Download Analytic calculation of the nonzero fast wave reflection coefficient

Analytic calculation of the nonzero fast wave reflection coefficient
from the tunneling equation with absorption
C. S. Ng and D. G. Swanson
Physics Department, Auburn University, Auburn, Alabama 36849
(Received 17 September 1993; accepted 16 November 1993)
An analytic series expression of the nonzero fast wave reflection coefficient has been found from
the tunneling equation which models the wave propagation in weakly inhomogeneous media
near a back-to-back resonance-cutoff pair. Explicit calculations are done for both second ion
cyclotron and second electron cyclotron harmonic cases. Numerical values of.this expression are
compared to those calculated by solving an integral equation iteratively. They, agree well with
each other whenever both the series and the iteration converge. This result is based on a new
method of evaluating integrals derived from the integral equation without having to solve the
equation itself. This new method may have more applications and deserves further research and
development.
1.INTRODUCTION
Waves propagating in weakly inhomogeneous media
through a back-to-back resonance-cutoff region, in general,
experience effects of transmission, reflection, and mode
conversion with or without absorption. These phenomena
converslon
~ ~ ~ ~ ~ ~ ~ ~~~icl
can be modeled by different kinds of high order ordinary
these is
a fourth order equation of the form (e.g., Refs. 14)
f lD+) 2Z(fnr +f)
(I)
+ yf =°
with A and y being real constants. We call this the tunneling equation without absorption. This equation can model
many different physical situations. We will explicitly consider two of them: (i) second ion cyclotron harmonic with
y>-1; (ii) second electron cyclotron harmonic with r
<-1. We can generalize (1) to include localized
absorption:3 .5
4'
!l)A2Z(4/44j)
+yr=htz)('
++),
(2)
where h (z) is the absorption function which generally
must fall off at least as fast as z- I as Iz }
oo0.Analytic
questions athe scattering parameters i.e Analtic
expressions for the scattering parameters, i.e., transmission, reflection, and mode conversion coefficients, for Eq.
(1) have long been known. Solutions for Eq. (2), and thus
scattering parameters, can be found by solving an integral
equation iteratively, making use of the numerical solutions
of Eq. (1). These scattering parameters are expressed in
terms of integrals involving h (z), solutions of Eqs. (1) and
(2) along the z-axis.
It has, been pointed out that odd order derivatives
should be added to Eqs. (1) and (2), with a much more
complicated expression in the right hand side of Eq. (2), in
order to conserve energy. 3 However, from previous numerical results the scattering parameters found by both sets of
equations are close, especially for the reflection coefficient
R2 ? So we will use the simpler equations for our analysis.
We believe that the method we develop here should also
work for the energy conserving equations, since the two
sets of equations are very similar.
Phys. Plasmas 1 (4), April 1994
Recently, by extending solutions of Eqs. (I) and (2)
into the complex z-plane, 6some of these integrals were
shown to be zero identically. Thus, fast wave transmission
official
entsa
al To each othe promote
sides and the fast wave reflection coefficient (which is idenzero fro theou
side which encounters resonance
typically zero) from the side which encounters the resonance
before the cutoff, were shown to be independent of absorption. However, analytic expressions for other scattering parameters which do change with absorption have been unknown so far. Here, we go further along this direction and
develop a new method to calculate some of those nonzero
integrals analytically.
We note that the above method is not the only approach used to solve the mode conversion problem. Other
methods have been developed, including, e.g., direct numerical integration, 7 finite element, 8 finite difference,9
phase space methods and order reduction.' 1 -1 4 Partial"3
or approximate expressions1 1 4 of the scattering parameters have also been reported. However, we will concentrate
on the above integral equation method of solving Eq. (2)
here.
We emphasize that the analytical expression for the
relcinofiintgvnheiseatntesnetato
reflection coefficient given here is exact in the sense that no
further approximation is used after we accept Eqs. (1) and
()
hc
r nagnrlmteaia
omta
a
(2), which are in a general mathematical form that may
represent many other physical situations. In all other semianalytic methods, additional approximations are made before evaluating the reflection coefficient.
We have been successful in calculating the other fast
wave reflection coefficient R2 , which is nonzero, for the
above two cases (i) and (ii). It seems that this method,
with further development, may also be useful in calculating
other scattering parameters or integrals of similar nature
for other problems.
