To Begin: A Cartoon in 2d

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2 To Begin: A Cartoon in 2d r k scattering planewave r r r r H, ~ e ikx ( ωt) r k = ω/ c= 2π λ

3 To Begin: A Cartoon in 2d r k a planewave r r r r H, ~ e ikx ( ωt) r k = ω/ c= 2π λ for most λ, beam(s) propagate through crystal without scattering (scattering cancels coherently)...but for some λ (~ 2a), no light can propagate: a photonic band gap

4 Photonic Crystals periodic electromagnetic media D 2-D 3-D periodic in one direction periodic in two directions periodic in three directions with photonic band gaps: optical insulators (need a more complex topology)

5 Photonic Crystals periodic electromagnetic media can trap light in cavities 3D Photonic Crystal with Defects and waveguides ( wires ) magical oven mitts for holding and controlling light with photonic band gaps: optical insulators

6 Photonic Crystals periodic electromagnetic media High index of refraction Lo w ind ex of refraction 3D Photonic Crystal But how can we understand such complex systems? Add up the infinite sum of scattering? Ugh!

7 A mystery from the 19th century e conductive material r e + current: r r =σ + conductivity (measured) mean free path (distance) of electrons

8 A mystery from the 19th century e r e crystalline conductor (e.g. copper) s of periods! current: r r =σ conductivity (measured) mean free path (distance) of electrons

9 A mystery solved 1 electrons are waves (quantum mechanics) 2 waves in a periodic medium can propagate without scattering: Bloch s Theorem (1d: Floquet s) The foundations do not depend on the specific wave equation.

10 Time to Analyze the Cartoon r k a planewave r r r r H, ~ e ikx ( ωt) r k = ω/ c= 2π λ for most λ, beam(s) propagate through crystal without scattering (scattering cancels coherently)...but for some λ (~ 2a), no light can propagate: a photonic band gap

11 algebraic interlude algebraic interlude completed I hope you were taking notes* [ *if not, see e.g.: oannopoulos, Meade, and Winn, Photonic Crystals: Molding the Flow of Light ]

12 a 2d periodicity 1 frequency ω (2πc/a) = a / λ Photonic Band Gap TM bands irreducible Brillouin zone M r k Γ X 0 Γ X M Γ TM H

13 2d periodicity z (+ 90 rotated version) Photonic Band Gap z 0.2 TM bands Γ X M Γ + TM H

14 a 2d periodicity 1 frequency ω (2πc/a) = a / λ Photonic Band Gap TM bands T bands irreducible Brillouin zone M r k Γ X 0 Γ X M Γ TM H T H

15 2d photonic crystal: T gap a T gap T bands M K 0.1 TM bands Γ 0 Γ M K Γ

16 I. 3d photonic crystal: complete gap II % gap L' U' X Γ U'' W K' U W' L K z I: rod layer II: hole layer 0 U L Γ X W K [ S. G. ohnson et al., Appl. Phys. Lett. 77, 3490 (2000) ]

17 Photonic Crystals: Periodic Surprises in lectromagnetism A Defective Lecture cavity waveguide

18 The Story So Far a Waves in periodic media can have: propagation with no scattering (conserved k) photonic band gaps (with proper ε function) igenproblem gives simple insight: Bloch form: r r 1 r r r ( + ik ) ( + ik ) H ε ˆΘ r k r r r r r H = e ikx ( ωt) Hr( x) r k r ( ) = ω n k c Hermitian > complete, orthogonal, variational theorem, etc. 2 r H r k k ω band diagram Photonic Band Gap TM bands k

19 Applications of Bulk Crystals using near-band-edge effects Zero group-velocity dω/dk: distributed feedback (DFB) lasers divergent dispersion (i.e. curvature): Superprisms [Kosaka, PRB 58, R10096 (1998).] super-lens Veselago (1968) object im age negative group-velocity or negative curvature ( eff. mass ): Negative refraction, Super-lensing source ima g e negat ive refraction mediu m [ C. Luo et al., Appl. Phys. Lett. 81, 2352 (2002) ]

20 Cavity Modes Help!

21 Cavity Modes finite region > discrete ω

22 Cavity Modes: Smaller Change

23 Cavity Modes: Smaller Change 0.6 Bulk Crystal Band Diagram 0.5 L frequency (c/a) Photonic Band Gap Γ X M Γ r k Γ M X

24 Cavity Modes: Smaller Change Defect Crystal Band Diagram L Defect bands are shifted up (less ε) with discrete k # λ ~ 2 L ( k ~ 2π/ λ) frequency (c/a) Photonic Band Gap escapes: Γ X M Γ r k Γ M X k ~ π / L k not conserved at boundary, so not confined outside gap confined modes

25 Single-Mode Cavity 0.6 Bulk Crystal Band Diagram 0.5 A point defect can push up a single mode from the band edge field decay ~ ω ω 0 curvature frequency (c/a) Photonic Band Gap Γ X M Γ r k Γ M X ω ω 0 (k not conserved)

26 Single -Mode Cavity 0.6 Bulk Crystal Band Diagram 0.5 A point defect can pull down a single mode frequency (c/a) Photonic Band Gap here, doubly-degenerate (two states at same ω) 0 Γ X M Γ r k X Γ M X (k not conserved)

27 Tunable Cavity Modes 0.5 Air Defect frequency (c/a) air bands dielect ric bands Radius of Defect (r/ a) Dielectri c Defect z : monopole dipole

28 Tunable Cavity Modes band #1 at M band #2 at X s z : multiply by exponential decay monopole dipole

29 a Defect Flavors

30 Projected Band Diagrams 1d periodicity conserved k! r k Γ M X not conserved conserved So, plot ω vs. k x only project Brillouin zone onto Γ X: gives continuum of bulk states + discrete guided band(s)

31 Air-waveguide Band Diagram band gap frequency (c/a) states of the bulk crystal wavenumber k (2π/a) any state in the gap cannot couple to bulk crystal > localized

32 (Waveguides don t really need a complete gap) Fabry-Perot waveguide: We ll exploit this later, with photonic-crystal fiber

