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Lecture 5: Wave Propagation in Material Media
5.1. Basic Equations and Propagation Characteristics of Plane Waves
Because above was shown that both cylindrical and spherical waves can be
represented by plane waves, we consider below uniform plane waves in a material
medium, taking into account formulas and equations described in Lectures 1 and 2 for
isotropic homogeneous infinite medium where the material medium characteristics,
 ,  ,  , are constant and do not depend on coordinates. In this case the constitutive
relations are
J  E,
D  E,
B  H
(5.1)
and the Maxwell’s equations can be written as
B
H
 
t
t
D
E
H  J
 E  
t
t
E  
(5.2)
Without any loss of generality, let us consider one-dimensional case, when wave
propagates along z-axis, that is,
E  E x ( z, t )i x
H = H y ( z, t )i y
(5.3)
For this case the corresponding wave equations are:
H y
E x
 
z
t
(5.4)
H y
z
 E x  
E x
t
For time dependence e jt by introducing so-called phasors (see Lecture 1)
~
E x ( z , t )  Re E x ( z )e jt
~
H y ( z , t )  Re H y ( z )e jt




one can obtain from (5.4) the corresponding phasor equations:
(5.5)
2
~
E x
~
  jH y
z
~
H y
~
~
~
  E x  jE x     j  E x
z
(5.6)
By differentiation of the first equation in (5.6) and by use the second equation in the
system (5.6) we get
~
~
H y
2 Ex
~
~
  j
 j  j E x   2 E x
2
z
z
(5.7)
where

j   j 
(5.8)
is the propagation characteristic describing the propagation properties of EM-wave in
the concrete medium, that is, losses of wave energy and dispersion (see Leture 1).
The solution of (5.7) is follows
~
E x  Ae  z  Be  z
(5.9)
where A and B are constant which can be obtained from the corresponding boundary
conditions; the propagation parameter is complex and can be written as
    j
(5.10)
Here  describes the attenuation of EM-wave amplitude, that is, the wave energy
losses,  describes the phase velocity of the plane wave in the material media.
As an example, we show in Fig. 5.1 attenuation of wave for two kinds of 
(e.g., for two time instants, t1 and t 2 ). A full description of their properties, we will
show later.
Fig. 5.1
3
Now we can present the magnetic field phasor component in the same manner,
as the electric field by use (5.6) and (5.9):
~

1 E x
~
Hy  

( Ae  z  Be  z )
j z
j
  j ( Ae  z  Be  z ) 
j
(5.11)
1
( Ae  z  Be  z )

where
j
  j

(5.12)
is the intrinsic impedance of the medium, which is also complex. In free space, when

the material parameters are constants, i.e., dielectric constant  =  =8.854 10-12 F/m

and magnetic constant  =  = 4 10-7 H/m, but conductivity   0 (see Lecture 1),
and   0  120   377  . Then the wave velocity in free space, c 

1
, is
 0 0
also constant and equal 3 108 m/s.
Solutions (5.9) and (5.11) can be concretized by use the corresponding
boundary conditions. But this is not a goal of our future analysis. We show the reader
how the properties of the material medium change propagation conditions within it.
For this purpose we discuss further propagation parameters (5.8), (5.10) and (5.12)
associated with plane waves (5.9) and (5.11). As it follows from (5.8) and (5.10)
 2  (  j) 2  j  j
from which
 2   2   2 
2  
(5.13)
Squaring now first and second equations in (6.13), adding them and taking the square
root we finally have
 
      1   
  
2
2
2
2
Adding now the first equation from (6.13) and (6.14), we get
 
2       1   
  
2
or
2
2
2
4

 2 
2

 
1 
  1







2
1/ 2
(5.15a)
In the same manner
 
2 2   2    2  1   
  
2
or
2

 2 
 

1 
  1


2
  


1/ 2
(5.15b)

is defined as the loss tangent, which really is the ratio of the

~
magnitude of the conduction current density, | j c |~ E x , to the magnitude of the
In (5.15) parameter
~
displacement current density, | j d |~ E x , in the concrete material medium. As is
clear, this parameter is not simple function of  , because in the dispersion material
media  and  also depend on frequency  .
The phase velocity is described by the propagation parameter  along the
direction of propagation, which is defined by (6.15b):
v ph

 

