Chapter 7. Plane Electromagnetic Waves and Wave
Propagation
7.1 Plane Monochromatic Waves in Nonconducting Media
One of the most important consequences of the Maxwell equations is the equations for
electromagnetic wave propagation in a linear medium. In the absence of free charge and
current densities the Maxwell equations are
(7.1)
The wave equations for
and
are derived by taking the curl of
and
(7.2)
For uniform isotropic linear media we have
and
complex functions of frequency . Then we obtain
, where and
are in general
(7.3)
Since
(
)
and, similarly,
,
(7.4)
Monochromatic waves may be described as waves that are characterized by a single frequency.
( )
Assuming the fields with harmonic time dependence
, so that ( )
and
( )
( )
we get the Helmholtz wave equations
(7.5)
1
Plane waves in vacuum
Suppose that the medium is vacuum, so that
and
. Further, suppose ( ) varies
in only one dimension, say the -direction, and is independent of and . Then Eq. 7.5
becomes
( )
(7.6)
( )
where the wave number
. This equation is mathematically the same as the harmonic
oscillator equation and has solutions
(7.7)
( )
where
is a constant vector. Therefore, the full solution is
(
(
)
(7.8)
)
This represents a sinusoidal wave traveling to the right or left in the -direciton with the speed
of light . Using the Fourier superposition theorem, we can construct a general solution of the
form
( )
(
)
(
)
(7.9)
Plane waves in a nonconducting, nonmagnetic dielectric
In a nonmagnetic dielectric, we have
and the index of refraction
√
( )
(7.10)
( )
We see that the results are the same as in vacuum, except that the velocity of wave
propagation or the phase velocity is now
instead of . Then the wave number is
(7.11)
( )
( )
Electromagnetic plane wave of frequency
and wave vector
Suppose an electromagnetic plane wave with direction of propagation to be constructed,
where is a unit vector. Then the variable in the exponent must be replaced by
, the
projection of in the direction. Thus an electromagnetic plane wave with direction of
propagation is described by
( )
(7.12)
( )
where and are complex constant vector amplitudes of the plane wave.
wave equations (Eq. 7.5), therefore the dispersion relation is given as
(
)
2
and
satisfy the
(7.13)
Let us substitute the plane wave solutions (Eq. 7.12) into the Maxwell equations. This
substitution will impose conditions on the constants, , and , for the plane wave functions
to be solutions of the Maxwell equations. For the plane waves, one sees that the operators
Thus the Maxwell equations become
(7.14)
where
. The direction
equations demand that
and frequency
are completely arbitrary. The divergence
(7.15)
This means that and are both perpendicular to the direction of propagation . The
magnitude of is determined by the refractive index of the material
(7.16)
Then
is completely determined in magnitude and direction
(7.17)
√
Note that in vacuum (
),
in SI units. The phase velocity of the wave is
.
Energy density and flux
The time averaged energy density (see Eq. 6.94) is
(
)
(
)
This gives
| |
| |
(7.18)
The time averaged energy flux is given by the real part of the complex Poynting vector
(
)
Thus the energy flow is
√ | |
| |
3
(7.19)
7.2 Polarization and Stokes Parameters
There is more to be said about the complex vector amplitudes and . We introduce a righthanded set of orthogonal unit vectors (
), as shown in Fig. 7.1, where we take to be
the propagation direction of the plane wave. In general, the electric field amplitude can be
written as
(7.20)
where the amplitudes
and
are arbitrary complex numbers. The two plane waves
(7.21)
with
(7.22)
(if the index of refraction is real, and have the same phase) are said to be linearly
polarized with polarization vectors and . Thus the most general homogeneous plane wave
propagating in the direction
is expressed as the superposition of two independent
plane waves of linear polarization:
(
)
(
)
(7.23)
Fig 7.1
It is convenient to express the complex components in polar form. Let
(7.24)
Then, for example,
(
that is,
is the phase of the -field component in the
)
(7.25)
-direction. It is no restriction to let
(7.26)
4
since
merely dictates a certain choice of the origin of . With this choice,
(
(
)
)
(
(7.27)
)
or the real part is
(
)
(
)
(
)
(7.28)
The -field is resolved into components in two directions, with real amplitudes and ,
which may have any values. In addition the two components may be oscillating out of phase by
, that is, at any given point , the maximum of in the -direction may be attained at a
different time from the maximum of in the -direction.
