Tailoring the amplification of attosecond pulse through detuned X-ray FEL undulator

We demonstrate that the amplification of attosecond pulse in Xray free electron laser (FEL) undulator can be tailored. The characteristic of the amplification of an isolated attosecond pulse in the FEL undulator is investigated. An isolated 180 attoseconds full width half maximum (FWHM) pulse at 1.25 nm with a spectral bandwidth of 1% is injected into an undulator. The simulation results show that for a direct seeding of 3MW, the seed is amplified to the peak power of 106 GW (40 μJ, an output pulsewidth of 383 attoseconds) in the presence of a detuning at FEL resonance condition in 100-m long undulator. We note that the introduction of detuning leads to the better performance compared to the case without detuning: shorter by 15.5% in a pulse-width and higher by 76.6% in an output power. Tapering yields a higher power (116% increases in the output power compared to the case without detuning) but a longer pulse (15.4% longer in the pulse-width). It was observed that ± Δλr/8 (Δλr/λr ~1%) is the maximum degree of detuning, beyond which the amplification becomes poor: lower in the output power and longer in the pulse duration. The minimum power for a seed pulse needs to be higher than 1 MW for the successful amplification of an attosecond pulse at 1.25 nm. Also, the electron beam energy-spread must be less than 0.1% for a suitable propagation of attosecond pulse along the FEL undulator under this study. ©2015 Optical Society of America OCIS codes: (140.2600) Free-electron lasers (FELs); (140.7215) Undulator radiation; (140.7240) UV, EUV, and X-ray lasers; (320.5550) Pulses. References and links 1. M. Hentschel, R. Kienberger, Ch. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heinzmann, M. Drescher, and F. Krausz, “Attosecond metrology,” Nature 414(6863), 509–513 (2001). 2. F. Krausz and M. Ivanov, “Attosecond Physics,” Rev. Mod. Phys. 81(1), 163–234 (2009). 3. G. Sansone, F. Calegari, and M. 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Introduction
Attosecond science has emerged as an important research area of ultrafast phenomena during the past decade [1] as it provides a direct access to capturing, measuring and controlling the electronic dynamics in atoms, molecules and condensed matters [2][3][4][5][6][7][8][9].However, high-energy isolated attosecond pulses required for the most intriguing nonlinear attosecond experiments spectroscopy are still lacking.Therefore, techniques are to be further developed to make a powerful isolated attosecond pulse in the x-ray range (1 keV and higher).
Several techniques have been proposed to achieve attosecond pulses in a free electron laser (FEL) [10][11][12][13][14][15][16][17][18][19].The shortest pulse-width generated in the FEL are limited by the FEL coherence length, / 4 c r l λ πρ = [20] where r λ is the radiation wavelength and ρ is the FEL parameter.For 1.25 nm wavelength, the shortest pulse-widths supported by the FEL is / c l c ≥ 200 attoseconds.In a self-amplified spontaneous emission (SASE) operation, the longitudinal coherence has been known to be poor [21], while, in a seeded FEL case, the seed pulse imprints its coherence properties onto the electron beam, resulting in a better longitudinal coherence.All the more, the spectral properties obtained in a seeded FEL are strongly related to the spectral properties of the seed pulse, leading to the highly stable output spectra as demonstrated [22].All of these features are due to the efficient amplification because the seed power is in two orders of magnitude higher than the initial shot noise in an electron beam.
A number of high-harmonic generation (HHG) sources for seeding the FEL are discussed in the past [23][24][25].The first seeding of an HHG source into the FEL has been demonstrated at the SPring-8 compact SASE source at 160 nm [26].Later, the direct seeding at 61.2 nm has been performed at the same facility [27].In the ðrst stage of FLASH FEL, the direct seeding at the 21st harmonic of an 800 nm drive-laser (38.1 nm) has been demonstrated [28].The recent generation of photons at 1.6 keV, using laser-HHG, showed a potential for attosecond pulse at high photon energies [29][30][31][32][33]. Therefore, we consider the injection of an isolated attosecond pulse at 1 keV into the FEL and investigate the amplification characteristics by the FEL.
For maximum interaction in FEL, the resonance condition, ( )( ) + is maintained throughout the undulator, where r λ is the radiation wavelength, u λ the undulator period, γ the electron-beam relativistic factor and K is the wiggler parameter.However, the electron beam loses its energy during lasing in an undulator.Therefore, the FEL resonance wavelength is shifted, resulting in a poor FEL efficiency due to this mismatch.To maintain the resonance condition, throughout the amplification, tapering is done along the undulator [34] to compensate this mismatch, increasing an output.In this paper, we investigate the effect of detuning to support the propagation and amplification of a short pulse throughout the FEL undulator and demonstrate the capability of tailoring the amplification for the first time.The 100 m long undulator considered in this study is divided into 30 undulator modules.Detuning in the FEL resonance wavelength is applied to these modules in different ways, as discussed in the section II.This work mainly concentrates on the amplification characteristics of a seeded attosecond pulse with the variation of detuning in the FEL resonance wavelength.In the section II, the simulation model is presented.In the section III, the simulation parameter and conditions are explained.The influence of different detunings on the seed pulse-width and the efficiency of amplification are presented.In addition, the minimum threshold to a seed power and the requirement of electron beam energy-spread are also studied in view of radiation power and pulse-width.Finally in the section IV, a conclusion is presented.

