PFC/JA-84-25
ON THE THEORY OF PAIRWISE COUPLING EMBEDDED IN
MORE GENERAL LOCAL DISPERSION RELATIONS* by
A. Bers and L. Harten
* This work was supported in part by DOE Contract DE-AC02-78ET-51013 and in part by NSF Grant ECS-82-13430.

ON THE THEORY OF PAIRWISE COUPLING EMBEDDED IN MORE
GENERAL LOCAL DISPERSION RELATIONS
V. FUCHS
Projet Tokamak, Institut de Recherche d'Hydro-Oudbec,
Varennes, Qu6bec, Canada JOL 2PO
A. BERS and L. HARTEN
Plasma Fusion Center, Massachusetts Institute of Technology,
Cambridge, Massachusetts 02139, U.S.A.
Earlier work on the mode conversion theory by Fuchs, Ko and Bers is detailed and expanded upon, and its relation to energy conservation is discussed. Given a local dispersion relation D(w; k,z) = 0, describing stable waves excited at an externally imposed frequency w, a pairwise mode-coupling event embedded therein is extracted by expanding D(k,z) around a contour k
= kc(z) given by aD/ak = 0. The branch points of
D(k,z) = 0 are the turning points of a second-order differential-equation representation. In obtaining the fraction of mode-converted energy, the connection formula and conservation of energy must be used together. Also, proper attention must be given to distinguish cases for which the coupling disappears or persists upon confluence of the branches, a property which is shown to depend on the forward (vgvph>O) or backward (vgvph<O) nature of the waves. Examples occurring in ion-cyclotron and lower-hybrid heating are presented, illustrating the use of the theory.

2
I. INTRODUCTION
A magnetized homogeneous plasma can support a multitude of wave modes which, in a linear approximation,can be described by a dispersion relation
D (w,k) = 0, w and k being respectively the generally complex frequency and wave number. Any two modes, say 1 and 2, may linearly couple and energy will be exchanged between them, if the linear resonance conditions wl=w
2 kl=k2 are satisfied.1 In a spatially nonuniform plasma the wave number varies at least in one direction and, if a frequency wo is impressed by an external source of electromagnetic radiation (as is the case in radiofrequency heating experiments), then under the conditions of the eikonal approximation, the dispersion relation will respect the (weak) nonuniformity in a local sense
D(wpk,z) = 0. (1)
The frequency condition is now satisfied for all the branches so that the coupling of modes occurs at branch points of the mapping k = k(z) obtained from the dispersion relation. Since k is generally complex, the spatial variable z is understood to be generally complex also, remembering that the branches k(z) of the dispersion relation recover their original physical meaning on the real axis x = Re z. We recognize the fundamental difference between the spatially uniform and nonuniform situations in that coupling is exceptional in the former whereas it occurs by necessity in the latter, as is clear from the fact that a mapping with two branches must possess at least one branch point. For plasma heating, the physical implications of this fact was explored by Stix
2 and thereafter many workers
3 have analyzed

3 the transfer of enerqy from one mode to another with emphasis on the conversion of incident radio-frequency (RF) eneroy to plasma modes that are susceptible to kinetic damping processes.
With regard to the analysis of one particular mode-coupling event, it is then natural to ask whether the relevant information can be extracted from a more complicated dispersion relation describing more than one coupling.
The more general problem then.concerns the analytic continuation of the incoming branch through all branch points accessible to it, thus ultimately solving the full dispersion relation problem with respect to a particular
"incoming" boundary condition. Needless to say, if decomposition of the mode-coupling problem were possible in the sense stated above, it would greatly simplify the analysis, effectively reducing-it to a series of second-order problems. This problem of second-order coupling embedded in higher-order systems was addressed in a previous study (hereafter referred to as FKB) and more recently, from a different point of view, by Cairns and
Lashmore-Daviess (hereafter referred to as CLD). In their work CLD also discuss the issue of energy conservation in the formalism, and raise doubts as to whether the FKB embedded second-order differential equation will necessarily conserve energy in the coupling process. In other words, the question is whether the FKB formalism yields the correct transport coefficients for transmission, reflection and mode conversion in a particular coupling event.
The issue is obviously a serious one and we undertake, therefore, in the present work to show how the transmission/mode-conversion coefficients are

4 determined within the framework of the FKB theory. It is useful in this context to outline the basic differences between the two approaches, for this will shed light on the difficulties CLD encountered when trying to apply the FKB theory to their model dispersion relation. In the formalism
CLD represent their second-order dispersion relation by a system of two coupled first-order differential equations. This immediately limits the scope of couplings that are tractable within the formalism to a class for which a confluence of the branches results in their decoupling and, consequently, complete transmission of the incident mode. Furthermore, CLD chose a form of the coupled-mode equations which ab initio satisfies energy flow conservation. Applying their results to cases of conservative
(i.e. dissipation-free) couplings they find, of course, that the transmission/mode-conversion coefficients satisfy energy-flow conservation. The FKB formalism for its part, is not restricted to conservative couplings, so that if one wishes to apply the formalism in a loss-free situation one must bring in the equation for energy flow conservation in an appropriate way. Furthermore, FKB is not restricted to the class of coupling events for which transmission is complete at confluence of the branches (or branch points). In fact, the FKB formalism is more naturally suited for the description of coupling events in which the incident energy, at confluence of the branch points (and in a loss-free situation), will split evenly between the transmitted and mode-converted (or reflected) branches. For this particular class of couplings the connection formula is immediately identifiable with energy flow conservation, whereas in the first case one obtains a different connection formula.
We will show that the correct connection formula together with energy flow conservation

5 always yields the appropriate transmission/mode-conversion coefficients.
Since these two classes of mode-coupling problems lead to different connection formulae and consequently, also, different transmission/modeconversion coefficients, it is obviously important to be able to determine the class to which any particular coupling event belongs. This will be shown to depend on the forward (Vg vph>) or backward (vg vph<O) nature of the incident, transmitted and mode-converted (or reflected) waves.
The paper is organized as follows. In Section II we review the FKB embedded equations. Boundary conditions for coupled modes are discussed in
Section III while in Section IV we describe connection formulae and the transmission/mode-conversion coefficients. Finally, Section V summarizes our results. Each of the three Appendices treat a particular coupling problem, illustrating the use of our theory for both types of connection formulae.

