SSMB Proof-of-Principle Experiments

Abstract

To make SSMB a real option for future photon source facility, a crucial step is to experimentally demonstrate its working mechanism. In this chapter, we report the first successful experimental demonstration of the SSMB mechanism. Parts of the work presented have been published in Refs. [1, 2].

To make SSMB a real option for future photon source facility, a crucial step is to experimentally demonstrate its working mechanism. In this chapter, we report the first successful experimental demonstration of the SSMB mechanism. Parts of the work presented have been published in Refs. [1, 2].

5.1 Strategy of the PoP Experiments

5.1.1 Three Stages of PoP Experiments

Considering the fact that it is a demanding task to realize SSMB directly in an existing machine, part of the reasons we have analyzed in previous chapters, among them the most fundamental one is the large quantum diffusion of bunch length in rings not optimized for SSMB, the SSMB PoP experiment has been divided into three stages as shown in Fig. 5.1. Some brief descriptions of the three stages are as follows.

• Phase I: a single-shot laser is fired to interact at the undulator with the electron beam stored in a quasi-isochronous ring. The modulated electron beam becomes microbunched at the same place of modulation after one complete revolution in the ring and this microbunching can preserve for several revolutions. By doing this experiment, we want to confirm that the optical phases, i.e., the longitudinal coordinates, of electrons can be correlated turn-by-turn in a sub-laser-wavelength precision. The realization of SSMB relies on this precise turn-by-turn phase correlation.

• Phase II: on the basis of Phase I, we replace the single-shot laser with a high-repetition phase-locked one to interact with the electrons at the undulator turn after turn. In this stage, we want to establish stable microbuckets and sustain the microbunching in the microbuckets to reach a quasi steady state.

• Phase III: Phase II is very close to, but is still not, the final SSMB as a true SSMB means the balance of excitation and damping. However, the requirement of a true SSMB on the magnet lattice is demanding, especially the quantum diffusion of longitudinal coordinate z as analyzed in Sect. 2.1. Therefore, this final stage is more likely to be realized in a dedicated ring designed for SSMB, which is also one of the key ongoing tasks of the SSMB task force [3].

Below, we use PoP I, II, III to represent the three stages of the experiment. The key words of the three stage experiments are summarized as follows.

• PoP I: microbunching based on stored electron bunch, turn-by-turn phase correlation;

• PoP II: bounded motion in microbuckets, quasi steady state;

• PoP III: balance of diffusion and damping, real steady state.

These three stages each have their own significance and are all important for the SSMB development. Among them, Phase I is from 0 to 1, and is the most important one from conceptual viewpoint. Recently, we have successfully performed the PoP I and demonstrated the mechanism of SSMB at the Metrology Light Source (MLS) of Physikalisch-Technische Bundesanstalt (PTB) in Berlin [1, 2]. The experiment is a collaboration work of Tsinghua, Helmholtz-Zentrum Berlin (HZB) and PTB.

Fig. 5.1

Three stages of the SSMB PoP experiments: from single-shot to multiple shots to infinite shots laser pulse; from short-lived to quasi-steady-state to real steady-state microbunching

5.1.2 Metrology Light Source Storage Ring

The MLS is a storage ring optimized for quasi-isochronous operation [4,5,6], thus an appropriate testbed for SSMB physics investigation and PoP experiments. However, the partial phase slippage of the MLS is large as the bending angle of each dipole is large ($\frac{\pi }{4}$) and the dispersion magnitude inside the dipoles is also large, so it is not feasible to realize true SSMB, i.e., PoP III, directly at the MLS. Therefore, the SSMB PoP experiment has been divided into three stages as introduced just now, and PoP I and II are what we have performed and plan to conduct at the MLS. Some basic parameters of the MLS are shown in Table 5.1. The lattice optics of the MLS used in the SSMB PoP experiments are shown in Fig. 5.2.

Table 5.1 Basic parameters of the MLS lattice

Fig. 5.2

(Figure from Ref. [1])

The MLS quasi-isochronous magnet lattice used to generate microbunching. The magnet lattice and the key are shown at the top. The curves are the model horizontal (red) and vertical (blue) $\beta$-functions and the horizontal dispersion $D_{x}$ (green). Operating parameters of the ring: beam energy, $E_{0}$ = 250 MeV; relative energy spread, $\sigma _{\delta }=1.8\times 10^{-4}$ (model); horizontal emittance, $\epsilon _{x}$ = 31 nm (model); horizontal betatron tune, $\nu _{x}=3.18$ (model and measured); vertical betatron tune, $\nu _{y}=2.23$ (model and measured); horizontal chromaticity, $\xi _{x}=-0.5$ (measured). Note that this optics is different from that used in Sect. 2.1.3 for the simulation of partial phase slippage effect.

5.2 PoP I: Turn-by-Turn Laser-Electron Phase Correlation

5.2.1 Experimental Setup

Figure 5.3 shows the schematic setup of the SSMB PoP I experiment. A horizontally polarized laser pulse (wavelength, $\lambda _{L}$ = 1064 nm; pulse length, full-width at half-maximum, FWHM $\approx$ 10 ns; pulse energy, $\approx$ 50 mJ) is sent into a planar undulator (period, $\lambda _{u}$ = 125 mm; total length, $L_{u}=4$ m) to co-propagate with the electron bunches (energy, $E_{0}$ = 250 MeV) stored in the MLS storage ring (circumference, $C_{0}$ = 48 m). To maximize the laser-electron energy exchange, the undulator gap is chosen to satisfy the resonance condition $\lambda _{s}=\lambda _{L}$, where $\lambda _{s}=\frac{1+K^{2}/2}{2\gamma ^{2}}\lambda _{u}$ is the central wavelength of the spontaneous undulator radiation, with $\gamma \propto E_{0}$ being the Lorentz factor and $K=\frac{eB_{0}}{m_{e}ck_{u}}=0.934\cdot B_{0}[\text {T}]\cdot \lambda _{u}[\text {cm}]$ being the dimensionless undulator parameter, determined by the undulator period and magnetic flux density. This laser-electron interaction induces a sinusoidal energy modulation pattern on the electron beam with a period of the laser wavelength. Because particles with different energies have slightly different revolution periods, after one revolution in the ring, the energy-modulated electrons shift longitudinally with respect to each other, clumping towards synchronous phases and forming microbunches. The formed microbunches can last several revolutions in the ring. The coherent undulator radiation generated from the microbunches, detected by a high-speed photodetector with a photodiode, confirms microbunching.

Fig. 5.3

(Figure from Ref. [1])

Schematic of the experimental setup. The stored 250 MeV electron bunches are energy-modulated by a 1064 nm wavelength laser in an undulator, and become microbunched after one complete revolution in the 48 m circumference quasi-isochronous storage ring. This formed microbunching can then preserve for multiple turns in the ring. Each time the microbunching going through the undulator, narrowband coherent radiation will be generated. The undulator radiation is separated into the fundamental and second harmonics by dichroic mirrors, and sensitive photodiodes are used as the detectors. Narrow band-pass filters can be inserted in front of the photodetectors to pick out the narrowband coherent radiation generated from the microbunching.

