Cyclotron Maser Radiation from an Inhomogeneous Plasma R. A. Cairns, 1 I. Vorgul, 1 and R. Bingham 2,3 1 University of St Andrews, School of Mathematics and Statistics, St Andrews, Fife, Scotland, KY16 9SS 2 STFC Rutherford Appleton Lab, Space Science & Technology Dept, Didcot, Oxon, OX11 0QX, United Kingdom 3 The Physics Department, University of Strathclyde, Glasgow, G4 0NG, United Kingdom (Received 9 September 2008; published 20 November 2008) 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. DOI: 10.1103/PhysRevLett.101.215003 PACS numbers: 52.35.g Cyclotron maser instabilities are of importance in vari- ous astrophysical contexts such as planetary radio emis- sion, for example, auroral kilometric radiation (AKR) [1– 3], solar decimetric radiation [4], and in recently discov- ered emissions from rotating stars including periodic emis- sion [5–7], astrophysical shocks [8], and blazars [9]. It has been argued in a number of previous papers that AKR and similar phenomena are likely to be produced by a distri- bution with a horseshoe or crescent shape in velocity space [10–12] and we have collaborated in setting up an experi- ment in which an electron beam moving along converging magnetic field lines produces a horseshoe distribution [13,14]. This experiment has produced radiation for which the spectral properties and the conversion efficiency from beam energy to radiation match those observed in AKR. In this paper we look at some effects of plasma nonuniformity on cyclotron maser instabilities. One long-standing prob- lem in studies of cyclotron masers is how radiation, gen- erally observed to be generated below the local cyclotron frequency, gets onto the branch of the dispersion relation which connects to vacuum propagation. It evidently does, since the AKR radiation is observed by satellites above the wave generation region, although the cold plasma disper- sion curve goes to a cutoff at or above the cyclotron frequency and does not connect to the branch below the cyclotron frequency [15]. We shall show that the topology of the dispersion curves may be different in a plasma with an electron distribution showing a cyclotron maser insta- bility and look at some properties of localized instabilities in an inhomogeneous plasma. Because of the complexity of the dispersion relation for a horseshoe distribution [10,16] we consider a system with a simple ring distribu- tion of the form fð k ; ? Þ¼ n 0 ð k Þð ?   0 Þ. Such a distribution has been considered in the literature on gyro- trons [17,18] (and references therein), and also in the space physics literature [8]. While it might be a reasonable representation of reality in the gyrotron, a monoenergetic distribution of this sort is perhaps less likely in the space plasma context. It is, however, simpler to treat since it gives rise to a simple algebraic dispersion relation. We shall show that in a nonuniform plasma it has some interesting and perhaps surprising properties which may cast some light on how more complex cyclotron maser instabilities might behave. The dispersion relation for the ring distribution can readily be obtained by sustituting the ring distribution into the general dispersion relation as given, for example, by Stix [15]. Since relativistic effects play a crucial role in cyclotron maser instabilities the relativistic dielectric ten- sor elements must be used. We also make the approxima- tion that the wave frequency is around the cyclotron frequency. The result is, for propagation parallel to the ambient magnetic field, ð! 2  k 2 c 2 Þ  ð!  Þ 2 þ ! 2 pe T 2   ! 2 pe  !ð!  Þ¼ 0; (1) and for perpendicular propagation k 2 c 2 ¼ ! 2  " k  " 2 ? " k  ; (2) FIG. 1. The dispersion relation for parallel propagation for a ring distribution with  ¼ 1:02 and ! ce ! pe ¼ 10. PRL 101, 215003 (2008) PHYSICAL REVIEW LETTERS week ending 21 NOVEMBER 2008 0031-9007= 08=101(21)=215003(4) 215003-1 Ó 2008 The American Physical Society