Section 33–3 Birefringence

(Birefringent crystals / Birefringent waveplate / Induced birefringence)

 

In this section, Feynman discusses birefringent crystals, birefringent waveplate, and induced birefringence. To be consistent, it could also be titled “polarization by birefringence” or “polarization by absorption” because the previous section is about “polarization by scattering of light” and the next section is related to “polarization by reflection.”

 

1. Birefringent crystals:

“Let us call the direction of the axes of the molecules the optic axis. When the polarization is in the direction of the optic axis the index of refraction is different than it would be if the direction of polarization were at right angles to it (Feynman et al., 1963, p. 33–3).”

 

Feynman defines optic axis as the direction of the axes of the molecules whereby the refractive index is different than it would be if the direction of polarization were at right angles to it. However, a more precise definition of the optic axis of a birefringent crystal should include atomic arrangement, refractive index, and velocities of light. Firstly, the optic axis is the axis of symmetry with respect to the arrangement of atoms, i.e., it lies in a plane of symmetry for crystals with certain symmetries. Secondly, light moving through this axis would feel the same refractive index (or binding force in x and y direction) independent of the polarization direction. Thirdly, the velocities of ordinary and extraordinary rays are different in all directions of light propagation except in the optic axis. In other words, the optic axis is not a line, but a direction whereby light can pass through the crystal without birefringence (double refraction).

 

“Such a substance is called birefringent. It has two refrangibilities, i.e., two indexes of refraction, depending on the direction of the polarization inside the substance (Feynman et al., 1963, p. 33–3).”

 

Feynman explains that a birefringent substance has two refractive indexes (or indices) that depend on the polarization direction inside the substance. One may clarify that birefringence is a phenomenon of two refracted light rays in an optically anisotropic medium where the interaction of light (or refractive index) depends on the polarization state and direction of propagation of light. In essence, it is a splitting of incident light into two parallel rays of perpendicular polarization through a uniaxial (single optic axis) crystal provided the light is not parallel to the optic axis. Specifically, hexagonal, and tetragonal crystals have two refractive indices (uniaxial), but orthorhombic, monoclinic, and triclinic crystals have three refractive indices (biaxial) (See Fig. 1). Mathematically, birefringence is a measure of the difference between two refractive indices for two different directions of light propagation.

Fig. 1

Source: https://viva.pressbooks.pub/petrology/chapter/2-8-interference-figures-part-2/

Note: In the UK, there are seven categories of crystal systems: cubic, tetragonal, hexagonal, trigonal, orthorhombic, monoclinic, and triclinic. In the US, there are six categories because the trigonal system is considered to be a subset of the hexagonal system.

 

“Such a substance is called birefringent. It has two refrangibilities, i.e., two indexes of refraction, depending on the direction of the polarization inside the substance (Feynman et al., 1963, p. 33–3).”

 

Instead of using the phrase direction of the polarization, we may state the optic axis inside the birefringent crystal. Furthermore, natural light has two linear polarizations that may result in ordinary and extraordinary rays. For example, we can cut and polish a calcite crystal so that its optic axis is normal to its front and back surfaces (Hecht, 2002). If the incident light is normal to the front surface of the crystal, it will not be split into ordinary and extraordinary rays. If we cut and polish the calcite crystal so that its optic axis is in the plane of the front surface, we can observe ordinary and extraordinary rays because the light would feel two different refractive indices within the crystal. Feynman discusses ordinary and extraordinary rays at the end of the lecture, but some physics teachers may not agree with this teaching sequence.

 

2. Birefringent waveplate:

“Since the x- and y-polarizations travel with different velocities, their phases change at a different rate as the light passes through the substance... If the thickness of the plate is just right to introduce a 90^o phase shift between the x- and y-polarizations, as in Fig. 33–2(c), the light will come out circularly polarized. Such a thickness is called a quarter-wave plate, because it introduces a quarter-cycle phase difference between the x- and the y-polarizations (Feynman et al., 1963, p. 33–3).”

