lecture musumeci a2 - slac conference services | · pdf file · 2013-07-22preferred...

Electron Sourcery

SSSEPBSLAC, July 22‐26th 2013

P. MusumeciUCLA

Department of Physics and Astronomy

Acknowledgements

J. B. RosenzweigD. DowellX. J. WangC. BrauJ. LuitenF. SannibaleD. FilippettoB. CarlstenW. Graves

P. PiotJ. SchmergeK. JensenP. HommelhoffC. PellegriniM. FerrarioR. LiB. ReedM. Krasilnikov

F. StephanS. SchrieberC. VicarioL. CultreraI. Bazarov

Material from these lectures has been liberally taken from talks/papers/lectures/proceedings/notes from a large number of people which I thank and acknowledge here

P. Musumeci, SSSEPB, SLAC, July 2013

Course objective• Lecture 2: Generation of electron beams: physics of photocathodes, photoinjectors, alternative sources using ultra‐cold gases or plasmas, fundamental limits on beams phase space density.

• Disclaimer. Extremely wide topic. Hard to cover exhaustively in a full semester class. Impossible in 3 hours.

• The declared goal of the class is to increase breadth more than depth

• So concepts, figures, tables and references are preferred over equations and derivations


Outline/Syllabus• Electron source applications• Electron source figures of merit• Cathode physics

– Photoemission. Thermal emittance and quantum efficiency– Child‐Lamguir law and space charge limits– Maximum achievable brightness

• Photoinjector gun types• Beam Dynamics in RF guns

– Longitudinal equation of motion– Transverse RF effects: rf focusing and emittance.

• Space charge effects– Cylindrical beams– Emittance oscillations– Scaling laws– Blow‐out regime, pancakes and cigars

• Magneto Optical Trap‐based ultracold sources• Nanotips based e‐sources


Short wavelength FELs

• Demands:– High brightness beams– Ultralow normalized emittance (~ 1 um)– High current (kAmp)– Beam charge ranging from 10 pC ‐ 1 nC

Progress in electron sources in the last 20 years is widely recognized as one of the enabling technologies for the XFEL development.


High average power applications• High average power FELs

– Soft X‐ray. – IR

• THz radiation sources– Beam based THz generation

• Energy Recovery Linacs

• Requirements:‒ mAmp average currents‒ MHz or higher repetition rates‒ Low emittances

Next Generation Light Source (LBNL)

Cornell ERL

Jefferson Lab FEL (JLAB)


Electron diffraction and microscopy

FRED DTEM

# of particles 105 108

Bunch length 100 fs 10 ns

Emittance 0.1 um few nm

Femtosecond Relativistic Electron Diffraction

Dynamic Transmission Electron Microscope


Inverse Compton Scattering sources

• Small emittances to allow focusing to micron‐spot sizes• High charge to increase x‐ray yield• Short bunches for ultrafast x‐ray pulses• Repetition rate to match laser repetition rate

from ELI‐NP project


Injectors for HEP machines?• Polarized electron sources• Flat beams• High gradient laser based accelerators• Very low timing jitter for laser interaction• Ultrashort bunch lengths• Small emittances to fit small acceptances

2 um

Laser driven dielectric structure

Plasma wave accelerator


1 pC 10 pC 100 pC 1 nC

107 108 109 1010For single shot electron diffraction and microscopy the number of particles per bunch should be enough to allow sufficient contrast in the diffraction pattern/image. For a diffraction pattern of a single crystal, we only need 105particles. At least 108 particles are required to form an image.

1‐10 pC per bunch enable the generation of beams with smaller transverse and longitudinal normalized emittances. The resulting improved beam quality allows for shorter FEL gain lengths at a relatively moderate electron beam energy.

The main operational mode for X‐Ray FELs relies on a charge/bunch of a few 100s pC, a satisfactory tradeoff between the number of photons/pulse and a moderate

transverse emittance

Experiments in FELs and ERLs requiring large number of photons per pulse or very narrow transform limited photon bandwidth in seeded FEL schemes require longer bunches and hence higher charges per bunch that can approach the nC.

