Photonic Crystals 1 - Overview & Examples
-
Upload
samarth-negi -
Category
Documents
-
view
230 -
download
1
Transcript of Photonic Crystals 1 - Overview & Examples
-
7/28/2019 Photonic Crystals 1 - Overview & Examples
1/53
Photonic Crystals
-
7/28/2019 Photonic Crystals 1 - Overview & Examples
2/53
Photonic Crystals
From Wikipedia:
Photonic Crystalsare periodic optical
nanostructures that are designed to affect the
motion of photons in a similar way thatperiodicity of a semiconductor crystal affects the
motion of electrons. Photonic crystals occur in
nature and in various forms have been studiedscientifically for the last 100 years.
-
7/28/2019 Photonic Crystals 1 - Overview & Examples
3/53
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 opticalphenomena 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 photoniccrystals 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.
-
7/28/2019 Photonic Crystals 1 - Overview & Examples
4/53
Photonic Crystals:A New Frontier in Modern Optics
MARIAN FLORESCU
NASA Jet Propulsion LaboratoryCalifornia Institute of Technology
-
7/28/2019 Photonic Crystals 1 - Overview & Examples
5/53
-
7/28/2019 Photonic Crystals 1 - Overview & Examples
6/53
Two Fundamental Optical Principles
Localization of Light
S. John, Phys. Rev. Lett. 58,2486 (1987)
Inhibition of Spontaneous EmissionE. Yablonovitch, Phys. Rev. Lett. 58 2059 (1987)
Photonic crystals: periodic dielectric structures.
interact resonantly with radiation with wavelengths comparable to theperiodicity 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
Guide and confine light without losses Novel environment for quantum mechanical light-matter interaction
A rich variety of micro- and nano-photonics devices
-
7/28/2019 Photonic Crystals 1 - Overview & Examples
7/53
Photonic Crystals History
1987: Prediction of photonic crystals
S. John, Phys. Rev. Lett. 58,2486 (1987), Stronglocalization of photons
in certain dielectric superlatticesE. Yablonovitch, Phys. Rev. Lett. 582059 (1987), I nhi bited spontaneous
emissionin 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 ofmicrowave 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)
-
7/28/2019 Photonic Crystals 1 - Overview & Examples
8/53
Photonic Crystals- Semiconductors of Light
Semiconductors
Periodic array of atoms
Atomic length scales
Natural structures
Control electron flow
1950s electronic revolution
Photonic Crystals
Periodic variation of dielectric
constant
Length scale ~
Artificial structures
Control e.m. wave propagation
New frontier in modern optics
N t l Ph t i C t l
-
7/28/2019 Photonic Crystals 1 - Overview & Examples
9/53
Natural opals
Natural Photonic Crystals:Structural Colours through Photonic Crystals
Periodic structure striking colour effect even in the absence of pigments
-
7/28/2019 Photonic Crystals 1 - Overview & Examples
10/53
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
J. Wijnhoven & W. Vos, Science (1998)S. Lin et al., Nature (1998)
Woodpile structure Inverted Opals
Artificial Photonic Crystals
-
7/28/2019 Photonic Crystals 1 - Overview & Examples
11/53
Photonic Crystals
complex dielectric environment that controls the flow of radiation designer vacuum for the emission and absorption of radiation
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
-
7/28/2019 Photonic Crystals 1 - Overview & Examples
12/53
Defect-Mode Photonic Crystal Microlaser
Photonic Crystal Cavity formed by a point defect
O. Painteret. al., Science (1999)
-
7/28/2019 Photonic Crystals 1 - Overview & Examples
13/53
3D Complete Photonic Band Gap
Suppress blackbody radiation in the infrared and redirect and enhance thermal energy into visibl
Photonic Crystals Based Light Bulbs
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
Light bulb efficiency may raise from 5 percent to 60 percent
3D Tungsten Photonic
Crystal Filament
Solid Tungsten Filament
-
7/28/2019 Photonic Crystals 1 - Overview & Examples
14/53
Solar Cell Applications
Funneling of thermal radiation of larger wavelength (orange area) to thermal radiationof shorter wavelength (grey area).
Spectral and angular control over the thermal radiation.