33 So What?

34 Review: Why no scattering? forbidden by Bloch (k conserved) forbidden by gap (except for finite-crystal tunneling)

35 Benefits of a complete gap broken symmetry > reflections only effectively one-dimensional

36 Lossless Bends [ A. Mekis et al., Phys. Rev. Lett. 77, 3787 (1996) ] symmetry + single-mode + 1d = resonances of 100% transmission

37 Waveguides + Cavities = Devices tunneling Ugh, must we simulate this to get the basic behavior? No! Use coupling-of-modes-in-time (coupled-mode theory) [H. Haus, Waves and Fields in Optoelectronics]

38 algebraic interlude algebraic interlude completed I hope you were taking notes

39 A Menagerie of Devices λ λ 1.55 microns

40 Wide-angle Splitters [ S. Fan et al.,. Opt. Soc. Am. B 18, 162 (2001) ]

41 Waveguide Crossings [ S. G. ohnson et al., Opt. Lett. 23, 1855 (1998) ]

42 Waveguide Crossings x1 throughput empty 3x3 empty 3x3 0 1x10 0 1x10-2 empty 1x1 5x5 5x5 1x1 crosstalk 1x10-4 1x10-6 3x3 5x5 1x10-8 1x frequency (c/a)

43 Channel-Drop Filters [ S. Fan et al., Phys. Rev. Lett. 80, 960 (1998) ]

44 Channel-Drop Filters

45 Cavities + Cavities = Waveguide tunneling coupled-cavity waveguide (CCW/CROW): slow light + zero dispersion [ A. Yariv et al., Opt. Lett. 24, 711 (1999) ]

46 nhancing tunability with slow light [ M. Soljacic et al.,. Opt. Soc. Am. B 19, 2052 (2002) ]

47 Waveguides + Cavities + Nonlinearity Kerr nonlinearity: suppose index n > n + n 2 2

48 Bistable Transmission: Optical Diode [ M. Soljacic et al., Phys. Rev. Rapid Comm. 66, (R) (2002) ] 1.4 x % 20% 1.2 all-optical logic gates, amplification, rectification, switching output power x input power

49 Uh oh, we live in 3d C rod layer A B hole layer (fcc crystal)

50 2d-like defects in 3d [ M. L. Povinelli et al., Phys. Rev. B 64, (2001) ] modify single layer of holes or rods

51 3d projected band diagram 0.5 3D Photonic Crystal 2D Photonic Crystal frequency (c/a) frequency (c/a) TM gap wavevector kx (2π/a) wavevector kx (2π/a)

52 2d-like waveguide mode 3D Photonic Crystal 2D Photonic Crystal x -1

53 2d-like cavity mode

54 The Upshot To design an interesting device, you need only: symmetry + single-mode (usually) + resonance + (ideally) a band gap to forbid losses Oh, and a full Maxwell simulator to get Q parameters, etcetera.

55 Photonic Crystals: Periodic Surprises in lectromagnetism Those Clever xperimentalists Fabrication of Three-Dimensional Crystals

56 The Mother of (almost) All Bandgaps The diamond lattice: fcc (face-centered-cubic) with two atoms per unit cell (primitive) a Recipe for a complete gap: fcc = most-spherical Brillouin zone + diamond bonds = lowest (two) bands can concentrate in lines Image:

57 The First 3d Bandgap Structure K. M. Ho, C. T. Chan, and C. M. Soukoulis, Phys. Rev. Lett. 65, 3152 (1990). frequency (c/a) (c/a) X 11% gap G W U L K U L G X W K X for gap at l = 1.55µm, sphere diameter ~ 330nm overlapping Si spheres MPB tutorial,

58 Make that? Are you crazy? maybe! fabrication schematic carefully stack bcc silica & latex spheres via micromanipulation dissolve latex & sinter (heat and fuse) silica make Si inverse (12% gap) [ F. Garcia-Santamaria et al., APL 79, 2309 (2001) ]

59 Make that? Are you crazy? maybe! [ F. Garcia-Santamaria et al., Adv. Mater. 14 (16), 1144 (2002). ] 5µm 5µm dissolve latex spheres 4-layer [111] silica diamond lattice 6-layer [001] silica diamond lattice

60 Fortunately, there are easier ways.

61 Layer-by-Layer Lithography Fabrication of 2d patterns in Si or GaAs is very advanced (think: Pentium IV, 50 million transistors) inter-layer alignment techniques are only slightly more exotic So, make 3d structure one layer at a time Need a 3d crystal with constant cross-section layers

62 A Layered Structure We ve Seen Already (diamond-like: rods ~ bonds ) C rod layer A B hole layer [ S. G. ohnson et al., Appl. Phys. Lett. 77, 3490 (2000) ] Up to ~ 27% gap for Si/air

63 Making Rods & Holes Simultaneously side view substrate Si top view

64 Making Rods & Holes Simultaneously expose/etch holes A A A A substrate A A A A A A A A A A A A A A A A A A A A A

65 Making Rods & Holes Simultaneously backfill with silica (SiO 2 ) & polish substrate A A A A A A A A A A A A A A A A A A A A A A A A A

66 Making Rods & Holes Simultaneously deposit another Si layer layer 1 A A A A substrate A A A A A A A A A A A A A A A A A A A A A

67 Making Rods & Holes Simultaneously dig more holes offset & overlapping layer 1 B B B B A A A A substrate B B B B A A A A B B B A A A B B B B A A A A B B B A A A B B B B A A A A B B B A A A

68 Making Rods & Holes Simultaneously backfill layer 1 B B B B A A A A substrate B B B B A A A A B B B A A A B B B B A A A A B B B A A A B B B B A A A A B B B A A A

69 Making Rods & Holes Simultaneously etcetera (dissolve silica when done) layer 1 layer 2 substrate layer 3 A A A A C C C C B B B B A A A A one period C A B C A B C A B C A B C A B C A B C A B C C A B C A B C A B C A B C A B C A B C A B C C A B C A B C A B C A B C B C B C B C A A A

