Generating intense fully coherent soft x-ray radiation based on a laser-plasma accelerator

Chao Feng; Dao Xiang; Haixiao Deng; Dazhang Huang; Dong Wang; Zhentang Zhao

doi:10.1364/OE.23.014993

X-ray free-electron lasers (FELs) based on electron linear accelerators (linacs) hold the promise for opening up new frontiers of ultra-fast and ultra-small sciences at the atomic length scale. This has been proved with the operation of the x-ray FEL facilities such as Free-Electron laser in Hamburg (FLASH) [1], Linac Coherent Light Source (LCLS) [2], Spring-8 Angstrom Compact free-electron Laser (SACLA) [3], and Trieste FERMI [4]. However, x-ray FEL facilities are always so large and expensive that only a few countries can afford them.

Over the past decades, the concept of laser-plasma accelerator (LPA) has been evolved from an idea into a technology ready for first applications. The development of high-power lasers provide the way for the experimental generation of quasi-monoenergetic electron bunches with the accelerating gradient 3 orders of magnitude higher than conventional accelerators [5–10]. Thus it makes the LPAs quite attractive for the next generation of compact light sources applications, such as the soft x-ray undulator radiation sources or eventually high-gain FELs. The electron beams from LPAs are usually characterized by high energy at GeV level, high peak current above 10 kA, and low transverse emittance less than 0.1 μm. However, the large energy spread up to a few percent level limits its application in high-gain FELs. Various great efforts have been made to reduce the intrinsic energy spread in LPA community through various external injection schemes [11–16]. Meanwhile, in the FEL community, people are focusing on compensating the large energy spread effects in high-gain FELs by bunch decompression [17] or using a particularly designed undulator with transverse gradient [18].

Here, we will consider an alternative method based on the coherent harmonic generation (CHG) technique [19], which has been widely used in storage rings and high-gain FELs [20–23]. We note that if the beam can be pre-bunched with high bunching factor, intense radiation can be directly produced with a relatively short undulator and the exponential gain process of the FEL may not be needed. This proposal makes full use of the high peak current and low emittance of the electron beams from LPA while effectively mitigating the detrimental effect from the large energy spread in the meantime.

The standard CHG scheme consists of two short undulators separated by a small chicane. A seed laser with wavelength of $λ_{s}$ is used to interact with the electrons in the first undulator, so-called modulator, to generate sinusoidal energy modulation in the electron beam longitudinal phase space. Such energy modulation then develops into an associated density modulation when electrons traverse the small chicane, so-called dispersion section. Since the micro-bunched electron beam shows Fourier harmonic components, it is possible to produce fully coherent FEL radiation in the second undulator, so-called radiator, at higher harmonics of the seed. The output radiation peak power of CHG is strongly coupled with the peak current, bunching factor and transverse beam size of the electron beam but nearly independent of the electron beam energy spread [19]. The electron beams from LPA naturally have high peak current and small transverse emittance, and introducing sufficient microbunching at optical wavelength would greatly increase the CHG power output. However, generating the microbunching at nth harmonic of the seed requires the energy modulation amplitude to be approximately n times larger than the initial beam energy spread, which is very challenging for the case of LPA beam with large inherent large energy spread.

An alternative method for introducing strong density modulation into the electron beam with large initial energy spread relays on the so called angular modulation [23, 24]. It has been proposed [23] to use an intense seed laser operating at TEM01 mode to interact with the electron beam in the modulator for modulating the beam’s transverse divergence at the seed wavelength. After that, a particularly designed chicane is used for converting the angular modulation into longitudinal density modulation. Simulations show that ultra-high harmonic radiation in the soft x-ray region can be generated by using this technique. However, the required peak power of the seed laser pulses is very high (~TW) and a relatively complex beamline is generally needed for the realization of this technique. What’s more, to the best of our knowledge, it’s very challenging to operate an UV laser with the TEM01 mode.

In this paper, we propose a novel and easy-to-implement method to introduce strong coherent microbunching into the electron beam from LPA by using a wavefront-tilted seed laser. It is found that, besides the energy modulation, seed laser with wavefront-tilt can also introduce angular modulation at the same time, and this angular modulation can be simply converted into density modulation by a weak dipole magnet. The density modulation efficiency is not limited by the angular modulation amplitude but nearly only decided by the initial transverse electron beam parameters. So this technique provides a convenient way to enhance the high harmonic bunching of an electron beam with large intrinsic energy spread. With representative realistic beam parameters of LPA, we show the possibility of generating fully coherent femtosecond soft x-ray radiation with peak power up to GW level in a very short radiator.

