Electromagnetic Engineering MAPTele 2008-2009
68
Faculdade de Engenharia Inês Carvalho Assistant Professor Faculdade de Engenharia, Universidade do Porto www.fe.up.pt/~mines/ [email protected] Electromagnetic Engineering MAPTele 2008-2009 Electromagnetic waves Propagation Incidence Waveguides Transmission lines Radiation Program
Transcript of Electromagnetic Engineering MAPTele 2008-2009
Faculdade de Engenharia
Inês Carvalho
Assistant ProfessorFaculdade de Engenharia, Universidade do Porto
www.fe.up.pt/~mines/[email protected]
Electromagnetic EngineeringMAPTele 2008-2009
Electromagnetic waves
Propagation
Incidence
Waveguides
Transmission lines
Radiation
Program
EE 0809Incidence 2
Faculdade de EngenhariaEE – Propagation
Studying material
• any good book on electromagnetism:
“Microwave Engineering”, David Pozar, Addison-Wesley
http://paginas.fe.up.pt/~mines/EE/
• transparencies of lectures
• problems on incidence, waveguides and transmission lines
“Field and Wave Electromagnetics”, David K. Cheng, Addison-Wesley
…
EE 0809Incidence 3
Faculdade de EngenhariaEE – forthcoming lectures
Waveguides
Transmission Lines
à next week
Plane wave incidence à today
à 6/2 and 13/2
Electromagnetic Waves
Radiation
EE 0809Incidence 4
Faculdade de EngenhariaPlane wave incidence
Contents
normal incidence
reflection and transmission coefficients
stationary wave
interface with perfect conductor
multiple interfaces
oblique incidence
perpendicular and parallel polarizations
reflection and transmission coefficients
interface with perfect conductor
total internal reflection
EE 0809Incidence 5
Faculdade de EngenhariaMaxell’s equations in lossless media
tHE
∂∂−=×∇
rrµ
0=⋅∇ Er
tEH
∂∂=×∇
rrε
0=⋅∇ Hr
à electric field
à magnetic field
à dielectric permitivitty
à magnetic permeabilityµε
H
Er
r
( )0εεε r=
( )0µµµ r=m/s103
1 8
00
×== cεµ
2
22
tH
H∂
∂=∇
rrµε
2
22
tE
E∂∂
=∇rr
µεwave equations
in lossless media
EE 0809Incidence 6
Faculdade de EngenhariaPlane electromagnetic waves – temporal expressions
Plane wave in lossless media
linearly polarized along x
propagating along +z
( )xkztEE ˆcos0 −= ωr
εµω=k is the wavenumber
Note: they are solutions of wave
equations in lossless media
ω is the angular frequency
0E is the amplitude
( )tωcos
t
( )kzcos
z
fT /1= λ ( )kzt −ωcos
z
direction of propagation
εµ1=fv is the phase velocity
=
λπ2k
( )fπω 2=
EE 0809Incidence 7
Faculdade de EngenhariaHelmholtz’s equations
( ) ( )φω += tXtx cos0
022 =+∇ EkErr
022 =+∇ HkHrr
Helmholtz’s equations
( )µεω=k
2
22
tH
H∂
∂=∇
rrµε
2
22
tE
E∂∂
=∇rr
µε
wave equations
phasor notation:
( ) ( ) φω += tjeXtx 0Re
φjeXX 0=
tjXe ωRe=
phasor is
Note:
important
( ) ( )θθθ sincos je j +=
( )dt
tdxphasor of Xjωis
EE 0809Incidence 8
Faculdade de EngenhariaPlane electromagnetic waves – in phasor notation
Plane waves in lossless media
erajk peEE n ˆˆ
0
rr ⋅−=
εµω=khrajk peHH n ˆˆ
0
rr ⋅−=
na
is the wavenumber
and indicate the polarization directionep hp
wave propagating along the direction
Note: the polarization direction of the electromagnetic wave is defined by
the polarization direction of the electric field
EE 0809Incidence 9
Faculdade de EngenhariaPlane electromagnetic waves – phasors
Electric and magnetic fields are perpendicular to
each other and to the direction of propagation
transverse electromagnetic (TEM) wave
( )EaH n
rr×= ˆ1
η Ω=εµ
η
( )HaE n
rr×−= ˆη
Note:
is the intrinsic impedance of the medium
EE 0809Incidence 10
Faculdade de Engenharia
( )EaH n
rr×= ˆ1
η
Plane electromagnetic waves – phasors
( )HaE n
rr×−= ˆη
example 1
erajk peEE n ˆˆ
0
rr⋅−=
hrajk peHH n ˆˆ
0
rr⋅−=
Write the electric and magnetic field phasors of a 3 GHz TEM wave propagating along the +z
direction in free space. The wave is linearly polarized along the x direction and the electric field
amplitude is 10 V/m
V/m100 =E
zan ˆˆ +=
xpe ˆˆ =
zzyyxxr ˆˆˆ ++=r
V/mˆ10 20 xeE zj π−=r
rad/m20200 ππεµω ===
cfk Ω=== π
εµ
ηη 1200
00
( ) A/mˆ12
1ˆ1 20 yeEaH zjn
π
πη−=×=
rr
EE 0809Incidence 11
Faculdade de Engenharia
( )EaH n
rr×= ˆ1
η
Plane electromagnetic waves – phasors
( )HaE n
rr×−= ˆη
example 2
erajk peEE n ˆˆ
0
rr⋅−=
hrajk peHH n ˆˆ
0
rr⋅−=
The electric field phasor of a TEM wave propagating in a nonmagnetic medium with dielectric
constant 4 is given by
Find the frequency of the wave and obtain the magnetic field phasor.
