Investigation of the defect state in hexagonal photonic crystals 作者：鄭麗蓮 系級：光電學系四年級 學號：D9636273 開課老師：蔡雅芝 課程名稱：專題研究(二) 開課系所：光電學系 開課學年： 99 學年度 第 1 學期
(plane wave expansion method)
(Supercell) (photonic crystal)
Abstract
We employed the plane-wave expansion method and the concept of supercell to study the defect state in two dimension photonic crystal of hexagonal lattice. The defect is defined by a different radius or dielectric constant in an otherwise perfect photonic crystal. We calculated the band structures to see how the defect modes evolve with these parameters. The density of states was also worked out to estimate the corresponding Q factor, which can also be perceived in the localized field distribution. These approaches can be easily adapted to systems containing an arbitrary defect. They can be used to analyze the photonic crystal fiber after being extended to the off-plane calculations in the future.
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? ?? ? [1] (opal) (seamouse) (hexagonal) [2] Fig.1 (a) (b)
1989 Yablonovitch Gmitter Al2O3
[3]
(Density of state)
Q (quality factor) (transmission spectra)
[4] (Photonic crystal fiber, PCF)[5]
( )
(defect)
( )
(reciprocal lattice space)
(Brillouin zone) (Bloch function)
(Fourier transform)[6]
( Fig.2[7] )
(plane
wave expansion method, PWE)[8] (transfer matrix
method)[9] (finite difference time domain method,
FDTD)[10] [11]
(Maxwell’s equations) MKS 0 ) , ( = ⋅ ∇ D rv t v (1) 0 ) , ( = ⋅ ∇ B rv t v (2) t t r B t r E ∂ ∂ − = × ∇ ( , ) ( , ) v v v v (3) t t r D t r H ∂ ∂ = × ∇ ( , ) ( , ) v v v v (4) E v H v D v B v
(permittivity of free space) ε
( )
rv,ω (dielectric function)( )
ω ε rv,( )
r,ω µ0H( )
r,ω B v v v v = (5)( )
r,ω ε0ε( ) ( )
r E r,ω D v v v v v = (6){
( )}
( ) ) ( 1 2 2 r E c r E r v v v v v ω ε ∇× ∇× = (7) ) ( ) ( ) ( 1 2 2 r H c r H r v v v v v ω ε = × ∇ × ∇ (8) ) ( ) (r T rv v v ε ε + = (9) 3 3 2 2 1 1a u a u a u T v v v v + + = ai v i u ) ( 1 ) ( 1 r T r v v v ε ε + = (10) 1/ε(rv)∑
⋅ = G r G i G r) ( )exp( ) ( 1 v v v v κ ε (11) 3 3 2 2 1 1b v b v b v G v v v v + + = bi v av b v k j i k j i a a a a a b v v v v v v × ⋅ × =2π (12)(G) v κ r d r G i r V G V v v v v v ) exp( ) ( 1 1 ) ( 0 0 ⋅ − ∫ = ε κ (13) 0 V (G) v κ (structure factor) (Bloch Theorem)
∑
+ = G r G k i G E r E( ) ( )exp[ ( )v] v v v v v v (14)∑
+ = G r G k i G H r H( ) ( )exp[( )v] v v v v v v (15) (11) (14) (15) (7) (8) ) ( )} ( ) {( ) ( ) ( 2 2 G E c G E G k G k G G kn kn n k G v v v v v v v v v v v v v ω κ − ′ + ′ × + ′ × ′ = −∑
′ (16) ) ( )} ( ) {( ) ( ) ( 2 2 G H c G H G k G k G G kn kn kn G v v v v v v v v v v v v v ω κ − ′ + ′ × + ′ × ′ = −∑
′ (17) k v(first Brillouin zone) n (band
index) (14) (15) (equivalent) 2 2 / c n k v ω xy ) (rv ε z E(rv) v ) (r H v v z
) ( // r E v v ) ( // r H v v (3) (4) (5) (6)
( )
// 2 2 // 2 2 2 2 // ) ( ) ( 1 r E c r E y x r z z v v v ω ε = ∂ ∂ + ∂ ∂ − (18)( )
( )
2( )
