Photonic Crystals

Photonic Crystals

Photonic Crystals

From Wikipedia:

“Photonic Crystals are periodic optical nanostructures that are designed to affect the motion of photons in a similar way that periodicity of a semiconductor crystal affects the motion of electrons. Photonic crystals occur in nature and in various forms have been studied scientifically for the last 100 years”.

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 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 the wavelength 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.

Photonic Crystals: A New Frontier in Modern

Optics

Photonic Crystals: A New Frontier in Modern

Optics

MARIAN FLORESCU

NASA Jet Propulsion Laboratory

California Institute of Technology

MARIAN FLORESCU

NASA Jet Propulsion Laboratory

California Institute of Technology

“ If only were possible to make materials in which electromagnetically waves cannot propagate at certain frequencies, all kinds of almost-magical things would happen”

“ If only were possible to make materials in which electromagnetically waves cannot propagate at certain frequencies, all kinds of almost-magical things would happen”

Sir John Maddox, Nature (1990)Sir John Maddox, Nature (1990)

Two Fundamental Optical Principles

• Localization of LightLocalization of LightS. John, Phys. Rev. Lett. 58,2486 (1987)

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

Two Fundamental Optical Principles

• Localization of LightLocalization of LightS. John, Phys. Rev. Lett. 58,2486 (1987)

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

Photonic crystals: periodic dielectric structures. interact resonantly with radiation with wavelengths comparable to the

periodicity 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: periodic dielectric structures. interact resonantly with radiation with wavelengths comparable to the

periodicity 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 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

Guide and confine light without losses Novel environment for quantum mechanical light-matter interaction A rich variety of micro- and nano-photonics devices

Photonic Crystals HistoryPhotonic Crystals History 1987: Prediction of photonic crystals S. John, Phys. Rev. Lett. 58,2486 (1987), “Strong localization of photons in certain dielectric superlattices” E. Yablonovitch, Phys. Rev. Lett. 58 2059 (1987), “Inhibited spontaneous emission in 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 of microwave 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)

1987: Prediction of photonic crystals S. John, Phys. Rev. Lett. 58,2486 (1987), “Strong localization of photons in certain dielectric superlattices” E. Yablonovitch, Phys. Rev. Lett. 58 2059 (1987), “Inhibited spontaneous emission in 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 of microwave 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)

Photonic Crystals- Semiconductors of Light Photonic Crystals- Semiconductors of Light

Semiconductors

Periodic array of atoms

Atomic length scales

Natural structures

Control electron flow

1950’s electronic revolution

Semiconductors

Periodic array of atoms

Atomic length scales

Natural structures

Control electron flow

1950’s electronic revolution

Photonic Crystals

Periodic variation of dielectric constant

Length scale ~

Artificial structures

Control e.m. wave propagation

New frontier in modern optics

Photonic Crystals

Periodic variation of dielectric constant

Length scale ~

Artificial structures

Control e.m. wave propagation

New frontier in modern optics

Natural opalsNatural opals

Natural Photonic Crystals: Natural Photonic Crystals: Structural Colours through Photonic CrystalsStructural Colours through Photonic Crystals

Natural Photonic Crystals: Natural Photonic Crystals: Structural Colours through Photonic CrystalsStructural Colours through Photonic Crystals

Periodic structure striking colour effect even in the absence of pigmentsPeriodic structure striking colour effect even in the absence of pigments

Requirement: overlapping of frequency gaps along different directionsRequirement: overlapping of frequency gaps along different directions High ratio of dielectric indicesHigh ratio of dielectric indices Same average optical path in different mediaSame average optical path in different media Dielectric networks should be connectedDielectric networks should be connected

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

Woodpile structureWoodpile structure Inverted OpalsInverted Opals

Artificial Photonic CrystalsArtificial Photonic Crystals

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

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

Photonic Crystals: Opportunities

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

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

Defect-Mode Photonic Crystal MicrolaserDefect-Mode Photonic Crystal MicrolaserDefect-Mode Photonic Crystal MicrolaserDefect-Mode Photonic Crystal Microlaser

Photonic Crystal Cavity formed by a point defectPhotonic Crystal Cavity formed by a point defect

O. Painter et. al., Science (1999) O. Painter et. al., Science (1999)

3D Complete Photonic Band Gap3D Complete Photonic Band Gap

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

3D Complete Photonic Band Gap3D Complete Photonic Band Gap

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

Photonic Crystals Based Light BulbsPhotonic Crystals Based Light Bulbs

S. Y. Lin et al., Appl. Phys. Lett. (2003)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” 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 percentLight bulb efficiency may raise from 5 percent to 60 percent Light bulb efficiency may raise from 5 percent to 60 percentLight bulb efficiency may raise from 5 percent to 60 percent

3D Tungsten Photonic Crystal Filament

3D Tungsten Photonic Crystal Filament

Solid Tungsten FilamentSolid Tungsten Filament

Solar Cell ApplicationsSolar Cell Applications

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

– Spectral and angular control over the thermal radiation.Spectral and angular control over the thermal radiation.

