photonic bandgap materials - mcgill university

Photonic Bandgap andPhotonic Bandgap and Electromagnetic Metamaterials

Andrew Kirk

andrew kirk@mcgill [email protected]

Department of Electrical and Computer Engineering

McGill Institute for Advanced Materials

1Photonic bandgap and metamaterialsA Kirk 11/24/2008

References and further readingReferences and further reading

• Steven Johnson (MIT), Tutorial slides and notes:Steven Johnson (MIT), Tutorial slides and notes: http://ab‐initio.mit.edu/photons/tutorial/

• J‐M Lourtioz, Photonic Crystals: Towards Nanoscale, yDevices, Springer, 2005

• Maksim Skorobogatiy (Ecole Polytechnique), course g y y qin photonic crystals


ContentsContents

• Background: EM wavesBackground: EM waves

• Photonic crystalsB d ff t– Bandgap effects

– Resonant cavities

– Superprisms

– Negative index properties

• Metamaterials and optical cloaking

• Fabrication


After attending this class, you should b blbe able to:

• Explain what is meant by a photonic bandgapp y p g p• Explain what is meant by an electromagnetic metamaterial

• Estimate the critical dimensions for photonic bandgap structures, as a function of wavelength

• Describe the way that photonic bandgap• Describe the way that photonic bandgapstructures can be used to modify the reflective, refractive or dispersive properties of materials

• Describe the common fabrication processes for photonic bandgap and metamaterials


Harmonic WavesHarmonic Waves

U0

z/2 /2

-U0

0

Ti2 12

)cos(, 0 kztUtzU

• Wavelength : distance needed to recover the same phase

Space representation

Time representationk

2

f12

fc

Wavelength : distance needed to recover the same phase.• Wave period : time needed to recover the same phase.• Phase of the wave: state at that point in space‐time• Wave frequency : inverse of the period• Wave frequency : inverse of the period.• Phase velocity: rate at which constant phase propagates


An electromagnetism reviewAn electromagnetism reviewy

xLight is an electromagnetic wave:

• Orthogonal electric (E) and magnetic (H) fields H

xLight is an electromagnetic wave:

• Wave equation:E z

22

2

1t

EE

22

2

1t

HH

• Phase velocity: which is 3x108 m/s in vacuum1pv

t t

• Material properties:

• Permittivity describes electric field response• Permeability describes magnetic field response• Refractive index n is relative speed of light 1/ pn v


Transverse EM waves (plane waves)Transverse EM waves (plane waves)

W f l f hWavefronts: planes of constant phase Have infinite extent

E0

-E0

0

kk

How do we describe a wave that does not travel on the zHow do we describe a wave that does not travel on the z‐axis ?

Introduce the wave vector kIntroduce the wave vector k


Transverse EM wavesTransverse EM waves

k W t ( 1)

Write the wave as:

k

k = Wave vector (m‐1)

k a k a k a k

0, , , expx y z t t E E k r

R

P

an

k a xkx a yky azkz

k 2 2

2 kx2 ky

2 kz2 2

0 zu y

What is the plane for which k R is constant ?What is the plane for which k.R is constant ?

Plane of constant phase WAVEFRONT

k is always normal to wave‐front9Photonic bandgap and metamaterialsA Kirk 11/24/2008

Dispersion relationDispersion relation

• Relationship between wave number and frequency• For an anisotropic medium, we need to consider the wave‐vector k

2 2 2c k

pv k

ency

l l l d f

Freq

ue Group velocity is local gradient of dispersion curve

gdvdk

k

In free‐space, dispersion relation is linear

Wave‐vectorg dk


Photonic crystals MultilayerPhotonic crystals

• Also known as photonic b d l

1‐D

aperiod

bandgap materials• Periodic dielectric and/or metallic structures

n1n2metallic structures

• Period is typically close to the wavelength of light

22‐D

Hexagonalg g

• Can be 1‐D, 2‐D, 3‐D structures

i d f

3‐DHexagonal

• May contain defects, heterostructures and other more complex inclusionsp

Woodpile11Photonic bandgap and metamaterialsA Kirk 11/24/2008

Scattering from periodic structuresScattering from periodic structures

• Coherent illumination with wavelength 0

• Each plane scatters waves which i t f t ti l interfere, constructively or destructively

• Reflection maxima occur at aangles given by Bragg’s law:

2 cosa m

• Or equivalently

2 cosa m

m a k

0( / )effn 12Photonic bandgap and metamaterialsA Kirk 11/24/2008

Floquet‐Bloch theoremFloquet Bloch theorem

• Infinitely periodic structure– Implies that EM fields will have same periodicity– Fields can be expanded as a summation of Bloch waves (i.e.

