Waveguide, cavities, linear accelerators and RFQs · Waveguide, cavities, linear accelerators and RFQs Lars Hjorth Præstegaard Aarhus University Hospital Aarhus University Hospital,

1

Waveguide, cavities,

linear accelerators and RFQs

Lars Hjorth Præstegaard

Aarhus University Hospital

Aarhus University Hospital, Århus Sygehus

Outline

• Waveguides

• Cavities

• Linear accelerators (linacs)

– Accelerating structures

– Traveling and standing wave acceleration

– Longitudinal dynamics

– Beam loading

• RFQ linear accelerator

2

Waveguides


Electromagnetic wave in free space

Electromagnetic wave in free space:

Transverse electromagnetic wave

(TEM wave)

No electric field in direction of wave propagation: No acceleration

Both electric and magnetic field perpendicular to

direction of wave propagation (z axis)

: Angular frequency (2f).

z: Wavelength

kz : Wave number (2/).

c: Speed of light

Dispersion relation for TEM wave (propagation along z axis) :

Phase velocity ≡ /kz = c

Group velocity ≡ d/dkz = c

≡ 2f = 2c/z

≡ kzc

Dispersion relation: Relation between and kz(z)

Planar wave:

E(z,t)=E0Exp(i(t+kzz))

wave in +z dir. wave in -z dir.

3


z

z

Waveguide: Boundary conditions

Wave propagation

in +z direction

Solutions with a longitudinal field component must

be present in a waveguide (internal reflection)

Maxwell + boundary conditions No TEM wave in hollow waveguide

n

Conductor boundary conditions:

Conductor = equipotential: Ez=0

Conductor skin depth << : Bz/n=0


Waveguide: Solutions of wave equation

Wave equation for EZ (see appendix):

0,01

2

2

2

surfacezz EEtc

0,01

2

2

2

surface

zz

n

BB

tc

TE solution (Ez=0) exist:

Not suited for particle

acceleration

Wave equation for Bz:

TM solution (Bz=0) exist:

Suited for particle

acceleration

n TM: Transverse magnetic

TE: Transverse electric

From Ez and Bz, E and B

can easily be determined

4


Waveguide: Disp. relation for TM wave

Ansatz: zkωti

zzzex,yEr,tE

Oscillation with angular frequency

Propagation in +z direction

Wavelengthz=2/kz()

Physical field: Real component

yxEc

kyxE zzzt ,,

2

22

TM eigenvalue equation:

Ansatz

+ TM wave equation

Dispersion relation for TM

waveguide modes:

22

,2

2

znc kkc

Generator

frequency

Free space

solution


No wave propagation below cut-

off frequency c,1 ≡ ckc,1 : Stopband

n'th eigenvalue (infinite number of

eigenvalues: waveguide modes)

n=1

n=2

n=3

kz

-kc,n2

2

2

2

2

yxtEz,surface=0


Waveguide: Phase and group velocity

Phase velocity and group velocity:

12

2

, z

nc

z k

kc

k

12

2

,

z

ncz

g

k

k

c

k

> c < c

Generator

frequency

Free space

solution


Stopband

n=1

n=2

n=3

kz

5


Rectangular waveguides


Rectangular waveguide: TM modes

Application:

Transport of microwaves from source

to application (accelerator, radar,

radio broadcasting etc.)

Ansatz: yfxfyxE yxz ,

Ansatz +

TM eigenvalue equation

2

2

2

2

2 11c

y

y

x

x

ky

yf

yfx

xf

xf

yfky

yfxfk

x

xfyy

y

xxx 2

2

22

2

2

,

Eigenvalue (= constant)

-kx2 -ky

2

Eigenvalue equations for fx(x) and fy(y)

Ez,surface=0

Ez,surface=0

6


22

222

b

n

a

mkkk yxmnc

,


Solutions to eigenvalue equations for fx(x) and fy(y) :

xkBxkAxf xxx cossin

ykDykCyf yyy cossin Ez,surface=0

B=D=0 (fx(a)=fy(b)=0)

kxa = m , kyb = n

TMnm modes (Bz=0):

Color: Strength of Ez

22

b

n

a

mcck mnc

TM

mnc ,,

m,n=1,2,3,..

y

b

nx

a

mACyfxfyxE yxz

sinsin,

Cut-off frequency:



GHz .m .m .

