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.