2

2




 1     1
  


1/ 2
(5.16)
The dispersion properties follow from dependence on the frequency of the wave phase
velocity v ph  v ph () . Thus waves with different frequencies   2 f travel with
different phase velocities. In the same manner the wavelength in the medium is
dependent on the frequency of the wave:
2

2
2 


 1     1


  

f  

1/ 2
(5.17)
In view of attenuation of the wave with distance, the field variations with distances are
not pure sinusoidal, as in free space (see Lecture 1). In other words, the wavelength is
not exactly equal to the distance between two consecutive positive (or negative)
extrema. It is equal to the distance between two alternative zero crossings.
5
The intrinsic impedance, as follows from (5.9) and (5.11), is the ratio of the
amplitude of the electric field phasor to the amplitude of the magnetic field phasor,
i.e.,
~
E x  ,
~ 
H y  ,
for the (  ) wave
for the (  ) wave
(5.18)
From (6.8) and (6.12) is follows that
  j

   j

(5.19)
from which one immediatly has
 
  Re  
 
 
Im 
  
1


j

1
(5.20)
We can now present formulas (5.15a) and (5.15b) using general presentation of  in
the complex form, that is,    ' j " . If so,
 2  (  j ) 2  j   j "   2  '
(5.21)
where now
2

 2 ' 
    " 


1 
  1
2


  ' 


1/ 2
(5.22a)
and
2

 2 ' 
    " 


1 
  1
2


  ' 


1/ 2
(5.22b)
From general formulas (6.8), (6.15a), (6.15b) and (6.17) follow several special cases
for different kinds of material media. We also will use general complex presentation
of the dielectric permittivity and the corresponding formulas (5.21)-(5.22b). Let us
consider them.
6
5.2. Propagation of Plane Wave in the Perfect Dielectric Medium
Perfect dielectrics are characterized by   0 . Then from (5.8) we get
    j 
jj  j 
(5.23)
that is, the propagation parameter is purely imaginary. If so, we finally have
  0,
   
(5.24a)

1



(5.24b)

2
1


f 
(5.24c)

j

j
(5.25)
v ph 
and
Further


Thus the wave in the perfect dielectric medium propagates without attenuation
(   0 ) and with E and H in phase, as in free space, but with  0 replaced by  and
 0 replaced by  . In terms of the relative permittivity  r   /  0 and the relative
permeability  r   /  0 of the perfect dielectric medium, the propagation parameters
are:
  0  r  r 
2
rr
0
c
rr
v ph 

0
rr
  0
r
r
(5.26a)
(5.26b)
(5.26c)
(5.26d)
Here all parameters denoted by subscript “0” for free space have been introduced
above.
7
5.3. Propagation of Plane Wave in the Imperfect Dielectric Medium
This medium is characterized by   0, but  /   1, that is the magnitude of the
~
conduction current density, | j c |~ E x , is small compared to the magnitude of the
~
displacement current density, | j d |~ E x . Using the following expansion
(1  x ) n  1  nx 
n(n  1) 2
x ...
2!
we can rewrite (5.8) as:

j  j  j  1  j



 
2 
2 

1


j


1





2   8 2  2 
 8 2  2 
(5.27)
So that
 
2 
1 

2   8 2  2 
(5.28a)

2 
    1 

 8 2  2 
(5.28b)

In the same manner


j

  j
j 

1  j 
j 
 
 
3 2 
 
1

j


2 2
  8  
2 
1/ 2
(5.29)
v ph

1 
2 
 
1  2 2 

  8  
(5.30a)

2
1 
2 

1  2 2 

f   8  
(5.30b)
In all expressions all terms with power higher than two have been neglected, since
 /   1. As follows from (5.28)-(5.30), for all practical purposes (  /  is not
higher than 0.1), the only significant feature different from the perfect dielectric case
is the attenuation  .
8
We do the same procedure, but taking into account the complexity of the
dielectric permittivity too, based on formulas (5.22a) and (5.22b). Finally, we get
  "
' 2

(5.31a)

 "2 
    ' 1  2 
8 '

(5.31b)

