(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 90o 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 90o 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 45o
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 45o 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.
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