Electromagnetic Wake-Fields in Optical Accelerator

Scaling Laws of Wake-Fields in Optical Structures

Scaling Laws of Wake-Fields in Optical Structures

Samer BannaSupervised by:

Prof. L. Schächter & Prof. D. Schieber

Technion – Israel Institute of TechnologyDepartment of Electrical Engineering

2

OutlineOutlineOutlineBackground & Motivation

Optical AcceleratorsWake-Fields

Assumptions & approach to solutionWake-field in the vicinity of dielectric body

Radiated energyFrequency dependence of dielectric coefficient

Wake-field due to geometric discontinuityFrequency-domain versus time-domain solutionRadiated energy

Wake-field due to surface roughnessSimple model: quasi-periodic structure with random perturbationRadiated energy: average value and standard-deviation

Summary

3

Background: AcceleratorBackground: AcceleratorBackground: Accelerator

Foundation of nature (constituents of matter)

Particle physics

X – ray sources

Medical Diagnostics and treatment

Applications:

Fermilab AcceleratorAccelerator: A device used to produce high-energy (high-speed) particles by converting EM energy into kinetic energy.

4

Background: Beam-Wave InteractionBackground: BeamBackground: Beam--Wave InteractionWave Interaction

e-beam

EM-Wave EM-Wave

e-beam

1I 2 1I I<

1β 2 1β β>

Interaction

The EM wave’s phase velocity

The electrons’average velocity=

5

Motivation: Optical AcceleratorsMotivation: Optical AcceleratorsMotivation: Optical Accelerators

Cost consideration

2E Pω∝High-energy requirement

Compact accelerators based on optical structures

CompactnessHigh

frequencyHigh-gradient

6

Motivation: Lawson-Woodward TheoremMotivation: LawsonMotivation: Lawson--Woodward TheoremWoodward Theorem

No net energy gain in vacuum using optical fields

Infinite interaction region.

No walls or boundaries are present.

Highly relativistic electron on the acceleration path.

No static electric or magnetic fields present.

Non-linear effects are neglected.

7

Motivation: Lawson-Woodward TheoremMotivation: LawsonMotivation: Lawson--Woodward TheoremWoodward Theorem

Acceleration: interaction should be limited in space or time

Lawson-Woodwardtheorem conditions

No acceleration in vacuum Optical

acceleration structures

8

Motivation: Wake-FieldsMotivation: WakeMotivation: Wake--FieldsFields

Optical Acceleration Structures: Laser acceleration of electrons in vacuum entails relativistic motion of electrons in the vicinity of metallic or dielectric structures of sub-micron size.

It is important to determine:Characteristics of the wake-field.The impact of the wake-field on the bunch properties.Impact of radius of curvature.Wake-field of bunches moving/leaving geometric discontinuities.

Wake-field due to surface roughness in optical structures.Temperature consideration.Nonlinear effects.

9


Deceleration Force

WakeWake--FieldFieldBeam Characteristics Bunch Stability

Transverse Force

10


We focus on an analytic or quasi-analytic solutions, in order to have better understanding of the wake-field properties.

A bunch moving in a free space generates a broad spectrum of evanescent waves.

In the presence of an obstacle the evanescent waves are diffracted, carrying away energy.

11

AssumptionsAssumptionsAssumptions

An external force is acting on the moving bunch keeping its velocity constant.

Transverse motion of the moving bunch is neglected.

12

Approach to SolutionApproach to SolutionApproach to Solution

A moving bunch generates a current

density J

Excites the magnetic vector

potential A Effect of

an obstacle

A(p) - Primary fieldnon-homogeneous

wave equation

A(s) - Secondary fieldhomogeneous

wave equation

Boundary conditionsNo obstacle

13

Line Charge Moving in the Vicinity of a Dielectric CylinderLine Charge Moving in the Vicinity of a Dielectric CylinderLine Charge Moving in the Vicinity of a Dielectric Cylinder

h 2R

x

yFree space

Infinite cylinder

Constant velocity

( )00 , µε

14

Primary FieldPrimary FieldPrimary Field

)exp()(2

);,( xc

jhyyxJx βωδ

πλω −−−=

( ) 02 2 2

( )

exp sin sgn( )cos4

exp sgn( )4

p

pz

jE j x y h y hc

H j x y h y hc

ϕλµ ω ω ωϕ ϕπ β γ β γβ

λ ωπ β

− −= − −Γ − + − Γ

−= − −Γ − −

21 1;

