# Synchrotron Radiation in Periodic Magnetic Fields of FEL Undulators—Theoretical Analysis for Experiments

## Abstract

**:**

## 1. Introduction

## 2. Spontaneous UR intensity and Spectrum Distortions

_{0}~1 Tesla. The proposed analytical formalism, however, is not limited to such weak fields H

_{d}/H

_{0}~10

^{−4}, but allows arbitrary strengths.

_{H}, which describes the effect of the non-periodic magnetic components. The purely periodic terms in the exponential of the radiation integral are collected and form the generalized Bessel-type functions ${J}_{n}^{m}\left({\xi}_{i}\right)$, which naturally arise in the following form:

_{d}is accumulated along the undulator length $L={\lambda}_{u}N$, where N is the total number of periods, and quantified by the normalized bending angle θ

_{H}:

_{d}bends the electron trajectory into the effective angle γθ

_{H}and causes synchrotron radiation from much wider curve, than that of the electron oscillations along the undulator periods. However, its effect in long undulators should not be underestimated, as we will show in what follows. The non-periodic magnetic components in the exponential of the radiation integral compose the following ad-hoc generalized Airy-type function in the integral form:

_{n}in (21). Upon computing the radiation integral we get the UR intensity:

_{d}, written in terms of the bending angle θ

_{H}. For the odd UR harmonics n = 1,3,5,… mainly the Bessel coefficient ${f}_{n}^{1}$ determines the UR intensity (20). The resonance of the UR has an infrared shift respectively to the ideal value at v

_{n}= 0:

_{d}= κH

_{0}, for $\phi =\pi $ we get the angle $\tilde{\theta}$, in which the infrared shift of the received radiation is compensated: ${\nu}_{n}=0$ for $\tilde{\theta}=\frac{2\pi}{3}\frac{k}{\gamma}\text{}N\kappa =\sqrt{3}{\theta}_{H}$; proper UR resonances are ${\omega}_{n}\text{}\cong 2n{\omega}_{0}{\gamma}^{2}/\left(1+(\varpi {k}^{2}/2)+0.27{(\gamma {\theta}_{H})}^{2}\right)$. The examples of the UR lines for the PAL-XFEL [39] undulator with N = 194 periods, k = 1.87, period λ

_{u}= 2.57 cm, length L = 5 m and the electron energy spread σ

_{e}= 1.8 × 10

^{−4}, is shown in Figure 1a for γθ = 0 and Figure 1b for γθ ≠ 0 in the presence of the non-periodic magnetic component H

_{d}. In Figure 1a, the field H

_{d}causes an infrared shift and broadens the spectrum line, viewed in zero angle γθ = 0. In Figure 1b, note that the same field H

_{d}can improve the shape of the spectrum line viewed in the angle γθ = 0.067. Note in Figure 1b as the initial detuning, caused by the off-axis angle γθ = 0.07, reduces if the undulator is affected by the field H

_{d}= κH

_{0}~10

^{−4}H

_{0}; a further increase of κ broadens the UR line. For the on the axis case, γθ = 0, the effect of H

_{d}is purely detrimental (see Figure 1a).

_{d}and their interplay are also shown for the fundamental harmonic in Figure 2. In Figure 2a, the symmetric ideal UR line of the fundamental tone is described by the sinc(ν

_{n}/2) function for γθ = 0, H

_{d}= 0; it shows a red shift, if viewed in the angle γθ ≠ 0; the angles γθ > 0.1 cause a significant shift down from ν

_{n}= 0 and the intensity slightly decreases.

_{n}= 0 (see Figure 2a). In Figure 2b we demonstrate the spectral line of the fundamental tone, broadened and red-shifted in -π by the constant magnetic field H ≈ H

_{0}× 10

^{−4}, if viewed on the axis, γθ = 0. In Figure 2b, the spectral line reassumes a more distinct shape with the increase of the off-axis angle γθ from zero to ~0.1. This demonstrates that the non-periodic magnetic component κH

_{0}and the off-axis angle γθ can compensate each other’s effect on the UR. In the presence of the field H

_{d}≈ 10

^{−4}H

_{0}, the spectrum line gets narrower and the red shift is smaller for the same angles γθ ≈ 0.1 as shown in Figure 2b.

_{⟷}0.05–0.1. In a long undulator, this angle can be generated by rather weak magnetic field H

_{d}: for example, the bending angle $\gamma {\theta}_{H}=0.05$ can have noticeable effect on the FEL performance at LCLS, where the off-axis target deviation was 5 μm. LCLS undulator length, L = 3.4 m, is translated to the field strength H = 0.44 Gauss. The need to screen out such fields was pointed out in [18,19]. Moreover, at the PAL-XFEL the field of the Earth, ~0.5 Gauss, can induce the angle $\gamma {\theta}_{H}\approx 0.08$ in the 5 m long undulator, and cause even stronger deviation of the electron trajectories.

_{e}= 10

^{−4}. The contribution to the second harmonic intensity due to the constant field κH

_{0}(24) accounting for the off-axis angle θ is shown in Figure 3. The shape of the spectrum line is given by ∂S/∂v

_{n}. The term ${f}_{n}^{3}$ increases with the increase of the bending field H

_{d}. The interplay with the angles γθ (see Figure 3b) limits the increase of ${f}_{n}^{3}$ and determines its behavior for the stronger field H

_{d}> 1.5 × 10

^{−4}Gauss. This latter value depends on γθ and on the undulator parameters; for θ = 0, the Bessel coefficient ${f}_{n}^{3}$ grows further for increasing κH

_{0}(see Figure 3a).

_{0}; the associated spectrum line shape is described by the function S. The comprehensive contribution of all terms to the normalized intensity of the second harmonic, I

_{2}, is shown in Figure 5; the maximum intensity is at H

_{d}≈ 0.7 × 10

^{−4}Gauss. The decrease of the UR intensity, caused by γθ~0.067, is compensated by the field H

_{d}≈ 0.7 × 10

^{−4}Gauss, and at this point, the second harmonic intensity is at its maximum.

_{e}, is accounted for by the convolution $\underset{-\infty}{\overset{\infty}{\int}}\frac{{d}^{2}I\left({\nu}_{\mathrm{n}}+4\pi nN\epsilon \right)}{d\omega d\mathsf{\Omega}\sqrt{2\pi}{\sigma}_{e}}{e}^{-\frac{{\epsilon}^{2}}{2{\sigma}_{e}^{2}}}\text{}d\epsilon$. The effect of the energy spread on the UR harmonics is purely detrimental and it causes symmetric broadening of the spectral lines. In this context it is important to underline that high radiation harmonics are more sensitive to the energy spread and to other loss factors, than the fundamental harmonic. Weak, but detectable at low energy spread, ${\sigma}_{e}\approx {10}^{-4}$, FEL harmonics can be almost totally suppressed, if the energy spread increases to ${\sigma}_{e}\approx {10}^{-3}$. The relevant example of SACLA radiation will be considered in what follows.

