Post on 21-Apr-2022
Novel electromagnetic structures for high frequency acceleration
(Part 4-6)
Rosa Letizia
Lancaster University/ Cockcroft Institute
r.letizia@lancaster.ac.uk
Cockcroft Institute, Spring term, 13/03/17
Lecture 4 – Introduction to Metamaterials• Novel electromagnetic properties• Dispersion engineering• Metamaterials loaded waveguides• Potential application in accelerators
Lecture 5 – Introduction to Plasmonic waveguiding• Surface plasmon polariton• Dispersion • Plasmonic waveguide• Accelerators applications
Lecture 6 – Introduction to Finite Difference Time Domain• Time domain computational modelling• Finite Difference Time Domain• Examples
Lectures outline
CI lectures, Spring term 2
“Beyond” (Meta) materials
http://metamaterials.duke.edu/research/metamaterials/retrieval
CI lectures, Spring term 3
The term “Metamaterial” is first coined by Rodger Walser as (2001/2):
‘Macroscopic composites having man-made, three dimensional, periodic cellular architecture designed to produce an optimised combination, not available in nature, of two or more responses to specific excitation.’
Meta-materials
Lycurgus Cup – British Museum
CI lectures, Spring term 4
Atomic
Meta-materials
Photonic Crystals
period ≈ λ
period<<λ
period ≈ λ/10
λ
Dispersion engineering
8CI lectures, Spring term 5
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 an effectively
continuous medium
CI lectures, Spring term 6
Artificial electromagnetic materials
Negative refractive index materials do not exist in nature. These type of materials were first theoretically introduced by Veselago (1968) but only in 90’s Pendry/Smith showed how to 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 or negative index materials, NIM).
Reverse Snell’s law of refraction Reverse Cherenkov effect Reverse Doppler effect
CI lectures, Spring term 7
Negative Refractive index
MTMs obey Snell’s law: 2211 sinsin nn
02
rr
rr
j
j
rrrrn
CI lectures, Spring term 8
Negative Refraction
Negative refraction takes place at the interface between “normal” media and a double negative material
Designing the EM response
CI lectures, Spring term 9
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
1
n
CI lectures, Spring term 10
Focusing - Perfect lens
CI lectures, Spring term 11
Negative refraction
Negative permittivity
Bulk metal – Drude response
PLASMA FREQUENCY
When ɣ = 0 and ω < ωp ε < 0
ip
2
1
mNe
p0
22
(frequency with which the
collection of free electrons (plasma) oscillates in the presence of an external driving field)
Limitations:
- ωp is typically in the ultraviolet range
- For ω < ωp the response is dominated by the imaginary component high losses
CI lectures, Spring term 12
Artificial negative permittivity – Wire medium
Bulk metal – Drude response
Artificial negative permittivity – “Wire medium” (Pendry et al. 1998)
The plasma frequency depends critically on density and mass of the collective electronic motion
aeEE kzti
,0
a
ip
2
1
The whole structure appears as an effective medium whose electrons (in the wires) move in the +z direction.
2
2
arNNeff
<< N (electron density within each wire)
rarNemeff /ln5.0 22
0 >> m (free electron mass)
raac
meN
eff
effp
/ln
22
2
0
2
2
2ppf << metal plasma freq.
CI lectures, Spring term 13
Negative permeability –Split ring resonator
Negative permeability does not normally occur in natural materials.
Pendry at al. suggested that an array of ring resonators (SRRs) could respond to the magnetic component of the light.
)(
0
kxtieHH
0
2
0 HriU
lIr 2
0
lr
IL
2
0
FLLarM
2
2
IMiLiciRU
CI lectures, Spring term 14
where l = distance in x between SRRs (solenoid approximation) and
R is the ohmic resistance of each ring.
Negative permeability –Split ring resonator
Negative permeability does not normally occur in natural materials.
