Transcript PowerPoint 簡報

Superprism effect and EBG
antenna applications
Reporter:楊岡昀
Date:2010/5/20
Outlines
•
•
•
•
Floquet-Bloch transform
Some practical structure of PC
Superprism effect
Antenna applications
Floquet-Bloch transform
• Some hypotheses:
 U  x  dx dx
1
2

• Size of PC is much larger than the wavelength
• The surface of the PC can be considered as a
plane
The Floquet-Bloch theorem
• Any function satisfying  U  x  dx dx   can be
represented as F  x    F  k , x  e dk
d d


F
k
,
x

F
x

n
d

n
d
e





• And
 2 
• This is Floquet-Bloch transform
1
2
jk// x
//
//
B//
1
//
 jk//  n1d1  n2 d 2
2
2
n1 , n2 Z
1 1
2
2
Proof
 
F  x    Fˆ k// e jk// x dk// 

 
F k// , x 
n1 , n2Z

 
B// n1 , n2Z


j k n d  n d   x
Fˆ k//  n1d1  n2d2 e // 1 1 2 2 dk//   F k// , x e jk// x dk//


B//
j  n d  n d   x
Fˆ k//  n1d1  n2d 2 e 1 1 2 2
• Use reverse Fourier transform we have


F k// , x 
d1  d 2
 2 
2
 F x  n d
n1 , n2 Z
1 1

 n2 d 2 e

 jk//  n1d1  n2 d 2


Field coupling
• Consider a incident wave Ei and refraction
wave Er and transmission wave Et
Ei  E0eit k3x3 k// x//
Er  E '0 ei 't k '3 x3 k '// x//
   '   ''
k//  k '//  k ''//
• Dispersion relation:
  f  k// , k3 
Et  E ''0 ei ''t k ''3 x3 k ''// x//
Field coupling
Some practical structure of PC
• The blue triangle in the
figure is the first
Brillouin’s zone
Brillouin’s zone
• For 1D condition, Ɛ(x+a)=Ɛ(x) the first




k

Brillouin’s zone is a a
• Extend this concept to 3D
Some practical structure of PC
Some practical structure of PC
Superprism effect
• Group velocity effect
• Group velocity change
sharply
Superprism effect
• Chromatic dispersion
effect
• The amp also change
with the freq
A  k1  
W 
e
2 
 k1  k0 sin  2 W 2
4
Antenna applications
• We can use PC to
control the propagation
direction of antenna
Antenna applications
• The experiment result :
More applications of PC
• PCF(Photonic crystal
fibers)
• Invisible cloak
References
• METAMATERIALS Physics and Engineering
Explorations, NADER ENGHETA/RICHARD W.
ZIOLKOWSKI
• Semiconductor Physics and Devices, D.A.
Neamen
• Handout of the lesson 光子晶體 (Photonic
Crystals), which is held by Professor 邱奕鵬