# Spontaneous and Stimulated Undulator Radiation in Symmetric and Asymmetric Multi-Periodic Magnetic Fields

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Numerical Approach to UR Calculations with the SPECTRA Code

_{p}or the radiated power dP per unit of surface dS in a relative spectral interval dω/ω through the following convolution [46]:

_{1}is the fundamental UR frequency and N is the number of undulator periods;

_{u}is the undulator period, φ is the polar angle and [46]

_{ε}is the spread. The usual approximation for the radiation studies is the far zone, where $\left|Z-z\right|\gg \left|X-x\right|,\text{}\left|Y-y\right|$ and the distance between the electron and the observer is much longer than the undulator size; although it is possible to include near zone in the SPECTRA calculations, this noticeably slows down the process. We use the far zone approximation in what follows.

_{N}, which allows for a fast Fourier transform algorithm for the convolution. In SPECTRA code, fast oscillating integrands occur, and they must be computed many times for precision. To help with this task, the whole region of integration over the time of the electron motion in the undulator is divided into several sections, within which the functions are approximated by third order polynomials. If the function $g(t)$ can be approximated by an n-th order polynomial in the region of integration, then its n-th derivative ${d}^{n}g/d{t}^{n}$ is constant. By integrating the function $G={\displaystyle \int g(t){e}^{i\omega t}dt}$ by parts n times, we get

_{N}of Equation (2) has little effect on the shape of the spectrum. In this case, SPECTRA omits the convolution and saves time without sacrificing accuracy. For our calculations, we used the “Energy Dependence → Angular Flux Density” option, where the above-described integrals were computed. For further details and options of SPECTRA, see [46].

## 3. Bessel Factors for UR Harmonics in Multiperiodic Magnetic Fields

_{0}is the amplitude of the magnetic field on the undulator axis, directed along z. The above model, complemented by the hyperbolic trigonometric functions (see [51,52]), ${H}_{y}={H}_{0}\mathrm{sin}({k}_{\lambda}z)\mathrm{cosh}({k}_{\lambda}y)$ and ${H}_{z}={H}_{0}\mathrm{cos}({k}_{\lambda}z)\mathrm{sinh}({k}_{\lambda}y)$, describes the magnetic field in the whole gap between the undulator magnets and satisfies Maxwell equations. The off-axis position of electrons in a finite size beam causes betatron oscillations, the split of the radiation lines, and even UR harmonics on the undulator axis. The even UR harmonics also appear due to the off-axis angles, in which the radiation is viewed in the whole section of the beam, and due to the distortions induced by non-periodic magnetic fields. The above-mentioned effects are known; they have recently been readdressed in the context of multi-harmonic undulators in [52,53,54,55,56,57,58,59,60]. Here, we focused on the effect of the field harmonics on the spontaneous UR and high-gain FEL radiation; we analyzed the radiation harmonic intensities and their Bessel factors. The latter determine the universal scaling FEL parameter, the so-called Pierce parameter [15,16]:

^{2}] is the current density, γ is the relativistic factor, $i\cong 1.7045\times {10}^{4}$ is the Alfven current constant (A), ${k}_{eff}$ is the effective undulator parameter that reduces to $k={H}_{0}{\lambda}_{u;x}e/2\pi {m}_{e}{c}^{2}$$\cong {\lambda}_{u}\left[cm\right]{H}_{0}\left[kG\right]/10.7$ in common planar undulators, and ${f}_{n}$ is the Bessel factor for the n-th UR harmonic. The increase of the Pierce parameter, defined by Equation (8), shortens the FEL gain length, ${L}_{n,g}\cong {\lambda}_{u}/\left(4\pi \sqrt{3}{n}^{1/3}{\rho}_{n}\right)$, the saturation length, and the FEL itself. When the fundamental frequency of a FEL saturates, higher harmonics growth terminates, thus reaching its saturated powers (see, for example, [28,29]):

