Download Chapter 33 The Nature And Propagation Of Light

Chapter 33
The Nature and Propagation of Light
by C.-R. Hu
Light is a transverse wave of the electromagnetic field. In 1873, James C.
Maxwell predicted it from the Maxwell equations. The speed of all electromagnetic waves in vacuum are the same, and is equal to:
c = 2.99792458×108 m/s ≈ 3×108 m/s .
Electromagnetic waves become visible to a human eye if its wavelength lies
from about 400 nm (violet limit) to about 700 nm (red limit). The corresponding frequency range is from about 430 THz (red limit) to about 750
THz (violet limit). (THz stands for terahertz, or 1012 Hertz.)
When the length scales of all objects, gaps, and holes involved are much larger than the light wavelength, one can treat light as made of light rays
perpendicular to the wave fronts (where the crests and troughs of the wave
are located). Light rays travel in straight lines in a uniform medium. They
change direction only at the interface of two different optical media of
different indices of refraction (n), giving rise to reflection and refraction.
Light rays will be curved if n changes continuously. It can be viewed as
many mini-refractions, with little reflection in each step.
1. The law of reflection
surface normal
incident beam
reflected beam
Plane of incidence
θi θr
reflecting surface
a.
b.
c.
The plane of incidence is a plane containing the incident
beam and the surface normal.
The reflected beam lies in the plane of incidence and is on the
other side of the surface normal as the incident beam.
θi = θr
where
θ i is the angle of incidence, or the incident angle;
θ r is the angle of reflection.
2. Index of refraction
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The speed of light in an optical material, v = 1/
εµ ,
is slower than
that in vacuum, c =1/ ε0 µ0 , by the factor n = KK m ,where
K ≡ ε / ε 0 is called the dielectric constant, and K m ≡ µ / µ 0 is called
the relative permeability. n is called the index of refraction of an
optical material, so that
v= c/n
(speed of light in an optical material).
Most materials are non-magnetic, with K m = 1. Also, most materials
have K > 1. Thus most materials have n > 1 , and v < c .
3. The law of refraction, or Snell’s law
incident beam
surface normal
na
nb
θa
reflected beam
flat interface
θb
refracted beam
When a light beam, initially propagating in a medium with an index of
refraction n 1 , is incident on a flat interface with a different medium of an
index of refraction n 2 , then the light beam will be partially reflected in
accordance with the law of reflection, and partially transmitted into the
second medium with a change of direction. This is called refraction.
The law of refraction, or Snell’s law, is made of two statements:
Statement 1: The incident beam, the reflected beam, the refracted beam,
and the surface normal, all lie in one plane, which is the plane of incidence.
Statement 2: Let θ a be the angle of incidence (i.e., the angle between the
incident beam and the surface normal), and let θ b be the angle of
refraction (i.e., the angle between the refracted beam and the surface
normal), then these two angles are related by the equation:
n a sinθ a = n b sinθ b
(the law of refraction, or Snell’s law).
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4. Consequences of the Snell’s law
(4a) When light crosses an interface between two different optical media,
its frequency does not change, but its wavelength changes, since
λ a = v a/f = c/na f,
and
λ b = v b /f = c/nb f.
Clearly, λ a = λ / n a , and λ b = λ / n b , where
of light in vacuum with the same frequency.
λ = c / f is the wavelength
(4b) If n a > n b , that is, if light is incident from a denser medium, then we
can satisfy sin θ a = n b / n a , which implies sin θ b = 1, or θ b = 90° . This
particular θ a is called the critical angle for total reflection. That is:
sin θ crit = n b / n a
(the critical angle for total reflection).
Clearly, this critical angle exists only on the side with the larger n, so that
n a > n b, because sin θ crit is always less than unity.
For θ a > θ crit , refraction is no longer possible, so the incident beam
becomes 100% reflected. This is what is meant by total reflection.