,

, 65030

1

060

1

222

2222

11

11

c

b

n

a

mcf

TM

cTM

c

Example:

Lowest TM cut-off frequency in rectangular waveguide (a=6 cm, b=3

cm):

Wavelength of TM11 wave at cut-off frequnecy (5.6 GHz):

TM11, 5.6 GHz = c/5.6 GHz = 5.4 cm (for generator)

The lowest cut-off frequency has a corresponding wavelength in free

space comparable to the transverse dimensions of the waveguide

½

,,

½

,11 2

2

11

2

2

112

112

2

222

cck

ck

TM

c

TM

cTM

c

z

TM

22

2

2

znc kkc

,

(in waveguide)

http://www.comsol.com/showroom/gallery/images/1421_l.jpg

7


Rectangular waveguide: TE modes

TEnm modes (Ez=0):

Ansatz for Bz + TE wave equation + Bz/n,surface=0

n 22

222

b

n

a

mkkk yxmnc

,

y

b

nx

a

mByxBz

coscos, 0

22

b

n

a

mcck mnc

TE

mnc ,,

m,n=0,1,2,... (but (m,n) (0,0))

Mode TE01 and TE10 have a lower cut-off

frequency than TM modes

TE01 and TE10 modes are used for transport

of microwave power (no mode mixing)

kz=0



GHz .m .

,

, 52060

1

222

222

10

10

c

b

n

a

mcf

TE

cTE

c

Example: Lowest TE cut-off frequency in waveguide (a=6 cm,

b=2 cm):

TE10

TE01

TM11, TE11 a=6 cm, b=4 cm

Mode mixing

No mode mixing

TE20

8



1 possible mode

2 possible modes

Mode mixing in bend

(change of geometry)

a=6 cm, b=2 cm

a=6 cm, b=2 cm


Circular waveguide: TM modes

Ansatz for circular waveguide:

TM eigenvalue equation in cylindrical coordinates:

D

z

m=0,1,2,.. (number of azimuthal oscillations) mrRrEz cos,

D: Diameter

Ez,surface=0

2

2

22

2 11

rrrrt

,,

2

22

rEc

krE zzzt

-kc,n2

http://en.wikipedia.org/wiki/Image:Cylindrical_coordinates2.svg

9


mnc xD

k 2


rRc

krRr

m

rrrz

2

22

22

2 1

Eigenvalue: -kc2 Solution for kc

2>0:

rkJArR cm

Boundary condition:

022

DkJA

DR c

m

n'th root of Jm

D

xk mn

mnc

2,

n=1 n=2

Dkc

405.2*2 :mode TM 01,01

n=3

Rsurface=0

D

cxmnTM

mnc

2,

n=1,2,3,...

Ansatz +

TM eigenvalue

equation

Bessel functions of order m: m=1,2,3,...



Ez=0 at surface

kzti

cz erkAJtzrE 01,0,,,

TM01 mode:

Magnetic field:

http://upload.wikimedia.org/wikipedia/commons/c/c1/BesselJ_plot.svg

10

Cavities


Cavities: TM modes

Anzats for TM modes moving in both -z and +z direction:

ti

z

zkzk

z eyxEBeAetE zz ,, r

Oscillation with angular frequency

Wavelengthz=2/kz()

Propagation in both -z and +z directions

Physical field: real component

Total reflection at cavity end at z=0 A = B

ti

z

zikzik

z eyxEeeAtE zz ,, r

ti

zz eyxEzkA ,cos2

Standing wave

D

z

l

Waveguide

with end faces Cavity ≡

D: Diameter

l: Length af cavity

Cavity end at z=0

11


Cavities: TM modes

Boundary condition at cavity ends (Et=0)

0sin,,,,,0,, ti

ztt elkyxCtlyxEtyxE

Requires some

derivation

Transverse electric field of standing wave:

ti

zt ezkyxCtzyxE sin,,,,

kzl = q q

lz

2

Only certain resonant frequencies are present in a cavity

Dispersion relation for cavity:

q=0,1,2,...

