1. Chapter 8 Plane Electromagnetic Waves
Plane waves in perfect dielectric
Plane waves in conducting media
Polarizations of plane waves
Normal incidence on a planar surface
Plane waves in arbitrary directions
Oblique incidence at boundary
Plane waves in anisotropic media

2. 1. Wave Equations
2. Plane Waves in Perfect Dielectric
3. Plane Waves in Conducting Media
4. Polarizations of Plane Waves
5. Normal Incidence on A Planar Surface
6. Normal Incidence at Multiple Boundaries
7. Plane Waves in Arbitrary Directions
8. Oblique Incidence at Boundary between Perfect Dielectrics
9. Null and Total Reflections
10. Oblique Incidence at Conducting Boundary
11. Oblique Incidence at Perfect Conducting Boundary
12. Plane Waves in Plasma
13. Plane Waves in Ferrite

3. 1. Wave Equations
In infinite, linear, homogeneous, isotropic media, a time-varying electromagnetic field satisfies the following equations:
which are called inhomogeneous wave equations,and
where is the impressed source.

4. The relationship between the charge density  (r, t) and the conduction current is
In a region without impressed source, J ' = 0. If the medium is a perfect dielectric, then,  = 0 . In this case, the conduction current is zero, and  = 0. The above equation becomes
Which are called homogeneous wave equations.
To investigate the propagation of plane waves, we first solve the homogeneous wave equations.

5. For a sinusoidal electromagnetic field, the above equation becomes
which are called homogeneous vector Helmholtz equations, and here
In rectangular coordinate system, we have
which are called homogeneous scalar Helmholtz equations.
All of these equations have the same form, and the solutions are similar.

6. In a rectangular coordinate system, if the field depends on one variable only, the field cannot have a component along the axis of this variable.
If the field is related to the variable z only, we can show
Since the field is independent of the variables x and y, we have

7. Due to , from the above equations we obtain
Considering
Substituting that into Helmholtz equations:
We find

8. 2. Plane Waves in Perfect Dielectric
In a region without impressed source in a perfect dielectric, a sinusoidal electromagnetic field satisfies the following homogeneous vector Helmholtz equation
Where
If the electric field intensity E is related to the variable z only, and independent of the variables x and y, then the electric field has no z-component.
Let , then the magnetic field intensity H is

9. Due to
We have
From last section, we know that each component of the electric field intensity satisfies the homogeneous scalar Helmholtz equation. Considering , we have
which is an ordinary differential equation of second order, and the general solution is
The first term stands for a wave traveling along the positive direction of the z-axis, while the second term leads to the opposite .

10. The instantaneous value is
Here only the wave traveling along with the positive direction of z-axis is considered
where Ex0 is the effective value of the electric field intensity at z = 0 .
The instantaneous value is
Ez(z, t) z O 
An illustration of the electric field intensity varying over space at different times is shown in the left figure.
t1 = 0
The wave is traveling along the positive z-direction.

11. where  t accounts for phase change over time, and kz over space
where  t accounts for phase change over time, and kz over space. The surface made up of all points with the same space phase is called the wave front.
Here the plane z = 0 is a wave front, and this electromagnetic wave is called a plane wave.
Since Ex(z) is independent of the x and y coordinates, the field intensity is constant on the wave front. Hence, this plane wave is called a uniform plane wave.

12. The time interval during which the time phase (t) is changed by 2 is called the period, and it is denoted as T. The number of periods in one second is called the frequency, and it is denoted as f.
Since , we have
The distance over which the space phase factor (kr) is changed by 2 is called the wavelength, and it is denoted as .
Since , we have
The frequency describes the rate at which an electromagnetic wave varies with time, while the wavelength gives the interval in space for the wave to repeat itself.
And we have
The constant k stands for the phase variation per unit length, and it is called the phase constant, and the constant k gives the numbers of full waves per unit length. Thus k is also called the wave number.

13. The speed of phase variation vp can be found from the locus of a point with the same phase angle. Let , and nothing that
, then the phase velocity vp is
Considering , we have
In a perfect dielectric, the phase velocity is governed by the property of the medium.
Consider the relative permittivities of all media with , and with relative permeability The phase velocity of a uniform plane wave in a perfect dielectric is usually less than the velocity of light in vacuum.
It is possible to have Therefore , the phase velocity must not be the energy velocity.

14. From the above results, we find
The frequency of a plane wave depends on the source, and it is always the same as that of the source in a linear medium. However, the phase velocity is related to the property of the medium, and hence the wavelength is related to the property of the medium.
We find
where
where 0 is the wavelength of the plane wave with frequency f in vacuum.
Since , , and Namely, the wavelength of a plane wave in a medium is less than that in vacuum. This phenomenon may be called the shrinkage of wavelength.

