Photonic crystals Semi-conductor crystals for light The smallest dielectric lossless structures to control whereto and how fast light flows Femius Koenderink Center for Nanophotonics AMOLF, Amsterdam f.koenderink@amolf.nl
Definition Definition: A photonic crystal is a periodic arrangement with a ~ λ of a dielectric material that exhibits strong interaction with light - large ε
Examples 1D: Bragg Reflector 2D: Si pillar crystal 3D: Colloidal crystal
Bragg diffraction Bragg mirror Antireflection coatings (Fresnel equations) Bragg s law: 2n average d cos(θ) = mλ Each succesive layer gives phase-shifted partial reflection Interference at Bragg condition yields 100% reflection 4
Conclusions 1D stack Light can propagate into/through 1D stacks, except around - a stop gap around the Bragg condition - in a frequency interval of width standing wave in n 2 n 1 n 2 n 1 n 2 n 1 n 2 n 1 Stop gap standing wave in n 1 0 π/a At the band edge: standing waves that stand still In the band edge: no propagating states at all 5
Frequency [ωd/2πc] Periodicity d=d 1 +d 2 k i =n i ω/c - frequency and index contrast K sets phase increment per unit cell exp(ikd) Κ [π/d] 6 M=(M 1.M 2 ) N
Dielectric mirror 12 layers Reflectivity d/λ R arbitrarily close to 100%, independent of index contrast (N-1) end-facet Fabry Perot fringes 7
2D and 3D lattices Dielectric constant is periodic on a 2D/3D lattice a 2 Bravais lattices 2D: square, hexagonal, rectangular, oblique, rhombic a 1 3D: sc, fcc, bcc, etc (14 x) 8
Reciprocal lattice Reciprocal lattice with property a 2 a 1 Special wave vectors of scale b ~ 2π/a 9
Example of reciprocal lattice vectors Note how: 10
Bloch s theorem Suppose we look for solutions of: with periodic ε(r) Bloch s theorem says the solution must be invariant up to a phase factor when translating over a lattice vector Equivalent: Phase factor Truly periodic Note how k and k+g are really the same wave vector 11
Note how k and k+g are really the same wave vector Band structure 12
Folding bands of 1D system frequency ω 2π/a -π/a 2π/a 0 π/a wave vector k Bloch wave with wave vector k is equal to Bloch wave with wave vector k+m2π/a 13
Formal derivation in 3D Wave eq: Bloch: Substitute and find: The structure is that of an infinite dimensional linear problem frequency acts as the eigenvalue 14
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Dispersion relation of vacuum -folded frequency ω 2π/a -π/a 0 π/a wave vector k Folded Free dispersion relation 16
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Nearly free dispersion frequency ω 2π/a -π/a 0 π/a wave vector k Crossing -> anticrossing upon off-diagonal coupling Compare QM: degeneracies are lifted by perturbation 18
More complicated example FCC crystal close packed connected spheres 1 st Brillouin zone (bcc cell) Wedge: irreducible part 19
Folded bands almost of vacuum Example: n=1.5 spheres, (26% air) a/λ k 1 st Brillouin zone (bcc cell) Wedge: irreducible part 20
Folded bands almost of vacuum Example: n=1.5 spheres, (26% air) Spagghetti Diffractive / Photonic crystal a/λ Effective medium / metamaterial k 21
Folded bands almost of vacuum Example: n=1.5 spheres, (26% air) 2n average d cos(θ) = mλ 1/cos(θ) shift a/λ Bragg gap at normal incidence to 111 planes k 22
Wider bands Reversing air & glass to reduce the mean epsilon a/λ Note (1): band shifts up lower effective index (2): Relative gap broadens 23
Wider bands Replacing n=1.5 by n=3.5, keeping ~ 80% air `airholes A true band gap FOR ALL wave vectors opens up 24
Band gap for air spheres in Si We counted all the eigenstates in the 1 st Brillouin zone Note (1) a true gap, and (2) regimes of very high state density 25
Si 3D photonic crystals 1. Colloids stack in fcc crystals 2. Silicon infilling with CVD 3. Remove spheres Vlasov/Norris Nature 2001 Technique pioneered at UvA (Vos & Lagendijk, 1998) 26
Woodpile crystals Silicon repeated stacking, folding Sandia, Kyoto GaAs (also n=3.5) robotics in SEM 27
2D crystals Si or GaAs membranes Very thin (200 nm) Kyoto, DTU, Wurzburg, Si posts in air (AMOLF) Zijlstra, van der Drift, De Dood, and Polman (DIMES, FOM)
