Cyclotron maser radiation from an inhomogeneous plasma

Cyclotron maser radiation is important in both laboratory devices such as gyrotrons and in space physics applications to phenomena such as auroral kilometric radiation. To understand the behavior, especially in the latter case where there is generally a localized region of instability, requires an understanding of how such instabilities behave in an inhomogeneous plasma. Here we consider, for simplicity, a simple ring distribution of electrons in either a step function variation of magnetic field or a continuous gradient. In each case we show that there can exist localized regions of instability from which waves, growing in time, can be radiated outwards.