Received 10 February 2006, in final form 13 February 2006Published 18 May 2006Online at stacks.iop.org/PPCF/48/869
AbstractPeculiarities of the ordinary to extraordinary wave conversion in the electroncyclotron frequency range near the cut-off surfaces in magnetized plasmasare analysed in a two-dimensionally inhomogeneous simplified tokamak-likegeometry. It is demonstrated that the mode conversion may be of essentiallytwo-dimensional nature even neglecting the curvature of the cut-off surfaces;for the latter case a set of reduced wave equations in the transformation regionis derived and solved analytically. Transformation coefficients for quasi-opticalbeams with a finite transverse distribution of rf field are obtained as well as arecipe for synthesis of the optimal (perfectly converted) beam. Applicabilitylimits of a conventional one-dimensional theory are finally discussed.
1. Introduction
The problem of linear mode conversion in the electron cyclotron resonance frequency rangeis widely discussed in connection with high frequency plasma heating and diagnostics inoptimized stellarators and spherical tokamaks. Such devices are characterized by high plasmadensity confined at comparatively low magnetic field in which either incident electromagneticwaves could not propagate, or its resonant absorption is inefficient. One of the most promisingways to overcome this difficulty is based on the conversion of electromagnetic waves intoelectron Bernstein (EB) waves, which have no density cut-offs and are heavily damped in a widerange of cyclotron harmonics numbers. In large toroidal traps EB waves may be effectivelyexcited via the so-called OXB conversion process proposed in [1–3] and demonstrated in anumber of experiments, see e.g. [4–8]. In this scheme, launched from the low field side anordinary (O) wave transforms into an extraordinary (X) wave in the vicinity of the O-modecut-off surface; the slow X wave then propagates towards the upper hybrid resonance (UHR)layer where it finally transforms into an EB wave. The efficiency of this process is mainly
determined by the efficiency of the O to X mode conversion, which is very sensitive to thevalue of the parallel refractive index of the wave with respect to a static magnetic field.
Most of the analytical results in the theory for OXB conversion are obtained in a slabgeometry, in which the only spatial inhomogeneity is related to the constant direction of theplasma density gradient [2, 3, 9–12]. Although the importance of non-slab inhomogeneityfor conversion of electromagnetic waves to EB waves in the vicinity of the UHR has beenpreviously recognized [13], less attention has been paid to the investigation O–X couplingnear the cut-off surface taking into account 2D non-uniformity in a tokamak geometry (or 3Dnon-uniformity typical of stellarators). Such analysis has been initiated in [14], where it wasfound that in a two-dimensional case O–X conversion occurs in the essentially wider rangeof incident beam parameters as compared with a perpendicularly stratified one-dimensionalmodel in which effective transformation is possible only in a narrow range of the parallelrefractive index values. Moreover, all solutions found in [14] correspond to waves that aretransmitted with total mode conversion and no mode reflection, although conclusion about thetotal absence of a reflected O mode from the transformation region in the non-slab geometryseems to be quite doubtable. In the present communication we propose an alternative theoryof O–X coupling in a model 2D geometry based on physically clear simplifications of theproblem allowing for an analytical solution which includes in particular reflected waves, thusbeing free of the above-mentioned limitations of the paper [14].
The paper is organized as follows. In the next section a qualitative picture of the effectswhich cannot be described within a common one-dimensional theory of O–X conversion isgiven and the model 2D distributions of plasma density and magnetic field which might be ofinterest for tokamak applications are specified. In section 3 a reduced set of wave equations isobtained and solved for rf field intensity in the conversion region. Some practical applicationsof the obtained solution are analysed in section 4 where transformation coefficients and thestructure of an optimal (zero reflection) beam are investigated. In section 5 a transition fromthe two- to one-dimensional theory is analysed in detail, and applicability conditions of theone-dimensional modelling are pointed out. The conclusions to the paper are summarized inthe last section.
2. Two- versus one-dimensional mode conversion
The major features of the O–X conversion process may be characterized qualitatively withinthe geometrical optics approximation [1–3, 9–12]. The dispersion relation of cold magnetizedplasma [15] may be presented in the following form:
|| ; N⊥ and N|| are components of the wave refractive index perpendicularand parallel to the external magnetic field;
ε± = 1 − ω2pe
ω(ω ± ωce), ε|| = 1 − ω2
pe
ω2(2)
are the elements of the cold plasma dielectric tensor in the Stix frame [15]; ωce and ωpe arecorrespondingly, the electron cyclotron and plasma frequencies. The O–X conversion occursin the vicinity of the O mode and the slow X mode cut-offs (ε|| = 0 and ε+ = N2
Influence of 2D inhomogeneity on mode conversion in magnetized plasmas 871
Figure 1. O–X transformation in 1D geometry (plasma density and magnetic field strength arevarying along the x-axis, magnetic field is directed along the z-axis). The evanescent region isdashed.
Figure 2. O–X transformation in 2D geometry (plasma density and magnetic field strength arevarying in the xy-plane, magnetic field is directed along the z-axis). The evanescent region isdashed.
