Electromagnetic Field Theory 2 - cvut.cz

Lukas Jelinek

[email protected]

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



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



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



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



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



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



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



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



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



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



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


are orthogonal to

propagation direction



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



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


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



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



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



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



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



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 ±



Angular Spectrum Representation

21 / XXX


( )( )

( ) ( )

( )( )

( ) ( )

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



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



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



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



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= =



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



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



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 ∑



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



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



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


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



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



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



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]



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



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



Field in Radiation Zone – General Case FD

38 / XXX


( ) ( )

( ) ( )

( ) ( )( )

( )

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 ′∧≫ ≫



Field in Radiation Zone – General Case TD

39 / XXX


( )

( ) ( )

( ) ( )( )

( )

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



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



Angular Spectrum Representation (Sources)

41 / XXX


( )( ) ( )( ) ( )

, ,

,

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 , , , , , ,


( ) jˆ , , ,

ˆ e2

zx y z k z

z

k k k

k

ω=J

G∓

∓

Valid only outside the

source region




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 → ∞



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


( ) ( ) , , ,ˆ , , , , , ,



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= − −



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



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= =



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



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



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

=

==



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



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



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



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→ →= =


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



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



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

∂ ∂= −

∂ ∂

∂ ∂= −

∂ ∂



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

⊥ ⊥

⊥ ⊥

∆ + =

∆ + =



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



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



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



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



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



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



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



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=


are 90° out of phase



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

[email protected]

Department of Electromagnetic Field

Czech Technical University in Prague

Czech Republic

Ver. 2019/05/06

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Electromagnetic Field Theory 2 - cvut.cz