70 Making Rods & Holes Simultaneously etcetera layer 2 layer 3 A A A A C C C C one period layer 1 B B B B hole layers A A A A substrate C A B C A B C A B C A B C A B C A B C A B C C A B C A B C A B C A B C A B C A B C A B C C A B C A B C A B C A B C B C B C B C A A A

71 Making Rods & Holes Simultaneously etcetera layer 2 layer 3 A A A A C C C C one period layer 1 B B B B rod layers A A A A substrate C A B C A B C A B C A B C A B C A B C A B C C A B C A B C A B C A B C A B C A B C A B C C A B C A B C A B C A B C B C B C B C A A A

72 A More Realistic Schematic [ M. Qi, H. Smith, MIT ]

73 7-layer -Beam Fabrication [ M. Qi, H. Smith, MIT ]

74 X-ray Interference Lithography [ M. Qi, H. Smith, MIT ] The Good Large area: up to 10x10cm! Cheap ($50k vs. $500k for e-beam) Nearly perfect periodicity High resolution The Ugly Layer alignment still tricky no defects: use e-beam locally non-rectangular more tricky

75 From Rectangular to Hexagonal [ M. Qi, H. Smith, MIT ]

76 an earlier design: (& currently more popular) The Woodpile Crystal [ K. Ho et al., Solid State Comm. 89, 413 (1994) ] [ H. S. Sözüer et al.,. Mod. Opt. 41, 231 (1994) ] (diamond-like, bonds ) Up to ~ 17% gap for Si/air [ Figures from S. Y. Lin et al., Nature 394, 251 (1998) ]

77 1.25 Periods of the Woodpile (4 log layers = 1 period) [ S. Y. Lin et al., Nature 394, 251 (1998) ] Si gap UV Stepper: e-beam mask at ~4x size + UV through mask, focused on substrate Good: high resolution, mass production Bad: expensive ($20 million)

78 Woodpile by Wafer Fusion 2nd substrate + logs, rotated 90 and flipped substrate + first log layer [ S. Noda et al., Science 289, 604 (2000) ]

79 Woodpile by Wafer Fusion fuse wafers together substrate + first log layer [ S. Noda et al., Science 289, 604 (2000) ]

80 Woodpile by Wafer Fusion dissolve upper substrate substrate + first log layer [ S. Noda et al., Science 289, 604 (2000) ]

81 Woodpile by Wafer Fusion double, double, toil and trouble [ S. Noda et al., Science 289, 604 (2000) ]

82 It s only wafer-thin. [ M. Python ] [ S. Noda et al., Science 289, 604 (2000) ]

83 Woodpile Gap from µm [ S. Noda et al., Science 289, 604 (2000) ]

84 Finally, a Defect! [ S. Noda et al., Science 289, 604 (2000) ]

85 Stacking by Micromanipulation [ K. Aoki et al., Appl. Phys. Lett. 81 (17), 3122 (2002) ] microsphere into hole break off suspended layer lift up and move to substrate tap down holes onto spheres spheres enforce alignment goto a;

86 Stacking by Micromanipulation [ K. Aoki et al., Appl. Phys. Lett. 81 (17), 3122 (2002) ]

87 Yes, it works: Gap at ~4µm [ K. Aoki et al., Nature Materials 2 (2), 117 (2003) ] 20 layers 50nm accuracy: (gap effects are limited by finite lateral size) 1µm

88 Hey, forget these FCC crystals! simple-cubic lattice [ S.-Y. Lin et al., OSA B 18, 32 (2001). ] (UV stepper, Si/air) Whoops! only a 5% gap = 3.2µm

89 A Metal Photonic Crystal [. G. Fleming et al., Nature 417, 52 (2002) ] Start with Si woodpile in SiO 2 dissolve Si with KOH fill with Tungsten via chemical vapor deposition (CVD) (on thin TiN layer) dissolve SiO 2 with HF

90 Thermal properties of metal crystal [. G. Fleming et al., Nature 417, 52 (2002) ] T R absorption Kirchoff s Law: a good absorber is a good emitter modify thermal emission! solar cells light bulbs

91 enough layer-by-layer already!

92 Two-Photon Lithography 2 hn = D 2-photon probability ~ (light intensity) 2 e 0 hn hn photon photon 3d Lithography Atom lens dissolve unchanged stuff (or vice versa) some chemistry (polymerization)

93 Lithography is a Beast [ S. Kawata et al., Nature 412, 697 (2001) ] l = 780nm resolution = 150nm 7µm (3 hours to make) 2µm

94 For a physicist, this is cooler [ S. Kawata et al., Nature 412, 697 (2001) ] 2µm (300nm diameter coils, suspended in ethanol, viscosity-damped)

95 A Two-Photon Woodpile Crystal [ B. H. Cumpston et al., Nature 398, 51 (1999) ] (much work on materials with lower power 2-photon process) Arbitrary lattice No mask Fast/cheap prototyping Difficult topologies [ fig. courtesy. W. Perry, U. Arizona ]

96 Mass-production, pretty please?

97 Two-Photon Holographic Lithography [ D. N. Sharp et al., Opt. Quant. lec. 34, 3 (2002) ] Four beams make 3d-periodic interference pattern k-vector differences give reciprocal lattice vectors (i.e. periodicity) two-photon material (1.4µm) beam polarizations + amplitudes (8 parameters) give unit cell

98 Two-Photon Holographic Lithography [ D. N. Sharp et al., Opt. Quant. lec. 34, 3 (2002) ] 10µm huge volumes, long-range periodic, fcc lattice backfill for high contrast

99 Two-Photon Holographic Lithography [ D. N. Sharp et al., Opt. Quant. lec. 34, 3 (2002) ] [111] cleavages simulated structure 5µm [111] closeup 1µm 1µm titania inverse structure

100 Mass-production II: Colloids (evaporate) silica (SiO 2 ) microspheres (diameter < 1µm) sediment by gravity into close-packed fcc lattice!

