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nowadays the electron bernstein waves ( ebws ) having no density cut - offs and effectively damped even at high electron cyclotron harmonics are considered as the most promising candidate to provide an auxiliary heating and current drive in a dense plasma of a spherical tokamaks and stellarators  @xcite . the ebw could be excited via direct conversion of x mode to bernstein mode in a vicinity of the upper hybrid resonance ( uhr ) or via so - called o - x - b scheme . the efficiency of last scheme , as it was demonstrated theoretically  @xcite in the frame of 1d slab model , is determined by the efficiency of o to x mode conversion , which can reach 100 percent value at the certain parallel refractive index being constant in slab geometry . due to , in a real spherical tokamak s configurations , where the poloidal inhomogeneity of the magnetic field is important , both the parallel refractive index is no longer constant and the components of the dielectric tensor are functions of two co - ordinates , an analysis of the full - wave equations in the frame of 2d model is important . the first attempt to consider 2d model of ox conversion has been undertaken a couple of years ago  @xcite . the main conclusion provided by author of  @xcite concerning the absence of the o mode reflection from the o - mode cut - off surface seems to be quite doubtful and lacks support from the last two papers devoted to this topic  @xcite . unfortunately , the ox mode conversion in  @xcite has been considered in the frame of the oversimplified model ignoring as it does both the poloidal magnetic field and , as result , varying of the parallel refractive index on the magnetic surface . these effects were taken into account in  @xcite , where the the explicit expressions for the ox and xo conversion coefficients have been obtained simultaneously with  @xcite but for @xmath0 ( @xmath1 and @xmath2 are the poloidal and toroidal components of the magnetic field and @xmath3 is an angle between the toroidal direction and the magnetic field ) .    unlike the assumption used in  @xcite , @xmath4 , in spherical tokamak s plasma @xmath3 is no longer small as it is demonstrated in figure 1 for a typical mast tokamak discharge . therefore , straightforward using the results of  @xcite , without mentioning paper  @xcite , to describe ox conversion in spherical tokamaks seems to be overhasty . because of the intensive efforts are paid to provide an auxiliary heating in spherical tokamaks using oxb scheme , calculation of the ox conversion coefficient in the realistic 2d model is of great interest . in this paper a reduced set of the partial differential wave equations valid in a vicinity of the o - mode cutoff and accounting for the magnetic field 2d inhomogeneity with no restriction to an angle between the toroidal direction and the magnetic field are derived . a solution to the reduced set of equations has been obtained and it properties are considered in details . there are three effects that remain beyond the scope of the present paper . we neglect , first , the curvature of the magnetic field line at the magnetic surface because of its local radius @xmath5 is considerably greater than the beam radius @xmath6 , second , the curvature of the magnetic flux surfaces assuming high localization of the conversion region , third , the magnetic field shear which is not important for ox conversion  @xcite . we restrict ourselves to the case of not extremely strong plasma density inhomogeneity @xmath7 when geometrical optics can be applied except possibly near cut - off layer or possibly mode conversion layer . one introduces two cartesian co - ordinate systems @xmath8 and @xmath9 with distances scaled in the units @xmath10 and their origin located at the o - mode cut - off surface ( figure 2 ) . the co - ordinates @xmath11 , @xmath12 and @xmath13 imitate the flux surface label , the poloidal and the toroidal co - ordinates , respectively , and axes @xmath14 and @xmath15 are along the magnetic field and perpendicular to it on the o - mode cut - off surface , respectively ( figure 3 ) . the transformation from @xmath12 , @xmath13 components to @xmath16 , @xmath17 components convenient to represent the maxwellian equations is @xmath18 one introduces the set of the wave equations for a monochromatic wave @xmath19 as @xmath20 where @xmath21 , @xmath22 @xmath23 @xmath24 @xmath25 , @xmath26 and @xmath27 are electron langmuir and cyclotron