Transcript Lecture 45

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 omnidirectional 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
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”
Sir John Maddox, Nature (1990)
Photonic Crystals
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.
Two Fundamental Optical Principles
• Localization of Light
S. John, Phys. Rev. Lett. 58,2486 (1987)
• Inhibition of Spontaneous Emission
E. Yablonovitch, Phys. Rev. Lett. 58 2059 (1987)
 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 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)
Photonic Crystals- Semiconductors of Light
Semiconductors
Photonic Crystals
Periodic array of atoms
Periodic variation of dielectric
constant
Atomic length scales
Length scale ~ 
Natural structures
Artificial structures
Control electron flow
Control e.m. wave propagation
1950’s electronic revolution
New frontier in modern optics
Natural Photonic Crystals:
Structural Colours through Photonic Crystals
Natural opals
Periodic structure  striking colour effect even in the absence of pigments
Artificial Photonic Crystals
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
Woodpile structure
S. Lin et al., Nature (1998)
Inverted Opals
J. Wijnhoven & W. Vos, Science (1998)
Photonic Crystals: Opportunities
 Photonic Crystals
 complex dielectric environment that controls the flow of radiation
 designer vacuum for the emission and absorption of radiation
 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 Microlaser
Photonic Crystal Cavity formed by a point defect
O. Painter et. al., Science (1999)
Photonic Crystals Based Light Bulbs
C. Cornelius, J. Dowling, PRA 59, 4736 (1999)
“Modification of Planck blackbody radiation by photonic band-gap structures”
3D Complete Photonic Band Gap
Suppress blackbody radiation in the infrared and redirect and enhance thermal energy into visible
Solid Tungsten Filament
3D Tungsten Photonic
Crystal Filament
S. Y. Lin et al., Appl. Phys. Lett. (2003)
 Light bulb efficiency may raise from 5 percent to 60 percent
Solar Cell Applications
– Funneling of thermal radiation of larger wavelength (orange area) to thermal radiation
of shorter wavelength (grey area).
– Spectral and angular control over the thermal radiation.
Foundations of Future CI
Cavity all-optical transistor
Iin
Photonic crystal all-optical transistor
Iout
χ (3)
IH
H.M. Gibbs et. al, PRL 36, 1135 (1976)
 Fundamental Limitations
 switching time • switching intensity =
constant
 Incoherent character of the switching
 dissipated power
 Operating Parameters




Holding power:
5 mW
Switching power: 3 µW
Switching time:
1-0.5 ns
Size:
500 m
Pump Laser
Probe Laser
M. Florescu and S. John, PRA 69, 053810 (2004).
 Operating Parameters
 Holding power:
 Switching power:
 Switching time:
 Size:
10-100 nW
50-500 pW
< 1 ps
20 m
Single Atom Switching Effect
 Photonic Crystals versus Ordinary Vacuum

Positive population inversion

Switching behaviour of the atomic inversion
M. Florescu and S. John, PRA 64, 033801 (2001)
Quantum Optics in Photonic Crystals
 Long temporal separation between incident laser photons
 Fast frequency variations of the photonic DOS


Band-edge enhancement of the Lamb shift
Vacuum Rabi splitting
T. Yoshie et al. , Nature, 2004.
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 achieve
prescribed DOS at the ion position
M. Florescu et al., EPL 69, 945 (2005)
CI Enabled Photonic Crystal Design (I)
Photo-resist layer exposed to multiple laser beam interference
that produce a periodic intensity pattern
10 m