Fourier series)– This is Floquet‐Bloch Theorem

• Structure has reciprocal lattice, with basis vectors Gj• Fields are written:Fields are written:

exp . exp .G

j j u k r V G k r

• Where V derives from the Fourier series• Bloch waves are conserved as they move through the

lattice


Brillouin zonesBrillouin zones• In periodic structure, all of dispersion curve

a

can be mapped into the irreducible Brillouinzone a k is periodic k+2/a isperiodperiod k is periodic k+2/a is

equivalent to k n x n x a Limit of zero modulation

cy

cy

Limit of zero modulation

Freq

uenc

Freq

uenc

kWavevector Wavevector 2ka

0‐0.5 0.5


Brillouin zonesBrillouin zones• In periodic structure, all of dispersion curve can be mapped into the irreducible Brillouinzone a k is periodic k+2/a isperiod k is periodic k+2/a is

equivalent to k n x n x a Finite index difference

cy

cy

Finite index difference

Freq

uenc

Freq

uenc Bandgap

kWavevector Wavevector 2ka

0‐0.5 0.5


Degeneracy splitting at band‐edgeDegeneracy splitting at band edgeFinite index differenced m

ency

Bandgap

a

period

Freq

ue

ka00 5 0 5

Bandgap

EM waves

• Bandgap width is approximately

Wavevector 2ka

0‐0.5 0.5 n x n x a

d• Bandgap width is approximately

• State concentrated in higher index has lower frequency

0

d

m

State concentrated in higher index has lower frequency


BandgapBandgap

• Bandgap: Frequency bandFinite index difference

Bandgap: Frequency band for which propagation is forbidden

ency

Bandgap• All 1‐D photonic crystals have a bandgap Fr

eque

ka00 5 0 5

Bandgap

• Only some 2‐D and 3‐D crystals have complete b d

Wavevector 2ka

0‐0.5 0.5

bandgaps

Photonic bandgap and metamaterialsA Kirk 11/24/2008 17

Square 2‐D lattice (From Johnson)a Rods in air

a/

0.8

0.9

1

(2πc/a) =

0 5

0.6

0.7

uency

(0.3

0.4

0.5

Photonic Band Gapfreq

0

0.1

0.2TM bands=12:1

ETM

M

X M irreducible Brillouin zone

k

gap for

HTM Xk n > ~1.75:1

A Kirk 11/24/2008 18Photonic bandgap and metamaterials

Electric field distribution (from Johnson)

0.8

0.9

1

E

0 5

0.6

0.7

Ez

0.3

0.4

0.5

Photonic Band Gap(+ 90° rotated version)

0

0.1

0.2TM bands

Ez

ETM

X M

gap for

HTM– + n > ~1.75:1


3‐D photonic crystal: complete gap , =12:1I.

0.8

II.

0.6

0.7

21% gap

0.3

0.4

0.5

L'

z

0.1

0.2

LK'

W

U'XU'' U

W' K

UÕ L X W K0I: rod layer II: hole layer

gap for n > ~4:1[ S. G. Johnson et al., Appl. Phys. Lett. 77, 3490 (2000) ]


Properties of Bulk Crystals(from Johnson)

(cartoon)band diagram (dispersion relation)

ncy photonic band gap

d /dk l l h

backwards slope:negative refraction

ved freq

ue d/dk 0: slow light(e.g. DFB lasers)

synthetic mediumf i

conserv

strong curvature:super‐prisms, …

(+ negative refraction)

for propagation

conserved wavevector k






d /dk l l h


ved freq


synthetic mediumf i

conserv



for propagation




Bandgap effectsBandgap effects• Make use of bandgap to confine light

within `defect` waveguideStandard bendwithin defect waveguide

• Hexagonal crystal structure typically used

bend

• Vertical confinement achieved via total internal reflection (i.e. Conventional guiding)

