Harmonics In Free-electron Laser Research Articles

Suppressing of even harmonics in free electron lasers by modifying the energy spread of the beam

Using an analytical description of the evolution of the power of harmonics in free electron laser (FEL) undulators, we study the effect of electron beam parameters on the main characteristics of FEL radiation: the gain length, spectrum, and harmonic power. The possibility of suppressing even harmonics of the FEL as a light source in the study of materials and surfaces by analyzing the second harmonic nonlinear response (SHG) is being studied. The main factors influencing the radiation of even FEL harmonics are revealed; the influence of the beam cross section and the spread of electron energies on their radiation is analyzed. An example of a LEUTL FEL with visible light emission, which is used in studies of the optical properties of materials and the chemical state of molecules on surfaces and interfaces between media, is considered. In this context, the possibility of suppressing the second FEL harmonic is analyzed.

Suppression of Even Harmonics in Free-Electron Lasers by Modifying the Energy Spread of the Beam

Suppression of Even Harmonics in Free-Electron Lasers by Modifying the Energy Spread of the Beam

Beam Asymmetry, Divergence and Energy Spread Effects on the Radiation from Planar Undulators

The theoretical study of the effect of electron beam parameters, in particular, the emittance and its asymmetry on the radiation from relativistic electrons in undulators is conducted both analytically and numerically. The reasons for the odd and even harmonic generation and radiation are explored. The difference in the underlying physical reasons for the spontaneous and stimulated radiation of harmonics in free electron lasers (FELs) is elucidated. The generalized forms of the special functions of the Bessel and Airy type are employed to account analytically for the off-axis and angular effects together with the effect of the beam energy spread. A comparative analysis of the radiation spectra for undulators with different beams is performed. The examples of the radiation at SPARC and LEUTL are given. The effect of the asymmetry of the beam on the radiation properties is analyzed. The alternative theoretical approaches of other authors are also employed for the analytical calculation of the harmonic powers in FELs. The results are compared with existing experimental data.

Analysis of the spectral properties of free electron lasers in X-RAY and other bands

Analytical formalism for the description of free electron laser (FEL) spectrum is proposed with account for the electron beam size, divergences, beam energy spread, the off-axis deflection and the periodic harmonic and non-periodic constant magnetic field components in undulators. The exact analytical expressions for the spontaneous undulator radiation (UR) are obtained in terms of generalized Bessel and Airy functions, which describe the spectrum line shape and intensities. The obtained analytical formulae distinguish the effects of each installation and beam parameter on the FEL harmonic radiation. The enhanced phenomenological model is employed for the FEL modeling; it describes the harmonic powers evolution along the undulators and gradual saturation around the saturation length with account for the saturated powers oscillations. With this model, we study the generation of FEL harmonics, including even, in several instances of FEL experiments at LCLS, LEUTL and PAL-XFEL. We analytically demonstrate the even harmonics reasons in the LCLS and LEUTL FEL experiments. The spectrum line split due to the beam size is demonstrated analytically and compared with experimentally measured spectral characteristics. The PAL-XFEL experiment is analyzed and compared with the LCLS experiment; the possibility of high harmonic radiation in it is explored. The modeling results fully agree with the measured values in the studied FELs. The analytical formalism shows physical background for each harmonic in every FEL. It allows the analysis of the radiation and beam alignment in projected, built and working FELs and it can help minimizing losses and correcting magnetic fields as well as projecting experiments with FELs.

Harmonics Generation in Experiments with Free-Electron Lasers in the X-Ray Wavelength Range: a Theoretical Analysis

The generation of undulator X-ray radiation harmonics in free-electron lasers (FELs), which has been observed in different experiments, has been given an analytical explanation. Expressions for spectrum lines and radiation intensity are written in explicit form in terms of generalized Bessel and Airy functions with regard to the spreads of electron energy, electron beam size, emittance, spectrum line splitting, and magnetic field constant components. The theory proposed here adequately explains the spectrum and intensity of radiation harmonics in all experiments with a SACLA FEL, a Linac coherent light source (LCLS) based on an FEL, PAL-XFEL, SwissFEL, and other equipment conducted under different conditions in a wide wavelength range, including hard X-ray radiation.

