Chap. 3 Electromagnetic Theory and Light

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Nov 16, 2013 (3 years and 8 months ago)

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-

1

Chap. 3 Electromagnetic Theory and Light

Light possesses both wave
-
particle manifestations.


Classical electrodynamics based on Maxwell’s electromagnetic theory unalterably leads to
the picture of a continuous transfer of energy by way of
electromagnetic waves
.

Quantum electrodynamics describes electromagnetic interaction and the transport of
energy in terms of massless elementary “particles” known as
photons
, which are localized
quanta of energy.

One of the basic tenets of quantum mechanics is that both light and material objects each
display both wave
-
particle properties.

In physical optics light is treated as an electromagnetic wave.


3.1 Maxwell’s equations


The simplest statement of Maxwell’s equations governs the behavior of the electric
and magnetic fields in free space.


Maxwell’s equations are generalization of experimental results.

3
-

2






















A
A
c
A
c
A
S
d
B
S
d
E
S
d
t
E
l
d
B
S
d
t
B
l
d
E
(3.4)

0
)
3
.
3
(
,
0
)
2
.
3
(
,
)
1
.
3
(
,
0
0














where and are the permeability and the electric permittivity of free space,
respectively.


It should be noted that except for a multiplicative scalar, the electric and magnetic
fields appears in above equations with a remarkable symmetry. The mathematical
symmetry implies a good deal of physical symmetry.


Maxwell’s equations tell us that
a time
-
varying magnetic field generates an
electric field and a time
-
varying electric field generates a magnetic field
.


Maxwell’s equations above can be written in differential form by using following
two theorems from vector calculus.

0

0

3
-

3




(3.6)




theorem
Stocke

and
(3.5)




theorem
divergence

Gauss













A
C
A
V
S
d
F
l
d
F
dV
F
S
d
F







)
10
.
3
(
0
)
9
.
3
(
0
)
8
.
3
(
)
7
.
3
(
0
0

















B
E
t
E
B
t
B
E








By applying theorem (3.5) to Eqs. (3.1) and (3.2) and applying theorem (3.6) to Eqs
(3.3) and (3.4), we obtain the following differential equations

)
12
.
3
(
)
11
.
3
(
Z
E
y
E
x
E
E
E
E
E
z
y
x
k
j
i
E
z
y
x
z
y
x

























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-

4

The consequent equations for free space are in detail as follows:

)
16
.
3
(
,
0
)
15
.
3
(
,
0
)
(
,
)
14
.
3
(
)
(
,
)
(
,
)
(
,
)
13
.
3
(
)
(
,
)
(
,
0
0
0
0
0
0





































































z
B
y
B
x
B
z
E
y
E
x
E
iii
t
E
y
B
x
B
ii
t
E
x
B
z
B
i
t
E
z
B
y
B
iii
t
B
y
E
x
E
ii
t
B
x
E
z
E
i
t
B
z
E
y
E
z
y
x
z
y
x
z
x
y
y
z
x
x
y
z
z
x
y
y
z
x
x
y
z






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-

5

3.2 Electromagnetic waves


Maxwell’s equations for free space can be manipulated into the form of two vector
expressions:

Taking the curl of Eq. (3.7)

)
19
.
3
(
have

we
)
18
.
3
(
have

e
.
.
)
(
.
.
)
17
.
3
(
)
(
)
(
2
2
0
0
2
2
2
0
0
2
2
2
0
0
)
8
.
3
.(
2
)
9
.
3
.(
2
t
B
B
Similarly
t
E
E
W
t
E
S
H
R
E
E
E
S
H
L
B
t
E
Eq
Eq






















































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-

6

Each component of the electromagnetic field ( ) therefore
obeys the scalar differential wave equation

z
y
x
z
y
x
B
B
B
E
E
E
,
,
,
,
,
)
21
.
3
(
,
1
2
2
2
2
2
2
2
2
2
t
v
z
y
x















provided that

)
22
.
3
(
.
1
0
0



v
If we substitute the values of and into Eq. 3.22, the predicted speed of all
electromagnetic waves travelling in free space would then be c= 3 x 10
8

m/s. This
theoretical value was in remarkable agreement with the previously measured speed
of light.

