Transcript metal staircase house work

Photonic crystal and metamaterials
and accelerator applications
Dr Rosa Letizia
Lancaster University/ Cockcroft Institute
[email protected]
Spring term, Lecture 3 & 4
Cockcroft Institute, 23/04/12
New accelerators concepts
 New regimes of particle interactions are probed using particle beams of higher energy.
 In the past, particle energy was increased by simply increasing the physical size of the
accelerator.
 Alternatively, beam energy can be improved by increasing the gradient in linear
accelerators.
 Advances in fundamental accelerator technology may allow to build new compact and
less expensive accelerators for uses in:
 material science
 condensed matter
 medicine (radiation treatment and isotope production)
Cockcroft Institute, Spring term, 23/04/12
New accelerators concepts
Structure-based laser-driven acceleration in vacuum
• Power source: Lasers instead of microwave klystrons  higher E-fields
P  1TW
  1m
P  100MW
  10cm
• To use a laser for accelerating a particle beam:
1. We must have E-field in the direction of propagation of particle beam
2. The mode must have phase velocity in that direction equal to the speed of light in
vacuum for phase matching with the particle bunch
Conventional RF
technology scaled down?
Iris-loaded structure
Cockcroft Institute, Spring term, 23/04/12
• Metal losses at optical
frequencies
• Manufacturing a circular diskloaded waveguide on such a
small scale poses a serious
challenge
Laser/electron energy exchange
•
Laser-driven particle accelerator in structure loaded vacuum (waveguides, semiopen free-space structures, ect)
• 2005 – first experiment to demonstrate laser acceleration mechanism could be
achieved in semi-infinite vacuum
λ =800 nm
Up to 30KeV modulation over
1000 µm (40 MeV/m)
[T. Plettner et al., PRL, 95, 134801, 2005]
Cockcroft Institute, Spring term, 23/04/12
Phase stable net acceleration of e- from a
two-stage optical accelerator
2008 - Demonstration optical bunching and acceleration (acceleration in 2 stages):
1. Slice the electron beam in microbunches spaced by the optical period
2. Optical acceleration by inverse transition radiation (ITR) mechanism
[C.M.S. Sears et al., Phys Rev. ST
Accel. Beams 11, 101301, 2008]
Cockcroft Institute, Spring term, 23/04/12
Photonic crystals recap
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 (dielectric atoms)
• allowed energy (wavelength) bands arise
Cockcroft Institute, Spring term, 23/04/12
Pillbox resonator vs PBG cavities
PBG cavity, TM01 - like mode
Pillbox cavity, TM01 mode
For filling factor f = r/a > 0.2, the PBG structure confines higher order modes (TM11like mode)
f = 0.30
f = 0.15
TM01
TM11
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TM01
TM11
PBG cavity for RF
• An initial experimental work has been directed toward the use of the photonic crystal technology in the
context of particle acceleration [1]
- Operating at 17 GHz
- Gradient: 35 MV/m
• Successful fabrication and use of a PhC structure in a particle accelerator, whose schematic of the
experimental setup and of the PhC structure are shown in figure.
• PhC structures are promising candidate for future accelerator applications because of their ability to
effectively damp high order modes and thus suppress wake field generation.
[1] E.I. Smirnova et al., Phys. Rev. Lett., Vol. 95, pp. 074801, Aug. 2005.
Cockcroft Institute, Spring term, 23/04/12
Photonic acceleration concepts
 Transverse size of the waveguide must be the order of a wavelength (1 µm)
 The laser field co-propagates with the particle beam with a phase velocity equal to
speed of light in vacuum
 Particle beam must form short optical bunches which have only small phase extent
within a laser oscillation
 PBG waveguides are transmission-mode structures
 Ultrafast pulses (~ 1 ps) so material can sustain larger fields
 Accelerating segments in waveguide (individual segment length ~ 100µm – 1mm)
 Wakefields limit the amount of charge we can accelerate
Cockcroft Institute, Spring term, 23/04/12
The individual segments in DLA
• high-gradient
(> 200 MV/m)
• compactness (micron-scale)
• low cost
•(higher breakdown thresholds, 1-5 GV/m)
Si woodpile PhC waveguide
Glass hollow core PhC fiber
[B. Cowan, 2006]
Cockcroft Institute, Spring term, 23/04/12
[R. Noble, 2007]
Double grating (quartz)
[T. Plettner 2009]
Woodpile structure
[B. Cowan., Phys Rev. ST Accel.
Beams 11, 011301, 2008]
3 MV
1.00
2 MV
Si Eacc=337 MV/m
@1550nm(1.94e5GHZ)
Gradient (GeV/m)
Trapping
Breakdown
Pulsed Heating
120 K
40 K
CLIC
0.10
NLC
HRC MIT
SLC
0.01
1.0
10.0
Frequency (GHz)
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100.0
DLA concept
Laser
Electron gun
Image credit: Chris MacGuinness (SLAC)
Cockcroft Institute, Spring term, 23/04/12
Dielectric Breakdown
Laser threshold damage measurement of bulk materials gives a good sense of the
performance of the accelerating structure but realistically that is an upper performance
limit (e.g. Gratings reduce the threshold limits of the material).
λ=800nm
Silicon
[Proc. SPIE 6720, 67201M-1]
[M. Mero, et. al. Phys. Rev. B 71 115109 (2005)]
Cockcroft Institute, Spring term, 23/04/12
New directions in PhC cavities
 PhCs offer a unique way to create resonant cavities for a number of very diverse
fields and recently they have been considered also for accelerator applications.
 By strategically choosing the geometrical parameters of the PhC, it is possible to
realise devices, and in particular resonant cavities, for virtually any range of
frequencies.
 By engineering a defect in an otherwise perfect lattice of a PhC, it is possible to
design a resonant cavity that can sustain resonant modes with field profiles with fixed
shapes.
Cockcroft Institute, Spring term, 23/04/12
New directions in PhC cavities
However, PhC cavities can be highly overmoded thus strategies are needed to
completely remove (or at least to highly suppress) higher frequency resonant modes
fn = 0.38
Q  1200
Q  70
fn = 0.27
Q  500
Q  400
Cockcroft Institute, Spring term, 23/04/12
Photonic quasi-crystal
 “Photonic quasicrystals” (PQCs) are based on the so-called aperiodic-tiling geometries
characterised by weak rotational symmetry of “noncrystallographic” type.
 In PQCs, the EM response can be strongly dependent on the lattice short-range
configuration and an additional degree of freedom for design can come from aperiodicity.
 In terms of quality factor, the quasycristal PhC cavity outperform the counterpart
hexagonal lattice PhC*
 A prototype of a PQC has been fabricated and tested and it has been shown that hybriddielectric structures can be successfully exploited for the design of high-gradient
accelerators*
* E. Di Gennaro, et al., Appl. Phys. Lett., 93, 164102 (2008)
Cockcroft Institute, Spring term, 23/04/12
Photonic crystal cavities
In a work presented by Tanabe et al. [1] a PhC cavity with a quality factor Q in the order of 106 has
been designed and realised for all-optical switching.
By strategically shifting the position of the rods surrounding the PhC cavity it has been possible to
realise a very high-Q resonant cavity
In the same work it has
been shown that by
exploiting
nonlinear
phenomena, and with the
help of an external pump
signal, it is possible to
dynamically vary the Q
factor of the cavity
[1] T. Tanabe, et al., Nature Photonics, 1, 49-52, Jan. 2007
Cockcroft Institute, Spring term, 23/04/12
Dielectric photonics: key
benefits
 low losses