frequencies , respectively . since a domain in a vicinity of o - mode cut off surface beyond the tokamak mid - plane is of interest , one can expand the plasma parameters , namely density @xmath28 and magnetic field modulus @xmath29 , into the taylor serious at @xmath30 @xmath31 where @xmath32 are parameters being the first order quantity ( @xmath33 ) and @xmath34 is one being the second order quantity ( @xmath35 ) . in order to study the properties of the waves in the mode conversion region we keep in mind that the component @xmath36 is small compared to two others @xmath37 there . omitting in ( 2 ) the terms being higher order quantity than the first one obtains @xmath38 due to tokamak symmetry we may assume that the wave fields vary as @xmath39 , where @xmath40 being constant is large . we would like to construct a solution @xmath41 of the system ( 6 ) . to this end we look to develop the integral representation for required solutions of laplace integral type @xmath42 where @xmath43 is assumed analytic in some domain and the path of integration in the @xmath44 plane is such that the integrand vanishes rapidly at the ends of the contour @xmath45 or at infinity . next    @xmath46    where @xmath47 and @xmath48 we can not easily deal with the system ( 8) as it stands after transformation , but recalling the conclusions of 1d theory  @xcite that the effective conversion is possible if both @xmath49 @xmath33 and @xmath50 @xmath51 we may reduce it to a simpler form . expanding @xmath52 around @xmath53 and @xmath54 around @xmath55 we have found that @xmath56 and @xmath57 . keeping in ( 8) terms being the first order quantity one has @xmath58 where the explicit expressions for @xmath59 and @xmath60 could be found by varying ( 9 ) . making the backward transformation with respect to @xmath61 yields @xmath62 where the electric field s components @xmath63 obey the system of equations    @xmath64    introducing new notations @xmath65 @xmath66 we read ( 13 ) in the compact form @xmath67    in order to make the first step in analysis of the electric field components behavior in the conversion region , we consider in the next section wkb solution to the system ( 14 ) , namely we focus on the quality analysis of the ray trajectory along which the energy of the incident beam of the ordinary ( or extraordinary ) waves is carried over . although in the mode conversion region the wkb approximation breaks down and a local full wave equations ( 14 ) to be solved , the ray trajectory analysis could be useful . the ray representation of the wave field in four dimensional ray phase space @xmath68 @xmath69 is governed by hamilton s equations : @xmath70 where @xmath71 is the local dispersion relation and @xmath72 denotes the orbit parameter . since the general picture of the ray behavior in the two - dimensional subspace @xmath73 is of interest , let us re - parameterize @xmath74 , where @xmath75 . substituting ( 16 ) into ( 15 ) we obtain @xmath76 where @xmath77 . the equations constituting ( 17 ) are coupled with @xmath78 being the coupled coefficient . one can introduce the normal co - ordinates @xmath79 reducing the system of equations ( 17 ) to two independent ones . the set of linear independent solutions for either of the two is @xmath80 the expressions ( 18 ) , which show the projections of hamiltonian rays on the ( @xmath81 ) plane , deserves a few comments . first , note the oscillatory behavior along the @xmath82 direction , while the @xmath83-motion displays the influence of a retarding force . second , due to @xmath84 oscillations along the @xmath82 direction are not to be clear - cut . third , we can anticipate the full wave solution to the set of equations ( 14 ) could be represented as the superposition of the eigenmodes intrinsic to the confining potential ( @xmath85 ) in the @xmath82 direction leading to oscillatory @xmath82 motion and the superposition of the linear independent parabolic cylinder functions along @xmath83 direction , which proper for the effective potential @xmath86 . the full - wave solution to the set of equations ( 14 ) is studied in the next section . we would like to obtain the appropriate particular solution of the system ( 14 ) matching to the wkb solution outside the mode conversion region @xmath87 where * e * is the polarization vector , @xmath88 is a set of eigen functions to be found , @xmath89 and @xmath90 correspond to incident o mode ( ox conversion ) and incident x mode ( xo conversion ) . with that end in view