2.5 Numerical results
2.5.4 CSR in a wiggler
2.5 Numerical results
R0 = 100 m and magnetic field wavelength λw = 2π/kw = 1 m, and the number of periods Nu = 10. The chamber width and height are set to be a = 10 cm and b = 2 cm. The results are shown in Figs. 2.22(a) and 2.22(b). The wake potentials with rms bunch length of 0.5 mm corresponding to the impedances are plotted in Fig. 2.22(c). It turns out that the CWR with chamber shielding can differ remarkably from the free-space model.
By enlarging the chamber height, the shielding effect of up- and down-side chamber walls can be tested. This is demonstrated in Figs. 2.23. In these
calcu-0 2 4 6 8 10
0 5 10 15
kHmm-1L
ReZHWL
(a) Real part
0 2 4 6 8 10
-5 0 5 10 15 20
kHmm-1L
ImZHWL
(b) Imaginary part
-5 0 5 10 15 20
-0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6
z (mm) -W°HVpCL
(c) Wake potential
Figure 2.23: CSR impedance and wake potential of a wiggler withb= 2,5,10 cm.
The gaussian bunch length σz = 0.5 mm with bunch head to the left side. Blue solid lines: b= 2 cm; red dashed lines: b = 5 cm; green dashed lines: b= 10 cm;
black solid lines: free-space model.
lations, the chamber height is varied as b = 2,5,10 cm and all other parameters are kept the same as in the previous example. Both the impedance and wake potential tend to be close to that of free-space model while the chamber height is enlarged.
2.5 Numerical results
Next the effect of wiggler length is tested. This is demonstrated in Figs. 2.24 and 2.24(c). In these calculations, the chamber height is set to beb= 10 cm and the number of periods, which correspond to total length of the wiggler, is varied as Nu = 1,4,8. All other parameters are kept the same as in the previous examples.
It is seen that as the wiggler becomes longer, the impedance spectrum changes from smooth curve to that with many narrow peaks at low frequency part. And the narrow peaks in impedance cause fluctuations in the wake potential at the tail parts.
0 2 4 6 8 10
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
kHmm-1L
ReZHWL
(a) Real part
0 2 4 6 8 10
0.0 0.5 1.0 1.5 2.0
kHmm-1L
ImZHWL
(b) Imaginary part
-5 0 5 10 15 20
-0.08 -0.06 -0.04 -0.02 0.00 0.02 0.04 0.06
z (mm) -W°HVpCL
(c) Wake potential
Figure 2.24: CSR impedance and wake potential of a wiggler with the total length varied by setting Nu = 1,4,8. The impedances and wake potentials have been normalized by the number of period for convenience of comparison. The gaussian bunch length σz = 0.5 mm with bunch head to the left side. Blue solid lines: Nu = 1; red dashed lines: Nu = 4; green dashed lines: Nu = 8; black solid lines: free-space model.
In Ref. [89], the real part impedance in a rectangular chamber was calculated
analytically using mode expansion method ReZ(k) = 4Z0
abR20 X∞ m=0
X∞ p=1
k (1 +δm0)kz
sin2((k−kz−kw)Lw/2)
(k−kz)2−kw2 , (2.90) where kz =p
k2−α2mp with αmp = p
k2x+ky2, kx = mπ/a and ky =pπ/b. The quantity Lw =Nu(2π/kw) is the total length of the wiggler. The summation in Eq. (2.90) goes over the even values of m and the odd values of p. Equation (2.90) also indicates the resonance condition
k−kz−kw = 0, (2.91)
which shows that the resonant peaks in impedance should appear at kmp = α2mp+kw2
2kw
. (2.92)
The resonant peaks in Fig. 2.22(a) do agree with the above equation. The impedance for a wiggler in free space is given in Ref. [90]
Z(k) = 1
4Z0Lwkkw
k0
1− 2i
π
log4k k0
+γE
, (2.93)
where k0 = 4kw3R20 is the fundamental radiation wavenumber with the wiggler parameter K ≫ 1. The quantity γE ≈ 0.577 is the Euler constant. One sees that the numerical results again agree well with the analytic formula Eq. (2.90) but disagree with Eq. (2.93). Specially, the imaginary part does not show the property of linear slope in the limit of low frequency which is predicted by the free space model.
In the last demonstration, the CSR impedance of one wiggler section in the SuperKEKB low energy ring was calculated. The wiggler field is approximated by a series of short hard-edge bending magnets with opposite polarity interleaved with drift chambers. The cross-section of the chamber is square with the size of 90 mm. The wiggler section consists of 15 identical super-periods and its total length is 141.4 m. The bending radius as function of s for one super-period is plotted in Fig. 2.25. The calculated impedance is shown in Figs. 2.26(a) and 2.26(b).
The wake potentials with rms bunch length of 0.5 mm corresponding to the impedances are plotted in Fig. 2.26(c).
2.5 Numerical results
0 2 4 6 8
-20 -10 0 10 20
sHmL
RHmL
Figure 2.25: Bending radius as function of sfor one super-period in the wiggler section of SuperKEKB low energy ring.
0 2 4 6 8 10
0 2 4 6 8 10 12 14
kHmm-1L
ReZHkWL
(a) Real part
0 2 4 6 8 10
-5 0 5 10
kHmm-1L
ImZHkWL
(b) Imaginary part
0 5 10 15 20
-200 -100 0 100
z (mm) -W°HVpCL
(c) Wake potential
Figure 2.26: CSR impedance and wake potential of the wiggler section in the SuperKEKB low energy ring. The gaussian bunch lengthσz = 0.5 mm with bunch head to the left side. Blue solid lines: 15 periods; red dashed lines: 1 super-period. For convenience of comparison, the values of red dashed curves have been timed by a factor of 15.
3
Coherent wiggler radiation
3.1 Introduction
In the main rings of KEKB and SuperKEKB, wigglers are used to shorten the damping time. In a damping ring where wigglers are used for main pur-pose of radiation damping, coherent radiation in the wigglers can also con-tribute impedance. The impedance from undulator or wiggler was first studied in Refs. [91, 92]. Simple formula was found for an infinite long wiggler in free space [90] and applied to the instability analysis in a storage ring [93].
In the light-source community, usually one is more interested in the radiation power emitted from accelerated particle in a wiggler or an undulator. A wiggler is characterized byK ≫1 and typically an undulator has the feature ofK ≪1 [19].
The analysis of radiation is usually based on Li´enard-Wiechert potentials [19]
derived in free space. In the presence of uniform waveguide, it was treated using mode expansion methods, which is intimately related to Green’s function (for examples, see Refs. [94, 95, 96, 97, 98, 99, 100]). The radiation in finite-length undulator was also studied based on Green’s function method [91, 101]. Undulator radiation fields and impedance in the presence of round waveguide were calculated using paraxial Green’s functions in Refs. [102, 103].
In Chapter 2, the coherent wiggler radiation (CWR) impedance was calcu-lated using numerical code CSRZ. Therein, the chamber was approximated to be wiggling in the horizontal plane. In practice, the general theory based on dyadic Green’s functions was applicable to this problem [91, 94]. Using dyadic Green’s
functions, a general expression for the electric fields excited by a single electron, including both space-charge and radiation fields, can be obtained. This is exactly the main goal of this chapter.
The rest of this chapter is organized as follows. In Section 3.2, general the-orems and formulas are discussed for purpose of preparation. For a test of the eigenfunction method, the space-charge fields and impedance is treated in Sec-tion 3.3. The main part of this chapter, SecSec-tion 3.4, is devoted to an attempt to find the full solution for the CWR fields and impedance.