Implications of beam filling patterns on the design of recirculating Energy Recovery Linacs

Recirculating energy recovery linacs are a promising technology for being able to deliver high power particle beams ($\sim$GW) while only requiring low power ($\sim$kW) RF sources. This is achieved by decelerating the used bunches and using the energy they deposit in the accelerating structures to accelerate new bunches. We present studies of the impact of the bunch train filling pattern on the performance of the accelerating RF system. We perform RF beam loading simulations under various noise levels and beam loading phases with different linac topologies. We also present a mathematical description of the RF system during the beam loading, which can identify optimal beam filling patterns under different conditions. The results of these studies have major implications for design constraints for future energy recovery linacs, by providing a quantitative metric for different machine designs and topologies.


A. Introduction into ERLs
There is an increasing interest in Energy Recovery Linacs worldwide due to their unique promise of combining the high-brightness electron beams available from conventional linacs with the high average powers available from storage rings. Applications requiring this step-change in capability are coming to the fore in a wide variety of fields, for example high energy particle physics colliders [1], high luminosity colliders for nuclear physics [2], free-electron laser drivers for academic and industrial purposes [3,4], and inverse Compton scattering sources [5,6].
Historically, an effective method to cost-optimise an electron linac (where beam dynamics restrictions allow) is to implement recirculation [7,8], i.e. accelerating the beam more than once within the same RF structures. Analogously, one may implement recirculation in an ERL, accelerating and decelerating within the same structures. This has been successfully demonstrated in the normal-conducting Novosibirsk infrared FEL [9]. There are a number of GeV scale user facilities proposed that are therefore based upon recirculating superconducting ERLs [1,10,11], and two test facilities are currently attempting such a multi-turn ERL demonstration [12,13].
It is thus timely to explore the implications of this relatively new accelerator class. Unlike a linac or storage ring, there is large number of degrees of freedom in the basic accelerator topology. For example one may choose a dogbone or racetrack layout, subsequent accelerating pass may be transported in common or separate beam transport, and decelerating passes may be transported pairwise with their equivalent accelerating beam in common or separate transport [14][15][16].
In this article we explore the consequence of these choices on the most important aspect of an ERL-based user facility, the RF stability. Specifically, we consider all possible beam filling patterns in an N-pass recirculating ERL and their interaction with the accelerator lowlevel RF control system. We show that there are optimal choices, and note which topologies allow these optima to be chosen.
It is vital that this analysis is performed during the design stage of an ERL-based facility as it fixes the pass-topass path length required in the recirculation transport at the scale of multiples of the fundamental RF wavelength, typically many metres, therefore any path length variability built in to allow pass-to-pass RF phase variation cannot correct for this macro scale requirement. Similarly, transverse phase advance manipulations that are capable of mitigating BBU thresholds [17] would not be effective against sub-optimal filling pattern generated instabilities.
We first introduce beam filling and beam loading patterns, and describe how they affect cavity voltage. We then describe an analytical model of beam loading and use this to make predictions about the system. The next section describes beam loading simulations while varying different parameters such as the signal-to-noise ratio (S/N ) and synchronous phase. We will expand these studies to more general topologies in the section IV and compare all the simulations results in the section V.

B. Filling patterns
In this article, we note that the topology of the recirculating ERL can impact the filling pattern or ordering of the bunches. We start with a simple recirculating ERL with single arc on two sides as shown in Fig. 1 and discuss more complex setup later on. The injected bunches in this simple ERL maintain their order until the extraction. As an example, We consider a 6-turn ERL with 3 acceleration and 3 deceleration turns. If all the bunches are injected in one turn, the cavity voltage will decrease or increase drastically, compared to the case where only one bunch is injected per turn. In the second scenario, the accelerating and decelerating bunches are alternated as shown in Fig. 2. Here, 3 decelerating bunches are followed by 3 accelerating ones. The acceleration takes energy from the cavity and thus decrease cavity voltage and vise versa. So, mixing accelerating bunches with decelerating bunches can minimize cavity voltage fluctuation. These 6 bunches form a bunch train. Bunch trains are repeated and fill up the ERL . During the operation, one bunch per train per turn is extracted and replaced by a new bunch.
Firstly, It is necessary to define various quantities for use in our analysis. The "bunch number" is the order in which bunches are injected into a bunch train over N turns, for example bunch 1 (or 1st bunch) is injected on turn 1, bunch 2 (or 2nd bunch) is injected on turn 2 and so on. We can give notation of filling pattern by describing which bunch goes to which RF bucket. The number indicates the bunch number and its position in the vector indicates the RF bucket number. Usually, not all the RF cycles are filled by bunches, but one bunch occupies N RF cycles. These N RF cycles is called a "RF bucket". In a N -turn ERL, 1 train occupies N RF buckets. The filling pattern of Fig. 2 is a a 6-element vector [1 2 3 4 5 6]. We will also use "pattern number" for brevity instead of 6-element vector indicated by "i" ranges from 1 to 120 for 6-turn ERL. The pattern number i is related to the 6-element vector V (i) as (1) As there are many trains in a ring, without losing the generality we can name RF bucket of the 1st bunch as the 1st bucket, i.e. the 1st bunch will always be in the 1st RF bucket. We can vary the order of the other 5 bunches to get different filling patterns.

