electromagnetic field theory 2 · 2019. 5. 20. · ctu-fee in prague, department of electromagnetic...
TRANSCRIPT
Lukas Jelinek
Department of Electromagnetic Field
Czech Technical University in Prague
Czech Republic
Ver. 2019/05/06
Electromagnetic Field Theory 2(fundamental relations and definitions)
CTU-FEE in Prague, Department of Electromagnetic Field
Electromagnetic Field Theory 2INTRODUCTION
Maxwell(’s)-Lorentz(’s) Equations
2 / XXX
( ) ( ) ( )( ) ( ) ( )( )
0
0
, , ,
, , ,
t t t
t t t
ε
µ
= +
= +
D r E r P r
B r H r M r
Absolute majority of things happening around us is described by these equations
( ) ( ) ( )
( ) ( )
( )( ) ( )
,, ,
,,
, 0
, ,
tt t
tt
tt
t
t tρ
∂∇× = +
∂∂
∇× = −∂
∇⋅ =
∇ ⋅ =
D rH r J r
B rE r
B r
D r r
( ) ( ) ( ) ( ) ( ), , , , ,t t t t tρ= + ×f r r E r J r B r
Equations of motion
for fields
Equation of motion
for particles
Interaction with materials
CTU-FEE in Prague, Department of Electromagnetic Field
Electromagnetic Field Theory 2INTRODUCTION
Boundary Conditions
3 / XXX
Normal
pointing to
region (1)
( ) ( ) ( )1 2, , 0t t ⋅ − = n r B r B r
( ) ( ) ( ) ( )1 2, , ,t t t × − = n r H r H r K r
( ) ( ) ( ) ( )1 2, , ,t t tσ ⋅ − = n r D r D r r
( ) ( ) ( )1 2, , 0t t × − = n r E r E r
CTU-FEE in Prague, Department of Electromagnetic Field
Electromagnetic Field Theory 2INTRODUCTION
Electromagnetic Potentials
4 / XXX
( ) ( )
( ) ( ) ( ), ,
,, ,
t t
tt t
tϕ
= ∇×
∂= −∇ −
∂
B r A r
A rE r r
( ) ( ) ( ),, ,
tt t
t
ϕσµϕ εµ
∂∇ ⋅ = − −
∂
rA r r
Lorentz(‘s)
calibration
CTU-FEE in Prague, Department of Electromagnetic Field
Electromagnetic Field Theory 2INTRODUCTION
Wave Equation
5 / XXX
( ) ( ) ( ) ( )2
source2
, ,, ,
t tt t
t tσµ εµ µ
∂ ∂∆ − − = −
∂ ∂
A r A rA r J r
Material parameters are assumed
independent of coordinates
CTU-FEE in Prague, Department of Electromagnetic Field
Electromagnetic Field Theory 2INTRODUCTION
Poynting(’s)-Umov(’s) Theorem
6 / XXX
Power supplied
by sources
( ) 2 2 2
source
1d d d d
2V S V V
V V Vt
σ ε µ∂ − ⋅ = × ⋅ + + + ∂∫ ∫ ∫ ∫E J E H S E E H
Energy balance in an electromagnetic system
Power passing the
bounding envelope
Heat losses
Energy storage
CTU-FEE in Prague, Department of Electromagnetic Field
Electromagnetic Field Theory 2INTRODUCTION
Frequency Domain
7 / XXX
( ) ( ) j1 ˆ, , e2
tt dωω ωπ
∞
−∞
= ∫F r F r
Time derivatives reduce to
algebraic multiplication
( ) ( ) jˆ , , e tt dtωω
∞−
−∞
= ∫F r F r
Frequency domain helps us to remove explicit time derivatives
( ) ( )
( ) ( )
,ˆj ,
ˆ, ,
t
t
t
r rξ ξ
ω ω
ω
∂↔
∂
∂ ∂↔
∂ ∂
F rF r
F r F r
Spatial derivatives are
untouched
( ),t ∈F r ℝ ( )ˆ ,ω ∈F r ℂ
CTU-FEE in Prague, Department of Electromagnetic Field
Electromagnetic Field Theory 2INTRODUCTION
Phasors
8 / XXX
( ) ( )*ˆ ˆ, ,ω ω− =F r F r
Reduced frequency domain representation
( ) ( ) j
0
1 ˆ, Re , e tt dωω ωπ
∞
= ∫F r F r
CTU-FEE in Prague, Department of Electromagnetic Field
Electromagnetic Field Theory 2INTRODUCTION
Maxwell(’s) Equations – Frequency Domain
9 / XXX
We assume linearity of material relations
( ) ( ) ( )
( ) ( )
( )
( ) ( )
ˆ ˆ ˆ, , j ,
ˆ ˆ, j ,
ˆ , 0
ˆ ,ˆ ,
ω ω ωε ω
ω ωµ ω
ω
ρ ωω
ε
∇× = +
∇× = −
∇⋅ =
∇ ⋅ =
H r J r E r
E r H r
H r
rE r
CTU-FEE in Prague, Department of Electromagnetic Field
Electromagnetic Field Theory 2INTRODUCTION
Wave Equation – Frequency Domain
10 / XXX
( ) ( ) ( ) ( )sourceˆ ˆ ˆ, j j , ,ω ωµ σ ωε ω µ ω∆ − + = −A r A r J r
Helmholtz(‘s) equation
CTU-FEE in Prague, Department of Electromagnetic Field
Electromagnetic Field Theory 2INTRODUCTION
