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This book presents a comprehensive description of the physics of free-electron lasers starting from the fundamentals and proceeding through detailed derivations of the equations describing electron trajectories, and spontaneous and stimulated emission. Linear and nonlinear analyses are described, as are detailed explanations of the nonlinear simulation of a variety of configurations including amplifiers, oscillators, self-amplified spontaneous emission, high-gain harmonic generation, and optical klystrons. Theory and simulation are anchored using comprehensive comparisons with a wide variety of experiments.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract
The basic concepts underlying the free-electron laser are presented including the resonance condition, the amplification process, gain and saturation efficiency, and electron beam quality requirements. The fundamental configurations such as master oscillator power amplifiers (MOPAs), oscillators and regenerative amplifiers (RAFELs), optical klystrons and high-gain harmonic generation (HGHG), and self-amplified spontaneous emission (SASE) are introduced. While all current free-electron lasers operate in the classical regime, the condition under which quantum mechanical effects become important is discussed. Finally, a discussion of the history and current status of free-electron laser experiments, facilities, and potential applications is given.
H. P. Freund, T. M. Antonsen

Chapter 2. The Wiggler Field and Electron Dynamics

Abstract
The electron trajectories in the external magnetostatic fields in free-electron lasers are fundamental to any understanding of the operational principles and have been the object of study for a considerable time. The concept relies upon a spatially periodic magnetic field, called either a wiggler or undulator, to induce an oscillatory motion in the electron beam, and the emission of radiation is derived from the corresponding acceleration. The specific character of the wiggler field can take on a variety of forms exhibiting both helical and planar symmetries. The most common wiggler configurations that have been employed to date include helically symmetric fields generated by bifilar current windings and linearly symmetric fields generated by alternating stacks of permanent magnets. However, wiggler fields generated by rotating quadrupole fields (helical symmetry) and pinched solenoidal fields (cylindrical symmetry) have also been considered. This chapter deals with the single-particle trajectories in both helical and planar wigglers in both one and three dimensions. Detailed expressions for the trajectories are given, and concepts such as betatron oscillations are discussed.
H. P. Freund, T. M. Antonsen

Chapter 3. Incoherent Undulator Radiation

Abstract
The spontaneous synchrotron radiation produced by individual electrons executing undulatory trajectories in a magnetostatic field is incoherent and is the radiation mechanism used in synchrotron light sources. The magnetostatic field in these devices are formally identical to those employed in free-electron lasers but are commonly referred to as undulators rather than wigglers. The reason for this is that electron synchrotrons produce high-energy electron beams which permit the use of extremely long-period undulations. The use of long-period undulations makes possible the production of relatively large-amplitude magnetostatic fields which are required to ensure the production of a relatively high radiation intensity from this incoherent mechanism. In contrast, since the free-electron laser relies on a coherent emission process, the wiggler magnets employed can be of shorter periods and lower amplitudes. However, incoherent synchrotron radiation is produced in free-electron lasers as well. In this chapter, we present a derivation of the spontaneous undulator radiation emitted as individual electrons propagate through the wiggler. However, it should be remarked that this is only one part of the process. The spontaneously emitted photons can stimulate the emission of more photons or be reabsorbed. The complete physics must include both the spontaneous and stimulated emission mechanisms. Aspects of the stimulated emission including both the linear instability and nonlinear saturation of that instability are presented in succeeding chapters.
H. P. Freund, T. M. Antonsen