15. Using , we find
where
In perfect dielectrics, the electric field and the magnetic field of a uniform plane wave are in phase, and both have the same spatial dependence, but the amplitudes are constant. z Ex
The left figure shows the variation of the electric field and the magnetic field in space at t = 0. Hy

16. The ratio of the amplitude of electric field intensity to that of magnetic field intensity is called the intrinsic impedance, and is denoted as Z as given by
The intrinsic impedance is a real number.
In vacuum, the intrinsic impedance is denoted as Z0
The above relationship between the electric field intensity and the magnetic field intensity can be written in vector form as follows: Ex Hy z Or

17. The electric field and the magnetic field are transverse with respect to the direction of propagation and the wave is called a transverse electromagnetic wave, or TEM wave.
We will encounter non-TEM wave that has the electric or the magnetic field component in the direction of propagation.
A uniform plane wave is a TEM wave. Only non-uniform waves can be non-TEM waves, and TEM waves are not necessarily plane waves.
From the electric field intensity and the magnetic field intensity found, we can find the complex energy flow density vector Sc as
The complex energy flow density vector is real, while the imaginary part is zero. It means that the energy is traveling in the positive direction only,

18. We construct a cylinder of long l and cross-section A along the direction of energy flow, as shown in the figure. l S A
Suppose the distribution of the energy is uniform in the cylinder. The average value of the energy density is wav , and that of the energy flow density is Sav.
Then the total energy in the cylinder is wav Al , and the total energy flowing across the cross-sectional area A per unit time is Sav A.
If all energy in the cylinder flows across the area A in the time interval t, then
Obviously, the ratio stands for the displacement of the energy in time t, and it is called the energy velocity, denoted as ve. We obtain

19. Considering and , we find
The wave front of a uniform plane wave is an infinite plane and the amplitude of the field intensity is uniform on the wave front, and the energy flow density is constant on the wave front. Thus this uniform plane wave carries infinite energy. Apparently, an ideal uniform plane wave does not exist in nature.
If the observer is very far away from the source, the wave front is very large while the observer is limited to the local area, the wave can be approximately considered as a uniform plane wave.
By spatial Fourier transform, a non-plane wave can be expressed in terms of the sum of many plane waves, which proves to be useful sometimes

20. Example. A uniform plane wave is propagating along with the positive direction of the z-axis in vacuum, and the instantaneous value of the electric field intensity is
Find: (a) The frequency and the wavelength.
(b) The complex vectors of the electric and the magnetic field
intensities.
(c) The complex energy flow density vector.
(d) The phase velocity and the energy velocity.
Solution: (a) The frequency is
The wavelength is

21. (b) The electric field intensity is
The magnetic field intensity is
(c) The energy flow density vector is
(d) The phase and energy velocities are

22. 3. Plane Waves in Conducting Media
If   0 , the first Maxwell’s equation becomes
If let
Then the above equation can be rewritten as
where e is called the equivalent permittivity.
In this way, a sinusoidal electromagnetic field then satisfies the following homogeneous vector Helmholtz equation:

23. Let
We obtain
If we let as before, and , then the solution of the equation is the same as that in the lossless case as long as k is replaced by kc, so that
Because kc is a complex number, we define
We find

24. In this way, the electric field intensity can be expressed as
where the first exponent leads to an exponential decay of the amplitude of the electric field intensity in the z-direction, and the second exponent gives rise to a phase delay.
The real part k is called the phase constant, with the unit of rad/m, while the imaginary part k is called the attenuation constant and has a unit of Np/m.
The phase velocity is
It depends not only on the parameters of the medium but also on the frequency.
A conducting medium is a dispersive medium.

25. The wavelength is
The wavelength is related to the properties of the medium, and it has a nonlinear dependence on the frequency.
The intrinsic impedance is
which is a complex number.
Since the intrinsic impedance is a complex number, and it leads to a phase shift between electric field and the magnetic field.

26. The magnetic field intensity is
The amplitude of the magnetic field intensity also decreases with z, but the phase is different from that of the electric field intensity.
Since the electric and the magnetic field intensities are not in phase, the complex energy flow density vector has non-zero real and imaginary parts. This means that there is both energy flow and energy exchange when a wave propagates in a conductive medium. Ex Hy z

27. Two special cases ：
(a) If , as in an imperfect dielectric, the approximation
Then
The electric and the magnetic field intensities are essentially in phase. There is still phase delay and attenuation in this case. The attenuation constant is proportional to the conductivity .
(b) If , as in good conductors, we take

28. Then
The electric and the magnetic field intensities are not in phase, and the amplitudes show a rapid decay due to a large . In this case, the electromagnetic wave cannot go deep into the medium, and it only exists near the surface. This phenomenon is called the skin effect.
The skin depth  is the distance over which the field amplitude is reduce by a factor of , mathematically determined from
The skin depth is inversely proportional to the square root of the frequency f and the conductivity .