With these conditions one can obtain the approximate solution of the dispersion relation (1) ina vicinity of the transformation region [12]
N2⊥ ≈ 2ε||(ε+ − N2
||)/ε+. (4)
In a transparency region this solution describes a left polarized electromagnetic wave.According to usual notation, this wave is attributed to the O mode for ε|| > 0 and ε+ > N2
|| ,or to the X mode for ε|| < 0 and ε+ < N2
|| (correspondingly, plasma density is less or greaterthan the critical density). The region with ε||(ε+ −N2
||) < 0 is evanescent for the left polarizedwaves.
The situation typical of a one-dimensional approximation is illustrated in figure 1. Here thepropagation regions for the O and X waves with a fixed parallel refractive index are separatedby a slab evanescent region bounded by the parallel cut-off planes. Mode conversion actuallyoccurs as tunnelling of the electromagnetic radiation throughout the evanescent region; thusthe transformation efficiency is defined by its width, which for a plane wave is dependent onthe parallel refractive index N||. There is an optimal value of N|| at which both cut-off planesmerge (so that the evanescent region vanishes) corresponding to the case of full (no reflection)mode conversion.
As shown schematically in figure 2 the situation may be topologically different in 2D or3D geometry when cut-off surfaces intersect along a line. For geometro-optical rays crossingthis line, the evanescent region is absent in what formally corresponds to the total modeconversion. Opposite to the one-dimensional model, such rays exist in a continuous range of
872 E D Gospodchikov et al
Figure 3. Cut-off surfaces ε|| = 0 and ε+ = N2|| in a poloidal cross-section of a tokamak. The
regions of O–X transformation are marked by arrows.
N|| since variation of the parallel refractive index results first of all in some spatial shift of the‘transparency’ line, but not in its disappearance. Of course the geometro-optical descriptionfails in the vicinity of the transformation region where N⊥ → 0. However, as shown below fullwave treatment of the problem yields essentially the same result: for any value of the parallelrefractive index which corresponds to the crossing of the O and X mode cut-off surfaces onecan find an optimal beam that is perfectly converted (partially converted wave structures arefound as well opposite to the results of paper [14]). It should also be noted that the geometryshown in figure 2 does not posses symmetry with respect to the interchange of the surfacesε|| = 0 and ε+ = N2
|| . This is quite natural since the gradients of density and magnetic fieldstrength have finite projections on the y-axis that break the left–right symmetry. The obviousconsequence is that in the considered geometry the transformation coefficients for O to X andX to O mode conversion may differ without violation of the reciprocity principle.
In a tokamak geometry, intersection of the cut-off surfaces naturally appears due todifferent spatial distributions of the magnetic field strength and plasma density in a toroidallyconfined plasma, so that the surfaces ε|| = const and ε+ = const are not parallel as shownin figure 3. The intersection of the cut-off surfaces (figure 2) is obviously more probablethan its parallel connection (figure 1) when an incident beam is close to the optimal modeconversion (i.e. cut-off surfaces come together in any way). This means that even neglectingthe curvature of these surfaces the transformation may essentially be two-dimensional in naturewhen it occurs outside the equatorial plane, see figure 3. For the sake of simplicity, a poloidalcomponent of the tokamak magnetic field is also neglected in the present paper. So thefollowing simplified model will be considered: the direction of the magnetic field is constant,but the magnetic field strength and the plasma density vary in the plane orthogonal to thedirection of the magnetic field. In such a situation, the parallel component of the refractiveindex is conserved as in the slab model. The cut-off surfaces are assumed to be plain in thelocal vicinity of transformation region.
Let us introduce a coordinate system with the z-axis directed along the magnetic field,the x-axis placed in the propagation region, and the y-axis placed in the evanescent region asshown in figure 2. To put it more precisely, the x-axis is directed along a bisector of the anglebetween ∇ε||(x, y) and ∇ε⊥(x, y) towards the plasma density increase. The propagation of awave beam in the positive direction of the x-axis corresponds to the O to X mode conversion,while the propagation in the negative direction corresponds to the X to O mode conversion.Formally, a unit vector along the x-axis may be defined as x = −(e+ + e||)/|e+ + e|||, where
Influence of 2D inhomogeneity on mode conversion in magnetized plasmas 873
e+ = ∇ε+/|∇ε+| and e|| = ∇ε||/|∇ε|||. The origin of the coordinate system (x, y) is placed atthe intersection line of the cut-off surfaces, i.e. at the point where the curves ε||(x, y) = 0 andε+(x, y) = N2
|| cross. Note that the position of the coordinate system origin depends on somereference value of the parallel component of refractive index N||.
It should be finally stressed again that the two effects remain beyond the scope of the presentcommunication. First, we neglect the curvature of the magnetic flux surfaces assuming highlocalization of the conversion region. Second, the parallel component of refractive index isnot constant when a poloidal component of the magnetic field is taken into account (since dueto a tokamak symmetry only the toroidal component of a refractive index is conserved). Itseems that both ignored effects cannot modify the topological structures presented in figures 1and 2; however they might be of importance for quantitative analysis of the O–X conversionespecially in connection with the high sensitivity of the process to the local variation of N||value.
