Theory of Microwave Instability and Coherent Synchrotron Radiation in Electron Storage Rings∗

FRXAA01 Proceedings of IPAC2011, San Sebastián, Spain

THEORY OF MICROWAVE INSTABILITY AND COHERENT SYNCHROTRON RADIATION IN ELECTRON STORAGE RINGS∗

Yunhai Cai† , SLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA

Abstract based on the linearized Vlasov equation of coasting beam. Here I = mc3/e = 17045 A is the Alfven current, Bursting of coherent synchrotron radiation has been ob- A α the momentum compaction factor, Zˆ(k) the impedance served and in fact used to generate THz radiation in many 2 −p /2 √ electron storage rings. In order to understand and control per unit length, and F0 = e / 2π. In general, the bursting, we return to the study of the microwave in- a = Ω/αckσδ, is a complex number and is to be solved stability. In this paper, we will report on the theoretical for a given value of k. One can see that there is a pole on understanding, including recent developments, of the mi- the real axis in the integral in Eq. (1). The correct treatment crowave instability in electron storage rings. The histori- of the pole leads to the Landau damping. Actually, one can cal progress of the theories will be surveyed, starting from ﬁrst evaluate the integral in the upper half plane and then the dispersion relation of coasting beams, to the work of analytically continue it into the lower half plane. The result Sacherer on a bunched beam, and ending with the Oide and of the integral is given by

Yokoya method of discretization. This theoretical survey 2 π −a /2 a will be supplemented with key experimental results over G(a)= 1+ ae (erfi( ) i). (2) − 2 √2 − the years. Finally, we will describe the recent theoretical r development of utilizing the Laguerre polynomials in the It is easy to see from the perturbation that the beam is un- presence of potential-well distortion. This self-consistent stable if Im[Ω] > 0. method will be applied to study the microwave instability driven the impedances due to the coherent synchrotron ra- CSR Impedance in Free Space diation. For electrons, orbiting on a circle with radius ρ inside bending magnets, the longitudinal wakefield due to the INTRODUCTION steady CSR in free space was given by Murphy, Krinsky, and Gluckstern[2] Over the past quarter century, there has been steady 4πρ1/3 progress toward smaller transverse emittances in electron W (z)= , (3) storage rings used for synchrotron light sources, from tens csr −(3z)4/3 of nm decades ago to the nm range recently. In contrast, for z > 0 and the wake vanishes when z < 0. Unlike a there is not much progress made in the longitudinal plane. conventional wake, the CSR force is acting on the electron For an electron bunch in a typical ring, its relative energy −3 ahead. Its corresponding impedance was actually found by spread σδ remains about 10 and its length σz is still in Faltens and Laslett[3] between 5 mm to 10 mm. Now the longitudinal emittance (σ σ ) becomes a factor of thousand larger than those in 2π Γ(2/3)(√3+ i) δ z Z (k) = ( ) (ρk)1/3, (4) the transverse dimensions. In this paper, we will address csr c 31/3 questions of: How short a bunch can be? What is the fun- where Γ is the Gamma function. damental limit? If there is a limit, is there any mitigation method? Since the synchrotron radiation is so fundamen- Microbunching tal in electron storage rings, let us start with the coherent synchrotron radiation (CSR). Using Zˆcsr(k) = Zcrs(k)/2πρ, Stupakov and Heifets[1] analyzed the dispersion relation of Eq. (1) and COASTING BEAM THEORY showed that the beam becomes unstable if 3/2 −iΩs/c+ikz kρ < 2.0Λ , (5) If one studies a perturbation Ψ1 = Ψ(δ)e for 2 2 an electron beam with an energy of E0 = γmc , a current where Λ = I/αγσδ IA. In this model, given a current, I, and a Gaussian distribution with a relative energy spread there is always an unstable mode when its wave number k σδ, one can derive (for example see[1]) is low enough. This stability condition was confirmed[4] experimen- cI Zˆ(k) ∞ dF /dp tally at the Advanced Light Source, where the evidence of 1= i ( ) 0 dp, (1) αγσ2I k p a microbunching in the bolometer signal was found when the δ A Z−∞ −

2011 by IPAC’11/EPS-AG — cc Creative Commons Attribution 3.0 (CC BY 3.0) bunch current ∗

c Work supported by the Department of Energy under Contract No.