C. Cavity voltage calculation
As the bunches pass through the linacs, they are either accelerated or decelerated by the RF field in the cavity. In doing so, energy is either put into or taken out of the cavity. The cavity voltage V cav is related to the stored energy U stored as with R Q being shunt impedance of the cavity divided by its Q-factor. For an accelerating cavity, the change in stored energy from a particle bunch passing through is Therefore, the change in cavity voltage from beam loading is given as where φ is the phase difference between the bunch and the RF and q bunch is the bunch charge. In general, the bunches will not necessarily pass through the cavity on-crest (maximum field) or on-trough (minimum field). When dealing with RF fields, it is convenient to consider the field as a complex number, where only the real part can interact with the beam at any moment in time. Indeed this implies that beam loading can only change the real component of the cavity voltage for any given phase.
In order for a recirculating ERL to operate stably over time, we require that the vector sum of the cavity voltage experienced by each bunch in a bunch train must equal zero, as shown Fig. 3. If this is not the case, then there will be a net change in stored energy in the cavity each train, reducing the overall efficiency of the ERL.
For now, we will neglect the phase of the bunches and only consider voltages as real numbers for brevity in the following mathematical description. Later we will consider off-crest beam loading cases by replacing binary notation with complex notation, i.e. by replace "1" and "0" by e iφ and e −iφ . We define a recirculating ERL to be at 'steady state' when all RF buckets in the machine are occupied. In this case, on any given turn, half the bunches in the train pass through the cavity at accelerating phases and half at decelerating phases. As cavity voltage experienced by all bunches in the train sum to zero, there is no net energy gain or loss over bunch train.
If we neglect the phase of the bunches and only consider bunches passing through the cavity on-crest and ontrough, then the change in cavity voltage due to beam loading from a bunch is simply ± q bunch 2 ω R Q cos (φ), from Eq. 4. Therefore in this case, every time a bunch passes through a linac, the cavity voltage is incremented or decremented by a fixed amount.

D. Beam loading pattern
Let us consider a 6-turn ERL. Table I shows how the buckets are occupied turn-by-turn for the filling patterns [1 2 3 4 5 6], [1 4 3 6 5 2], and [1 4 5 2 3 6]. If we use "0" and "1" to denote accelerated and decelerated bunches, respectively, we get beam loading patterns as shown in Table I. The accelerating bunches adds voltage to the cavity and vise versa. Now that we have defined the bunch filling pattern and showed how this is associated with a unique sequence of beam loading patterns, we should understand how this beam loading pattern affects the cavity voltage. Fig. 4 shows how the beam loading pattern can be translated into a change in cavity voltage.
For an ERL at steady state, the definition of "bucket 1" is arbitrary and can be one of N choices in a N -turn ERL; therefore there are (N − 1)! unique bunch filling patterns for a N -turn ERL. A 6-turn ERL can have 120 unique filling patterns. Each of these filling patterns is associated with a unique sequence of beam loading patterns. Beam loading patterns changes turn by turn and are periodic over N turns, as shown in Table. I. Fig. 5 shows beam loading patterns of two filling patterns over 6-turns. The red beam loading pattern has larger in cavity voltage fluctuation than blue one. This shows some filling patterns cause larger disturbances to the cavity voltage and RF system of the ERL than others. For a 6-turn ERL, we can evaluate the RF jitters associated with a specific beam filling pattern and use this to identify which patterns are optimal. In Table I, the beam loading increments have been normalised to ±1 rather than ± q bunch 2 ω R Q cos (φ) for brevity and clarity. For the remainder of the article, we will continue to use a normalised beam loading to help the reader understand the methodology.
Once a list of all unique bunch filling patterns is defined, we can determine the associated sequence of beam loading patterns, using the method described in Table I. To determine the normalised change in cavity voltage, we simply calculate the cumulative sum of the beam loading sequence. We define a specific filling pattern as F i , the associated beam loading pattern as B (F i ) and the normalised change in cavity voltage as δV given as We can use δV to estimate the RF stability performance of all patterns.