3. Approximate wave equation
The distribution of a monochromatic electromagnetic field in a cold magnetized plasma isgoverned by the well-known wave equation
rot rot E − k20 εE = 0, (5)
where k0 = ω/c and ε is a plasma dielectric tensor. In the Stix representation for the electricfield E± = (Ex ± iEy)/
√2, E|| = Ez:
E = (E+, E−, E||), ε = ε+ 0 0
0 ε− 00 0 ε||
. (6)
As mentioned before the parallel component of the refractive index N|| is conserved, sodependence of the electric field on the z-coordinate may be taken in the following form:
E(x, y, z) = F(x, y, N||) exp(ik0N|| z). (7)
Indeed, general distributions of the electric field may obtained by convolution of the partialsolutions in form (7) with any finite weight function over N||:
E(x, y, z) =∫
G(N||)F(x − x0(N||), y − y0(N||), N||) exp(ik0N|| z)dN||. (8)
Here x0(N||) and y0(N||) are responsible for the shift of the coordinate system origin with N||.To proceed further we assume that the radiation wavelength λ is much less than a
characteristic scale L of inhomogeneity of plasma parameters. Substituting equation (7) intothe general wave equation (4) and using the standard expansion procedure over the smallparameter λ/L one can obtain the following set of equations for the electric field componentsF+, F− and F||: √
2k0(ε+(x, y) − N2||)F+ = N||(i∂/∂x − ∂/∂y)F||,√
2k0 ε||(x, y)F|| = N||(i∂/∂x + ∂/∂y)F+,
F− = 0.
(9)
Here only the first-order terms are retained with respect to the small parameters defined by thefollowing inequalities:
1
k0N||
∣∣∣∣ 1
F+,||
∂F+,||∂x
∣∣∣∣ � 1,1
k0N||
∣∣∣∣ 1
F+,||
∂F+,||∂y
∣∣∣∣ � 1, |ε+ − N2|| | � 1, |ε||| � 1. (10)
874 E D Gospodchikov et al
The first two conditions represent slightly corrected modifications of the geometro-opticalcondition λ/L � 1 and the last two conditions are due to the vicinity of the region underconsideration to the optimal mode conversion point located at x = y = 0.
Equations (9) may be further simplified by expanding ε||(x, y) and ε+(x, y) in the vicinityof the conversion point; finally one obtains
(cos α · x ′ + sin α · y ′)A+ = − (i∂/∂x ′ − ∂/∂y ′)A||(cos α · x ′ − sin α · y ′)A|| = −(i∂/∂x ′ + ∂/∂y ′)A+
. (11)
Here new normalizations of the field components and coordinates are introduced as follows:
A+ = F+
|∇ε+|1/2, A|| = F||
|∇ε|||1/2, x ′ = x
L∇, y ′ = y
L∇,
L2∇ = N||
k0(2|∇ε+||∇ε|||)1/2. (12)
Having in mind that 1/|∇ε||| ≈ L and N2||/|∇ε+| ≈ (ωce/ω)L with L being a scale of plasma
density inhomogeneity in the vicinity of the conversion point, one can exclude N|| from thedefinition of the normalizing length: L∇ ≈ L (k0L)−1/2(ωce/2ω)1/4. Angle 2α is defined as anangle between the directions of ∇ε+ and ∇ε||, see figure 2. The terms proportional to sin α · y ′
on the left-hand sides of equations (11) are responsible for the two-dimensional nature of themode conversion. With α → 0 the set of equations (11) reduces to the standard reference waveequation of the one-dimensional theory [9, 10]; transition to this limiting case is consideredin more detail in section 5. Note that when α � λ/L the two-dimensional effects describedby (11) may be of the same order as the effects of cut-off surface curvature neglected in thepresent communication.
Let us start from the following partial solution of equations (11):(A+
A||
)partial
=(
+C1
−C1
)e−i cos α· x ′2/2−sin α· y ′2/2 + C2 ei cos α· x ′2/2+sin α· y ′2/2 (13)
with C1 and C2 being constants. Checking the polarization of the wave beam defined by thissolution, one can find that the first term corresponds to the beam propagating in the positivedirection along the x-axis, and the second term corresponds to the beam propagating in theopposite direction. Indeed, for the wave propagating in the positive and negative directionalong x, E+/E|| = ±(N⊥/
√2N||)(ε|| − N2)/(ε+ − N2) ≈ ∓√|∇ε|||/|∇ε+|; in our notation
this means that A+ = −A|| (the term ∝ C1) and correspondingly A+ = +A|| (the term ∝ C2).A general solution of wave equations may conveniently be found using the variation of
constants C1 and C2 in the form (13). Assuming that C1(x, y) and C2(x, y) are some arbitraryfunctions of x and y, and substituting (13) into wave equations (11), one obtains{
−ie−i cos α·x ′2−sin α·y ′2∂C1/∂x ′ = ∂C2/∂y
′
ie−i cos α·x ′2−sin α·y ′2∂C1/∂y
′ = ∂C2/∂x ′ . (14)
In principle, this set of equations is equivalent to (11); however the boundary conditions for thefunctions C1 and C2 are essentially simpler than for A+ and A|| because C1 and C2 correspondto wave structures propagating in the positive and negative directions of the x-axis, as in thepartial solution (13). More precisely, this separation should be formulated for projectionsof a group velocity vector on the x-axis in a region of geometrical optics where the WKBapproximation is valid, i.e. far enough from the transformation point.
