1

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

The wave equations for and are derived by taking the curl of and

For uniform isotropic linear media we have and , where and are in general

complex functions of frequency . Then we obtain

Since ( ) and, similarly, ,

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

(7.1)

(7.2)

(7.3)

(7.4)

(7.5)

2

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

( )

( )

where the wave number . This equation is mathematically the same as the harmonic

oscillator equation and has solutions

( )

where is a constant vector. Therefore, the full solution is

( ) ( )

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

( ) ( ) ( )

Plane waves in a nonconducting, nonmagnetic dielectric

In a nonmagnetic dielectric, we have and the index of refraction

( ) √ ( )

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

( ) ( )

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

( )

( )

where and are complex constant vector amplitudes of the plane wave. and satisfy the

wave equations (Eq. 7.5), therefore the dispersion relation is given as

(

)

(7.6)

(7.7)

(7.8)

(7.9)

(7.10)

(7.12)

(7.11)

(7.13)

3

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

where . The direction and frequency are completely arbitrary. The divergence

equations demand that

This means that and are both perpendicular to the direction of propagation . The

magnitude of is determined by the refractive index of the material

Then is completely determined in magnitude and direction

√

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

| |

| |

The time averaged energy flux is given by the real part of the complex Poynting vector

( )

Thus the energy flow is

√

| |

| |

(7.14)

(7.15)

(7.16)

(7.17)

(7.18)

(7.19)

4

7.2 Polarization and Stokes Parameters

There is more to be said about the complex vector amplitudes and . We introduce a right-

handed 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

where the amplitudes and are arbitrary complex numbers. The two plane waves

with

(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:

( ) ( )

It is convenient to express the complex components in polar form. Let

Then, for example,

( )

that is, is the phase of the -field component in the -direction. It is no restriction to let

Fig 7.1

(7.21)

(7.22)

(7.20)

(7.23)

(7.24)

(7.25)

(7.26)

5

since merely dictates a certain choice of the origin of . With this choice,

( ) ( )

( )

or the real part is

( ) ( ) ( )

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., , is best seen by

considering some special cases.

( )

( )

( ) ( )

Linearly polarized wave

If and have the same phase, i.e., ,

( ) ( )

( ) ( )

represents a linearly polarized wave, with its polarization vector with

( ) and a magnitude √

, as shown in Fig. 7.2.

If or , we also have linear polarization. For ,

( ) ( )

( ) ( )

is again linearly polarized.

(7.27)

(7.28)

(7.29)

Fig 7.2 -field of a linearly polarized wave

(7.30)

(7.31)

6

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 :

( ) ( )

( )

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.

Stokes Parameters

The two circularly polarized waves form a basis set for a general state of polarization. We

introduce the complex orthogonal unit vectors:

√ ( )

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.

(7.32)

(7.33)

7

They satisfy the orthonormal conditions,

{

Then the most general homogeneous plane wave propagating in the direction (Eq.

7.23) can be expressed as the superposition of two circularly polarized waves:

( ) ( ) ( )

where and are complex amplitudes.

When the ratio of the amplitudes is expressed as

the trace of the tip of the -vector is an ellipse as shown in Fig. 7.4. For , the ratio of

semimajor to semiminor axis is |( ) ( )|.

Stokes parameters

The polarization state of the general plane wave (Eq. 7.35)

( ) ( ) ( )

can be expressed by either ( ) or ( ). We can determine these complex coefficients

using Stokes parameters obtained by intensity measurements using polarizers and wave plates.

We express the complex components in polar form:

The Stokes parameters of the linear polarization basis

| | | |

| | | |

[ ] ( )

[ ] ( )

Fig 7.4 Electric field for an

elliptically polarized wave.

(7.34)

(7.35)

(7.37)

(7.38)

(7.39)

(7.36)

8

and of the circular polarization basis

| | | |

[ ] ( )

[ ] ( )

| | | |

The four parameters are not independent and satisfy the relation

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

becomes

(

)

( )

where we define a complex dielectric constant

(

)

Comparing Eq. 7.44 with Eq. 7.14, we can see that the transverse dispersion relation results in

√

where we define a complex refractive index

√

To interpret the wave propagation in the conducting medium, it is useful to express the

complex propagation vector as

(7.40)

(7.41)

(7.42)

(7.44)

(7.43)

(7.45)

(7.46)

(7.47)

9

Then the plane wave is expressed as

( ) ( ) ( )

This is a plane wave propagating in the direction with wavelength ; but it

decreases in amplitude, most rapidly in the direction .

