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Defect modes in multisection helical photonic crystals

Fei Wang and Akhlesh Lakhtakia

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Abstract

We examine the two defect modes in four-section helical photonic crystals (HPCs) that comprise three twist defects located at the intersections. The three twist defects are introduced by a single angle ϕt , but they are quantified differently by the jump angles across the successive sections. The two defect modes are localized at the different defect sites and can be either coupled or uncoupled to each other, depending on the value of ϕt . Both defect modes are excited by normally incident plane waves of different circular polarization states as the HPC thickness increases. When the two defect modes are uncoupled to each other, two co-handed reflection holes are present in the Bragg regime for small thickness, but they evolve into two stable cross-handed transmission holes for sufficiently large thickness. When the two defect modes are coupled to each other, however, three co-handed reflection holes appear around the center of the Bragg regime for small thickness, and they evolve into three cross-handed transmission holes as the thickness increases, and eventually all three co-handed transmission holes coalesce into one stable cross-handed transmission hole for sufficiently large thickness. Finally, the simultaneous occurrence of the two types of spectral holes at a single resonance wavelength can be realized for specific values of sample thickness when the two defect modes are uncoupled to each other.

©2005 Optical Society of America

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Figures (8)
Fig. 1.
Fig. 1. Schematic of the boundary value problem involving a four-section HPC with three inter-sectional twist defects. The half-spaces z ≤ 0 and z ≥ 4D are filled with a homogeneous, isotropic, dielectric medium of refractive index nhs . The HPC is excited by a normally incident CP plane wave from the half-space z ≤ 0.

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Fig. 2.
Fig. 2. Spectrums of the co-handed transmittance TRR (solid lines) and the cross-handed transmittance TRR (dashed lines), computed for a four-section HPC with εa = 2.62, εb = 3.18, εc = 2.72, χ = π/6, (thereby εd = 3.02), h = 1, ϕt = π/3, and Ω. = 200 nm. The refractive index n hs = ( ε c + ε d ) 2 was chosen to diminish the index-mismatch across the boundaries. [5] The values of vD , are (a) 10, (b) 20, (c) 40, and (d) 60. The transmittance spectrums of the four-section HPC exhibit two types of spectral holes in the Bragg regime λ 0 ∈ (660,695) nm of a defect-free HPC. Two reflection holes in the spectrum of Trr emerge at the resonance wavelengths λ 01 < λ 02 when VD is relatively small. As VD increases, the two co-handed reflection holes vanish and are replaced by two transmission holes in the spectrum of TLL when VD is sufficiently large. Also, the simultaneous occurrence of both types of spectral holes at λ 02 is observable in (c).

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Fig. 3.
Fig. 3. The relation of VD to |π/2 - ϕt | for the simultaneous occurrence of a co-handed reflection hole and a cross-handed transmission hole at the same wavelength (λ 02 ) in the four-section HPC. The solid line marked with diamond symbols is plotted for 0 < ϕt < π/2, while the dashed line marked with box symbols is plotted for π/2 < ϕt < π. See Fig. 2 for other parameters.

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Fig. 4.
Fig. 4. Distribution of the normalized energy density of the electromagnetic fields inside the four-section HPC with ϕt = π/3, computed for the resonance wavelengths (a1)–(a4) λ 0 = λ 02 and (b1)–(b4) λ 0 = λ 01 . The solid lines are for co-handed CP planewave incidence, while the dotted lines are for cross-handed CP planewave incidence. The values of VD are (a1, b1) 10, (a2, b2) 20, (a3, b3) 40, and (a4, b4) 60. The defect mode coupled to the ϕt -twist defect is evinced in (a1)–(a4) by the peaks at the sites of the two ϕt -twist defects, while the defect mode coupled to the - ϕt -twist defect is evinced in (b1)–(b4) by the peak at the site of the - ϕt -twist defect. The two defect modes are uncoupled to each other. See Fig. 2 for other parameters.

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Fig. 5.
Fig. 5. Same as Fig. 2, but for ϕt = π/2. The transmittance spectrum in (a) exhibits three resonance wavelengths: λ 0_ , λ0Br, and λ 0+ . As vD increases in (b) and (c), the three resonance wavelengths approach each other. Eventually, only one cross-handed transmission hole, located at λ0Br = 2n̄Ω, is present for sufficient large VD in (d).

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Fig. 6.
Fig. 6. Same as Fig. 4, but for ϕt = π/2 and the resonance wavelengths (a1–a4) λ 0 = λ0Br and(b1)–(b4) λ 0 = λ 0± . The defect mode coupled to the ϕt -twist defect is shown in (a1)–(a4) by the considerable peaks at the sites of the two ϕt -twist defects, while the defect mode coupled to all three twist defects (or the ±ϕt -twist defects) is shown in (b1)–(b4)bythe considerable peaks at the sites of all three twist defects. The two defect modes are coupled to each other. It is also noted that all three resonance wavelengths λ 0± and λ0Br are actually located in a single cross-handed transmission hole for sufficiently large VD , such as VD = 60. See Fig. 5 for other parameters.

