research papers
Numerical study of the second harmonic generation in FELs
^{a}Department of Theoretical Physics, Faculty of Physics, M. V. Lomonosov Moscow State University, Moscow 119991, Russian Federation
^{*}Correspondence email: am.kalitenko@physics.msu.ru
A numerical study of the effect of betatron oscillations on the second harmonic generation in freeelectron lasers (FELs) is presented. Analytical expressions for the effective coupling strength factors are derived that clearly distinguish all contributions in subharmonics and each polarization of the radiation. A threedimensional timedependent numerical FEL code that takes into account the main FEL effects and the individual contribution of each electron to the second harmonic generation is presented. Also, the X and Ypolarizations of the second harmonic are analyzed. The second harmonic was detected in experiments at the Advanced Photon Source (APS) Low Energy Undulator Test Line (LEUTL) and Linac Coherent Light Source (LCLS) in the soft Xray regime. The approach presented in the article can be useful for a comprehensive study and diagnostics of XFELs. In the paper, the LCLS and Pohang Accelerator Laboratory Xray FreeElectron Laser (PALXFEL) experiments are modeled. The simulation results are in a good agreement with the experimental data.
1. Introduction
In recent decades, the application of synchrotron radiation (Artcimovich & Pomeranchuk, 1945) as a freeelectron laser (FEL) has been of increasing interest, largely due to the improved of Xrays. As synchrotron radiation (Meinert et al., 2010), FELs can be used to study various materials, crystals and nanostructures. Radiation in the FEL occurs due to the passage of an electron beam through a periodic magnetic structure called an undulator, where the interaction of electrons with radiation causes electrons to form microbunches where the electrons are spaced by the radiation wavelength (McNeil & Thompson, 2010; Pellegrini et al., 2016; Huang & Kim, 2007).
The idea of an undulator, with the charges performing small transversal oscillations in a spatially periodic magnetic field and drifting at relativistic speeds along the undulator axis, was proposed by Ginzburg (1947). He also suggested that this radiation could be coherent if it were emitted from a series of electron bunches spaced by the radiation wavelength, with the bunch length being shorter than the radiation wavelength. Later in the mid20th century, Motz et al. (1953) proposed an FEL amplifier consisting of a sequence of dipole magnets with alternating polarity and uniform spacing along the axis.
In 1971, Madey pointed out that, although the radiation of accelerated electrons differs from that of conventional lasers, the bremsstrahlung of electrons in a periodic magnetic field can be caused by application of an external radiation field (Madey, 1971). Since the electrons were not bound to atoms and molecules, Madey called this device the freeelectron laser.
Singlepass highgain FELs do not require optical resonators, so their spectrum is not limited by the reflective properties of the mirrors [see, for example, Uehara et al. (1984)].
In addition to the dominant fundamental harmonic, FEL radiation contains higher harmonics. However, due to the finite size of the electron beam and betatron oscillations, even harmonics can contribute to the FEL radiation, as was observed in experiments (Tremaine et al., 2002; Biedron et al., 2002; Ratner et al., 2011). The strong focusing via the quadrupoles also causes betatron oscillation. In our simulations the effect of strong focusing on the second harmonic generation is also taken into account.
Second harmonic generation in crystals (Kleinman, 1962) is the main nonlinear optical effect. It is used to study the optical properties of new materials (Boursier et al., 2015), and therefore it is desirable to take into account the power of the FEL second harmonics. Multiharmonic radiation is also used for research purposes, especially if laser beams with different wavelengths can be obtained so that complete spatial overlapping is maintained even when both beams are focused to the diffractionlimited spot sizes. For example, higherharmonic lasers can be used in twophoton spectroscopy (Kaiser & Garrett, 1961; Perrella et al., 2013). The beam quality limits the number of harmonics that can effectively be generated at FELs, so even harmonics and their properties are important when designing lasers with multiple radiation harmonics.
In this paper, we derive new analytic expressions of the coupling factors for a planar undulator with consideration for averaged betatron oscillations. To study modern FELs in the Xray range, various numerical programs are used [see the comparative analysis by Biedron et al. (2000)]. In some XFELs the second harmonic of a high power can be recorded. This paper presents a numerical study of the contribution from betatron oscillations to the second harmonic generation; the numerical modeling of harmonics is performed and their contribution to the generation of FEL radiation is estimated.