)43(4ˆ zyrak n +=⋅ πr
zzyyxxr ˆˆˆ ++=r
V/mˆ2 )43(4 xeE zyj +−= πr
)ˆ4ˆ3(4ˆ zyak n += π1ˆ =na
122 m20434 −=+= ππk
zyzy
an ˆ8.0ˆ6.020
)ˆ4ˆ3(4ˆ +=+
=π
π
y
z
º1.53
EE 0809Incidence 12
Faculdade de Engenharia
( )EaH n
rr×= ˆ1
η
Plane electromagnetic waves – phasors
( )HaE n
rr×−= ˆη
example 2
erajk peEE n ˆˆ
0
rr⋅−=
hrajk peHH n ˆˆ
0
rr⋅−=
The electric field phasor of a TEM wave propagating in a nonmagnetic medium with dielectric
constant 4 is given by
Find the frequency of the wave and obtain the magnetic field phasor.
nc
fk rrπεµεµωεµω 2=== 00
Ω=== πεµ
εµ
εµ
η 600
0
r
rA/m)ˆ6.0ˆ8.0(
301 )43(4 zyeH zyj −= +− π
π
r
V/mˆ2 )43(4 xeE zyj +−= πr
1m20 −= πk
zyan ˆ8.0ˆ6.0ˆ +=
1,4 == rr µε
GHz5.1=f
EE 0809Incidence 13
Faculdade de EngenhariaIncidence of a TEM wave at a plane boundary
medium 1
( )111 ,, σµε
z
x
medium 2
( )222 ,, σµε
iθ
tθrθ
plane of incidence à xz plane
angle of incidence à iθ
plane containing the normal to the boundary surface and the vectorindicating the direction of propagationof the incident wave
nta
transmitted
zxa iini ˆcosˆsinˆ θθ +=
zxa ttnt ˆcosˆsinˆ θθ +=
zxa rrnr ˆcosˆsinˆ θθ −=
directions of propagation:
nia
incident
nrareflected
EE 0809Incidence 14
Faculdade de EngenhariaSnell’s laws – law of reflection
medium 1( )111 ,, σµε
z
x
medium 2( )222 ,, σµε
nia
nra
nta
iθ
tθrθ
wavefrontà region with constant phase
O
/O
B
/A
A
naplane waves à wavefronts are perpendicular to
ri OOOO θθ sinsin // =
/1
/1 OAkAOk =
points and have the same phase O A/O /A
phase= dist.⋅k
ri θθ =
points and have the same phase
fvk
ωµεω ==
EE 0809Incidence 15
Faculdade de EngenhariaSnell’s laws – law of refraction
medium 1( )111 ,, σµε
z
x
medium 2( )222 ,, σµε
nia
nra
nta
iθ
tθrθ
O
/O
B
/A
A phase= dist.⋅k
ti OOkOOk θθ sinsin /2
/1 =
2
1
sinsin
kk
i
t =θθ
1
2
f
f
v
v=
naplane waves à wavefronts are perpendicular to
O A
B/O
OBkAOk 2/
1 =
fvk
ωµεω ==
points and have the same phase
points and have the same phase
EE 0809Incidence 16
Faculdade de EngenhariaIndex of refraction
Index of refraction à ratio of the speed of light in freespace to that in the medium
fvc
n =
2
1
sinsin
nn
i
t =θθ
ß Snell’s law of refraction
Ex: lossless mediaεµ
1=fv 00 εµ
εµ=n
rr εµ=
1≥n
large n à low speed
1
2
sinsin
f
f
i
t
vv
=θθ
EE 0809Incidence 17
Faculdade de EngenhariaBoundary conditions
medium 1( )111 ,, σµε
na
medium 2( )222 ,, σµε
let be the normal to the interfacepointing from medium 2 to medium 1
na( ) 0ˆ 21 =−× EEan
rr
( ) Sn JHHarrr
=−× 21ˆ
( ) Sn DDa ρ=−⋅ 21ˆrr
( ) 0ˆ 21 =−⋅ BBan
rr
tan,2tan,1 EE =
norm,2norm,1 BB =
continuosnormB
continuostanE
0ifcontinuostan =SJHr
0ifcontinuosnorm =SD ρNote:
0and0 ≠≠ SSJ ρr
only in ideal conductors
EE 0809Incidence 18
Faculdade de EngenhariaBoundary conditions – ideal conductors
00
00
condcond
condcond
==
==
BD
HErr
rr
Ideal conductors à
0e0 ≠≠ SSJ ρr
σµπδ
f1
=
∞=σ
0=
Example
∞=2σ
( )21ˆ HHaJ nS
rrr−×=
( )21ˆ DDanS
rr−⋅=ρ
1ˆ Han
r×= taH ˆtan,1=
1ˆ Dan
r⋅= naD ˆnorm,1=
medium 1( )111 ,, σµε
nu
medium 2( )222 ,, σµε
skin depth
EE 0809Incidence 19
Faculdade de EngenhariaNormal incidence
normal incidence à 0=iθ
2
1
sinsin
nn
i
t =θθ
ri θθ =0== tr θθ
medium 1
z
xmedium 2
nra
yiE
r
iHr
nta
tEr
tHr
nia
rEr
rHr
incident à xeEE zjkii ˆ10
−=r
yeE
H zjkii ˆ1
1
0 −=η
r
reflected à xeEE zjkrr ˆ10
+=r
yeE
H zjkrr ˆ1
1
0 +−=η
r
transmitted à xeEE zjktt ˆ20
−=r
yeE
H zjktr ˆ2
2
0 −=η
r
( )xeEeE zjkr
zjki ˆ11
00+− +=
yeE
eE zjkrzjki ˆ11
1
0
1
0
−= +−
ηη
ri EEErrr
+=1
ri HHHrrr
+=1
medium 1
medium 2
EE 0809Incidence 20
Faculdade de EngenhariaNormal incidence – reflection and transmission coefficients
boundary conditions à
medium 1
z
x
medium 2
yiEr
nta
tEr
tHr
iHr
nia
nra
rEr
rHr
continuoustanE
continuoustanH ( )0if =SJr
1Er ( )xeEeE zjk
rzjk
i ˆ1100
+− +=
1Hr
yeEeE zjkrzjki ˆ11
1
0
1
0
−= +−
ηη
medium 1
medium 2
xeEE zjkt ˆ202
−=r
yeE
H zjkt ˆ2
2
02
−=η
r
2
0
1
0
1
0
000
ηηηtri
tri
EEEEEE
=−
=+
at z=0 à21
21
HH
EErr
rr
=
=
12
2
0
0
12
12
0
0
2ηη
ηηηηη
+=
+−
=
i
t
i
r
EEEE
EE 0809Incidence 21
Faculdade de EngenhariaNormal incidence – reflection and transmission coefficients
medium 1
z
x
medium 2
nra
yiEr
iHr
nta
tEr
tHr
nia
rEr
rHr
( )xeEeEE zjki
zjki ˆ11
001+− Γ+=
r
1Hr
medium 1
medium 2
12
12
0
0
ηηηη
+−=
i
r
EE
12
12
ηηηη
+−
=Γ
12
22ηη
ητ
+=
12
2
0
0 2ηη
η+
=i
t
EE
transmission coefficient
reflection coefficient
τ=Γ+1
Note
1.