In the next section, we will review the integral equation method and thus the integral expressions of the scattering parameters. In Sec. III, we will discuss the basic
ideas of this method and show explicitly how to calculate
R 2 analytically for the two cases. We will compare results
from this method to those calculated by the integral equation method in Sec. IV.
1070-664X/94/1(4)/615/7/$6.00
y 1994 American Institute of Physics -
915
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i (k 3 k
g(k) =-ikc+- ,+)p+
z
11.INTEGRAL EQUATION
Equation (I) can be solved exactly by using the
method of Laplace: 4
(i-a)ln(k+lI)
+ (i+a)ln(k-1)
fj(z)= Cfr [expzg(k)] dk,
where a = (I+ y)/2A2. With a suitable choice of constants
ci, we can express the asymptotic behavior of fj (note
that the definitions in this paper are in general different
from those in previous literatures 4 ) as
where the rj are contours in the complex k-plane which
end at infinity with approach angles of -r/6, 51r/6, or 3ir/2,
and
(
0
{1 R 2
0
o
f,\
T, 0 c, \f+\
0
C2 4
f
C3 2
0
C34
a+
f
C42
I
R4
a_-
f4
(
-
0-z
}
(3)
j=1,2,3,4,
Z- 0°
(fzZ
>
RI
I
C13
T2
0
C2 3
C3 1
0
R3
I S-I.
C4 1 0 C43
(4)
for y> -1, and
R
1
0 C:4 f_
T2
0
0
C24 f+
C31
0
0
C34
C4 ,
0
1
R4
-c0z
fz
Z- 00
U+
\ a/
0 T,
C13
I
R2
C23
0
C32
R3
0
C42
C43
I
S
(5)
for y < -1, with scattering parameter given by
T,
C13
c14
0
g
-t
-T2 Rz
C23
C24 -S(°)
g
e2
gg
31
C3 2
R3
C34
_ - g g
C41
C4 2
C4 3
R4
0
RI
Su=C
whereg=exp(-7), q = rTaj,
I
r-4
-ire-n/2
f
'-
=>r()-epI=I ii|j+a
i,:id~
~
si=sgn(a), ,4,
4
2
(6)
g2
g
g
1/21
=I-g 2 , and
1
In 2+z+cr In jzj |
zil
n
3
IzI
_
-
Fj(y)h(y)PIk(Y)dY,
(8)
f 0 -=f 3 -gf,,
for y>-1, fo-8g(f 2 +gf 3 ),
f,/g, for y < -1. Equation (7) can be solved
iteratively using bk=fk as the first trial functions with
fk calculated numerically by Eq. (3). This can be done for
the three physical solutions, k = 1,2,3, if h(z) fall off at
least as fast as z- I as IzI - oo . For the k=4 solution, this
0 fast enough as z
can be done only if h(z)a+(z)
- 0. Numerically, this method converges in general,
but may become divergent when the absorption is very
large. After solving Eq. (7), scattering parameters can be
found, making use of Eqs. (4)-(6), by
f
___4
f10
F.=f'±f,, \I'j=0j7+ 3 1,
and
_Z /2
(7 = pjt] rep Z23/2+
2 3.7xp|_
12
* 2I 2
with
II,1
Xp~di|2AZ3/2~-Z/xi2tZ 2
_=-sgn(a)
jk=
an
2+
5 =f 4 -
-
Only I',, f2, f 3 are physically allowed in an unbounded
region. Using these we can find an integral equation that
solves Eq. (2 ):3'5
'=7fk1z IfI2
Only
|
k+f4ok+f2,-k+fosk
arephsicllr<lowdiaunoude-1,
f2
k
Sjk=S5k=1
for
4 (r+l 1) > 0,
with
where
816
-1,
Phys. Plasmas, Vol. 1, No. 4, April 1994
C. S. Ng and D. G. Swanson
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t
~~~
3~~~
y~~ooe
Y-
Y oo'7
FIG. 1. Integration contours of 122 for the y> -I case.
Ijk
A
Fj(z)h(z)'Pk(z)dz,
FIG. 2. Integration contours of I12 for the y < - I case.
(9)
where we have already used the fact that Ijk= I.4 - In particular, the integral expression for R 2 is R 2 =g F'22. In
the next section, we will show how to calculate this integral
I22 analytically, without the need of solving Eq. (3) or Eq.
(7) numerically.
