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Photonic Crystals
and Metamaterials and
accelerator applications
Rosa Letizia
Lancaster University/ Cockcroft Institute
[email protected]
Cockcroft Institute, Spring term, 16/04/12
Lectures outline
Lecture 1 – Introduction to Photonic Crystals
• What is a photonic crystal
• Bandgap property
• Intentional defects in Photonic Crystals
• Photonic Crystal applications examples
Lecture 2 – Metamaterials
• What is a metamaterial
• Effective parameters from periodic unit cells
• Permittivity and permeability models
• Retrivial technique
• Modelling and characterisation of metamaterials
Lecture 3 – Accelerator applications
• Photonic crystal resonant cavities
• Dielectric laser accelerators
• Metamaterials accelerating waveguides
Lecture 4 – Computational Photonics
• Introduction to Finite Difference Time Domain
• Computational challenges
Cockcroft Institute, Spring term, 16/04/12
Advanced functional materials
Predesigned electromagnetic properties
•
Overcoming the limitations of natural materials by means of “function through
structure” concept
– Photonic Crystals technology
– Metamaterials
•
Engineering of the geometry of the structure allows for creation of “artificial materials”
for unusual EM responses
•
Scalability
•
Interference lithography (IL) holds the promise of fabricating large-area, defect-free 3D
structures on the sub-micrometer scale both rapidly and cheaply
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What are Photonic crystals (PhCs)?
1-D
2-D
3-D
PhC
PhC
PhC
• Electronic crystal – a familiar analogy
• a periodic array of atoms forms a lattice
• lattice arrangement defines energy bands
• The OPTICAL ANALOGY – Photonic Band Gap (PBG) crystal
• a periodic array of optical materials forms a lattice
• allowed energy (wavelength) bands arise
atoms in diamond structure
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dielectric spheres, diamond lattice
PhCs exist in nature
Iridescence from butterfly wing
The colours produced are not the results
of the presence of pigments.
The mixing of photonic structures and
organic pigments will vary the shades
we see
[ P. Vukosic et al.,
Proc. Roy. Soc: Bio.
Sci. 266, 1403
(1999) ]
3µm
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[ also: B. Gralak et al., Opt. Express 9, 567 (2001) ]
Bragg’s diffraction law
The condition that defines constructive interference:
Interference between two diffracted waves from a series of atomic planes separated by d.
m  2  d  sin 
m  1,2,3..
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Maxwell’s equations in periodic
media
First studied in 1982 by Bloch who extended a theorem developed by Floquet in 1883 for
the 1D case.
BLOCH’S THEOREM: waves in a periodic material can propagate with no scattering and their
behaviour is ruled by a periodic envelop function which is multiplied by a plane wave.
(for most λ, scattering cancels coherently)



B
 E  
t



D
 H 
t
 
 D  0
 
 B  0


D   0 r E


B  0 H
 1     2 
  H    H

c
 r    r  a

H n,k r   e jkr un,k r 
Plane wave
Has the periodicity of the crystal lattice
BLOCH MODES
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Maxwell’s equations in periodic
media
• u n, k r  is given by a finite unit cell so ωn(k) is discrete (the
dispersion relation is organised in bands defined by the index n)
•The solutions of the wave equation:
• The inverse of the dielectric constant and the Bloch modes are expanded in
Fourier series upon the reciprocal vector of the lattice, G
 r  
1
 G e jk r
 r  G

