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

• Dowell’s equations of QE & emittance for bulk emission

QE(ω) ≈ 1 − R(ω) 1 + λopt λe−e(ω) (~ω − φeff)2 8φeff(EF + φeff)

εn,x = σx r ~ω − φeff 3mc2

What causes the emittance growth

Dowell, 2009, PRST

Motivation

• Dowell’s equations of QE & emittance for bulk emission

QE(ω) ≈ 1 − R(ω) 1 + λopt λe−e(ω) (~ω − φeff)2 8φeff(EF + φeff)

εn,x = σx r ~ω − φeff 3mc2

What causes the emittance growth

Dowell, 2009, PRST

QE: Theory vs. Measurement

1x10-3 1x10-4 1x10-5 1x10-6 1x10-7 1x10-2 1x10-8 Quantum Efficiency

180 200 220 240 260 280 300

Theory at 0 MV/m

Theory at 50 MV/m

Experiment Wavelength (nm)

Dowell, 2006, PRST

Motivation

• Dowell’s equations of QE & emittance for bulk emission

QE(ω) ≈ 1 − R(ω) 1 + λopt λe−e(ω) (~ω − φeff)2 8φeff(EF + φeff)

εn,x = σx r ~ω − φeff 3mc2

What causes the emittance growth

Dowell, 2009, PRST

Emittance: Theory vs. Measurement

Qian, 2012, Ph.D.

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. µm/mm 0.92 0.9 0.68 0.54 0.41 theory therm. emit. µm/mm / 0.54 0.64 0.54 0.43 copper work function µm/mm 4.59 4.65 4.3 4.3 4.3

photon electron

VACUUM METAL

x z O

BASE PLANE

x' z' O' P'

Motivation

How people deal with the surface roughness effect

ε2

x ≈ ε2 D,x

✓ 1 + 1 2ξ2 ◆ ε2

x ≈ ε2 D,x

✓ 1 + 3πe 4 · a2kE ~ω − φeff ◆

Ex = Eξ · e−kz sin kx Ez = E

• 1 + ξe−kz cos kx
• ξ = ak

slope effect field effect

R = a sin(kx)

surface morphology field distribution

Motivation

Diffjculties in 3D case calculation

• Initial electron phase-space

distribution (slope effect)

• EM field on an arbitrary

surface (field effect)

Motivation

Diffjculties in 3D case simulation

• Generate initial electron

samples (slope effect)

• Simulation of the EM field

near a real-life rough surface (field effect)

Motivation

Diffjculties in 3D case simulation

3-step Model

• 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

barrier 
 Could sampling by applying the Monte-Carlo method.

Dowell, 2009, PRST

• Generate initial electron

samples (slope effect)

• Simulation of the EM field

near a real-life rough surface (field effect)

• Generate initial electron

samples (slope effect)

• Simulation of the EM field

near a real-life rough surface (field effect)

Motivation

Diffjculties in 3D case simulation

3-step Model

• 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

barrier 
 Could sampling by applying the Monte-Carlo method.

Dowell, 2009, PRST

Monte-Carlo Sampling

Generate s ~ Exp(λ), E ~ U(EF−ħω, EF), θ′ ~ U(0, π/ 2), φ′ ~ U(0, 2π), where 1/λ = 1/λopt+1/λe-e, then apply the filter condition (E+ħω)cos2θ′ ≥ φeff.


However the sampling efficiency is quite low (~ 1e-4) because of the low QE of metals.

z ' y ' x' O'

VACUUM

P

z y x O

METAL

Motivation

Diffjculties in 3D case simulation

Meshing on the Rough Surface

The rms amplitude of the surface roughness is ~ 10 nm, the average rms wavelength is ~ 10 µm, and the size of the laser spot is ~ 1 mm. Meshing would be too memory-consuming.


Unrealistic to do this in EM field simulation code.

• Generate initial electron

samples (slope effect)

• Simulation of the EM field

near a real-life rough surface (field effect)

• Generate initial electron

samples (slope effect)

• Simulation of the EM field

near a real-life rough surface (field effect)

Motivation

How to deal with the diffjculties in 3D case

Utilize the Point Spread Function

By applying the Point Spread Function (PSF) of photocathode,

• ne could reveal a simple rule for the electron distribution on

the rough surface.


Thus the sampling efficiency could be significantly improved.

fp(px, py, pz) = Cp(θ)pz p p2

z + p2 m ·

q p2

x + p2 y + p2 z + p2 m

· H(pz)H(p2

M − p2 m − p2 x − p2 y − p2 z)

Cp(θ) = 1 − R(θ) 1 + λopt λe−e cos θ · 1 4πm~ω

• Generate initial electron

samples (slope effect)

• Simulation of the EM field

near a real-life rough surface (field effect)

Motivation

How to deal with the diffjculties in 3D case

Approximate Formula for the Electric Potential

For 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.

