Computational Electromagnetics: from Metamaterials to Particle Accelerators
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1 Computational Electromagnetics: from Metamaterials to Particle Accelerators Arya Fallahi Ultrafast optics and X-ray Division 1. July 213
2 2/44 Outline Ø Frequency Selective Surfaces (FSS) Analysis techniques: Diffraction Analysis Basis Functions Dispersion Analysis Design and optimization of FSS Ø Graphene metasurfaces Ø Particle accelerators Ø Bunch acceleration in THz waveguides Bunch Compression in THz waveguides Conclusion
3 Frequency Surfaces FrequencySelective Selective Surfaces (FSS) Reflector antennas Radome design 3/44 3/3 Polarizers Beam splitters
4 4/44 4/3 Frequency Selective Surfaces A 2D array of patches printed on a grounded substrate can perform as 1. artificial magnetic conductor (AMC) 2. radar absorbing surface patches lossy substrate PEC F. Yang et. al, MTT-47, 1999 D. Kern et. al, Microwave & Opt. Tech. Lett., 23
5 5/44 Diffraction Analysis of FSS FSS with homogeneous substrate (MoM) Boundary condition on the patch: From the theory of Green s function known variable unknown variable Method of Moments (MoM) known operator
6 Diffraction Analysis of FSS 6/44
7 7/44 Diffraction Analysis of FSS z FSS with periodic substrate (MoM/TL) incident wave Solved using MoM Lx Ly y periodic substrate x Computed using the coupled multiconductor transmission line (TL) model arbitrarily shaped patches coupled multiconductor transmission lines MoM/TL method Fallahi et al. Elsevier Metamaterials, Oct. 29.
8 8/44 Diffraction Analysis of FSS FSS with periodic substrate (MoM/TL)
9 9/44 Diffraction Analysis of FSS FSS with periodic substrate (MoM/TL) Better control over the substrate properties plane-wave patches Shifting resonance frequencies Increasing or decreasing the number of resonances Coupling of different diffraction orders within the substrate periodic substrates homogenous substrates More degrees of freedom in the design
10 1/44 Basis Functions The most critical point concerning the selection of basis functions: The normal component of the current at the boundary should vanish. Subdomain basis functions Rooftop basis functions Surface patch basis functions
11 11/44 Basis Functions Entire domain basis functions Boundary Integral Resonant Mode Expansion (BI-RME) PEC Ht Transverse magnetic fields of the guided modes Hn= Entire domain basis functions TM mode TE mode Fallahi et al. IEEE TAP - 58, March 21.
12 12/44 Basis Functions Largely overlapping subdomain basis functions Problems with the MoM/BI-RME: 1) Long computation time for basis functions 2) Sophisticated implementation of BI-RME Fallahi et al. IEEE MTT 21
13 13/44 Basis Functions Largely overlapping subdomain basis functions
14 Dispersion Analysis of FSS 14/44
15 Dispersion Analysis of FSS 15/44 Energy coupling technique z p x prism y air h(k z ) air r r j r (a) (b) For a grounded substrate, I=R for all frequencies except the resonance points.
16 Dispersion Analysis of FSS 16/44 Energy coupling technique Band diagram is obtained by finding the minima of the reflection spectrum. Fallahi et al. Antenna & Prop. Symp. July 28
17 Designing FSS Structures 17/44 Efficient procedures for FSS optimization Unit cell is divided to an N N grid, and encoded to a binary string. no patch metallic patch 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 1 times to obtain reliable statistical data about its efficiency.
18 Designing FSS Structures 18/44 Efficient procedures for FSS optimization Probabilities of finding global optimum in percent M STAT av MGA MGA MGA MUT MUT MUT RHC Fallahi et al. IEEE TAP - 56, May 28. Fallahi et al. CTN Journal, April 28.
19 Design of Thin Radar Absorbers 19/44 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
20 Implementation of Radar Absorbers 2/44 Fabrication of radar absorbers Characterization of radar absorbers
21 Implementation of Radar Absorbers Homogeneous absorber 21/44 FSS absorber
22 Implementation of Radar Absorbers 22/44 Perforated FSS absorber Fallahi et al. IEEE TAP 21
23 23/44 Graphene Metasurfaces Dynamic Frequency Selective Surface periodic arrangement of metals in a surface with a dynamic response Varactor diodes Liquid crystals MEMS switches
24 24/44 Graphene Metasurfaces A promising solution to the mentioned problems is graphene 2D atomic lattice Electrons behaving as massless Dirac Fermions Large electron mobilities Transparent conductivity Large nonlinear Kerr effect Electrically tunable conductivity A 2D honeycomb lattice made of carbon atoms Electromagnetic properties of patterned graphene
25 25/44 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.
26 Modified PMoM for Graphene Metasurface 26/44 In the periodic Method of Moments, we work in the spectral domain:
27 Examples of Graphene Metasurfaces 27/44 L=1mm, D=7.5mm and d=1.25mm A. Fallahi and J. Perruisseau-Carrier, PRB, 212.
28 28/44 Examples of Graphene Metasurfaces E=V/nm E=2V/nm E=2V/nm Increasing the electric biasing causes the increase in the graphene conductivity and this in turn strengthens the resonances between adjacent patches.
