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Fig. 33-7. The upper diagram shows the path of the ordinary ray
through a doubly refracting crystal. The extraordinary ray Is shown In
the lower diagram. The optic axIs lies in the plane of the paper.
Anomalous refraction is a particular case of the same birefringence that we
considered earlier. Anomalous refraction comes about when the optic axis, the
long axis of our asymmetric molecules, is no parallel to the surface of the crystal.
In Eig. 33-7 are drawn two pieces of birefringent material, with the optic axis as
shown. In the upper figure, the inecident beam falling on the material is linearly
polarized in a direction perpendicular to the optic axis of the material. When
this beam strikes the surface of the material, each point on the surface acts as a
source oŸ a wave which travels into the crystal with velocity ø¡, 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
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Just the ordinary behavior we would expect, and this ray ¡s called the ordinar
In the lower fñgure the linearly polarized light falling on the crystal has its
direction of polarization turned through 907, so that the optic axis lies in the
plane of polarization. When we now consider the little waves originating at any
point on the surface of the crystal, we see that they do not spread out as spherical
waves. Light travelling along the optic axis travels with velocity ø¡ because
the polarization is perpendicular to the optic axis, whereas the light travelling
perpendicular to the optic axis travels with velocity 0i because the polarizatlon
is parallel to the optic axis. In a birefringent material 0 z# 0¡, and in the figure
0Ịị < 0L. Á more complete analysis will show that the waves spread out on the
surface of an ellipsoid, with the optic axis as major axis of the ellipsoid. “The
envelope of all these elliptical waves 1s the wavefront which proceeds through
the crystal in the đirection shown. Again, at the back surface the beam will be
defected 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
Snells law, but goes in an extraordinary direction. It ¡is therefore called the
cztraordinar4J ray,
'When an unpolarized beam strikes an anomalously refracting crystal, i% is
separated into an ordinary ray, which travels straight through in the normal
mamner, and an extraordinary ray which is displaced as it passes through the
crystal. 'Phese two emergent rays are linearly polarized at right angles to each
other. 'Phat this is true can be readily demonstrated with a sheet of polaroid
to analyze the polarization of the emergent rays. We can also demonstrate that
our interpretation of this phenomenon 1s correct by sending linearly polarized
light into the crystal. By properly orienting the direction of polarization of the
incident beam, we can make this light go straight through without splitting, or
we can make it go through without splitting but with a displacement.
W© have represented all the various polarization cases in Figs. 33-I and 33-2
as superpositions of two special polarization cases, namely + and ø in various
amounts and phases. Other pairs could equally well have been used. Polarization
along any two perpendicular axes 4, ˆ inclined to #z and would serve as well [for
example, any polarization can be made up of superpositions of cases (a) and (e)
of Eig. 33-2]. It is interesting, however, that this idea can be extended to other
cases also. For example, any neør polarization can be made up by superposing
suitable amounts at suitable phases of right and left c#cular polarizations [cases
(c) and (g) of Fig. 33-2], since two equal vectors rotating in opposite directions
--- Trang 586 ---
Fig. 33-8. Two oppositely rotating vectors of equal amplitude add to
produce a vector In a fixed direction, but with an oscillating amplitude.
add to give a single vector oscillating in a straight line (Eig. 33-6). TỶ the phase
of one is shifted relative to the other, the line is inclined. 'Thus all the pictures
of Eig. 33-1 could be labeled “the superposition of equal amounts of right and
left circularly polarized light at various relative phases.” As the left slips behind
the right in phase, the direction of the linear polarization changes. 'Therefore
optically active materials are, in a sense, birefringent. Their properties can be
described by saying that they have diferent indexes for right- and left-hand
circularly polarized light. Superposition of right and left circularly polarized light
of diferent intensities produces elliptically polarized light.
Circularly polarized light has another interesting property—it carries øngulœr
momentum (about the direction of propagation). To illustrate this, suppose that
such light falls on an atom represented by a harmonic oscillator that can be
displaced equally wellin any direction in the plane ø. Then the z-displacement of
the electron will respond to the #„ component of the feld, while the -component
responds, equally, to the equal 2 component of the fñield but 90 behind in phase.
That is, the responding electron goes around in a circle, with angular velocity œ,
in response to the rotating electric field of the light (Fig. 33-9). Depending on
the damping characteristics of the response of the oscillator, the direction of the
displacement œ of the electron, and the direction of the force q¿#⁄ on it need not
be the same but they rotate around together. The # may have a component at
right angles to ø, so work is done on the system and a torque 7 is exerted. The
work done per second is 7w. Ôver a period of time 7' the energy absorbed is 7uT,,
while 77' is the angular momentum delivered to the matter absorbing the energy.
We see therefore that œ bewm oƒ right círcularlụ polarized light contaimimng a total
--- Trang 587 ---
Fig. 33-9. A charge moving In a circle in response to circularly polarized
light.
energu Ê carries an œnguÏar mmormnentum (uuith 0ector dárected œlong the đireclion
oƒ propagation) Ê/œ. 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, —Ê/œ.
--- Trang 588 ---
Miolqfitrsffc ifocés rrẻ Haclf(ffort
34-1 Moving sources
In the present chapter we shall describe a number of miscellaneous efects
in connection with radiation, and then we shall be finished with the classical
theory of light propagation. In our analysis of light, we have gone rather far and
Into considerable detail. "The only phenomena of any consequence associated
with electromagnetic radiation that we have not discussed is what happens If
radiowaves are contained in a box with reflecting walls, the size of the box
being comparable to a wavelength, or are transmitted down a long tube. The
phenomena of so-called cauify resonators and uaueguzdes we shal] discuss later;
we shall frst use another physical example—sound——=and then we shall return to
this subject. Except for this, the present chapter is our last consideration of the
classical theory of light.