_{β}(27) is much lower than the UR frequency ${\omega}_{n}\cong \frac{4\pi cn{\gamma}^{2}}{{\lambda}_{u}{\left(1+\left({k}^{2}/2\right)\right)}^{}}$; their ratio is roughly the inverse of the relativistic factor: $\frac{{\omega}_{\beta}}{{\omega}_{n}}\cong \frac{k\delta}{\sqrt{2}n\gamma}\propto \frac{1}{\gamma}.$ This explains the high interest to this topic already in early SR and UR experiments, where relatively low-energy electron beams were used and the contribution of the betatron oscillations was considerable. For the intensity of the UR harmonic n, accounting for the betatron oscillations, we get the following expression:

_{0}is the off-axis position of the electron in the beam and θ is off-axis angle. The summation series $\sum}_{p=-\infty}^{+\infty$ over p describe the account for all subharmonics p of the harmonic n. In real devices, finite number q of the subharmonics contribute: $\sqrt{{\displaystyle {\sum}_{p=-q}^{+q}{\tilde{J}}_{p}^{2}}}\cong 1$; where q describes the degree of the split of the harmonic n and depends on the beam parameters; it varies strongly from one installation to another. Some examples will be considered in the following section, where we model some FEL experiments. The subharmonics are distant at the betatron frequency ω

_{β}. In the relativistic beams, γ>>1, this split of the UR lines due to the betatron oscillations is small: ${\omega}_{\beta}\propto {\omega}_{n}/\gamma $. The even UR harmonics appear on the undulator axis due to the betatron oscillations [9,40,41,42,43,44]; proper Bessel coefficient expectably differs from that in [44] only in Bessel functions due to different undulator field (2):

^{−2}, in comparison with other Bessel coefficients: ~0.15–0.8 in (22), (23); usually ${f}_{n,p;y}^{4}$ do not exceed ${f}_{n=1,3,5}^{1,2}$. However, the split of the spectrum lines due to the betatron oscillations can be considerable and it strongly depends on the parameters of the installation and on the beam. Some examples are given in the context of the modeling of FELs in the following section. Beam sizes vary from ~0.2 mm to ~20 μm in modern FELs; beam deviations from the axis are usually small; for example, they are ~5–25 μm on one gain length, L

_{g}= 1.6–3.5 m, in the LCLS FEL experiments [18,19,20]. However, the off-axis deviation of ultrarelativistic electrons in just ~10 μm in one undulator section length, ≅3 m, can cause the effective angle γθ~0.1 and noticeable effects. In what follows we will analyze in detail the harmonic generation in SACLA and PAL-XFELs and compare them with some other user facilities, such as LCLS.

## 3. Analysis of the Harmonic Generation in Some FEL Experiments

#### 3.1. SACLA FEL Experiment

^{−4}and betatron value 30 m in [69], while in [61] we find the energy spread (in projection) <10

^{−3}and β

_{x,y}= 22 m. One order of magnitude difference in the energy spread together with the change in β from 30 m to 22 m has very strong effect of the FEL radiation: the saturation length can vary from ~20 m to ~60 m, the saturated powers change etc.

_{u}= 1.8 cm long. The coherent radiation is generated at the fundamental λ~1–12 nm; the details are available in [62]. Despite explicit description, [62] does not contain any data on the power evolution, saturation and gain lengths, although the beam and radiation characteristics are well specified. We have studied the instance of this experiment with the maximum possible value of k = 2.1, the electron energy E = 780 MeV and beta-functions β

_{x}= 6 m, β

_{y}= 4 m [62]. The current I = 300 A was calculated by the authors of [62] for the bunch charge 0.23 nC and the bunch length τ

_{e}≈0.7 ps (we get 330 A though). There is a great deal of uncertainty with regard to the values of the energy spread and the emittance; the energy spread per slice is not given, the projected value, σ

_{e}

^{projected}= 0.6%, is well too high as compared with other installations, such as LCLS [19] etc.; this lack of definite data for experiments also includes the emittance ε

^{n}

_{x}

_{,y}: the reported data vary between 0.5–3 mm × mrad [62]. We suppose that most of it is in the projection that is due to transverse centroid shifts along the bunch and the time-sliced values after the injector are well preserved. Reassuming [62], we adopt the data simulation in Table 1, which yields the FEL power evolution, demonstrated in Figure 6.

_{1}≈ 0.2 GW, coinciding with the value ${P}_{\mathrm{max}}={E}_{\gamma}/{\tau}_{rad}$ = 0.2 GW, for the measured fundamental energy [62] E

_{γ}≈ 0.1 mJ ± 13% for the FEL radiation pulse duration ${\tau}_{rad=}{\tau}_{e}\sqrt{2\pi {L}_{g}/{L}_{s}}$ = 0.5 ps, emitted from the electron bunch with the root mean square r.m.s. length τ

_{e}= 0.7 ps. For the third FEL harmonic (see green dashed line in Figure 6) we get ≈0.3% power of the fundamental in agreement with [62]. Note that the second harmonic level is very low. The simulated gain length is L

_{g}≈ 1 m and the saturated length is L

_{s}≈ 13 m; other data are collected in Table 1.

^{−3}, comparable with the natural UR line width 1/2N ≈ 2 × 10

^{−3}. Theoretical estimation of the relative radiation line bandwidth in SASE FEL after the gain-narrowing in the exponential growth yields similar value $\mathsf{\Delta}\lambda /\lambda \approx \sqrt{\rho {\lambda}_{u}/{L}_{s}}$≈ 0.15%, close to the FEL scaling parameter ρ ≈ 0.0016 (see Table 1). Superposition of the randomly distributed over the length of the electron bunch wave trains with the coherence length ${l}_{c}={\lambda}^{2}/\mathsf{\Delta}\lambda $~6 μm gives the coherence time ${t}_{c}={\lambda}^{2}/(c\text{}\mathsf{\Delta}\lambda )$~0.02 ps. The number of the coherence regions in the radiation pulse is therefore ${\tau}_{rad}/{t}_{c}$~ 20.