6
II. EMBEDDED EQUATIONS AND COLPLING POTENTIAL
In order to estimate the energy flow through various branches of a local dispersion relation, use can be made of the system of embedded second-order differential equations linked by the analytic continuation of the incident branch through the accessible branch points, as described in Ref. 4. We will begin by briefly recalling the concept of an embedded equation and the coupling potential, and discus.sing its limitations.
First it must be possible to select branch and saddle points of the mapping
(1), equivalently written as k = k(z), (2) that correspond to a particular pairwise coupling event. The branch points zb and saddle points k. satisfy the simultaneous system of equations
D(k,z) = 0, 3D/ak = 0, subject to the condition of pairwise coupling, which is aD/ak
2
* 0 at
(3)
(ks,zb)- We emphasize that the conditions (3) are not completely sufficient since, by definition, a saddle point of the mapping (2) is that for which dk/dz diverges. We must therefore verify that simultaneously with (3) BD/3z * 0 also is satisfied at (ks,Zb). In order to identify a certain group of critical points zb and ks with a particular coupling event, we take the position that two coupled branches are characterized by one of the branches k = kc(z) of the mapping
3D/ak = Dk = 0. (4)

7
By definition, then, the corresponding branch points are given by
D[kc(z), z] = 0. More generally, if D(k, z) = 0 possesses n branches, then Dk = 0 has n-1 branches, termed "mode boundaries".
4
The term is appropriate because the branches of the dispersion relation have no common points with k
= kc(z) other than the branch and saddle points themselves. Hence two coupled branches lie on either side of a line k = kc(z), as illustrated in Fig. 1. It follows that the embedded dispersion relation describing the pairwise coupling can be obtained by expanding the full dispersion relation around the line k = kc(z), which gives
D(kc) + 1 (k kc)
2
Dkk
subject to the condition Dkk(kc) * 0 in the coupling region. As
(5) z + zb, also kc + ks and since, in addition, Dkk(ks) # 0 for pairwise coupling, we see that Eq. (5) is satisfied very well in the vicinity of the branch points. Outside the coupling region we have propagating waves incident, transmitted and mode-converted for example so that the coupling region is typically a barrier between two branch points. The validity condition thus reduces to demanding that Dkk(kc) not vanish between the turning points.
If the condition of validity is not satisfied, at least two more terms in the Taylor development must be added, amounting to the admission that other branches (such as the reflected branch of the incident mode) interfere with the pairwise coupling process and that a simple second-order theory based on regular turning points would be inadequate. A different point of view

8 enforces the above ramification of violating the validity condition.
Noting that Dkk(kc) = 0 and Dk(kc) = 0 together define the branch points of the mapping k
= kc(z), it is then clear that in the vicinity of points where Dkk(kc) = 0 there must be more than one line k = kc(z) and, consequently, more than two branches of the dispersion relation since each line k
= kc(z) is associated with exactly two of these. However, it is not necessarily true that at least a fourth-order equation is required appropriate Budden equation when the validity condition in question is violated.
Supposing now that the embedding (5) is valid, how can it be used to determine the power flow in the coupled branches or, equivalently, what is the form of the associated second-order differential equation? We simply require that the turning points of the differential equation be the branch points of the dispersion relation (5). Since D[kc(zb)] = 0, it is natural to take an equation of the form
Y'' + Q (z)Y = 0, where the coupling potential Q is
0 =
-2D/Dkkkc(z).
(6)
(7)
The function 0 defined as
= Y exp (-ifkcdz) (8)

9 obviously satisfies
D(-i d dz z) =0, and, since Q(zb) = 0, we indeed have the desired representation. Note
(9) that Eq. (6) with the potential Q is immediately obtained from (5) upon rotating the axes, k' = k-kc, with the subsequent usual identification
+ -id/dx. The transformation k' = k-kc obviously amounts to going from the axes k and x of Fig. 1 to x' = kc(x) and k' = k-kc, and it is in these that Eq. (6) is formulated. An important consequence of the transformation is that the phase velocity of a branch lying below kc will change sign in the transformation if the branch was originally in the upper half-plane. Hence in Fig. 1, the phase velocities on branches F and H will change sign, but will remain the same on E and G. This is important for the imposition of correct boundary conditions for Eq. (6), a subject dealt with in detail in the next section.

10
III. BOUNDARY CONDITIONS FOR COUPLED MODES
A mode-coupling event is understood here to be a process in which energy can be transferred from one branch of the local dispersion relation to another in such a way that, outside the coupling region, only three distinct waves (e.g. the incident and transmitted waves associated with the incoming mode, and the mode-converted wave associated with the other mode) at the most can be identified. This definition encompasses not only the phenomenon of mode conversion between physically truly distinct modes, such as fast Alfvdn to ion-cyclotron, extra-ordinary to electron-Bernstein or thermal-ion to ion-Bernstein, but also coupling of branches of the same physical mode, such as specular reflection of electromagnetic radiation from a barrier.
Any coupling between two modes can then be described by a group of modecoupling events thus defined. For example, if we can identify four waves in the asymptotic region, an incident, transmitted, reflected and modeconverted, then, as is demonstrated in Appendices B and C, we are typically dealing with two coupling events involving an incident, transmitted and mode-converted wave, plus reflection at a cutoff.
Henceforth, we limit our attention to the problem of energy flow in an elementary coupling event involving three waves at the most [incident, transmitted and mode-converted (or reflected)], an event that can be described by a second-order differential equation of the type (6).
In the asymptotic regions, far away from the branch points (centered for

11 convenience around x = 0), the uncoupled, propagating waves can be described by the WKB solutions 7
Y
1
,
2 x
-i
= Q exp (±i f Q dx), r where r is some reference point. On x we will write
Y+ = GY
1
+ + HY
2
+ and we assume that, upon tracing to x<0 the solution Y+ becomes
Y_ = EY
1
.+ FY
2
--
(10)
(11)
(12)
In Eqs. (11) and (12) the subscripts ± refer to the sign of the spatial variable, so that Y
2
-, for example, signifies the wave Q exp(-i f
Qidx) in the region x<0.
Ultimately we would like to express the transmission/mode-conversion coefficients in terms of the amplitudes E, F, G and H but first the boundary conditions to be imposed on the WKB waves must be formulated.
This amounts to being able to identify the WKB waves Y
1
± and Y
2
± with the incident, transmitted and mode-converted (or reflected) waves as given
by the local dispersion relation. The asymptotic (i.e. uncoupled) waves of the local dispersion relation are specified by their asymptotic wave numbers k, the group velocities vg = 3< /k, and the phase velocities vph = w/k. Since the waves are excited at an externally imposed (and stable) frequency, k alone suffices to specify the phase velocity and we shall agree that k and vph have the same sign. We recall that the phase velocities are to be specified in the transformed reference system k',x' of
Fig. 1. As far as the WKB waves are concerned, all we know at the outset