-
7/28/2019 Photonic Crystals 1 - Overview & Examples
15/53
Fundamental Limitations
switching time switching intensity =
constant Incoherent character of the switching dissipated power
Foundations of Future CI
Cavity all-optical transistor
(3)
IoutIin
IH
H.M. Gibbs et. al, PRL 36, 1135 (1976)
Operating Parameters
Holding power: 5 mW
Switching power: 3 W
Switching time: 1-0.5 ns Size: 500 m
Photonic crystal all-optical transistor
Probe Laser
Pump Laser
Operating Parameters
Holding power: 10-100 nW
Switching power: 50-500 pW
Switching time: < 1 ps
Size: 20 m
M. Floresc u and S. Joh n, PRA 69, 053810 (2004).
-
7/28/2019 Photonic Crystals 1 - Overview & Examples
16/53
Single Atom Switching Effect Photonic Crystals versus Ordinary Vacuum
Positive population inversion
Switching behaviour of the atomic inversion
M. Flor escu and S. John , PRA 64, 033801 (2001)
-
7/28/2019 Photonic Crystals 1 - Overview & Examples
17/53
Long temporal separation between incident laser photons
Fast frequency variations of the photonic DOS
Band-edge enhancement of the Lamb shift
Vacuum Rabi splitting
Quantum Optics in Photonic Crystals
T. Yoshi e et al. , Natu re, 2004.
Foundations for Future CI
-
7/28/2019 Photonic Crystals 1 - Overview & Examples
18/53
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 achieveprescribed DOS at the ion position
M. Florescu et al., EPL 69, 945 (2005)
-
7/28/2019 Photonic Crystals 1 - Overview & Examples
19/53
M. Campell et al. Nature, 404, 53 (2000)
CI Enabled Photonic Crystal Design (I)
Photo-resist layer exposed to multiple laser beam interferencethat produce a periodic intensity pattern
3D photonic crystals fabricated
using holographic lithography
Four laser beams interfere to form a
3D periodic intensity pattern
10 m
O. Toader, et al., PRL 92, 043905 (2004)
-
7/28/2019 Photonic Crystals 1 - Overview & Examples
20/53
O. Toader & S. John, Science (2001)
CI Enabled Photonic Crystal Design (II)
-
7/28/2019 Photonic Crystals 1 - Overview & Examples
21/53
S. Kennedy et al., Nano Letters (2002)
CI Enabled Photonic Crystal Design (III)
-
7/28/2019 Photonic Crystals 1 - Overview & Examples
22/53
Transport
Properties:
Photons
ElectronsPhonons
Photonic Crystals
Optical Properties
RethermalizationProcesses:
PhotonsElectrons
Phonons Metallic (Dielectric)
Backbone
Electronic
Characterization
Multi-Physics Problem:
Photonic Crystal Radiant Energy Transfer
-
7/28/2019 Photonic Crystals 1 - Overview & Examples
23/53
Summary
Designer Vacuum:Frequency selective control of
spontaneous and thermal emission
enables novel active devices
PBG materials: Integrated optical micro-circuits
with complete light localization
Photonic 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
CI Enabled Photonic Crystal Research and Technology:
Photonic materials by design
Multiphysics and multiscale analysis
-
7/28/2019 Photonic Crystals 1 - Overview & Examples
24/53
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 bydefining 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 ofwavelengths 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 thewavelength 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.
Ph t i C t l
-
7/28/2019 Photonic Crystals 1 - Overview & Examples
25/53
Photonic Crystals:Periodic Surprises in Electromagnetism
Steven G. Johnson
MIT
-
7/28/2019 Photonic Crystals 1 - Overview & Examples
26/53
To Begin: A Cartoon in 2d
planewave
E,H
~ei(kxt)
k / c 2
k
scattering
-
7/28/2019 Photonic Crystals 1 - Overview & Examples
27/53
To Begin: A Cartoon in 2d
planewave
E,H
~ei(kxt)
k / c 2
k
formost , beam(s) propagate
through crystal without scattering
(scattering cancels coherently)
...but forsome (~ 2a), no light can propagate: a photonic band gap
a
-
7/28/2019 Photonic Crystals 1 - Overview & Examples
28/53
1887 1987
Photonic Crystals
periodic 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)
-
7/28/2019 Photonic Crystals 1 - Overview & Examples
29/53
Photonic Crystals
periodic electromagnetic media
with photonic band gaps: optical insulators
magical oven mitts for
holding and controlling light
3D Photonic Crysta l with De fe c ts
can trap light in cavities and waveguides(wires)
-
7/28/2019 Photonic Crystals 1 - Overview & Examples
30/53
Photonic Crystals
periodic electromagnetic media
But how can we understand such complex systems?
