3D Photonic Crystals: Making a Cage for Light

3D Photonic Crystals: Making a Cage for Light Willem Vos 0.44 µ m Complex Photonic Systems ( COPS ) Fac. Technische Natuurkunde & MESA + Universiteit Twente (UT), Enschede www.photonicbandgaps.com

Outline of the lectures 1. Introduction ( the big picture ) 2. Photonic band gap formation 3. Making photonic crystals 4. Disorder: optical & structural 5. Spontaneous emission control 6. Discussion, conclusions

Femius Koenderink Peter Lodahl Tijmen Euser Ad Lagendijk Many thanks to: Lydia Bechger Martijn Wubs former group members: Juan Galisteo Lòpez (Madrid), Patrick Johnson (Utrecht), Dirk Riese (Daimler), Henry Schriemer (start-up), Henry van Driel (Toronto), Mischa Megens (Philips), Judith Wijnhoven (Utrecht), Michiel Thijssen (Elsevier)

And many thanks to: F. van Driel, D. Vanmaekelbergh, J. Kelly (Utrecht) E. Flück, L. Kuipers, N. van Hulst (Twente) D. Wiersma, S. Gottardo (Firenze) J. Farthing, W. Buhro (Washington Univ.) F. Roozeboom, R. Balkenende (Philips), I. Setija (ASML), W. Tjerkstra, M. de Boer (Twente) S. Kole, T. Figg, K. Michielsen, H. deraedt (Groningen) R. Sprik (Amsterdam) Support: FOM and NWO

1. Introduction What is a photonic crystal Compare photonic & electronic systems What is a photonic band gap How to envision a cage for light

Dielectric composite Photonic crystal Refractive index n varies light scatters Photonic : a λ interference, Bragg diffraction a <<λ : effectively homogeneous system (e.g. water) a >>λ : geometrical optics limit (e.g. the audience: you!) λ a

Maxwell Schrödinger equation Eigenvalue equation, with momentum p, potential V, wavefunctionψ : (p 2 +V) ψ = Eψ [neglect spin, polarization] Light: eigenvalues E = ω 2 > 0, r 2 potential = ( 1 ε ( )) ω V light spatial variation of ε [ = (refractive index n) 2 ] controls the light potential. Light & electrons most similar in stationary cases. Lagendijk & van Tiggelen, Phys. Rep. 270 (1996) 143.

Photons & electrons: differences Since V light ω 2 : - No localized light at low E or ω. - Different dynamics (> 1 energy level involved). Some other differences: property photons electrons mass? DOS dispersion no DOS ω 2 ω k yes DOS ω ½ ω k 2 charge? (interact) no yes statistics Bose Fermi

Goal: photonic band gap (PBG) Strong interaction light+crystal: diffraction in every direction gap in the DOS Density of states ("DOS") Spontaneous emission inhibited Light localized (near defects) Also: change dispersion forces (e.g. van der Waals) 0 0 photonic band gap crystal free space band gap Frequency ~ ω 2 E. Yablonovitch, S. John (1987)

Perfect crystal, light with frequencies inside photonic band gap: it s dark! Cage for photons (1/3) Incident light is reflected for all directions.

Perfect crystal, let s place a light source ( ) inside and let s excite it: what happens?? Cage for photons (2/3) source

Perfect crystal, light source ( ) inside: it s still dark! Cage for photons (2/3) source, still dark We will show the first observation of such an effect!

Crystal with defect & source: Source emits, but light cannot enter the bulk. Photon localized near the defect. Cage for photons (3/3) Cage for photons.

Complete control over photons Photonic crystals: analogue of semiconductors Control emission of light sources Control propagation of light Conceivable applications: - efficient miniature LEDs and lasers - optical circuits? - transistors for photons?? - controlled photo-chemistry?

2. Photonic band gap formation Interaction light + crystal Bragg diffraction From Bragg diffraction to photonic band gaps

Interaction light & photonic material Photonic crystal, consist of e.g. 2 materials with indices n 1, n 2 What does light feel?? Vos et al., NATO-ASI (2001) 191 & Phys. Rev. B 53 (1996) 16231

Interaction light & photonic material Photonic strength : Ψ = 4πα v 3φ m m 1 m=n 1 /n 2 φ = volume fraction 2 2 + 2 Vos et al., NATO-ASI (2001) 191 & Phys. Rev. B 53 (1996) 16231

Interaction light & photonic material Photonic strength : Ψ = 4πα v 3φ m m 1 m=n 1 /n 2 φ = volume fraction 2 Ψ also describes atomic lattices 2 + 2 band gap: Ψ > ~0.15-0.2 m > 2.8 (fcc) m > 2 (diamond) Vos et al., NATO-ASI (2001) 191 & Phys. Rev. B 53 (1996) 16231

