photonic crystals - amolf · •photonic crystals diffract light, just like x-ray diffraction...
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Photonic crystals
Femius KoenderinkCenter for NanophotonicsAMOLF, [email protected]
Semi-conductor crystals for light
The smallest dielectric – lossless structures to control whereto and how fast light flows
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Definition
Jan 2010, UU 2
Definition:
A photonic crystal is a periodic arrangement with a ~ lof a dielectric materialthat exhibits strong interaction with light - large De
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Bragg diffraction
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Bragg mirrorAntireflection coatings (Fresnel equations)
Bragg’s law:
2naveragedcos() ml
Each succesive layer gives phase-shifted partial reflection
Interference at Bragg’ conditionyields 100% reflection
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d/l
Refl
ecti
vit
y
Dielectric mirror
R arbitrarily close to 100%, independent of index contrast
(N-1) end-facet Fabry Perot fringes
12 layers
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Folding bands of 1D system
freq
uen
cy ω
wave vector k
0 π/a-π/a
Wave vector K is equal to K+m2p/a“Bloch theorem”
2p/a 2p/a
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Defects trap light
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1D stack calculation Semiconductor micropillar
N layers, defect [thicker high index layer], again N layers
Localized resonance caught between mirrors
“Defect state” compare semiconductor donors & acceptors
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2D and 3D lattices
• Dielectric constant is periodic on a 2D/3D lattice
a1
a2
Bravais lattices2D: square, hexagonal, rectangular, oblique, rhombic
3D: sc, fcc, bcc, etc (14 x)
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Reciprocal lattice
a1
a2
Reciprocal lattice withproperty
Special wave vectors ofscale b ~ 2p/a
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Example of reciprocal lattice vectors
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Note how:
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Bloch’s theorem
Suppose we look for solutions of:with periodic e(r)
Bloch’s theorem says the solution must be invariant up toa phase factor when translating over a lattice vector
Truly periodicPhase factor
Equivalent:
1. Note how k and k+G are really the same wave vector 2. Note how this “k” is exactly the K in the 1D eigenvalue eiKd
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Note how k and k+G are really the same wave vector
‘Band structure’
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Folding bands of 1D system
freq
uen
cy ω
wave vector k
0 π/a-π/a
Bloch wave with wave vector k is equal to Bloch wave with wave vector k+m2p/a
2p/a 2p/a
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Formal derivation in 3D
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Wave eq:
Bloch:
Substitute and find:
The structure is that of an infinite dimensional linear problem
frequency acts as the eigenvalue
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Dispersion relation of vacuum -folded
freq
uen
cy ω
wave vector k
0 π/a-π/a
Folded “Free dispersion relation”
2p/a
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Nearly free dispersion
freq
uen
cy ω
wave vector k
0 π/a-π/a
Crossing -> anticrossing upon off-diagonal couplingCompare QM: degeneracies are lifted by perturbation
2p/a
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More complicated example
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FCC crystalclose packed connected spheres
1st Brillouin zone (bcc cell)Wedge: irreducible part
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Folded bands – almost of vacuum
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1st Brillouin zone (bcc cell)Wedge: irreducible part
k
a/l
Example: n=1.5 spheres, (26% air)
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Folded bands – almost of vacuum
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k
a/l
Example: n=1.5 spheres, (26% air)
Effective medium /
metamaterial
Diffractive / Photonic crystal
Spagghetti
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Folded bands – almost of vacuum
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k
a/l
Example: n=1.5 spheres, (26% air)
Bragg gap atnormal incidence to 111 planes
2naveragedcos() ml
1/cos() shift
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Wider bands
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a/l
Reversing air & glass to reduce the mean epsilon
Note (1): band shifts up – lower effective index(2): Relative gap broadens
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Wider bands
Replacing n=1.5 by n=3.5, keeping ~ 80% air `airholes’A true band gap FOR ALL wave vectors opens up
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Band gap for air spheres in Si
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We counted all the eigenstates in the 1st Brillouin zoneNote (1) a true gap, and (2) regimes of very high state density
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Si 3D photonic crystals
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1. Colloids stack in fcc crystals2. Silicon infilling with CVD3. Remove spheres
Vlasov/Norris Nature 2001Technique pioneered at UvA(Vos & Lagendijk, 1998)
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Woodpile crystals
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Silicon – repeated stacking, foldingSandia, Kyoto
GaAs (also n=3.5) – robotics in SEM
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Zijlstra, van der Drift, De Dood, and Polman (DIMES, FOM)
2D crystals
Si or GaAs membranesVery thin (200 nm)Kyoto, DTU, Wurzburg,…
Si posts in air (AMOLF)
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Dielectric rod structure
TM means E out of plane
Snapshots of field at band edges
(k = G/2 ) for G=X =2p/a(1,0)
G=M =2p/a(1,1)
Note the phase increment due to k
Note field concentration in air (band 2)
rod (band 1)
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Why all the effort for just a lot of math?
