**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 5. Electrodynamics – III
(A nanobio-technológia fizikai alapjai )
(Elektrodinamika)
Árpád I. CSURGAY,
Ádám FEKETE, Kristóf TAHY, Ildikó CSURGAYNÉ
Table of Contents
5. Electrodynamics – III
1. Electromagnetic Field in Cavities 2. Modal Expansion of the EM Field 3. The quality factor (Q) of cavities
1. Electromagnetic Fields in Cavities
An ideal cavity is a three dimensional, finite volume, filled with vacuum, surrounded by a metal wall of perfect conductivity.
The state variable of the electromagnetic field inside the cavity is the vector potential satisfying the wave equation
and the electric and magnetic fields can be calculated as
Let us look for the solutions as
2
0 0
2 2 2
1 1
( , ) :t , )t , );t
c t c
A r A(r A(r
, )t , );t , )t ( , ).t t
E(r A(r B(r A r
) A(r )
( )
,
A(r t f t
The left hand side of the equation depends only on r, the right hand side only on t, thus both sides should be constant
Thus the solution of the Maxwell’s equations has been reduced to a time-independent eigen-value problem, and a time-dependent ordinary differential equation.
2 2
2 2 2 2
1 d ( ) 1 1 d ( )
( ) ) ) )
d ) ( ) d
f t f t
f t A(r A(r c t A(r f t c t A(r
2
2
2 2
1 1 d ( )
) ) ( ) d
f t k
f t c t
A(r
A(r
2
2 2 2 2
2
d ( )
) ); ( ) ( )
d
k
k k k k k
k f t k c f t f t
t
A (r A (r
The fields calculated from the vector potential should satisfy the boundary conditions at the walls of the cavity.
The eigen-value problem has nontrivial solutions for the vector potential only for discrete values of k:
Each k specifies a resonant frequency
The solutions of the eigen-value problem can be normalized as
2
1 2
) ) , ,...
k k k k k k
A (r A (r
2
2 2
d ( ) d ( )
k
k k k
f t f t k c
t
d 1k V
V
A r 2and it can be shown that for normalized vector potentials
The normalized solutions of the eigenvalue problem define the
“cavity modes”. The cavity modes form a complete ortho-
normal set, and the general solution can be given as the sum of orthonormal modes
The electric and magnetic fields of a mode are , ) k ( ) k ).
k
t
f tA(r A (r
k ( )
d 2V
V k
A r 2d ( )
, ) ); , ) ( ) ( ).
d
k
k k k k k
t f t t f t
t
E (r A (r B (r A r
The electric and magnetic energies stored in a mode can be calculated as
Because of the normalizations
2 2
0
0
1 1
( , ) d ; , ) d .
2 2
Ek k Mk k
V V
E t V E t V
E r
B (r
2 2 2
0 0
2 2 2 2
0 0
1 1
( ) ) d ( )
2 2
1 ( ) ( ) d ( )
2 2
Ek k k k
V
Mk k k k
V
E f t V f t
E f t V k f t
A (r
A r
The Lagrangian of a cavity mode is
and the Euler equation belonging to this Lagrangian gives back the dynamic equation of a harmonic oscillator
2
2 2
0
0
1
2 2
k Ek Mk k k
L E E f k f
2 2
2 2
0 0
d ( )
( ) ( )
d
k
k k
f t k
f t f t
t
2. Modal Expansion of the Cavity EM-Field
Example 1: Electromagnetic field between two plan-parallel mirrors (one- dimensional cavity)
The eigen-value problem A (rk ) k2A (rk ) A (rk ) A xk( )j
2
2
1 2
2
( ) ( ) ( ) sin cos
k
k k
d A x
k A x A x c kx c kx
dx
d ( )
, ) ) (0) ( ) 0
d
k
k k k k
t f t A A d
t
E (r A (r
2
1
(0) 0 0
( ) 0 , 1, 2,...
( ) sin
n
k
k n
k
A c
A d kd n k n n
d
A x c n x
d
y
x x d
0 x
2 2 2 1
0 0
( )d sin xd 1 ( ) 2 sin , 1, 2,...
d d
k n
n n
A x x c x A x x n
d d d
2
2 2
d ( ) 2 d ( )
, ) sin ; ( )
d d
k n
n n n n n
f t n x f t
x t f t k c
t d d t
E ( j
, ) sin( ) 2 sin
n n n n
n x
x t F t
d d
E ( j
E1
E2
E
x d x
0 x
Example 2: Electromagnetic field in cylindrical cavities
A waveguide of length L, closed by two metal plates at the ends, form a cylindrical cavity (coaxial, rectangular, circular, elliptical, etc.) We can construct the EM-field in the cavity from the cross-sectional waveguide modes, because in the longitudinal direction the field will be sinusoidal. If the eigen-value of the waveguide mode is k, then the wavelength along the waveguide is
To every waveguide mode there are infinite number of cavity modes with length
2 0 0 / 1
gk 2
k
/ 2, 1, 2,...
L gk
In rectangular cavities with cross section a x b and length , the resonant wavelengths for integers
are
At resonance the stored electric and magnetic energies in the cavity are equal. If we perturb the shape of the cavity or the dielectric material filling the cavity, the equality does not hold anymore at the same frequency. The resonant
frequency of the cavity is perturbed.
1, 2,...; 1, 2,...; 1, 2,...; if 0; 0
m n m n
2 2 2
2
res
res
c
f m n
a b L
3. The quality factor (Q) of cavities
In real cavities the walls dissipate energy, thus resonant vibrations will be attenuated. Let us generate a resonant mode at t = 0, and observe the transient. If the losses are just perturbations, then the field can be approximated as
In a “good” (close to ideal) cavity
The ratio is defined as the “quality factor” (Q) of the cavity:
/ 2 . Q res
res / 2
/ 1.
res
0 ( j )res t
E t E e
