Emittance Growth Caused by Surface Roughness

Transcription

1 Emittance Growth Caused by Surface Roughness he hang, Chuanxiang Tang Tsinghua University, Beijing Oct. 17th, 2016

2 Motivation What causes the emittance growth Dowell s equations of QE & emittance for bulk emission Dowell, 2009, PRST QE(!) 1 R(!) 1+ e ot e(!) (~! e ) 2 8 e (E F + e ) " n,x = x r ~! e 3mc 2

3 FIG. 8. (Color) Plot comaring the measured QE at low field (oints) and comuted QE s using Eq. (19) and the arameters listed in Table I at low (red) and high (blue) alied fields. Motivation What causes the emittance growth Dowell s equations of QE & emittance for bulk emission Dowell, 2009, PRST QE(!) 1 R(!) 1+ e ot e(!) QE: Theory vs. Measurement r ~! e " n,x = x 3mc 2 1x10-2 1x10-3 (~! e ) 2 8 e (E F + e ) In situ CLEANING OF METAL CATHODES USING A... Phys. Rev. ST Accel. Beams 9, (2006) Quantum Efficiency 1x10-4 1x10-5 1x10-6 1x10-7 1x10-8 Dowell, 2006, PRST Wavelength (nm) Theory at 0 MV/m Theory at 50 MV/m Exeriment

4 Motivation What causes the emittance growth Dowell s equations of QE & emittance for bulk emission Dowell, 2009, PRST QE(!) 1 R(!) 1+ ot e(!) (~! e ) 2 8 e (E F + e ) Emittance: e Theory vs. Measurement r ~! e " n,x = x 3mc 2 Parameter Unit BNL SLAC PSI Qian, 2012, Ph.D. laser wavelength mm gun gradient MV/m / / / gun hase deg / 15 / / / launch electric field MV/m / measured therm. emit. µm/mm theory therm. emit. µm/mm / coer work function µm/mm

5 Motivation How eole deal with the surface roughness effect VACUUM hoton P' z O x electron z' O' x' BASE PLANE METAL surface morhology field distribution sloe effect field effect R = a sin(kx) E x = E e kz sin kx E z = E 1+ e kz cos kx = ak " 2 x " 2 D,x " 2 x " 2 D,x e 4 a 2 ke ~! e

6 Initial electron hase-sace distribution (sloe effect) EM field on an arbitrary surface (field effect) Motivation Difficulties in 3D case calculation

7 Generate initial electron samles (sloe effect) Motivation Difficulties in 3D case simulation Simulation of the EM field near a real-life rough surface (field effect)

8 Motivation Difficulties in 3D case simulation Generate initial electron samles (sloe effect) Simulation of the EM field near a real-life rough surface (field effect) 3-ste Model 1. Absortion of the hoton with energy hv 2. Migration including e-e scattering to the surface 3. Escae for electrons with kinematics above the barrier Could samling by alying the Monte-Carlo method. Dowell, 2009, PRST

9 Motivation Difficulties in 3D case simulation Generate initial electron samles (sloe effect) Monte-Carlo Samling Simulation of the EM field near a real-life rough surface (field effect) Generate s ~ Ex(λ), E ~ U(E F ħω, E F ), θ ~ U(0, π/ 2), φ ~ U(0, 2π), where 1/λ = 1/λ ot +1/λ e-e, then aly the filter condition (E+ħω)cos 2 θ φ eff. However the samling efficiency is quite low (~ 1e-4) because of the low QE of metals. 3-ste Model 1. Absortion of the hoton with energy hv 2. Migration including e-e scattering to the surface 3. Escae for electrons with kinematics above the barrier Could samling by alying the Monte-Carlo method. Dowell, 2009, PRST METAL electron hoton x x' P y O global z y' local O' z' VACUUM

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

11 Motivation How to deal with the difficulties in 3D case Generate initial electron samles (sloe effect) Simulation of the EM field near a real-life rough surface (field effect) Utilize the Point Sread Function By alying the Point Sread Function (PSF) of hotocathode, one could reveal a simle rule for the electron distribution on the rough surface. Thus the samling efficiency could be significantly imroved. f ( x, y, z )= C ( ) z 2 z + 2 m q 2 x + 2 y + 2 z + 2 m C ( ) = H( z )H( 2 M 2 m 2 x 2 y 2 z) 1 R( ) 1+ ot e e cos 1 4 m~!