Beams for HEP and plasma wakefield drivers require very

large charges, over 1 nC

Particles per bunch. Charge. One of the most important parameter of a (pulsed) electron source. Peak current is not preserved since the bunch length can be controlled using

compression techniques.

Beam brightness: early historyIntuitively the peak current is an important parameter of thebeam, but it is also important to be able to collimate and focus thebeam.

In 1939 von Borries and Ruska (Nobel prize in Physics for theinvention of the Electron Microscope) introduced the so‐calledbeam brightness (“Richstrahlwert”) defined as:

Empirically constant along the microscope column. The smaller the spot the larger the divergence.

This definition still holds today in the field of electron microscopy with peak numbers of Bmicr up to 1013 Amps/m2/sr

Ω

E. Ruska1986 Nobel prize


5D beam brightnessThe 5D brightness is often used in FEL community to compare electron sources and it is the relativistic analog of the microscope brightness

For example it enters directly in FEL parameter The main difference with electron microscopy brightness is the use of normalized

emittances to take into account relativistic effects.

2~

2

ρ is the FEL parameter. It is important to have it as high as possible(~10‐3), since:


For high average power applications typically we are interested in the total amount of charge per second. For these applications people often use the 4D brightness

where the longitudinal beam properties are not considered. The average 5D brightness will be B4D f where f is the repetition rate of the accelerator.

Another reason to introduce the 4D brightness is due to the development of bunch compressors which can reduce the bunch length by many times, increasing the final current (and thus 5D brightness).

Average and 4D beam brightness


Liouville’s theorem

Liouville theorem states that for Hamiltonian systems the phase space density stays constant.

As long as the particle dynamics in the beamline elements (transport optics, accelerating sections) can be described by Hamiltonian functions (no binary collisions, stochastic processes, etc. ), the phase space density will stay constant throughout the accelerator.

The phase space density obtained at the electron source is a critical parameter.

It follows that research in optimizing electron sources is crucial in the field of particle accelerator. It makes sense to classify the various electron sources based on the 6D phase space density at the source.

In practice 6D brightness is hard to measure !


6D beam brightnessThe meaningful quantity to use to describe electron sources should then be the 6D beam brightness defined as

66

0

nxnynz

where V6D is the volume occupied by the beam in the 6D phase space (x, px, y, py, z, pz).

V6D is proportional to the product of the 3 rms normalized emittances nxnynz• Neglecting factors of and m0c• RMS emittances are conserved only when linear forces act on the

distribution.• There might be regions of the beam (usually the center) where the phase

space density is higher. Filtering particles increases the brightness, at the expenses of beam charge.

• Ratio between peak center brightness and rms beam brightness typically depends on the beam distribution. (for 3D gaussian F = 1/8)


Brightness quantum limitPauli exclusion principle prevents electrons from being in the same quantum state.

Since the elementary quantum of phase space area is set by Heisenberg uncertainty principle, we find a quantum limit for the maximum beam brightness

No beam can ever beat this limit. In practice state‐of‐the‐art electron sources, as we will see, do not even come close and have B6D ~ 10‐4 Bquantum or worse.

We can introduce the Degeneracy Factor representing the number of particles per elementary volume of phase space

20

32

3

/

~1025 / 2

M. B. Callaham, IEEE J. quantum electronics, 24:1958, 1988

W. PauliNobel prize 1945

W. HeisenbergNobel prize 1932


Number of electrons per transverse coherence area

• A coherent photon beam has all the particles in the same quantum state.

• A coherence electron beam will have degeneracy factor approaching 1

• Beam coherence is important for some applications such as for example coherent diffractive imaging.

• In particular no coherent imaging is possible if the number of electrons per coherent transverse area of the beam is smaller than one.

• TEM solution is to use very long pulse (very large longitudinal emittance) to recover transverse coherence.

