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
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