2. Undulator radiation in a planar singlefrequency undulator with consideration of betatron oscillations
Let us consider the periodic field in the gap of a planar singlefrequency undulator [see, for example, Scharlemann (1985)],
The laws of electron motion in the field of the magnet configuration (1) are
where
and is an average velocity in the direction x. Betatron oscillations result from both strong and weak focusing. The contribution from strong focusing via quadrupole sections to the second harmonic generation is taken into account via velocities , .
is the energy radiated per solid angle per unit frequency interval by an undulator field and is calculated from the Lienard–Wiechert integral (Jackson, 1975where e is the c is the speed of light in a vacuum, ω is the radiation frequency, is the electron trajectory, β is the reduced electron velocity, and is the observation unit vector, which can be written as
where θ is the observation angle and φ is the azimuthal angle.
It can be easily shown that
can be expressed asThe coupling factors are then
with resonant frequencies
For convenience, the following functions are introduced,
with the following arguments,
In our case, the resonant frequencies split due to betatron oscillations, n is the number of the main harmonic, m is the number of the betatron oscillations (sub)harmonic. However, since the period and amplitude of betatron oscillations are small, the splitting in the radiation of odd harmonics n = 1, 3,… is not noticeable. Betatron oscillations can also contribute to the generation of even harmonics n = 2, 4,…, Ypolarized radiation splits into several subharmonics m = 0, ±1,…. The contribution of Ypolarization is considered theoretically (Xie, 2002; Geloni et al., 2007) in another formalism.
The main contribution to the generation of even harmonics is expressed by the term . A similar argument in a different formalism with series expansion was derived by Reiche & Musumeci (2007); the Bessel coefficients for the second harmonic in Reiche (1999) equals . However, a term also gives a contribution. For the LCLS experiment, this contribution increases the coupling factor by about 30%, and the square of this coefficient by 70%. For odd harmonics, the term can be neglected, but for even harmonics we cannot do this. The contribution to the generation of even harmonics caused by weak focusing is orders of magnitude smaller than the effect of strong focusing. The second harmonic in planar singlefrequency undulators is the result of mixing wiggler oscillations with betatron motion caused by strong focusing.
3. Simulation of experiments
The second harmonic of FEL radiation was registered in several experiments, for example the Advanced Photon Source's (Biedron et al., 2002) LCLS (Linac Coherent Light Source) in the soft Xray regime (Ratner et al., 2011). The evaluation of second harmonic generation was made in the modification of the numerical code GENESIS1.3 (Reiche & Musumeci, 2007), but GENESIS's coupling coefficients have a different form from the above formulas and do not consider Ypolarization.
We designed a numerical program for FEL simulation with consideration of the second harmonic in the modern C++ programming language using parallel programming. Our threedimensional numerical code is based on a mathematical model that is also used in such codes as GENESIS (Reiche, 1999) and GINGER (Fawley, 2002). In the case of even harmonics, the coupling coefficient is different for each particle. It depends on the position and momentum, so the developed numerical code assigns each particle its own coupling coefficient.
For Ypolarized radiation, there is a split of the second harmonic at the betatron frequency Ω_{β} = , and for modern lasers the split of the second harmonic occurs mainly into two subharmonics, which are considered as separate harmonics, see Fig. 1. Fig. 1 shows the splitting of the Ypolarized second harmonic into subharmonics m = 0, ±1,…. For Xpolarized radiation of the first harmonic, the split also occurs, see Fig. 2. However, the power of the harmonic radiation is proportional to the square of the coupling coefficient T_{n}^{2}; in addition, the electron beam has a Gaussian distribution, and, therefore, the contribution is mainly given by the electrons in the center of the beam.
Simulation of the LCLS and PALXFEL experiments is presented below. The center of the electron beam is located on the undulator axis and has a Gaussian distribution in all simulations.
3.1. Simulation of the LCLS experiment in the soft Xray regime
In the experiment (Ratner et al., 2011) with XFEL LCLS (Emma et al., 2010), higher harmonics were recorded, namely the second and third ones. In the soft Xray experiment, the spatial distribution of the second harmonic intensity is also obtained. The wavelength of the first harmonic is 1.5 nm. Some relevant simulation data are collected in Table 1.

