plane electromagnetic waves
TRANSCRIPT
Electromagnetics Plane Waves
Plane Electromagnetic Waves
Plane waves in lossless mediaPlane waves in lossy media
Group velocity
C-N Kuo, Winter 2004 1
Group velocityFlow of electromagnetic power
Poynting vectorIncidence at conducting boundaryIncidence at dielectric boundary
Incidence at multiple dielectric interfaces
Electromagnetics Plane Waves
Energy Transmission
• Energy stored in the static electric and magnetic fields.
• Electromagnetic waves carry with the electromagnetic power.
• Energy is transported through space (media) to distant receiving points by electromagnetic waves.
C-N Kuo, Winter 2004 2
electromagnetic waves.
01
2
2
22 =
∂∂−∇
t
E
cE
Electromagnetics Plane Waves
Maxwell’s Equation
Law Circuital sAmpere'
Law sFaraday' t
-E
ceSignifican Form Integral Form alDifferenti
•∂+=•∂+=×∇
Φ−=•∂∂=×∇
∫ ∫
∫c
dsD
IdlHD
JH
dt
ddlE
B
C-N Kuo, Winter 2004 3
charge Magnetic Isolated No 0 0
Law sGauss'
Law Circuital sAmpere'
=•=•∇
=•=•∇
•∂∂+=•
∂∂+=×∇
∫
∫
∫ ∫
S
S
c S
dsBB
QdsDD
dst
DIdlH
t
DJH
ρ
Electromagnetics Plane Waves
t
HE
∂∂−=×∇ µ
t
EH
∂∂=×∇ ε
For a source free medium, EM relations:
0=•∇ E 0=•∇ H
)( 2∂−=×∇∂−=×∇×∇ EHE µεµ
C-N Kuo, Winter 2004 4
01
/1c
0
)(
)(
2
2
22
2
22
22
2
=∂∂−∇⇒=
=∂∂−∇⇒
−∇=∇−•∇∇=×∇×∇∂∂−=
∂×∇∂−=×∇×∇
t
E
cEif
t
EE
EEEE
t
E
t
HE
µε
µε
µεµ
01
01
2
2
22
2
2
22
=∂
∂−∇
=∂∂−∇
t
H
cH
t
E
cE
Electromagnetics Plane Waves
Time-Harmonic Fields
– Arbitrary periodic time functions can be expanded into Fourier series of harmonic sinusoidal components; and transient nonperiodic functions can be expressed as Fourier Integral
– Maxwell’s equations are linear differential equations, sinusoidal time variations of source functions of a given frequency will produce sinusoidal variations of E and H with the same frequency in the steady state. Thus, for source functions with an arbitrary time dependence, electrodynamics fields can be source functions.
C-N Kuo, Winter 2004 5
arbitrary time dependence, electrodynamics fields can be source functions.
0
free) source(0
]),,(Re[),,,(
=•∇
==•∇
=×∇−=×∇
=
H
E
EjH
HjE
ezyxEtzyxE tj
ερ
ωεωµ
ω
Electromagnetics Plane Waves
Source-Free Fields• For free-space (simple, non-conducting source-free medium), Helmholtz’s
equation
– where free-space wavenumber
022 =+∇ EkE o
rr
022 =+∇ HkH o
rr
sec)/(1031
where 8 mcc
koo
ooo ×≅===εµ
ωεµω
C-N Kuo, Winter 2004 6
• In Cartesian coordinates, each field component follows this equation. Consider the E field component in the x direction.
c ooεµ
00),,()( 2
2
2 waveplane uniform2
2
2
2
2
2
2
=+∂
∂ →=+∂∂+
∂∂+
∂∂
xox
xo Ekz
EzyxEk
zyx )(),,( zEzyxE xx =
→=o
okλπ2
oo k
πλ 2=
Consider uniform plane wave with z-dependence
Electromagnetics Plane Waves
Plane Waves in Lossless Media
• Solutions of wave equation are traveling waves (unique solution).
zjko
zjko
xxx
oo eEeE
zEzEzE+−−+
−+
+=
+= )()()(
C-N Kuo, Winter 2004 7
• Constants are determined by B.C.
• Instantaneous fields
)cos(),( zktEtzE oox −= ++ ω
+z direction -z direction
Electromagnetics Plane Waves
Traveling Waves
• At successive times the curve effectively travels in the positive z direction.• For a point of a particular phase (fixed phase),
constzkt o =−ω
C-N Kuo, Winter 2004 8
• Phase velocity in free space the velocity of propagation of an equi-phase front.
ckdt
dzu
ooop ====
εµω 1
Electromagnetics Plane Waves
Magnetic Field
• Associated magnetic field can be determined by curl E.– Consider only Ex component exists.
)ˆˆˆ(
00)(
00
ˆˆˆ
+++
+
++−=∂∂=×∇ zzyyxxo
x
zyx
HaHaHaj
zEz
aaa
E ωµ
C-N Kuo, Winter 2004 9
• Intrinsic impedance for free-space.
00)(x zE
+−+−++
+ ==∂∂
−=
∂∂
−= x
o
zjko
o
ozjko
o
x
oy EeE
keE
zjz
zE
jH oo
ηωµωµωµ1
)(1)(1
)(1
)( zEzH xo
y−− −=
ηnote:
)( 377120 Ω≈≅== πεµωµη
o
o
o
oo k
(real number)
Electromagnetics Plane Waves
Example 8-1
• Uniform plane wave propagates in a lossless simple medium in the +z-direction.
– Write the instantaneous expression for E for any t and z.
– Write the instantaneous expression for H.
C-N Kuo, Winter 2004 10
for H.– Determine the locations where Ex is
a positive maximum when t=10^-8(s)
xxEaE ˆ=r
0,1,4 === σµε rr
8
1&10at t )/(10
,100
8-4max, ===
=
− zmVE
MHzf
x
Electromagnetics Plane Waves
Example 8-1 (cont’)• wavenumber and wavelength.
• Expression for E field.
)/(3
44
103
102
8
8
mradc
k rro
ππεµωµεω =××===
+−== − ψωr
)(2
32m
k== πλ
C-N Kuo, Winter 2004 11
)]8
1(
3
4102cos[10ˆ
)63
4102cos(10ˆ
)(68
08
10102 @ maximum
)cos(10ˆˆ
84
84
8
4
−−×=
+−×=
==⇒=+−××
+−==
−
−
−
zta
ztaE
radk
k
kztaEaE
x
x
xxx
ππ
πππ
πψψπ
ψω
r
r
Electromagnetics Plane Waves
Example 8-1 (cont’)
• Expression for H field.