// 2 // // // // ) ( 1 1 ) ( 1 r H c r H y r y x r x r z z v v v v v ω ε ε ε = ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ − (19) (18) (19) ) ( ) ( ) ( 2 2 // 2 // // // // E G c G E G k G G z z G v v v v v v v v v ω κ − ′ + ′ ′ =∑
′ (20)(
) (
)
( ) ( ) ) ( 2 2 // // // // // // // E G c G H G k G k G G z z G v v v v v v v v v v v ω κ − ′ + ⋅ + ′ ′ =∑
′ (21) // k v // G v xy // k v [11] a r rd()
a
ε εb εd Λ
(lattice constant) a Fig.3
Fig.4(a) 1 2 3 ( , 0), ( , ) 2 2 a ar = a ar = a (22) ) 3 2 , 0 ( 2 ), 3 1 , 1 ( 2 2 1 a b a b = π − = π r v (23) Fig.4(b) ΓMK [12] Fig.3 36
) (G// v κ ( ) Fig.5 Fig.5 10 10× (a) (b) Fig.4(a) (b) ΓMK [13] Fig.5
fast method [14] ) (rv ε (13) r d r G i r V G V v v v v v ) exp( ) ( 1 ) ( 0 0 ⋅ − ∫ = ε η (24) ) (G v η ) ( ) ( 1 G G v v κ η− = (25) ) (G v κ (20) (21) [12]
ρ (density of photon states, DOS)
[15]
( )
(
)
∑
− = BZ k 1 v ω ω δ ρ (26) (24) :187750 30100 0.054% 1687 967 0.29% Fig.6(a) 1.590~2.623 Fig.6(b) 6 1/6 6
rd = 0 rd = 0.5 Fig.7 [4] rd 0.38 1 = b ε εb =20
Fig.8 Fig.7 Fig.8
(donor-type) (acceptor- type) (a) (b) Fig.6 (a) ra =0.2 13 = a ε εb =1 (b) 3 . 0 = d r N o rm a liz e d f re q u e n c y ( ωa/2 πc) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 K M Γ K N o rm a liz e d f re q u e n c y ( ω a /2 π c ) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 K M Γ K
Fig.8 εd εd 0 2 4 6 8 10 12 14 16 18 20 N o rm a liz e d f re q u e n c y ( ωa/2 πc) 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0
Band
Band
Fig.7 rd rd / Λ 0.0 0.1 0.2 0.3 0.4 0.5 N o rm a liz e d f re q u e n c y ( ω a /2 π c ) 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0Band
Band
(DOS) Fig.9 (peak) (Fig.10 ) (rd = 0.08 ) (rd = 0.17 ) N o rm a liz e d f re q u e n c y ( ω a /2 π c ) 0.0 0.5 1.0 1.5 2.0 2.5 K M Γ K
peak
Fig.9 rd = 0.08 DOS[a.v.]6 DOS Fig.10 (a) rd=0.08 Γ (b) rd=0.17 Γ X Data -0.4 -0.2 0.0 0.2 0.4 Y D a ta -0.4 -0.2 0.0 0.2 0.4 (a) -3 -2 -1 0 1 2 3 4 5 6 7 8 9 X Data -0.4 -0.2 0.0 0.2 0.4 Y D a ta -0.4 -0.2 0.0 0.2 0.4 (b)
[1] , “ ”, ( ) 1999 8
[2] , http://nano.nchc.org.tw/photonic/photonic.php
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[4] Kazuaki.Sakoda, Optical Properties of Photonic Crystals, Chapter5, Springer, 2001.
[5] Hollow-Core, Photonic Crystal Fibers, THORLABS. [6] Charles Kittel, Introduction to Solid State Physics, 2005.
[7] , “ ”, , , 2010
[8] K. M. Ho et al., "Existence of a photonic gap in periodic dielectric structures,"
Phys. Rev. Lett. 65, 3152 (1990).
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Phys. Rev. Lett. 69, 2638 (1992).
[10] J. D. Jaonnopoulos et al., "Photonic crystals," Solid State Commu. 102, 165 (1997).
[11] , “ ”, , ,
94 7
[12] , , ,
97 7
[13] M. Fujita et al, “Simultaneous Inhibition and Redistribution of Spontaneous Light Emission in Photonic Crystals”, Science 308, 1296 (2004).
[14] K. M. Ho, C. T. Chan, and C. M. Soukoulis, Phys. Rev. Lett. 65, 3152 (1990). [15] Busch Kurt, Lölkes Stefan, Wehrspohn Ralf B., Föll Helmut (Hrsg.), “Photonic
