Lifetime increase using passive harmonic cavities in synchrotron light sources ∗

eScholarship provides open access, scholarly publishing services to the University of and delivers a dynamic research platform to scholars worldwide. Abstract Harmonic cavities have been used in storage rings to increase beam lifetime and Landau damping by lengthening the bunch. The need for lifetime increase is particularly great in the present generation of low to medium energy synchrotron light sources where the small transverse beam sizes lead to relatively short lifetimes from large–angle intrabeam (Touschek) scattering. We review the beam dynamics of harmonic radiofrequency (RF) systems and discuss optimization of the beam lifetime using passive harmonic cavities.

elastic collisions of electrons within the bunch have a finite probability of transferring enough longitudinal momentum to each electron such that they no longer are within the momentum acceptance of the storage ring and are lost. This process is particularly important for storage rings such as the Advanced Light Source (ALS) because of the high density of electrons resulting from the small transverse beam size.
To improve the Touschek lifetime one can increase the momentum acceptance and/or lower the bunch charge density. The momentum acceptance is usually limited by the storage ring lattice and practically cannot be greatly improved. The vertical beam density can be varied by increasing the betatron coupling, as is currently done in the ALS, or vertical betatron oscillations can be excited, effectively diffusing the vertical beam distribution and increasing the beam size [1]. However, both of these reduce the photon beam brightness.
The longitudinal density can be increased by exciting coherent synchrotron oscillations [2].
However, this increases the average beam energy spread, which is undesirable for its effect on undulator harmonics. The longitudinal charge density can also be decreased by defocussing the bunch at its center using a harmonic RF system.
A particularly attractive option for improving the lifetime is to lengthen the bunches using a passive harmonic cavity. The voltage in a passive cavity is generated by the beam itself and thus an external RF source is not required. Because the harmonic cavities do not accelerate the beam, there are interesting arguments for the use of either normal or superconducting cavities. However, the main difficulty of a passive harmonic system is that the effect can vary greatly with total beam current. This paper presents the general considerations for optimizing the beam lifetime using passive harmonic cavities, with the ALS as an example. An introduction to harmonic RF systems and a review of the longitudinal beam dynamics with a harmonic RF voltage are given in Section II. We discuss optimization of the lifetime as a function of cavity tuning and beam current using either normal or superconducting cavities in Section III. Conclusions are given in Section IV.

II. INTRODUCTION TO HARMONIC CAVITIES
Consider the voltage seen by the bunch generated by the main RF system as shown in Figure 1. Near the bunch center, the restoring force of the RF voltage is approximately linear.
Given a gaussian energy spread, the resulting longitudinal distribution is also gaussian.
If another voltage is added to the main RF voltage with an amplitude and phase such that the slope at the bunch center is zero, the energy distribution is unaffected but the bunch lengthens and the peak charge density decreases and the lifetime improves. This also reduces the effect of intrabeam scattering which tends to increase the emittance. To achieve cancellation on every RF cycle, the frequency of the secondary voltage must be a higher harmonic of the main RF voltage.
A higher harmonic cavity has several other benefits to machine operation. When the phase of the harmonic voltage is adjusted such that the bunch lengthens, there is an increase in the spread of synchrotron frequencies within the bunch. This spread can help in damping coherent instabilities such as the longitudinal coupled bunch instabilities through Landau damping. For this reason, harmonic cavities are often called "Landau" cavities. The decrease in peak current and synchrotron tune spread is also useful in raising the threshold for single bunch instabilities. Another benefit is that the phase of the harmonic voltage can be adjusted such that the bunch is shortened. This mode of operation may be of interest to a select group of users for whom lifetime is not the primary concern.
To produce the harmonic voltage, a second RF cavity system is installed in the ring with a resonant frequency several times the main RF cavity. The voltage in the harmonic cavity is generated either by an external generator (i.e. active cavity) or by the beam itself (passive cavity). Both active and passive harmonic cavities have been used in other synchrotron light sources [4,6]. Typically the harmonic RF system is introduced in passive mode and subsequently upgraded to active mode. One of the limitations of operating in passive mode is that the optimum bunch lengthening conditions can only be reached at one value of the beam current. We present the required impedance to operate passively as well as the expected effect on bunch length as a function of beam current.

