new results in theory of csr wakefields

New results in theory of CSR wakefields

Gennady Stupakov

TWIICE 2 Workshop, February 8, 2016

1

Acknowledgements

This work has been done in collaboration with Demin Zhou from KEK. Itis documented at SLAC-PUB-16459 (arXiv:1601.04008).

2

Outline of the talk

Introduction and motivation

Analytical models of CSR wake

Divergence of the wake for a short bend and transition radiation

Integrated CSR impedance Z(k) and parallel-plates shielding model

CSRZ computer code

New results for CSR wake of a kink, bend and an undulator withparallel-plates shielding

Conclusions

3

Motivation

There are several analytical models of CSR wakefields.The simplest one is the CSR wake for circular motion in free space1. Thetheory assumes

1 Free space

2 Ultra-relativistic beam, v = c or γ = ∞.

3 Filament beam, σ⊥ = 0.

From 2 and 3 it follows that the beam longitudinal profile is frozen anddoes not change with time (no bunch compression).

θThe wake is determined bythe radiation at angles θr ∼

(λ/ρ)1/3 where ρ is the bend-ing radius, λ = λ/2π ∼ σz.

1Derbenev, Rossbach, Saldin and Shiltsev, DESY FEL Report TESLA-FEL 95-05, (1995); Murphy, Krinsky, and

Gluckstern, Part. Accel. 57, 9 (1997).

4

CSR for a circular orbit in free space

- - -

/σ

σ/ρ/

Positive wake corresponds tothe energy loss, negativewake–energy gain.

This plot is universal, all the parameters of the problem (σz—the bunchlength, ρ—the bending radius) are in the scalings.

Formation length of the wake: l‖ ∼ (24ρ2σz)1/3 ∼ λ/θ2r .

CSR impedance per unit length

Zvac(k) = (1.63+ 0.94i)k1/3

cρ2/3

k = ω/c.5

CSR for circular orbit with shielding by parallel plates

2hCalculated in the paper by Murphy etal.2 The same assumptions as beforeexcept for the free space.

-

-

ρ(/ρ)/

//

Plot of ReZ/ReZvac (red) andImZ/ImZvac (blue). The wakegets suppressed when h . l⊥ =(λ2ρ)1/3 ∼ λ/θr—the transverse co-herence size.

Using this impedance we can calculate the wake of a bunch with givenlongitudinal charge distrbitution.

2J. B. Murphy, S. Krinsky, and R. L. Gluckstern, Part. Accel. 57, 9 (1997).

6

Bend of finite length in free space

The wake was derived in3, the limit v = c was obtained in4. The v = c option isimplemented in the computer code elegant.

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Z [m]

−⟨∆E⟩

and

⟨(∆E−

⟨∆E⟩

)2 ⟩1/

2 [MeV

]

⟨(∆E−⟨∆E⟩)2⟩1/2

−⟨∆E⟩

bend magnet

L 0 = (2

4σsR

2 )1/3

Radiation induced energy loss andenergy spread: L = 50 cm, ρ = 1.5 m,σz = 50 µm, Q = 1 nC, E = 150 MeV.

3E. L. Saldin, E. A. Schneidmiller, and M. V. Yurkov, Nuclear Instruments and Methods A398, 373 (1997).

4G. Stupakov and P. Emma, PAC 2002, p. 1479 (2002).

7

Bend of finite length in free space

It has not been realized until recently that the free space model, in thelimit v = c, gives an infinite 〈∆E〉 (and 〈(∆E− 〈∆E〉)2〉1/2) if integratedto z = ∞. This is an essential drawback of the model that makes theresults of the calculations dependent on how much space is allocated forthe trajectory after the bend.

One of the goals of this work is to understand the origin of thisdivergence and to develop a model that gives a finite integrated wakefieldfor the bunch. We want to derive formulas for the integrated impedanceZ(k) that related the energy change ∆E(z) inside a bunch withlongitudinal distribution λ(z) after passing through the system:

∆E(z) = −e2c

∫∞−∞ dkZ(k)λ(k)eikz

where λ(k) = 12π

∫∞−∞ dz ′e−ikz ′λ(z ′).