Add up the infinite sum of scattering? Ugh!
3D Photo nic C rysta l
Hig h ind e x
o f re fra c tio n
Lo w ind e x
o f re fra c tio n
-
7/28/2019 Photonic Crystals 1 - Overview & Examples
31/53
A mystery from the 19th century
e
e
E
+
+
+
+
+
JEcurrent:conductivity (measured)
mean free path (distance) of electrons
conductive material
-
7/28/2019 Photonic Crystals 1 - Overview & Examples
32/53
A mystery from the 19th century
e
e
E
+
JEcurrent:conductivity (measured)
mean free path (distance) of electrons
+ + + + + + +
+ + + + + + + +
+ + + + + + + +
+ + + + + + + +
crystalline conductor(e.g. copper)
10sof
periods!
-
7/28/2019 Photonic Crystals 1 - Overview & Examples
33/53
A mystery solved
electrons are waves (quantum mechanics)1
waves in aperiodic medium can
propagate without scattering:
Blochs Theorem (1d: Floquets)
2
The foundations do not depend on the specificwave equation.
-
7/28/2019 Photonic Crystals 1 - Overview & Examples
34/53
Time to Analyze the Cartoon
planewave
E,H
~ei(kxt)
k / c 2
k
formost , beam(s) propagate
through crystal without scattering
(scattering cancels coherently)
...but forsome (~ 2a), no light can propagate: a photonic band gap
a
-
7/28/2019 Photonic Crystals 1 - Overview & Examples
35/53
Fun with Math
E 1c
tH i
cH
H 1
c
t
EJ i
c
E0
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
-
7/28/2019 Photonic Crystals 1 - Overview & Examples
36/53
Hermitian Eigenproblems
1
H c
2
H
eigen-operator eigen-valueeigen-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)
-
7/28/2019 Photonic Crystals 1 - Overview & Examples
37/53
Periodic Hermitian Eigenproblems[ G. Floquet, Sur les quations diffrentielles linaries coefficients priodiques,Ann. cole Norm. Sup. 12, 4788 (1883). ]
[ F. Bloch, ber die quantenmechanik der electronen in kristallgittern,Z. Physik52, 555600 (1928). ]
if eigen-operator is periodic, then Bloch-Floquet theorem applies:
H(x
,t)
ei kxt
Hk(x
)can choose:
periodic envelopeplanewave
Corollary 1: kis conserved, i.e.no scattering of Bloch wave
Corollary 2: given by finite unit cell,
so are discrete n(k)
Hk
-
7/28/2019 Photonic Crystals 1 - Overview & Examples
38/53
Periodic Hermitian EigenproblemsCorollary 2: given by finite unit cell,
so are discrete n(k)
Hk
1
2
3
k
band diagram (dispersion relation)
map of
what states
exist &
can interact
?range ofk?
d
-
7/28/2019 Photonic Crystals 1 - Overview & Examples
39/53
Periodic Hermitian Eigenproblems in1d
1
2
1
2
1
2
1
2
1
2
1
2
(x) = (x+a)
H(x) eikxHk(x)
a
Considerk+2/a: ei(k 2
a)x
Hk2
a
(x) eikx ei
2
ax
Hk 2
a
(x)
periodic!