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

In this work, the radiation from electrons in multi-periodic undulator fields with symmetric and asymmetric harmonics was analyzed using generalized Bessel functions formalism. The asymmetric, symmetric, and anti-symmetric periodic magnetic fields with harmonics were studied in order to get the enhanced radiation of the high harmonics of undulator radiation (UR). The effect on the spontaneous and stimulated UR was explored. The exact integral forms for the Bessel coefficients were obtained for undulators with general symmetric and asymmetric field harmonics. Spectral properties of the radiation from several configurations of the undulator fields with harmonics were compared with each other. The resulting spontaneous UR spectrum and harmonic intensities were obtained analytically in the form of integrals and compared with the respective results that were obtained numerically with SPECTRA program. The dimensionless scaling parameter of a free electron laser (FEL)—the Pierce parameter (ρ)—was computed and compared for the different considered undulators. We studied the differences in the behavior of the high-gain single pass FEL harmonics and the spontaneous UR harmonics in the same undulators. The undulators with variable deflection parameter (k) were considered. The effect of the k parameter (deflection parameter for a common planar undulator) on the spontaneous UR and on the high-gain FEL radiation was explored. In this context, an experiment with variable strength undulators at FLASH 2 FEL was analyzed; the shorter saturated length in high harmonic self-seeding (HHSS) regime vs. self-amplified spontaneous emission (SASE) is explained.

Analytical account for the off-axis effects and undulator field harmonics in FELs

We give analytical description of the spontaneous and stimulated free electron laser (FEL) harmonic generation in multiperiodic fields, accounting for the off-axis effects and field harmonics in undulators. Exact analytical expressions for the Bessel coefficients involve generalized forms of Bessel- and Airy-type functions and describe also the effect of the non-periodic magnetic components in various elliptic and planar undulators with field harmonics. They allow clear identification of proper contributions in the UR harmonic intensities and polarizations. Advanced phenomenological model of the FELs harmonic powers provides analytical description of relevant effects in FEL harmonic radiation, verified with FEL experiments. The analysis of the off-axis effects for the odd and even harmonic generation is performed for LEUTL and LCLS FEL experiments. The modelled harmonic powers perfectly fit the experimentally measured data. High levels of the 2nd harmonic are explained by the wide beam in LEUTL experiment and by the off-axis beam deviation in LCLS experiment.

Generation of Soft X-Ray Radiation in a Compact Free-Electron Laser with Harmonic Multiplication

Single-pass free-electron lasers (FELs) with harmonic multiplication in the X-ray range are studied theoretically with the objective of producing intense radiation with the minimum possible FEL length, beam energy, and seed laser frequency. A cascade FEL with a two-frequency undulator for electron bunching at wavelengths of higher harmonics is proposed. The dynamics of radiation power in a FEL with harmonic multiplication cascades is investigated based on a well-proven phenomenological FEL model that covers all major losses for each harmonic in each FEL cascade. The effect of the beam quality on X-ray generation in a FEL is examined. It is demonstrated that electron beams with energy spread σe ≤ 0.0002 are required for efficient use of harmonics in a cascade X-ray FEL. The evolution of power of harmonics in several cascade FELs at wavelengths of ~2–3 nm with multiplication of seed harmonics at a wavelength of 13.51 nm (Mo/Si reflection peak) and 11.43 nm (MoRu/Be reflection peak) is studied. It is demonstrated that the power increases to ~1 GW at a FEL length up to 40 m. In addition, the operation of multicascade FELs with a seed F2 excimer 157-nm UV laser and harmonics of lasers with CO2 and C2H2 at 30 nm with a beam of electrons with an energy of ~0.6 GeV and a current of ~1 kА is studied. The power of the modeled X-ray radiation of these FELs at λ = 2.5 and 3.3 nm increases to ~0.5 GW at ~30 m.

Генерация мягкого рентгеновского излучения в компактном лазере на свободных электронах с умножением гармоник

AbstractSingle-pass free-electron lasers (FELs) with harmonic multiplication in the X-ray range are studied theoretically with the objective of producing intense radiation with the minimum possible FEL length, beam energy, and seed laser frequency. A cascade FEL with a two-frequency undulator for electron bunching at wavelengths of higher harmonics is proposed. The dynamics of radiation power in a FEL with harmonic multiplication cascades is investigated based on a well-proven phenomenological FEL model that covers all major losses for each harmonic in each FEL cascade. The effect of the beam quality on X-ray generation in a FEL is examined. It is demonstrated that electron beams with energy spread σ_ e ≤ 0.0002 are required for efficient use of harmonics in a cascade X-ray FEL. The evolution of power of harmonics in several cascade FELs at wavelengths of ~2–3 nm with multiplication of seed harmonics at a wavelength of 13.51 nm (Mo/Si reflection peak) and 11.43 nm (MoRu/Be reflection peak) is studied. It is demonstrated that the power increases to ~1 GW at a FEL length up to 40 m. In addition, the operation of multicascade FELs with a seed F _2 excimer 157-nm UV laser and harmonics of lasers with CO_2 and C_2H_2 at 30 nm with a beam of electrons with an energy of ~0.6 GeV and a current of ~1 kА is studied. The power of the modeled X-ray radiation of these FELs at λ = 2.5 and 3.3 nm increases to ~0.5 GW at ~30 m.

