Two-frequency Undulator Research Articles

Erratum to: Improved Generation of Higher Harmonics and Suppression of the Lowest Harmonics in an X-Ray FEL with a Two-Frequency Undulator

The authors apologize for the misprints in the article “Improved Generation of Higher Harmonics and Suppression of the Lowest Harmonics in an X-Ray FEL with a Two-Frequency Undulator” by K. Zhukovsky in Moscow University Physics Bulletin, 2018, Vol. 73, No. 5, pp. 462–469, DOI: https://doi.org/10.3103/S0027134918050193, and communicate the following corrected formula (5) for the Bessel coefficients fn of the two-frequency undulator:

Erratum to: Improved Generation of Higher Harmonics and Suppression of the Lowest Harmonics in an X-Ray FEL with a Two-Frequency Undulator

Erratum to: Improved Generation of Higher Harmonics and Suppression of the Lowest Harmonics in an X-Ray FEL with a Two-Frequency Undulator

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.

Phenomenological and numerical analysis of power evolution and bunching in single-pass X-ray FELs.

The harmonic power and bunching evolution in X-ray single-pass free-electron lasers (FELs) is modelled and the harmonic generation in a phase-shifted two-frequency FEL is explored. The advanced phenomenological FEL model, which is validated numerically and experimentally, is employed. The model accounts for major losses for each harmonic individually; it is compared with reported experimental data and with PERSEO numerical simulations, which are performed here for a variety of experiments. The latter cover the radiation wavelength range 0.15-300 nm. The phenomenological description is based on a few key FEL parameters: electron beam section, current, energy and its spread and divergence. The model is employed for modelling harmonic bunching and power evolution in a phase-shifted X-ray FEL with a two-frequency undulator, where lower harmonics with numbers less than nth are suppressed by the electron-photon phase shift of kπ/n, k = 2, 4, …, between the undulator sections. The benefits of the two-frequency phase-shifted FEL are highlighted. FEL-induced energy spread is shown to be three times lower than in a FEL without the phase-shift. The high-power harmonic and sub-harmonic radiation in such a FEL is demonstrated. In particular, powerful ∼14 GW X-ray radiation at λ5 = 0.15 nm from electrons with energy of 5.47 GeV and beam current ∼3.66 kA is possible in a two-frequency phase-shifted FEL at 30 m; this constitutes half of a FEL length where a common planar undulator radiates the same wavelength and power at the fundamental harmonic. Moreover, about a three times lower energy spread is induced by the dominant fifth harmonic, and the harmonic power can be thousands of times higher than in a common planar undulator FEL.

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.

Improved Generation of Higher Harmonics and Suppression of the Lowest Harmonics in an X-Ray FEL with a Two-Frequency Undulator

Improved Generation of Higher Harmonics and Suppression of the Lowest Harmonics in an X-Ray FEL with a Two-Frequency Undulator

Two-frequency undulator in a short SASE FEL for angstrom wavelengths

We perform a theoretical study and modeling of harmonic evolution in an x-ray single-pass free-electron laser (FEL) with a two-frequency undulator and phase shifters. For this purpose a phenomenological FEL model is developed and validated; it accounts for all losses individually for each harmonic. The model was tested with FEL experiments, covering radiation wavelengths ranging from 0.15–500 nm. An example of the LCLS experiment modeling with 1.5 angstrom radiation is presented. The phenomenological description is based on a few main FEL parameters: electron beam current, energy and its spread, and emittance. The results match well with measurements and numerical simulations in various conditions. We model the harmonic bunching and power evolution in an x-ray FEL with a two-frequency undulator, in which the first and third harmonics are suppressed between the undulator sections by phase shifts of kπ/5, k = 2, 4,… of the electrons with respect to the photons. The advantages of the two-frequency undulator in such phase-shifted FELs are highlighted. The possibility to produce intense x-ray radiation by the dominant fifth harmonic in the linear regime is demonstrated. The radiation at subnanometric wavelength λ5 = 0.15 nm from electrons with E = 5.47 GeV energy and I ∼ 3.66 kA current already reaches ∼14 GW power at 30 m. This is shorter than the saturation length of the common undulator FEL with its fundamental harmonic at the same wavelength. The power boost and the reduction of the saturation length are complemented by 2.5 times lower electron energy and 3 times lower FEL-induced energy spread. The harmonic power increase can be up to 104 times greater than that in a common planar undulator FEL.

Two-Frequency Undulators for Generation of X-Ray Radiation in Free-Electron Lasers

We present a theoretical study and a computer simulation of characteristics of the undulator radiation in single-pass free-electron lasers (FELs). Using a phenomenological model describing the dynamics of the radiated power in FELs with allowance for the basic loss, we study generation of harmonics in the X-ray range in a FEL with a two-frequency undulator. We study the possibility to achieve a hundredfold increase in the radiation intensity of the nth harmonic in a FEL, in which the electron-phase shift by kπ/n with respect to photons occurs between undulator sections, where k = 2, 4, . . . . The advantages of using a two-frequency undulator in a single-pass FEL and the possibility of generating the high-power X-ray radiation by the FEL at the harmonic wavelengths 2.3–3.3 nm in the linear regime are demonstrated. The FEL is compared with the two-frequency undulator and the conventional plane undulator. Additionally, generation of radiation having a power of tens of megawatts is studied at the wavelength λ ≈ 3.27 nm in a multistage FEL with a length of 40 m, an off-the-shelf excimer ultraviolet seed laser, which operates at a wavelength of 157 nm, and an electron beam having an energy of about 0.6 GeV.