Heat Balance in Time-Harmonic Steady State
11 / XXX
Cycle mean
2
source
2* *
source
d d d
1 1 1ˆ ˆ ˆ ˆ ˆRe d Re d d2 2 2
V S V
V S V
V V
V V
σ
σ
− ⋅ = × ⋅ +
− ⋅ = × ⋅ +
∫ ∫ ∫
∫ ∫ ∫
E J E H S E
E J E H S E
Valid for general periodic steady state
Valid for time-harmonic steady state
CTU-FEE in Prague, Department of Electromagnetic Field
Electromagnetic Field Theory 2INTRODUCTION
Plane Wave
12 / XXX
The simplest wave solution of Maxwell(‘s) equations
( ) ( )
( ) ( )
( )
( )
( )
j
0
j
0
0
0
2
ˆ , e
ˆ , e
0
0
j j
k
kk
k
ω ω
ω ωωµ
ω
ω
ωµ σ ωε
− ⋅
− ⋅
=
= ×
⋅ =
⋅ =
= − +
n r
n r
E r E
H r n E
n E
n H
Wave-number
Unitary vector representing
the direction of propagation
Electric and magnetic fields
are mutually orthogonal
Electric and magnetic fields
are orthogonal to
propagation direction
CTU-FEE in Prague, Department of Electromagnetic Field
Electromagnetic Field Theory 2INTRODUCTION
Plane Wave Characteristics
13 / XXX
General isotropic
material
( )
f
j j
Re 0; Im 0
2
Re
Re
1
Im
k
k k
k
vk
Zk
k
ωµ σ ωε
πλ
ω
ωµ
δ
= − +
> <
=
=
=
= −
0
0
f 0
0
0 0
0
Re 0; Im 0
377
kc
k k
c
f
v c
Z c
ω
λ
µµ
ε
δ
=
> =
=
=
= = ≈ Ω
→ ∞
Vacuum
CTU-FEE in Prague, Department of Electromagnetic Field
Electromagnetic Field Theory 2INTRODUCTION
Cycle Mean Power Density of a Plane Wave
14 / XXX
( ) ( ) ( )2 2 Im
0
Re1, , e
2
kk
t t ωωµ
⋅ × = n r
E r H r E n
Power propagation coincides with
phase propagation
CTU-FEE in Prague, Department of Electromagnetic Field
Electromagnetic Field Theory 2FREE WAVES
Source Free Maxwell(’s) Equations in Free Space
15 / XXX
Fourier‘s transform leads to simple algebraic equations
( ) ( ) ( ) ( ) ( )
( ) ( ) ( )
( )( )
,, ,
,,
, 0
, 0
tt t t t
tt
t tt
t
t
σ ε
µ
∂∇× = ∗ + ∗
∂∂
∇× = − ∗∂
∇ ⋅ =
∇ ⋅ =
E rH r E r
H rE r
H r
E r
( ) ( ) ( )( )( ) ( ) ( ) ( )
( ) ( ) ( )
( )( )
22 ˆ ˆˆj j
ˆ ˆ, ,ˆ ˆj
ˆ ˆ, ,ˆ
ˆ , 0
ˆ , 0
k ωµ ω σ ω ωε ω
ω ωωε ω σ ω
ω ωωµ ω
ω
ω
= = − +
= ×−
= − ×
⋅ =
⋅ =
k
kE k H k
kH k E k
k E k
k H k
( )( )
( ) ( )j
4
,
1 ˆ, , e d d2
tt
ω
ω
ω ω
π
⋅ += ∫k r
k
F r F k k
CTU-FEE in Prague, Department of Electromagnetic Field
Electromagnetic Field Theory 2FREE WAVES
Spatial Wave Packet
16 / XXX
General solution to free-space Maxwell‘s equations
( )( )
( ) ( )( )
( )
j
03
0
1 ˆ, e d2
ˆ 0
t
tω
π
⋅ +=
⋅ =
∫kk r
k
F r F k k
k F k
This can be electric or magnetic
intensity
( )ω ω= k( ) ( ) ( )( )22 ˆ ˆˆj jk ωµ ω σ ω ωε ω= = − +k
CTU-FEE in Prague, Department of Electromagnetic Field
Electromagnetic Field Theory 2FREE WAVES
Spatial Wave Packet in Vacuum
17 / XXX
The field is uniquely given by initial conditions
( ) ( )( ) ( )
*
0 0
*
0 0
ˆ ˆ
ˆ ˆ
− +
+ −
= − = −
F k F k
F k F k
( ) 0cω = ±k k
( )( )
( ) ( )
( ) ( ) ( )
( ) ( ) ( )
0 0j jj
0 03
j
0
0 0
j
0
0 0
1 ˆ ˆ, e e e d2
,1 1ˆ ,0 e d2 j
,1 1ˆ , 0 e d2 j
c t c t
t
t
t
t
tc
t
tc
π
−⋅ + −
+ − ⋅
=
− − ⋅
=
= +
∂ = +
∂ ∂ = −
∂
∫
∫
∫
F
F
k kk r
k
k r
r
k r
r
F r F k F k k
rF k F r r
k
rF k F r r
k
( ) ( )0 0ˆ ˆ 0+ −⋅ ⋅ =k F k = k F k
CTU-FEE in Prague, Department of Electromagnetic Field
Electromagnetic Field Theory 2FREE WAVES
Spatial Wave Packet in Vacuum
18 / XXX
Electric and magnetic field are not independent
( ) 0cω = ±k k
( )( )
( ) ( )
( )( )
( ) ( )
( ) ( )
0 0
0 0
j jj
3
j jj
3
0
1 ˆ ˆ, e e e d2
1 ˆ ˆ, e e e d2
ˆ ˆ 0
c t c t
c t c t
t
tZ
π
π
−⋅ + −
−⋅ + −
+ −
= +
= − × −
⋅ = ⋅ =
∫
∫
k
k
k kk r
k kk r
E r E k E k k
kH r E k E k k
k
k E k k E k
CTU-FEE in Prague, Department of Electromagnetic Field
Electromagnetic Field Theory 2FREE WAVES
Vacuum Dispersion
19 / XXX
1D waves in vacuum propagate without dispersion