The layout for the proposed scheme is shown in Fig. 1. A seed laser with wavefront-tilt is overlapped with the electron beam from LPA in the modulator for introducing angular modulation. After that, a weak dipole is adopted as the dispersion section for converting the angular modulation into density modulation. The electron beam is then sent through a radiator that resonates at high harmonic of the seed for the generation of short wavelength radiation. The relative wavefront-tilt of the seed laser to the electron beam can be realized either by sending the laser pulse through a prism [25] or reflecting the laser pulse with a grating. However, we found that the simplest way for generating the required wavefront-tilt is injecting a normal laser pulse into the modulator with a very small angle $θ$ .

Fig. 1 Schematic layout of the CHG beam line for the proposed technique.

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Here we assume a vertically polarized seed laser pulse with a relatively uniform transverse intense distribution in the area of laser-electron beam interaction. The relative wavefront-tilt angle of the seed laser to the electron beam is $θ$ in the horizontal direction. The electric field of the seed laser at time $t$ can be expressed as

E_{y} = E_{0} \sin (ω t + k_{z} z + k_{x} x + φ), E_{x} = E_{z} = 0,

where

E_{0}

,

ω

,

k_{z}

and

φ

are the seed laser field-amplitude, angular frequency, wave number in the longitudinal direction, initial phase and

k_{x} = k_{z} \tan (θ)

is the wave number in the horizontal direction due to the relative wavefront-tilt. The associated magnetic field is obtained from Maxwell’s equation:

\nabla \times E = - \frac{\partial B}{\partial t}, B_{x} = - \frac{E_{y} \cos θ}{c}, B_{z} = \frac{E_{y} \sin θ}{c},

where

c

is the speed of light. One can find from Eq. (2) a magnet field

B_{z}

in the longitudinal direction, which will introduce an angular kick to the electrons. As the needed wavefront-tilt angle should be very small, here after we adopt the approximate of

\sin θ \approx \tan θ \approx θ

, and we define

τ = θ

as the wavefront-tilt parameter.

Consider a planar undulator with a sinusoidal magnetic field in the horizontal direction and a period length of $λ_{u}$ . In the laboratory frame, the electron’s transverse velocity induced by the undulator magnet can be written as

υ_{y} = - \frac{K c}{γ} \cos (ω_{u} t),

where

K = e B_{0} / k_{u} m c

,

e

and

m

are the charge and mass of the electron,

B_{0}

is the peak magnetic field of the undulator,

k_{u} = 2 π / λ_{u}

,

γ

is the relativistic electron energy and

ω_{u}

is the angular frequency of the orbit. Under the resonant condition, the electron’s energy change after the interaction with the seed laser field can be found by solving the equation [19, 26]:

\frac{Δ γ}{γ} = \frac{e}{γ m c^{2}} \int_{0}^{L_{m} / c} E_{y} \cdot υ_{y} d t,

where

L_{m} = N λ_{u}

is the length of the modulator, and

N

is the period number of the modulator. The divergence change of the electron due to the angular kick from the wavefront-tilted seed laser can be calculated by

Δ x' \approx \frac{Δ υ_{x}}{c} = \frac{1}{c} \int_{0}^{L_{m} / c} \frac{e}{m γ} B_{z} \cdot υ_{y} d t = \frac{τ e}{γ m c^{2}} \int_{0}^{L_{m} / c} E_{y} \cdot υ_{y} d t,

Comparing Eqs. (4) and (5) we find that

Δ x' \approx τ \frac{Δ γ}{γ},

which means that the angular modulation amplitude is about

τ

times of the energy modulation amplitude. If we define the angular modulation amplitude as

A_{a}

and energy modulation amplitude as

A_{m}

, then the energy modulation and angular modulation induced by the seed laser can be written as

Δ γ (x, s) / γ = A_{m} \sin (k_{z} s + k_{x} x), Δ x' (x, s) \approx A_{a} \sin (k_{z} s + k_{x} x),

where

s

is the longitudinal position of the electron,

A_{a} = τ A_{m}

. We will show below that a very small angular modulation is sufficient for significantly enhancing the ultra-high harmonic bunching factor of CHG by using this technique. Furthermore, the bunching factor can be made almost independent of the intrinsic energy spread and the intrinsic transverse divergence, which is quite different from the conventional CHG technique.

