Harmonic generation in a terawatt x-ray free-electron laser

Terawatt x-ray free-electron lasers (XFELs) represent the frontier in further development of x-ray sources and require high current densities with strong transverse focusing. In this paper, we investigate the implications/potentialities of TW XFELs on the generation of harmonics at still shorter wavelengths and higher photon energies. The simulations indicate that significant power levels are possible at high harmonics of the XFEL resonance and that these XFELs can be an important coherent source of hard x-rays through the gamma ray spectrum. For this purpose, we use the MINERVA simulation code which self-consistently includes harmonic generation. Both helical and planar undulators are discussed in which the fundamental is at 1.5 Å and study the associated harmonic generation. While tapered undulators are needed to reach TW powers at the fundamental, the taper does not enhance the harmonics because the taper must start before saturation of the fundamental, while the harmonics saturate before this point is reached. Nevertheless, the harmonics reach substantial powers. Simulations indicate that, for the parameters under consideration, peak powers of the order of 180 MW are possible at the fifth harmonic with a photon energy of about 41 keV and still high harmonics may also be generated at substantial powers. Such high harmonic powers are certain to enable a host of enhanced applications.

variety of different polarizations.Time-dependence is treated using a breakdown of the electron bunch and the optical pulse into temporal slices each of which is one wave period in duration.The optical slices are allowed to slip ahead of the electron slices.MINERVA integrates each electron and optical slice from z → z + ∆z and the appropriate amount of slippage can be applied after each step or after an arbitrary number of steps by interpolation.
Particle orbits are integrated using the full Lorentz force equations in the complete optical and magnetostatic fields (undulators, quadrupoles and dipoles).It is important to remark that the use of the full Lorentz orbit analysis allows MINERVA to self-consistently treat both the entry/exit tapers of undulators, and harmonic generation.In particular, the harmonic treatment has been validated by comparison with a SASE FEL at ENEA Frascati [16] and a tapered amplifier experiment at Brookhaven National Laboratory [17].Thus, MINERVA self-consistently tracks the particle distribution and optical field through the undulator line and includes optical guiding and diffraction and the associated phase advance of the optical field relative to the electrons throughout.
In this paper, the steady-state simulations discussed previously [11] are extended to include the 2 nd through the 5 th harmonics.To that end, 248,832 macroparticles are included to ensure convergence.The electron beam parameters correspond to those in the LCLS in which the energy is 13.64 GeV, the current is 4000 A, the emittances are 0.3 mm-mrad in both the x-and y-directions, and the rms energy spread is 0.01%.Both helical and planar undulators are considered herein with a fundamental resonance at 1.5 Å but the FODO lattice is the same for both undulator lines; specifically, the strong focusing lattice with a cell length of 2.2 m using quadrupoles with a field gradient of 26.4 kG/cm and a length of 7.4 cm [11] corresponding to a beam radius of about 7.1 µm and a current density of 2.5 GA/cm 2 .The Twiss parameters are chosen to match the electron beam into this lattice.This compares with a current density of about 210 MA/cm 2 achieved in the LCLS.Given the strength of the FODO lattice, the beam propagation is only weakly affected by the focusing properties of the planar or helical undulators.Steady-state simulations are adequate since at this wavelength slippage over 100 m of the undulator line is less than 1.5 fs.The LCLS beam is characterized by a flat top temporal profile with a full duration of about 83 fs; hence, slippage is expected to be negligible.Let us first consider the planar undulator line.The evolution of the beam envelope in the x-and y-directions is shown in Fig. 1.Here we model flat pole face undulators with a maximum on axis field of 12.49 kG, a period of 3.0 cm and a length of 113 periods (3.39 m) with one period entry/exit tapers.The undulator line consists of 25 segments for an overall length, including the drift spaces between the undulators, of 99.7 m.Based on the estimate developed by Ming Xie [18], we expect a saturated power of about 60 GW.A step down-taper was used to enhance the power in the fundamental with a drop of about 0.2% from one undulator to the next.The optimal start taper point differed between the SASE and self-seeded cases.The start-taper point for the SASE case was found to be the ninth undulator and the fundamental power reached 0.63 TW at the end of the undulator line, which corresponds to an enhancement by a factor of about ten relative to the untapered saturated power.The evolution of the powers at the fundamental and the 2 nd -5 th harmonics is shown in Fig. 2. It should be noted that this figure corresponds to only a single noise seed.A more complete study would describe an average over at least 15 noise seeds; however, this is enough to show the potential of harmonic generation.It is clear from the figure that (1) the harmonics remain at low power levels until the fundamental power reaches the 1 -10 MW range and the NHG mechanism becomes effective, and (2) the harmonics are not enhanced by the taper.This is because the harmonics saturate before the fundamental and are not in proper phase for the taper to be effective A summary of the wavelength, photon energy and output power for the harmonics from the planar undulator line under SASE is shown in Table 1.Observe that, as expected for harmonics from planar undulators, the odd harmonics are more strongly generated than the even harmonics.The most strongly generated harmonic is the 3 rd and is found at a peak power of 498 MW.The 5 th harmonic, with a photon energy of 41.33 keV, is much reduced in comparison but is still found at a substantial peak power of 172 MW.It is expected that the odd harmonics peak on-axis while the even harmonics peak off-axis, and this is what we find in simulation.However, the NHG mechanism is driven by high fundamental powers which are peaked on-axis and it might be expected that a substantial component of the even harmonic power is also found on-axis, and this is what is found.The dominant modes for the 2 nd harmonic at the end of the undulator line are the TEM0,0 at 0.632 MW, the TEM1,0 at 1.73 MW, the TEM2,0 at 0.262 MW, the TEM0,2 at 0.104 MW, the TEM3,0 at 2.33 MW and the TEM1,2 at 0.723 MW.Hence, although the dominant modes are peaked off-axis, there is a local maximum on-axis.A similar result is found for the 4 th harmonic.The evolution of the powers at the fundamental and harmonics for the self-seeded planar undulator line is shown in Fig. 3.In this case the fundamental reaches a peak power level of 0.51 TW.While this is somewhat less than that found for the SASE example, it should be noted that the enhancement due to a tapered undulator line is extremely sensitive to the start-taper point and it is difficult to precisely choose the optimal start-taper point in a steptapered undulator line.Thus, the enhancement will be sensitive to the fluctuations in the SASE power or to the large fluctuations in the self-seeded power.
A summary of the peak powers at the harmonics for the self-seeded example is shown in Table 2.As in the SASE example, the odd harmonics achieve higher powers and the 3 rd harmonic power reaches 1.135 GW.
Turning to simulations of harmonic generation in helical undulators, it should be noted that the harmonics in helical undulators are generated by a completely different mechanism than is the case for planar undulators [19].In planar undulators, harmonics are generated due to fluctuations on the axial velocity.In contrast, the axial velocity is constant in helical undulators and harmonics are generated by a phase resonance.The fundamental is found when the polarization rotates through 2π radians in one undulator period.The h th harmonic is generated when the polarization rotates through 2hπ radians in one undulator period.This emission is into off-axis modes for the incoherent synchrotron radiation and for the linear instability and is generally weaker than is found in planar undulators; however, since it is a phase resonance it is not so sensitive to the magnitude of the undulator field which governs the harmonic generation in planar undulators.Just as in the case of the even harmonics is planar undulators, there is an on-axis component of the harmonics in helical undulators due to the NHG mechanism.
The helical undulators we use have a peak on-axis field of 16.135 kG, a period of 2.0 cm and are 46 periods in length with one period in the entry/exit taper.The optimal down-taper was found to be a drop of 0.08% from segment to segment.The evolution of the powers in the fundamental and harmonics for the SASE example in the step tapered helical undulator line is shown in Fig. 4. It is evident that the fundamental reaches a peak power of 0.93 TW.It is estimated that the saturated power in a uniform undulator line will reach about 82 GW [18], so the taper has resulted in an enhancement by more than a factor of ten.This is greater than for the planar undulator line and is not surprising as the interaction in a helical undulator is typically stronger than in an equivalent planar undulator.Also, it is seen that, once again, the taper does not enhance the harmonics.A summary of the harmonic powers at the harmonics is shown in Table 3.Note that unlike in the planar undulator line the harmonic powers decrease monotonically with increasing harmonic number because there is no distinction between the even and odd harmonics in the phase resonance.Also note that while the fundamental power is higher with the helical undulators, the harmonic powers are reduced relative to the planar undulator line.Just as in the case of the even harmonics in the planar undulators, the NHG mechanism is driven by high fundamental powers which are peaked on-axis and there is a substantial component of the harmonic power is also found on-axis.For example, the dominant modes for the 2 nd harmonic at the end of the undulator line are the TEM0,0 at 2.74 MW, the TEM1,0 at 1.68 MW, the TEM0,1 at 5.28 MW, the TEM0,2 at 5.93 MW, the TEM0,2 at 7.04 MW and the TEM0,3 at 1.47 MW.Hence, although the dominant modes are peaked off-axis, there is a local maximum onaxis.A similar result is found for the other harmonics.