E. Low level RF system
We model Low level RF (LLRF) system as a proportional-integral (PI) controller [18][19][20]. In the PI controller, the LLRF system first calculates the error u voltage, which is difference between actual cavity voltage V actual with set-point voltage V set Then, two types of corrections are made, namely the proportional V pro and integral term corrections V int . The proportional term correction is calculated based on the previously measured dV and proportional gain G p , given as The integral term correction is calculated integrating over on all the previously measured dV and integral term gain G i , given as where t is the time measurement took place. The proportional and integral term corrections address fast and slow changes, respectively. The set-point voltage can be constant (static set-point) or can change over time (dynamic set-point). A dynamic set-point can be useful in order to improve RF stability in a recirculating ERL because it prevents the LLRF system from competing with the beam loading voltage in the cavity. If the LLRF feedback system can adjust its set-point voltage according to the anticipated beam loading, then it has a "dynamic set-point" voltage. In this case, the feedback system only amplifies noise. If the set-point is static, LLRF system will treat beam loading as noise and amplify it as well.

A. Variations in cavity voltage
If we consider the effects of beam loading and noise, the cavity voltage, V cav , can be expressed as: where V 0 is the steady state cavity voltage, which we will assume to be time-independent, V b is the voltage contribution due to beam loading, and V n is the voltage contribution due to all noise sources in the system. We shall assume that noise originates from the electronics in the low-level RF system (LLRF), which in turn introduces noise to the cavity voltage. How the noise propagates through the RF system depends on the behaviour of the LLRF system as well as the beam loading patterns, but the noise voltage in the cavity can be defined as where S/N is the voltage signal to noise ratio and α RF is a constant of proportionality, which depends on the parameters of the system. From Eq. 9, we can obtain an expression for the cavity voltage squared: We shall assume that V b and V n are independent variables and that V 0 is constant, therefore from Eq. 9 and 11, we obtain expressions for the mean and standard deviation of the cavity voltage.
If V b and V n have zero mean, then Eq. 12 produces the expected result that V cav = V 0 . Because noise and beamloading is independent, Therefore, From Eqs. 10 and 14, we can express the noise on the cavity voltage as The σ V b is pattern specific, and depends on topology of the ERL as well as the expected beam jitters. The voltage fluctuation due to the beam loading and given by where σ Vpattern is RMS fluctuation of the normalized beam loading pattern over all turns of the machine. The σ Vpattern for all 120 patterns is shown in Fig. 6 for a 6turn ERL, where we have assumed a First-In-First-Out (FIFO) topology, where the order of the bunch train does not change turn by turn. One can see clearly that σ Vpattern varies by approximately a factor of 2 depending on the choice of filling pattern.

B. Variations in amplifier power
From [21], the cavity voltage can be determined from an envelope equatioṅ Where ω 0 is the resonant frequency of the cavity, ω is the amplifier drive frequency, Q L and Q e are the loaded and external Q-factors respectively and F is the forward power from the amplifier expressed as a voltage as If we assume that the cavity is driven at the resonant frequency and that the cavity is at steady state, then from Eq. 17, we obtain thus From Eqs. 11 and 20, we obtain Note that for the beam loading terms, we now use V β rather than V b . This is because the LLRF feedback algorithm determines the power required to maintain a stable cavity voltage. If we implement a static set point algorithm, then V β = V b , if a dynamic set point algorithm is used then V β = δV b , which is an error residual when subtracting the expected beam loading voltage from the real value. This error residual depends on pattern number, LLRF algorithm, gains and other factors.
We should note that for the amplifier power, the noise has a simpler relationship to the signal to noise ratio than the noise observed on the cavity voltage (Eq. 10) because the noise on the amplifier is the measured noise amplified by the proportional gain of the LLRF, so If we assume that V β and V n are independent and zero mean, then Eq. 21 can be simplified as: By a similar method, we can also determine the standard deviation on the amplifier power as For low signal to noise ratios, the first terms dominates, whereas for high signal to noise ratios, we encounter a noise floor due to either beam loading (static set-point) or a residual error (dynamic set-point); this noise floor will be pattern dependent. For the first term, note that it is independent of beam loading pattern and therefore, for lower signal to noise ratios, we expect σ Pamp to be independent of beam loading pattern.

