Suppression of Microbunching Instability in the Linac Coherent Light Source

A microbunching instability driven by longitudinal space charge, coherent synchrotron radiation, and linac wakefields is studied for the linac coherent light source (LCLS) accelerator system. Since the uncorrelated (local) energy spread of electron beams generated from a photocathode rf gun is very small, the microbunching gain may be large enough to significantly amplify rf-gun generated modulations or even shot-noise fluctuations of the electron beam. The uncorrelated energy spread can be increased by an order of magnitude to provide strong Landau damping against the instability without degrading the free-electron laser performance. We study different damping options in the LCLS and discuss an effective laser heater to minimize the impact of the instability on the quality of the electron beam. Submitted to Phys. Rev. ST Accel. Beams ∗Work supported by Department of Energy contract DE–AC03–76SF00515 and W-31-109-ENG-38. †Email: zrh@slac.stanford.edu


INTRODUCTION
In order to reach the desired electron peak current capable of inducing the collective free-electron laser (FEL) instability in the x-ray regime [1, 2], the pulse length of a low-emittance electron bunch generated from the photocathode rf gun is magnetically compressed in the linear accelerator by more than one order of magnitude.Numerical and theoretical investigations of high-brightness electron bunch compression lead to a coherent synchrotron radiation (CSR) microbunching that can significantly degrade the beam quality [3,4,5,6].Recently, Saldin et al. pointed out that the longitudinal space charge (LSC) field can be the main effect driving the microbunching instability in the TESLA test facility (TTF) (phase 2) linac [7].In addition, significant LSC-induced energy modulation in the DUV-FEL linac has been experimentally characterized using an rf zero-phasing method [8].Because the microbunching instability is very sensitive to the uncorrelated (local) energy spread of the electron beam, increasing it within the FEL tolerance can provide strong Landau damping against the instability.In this paper, we study the suppression of the microbunching instability driven by LSC, CSR, and linac wakefields in the linac coherent light source (LCLS) using an effective laser heater.

MICROBUNCHING INSTABILITY
The mechanism for microbunching instability is similar to that in a klystron amplifier [4].A high-brightness electron beam with a small amount of longitudinal density modulation can create self-fields that lead to beam energy modulation.Since a magnetic bunch compressor (usually a chicane) introduces path length dependence on energy, the induced energy modulation is then converted to additional density modulation that can be much larger than the initial density modulation.This amplification process (the gain in microbunching) is accompanied by a growth of energy modulation and a possible growth of emittance if significant energy modulation is induced in a dispersive region such as the chicane.Thus, the instability can be harmful to FEL performance, which depends critically on the high quality of the electron beam.
The initial electron density modulation is most likely caused by the intensity fluctuation on the drive laser that produces the electron beam from the photocathode.The electrons repel each other in the higher density regions and initiate the space charge oscillation between density and energy modulations in the low-energy section of a photoin-jector.As a result, the initial density modulation at the injector end may be reduced by a factor of a few, while noticeable energy modulation can be accumulated in the injector [9].Start-to-end simulations including the injector modulation dynamics are carried out to specify the tolerable drive-laser modulation level [10].In this paper, we neglect the injector modulation dynamics for simplicity and focus on the amplification of only small density modulations starting from the injector end.
At the end of the LCLS photoinjector (at 135 MeV in Fig. 1), the electrons are too relativistic to have any relative longitudinal motion in the linac.Thus, the electron density modulation is frozen while the energy modulation is accumulated in the linac.After a bunch compressor, the gain in density modulation for a Gaussian energy distribution is [4] where I 0 and I A (≈ 17 kA) are the initial and Alfven current, k 0 = 2π/λ 0 and k f = k 0 /|1 + hR 56 | are the initial and compressed modulation wavenumber, h is the (linear) chirp, R 56 is the momentum compaction of the chicane, L is the linac length, Z 0 = 377 Ω is the free space impedance, and σ δ is the relative uncorrelated energy spread prior to the chicane.The longitudinal impedance Z(k 0 ) per unit length includes geometric wakefields and LSC given by [11,12] where r b is the radius of the uniform cross section and is approximately the sum of rms beam sizes in both transverse planes for a Gaussian or parabolic cross section, and K 1 is the modified Bessel function.Effects of the vacuum chamber are ignored for these very short modulation wavelengths.We have also neglected a small transverse variation of the LSC field that can contribute to a slight increase of the local energy spread.The LSC impedance is implemented in the numerical tracking code elegant [13].
Both photoinjector simulations and measurements [14] show an uncorrelated energy spread about 3 keV (rms) at ≥ 1 nC charge, yielding σ δ = 1.2×10 −5 at the BC1 energy of 250 MeV.With such a small intrinsic energy spread, the peak gain including CSR amplification [5,6] after BC1 can be on the order of 100.With two bunch compressors designed for the LCLS, the total gain after BC2 can be ∼ 10 4 and may even amplify shot-noise fluctuations [15].
The very large gain in density modulation at these short wavelengths can be suppressed by increasing the uncorre- lated energy spread of the electron beam.Note the uncorrelated energy spread after compression and acceleration is less than 1 × 10 −5 at the undulator (14 GeV).Since the FEL parameter ρ ≈ 5 × 10 −4 for the LCLS when the fundamental radiation wavelength is 1.5 Å, a factor of 10 to 15 increase in uncorrelated energy spread has a rather minimal impact on the FEL performance.Taking into account that quantum fluctuations of spontaneous radiation in a 130-m undulator can increase the rms energy spread to ∼ 2 × 10 −4 [1], the average power gain length is almost independent of the slice (over FEL slippage length) energy spread σ δ f up to 1 × 10 −4 .However, for σ δ f > 1 × 10 −4 , the FEL gain length and hence the saturation length starts to increase much faster.Thus, the tolerable rms energy spread at the undulator entrance is about 1 × 10 −4 or 1.4 MeV.