2. 1≤Γ
3. 0≥τ
4.
xeEE zjki ˆ202
−=τr
( )xeEeE zjkr
zjki ˆ11
00+− +=
yeEeE zjkrzjki ˆ11
1
0
1
0
−= +−
ηη
xeEE zjkt ˆ202
−=r
yeE
H zjkt ˆ2
2
02
−=η
r
1Er
EE 0809Incidence 22
Faculdade de EngenhariaNormal incidence – stationary wave
medium 1
z
x
medium 2
nra
yiEr
iHr
nta
tEr
tHr
nia
rEr
rHr
( )xeEeEE zjki
zjki ˆ11
001+− Γ+=
r
ηηηη
+−
=Γ2
12
ηηη
τ+
=2
22
( )[ ]xeeEE zjkzjki ˆ1101
+− Γ+Γ−= τr
τ=Γ+1
( )xeeExeEE zjkzjki
zjki ˆˆ 111
001−+− −Γ+=τ
r
( )2
sinjxx eej
x−−
=
( )xzkEjxeEE izjk
i ˆsin2ˆ 10011 Γ+= −τ
r
propagating wave stationary wave
z
EE 0809Incidence 23
Faculdade de EngenhariaNormal incidence – maxima and minima
medium 1
z
x
medium 2
nra
yiEr
iHr
nta
tEr
tHr
nia
rEr
rHr
( )xeEeEE zjki
zjki ˆ11
001+− Γ+=
r
ηηηη
+−
=Γ2
12
ηηη
τ+
=2
22
( )[ ] ( )( )21
2101 2sin2cos1 zkzkEE i +Γ++Γ+= ΓΓ θθ
r
( )xzkEjxeEE izjk
i ˆsin2ˆ 10011 Γ+= −τ
r
maxima:
( )xeeE zkjzjki ˆ1 11 20
+− Γ+=
( )zkEi 12
0 2cos21 +Γ+Γ+= Γθ
ΓΓ=Γ θje
minima:( ) 12cos 1 +=+Γ zkθ ( ) 12cos 1 −=+Γ zkθ
( )πθ nk
zMAX 221
1
+−= Γ( )[ ]πθ 12
21
1min ++−= Γ n
kz
( )Γ+= 101 iMAXEE
r ( )Γ−= 10min1 iEEr
EE 0809Incidence 24
Faculdade de EngenhariaNormal incidence – incidence on ideal conductor
medium 1
z
x
medium 2
nra
yiEr
iHr
nta
tEr
tHr
nia
rEr
rHr
( )xeEeEE zjki
zjki ˆ11
001+− Γ+=
r
ηηηη
+−
=Γ2
12
ηηη
τ+
=2
22
if medium 1 is lossless
maxima:
minima:
( )π1221
1
+= nk
zMAX
1min k
nz
π=
01 2 iMAXEE =
r
0min1 =E
r
if medium 2 is ideal conductor
( )01 =σ
ωεσ
εµ
ηj−
=1
02 =η( )∞=2σ0
1=
−=Γτ
02 =Er
( )xeeEE zjkzjki ˆ1101
+− −=r
( )xzkjEi ˆsin2 10−=no propagating wave, only stationary wave
and
and
EE 0809Incidence 25
Faculdade de Engenharia
∫ ⋅=A
adSPrr
avav
average power carried by the wave across surface A:
Average power carried by electromagnetic wave
*av Re
21
HESrrr
×=
ndaad ˆ=r
à average Poynting’s vector
à perpendicular to surface A and with infinitesimal absolute value
A
adr
EE 0809Incidence 26
Faculdade de Engenharia
TEM waves
average Poynting’s vector points along the propagation direction
naHE ˆ⊥⊥rr
naES ˆ1Re21
*
2
av
=η
rr
EaH n
rr×= ˆ1
η
( )∗××=× EaEHE n
rrrrˆ1
**
η( ) ( ) nn aEEaEE ˆˆ1 **
* ⋅−⋅=rrrr
η naE ˆ1 2
*
r
η=
( ) ( ) ( )BACBCACBArrrrrrrrr
⋅−⋅=××
Average Poynting’s vector for TEM waves
*av Re
21
HESrrr
×=
EE 0809Incidence 27
Faculdade de Engenharia
TEM waves in lossless media
naHE ˆ⊥⊥rr
naES ˆ1Re21
*
2
med
=η
rr
EaH n
rr×= ˆ1
η
Average power for TEM waves in lossless media
η is real
erajk peEE n ˆˆ
0
rr⋅−=
( )220
av W/mˆ2 naESη
=r
∫ ⋅=A
adSPrr
avav
nadaad ˆ=rLet A be the surface
perpendicular to the propagating direction
( )W2
2
av AE
P o
η=( )Wav ASP av
r=
avSr
is constant
EE 0809Incidence 28
Faculdade de EngenhariaAverage incident, reflected and transmitted power
∫ ⋅=A
adSPrr
avav AEo
η2
2
=ASav
r=
medium 1
z
x
medium 2
nra
yiEr
iHr
nta
tEr
tHr
nia
rEr
rHr
ηηηη
+−
=Γ2
12
ηηη
τ+
=2
22
00 ir EE Γ= 00 it EE τ=
AE
P io
1
2
iav, 2η=
AE
P ro
1
2
rav, 2η=
AE
P to
2
2
tav, 2η=
2
iav,
rav, Γ=P
P
2
iav,
tav, τ≠P
Pimportant
2
iav,
tav, 1 Γ−=P
Ptav,rav,iav, PPP +=
EE 0809Incidence 29
Faculdade de EngenhariaNormal incidence – multiple interfaces
medium 1
z
xmedium 2
y
medium 3
0=z dz =
M M M
xeEE zjki ˆ1
0−=
r
interface 1 à 2 :12
1212 ηη
ηη+−
=Γ12
212
2ηη
ητ
+=
interface 2 à 1 :21
2121 ηη
ηη+−
=Γ21
121
2ηη
ητ
+=
12Γ−=
interface 2 à 3 :23
2323 ηη
ηη+−
=Γ32
323
2ηη
ητ
+=
all waves are linearly polarized along x
and
and
and
1. media 1 and 3 à infinite
reflection and transmission coefficients:
2.
