Strongly Localized Photonic Mode in 2D Periodic Structure Without Bandgap
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1 Strongly Localized Photonic Mode in D Periodic Structure Without Bandgap V. M. APALKOV M. E. RAIKH Physics Department, University of Utah The work was supported by: the Army Research Office under Grant No. DAAD ; the Petroleum Research Fund under Grant No AC6NSF
2 Localization of Electrons Disordered Media Length scale: - mean free path, l - a step of diffusion motion - de Broglie wave length, π λ = me diffusion l π Localization: l ~1 λ (Ioffe-Regel criterion) tail state P.W. Anderson, Absence of Diffusion in Certain Random Lattices, Phys. Rev. 19, 149 (1958) Potential Well Length scale of the potential, a Localization: λ ~ ψ ( x) + V( x) ψ( x) = Eψ( x) x a a V( x) E< x localized state
3 Schroedinger-Maxwell Analogy Electrons ψ( r) + V( r) ψ( r) = Eψ( r) Photons Er ( ) δε( rer ) ( ) = εer ( ) c c ε c δε c The electron can have a negative energy, can be trapped in deep potentials Frequency-dependent potential Energy is positive, > δε (unlike plasmons) ε
4 Electrons Localization Criteria for Electrons and Photons V( rv ) ( r ) =Γδ( r r ) Golden Rule = ImEk = π ( V( r) ) δ( E kk k E, k ) τ ~ ΓgE ( ) ge ( )~ E 1/ localization criterion for low enough E photons l 4 ~ λ l (Rayleigh law) electron density of states l 1 E ~ ~ λ ΓgE ( ) ( λ l) E 1/ is satisfied λ k l Photons δε( r) δε( r ) =Γδ( r r ) g electrons = Im( k ) = 3 k k kk, = π δε( r) δ ε ε c c c 3 g( ) ~ Γ ε c ( )~ localization criterion satisfied l ~const localized states, photon density of states l ε 1 ~ ~ λ Γ g( ) λ l ( λ l) 3 can not be
5 Localization Criterion for Photons g Photons ( )~ λ Γ ~ g( ) l ε free space value Scattering resonances (Mie resonances, Bragg resonances) strongly modify photon density of states l g weak disorder Rayleigh geometric ray optics l 4 ~ λ strong scattering (resonance) λ Bragg Resonance Photonic Crystal pseudogap, strong localization Geometric optics S. John, Phys. Rev. Lett. 58, 486 (1987) Rayleigh frequency
6 Photon Localization: Photonic Band Gap Materials Two Fundamental Optical Principles: Localization of Light - S. John, Phys. Rev. Lett. 58, 486 (1987) Inhibition of Spontaneous Emission - E. Yablonovitch, Phys. Rev. Lett. 58, 59 (1987) g pseudogap, strong localization no emission, if lies in the gap. δε Photonic crystal periodic modulation of dielectric constant 1D gap a lattice constant R π / a π / a k
7 Photonic Band Gap Materials: Complete Bandgap Complete bandgap: The frequency domain where the propagation of light is completely forbidden Requires high contrast of the dielectric constant, > 1:1 Difficult to achieve g Strongly localized photon modes Disorder-induced localized mode g Localized modes Point-defect induced localized in-gap mode acts as a high-q resonator g
8 Complete Bandgap: Computational Demonstration 3D 9 citations in PR.. Periodic lattice of dielectric spheres Diamond structure complete photonic bandgap Contrast of dielectric constant > 4:1 The MIT Photonic-Bands package:
9 Photonic Crystals 1D D 3D specially designed specially designed specially designed Natural assembly of colloidal microspheres: - opals - inverted opals - structural defects destroy the bandgap
10 Synthetic Opals: Thin Layers 3D Silica (SiO ) microspheres - sediment by gravity into close-packed fcc lattice Nature (London) 414, 89 (1) Silica opal contrast ~ 1.5:1 Thick layers structural defects - filled with silicon (Si) - silica template removed by wet etching 7 layers Theory black Experiment red/blue V.N. Astratov, et. al., PRB 66, () Inverted silicon opal contrast ~ 1:1
11 Complete Bandgap: Inverted Opals micrometers 3D Nature (London) 45, 437 () fcc lattice of air sphere in silicon: contrast ~ 1:1 (theoretical threshold for complete bandgap ~ 8:1) calculations experiment (many structural defects which destroy bandgap)