III. BEAM LOADING SIMULATION
The cavity voltage fluctuation can be simulated by simulating beam loading and its interaction with RF system [21].

A. Static and dynamic set-points
Before running simulations, it is important to determine the set-point voltage of LLRF system. As we mentioned earlier, there are two types of set-point voltages: dynamic and static set-points. During the beam loading, the cavity voltage fluctuates but the net beam loading of a train is zero and voltage will return to nominal voltage. So, there is no need for LLRF correction for beam loading. The dynamic set-point is designed to exclude beam loading correction. In static set-point, however, the LLRF system treats beam loading as noise, tries to correct to the oscillatory beam loading, and thus becomes unstable. Therefore, the dynamic set-point is better than static set-point as it creates less cavity voltage fluctuation and requires much less amplifier power. This is also confirmed by simulations shown in Fig. 7.

B. Simulation parameters
The simulation parameters are shown in Table II. We simulated 6-turn ERL, so there are 6 bunches in the train. The bunch charge was set high to increase the affect of the beam loading and to allow us to explore the behaviour of the RF system under extreme conditions. The circumference is set to 360, so number of RF cycles in the ring would be 1200 for a 1 GHz RF frequency. We set 1 RF bucket is 10 RF cycles, so 20 trains fill up the ring. New bunches replaced old bunches, until total of 96 turns are tracked, which is about 121 µs time duration. We scanned through all the 120 filling patterns of 6-turn ERL.

Comparison of optimal and non-optimal patterns
Firstly, we have looked at the affect of beam loading pattern on the cavity voltage and amplifier power. As show in Fig. 8 Increasing the S/N reduced cavity voltage fluctuation slightly and amplifier power significantly. Simulation results confirmed that certain patterns are better from the perspective of cavity voltage jitters, RF stability, and power requirements.

Noise scan
We observed the cavity voltage jitters and amplifier power is reduced when S/N is increased. To investigate noise dependence, we have performed simulations with filling patterns [1 4 3 6 5 2] and [1 2 3 4 5 6] by varying S/N . The results are shown in Fig. 9 for (a) σ Vcav , (b) σ Pamp , and (c) average P amp .
In Fig. 9 (a), we see that the σ Vcav is more sensitive to the filling pattern than S/N . In other words, σ Vcav is dominated by filling pattern. σ Vcav reaches pattern specific limit σ V b around 10 3 , so S/N needs to larger than 10 3 to minimize cavity voltage jitters. In Fig. 9 (b) and (c), we see σ Pamp and average P amp are sensitive to noise than filling pattern. To minimize power consumption P amp around to 11.15 kW, the S/N has to be larger than 10 4 . Two patterns has similar amplifier power fluctuations σ Pamp up to S/N = 10 5 . Beyond this point, σ Pamp reach filling pattern specific floors.
The analytical model underestimates P amp as shown in Fig. 9 (b) at high noise. As the noise increase, the amplifier starts to have saturation. In this case, the proportional term can't provide sufficient power. As the power shortage build up, the integral term will start to make correction and add power the cavity. The simulation can model the proper PI controller and have integral term. But the analytical doesn't have the integral term and thus can't include the power from integral term. This will cause analytical model to fail at very high noise levels and accounts for the difference between the analytic model and simulation

Cavity voltage
The cavity voltages jitters σ Vcav of all 120 filling patterns are shown in Fig. 10. We see that σ Vcav is different when different set-points are used.
withV i being the average voltage of i th turn, and N t being number of turns. In this case, we averaging voltage over one turn and getV i first, then calculating the RMS of these N t turns. As shown in Fig. 10 (a), the FOM roughly overlaps with simulation. Although, the FOM doesn't predict jitters exactly, but it can find optimal pattern quickly without simulations. For dynamic setpoint, the FOM is Eq. 15. The theoretical prediction matches simulation results exactly for S/N = 1 × 10 12 as shown in Fig. 10 (b). We see the dynamic set-point give smaller jitters. The patterns [1 4 3 6 5 2] and [1 4 5 2 3 6] (pattern number 60 and 61) are optimal in both set-points. Optimal pattern has 2−3 times less cavity voltage jitters than worst patterns.