( ) ( ) ( )
( ) ( ) ( )
0 0
0 0 0
0
,
1,
z t z c t z c t
z t z c t z c tZ
+ −
+ −
= + + −
= − × + − − z
E E E
H E E
( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
2
0 0
ˆ ˆ ˆ2
ˆ ˆ 0
z x y
z z
k k k
k k
π δ δ+ − +
+ −
= =
⋅ = ⋅ =
E k
z z
E k E
E E1D problem
no dispersion
Propagation in one direction
CTU-FEE in Prague, Department of Electromagnetic Field
Electromagnetic Field Theory 2FREE WAVES
Vacuum Dispersion
20 / XXX
2D and 3D waves in vacuum always disperse = change shape in time
In general this term does not
represent translation
Waves propagating in all directions
( )( )
( ) ( )0 0j jj
3
1 ˆ ˆ, e e e d2
c t c tt
π
−⋅ + − = + ∫k
k kk rE r E k E k k
( )0, ,x y z c t ±
CTU-FEE in Prague, Department of Electromagnetic Field
Electromagnetic Field Theory 2FREE WAVES
Angular Spectrum Representation
21 / XXX
General solution to free-space Maxwell‘s equations
( )( )
( ) ( )
( )( )
( ) ( )
2 2 2
2 2 2
j j
03
, ,
j j
03
, ,
0
1 ˆ, , 0, e , , e d d d2
1 ˆ, , 0, e , , e d d d2
ˆ 0
x y x y
x y
x y x y
x y
k x k y t k k k z
x y x y
k k
k x k y t k k k z
x y x y
k k
x y z t k k k k
x y z t k k k k
ω
ω
ω
ω
ω ω
π
ω ωπ
+ + − −
+ + − − −
< =
> =
⋅ =
∫
∫
E E
E E
k E
2 2 2
z x yk k k k= ± − −( ) ( ) ( )( )2
2 ˆ ˆˆj jk ωµ ω σ ω ωε ω= = − +k
Im 0zk <
( ) ( )0 0ˆ ˆ, , , ,
x y x yk k k k
Zω ω= − ×
kH E
k
( ) ( ) , ,0ˆ , , , , 0,
x tx y yk k x y tω =E EF
CTU-FEE in Prague, Department of Electromagnetic Field
Electromagnetic Field Theory 2FREE WAVES
Propagating vs Evanescent Waves
22 / XXX
Field picture losses it resolution with distance from the source plane
2 2 2
x yk k k+ >
2 2 2
x yk k k+ <
These waves propagate and
can carry information to far distances
These waves exponentially decay in amplitude
and cannot carry information to far distances
CTU-FEE in Prague, Department of Electromagnetic Field
Electromagnetic Field Theory 2FREE WAVES
Paraxial Waves
23 / XXX
2 2 2
x yk k k+ ≪( )0
ˆ , ,x yk k ωE ( )2 2 2 2 21
2x y x yk k k k k k
k− − ≈ − +
( )( )
( ) ( ) ( )2 21jj2
03
, ,
0
1 ˆ, , 0, e , , e d d d2
ˆ 0
x yx y
x y
k k zk x k y kz tk
x y x y
k k
x y z t k k k kω
ω
ω ωπ
++ − +> =
⋅ =
∫E E
k E
0 0ˆ 0⋅ ≈z E
Propagates almost as a planewave
CTU-FEE in Prague, Department of Electromagnetic Field
Electromagnetic Field Theory 2FREE WAVES
Gaussian Beam
24 / XXX
Approximates radiation of sources large in comparison to wavelength
( ) ( )2 2 20
12 4
0 0 0ˆ , ,
x yw k k
x yk k w eω π
− +
⊥ ⊥= AE
Gaussian profile
in amplitude
( ) ( )( ) ( )
2 2 2
2
2
20
jarctan j0
0
j1, , 0, e de
2e e
RR
zz zt
zw z w zz c
x y x y
wx y z t
w z
ω
ω
ωπ
−
+ +−
⊥ ⊥
−
> = ∫E A
Planewave-like
propagation
Paraxial
corrections
CTU-FEE in Prague, Department of Electromagnetic Field
Electromagnetic Field Theory 2FREE WAVES
Gaussian Beam
25 / XXX
Beam divergence
0
2
w
λ
πΘ =
( )2
01
R
zw z w
z
= +
Half-width of the beam
Rayleigh‘s distance
2
2 0
0
1
2R
wz kw
π
λ= =
CTU-FEE in Prague, Department of Electromagnetic Field
Electromagnetic Field Theory 2FREE WAVES
Gaussian Beam – Time-Harmonic Case
26 / XXX
( ) ( ) ( )( )2
2
2
* 0
0 0 2
2
1 ˆ ˆRe , , , , , ,2
w ze
wx y z x y z S
w z
ρ
ω ω
− = × = zS E H
86.5 % of power flows
through the beam width
( )2
01
R
zw z w
z
= +
Power density at origin
CTU-FEE in Prague, Department of Electromagnetic Field
Electromagnetic Field Theory 2FREE WAVES
Material Dispersion
27 / XXX
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
, , d
, , d
, , d
t t
t t
t t
ε τ τ τ
µ τ τ τ
σ τ τ τ
∞
−∞
∞
−∞
∞
−∞
= −
= −
= −
∫
∫
∫
D r E r