Diamond has the largest n (= 2.417). So it can be cut easily into a shape
to give a great deal of total reflection on its back side. This explains
diamond jewelry’s brilliance. Other applications of total reflection include
totally-reflecting prisms inside binoculars, and optic fibers.
(4c) The index of refraction n varies with frequency of light slightly. This
phenomenon is called dispersion. This fact can be used to separate different frequency components of light using a prism. It also explains the origin
of rainbows (water droplets separating the different frequency components
of sunlight).
sun light
Mixture of
frequencies
red
n(f )
prism
violet
violet
red
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water
droplet
Sun light and lamp light are examples of “white light”, which refers to light
which has practically all visible and lower frequency components, and
usually also a range of higher frequency components.
Optical fiber uses total internal reflection to make light follow the path of a
curved fiber.
5. Polarization
Light is a transverse wave. That is, the electric and magnetic fields in a
light wave are both perpendicular to the direction of the light wave. It can
then have two mutually perpendicular choices:
G
G
E
E
or
These are the two possible linear (or planar) polarizations of an electromagnetic (or light) wave. (The first direction is arbitrarily chosen, as long as
it is perpendicular to the propagation direction of the wave. The second
direction must be perpendicular to the first and to the propagation direction
of the wave.) A light wave linearly-polarized in any other direction can be
decomposed into these two linear polarizations, in the same wave that a vector is decomposed into x- and y- components:
This is because an electric field is a vector.
y
G
A Polaroid sheet has an intrinsic direction, It allows
E
only light polarized in this direction to get through.
Light from ordinary sources like a fluorescent or
θ
x
incandescent light bulb is unpolarized, meaning
that it is an incoherent mixture of both polarizations
of equal intensities. A Polaroid sheet is a polarizer, in the sense that unpolarized light, after going through a Polaroid sheet, becomes polarized in
the direction determined by the intrinsic direction of the Polaroid sheet, and
its intensity is cut down by a factor of 2 (because the Polaroid sheet has
blocked the other polarization). (The blocking is not perfect in most commercial Polaroid sheets.) They are called polarization filters.
If the incoming light is already linearly polarized in the direction Ê1 (a unit
G
vector in the direction of E1 ), and it is sent through a Polaroid sheet with an
intrinsic direction Ê2 (another unit vector), Then the electric field of the
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incoming light in the direction Ê1 must be projected into the direction Ê2 ,
leading to a factor Ê1 · Ê2 = cos θ, where θ is the angle between the two unit
vectors. As light intensity is proportional to the square of the electric field,
the light coming out of the Polaroid sheet will be cut down by a factor cos2 θ
from the intensity of the incoming light. Hence the formula
I = I 0 cos 2θ .
(Malus’s law)
If the out-coming light is sent through another Polaroid sheet with an intrinsic direction Ê3 , Then the electric field will be cut down by another factor
Ê2 · Ê3 = cos θ ', because the electric field is projected again. The intensity is
then cut down by another factor of cos 2 θ '. The direction of polarization of
the out-coming light is always determined by the intrinsic direction of the
last Polaroid sheet it has gone through.
Light reflected from a surface will in general be partially polarized in the
direction perpendicular to the incident plane. (For incident angle θ i = 0, the
reflected light is unpolarized. For the special incident angle θ p obeying
tan θ p = n2 / n1,
(the Brewster’s law)
the reflected light is 100% polarized in the direction perpendicular to the
incident plane. This special incident angle is called the Brewster angle, and
has the notation θ p, since it is also called the polarization angle. It can be
easily shown that, at this angle, the refracted light is perpendicular to the
reflected light. (Simply use the above equation and the Snell's law.)
Applications:
1. Polarized sun glasses to eliminate reflected glare (because reflected light
is mostly polarized with electric field perpendicular to the incident plane);
2. liquid crystal display (LCD), which is used as, for example, computer
screen.