)]8
1(
3
4102cos[
60
10ˆ
ˆˆ),(
84
−−×=
==
−
zta
EaHatzH
x
yyyy
πππ
ηr
)(60 Ω== πε
ηηr
o
C-N Kuo, Winter 2004 12
• At t=10^-8 (sec) maximum of cosine function occurs at
)]8
(3
102cos[60
ˆ −−×= ztax ππ
πππ nzt 2)8
1(
3
4)10(102 48 ±=−−=×× −
0,1,2n 8
13
2
3
8
13K=±=±=⇒ λnnz
Electromagnetics Plane Waves
Transverse Electromagnetic Waves
• A transverse electromagnetic (TEM) wave is a wave for which the electric and magnetic field vectors lie in a plane which is transverse or perpendicular to the axis of propagation.
• Phasor expression for a general form.Er
Hr
jkzoeEzE −=rr
)(
C-N Kuo, Winter 2004 13
H
22222 where),,( kkkkeeeEzyxE zyxzjkyjkxjk
ozyx ==++= −−− µεω
rr
022 =+∇ EkErr
0),,()( 22
2
2
2
2
2
=+∂∂+
∂∂+
∂∂
zyxEkzyx
r
Electromagnetics Plane Waves
Constant Phase Plane
• Define wavenumber vector.
nzzyyxx akkakakak ˆˆˆˆ =++=r
rrrrrrr
•−•−
zayaxaR zyx ˆˆˆ ++=r
C-N Kuo, Winter 2004 14
• Constant phase plane.
Rajko
Rkjo
neEeERErrr
rrrr
•−•− == ˆ)(
constantˆ =•=• RakRk n
rrr
Electromagnetics Plane Waves
Characteristic of Plane-wave Solution
• In charge-free region,
• Since
0)(0 =∇•⇒=•∇ •− Rjko eEErr
zjkyjkxjkzzyyxx
zjkyjkxjkzyx
Rjk
eeekakakaj
eeez
ay
ax
ae
zyx
zyx
−−−
−−−•−
++−=
∂∂+
∂∂+
∂∂=∇
)ˆˆˆ(
) )(ˆˆˆ()(
C-N Kuo, Winter 2004 15
Rjkneajk •−−= ˆ
0ˆ0)ˆ( =•⇒=−• •−no
Rjkno aEeajkE
rr E field is transverse to the direction of propagation.
Electromagnetics Plane Waves
TEM Waves
• Uniform plane wave is a TEM wave with E and H field perpendicular to each other.
• Express H field in terms of E field.
)(ˆ1
)(1
)( REaRERH n
rrrrrr
×=×∇−=ηωµ
Rajkon
neEaRHr
rrr
•−×= ˆ)ˆ(1
)(η
C-N Kuo, Winter 2004 16
• Express E field in terms of H field.
)(ˆ)()( REaREj
RH n ×=×∇−=ηωµ on eEaRH ×= )ˆ()(
η
RkjoeHRH
rrrrr
•−=)(
)(ˆ)(1
)( RHaRHj
RE n
rrrrrr
×−=×∇= ηωε
Electromagnetics Plane Waves
Polarization• The polarization of a uniform plane wave describes the time-varying behavior
of the electric field intensity vector at a given point in space.
• Linear polarization.– Field is fixed in one direction (in space quadrature, but in time phase).
• Circular polarization.– Superposition of two linearly polarized waves (both in space and time quadrature).
C-N Kuo, Winter 2004 17
– Superposition of two linearly polarized waves (both in space and time quadrature).
• General expression.
90by leads assume(*ˆˆ
)(ˆ)(ˆ)(0
122010
21
(z)E(z)EejEaeEa
zEazEazEjkz
yjkz
x
yx
−− −=
+=r
)2
cos(ˆ)cos(ˆ
)](ˆ)(ˆRe[),(
2010
21
πωω
ω
−−+−=
+= −
kztEakztEa
ezEazEatzE
yx
tjyx
r
time harmonic field
instantaneous field
Electromagnetics Plane Waves
Circular Polarization
• Observe field at z=0.
)sin(ˆ)cos(ˆ
),0(ˆ),0(ˆ),0(
2010
21
tEatEa
tEatEatE
yx
yx
ωω +=
+=r
angular velocity ω
C-N Kuo, Winter 2004 18
angular velocity ω
Electromagnetics Plane Waves
Circular Polarization (cont’)
• Right-hand (positive circularly polarized wave)– Finger of the right hand follow the direction of the rotation of E, the thumb points to
the direction of wave propagation.– Counterclockwise direction.
jkzjkz
yx
ejEaeEa
zEazEazE−− −=
+= 21
ˆˆ
)(ˆ)(ˆ)(r
C-N Kuo, Winter 2004 19
• Left-hand (negative circularly polarized wave)– Clockwise direction.
jkzy
jkzx ejEaeEa −− −= 2010 ˆˆ
jkzy
jkzx
yx
ejEaeEa
zEazEazE−− +=
+=
2010
21
ˆˆ
)(ˆ)(ˆ)(r
Electromagnetics Plane Waves
Linear Polarization
• If E2(z) and E1(z) are in space quadrature, but in time phase.
tEaEatE yx ωcos)ˆˆ(),0( 2010 +=r
C-N Kuo, Winter 2004 20
• General expression.
• Note: the reception of transmitted signal must be coincident with the polarization.
)cos(ˆ)cos(ˆ),( 2010 kztEakztEatzE yx −+−= ωωr
Electromagnetics Plane Waves
Example 8-3
• Decomposition of linearly polarized plane wave into a right-hand circularly polarized wave and a left-hand circularly polarized wave of equal amplitude.
jkzyx
jkzyx
jkzx
eajaE
eajaE
eEazE
−−
−
++−=
=
)ˆˆ(2
)ˆˆ(2
ˆ)(
00
0
r
C-N Kuo, Winter 2004 21
22
right-hand circular polarization left-hand circular polarization
Electromagnetics Plane Waves
Plane Waves in Lossy Media
• In a source-free lossy medium, the homogeneous vector Helmholtz’s equation:
022 =+∇ EkE c
ωεσµεωεεµωµεω
jjk cc +=−== 1)"'(
C-N Kuo, Winter 2004 22
• Define propagation constant
ωεσµεω
εεµεω
µεωγ
jjjj
jjk cc
+=−=
==
1'
"1'
βαγ j+=
attenuation constant
phase constant
Electromagnetics Plane Waves
Low-Loss Dielectrics
• Low-loss dielectric is a good but imperfect insulator with a nonzero equivalent conductivity.
])'
"(
8
1
'2
"1[' 2
εε
εεµεωβαγ +−≅+= jjj
(Np/m) '2
"
εµωεα ≅
C-N Kuo, Winter 2004 23
(rad/m) ])'
"(
8
11['
(Np/m) '2
2
εεµεωβ
εα
+≅
≅
)'2
"1(
')
'
"1(
'2/1
εε
εµ
εε
εµη jjc +≅−= −
complex impedance impliesE and H fields are not in timephase.
