1. Dr. Rakhesh Singh Kshetrimayum
6. Plane waves reflection from media
interface
Dr. Rakhesh Singh Kshetrimayum
3/25/20141 Electromagnetic FieldTheory by R. S. Kshetrimayum
2. 6.1 Introduction
Plane waves reflection
from a media interface
Normal
incidence
TE
Oblique
incidencePerfect
conductor
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum2
TE
Lossless medium
Good
conductor
Fig. 6.1 Plane waves reflection from media interface
Lossy conducting
medium
TM
Brewster angle
Total internal reflection
Effect on
polarization
3. 6.1 Introduction
Till now, we have studied plane waves in various medium
Let us try to explore how plane waves will behave at a media
interface
In practical scenarios of wireless and mobile
communications,
radio wave will reflect from
walls &
other obstacles on its path
When a radio wave reflects from a surface,
the strength of the reflected waves is less than that of the
incident wave
3/25/20143 Electromagnetic FieldTheory by R. S. Kshetrimayum
4. 6.1 Introduction
The ratio of the two (reflected wave/incident wave) is
known as the‘reflection coefficient’ of the surface
This ratio depends on the
conductivity (σ),
permittivity (ε) and
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum4
permittivity (ε) and
permeability (µ)
of the material that forms the reflective surface
as well as material properties of the
air
from which the radio wave is incident
5. 6.1 Introduction
Some part of the wave will be transmitted through the
material
How much of the incident wave has been transmitted
through the material is
also dependent on the material parameters mentioned above
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum5
also dependent on the material parameters mentioned above
It is given by another ratio known as‘transmission
coefficient’
It is the ratio of the transmitted wave divided by the incident
wave
6. 6.1 Introduction
In plane wave reflection from media interface,
what we will be doing is
basically writing down the
electric and
magnetic field expressions
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum6
magnetic field expressions
in all the regions of interest and
apply the boundary conditions
to get the values of the
transmission and
reflection coefficients
7. 6.1 Introduction
One of the possible applications of such an exercise is in
wireless communication,
where we have to find the
transmission and
reflection coefficients of
multipaths
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum7
multipaths
Electromagnetic waves are often
reflected or
scattered or
diffracted
at one or more obstacles before arriving at the receiver
8. 6.2 Plane wave reflection from media
interface at normal incidence
For smooth surfaces,
EM waves are reflected;
For rough surfaces,
EM waves are scattered;
For edges of surfaces,
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum8
For edges of surfaces,
EM waves are diffracted
Fig. 6.2 depicts multipath for a mobile receiver in the car from
direct,
reflected and
diffracted waves
of the signal sent from the base station
9. 6.1 Introduction
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum9
Fig. 6.2 Illustration of reflected and diffracted wave in mobile
communications (this chapter’s study will be very helpful in
understanding wireless and mobile communications)
10. 6.2 Plane wave reflection from media
interface at normal incidence
We will consider the case of normal incidence,
when the incident wave propagation vector is along the normal
to the interface between two media
6.2.1 Lossy conducting medium
We will assume plane waves with electric field vector
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum10
We will assume plane waves with electric field vector
oriented along the x-axis and
propagating along the positive z-axis without loss of
generality
For z<0 (we will refer this region as region I and it is
assumed to be a lossy medium)
11. 6.2 Plane wave reflection from media
interface at normal incidence
Let us do a more generalized analysis by assuming this region
I as a lossy medium
It could be a lossless medium like free space
The incident electric and magnetic fields can be expressed as
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum11
1
ˆ z
i oE xE e γ−
=
r
1
1
1
ˆ z
i oH y E e γ
η
−
=
r
12. 6.2 Plane wave reflection from media
interface at normal incidence
where η1 is the medium 1 wave impedance and
Eo is the arbitrary amplitude of the incident electric field
The expression for intrinsic wave impedance and propagation
constant
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum12
( )
1 1 1 1
1 1 1 1 1 1
1 1 1 11 1 1
; 1
j j j j
j j
jj j
ωµ ωµ ωµ σ
η γ α β ω µ ε
γ ωε σ ωεωµ ωε σ
= = = = + = −
++
13. 6.2 Plane wave reflection from media
interface at normal incidence
iE
r
E
r
2
t
γ
r
1
i
γ
r
iH
r
tE
r
tH
r
iE
r
r
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum13
2 2 2, ,ε µ σ
rE
r
1 1 1, ,ε µ σ
1
r
γ
r
rH
r
rE
r
Fig. 6.3A plane EM wave is incident from region I or medium 1 for z<0
14. 6.2 Plane wave reflection from media
interface at normal incidence
Convention:
a circle with a dot in the center an arrow pointing
perpendicularly out of the page and
a circle with a cross an arrow pointing perpendicularly into
the page)
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum14
the page)
Notation for fields:
subscript i incident, r reflected and t transmitted
15. 6.2 Plane wave reflection from media
interface at normal incidence
Note that the incident wave from region I will be partially
reflected and transmitted at the media interface between two
regions
For reflected wave, z<0, wave direction is in the –z axis and