Optimized bend

• Advantage over conventional index guiding is small size

• However scattering loss is higher (4• However scattering loss is higher (4 dB/cm reported)

• Operating bandwidth is typically <30 nm

Watanabe 2007

A Kirk 11/24/2008 Photonic bandgap and metamaterials 23

Thermo‐optic Mach‐Zender modulatorThermo optic Mach Zender modulator

Camargo 2006A Kirk 11/24/2008 Photonic bandgap and metamaterials 24

Resonant cavity structuresResonant cavity structures

• 2‐D bandgaps can be used to define very small cavities

Photonic crystal cavity with displaced holes (marked by letters A, B and C), fabricated in SOI [Asano 2006]

• Applications include single photon lasers• Require high Q‐factor and small volume (i.e. large Q/V)


Heterostructure cavitiesHeterostructure cavities


Asano 2006




d /dk l l h


ved freq


synthetic mediumf i

conserv



for propagation




Photonic crystal superprismPhotonic crystal superprism

• Photonic crystal operated o o c c ys a ope a edat wavelength above bandgap

• Periodic structure significantly modifies dispersiondispersion

• Can be used to form optical multiplexer/demultiplexerp / p

• Potential for very compact devices

Kosaka 1999A Kirk 11/24/2008 28Photonic bandgap and metamaterials

S‐vector and k‐vector effectsS vector and k vector effectsEffective index diagram

n1 n2

1‐D photonic crystal nx=kx/k0

n1 n2

xkr

Sr(1)

kr(2)Sr(2)

ki Sr kr(1)1

ki

kz

• S‐vector: Group velocity dispersion

z2 nz=kz/k0

– Beam is directed along normal to dispersion surface

• k‐vector: Phase velocity dispersion– Wavefront is refracted according to effective indexg

• Which effect is most useful? Which lattice is best – 1‐D, 2‐D, hexagonal, square?A Kirk 11/24/2008 29Photonic bandgap and metamaterials

S‐vector and k‐vector superprismsS vector and k vector superprisms

S‐vector device k‐vector device

D i bj ti

Spatial separation within photonic crystal

Angular dispersionCan make use of non‐parallel edges

• Design objectives:– High spectral resolution– Small device areaSmall device area


Material systemMaterial system• All designs were carried out for silicon‐on‐insulator (SOI)insulator (SOI)

• Design wavelength 1550 nm• Plane wave expansion method analysis (3 D)

Air

• Plane wave expansion method analysis (3‐D)

Si n=3.45 0.2‐0.5 m

Air

SiO2

S

n=1.46

n 3.45

3 m

0.2 0.5 m

SiA Kirk 11/24/2008 31Photonic bandgap and metamaterials

k‐vector scales to DWDM Resultsk-vector designs(DWDM 32 channels x 100 GHz)

1-D 2-D square 2-D hexagonal

P i d ( ) 314 314 359Period (nm) 314 314 359

Hole size (nm) 204 241 211

Prism area (mm2) 0.0135 0.0125 0.016Lattices equalOnly n is important

S-vector designs(CWDM 4 channels x 20

1-D 2-D square

2-D hexagonal

nm)

Period (nm) 251.7 246.1 283.4

Hole size (nm) 176 7 171 1 208 4Hole size (nm) 176.7 171.1 208.4

Prism area (mm2) 0.0068 1.01 2.42

1‐D significantly more compact due to lower band curvatureg f y p

Layout for a 3X32 DWDM k‐vector multiplexery p





d /dk l l h


ved freq


synthetic mediumf i

conserv



for propagation




Negative refractive index materialsNegative refractive index materials

Ortwin Hess, ‘Optics: Farewell to Flatland’, Nature 455, 299‐300(18Ortwin Hess, Optics: Farewell to Flatland , Nature 455, 299 300(18 September 2008)


Strange properties of negative index materialsg p p g

Positive index material

Negative index material

sin sini i t tn n Incident ray

iin

tn

Air Incident rayi

in

tn

Air

Transmitted rayt Transmitted rayt

Flat lens (originallyFlat lens (originally proposed by Pendry)


Negative index regionNegative index region

• Periodic arrays of air holes in dielectrics canPeriodic arrays of air holes in dielectrics can also have n=‐1

Imaging properties of dielectric photonic crystal slabs for large object distancesGuilin Sun, Aju S. Jugessur, and Andrew G. Kirk

Optics Express, Vol. 14, Issue 15, pp. 6755‐6765 A Kirk 11/24/2008 37Photonic bandgap and metamaterials

Electromagnetic metamaterialsElectromagnetic metamaterials

• So far we have discussed periodic dielectric pstructures

• We have modulated but not • What happens if we modulate both?