A. Longitudinal beam dynamics with a harmonic voltage
The equations of motion of a single electron are where z is the longitudinal coordinate of the electron, ε is the fractional energy deviation from a synchronous electron, α is the momentum compaction, and E is the energy. The combined voltage from the main and harmonic RF system is given by where k is the relative harmonic voltage to the main RF voltage, φ s is the synchronous phase, φ h is the relative harmonic phase, and n is the harmonic. U 0 is the energy loss per turn. Note that this includes any energy dissipated in the harmonic cavities should they be operated passively.
Neglecting collective effects, an electron bunch has a Gaussian distribution in energy due to the emission of synchrotron radiation. The longitudinal density distribution of the bunch is determined from the energy distribution in the potential well formed by the total RF voltage. The density distribution is given by [10,11] where ρ is a normalization constant such that ρ(z)dz = 1. The potential, Φ(z), is given by where C is the circumference.
Using the above, the longitudinal bunch distribution can be shaped by varying the relative amplitude and phase of the harmonic voltage and thus the potential. To lengthen the bunch, the harmonic amplitude and phase should be adjusted to cancel the slope of the main RF voltage at the bunch center. The main RF voltage is perfectly cancelled when Φ(z) = 0 and Φ (z) = 0. The harmonic voltage and phase at this condition are given by and The synchronous phase angle of the beam with respect to the main RF voltage, φ s , is given by The potential well and bunch distribution for the main RF voltage and with the harmonic  Short-range wake voltages generated by the broadband impedance of the ring also affect the bunch lengthening, particularly near optimum flattening of the potential well when the bunch shape is most sensitive to small perturbations. However, we expect these effects to be relatively small in modern light sources which operate with low current/bunch in multibunch mode and have low impedance vacuum chambers. Furthermore, as described in section III, the passive cavities cannot be operated near optimum bunch lengthening for most of a fill.
Therefore, we neglect these effects in this paper.
If the the harmonic voltage shown in Figure 1 were phased such that the slopes of the main and harmonic voltages added rather than cancelled, the bunch would be shortened.
This mode of operation may be of interest to some users where lifetime is a secondary concern.

B. Touschek lifetime
To calculate the improvement in Touschek lifetime, consider the expression for the Touschek loss rate [12] where vσ is the probability for scattering beyond the RF acceptance rf and ρ is the volume charge density of the bunch. The relative lifetime change due to the change in the inte- , so it is necessary to include the effect of the harmonic voltage on the RF acceptance, ε rf . The ratio of lifetimes with and without harmonic voltage can be found where ε hc and ρ hc are the RF acceptance and longitudinal density with the harmonic voltage.
Shown in Figure 3 are the separatrices for the harmonic voltage at optimum lengthening and the main RF voltage only indicating the small reduction in RF acceptance with the harmonic voltage. Assuming the momentum acceptance is given by the RF acceptance, this corresponds to a 6-7% reduction in the lifetime improvement that one expects from the ratio of the integral charge densities. This effect tends to increase for lower radiation loss/turn.

C. Passive cavity operation
In passive mode, the voltage to modify the longitudinal distribution is generated by the beam itself. The tuning of the cavity for the case for bunch lengthening is illustrated in Assuming equal filling of all RF buckets and a cavity bandwidth small compared with the RF frequency, the beam-induced voltage is given by where R s is the cavity shunt impedance. The harmonic cavity tuning angle, ψ h , is given by and the bunch form factor, F , is given by From inspection of Eqs. 1 and 3, the harmonic phase angle, φ h is related to the tuning angle, ψ h , by Thus the amplitude and phase of the harmonic voltage are adjusted by tuning the resonant frequency of the passive harmonic cavity. However, for a fixed cavity Q and shunt impedance, the amplitude and phase cannot be independently adjusted. Thus the optimum lengthening conditions can be achieved at only a single beam current. This is one of the primary limitations of passive operation.
The minimum shunt impedance to reach optimum flattening is given by Because the amount of required shunt impedance to flatten the bunch strongly influences the cost of the system, it is useful to discuss Eq. 15 further.  [13]. Using this cavity, we calculate that we can achieve optimum flattening using 3 cells at 320 mA. At a beam energy of 1.5 GeV, where the loss/turn is much smaller, we estimate we require 12 cells. Given the range of requirements, we installed 5 cells in the ALS. Results of the commissioning are described elsewhere [14].
Because the harmonic cavity is tuned above the RF beam harmonic, it can potentially excite the Robinson instability. This mechansism is well understood and has been discussed extensively in the literature [15]. For the studies in this paper, we have considered this instability and simply state that it is not of concern for the range of conditions we have studied.

III. LIFETIME OPTIMIZATION WITH PASSIVE CAVITIES
We define the optimal effect of the passive harmonic cavities as the maximization of the integrated current over the course of a fill. This section presents the results of numerical calculations of the lifetime with the aim of finding the operational conditions which optimize the cavities. We use the nominal ALS parameters at 1.9 GeV with a main RF voltage of 1.1 MV and assume an ideal case of a symmetric fill pattern. We also assume that the lifetime is Touschek-dominated. The result is not as trivial as it seems because of the strong dependence the bunch length and lifetime on beam current, especially when operating the cavities near optimum bunch lengthening. Although the specific results described below are not directly applicable to other rings, we believe that the general trends are relevant.
Although the ALS design uses a normal conducting (NC) cavity, we also present results for a superconducting (SC) cavity. The beam current is assumed to vary from 0.4 to 0.2 A over a fill. For the NC cavity, we use 3 ALS cells with R/Q = 80.4Ω and Q = 21000 [13] with the restriction that the power dissipated in each cell is less than 10 kW. For the SC cavity, we use a single cell with the design parameters of the harmonic cavity currently under construction at Elettra with R/Q = 87Ω and Q = 1.5e8 [16].

