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Abstract
Coherence time is one of the fundamental characteristics of light sources. Methods based on autocorrelation have been widely applied from optical domain to soft X-rays to characterize the radiation coherence time. However, for the hard X-ray regime, due to the lack of proper mirrors, it is extremely difficult to implement such autocorrelation scheme. In this paper, a novel approach for characterizing the coherence time of a hard X-ray free-electron laser (FEL) is proposed and validated numerically. A phase shifter is adopted to control the correlation between X-ray and microbunched electrons. The coherence time of the FEL pulse can be extracted from the cross-correlation. Semi-analytical analysis and three-dimensional time-dependent numerical simulations are presented to elaborate the details. A coherence time of 218.2 attoseconds for 6.92 keV X-ray FEL pulses is obtained in our simulation based on the configuration of Linac Coherent Light Source. This approach provides critical temporal coherence diagnostics for X-ray FELs, and is decoupled from machine parameters, applicable for any photon energy, radiation brightness, repetition rate and FEL pulse duration.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Corrections
3 April 2020: Typographical corrections were made to the author affiliations and abstract.
1. Introduction
Hard X-ray free-electron laser (FEL) [1–5] opens the door to a new era of X-ray experiments in various research fields, e.g., physics [6], chemistry [7], life science [8] and material science [9]. Combined with ultra-short duration, high resolution, high photon flux, hard X-ray FELs become powerful tools to capture simultaneous information on atomic spatial structure and ultra-fast dynamics. At present, most of the hard X-ray FELs are operating in self-amplified spontaneous emission (SASE) scheme, in which spontaneous radiation from the electron beam is amplified along the magnetic field in undulators. Due to starting from electron shot noise, SASE FELs usually are generated with imperfect temporal coherence corresponding to temporally isolated spikes [10]. As one of the fundamental characteristics of the X-ray pulse, pre-known information about X-ray coherence time would potentially benefit experiments including but not limited to, ionization dynamics [6], spectro-holography [11], nonlinear mixing-wave experiment [12], etc. Hence, there is a growing demand for the community to develop a time-domain coherence time characterization method for the ultra-fast hard X-ray pulses.
For FEL coherence time characterization, the most straightforward way is to directly implement conventional optical method, autocorrelation, which is a widely used method for coherence time characterization for optical radiation pulses. Typically, the radiation pulse is split into two identical pulses, then these two pulses travel through different optical path and the delay between them is controlled by optical mirrors, eventually these two pulses are recombined. The output pulse energy versus delay shows the information of the input pulse coherence time. It’s clear that the autocorrelation itself has a strong dependence on the mirrors to control the optical delays, which, in turn, limits the photon energy range of its application. It has been proven that a combination of laser beam splitter and mirror based optical delay can be used to implement autocorrelation to characterize the FEL coherence time in the extreme ultraviolet and soft X-ray regime [13,14]. However, different from its implementation in these two frequency regimes, the autocorrelation is difficult to realize in hard X-ray regime. Although there are existing grazing incidence mirrors for hard X-rays, the small incidence angle makes them unpractical to be utilized for conventional autocorrelation setups. Experiments employing crystals as mirrors to generate an effective delay has been carried out to measure the coherence time of the monochromatic hard X-ray pulses [15]. However, it is not appropriate to directly implement this method to SASE FEL coherence time characterization. Compared with monochromatic X-ray pulse, SASE FEL has much shorter coherence time and wider bandwidth. Due to Bragg diffraction, these crystals introduce a strong purification on the spectrum of the X-ray pulse, leading to a considerable overestimation of the intrinsic SASE FEL coherence time. Hence, currently, there is no effective method to characterize the coherence time of SASE FEL in time domain. With the development of single-shot transmissive spectrometer, one can measure the FEL spectrum shot by shot [16,17] and according to the bandwidth to estimate the coherence time. However, due to FEL frequency chirp and the electron imperfectness in a real machine, e.g., energy chirp, unexpected dispersion [18], the estimated coherence time may deviate from the actual coherence time. Meanwhile, X-ray pulse duration characterization method based on FEL dynamics has been well established recently [19,20], in which the cross-correlation between ‘fresh’ electrons and X-rays has been used to characterize the X-ray pulse length. However, in this case, the coherence time information is not included in the cross-correlation method since the electrons are almost ‘fresh’.