1cos ; sin

c

x r y r

ωγγββ

ϕ ϕ

≡ Γ ≡−

= =

15

Secondary FieldSecondary FieldSecondary Field

(2)

( )

(2)

( )0

H ( , ; )

J

H ( , ; )

J

n ns jn

zn

n n r

n ns jn

nn r n r

A r r Rc

H r eB r r R

c

A r r Rc

E r j eB r r R

c

ϕ

ϕϕ

ω

ϕ ωω ε

ω

ϕ ω ηωε ε

∞

=−∞

∞

=−∞

> = <

> = <

∑

∑

0 0 0η µ ε≡

16

Longitudinal Impedance - SpectrumLongitudinal Impedance Longitudinal Impedance -- SpectrumSpectrum

||dW Zd

=Ω

20

//

zWWλ πε

∆=

||0

|| 21 Z

hZ

η≡

||

2

||

1 ( , ; ) exp( )

/ ( , ; ) exp( ) 2

1 2

x

z x

z

Z dx E x h j xc

W d dx E x h j xc

dW Zd

ωωλ β

λ ωω ωπ β

λω π

∞

−∞

∞ ∞

−∞ −∞

= −

∆ = −

⇒ =∆

∫

∫ ∫

⇒h

cω

=Ω

17


0.0

0.5

1.0

1.5

2.0

0 25 50 75 100

Z ||

Ω

γβ =1000ε = 2r

R/h = 0.9R/h = 0.5

( ), peakpeak ZΩ

0.0

0.2

0.4

0.6

0.8

1.0

0 25 50 75 100Z ||

Ω

R/h = 0.5ε = 2r

γβ = 150γβ = 60γβ = 30

hcω

=Ωhcω

=Ω

||0

|| 21 Z

hZ

η≡

18


Scaling Laws

peakRZh

∝ ( )1/3peakZ γβ∝ peak

hR

Ω ∝

Ω∝

1||Z

1R hΩ >>

19

Radiated EnergyRadiated EnergyRadiated Energy

Ω

Ω−

Ω

Ω

Ω−

Ω

=Ω

ΩΩ

=Ω

ΩΩ

Ω=

Ω−

∞

−∞=

∞

−∞=

∞

∑∑∫

hRJ

hRH

hRJ

hRH

ehRJU

hRJ

jV

A

AdWdAdW

rnnrrnn

rnnrnrn

n

nn

nn

εεε

εεεβ

γβ

)2()2(

22

0

)(

)(1)(1

Analytic Functions

112/

2/

)1(21)1(

21;

11

+−− −++=

Ω

−+

= nnnjn

n

n

n UjUjVehRJU

γγγγ π

20

Radiated EnergyRadiated Energy

0

10

20

30

40

50

0 500 1000 1500 2000

W

γ−1

R/h = 0.9

R/h = 0.5

R/h = 0.3

ε = 2rε = 10r

2 250γ< < 1000γ >

2

2

2 hbR RW a eh

− ≈

11.5 2a< <

10.05 0.15b< <17 10a< <

3 614 10 2 10b× < < ×

20.1 / 0.9

r

R hε >

< <

[ ]1 1ln ( 1)W a b γ≈ −

5 100; 4rγ ε< < =

25 15a< <

243 10 0.3b−× < <

21

Angular Distribution of RadiationAngular Distribution of Radiation

150

120 60

30

180 0

210 330

240 300270

90

0.1

0.2

0.30.4

0.5γβ =γβ =γβ =

1.5 0.5 0.3

150

120 60

180 0

210 330

240 300270

90

200

300

500

400

100

30γβ =γβ =γβ =

1000 100 20

Low Energies High Energies

22

Transverse KickTransverse Kick

∫∞

∞−

=−∞=−∞== yyyy dtFtPtPP )()(∆

02∆∆2

∆ηλ

π

z

yy

PP ≡

2 30∆2

zPyλ η

π γ∆

≈ ×

1γ >>

0

5

10

15

20

0 20 40 60 80 100γ−1

( γ ∆

P y )