Polarization
A detailed picture of the oscillating -field at a certain point, e.g.,
considering some special cases.
(
)
( )
( )
(
, is best seen by
)
(7.29)
Linearly polarized wave
If
and
have the same phase, i.e.,
,
( ) (
( ) (
(7.30)
)
)
represents a linearly polarized wave, with its polarization vector
(
) and a magnitude
√
with
, as shown in Fig. 7.2.
Fig 7.2 -field of a linearly polarized wave
If
or
, we also have linear polarization. For
( ) (
( ) (
,
)
)
is again linearly polarized.
5
(7.31)
Elliptically polarized wave
If and have different phases, the wave of Eq. 7.27 is elliptically polarized. The simplest
case is circular polarization. Then
and
:
(
(
)
)
(
(7.32)
)
At a fixed point in space, the fields are such that the electric vector is constant in magnitude,
but sweeps around in a circle at a frequency , as shown in Fig. 7.3. For
,
√
(
), the tip of the -vector traces the circular path counterclockwise. This wave is
called left circularly polarized (positive helicity) in optics. For
,
√
(
),
same path but traced clockwise, then the wave is called right circularly polarized (negative
helicity). For other values of , we have elliptical polarization for the trace being an ellipse.
Fig 7.3 Trace of the tip of the -vector (
) at a given point in space as a function of time. The
propagation direction is point toward us. The traces for
and are linearly polarized. The traces
for
and
are left and right circularly polarized, respectively.
Stokes Parameters
The two circularly polarized waves form a basis set for a general state of polarization. We
introduce the complex orthogonal unit vectors:
√
(
6
)
(7.33)
They satisfy the orthonormal conditions,
(7.34)
{
Then the most general homogeneous plane wave propagating in the direction
7.23) can be expressed as the superposition of two circularly polarized waves:
(
where
and
)
(
)
(
)
(Eq.
(7.35)
are complex amplitudes.
Fig 7.4 Electric field for an
elliptically polarized wave.
When the ratio of the amplitudes is expressed as
(7.36)
the trace of the tip of the -vector is an ellipse as shown in Fig. 7.4. For
semimajor to semiminor axis is |(
) (
)|.
, the ratio of
Stokes parameters
The polarization state of the general plane wave (Eq. 7.35)
(
)
(
)
(
)
(7.37)
). We can determine these complex coefficients
can be expressed by either (
) or (
using Stokes parameters obtained by intensity measurements using polarizers and wave plates.
We express the complex components in polar form:
(7.38)
The Stokes parameters of the linear polarization basis
| |
| |
|
|
[
[
|
|
(7.39)
]
]
(
(
7
)
)
and of the circular polarization basis
|
|
|
[
[
|
|
|
(7.40)
]
]
|
(
(
)
)
|
The four parameters are not independent and satisfy the relation
(7.41)
7.3 Plane Monochromatic Waves In Conducting Media
In a conducting medium there is an induced current density in response to the -field of the
wave. The current density J is linearly proportional to the electric field (Ohm’s law, Eq. 5.21):
The constant of proportionality is called the conductivity. For an electromagnetic plane wave
with direction of propagation (Eq. 7.12) described by
(
(
)
)
the Maxwell equation
(7.42)
becomes
(
( )
)
(7.43)
where we define a complex dielectric constant
(
)
(7.44)
Comparing Eq. 7.44 with Eq. 7.14, we can see that the transverse dispersion relation results in
(7.45)
√
where we define a complex refractive index
√
(7.46)
To interpret the wave propagation in the conducting medium, it is useful to express the
complex propagation vector as
(7.47)
8
Then the plane wave is expressed as
(
)
(
(
)
)
This is a plane wave propagating in the direction
with wavelength
decreases in amplitude, most rapidly in the direction .