The symplectic longitudinal dynamics of the the above experiment processes can be modeled by

$$\begin{aligned} {\left\{ \begin{array}{ll} &{}\delta _{1}=\delta _{0}+A\sin (k_{L}z_{0}),\\ &{}z_{1}=z_{0}-\eta C_{0}\delta _{1}, \end{array}\right. } \end{aligned}$$

(5.1)

for the first revolution with laser modulation, and

$$\begin{aligned} {\left\{ \begin{array}{ll} &{}\delta _{m+1}=\delta _{m},\\ &{}z_{m+1}=z_{m}-\eta C_{0}\delta _{m+1}, \end{array}\right. } \end{aligned}$$

(5.2)

for the later revolutions. This demonstration proves that the longitudinal dynamics described by the one-turn map Eq. (5.1) can be extrapolated from the RF wavelength (metre scale) to laser wavelength (micrometre scale) for a stored electron beam, thus validating the SSMB microbunching mechanism.

5.2.2 Physical Analysis of Microbunching Formation

5.2.2.1 Storage Ring

Operation energy The above models Eqs. (5.1) and (5.2), however, do not consider the non-symplectic, transverse-longitudinal coupling and nonlinear lattice dynamics, which all could lead to degradation of the microbunching. It turns out that the first non-symplectic dynamics we need to account for is the synchrotron radiation. As we know, when a relativistic electron is subjected to an acceleration normal to its velocity exerted by a bending magnet, it radiates electromagnetic energy. This radiation is characterized by the quantum nature of the photon emission process. The photon energy and emission place or time are both stochastic, giving rise to changes on particle energy (instantly) and the longitudinal coordinate z (non-instantly), as analyzed before in Sect. 2.1. Of special interest in the experiment is the root-mean-square (RMS) quantum diffusion of z in one turn $d_{z}$. According to Eq. (2.41), we have

$$\begin{aligned} d_{z}=\sqrt{\langle z^2\rangle -\langle z\rangle ^2}=\sqrt{\langle F^{2}\rangle \langle \mathcal {N}\rangle \left\langle \frac{u^{2}}{E_{0}^{2}}\right\rangle }, \end{aligned}$$

(5.3)

with $F(s_{2},s_{1})\equiv -\tilde{\eta }(s_{2},s_{1})C_{0}$. For the MLS quasi-isochronous magnet lattice used in the PoP experiment as shown in Fig. 5.2, $d_{z}$ is as large as 260 nm at its standard operation energy of 630 MeV, deteriorating the sub-micrometre microbunching considerably. Therefore, the beam energy needs to be lowered to mitigate this diffusion, as $\sqrt{\langle \mathcal {N}\rangle \left\langle \frac{u^{2}}{E_{0}^{2}}\right\rangle }\propto \gamma ^{2.5}$. At the same time, a lower beam energy gives a smaller energy spread and is also beneficial for microbunching, as the smearing from the natural uncorrelated energy spread becomes smaller. Nevertheless, the beam energy cannot be too low, otherwise the beam parameters and lifetime could be profoundly affected by scattering among particles [7,8,9]. An electron beam energy of 250 MeV is adopted in the experiment to balance these issues. At $E_{0}$ = 250 MeV, we have $d_{z}=26$ nm and $\sigma _{\delta }=1.8\times 10^{-4}$.

Phase slippage factor Because the laser wavelength is much smaller than that of an RF wave, the phase slippage factor $\eta$ needs to be ultrasmall. That is, the ring should be quasi-isochronous to allow turn-by-turn stabilization of the electron optical phases, i.e., the longitudinal coordinates, for particles with different energies. More quantitatively, the RMS spread of z in one turn that arises from the uncorrelated electron energy spread should be adequately smaller than the laser wavelength,

$$\begin{aligned} \Delta z_{\text {ES}}=|\eta C_{0}\sigma _{\delta }|\le \lambda _{L}/2\pi . \end{aligned}$$

(5.4)

To fulfill this requirement, the phase slippage factor of the MLS was lowered to $\eta \approx -2\times 10^{-5}$, which is three orders of magnitude smaller than its standard value of $3\times 10^{-2}$. By implementing these parameters, $\Delta z_{\text {ES}}=|\eta C_{0}\sigma _{\delta }|\approx 0.17\ \upmu$m (about 0.6 fs), enabling the formation and preservation of sub-micrometre microbunching.

Such a quasi-isochronous magnet lattice is achieved by tailoring the horizontal dispersion functions $D_{x}$ around the ring so that a particle with non-ideal energy travels part of the ring inwards and part of the ring outwards compared to the reference orbit, thus having a revolution period nearly the same as that of the ideal particle. The tailoring of $D_{x}$ is accomplished by adjusting the (de)focusing strengths of the quadrupole magnet. The operation of the MLS as a quasi-isochronous ring also benefits from the optimization of the sextupole and octupole nonlinear magnet schemes to control the higher-order terms of the phase slippage [5, 6], which affect both the equilibrium beam distribution in the longitudinal phase space before the laser modulation and the succeeding microbunching evolution as analyzed in Sect. 2.2.1.

In the experiment, the value of the small phase slippage factor is quantified by measuring the electron orbit offsets while slightly adjusting the RF frequency up and down at a beam position monitor (BPM), where $D_{x}$ is large. From the offsets and $D_{x}$, the phase slippage-dependent electron energy shifts caused by the RF frequency adjustment can be derived, as well as the phase slippage factor. The $D_{x}$ value at the BPM is acquired using the same method in a reversible way, that is, based on a known phase slippage factor. This is done at a larger phase slippage factor, at which its value can be determined from its relation to the synchrotron oscillation frequency of the electron beam as given in Eq. (2.28). The synchrotron oscillation frequency can be measured accurately at a phase slippage factor such as $-5\times 10^{-4}$, and the model confirmed that the relative change of $D_{x}$ at the highly dispersive BPM is small ($<4\%$) when the phase slippage factor is reduced from $-5\times 10^{-4}$ to the desired $-2\times 10^{-5}$ by marginally changing the quadrupole magnet strengths.

Bunching factor With the electron beam evolved according to Eq. (5.1) for one revolution in the ring, the bunching factor at the n-th laser harmonic as analyzed in Sect. 2.2.1 is

$$\begin{aligned} b_{n}=J_{n}(nk_{L}\eta C_{0} A)\text {exp}\left[ -\frac{(nk_{L}\eta C_{0}\sigma _{\delta })^{2}}{2}\right] , \end{aligned}$$

(5.5)

where $J_{n}$ is the n-th order Bessel function of the first kind. The coherent radiation power at the n-th harmonic is proportional to the bunching factor squared, $P_{n,\text {coh}}\propto |b_{n}|^{2}$. With the dynamics in the following turns modeled by Eq. (5.2), for the m-th revolution, we just need to replace the $\eta C_{0}$ in Eq. (5.5) with $m\eta C_{0}$.

The maximum reachable bunching factor becomes larger with the decrease of $\eta$, given that the optimal A can always be realized. $\eta =-2\times 10^{-5}$ is approximately the present lowest reachable value at the MLS, so here below we use this $\eta$ for the analysis. As we will explain soon, our signal detection at first focuses on the second harmonic of the undulator radiation, since it is easier to mitigate the impact of the modulation laser on signal detection compared to the fundamental frequency. As shown in Fig. 5.4, given $\eta =-2\times 10^{-5}$, a modulation strength of $A\approx 1.5\sigma _{\delta }$ results in the maximum bunching at the second harmonic. Correspondingly, given $A=1.5\sigma _{\delta }$, the optimal $\eta$ for the fundamental frequency and second harmonic bunching is a bit smaller than $2\times 10^{-5}$. Note that the optimized conditions for the fundamental frequency and second harmonic bunching are different. Figure 5.5 shows the bunching factor evolution with respect to the revolution number, with $A=1.5\sigma _{\delta }$ and $|\eta |=2\times 10^{-5}$. As can be seen, the second harmonic bunching can last only one turn, while the fundamental frequency bunching can last about three turns or even more if $\eta$ becomes smaller, as also can be seen in the right part of Fig. 5.4. These expectations have also been confirmed in the experiment as will be presented soon.