Dielectric rod structure TM means E out of plane Snapshots of field at band edges (k = G/2 ) for G=X =2π/a(1,0) G=M =2π/a(1,1) Note the phase increment due to k Note field concentration in air (band 2) rod (band 1) 29
Why all the effort for just a lot of math? What we have seen so far: Photonic crystals diffract light, just like X-ray diffraction Unlike X-ray diffraction, the bandwidth is ~ 20%, not 10-4 Light has a nontrivial band structure What is so great: A nontrivial band structure means control over how fast light travels, and how it refracts A true band gap expels all modes Complete shielding against radiative processes Line and point defects would be completely shielded traps for light 30
2D crystal 1. In a 2D crystal polarization splits into TE and TM Photonic bandgap 2. Band structure is for in-plane k-only 3. Light-line separates bound from leaky 31
Line defects frequency w (units 2 p c/a) 0.35 0.3 0.25 0.2 crystal modes waveguide modes crystal modes 0.15 0 0.1 0.2 0.3 0.4 0.5 wavevector k (units A single line surrounded by a full band gap guides light Light at the line is forbidden from escaping into the crystal π /a) 32
Line defects and bends Line defect, bend Exceptionally tight 90 o bend Potential for very small low-loss chips E 33
Measurement of guiding & bending Sample: AIST Japan Meas: AMOLF 34
January 2007 35 Cavity in experiment Free standing GaAs membrane 250 nm thick, 800 µm long, 30 µm wide 1 row of holes missing Lattice spacing a=410 nm Pink areas: a=400 nm Song, Noda, Asano, Akahane, Nature Materials
Why a cavity? January 2007 36
January 2007 37 Simulated mode Mode intensity Cycle averaged E 2 Mode volume 1.2 (λ/n GaAs ) 3 Exceptionally small cavity Very high Q, up to 10 6
January 2007 38 Narrow cavity resonance 125 100 Picked up by tip Few µm above cavity Q =(1±0.5) 10 5 Lorentz Q =88000 Counts 75 50 25 Tip: pulled glass fiber ~ 100 nm 0 1565.0 1565.2 1565.4 Wavelength (nm) Laser: grating tunable diode laser 20 MHz linewidth (10-7 λ) around 1565 nm
Refraction k n1 n2 n 1 ω/c n 2 ω/c Generic solution steps: 1) Plane waves in each medium 2) Use k conservation to find allowed waves 3) Use causality to keep only outgoing waves -> refracted k 4) Match field continuity at boundary to find r and t
Folding bands of 1D system frequency ω 2π/a -π/a 2π/a 0 π/a wave vector k Bloch wave with wave vector k is equal to Bloch wave with wave vector k+m2π/a 40
January 2007 41 Harrison s construction In 2D free space, the dispersion ω=c k looks like a circle at any ω Periodic system: repeated zone-scheme brings in new bands At crossings: coupling splits bands
January 2007 42 Observation of band folding Angle resolved map of Fluorescence from a corrugated Waveguide Common application: LED light extraction
January 2007 43 Observation of band folding Angle resolved map of Fluorescence from a corrugated Waveguide Slow-light at Bragg condition plasmonic crystal laser
January 2007 44 Measurement Folded band structure of a surface plasmon, probed at one λ
January 2007 45 Refraction A single incident beam can split into multiple refracted beams Group velocity = direction of energy flow not along k
January 2007 46 Refraction Superprism: exceptional sensitivity to incidence θ and λ Supercollimation: exceptional a-sensitivity to incidence θ and λ
Super collimation A tightly focused beam has many k, and should diffract Supercollimation: beam stays collimated because v g is flat Note: there is no guiding defect here Shanhui Fan APL (2003) 47
Superprisms Application: a minute change in ω is a huge change in θ Wavelength demultiplexer note the negative refraction 48
Conclusions Diffraction Photonic crystals diffract light, just like X-ray diffraction Unlike X-ray diffraction, the bandwidth is ~ 20%, not 10-4 Propagation Light has a nontrivial band structure similar to e- band-structure Dispersion surfaces are like Fermi surfaces Light has polarization. Photons do not interact with photons Band structure controls refraction and propagation speed Band gap: light does not enter. No states in the crystal Defects line defects guide light Point defects confine light for up to 10 6 optical cycles in a λ 3 volume 49