])(8
11[
1 2'
"
' εε
µεβω −≅=pu
Electromagnetics Plane Waves
Good Conductors
• A good conductor is a medium for which 1>>ωεσ
ωµσωµσωεσµεω
ωεσµεωγ
2
11
jj
jj
jj
+==≈+=
σµσπβα , ff ∝==
C-N Kuo, Winter 2004 24
σµσπβα , ff ∝==
σα
σµπ
σωµ
εµη )1()1( j
fj
j
cc +=+=≅=
magnetic field intensity lags behind the electric field intensity by 45 deg.
µσω
βω 2≅=pu
Electromagnetics Plane Waves
Skin Depth
• At very high frequencies the attenuation constant tends to be very large for a good conductor. The amplitude of a wave will be attenuated by the factor
• The distance through which the amplitude of a traveling plane wave
ze α−
C-N Kuo, Winter 2004 25
• The distance through which the amplitude of a traveling plane wave decreases by a factor of , or 0.368, is called the skin depth, or the depth of penetration of a conductor.
1−e
(m) 11
µσπαδ
f==
(m) 2
1
πλ
βδ == δ
1−epenetration depth
conductor interface
Electromagnetics Plane Waves
Skin Depth of Various Materials
C-N Kuo, Winter 2004 26
Electromagnetics Plane Waves
Current Distribution
• At high frequencies, electric current distribution is no longer uniform due to the effect of skin depth.
• Cross-sectional view of a good conductor.
C-N Kuo, Winter 2004 27
high frequency
Electromagnetics Plane Waves
Ionized Gases
• Gases become ionized due to high energy excitation, e.g., ionosphere.
• The electron and ion densities in the individual ionized layers of the ionosphere are essentially equal called plasmas.
• When electromagnetic waves passing through the ionosphere, the electrons
C-N Kuo, Winter 2004 28
• When electromagnetic waves passing through the ionosphere, the electrons are accelerated by the electric fields more than positive ions.
• Assume the intensity of plasmas is low and ignore the collisions between the electrons and the gas atoms.
Electromagnetics Plane Waves
Ionized Gases
• An electron of charge –e and mass m in a time-harmonic electric field E in the x-direction at an angular frequency ω experiences a force –eE.
• Such a displacement gives rise to an electric dipole moment:
Exxx
E2
22
2
ωω
m
em
dt
dme =⇒−==−
xp e−=2Ne
C-N Kuo, Winter 2004 29
• ωp is called plasma angular frequency
)( )1(
)1(
0
2
20
02
2
00
εω
ωω
ε
εωεε
m
NeE
Em
NePED
pp =−=
−=+=
)1()1(
2
1
2
2
2
02
2
0
0
2
f
f
m
Nef
ppp
pp
−=−=
==
εωω
εε
εππω
Electromagnetics Plane Waves
Ionized Medium
• Consider wave propagation in an ionized medium:
• At ω=ω , β=0. no propagation.
22 )/(1)/(1 ωωωµεωβ ppo cff −=−=
C-N Kuo, Winter 2004 30
• At ω=ωp, β=0. no propagation.
• At ω=ωp, propagation is possible.
cc
up
p ≥−
==2)/(1 ωωβ
ω
ccdd
u pg ≤−== 2)/(1/
1 ωωωβ
2cuu gp =•
Electromagnetics Plane Waves
Wave Velocity
• Phase velocity is defined as the velocity of propagation of an equi-phase wavefront.• Group velocity is defined as the velocity propagation of the wave-packet envelope.
ωβ ddug /
1=
C-N Kuo, Winter 2004 31
ωββω
/
1==pu
Electromagnetics Plane Waves
Time-Harmonic Traveling Waves
• Consider a wave packet that consists of two traveling waves having equal amplitude and slightly different angular frequencies.
)cos()cos(2
])()cos[(])()cos[(),(
ztztE
ztEztEtzE
ooo
oooooo
βωβωββωωββωω
−∆−∆=∆−−∆−+∆+−∆+=
C-N Kuo, Winter 2004 32
• Phase velocity:
• Group velocity:
amplitude of slow variation with an angular frequency ∆ω
ωββωβω
∆∆=
∆∆==⇒=∆−∆
/
1.
dt
dzuconstzt g
βωω ===
kdt
dzup ωβ dd
ug /
1=
Electromagnetics Plane Waves
ω−β Diagram
• For plane waves in a lossless medium, phase constant β is a linear function of ω. As a consequence, phase velocity is a constant.
Fast wave region
C-N Kuo, Winter 2004 33
• In general, phase constant is not a linear function of ω.
• Dispersion: waves of the component frequencies travel with different phase velocities, causing a distortion in the signal wave shape.
Slow wave region
Electromagnetics Plane Waves
Ionized Medium
• Consider wave propagation in an ionized medium:
• At ω=ω , β=0. no propagation.
22 )/(1)/(1 ωωωµεωβ ppo cff −=−=
C-N Kuo, Winter 2004 34
• At ω=ωp, β=0. no propagation.
• At ω=ωp, propagation is possible.
cc
up
p ≥−
==2)/(1 ωωβ
ω
ccdd
u pg ≤−== 2)/(1/
1 ωωωβ
2cuu gp =•
Electromagnetics Plane Waves
General Relation
• General relation between the group and the phase velocities.
ωωω
ωωβ
d
du
uuud
d
d
d p
ppp2
1)( −==
ωω
d
du
u
uu
p
p
pg
−=
1
C-N Kuo, Winter 2004 35
• Three possible cases:– No dispersion (independent of frequency)
– Normal dispersion
– Anomalous dispersion
0=ωd
dup
0<ωd
dup
pg uu =
pg uu <
0>ωd
dup
pg uu >
Electromagnetics Plane Waves
Flow of Electromagnetic Power• Relation between the rate of energy transfer and the electric and magnetic
field intensities associated with a traveling electromagnetic wave.
t
DJH
t
BE
∂∂+=×∇
∂∂−=×∇
r
rr
r
r
t
DEJE
t
BHHEEHHE
∂∂⋅−⋅−
∂∂⋅−=×∇⋅−×∇⋅=×⋅∇
r
rrr
r
rrrrrrr
)()()(
C-N Kuo, Winter 2004 36
• In a simple medium,
tt ∂∂
2
2
2
)2
1(
)(
)2
1(
)(
2
1)(
EJE
Ett
EE
t
DE
Htt
HH
t
HH
t
BH
σ
εε
µµµ
=⋅∂∂=
∂∂⋅=
∂∂⋅
∂∂=
∂⋅∂=
∂∂⋅=
∂∂⋅
rr
r
r
r
r
rrr
r
r
r
Electromagnetics Plane Waves
Flow of Electromagnetic Power
• The integral form
222 )2
1
2
1(
)(
EHEt
t
DEJE
t
BHHE
σµε −+∂∂−=
∂∂⋅−⋅−
∂∂⋅−=×⋅∇
r
rrr
r
rrr
C-N Kuo, Winter 2004 37
∫∫∫ −+∂∂−=⋅×
VVSdvEdvHE
tsdHE 222 )
2
1
2
1()( σµεr
rr
Time-rate of change of the energy stored in the electric and magnetic fields. rate of decrease of the electric and magnetic energies stored
The Ohmic power dissipated in the volume
Electromagnetics Plane Waves
Poynting Vector
• To be consistent with the law of conservation of energy, the LHS must equal to the power leaving the volume through the surface.
