Generating high repetition rate X-ray attosecond pulses in a diffraction limited storage ring

To steer and track electron motion in atoms, molecules, and nanostructures, light pulses with attosecond duration and high repetition rate are required. In this paper, we use the angular dispersion-induced microbunching scheme and a few-cycle laser within a straight section (a few meters) of a diffraction-limited storage ring to generate a coherent high-flux attosecond pulse in the water window region. Simulation results based on the Southern Advanced Photon Source indicate that the proposed method can generate a chirp-free Fourier transform limited pulse with a minimum duration of 50 as, a maximum repetition rate of a few MHz, and a maximum average flux of about \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$4.4\times 10^{11}$$\end{document}4.4×1011 photons/s/1%Bw.

In this paper, we propose a novel method that combines angular dispersion-induced microbunching (ADM) [18][19][20][21] and a few-cycle laser to generate attosecond pulses in a DLSR.The ADM method takes advantage of the low emittance in the vertical direction of the storage ring, which can achieve microbunching in a single modulation and can be implemented in a straight section.In addition, the ADM method requires less laser power to achieve the same performance as EEHG and other similar methods [22][23][24][25][26][27][28] .
Note that after a modulation, the next modulation cannot be performed immediately due to the reduced beam quality.Therefore, the repetition rate of the proposed method (and other similar methods) is limited by the recovery time of the beam quality compared to the steady-state microbunching (SSMB) 29,30 or the reversible modulation method 20 .However, there is no suitable method to implement the SSMB and the reversible modulation method in a normal DLSR.To further increase the repetition rate, considering that the few-cycle laser modulates only a small fraction of the electron beam, we appropriately design the time delay between the laser and beam to modulate the fresh part of the electron beam at different turns.In this way, the repetition rate can be increased to the MHz range.
In addition, the combination of ADM and a few-cycle laser has the following additional advantages: Firstly, the radiation pulse generated by the microbunched beam is longitudinally coherent and has no chirp, making it close to the Fourier transform limit (FTL).This characteristic distinguishes it from the pulse generated by an HHG source, which has a natural chirp and requires additional chirp compensation elements (such as inert gases or metals) to approach an FTL pulse 31 .Secondly, the 100 ps electron beam length of the storage ring provides a large arrival time tolerance for the few-cycle laser.Thirdly, since the modulation interval of the few-cycle laser is only about one ten-thousandth of the beam length, the perturbation of the beam parameters by a single modulation is minimal.As a result, the radiation brilliance of other insertion devices (IDs) is not affected for the low repetition rate case.
To demonstrate the performance, we apply the proposed method to the Southern Advanced Photon Source (SAPS) 32 , as an example, which is a DLSR in the design phase and planned to be built in the Guangdong Province of China, adjacent to the China Spallation Neutron Source 33 .
The ADM structure, located in one of the straight sections of SAPS as illustrated in Fig. 1, initially couples the electron beam in both transverse and longitudinal directions using a vertical dipole.Subsequently, the beam undergoes energy modulation through interaction with a short wiggler (called a modular) and a few-cycle laser.Finally, a dogleg consisting of two vertical dipoles with equal strength but opposite deflection angles imparts transverse and longitudinal dispersion that converts the energy modulation to density modulation, resulting in microbunching and a large local peak current in the electron beam.The modulated beam then passed through an undulator (called a radiator) to generate a coherent attosecond pulse.After this process, the radiated electron beam passes through four dipole magnets that are specially designed in position and strength to eliminate the vertical dispersion and allow the electron beam to return to the ring.

