Most of the analytical results in the theory of OX conversion are obtained in a slab geometry, in which the magnetic field is uniform and ID spatial inhomogeneity is related to a constant plasma density gradient [1][2][3][4]. For conventional OXB scheme in toroidal systems this approximation is generally valid for the nearequatorial launch of the 0 mode when plasma density and magnetic field gradients are parallel to each other in the transformation region. With the off-equatorial launch, the density and magnetic field gradients are not parallel what requires at least 2D consideration of the problem. The first attempt for such treatment has been done in [5]. More extended analytical study of the OX wave coupling in 2D geometry has been performed in our works [6,7], which are briefly summarized in the current communication.
Reduced wave equationsWe consider below the simplified model of 2D plasma inhomogeneity in which the magnetic field is directed along z-axis and both density and magnetic field gradients are transverse to this direction and are not parallel to each other). In such a situation, the parallel component of refractive index N11 with respect to the magnetic field is conserved like in the slab model. In the vicinity of transformation region an approximate dispersion relation for the 0 and X waves takes the form [4]: Arl 2 ;:1 c (£+, -! N,2 +'( ) where Stix notations 2I =1-Owe 1( (w Wce )), v 1 = 1-/ 02 are used [17]. The two cut-off surfaces, l , (x, y) = 0 and e, (x, y) = Nil, are specified by Eq.(1). In 2D model these surfaces (which may be considered as being plane in the vicinity of the transformation region) cross so that the position of the crossing point in xy plane is dependent on Nil . Let the y-axis be directed towards the "strongest evanescency" direction, the x-axis be directed transverse to y-axis in the direction of plasma density increase, the origin of xy co-ordinate system be placed in the crossing point of the cut-off surfaces. Then, propagation of a wave beam in the positive direction of x-axis corresponds to the OX mode conversion while the propagation in the opposite direction corresponds to the XO mode conversion. Let us start with the standard equation for a monochromatic electromagnetic field: rot rotE -k',£ E = 0,with ko = w / c and £ being the dielectric tensor. Stix presentation is used below in which E = (E+, E , Ell)