we introduce two possible functional substitutions  @xcite providing the identity of the equations constituting ( 14 ) @xmath91 with @xmath92 and @xmath93 being required functions that satisfy the equations @xmath94 let us use the transformation @xmath95 where @xmath96 @xmath97 and @xmath98 , that makes the potential in the brackets in ( 22 ) symmetrical @xmath99 if @xmath100 ( see figure 4 ) then @xmath101 . seeking the solutions to ( 24 ) in the form @xmath102 where the definition for @xmath103 is given by @xmath104 and @xmath105 are hermitian polynomials , we obtain the equation for @xmath106 and @xmath107 @xmath108 where @xmath109 are important for subsequent analysis we list theirs below expressing its argument @xmath110 explicitly @xmath111    the solutions @xmath106 and @xmath107 to ( 27 ) matching to the wkb solutions ( 19 ) for the incident ordinary and extraordinary waves ( see appendix a ) are @xmath112 where @xmath113 are an arbitrary constants , which we choose so that ( 20 ) and ( 21 ) fit the incident wkb wave outside the conversion layer . as it is demonstrated in appendix a @xmath114 and @xmath115 correspond to the incident ordinary and extraordinary waves , respectively . the parameter @xmath116 has meaning the length of an evanescence layer ( see the equation ( a.3 ) ) . its value depends on the mode s number @xmath117 and combination of the parameter s @xmath110 sign and the direction of the process ( see ( 28 ) ) . that is , first , either of the mode tunnels through an evanescence layer with the efficiency inherent itself , second , the conversion efficiency being in 1d model the invariant under the conversion s direction reversal and the poloidal position of the incident beam changing has no longer the same property in 2d inhomogeneous plasma . being mentioned for the first time in  @xcite for the oversimplified plasma model ignoring the poloidal magnetic field and confirmed in  @xcite for the reasonable plasma model accounting for the poloidal magnetic field @xmath118 this property depends entirely on the essential two - dimensional character of the waves .    inserting ( 29 ) and ( 30 ) into ( 20 ) , ( 21 ) and ( 11 ) after simple algebra we obtain an integral representation of solution to ( 6 ) @xmath119 where upper signs at @xmath120 correspond to the incident ordinary wave and lower signs correspond to the incident extraordinary wave . the explicit expressions for @xmath121 and @xmath122 are @xmath123 and @xmath124    while outside the mode conversion region at @xmath125 @xmath126 and @xmath127 match to the geometrical optics solutions that correspond to the wkb incident and conversed waves ( compare the first terms in the r.h.s . of ( a.1 ) and ( a.2 ) , ( a.6 ) and ( a.7 ) ) , the wkb asymptotics of @xmath128 and @xmath129 correspond to the wkb wave reflected from the evanescence layer ( see ( a.5 ) ) . it is possible to obtain asymptotic expansion of the solution for @xmath130 . for the incident ordinary wave at @xmath131 we have @xmath132 last expressions recover the expressions for the electric field components derived in  @xcite for plasma with @xmath118 . for usual spherical tokamak s configuration o mode cut - off layer and the x mode cut - off layer cross at the small angle , i.e. @xmath133 . keeping only terms in ( 23 ) being zero and first order quantity with respect to the parameter @xmath78 we read it as @xmath134 then we simplify @xmath135 where @xmath136 and @xmath12 is scaled in the units @xmath137 . comparing the asymptotic representation of @xmath138 for the argument @xmath139 and asymptotic representation of @xmath138 for the argument @xmath140 ( see appendix a ) we obtain an explicit expressions for the conversion coefficient , @xmath141 , ( in energy ) and reflection coefficient , @xmath142 , where the subscribes @xmath143 and @xmath144 correspond to the direction of the process @xmath145 and @xmath146 the coefficient @xmath147 is given by the expression @xmath148 where @xmath149 means the complex conjugation , @xmath150 is given field distribution in the incident beam in the wkb region , @xmath11 and @xmath13 are scaled in @xmath151 and @xmath152 is a normalizing coefficient proportional to the beam intensity . the expressions ( 38 ) and ( 39 ) derived similar to ones obtained in  @xcite excepting the formula for @xmath153 given by ( 40 ) and deserve one comment . noting @xmath154 at the fixed @xmath78 , we could read the reciprocal relation for the conversion coefficients in 2d inhomogeneous plasma as @xmath155 where the argument @xmath156 is presented explicitly . for the first time the property ( 41 ) was mentioned in  @xcite . in @xcite it was proved for plasma with poloidal magnetic field @xmath157 .    