101 Mass-production II: Colloids

102 Inverse Opals [ figs courtesy D. Norris, UMN ] fcc solid spheres do not have a gap but fcc spherical holes in Si do have a gap sub-micron colloidal spheres 3D Template (synthetic opal) Infiltration complete band gap Remove Template Inverted Opal ~ 10% gap between 8th & 9th bands small gap, upper bands: sensitive to disorder

103 In Order To Form a More Perfect Crystal [ figs courtesy D. Norris, UMN ] 65C meniscus 1 micron silica spheres in ethanol silica 250nm evaporate solvent 80C Convective Assembly [ Nagayama, Velev, et al., Nature (1993) Colvin et al., Chem. Mater. (1999) ] Heat Source Capillary forces during drying cause assembly in the meniscus xtremely flat, large-area opals of controllable thickness

104 A Better Opal [ fig courtesy D. Norris, UMN ]

105 Inverse-Opal Photonic Crystal [ fig courtesy D. Norris, UMN ] [ Y. A. Vlasov et al., Nature 414, 289 (2001). ]

106 Inverse-Opal Band Gap good agreement between theory (black) & experiment (red/blue) [ Y. A. Vlasov et al., Nature 414, 289 (2001). ]

107 Mass-Production? What about defects? (Remember cavities, waveguides?) Answer: fabricate bulk crystal via mass production + two-photon lithography for defects (Use confocal microscopy to see what you are doing, i.e. alignment) [ Inverse opals: P. Braun, UIUC; Holography: A. Turberfield, Oxford. ]

108 Mass-Production III: Block (not Bloch) Copolymers two polymers can segregate, ordering into periodic arrays periodicity ~ polymer block size ~ 50nm (possibly bigger) [ Y. Fink, A. M. Urbas, M. G. Bawendi,. D. oannopoulos,. L. Thomas,. Lightwave Tech. 17, 1963 (1999) ]

109 Block-Copolymer 1d Crystal CdSe nanocrystals for higher index (with surfactant to attract particles to one phase) (UV bandgap) [ Y. Fink, A. M. Urbas, M. G. Bawendi,. D. oannopoulos,. L. Thomas,. Lightwave Tech. 17, 1963 (1999) ]

110 Block-Copolymer 1d Visible Bandgap / homopolymer Flexible material: bandgap can be shifted by stretching it! reflection for differing homopolymer % dark/light: polystyrene/polyisoprene n = 1.59/1.51 [ A. Urbas et al., Advanced Materials 12, 812 (2000) ]

111 Block-Copolymer 2d Crystal [ Y. Fink, A. M. Urbas, M. G. Bawendi,. D. oannopoulos,. L. Thomas,. Lightwave Tech. 17, 1963 (1999) ]

112 Be GLAD: ven more crystals! GLAD = GLancing Angle Deposition 15% gap for Si/air diamond-like with broken bonds doubled unit cell, so gap between 4th & 5th bands [ O. Toader and S. ohn, Science 292, 1133 (2001) ]

113 GLAD it works? seed posts glancing-angle Si only builds up on protrusions evaporated Si rotate to spiral Si [ S. R. Kennedy et al., Nano Letters 2, 59 (2002) ]

114 GLAD it works! [ S. R. Kennedy et al., Nano Letters 2, 59 (2002) ]

115 A new twist on layer-by-layer start with an old layer-by-layer modify layering slightly (14% gap for Si/SiO 2 /air) (don t forget the holes) [ S. Fan et al., Appl. Phys. Lett. 65, 1466 (1994) ] [ S. Kawakami et al., Appl. Phys. Lett. 74, 463 (1999) ]

116 Auto-cloning Competition between 3 processes clones shape of substrate neutral atoms ions diffuse deposition leaves trenches (shadows) bias sputtering cuts corners (prefers 60 ) re-deposition fills trenches so, only planar patterning is in substrate only drilling needs alignment minimize etch roughness [ S. Kawakami et al., Appl. Phys. Lett. 74, 463 (1999) ]

117 Auto-cloned Photonic Crystal [. Kuramochi et al., Opt. Quantum. lec. 34, 53 (2002) ]

118 Yablonovite [. Yablonovitch, T. M. Gmitter, and K. M. Leung, Phys. Rev. Lett. 67, 2295 (1991) ] diamond-like fcc crystal earliest fabrication-amenable alternative to diamond spheres [ image: ] (Topology is very similar to 2000 layer-by-layer crystal)

119 Making Yablonovite electrochemical + focused-ion-beam (FIB) etching (deep vertical holes) Si l=3.1µm [ A. Chelnokov et al., Appl. Phys. Lett. 77, 2943 (2000) ]

120 in short: Those experimentalists are damned clever* * either that, or they are out of their minds

121 Photonic Crystals: Periodic Surprises in lectromagnetism Complete Band Gaps: You can leave home without them.

122 How else can we confine light?

123 Total Internal Reflection n o n i > n o rays at shallow angles > θ c are totally reflected Snell s Law: n i sinθ i = n o sinθ o θ o θ i sinθ c = n o / n i < 1, so θ c is real i.e. TIR can only guide within higher index unlike a band gap

124 Total Internal Reflection? n o n i > n o rays at shallow angles > θ c are totally reflected So, for example, a discontiguous structure can t possibly guide by TIR the rays can t stay inside!

125 Total Internal Reflection? n o n i > n o rays at shallow angles > θ c are totally reflected So, for example, a discontiguous structure can t possibly guide by TIR or can it?

126 Total Internal Reflection Redux n o n i > n o ray-optics picture is invalid on λ scale (neglects coherence, near field ) Snell s Law is really conservation of k and ω: k i sinθ i = k o sinθ o (wavevector) k = nω/c (frequency) θ o θ i translational symmetry k conserved!

127 Waveguide Dispersion Relations i.e. projected band diagrams ω light cone projection of all k in n o light line: ω = ck / n o ω = ck / n i higher-order modes at larger ω, β higher-index core pulls down state (a.k.a. β) k n weakly guided (field mostly in n o o ) ( ) n i > n o

128 Strange Total Internal Reflection Index Guiding light cone 0.35 frequency (c/a) a wavenumber k (2π/a) Conserved k and ω + higher index to pull down state = localized/guided mode.