0

0

The Laplacian, , operates on each component of and , so that the two vector
equations (3.18) and (3.19) actually represent a total of 6 scalar equations. One of
these expressions, in Cartesian coordinates, is

2

E

B

)
20
.
3
(
2
2
0
0
2
2
2
2
2
2
t
E
z
E
y
E
x
E
x
x
x
x













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7


The experimentally verified transverse character of light should be explained within the
context of the electromagnetic theory. To that end, consider the fairly simple case of a
plane wave propagating in the positive x
-
direction and write as . Eq. (3.15) is
reduced to

j

)
,
(


t
x
E
E
y

)
24
.
3
(
.
t
B
x
E
z
y






This implies that the time
-
dependent
B
-
field
can only have a component in the z
-
direction.
Clearly then, in free space, the plane
electromagnetic wave is indeed transverse, as
shown in Fig. 3.1. Now let’s write

)
,
(
t
x
E
E



)
23
.
3
(
0



x
E
x
)
25
.
3
(
],
)
/
(
cos[
)
,
(
0





c
x
t
E
t
x
E
y
y
For a progressive wave, the solution of (3.23) is E
x
=0. So the electric component
must be perpendicular to the propagation direction, x. Let’s orient the coordinate axes
so that the electric field is parallel to the y
-
axis: . From Eq. (3.13
III), it follows that

Fig. 3.1 Field configuration in a plane
harmonic electromagnetic wave.

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-

8


The associated magnetic field can be
found by directly integrating Eq. (3.25), that
is,


.
]
)
(
cos[
1
]
)
(
sin[
0
0
c
E
c
x
t
E
c
dt
c
x
t
c
E
dt
x
E
B
y
y
y
y
z



















(3.26)


Clearly, and have the same time

y
E
z
B
dependence, and are in phase at all points in space.
Moreover, and are mutually perpendicular, and their cross
-
product, ,
points in the propagation direction, as shown in Fig.3.2.


It should be noted that plane waves are not only solutions to Maxwell’s
equations. As we saw in the previous chapter, the differential wave equation allows
many solutions including spherical waves.

)
,
(
ˆ
t
x
E
j
E
y


)
,
(
ˆ
t
x
B
k
B
z


E

B

B
E



Fig. 3.2 Orthogonal harmonic E
-
field
and B_field.

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9

3.3 Energy


Energy density
, , which is the radiant energy per unit volume, is given by

)
27
.
3
(
2
0
E
u


u
From Eqs. (3.27) and (3.28), we have

)
28
.
3
(
.
1

2
0
B
u
or


)
31
.
3
(
.
1
0
B
E
S






It’s along the wave propagation direction. Its SI unit is watt per square meter ( ).

2
m
W
The amount of energy transported during a unit time and through a unit area
perpendicular to the transport direction is (suppose the wave travels through an area
of A and with a speed of c and with a time duration of )

(3.30)



EB
1



)


(

0






uc
A
t
A
t
c
u
S
t

)
29
.
3
(
1
0
EB
c
u


The corresponding vector is called
Poynting vector:

3
-

10

Where E
0

is the peak magnitude of E. Within a linear, homogeneous, isotropic
medium, the expression for the irradiance becomes

)
34
.
3
(
2
1
2
0
2
vE
E
v
I




For a point light source, its irradiance is proportional to . This is well
-
known
inverse
-
square law. Fig. 3.3 shows that a point source emits electromagnetic
waves uniformly in all directions. Let us assume that the energy of the waves is

2
1
r
)
33
.
3
(
2
1
2
0
0
2
0
cE
E
c
S
I






The time
-
averaged value of the magnitude of the Poynting vector, symbolized by


, is a measure of the significant quantity known as the
irradiance
, .

S
I
E and B are so closely coupled to each other that we need to deal with only one of
them. Using from Eq. 3.26, we can rewrite Eq. 3.30 as

c
E
B

)
32
.
3
(
1
2
0
2
0
cE
E
c
S




3
-

11

conserved as they spread from the source. Let
us also center an imaginary sphere of radius
on the source, as shown in Fig. 3.3. If the power
of the source is , the irradiance at the
sphere must then be

r
s
P
I
)
35
.
3
(
.
4
2
r
P
I
s


Fig. 3.3 A point source emits light
isotropically.

In Quantum theory, light possesses quantum energy








ev,

of
unit

in

is

)
36
.
3
(

/
12400
/



hc
h
is in unit of angstrom, h is Plank constant.

3
-

12

3.4 Electromagnetic
-
photon spectrum


Although all forms of electromagnetic radiation propagate with the same speed in
vacuum, they differ in frequency and wavelength. Fig. 3.4 plots the vast electromagnetic
spectrum. The frequency range for whole electromagnetic spectrum is from a few Hz to
10
22

Hz. The corresponding wavelength range is from many kilometers to 10
-
14

m.


Radiofrequency waves
: a few Hz to 10
9

Hz


Microwaves
: 10
9

Hz to about 3 x 10
11

Hz.