• Poynting vector is defined to represent the power flow per unit area (leaving the enclosed volume)
)(W/m 2HEΡrrr
×= in the direction normal
C-N Kuo, Winter 2004 38
• Poynting’s theorem
)(W/m 2HEΡ ×=
∫∫∫ ++∂∂=⋅−
VV meSdvPdvww
tsdP σ)(r
r
in the direction normal to both E and H
this term vanishes if the region of concern is loseless
this term vanishes in a static situation
Electromagnetics Plane Waves
Example 8-7
• Find the Poynting vector on the surface of a long, straight conducting wire (of radius b and conductivity σ) that carries a direct current I.
• DC situation; uniform current distribution over the cross-sectional area.
surface on the ˆˆ ˆI
aHI
aJ
EI
aJ =⇒===r
r
rr
C-N Kuo, Winter 2004 39
• To verify Poynting theorem.
surface on the 2
ˆˆ ˆ22 b
aHb
aEb
aJ zz πσπσπ φ=⇒===
32
2
2ˆ
b
IaHEΡ r σπ
−=×=rrr
RIb
Ibb
IdsaPsdP
S rS
22
232
2
)(22
ˆ ===⋅−=⋅− ∫∫ σππ
σπl
l
rr
r
Electromagnetics Plane Waves
• Where the previous equation, the resistance of a straight wire, R=l/σS, has been used. The above result affirms that the negative surface integral of the Poynting vector is exactly equal to the I2R ohmic power loss in
C-N Kuo, Winter 2004 40
exactly equal to the I2R ohmic power loss in the conducting wire.
Electromagnetics Plane Waves
Instantaneous Power Densities• Care with nonlinear operations on phasor expressions.
)cos(||
ˆ])(Re[),(
)cos(ˆ])(Re[),(
ηαω
ω
θβωη
βω
−−==
−==
− zteE
aezHtzH
ztEaezEtzE
zoy
tj
oxtj
rr
rr
])()(Re[])(Re[])(Re[ tjtjtj ezHzEezHezE ωωωrrrr
×≠×
C-N Kuo, Winter 2004 41
• Note:
• Instantaneous power density.
])()(Re[])(Re[])(Re[ tjtjtj ezHzEezHezE ωωωrrrr
×≠×
])(Re[])(Re[),(),(),( tjtj ezHezEtzHtzEtzP ωωrrrrr
×=×=
Electromagnetics Plane Waves
Instantaneous Power Densities• Power density vector.
)]22cos([cosˆ
)cos()cos(ˆ
])(Re[])(Re[),(
22
22
α
ηα
ωω
θβωθ
θβωβωη
−−+=
−−−=
×=
−
−
zteE
a
ztzteE
a
ezHezEtzP
zo
zoz
tjtjrrr
C-N Kuo, Winter 2004 42
• On the other hand.
)]22cos([cosˆ 2ηη
α θβωθη
−−+= − zteE
a zoz
)cos(ˆ])()(Re[ 22
ηαω θβω
η−−=× − zte
EaezHzE zo
ztj
rr
not the same!
Electromagnetics Plane Waves
Average Power Densities
• As far as the power transmitted by an electromagnetic wave is concerned, the average value is a more significant quantity than the instantaneous value.
• Make use of this identity.
ηα θ
ηcos
2ˆ),(
1)( 2
2
0
zoz
T
av eE
adttzPT
zP −== ∫rr
C-N Kuo, Winter 2004 43
• Make use of this identity.
])()()()(Re[2
1
])(Re[])(Re[),(
2* tj
tjtj
ezHzEzHzE
ezHezEtzP
ω
ωω
rrrr
rrr
×+×=
×=
]Re[2
1)(
2
1)(
2
1]Re[]Re[ *** BABABBAABA ×+×=+×+=×
)]()(Re[2
1),(
1)( *
0zHzEdttzP
TzP
T
av
rrrr
×== ∫
Electromagnetics Plane Waves
Bounded Region
• In practice, waves propagate in bounded regions where several media with different constitutive parameters are present.
• When an electromagnetic wave
C-N Kuo, Winter 2004 44
traveling in one medium impinges on another medium with a different intrinsic impedance, it experiences a reflection.
• Two cases will be considered: – normal incidence– oblique incidence.
(lossless) (perfect conductor)
Electromagnetics Plane Waves
Normal Incidence
• Assume the phasors of the incident fields inside medium 1.
zjioyi
zjioxi
eE
azH
eEazE
1
1
1
ˆ)(
ˆ)(
β
β
η−
−
=
=take z=0 Eio is the magnitude of field intensity @ z=0.
)()()( zHzEzP ×=
C-N Kuo, Winter 2004 45
• Inside medium 2 (PEC), both electric and magnetic fields vanish. no wave is transmitted across the boundary into the z>0 region.
• The incident wave is reflected, giving rise to a reflected wave (Er, Hr).
)()()( zHzEzP iii ×=
zjroxr eEazE 1ˆ)( β+=
Electromagnetics Plane Waves
Normal Incidence
• The total field in medium 1.
• Boundary condition (continuity of the tangential components) at z=0 demands that
)(ˆ)()()( 111
zjro
zjioxri eEeEazEzEzE ββ +− +=+=
iororoiox EEEEEaE −=⇒==+= 0)0()(ˆ)0( 21
C-N Kuo, Winter 2004 46
• The magnetic field intensity Hr of the reflected wave is related to Er:
iororoiox EEEEEaE −=⇒==+= 0)0()(ˆ)0( 21
zEjaeeEazE ioxzjzj
iox 11 sin2ˆ)(ˆ)( 11 βββ −=−= +−
zjiy
zjry
rzrnrr
eE
aeEa
zEazEazH
11
1
00
1
11
1
)(1
)(1
)(
ββ
ηη
ηη
×=×−=
×−=×=
Electromagnetics Plane Waves
• No average power is associated with the total eletromagnetic wave in medium 1, since are E1(z) in H1(z) phase quadrature.