Here, we further advance this scheme to characterize the coherence time of hard X-ray FEL pulses by cross-correlating the X-ray pulse and the microbunched electron beam. In this method, a phase shifter is employed to introduce an arbitrary delay between the microbunched electrons and the X-ray pulse with the aim to control their correlation. Benefited from the FEL dynamics, the temporal coherence characteristics of the X-ray pulses are mapped to the cross-correlation, which is measurable. Then the corresponding coherence time can be obtained by decoding the information from the measured cross-correlation. In the following, the physics interpretation and the semi-analytical analysis of the coherence time characterization approach would be presented in the section of method. In the section of simulation, we would show the 3D time-dependent simulation result of this approach.
2. Method
In this section, the physics interpretation and semi-analytical analysis of the approach we proposed will be introduced. Figure 1 illustrates the configuration of this characterization method and depicts the essence of how it works. In this approach, the whole measurement system can be considered as three parts. The SASE FEL to be measured is generated in the first part with a few sections of undulator. The second part is the phase shifter used to control the delay between the X-ray pulse and the microbunched electrons to introduce the correlation. Then the third part, a short radiator, works on converting the correlation to X-ray pulse energy, which is measurable.
Fig. 1. Schematic description of hard X-ray SASE FEL coherence time characterization based on cross-correlation between microbunched electrons and X-ray pulse. Hard X-ray SASE FEL is generated in the before-delay undulators, then a phase shifter is employed to generate a relative delay between the electrons and X-ray, and finally the after-delay undulator converts the correlation to X-ray pulse energy.
In a SASE FEL, a bunch of highly relativistic electrons are injected into a periodically varying magnetic field, known as undulator. While traveling through the undulator, the electrons are forced to wiggle transversely and emit spontaneous radiation. This process produces ultra-short bursts of X-rays at the wavelength,
where $\lambda _{u}$ is the undulator period, $\gamma$ is the electron beam Lorentz factor and $a_{u}$ is the RMS undulator parameter.Due to the on-axis speed difference between the electrons and X-rays, there is a natural slippage between the electrons and X-rays (slippage effect). While the electron beam traveling through the undulator, the slippage effect gradually becomes more evident. This increasing slippage $l_{s}$ between X-rays and electrons determines the cooperate-interaction range of them and develops coherence spikes. Hence, within cooperate-interaction range, the X-ray and the electrons have a strong correlation [21]. Since these coherence spikes are all evolved from shot noise, X-rays and electrons in different coherence spikes will not have much correlation. It is now clear that the coherence time of the FEL pulse is determined by the length of the temporal coherence spike. In spite of some coherence enhanced schemes [22–24], the length of the coherent spike is usually much smaller than the pulse length of hard X-ray FELs, which leads to imperfectness on temporal coherence. For example, LCLS, operating in the mode of hard X-ray SASE FEL, generates X-ray pulses with tens of coherence spikes [25–27].
By employing a phase shifter to control the delay between the microbunched electrons and the X-rays, we can control the correlation between them. Then, additional undulators after the phase shifter, about one FEL gain length, can be used to convert the correlation to X-ray pulse energy, which can be measured by gas detector (shot-to-shot FEL pulse energy measurement device [28]). It is expected that, the pulse energy should oscillate according to the delay with a decaying envelop amplitude. The pulse energy oscillation comes from the phase mismatch between the microbunched electrons and X-ray pulse in the scale of X-ray wavelength, similar to a two-wave interference model. The decaying amplitude comes from the decreasing correlation, in the scale of coherence time. According to this, we can infer that there is an oscillation amplitude diminishing point after which the amplitude is negligible. This feature of SASE FEL can be used to experimentally characterize the coherence time of hard X-ray pulses.
