Download Chapter 3. EM-wave Propagation Above the Terrain

Lecture 4: Reflection of Optical Waves from Intersections
4.1. Boundary Conditions
The simplest case of wave propagation over the intersection between two media is
that where the intersection surface can be assumed as a flat perfectly conductive.
If so, for a perfectly conductive flat surface the total electric field vector is equal to
zero, i.e., E  0 . In this case, as was shown above in Lecture 2, the tangential
component of electric field vanishes at the perfectly conductive flat ground surface,
that is,
E  0
(4.1)
Consequently, as follows from Maxwell’s equation   E( r)  iH( r) (see Lecture 1
for the case of   1 and B  H ), at such a flat perfectly conductive ground surface
the normal component of the magnetic field also vanishes, i.e.,
Hn  0
(4.1a)
As also follows from Maxwell’s equations (1.1), the tangential component of
magnetic field does not vanish because of its compensation by the surface electric
current. At the same time the normal component of electric field is also compensated
by pulsing electrical charge at the ground surface. Hence by introducing the Cartesian
coordinate system, one can present the boundary conditions (4.1)-(4.1a) at the flat
perfectly conductive ground surface as follows:
E x ( x, y, z  0)  E y ( x, y, z  0)  H z ( x, y, z  0)  0
4.2.
(4.2)
Main Reflection and Refraction formulas
As was shown above, the influence of a flat terrain on wave propagation leads to
phenomena such as reflection. Because all kinds of waves can be represented by
means of the concept of the plane waves, let us obtain the main reflection and
2
refraction formulas for a plane wave incident on a plane surface between two media,
the atmosphere and the earth, as shown in Fig. 4.1. The media have different dielectric
properties which are described above and below the boundary plane z=0 by the
permittivities and permeabilities  1 ,  1 and  2 ,  2 , respectively.
Fig. 4.1
Without reducing the general problem, let us consider a plane wave with wave vector
k and frequency   2 f
incident from a medium described by parameters
 1 and  1 . The reflected and refracted waves are described by wave vectors k 1 and
k 2 , respectively. Vector n is a unit normal vector directed from medium (  2 ,  2 )
into medium (  1 ,  1 ). According to relations between electrical and magnetic
components which follow from Maxwell’s equations, the incident wave can be
represented as follows:
E  E 0 expik  x    t  ,
H
1
kE
 1  | k|
(4.3)
The same can be done for the reflected wave
E1  E01 expik 1  x    t ,
and for the refracted wave
H
1
1 
k 1  E1
|k1|
(4.4)
3
E2  E02 expik 2  x    t ,
H
2
2 
k 2  E2
|k 2 |
(4.5)
The values of the wave vectors are related by the following expressions:
| k| | k 1 |  k 

 1 1 ,
c
|k 2 |  k 2 

22
c
(4.6)
From the boundary conditions that were described earlier by (4.1)-(4.2), one can easily
obtain the condition of the equality of phase for each wave at the plane z=0:
(k  x) z  0  (k 1  x) z  0  (k 2  x) z  0
(4.7)
which is independent of the nature of the boundary condition. Equation (3.37)
describes the condition that all three wave vectors must lie in a same plane. From this
equation it also follows that
k sin  0  k1 sin 1  k 2 sin  2
(4.8)
which is the analogue of Snell’s law:
 1 1 sin  0   2  2 sin  2
(4.9)
Moreover, because | k 0 | | k 1 | , we find  0   1 ; the angle of incidence equals the angle
of reflection.
It also follows from the boundary conditions that the normal components of
vectors D and B are continuous. In terms of the field presentation, these boundary
conditions at the plane z=0 can be written as
 E
1
0

 E1    2 E 2  n  0
k  E 0  k 1  E 1  k 2  E 2   n  0
E 0  E 1  E 2   n  0
(4.10)
1

1
 k 2  E 2   n  0
 k  E 0  k 1  E1  
2
 1

Usually in applying these boundary conditions for estimating the influence of the flat
ground surface on wave propagation over terrain, it is convenient to consider two
separate situations: The first one is when the vector of the wave’s electric field
component E is perpendicular to the plane of incidence (the plane defined by vectors
k and n ), but the vector of the wave’s magnetic field component H lies in this plane
4
(see Fig. 4.2). The second one is when the vector of the wave’s electric field
component E is parallel to the plane of incidence, but the vector of the wave’s
magnetic field component H is perpendicular to this plane (see Fig. 4.3).
Fig. 4.2
Fig. 4.3
5
In the literature which describes wave propagation aspects, they are usually called the
TE-wave (transverse electric) and the TM-wave (transverse magnetic), or waves with
vertical and horizontal polarization, respectively. We will derive the reflection and
refraction coefficients for the case of an incident plane wave with linear polarization;
the general case of arbitrary elliptic polarization can be obtained by use of the
appropriate linear combinations of the two results, following the approach presented
in Lecture 2.
First of all, we consider the incident plane linearly polarized wave with its
electric field perpendicular to the plane of incidence (TE-wave), as shown in Fig. 4.2.
The orientations of the magnetic field components of the incident, reflected and
refracted waves, Hi , i  0, 1, 2, are chosen to give a positive flow of energy in the
direction of wave vectors k , k 1 and k 2 , respectively. Since the electric fields are all
parallel to the boundary surface, the first boundary condition in (4.10) yields nothing.
The third and fourth conditions in (4.10) give
E 0  E1  E 2  0
(4.11)
1