B r H r
J r E r
Causality requirement
Even single planewave undergoes time dispersion when materials are present
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
ˆ ˆˆ, ,
ˆ ˆˆ, ,
ˆ ˆˆ, ,
ω ε ω ω
ω µ ω ω
ω σ ω ω
=
=
=
D r E r
B r H r
J r E r
( ) 0,ε τ τ→ → ∞( ) 0, 0ε τ τ= <
Stability requirement
CTU-FEE in Prague, Department of Electromagnetic Field
Electromagnetic Field Theory 2FREE WAVES
Lorentz‘s Dispersion Model
28 / XXX
( )2
p
0 2 2
0
1j
ωε ω ε
ω ω ω
= + − + Γ
Losses
Dispersion model able to describe vast amount of natural materials
Resonance
( ) ( ) ( ) ( )2
2 2
0 0 p2
t tt t
ttω ε ω
∂ ∂+ Γ + =
∂∂
P PP E
Coupling
strength
( )2
p,
0 2 2
0,
1j
i
i i i
ωε ω ε
ω ω ω
= + − + Γ ∑
CTU-FEE in Prague, Department of Electromagnetic Field
Electromagnetic Field Theory 2FREE WAVES
Drude‘s Dispersion Model
29 / XXX
Special case of
Lorentz‘s dispersion
Dispersion model describing neutral plasma
2
p
0 1j
0
2 0p
0
0
2
p
0 21 1
≪
0
1 j
Permittivity model Conductivity model
Collisionless plasma
CTU-FEE in Prague, Department of Electromagnetic Field
Electromagnetic Field Theory 2FREE WAVES
Appleton‘s Dispersion Model
30 / XXX
Dispersion model describing magnetized neutral plasma
Direction of magnetization
( ) ( ) ( )2
2
c 0 0 p2
t tt
ttω ε ω
∂ ∂+ × =
∂∂z
P PE
( )
( )
2 2
p c p
2 2 2 2
c c
2 2
c p p
0 2 22 2
cc
2
p
2
j1 0
jˆ 1 0
0 0 1
ω ω ω
ω ω ω ω ω
ω ω ωε ε
ω ωω ω ω
ω
ω
− − − − = − −− −
Tˆ ˆε ε≠
Plasma frequency
Cyclotron frequency
Propagation in opposite
directions is not the same
CTU-FEE in Prague, Department of Electromagnetic Field
Electromagnetic Field Theory 2FREE WAVES
Propagation in Appleton‘s Dispersion Model
31 / XXX
Dispersion model describing magnetized neutral plasma
Fundamental modes
are circularly polarized
waves
( )
22
p
2
0 c
1
ˆ ˆj
z
x y
k
k
E E
ω
ω ω ω= −
±
= ∓
Planewave propagation
along magnetization
j
0
0
ˆ e
0
zk z=
⋅ =
E E
k E
CTU-FEE in Prague, Department of Electromagnetic Field
Electromagnetic Field Theory 2RADIATION
Radiation
32 / XXX
Only accelerating charges can radiate
Microscopic
charge velocity
( ),0
t
t
∂≠
∂
J r( )0
t
t
∂≠
∂
v
Macroscopic
current density
( ) ( )( ), , d d 0S
t t t
∞
−∞
× ⋅ ≠∫ ∫ E r H r SAny surface
circumscribing
the sources
CTU-FEE in Prague, Department of Electromagnetic Field
Electromagnetic Field Theory 2RADIATION
Time-Harmonic Electric Dipole
33 / XXX
Elementary source of radiation
( ) ( ) ( ) ( ) ( )0ˆ ,
zp x y zω ω δ δ δ= zP r
( ) ( ) ( )
( ) ( )
( ) ( )
j2
0 0 0
j3
0 0
j3
0 0 0 02 2 2 2
eˆ , j cos sin4
j eˆ , sin 14
j 1 j 1 eˆ , 2 cos 1 sin4
kr
z
kr
z
kr
z
Z k pkr
c k pkr kr
Z c k pkr kr krk r k r
ω θ θ θ ωπ
ω ϕ θ ωπ
ω θ θ θ ωπ
−
−
−
= −
= − +
= + + − + +
A r r
H r
E r r
( ), 0ρ ω ≈r
CTU-FEE in Prague, Department of Electromagnetic Field
Electromagnetic Field Theory 2RADIATION
Time-Harmonic Electric Dipole - Field Zones
34 / XXX
Static, quasi-static and fully dynamic terms all appear in the formula
( ) ( )j
3
0 0 0 02 2 2 2
j 1 j 1 eˆ , 2 cos 1 sin4
kr
zZ c k p
kr kr krk r k rω θ θ θ ω
π
− = + + − + + E r r
Static zone
Induction zone
Radiation zone
CTU-FEE in Prague, Department of Electromagnetic Field
Electromagnetic Field Theory 2RADIATION
Time-Harmonic Electric Dipole - Radiation Zone
35 / XXX
Farfield has a planewave-like geometry
( ) ( ) ( ) ( ) ( )0ˆ ,
zp x y zω ω δ δ δ= zP r
( ) ( )
( ) ( )
( )
j3
0 0 0
0
0
2*
0
0
eˆ , sin4
1ˆ ˆ, ,
1 1ˆ ˆ ˆRe ,2 2
kr
zZ c k p
kr
Z
Z
ω θ ω θπ
ω ω
ω
−
∞
∞ ∞
∞ ∞ ∞ ∞
≈ −
≈ ×
= × =
E r
H r r E r
S E H E r r
0ˆ 0∞
⋅ ≈r E
( )2 4
20 0
rad 12 z