Chapter 4. Coherent Emission: Linear Theory

Abstract
In order to give rise to stimulated emission, it is necessary for the electron beam to respond in a collective manner to the radiation field and to form coherent bunches. This can occur when a light wave traverses an undulatory magnetic field such as a wiggler because the spatial variations of the wiggler and the electromagnetic wave combine to produce a beat wave, which is essentially an interference pattern. It is the interaction between the electrons and this beat wave which gives rise to the stimulated emission in free-electron lasers. In the case of a magnetostatic wiggler, this beat wave has the same frequency as the light wave, but its wavenumber is the sum of the wavenumbers of the electromagnetic and wiggler fields. As a result, the phase velocity of the beat wave is less than that of the electromagnetic wave, and it is called a ponderomotive wave. Since the ponderomotive wave propagates at less than the speed of light in vacuo, it can be in synchronism with electrons that are limited by that velocity. Our purpose in this chapter is to give a detailed discussion of the free-electron laser as a linear gain medium as well as to provide a comprehensive derivation of the relevant formulae for the gain in various configurations in both the idealized one-dimensional and the realistic three-dimensional limits. To this end, we derive the expressions for the gain in both the low- and high-gain regimes. The low-gain regime is relevant to short-wavelength free-electron laser oscillators driven by high-energy but low-current electron beams. In contrast, the results in the high (exponential)-gain regime are usually described in terms of a dispersion equation and are appropriate to free-electron laser amplifiers and SASE driven by intense relativistic electron beams.
H. P. Freund, T. M. Antonsen

Chapter 5. Nonlinear Theory: Guided-Mode Analysis

Abstract
The self-consistent nonlinear theory of the free-electron laser describes the interaction through the linear regime and includes the saturation of the growth mechanism. Saturation can occur through a variety of mechanisms. For an ideal beam that is both monoenergetic and vanishing pitch-angle spread, saturation occurs by means of electron trapping in the ponderomotive potential. In the thermal regime, saturation occurs by a different process. In this case, the axial energy spread of the beam (which can arise due to either a distribution in the total energy of the beam electrons or pitch-angle spread) gives rise to a broadband emission spectrum. As a result, a quasilinear saturation mechanism is operative in which the beam undergoes turbulent diffusion in momentum space. The growth rate in this regime is proportional to the slope of the distribution function; turbulent diffusion acts to form a plateau in momentum space that flattens out the distribution of the beam. As a result, the axial energy spread of the beam increases, and the instability is quenched when the slope of the distribution falls to zero. However, the saturation efficiency in the thermal regime is greatly reduced relative to that found for a sufficiently cold beam in which saturation occurs through the particle-trapping mechanism. We shall focus attention on the latter case in this chapter. This chapter will describe the development of slowly varying envelope approximation (SVEA) formulations in the steady-state regime, as well as the application of the analyses to the description of the fundamental physics of the nonlinear saturation mechanism.
H. P. Freund, T. M. Antonsen

Chapter 6. Nonlinear Theory: Optical Mode Analysis

Abstract
The previous chapter dealt with the nonlinear theory in the steady-state regime based on the slowly varying envelope approximation (SVEA). Most of the time-dependent free-electron laser simulation codes that are in use at the present time deal either with an extension of the SVEA in order to solve the wave equation or a particle-in-cell simulation where Maxwell’s equations are solved using a finite-difference time-domain (FDTD) algorithm. The time-dependent formulation presented in this chapter is an extension of the SVEA, in which the SVEA is extended by allowing the slowly varying amplitude to vary in both axial position and time. A time-dependent formulation is necessary to simulate short-wavelength free-electron lasers employing radio-frequency linear accelerators (RF linacs) or storage rings. RF linacs produce high-energy beams with picosecond pulse times and bunch charges of at most several nano-Coulombs. In X-ray free-electron lasers, the actual bunch charge used is about 250 pC or less. Since the growth rate depends upon the peak current, it is desirable to produce bunches with peak currents of several hundred to several thousand amperes, and this requires compression of the bunch to sub-picosecond pulse times. As a result, the slippage of the optical field relative to the electrons can be significant. In addition to describing the slippage of the optical pulse, time dependence is also needed to study the spectral properties of the optical field such as the temporal coherence, linewidth, sideband production, etc. Furthermore, in contrast to the guided-mode analysis used for the steady-state formulation presented in the preceding chapter, the three-dimensional formulations presented in this chapter make use of superpositions of Gaussian optical modes to represent the radiation fields.
H. P. Freund, T. M. Antonsen