 

According to Feynman, a quarter-wave plate is a plate whose thickness is just right to introduce a 90^o phase shift between the x- and y-polarizations. In general, waveplates are made of birefringent materials that have a difference in refractive index between two orthogonal principal axes, commonly known as fast axis and slow axis. The fast axis of the waveplate has a lower refractive index, resulting in a faster velocity for polarized light moving in this direction. Conversely, the slow axis has a higher refractive index, resulting in a slower velocity for light moving parallel to this axis. In a positive uniaxial crystal, the fast axis (ordinary ray) is perpendicular to the optic axis, while the slow axis (extraordinary ray) coincides with the optic axis. However, this section is potentially confusing because of many terms related to the axis: molecular axis, axis of the polaroid, axis of linearly polarized light, beam axis, pass axes of the polaroid sheets, and optic axes of the cellophane.

“Polaroid, which we will discuss later in more detail, has the useful property that it transmits light that is linearly polarized parallel to the axis of the polaroid with very little absorption, but light polarized in a direction perpendicular to the axis of the polaroid is strongly absorbed (Feynman et al., 1963, p. 33–4).”

 

Polarization by absorption: A polaroid transmits polarized light that is parallel to the polaroid’s axis and absorbs polarized light that is perpendicular to the polaroid’s axis. The absorption of light is dependent on the molecular structure (e.g., needle-like crystals or long molecular chains) of the polarizer because the electrons oscillate more easily in the longer parts of the molecules. The oscillation of electrons along the long molecular chains may result in collisions with other molecules and re-emission of light in all directions. In the Audio Recordings* [23 min: 00 sec] of this lecture, Feynman says: “Different colors have different indices,” but this sentence could be edited instead of omitted. For example, different light frequencies have different refractive indices and absorption bands depending on the optic axis and direction of light propagation (see Fig. 2 below).

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

Fig. 2 (Hecht, 2002)

3. Induced birefringence:

“So when a stress is applied to certain plastics, they become birefringent, and one can see the effects of the birefringence by passing polarized light through the plastic (Feynman et al., 1963, p. 33–5).”

 

According to Feynman, we can see the effects of the birefringence by passing polarized light through certain plastics when a stress is applied. We may explain stress-induced birefringence (photoelastic effect) from the perspective of random polymer chains, optic axis, and operating procedure. Firstly, the applied stress (tension or compression) aligns the random polymer chains and inherent uniaxial anisotropy (related to refractive indices) of the chain structure. Secondly, the optic axis is in the direction of the applied stress and the induced birefringence is proportional to the stress. Last but not least, the optic axis of a stressed sample should be oriented at 45^o with respect to the polarizer and analyzer such that the maximum brightness for the sample can be observed.

 

“So we have an electrical switch for light, which is called a Kerr cell. This effect, that an electric field can produce birefringence in certain liquids, is called the Kerr effect (Feynman et al., 1963, p. 33–5).”

 

Feynman relates the Kerr effect to an electric field that can induce birefringence in certain liquids. Feynman’s explanation could be elaborated from the perspective of randomly oriented molecules, optic axis, and operating procedure. Simply put, light is passed through a liquid or glass (randomly arranged and oriented molecules) that is contained in a Kerr cell. Kerr effect means that an isotropic material can behave like a uniaxial crystal when a voltage is applied such that an optical axis is made parallel to the electric field. In the Audio Recordings* [26 min: 30 sec] of this lecture, Feynman says: “I put the birefringence at 45 degrees and I change it and so on and the light comes through,” but this sentence could be edited and illustrated by the diagram as shown below (see Fig. 3). However, induced birefringence is a “symmetry breaking” phenomenon that may also be thermally induced, flow-induced, or laser-induced.

 

Fig. 3

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

 

Review Questions:

1. How would you define the optic axis of a birefringent crystal?

2. Would you describe a birefringent waveplate using terms such as molecular axis, axis of the polaroid, beam axis, pass axes, and optic axes (or fast axis and slow axis)?

3. How would you explain the optic axis of a sample should be oriented at 45^o with respect to the polarizer and analyzer such that induced birefringence can be observed?

 

The moral of the lesson: a birefringent substance, natural or induced, is due to the presence of an optic axis (axis of symmetry), whereby the refractive index is dependent on the direction of light propagation.

 

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