G
G G '
a
k

a
Brillouin Zone

H n , k r    uGn , k e j ( k  G )r


k  G'  k  G   u
n, k
G
G
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
n k 2
c2
uGn,'k ,
Origin of the bandgap
A complete photonic bandgap is a range of frequencies ω in which there are no
propagation solutions (real k) of Maxwell’s equations for any vector k and it is
surrounded by propagation states above and below the forbidden gap
 x    x  a 
k   ck
a=0  usual dispersion relation.
States can be defined as Bloch
functions and wavevectors  the
bands for k>π/a are translated
‘folded’ into the first Brillouin zone
Cockcroft Institute, Spring term, 16/04/12
The degeneracy is broken, the
shift of the bands leads to the
formation of bandgap
A 1D PhC – Quarter-wave stack
•All 1D periodicity in space give rise to a bandgap for any contrast of the refractive
index.
•Smaller the contrast and smaller is the size of the badgap
•A peculiar case: quarter-wave stack can maximise the size of its photonic
bandgap by making the all reflected waves from the layers exactly in phase one
with each others at the midgap frequency
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2D Photonic Crystals
Rectangular lattice
Hexagonal lattice
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2D Photonic Crystals
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PhC and defects
Breaking the periodicity: Point defect
Cavity
Joannopoupos, Photonic Crystals Molding the flow of light: jdj.mit.edu/book
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PhC and defects
Line defect
Waveguide
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PhC and defects
BENEFITS
• EM fields highly confined in defect regions
• enhanced interaction EM excitation – structure (e. g. enhanced nonlinear
effects for frequency conversion applications)
• Scaled-down optical devices
• Integration of multiple optical functionalities on single platform (all-optical
circuits)
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PhC fibers
• Wide single mode wavelength range
• large effective mode area
• anomalous dispersion at visible and near IR wavelengths
• Hollow core is allowed (no limited by material absorption)
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[ R. F. Cregan et al., Science 285, 1537 (1999) ]
[ B. Temelkuran et al., Nature 420, 650 (2002)
]
3D PhC – Woodpile structure
Unit cell: 4 layers
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PhC applications
PhC-based optical filter*
*R. Letizia, and S. S. A. Obayya, IET
Optoelectronics, vol. 2, n. 6, pp. 241253, 2008.
Cockcroft Institute, Spring term, 16/04/12
PhC applications
Photonic wire (PhW)
y
x
w = 0.38 m
a = 0.3162 m
r = 0.23a
nnl = 1.43 x 10-17 m2/W
sat = 0.31
D Pinto, and S S A Obayya, IET Optoelectron., vol. 2, no. 6, pp. 254261, 2008
Cockcroft Institute, Spring term, 16/04/12
c=a
n1 = 1.0
n2 = 3.48
a1 = 262 nm
r1 = 51.93 nm
a2 = 280 nm
r 2= 63.94 nm
PhC applications
Linear
res = 1.536 m
Q = 427
T = 0.66
Nonlinear
res = 1.543 m
Q = 424
T = 0.52
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PhC applications
Enhanced Second Harmonic Generation
Fundamental frequency (TM)
SH frequency
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SH frequency (TE)
Metamaterials (MTMs)
• Control the flow of EM wave in unprecedented way
• The design relies on inclusions and the new properties emerge due the specific
interactions with EM fields
• These designs can be scaled down and the MTM really behaves as a effectively
continuous medium
• Composite by elements as
materials are composite of atoms
• METAMATERIAL represents
the “next” level of organisation of
the matter  the prefix “META”
originates from Greek work “µετα”
which means “beyond”
Cockcroft Institute, Spring term, 16/04/12
MTM applications: where we are
Industrial Applications
• Information and Communication technologies
• Space & Security and defence
• Health
• Energy
• Environmental
Device already realised
• Sensors
• Superlensing
• Cloaking
• Light Emitting Diodes/ cavities for low threshold Lasers
Cockcroft Institute, Spring term, 16/04/12
MTM for cloaking
Metamaterials are also currently a basis for building a cloaking device. A possibility of a
working invisibility cloak was demonstrated in 2006 [1]. Following this result, an intense
research work has been spent in order to build a cloaking device at optical frequencies [2 - 4].
[1] D. Schurig, et al., Science, 314, 977-980, 2006.
[2] A. Greenleaf, et al., Phys. Rev. Lett., 102, 183901 (1-4), 2007.
[3] X. Zhang, et al., Opt. Express, 16, 11764-11768, 2008.
[4] R. Liu, et al., Science, 323, 366-369, 2009.
Cockcroft Institute, Spring term, 16/04/12
Important concepts
Backward-wave materials
•Negative refractive index materials do not exist in nature. These
type of materials were first theoretically introduced by Veselago
but only in 90’s Pendry showed how physically realise them.
•Backward wave media are materials in which the energy
velocity direction is opposite to the phase velocity direction. In
particular this takes place in isotropic materials with negative
permittivity and permeability (double negative materials
DNM).
Negative refraction takes place at the interface
between “normal” media and DNM
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Negative refractive index
r  j r
r  j r
MTMs obey Snell’s law:
n   r r    r r
n1 sin 1  n2 sin 2
2  0
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Superlens
•Negative refraction can be used to focus light. A flat slab of material will produce two
focal points, one inside the slab and the other one outside.
• UNUSUAL focussing properties.
• no reflection from surfaces
• aberration-free focus
• free from wavelength restriction on resolution
  1
  1
n  1
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Coupling of unit cells
•Effective continuous behaviour for wavelengths much larger than the periodicity of
the inclusions (λ > 10*a)
• For shorter wavelengths? (No longer effective medium but Photonic bandgap
effect)
• Unlimited possibilities for the design of inclusions (unit cells)
Induced current
Induced current in the
neighbouring cell
conventional material
Magnetic field from
incoming light
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Induced magnetic
field
metamaterial
A bit of theory
Governing equations describing EM behaviour of materials




