Linear theory of microwave instability in electron storage rings

The well-known Haissinski distribution provides a stable equilibrium of longitudinal beam distribution in electron storage rings below a threshold current. Yet, how to accurately determine this threshold, above which the Haissinski distribution becomes unstable, is not firmly established in theory. In this paper, we will show how to apply the Laguerre polynomials in an analysis of this stability that are associated with the potential-well distortion. Our approach provides an alternative to the discretization method proposed by Oide and Yokoya. Moreover, it reestablishes an essential connection to the theory of mode coupling originated by Sacherer. Our new and self-consistent method is applied to study the microwave instability driven by commonly known impedances, including coherent synchrotron radiation in free space.

Figure 1

Comparison of Haissinski distribution at the threshold current $I_n^{\mathrm{th}}=0.042\mathrm{pC}/\mathrm{V}$ in the SLC damping ring to a Gaussian distribution.

Figure 2

Wakefield (left) and impedance (right) for the SLC damping ring for ${\sigma }_z=4.946\mathrm{mm}$ (bunch length at zero current).

Figure 3

Spectrum functions at the threshold current $I_n^{\mathrm{th}}=0.042\mathrm{pC}/\mathrm{V}$ in the SLC damping ring. For the Gaussian model, their imaginary parts should be vanishing for the $l=2$ mode.

Figure 4

Tune shift (left) and growth rate (right) of the most unstable mode in the SLC damping ring.

Figure 5

Mode analysis using the Gaussian model. Solid lines represent the coherent tune shifts [$\mathrm{Re}(\Omega /{\omega }_s)$] and dashed lines are for the growth rates [$\mathrm{Im}(\Omega /{\omega }_s)$] of the unstable modes. For a given azimuthal mode number, only the radial modes with the largest and smallest tune shifts are shown.

Figure 6

Real parts (left) and imaginary parts (right) of all unstable modes driven by CSR in free space. Each dot represents an unstable mode.

Figure 7

Threshold of microwave instability as a function of number of radial modes expanding with the Laguerre polynomials.

Figure 8

Coupling of the radial modes: quadrupoles on the left and sextupoles on the right.

Figure 9

Growth rates of all modes at various currents using the discretization method for the CSR impedance in free space.

Figure 10

Double-peak distribution on the left and its corresponding double-well potential on the right for the $Q=1$ broadband resonance model with ${\nu }_r={\omega }_r{\sigma }_z/c=0.5$ and $\xi =I_{n}R{\omega }_r=18$.

Figure 11

Threshold as a function of ${\nu }_r={\omega }_r{\sigma }_z/c$ in the $Q=1$ broadband impedance model with various methods.

Figure 12

The potential well at the threshold current $I_n^{\mathrm{th}}=0.042\mathrm{pC}/\mathrm{V}$ in the SLC damping ring. Note that most distortion occurs near the bottom of the well.

Figure 13

Incoherent tune shift as a function of $K$ at the threshold current $I_n^{\mathrm{th}}=0.0420\mathrm{pC}/\mathrm{V}$ in the SLC damping ring.