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Ultrafast Optics and XRay Division
Computational Electromagnetics: from Metamaterials to Particle
Accelerators
Arya FallahiUltrafast optics and Xray Division
1. July 2013
Ø Frequency Selective Surfaces (FSS)● Analysis techniques:
● Diffraction Analysis● Basis Functions ● Dispersion Analysis
● Design and optimization of FSSØ Graphene metasurfacesØ Particle accelerators
● Bunch acceleration in THz waveguides● Bunch Compression in THz waveguides
Ø Conclusion
Ultrafast Optics and XRay Division
Outline 2/44
Reflector antennas
Radomedesign
Polarizers
Beamsplitters
Frequency Selective Surfaces (FSS) 3/30Frequency Selective Surfaces
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A 2D array of patches printed on a grounded substrate can perform as
1. artificial magnetic conductor (AMC)2. radar absorbing surface
F. Yang et. al, MTT47, 1999D. Kern et. al, Microwave & Opt. Tech. Lett., 2003
PEC
lossy substrate
patches
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Ultrafast Optics and XRay Division
Frequency Selective Surfaces 4/44
FSS with homogeneous substrate (MoM)
Boundary condition on the patch:
Diffraction Analysis of FSS
From the theory of Green’s function
known variable
known operator
unknown variable Method of Moments (MoM)
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Diffraction Analysis of FSS
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MoM/TL method
Diffraction Analysis of FSS
FSS with periodic substrate (MoM/TL)
Solved using MoM
Computed using the coupled multiconductor transmission
line (TL) model
Fallahi et al. Elsevier Metamaterials, Oct. 2009.
incident wave arbitrarily shaped
patches
periodic substratex
y
z
Lx
Ly
coupled multiconductor
transmission lines
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Diffraction Analysis of FSS
FSS with periodic substrate (MoM/TL)
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• Better control over the substrate properties
• Shifting resonance frequencies
• Increasing or decreasing the number of resonances
• Coupling of different diffraction orders within the substrate
patches
homogenoussubstrates
periodicsubstrates
planewave
Diffraction Analysis of FSS
FSS with periodic substrate (MoM/TL)
Ultrafast Optics and XRay Division
More degrees of freedom in the design
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Ultrafast Optics and XRay Division
Subdomain basis functions
Rooftop basis functions Surface patch basis functions
Basis Functions
The most critical point concerning the selection of basis functions:
The normal component of the current at the boundary should vanish.
10/44
PEC Ht
Hn=0
Entire domain basis functions
Fallahi et al. IEEE TAP  58, March 2010.
Transverse magnetic fields of the guided modes
Entire domain basis functions
Boundary Integral Resonant Mode Expansion (BIRME)
TM mode TE mode
Ultrafast Optics and XRay Division
Basis Functions 11/44
Largely overlapping subdomain basis functions
Problems with the MoM/BIRME:1) Long computation time for basis functions2) Sophisticated implementation of BIRME
Fallahi et al. IEEE MTT 2010
Ultrafast Optics and XRay Division
Basis Functions 12/44
Largely overlapping subdomain basis functions
Ultrafast Optics and XRay Division
Basis Functions 13/44
Dispersion Analysis of FSS
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For a grounded substrate, I=R for all frequencies except the resonance points.
r r rj
air air
p
prism
( )zh k
(a) (b)
z
xy
Energy coupling technique
Dispersion Analysis of FSS
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Band diagram is obtained by finding the minima of the reflection spectrum.
Energy coupling technique
Fallahi et al. Antenna & Prop. Symp. July 2008
Dispersion Analysis of FSS
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Unit cell is divided to an N×N grid, and encoded to a binary string.
Fitness function is defined based on the reflection spectrum from the FSS.
A bruteforce simulation of all the possible cases is done.
Each optimization algorithm is run 1000 times to obtain reliable statistical data about its efficiency.
metallic patchno patch
Efficient procedures for FSS optimization
Designing FSS Structures
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Probabilities of finding global optimum in percent
Efficient procedures for FSS optimization
Fallahi et al. IEEE TAP  56, May 2008.
