eleg 648 plane waves ii mark mirotznik, ph.d. associate professor the university of delaware email:...
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ELEG 648Plane waves II
Mark Mirotznik, Ph.D.Associate Professor
The University of DelawareEmail: mirotzni@ece.udel.edu
Uniform Plane Waves: Propagation in Any Arbitrary Direction
E
H
y
x
z
zzyyxxo
zyx
zzyyxx
zjyjxjo
EaEaEaE
zayaxar
aaa
eeeErE zyx
ˆˆˆ
ˆˆˆ
ˆˆˆ
)(~
222zyx
222
22
)sin(zyx
yx
22)cos(
yx
y
Uniform Plane Waves: Propagation in Any Arbitrary Direction
E
H
y
x
z
zzyyxxo
zyx
zzyyxx
rjo
EaEaEaE
zayaxar
aaa
eErE
ˆˆˆ
ˆˆˆ
ˆˆˆ
)(~
222zyx
222
22
)sin(zyx
yx
22)cos(
yx
y
Uniform Plane Waves: Propagation in Any Arbitrary Direction
0
)(~
o
rjo
E
eErE
E
H
y
x
z
Since E and are at right anglesfrom each other.
EaEa
EaE
EeE
Eejj
Eej
eEj
Ej
H
rjo
orj
orj
rjo
ˆ1
ˆ
ˆ1
11
11
1~1~
where 222
ˆˆˆˆ
zyx
zzyyxx aaaa
and
Uniform Plane Waves: Propagation in Any Arbitrary Direction
aaa
aaH
aaE
aErtE
trHEarH
aErtEtrEeErE
HE
Ho
Eo
Hoo
o
Eoorj
o
00
00
)cos(),(1
)(~
)cos(),()(~
Observation 1. E, H and vectors are pointing in orthogonal directions.
Summary and Observations:Frequency Domain Time Domain
Observation 2. E and H are in phase with each other, however, H’s magnitude is smaller by the amount of the wave impedance
Uniform Plane Waves: Propagation in 2D
E
H
x
y
22
)sin(
)cos(
)(~
yx
y
x
yjxjo
yx eeErE
Can we write this a bit more compact?
Uniform Plane Waves: Propagation in 2D
E
H
x
y
22
)sin(
)cos(
)(~
yx
y
x
yjxjo
yx eeErE
Can we write this a bit more compact?
yyxxo
yx
yyxx
rjo
EaEaE
yaxar
aa
eErE
ˆˆ
ˆˆ
ˆˆ
)(~
Uniform Plane Waves: Propagation in 2D
E
H
x
y
What about the polarization of E?
yyxxo
yx
yyxx
rjo
EaEaE
yaxar
aa
eErE
ˆˆ
ˆˆ
ˆˆ
)(~
Uniform Plane Waves: Propagation in 2D
E
H
x
y
What about the polarization of E?
0
)(~
o
rjo
E
eErE
yyxxo
yx
yyxx
rjo
EaEaE
yaxar
aa
eErE
ˆˆ
ˆˆ
ˆˆ
)(~
Uniform Plane Waves: Propagation in 2D
E
H
x
y
What about the polarization of E?
0
)(~
o
rjo
E
eErE
yyxxo
yx
yyxx
rjo
EaEaE
yaxar
aa
eErE
ˆˆ
ˆˆ
ˆˆ
)(~
Uniform Plane Waves: Propagation in 2D
E
H
x
y
What about the polarization of E?
)cos(ˆ)sin(ˆ
)(~
oyoxo
rjo
EaEaE
eErE
Two cases
E
H
x
y
Parallel Polarization Perpendicular Polarization
ozo
rjo
EaE
eErE
ˆ
)(~
Uniform Plane Waves: Propagation in 2D
E
H
x
y
What about H?
o
o
o
o
o
o
rjo
rjo
H
H
E
E
H
E
eHrH
eErE
0
0
)(~
)(~
yyxxo
yx
yyxx
rjo
EaEaE
yaxar
aa
eErE
ˆˆ
ˆˆ
ˆˆ
)(~
o
o
o
EH
Uniform Plane Waves: Propagation in 2D
E
H
x
y
What about the polarization of H?