Four laser beams interfere to form a
3D periodic intensity pattern
O. Toader, et al., PRL 92, 043905 (2004)
3D photonic crystals fabricated
using holographic lithography
M. Campell et al. Nature, 404, 53 (2000)
CI Enabled Photonic Crystal Design (II)
O. Toader & S. John, Science (2001)
CI Enabled Photonic Crystal Design (III)
S. Kennedy et al., Nano Letters (2002)
Multi-Physics Problem:
Photonic Crystal Radiant Energy Transfer
Photonic Crystals
Optical Properties
Rethermalization
Processes:
Transport
Properties:
Photons
Electrons
Phonons
Metallic (Dielectric)
Backbone
Electronic
Characterization
Photons
Electrons
Phonons
Summary
Photonic Crystals: Photonic analogues of semiconductors that
control the flow of light
PBG materials: Integrated optical micro-circuits
with complete light localization
Designer Vacuum:
Frequency selective control of
spontaneous and thermal emission
enables novel active devices
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
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 omnidirectional 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
k
scattering
planewave
E, H ~ e
i (k x t )
k  /c 
2

To Begin: A Cartoon in 2d
k
a
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
planewave
E, H ~ e
i (k x t )
k  /c 
2

for most , beam(s) propagate
through crystal without scattering
(scattering cancels coherently)
...but for some  (~ 2a), no light can propagate: a photonic band gap
Photonic Crystals
periodic electromagnetic media
1887
1987
1-D
2-D
periodic in
one direction
periodic in
two directions
3-D
periodic in
three directions
with photonic band gaps: “optical insulators”
(need a
more
complex
topology)
Photonic Crystals
periodic electromagnetic media
can
cavities
3D
Photrap
to niclight
C rystain
l with
De fe c ts
and waveguides (“wires”)
magical oven mitts for
holding and controlling light
with photonic band gaps: “optical insulators”
Photonic Crystals
periodic electromagnetic media
Hig h in d e x
o f re fra c tio n
Lo w ind e x
o f re fra c tio n
3D Pho to nic C rysta l
But how can we understand such complex systems?
Add up the infinite sum of scattering? Ugh!
A mystery from the 19th century
conductive material
+
+
e–
+
e–
E
+
current:
+
J  E
conductivity (measured)
mean free path (distance) of electrons
A mystery from the 19th century
crystalline conductor (e.g. copper)
+ + + + + + + +
e–
e–
E
+
+
+
+
+
+
+
+
+
+
+
+
+
+
10’s
+
of
+ periods!
+
+
+
+
+
+
+
+
current:
J  E
conductivity (measured)
mean free path (distance) of electrons
A mystery solved…
1
electrons are waves (quantum mechanics)
2
waves in a periodic medium can
propagate without scattering:
Bloch’s Theorem (1d: Floquet’s)
The foundations do not depend on the specific wave equation.
Time to Analyze the Cartoon
k
a
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
planewave
E, H ~ e
i (k x t )
k  /c 
2

for most , beam(s) propagate
through crystal without scattering
(scattering cancels coherently)
...but for some  (~ 2a), no light can propagate: a photonic band gap
Fun with Math
1

 E  
H i H
c t
c
0 
1
 H  
E  J  i E
c t
c
First task:
get rid of this mess
dielectric function (x) = n2(x)
 
    H    H
 c 

1
eigen-operator
2
eigen-value
+ constraint
 H  0
eigen-state
Hermitian Eigenproblems
 
    H    H
 c 

1
eigen-operator
2
eigen-value
+ constraint
 H  0
eigen-state
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:
can choose:

i k x t
H(x ,t)  e
planewave

Hk (x )
periodic “envelope”
Corollary 1: k is conserved, i.e. no scattering of Bloch wave
Corollary 2: H k given by finite unit cell,
so  are discrete n(k)
Periodic Hermitian Eigenproblems
Corollary 2: H k given by finite unit cell,
so  are discrete n(k)
band diagram (dispersion relation)
3

map of
what states
exist &
can interact
2
1
k
?
range of k?
Periodic Hermitian Eigenproblems in 1d
ikx
H(x)  e Hk (x)
1 2  1 2 1 2 1 2 1 2 1 2
a
Consider k+2π/a: e
i(k 
2
)x
a
(x) = (x+a)
2


i
x
ikx
H 2  (x)  e e a H 2  (x)
k
k




a
a
k is periodic:
k + 2π/a equivalent to k
“quasi-phase-matching”
periodic!
satisfies same
equation as Hk
= Hk
Periodic Hermitian Eigenproblems in 1d
1  2 1 2 1 2 1 2 1 2 1 2
k is periodic:
k + 2π/a equivalent to k
“quasi-phase-matching”
a
(x) = (x+a)

band gap
–π/a
0
π/a
irreducible Brillouin zone
k
Any 1d Periodic System has a Gap
[ 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 with
a uniform (1d) medium:
1