○ 1/6 2 DE-AC02-76SF00515. π αγσδ IAσz † Ib > , (6) [email protected] √2ρ1/3λ2/3 05 Beam Dynamics and Electromagnetic Fields

Copyright 3774 D03 High Intensity in Circular Machines Proceedings of IPAC2011, San Sebastián, Spain FRXAA01 for the wavelengths at λ = 2 mm and λ = 3.2 mm at var- The scaled impedances written in Eqs. (8) and (10) are ious beam energies. This expression can be derived from plottedin Fig 1. As onecansee inthe ﬁgure, fortheparallel Eq. (5) provided, plate model, its real part becomes zero near kh3/2/ρ1/2 = 2 at the low end of frequency. Clearly, there is a strong I = √2πρIb/σz, (7) shielding effect at the long wavelength by the metal plates. At the end of short wavelength, its impedance is asymptot- which is a result of identifying the peak current of the ically approaching the impedance in free space as it should bunched beam with . I be.

CSR Impedance with Shielding Stability Analysis with Shielding The calculation of the exact impedance with the shield- Using the impedance defined in Eq. (8), we analyzed the ing from two parallel metal plates, separated by a dis- dispersion relation of Eq. (1) for various values of a scaled tance h, was carried out by Warnock[5] in terms of current, S = Ih/αγσ2I ρ, as shownin Fig. 2. The disper- the Bessel functions. Utilizing the uniform asymptotic δ A sion curves clearly show the shielding effect at the low end expansion[6] of the Bessel functions, one can approximate of kh3/2/ρ1/2. As one can see, the beam is stable for all the impedance with the Airy functions Ai and Bi, values of the wave number when S < 6/π. The threshold th 3 1/3 of the instability occurs at the current of S =6/π with ρ Z(n) 16π 2 h 3/2 −4/3 ( )( )csr p = [n( ) ] h n c ρ kth =5.7ρ1/2/h3/2. (11) [Ai′(u)Ci′(u)+ uAi(u)Ci(u)], (8) × p>=1,3,... Above the threshold, the lowest unstable wave number is X also proportional to ρ1/2/h3/2; However, its coefficient where n = kρ, Ci = Ai iBi, and u is defined as varies as a function of the current and is not a fixed value − as suggested in many previous publications. π2p2 h 3/2 −4/3 (9) u = 2/3 [n( ) ] . 2 ρ 2 Note that the dependency of n on the left side of the equa- 0 L

3/2 tion is all through n(h/ρ) . This kind of scaling law was Σ c Α 2 1 proposed as an approximatedproperty by Murphy, Krinsky, Ρ

H 2 2 3 h and Gluckstern[2]. In fact, one can show that Eq. (8), gives D @ Im the impedance that corresponds to their wakeﬁeld, which 4 was used in our recent simulations[7].

6 150 0 2 4 6 8 10

kh32Ρ12

100 Figure 2: Scaled imaginary part of the dispersion curves ] Ω 50 with three values of the scaled current S =1, 6/π, 3 in the color of blue, black, and red respectively.

/h)Z(n)/n [ 0 ρ ( real (parallel plates To compare the theory with the observations, we need to imag (parallel plates) −50 real (free space) use Eq. (7) to rewrite the instability condition, S > 6/π, in imag (free space) terms of the bunch current,

−100 0 5 10 15 20 √ 2 3/2 1/2 3 2αγσδ IAσz kh /ρ I > . (12) b π3/2h Figure 1: Scaled Z(n)/n for the CSR impedances in parallel Note that the threshold does not depend on ρ. plates and in free space.

Moreover, the same scaling property holds for the CSR Table 1: Measurements in the NSLS VUV Ring impedance in free space as well; formally Eq. (4) can be Parameter Measurement Theory rewritten as, Threshold wavelength, λth 7.0mm 6.9mm Threshold current, Ith 100mA 134mA ρ Z(n) 2π Γ(2/3)(√3+ i) b

( )( ) = ( ) 2011 by IPAC’11/EPS-AG — cc Creative Commons Attribution 3.0 (CC BY 3.0) h n csr c 31/3 c h A comparison of the theory, using Eqs. (11) and (12), ○ [n( )3/2]−2/3. (10) × ρ with the observation[8] in the VUV ring of the National 05 Beam Dynamics and Electromagnetic Fields

D03 High Intensity in Circular Machines 3775 Copyright FRXAA01 Proceedings of IPAC2011, San Sebastián, Spain