Br , t 
Maxwell’s equations in time domain




E
r
,
t



t

 
 

Dr , t   
 J r , t 
  H r , t  

t
  

  Dr , t    r , t 
 
  Br , t   0
Medium response description:
 
 
 
Dr , t    0 E r , t   Pr , t 
 
 
 
Br , t    0 H r , t   M r , t 


t
 
 

Pr , t    0  Rr , t  t E r , t dt 

Frequency domain makes it easier:
Constitutive relations
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 
 

Pr ,     0  e r ,  E r ,  
 
 

M r ,     m r ,  H r ,  
Resonances in permittivity and
permeability
Lorentz model for dielectrics
P  
0 f
E     0   E  
2
2
e     j
•Where f is a phenomenological
strength of the resonance and γ is the
damping factor
•Amplitude is depending on the
detuning of the frequency from the
resonance.
•The lower the damping and the sharper
the resonance, the resonance peak will
also be higher and the transfer of energy
from the illuminating wave to the dipole
will be strongest directly at the resonance
Relative permittivity:
   1    
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Resonances in permittivity and
permeability
Drude model for metals
A special case of Lorentz model:
        0
 p2
 2  j
At low frequencies the
response is dominated by the
imaginary part
    
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j

Equivalent circuits of MTMs
Example: permeability of artificial magnetic structures: the response is due to
resonant oscillating currents. Equivalent circuit  equivalent RLC circuit and apply
the circuit theory equations
H  H 0 e j t
U L  U C  U R  U ind
1
Idt  IR  

C
U
 lt
I
I
I  R 
 ind   2 0 b H 0e jt
L LC
L
L
LI 
I0  
 0lt b / L 
H0
2
  1 LC  j R L
2
   / 0  1   m  
M  N LC V lt b I
M  0  m  H


 1 VUC  0 lt b 2 L  2 
    0 1 

2
2




j

LC


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LC  1 LC
 R L
Designing the EM response
• MTM properties are mainly due to the cellular architecture and also depend on the PCB
substrate.
• Great flexibility to control the EM propagation through MTMs
• Material properties are characterised by an electric permittivity ε and a magnetic
permeability µ.
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EM modelling of MTMs
In order to verify the interesting MTM functionalities we need to retrieve the effective
parameters in order to know how MTM affects light propagation.
Exact material
configuration (geometry
+ parameters of all
constituents)
E, H field distribution
in every unit cell with
periodic excitation
MODEL
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S-parameters
(Reflection and
Transmission
coefficients)
Retrivial method
S-parameters are defined in terms of reflection coefficient R and transmission coefficient T:
S11  R
S 21  Te jk0 d
Z 
n
1  S11 2  S212
1  S11 2  S212
  

e jnk0 d 
   
1
Im ln e jnk0 d  2m  j Re ln e jnk0 d
k0 d
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
S 21
Z 1
1  S11
Z 1
A numerical example
Negative index MTM (unit cell): SRR for magnetic resonance and wire for electric
resonance (copper on FR4 substrate)
S-parameters
computed by CST
Microwave Studio
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A numerical example
Calculated permittivity and
permeability from S-parameters:
Smith, Physical Review E, 71, 036617, (2005)
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Characterisation
Which post-processing can be applied to obtain effective material parameters?
EM characterisation of different types of MTM is still a challenge.
Most known measurement techniques for linear EM characterisation of nano-structured
layers and films:
Techniques
Measurement
equipment
Direct results
Spectroscopy (optical range)
Precision spectrometer
Absolute values of S-parameters
of a layer (film)
Interferometry (optical range)
Precision interferometer
Phase of S-parameters of a layer
(film)
THz time domain spectrometry
(optical range)
Detection of the phase change
of THz wave passing through
the MTM sample, compared to
reference
Real and imaginary part of
effective refractive index via
phase measurements
Free space techniques (RF
range)
Receiving antenna + vector
network analyser
Complex S-parameters of the all
set up between input and output
ports
Waveguide techniques (RF
range)
Receiving probe + vector
network analyser
Complex S-parameters of the all
set up between input and output
ports
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