LRiLCFlHI
larIM d
2
0
2
2
11,
LRiLCF
MBB
dr
2
0
0
111
CI lectures, Spring term 15
LCm1
0 Resonance frequency of the Lorentzian variation of the medium’s effective permeability
FLCmp
1
1 Plasma frequency
(in the direction x)
CI lectures, Spring term 16
[Smith et al, Phys Rev Letts, 84, (2000)][Antipov et al, Prooc. IPAC (2007)]
Left-handed material
CI lectures, Spring term 17
Left-handed material
NIR range (X. Zhang et al., 2008)
microwave range (Smith et al., 2000) microwave range (Smith et al., 2004)
djkTeS
RS0
21
11
2
21
2
11
2
21
2
11
1
1
SSSSZ
djnkdjnk ejmedk
n 00 lnRe2lnIm1
0
1
11 11
210
ZZS
Se djnk
S-parameters are defined in terms of reflection coefficient R and transmission coefficient T:
CI lectures, Spring term 18
Retrieving the effective parameters
Negative index MTM (unit cell): SRR for magnetic resonance and wire for electric resonance (copper on FR4 substrate)
CI lectures, Spring term 19
A numerical example
[Smith, Physical Review E, 71, 036617, (2005)]
Calculated permittivity and permeability from S-parameters:
CI lectures, Spring term 20
A numerical example
Negative index MTM (unit cell): SRR for magnetic resonance and wire for electric resonance (copper on FR4 substrate)
Param. Lx Ly lx ly tm tsµm 150 97 68 19 4.9 14
CI lectures, Spring term 21
Fishnet metamaterial
[R. Letizia, Proc. of UCMMT, IEEE, (2013)]
NIM, TM-like
CI lectures, Spring term 22
Fishnet metamaterial
Ɛ<0, µ<0
kk
SSS
k
10 periods of fishnet single layer
.
.
.
.
.
.
NIM
claddingcladding
Below cut-off propagation and lateral mode confinement without side walls
Metamaterial waveguides
Split Ring Resonators µ < 0Metal wire ε < 0
CI lectures, Spring term 23
CI lectures, Spring term 24
Interactions with particle beam
No need for band-folding given by longitudinal periodicity: fundamental spatial harmonic interacts with sub-relativistic electron beam
Possibility of isotropic negative index in bulk metamaterials
Reverse Cherenkov radiation
[Antipov, J. Appl. Phys. 104, 014901 (2008)]
Metamaterials can be used to generate wakefield (Cherenkov radiation) in acceleratingstructures (artificial dielectric).
Slow backwards waves are produced for NIM loaded waveguide.
2008, Antipov – first test of electron beam through a NIM medium
CI lectures, Spring term 25
MTM-loaded waveguides
Transmission through only wires arrayTransmission through only SRRs array
[1] Antipov, J. Appl. Phys. 104, 014901 (2008) CI lectures, Spring term 26
MTM-loaded waveguides
BEAM = 0.25 nC, 3 mm (10 ps) (longitudinal size)
[Antipov, J. Appl. Phys. 104, 014901 (2008)]
CI lectures, Spring term 27
MTM-loaded waveguides
CI lectures, Spring term 28
MTM-loaded waveguides
2009 – An all-metallic proposal:Left-handed material based on CSRR sheets (spacing = 26 mm)
[M A Saphiro, et al., Proceedings of PAC09, Vancouver, Canada (2009)]
MTM-loaded waveguides
Width 8 mm
Outer ring slot length 6.6 mm
Slot width 0.8 mm
Inner ring slot length 4.6 mm
Split width 0.3 mm
Thickness 1 mm
TM-like NIM metamaterial mode
[E. Sharples, R. Letizia, Journal of Instrumentation, 2014
and IEEE Trans. Plasma Science, 2016]
Long. Wake impedance = 55 kΩ/m, shunt impedance = 21 MΩ/m, R/Q = 4 kΩ/m
MTM-Waveguide mode
9CI lectures, Spring term
Wakefields simulations
10
[E. Sharples, R. Letizia, Journal of Instrumentation, 2014
and IEEE Trans. Plasma Science, 2016]
CI lectures, Spring term
Beam-wave interaction
CI lectures, Spring term 31
CI lectures, Spring term 32
[A. Danisi at al., 2014]
MTMs in accelerators studies
Great need for designing MTMs tailored for high power applications
MTMs are by nature narrowband highly sensitive to changes in the material parameters and in their mechanical shapes due to thermal effects
Another issue is the potential for breakdown induced by edge effects and high circulating currents in the metallic elements embedded in the background medium
Choice of the geometry of the elements and the materials involved are central in the design of MTMs
CI lectures, Spring term 33
MTMs for high power applications
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
CI lectures, Spring term 34
Characterisation
35
Novel electromagnetic structures for high frequency acceleration
(Part 5)
Rosa Letizia
Lancaster University/ Cockcroft Institute
r.letizia@lancaster.ac.uk
Cockcroft Institute, Spring term, 13/03/17
CI lectures, Spring term 36
Surface Plasmon Polariton wave Solutions of Maxwell equations for metal/dielectric interface
Excitation of a coupled state between photons and plasma oscillations at the interface between a metal and a dielectric
Perfectly flat surfaces support non-radiative (bound) SPPs, field on both sides is evanescent
Longitudinal surface wave εd
εmmetal
dielectric
CI lectures, Spring term 37
[M. L. Brongersma, and V. M. Shalaev, Science, 328 (2010)]
Plasmonics
CI lectures, Spring term 38
[M. L. Brongersma, Faraday Discuss., 178, 9 (2015)]
Evolution of plasmonics
CI lectures, Spring term 39
Surface Plasmon Polaritons
Plasmon: Elementary excitation.