Infrared
: 3 x 10
11

Hz to 4 x 10
14

Hz.


The infrared (IR) is often subdivided into four regions: the near IR (780
-



3000 nm), the intermediate IR (3000
-
6000 nm), the far IR (6000
-
15,000 nm),


and the extreme IR (15,000 nm
-
1.0 mm).

Light
: 3.84 x 10
14

to 7.69 x 10
14

Hz.


An narrow band of electromagnetic waves could be seen by human eye. Color is


not a property of light itself but a manifestation of the electrochemical sensing


system
-
eye, nerves, and brain.

3
-

13

Fig. 3.4 Electromagnetic
-
photon spectrum.

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-

14

Table 3.1. Frequency and vacuum wavelength ranges for
various colors

Ultraviolet
: 8 x 10
14

Hz to 3.4 x 10
16

Hz.

X
-
rays
: 2.4 x 10
16

Hz to 5 x 10
19

Hz.

Gamma rays
: 5 x 10
19

Hz to 2.5 x 10
33

Hz.

3
-

15

590
600
610
620
630
640
650
660
670
0
20
40
60
80
100
120
GaInP
2


Emission intensity (a.u.)
Wavelength (nm)
Fig. 3.5 Spectra of sunlight and the light from
a tungsten lamp.

Fig. 3.6 Emission spectrum of GaInP
2

semiconductor under excitation of a He
-
Cd laser..

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-

16

3.5 Light in matter

3.5.1 Dispersion


In a homogeneous, isotropic dielectric, the phase velocity of light propagation becomes

)
37
.
3
(
.
1


v
The ratio of the speed of electromagnetic wave in vacuum to that in matter is known as
the
absolute index of refraction

and is given by

n
)
38
.
3
(
.
0
0





v
c
n
For most materials, is generally equal to 1. So, the expression of becomes

n
)
39
.
3
(
.
/
0



n
0
/


Actually, n is frequency
-
dependent, so called
dispersion
. When a dielectric is
subject to an applied electric field
E,

the internal charge distribution is distorted,
which generates electric dipole moment
p
=L
q
,
with
L

the position vector from the
negative charge
-
q to the positive charge q.
The dipole moment per unit volume is
called the electric polarization
P.
with

3
-

17

)
40
.
3
(
)
(
0
E
P






Fig. 3.8 shows the dipole formation and a oscillator
model for the vibration of electrons under an
E
-
field.
The negative electrons are fastened to a stationary
positive nucleus. The natural frequency of the spring
is with k and m being the spring constant
and the electron mass. The force (F
E
) exerted on an
electron of charge q by a harmonic wave E(t) with
frequency is

Fig. 3.8 (a) Distortion of the
electron cloud in response to an
applied
E
-
field. (b) The
mechanical oscillator model for an
isotropic medium

m
k
/
0



)
41
.
3
(
)
cos(
)
(
0
t
qE
t
qE
F
E



Newton’s second law provides the equation of
motion with the second term the restoring force.

)
43
.
3
(
)
(
)
(
/
)
(
)
cos(
)
(
)
42
.
3
(
)
cos(
2
2
0
2
2
2
0
0
0
t
E
m
q
t
x
have
we
t
x
t
x
Let
dt
x
d
m
x
m
t
qE










For medium with electron density N
the electric polarization P is

3
-

18

)
44
.
3
(
)
1
(
1
)
(
)
39
.
3
.(
)
(
/
)
(
/
)
(
)
40
.
3
.(
)
(
)
(
/
)
(
2
2
0
0
2
2
2
2
0
2
0
0
2
2
0
2





















m
Nq
n
Eq
From
m
N
q
t
E
t
p
Eq
from
t
mE
N
q
t
P
Fig. 3.9 Index of refraction versus
wavelength and frequency for several
important optical crystals.

Eq. (3.44) indicates dispersion.
When n>1; When
n<1. Fig 3.9 shows dispersion
of materials.

0



0



3
-

19

3.5.2 Electric dipole radiation

The electric dipole moment oscillates under the electric filed as

)
45
.
3
(
)
cos(
)
(
0
t
p
t
p


)
45
.
3
(
sin
32
)
(
2
2
0
3
2
4
2
0
r
c
p
I






This oscillation emits radiation. The irradiance (radiated
radially outward from the dipole) is given by

Fig 3.10 shows the dipole radiation. Notice that
the irradiance is inversely proportional to the
distance r. The maxima occurs in the direction
of . There is no radiation along the
dipole axis ( ).

o
90


o
0


Fig. 3.10 Field orientation for
an oscillating electric dipole