12 Photonic Band Gap Materials: Specially Designed 3D S. Fan, et. al., Appl. Phys. Lett. 65, 1466 (1994) M. E. Povinelli, et. al., Phys. Rev. B 64, (1) O. Toader, S. John, Science, 9, 1133 (1) A. Chutinan, S. John, and O. Toader, Phys. Rev. Lett. 9, 1391 (3)
13 Photonic Band Gap Materials D Two dimensionally periodic structures of finite height (photonic-crystal slabs) Light is confined by a combination of an in-plane photonic band gap and out-of-plane index guiding. Advantage easy to manufacture complete band gap: contrast ~ 1:1 Defect-induced localized in-gap mode acts as a high-q resonator M. Meier, et. al., Appl. Phys. Lett. 74, 7 (1999) g Localized modes
14 Defects in Periodical Structures a 1D D Point-like defect na Phase slip - linear defect (periodicity interruption) na
15 Point Defect-Induced Localized Mode in D Science 84, 1819 (1999). Two-Dimensional Photonic Band-Gap Defect Mode Laser g O. Painter 1, R. K. Lee 1, A. Scherer 1, A. Yariv 1, J. D. O Brien, P. D. Dapkus, I. Kim 1 California Institute of Technology; University of Southern California. Defect-induced localized state inside the gap acts as a high-q resonator InGaAs 515 nm Power laser line spontaneous emission spectrum 48 nm λ Hexagonal array of air holes (radius 18 nm) High dielectric contrast 1:1
16 D Photonic Crystal: Weak Contrast of Dielectric Constant (solid organic gain media) dielectric contrast 1.7 : 1.46 : 1 no complete bandgap organic substrate air
17 Defects in Periodical Structures a 1D D Point-like defect na Phase slip - linear defect (periodicity interruption) na
18 Linear Defect-Phase Slip a 1D case a d phase slip (stacking fault) d < a δε( x)~cos( π x a+ φ1) δε( x)~cos( π x a+ φ) φ φ = πd a 1
19 Phase Slip Localized State, 1D a d δε( x) = cos( π x a + φ ) 1 δε( x) = cos( π x a + φ ) x < x > E ( x) δε( xe ) ( x) = εe ( x) c c Solution localized mode: E x e e e γ x iσx iσx ( ) = ( β + µ ) gap Ω = ε 3/ cσ σ = π / a Ω= = cσ ε σ σ k Ω=± 1 φ φ1 Ωcos γ = 1 Ω φ φ1 sin 1/ сε
20 Phase Slip Localized State, 1D a d localized mode, d =.5 a ( ) γ x E cos( ) x e σ x gap Ω σ σ k Frequency (energy) of localized mode, 1 φ φ1 Ω= =± Ωcos Ω spatial extension of localized mode, 1 с sin φ φ = γ Ω 1 1 Ω ( π /-phase slip) d =.5 a 1 γ Ω φ φ 1 d =.5 a ( π /-phase slip) c Ω.5 1 d/a
21 Phase Slip Localized Mode, 1D Nature (London) 39, 143 (1997) Dielectric contrast - 1:1 Transmission λ
22 Localization of Photons in D Crystals with Incomplete Bandgap V.M. Apalkov and M.E. Raikh, Phys. Rev. Lett. 9, 5391 (3) Strong localization of a photon can be achieved in D photonic crystals with low contrast of dielectric constant A long-living photon mode exists only in D photonic crystal with a certain MAGIC GEOMETRY of a unit cell Conventional approach g strongly localized mode g Our result complete bandgap no bandgap
23 Two Phase Slips, D low contrast of dielectric constant two phase slips localized mode y x
24 Localized Mode k y delocalized modes σ = const localized mode σ k x g gap E x y e e x y γ x γ y (, ) = cos( σ )cos( σ ) no bandgap E E = const y x
25 Strongly Localized Mode: Magic Geometry of a Unit Cell Fourier harmonics of ε( x, y) : radius of cylinders = J ( σr) J ( 3/ σr) 11 1 surface of equal frequency 1 k y 1 σ 11 σ delocalized modes k x π σ = a localized mode LEAKAGE 11 = 3/ σr u 3.8 Rс с.43a Q-factor: J ( u ) = 1 11 Magic geometry of a unit cell: Q = =? Im gap R 11 =.3 R
26 Magic Crystal: Localized Mode low contrast of dielectric constant y localized mode x two phase slips Q-factor: Q = =? Im
27 D Photonic Crystal: Weak Contrast of Dielectric Constant (solid organic gain media) dielectric contrast 1.7 : 1.46 : 1 no complete bandgap organic substrate air
