Photonic Crystals 1 - Overview & Examples

7/28/2019 Photonic Crystals 1 - Overview & Examples

1/53

Photonic Crystals


2/53

Photonic Crystals

From Wikipedia:

Photonic Crystalsare periodic optical

nanostructures that are designed to affect the

motion of photons in a similar way thatperiodicity of a semiconductor crystal affects the

motion of electrons. Photonic crystals occur in

nature and in various forms have been studiedscientifically for the last 100 years.


3/53

Wikipedia Continued Photonic crystals are composed of periodic dielectric or metallo-dielectric

nanostructures that affect the propagation of electromagnetic waves (EM) in the

same way as the periodic potential in a crystal affects the electron motion by

defining allowed and forbidden electronic energy bands. Photonic crystals

contain regularly repeating internal regions of high and low dielectric constant.

Photons (as waves) propagate through this structure - or not - depending on

their wavelength. Wavelengths of light that are allowed to travel are known as

modes, and groups of allowed modes form bands. Disallowed bands of

wavelengths are called photonic band gaps. This gives rise to distinct opticalphenomena such as inhibition of spontaneous emission, high-reflecting omni-

directional mirrors and low-loss-waveguides, amongst others.

Since the basic physical phenomenon is based on diffraction, the periodicity of

the photonic crystal structure has to be of the same length-scale as half the

wavelength of the EM waves i.e. ~350 nm (blue) to 700 nm (red) for photoniccrystals operating in the visible part of the spectrum - the repeating regions of

high and low dielectric constants have to be of this dimension. This makes the

fabrication of optical photonic crystals cumbersome and complex.


4/53

Photonic Crystals:A New Frontier in Modern Optics

MARIAN FLORESCU

NASA Jet Propulsion LaboratoryCalifornia Institute of Technology


5/53


6/53

Two Fundamental Optical Principles

Localization of Light

S. John, Phys. Rev. Lett. 58,2486 (1987)

Inhibition of Spontaneous EmissionE. Yablonovitch, Phys. Rev. Lett. 58 2059 (1987)

Photonic crystals: periodic dielectric structures.

interact resonantly with radiation with wavelengths comparable to theperiodicity length of the dielectric lattice.

dispersion relation strongly depends on frequency and propagation direction

may present complete band gaps Photonic Band Gap (PBG) materials.

Photonic Crystals

Guide and confine light without losses Novel environment for quantum mechanical light-matter interaction

A rich variety of micro- and nano-photonics devices


7/53

Photonic Crystals History

1987: Prediction of photonic crystals

S. John, Phys. Rev. Lett. 58,2486 (1987), Stronglocalization of photons

in certain dielectric superlatticesE. Yablonovitch, Phys. Rev. Lett. 582059 (1987), I nhi bited spontaneous

emissionin solid state physics and electronics

1990: Computational demonstration of photonic crystal

K. M. Ho, C. T Chan, and C. M. Soukoulis, Phys. Rev. Lett. 65, 3152 (1990)

1991: Experimental demonstration ofmicrowave photonic crystals

E. Yablonovitch, T. J. Mitter, K. M. Leung, Phys. Rev. Lett. 67, 2295 (1991)

1995: Large scale 2D photonic crystals in Visible

U. Gruning, V. Lehman, C.M. Englehardt, Appl. Phys. Lett. 66 (1995)

1998: Small scale photonic crystals in near Visible; Large scale

inverted opals

1999: First photonic crystal based optical devices (lasers, waveguides)


8/53

Photonic Crystals- Semiconductors of Light

Semiconductors

Periodic array of atoms

Atomic length scales

Natural structures

Control electron flow

1950s electronic revolution

Photonic Crystals

Periodic variation of dielectric

constant

Length scale ~

Artificial structures

Control e.m. wave propagation

New frontier in modern optics

N t l Ph t i C t l


9/53

Natural opals

Natural Photonic Crystals:Structural Colours through Photonic Crystals

Periodic structure striking colour effect even in the absence of pigments


10/53

Requirement: overlapping of frequency gaps along different directions

High ratio of dielectric indices Same average optical path in different media

Dielectric networks should be connected

J. Wijnhoven & W. Vos, Science (1998)S. Lin et al., Nature (1998)

Woodpile structure Inverted Opals

Artificial Photonic Crystals


11/53

Photonic Crystals

complex dielectric environment that controls the flow of radiation designer vacuum for the emission and absorption of radiation

