Plane EM waves - San Jose State UniversityElectromagnetic Waves EE142 Dr. Ray Kwok •reference:...
Transcript of Plane EM waves - San Jose State UniversityElectromagnetic Waves EE142 Dr. Ray Kwok •reference:...
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PlaneElectromagnetic Waves
EE142Dr. Ray Kwok
•reference:
Fundamentals of Engineering Electromagnetics, David K. Cheng (Addison-Wesley)
Electromagnetics for Engineers, Fawwaz T. Ulaby (Prentice Hall)
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Plane EM Wave - Dr. Ray Kwok
Source-free Maxwell Equations
t
EH
0Ht
HE
0E
∂
∂ε=×∇
=⋅∇∂
∂µ−=×∇
=⋅∇
rr
r
rr
r
t
EJH
0Ht
HE
E
f
f
∂
∂ε+=×∇
=⋅∇∂
∂µ−=×∇
ρ=⋅∇ε
rrr
r
rr
r
in source-free medium
(homogeneous,
linear, isotropic)
ρ = 0, J = 0
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Plane EM Wave - Dr. Ray Kwok
Wave Equation
( ) ( )
( ) ( )
2
22
22
2
2
t
EE
EEEE
t
EH
tt
HE
t
EH
t
HE
∂
∂µε=∇
−∇=∇−⋅∇∇=×∇×∇
∂
∂µε−=×∇
∂
∂µ−=
∂
∂µ−×∇=×∇×∇
∂
∂ε=×∇
∂
∂µ−=×∇
rr
rrrr
rr
rr
rr
rr
plane wave equation with v2 = 1/µε
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Plane EM Wave - Dr. Ray Kwok
Traveling Wave
2
2
22
2
2
2
2
2
2
t
f
v
1
x
f
)u("fvt
u)u("vf
t
f
)u('vft
u)u('f
t
f
)u("fx
u)u("f
x
f
)u('fx
u)u('f
x
f
)u(f)vtx(f
∂
∂=
∂
∂
=∂
∂±=
∂
∂
±=∂
∂=
∂
∂
=∂
∂=
∂
∂
=∂
∂=
∂
∂
≡± reverse / forward traveling wave
wave equation
( )
( )
( )
( )rktf
kxtf)vtx(f
kxtk
1
ft2kxk
1
vt2
x2
k
1vtx
rr⋅−ω
±ω=±
±ω±=
π±=
λ
π±
λ
π=±
note:
(3D)
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Plane EM Wave - Dr. Ray Kwok
Plane Wave
µε==
ω
µεω=
µεω−=−
ω−=∂
∂
−=∇
=
∂
∂µε=∇
⋅−ω
1v
k
k
k
Et
E
EkE
eE)t,r(E
t
EE
22
2
2
2
22
)rkt(j
o
2
22
rr
rr
rrr
rr
rr
)rkt(j
o
2
22
eH)t,r(H
t
HH
rrrrr
rr
⋅−ω=
∂
∂µε=∇
Similarly for B or H field
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Plane EM Wave - Dr. Ray Kwok
Source-free EM wave)kzt(j
oeEx)t,r(E −ω=rr
)kzt(jo
)kzt(j
o
)kzt(j
o
ekE
y)t,r(H
ejkEyE
HjE
00eE
zyx
zyx
E
−ω
−ω
−ω
ωµ=
−=×∇
ωµ−=×∇
∂
∂
∂
∂
∂
∂=×∇
rr
r
rr
r
V/m in vacuum with no current source.
What is H ?
)kzt(jo
)kzt(jo
)kzt(jo
)kzt(jo
)kzt(j
o
ev
Ey)t,r(B
ev
EyHB
eE
y)t,r(H
1
v
1
f
1k
ekE
y)t,r(H
ejkEyE
−ω
−ω
−ω
−ω
−ω
=
µ
µ=µ=
η=
η≡
µ
ε=
µ
µε=
µ=
λµ=
ωµ
ωµ=
−=×∇
rr
rr
rr
rr
r
ηηηη = wave
Impedance
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Plane EM Wave - Dr. Ray Kwok
EM wave properties
)kzt(jo
)kzt(jo
)kzt(j
o
ev
Ey)t,r(B
eE
y)t,r(H
eEx)t,r(E
−ω
−ω
−ω
=
η=
=
rr
rr
rr
HEk
kHE
×=
⊥⊥rrr
(1) E & H are in phase.
ε
µ=η=
µε===
o
o
rro
o
H
E
c
n
cv
B
E
(2)
(3)
Index of refraction
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Plane EM Wave - Dr. Ray Kwok
Electromagnetic wave
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Plane EM Wave - Dr. Ray Kwok
Earlier exercise
)zt900cos(01.0x)t,r(H β+=rr
A/m in vacuum with no current source.