12 Motivation How to deal with the difficulties in 3D case Generate initial electron samles (sloe effect) Simulation of the EM field near a real-life rough surface (field effect) Aroximate Formula for the Electric Potential For gently undulating surface, there exist some aroximate formula for the electric otential distribution, which is roved to be accurate enough for our case. Therefore we could generate the fields much faster and cost much less memory. (x, y, z) =z dk x dk y R(k x,k y ) e j(k xx+k y y) kz

13 Modeling The PSF of the flat surface x Point f(s, Sread,,E,!) Function =(1 R) 1 ot ale ex The resonse (image) is the convolution of Dowell, 2009, PRST the oint sread function (PSF) and the source (object). 1 s ot H(E F E)H(E + ~! E F ) ~! + 1 e-e sin 4 electron y O hoton z Transform (s, θ, φ, E, ω) to (x, y, x, y, z ) 2 f(x, y, x, y, z )=Cex 4 2 z + 2 m q 2 x + 2 y x2 + y METAL VACUUM z q 2 x + 2 y + 2 z + 2 m (x y y x ) H( z )H(x x )H( 2 M 2 m 2 x 2 y 2 z) D(x, y) =I(x, y) f(x, y)

14 Modeling The PSF of the flat surface x f(s,,,e,!) =(1 R) 1 ot Dowell, 2009, PRST ale ex 1 s ot H(E F E)H(E + ~! E F ) ~! + 1 e-e sin 4 electron y O hoton z Transform (s, θ, φ, E, ω) to (x, y, x, y, z ) 2 f(x, y, x, y, z )=Cex 4 2 z + 2 m q 2 x + 2 y x2 + y METAL VACUUM z q 2 x + 2 y + 2 z + 2 m (x y y x ) H( z )H(x x )H( 2 M 2 m 2 x 2 y 2 z)

15 Modeling The PSF of the rough surface D P = I(x, y) f(x, y, x, y, z ) x=x0,y=y 0 METAL D P I(x 0,y 0 )f ( x, y, z ) electron hoton x x' P y O global z y' local O' z' VACUUM

16 Modeling The PSF of the rough surface D P = I(x, y) f(x, y, x, y, z ) x=x0,y=y 0 METAL D P I(x 0,y 0 )f ( x, y, z ) y x O electron global z y' x' local O' P z' hoton VACUUM The Momentum PSF The momentum PSF is the integration of PSF over the real-sace (x, y). f ( x, y, z )= C ( ) z 2 z + 2 m q 2 x + 2 y + 2 z + 2 m H( z )H( 2 M 2 m 2 x 2 y 2 z) C ( ) = 1 R( ) 1+ ot e e cos 1 4 m~!

17 Modeling The PSF of the rough surface D P = I(x, y) f(x, y, x, y, z ) x=x0,y=y 0 METAL D P I(x 0,y 0 )f ( x, y, z ) The Effect of the Incident Angle For a gently electron undulating surface, the effect hoton of the incident angle could be neglected. x O y global z y' x' local O' P z' VACUUM The Momentum PSF The momentum PSF is the integration of PSF over the real-sace (x, y). f ( x, y, z )= C ( ) = C ( ) z 2 z + 2 m q 2 x + 2 y + 2 z + 2 m H( z )H( 2 M 2 m 2 x 2 y 2 z) 1 R( ) 1+ ot e e cos 1 4 m~!

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

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

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

21 Modeling The electric field distribution near the rough surface (x, y, z) =z + dk x dk y C(k x,k y ) e j(k xx+k y y) (x, y, z) =z dk x dk y R(k x,k y ) e j(k xx+k y y) The Form of the Aroximate Potential The form is set to satisfy the Lalace s equation and the B.C. for infinity. kz kz z = d + φ = d Vacuum Cathode z = R(x, y) φ = 0

22 Modeling The electric field distribution near the rough surface (x, y, z) =z + (x, y, z) =z dk x dk y C(k x,k y ) e j(k xx+k y y) dk x dk y R(k x,k y ) e j(k xx+k y y) kz kz The B.C. at Surface When kr(x, y) << 1, the B.C. at surface would lead to C(k x, k y ) = -R(k x, k y ). (x, y, R(x, y)) = R(x, y)+ dk x dk y C(k x,k y ) e j(k xx+k y y) kr(x,y) R(x, y)+ dk x dk y C(k x,k y ) e j(k xx+k y y) (1 kr(x, y)) = dk x dk y (R(k x,k y )+C(k x,k y )) e j(k xx+k y y) + O(1) = O(1) when R(kx,k y )+C(k x,k y )=0