A. Normal conducting cavity
Shown in Figure 5 is a density plot of the lifetime improvement as a function of HC detuning and beam current. There is a clear maximum in the lifetime when the bunch is near its maximum length. The maximum occurs at a different detuning for each beam current, implying that one can maximize the integrated effect over a fill by varying the detuning as a function of beam current. Note that the maximum lifetime occurs at the theoretical maximum because at some current, Eq. 15 is satisfied.
Although there are many possible operating scenarios to optimize the lifetime performance over a fill, we explore the two simplest ones here. The first is to keep the cavities detuned a fixed amount as a function of beam current and the second is to detune the cavities as to maintain a constant cell voltage. Figure 6 shows the increase in integrated current relative to the case of no bunch lengthening for these two cases. It is clear that the fixed voltage realizes a substantial improvement over a fixed detuning. However, it may be more practical to operate with a fixed detuning for practical reasons such as mechanical limitations of tuners and tuning of cavity higher order modes.
One of the potential disadvantages of the passive cavity is that the bunch shape and synchronous phase are not kept constant over a fill. This is demonstrated in Fig. 7 which shows the evolution of bunch shape and position relative to the main RF phase over a fill with the cavites for the two cases described above. For the more optimal case of a fixed voltage, when the beam current is not near the optimum condition, the bunch position slips either away from the nominal synchronous phase. This may be a nuisance to some synchrotron radiation experiments which use the time structure of the bunch in the experiments. Fortunately, the phase slip occurs slowly over the course of a fill. There is much less change in the bunch shape and phase over a fill for the case of the fixed detuning. The conclusion is that when operating passive harmonic cavities closer to maximum bunch lengthening, the bunch shape and phase are much more sensitive to the beam current and can vary significantly.

B. Superconducting cavity
Shown in Figure 8 is a display of the lifetime improvement as a function of detuning and beam current for a single SC cell. The result is resembles those for the NC cavity but has several significant differences. First, the SC cavity must be tuned much closer to the beam harmonic to generate any voltage due to the narrow cavity bandwidth. Second, the maximum lifetime improvement is only about 60% of the improvement for the NC cavity.
This results from the fact that Eq. 15 is not well satisfied for the SC parameters. For example, the ideal harmonic phase for ALS conditions is about 85 degrees. Again, because of the narrow cavity bandwidth, it is difficult to achieve this phase and the ideal voltage at the same time. The third difference is the ability of the SC to lengthen the bunch at low beam currents due to the high impedance. This may be very useful to increase the lifetime in a single bunch mode at high current/bunch.
The increase in integrated current for the case of fixed detuning and fixed voltage are shown in Figure 9. The improvement at the optimum detuning or voltage is not as much as in the case of NC cavities for the same reason stated above. However, a calculation of the bunch shape and phase vs. current for either case shows very little change. Again, this is because the SC cavities are operated even further from maximum bunch lengthening.
One significant advantage of the SC harmonic cavity is that it has sufficient impedance to generate enough voltage to get similar lifetime improvements even at relatively low beam currents. This is not useful for low current multibunch mode since the Touschek lifetime is very long but rather for a filling mode with high current/bunch with low total current. The operational mode has become popular for a certain class of experiments at light sources which require large gaps between light pulses but unfortunately suffers from very poor lifetime.
Robinson stability should be analyzed for the type of operation.

IV. DISCUSSION AND CONCLUSIONS
We have reviewed the dynamics of higher harmonic RF systems in electron storage rings and its effect on the Touschek lifetime. We also presented the basic operational parameters of a passive harmonic system. The effects of the passive RF system were calculated over the range of operating currents for the case of a normal conducting and superconducting passive cavity and the passive system can provide substantial improvements in lifetime and integrated current.
One of the main disadvantages of the passive system is that optimal lifetime increase occurs only over a narrow range of beam currents for the case of NC cavities and is never achieved for SC cavities. When operated near maximum bunch lengthening, there is significant variation of the bunch shape and synchronous phase over the course of a fill. The harmonic cavities show better performance when detuned to maintain a constant voltage compared to fixed detuning. However, this may not be a practical limitation since it may be difficult for practical reasons such as voltage and tuner limitations to operate at optimum condition.
We have ignored the effects beam loading transients which will be addressed in a separate study. The authors would like to thank M. Eriksson, A. Hoffman, G. Lambertson,Å.
Andersson, and N. Towne for many useful discussions.