Origin of Photonic Band Gap (PBG)

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

Λ 2-D

1-D

Consider a two-dimensional photonic crystal p y

Bloch theorem

Bloch theorem

Bloch theorem

Bloch theorem

Bloch theorem

Bloch theorem

Photonic Bandstructure

Dispersion curve = Photonic band structure

Photonic Bandstructure

B d #2

Bandgap #2

S di

Bandgap (no transmission) Standing wave v_group=0

Long wavelength limit: effective index limit: 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 the For 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

nd Sin

λ = ⋅ θ

k 2 π π

B

~ 2 d

λ

B

k ⁼ λ ⁼ d

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 gap

Photonic band gap

Electron Energy gap

h2 ²

E 2 k

= hm

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

PBG formation

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

a k

a π

λ 2 = =

p y

wavelength corresponds to the periodicity

the Bragg effect prohibits photon 4. Standing waves transport no energy the Bragg effect prohibits photon

propagation. with zero group velocity

Dispersion relation

Dispersion curve = Photonic band structure

Dispersion relation

hi h i d t i l n₁: high index material

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

ω

n₁ n₂ n₁ n₂ n₁ n₂ n₁ standing wave in n₂

Air band

bandgap Stop band bandgap

standing wave in n₁

p

Dielectric band

0 π/a

g ₁

Dielectric band

π/a k

Dispersion Relation

Dispersion curve = Photonic band structure

Dispersion Relation

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

and 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 PBG

1. Stop band

1. Use of Stop Band

1. Stop Band:

Use PBG as high reflectivity Stop band Use 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 l

in a photonic crystal

(dispersion engineering with form birefringence)

Dielectric band with form birefringence)

2. Dielectric band

Remind the dispersion relation in bulk media

2. 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 li

dispersion 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-space

2. 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 band

from high-efficiency LEDs and 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 band

spectral 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 bandgap Waveguide modes exist within the bandgap.

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

Defects in PBG

4. Defect band

4. Defect band

4. Defect band

4. Defect band

3D Photonic band structure

3D Photonic materials

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

E. Yablonovitch, PRL(1989)

Artificial Phonic Structure

3D Photonic band structure

Artificial Phonic Structure

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

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

material.

Bragg diffraction through all electromagnetic region Bragg diffraction through all electromagnetic region

Natural Opals

Artificial Opal

3D Photonic band structure

Artificial Opal

Artificial opal sample (SEM Image)

Several cleaved planes of fcc structure are shown

Fabrication of artificial opals

3D 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 in

f l tti ABCABC ACBACB

p g

layers fcc lattice: ABCABC or ACBACB hcp lattice: ABABAB

Inverted Opals

3D Photonic band structure

Inverted Opals

Inversed opals obtain greater dielectric contrast than opals.

Band structure of diamond lattice

3D 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 Fibers

Photonic Crystal Fibers

PCF

PCF

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

PCF

PCF

PCF

PCF

PCF