 E 0  E1  cos  0  2 E 2 cos  2  0
1
2
while the second condition in (4.10), using Snell’s law (4.9), duplicates the third
condition. Now, from (4.11), we can obtain the amplitudes of the reflected and
refracted waves respectively:
1
( 2   2 ) 2  ( 1   1 ) 2 sin 2  0
2
| E1 | | E 0 |

 1   1 cos  0  1 ( 2   2 ) 2  ( 1   1 ) 2 sin 2  0
2
 1   1 cos  0 
| E 2 | | E 0 |
2 1   1 cos  0
1
 1   1 cos  0 
( 2   2 ) 2  ( 1   1 ) 2 sin 2  0
2
(4.12a)
(4.12b)
The same results can be obtained from (4.10) for the case of the TM-wave, when the
electric field vectors are parallel to the plane of incidence, as is shown in Fig. 4.3. The
6
boundary conditions for the normal component of vector D and for the tangential
components of vectors E and H lead to the first, third and fourth equations in (4.10),
from which follow:
( E 0  E 1 ) cos  0  E 2 cos  2  0
(4.13)
1

E 0  E1   2 E 2  0
1
2
The continuity of the normal components of the vector D , plus Snell’s law (4.9),
merely duplicates the second of equations (4.10). Therefore the amplitudes of the
reflected and refracted waves can be written as:
1
( 2   2 ) 2 cos  0   1   1 ( 2   2 ) 2  ( 1   1 ) 2 sin 2  0

| E1 | | E 0 | 2
1
( 2   2 ) 2 cos  0   1   1 ( 2   2 ) 2  ( 1   1 ) 2 sin 2  0
2
| E 2 | | E 0 |
2 1   1   2   2 cos  0
1
( 2   2 ) 2 cos  0   1   1 ( 2   2 ) 2  ( 1   1 ) 2 sin 2  0
2
For the real situation of wave propagation, it is usually permitted to put
(4.14a)
(4.14b)
1
 1.
2
introducing the relative permittivity (with respect to the air),  r   2 /  1 , we will
obtain, by use of (4.12) and (4.13) the expressions for the complex coefficients of
reflection (  ) and refraction (T) for waves with vertical (denoted by index V) and
horizontal (denoted by index H) polarization, respectively.
For vertical polarization:
RV | RV | e
jV
TV | TV | e jV 
'
For horizontal polarization:

 r cos 0   r  sin 2  0
 r cos 0   r  sin 2  0
2  r cos  0
 r cos  0   r  sin 2  0
(4.15a)
(4.15b)
7
RH | RH | e
j H
TH | TH | e j H 
'

cos 0   r  sin 2  0
cos 0   r  sin 2  0
2 cos  0
cos  0   r  sin 2  0
(4.16a)
(4.16b)
Here | V | , | H | , | TV | , | TH | and V ,  H , V' ,  'H are the modulus and phase of
the coefficients of reflection and refraction for vertical and horizontal polarization,
respectively. Dependence of the coefficient of reflection on the angle of incidence is
shown in Fig. 4.4 for two types of field polarization.
It is very important to note that for normal incidence of a radio wave on a flat
ground surface there is no difference between vertical and horizontal wave
polarization.
Fig. 4.4.
Thus, for  0  0, cos  0  1, sin  0  0, all the formulas above reduce to:
| E 1 | | E 0 |
| E 2 | | E 0 |
r  1
r  1
2
r  1
(4.17)
8
RV  RH 
r 1
r 1
(4.18)
2
TV  TH 
r 1
It should be noted that the results presented by (4.17) are correct only for  1   2 [6].
Moreover, for the reflected wave E 1 the sign convention is that for vertical
polarization (4.18). This means that if  2   1 there is a phase reversal of the reflected
wave. In the case of vertical polarization there is a special angle of incidence, called
the Brewster angle, for which there is no reflected wave. For simplicity we will
assume that the condition  1   2 is valid. Then from (4.15) it follows that the
reflected wave E 1 limits to zero when the angle of incidence is equal to Brewster’s
angle
 2 


 1
 0   Br  tan 1 
(4.19)
Another interesting phenomenon that follows from the presented formulas is called
total reflection. It takes place when the condition of  2   1 (or n2  n1 ) is valid. In
this case from Snell’s law (4.9) it follows that, if  2   1 , then