Let us consider for definiteness the case of the O to X mode conversion: a beam incidentto the transformation region is launched from the low plasma density side and propagatesin the positive direction of the x-axis towards the plasma density increase. In this case, the
Influence of 2D inhomogeneity on mode conversion in magnetized plasmas 875
function C1 describes the incident O wave in the limit x ′ → −∞ and the transmitted Xwave in the limit x ′ → +∞, while function C2 describes the reflected O wave in the limitx ′ → −∞ and vanishes in the limit x ′ → +∞. Bearing in mind that the obtained referencewave equations are valid in the localized vicinity of the transformation region, the limitsx ′ → ±∞ should be understood as fitting of wave solution asymptotics to the correspondingWKB solutions outside the transformation zone. In this sense we will also consider the ‘initial’condition for the incident beam field distribution over the perpendicular coordinate, Ainc
+ (y ′)and Ainc
|| (y ′) = −Ainc+ (y ′), corresponding formally to x ′ → −∞.
Excluding C2 and separating the variables in (14), one can obtain a set of ordinarydifferential equations describing the wave propagating in the positive direction of the x-axis:
d2
dx ′2 C1x − 2ix ′ cos αd
dx ′ C1x = λ C1x, (15)
d2
dy ′2 C1y − 2y ′ sin αd
dy ′ C1y = −λ C1y. (16)
Here C1(x′, y ′) = C1x(x
′) · C1y(y′). The boundary conditions for C1x must require
the waves travelling in a positive direction of the x-axis after the transformation region:C1x(+∞) → const and dC1x(+∞)/dx → 0. This leads to the following solution for C1x [10]:
C1x = A ei cos α·x ′2/2Diν(√
2 cos α · eiπ/4x ′), (17)
where A is an arbitrary constant, ν = λ/(2 cos α), λ is the eigennumber defined by equation(16) and Diν(ξ) is one of the functions of the parabolic cylinder [16, 17]. We are looking forlocalized wave structures in y ′-direction, hence equation (16) must be solved with the followingboundary conditions: C1y exp(− sin α ·y ′2/2) → 0 with y ′ → ±∞. Note that partial solution(13) does not satisfy this condition when α < 0. It should be stressed here that equations (11),(14) and (16) possess asymmetry with respect to a sign of the angle α related to the violationof the left–right symmetry mentioned in the previous section. For α > 0, equation (16) isconverted to a standard quantum oscillator equation with solutions given by a set of Hermitepolynomials Hn(ξ) which corresponds to a discrete spectrum of eigennumbers λ:
C1y = Hn(√
sin α · y ′), λ = 2n sin α, n = 0, 1, 2, · · · . (18)
For α < 0, after a substitution C1y = exp(− sin |α| · y ′2) C1y solutions of equation (16) againmay be expressed using Hermite polynomials as
C1y = e− sin |α|·y ′2Hn(
√sin |α| · y ′), λ = 2 (n + 1) sin |α|, n = 0, 1, 2, · · · . (19)
Function C2 corresponds to the reflected wave and may be obtained by basing it on the followingrelation resulting from the first equation in (14):
C2 = −i∫ y
e−ix ′2 cos α−y ′2 sin α ∂C1
∂x ′ dy ′, (20)
where the lowest limit of integration is −∞ if α > 0 and any zero of Hn+1(√
sin |α|y ′) ifα < 0 (chosen in such a way that C2 exp(sin α · y ′2/2) → 0 with y ′ → ±∞). After somestraightforward algebra, explicit expressions for C2 may be found based on the well knownproperties of functions Diν = Diν(
√2 cos α · eiπ/4x ′) and Hn = Hn(
√sin α · y ′) [17].
Substituting the obtained solutions for C1 and C2 back into equation (13), we obtain thegeneral solution of the reference wave equation (11):
A+,|| =∞∑
n=0
Ane− sin |α|·y ′2/2
{ ±HnDiνn+
√(i tan α)/2 H ′
nDiνn−1, α > 0±HnDiνn
− √(i tan |α|)/2 Hn+1Diνn−1, α < 0,
(21)
876 E D Gospodchikov et al
where An are arbitrary constants, H ′n = 2nHn−1 is a derivative of the Hermite polynomial, and
spectral numbers νn are defined as
νn ={
n tan α, α > 0
(n + 1) tan |α|, α < 0. (22)
Terms proportional to Diνncorrespond to waves propagating in the positive direction along x,
namely to the incident and converted waves. Terms proportional to Diνn−1 correspond to thereflected waves propagating in the negative x-direction.