129 0.5 A Hybrid Photonic Crystal: 1d band gap + index guiding frequency (c/a) light cone band gap range of frequencies in which there are no guided modes slow-light band edge wavenumber k (2π/a) a

130 A Resonant Cavity + photonic band gap index-confined increased rod radius pulls down dipole mode (non-degenerate)

131 A Resonant Cavity + photonic band gap index-confined The trick is to keep the radiation small (more on this later) frequency (c/a) light cone wavenumber k (2π/a) band gap ω k not conserved so coupling to light cone: radiation

132 Meanwhile, back in reality Air-bridge Resonator: 1d gap + 2d index guiding 5 µm d d = 632nm d = 703nm bigger cavity = longer λ [ D.. Ripin et al.,. Appl. Phys. 87, 1578 (2000) ]

133 Time for Two Dimensions 2d is all we really need for many interesting devices darn z direction!

134 How do we make a 2d bandgap? Most obvious solution? make 2d pattern really tall

135 How do we make a 2d bandgap? If height is finite, we must couple to out-of-plane wavevectors k z not conserved

136 A 2d band diagram in 3d Let s start with the 2d band diagram. This is what we d like to have in 3d, too! frequency (c/a) Square Lattice of Dielectric Rods (ε = 12, r=0.2a) TM bands T bands wavevector

137 A 2d band diagram in 3d Let s start with the 2d band diagram. This is what we d like to have in 3d, too! 3D Structure: No! When we include out-of-plane propagation, we get: wavevector frequency ω ω+δω frequency (c/a) projected band diagram fills gap! Square Lattice of Dielectric Rods (ε = 12, r=0.2a) but this empty space looks useful TM bands T bands wavevector

138 Photonic-Crystal Slabs 2d photonic bandgap + vertical index guiding [ S. G. ohnson and. D. oannopoulos, Photonic Crystals: The Road from Theory to Practice ]

139 Rod-Slab Projected Band Diagram frequency (c/a) Square Lattice of Dielectric Rods (ε = 12, r=0.2a, h=2a) 0 light cone odd (TM-like) bands even (T-like) bands Γ X M Γ Γ The Light Cone: All possible states propagating in the air The Guided Modes: Cannot couple to the light cone > confined to the slab Thickness is critical. Should be about λ/2 (to have a gap & be single-mode) M X

140 Symmetry in a Slab 2d: TM and T modes r r mirror plane z = 0 slab: odd (TM-like) and even (T-like) modes Like in 2d, there may only be a band gap in one symmetry/polarization

141 Slab Gaps Square Lattice of Dielectric Rods (ε = 12, r=0.2a, h=2a) Triangular Lattice of Air Holes (ε = 12, r=0.3a, h=0.5a) frequency (c/a) light cone odd (TM-like) bands even (T-like) bands light cone odd (TM-like) bands even (T-like) bands Γ X M Γ Γ M K Γ TM-like gap T-like gap

142 Substrates, for the Gravity-Impaired (rods or holes) superstrate restores symmetry substrate substrate breaks symmetry: some even/odd mixing kills gap BUT with strong confinement (high index contrast) mixing can be weak extruded substrate = stronger confinement (less mixing even without superstrate

143 xtruded Rod Substrate high index S. Assefa, L. A. Kolodziejski

144 Air-membrane Slabs who needs a substrate? AlGaAs 2µm [ N. Carlsson et al., Opt. Quantum lec. 34, 123 (2002) ]

145 Optimal Slab Thickness ~ λ/2, but λ/2 in what material? effective medium theory: effective ε depends on polarization gap size (%) even (T-like) gap T sees <ε> ~ high ε odd (TM-like) gap TM sees <ε -1 > -1 ~ low ε slab thickness (a)

146 Photonic-Crystal Building Blocks point defects (cavities) line defects (waveguides)

147 A Reduced-Index Waveguide frequency (c/a) (r=0.10a) (r=0.12a) (r=0.14a) (r=0.16a) (r=0.18a) We cannot completely remove the rods no vertical confinement! Still have conserved wavevector under the light cone, no radiation wavevector k (2π/a) (r=0.2a) Reduce the radius of a row of rods to trap a waveguide mode in the gap.

148 Reduced-Index Waveguide Modes x x y x z z y z x z y 1 +1 z y 1 +1 Hz

149 xperimental Waveguide & Bend bending transmission efficiency caution: can easily be multi-mode SiO 2 GaAs AlO 1µm 1µm [. Chow et al., Opt. Lett. 26, 286 (2001) ] band gap experiment theory waveguide mode wavelength (µm)

150 Inevitable Radiation Losses whenever translational symmetry is broken e.g. at cavities, waveguide bends, disorder coupling to light cone = radiation losses ω (conserved) k is no longer conserved!

151 All Is Not Lost A simple model device (filters, bends, ): Qr Qw 1 Q = 1 + Qr 1 Qw We want: Q r >> Q w Q = lifetime/period = frequency/bandwidth 1 transmission ~ 2Q / Q r worst case: high-q (narrow-band) cavities

152 Semi-analytical losses A low-loss strategy: Make field inside defect small = delocalize mode Make defect weak = delocalize mode r r t rr rr r x ( ) = G( xx, ) x ( ) ε( x ) ω defect far-field (radiation) Green s function (defect-free system) near-field (cavity mode) defect

153 Monopole Cavity in a Slab Lower the ε of a single rod: push up a monopole (singlet) state. decreasing ε (ε = 12) Use small ε: delocalized in-plane, & high-q (we hope)

154 Delocalized Monopole Q 1,000,000 ε=11 100,000 Q r 10,000 ε=10 ε=9 1,000 ε=8 ε= mid-gap 0.1 frequency above band edge (c/a) ε=6

155 Super-defects Weaker defect with more unit cells. More delocalized at the same point in the gap (i.e. at same bulk decay rate)