Undulator lattice and FODO (focus-drift-defocus-drift) lattice
Korea is currently building a hard x-ray FEL facility, PAL-XFEL that will generate X-ray in a range of 0.1 to 10 nm in a SASE operation mode [35].For our simulation study, we adopt the undulator design of PAL-XFEL.Fig. 1.Schematic layout of the undulator system considered in this study.The 2nd undulator section is repeatedly added to make a ~100 m-long undulator system.
In the PAL-XFEL undulator design, a planar undulator lattice is considered with each undulator module length being 100λ u , which is 3.4 meter for λ u (the undulator period) of 34 mm.30 undulator modules are considered in our simulation and the total length of the undulator is ~100 m. Between undulator modules, there is a drift space used for diagnostic and focusing.To establish the maximum overlap between an electron beam and a radiation in the undulator, the transverse and longitudinal matching between the electron beam and the radiation should be properly done.The electron beam expands in a free space; to keep the electron beam size nearly constant and maximize the FEL interaction in a long undulator, the external focusing by quadrupole magnets is used.This focusing is provided by FODO lattice for the stable propagation of the electron beam through the undulator.But the decrease of the electron beam size below the radiation beam size degrades the FEL performance, because the increase of angular spread introduces a spread in the resonance wavelength.For the maximum interaction between an electron beam and the radiation, the Rayleigh length (rms) electron beam-size and radiation wavelength, respectively.All input parameters were optimized using relevant formulas [36] and presented below in Table 1.In our simulation, a Gaussian pulse of 180 attoseconds pulse duration (FWHM) centered at 1.25 nm with a bandwidth of Δλ/λ ~1% is injected into the undulator.For most of the simulations, the seed energy of 3MW (0.54 nJ/pulse) is used.Such a seed pulse energy used in the simulation is not currently available.To produce such a seed pulse, long wavelength light sources are promising candidates.To our knowledge, HHG at 1.6 keV and in the water windows have been demonstrated by using mid-infrared fs laser [30] and infrared laser pulses [31], respectively.The pulse energy as high as 1.3 µ J with 500 as pulse duration in a few 10 eV region has been achieved [32].Hence, with the rapid advance in laser based technology, attosecond pulse of several 100 pJ at high photon energies may become available soon.
Proper characterization of these attosecond seed pulses is necessary as the temporal and spectral properties are directly imprinted on the electron beam.The lack of any measurement technique capable of such seed pulses at the laser-electron interaction region is a drawback.However, since the period of 1.25 nm light is 4.1 as, there are about 45 cycles in 180 as pulse duration.Hence the Gaussian temporal shape for an attosecond seed pulse is a good approximation.
For an effective seeding, an attosecond pulse is injected coincidently with an electron beam of spatial Gaussian profile at the entrance of the undulator.