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

– Spectral and angular control over the thermal radiation.Spectral and angular control over the thermal radiation.

Fundamental LimitationsFundamental Limitations switching time switching time • • switching intensity = switching intensity =

constantconstant Incoherent character of the switching Incoherent character of the switching

dissipated power dissipated power

Fundamental LimitationsFundamental Limitations switching time switching time • • switching intensity = switching intensity =

constantconstant Incoherent character of the switching Incoherent character of the switching

dissipated power dissipated power

Foundations of Future CIFoundations of Future CIFoundations of Future CIFoundations of Future CI

Cavity all-optical transistorCavity all-optical transistor Cavity all-optical transistorCavity all-optical transistor

(3)χ

IoutIoutIinIin

IHIH

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

Operating ParametersOperating Parameters Holding power: 5 mWHolding power: 5 mW Switching power: 3 µWSwitching power: 3 µW Switching time: 1-0.5 ns Switching time: 1-0.5 ns Size: Size: 500 500 m m

Operating ParametersOperating Parameters Holding power: 5 mWHolding power: 5 mW Switching power: 3 µWSwitching power: 3 µW Switching time: 1-0.5 ns Switching time: 1-0.5 ns Size: Size: 500 500 m m

Photonic crystal all-optical transistorPhotonic crystal all-optical transistor Photonic crystal all-optical transistorPhotonic crystal all-optical transistor

Probe LaserProbe Laser

Pump LaserPump Laser

Operating ParametersOperating Parameters Holding power: 10-100 nWHolding power: 10-100 nW Switching power: 50-500 pWSwitching power: 50-500 pW Switching time: < 1 ps Switching time: < 1 ps Size: Size: 20 20 mm

Operating ParametersOperating Parameters Holding power: 10-100 nWHolding power: 10-100 nW Switching power: 50-500 pWSwitching power: 50-500 pW Switching time: < 1 ps Switching time: < 1 ps Size: Size: 20 20 mm

M. Florescu and SM. Florescu and S. John, PRA . John, PRA 6969, 053810 (2004)., 053810 (2004).

Single Atom Switching EffectSingle Atom Switching EffectSingle Atom Switching EffectSingle Atom Switching Effect

Photonic Crystals versus Ordinary VacuumPhotonic Crystals versus Ordinary Vacuum

Positive population inversionPositive population inversion Switching behaviour of the atomic inversionSwitching behaviour of the atomic inversion

Photonic Crystals versus Ordinary VacuumPhotonic Crystals versus Ordinary Vacuum

Positive population inversionPositive population inversion Switching behaviour of the atomic inversionSwitching behaviour of the atomic inversion

M. Florescu and SM. Florescu and S. John, PRA 64, 033801 (2001). John, PRA 64, 033801 (2001)

Long temporal separation between incident laser photonsLong temporal separation between incident laser photons

Fast frequency variations of the photonic DOSFast frequency variations of the photonic DOS Band-edge enhancement of the Lamb shiftBand-edge enhancement of the Lamb shift Vacuum Rabi splittingVacuum Rabi splitting

Long temporal separation between incident laser photonsLong temporal separation between incident laser photons

Fast frequency variations of the photonic DOSFast frequency variations of the photonic DOS Band-edge enhancement of the Lamb shiftBand-edge enhancement of the Lamb shift Vacuum Rabi splittingVacuum Rabi splitting

Quantum Optics in Photonic CrystalsQuantum Optics in Photonic CrystalsQuantum Optics in Photonic CrystalsQuantum Optics in Photonic Crystals

T. Yoshie et al. T. Yoshie et al. , Nature, 2004., Nature, 2004.

Foundations for Future CI:Foundations for Future CI:Single Photon Sources Single Photon Sources

Foundations for Future CI:Foundations for Future CI:Single Photon Sources Single Photon Sources