_{e}= 0.0926%, following [63], where the upgraded RF system of SPRING 8 was described. Considering that the beam was alternatively sourced to BL3 and BL2 undulator lines, and the above spread was reported for E = 6 GeV, we assume that the spread should not increase in the experiment with the energy E = 7.8 GeV on BL3 line and 10 keV photons. Of course, it depends on the spreader, the optics, on whether the dispersion was closed etc., thus, the experimental conditions can be different; however, in the absence of explicitly reported data, we have to assume the first approximation of the only available data from [63,65]. Our simulation results are collected in Table 2; the computed saturated power is compared with that obtained from the measured in this experiment photon energy, E

_{γ}= 0.4–0.5 mJ, reported in [64] (see Figure 2c and Figure 3 in [64]).

_{e}= 20 fs and charge Q = 0.2 nC, the current I = 10 kA, and other data [64], we get the photon pulse duration τ

_{γ}≈13 fs, and the saturated powers of the fundamental and third harmonics as shown by the dashed lines after 50 m in Figure 8; they agree with our theoretical simulations. Horizontal dashed green and orange lines in the saturation region in Figure 8 trace the values 0.2% for the third and 0.03% for the second harmonics. Variation of the emittance, ±1 μm, and of energy spread, 0.08–0.1%, influences the gain and the saturation lengths and the third harmonic power; the fundamental power is less sensitive to it.

_{c}~0.7 fs, which means that less than 20 coherence regions are in τ

_{γ}≈ 13 fs photon pulse.

#### 3.2. POHANG FEL X-ray Experiments

_{1}= 1.52 nm was produced by the electrons with the energy E = 3 GeV and the energy spread ${\sigma}_{e}^{soft}=0.05\%$ (~five times higher than in LCLS), in the undulators with the total pure length ~40 m; the undulator parameter was k = 2. The hard X-ray radiation at λ

_{1}= 0.144 nm was generated by the electrons with the energy E = 8 GeV (vs. E~13 GeV in LCLS) with the energy spread ${\sigma}_{e}^{hard}=0.018\%$ (~two times higher than in LCLS) in the undulators with the deflection parameter k = 1.87 (vs. k = 3.5 in LCLS) of the total pure length 100 m. The undulator sections were 5 m long.

^{−10}m to be compared with λ

_{1}/4π = 1.2 × 10

^{−10}m. However, for the fifth harmonic we get λ

_{5}/4π = 2.5 × 10

^{−11}m and ε≅ λ

_{5}/π = 1 × 10

^{−10}m. The third harmonic could appear with the power rate ~0.7% of the fundamental, the second harmonic would have the power rate ~0.05%, as shown in Figure 12 and Figure 13. Our estimation for the third harmonic at PAL-XFEL in soft X-rays, ${P}_{3}/{P}_{1}$~0.7%, is roughly a half of that for a similar LCLS experiment, where ${P}_{3}/{P}_{1}$~1.3% for λ

_{3}= 0.5 nm [20] with similar radiation parameters. Thus, we can expect some weaker third harmonic at PAL-XFEL due to the smaller value of the undulator parameter k as compared with LCLS; moreover, the detrimental effect of the energy spread is higher for PAL-XFEL, ${\sigma}_{e}=0.0002\xf70.0005$, as compared with that in LCLS, where ${\sigma}_{e}=0.0001$. The second hard X-ray harmonic at PAL-XFEL is weak; high High harmonics were not registered in the PAL-XFEL experiments.

_{e}and Pierce parameters ρ

_{n}are as follows: ${\sigma}_{e}^{hard}=0.00018\cong {\rho}_{1}/2<{\rho}_{3}=0.00021>{\rho}_{5}=0.00015$,${\rho}_{2}=0.00006$; the relations between σ

_{e}and ρ

_{n}are close to those for soft X-ray radiation. Moreover, for hard X-rays the comparison of the pure emittance ε ≅ 3.5 × 10

^{−11}m with λ

_{3}/4π = 3.8 × 10

^{−12}m is not favorable for the third harmonic radiation: ε

_{x,y}≅ 10 × λ

_{3}/(4π). Unsurprisingly, high hard X-rays harmonics were not detected. However, if the energy spread and emittances are improved, then we can expect at the PAL-XFEL high harmonic generation as suggested in Figure 12 and Figure 13. The off-axis deviation of the beam in PAL-XFEL amounted to ~10 μm on one undulator length [39]. This causes the off-axis angle ~2 μrad, comparable with the divergence, ~2 μrad for soft X-rays and ~1 μrad for hard X-rays. In the soft X-ray experiment the deviation of the beam in few undulator segments reached 20 μm on one undulator length [39]; this induces the angle 4 μrad. However, the electron-photon interaction on one gain length must be considered with the angle $\overline{\theta}\cong 14$ μrad, far exceeding the beam deviation. The latter angle causes the 2nd FEL harmonic, whose power is estimated ~10

^{−4}of the fundamental (see Figure 12). For the hard X-ray experiment we get much smaller value $\overline{\theta}\cong 4$ μrad, and the beam must be kept on the axis more precisely.

_{1}= 1.5 nm is split in ~9 subharmonics. The total width of the line is Δλ~2.3 pm, the relative value is Δλ/λ~1.5 × 10

^{−3}. It is small, but it is higher than the respective value in the soft X-ray LCLS experiment, where Δλ/λ~5 × 10

^{−4}. For hard X-rays in PAL-XFEL we have to account for more subharmonics: p = −7,…,+7. Nevertheless, the line remains rather narrow even with account for this split; the subharmonics are close to each other because of the electrons are ultrarelativistic. The absolute width of the hard X-ray line is Δλ~0.14 pm and the relative width is Δλ/λ ≈ 1.0 × 10

^{−3}. Compared with the respective values in the LCLS experiment, Δλ/λ≈3 × 10

^{−5}, the spectrum lines for PAL-XFEL radiation appear wider by ~1–2 orders of magnitude.

## 4. Conclusions

^{−3}, Δλ~0.14 pm due to γ~1500 >> 1. However, this is >10 times wider than in the LCLS experiment at the same wavelength, where Δλ/λ ≈ 3 × 10

^{−5}. The radiation of high harmonics at PAL-XFEL is limited by a rather high energy spread: for hard X-rays ${\sigma}_{e}^{hard}\text{}=0.00018$, the Pierce parameters for the n = 1,2,3,5 harmonics are ${\rho}_{1}\cong 0.0004$, ${\rho}_{3}\cong 0.0002$, ${\rho}_{5}\cong 0.00015$, ${\rho}_{2}\cong 0.00006$; for soft X-rays the energy spread is ${\sigma}_{e}^{soft}=0.0005$, and the Pierce parameters are ${\rho}_{1}\cong 0.0010$, ${\rho}_{3}\cong 0.0006$, ${\rho}_{5}\cong 0.0004$, ${\rho}_{2}\cong 0.00007$. We have demonstrated possible theoretical harmonic radiation at PAL-XFEL; however, due to relatively high energy spread, we can hardly expect radiation of the harmonics higher than the third.