The state of the electron can be characterized by $\vec{g} = {(x x' y y' z δ)}^{T}$ in the 6-dimensional phase space, where $δ = γ - γ_{0}$ is the particle energy divergence [27]. After passing through a linear Hamiltonian system with transfer matrix of $R$ , the electron’s state will be changed to $\vec{g_{1}} = R \vec{g}$ . Hereafter we neglect the uncoupled motion in vertical plane. Without losing generality, consider an electron beam from LPA with initial horizontal rms beam size, divergence and energy spread of $σ_{x}$ , $σ_{x'}$ and $σ_{γ}$ , respectively, the initial Gaussian distribution of the beam can be written as

f_{0} (x, x', δ) = N_{0} / {(2 π)}^{3 / 2} \exp (- x^{2} / 2 σ_{x}^{2}) \exp (- x'^{2} / 2 σ_{x'}^{2}) \exp (- δ^{2} / 2 σ_{γ}^{2}),

where

N_{0}

is the number of electrons per unit length of the beam. After a drift section between the LPA and the modulator with length of

L_{1}

, the horizontal position is changed to

x_{1} = x + L_{1} x'

. Then the electron beam is sent into a short modulator for energy and angular modulations. A particle’s divergence and energy are related to its initial values by

x_{1}' = x' + A_{a} \sin (k_{z} s + k_{x} x)

and

δ_{1} = δ + A_{m} \sin (k_{z} s + k_{x} x)

according to Eq. (7), and the beam’s distribution evolves to

\begin{array}{l} f_{1} (s, x_{1}, x_{1}', δ_{1}) = N_{0} / (^{2 π) 3 / 2} \exp {- [^{x_{1} - L_{1} x_{1}' + L_{1} A_{a} \sin (k_{z} s + k_{x} x)] 2} / 2 σ_{x}^{2}} \\ \times \exp {- [^{x_{1}' - A_{a} \sin (k_{z} s + k_{x} x)] 2} / 2 σ_{x'}^{2}} \exp {- [^{δ_{1} - A_{m} \sin (k_{z} s + k_{x} x)] 2} / 2 σ_{γ}^{2}}, \end{array}

Here, we neglect the modulator length

L_{m}

due to its small value compared with

L_{1}

. After passage through another drift section between modulator and a dispersion section with length of

L_{2}

, the horizontal position is changed to

x_{2} = x_{1} + L_{2} x_{1}'

. Then the electron beam is sent into a dipole with linear transfer matrix elements of

R_{51} = - R_{26} \approx D

and

R_{52} = - R_{16} \approx D L_{3} / 2

, where

D

and

L_{3}

is the bending angle and length of the dipole, respectively. As the dipole is very weak, we ignore the

R_{56}

here for simplicity. In the following simulation part, we will consider all the elements of the transfer matrix of the dipole and the length of the modulator. The particle’s longitudinal position is changed to

s_{1} = s + D x_{2} + D L_{3} x_{1}' / 2

, and the electrons distribution evolves to

\begin{array}{l} f_{2} (s_{1}, x_{2}, x_{1}', δ_{1}) = N_{0} / (^{2 π) 3 / 2} \exp {- [^{x_{2} - (L_{1} + L_{2}) x_{1}' + L_{1} A_{a} \sin ς] 2} / 2 σ_{x}^{2}} \\ \times \exp {- [^{x_{1}' - A_{a} \sin ς] 2} / 2 σ_{x'}^{2}} \exp {- [^{δ_{1} - A_{m} \sin ς] 2} / 2 σ_{γ}^{2}}, \end{array}

where

ς = k_{z} (s_{1} - D x_{2} - D L_{3} x_{1}' / 2 + τ x_{2} - τ L_{2} x_{1}')

is the final phase of the electron at the entrance of the radiator. Integration of Eq. (10) over

x_{2}, x_{1}'

and

δ_{1}

gives the beam density distribution

N_{p}

as a function of the longitudinal position

s

, which can be expanded into Fourier series as

N_{p} (s) = N_{0} [1 + 2 \sum_{n = 1}^{\infty} b_{n} \sin (n k_{z} s + ϕ_{n})]