)cos(ˆ)sin(ˆ
)(~
oyoxo
rjo
HaHaH
eHrH
Two cases
E
H
x
y
Parallel Polarization Perpendicular Polarization
ozo
rjo
HaH
eHrH
ˆ
)(~
Reflection and Transmission
Write down the electric fields in the two regions(2 unknowns, R and T)
Reflection and Transmission
Next find the magnetic fields in each region
Reflection and Transmission
Apply boundary conditions
Reflection and Transmission
Reflection and Transmission
Write down the E field in both regions(4 unknowns, R, T, r and t)
Reflection and Transmission
Find the H field in both regions
Reflection and Transmission
Apply boundary conditions
2 equations and 4 unknowns
We need two more equations. How do we get them?
Reflection and Transmission
Reflection and Transmission
Reflection and Transmission
Reflection and Transmission
Reflection and Transmission
Reflection and Transmission
Angle of Incidence, Degrees
Ref
lect
ion
Coe
ffic
ient
Example: Reflection from an Ocean Interface
Reflection and Transmission from Dielectric Slabs
1. Normal Incidence
111
1
11
11,
222
2
22
22 ,
333
3
33
33 ,
z=0 z=d
Region I Region II Region III
Reflection and Transmission from Dielectric Slabs
Region I:
zjoRy
zjoincy
zjo
Rx
zjo
incx
eRE
H
eE
H
eERE
eEE
1
1
1
1
1
1
Region II:
zjzjIIy
zjzjIIx
eAeAH
eAeAE
22
22
212
21
1
Region III:
zjoTy
zjo
Tx
eTE
H
eETE
3
3
3
Boundary Conditions
z=0 z=d
000
000
z
IIyz
Ryz
incy
z
IIxz
Rxz
incx
HHH
EEE
dz
Tydz
IIy
dz
Txdz
IIx
HH
EE
Reflection and Transmission from Dielectric Slabs
Boundary Conditions
z=0 z=d
2
2
2
1
11
21
AAREE
AAREE
oo
oo
djodjdj
djo
djdj
eTE
eAeA
eTEeAeA
322
322
321
2
21
1
Four equations and four unknownsSolution for the Reflection Coefficient:
23
2323
12
1212
22312
22312
2
2
1
dj
dj
e
eR
Reflection and Transmission from Dielectric Slabs
Special Cases
23
2323
12
12122
2312
22312
2
2
1
dj
dj
e
eR
I. Half Wavelength Thickness Slab
""0
1
22
,2,1,0,
231231
2312
2312
2
2
2
2
reflectionnoR
thenif
R
nnn
d
nnd
o
111
1
11
11,
222
2
22
22 ,
z=0 z=2/2
Region I Region II Region III
111
1
11
11,
Reflection and Transmission from Dielectric Slabs
Special Cases
23
2323
12
12122
2312
22312
2
2
1
dj
dj
e
eR
II. Quarter Wavelength Thickness Slab
""0
1
442
2
2312312
2312
2312
2
2
2
2
reflectionnoR
thenif
R
d
d
o
111
1
11
11,
222
132
22 ,
z=0 z=2/4
Region I Region II Region III
333
3
33
33 ,
Reflection and Transmission from Dielectric Slabs: Example
23
2323
12
12122
2312
22312
2
2
1
dj
dj
e
eR
377
,
01
11
o
oo
92
222
132
2
2
10776.5
5.217
3
o
o
z=0 z=m
Region I Region II Region III
67.125
,9
03
33
o
oo
268.05.21767.125
5.21767.125268.0
3775.217
3775.2172312
9
9
2
2
10776.5
10776.5
22312
22312
0716.01
)1(267.0
1
j
j
dj
dj
e
e
e
eR
Reflection and Transmission from Dielectric Slabs: Example
9
9
2
2
10776.5
10776.5
22312
22312
0716.01
)1(267.0
1
j
j
dj
dj
e
e
e
eR
Frequency, MHz
|R|
How do we broaden the bandwidth around the zero reflection point?