Optimization and performance
In the ADM method, the conditions for generating microbunching are 18 , where b is the deflection angle of the first dipole, η and r 56 are the vertical and longitudinal dispersion of the dogleg, respectively.The parameter h, also known as the local energy chirp, is determined by the energy modulation intensity A and the wave number k L of the seed laser: For a given set of laser parameters and operating under the ADM optimal conditions (Eq.1), the dogleg vertical dispersion η is constrained by the geometric layout of the ADM structure.As shown in Fig. 1, all dipole elements (each with a length of L b , except for the mid-dipole which has a length of 2L b ) are symmetrically distributed in a straight section (with a length of L) of the ring, ensuring that the beam orbit returns to the horizontal plane.
To find the boundaries of the η , assuming a truncation of the dipole deflection angle θ up to the second order term, the r 56 and η ( > 0 ) of the dogleg can be expressed as follows: here, L 2 represents the distance between the two dipoles of the dogleg.
(1) 1 + hr 56 = 0 r 56 + bη = 0 , Given the values of r 56 and η , Eq. ( 3) can be solved to determine two sets of θ and L 2 .The solution with the smaller absolute value of θ is selected: To guarantee that the total space occupied by the dipole, modulator, dogleg, radiator, and the half mid-dipole is less than L/2, the inequality 0 ≤ L 2 < (L/2 − L 3 − 4L b − L 1 ) must hold, where L 1 and L 3 are the distances between the first dipole of the ADM section and the first dipole of the dogleg, and between the second dipole of the dogleg and the middle dipole of the ADM section, respectively.By substituting the expression for L 2 from Eq. ( 4) and ensuring a real result for the square root, it can be deduced that η must satisfy the following condition: where . Note that the upper bound of η increases as the length of the straight section grows, while the finite length of the straight sections in the storage ring directly limits the optimization space of η.
After the modulation with a few-cycle laser, a current spike is generated in the electron beam.When Eq. ( 1) is satisfied, the local peak current I can be expressed as 34 , with where I 0 is the peak current of the beam before modulation, m is the harmonic number of the seed laser, γ y and ǫ y are the vertical Twiss function and emittance, respectively.For large I/I 0 , one needs a large f(m), which implies small η and ǫ y .
To characterize the intensity of the microbunching, we adopt the local bunching factor for the harmonic n, which is defined as 35 , With Eqs. ( 6) and (7), we obtain the explicit expression for the local bunching factor of the ADM: For a given laser wavelength of 800 nm, we calculate the local bunching factor at a harmonic number of 200 (corresponding to the wavelength at 4 nm) under different dogleg dispersion and vertical emittance.The results are shown in Fig. 2. The bunching factor has two symmetrical peaks, and the peaks increase with decreasing vertical emittance and dogleg dispersion.The change rate of the bunching factor is more sensitive to dispersion compared to emittance.Moreover, the bunching factor with sufficient values covers an interval of about 0.4 fs, which guarantees the generation of an attosecond pulse.
In order to obtain high radiation power, a sufficiently large I/I 0 and local bunching factor are required.They all need low η and ǫ y .Note that for a given energy modulation intensity A, the acceptable range of η is determined by Eq. ( 5).All parameters of the dipole in the ADM section are determined by Eqs.(1)-( 4) for a given η in the acceptable range.When the betatron coupling of the ring is defined, all dipoles of the ADM section specify the vertical emittance ǫ y .Thus ǫ y can be considered as a function of η.
To obtain a small ǫ y , it is necessary to control the betatron coupling to a small level, which we assume to be 0.4%.This can be achieved using a suitable beam control technique 36 .Furthermore, to take into account the intrabeam scattering (IBS) effect, we adopt the completely integrated modified Piwinski approximation 37 method, which incorporates the vertical dispersion and betatron coupling to calculate the equilibrium parameters of the ring.
In SAPS, the straight section length L = 6 m, and to provide enough space for the modular and the radiator, we set L 1 and L 3 to 0.7 m and 1 m, respectively.The dipole length L b is 0.2 m and the laser wavelength is 800 nm.Other SAPS parameters are listed in Table 1.
Taking the energy modulation intensity A as 0.4%, which is about 3 times of the initial beam energy spread with IBS, this value defines the acceptable η range as (3.1 mm, 5.3 mm).In this range, the vertical chromatic H function decreases with increasing η , leading to a decrease in the vertical emittance with increasing η .The cor- responding I/I 0 and the peak of the local bunching factor have optimum values obtained at η = 4.8 mm of 29 and 0.17, respectively, as shown in Fig. 3. Additionally Fig. 3 presents the vertical dispersion function ( D y ) and the vertical chromatic H function ( H y ) of the ADM section under different η .It is evident from the figure that η exhibits a negative correlation with both vertical dispersion and H y .At the optimum, the equilibrium emittances and the energy spread of the ring are ǫ x = 100 pm, ǫ y = 1.4 pm and 0.134%, respectively.The optimal parameters of the ADM section are given in Table 2.
Performance.We adopted GENESIS 38 and ELEGANT 39 codes to perform the laser modulation and beam tracking in the ADM beamline.After interacting with a near single cycle 800 nm laser, which is available with the state of the art technology 40 , within a 0.4 m long wiggler with one period, the electron beam passes through the dogleg, then a microbunching is generated.The local current is amplified by 24 times and the peak local bunching factor is about 0.16, as shown in Fig. 4, which are in a great agreement with the theoretical optimal results.The modulated beam is capable of producing a coherent pulse with a wavelength of 4 nm and varying FWHM pulse duration t within the radiator, depending on the period number N. The radiation simulations are carried out by GENESIS with a self-consistent method 41 .Figure 5 shows that t is directly proportional to N. In order to achieve a pulse duration of less than 100 as, it is necessary to ensure that N is less than 10.However, small N results in lower photon flux and larger FWHM bandwidth (Bw).For instance, when N = 4 , the pulse duration can be as short as 50 as, but the flux is more than an order of magnitude lower than that of N = 20 .The FWHM pulse duration, Bw and photon flux of three cases with different N are shown in Table 3 for comparison.
It is well known that attosecond pulse form HHG has an intrinsic chirp called atto-chirp 42 .This chirp limits the duration of the pulse and needs additional compensation methods.Typically, the atto-chirp is compensated by shooting the attosecond pulse through thin metal foils or neutral gases depending on the wavelength of the pulse 31 .The transmittance of these materials is not perfect, pulse energy will be lost in the material which may induce thermal loading problems and limit the repetition rate of the attosecond pulse.In contrast, the proposed method generates chirp-free attosecond pulses, as evidenced by the Wigner distributions of the pulses shown    2. www.nature.com/scientificreports/ in Fig. 5.In other words, these pulses are Fourier transform limited.As shown in Table 3, the time-bandwidth products are close to the Fourier transform limited ( ∼1.83 fs • eV).Note that the few-cycle laser modulates only a small portion of the beam, about 0.01%.The vertical emittance and energy spread of this modulated portion increase from 1.4 pm to 33 pm and from 0.134% to 0.31%,   respectively.As shown in Fig. 6, these parameters quickly return to their equilibrium values with the help of radiation damping.This happens within about four times the damping time, i.e., 60 ms.Given the presence of 405 bunches in the ring, the repetition rate can reach up to 6.75 kHz, provided that each bunch is modulated only once in each 60 ms recovery period.The average flux for pulses of 50 as, 122 as, and 252 as FWHM duration are 1.28 × 10 8 , 6.4 × 10 8 , and 2.29 × 10 9 photons/s/1%Bw, respectively.