8

Edge radiation and infinite total impedance

The physical mechanism behind the divergence of the integrated wakeand impedance is the edge radiation:

θ

θ

The intensity of the edge radia-tion per unit frequencyω per unitsolid angle Ω in free space (fromone edge)

dP

dωdΩ∝ θ2

(θ2 + γ−2)2

It is localized at angles θ ∼ 1/γ and the total spectral energy is

dP

dω=

∫dΩ

dP

dωdΩ∝

∫π/20

2πθdθθ2

(θ2 + γ−2)2∝ lnγ

In the limit γ→ ∞ the radiation power dPdω → ∞, and ReZ ∝ dP

dω .

9

What makes the impedance finite in reality?

Two mechanisms make the impedance finite: the finite γ and theshielding by metal walls of the vacuum chamber.The edge radiation in free space is localized at angles θr ∼ 1/γ. In avacuum chamber of transverse size a, the modes with k⊥ . 1/a do notpropagate, so the angles of radiation cannot be smaller thank⊥/k ∼ 1/ka. Hence, in comparison with finite γ, shielding plays adominant role if

1

ka∼σz

a&1

γ

Numerical example: σz = 100 µm, a = 2 cm,

σz

a=100 µm

2 cm=

1

200

hence for γ & 200 the shielding is more important.In this work we assume that the shielding is dominant,

ka γ

and take the limit γ→ ∞.10

Free-space CSR impedance

We first derive an expression for theintegrated radiation impedance Z(k)using retarded potentials5 with as-sumptions 1-4 (that is in free space).The integration goes along the beamtrajectory, s is the path length

β()

β()

τ=|()-()|

Z(k) =ik

c2

∫∞−∞ ds

∫∞−∞

ds ′

τ(s, s ′)(1− β(s) · β(s ′))eik(cτ(s,s

′)−s+s ′)

5L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields (Pergamon, London, 1979), 4th ed.

11



β()

β()

τ=|()-()|

Z(k) =ik

c2

∫∞−∞ ds

∫s−∞

ds ′

τ(s, s ′)(1− β(s) · β(s ′))eik(cτ(s,s

′)−s+s ′)

We know that radiation propagates ahead of the source, so we can replace theupper limit in the integral by s (this approximation is valid for short bunches,σz ρ).


11



β()

β()

τ=|()-()|

Z(k) =ik

c2

∫∞−∞ ds

∫s−∞

ds ′

τ(s, s ′)(1− β(s) · β(s ′))eik(cτ(s,s

′)−s+s ′)


The integral converges only if |β(s)|, |β(s ′)| = 1, because of the singularity ats→ s ′.


11



β()

β()

τ=|()-()|

Z(k) =ik

c2

∫∞−∞ ds

∫s−∞

ds ′

τ(s, s ′)(1− β(s) · β(s ′))eik(cτ(s,s

′)−s+s ′)


The integral converges only if |β(s)|, |β(s ′)| = 1, because of the singularity ats→ s ′.

Analysis shows that, unfortunately, the integral diverges at s→ ±∞ if the orbithas half-infinite straight lines (like in the case of finite bend).


11

Parallel-plates model

Using method of images and v = c

Z(k) =ik

c2

∫∞−∞ ds

∫s−∞ ds

′∞∑

m=−∞(−1)m1− β(s) · β(s ′)τm(s, s ′)

e−ik(s−s′−cτm(s,s ′))

τm = |r(s) − r(s ′) +may|

In principle, this formula allows to compute the impedance with trajectories thatare straight lines at ±∞. The summation should be carried out first, before theintegration. However, a better convergence is achieved through rearranging thesummation.

12

Carrying out summation over images

Z(k) = −2πk

ac

∞∑p=0

∫∞−∞ ds

∫s−∞ ds

′H(1)0

(ckτ(s, s ′)

√1−

(2p+ 1)2π2

k2a2

)× (1− β(s) · β(s ′))e−ik(s−s

′)

where H(1)0 (x) is the Hankel function. Only small number of lower values of p is

needed in practical calculations. Assuming that: a) the argument of H(1)0 (x) is

large (σz ρ), b) ka 1 (or σz a), and c) the orbit has a small angle withthe z axis (small deflection angle) we can simplify

Z(k) = (i− 1)2√πk

ac

∞∑p=0

∫∞−∞ dz

∫z−∞ dz

′ 1− β(z) · β(z ′)√z− z ′

× exp

(−i(z− z ′)

(2p+ 1)2π2

2ka2+ ik(cτ(z, z ′) − s(z) + s(z ′))

]The recipe now is: for a given orbit find s(z), τ(z, z ′) and β(z) and calculatethe integrals.