φ(x, y, z) = z − Z dkxdkyR(kx, ky) · ej(kxx+kyy)−kz

Modeling

The PSF of the fmat surface

z y x O

photon electron

METAL VACUUM

Transform (s, θ, φ, E, ω) to (x, y, px, py, pz)

f(s, θ, φ, E, ω) =(1 − R) 1 λopt exp  −s ✓ 1 λopt + 1 ¯ λe-e ◆ · H(EF − E)H(E + ~ω − EF) ~ω sin θ 4π f(x, y, px, py, pz) =C exp 2 4− p p2

z + p2 m

q p2

x + p2 y

· p x2 + y2 λ 3 5 · pz q p2

x + p2 y + p2 z + p2 m

· δ(xpy − ypx) · H(pz)H(xpx)H(p2

M − p2 m − p2 x − p2 y − p2 z)

Dowell, 2009, PRST

Point Spread Function

The response (image) is the convolution of the point spread function (PSF) and the source (object).

D(x, y) = I(x, y) ∗ f(x, y)

Modeling

The PSF of the fmat surface

z y x O

photon electron

METAL VACUUM

Transform (s, θ, φ, E, ω) to (x, y, px, py, pz)

f(s, θ, φ, E, ω) =(1 − R) 1 λopt exp  −s ✓ 1 λopt + 1 ¯ λe-e ◆ · H(EF − E)H(E + ~ω − EF) ~ω sin θ 4π f(x, y, px, py, pz) =C exp 2 4− p p2

z + p2 m

q p2

x + p2 y

· p x2 + y2 λ 3 5 · pz q p2

x + p2 y + p2 z + p2 m

· δ(xpy − ypx) · H(pz)H(xpx)H(p2

M − p2 m − p2 x − p2 y − p2 z)

Dowell, 2009, PRST

Modeling

The PSF of the rough surface

z' y ' x' O'

photon

VACUUM

P

electron

z y x O

METAL

l

• c

a l global

D

• P = I(x, y) ∗ f(x, y, px, py, pz)
• x=x0,y=y0

D

• P ≈ I(x0, y0)fp(px, py, pz)

Modeling

The PSF of the rough surface

z' y ' x' O'

photon

VACUUM

P

electron

z y x O

METAL

l

• c

a l global

D

• P = I(x, y) ∗ f(x, y, px, py, pz)
• x=x0,y=y0

D

• P ≈ I(x0, y0)fp(px, py, pz)

The Momentum PSF

The momentum PSF is the integration of PSF over the real-space (x, y).

fp(px, py, pz) = Cp(θ)pz p p2

z + p2 m ·

q p2

x + p2 y + p2 z + p2 m

· H(pz)H(p2

M − p2 m − p2 x − p2 y − p2 z)

Cp(θ) = 1 − R(θ) 1 + λopt λe−e cos θ · 1 4πm~ω

The Momentum PSF

The momentum PSF is the integration of PSF over the real-space (x, y).

fp(px, py, pz) = Cp(θ)pz p p2

z + p2 m ·

q p2

x + p2 y + p2 z + p2 m

· H(pz)H(p2

M − p2 m − p2 x − p2 y − p2 z)

Cp(θ) = 1 − R(θ) 1 + λopt λe−e cos θ · 1 4πm~ω

Modeling

z' y ' x' O'

photon

VACUUM

P

electron

z y x O

METAL

l

• c

a l global

D

• P = I(x, y) ∗ f(x, y, px, py, pz)
• x=x0,y=y0

D

• P ≈ I(x0, y0)fp(px, py, pz)

The PSF of the rough surface

The Effect of the Incident Angle

For a gently undulating surface, the effect of the incident angle could be neglected.

Modeling

The slope effect on the rough surface

fp(px, py, pz) = Cp0 · pz p p2

z + p2 m ·

q p2

x + p2 y + p2 z + p2 m

Cathode Surface

Modeling

The slope effect on the rough surface

fp(px, py, pz) = Cp0 · pz p p2

z + p2 m ·

q p2

x + p2 y + p2 z + p2 m

Cathode Surface

Modeling

The slope effect on the rough surface

fp(px, py, pz) = Cp0 · pz p p2

z + p2 m ·

q p2

x + p2 y + p2 z + p2 m

Cathode Surface

Modeling

The electric fjeld distribution near the rough surface

φ(x, y, z) = z + Z dkxdkyC(kx, ky) · ej(kxx+kyy)−kz φ(x, y, z) = z − Z dkxdkyR(kx, ky) · ej(kxx+kyy)−kz The Form of the Approximate Potential

The form is set to satisfy the Laplace’s equation and the B.C. for infinity.

z = R(x, y) φ = 0 z = d → +∞ φ = d Cathode Vacuum

Modeling

The electric fjeld distribution near the rough surface

φ(x, y, z) = z + Z dkxdkyC(kx, ky) · ej(kxx+kyy)−kz φ(x, y, z) = z − Z dkxdkyR(kx, ky) · ej(kxx+kyy)−kz The B.C. at Surface

When kR(x, y) << 1, the B.C. at surface would lead to 
 C(kx, ky) = -R(kx, ky).