note
EE 0809Incidence 30
Faculdade de EngenhariaNormal incidence – multiple inerfaces
z0=z dz =
zjkeE 10
−
medium 1 medium 2 medium 3
zjkdkj eeE 1220122321
+−Γ ττ
( )0E
( )012EΓ
( )012Eτ zjkeE 2012
−τ ( )djkeE 2012
−τ( )dzjkdjk eeE −−− 32
01223ττ
( )dzjkdjk eeE −+−Γ 2201223τ( )dkjeE 22
01223−Γ τ
zjkeE 1012
+Γ
zjkdkj eeE 2220122321
−−ΓΓ τ ( )dzjkdkj eeE −−−ΓΓ 323012232123 ττ
( )dzjkdkj eeE −+−ΓΓ 223012
22321 τ
zjkdkj eeE 124012
2232121
+−ΓΓ ττ
zjkdkj eeE 224012
223
221
−−ΓΓ τ ( )dzjkdkj eeE −−−ΓΓ 325012
223
22123 ττ
( )dzjkdkj eeE −+−ΓΓ 225012
323
221 τ
M M M
EE 0809Incidence 31
Faculdade de EngenhariaNormal incidence – multiple interfaces
medium 1
z
medium 2 medium 3
0=z dz =
M M M
medium 1
z
medium 2 medium 3
0=z dz =
zjkeE 11
−+
zjkeE 11
+−
zjkeE 22
−+
zjkeE 22
+−
zjkeE 33
−+
01 EE =+
L+ΓΓ+ΓΓ+Γ+Γ= −−−− dkjdkjdkj eEeEeEEE 222 6012
323
22121
4012
2232121
201223210121 ττττττ
( )L+ΓΓ+ΓΓ+Γ+Γ= −−− dkjdkjdkj eeeEE 222 4223
221
22321
20231221012 1ττ
( )∑+∞
=
−− ΓΓΓ+Γ=0
22321
20231221012
22
n
ndkjdkj eeEE ττ02
2312
22312
2
2
1E
ee
dkj
dkj
−
−
ΓΓ+Γ+Γ
=022321
2231221
122
2
1E
ee
dkj
dkj
ΓΓ−Γ
+Γ= −
−ττ
EE 0809Incidence 32
Faculdade de EngenhariaNormal incidence – multiple interfaces
medium 1
z
medium 2 medium 3
0=z dz =
M M M
medium 1
z
medium 2 medium 3
0=z dz =
zjkeE 11
−+
zjkeE 11
+−
zjkeE 22
−+
zjkeE 22
+−
zjkeE 33
−+
L+ΓΓ+ΓΓ+ΓΓ+= −−−−+ djkdkjdjkdkjdjkdkjdjkdjk eeEeeEeeEeeEE 32323232 7012
323
32123
5012
223
22123
3012232123012233 ττττττττ
( ) ( )L+ΓΓ+ΓΓ+ΓΓ+= −−−− dkjdkjdkjdkkj eeeeE 22223 6323
321
4223
221
2232101223 1ττ
( ) ( )∑∞
=
−− ΓΓ=0
2232101223
223
n
ndkjdkkj eeEττ( )
022312
12232
23
1E
ee
dkj
dkkj
−
−
ΓΓ+=
ττ
EE 0809Incidence 33
Faculdade de EngenhariaNormal incidence at multiple interfaces – alternative approach
medium 1
z
xmedium 2
y
medium 3
0=z dz =
M M M
medium 1
z
xmedium 2
y
medium 3
0=z dz =
( )ye
Ee
EH
xeEeEE
zjkzjk
zjkzjk
ˆ
ˆ
11
11
1
1
1
11
111
−=
+=−
−+
−−+
ηη
r
rmedium 1
zjkeE 11
−+
zjkeE 11
+−
zjkeE 22
−+
zjkeE 22
+−
zjkeE 33
−+
( )ye
Ee
EH
xeEeEE
zjkzjk
zjkzjk
ˆ
ˆ
22
22
2
2
2
22
222
−=
+=−
−+
−−+
ηη
r
rmedium 2
yeE
H
xeEE
zjk
zjk
ˆ
ˆ
3
3
3
33
33
−+
−+
=
=
η
r
rmedium 3
EE 0809Incidence 34
Faculdade de EngenhariaMultiple interfaces – boundary conditions
medium1
z
x
medium 2
y
medium 3
0=z dz =
dielectric media à 0=SJr
continuoustanE
continuoustanH
0=z
2
22
1
11
2211
ηη
−+−+
−+−+
−=−
+=+
EEEE
EEEE
dz =
3
3
2
22
322
322
322
ηη
djkdjkdjk
djkdjkdjk
eEeEeE
eEeEeE
−+−−+
−+−−+
=−
=+
4 equations4 unknowns ( )+−+−
3221 ,,, EEEE
assuming 01 EE =+
EE 0809Incidence 35
Faculdade de EngenhariaMultiple interfaces – boundary conditions
medium 1
z
x
medium 2
y
medium 3
0=z dz =
2
22
1
11
2211
ηη
−+−+
−+−+
−=−
+=+
EEEE
EEEE
3
3
2
22
322
322
322
ηη
djkdjkdjk
djkdjkdjk
eEeEeE
eEeEeE−+−−+
−+−−+
=−
=+
01 EE =+
022312
22312
12
2
1E
ee
E dkj
dkj
−
−−
ΓΓ+Γ+Γ
=
( )
022312
12233
2
23
1E
ee
E dkj
dkkj
−
−+
ΓΓ+=
ττ
previous expressions, as expected
EE 0809Incidence 36
Faculdade de EngenhariaMultiple interfaces – examples
medium 1
z
x
medium 2
y
medium 3
0=z dz =
Elimination of reflections at interface 1à 3 by inserting medium 2
Examples
•Eliminations of reflexes in lenses
•Attenuation of radar echoes (invisible airplanes)
•…
à medium 2 is a matching device
EE 0809Incidence 37
Faculdade de EngenhariaMultiple interfaces – λ /4 transformer
medium 1
z
x
medium 2
y
medium 3
0=z dz =
022312
22312
12
2
1E
ee
E dkj
dkj
−
−−
ΓΓ+Γ+Γ
=
( )
022312
12233
2
23
1E
ee
E dkj
dkkj
−
−+
ΓΓ+=
ττ
eliminate reflections à 01 =−E 122
232 Γ−=Γ − dkje
( ) ( )( ) ( ) 0111 222 222
2312
231 =−−+−+− −−− dkjdkjdkj eee ηηηηηη
λ /4 transformer :
122 −=− dkje integerodd,2 2 mmdk π=
312 ηηη =
integer odd,4
