Emittance Growth Caused by Surface Roughness Zhe Zhang, Chuanxiang - PowerPoint PPT Presentation

Emittance Growth Caused by Surface Roughness Zhe Zhang, Chuanxiang Tang Tsinghua University, Beijing Oct. 17th, 2016

Motivation What causes the emittance growth • Dowell’s equations of QE & emittance for bulk emission Dowell, 2009, PRST ( ~ ω − φ e ff ) 2 1 − R ( ω ) QE( ω ) ≈ λ opt 8 φ e ff ( E F + φ e ff ) 1 + λ e − e ( ω ) r ~ ω − φ e ff ε n,x = σ x 3 mc 2

Motivation What causes the emittance growth • Dowell’s equations of QE & emittance for bulk emission Dowell, 2009, PRST ( ~ ω − φ e ff ) 2 1 − R ( ω ) QE( ω ) ≈ λ opt 8 φ e ff ( E F + φ e ff ) 1 + λ e − e ( ω ) QE: Theory vs. Measurement r ~ ω − φ e ff ε n,x = σ x 1x10 -2 3 mc 2 1x10 -3 Quantum Efficiency 1x10 -4 1x10 -5 1x10 -6 1x10 -7 Dowell, 2006, PRST 1x10 -8 180 200 220 240 260 280 300 Wavelength (nm) Theory at 0 MV/m Theory at 50 MV/m Experiment

Motivation What causes the emittance growth • Dowell’s equations of QE & emittance for bulk emission Dowell, 2009, PRST ( ~ ω − φ e ff ) 2 1 − R ( ω ) QE( ω ) ≈ λ opt 8 φ e ff ( E F + φ e ff ) 1 + λ e − e ( ω ) Emittance: Theory vs. Measurement r ~ ω − φ e ff Qian, 2012, Ph.D. ε n,x = σ x 3 mc 2 Parameter Unit BNL SLAC PSI laser wavelength mm 266 253 261 272 282 gun gradient MV/m 95 115 / / / gun phase deg / 15 / / / launch electric field MV/m / 30 25 25 25 measured therm. emit. µ m/mm 0.92 0.9 0.68 0.54 0.41 theory therm. emit. µ m/mm / 0.54 0.64 0.54 0.43 copper work function µ m/mm 4.59 4.65 4.3 4.3 4.3

Motivation How people deal with the surface roughness effect photon VACUUM P' z' electron x' z BASE PLANE x O' O METAL surface morphology R = a sin( kx ) E x = E ξ · e − kz sin kx field distribution 1 + ξ e − kz cos kx � � E z = E ξ = ak ✓ ◆ 1 + 1 ε 2 x ≈ ε 2 2 ξ 2 slope effect D,x a 2 kE ✓ ◆ 1 + 3 π e ε 2 x ≈ ε 2 field effect · D,x 4 ~ ω − φ e ff

Motivation Dif fj culties in 3D case calculation • Initial electron phase-space distribution (slope effect) • EM field on an arbitrary surface (field effect)

Motivation Dif fj culties in 3D case simulation • Generate initial electron samples (slope effect) • Simulation of the EM field near a real-life rough surface (field effect)

Motivation Dif fj culties in 3D case simulation • Generate initial electron 3-step Model samples (slope effect) 1. Absorption of the photon with energy hv 2. Migration including e-e scattering to the surface 3. Escape for electrons with kinematics above the • Simulation of the EM field barrier 
 near a real-life rough surface Could sampling by applying the Monte-Carlo (field effect) method. Dowell, 2009, PRST


 Motivation Dif fj culties in 3D case simulation • Generate initial electron 3-step Model samples (slope effect) 1. Absorption of the photon with energy hv 2. Migration including e-e scattering to the surface 3. Escape for electrons with kinematics above the Monte-Carlo Sampling • Simulation of the EM field barrier 
 Generate s ~ Exp( λ ), E ~ U(E F −ħω , E F ), θ ′ ~ U(0, π / near a real-life rough surface 2), φ ′ ~ U(0, 2 π ), where 1/ λ = 1/ λ opt +1/ λ e-e , then Could sampling by applying the Monte-Carlo apply the filter condition (E+ ħω )cos 2 θ ′ ≥ φ eff . 
 (field effect) method. However the sampling efficiency is quite low (~ Dowell, 2009, PRST 1e-4) because of the low QE of metals. METAL electron photon P x x' z ' local global z VACUUM O' O y ' y


 Motivation Dif fj culties in 3D case simulation • Generate initial electron Meshing on the Rough Surface samples (slope effect) The rms amplitude of the surface roughness is ~ 10 nm, the average rms wavelength is ~ 10 µm, and • Simulation of the EM field the size of the laser spot is ~ 1 mm. Meshing would be too memory-consuming. 
 near a real-life rough surface (field effect) Unrealistic to do this in EM field simulation code.