. And the bunching factor at nth harmonic of the seed can be written as

b_{n} = J_{n} [- n k_{z} A_{a} D (L_{2} + L_{3} / 2)] \exp {- (1 / 2) {[n k_{z} (a σ_{x} + b σ_{x'})]}^{2}},

where

a = τ + D

,

b = τ L_{1} + D L

and

L = L_{1} + L_{2} + L_{3} / 2

. For the harmonic number

n > 4

, the maximal value of the Bessel function in Eq. (12) is about

0.67 / n^{1 / 3}

and is achieved when its argument is equal to

n + 0.81 n^{1 / 3}

. For a given value of

A_{a}

, the optimized strength of the dispersion is

D = - (n + 0.81 n^{1 / 3}) / n k_{z} A_{a} (L_{2} + L_{3} / 2)

. The maximal value of the second term in the right-hand side of Eq. (12) will be achieved either by making

a = 0

for a small

σ_{x'}

, or

b = 0

for a small

σ_{x}

. For the LPA case, one should choose

b = 0

, i.e.

τ = - D L / L_{1}

, as the initial transverse beam size is very small (<1μm). Under this condition, the maximal bunching factor of the proposed technique only relies on the initial transverse beam size and is independent of transverse angular divergence and beam energy spread when we ignore the

R_{56}

of the dipole. Thus makes the maximal value of the nth harmonic bunching factor approach

0.67 / n^{1 / 3}

, which is much larger than a conventional CHG. We will show below that the effects of

R_{56}

on the bunching factor will become significant only at ultra-high harmonics.

To illustrate the physical mechanism and give a possible application with realistic parameters, three-dimensional simulations have been carried out for the generation of coherent soft x-ray radiation at 13 nm by using this proposed technique. Here we consider a LPA operating at 1 GeV with normalized emittance of $γ_{0} ε_{x, y} ~ 0.1 μ m$ in both horizontal and vertical directions, intrinsic relative energy spread of 1%, rms bunch length of 3 fs and a peak current of 10 kA. The intrinsic transverse beam size at the exit of the LPA is assumed to be about $σ_{x, y} ~ 1.0 μ m$ , which makes the intrinsic transverse divergence of about $σ_{x', y'} ~ 50 μ r a d$ . After a drift section with length of 3 m, the electron beam is sent into a short modulator with 2 periods and a period length of 8 cm. Simulations for the modulation process was done with a three-dimensional algorithm [28] based on the fundamentals of electrodynamics when considering the appearance of electric and magnet fields of a wavefront-tilted seed laser beam according to Eqs. (1) and (2). The peak power of the 260 nm seed laser is about 5 GW, the FWHM pulse length is around 30fs and the laser waist size in the modualtor is over 200 μm. The drift length between the modulator and the dispersion section is 1 m.

According to the optimized method mentioned above, to enhance the 20th harmonic bunching factor of the seed, we get the optimal dispersion strength and wavefront-tilt parameters are $R_{51} = 0.0037$ and $τ \approx - 0.0049$ . The length of the dipole is chosen to be 5 cm, which makes other elements of the transfer matrix of the dipole $R_{12} = 0.05 m$ , $R_{16} = - R_{52} = - 9.17 \times 10^{- 5} m$ and $R_{56} = - 1.12 \times 10^{- 7} m$ . With these parameters, Fig. 2 gives simulation results for electron distributions in $x - x' - z$ space at the exit of the modulator. One can find that an angular modulation has been imprint on the electron beam with a wavefront-tilt just as the seed laser. This wavefront-tilt in the angular modulation can be corrected by the following weak dipole, and the coherent micro-bunching will form simultaneously as Fig. 3 (a) shows. The energy modulation amplitude and angular modulation amplitude induced by the seed laser are about 3 MeV and 13 μrad, which are much smaller than the intrinsic values of the electron beam. Thus one cannot find the sinusoidal curve in $x' - z$ space but only the strong microbunching around the zero-crossing phase of the seed laser. Figure 3(b) gives the corresponding bunching factor for various harmonic numbers. The bunching factor distribution from simulation (blue solid) fit well with the one-dimensional analytical results (red dot) that was calculated by Eq. (12) for small harmonic number. However, the difference becomes prominent as $n$ exceeds 20. This deviation is mainly caused by the $R_{56}$ of the dipole, which can be decreased by increasing the angular modulation amplitude or using a longer drift section between the modulator and the dipole. The bunching factor at 20th harmonic is over 15%, which is sufficient for producing intense high harmonic radiation.

