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Photonic crystals

Femius Koenderink Center for Nanophotonics AMOLF, Amsterdam f.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

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Definition: A photonic crystal is a periodic arrangement with a ~ λ of a dielectric material that exhibits strong interaction with light - large ∆ε

1D: Bragg Reflector

Examples

2D: Si pillar crystal

3D: Colloidal crystal

Bragg diffraction

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Bragg mirror Antireflection coatings (Fresnel equations)

Bragg’s law:

2naveragedcos(θ) = mλ

Each succesive layer gives phase-shifted partial reflection Interference at Bragg’ condition yields 100% reflection

Conclusions 1D stack

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n1 n2 n1 n1 n1 n2 n2

Stop gap

0 π/a

standing wave in n1

standing wave in n2

Light can propagate into/through 1D stacks, except around - a stop gap around the Bragg condition - in a frequency interval of width

At the band edge: standing waves that stand still In the band edge: no propagating states at all

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Κ [π/d]

Freq

uenc

y [ω

d/2π

c]

M=(M1 .M2)N

Periodicity d=d1+d2 ki=niω/c - frequency and index contrast K sets phase increment per unit cell

exp(iKd)

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d/λ

Refl

ecti

vity

Dielectric mirror

R arbitrarily close to 100%, independent of index contrast (N-1) end-facet Fabry Perot fringes

12 layers

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2D and 3D lattices

• Dielectric constant is periodic on a 2D/3D lattice

a1

a2

Bravais lattices 2D: square, hexagonal, rectangular, oblique, rhombic 3D: sc, fcc, bcc, etc (14 x)

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Reciprocal lattice

a1

a2

Reciprocal lattice with property

Special wave vectors of scale b ~ 2π/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 ε(r)

Bloch’s theorem says the solution must be invariant up to a phase factor when translating over a lattice vector

Truly periodic Phase factor

Equivalent:

Note how k and k+G are really the same wave vector

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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

uenc

y ω

wave vector k

0 π/a -π/a

Bloch wave with wave vector k is equal to Bloch wave with wave vector k+m2π/a

2π/a 2π/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

uenc

y ω

wave vector k

0 π/a -π/a

Folded “Free dispersion relation”

2π/a

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Nearly free dispersion

freq

uenc

y ω

wave vector k

0 π/a -π/a

Crossing -> anticrossing upon off-diagonal coupling Compare QM: degeneracies are lifted by perturbation

2π/a

More complicated example

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FCC crystal close 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/λ

Example: n=1.5 spheres, (26% air)

Folded bands – almost of vacuum

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k

a/λ

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/λ

Example: n=1.5 spheres, (26% air)

Bragg gap at normal incidence to 111 planes

2naveragedcos(θ) = mλ1/cos(θ) shift

Wider bands

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a/λ

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 zone Note (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 crystals 2. Silicon infilling with CVD 3. Remove spheres

Vlasov/Norris Nature 2001 Technique pioneered at UvA (Vos & Lagendijk, 1998)

Woodpile crystals

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Silicon – repeated stacking, folding Sandia, Kyoto

GaAs (also n=3.5) – robotics in SEM

Zijlstra, van der Drift, De Dood, and Polman (DIMES, FOM)

2D crystals

Si or GaAs membranes Very 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 =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)

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 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

2D crystal

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1. In a 2D crystal polarization splits into TE and TM 2. Band structure is for 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 light Light at the line is forbidden from escaping into the crystal

0 0.1 0.2 0.3 0.4 0.5 0.15

0.2

0.25

0.3

0.35

wavevector |k| (units π /a)

freq

uenc

y w

(un

its 2

p c

/a) crystal

modes

crystal modes

waveguide modes

Line defects and bends

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Line defect, bend

E

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

Measurement of guiding & bending

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Sample: AIST Japan Meas: AMOLF

January 2007 35

Free standing GaAs membrane 250 nm thick, 800 µm long, 30 µm 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 36

Why a cavity ?

January 2007 37

Simulated mode

Mode intensity Cycle averaged |E|2

Mode volume 1.2 (λ/nGaAs)3

Exceptionally small cavity Very high Q, up to 106

January 2007 38

Narrow cavity resonance

Tip: pulled glass fiber ~ 100 nm Laser: grating tunable diode laser 20 MHz linewidth (10-7 λ) around 1565 nm

1565.0 1565.2 1565.40

25

50

75

100

125

Coun

ts

Wavelength (nm)

Picked up by tipFew µm above cavity Q =(1±0.5) 105

Lorentz Q =88000

Refraction

n1 n2

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

n1ω/c n2ω/c

k||

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Folding bands of 1D system

freq

uenc

y ω

wave vector k

0 π/a -π/a

Bloch wave with wave vector k is equal to Bloch wave with wave vector k+m2π/a

2π/a 2π/a

Harrison’s construction

January 2007 41

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

Observation of band folding

January 2007 42

Angle resolved map of Fluorescence from a corrugated Waveguide Common application: LED light extraction

Observation of band folding

January 2007 43

Angle resolved map of Fluorescence from a corrugated Waveguide Slow-light at Bragg condition – plasmonic crystal laser

Measurement

January 2007 44

Folded band structure of a surface plasmon, probed at one λ

Refraction

January 2007 45

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

Refraction

January 2007 46

Superprism: exceptional sensitivity to incidence θ and λ Supercollimation: exceptional a-sensitivity to incidence θ and λ

Super collimation

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A tightly focused beam has many ∆k, and should diffract Supercollimation: beam stays collimated because vg is flat Note: there is no guiding defect here

Shanhui Fan APL (2003)

Superprisms

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Application: a minute change in ω is a huge change in θ ‘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 λ3 volume