• Instantaneous electric and magnetic field intensities:
zE
azHzHzH iyri 1
1
01 cos2)()()( β
η=+=
( ) tzEaezEtzE ioxtj ωβω sinsin2ˆ)(Re),( 111 ==
C-N Kuo, Winter 2004 47
( ) tzE
aezHtzH iox
tj ωβη
ω coscos2ˆ)(Re),( 11
11 ==
),(H of zero),,(E of maxi. ........0,1,2,....n
,4
1)-(2nzor ,1)-(2n z While
),(H of maxi. .......0,1,2,....n
),(E of zero ,2
-nzor ,-n z While
11
1
1
11
tztz
tz
tz
=
+=+=
=
==
λπβ
λπβ
Electromagnetics Plane Waves
Standing Wave
C-N Kuo, Winter 2004 48
Electromagnetics Plane Waves
Oblique Incidence at a Plane Conducting Boundary
• Perpendicular polarization, Ei is perpendicular to the plane of incidence. (θi, angle of incidence).
izixni aaa θθ cossin +=)cossin(
0011),( iini zxj
iyRaj
iyi eEaeEazxE θθββ +−•− ==
C-N Kuo, Winter 2004 49
[ ]
[ ] )cossin(
1
0
1
00
1sincos
),(1
),(
),(
ii zxjizix
i
inii
iyiyi
eaaE
zxEazxH
eEaeEazxE
θθβθθη
η+−+−=
×=
==
Electromagnetics Plane Waves
• For the reflected wave:
.reflection of angle theis where
cossin
r
rzrxnr aaa
θθθ −=
zxjryr
xExExE
eEazxE rr θθβ
+==
= −−
)0,()0,()0,(
:0zat
),( )cossin(0
1
C-N Kuo, Winter 2004 50
• Snell’s law of reflection: the angle of reflection equals to the angle of incidence.
riri
xjr
xjiy
ri
EE
eEeEa
xExExEri
θθ
θβθβ
=−=→
=+=
+=−−
and
0)(
)0,()0,()0,(
00
sin0
sin0
1
11
Electromagnetics Plane Waves
)cossin(0
1),( ii zxjiyr eEazxE θθβ −−−=
[ ]
[ ] )cossin(
1
0
1
1sincos
),(1
),(
ii zxjizix
i
rnrr
eaaE
zxEazxH
θθβθθη
η−−−−=
×=
E
C-N Kuo, Winter 2004 51
i
iii
xjiiy
xjzjzjiy
ri
ezEja
eeeEa
zxEzxEzxE
θβ
θβθβθβ
θβ sin10
sincoscos0
1
1
111
)cossin(2
)(
),(),(),(
−
−−
−=
−=
+=
])cossin(
sin)coscos(
cos[2),(
sin1
sin1
1
01
1
1
i
i
xji
izxj
i
ixi
ez
jaez
aE
zxH
θβ
θβ
θβθθβ
θη
−
− +
−=
Electromagnetics Plane Waves
• Along the z-direction, E1y and H1x maintain standing –wave patterns. No average power is propagated.
• Along the x-direction, E1y and H1z are in both time and space phase and propagate with a phase velocity.
• The propagating wave in x-direction is a nonuniform plane wave because its amplitude varies with x.
ixx
iixx
uu
θλ
βπλ
θθβω
βω
sin
2
sinsin
1
11
1
111
==
===
C-N Kuo, Winter 2004 52
varies with x.• When
• E1=0 for all x since
• A transverse electric (TE) wave (E1x=0) would bounce back and forth between the conducting planes and propagate in the x-direction.
.....................3,2,1
,cos2
cos1
1
=
−==
m
mzz ii πθλπθβ
0)cossin( 1 =iz θβ
Electromagnetics Plane Waves
:OA2 toequalh wavelengtguide a has
waveguide-parallel in the wave travelingThe
cos2
2'
"
1
1
1
==
==
θλ
βπλ
i
bOA
OA
C-N Kuo, Winter 2004 53
axis x along gpropagatin waveno ,0
sinsin
OA'2OA2
:OA2 toequalh wavelengtguide a has
i
11"
"
=
>===
θ
λθ
λθ
λii
g
Electromagnetics Plane Waves
H-Polarization Incidence
• Incidence wave
• Reflective wave
[ ])cossin(
1
0
)cossin(0
1
1
),(
sincos),(
ii
ii
zxjiyi
zxjizixii
eE
azxH
eaaEzxE
θθβ
θθβ
η
θθ+−
+−
=
−=
C-N Kuo, Winter 2004 54
[ ])cossin(
1
0
)cossin(0
1
1
),(
sincos),(
rr
ii
zxjryr
zxjrzrxrr
eE
azxH
eaaEzxE
θθβ
θθβ
η
θθ−−
−−
−=
+=
riri
xjrr
xjii
rxix
EE
eEeE
xExEri
θθθθ θβθβ
=−=→=+
=+=
−−
and
0)cos()cos(
0)0,()0,(
:0zat
00
sin0
sin0
11
Electromagnetics Plane Waves
i
iii
iii
xjiiziixi
xjzjzjiiz
xjzjzjiix
ri
ezazjaE
eeeEa
eeeEa
zxEzxEzxE
θβ
θβθβθβ
θβθβθβ
θβθθβθθθ
sin110
sincoscos0
sincoscos0
1
1
111
111
)]coscos(sin)cossin(cos[2
)(sin
)(cos
),(),(),(
−
−−
−−
+−=
−−
−=
+=
ri zxHzxHzxH1 ),(),(),( +=
C-N Kuo, Winter 2004 55
ixji
iy
ri
ezE
a θβθβη
sin1
1
0
1
1)coscos(2 −=
Electromagnetics Plane Waves
• Along the z-direction, E1x and H1y maintain standing –wave patterns. No average power is propagated.
• Along the x-direction, E1z and H1y are in both time and space phase and propagate with a phase velocity.
• The propagating wave in x-direction is a nonuniform plane wave because its amplitude ix
x
iixx
uu
θλ
βπλ
θθβω
βω
sin
2
sinsin
1
11
1
111
==
===
C-N Kuo, Winter 2004 56
• The propagating wave in x-direction is a nonuniform plane wave because its amplitude varies with x.
• When
• E1x=0 for all x since
• A transverse magnetic (TM) wave (H1x=0) would bounce back and forth between the conducting planes and propagate in the x-direction.