156 Super-Defect vs. Single-Defect Q 1,000,000 ε=11.5 G 100,000 ε=11 ε=11 G Q r 10,000 ε=10 G ε=10 ε=9 G ε=9 ε=8 G 1,000 ε=8 ε=7 ε=7 G mid-gap 0.1 frequency above band edge (c/a) ε=6

157 Super-Defect vs. Single-Defect Q 1,000,000 ε=11.5 G 100,000 ε=11 ε=11 G Q r 10,000 ε=10 G ε=10 ε=9 G ε=9 ε=8 G 1,000 ε=8 ε=7 ε=7 G mid-gap 0.1 frequency above band edge (c/a) ε=6

158 Super-Defect State (cross-section) ε = 3, Q rad = 13,000 z (super defect) still ~localized: In-plane Q is > 50,000 for only 4 bulk periods

159 (in hole slabs, too) Hole Slab ε=11.56 period a, radius 0.3a thickness 0.5a Q = 2500 near mid-gap ( freq = 0.03) Reduce radius of 7 holes to 0.2a Very robust to roughness (note pixellization, a = 10 pixels).

160 How do we compute Q? (via 3d FDTD [finite-difference time-domain] simulation) 1 excite cavity with dipole source (broad bandwidth, e.g. Gaussian pulse) monitor field at some point extract frequencies, decay rates via signal processing (FFT is suboptimal) [ V. A. Mandelshtam,. Chem. Phys. 107, 6756 (1997) ] Pro: no a priori knowledge, get all ω s and Q s at once Con: no separate Q w /Q r, Q > 500,000 hard, mixed-up field pattern if multiple resonances

161 How do we compute Q? (via 3d FDTD [finite-difference time-domain] simulation) 2 excite cavity with narrow-band dipole source (e.g. temporally broad Gaussian pulse) source is at ω 0 resonance, which must already be known (via 1 ) measure outgoing power P and energy U Q = ω 0 U / P Pro: separate Q w /Q r, arbitrary Q, also get field pattern Con: requires separate run 1 to get ω 0, long-time source for closely-spaced resonances

162 Can we increase Q without delocalizing?

163 Semi-analytical losses Another low-loss strategy: exploit cancellations from sign oscillations r r t rr rr r x ( ) = G( xx, ) x ( ) ε( x ) ω defect far-field (radiation) Green s function (defect-free system) near-field (cavity mode) defect

164 Need a more compact representation Cannot cancel infinitely many (x) integrals Radiation pattern from localized source use multipole expansion & cancel largest moment

165 Multipole xpansion [ ackson, Classical lectrodynamics ] radiated field = dipole quadrupole hexapole ach term s strength = single integral over near field one term is cancellable by tuning one defect parameter

166 Multipole xpansion [ ackson, Classical lectrodynamics ] radiated field = dipole quadrupole hexapole peak Q (cancellation) = transition to higher-order radiation

167 Multipoles in a 2d example + photonic band gap index-confined increased rod radius pulls down dipole mode (non-degenerate) as we change the radius, ω sweeps across the gap

168 r = 0.35a r = 0.375a Q = 1,773 30,000 25,000 2d multipole cancellation Q = 28,700 Q 20,000 15,000 r = 0.40a 10,000 Q = 6,624 5, frequency (c/a)

169 cancel a dipole by opposite dipoles cancellation comes from opposite-sign fields in adjacent rods changing radius changed balance of dipoles

170 3d multipole cancellation? quadrupole mode 2000 enlarge center & adjacent rods vary side-rod ε slightly for continuous tuning (balance central moment with opposite-sign side rods) 1500 Q 1000 ( z cross section) gap bottom frequency (c/a) gap top

171 3d multipole cancellation Q = 408 Q = 1925 Q = 426 far field 2 near field z nodal planes (source of high Q)

172 An xperimental (Laser) Cavity [ M. Loncar et al., Appl. Phys. Lett. 81, 2680 (2002) ] elongate row of holes cavity longation p is a tuning parameter for the cavity in simulations, Q peaks sharply to ~10000 for p = 0.1a (likely to be a multipole-cancellation effect) * actually, there are two cavity modes; p breaks degeneracy

173 An xperimental (Laser) Cavity [ M. Loncar et al., Appl. Phys. Lett. 81, 2680 (2002) ] elongate row of holes H z (greyscale) cavity longation p is a tuning parameter for the cavity in simulations, Q peaks sharply to ~10000 for p = 0.1a (likely to be a multipole-cancellation effect) * actually, there are two cavity modes; p breaks degeneracy

174 An xperimental (Laser) Cavity [ M. Loncar et al., Appl. Phys. Lett. 81, 2680 (2002) ] cavity (InGaAsP) Q ~ 2000 observed from luminescence quantum-well lasing threshold of 214µW (optically 1% duty cycle)

175 How can we get arbitrary Q with finite modal volume? Only one way: a full 3d band gap Now there are two ways. [ M. R. Watts et al., Opt. Lett. 27, 1785 (2002) ]

176 The Basic Idea, in 2d start with: junction of two waveguides ε 1 ε 1 ' ε 2 < ε 1 ε 2 ' < ε 1 ' No radiation at junction if the modes are perfectly matched

177 Perfect Mode Matching requires: same differential equations and boundary conditions ε 1 ε 1 ' ε 2 < ε 1 ε 2 ' < ε 1 ' Match differential equations ε 2 ε 1 = ε 2 ' ε 1 ' closely related to separability [ S. Kawakami,. Lightwave Tech. 20, 1644 (2002) ]

178 Perfect Mode Matching requires: same differential equations and boundary conditions ε 1 ε 1 ' ε 2 < ε 1 ε 2 ' < ε 1 ' Match boundary conditions: field must be T ( out of plane, in 2d) (note switch in T/TM convention)

179 T modes in 3d for cylindrical waveguides, azimuthally polarized T 0n modes

180 A Perfect Cavity in 3d (~ VCSL + perfect lateral confinement) Perfect index confinement (no scattering) + 1d band gap = 3d confinement R N layers defect N layers