2 mmdλ
=
wavelength in medium 2
EE 0809Incidence 38
Faculdade de Engenharia
pp f
v2,2 =λ
δδ2
2
cos1cosF
F+
=
λ /4 transformer – different wavelength
medium 1
z
x
medium 2
y
medium 3
0=z dz =
022312
22312
12
2
1E
ee
E dkj
dkj
−
−−
ΓΓ+Γ+Γ
=
( )
022312
12233
2
23
1E
ee
E dkj
dkkj
−
−+
ΓΓ+=
ττpf
vd
42=
velocity in medium 2
12
1212 ηη
ηη+−
=Γ13
13
ηη
ηη
+
−=312 ηηη =
23
2323 ηη
ηη+−
=Γ13
13
ηη
ηη
+
−=
Γ=Γ=Γ 2312
pf à design frequency
frequency f ( )dkj
dkj
ee
EE
2
2
22
2
0
1
11
−
−−
Γ++Γ
=
where
*2 zzz =
( )( )( )dk
dk
E
E
PP
224
22
20
2
1
inc
ref
2cos212cos12
Γ+Γ++Γ
==−
2
212
Γ−Γ
=F
dk2=δpf
f2π
=
EE 0809Incidence 39
Faculdade de Engenharia
δδ2
2
inc
ref
cos1cosF
FPP
+=
λ /4 transformer – different wavelength
medium 1
z
x
medium 2
y
medium 3
0=z dz =
where2
212
Γ−Γ
=Fpff
2π
δ =and
0=Γ 0=F
1→Γ ∞→F
0=f 0=δ
∞→f ∞→δ
0 1 2 3 4 5 6 7 8 9 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
inc
ref
PP
100=F
10=F
1=F
1.0=F
δ2π
23π
25π
EE 0809Incidence 40
Faculdade de EngenhariaProblem
formulae
Note: 0conductor =η
EE 0809Incidence 41
Faculdade de EngenhariaProblem
A plane electromagnetic wave with the following electric field phasor propagates in air and is incident normally on an interface at with a nonmagnetic medium with refractive index that occupies the region . Find
( )V/mˆ10 4 xeE zji
π−=r
22 =n0>z
a) the reflection and transmission coefficients;b) the electric field phasors of the reflected and transmitted waves;c) the average Poynting vectors of the incident, reflected and transmitted waves;d) the fraction of incident power that is transmitted to medium 2.
formulae
0=z
EE 0809Incidence 42
Faculdade de EngenhariaProblem
formulae
δδ2
2
inc
ref
cos1cosF
FPP
+=
2
212
Γ−Γ
=Fpff
2π
δ =312 ηηη =
EE 0809Incidence 43
Faculdade de EngenhariaOblique incidence of a TEM wave at a plane boundary
TEM waveà naHE ˆ⊥⊥rr
ii HErr
and are in plane na⊥
medium 1( )111 ,, σµε
z
x
medium 2( )222 ,, σµε
iθ
tθrθ ntatransmitted
nia
incident
nra
reflected
2. parallel to plane of incidence
iEr
iEr
yeEE rajkii
ni ˆˆ0
1rr
⋅−=
( )zxeEE iirajk
iini ˆsinˆcosˆ
01 θθ −= ⋅−
rr
perpendicular polarization
parallel polarization
( ) yeEzxeEE rajkiii
rajkii
nini ˆˆsinˆcos ˆ2,0
ˆ1,0
11rrr
⋅−⋅− +−= θθ
perpendicular polarizationparallel polarization
1. normal to plane of incidence
General case:
EE 0809Incidence 44
Faculdade de EngenhariaPerpendicular and parallel polarizations – convention
perpendicular polarization
medium 1
z
x
medium 2
iθ
tθrθ
iEr
iHr
nia
nra
y
rEr rH
r
nta
tEr
tHr
medium 1
z
x
medium 2
iθ
tθrθ
iEr
iHr
nia
nra
y
rEr
rHr
nta
tEr
tHr
parallel polarization
Components of parallelto the interface keep their direction
tri EEErrr
e,
EE 0809Incidence 45
Faculdade de EngenhariaPerpendicular polarization – electric and magnetic fields
medium 1
z
x
medium 2
iθ
tθrθ
iEr
iHr
nia
nra
y
rEr rH
r
nta
tEr
tHr
incident
zxa iini ˆcosˆsinˆ θθ +=
yeEE rajkii
ni ˆˆ0
1rr ⋅−=
inii EaHrr
×= ˆ1
1η ( )xzeE
iirajki ni ˆcosˆsinˆ
1
0 1 θθη
−= ⋅−r
reflected
zxa rrnr ˆcosˆsinˆ θθ −=
yeEE rajkrr
nr ˆˆ0
1rr ⋅−=
rnrr EaHrr
×= ˆ1
1η( )xze
Eii
rajkr nr ˆcosˆsinˆ
1
0 1 θθη
+= ⋅−r
zx ii ˆcosˆsin θθ −=
transmitted
zxa ttnt ˆcosˆsinˆ θθ +=
yeEE rajktt
nt ˆˆ0
2rr ⋅−=
tntt EaHrr
×= ˆ1
2η( )xze
Ett
rajkt nt ˆcosˆsinˆ
2
0 2 θθη
−= ⋅−r
relationships between
obtained from boundary conditions000 and, tri EEE
EE 0809Incidence 46
Faculdade de EngenhariaPerpendicular polarization – electric and magnetic fields
medium 1
z
x
medium 2
iθ
tθrθ
iEr
iHr
nia
nra
y
rEr rH
r
nta
tEr
tHr
boundary conditions àcontinuoustanE
continuoustanH ( )0if =SJr
at 0=z tri EEErrr
=+
txrxix HHH =+
xjkt
xjkr
xjki
tii eEeEeE θθθ sin0
sin0
sin0
211 −−− =+
2
sin0
1
sin0
