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Photonic crystals

Femius KoenderinkCenter for NanophotonicsAMOLF, Amsterdamf.koenderink@amolf.nl

Semi-conductor crystals for light

The smallest dielectric – lossless structures to control whereto and how fast light flows

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

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

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

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

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

More complicated example

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FCC crystalclose packed connected spheres

1st Brillouin zone (bcc cell)Wedge: irreducible part

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)

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

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

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

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

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)

Woodpile crystals

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Silicon – repeated stacking, foldingSandia, Kyoto

GaAs (also n=3.5) – robotics in SEM

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)

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

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

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

Line defects and bends

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Line defect, bend

E

Exceptionally tight – 90o bendPotential for very small low-loss chips

Measurement of guiding & bending

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Sample: AIST JapanMeas: AMOLF

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

January 2007 35

Why a cavity ?

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

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

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

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

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

Measurement of gap

January 2007 42

Folded band structure of a surface plasmon crystal

Probed dispersion at a single l

Refraction

January 2007 43

A single incident beam can split into multiple refracted beamsGroup velocity = direction of energy flow not along k

Refraction

January 2007 44

Superprism: exceptional sensitivity to incidence and lSupercollimation: exceptional non-sensitivity to incidence and l

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

Superprisms

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Application: a minute change in w is a huge change in ‘Wavelength demultiplexer’ – note the negative refraction

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