for illustration , in figure 5 the conversion coefficient , @xmath158 , of an incident beam with transversal distribution @xmath159 calculated with the use of 1d model  @xcite ( dashed curve ) and the formula ( 38 ) ( solid curves ) for @xmath160 and different @xmath3 versus the dimensionless radius , @xmath161 , is shown . calculations are executed under conditions of ebw heating experiment in mast @xmath162 , @xmath163 , @xmath164 , @xmath165 . rather strong variation of dependence presented in figure 5 with variation of @xmath3 is observed . the mode conversion of o mode to x mode have been examined in a spherical tokamak geometry for plasma with a cold plasma dielectic tensor . we have assumed the wavelengths are much shorter than the equilibrium plasma gradient length , @xmath7 . thus , we expect wkb approximation to apply except possibly near o - mode cut - off surface or mode conversion layer . expanding wave equations in the region near the intersection of o and x mode cut - off surfaces , one finds reduced wave equations appropriate for this region which depends on two coordinates , @xmath11 and the poloidal angle @xmath12 . we seek the required solution to the set of equations so that it matches to the wkb solution outside the mode conversion region . after functional substitution we have found the integral representations of the required solution .    to the case typical of a spherical tokamak s configuration for which the o mode cut - off layer and the x mode cut - off layer cross at the small angle we have been simplified the obtained integral representations of solution and found the wave fields and the conversion coefficients explicitly . we have shown that at the fixed poloidal position , i.e. fixed @xmath78 , the conversion coefficients obey the reciprocial relation ( 41 ) as it was shown earlier in the frame of the model neglecting the poloidal magnetic field  @xcite and of the model in which the poloidal magnetic field is assumed to be small , @xmath118 .    the importance of 2d effects has been demonstrated by the example under the mast conditions . the work was supported by rfbr grants 04 - 02 - 16404 , 06 - 02 - 17212 we pay our attention to the functions of the parabolic cylinder with different arguments and indices @xmath166 and @xmath167 . let us begin from asymptotic expressions for the first one at two limits @xmath125 which describe the corresponding wkb solutions beyond the conversion layer . when the argument @xmath83 tends to @xmath168  @xcite @xmath169 while at @xmath170 @xmath171 the second term in ( a.2 ) is much smaller ( with factor @xmath172 ) compared to the first one . therefore the function @xmath166 in the wkb sense describes the wave approaching an evanescence layer from @xmath173 and conversed wave propagating from an evanescence layer to @xmath174 . comparing the coefficients at @xmath175 in ( a.1 ) and ( a.2 ) one obtains the conversion coefficient ( in energy ) for the wave described by the parabolic cylinder equation @xmath176 farther we obtain the derivative of the parabolic cylinder function @xmath177 the first term in the r.h.s . of ( a.4 ) , which has the asymptotics  @xcite @xmath178 describes the reflected wave .    comparing the asymptotic expansions for the parabolic cylinder function @xmath167  @xcite @xmath179 and @xmath180 we note that it describes the wave which incident on an evanescent layer from @xmath181 ( see the first term in ( a.5 ) ) and propagates from an evanescent layer to the plasma boundary ( @xmath170 ) ( see ( a.6 ) ) . 10 laqua h p , erckmann v , hartfuss h j and w - 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in this paper a reduced set of the partial differential wave equations valid in the conversion layer close to o - mode cutoff surface and accounting for the magnetic field 2d inhomogeneity with no restriction to an angle between the toroidal direction and the magnetic field direction is derived . 
 an integral representation of a solution to the derived set of equation is given . 
 for the particular case of small angle between o mode cut - off surface and x mode cut - off surface an explicit expressions for both the electric field components and the conversion coefficients are obtained and its properties are considered in details .