28 σ Two Phase Slips - Quasilocalized Mode δε( x, y) = δε( x + na, y+ ma) = cos( σnx)cos( σmy) = δε( x, y) = δε ( x) + δε ( y) + δε ( x, y) = δε ( 1 x ) 1 11 = cos( σ x) + cos( σ y) + cos( σ x)cos( σ y) σ σ k y 1 1 nm, nm, = δε ( x) + δε ( y) + cos( σ x)cos( σ y) + cos( σnx)cos( σmy) 1 11 nm, nm, > ( nm, ) (1,1) δε ( 11 x, y ) σ separable part δε ( y ) k x δε ( 1 x ) + phase slip δε ( y ) + phase slip } } δε ( 11 x, y ) a/ a/ 11 = a/ a/ a/ a/ 1 = a/ a/ - localization in x-direction - localization in y-direction - destroys localization dx dyδε( x, ye ) e dx dyδε( x, ye ) iσx iσ y iσx π σ = a
29 Q-factor of Quasilocalized Mode: Higher-Order Corrections E( x, y) δε( x, ye ) ( x, y) = εe( x, y) c c с δε( x, y) = U ( x, y) = cos( σnx)cos( σmy) nm, nm, nm, > с nm, > ( ) ps ( ) E( x, y) U ( x) + U ( y) E( x, y) U ( x, ye ) ( x, y) =κ E( x, y) 1 ( ) ps U x y = U x+ dx x y+ dy y ( ) (, ) sign( ), sign( ) U ( x) = U ( x) ( ps) 1 n, n> U ( y) = U ( y) ( ps), m m> pert HE ˆ ( x, y) Hˆ perte( x, y) U ( x, y) = U ( x, y) ( ps) ( ps) pert nm, nm, > HE ˆ ( x, y) = κ E ( x, y) κ = E H E = (1) pert for magic crystals E Imκ x
30 Q-factor of Quasilocalized Mode: Higher-Order Corrections κ = ˆ ( ˆ ) ˆ = () 1 E Hpert κ H Hpert E ˆ = (, ) ( ps) H pert U1,1 x y E Hˆ E κ pert κ µ µ ( ) κ = κ = π E Hˆ E δ κ κ Im Im () pert µ µ µ () κ = E µ Imκ resonant term for magic crystals x Hˆ Higher-order corrections: pert U E E U ( ps) ( ps) mn, p, p p, p m, n x y x y mnm,,, n p, p p, p 1 1 κ κ x y x y 1 1
31 Q - factor of the Quasilocalized Mode. Q Im R R C δε δε α δε = = + α 1 c 1 ε a ε ε 1. С 11/ uj ( u) = u J1 J ( u ) = u = α 4 3 nm, n+ 1, m+ 1 nm, = uj ( u) nm, > n + n+ m + m α 7 F + F + F nm, n+ 1, m = α uj ( u) nm, > n + n+ m + m F F R F = π J ( q R ) c nm 1 nm c qnm π qnm = n + m a. δε R= Rc + α1 a ε fine tuning of R (Fano resonance) Q ε 6ε =.4 1 C α δε δε max HIGH-Q mode
32 Q - factor of the Quasilocalized Mode Q =.3 ε δε R R c 5δε + 61 a ε Q ~ ε δε 3 δε ε 3 R R c a
33 Weakly disordered media Photonic crystal without bandgap Q=5 Q=1 6
34 Magic Crystal and Localized Mode: Example ε1 1.5 hole, ε = 1 D R
35 Long-living Quasilocalized Mode: Fine Tuning For R=R c 11 = π qnm = n + m a π irq nm cosϕ Fine tuning of the shape of cylinders 11 = and nm = (for all nm, > ) Q = ( ( ) ) nm = δε dϕ rdre θ R ϕ r π J1( qnmr) dϕ irq R Ap e qnmr p π nm = πδε nm R( ϕ ) = R + R Ae p cosϕ+ ipϕ J1( qnmr) nm = πδε R AJ p p( qnmr) = qnmr p p ipϕ
36 Light Localization in 3D Photonic Crystals with Incomplete Bandgap Face Centered Cubic (FCC) lattice (closed packed) introduce three phase slips along the major axes separable part of δε ( x, yz, ) localized mode δε ( x) + δε ( y) + δε ( z) = cos( σ x) + cos( σ y) + cos( σ z) two diagonal components, and, destroy localization magic crystal: 11 = 111 = specific feature of FCC lattice 11 magic crystal: only one condition 111 = composite particles R1 ε1, R1 rdr ε( r) ε sin 3σr = ε, R 111 σ = π R 1 ( ) ( ) R.86R 1 ε 8.5 ε1 1
37 Magic Crystal and Localized Mode: Example ε1 1.5 hole, ε = 1 D R
38 Weakly disordered media Photonic crystal without bandgap Q=5 Q=1 6
39 Photon Localization Disordered medium The main problems: realization of strong enough scattering metallic particles (Mie resonances) semiconductor particles (GaAs, GaP) with very large refractive index observation of photon localization exponential scaling of transmission coefficient rounding of the top of the backscattering cone variance of relative fluctuations absorption Photonic crystals (background medium) strongly modify the photon density of states density of states becomes similar to the electron density of states disorder-induced localized states custom-made defects and in-gap localized modes
40 Linear Defect - Phase Slip: 1D a a d phase slip (stacking fault) d < a (d > a) = xa+ 1 φ d a δε x = π xa+ φ φ1 = π δε( x) cos( π φ ) ( ) cos( ) gap Ω = ε 3/ cσ 1 φ φ1 Ω= =± Ωcos Ω φ φ 1 E x e x ( ) γ x cos( ) σ 1 Ω φ φ1 γ = sin 1/ сε σ σ = π a k Ω d =.5 a (or d = 1.5 a) ( π /-phase slip)
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