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Photonic Crystals (Photonic Band-gap Materials)

and Nanophotonics

Y. Tzeng Auburn University Auburn, Alabama

USA July 2003

http//ab-initio.mit.edu/photons/

photonic crystals (also known as photonic

band-gap materials). Photonic crystals are

periodic dielectric structures that have a band

gap that forbids propagation of a certain

frequency range of light. This property enables

one to control light with amazing facility and

produce effects that are impossible with

conventional optics. Photonic crystals are

described exactly by Maxwell's Equations, which

we can (and do) solve by the application of

massive computational power.

Introduction The MIT Photonic-Bands (MPB)

package is a free program for computing the band

structures (dispersion relations) and

electromagnetic modes of periodic dielectric

structures, on both serial and parallel

computers. It was developed by Steven G. Johnson

at MIT in the Joannopoulos Ab Initio Physics

group. This program computes definite-frequency

eigenstates of Maxwell's equations in periodic

dielectric structures for arbitrary wavevectors,

using fully-vectorial and three-dimensional

methods. It is especially designed for the study

of photonic crystals (a.k.a. photonic band-gap

materials), but is also applicable to many other

problems in optics, such as waveguides and

resonator systems. (For example, it can solve for

the modes of waveguides with arbitrary

cross-sections.)

http//ab-initio.mit.edu/mpb/

http//people.bu.edu/theochem/rabani.pdf

http//people.bu.edu/theochem/rabani.pdf

http//people.bu.edu/theochem/rabani.pdf

In a photonic crystal the dielectric constant is

periodic, ie., e(r) e(rR). In the band

structure calculation using the Schrodingers

equation for a silicon crystal, the potential is

periodic, i.e., u(r) u(rR).

http//ab-initio.mit.edu/photons/

http//ab-initio.mit.edu/photons/

For a range of frequencies, there is no solution

to the equation having a real value of k. That

is, there is not a real plane wave solution to

the equation. These frequencies are forbidden

and form a band gap in the k vs ? disperion plot.

In analog to semiconductor crystals, this

periodic media is called a photonic crystal or a

photonic bandgap material.

http//ab-initio.mit.edu/photons/

http//ab-initio.mit.edu/photons/

http//ab-initio.mit.edu/photons/

http//ab-initio.mit.edu/photons/

http//ab-initio.mit.edu/photons/

http//ab-initio.mit.edu/photons/

http//ab-initio.mit.edu/photons/

Resonant Cavities By making point defects in a

photonic crystal, light can be localized, trapped

in the defect. The frequency, symmetry, and other

properties of the defect mode can be easily tuned

to anything desired.

Such a point defect, or resonant cavity, can be

utilized to produce many important effects. For

example, it can be coupled with a pair of

waveguides to produce a very sharp filter

(through resonant tunnelling). Point defects are

at the heart of many other photonic crystal

devices, such as channel drop filters. Another

application of resonant cavities is enhancing the

efficiency of lasers, taking advantage of the

fact that the density of states at the resonant

frequency is very high (approaches a delta

function).

By changing the size or the shape of the defect,

its frequency can easily be tuned to any value

within the band gap. Moreover, the symmetry of

the defect can also be tuned. By increasing the

amount of dielectric in the defect, one can pull

down higher-order modes, corresponding to the s,

p, d, etcetera states in atoms. Here, we see a

p-like state in a two-dimensional photonic

crystal (square lattice of rods). The defect was

created by increasing the radius of the center

rod by 50.

http//ab-initio.mit.edu/photons/resonant-cavities

.html

http//ab-initio.mit.edu/photons/

http//ab-initio.mit.edu/photons/

Waveguide Bends in Photonic Crystals In

conventional waveguides, such as fiber-optic

cables, light is confined by total internal

reflection (also known as index confinement, a

more accurate term when the guide diameter is on

the order of the wavelength). One of the

weaknesses of such waveguides, however, is that

creating bends is difficult. Unless the radius of

the bend is large compared to the wavelength,

much of the light will be lost. This is a serious

problem in creating integrated optical

"circuits," since the space required for

large-radius bends is unavailable.

Photonic crystal waveguides operate on a

different principle. A linear defect is created

in the crystal which supports a mode that is in

the band gap. This mode is forbidden from

propagating in the bulk crystal because of the

band gap. (That is, waveguides operate in a manner

similar to resonant cavities, except that they

are line defects rather than point defects.)