sin0
211 coscoscosη
θη
θθ θθθ xjktt
xjkir
xjkii
tii eEeEeE −−−
−=+−
ti kk θθ sinsin 21 =
000 tri EEE =+
( ) tt
iriE
EE θη
θη
coscos1
2
000
1
=−
EE 0809Incidence 47
Faculdade de EngenhariaPerpendicular polarization – reflection and transmission coefficients
medium 1
z
x
medium 2
iθ
tθrθ
iEr
iHr
nia
nra
y
rEr rH
r
nta
tEr
tHr
000 tri EEE =+
( ) tt
iriE
EE θη
θη
coscos1
2
000
1
=−
ti
ti
i
r
EE
θηθηθηθη
coscoscoscos
12
12
0
0
+−
=
ti
i
i
t
EE
θηθηθηcoscos
cos2
12
2
0
0
+=
ti
ti
θηθηθηθη
coscoscoscos
12
12
+−
=Γ⊥
ti
i
θηθηθη
τcoscos
cos2
12
2
+=⊥
reflection coefficient
transmission coefficient
EE 0809Incidence 48
Faculdade de EngenhariaPerpendicular polarization – reflection and transmission coefficients
medium 1
z
x
medium 2
iθ
tθrθ
iEr
iHr
nia
nra
y
rEr rH
r
nta
tEr
tHr
ti
ti
θηθηθηθη
coscoscoscos
12
12
+−
=Γ⊥
ti
i
θηθηθη
τcoscos
cos2
12
2
+=⊥
reflection coefficient
transmission coefficient
note
1. ⊥⊥ =Γ+ τ1 (as in normal incidence)
2. it is possible that 0=Γ⊥ ti θηθη coscos 12 =
⊥= Bi θθ(Brewster’s angle)
ti nn θθ sinsin 21 =
( )221
12212
11
sinµµ
εµεµθ
−−
=⊥B
3. if medium 2 is ideal conductor, 02 =η0
1
=
−=Γ
⊥
⊥
τ
21 µµ ≠only when
EE 0809Incidence 49
Faculdade de EngenhariaElectric field in medium 1 – perpendicular polarization
ri EEErrr
+=1
⊥⊥ =Γ+ τ1
( )[ ] yeeeEE xjkzjkzjki
iii ˆsincoscos01
111 θθθτ −⊥
−⊥⊥ Γ+Γ−=
r
wave propagating along
wave propagating along x,
with z dependent amplitude
medium 1
z
x
medium 2
iθ
tθrθ
iEr
iHr
nia
nra
y
rEr rH
r
nta
tEr
tHr
yeEyeE rajkr
rajki
nrni ˆˆ ˆ0
ˆ0
11rr ⋅−⋅− +=
zxra
zxra
iinr
iini
θθ
θθ
cossinˆcossinˆ
−=⋅
+=⋅rr
( ) yeeeEyeE xjkzjkzjki
rajki
iiini ˆˆ sincoscos0
ˆ0
1111 θθθτ −−⊥
⋅−⊥ −Γ+=
r
( ) yezkEjyeE xjkii
rajki
ini ˆcossin2ˆ sin10
ˆ0
11 θθτ −⊥
⋅−⊥ Γ+=
r
nia
EE 0809Incidence 50
Faculdade de EngenhariaMaxima and minima in medium 1 – perpendicular polarization
medium 1
z
x
medium 2
iθ
tθrθ
iEr
iHr
nia
nra
y
rEr rH
r
nta
tEr
tHr
yeEyeEE rajkr
rajki
nrni ˆˆ ˆ0
ˆ01
11rrr
⋅−⋅− +=
zxra
zxra
iinr
iini
θθ
θθ
cossinˆcossinˆ
−=⋅
+=⋅rr
( )( )yeeEE zkjzkxkji
iii ˆ1 cos2cossin01
111 θθθ +⊥
+− Γ+=r
( )[ ] ( )[ ]212
101 cos2sincos2cos1 zkzkEE iii θθθθ +Γ++Γ+= Γ⊥Γ⊥
r
( )zkE ii θθ cos2cos21 12
0 +Γ+Γ+= Γ⊥⊥
maxima: minima:
( )πθθ
nk
zi
MAX 2cos21
1
+−= Γ( )[ ]πθ
θ12
cos21
1min ++−= Γ n
kz
i
( )⊥Γ+= 101 iMAX
EEr ( )⊥Γ−= 10
min1 iEE
r
Γ⊥⊥ Γ=Γ θje
EE 0809Incidence 51
Faculdade de EngenhariaIncidence on ideal conductor – perpendicular polarization
medium 1
z
x
ideal conductor
iθ
rθ
iEr
iHr
nia
nra
y
rEr rH
r
( )yeeeEE zjkzjkxjki
iii ˆcoscossin01
111 θθθ −= −−r
maxima: minima:( )
iMAX k
nz
θπ
cos212
1
+=
ikn
zθ
πcos1
min =
01 2 iMAX
EE =r
0min
1 =Er
02 =η0
1=
−=Γ
⊥
⊥
τ
02 =Er
( )yzkeEj ixjk
ii ˆcossin2 1
sin0
1 θθ−−=
If medium 2 is ideal conductor
wave propagating along x,
with z dependent amplitude
EE 0809Incidence 52
Faculdade de EngenhariaParallel polarization – electric and magnetic fields
incident
zxa iini ˆcosˆsinˆ θθ +=
( )zxeEE iirajk
iini ˆsinˆcosˆ
01 θθ −= ⋅− rr
yeE
H rajkii
ni ˆˆ
1
0 1rr
⋅−=η
reflected
zxa rrnr ˆcosˆsinˆ θθ −=
yeE
H rajkrr
nr ˆˆ
1
0 1rr
⋅−−=η
zx ii ˆcosˆsin θθ −=
transmitted
zxa ttnt ˆcosˆsinˆ θθ +=
( )zxeEE ttrajk
ttnt ˆsinˆcosˆ
02 θθ −= ⋅− rr
yeE
H rajktt
nt ˆˆ
2
0 2rr
⋅−=η
relationships between
obtained from boundary conditions000 and, tri EEE
medium 1
z
x
medium 2
iθ
tθrθ
iEr
iHr
nia
nra
y
rEr
rHr
nta
tEr
tHr
( )zxeEE iirajk
rrnr ˆsinˆcosˆ
01 θθ += ⋅− rr
EE 0809Incidence 53
Faculdade de EngenhariaParallel polarization – electric and magnetic fields
meio 1
z
x
meio 2
iθ
tθrθ
iEr
iHr
nia
nra
y
rEr rH
r
nta
tEr
tHr
boundary conditions àcontinuoustanE
continuoustanH ( )0if =SJr
at 0=z txrxix EEE =+
tri HHHrrr
=+