Detuning
To study the detuning effect, the undulator considered in study is divided in several sections.As shown in Fig. 1, the first 10 meter-long section has three undulator modules and the second 14 meter-long section has four undulator modules.The second undulator section is repeatedly added to make ~100 meter long undulator system.Each undulator module length is 3.4 m, which is 100 λ u for λ u = 3. 4 cm.
The first three undulator modules are tuned at λ r = 1.25 nm to maximize the initial bunching process.The remaining twenty seven undulator modules are detuned with respect to λ r = 1.25 nm to observe the effect of the detuning on the temporal characteristics and saturation behavior of the output radiation.These undulator modules are detuned to λ r ± nΔλ r where n is a fractional number and Δλ r = 1% λ r equivalent to 0.0125 nm.A series of simulations for differently detuned undulator configurations were performed.In Table 2, we only show a limited number of configurations.The first undulator configuration UC1 has no detuning.The second configuration UC2 is detuned alternatively by ± Δλ r /16 after 10 meter.
the configuration UC3 has a little larger degree of detuning compared to UC2, and UC4 has even larger detuning compared to UC2.
A series of simulations were carried out using the simulation parameters in Table 1 by GENESIS, the three dimensional time dependent simulation code [37] for all the undulator configurations shown in Table 2.

Undulator detuning effect on the output power and pulse width of a seed pulse
In this section, we present the simulation results of seeded FEL for zero-detuned undulator and finite-detuned undulator configurations.In addition, we also discuss a tapered undulator case.Except for the detuning parameters in undulator modules, all the electron beam parameters and undulator parameters are the same as shown in Table 1 for all the configurations.
. Fig. 2. Peak pulse-power at various positions along the undulator for different undulator configurations as shown in Table 2.
We compare the radiations powers among these configurations.Figure 2 shows the maximum radiation power obtained at different positions along the undulator for different detuning cases.For zero detuning (UC1, the black line) the output power at the undulator exit is 60 GW, corresponding to a pulse energy of 27 μJ (or a photon flux of 1.6 × 10 11 photon/pulse).When a detuning of ± Δλ r /16λ r ≈ ± 0.0625% is introduced (UC2), the output power at the undulator exit is increased to 79 GW (blue-line).In the configuration UC3, where the detuning is ± 2Δλ r /16λ r ≈ ± 0.125%, the output power at the undulator exit reaches to 106 GW (red-line), the corresponding FEL pulse energy is 40 μJ (or a photon flux of 2.5 × 10 11 photon/pulse).For another configuration UC4, we introduce a detuning of ± 3Δλ r /16λ r ≈ ± 0.1875% (green line) slightly higher that for UC3 case.We find the amplification becomes poor: the output power at the undulator exit is 97 GW (green-line) lower than that in the configuration UC3.We have also explored the tapering for a given undulator.We applied the linear tapering along the undulator configuration UC1, starting from the undulator entrance.For a smooth taper profile K (z) along the undulator, Fig. 2 clearly shows that the tapering increases the radiation power by maintaining the resonance condition (pink line).The output power at the undulator exit is 130 GW, the increase of 116% compared to the zero detuning case (60 GW) and 22% larger than the UC3 case (106 GW).In the tapering case, the FEL pulse energy and photon flux are 67 μJ, and 4 × 10 11 photon/pulse, respectively.Fig. 3. Pulse-width at various positions along the undulator for different undulator configurations as shown in Table 2.
Figure 3 shows the change of the pulse-width of the seeded attosecond pulse along the undulator for different detuning cases shown in Table 2.For zero detuning (black-line), the seed pulse-width broadens initially as the interaction between electron bunch and radiation proceeds along the undulator up to the position of 25 meter from the undulator entrance.This broadening is mainly caused due to the gain narrowing process.Then it starts to decrease till the saturation point.The FEL pulse-width at z = 25 m is 770 attoseconds and decreases down to 413 attoseconds at z = 60 m, shortened by 46%.After the saturation point, the pulse-width becomes almost constant till the undulator exit.The output pulse-width at the undulator exit is 453 attoseconds.Now, we examine different detuning cases.For a detuning of ± Δλ r /16λ r ≈ ± 0.0625% (UC2), as shown in Fig. 3, the pulse-width variation up to 40 meter is almost similar to zero detuning case (UC1) and the output pulse-width of 413 