Aberration correction

Courtesy of B. ReedP. Musumeci, SSSEPB, SLAC, July 2013

Single‐electron beam brightness

• To avoid space charge effects, in some applications beams formed by a single electron are used.

• But a single electron beam does not mean a quantum degenerate beam !• Brightness is still limited by the electron source type.• Beam parameters in this case have statistical meaning.

F. O. Kirchner, S. Lahme, F. Krausz and P. Baum. New J. Phys. 15, 063021 (2013).


Cathode physics• Cathodes are obviously a fundamental part of electron sources. The

gun performance heavily depends on cathodes.• Photocathodes are most commonly used in present injector

systems with one notable exception (SACLA XFEL uses a thermoionic cathode)

• In the lower charge regime the ultimate brightness performance of the linac is set by the cathode intrinsic emittance.

• In high repetition rates photo‐sources high quantum efficiency photocathodes are required to operate with present laser technology

• Advanced cathodes is an active area of research (photoassistedfield emission, needle arrays, photo‐thermoionic, diamond amplifier, nanopatterned cathodes, …)

• Important References:– D. Dowell et al. Cathode R&D for future light sources. Nucl Instrum

Methods Phys Res A 622:13 (2010) with many references– Photocathode Physics for Photoinjectors workshop series.


Quantum efficiency• Quantum efficiency: Number of photo‐emitted electrons per photon

impinging on the cathode.– High QE at the longest possible wavelength. – Fast response time: <100 ps– Uniform emission


Thermal/Intrinsic emittance• Thermal emittance: Spread in velocities of emitted electrons

– As low as possible– Atomically flat surface: ~few nm p‐p, to minimize emittance growth

due to surface roughness and space charge– Tunable, controllable with photon wavelength– Better at cryogenic temperatures?

• Lifetime, survivability, robustness, operational properties– Require >1 year of operating lifetime– reasonable vacuum level: 10‐10 Torr range– Easy, reliable cathode cleaning or rejuvenation or re‐activation– Low field emission at high electric fields– Reliable installation and replacement system (load lock)


Common photocathodes

Electron initial distributionElectron are fermions. Thus, the probability of occupied single‐particle energy states in a crystalline solid obey the Fermi‐Dirac distribution:

where Ef is the Fermi level (metal physics).For high energy tails (E>Ef) the FD distribution and the Maxwell‐Boltzmann distribution coincide, and the simpler MB can be used

the MB distribution is the approximation of FD and BE distr. in case of high energies and low densities (interactions between particles negligible)

In calculating the probability of electron escape from a material, the specific emission process determines what distribution to use.The MB distribution (high energy tails) is used for thermionic emission, while field and photo‐emission calculations use the FD distribution, since the excited electrons come from energy levels below the Fermi level

Metals vs. semiconductors

K. Flottmann, Tesla FEL‐report 1997‐01

Ef=EG/2

simplified semiconductor energy bands:

The electronic properties of a semiconductor are dominated by the highest partially empty band and the lowest partially filled band, it is often sufficient to only consider those bands. Electron affinity is defined only for semiconductors and is the energy difference between the Vacuum level and the beginning of the conduction band

T~0KT>>0K

the Fermi level for metalsis in the conduction band

photoemission in semicond.

thermionic em. in semicond.

Effective potential•image charge

vacuummetal

•Schottky effectapplied ext. field lower the barrier

x x

e‐e+

3 different emission processes:

1) Thermionic emission2) Photo‐emission

3) Field emission

Require electrons to overcome the barrier;

Consequence of the Schottky effect. Electrons tunnelling thebarrier. Very fast dependence on appl. field. DARK CURRENT

Thermionic emissionElectron kinetic energy higher than the effective potentialMB distribution may be used in the calculation of the electron current density

Richardson‐Dushman equation

•The velocity spread of emitted thermo‐electrons is given by:

We’ll use this result to calculate the emittance of emitted beam

electron kinetic energy in the direction normal to the surface> work function

•if we have

Operating temperature 1500 KSo excess kinetic energy ~ 0.5 eV

A is Richardson constantLimit is ~ 1 A/mm2

Field emission

Strongly depends on applied electric fieldmaterial characteristics surface roughness

L.W.Nordheim, Proc. R. Soc. Lond. A 1928 121, 626‐639 doi: 10.1098/rspa.1928.0222

In rf fields is localized at high E field phases, and is then transported trough the machine

work function

Fowler‐Nordheim equation for the field emitted current

I [A ] 1.56 106A

[m 2 ]E[V / m ]

2

[eV ]

104.52 1 2

e

6.53109 3 2

E

enhancement factorA effective area; work function

Photoemission physics• Three step process

1. Photon Absorption Optical depth (~14nm for UV light on copper)

2. Electron Transport Electron‐electron (phonon) scattering; MFP ~5‐6nm for 4.5eV exited electrons

3. Vacuum Boundary Crossing maximum angle of emission; pz


Electron refractionElectrons need to have enough kinetic energy in the direction normal to the exit surface in order to escape the material:

Refraction law for electrons at the metal‐vacuum boundary.usual numbers for θinmax<~ 10deg , θoutmax ~90deg

Courtesy of D. Dowell

QE for metals

1

2

01 2

1 0

(cos )

( ) 1 ( )(cos )

F

F eff

F eff

F

F

E

EE E

e e E

E

ddE d

QE R Fd ddE

Step1:OpticalReflectivity~40%formetals~10%forsemi‐conductorsOpticalAbsorptionDepth~120angstroms

Fraction~0.6to0.9

Step2:TransporttoSurfacee‐escattering esp.formetalse‐phononscatteringsemiconductorsFraction~0.2

Step3:EscapeoverthebarrierE istheelectronenergyEF istheFermiEnergyeff istheeffectiveworkfunction

eff W Schottky

•Sumoverthefractionofoccupiedstateswhichareexcitedwithenoughenergytoescape,Fraction~0.04

•AzimuthallyisotropicemissionFraction 1

•Fractionofelectronswithinmaxinternalangleforescape,Fraction~0.01

QE(Cu) ~ 0.5*0.2*0.04*0.01*1 = 4x10‐5Courtesy of D. Dowell,

D. H. Dowell et al., PRST‐AB 9, 063502 (2006)

Cathode emittance

Ef Ef+ϕeff Ef+hνEf‐hν+ϕeff

Scaling of QE and thermal emittance !Expand integrals for small h‐eff

The spread in momentum can be computed in the same way

QE~ Φ 2 εnx~ Φ

So chasing the work function with the laser frequency is useful to increase QE, but not to decrease thermal emittance

ee

nnn

nn kTenhFTIR

heAJ 021

Multiphoton photoemission• “Violating” Einstein photoelectric effect…

o For a given metal and frequency of incident radiation, the rate at which photoelectrons are ejected is directly proportional to the intensity of the incident light.

o For a given metal, there exists a certain minimum frequency of incident radiation below which no photoelectrons can be emitted. This frequency is called the threshold frequency

• …..and few years of RF photoinjector common practice spent on “getting ready the UV on the cathode”

• Two or more small photons can do the job of a big one.• Generalized Fowler‐Dubridge theory: photoelectric current

can be written as sum of different terms. n nJJ where

Fowler functionselects the dominant n ‐order of the process


0 100 200 300 400 500 600 700 800100

150

200

250

300

350

Charge autocorrelation Polarization gating autocorrelation

Time (fs)

Aut

ocor

rela

tion

sign

al (a

rb. u

nits

)

Save laser energy. Use IR photons on the cathode

Question: Why hasn’t this been done before? Recent interest in pancake regime. Ultrashort beam at cathode => uniformly filled

ellipsoidal beam. Very high extraction field in RF photoinjector: away from space‐charge induced

emission cutoff. (Early experiments using low gradient setups.) Damage threshold few 100 GW/cm2 at sub‐100 fs pulse lengths. AR coating on the cathode improves charge yield. (at Pegasus 2 J of 800 nm ‐> 50 pC )

BUT conversion efficiency is ~10%

Autocorrelation of two IR pulses on the cathode shows promptness of emission.

