Transcript Prezentacja programu PowerPoint

Quantum Dots in Photonic Structures
Lecture 4: Photonic crystals
Jan Suffczyński
Wednesdays, 17.00, SDT
Projekt Fizyka Plus nr POKL.04.01.02-00-034/11 współfinansowany przez Unię Europejską ze środków Europejskiego
Funduszu Społecznego w ramach Programu Operacyjnego Kapitał Ludzki
Plan for today
1. DBR mirrors
2. Twodimensional
photonic
crystals – band
structure
3. Twodimensional
photonic crystal
– fabrication
methods
• Refractive index:
Reminder
𝑫 = 𝜀0 𝜀𝑟 (𝜔)𝑬
Polarization by EM wave
Complex dielectric function
• Refractive index:
Reminder
𝑫 = 𝜀0 𝜀𝑟 (𝜔)𝑬
Polarization by EM wave
Complex dielectric function
Simultaneous description of
refraction and absorption
𝜀𝑟 (𝜔) = 𝑛 𝜔 + 𝑖 ∙ 𝜅(𝜔)
Speed light in a medium:
𝑐
𝑣=
𝑛
Dispersion
After: András Szilágyi
Reminder
• Photonic crystal: periodic arrangement
of dielectric (or metallic…) objects
– periodic refractive index contrast
– the period comparable to the wavelength
of light in the material.
• 1D photonic crystal:
– Distributed Bragg Reflector (DBR)
– Example calculation:
Transfer Matrix Method
Bragg mirror from the University of Warsaw
Low n: 20 x
Superlattice 1:1
Cd0.86Zn0.14Te
MgTe
20 stack
DBR
High n: Cd0.86Zn0.14Te
Cd0.86Zn0.14Te buffer
GaAs substrate
1.0
Experimental data
Fit
0.9
1 mm
reflectivity
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
650
700
750
800
850
900
Wavelength (nm)
950
1000
1050
SEM: T. Jakubczyk
J.-G. Rousset
Refractive index engineering
Refractive index
Energy gap
MgTe
ZnTe
CdTe
6.1
6.2
6.3
6.4
Lattice constant (A)
6.5
For a good DBR we need a pair of materials that have:
• large refractive index contrast Δn = nhigh-nlow
• lattice paramters as close as possible
Bragg mirror lattice matched to CdTe
The structure
15 par
ZnTe 53nm
superlattice
18 periods
ZnTe 53 nm
ZnTe 0.7 nm
superlattice
MgTe 0.9 nm
ZnTe buffer
1000 nm
ZnTe 0.7 nm
MgSe 1.3 nm
ZnSe 62 nm
GaAs
1 μm
W. Pacuski, UW
Bragg mirror lattice matched to CdTe
The structure
15 par
ZnTe 53nm
supersieć
18 powtórzeń
ZnTe 53 nm
ZnTe 0.7 nm
supersieć
MgTe 0.9 nm
ZnTe buffer
1000 nm
ZnTe 0.7 nm
MgSe 1.3 nm
ZnSe 62 nm
GaAs
W. Pacuski, UW
Bragg mirror lattice matched to CdTe
The structure
15 par
ZnTe 53nm
ZnTe 0.7 nm
supersieć
18 powtórzeń
ZnTe 53 nm
ZnTe 0.7 nm
supersieć
MgTe 0.9 nm
MgTe 0.9 nm
ZnTe 0.7 nm
ZnTe buffer
1000 nm
ZnTe 0.7 nm
MgSe 1.3 nm
MgSe 1.3 nm
ZnSe 62 nm
ZnTe 0.7 nm
MgTe 0.9 nm
GaAs
ZnTe 0.7 nm
MgSe 1.3 nm
1 nm
W. Pacuski, UW
DBR mirror and DBR cavity reflectivity
DBR
Refractive index
500
3.0
Wavelength (nm)
700
800
600
900
1000
ZnTe
n = 0.53
2.5
SL
1.0
Reflectivity
Experiment
Calculation
>99%
0.8
0.6
0.4
0.2
Microcavity
1.0
Experiment
Calculation
Reflectivity
0.8
0.6
0.4
0.2
0
500
600
700
800
Wavelength (nm)
900
1000
W. Pacuski, UW
DBR mirror and DBR cavity reflectivity
DBR
Refractive index
500
3.0
Wavelength (nm)
700
800
600
900
1000
ZnTe
n = 0.53
2.5
SL
1.0
Reflectivity
Experiment
Calculation
>99%
0.8
0.6
Q = λ/Δλ = 3600
0.4
0.2
Microcavity
1.0
Experiment
Calculation
Reflectivity
0.8
0.6
0.4
0.2
0
500
600
700
800
Wavelength (nm)
900
1000
CdTe based microcavity – 60 pairs
1.0
0.8
0.6
0.4
0.2
0
500
550
600
 (nm)
But sometimes technology makes jokes…
650
700
750
Planar cavity with DBR mirrors
Stop band Δλ=(n1-n2)/π(n1+n2)
Reflectivity
Δθ ~ 20o typically in the
case og GaAs/AlAs DBR
Br
Antinode of the field in
the center of the cavity
λ-cavity
Electric field distribution
Cavity mode
Exponential decay of the
stationary field from the
center of the cavity
Towards 2D and 3D photonic crystals
Low index of refraction
High index of refraction
3D photonic crystal
Photonic crystals – how it works?