What is f, λ, direction of propagation, E ?
ω = 2πf = 900, f = 143 Hz
λ = c/f = (3 x 108)/(143) = 2094 km
propagate to the –z direction
EjHrr
ωε=×∇
E(r) = ay 3.77 cos[900t + 3(10-6)z] V/m
HEk
kHE
×=
⊥⊥rrr
(1) E & H are in phase.
η=
=
o
o
o
o
H
E
vB
E
(2)
(3)
ηo = 377 Ω
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Plane EM Wave - Dr. Ray Kwok
Example)1.0zt10sin()02.0y01.0x()t,r(H 9 −β+−=
rrA/m. What is E ?
β = ω/c = 109/(3 x 108) = 3.33 rad/m
(1) E & H are in phase.
HEk
kHE
×=
⊥⊥rrr
(2)
η=
=
o
o
o
o
H
E
vB
E(3)
ηo = 377 Ω
)1.0z33.3t10sin(E)t,r(E9
o −+=rrr
V/m
y447.0x894.05
yx2E +=
+=x
y
H
E
Ho2 = 0.012 + 0.022 → Ho = 0.022 = Eo/ηo = Eo/377
Eo = 8.3 V/m
)1.0z33.3t10sin()7.3y4.7x()t,r(E
)1.0z33.3t10sin()447.0y894.0x(3.8)t,r(E
9
9
−++=
−++=rr
rr
V/m
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Plane EM Wave - Dr. Ray Kwok
Exercise)yz3t10cos(20x)t,r(E 8 −+=
rrV/m in non-magnetic material.
What is H ?
v = ω/k = 108/√10 = 3.16 x 107 m/s
v = c/n = c/√εr → εr = (c/v)2 = 90
η = √(µ/ε) = ηo/√εr = 377/√90 = 39.7
(1) E & H are in phase.
HEk
kHE
×=
⊥⊥rrr
(2)
)yz3t10cos(H)t,r(H 8
o −+=rrr
A/m
Ho = Eo/η = 20/39.7 =0.5
)yz3t10cos()158.0z475.0y()t,r(H
)yz3t10cos()316.0z949.0y(5.0)t,r(H
8
8
−++−=
−+−−=rr
rr
A/m
z316.0y949.010
zy3H −−=
−−=
y
z
k
E
H
η=
=
o
o
o
o
H
E
vB
E(3)
ηo = 377 Ω
yz3k +−=r
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Plane EM Wave - Dr. Ray Kwok
EM wave properties - generalized
HEk
kHE
×=
⊥⊥rrr
(1) E & H are in phase.
ε
µ=η=
µε===
o
o
rro
o
H
E
c
n
cv
B
E
(2)
(3)
Index of refraction
Ek1
Hrr
×η
=
kHE ×η=rr
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Plane EM Wave - Dr. Ray Kwok
Earlier exercise (free space)
)1.0zt10sin()02.0y01.0x()t,r(H 9 −β+−=rr
A/m. What is E ?
β = ω/c = 109/(3 x 108) = 3.33 rad/m
)1.0z33.3t10sin()54.7x77.3y(E
)z()1.0ztsin()02.0y01.0x(377E
kHE
9 −++=
−×−β+ω−=
×η=
r
r
rr
V/m.
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Plane EM Wave - Dr. Ray Kwok
Earlier exercise)yz3t10cos(20x)t,r(E 8 −+=
rrV/m in non-magnetic material.
What is H ?
A/m
yz3k +−=r
( )[ ]
( ) ( )yz3t10cos159.0z478.0yH
yz3t10cos20x10
yz3
7.39
1H
Ek1
H
8
8
−+−−=
−+×
+−=
×η
=
r
r
rr
v = ω/k = 108/√10 = 3.16 x 107 m/s
v = c/n = c/√εr → εr = (c/v)2 = 90
η = √(µ/ε) = ηo/√εr = 377/√90 = 39.7
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Plane EM Wave - Dr. Ray Kwok
Poynting Theorem
( )
( )
( )
( )HEt
HHHE
HEEHHE
t
EEHEJE
t
EJH
dVJEdt
dW
dtJEVdtvEVdW
dtvEqdW
dtvBvEqdFdW
f
rrr
rrr
rrrrrr
rrrrrr
rrr
rr
rrrr
rr
rrrrlrr
×⋅∇−
∂
∂µ−⋅=×∇⋅
×∇⋅−×∇⋅=×⋅∇
∂
∂⋅ε−×∇⋅=⋅
∂
∂ε+=×∇
⋅=
⋅δ=⋅ρδ=
⋅δ=
⋅×+δ=⋅=
∫
Work done on a charge δδδδq by EM force:
( )
( )
( )
( )
adSt
U
dt
dW
adHEdVt
E
t
H
2
1
dt
dW
dVHEdVt
E
t
H
2
1
dt
dW
HEt
E
2t
H
2JE
t
EEHE
t
HHJE
EM
2
o
2
22
22
rr
rrrrr
rrrr
rrrr
rr
rrrr
rrrr
∫
∫∫
∫∫
⋅−∂
∂−≡
⋅×−
∂
∂ε+
∂
∂µ−=
×⋅∇−
∂
∂ε+
∂
∂µ−=
×⋅∇−∂
∂ε−
∂
∂µ−=⋅
∂
∂⋅ε−×⋅∇−
∂
∂⋅µ−=⋅
(math)
Poynting Vector
Power flow out of the enclosed surface
EM power stored in the volume
The decrease of the EM energy stored is to do work on
a charge or due to the net energy escape through the
surface.