12 is that the waves Yj and Y
2 have oppositely oriented phase velocities.
And, of course, the WKB waves carry no information at all about group velocities. Any preferred orientation vph of either Yj or Y
2 is a matter of convention, depending on the time factor exp(±it). If we agree to take exp(-iwt), then the phase velocity of a WKB wave is defined as vph =-
Wa/t
,-i
Wa/ax x t f -dx r
(13)
Hence vph = t- -
-~ ,
QI k where k = ±Q results from the local dispersion relation k
2
= Q of
(14)
Eq. (6). The WKB wave Yj thus propagates to the right, and the wave Y
2 to the left (see also Heading's rule, p. 75 of Ref. 7).
Let us now fix the boundary condition for the incident wave by requiring an incoming mode from the left. What we mean, of course, is that its group velocity is oriented toward the right, since that determines the orientation of energy flow in the wave. If the wave is forward, i.e. if vgvph >0, then we identify it with EYi. and, if it is backward, i.e.
(vgvph<0), then FY
2
- is the appropriate branch. Since the transmitted and incident waves must have identically oriented group velocities, it follows that GY
1
+ will represent a transmitted forward wave and HY
2
+ a transmitted backward wave. Two examples will serve to clarify this.

13
Consider first the electromagnetic local dispersion relation
C2k2 = 2 _ .2(x) =Q(x) c p
2
, where we take a parabolic plasma density profile, i.e.
= W2 (-x p po
(15)
2
/a
2
), to form a barrier causing some reflection of the incident energy. Since
vg vph
= c2 holds for the dispersion relation (15), this case can be therefore taken to represent a coupling where all three waves are forward.
The dispersion relation (15) describes two modes w = ±ck propagating in free space (up = 0) independently of each other. A plasma couples these two branches, as shown in Fig. 2, causing reflection for frequencies w that fall into and are close to the stop band between -wp and wp. The forward nature of all the branches is evident from the plot since
k aw/ak >0 everywhere except at k = 0. The branches k = k(x) for fixed w are portrayed in Fig. 3. In incidence from the left, the transmitted wave is represented by the branch G (i.e. k>0) on x>0, in accordance with the results of the preceding discussion.
As a second example, we take the class of mode-conversion problems considered by Cairns and Lashmore-Davies .
They discuss the dispersion relation
(u-wd) (u -w2)
= which describes the coupling of two modes w = un,
2
(x,k) in a region around the point x = xm, where the frequencies and wave numbers match,
(16) i.e. where wl(xmpkm)
=
2(xmgkm)
= m (17)

14
An incident mode, say w
1
, will then couple on account of a non-zero %m to
W = W2- Let us assume, together with Cairns and Lashmore-Davies5, that the functions w, and W2 can be expanded to first order in k and x around
W , whereupon Eq. (16) becomes
( Q-a 6-b () ( Q-f -g E) =
,(18) with
wM, 6 = k-km, X-XM (19)
Away from the coupling region situated around
E=0, the coupling constant no can be neglected, and the two modes are approximately (Fig. 4) a + bE Q, 0 f 6 + gC 2 (20)
It is thus clear that near resonance (Q=0) each mode has oppositely oriented phase velocities on either side (i.e. >0 and '<0) of the matching point. It follows that, in contrast to the preceding example, the incident and transmitted waves cannot both be forward or backward.
With this in mind, we now consider the local dispersion relation (18) at perfect resonance (a 0). The resulting branches k1,2
( -
2 a f r af
(21) kc = km b+
2 a f
), are shown in Figs. 5a and 5b for the cases, respectively, of antiparallel
(af< 0) and parallel (af >0) asymptotic group velocities (a and f). In the rotated frame x' = kc(x), k' = k-kc, the barriers are reduced to the

15 standard type of Fig. 3, and correspond to the coupling potential Q
O(E) =
2b
2
(- )
4 a f
+ af
(22)
(
The important difference now is that, if the incident wave is forward, then the transmitted one is backward, represented by the branch H (i.e. k'< 0) rather than G.
We conclude this section by pointing out that any coupling event between two modes can be represented by one of the three cases depicted respectively in Figs. 3, 5a and 5b. There are a number of distinctions between these cases but only one is fundamental in nature. There is a difference between the two cases of Fig. 5 with respect to the mutual orientation of qroup velocities, which leads to different mode-converted branches in the two cases, but the basic distinction lies in the dissimilarity between the type of coupling in Fig. 3 and that of Fig. 5, characterized by different transmitted branches. This, as will be seen in the next section, results in different connection formulae and transmission/mode-conversion coefficients in the two cases.

16
IV. ENERGY CONSERVATION AND CONNECTION FORMULAE
A valuable property of the WKB solutions (10) of Eq. (6) is that the quadratic quantity Im[Y
1
*
2
,
2
/dx)] is conserved when 0 is real. It is this fact that allows an interpretation of the coefficients E, F, G, H entering into the solutions (11) and (12) in terms of energy flows. More specifically, it is easily shown that Eq. (6) is equivalent to d Im(y* dy) dx dx
-YY* Im Q,
(23) so that Im Q = 0 implies the existence of a first integral Im(Y*Y') const. Upon substitution of solutions (11) and (12) into this equation, we obtain
E12 _ , 2 1,G 2 - JH 2
24)
For the case of conservative coupling, the incident energy flow, I, must eventually appear in the asymptotic regions as transmitted, T, and mode-converted, M, (or reflected, R) energy flows,
I = T + M (or R)
(25)
The relationships between the quantities that enter into the energy flow conservation equation (25) and the moduli that enter into the connection formula (24) are in general not self-evident. We will consider each of the couplings discussed in the preceding section separately. The three possibilities are summarized in Fig. 6, with E being the incident, and a forward, wave. Identifying the energy flow on branch E as E
1 2
, we then have I = E1 2
.
Since the transmitted wave is the same type of mode as the