Polaritons: Coupled state between an elementary excitation and a photon.
Plasmon polariton: coupled state between a plasmon and a photon.
Plasmons:
Free electrons in metal
Longitudinal density fluctuation (plasma oscillations) at frequency
Volume plasmon polariton propagate through the volume for frequency for:
Surface plasmon polariton propagate along a metal interface for:
32310 cmnp
p
20 p
Considering the metal as a free electron gas:
2
2
1)(
p
Dispersion relation for EM waves in the electron gas (volume or bulk plasmons):
222 kc
2222 kcp
xck
2222
xp kc
p
xk
Only modes for: p
CI lectures, Spring term 40
Drude model:
Bulk plasmon in metals
)()0,,0(
tzkxkiydd
zdxdeHH
)(),0,(
tzkxkizdxdd
zdxdeEEE
)()0,,0(
tzkxkiymm
zmxmeHH
)(),0,(
tzkxkizmxmm
zmxmeEEE
Longitudinal surface wave εd
εmmetal
dielectricz>0
z<0
zddzmm EE
ydym HH
xdxm EE z=0
xdxm kk
SPP dispersion
0
0
zx
y
HHE
CI lectures, Spring term 41
xddydzd
xmmymzmxiiyizi EHk
EHkEHk
ydym
xdxm
HHEE
ydd
zdym
m
zm HkHk
d
zd
m
zm kk
Condition for the SP to exist
Considering that for any EM wave:
2
22
ckk izix
2
2
ziiSP kc
k
(1)
(2)
Thus using (1) and (2):
dm
dmSP c
k
Dispersion relation
SPP dispersion
CI lectures, Spring term 42
CI lectures, Spring term 43
Surface Plasmon modes Optical frequencies with small λSP Sub-wavelength confinement
xd
kc
2222
xp kc
xdm
dm ck
Light in dielectric
Plasmon polariton
Surface plasmon polaritondpSP
1
1
p
xk
Radiative modes
Bound modes
CI lectures, Spring term 44
Drude-like materials
Noble metals (Ag, Au, Cu, Al …)
Plasma frequency in the UV range
Applications from visible range to the mid-IR
M. A. Ordal et al., Appl. Opt., 22, 7 (1983)
Doped silicon
Plasma frequency in the IR range
J. C. Ginn et al., J. Appl. Phys. 110, 4, (2011)
Oxides and nitrides
Oxides: near-IR range
Transition metal nitrides (TiN): visible range
G. V. Naik et al., Opt. Material Express, 1, 6, (2011)
Graphene
IR frequency range
A. Vakil and N. Engheta, Science, 332, 6035 (2011)
CI lectures, Spring term 45
ip
2
2
1)(
''2
1
xx k
L
1
'''
m
mxxspp c
ikkk
Propagation Length
Considering lossy Drude model for metal:
SPP propagation length
CI lectures, Spring term 46
Excitation of Surface Plasmon modes
How to couple light to SPPs?
Different techniques are possible:
High energy electrons
Kretchman geometry
Otto configuration
Gratings Nanoslits …
photonxspp kk
Kretchmann geometry
SPPs at a metal/air interface can be excited from a high refractive index medium
SiO2 prism creates evanescentwave by total internal reflection.
Strong coupling between parallel component of khigh and kSPP.
Reflected wave reduced in intensity indicate coupling with SPP.
SPPs at the metal/glass interface cannot be excited.
