Photonic Bandgap Materials - McGill University
Transcript of 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
2Photonic bandgap and metamaterialsA Kirk 11/24/2008
ContentsContents
• Background: EM wavesBackground: EM waves
• Photonic crystalsB d ff t– Bandgap effects
– Resonant cavities
– Superprisms
– Negative index properties
• Metamaterials and optical cloaking
• Fabrication
3Photonic bandgap and metamaterialsA Kirk 11/24/2008
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
4Photonic bandgap and metamaterialsA Kirk 11/24/2008
5Photonic bandgap and metamaterialsA Kirk 11/24/2008
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
6Photonic bandgap and metamaterialsA Kirk 11/24/2008
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
7Photonic bandgap and metamaterialsA Kirk 11/24/2008
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
8Photonic bandgap and metamaterialsA Kirk 11/24/2008
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
10Photonic bandgap and metamaterialsA Kirk 11/24/2008
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
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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
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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
15Photonic bandgap and metamaterialsA Kirk 11/24/2008
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
16Photonic bandgap and metamaterialsA Kirk 11/24/2008
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
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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
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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
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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) ]
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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
(+ negative refraction)
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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
(+ negative refraction)
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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
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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)
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Heterostructure cavitiesHeterostructure cavities
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Asano 2006
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
(+ negative refraction)
A Kirk 11/24/2008 27Photonic bandgap and metamaterials
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
A Kirk 11/24/2008 30Photonic bandgap and metamaterials
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
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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
(+ negative refraction)
A Kirk 11/24/2008 34Photonic bandgap and metamaterials
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)
A Kirk 11/24/2008 35Photonic bandgap and metamaterials
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)
A Kirk 11/24/2008 36Photonic bandgap and metamaterials
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
A Kirk 11/24/2008 Photonic bandgap and metamaterials 38
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
A Kirk 11/24/2008 39Photonic bandgap and metamaterials
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
A Kirk 11/24/2008 40Photonic bandgap and metamaterials
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
A Kirk 11/24/2008 41Photonic bandgap and metamaterials
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
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Measurement systemMeasurement system
ProbeProbe
Cloak
Microwave source (8.5 GHz wavelength is 3 5GHz, wavelength is 3.5 mm)
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Results: Electric field patternsSimulation: ideal materials Simulation: actual materials
Experiment: uncloaked copper cylinder Experiment: cloaked copper cylinder
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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
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Simulated results (at 632 nm)Simulated results (at 632 nm)
Cloaked
Uncloaked
As before, this does not provide impedance matching, so reflections still occur
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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
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ResultsResults
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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
A Kirk 11/24/2008 49Photonic bandgap and metamaterials
ResultsResults
J Valentine et al. Nature 000, 1‐4 (2008) doi:10.1038/nature07247
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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
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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
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