Fallahi et al. CTN Journal, April 2008.
M 1 2 3 4 5 6 7 8 av
STAT 7.84 7.55 8.44 7.74 10.1 9.57 8.88 8.77 8.61
MGA0 11.9 13.1 15.4 17.2 20.5 19.8 19.9 17.3 16.9
MGA1 11.7 14.4 17.7 20.1 22.9 23.6 24.2 23.6 19.8
MGA2 20.6 22.0 20.4 18.9 27.6 30.3 29.4 28.0 24.7
MUT0 6.86 6.99 7.81 7.42 7.59 7.96 8.11 8.11 7.54
MUT1 25.7 26.9 25.1 23.6 23.8 23.5 23.2 21.3 24.1
MUT2 27.3 27.3 25.5 23.6 24.3 24.5 22.6 19.8 24.4
RHC 97.6 85.5 42.9 26.7 23.7 23.5 21.7 20.0 42.7
Designing FSS Structures
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➢ For the patch layer the unit cell is encoded into a 14 by 14 grid.
➢ It is demonstrated that considering one hole and optimizing its dimension leads to better absorbers.
FSS absorber Perforated FSS absorber
Ultrafast Optics and XRay Division
Design of Thin Radar Absorbers 19/44
Fabrication of radar absorbers
Characterization of radar absorbers
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Implementation of Radar Absorbers 20/44
Homogeneous absorber FSS absorber
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Implementation of Radar Absorbers 21/44
Perforated FSS absorber
Fallahi et al. IEEE TAP 2010
Ultrafast Optics and XRay Division
Implementation of Radar Absorbers 22/44
Ultrafast Optics and XRay Division
Graphene Metasurfaces
Dynamic Frequency Selective Surfaceperiodic arrangement of metals in a surface with a dynamic response
Liquid crystalsVaractor diodes MEMS switches
23/44
A promising solution to the mentioned problems is graphene
A 2D honeycomb lattice made of carbon atoms
➢2D atomic lattice➢Electrons behaving as massless Dirac Fermions➢Large electron mobilities➢Transparent conductivity➢Large nonlinear Kerr effect…
Electrically tunable conductivity
Electromagnetic properties ofpatterned graphene
Ultrafast Optics and XRay Division
Graphene Metasurfaces 24/44
Ultrafast Optics and XRay Division
Graphene Conductivity
For modeling patterned graphene surfaces, graphene conductivity is needed.
Kubo formalism
The conductivity is dispersive, anisotropic, bias dependent and most important of all, it is an operator.
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Modified PMoM for Graphene Metasurface
In the periodic Method of Moments, we work in the spectral domain:
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Examples of Graphene Metasurfaces
L=10mm, D=7.5mm and d=1.25mm
A. Fallahi and J. PerruisseauCarrier, PRB, 2012.
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Ultrafast Optics and XRay Division
E0=0V/nm E0=2V/nm E0=20V/nm
Increasing the electric biasing causes the increase in the graphene conductivity and this in turn strengthens the resonances between adjacent patches.
Examples of Graphene Metasurfaces 28/44
Ultrafast Optics and XRay Division
Problems with the presented example:
Ø Effective performance needs very high bias electric fieldsØ How this electrostatic biasing can be implemented without disturbing
the FSS?Ø How the bias fields should be applied throughout the FSS?
Ø Solution: Double graphene layers with DCconnected patches
Examples of Graphene Metasurfaces 29/44
Ultrafast Optics and XRay Division
L = 5 µm, l = 0.5 µm, d = 1.25 µm, and D = 1.5 µm and the dielectric thickness is t = 50 nm.
Examples of Graphene Metasurfaces 30/44
The concept of oscillating field accelerators was mainly proposed in 1920s.
A number of cavities with designed lengths are placed along the acceleration line and the electron beam gains energy from the cavity fields.
This has been the basic concept of the various modern accelerators.
Ultrafast Optics and XRay Division
Particle Accelerators 31/44
Recently, there has been many studies on optical acceleration of particles.
