K. Zhukovsky

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/64439

#### **Abstract**

We review the synchrotron type radiation sources with focus on undulator and freeelectron laser (FEL) schemes, aimed on working in X-ray range and ultra-short time interval. Main FEL schemes, useful for generation of high frequency radiation, extending to X-rays, are presented. High harmonic generation is explored. The advantages and disadvantages of single pass and of multipass designs are discussed. The viable ways to reduce the duration of the pulse, with the goal to generate femtosecond pulses, are indicated. Future developments of X-ray FELs (X-FELs) and the ways to improve the quality of the FEL radiation in this context are discussed.

**Keywords:** undulator radiation, harmonics generation and broadening, homogeneous and inhomogeneous losses, free-electron laser

#### **1. Introduction**

Synchrotron radiation (SR) and undulator radiation (UR) have been attracting researcher's attention for more than half a century. The reasons for that varied with time passing as the challenges for the scientists evolved and the technical progress stepped forward. UR was predicted [1] and then discovered [2] in the middle of the twentieth century. During the following 70 years, the study of the radiation, emitted by ultra-relativistic electrons, was performed, and the SR theory was refined; extensive theoretical studies of the electron motion in periodic magnetic fields of various configurations have been performed [3–7]. Now UR is

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again in focus due to the request for coherent X-ray sources [8], while free-electron lasers (FELs) extend to X-range [9]. Both, SR and UR are due to the radiation of relativistic electrons, executing curved trajectories [10]. The difference between them is in the length on which the radiation is formed: short part of the circle for UR and the full length of the undulator for the UR. This determines the fundamental difference in the quality of the radiation obtained from these two sources: short pulses with very broad spectrum for the SR and relatively longlasting radiation bursts with narrow spectrum for the UR. Nowadays, the research frontier is represented by studies of ultra short attosecond time intervals and Röntgen range [11, 12]. To achieve these characteristics, the devices require extremely high quality and intense magnetic fields, long undulators with many periods. To obtain high frequency radiation, sometimes undulator periodic structures with double or even triple period are used [13–18], facilitating control over high harmonics and regulating their emission [6, 19]. Maintaining best quality UR line is important. Nowadays, electron accelerators provide ultra fast and high coherent electron beams and intense emission even in Röntgen range. Modern undulators allow for the harmonic emission regulation [19], which can be achieved by superimposing different periodic magnetic fields in many period undulators. It should be noted that in long undulators the distortions of the magnetic field and of unavoidable inhomogeneities of periodic magnetic field have very strong effect on the operation of the devices. High gain in undulators is essential in FEL with self-amplified spontaneous emission (SASE), with highgain harmonic generation (HGHG), and in other modern schemes. The quality of the electron beams, fed into the undulators, is extremely important for such new FELs [20], in particu‐ lar at high frequencies. The requirements for the periodicity of the field are particularly rigid in long undulators [21]. Deviation from the ideal oscillatory trajectories of electrons results in degradation of the beam quality and of the FEL output in terms of power, brightness FEL gain. Therefore, the emitted UR can be exploited for diagnosing the quality of the undula‐ tor itself.

The radiation spectrum lines from undulators inevitably broaden due to a number of rea‐ sons, first of all due to the electron beam energy spread and the beam divergency, as well as due to inhomogeneity of the periodic magnetic field in undulators. They may have internal or external origin [22–26], but their presence is eminent also due to the fact that the ideal *H* <sup>→</sup> <sup>=</sup>*H*0*sin*(2*π<sup>z</sup>* / *<sup>λ</sup>*) periodic magnetic field simply does not satisfy Maxwell equations. The electron energy spread is the most common detrimental factor; some researchers even concluded that the spectral properties of higher UR harmonics should be limited only by the electron beam properties and not by the undulators [24]. The role of the divergence was underlined, for example, in Refs. [22, 27]. At the same time the constant magnetic field shifts the resonance frequencies and causes loss of intensity [28–31].