_{e}is the power of the electron beam. Higher values of the Pierce parameter ρ yield stronger amplification of harmonics and less strict requirements to the energy spread in an FEL. To maximize ρ for a given electron current and energy, the value of ${\kappa}_{n}$ in Equation (8) can be increased by using special undulators with possibly large Bessel factors f

_{n}, or by making an undulator with long period λ

_{u}and large deflection parameter k. However, the higher the values of λ

_{u}and k, the longer the UR harmonic resonance wavelength becomes:

_{n}) for high harmonic powers and their fast growth. Moreover, use of UR harmonics allows for lower electron energy than that needed for the fundamental frequency at the same wavelength. This is particularly important for the X-ray band, where all installation parameters are at their extreme values. The n-th harmonic of spontaneous UR from an undulator with N main periods has the following intensity:

_{u}requires renormalization that accounts for different k values. Indeed, $\frac{{d}^{2}{I}_{1,n}}{d\omega d\Omega}\propto \frac{{L}^{2}{H}_{1}{}^{2}{n}^{2}{\left|{f}_{n}\right|}^{2}}{{\left(1+\left({k}_{1,eff}^{2}/2\right)\right)}^{2}}$ (see Equation (11)), and the ratio of the intensity of the harmonic n of one undulator to the intensity of the harmonic m of another undulator for the same undulator length is

_{1,n}and f

_{2,m}are, respectively, the Bessel factors for the n-th harmonic of the first undulator and for the m-th harmonic of the second undulator with their respective deflection parameters k

_{1,2}, ${k}_{i,eff}^{2}={k}_{i}{}^{2}\varpi $.

_{u,x}is denoted by ${\lambda}_{u}\equiv {\lambda}_{u;x}$ for conciseness. The UR resonances λ

_{n}for the undulator with multiple periods are determined by usual Formula (10), where the field harmonics p, h, and l, with the amplitudes d, d

_{1}, and d

_{2}, respectively, yield $\varpi =1+{\left(d/p\right)}^{2}+{\left({d}_{1}/h\right)}^{2}+{\left({d}_{2}/l\right)}^{2}$. The Bessel factors for the UR harmonics from the undulator of Equation (13) read as follows [55,56,57,58,59,60]:

_{4}, and thus also in all other arguments ξ

_{i}, reported below; the off-axis angle θ, ${\theta}^{2}={\theta}_{x}^{2}+{\theta}_{y}^{2}$, ${\theta}_{x}={\mathrm{tan}}^{-1}\left(X/Z\right)$, ${\theta}_{y}={\mathrm{tan}}^{-1}\left(Y/Z\right)$, and the polar angle φ are also present in the following first arguments:

_{i}in Equation (20) are given by Equations (17) and (18), except for the different sign of ${\xi}_{6}=+{\xi}_{4}{d}_{2}^{2}/{l}^{3}$ in Equation (17). Because of the change of the H

_{y}component in Equation (19) with respect to Equation (13), the Bessel factor f

_{n}

_{;x}for the undulator field of Equation (19) differs from that in Equation (14); the Bessel factors f

_{n}

_{;x}and f

_{n}

_{;y}for the undulator field of Equation (19) read as follows:

## 4. Radiation from Elliptic and Planar Undulators with Field Harmonics

_{u}= 2.8 cm and k = 2.133. It is worth remembering that the relative spectral distribution of UR is determined by the undulator and not by the beam, which is different from SR, where the electron energies determine the harmonic ratios and the overall spectrum shape. The small beam emittance (see Table 1) mainly affected the even harmonics. The third y-field harmonic in Equation (23) determined the x-polarization of the radiation in Figure 1; there, the fundamental frequency was absent, and the total and relative radiation powers for the third harmonic at the wavelength λ

_{3}= 183 nm were thus increased. The Bessel factors are f

_{y;n =}

_{1,3,5}≈ {0.81, 0.33, 0.15} and f

_{x;n=}

_{1,3,5}≈ {0.06, 0.25, 0.18}.