Synchrotron Light Sources at BNL is presented in Table 1. ∞ ∞ dK′G (K,K′)P (K′). (15) As one can see, the agreement in the wavelength of the un- l,m m m=−∞ 0 stable mode is excellent. The calculated threshold is 30% X Z higher than the measured value. Its kernel is given by

−(K+Vmin) BUNCHED BEAM THEORY ′ √2lInce Gl,m(K,K )= − √πκσz In a storage ring, electrons inside a bunched beam ex- ∞ Z(ν/σz) ∗ ′ ecute synchrotron oscillation at frequency fs = νsfrev, Im[ dν h (ν,K)h (ν,K )], (16) × ν l m where frev is the revolution frequency. The synchrotron Z0 tune νs is given by where 2π dφ −ilφ+iνq(φ,K) αfrf eVrf hl(ν,K)= e . (17) ν = ( )cos φ , (13) 0 2π s 2πf E s Z s rev 0 Here we have the normalized current

where frf is the RF frequency. The RF voltage Vrf is nec- reNb essary to compensate the energy loss U due to the syn- In = , (18) 0 2πν γσ chrotron radiation in every turn of the circulation, namely s δ U = eV sin φ . The equilibrium bunch distribution is a 2 2 0 rf s where re = e /mc is the classic radius of electron and Gaussian. The bunch length is given by[9] Nb the bunch population. It can be rewritten in terms of the bunch current Ib, σz = αcσδ/ωs, (14) σ I where ω =2πf . z b s s In = 2 , (19) αγσδ IA 0.5 ξ=0.1 using Eq. (14). κ, V , ω(K), and q(φ, K) can be deter- 0.45 ξ=0.3 min ξ 0.4 =0.5 mined by the underling Haissinski distribution. Since the Haissinski solution is a function of the current, they depend (q) 0.35 λ on I as well. 0.3 n Given a current I , one needs to solve P (K) for all l 0.25 n l along with Ω. When Im[Ω] > 0, the beam is unstable. One 0.2 method to solve the integral equations is to discretize[11] 0.15 Bunch Distribution: the variable K. Here we will present an alternative. 0.1 0.05 Polynomial Expansion 0 −8 −6 −4 −2 0 2 4 6 8 |l| q=z/σ z Using the generalized Laguerre polynomials Lα (K), we decompose

Figure 3: Haissinski distributions at various scaled cur- ∞ rents, ξ = 0.1, 0.3, 0.5, for the CSR in free space. The −K α (l) Pl(K)= e al fα (K), (20) head of the bunch is to the right. α=0 X When there are longitudinal wakefields in the storage where ring, the equilibrium becomes a Haissinski distribution[10] α! at a sufficiently low bunch current. For the wakefield driven f (l)(K)= K|l|/2L|l|(K). (21) by the CSR in free space in Eq. (3), the Haissinski dis- α ( l + α)! α s | | tributions are shown in Fig. 3. To study the stability at a higher current, one can make a small perturbation near Applying the orthogonal and normal condition of the La- the Haissinski distribution and then analyze the linearized guerre polynomials, we reduce the Sacherer integral equa- Vlasov equation for the perturbation. This leads to an set tions to a set of linear equations, of integral equations[11, 12]. Ω ∞ ∞ al = M α,βam, (22) ω α l,m β Sacherer Integral Equation s m=−∞ X βX=0 For all azimuthal mode number l = , ... , we have −∞ ∞ for l = , ... and α =0, ... .

2011 bythe IPAC’11/EPS-AG — cc Creative Commons Attribution 3.0integral (CC BY 3.0) equation −∞ ∞ ∞ Clearly, it becomes an eigen value problem. Ω/ωs is the c ○ Ω ω(K) eigen value. In fact, M is a real matrix. When the current ( l )Pl(K)= is small, all eigen values are real and therefore the beam is ωs − ωs 05 Beam Dynamics and Electromagnetic Fields

Copyright 3776 D03 High Intensity in Circular Machines Proceedings of IPAC2011, San Sebastián, Spain FRXAA01 stable. It becomes unstable, when the ﬁrst pair of complex 3 value emerges as the current increases. VFP simulation 2.5 Bunched beam theory The matrix elements are given by Coasting beam theory