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Fig. 7.
Fig. 7. Spectrums of the co-handed transmittance TRR (solid lines) and the cross-handed transmittance TLL (dashed lines), computed for a six-section HPC with five intersectional twist defects that are quantified by the jump angles φj+1 - φj = (-1)j+1 ϕtfor 1 ≤ j ≤ 5. The angle フt = π/4, and the values of VD are (a) 10, (b) 20, (c) 42.5, and (d) 50. Three co-handed reflection holes emerge at the resonance wavelengths λ 01 (1) < λ 02 when VD is relatively small. As VD increases, all three co-handed reflection holes vanish and are replaced by two cross-handed transmission holes at the resonance wavelengths λ 01 < λ 02 when VD is sufficiently large. Two defect modes have been identified accordingly: One is coupled to the ϕt -twist defect so that it is responsible for the spectral hole at λ 02 ; the other is coupled to the - ϕt -twist defect so that it is responsible for the spectral hole(s) at λ 01 (λ 01 (1) and λ 01 2)). Therefore, the two defect modes are uncoupled to each other. The simultaneous occurrence of the two types of spectral holes at the same resonance wavelength (λ 02 ) is also observable in (c). See Fig. 2 for other parameters.

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Fig. 8.
Fig. 8. Same as Fig. 7, but for ϕt = π/2. Five co-handed reflection holes emerge at the resonance wavelengths λ0−(1) < λ0−(2) < λ 0+ Br < λ 0+ (1) λ λ 0+ (2) when vD is relatively small. As VD increases, all five co-handed reflection holes wane and are replaced by five cross-handed transmission holes, and eventually they coalesce into one stable cross-handed transmission hole for sufficiently large VD . Two defect modes have also been identified: One is coupled to the ϕt -twist defect so that it is responsible for the spectral hole at ϕt ; the other is coupled to both the ϕt - and the - ϕt -twist defects so that it is responsible for the other four spectral holes at λ 0± (2) and λ 0± (2). Therefore, the two defect modes are coupled to each other, which leads to the degeneracy of all the spectral holes for sufficiently large vD . It is also true that the two defect modes are actually localized in a single cross-handed transmission hole for sufficiently large VD , such as vD = 50.

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Tables (1)

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Table 1. The value of δϕt for various values of vD calculated by (41). See Fig. 2 for other parameters.

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Equations (44)