ε = 4r

R/h = 0.7

R/h = 0.5

R/h = 0.3

23

Frequency Dependence of Frequency Dependence of Frequency Dependence of rε

0.0

0.2

0.4

0.6

0.8

1.0

0 20 40 60 80 100

Z ||

Ω

R/h=0.5

ε =2r

γβ=100

Laser pulse

; ( )

1; r C

rC

εε

Ω < ΩΩ = Ω > Ω

)200(2)0( nmWW cc =≈= λλ

pulselaser 06.1 mµ100peak nmλ ≈

c0.65W ≈ Ω4rε =

/ 0.5R h =

24

Finite Size BunchFinite Size BunchFinite Size Bunch

0

2

4

6

0 10 20 30 40 50Ω

Z ||(∆x ,

∆ y )

ε =4rR/h=0.5

γβ=10

0

5

10

15

0 10 20 30 40 50Ω

Z ||(∆x ,

∆y )

/ Z||(∆

x = 0

, ∆ y=

0) ε =4rR/h=0.5

γβ=10

rεR

βc

h

x∆

y∆

( )( )( )5.0,5.1

5.1,5.1

5.1,5.0

=∆=∆

=∆=∆

=∆=∆

yx

yx

yx ( )( )( )5.0,5.1

5.1,5.1

5.1,5.0

=∆=∆

=∆=∆

=∆=∆

yx

yx

yx

/x x R∆ = ∆ /y y R∆ = ∆

25

Finite Size BunchFinite Size BunchFinite Size Bunch

rεR

βc

h

x∆

y∆

0

10

20

30

40

0.0 0.5 1.0 1.5 2.0∆

y/R

W(∆

x , ∆ y )

/ W(∆

x=0 ,

∆ y= 0) ε =4r

R/h=0.5

γβ=10

5.10.15.0

=∆

=∆

=∆

x

x

x

Radiated energy increases with the transverse width and decreases with the longitudinal length.

Radiated energy is ∆y independent for γβ >>1 and decreases dramatically with the longitudinal length for γβ >>1.

26

Electron Bunch Traversing a Geometric DiscontinuityElectron Bunch Traversing a Geometric DiscontinuityElectron Bunch Traversing a Geometric Discontinuity

22R

1RbR2

0v

Free space

Symmetric Discontinuity

Constant velocity

( )00 , µε

27

WG Discontinuity: Frequency-Domain SolutionWG Discontinuity: FrequencyWG Discontinuity: Frequency--Domain SolutionDomain Solution

Green’s Function

( ) )0','()',0'|,0(1

20

)2(∑∞

=

Γ−<=<>ν

ννντ zeRrpJzrrzrzG

( ) ( )( ) [ ]

( ) )0','(

421

')',0'|,0(

110

1

'

)1(21

21

1010

)1(

)1(

∑

∑∞

=

Γ

∞

=

−Γ−

<+

Γ

=<<

s

zss

s

zz

ss

ss

s

s

eRrpJzr

epJR

RrpJRrpJrzrzG

ρ

πHomogeneous Solution

( ) ( ) ( ) ( )222

)2(221

)1( ; cRpcRp sss ωω ν −=Γ−=Γ

22R

1RbR2

0v

' 0z <

( )( )zj

z err

qzrJ 0v2 )(

2);,( ωδ

πω −−=

Non-Homogeneous Solution

28

WG Discontinuity: Frequency-Domain SolutionWG Discontinuity: FrequencyWG Discontinuity: Frequency--Domain Solution

Er: continuity over the cross section

Hϕ: continuity across the aperture+

Bessel functions orthogonality

Matrix Formulation

0=z

Domain Solution

Boundary Conditions

( ) ( )( ) ( )[ ] )1(1)(

)1(1)(

gYZIYZIIYT

gYZIYZIR

−++=

−+=−−

−−

)1()1(

)(

)(

)0','(

)0','(

)0','(

gzrg

Tzr

Rzr

s

s

→<

→<

→<−

−

ντ

ρ

( ) ( ) ( )21

0

1121

212

1)1(

)2(

,

2

21)( RrpJRrpdrrJRpJR

RppZ

R

ssss

s ννν

νω ∫Γ

Γ=

Single Frequency

( )( )