Chapter 7. Sideband Instabilities

Abstract
This chapter deals with the theory of sideband growth in free-electron lasers. The growth of sidebands of the primary signal in free-electron lasers can occur after the bulk of the electron beam becomes trapped in the ponderomotive potential formed by the beating of the wiggler and radiation fields. Trapped electrons execute an oscillatory bounce motion in the ponderomotive well, and the waves formed by the beating of this oscillation with the primary signal are referred to as the sidebands. The difficulties imposed by the growth of sidebands are that they can compete with and drain energy from the primary signal as well as considerably broaden the output spectrum. Sideband control, therefore, is an important consideration for free-electron laser configurations that operate in the trapped particle regime. Examples of such systems include (1) tapered wiggler configurations designed to trap the beam at an early stage of the interaction and then extract a great deal more energy from the beam over an extended interaction length and (2) oscillators which run at sufficiently high power over an extended pulse time that the electron beam becomes trapped upon entry to the wiggler.
H. P. Freund, T. M. Antonsen

Chapter 8. Coherent Harmonic Radiation

Abstract
In this chapter, we discuss harmonic generation in free-electron lasers. Harmonic generation in free-electron lasers occurs by both linear and nonlinear mechanisms. The principal difficulty with linear harmonic generation, however, is that the beam quality requirement associated with the growth rate of the linear instability increases with the harmonic number. Nonlinear harmonic generation is driven parasitically off of a high-power fundamental and is less sensitive to the electron beam quality than the linear instability. Harmonics are not commonly seeded in free-electron lasers. The growth of harmonic radiation starts by a combination of shot noise on the electron beam and the initiation of harmonic bunching due to the growth of power at the fundamental. This gives rise to a rapid initial growth of the harmonic power which quickly rolls over to the slower growth associated with the linear instability. The linear instability drives harmonic growth until the fundamental reaches a power level necessary for the nonlinear mechanism to take over, after which the harmonic power grows exponentially with a growth rate that scales as the product of the harmonic number and the growth rate of the fundamental. In this regime, the harmonic grows substantially faster than the fundamental until the harmonic saturates. Saturation of the harmonic typically occurs slightly prior to that of the fundamental due to over-bunching of the electrons at the harmonic wavelengths.
H. P. Freund, T. M. Antonsen

Chapter 9. Oscillator Configurations

Abstract
The gain mechanism can also be used as an oscillator by feedback of a portion of the output signal. An oscillator may be self-excited in the sense that radiation will spontaneously grow from noise if the gain on traversing the wiggler exceeds the losses. An oscillator may also be mode-locked by the injection of a large-amplitude signal with a frequency within the gain band. Since the gain in the interaction region increases with beam current, there is a threshold value of current (known as the start current) below which the gain is exceeded by the losses in the cavity and the radiation will not grow. If the beam current exceeds this threshold, spontaneous noise in the electron beam will be amplified and will grow exponentially in time until the radiation in the oscillator reaches saturation. There are a number of reasons for constructing an oscillator as opposed to an amplifier. First, it may be that no source is available to drive an amplifier. Second, the amplification in one pass through the wiggler may be too small. An amplifier with a small gain is of little practical value. However, if most of the output signal can be fed back to the input, a useful oscillator can be constructed. As a general rule, one finds that shorter-wavelength devices which rely on relatively high-energy but low-current electron beam sources fall into this category and tend to be oscillators because of their inherent low gain. In this chapter we discuss the general theory underlying free-electron laser oscillators.
H. P. Freund, T. M. Antonsen

Chapter 10. Oscillator Simulation

Abstract
Building on the general theory of free-electron laser oscillators discussed in Chap. 9, we discuss the simulation of free-electron laser oscillators in this chapter. This involves using the simulation procedures discussed in Chap. 6 to simulate the interaction in the wiggler in concert with an optics propagation code to simulate the propagation of the optical field through the resonator. General features of the procedure are discussed, and examples showing the validation of the process are provided.
H. P. Freund, T. M. Antonsen