Fig. 2 Electrons distribution in $x - x' - z$ space after the angular modulation.

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Fig. 3 $x' - z$ space distribution in 5 seed laser wavelengths (a) and the corresponding bunching factor at various harmonic numbers (b) at the entrance to the radiator.

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After the dipole, the electron beam is sent into the radiator undulator with period length of 3 cm and total length of 1.5 m. The CHG process was simulated by the time-depended mode of GENESIS [29]. Figure 3 summarizes the FEL simulation results at wavelength of 13 nm. For comparison purpose, simulations were also performed for self-amplified spontaneous emission (SASE) [30, 31] case with the same electron beam at the entrance to the modulator of CHG. Figure 4(a) shows the simulated FEL peak power growth. The large bunching factor offered by the angular modulation is responsible for the initially steep quadratic power growth. The significant enhancement in FEL performance is clearly seen. After 1 m long of the radiator, the output peak power is around 1 GW, which is over 3 orders of magnitude higher than the SASE case at the same position. This output power is already comparable to the saturation power of a soft x-ray facility based on the conventional linear accelerator.

Fig. 4 (a) FEL peak power gain curves for the proposed CHG (solid blue) and SASE (dashed red) and corresponding typical single shot spectra at the exit of the radiator. The insert in (b) plots the transverse spot for CHG FEL at the same position.

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Because of the microbunching is generated through electron beam manipulation in $x' - z$ space, the output radiation wavelength is naturally immune to the residual energy chirp in the electron beam. Thus makes the output spectrum much cleaner than a conventional CHG. The output FEL spectra and radiation spot for the CHG case are illustrated in Fig. 4(b). The output bandwidth of CHG is about 0.3%, which is quite close to the transform-limited considering the 3 fs pulse length, and is over 10 times narrower than that of the SASE case.

The performance of the proposed technique may be affected by some practical issues of the machine, such as the shot-to-shot fluctuations of the laser power and incidence angle, and the magnetic field instability of the dipole. The sensitivity of the 20th harmonic bunching factor to the instability of the machine has been studied by introducing normal random fluctuations of the laser power of 5%, incidence angle of 0.1%, and dipole magnetic field of 0.01%. These values are feasible with state-of-the-art laser and accelerator technologies. The resulting 1000 shots of the fluctuations of the 20th harmonic bunching factor are shown in Fig. 5. One can find that the 20th harmonic bunching factor can be well maintained over 14%. These results demonstrate that the proposed CHG scheme is able to generate fully coherent soft x-ray radiation with good transverse mode and high intensity by fully using the advantage of the electron beam from LPA. This kind of university-laboratory scale soft x-ray light source would allow one to perform many challenging experiments which require narrow-bandwidth short wavelength radiation.

Fig. 5 20th harmonic bunching factor at various shots for random fluctuations of the laser power of 5%, incidence angle of 0.1%, and dipole magnetic field of 0.01%.

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Besides LPA, the proposed scheme also has other potential applications involving the use of electron beam with large intrinsic energy spread from storage rings, especially the ultimate storage rings. And it may also be used to enhance high harmonic bunching and improve the output spectrum of a seeded high gain FEL. In order to extend the proposed scheme to shorter wavelength, e.g. in the “water window” or even sub-nanometer wavelength regions, one may need to increase the seed laser power and reduce the horizontal emittance of the electron beam from LPA for further increasing the harmonic up-conversion efficiency. Further investigations on these topics are ongoing.

Acknowledgments

The authors would like to thank L.H. Yu (BNL), B. Li, T. Zhang and Z. Huang (SLAC) for helpful discussions and useful comments. This work is supported by the Major State Basic Research Development Program of China (2011CB808300 and 2015CB859700) and the National Natural Science Foundation of China (11475250, 11175240, 11275253, and 11322550).

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Generating intense fully coherent soft x-ray radiation based on a laser-plasma accelerator

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