The demand for radiation with specific properties and high requirement to the UR and undulator quality stimulated analytical study of their spectral properties [32–39]. There appear generalized Bessel and Airy functions naturally, as well as in other mathematical problems of radiation, emitted by charges, executing complicated oscillating trajectories. The mathemati‐ cal apparatus of inverse differential operators and orthogonal polynomials was developed for treating broad spectrum of physical problems, which include radiation and propagation of electron beams [40–45]. The contributions of all sources of broadening in various undulator schemes were analyzed by means of precise analytical treatment of the UR, employing extended forms of special functions of Airy and Bessel types [34–37, 39, 42]. In these works, the role of the various broadening terms, accounting for the real size and the emittance of the electron beam, for the energy spread and for the constant field component, was explored. It was shown that the undulator length has strong detrimental effect on the spontaneous harmonic emission; partial compensation of the beam divergences by constant magnets was also demonstrated.

again in focus due to the request for coherent X-ray sources [8], while free-electron lasers (FELs) extend to X-range [9]. Both, SR and UR are due to the radiation of relativistic electrons, executing curved trajectories [10]. The difference between them is in the length on which the radiation is formed: short part of the circle for UR and the full length of the undulator for the UR. This determines the fundamental difference in the quality of the radiation obtained from these two sources: short pulses with very broad spectrum for the SR and relatively longlasting radiation bursts with narrow spectrum for the UR. Nowadays, the research frontier is represented by studies of ultra short attosecond time intervals and Röntgen range [11, 12]. To achieve these characteristics, the devices require extremely high quality and intense magnetic fields, long undulators with many periods. To obtain high frequency radiation, sometimes undulator periodic structures with double or even triple period are used [13–18], facilitating control over high harmonics and regulating their emission [6, 19]. Maintaining best quality UR line is important. Nowadays, electron accelerators provide ultra fast and high coherent electron beams and intense emission even in Röntgen range. Modern undulators allow for the harmonic emission regulation [19], which can be achieved by superimposing different periodic magnetic fields in many period undulators. It should be noted that in long undulators the distortions of the magnetic field and of unavoidable inhomogeneities of periodic magnetic field have very strong effect on the operation of the devices. High gain in undulators is essential in FEL with self-amplified spontaneous emission (SASE), with highgain harmonic generation (HGHG), and in other modern schemes. The quality of the electron beams, fed into the undulators, is extremely important for such new FELs [20], in particu‐ lar at high frequencies. The requirements for the periodicity of the field are particularly rigid in long undulators [21]. Deviation from the ideal oscillatory trajectories of electrons results in degradation of the beam quality and of the FEL output in terms of power, brightness FEL gain. Therefore, the emitted UR can be exploited for diagnosing the quality of the undula‐

The radiation spectrum lines from undulators inevitably broaden due to a number of rea‐ sons, first of all due to the electron beam energy spread and the beam divergency, as well as due to inhomogeneity of the periodic magnetic field in undulators. They may have internal or external origin [22–26], but their presence is eminent also due to the fact that the ideal

<sup>→</sup> <sup>=</sup>*H*0*sin*(2*π<sup>z</sup>* / *<sup>λ</sup>*) periodic magnetic field simply does not satisfy Maxwell equations. The electron energy spread is the most common detrimental factor; some researchers even concluded that the spectral properties of higher UR harmonics should be limited only by the electron beam properties and not by the undulators [24]. The role of the divergence was underlined, for example, in Refs. [22, 27]. At the same time the constant magnetic field shifts

The demand for radiation with specific properties and high requirement to the UR and undulator quality stimulated analytical study of their spectral properties [32–39]. There appear generalized Bessel and Airy functions naturally, as well as in other mathematical problems of radiation, emitted by charges, executing complicated oscillating trajectories. The mathemati‐ cal apparatus of inverse differential operators and orthogonal polynomials was developed for treating broad spectrum of physical problems, which include radiation and propagation of

the resonance frequencies and causes loss of intensity [28–31].

tor itself.

200 202High Energy and Short Pulse Lasers

*H*