29 Examples of Graphene Metasurfaces 29/44 Problems with the presented example: Effective performance needs very high bias electric fields Ø How this electrostatic biasing can be implemented without disturbing the FSS? Ø How the bias fields should be applied throughout the FSS? Ø Ø Solution: Double graphene layers with DC-connected patches
30 Examples of Graphene Metasurfaces 3/44 L = 5 µm, l =.5 µm, d = 1.25 µm, and D = 1.5 µm and the dielectric thickness is t = 5 nm.
31 Particle Accelerators 31/44 The concept of oscillating field accelerators was mainly proposed in 192s. 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.
32 Particle Accelerators 32/44 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.
33 33/44 Particle Accelerators The main challenges in this regime: 1) Large loss of the metals preclude the design of high-q cavities 2) The very limited available THz sources 3) 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.
34 34/44 Bunch Acceleration in THz Waveguides Side view Dielectric r Vacuum Metal Electron bunch THz pulse 1 E z E r H 1.2 6,8 4, M a g n e tic F ie ld M a g n itu d e ( T ) End view E le c tr ic F ie ld M a g n itu d e ( G V /m ) 2 V a c u u m R a d iu s ( m ) Uniform longitudinal field at β=1
35 Bunch Acceleration in THz Waveguides 35/44 Optimization Very many parameters are involved in the acceleration level of electrons, including vp, vg, GVD, α, z and ψ. 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.
36 36/44 Bunch Acceleration in THz Waveguides Optimization 6GHz operation frequency F in a l E n e r g y ( M e V ) Optimum frequency of optical rectification is assumed to be 6GHz D i e l e c t r i c T h ic k n e s s ( m ) 45 Vacuum radius 38µm Dielectric thickness 32µm
37 Bunch Acceleration in THz Waveguides 37/44 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 =.1% Initial bunch spread is a cube of 3µm 3µm 3µm
38 38/44 Bunch Acceleration in THz Waveguides Dispersion curve 1.5 / k F re q u e n c y (T H z ).8 1 M e a n b e a m e n e rg y (M e V ) Energy of a bunch (2mJ THz pulse) 6 1MeV to 1 MeV % energy spread.1. 5 w it h o u t s p a c e - c h a r g e w it h s p a c e - c h a r g e D is t a n c e ( m m ) w it h o u t s p a c e - c h a r g e w it h s p a c e - c h a r g e D is t a n c e ( m m ) 35 o Temperature of the coating C o a tin g T e m p r a tu r e ( C ) B e a m e n e r g y d e v ia tio n ( M e V ) Energy spread in a bunch 1 3 No melting of the coating D is t a n c e f r o m t h e W a v e g u id e E n t r a n c e ( c m )
39 39/44 Bunch Acceleration in THz Waveguides w it h o u t s p a c e - c h a r g e w it h s p a c e - c h a r g e D is t a n c e ( m m ) Electron is injected in a point within the pulse It is not possible practically A rule of thumb Practically achievable results are 1% lower than the theoretical optimum point. M e a n b e a m e n e rg y (M e V ) 1 Acceleration to 8.5MeV 1 w it h s p a c e - c h a r g e w it h o u t s p a c e - c h a r g e D is t a n c e ( m m ) 4 9% energy spread B e a m e n e r g y d e v ia tio n ( M e V ) M e a n b e a m e n e rg y (M e V ) Energy of a bunch (2mJ THz pulse).8 w it h s p a c e - c h a r g e w it h o u t s p a c e - c h a r g e D is t a n c e ( m m ) 4
40 Bunch Acceleration in THz Waveguides Without space-charge: unlimited With space-charge: limited What is the limit? We stop a macro-particle as soon as it hits the walls of the accelerator. 1 M e a n b e a m e n e rg y (M e V ) Amount of charge we can accelerate in the waveguide? 4/ p C b u n c h c h a rg e 1 6 p C b u n c h c h a rg e 1 6 p C b u n c h c h a rg e D is t a n c e ( m m ) 16pC is by far more than what we need.
41 41/44 Bunch Compression in THz waveguides Rectilinear Compression Fback Ffront Compression and deceleration Fback Ffront Maximum Compression Fback Ffront Compression and acceleration
42 42/44 Bunch Compression in THz waveguides 5 times bunch compression 1 w it h s p a c e - c h a r g e w it h o u t s p a c e - c h a r g e z ( m ) D is t a n c e ( m m ) 3 B e a m e n e r g y d e v ia tio n ( M e V ) Ø Mean initial energy = 1MeV Ø Initial energy spread =.1% Ø Initial bunch spread is a cube of 3mm 3mm 3mm Ø 2mJ 1 cycle THz pulse M e a n b e a m e n e rg y (M e V ) Acceleration to 3MeV 5 w it h s p a c e - c h a r g e w it h o u t s p a c e - c h a r g e D is t a n c e ( m m ) 3 7% energy spread.8 w it h s p a c e - c h a r g e w it h o u t s p a c e - c h a r g e D is t a n c e ( m m ) 3
43 43/44 Conclusion 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 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
44 44/44 Acknowledgements ETHZ EPFL DESY Thank you for your attention
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