Consequently, when  0   0 kr the reflection angle  1  , where
2
 
 0 kr   c  sin 1  2 
 1 
 1   0 .
(4.20)
For waves incident at the surface (this case is realistic for ferroconcrete building’s
wall surfaces) under the critical angle  0   0kr   c there is no refracted wave within
the second medium; the refracted wave is propagated along the boundary between the
first and second media and there is no energy flow across the boundary of these two
media.
Therefore, this phenomenon called in literature total internal reflection (TIR),
and the smallest incident angle  0 for which we get TIR, is called the critical angle
 0   0kr   c .
9
4.3.
Analysis of Total Internal Reflection in Optics
Now, introducing the index of rays refraction for both media which can be defined as
n1  1  1 and n2   2   2 ,
we can in the same manner consider the ray
reflection from intersection of two media by introducing instead of propagation
parameters (see above). If so, the main reflection formulas (4.12) and (4.14) can be
rewritten as:
1) For TE-waves, as shown in Fig. 4.2, we immediately have from (4.12) the
amplitude of reflected and refracted rays respectively [1-3]:
1
n22  n12 sin 2  0
2
| E1 | | E 0 |

n1 cos  0  1 n22  n12 sin 2  0
2
n1 cos  0 
| E 2 | | E 0 |
2n1 cos  0

n1 cos  0  1 n22  n12 sin 2  0
2
(4.21a)
(4.21b)
For TM-waves, as shown in Fig. 4.3, the amplitudes of the reflected and refracted rays
can be written as [1-3]:
1 2
n2 cos  0  n1 n22  n12 sin 2  0
2
| E1 | | E 0 |
1 2
n2 cos  0  n1 n22  n12 sin 2  0
2
2n1n2 cos  0
| E 2 | | E 0 |
1 2
n2 cos  0  n1 n22  n12 sin 2  0
2
where, once more, n12   1 1 and n22   2  2 .
(4.22a)
(4.22b)
10
We can rewrite now a Snell’s, presented above for  0   1 , as [1-3] (see also
the geometry of the problem shown in Fig. 4.1):
n1 sin 1  n2 sin  2
(4.23)
or
sin  0  sin 1 
n2
sin  2
n1
(4.24)
If the second medium is less optically dense than the first medium which consist the
incident ray with amplitude | E 0 | , that is, n1  n2 , from (4.24) follows that
sin  0 
n2
n1
or
n1
sin  0  1
n2
(4.25)
The value of incident angle  0 for which (4.25) becomes true is known as a critical
angle, which was introduced above. We now define its meaning by use a ray concept
[4]. If a critical angle is determined by
sin  kr  sin  c 
n2
n1
(4.26)
then for all values of incident angles  0   kr   c the light is totally reflected at the
boundary of two media. This phenomenon is called in ray theory the total internal
reflection (TIR) of rays, the effect, which is very important in light propagation in
fiber optics. Figure 4.5 shows effect of total reflection when  0   c .
We also can introduce another main parameters usually used in fiber optic
communication (see Lecture 8). Thus, the effective index of refraction is defined as:
neff  n1 sin  0
(4.27)
11
When the incident ray angle  0  90 0 , neff  n1 , and when  0   c , neff  n 2 .
Fig. 4.5
In fiber optics there is another parameter usually used, called numerical aperture of
fiber optic guiding structure, denoted as N.A.,
N . A.  n1 sin  c  sin  a
(4.28)
where 2   a is so-called the angle of existence of full communication [5, 6], when
total internal reflection occurs in fiber optic structure.
Accounting for
cos 2   1  sin 2  , we finally get

N . A.  n12  n22

1/ 2
(4.29)
Sometimes, in fiber optic physics, designers used the parameter, called relative
refractive index difference:

n
2
1
 n22
2  n12
  N.A
2
2  n12
(4.30)
Using above formulas, we can find relations between these two engineering
parameters (we will talk about them in Lecture 8 discussing about fiber optic
parameters and properties):
N . A.  n1  ()1 / 2
(4.31)
12
Example 1: Let us consider that n1  1.45 and   0.02 (2%) . Find: N.A.
and 2   a .
Solution
1) According to (4.31)
N. A.  n1  ()1/ 2  sin 1  a
Then:
N . A.  n1  ()1 / 2  1.45  2  0.02  0.29
2)
N. A.  sin 1  a  sin 1 0.29  15.660
Then:
2   a  2  15.660  31.330
Let us now explain the total internal reflection from another point of view. When we
have total internal reflection, we should assume that there would be no electric field in
the second medium. This is not the case, however. The boundary conditions presented
above require that the electric field be continuous at the boundary, that is at the
boundary the field in region1 and region 2 must be equal. The exact solution shows
that due to total internal reflection we have in region 1 standing waves caused by
interference of incident and fully reflected waves, whereas in region 2 a finite electric
field decays exponentially away from boundary and carries no power into the second
medium. This wave called an evanescent field (see Fig. 4.6).
This field attenuate away from boundary as
E  exp z
(4.32)
where the attenuation factor equals