high damage threshold

enhancement of light-matter interactions

single mode operation in over-moded structures

flexibility in design

tunability of semiconductors electrical properties
Cockcroft Institute, Spring term, 23/04/12
Periodicity
a
Photonic
bandgap
MTMs wave-particle interactions
2008, Antipov – first beam test
 Metamaterials can be used to generate wakefield (Cherenkov radiation) in accelerating
structures.
 Slow backwards waves are produced for NIM loaded waveguide.
[1] Antipov, J. Appl. Phys. 104, 014901 (2008)
Cockcroft Institute, Spring term, 23/04/12
MTMs wave-particle interactions
Transmission through the only
SRRs array
[1] Antipov, J. Appl. Phys. 104, 014901 (2008)
Cockcroft Institute, Spring term, 23/04/12
Transmission through the only
wires array
MTMs wave-particle interactions
Particle beam test
[1] Antipov, J. Appl. Phys. 104, 014901 (2008)
Cockcroft Institute, Spring term, 23/04/12
BEAM = 0.25 nC
3 mm (10 ps)
(longitudinal size)
MTMs wave-particle interactions
MTMs mediate energy exchange beam-wave



Complementary SRR (CSRR)  Electric coupling for particle acceleration and generating
radiation
The CSRR MTM has an accelerating mode with longitudinal E-field parallel to planar
CSRR sheet. This mode has also NRI.
To excite the accelerating mode we need to send light with resonant frequency of the mode
CSRR unit
cell
Cockcroft Institute, Spring term, 23/04/12
[* M A Saphiro, et al., Proceedings of PAC09, Vancouver, Canada (2009)]
MTMs for High Power applications