Since Iz | co on the contour, all terms in this asymptotic
series may be kept. Although the integrand is in general a
divergent series in z-" for any finite z, the series obtained
after integration may become convergent for certain absorption function h(z). First of all, it is easy to expand
h(z) in asymptotic series,
hz
III. INTEGRATION
To perform the integration of I22, we must first specify
the absorption function h(z). For case (i), the second ion
cyclotron harmonic with y
-1,
,n=1
Y
fo
-
y+1)>0
where y-z-z0 . This can be done by inverting the two
asymptotic series,
i
17r(n+ 1/2)
h(z) =ii12K[g-I/Z(-a) ];
and for case (ii), the second electron cyclotron harmonic
with y <-1,
2
h (z) =A K[g-/F
7 /2 (~-
7/2)],
(12)
and
(-l)'r(n+q)
1
n
(10)
with the plasma dispersion-function Z(g)=i+i w(g),
where w is the error function for complex argument, 15 and
F7/ 2 being the relativistic plasma dispersion function ,16,17
and g= (z-zo)/K, zo= -r,2, where K is a real parameter characterizing the strength of absorption. Note that
both Z(g) and F7/ 2 (g) are analytic function and have zeros only in the lower half g-plane. Also, solutions Fk and
6
1
Then,
f 'k have been shown to be analytic everywhere.
following Ref. 6, we can change the path of integration of
Ijk, defined in Eq. (9), to the semicircles Co:
co and 0 from -'w to 0 for case
z=R exp(iO) with R
(i), see Fig. 1, and 0 from 1r to 0 for case (ii), see Fig. 2.
In Ref. 6, solutions fk and Opk have been extended to the
complex z-plane, and it was shown that on these new contours C,, both fl and l&1 ccexp ( r iz) and are thus exponentially small for both cases of - (y + 1) > 0, and that
f2 and iP2 c exp ( - iz) and are thus exponentially large. It
was then concluded that 111=I12=I21=0, which means
that T 1=T 2 =exp(-s7), and R 1=0, independent of
absorption. 6 Our approach here to calculate I22 is to expand the integrand F2hT 2 in an asymptotic series on C,.
(11)
r (q)g'
(13)
on the path of integration C;. First, we need to shift the
argument of the F7,/2 function in Eq. (10) from g-7/2 to
A. Let,
A,(q)
A
q(¢_q) =
' ng
q
The coefficients An(q) can be founding Eq. (13),,by
n
1
An(q)=-F-(Y, (-i),r(q+m-1)C'_-mq
m=1
7m,:
-
Phys. Plasmas, Vol. 1, No. 4, April 1994
where Cn'= n/ml (n - m)! is the binomial coefficients. The
first few values of An areA 1 = 1, A 2 =0, A 3 = q;Then we can
invert Fq by,
_n= 2 C~q
Fq(~-q)
(14)
where
In
C2.+ 1/2.1/2(q)= I
n=~1
[-A3 (q)I nD2Cm n) +1/2kL/(")
C. S. Ng and D.G. Swanson
817
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for -(y
D,,(q,m) =
Z D.-k(qsmk=O
+ 1) > 0, where
) Dk(qI ),
n=
.
D.(q,l ) =A.+3(q)1A3(q).
Finally we can get the coefficients hn of Eq. (11) by
We can also invert the asymptotic series of Z( -g) of Eq.
( 12), using calculations similar to Eqs. ( 14) and ( 15), to
get h+.
To find the asymptotic series for f2, and 4l2, we first
note that only the part of them with leading behavior
exp 4 i(y+ a In y) on C, contribute to the integral I22, so
21,riAI
2 2=
f
(17)
(15)
An=-A~e~l~n~l(7/2).
F2 hq 2dz= fc F,(y)h(y)'I',(y)dy,
(16)
with
C =
-ze
ira/2
[a-4i(n-I)][a
,
',+I,~:=12
afzk = afl- 1+ [a-~i(n-1)1a' l~k-,
(:Fnab11
E
Y
ye+2i(Y+ajny)dy,
f
(19)
n-2
for n>2.