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Phenomenological Model of Harmonic Power Evolution in High-Gain FELs

_{n}reads accounting for the diffraction as follows [3,4,5,6,23,45,46,47,48,49,50,51,52,53]:

^{2}] is the current density, $\mathsf{\Sigma}=2\pi {\sigma}_{x}{\sigma}_{y}$ is the beam section, ${\sigma}_{x,y}=\sqrt{{\epsilon}_{x,y}{\beta}_{x,y}}$ are the sizes of the beam, ${\epsilon}_{x,y}={\sigma}_{x,y}{\theta}_{x,y}$ are the emittances, ${\beta}_{x,y}={\epsilon}_{x,y}/{\theta}_{x,y}^{2}$ are the betatron average values, ${\theta}_{x,y}$ are the divergences, $i\cong 1.7045\times {10}^{4}$ is the constant of Alfven current [A], ${k}_{eff}=k\sqrt{\varpi}$ (see (3) for $\varpi $) is the effective undulator parameter, which reduces for the common planar undulator to $k=\frac{e{H}_{0}{\lambda}_{u}}{2\pi m{c}^{2}}\approx 0.934{H}_{0}{\lambda}_{u}[\mathrm{T}\cdot \mathrm{cm}]$, ${H}_{0}$ is the magnetic field amplitude on the undulator axis, ${f}_{n}$ is the Bessel factor for the n-th UR harmonic. The Bessel factors f

_{n}in the general case of the two-dimensional field with harmonics (2) are given by (22)–(24) and (31) accounting for the finite beam size effects and constant magnetic components, which cause even harmonics. We assume the fundamental harmonic is not suppressed and it dominates. The saturated n-th harmonic power can be calculated accounting for the loss factors following [53]: ${P}_{n,F}=\sqrt{2}{P}_{e}{\eta}_{n}{\eta}_{1}{\chi}^{2}{\rho}_{1}^{}{f}_{n}^{2}/\left({n}^{5/2}{f}_{1}^{2}\right)$, where ${P}_{e}={I}_{0}E$ is the beam power, I

_{0}is the beam current [A], E is the electron energy [eV], ρ

_{1}is the Pierce parameter. The gain length for the n-th harmonic is ${L}_{n,g}\cong {\mathsf{\Phi}}_{n}{\lambda}_{u}/\left(4\pi \sqrt{3}{n}^{1/3}\chi {\rho}_{n}\right)$, where ${\lambda}_{u}$ is the undulator period, Φ

_{n}η

_{n}are the loss factors. For the fundamental tone we denote ${L}_{1,g}\equiv {L}_{g}$; the fundamental tone saturation length is ${L}_{s}\cong 1.07{L}_{1,\text{}g}\text{}\mathrm{ln}\left(9{P}_{1,F}/{P}_{1,0}\right)$.

_{n}; the beam energy spread ${\sigma}_{e}$ and the emittances ${\epsilon}_{x,y}$ prolong the gain ${L}_{n,g}$ and reduce the saturated powers ${P}_{n,F}$ for harmonics. For stable FEL amplification, weak conditions σ

_{e}○ρ

_{n}/2, ε

_{x,y}○λ

_{n}/4π should be fulfilled (see, for example, [3,4,5,6,23,45,46,47,48,49,50]); however, failure to satisfy them exactly, especially in X-ray band, does not mean these harmonics will not be radiated at all.

_{0,1}, the n-th harmonic in nonlinear generation begins to saturate at the power level ${\tilde{P}}_{F}={{P}_{F}|}_{{\eta}_{n}\to {\tilde{\eta}}_{n}}$ and saturates with oscillations around the power ${\overline{P}}_{n,f}$. In an elliptic undulator two polarizations are radiated and the effective Pierce parameter is modified accordingly (see, for example, [16]). The above analytical model of the FEL harmonic power evolution describes independent and induced harmonic contributions, multistage harmonic saturation, power oscillations, all major losses and different sensitivity of the photon-electron interaction at different harmonic wavelengths; it agrees with the available results of FEL experiments in a wide range of conditions and radiated wavelengths.

