“
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.

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
Natural structures
Control electron flow
1950’s electronic revolution
Length scale ~ 
Artificial structures
Control e.m. wave propagation
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 Inverted Opals
S. Lin et al., Nature (1998) J. Wijnhoven & W. Vos, Science (1998)

 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
I in
I out
Photonic crystal all-optical transistor
χ (3)
I
H
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
Probe Laser
Pump Laser
M. Florescu and S. John, PRA 69 , 053810 (2004).

Operating Parameters

Holding power: 10-100 nW
 Switching power: 50-500 pW

Switching time: < 1 ps
 Size: 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
Four laser beams interfere to form a
3D periodic intensity pattern
O. Toader, et al., PRL 92, 043905 (2004)
10
 m

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:
Photons
Electrons
Phonons
Metallic (Dielectric)
Backbone
Electronic
Characterization
Transport
Properties:
Photons
Electrons
Phonons

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

• “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:
Steven G. Johnson
MIT

k scattering planewave
E , H ~ e i ( k  x   t ) k   / c 
2 


a k
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • • k planewave
E , H ~ e i ( k  x   t )
  / c 
2

 for most  , beam(s) propagate through crystal without scattering
(scattering cancels coherently )
...but for some  (~ 2 a ), no light can propagate: a photonic band gap

1887
Photonic Crystals
1-D periodic electromagnetic media
1987
2-D 3-D periodic in one direction periodic in two directions periodic in three directions with photonic band gaps: “ optical insulators ”
(need a more complex topology)

Photonic Crystals periodic electromagnetic media can trap light in cavities
3D Ph o to n ic C rysta l w ith De fe c ts and waveguides magical oven mitts for holding and controlling light
(“wires”) 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 in d e x o f re fra c tio n
3D Ph o to n ic C rysta l
But how can we understand such complex systems?
Add up the infinite sum of scattering? Ugh!

e – e – conductive material
+
+
+
+
+
E current:
J   E conductivity (measured) mean free path (distance) of electrons

e – e – crystalline conductor (e.g. copper)
+ + + + + + + +
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
10’s of periods!
+ + + + + + + +
E current:
J   E conductivity (measured) mean free path (distance) of electrons

1 electrons are waves (quantum mechanics)
2 waves in a periodic medium can propagate without scattering :
The foundations do not depend on the specific wave equation .

a k
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • • k planewave
E , H ~ e i ( k  x   t )
  / c 
2 
 for most  , beam(s) propagate through crystal without scattering
(scattering cancels coherently )
...but for some  (~ 2 a ), no light can propagate: a photonic band gap

 E  
 H   c
1 c
1

 t

 t
H  i
 c
H
0
E  J  i
 c
 E
First task: get rid of this mess dielectric function  ( x ) = n 2 ( x )
 

1
  H  eigen-operator


 c
 2

H eigen-value
+ constraint
 H  0 eigen-state

 

1
  H  eigen-operator


 c
 2

H 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)

[ 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:
H
x
t
 e

 x   t

H k
x
planewave periodic “envelope”
Corollary 1: k is conserved , i.e.
no scattering of Bloch wave
H unit cell , so  are discrete  n
( k )


H unit cell , so  are discrete  n
( k ) band diagram (dispersion relation)

3

2 map of what states exist & can interact

1 k
?
range of k ?

Periodic
1d

1

2

1

2

1

2

1

2

1

2

1

2
H ( x )  e ikx H k
( x )
Consider k +2π/ a : a e i ( k 
2  a
) x
H k 
2 a

 ( x ) =  ( x + a )
( x )  e ikx


  e i
2  a x
H k 
2  a
( x )


 k is periodic : k + 2π/ a equivalent to k
“quasi-phase-matching” periodic!
satisfies same equation as H k
= H k

Periodic
1d k is periodic : k + 2π/ a equivalent to k
“quasi-phase-matching”

1

2

1

2

1

2

1

2

1

2

1

2 a
 ( x ) =  ( x + a )
 band gap k
–π/ a 0 π/ a irreducible Brillouin zone

Any 1d
[ 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

  k

1
0 k

Any 1d
[ 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

1
 ( x ) =  ( x + a ) bands are “folded” by 2π/ a equivalence
 a
–π/ a 0 e

 a x
, e

 a x
 cos


 a x


, sin
 
 a k
π/ a x



Any 1d
[ 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 sin
 
 a cos


 a x

 x



1 a
0 π/ a x = 0
 ( x ) =  ( x + a )

Any 1d
[ 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  sin
 
 a cos


 a x

 x



1 a

2

1

2
 ( x ) =  ( x + a )

1

2

1

2

1

2

1

2
0 π/ a x = 0

Any 1d
[ 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  band gap
Splitting of degeneracy: state concentrated in higher index ( 
2
) has lower frequency sin
 
 a cos


 a x

 x



1 a

2

1

2
 ( x ) =  ( x + a )

1

2

1

2

1

2

1

2
0 π/ a x = 0


• In general, eigen-frequencies satisfy Variational Theorem :

1
k
2 
E
1
  E
1
 0
 
  ik

 E
1
  E
1
2
2 c 2
“kinetic” inverse
“potential”

2
k
2 
E
2

  E
2
 E
1
*  E
 0
2
 0
bands “want” to be in high
…but are forced out by orthogonality
–> band gap (maybe)


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 ]

a irreducible Brillouin zone
M k
G
X
2d periodicity,  =12:1
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
G
Photonic Band Gap
TM
E
H
X
TM bands
M gap for n > ~1.75:1
G

E z
E z
–
2d periodicity,  =12:1
(+ 90° rotated version)
0.3
0.2
0.1
0
G
1
0.9
0.8
0.7
0.6
0.5
0.4
Photonic Band Gap
M
TM bands
+
TM
E
H
X gap for n > ~1.75:1
G

a irreducible Brillouin zone
M k
G
X
2d periodicity,  =12:1
0.3
0.2
0.1
0
G
1
0.9
0.8
0.7
0.6
0.5
0.4
Photonic Band Gap
TM bands
TE bands
M
TM
E
H
X
TE
H
E
G

2d photonic crystal: TE gap ,  =12:1
TE bands
TM bands
TE
H
E gap for n > ~1.4:1

II.
,  =12:1
I.
I: rod layer II: hole layer
0.8
0.7
0.6
0.5
21% gap
0.4
z
0.3
0.2
L'
U'
X
G
U'' U
W' K
W
L
K'
0.1
0
U Õ L
G gap for n > ~4:1
X W K
[ S. G. Johnson et al.
, Appl. Phys. Lett.
77 , 3490 (2000) ]

MIT Photonic-Bands ( MPB ) package: http://ab-initio.mit.edu/mpb on Athena: add mpb