c Z kP p ω
π=
Radiated power [W]
CTU-FEE in Prague, Department of Electromagnetic Field
Electromagnetic Field Theory 2RADIATION
Time-Harmonic Electric Dipole – General Case
36 / XXX
Elementary source of radiation
( ) ( ) ( )ˆ ˆ,ω ω δ ′= −P r p r r
( )
( )
j3
0
j3
0 0 2 2
1 eˆ ˆ, 1j 4
1 j eˆ ˆ ˆ ˆ, 34
kR
kR
c kR kR kR
Z c kR R R R kR kRk R
ωπ
ωπ
−
−
= × +
= − × × + ⋅ − +
RH r p
R R R RE r p p p
R ′= −r r
′= −R r r
CTU-FEE in Prague, Department of Electromagnetic Field
Electromagnetic Field Theory 2RADIATION
General Radiator
37 / XXX
Superposition of dipole fields
( ) ( )ˆ , , ,tωJ r J r
( ) ( )
( )
0j
0
00
eˆ ˆ, , d4
,
, d4
k R
V
V
VR
Rt
ct V
R
µω ω
π
µ
π
−
′
′
′ ′=
′ − ′=
∫
∫
A r J r
J r
A r
R ′= −r r
time
retardation
CTU-FEE in Prague, Department of Electromagnetic Field
Electromagnetic Field Theory 2RADIATION
Field in Radiation Zone – General Case FD
38 / XXX
Farfield has a planewave-like geometry
( ) ( )
( ) ( )
( ) ( )( )
( )
0 0
jj0
0
0
0 0
22
0 0
0
eˆ ˆ, , e d4
j ˆˆ , ,
ˆˆ , j ,
1 ˆ ,2
krk
V
Vr
Z
Z
µω ω
π
ωω ω
ω ω ω
ω ω
−′⋅
∞′
∞ ∞
∞ ∞
∞ ∞
′ ′≈
≈ − ×
≈ × ×
= ×
∫ r
r
r r
r r
rA r J r
H r A r
E r A r
S A r
( )0 0ˆ ˆZ∞ ∞≈ − ×rE H
1kR r r ′∧≫ ≫
CTU-FEE in Prague, Department of Electromagnetic Field
Electromagnetic Field Theory 2RADIATION
Field in Radiation Zone – General Case TD
39 / XXX
Farfield has a planewave-like geometry
( )
( ) ( )
( ) ( )( )
( )
0
0 0
0
0
0 0
2
0 0
0
, , d4
1, ,
, ,
1,
V
rt t V
r c c
t tZ
t t
tZ
µ
π∞′
∞ ∞
∞ ∞
∞ ∞
′⋅ ′ ′≈ − +
≈ − ×
≈ × ×
≈ ×
∫r
r A
r r A
r r
rA r J r
H r r
E r r
S A r
ɺ
ɺ
ɺ
( )0 0Z∞ ∞≈ − ×rE H
CTU-FEE in Prague, Department of Electromagnetic Field
Electromagnetic Field Theory 2RADIATION
Radiation Zone = Rays
40 / XXX
Radiation diagram is formed by Fourier(`s) transform of sources
( ) ( ) 0 0
jj0 eˆ ˆ, , e d
4
krk
V
Vr
µω ω
π
−′⋅
∞′
′ ′≈ ∫ r rA r J r
( ) ( )0
1
j
0
0 0
eˆ, ,4
k r
t krω
µω
π
−−
∞
≈
−
rA r JF
4D Fourier(´s) transform
( ) ( ) , , ,ˆ , , , , , ,
x y z tx y zk k k x y z tω =J JF
CTU-FEE in Prague, Department of Electromagnetic Field
Electromagnetic Field Theory 2RADIATION
Angular Spectrum Representation (Sources)
41 / XXX
General solution to free-space Maxwell‘s equations
( )( ) ( )( ) ( )
, ,
,
1
0
,
1
maxˆ, , , , ,
min
maxˆ, , , , , , ,
min
x y
x y
k k x y z
k k x y z x y z
z zx y t k k k
z z
z z Zx y t k k k k k k
z z k
ω
ω
−
−
′> = × ′< ′> = × × ′<
H G
E G
∓
∓ ∓
F
F
2 2 2
z x yk k k k= − −
Im 0zk <
( ) ( ) , , ,ˆ , , , , , ,
x y z tx y zk k k x y z tω =J JF
( ) jˆ , , ,
ˆ e2
zx y z k z
z
k k k
k
ω=J
G∓
∓
Valid only outside the
source region
4D Fourier(´s) transform
CTU-FEE in Prague, Department of Electromagnetic Field
Electromagnetic Field Theory 2RADIATION
Angular Spectrum in Radiation Zone
42 / XXX
Farfield is made of propagating planewaves
( ) ( ) ,
1 jˆ, , 0, , e z
x y
k z
k k x yx y z k kω − −> =F LF
Stationary phase method
0
, ,x y z
r
=r2 2 2
0
Im 0
z x y
z
k k k k
k
= − − <
( ) ( )0
j
0 0
0 0 0 0
j eˆ, ,2
k r
z
x y
k rk r k r
rω
π
−
∞ = − −F r L
( )( )
( )0 0 0 0
2 2 2
j
2
1 ˆ, , e d d2
yx zx y z
x y
kk kk r r r r
k k k
x y x y
k k k
k k k kω
π
+ − ∞
> +
= ∫F r L
0k r → ∞
0k r → ∞
CTU-FEE in Prague, Department of Electromagnetic Field
Electromagnetic Field Theory 2RADIATION
Angular Spectrum in Radiation Zone
43 / XXX
Farfield is made of propagating planewaves
0
, ,x y z
r
=r
( ) ( )
( ) ( )
0
0
j
0
0 0 0
j
0 0
0 0
1
0 0
1j eˆ, ,4
j eˆ, ,4
k r
k r
kt k
r
k Zt k
r
ω
ω
ωπ
ωπ
−−
−
∞
−
∞
≈ − × −
≈
× ×
−
H r r J r
E r r r J r
F
F
4D Fourier(´s) transform
( ) ( ) , , ,ˆ , , , , , ,