We estimated the radiation power of higher harmonics at the LCLS using the derived coupling coefficients, which are used in our threedimensional program. Also, the second harmonic polarization analysis was performed. The Ypolarization of the second harmonic is caused by weak focusing in the undulator, and for the experiment (Ratner et al., 2011) we estimate it at 3% of the second harmonic power. The results of numerical simulations give the following ratios between harmonic powers: for the third harmonic P_{3}/P_{1} ≃ 3%, for the fifth harmonic P_{5}/P_{1} ≃ 0.3%. It was mentioned by Ratner et al. (2011) that the fifth harmonic was not separated from the third one due to its small value, which simplified the study. The power ratio for the second harmonic is P_{2}/P_{1} ≃ 0.04%, which is within the experimental values of P_{3}/P_{1} ≃ 2.0–2.5%, P_{2}/P_{1} ≃ 0.05–0.06%. The lower power value for the third harmonic may be a consequence with the diffraction limit λ_{3}/4π = 4 × 10^{−11} m rad, and the emittance ɛ = 4.8 × 10^{−11} m rad, because by the end of saturation the emittance increases. However, an increase in the emittance leads to an increase in the power of the second harmonic. It should also be noted that the exact dynamics of electrons in the beam is unknown, so the power of the second harmonic was calculated based on experimental data.
Fig. 3 shows the evolution of the harmonic radiation power and electron bunching in the FEL in the selfamplified (SASE) regime. The first harmonic n = 1 is represented by a black line, n = 2 by a red line, n = 3 by a blue line and n = 5 by a green line. The higher odd harmonics follow the principles of nonlinear growth (Huang & Kim, 2001). The second harmonic also has a nonlinear growth, but its generation also depends on the dynamics and position of each electron in the beam.
The spatial distribution of the second harmonic intensity is shown below in Fig. 4. The distribution has a double symmetric structure; some asymmetry is due to the beam asymmetry. Fig. 4 shows the Xpolarization, which gives the prevailing contribution. In our simulations, we assume that the electrons in the beam have a Gaussian distribution.
We performed a comparative analysis of our program with GENESIS (Reiche, 1999) with the parameters of LCLS in the soft Xray experiment. For clarity, we compared the harmonics in the amplifier mode. As can be seen in Fig. 5, the fundamental harmonics of the radiation are almost exactly the same; the second harmonics differ by an order of magnitude. Our program estimates the second harmonic power at P_{2} = 2.9 MW, the GENESIS estimate is 0.55 MW, and for the first harmonic we have P_{1} ≃ 8 GW. We also compared the results in the SASE regime, with magnitudes and power ratios similar to Fig. 5. It should be noted that our program considers the interaction of electrons with radiation of the first and higher harmonics with numbers n = 2, 3, 5, which may be one of the reasons for the slightly accelerated growth of harmonics and a greater increase in the second harmonic power.
3.2. Simulation of the PALXFEL experiment in the soft Xray regime
FEL experiments for soft and hard Xray radiation were performed at the Pohang Accelerator Laboratory Xray Freeelectron Laser (PALXFEL); radiation with the fundamental harmonic wavelengths λ = 1.52 nm and λ = 0.144 nm was obtained. The experiments have been well documented; among other data, Kang et al. (2017) reported the evolution of harmonic power. We analyzed the generation of harmonics in the soft Xray regime. Some relevant simulation data are collected in Table 2. The results are shown in Fig. 6 and discussed below.

PALXFEL as well as LCLS are FELs capable of generating radiation with a wavelength of the order of 1 Å. The undulator parameter K = 2 in the PALXFEL experiment was lower than that in the LCLS with K = 3.5, while the electron beam had a lower energy for the same generated wavelength. The soft Xray radiation at λ = 1.52 nm was created by electrons with the energy E = 3 GeV; the energy spread in the simulation was set to σ_{e} = 0.05%. The length of the undulator section is 5 m, the distance between the sections is 1 m, the undulator period is λ_{u} = 3.4 cm, and the parameter K = 2. Higher harmonics were not recorded in the experiment. A smaller undulator parameter K in PALXFEL, compared with in LCLS, results in lower values of the coupling coefficients for odd higher harmonics.