Fig. 5.4

Left: impact of the energy modulation strength ($A\propto \sqrt{P_{L}}$) on the bunching factor $|b_{n}|$ at the fundamental and second harmonic, with $\eta =-2\times 10^{-5}$. Right: impact of the phase slippage factor $\eta$ on the bunching factor $|b_{n}|$ at the fundamental and second harmonic, with $A=1.5\sigma _{\delta }$

Fig. 5.5

The evolution of the bunching factor $|b_{n}|$ at the fundamental and second harmonic with respect to the revolution number, with $A=1.5\sigma _{\delta }$ and $\eta =-2\times 10^{-5}$

Chromatic $\mathcal {H}_{x}$ function and chromaticity $\xi _{x}$ Apart from the longitudinal beam dynamics, the coupling of the particle betatron oscillation to the longitudinal dimension is also critical. The reason is based on the fact that the horizontal beam width at the undulator is about $600\ \upmu$m (model value), three orders of magnitude larger than the sub-micrometre longitudinal structures that we aim to produce. Because the vertical emittance is much smaller than the horizontal one in a planar x-y uncoupled storage ring, in the following we consider only the impact of the horizontal betatron oscillation.

According to Eq. (3.13), for a periodic system, the RMS bunch lengthening of an electron beam longitudinal slice after m complete revolutions in the ring, due to betatron oscillation, is

$$\begin{aligned} \Delta z_{B,m}=2\sqrt{\epsilon _{x}\mathcal {H}_{x}}\left| \sin (m\pi \nu _{x})\right| . \end{aligned}$$

(5.6)

With this bunch lengthening taken into account, the bunching factor at the n-th laser harmonic after m revolutions will be

$$\begin{aligned} b_{n,m}=J_{n}(nk_{L}m\eta C_{0} A)\text {exp}\left[ -\frac{(nk_{L}m\eta C_{0}\sigma _{\delta })^{2}+\left( nk_{L}2\sqrt{\epsilon _{x}\mathcal {H}_{x}}\left| \sin (m\pi \nu _{x})\right| \right) ^{2}}{2}\right] . \end{aligned}$$

(5.7)

The relative bunching factor reduction due to the non-zero $\mathcal {H}_{x}$ at the undulator can thus be defined as

$$\begin{aligned} R_{n,m}(\mathcal {H}_{x})=\text {exp}\left[ -\frac{\left( nk_{L}2\sqrt{\epsilon _{x}\mathcal {H}_{x}}\left| \sin (m\pi \nu _{x})\right| \right) ^{2}}{2}\right] . \end{aligned}$$

(5.8)

Fig. 5.6

Influence of $\mathcal {H}_{x}$ at the undulator on the bunching factor one turn after laser modulation, with $\epsilon _{x}$ = 31 nm (model) and $\nu _{x}=3.18$

Putting in $\epsilon _{x}$ = 31 nm (model) and $\nu _{x}=3.18$ (model and measured), we need $\mathcal {H}_{x}\le 0.8\ \upmu$m at the undulator to have $\Delta z_{B,1}\le \lambda _{L}/2\pi$. Figure 5.6 shows the bunching factor reduction at the fundamental frequency and the second harmonic one turn after the laser modulation as a function of the $\mathcal {H}_{x}$ at the undulator. As can be seen, the second-harmonic bunching is even more sensitive to the $\mathcal {H}_{x}$ at the undulator. This stringent condition on $\mathcal {H}_{x}$ (note that $\mathcal {H}_{x}$ at other places of the ring is typically $\ge$0.1 m) is satisfied by fine-tuning the quadrupole magnet (de)focusing strengths to correct the dispersion $D_{x}$ and dispersion angle $D_{x}'$ at the undulator to the level of millimetre and 0.1 mrad, respectively (see Fig. 5.2).

In addition to the linear-order oscillating bunch lengthening, as discussed in Sect. 3.2, the betatron oscillation also produces an average path lengthening or shortening (second-order effect) described by the formula

$$\begin{aligned} \Delta C_{B}=-2\pi J_{x}\xi _{x}, \end{aligned}$$

(5.9)

with $\Delta C_{B}$ being the average change of the particle recirculation path length, and $\xi _{x}=d\nu _{x}/d\delta$ being the horizontal chromaticity of the ring. Because different particles have different betatron oscillation amplitudes (actions), this effect results in a loss of synchronization between particles and degrades microbunching. Moreover, it broadens the equilibrium energy spread and distorts the beam from the Gaussian form before the laser modulation as investigated in Sect. 3.2, which also affects the microbunching. Therefore, the horizontal chromaticity should be small, to moderate its detrimental outcome, and simultaneously sufficient to suppress collective effects such as the head-tail instability [10]. As a consequence, a small negative chromaticity is used in the experiment.

5.2.2.2 Modulation Laser

Long-pulse laser A long-pulse laser (FWHM $\approx$ 10 ns) has been used to simplify the experiment by avoiding a dedicated laser-electron synchronization system, as the shot-to-shot laser timing jitter is $t_{\text {jitter}}\le$ 1 ns (RMS). According to Eq. (5.5) and Fig. 5.4, for a given phase slippage factor and harmonic number, there is an optimal laser-induced energy modulation amplitude A ($A\propto \sqrt{P_{L}}$ with $P_{L}$ the laser power) that gives the maximum bunching factor. The laser used in the experiment (Beamtech Optronics Dawa-200) has multiple longitudinal modes, and its temporal profile has several peaks and fluctuates considerably from shot to shot (see Fig. 5.7). Therefore, the laser-induced electron energy modulation amplitudes are different from shot to shot and from bunch to bunch. When the modulation amplitude matches the phase slippage factor, the energy-modulated electrons are properly focused at synchronous phases, which gives optimal microbunching. For some of the shots, the laser intensity is higher or lower than the optimal value, and the electrons are then over-focused or under-focused, giving weaker microbunching and less coherent radiation. As we will see soon, this explains the large shot-to-shot fluctuation of the coherent amplified signals shown in Fig. 5.16c, e.

Fig. 5.7

(Figure from Ref. [1])

Fluctuating temporal profiles of the multilongitudinal-mode laser. a, Temporal profiles of two example consecutive laser shots (red and blue) and the averaged waveform of 200 consecutive laser shots (black). b, Statistical distribution of the laser power at t = 0 ns in a for $10^{4}$ consecutive laser shots, where the red curve is a gamma distribution fit. Laser: compact Nd:YAG Q-switched laser (Beamtech Optronics Dawa-200). Detector: ultrafast photodetectors (Alphas UPS-40-UVIR-D; rise time < 40 ps). Measurement system: digital oscilloscope (Teledyne LeCroy WM825Zi-B; bandwidth 25 GHz; sample rate 80 billion samples per second).