R lt i t t i l C h ti t• Result is a metamaterial: Can have properties not found in nature

• Typically the modulation is on a period smallerTypically the modulation is on a period smaller than the wavelength of light, so effect is not due to interference


Example: Optical cloakingExample: Optical cloaking

• If we could bend light around an object, we could make it g j ,invisible

• This is called ‘Optical cloaking’

• In Pendry’s 2006 ‘Science’ paper he works out what electromagnetic properties this cloak should have


Material requirementsMaterial requirements• Pendry showed that the material needs to anisotropic• The permittivitity and permeability (for a cylinder) must be given by:

2

1r r

r Rr

1

rr R

2

2 1

2 1z z

R r RR R r

r r

4

5

ability

rr rr z

Natural materials do not usually have =

2

3

, tivity, perme

1 2 1 4 1 6 1 8 2

1 r, r

z, z

Perm

ittr

1.2 1.4 1.6 1.8 2Radius, r


Engineering a microwave cloakEngineering a microwave cloak

• Only 5 months after Pendry’s paper, the first experimental y y p p , pdemonstration of electromagnetic cloaking was published

• The research was led by D.R.Smith at Duke University


How it was doneHow it was done• Cloak is made of many split ring

resonators

• Curved conductors with a precisely calculated inductance

• This is a metamaterial for microwaves (3.5 mm wavelength)

• In order to simplify the experiment, the structure guided EM waves in the gcorrect direction but did not provide the full impedance matching

• Therefore it still reflected some radiation


Measurement systemMeasurement system

ProbeProbe

Cloak

Microwave source (8.5 GHz wavelength is 3 5GHz, wavelength is 3.5 mm)


Results: Electric field patternsSimulation: ideal materials Simulation: actual materials

Experiment: uncloaked copper cylinder Experiment: cloaked copper cylinder


Electromagnetic cloaking for lightElectromagnetic cloaking for light• In April 2007, in a letter in Nature, Vladimir Shalaev at Purdue showed (by

simulation) that optical cloaking should possible by using metallic wires in a di l i didielectric medium


Simulated results (at 632 nm)Simulated results (at 632 nm)

Cloaked

Uncloaked

As before, this does not provide impedance matching, so reflections still occur


Demonstration of negative index optical i lmaterials

• An experimental demonstration is probably not far offp p y

• The Purdue group have already demonstrated negative refractive index optical materials

Silver gratings separated by 38 nm of alumina


ResultsResults


3‐D negative refractive index materials3 D negative refractive index materials• Researchers at Berkeley have recently demonstrated a 3‐D negative

index optical meatmaterial (Nature 455 September 2008):

Silver and magnesium fluoride ‘fishnet’ structure

Layers are 80 nm thick and period is 860 nm


ResultsResults

J Valentine et al. Nature 000, 1‐4 (2008) doi:10.1038/nature07247


FabricationFabrication

• Fabrication of photonic crystal structures isFabrication of photonic crystal structures is typically achieved via nanolithography

• Electron beam lithography is often employed• Electron‐beam lithography is often employed

• Deep‐UV optical lithography is also suitable ( d ffi i f d i )(and more efficient for mass‐production)

• Focused ion‐beam etching is also used

• Typical materials are silicon‐on‐insulator (SOI), silicon and III‐V semiconductors


SummarySummary

• Photonic crystals are materials withPhotonic crystals are materials with periodically modulated permittivity (on a scale of the wavelength)of the wavelength)

• This modifies the reflective, dispersive and refractive propertiesrefractive properties

• Metamaterials typically have modulated bili i ddi i i i i dpermeability, in addition to permittivity, and

do not operate via interference


photonic bandgap materials - mcgill university

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