(7.48)
; but it
7.4 Reflection and Refraction of Electromagnetic Waves at a Plane Interface
between Dielectrics
Normal Incidence
We begin with the simplest possible case: a plane wave normally incident on a plane dielectric
interface. We will see that the boundary conditions are satisfied only if reflected and
transmitted waves are present.
Fig 7.5 Reflection and
transmission at normal
incidence
Fig. 7.5 describes the incident wave (
) travelling in the z-direction, the reflected wave
(
) travelling in the minus z-direction, and the transmitted wave (
) travelling in the zdirection. The interface is taken as coincident with the -plane at
, with two dielectric
media with the indices of refraction, for
and for
. The electric fields, which are
assumed to be linearly polarized in the -direction, are described by
(
{
)
(
(7.49)
)
(
)
where
(7.50)
From Eq. 7.17,
9
Therefore, the magnetic fields associated with the electric fields of Eq. 7.49 are given by
(
)
(
{
(
(7.51)
)
)
Clearly the reflected and transmitted waves must have the same frequency as the incident
wave if boundary conditions at
are to be satisfied for all . The -field must be
continuous at the boundary,
(7.52)
The -field must also be continuous, and for nonmagnetic media (
-field:
(
)
Eqs. 7.52 and 7.53 can be solved simultaneously for the amplitudes
incident amplitude :
), so must be the
(7.53)
and
in terms of the
(7.54)
The Fresnel coefficients for normal incidence reflection and transmission are defined as
(7.55)
For
, there is a phase reversion for the reflected wave.
What is usually measureable is the reflected and transmitted average energy fluxes per unit
area (a.k.a., the intensity of EM wave) given by the magnitude of the Poynting vector
(7.56)
|
|
| |
We define the reflectance
intensities
and the transmittance
| |
(
for normal incidence by the ratios of the
)
| |
With the Fresnel coefficients given by Eq. 7.55,
and
(7.57)
(
)
satisfy
(7.58)
for any pair of nonconducting media. This is an expression of energy conservation at the
interface.
10
Oblique incidence
We consider reflection and refraction at the boundary of two dielectric media at oblique
incidence. The discussion will lead to three well-known optical laws: Snell’s law, the law of
reflection, and Brewster’s law governing polarization by reflection. Fig. 7.6 depicts the situation
that the wave vectors, , , and , are coplanar and lie in the -plane. The media for
and
have the indices of refraction, and , respectively. The unit normal to the
boundary is . The plane defined by and is called the plane of incidence, and its normal is in
the direction of
.
Fig 7.6 Reflection and transmission at
oblique incidence. Incident wave strikes
plane interface between different media,
giving rise to a reflected wave
and
refracted wave .
The three plane waves are:
Incident
(
)
(7.59)
Refracted
(
)
(7.60)
Reflected
(
)
(7.61)
where
(7.62)
Phase matching on the boundary
Not only must the refracted and reflected waves have the same frequency as the incident
wave, but also the phases must match everywhere on the boundary to satisfy boundary
conditions at all points on the plane at all times:
(7.63)
(
)
(
)
(
)
11
This condition has three interesting consequences. Using the vector identity
and
(
on the boundary, we obtain
)
(
(
(7.64)
)
(7.65)
)
We substitute this into Eq. 7.63,
[
(
)]
and similarly for the other members of Eq. 7.63. Since
Eq. 7.63 can hold if and only if
(
) (
(7.66)
)
is an arbitrary vector on the boundary,
(7.67)
This implies that
(i)
All three vectors, ,
incidence;
(ii)
Law of reflection: |
(iii)
Snell’s Law: |
and
|
|
|
, lie in a plane, i.e.,
|
and
|
lie in the plane of
, thus
|
(7.68)
, thus
(7.69)
Boundary conditions and Fresnel coefficients
At all points on the boundary, normal components of
and are continuous. The boundary conditions at
[ (
()
( )
and and tangential components of
are
)
]
]
[
( )
[
]
( ) [ (
)
]
(7.70)
In applying the boundary conditions it is convenient to consider two separate situations: the
incident plane wave is linearly polarized with its polarization vector (a) perpendicular (spolarization) and (b) parallel (p-polarization) to the plane of incidence (see Fig. 7.7). For
simplicity, we assume the dielectrics are nonmagnetic (
).