x y z tx y zk k k x y z tω =J JF
CTU-FEE in Prague, Department of Electromagnetic Field
Electromagnetic Field Theory 2PLANAR BOUNDARIES
Planar Material Boundary
44 / XXX
Field is composed of incident, reflected and transmitted waves
Boundary is at z = 0
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
1 1
1 1
2 2
j j
, 1 1
1 1 1 1j j
, 1 1
1 1
j j
, 2 2
1
1
1
0 , , e , , e
, , , , , , , ,0 e e
0 , , e , , e
z z
x y
z z
x y
z z
x y
k z k z
k k x y x y
x y z x y x y z x yk z k z
k k
k z k z
k k x y x y
z k k k k
k k k k k k k k k kz Z Z
k k
z k k k k
ω ω
ω ω
ω ω
−+ −
+ −
−
−
−
−
−+ −
< = +
− × × < = +
+
> =
H H H
H HE
H H H
F
F
F
( ) ( ) ( )2 2
2 2 2 2j j
, 2 2
2
1
2
, , , , , , , ,0 e ez z
x y
x y z x y x y z x yk z k z
k k
k k k k k k k k k kz Z Z
k k
ω ω+ −−−
− × × >
= +
H HE F
Incident wave
Incident wave Reflected (1 1) / Transmitted (2 1) wave
Reflected (2 2) / Transmitted (1 2) wave
2 2 2
z x yk k k k= − −
CTU-FEE in Prague, Department of Electromagnetic Field
Electromagnetic Field Theory 2PLANAR BOUNDARIES
Planar Material Boundary – Boundary Conditions
45 / XXX
Relations valid for both propagative and evanescent waves
are equal on both sides
( ) ( )
( ) ( )
1 1 1 1
1 1
1 1
2 2 2 2
2 2
2 2
0 0
0 0
, , , ,
, , , ,
x y z x y z
x y z x y z
k k k k k kZ Z
k k
k k k k k kZ Z
k k
+ −
+ −
− × × + =
− × ×
× ×
× ×+
z z
z z
H H
H H
0 0 0 01 1 2 2
+ − + −+ = +× × × ×z z z zH H H H
2 2, , 0x y zk k k ± ⋅ = H∓
2 2 2
Im 0
z x y
z
k k k k
k
= − − <
1 1, , 0x y zk k k ± ⋅ = H∓
,x yk k
CTU-FEE in Prague, Department of Electromagnetic Field
Electromagnetic Field Theory 2PLANAR BOUNDARIES
Perpendicular Incidence – Matrix Form
46 / XXX
Matrices with the use across the electrical engineering
Scattering matrix
(experiments)
2 1
2 1
1 1
2 2
E ET
E E
E ES
E E
+ +
− −
− +
+ −
=
=
1 2 1 2
1 2 1 21
2 1 1
2 1 22 1
1
2
21
2
Z Z Z ZT
Z Z Z ZZ
Z Z ZS
Z Z ZZ Z
+ − = − +
− = −+
21
1222
22
1112
11
det
det1
1
TS
T TT
S ST
SS
− = − = −
Transmission matrix
(multilayer cascade)
0x yk k= =
CTU-FEE in Prague, Department of Electromagnetic Field
Electromagnetic Field Theory 2PLANAR BOUNDARIES
Perpendicular Incidence – Interesting Cases
47 / XXX
Technically important special cases
Transparent dielectric layer
2
2d Nλ
=
( )0 1 1 2 2k n d n d Nπ+ =
Bragg‘s mirror
(dielectric mirror)
1 0 0 1,
0 1 1 0T S
− − = = − −
Alternating dielectric layers
Wavelength inside the slab
CTU-FEE in Prague, Department of Electromagnetic Field
Electromagnetic Field Theory 2PLANAR BOUNDARIES
Oblique Incidence – TM / TE Case
48 / XXX
Snell‘s law is a property of k-vectors
0x x
y
k k
k
=
=
Snell‘s law of refraction
1 inc 2 trans
1 inc 1 refl
sin sin
sin sin
k k
k k
α α
α α
=
=
The angle of incidence equals
to the angle of reflection
CTU-FEE in Prague, Department of Electromagnetic Field
Electromagnetic Field Theory 2PLANAR BOUNDARIES
Oblique Incidence – TM Case
49 / XXX
Generalization of reflection and transmission to oblique incidence
2 2
2 12 2
2 1TM 1
1 12 2
1
2 12 2
2 1
2
2 2
2TM 2
1 22 2
1
2 12 2
2 1
1 1
1 1
2 1
1 1
x x
x
x x x
x
x
x x x
k kZ Z
k kER
E k kZ Z
k k
kZ
kET
E k kZ Z
k k
−
→ +
+
→ +
− − −
= =
− + −
−
= =
− + −
0
0
0
y
x
z
k
H
H
=
==
CTU-FEE in Prague, Department of Electromagnetic Field
Electromagnetic Field Theory 2PLANAR BOUNDARIES
Oblique Incidence – TM Case
50 / XXX
Can be used for polarizing unpolarizaed light beams
( )( )
2 2 1 1 2
2 21 1 1 2
xk
k
ε µ ε µ ε
µ ε ε
−=
−
Vanishing reflection on a boundary
0
0
0
y
x
z
k
H
H
=
==
TM
1 10R → =
Brewster‘s angle