Improvement of repetition rate
When each bunch in the ring undergoes only one modulation during the 60 ms recovery period, the corresponding repetition frequency is 6.75 kHz.It is worth noting that during each modulation, only a small part of the beam (with beam length σ b ) is modulated by the few-cycle laser (with duration σ l ).If multiple modulations of different parts of the bunch are carried out during each recovery period, the repetition rate can be increased to a maximum of approximately σ b /σ l times.
However, after each modulation, the particles undergo a transverse betatron oscillation and are affected by the momentum compaction effect so that the modulated particles have a path length variation after one revolution.The equivalent path length difference of the modulated electron beam can be estimated as 43 where α p is the momentum compaction factor, C is the circumference of the ring, ξ x,y are the chromaticities of the ring, σ δ and ǫ x,y are energy spread and emittances of the modulated beam.The estimated value of σ L is about 63 µ m.Therefore, the required delay distance for the laser must exceed 63 µm.
Furthermore, it is important to consider the Gaussian longitudinal distribution of the beam, as a delay distance that is too large can result in a decrease in local current.Figure 7 shows that when a delay length of 0.1 mm is selected and 200 modulations are performed on the beam, the corresponding reduction in local current is maximally 6%, and there is a decrease in radiation power of approximately 10%.This suggests that multiple modulations generate radiation pulses with a variation magnitude of less than 10%.In this way, the repetition rate can be increased to 1.35 MHz and the flux per second of the radiation pulses with pulse FWHM durations of 50 as, 122 as, and 252 as can reach 2.47 × 10 10 , 1.23 × 10 11 , and 4.4 × 10 11 photons/s/1%Bw, respectively.
In addition, the degradation of the beam parameters of the modulated beam after multiple modulations is more severe than that after a single modulation.This degradation significantly affects the brilliance of other IDs.It is noteworthy that the beam parameters of the modulated part of the beam gradually recover to equilibrium within 60 ms following modulation.As a result, estimating the overall degradation of beam parameters can be challenging.
To simplify our analysis, we assume that the beam parameters of the modulated portion after a single modulation represent the parameters of the entire beam when calculating the brilliance change.As shown in Fig. 7, the highest brilliance of an ID in the ring decreased by approximately 72%.Although the actual decrease, which is less than 72%, is still significant and cannot be ignored.Therefore, with the proposed method, high repetition rate attosecond experiments cannot be conducted simultaneously with other high-brightness experiments.(10)    To achieve a repetition rate of MHz, it is necessary to employ a few-cycle laser at the MHz level.The development of advanced laser technology has made it possible to achieve an ultra-short high-energy pulse laser with a repetition rate up to and even beyond 1 MHz using the optical parametric chirped-pulse amplification (OPCPA) scheme 44,45 .