Ψ vs. volume fraction of dielectric spheres. Bend-over: form factor decreases. Ψ vs. frequency for a crystal of 2-level atoms (all positions filled). Photonic strength Vos et al., J.Phys.CM 8 (1996) 9503 NATO-ASI (2001) 191 4πα/V 4πα/v 0.2 0.1 0.0 0 20 40 60 80 1.5 1.0 0.5 0.0 m=2.8 m=1.2 m=1.2 our expts. density (vol%) -20-10 0 10 20 (ω-ω Resonance )/Γ p:/data/optica/colloidxtal97/gaps97_2002.opj P:/theory/atom_lattice_2002.OPJ

Bragg diffraction: interference. Simple Bragg diffraction Diffraction condition: mλ hkl = 2 d hkl sinθ [m = integer] Only one wavelength diffracted? [pretty useless!] But wait, no! Let s consider simple case: sinθ=1...

Simple case: 1-D stack. Where is energy concentrated? Wave(1) mostly in medium n 1 Wave(2) mostly in medium n 2 Bragg: standing waves (i) n 1, n 2 = refractive indices λ=2d

Bragg: standing waves (ii) Recall for (light) waves: frequency = speed of light refractiveindex wave in n low high frequency wave in n high low frequency (n low <n high ) 2 different frequencies for one and the same wavelength (λ=2d) wavelength 1

Standing waves: - stop gap, that is, gap in 1 direction - 2 coupled waves (incident+diffracted) Standing waves: stop gap Frequency ω photonic crystal free space stop gap Stop gap width: 0 ω/ω 0 Ψ increases with Ψ, refr. index contrast. Bragg wave in n low wave in n high π/a 2π/a Wave vector k Group velocity = slope, tends to 0 at edge of stop band [Imhof et al., PRL 83 (1999) 2942] n:/tekst/present/praatjes/plaatjes/2bands_2.opj

What you see: opalescence Inverse opals: green: λ ~ 560 nm d ~ 420 nm red: λ ~ 680 nm d ~ 490 nm Wavelength lattice spacing Optical Bragg diffraction Thijssen et al., Phys. Rev. Lett. 83 (1999) 2730

PBG = sum of Bragg conditions? (schematic example) Square lattice: Bragg conditions in different directions. Frequency 0 0 30 60 90 Naive picture: photonic band gap is the overlap of all stop gaps. 3 2 1 01 stop gap 10 stop gap Bragg: ~1/sin(θ) Propagation direction (deg.) p:/present/praatjes/plaatjes/simplpbg_brgg.opj Gap?

3D crystal. Low bands: ω/c ~ k/n avg Why does a band gap form? Why are high bands flat & form a band gap?? ω/c [2π/a] What is the connection with Bragg diffraction?? 0.9 0.6 0.3 0.0 Let s look at an experiment... inverse opal, ε=11, 74 vol% (F. Koenderink) X ω/c=k/n avg U L Γ X W K Wavevector k p:/projects/switch/plaatjes/may18run123switching.opj

Not so simple Bragg diffraction Bragg reflectivity: hkl=111 peak vs. angle Why double peaks with avoided crossing? Multiple diffraction: 111 condition is modified by another one (hkl=200). Reflectivity (%) van Driel&Vos, Phys. Rev. B 62 (2000) 9872 40 0 15 0 10 0 s-pol. p-pol. 35 o 45 o 55 o 10000 14000 1/wavelength (cm -1 )

Half-heights are edges of stop gaps. Multiple diffraction: 3 coupled waves, bands repel, form flat bands. Flat bands Frequency (cm -1 ) 14000 12000 10000 8000 0 40 80 Photonic band gap formation is an extreme case of multiple diffraction; flat bands for all directions. 200 000+111 van Driel&Vos, Phys. Rev. B 62 (2000) 9872; NATO-ASI (2001) 191. flat! expt. theory: bands Angle of incidence (degrees) l:/wvos/biorad/sp38peak(angle).opj

fcc: expect PBG near 2 nd order Even more coupled waves Increase frequency, more modes nearly degenerate many-wave coupling. 2nd order : >>15 modes in the mix! Vos & van Driel, Phys. Lett. A 272 (2000) 101 Reflectivity (%) 40 20 Wave vector 0 Γ L 1 st order: (111) "2 nd order" 2*ω 111 10000 20000 Frequency (cm -1 ) p:/theory/vandriel/000405.opj

3. Make photonic crystals Fabrication: requirements Various ways to make 3D crystals Self-organization and inverse opals

Required for photonic band gap (i) High refractive index contrast m=n 1 /n 2 > 2 3 some optical refractive indices: air 1.0 glass 1.5 diamond 2.3 water 1.3 plastics 1.6 titania 2.7 Si, GaAs 3.5 (cf. for x-rays usually 0.9999< n <1) Small volume fraction φ of high index-material: structure optimal if optical path length (n d) is same in low & high index materials. Majority of holes is best!