Chapter 11. Wiggler Imperfections

Abstract
The free-electron laser operates by the coherent axial bunching of electrons, and the interaction is extremely sensitive to random imperfections in the wiggler field. Planar wigglers can easily exhibit a random rms fluctuation of 0.5% from pole to pole. This yields a velocity fluctuation that causes a phase jitter that also detunes the wave-particle resonance. In this chapter, we explore the effects of wiggler imperfections on free-electron laser performance and compare the effects of wiggler imperfections with those of an axial energy spread.
H. P. Freund, T. M. Antonsen

Chapter 12. X-Ray Free-Electron Lasers and Self-Amplified Spontaneous Emission (SASE)

Abstract
In this chapter, we consider X-ray free-electron lasers and self-amplified spontaneous radiation (SASE). Because there are no suitable seed lasers at these wavelengths and because the development of X-ray optics has not reached a point which makes oscillator configurations robust, the development of X-ray free-electron lasers has relied on SASE where shot noise on the electron beam grows to saturation in a single pass through a long undulator. Because this requires extremely high peak currents in order to enhance the exponential gain, extreme bunch compression is required prior to the injection of the electron beam into the wiggler. Also, since long wigglers are needed, the wiggler line is composed of multiple wiggler segments separated by quadrupoles to provide for strong focusing of the electron beam. In this chapter, we discuss the equivalent noise power for the start-up of SASE, magnetic chicanes for bunch compression, focusing/defocusing (FODO) lattices, simulation of shot noise, comparison between SASE and master oscillator power amplifiers (MOPAs), phase matching between wiggler segments, and phase shifters, and we give comparisons between the simulation procedures discussed in Chap. 6 with SASE experiments.
H. P. Freund, T. M. Antonsen

Chapter 13. Optical Klystrons and High-Gain Harmonic Generation

Abstract
Optical klystrons have been in use for decades, and the first use was in an ultraviolet free-electron laser oscillator. An optical klystron (OK) is composed of a modulator wiggler that imposes a velocity modulation on the electrons followed by a magnetic dispersive section that enhances the modulation prior to injection into a radiator wiggler that takes the interaction with the modulation-enhanced electrons to saturation. The magnetic dispersive element is, typically, a three- or four-dipole chicane. The radiator can be tuned to the fundamental or a harmonic of the modulator, in which case the interaction is referred to as high-gain harmonic generation (HGHG). In this chapter, we discuss these concepts and simulations of representative configurations.
H. P. Freund, T. M. Antonsen

Chapter 14. Electromagnetic-Wave Wigglers

Abstract
The physical mechanism in the free-electron laser depends upon the propagation of an electron beam through a periodic magnetic field. Both incoherent and coherent radiation results from the undulatory motion of the electron beam in the external fields which permits a wave-particle coupling to the output radiation. Coherent radiation depends upon the stimulated emission due to the ponderomotive wave formed by the beating of the radiation and wiggler fields. The wiggler field itself may be either magnetostatic or electromagnetic in nature. Although the bulk of experiments as of this time have relied upon magnetostatic wigglers with either helical or planar polarizations, the fundamental principle has also been demonstrated in the laboratory using a large-amplitude electromagnetic wave to induce the requisite undulatory motion in the electron beam. In this chapter, we discuss the fundamental theory of free-electron lasers with electromagnetic-wave wigglers.
H. P. Freund, T. M. Antonsen

Chapter 15. Chaos in Free-Electron Lasers

Abstract
Researchers in the field of chaos are concerned with the basic properties of the solutions of systems of nonlinear equations. This interest stems from the fact that almost every physical system can be described at some level of approximation by a system of nonlinear equations. The development of this field has led to several general conclusions about nonlinear systems. On the one hand, even the simplest deterministic nonlinear systems can exhibit behavior that is complicated and appears to be random. This behavior has been termed chaos. On the other hand, the chaotic behavior of much more complicated systems often seems to follow the same rules as the simple systems. Thus, there is order in the chaos. In this chapter, we introduce some basic concepts regarding chaos and nonlinear dynamics before going on to a discussion of the application of these concepts to the physics of free-electron lasers.
H. P. Freund, T. M. Antonsen

Backmatter

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