We simulated the PALXFEL experiment using our numerical code. The evolution of harmonic power is presented in Fig. 6, the black line represents the first harmonic n = 1, red represents the second harmonic n = 2, and blue the third harmonic n = 3. The saturation power of the first harmonic is achieved at the length of ∼42 m as in the measurements (Kang et al., 2017). The diffraction limit for a Gaussian photon beam with a wavelength of λ is ɛ_{r} = λ/4π. For the first harmonic, we have λ_{1}/4π = 1.2 × 10^{−10} m rad, and emittance ɛ = 9.5 × 10^{−11} m rad. The numerical program gives the following estimation for the power of the second and third harmonics: the power ratio P_{2}/P_{1} ≃ 0.2%, the Ycomponent of the radiation of the second harmonic is ∼6.5%; for the third harmonic we have P_{3}/P_{1} ≃ 1.1%, the diffraction limit is not satisfied, so we should expect a decrease in the radiation power of higher harmonics. As can be seen, the third harmonic is relatively weaker than in the LCLS experiment, since the undulator parameter is smaller, which leads to lower coupling coefficients. The second harmonic also has a smaller relative contribution to radiation compared with the LCLS experiment. In Fig. 6 the bottom graph is the evolution of bunching – the bunching coefficient for the second and third harmonics is relatively smaller than at the LCLS. The spatial density of the second harmonic distribution, for Xpolarization, is shown in Fig. 7 – it also has a double lobe structure.
4. Conclusions and discusions
We have obtained analytic expressions for the coupling factors in a planar undulator, taking into account betatron oscillations. These factors differ from those of Reiche & Musumeci (2007) by another term, which increases the discussed coupling factors by ∼30%. This allows the power of even harmonics to be calculated more precisely. We have written a program that simulates the generation of harmonics with n = 1, 2, 3, 5 harmonic numbers, and for the second harmonic the contribution of Ypolarization is considered.
The obtained rigorous theoretical results can be used to study radiation in FELs operating in a wide range of wavelengths, from infrared to Xray. We studied an experiment with LCLS in the soft Xray region (Ratner et al., 2011), in which the second and third harmonics of radiation were registered. We compared our results with the GENESIS simulation, which showed that in our program the second harmonic has a greater contribution to radiation. Our numerical program with integrated derived analytical expressions gives the result P_{2}/P_{1} ≃ 0.04%, which indicates a good agreement with the experiment. We also analyzed the generation of higher harmonics at the PALXFEL (Kang et al., 2017) experiment at the POHANG laboratory. We suppose that the higher harmonics should be weaker than at LCLS due to the smaller undulator parameter K. We have demonstrated a possible theoretical emission of harmonics at PALXFEL, but the beam parameters hardly allow us to expect the detection of harmonics above the third one. In experiments, the spatial power distributions of the second harmonic were obtained, Figs. 4 and 7. They have a double lobe structure, as was recorded by Ratner et al. (2011). Our formalism can also be applied to other magnetic field configurations, for example for asymmetric elliptic undulators (Kalitenko & Zhukovskii, 2020) and twofrequency undulators (Zhukovsky & Kalitenko, 2019).
The results obtained using the theoretical formalism and numerical code developed by us are consistent with the experiments and refine the previous expressions for the coupling factors of even harmonics. The presented formalism makes it possible to carry out more precise current and planned FEL experiments for examination of new materials, nanostructures, and to evaluate the et al., 2020).
and harmonic generation at existing and constructed FELs, for example (OwadaAPPENDIX A
Simulation parameters in the experiments
This section provides an overview of the simulation parameters of the experiments presented in the article. The simulations start in the center of the first quadrupole, which simplifies the matching of the electron beam (α_{x}, α_{y} = 0). See Table 3 and Table 4.


In the manual by Kang et al. (2014) it was stated that the beta function can have a deviation of 73% in the soft Xray regime; our estimates of the deviation of the beta function have the same order, but a smaller value, which satisfies the experimental data.
Acknowledgements
The author is grateful to Docent P. I. Pronin, Professor A. V. Borisov and Professor V. Ch. Zhukovsky for useful discussions.
Funding information
Funding for this research was provided by: Theoretical Physics and Mathematics Advancement Foundation `BASIS' (grant No. 1926821).