_{0}and the only period λ

_{u}/3, in contrast with the bi-harmonic undulator with two periods λ

_{u}and λ

_{u}/3 in Equations (23) and (24). The considered undulator field read as follows:

_{n}(Equation (10)), and the comparison could not be done at the same identical wavelengths. Moreover, the shape of the UR spectrum was determined by the effective deflection parameters k; since k was different for Equation (25) from that for Equations (23) and (24), we got different overall spectrum shapes. Thus, we could only estimate, for example, the radiation intensity for the seventh harmonic of the undulator with the field of Equation (24) and compare it with that of the fundamental frequency of the undulator with the field of Equations (25) for some given field amplitude and undulator length.

_{1}= 549 nm; the fundamental wavelength in the field of Equation (25) was λ

_{1}= 65 nm. For a correct comparison, we accounted for the renormalization factor of Equation (12): ${(1+({k}_{1}^{2}(1+{(1/3)}^{2})/2))}^{2}/{(1+({k}_{2}^{2}/2))}^{2}$, where k

_{2}= 0.711 and k

_{1}= 2.133. While accounting for the energy spread ${\sigma}_{e}=0.1\%$, (see Table 1), we computed and show in Figure 3 the spectrum of the UR in the field of Equation (25).

_{1}≈ 65 nm of the planar undulator of Equation (25) (see Figure 3) with that of the ninth harmonic λ

_{9}≈ 61 nm of the elliptic undulator of Equation (24) (see Figure 2) showed that the fundamental frequency of the planar undulator of Equation (25) was stronger than the ninth harmonic of the elliptic undulator of Equation (24). The difference was roughly one order of magnitude (compare the height of the red bar in Figure 3 with that of magenta bars in Figure 2); the ratio was higher for the harmonics, radiated in the pure sinusoidal symmetric field of Equation (23).

_{7}= 78 nm and λ

_{9}= 61 nm, while for the fundamental tone of the planar undulator of Equation (25) at λ = 65 nm, we got ${\kappa}_{n=1}^{planar}=0.034$. These values were comparable with each other, and thus we conclude that the fundamental frequency of the planar undulator with the field of Equation (25) was amplified in an FEL approximately as well as the harmonics at close wavelengths of ~60–80 nm in the elliptic undulator with the field of Equation (24). We only considered the independent amplification of the harmonics and omitted the nonlinear term induced by the fundamental frequency of the elliptic undulator of Equation (24).

_{u}= 2.3 cm, and k = 2.216. The amplitude of the third field harmonic was small: d = 0.0825. An experiment KAERI [53] was conducted with low energy electrons with E = 6.5 MeV. The radiation wavelength of the fundamental tone was λ = 0.42 mm. The data of the accelerator and the light source are collected in Table 2. The harmonic intensities for the UR in the field of Equation (26) were reported in [53]; the fifth UR harmonic content on the undulator axis was ≈2% on the background of noise and second harmonic in the asymmetric wide beam. Our numerical calculations with the SPECTRA code had good agreement with the data in [53], especially for odd harmonics both on and off the axis (see Figure 4); we got some weaker even harmonics and a clearer picture with less noise and less split UR lines than in [53]; the analytical results looked similar and are omitted for conciseness.

_{0}in Equation (27) and other parameters of the undulator and the beam from Table 1. Then, dependently on the field harmonic phase, h = ±3, we got the results shown in Figure 6. Note that for d = +0.3, the third UR harmonic was enhanced, and for d = −0.3, it was weakened compared with the radiation from a common planar undulator, where d = 0. Evidently, d = +0.3 in Equation (27) enhanced the radiation of the third UR harmonic; the latter had y-polarization. A comparison of the spectrum of the planar undulator with the field of Equation (27) with the spectrum of the elliptic undulator with the field of Equation (26) showed that the third UR harmonic in Equation (27) had approximately the same wavelength as the fifth UR harmonic in Equation (26). However, the third UR harmonic, emitted in the planar field of Equation (27) (see green bars in Figure 6) was much stronger than the fifth UR harmonic emitted in the elliptic field of Equation (26) (see blue bar in Figure 5) for the same amplitude d of the field harmonic.