Due to small cross section of optical beams, acceleration of a very small amount of charge is feasible using laserplasmon acceleration.
THz acceleration of particles seems to be a good candidate.
Ultrafast Optics and XRay Division
Particle Accelerators 32/44
The main challenges in this regime:
1) Large loss of the metals preclude the design of highQ cavities2) The very limited available THz sources3) The dissipated energy in the metallic walls are much larger than RF
domain.
The second difficulty is recently tackled by using optical rectification techniques to design THz sources.
Short THz pulses can be efficiently generated from a reasonable laser intensity.
Ultrafast Optics and XRay Division
Particle Accelerators 33/44
Ultrafast Optics and XRay Division
Bunch Acceleration in THz Waveguides
0 1 0 0 2 0 0 3 0 0 4 0 00
0 , 4
0 , 8
1 . 2
1 . 6
2
V a c u u m R a d i u s ( m )
Ele
ctric
Fie
ld M
agni
tude
(G
V/m
)
0 1 0 0 2 0 0 3 0 0 4 0 00
2
4
6
8
1 0
Mag
netic
Fie
ld M
agni
tude
(T
)  Ez

 Er

 H

Uniform longitudinal field at β=1
End view Side view
Dielectric
Vacuum Metal Electron bunch THz pulse
r
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Ultrafast Optics and XRay Division
Optimization
Very many parameters are involved in the acceleration level of electrons, including v
p,
vg, GVD, α, z
0 and ψ
0 .
The best fitness function for optimization is the final energy of the electron.
The problem of one electron is solved numerically,
and the final energy of the electron is used as the figure of merit for the group of parameters.
Bunch Acceleration in THz Waveguides 35/44
Ultrafast Optics and XRay Division
Optimization
600GHz operation frequency
2 0 2 5 3 0 3 5 4 0 4 58
8 . 5
9
9 . 5
1 0
1 0 . 5
D i e l e c t r i c T h i c k n e s s ( m )
Fin
al E
nerg
y (M
eV)
Vacuum radius380µm
Dielectric thickness32µm
Optimum frequency of optical rectification is assumed to be 600GHz.
Bunch Acceleration in THz Waveguides 36/44
Ultrafast Optics and XRay Division
Electron bunch acceleration
● CashKarp RungeKutta method is used for efficient update of electron motion with time.
● BoxMuller method is used for generating a bunch of electrons with Gaussian distribution.
● We can not consider all the particles, so we use macroparticles.
● We consider the following initial condition:
● Mean initial energy = 1MeV● Initial energy spread = 0.1%● Initial bunch spread is a cube of 30µm×30µm×30µm
Bunch Acceleration in THz Waveguides 37/44
Ultrafast Optics and XRay Division
0 0 . 2 0 . 4 0 . 6 0 . 8 10
0 . 5
1
1 . 5
/k
0
F r e q u e n c y ( T H z )
Dispersion curve Energy of a bunch (20mJ THz pulse)
0 1 0 2 0 3 00
2
4
6
8
1 0
D i s t a n c e ( m m )
Mea
n be
am e
nerg
y (M
eV)
w i t h o u t s p a c e  c h a r g e w i t h s p a c e  c h a r g e
Energy spread in a bunch
0 1 0 2 0 3 00
0 . 0 5
0 . 1
0 . 1 5
0 . 2
0 . 2 5
D i s t a n c e ( m m )
Bea
m e
nerg
y de
viat
ion
(MeV
)
w i t h o u t s p a c e  c h a r g e w i t h s p a c e  c h a r g e
0 2 4 6 8 1 01 5 0
2 0 0
2 5 0
3 0 0
3 5 0
D i s t a n c e f r o m t h e W a v e g u i d e E n t r a n c e ( c m )
Coa
ting
Tem
prat
ure
( o C)
Temperature of the coating
1MeV to 10 MeV
2% energy spread
No melting of the coating
Bunch Acceleration in THz Waveguides 38/44
Ultrafast Optics and XRay Division
Energy of a bunch (20mJ THz pulse)
0 1 0 2 0 3 00
2
4
6
8
1 0
D i s t a n c e ( m m )
Mea
n be
am e
nerg
y (M
eV)
w i t h o u t s p a c e  c h a r g e w i t h s p a c e  c h a r g e
Electron is injected in a point within the pulse
It is not possible practically
0 1 0 2 0 3 0 4 00
2
4
6
8
1 0
D i s t a n c e ( m m )
Mea
n be
am e
nerg
y (M
eV)
w i t h s p a c e  c h a r g e w i t h o u t s p a c e  c h a r g e
Acceleration to 8.5MeV
0 1 0 2 0 3 0 4 00
0 . 2
0 . 4
0 . 6
0 . 8
D i s t a n c e ( m m )
Bea
m e
nerg
y de
viat
ion
(MeV
)
w i t h s p a c e  c h a r g e w i t h o u t s p a c e  c h a r g e
9% energy spread
A rule of thumb
Practically achievable results are 10% lower than the theoretical optimum point.