29. The skin depths at different frequencies for copper
f MHz
0.05 1
 mm
29.8
0.066
The skin depth deceases with increasing frequency.
The frequency for sets the boundary between an imperfect dielectric and a conductor, and it is called crossover frequency.
Several crossover frequencies for different materials:
Media
Frequencies MHz
Dry Soil
(Short Wave)
Wet Soil
(Short Wave)
Pure Water
(Medium Wave)
Sea Water
890 (Super Short Wave)
Silicon
15 (Microwave)
Germanium
11 (Microwave)
Platinum
16.9 (Light Wave)
Copper
104.41016 (Light Wave)
stands for the ratio of amplitude of the conduction current to that of the displacement current.
In imperfect dielectrics the dis-placement current dominates, while the converse is true for a good conductor.

30. The attenuation of a plane wave is caused by the conductivity , resulting in power dissipation, and conductors are called lossy media. Dielectrics without conductivity are called lossless media.
Besides conductor loss there are other losses due to dielectric polarization and magnetization. As a result, both permittivity and permeability are complex, so that and
The imaginary part stands for dissipation, and they are called dielectric loss and magnetic loss, respectively.
For non-ferromagnetic media, the magnetization loss can be neglected.
For electromagnetic waves at lower frequencies, dielectric loss can be neglected.

31. Example. A uniform plane wave of frequency 5MHz is propagating along the positive direction of the z-axis. The electric field intensity is in the x-direction at , with an effective value of 100(V/m). If the region is seawater, and the parameters are , find:
(a) The phase constant, the attenuation constant, the phase velocity, the wavelength, the wave impedance, and the skin depth in seawater.
(b) The instantaneous values of the electric and the magnetic field intensities, and the complex energy flow density vector at z = 0.8m.
Solution: (a)
The seawater can be considered as a good conductor, and the phase constant k' and the attenuation constant k" are, respectively,

32. The phase velocity is
The wavelength is
The intrinsic impedance is
The skin depth is
(b) The complex vector of the electric field intensity is
The complex vector of the magnetic field intensity is

33. The instantaneous values of the electric and the magnetic field intensities at z = 0.8m as
The complex energy flow density vector as
The plane wave of frequency 5MHz is attenuated very fast in seawater. Therefore it is impossible to communicate between two submarines by using the direct wave in seawater.

34. 4. Polarizations of Plane Waves
The time-varying behavior of the direction of the electric field intensity is called the polarization of the electromagnetic wave.
Suppose the instantaneous value of the electric field intensity of a plane wave is
Obviously, at a given point in space the locus of the tip of the electric field intensity vector over time is a straight line parallel to the x-axis. Hence, the wave is said to have a linear polarization.
The instantaneous value of the electric field intensity of another plane wave of the same frequency is
This is also a linearly polarized plane wave, but with the electric field along the y-direction.

35. If the above two orthogonal, linearly polarized plane waves with the same phase but different amplitudes coexist, then the instantaneous value of the resultant electric field is
The time-variation of the magnitude of the resultant electric field is still a sinusoidal function, and the tangent of the angle  between the field vector and the x-axis is Ey Ex E y x  O Ey Ex E y x  O Ey Ex E y x  O
The polarization direction of the resultant electric field is independent of time, and the locus of the tip of the electric field intensity vector over time is a straight line at an angle of  to the x-axis. Thus the resultant field is still a linearly polarized wave.

36. If two orthogonal, linearly polarized plane waves of the same phase but different amplitudes are combined, the resultant wave is still a linearly polarized plane wave.
Conversely, a linearly polarized plane wave can be resolved into two orthogonal, linearly polarized plane waves of the same phase but different amplitudes.
If the two plane waves have opposite phases and different amplitudes, how about the resultant wave?
If the above two linearly polarized plane waves have a phase difference of , but the same amplitude Em, i.e.

37. Then the instantaneous value of the resultant wave is
The direction of the resultant wave is at an angle of  to the x-axis, and
i.e.
At a given point z the angle  is a function of time t. The direction of the electric field intensity vector is rotating with time, but the magnitude is unchanged. Therefore, the locus of the tip of the electric field intensity vector is a circle, and it is called circular polarization.

38. Ey Ex E y x  O
Right z y x O
Left
The angle  will be decreasing with increasing time t . When the fingers of the left hand follow the rotating direction, the thumb points to the propagation direction and it is called the left-hand circularly polarized wave.

39. If Ey is lagging behind Ex by , the resultant wave is at an angle of
to the x-axis. At a given point z, the angle  will be increasing with increasing time t. The rotating direction and the propagation direction ez obey the right-hand rule, and it is called a right-hand circularly polarized wave.
Two orthogonal, linearly polarized waves of the same amplitude and phase difference of result in a circularly polarized wave.
Conversely, a circularly polarized wave can be resolved into two orthogonal, linearly polarized waves of the same amplitude and a phase difference of
A linearly polarized wave can be resolved into two circularly polarized waves with opposite senses of rotation, and vice versa.