m=I
To evaluate the y-integrals in Eq. (19), we change the
integration contour again. Now, there is only one pole y
= 0 in the integrand, and there is also a branch cut. We
choose the branch cut to be from y = 0 to 4 ico for the two
cases (see Figs. 1 and 2). Then we change the integration
contours to go along the semicircles with infinite radius at
the other side of the real axis, opposite to C+, and go
around the branch cut (see Figs. I and 2) so that the
analytic continuation is still valid. The contributions from
integrating along the two quarters of the semicircles are
zero due to the exp - 2iy factor. Integration along the path
going around the branch cut can be found by making use of
the Hankel's contour integral, 15
Jo(e-"t)
818
--*-tdt=
_2rri)
Phys. Plasmas, Vol. 1, No. 4, April 1994
ia
where the contour CH starts at t= ocexp(Oi), come along
the positive real axis, turn around the origin counterclockwise, then go back along the positive real axis to
tc= oexp(27ri). The y-integral in Eq. ( 19) can be changed
to this form by means of a variable transformation
t=2yexp(3-ri/2) for y> -1, and t=2yexp(iri/2) for y
< - 1. Thus, we can express 122, or R 2 in a series expression,
i
for n>3,
n-I
=a
n- I
((818)
for -(y + 1) > 0, where
XYn,
m=1
-
,
k==2,3,4,
and aOk= 1. The calculation for a is similar to Eqs. ( 18),
with hk =0. Substituting Eqs. (11) and (17) into Eq.
(16), we get
27ri)L2 2 2= C
I
a~tn
I~;2 +ZOjJ
The coefficients a' and &< can be found by substituting
the asymptotic series (17), making use of Eq. (8), into the
two differential equations (1) and (2), requiring that all
terms vanish and that they satisfy boundary conditions Eq.
(4) or (5). The result is
zi:
4-4-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
a I=
-4
''
FTia
Trge =t=i(2b-4/3A2)'j2
R2=0-4- -
2 aAr(4-ict)
00 Y,,' ( =I- 20 '
1
n=3 nnTNa)
'
(20)
for 4 (ry 1) >0. This expression is analytic in the sense
that all coefficients r; can be calculated exactly by some
algebraic recurrence formula, although they are in general
rather lengthy to be written out explicitly. This series is
useful only if it is convergent, unless we know how to sum
it analytically. However, due to the complicated nature of
the coefficients, it is beyond the scope of this paper to study
the convergence analytically. In the next section, we will
present numerical results showing that it does converge, in
many cases, to numerically the same values found by solving the integral equation (7) iteratively.
IV. COMPARISONS
We do our calculations with a deuterium plasma. Table I compares calculations of R2 for case (i), the second
ion cyclotron harmonic with y> - 1, for plasma parameters: electron density np, characteristic length of the inhomogeneity L, magnetic field strength B, and plasma temC. S. Ng and D. G. Swanson
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TABLE 1. Case (i) with ne=2X leX m- 3 , L=3 m, B=3 T, T=1000
eV.
oq,~~~~~~~~~~l
kil(m
1
2
3
4
5
6
7
8
9
10
11
12
13
K
P.(%)
P,,(%)
q,
qa
0.2837
0.5668
0.8488
1.1292
1.4073
1.6828
1.9550
2.2234
2.4875
2.7468
3.0009
3.2492
3.4913
60.495
37.427
16.903
5.6244
1.4016
0.26916
0.041933
5.6902e-3
9.0586e-4
1.6609e-4
3.3683e-5
-4.e-6
diverge
60.498
37.432
16.905
5.6237
1.4014
0.27041
0.043239
6.4453e-3
1.0059e-3
1.6403e-4
2.4967e-5
2.99e-6
2.1e-7
0.9896
0.9891
0.9881
0.9862
0.9826
0.9759
0.9639
0.9450
0.9014
0.8497
0.7986'
~0.77
diverge
0.9893
0.9889
0.9881
0.9863
0.9826
0.9751
0.9599
0.9324
0.8929
0.8505
0.8152
0.794
0.79
1)
~
0.2 -
-
a
0.:1-
I
q
-
'I
.3
2
1
FIG. 3. Plot of 1-q ., I -q,,, and 1-qf versus K, where q. and q, are the
values in Table I, and qf are calculated by Eq. (22).
perature T, for different parallel wave vectors k1l. P, is
jR212 calculated by solving the integral equation (7) iteratively. Pa is that calculated by the series in Eq. (20).
Table II compares these for case (ii), the second electron
cyclotron harmonic with y < - 1, where X is the square of
the ratio of the plasma frequency to the wave frequency.
Previously, it was found numerically that
with q
R2=e~exp(-qK)
with q
1 for case-(i)
7 for case (ii).
