(Linearly polarized / Elliptically polarized / Unpolarized)
In this
section, Feynman discusses linearly polarized light, elliptically polarized
light, and unpolarized light.
1. Linearly polarized:
“Light is linearly
polarized (sometimes called plane polarized) when the electric field
oscillates on a straight line; Fig. 33–1 illustrates linear
polarization (Feynman et
al., 1963, p. 33–2).”
Feynman explains that light is linearly polarized (or
plane polarized) provided its electric field oscillates on a straight line. Historically,
Étienne-Louis Malus (1811) defines plane of polarization as the plane that contains
the direction of propagation and the magnetic field. In the Audio Recordings* [9 min: 00 sec] of this lecture, Feynman says: “historically, they did not know which field
is the right one,” i.e., the term plane
of polarization is confusing
due to the older literature. Thus, some physicists define plane of vibration
using the orientation of electric field because most materials interact more
strongly with electric fields as compared to magnetic fields. The plane of vibration
is not definitely constant at any point, but if it rotates uniformly,
the light may be described as circularly (or helically) polarized.
*The Feynman Lectures Audio Collection: https://www.feynmanlectures.caltech.edu/flptapes.html
“The motion in a straight
line is a particular case corresponding to a phase difference of zero (or an
integral multiple of π); motion in a circle corresponds to equal amplitudes
with a phase difference of 90o (or any odd integral
multiple of π/2) (Feynman et al., 1963, p. 33–1).”
We can
define polarized light using E = (Acos
qx)ei(kz
- wt) x + (Asin
qy)ei(kz
- wt) y. According
to Feynman, linearly polarized
light is a special case corresponding to a phase difference Dq = qy − qx = nπ, n = 0, 1, 2, 3…; circularly polarized light corresponds to equal
amplitudes with a phase difference of π/2, 3π/2, or equivalent. In addition, any linearly
polarized light can be formed from two different circularly polarized
lights because the “left-handed rotation” component can be canceled by the
“right-handed rotation” component. In the Audio Recordings [11 min: 00 sec], Feynman says: “suppose we have a left-hand circular and right-hand
circular… adding two circular polarized lights in phase and equal
amplitude, we get linearly polarized light.” This
explanation is edited and shifted to section 33-5: “any linear polarization can be made up by
superposing suitable amounts at suitable phases of right and left circular polarizations...
(Feynman et al., 1963).”
“The real
and imaginary components of a complex electric field vector are only a
mathematical convenience and have no physical significance (Feynman et
al., 1963, p. 33–2).”
The above sentence is potentially misleading on the
real component of a complex electric field. In the Audio Recordings [3 min: 10 sec],
Feynman says: “we have
been using diagrams in which the real part of complex number to represent the
electric fields and the imaginary part in this complex number diagram is merely
a construction for the analysis of the electric fields... The diagrams on the
board today are not complex number diagrams, but the real part of the electric
fields in the x and y direction.” Specifically, the real component of a complex
electric field does have physical significance because it can be measured. The
complex electric field can also be represented by a column or row vector. In a lecture on QED,
Feynman (1985) explains that a complex
number is equivalent to an imaginary stopwatch hand.
2. Elliptically
polarized:
“We have considered linearly, circularly, and elliptically
polarized light, which covers everything except for the case of unpolarized
light (Feynman et
al., 1963, p. 33–2).”
Feynman distinguishes three kinds of polarized
light: linearly polarized, circularly polarized, and elliptically polarized. Alternatively,
Jackson (1999) defines linearly polarized light as Ex and Ey have the same phase, and q = tan-1 Ey/Ex; if Ex and Ey have different phases,
it is described as an elliptically polarized light. However, some consider linearly
and circularly polarized light as special cases of elliptically polarized
light. This is because these three kinds of polarized light are based on the
same formula E = (Acos qx)ei(kz
- wt) x + (Asin
qy)ei(kz
- wt) y, but some include the
conditions: qy-qx = 0o, 180o means linear
polarization and q = 90o, 270o means circular
polarization. In a sense, some prefer to unify linearly, circularly, and
elliptically polarized into a single category, however, it is convenient to
describe how a linearly polarized light becomes circularly or elliptically
polarized light through a waveplate.
Some
may define an elliptically polarized light as an electromagnetic wave in which
the tip of its electric field vector traces an elliptical locus in space. Additionally,
elliptical
polarization corresponds to the case where two components
of the electric field are not equal and differ in phase by an angle (e.g., qy - qx.¹ 0o, 90o, 180o, 270o). Elliptically polarized light can be produced by
combining two linearly polarized lights having a phase difference other than 90o or 270o. One may
distinguish an elliptically polarized light as right-hand or left-hand just
like circularly polarized light. In the
Audio Recordings [10 min: 05 sec], Feynman says: “This is
right-hand circular, it is the way a right-hand screw would go as light comes
out. One of those nonsensical, difficult matter of notations that has nothing
to do with physics.”
3. Unpolarized
light:
“If the polarization changes
more rapidly than we can detect it, then we call the light unpolarized, because
all the effects of the polarization average out. None of the interference
effects of polarization would show up with unpolarized light. But as we see
from the definition, light is unpolarized only if we are unable to find
out whether the light is polarized or not (Feynman et al., 1963, p. 33–2).”
Feynman explains that
light is unpolarized because of our inability to find out
whether it is polarized or not. Thus, unpolarized light is a misnomer, but it means that the instantaneous
direction of polarization can vary rapidly in time between 0 and 2π. One may
prefer the term randomly polarized, but the so-called unpolarized light can be
represented in terms of two arbitrary, orthogonal, linearly polarized light waves
of equal amplitudes in which the relative phase difference varies rapidly and
randomly. However, light is neither completely
polarized nor completely randomly polarized, but it is somewhere in between. We
may describe light as partially polarized because it is related to the
superposition of certain amounts of completely polarized and randomly polarized
light.
Note: In the Audio Recordings [14 min: 05 sec] of this lecture, Feynman says: “unpolarized
light is only an approximate idea.”
“Remember
that one atom emits during 10−8 sec, and if one atom emits a certain
polarization, and then another atom emits light with a different polarization, the
polarizations will change every 10−8 sec (Feynman et
al., 1963, p. 33–2).”
Feynman could have clarified why the polarizations will change every 10−8 sec. Firstly, we do not completely understand what really occurs during the atom-transition interval of 10-8 s. We may use the term wave packet (wave pulse) instead of polarization or infinite wave train due to the very short duration of light emission. To be more precise, each excited atom emits a polarized wave packet and then transits to a lower energy state or ground state within a duration of the order of 10-8 s to 10-9 s. Furthermore, a light source consists of a very large number of randomly oriented atoms that are in random motion. Thus, new randomly polarized wave packets are continuously being emitted, and the sum of the so-called polarization changes in a completely unpredictable manner (instead of changing every 10−8 sec).
Review Questions:
1. How would you define linearly
polarized light?
2.
Would you consider linearly polarized light to be a particular case of
elliptically polarized light?
3. How would you explain the polarizations will change every 10−8
sec?
The moral of the lesson: the term linearly
polarized light may be used if Ex and Ey are
in phase, the term elliptically polarized light may be used if Ex and Ey are out of phase, but the term unpolarized could be replaced by randomly
polarized.
References:
1. Feynman, R. P. (1985). QED: The strange theory of light and matter.
Princeton: Princeton University Press.
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. Jackson, J. D. (1999). Classical Electrodynamics (3rd ed.). John Wiley & Sons, New York.
4. Malus, E. L. (1811). Mémoire sur de nouveaux phénomenes d’optique. Journal de physique, 72, 393-398.