the work-energy theorem
for electrodynamics
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Plane EM Wave - Dr. Ray Kwok
Poynting Vector
HESrrr
×≡
Magnitude gives power intensity = power/area
Direction gives direction of propagation
2
o
2
o
ave
ave
H22
ES
*]HE[e2
1S
η=
η=
×ℜ≡rrr
This is similar to P = ½ (V2/Z) = ½ (I2Z)
Time average
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Plane EM Wave - Dr. Ray Kwok
ExampleIf solar illumination is characterized by a power density of 1 kW/m2 at Earth’s surface,
find (a) the total power radiated by the sun,
(b) the total power intercepted by Earth, and
(c) the electric field of the power density incident upon Earth’s surface, assuming that
all the solar illumination is at a single frequency. The radius of Earth’s orbit around the
sun, Rs, is approximately 1.5 x 108 km, and Earth’s mean radius RE is 6380 km.
(a) Average power radiated Psun = Save(4πRs2) = (103)(4π)(1.5 x 1011)2 = 2.8 x 1026 W
(b) Average power intercepted PE = Save(πRE2) = (103)(π)(6.38 x 106)2 = 1.28 x 1017 W
(c) Electric field intensity …
870)10)(377(2S2E
2
ES
3
aveoo
2
o
ave
==η=
η=
V/m
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Plane EM Wave - Dr. Ray Kwok
Linear Polarization
)ztcos(Ex)t,r(E 1o1 β−ω=rr
x
y Defined by the direction of E-field. For example:
Horizontal polarization
)ztcos(Ey)t,r(E 2o2 β−ω=rr
Vertical polarization
x
y
21TOTAL EEErrr
+=ETOTAL is still linearly polarized
If Eo1 = Eo2, then ETOTAL is at 45 degrees.
A linear combination of these….
E1
E2
Any EM wave can be expressed as a linear combination ofhorizontal & vertical polarized field.
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Plane EM Wave - Dr. Ray Kwok
Circular Polarization
)ztcos(Ex)t,r(E 1o1 β−ω=rr
21T EEErrr
+=
x
y
If E1 & E2 are 90 degrees out-of-phase, & Eo1 = Eo2:
Horizontal polarization
)ztsin(Ey)t,r(E 2o2 β−ω=rr
Vertical polarization
ET(t=0)
ET(t>0)RHCP
Right-Hand Circularly Polarized
)zt(j
o
)2/zt(j
o
)zt(j
o
oo
eE)yjx()t,r(E
eEyeEx)t,r(E
)ztsin(Ey)ztcos(Ex)t,r(E
β−ω
π−β−ωβ−ω
−=
+=
β−ω+β−ω=
rr
rr
rr
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Plane EM Wave - Dr. Ray Kwok
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Plane EM Wave - Dr. Ray Kwok
LHCP
x
y
ET(t=0)
ET(t>0)
)zt(j
o
)2/zt(j
o
)zt(j
o
oo
eE)yjx()t,r(E
eEyeEx)t,r(E
)ztsin(Ey)ztcos(Ex)t,r(E
β−ω
π+β−ωβ−ω
+=
+=
β−ω−β−ω=
rr
rr
rr
LHCP
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Plane EM Wave - Dr. Ray Kwok
Circular polarization in nature
Rose Chafer (a beetle)& others…
Reflection from its surface is LHCP.,due to the helical molecular structureof their shells.
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Plane EM Wave - Dr. Ray Kwok
Mantis Shrimp (Stomatopod Crustacean)
Able to sense LHCP & RHCP,
over a wide spectrum of color,
better than man-made instrument.
Most complex eye structure in
animal kingdom.
Fastest animal (striking) & produces
over 200 lbs of striking force !!!
Nature Photonics 3, 641-644 (2009)Roberts, Chiou, Marshall & Cronin,
Sheila Patek at TED.com
a popular dish known as
"pissing shrimp" (攋尿蝦)
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Plane EM Wave - Dr. Ray Kwok
Circular polarization in starlight
Most starlight are slightly circularly polarized.