xjktt
xjkir
xjkii
tii eEeEeE θθθ θθθ sin0
sin0
sin0
211 coscoscos −−− =+
2
sin0
1
sin0
sin0
211
ηη
θθθ xjkt
xjkr
xjki
tii eEeEeE −−−
=−
ti kk θθ sinsin 21 =
( ) ttiri EEE θθ coscos 000 =+
( )2
000
1
1ηη
tri
EEE =−
medium 1
z
x
medium 2
iθ
tθrθ
iEr
iHr
nia
nra
y
rEr
rHr
nta
tEr
tHr
EE 0809Incidence 54
Faculdade de EngenhariaParallel polarization – reflection and transmission coefficients
it
it
i
r
EE
θηθηθηθη
coscoscoscos
12
12
0
0
+−
=
it
i
i
t
EE
θηθηθηcoscos
cos2
12
2
0
0
+=
reflection coefficient
transmission coefficient
medium 1
z
x
medium 2
iθ
tθrθ
iEr
iHr
nia
nra
y
rEr
rHr
nta
tEr
tHr
( ) ttiri EEE θθ coscos 000 =+
( )2
000
1
1ηη
tri
EEE =−
it
it
θηθηθηθη
coscoscoscos
12
12| | +
−=Γ
it
i
θηθηθη
τcoscos
cos2
12
2| | +
=
EE 0809Incidence 55
Faculdade de EngenhariaParallel polarization – reflection and transmission coefficients
note
1.
=Γ+
i
t
θθ
τcoscos
1 | || |
2. it is possible that 0| | =Γ it θηθη coscos 12 =
||Bi θθ =(Brewster’s angle)
ti nn θθ sinsin 21 =
( )221
2112| |
2
11
sinεε
εµεµθ
−−
=B
3. If medium 2 is ideal conductor, 02 =η0
1
| |
| |
=
−=Γ
τ
21 µµ =when
reflection coefficient
transmission coefficient
it
it
θηθηθηθη
coscoscoscos
12
12| | +
−=Γ
it
i
θηθηθη
τcoscos
cos2
12
2| | +
=
( )21| |
11
sinεε
θ+
=B
medium 1
z
x
medium 2
iθ
tθrθ
iEr
iHr
nia
nra
y
rEr
rHr
nta
tEr
tHr
EE 0809Incidence 56
Faculdade de EngenhariaElectric field in medium 1 – parallel polarization
ri EEErrr
+=1
medium 1
z
x
medium 2
iθ
tθrθ
iEr
iHr
nia
nra
y
rEr
rHr
nta
tEr
tHr
( ) ( )zxeEzxeE iirajk
riirajk
inrni ˆsinˆcosˆsinˆcos ˆ
0ˆ
011 θθθθ ++−= ⋅−⋅−
rr
=Γ+
i
t
θθ
τcoscos
1 ||||
( )
( ) ( )zxeEzxeE
zxeE
iirajk
iiirajk
i
iirajk
ii
t
nrni
ni
ˆsinˆcosˆsinˆcos
ˆsinˆcoscoscos
ˆ0| |
ˆ0| |
ˆ0| |
11
1
θθθθ
θθθθ
τ
+Γ+−Γ−
−=
⋅−⋅−
⋅−
rr
r
( )
( )( ) zeeeE
xeeeE
zxeEE
izjkzjkxjk
i
izjkzjkxjk
i
iirajk
ii
t
iii
iii
ni
ˆsin
ˆcos
ˆsinˆcoscoscos
coscossin0| |
coscossin0| |
ˆ0| |1
111
111
1
θ
θ
θθθθ
τ
θθθ
θθθ
+Γ+
−Γ+
−=
−
−
⋅−rr
( )
( )( ) zzkeE
xzkeEj
zxeE
iixjk
i
iixjk
i
iirajk
ii
t
i
i
ni
ˆsincoscos2
ˆcoscossin2
ˆsinˆcoscoscos
1sin
0| |
1sin
0| |
ˆ0| |
1
1
1
θθ
θθ
θθθθ
τ
θ
θ
−
−
⋅−
Γ+
Γ+
−=r
EE 0809Incidence 57
Faculdade de EngenhariaElectric field in medium 1 – parallel polarization
waves propagating along x,
with z dependent amplitudes
wave propagatingalong nia
medium 1
z
x
medium 2
iθ
tθrθ
iEr
iHr
nia
nra
y
rEr
rHr
nta
tEr
tHr
( )zxeEE iirajk
ii
t ni ˆsinˆcoscoscos ˆ
0| |11 θθ
θθ
τ −= ⋅−rr
( ) xzkeEj iixjk
ii ˆcoscossin2 1
sin0| |
1 θθθ−Γ+
( ) zzkeE iixjk
ii ˆsincoscos2 1
sin0||
1 θθθ−Γ+
EE 0809Incidence 58
Faculdade de EngenhariaMaxima and minima in medium 1 – parallel polarization
( )( ) iraajkrajk
ixnrnini eeEE θcos1 ˆˆ
||ˆ
0111
rr⋅−⋅− Γ+=
( )[ ] ( )[ ]21| |
21| |01 cos2sincos2cos1cos zkzkEE iiiix θθθθθ +Γ++Γ+= ΓΓ
( )zkE iii θθθ cos2cos21cos 1| |2
| |0 +Γ+Γ+= Γ
maxima: minima:
( )πθθ
nk
zi
MAX 2cos21
1
+−= Γ ( )[ ]πθθ
12cos21
1min ++−= Γ n
kz
i
( )| |01 1cos Γ+= iiMAXEE θ ( )| |0min1 1cos Γ−= iix EE θ
ΓΓ=Γ θje||||
medium 1
z
x
medium 2
iθ
tθrθ
iEr
iHr
nia
nra
y
rEr
rHr
nta
tEr
tHr
ri EEErrr
+=1 ( ) ( )zxeEzxeE iirajk
riirajk
inrni ˆsinˆcosˆsinˆcos ˆ
0ˆ
011 θθθθ ++−= ⋅−⋅−
rr
( ) ( ) zeEeExeEeE irajk
rrajk
iirajk
rraj
inrn inrn i ˆsinˆcos ˆ
0ˆ
0ˆ
0ˆ
01111 θθβ
rrrr⋅−⋅−⋅−⋅− +−++=
( ) izkjrajk
ixini eeEE θθ cos1 cos2
||ˆ
0111 Γ+= ⋅−
r
EE 0809Incidence 59
Faculdade de Engenharia
ideal conductor
Incidence on ideal conductor – parallel polarization
maxima of E1x:
( )i
MAX kn
zθπ
cos212
1
+=
ikn
zθ
πcos1
min =
iiMAXx EE θcos2 01 = 0min1 =xE
02 =η0
1
||
||
=
−=Γ
τ
02 =Er
if medium 2 is ideal conductor
waves propagating along x,
with z dependent amplitudes medium 1
z
x
iθ
rθ
iEr
iHr
nia
nra
y
rEr