the reflected electric field is expressed as
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum15
the reflected electric field is expressed as
where the Γ is the reflection coefficient
1
ˆ z
r oE x E e γ+
= Γ
r
16. 6.2 Plane wave reflection from media
interface at normal incidence
The reflection coefficient is defined as the ratio of amplitude
of the reflected electric field divided by amplitude of the
incident electric field as follows:
r
i
E
E
Γ =
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum16
The reflected magnetic field can be obtained from the
reflected electric field using the Maxwell’s curl equations as
iE
17. 6.2 Plane wave reflection from media
interface at normal incidence
1
1
1 1 1
ˆ ˆ ˆ
0 0
r r
r r
r
z
o
E j H
x y z
E j E j
H
j x y z
E e γ
ωµ
ωµ ωµ ωµ
+
∇× = −
∇× ∇× ∂ ∂ ∂
⇒ = − = =
∂ ∂ ∂
Γ
r r
r r
r
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum17
( ) ( ) ( )1 1 11
1
1 1 1
0 0
ˆ ˆ ˆ
o
z z z
o o o
E e
j j
y E e y E e y E e
z j
γ γ γγ
γ
ωµ ωµ ωµ
+ + +
Γ
∂
= Γ = Γ = − Γ
∂
1
1
ˆ z
r oH y E e γ
η
+Γ
∴ = −
r
18. 6.2 Plane wave reflection from media
interface at normal incidence
Therefore, the Poynting vector for reflected wave of region I
is
shows power is traveling in the -z axis for the reflected wave
2 2* 2
*
1
ˆz
r r r o
z
S E H E e α
η
+
= × == − Γ
r r r
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum18
shows power is traveling in the -z axis for the reflected wave
For transmitted wave, z>0, wave is propagating in lossy
medium 2 (we will refer this region as region II),
19. 6.2 Plane wave reflection from media
interface at normal incidence
where η is the wave impedance of lossy medium 2 and τ is
z
ot eExE 2ˆ γτ −
=
r
zo
t e
E
yH 2
2
ˆ γ
η
τ −
=
r
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum19
where η2 is the wave impedance of lossy medium 2 and τ is
the transmission coefficient
intrinsic wave impedance and propagation constant are
20. 6.2 Plane wave reflection from media
interface at normal incidence
transmission coefficient is defined as the ratio of amplitude of
the transmitted electric field divided by amplitude of the
( )
2 2 2 2
2 2 2 2 2 2
2 2 2 22 2 2
; 1
j j j j
j j
jj j
ωµ ωµ ωµ σ
η γ α β ω µ ε
γ ωε σ ωεωµ ωε σ
= = = = + = −
++
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum20
the transmitted electric field divided by amplitude of the
incident electric field as follows:
t
i
E
E
τ =
21. 6.2 Plane wave reflection from media
interface at normal incidence
Our intention here is to find the transmission and reflection
coefficients
Let us rewrite the fields in the two regions: region I (z<0)
and region II (z>0) (seeTable 6.1) and
Apply the boundary conditions to obtain the two unknown
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum21
Apply the boundary conditions to obtain the two unknown
coefficients
22. 6.2 Plane wave reflection from media
interface at normal incidence
Region I (lossy medium 1) Region II (lossy medium 2)
Table 6.1 Fields in the two lossy regions (normal incidence)
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum22
1
ˆ z
i oE xE e γ−
=
r
1
1
1
ˆ z
i oH y E e γ
η
−
=
r
1
ˆ z
r oE x E e γ+
= Γ
r
1
1
ˆ z
r oH y E e γ
η
+Γ
= −
r
z
ot eExE 2ˆ γτ −
=
r zo
t e
E
yH 2
2
ˆ γ
η
τ −
=
r
23. 6.2 Plane wave reflection from media
interface at normal incidence
This is basically boundary value problem with boundary
conditions at z=0
Note that total electric and magnetic fields (both incident
and reflected) are tangential to the interface at z=0
Similarly, the transmitted electric and magnetic field are also
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum23
Similarly, the transmitted electric and magnetic field are also
tangential to the interface at z=0
Also note that there are no surface current density at the
interface
Hence, the tangential components of electric and magnetic
fields must be continuous at the interface z=0
24. 6.2 Plane wave reflection from media
interface at normal incidence
Therefore,
( ) 0 00
1 1i r t z
E E E E Eτ τ
=
+ = ⇒ + Γ = ⇒ + Γ =
r r r
0 0
0
1 2 1 2
1 1
i r t
z
H H H E E
τ τ
η η η η=
− Γ − Γ
+ = ⇒ = ⇒ =
r r r
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum24
Therefore,
Hence,
( ) ( ) 2 1
2 1
1 2 1 2 2 1
1 1 1
1 1
η ητ
η η
η η η η η η
−− Γ − Γ + Γ
= ⇒ = ⇒ − Γ = + Γ ⇒ Γ =
+
2 1 2
2 1 1 2
2
1 1
η η η
τ
η η η η
−
= + Γ = + =
+ +
25. 6.2 Plane wave reflection from media
interface at normal incidence
6.2.2 Lossless medium
If the regions are lossless dielectric,
then, σ = 0 and µ and ε are real quantities
The propagation constant for this case is purely imaginary
and can be written as
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum25
and can be written as
γ = jβ =jω√(µε)
The wave impedance of the dielectric is
η= = = = =jωµ
γ
j
j
ωµ
β
ωµ
ω µε
µ
ε r
r
ε
µ
η0
26. 6.2 Plane wave reflection from media
interface at normal incidence
For a lossless medium,
η is real, so,
both Γ and τ are real
Electric and magnetic fields are in phase with each other
The wavelength in the dielectric is
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum26
The wavelength in the dielectric is
= =
And the phase velocity is
= = =
where c is the speed of light in free space
λ 2π
β
2π
ω µε
pv ω
β µε
1
rr
c
εµ
27. 6.2 Plane wave reflection from media
interface at normal incidence
Table 6.2 Fields in the two lossless regions (normal incidence)
Region I (lossless medium 1) Region II (lossless medium 2)
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum27
1
ˆ j z
i oE xE e β−
=
r 1
1
1
ˆ j z
i oH y E e β
η
−
=
r
1
ˆ j z
r oE x E e β+
= Γ
r 1
1
ˆ j z
r oH y E e β
η
+Γ
= −
r
2
ˆ j z
t oE x E e β
τ −
=
r
2
2
ˆ j zo
t
E
H y e β
τ
η
−
=
r
28. 6.2 Plane wave reflection from media
interface at normal incidence
Applying the boundary conditions like before,
we can get the expression for
reflection and
transmission coefficients,
we will get the same expression as before
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum28
we will get the same expression as before
2 1
2 1
;
η η
η η
−
Γ =
+
2
1 2
2η
τ
η η
=
+
29. 6.2 Plane wave reflection from media
interface at normal incidence
Poynting vector in region I
( ) ( ) ( ) ( )1 1 1 1
* *
* *
1 1 1 0 0
0 0
1 1
ˆ ˆ ˆ ˆj z j z j z j z
tot tot i r i r
E E
S E H E E H H E e x E e x e y e yβ β β β
η η
− + −
= × = + × + = + Γ × −Γ
r r r r r r r
2 2 2 2 2 22 2 2j z j zβ β− + Γ Γ Γ Γ
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum29
( ) ( )
( ) ( )
1 1
1 1
2 2 2 2 2 22 2 2
0 0 0 0 0 0 2 22
1 1 1 1 1 1
2 2
0 02
1
1 1
ˆ ˆ1
ˆ1 2 sin 2
j z j z
j z j z
E E e E e E E E
z e e z
E E
j z z
β β
β β
η η η η η η
β
η η
− +
+ −
Γ Γ Γ Γ
= − + − = − Γ + −
Γ
= − Γ −
30. 6.2 Plane wave reflection from media
interface at normal incidence
Poynting vector in region II
( )2 2
* 22
02 * 0
0
2 2
ˆ ˆ ˆj z j z
t t
EE
S E H x E e e y zβ β ττ
τ
η η
− −
= × = × =
r r r
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum30
( )
( ) ( ) ( )
2 2
2 12 2 1 2
2 2 2
12 1 2 1 2 1
4 41
1 1
η η η η η
ηη η η η η η
− −Γ
− Γ = − = ⇒ =
+ + +
Q
( )
( ) 222 2 2
02 22 2
2
2 1 1 2 12 1
12 4 1
ˆ
E
S z
η η τ
τ τ
η η η η ηη η
− Γ− Γ
= ⇒ = ⇒ = ∴ =
+ +
r
31. 6.2 Plane wave reflection from media
interface at normal incidence
Power conservation
Compare the time average power flow in the two regions
For z <0, the time average power flow through 1m2 cross
section is
( )
2
1 1 1− Γr
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum31
And for z >0, the time average power flow through 1m2
cross section is
( )
2
21 1
0
1
1 1 1
ˆRe
2 2
avgS S z E
η
− Γ
= • =
r
( )
2
22 2
0
1
1 1 1
ˆRe
2 2
avgS S z E
η
− Γ
= • =
r
32. 6.2 Plane wave reflection from media
interface at normal incidence
Hence,
So real power is conserved
6.2.3 Good conductor:
If the region II (z >0) is a good (but not perfect) conductor
1 2
avg avgS S=
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum32
If the region II (z >0) is a good (but not perfect) conductor
and
region I is a lossless medium like free space,
the propagation constant for medium 2 is
33. 6.2 Plane wave reflection from media
interface at normal incidence
where δ is the skin depth for the medium 2,
which is a good conductor
Similarly, the intrinsic impedance of the conducting medium
( ) ( )2 2
2 2 2 2
1
1 1
2 s
j j j
ωµ σ
γ α β
δ
= + = + = +
2
s
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum33
Similarly, the intrinsic impedance of the conducting medium
2 (see also example 6.2) simplifies to
( )
( )
( )( )
( )
( )
( )
( )
2
22 2 2
2
2 22
2 2 2 2
2
2 2
1 1
(1 )
2
1 1 12 2 2
1
1 1 2 21
j j j j
j j
j
j j j j
j
j j j
ωµωµ ωµ ωµ
η
γ ωµ σ σωµ σ
ωµ ωµ ωµ ωµ
σ σ σ σ
= = = =
+ +
+
− + +
= = = = +
+ − −
34. 6.2 Plane wave reflection from media
interface at normal incidence
Now, the wave impedance of medium 2 is complex, with a
phase angle of 45o so the electric
and magnetic field will be 45o out of phase and Γ and τ will
be complex.
Now, let us write down the field expressions in region I and
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum34
Now, let us write down the field expressions in region I and
II for lossless dielectric (region I) and good conductor
(region II) interface
35. 6.2 Plane wave reflection from media
interface at normal incidence
Table 6.4 Fields in the two regions (normal incidence)
Region I (lossless medium) Region II (good conductor)
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum35
1
ˆ j z
i oE xE e β−
=
r 1
1
1
ˆ j z
i oH y E e β
η
−
=
r
1
ˆ j z
r oE x E e β+
= Γ
r
1
1
ˆ j z
r oH y E e β
η
+Γ
= −
r
2
ˆ z
t oE x E e γ
τ −
=
r
2
2
ˆ zo
t
E
H y e γ
τ
η
−
=
r
36. 6.2 Plane wave reflection from media
interface at normal incidence
Applying the boundary conditions like before,
we can get the expression for
reflection and
transmission coefficients,
we will get the same expression as before
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum36
we will get the same expression as before
2 1
2 1
;
η η
η η
−
Γ =
+
2
1 2
2η
τ
η η
=
+
37. 6.2 Plane wave reflection from media
interface at normal incidence
Note that we have to use the previous expressions for
intrinsic wave impedances for lossless medium and good
conductor
1 ;
µ
η
ε
= ( ) 2
2 1
2
j
ωµ
η
σ
= +
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum37
Power conservation
let us find out the Poynting vector in the two regions
Noting that the field in region I comprise of incident and
reflected wave,
we can write the Poynting vector for z <0
ε ( )2
2
1
2
jη
σ
= +
38. 6.2 Plane wave reflection from media
interface at normal incidence
For z >0, only transmitted fields exist and we can write
( ) ( ) ( ) ( )
( )
1 1 1 1
1 1
* *
* *
1 1 1 *0 0
0 0
1 1
2
20 2 2*
1
ˆ ˆ ˆ ˆ
ˆ1
j z j z j z j z
tot tot i r i r
j z j z
E E
S E H E E H H E e x E e x e y e y
E
e e z
β β β β
β β
η η
η
− + −
− +
= × = + × + = + Γ × − Γ
= − Γ + Γ − Γ
r r r r r r r
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum38
For z >0, only transmitted fields exist and we can write
down the Poynting vector as
( )
*
2 2* *
02 * 20
0 * *
2 2
ˆ ˆ ˆz z z
t t
EE
S E H x E e e y z eγ γ αττ
τ
η η
− − −
= × = × =
r r r
39. 6.2 Plane wave reflection from media
interface at normal incidence
At z=0,
( )
2
201 *
1
ˆ1
E
S z
η
= − Γ + Γ − Γ
r
( )
( )
2 2 2 2
20 0 2 02 *4
ˆ ˆ ˆ1
E E E
S z z z
τ η
= = = − Γ + Γ − Γ
r
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum39
Complex power is conserved