P. Musumeci et al., Phys. Rev. Lett, 104, 084801 (2010)


Surface‐plasmon assisted photoemission from a nano‐structured cathodeR. K. Li UCLA

Summary: Fabricating nano‐structures on a flat pure Cusurface. The nano‐structures have dimensions on the sameorder of ir laser wavelength (800 nm). Enhanced EMdensity leads to great enhancement of charge yield.

(1) 3D simulation geometry reflectivity

~10 nm FWHM

25 um square125 um

(2) SEM images of fabricated nanohole array

800 nm

(3) measured charge yield map

• installed in S‐band rf gun and conditioned to > 70 MV/m• > 100 times more charge compared to flat surface• prompt emission and 3‐photon process• thermal emittance better than 2 mm‐mrad/mm rms

(5) summary of experiment results

(6) next step• on single crystal substrate• optical damage threshold• nm‐beamlet dynamics

(4) how to measure ultralow emittance

imaging single electron Measured 20‐100 nm emit.P. Musumeci, SSSEPB, SLAC, July 2013

Space charge effects in emissionChild Langmuir Law

As emitted particles come out from the cathode they create their own electric field. This field in the beam tail field is opposed to the external field, growing with the

extracted charge. The effective total potential is distorted by this field. The potential distortion creates asymmetries in the electron beam (tails) and sets a

maximum extractable current in the steady state regime The maximum current is given by the Child‐Langmuir law

‐ +V

‐‐

‐ ‐‐+

+++

Fext

FSCcathode anode

00

anode

z

V

z=ddP. Musumeci, SSSEPB, SLAC, July 2013

Child Langmuir lawThe Child‐Langmuir law expresses how the steady‐state current varies with both the gap distance and bias potential of the parallel plate system.It is a 1D derivation, assuming initial electron velocity equal to 0 and non relativistic motion


Child Langmuir Law for short beams2 cases:R > ze pancake aspect ratioR < ze cigar aspect ratio

Pancake aspect ratio caseMaximum surface charge density set by the cathode extraction field.

Cigar aspect ratio caseOnly a small part of the beam contributes to the space charge field and higher charge can be extracted.

2 0 0

Q

2 ∝ 22 ∝ 0


Time dependent behaviourWhat happens if we force more current density than CL limit into the diode?

As injected current increase the space charge potential increases, slowing down particles.If enough current is injected, the SC potential energy becomes higher than the particle kinetic energy and some particles will eventually stop and reflect back to the photocathode (Virtual cathode).The reflection weaken the SC potential and the emission starts againThis phenomena is called VIRTUAL CATHODE INSTABILITY.

D. Dowell et al., Phys. Plasmas, Vol. 4, No. 9, September 1997

Virtual cathode instability observedin a photo‐injector

The oscillation frequency is a function of the current densityand the e‐beam momentum (plasma frequency)


Maximum Brightness for photoinjectorsUsing the space charge limited charge density, we can find a limit on the transverse emittanceand then on the beam brightness.

For pancake aspect ratios, the maximum transverse brightness does not depend on the beam charge!!! It only depends on the acceleration field and on the emission process.

The above brightness limit, such as the minimum emittance values calculated in the previous slide, consider a beam extraction at the onset of the virtual cathode formation. In practice, extracting beams with charge density close to the SCL limit may lower the overall brightness of the beam

I. Bazarov et al., PRL 102, 104801 (2009)

,


Brightness limitations and core emittance Approaching the SCL the accelerating electric field is significantly reduced in the cathode region. Particles in the back of the beam experience lower fields and non linear potentials. The formation of an asymmetric tail and the non linearity of involved space charge fields lower the

beam brightness.

simulated bunch length at 1mm from the photocathode for the CORNELL photo‐gun(DC, ~2.7MV/m acc. field)

The two beams have the same starting aspect ratio.