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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 howfast 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
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2D crystal
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1. In a 2D crystalpolarization splitsinto TE and TM
2. Band structure isfor in-plane k-only
3. ‘Light-line’ separates bound from leaky
Photonic bandgap
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Line defects
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A single line surrounded by a full band gap guides lightThe band gap forbids it from escaping into the crystal
0 0.1 0.2 0.3 0.4 0.50.15
0.2
0.25
0.3
0.35
wavevector |k| (units p/a)
freq
uen
cy
w(u
nit
s 2
pc/
a)
crystalmodes
crystalmodes
waveguidemodes
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Line defects and bends
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Line defect, bend
E
Exceptionally tight – 90o bendPotential for very small low-loss chips
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Measurement of guiding & bending
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Sample: AIST JapanMeas: AMOLF
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January 2007 34
Free standing GaAs membrane250 nm thick, 800 mm long, 30 mm wide
1 row of holes missing
Cavity in experiment
Lattice spacing
a=410 nm
Pink areas: a=400 nm
Song, Noda, Asano, Akahane, Nature Materials
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January 2007 35
Why a cavity ?
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January 2007 36
Simulated mode
Mode intensityCycle averaged |E|2
Mode volume 1.2 (l/nGaAs)3
Exceptionally small cavityVery high Q, up to 106
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January 2007 37
Narrow cavity resonance
Tip: pulled glass fiber ~ 100 nm
Laser: grating tunable diode laser20 MHz linewidth (10-7 l) around 1565 nm
1565.0 1565.2 1565.40
25
50
75
100
125
Co
unts
Wavelength (nm)
Picked up by tip
Few mm above cavity Q =(10.5) 105
Lorentz Q =88000
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Refraction
n1 n2
Generic solution steps:1) Plane waves in each medium2) Use k|| conservation to find allowed waves3) Use causality to keep only outgoing waves -> refracted k 4) Match field continuity at boundary to find r and t
n1w/cn2w/c
k||
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Folding bands of 1D system
freq
uen
cy ω
wave vector k
0 π/a-π/a
Bloch wave with wave vector k is equal to Bloch wave with wave vector k+m2p/a
2p/a 2p/a
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Harrison’s construction
January 2007 40
In 2D free space, the dispersion w=c|k| looks like a circle at any w
Periodic system: repeated zone-scheme brings in new bands
At crossings: coupling splits bands
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Observation of band folding
January 2007 41
Angle resolved map of fluorescence from a corrugated
emitted into modes of the 2D plane,
Common application: LED light extraction
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Measurement of gap
January 2007 42
Folded band structure of a surface plasmon crystal
Probed dispersion at a single l
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Refraction
January 2007 43
A single incident beam can split into multiple refracted beamsGroup velocity = direction of energy flow not along k
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Refraction
January 2007 44
Superprism: exceptional sensitivity to incidence and lSupercollimation: exceptional non-sensitivity to incidence and l
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Super collimation
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A tightly focused beam has many Dk, and should diffractSupercollimation: beam stays collimated because vg is flatNote: there is no guiding defect here
Shanhui Fan, Stanford
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Superprisms
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Application: a minute change in w is a huge change in ‘Wavelength demultiplexer’ – note the negative refraction
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Conclusions
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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 106 optical cycles in a l3 volume