rHr
( ) ( )[ ]zeexeeEE irajkrajk
irajkrajk
inrninrni ˆsinˆcos ˆˆˆˆ
011111 θθ
rrrrr⋅−⋅−⋅−⋅− +−−=
( )( ) zzkeE
xzkeEj
iixjk
i
iixjk
i
i
i
ˆsincoscos2
ˆcoscossin2
1sin
0
1sin
0
1
1
θθ
θθθ
θ
−
−
−
−=
minima of E1x:
EE 0809Incidence 60
Faculdade de Engenharia
idealconductor
Metallic waveguides
medium 1
z
x
iθ
rθ
y
perpendicular polarization:ik
nz
θπ
cos1
=01 =Er
( )yzkeEjE ixjk
ii ˆcossin2: 1
sin01
1 θθ−−=⊥r
( )( ) zzkeE
xzkeEjE
iixjk
i
iixjk
i
i
i
ˆsincoscos2
ˆcoscossin2:||
1sin
0
1sin
01
1
1
θθ
θθθ
θ
−
−
−
−=r
at
parallel polarization:ik
nz
θπ
cos1
=01 =xE at
for both polarizations, a conducting plane parallel
to the xy plane could be inserted at
without modifying electric field in medium 1ik
nz
θπ
cos1
= ikn
zθ
πcos1
=
EE 0809Incidence 61
Faculdade de Engenharia
idealconductor
Metallic waveguides
medium 1
z
x
iθ
rθ
y
i
nz
θβπ
cos1
=
electromagnetic wave is guided
by the two conducting surfaces
metallic waveguide
could it be possible to guide an electromagnetic
wave with only dielectric media?
EE 0809Incidence 62
Faculdade de EngenhariaDielectric waveguides
dielectric 1
iθ
rθ
general case:
in each incidence part of the wave is transmitted to dielectric 2
after some distance, the electromagnetic wave in medium 1
is considerably attenuated
the solution would be to have no
energy transmitted to medium 2
dielectric 2dielectric 2
generally, dielectric media don’t guide
electromagnetic waves efficiently
is this possible?
EE 0809Incidence 63
Faculdade de EngenhariaTotal internal reflection
medium 1
z
x
medium 2
iθ
tθrθ
nta transmitted
nia
incident
nra
reflectedti nn θθ sinsin 21 =
21 nn >
Snell’s law of refraction:
it θθ >
Critical angle:
no transmitted wave to
medium 2ci θθ ≥
º90thatsuch == tic θθθ1
2arcsinnn
c =θ
Total internal reflection
cit nn
nn
θθθ sinsinsin2
1
2
1 ≥= 1sin ≥tθ 1sinsin1cos 22 −±=−±= ttt j θθθ
EE 0809Incidence 64
Faculdade de EngenhariaTotal internal reflection
medium 1
z
x
medium 2
iθ
tθrθ
nta transmitted
nia
incident
nra
reflectedTotal internal reflection:
ti
ti
θηθηθηθη
coscoscoscos
12
12
+−
=Γ⊥it
it
θηθηθηθη
coscoscoscos
12
12|| +
−=Γ
1sin ≥tθ
1sincos 2 −±= tt j θθ
Lossless and nonmagnetic media:εµ
η 0=
rrrn εεµ ==
n0η
=
ti
ti
nnnn
θθθθ
coscoscoscos
21
21
+−
=Γ⊥
Reflection coefficients:
it
it
nnnn
θθθθ
coscoscoscos
21
21|| +
−=Γ
( )( ) 1sincos
1sincos
22
21
22
21
2
1
2
1
−±
−=Γ⊥
inn
i
inn
i
jnn
jnn
θθ
θθ m ( )( ) 1sincos
1sincos
22
12
22
12||
2
1
2
1
−±
−±−=Γ
inn
i
inn
i
jnn
jnn
θθ
θθ1| | =Γ=Γ⊥
EE 0809Incidence 65
Faculdade de EngenhariaTotal internal reflection – evanescent waves
medium 1
z
x
medium 2
iθ
tθrθ
n ta transmitted
n ia
incident
nrareflected
1sincos 2 −±= tt j θθ
zxa ttnt ˆcosˆsinˆ θθ +=
rajk nter
⋅− ˆ2
wave propagating along +x
amplitude decreasing exponentially with z
tt xjkzk ee θθ sin1sin 22
2 −−−
( )zxjk tte θθ cossin2 +−
spatial variation of fields in medium 2:
evanescent fields
EE 0809Incidence 66
Faculdade de EngenhariaProblem
(a) the direction of propagation of the incident wave and the angle of incidence;(b) the magnetic field phasor of the incident wave;(c) the propagation directions of the reflected and transmitted waves.(d) the electric field phasors of the reflected and transmitted waves(e) the fraction of the incident power that is transmitted to the dielectric
formulae
The electric field of a uniform plane wave propagating in air is given by .ˆ )34(0
zyjeExE +−= πr
This wave is incident on an interface with a dielectric medium with refractive index 2 that occupies the region z>0.
Find
EE 0809Incidence 67
Faculdade de EngenhariaProblem
formulae
EE 0809Incidence 68
Faculdade de EngenhariaProblem
formulae
Lossless media:
