.