αβ αβ αβ 2 M = l(δlmO C ) (23) 4/3 z lm (l) lm σ − / 1/3 1.5 ρ and n = I ξ ∞ 1 αβ ω(K) −K (l) (l) O(l) = dK e fα (K)fβ (K), (24) ω 0.5 Z0 s

0 0 1 2 3 4 5 6 √ −Vmin αβ 2Ince χ=σ ρ1/2/h3/2 Clm = z √πκσz ∞ Z(ν/σ ) ∗ th Im[ dν z gα(ν)gβ (ν)], (25) Figure 5: Threshold ξ as a function of the shield- × ν l m ing parameter χ. The circles are the result of the VFP Z0 simulation[7]. where ∞ α −K (l) The solid line in the figure is plotted using the formula, gl (ν)= dKe fα (K)hl(ν,K). (26) 2/3 3/2 Z0 ξ = 3√2χ /π . It can be derived from the coast- Now, the problem of analyzing the instability of bunched ing beam threshold in Eq. (12) using the definitions of ξ beam is reducedto first evaluating the integralsin Eqs. (17), and χ and Eq. (19). Clearly, the agreement between the (26), (24), and (25), and then finding the eigen values and simulation and the coasting beam theory is excellent when eigen vectors of the matrix M. χ > 2. Finally, the bunched beam theory confirms the dip at χ =0.25, seen first in the simulation. th Instability Driven by the CSR in Free Space Finally, for the theoretical ξ , one may simplify the result in the figure to For the impedance given by Eq. (4), a complete analysis th of the stability of the Haissinski distribution as a function ξ (χ)=0.5+0.34χ, (27) of the current was carried out[12]. As shown in Fig 4, the except the dip at χ = 0.25. This linear relation was first threshold of the instability is at ξth =0.482. obtained by fitting to the result of the VFP simulation[7].

0.05 0.04 COMPARISON TO THE 0.03 MEASUREMENTS 0.02

) Although the bursting phenomenonat THz was observed

s 0.01 ω /

Ω under various momentum compaction factors, RF voltages, 0 bunch lengths, and energies in many synchrotron light

Imag( −0.01 sources, its threshold current Ith satisﬁes a simple scaling −0.02 b law[13] with respect to the bunch length σ (in MKS units), −0.03 z

−0.04 2 7/3 c Z0 th 1/3 −0.05 σz = Ib ρ /(Vrf frf frev), (28) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 2πF 31/3 ξ=I ρ1/3/σ4/3 n z where F is a constant. Note that σz is the bunch length at zero current not at the threshold current. This scaling prop- Figure 4: Imaginary part of the eigen values for Ω/ωs as erty with F = 7.456 was derived[14] based on the coast- 1/3 4/3 a function of the scaled current, ξ = Inρ /σz , for the ing beam theory developed by Stupakov and Heifets[1]. It impedance driven by the CSR in free space. agrees very well with the BESSY II measurement[15]. A similar equation can be derived easily from the 1/3 4/3 bunched beam theory. Starting from ξ = Inρ /σz and Threshold of the CSR Instability with Shielding using Eqs. (19), (14), and (13), we obtain (in MKS units)

Similar analysis is carried out using the impedance de- 2 7/3 c Z0 1/3 ﬁned in Eq (8). Because of the additional parameter σz = Ibρ /(Vrf cos φsfrf frev). (29) h 8π2ξ and the scaling property in the impedance formula, the 2011 by IPAC’11/EPS-AG — cc Creative Commons Attribution 3.0 (CC BY 3.0) c threshold becomes a function of the shielding parameter, Here we have used Z0 =4π/c to change fromCGS units to ○ 1/2 3/2 χ = σzρ /h , as shown in Fig. 5 as the green stars. MKS units. So far, this is merelya general relation between 05 Beam Dynamics and Electromagnetic Fields