Harmonic
The evolution of the powers at the fundamental and harmonics for the self-seeded planar undulator line is shown in Fig. 5.In this case the fundamental reaches a peak power level of 0.20 TW.Again, as in the case of the planar undulator line, while this is somewhat less than that found for the SASE example, the reason is that it the step-taper represents a coarse optimization.A summary of the harmonic powers is given in Table 4 In summary, the development of TW XFELs will give rise to a host of new research opportunities.An earlier work [11], it was demonstrated that terawatt powers in an XFEL requires extremely strong FODO lattices to achieve high current densities.In this paper, the focus has been on the generation of harmonics in these XFELs.Both helical and planar step-tapered undulator lines are studied.It was shown that while the step-taper effectively enhances the fundamental powers, the harmonics saturate before the fundamental and before the optimal start of the step-taper so that the harmonic power is unaffected by the taper.While the helical undulators produce higher fundamental powers than equivalent planar undulators, the planar undulators give rise to higher harmonic powers although strong harmonic generation is found for both configurations.A significant result was that a terawatt XFEL using a planar undulator line is capable of potentially generating hundreds of megawatts of 41 keV photons and substantial powers at still higher harmonics which are much higher than what has been previously achieved at similar wavelengths.These harmonics are tunable across the K-edges of several materials [20].This opens the possibility of enhanced biomedical K-edge subtraction imaging and x-ray absorption spectroscopy [21].

Fig. 2 :
Fig. 2: Evolution of the fundamental and harmonics in the tapered, planar undulator line (SASE).

Fig. 3 :
Evolution of the fundamental and harmonics in the tapered, planar undulator line (self-seeded).

Fig. 4 :
Evolution of the fundamental and harmonics in the tapered, helical undulator line (SASE).

Fig. 5 :
Fig. 5: Evolution of the fundamental and harmonics in the tapered, helical undulator line (self-seeded).

Table 1 :
Harmonic number, wavelength, photon energy and power in the planar undulator line (SASE).

Table 3 :
Harmonic powers from the helical undulator line after SASE.

Table 4 :
. Harmonic powers from the helical undulator line with self-seeding.