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 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 z-

direction. 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

{

( )

( )

( )

where

From Eq. 7.17,

(7.48)

Fig 7.5 Reflection and

transmission at normal

incidence

(7.49)

(7.50)

10

Therefore, the magnetic fields associated with the electric fields of Eq. 7.49 are given by

{

( )

( )

( )

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,

The -field must also be continuous, and for nonmagnetic media ( ), so must be the

-field:

( )

Eqs. 7.52 and 7.53 can be solved simultaneously for the amplitudes and in terms of the

incident amplitude :

The Fresnel coefficients for normal incidence reflection and transmission are defined as

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

| |

| |

We define the reflectance and the transmittance for normal incidence by the ratios of the

intensities

| | (

)

| |

( )

With the Fresnel coefficients given by Eq. 7.55, and satisfy

for any pair of nonconducting media. This is an expression of energy conservation at the

interface.

(7.51)

(7.52)

(7.53)

(7.54)

(7.55)

(7.56)

(7.57)

(7.58)

11

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 .

The three plane waves are:

Incident

( )

Refracted

( )

Reflected

( )

where

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:

( ) ( ) ( )

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 .

(7.62)

(7.59)

(7.60)

(7.61)

(7.63)

12

This condition has three interesting consequences. Using the vector identity

( ) ( )

and on the boundary, we obtain

( )

We substitute this into Eq. 7.63,

[ ( )] ( ) ( )

and similarly for the other members of Eq. 7.63. Since is an arbitrary vector on the boundary,

Eq. 7.63 can hold if and only if

This implies that

(i) All three vectors, , and , lie in a plane, i.e., and lie in the plane of

incidence;

(ii) Law of reflection: | | | | , thus

(iii) Snell’s Law: | | | | , thus

Boundary conditions and Fresnel coefficients

At all points on the boundary, normal components of and and tangential components of

and are continuous. The boundary conditions at are

( ) [ ( )

]

( ) [

]

( ) [

]

( ) [ (

)

]

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 (s-

polarization) 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

and

( )

while (ii), using Snell’s law, duplicates (iii). With Eqs. 7.71 and 7.72, we obtain the s-pol Fresnel

coefficients,

(7.64)

(7.65)

(7.66)

(7.67)

(7.68)

(7.69)

(7.70)

(7.71)

(7.72)

13

√

and

√

√

where, using Snell’s law, we could write

√ ( )

(b) p-polarization

The -fields are normal to , therefore (ii) in Eq. 7.70 is automatically satisfied. (iii) and (iv) give

( )

and

( )

while (i), using Snell’s law, duplicates (iv). With Eqs. 7.76 and 7.76, we obtain the p-pol Fresnel

coefficients,

√

and

√

√

For normal incidence, ( ) ( ), because we assign opposite directions

for and for p-polarization.

Fig 7.7 Reflection and refraction with polarization (a) perpendicular (s-polarization) and (b) parallel (p-polarization) to the plane of incidence

(7.76)

(7.77)

(7.78)

(7.75)

(7.73)

(7.74)

(7.79)

14

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

( )

Similarly,

( )

( )

( ) ( )

and

( )

( )

Brewster’s angle and total internal reflection

We next consider the dependence of and on the angle of incidence, using the Fresnel

coefficients.

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

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.

(7.80)

(7.81)

(7.82)

(7.83)

(7.84)

15

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

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 7.8 Reflectance for s- and

p-polarzation at an air-glass

interface. Brewster’s angle is

Fig 7.9 Reflectance for s- and

p-polarzation at an air-glass

interface. Brewster’s angle is

and the critical

angle is

(7.85)

16

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.

√(

)

Then Eqs. 7.74 and 7.79 indicates that and both take the form

where and 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:

( )

(

)

where

√

√

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.

(7.86)

(7.87)

(7.88)

17

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:

(

)

7.5 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

( )

( )

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

( )

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 as , the

dipole moment contributed by one electron is

Fig 7.10 Geometrical interpretation of

the Goos-Hänchen effect, the lateral

displacement of a totally internally-

reflected beam of radiation because of

the penetration of the evanescent wave

into the region of smaller index of

refraction.