So far the electric field vector oscillates in a fixed direction perpendicular to
the light direction. That is called linear or planar polarization. But two
electromagnetic waves can occupy the same location without interferring
each other. (This is called the principle of superposition.) So one can
consider two mutually-perpendicular linearly polarized light waves occupying the same location. The electric fields of these two waves should then be
added up vectorially (i.e., as vectors). Assuming that the wavelengths and
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directions of propagation of these two waves are the same, the result of the
vector sum will depend on their amplitudes and the relative phase of the two
waves.
If the two waves have the same amplitude and are in phase, the result is a
wave linearly polarized in the direction which is at 45 D with the original two.
G
E1
G
E2
But if the two waves are 90 D out of phase, and still have the same amplitudes, then they add up to a circularly-polarized light wave:
G
E1
G
E2
That is, it you look against the direction of the wave, the electric field vector
either rotates clockwise, called right circularly polarized wave, or counterclockwise, called left circularly polarized wave.
If the amplitudes of the two waves are not equal, or if the relative phase of
the two waves is neither zero nor 90 D , then the tip of the electric field vector
will draw an ellipse rather than a circle, and is called an elliptically polarized wave.
Of course, the ellipse can also appear as:
or
, etc.,
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and the electric field vector can again rotate in either directions (with its
length changing at the same time).
In all cases discussed above, only the electric field vector has been drawn.
The magnetic field vector is always present, and is always related to the
direction of propagation and the electric field vector by the right hand rule:
G
E
G
B
the direction of propagation
Unpolarized light, after scattering by an atom or a dust particle, also becomes at least partially polarized. If the scattering angle is 90 D , then the
scattered light is purely linearly polarized, in the direction perpendicular to
the scattering plane (the plane containing the directions of the incoming light
and the scattered light):
the scattering plane
because the other direction of the electric field in the incoming wave is now
along the direction of the scattered wave, and is unacceptable, because the
light wave is a transverse wave.
6. Huygen’s principle
This principle is a general property of all
waves, and can be easily demonstrated
with water or sound wave. Imaging that a
planar sound wave with straight wave fronts
is generated on the left of a long straight
corridor. The wave propagates to the right
but is stopped by a wall with a small hole
at its center. Then to the right of the wall
you will have a spherical sound wave
emanating from the hole in the wall. This
is the essence of the Huygen’s principle.
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wave
front
at a
earlier
time
wave
front
at a
later
time
Namely, any point of a wave front is the source of a spherical wave.
If there is no wall present, then indeed every point of a wave front is the
source of a spherical wave, and all these spherical waves emanated from all
points of the same wave front define a new wave front by their envelop.
If the radii of all of these spherical waves are equal to the speed of the wave
at the respective locations times ∆t, then the new wave front obtained from
the envelop is the wave front at time ∆t later. When a plane EM wave
reaches an interface of two different optical media at an angle θ1, one end of
a wave front touches the interface first.
Since the speed of light is different in the
two media (c/n1 vs. c/n2), the radii of
θ1
spherical waves drawn on the two ends
of that wave front are therefore different.
n1
(In the figure shown, n2 > n1, so the spheθ1
rical wave drawn on the left is smaller,
θ2
since it is in the lower medium in which
n2
the speed of light c/n2 is smaller.) As a
result the direction of light becomes refθ2
racted when light crosses the interface between the two media. From the figure it
(This figure assumes n2 > n1 .)
can be seen that sin θ1 = (c/n1) ∆t /d,
where d is the distance between the two ends of the wave front, and
sin θ2 = (c/n2) ∆t /d. Thus one obtains the Snell’s law:
n1 sin θ1 = n2 sin θ2 for refraction. (Both are equal to c ∆t / d .)
If n varies in the direction perpendicular to the direction of propagation of
the light wave, then the successive wave fronts are not parallel to each other,
leading to curved light rays and path:
n decreases
because (c / n) ∆t is longer for smaller n. This is how mirage is formed.
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