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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 , n2Z
B// n1 , n2Z
j k n d n d x
Fˆ k// n1d1 n2d2 e // 1 1 2 2 dk// F k// , x e jk// x dk//
B//
j n d n d x
Fˆ k// n1d1 n2d 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 E0eit 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