Then the time average power flow through a 1m2 cross
section can be calculated as follows
Since
( )
( )2*
2 11 2
ˆ ˆ ˆ1S z z z
η ηη η
= = = − Γ + Γ − Γ
+
40. 6.2 Plane wave reflection from media
interface at normal incidence
is purely an imaginary number, therefore,
1 12 2* j z j z
e eβ β− +
−Γ + Γ
( )
2
21 1
0
11 1
ˆRe
2 2
avgS S z E
η
− Γ
= • =
r
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum40
which shows power balance at z=0 only
( ) 0
1
ˆRe
2 2
avgS S z E
η
= • =
( ) z
avg eEzSS α
η
2
2
2
2
0
22 1
2
1
ˆRe
2
1 −Γ−
=•=
r
41. 6.2 Plane wave reflection from media
interface at normal incidence
Note that the Poynting vector for z>0,
this power density in the lossy conductor decays
exponentially according to the attenuation factor
6.2.4 Perfect conductor:
Now assume that the region II (z >0) contains a perfect
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum41
Now assume that the region II (z >0) contains a perfect
conductor and region I is a lossless medium
Let us rewrite the expressions for propagation constant and
wave impedance of a good conductor and deduce the
expressions for perfect conductor
42. 6.2 Plane wave reflection from media
interface at normal incidence
Perfect conductor implies σ →∝, and
( ) ( )2 2
2 2 2 2
1
1 1
2 s
j j j
ωµ σ
γ α β
δ
= + = + = +
( ) 2
2
2
1
2
j
ωµ
η
σ
= +
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum42
Perfect conductor implies σ2→∝, and
correspondingly, α2→∝, δs
2→0, η2→0
Note that
Therefore, τ→0 and Γ→-1
2 2 1
1 2 2 1
2
,
η η η
τ
η η η η
−
= Γ =
+ +
43. 6.2 Plane wave reflection from media
interface at normal incidence
Table 6.5 Fields in the two regions (normal incidence)
Region I (lossless medium) Region II (perfect conductor)
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum43
1
ˆ j z
i oE xE e β−
=
r
1
1
1
ˆ j z
i oH y E e β
η
−
=
r
1
ˆ j z
r oE x E e β+
= Γ
r 1
1
ˆ j z
r oH y E e β
η
+Γ
= −
r
0tE ≅
r
0tH ≅
r
44. 6.2 Plane wave reflection from media
interface at normal incidence
The field for z >0 thus decay infinitely fast and
are identically zero in the perfect conductor as we have seen
in example 6.2 as well
The perfect conductor can be thought of as‘shorting out’ the
incident electric field
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum44
incident electric field
Now let us write down the field expressions in region I and II
for lossless dielectric and perfect conductor interface (see
Table 6.5)
45. 6.2 Plane wave reflection from media
interface at normal incidence
Power conservation
In order to see the power conservation,
let us find out the Poynting vector in the two regions
Noting that the field in region I comprise of
incident and
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum45
incident and
reflected wave,
we can write the Poynting vector for z <0 (Γ=-1)
( )1 11
0 0 1
ˆ ˆ2 sinj z j z
tot i rE E E xE e e x jE zβ β
β−
= + = − = −
r r r
46. 6.2 Plane wave reflection from media
interface at normal incidence
Therefore, the Poynting vector for z<0 is
( )1 11 0 0
1
1 1
ˆ ˆ2 cosj z j z
tot i r
E E
H H H y e e y zβ β
β
η η
−
= + = + =
r r
( )
2
4 Er r
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum46
which has a zero real part and
thus indicates that no real power is delivered to the perfect conductor
Fields in region II is anyway zero and hence power is also zero
( )
2
* 01 1 1
1 1
1
4
ˆ sin costot tot
E
S E H zj z zβ β
η
= × = −
r r
47. 6.3 Plane wave reflection from media
interface at oblique incidence
We will consider the problem of a plane wave
obliquely incident on a plane interface
between two lossy conducting regions
We will first consider two particular cases of this problem as
follows:
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum47
follows:
the electric field is in the xz plane (parallel polarization)
the electric field is in normal to the xz plane (perpendicular
polarization)
48. 6.3 Plane wave reflection from media
interface at oblique incidence
Any arbitrary incident plane wave can be expressed
as a linear combination of these two principal polarizations
The plane of incidence is that plane containing
the normal vector to the interface and
the direction of propagation vector of the incident wave
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum48
the direction of propagation vector of the incident wave
49. 6.3 Plane wave reflection from media
interface at oblique incidence
For Fig. 6.4, this is the xz plane
For perpendicular polarization (TE),
electric field is perpendicular to the plane of incidence
For parallel polarization (TM),
electric field is parallel to the plane of incidence
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum49
electric field is parallel to the plane of incidence
50. 6.3 Plane wave reflection from media
interface at oblique incidence
tt HE
rr
,
z
x
rr HE
rr
,
rθ tθ
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum50
Fig. 6.5 Oblique incidence of plane EM wave at a media interface
1 1 1
, ,ε µ σ 2 2 2, ,ε µ σ
z
Region II
ii HE
rr
,
Region I
iθ
rθ tθ
nˆ
51. 6.3 Plane wave reflection from media
interface at oblique incidence
6.3.1 Perpendicular polarization (TE):
In this case, electric field vector is perpendicular to the xz
plane,
Hence, it will have component along the y-axis
Since the electric field is transversal to the plane of incidence
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum51
Since the electric field is transversal to the plane of incidence
They are also known transverse electric (TE) waves
52. iθ rθnˆ nˆ zˆ
xˆxˆ
zˆ
iθγ cos1
iθγ sin1
rθγ cos1
rθγ sin11
i
γ
r 1
r
γ
r
xˆ
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum52
tθnˆ zˆ
2 cos tγ θ
2 sin tγ θ 2
t
γ
r