The above process can be described with the well-known one-dimensional model analytically. Following the method in Ref. [25], universally scaled collective variables are introduced to describe the FEL dynamics. The normalized radiation field is defined as $A=E/(4\pi m_ec^{2} \gamma \rho n_{e})^{1/2}$ where $E$ is the radiation field, $m_e$ is the electron mass, $c$ is the speed of light, $n_{e}$ is the electron beam density and $\rho =(a_{u}\omega _{p}/4ck_{u})^{2/3}/{\gamma }$ is the Pierce parameter, with $\omega _{p}=(4\pi e^{2}n_{e}/m)^{1/2}$, the plasma frequency and $k_{u}=2\pi /\lambda _{u}$, the undulator wave number. The bunching factor can be defined as $B=\frac {1}{N_{\lambda }}\sum _{j=1}^{N_{\lambda }} e^{-i\theta _{j}}$, where $\theta _{j}=(k_{u}+k_{r})z-ck_{r}t_{j}$ is the ponderomotive phase, $k_{r}$ is the radiation wave number, $z$ is the coordinate along the undulator axis, $t_{j}$ is time and $N_{\lambda }$ is the number of electrons within one radiation wavelength $\lambda _{r}$. Considering the part before the phase shifter, it is a typical high-gain SASE FEL process. Since we mainly care about the phase relationship between the radiation and microbunching in the time domain, the steady state model in the following is good enough to describe the FEL dynamics,
Normally, the phase shifter introduces two effects: (1) time delay, due to the electron beam going though a bypass (2) bunch compression effect, because of the in-chicane path difference between electrons with different energy. The path difference will change the relative longitudinal position between electrons, which will potentially change the microbunching structure of the electron beam. In our case, the requirement of the delay time is at a magnitude of hundreds of attoseconds, and energy difference between electrons induced by the FEL process is not strong, so that the longitudinal position variation of electrons are pretty small compared to the radiation wavelength. Hence, in the theoretical analysis, we only consider the time delay effect. If we denote the radiation field and bunching factor at the end of the first part of the undulator, which is also the phase shifter entrance, as $A_{s}(\overline {z}_{p},t)$ and $B_{s}(\overline {z}_{p},t)$, where $\overline {z}_{p}$ is the position of the phase shifter, the radiation field and bunching factor at the end of the phase shifter will be $A_{s}(\overline {z}_{p},t)$ and $B_{s}(\overline {z}_{p},t+\Delta t)$ respectively, where $\Delta t$ is the delay induced by the phase shifter. Here, we suppose the phase shifter length is negligible. For the after-delay part, we use a short radiator to convert the correlation between the microbunched electron beam and the X-ray to X-ray pulse energy. We can regard the process in the after-delay part as coherent emission, since the radiator length is not sufficiently long for high-gain process. During coherent emission, the electron beam bunching factor can be regarded as a constant and the radiation field keeps increasing. Namely, only the first equation in Eq. (2) is valid with the bunching term as a constant. In this way, we get the radiation field $A_{c}(\overline {z},t)$ at the end of the radiator as,
Furthermore, according to Eq. (6) and Eq. (7), one can easily find that,
Here, we present a case with the X-ray photon energy being 6.92 keV and $\sigma _{t}$ in Eq. (5) equals to 50.59 attoseconds (about 85 $\lambda _r$), to illustrate the analysis above. According to Eq. (8), the corresponding coherence time is 179.36 attoseconds. Supposing that $A_{j}$ comes from Rayleigh distribution, $\phi _{j}$ comes from uniform distribution and the modes number $n$ is 50, we can obtain the temporal profile of the output X-rays at the end of the radiator [29]. The corresponding power profile (first several coherence spikes) is shown in Fig. 2(a). After integrating the power profile along $t$ with each specific $\Delta t$, we can plot Eq. (6), which is shown in Fig. 2(b). As mentioned in Eq. (7), the variance of the output pulse energy can be used to extract the coherence time information. Figure 2(c) presents the moving variance of the output pulse energy (blue) and the corresponding Gaussian fitting (red). The reconstructed coherence time based on the proposed method is about 174.86 attoseonds, which is pretty close to the coherence time obtained by initial parameters: 179.36 attoseconds as mentioned above.