Frequency, MHz
|R|
One Solution is Multiple Dielectric Layers
111
1
11
11,
222
2
22
22 ,
333
3
33
33 ,
4
4
5
5
6
6
Reflection and Transmission from Dielectric Slabs
1. Oblique Incidence ( Parallel Polarization)
111
1
11
11,
222
2
22
22 ,
333
3
33
33 ,
z=0 z=d
Region I Region II Region III
i
t
r II
II
Reflection and Transmission from Dielectric Slabs
Region I:
])sin(1)cos(1[])sin(1)cos(1[
])sin(1)cos(1[])sin(1)cos(1[
])sin(1)cos(1[
11
11
11
])sin()cos([
)sin(ˆ)cos(ˆ~
)sin(ˆ)cos(ˆ~
ˆ~
ˆ~
yizijyizij
yizijyizij
yizij
ii
eRE
aeRE
aH
eE
aeE
aH
eREaE
eEaE
oizi
oy
R
oizi
oy
inc
oxR
yzjox
inc
Region II:
zy
zyII
yzjyzjIIx
aaeA
aaeA
H
eAeAE
yzj
yzj
ˆ)sin(ˆ)cos(
ˆ)sin(ˆ)cos(~
222
2
222
1
])sin()cos([2
])sin()cos([1
])2sin(2)2cos(2[
])2sin(2)2cos(2[
22222222
Reflection and Transmission from Dielectric Slabs
Region III:
])sin(3)cos(3[])sin(3)cos(3[
])sin(3)cos(3[
33
3
)sin(ˆ)cos(ˆ~
ˆ~
ytztjytztj
ytztj
eTE
aeTE
aH
eTEaE
oiz
oy
T
oxT
Boundary Conditions
z=0 z=d
000
000
z
IIyz
Ryz
incy
z
IIxz
Rxz
incx
HHH
EEE
dz
Tydz
IIy
dz
Txdz
IIx
HH
EE
Phase Matching Conditions
)sin()sin(
)sin()sin(
322
221
t
i
Reflection and Transmission from Dielectric Slabs
Six Equations and Six Unknowns
)sin()sin(
)sin()sin(
322
221
t
i
dtjdjdj
dtjdjdj
eT
EeA
eA
TeEeAeA
AAREE
AAREE
to
o
oo
oo
)cos(3)2cos(2)2cos(2
)cos(3)2cos(2)2cos(2
)cos()cos()cos(
)cos()cos(
32
2
22
2
1
21
22
22
2
1
11
21
Reflection and Transmission from Dielectric Slabs: Solution (parallel polarization)
dj
dj
e
eR
2
2
22312
22312
1
)cos()cos(
)cos()cos(
)cos()cos(
)cos()cos(
))sin((sin
))sin((sin
322
32223
221
22112
23
21
2
112
22
t
t
i
i
t
i
o
*note we have assumed all non-magneticmaterials here
Reflection and Transmission from Dielectric Slabs: Solution (perpendicular polarization)
dj
dj
e
eR
2
2
22312
22312
1
)cos()cos(
)cos()cos(
)cos()cos(
)cos()cos(
))sin((sin
))sin((sin
223
22323
212
21212
23
21
2
112
22
t
t
i
i
t
i
o
*note we have assumed all non-magneticmaterials here
Reflection and Transmission from Dielectric Slabs: Example
377
,
01
11
o
oo
92
222
132
2
2
10776.5
5.217
3
o
o
z=0 z=m
Region I Region IIRegion III
67.125
,9
03
33
o
oo
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