Enabling Linear Optical Quantum Computing and Quantum CryptographyEnabling Linear Optical Quantum Computing and Quantum Cryptography

fully deterministic pumping mechanismfully deterministic pumping mechanism very fast triggering mechanism very fast triggering mechanism accelerated spontaneous emission accelerated spontaneous emission PBG architecture design to achieve PBG architecture design to achieve

prescribed DOS at the ion positionprescribed DOS at the ion position

Enabling Linear Optical Quantum Computing and Quantum CryptographyEnabling Linear Optical Quantum Computing and Quantum Cryptography

fully deterministic pumping mechanismfully deterministic pumping mechanism very fast triggering mechanism very fast triggering mechanism accelerated spontaneous emission accelerated spontaneous emission PBG architecture design to achieve PBG architecture design to achieve

prescribed DOS at the ion positionprescribed DOS at the ion position

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

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

CI Enabled Photonic Crystal Design (I)CI Enabled Photonic Crystal Design (I)CI Enabled Photonic Crystal Design (I)CI Enabled Photonic Crystal Design (I)

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

3D photonic crystals fabricated using holographic lithography3D photonic crystals fabricated using holographic lithography

Four laser beams interfere to form a 3D periodic intensity pattern

Four laser beams interfere to form a 3D periodic intensity pattern

10 m

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

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

CI Enabled Photonic Crystal Design (II)CI Enabled Photonic Crystal Design (II)CI Enabled Photonic Crystal Design (II)CI Enabled Photonic Crystal Design (II)

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

CI Enabled Photonic Crystal Design (III)CI Enabled Photonic Crystal Design (III)CI Enabled Photonic Crystal Design (III)CI Enabled Photonic Crystal Design (III)

Transport Properties:

Photons ElectronsPhonons

Transport Properties:


Photonic CrystalsOptical PropertiesPhotonic CrystalsOptical Properties

RethermalizationProcesses:


RethermalizationProcesses:


Metallic (Dielectric)Backbone Electronic

Characterization

Metallic (Dielectric)Backbone Electronic

Characterization

Multi-Physics Problem:Multi-Physics Problem:Photonic Crystal Radiant Energy Photonic Crystal Radiant Energy

TransferTransfer

Summary Summary

Designer Vacuum: Frequency selective control of

spontaneous and thermal emission enables novel active devices

Designer Vacuum: Frequency selective control of

spontaneous and thermal emission enables novel active devices

PBG materials: Integrated optical micro-circuits with complete light localization

PBG materials: Integrated optical micro-circuits with complete light localization

Photonic Crystals: Photonic analogues of semiconductors that control the flow of lightPhotonic 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

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

CI Enabled Photonic Crystal Research and Technology: Photonic “materials by design” Multiphysics and multiscale analysis

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 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 the wavelength 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.

Photonic Crystals:Periodic Surprises in Electromagnetism

Steven G. Johnson

MIT

To Begin: A Cartoon in 2d

planewave

E ,

H ~ ei (

k x t )

k / c

2

k

scattering

To Begin: A Cartoon in 2d

planewave

E ,

H ~ ei (

k x t )

k / c

2

k

• • •

• • •

• • •

• • •

• • •

• • •

• • •

• • •

• • •

• • •

••

•

••

•

••

•

••

•

••

•

••

•

••

•

••

•

••

•

••

•

••

•

••

•

••

•

••

•

for most , beam(s) propagatethrough crystal without scattering(scattering cancels coherently)

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

a

1887 1987

Photonic Crystalsperiodic 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)


with photonic band gaps: “optical insulators”

magical oven mitts forholding and controlling light

3D Photonic Crystal with Defectscan trap light in cavities and waveguides (“wires”)


But how can we understand such complex systems?Add up the infinite sum of scattering? Ugh!

3D Photonic Crystal

High indexof refraction

Low indexof refraction

A mystery from the 19th century

e–

e–

E

+

+

+

+

+

J

E current:

conductivity (measured)

mean free path (distance) of electrons

conductive material

A mystery from the 19th century

e–

e–

E

+

J

E current:

conductivity (measured)

mean free path (distance) of electrons

+ + + + + + +

+ + + + + + + +

+ + + + + + + +

+ + + + + + + +

crystalline conductor (e.g. copper)

10’sof

periods!

A mystery solved…

electrons are waves (quantum mechanics)1

waves in a periodic medium can propagate without scattering:

Bloch’s Theorem (1d: Floquet’s)

2

The foundations do not depend on the specific wave equation.