2

222

2

2

l

qk

cc

1. Also valid for TE mode, except that q=0 do not exist (Et=0)

2. Resonant frequency of q=0 mode is independent of length

of cavity


Cylindrical cavity: TM modes

D

z

l

Cylindrical cavity:

Preferred design for

acceleration of charged

particles)

2

222

2

222 2

l

q

D

xc

l

qkc mn

c

TM

mnq

Resonant frequencies of TM modes:

TM010 mode:

[cm] D

GHz .952201010

D

cxf TM

Note:

Wille specifies r≡c/fr=2c/r. This is

the generator wavelength

corresponding to resonant frequency

(different to the wavelength of the

wave in the cavity)

12


Cylindrical cavity: TM010 mode

Example: DORIS Cavity (D=462 mm, l=276 mm):

MHz .cm 46.2

GHz .7496

952201010

D

cxf TM

Tuning

plunger

Coupling loop:

Coupling of transmission

line (waveguide or coaxial

cable) to cavity mode

Cavity tuning:

Adjustment of plunger

position (plunger in

increased resonant

frequency to 500 MHz)

Waveguide

(500 MHz TE01 mode)

Coaxial waveguide +

Coupling loop


Cylindrical cavity: ASTRID cavity

13


Cavity: Circuit model

Only specific frequencies can exist in cavity

Large amplitude decrease for small

frequency deviation

Cavity behaves as an electrical resonator

(high Q value)

sRLC

11

1

1

iiZ

1

0

0

01 iQ

RsZRs: Shunt impedance

Q0 = R00C

0=1/(LC)

Steady state:

ss

RFR

Vt

R

VtItVP

2

2

022

0

period RF

period RFcos

sRFRPV 20

titi eVtZItVeItI 00 ,

At resonance (Z=Rs):

Physical field:

Real component

Linacs:

Accelerating structures

14


Linear acceleration in a waveguide

Phase velocity in a TM waveguide:

No net transfer of energy to particles

Wave crest desynchronizes with the particle

12

2

, z

nc

z k

kc

k

> c

Linear accelerator (linac): Acceleration along a linear path

Goal:

Transfer of energy from

microwave source to the particle

beam


Disk-loaded waveguide

Reflections from discs

Disk-loaded waveguide: Addition of discs to the waveguide:

Positive interference of reflections from neighbor disks if

Disk-loaded waveguide with period d

Phase advance = kz*2d 2 kzd=

Large perturbation of waveguide dispersion relation for kzd≈

Standing wave behavior for kzd= (no energy transport)

Group velocity close to zero for kzd

15


Disk-loaded waveguide: Dispersion

Dispersion relation for disk-loaded waveguide:


Passband:

Large for large cell-

to-cell coupling

Electric

coupling via

beam iris

kzd

Disks Frequency exist for which = c

Acceleration of relativistic electrons (v=c)


Disk-loaded waveguide: Modes

Loss-free propagation: kzd = 2/p p=1,2,3,...

Modes used for particle acceleration:

0 mode: kzd=2 (p=1)

mode: kzd= (p=2)

2/3 mode: kzd=2/3 (p=3)

/2 mode: kzd=/2 (p=4)

Small holes in discs

Positive interference: phase advance = kz*pd = 2

p=2

Scattering of forward wave (hole=source of wave propagation)

16

Linacs:

Traveling wave (TW)

acceleration


Traveling wave (TW) acceleration

• Disk-loaded waveguide (slowed-down wave)

• TM wave and beam travels synchronous

• Input of RF power at first cell

• Output of RF power at last cell

• Injection of beam along axis of waveguide

RF load

Beam source

Resistive loss in walls

+ Energy transfer to beam

Reduction of microwave

power along waveguide

RF power in

17


TW acceleration: Power dissipation in walls

Change of power along dz: dP: Change of power along dz

Ez0: Axial electric field amplitude

l: Length of TW structure

Zs: 2Rs/l: Shunt impedance per unit length

TW power:

dzdP

w

dP

wdzQ

loss Power

storedEnergy

wzP g

Q factor:

w: Stored energy per unit length

g: Group velocity

(z): Field attenuation per unit length

Attenuation of electric field:

zEz

dz

zdEz

z0

0

dz

Z

E

l

dzR

dzEdP

s

z

s

z

2

0

2

0

2

dzdP

EZ z

s

2

0


TW acceleration: Power dissipation in walls

Definition of Zs, Q, and P 20 zEZ

QzP z

s

g

zQzP

dzzdPz

g

22

Attenuation of TW power:

zPzdz

zdEzE

Z

Q

dz

zdPz

s

g

22 0

Definition of

Q and P

zPzZzPQ

ZzE s

g

sz

2

2

0 Equation

for P(z) above

Definition of Zs,

P, Q and

18


l

elqEdzeEqW

l

z

l

z

z

1

00 0

0

0

TW acceleration: Constant impedance

Constant impedance accelerating structure:

Uniform cell geometry

Q, Zs, g and do not depend on z

z

z

z

eE

zE

00

0

Energy gain for synchronous particle at wave crest:

l: Length of TW structure

Maximum of K (l=1.26):

Ez0(l) = 0.28Ez0(0) , P(l) = 0.08 P(0) (remaining power in load)

Low l: High power loss in load

High l: High power dissipation to walls

Tuning of K by changing g (disk aperture)

Importance of shunt impedance K (typically K≈0.8)

lPZql

es

l

01

2

K = 0.903

Equation

for Ez02


TW acceleration: Constant gradient

Constant gradient accelerating structure:

Ez0 does not depend on z (change of structure geometry):

• g and depends on z (sensitive to structure geometry)

• Q and Zs=Ez02/(-dP/dz) do not depend on z (approximately correct)

dP(z)/dz = constant z

l

PlPPzP

00

llP

P

dzzzP

zdP

00

2

Use of equation for attenuation of TW power:

zPz

dz

zdP2

l

dzzePlP0

2 ,0

19


TW acceleration: Constant gradient

Energy gain for synchronous particle at wave crest:

lqEdzEqW z

l

z 00 0

0

0 Tuning of K by changing g (disk aperture)

Importance of shunt impedance

22

0 10 e

l

PZ

dz

zdPZzE s

sz

Maximum of K (=):

21

0 el

P

dz

zdP

2110 e

l

zPzP

K (typically K≈0.8)

K = 1

lPZeq s 01 2

Expression

for Zs

Ez0(l) = P(l) = 0 (infinite filling time as g=0)


TW acceleration: Stanford linear accelerator

50 GeV electrons

932 disc-loaded sections of 3.05 m

Water cooling

RF input

Discs

20


TW acceleration: Stanford linear accelerator

Constant gradient:

Energy: 50 GeV electrons (3 km linac with 932 linac sections)

• Uniform power dissipation

• Lower peak surface electric field

Accelerating structure: • 2/3 mode (large group velocity short fill time for

pulsed operation)

• Period (d): 35 mm

• Structure length: 3.05 m

• Iris (hole) tapering: 26 mm to 20 mm

• Cavity radius tapering: 84 mm to 82 mm

=0.57 (K=0.82): Compromise between high energy gain and short filling time

Linacs:

Standing wave (SW)

acceleration

21


SW acceleration

1. Full reflection of traveling waves at structure ends

Disk-loaded waveguide with reduced apertures at ends:

Standing waves

Different to TW structure (l1)

Synchronous filling of all cavities 2. Low field attenuation (l<<1)

d


SW acceleration: Modes

Family of N+1 normal modes (q=0, 1, 2,..., N)

Characterized by the phase advance per cavity: kzd=q/N

Highly resonant structure

Structure with N+1 cavities behaves as N+1 coupled

harmonic oscillators:

oscillator 1 oscillator 2 oscillator 3

coupling coupling

Dispersion curve:

kd

22


0 1 2 3 4 5 6

q=3 (pi/2 mode)

q=4 (2*pi/3 mode)

q=6 (pi mode)

SW acceleration: Energy gain

tN

qnEE nz

coscos0,

Axial electric field at nth cavity for N+1 cavities:

E0: Amplitude of electric field

n: Cavity number (0,1,2,...,N)

q: Mode number (0,1,2,...,N)

q/N: Phase advance per cell 7 cavities (N=6):

Other modes (q=1,2 and 5) provides ineffective acceleration

(small number of cavities with max. field strength)