Solution (21) may be rearranged in the following way:
A+,|| =∞∑
n=0
Anhn(y′)Diνn
{±1 +√
(i tan α)/2 H ′nDiνn−1/(HnDiνn
), α > 0±1 − √
(i tan |α|)/2 Hn+1Diνn−1/(HnDiνn), α < 0,
(23)
where functions
hn(y′) = sin1/4 |α|
π1/4√
2nn!e− sin |α|·y ′2/2Hn(
√sin |α| · y ′) (24)
form an orthonormal basis with the scalar product∫ +∞−∞ hnhmdy ′ = δnm. This fact essentially
simplifies the calculation of coefficients An in solution (23). Indeed, from (23) one can obtainthat coefficients, An, are defined as projections of the ‘C1 field component’ to the basis functionshn:
An = 1
D0iνn
∫ ∞
−∞
A+(x′0, y
′) − A||(x ′0, y
′)2
hn(y′) dy ′, (25)
where D0iνn
= Diνn(√
2 cos α · eiπ/4x ′0) and x ′
0 is an arbitrary point, An is independent of x ′0.
This formula may be further simplified when x ′0 → −∞ in the sense of fitting to a proper
WKB solution, i.e. when all Diνncorrespond to a beam incident to the transformation region:
An = 1
D0iνn
∫ ∞
−∞Ainc
+ (y ′) hn(y′) dy ′ (26)
where Ainc+ (y ′) is the given field distribution in the incident beam in the WKB region.
Relations (8), (12), (23) and (26) provide the complete solution for the waves tunnellingthrough and reflected from the evanescent region. This solution has been extensively checkedwith the Mathematica® software. The case of X to O mode conversion when the incidentbeam propagates in the negative direction of the x-axis (i.e. in the direction of plasma densitydecrease) may be studied in the same way. Moreover, from symmetry properties of equations(11) it follows that changing of propagation direction along the x-axis is equivalent to changein a sign of the angle α and a direction of the y-axis: O to X conversion with α ≷ 0 is thesame as X to O conversion with α ≶ 0. In the next sections we demonstrate some practicalapplications of the obtained solution.
4. Transformation coefficients and the optimal beam structure
In this section we consider transformation coefficients for an intensity of the wave beam afterpassing though the conversion region. This allows neglecting the phase relations between theincident and transmitted wave structures, that significantly simplifies calculations.
First of all, we show that solution (23) is consistent with the energy conservation law. Letus introduce the ‘partial’ transformation (Tn) and reflection (Rn) coefficients corresponding to
Influence of 2D inhomogeneity on mode conversion in magnetized plasmas 877
a separate nth term in the sum (23) over the basis functions. These coefficients can be obtainedfrom the following asymptotics of parabolic cylinder functions [17]
Diν(z) = ziνe−z2/4 +
√2π σ
�(−iν)z−iν−1eπν+z2/4 + O(|z|−2) when |z| → ∞, (27)
where σ depends on arg z, in particular, when z = √2 cos αeiπ/4x ′ one may find that σ = 1 if
x ′ < 0 and σ = 0 if x ′ > 0. The first term in (27) is leading when x ′ → ±∞, therefore thetransformation coefficient can be determined as
Tn =∣∣∣∣Diνn
(+∞)
Diνn(−∞)
∣∣∣∣2 = |(−1)iνn |2 = exp(−2πνn), (28)
with νn given by (22). However, the reflection coefficient is defined by the second term of theasymptotics (27), which becomes dominating in the derivative of Diν over x ′. Actually, from(21) one might see that
Rn =∣∣∣∣Diνn−1(−∞)
Diνn(−∞)
∣∣∣∣2[
tan |α|2
∫ +∞−∞e− sin |α|·y ′2
/2{H ′2n or H 2
n+1}(√
sin |α|y ′)dy ′∫ +∞−∞ e− sin |α|·y ′2
/2H 2n (
√sin |α| · y ′)dy ′
]. (29)
The term in square brackets appears due to the different y-structures of the incident and reflectedbeams; H ′2
n and H 2n+1 correspond to positive and negative signs of the angle α. It may be shown
that this term is equal to νn. Taking into account the asymptotics (27) one gets
Thus, the energy flux defined by solution (23) is conserved. Note that the partial transformationand reflection coefficients are independent of N||.
Due to the orthogonality of the basis functions (24), there is no interference betweendifferent terms in the sum (23) when calculating the transformed wave intensity—eachnth term is converted independently with the transformation coefficient (28). Therefore,the transformation coefficient TN|| corresponding to a wave structure with a fixed N|| andan arbitrary field distribution over y ′, can be obtained as a weighted sum over the partialtransformation coefficients
TN|| = 1
P0
∞∑n=0
∣∣D0iνn
An
∣∣2 exp(−2πνn), P0 =∞∑
n=0
|D0iνn
An|2 =∫ +∞
−∞|Ainc
+ (y ′)|2dy ′ (31)
where coefficients An are defined by (26) and P0 is a normalizing coefficient proportional tothe beam intensity. Note that An are introduced in such a way that P0 is independent of theangle α and the parallel refractive index N||. The transformation coefficient does not dependexplicitly on N||; however the origin of the y ′-coordinate depends on N|| therefore it must betaken into account in the summation over N||.