1
21
1 2
1xk
k
ε
ε
− = +
Simplification for pure dielectrics
CTU-FEE in Prague, Department of Electromagnetic Field
Electromagnetic Field Theory 2PLANAR BOUNDARIES
Oblique Incidence – TE Case
51 / XXX
2 2
2 12 2
1 1 2TE
1 12 2
1
2 12 2
1 2
2
2 2
2 1TE
1 22 2
1
2 12 2
1 2
1 1
1 1
2 1
1 1
x x
y
y x x
x
y
y x x
k kZ Z
E k kR
E k kZ Z
k k
kZ
E kT
E k kZ Z
k k
−
→ +
+
→ +
− − −
= =
− + −
−
= =
− + −
0
0
0
y
x
z
k
E
E
===
Generalization of reflection and transmission to oblique incidence
CTU-FEE in Prague, Department of Electromagnetic Field
Electromagnetic Field Theory 2PLANAR BOUNDARIES
Oblique Incidence – TE Case
52 / XXX
Unrealistic scenario for natural materials
( )( )
2 2 1 1 2
2 21 1 1 2
xk
k
µ ε µ ε µ
ε µ µ
−=
−
0
0
0
y
x
z
k
E
E
===
TE
1 10R → =
Brewster‘s angle
Vanishing reflection on a boundary
1
21
1 2
1xk
k
µ
µ
− = +
Simplification for pure magnetics
CTU-FEE in Prague, Department of Electromagnetic Field
Electromagnetic Field Theory 2PLANAR BOUNDARIES
Oblique Incidence – Total Reflection
53 / XXX
Valid for both, the TM and the TE case
2 2
1 1 1
1xk k n
k k n> = <
TM TE
1 1 1 11R R→ →= =
CTU-FEE in Prague, Department of Electromagnetic Field
Electromagnetic Field Theory 2GUIDED WAVES
Guided TEM Wave
54 / XXX
Generalization of a planewave
( ) ( )
( ) ( )
j
j
ˆ , , , e
ˆ , , , e
kz
kz
x y
x y
ω ω
ω ω
−⊥
−⊥
=
=
E r E
H r H
Wave propagation identical to a planewave
( )2 j +jk ωµ σ ωε= −
( )0ˆ ˆk
ωµ= ×H z E
0
0⊥ ⊥
⊥ ⊥
∆ =
∆ =
E
H
Geometry of a planewave
0⊥× =n E
Boundary condition
on the conductor
CTU-FEE in Prague, Department of Electromagnetic Field
Electromagnetic Field Theory 2GUIDED WAVES
Circuit Parameters of the TEM Wave
55 / XXX
( )
( )
pul
0
pul
0
ˆ d
ˆ d
l
B
A
QkI
Uk
ωωµ ε
ωµω
µ
⊥
⊥
= ⋅ = ⋅
Φ= − ⋅ = ⋅
∫
∫
H l
E l
Enclosing conductor
( ) ( )
( ) ( )
j
0
j
0
ˆ ˆ, e
ˆ ˆ, e
kz
kz
U z U
I z I
ω ω
ω ω
−
−
=
=
( )( )0 pul pul
TRL
pul pul0
phase
pul pul
ˆ
ˆ
1 1
U L LZ
k C k CI
vC L
ω ωµ ε ωµ
µω
εµ
= = ⋅ = ⋅ =
= =Per unit length
Velocity of phase
propagation
Between conductors
CTU-FEE in Prague, Department of Electromagnetic Field
Electromagnetic Field Theory 2GUIDED WAVES
The Telegraph Equations
56 / XXX
Circuit analog of Maxwell’s equations
( ) ( )
( ) ( )
pul
pul
, ,
, ,
U z t I z tL
z t
I z t U z tC
z t
∂ ∂= −
∂ ∂
∂ ∂= −
∂ ∂
CTU-FEE in Prague, Department of Electromagnetic Field
Electromagnetic Field Theory 2GUIDED WAVES
Guided TE and TM Waves
57 / XXX
TEM mode must be completed with TE and TM modes to form a complete set
( )
( )( )
02
02
1j j
1j +j
z z z
z z z
k E Hk
k H Ek
ωµ
σ ωε
⊥ ⊥ ⊥⊥
⊥ ⊥ ⊥⊥
= − ∇ − ×∇
= − ∇ + ×∇
E z
H z
( ) ( ) ( )
( ) ( ) ( )
j
0
j
0
ˆ , , , e
ˆ , , , e
z
z
k z
z
k z
z
E
H
ω ω ω
ω ω ω
−
⊥ ⊥ ⊥
−
⊥ ⊥ ⊥
= +
= +
E r E r z r
H r H r z r
Wave propagation differs
from a planewave
2 2 2
zk k k
⊥= −
2
2
0
0
z z
z z
E k E
H k H
⊥ ⊥
⊥ ⊥
∆ + =
∆ + =
CTU-FEE in Prague, Department of Electromagnetic Field
Electromagnetic Field Theory 2GUIDED WAVES
PEC Waveguides – pure TE, TM modes
58 / XXX
ˆ 0× =n E
Boundary condition
on the conductor
Wavenumbers form a discrete set
Modes are orthogonal in waveguide cross-section
Modes form a complete set in waveguide cross-section
0k⊥>
TE / TM
0
TE TM j,
j
z
z
Z
kZ Z
k
H z E
Impedances differ
from those of a
planewave
TEM mode must be completed with TE and TM modes to form a complete set
CTU-FEE in Prague, Department of Electromagnetic Field
Electromagnetic Field Theory 2GUIDED WAVES
PEC Waveguides – modal orthogonality
59 / XXX
Waveguide modes form an orthogonal set