The interstellar medium (low density dust) can produce circularly polarized
(CP) light from unpolarized light by sequential scattering from elongated
interstellar grains (with complex index of refraction) aligned in different
directions. One possibility is twisted grain alignment along the line of sight
due to variation in the galactic magnetic field; another is the line of sight
passes through multiple clouds.
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Plane EM Wave - Dr. Ray Kwok
Polarizer
θunpolarized
Io
polarized
I1 = ½ Io
polarized
I2 = I1cos2θ = ½ Iocos2θ
complex n
Linearly polarized
intensity
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Plane EM Wave - Dr. Ray Kwok
Elliptical Polarization
)zt(j
2o1o
2o1o
e)EyjEx()t,r(E
)ztsin(Ey)ztcos(Ex)t,r(E
β−ω=
β−ω±β−ω=
mrr
rrIf Eo1 ≠ Eo2
Or, if the phase difference is not 90 degrees
RHEP
LHEP
)zt(j
o
j
)zt(j
o
)zt(j
o
eE)eyx()t,r(E
eEyeEx)t,r(E
β−ωδ
δ+β−ωβ−ω
+=
+=rr
rr
RHEP (δ < 0)
LHEP (δ > 0)
Special cases:
if δ = −π/2, → RHCP
if δ = π/2, → LHCP
x
y
ET(t=0)
LHEP
ET(t>0)
E02
E01
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Plane EM Wave - Dr. Ray Kwok
Polarization Generalization)zt(j
y
)zt(j
x eayeax)t,r(E δ+β−ωβ−ω +=rr
0
0
cos)2tan(2tan
sin)2sin(2sin
a
atan
R
1tan
a
aR
o
o
x
y
o
<γ
>γ
δψ=γ
δψ=χ
≡ψ
±≡χ
≡η
ξaxial ratio ≥ 1= 1 circular
= ∞ linear
ellipticity angle
χ > 0 when sinδ > 0 → LHχ < 0 when sinδ < 0 → RH
auxiliary angle
0 ≤ ψο ≤ π/2
−π/4 ≤ χ ≤ π/4
rotation angle −π/2 ≤ γ ≤ π/2
if cosδ > 0
if cosδ < 0
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Plane EM Wave - Dr. Ray Kwok
Polarization States
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Plane EM Wave - Dr. Ray Kwok
e.g. what’s the polarization?
( )( ) )6/kzt(j105j
)6/kzt(j12/7j
)4/3kzt(j)6/kzt(j
)2/4/kzt(j)6/kzt(j
)2/4/kzt(j)6/kzt(j
ee4y3x)t,z(E
ee4y3x)t,z(E
e4ye3x)t,z(E
e4ye3x)t,z(E
e4ye3x)t,z(E
o π+−ω
π+−ωπ
π+−ωπ+−ω
π+π−π+−ωπ+−ω
π−π+−ωπ+−ω
+=
+=
+=
+=
−=
r
r
r
r
r)4/kztsin(4y)6/kztcos(3x)t,z(E π+−ω−π+−ω=
r
o
oo
o
o
oo
o
o
o
x
y
o
69
)105cos()106tan(cos)2tan(2tan
34
)105sin()106sin(sin)2sin(2sin
53
3
4
a
atan
−=γ
=δψ=γ
=χ
=δψ=χ
=ψ
=≡ψ00256.0)105cos(cos
LH00966.0)105sin(sin
105
o
o
o
<γ⇒<−==δ
⇒>χ⇒>==δ
=δ
x
y
ay=4
ax=3
χχχχ
γ γ γ γ = −−−− 69o
ψψψψo
LHEP
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Plane EM Wave - Dr. Ray Kwok
Linear combination
)ztcos(Ex)t,r(E o β−ω=rr
Any LP EM wave can be written as linear combination of RHCP + LHCP
( ) ( )[ ]
2
EEE
eEEyjEEx)t,r(E
)t,r(E)t,r(E)t,r(E
eE)yjx()t,r(E
eE)yjx()t,r(E
o12
)zt(j
1221T
LHCPRHCPT
)zt(j
2LHCP
)zt(j
1RHCP
==
−++=
+=
+=
−=
β−ω
β−ω
β−ω
rr
rrrrrr
rr
rr
e.g.
More generally, any EM wave can be written as linear combination of RHEP + LHEP
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Plane EM Wave - Dr. Ray Kwok
ExerciseThe amplitudes of an elliptically polarized plane wave traveling in a lossless,
nonmagnetic medium with εr = 4 and Hxo= 8 mA/m, and Hyo= 6 mA/m. Determine the average power flowing through an aperture in the y-z plane if its area is 20 m2.
η = 60π = 188.5 ΩSave = 9.43 mW/m2 along the x-axis
Pave = 0.19 W