Investigation of Propagation and Coupling of Electromagnetic Waves in a Two-dimensional Inhomogeneous Plasma M. K. SALEM, F. M. AGHAMIR 1 , B. Sh. SABZEVARI 2 and M. GHORANNEVISS Plasma Physics Research Center, Science and Research Campus, Islamic Azad University, Tehran, Iran 1 Physics Department, University of Tehran, Tehran, Iran 2 Physics Department, University of Isfehan, Isfehan, Iran (Received July 2, 2003) The propagation of electromagnetic waves in a two dimensional inhomogeneous plasma and applied magnetic field is studied. The generalized eikonal equations along with two dimensional Hamiltonian equations are used to reach at a system of transport equations for two separate modes. The reflection of electromagnetic waves in a direction perpendicular to the applied magnetic field, as a mean of mode coupling, is investigated. Application of this model to a simple practical example is presented to check its merit. It is found that the difference of this approach with the one dimensional case is in the number of cut off points wave encounters. KEYWORDS: electromagnetic waves, inhomogeneous plasma, Hamiltonian equations, mode coupling, cut off points DOI: 10.1143/JPSJ.73.3044 1. Introduction The propagation and coupling of electromagnetic waves in plasma plays an important role in physics of ionosphere, 1) in various heating schemes of plasma 2) and, in particular, in linear mode conversion. 3,4) Nearly all recent works concern- ing coupling of electromagnetic waves in plasmas have been restricted to spatial symmetries in media. In physics of ionosphere, plasmas have been considered plane-stratified, i.e., the plasma medium is inhomogeneous in one direction. 1) However, the variation of the curvature of the ionosphere can be taken into account, and as a result density can be considered as a function of two independent variables. In the resonant absorption of radio frequency heating of plasmas, antennas are used for the launch of an electro- magnetic wave and mode conversion takes place in certain regions called the resonant layers. 4,5) In almost all cases the inhomogenity of layers is considered as a one dimensional problem; however, the toroidal geometry of tokamaks constitutes the variation of the curvature of these layers as well as that of the magnetic field. This is indicative of the notion that inhomogenity of the medium and the magnetic field could be dependent upon more than one dimension. Bernstein and Friedland, 6) and Kravtsov, 7) studied the propagation of the electromagnetic wave in four dimensional space and time varying plasmas, where the coupling of waves were not considered. Kaufmann and Friedland, 8) worked on four dimensional mode conversion where only some special cases of coupling of electromagnetic waves by plasma were considered. In a more recent study, the coupling of electromagnetic waves in slowly varying inhomogeneous, nonstationary, anisotropic absorbing plas- mas was studied by Sabzevari. 9) The formalism is valid for propagation and coupling of electromagnetic waves in any kind of media, with general geometries of space-time and magnetic field configurations. In the present study, the case where a stationary plasma and the magnetic field are inhomogeneous in two spatial directions is considered. The geometrical optics approxima- tion has been the most successful general approach to the problem. One usually applies this approximation in a case where inhomogenity is weak on the scale of the character- izing typical wavelength. Therefore, the only restricting condition is that the properties of the medium should be slowly varying in two directions. In the following section, a generalized eikonal Maxwell’s curl equations along with constitutive relations are trans- formed into a coupled system of six linear, partial differ- ential equations for the amplitude components of the electromagnetic field. The two-dimensional Hamiltonian equations are used to reach at a system of transport equations for the amplitudes. Coalescence of two eigenvalues at a coupling region leads to a system of two coupled differential equations for the two amplitudes. Application of this method for a simple model and the differences of this model with the one-dimensional case are discussed in §3. Conclusions are drawn in the last section. 2. Basic Equations In this section, we will discuss the eikonal Maxwell’s system of equations and also transport equations for amplitude. The discussion will be based on the general formalism used in ref. 9. For the sake of completeness, some of the results of which will be repeated here. In our model, a weakly inhomogeneous, two dimensional stationary plasma has been considered. A small amplitude electromagnetic wave supported by this system obeys Maxwell’s equations: r H ¼ @ @t D þ J ð1aÞ r E ¼ @B @t ð1bÞ These equations have to be supplemented with constitutive relations which can be expressed by integral relations for slowly varying media. For example Ohm’s law is written as 10) JðXÞ¼ Z d 4 X 0 ' ker ðX  X 0 ; rÞ EðX 0 Þ ð2Þ Journal of the Physical Society of Japan Vol. 73, No. 11, November, 2004, pp. 3044–3050 #2004 The Physical Society of Japan 3044