Below, we see the dispersion relation for the

guided mode created in a 2d photonic crystal

(square lattice of rods) by removing a row of

rods

http//ab-initio.mit.edu/photons/bends.html

Single mode optical waveguide in photonic

crystals

http//www.ntt.co.jp/RD/OFIS/active/2002pdfe/ct34_

e.pdf

One-dimensionally Periodic Structures By adding

a periodic structure to a conventional waveguide,

it is possible to create a one-dimensionally

periodic photonic crystal. Such structures can be

used as high-Q filters.

One-Dimensional Photonic Crystals Below is an SEM

image of a air bridge structure fabricated (and

imaged) by K. Y. Lim, G. Petrich, and L.

Kolodziejski in CMSE. This is a conventional

waveguide, surrounded by air, into which a

regular set of holes have been punched (the

center to center spacing between the holes is 1.8

µm). The periodic structure provided by these

holes creates a band gap, a one-dimensionally

periodic photonic crystal (often referred to as

simply a "one-dimensional" photonic crystal,

although it exists in three dimensions). In the

center of the bridge, a slight defect in the

crystal has been formed by leaving extra space

between the holes. This forms a resonant cavity

which can be used as a filter. (This is a

prototype design for 4.5 µm light later designs

have been scaled down for wavelengths of 1.5 µm.)

http//ab-initio.mit.edu/photons/1d-crystal.html

http//ab-initio.mit.edu/photons/

http//ab-initio.mit.edu/photons/

http//ab-initio.mit.edu/photons/

http//ab-initio.mit.edu/photons/

http//ab-initio.mit.edu/photons/

http//ab-initio.mit.edu/photons/

The Color of Shock Waves in Photonic Crystals

New physical effects occur when light interacts

with a shock wave propagating through a

one-dimensional photonic crystal. These new

phenomena include frequency shifts of light

across the photonic crystal bandgap, the

bandwidth narrowing of an arbitrary input signal

with 100 efficiency. It is also possible to slow

down the speed of light propagation by orders of

magnitude.

Movie http//ab-initio.mit.edu/photons/shocked_PC

/fastup_final_thumb

http//ab-initio.mit.edu/photons/shocked_PC/shocke

d_PC.html

http//ab-initio.mit.edu/photons/

http//ab-initio.mit.edu/photons/

http//ab-initio.mit.edu/photons/

http//ab-initio.mit.edu/photons/

http//ab-initio.mit.edu/photons/

http//ab-initio.mit.edu/photons/

http//ab-initio.mit.edu/photons/

Photonic Crystals Periodic Surprises in

Electromagnetism Steven G. Johnson a one-week

seminar (five 1.5-hour lectures) MIT MRS

Chapter, 2003 IAP tutorial series, organized by

Ion Bita This course introduced light

propagation in periodic systems, photonic

crystals and band gaps, localized defect states,

3d fabrication technology, hybrid structures and

index guiding, and photonic-crystal fibers, among

other topics. This crash course will introduce

Bloch's theorem for electromagnetism, photonic

band gaps, the confinement of light in novel

waveguides and cavities by synthetic optical

"insulators," startling sub-micron fabrication

advances, exotic optical fibers, and will upend

what you thought you knew about total internal

reflection. We will focus less on gory

differential equations than on high-level

approaches such as linear algebra, variational

theorems, conservation laws, and coupled-mode

theory the course should be accessible to anyone

with a grasp of basic electromagnetism and who

does not quake in fear at the word "eigenvalue."

http//ab-initio.mit.edu/photons/tutorial/

http//ostc.physics.uiowa.edu/prineas/Poster2.pdf

http//nano.sandia.gov/pdf_docs/CINT_photon20-20

allpdfversion.pdf

http//nano.sandia.gov/pdf_docs/CINT-all-about.pdf

http//nano.sandia.gov/NCINTphotonics.htm

http//users.ece.gatech.edu/alan/11-27-Yeo20Phot

onic20Bandgap20Fabrication.pdf

Resonant photonic bandgap structures http//ostc.p

hysics.uiowa.edu/prineas/Poster2.pdf Fabrication

of photonic bandgap structures http//users.ece.ga

tech.edu/alan/11-27-Yeo20Photonic20Bandgap20Fa

brication.pdf Finite photonic crystal theory and

simulation http//www.ifh.ee.ethz.ch/erni/PDF_Pap

er/NCCR_Tut_9C_DEF.pdf MIT photonics

tutorial http//ab-initio.mit.edu/photons/tutorial

/photonic-intro.pdf Perfect Channel Drop

Filters http//ab-initio.mit.edu/photons/ch-drop.h

tml Movie for MIT photonic crystal tutorial

5 http//ab-initio.mit.edu/photons/tutorial/fink-

10.6.wmv

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