Now, if we will introduce the refractive index n  n' jn" , where n'   ' /  0 and
n"   " /  0 [5, 6], we will get in the case of a low-loss dielectric (or "imperfect"
dielectric) that

n"
c
and n"  n'
"
2 '
(5.32)
5.4. Propagation of Plane Wave in Good Conductor Medium
Good conductors are characterized by  /   1, the opposite of imperfect
~
~
dielectrics. In this case: | j c |~ E x | j d |~ E x . Here we obtain

j  j 
 e
j

4
j
(5.33)
 f 1  j 
So that
  f
(5.34)
  f
In the same manner we obtain that

j

  j
j



 j 4
e 

f
1  j 

(5.35)
and



4 f

(5.36a)
2


4
f
(5.36b)
v ph 

9
As clear seen, most parameters of propagation are proportional to f 1/ 2 , when  and
 are constant. This behavior is much different from the above case of imperfect
dielectric.
We can also define a skin depth as a distance at which the field is attenuated
by a factor e 1 or 0.368. This distance equals 1/  and is denoted by the symbol  :

1


1
f
(5.37)
Phenomenon of concentration of the wave field energy near to the skin of the
conductor is known as the skin effect.
Example 1: Find expression of skin depth for cupper and intrinsic impedance.
Solution
1) The skin depth for copper is equal to
 
1
f 4  10 7 1s   5.8  10 7  mS 

0.066
f
m 
2) The amplitude of intrinsic impedance is equal to
|  |
2f  4  10 7
 3.69  10 7
7
5.8  10
f, 
Thus, the intrinsic impedance of copper has a low magnitude as 0.369  even at the
high frequency of 1012 Hz . Thus in copper the field is attenuated by a factor e 1 in a
distance of 0.066 mm even at the low frequency of 1 MHz , resulting in the
concentration of the field energy near to the skin of the conductor.
We will show now other feature, which follows from (5.35). In fact, as
follows, the intrinsic impedance of good conductor has a phase ange of 450 . Hence
the components E and H of the EM-field in such a medium are out of phase by 450 .
The amplitude of intrinsic impedance is given by
| | 
f
1  j  

2f

(5.38)
From (5.23) and (5.34) it follows that the magnitude of the intrinsic impedance of
conductor is less than that of dielectric for the same  and  . In fact,
| cond | 
2f
 



| diel |

 

(5.39)
10
and because of

 1 we finally have that | cond | | diel | . Thus, for the same

parameters  and  , as well as for the constant E , the H component inside a
dielectric much less than that inside the conductor.
Using again relationships between the complex dielectric permittivity and the
refractive index, n  n' jn" , where n'   ' /  0 and n"   " /  0 , we can obtain for
a good conductor the following formulas

n"
c


2
and n"