17 incident one, its energy flow is likewise given by the modulus of its amplitude, i.e. T = 1G1 2 in case i), and T = H 1 2 in cases ii) and iii).
In case i), where all three waves are forward, Eq. (24) becomes
= F 2 + G12 (26) and upon comparison with (25), it then follows that M (or R)
= jF
2
.
In case i), therefore, all the moduli are readily identifiable with energy flows.
When not all three waves are forward (or backward), as in cases ii) and iii) of Fiq. 6, the energy flow in the mode-converted wave can no longer be identified with the modulus of its amplitude. In case ii), Eq. (24) gives
I = F
2
H 2 (27) which we write in the form
I = F1
2 21H 2 + T, (28) yielding upon comparison with (25), the mode-conversion coefficient
M = F 1 2 21H
2
(29)
We proceed similarly as in case iii) of Fig. 6 to obtain
M = G 2 21H 1
2
.
(30)
All that remains to be done is to express the coefficients E, F, G, H in explicit form, and to demonstrate that the expressions (29) and (30) for M do, in fact, equal I-T. We note therefore that Eq. (6), for a real potential Q(x) with two turning points x, and x
2 and anti-Stokes lines emanating outward along the real axis, can be approximated by the model

(parabolic cylinder or Weber) equation d&2
+ ( - a)Y = 0,
4 where & = (X-Xm) /2h, and
2
4
(x
1
-x
2
) 2 xm = k(xl + X2), a = h(x,
x
2
)
2
8
(31)
(32)
The asymptotic solutions of Eq. (31) can be written using Darwin's expansions9 where
Y+ =/ q
(Aq cosp + B sin*) evr
Y.
= /3
(A sin* q a
(33a)
(33b)
(34)
=
+ vi + 6,
4 and
1 C ( -
4 a cosh-
2/a
,
1 ;
4
2-4a)
a sinh-
1
2
7
C
7a
,
.
(35)
The exact form of vi and vr is not relevant to the analysis. We may therefore write
Y + (Aq + B/i)eil*+ (Aq + B/i)e-i *
Y- (A/i + Bq)ei* + (-A/i
+ Bq )e-i*
(36a)
(36b)
18

19
Identifying these solutions with expressions (11) and (12) immediately gives
E = -A/i + Bq , F = A/i + Bq
G = Aq + B/i ,
= Aq B/i
(37a)
(37b)
An additional relation between the coefficients A and B arises upon applying the appropriate boundary condition. Let us successively consider the three cases of interest in Fig. 6. i) H = 0 i.e. B = iqA
E = Ai(1+q
2
), F = -Ai(1-q
2
), G = 2Aq, so that
T = G'E
2 exp (-2 ra)
1 + exp(-2ra)
M (or R) = IF/El2 ii
2
=
1
1 + exp (-2ina)
(38)
(39a)
(39b) ii) G = 0 i.e. B = -iqA, 5>0
E = Ai(1-q
2
), F = -Ai(1+q
2
), H = 2Aq, so that
T =
IH/E1 2
= exp(-2wa), and from Eq. (29)
M = IF/E 2
- 2 1H/E1 2
= 1 - exp(-2 na)
(40)
(41a)
(41b)
I

20 iii) F = 0 i.e. A_= -iqB, a<0
E = 2qB, G -Bi(1+q
2
), H = Bi(1-q
2
), so that
T = IH/E 2 exp(2ira), and from (30)
M = IG/E1
2
21 H/E 12 = 1 - exp(2ra).
The two cases ii) and iii) can be united by noting that
T = exp (-2T aI ).
(42)
(43a)
(43b)
(44)
In particular, for the coupling potential (22), corresponding to the problem studied by Cairns and Lashmore-Daviess, we obtain exactly the same result:
T =
2 r
bfj
.
It is of practical interest to note that the above results can be obtained from the properly traced WKB solutions (11) and (12). We find that the exponential factors in the transmission/mode-conversion coefficients are thereby replaced by exp (-i
X2 fQ dx) ,
X1
(46) where the integration is performed along a Stokes line between the relevant turning points 8,10 (if, for example, Q = x
4
-a
4
, the relevant turning points are x
1
,
2
= ±a).

Examples covering all three types of boundary conditions are given in appendix.
21

22
V._ CONCLUSION
We will conclude with some remarks of a more general nature on our method and its relation to other mode-coupling theories. The simplification we strive for in the present analysis of wave energy flow or wave transmission properties, in a plasma described by a many-valued dispersion relation, is based on decomposition of the dispersion relation into elementary pairwise coupling events, combined with the process of analytic continuation of the incoming branch through the accessible branch points.4 The next step is the solution of the energy flow problem for a single, embedded, pairwise coupling event.
The method is best illustrated by the example of Appendix B, where we examine the model fourth-order mode-conversion-with-tunneling equation, studied previously in detail by Laplace transform techniques 11 9 as well as by Cairns and Lashmore-Davies
5 within their second-order formalism. The example is valuable in demonstrating, as CLD did also, that a second-order theory can adequately deal with a situation where tunneling occurs, even when the incoming mode encounters the cutoff-resonance combination from the low-field side, i.e. the side from which the cutoff is accessible. While it is true that a single second-order equation cannot cope with such a situation unless it is a Budden equation (the modeconverted wave being implicitly represented by the divergent branch), analysis of the coupling diagram, Fig. 9, shows that in incidence from the low-field side the coupling between the two modes can be represented by three separate coupling events: two between the fast and the ion-Bernstein

23 waves, and one between oppositely propagating ion-Bernstein waves a reflection at the cutoff. The resulting composite mode-conversion/ reflection coefficients are identical to those obtained from the Laplacetransform treatment of the fourth-order equation directly.
The analysis of Appendix B thus shows unequivocally that a cutoff-resonance combination is not inseparable, requiring by necessity a Budden-equation treatment, or a fourth-order treatment when the resonance is resolved by thermal effects. We wish to stress, however, that the essential feature of both second-order theories in question is that no information contained in the full dispersion relation is lost in the process of extraction of the second-order coupling event, which is really why the methods yield the correct transmission properties.
The fundamental issue in the formulation of second-order theories, and one in which the two in question differ, is the ability to extract an elementary coupling event, and to subsequently formulate an appropriate differential-equation representation for analysis of the energy flow. In the CLD theory, the extraction is performed by identifying each of the two coupled modes in its uncoupled form (i.e. asymptotic form far away from the coupling region), and factoring the dispersion relation in such a way that it will decompose into these modes when an appropriate coupling parameter is allowed to vanish.