satisfies same
equation asHk=Hk
kis periodic:
k+ 2/a equivalent to kquasi-phase-matching
1d
-
7/28/2019 Photonic Crystals 1 - Overview & Examples
40/53
band gap
Periodic Hermitian Eigenproblems in1d
1
2
1
2
1
2
1
2
1
2
1
2
(x) = (x+a)a
kis periodic:k+ 2/a equivalent to kquasi-phase-matching
k
0 /a/a
irreducible Brillouin zone
1d P i di S h G
-
7/28/2019 Photonic Crystals 1 - Overview & Examples
41/53
Any1d Periodic System has a Gap
1
k
0
[ Lord Rayleigh, On the maintenance of vibrations by forces of double frequency, and on the propagation ofwaves through a medium endowed with a periodic structure, Philosophical Magazine 24, 145159 (1887). ]
Start with
a uniform (1d) medium:
k
1
1d P i di S h G
-
7/28/2019 Photonic Crystals 1 - Overview & Examples
42/53
Any1d Periodic System has a Gap
1
(x) = (x+a)a
k
0 /a/a
[ Lord Rayleigh, On the maintenance of vibrations by forces of double frequency, and on the propagation ofwaves through a medium endowed with a periodic structure, Philosophical Magazine 24, 145159 (1887). ]
Treat it as
artificially periodic
bands are foldedby 2/a equivalence
e
ax
, e
ax
cos
ax
, sin
ax
1d P i di S h G
-
7/28/2019 Photonic Crystals 1 - Overview & Examples
43/53
(x) = (x+a)a1
Any1d Periodic System has a Gap
0 /a
[ Lord Rayleigh, On the maintenance of vibrations by forces of double frequency, and on the propagation ofwaves through a medium endowed with a periodic structure, Philosophical Magazine 24, 145159 (1887). ]
sin
ax
cos
ax
x = 0
Treat it as
artificially periodic
A 1d P i di S h G
-
7/28/2019 Photonic Crystals 1 - Overview & Examples
44/53
(x) = (x+a)a12 12 12 12 12 12
Any1d Periodic System has a Gap
0 /a
[ Lord Rayleigh, On the maintenance of vibrations by forces of double frequency, and on the propagation ofwaves through a medium endowed with a periodic structure, Philosophical Magazine 24, 145159 (1887). ]
Add a small
real periodicity2 = 1 + D
sin
ax
cos
ax
x = 0
A 1d P i di S h G
-
7/28/2019 Photonic Crystals 1 - Overview & Examples
45/53
band gap
Any1d Periodic System has a Gap
0 /a
[ Lord Rayleigh, On the maintenance of vibrations by forces of double frequency, and on the propagation ofwaves through a medium endowed with a periodic structure, Philosophical Magazine 24, 145159 (1887). ]
Add a small
real periodicity2 = 1 + D
sin
ax
cos
ax
(x) = (x+a)a12 12 12 12 12 12
x = 0
Splitting of degeneracy:state concentrated in higher index (2)
has lower frequency
-
7/28/2019 Photonic Crystals 1 - Overview & Examples
46/53
Some 2d and 3d systems have gaps
In general, eigen-frequencies satisfy Variational Theorem:
1(k)
2 minE1
E1 0
ik E12
E1
2
c
2
2(k)2 minE2E2 0
E1* E2 0
" "
kinetic
inverse
potential
bands want to be in high-
but are forced out by orthogonality
>band gap (maybe)
-
7/28/2019 Photonic Crystals 1 - Overview & Examples
47/53
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
-
7/28/2019 Photonic Crystals 1 - Overview & Examples
48/53
2d periodicity, =12:1
E
HTM
a
frequency
(2c/a)
=a/
G X
M
G X M Girreducible Brillouin zone
k
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Photonic Band Gap
TM bands
gap for
n > ~1.75:1
2d periodicity =12:1
-
7/28/2019 Photonic Crystals 1 - Overview & Examples
49/53
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Photonic Band Gap
TM bands
2d periodicity, =12:1
E
HTM
G X M G
Ez
+
Ez
(+ 90 rotated version)
gap for
n > ~1.75:1
2d periodicity =12:1
-
7/28/2019 Photonic Crystals 1 - Overview & Examples
50/53
2d periodicity, 12:1
E
H
E
H
TM TE
a
frequency
(2c/a)
=a/
G X
M
G X M Girreducible Brillouin zone
k
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Photonic Band Gap
TM bands
TE bands
-
7/28/2019 Photonic Crystals 1 - Overview & Examples
51/53
2d photonic crystal: TE gap, =12:1
TE bands
TM bands
gap forn > ~1.4:1
E
H
TE
3d h t i t l l t 12 1
-
7/28/2019 Photonic Crystals 1 - Overview & Examples
52/53
3d photonic crystal: complete gap , =12:1
U L G X W K
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0
21% gap
L'
L
K'
G
W
U'
X
U'' UW' K
z
I: rod layer II: hole layer
I.
II.
[ S. G. Johnson et al.,Appl. Phys. Lett.77, 3490 (2000) ]
gap forn > ~4:1
-
7/28/2019 Photonic Crystals 1 - Overview & Examples
53/53
You, too, can compute
photonic eigenmodes!
MIT Photonic-Bands (MPB) package:
http://ab-initio.mit.edu/mpb
on Athena:
add mpb