SPPhigh kk
CI lectures, Spring term 47
CI lectures, Spring term 48
Predicted by Ritchie, Phys. ReV. 1957, 106 (5), 874-881, SPP modes can be excited by perpendicular electron beam (50 keV)
Beam excitation of Surface Plasmon modes
[M. V. Bashevoy et al., Nano Letts, (2006)]
[S. Gong et al., Optics Express, (2014)]
CI lectures, Spring term 49
SPP modes can also be excited by parallel electron beam
Example: 100 keV electron bunch
Beam excitation of Surface Plasmon modes
Considering contribution of losses, SPPs decay within the propagation length
CI lectures, Spring term 50
Tunability of plasmonic properties
[A. Boltasseva and H. Atwater, Science, 331, (2011)]
Metal-Insulator-Metal plasmonic waveguide
Similarly to the case of the open grating, a single interface does not support luminous phase velocities (vph = w/kx= c).
Coupling of the SPP modes from the two single interfaces tri-layer system.
Sub-wavelength propagation MIM and IMI plasmonic waveguides.
Miniaturisation of conventional optical waveguides.
CI lectures, Spring term 51
D D
CI lectures, Spring term 52
Accelerating modes for relativistic beams are found coupling two interfaces in the metal-insulator-metal configuration.
Symmetric and antisymmetric SPP modes
Longitudinal luminous modes
Transverse sub-luminous modes
Properties:
Metal or Drude-like material Plasmonic resonances make metal suitable at very high frequencies Easy fabrication Narrow gap (maximises the ) Dispersion engineering Tunable via material properties
CI lectures, Spring term 53
Surface wave accelerators?
EEacc /
GZO-Air-GZO
fsp
[R. Letizia, D. Pinto, IEEE Journal of Lightwave Technology, (2014)]
Surface wave accelerators?
CI lectures, Spring term 54
CI lectures, Spring term 55
IR-range surface wave accelerator (Shvets 2012)
[Neuner et al., PR-STAB, 15, 031302 (2012)]
Silicon carbide: Negative dielectric constant in mid-IR range Compatible with short pulse high power CO2 lasers at 9.6 µm <λ <11.3 µm Operation in hostile and high temperature environment (>1000° C) High electrical breakdown (DC threshold ~300 MV/m)
CI lectures, Spring term 56
Shift of λ luminous of longitudinal mode (solid line) as gap size increases
IR-range surface wave accelerator (Shvets 2012)
[Neuner et al., PR-STAB, 15, 031302 (2012)]
CI lectures, Spring term 57
Plasmonic metasurface (Bar-Lev/Scheuer 2014)
λ = 800 nm
[Bar-Lev and Scheuer, PR-STAB, 17, 121302, (2014)]
Laser driven plasmonic metasurface-based particle acceleration: Plasmonic nanoantennas to enhance EM energy concentration in a small volume large
enhancement of the local field Combination of resonant structure (metasurface) with ultra-short pulse operation Under symmetrical excitation, G =2.64 E0
Au films on dielectric substrate can attain similar damage threshold than SiO2 under fs pulses. Max unloaded gradient estimated as 4.64 GV/m for 100 fs pulses (higher for shorter pulses).
CI lectures, Spring term 58
Plasmonic approach to negative refraction
[Verhagen et al., Phys Rev Letts, 105 (2010)][Lezec et al., Science,316, (2007)]
Coupling of MIM waveguides, isotropic negative refraction of TM waves.
59
Novel electromagnetic structures for high frequency acceleration
(Part 6)
Rosa Letizia
Lancaster University/ Cockcroft Institute
r.letizia@lancaster.ac.uk
Cockcroft Institute, Spring term, 13/03/17
Allen Taflove, 1998
CI lectures, Spring term 60
Modelling – Finite Difference Time Domain
The most common method for thesimulation of optical and microwave devices.
It relies on the finite difference formulationin order to calculate derivatives in space andin time.
It can handle a wide variety of photonic andmicrowave devices.
Its formulation is relatively simple and adaptswell on parallel processing
Simulation time can be long because of themaximum limit imposed to the time step.
Develop time-dependent integral or Maxwell’s curl equations.
Discretise the equations in space and in time by means of an appropriategrid in space, and suitable basis and testing functions.
Derive a set of equations that relate unknown with known quantities(starting from an initial value that usually is given by the source field).
Generate a numerical solution of this initial-value problem in space andtime.