References
Artcimovich, A. L. & Pomeranchuk, I. J. (1945). J. Phys. USSR, 9, 267. Google Scholar
Biedron, S. G., Chae, Y. C., Dejus, R. J., Faatz, B., Freund, H. P., Milton, S. V., Nuhn, H.D. & Reiche, S. (2000). Nucl. Instrum. Methods Phys. Res. A, 445, 110–115. Web of Science CrossRef CAS Google Scholar
Biedron, S. G., Dejus, R. J., Huang, Z., Milton, S. V., Sajaev, V., Berg, W., Borland, M., Den Hartog, P. K., Erdmann, M., Fawley, W. M., Freund, H. P., Gluskin, E., Kim, K. J., Lewellen, J. W., Li, Y., Lumpkin, A. H., Moog, E. R., Nassiri, A., Wiemerslage, G. & Yang, B. X. L. (2002). Nucl. Instrum. Methods Phys. Res. A, 483, 94–100. Web of Science CrossRef Google Scholar
Boursier, E., Segonds, P., Debray, J., Inácio, P. L., Panyutin, V., Badikov, V., Badikov, D., Petrov, V. & Boulanger, B. (2015). Opt. Lett. 40, 4591–4594. Web of Science CrossRef CAS PubMed Google Scholar
Emma, P., Akre, R., Arthur, J., Bionta, R., Bostedt, C., Bozek, J., Brachmann, A., Bucksbaum, P., Coffee, R., Decker, F.J., Ding, Y., Dowell, D., Edstrom, S., Fisher, A., Frisch, J., Gilevich, S., Hastings, J., Hays, G., Hering, Ph., Huang, Z., Iverson, R., Loos, H., Messerschmidt, M., Miahnahri, A., Moeller, S., Nuhn, H.D., Pile, G., Ratner, D., Rzepiela, J., Schultz, D., Smith, T., Stefan, P., Tompkins, H., Turner, J., Welch, J., White, W., Wu, J., Yocky, G. & Galayda, J. (2010). Nat. Photon. 4, 641–647. Web of Science CrossRef CAS Google Scholar
Fawley, W. M. (2002). Phys. Rev. ST Accel. Beams, 5, 070701. Web of Science CrossRef Google Scholar
Geloni, G., Saldin, E., Schneidmiller, E. & Yurkov, M. (2007). Opt. Commun. 271, 207–218. Web of Science CrossRef CAS Google Scholar
Ginzburg, V. L. (1947). Izv. Acad. Sci. USSR (Phys.), 11, 1651. Google Scholar
Huang, Z. & Kim, K.J. (2001). Nucl. Instrum. Methods Phys. Res. A, 475, 112–117. Web of Science CrossRef CAS Google Scholar
Huang, Z. & Kim, K.J. (2007). Phys. Rev. ST Accel. Beams, 10, 034801. Web of Science CrossRef Google Scholar
Jackson, J. D. (1975). Classical Electrodynamics. New York: Wiley. Google Scholar
Kaiser, W. & Garrett, C. G. B. (1961). Phys. Rev. Lett. 7, 229–231. CrossRef CAS Web of Science Google Scholar
Kalitenko, A. M. & Zhukovskii, K. V. (2020). J. Exp. Theor. Phys. 130, 327–337. Web of Science CrossRef CAS Google Scholar
Kang, H., Min, C., Heo, H., Kim, C., Yang, H., Kim, G., Nam, I., Baek, S. Y., Choi, H., Mun, G., Park, B. R., Suh, Y. J., Shin, D. C., Hu, J., Hong, J., Jung, S., Kim, S., Kim, K., Na, D., Park, S. S., Park, Y. J., Han, J., Jung, Y. G., Jeong, S. H., Lee, H. G., Lee, S., Lee, S., Lee, W., Oh, B., Suh, H. S., Parc, Y. W., Park, S., Kim, M. H., Jung, N., Kim, Y., Lee, M., Lee, B., Sung, C., Mok, I., Yang, J., Lee, C., Shin, H., Kim, J. H., Kim, Y., Lee, J. H., Park, S., Kim, J., Park, J., Eom, I., Rah, S., Kim, S., Nam, K. H., Park, J., Park, J., Kim, S., Kwon, S., Park, S. H., Kim, K. S., Hyun, H., Kim, S. N., Kim, S., Hwang, S., Kim, M. J., Lim, C., Yu, C., Kim, B., Kang, T., Kim, K., Kim, S., Lee, H., Lee, H., Park, K., Koo, T., Kim, D. & Ko, I. S. (2017). Nat. Photon. 11, 708–713. Web of Science CrossRef CAS Google Scholar