Fig. 6.6Wave propagation vector for (a) incident (b) reflected and (c) transmitted
EM waves at oblique incidence
53. 6.3 Plane wave reflection from media
interface at oblique incidence
Let us assume that the incident wave propagates in the first
quadrant of xz plane without loss of generality and
(incident propagation vector) makes an angle θi with
the normal (see Fig. 6.6 (a))
1
i
γ
r
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum53
( ) ( ) ( )'
1 1 1 1 1 1
ˆ ˆˆ ˆcos sin cos sin cos sini
i i i i i iz z x zz xx z x z xγ γ θ γ θ γ θ γ θ γ θ θ• = + • + = + = +
r r
( )1 cos sin
0
ˆi iz x
iE E e y
γ θ θ− +
=
r
1
1
i
i i i
E
E j H H
j
ωµ
ωµ
∇×
∇× = − ⇒ =
−
r
r r r
Q
54. 6.3 Plane wave reflection from media
interface at oblique incidence
( )
( ) ( )
( ) ( )
1 1
1
1 1
cos sin cos sin
0
1 1
cos sin
0
cos sin cos sin
0
ˆ ˆ ˆ
1
ˆ ˆ
0 0
ˆ ˆ
i i i i
i i
i i i i
z x z x
z x
z x z x
x y z
E e e
x z
j x y z j z x
E e
E e e
x z
γ θ θ γ θ θ
γ θ θ
γ θ θ γ θ θ
ωµ ωµ
− + − +
− +
− + − +
∂ ∂ ∂ ∂ ∂
= = − + − ∂ ∂ ∂ − ∂ ∂
∂ ∂
= −
( )
{ }1 cos sin0 1
ˆ ˆcos sini iz xE
e x z
γ θ θγ
θ θ− +
= − +
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum54
0
1
ˆ ˆ
E e e
x z
j z xωµ
∂ ∂
= − ∂ ∂
( )
{ }
( )
( )
1
1
cos sin0 1
1
cos sin0
1
ˆ ˆcos sin
ˆ ˆcos sin
i i
i i
z x
i i
z x
i i
E
e x z
j
E
e x z
γ θ θ
γ θ θ
γ
θ θ
ωµ
θ θ
η
− +
− +
= − +
= − +
55. 6.3 Plane wave reflection from media
interface at oblique incidence
Let us assume that the reflected wave propagates in the
second quadrant of xz plane and
(reflected propagation vector) makes an angle θr with
the normal (see Fig. 6.6 (b))
1
r
γ
r
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum55
( ) ( ) ( )1 1 1 1 1 1
ˆ ˆˆ ˆ' cos sin cos sin cos sinr
r r r r r rz z x zz xx z x z xγ γ θ γ θ γ θ γ θ γ θ θ• = − + • + = − + = − +
r r
( )1 cos sin
0
ˆr rz x
r TEE E e y
γ θ θ− − +
= Γ
r
56. 6.3 Plane wave reflection from media
interface at oblique incidence
Note that and will have the same magnitude
since both the waves are still in the same region I,
only their direction changes
Since the Poynting vector must be negative like the previous
case of normal incidence,
1
r
γ
r
1
i
γ
r
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum56
case of normal incidence,
You could also use the Maxwell’s curl equation below to find
this
( )
( )1 cos sin0
1
ˆ ˆcos sinr rz x
r TE r r
E
H e x z
γ θ θ
θ θ
η
− − +
= Γ +
r
1ωµj
E
H r
r
−
×∇
=
r
r
57. 6.3 Plane wave reflection from media
interface at oblique incidence
The transmitted fields will have similar expression with the
incident fields except
that now the θi should be replaced by θt (angle that transmitted
propagation vector makes with the normal),
γ1 should be replaced by γ2 (wave is in region II now) and
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum57
γ1 should be replaced by γ2 (wave is in region II now) and
multiplication by (transmission coefficient)
The transmitted fields are ( )2 cos sin
0
ˆ t tz x
t TEE yE e
γ θ θ
τ − +
=
r
( )
( )2 cos sin0
2 2
ˆ ˆcos sint tz xt TE
t t t
E E
H e x z
j
γ θ θτ
θ θ
ωµ η
− +∇×
= = − +
−
r
r
58. 6.3 Plane wave reflection from media
interface at oblique incidence
Table 6.5 Fields in two regions (oblique incidence:
perpendicular polarization)
Region I (lossy medium 1) Region II (lossy medium 2)
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum58
( )1 cos sin
0
ˆi iz x
iE E e y
γ θ θ− +
=
r
( )
( )1 cos sin0
1
ˆ ˆcos sini iz x
i i i
E
H e x z
γ θ θ
θ θ
η
− +
= − +
r
( )1 cos sin
0
ˆr rz x
r TEE E e y
γ θ θ− − +
= Γ
r
( )
( )1 cos sin0
1
ˆ ˆcos sini iz xTE
r r r
E
H e x z
γ θ θ
θ θ
η
− +Γ
= +
r
( )2 cos sin
0
ˆ t tz x
t TEE yE e
γ θ θ
τ − +
=
r
( )
( )2 cos sin0
2
ˆ ˆcos sint tz xTE
t t t
E
H e x z
γ θ θτ
θ θ
η
− +
= − +
r
59. 6.3 Plane wave reflection from media
interface at oblique incidence
Equating the tangential components of electric field
(electric field has only Ey component and it is tangential at the
interface z=0) and
magnetic field
(magnetic field has two components: Hx and Hz and only Hx is
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum59
(magnetic field has two components: Hx and Hz and only Hx is
tangential at the interface z=0)
at z=0 gives 1 21sin sinsini trx xx
TE TEe e eγ θ γ θγ θ
τ− −−
+ Γ =
1 21sin sinsin
1 1 2
1
cos cos cosi trx xxTE TE
i r te e eγ θ γ θγ θ τ
θ θ θ
η η η
− −−Γ−
+ = −
60. 6.3 Plane wave reflection from media
interface at oblique incidence
If Ex and Hy are to be continuous at the interface z = 0 for all
x,
then, this x variation must be the same on both sides of the
equations (also known as phase matching condition)
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum60
tri θγθγθγ sinsinsin 211 ==
1 2; sin sini r i tθ θ γ θ γ θ⇒ = =
61. 6.3 Plane wave reflection from media
interface at oblique incidence
The first is Snell’s law of reflection
which states that the angle of incidence equals the angle of
reflection
The second result is the Snell’s law of refraction
(refraction is the change in direction of a wave due to change in
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum61
(refraction is the change in direction of a wave due to change in
velocity from one medium to another medium)
Also note that refractive index of a medium is defined as
0 0
0 0
r r
r r
p
c
n
v
µ ε µ ε
µ ε
µ ε
= = =
62. 6.3 Plane wave reflection from media
interface at oblique incidence
hence, for a lossless dielectric media,
Now we can simplify above two equations by applying Snell’s
2 2 22 2 1 2
1 1 2 11 1 1
sin
sin
i
t
v n
v n
µ ε εθ γ β
θ γ β µ ε ε
= = = = = =
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum62
Now we can simplify above two equations by applying Snell’s
two laws as follows
1 TE TEτ+Γ =
