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 with a periodic potential, photonic crystals exhibit a band gap

1 2 1 2 1 2 1 2 1 2 1 2

(x) = (x+a)a

2-D

periodic intwo directions

3-D

periodic inthree di rections

1-D

periodic inone direction

Johnson, S. G., “Photonic Crystals: Periodic Surprises in Electromagnetism”

Motivation

Perfect waveguide bendsPerfect channel-drop filtersNegative refraction

Perfect lens/lithography

Resonant cavitiesOptical logic

All-optical transistors

Yokohama National University/Baba Research Lab

Origin of Photonic Band Gap

E 1

c

t

H i

c

H

H

1

c

t

E

J i

c

E

1

H

c

2 H

eigen-operator

Faraday’s Law

Ampere’s Law

eigenvalue

Schrödinger-likeMaxwell 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

Y. A. Vlasov, et al., Nature 414, 289-293 (2001).

Self-assembled silica opals grown on silicon substrate

LPCVD is used to fill opal template with silicon; wet

etching yields inverse opal silicon structure

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 (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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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.