(a) s-polarization
The -fields are normal to , therefore (i) in Eq. 7.70 is automatically satisfied. (iii) and (iv) give
(7.71)
and
(
)
(7.72)
while (ii), using Snell’s law, duplicates (iii). With Eqs. 7.71 and 7.72, we obtain the s-pol Fresnel
coefficients,
12
√
(7.73)
√
(7.74)
and
√
where, using Snell’s law, we could write
√
(
(7.75)
)
(b) p-polarization
The -fields are normal to , therefore (ii) in Eq. 7.70 is automatically satisfied. (iii) and (iv) give
(
)
(7.76)
(
)
(7.77)
and
while (i), using Snell’s law, duplicates (iv). With Eqs. 7.76 and 7.76, we obtain the p-pol Fresnel
coefficients,
(7.78)
√
and
√
(7.79)
√
For normal incidence,
for and
for p-polarization.
(
) (
), because we assign opposite directions
Fig 7.7 Reflection and refraction with polarization (a) perpendicular (s-polarization) and (b) parallel (ppolarization) to the plane of incidence
13
For certain purposes, it is more convenient to express the Fresnel coefficients in terms of the
incident and refraction angles, and only. Using the Snell’s law,
, we can
write
then
(
)
(
(
)
)
(7.80)
Similarly,
(
)
(
(
)
)
(7.81)
(
)
(7.82)
and
(7.83)
Brewster’s angle and total internal reflection
We next consider the dependence of
coefficients.
and
on the angle of incidence, using the Fresnel
Brewster angle
We see that in Eq. 7.88 vanishes when
. Using Snell’s law, we can determine
Brewster’s angle
at which the p-polarized reflected wave is zero:
(
)
or
(7.84)
Polarization at the Brewster angle is a practical means of producing polarized radiation. If a
plane wave of mixed polarization is incident on a plane interface at the Brewster angle, the
reflected radiation is completely s-polarized. The generally lower reflectance for p-polarized
lights accounts for the usefulness of polarized sunglasses. Since most outdoor reflecting
surfaces are horizontal, the plane of incidence for most reflected glare reaching the eyes is
vertical. The polarized lenses are oriented to eliminate the strongly reflected s-component. Fig.
7.8 shows and
as a function of with
and
, as for an air-glass interface.
The Brewster angle is
for this case.
14
Fig 7.8 Reflectance for s- and
p-polarzation at an air-glass
interface. Brewster’s angle is
Total internal reflection
There is another case in which
. Eqs. 7.74 and 7.79 indicates that perfect reflection
occurs for
. The incident angle for which
is called the critical angle,
.
From Snell’s law
(7.85)
can exist only if
, i.e., the incident and reflected waves are in a medium of larger index
of refraction than the refracted wave.
Fig 7.9 Reflectance for s- and
p-polarzation at an air-glass
interface. Brewster’s angle is
and the critical
angle is
15
For waves incident at , the refracted wave is propagated parallel to the surface. There can be
no energy flow across the surface. Hence at that angle of incidence there must be total
reflection. For incident angles greater than the critical angle
, Snell’s law gives
This means that
is a complex angle with a purely imaginary cosine.
(7.86)
√(
Then Eqs. 7.74 and 7.79 indicates that
where
and
and
)
both take the form
are real, therefore,
| |
|
|
The result is that
for all
. This perfect reflection is called total internal
reflection. The meaning of this total internal reflection becomes clear when we consider the
propagation factor for the refracted wave:
(
(
)
)
(7.87)
where
√
√
(7.88)
With the wavelength of the radiation, . This shows that, for
, the refracted wave is
propagating only parallel to the surface and is attenuated exponentially beyond the interface.