13

Wakes known in literature and new wakes

With this approach we easily reproduce CSR wakes known in theliterature:

1. Circular orbit in free space

2. Infinitely long wiggler with K 1 in free space6

3. Circular orbit between two parallel metallic plates

(no need for image charges in cases 1 and 2).New results (with parallel-plates shielding)

Impedance of a kink

CSR impedance of a bend of finite length

Impedance of a finite length undulator

6J. Wu, T. Raubenheimer, and G. Stupakov, Phys. Rev. ST Accel. Beams 6, 040701 (2003).

14

Comparison with CSRZ code

We compared our analytical results with CSRZ code7. The code solves theparabolic equation in the frequency domain in a curvilinear coordinate systemx, y, s,

∂E⊥∂s

=i

2k

(∇2⊥E⊥ − 4πe∇⊥ n0 +

2k2x

ρ(s)E⊥

)where E⊥ = (Ex, Ey) is the transverse electric field and k is the wavenumber.The radius of curvature of the reference orbit ρ(s) is allowed to arbitrary varyalong s. The parabolic equation is valid in the limit of high frequencies (shortbunches).Within the paraxial approximation, the longitudinal electric field can beapproximated by

Es =i

k

(∇⊥ · E⊥ −

4π

cjs

), Z‖(k) = −

1

Q

∫∞−∞ Es(xc, yc, s)ds

where js = enc is the current density, (xc, yc) denotes the center of the beamin the transverse plane and Q is the charge of the beam.

7D. Zhou et al., Japanese Journal of Applied Physics 51(1R), 016401 (2012).

15

Comparison with CSRZ code

The boundary conditions for thefield correspond to a metal sur-face of a rectangular cross sectionwith a given aspect ratio b/a.

We set the vertical dimension of the vacuum chamber in the code equalto the gap between the plates. To suppress the effect of the vertical wallsof the chamber, we choose a large aspect ratio b/a which shifts thevertical walls further away from the beam orbit. We found that a goodagreement with the parallel plates model can be achieved if the aspectratio b/a & 3.

16

Impedance of a kink

“Kink” is an orbit of a short magnet,such that the length of the magnetcan be neglected, but the deflectionangle θ0 is finite.We assume θ0 1.

θ

Radiation of a point charge moving on such an orbit can be related to thelow-frequency limit of the bremsstrahlung radiation; it is studied in detail inclassical8 and quantum9 electrodynamics.

CSR impedance if a kink is given by the analytical formula:

Z =Z0

2π

[ψ(0)

(1

2+ikaθ0

4π

)+ψ(0)

(1

2−ikaθ0

4π

)− 2ψ(0)

(1

2

)]with ψ(0)(x) = Γ ′(x)/Γ(x) the polygamma function of order zero and Γ(x) thegamma function. One can show that ImZ = 0.

8J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1999), 3rd ed.

9M. E. Peskin and D. V. Schroeder, An introduction to quantum field theory, (Westview Press Reading (Mass.), Boulder

(Colo.), 1995).

17

Impedance of a kink

Plot of the impedanceof a kink. In the limitk→ ∞, Z ∝ lnk.

Comparison with CSRZ (solid lines—thecode, dots—theory). The code com-putes a short bending magnet, L = 1cm, ρ = 1 m; a = 2 cm with the aspectratio b/a = 5. Analytical calculationsused the same a = 2 cm and the bend-ing angle θ0 = L/ρ = 0.01. The lastpoint on the plot corresponds to the di-mensionless parameter w = kaθ0 = 2.

(-)

()

18

θ

/

Bending magnet of finite length with shielding

We assume θ0 1. There are 4 regions of integration that should betreated separately.

