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.