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

hence the choice of the geometry of the elements and the materials involved are central in
the design of MTMs
Cockcroft Institute, Spring term, 23/04/12
Time domain simulations
Allen Taflove, 1998
Cockcroft Institute, Spring term, 23/04/12
Modelling Techniques
Finite Difference Time Domain (FDTD)
 The most common method for the simulation of optical and microwave
devices.
 It relies on the finite difference formulation in order to calculate derivatives in
space and in time.
 It can handle a wide variety of photonic and microwave devices.
 Its formulation is relatively simple
 Simulation time can be long because of the maximum limit imposed to the time
step.
Cockcroft Institute, Spring term, 23/04/12
Finite differences
Central difference scheme (1D case)
f x0  x   f x0  x 
f x0  
2x
If spatial derivatives are approximated by trigonometric functions Chebyshev
polynomials, then we obtain Pseudospectral Time Domain method (PSTD – Liu,
1997)
Cockcroft Institute, Spring term, 23/04/12
Modelling Techniques
Finite Difference Time Domain (FDTD)
•
Maxwell’s equations
B
 E  
t
 H 
D
J
t
D  0
B  0
D   0 E
B   0 H

 E r , t   E y x , t 


 H r , t   H z x , t 
TEM wave
Ey , H z
 H z t 
1 E y



0  x
 t

 E y t   1 H z

 0 x
 t
*A. Taflove, and S. C. Hagness, Computational Electrodynamics: the finite-difference time-domain method, Artech House,
2005
Cockcroft Institute, Spring term, 23/04/12
Finite differences
Uniform grid:
xi 
xi  i
h, i  0,1,2...
ui  u xi 
dui ui 1  ui

 Oh   D  ui  Oh 
dx
h
dui ui  ui 1

 Oh   D ui  Oh 
dx
h
dui ui 1  ui 1
2
2

 O2h   D C ui  O2h 
dx
2h
Cockcroft Institute, Spring term, 23/04/12
Staggering and leapfrogging
Marching of the fields:
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Yee’s cell
Discretisation in space is performed on the Yee’s cell*
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Leapfrog scheme
Discretisation in time is obtained following the leapfrog arrangement*
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Modelling Techniques
Finite Difference Time Domain (FDTD): 2D- formulation
• Applying Yee’s cell and leapfrog arrangement to Maxwell’s equations a set of
discretised equations are derived (2-dimension TE mode)
n 1 / 2
x i 1 / 2 , j 1
H |
n 1 / 2
x i 1 / 2 , j 1
H |
n
n
t  E y |i 1/ 2, j 3 2  E y |i 1/ 2, j 1 2 


 
z

H z |in11, /j21 2  H z |in11,/j21 2
n 1
y i , j 1 / 2
E |
E |
n
y i , j 1/ 2
n
n
t  E y |i 3 2, j 1 2  E y |i 1/ 2, j 1 2 


 
z

n 1 / 2
n 1 / 2
t  H x |i , j 1  H x |i , j

 
z
Cockcroft Institute, Spring term, 23/04/12
 t  H z

  


n 1 2
i 1 2 , j 1 2
 Hz
x
n 1 2
i 1 2 , j 1 2




Numerical dispersion
• Numerical dispersion means dependence of phase velocity on wavelength,
direction of propagation and lattice discretisation.
• It can lead to pulse distorsion, artificial anisotropy and pseudorefraction.
2
 k y y   1
1
 k x x   1
 k z z   1
 t 


sin

sin

sin

sin




 x
  y  2   z
  c  t  2 
2
2

 

 




 
2
2
2
The numerical stability of the method is
ensured by Courant-Friedrichs-Levy
(CFL) condition
1
t 
c
1
1

x 2 z 2
•c is the light velocity
•x is the discretisation along x direction
•z is the discretisation along z direction
Cockcroft Institute, Spring term, 23/04/12
Meshing and staircase problem
•
Grids: collocated or staggered, rectangular or hexagonal, orthogonal,
nonorthogonal or curvilinear, structured or unstructured, regular or irregular
Cockcroft Institute, Spring term, 23/04/12
Meshing and staircase problem
•
Grids: collocated or staggered, rectangular or hexagonal, orthogonal,
nonorthogonal or curvilinear, structured or unstructured, regular or irregular
structured mesh
unstructured mesh
Cockcroft Institute, Spring term, 23/04/12
Truncation of the numerical mesh
• Periodic Boundary Conditions (PBC)
• Absorbing Boundary Conditions (ABC)
• analytical (Mur’s ABC)
• perfectly matched layers (PML)
• PEC, PMC
Uniaxial Perfectly Matched Layers (UPML)
Cockcroft Institute, Spring term, 23/04/12
UPML
Gednay, 1996
 s y sz

 sx

   0


 0

  E   j 0  H
  H  j 0 E
Bx
1
n
2
i, j 
1
2

n
n
t
 Bx 1 
Ey
 Ey
i, j
i, j 
z j i , j 1
2
1
n
2

0
sx sz
sy
0
si  k i 

0 


0 

sx s y 
s z 
i
j 0
 2 0   z t  n  12  2 0   x t  1 n  12  2 0   x t  1 n 12
 H x 1  
 Bx 1  
 Bx 1
H x 1  
i, j 
i
,
j

i
,
j

i, j
 2 0   z t 
2
2  2 0   z t  0
2  2 0   z t   0
2
n
Dy
Ey
1
2
n 1
i, j
n1
i, j
1
1
1
n 
n
n 