Velocity of particle synchronous with mode q (N+1 modes):

q

Nd

Ndqkz

s

Energy gain for particle at cavity

n=0 at t=0):

N

n

N

n

nz dN

qnEedEeW

0

2

0

0

, cos

backward wave

0, modes: Energy gain doubled due to backward wave

Similar shunt impedance as that of TW structure

Nqn

ndt sn

NqdEN

NqdEN

N

qndEe

N

n ,0,1

0,2

22

cos12

0

0

0

0

23



SW acceleration with mode:

Particle in every

second cavity


SW acceleration: /2-mode

-mode: Large energy gain

Short fill time (large group velocity)

Insensitive to geometrical errors

Low energy gain

/2-mode:

/2 mode

bi-periodic /2 mode

Coupling cavity

Biperiodic /2-mode SW structure:

All advantages for /2 and modes

Looks like -

mode for beam

24


SW acceleration: Medical linac

Varian 600c biperiodic /2-mode SW structure:

microwaves in coupling cavity

Normal cavity Pulsed operation


SW acceleration: Medical linac

Varian TrueBeam biperiodic /2-mode SW structure:

Coupling

cavity

25


TW/SW acceleration: Choice of frequency

Low frequency:

Large mechanical tolerances

Large beam aperture

Wake fields 2 (long.), 3 (transv.)

Stored energy per unit length (-2)

High frequency:

Efficient acceleration (Zs½)

Higher threshold for breakdown

High accelerating gtradient

LEP cavities: 350 MHZ

Gradient: 6 MV/m

CLIC cavities: 30 GHZ

Gradient: 150 MV/m

Linacs:

Longitudinal dynamics

26


Longitudinal dynamics: TW acceleration

21mcW

Optimum: Inject particles on blue curve

Wave phase velocity = c:

Winj

Definition of :

Longitudinal focusing:

• Longitudinal focusing:

• Bunching: Electrons injected with

inappropriate phase (red) are lost

=0: Early arrival of

electron in cavity

Optimum asymptotic phase:

max=-/2 (max. field strength)

Early arrival (●) Lower electric field


Longitudinal dynamics: TW acceleration

Bunching + longitudinal focusing:

• Continuous beam is transformed to train of narrow bunches

• Low energy spread

Continuous beam bunch

Capture efficiency: Percentage of electrons captured in a bunch.

Depends on

• Electric field strength (structure geometry, input microwave power)

• Injection energy

• Injection phase

After section of linac with spped < c (buncher):

Electron speed is very close to the speed of light

Fixed phase of particles relative to the electric field

27

Linacs:

Beam loading


Beam loading

A bunched beam moving through a cavity:

Induced charges in the cavity walls

Induced fields in the cavity:

• Modification of the field of the acc. mode (beam loading)

• Creation of higher order modes ( power loss, beam

instabilities)

Energy conservation:

Beam loading The field of the acc. mode is reduced

corresponding to the transfer of energy to the beam.

High beam current Reduced field Reduced beam energy

28


Beam loading in SW and TW structures

SW structure:

Constant gradient TW structure:

Small change

of initial

electric field

Large change

of initial

electric field

Radio frequency

quadrupole (RFQ)

linear accelerator

29


RF quadrupole (RFQ) linac

RFQ linac:

• Bunching of beam

• Focusing of beam

• Acceleration of beam

• Pre-accelerator for ion linacs

• Acceleration of low velocity ion

beams: 0.01-0.06 c

• Almost 100 % capture of continuous

ion beam

• Electric focusing: Efficient for low

velocities

• Velocity-independent focusing

(focusing by electric field)


RFQ: Focusing and acceleration

Transverse component of E : Alternating focusing

Longitudinal component of E: Acceleration

ion

Period: L

Vanes with longitudinal

modulated tip

TE210-mode cavity • Increased field strength

• Long. field component (also

on axis)

+ =

30



Quasistatic approximation:

• Ignore induced electric field (Faraday's law)

(Good approximation when a, ma<<)

• Ignore magnetic field (low near z-axis)

011

2

2

2

2

2

z

UU

rr

Ur

rr

: Angular frequency

: Initial phase of potential tzrUtzrUdep sin,,,,,

Electric field described by time-dependent scalar potential:

U(r,,z) is a solution of Laplace's equation:

zrUzr ,,,, E



General solution to Laplace's equation:

n l

nln

n

n

n

lkznlkrIAV

nrAV

zrU

cos2cos2

2cos2

,,

2

0

2

0

l+n=2p+1, p=0, 1, 2,...