.....................3,2,1
,cos2
cos1
1
=
−==
m
mzz ii πθλπθβ
0)cossin( 1 =iz θβ
Electromagnetics Plane Waves
Normal Incidence at Plane Dielectric Boundary
• Incidence Wave
• Reflective Wave
zjioyi
zjioxi
eE
azH
eEazE
1
1
ˆ)(
ˆ)(
β
β
η−
−
=
=
C-N Kuo, Winter 2004 57
• Transmitted Wave
zjroy
rzr
zjroxr
eE
azE
azH
eEazE
1
1
11
ˆ)(
ˆ)(
ˆ)(
β
β
ηη−=−=
=
)0(,ˆ)(
ˆ)(
ˆ)(
2
2
22
===×=
=
−
−
zEEeE
azE
azH
eEazE
ttozjto
yt
zt
zjtoxr
β
β
ηη
Electromagnetics Plane Waves
• B.C. :Tangential components of the electric and magnetic field intensities must be continuous:
000)0()0()0( tritri EEEEEE =+→=+
2
000
1
)(1
)0()0()0(ηη
tritri
EEEHHH =+→=+
012
120 ir EE
ηηηηη
+−=
C-N Kuo, Winter 2004 58
• Definitions:
012
20
2it EE
ηηη+
=
12
12
0
0
12
2
0
0
t coefficien Reflection
2t coefficienon Transmissi
ηηηη
ηηητ
+−==Γ
+==
i
r
i
t
E
E
E
E
τ 1=+Γ
Electromagnetics Plane Waves
• Γ and τ may themselves be complex in the general case. A complex Γ or τsimply means that a phase shift is introduced at the interface upon reflection or transmission:
• Example: if medium 2 is a perfect conductor, η2=0. Then, Γ =-1 and τ=0. Thus, Er0=-Ei0., and Et0=0. The incident wave will be totally reflected, and a standing wave will be produced in medium 1.
• If medium 2 is not a perfect conductor, partial reflection will result.
)()()( zEzEzE +=
C-N Kuo, Winter 2004 59
wavestanding:2
wave traveling:
)]sin2([
)]()1[(
)(
)()()(
0
0
10
0
0
1
1
111
11
i
i
zjix
zjzjzjix
zjzjix
ri
ΓE
E
zjeEa
eeeEa
eeEa
zEzEzE
τβτ β
βββ
ββ
Γ+=
−Γ+Γ+=
Γ+=
+=
−
−−
−
Electromagnetics Plane Waves
• Γ >0, (η2 > η1):– The maximum value of E1(z) is Ei0(1+Γ) occurs when 2β1zmax=-2nπ (n=0,1,2,3…)– zmax= -nπ/ β1=-nλ1/2, n=0,1,2,3……– The minimum value of E1(z) is Ei0(1-Γ) occurs when 2β1zmin=-(2n+1)π
(n=0,1,2,3…)– zmax= -(2n+1)π/ 2β1=-(2n+1)λ1/4, n=0,1,2,3……
• Γ <0, (η2 < η1):
)1()( 11 201
zjzjix eeEazE ββ Γ+= − )1()( 11 2
1
01
zjzjiy ee
EazH ββ
ηΓ−= −
C-N Kuo, Winter 2004 60
2 1
– The maximum value of E1(z) is Ei0(1-Γ) occurs when 2β1zmax=(2n+1)π(n=0,1,2,3…)
– zmax=-(2n+1)π/ 2β1=-(2n+1)λ1/4, n=0,1,2,3……– The minimum value of E1(z) is Ei0(1+Γ) occurs when 2β1zmin=-2nπ (n=0,1,2,3…)– zmax= -nπ/ β1=-nλ1/2, n=0,1,2,3……
• In a dissipationless medium, Γ is real; and H1(z) will be a minimum at locations where E1(z) is a maximum, and vice versa.
• In medium 2:zj
iyzj
ix eEazHeEazE 220
2202 )(,)( ββ
ηττ −− ==
Electromagnetics Plane Waves
Standing Wave Ratio
• Definition: the ratio of the maximum value to the minimum value of the electric field intensity of a standing wave.
1
1
1
min
max
−=Γ↔
Γ−Γ+
==
S
E
ES
C-N Kuo, Winter 2004 61
• While the value of Γ ranges from –1 to +1, the value of S ranges from 1 to ∝.• The standing wave ratio in decibels is 20 log10 S.
– S=2→20log102=6.02 dB. Γ =1/3– SWR:2dB →S=1.26, Γ =0.015
1
1
+−=Γ↔
S
S
Electromagnetics Plane Waves
Normal Incidence at Multiple Dielectric Interfaces
• X-polarized incident field:
• The reflected field in medium 1:– The field reflected from the interface at z=0 as
the incident wave impinges on it
)( 11001
zjr
zjix eEeEaE ββ += −
C-N Kuo, Winter 2004 62
the incident wave impinges on it– The field transmitted back to medium 1 from
medium 2 after a first reflection from the interface at z=d
– The field transmitted back into medium 1 from medium 2 after a second reflection at z=d
– And so on………– How to determine it?
Electromagnetics Plane Waves
• Er0 can be solved base the following procedure:– Write down the electric and magnetic field intensity vectors in all three regions– Apply the B.C. to solve the equations.
• )( 11001
zjr
zjix eEeEaE ββ += −
)(1
1100
11
zjr
zjiy eEeEaH ββ
η−= −
zjzj eEeEaE 22 )( ββ −−+ += )()( d,zAt
)0()0(
)0()0( 0,zAt
)continuouscomponent l(tangentia .
21
21
dEdE
HH
EE
sBC
=====
C-N Kuo, Winter 2004 63
zjy
zjx
zjzjy
zjzjx
eEaH
eEaE
eEeEaH
eEeEaE
3
3
22
22
32
2
33
222
2
222
1
)(1
)(
β
β
ββ
ββ
η
η
−+
−+
−−+
−
=
=
−=
+=
)()(
)()( d,zAt
32
32
dHdH
dEdE
===
Electromagnetics Plane Waves
Wave Impedance of The Total Field
• Wave Impedance of the total field: the ratio of the total electric field intensity to the total magnetic field intensity at any plane parallel to the plane boundary.
• For instance, a z-dependent uniform plane wave as shown in the previous figure, we write, in general:
)( Total
)( Total)(
zH
zEzZ x=
C-N Kuo, Winter 2004 64
• For previous case (normal incidence)
)( Total)(
zHzZ
y
=
zjzj
zjzj
zjzji
zjzji
y
x
ee
ee
eeE
eeE
zH
zEzZ
11
11
11
11
1
1
0
0
1
11
)(
)(
)(
)()(
ββ
ββ
ββ
ββ
η
η
Γ−Γ+=
Γ−
Γ+==
−
−
−
−
Electromagnetics Plane Waves
• Which correctly reduces to η when η2=η . In that case there is no
22
12
1211
111211
1
11 sincos
sincos
)(
)()(
11
11
ηηηη
βηβηβηβηηη ββ
ββ
+−=Γ
++=
Γ−Γ+=
−−=−
−=
−
−
ljl
ljl
ee
ee
lH
lElZ
lZ
ljlj
ljlj
y
x
C-N Kuo, Winter 2004 65
• Which correctly reduces to η1 when η2=η1. In that case there is no discontinuity at z=0; hence there is no reflected wave and the total-field wave importance is the same as the intrinsic impedance of the medium.