LASER HEATER
The microbunching instability is predominantly driven by LSC at low-energy section of the linac (<1 GeV) and is not effectively suppressed by a superconducting wiggler that increases the uncorrelated energy spread at BC2 (at 4.5 GeV as shown in Fig. 1) [15].At energies less than about 1 GeV, uncorrelated energy spread cannot be easily increased by quantum fluctuations of synchrotron radiation.Nevertheless, resonant laser-electron interaction in a short undulator induces rapid energy modulation at the optical frequency, which can be used as an effective energy spread for beam "heating" [7,16].
Suppose a Gaussian laser beam co-propagates with a round electron beam at the energy γ 0 mc 2 (=135 MeV) in a planar undulator of length L u , which is short compared to both the Rayleigh length Z R of the laser and the beta functions β x,y of the electrons.The energy modulation amplitude of the resonant FEL interaction is where P L is the peak laser power, P 0 = I A mc 2 /e ≈ 8.7 GW, K is the undulator parameter, [JJ] is the Besselfunction factor, r is the radial position of the electron, and σ r is the rms laser spot size.Table 1 lists the main laser heater parameters under design at the end of the LCLS photoinjector (see Fig. 1).Two sets of laser spot size and Figure 2: Electron energy distribution after the laser heater for a large laser spot (blue) and for a matched laser spot (red).The laser powers are given in Table 1 so that the rms energy spread ≈ 40 keV for both distributions.peak power are considered, both of which increase the rms energy spread from 3 keV to about 40 keV.After a total compression factor of about 30, the slice rms energy spread should be about 1.2 MeV or σ δ f ≈ 0.9 × 10 −4 at the undulator entrance (at 14 GeV) in the absence of impedance effects.The necessary laser power (37 MW) for the large laser spot size (σ r = 1.5 mm) is still a small fraction of the available power of the Ti-Sapphire laser that drives the photocathode rf gun and hence can be extracted from it.
Assuming initially Gaussian distributions in energy and in transverse coordinates, we obtain the modified energy distribution after the laser heater as shown in Fig. 2 for  σ r σ x (when the laser spot size is much larger than the electron beam size) and σ r ≈ σ x (when the laser spot size is matched to the e-beam size).A large laser spot size may be useful to establish the initial laser-electron interaction.However, the resulting energy modulation amplitude is almost the same for all electrons, and the energy profile is a double-horn distribution (the blue curve in Fig. 2).The two sharp spikes at ∆γ 0 ≈ ±∆γ L (0) act like two separate cold beams that do not contribute much to suppressing the instability.For σ r ≈ σ x , the off-axis electrons experience smaller modulation with smaller laser field than the on-axis ones (see Eq. ( 3)).As a result, the "heating" is more uniform in terms of the energy distribution (the red curve in Fig. 2), and we expect better Landau damping.The gain in Eq. ( 1) is reduced by a suppression factor  S L (k f R 56 ∆γ L (0)/γ, σ r /σ x ) [15], where For |A| = |k f R 56 ∆γ L (0)/γ| 1, the Bessel functions J 0,1 (A) ∼ |A| −1/2 .Thus, a laser heater with a large laser spot size (B 1) has S L ∼ |A| −1/2 and suppresses the gain weakly, while a laser heater with a matched spot size (B = 1) has S L ∼ |A| −3/2 and is more effective at smearing the instability at short wavelengths.
Figure 3 shows that the BC1 gain computed from the linear theory agrees reasonably with elegant simulations using two sets of laser spot size and peak power given in Table 1.The theoretical gain after both compressors at short wavelengths (λ 0 ≤ 60 µm) can still be very high (∼ 100) for a laser heater with a large spot size because of its ineffective Landau damping at these wavelengths.Starting with 1% initial density modulation, elegant simulations show reduced gain at these very short wavelengthes as the density modulation after BC2 is not sinusoidal, but the slice energy spread can still increase as a result of the distorted longitudinal phase space (see Fig. 4). Figure 5 shows the slice energy spread of the bunch core at the undulator entrance without a laser heater and in presence of a laser heater with two different spot sizes.Thus, a laser heater with a large laser spot allows the growth of short-wavelength modulations that increases the slice energy spread at the undulator entrance, while a laser heater with a matched laser spot effectively suppresses the instability and does not change the slice energy spread above the design goal (about 1×10 −4 ).
Finally, the laser heater can be embedded in a weak chicane (R 56 ≈ 3 mm) to allow convenient laser-electron interaction with no crossing angle and to provide a useful temporal washing effect that completely smears the laserinduced 800-nm energy modulation [15].The implementation of the laser heater in the LCLS photoinjector beam line is described elsewhere [17] in these proceedings.

Figure 1 :
Figure 1: LCLS accelerator system layout with a laser heater at 135 MeV or a SC wiggler at 4.5 GeV.

Figure 3 :
Figure 3: Microbunching gain after BC1 as a function of the initial modulation wavelength λ 0 for a laser heater with a large laser spot (blue) and with a matched laser spot (red).

Figure 4 :
Figure 4: Central portion of the longitudinal phase space without a laser heater (upper), in presence of a laser heater with σ r = 1.5 mm (middle) and with σ r = 175 µm (lower) at the undulator entrance.Curves offset vertically for clarity.Simulations are seeded with 1% initial density modulation at λ 0 = 30 µm.

Figure 5 :
Figure5: Slice rms energy spread σ δ f at the undulator entrance at 14 GeV for 1% initial density modulation without a laser heater (black), in presence of a laser heater with a large spot size (blue) and with a matched spot size (red).