If the factorization is directly identifiable with the CLD model equations
(obtained by expanding the dispersion relation around the mode resonance, or "crossing", point), the method immediately yields the transmission coefficient. Otherwise, the individual factors have to be expanded to first order in k and x around the uncoupled-mode crossing point to obtain the model representation.
In FKB, without first examining the nature of the two coupled modes, the
"embedded" dispersion relation is extracted by expanding the dispersion relation around the natural branch cut of the two modes. A second-order differential equation having the correct turning points and appropriate boundary conditions is then directly attributed to it. This does not restrict the class of couplings that can be treated by FKB to modes that decouple upon confluence of the branch points (the class of couplinos considered by CLD). Furthermore, in contrast to CLD, where energy flow conservation is built into the form of the model coupled-mode equations, the
FKB equation has to be supplemented, in a loss-free situation, by the energy flow conservation law.
The transmission/mode-conversion coefficients can then be obtained from the asymptotic (or WKB) solution of the FKB second-order equation. The transmitted energy is always given by the amplitude of the transmitted WKB wave but the converted fraction also depends on the type of coupling. In the first class of couplings [case (i)], of modes that do not decouple upon confluence of the branch points, the connection formula is equivalent to

25 energy flow conservation, and the converted fraction is given by the amplitude of the corresponding WKB wave. In the second class of couplings
[cases (ii) and (iii)], the WKB solutions are not directly related to the converted mode and in obtaining the converted fraction energy the connection formula and energy conservation law must be used together.
To sum up, we examined the transmission in a loss-free plasma of an incoming mode which can couple, and thereby transfer energy, to other plasma modes. The formalism discussed herein permits the extraction and analysis of energy flow in one elementary coupling event at a time, thus dispensing with the necessity of solving the full higher-order problem.
The generalization of the method to a situation with loss, such that not all of the incident energy ends up on other branches, runs into complications because of the absence, in general, of a first integral the connection formula. We will address this problem in a separate work.

ACKNOWLEDGMENTS
We would like to thank Dr. T.W. Johnston for valuable discussions, and Dr.
C.N. Lashmore-Davies and Dr. R.A. Cairns for stimulating comments on our
DOE Contract DE-AC02-78ET-51013 and NSF Grant ECS-82-13430.
26

27
APPENDIX A LOWER-HYBRID ION HEATING IN TOKAMAKS
This is a case which cannot be treated by the CLD method and for which the
Laplace transform method would be very difficult to apply.
We start from the two-dimensional (xiii, zJ.) kinetic electrostatic dispersion relation 14, 15
.
k2 + k2 + x z a=e, i v 2 ta a
(Hak)
=
0, where we define v 2 ta
2k T
$ a m a k
2 v 2 x ta , V2
2w2 a ca k
2
V 2 z ta
2w2 ca
(Al)
(A2)
In the lower-hybrid range w <<<<W , and for
Xe << 1, ve<< 1, i>>1, vi <<1, the electronic and ionic contributions Ha can be approximated by
1 5
(A3),
H vte e te (k2 ce
-(
Z
21
A4)
H. 2 + 2c whr
[Z(;) - iir e~C + v./2
(-n v/2
, (A5) where c =a/k 2 ti' n i - n, Z is the plasma dispersion function, and the integer n is the order of the ion-cyclotron harmonic. When the plasma is not very hot, i.e. when C >>1, we may simplify using the asymptotic expansion

28
Z
(
(1 + -- + where
Y
0 , Im ;>0
1 , Im;=0
2 , Im ;<0.
ia e
The dispersion relation (Al) thereby becomes
_3
2 pi + 1
4
W 2
2
1 + pe ce2
2
2 i
+ n.2 v2 z ti
2
Pe)
+2;e
2
W2
-2
7 L
(1) i /r + v/27r
Z v V2
=0.
(A6)
(A7)
(A8)
At the plasma edge the incoming mode thus satisfies k x
2
=
k
2
2
K z Ki
(A9) and, as the wave propagates inward, k increases and we have to retain the warm-plasma correction = 1/;4 giving rise to another branch, the warm-ion mode. Additional terms of the asymptotic expansion contribute only imaginary roots, i.e. roots that are immediately damped out. The absorption of incoming energy by the plasma is due to ion-Landau damping of the warm-ion branch and to cyclotron damping of the ion-Bernstein branch associated with the cyclotron term in Eq. (A). Since the ion-Bernstein branch is energized by coupling with the warm-ion mode, the absorption

29 process is ultimately controlled by mode conversion of the incoming lower-hybrid branch to the warm-ion branch. The effect of the ion-cyclotron harmonics on the lower-hybrid and warm-ion branches is illustrated in Fig. 7, where comparison is made with the straight ion-orbit approximation (i.e. unmagnetized ions). We see that the principal features of the configuration, including the position of the mode-conversion points, are not determined by the cyclotron harmonics. Since ion-Landau damping also has little effect on the real parts of the roots, we may assume for now that the -mode-conversion process itself is adequately described by EQ.
(A8) without the ion-Landau and cyclotron terms.
We therefore condider the dispersion relation (k = k) k 2f(x)k
2
+ b
2 = 0, where
(Alo) f(x)
3
W2_ vt a n(x)
[1 a n(x)(1
y
2
)]
Al1) b m.
.
2 k
2
2
3 me Vt2 z , a
-
-
2 -_
, y
2-
2
__
W eci and n(x) 1-x
2
/a 2
(A12) is the normalized ion density profile with a the tokamak minor radius. For a linear density profile, centered around the resonance point K
1
= 0, Eq.
(AlO) can be transformed to the standard Stix form
2 k - 2xk
2
+ b
2 = 0.
(A13)

30
In this case, complete mode conversion is predicted.
2
For a parabolic profile the situation is qualitatively different, since the symmetrical configuration introduces a barrier between the mode-conversion points on each side of the profile, makinq transmission of the lower-hybrid mode possible. Roughly speaking, the mode-conversion efficiency is proportional to the barrier width 16 17 but we will see that owing to a large parameter wa/vti in the problem, there is a threshold for mode conversion in terms of a minimum required
10
(i.e. density), k, and toroidal magnetic field. We now proceed with the analysis. First, for the Stix problem (A13) we have k 112
= X ± (x
4
, (A14) indicating the absence of propagating branches in the region x <0. Hence an incoming wave is completely turned around (i.e. mode-converted) at the branch point xb = /b. Reflection in the form of its own oppositely propagating branch is not possible since the wave number k = 0 corresponding to this process is not a solution of the dispersion relation
(A13). In the more general parabolic case (A10), we have k212 = f(x) ± [f
2
(x) b2] indicating regions of propagation only where f(x)>0, i.e. where
1 a n(x)(1 y
2
) = Kj>0.
(A15)
(A16)
Since from (A12) the normalized peak density is n(O) = 1, we see that in principle, when KjO>0, propagation is possible everywhere, with two real turning (branch) points at f(x b) = b. (A17)