The attenuation occurs within a few wavelengths of the boundary except for
.
Goos-Hänchen effect
An important side effect of total internal reflection is the propagation of an evanescent wave
across the boundary surface. Essentially, even though the entire incident wave is reflected back
into the originating medium, there is some penetration into the second medium at the
boundary. The evanescent wave appears to travel along the boundary between the two
materials. The penetration of the wave into the “forbidden” region is the physical origin of the
Goos-Hänchen effect: If a beam of radiation having a finite transverse extent undergoes total
internal reflection, the reflected beam emerges displaced laterally with respect to the
prediction of a geometrical ray refected at the boundary.
16
Fig 7.10 Geometrical interpretation of
the Goos-Hänchen effect, the lateral
displacement of a totally internallyreflected beam of radiation because of
the penetration of the evanescent wave
into the region of smaller index of
refraction.
Fig. 7.10 shows a geometrical interpretation of the Goos-Hänchen effect. We can estimate the
displacement
. Rigorous calculation shows that depends on the polarization of
the incident radiation:
(
(7.89)
)
7.5 Frequency Dispersion in Materials
How an EM wave propagates in a linear material medium is determined entirely by the optical
constants,
and , where the complex index of refraction is
depending only on
and . In general, ( ) and ( ) depend on the frequency of the wave, varying widely in the
range from d-c to x-rays. Dispersion is the phenomenon in which the phase velocity of a wave
depends on its frequency. Media having such a property are termed dispersive media. The
dispersion relation of an EM wave in a dispersive medium is expressed as
( )
(7.90)
( )
Drude-Lorentz harmonic oscillator model
All ordinary matters are composed of electrons and nuclei. The bound electrons can be treated
as harmonic oscillators. For generality we make it a damped harmonic oscillator. When an EM
wave is present, the oscillator is driven by the electric field of the wave. The response of the
medium is obtained by adding up the motions of the electrons. The equation of motion for an
electron of charge
and acted on by an electric field ( ) is
(
(7.91)
)
where the damping constant has the dimensions of frequency. The amplitude of oscillation is
small compared to the spatial variation of the field (e.g., size of atom
nm). Assuming that the field varies harmonically in time with frequency
dipole moment contributed by one electron is
as
, the
(7.92)
17
If there are molecules per unit volume with electrons per molecule, and there are
electrons per molecule with binding frequency
and damping constant , then the dielectric
constant is given by
( )
(7.93)
( )
( )
∑
where the oscillator strengths
satisfy the sum rule,
(7.94)
∑
Resonant absorption and anomalous dispersion
In a dispersive medium (nonmagnetic), plane waves are expressed as
(
(7.95)
( )
)
with the complex wave number
( )
Writing
(7.96)
( )
√
in terms of its real and imaginary parts,
(7.97)
with the attenuation constant or absorption coefficient , Eq. 7.95 becomes
(
(
)
)
(7.98)
Evidently the wave is exponentially attenuated because the damping absorbs energy. The
intensity of the wave ( | ( )| ) falls off as
. The relation between (
) and is
(7.99)
Fig 7.11 Real and imaginary parts of
the dielectric constant in the
neighborhood of a resonance. The
region of anomalous dispersion is
the frequency interval where
absorption occurs.
18
The general features of the real and imaginary parts of ( ) around a resonant frequency are
( ) (or the index of refraction with small ) rises
shown in Fig. 7.11. Most of the time
gradually with increasing frequency (normal dispersion). However, in the immediate
( ) drops sharply. Because this behavior is atypical, it is
neighborhood of a resonance
called anomalous dispersion. Notice that the region of anomalous dispersion coincides with the
region of maximum absorption.
Drude model: Electric conductivity at low frequencies
If the density of free electrons (i.e.,
( )
where
in Eq. 7.93) is
,
(7.100)
( )
(
)
( ) is the contribution of the bound electrons. With the Ohm’s law
and
where the fields are harmonic in terms of
, the Maxwell-Ampere equation
becomes
(
)
(7.101)
Comparing Eq. 7.101 with Eq. 7.100, we obtain an expression for the Drude conductivity:
( )
(7.102)
(
)
where the scattering time
(7.103)
and the d-c conductivity
(
)
The scattering times of the common metals are on the order of
for
Hz.