Power and Rayleigh length The electric field of a TEM00 mode Gaussian laser beam is [11]

$$\begin{aligned} \begin{aligned} E_{x}&=E_{x0}e^{ik_{\text {L}}z-i\omega t+i\phi _{0}}\frac{1}{1+i\frac{z}{Z_{R}}}\text {exp}\left[ i\frac{k_{\text {L}}Q}{2}(x^2+y^2)\right] ,\\ E_{z}&\approx -E_{x}x, \end{aligned} \end{aligned}$$

(5.10)

with $Z_{R}=\frac{\pi w_{0}^{2}}{\lambda _{L}}$ the Rayleigh length, $w_{0}$ the beam waist radius, and $Q=\frac{i}{Z_{R}(1+\frac{z}{Z_{R}})}$. The relation between $E_{x0}$ and the laser peak power is given by

$$\begin{aligned} P_{L}=\frac{E_{x0}^{2}Z_{R}\lambda _{L}}{4Z_{0}}, \end{aligned}$$

(5.11)

in which $Z_{0}=376.73\ \Omega$ is the impedance of free space. The electron wiggles in an undulator according to

$$\begin{aligned} x(z)=\frac{K}{\gamma k_{u}}\sin (k_{u}z), \end{aligned}$$

(5.12)

and the laser-electron exchange energy according to

$$\begin{aligned} \frac{dW}{dt}=ev_{x}E_{x}+ev_{z}E_{z}. \end{aligned}$$

(5.13)

Assuming that the laser beam waist is in the middle of the undulator, and when $x,y\ll w(z)$, which is the case for SSMB PoP I, we drop the $\text {exp}\left[ i\frac{k_{\text {L}}Q}{2}(x^2+y^2)\right]$ in the laser electric field. Further, when $Z_{R}\gg \lambda _{u}$, we can also drop the contribution from $E_{z}$ on the energy modulation. The integrated modulation voltage induced by the laser in the planar undulator, normalized by the electron beam energy, is then [11]

$$\begin{aligned} \begin{aligned} \frac{eV_{\text {mod}}}{E_{0}}&=\frac{e[JJ] K}{\gamma ^{2}mc^{2}}\sqrt{\frac{4P_{L}Z_{0}Z_{R}}{\lambda _{L}}}\tan ^{-1}\left( \frac{L_{u}}{2Z_{R}}\right) , \end{aligned} \end{aligned}$$

(5.14)

in which $[JJ]=J_{0}(\chi )-J_{1}(\chi )$ and $\chi =\frac{K^{2}}{4+2K^{2}}$.

Fig. 5.8

Integrated modulation voltage of a laser modulator as a function of the ratio between laser Rayleigh length and the modulator undulator length

As can be seen from the above formula, when $\frac{L_{u}}{Z_{R}}$ is kept constant, then $V_{\text {mod}}\propto \sqrt{Z_{R}}\propto \sqrt{L_{u}}$. In our case, $L_{u}$ is fixed, then as shown in Fig. 5.8, to maximize $V_{\text {mod}}$ we need $\frac{Z_{R}}{L_{u}}=0.359\approx \frac{1}{3}$. On the other hand, the modulation strength does not depend on $Z_{R}$ sensitively when $Z_{R}$ is larger than this optimal value. To make the laser beam waist larger than the electron beam and thus induce the same energy modulation on different electrons, a larger Rayleigh length might be in favored in our case. For example if $Z_{R}=2L_{u}$, then in order to induce an energy modulation depth of $A=1.5\sigma _{\delta }$, we can calculate that the laser power required is 430 kW. Considering the non-ideal conditions and the fact that the laser used contains higher-order Gaussian modes, one order of magnitude higher laser power might be required in the actual case.

Fig. 5.9

Example current profile and bunching factor spectrum one turn after the laser modulation in SSMB PoP I, obtained from ELEGANT [12] tracking. Parameters used: $\sigma _{t}=100$ fs, $\epsilon _{x}=31$ nm, $\epsilon _{y}=\frac{1}{10}\epsilon _{x}$. $1\times 10^{6}$ particles are simulated, meaning 160 fC for a one-to-one correspondence

5.2.2.3 Microbunching Simulation

Based on the above parameters, we have conducted the simulation of microbunching formation in the storage ring. The beam current and bunching factor one turn after the laser modulation is shown in Fig. 5.9. As can be seen from the comparison with Fig. 5.5, the simulation agrees with theory well.

5.2.3 Microbunching Radiation Calculation

Now we evaluate what radiation we can obtain from the formed microbunching. The numerical calculation of incoherent and coherent undulation radiation shown in this section are obtained using SPECTRA [13]. The beam energy and undulator parameters used are those in our SSMB proof-of-principle experiment, i.e., $E_{0}=250$ MeV, $\lambda _{L}=\lambda _{0}=1064$ nm, $\lambda _{u}=125$ mm, $K=2.5$, $N_{u}=32$. The results are also used to compare with the theoretical formulas presented in Chap. 4. Note that the numerical calculation of coherent radiation with a 3D charge distribution is usually time-consuming. This is also one of the motivations for us in Chap. 4 to develop the simplified analytical formulas with the main physics accounted for.

The left part of Fig. 5.10 shows the incoherent undulator radiation flux of $10^{6}$ electrons (0.16 pC) versus the opening angle of a circular aperture placed in the forward direction of electron traveling. As can be seen, with the increase of the aperture opening angle, the red-shifted part of the radiation grows. For the total flux, there are sharp spikes near the odd harmonics and no clear spikes near the even harmonics. This is due to the fact that there is no on-axis radiation at the even harmonics. Also note that with the change of the aperture opening angle, there are jumps in the flux at a specific frequency $n\omega _{0}$. This is due to the fact that the red-shifted radiation of higher harmonics $m>n$ can contribute to the flux at $\omega =n\omega _{0}$ when the aperture is large enough.

Fig. 5.10

Left: incoherent radiation photon flux of $10^{6}$ electrons within a circular aperture placed in the forward direction, whose largest opening angles are $\theta _{\text {max}}$. Right: coherent radiation photon flux of $10^{6}$ electrons within the circular aperture whose largest opening angle $\theta _{\text {max}}=8$ mrad. The beam current and bunching factor spectrum used in the calculation are shown in Fig. 5.9. Other related parameters: $E_{0}=250$ MeV, $\lambda _{0}=1064$ nm, $\lambda _{u}=125$ mm, $K=2.5$, $N_{u}=32$

Now we calculate the coherent radiation of the laser modulation-induced microbunched beam, using an RMS bunch length of 100 fs ($\sigma _{z}=30\ \upmu$m). An example beam current and bunching factor spectrum of the laser modulation-induced microbunched beam are shown in Fig. 5.9. We remind the readers that the bunch length in the real machine is typically longer than that used in the calculation here. Therefore, the coherent radiation is even more narrowbanded than presented in the example calculation in this section.

Fig. 5.11

Coherent radiation photon flux of $10^{6}$ electrons versus opening angle $\theta _{\text {max}}$ of the circular aperture for the first two harmonics, with $\sigma _{\bot }=0\ \upmu$m (up), $\sigma _{\bot }=100\ \upmu$m (middle) and $\sigma _{\bot }=400\ \upmu$m (bottom), respectively. The beam current and bunching factor spectrum used in the calculation are shown in Fig. 5.9. Other related parameters: $E_{0}=250$ MeV, $\lambda _{0}=1064$ nm, $\lambda _{u}=125$ mm, $K=2.5$, $N_{u}=32$

Fig. 5.12

Total radiation power as a function of the observation time gathered within a circular aperture with $\theta _{\text {max}}=1$ mrad, for $\sigma _{\bot }=0,100,400\ \upmu$m, respectively. The beam current and bunching factor spectrum used in the calculation are shown in Fig. 5.9. Other related parameters: $E_{0}=250$ MeV, $\lambda _{0}=1064$ nm, $\lambda _{u}=125$ mm, $K=2.5$, $N_{u}=32$

