7-2E. Photonic crystals

Purdue Univ, Prof. Shalaev, http://cobweb.ecn.purdue.edu/~shalaev/

Univ Central Florida, CREOL, Prof Kik, http://sharepoint.optics.ucf.edu/kik/OSE6938I/Handouts/Forms/AllItems.aspx

3-D

Λ2-D

1-D

Consider a two-dimensional photonic crystalp y

Bloch theorem

Bloch theorem

Bloch theorem

Bloch theorem

Bloch theorem

Bloch theorem

Photonic BandstructureDispersion curve = Photonic band structure

Photonic Bandstructure

B d #2Bandgap #2

S diBandgap (no transmission) Standing wavevgroup=0

Long wavelengthlimit: effective indexlimit: effective index

Dispersion curve = Photonic band structure

Remind the Dispersion Curve of Slab Waveguide

Dispersion curve = Photonic band structure

Remind the Dispersion Curve of Slab Waveguide

Because guiding modes redistribute themselves with

Band structure

frequency, for small ω, the dispersion curve of guiding modes approaches the cladding line;

For large ω, it approaches theFor large ω, it approaches the core line.

Dispersion curve = Photonic band structure

Photonic band gap

Origin of Photonic Band Gap (PBG)

Light in 1-D photonic crystal

Origin of Photonic Band Gap (PBG)

H L H L H L

Photonic band gap

B R fl ti

Photonic band gap

Bragg Reflection

2 ( )B Bnd Sinλ θ= ⋅2k π π

~ 2B dλ BB

kdλ

= =

B Diff ti

Photonic band gap

Bragg Diffraction

Wavelength does not correspond to the period

Reflected waves are not in phase

Wavelength corresponds to the period.

R fl t d i h Reflected waves are not in phase.

Wave propagates through.Reflected waves are in phase.

Wave does not propagate inside.

Electron Energy gapPhotonic band gap

Electron Energy gap

2h 2

2E k

m=h

Gap in energy spectra of electrons arises in periodic structureGap in energy spectra of electrons arises in periodic structure

PBG formationPhotonic band gap

1. Dispersion curve for free space 3. At the band edges, standing waves form, with the energy being either in the high or the low index regions

2. In a periodic system, when half the

aka πλ ==2

p ywavelength corresponds to the periodicity

the Bragg effect prohibits photon4. Standing waves transport no energy

the Bragg effect prohibits photon propagation.

with zero group velocity

Dispersion relation

Dispersion curve = Photonic band structure

Dispersion relationhi h i d t i ln1: high index material

n2: low index material4. Standing waves transport no energy with zero group velocityω

n1 n2 n1 n1 n1n2 n2standing wave in n2Air band

bandgapStop band bandgap

standing wave in n1

p

Dielectric band

0 π/a

g 1Dielectric band

kπ/a

Dispersion Relation

Dispersion curve = Photonic band structure

Dispersion RelationPlot the dispersion curves for both the positive and the negative sides, and then shift the curve segments with |k|>π/a upward or downwardand then shift the curve segments with |k|>π/a upward or downward one reciprocal lattice vectors.

This reduced range of wave vectors is called the “Brillouin zone”This reduced range of wave vectors is called the Brillouin zone

2-D Photonic Crystals

1. In 2-D PBG, different layer spacing, a, can be met along different direction. Strong interaction occurs when λ/2 = a.direction. Strong interaction occurs when λ/2 a.

2. PBG (Photonic band gap) = stop bands overlap in all directions( g p) p p

B d Di

2D Photonic band structure

Band Diagram

Air band

Stop band

Dielectric band

2D Photonic band structure

2D Photonic band structure

2D Photonic band structure

2D Photonic band structure

Four Possible Functionalities of PBG1. Stop band

1. Use of Stop Band

1. Stop Band: Use PBG as high reflectivity Stop bandUse PBG as high reflectivity omni-directional mirror (PBG waveguides)( g )

2 Use of Dielectric Band

2. Dielectric band

2. Use of Dielectric Band

2. Dielectric Band: Uses the strong dispersion available i h t i t lin a photonic crystal(dispersion engineeringwith form birefringence)

Dielectric bandwith form birefringence)

2. Dielectric band

Remind the dispersion relation in bulk media2. Dielectric band

1. In a homogeneous material in absence of material dispersion n(ω)=constant =n, the di i di i i l t i ht lidispersion diagram is simply a straight line: ω=kc/n.

2. In 2D systems, one can think of this line as a cone. For a given frequency ω, this cone becomes a constant frequency circle.

2. Dielectric band

ky

kx

Wave propagation in k-space2. Dielectric band

Real spacep

The wave vector diagram tells us the direction and magnitude of the refracted and reflected beams. Their direction is normal to the iso-frequency curve and corresponds to Snell’s law.

2. Dielectric band

2. Dielectric band

2. Dielectric band

2. Dielectric band

3. Air band

3. Use of Air Band

3. Air Band : Couples to radiative modes for light extraction from high-efficiency LEDs Air bandfrom high-efficiency LEDsand fiber coupling.

3. Air band

4 Use of Defect Band

4. Defect band

4. Use of Defect Band

4. Defect Band : Couples to waveguide/cavity modes for Defect bandspectral control such as PBG point defect laser or PBG line defect filter, etc.

Line Defect PBG Waveguide 4. Defect band

Defect modes in stop band

Dispersion diagram of W1 line-defect photonic crystal waveguide:Waveguide modes exist within the bandgapWaveguide modes exist within the bandgap.

Photons are prohibited in the 2D PBG,which lead to lossless confinement ofwhich lead to lossless confinement of photons in the line defect area.

Defects in PBG4. Defect band

4. Defect band

4. Defect band

4. Defect band

4. Defect band

3D Photonic band structure

3D Photonic materialsS.Noda, Nature (1999) K. Robbie, Nature (1996)

E. Yablonovitch, PRL(1989)

Artificial Phonic Structure

3D Photonic band structure

Artificial Phonic StructureE.Yablonovitch et al., PRL (1987, 1991)

Fabrication of artificial fcc material and band gap structure for suchand band gap structure for such

material.

Bragg diffraction through all electromagnetic regionBragg diffraction through all electromagnetic region

Natural Opals

Artificial Opal3D Photonic band structure

Artificial Opal

Artificial opal sample (SEM Image)Several cleaved planes of fcc structure are shown

Fabrication of artificial opals3D Photonic band structure

Fabrication of artificial opals

There are 3 in-layer position

Silica spheres settle in close packed hexagonal

A – red; B – blue; C –green;Layers could pack inf l tti ABCABC ACBACBp g

layers fcc lattice: ABCABC or ACBACBhcp lattice: ABABAB

Inverted Opals3D Photonic band structure

Inverted Opals

Inversed opals obtain greater dielectric contrast than opals.

Band structure of diamond lattice3D Photonic band structure

Ph i b d f di d l i ( f i i d 3 45)Photonic band structure of diamond lattice (refractive index ~3.45)John et. al. PRE (1998)

PCF

Photonic Crystal FibersPhotonic Crystal Fibers

PCF

PCF

The fiber supports a single mode over the range of at least 458-1550nm!

PCF

PCF

PCF

PCF

PCF