PETER PAZMANY CATHOLIC UNIVERSITYConsortiummembersSEMMELWEIS UNIVERSITY, DIALOG CAMPUS PUBLISHER

**Molekuláris bionika és Infobionika Szakok tananyagának komplex fejlesztése konzorciumi keretben

***A projekt az Európai Unió támogatásával, az Európai Szociális Alap társfinanszírozásával valósul meg.

Development of Complex Curricula for Molecular Bionics and Infobionics Programs within a consortial* framework**

Consortium leader

PETER PAZMANY CATHOLIC UNIVERSITY

Consortium members

SEMMELWEIS UNIVERSITY, DIALOG CAMPUS PUBLISHER

The Project has been realised with the support of the European Union and has been co-financed by the European Social Fund ***

PHYSICS FOR NANOBIO-TECHNOLOGY

Principles of Physics for Bionic Engineering

Chapter 4. Electrodynamics – II

(A nanobio-technológia fizikai alapjai )

(Elektrodinamika)

Árpád I. CSURGAY,

Ádám FEKETE, Kristóf TAHY, Ildikó CSURGAYNÉ

Table of Contents

4. Electrodynamics - II

1. Plane Wave Reflection and Refraction 2. Wave-guides

3. Electromagnetic Radiation 4. Antennas

1. Plane Wave Reflection and Refraction

Normal incidence plane-wave reflection and transmission between two conductive media

Boundary at

2 2

2,







z y

x

 0 z

i i

H E

t t

H E

r r

H E Reflected wave

Transmitted wave Incident wave

1 1 1,







Boundary conditions at

Incident wave Reflected wave Transmitted wave

m z i

y

z m

i x

E e H

e E E

1 1

1 1 1







 









m z r

y

z m

r x

E e H

e E E

1 1

1 1 1

















m z t

y

z m

t x

E e H

e E E

2 2

2 2 2







 





















₁  _m1  _m2

m E E

E

2 2 1

1 1

1

 







  ^m  ^m

m E E

E

 0 z

Transmission Reflection



 



 

 ^



2 1 2

1 1

2





m

m E

E 



 



 

 ^



2 1 2

1 1

2





m

m E

E

2 2

1 1 2

m 2

m

T E

E



 



  



1 2 1

1 2 1

m m

R E

E

 

 





  

 1 R T

Plane wave reflection and refraction Plane wave propagation at arbitrary angle

Wave impedance of the medium const ,     



 n β r n r

β





( )  _m e^{ }j^{β r}

E r E

j

( ) m

( ) ^β _ e

 

    

 

n E r n E β r

H r

r



 







Equiphase plane

r

Direction of propagation

z

x y

1 1 1 1

 

 

2 2 2 2

 

 



E in plane of incidence E normal to plane of incidence

βi

i β n

β  ₁  β_r 



t nβt

β 



( ) j ⁱ

i i

m e^{ }

  ^{β r} E r E

j

( )

i i

βi m

i e



 

 



n E β r

H r

( ) j ^r

r r

m e^{ }

  ^{β r} E r E

j

( )

r r

βr m

r e



 

 



n E β r

H r

( ) j ^t

t t

m e^{ }

  ^{β r} E r E

 

i i i

x z

E E e^{ }

     ^{β r}

E i j ^{Ei }





E in plane of incidence

i

 

 

j

j sin cos

cos sin

i

i i

i i i

x z

x z

i

m i i

E E e

E

 

 

 

     

    

E i k β r

i k

 

1

j

1

j sin cos

j

1 1

( )

sin 0 cos

cos 0 sin

i

i i

i

i

βi m

i

i r

x z

m m

i i

i i

e

E E

e e ^{  }



 

 

 

 

 

 

 

 

   



β r

β r

n E

H r

i j k

j

Boundary conditions for

Continuity of the tangential magnetic field

Snell’s Law

x z  0, 

     

1 1 2

j sin j sin j sin

1 1 2

i r t

i r t

m x m x m x

E E E

e ^{ } e ^{ } e ^{ }

  

  

    

     

1 1 2

1 2

sin sin sin

sin sin

r i

i r t

i t

x x x

 

     

   

 

    

i

t





 

2

 1

1 1 2

i r t

m m m

E E E







Continuity of the tangential electric field

 Two unknowns – two equations



 

 



i r t

i r t

m m m

E







1 2

2 cos

cos

i t t

m m

i

E

 

 

 

   

 





^

^

^

^

E Normal to Plane of Incidence

1 2

1 2

cos cos

cos cos

r i t i i

m m m

i t

E

   

   

     



1 2

1 2

cos cos

cos cos

r

m i t

i

i t

m

E

R E

   

   

  



2

1 2

2 cos

cos cos

t

m i

i

i t

m

E

T E

 

   

  

  

2 1

2 1

cos cos

cos cos

r

m i t

i

m i t

R E

E

   

   

 



  



2

2 1

2 cos

cos cos

t

m i

i

m i t

T E

E

 

   

 