At first, we ignore the influence of electron beam’s transverse dimension, i.e., a thread beam is assumed. The coherent radiation spectrum of $10^{6}$ electrons is shown in the right part of Fig. 5.10. As can be seen, there is narrowband coherent radiation only at the modulation laser harmonics, which fits with expectation due to the fact that there is only notable bunching factor at the laser harmonics. More closer look of the first two harmonics versus the aperture opening angle are shown in the upper part of Fig. 5.11. As can be seen, indeed the relative bandwidth of the coherent radiation at the fundamental frequency and second harmonic are $1\%$ and $0.5\%$, respectively, agreeing with the values calculated from Eq. (4.21). In addition, the flux of the fundamental mode $H=1$ at the fundamental frequency agrees reasonably well with that according to Eq. (4.48), i.e., $\mathcal {F}_{1}(\omega =\omega _{0},\sigma _{\bot }=0\ \upmu {\text {m}})=3.4\times 10^{7}\ \text {(photons/pass/0.1\% b.w.)}$. In other words, the amplification factor of the flux at $\omega =H\omega _{0}$ is indeed $N_{e}^{2}|b_{z,H}|^{2}$ when $\sigma _{\bot }=0\ \upmu$m. Also note that the jumps of the flux with the change of aperture opening angle as we commented just now.

Now we investigate the impact of transverse electron beam sizes on the coherent radiation. As can be seen from the middle and bottom parts of Fig. 5.11, which correspond to a transverse electron beam size of 100 $\upmu$m and 400 $\upmu$m, respectively, and the comparison with the upper part, the transverse sizes of the electron beam suppress the coherent radiation. And the calculated fluxes at the fundamental frequency agrees well with those predicted by Eq. (4.49), i.e., $\mathcal {F}_{1}(\omega =\omega _{0},\sigma _{\bot }=100\ \upmu {\text {m}})=3.1\times 10^{7}\ \text {(photons/pass/0.1\% b.w.)}$ and $\mathcal {F}_{1}(\omega =\omega _{0},\sigma _{\bot }=400\ \upmu {\text {m}})=1.6\times 10^{7}\ \text {(photons/pass/0.1\% b.w.)}$. The suppression from transverse beam size is even more significant at the higher harmonics and the suppression factors agree with those predicted according to the transverse form factor Eq. (4.32). Also note that different from that of incoherent radiation, when $\sigma _{\bot }=100\ \upmu$m or $400\ \upmu$m, there is no visible jump of the flux with the aperture opening angle $\theta _{\text {max}}$ grown from 1 mrad to 8 mrad. This is because that the off-axis red-shifted coherent radiation of higher modes are suppressed now.

We have also confirmed our derivation of the coherent radiation power by comparing it with simulation. As shown in Fig. 5.12, the calculated peak powers of coherent radiation with different transverse electron beam sizes also agree well with the theoretical predictions from Eq. (4.46), i.e., $P_{1,\text {peak}}(\sigma _{\bot }=0\ \upmu \text {m})=363$ W, $P_{1,\text {peak}}(\sigma _{\bot }=100\ \upmu \text {m})=332$ W, $P_{1,\text {peak}}(\sigma _{\bot }=400\ \upmu \text {m})=168$ W.

From the calculation and analysis, we know that the coherent radiation from the formed microbunching is mainly at the fundamental frequency and second harmonic of the modulation laser, and in the forward direction. The coherent radiation is narrowbanded, and stronger than the incoherent radiation. These calculations and observation are the basis for our signal detection scheme.

5.2.4 Signal Detection and Evaluation

After investigating the microbunching formation beam dynamics and radiation characteristics of the formed microbunching in the above sections, now we consider how we can measure and evaluate the signals.

Fig. 5.13

(Figure from Ref. [1])

Evaluation of bunch charge based on the stripe line measurement. Blue dots are the measurement results with the systematic offset subtracted and the red curve is a fit by the sum of two exponential functions, $Q(t)=Q_{1}\text {exp}(-t/\tau _{1})+Q_{2}\text {exp}(-t/\tau _{2})$, performed at different time intervals, with the fit results connected by a smoothed line.

Measurement and evaluation of bunch charge The bunch-by-bunch charge (current) in the experiment is measured by a single-bunch current monitor [14], which analyses the electron beam-induced RF signals from a set of four stripline electrodes (3 GHz bandwidth). To minimize the influence of neighbouring bunches on the signal, the pulse response of the electrodes is reshaped by a 500-MHz low-pass filter. The current calibration of the monitor is conducted using a parametric current transformer [4] at higher current, and the linearity of the system at lower current is checked with the signal of the photodiode illuminated by synchrotron radiation. During the current decay in the experiment, one data point of the result given by the monitor is saved every second for each individual bunch. The averaged measurement of ten unfilled bunches preceding the homogeneous filled bunches (10 ns time gap in between) is used as the systematic offset. To smooth the measurement noise and at the same time account for the change of the beam lifetime, the time evolution of the offset-removed data points is then fitted by the sum of two exponential functions, $Q(t)=Q_{1}\text {exp}(-t/\tau _{1})+Q_{2}\text {exp}(-t/\tau _{2})$, at different time intervals, with the fit results connected smoothly. One example evaluation of the bunch charge measurement result is presented in Fig. 5.13. Based on the evaluated data, we obtain a linear bunch-charge dependence of the broadband incoherent signal that is detected by the photodetector without the 3-nm-bandwidth band-pass filter, as shown in Fig. 5.14, confirming the reliability of the bunch-charge measurement and evaluation method.

Fig. 5.14

(Figure from Ref. [1])

Linear dependence of the broadband incoherent undulator radiation on the bunch charge. a, Results corresponding to individual laser shots; the shading (light red) represents $3\sigma$ of the detection noise. b, The result after 200-consecutive-laser-shot averaging. The blue dots are the experimental data of a bunch not modulated by the laser and the red curves are linear fits.

If the ring works in single-bunch mode, there is a more direct and accurate method of bunch charge measurement based on the measurement of synchrotron radiation strength using a photodiode. Both methods have been used in our experiments.

Detection and evaluation of undulator radiation The long-pulse laser (FWHM $\approx$ 10 ns) is used to simplify the experiment by avoiding a dedicated laser-electron synchronization system, given that the shot-to-shot laser timing jitter is $t_{\text {jitter}}\le$ 1 ns (RMS). However, the photodetector (Femto HSPR-X-I-1G4-SI; rise/fall time, 250 ps) becomes saturated and even damaged by the powerful laser (Beamtech Optronics Dawa-200) if it is placed in the path of the laser. To address this issue, the undulator radiation is separated into the fundamental and second harmonics by appropriate dichroic mirrors (Thorlabs Harmonic Beamsplitters HBSY21/22), as shown in Fig. 5.3, and the signal detection at first focuses on the second harmonic with the wavelength centred at 532 nm. The photodetector output voltage, which is proportional to the radiation power, is then measured by a digital oscilloscope (Tektronix MSO64:6-BW-4000; bandwidth, 4 GHz; sample rate, 25 billion samples per second). Later we will also present the result of the 1064 nm radiation by implementing Pockels cells to block the modulation laser and let pass the radiation in the following turns.