(7.104)
s, thus ( )
High-frequency limit: plasma frequency
At frequencies far above the highest resonant frequency Eq. 7.93 becomes
(7.105)
( )
where the plasma frequency is defined as
(7.106)
19
Some typical electron densities and plasma frequencies are listed below.
(
)
( )
Metal
Semiconductor (doped)
Semiconductor (pure)
Ionosphere
The dispersion relation is
√ ( )
(7.107)
√
or
(7.108)
For
, is pure imaginary, therefore the light exponentially decays and penetrates only a
very short distance in the medium. The plasma frequencies of common metals are in the UV,
and hence the visible light is almost entirely reflected from metal surfaces and the metals
suddenly become transparent in UV.
7.6 Wave Propagation in a Dispersive Medium
Wave packet
A wave packet or a pulsed electromagnetic wave is spatially and temporally localized.
Fig 7.12
Fourier integral transform
From the basic solutions of Eq. 7.12 a plane wave takes the form
(
(7.109)
( )
)
and the superposition principle leads to the a general solution
(
)
√
( )
( )
∫
(7.110)
The amplitude ( ) is given by
( )
√
(
∫
20
)
(7.111)
Form of the wave packets
(i)
Square wave packet
Fig 7.13
The amplitude ( ) of the normalized square wave shown in Fig. 7.13 is
( )
√
∫
( )
√
[
√
]
√
∫
(
)
(
√
)
(7.112)
As the pulse length becomes small, i.e., more tightly localized, then
, which is the
bandwidth of ( ), becomes larger. The pulse length and the bandwidth have the relation
(7.113)
(ii)
Gaussian wave packet
Fig 7.14
The normalized Gaussian wave packet shown in Fig 7.14 is expressed as
( )
(7.114)
√
The amplitude ( ) of the normalized square wave shown in Fig. 7.14 is
( )
√
√
∫
( )
√
∫
√
∫
21
(
)
(7.115)
The pulse length and the bandwidth have the inequality relation
(7.116)
(iii)
Gaussian pulse in the time domain
Fig 7.15
The time-bandwidth product is
(7.117)
Phase vs. Group velocity
If the distribution ( ) is sharply peaked around some value
expanded around :
( )
|
(
, the frequency ( ) can be
(7.118)
)
Then the field amplitude takes the form
[
(
)
(
)|
]
√
(
( )
∫
(
)|
[
)
(
[
(
)|
)| ]
]
(7.119)
The pulse travels with a velocity, called the group velocity:
|
22
(7.120)
The phase velocity is the speed of the individual wave crests, whereas the group velocity is the
speed of the wave packet as a whole, i.e., the speed of the envelope propagation. For light
waves the dispersion relation between and is given by
( )
(7.121)
( )
The phase velocity is
( )
(7.122)
( )
The group velocity is
(
|
|
(
( ))|
(
)
( )
|
(
)
)
( )
|
(7.123)
Gaussian pulse propagation through a uniform, lossless, and dispersive medium
We assume the dispersion relation
( )
are
and
The group and the phase velocities at
A Gaussian pulse
(
located at
amplitude is
at
)
√
is propagating in the
direction. The corresponding Fourier
(
√
( )
)
The Gaussian pulse at a later time is
(
)
( )
∫
√
√
(
(
)
)
(
∫
√
(
)
)
(
∫
(
)
(
)(
)
)
(
[
(
)
23
)(
(
(
)
)
]
)
The pulse envelop is
| (
)|
[
(
)
(
(
)
]
)
Fig 7.16



The peak moves with group velocity.
The packet width becomes larger with time.
The pulse energy is preserved during the propagation.
Fig 7.17. Optical pulse broadening through propagating.
24
7.7 Causality and Kramers-Kronig Relations
25