Amplifier power results
The required average amplifier powers P amp for different patterns and different S/N are given in Fig. 11. We see that the average P amp is reduced from 28 kW to 11.13 kW, when the S/N increased from 7.1 × 10 3 to 7.1 × 10t. When S/N reduced further, the P amp is reduced to minimum of 11.147 kW, which is the resistive power loss. This shows that ERLs can be operated with very low power, when S/N is sufficiently high.

D. Property of optimal patterns
In Fig. 12, we compared cavity voltage of optimal and non-optimal patterns, indicated by blue and red lines respectively.  Table I. We observe their two consecutive bits are in either updown (10) or down-up (01) pairs. Such combinations limit cumulative sum of beam loading pattern to a range of [−1, 1], and thus minimizes jitters. We also see 1 pair flips ("1" and "0" switch positions) per turn. The change from "0" to "1" (acceleration to deceleration) happens in 3rd to 4th turn transition and the change from "1" to "0" is the new bunch replacing the extracted bunch. Therefore, in optimal patterns, consecutive pairs are made up by bunches that are 3 turns apart like  Fig. 12. Both patterns have same fluctuation range, but the turn average of the DSPO is larger in the 1st, 4th, and 7th turns. So, σ Vturn of pattern DSPO is larger, which makes it non-optimal for static set-points according to Eq. 25.

E. Off-crest beam loading
So far, we have studied the effects of beam loading for on-crest phases. In many applications, such as FELs and ERLs, bunches are compressed to get high peak current. Beams often pass through the RF system off-crest to introduce an energy chirp for bunch compression [22,23]. In recirculating ERLs, we want to minimize the net beam loading of a train, so the in-phase (I) and quadrature phase (Q) components of the beam loading from a train should sum to approximately zero, i.e. the vector sum of the voltage changes sums to zero for the bunch train. By doing so, the amplitude and phase of the cavity voltage changes minimally after a train. This implies that the phase and amplitude perturbations from beam loading cancel out over a bunch train, as shown in Fig. 13. Here, by "mirror turns" we meant turns that has same energy but the bunch phase is offset by π radians. In 6-turn ERLs, turn 1 and 6, 2 and 5, and 3 and 4 are mirror turns. Mirror bunches have same energy and off-set angles as shown in Fig. 13, so their vector sum is zero. In Fig. 13, φ 1 is the off phase angle of 1st and 6th turns; φ 2 is the off phase angle of 2nd and 5th turns; φ 3 is the off phase angle of 3rd and 4th turns.

Phase angle jitters
We have estimated off-crest cavity voltage phase fluctuation for 120 patterns of the 6-turn ERL and results are given in Fig. 14. The S/N was set to 10 12 to turn off the noise. We have simulated two sets of off-set angles φ 1,2,3 = 20 • , −20 • , 0 • and φ 1,2,3 = 20 • , −10 • , −9.7 • . We see that: (1) phase jitters is pattern dependent; (2) phase jitters is off-phase angle dependent; (3) in the worst case scenario, the RMS cavity phase jitters is less than 0.03 • , even at fairly large off-set angles. (4) the jitters in the on-crest case is ignorable.

Cavity voltage and amplifier power jitters
We have also estimated cavity voltage and amplifier power jitters and results are given in Fig. 15. The difference in on-and off-crest cases are insignificant. The average amplifier power is the same as on-crest case, which is about 11.15 kW for all filling patterns.