Required for photonic band gap (ii) Low absorption [multiple scattering crucial] Extended in 3 dimensions Materials interconnected [helps great deal] Ordered structure: size variations R << R avg., displacements u << a lattice

3D photonic crystal fabrication Colloids, opals, inverse opals [see poster Lydia Bechger] Layer by layer fabrication (e.g. Sandia, Kyoto) Etching, holography (e.g. Halle, Oxford) [see also poster Tijmen Euser] robotics, etc.

Template-assisted assembly (1/3) Make a template: self-assembly of colloids, slow sedimentation, dried: opal electron microscopy - f.c.c. (111) face - latex colloids - particle size R/R avg. < 2% 1.66 µ m

Template-assisted assembly (2/3) Infiltrate high-index material (TiO 2, GaAs, Ge,...) by: sol-gel, c.v.d., polymerization, hydrolysis, electrodeposition,... see: Velev et al, Nature 389 (1997) 447 Imhof & Pine, Nature 389 (1997) 948 Holland et al, Science 281 (1998) 538 Wijnhoven & Vos, Science 281 (1998) 802 Zakhidov et al., Science 281 (1998) 897 Blanco et al, Nature 405 (2000) 437 & many many many many others!

Template-assisted assembly (3/3) Remove template (etch, burn) air-sphere crystal or inverse opal TiO 2 (m=2.7) with air spheres (holes) Windows connect to neighbor spheres ~ 12 vol% TiO 2 Wijnhoven et al., Chem. Mater. 13 (2001) 4486; Science 281 (1998) 802

Germanium electrodeposition Electrodeposited Ge: m=4.0 (E gap =0.7 ev λ>1900 nm) Voids between spheres are completely infiltrated (26 vol%) Ge via other routes: Miguez et al. (2000, 2001). Van Vught et al., Chem. Comm (2002) 2054. see also poster Lydia Bechger.

New: assembly on wafer Thin-film evaporation: grow opal on Si-wafer, large ordered domains Infiltrate opal by c.v.d., Si inverse opal on Si-wafer Vlasov et al, Nature 414 (2001) 289

MESA + : laser interference lithography 2D crystal slabs, defined by interference pattern in photo resist. Large structures ~1.5 cm 2 in Si and SiO 2. See poster Tijmen Euser and: Vogelaar et al., Adv. Mater. 13 (2001) 1551

3D photonic crystal challenges Colloids, opals, inverse opals How to engineer defects: waveguides, cavities? Systems grow as big pieces that contain cracks. What is the domain size: individual parts or whole piece? [How to improve?]

3D photonic crystal challenges Colloids, opals, inverse opals Layer by layer fabrication (e.g. Sandia, Kyoto) Flexible, but how to make thick structures? Etching, holography (e.g. Halle, Oxford) Great, but how to get desired 3D symmetry? How to surmount these issues?? A nice challenge for young people like you!!

Introduction Characterize disorder: 4. Disorder Consequences for light propagation positional variation size variation roughness Reduce effects of disorder on optical experiments

Inverse opals: R/R < 5% u/a < 2% otherwise no PBG [Li&Zhang, 2000] Some disorder allowed Diamond structure: more relaxed, but also limits. Disorder also important to slabs, waveguides, etc. We must know, so let s characterize it...

Small angle x-ray scattering (SAXS), at the ESRF: Probe 3D structure ESRF x-rays sample probe lengths up to 1 µm probe bulk (unlike SEM) not susceptible to multiple scattering (unlike light scattering) sensitive to long-range order & variations see: Langmuir 12 (1997) 6004, 6120; Phys. Rev. Lett. 85 (2000) 5460 detector

Colloidal crystal structure factor Parent of inverse opal Analyse S(Q): peaks: structure, fcc, [a fcc =335.5 (5) nm] <5% other stackings decay at high Q: Debye-Waller factor, displacements u [ u=33.6(8) nm] Structure Factor Megens&Vos, Phys. Rev. Lett. 86 (2001) 4855 3 2 1 Experiment model Bragg reflections formfactor minima 0 0.00 0.01 0.02 Scattering Vector (nm -1 ) /prl_dw/plots/pcollsc.opj