a>> incoherent scattering
a
a~ coherent scattering
a
a<<
averaging
a
Photonic crystals
2D, 3D photonic bandgap?
J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade,
Photonic Crystals: Molding the Flow of Light
Dispersion relation and origin of the
band gap in 1D
Consider medium of refractive
index n1
and light of wavelenght of   2a
frequency ω
Light line:
free space
  k c
Light line:
medium
c
k
n1
standing wave in n1
0
π/a
wave vector k
n1
Dispersion relation and origin of the
band gap in 1D
Consider the stack of layers of
refractive indeces n1 and n2
and light of wavelenght of   2a
frequency ω
n1 n2
0
n1
n2
n1
n2
n1
a
π/a
wave vector k
n1: high index material
n2: low index material
Dispersion relation and origin of the
band gap in 1D
c
k
n
Consider the stack of layers of
refractive indeces n1 and n2
and light of wavelenght   2a
frequency ω
standing wave in n2
n1 n2
n1
n2
n1
n2
n1
standing wave in n1
0
π/a
wave vector k
n1: high index material
n2: low index material
Dispersion relation and origin of the
band gap in 1D
c
k
n
Consider the stack of layers of
refractive indeces n1 and n2
and light of wavelenght   2a
frequency ω
standing wave in n2
n1 n2
n1
n2
n1
n2
n1
bandgap
standing wave in n1
0
π/a
wave vector k
n1: high index material
n2: low index material
frequency ω
-π/a
0
π/a
wave vector k
Bloch wave with
wave vector k is
equal to Bloch wave
with wave vector
k+m2p/a:
modified slide from Rob Engelen
Band diagram
k is periodic:
k + 2π/a is equivalent to k
-2π/a
-π/a
0
π/a
wave vector k
frequency ω
frequency ω
Band diagram
2π/a
-2π/a
-π/a
0
π/a
wave vector k
2π/a
This is the first Brillouin zone
modified slide from Rob Engelen
Band diagram – one more view
Anticrossing of
modes leading
to formation of
the band gap
Another look- Bragg scattering
conditions
When a wave impinges on a crystal it will be reflected at a
particular set of lattice planes characterized by its reciprocal lattice
vector g only if the so-called Bragg condition is met
k  k  g
If the Bragg condition is not met, the incoming wave just moves
through the lattice and emerges on the other side of the crystal
(when neglecting absorption)
Photonic crystals - introductory example
from the prevous lecture
1. Bragg scattering
Regardless of how small the reflectivity r is from an individual
scatterer, the total reflection R from a semi infinite structure:
Complete reflection when:
 Propagation of the light in crystal inhibited when Bragg condition satisfied
 Origin of the photonic bang gap
Reciprocal lattice
Dispersion relation for 2D photonic crystal
2D square lattice
Dispersion relation for 2D photonic crystal
2D hexagonal lattice
Band gap:
no propagation possible at that frequency
density of optical states (DOS) is 0
Dispersion relation for 2D photonic crystal vs
transmission
Photonic crystals in Nature
Sea mouse
Opal
McPhedran et al.