23 Modeling The electric field distribution near the rough surface (x, y, z) = z + (x, y, z) = z dkx dky C(kx, ky ) ej(kx x+ky y) j(k Gently Undulating Surface x x+ky y) dkx dky R(k x, ky ) e kz kz A gently undulating surface should mean that: The B.C. at Surface rms(r) rms(k) 1 When kr(x, y) << 1, the B.C. at surface would lead to C(kx, ky) = -R(kx, ky). (x, y, R(x, y)) = R(x, y) + dkx dky C(kx, ky ) ej(kx x+ky y) kr(x,y) R(x, y) + dkx dky C(kx, ky ) ej(kx x+ky y) (1 kr(x, y)) = dkx dky (R(kx, ky ) + C(kx, ky )) ej(kx x+ky y) + O(1) = O(1) when R(kx,ky )+C(kx,ky )=0

24 Modeling The electric field distribution near the rough surface (x, y, z) =z + (x, y, z) =z dk x dk y C(k x,k y ) e j(k xx+k y y) dk x dk y R(k x,k y ) e j(k xx+k y y) kz kz Aroximate Formula for the Electric Field E x = j dk x dk y k x R(k x,k y ) e j(k xx+k y y) kz E y = j dk x dk y k y R(k x,k y ) e j(k xx+k y y) kz E z = 1 dk x dk y kr(k x,k y ) e j(k xx+k y y) kz

25 Theory 3D arbitrary surface Saturated Transverse Momentum 1 A = ee/m r A 0 = m 2 r A = jm 2 E x z dz dk x dk y k x k R(k x,k y ) e j(k xx+k y y) " 2 = hx 2 ih 2 xi hx x i 2 Roughness Emittance on Arbitrary Gently Undulating Surface " 2 x = " 2 D,x x R + * r A 0 x R + jm 2 dk x dk y k x k R(k x,k y ) e j(k xx+k y y) 0 x 2!

26 Initial electron beam samling EM field generation Simulation The rinciles Motion equation integration

27 Initial electron beam samling EM field generation Simulation The rinciles Average Number of Attemts to Produce an Acceted Samle Motion equation integration The samles generated by rejective method obey the geometry distribution. N G() m = 2m(E F + e ) M = 2m(E F + ~!) E(N) = 1+ m 2 M

28 Initial electron beam samling EM field generation Simulation The rinciles -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. d x [kev/c] dz [nm] Motion equation integration = E0 [MV/m] z [kev/c] dx [µm] dz [nm] = x [kev/c] z [kev/c] Êx(x, y, z) -layers Electron Cathode Surface

29 Simulation Figure 2: Validation of the accuracy of the analytical electric otential. We take the rofile of the 3-D surface shown in Fig. 3 along x = µm, the surface rofile is marked by the black bold curve. The white bold curve in the lot is the zero otential contour of the analytical electric otential in the y-z lane at x = µm, which is calculated by the equations. In the lot, the surface rofile and the zero otential contour are mostly overlaed, therefore we conclude that the analytical electric otential is quite accurate. The simulation configuration NUMERICAL SIMULATION Simulation configuration Our simulation configuration is shown in Fig. 3, the details could be found in the cation. Table 1: Parameters used in numerical simulation. Parameter Value Unit Descrition uniform nm µm µm µm laser wavelength laser transverse distribution laser transverse radius laser incident x center laser incident y center EF coer MV/m ev ev ev material of the cathode effective electric field strength work function effective work function Fermi energy N zi zf dz nm nm nm number of articles simulation starting osition simulation ending osition simulation z ste l l-dist rl xl yl mat E0 w eff Figure 3: Simulation configuration. The yellow-black background shows the morhology of the surface we emloyed in the simulation, while the blue-red sot in the center describes the transverse intensity distribution of the laser. Simulation results In Fig. 5, we comared the emittance growth factor between several 2-D sinusoidal surface cases and the 3-D random surface case. We choose different sets of satial eriod ( in the legend) and amlitude a for 2-D sinusoidal cases. The details are described in the cation. It s obvious that the emittance growth factor for 2-D macro roughness case (dotted magenta curve) is significantly larger than the 3-D case under the same alied electric field strength. However,

30 Simulation The hase-sace & emittance evolution

31 Simulation Comarison between 2D and 3D result " 2 n,x = 2 x " ~! e 3mc 2 + e2 mc 2 R2 qe q #

32 Summary Pros & Cons Reveal a simle rule for the initial hase-sace distribution due to sloe effect Predict the emittance growth / hase-sace evolution based on the cathode morhology Show the roughness tolerance for a given emittance growth uer limit NOT include the surface emission effect NOT consider the ossible SPPs (Surface Plasmon Polaritons) generation NOT consider the microwave smoothing effect NOT consider the effective work function variation due to the surface roughness

33 Thanks Questions or Comments