')1(

21

21

10)1( )1(

41

21

')0','( z

ss

ss

sepJR

RrpJzrg Γ

Γ=<

π ' 0z <

Infinite size Matrices

29


( ) ( )( ) [ ]

( ) )0','(

421

')',0'|,0(

120

1

'

)2(21

22

2020

)2(

)2(

∑

∑∞

=

Γ−

∞

=

−Γ−

>+

Γ

=><

ν

νν

ννν

νν

ν

ν

τ

π

z

zz

eRrpJzr

epJR

RrpJRrpJrzrzG

Homogeneous Solution

Non-Homogeneous Solution

( )( )zj

z err

qzrJ 0v2 )(

2);,( ωδ

πω −−=

22R

1RbR2

0v

' 0z >( ) )0','()',0'|,0(

120

)1(∑∞

=

Γ>=>>s

zss

seRrpJzrrzrzG ρ

Green’s Function

( ) ( ) ( ) ( )222

)2(221

)1( ; cRpcRp sss ωω ν −=Γ−=Γ

30

WG Discontinuity: Frequency-Domain SolutionWG Discontinuity: FrequencyWG Discontinuity: Frequency--Domain Solution

Er: continuity over the cross section

Hϕ: continuity across the aperture+

Bessel functions orthogonality

Matrix Formulation

0=z

Domain Solution

Boundary Conditions

Single Frequency ' 0z >

( )

( )

(2) (2)

( ', ' 0)

( ', ' 0)

( ', ' 0)

s

s

r z R

r z T

g r z gν

ρ

τ

+

+

> →

> →

> →

( ) ( ) ( )21

0

1121

212

1)1(

)2(

,

2

21)( RrpJRrpdrrJRpJR

RppZ

R

ssss

s ννν

νω ∫Γ

Γ=

( )( )

')2(

21

22

20)2( )2(

41

21

')0','( zepJR

RrpJzrg ν

νν

νν π

Γ−

Γ=>

( ) ( )( ) ( )

1( ) (2)

1( ) (2)

R I ZY I ZY g

T Y I I ZY I ZY g

−+

−+

= + −

= + + −

Infinite size Matrices

31


Single Frequency

Solution

Infinite size matrix inversion

Truncation Problems

Pulse Approaching

The DiscontinuityWide

Spectrum

Time-domain solution is preferable

Linear independent function+

Infinite size matrix inversion for each

frequency

Scalar solution

without truncation problem

Bessel function orthogonality

32

WG Discontinuity: Time-Domain SolutionWG Discontinuity: TimeWG Discontinuity: Time--Domain SolutionDomain Solution

22R

1RbR2

0v

,10

,20

Ωv

,1 011( )

Ωv

,2 021

; 0, 0

( , ; )

; 0, 0

s

zt

s ssp

zzt

rJ p e t zR

A r z t

rJ p e t zR

ν

ν νν

α

α

− −∞

=

− −∞

=

< < =

> >

∑

∑

Primary Field

( ) ( )2

, , 2 20 , 1

02

1Ω ;2 Ω

v1 ; ; , ; 1, 21

ii j i j

j j i j i

p c qR R J p

i s jc

γβ α γβπε

γ β νβ

−= =

= = = =−

33

tse 1,Ω

Long.-dependence

>

<

+

−

0 ;

0;2

1

2

2

01, v

Ω

ze

ze

Rp

Rpz

z

s

s

γβν

>

<

0;

0;

20

10

zR

rpJ

zR

rpJ s

ν

22R

1RbR2

0vFormulation: Secondary Field t<0Formulation: Secondary Field t<0Formulation: Secondary Field t<0

Metallic

BoundariesRadial-dependence

Primary Field

01, v

Ω

zts

e−− Time-dependence

34

02, v

Ω

zt

e−− ν Time-dependence

Metallic

Boundaries Long.-dependence

22R

1RbR2

0v

te 2,Ων−

,20

2 2

1 2

Ωv ; 0

; 0s

z

p pzR R

e z

e z

ν

νγβ

−

+

>

<

>

<

0;

0;

20

10

zR

rpJ

zR

rpJ s

ν

Formulation: Secondary Field t>0Formulation: Secondary Field t>0Formulation: Secondary Field t>0