(7.89)

(7.90)

(7.91)

(7.92)

18

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

( ) ( )

( )

∑

where the oscillator strengths satisfy the sum rule,

∑

Resonant absorption and anomalous dispersion

In a dispersive medium (nonmagnetic), plane waves are expressed as

( ) ( )

with the complex wave number

( ) √ ( )

Writing in terms of its real and imaginary parts,

with the attenuation constant or absorption coefficient , Eq. 7.95 becomes

( ) ( )

Evidently the wave is exponentially attenuated because the damping absorbs energy. The

intensity of the wave ( | ( )| ) falls off as . The relation between ( ) and is

(7.93)

(7.94)

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.

(7.95)

(7.96)

(7.97)

(7.98)

(7.99)

19

The general features of the real and imaginary parts of ( ) around a resonant frequency are

shown in Fig. 7.11. Most of the time ( ) (or the index of refraction with small ) rises

gradually with increasing frequency (normal dispersion). However, in the immediate

neighborhood of a resonance ( ) drops sharply. Because this behavior is atypical, it is

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., in Eq. 7.93) is ,

( ) ( )

( )

where ( ) 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

(

)

Comparing Eq. 7.101 with Eq. 7.100, we obtain an expression for the Drude conductivity:

( )

( )

where the scattering time

and the d-c conductivity

( )

The scattering times of the common metals are on the order of s, thus ( )

for Hz.

High-frequency limit: plasma frequency

At frequencies far above the highest resonant frequency Eq. 7.93 becomes

( )

where the plasma frequency is defined as

(7.100)

(7.101)

(7.102)

(7.103)

(7.104)

(7.105)

(7.106)

20

Some typical electron densities and plasma frequencies are listed below.

( ) (

)

Metal

Semiconductor (doped)

Semiconductor (pure)

Ionosphere

The dispersion relation is

√ ( ) √

or

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.

Fourier integral transform

From the basic solutions of Eq. 7.12 a plane wave takes the form

( ) ( )

and the superposition principle leads to the a general solution

( )

√ ∫ ( ) ( )

The amplitude ( ) is given by

( )

√ ∫ ( )

Fig 7.12

(7.107)

(7.108)

(7.109)

(7.110)

(7.111)

21

Form of the wave packets

(i) Square wave packet

The amplitude ( ) of the normalized square wave shown in Fig. 7.13 is

( )

√ ∫ ( )

√ ∫

√ [

] √

( )

√

( )

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

(ii) Gaussian wave packet

The normalized Gaussian wave packet shown in Fig 7.14 is expressed as

( )

√

The amplitude ( ) of the normalized square wave shown in Fig. 7.14 is

( )

√ ∫ ( )

√ ∫

( )

√

∫

√

Fig 7.13

(7.112)

Fig 7.14

(7.114)

(7.115)

(7.113)

22

The pulse length and the bandwidth have the inequality relation

(iii) Gaussian pulse in the time domain

The time-bandwidth product is

Phase vs. Group velocity

If the distribution ( ) is sharply peaked around some value , the frequency ( ) can be

expanded around :

( )

|

( )

Then the field amplitude takes the form

( ) [ ( )| ]

√ ∫ ( ) [ ( )| ]

( (

)|

) [ ( )| ]

The pulse travels with a velocity, called the group velocity:

|

(7.116)

Fig 7.15

(7.117)

(7.118)

(7.119)

(7.120)

23

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

( )

( )

The phase velocity is

( )

( )

The group velocity is

|

|

( ( ))|

( )

( )

|

( )

( ) ( )

|

Gaussian pulse propagation through a uniform, lossless, and dispersive medium

We assume the dispersion relation

( )

The group and the phase velocities at are

and

A Gaussian pulse

( )

√

located at at is propagating in the direction. The corresponding Fourier

amplitude is

( ) √

( )

The Gaussian pulse at a later time is

( )

√ ∫ ( ) ( )

√

∫

( )

( )

√

( ) ∫

(

)( )

( )( )

( ) (

)

(

)

[ ( )

(

)]

(7.123)

(7.122)

(7.121)

24

The pulse envelop is

| ( )|

(

)

[ ( )

(

)]

The peak moves with group velocity.

The packet width becomes larger with time.

The pulse energy is preserved during the propagation.

Fig 7.16

Fig 7.17. Optical pulse broadening through propagating.

25

7.7 Causality and Kramers-Kronig Relations