Photonic Crystals and Negative Refraction

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Photonic Crystals and Negative Refraction

• Dane Wheeler
• Jing Zhang

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Introduction

• Photonic crystals are materials with a periodic
  dielectric constant in one, two, or three
  dimensions
• Like semiconductors with a periodic potential,
  photonic crystals exhibit a band gap

Johnson, S. G., Photonic Crystals Periodic
Surprises in Electromagnetism
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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
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Origin of Photonic Band Gap
Faradays Law
Amperes Law
Schrödinger-like Maxwell equation
eigen-operator
eigenvalue
eigen-state
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Photonic Band Structure

• Solution leads to photonic band structure

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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 Maxwells equations
  each field has a definite frequency
• Offers computational advantages over time-domain
  approaches

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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).
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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).
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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,
24802483 (1996).
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3D PhC based on etched DBR

• X-Y plane Triangular array of holes
• Z direction Distributed Bragg Reflectors
  (DBR)
• Materials GaAs (?1 14.44 ) and oxidized AlAs
  (?2 2.25) for large contrst.
• Dimension Data R/a is 0.275 and l1 / l2 is
  1.69 for a common band gap.

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Computation methods for band structures

• Plane Wave Expansion (PWE)
• Modified
  Maxwells 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

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Calculated band structure with PWE method
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Computation with MPB program for same structure
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Refractive index calculated from Band Structure

• Central Dilemma d?/dk lt 0
• Vg k lt 0 as Vg
  ?? / ?k
• Left Handed Material E ? H k lt 0
• E ? H Poynting Vector, describing the
  magnitude and
• direction of the flow of
  energy.
• Refractive Index n sign(Vg k) c k / ?

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Frequency contour in k space

• Central Dilemma
• Convergent frequency contour in k
  space
• gives negative refractive index.

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Refractive index and corresponding band structure
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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.