Discussion and conclusion
We propose a method that combines ADM and a few-cycle laser to generate a coherent attosecond pulse in a DSLR.Compared with the EEHG-based method, the proposed method can be applied to a single straight section with only one energy modulation, which makes it easier to synchronize the laser and electron beam and requires less laser power.The proposed method can achieve a repetition rate of 6.75 kH for the attosecond pulse during single modulation of a given electron beam.By introducing a suitable time delay between the laser and the beam, the modulation can be performed repeatedly on an electron beam, thereby increasing the repetition rate to 1.35 MHz.
Of course, the vertical dispersion introduced by the ADM section in the straight section breaks the periodicity of the storage ring and has some effects on the beam dynamics.The calculations show that both the dynamic aperture (DA) and the local momentum apertures (MA) are reduced.But, the DA reductions of less than 2% and the MA reductions of less than 25% have a negligible impact on the daily operation of the storage ring.
In addition, our work shows a linear relationship between the number of undulator periods and the attosecond pulse duration.Different pulse durations can be obtained by using segmented undulators and adjusting the gap to control the number of periods.However, the phase shift between different segments may cause a reduction in the radiated power, which deserves further investigation.Note that when calculating the pulse duration for the undulator with different periods, we did not consider the case of the undulator with less than 4 periods, because their fluxes are too low with the current ADM parameters.With higher laser power, increasing the energy modulation depth to ∼0.8% (6 times the initial energy spread) and using an undulator with only 2 periods, this method is able to produce a shorter pulse of ∼ 25 as (FWHM) with a flux of ∼ 10 4 photons/pulse/1%Bw.
At a repetition rate of 1.35 MHz, the multi-modulation leads to significant degradation of the beam quality, resulting in a noticeable reduction of the ID brilliance.How to further compensate for this degradation is currently an open question.Nonetheless, one can mitigate the degradation of the electron beam by reducing the energy modulation intensity, but this also implies a reduction of the photon flux of the attosecond pulses.Further research is needed to find a more feasible solution.

1 3 (,Figure 1 .
Figure 1.Layout of the ADM section in one of the straight sections of SAPS.

Figure 2 .
Figure 2. Local bunching factor of harmonic 200 for different η and ǫ y : low η and ǫ y resulting large bunching factor and the bunching factor is more sensitive to η compared to ǫ y .

Figure 3 .
Figure 3. Top: Vertical emittance, peak current and peak local bunching factor for different η .Bottom: Vertical dispersion ( D y ) and vertical chromatic H function ( H y ) of the ADM section under different η .The element distribution shown in the figure is only for the case of η = 4.8 mm.In other cases, the position of some elements is slightly different.The undulator parameters are given in Table2.

Figure 4 .
Figure 4. longitudinal phase space, I/I 0 and local bunching factor of the beam after density modulation.

Figure 5 .
Figure 5. Top: Performance of the radiation generated by the radiator with varying period numbers.From left to right: Flux per pulse, FWHM pulse duration, and FWHM bandwidth.Bottom: Wigner distribution and time spectrum profile for the output of N = 4 (left) and 20 (right), respectively.

Figure 6 .
Figure 6.The vertical emittance and energy spread evolution of the modulated local beam after the radiation.

Figure 7 .
Figure 7. Left: Variation of peak current, peak radiation power, FWHM pulse duration, and FWHM bandwidth at different longitudinal positions in the beam.Right: Comparisons of brilliance between SAPS with and without ADM section, calculated under a 4.8 m length, k = 1.1, and 16 mm period undulator.

Table 1 .
Main parameters of the SAPS.

Table 2 .
The optimal parameters of the ADM elements.

Table 3 .
Performance of the radiation for three different periods.