181 A Perfectly Confined Mode ε 1, ε 2 = 9, 6 λ/2 core ε 1 ', ε 2 ' = 4, 1 energy density, vertical slice

182 Q limited only by finite size ~ fixed mode vol. V = (0.4λ) 3 N = 10 Q G G G G G G G N = 5 G 100 G G cladding R / core radius

183 Q-tips Three independent mechanisms for high Q: Delocalization: trade off modal size for Q Q r grows monotonically towards band edge Multipole Cancellation: force higher-order far-field pattern Q r peaks inside gap New nodal planes appear in far-field pattern at peak Mode Matching: allows arbitrary Q, finite V Requires special symmetry & materials

184 Forget these devices I just want a mirror. ok

185 Projected Bands of a 1d Crystal (a.k.a. a Bragg mirror) ω incident light k conserved 1d band gap modes in crystal TM light line of air ω = ck T (normal incidence) k

186 Omnidirectional Reflection [. N. Winn et al, Opt. Lett. 23, 1573 (1998) ] in these ω ranges, there is no overlap between modes of air & crystal all incident light (any angle, polarization) is reflected from flat surface ω modes in crystal TM light line of air ω = ck T needs: sufficient index contrast & n hi > n lo > 1 k

187 Omnidirectional Mirrors in Practice [ Y. Fink et al, Science 282, 1679 (1998) ] Index ratio, n 2 / n contours of omnidirectional gap size λ/λ mid 50% 40% 30% 20% 10% 0% Reflectance (%) Reflectance (%) Te / polystyrene normal 45 0 s 45 0 p 80 0 s 80 0 p Smaller index, n Wavelength (microns)

188 Photonic Crystals: Periodic Surprises in lectromagnetism A Long and Winding Road Photonic-Crystal Fibers 1/31/02 INSPC literature search: hits hits hits Bloch s theorem is more important than Maxwell s equations ;^)

189 Optical Fibers Today (not to scale) more complex profiles to tune dispersion high index doped-silica core n ~ 1.46 losses ~ 0.2 db/km (amplifiers every km) silica cladding n ~ 1.45 LP 01 confined mode field diameter ~ 8µm protective polymer sheath [ R. Ramaswami & K. N. Sivarajan, Optical Networks: A Practical Perspective ] but this is ~ as good as it gets

190 The Glass Ceiling: Limits of Silica Loss: amplifiers every km limited by Rayleigh scattering (molecular entropy) Nonlinearities: after ~100km, cause dispersion, crosstalk, power limits (limited by mode area ~ single-mode, bending loss) Polarization-mode dispersion (PMD): pulse spreading! same speed: single-mode fiber random stress, imperfections different speeds: Long Distances High Bit-Rates Dense Wavelength Multiplexing (DWDM)

191 Breaking the Glass Ceiling: Hollow-core Fibers 1000x better loss/nonlinear limits (from density) 1d crystal Bragg fiber [ Yeh et al., 1978 ] + omnidirectional = OmniGuides Photonic Crystal 2d crystal PCF [ Knight et al., 1998 ]

192 Breaking the Glass Ceiling: Hollow-core Fibers There seems then no future whatsoever for cylindrical silica-based Bragg fibers for optical communications, however good the fabrication techniques may become in the future. Bragg fiber [ Yeh et al., 1978 ] + omnidirectional = OmniGuides [ N.. Doran and K.. Blow,. Lightwave Tech. 1, 588 (1983) ] (fortunately, a completely erroneous mathematical analysis but it may be true that silica-based Bragg fibers are of limited use.)

193 Omnidirectional Bragg Mirrors a 1d crystal can reflect light from all angles and polarizations [ Winn, Fink et al. (1998) ] it behaves like a metal (but at any wavelength) perfect metal

194 OmniGuide Fibers omnidirectional mirrors c.f. Photonic Bandgap Fibers & Devices MIT (also a Cambridge MA start-up: [ S. G. ohnson et al., Opt. xpress 9, 748 (2001) ]

195 Hollow Metal Waveguides, Reborn metal waveguide modes OmniGuide fiber gaps frequency ω 1970 s microwave Bell Labs wavenumber β wavenumber β

196 Hollow Metal Waveguides, Reborn metal waveguide modes OmniGuide fiber modes frequency ω 1970 s microwave Bell Labs wavenumber β wavenumber β modes are directly analogous to those in hollow metal waveguide

197 An Old Friend: the T 01 mode lowest-loss mode, just as in metal r (near) node at interface = strong confinement = low losses from metal: optimal R ~ 10λ R Here, use R=13µm for λ=1.55µm n=4.6/1.6 (any omnidirectional is similar)

198 T 01 vs. PMD non-degenerate mode, so cannot be split r i.e. immune to birefringence i.e. PMD is zero

199 Let s Get Quantitative but what about the cladding? Gas can have low loss & nonlinearity some field penetrates! & may need to use very bad material to get high index contrast

200 Let s Get Quantitative Absorption (& Rayleigh Scattering) = small imaginary ε Nonlinearity = small ε ~ 2 Acircularity, bending, roughness, = small perturbations Hard to compute directly use Perturbation Theory

201 Perturbation Theory Given solution for ideal system compute approximate effect of small changes solves hard problems starting with easy problems & provides (semi) analytical insight

202 Perturbation Theory for Hermitian eigenproblems given eigenvectors/values: Ôu = uu find change u & u for small Ô Solution: expand as power series in Ô () 1 ( 2) u= 0+ u + u + u Ou ˆ u () 1 = uu & u () 1 = 0+ u + (first-order is usually enough)

203 Perturbation Theory for electromagnetism ω () 1 r r c 2 H Θˆ H = r r 2ω HH ω r ε = r 2 ε 2 2 e.g. absorption gives imaginary ω = decay! β () 1 = ω () 1 / v v g g = d ω dβ

204 Suppressing Cladding Losses Mode Losses 1x10-2 / Bulk Cladding Losses: Large differential loss 1x10-3 H 11 T 01 cladding loss strongly suppressed! 1x10-4 T 01 (like ohmic losses) 1x λ (µm)