Mean square displacements Parent of inverse opal u vs. density: common curve for various kinds of colloids. u/a < 4%, u rms /a fcc (%) in (inverse) opals probably much less. Megens&Vos, Phys. Rev. Lett. 86 (2001) 4855 0 0.0 0.5 1.0 10 5 theory: Hard spheres melting density? close packed limit ** opals ** (ρ-ρ melt )/(ρ closepacked -ρ melt ) n:/papers/prl_dw/plots/lindmn.opj Li

Inverse opal single crystals Pattern on ccd camera fcc, a fcc =860 nm Many diffraction spots excellent long-range order Further analysis: R/R < 3%, surface roughness < 10 nm. Wijnhoven et al., Chem. Mater. 13 (2001) 4486

Is disorder general? Our results: u/a << 4%, R/R < 3% Similar figures from other work, completely different structures & techniques: R/R < 4% [Baba, 2000] u/a < 1%, R/R < 5% [Notomi, 2001] u/a < 7% [Noda, 2002] Does the whole field share similar limitations??

Disorder multiple light scattering Disorder multiple scattering, light becomes diffuse. Gauge disorder: mean free path l, direction randomized. l scat : avg. distance between scattering events (step in random walk) Typically you ll want: l Bragg < L < l, l absorb sample size L absorption length l absorb Bragg attenuation length l Bragg l scat l l l scat

Total transmission: TT l / L Inverse opals: l = 15-20 µm (> 30 unit cells) Indeed l Bragg < l: crystal properties ok. [Also: enhanced backscatter cone] Measure the mean free path Total Transmission ["TT"] 0.0 0 20 40 60 0.4 0.2 I 0 Sample I t Detector Sample thickness (µm) Integrating sphere Gomez Rivas et al., Europhys. Lett. 48, 22-28 (1999)

From structural disorder to optics Cross section (l density) -1 Model: scatter off size & positional disorder, (u+r) = 5%. Much improved fabrication needed for l >20 µm!! Koenderink et al., Phys. Lett. A 268 (2000) 104 0.2 scattering cross-section, Q App /πr 2 Opals (u+r)=5% 0.0 0 1 2 3 4 size parameter, 2πR/λ

Focus laser beams to x < 10 µm & align rotation axes x < 10 µm Angle-resolved, calibrated reflectivity. Avoid disorder: focus! O 1 xtal det. Galisteo López & Vos, Phys. Rev. E 66 (2002) 036616 telesc. chop BS O 2 L 1 L 2 L 3 2q

Optical reflectivity on single-domains Inverse opals, L-gap. R up to 95%! Mosaic spread ±3 Domains 200 µm Reflectivity (%) 100 Data agree with prior sample angle (degrees) expts. using coarse beams: small inhomogeneous broadening (<8 % relative). Galisteo López & Vos, Phys. Rev. E 66 (2002) 036616 50 0 a/λ=0.77, various spots, normal incidence -8-4 0 4 8 c:/users/vos/papers/pre_focus/origins/figure3_1.opj

Angular effects Role of disorder 5. Spontaneous emission Efficiency of the light sources Local density of states [see poster Femius Koenderink]

Spontaneous emission 1. Emission redirected Spectra change w. angle 2. Angle independent density of states emitted power excited lifetime Experiments: dope inverse opals with laser dye and bleach surface of the crystals. Megens et al, PRL 83 (1999) 5401; PRA 59 (1999) 4727; JOSA B16 (1999) 1403

Angle dependent emission Inverse opals [a=500 nm]: spectra change strongly, wide stop bands. [α=angle with surface normal] Why is attenuation in the stop bands limited? I Normalized I Relative 1.0 0.5 0.0 1.0 Schriemer et al., Phys. Rev. A 63 (2001) 011801 Wavelength (nm) 45 25 α = 0 60 0.0 14000 16000 18000 0.5 714 625 S 1 556 S 2 Frequency (cm -1 ) (A) (B) n:/wvos/tekst/pra_emit/plots.opj

Effect of disorder Emission generated inside, diffuses on distances > l = 15 µm (= 50 d 111 ), samples all directions. Stop gaps determined near exit surface attenuation: 1-L B /l ~ 50 to 80 %, agrees with observations. Schriemer et al., Phys. Rev. A 63 (2001) 011801 l: mean free path L B : Bragg attenuation