## References

- Ginzburg, V.L. On the radiation of microradiowaves and their absorbtion in the air. Isvestia Akad. Nauk. SSSR (Fizika)
**1947**, 11, 1651. [Google Scholar] - Motz, H.; Thon, W.; Whitehurst, R.N.J. Experiments on radiation by fast electron beams. Appl. Phys.
**1953**, 24, 826. [Google Scholar] [CrossRef] - McNeil, B.W.J.; Thompson, N.R. X-ray free-electron lasers. Nat. Photonics
**2010**, 4, 814. [Google Scholar] [CrossRef] - Pellegrini, C.; Marinelli, A.; Reiche, S. The physics of X-ray free-electron lasers. Rev. Mod. Phys.
**2016**, 88, 015006. [Google Scholar] [CrossRef] - Schmüser, P.; Dohlus, M.; Rossbach, J.; Behrens, C. Free-Electron Lasers in the Ultraviolet and X-ray Regime. In Springer Tracts in Modern Physics; Springer: Cham, Switzerland, 2014; Volume 258. [Google Scholar]
- Huang, Z.; Kim, K.J. Review of X-ray free-electron laser theory. Phys. Rev. ST-AB
**2007**, 10, 034801. [Google Scholar] [CrossRef] - Margaritondo, G.; Ribic, P.R. A simplified description of X-ray free-electron lasers. J. Synchrotron Rad.
**2011**, 18, 101–108. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Margaritondo, G. Synchrotron light: A success story over six decades. Riv. Nuovo Cim.
**2017**, 40, 411–471. [Google Scholar] - Bagrov, V.G.; Bisnovaty-Kogan, G.S.; Bordovitsyn, V.A.; Borisov, A.V.; Dorofeev, O.F.; Ya, E.V.; Pivovarov, Y.L.; Shorokhov, O.V.; Zhukovsky, V.C. Synchrotron Radiation Theory and Its Development; Bordovitsyn, V.A., Ed.; Word Scientific: Singapore, 1999; p. 447. [Google Scholar]
- Margaritondo, G. Characteristics and Properties of Synchrotron Radiation. In Synchrotron Radiation; Mobilio, S., Boscherini, F., Meneghini, C., Eds.; Springer: Berlin/Heidelberg, Germany, 2015. [Google Scholar]
- Dattoli, G.; Ottaviani, P.L. Semi-analytical models of free electron laser saturation. Opt. Commun.
**2002**, 204, 283–297. [Google Scholar] [CrossRef] - Zhukovsky, K.; Kalitenko, A. Phenomenological and numerical analysis of power evolution and bunching in single-pass X-ray FELs. J. Synchrotron Rad.
**2019**, 26, 159–169. [Google Scholar] [CrossRef] [PubMed] - Zhukovsky, K.; Kalitenko, A. Analysis of harmonic generation in planar undulators in single-pass free electron lasers. Russ. Phys. J.
**2019**, 61, 153–160. [Google Scholar] - Zhukovsky, K.; Kalitenko, A. Phenomenological and numerical analysis of power evolution and bunching in single-pass X-ray FELs. Erratum. J. Synchrotron Rad.
**2019**, 26, 605–606. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Zhukovsky, K. Two-frequency undulator in a short SASE FEL for angstrom wavelengths. J. Opt.
**2018**, 20, 095003. [Google Scholar] [CrossRef] - Zhukovsky, K. Analysis of harmonic generation in planar and elliptic bi-harmonic undulators and FELs. Results Phys.
**2019**, 13, 102248. [Google Scholar] [CrossRef] - Zhukovsky, K.V. Analytical Description of Nonlinear Harmonic Generation Close to the Saturation Region in Free Electron Lasers. Mosc. Univ. Phys. Bull.
**2019**, 74, 480–487. [Google Scholar] [CrossRef] - Emma, P. First lasing of the LCLS X-ray FEL at 1.5 Å. In Proceedings of the PAC09, Vancouver, BC, Canada, 4–8 May 2009. [Google Scholar]
- Emma, P.; Akre, R.; Arthur, J.; Bionta, R.; Bostedt, C.; Bozek, J.; Brachmann, A.; Bucksbaum, P.; Coffee, R.; Decker, F.-J.; et al. First lasing and operation of an angstrom-wavelength free-electron laser. Nat. Photonics
**2010**, 4, 641–647. [Google Scholar] [CrossRef] - Ratner, D.; Brachmann, A.; Decker, F.J.; Ding, Y.; Dowell, D.; Emma, P.; Frisch, J.; Huang, Z.; Iverson, R.; Krzywinski, J.; et al. Second and third harmonic measurements at the linac coherent light source. Phys. Rev. ST-AB
**2011**, 14, 060701. [Google Scholar] [CrossRef] [Green Version] - Biedron, S.G.; Dejusa, R.J.; Huanga, Z.; Miltona, S.V.; Sajaeva, V.; Berga, W.; Borlanda, M.; den Hartoga, P.K.; Erdmanna, M.; Fawleyc, W.M.; et al. Measurements of nonlinear harmonic generation at the Advanced Photon Source’s SASE FEL. Nucl. Instrum. Meth. Phys. Res.
**2002**, 483, 94–100. [Google Scholar] [CrossRef] [Green Version] - Milton, S.V.; Gluskin, E.; Arnold, N.D.; Benson, C.; Berg, W.; Biedron, S.G.; Borland, M.; Chae, Y.C.; Dejus, R.J.; den Hartog, P.K.; et al. Exponential Gain and Saturation of a Self-AmpliÞed Spontaneous Emission Free-Electron Laser. Science
**2011**, 292, 2037–2041. [Google Scholar] [CrossRef] [Green Version] - Huang, Z.; Kim, K.-J. Nonlinear Harmonic Generation of Coherent Amplification and Self-Amplified Spontaneous Emission. Nucl. Instr. Meth.
**2001**, 475, 112. [Google Scholar] [CrossRef] [Green Version] - Zhukovsky, K.V. Effect of the 3rd undulator field harmonic on spontaneous and stimulated undulator radiation. J. Synchrotron Rad.
**2019**, 26, 1481–1488. [Google Scholar] [CrossRef] [Green Version] - Zhukovsky, K.V. Generation of UR Harmonics in Undulators with Multiperiodic Fields. Russ. Phys. J.
**2019**, 62, 1043–1053. [Google Scholar] [CrossRef] - Alexeev, V.I.; Bessonov, E.G. On some methods of generating circularly polarized hard undulator radiation. Nucl. Instr. Meth.
**1991**, 308, 140. [Google Scholar] - Dattoli, G.; Mikhailin, V.V.; Ottaviani, P.L.; Zhukovsky, K. Two-frequency undulator and harmonic generation by an ultrarelativistic electron. J. Appl. Phys.
**2006**, 100, 084507. [Google Scholar] [CrossRef] - Zhukovsky, K. Analytical account for a planar undulator performance in a constant magnetic field. J. Electromagn. Waves Appl.
**2014**, 28, 1869–1887. [Google Scholar] [CrossRef] - Dattoli, G.; Mikhailin, V.V.; Zhukovsky, K. Undulator radiation in a periodic magnetic field with a constant component. J. Appl. Phys.
**2008**, 104, 124507-1–124507-8. [Google Scholar] [CrossRef] - Dattoli, G.; Mikhailin, V.V.; Zhukovsky, K.V. Influence of a constant magnetic field on the radiation of a planar undulator. Mosc. Univ. Phys. Bull.
**2009**, 64, 507–512. [Google Scholar] [CrossRef] - Zhukovsky, K.; Srivastava, H. Operational solution of non-integer ordinary and evolution-type partial differential equations. Axioms
**2016**, 5, 29. [Google Scholar] [CrossRef] - Zhukovsky, K.V. Solving evolutionary-type differential equations and physical problems using the operator method. Theor. Math. Phys.
**2017**, 190, 52–68. [Google Scholar] [CrossRef] - Zhukovsky, K.V. Operational solution for some types of second order differential equations and for relevant physical problems. J. Math. Anal. Appl.
**2017**, 446, 628–647. [Google Scholar] [CrossRef] - Dattoli, G.; Srivastava, H.M.; Zhukovsky, K.V. Orthogonality properties of the Hermite and related polynomials. J. Comput. Appl. Math.