Radial-dependence

Primary Field

35

>

<

=<

∑

∑

∞

=

+

−

∞

=

1,

Ω

20,

1

ΩΩ

10

)(

0;

0;

)0;,( 2

1

2

21,

01,

1,

ν

γβ

νν

ν

τ

ρ

s

Rp

Rpz

ts

s

vz

tss

sz

zeeRrpJ

z eeRrpJ

tzrAs

s

ss

22R

1RbR2

0vSecondary FieldSecondary FieldSecondary Field

>

<

=>

∑

∑∞

=

−−

∞

=

+

−

1

ΩΩ

20

1,

Ω

10,

)(

0;

0;)0;,(

02,

2,

2

2

2

12,

νν

ν

γβ

ν

νν

ν

ν

τ

ρ

z eeRrpJ

z eeRrpJ

tzrAvz

ts

s

Rp

Rpz

tss

sz

s

36

∞<<= tz 0;0Er : continuity over the cross section

Hϕ : continuity across the aperture

22R

1RbR2

0vBoundary ConditionsBoundary ConditionsBoundary Conditions

0;0 <<∞−= tzEr : continuity over the cross section

Hϕ : continuity across the aperture

are determinedss ρτ ν & ,

are determinedνν τρ & ,s

te 2,Ων−

Linearly independent

functions

tse 1,Ω

Linearly independent

functions

( )RrpJ s /0

Orthogonal set of

functions

37

22R

1RbR2

0vMagnetic FieldMagnetic FieldMagnetic Field

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

2 12, 0.5 , 0bR R Rγ = = =

10.25Rtc

=−cRt 125.0=

38

Finite Size EffectFinite Size EffectFinite Size Effect

z∆

bR

( ) ( )|v|222v);,( 02

0 tzUrRUR

qtzrJ zbb

z −−∆−−=π

0 0( )