31
If, on the other hand, KjO<0, there are two points xr for which f(x)< 0 when xr < x < x r) will become clear later on in the analysis. For now, let us assume that there is no resonant layer in the plasma, i.e. K
1 0
>0. The branch point condition
(A17) can be written as
(1 + Vz y
2
)a n(x) z
= 1, v (k v/w)(6mi/me) z z ti so that the turning points are
(A18) xb1,2
a1 + vt y
2) .
(A19)
The wave number k n Wuc is determined by the wave guide (n is usually z z z somewhere between 2 and 10 for accessibility and good plasma wave guide matching).
Next, we determine
Dk and Dkk
Dk
=
4k (k
2
f) , Dkk =
12k 4f. (A20)
The condition Dk =
0 gives k 0, k
2 = f. (A21)
The root kC = 0 does not satisfy the dispersion relation and hence specular reflection is absent. We thus conclude that, unless they are both launched from the outside, only one of the two oppositely propagating branches of a mode can be present. The other solution k
2
C
=
2 -C k
2 =
f characterizes the

mode coupling. When substituted into the dispersion relation, k
2
C
= f gives
D(k ) = b
2
_
2
0, (A22) the condition for branch points. Finally, we find
Dkk (k 8f(x). (A23)
32
Under the given assumption K
1 0
>0 implying f(x)>0 everywhere, we can thus use the pairwise embedding (5). If K
1 o<0 however, there exist two points xr such that f(xr)
=
0 situated between the turning points.
In this case, then, the embedding approximation cannot be used so that from now on we will use the assumption KjI>0.
The differential equation corresponding to the embedding (A5) is d
2
+ 0
(A24)
Factoring out the relevant turning points gives the potential in the form
1 ~2
12 v
2 ti a2
[f(x) + b] (1+v y
2
) n(x) f(x) (x-xb) (xxb2(2
Following the prodedure outlined by the transformations (32), we translate to the median xm (Xb + xb)
0 obtaining,
(A26)
2
12 v a
([f(0) + b (1+vz- n(0) f(0)
2 a(1+v Y2) a(1+v- Y
2
) a (A27)

33
At this point, to determine the transmission and mode-conversion coefficients, we have to know the appropriate boundary conditions. It is known that the lower-hybrid mode is backward and that the warm-ion mode is forward, as indicated in Fig. 8. With respect to the k contour, however, all three waves, the incident, transmitted and mode-converted, are backward, which is equivalent (insofar as the connection formula is concerned) to having all three waves forward. Boundary conditions of the type i) of Section IV thus apply.
The transmission and mode-conversion coefficient is therefore e-2nOM
1 + e-2i
1
1f + e-2 na where
[(1+vz
2 )(1
a + av z+ W 2) a(1 + V z 2A
48
(1
+ 2) a( + V- Y
2
)
(A28)
Within the confines of the given assumption KLo>O, there are two parameter ranges where the transmission/reflection coefficients exhibit critical behavior. The first occurs in the limit of KjO=0. Since by necessity
$ is positive and diverges when Kjo = 0, mode conversion is complete, in agreement with Fidone and Paris 18, whose full-wave treatment considers the case of KjLO = 0. The second occurs around a(1+v
2
) = 1, which is where 8 changes sign, and where the branch points merge to change from real to imaginary. We easily ascertain that around a(1+v -y
2
) = 1 we can write
/ 1,6
24 vti
[a(1+v -y z
2
) -
1]. (A30)

34
If the parameter wa/vti is very large, as is the case in present-day lower-hybrid heating experiments (wa/vti 104), then the energy transport coefficients (A28) will behave as step functions of the variable a(1+v
2
). Ion heating will appear to have a threshold, suddenly disappearing when the ion density and/or nz and/or toroidal magnetic field fall below values given by a(1+vz- y
2
) 1.
In the light of this result, the effect of the real part of the cyclotron term in Eq. (AB) should be specifically examined. If the mode-conversion points are close to the plasma center, the nearest ion-cyclotron harmonic might critically influence the transmission properties.

35
APPENDIX B ION-CYCLOTRON AND ION-ION HYBRID MODE CONVERSION
This, our principal example, shows that care must be exercised in the selection of embedding for a given configuration, and that the appropriate tool for coupling-pattern recoanition is Rek plotted versus x, rather than the more standard k2 versus x.
Let us examine the dispersion relation k4 X 2 xk
2
+ X
2 x
a
-
0 (81) which typifies the warm-plasma version of a back-to-back cutoff-resonance
(Budden) situation and can thus serve as a model for a number of modeconversion problems. When a = 2/3, for example, the dispersion relation describes the conversion of a fast Alfv6n wave to an ion-Bernstein wave at the second ion-cyclotron harmonic (w = 2w a coupling considered in detail by Ngan and Swanson l who employ Laplace transform/asymptotic expansion techniques, and to whose work we refer the reader for details on the meaning of the various parameters.
When a is negative and large, a << -1, the dispersion relation describes ion-ion hybrid heating13, a case also discussed in Refs. 11 and 12. The second-order "crossing-point" approximation of the dispersion relation (81) was discussed by Cairns and Lashmore-Davies
5
.
We proceed as follows. The solutions of Eq. (B1) are k212
1 x2x ±
(1
4x
12-2 4
_
2 x + a) (82)

36
An obvious pair of branch points is, therefore, x b
= 2
X2
[1 ± (1-a)
with corresponding saddle points k21 (1-a) .
(83)
(B4)
All branch points must satisfy the condition
Dk 2k(2k
2
_ X2x) -= 0, which has the two roots k1 ci
0, k2 c2 2
X2X.
(B5)
(B6)
Substituted into the dispersion relation, the root kc2 gives the branch and saddle points (B3) and (B4) respectively, and k ci 0 corresponds to a branch point (cutoff) at x = a/X
2
, (B7) always situated to the left of the branch points (83).
The branches of the dispersion relation for both cases of interest are shown in Figs. 9 and 10. The two k
2 versus x plots look alike and do not reveal much difference between the two cases except for showing that in the ion-ion hybrid case it is the fast rather than the slow mode which can experience a cutoff. More relevant, as far as wave propagation is concerned, are the actual wave numbers k. Shown are the plots of Re k versus x, but with Im k omitted, since this information is not needed in the present analysis. We thus see that the second-harmonic case exhibits a