2

n12 sin 2  i  n22
(4.33)
13
At can be seen from (4.31), at the critical angle  i   c   0 , and attenuation
increases as the incident angle increases beyond the critical angle defined by (4.31).
Because  is so small near the critical angle, the evanescent fields penetrate deeply
beyond the boundary but do so less and less as the angle increases.
Fig. 4.6
However, behavior of Fresnel’s formulas (4.21)-(4.22) depends on boundary
conditions. Thus if the fields are to be continuous across the boundary, as required by
Maxwell’s equations, there must be a field disturbance of some kind in the second
media (see Fig. 4.1). To investigate this disturbance we can use Fresnel’s formulas.
We, first of all rewrite cos  2  (1  sin 2  2 ) 1/ 2 . For  2   kr we can present sin  2 by
use some additional function sin  2  cosh  , which can be more than unit. If so,
cos  2  j (cosh 2   1) 1/ 2   j sinh  . Hence we can write the field component in the
second medium to vary as (for nonmagnetic materials  1   2   0 )
x cosh   jz sinh   
 
exp j t  n2



c

(4.34a)
14
or
x cosh   
 
 n z sinh  
exp  2

 exp j t  n2


c
c 
 
(4.34b)
This formula represents a ray traveling in the x-direction in the second medium (that
is, parallel to the boundary) with amplitude decreasing exponentially in the z-direction
(at right angles to the boundary). The rate at which the amplitude decreases with z can
be written
 2z sinh  
exp 

2


(4.35)
where is a wavelength of the light in the second medium. As seen, the wave attenuate
significantly (~ e 1 ) over distances z of about  2 .
Example
2:
At
the
glass-air
interface,
the
critical
angle
of
is
 krr  sin 1 (1 / 15
. )  418
. 0 . For a light in the glass incident on the glass-air boundary
at 60 0 we find that sinh   164
. . Hence the amplitude of the wave in the second
medium is reduced by a factor of 5.4  10 3 in a distance of only one wavelength,
which is of the order of 1m = 10 -6 m .
Even though the wave is propagating in the second medium, it transports no
light energy in a direction normal to the boundary. All the light is totally internally
reflected (TIR) at the boundary. The fields, which exist in the second medium, give a
Poynting vector which averages zero in this direction over one oscillation period of
the light wave. The guiding effect is based on TIR phenomenon: all energy transport
occurs along the boundary of two media after TIR, without any penetration of light
energy inside the intersection.
15
Moreover, according to (4.21)-(4.22) and (4.31), we can note that the totally
internal reflected (TIR) wave undergoes a phase change which depends on both the
angle of incidence and the field polarization. This directly follows from derivations of
Fresnel’s equation (4.21)-(4.22). Namely, for TM wave (i.e., E || -polarization) from
(4.22) for  1   2   0 and from above mentioned it follows that
E1
jn sinh   n2 cos  0
 1
E0
jn1 sinh   n2 cos  0
(4.36)
This complex number provides the phase change on TIR as  p where for TM wave
( E || -polarization) we have

2
2
2
  p  n1 n1 sin  0  n2
tan  
n22 cos  0
 2

1/ 2
(4.37a)
and for TE wave ( E  -polarization)

n12 sin 2  0  n22
 s 
tan  
 2
n1 cos  0

1/ 2
(4.37b)
We note also that there is a close relationship between the light wave phase changes at
TIR for both kinds of waves, that is,
p 
 
tan   n12 tan s 
 2
 2
(4.38)
and that

2
2
2
  p   s  cos  0 n1 sin  0  n2
tan

n1 sin 2  0
 2 

1/ 2
(4.39)
The variations of phase changes,  p and  s , and their difference,  p   s , as a
function of the incident angle  0 are shown in Fig. 4.7. It is clear that the polarization
state of light undergoing TIR will changed as a result of the differential phase change
16
 p   s . By choosing  0 appropriately and perhaps using two TIRs, it is possible to
produce any required final polarization state from any given initial state. It is
interesting to note that the reflected ray in TIR appears to originate from a point,
which is displaced along the boundary from the point of incidence.
Fig. 4.7
This is consistent with the incident ray’s is being reflected from a parallel plane which
lies a short distance within the second boundary (see Fig. 4.8).
Fig. 4.8
This view is also consistent with the observed phase shift, which is now regarded as
being due to the extra optical path traveled by the ray. The displacement is known as
17
the Goos-Hanchen effect and provides an entirely consistent alternative expansion of
TIR [4].
Bibliography
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