EXAMPLE: Emittance and Brightness of a beam at the edge and below the SCL (CORNELL):


Various types of photoguns


Outline• Electron source applications• Electron source figures of merit• Cathode physics

– Photoemission. Thermal emittance and quantum efficiency– Child‐Lamguir law and space charge limits– Maximum achievable brightness

• Photoinjector gun types– RF– DC– SRF

• Beam Dynamics in RF guns– Longitudinal equation of motion– RF effects on the beam– Scaling laws

• Space charge effects– Laminar flow and envelope equation with space charge– Ferrario working point– Blow‐out, pancakes and cigars

• Magneto Optical Trap‐based ultracold sources• Nanotips based e‐sources


RF gunso RF guns are widely used to produce high peak current electron beamso Last generation of low repetition rate normal conducting guns LCLS‐like, works at GHz o frequencies (typically 3GHz).o The cavity structure is n+1/2 cell (n=1,3…)o Standing wave structure, TM mode to accelerate the beamo The cathode is placed in the half cell (max field)

first order approx. from Maxwell equations solution for fundamental accelerating mode in a pillbox


Electron capture and accelerationLength needed by the acc. electron to gain 1 unit of .

Capture parameter. It should be greaterthan 1, so that electrons become relativistic withinhalf the rf period, and are thus captured by the forward wave of the field

Intuitively, an electron is captured when it gains enough momentum from the wave to “surf” it until the exit of the cavity, without slipping back todecelerating phases.

Shorter wavelengths require higher fields!At LCLS α ~ 2; Lc= 4 mm

The fast acceleration peculiar of rf guns is crucial inmaintaining the beam characteristics set by the laserbeam (current, shape, starting emittance…). It greatlydiminishes the effect of space charge forces (later)


Longitudinal equations of motion

Forward wave

Backward wave

Using dβ

These equations can be numerically solved.

But before we do that, we can make some analytical approximation to help us understand better the dynamics.


Kim’s approximationAt first order we can calculate the energy assuming the phase to be constant

main contribution to phase slippage very close to the cathode, z ~ 0

12 sin 0

1 1 0

when the particles are ultrarelativistic , there is no more slippage

and then solve for the phase slippage

→1

2 sin 00

Kim vs. Numerical solution

1.5 cell gun 120 MV/m

0 10 20 30 40 50 60 70 80 900

153045607590

105120135

Final phase vs. initial phase


Energy and bunching in RF gunWe can see immediately the bunching properties of the RF gun

sin2cot11

dd

o

f

1 sin12 cos cos 2

0 10 20 30 40 50 60 70 80 90

6

8

10

12Energy vs. phase

0 10 20 30 40 50 60 70 80 900

0.2

0.4

0.6

0.8

1Compression factor vs. phase

Max energy from 1.5 gun at 120 MV/m is at 63 degrees.For 1.6 cell guns is at 30 degrees


RF effects on the beam: transverse phase space

We now try to calculate the effect of radial rf fields on transv .emittance at the exit.TM01‐like mode (Ez ,Er , Bθ). We calculate radial field close to axis, in the approx. that Ez does not depends on r.

Er/r

Ez

x

x’

This represents a defocussing kick when 0 < <

mMVbandSmradpr r / 100 , @ 10

Thus it usually will be necessary to focus the beam immediately after leaving the cavity. P. Musumeci, SSSEPB, SLAC, July 2013

Transverse RF emittance

Rewriting pr in Cartesian coordinates:

It gives the phase space distribution: a collection of lines with different slopes corresponding to different

The normalized transverse emittance is:

By inserting px we obtain


sin sin cos 2

2sin

sin2 sin2 2cos sin 2

2cos2 sin2

writing and assuming that is small

one obtains from:

so that

which has a minimum for <>=/2For Gaussian distributions


Solenoid focusingin a solenoid lens the beam gets focused and rotated:

<0, focusing •parallel beam at the entrance,•thin lens, r~const inside the lens

solving the paraxial equation we find, for a lens of length L:

The fringe field (divergence of B=0) is the key for beam focusing.The particle interact with Br in the fringe and acquire a transverse velocity that inside the solenoid couples with Bz generating a net transverse force

used in the derivation of paraxial equation

Magnetic EmittanceIf particles born in a region where static magnetic field is present, a transverse momentumis added to their intrinsic one, that leads to an emittance increase. The conservation of canonical momentum leads to the Busch’s theorem:

For a constant magnetic field Bz =B0 in the considered surface S: initial uniform distributionwith xrms=r0/2

Also, out from the magnetic field, the beam will keep a net defocusing kick x’:

Larmor Frequency, ωL

Aberrations and emittance growth in solenoids:chromatic effects

Because of rf (and space charge) defocusing it is important to focus the beam right after the gunwith strong magnetic fields. Solenoids are usually used at low energies, because of their abilityof focusing at the same time in both transverse planes.By comparing the centrifugal and the Lorentz force on a particle with an external magnetic fieldwe find the magnetic rigidity:

the bending radius ρ is energy dependent. Substituting into the formula for k we find:

the sol. focal length is energy dependent (1/f=ksin(kL))

D.H. Dowell, US Particle Accelerator School, June 14‐18, 2010, Boston, MA.

it can be shown that the associated emittance increase is:

Aberrations and emittance growth in solenoids:geometric effects

61

“Pincushion” aberrationat the beam waist

~50cm after the solenoid

4th order fit of the aberrationemittance vs. rms beam size

at the solenoid

Leff =19.35 cmBsol = 2.5kG

Initial beam has zero divergence (zero emittance) and is a uniform radial distribution, radius = 2x

Example for aLCLS like sol parameters:

D.H. Dowell, Beam Physics At, Near and Far from the Cathode, Max Zolotorev’s 70th Birthday,LBNL, nov. 2011

All magnetic field solenoids exhibit a 3rd order angular aberration also known as the spherical aberration

The fields producing this aberration are usually located at the ends of the solenoid. The aberration depends upon the second derivative of the axial fieldSmall contribution for usual beam sizes (1‐2mm)

Emittance growth in solenoids:quadrupole fields due to lack of symmetry

quadsolenoid

, , ,

sin 2x qs x sol y sol

q

KLf

Measured magnetic fields, LCLS

D.H. Dowell, Beam Physics At, Near and Far from the Cathode, Max Zolotorev’s 70th Birthday,LBNL, nov. 2011

The quadrupole phase angle is the angular rotation of the poles relative to an aligned quadrupole

quadrupole rotation + solenoid rotation

Solenoid emittance growth

D.H. Dowell, US Particle Accelerator School, June 14‐18, 2010, Boston, MA.

Relativistic paraxial ray equation1. write the equations of motions in cylindrical coordinates2. consider axially symmetric fields; expand fields in power series, keep the first order3. find the electric and magnetic field relations at the first order, and substitute in eq of motion

change in slope of particle trajectory

“adiabatic damping” oftransverse particle angles

focusing/defocusing oftransverse electric field

static magnetic field centrifugal force related to a non zeroangular momentum

if we introduce the Larmor frame which is a rotating frame with angular velocity ωL=dϑ/dt, then we have pϑ=0. We can then solve the equation without the last term, and then calculate the particle angle respect to the stationary frame:

Transverse envelope equation w/o space charge

r r

2 (kr

' '22 )r

n2

2 2r3 0

We now want to write the equation for the beam envelope. It can be derived from theparaxial equation (single particle) by averaging over the distribution:

multiply by rand average

Using the relations:

The envelope equation describes the evolution of transverse electron beam momentsin hypothesis of linear fields. In the same hypothesis later we will add a term coming fromself field forces. The emittance enters in the equation as a source of pressure.

lecture musumeci a2 - slac conference services | · pdf file · 2013-07-22preferred...

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