φ (x, y, R(x, y)) = R(x, y) + Z dkxdkyC(kx, ky) · ej(kxx+kyy)−kR(x,y) ≈ R(x, y) + Z dkxdkyC(kx, ky) · ej(kxx+kyy) (1 − kR(x, y)) = Z dkxdky(R(kx, ky) + C(kx, ky)) · ej(kxx+kyy) + O(1) = O(1)

• when R(kx,ky)+C(kx,ky)=0

Modeling

The electric fjeld distribution near the rough surface

φ(x, y, z) = z + Z dkxdkyC(kx, ky) · ej(kxx+kyy)−kz φ(x, y, z) = z − Z dkxdkyR(kx, ky) · ej(kxx+kyy)−kz The B.C. at Surface

When kR(x, y) << 1, the B.C. at surface would lead to 
 C(kx, ky) = -R(kx, ky).

φ (x, y, R(x, y)) = R(x, y) + Z dkxdkyC(kx, ky) · ej(kxx+kyy)−kR(x,y) ≈ R(x, y) + Z dkxdkyC(kx, ky) · ej(kxx+kyy) (1 − kR(x, y)) = Z dkxdky(R(kx, ky) + C(kx, ky)) · ej(kxx+kyy) + O(1) = O(1)

• when R(kx,ky)+C(kx,ky)=0

Gently Undulating Surface

A gently undulating surface should mean that:


rms(R) · rms(k) ⌧ 1

Modeling

The electric fjeld distribution near the rough surface

φ(x, y, z) = z + Z dkxdkyC(kx, ky) · ej(kxx+kyy)−kz φ(x, y, z) = z − Z dkxdkyR(kx, ky) · ej(kxx+kyy)−kz Approximate Formula for the Electric Field Ex = j Z dkxdky · kxR(kx, ky) · ej(kxx+kyy)−kz Ey = j Z dkxdky · kyR(kx, ky) · ej(kxx+kyy)−kz Ez = −1 − Z dkxdky · kR(kx, ky) · ej(kxx+kyy)−kz

Theory

3D arbitrary surface

ε2 = hx2ihp2

xi hxpxi2

Roughness Emittance on Arbitrary Gently Undulating Surface

ε2

x = ε2 D,x

2 6 6 6 6 6 4 1 − ⌦ ∂2

xR

↵ + * p0

z · ∂xR + jm

r πA 2 · Z dkxdky kx √ k R(kx, ky) · ej(kxx+kyy) !2+ ⌦ p0

x 2↵

3 7 7 7 7 7 5

Saturated Transverse Momentum p∞ − p0 = m r A 2 · Z − Ex √z dz = −jm r πA 2 · Z dkxdky kx √ k R(kx, ky) · ej(kxx+kyy)

A = eE/m

• Initial electron beam sampling
• EM field generation
• Motion equation integration

Simulation

The principles

• Initial electron beam sampling
• EM field generation
• Motion equation integration

Simulation

The principles

Average Number of Attempts to Produce an Accepted Sample

The samples generated by rejective method obey the geometry distribution.

E(N) = π ✓ 1 + pm pM ◆ ≈ 2π pm = p 2m(EF + φeff) pM = p 2m(EF + ~ω) N ∼ G(p)

• Initial electron beam sampling
• EM field generation
• Motion equation integration

Simulation

The principles

Z-based Motion Equations

We choose the z-based motion equations because the E-field is calculated by z-layer, z-based motion could guarantee the accuracy.

dpx [keV/c] dz [nm] = 511 × 10−6 · E0 [MV/m] pz [keV/c] · ˆ Ex(x, y, z) dx [µm] dz [nm] = px [keV/c] pz [keV/c] · 1 × 10−3

Cathode Surface Z-layers Electron

Simulation

The simulation confjguration

details Table 1: Parameters used in numerical simulation. Parameter Value Unit Description λl 266.0 nm laser wavelength l-dist uniform

• laser transverse distribution

rl 20.0 µm laser transverse radius xl 80.0 µm laser incident x center yl 60.0 µm laser incident y center mat copper

• material of the cathode

E0 50.0 MV/m effective electric field strength φw 4.31 eV work function φeff 4.04 eV effective work function EF 7.0 eV Fermi energy N 10000

• number of particles

zi nm simulation starting position zf 5000.0 nm simulation ending position dz 10.0 nm simulation z step

Simulation

The phase-space & emittance evolution

Simulation

Comparison between 2D and 3D result

¯ ε2

n,x = σ2 x ·

" ~ω − φeff 3mc2 + πe2 mc2 · R2

qE

λq #

Summary

• Reveal a simple rule for the

initial phase-space distribution due to slope effect

• Predict the emittance growth /

phase-space evolution based

• n the cathode morphology
• Show the roughness tolerance

for a given emittance growth upper limit

• NOT include the surface

emission effect

• NOT consider the possible

SPPs (Surface Plasmon Polaritons) generation

• NOT consider the microwave

smoothing effect

• NOT consider the effective

work function variation due to the surface roughness

Pros & Cons

Thanks

Questions or Comments