Kang, H.S., Kim, D. E. & Ko, I. S. (2014). PALXFEL Technical Design Report. Pohang Accelerator Laboratory, Pohang, South Korea. Google Scholar
Kleinman, D. A. (1962). Phys. Rev. 126, 1977–1979. CrossRef CAS Web of Science Google Scholar
Madey, J. M. J. (1971). J. Appl. Phys. 42, 1906–1913. CrossRef CAS Web of Science Google Scholar
McNeil, B. W. J. & Thompson, N. R. (2010). Nat. Photon. 4, 814–821. Web of Science CrossRef CAS Google Scholar
Meinert, C., Filippi, J., Nahon, L., Hoffmann, S. V., D'Hendecourt, L., De Marcellus, P., Bredehöft, J. H., Thiemann, W. H. & Meierhenrich, U. J. (2010). Symmetry, 2, 1055–1080. Web of Science CrossRef CAS Google Scholar
Motz, H., Thon, W. & Whitehurst, R. N. J. (1953). J. Appl. Phys. 24, 826–833. CrossRef Web of Science Google Scholar
Owada, S., Fushitani, M., Matsuda, A., Fujise, H., Sasaki, Y., Hikosaka, Y., Hishikawa, A. & Yabashi, M. (2020). J. Synchrotron Rad. 27, 1362–1365. Web of Science CrossRef IUCr Journals Google Scholar
Pellegrini, C., Marinelli, A. & Reiche, S. (2016). Rev. Mod. Phys. 88, 015006. Web of Science CrossRef Google Scholar
Perrella, C., Light, P. S., Anstie, J. D., Stace, T. M., Benabid, F. & Luiten, A. N. (2013). Phys. Rev. A, 87, 013818. Web of Science CrossRef Google Scholar
Ratner, D., Brachmann, A., Decker, F. J., Ding, Y., Dowell, D., Emma, P., Fisher, A., Frisch, J., Gilevich, S., Huang, Z., Hering, P., Iverson, R., Krzywinski, J., Loos, H., Messerschmidt, M., Nuhn, H. D., Smith, T., Turner, J., Welch, J., White, W. & Wu, J. (2011). Phys. Rev. ST Accel. Beams, 14, 060701. Web of Science CrossRef Google Scholar
Reiche, S. (1999). Nucl. Instrum. Methods Phys. Res. A, 429, 243–248. Web of Science CrossRef CAS Google Scholar
Reiche, S., Musumeci, P. & Goldammer, K. (2007). Proceedings of the 22nd Particle Accelerator Conference (PAC07), 25–29 June 2007, Albuquerque, NM, USA, pp. 1269–1271. TUPMS038. Google Scholar
Scharlemann, E. T. (1985). J. Appl. Phys. 58, 2154–2161. CrossRef Web of Science Google Scholar
Tremaine, A., Wang, X. J., Babzien, M., BenZvi, I., Cornacchia, M., Nuhn, H.D., Malone, R., Murokh, A., Pellegrini, C., Reiche, S., Rosenzweig, J. & Yakimenko, V. (2002). Phys. Rev. Lett. 88, 204801. Web of Science CrossRef PubMed Google Scholar
Uehara, Y., Fujiwara, E., Yamanaka, M., Tsuchida, K., Fujita, J., Kato, Y., Sasaki, W. & Saito, S. (1984). Opt. Lett. 9, 539–541. CrossRef PubMed CAS Web of Science Google Scholar
Xie, M. (2002). Nucl. Instrum. Methods Phys. Res. A, 483, 527–530. Web of Science CrossRef CAS Google Scholar
Zhukovsky, K. & Kalitenko, A. (2019). J. Synchrotron Rad. 26, 159–169. Web of Science CrossRef CAS IUCr Journals Google Scholar
© International Union of Crystallography. Prior permission is not required to reproduce short quotations, tables and figures from this article, provided the original authors and source are cited. For more information, click here.