40. If two orthogonal, linearly polarized plane waves Ex and Ey have different amplitudes and phases
The components Ex and Ey of the resultant wave satisfy the following equation  y x E
x '
y '
Ey m
Ex m
which describes an ellipse. At a given point z, the locus of the tip of the resultant wave vector over time is an ellipse, and it is called an elliptically polarized wave.

41. If  < 0 , Ey lags behind Ex , and the resultant wave vector is rotated in the counter-clockwise direction. It is a right-hand elliptically polarized wave.
If  > 0 , then the resultant wave vector is rotated in the clockwise direction, and it is a left-hand elliptically polarized wave.
The linearly and the circularly polarized waves can both be considered as the special cases of the elliptically polarized wave. Since all polarized waves can be resolved into linearly polarized waves, only the propagation of linearly polarized wave will be discussed.
The propagation behavior of an electromagnetic wave is a useful property with many practical applications.
Since a circularly polarized electromagnetic wave is less attenuated by rain, it is used in all-weather radar.

42. In wireless communication systems, the polarization of the receiving antenna must be compatible with that of the wave to be received.
In mobile satellite communications and globe positioning systems, because the position of the satellite changes with time, circularly polarized waves should be used.
some microwave devices use the polarization of the wave to achieve special functions, as found in ferrite circulators, ferrite isolators, and others.
Stereoscopic film is taken by using two cameras with two orthogonally polarized lenses. Hence, the viewer has to wear a pair of orthogonally polarized glasses to be able to see the three-dimensional effect.

43. 5. Normal Incidence on A Planar Surface
Consider an infinite planar boundary between two homogeneous media with the parameters of the media and
111
222 z x y
Let the boundary coincides with the plane z = 0. As an x-directed, linearly polarized plane wave is normally incident on the boundary from medium ①, a reflected and a transmitted wave are produced at the boundary.
S r
S i
S t
Since the tangential components of the electric field intensities must be continuous at any boundary, the sum of the tangential components of the electric field intensities of the incident and the reflected waves is equal to that of the transmitted wave.

44. where , , are the amplitudes at the boundary, respectively.
The polarization cannot be changed during the reflection and the transmission of a linearly polarized wave.
Assume that the electric field intensities of the incident, the reflected, and the transmitted waves are given in the figure, and they can be expressed as follows:
111
222 z x y
S i
Incident wave
S r
Reflected wave
S t
Transmitted wave
where , , are the amplitudes at the boundary, respectively.

45. The magnetic field intensities as
Incident wave
Reflected wave
Transmitted wave
The tangential components of electric field intensities must be continuous at any boundary. Consider there is no surface current at the boundary with the limited conductivity. Hence the tangential components of the magnetic field intensities are also continuous at the boundary z = 0.
We have

46. We find
The ratio of the electric field components of the reflected wave to that of the incident wave at the boundary is defined as the reflection coefficient, denoted as R. The ratio of the electric field components of the transmitted wave to that of the incident wave at the boundary is called the transmission coefficient, denoted as T.
We obtain
In medium ①, the resultant electric and magnetic field intensities are

47. Two special boundaries as follows:
(a) If medium ① is a perfect dielectric and medium ② is a perfect electric conductor , the intrinsic impedances of the media, are respectively,
We find
All of the electromagnetic energy is reflected by the boundary, and no energy enters into medium ②. This case is called total reflection.
The refection coefficient R = 1 implies that at the boundary so that the electric fields of the reflected and the incident waves have the same amplitude but opposite phase, resulting in a total electric field that is zero at the boundary.

48. The propagation constant for medium ① is , and in medium ① the complex vector of the resultant electric field is
The instantaneous value is
In medium ① the phase of the resultant electric field is dependent on time only, and the amplitude has a sinusoidal dependence on z. when
the electric field is zero all the time.
At the amplitude of the electric field is always the largest at any time.
This means that the spatial phase of the resultant wave is fixed, and the amplitudes vary in a proportionate manner. This plane wave is not traveling, but stays at a fixed location with the field intensities varying periodically as time progresses. It is called a standing wave.

49. Ex 0>0 z 1 O
1 = 0
2 = 
t1 = 0
Ex 0>0 1 z
1 = 0
2 =  O
Ex 0>0 1 z
1 = 0
2 =  O
Ex 0>0 1 z
1 = 0
2 =  O
Ex 0>0
t1 = 0 1 z
1 = 0
2 =  O
The location at which the amplitude is always zero are called wave node, while the location at which the amplitude is always maximum are called wave loop.
The plane wave in an open perfect dielectric as discussed previously is called a traveling wave, as opposed to a standing wave. The phase of a traveling wave is progressing in the propagation direction, while that of a standing wave does not move in space.

50. The complex vector of the resultant magnetic field intensity is
The instantaneous value is
Hy 0 z 1 O
1 = 0
2 =  y
The resultant magnetic field is still a standing wave, but the positions of the wave nodes and the wave loops are opposite to that of electric field.