Fig. 2. SASE FEL temporal coherence length measurement based on analytical formula. (a) Temporal power profile of the first several coherence spikes. (b) Pulse energy vs. phase shifter induced delay. (c) The moving variance obtained by Eq. (7) together with its fitting curve.
3. Simulation
To further demonstrate the reliability of this method, we use GENESIS 1.3 (version 2), a well-benchmarked three-dimensional time-dependent FEL simulation code [30], to simulate the FEL process. For the phase shifter part, both phase space bunch compression and time-delay effects have been carefully considered. Technically, for the particle distribution manipulation, we model the chicane with its first order transport matrix, which is a build-in element in GENESIS 1.3. For the time delay effect, since the electron beam and radiation field are sliced in GENESIS 1.3, we have to separate the time delay into two parts: integer (wavelength) delay and fractional (wavelength) delay. The simulation time window of GENESIS 1.3 is moving with the electron beam, so that the integer delay can be realized by adding zero-intensity field at the tail of three-dimensional radiation field. For the fractional delay, we deal with it by adding a corresponding phase to each particle. The simulation configuration is based on the normal LCLS SASE FEL parameters [31]. The hard X-ray self-seeding chicane, formed by four dipoles, is employed as the phase shifter to generate a bypass for the electron beam. The key simulation parameters are presented in Table 1.
Table 1. Key parameters for hard X-ray FEL pulse coherence time characterization
The minimum step size of the delay generated by the chicane at LCLS is about 0.875 attosecond for 12.48 GeV electron beam. Thus, to be close to future experiment, in our simulation, the step of scan is set to 0.875 attosecond and the total scan range is set to 520 attoseconds according to the coherence time estimation based on analytical works [29]. In actual FEL facilities, the location of the phase shifter and gas detector are always fixed. As mentioned before, to maximize the pulse energy oscillation contrast, the after-delay radiator length should be around one FEL gain length. Normally, the undulator system after the chicane is much longer than that. Thus, to control the after-delay FEL interaction length, we use quadrupole magnets to distort the electron beam orbit after the radiator with a designed length. Then, after that quadrupole, the coupling between the X-ray and electrons is extremely weak. Namely, the after-delay radiator is controlled to be proper to maximize the pulse energy oscillation contrast. In Fig. 3, we present a simulation comparison between distorted orbit case and normal orbit case with the parameters in Table 1. In this simulation, the electrons and X-rays still have one undulator section to perform intense interaction after the phase shifter. Right after that undulator section, quadrupole magnet is tuned to a large value to destroy the electron beam orbit. In Fig. 3, it is pretty clear that the FEL gain has been interrupted after the orbit-distortion quadrupole magnet. Thus, the after-delay FEL interaction length is controlled to be one undulator section.
Fig. 3. Simulation comparison between distorted electron beam orbit case and normal electron beam orbit case. Radiation power as a function of the length electrons going through in the undulator.