2
(5.40)
5.5. Polarization and Magnetization of the Materials
Relationship between outer and inner fields in the materials, dielectric or conductive,
can be explain by the orientation of electrical and magnetic "dipoles" inside the
material.
Thus, in the case of dielectric material, each molecule/atom is oriented
randomally (because the thermal temperature is differing from the absolute zero
temperature. 0K  2730 C , (as shown in Fig. 5.2a). If a material is placed between
two electrodes, that are separated by a distance a, an outer electric field is applied
between the two electrodes, which leads to reorientation of the molecules/atoms, as
electrical dipoles, along the outer electric field (as shown by Fig. 5.2b).
Fig. 5.2
11
This field we will call the applied field and will denote it E a . The reorganization of
the molecules/atoms in a material due to application of an outer electric field creates a
polarized charge inside the material at the two edges, as shown in Fig. 5.3. The
density of such charge is  P . This polarization charge creates a polarization field with
momentum P, according to which the intrinsic (inner) electric field is create inside the
material. We will call this field the secondary field and will denote it E s .
Fig. 5.3
Thus we can now present the total field E as a vector sum of two fields, the applied
and the secondary, that is, E  E a  E s , as is simply sketched in Fig. 5.4.
Applied field
Ea
Total field
+
+
Dielectric
E = Ea + Es
Secondary field
Es
Fig. 5.4
Polarization
12
We can, therefore, relate the vector of induction of the total electric field in a material,
D, recall as a vector of displacement of polarization charge density or the electric flux
density, with the polarization vector P and present the first as
D  E   0 r E a  E s    0 r E a  P
(C / m 2 )
(5.41)
In many materials, the vector of polarization P is linearly proportional to the applied
electric field E a , that is,
P   0 r  e E a
(5.42)
where  e is the electric susceptibility. This parameter characterizes the resistance of
electrical dipoles against the outer applied field, i.e., their possibility to be directed
along the desired direction by this outer electric field, because they create their
intrinsic (inner) field, which is resisted to the outer field. The expressions (5.41) and
(5.42) apply only for linear and isotropic materials. As for plasma or crystals, there we
should use instead of value of dielectric constant its tensor (see Lecture 1).
It is clear seen from relations (5.41) and (5.42) that E s   e E a and
P   0 r E s . Here, as above,  r is the relative dielectric constant for a material. In the
vacuum,  e =0 and  r =1 by definition.
The same random orientation of magnetic dipoles occurs in a material for the
thermal temperature differing from absolute zero temperature (as shown in Fig. 5.5a).
Fig. 5.5a
Fig. 5.5b
If now we will put a material in the external (outer) magnetic field, each dipole
with its own magnetic momentum m is in opposition to this external field. This field
we will call the applied field and will denote it B a . All such dipoles create, finally,
13
the domain structure, where the magnetic momentum of each domain consists
moments of all atoms inside each of them and pointed in Fig. 5.5b in the same
direction. Intrinsic magnetic field created by all domains is opposite the applied
magnetic field. We will call this field the secondary field and will denote it B s . Thus
we can now present the total field E as a vector sum of two fields, the applied and the
secondary, that is, B  B a  B s , as is simply sketched in Fig. 5.6.
Applied field
Ba
Total field
+
+
B = Ba + Bs
Secondary field
Magnetic
material
Magnetization
Bs
Fig. 5.6
If we will now introduce the total momentum of magnetization of all domains
inside the material, denoted it as M, we can, therefore, relate the vector of induction of
the total magnetic field in a material, B, with the momentum M and present the first
as
B  H total   0  r H a  H s    0  r H a  M (Tesla )
(5.43)
In many materials, the vector of polarization M is linearly proportional to the applied
magnetic field H a , that is,
M  0 r  mH s
(5.44)
where  m is the magnetic susceptibility. Then,
B  H total   0 (1   m )H total   0  r H total
(5.45)
14
This parameter characterizes the resistance of magnetic dipoles of molecules/atoms
[as shown in Fig. 5.7a] or domains [as shown in Fig. 5.7b] against the external applied
field, i.e., their possibility to be directed along the desired direction by this external
magnetic field, because they create their intrinsic (inner) magnetic field, which is
opposite the outer field and characterized by a total magnetized momentum M. The
parameter of the magnetic susceptibility determines three classes of materials:
diamagnetic (when  m is negative and very small – typical value for  m  10 5 ),
paramagnetic (when  m is positive and very small – typical value for  m  10 5 ),
and ferromagnetic (when  m is positive and very large– typical values, for example,
for pure iron is  m  4  103 and can achieve 104 for strong ferromagnetic).
Fig. 5.7
Much convenient, in practical point of view, to use normalized permeability  r
instead of magnetic susceptibility  m , since  r  1   m . We should note that Fig.
5.7a presents situation occurring in paramagnetic and diamagnetic materials, whereas
Fig. 5.7b presents situation with ferromagnetic material.
It is clear seen from relations (5.43) and (5.44) that H s   m H a and
M   0  r H s . In Table 5.1 most useful diamagnetic, paramagnetic and ferromagnetic
materials are presented.
Table 5.1: Relative Permeability of Different Materials
15
It is clear seen that except ferromagnetic materials,  m  0 and  r is closed to unit,
whereas for ferromagnetic materials, such as nickel, steel, iron, ferrite, it can achieve
from hundreds to thousands units.
5.6. Summary
In Fig. 5.8, we summarize all results obtained and discussed above. As follows from
presented there, as  varies from 0 to  , a material medium is classified as:
* perfect dielectric for   0 ,
* imperfect dielectric for   0, but  /   1,
* good conductor for  /   1,
* perfect conductor for    .
The same classification can be done in “dispersive” parameter f (for   0 ):
* imperfect dielectric for f  f q 
* good conductor for f  f q 