Let us now consider an incident wave beam with the arbitrary distribution over both thetransverse coordinates, Einc(y ′, z), corresponding to x ′ → −∞. Exploiting the orthogonalityof the basis functions hn(y
′) and Fourier harmonics over z, the transformation coefficient maybe obtained as a weighted sum of the partial transformation coefficients corresponding to fixedn and N||:
T =1
P0
∞∑n=0
exp(−2πνn)
∫ ∞
−∞
∣∣∣∣∫ ∞
−∞
∫ ∞
−∞e−ik0N||zhn(y
′ − y ′0(N||))Einc(y ′, z)dy ′dz
∣∣∣∣2 dN||. (32)
878 E D Gospodchikov et al
Here the weight function is represented by the internal integrals over y ′ and z that project theinitial field distribution to the basis functions; P0 is a norm calculated as
P0 =∞∑
n=0
∫ ∞
−∞
∣∣∣∣∫ ∞
−∞
∫ ∞
−∞e−ik0N||zhn(y
′ − y ′0(N||))Einc(y ′, z)dy ′dz
∣∣∣∣2 dN||
= 1
2π
∫ ∞
−∞
∫ ∞
−∞|Einc(y ′, z)|2dy ′dz, (33)
and y ′0(N||) = −(N|| − N0
||)N0||/(L∇|∇ε+| sin α). We recall that displacement y ′
0(N||) inequations (32) and (33) reflects the N|| dependence of the crossing point position of thecurves ε||(x, y) = 0 and ε+(x, y) = N2
|| . Note that the similar shift x ′0(N||) in the direction
of beam propagation results in an additional phase shift that does not affect integrationin (32) and (33).
Several comments are made concerning the physical meaning of equations (28)–(32).First of all, ν0 = 0 when α > 0 for O to X conversion (incident beam propagation to theplasma density increase) or α < 0 for X to O conversion (propagation to the plasma densitydecrease). This means that for every N|| there is a field distribution corresponding to theperfect mode conversion. The spatial structure of such optimal field distribution with a fixedN|| should be proportional to the first basis function h0(y
′), in this case TN|| = T0 = 1.Returning to the dimensional coordinates the optimal field distribution may be presented as aGaussian beam
where L∇ is defined by equation (12). The absence of a unique value for the parallel refractiveindex corresponding to the total conversion is an essentially two-dimensional effect foundin [14] and discussed qualitatively in section 2. Note however, that solutions with strictly zeroreflection are absent when ν0 �= 0, i.e. α > 0 for X to O conversion or α < 0 for O to Xconversion. The reflected wave also appears when the structure of the beam is not ‘optimal’,e.g. the n > 0 terms are present in the solution (23).
Based on equation (34) and taking into account displacement y ′0(N||) one can construct
an optimal distribution corresponding to the perfectly transformed (with zero reflection) beamwith a finite spectrum over N||:
Einc(y, z) =∫ ∞
−∞G(N||) exp
− 1
2L2opt
(y − N|| − N0
|||∇ε+| sin α
N0||
)2
+ iN||k0z
dN||. (35)
Note that within our approximations, the width of the optimal beam Lopt ≈L (k0L)−1/2(ωce/2ω)1/4 sin−1 |α| does not depend on N||. Taking as an example a Gaussianweight function over N||, G ∝ exp[−(N|| − N0
||)2k2
0L2||/2], one obtains that the optimal incident
beam may be formed outside the transformation region by an astigmatic phase-modulatedGaussian distribution
Einc(y, z) ∝ exp(−y2/2L2y − z2/2L2
z + iyz/2L2yz + iN0
||k0z) (36)
with parameters defined as
L2y = ηL2
opt, L2z = ηL2
||, L2yz = sign α · η|∇ε+|k0L
2||L
2∇ /N0
|| , (37)
with η = 1 + (N0||)
2(|∇ε+|k0L||L∇)−2 sin−1 |α|.Considering that changing of the propagation direction along x is equivalent to change of
sign α, one can obtain the following relation between the partial coefficients for O to X and X
Influence of 2D inhomogeneity on mode conversion in magnetized plasmas 879
to O conversion (for propagation to and from a dense plasma):
T OXn / T XO
n = exp(2π tan α). (38)
Since this relation is not dependent on n and N||, it remains true for the generalizedtransformation coefficients (31) and (32). The transformation coefficient must be not greaterthan unity; therefore, from the latter equation one can make sure once more that depending onsign α the perfect conversion may only be realized either in positive, or in negative directionalong x-axis.
5. Transition to one-dimensional case
From geometrical considerations it follows that the analysed two-dimensional model mustreduce to the one-dimensional slab model in the limiting case of α → 0 with all otherparameters being constant. From equation (38) one might suggest that the necessary conditionfor the slab approximation is α � 1 (then T OX ≈ T XO). However, as shown below α � 1is not a sufficient condition. One might identify the non-trivial nature of the transition to theone-dimensional case from the structure of the transformation coefficients (31) and (32), inwhich all partial transformation coefficients tend to be equal, Tn → 1 and An → 0 withα → 0, resulting in a contribution of a large number of terms in the total transformationcoefficient.