**2005**, 182, 165–172. [Google Scholar] [CrossRef] [Green Version] - Dattoli, G.; Srivastava, H.M.; Zhukovsky, K. Operational Methods and Differential Equations with Applications to Initial-Value problems. Appl. Math. Comput.
**2007**, 184, 979–1001. [Google Scholar] [CrossRef] - Zhukovsky, K.V. The operational solution of fractional-order differential equations, as well as Black-Scholes and heat-conduction equations. Mosc. Univ. Phys. Bull.
**2016**, 71, 237–244. [Google Scholar] [CrossRef] - Zhukovsky, K. Operational Approach and Solutions of Hyperbolic Heat Conduction Equations. Axioms
**2016**, 5, 28. [Google Scholar] [CrossRef] [Green Version] - Gould, H.W.; Hopper, A.T. Operational formulas connected with two generalizations of Hermite polynomials. Duke Math. J.
**1962**, 29, 51–63. [Google Scholar] [CrossRef] - Kang, H.-S.; Chang-Ki, M.; Hoon, H.; Changbum, K.; Haeryong, Y.; Gyujin, K.; Inhyuk, N.; Soung, Y.B.; Hyo-Jin, C.; Geonyeong, M.; et al. Hard X-ray free-electron laser with femtosecondscale timing jitter. Nat. Photonics
**2017**, 11, 708–713. [Google Scholar] [CrossRef] - Alferov, D.F.; Bashmakov Yu, A.; Bessonov, E.G. Undulator radiation. Sov. Phys. Tech. Phys.
**1974**, 18, 1336. [Google Scholar] - Alferov, D.F.; Bashmakov, U.A.; Cherenkov, P.A. Radiation from relativistic electrons in a magnetic undulator. Usp. Fis. Nauk.
**1989**, 32, 200. [Google Scholar] - Vinokurov, N.A.; Levichev, E.B. Undulators and wigglers for the production of radiation and other applications. Phys. Usp.
**2015**, 58, 917–939. [Google Scholar] [CrossRef] - Dattoli, G.; Renieri, A.; Torre, A. Lectures on the Free Electron Laser Theory and Related Topics; World Scientific: Singapo, 1993. [Google Scholar]
- Prakash, B.; Huse, V.; Gehlot, M.; Mishra, G.; Mishra, S. Analysis of spectral properties of harmonic undulator radiation of anelectromagnet undulator. Optik
**2016**, 127, 1639–1643. [Google Scholar] [CrossRef] - Saldin, E.L.; Schneidmiller, E.A.; Yurkov, M.V. The Physics of Free Electron Lasers; Springer: Berlin/Heidelberg, Germany, 2000. [Google Scholar]
- Bonifacio, R.; Pellegrini, C.; Narducci, L. Collective instabilities and high-gain regime in a free electron laser. Opt. Commun.
**1984**, 50, 373. [Google Scholar] [CrossRef] [Green Version] - Bonifacio, R.; De Salvo, L.; Pierini, P. Large harmonic bunching in a high-gain free-electron laser. Nucl. Instrum. A
**1990**, 293, 627. [Google Scholar] [CrossRef] - Huang, Z.; Kim, K.-J. Three-dimensional analysis of harmonic generation in high-gain free-electron lasers. Phys. Rev. E
**2000**, 62, 7295. [Google Scholar] [CrossRef] [PubMed] - Saldin, E.L.; Schneidmiller, E.A.; Yurkov, M.V. Study of a noise degradation of amplification process in a multistage HGHG FEL. Opt. Commun.
**2002**, 202, 169–187. [Google Scholar] [CrossRef] - Shatan, T.; Yu, L.-H. High-gain harmonic generation free-electron laser with variable wavelength. Phys. Rev. E
**2005**, 71, 046501. [Google Scholar] [CrossRef] [PubMed] - Dattoli, G.; Giannessi, L.; Ottaviani, P.L.; Ronsivalle, C. Semi-analytical model of self-amplified spontaneous-emission free-electron lasers, including diffraction and pulse-propagation effects. J. Appl. Phys.
**2004**, 95, 3206–3210. [Google Scholar] [CrossRef] - Dattoli, G.; Ottaviani, P.L.; Pagnutti, S. Nonlinear harmonic generation in high-gain free-electron lasers. J. Appl. Phys.
**2005**, 97, 113102. [Google Scholar] [CrossRef] - Dattoli, G.; Ottaviani, P.L.; Pagnutti, S. Booklet for FEL Design; ENEA Pubblicazioni: Frascati, Italy, 2007. [Google Scholar]
- Zhukovsky, K.V. Generation of Coherent Radiation in the Near X-ray Band by a Cascade FEL with a Two-Frequency Undulator. Mosc. Univ. Phys. Bull.
**2018**, 73, 364–371. [Google Scholar] [CrossRef] - Zhukovsky, K. Compact single-pass X-ray FEL with harmonic multiplication cascades. Opt. Commun.
**2018**, 418, 57–64. [Google Scholar] [CrossRef] - Zhukovsky, K. Soft X-ray generation in cascade SASE FEL with two-frequency undulator. EPL
**2017**, 119, 34002. [Google Scholar] [CrossRef] - Zhukovsky, K.V. Generation of X-Ray Radiation in Free-Electron Lasers with Two-Frequency Undulators. Russ. Phys. J.
**2018**, 60, 1630–1637. [Google Scholar] [CrossRef] - Zhukovsky, K. Generation of coherent soft X-ray radiation in short FEL with harmonic multiplication cascades and two-frequency undulator. J. Appl. Phys.
**2017**, 122, 233103. [Google Scholar] [CrossRef] - Zhukovsky, K. High-harmonic X-ray undulator radiation for nanoscale-wavelength free-electron lasers. J. Phys. D
**2017**, 50, 505601. [Google Scholar] [CrossRef] - Zhukovsky, K.; Potapov, I. Two-frequency undulator usage in compact self-amplified spontaneous emission free electron laser in Roentgen range. Laser Part. Beams
**2017**, 35, 326. [Google Scholar] [CrossRef] - Tetsuya, I.; Hideki, A.; Takao, A.; Yoshihiro, A.; Noriyoshi, A.; Teruhiko, B.; Hiroyasu, E.; Kenji, F.; Toru, F.; Yukito, F.; et al. A compact X-ray free-electron laser emitting in the sub-ångström region. Nat. Photonics
**2012**, 6, 540–544. [Google Scholar] - Owada, S.; Togawa, K.; Inagaki, T.; Hara, T.; Tanaka, T.; Joti, Y.; Koyama, T.; Nakajima, K.; Ohashi, H.; Senba, Y.; et al. Soft X-ray free-electron laser beamline at SACLA: The light source, photon beamline and experimental station. J. Synchrotron Rad.
**2018**, 25, 282–288. [Google Scholar] [CrossRef] - Ego, H. RF system of the SPring-8 upgrade project. In Proceedings of the IPAC2016, Busan, Korea, 8−13 May 2016. [Google Scholar]
- Tono, K.; Hara, T.; Yabashi, M.; Tanaka, H. Multiple-beamline operation of SACLA. J. Synchrotron Rad.
**2019**, 26, 595–602. [Google Scholar] [CrossRef] [Green Version] - Ichiro, I. Generation of narrow-band X-ray free-electron laser via reflection self-seeding. Nat. Photonics
**2019**, 13, 319–322. [Google Scholar] - Tono, K.; Togashi, T.; Inubushi, Y.; Sato, T.; Katayama, T.; Ogawa, K.; Ohashi, H.; Kimura, H.; Takahashi, S.; Takeshita, K.; et al. Beamline, experimental stations and photon beam diagnostics for the hard X-ray free electron laser of SACLA. New J. Phys.
**2013**, 15, 083035. [Google Scholar] [CrossRef] - Yabashi, M. Overview of the SACLA facility. J. Synchrotron Rad.
**2015**, 22, 477–484. [Google Scholar] [CrossRef] [Green Version] - Yabashi, M.; Tanaka, H.; Tono, K.; Ishikawa, T. Status of the SACLA Facility. Appl. Sci.
**2017**, 7, 604. [Google Scholar] [CrossRef] [Green Version] - Takashi, T.; Shunji, G.; Toru, H.; Takaki, H.; Haruhiko, O.; Kazuaki, T.; Makina, Y.; Hitoshi, T. Undulator commissioning by characterization of radiation in X-ray free electron lasers. Phys. Rev. ST-AB
**2012**, 15, 110701. [Google Scholar]