attoseconds at the undulator exit, shorter than the zero detuning (UC1).In another configuration UC3 of a detuning of ± 2Δλ r /16λ r ≈ ± 0.125%, the output pulse-width at the undulator exit becomes as short as 383 attoseconds (red line in Fig. 3).For another configuration (UC4), where we introduce a detuning of ± 3Δλ r /16λ r ≈ ± 0.1875%, one can see that the pulse-width again slightly increases after 60 meter along the undulator and the output pulse-width at the undulator exit is 403 attoseconds (green line).The effect of a tapering on the pulse-width is also studied.We find that the pulse-width after the exponential growth regime starts to decrease and ends up with a longer pulse-width compared to those in other undulator configurations: At the exit, a pulse width of 523 attoseconds is obtained, which is 16% larger than the zero detuning case (453 attoseconds) and 37% larger than UC3 case (383 attoseconds).In the exponential growth regime, the behavior of the pulse-width in a tapering case is similar to those in other configurations but near the saturation and afterwards, the detuning is more effective in reducing the pulse-width.1.
To better understand the change of the pulse-width, we examine the temporal and spectral profiles for UC3 shown in Fig. 4. Figure 4 λ πρ = in our case, the amplification is in the exponential growth regime after a distance of 5 to 7 m.Therefore, the high amplification is expected in the first 10 meter undulator.The gain at the resonance wavelength is larger than that at off-resonance wavelengths.As the amplification continues, the spectral bandwidth effectively decreases as shown by corresponding spectra in Fig. 4(e) and 4(f) (gain narrowing) and the pulse width increases.The shortening in pulse-width can be explained on the basis of interference between SASE field and amplified seed field.Near a saturation point and afterwards, the spectral bandwidth gets larger which supports in principle the short pulse and SASE radiation field becomes significant, as clearly shown in Fig. 4(c) and 4(d): the spikes (black dashed line) are SASE radiation.The field of SASE radiations interferes with the field of the amplified seed pulse.This interference leads to the shortening of the main seed pulse.
The above results show that the optimization of detuning in an undulator may lead to even a shorter pulse-width, indicating that a fractional change in the FEL resonance condition changes the FEL pulse-width effectively.Hence demonstrate that the detuning in the resonance condition can be a parameter to manipulate the radiation pulse-width and spectrum.
To check whether the main pulse inside the undulator is Fourier-transform limited pulse, we estimate the time-bandwidth product v t Δ Δ along the undulator at different locations for UC3 case.For a Gaussian pulse, the minimum time-bandwidth product is v t Δ Δ ≈0.44.While the SASE radiation temporal and spectral profile give us v t Δ Δ = 8.64, in the seeding case, the time-bandwidth product is found close to 0.44 along the undulator as shown in Table 3.Here, the variation in the time band-width product can be understood based on Fig. 4; as the radiation generation progresses along the undulator, in the initial exponential regime of power growth, the FEL spectrum gets narrower up to z = 40 m due to limited FEL bandwidth.Toward the saturation regime, the SASE radiation becomes effective and bandwidth becomes larger.Then the interference of the SASE with the main radiation pulse leads to shorter pulses.Hence the time bandwidth product does not remain constant over the whole length of the undulator.

Effect of seeding power on attosecond pulse amplification
To optimize the minimum requirement of the seed pulse power to support attosecond pulse propagation along the undulator, a series of simulations for the input threshold has been done.The dependence of the output pulse-width and power on the seeding power is presented in Fig. 5. pulse-width and its power are obtained at z = 30 m for zero detuning case (UC1).Both are found fairly sensitive to the seeding power.The red line shows the output pulsewidth and the black line represents the output power.These results give a rough idea of the required power of attosecond seed pulse at 1.25 nm.The output power continuously increases with the increase of the seed power.For a power less than 1 MW, the pulse-width becomes larger than 1 femtosecond within a propagation of 30 meter in the undulator.Mainly, the SASE radiation amplification overtakes the main seed-pulse amplification for seed power less than 1 MW.Hence, a minimum seed power of 1 MW would be required for proper amplification of attosecond pulse at 1.25 nm radiation for the undulator configurations under this study.