As Eq. (10) indicates, multi-shot average is essential to reduce the impact of the stochastic effects in SASE scheme. Nine cases with different shot-noise generator for each specific delay were simulated. Although it is apparent that more cases with different startup shot-noise will increase the simulated measurement precision, thousands of time-dependent GENESIS simulation is really computational expensive and nine cases are enough to show the expected physics. The simulation result is presented in Fig. 4, in which Fig. 4(a) illustrates the radiation power profile, Fig. 4(b) shows the pulse energy versus delay and Fig. 4(c) presents the moving variance of pulse energy corresponding to the delay. We can find out that there are many time-domain isolated coherence spikes within a SASE FEL pulse in Fig. 4(a). And in Fig. 4(b), light gray points represent for simulations with different noise at certain delays and the black line is the average radiation energy over different shot-noise cases, which indicates that if the delay is quite small the pulse energy oscillation mainly comes from the varying correlation between the microbunched electrons and the X-rays, and if the delay is sufficiently long, the pulse energy fluctuation is dominated by shot-noise. As Eq. (7) indicates, we use Gaussian function to fit the pulse energy oscillation amplitude (blue line in Fig. 4(c)) and then the fitting curve is shown as the red line in Fig. 4(c), with a corresponding fitted $\sigma _{t}=$61.55 attoseconds. According to Eq. (8), we can find that the reconstructed hard X-ray FEL coherence time under our simulation configuration is 218.2 attoseconds. For comparison, we dump the three-dimensional radiation field generated by GENESIS 1.3 and calculate the coherence time according to its definition,
Fig. 4. Simulation result of the coherence time characterization method. (a) Radiation power profile. (b) Pulse energy versus delay. (c) The moving variance obtained by Eq. (7) together with its fitting curve.
One may notice that, the fitted Gaussian function in Fig. 4(c) differs from our theoretical prediction, i.e., not centered at zero. This difference comes from the slippage effect in the after-delay undulator. Suppose the chicane induced time-delay is $\Delta t_1$ and the slippage effect induced time-delay is $\Delta t_2$, so that the total delay is $\Delta t=\Delta t_1 + \Delta t_2$. In Sec. Method, $\sigma ^2_{W(\Delta t)}$ is a Gaussian function (centering at zero) of the total delay $\Delta t$. Hence if the slippage effect in the after-delay undulator comes in, when we are doing Gaussian fitting with chicane induce delay $\Delta t_1$ and pulse energy oscillation amplitude, it is clear that the center of the Gaussian function would be -$\Delta t_2$. However, the center shift does not affect the result of our coherence time characterization since we only care the standard deviation of the fitted Gaussian function. Furthermore, in a real machine, electron-beam parameter jitter, e.g. peak current and emittance jitter, may lead to the fine pulse energy oscillation amplitude deviate from our semi-analytical analysis, Eq. (7). However, if the jitter is not very strong, the pulse oscillation amplitude which we extract from measured pulse energy would just fluctuate around its expected value. The jitter’s impact on the coherence time characterization can be effectively suppressed by the Gaussian fitting.
4. Conclusion and discussion
In conclusion, a novel approach for hard X-ray FEL coherence time characterization is proposed and its feasibility is investigated by semi-analytical method and three-dimensional time-dependent numerical simulations. Our work shows that hard X-ray FEL coherence time information can be obtained by measuring the cross-correlation between microbunched electrons and X-ray pulse. Our method fills the vacancy of coherence time characterization in time domain for hard X-ray FELs. It provides powerful diagnostic tools for hard X-ray FEL temporal coherence monitoring. Researches working on ultra-fast dynamics, structure imaging and spectrum reconstruction would potentially benefit from the pre-known coherence time measured before experiments. It is worthwhile to point out that the approach we introduced is also applicable to seeded FEL, either external seeded [32] or self-seeding [33]. Normally, it is presumed that a seeded FEL will have fully temporal coherence; yet, due to SASE components, or electron bunch imperfectness, the seeded FEL temporal coherence can be degraded [18]. The approach introduced in this manuscript can then be adopted to check this situation. This approach focuses on average temporal coherence time characterization. Namely, the measurement result shows the overall longitudinal property of radiation pulse based on the existing FEL setup. Moreover, the method builds upon the underlining FEL physics, which means it be can be applied to any high-gain FEL regime, like high repetition facilities [34] and ultra-fast FEL scheme [35].
Funding
U.S. Department of Energy (DE-AC02-76SF00515); U.S. DOE Office of Science Early Career Research Program (FWP-2013-SLAC-100164).
Acknowledgment
G.Z. thanks Weilun Qin, Jingyi Tang, Haoyuan Li, Zhibin Sun and Zhengxian Qu for useful discussion.
Disclosures
The authors declare no conflicts of interest.
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