= 2 na sin ⋅ ( θ

(1)

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 Applications 28. Selected Modern Applications

(Optical crystal 과는 다른 개념이다.)

(2)

Λ

1-D

2-D

3-D

(3)

(4)

We know the origin of electronic Energy Band Gaps

Æ Gap in energy spectra of electrons arises in periodic structure Æ Origin of energy gap : Bragg reflection of electron waves

2 2

E 2 k

= =m

Conduction band

Valence band Band gap

Energy of free electrons Electron energy in crystal

Periodic lattice structure

a ~ nm

(5)

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

(6)

Bragg reflection in crystals

Incident wave

Wave

wave is such that

Origin of the energy band gap

(7)

We know the Bragg condition :

λ

_B

= 2 na sin ⋅ ( θ

_B

)

B

2 na

λ = 2

B

k a

π π

= λ =

If θ ^{= 90 deg.}

a

(8)

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

Dispersion relation of a EM wave in free space

ω

c k ω = n

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

(9)

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

0 π/a

ω

k

(10)

Natural Opals

(11)

a k

a π

λ 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 formation

Photonic band gap

(12)

n₁ n₂ n₁ n₂ n₁ n₂ n₁

Dispersion relation

n₁: high index material n₂: low index material bandgap

0 π/a

standing wave in n₁ standing wave in n₂

4. Standing waves transport no energy with zero group velocity

ω

k

Dispersion curve = Photonic band structure

(13)

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

(14)

2D Photonic band structure

(15)

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

(16)

Band Diagram

Stop band Air band

Dielectric band

(17)

(18)

(19)

Four Possible Functionalities of PBG 1. Use of Stop Band

1. Stop Band:

Use PBG as high reflectivity omni-directional mirror

(PBG waveguides)

Stop band

1. Stop band

(20)

2. Dielectric Band: Uses the strong dispersion available in a photonic crystal

(dispersion engineering with form birefringence)

Dielectric band

2. Use of Dielectric Band

2. Dielectric band

(21)

(22)

(23)

3. Use of Air Band

3. Air band

3. Air Band : Couples to radiative modes for light extraction

from high-efficiency LEDs and fiber coupling.

Air band

(24)

3. Air band

(25)

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

(26)

Defects in PBG

4. Defect band

(27)

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

(28)

(29)

4. Defect band

(30)

3D Photonic materials

S.Noda, Nature (1999) K. Robbie, Nature (1996)

E. Yablonovitch, PRL(1989)

(31)

Artificial Phonic Structure

E.Yablonovitch et al., PRL (1987, 1991)

Fabrication of artificial fcc material and band gap structure for such

material.

(32)

Artificial Opal

Artificial opal sample (SEM Image)

Several cleaved planes of fcc structure are shown

(33)

Fabrication of artificial opals

Silica spheres settle in close packed hexagonal

layers

There are 3 in-layer position A – red; B – blue; C –green;

Layers could pack in

fcc lattice: ABCABC or ACBACB hcp lattice: ABABAB

(34)

Inverted Opals

Inversed opals obtain greater dielectric contrast than opals.

(35)

PCF

Photonic Crystal Fibers

(36)

PCF

(37)

PCF

(38)

PCF