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

28. Selected Modern Applications28. Selected Modern Applications

(Optical crystal 과는 다른 개념이다.)

Λ

1-D

2-D

3-D

We know the origin of electronic Energy Band Gaps

Gap in energy spectra of electrons arises in periodic structureOrigin of energy gap : Bragg reflection of electron waves

22

2E k

m= Conduction band

Valence band

Band gap

Energy of free electrons Electron energy in crystal

Periodic lattice structure

a ~ nm

Wavelength does not correspond to the period

Reflected waves are not in phase.

Wave propagates through.

Wavelength corresponds to the period.

Reflected waves are in phase.

Wave does not propagate inside.

Bragg reflection = Bragg diffraction = Bragg scattering

Bragg reflection in crystals

Incident wave

Wavewave is such that

Origin of the energyband gap

We know the Bragg condition : 2 ( )B Bna sinλ θ= ⋅

2B naλ =2

BB

ka

π πλ

= =If θ = 90 deg.

a

In same way, we may define a new terminology : Photonic Band Gap (PBG)

Dispersion relation of a EM wave in free space

ω

c kn

ω =

H L H L H L

Bragg reflection from a periodic index structure

Photonic band gap (PBG)

PBG

Photons with energy in the PBG does not propagate inside the structure.

a ~ wavelength

Therefore, the Photonic crystals mean

Air band

Dielectric band

Band Gap

periodic structures with photonic band gaps (PBG)and their lattice constants are comparable to wavelength

0π/a

ω

k

Natural Opals

aka πλ ==2

1. Dispersion curve for free space

2. In a periodic system, when half the wavelength corresponds to the periodicity

the Bragg effect prohibits photon propagation.

3. At the band edges, standing waves form, with the energy being either in the high or the low index regions

4. Standing waves transport no energy with zero group velocity

PBG formationPhotonic band gap

n1 n2 n1 n1 n1n2 n2

Dispersion relation

n1: high index materialn2: low index material

bandgap

0π/a

standing wave in n1

standing wave in n2

4. Standing waves transport no energy with zero group velocityω

k

Dispersion curve = Photonic band structure

Dispersion Relation

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

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

Dispersion curve = Photonic band structure

2D Photonic band structure

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.

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

Band Diagram

Stop band

Air band

Dielectric band

2D Photonic band structure

Four Possible Functionalities of PBG1. Use of Stop Band

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

Stop band

1. Stop band

2. Dielectric Band: Uses the strong dispersion available in a photonic crystal(dispersion engineeringwith form birefringence)

Dielectric band

2. Use of Dielectric Band

2. Dielectric band

2. Dielectric band

2. Dielectric band

3. Use of Air Band

3. Air band

3. Air Band : Couples to radiativemodes for light extraction from high-efficiency LEDsand fiber coupling.

Air band

3. Air band

4. Use of Defect Band

4. Defect band

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

Defect band

Defects in PBG4. Defect band

Line Defect PBG Waveguide

Waveguide modes exist within the bandgap.

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

Defect modes in stop band

4. Defect band

4. Defect band

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

3D Photonic band structure

E. Yablonovitch, PRL(1989)

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

Fabrication of artificial fcc material and band gap structure for such

material.

3D Photonic band structure

Artificial Opal

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

3D Photonic band structure

Fabrication of artificial opals

Silica spheres settle in close packed hexagonal

layers

There are 3 in-layer positionA – red; B – blue; C –green;Layers could pack infcc lattice: ABCABC or ACBACBhcp lattice: ABABAB

3D Photonic band structure

Inverted Opals

Inversed opals obtain greater dielectric contrast than opals.

3D Photonic band structure

PCF

Photonic Crystal Fibers

PCF

PCF

PCF