Angle dependent emission Inverse opals [a=500 nm]: spectra change strongly, wide stop bands. [α=angle with surface normal] 2 nd gap at large angles: multiple Bragg diffraction. I Normalized I Relative 0.0 1.0 Schriemer et al., Phys. Rev. A 63 (2001) 011801 1.0 0.5 0.5 714 Wavelength (nm) 45 25 α = 0 625 60 S 1 556 S 2 0.0 14000 16000 18000 Frequency (cm -1 ) (A) (B) n:/wvos/tekst/pra_emit/plots.opj

Emission: multiple diffraction Symbols: half-heights of stop bands Avoided crossing: multiple diffraction (000) + (111) + (200) Frequency (cm -1 ) 22000 18000 14000 Flat bands: range of modified DOS... Schriemer et al., Phys. Rev. A 63 (2001) 011801 band structure calcs Bragg's law S 2 S 1 0 30 60 α (degrees) n:/wvos/tekst/pra_emit/plots.opj

Importance of the efficiency of sources Continuous wave experiment: photon emission rate: Γ=(W /γ tot ) γ rad [W=pump rate] Low efficiency source: γ tot = γ rad + γ non-rad γ non-rad Γ γ rad LDOS Efficient source: γ tot γ rad Γ W (i.e., you get as much out as you pump!) [To get LDOS, measure lifetime τ=1/γ rad time-resolved] Dye on TiO 2 is quenched do cw experiments Koenderink et al., Phys. Rev. Lett. 88 (2002) 143903

Angle independent emission α > 60 o : spectra without stop bands. Reference: other crystals in long wavelength limit, same chemistry. Frequency of inhibition (a fcc ) -1 Inhibited total emission, vacuum fluctuations! [first observation in photonic crystals] Koenderink et al., Phys. Rev. Lett. 88 (2002) 143903

Disorder helps! Emission generated inside, diffuses on lengths > l = 15 µm (= 50 d 111 ), samples all directions. Emission observed represents total emission! Koenderink et al., Phys. Rev. Lett. 88 (2002) 143903 l: mean free path L B : Bragg attenuation

Inhibition also for different dye. Density of states Frequency range agrees with theory (DOS). Probe local DOS: strong change. Scaled intensity Intensity ratio 1.0 0.5 0.0 1.0 Koenderink et al., Phys. Rev. Lett. 88 (2002) 143903 Reference Crystal total DOS (calc.) 0.5 0.0 12500 15000 17500 Frequency (cm -1 )

DOS: gaps inhibited emission DOS for fcc inverse opal peaks enhanced emission, if in a few modes enhanced lasing? DOS (per unit volume) 6 fcc close-packed air spheres 0 0.0 0.2 0.4 0.6 0.8 1.0 Our experiment near 1st order diffraction range; bigger effects at higher frequencies? 4 2 m=3.45 (~ GaAs) (calc.: F. Koenderink) range of our expt. Frequency in units 2π/a

Time-resolved experiment Total decay rate: crystal & reference the same. Why?? Dye in semiconductor: low quantum efficiency. Lifetime determined by non-radiative decay, must improve quantum efficiency. Koenderink et al., Phys. Rev. Lett. 88 (2002) 143903

6. Discussion Can true PBGs exist?? How to probe a PBG

Can true PBG s exist?? DOS = 0 only if: - infinitely large crystal - no absorption - near perfect ( R< 5%, u< 2% [Li&Zhang, 2000]) So in practice no true PBG? How to practically define a PBG? DOS pbg (ω) < ε ε = small number, chosen for your application/ experiment to work. see: Wubs & Lagendijk, Phys. Rev. E 65 (2002) 046612

How to probe a PBG (1) External experimental probes: e.g. observe omnidirectional reflectivity. But there may be uncoupled or silent modes, so always need theory to justify... Wouldn t it be great to develop purely experimental probes? Need probe inside the crystal. Megens et al., J. Opt. Soc. Am. B16 (1999) 1403

Place light sources inside the crystal. What to expect? How to probe a PBG (2) 1) PBG forms stop band with complete attenuation on top of usual stop bands. 2) Very long radiative lifetime. Megens et al., J. Opt. Soc. Am. B16 (1999) 1403

Conclusions 3D photonic crystals: control density of states, vacuum fluctuations. Photonic band gap: extreme multiple diffraction. Inverse opals: effective photonic crystals. Characterize disorder parameters. First observation of inhibited total spontaneous emission; also directional stop bands.

Outlook Demonstrate band gaps experimentally! Fabrication: reduce disorder. Engineer controlled defects in 3D structures. Combine understanding order + disorder. Dynamic quantum optics (e.g. lifetime). Experiments on photonic band gap switching. Johnson et al., Phys. Rev. B 66 (2002) 081102

The End thank you!