General steps of time domain modelling
CI lectures, Spring term 61
Modelling in time domain
Central difference scheme (1D case)
xxxfxxfxf
2
000
If spatial derivatives are approximated by trigonometric functions Chebyshevpolynomials, then we obtain Pseudospectral Time Domain method (PSTD – Liu, 1997)
CI lectures, Spring term 62
Finite differences
Maxwell’s equations
JtDH
tBE
Finite Difference Time Domain (FDTD)
*A. Taflove, and S. C. Hagness, Computational Electrodynamics: the finite-difference time-domain method, Artech House, 2005
0 D 0 B
ED 0 HB 0
txHtrH
txEtrE
z
y
,,
,,
xH
ttE
xE
ttH
zy
yz
0
0
1
1
TEM wave
zy HE ,
CI lectures, Spring term 63
Modelling techniques
ix
...2,1,0,
ihixi
Uniform grid:
ii xuu
hOuDhOh
uudxdu
iiii
1
hOuDhOhuu
dxdu
iiii
1
2211 222
hOuDhOhuu
dxdu
iCiii
CI lectures, Spring term 64
Finite differences
Marching of the fields:
CI lectures, Spring term 65
Staggering and leapfrogging
Discretisation in space is performed on the Yee’s cell:
CI lectures, Spring term 66
Yee Algorithm
Discretisation in time is obtained following the leapfrog arrangement*:
CI lectures, Spring term 67
Leapfrog scheme
Applying Yee’s cell and leapfrog arrangement to Maxwell’s equations a set of discretised equations are derived (2-dimension TE mode)
zEEtHH
njiy
njiyn
jixn
jix21,2/123,2/12/1
1,2/1
2/1
1,2/1
||||
x
HHtzHHtEE
n
jizn
jizn
jixn
jixnjiy
njiy
21
21,21
21
21,21
2/1
,
2/1
1,
2/1,
1
2/1,
||||
zEEtHH
njiy
njiyn
jizn
jiz21,2/121,232/1
21,1
2/1
21,1
||||
CI lectures, Spring term 68
2D-FDTD formulation
Numerical dispersion means dependence of phase velocity on wavelength, direction of propagation and lattice discretisation.
It can lead to pulse distorsion, artificial anisotropy and pseudo-refraction.
22
2sin
1
2sin
1
2sin
1
ttc
zkz
xkx
zx
The numerical stability of the methodis ensured by Courant-Friedrichs-Levy(CFL) condition
22
11
1
zxc
t
c is the light velocity x is the discretisation along x direction z is the discretisation along z direction
CI lectures, Spring term 69
Numerical dispersion
Grids: collocated or staggered, rectangular or hexagonal, orthogonal, non-orthogonal or curvilinear, structured or unstructured, regular or irregular
CI lectures, Spring term 70
Meshing and staircase problem
structured mesh
unstructured mesh
CI lectures, Spring term 71
Meshing and staircase problem
FDTD – Periodic Boundary Conditions (PBC)
λ = 438.9 nm λ = 235.8 nm
A possible solution: FVTD - PBC
CI lectures, Spring term 72
Silver nanorod modes: the unit cell
a = 200 nm
r = 0.25 a
λ = 600 nm
2D plasmonic waveguide on Silver nanorods in hexagonal lattice
The Silver complex dielectric constant is modelled by Drude–Critical Points model and solved into Auxiliary Differential Equation (ADE) scheme
Real part of ε Imaginary part of ε
Silver nanorods waveguide
CI lectures, Spring term 73
Periodic Boundary Conditions (PBC) Absorbing Boundary Conditions (ABC)
analytical (Mur’s ABC) perfectly matched layers (PML)
PEC, PMC
Uniaxial Perfectly Matched Layers (UPML)
CI lectures, Spring term 74
Truncation of the computational window
Gednay, 1996
HjE 0
EjH 0
z
yx
y
zx
x
zy
sss
sss
sss
00
00
00
0
jks i
ii zyxi ,, n
jiyn
jiyj
n
jix
n
jix EEztBB
,1,
2
1
2
1,
2
1
2
1,
2
1
2
1,
00
02
1
2
1,
0
02
1
2
1,
1
2
2
2
2
n
jixz
xn
jixz
zn
jix BttH
ttH
2
1
2
1,
00
0 1
2
2
n
jixz
x Btt
2
1
,2
11
2
1
,2
11
0
02
1
2
1,
2
1
2
1,
0
0
,0
01
, 2
21
2
21
2
2 n
jz
n
jzz
ix
n
jix
n
jixz
jz
n
jiyz
zn
jiy HHt
th
HHt
th
DttD
n
jiy
n
jiyx
n
jiyx
xn
jiy DDt
EttE
,
1
,0
0
,0
01
,
1
2
2
2
2
CI lectures, Spring term 75
Uniaxial Perfectly Matched Layers (UPML)
Comparison of different boundary schemes to truncate an optical waveguide
CI lectures, Spring term 76
Truncation of the computational window
Def
ine
Top
olo
gy Dielectric constant on a grid
Def
ine
Mes
hin
g Interval step in space (depends on frequency , geometry complexity)
Sim
ula
tio
n p
aram
eter
s
Calculation of interval in time, decision on length of simulation
CI lectures, Spring term 77
Implementation
detector point
A wide broadband plane wave pulse is sent towards the photonic crystal
The time-domain variations of the incident and transmitted fields are stored for all timesteps of the simulation at some detectors
CI lectures, Spring term 78
Determination of the photonic bandgap
The FFT of the stored incident and trasmitted fields is performed and the transmissioncoefficient is calculated
fNmin 0.25
fNmax 0.42
PBG = fNmax - fNmin [0.42 - 0.25] = 0.17
The photonic bandgap (PBG) is determined by the range of normalised frequencies forwhich the transmission is zero.