205 Suppressing Cladding Nonlinearity Mode Nonlinearity* / Cladding Nonlinearity: 1x10-6 Will be dominated by nonlinearity of air 1x10-7 ~10,000 times weaker nonlinearity than silica (including factor of 10 in area) 1x10-8 T 01 * nonlinearity = β (1) / P 1x λ (µm)

206 Absorption & Nonlinearity Scaling 1x10-2 1x10-3 1x10-4 cladding absorption ~ 1/R 3 (like metal ohmic loss!) 1x10-5 1x10-6 1x10-7 B B B B B B B B B B B cladding nonlinearity ~ 1/R 5 B B B B 1x10-8 B B B B B B B B B 1x core radius (µm)

207 Radiation Leakage Loss (17 layers) Finite # layers: modes are leaky loss decreases exponentially with number of layers (~ 1/R 3 ) leakage loss (db/km) 1x10 5 1x10 4 1x10 3 1x10 2 1x10 1 1x10 0 1x10-1 1x10-2 1x10-3 1x λ (µm)

208 ther Losses Acircularity & Bending tricky main effect is coupling to lossier modes, but can be ~ 0.01 db/km with enough (~50) layers Surface Roughness suppressed like absorption

209 Acircularity & Perturbation Theory (or any shifting-boundary problem) ε 2 ε = ε 1 ε 2 ε 1 ε = ε 2 ε 1 just plug ε s into perturbation formulas? FAILS for high index contrast! beware field discontinuity fortunately, a simple correction exists [ S. G. ohnson et al., PR 65, (2002) ]

210 Yes, but how do you make it? 1 find compatible materials (many new possibilities) [ figs courtesy Y. Fink et al., MIT ] 2 Make pre-form ( scale model ) 3 chalcogenide glass, n ~ polymer (or oxide), n ~ 1.5 fiber drawing

211 Fiber Draw MIT building 13, constructed ~6 meter (20 feet) research tower [ figs courtesy Y. Fink et al., MIT ]

212 A Drawn Bandgap Fiber [ figs courtesy Y. Fink et al., MIT ] Photonic crystal structural uniformity, adhesion, physical durability through large temperature excursions white/grey = chalco/polymer

213 Band Gap Guidance Transmission window can be shifted by scaling (different draw speed) Wavevector original (blue) & shifted (red) transmission: Transmission (arb. u.) [ figs courtesy Y. Fink et al., MIT ] Wavenumber (cm -1 ) 2000

214 High-Power Transmission at 10.6µm (no previous dielectric waveguide) Polymer ~ 50,000dB/m -3.0 waveguide losses ~ 1dB/m Transmission (arb. u.) Log. of Trans. (arb. u.) slope = db/m R 2 = Length (meters) [ B. Temelkuran et al., Nature 420, 650 (2002) ] cool movie Wavelength (µm) [ figs courtesy Y. Fink et al., MIT ]

215 nough about MIT already

216 2d-periodic Photonic-Crystal Fibers [R. F. Cregan et al., Science 285, 1537 (1999) ] air holes a Not guided via 2d T bandgap: silica (usually) TM bands T bands Γ M K Γ β wavenumber breaks mirror plane, so no pure T/TM polarizations

217 PCF Projected Bands [. Broeng et al., Opt. Lett. 25, 96 (2000) ] ω (c/a) (not 2πc/a) bulk crystal continuum air light line band gap fingers appear! β (a 1 )

218 PCF Guided Mode(s) [. Broeng et al., Opt. Lett. 25, 96 (2000) ] ω (c/a) (not 2πc/a) fundamental & 2nd order guided modes bulk crystal continuum air light line fundamental air-guided mode β (a 1 )

219 xperimental Air-guiding PCF Fabrication (e.g.) silica glass tube (cm s) (outer cladding) fiber draw ~50 µm ~1 mm fuse & draw

220 xperimental Air-guiding PCF [ R. F. Cregan et al., Science 285, 1537 (1999) ] 10µm 5µm

221 xperimental Air-guiding PCF [ R. F. Cregan et al., Science 285, 1537 (1999) ] transmitted intensity after ~ 3cm ω (c/a) (not 2πc/a)

222 Air-guiding PCF Losses Best reported results: 13dB/km over ~ [ Corning, COC 2002 ]

223 Index-Guiding PCF & microstructured fiber: Holey Fibers solid core holey cladding forms effective low-index material Can have much higher contrast than doped silica strong confinement = enhanced nonlinearities, birefringence, [. C. Knight et al., Opt. Lett. 21, 1547 (1996) ]

224 Holey Projected Bands, Batman! (Schematic) ω (c/a) (not 2πc/a) bulk crystal continuum guided band lies below crystal light line band gaps are unused β (a 1 )

225 Holey Fiber PMF (Polarization-Maintaining Fiber) birefringence B = βc/ω = (10 times B of silica PMF) Loss = µm over 1.5km no longer degenerate with Can operate in a single polarization, PMD = 0 (also, known polarization at output) [ K. Suzuki, Opt. xpress 9, 676 (2001) ]

226 Nonlinear Holey Fibers: Supercontinuum Generation (enhanced by strong confinement + unusual dispersion) e.g nm white light: from 850nm ~200 fs pulses (4 n) [ W.. Wadsworth et al.,. Opt. Soc. Am. B 19, 2148 (2002) ]

227 ndlessly Single-Mode [ T. A. Birks et al., Opt. Lett. 22, 961 (1997) ] at higher ω (smaller λ), the light is more concentrated in silica so the effective index contrast is less and the fiber can stay single mode for all λ!

228 Low Contrast Holey Fibers [. C. Knight et al., lec. Lett. 34, 1347 (1998) ] The holes can also form an effective low-contrast medium i.e. light is only affected slightly by small, widely-spaced holes This yields large-area, single-mode fibers (low nonlinearities) ~ 10 times standard fiber mode diameter but bending loss is worse

229 Holey Fiber Losses Best reported results: 0.58 [ Blaze Photonics ]

230 The Upshot: The Upshot Potential new regimes for fiber operation, even using very poor materials. The Story of Photonic Crystals Finding Materials > Finding Structures