**Figure 1.**The PAL-XFEL fundamental UR line shape as the function of the constant magnetic field H

_{y}: (

**a**)—on-the-axis case γθ = 0, (

**b**) off-axis angle γθ = 0.067; the undulator has k = 1.87, period λ

_{u}= 2.57 cm, length L = 5 m, N = 194 periods.

**Figure 2.**The PAL-XFEL fundamental UR line shape as the function of the off-axis angle γθ; (

**a**)—no constant magnetic component, κ = 0, (

**b**)—in the presence of the constant magnetic component H

_{d}= 0.5 Gauss; the undulator has k = 1.87, period λ

_{u}= 2.57 cm, length L = 5 m, N = 194 periods.

**Figure 3.**The contribution of the constant magnetic field term ${f}_{2}^{3}$ to the UR line of the second UR harmonic, n = 2, of the PAL-XFEL undulator with k = 1.87, period λ

_{u}= 2.57 cm, length L = 5 m, N = 194 periods: (

**a**)—on the axis case, γθ = 0, (

**b**)—the off-axis angle γθ = 0.067.

**Figure 4.**The contribution of the terms ${f}_{2}^{1}$ and ${f}_{2}^{2}$ to the UR line of the second UR harmonic, n = 2, of the PAL-XFEL for the undulator with k = 1.87 period 2.57 cm, length 5 m, N = 194 periods, off-axis angle γθ = 0.067.

**Figure 5.**The total contribution of all terms ${f}_{2}^{1,2,3}$ to the normalized intensity I

_{2}of the harmonic n = 2 accounting for the functions S and $\partial S/\partial {\nu}_{n}$, giving the line shape to 2nd UR harmonic of the PAL-XFEL undulator with k = 1.87, period λ

_{u}= 2.57 cm, length L = 5 m, N = 194 periods for the off-axis angle γθ = 0.067.

**Figure 6.**Evolution of the harmonic power in the SACLA experiment for E = 780 MeV, λ

_{1}= 12.4 nm, σ

_{e}= 1.6 × 10

^{−3}, I

_{0}= 300 A. The harmonics are color coded: n = 1—red solid, n = 3—green dashed, n = 5—blue dotted. The experimental values of the harmonic powers are denoted by the colored dot-dashed lines on the right.

**Figure 7.**Split of the fundamental radiation line λ = 12.4 nm for SACLA experiment, as a function of the distance Δ from the electron beam axis.

**Figure 8.**Evolution of the harmonic power in the SACLA FEL experiment for E = 7800 MeV, λ

_{1}= 0.124 nm, σ

_{e}= 9.26 × 10

^{−4}, I

_{0}= 10 kA. The harmonics are color coded: n = 1—red solid, n = 2—orange dot-dashed, n = 3—green dashed, n = 5—blue dotted. The experimental values of the harmonic powers are denoted by the colored dotted lines on the right.

**Figure 9.**Split of the fundamental spectrum line λ = 0.124 nm for SACLA BL3 line, as a function ofthe distance Δ from the electron beam axis.

**Figure 10.**Total contribution of 11 subharmonics with p = −5…+5, factorizing the Bessel factors f

_{n}as a function of the distance Δ from the electron beam axis and electron-photon interaction angle θ.

**Figure 11.**Evolution of the harmonic power in the SACLA experiment for E = 7 GeV, λ

_{1}= 0.124 nm, σ

_{e}= 8.7 × 10

^{−4}, I

_{0}= 3.5 kA, β

_{x,y}= 22 m, ε

^{n}= 0.6π mm × mrad. The harmonics are color coded: n = 1—red solid, n = 3—green dashed. The experimental values of the harmonic powers are denoted by the dots and by colored areas on the right.

**Figure 12.**The harmonic power evolution along the undulators at PAL-XFEL for soft X-rays, λ

_{1}= 1.52 nm. The experimental average values are shown by dots, following the data in [39]. The harmonics are color coded: n = 1—red solid, n = 2—orange dot-dashed, n = 3—green dashed, n = 5—blue dotted.

**Figure 13.**The harmonic power evolution along the undulators at PAL-XFEL for hard X-rays, λ

_{1}= 0.144 nm. The experimental average values are shown by dots, following the data in [39]. The harmonics are color coded: n = 1—red solid, n = 2—orange dot-dashed, n = 3—green dashed, n = 5—blue dotted.

Beam parameters: relativistic factor γ = 1526, beam power P _{E} = 234 GW, current I_{0} = 300 A, current density J = 2.9 × 10^{10} A/m^{2}, beam section ∑ = 1.03×10^{−8} m^{2}, emittances $\gamma {\epsilon}_{x,y}^{}\approx 0.5$ μm, β_{x} = 6 m, β_{y} = 4 m, beam size ${\sigma}_{x,y}$≈40 μm, divergence ${\theta}_{div}$≈ 8 μrad, $\theta ={\sigma}_{photon}/{L}_{gain}$ ~40 μrad, γθ ≈ 0.06, energy spread (per slice) σ_{e} = 1.6 × 10^{−3} | |||||

Undulator parameters: k = 2.1, λ_{u} = 1.8 cm, N = 259, section length 4.66 m | |||||

Calculated FEL properties: saturated length L_{s} = 13 m, gain length L_{gain} = 1.1 m, radiation beam size ${\sigma}_{photon}\approx \sqrt{{\sigma}_{x,y}\sqrt{{\lambda}_{1}{L}_{g}/4\pi}}$ ≅ 36 μm | |||||