 Motivation How to deal with the dif fj culties in 3D case • Generate initial electron samples (slope effect) • Simulation of the EM field Utilize the Point Spread Function near a real-life rough surface By applying the Point Spread Function (PSF) of photocathode, one could reveal a simple rule for the electron distribution on (field effect) the rough surface. 
 Thus the sampling efficiency could be significantly improved. C p ( θ ) p z f p ( p x , p y , p z ) = q p p 2 z + p 2 p 2 x + p 2 y + p 2 z + p 2 m · m · H ( p z ) H ( p 2 M − p 2 m − p 2 x − p 2 y − p 2 z ) 1 − R ( θ ) 1 C p ( θ ) = · 1 + λ opt 4 π m ~ ω cos θ λ e − e


 Motivation How to deal with the dif fj culties in 3D case • Generate initial electron samples (slope effect) • Simulation of the EM field near a real-life rough surface (field effect) Approximate Formula for the Electric Potential F or gently undulating surface, there exist some approximate formula for the electric potential distribution, which is proved to be accurate enough for our case. 
 Therefore we could generate the fields much faster and cost much less memory. Z dk x dk y R ( k x , k y ) · e j ( k x x + k y y ) − kz φ ( x, y, z ) = z −

Modeling The PSF of the fm at surface ✓ 1 f ( s, θ , φ , E, ω ) =(1 − R ) 1  1 ◆� Point Spread Function exp + − s ¯ λ opt λ opt λ e-e The response (image) is the convolution of Dowell, 2009, PRST x the point spread function (PSF) and the · H ( E F − E ) H ( E + ~ ω − E F ) sin θ source (object). 4 π ~ ω Transform (s, θ , φ , E, ω ) to (x, y, p x , p y , p z ) electron photon 2 3 O z x 2 + y 2 p p p 2 z + p 2 m f ( x, y, p x , p y , p z ) = C exp y · 4 − 5 q λ p 2 x + p 2 y p z · δ ( xp y − yp x ) · q p 2 x + p 2 y + p 2 z + p 2 METAL VACUUM m · H ( p z ) H ( xp x ) H ( p 2 M − p 2 m − p 2 x − p 2 y − p 2 z ) D ( x, y ) = I ( x, y ) ∗ f ( x, y )

Modeling The PSF of the fm at surface ✓ 1 f ( s, θ , φ , E, ω ) =(1 − R ) 1  1 ◆� exp + − s ¯ λ opt λ opt λ e-e Dowell, 2009, PRST x · H ( E F − E ) H ( E + ~ ω − E F ) sin θ 4 π ~ ω Transform (s, θ , φ , E, ω ) to (x, y, p x , p y , p z ) electron photon 2 3 O z x 2 + y 2 p p p 2 z + p 2 m f ( x, y, p x , p y , p z ) = C exp y · 4 − 5 q λ p 2 x + p 2 y p z · δ ( xp y − yp x ) · q p 2 x + p 2 y + p 2 z + p 2 METAL VACUUM m · H ( p z ) H ( xp x ) H ( p 2 M − p 2 m − p 2 x − p 2 y − p 2 z )

Modeling The PSF of the rough surface � � P = I ( x, y ) ∗ f ( x, y, p x , p y , p z ) D � � x = x 0 ,y = y 0 � P ≈ I ( x 0 , y 0 ) f p ( p x , p y , p z ) D � METAL electron photon P x x' z' l a c o l global z VACUUM O' O y ' y

Modeling The PSF of the rough surface � � P = I ( x, y ) ∗ f ( x, y, p x , p y , p z ) D � � x = x 0 ,y = y 0 � P ≈ I ( x 0 , y 0 ) f p ( p x , p y , p z ) D � METAL The Momentum PSF electron photon The momentum PSF is the integration of PSF over P the real-space (x, y). x x' C p ( θ ) p z f p ( p x , p y , p z ) = z' q l a p c o p 2 z + p 2 p 2 x + p 2 y + p 2 z + p 2 l global z m · VACUUM m O' O y ' · H ( p z ) H ( p 2 M − p 2 m − p 2 x − p 2 y − p 2 z ) y 1 − R ( θ ) 1 C p ( θ ) = · 1 + λ opt 4 π m ~ ω cos θ λ e − e

Modeling The PSF of the rough surface � � P = I ( x, y ) ∗ f ( x, y, p x , p y , p z ) D � � x = x 0 ,y = y 0 � P ≈ I ( x 0 , y 0 ) f p ( p x , p y , p z ) D � METAL The Effect of the Incident Angle The Momentum PSF For a gently undulating surface, the effect of the electron photon The momentum PSF is the integration of PSF over incident angle could be neglected. P the real-space (x, y). x x' C p ( θ ) p z f p ( p x , p y , p z ) = z' q l a p c o p 2 z + p 2 p 2 x + p 2 y + p 2 z + p 2 l global z m · VACUUM m O' O y ' · H ( p z ) H ( p 2 M − p 2 m − p 2 x − p 2 y − p 2 z ) y 1 − R ( θ ) 1 C p ( θ ) = · 1 + λ opt 4 π m ~ ω cos θ λ e − e

Modeling The slope effect on the rough surface Cathode Surface C p 0 · p z f p ( p x , p y , p z ) = q p p 2 z + p 2 p 2 x + p 2 y + p 2 z + p 2 m · m

Modeling The slope effect on the rough surface Cathode Surface C p 0 · p z f p ( p x , p y , p z ) = q p p 2 z + p 2 p 2 x + p 2 y + p 2 z + p 2 m · m

Modeling The slope effect on the rough surface Cathode Surface C p 0 · p z f p ( p x , p y , p z ) = q p p 2 z + p 2 p 2 x + p 2 y + p 2 z + p 2 m · m

Modeling The electric fj eld distribution near the rough surface Z dk x dk y C ( k x , k y ) · e j ( k x x + k y y ) − kz φ ( x, y, z ) = z + Z dk x dk y R ( k x , k y ) · e j ( k x x + k y y ) − kz φ ( x, y, z ) = z − The Form of the Approximate Potential The form is set to satisfy the Laplace’s equation and the B.C. for infinity. z = d → + ∞ φ = d Vacuum z = R(x, y) φ = 0 Cathode