Generation of Coherent Radiation in the Near X-Ray Band by a Cascade FEL with a Two-Frequency Undulator

Generation of X-ray radiation in a cascade self-amplified spontaneous emission free-electron laser (SASE FEL) using the harmonics of a two-frequency undulator is studied. The advanced phenomenological model of a one-pass FEL that accounts for the main losses in real FELs is presented: the electron energy spread in the beam, the beam divergence, diffraction, and the fact that emission losses are greater at higher harmonics than in the main frequency range. The FEL mathematical model was performed using the Mathematica software and calibrated within the experiment carried out at the operating SPARC facility via complex three-dimensional numerical simulations. The phenomenological model is used to analyze FEL dynamics for generation of a high-energy X-ray emission at a relatively short length. It is proposed to use a two-frequency undulator for the initial electron grouping and subsequent frequency multiplication in a cascade FEL with higher harmonic amplification (HGHG). The advantages of the two-frequency undulator are presented for electron grouping at higher harmonics of the undulator radiation (UR). The operation of several types of FEL is simulated with amplification of the seed laser wave frequency in two and three cascades to generate the soft X-ray radiation. A seed laser with a wavelength of 11.43 nm corresponding to the peak reflectivity of mirror coatings with MoRu/Be is proposed for generating the intensive X-ray laser radiation with λ ~ 1.27–3.37 nm. Here, the intensive radiation power reaches 50 MW at a length of only 35 meters; the radiation shows good temporal coherence corresponding to the performance of a low-power seed laser with a lower frequency.

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.

Generation of X-Ray Radiation in Free-Electron Lasers with Two-Frequency Undulators

The possibility of creating an x-ray laser with a two-frequency undulator with the help of the phenomenological model of a single-pass free-electron laser (FEL) that takes into account the beam characteristics and the undulator field is studied. The model is used to analyze the FEL dynamics, and the simulation results are compared with the experimental data. The advantages of using a two-frequency undulator in the single-pass FEL and the possibility of generating powerful x-ray radiation in it are demonstrated. The evolution of power of higher-harmonic generation in the FEL and the bunching coefficient at wavelengths of these harmonics are studied. The possibility of their further amplification until saturation in a cascade FEL matched to higher harmonics of the two-frequency undulator is demonstrated.

Generation of coherent soft X-ray radiation in short FEL with harmonic multiplication cascades and two-frequency undulator

The generation of powerful nanometer x-ray radiation in cascaded high-gain free-electron lasers (FELs) with two-frequency undulator is studied. An advanced phenomenological model of single pass FEL is developed, which describes both linear and nonlinear harmonic generation from seed and initial shot noise; it includes calibrated corrections for all loss factors, specific for each FEL harmonic in each cascade. In this way, it differs from other phenomenological models, which account for the energy spread or divergence equally for all FEL harmonics. The new model, calibrated with a FEL experiment and numerical 3D simulations, is employed for the analysis of the FEL power dynamics to obtain maximum high-harmonic power in the X-ray band at the shortest possible FEL length with a low seed frequency and electron energy. The advantages of the two-frequency undulators in FEL prebunchers are demonstrated; their use in the cascaded FELs is proposed. Several such FELs with two-frequency undulators are simulated. The nanometer X-ray radiation production is studied in high-gain cascaded FELs with the seed wavelengths, matching peak reflectivities of MoRu/Be and Mo/Si. Dozens of megawatts of X-ray radiation are modeled at ≈40 m in a multi-cascade FEL, seeded from commercially available F2 excimer laser.

High-harmonic x-ray undulator radiation for nanoscale-wavelength free-electron lasers

We explore the undulator radiation (UR) from relativistic electrons in two-frequency undulators intended for use in x-ray free-electron lasers (FELs). We use an exact analytical account for the major sources of spectral line broadening of the spontaneous UR to describe the UR spectrum, intensity and emission line shape. We also show possible partial compensation of the off-axis divergence by ad hoc imposed weak transverse constant magnetic components. The harmonic radiation of the two-frequency undulator is studied. High-harmonics amplification in the two-frequency undulator is underlined for possible use in x-ray FELs and the generation of powerful x-ray radiation is demonstrated. The evolution of high harmonic power is studied with the help of the developed phenomenological self-amplified spontaneous emission FEL model. The performance of a high-gain harmonic generation FEL with a two-frequency undulator–prebuncher is explored. Harmonic multiplication cascades are introduced in FELs to obtain nanometer x-ray FEL radiation in reduced size FELs at ~30 m with seed from commercially available excimer laser and ~0.5 GeV electrons.