 



Parallel Perpendicular

Comparison between reflection and transmission for parallel and perpendicular polarizations

i

r





2 1 2

2 1 1 2

1

0 2 1

sin sin

































 i 

t

2 1

1 2

cos cos

cos cos

r

m t i

i

i t

m

R E

E

   

   

  



2 1

2 1

cos cos

cos cos

r

m i t

i

m i t

R E

E

   

   

 



  



2

1 2

2 cos

cos cos

t

m i

i

i t

m

T E

E

 

   

  

  

2

2 1

2 cos

cos cos

t

m i

i

m i t

T E

E

 

   

 



 



For parallel polarization at a certain angle no reflection occurs

Brewster-angle

For both polarizations the transmitted wave’s angle can be bigger than 90^° , and total reflection occurs

At the critical angle

2 1 B

1 B 2

cos cos

cos cos 0

r

m t i

i

i t

m

E

R E

   

   

   



2

2 1 B Brewster

1

cos _t cosθ_i tan _i



   

  



1

1 2

2

sin _t ε sin _{i t i}

ε ε θ θ θ θ

   ε  

900

2 π/ 

t 

θ 2 1

1

critical sin ε / ε θ

θ_i   ^

Geometrical Optics: Ray Optics Model

It assumes light travels in rays, assumes geometric propagation, reflections, refractions.

Prism Optical fibre

Primary Rainbow Brewster

angle

2. Wave-guides Plane waves

 TEM mode

“Two-wire” and multi-wire wave-guides

TEM mode propagation is possible

E.g. Coaxial waveguide, Microstrip waveguide

Wave-guides

 TM and TE modes

“Single-wire” wave-guides Propagation in TEM mode is

not possible! TM and TE modes are Possible!

E.g. Rectangular wave-guide, Circular wave-guide, etc.

Example 2: Rectangular Wave-guide

“Single-wire” wave-guide Propagation only for waves

small enough wavelength is possible

Let us assume ideal metallic walls, and sinusoidal time-harmonic field (Complex amplitudes)

a

 2



TM modes TE modes

0

Hz Ez(x, y,z)  X(x)Y(y)e^z

 0

Ez H_z(x, y,z)  X(x)Y(y)e^z

y

x a

b

To solve for the z components of E and H, we start with Maxwell Equations

j j

j j

 

 

        

      

E H E H

H E H E

H H

H

E E

E









j j

j j

2 2

































0 0







 H

E

 





2 2

















H E













 

 

ˆ

2 2

2 2

















z z

H E













2 2 2

2 2

2 2

z y

x 

 



 



 



TM modes





z y

E_z x



 



 



 







  

0

Hz E_z(x, y,z)  X(x)Y(y)e

0 )

d ( d d

d _{2 2}

2 2 2

2    X Y 

y X Y x

Y X    ( ) 0

d d 1 d

d

1 _{2 2}

2 2 2

2      

y Y Y

x X X

2 2

2

d d

1 M

x X

X   ² ₂ ²

d d

1 N

y Y

Y   M²  N² ( ² ²)  0

Ny D

Ny C

Y Mx

B Mx A

X  sin  cos  sin  cos





E_z   

0 0

 x

Ez E_{z xa}  0 E_{z y0}  0 E_{z yb}  0

 0

 D

B E_z  A'sinMxsinNy

,...

1,2,3 π

0 sin

sin '

0       

  E A Ma Ny Ma m m

E_{z x a z}

...

1,2,3, 0 π

sin sin

'

0       

  E A Mx Nb Nb n n

E_{z y b z}

z

z y e

b x n

a A m

z y x

E  ^



 



 



 



  π

π sin sin

' )

, , (

TM_mn mode

If the wave propagates,

„Cut off” frequency

If imaginary (no decay).

π π

( , , ) 'sin sin ^z

z

m n

E x y z A x y e

a b



    

    

   

2 2 2 2

( ) 0

M N   

     

2 2

π π 2

m n

a b

  ^ ^ ^ ^  

2 2

2 π π

j _mn j m n

a b

   ^{   }

     

2 2

2 π π

m n 0

a b

  ^ ^ ^ ^ 

   

2 2

2 π π

c 0

m n

a b

  ^ ^ ^ ^ 

   

2 2

,

1 π π

c mn 2

m n

f   a b

   

    



 j

, 

 f_c f

Wave propagation in TM mode

If imaginery (no decay)

2 2 2

2 π π ,

1 ^{c mn}

mn

m n f

a b f

    ^ ^ ^ ^    ^ ^

2 ,

2

1

mn

mn c mn

λ λ

β f

f

  

 

  

 

2 ,

TM_mn ^{x mn} 1 ^{c mn}

y

E β μ f

η H ωε ε f

 

     

 



 j

, 

 f_c f

TM_mn mode ^H_z ^0 _{'sin sin}

Ez A x y e

a b

    

   

j

2 2

j π

π π

a 'cos sin

π π

a

mn

mn

z x

m

m n

E A x y e

a b

m n

b







 