Artificial PC production:
Layer-by-Layer Lithography
• Fabrication of 2d patterns in Si or GaAs is very advanced
(think: Pentium IV, 50 million transistors)
…inter-layer alignment techniques are only slightly more exotic
So, make 3d structure one layer at a time
Need a 3d crystal with constant cross-section layers
A Schematic
[ M. Qi, H. Smith, MIT ]
Making Rods & Holes Simultaneously
Steven G. Johnson, MIT
side view
substrate
Si
top view
Steven G. Johnson, MIT
Making Rods & Holes Simultaneously
expose/etch
holes
A
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substrate
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Steven G. Johnson, MIT
Making Rods & Holes Simultaneously
backfill with
silica (SiO2)
& polish
A
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substrate
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Steven G. Johnson, MIT
Making Rods & Holes Simultaneously
deposit another
Si layer
layer 1
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substrate
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Steven G. Johnson, MIT
Making Rods & Holes Simultaneously
dig more holes
offset
& overlapping
layer 1
B
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substrate
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Steven G. Johnson, MIT
Making Rods & Holes Simultaneously
backfill
layer 1
B
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substrate
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Steven G. Johnson, MIT
Making Rods & Holes Simultaneously
etcetera
(dissolve
silica
when
done)
layer 3
A
C
layer 2
layer 1
A
A
C
A
C
B
C
B
A
one
period
B
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A
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substrate
C
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Steven G. Johnson, MIT
Making Rods & Holes Simultaneously
etcetera
layer 3
A
C
layer 2
layer 1
hole layers
A
A
C
A
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B
C
B
A
one
period
B
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A
substrate
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B
A
B
C
Steven G. Johnson, MIT
Making Rods & Holes Simultaneously
etcetera
layer 3
A
C
layer 2
layer 1
rod layers
A
A
C
A
C
B
C
B
A
one
period
B
A
B
A
A
substrate
C
B
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Steven G. Johnson, MIT
7-layer E-Beam Fabrication
5 mm
[ M. Qi, et al., Nature 429, 538 (2004) ]
Three-dimensional Si photonic crystal
S.-Y. Lin et al., Nature 394, 251 (1998)
Y. A. Vlasov et al., Nature 414, 289 (2001)
Two-Photon Lithography
2-photon probability ~ (light intensity)2
3d Lithography
lens
some chemistry
(polymerization)
…dissolve unchanged stuff
(or vice versa)
Lithography – the best friend of a man
 = 780nm
resolution = 150nm
7µm
(3 hours to make)
2µm
S. Kawata et al., Nature(2001)
Holographic Lithography
Four beams make 3d-periodic interference pattern
k-vector differences give reciprocal lattice vectors (i.e. periodicity)
absorbing material
(1.4µm)
beam polarizations + amplitudes (8 parameters) give unit cell
[ D. N. Sharp et al., Opt. Quant. Elec. 34, 3 (2002) ]
Holographic Lithography
[ D. N. Sharp et al., Opt. Quant. Elec. 34, 3 (2002) ]
10µm
huge volumes, long-range periodic, fcc lattice…backfill for high contrast
Colloidal photonic crystals
Colvin, MRS Bulletin 26, (2001)
3D Bandgap Structure
frequency (c/a)
K. M. Ho, C. T. Chan, and C. M. Soukoulis, Phys. Rev. Lett. 65, 3152 (1990).
11% gap
X
for gap at λ = 1.55µm,
sphere diameter ~ 330nm
G
W
U L
K
overlapping Si spheres
MPB tutorial, http://ab-initio.mit.edu/mpb
Experimental Air-guiding Photonic
Crystal Fiber
10µm
[ R. F. Cregan et al., Science (1999) ]
5µm
How to create a cavity within
a photonic crystal?
Reminder: Bragg mirror cavity
Two equivalent views:
1. Cavity = mirror + resonator+ mirror
2. Cavity = mirror with a „good” defect
Single-Mode Cavity
A point defect
can push up
a single mode
from the band edge
frequency (c/a)
Bulk Crystal Band Diagram
w
G
X
M
M

X
w0
G
“Single”-Mode Cavity
A point defect
can pull down
a “single” mode
frequency (c/a)
Bulk Crystal Band Diagram
…here, doubly-degenerate
G
X
M
X

M
X
G
frequency (c/a)
Tunable Cavity Modes
Ez:
monopole
dipole
channel-drop filters
high-transmission
sharp bends
waveguide splitters
resonant filters
“1d” Waveguides + Cavities = Devices
Outlook: Optical Microcavities
Vahala, Nature
424, 839 (2003)
Microcavity characteristics: Quality factor Q, mode volume V