Let us consider an incident wave beam with a fixed parallel component of refractiveindex, N||, and an arbitrary field distribution over the y-coordinate, A(y ′) = Ainc
+ (y ′). Thetransformation coefficient (31) may be expressed as
TN|| = 1
P0
∞∑n=0
Tn
∫ +∞
−∞A(y1)h(y1)dy1
∫ +∞
−∞A∗(y2)h (y2) dy2
= 1
P0
∫ +∞
−∞
∫ +∞
−∞A (y1) A∗ (y2) W (y1, y2) dy1dy2, (39)
where
W (y1, y2) =∞∑
n=0
Tnhn (y1) hn (y2) =√
sin |α|π
exp
(− sin |α|y
21 + y2
2
2− 2πν0
)
×∞∑
n=0
1
n!
(exp(−2π tan |α|)
2
)n
Hn
(√sin |α| · y1
)Hn
(√sin |α| · y2
). (40)
One can perform summation over n in the latter expression using the Mehler’s formula (seee.g. (22) in chapter 10.13 of [17]):
W (y1, y2) =
=√
sin α
π [1 − exp(−4π tan α)]exp
[(2y1y2 − (y2
1 + y22 ) cosh(2π tan |α|)) sin |α|
2 sinh(2π tan |α|)
]. (41)
It should be noted that this expression is valid for any sign of the angle α and W(α) =W(−α) e2π tan α in accordance with equation (38). Equations (39) and (41) provide analternative form of the two-dimensional transformation coefficient, which is in some cases
880 E D Gospodchikov et al
more convenient than the direct sum (31) over the contributions of the basis functions1. Inpractical applications one should have in mind that y1 and y2 are dimensionless variablesnormalized with the scale L∇ .
With the forms (39) and (41) one performs a transition to a particular one-dimensionalcase in which the parallel component of the refractive index is equal to its optimal value,N|| = ω
1/2ce (ω + ωce)
−1/2 [9,10]. The fixed value of N|| in the α → 0 limiting case is imposedby the geometry, in which the cut-off surfaces are always crossing. The norm P0 does notdepend on α; therefore it remains the same in one- and two-dimensional cases. ExpandingW(y1, y2) over the small parameter α, one obtains (α � 1):
TN|| =1
2π P0
∫ +∞
−∞
∫ +∞
−∞A (y1) A∗ (y2) exp
− 1
4π(y1 − y2)
2 + α2∑
i,j=1,2
κij yiyj
dy1dy2, (42)
with κij being constants of the order of unity. The first term in the square brackets correspondsto the essentially one-dimensional result for the incident wave beam with the optimal parallelcomponent of refractive index and an arbitrary field distribution over the y-coordinate. Indeed,the partial transformation coefficient corresponding to a fixed perpendicular component Ny
of the refractive index is T1D, Ny= exp(−π k2
0L2∇N2
y ) [9, 10, 12]. The total transformationcoefficient in the one-dimensional case may be rearranged using the same technique as thatused in (39) and shown as
T1D = 1
P0
∫ +∞
−∞e−π k2
0L2∇N2
y
∣∣∣∣∫ +∞
−∞A(y ′)e−iNyk0L∇y ′
dy ′∣∣∣∣2 dNyk0L∇
2π
= 1
2π P0
∫ +∞
−∞
∫ +∞
−∞A(y1)A
∗(y2)
∫ +∞
−∞e−π k2
0L2∇N2
y −iNyk0L∇ (y1−y2)d(Nyk0L∇) dy1dy2
= 1
2π P0
∫ +∞
−∞
∫ +∞
−∞A(y1)A
∗(y2) exp
[− 1
4π(y1 − y2)
2
]dy1dy2. (43)
The second term in the square brackets in (42) corresponds to the two-dimensional correctionsto the one-dimensional result. Introducing a characteristic width of the beam in the y-direction, Ly , one may formulate a condition when the second term can be neglected asα2L2
y/L2∇ � 1. Thus, we come to the following applicability condition of the one-dimensional
approximation:
|α| � min[1, L∇/Ly
] = min[1,(L/Ly
)(k0L)−1/2 (ωce/2ω)1/4
]. (44)
The one-dimensional model is not applicable for sufficiently wide beams with Ly � L∇/|α|.Within the one-dimensional approach it may be shown that this condition is equivalent to the
1 The same idea may be used in calculating the sum over n directly in the solution (23) at least for the wavespropagating in the positive x-direction. In this case, equation (23) can be converted to the following mappingA+,||(x1, y1) = ± ∫ +∞
−∞ W (x1, y1, x2, y2) A+,||(x2, y2)dy2 in which the kernel has a quite simple form provided thatboth x1 and x2 lie in the WKB region.
W =√
sin α
π [1 − exp(−2π tan α)]exp
[i (x2
2 − x21 ) cos α
2+
(2y1y2 − (y2
1 + y22 ) cosh(π tan |α|)) sin |α|
2 sinh(π tan |α|)
]
with π = i log(x2/x1) (π = π when x2 = −x1 and x1 < 0). Such mapping defines the coupling between the incidentand transformed waves which may be used in ‘jumping’ over O–X conversion (non-WKB) region in ray or beamtracing calculations. This provides a bridge between the proposed theory and applications. Further development ofthis idea, which came to the authors after a referee’s report, will be published elsewhere.