_{n}can be found, for example, in [28,31,32,34,40].

## 5. Harmonic Radiation in Undulators with Variable k Parameter

_{0}< 2.5 kA, the emittances are ε

_{x,y}= 1.4 μm × rad, and the Twiss parameters are β

_{x,y}= 6 m. The undulator period is λ

_{u}= 3.14 cm, the deflection parameter k varies in the range of $k\in \left[0.7-2.8\right]$, and each undulator section is 2.5 m long. The radiated FEL wavelengths are in the range of $\lambda \in 4-90$ nm. A variable k parameter allows for the use of undulators for the SASE FEL and for harmonic self-seed (HHSS) in the high gain harmonic generation (HGHG) FEL, where the last undulator sections are tuned to the high harmonic of the first sections. The range of variation of k allows for the amplification of the third harmonic of the first sections in the last sections. Consider one of the FEL experiments at FLASH 2 [63] with a beam current I

_{0}= 600 A, an electron energy E = 757 MeV, an energy spread σ

_{e}= 0.66×10

^{−3}, an undulator parameter for the first sections k = 2.687, and an undulator parameter for the last sections k = 1.032; the fundamental wavelength from the buncher was λ = 33 nm. The undulator cascades of the amplifier with k = 1.032 were tuned to the third harmonic λ

_{3}= 11 nm of the first sections. We computed the spontaneous and the stimulated UR for it.

_{3}= 11 nm of the buncher for k = 2.7, the Pierce parameter was practically the same as that for the fundamental frequency at the same wavelength λ = 11 nm of the amplifier, where k = 1: ${\kappa}_{3}^{k=2.7}\text{}\approx $${\kappa}_{1}^{k=1}\text{}\approx $ 0.093 and ${\rho}_{n=3}^{k=2.7}\approx {\rho}_{n=1}^{k=1}\approx 0.0013$. Thus, the independent amplification of the third FEL harmonic of the undulator with k = 2.7 was the same as the amplification of the fundamental frequency of the undulator with k = 1; this was different from the spontaneous UR results in Figure 7. In addition to the independent harmonic generation, there was a contribution induced by the fundamental tone on the third harmonic wavelength that helped to amplify the third harmonic at 11 nm in the undulator with k = 2.7, and the radiation power at 11 nm grew faster towards the end of the buncher. However, the saturation of the fundamental frequency did not allow for the further growth of the third FEL harmonic; moreover, the fundamental tone induced the energy spread towards the end of the FEL. To avoid this negative effect, the buncher was cut far from saturation, where ${\sigma}_{e}=\text{}0.00067\le \rho /2$ and amplification in the following cascades was ensured. The fact that the buncher was cut after the fourth section [63] meant that our theoretical estimations agreed with those of the engineers at FLASH 2.

_{1}= 11 nm was radiated.

## 6. Results and Conclusions

- Elliptic undulators with field harmonics are not the best choice for harmonic generation. This includes undulators with both symmetric and asymmetric elliptic fields with third harmonics along with one and two coordinates. These undulators provide elliptic polarizations of the radiation, but their UR harmonic content is inferior to that of a planar undulator with the same harmonic in the magnetic field. Moreover, because of the main field harmonic is present alongside both coordinates in helical undulators, their spectrum is lower than that of a planar UR.
- A helical undulator with an antisymmetric third field harmonic has a noticeable fifth UR harmonic and a very weak third harmonic in the spectrum. However, this fifth UR harmonic is not enough strong for practical use. In an FEL, it is mainly induced by the fundamental frequency and barely reaches 0.01–0.1% of content. A planar undulator with the same magnetic field along one axis radiates the third UR harmonic at the wavelength, similar to that of the fifth UR of a helical undulator. However, the third harmonic of the planar undulator is much stronger than the fifth of the helical with field harmonics.
- The third UR harmonic of a planar undulator with kτ 1.5 can be enhanced by the third field harmonic if the latter comes in phase with the main undulator field.