51. The electric field and the magnetic field have a phase difference of
The electric field and the magnetic field have a phase difference of Hence, the real part of the energy flow density vector is zero, and it has the imaginary part only.
It means that there is no energy flow in space, and the energy is converted between the electric field and the magnetic field.
In medium ① immediately to the left of the boundary at z = 0, the resultant magnetic field is , but inside medium ② it is
. Therefore, in this case the tangential components of the magnetic field intensities are discontinuous at the boundary. There exists a surface current JS at the boundary, with a surface current density given by

52. (b) If medium ① is a perfect dielectric  = 0 and medium ② is a conductor with finite conductivity. The intrinsic impedance and the propagation constant of medium ① are given by
The reflection coefficient is
where is the amplitude of R, and  is its argument.
The resultant electric field intensity in medium ① is
If , which corresponds to a maximum for the electric field, located at , the magnitude of the electric field intensity is given by

53. Since , the amplitude of the electric field lies in .
If , which corresponds to a minimum for the electric field at the locations , the magnitude of the electric field intensity is given by O 1 z
Since , the amplitude of the electric field lies in
The pattern of the standing wave is shown in the left figure.
The distance between two maxima (or minima) on a standing wave is
The ratio of the maximum value to the minimum value for the electric field intensity is called the standing wave ratio S, i.e.

54. For the case of total reflection, .
If , then , corresponding to the absence of the reflected wave. This boundary without reflection is called a matched boundary.
Hence, the range of the standing wave ratio is
If two media are perfect dielectric, the maximum point is at the boundary when , and if the minimum point is at the boundary .
In this case, this standing wave is different from the full standing wave, and its pattern can be thought to be made up of a full standing wave together with a traveling wave.

55. Example. Two media with the parameters, , ;
, are joined at a planar boundary. A right-hand circularly polarized plane wave is incident on the boundary from medium ①. Find the reflected and the transmitted waves and their polarizations.
111
222 z x y
S t
S r
S i
Solution: Select rectangular coordinate system, and let the boundary be placed at the plane
Since the incident wave is a right-hand circularly polarized wave, the incident wave can be expressed as
The reflected wave and the transmitted wave are, respectively,

56. The reflection and the transmission coefficients are
Because the y-components of the reflected and the transmitted waves are still lagging behind the x-components, but the propagating direction of the reflected wave is always is along the negative z-direction, and it becomes left-hand circularly polarized wave.
The propagating direction of the transmitted wave is in the positive z-direction, and it is still the right-hand circularly polarized wave.

57. 6. Normal Incidence at Multiple Boundaries
As an example, consider a three-layer medium first.
Zc1
Zc2
Zc3 -l O z ① ② ③
There are multiple reflections and transmissions between the two boundaries.

58. Based on the solution of one-dimensional wave equation, we can consider there are two waves in medium ① or ② only, one is propagating along with the positive z-direction denoted as and , and another is in the negative z-direction denoted as and . In medium ③ there is only one wave propagating along the positive z-direction.
Hence, the electric fields in the media can be expressed as follows:

59. The magnetic fields are, respectively

60. Since the tangential components of the electric field intensities must be continuous at z = 0 and z = l , we have
Because the tangential components of the magnetic field intensities must be continuous at the two boundaries also, we have
The quantity is given, while , , , and are unknown, which can be found from the four equations above.

61. For an n-layered medium, since the incident wave is known, and there is only a transmitted wave in the n-th layer, the total number of unknowns are The n-layered medium has boundaries, with which we can construct equations.Therefore, all unknowns can be determined.
If we only need to find the total reflection coefficient at the first boundary, then using the input wave impedance, one can simplify the process.
For the above example, the ratio of the resultant electric field to the resultant magnetic field at a point (z) in medium ② is defined as the input wave impedance at the point, and it is denoted as Zin, given by

62. The resultant electric field in medium ② is
where
The resultant magnetic field in medium ② can be expressed as
We find
Since the resultant electric and magnetic fields should be continuous at the boundary , we have

63. The total reflection coefficient at the first boundary is defined
where
For medium ①, media ② and ③ can be considered as one medium with the wave impedance Zin(l) . If the thickness and the parameters of medium ②, and the parameters of medium ③ are known, then the input wave impedance Zin(l) can be found.
The approach of input wave impedance is in fact similar to the network method in circuit theory. The total effect of the media needs to be considered collectively instead of the inside of the multi-layered medium.

64. n-layered medium
Zc1
Zc2
Zc3
(n-2)
(n-1)
(3)
(2)
(1)
Zc(n-2)
Zc(n-1)
Zc n
First we calculate the input wave impedance toward the right at the (n2)-th boundary, and then for the (n2)-th medium, the (n1)-th and the n-th media can be replaced by a medium with wave impedance
Similarly, the input wave impedance toward the right at each boundary can be found one by one from the right to the left. Finally, we can find the total reflection coefficient.

65. Z1 Zn Z3 Z2
Zn-1
Zn-2 Z1 Z3 Z2
Zn-2 Z3 Z1 Z2 Z1 Z1 Z2

66. To eliminate the reflection, is required, and we find
Example. The wave impedances of two perfect dielectrics are Z1 and Z2 , respectively. In order to eliminate the reflection, a piece of perfect dielectric of the thickness of one-quarter wavelength (in the middle dielectric) is inserted between the two dielectrics. Find the wave impedance of the middle dielectric.
Solution. First we calculate the input wave impedance to the right at the first boundary. Consider Z1 Z Z2 ② ①
We find
To eliminate the reflection, is required, and we find
The approach of input wave impedance is a kind of impedance transform.

67. We know that the input wave impedance is
The change of input wave impedance is the same as that of a tangent function. The period of change is , and the sandwiched dielectric with a thickness of one half-wavelength or integral times of half-wavelength has no action of impedance transform. 。
In microwave circuits, transmission line of one quarter wavelength is used to transform the impedance. In this case, it will extend the total length of the transmission line, but the impedance match can be realized.
This transform is applicable for a certain frequency, resulting in narrow-band operation.

68. If the relative permittivity of a dielectric is equal to the relative permeability so that  r =  r , the wave impedance will be equal to that of vacuum.
When such a dielectric slab is placed in air, and a plane wave is normally incident upon the surface of the slab, there will be no reflection regardless of the slab thickness. In other words, for the electromagnetic wave it is “transparent”.
If this dielectric slab is used as a radom, it should have perfect performance. However, the relative permittivity and the relative permeability are usually not of the same order of magnitude. Recently, a newly developed magnetic material can approximately meet this requirement.

69. z y x 7. Plane Waves in Arbitrary Directions
Suppose the propagating direction of a plane wave is eS , and it is perpendicular to the wave front of the plane wave as shown in the figure.    z y x d eS P0 E0
Wave front
Let the distance between the origin and the wave front be d, and the electric field intensity at the origin be E0 , then we can obtain the electric field intensity at a point P0 on the wave front as 
P(x, y, z) r
If let P be a point on the wave front, and its coordinates are (x, y, z), then the position vector r of the point P is

70. z y x We have eS and P0 If let d P(x, y, z)   Then   E0r
and
If let
Then
which is the expression for a plane wave traveling in an arbitrary direction.
Here k is called the wave vector, and its direction is the propagating direction and the magnitude is equal to the propagation constant k.
If the wave vector k makes an angle of , ,  with the x, y, z-axes, respectively, then the propagating direction eS can be written as
And the wave vector is

71. If let
Then the wave vector k can be expressed as
The electric field intensity can be expressed at any point as or
Since , should satisfy the following equation
Obviously, only two of are independent.

72. In the source-free region, the uniform plane wave traveling in the direction of k should satisfy the following equations: S E H
The real part of the complex energy flow density vector Sc as
Considering , we have

73. Example. In vacuum a plane wave is TEM wave, and the electric field intensity is
Where is a constant.
Find out: (a) If it is a uniform plane wave. (b) The frequency and the wavelength of the plane wave. (c) The y-component of the electric field. (d) The polarization of the plane wave.
Solution: The electric field intensity is given as
In view of this, the propagating path is in the xy-plane, and the wave front is parallel to the z-axis. Because the amplitude of the field intensity is related to the variable z, it is an inhomogeneous plane wave.

74. From We find Due to ，we obtain .
The resultant field of the x-component and the y-component of the electric field intensity is a linearly polarized wave. Together with the z-component with the different phase and amplitude, they are combined into an elliptically polarized wave. Since the component lags behind the component Ez, the resultant wave is a right-hand elliptically polarized wave. x y z k
Wave front
(Ex + Ey)
(Ex+Ey +Ez) Ez

75. 8. Oblique Incidence at Boundary between Perfect Dielectrics
When a plane wave is incident upon a plane boundary, reflection and transmission will take place. Since the direction of the transmitted wave is different from that of the incident wave, this transmitted wave is called a refracted wave. The planes containing the incident ray, the reflected ray, the refracted ray and the normal to the boundary are called the plane of incidence, the plane of reflection, the plane of refraction.
1 1
2 2 x z y
Normal
Reflected wave r i
Incident wave t
Refracted wave

76. (a) All of the incident ray, the reflected ray, and the refracted ray are in the same plane. (b) The angle of incidence  i is equal to the angle of reflection  r . (c) The angle of incidence  i and the angle of refraction  t satisfy the following relationship:
The three conditions above are called Snell’s Law, and
Suppose the plane of incidence be in the xz-plane, then the electric field intensity of the incident wave can be written as
The reflected and the refracted waves can be expressed, respectively, as

77. Since the tangential components of the resultant electric field intensities must be continues at z = 0, giving
which holds for any x and y, thus the corresponding coefficients in each exponent should be equal, so that
We have ，i.e.
which states that the reflected ray and the refracted ray are in the xz-plane.

78. Considering , , , we have
states that the phases of the reflected and the refracted waves are varying along the boundary in synchronism with the phase of the incident wave, and it is called the condition for phase matching.
Snell’s law is used widely in practice. For example, the bottom surfaces of stealth planes B2 and F117 are almost a flat plane, so that the echo wave of the radar is reflected forward so that a mono-static radar cannot receive it. The stealthy behavior of the flights is therefore realized.

79. For the oblique incidence, the reflection and the transmission coefficients are related to the polarization of the incident wave.
The plane wave whose direction of electric field is parallel to the plane of incidence as parallel polarized wave, and that being perpendicular to the plane of incidence is called perpendicular polarized wave, i r t
1 1
2 2
E i
E t
E r
H i
H r
H t z x O i r t
1 1
2 2
E i
E t
E r
H i
H r
H t z x O
Parallel
Perpendicular
The polarization of a parallel or perpendicular polarized wave remains unchanged after reflection and refraction.

80. The reflection and the transmission coefficients
For the parallel polarized wave, we have
Considering the condition for phase matching, we obtain
From the boundary condition of the tangential components of the magnetic fields to be continuous at the boundary, similarly we find
Based on the definitions of the reflection and the transmission coefficients at the boundary, we find the reflection coefficient and the transmission coefficient for the parallel polarized wave, respectively, as

81. For the perpendicular polarized wave, we obtain
If , it is called glancing incidence.
In this case, and no matter what kind of polarization and media may be.
It means that the incident wave is reflected at all, and the reflected and the incident waves have the same amplitude but opposite in phase. In other words, for the glancing incidence on any boundary, the reflection coefficients of the plane waves with any polarization are equal to −1.

82. When we view very slantingly the surface of an object, it appears to be quite bright.
This phenomenon leads to the blind area of the radar, and the radar is unable to detect the lower objects.

83. 9. Null and Total Reflections
For most media, we have and

84. Due to
Hence, if the angle of incidence satisfied the following condition:
then , and it means that all of the energy of the incident wave will enter medium ②, while the reflected wave vanishes. This phenomenon is called null reflection or total transmission.
The angle of incidence at which null reflection arises is called the Brewster angle denoted as B , and we find

85. We know
The reflection coefficient only if Hence, the perpen-dicularly polarized wave cannot have null reflection.
A plane wave with an arbitrary polarization can be resolved into a parallel polarized wave and a perpendicular polarized wave.
When a beam of unpolarized light is incident on the boundary at Brewster angle, the reflected wave becomes a perpendicular polarized wave since the parallel polarized portion will not be reflected. In optical engineering, a polarized light can be obtained this way.

86. If , then , and this phenomenon is called total reflection.
From Snell’s law , we can see that if the above equation is satisfied, then the angle of refraction will be
When the angle of incident is greater than the angle at which total reflection happens, total reflection still exists.

87. The angle at which total reflection just happens is called the critical angle, and it is denoted as c , and
Since the function , total reflection will happen only if
. Namely, that happens only if a plane wave is incident on a medium with a lower permittivity than that of the current medium. x z
During total reflection the refracted wave is c
We have
The greater the ratio or the angle of incidence is, the faster the amplitude will be decreased in the positive z-direction.

88. One kind of fiber optic consists of two layered dielectrics with different permittivities, and the permittivity of the interior core is greater than that of exterior coating. When light is incident on the interface from the interior core at , total reflection will happen. 2
Surface wave 1
Since there is a surface wave along the surface of the exterior coating, the optical fiber has to be clad from the outside. This arrangement leads to the composite structure of an optical cable.

89. All of the results obtained above hold under the premise of .
If and , only the perpendicular polarized wave has null reflection.
If and , both polarized waves can have null reflection.
Example. Suppose the parameters of the medium in the region
are , and in If the electric field intensity of the incident wave is
Find: (a) The frequency.
(b) The angles of reflection and refraction.
(c) The reflected and the refracted waves.

90. y i r t
1 1
2 2 z x
Solution: The incident wave can be resolved into a perpendicular polarized wave and a parallel polarized wave, i.e.
Where
Due to
We find

91. From we obtain
Then

92. The electric field intensity of the reflected wave is , here
The electric field intensity of the refracted wave is , here
Note that the change of the propagating directions of the reflected and the refracted waves in the calculations.

93. 10. Oblique Incidence at Conducting Boundary
Suppose medium ① is a perfect dielectric, while medium ② is a conducting medium, i.e.
For medium ② we use the equivalent permittivity. Namely, let
Then the intrinsic impedance of medium ② is
Since Zc2 is a complex number, both the reflection and the refraction coefficients are complex numbers, the null reflection and the total reflection conditions cannot arise.