Effect of electron beam energy-spread on attosecond pulse amplification
In the FEL, the radiation power is sensitive to the energy spread of an electron beam.The uncorrelated electron beam energy spread ΔE/E should be smaller than the FEL parameter ρ.The energy spread of a typical electron bunch is 0.06%.The simulation was carried for different energy spreads (ΔE/E): 0.06%, 0.1%, and 0.15%.For a larger energy spread, the output profile gets worse; moreover, the pulse-width becomes in the order of femtosecond, with a net decrease in the output power.It can be understood because the larger the beam energy spread, the larger the slippage between electron beam and radiation.Therefore the radiation profile becomes worse at saturation and SASE radiation overtakes the main seed radiation peak.We found that ΔE/E = 0.1% is the maximum value for an energy spread to support the attosecond pulse propagation at 1.25 nm.It is observed that the energy spread ΔE/E less than 0.1% yields a strong improvement in the final output power.
Another issue in implementing direct seeding in the FEL is that the seed pulse needs to be properly synchronized with the electron beam.In the PAL-XFEL, the electron bunch length is longer than 100 femtosecond.We considered 20 femtosecond long electron bunch in this work to minimize the computation cost and time.But in actual experiment of PAL-XFEL, the electron bunch length is longer than 100 femtosecond and the timing jitter of ~50 femtosecond is expected.The seeding a light pulse into an electron bunch can be done.The current distribution of the real electron bunch is designed to be uniform within ± 20%.Our simulation results show that the current fluctuation in e-bunches with ± 20% leads to the power fluctuation of ± 10%.

Summary
The amplification of a seeded attosecond pulse of a 180 attoseconds FWHM, and 3 MW peak power with a spectral bandwidth of 1% is studied in an undulator which has a set of alternatively detuned undulator modules.In a typical FEL amplifier, to achieve maximum output power, it is essential to match the resonance condition.Our study shows that a detuning in the resonance wavelength can tailor the attosecond pulse-width and can improve the output power.The results show that for zero detuning, a seeded attosecond pulse of a 180 attoseconds FWHM, and 3 MW was amplified to 453 attoseconds, 60 GW peak power (pulse energy 27 μJ, 1.68 × 10 11 number of photons per pulse) after the propagation of a 100 meter long undulator.On the other hand, for a detuning of ± Δλ r /8, the same seed pulse was amplified to 383 attoseconds, 106 GW peak power (pulse energy 40 μJ, 2.5 × 10 11 number of photons per pulse).For a larger detuning of ± 3Δλ r /16 in undulator, the amplification becomes poorer in output power and pulse duration gets broader.A tapered undulator can also amplify an attosecond pulse; however, the final pulse-width is longer than that in an optimized detuning case.Moreover, it was found that the seed power needs to be bigger than 1 MW for a 180 as pulse at 1.25 nm.Also, the electron-beam energy-spread should be lower than 0.1% for a suitable amplification of attosecond pulse.
In summary, an optimal choice of in the resonance wavelength can improve the amplification characteristics.Therefore, the detuning and its impact on the attosecond pulse amplification need more attention.
as long as the total undulator length, where x σ and r λ are the root-mean-square
(a)-4(d) show temporal profiles at different locations along the undulator, and the corresponding spectra are shown in Fig. 4(e)-4(h).The red line is for seeding and the black line is for SASE without seeding.A 180 as FWHM, 3 MW radiation pulse is injected as a seed.In Fig. 4(a) and 4(b), SASE radiation result is multiplied by a factor of 10 for an easier comparison with seeded FEL results.From Fig. 4 (a)-4(d) (black lines), one can see that SASE produces a long pulse (10 femtoseconds or longer) with several spikes.Because of a short gain length ( )

Table 3 .
Time bandwidth product of radiation pulse at different position along

Fig. 5 .
Fig. 5.The change of the output power and pulse duration of attosecond seed pulse at z = 30 with respect to the change of seed power.

Table 1 . FEL Simulation parameters
g L ~2 m