1 1 2
cos cos
cosi r TE
TE t
θ θ τ
θ
η η η
− + Γ = −
63. 6.3 Plane wave reflection from media
interface at oblique incidence
The above two equations has two unknowns τTE and ГTE and
it can be solved easily as follows
Therefore,
2
1 1
cos cos
cos
i r
TE TE
t
θ θ η
τ
η η θ
= −Γ
1 TE TEτ+Γ =
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum63
Therefore,
22 2 2
1 1 1 1
1 2 2 1 2 1
1 1 1 2
cos coscos cos
1 1 1
cos cos cos cos
cos cos cos cos cos cos
cos cos cos cos
i ir r
TE TE TE
t t t t
t r i t i t
TE TE
t t t r
θ η θη η θ η θ
η θ η θ η θ η θ
η θ η θ η θ η θ η θ η θ
η θ η θ η θ η θ
+ Γ = −Γ ⇒ Γ + = −
+ − −
⇒ Γ = ⇒ Γ =
+
64. 6.3 Plane wave reflection from media
interface at oblique incidence
Hence, the reflection and transmission coefficients for
oblique incidence (Fresnel coefficients) for perpendicular
2 1 1 2 2 1 2
1 2 1 2 1 2
cos cos cos cos cos cos 2 cos
1 1
cos cos cos cos cos cos
i t t r i t i
TE TE
t r t r t r
η θ η θ η θ η θ η θ η θ η θ
τ
η θ η θ η θ η θ η θ η θ
− + + −
∴ = + Γ = + = =
+ + +
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum64
oblique incidence (Fresnel coefficients) for perpendicular
polarization are given as follows:
For normal incidence, it is a particular case and put
2 1
2 1
cos cos
cos cos
i t
TE
i t
η θ η θ
η θ η θ
−
Γ =
+
2
2 1
2 cos
cos cos
i
TE
i t
η θ
τ
η θ η θ
=
+
0=== tri θθθ
65. 6.3 Plane wave reflection from media
interface at oblique incidence
6.3.2 Parallel Polarization (TM):
In this case, electric field vector lies in the xz plane
Since the magnetic field is transversal to the plane of
incidence such waves are also called as transverse magnetic
(TM) waves
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum65
(TM) waves
So let us start with H vector which will have only y
component,
( )ii xz
i e
E
yH θθγ
η
sincos
1
0 1ˆ +−
=
r
66. 6.3 Plane wave reflection from media
interface at oblique incidence
1
1
i
i
H
H j E E
j
ωε
ωε
∇×
∇× = ⇒ =
r
r r r
Q
( ) ( )1 1cos sin cos sin
ˆ ˆ ˆ
1 i i i iz x z x
x y z
E e e
γ θ θ γ θ θ− + − + ∂ ∂ ∂ ∂ ∂
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum66
( )
( ) ( )
( )
{ } ( )
1 1
1
1 1
cos sin cos sin
0
1 1 1
cos sin0
1
cos sin cos sin0 1
0
1 1
1
ˆ ˆ
0 0
ˆ ˆˆ ˆcos sin cos si
i i i i
i i
i i i i
z x z x
z x
z x z x
i i i
E e e
x z
j x y z j z x
E
e
E
e x z E e x z
j
γ θ θ γ θ θ
γ θ θ
γ θ θ γ θ θ
ωε ωε η
η
γ
θ θ θ
ωε η
− + − +
− +
− + − +
∂ ∂ ∂ ∂ ∂
= = − + ∂ ∂ ∂ ∂ ∂
= − = −( )n iθ
67. 6.3 Plane wave reflection from media
interface at oblique incidence
Similar to the previous case of parallel polarization, we can
write the reflected magnetic field vector as
The transmitted fields will have similar expression with the
( )1 cos sin0
1
ˆ ˆ r rz xTM
r
E
H y ye
γ θ θ
η
− − +Γ
= −
r
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum67
The transmitted fields will have similar expression with the
incident fields except that
now the θi should be replaced by θt (angle that transmitted
propagation vector makes with the normal),
γ1 should be replaced by γ2 (wave is in region II) and
multiplied by a factor (transmission coefficient forTM case)
68. 6.3 Plane wave reflection from media
interface at oblique incidence
Table 6.6 Fields in two regions (oblique incidence: parallel
( )
( )2
cos sin
0
ˆ ˆcos sint tz x
t TM t tE E e x z
γ θ θ
τ θ θ
− +
= −
r
( )2 cos sin0
2
ˆ t tz xTM
t
E
H y e
γ θ θτ
η
− +
=
r
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum68
Table 6.6 Fields in two regions (oblique incidence: parallel
polarization) {next page}
69. 6.3 Plane wave reflection from media
interface at oblique incidence
Region I (lossy medium 1) Region II (lossy medium 2)
( )
( )1 cos sin
0
ˆ ˆcos sini iz x
i i iE E e x z
γ θ θ
θ θ− +
= −
r
( )Er
( )
( )2
cos sin
0
ˆ ˆcos sint tz x
t TM t tE E e x z
γ θ θ
τ θ θ
− +
= −
r
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum69
( )ii xz
i e
E
yH θθγ
η
sincos
1
0 1ˆ +−
=
r
( )
{ }1 cos sin
0
ˆ ˆcos sinr rz x
r TM r iE E e x z
γ θ θ
θ θ− − +
= Γ +
r
( )1 cos sin0
1
ˆ ˆ r rz xTM
r
E
H y ye
γ θ θ
η
− − +Γ
= −
r
( )2 cos sin0
2
ˆ t tz xTM
t
E
H y e
γ θ θτ
η
− +
=
r
70. 6.3 Plane wave reflection from media
interface at oblique incidence
Equating the tangential components of magnetic field
(magnetic field has only Hy component and it is tangential at the
interface z=0) and
electric field
(electric field has two components: Ex and Ez and only Ex is
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum70
(electric field has two components: Ex and Ez and only Ex is
tangential at the interface z=0)
at z=0 gives
1 21sin sinsin
cos cos cosi trx xx
i TM r TM te e eγ θ γ θγ θ
θ θ τ θ− −−
+ Γ =
21 1 sinsin sin
1 1 2
ti r xx x
TM TMe ee γ θγ θ γ θ
τ
η η η
−− −
Γ
− =
71. 6.3 Plane wave reflection from media
interface at oblique incidence
Therefore,
which is Snell’s law of reflection and refraction and this
tri θγθγθγ sinsinsin 211 ==
1 2; sin sini r i tθ θ γ θ γ θ⇒ = =
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum71
which is Snell’s law of reflection and refraction and this
implies that
cos cos cosi TM r TM tθ θ τ θ+ Γ =
( )
1 2
1
1 TM
TM
τ
η η
− Γ =
72. 6.3 Plane wave reflection from media
interface at oblique incidence
The above two equations has two unknowns τTM and ГTM and
it can be easily solved as follows
Therefore,
cos cos
cos
i TM r
TM
t
θ θ
τ
θ
+ Γ
= ( )2
1
1TM TM
η
τ
η
= − Γ
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum72
Therefore,
( ) ( )2 2
1 1
2 1 1 22 2
1 1 1 1
2 1
2
cos cos cos cos
1 1
cos cos cos
cos cos cos cos coscos
cos cos cos cos
cos cos
cos
i TM r i TM r
TM TM
t t t
i t r i tr
TM TM
t t t t
t i
TM
θ θ θη η θ
θ η η θ θ
θ η θ η θ η θ η θη θ η
η θ θ η η θ η θ
η θ η θ
η θ
+Γ Γ
= −Γ ⇒ −Γ = +
