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Photonic Crystals and
Negative Refraction
Dane Wheeler
Jing Zhang
Introduction
 Photonic crystals are
materials with a
periodic dielectric
constant in one, two,
or three dimensions
 Like semiconductors
e 1 e 2 e 1 e 2 e1 e2 e1 e2 e1 e2 e1 e2
with a periodic
potential, photonic
crystals exhibit a
a
e(x) = e(x+a)
band gap
1-D
2-D
periodic in
one direction
periodic in
two directions
3-D
periodic in
three di rections
Johnson, S. G., “Photonic Crystals: Periodic Surprises in Electromagnetism”
Motivation
Perfect waveguide bends
Perfect channel-drop filters
Negative refraction
Perfect lens/lithography
Resonant cavities
Optical logic
All-optical transistors
Yokohama National University/Baba Research Lab
Origin of Photonic Band Gap
1

 E  
H i H
c t
c
1

 H  e
E  J  i eE
c t
c
 
    H    H
 c 
e
1
eigen-operator
2
eigenvalue
Faraday’s Law
Ampere’s Law
Schrödinger-like
Maxwell equation
eigen-state
Photonic Band Structure
Solution leads to photonic band structure
Previous Work - MPB
MIT has developed the Photonic Bands
(MPB) package to calculate photonic band
structures
MPB takes frequency domain approach to
calculating eigenstates of Maxwell’s
equations – each field has a definite
frequency
Offers computational advantages over
time-domain approaches
Previous Work – Negative Refraction
Cubukcu, et al. have experimentally
demonstrated negative refraction by a
photonic crystal
Structure is a square array of alumina rods
in the air
Cubukcu, et al., Nature 423, 604-605 (2003).
3D Structures – Inverse Opal
Self-assembled silica opals
grown on silicon substrate
LPCVD is used to fill opal
template with silicon; wet
etching yields inverse opal
silicon structure
Y. A. Vlasov, et al., Nature 414, 289-293 (2001).
3D Structures – Wire Mesh
 Copper wire diamond
mesh structure
 Exhibits microwave
band gap
 Also exhibits cutoff
frequency around 6-7
GHz
 Able to produce large
crystals – 18 x 18 x 7
cm (1 cm bonds)
D. F. Sievenpiper, et al., Phys. Rev. Lett. 76, 2480–2483 (1996).
3D PhC based on etched DBR
 X-Y plane: Triangular array of
holes
Z direction: Distributed Bragg
Reflectors (DBR)
 Materials: GaAs (e1 = 14.44 )
and oxidized AlAs (e2 = 2.25)
for large contrst.
 Dimension Data: R/a is
0.275 and l1 / l2 is 1.69 for a
common band gap.
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
Computation methods for band structures
 Plane Wave Expansion (PWE):
Modified Maxwell’s Equation
H field expanded in plane waves
Eigen function to obtain band structure
 FDTD: Finite Difference Time Domain
Electro-magnetic fields calculated at a given
instant in time
Calculated band structure with PWE method
Computation with MPB program for same structure
Refractive index calculated from Band
Structure
Central Dilemma:
d/d|k| < 0
Vg · k < 0 as Vg = ∂ / ∂k
Left Handed Material: E  H · k < 0
E  H: Poynting Vector, describing the magnitude and
direction of the flow of energy.
Refractive Index: n = sign(Vg · k) c |k| / 
Frequency contour in k space
Central Dilemma:
Convergent frequency contour in k space
gives negative refractive index.
Refractive index and corresponding band
structure
Summary
 Photonic crystals modulate light by modulating periodic
structure and consequently photonic band diagram.
 Advantages of photonic crystals:
- Can be fabricated with wide range of materials.
- Structure possibilities are limited only by human
imagination
 Wide applications
 Novel 3D photonic crystal structure can exhibit
overlapping band gaps along main crystal axes.
 Negative refractive index exists within certain frequency
range.