• If the plane boundary is perfectly conducting, η2 =0 and Γ=-1, and
Which is the same as the input impedance of a transmission line of length l that has a characteristic impedance η1 and terminates in a short circuit.
ljlZ 111 tan)( βη=−
Electromagnetics Plane Waves
Impedance Transformation with Multiple Dielectrics
• The total field in medium 2 is the result of multiple reflections of the two boundary planes at z=0 and z=d; but it can be grouped into a wave traveling in the +z direction and another traveling in the –z direction. The wave impedance of the total field in medium 2 at the left-hand interface z=0 can be found from the right side by replacing η2 by η3 , η1 by η2 , β1 by β2 , and l by d. thus,
djdZ 2223 sincos
)0(βηβηη +=
C-N Kuo, Winter 2004 66
• The effective reflection coefficient at z=0 for the incident wave in medium 1 is
• Γ0 differs from Γ only in that has been replaced by Z2(0)
djd
djdZ
2322
222322 sincos
sincos)0(
βηβηβηβηη
++=
12
12
0
0
0
00 )0(
)0(
ηη
+−=−==Γ
Z
Z
H
H
E
E
i
r
i
r
Electromagnetics Plane Waves
Example
• A dielectric layer of thickness d and intrinsic impedance η2 is placed between media 1 and 3 having intrinsic impedances η1 and η3 , respectively. Determine d and η2 such that no reflection occurs when a uniform plane wave in medium 1 impinges normally on the interface with medium2.
• Sol:
)sincos()sincos(
)0(or ,0,0 120
+=+==Γ=
djddjd
Zz
βηβηηβηβηηη
C-N Kuo, Winter 2004 67
......3,2,1,04
)12(dor 2
)12(
0cosor
sinsin and
coscos
)sincos()sincos(
22
213
23122
2
2123
2322122232
=
+=+=→
==→=
=→+=+
n
nnd
d
dd
dd
djddjd
λπβ
βηηβηηβη
βηβηβηβηηβηβηη
Electromagnetics Plane Waves
• When η1= η3 , we require d=nλ2/2, n=0,1,2…. That the thickness of the
......3,2,1,02
dor
0inor
sinsin hand,other On the
22
2132
23122
2
=
==→
===→=
n
nnd
ds
dd
λπβ
βηηηβηηβη
C-N Kuo, Winter 2004 68
• When η1= η3 , we require d=nλ2/2, n=0,1,2…. That the thickness of the dielectric layer be a multiple of a half-wavelength in the dielectric at the operating frequency. Such a dielectric layer is referred to as a half-wave dielectric windows. Since λ2 =up2/f=1/f√µε, where f is the operating frequency, a half-wave dielectric window is a narrow-band device.
• When η1≠ η3 . We require η2 = √ η1 η3, and d=(2n+1)λ2/4, n=0,1,2….• When media 1 and 3 are different, η2 should be the geometric mean of η1 and
η3 , and d should be an odd multiple of a quarter wavelength in the dielectric layer operating frequency in order to eliminate reflection. Under these conditions the dielectric layer acts like a quarter-wave impedance transformer.
Electromagnetics Plane Waves
Oblique Incidence at a Plane Dielectric Boundary
Since incidence. of plane with thely,respective
wave,ed transmittandreflected, incident,
theof phase)constant of (surface wavefronts
theofon intersecti theare BO',A'O' AO, Line
C-N Kuo, Winter 2004 69
.reflection of law sSnell' iswhich
incidence, of angle the toequal is reflection of angle The
sin'sin' ,velocity
phase same with the1 mediumin propagate
wavesreflected theandincident both the
Since incidence. of plane with thely,respective
1
ir
irp OOOO'AOOA'u
θθθθ
=→
=→=
Electromagnetics Plane Waves
2 and 1 mediafor refraction of indices theare and where
sin
sin
sin
sin
'
'
'
'
:2 mediumIn
21
2
1
2
1
1
2
1
2
12
nn
n
n
u
u
u
u
OO
OO
AO
OB
u
AO
u
OB
p
p
i
t
p
p
i
t
pp
===
==→=
ββ
θθ
θθ
C-N Kuo, Winter 2004 70
2 and 1 mediafor refraction of indices theare and where 21 nn
• Snell’s law of refraction: at an interface between two dielectric media, the ratio of the sine of the angle of refraction (transmission) in medium2 to the sine of the angle of incidence in medium1 is equal to the inverse ratio of indices of refraction n1/n2.
2
1
2
1
2
1
2
1
021
sin
sin
media, cnonmagnetiFor
:2 mediumIn
ηη
εε
εε
θθ
µµµ
====
==
n
n
r
r
i
t
Electromagnetics Plane Waves
Total Reflection
)(sinsin
angle. critical thecalled is )2
:reflection totalof
threshold the toingcorrespond(which angle incidence :
sin2
:Setting
2121
1
2
n
nc
t
c
ct
−− ==
=
=→=
εεθ
πθ
θεεθπθ
C-N Kuo, Winter 2004 71
)(sinsin11 nc ==
εθ
1sinsin1cos
1sinsin
If
?
2
2
12
2
1
−±=−=
>=→
>
itt
it
ci
j θεεθθ
θεεθ
θθ
Electromagnetics Plane Waves
ixixjze
c
zx-jβR a-jβ
tttztxnt
nt
βββe
ee
H Eaaa
a
x
ttnt
θεεθ
εεα
θθ
θθ
βα
θθ
sin,1sin,
For
spatiallyvary andBoth ,cossin
r unit vecto the2, mediumIn
2
122
2
2
122
i
)cossin(
22
22
=−=→
>=
+=
−−
+•
C-N Kuo, Winter 2004 72
• The positive sign has been abandoned because it would lead to the impossible result of an increasing field as z increase.
• For θi> θc an evanescent wave exists along the interface (in the x-direction), which is attenuated exponentially in medium 2 in the normal direction (z-direction). This wave is tightly bound to the interface and is called a surface wave. Obviously, it is a nonuniform plane wave. No power is transmitted into medium 2 under these conditions.
Electromagnetics Plane Waves
Example
• The permittivity of water at optical frequencies is 1.75ε0. It is found that an isotropic light source at a distance d under water yields an illuminated circular area of a radius 5 m. Determine d.
nw 32.175.1 water ofindex refractive The ==
C-N Kuo, Winter 2004 73
mO'P
d
n
mO'P
oc
o
wc
32.42.49tan
5
tan Thus,
2.49)32.1
1(sin)
1(sin
5 area, dilluminate of radius The
11
===
===
=
−−
θ
θ
Electromagnetics Plane Waves
Example
• A dielectric rod or fiber of a transparent material can be used to guide light or an electromagnetic wave under the conditions of total internal reflection. Determine the minimum dielectric constant of the guiding medium so that a wave incident on one end at any angle will be confined within the rod until it emerges from the other end.