An example radiation waveform of the second harmonic is shown in Fig. 5.16. To avoid the impact of the signal waveform offset caused by stray laser light, the data analysis takes the peak-to-peak value of the photodetector output voltage as a measure of the radiation power. The coherent radiation power that corresponds to each individual laser shot, obtained during a time interval with a decaying beam current, is presented in Fig. 5.15a, where the modest contribution on the measured quantity from the small amount of incoherent radiation transmitted through the 3-nm-bandwidth band-pass filter has been eliminated. As can be seen, the coherent signal fluctuates considerably from shot to shot. This is attributable to the shot-to-shot fluctuation of the laser intensity profile (see Fig. 5.7) and the measurement noise. Despite the fluctuation, quadratic functions fit reasonably to the lower and upper bounds of the data points, which correspond to the cases of minimum and maximum bunching factors induced by the fluctuating laser, respectively. When performing the fits, we took into account that the measured quantity is the real radiation signal convoluted with the detection noise. The impact of this noise, obtained by analysing the measurement result of the unfilled bunches, on the bounds of the measured data points is visualized as shading in Fig. 5.15a for the coherent signal and in Fig. 5.14a for the incoherent signal. To smooth this shot-to-shot fluctuation, a 200-consecutive-laser-shot averaging is conducted and the results are presented in Figs. 5.14b and 5.15b for the narrowband coherent and broadband incoherent signals, where a quadratic and a linear fit have been performed, respectively.

Fig. 5.15

(Figure from Ref. [1])

Quadratic dependence of the narrowband coherent undulator radiation generated from microbunching on the bunch charge. a, Results corresponding to individual laser shots; the shading (light red and grey) represents $3\sigma$ of the detection noise. b, The result after 200-consecutive-laser-shot averaging; the plot is the same as Fig. 5.17 and is presented again here for comparison with a and with the incoherent signal in Fig. 5.14. The blue dots represent the experimental data and the red curves are quadratic fits.

Fig. 5.16

(Figure from Ref. [1])

Waveforms of the undulator radiation produced from a homogeneous stored bunch train. a, b, Radiation one turn before the laser shot. The photodetector output voltage is proportional to the radiation power. c, d, Radiation one turn after the laser shot, from the same bunches as those of a and b, where the central five bunches are modulated by the laser pulse. The offset and general slight decreasing trend of the waveforms are due to the photodetector being saturated by stray light from the modulation laser one revolution before and not having completely recovered. e, f, Radiation one turn after the laser shot, obtained with a narrow band-pass filter (centre wavelength, 532 nm; bandwidth, 3 nm FWHM) placed in front of the photodetector, with bunch filling and charge similar to those in a to d.

5.2.5 Experimental Results

5.2.5.1 Second Harmonic Radiation

Figure 5.16 shows the typical measurement results of the second-harmonic undulator radiation emitted from a homogeneous stored bunch train, with a charge of about 1 pC per bunch and a time spacing of 2 ns, supplied by the 500 MHz RF cavity at the MLS. The spikes in the waveforms are the signals of different bunches. The left and right panels show the results corresponding to 2 and 40 consecutive laser shots, respectively. To smooth the measurement noise and signal fluctuation, the waveforms in the right panels have been averaged. Figure 5.16a, b shows the signals one turn before the laser shot, which correspond to the incoherent radiation and reflect the homogeneous bunch filling pattern. Figure 5.16c, d shows the radiation one turn after the laser shot, from the same bunches as those in Fig. 5.16a, b. The five larger spikes at the centre correspond to the bunches modulated by the laser. The enhanced signals of these five spikes indicate the formation of microbunching and the generation of coherent radiation from the laser-modulated bunches.

The laser used in the experiment has multiple longitudinal modes, and its temporal profile has several peaks and fluctuates considerably from shot to shot (see Fig. 5.7). Therefore, the laser-induced electron energy modulation amplitudes are different from shot to shot and from bunch to bunch. When the modulation amplitude matches the phase slippage factor, the energy-modulated electrons are properly focused at synchronous phases, which gives optimal microbunching. For some of the shots, the laser intensity is higher or lower than the optimal value, and the electrons are then over-focused or under-focused, giving weaker microbunching and less coherent radiation. This explains the shot-to-shot fluctuation of the coherent amplified signals shown in Fig. 5.16c, e.

As analyzed before, the microbunching coherent radiation is much narrowbanded compared to the incoherent radiation. To confirm that the amplified radiation is due to microbunching, we tested this narrowband feature of the coherent radiation. A band-pass filter (Thorlabs FL532-3; centre wavelength, 532 nm; bandwidth, 3 nm FWHM) was inserted in front of the detector. The radiation one turn after the laser shot is shown in Fig. 5.16e, f, which was obtained with a bunch filling and charge similar to that of Fig. 5.16c, d. From the comparison between Fig. 5.16e and c (Fig. 5.16f and d), we can see that the broadband incoherent signals are nearly completely blocked by the filter, whereas the amplified part is not affected much, confirming that the amplification is the narrowband coherent radiation generated by the microbunches.

Finally, we investigated the dependence of the coherent radiation on the bunch charge. To mitigate collective effects such as intrabeam scattering and head-tail instability, which could change the electron beam parameters, this investigation was conducted at low beam current, and the coherent signal was optimized by fine-tuning the machine to ensure a sufficient signal-to-noise ratio. Because the longitudinal radiation damping time in the experiment was 180 ms, we operated the laser at 1.25 Hz repetition rate to ensure that the electron bunches had time to recover their equilibrium parameters before each laser shot. The 3-nm-bandwidth band-pass filter was inserted to block the incoherent radiation, and the coherent signal corresponding to each individual laser shot was saved, with the beam current decaying naturally until the signal was at the detection noise level. The measurement results of the bunch closest to the laser temporal centre ($t = 0$ ns in Fig. 5.16) are used for quantitative analysis as introduced above. To lessen the impact of the laser temporal profile fluctuation and measurement noise, a 200-consecutive-laser-shot averaging is performed to obtain the data point for each bunch charge. The coherent undulator radiation power versus the single-bunch charge is shown in Fig. 5.17, where a quadratic function fits well to the experiment data. The quadratic bunch charge dependence, together with the narrowband feature of the coherent radiation, demonstrates unequivocally the formation of microbunching.

Fig. 5.17

(Figure from Ref. [1])

Quadratic dependence of the coherent undulator radiation generated from microbunching on the bunch charge. The blue dots represent the experimental data and the red curve is a quadratic fit. Each data point represents the averaged result of 200 consecutive laser shots. The error bars denote the standard deviation of the averaged results when the averaging time window shifts for $\pm 100$ consecutive laser shots from the corresponding data point.

Fig. 5.18

(Refer to J. Feikes’ talk in IPAC2021 and Ref. [2] for more details)

Raw data of the multi-turn microbunching preservation experiment result. The signal is for the fundamental-mode undulator radiation, i.e., 1064 nm.

5.2.5.2 Fundamental Frequency Radiation

The above deciding experimental results were obtained in the year of 2020, and have been published in Ref. [1]. After that, there are two main upgrades on the experimental setup. First, the multi-longitudinal-mode laser has been replaced by a single-longitudinal-mode one (Amplitude Surelite I-10). Second, Pockel Cells have been installed along the signal detection optical path to block the modulation laser and let pass the radiation in later revolutions, thus allowing the detection of the fundamental frequency radiation [15]. As shown in our analysis, we expect that the coherent radiation at the fundamental frequency is much stronger than that at the second harmonic and the microbunching can last multiple turns. These expectations have been confirmed in our following experimental investigations at the MLS [2].