)θ

)

)

)

19


Four contributions to the impedance, Z = Z1 + Z2 + Z3 + Z4, with each onebeing a sum of 1D integrals. They are numerically computed in Mathematica

Z1(k) = (i− 1)27/23

√Q

c√π

∞∑p=0

∫ l0

dξ (l− ξ)ξ3/2 exp[−iξ3 − iξQ(2p+ 1)2

]

Z2(k) = (i− 1)23/2√π

kacθ0e−iu/3

∞∑p=0

∫uZ

0

dτe−iτF

((τ+

1

2u

)2, q, τ+ u

)

Z3(k) = (i− 1)23/2√π

d2kacθ0

∞∑p=0

∫u0

τ2dτ exp

(−iτ3

3d2

)F

(τ4

4d2, q, τ

)

Z4(k) = (i− 1)23/2√π

d2kacθ0e−iu/3

∞∑p=0

∫u0

τ2dτ exp

(iτ+ i

(u− τ)3

3d2− iτ2

d

)

×[F

(τ2

4d2, q, τ

)− F

(τ2

4d2, q, τ+ uZ

)]

F(a, b, c) =√πe−2

√ab

2√ib

[erf(√

−ia−√ibc√

c

)+ e4

√aberfc

(√−ia+

√ibc√

c

)+ 1]

20


Impedance of a bend and comparison with CSRZ (solid lines—the code,dots—theory).

-

(-)

()

(-)

()

Left pane: bending magnet with L = 20 cm, ρ = 5 m, a = 2 cm, b/a = 5; rightpane: L = 55 cm, ρ = 12.94 m, a = 2 cm, b/a = 4. The second magnet hasparameters of the magnets in BC2 of the LSLS-II FEL.

21

Practical application: radiation power from LCLS-II BC2

θ = 42.5 mrad

LB = 0.55 m

L = 2.4 m

10 cm

BC2 bend #4

cooled absorber? (~100 W)

CSR Trap for BC2 ?

Q = 300 pC EBC2 = 1.6 GeV σz = 0.025 mm fmax = 0.929 MHz σx = 0.020 mm (at BC2 4th bend exit) σy = 0.16 mm (at BC2 4th bend exit)

CM16 BC2

CYC21 CXC21 CYC22 CXC22 CYC23 CXC23

collimator full gaps = 2.8 mm for all

z = 0 30.7 m 36.72m 42.2 m 48.2 m 54.2 m 60.2 m 78.1 m

Bending radius ρ = 12.9 m.22

Cartoon by P. Emma.

Practical application: radiation power from LCLS-II BC2

Integrated CSR wakefield for the last magnet of LCLS-II: L = 55 cm,ρ = 10 m, a = 3.2 cm.

- -

-

-

-

(μ)

(/)

The radiation power coming out from the magnet: 80 W.

23

Radiation impedance of a wiggler

Consider a plane wiggler with Nwperiods, the period length λw andthe undulator parameter K 1.We introduce θ0 = K/γ 1 andkw = 2π/λw.

Z(k) in this case is a sum of three terms: Z = Z1 + Z2 + Z3 with eachterm being a sum of 1D integrals.

(-)

()

The wiggler has one period,Nw = 1,the period length λw = 1 m and theangle θ0 = 1.6 × 10−2. The gapa = 2 cm.

24

Wiggler of infinite length—comparison with CSRZWe calculated the radiation impedance for NSLS-II damping wigglers10:Nw = 70, λw = 10 cm, K = 16.8. The beam energy is 3 GeV,θ0 = 1.86× 10−3, a = 11.5 mm. In the simulations the horizontal size b = 3a.

-

-

-

-

-

(-)

()

-

-

(-)

()

The impedance is dominated by the resonances with the waveguide modes11

k− kw =

√k2 −

π2n2

a2−π2m2

b2

where n is an odd and m is an even number.10

http://www.bnl.gov/nsls2/project/PDR, BNL (2007).11

G. Stupakov and D. Zhou, Preprint SLAC-PUB-14332, SLAC (2010).

25

Comparison of the wake for the NSLS-II wiggler

Using the impedance we computed the wake for a short bunch σz = 0.5mm—black is analytical theory, blue is the CSRZ wake with b/a = 3 andmagenta is the CSRZ wake with b/a = 16.

- - - -

-

-

()

(/)

26

Summary and discussion

We developed a general theory that allows to compute theintegrated radiation wakefield for a wide range of geometries in themodel of parallel-plates shielding. The impedance is given by aninfinite sum of integrals along the beam orbit.

In calculation of CSR wakes for finite length systems the transitionradiation is important and sometimes is dominant.

The parallel plates shielding model is a good approximation forrectangular cross section with the aspect ratio > 3.

In future work, using the same approach, we plan to compute theenergy change of the beam accumulated from −∞ to the currentlocation, ∆E(z, s).It is very desirable to also include into theory the compressionwork—a reversible energy change of the beam due to variation of itslength and the transverse size.

27

new results in theory of csr wakefields

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