− − −
⇒ Γ − − = − ⇒ Γ =
−
⇒ Γ = 2 1
1 2 1
cos cos
cos cos cos
t i
t r t i
η θ η θ
η θ η θ η θ
−
=
+ +
73. 6.3 Plane wave reflection from media
interface at oblique incidence
Hence, the reflection and transmission coefficients for oblique
( ) 2 1 2 1 2 12 2 2
1 1 2 1 1 2 1
1 22
1 2 1 2 1
cos cos cos cos cos cos
1 1
cos cos cos cos
2 cos 2 cos
cos cos cos cos
t i t i t i
TM TM
t i t i
i i
t i t i
η θ η θ η θ η θ η θ η θη η η
τ
η η η θ η θ η η θ η θ
η θ η θη
η η θ η θ η θ η θ
− + − +
∴ = −Γ = − =
+ +
= =
+ +
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum73
Hence, the reflection and transmission coefficients for oblique
incidence (Fresnel coefficients) for parallel polarization are given as
follows:
Note that cosine terms multiplication in the above equations is
different from the previous expression forTE waves
2 1
2 1
cos cos
cos cos
t i
TM
t i
η θ η θ
η θ η θ
−
∴Γ =
+
2
2 1
2 cos
cos cos
i
TM
t i
η θ
τ
η θ η θ
=
+
74. 6.3 Plane wave reflection from media
interface at oblique incidence
6.3.3 Brewster angle:
For Γ=0, the angle of incidence is known as Brewster angle
(θB)
This means at this angle of incidence of the EM wave from
region I,
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum74
region I,
there will be no reflection from region II,
all the EM waves will be absorbed
75. 6.3 Plane wave reflection from media
interface at oblique incidence
For parallel polarization,
2 1
2 1
cos cos
0
cos cos
t i
TM
t i
η θ η θ
η θ η θ
−
Γ = =
+
TM
i Bθ θ=
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum75
Also we have from 2nd Snell’s law
2 1cos cos TM
t Bη θ η θ⇒ =
t
TM
B θγθγ sinsin 21 =
76. 6.3 Plane wave reflection from media
interface at oblique incidence
Using the above two equations, we can get (see textbook)
−
= 22
2
2
2
1
2
1
sin
γη
η
η
θTM
B
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum76
For lossless media, we have,
1
1 1 1 1
1 1
2
2 2 22 2
2 2 2 2 1 21 1 1 2
2 2 2 2 2
2 12 1 2 1 1 2 1 2 2 2 2
1 2 2 2
1
; sin TM
B
µ ε
µ γ µ µ ε µ ε µ εη γ ε µ
θ
µ ε µ εη γ µ µ ε µ ε µ γ ε µ
ε µ ε µ
−
= = = = ∴ =
−
Q
− 2
2
2
1
2
2
2
1
γ
γ
η
ηB
77. 6.3 Plane wave reflection from media
interface at oblique incidence
For magnetic media,
is of the form 1/0 and Brewster angle does not
exist
21 εε =
TM
Bθ2
sin
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum77
exist
But, for dielectric media, it exist and can be calculated as
follows
2
1 2 2 2
2 1 2 1 1 1
1 2
1
sin tanTM TM
B B
n
n
ε
ε ε ε
θ θ
ε ε ε ε ε
ε ε
−
⇒ = = ⇒ = =
+−
78. 6.3 Plane wave reflection from media
interface at oblique incidence
For perpendicular polarization,
Similarly, Brewster’s angle doesn’t exist for perpendicular
polarization at oblique incidence for a non-magnetic or
dielectric medium (see textbook for derivation)
For magnetic medium
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum78
For magnetic medium
( )
( ) ( )
2
2 1 22 2 2 2 2
2 2
1 2 1 2 1 11 2
sin sin ;tanTE TE TE
B B B
n
n
µ µ µ µ µ µ
θ θ θ
µ µ µ µ µµ µ
−
∴ = = ⇒ = = =
+ +−
79. 6.3 Plane wave reflection from media
interface at oblique incidence
Brewster angle exist for oblique incidence (perpendicular
polarization) for interface between two magnetic media
An incident wave consisting of both polarizations will have
reflected wave for only one polarization for Brewster angle
Brewster angle is also known as Polarizing angle (used in
optics and lasers)
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum79
Polarizing angle
optics and lasers)
In other words, an incident wave which is composed of both
TE andTM waves will have reflectedTE (TM) waves only for
dielectric (magnetic) media interface
Thus, a circularly polarized incident wave at polarizing angle
will become a linearly polarized wave
80. 6.3 Plane wave reflection from media
interface at oblique incidence
6.3.4Total internal reflection:
Used in optical fibers
Light strikes an angle of incidence greater than critical angle,
all of the light energy will be reflected back, and
none of them will be absorbed
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum80
none of them will be absorbed
This critical angle is related to the refractive index of the two
media and is given by
where are the refractive index of medium 2 and 1
respectively
1 2
1
sinc
n
n
θ −
=
81. 6.3 Plane wave reflection from media
interface at oblique incidence
In fact, this can be observed for both the polarizations we
have discussed above
From Snell’s Second law, we have,
2
1 2
sin
sin sin sin t
i t i
γ θ
γ θ γ θ θ
γ
= ⇒ =
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum81
If we consider non-magnetic materials (dielectrics) of , in
that case,
1 2
1
sin sin sini t iγ θ γ θ θ
γ
= ⇒ =
2
1
sin
sin t
i
ε θ
θ
ε
⇒ =
82. 6.3 Plane wave reflection from media
interface at oblique incidence
Hence, for
(it means the transmitted wave travels along the interface, for
our case, it is x-axis) which implies that
,
2
i c t
π
θ θ θ= =
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum82
our case, it is x-axis) which implies that
2 1 2
11
sin sinc c
ε ε
θ θ
εε
−
⇒ = ⇒ =
83. 6.3 Plane wave reflection from media
interface at oblique incidence
We can write the above relation as
Note that to have a real value of critical angle
1 1 1 10 2 0 22 2 2 2
1 0 1 0 1 1 1 1
sin sin sin sinr r r r
c
r r r r
n
n
µ µ ε εε µ ε
θ
ε µ µ ε ε µ ε
− − − −
= = = =
ε ε>
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum83
Note that to have a real value of critical angle
Besides, we have assumed that for a dielectric,
1 2ε ε>
1 2 1r rµ µ= =
84. 6.3 Plane wave reflection from media
interface at oblique incidence
What will happen when the angle of incidence is greater than
the critical angle?