C-N Kuo, Winter 2004 74
satisfied. becan quartz and glass
22least at be tomedium guiding theofconstant dielectric the
sin1 :requireswhich
1sin
11sin
1sin
,sincos2
,sinsin
1
21
11
02
11
1
1
=→
+≥
=≥−→=
≥→−=
≥
n
ir
rr
ri
ri
r
t
ctt
c
θε
εεεθ
εθ
εθ
θθθπθ
θθ
Electromagnetics Plane Waves
Perpendicular Polarization• Incidence wave
• Reflective wave
)cossin(
1
0
)cossin(0
1
1
)sincos(),(
),(
ii
ii
zxjizix
ii
zxjiyi
eaaE
zxH
eEazxE
θθβ
θθβ
θθη
+−
+−
+−=
=
C-N Kuo, Winter 2004 75
• Transmitted wave
)cossin(
1
0
)cossin(0
1
1
)sincos(),(
),(
rr
rr
zxjrzrx
rr
zxjryr
eaaE
zxH
eEazxE
θθβ
θθβ
θθη
−−
−−
+=
=
)cossin(
2
0
)cossin(0
2
2
)sincos(),(
),(
tt
rr
zxjtztx
tt
zxjtyt
eaaE
zxH
eEazxE
θθβ
θθβ
θθη
+−
+−
+−=
=
Electromagnetics Plane Waves
• B.C.: tangential components of E and H be continuous at the boundary z=0
tri xjt
xjr
xji
tri
eEeEeE
EEEθβθβθβ sin
0sin
0sin
0211
)0()0()0(−−− =+→
=+
tri xjtt
xjrr
xjii
tri
eEeEeE
HHH
θβθβθβ θη
θθη
sin0
2
sin0
sin0
1
211 cos1
)coscos(1
)0()0()0(
−−− −=+−→
=+
C-N Kuo, Winter 2004 76
• For all x, all three exponential factors that are functions of x must be equal (phase matching)
• Snell’s law of reflection and refraction
ηη 21
trixxx θβθβθβ sinsinsin
211==
2
1
2
1
1
2
sin
sin
n
n
u
u
p
p
i
t ===ββ
θθ
riθθ =
Electromagnetics Plane Waves
ttiritri
tritri
EEEEEE
HHHEEE
θη
θη
cos1
cos)(1
and
)0()0()0( and )0()0()0(
02
001
000 =−=+→
=+=+
it
ii
ii
i
r
E
Eηη
θη
θη
θηθηθηθη coscos
coscos
coscos
12
12
12
12
0
0
+
−=
+−==Γ⊥
C-N Kuo, Winter 2004 77
• When θi=0, making θr= θt=0 these expressions reduce to those for normal incidence
it
iiiEθ
ηθ
ηθηθηcoscos
coscos 12120 ++
it
i
ii
i
i
t
E
E
θη
θη
θηθηθη
θητ
coscos
cos2
coscos
cos2
12
2
12
2
0
0
+=
+==⊥
Electromagnetics Plane Waves
• If medium 2 is a perfect conductor, η2 =0. We have
• Now, we inquire whether there is a combination of η1 , η2 and θi, which makesfor no reflection. Denoting this particular θi by θB⊥
⊥⊥ =Γ+ τ1
)0(,0,,1 000 ==−=−=Γ ⊥⊥ tir EEE τ
0=Γ⊥
C-N Kuo, Winter 2004 78
for no reflection. Denoting this particular θi by θB⊥0=Γ⊥
larperpendicu of case for the reflection no of angleBrewster thecalled is angle The
)(1
1sin
sin1sin1cos
sincos
221
12212
222
212
12
⊥
⊥
⊥
−−=
−=−=→
=
B
B
itt
tB
n
n
θµµ
εµεµθ
θθθ
θηθη
Electromagnetics Plane Waves
• For nonmagnetic media, µ1= µ2 = µ0→ θB⊥ does not exist.• In the case of ε1= ε2, µ1≠ µ2 :
)(1
1sin
21 µµθ
+=⊥B
C-N Kuo, Winter 2004 79
Electromagnetics Plane Waves
Parallel Polarization
• Incidence wave
• Reflective wave
)cossin(
1
0
)cossin(0
1
1
),(
)sincos(),(
ii
ii
zxjiyi
zxjizixii
eE
azxH
eaaEzxE
θθβ
θθβ
η
θθ+−
+−
=
−=
C-N Kuo, Winter 2004 80
• Transmitted wave
)cossin(
1
0
)cossin(0
1
1
),(
)sincos(),(
rr
rr
zxjryr
zxjrzrxrr
eE
azxH
eaaEzxE
θθβ
θθβ
η
θθ−−
−−
−=
+=
)cossin(
2
0
)cossin(0
2
2
),(
)sincos(),(
tt
rr
zxjtyt
zxjtztxtt
eE
azxH
eaaEzxE
θθβ
θθβ
η
θθ+−
+−
=
−=
Electromagnetics Plane Waves
• B.C.: tangential components of E and H be continuous at the boundary z=0, also lead to snell’s law of reflection and refraction:
1
)(1
,coscos)(
)0()0()0( and )0()0()0(
02
001
000 trittiri
tritri
EEEEEE
HHHEEE
ηηθθ =−=+→
=+=+
itrE θηθη coscos 120 −==Γ
C-N Kuo, Winter 2004 81
it
it
i
r
E
E
θηθηθηθη
coscos
coscos
12
12
0
0// +
−==Γ
it
i
i
t
E
E
θηθηθητcoscos
cos2
12
2
0
0// +
==
=Γ+
i
t
θθτ
cos
cos1 ////
Electromagnetics Plane Waves
• If medium 2 is a perfect conductor, η2 =0. We have
• Now, we inquire whether there is a combination of η , η and θ , which makes
vanish.conductor surface on the field E total theofcomponent l tangentia themaking
,0,1 //// =−=Γ τ
sunglass. Polaroid 0except allfor ii
2
//
2 →=Γ>Γ⊥ θθ
C-N Kuo, Winter 2004 82
• Now, we inquire whether there is a combination of η1 , η2 and θi, which makesfor no reflection. Denoting this particular θi by θB⊥0// =Γ
parallel of case for the reflection no of angleBrewster thecalled is angle The
)(1
1sin
sincos
//
221
2112//
2
//12
B
B
Bt
θεε
εµεµθ
θηθη
−−=
=
Electromagnetics Plane Waves
• µ1= µ2
)(1
1sin
21
// εεθ
+=B
== −−
1
21
1
21// tantan
n
nB ε
εθ
C-N Kuo, Winter 2004 83
• Because of the difference in the formulas for Brewster angles for perpendicular and parallel polarizations, it is possible to separate these two types of polarization in an unpolarized wave.