94. The modified Snell’s refraction law is
The equiamplitude surface and the wave front are not in the same plane. Hence, it is a non-uniform plane wave.
The modified Snell’s refraction law is i r
 1  1
 2  2  2 z x
Equiamplitude surface
Wave front t
If , we have
i.e.
when a plane wave is incident on the sea surface from the air, if the seawater can be considered to be the good conductor for the given frequency, the refracted wave propagates almost vertically downward regardless of the angle of incidence.
For communication with submarines, the main lobe of the receiving antenna pattern must be directed upward.

95. 11. Oblique Incidence at Perfect Conducting Boundary
Suppose medium ① is a perfect dielectric, and medium ② is a perfect conductor, so that
Then the intrinsic impedance of medium ② is
And the reflective coefficients are
It means that when a plane wave is incident on the surface of a perfect conductor, total reflection always happens regardless of the angle of incidence and the polarization.
Since the reflection coefficient is related to the polarization of the incident wave. Consequently the distribution of the field in the above half-space depends on the polarization.

96. For the parallel polarized wave, the x-component of the resultant electric field is
Consider , , we have
In the same way, the z-component of the resultant electric field and the resultant magnetic field respectively are
The phase of the resultant wave is changed with the variable x, while the amplitude is related to the variable z. Hence, the resultant wave is a non-uniform plane wave traveling in the positive x-direction.

97. The amplitude of the component Ex is
Since the direction of the component of the electric field coincides with the propagating direction, the resultant field is a non-TEM wave. Only the magnetic field intensity is perpendicular to the propagating direction, and it is called transverse magnetic wave or TM wave. O
1 = 0
2 =  x z
The amplitude of the component Ex is Ex
The amplitudes of Ex varies with the variable z sinusoidally.
while the amplitudes of Ez and Hy have a cosine dependence.
There is a standing wave in the z-direction, while a traveling wave is propagating in the x-direction.

98. The complex energy flow density vector of the resultant wave is
and
The energy flow is along the x-direction, while there is only energy exchanged in the z-direction.
If an infinite perfectly conducting plane is placed at , the original fields are not affected since Ex= 0 at these locations. It means that there can be TM wave between two parallel, infinite, perfectly conducting planes.

99. Infinite, perfectly conducting planeEx O
1 = 0
2 =  x z
Infinite, perfectly conducting plane  E H S x
TM Wave

100. For the perpendicular polarized wave, we have
It is still a non-uniform plane wave traveling in the x-direction. However, only the electric field is perpendicular to the propagating direction, and it is called transverse electric wave or TE wave.
The amplitudes of Ey and Hz vary with the variable z sinusoidally, while the amplitude of Hx has a cosine dependence.
If an infinite perfect conducting plane is placed at , the original fields are not affected since at these locations. It means that there can be TE waves between two parallel, infinite, perfectly conducting planes.

101. If two more perfectly conducting planes are placed perpendi-cularly to the y-axis, it does affect the original fields as well since is perpendicular to these planes. In this way, a TE wave can exist in a rectangular metal tube consisting of the four perfectly conducting planes. Ey
1 = 0
2 =  y z  E H S x
TE wave
We will see that a rectangular or circular waveguide can transmit TE and TM waves only, but TEM wave.

102. Using , we find the x -component of the magnetic field is
Example. A perpendicular polarized plane wave is incident on an infinite perfectly conducting plane with an angle of incidence i from the air. If the amplitude of the electric field of the incident wave is , find the surface electric current density on the conducting surface and the average energy flow density vector in air. i r
 0  0
  
E i
E r
H i
H r z x O
Solution: Let the boundary be at the plane , Then the surface electric current density JS is
Using , we find the x -component of the magnetic field is
and

103. The average of the energy flow density vector is
We know that
Find

104. 12. Plane Waves in Plasma
The plasma is a kind of ionized gas, consisting of electrons, positive ions and neutral molecules. The ionosphere in the range 60~2000km above the earth is a plasma.
Under the influence of a steady magnetic field, a plasma will exhibit anisotropic electric behavior, with the permittivity having up to 9 elements. Hence, in the terrestrial magnetic field the ionosphere appears in electric anisotropy.
The permittivity is
The terrestrial magnetic flux density is about 0.03~0.07mT.

105. Ionosphere
1000km O3
Stratosphere
60km
Troposphere
12km

106. Ionosphere Earth E(t1) E(t2) Birefringence
The polarization direction of the plane wave will be modified.
E(t1)
E(t2)

107. 13. Plane Waves in Ferrite
Ferrite is a kind of magnetic material. Its permeability is very large and the relative permittivity  r = 2~35, but the conductivity is about 10-4~1 (S/m).
Under the influence of a steady magnetic field, a ferrite will display anisotropic magnetic behavior.
When a plane wave is propagating in ferrite, the birefringence and the modification of the polarization direction of plane wave will also happen. The change in the polarization direction is used in microwave devices.