1 0U

ξξ

ξ≤

= >

0

0

0

0

1. 0;2v

2. 0;2v

3. 0;2v

4. 0;2v

z

z

z

z

z t

z t

z t

z t

∆< < −

∆< <

∆> >

∆> <

22R

1RbR2

0vz∆

39

-30

-25

-20

-15

0.2 0.4 0.6 0.8 1.0

[W(∆

z ) / W

(∆z=

0)] d

B

R2

γ = 100

∆ = 2.0z

Rb= 0.10, 0.15

Rb= 0.10, 0.15

Rb= 0.10, 0.15

∆ = 0.7z

∆ = 0.3z

-40

-30

-20

-10

0.0 0.1 0.2 0.3 0.4 0.5

[W(∆

z) / W

(∆z=

0)] d

B

Rb

γ = 100R

2 = 0.5

∆ = 0.7z

∆ = 0.3

zz

∆ = 2.0z

∆ = 5.0z

WR

qW1

10

2

2

−

≡

πε

22R

1RbR2

0vz∆

1/z z R∆ = ∆1/b bR R R=

2 2 1/R R R=

Radiated EnergyRadiated EnergyRadiated Energy

40

Radiated Energy22R

1RbR2

0vz∆

WR

qW1

10

2

2

−

≡

πε

1/z z R∆ = ∆1/b bR R R=

2 2 1/R R R=

-30

-20

-10

0

0 50 100 150 200 250

[W(∆

z ) / W

(∆z=

0)] d

B

γ−1

∆ = 2.0 = 1.0 = 0.5 = 0.3

zR

b= 0.1

R2= 0.5

0.95

0.96

0.97

0.98

0.99

1.00

0

0.01

0.02

0.03

0.04

0 50 100 150 200 250

WL /

W

WR / W

γ−1

R2 = 0.5 ∆

= 0.3zR

b = 0.3

Rb = 0.1

Rb = 0.1

Rb = 0.3

Radiated EnergyRadiated Energy

41

22R

1RbR2

0vz∆Radiated EnergyRadiated EnergyRadiated Energy

( ) [ ]25.01

212

38.01 )/(cosh

/1/11

RRR

RRW

zb ∆−

≈γγ

WR

qW1

10

2

2

−

≡

πε

1

12

1. 02. 03. 10

b

RzR R R

γ

≤ ∆ <

< < <

>

2 1

1. 0 Point Charge2. 0 Uniform Waveguide3. 0 Ultra-Relativistic

bR WR R W

Wγ

→ ⇒ → ∞

→ ⇒ →

→ ∞ ⇒ →

42

-50

-40

-30

-20

-10

0

0 50 100 150 200 250

[W] d

B

γ−1

R2= 0.5∆ = 0.5z

Rb= 0.1

ExactApproximate

-60

-50

-40

-30

-20

-10

0

0 1 2 3 4 5

[W] d

B

∆

R2= 0.5γ = 100

Rb= 0.1

Exact

Approximate

z

22R

1RbR2

0vz∆Exact vs. ApproximateExact vs. ApproximateExact vs. Approximate

43

WG Discontinuity: Time-Domain SolutionWG Discontinuity: TimeWG Discontinuity: Time--Domain SolutionDomain Solution

Time-domain analytic solution of the evanescent field in the region of the discontinuity. The solution uses the orthogonality of the Bessel functions and the fact that exponential functions describing the temporal behavior are linearly independent.Simulations based on this formulation enabled the development of an analyticexpression for the energy required to maintain the velocity of the bunch constant.

22R

1RbR2

0v

44

Surface Roughness: MotivationSurface Roughness: MotivationSurface Roughness: Motivation

?1accuracy A≈

Optical1 mλ µ≈1cmλ ≈

1accuracy mµ≈

X-band

Superposition of plane waves.Broad frequency spectrum.An assumption of small

perturbation is not necessarily relevant.

DimensionsConstrains Source Properties

Atomic level engineering.

The size of the bunch as well as the roughness of the structure are anticipated to be of the same order of magnitude.

45

v0

Roughness parameter

Partial reflection

Total charge Q

ConfigurationConfigurationConfiguration

Unif 0,ng δ ∼

in

ext, 11ext, 1 1

int intt

1 ; ; 0;2 2n n

n n n nn nn n n

R d d dzR d z z zg

Rg g

R R+

+≡ = + ≡ = ≡ = = + + +

46

Wake-Field WakeWake--Field Field

( ) ( )sinsinc

ξξ

ξ≡Current Density

( )( )

( ) 02 2

0

2 1, ;22

vsincv

zj

z b zb

QJ r z U R r eR

ωωωπ

− = − − ∆

( )1; 00; 0

U

ζζ

ζ>

≡ ≤Magnetic Vector Potential

( ) ( )2

20, ; , ;z zA r z J r z

cω ω µ ω

∇ + = −

47


intr R<

( ) ( ) ( ) ( ) ( )0 00

, | ', ', ; 2 ' ' ' ', ';

homogeneous solutionNon-homogeneous solution

IbR

jkzz zG r z rA r z dr r dz J r z dkAz k e rω πµ ω

∞ ∞−

−∞ −∞

= + Γ∫ ∫ ∫

( ) ( ) ( )'' ', || ', jk z zkG r z r d rez gk r

∞− −

−∞

= ∫

22 2k

cω Γ ≡ −

( )( )

( ) ( )( ) ( )

0 02

0 0

' ; 0 '1' ; '2

| 'I K

K Ik

r r r r

r rr

rg r

rπ

Γ Γ ≤ ≤= Γ Γ ≤ < ∞

Green’s function in a boundless space

48


( )( )

( )10 2

0 0

21 122

Isinc

v vz

uQ k

uB k ω ωµ δ

π

= − ∆ −

( ) ( ) ( ) ( ) ( )0 0, ; I K jkzz b BA r R z dk A k kr r eω

∞−

−∞

> = Γ + Γ ∫

( ) ( ) 0,, ; T nnz nA r z r

cDω ω ω=

0vbu Rω

γ≡

intbR r R< <

nth groove

0 0 , 00 ,, 0ext extT J Y Y Inn nr R r Rc cc c

rc

ω ω ωω ω ≡ −

Single mode

approximation

49

Wake-Field : Continuity of Ez & HϕWakeWake--Field : Continuity of Field : Continuity of EEzz & & HHϕϕ

intRcω

Ω ≡

intR∆ ≡ Γ

( ) ( ) ( ) ( ) ( ) ( ) ( )( )