Fig. 5.19

The bunch-charge scaling of the coherent undulator radiation signals of the first three turns after the laser shots. Up: each data point corresponds to the radiation signal strength after each single laser shot. Bottom: average of the radiation signal from 40 consecutive laser shots. The saturation level of the detector is about 2 V

Figure 5.18 shows the typical experimental results of the multi-turn coherent radiation at the fundamental frequency. A narrowband-pass filter has been inserted to select the narrowband coherent radiation. Figure 5.19 is the more quantitative data analysis of the signal of the first three turns after each laser shots. The bunch charge has now also been obtained in a more accurate way by using the incoherent synchrotron radiation signal measured by a photodiode. Several important observations are in order concerning the experiment results:

• First, signals of all three turns have shown nice quadratic bunch charge fits, confirming again the formation of microbunching and coherent radiation generation from it.

• Second, we mentioned in the above section and also in Ref. [1] that the huge shot-to-shot coherent radiation signal fluctuation obtained before the upgrades is mainly due to the laser profile fluctuation arising from its multi-longitudinal-mode nature. This argument has also been confirmed by the results shown in Fig. 5.19. From the comparison of Figs. 5.15a and 5.19, we can see that the shot-to-shot coherent signal with the present single-longitudinal-mode laser is much more stable.

• Third, the signal deviates from the quadratic scaling at high current and starts to saturates about 1.3 pC, which we believe is due to the influence of collective effects.

More in-depth investigations on the multi-turn microbunching is still ongoing and will be reported in the future [2].

5.2.6 Summary

In conclusion, we have demonstrated the mechanism of SSMB for the first time in a real machine. This demonstration represents the first milestone towards the implementation of an SSMB-based high-repetition, high-power photon source. As great as the experimental results are, to avoid confusion, here we make clear that here we do not report an actual demonstration of SSMB, but rather a demonstration of the mechanism by which SSMB will eventually be attained. First, the formation of microbunching after one complete revolution of a laser-modulated bunch in a quasi-isochronous ring and the maintenance of microbunching for multiple turns demonstrate the viability of a turn-by-turn electron optical phase correlation with a precision of sub-laser wavelength. Second, this microbunching is produced on the stored electron bunch, the equilibrium parameters and distribution of which before the laser modulation are defined by the same storage ring as a whole. The combination of these two crucial factors establishes a closed loop to support the realization of SSMB, provided that a phase-locked laser interacts with the electrons turn by turn.

5.3 PoP II: Quasi-steady-State Microbunching

On the basis of the PoP I, the next step is to replace the single-shot laser by a high-repetition phase-locked one to interact with the electrons turn-by-turn. By doing so, we want to form stable microbuckets to constrain the microbunching in it to reach a quasi steady state. This is the SSMB PoP II as introduced in the beginning of this chapter.

5.3.1 Phase-Mixing in Buckets

To reach a quasi steady state, the particles need to do synchrotron oscillations to reach phase mixing in the microbuckets, as a result of longitudinal amplitude dependent tune spread of the electron beam. Here we present a remarkable feature of phase mixing or filamentation in RF or optical buckets. As we will see soon, given an initial DC mono-energetic beam, there will be an equilibrium phase space distribution after phase mixing in the bucket. We find that in this final steady state, the beam current distribution has little dependence on the bucket height. This feature is favorable for the SSMB PoP II, as the requirement on the modulation laser power can then be much relaxed compared to PoP I. This effect is also of relevance to the injection process of the final real SSMB storage ring.

As the phase mixing is a rather fast process compared to radiation damping, we consider only the symplectic dynamics in this section for simplicity. The symplectic longitudinal dynamics of a particle in a storage ring with a single RF system, in SSMB a laser modulator, can be modeled by the well-known “standard map” [16]

$$\begin{aligned} {\left\{ \begin{array}{ll} &{}I_{n+1}=I_{n}+K\sin {\theta _{n}},\\ &{}\theta _{n+1}=\theta _{n}+I_{n+1}, \end{array}\right. } \end{aligned}$$

(5.15)

in which

$$\begin{aligned} \theta =k_{\text {RF}}z,\ I=R_{56}k_{\text {RF}}\delta ,\ K=\frac{V_{\text {RF}}}{E_{0}}R_{56}k_{\text {RF}}, \end{aligned}$$

(5.16)

with $R_{56}=-\eta C_{0}$. Note that K in this section is not the undulator parameter. Equation (5.15) can be described with the pendulum Hamiltonian driven by a periodic perturbation

$$\begin{aligned} H(I,\theta ,t)=\frac{1}{2}I^{2}+K\cos {\theta }\sum _{n=-\infty }^{\infty }\cos (2\pi nt). \end{aligned}$$

(5.17)

The dynamics is given by a sequence of free propagations interleaved with periodic kicks. For $K\ne 0$, the dynamics is non-integrable and chaotic. But for a K much smaller than 1, which is the case for usual storage rings working in the longitudinal weak focusing regime, the motion is close to integrable and the differences in Eq. (5.15) can be approximately replaced by differentiation, and the Hamiltonian Eq. (5.17) can be replaced by a pendulum Hamiltonain

$$\begin{aligned} H=\frac{1}{2}I^{2}+K\cos {\theta }. \end{aligned}$$

(5.18)

The separatrix of the pendulum bucket is $H=K$ with a bucket half-height of $2\sqrt{K}$. Or in unit of $\delta$, the bucket half-height is

$$\begin{aligned} \delta _{\frac{1}{2}}=\frac{2\sqrt{V_{\text {RF}}R_{56}k_{\text {RF}}}}{|R_{56}k_{\text {RF}}|}\approx \frac{1}{\beta _{z\text {S}}k_{\text {RF}}}, \end{aligned}$$

(5.19)

where $\beta _{z\text {S}}$ is the longitudinal beta function at the RF as analyzed in Sect. 2.1.2. The synchrotron tune is

$$\begin{aligned} \nu _{s}\approx -\text {sgn}(K)\frac{\sqrt{K}}{2\pi }. \end{aligned}$$

(5.20)

When K is large, the strongly chaotic dynamics can also be used for interesting applications, for example applying the bucket purification to generate short bunches as proposed in Ref. [17].

Figure 5.20 shows a simulation result of the evolution in the longitudinal phase space of a mono-energetic DC beam after injection into RF or optical buckets described by Eq. (5.15). We have chosen to observe the beam in the middle of the RF kick so the beam distribution in the longitudinal phase space is upright. As can be seen, there is a steady-state beam distribution due to phase mixing in the bucket. Note that the bucket center is at $\theta =\pi$ when $K>0$. If $K<0$, then the bucket center will be at $\theta =0$.

Fig. 5.20

Phase mixing (filamentation or decoherence) of a mono-energetic particle beam trapped in RF or optical bucket with $K=0.01$. Up: particle distribution in longitudinal phase space, with red curves being the separatrices. Bottom: the corresponding beam current distribution

As the longitudinal form factor, thus the coherent radiation power, depends more directly on the beam current (namely the longitudinal coordinate z of the electrons) rather than the energy spread, now we try to get an analytical formula for the steady-state beam current. For convenience, we shift the bucket center to the origin, which means $\theta -\pi \rightarrow \theta$, or a sign change of K. What we want to know is the steady-state distribution of $\theta$, i.e., $f(\theta ,t\rightarrow \infty )$. In action-angle $(\phi ,J)$ phase space, the distribution function evolves according to

$$\begin{aligned} f(\phi ,J,t)=f(\phi -\omega (J)t,J,0). \end{aligned}$$

(5.21)

When there is a tune dependence $\omega (J)$ on J, then in the limit of $t\rightarrow \infty$, the steady-state distribution depends only on the initial distribution of action J as a result of phase mixing,

$$\begin{aligned} f(\phi ,J,t\rightarrow \infty )=\frac{1}{2\pi }\int _{0}^{2\pi }f(\phi ,J,t=0)d\phi . \end{aligned}$$

(5.22)

The final angle for each action J will uniformly distributed in $[0,2\pi )$. As shown in Fig 5.21, after reaching the steady state, the percentage of the particles with $\theta \le x$ for $0<x<\pi$ is

$$\begin{aligned} P(\theta \le x)=\frac{x}{\pi }+\int _{x}^{\pi }\left( 1-\frac{\phi (x,I(x,\beta ))}{\pi }\right) \frac{1}{\pi }d\beta , \end{aligned}$$

(5.23)

in which $I(x,\beta )$ represents the I-coordinate of a point on the $(\phi ,J)$ phase space trajectory traversing $(\beta ,0)$ with a $\theta$-coordinate of x.