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ε ¯ ¯ ( z ) = S ¯ ¯ z ( z ) · S ¯ ¯ y ( χ ) [ ε a u ̅ z u ̅ z + ε b u ̅ x u ̅ x + ε c u ̅ y u ̅ y ] · S ¯ ¯ y − 1 ( χ ) · S ¯ ¯ z − 1 ( z ) , 0 < z < 4 D .
S ¯ ¯ y ( χ ) = u ̅ y u ̅ y + ( u ̅ x u ̅ x + u ̅ z u ̅ z ) cos χ + ( u ̅ z u ̅ x − u ̅ x u ̅ z ) sin χ
S ¯ ¯ z ( z ) = ( u ̅ x u ̅ x + u ̅ y u ̅ y ) cos [ π Ω z + φ j ] + u ̅ z u ̅ z + h ( u y ̅ u x ̅ − u x ̅ u y ̅ ) sin [ π Ω z + φ j ] ,
( j − 1 ) D < z < jD , 1 ≤ j ≤ 4 ,
φ 1 = φ 3 = 0 φ 2 = φ 4 = ϕ t } ;
φ 2 − φ 1 = ϕ t φ 3 − φ 2 = − ϕ t φ 4 − φ 3 = ϕ t } .
ε d = ε a ε b ε a cos 2 χ + ε b sin 2 χ
v D = D 2 Ω .
E ̅ inc ( z ) = ( a L u ̅ + + a R u ̅ − ) e ik 0 n hs z , z ≤ 0 ,
E ̅ ref ( z ) = ( r L u ̅ − + r R u ̅ + ) e ik 0 n hs z , z ≤ 0 ,
E ̅ trs ( z ) = ( t L u ̅ + + t R u ̅ − ) e i k 0 n hs ( z − 4 D ) , z ≥ 4 D ,
[ f ̅ exit ] = [ M ¯ ¯ ] [ f ̅ entry ]
[ f ̅ entry ] = 1 2 [ ( r R + r L ) + ( a L + a R ) i [ ( r R − r L ) + ( a L − a R ) ] i [ ( r R − r L ) − ( a L − a R ) ] n hs η 0 − [ ( r R + r L ) − ( a L + a R ) ] n hs η 0 ]
[ f ̅ exit ] = 1 2 [ ( t L + t R ) i ( t L − t R ) -i ( t L − t R ) n hs η 0 ( t L + t R ) n hs η 0 ] ,
[ M ¯ ¯ ] = [ M ¯ ¯ 4 ] [ M ¯ ¯ 3 ] [ M ¯ ¯ 2 ] [ M ¯ ¯ 1 ] ,
[ M ¯ ¯ j ] = [ B ¯ ¯ ( 2 jπv D + φ j ) ] exp ( i [ P ¯ ¯ ] D ) [ B ¯ ¯ ( 2 ( j − 1 ) πv D + φ j ) ] − 1 , 1 ≤ j ≤ 4 ,
[ P ¯ ¯ ] = [ 0 − ih π Ω 0 2 π η 0 λ 0 ih π Ω 0 − 2 π η 0 λ 0 0 0 − 2 π λ 0 η 0 ε c 0 − ih π Ω 2 π λ 0 η 0 ε d 0 ih π Ω 0 ] ,
[ B ¯ ¯ ( φ ) ] = [ cos φ − h sin φ 0 0 h sin φ cos φ 0 0 0 0 cos φ − h sin φ 0 0 h sin φ cos φ ] .
R uv = ∣ r u ∣ 2 ∣ a v ∣ 2 T uv = ∣ t u ∣ 2 ∣ a v ∣ 2 } , u , v = L , R .
n hs = ( ε c + ε d ) 2 .
γ ( z ) = ( 1 2 ) Re [ ε 0 E ̅ ( z ) · ε ¯ ¯ * ( z ) · E ̅ * ( z ) + μ 0 H ̅ ( z ) · H ̅ * ( z ) ]
[ E ˜ ̅ exit ] = [ W ¯ ¯ 0 ] [ E ˜ entry ̅ ] ,
[ E ˜ ̅ entry ] = [ E L + , E R + , E L − E L − ] z entry T ,
[ E ˜ ̅ exit ] = [ E L + , E R + , E L − E L − ] z exit T ,
[ W ¯ ¯ 0 ] = [ P − 0 0 Q − 0 P + Q + 0 0 Q + * P + * 0 Q − * 0 0 P − * ] .
P ± = e ± ihπD Ω [ cosh ( Δ ∓ D ) + i ( k ∓ hπ Ω ) Δ ∓ sinh ( Δ ∓ D ) ] ,
Q ± = e ± ihπ Ω [ ik δ Δ ∓ sinh ( Δ ∓ D ) ] ,
k = k 0 n ̄ k δ = kb n = ( ε c + ε d ) 2 b = ( ε d − ε c ) 2 ( ε c + ε d ) Δ ± = k δ 2 − ( k±hπ Ω ) 2 } .
[ W ¯ ¯ hpc ] = [ W ¯ ¯ ϕt ] [ W ¯ ¯ 0 ] [ W ¯ ¯ ϕt ] [ W ¯ ¯ 0 ] ,
[ W ¯ ¯ ϕ t ] = [ ℜ ¯ ¯ ( ϕ t ) ] [ W ¯ ¯ 0 ] [ ℜ ¯ ¯ ( ϕ t ) ] − 1 .
[ ℜ ¯ ¯ ( ϕ t ) ] = [ [ Y ¯ ¯ ] † [ R ¯ ¯ ( ϕ t ) ] [ Y ¯ ¯ ] [ 0 ¯ ¯ ] [ 0 ¯ ¯ ] [ Y ¯ ¯ ] † [ R ¯ ¯ ( ϕ t ) ] [ Y ¯ ¯ ] ] .
[ ℜ ¯ ¯ ( ϕ ) ] = [ cos ϕ − sin ϕ sin ϕ cos ϕ ]
[ Y ¯ ¯ ] = 1 2 [ 1 1 i − i ] ;
P ˜ + + e i 2 ϕ t P ˜ + * = 0
P ˜ + 2 + ( P ˜ + * ) 2 + 2 ∣ Q ˜ + ∣ 2 cos 2 ϕ t = 0 ,
P ˜ + = P + δ h , 1 + P − δ h , − 1 Q ˜ + = Q + δ h , 1 + Q − δ h , − 1 }
β = k − π Ω k δ .
λ 0 Br = 2 n ̄ Ω
β 0 ≈ − sinn ( b ) cos ϕ t ;
β ± ≈ ±sign ( b ) cos 2 ϕ t + 1 2 sinh 2 ( 2 π ∣ b ∣ v D ) .
δ ϕ t = 1 2 sinh ( 2 π ∣ b ∣ v D ) .
δ ϕ t ≈ 2 e − 2 π ∣ b ∣ v D .
λ 0 1 ≈ 2 n ̄ Ω [ 1 + ∣ b ∣ cos ( π − ϕ t ) ] = 2 n ̄ Ω ( 1 − ∣ b ∣ cos ϕ t ) = λ 0 Br ( 1 − ∣ b ∣ cos ϕ t )
λ 0 2 ≈ 2 n ̄ Ω ( 1 + ∣ b ∣ cos ϕ t ) = λ 0 Br ( 1 + ∣ b ∣ cos ϕ t ) ,
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