Photonic Crystals: Opportunities

Passive devices

dielectric mirrors for antennas

micro-resonators and waveguides

Active devices

low-threshold nonlinear devices

microlasers and amplifiers

efficient thermal sources of light

Integrated optics

controlled miniaturisation

pulse sculpturing


12/53

Defect-Mode Photonic Crystal Microlaser

Photonic Crystal Cavity formed by a point defect

O. Painteret. al., Science (1999)


13/53

3D Complete Photonic Band Gap

Suppress blackbody radiation in the infrared and redirect and enhance thermal energy into visibl

Photonic Crystals Based Light Bulbs

S. Y. Lin et al., Appl. Phys. Lett. (2003)

C. Cornelius, J. Dowling, PRA 59, 4736 (1999)

Modification of Planck blackbody radiation by photonic band-gap structures

Light bulb efficiency may raise from 5 percent to 60 percent

3D Tungsten Photonic

Crystal Filament

Solid Tungsten Filament


14/53

Solar Cell Applications

Funneling of thermal radiation of larger wavelength (orange area) to thermal radiationof shorter wavelength (grey area).

Spectral and angular control over the thermal radiation.


15/53

Fundamental Limitations

switching time switching intensity =

constant Incoherent character of the switching dissipated power

Foundations of Future CI

Cavity all-optical transistor

(3)

IoutIin

IH

H.M. Gibbs et. al, PRL 36, 1135 (1976)

Operating Parameters

Holding power: 5 mW

Switching power: 3 W

Switching time: 1-0.5 ns Size: 500 m

Photonic crystal all-optical transistor

Probe Laser

Pump Laser

Operating Parameters

Holding power: 10-100 nW

Switching power: 50-500 pW

Switching time: < 1 ps

Size: 20 m

M. Floresc u and S. Joh n, PRA 69, 053810 (2004).


16/53

Single Atom Switching Effect Photonic Crystals versus Ordinary Vacuum

Positive population inversion

Switching behaviour of the atomic inversion

M. Flor escu and S. John , PRA 64, 033801 (2001)


17/53

Long temporal separation between incident laser photons

Fast frequency variations of the photonic DOS

Band-edge enhancement of the Lamb shift

Vacuum Rabi splitting

Quantum Optics in Photonic Crystals

T. Yoshi e et al. , Natu re, 2004.

Foundations for Future CI


18/53

Foundations for Future CI:

Single Photon Sources

Enabling Linear Optical Quantum Computing and Quantum Cryptography

fully deterministic pumping mechanism

very fast triggering mechanism

accelerated spontaneous emission

PBG architecture design to achieveprescribed DOS at the ion position

M. Florescu et al., EPL 69, 945 (2005)


19/53

M. Campell et al. Nature, 404, 53 (2000)

CI Enabled Photonic Crystal Design (I)

Photo-resist layer exposed to multiple laser beam interferencethat produce a periodic intensity pattern

3D photonic crystals fabricated

using holographic lithography

Four laser beams interfere to form a

3D periodic intensity pattern

10 m

O. Toader, et al., PRL 92, 043905 (2004)


20/53

O. Toader & S. John, Science (2001)

CI Enabled Photonic Crystal Design (II)


21/53

S. Kennedy et al., Nano Letters (2002)

CI Enabled Photonic Crystal Design (III)


22/53

Transport

Properties:

Photons

ElectronsPhonons

Photonic Crystals

Optical Properties

RethermalizationProcesses:

PhotonsElectrons

Phonons Metallic (Dielectric)

Backbone

Electronic

Characterization

Multi-Physics Problem:

Photonic Crystal Radiant Energy Transfer


23/53

Summary

Designer Vacuum:Frequency selective control of

spontaneous and thermal emission

enables novel active devices

PBG materials: Integrated optical micro-circuits

with complete light localization

Photonic Crystals: Photonic analogues of semiconductors that

control the flow of light

Potential to Enable Future CI:

Single photon source for LOQC

All-optical micro-transistors

CI Enabled Photonic Crystal Research and Technology:

Photonic materials by design

Multiphysics and multiscale analysis


24/53

Wikipedia Continued Photonic crystals are composed of periodic dielectric or metallo-dielectric

nanostructures that affect the propagation of electromagnetic waves (EM) in the

same way as the periodic potential in a crystal affects the electron motion bydefining allowed and forbidden electronic energy bands. Photonic crystals

contain regularly repeating internal regions of high and low dielectric constant.