(2)
We also list the q 0 and qa values, calculated by each
method,;
defined
by
P-=g4exp( - 2qgK2),
and
P.=keep -2qnK 2 ). They are also plotted in Figs. 3 and 4.
We first note that these values calculated by both
methods agree with each other very well as long as both
methods converge. We have also compared results from
these two methods for many other cases over a broad range
of parameters and they all agree very well. It has been
known previously that the iteration method of solving Eq.
(7) becomes divergent for large absorption (i.e., large K)
for both cases. It seems that the series method using Eq.
(20) for case (i) is still convergent after the iteration
method diverges. However, it also has numerical problems
for large K, due to the fact that we can only sum a finite
number of terms and that the computer has a limited
power to handle large numbers. On the other hand, the
series method becomes divergent faster than the iteration
method for case (ii). Actually, it seems that there is a
convergence radius of K < 0.5 for the series. The reason for
this difference is that the asymptotic series of the relativistic plasma dispersion function F7 /2 of Eq. (13) diverges
faster than that of the plasma dispersion function Z of Eq.
(12).
Second, we see that the approximation Eq. (21) is
good, but not exact. Before the development of this series
method, the numerical values found by solving the integral
equation iteratively subject to many different kinds of numerical errors, which are difficult to analyze, within the
complicated algorithm to calculate fk and 1Obk.Therefore,
there was an uncertainty about whether Eq. (21) is really
exact, and whether the deviations from the rule in the
numerical results are just numerical errors. However, now
that the series method is analytic in principle, its results are
much more reliable and to the extent that they agree with
those calculated by the iteration method, this uncertainty
no longer exists. Moreover, the series method provides
some analytic explanation of Eq. (21), at least in extreme
~~~.
. . . . . . . . . .
. -. .I
3
T(eV)
50
100
150
200
250
300
350
400
450
500
550
A2
K
P.(%)
93.15
46.58
31.05
23.29
18.63
15.53
13.31
11.64
10.35
9.315
8.468
0.04736
0.09471
0.1421
0.1894
0.2368
0.2841
0.3315
0.3789
0.4262
0.4736
0.5209
0.15192
0.53250
0.99350
1.3983
1.6648
1.7766
1.7568
1.6496
1.4946
1.3215
1.1503
Pa%)
0.15189
0.53254
0.99340
1.3983
1.6646
1.7755
1.7564
1.6492
1.4941
-2
diverge
Phys. Plasmas, Vol. 1, No. 4, April 1994
n
a
6.9114
6.8615
6.6881
6.4770
6.2201
5.9345
5.6381
5.3328
5.0287
4.7341
4.4494
6.9608
6.8574
6.6927
6.4767
6.2212
5.9382
5.6391
5.3336
5.0295
-4
diverge
7-qu
o
TABLE I. Case (ii) withX=0.114, n,=4Xl109 m-3 , L=0.15 m, B=3
T.
x:7:- qrz,,
2
, :7-qf
5-X
1
,5
L.
,,~-5
-5-'
0.0
-
0.1
0.2
0.3
0.4
0.5
A;
FIG. 4. Plot of 7-q,
7- q., and 7 -qf versus K, where q. and q. are the
values in Table II, and qf are calculated by Eq. (23).
C. S. Ng and D. G. Swanson
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cases. Let us consider large /2 , small A1,and small K cases.
It is not hard to see that in these cases, we only need to
keep the n = 3 terms in the series in Eq. (20). Note that
2
Y3 =2a2A ,,K
and yT =14a2A22c, and using approximations
r'( + ia) z 4 Ilia, r(3
i-2ia) =2, g~ 1,e=~277,
we have
R ZtI(-K2) t eXPfK2)
R2
2e
(l
2
-7K
)
=
2
for case (i),
exp(-7Kc2)
for case (ii),
which is just Eq. (21). From numerical calculations, this
approximation is very good whenever the condition of
large A2, small 7, and small Kis satisfied, although it is also
good for some more general cases.
Since Eqs. (21) are good approximations already, it is
possible to find even better empirical formulas by fitting the
results found by the series method. For the ion cases, we
found that the q factor can be approximated by the following expression, so that IR21t 2 exp(-qua)
qf= 1- qe2 (ao+aiK 2 v),
(22)
where
ao=0.0065(1+0.131 w+0. 103w 2 ),
aI =exp( 17.5e-S5 w- 7+0.3w),
v=3.085( 1-0.1056w),
w= lOO,6iL,
and f3i=2yonikBT/B2 is the usual ,6 factor of the plasma.