Harmonic number | n = 1 | n = 2 | n = 3 | n = 4 | n = 5 |

Bessel coefficient f_{n} | 0.79 | 0.09 | 0.32 | 0.09 | 0.18 |

Pierce parameter ${\tilde{\rho}}_{n}$ | 0.0015 | 0.0003 | 0.0008 | 0.0003 | 0.0006 |

Harmonic wavelength λ_{n}, nm | 12.4 | 6.2 | 4.1 | 3.1 | 2.5 |

Saturated power P_{F,n},W | 1.9 × 10^{8} | — | 6 × 10^{5} | — | 3 × 10^{4} |

Beam parameters: relativistic factor γ = 15264, beam power P _{E} = 78 TW, current I_{0} = 10 kA, current density J = 3.04×10^{12} A/m^{2}, beam section ∑ = 3.29×10^{−9} m^{2}, emittances $\gamma {\epsilon}_{x,y}^{}\approx 0.4$μm, β_{x}_{,y} = 20m, beam size ${\sigma}_{x,y}$≈ 22μm, divergence ${\theta}_{div}$≈ 1.1 μrad, $\theta ={\sigma}_{photon}/{L}_{gain}$~ 9 μrad, γθ ≈ 0.14, energy spread σ _{e} = 0.926 × 10^{−3} | |||||

Undulator parameters: k = 2.1, λ_{u} = 1.8 cm, N = 277, section length 4.66 m | |||||

Calculated FEL properties: saturated length L_{s} = 38 m, gain length L_{gain} = 2.6 m, radiation beam size ${\sigma}_{photon}\approx \sqrt{{\sigma}_{x,y}\sqrt{{\lambda}_{1}{L}_{g}/4\pi}}$ ≅ 11 μm | |||||

Harmonic number | n = 1 | n = 2 | n = 3 | n = 4 | n = 5 |

Bessel coefficient f_{n} | 0.79 | 0.19 | 0.27 | 0.19 | 0.11 |

Pierce parameter ${\tilde{\rho}}_{n}$ | 0.00075 | 0.0003 | 0.00037 | 0.0003 | 0.0002 |

Harmonic wavelength λ_{n}, nm | 12.4 | 6.2 | 4.1 | 3.1 | 2.5 |

Saturated power P_{F,n},W | 1.9 × 10^{10} | 9 × 10^{6} | 5 × 10^{7} | 5 × 10^{6} | 1.6 × 10^{5} |

Beam parameters: γ = 5870, beam power P _{E} = 6.60 TW, current I_{0} = 2.2 kA, current density J = 1.246 × 10^{11} A/m^{2}, beam section ∑ = 1.766 × 10^{−8} m^{2}, emittances $\gamma {\epsilon}_{x,y}$= 0.55 μm, β = 30 m, beam size ${\sigma}_{x,y}$ = 53 μm, divergence ≈ 1.8 μrad, $\theta ={\sigma}_{photon}/{L}_{gain}$ ≈ 15 μrad, energy spread σ _{e} = 0.5 × 10^{−3} | |||||

Undulator parameters: k = 2, λ_{u} = 3.5 cm, section length 5 m | |||||

Calculated FEL properties: saturated length L_{s} = 31 m, gain length L_{gain} = 2.0 m, radiation beam size ${\sigma}_{photon}\approx \sqrt{{\sigma}_{x,y}\sqrt{{\lambda}_{1}{L}_{g}/4\pi}}$≈0.29 mm | |||||

Harmonic number | n = 1 | n = 2 | n = 3 | n = 4 | n = 5 |

Bessel coefficient f_{n} | 0.80 | 0.13 | 0.32 | 0.13 | 0.16 |

Pierce parameter ${\tilde{\rho}}_{n}$ | 0.0010 | 0.0003 | 0.0005 | 0.0003 | 0.0003 |

Harmonic wavelength λ_{n}, nm | 1.52 | 0.76 | 0.51 | 0.38 | 0.30 |

Saturated power P_{F,n},W | 8.2 × 10^{9} | 3.2 × 10^{6} | 5.4 × 10^{7} | 1.6 × 10^{6} | 2.0 × 10^{6} |

Beam parameters: γ = 15,660, beam power P _{E} = 20.0 TW, current I_{0} = 2,5 kA, current density J = 3.16 × 10^{11} A/m^{2}, beam section ∑ = 7.91 × 10^{−9} m^{2}, emittances $\gamma {\epsilon}_{x,y}\approx 0.55$μm, β ≈ 36 m, beam size ${\sigma}_{x,y}$ = 35 μm, divergence ≈ 1 μrad, $\theta ={\sigma}_{photon}/{L}_{gain}$≈4.5 μrad, energy spread σ _{e} = 0.18 × 10^{−3} | |||||

Undulator parameters: k = 1.87, λ_{u} = 2.571 cm, section length 5 m | |||||

Calculated FEL properties: saturated length L_{s} ~ 55 m, gain length L_{gain} = 3.4 m, radiation beam size ${\sigma}_{photon}\approx \sqrt{{\sigma}_{x,y}\sqrt{{\lambda}_{1}{L}_{g}/4\pi}}$≈15 μm | |||||

Harmonic number | n = 1 | n = 2 | n = 3 | n = 4 | n = 5 |

Bessel coefficient f_{n} | 0.82 | 0.09 | 0.31 | 0.09 | 0.16 |

Pierce parameter ${\tilde{\rho}}_{n}$ | 0.0004 | 0.00009 | 0.0002 | 0.00009 | 0.00014 |

Harmonic wavelength λ_{n}, nm | 0.144 | 0.072 | 0.048 | 0.036 | 0.029 |

Saturated power P_{F,n},W | 1.0 × 10^{10} | 2.0 × 10^{6} | 1.0 × 10^{8} | 1.0 × 10^{6} | 6.0 × 10^{6} |

© 2020 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Zhukovsky, K.
Synchrotron Radiation in Periodic Magnetic Fields of FEL Undulators—Theoretical Analysis for Experiments. *Symmetry* **2020**, *12*, 1258.
https://doi.org/10.3390/sym12081258

**AMA Style**

Zhukovsky K.
Synchrotron Radiation in Periodic Magnetic Fields of FEL Undulators—Theoretical Analysis for Experiments. *Symmetry*. 2020; 12(8):1258.
https://doi.org/10.3390/sym12081258

**Chicago/Turabian Style**

Zhukovsky, Konstantin.
2020. "Synchrotron Radiation in Periodic Magnetic Fields of FEL Undulators—Theoretical Analysis for Experiments" *Symmetry* 12, no. 8: 1258.
https://doi.org/10.3390/sym12081258