- For a fixed radiation wavelength, electron energy, and beam current, the Pierce parameter ρ for the fundamental frequency of an FEL with a planar undulator with kδ 1 was nearly the same as the values of ρ, as computed for the harmonics of an elliptic undulator with the matching wavelengths and kτ 2.5.
- In an FEL amplifier, there seemed to be little difference when using the fundamental frequency of a common planar undulator with low k, a third harmonic of a planar undulator with higher k, or a fifth or higher UR harmonic with a similar wavelength radiated in an elliptic undulator. The latter is more complicated and has higher costs.
- For the harmonic radiation, we advocate a planar undulator with symmetric third field harmonic; it gives a larger Pierce parameter for its UR harmonics compared to that of elliptic undulators at the same radiation lengths of matching harmonics.

- At FLASH 2, the third UR harmonic of the undulator with k = 2.7 resonated with the fundamental tone of the undulator with k = 1.
- Due to the enhanced harmonic radiation from the undulator with k = 2.7, it is best for a buncher in an FEL with harmonic multiplication cascades. For k = 1, the fundamental tone dominated, and k = 1 could be used in the amplifying cascades.

_{3}= 11 nm of the fundamental wavelength of λ

_{buncher}= 33 nm. Further amplification occurred in the undulators with k = 1 for the fundamental tone, resonating with the third harmonic of the buncher.

- The energy spread at the end of the buncher after the fourth cascade was lower than the Pierce parameter values: $2{\sigma}_{e}\le {\rho}_{n=3}^{k=2.7}\approx {\rho}_{n=1}^{k=1}\approx 0.0013$.
- The bunching at the third harmonic wavelength in the undulators with k = 2.7 was more efficient than that at the fundamental wavelength in the undulators with k = 1. This was due to the induced bunching from the fundamental frequency to the third harmonic of the buncher.
- The radiation pulse energy of the fundamental wavelength λ = 11 nm of the SASE FEL, where all undulators had k = 1, was lower along the FEL compared with the radiation energy of the harmonic self-seeded FEL, where the buncher sections with k = 2.7 grouped electrons at the harmonic wavelength λ
_{3}= 11 nm. - The power of the fundamental frequency in SASE regime rose later than the power of the harmonic from the buncher in the harmonic multiplication regime (HGHG, HHSS, or other). This is a general conclusion that applies to any installation with undulators with a variable k parameter.
- Our theoretical results for the harmonic radiation at FLASH 2 agreed with the reported values of the photon pulse energies.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The undulator radiation (UR) harmonic intensities for the undulator field of Equation (23); left bar charts—analytical results, where the harmonics are numbered and their intensities are dimensionless; right plots—numerical results of SPECTRA, where the harmonic photon energy and brightness are reported.

**Figure 2.**The UR harmonic intensities for the undulator field of Equation (24); left bar charts—analytical results, where the harmonics are numbered and their intensities are dimensionless; right plots—numerical results of SPECTRA, where the harmonic photon energy and brightness are reported.

**Figure 3.**The UR harmonic intensities for the field of Equation (25); left bar chart—analytical results, where the harmonics are numbered and their intensities are dimensionless; right plot—numerical results of SPECTRA, where the harmonic photon energy and brightness are reported.

**Figure 4.**Numerical results from SPECTRA for the UR harmonic brightness in KAERI FEL undulator [53] with the field of Equation (26), where d = 0.0825; on-axis—red line; 2° off the axis—blue line; and 4° off the axis—green line.

**Figure 5.**The UR harmonic intensities for the field of Equation (26) for d = 0.0825 in top plots and d = 0.3 in the bottom plots; left bar charts—analytical results, where the harmonics are numbered and their intensities are dimensionless; right plots—numerical results of SPECTRA, where the harmonic radiation energy and brightness are reported.

**Figure 6.**The UR harmonic intensities for the field of Equation (27), d = +0.3—upper plots; d = −0.3—lower plots; left bar charts—analytical results for dimensionless intensities of the harmonics n; right plots—numerical results of SPECTRA, where the harmonic radiation energy and brightness are reported.