,
2 

.
2 
But, dispersion properties of such materials, as plasma and ferromagnetic, are more
complicated, because  ,  and  can be also functions of frequency f.
16
Fig. 5.8
Example 2: Given e/m plane wave with the magnetic component, having field
strength B0  10 A / m and frequency f=600kHz, propagate in positive direction along
z-axis in cuprum medium with  r  1 and   5.8  107 S / m .
Find:  ,  , the depth of skin layer  , and B(z, t)
Solution
1) Since cuprum is a good conductor, according to formulas (5.34), we get
    f r  0  3.14  6  105  1  4  3.14  10 7  5.8  10 7  117  10 2 m 1
2) According to Example 1,

0.066
f
m 

0.066
66

 10 5  5.64  10 5 (m)  56.4m
2
117
117  10
 

3) B( z, t )  10  exp  117  10 2 z cos (2  6  10 5 )t  117  10 2 z u z ( A / m)
17
Above results enable us to point out that for the case of perfectly conductive medium,
when    , and as follows from (5.38)   0 . Thus there is no any penetration of
EM fields into the perfectly conductive material.
Example 3: In sea water with   4 S / m ,  '  81 0 , the frequency is 100
MHz. Find  and the attenuation (in dB/m) considering that transmission of wave is
proportional to   exp  z  . What will be, if the frequency will increase up to 10
THz?
Solution


1) Since  /  '  4 / 2  10 8 Hz  81  10 9 / 36  9  1 , we can consider seawater as
a good conductor. Then, using (5.40), we get


2

4  2  10 8  4  10 7
 39.7 m 1
2
2) Then, attenuation
1
1
Lt  10 log   10z log e  4.34  4.34  39.7  172.5 dB  m 1
z
z
3) Now, for f=10 THz, we have for seawater that n" 0.328 , and

n"
c

2  1013 Hz  0.328
 6.87  10 4 m 1
8
3  10 m / s
and
Lt  4.34  2.98  105 dB  m 1 .
Example 4: The absorption coefficient of glass at   10 m is   1.8cm1 .
Find the imaginary part of refractive index n".
Solution

n"
c
 1.8cm 1 
2

n"
from which we get

10 5 m  1.8  10 2 m 1
n" 

 2.9  10  4
2
2
18
5.7. Propagation Characteristics of Laser Beam in Material Media
Earlier we considered only plane waves, which are very specific case for optical
communication, since each wave, spherical or cylindrical, can be presented by plan
wave only far from the source. A more common case of beam energy spatial
distribution in optical communications, created by various optical sources (see Lecture
8), is the Gaussian beam. Its intensity distribution can be given by [5]:

I  I 0 exp  2(r / w2

(5.46)
where w is the initial width of circular spot of light of the source and I 0 is its output
initial intensity (namely, of the laser, see Lecture 8). This beam intensity distribution
is plotted in Fig. 5.9.
Fig. 5.9.
The radial distance from the origin of laser circular spot is r, therefore w is usually
called the spot size. It is clear that at r=w, I decrease with factor 1/e, that is,
I / I 0  e 2 for r=w. Far from the source, the electric field can be presented a
Gaussian plane wave traveling in the z direction (for 1-D case), that is,
E  E0 e r / w  e z e j (t kz )
2
(5.47)
19
Therefore, the intensity of such kind of wave equals
I  E  E *  E02 e 2r / w  e 2z
2
(5.48)
where E* is the complex conjugate of E. From (5.48) it follows that the peak intensity
at the center of the laser beam for any position z is E02e 2r / w  [along the propagation
2
path] and E 02 is the peak intensity at the origin (r=0, z=0).
Bibliography
[1] Jackson, J. D., Classical Electrodynamics, New York: John Wiley & Sons, 1962.
[2] Chew, W. C., Waves and Fields in Inhomogeneous Media, New York: IEEE Press,
1995.
[3] Elliott, R. S., Electromanetics: History, Theory, and Applications, New York:
IEEE Press, 1993.
[4] Kong, J. A., Electromagnetic Wave Theory, New York: John Wiley & Sons,
1986.
[5] Optical Fiber Sensors: Principles and Components. Ed. by J. Dakin and B.
Culshaw, Arthech House, Boston-London, 1988.
[6]
Palais,
J.
C.,
Optical
Communications,
in
Handbook:
Engineering
Electromagnetics Applications, Ed. by R. Bansal, New York: Taylor and Frances,
2006.
[7] Kopeika, N. S., A System Engineering Approach to Imaging, Washington: SPIE
Optical Engineering Press, 1998.