20

0.21 00

12

KT

II

N

z n nn

nDA k k B kπ

ω=

∆Ω= − Ω ℑ −

∆∆ ∆ ∑

( ) ( ) ( ),1

n m nm

m

N

D Sωτ ω ω=

=∑

( ) 1sinc

2n n

njkzk e kd ℑ =

( ) ( ) ( )*

0

1 1I

n nS dk B k k∞

−∞

= ℑΩ ∆∫

( ) ( ) ( ), 1, , 0, ,T Tn m n n m m n mτ ω δ χ≡ Ω − Ω

( ) ( ) ( )1, 1 0 , 1 0 ,ext extT J Y Y In n nR Rc cω ω Ω ≡ Ω − Ω

50

Power & EnergyPower & EnergyPower & Energy

Emitted Power

( ) ( )( )

0

( ) 2 , ; , ;bR

sz zP t rdr dzJ r z t E r z tπ

∞

−∞

= ∫ ∫

Secondary field due to metallic surface

Emitted Energy

( )2 2

0 int 0 int0

( ) Re4 4

Q QW dt P t d S W

R Rπε πε

∞ ∞

−∞

≡ = Ω Ω ≡

∫ ∫

Normalized spectrum

51

v0

int0.5bR R=

int int0.15 0.3zR R< ∆ <

Typical DimensionsTypical DimensionsTypical Dimensions

int 0.5 0 0.2R mµ δ≈ ⇒ ≤ ≤30 45 1 mµ÷ ≈Bunch:

Each data point is a result ofaveraging over 80 different distributions

for a given value of δ

52

Partial Reflection in GroovesPartial Reflection in GroovesPartial Reflection in Grooves

0.0

0.3

0.5

0.8

1.0

0 0.2 0.4 0.6 0.8

W /

N

|ZL|

∆z = 0.15

∆z = 0.20

∆z = 0.25

Rb= 0.5

δ = 10-4

γ = 104

0.0

0.3

0.5

0.8

1.0

0 0.2 0.4 0.6 0.8|Z

L|

∆z = 0.15

Rb= 0.5

δ = 10-1

γ = 104

∆z = 0.20

∆z = 0.25

Grooves’number intz z R∆ ≡ ∆

- Energy almost linear with the load impedance@ .

- Energy weakly dependent on the load impedance@ .

410δ −=

110δ −=

intb bR R R≡

Normalized impedance @ grooves’ end

53

Partial Reflection in GroovesPartial Reflection in GroovesPartial Reflection in Grooves

0.0

0.1

0.3

0.4

0.5

0 10 20 30 40 50Ω

Rb = 0.5

δ = 10-1

∆z = 0.25

γ = 104

N = 10

|ZL| = 0.70

|ZL| = 0.32

|ZL| = 0.10

- Spectrum’s peak close to .- Spectrum’s width determined by

the bunch’s spectrum: sinc2 shape.

410δ −=0Ω = - Increasing the load impedance shifts

the main peak of the spectrumtowards .

110δ −=

0Ω =

0.0

0.1

0.2

0.3

0.4

0.5

0 10 20 30 40 50

|Re[

S(Ω

)]|

Ω

Rb = 0.5

δ = 10-4

∆z = 0.25

γ = 104

N = 10

|ZL| = 0.70

|ZL| = 0.32

|ZL| = 0.10

54

- The spectrum per number of grooves is virtually the same.- The total emitted energy increases linearly with the number of grooves.

1 2LZ =

0.00

0.01

0.02

0.03

0.04

0.05

0 10 20 30 40 50

|Re[

S(Ω

)]| /

N

Ω

Rb= 0.50

∆z = 0.25

δ = 0.10

γ = 104N=4,6,8,10

0

1

3

4

5

0 2 4 6 8 10

WN

Rb = 0.50

∆z = 0.25

δ = 0.10

γ = 104

Number of GroovesNumber of GroovesNumber of Grooves

55

0

Re ( )W d SΩ Ω∞

≡ ∫

0.0

0.2

0.3

0.5

0.6

0.8

0 10 20 30 40 50

|Re[

S(Ω

)]|

Ω

∆z = 0.15

∆z = 0.20

∆z = 0.25

∆z = 0.30

Rb= 0.50

δ = 0.10

γ = 104

0.0

0.2

0.3

0.5

0.6

0.8

0 10 20 30 40 50Ω

Rb= 0.50

∆z= 0.30

δ = 0.10

γ = 104

maxavg+stdavgavg-stdmin

1 2LZ =SpectrumSpectrumSpectrum

- The main peak of the spectrum isalmost independent.