IV. SEQUENCE PERSEVERING ERL TOPOLOGY
So far we have only discussed about an over-simplistic recirculating ERL topology. For a recirculating linac to be an ERL, there has to be an extra path length to delay the bunch by 180 • phase to switch from accelerating FIG. 14: Cavity voltage phase jitters of off-crest beam loading for 120 patterns for 6-turn ERL.
mode to decelerating mode. The extra length can be in the form of longer arc length [24] or a chicane [25]. If it is extra arc, the topology has to changed from the "0" topology of Fig. 1 to the "8" topology of the Fig. 16.
More complicated topologies can be achieved by setting all the arcs to different lengths [10,26,27]. Here we would like to discuss "8" topology as an example to show that it can maintain up-down-up-down ([1 0 1 0 1 0]) optimal beam loading pattern for all trains and all turns by using filling pattern and delay scheme shown the Fig. 16.
Of course, one can maintain up-down-up-down patterns with more complicated topologies as well. This example should be sufficient for simple or complicated topologies as it can maintain up-down-up-down beam loading pattern and there is no difference from the RF system perspective.
In "8" topology of Fig. 16, all bunches go through a same arc, except the bunch transitioning from accelerating mode to decelerating mode. Transitioning bunch goes through the arc 6, which has extra length ∆L. The length of delay can be given as with n = 0, 1, 2, ..., m = 1, 2, ..., L train being length occupied by a bunch train, L bucket being length occupied by a RF bucket, and λ RF being a wave length of RF cycle. In the Eq. 26 case (1), the bunch flip phase but remains in the same train. The simple recirculating ERL described earlier sections is of case (1) and the beamline layout described in [25] can be an example. In case (2), the bunches not only flip phase, but also move to later trains. An example for case (2) is given in Fig. 17    Angal-Kalinin et al proposed [10] a similar filling pattern and delay mechanism for the purpose of separating low energy bunches to minimize Beam-Breakup (BBU) instability [28]. BBU is a main limiting factor for the ERL beam current [29] and we will investigate it further in a future study.

V. COMPARISON OF SIMULATION RESULTS
Simulations were performed for SP with on-and offcrest beam loadings. The results are given in Fig. 18. Simulations results of SP and FIFO with on-and off-crest beam loadings are compared in Tables III for S/N = 10 12 and IV for 7.1 × 10 3 . The S/N was set to 10 12 to observe the behaviour of the system without noise. The S/N was set to 7.1 × 10 3 to observe the behaviour of the system with moderate noise. In the tables, only optimal pattern results are compared.

A. Comparison of on-and off-crest
On-and off-crest beam loadings have different behaviours in SP and FIFO topologies. As shown in Fig. 18, the off-crest beam loading has larger the jitters in cavity voltage, phase and amplifier power for SP schemes. The off-crest beam loading also requires slightly more power than on-crest. So, if the ERL is only for on-crest accelerations, then SP scheme is preferable.
In the FIFO case, however, the off-crest beam loading doesn't increase the cavity voltage and amplifier power jitters as can be seen from Tables III and IV. There is small insignificant increase in phase jitters. So, if the ERL needs off-crest accelerations, then FIFO scheme is preferable.
B. Comparison of with and without noise some parameters are more sensitive to noise than others. As shown in Tables III and IV, when S/N is decreased from 10 12 to 7.1 × 10 3 , σ Vcav and P amp do not significantly change within this range. However, if S/N increases beyond this range, P amp does change notably, as seen from Fig. 11. The cavity phase jitter (σ φcav ) can be seen to be insensitive to S/N . The amplifier power jitter (σ Pamp ) is approximately 2 kW for S/N = 7.1 × 10 4 and for sequence preserving patterns and S/N = 10 12 , σ Pamp ∼ 80−90 W and for FIFO patterns, σ Pamp ∼ 1 W. It should be noted that the choice of topology, such as sequence preserving and FIFO schemes, will have a marked impact on the performance of the RF system.

VI. CONCLUSION
We studied recirculating ERL beam loading instabilities of different filling patterns under various noises, phases, and topologies by combining analytical model with simulations. Simulation results agreed with analytical predictions with some minor differences at very high or very low noises, possibly due to the non-linearity of system. These studies give us useful insight to ERL beam loading with different filling patterns, LLRF system and topologies.
Our studies show that ERL LLRF requires dynamic set-point voltage. The cavity voltage is more sensitive to the filling patterns than noise. The amplifier power fluctuation is more sensitive to noise than fill pattern. For our setup parameters, when S/N is increase to 10 4 or more, the average amplifier power can be reduced to around 11 kW. We have also introduced SP and FIFO topologies. We investigated off-crest beam loading and compared to on-crest cases under SP and FIFO topologies. Based on jitters and instabilities: if the ERL is only for on-crest accelerations, SP is preferable; if the ERL needs off-crest accelerations, then FIFO scheme is preferable.
It will be interesting study to investigate BBU instability for different filling patterns. This work has been done only 6-turn ERLs, but the theoretical construct and simulation can also be applied to higher or less turn numbers.