This approximation has been compared with the results of
the series method, for the range K2 < 4, so that IR2 12is at
least greater than 0.04% of that without absorption, i.e.,
e, and for k, < k, m,/2, where k, ma is
t the k, value when
the X-mode is cut off. The error in IR21 rarely exceeds
1%, and this happens only for the cases with w> 2. In all
cases, the error is smaller than 1% when IR2 1>i %. Without this correction, i.e., simply using Eq. (21), the error in
IR21 may exceed 15% for some cases. The qf factor calculated by Eq. (22) for the cases of Table I is also plotted
in Fig. 3. The error between qf and q, seems large in Fig.
3 for some cases. However, we should note that actual
difference is only about 2% in the q factor for the worst
cases, and that this happens only for 2> 4.
For the electron cases, the approximation is
qf=7-A [ I-e-a
so that IR2 I1;
2
2
],
(23)
exp ( - qyK2), where
A=3.602+2.413X,
a=- 13.6+17.04/(1
2X)0.0 922.
The comparison of this approximation with exact values is
limited by the convergence of the series method. The range
of jR21 is from 0.15% to 50% of t2, the reflection coefficient without absorption. The range of the X parameter is
from 0.114 to 0.457. The range of the temperature T is up
820
to 850 eV. Within these ranges, the error of this approximation in IR21 is less than 0.5%. The qf factor calculated
by Eq. (23) for the cases of Table II is also plotted in Fig.
4. We see that the approximation is good for the whole
range of data shown on Fig. 4. The large difference between q4and qf for the last data point is due to the error in
qg,, because the convergence of the series method is not
good for K near 0.5. However, the agreement between qf
and q, is still good, so the approximation Eq. (23) may
still be good beyond the K <0.5 limit.
Phys. Plasmas, Vol. 1, No. 4, April 1994
V. DISCUSSION
The fact that these two very distinct methods give essentially the same answers further confirms their own validity respectively, because if either one of these methods is
wrong, the chances for their answers to agree with each
other accidentally is extremely small.
The success of the series method is amazing and shows
how powerful complex analysis can be. In the original form
of the integral I22 in Eq. (9), we needed to know the values
of the solutions F 2 , T 2, and the absorption function h
along the real z-axis. The straightforward way to do this is
first to find F 2 (z) by a numerical path integration in the
complex plane using the method of Laplace like Eq. (3)
with a complicated integration and error control scheme.
Second, solve the integral equation (7) by doing numerical
integrations iteratively along the z-axis, with proper endpoint corrections, to find '+2(Z I),making use of numerical
methods to evaluate h(z) for z within different regions on
the z-axis. Then find 122 by numerical integration according to the definition Eq. (9). Now, the series method can
do the same integration correctly without having to know
F2 , 'I'2, and even h on the real z-axis. All it needs is their
analytic properties in the complex z-plane and their asymptotic expansions for large complex z values. Also, due to its
less complicated algorithm, the series method usually runs
much faster on a computer, if we do not care to know the
solutions themselves. This powerful idea may have other
applications to solve integrals of similar nature, which arise
either from the tunneling problem or other physical problems.
The main difficulty the series method faces is that it
does not converge for all cases. However, since it is essentially an analytical method, there is always a chance that
we may find a way to sum the series analytically or to
transform it into other convergent forms. Just like we can
sum +x+x 2+x 3+ ' ' ' as 1/(1-x), although the series
may be divergent itself. Note also that the iteration method
also has divergence problems, but the chance to "sum" it
analytically is much smaller, since it is basically numerical
in nature.
We have also tried to apply this method to calculate
the conversion coefficient C13, but the resulting series is
divergent. However, it is still possible that there is a way to
overcome this difficulty.
We conclude that the series method has been successful in the calculations of the fast wave reflection coefficient
from the tunneling equation and it has advantages in some
C. S. Ng and D. G. Swanson
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aspects. This also shows that the analytic method is powerful, and may have more applications and improvements.
ACKNOWLEDGMENT
Work supported by U.S. Department of Energy Grant
No. DE-FG05-85ER53206-93.
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C. S. Ng and D. G. Swanson
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