- The spectrum width decreases with the increase of . z∆

z∆

- At low freq. Significant difference between average and min or max spectrum

- At high freq. all curves coincide.

56

0.0

0.2

0.3

0.5

0.6

0.8

0 10 20 30 40 50

|Re[

S(Ω

)]|

Ω

δ = 0.20δ = 0.15δ = 0.10δ = 0.05

Rb = 0.50

∆z = 0.25

γ = 103

0.0

0.2

0.3

0.5

0.6

0.8

0 10 20 30 40 50Ω

γ = 103, δ = 0.15

γ = 101, δ = 0.15

Rb = 0.50

∆z = 0.25

γ = 103, δ = 0.05

γ = 101, δ = 0.05

1 2LZ =SpectrumSpectrumSpectrum

- The spectrum is weakly dependent

on .γ- Average spectrum increases with . - Main contribution from .20Ω <

δ

57

0.0

0.2

0.4

0.6

0.8

1.0

0.00 0.04 0.08 0.12 0.16 0.20δ

Rb= 0.50

∆z = 0.25

γ > 50γ = 10

2

0 int4QW W

Rπε≡

- Average energy per groove increases with .- More energy generated by shorter bunch.- The impact of is almost negligible.

δ

γ

0.0

0.2

0.4

0.6

0.8

1.0

0.00 0.04 0.08 0.12 0.16 0.20

W /

N

δ

Rb= 0.5

γ = 104∆

z = 0.15

∆z = 0.20

∆z = 0.25

∆z = 0.30

1 2LZ =Emitted EnergyEmitted EnergyEmitted Energy

58

0.00

0.03

0.06

0.09

0.12

0.15

∆z = 0.15

∆z = 0.20

∆z = 0.25

Approx.

0.00 0.04 0.08 0.12 0.16 0.20

∆W /

δ0.25

δ

Rb= 0.5

γ = 104

- Energy spread per groove increases weakly

with average .0.25δ

22i i

i

W WW

W

−∆ ≡

0.00

0.03

0.06

0.09

0.12

0.15

γ = 101

γ = 102

γ = 103

Approx.

0.00 0.04 0.08 0.12 0.16 0.20δ

Rb = 0.50

∆z = 0.25

1 2LZ =Emitted Energy: Standard-DeviationEmitted Energy: StandardEmitted Energy: Standard--DeviationDeviation

59

Number of modes - Using a single mode is sufficient for all practical

purposes.- Use of many modes introduces numerical noise.

0.00

0.01

0.02

0.04

0.05

0.06

0 10 20 30 40 50

|Re[

S(Ω

)]| /

Ν

Ω

Rb = 0.5

∆z = 0.3

γ = 104

M = 2,3,4

M = 1

Single Mode ApproximationSingle Mode ApproximationSingle Mode Approximation

60

Summary: Scaling-LawsSummary: ScalingSummary: Scaling--Laws

22 0.25

2int int

0 int

int

int int

0.15 tanh 121.2

4

451.4290.57 tanh

1 20.72 1 20.72z z

W W g gR RQ N

R

gR

R R

πε

∆ ∆

∆ ∆

− ×

× +

+ +

int2

int int0 int

451.4290.57 tanh

1 20.72 1 20.724

z z

gW R

Q NR RRπε∆ ∆

+

+ +×

Average Energy

average roughnessLaws

roughness std.Standard Deviation

61

Summary: Scaling-LawsSummary: ScalingSummary: Scaling--LawsLaws

2

0 int

24

W QN Rπε

×

Average EnergyPoint-Charge0z∆ =

Standard Deviation

22 0.252

0 int int

0.34

W W Q gN R Rπε

− ∆×

- Point-charge moving in a cylinder of radius Rint boredin a dielectric or metallic medium

- Point-charge moving in a cylindrical wave-guide withperiodic wall of arbitrary but azimuthally symmetric geometry.

Electromagnetic Wake-Fields in Optical Accelerator

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