V/2: Electrode potential

I2n(x): Modified Bessel function of order 2n

k=2/L

L: Period of structure modulation

Lowest order terms:

kzkrIArAV

zrU cos2cos2

,, 010

2

01

Electric QP potential

(focusing)

Acceleration

: Relativistic beta factor of

TRF: RF period

: Wavelength (in free space)

Synchronous acceleration: L = particle motion during one RF cycle

(cTRF = )

http://en.wikipedia.org/wiki/Image:Cylindrical_coordinates2.svg

31



Boundary condition on electrode:

2

2,0,0,0,V

maUaU

period=L=

mkaIkaIm

mA

00

2

2

10

1

2010201 1

1

akaIA

aA

Large m: Large acceleration

Small a: Large focusing

Focusing

Acceleration

+V/2



Electric field components:

kzkrIkAV

z

UE

rVAU

rE

kzkrIkArAV

r

UE

z

r

sin2

2sin1

cos2cos22

010

01

11001

Acceleration

(also on axis as I0(0)=1)

32

More examples


Superconducting TESLA cavities for ILC

Superconducting structure: Solid niobium (cooled by superfluid helium at 2 K)

Extremely low surface resistance, Q factor > 1010

Very low resistive power loss

Significant savings in primary electric power

ILC (International Linear Collider): 11 km long electron 250 GeV linac

11 km long positron 250 GeV linac

17000 9-cell superconducting structures!

33



Higher order modes (HOM)

couplers:

Damping of high frequency

eigenmodes excited by the beam

(cause beam instabilities).

Parameter Value

Accelerating structure Standing Wave

Accelerating Mode TM01, mode

Frequency 1.300 GHz

Qualification gradient 35.0 MV/m

Quality factor >1*1010

Length 1.038 m

Cell to cell coupling 1.87%

Iris diameter 70 mm 70 mm

Why multi-cell structure?

High active acceleration length in a

linac. However upper limit of cells

due to trapped modes, uneven field

distribution in the cells, high power

requirements on the input coupler



The key to high-gradient performance is the ultra-clean and defect-free

inner surface of the cavity.

Highest gradients to date for multi-cell cavities

Goal:

35 MV/m with a minimum

production yield of 80%

34

Appendix


01

0

2

2

2

EBE

BEEB

E

tct

tt

Maxwell's equations without any sources:

Wave equation

0 E

t

BE

0 B

tc

EB

2

1

01

2

2

2

B

tcWave equation for magnetic field:

Wave equation for electric field:

0 E

t

BE

tc

EB

2

1Laplace operator

2

2

2

2

2

2

zyxAAA ,

35


Literature

Wave guides:

• J. D. Jackson, Classical Electrodynamics, John Wiley & Sons.

• D.J. Griffiths - Introduction to Electrodynamics, Prentice-Hall.

• http://www.temf.tu-darmstadt.de/forschung_5/feldanimationen/index.de.jsp

Linear accelerators:

• E.A. Knapp et al., Coupled resonator model for standing

• wave tanks. Rev. Sci. Instr., v. 38, n. 11, p. 22, 1967

• E.A. Knapp et al., Standing wave accelerating structures for

• high energy linacs. Rev. Sci. Instr., v. 39, n. 7, p. 31, 1968

• Introduction to RF linear accelerators, Mario Weiss, CERN Accelerator School: 5th

General accelerator physics course, Jyväskylä, Finland, 7 - 18 Sep 1992

• Principles of RF linear accelerators, Thomas P. Wangler, John Wiley & Sons,1998

• Physics and technology of linear accelerator systems, World Scientific Publishing,

2003

• http://cas.web.cern.ch/cas/

• http://uspas.fnal.gov/lect_note.html

Documents

Waveguide, cavities, linear accelerators and RFQs · Waveguide, cavities, linear accelerators and RFQs Lars Hjorth Præstegaard Aarhus University Hospital Aarhus University Hospital,