Soft X-ray generation in cascade SASE FEL with two-frequency undulator

The generation of soft X-ray radiation in cascade self-amplified spontaneous emission free electron laser (SASE FEL) using harmonics of a two-frequency undulator is studied. An advanced phenomenological model of SASE FEL, accounting for the properties of the undulator and of the electron beam, including major losses, is employed. The account for the beam energy spread, divergence and diffraction is harmonic dependent, which is a step forward from the previous models and phenomenological approach, which consider all the losses independent of the harmonic number. The model is validated with the experimental measurements of the FEL harmonic generation and with relevant three-dimensional numerical simulations. The model is adopted for the description of cascading SASE FELs. With its help the dynamics of high-gain harmonic generation (HGHG) SASE FEL cascades is analyzed to obtain the maximum high harmonic power in the X-ray band with the lowest possible seed frequency and the shortest FEL length. The advantages of the two-frequency undulator for SASE FEL in this context are demonstrated and the possibility of its usage in cascading HGHG SASE FEL is shown. Several types of cascade FELs with two-frequency undulators are simulated. Frequency multiplication in two, three and four stage cascades is introduced to obtain soft X-ray radiation. The possibility of the generation of a powerful radiation, 75 MW at and a lower intensity 3rd harmonic, 100 kW at , is demonstrated in multi-cascade FELs at just 20–35 meters with the seed wavelengths and .

Two-frequency undulator usage in compact self-amplified spontaneous emission free electron laser in Roentgen range

AbstractThe generation of harmonics in two-frequency undulator in a self-amplified spontaneous emission free electron laser (SASE FEL) is studied in order to produce Roentgen radiation in a relatively compact sized installation. The dynamics of SASE FEL is analyzed with the help of the phenomenological model to obtain the maximum of the X-ray high-harmonic power. The model accounts for the properties of the undulator magnetic field and of the electron beam and includes the major sources of losses, such as the electron energy spread, etc. It is compared and calibrated with the existing data on a FEL experiment. The advantages of the two-frequency undulator for Roentgen SASE FEL are demonstrated and the possibility to generate powerful mild Roentgen radiation at already ~25 m length is shown. The evolution of the bunching coefficients for high harmonics is studied together with the evolution of the FEL-induced energy spread. The linear and non-linear regimes are explored for common and for two-frequency undulators The usage of the two-frequency undulator for cascade SASE FEL with high X-ray harmonic power and high-harmonic bunching coefficients with low-induced energy spread is proposed.

Emission and tuning of harmonics in a planar two-frequency undulator with account for broadening

AbstractWe analyze the undulator radiation from planar bi-harmonic undulators with account for all principle losses in real devices. The exact analytical expressions for the UR spectrum, intensity, and the line shape are obtained in terms of special functions. The interplay of various broadening contribution is elucidated and their role is explored. Suspension and boost of certain harmonics by fine tuning the undulator parameters is demonstrated. The constant non-periodic magnetic constituents are studied to compensate the divergency of the electronic beam. The examples of harmonic generation in various undulator schemes with single and double period magnetic field are explored. Influence of inhomogeneous and homogeneous broadening on these harmonics is demonstrated. The analysis is applied to evaluate the harmonics in self-amplified spontaneous emission free electron laser.

Harmonic generation by ultrarelativistic electrons in a planar undulator and the emission-line broadening

We present an analytical treatment of the undulator radiation (UR), accounting for major sources of the spectral-line broadening, for example, the beam energy spread, the emittance, and a constant magnetic field, additional to the ideal periodic fields. The examples of high harmonic generation in one- and two-frequency undulators are considered with account for homogeneous and inhomogeneous broadening in real devices. Qualitative estimations are given as well as the exact analytical expressions for the UR spectrum, and intensity are obtained and simplified in the case of high-energy electrons. The compensation of the divergency by the constant magnetic field is demonstrated in the analytical form. The application for free-electron lasers (FEL) is demonstrated and compared with the radiation of a FEL on the supposition of an ideal undulator.

Analysis of two frequency harmonic undulator radiation and gain

The spectral properties of radiation of a relativistic electron beam in a two frequency harmonic undulator are analytically investigated. The technique of generalized Bessel function is effectively used to present some of the novel features associated with the two frequency harmonic undulator radiation. An analytical expression for gain of the two frequency harmonic undulator (TFHU) is presented for comparison with the two frequency undulator (TFU).

Two-frequency undulator and harmonic generation by an ultrarelativistic electron

The spectrum of electromagnetic radiation of a relativistic electron moving in the magnetic field oscillating in two mutually transversal spatial directions with two different frequencies is analytically investigated. The spectral properties of radiation in the planar biharmonic undulator are also discussed. The technique based on the associated Bessel functions was effectively applied in our calculations. The effect of the magnetic field and the undulator parameters on the radiation of the fundamental as well as low and high harmonics was elucidated. It is demonstrated that the biharmonic undulator can be exploited to regulate the emission of certain selected harmonics and hence contribute to the development of the efficient devices with high extraction and narrow spectrum.