X-rays produced when highly relativistic electrons are accelerated along a curved trajectory, generally referred to as synchrotron radiation, have served as an important tool for studying the structure and dynamics of various atomic and molecular systems. The first dedicated synchrotron radiation facility was built in the 1970s using an electron storage ring, and since that time the demand for synchrotron radiation has steadily increased due to its high intensity, narrow angular opening, and broad spectral coverage. Over the past few decades the effectiveness of synchrotron radiation has been further advanced by improvements in storage ring design that led to an increase in the electron beam phase space density, and by the use of magnetic devices such as undulators that dramatically increase the X-ray brightness over traditional bending magnets. These developments have widened and deepened the reach of “photon sciences” around the globe.

Another revolutionary advance in X-ray generation was made with the development of X-ray free-electron lasers (FELs). The radiation produced in an FEL acts back on the electron beam in a positive feedback loop, resulting in X-rays with dramatically improved intensity and coherence over those produced with storage-ring based sources. The X-ray FEL became feasible thanks to improvements in linear accelerator technology in general, and in particular to advances in the injector (electron source).

High-brightness, high-energy electron beams from a linear accelerator can now drive a high-gain X-ray FEL amplifier in a long undulator. The gain can be so high that the initially incoherent undulator radiation evolves to an intense, quasi-coherent field known as self-amplified spontaneous emission (SASE). The SASE pulse can be made ultrashort by using an ultrashort electron bunch. With X-ray FELs, experimental techniques developed for traditional synchrotron light sources can be made much more efficient, and new areas of material, chemistry, and biology research, such as ultrafast dynamics, have become accessible to study.

In this chapter we delve more deeply into the 1D theory of the FEL. The 1D picture is sufficient to understand how an FEL works, since the essential FEL physics is longitudinal in nature. A free-electron laser acts as a linear amplifier in the small signal regime, and we will find that it is most easily analyzed theoretically in the frequency representation. Hence, we begin this section by deriving the Klimontovich equation describing the electron beam in the frequency domain, to which we add the Maxwell equation (3.68). We then apply these equations to the small-gain limit in Section 4.2, finding solutions that generalize those of Section 3.3. We then turn our attention to the high-gain FEL in Section 4.3, showing how the linearized FEL equations can be solved for arbitrary initial conditions using the Laplace transform. In particular, Section 4.3 covers self-amplified spontaneous emission (SASE) in some detail, because SASE provides the simplest way to produce intense X-rays. We derive the basic properties of SASE in the frequency domain, including its initialization from the fluctuations in the electron beam density (shot noise), its exponential gain, and its spectral properties. We then connect our analysis to the time domain picture via Fourier transformation, which helps complete the characterization of SASE's fluctuation properties. The chapter concludes with a discussion of how the FEL gain saturates in Section 4.4. We derive a quasilinear theory that describes the decrease in gain associated with an increase in electron beam energy spread, and show qualitatively how this is related to particle trapping. We also discuss tapering the undulator strength parameter after saturation to further extract radiation energy from the electron beam. Finally, we make a few comments on superradiance, focusing on the superradiant FEL solution associated with particle trapping that can support powers in excess of the usual FEL saturation power.