0

k
1
k
Any 1d Periodic System has a Gap
[ 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). ]
Treat it as
“artificially” periodic
bands are “folded”
by 2π/a equivalence
1
a
(x) = (x+a)



e

 cos 
 a
–π/a
0
π/a
a
x


x
,e

 
x, sin  x 

 a 
a
k
Any 1d Periodic System has a Gap
[ 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). ]
Treat it as
“artificially” periodic
a
1

0
 
sin  x 
 a 
 
cos  x 
 a 
π/a
x=0
(x) = (x+a)
Any 1d Periodic System has a Gap
[ 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). ]
Add a small
“real” periodicity
2 = 1 + D
(x) = (x+a)
1 2 1 2 1 2 1 2 1 2 1 2

0
a
 
sin  x 
 a 
 
cos  x 
 a 
π/a
x=0
Any 1d Periodic System has a Gap
[ 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). ]
Add a small
“real” periodicity
2 = 1 + D
state concentrated in higher index (2)
has lower frequency
a
(x) = (x+a)
1 2 1 2 1 2 1 2 1 2 1 2

 
sin  x 
 a 
 
cos  x 
 a 
band gap
0
Splitting of degeneracy:
π/a
x=0
Some 2d and 3d systems have gaps
• In general, eigen-frequencies satisfy Variational Theorem:
1(k )  min
2
E1
E1  0
 2 (k )  min
2


  ik  E1
E
1
2
2
“kinetic”
c
2
inverse
“potential”
" " bands “want” to be in high-
E2
E 2  0
*
 E1  E 2  0 …but are forced out by orthogonality
–> band gap (maybe)
algebraic interlude
algebraic interlude completed…
… I hope you were taking notes*
[ *if not, see e.g.: Joannopoulos, Meade, and Winn, Photonic Crystals: Molding the Flow of Light ]
2d periodicity,
picture.
=12:1
G
are
Qraphics
uickTim
needed
decom
e™
toand
see
pressor
athis
a
frequency  (2πc/a) = a / 
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
Photonic BandGap
0.2
TMbands
0.1
0
irreducible Brillouin zone
M
k
G
X
G
TM
X
E
H
M
G
gap for
n > ~1.75:1
2d periodicity,
picture.
=12:1
G
are
Qraphics
uickTim
needed
decom
e™
toand
see
pressor
athis
1
0.9
0.8
Ez
0.7
0.6
0.5
(+ 90° rotated version)
0.4
0.3
Photonic BandGap
0.2
TMbands
0.1
Ez
0
G
–
+
TM
X
E
H
M
G
gap for
n > ~1.75:1
2d periodicity,
picture.
=12:1
G
are
Qraphics
uickTim
needed
decom
e™
toand
see
pressor
athis
a
frequency  (2πc/a) = a / 
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
Photonic BandGap
TEbands
0.2
TMbands
0.1
0
irreducible Brillouin zone
M
k
G
X
G
TM
X
E
H
G
M
E
TE
H
2d photonic crystal: TE gap, =12:1
TE bands
TM bands
E
TE
H
gap for n > ~1.4:1
3d photonic crystal: complete gap , =12:1
I.
II.
0.8
0.7
0.6
21% gap
0.5
0.4
z
L'
0.3
U'
G
X
U'' U W K'
W' K L
0.2
0.1
I: rod layer
II: hole layer
0
UÕ
L
G
X
W
K
gap for n > ~4:1
[ S. G. Johnson et al., Appl. Phys. Lett. 77, 3490 (2000) ]
You, too, can compute
photonic eigenmodes!
MIT Photonic-Bands (MPB) package:
http://ab-initio.mit.edu/mpb
on Athena:
add mpb