Fig. 5.21

A plot to help better understand Eq. (5.23)

After getting $P(\theta \le x)$, the current distribution can then be calculated according to

$$\begin{aligned} f(\theta )=\frac{\partial {P}}{\partial {x}}\bigg |_{x=\theta }. \end{aligned}$$

(5.24)

However, $\phi (x,I(x,\beta ))$ has a complex form, and it is hard to get a simple analytical expression for $f(\theta )$. Here we simplify the discussion by approximating all the phase space trajectories in the bucket by ellipses to arrive at an analytical formula for $f(\theta )$. For an ellipse phase space trajectory, we have

$$\begin{aligned} \phi (x,I(x,\beta ))=\arccos {\frac{x}{\beta }}. \end{aligned}$$

(5.25)

Note that the result in Eq. (5.25) has no dependence on K. For a real RF or optical bucket, there is a dependence of $\phi (x,I(x,\beta ))$ on K, but the dependence is weak, especially for trajectories close to the bucket center. So we expect our approximated Eq. (5.25) is valid to a large extent. Substituting Eq. (5.25) into Eqs. (5.23) and (5.24), we have

$$\begin{aligned} \begin{aligned} f(\theta )=\frac{\partial {P}}{\partial {x}}\bigg |_{x=\theta } =\int _{\theta }^{\pi }\left( \frac{1}{\pi \sqrt{1-\left( \frac{\theta }{\beta }\right) ^{2}}}\frac{1}{\beta }\right) \frac{1}{\pi }d\beta =\frac{1}{\pi ^{2}}\ln \bigg |\frac{\pi +\sqrt{\pi ^{2}-\theta ^{2}}}{\theta }\bigg |. \end{aligned} \end{aligned}$$

(5.26)

Note that our simplified theoretically current distribution $f(\theta )$ is independent of K, which means the steady-state current distribution is independent of the bucket height.

Fig. 5.22

The steady-state current distribution after phase mixing in RF or optical bucket, with different K. In each simulation, $2\times 10^{5}$ particles have been tracked for $2\times 10^{5}$ turns. Also presented in the figure is the theoretical prediction given by Eq. (5.26)

Figure 5.22 shows the simulation result of the steady-state current distribution under different K, i.e., different bucket heights, and simultaneously our simplified theoretical distribution Eq. (5.26). As can be seen that indeed the steady-state current distribution has little dependence on the bucket height, and our simplified analysis is quite accurate. Note that the origin is not shifted in the plot.

The analysis reveals a remarkable feature of phase mixing in RF or optical bucket, i.e., the final steady-state current distribution after a mono-energetic beam getting trapped by RF or optical buckets has little dependence on the bucket height. This is helpful for our Quasi-SSMB experiment since it means the requirement on the modulation laser power is not that demanding. A bucket height several times of the natural energy spread is sufficient.

The above result is based on a constant RF voltage in the phase mixing process, it can be anticipated that more particles will be bunched closer to the bucket center phase when we increase K after the beam reach its steady-state distribution after phase mixing. Similar steps to the above section can be invoked for calculating the new steady-state current distribution. A transformation of the action when K changes is all that needed. Figure 5.23 shows the simulation result of the steady-state current distribution by increasing K in two consecutive steps from 0.001 to 0.03. As can be seen, the current are more concentrated to the center after the increase of K.

Fig. 5.23

The steady-state current distribution after phase mixing in RF or optical bucket, with an increase of K from 0.001 to 0.03 in two consecutive steps. In each step, $2\times 10^{5}$ particles have been tracked for $2\times 10^{5}$ turns

A discrete change of K can boost bunching as shown in Fig. 5.23. However, it is not without sacrifice, as the filamentation process will result in longitudinal emittance growth. This emittance increase is unwanted in some cases. As well-studied in RF gymnastics [18], an adiabatic change of RF voltage or lattice parameters can manipulate the bunch length while preserving the longitudinal emittance. Similar ideas can also be applied to boost microbunching with little emittance growth. A simulation of trapping of microbunch with K linearly ramped from $1\times 10^{-6}$ to $1\times 10^{-2}$ is shown in Fig. 5.24. Note the drastic difference between Figs. 5.24 and 5.20. The spirit of adiabatic buncher [19, 20] is the same with adiabatic trapping introduced here, for enhancing microbunching while preserving longitudinal emittance which is useful for FEL and inverse FEL. The adiabatic trapping mechanism can also be applied in the beam injection of the SSMB or other storage rings whose momentum aperture is of concern. It is interesting to note the connection of adiabatic trapping with the microbunching process in a high-gain FEL [21,22,23].

Fig. 5.24

Trapping of particles with a linear increase of K from $1\times 10^{-6}$ to $1\times 10^{-2}$ in $10^{4}$ turns. Up: particle distribution in longitudinal phase space, with red curves being the separatrices. Bottom: the corresponding beam current distribution

5.3.2 Experimental Parameters Choice

As the quantum diffusion of z is large for the MLS lattice (26 nm RMS per turn at 250 MeV corresponding to optics in Fig. 5.2), it is not feasible to realize true SSMB inside a 1064 nm wavelength microbucket at the MLS. Therefore, the goal of SSMB PoP II is to accomplish microbunching for 100 to 1000 consecutive turns to reach a quasi steady state. Based on the beam physics and noises analysis, the tentative experimental parameters choice is as shown in Table 5.2. We have conducted numerical simulations based on the parameters set, from which we observe that the typical evolution of electrons in PoP II experiment can be divided into several stages.

• I: with the modulation laser turned on, the bunching factor reaches the maximum after about one quarter of the synchrotron oscillation period;

• II: after several synchrotron oscillation periods, the whole microbuckets are filled with particles like that shown in the right part of Fig. 5.20 as a result of phase mixing. Bunching factor after reaching this quasi-steady state will be stable if there is no quantum excitation or other diffusion effects;

• III: due to quantum excitation and various diffusion effects, energy spread starts to increase and particles continue to leak out the microbuckets and begin to hit on the vacuum pipe and become lost. Bunching factor in this stage decreases;

• IV: after a while, all the particles are lost in the end.

Table 5.2 Tentative parameters of the Quasi-SSMB experiment to be conducted at the MLS

To make the experiment more realistic, each time we fire the laser, we only want to accomplish the above stages I and II, but avoid III and IV, i.e., to avoid particle loss, otherwise it will be too time-consuming to do the experiment. This is based on the fact that preparing the beam and storage ring state is time-consuming, while the particle can be lost in milli seconds with the laser keep firing. This is why we aim for preserving mircorbunching for $10^{3}$ turns, instead of $10^{6}$ turns or a longer time. The experiment is under preparation and more progress will be reported in the future. One thing worth mentioning is that the second-harmonic bunching in the quasi-steady state is negligible at the MLS, therefore, fundamental frequency radiation detection is needed in SSMB PoP II.