Multicascade X-Ray Free-Electron Laser with Harmonic Multiplier and Two-Frequency Undulator

The feasibility of generation of powerful x-ray radiation by a cascade free-electron laser (FEL) with amplification of higher harmonics using a two-frequency undulator is studied. To analyze the FEL operation, a complex phenomenological single-pass FEL model is developed and used. It describes linear and nonlinear generation of harmonics in the FEL with seed laser that takes into account initial electron beam noise and describes all main losses of each harmonic in each FEL cascade. The model is also calibrated against and approved by the experimental FEL data and available results of three-dimensional numerical simulation. The electron beam in the undulator is assumed to be matched and focused, and the dynamics of power in the singlepass FEL with cascade harmonic multipliers is investigated to obtain x-ray laser radiation in the FEL having the shortest length, beam energy, and frequency of the seed laser as low as possible. In this context, the advantages of the two-frequency undulator used for generation of harmonics are demonstrated. The evolution of harmonics in a multicascade FEL with multiplication of harmonics is investigated. The operation of the cascade FEL at the wavelength λ = 1.14 nm, generating ~30 MW already on 38 m with the seed laser operating at a wavelength of 11.43 nm corresponding to the maximal reflectivity of the multilayered mirror MoRu/Be coating is investigated. In addition, the operation of the multicascade FEL with accessible seed UVlaser operating at a wavelength of 157 nm (F2 excimer UV-laser) and electron beam with energy of ~0.5 GeV is investigated. X-ray radiation simulated in it at the wavelength λ ~ 3.9 nm reaches power of ~50 MW already at ~27 m, which is by two orders of magnitude shorter than 3.4 km of the x-ray FEL recently put into operation in Europe.

Compact single-pass X-ray FEL with harmonic multiplication cascades

The generation of X-ray radiation in cascaded single-pass free electron laser (FEL), which amplifies high harmonics of a two-frequency undulator, is studied. Power dynamics of FEL harmonics is explored with the help of the phenomenological model of a single pass FEL. The model describes both linear and non-linear harmonic generation, starting from a coherent seed laser and initial shot noise with account for main loss factors for each harmonic in each cascade individually: the energy spread and beam divergence, the coupling losses between FEL cascades, the diffraction etc. The model was validated with the experiment and with relevant 3-D simulations. It is employed for modeling the cascaded FELs with harmonic multiplication and analyzing the evolution of FEL harmonic power with the aim to obtain the maximum high harmonic power in the X-ray band at the shortest possible FEL length with the lowest possible seed frequency. The advantages of two-frequency undulators in HGHG FELs are elucidated. The requirements for the electron beam are studied; the need for low energy spread is evidenced: our evaluations yield σe<2×10−4. Several cascaded HGHG FELs with two-frequency undulators are modeled. Generation of soft X-ray radiation at λ=2.71nm, reaching ∼50 MW power with I0∼100 A in a cascaded FEL at just 40 m with 13.51 nm seed, matching peak reflectivity of Mo/Si, is demonstrated. The generation of 40 MW radiation power at λ=2.27nm with the beam current I0∼100 A, energy E=950MeV and the energy spread σe=2×10−4 is studied, using second and third harmonics in three-stage 45 m long FEL. The multistage FEL is modeled for generating radiation in nanometer band: ∼40 MW power at λ∼2.6 nm with I0∼175 A current in just ∼40 m long FEL with commercially available F2 excimer UV laser seed at 157 nm. The peak radiation power rises to ∼0.5 GW for ∼1 kA beam current.

Improvement of nonlinear harmonics in free electron laser with planar wiggler

Spontaneous emission of free electron laser with planar wiggler and ion-channel guiding is calculated analytically and possibility of emission at up-shifted wiggler or ion-channel betatron frequency and their harmonics has been found. To investigate the nonlinear odd harmonics, a set of self-consistent nonlinear differential equations that governs the evolution of radiation and electron beam are derived and solved numerically by Runge-Kutta method. Using the simulation code, gain improvement of third harmonic is studied in the range of microwave frequency by applying ion-channel guiding for a cold beam. It is shown that the combination of the ion-channel and a prebunched electron beam increases the amplitude of the third harmonic of the radiation and decreases its saturation length. The relation between the linear and nonlinear harmonics is discussed.