CI lectures, Spring term 79
Determination of the photonic bandgap
r=0.2a
nr
a=0.58652 m, nr =3.4
air
G
M
X
PBC
PBC
PB
CP
BC
Another method to determine the photonic band gap for a fixed photonic crystal latticeis to consider the unit cell of the crystal itself and applying the periodic boundaryconditions (PBCs)
CI lectures, Spring term 80
Determination of the photonic bandgap
To apply periodic boundary conditions, the electric and magnetic fields are splitted intosine and cosine components as:
RktRreRktRretre
RktRreRktRretre
RktreRktretRre
RktreRktretRre
cos;sin;;
sin;cos;;
cos;sin;;
sin;cos;;
212
211
212
211
where the subscripts 1 and 2 denote the sine and cosine components, respectively, Rstands for the lattice constant vector, and k stands for the wavevector. The aboveequations, together with the analogous for the magnetic fields, are inserted in the FDTDscheme to update the field components at periodic boundaries.
CI lectures, Spring term 81
Determination of the photonic bandgap
The excitation source takes the form of a current source whose expression is given as
where f0 is the modulation frequency, t0 is the time delay, T0 is the time-width, and y0,z0, are the centre of the source, and W0 is the space-width of the pulse.
The wavevector k is set according to the irreducible Brillouin zone.
The time-variation of the electric field is recorded at different observation points untilsteady-state is reached.
Upon using FFT, the spectrum of the time-domain response is obtained.
This spectrum presents peaks that correspond to all the eigenmodes compatible withperiodic boundary applied to terminate the unit cell.
2
0
0
2
0
0
2
0
0 222
02sin
W
zzW
yyT
tt
s eeetfJ
CI lectures, Spring term 82
Determination of the photonic bandgap
r=0.2a
nr
a=0.58652 m, nr=3.4
air
x
y
RktreRktretRre sin;cos;; 211
RktreRktretRre cos;sin;; 212
RktRreRktRretre sin;cos;; 211
RktRreRktRretre cos;sin;; 212
The phase shift is changed according to thewavevector k varied along the irreducibleBrillouin zone.
G
M
X
CI lectures, Spring term 83
Determination of the photonic bandgap
Nor
mal
ized
Freq
uenc
y(a
/)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
G MX G
r=0.2a
nr
a=0.58652 m, nr=3.4
air
G
M
X
Varying k along the Brillouin zone and extracting all the eighenmodes for each value ofk, it is possible to extract the photonic bandgap associated with the photonic crystal.
r=r=r 0.2a
nr
=0.58652 mm , nr=3.4
air
G
CI lectures, Spring term 84
Determination of the photonic bandgap
Fundamental frequency (TM)
SH frequency (TE)SH frequency
Enhanced Second Harmonic Generation
Nonlinear FDTD
CI lectures, Spring term 85
MEEP/MPB solvers: http://ab-initio.mit.edu/wiki/index.php/Meep_Introduction
FDTD++ (JFDTD): https://www.fdtdxx.com/about
A number of other 1D and 2D FDTD codes are available online,mainly in Matlab.
CI lectures, Spring term 86
FDTD codes
Both metamaterials and plasmonics provide newelectromagnetic properties which can modify the interactionwith particle beams.
Metamaterials explored in the microwave range.
Plasmonics can offer an alternative to dielectric photonicstructures for high frequency acceleration.
Choice of materials/geometry of inclusions is crucial.
Limitations need to be clearly identified.
CI lectures, Spring term 87
Conclusions