37 prominent barrier configuration formed by the pair of branch points (83), while in the ion-ion hybrid case a tunneling region is formed between the cutoff x
0 and the origin x
=
0. In both cases the arrows indicate incidence from the low-field side. In incidence from the high-field side, the cutoff is inaccessible, and the coupling is easier to analyze. The orientation of the group and phase velocities is in accordance with the forward/backward nature of the modes: the fast wave is known to be forward, whereas the ion-Bernstein branch is backward.
Let us begin with the second-harmonic case. The couplingdiagram Re k versus x immediately suggests the following transmission /mode-conversion properties. In incidence of a fast wave from the left, if on the first coupling event (on Barrier 1) the fraction T of incident energy is transmitted, then 1-T is mode-converted, propagates toward the cutoff and is totally reflected. The energy 1-T incident in the second coupling event
(Barrier 2) is in the form of the slow mode, and therefore its transmitted fraction T(1-T) is also in the slow mode, while its mode-converted fraction
(1-T)
2 is in the fast mode. The mode-conversion coefficient of the whole configuration is therefore M = T(1-T) and the reflection coefficient is
R = (1-T)
2
.
In incidence of a fast wave from the right, no energy can propagate toward the cutoff, so that simply M =
1-T, R = 0. All we need to determine now is the transmission coefficient T.
Proceeding according to Section IV, we first note that the transmitted wave in the frame k - k on Barrier 1 is backward while the other two waves c are forward. On Barrier 2 the situation is the other way around (the

38 transmitted wave being the slow mode) but equivalent to the first as far as the connection formula is concerned. The transmission coefficient is therefore of type (44) [the coupling being specifically of the type i)].
To form the coupling potential (7), we find
Dkk =
12k 2X
2 x , (88) so that further
Dkk(kc
2
) =
4X , (B9)
b1 and x The embedding is thus valid and the potential Q,
Q(kC
2
2
)
) = -2 D (k )
(2
b ) (810) upon performing transformations (32), gives the transmission coefficient
T
= exp (-T 2
A
2 in agreement with Ngan and Swanson 1 as well as Cairns and Lashmore-
Davies
5
.
(811)
We now proceed to the ion-ion hybrid case, portrayed in Fig. 10 and characterized by a large and negative parameter, a. We first note that, for such values of a, the transmission coefficient (B11) becomes
T2 = exp ( 1T jai / A
2
) , which is the appropriate result 11, 12, 13 for the ion-ion hybrid case.
(812)
This, however, must not lead us to the wrong conclusion that the second-order theory of the ion-ion hybrid case is merely a limit of the preceding second-harmonic one. In fact, what happens with increasing
1a2 is that the point x = 0 falls between the branch points xb±, and the

39 embedding around k = kC2 is thus invalidated. Since a barrier is simultaneously formed between x = x
0 and x
=
0, we see that the embedding k
=
kC2 is fortunately irrelevant, and that the "mode-boundary" k = k = 0 has to be taken in its place.
The new embedding is characterized by the potential
Q(k
C1
D(0)
)
2 -
DKK (0)
= x x
0 x
, where x
0
= a/X
2 is the cutoff. The embedded second-order differential
(B13)
H + (1 -
XO
-- ) = 0, x whose transmission coefficient is (812).
(814)

40
APPENDIX C ALFV N WAVE CYCLOTRON RESONANCE HEATING
White, Yoshikawa and Oberman
19 have shown that heating by the fast Alfvgn wave in configurations where the wave number changes predominantly along the direction of the magnetic field is only possible in the presence of a second ion species and in oblique incidence. The fast wave then couples to the ion-cyclotron branch associated with the lighter ion species, as shown in Fig. 11. White et al.19 approximate the dispersion relation in the vicinity of the pertinent resonance, in the form k + k
2
(1 _ f 2 + a/x)
-
(f
2
+ X2 + af
2
/x) = 0 , (C) valid when the lighter ions are a minority species and when N12<<1, where
NJ is the Alfv6n refractive index of the majority species.
For the purpose of the present analysis it suffices to say that x is a spatial variable directed along the magnetic field, and centered around the position where the field resonates with the lighter ion species. Further, f2 = 1/3 - NJ.
2
/2, a v
2
, where v
2 is the fraction of the minority species and a non-zero A = Nf/(2 + NI_
2
), couples the incoming fast wave to the resonant branch. It can be shown that the fast Alfv6n wave cannot propagate for N.
2
1, which places an absolute limit on the coupling in terms of Nj.
The branches of the dispersion relation (Cl) are k
2
1
,
2
= if
2 -1-
[(1 + f2 + a 3 2+
x x
(C2) with branch and saddle points determined by the conditions (25). We have
Dk= 2k[2k
2
(f
2 - 1 - a/x)], Dkk= 12k
2
- 2(f
2
- 1 - a/x), (C3)

41 so that the roots of Dk = 0 are kC1 0
, kC2 f 2
-
a/x),
The root kC1 goes with a cutoff at
0
2af2
A
2
+ f
2 while kC2 leads to the branch points
Xb± -a 1 + f b± (1 + f
2
2
+ 2iX
)2 + 4X2
(C4)
(C)
(C6)
It is now more difficult than it was in the preceding example to visualize the coupling between the two modes, since the branch points are complex and the barrier is of the "underdense" type. It is helpful, in this respect, to imagine that the branches switch identity at a point along the x-axis given by the real part of the branch point, that is, at xb =- b a(1 + f 2
)
____C_
2
(1 + f )2 + 4X
2
7)
The information gathered so far allows us to interpret the plot of Rek under consideration, then the cutoff x
0 lies to the left of the origin, and we have the configuration shown in Fig. 12. Hence, a fast wave incident from the left will partially mode-convert to the resonant branch, and some of its energy is transmitted to the right. The more complicated case, indicated in Fig. 12, is incidence from the right, since the cutoff is also accessible. As in the problem of Appendix B, we now have transmission, reflection and mode conversion, given respectively by T, (1-T)
2
, and

42
T(1-T). Again, group and phase velocities are marked respectively by solid and open arrows, indicating the forward nature of both modes in question
(evident from Fig. 11).
In the k kc frame of the first coupling, however, the transmitted wave is backward so that the coupling is of type iii). The second coupling is a mirror image of the first and therefore is also of type iii). It follows that the transmission coefficient is of type
(44).
In order to determine its explicit form, we go to the embedding approximation. The coupling is characterized by kc
2
, giving
Dkk (kc
2
) = 4(f
2
-1-a/x).
(C8)
This expression vanishes at x = a/(f
2
-1), well away from the segment connecting the branch points. We can thus form the coupling potential
Q -2 D(kc2
Dkk(k2)
[(f 2
8x
+1)x+a]
2 (f
2
+
2
-1-a/x)
4X
2 x
2 which, upon factoring out the branch points, gives
'
(f2+1)2+ 4X2 x f 2-1-a/xb
_-xb 2
4
2
2
21)2+4X2]2
C9)
C10)
We now substitute for xb from (C7) into the expression in front of the braces to obtain the transmission coefficient
T = exp(-w 2a a) ,
(C11) where
=
{ (1+f
2
) [(f
2
+1) 2 + 4X
2
] [f
2
(f
2
+1) + 2 X
2
] 1 -
(C12)