The effect of shot noise on the start up of the fundamental and harmonics in free-electron lasers

The problem of radiation start up in free-electron lasers (FELs) is important in the simulation of virtually all FEL configurations including oscillators and amplifiers in both seeded master oscillator power amplifier (MOPA) and self-amplified spontaneous emission (SASE) modes. Both oscillators and SASE FELs start up from spontaneous emission due to shot noise on the electron beam, which arises from the random fluctuations in the phase distribution of the electrons. The injected power in a MOPA is usually large enough to overwhelm the shot noise. However, this noise must be treated correctly in order to model the initial start up of the harmonics. In this paper, we discuss and compare two different shot noise models that are implemented in both one-dimensional wiggler-averaged (PERSEO) and non-wiggler-averaged (MEDUSA1D) simulation codes, and a three-dimensional non-wiggler-averaged (MEDUSA) formulation. These models are compared for examples describing both SASE and MOPA configurations in one dimension, in steady-state, and time-dependent simulations. Remarkable agreement is found between PERSEO and MEDUSA1D for the evolution of the fundamental and harmonics. In addition, three-dimensional correction factors have been included in the MEDUSA1D and PERSEO, which show reasonable agreement with MEDUSA for a sample MOPA in steady-state and time-dependent simulations.

Synthesis of C60 Polymer by Free Electron Laser Irradiation with Tunable Wavelength

The harmonics of free electron laser (FEL) were irradiated to the surfaces of compressed C60 and/or mixture of C60 and I2 in the vacuum. After the irradiation of FEL with the typical wavelength, 450 nm or 345 nm, the Raman peak of Ag(2)-derived vibration mode of C60 shifted to the lower energy side. This result suggested that polymerization reactions of C60 proceeded. The largest peak shift, ca. 10 cm−1 was obtained in the mixture powder irradiated by 345 nm irradiation. The decrease of the lattice parameter was confirmed by XRD analyses revealed that inter-molecular combinations took places. Furthermore the change of the crystal structure was closely studied by changing the wavelength of FEL in the range of 350–600 nm. The obtained results demonstrated that three-dimensional polymerization of C60 was effectively promoted by FEL irradiation and/or especially the effect of the photon-assisted hole-doping from iodine atoms to C60 molecules.

Free electron laser spectrum control with a quasiperiodic wiggler configuration

Recently, we have studied an unusual wiggler magnet configuration comprising two conventional periodic wigglers. It was shown that if the periods of the two constituent wiggler components are not integrally but irrationally related then considerable radiation is generated at frequencies that are irrational multiples of the fundamental radiation frequency. The generation of these irrationally related harmonics modifies considerably the basic free electron laser (FEL) mechanism and as such they may be used to manipulate the FEL spectrum. In this contribution, we consider the spontaneous emission of electrons in such a quasiperiodic wiggler structure with and without the electron energy spread effects. Numerical examples are used to demonstrate that significant radiation can be generated simultaneously at both the usual FEL harmonics and irrationally related frequencies. When the wiggler parameters are appropriately adjusted, the usual integrally related FEL harmonics are suppressed considerably. This feature may be used to relieve the mirror degradation problem in some ultraviolet and X-ray FELs.

Higher Harmonics in Free Electron Laser with a Two-Wavenumber Plane Wiggler Magnetic Field

The higher harmonics of the free electron laser (FEL) with a two-wavenumber plane wiggler magnetic field, which may induce a lot of higher harmonics, are investigated. Using a two-wavenumber wiggler comprised of two separate periodicities, kw and kw + Δk, numerical simulations show that the higher harmonic may be amplified and the first harmonic may be suppressed in this kind of FEL with an optimized parameter Δk. The coupling coefficient and gain coefficient in low-gain regime of the harmonic are also obtained.

Mechanisms for the production of harmonics in free electron lasers

Harmonics in the radiation of a free electron laser are useful for extending the range of tuning, may originate in spontaneous or parametric processes, and can take part in stimulated emission or amplification. These mechanisms exhibit interesting analogies with those of nonlinear optics. Apart from the well known nonparametric gain analogous to stimulated Brillouin scattering, they include third-harmonic generation and other four-wave mixing parametric processes, with spatial harmonics of the electron density taking the place of the higher-order terms in the nonlinear dielectric susceptibility. We investigate these mechanisms using a one-dimensional model.

Three-dimensional simulation of free-electron laser harmonics with FRED

FRED3D, a single-mode three-dimensional version of the free-electron laser (FEL) simulation code FRED, is being modified to follow the growth of signal components at the fundamental frequency and at even and odd harmonics. The wiggle-averaged particle and field equations for this multimode formulation are derived here, and their implementation in FRED3D is discussed.

The nonlinear wave equation for higher harmonics in free-electron lasers

The nonlinear wave equation and self-consistent pendulum equation are generalized to describe free-electron laser operation in higher harmonics; this can significantly extend their tunable range to shorter wavelengths. The dynamics of the laser field's amplitude and phase are explored for a wide range of parameters using families of normalized gain curves applicable to both the fundamental and harmonics. The electron phase-space displays the fundamental physics driving the wave, and we use this picture to distinguish between the effects of high gain and Coulomb forces.