Ultrafast Optics and X-Ray Division

Computational Electromagnetics: from Metamaterials to Particle

Accelerators

Arya FallahiUltrafast optics and X-ray 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

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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, MTT-47, 1999D. Kern et. al, Microwave & Opt. Tech. Lett., 2003

PEC

lossy substrate

patches

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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

plane-wave

Diffraction Analysis of FSS

FSS with periodic substrate (MoM/TL)

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More degrees of freedom in the design

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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.

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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 (BI-RME)

TM mode TE mode

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Basis Functions 11/44

Largely overlapping subdomain basis functions

Problems with the MoM/BI-RME:1) Long computation time for basis functions2) Sophisticated implementation of BI-RME

Fallahi et al. IEEE MTT 2010

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Basis Functions 12/44

Largely overlapping subdomain basis functions

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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 brute-force 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

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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

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Implementation of Radar Absorbers 22/44

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Graphene Metasurfaces

Dynamic Frequency Selective Surfaceperiodic arrangement of metals in a surface with a dynamic response

Liquid crystalsVaractor diodes MEMS switches

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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

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Graphene Metasurfaces 24/44

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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. Perruisseau-Carrier, PRB, 2012.

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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

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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 DC-connected patches

Examples of Graphene Metasurfaces 29/44

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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.

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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 laser-plasmon acceleration.

THz acceleration of particles seems to be a good candidate.

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Particle Accelerators 32/44

The main challenges in this regime:

1) Large loss of the metals preclude the design of high-Q 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.

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Particle Accelerators 33/44

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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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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

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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

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Electron bunch acceleration

● Cash-Karp Runge-Kutta method is used for efficient update of electron motion with time.

● Box-Muller method is used for generating a bunch of electrons with Gaussian distribution.

● We can not consider all the particles, so we use macro-particles.

● 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

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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

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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.

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Amount of charge we can accelerate in the waveguide?

Without space-charge: unlimitedWith space-charge: limited

16pC is by far more than what we need.

What is the limit?

We stop a macro-particle 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

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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

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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 X-Ray 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➢ Non-diagonal conduc/vity for B ≠ 0 and spatially-dispersive conductivity➢ Metal-graphene 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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Thank you for your attention

Acknowledgements

ETHZ EPFL DESY

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