Time to Analyze the Cartoon

planewave

E ,

H ~ ei (

k x t )

k / c

2

k

• • •

• • •

• • •

• • •

• • •

• • •

• • •

• • •

• • •

• • •

••

•

••

•

••

•

••

•

••

•

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•

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•

••

•

••

•

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•

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•

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•

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•

••

•

for most , beam(s) propagatethrough crystal without scattering(scattering cancels coherently)

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

a

Fun with Math

E 1

c

t

H i

c

H

H

1

c

t

E

J i

cE

0

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

Hermitian Eigenproblems

1

H

c

2 H

eigen-operator eigen-value eigen-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)

Periodic Hermitian Eigenproblems[ G. Floquet, “Sur les équations différentielles linéaries à coefficients périodiques,” Ann. École Norm. Sup. 12, 47–88 (1883). ]

[ F. Bloch, “Über die quantenmechanik der electronen in kristallgittern,” Z. Physik 52, 555–600 (1928). ]

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

H (

x , t)e

ik x t

H k (x )can choose:

periodic “envelope”planewave

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

Corollary 2: given by finite unit cell,so are discrete n(k)H

k

Periodic Hermitian EigenproblemsCorollary 2: given by finite unit cell,

so are discrete n(k)H

k

k

band diagram (dispersion relation)

map ofwhat states

exist &can interact

?range of k?

Periodic Hermitian Eigenproblems in 1d

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

(x) = (x+a)

H(x)eikx Hk(x)

a

Consider k+2π/a: ei(k2

a) x

Hk2

a

(x) e ikx ei2a

x

Hk2

a

(x)

periodic!satisfies sameequation as Hk

= Hk

k is periodic:

k + 2π/a equivalent to k“quasi-phase-matching”

band gap

Periodic Hermitian Eigenproblems in 1d

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

(x) = (x+a)a

k is periodic:

k + 2π/a equivalent to k“quasi-phase-matching”

k

0 π/a–π/a

irreducible Brillouin zone

Any 1d Periodic System has a Gap

1

k

0

[ Lord Rayleigh, “On the maintenance of vibrations by forces of double frequency, and on the propagation of waves through a medium endowed with a periodic structure,” Philosophical Magazine 24, 145–159 (1887). ]

Start witha uniform (1d) medium:

k

1


1

(x) = (x+a)a

k

0 π/a–π/a


Treat it as“artificially” periodic

bands are “folded”by 2π/a equivalence

ea

x, e

a

x

cosa

x

, sin

a

x

(x) = (x+a)a

1


0 π/a


sina

x

cosa

x

x = 0

Treat it as“artificially” periodic

(x) = (x+a)a

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


0 π/a


Add a small“real” periodicity

2 = 1 +

sina

x

cosa

x

x = 0

band gap


0 π/a


Add a small“real” periodicity

2 = 1 +

sina

x

cosa

x

(x) = (x+a)a

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

x = 0

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

has lower frequency

Some 2d and 3d systems have gaps

• In general, eigen-frequencies satisfy Variational Theorem:

1(k )2 min

E 1

E 10

ik

E 12

E 1

2

c 2

2(k )2 min

E 2

E 20

E1*E20

""

“kinetic”

inverse“potential”

bands “want” to be in high-

…but are forced out by orthogonality–> band gap (maybe)

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

E

HTM

a

freq

uenc

y

(2π

c/a)

= a

/

X

M

X M irreducible Brillouin zone

k

QuickTime™ and aGraphics decompressorare needed to see this picture.00.10.20.30.40.50.60.70.80.91Photonic Band GapTM bands

gap forn > ~1.75:1

QuickTime™ and aGraphics decompressorare needed to see this picture.00.10.20.30.40.50.60.70.80.91Photonic Band GapTM bands


E

HTM

X M

Ez

– +

Ez

(+ 90° rotated version)

gap forn > ~1.75:1


E

H

E

H

TM TE

a

freq

uenc

y

(2π

c/a)

= a

/

X

M

X M irreducible Brillouin zone

k

QuickTime™ and aGraphics decompressorare needed to see this picture.00.10.20.30.40.50.60.70.80.91Photonic Band GapTM bandsTE bands

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

TE bands

TM bands

gap for n > ~1.4:1

E

H

TE

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

UÕ L X W K

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0

21% gap

L'

LK'

W

U'XU'' U

W' K

z

I: rod layer II: hole layer

I.

II.

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

gap for n > ~4:1

You, too, can computephotonic eigenmodes!

MIT Photonic-Bands (MPB) package:

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

on Athena:

add mpb

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Photonic Crystals