Bunch Acceleration in THz Waveguides 39/44
Ultrafast Optics and XRay Division
Amount of charge we can accelerate in the waveguide?
Without spacecharge: unlimitedWith spacecharge: limited
16pC is by far more than what we need.
What is the limit?
We stop a macroparticle as soon as it hits the walls of the
accelerator.
Bunch Acceleration in THz Waveguides 40/44
0 1 0 2 0 3 00
2
4
6
8
1 0
D i s t a n c e ( m m )M
ean
beam
ene
rgy
(MeV
)
1 . 6 p C b u n c h c h a r g e 1 6 p C b u n c h c h a r g e 1 6 0 p C b u n c h c h a r g e
frontFbackFfrontFbackFbackF frontF
Ultrafast Optics and XRay Division
Bunch Compression in THz waveguides
Rectilinear Compression
Compression and deceleration
Maximum Compression
Compression and acceleration
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0 1 0 2 0 3 00
0 . 2
0 . 4
0 . 6
0 . 8
D i s t a n c e ( m m )
Bea
m e
nerg
y de
viat
ion
(MeV
)
w i t h s p a c e  c h a r g e w i t h o u t s p a c e  c h a r g e
7% energy spread
Ultrafast Optics and XRay Division
0 1 0 2 0 3 00
1
2
3
4
5
D i s t a n c e ( m m )
Mea
n be
am e
nerg
y (M
eV)
w i t h s p a c e  c h a r g e w i t h o u t s p a c e  c h a r g e
Acceleration to 3MeV
0 1 0 2 0 3 00
2 0
4 0
6 0
8 0
1 0 0
D i s t a n c e ( m m )
z (
m
)
w i t h s p a c e  c h a r g e w i t h o u t s p a c e  c h a r g e
50 times bunch compression
Bunch Compression in THz waveguides
Ø Mean initial energy = 1MeVØ Initial energy spread = 0.1%Ø Initial bunch spread is a cube of
30mm×30mm×30mmØ 20mJ 10 cycle THz pulse
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Conclusion
Ultrafast Optics and XRay Division
➢ MoM and MoM/TL for diffraction analysis of planar metamaterials➢ subdomain basis functions➢ entire domain basis functions➢ largely overlapping basis functions
➢ Energy coupling method for dispersion analysis
➢ Efficient procedures for the design of planar metamaterials
➢ PMoM is generalized for the simulation of periodic graphene metasurfaces:➢ Arbitrary number of layers➢ Arbitrary shapes for the unit cell configuration➢ Full vectorial and arbitrary angles of incidence➢ Simulation of a single cell of the periodic structure (Floquet)➢ discretization of conductive layers only➢ Both periodic and homogeneous substrates➢ Nondiagonal conduc/vity for B ≠ 0 and spatiallydispersive conductivity➢ Metalgraphene hybrid layers are recently implemented
➢ Compact size accelerators based on THz acceleration➢ THz waveguides for bunch acceleration➢ THz waveguides for bunch compression
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Ultrafast Optics and XRay Division
Thank you for your attention
Acknowledgements
ETHZ EPFL DESY
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