Photons (as waves) propagate through this structure - or not - depending on

their wavelength. Wavelengths of light that are allowed to travel are known as

modes, and groups of allowed modes form bands. Disallowed bands ofwavelengths are called photonic band gaps. This gives rise to distinct optical

phenomena such as inhibition of spontaneous emission, high-reflecting omni-

directional mirrors and low-loss-waveguides, amongst others.

Since the basic physical phenomenon is based on diffraction, the periodicity of

the photonic crystal structure has to be of the same length-scale as half thewavelength of the EM waves i.e. ~350 nm (blue) to 700 nm (red) for photonic

crystals operating in the visible part of the spectrum - the repeating regions of

high and low dielectric constants have to be of this dimension. This makes the

fabrication of optical photonic crystals cumbersome and complex.

Ph t i C t l


25/53

Photonic Crystals:Periodic Surprises in Electromagnetism

Steven G. Johnson

MIT


26/53

To Begin: A Cartoon in 2d

planewave

E,H

~ei(kxt)

k / c 2

k

scattering


27/53

To Begin: A Cartoon in 2d

planewave

E,H

~ei(kxt)

k / c 2

k

formost , beam(s) propagate

through crystal without scattering

(scattering cancels coherently)

...but forsome (~ 2a), no light can propagate: a photonic band gap

a


28/53

1887 1987

Photonic Crystals

periodic electromagnetic media

with photonic band gaps: optical insulators

2-D

periodic intwo directions

3-D

periodic inthree directions

1-D

periodic inone direction

(need a

more

complex

topology)


29/53

Photonic Crystals


with photonic band gaps: optical insulators

magical oven mitts for

holding and controlling light

3D Photonic Crysta l with De fe c ts

can trap light in cavities and waveguides(wires)


30/53

Photonic Crystals


But how can we understand such complex systems?

Add up the infinite sum of scattering? Ugh!

3D Photo nic C rysta l

Hig h ind e x

o f re fra c tio n

Lo w ind e x

o f re fra c tio n


31/53

A mystery from the 19th century

e

e

E

+

+

+

+

+

JEcurrent:conductivity (measured)

mean free path (distance) of electrons

conductive material


32/53

A mystery from the 19th century

e

e

E

+

JEcurrent:conductivity (measured)

mean free path (distance) of electrons

+ + + + + + +

+ + + + + + + +

+ + + + + + + +

+ + + + + + + +

crystalline conductor(e.g. copper)

10sof

periods!


33/53

A mystery solved

electrons are waves (quantum mechanics)1

waves in aperiodic medium can

propagate without scattering:

Blochs Theorem (1d: Floquets)

2

The foundations do not depend on the specificwave equation.


34/53

Time to Analyze the Cartoon

planewave

E,H

~ei(kxt)

k / c 2

k

formost , beam(s) propagate

through crystal without scattering

(scattering cancels coherently)

...but forsome (~ 2a), no light can propagate: a photonic band gap

a


35/53

Fun with Math

E 1c

tH i

cH

H 1

c

t

EJ i

c

E0

dielectric function (x) = n2(x)

First task:

get rid of this mess

1

H

c

2

H

eigen-operator eigen-value eigen-state

H 0+ constraint


36/53

Hermitian Eigenproblems

1

H c

2

H

eigen-operator eigen-valueeigen-state

H 0+ constraint

Hermitian for real (lossless)

well-known properties from linear algebra:

are real (lossless)

eigen-states are orthogonal

eigen-states are complete (give all solutions)


37/53

Periodic Hermitian Eigenproblems[ G. Floquet, Sur les quations diffrentielles linaries coefficients priodiques,Ann. cole Norm. Sup. 12, 4788 (1883). ]

[ F. Bloch, ber die quantenmechanik der electronen in kristallgittern,Z. Physik52, 555600 (1928). ]

if eigen-operator is periodic, then Bloch-Floquet theorem applies:

H(x

,t)

ei kxt

Hk(x

)can choose:

periodic envelopeplanewave

Corollary 1: kis conserved, i.e.no scattering of Bloch wave

Corollary 2: given by finite unit cell,

so are discrete n(k)

Hk


38/53

Periodic Hermitian EigenproblemsCorollary 2: given by finite unit cell,

so are discrete n(k)

Hk

1

2

3

k

band diagram (dispersion relation)

map of

what states

exist &

can interact

?range ofk?

d


39/53

Periodic Hermitian Eigenproblems in1d

1

2

1

2

1

2

1

2

1

2

1

2

(x) = (x+a)

H(x) eikxHk(x)

a

Considerk+2/a: ei(k 2

a)x

Hk2

a

(x) eikx ei

2

ax

Hk 2

a

(x)

periodic!

satisfies same

equation asHk=Hk

kis periodic:

k+ 2/a equivalent to kquasi-phase-matching

1d


40/53

band gap

Periodic Hermitian Eigenproblems in1d

1

2

1

2

1

2

1

2

1

2

1

2

(x) = (x+a)a

kis periodic:k+ 2/a equivalent to kquasi-phase-matching

k

0 /a/a

irreducible Brillouin zone

1d P i di S h G


41/53

Any1d Periodic System has a Gap

1

k

0

[ Lord Rayleigh, On the maintenance of vibrations by forces of double frequency, and on the propagation ofwaves through a medium endowed with a periodic structure, Philosophical Magazine 24, 145159 (1887). ]

Start with

a uniform (1d) medium:

k

1

1d P i di S h G


42/53


1

(x) = (x+a)a

k

0 /a/a


Treat it as

artificially periodic

bands are foldedby 2/a equivalence

e

ax

, e

ax

cos

ax

, sin

ax

1d P i di S h G


43/53

(x) = (x+a)a1


0 /a


sin

ax

cos

ax

x = 0

Treat it as

artificially periodic

A 1d P i di S h G


44/53

(x) = (x+a)a12 12 12 12 12 12


0 /a


Add a small

real periodicity2 = 1 + D

sin

ax

cos

ax

x = 0

A 1d P i di S h G


45/53

band gap


0 /a


Add a small

real periodicity2 = 1 + D

sin

ax

cos

ax

(x) = (x+a)a12 12 12 12 12 12

x = 0

Splitting of degeneracy:state concentrated in higher index (2)

has lower frequency


46/53

Some 2d and 3d systems have gaps

In general, eigen-frequencies satisfy Variational Theorem:

1(k)

2 minE1

E1 0

ik E12

E1

2

c

2

2(k)2 minE2E2 0

E1* E2 0

" "

kinetic

inverse

potential

bands want to be in high-

but are forced out by orthogonality

>band gap (maybe)


47/53

algebraic interlude completed

I hope you were taking notes*

algebraic interlude

[ *if not, see e.g.: Joannopoulos, Meade, and Winn,Photonic Crystals: Molding the Flow of Light]

2d periodicity =12:1


48/53

2d periodicity, =12:1

E

HTM

a

frequency

(2c/a)

=a/

G X

M

G X M Girreducible Brillouin zone

k

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Photonic Band Gap

TM bands

gap for

n > ~1.75:1



49/53

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Photonic Band Gap

TM bands

2d periodicity, =12:1

E

HTM

G X M G

Ez

+

Ez

(+ 90 rotated version)

gap for

n > ~1.75:1



50/53

2d periodicity, 12:1

E

H

E

H

TM TE

a

frequency

(2c/a)

=a/

G X

M

G X M Girreducible Brillouin zone

k

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Photonic Band Gap

TM bands

TE bands


51/53

2d photonic crystal: TE gap, =12:1

TE bands

TM bands

gap forn > ~1.4:1

E

H

TE

3d h t i t l l t 12 1


52/53

3d photonic crystal: complete gap , =12:1

U L G X W K

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0

21% gap

L'

L

K'

G

W

U'

X

U'' UW' K

z

I: rod layer II: hole layer

I.

II.

[ S. G. Johnson et al.,Appl. Phys. Lett.77, 3490 (2000) ]

gap forn > ~4:1


53/53

You, too, can compute

photonic eigenmodes!

MIT Photonic-Bands (MPB) package:

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

on Athena:

add mpb

Documents

Photonic Crystals 1 - Overview & Examples