The purpose of this chapter is:

1. ■ to explain how a backward wave oscillator (BWO) operates, from both a physical and a mathematical viewpoint;

2. ■ to demonstrate that a BWO cannot work with a smooth distribution;

3. ■ to estimate the minimum length of the generator region for the threshold condition to be met;

4. ■ to find the temporal growth rate for a step distribution and for a beam distribution; and

5. ■ to consider the effect of magnetic field inhomogeneity, e.g. as occurs along a geomagnetic flux tube.

The BWO generation regime is well-known in electronic devices, where wave generation is due to the interaction of a well-organized electron beam with man-made electrodynamic systems or with the external magnetic field (Johnson, 1955; Shevchik and Trubetskov, 1975; Wachtel and Wachtel, 1980; Ginzburg and Kuznetsov, 1981). The positive feedback inherent in such devices is due to the volume interaction of a ‘beam’ wave with an electromagnetic wave, the group velocities of the two interacting waves being in opposite directions. Thus, the absolute instability in the BWO generation regime does not need wave reflecting mirrors, and can be realized in length-limited systems. Such a regime is very attractive in space plasmas with their tremendous dimensions but may seem to be improbable there due to the need for having a well-organized charged particle beam. However, it turns out that such a ‘beam’ wave is not a rare phenomenon in space because that can appear on the step-like deformation of the distribution function (see Section 3.3).