D03 High Intensity in Circular Machines 3777 Copyright FRXAA01 Proceedings of IPAC2011, San Sebastián, Spain

the scaled current ξ and the bunch current Ib. In particular, see, the agreement is not perfect but reasonably good con- applying it to the threshold, we have (in MKS units) sidering that only the CSR impedance is included in the theory. c2Z 7/3 0 th 1/3 (30) σz = 2 th Ib ρ /(Vrf cos φsfrf frev), 8π ξ (χ) CONCLUSION where ξth(χ) is given by Eq. (27), ignoring the dip. 1/2 3/2 For a long bunch, χ = σzρ /h > 2, the coast- For very short bunches, the shielding parameter is so ing beam theory with the shielding impedance works well. th small that one can use ξ = 0.5 in Eq. (30) to make Eq. (12) should be used for estimating the threshold. a comparison with the threshold measurement. Such a When a bunch is short, χ < 2, the bunched beam the- measurement[16] was carried out at different momentum ory should be applied. According to Eq. (30), the beam compaction factors with the same RF voltage at ANKA. becomes unstable if The result is shown in Fig. 6. As one can see, the agree- 2 th 7/3 ment between the theory and the measurement is excellent. 8π ξ (χ)σz Vrf cos φsfrf frev (32) Ib > 2 1/3 . c Z0ρ

0.4 A shorter bunch is always more unstable. However, it is 0.35 data much better to reduce the bunch length with an increase in theory 0.3 RF voltage than with a decrease of the momentum com- 0.25 paction factor.

0.2 (mA) b I 0.15 ACKNOWLEDGEMENT 0.1 I would like to thank my colleagues: Karl Bane, Alex 0.05 Chao, Gennady Stupakov, and Bob Warnock for many 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 σ 0 (mm) stimulating discussions and their insights on the subject. I am also very grateful for the experimental data provided by J. Corbett, M. Klein, I. Martin, A.S. Muller, F. Sannibale, Figure 6: (courtesy of M. Klein) Measured bursting thresh- and G. Wustefeld. th old as as function of σz at ANKA. ξ =0.5 in Eq. (30) is used for the solid curve. REFERENCES

In most storage rings, cos φs is nearly equal to 1. As a [1] G. Stupakov and S. Heifets, Phys. Rev. ST Accel. Beams 5, result, Eqs. (28) and (30) are almost identical provided 054402 (2002). [2] J.B. Murphy, S. Krinsky, and R.L. Gluckstern, Part. Accel. F =4πξth(χ)/31/3. (31) 57, 9 (1997). In contrast to the coasting beam theory, F is not a constant [3] A. Faltens and L.J. Laslett, Part. Accel. 4, 151 (1973). but rather a function of the shielding parameter χ. Indeed, [4] J. Byrd et al. , Phys. Rev. Lett. 89, 224801 (2002). there is some experimental evidence shown in Table 2 to [5] R.L. Warnock, SLAC-PUB-5379 November (1990). support this claim. [6] F.W.J. Olver, Asymptotics and Special Functions, Academic Press, New York, (1974). Table 2: Comparison of Theory with Measurements at Var- [7] K.L.F. Bane, Y. Cai, and G. Stupakov, Phys. Rev. ST Accel. ious Synchrotron Light Sources Beams 13, 104402 (2010). Machine F (meas.) F (theory) χ [8] G.L. Carr et at., Nucl. Instrum. Methods Phys. Res. Sect. A 463, 387 (2001). BESSYII 7.46 5.84 0.48 MLS 3.40 5.23 0.29 [9] M. Sands, SLAC Report No. 121, (1979). ANKA 4.36 5.58 0.42 [10] J. Haissinski, Nuovo Cimenta B 18, 72 (1973). Diamond 2.90 1.48 0.25 [11] K. Oide and K. Yokoya, KEK Report No. 90-10, (1990). [12] Y. Cai, Phys. Rev. ST. Accel. Beams 14, 061002 (2011). In the table, the measured values of F were provided [13] G. Wustefeld et al., Proceedsings of IPAC’10, Kyoto, Japan, by Wustefeld for BESSY II and MLS, Klein for ANKA, p. 2508 (2010). and Martin for Diamond. The theoretical values are cal- [14] F. Sannibale et al., Phys. Rev. Lett. 93, 094801 (2004). culated using Eqs. (31) and (27), except for the Diamond [15] J. Feikes et al., Proceedings of EPAC 2004, Lucerne, Light Source, for which ξ =0.17 is used for the dip. More- Switzerland, p. 1954 (2004). 2011 byover, IPAC’11/EPS-AG — cc Creative Commons Attribution 3.0 (CC BY 3.0) the measurements at Diamond were carried out with c [16] M. Klein et al., Proceedings of PAC09, Vancouver, BC, ○ the negative momentum compaction factors. Therefore, for Canada, p. 4761 (2009). that one, its threshold has to be recalculated. As one can 05 Beam Dynamics and Electromagnetic Fields