      

        

z mn

y

e mn

b y x n

a A m

b n m

b n

E ^



j 2

2

cos π sin π

π ' a

π j π

 



 



 



 







 







 







 



 



j

2 2

j π

π π

a 'cos sin

π π

a

mnz y

m

m n

H A x y e

a b

m n

b







 

      

        

j

2 2

j π

π π

'sin cos

π π

a

mnz x

n

m n

H b A x y e

a b

m n

b







 

      

        

TE modes in rectangular wave guides

TE_mn mode ^E_z ^{ 0 H}_z ^{A'cos mπ x cos nπ y e j mnz}

a b



    

    

2 2

π π 2

mn

m n

γ ω εμ

a b

   

     

2 2

,

1 π π

c mn 2π

m n

f εμ a b

   

    

0TE 2

1 , mn

c mn

μ Z ε

f f



 

  

 

TM₁₀ mode

x y z 0

E  H  E 

j 10

0

cos π ^z

Hz H x e

a



  

  

j 10

0

j π

π sin

z y

E ωμa H x e

a



   

  

j 10

10

0

j π

π sin

z x

H β a H x e

a



   

  

Cut-off frequency

   

2 2

2 2

TM TE,

. 2

2 2

2 π

π 2

1 m n

a c a

n a

m c

b n a

m

f_cmn εμ   



 







 



 



 







 



 

 a f_c c

2

TE 10

,  _c^TE_,01 2 f_c10

a

f  c  _c^TE_,20 2 f_c10 a

f  c 

10 TM

TE, 11

, 5 5

2 ^c

c f

a

f  c 

10 TM

TE, 21

, 8 _c

c f

f 

/ _c10

c f

f

20 10,TE TE

TE01

11 11,TM TE

21 21,TM TE

3. Electromagnetic Radiation

Introduction of the scalar and vector potentials with Lorentz gauge , )t 0 , )t ( , )t

B(r   B(r  A r

, )t , )t , )t

t t

 

      

 

E(r B(r A(r

, )t , )t 0 t

  

 E(r   A(r    ^{, )t , )t}

 

t



   

E(r  A(r r





  

       

      

      

A B A

E A

Lorentz gauge

2

0

1 1

, )t , )t t)

c t μ

   

B(r  E(r J(r,

2

0

1 1

)

c t





 

 

          

A A A A J(r,

2

, )t 1 , )t 0 c ^t



  

A(r  (r

2

2 2

0

1 1

, )t , )t , )t

c t μ

    

A(r  A(r J(r

 

c t

 



   



r r r

Lorentz gauge has to be satisfied as well.

 

V

,

, d ;

4π

t r

μ c

t V

r

  

 

 





A r

 

0 V

1 ,

, d

4π

t r

t c V

ε r





  

 

 





r

ρ

 

  

          

        

        

J A B A

E A

Retarded Potentials

In static case we know the solutions of

In dynamic case the solution is the retarded potential

The Lorentz gauge joins the vector and scalar potentials , )t  , )t

J(r A(r





 

', '

, d '

4π _V '

t c

t μ V

  

  

 





J r r r

A r r r

t t '

c

  r r

   

2 0

0

' d '

4π _V '

μ V



     



A J A r

r r

4. Antennas

If we know the radiating currents, we can calculate the vector potential, utilizing the Lorentz gauge we know the scalar potential as well, thus we know the fields.

Radiated power

0

,

, ) d

4π _V

t r

μ c

t V

r

  

 

 

 



J(r A

0 0

div ε μ

t





A 

 

  

       

      

       A B A

E A

 

P 



In case of time-harmonic currents

Radiated power

 

0 , , , j

, j ) ( , j ) d d d

4π

kr

V

x y z ω e

ω x y z

r



 

J(r A r

0 0

1 jωε μ





 

      

 

      

     

A B A

E A

 

Re 1 d

A 2

P  ^ 

   





Example 2: The radiation of the Hertz dipole

j

( , , , )x y z t  I0( , )x y e ^t

J k

 

-j

0 0

. ( ) d

4π ( , )

d d d 4π

V

V

μ e

jω V

r I x y e

x y z r

 

  

 





kr

kr

A r J r

k





4π _V 4π

I e d I e x y x y z

r r

    ^{ }

  ^kr



A r k k

cos _r sin _θ

   

k e e

0

k  2

  

 r

 z

az

d

j 0

0 cos

4π

kr r

A μ I de

r 

  ₀ ^{0 j} sin

4π I de kr

A μ θ

 r

  

 

       

1 sin rot sin

1 1 1

sin

r r

φ

A A

r

A rA rA A

r r r r

 

 





  

  

  

 

     

   

 

       

 

           

B A A er

e e

0 0

0 



 



 



 ^



A

A A^r

   

r φ

A r

rA

r e

A

B 



 







 



 

 ^ 

rot 1 





 j 1 sin

4 ²

0 j

0 



 



 

 ^

r r

e k d

B I ^kr