For all angles greater than the critical angle,
using Snell’s second law, and
noting that waves are moving from a denser to a rarer dielectric
medium,
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum84
medium,
The value of should be imaginary
1 1 2
2 2
sin
sin 1 cos 1 sini
i c t t t
µ ε θ
θ θ θ θ θ
µ ε
> ∴ = > ⇒ = ± −Q
cos tθ
1 2 1r rµ µ= = 1 2ε ε>
85. 6.3 Plane wave reflection from media
interface at oblique incidence
It can be shown that (see textbook for derivation)
For z>0, since is an imaginary number, is
also purely imaginary
2 2
2 2
0 02 cos 2 cos* *
2 2
ˆ ˆsin cost tj z j z
t t
E E
S E H x e z eβ θ β θτ τ
θ θ
η η
− −
∴ = × = +
r r r
cos tθ *
cos tθ
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum85
also purely imaginary
Hence, the second term is imaginary, therefore, no real
power is flowing along z-axis in region II
The first term which has direction along x-axis is real and is
exponentially decaying with z
86. 6.3 Plane wave reflection from media
interface at oblique incidence
Hence,
When a wave is incident at an angle of incidence greater than
or equal to the critical angle from region I,
the wave will be total internally reflected in region I
Surface waves exist in the region II,
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum86
Surface waves exist in the region II,
which is propagating along x-axis (direction of power flow),
and
attenuating very fast along z-axis
People have used this for communications using submarines
87. 6.3 Plane wave reflection from media
interface at oblique incidence
6.3.5 Effects on polarization:
Since any arbitrary wave incident obliquely from region I can
be decomposed into linear combinations of TE andTM
waves and
we also know that the expressions for reflection and
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum87
we also know that the expressions for reflection and
transmission coefficients for parallel and perpendicular
polarizations are different
2 1
2 1
cos cos
;
cos cos
i t
TE
i t
η θ η θ
η θ η θ
−
Γ =
+
2 1
2 1
cos cos
cos cos
t i
TM
t i
η θ η θ
η θ η θ
−
Γ =
+
88. 6.3 Plane wave reflection from media
interface at oblique incidence
So, any circularly polarized incident wave will
become elliptically polarized after reflection or transmission
Similarly, a linearly polarized wave (note that linear
polarization could be expressed as linear combination of
RHCP and LHCP) may be
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum88
RHCP and LHCP) may be
rotated after reflection
89. 6.3 Plane wave reflection from media
interface at oblique incidence
For instance,
If the region II is a perfect conductor, η2=0, Г=-1 for both
TE andTM cases,
this means a RHCP wave after reflection will become LHCP
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum89
this means a RHCP wave after reflection will become LHCP
wave and vice versa
This is why circular polarization is not used for indoor
communication systems since we have many reflections inside a
room
90. 6.4 Summary
Plane waves reflection from a media interface
Normal incidence
TE
Lossless
Oblique incidence
Lossy conducting
medium
Perfect
conductor
TM
Brewster
angle
Effect on
polarization
2 1
2 1
;
η η
η η
−
Γ =
+
2
1 2
2η
τ
η η
=
+
τ→0 and Γ→-1
2 1
2 1
cos cos
cos cos
i t
TE
i t
η θ η θ
η θ η θ
−
Γ =
+
2
2 1
2 cos
cos cos
i
TE
i t
η θ
τ
η θ η θ
=
+
cos cosη θ η θ−
CP EP
3/25/2014Electromagnetic FieldTheory by R. S. Kshetrimayum90
Lossless
medium
Good conductor
Fig. 6.7 Plane waves reflection from media interface in a nutshell
angle Total internal
reflection
( )
2
21 1
0
1
1 1 1
ˆRe
2 2
avgS S z E
η
− Γ
= • =
r
1 2
avg avgS S=
2 1
2 1
cos cos
cos cos
t i
TM
t i
η θ η θ
η θ η θ
−
∴Γ =
+
2
2 1
2 cos
cos cos
i
TM
t i
η θ
τ
η θ η θ
=
+
CP EP
= −
1
21
sin
ε
ε
θc
1
2
1
2
tan
n
nTM
B ==
ε
ε
θ
1
2
1
2
tan
n
nTE
B ==
µ
µ
θ
1 1 2
2 2
sin
sin 1 cos 1 sini
i c t t t
µ ε θ
θ θ θ θ θ
µ ε
> ∴ = > ⇒ = ± −Q( ) z
avg eEzSS α
η
2
2
2
2
0
2 1
2
1
ˆRe
2
1 −+ Γ−
=•=
r