* dS
S C E E
* *
0 *
* *
*
1d d
1d d
S S
S S
S SZ
S SZ Z
zE H E E
H H E E
cross-section of
the waveguide
CTU-FEE in Prague, Department of Electromagnetic Field
Electromagnetic Field Theory 2GUIDED WAVES
PEC Waveguides – modal decomposition
60 / XXX
Any field within a waveguide can be composed of its modes
j
0
j
0
ˆ , , , e
ˆ , , , e
z
z
k z
z
k z
z
C E
C H
z
z
E r E r r
H r H r r
positive direction
negative direction
j
0
j
0
ˆ , , , e
ˆ , , , e
z
z
k z
z
k z
z
C E
C H
z
z
E r E r r
H r H r r
CTU-FEE in Prague, Department of Electromagnetic Field
Electromagnetic Field Theory 2RADIATION
PEC Waveguides – Field Sources
61 / XXX
Tangential fields within the cross-section fully define the field everywhere
( ) ( ) ( ) ( )
( ) ( )
2* *
j
*
ˆ ˆ, , , , d1
e2 , , d
zk zS
S
Z S
CS
β
β β β
β
β β
ω ω ω ω
ω ω
⊥ ⊥ ⊥ ⊥±±
⊥ ⊥ ⊥ ⊥
⋅ ± ⋅
=
⋅
∫
∫
E r E r H r H r
E r E r
transversal field of the
waveguide mode
Known field at arbitrary
cross-section of the waveguide
CTU-FEE in Prague, Department of Electromagnetic Field
Electromagnetic Field Theory 2RADIATION
PEC Waveguides – Field Sources
62 / XXX
This is how waveguide modes are excited
( ) ( )
( ) ( )
sˆ ˆ, , d
2 , , d
V
S
VZ
CS
β
β
β
β β
ω ω
ω ω
±
⊥ ⊥ ⊥ ⊥
⋅
= −⋅
∫
∫
E r J r
E r E r
∓
Valid to the right (+) or to the
left (-) of the source region
Full field of the
waveguide mode
transversal field of the
waveguide mode
Source current density
existing within the waveguide
CTU-FEE in Prague, Department of Electromagnetic Field
Electromagnetic Field Theory 2GUIDED WAVES
Dielectric Waveguides – mixed TE + TM modes
63 / XXX
Boundary condition
on dielectric interface
Finite number of guided modes
Continuum of radiating modes
Only combination of guided and radiating modes forms a complete
set in the waveguide cross-section
General field is not guided by a dielectric waveguide
1 2
1 2
ˆ ˆ 0
ˆ ˆ 0
× − =
× − =
n E E
n H H
CTU-FEE in Prague, Department of Electromagnetic Field
Electromagnetic Field Theory 2GUIDED WAVES
Cavity Resonators
64 / XXX
ˆ 0n
× =n E
Boundary condition
on the conductor
Eigenfrequencies form a discrete set
Modes are orthogonal in the volume of the cavity
Modes form a complete set in the volume of the cavity
0nω >
2
r r2
0
ˆ ˆ 0n
n nc
ωε µ∆ + =E E
Field in a lossless cavity forms an exception to the uniqueness theorem
Nontrivial solution only
exists in a lossless cavity
Master equation
Notice that field value is
defined on a closed surface
CTU-FEE in Prague, Department of Electromagnetic Field
Electromagnetic Field Theory 2GUIDED WAVES
Closed Waveguide as a Cavity Resonator
65 / XXX
2
2 2
r r2
0
zk k
c
α
α
ωε µ ⊥= +
Dispersion relation
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
0
0
ˆ , , sin j , cos
ˆ , j , cos , sin
z z z
z z z
k z E k z
k z H k z
α α α α α
α α α α α
ω ω ω
ω ω ω
⊥ ⊥ ⊥
⊥ ⊥ ⊥
= +
= +
E r E r z r
H r H r z r
Waveguide modes already solve the wave equation
z
pk
Lα
π=
Electric and magnetic fields
are 90° out of phase
CTU-FEE in Prague, Department of Electromagnetic Field
Electromagnetic Field Theory 2RADIATION
Excitation of a PEC Cavity
66 / XXX
Modes of a lossless cavity are used as a basis
( ) ( )( )
( ) ( )
( ) ( )
*
2 2 *
, d
d
V
V
V
Ck V
α
α
α α α
ωµ ω
ωω λ
⋅
= − ⋅− ⋅
∫
∫
A r J r
A r A r
Vector potential is expanded
into cavity modes ( ) ( )( ) ( )
2 0
0
α α α
α
λ∆ + =
× =
A r A r
n r E r
Modes of lossless cavity
( ) ( ) ( ), Cα α
α
ω ω=∑A r A r
Lukas Jelinek
Department of Electromagnetic Field
Czech Technical University in Prague
Czech Republic
Ver. 2019/05/06