This result agrees with neither White et al. 19 nor Cairns and Lashmore -
Davies,5 but all three results agree in the limit of Nj2<<1. The origin of the discrepancies for larger Nj
2 will be discussed elsewhere.
43

44
REFERENCES
1) A. Bers, in Plasma Physics Les Houches 1972, edited by C. Dewitt and
J. Peyraud (Gordon and Breach, New York, 1975), Chap. III.
2) T.H. Stix, Phys. Rev. Lett. 15, 878 (1965).
3) T.H. Stix and D.G. Swanson, Propagation and Mode Conversion for Waves in Nonuniform Plasmas, in Handbook of Plasma Physics (eds. M.N.
Rosenbluth and R.Z. Sagdeev), Vol. 1 Basic Plasma Physics (eds. A.A.
Galecv and R.N. Sudan) North Holland Publ. Co. 1983, Chap. 2, 4.
4) V. Fuchs, K.Ko, and A. Bers, Phys. Fluids 24, 1261 (1981).
5) R.A. Cairns and C.N. Lashmore-Davies, Phys. Fluids 26, 1270 (1983).
6) K.G. Budden, Radiowaves in Ionosphere (Cambridge University,
Cambridge, 1961), p. 476.
7) J. Heading, An Introduction to Phase Integral Methods (J. Wiley, New
York, 1962).
8) J. Heading, J. Phys. A6, 958 (1973).
9) J.C.P. Miller, in Handbook of Mathematical Functions, edited by M.
Abramowitz and I.A. Stegun (Nat. Bur. Stand. Applied Math. Ser. 55,
June 1964), p. 694.
10) F.W.J. Olver, Asymptotics and Special Functions (Academic Press, New
York, 1974), p. 515.
11) Y.C. Ngan and D.G. Swanson, Phys. Fluids 20, 1970 (1977).
12) D.J.D. Gambier and J.P.M. Schmitt, Phys. Fluids 26, 2200 (1983).
13) D.G. Swanson, Phys. Rev. Lett. 36, 316 (1976).
14) T.H. Stix, The Theory of Plasma Waves (McGraw-Hill, New York, 1962).
15) M. Brambilla, Plasma Phys. 18, 669 (1976).

16) V.S. Chan, S.C. Chiu, and G.E. Guest, Phys. Fluids 23, 1250 (1980).
17) K.Ko, V. Fuchs, and A. Bers in Proceedings of the Fourth Topical
Conference on Radio Frequency Plasma Heating, 9-10 February, 1981, The
University of Texas, Austin, Texas, paper C-17.
18) I. Fidone and R.B. Paris, Phys. Fluids 17, 1921 (1974).
19) R.B. White, S. Yoshikawa, and C. Oberman, Phys. Fluids 25, 384 (1982).

46
FIGURE CAPTIONS
Fig. 1. Typical topology of the branches of Eqs. (3) in the region of branch points (coupling region).
Fig. 2. The electromagnetic dispersion relation (15) parametrized with respect to w (x).
Fig. 3. The coupled branches of the dispersion relation Eq. (15) for w = wo and a double-valued (e.g. parabolic) w (x).
p
Fig. 4. Coupled modes with anti-parallel group velocities. Tracing of the dispersion relation (18) through the 2 = 0 i.e. w = w
0 line identifies the phase velocities away from the coupling region.
Fig. 5. Local dispersion relation (21): a) anti-parallel group velocities;
b) parallel group velocities.
Fig. 6. Classification of couplings in the k k frame. Solid arrows
indicate orientation of group velocity, open arrows the orientation of phase velocity. i) All three waves forward.
ii) Coupled modes with anti-parallel group velocities.
iii) Coupled modes with parallel group velocities.
Fig. 7. The lower-hybrid (LH) and warm-ion (WI) branches of the dispersion relation (A8) at conditions for lower-hybrid heating of tokamaks of ALCATOR type: a) Toroidal magnetic field BT = 8 (1 0.2x/a)
Tesla; b) Straight ion-orbit approximation (BT
+
0)
Fig. 9. The branches of the dispersion relation (B1) for fast-Alfvdn to ion-Bernstein mode conversion (a = 2/3).
Fig. 10. The branches of (B1) for ion-ion hybrid heating (a<<-1).

Fig. 11. The Alfv6n and ion-cyclotron branches of the cold plasma dispersion relation for two ionic species, characterized by i = eB/M c, M
2
<M
1
-
Fig. 12. The branches of the approximate dispersion relation (C).
47

k
E
F
-t
G
H
Figure 1

CL
/
/
/
/
/
//
/
/
/
/
/
/
/
~Z I /
\X
I
Io
1
2
K
Figure 2

E
WP max
F F
INC
k
1
0
*1* k
0
TP
H x x
Figure 3

22
Figure 4

k'
\
G k
E b)
F
\ i
0
It k
G C
F
0
Figure 5

REFL :
INC
F
INC
I
F x=O
G
im- T R
B=LqA
H,
B=- !qA
-7
1
INC
W.L)
A=-LqB G
H
sme~i*
*- TR
Figure 6

Cl
I I
-is
0
'a
-3 nz4
'a,
faa r.% -0.8
i.
/
-0.4
a.
x /a
0
0.4
0.8

C.'.
0
-J
)
I r4ky
1 i( RIk
___
NWI)----4I____
Wr
I t
-6iI
128 -0.4
x /a
0
0.8

LH-
KLO
+x
E w
V a
Vph
G x
LH
F
Figure. 8

2
w 2 wci Slow
C2
Fast
Slow
0
SI
ph
X
0
Xb- Xb+
Barrier i --
-i
xb-
Barrier
kC2
C2
U
Figure 9 x x
If

k k2
Fast
0
Xo b-xb+
INC
XXb-
0
Figure 10
Xb,+
MC

IC k
2
11
Slow
22
2
v
Figure 11
R

Cf)
C-)
C
0
-o
-~
C
C)
C-)
.J
U-
LU
C-)
0
'I,
0
U-
14
V
II
C-, z
'v
I
cv
C-,
4
0
I
C~) e
__________________
___________ A
0)
I.
C