**Figure 7.**The UR harmonic intensities for the FLASH 2 undulator; left bar charts—for k = 2.7; right plots—for k = 1.04.

**Figure 8.**Harmonic power evolution along the undulators in the free electron laser (FEL) FLASH 2 with energy E = 757 MeV, spread σ

_{e}= 0.5 MeV, current I

_{0}= 600 A, and charge Q = 0.25 nC. HGHG FEL for the 3rd harmonic at λ

_{1×3}= 11 nm. The harmonics are color-coded; prebuncher: λ

_{1}= 33 nm—red solid line; λ

_{3}= 11 nm—green long dashed line; λ

_{n}

_{=5}= 6.6 nm—blue dot dashed line; amplifier λ

_{n}

_{=1×3}= 11 nm—orange line; λ

_{n}

_{=3×3}= 3.7 nm—dashed green line; and self-amplified spontaneous emission (SASE) FEL n = 1 at λ

_{1}= 11 nm—red dotted line. The power of λ = 11 nm radiation was computed from the pulse energy, measured at 25 m; it is shown by the colored dots: E

_{γ}= 53 μJ for HGHG FEL—red dot, and E

_{γ}= 11 μJ the for SASE FEL—orange dot.

Accelerator | Light Source | ||||
---|---|---|---|---|---|

Variable | Value | Variable | Value | Variable | Value |

$\gamma $ | 300 | $\sigma $ | 1 × 10^{−3} | ${\lambda}_{u}$, cm | 2.8 |

E, MeV | 153.3 | ${\beta}_{x}$, m | 2.2 | L, cm | 210 |

${\u03f5}_{x}$, m×rad | 2.5 × 10^{−6} | ${\beta}_{y}$, m | 2.2 | ${N}_{u}$ | 75 |

${\u03f5}_{y}$, m×rad | 2.9 × 10^{−6} | ${\alpha}_{x}$ | 0 | $k$ | 2.133 |

${I}_{peak}$, A | 7.97 | ${\alpha}_{y}$ | 0 | ${k}_{Ef{f}_{x}}$ | 2.133 |

${k}_{Ef{f}_{y}}$ | 0.711 |

**Table 2.**Parameters of the undulator and the beam for the field of Equation (26) of the helical undulator [53].

Accelerator | Light Source | ||||
---|---|---|---|---|---|

Variable | Value | Variable | Value | Variable | Value |

$\gamma $ | 12.72 | $\sigma $ | 1 × 10^{−3} | ${\lambda}_{u}$, cm | 2.3 |

E, MeV | 6.5 | ${\beta}_{x}$, m | 0.4366 | L, cm | 69 |

${\u03f5}_{x}$, m×rad | 1.5 × 10^{−6} | ${\beta}_{y}$, m | 0.2875 | ${N}_{u}$ | 30 |

${\u03f5}_{y}$, m×rad | 0.3 × 10^{−6} | ${\alpha}_{x}$ | 2.223 | $k$ | 2.21622 |

${I}_{peak}$, A | 15.95 | ${\alpha}_{y}$ | 1.053 | ${k}_{Ef{f}_{x}}$ | 2.21706 |

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**MDPI and ACS Style**

Zhukovsky, K.; Fedorov, I.
Spontaneous and Stimulated Undulator Radiation in Symmetric and Asymmetric Multi-Periodic Magnetic Fields. *Symmetry* **2021**, *13*, 135.
https://doi.org/10.3390/sym13010135

**AMA Style**

Zhukovsky K, Fedorov I.
Spontaneous and Stimulated Undulator Radiation in Symmetric and Asymmetric Multi-Periodic Magnetic Fields. *Symmetry*. 2021; 13(1):135.
https://doi.org/10.3390/sym13010135

**Chicago/Turabian Style**

Zhukovsky, Konstantin, and Igor Fedorov.
2021. "Spontaneous and Stimulated Undulator Radiation in Symmetric and Asymmetric Multi-Periodic Magnetic Fields" *Symmetry* 13, no. 1: 135.
https://doi.org/10.3390/sym13010135