Influence of 2D inhomogeneity on mode conversion in magnetized plasmas 881
Figure 4. Transformation coefficient for the Gaussian beam Einc ∝ exp(−y2/2L2y + ik0N||Z) as a
function of Ly/L∇ for different angles α > 0 between the cut-off planes: the solid lines correspondto the results of the 2D model, the dashed line corresponds to the result of the 1D model.
non-negligible variation of the one-dimensional transformation coefficient within the beamaperture.
To illustrate the role of two-dimensional effects, in figure 4 we plot the transformationcoefficient for an arbitrary (not necessarily optimal) Gaussian beam in the y-direction,A ∝ exp(−y2/2L2
y). This coefficient depends on the ratio ζ = Ly/L∇ and the angle α.An exact expression for the transformation coefficient may be obtained from (39) and (41) asfollows:
TN|| =√
sin α
1 − exp(−4π tan α)
2ζ[1 + 2ζ 2 coth(2π tan |α|) sin |α| + ζ 4 sin2 |α|]1/2 . (45)
In the vicinity of an optimal beam width ζ ≈ sin−1/2 |α| the transformation coefficient of theGaussian beam is well approximated by the first few terms in sum (31):
TN|| ≈ 2 ζ sin1/2 |α|1 + ζ 2 sin |α|
(e−2πν0 +
1
2
(1 − ζ 2 sin |α|1 + ζ 2 sin |α|
)2
e−2πν2 + · · ·)
. (46)
In the one-dimensional limit (44) the transformation coefficient is TN|| ≈ ζ/√
ζ 2 + π
(corresponding to the optimal value of N||). One might see from figure 4 that in the vicinityof the optimal beam width the two-dimensional transformation coefficient is basically greaterthan its one-dimensional analogue (shown by a dashed line). Note that for the α = 5◦ casethe condition α < min[1, L∇/Ly] holds in the whole displayed domain, nevertheless thereis a noticeable difference between the one- and two-dimensional coefficients (about 20% inthe vicinity of the optimal beam width). This indicates the relative character of the inequality(44), which should be ‘strong’ in order to guarantee the applicability of the one-dimensionaltheory.
With all the above mentioned remarks one may find that for toroidal fusion devices thevalidity of the one-dimensional model is not well justified. For example, in a conventionaltokamak with circular cross-sections of flux surfaces and a large aspect ratio, the angle α andthe critical beam width L∇/|α| may be estimated as
sin |α| ≈ a sin |ϑ |(1 + ωce/ω)R
, L∇/|α| ≈(ωce
2ω
)1/4 (1 + ωce/ω)R√ak0 sin |ϑ | , (47)
882 E D Gospodchikov et al
where R is a major radius, a is a minor radius of the cut-off surface, and ϑ is a poloidalcoordinate (counted from the equatorial plane) of the O–X conversion zone. Taking typicalparameters such as R = 100 cm, a = 30 cm, ω = ωce = 2π · 140 GHz, one gets the criticalbeam width L∇/|α| ≈ 6 cm/ sin |ϑ |, which may be of the order of a typical rf beam widthprovided that the transformation region is far displaced from the equatorial plane (sin |ϑ | ∼ 1).The applicability of the one-dimensional theory in strongly shaped configurations, such asoptimized stellarators and spherical tokamaks, requires more detailed investigations (to bepublished elsewhere).
6. Summary
In this paper a theory of O–X transformation is developed taking into account the variation of amagnetic field strength on a flux surface within the two-dimensional model of magnetic field (B)and plasma density (ne) distributions in a tokamak. The set of approximate wave equations fordistribution of an electromagnetic field in a two-dimensionally inhomogeneous transformationregion is obtained and analysed for the particular case of ∇B⊥B and ∇ne⊥B. A full analyticalsolution including both transmitted and reflected waves is formulated in a convenient form as adiscrete sum over terms related to a specifically full orthogonal basis representing the electricfield in a wave beam. Based on this solution the efficiency of the O–X transformation is foundfor an arbitrary field distribution over the beam aperture. It is demonstrated that even neglectingthe curvature of the flux surfaces, the transformation may essentially be two-dimensional innature due to the finite angle between the surfaces of a constant magnetic field intensity and aconstant plasma density. The most pronounced features of the two-dimensional transformationare: (1) the existence of the ‘optimal’ beam exhibiting perfect transformation instead of the‘optimal’ plane wave specific for the one-dimensional geometry, and (2) the asymmetry intransformation efficiencies from O to X mode and backward. For a particular case in thetokamak geometry, the latter result may be reformulated in an alternative form: regions wheretotal transformation is possible for the similar beams propagating into and from the denseplasma are separated spatially (above and below the equatorial plane in figure 3). In principle,this statement may be checked experimentally by comparing BXO emission from the tworegions mentioned.
Acknowledgments
This work has been supported by Russian Foundation for Basic Researches (Grant 06-02-17081), Dutch–Russian cooperation program (NWO–RFBR Grant 047.016.016) and RussianScience Support Foundation.
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