 2 0   z t  n 1  2 0 t  n 12


2


t
1
0
 Dy  j 
 H x 1  H x 21   i 
 H z 12  H z 12 
 
i
,
j
i
,
j

i
,
j

1 , j
1 , j
hz  2 0   z t 
 2 0   z t 
2
2  hx  2 0   z t 
2
2 

n
 2   t  n  2 0  1 n1
 Dy  Dy
  0 x  E y  
i, j
i, j
i, j
 2 0   x t 
 2 0   x t  
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
i  x, y, z 
Truncation of the numerical mesh
Comparison of different boundary schemes to truncate an optical
waveguide
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Interval step
in space
(depends on
frequency ,
geometry
complexity)
Simulation parameters
Dielectric
constant on a
grid
Define Meshing
Define Topology
Practical implementation
Calculation of
interval in
time, decision
on length of
simulation
Determination of the Photonic
Band Gap
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 time
steps of the simulation at some detectors
detector point
Cockcroft Institute, Spring term, 23/04/12
Determination of the Photonic
Band Gap
The photonic
FFT of the
stored
incident
and trasmitted
is normalised
performed frequencies
and the transmission
bandgap
(PBG)
is determined
by the fields
range of
for which
coefficient
is calculated
the transmission
is zero.
fNmax  0.42
fNmin  0.25
PBG = fNmax - fNmin  [0.42 - 0.25] = 0.17
Cockcroft Institute, Spring term, 23/04/12
Determination of the Photonic
Band Gap
Another method to determine the photonic band gap for a fixed photonic crystal lattice is to
consider the unit cell of the crystal itself and applying the periodic boundary conditions
(PBCs)
PBC
PBC
PBC
a=0.58652 m, nr =3.4
nr
M
air
r=0.2a
G
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PBC
X
Determination of the Photonic
Band Gap
To apply periodic boundary conditions, the electric and magnetic fields are splitted into
sine and cosine components as
e1 r  R ; t   e1 r ; t cosk  R   e2 r ; t sin k  R 
e2 r  R ; t   e1 r ; t sin k  R   e2 r ; t cosk  R 
e1 r ; t   e1 r  R ; t cosk  R   e2 r  R ; t sin k  R 
e2 r ; t   e1 r  R ; t sin k  R   e2 r  R ; t cosk  R 
where the subscripts 1 and 2 denote the sine and cosine components, respectively, R
stands for the lattice constant vector, and k stands for the wavevector. The above
equations, together with the analogous for the magnetic fields, are inserted in the FDTD
scheme to update the field components at periodic boundaries.
Cockcroft Institute, Spring term, 23/04/12
Determination of the Photonic
Band Gap
• The excitation source takes the form of a current source whose expression is given as
J s  sin 2f 0 t e
 2  t t 0  



 T0 
2
e
 2  y  y0  



 W0 
2
e
 2  z  z0  



 W0 
2
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 until
steady-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 with
periodic boundary applied to terminate the unit cell.
Cockcroft Institute, Spring term, 23/04/12
Determination of the Photonic
Band Gap
a=0.58652 m, nr=3.4
e1 r  R ; t   e1 r ; t cosk  R   e2 r ; t sin k  R 
e2 r  R ; t   e1 r ; t sin k  R   e2 r ; t  cosk  R 
The phase shift is changed according to the
wavevector k varied along the irreducible
nr
Brillouin zone.
cosk  R   e2 r  R ; t sin k  R 
e1 r ; t   e1 r  R ; tM
y
air
x
Cockcroft Institute, Spring term, 23/04/12
r=0.2a
e2 r ; t   e1 r  R ; t sin k  R   e2 r  R ; t cosk  R 
G
X
Determination of the Photonic
Band Gap
It’s interesting to note that the photonic bandgap evaluated with the method of the plane
Varying
k along
the Brillouin
and extracting
all the
eighenmodes
for each
it
wave pulse
correspondes
with zone
the bandgap
extension
when
the wavevector
k is value
alongofthek, X
is
possibleontothe
extract
the photonic
direction
Brillouin
zone. bandgap associated with the photonic crystal.
0.7
0.6
Normalized Frequency (a/)
a=0.58652 m, nr=3.4
0.5
0.4
0.3
nr
0.2
0.1
air
0
G
Cockcroft Institute, Spring term, 23/04/12
M
r=0.2a
G
X
X
M
G