- Email: [email protected]
Diffraction of a wave bunch on a plasma cylinder
Diffrnction of a wave bunch
143
3.
TIKHONOV, N. A. Invariantsof the non-linear equation of oscillations. Zh. eks. teor. fiz., 66,1,109-115, 1974.
4.
KELDYSH, M. V. On the completeness of the e~enfunctions of some classes of non-selfconjugate linear operators. Usp. mot. rim&, t&4,15-41,197l.
D~FRA~ION
OF A WAVE BUNCH ON A PLASMA CYLINDER*
E. S. BIRGER, L. A. VAINSHTEIN and N. B. KONWKHOVA
Moscow (Received I 1 April 1975) A METHOD for the numerical solution of the twodimensional problem of the diffraction of an electroma~etic wave by a plasma cylinder, whose dielectric const~t is a fiction of the radius-vector r, being unity for large values of I, is described. In this problem the fields are represented by series of azimuthal harmonics, the corresponding radial functions being found by numerical integration of second-order linear differential equations with singular points. A calculation is made of the result of this averaging of the Green’s function over the coordinates of the source and point of observation, which corresponds to the irradiation of the plasma by an antenna creating Gaussian wave bunches, and to the reception of the diffraction field by a similar receive antenna. Numerical results are given, and the principal physical phenomena arising on the diffraction of a wave bunch by a plasma cylinder are discussed.
1. Statement of the problem
We consider the two-dimensional problem of the diffraction of an electromagnetic wave by a plasma cylinder, in which the dielectric constant e(r) is a monotonic~ increasing function of the distance from the axis of the cylinder, the cylinder being assumed of infinite length, and the field of the incident wave independent of the coordinate z along the axis of the cylinder. In numerical cal$ulations we describe a plasma cylinder (with the time relation a distribution of the complex dielectric constant E
(f-1=1-0
e-I”‘)
exp (4/a’>,
by
wo
where f is the distance from the axis of the cylinder, II is the effective radius of the cylinder, and 8 is a positive parameter. For 6> $ an important value is the critical radius rk for which E (tx) =O, for r>rA and e(r)O and r--u (In 0)‘“. However, the method is applicable for any function e(r), for which Im ~20 Im r,
If a wave polarized along the z-axis falls on the plasma cylinder, then in the cylindrical coordinate system r, cp,z the field has components E,=E,
fj,=---
and the function E(r, ip) satisfies the equation Vh. vfihhisl.Mat. mat. Fiz., 16,6,1526-1538,1976.
i $13 kr dip
Hi=--,
i SE k dr
144
E. S. Birger, L. A. Vainshteinand N. B. Konyukhova
+-
1 d’E
r2
-
+ k% (r) E=O
dcp”
(1.2)
everywhere with the exception of the sources (where a right side is present in (1.2)): k = w/c is the wave number, c is the velocity of light in a vacuum, where e = 1. If the wave is polarized perpendicular to the z-axis, the field will have components H,=H,
E,=LdH kera(p’
E,=-
i ,aH ke dr ’
where H(r, 9) is the solution of the equation -I- -
1 r’
-
d2H
+ kZE (r) H=O,
w
(1.3)
which is rather more complicated than Eq. (1.2), and also applicable only outside the sources of the field. Equations (1.2) and (1.3) follow from Maxwell’s equations for a space with cylindrical plasma. It is known that for these equations, supplemented by the radiation condition
(1.4) Green’s function exists [l] . As we shall see below this enables us to calculate the diffraction of a directed wave bunch by the plasma and the action of the diffracted bunch on the receive antenna, which is also directional. It must be pointed out that the two-dimensional problem of diffraction by a cylinder, inhomogeneous in the radial direction, has been discussed in many papers (see, for example, [2-6]), but even in the : of them directed towards numerical methods [3,4,6], the directivity of the receiving and transmitting antennas was not considered. Out statement of the problem corresponds to the experimental method [7] and in the limiting case of extremely narrow wave bunches and extremely short waves (ka-+ 00) it degenerates into the geometrical optics problem of the transillumination of a cylindrical plasma [8-lo]. However, we are interested in not too narrow bunches (the effective width of the bunch is, for example, a third to a quarter of the effective diameter 2a of the plasma) and not too short waves (ka = 10 to 20), when the numerical method explained below is most suitable for the solution of the problem.
2. Green’s function and the transmission coefficient In the absence of the plasma (E = I) Green’s function for Eqs. (1.2) and (1.3) is proportional to Hci) 0 (kd) , where H dl) is the Hankel function of the first kind, and R=[r’+F’-2rFcos
(cp-q)]“’
(2.1)
is the distance between the point of observation with the polar coordinates r, and the source (the -luminous filament) with coordinates r, 9. The function Hy) (kR) has at the origin (for R = 0)
Diffraction of a wave bunch
145
a logarithmic singularity and the following expansion in terms of the azimuthal harmonics:
I+”
x,.ly (kr) II,“’ (kr) I!2 v=*
(kR) =
cos v (cp-cp)
)
x0=1, (2.2)
x,=?.Cz=. . . =2, applicable for r
lcvRv (kr)R,“’
(kF)cos v(p-9).
(2.3)
v-0
in
which for r>F
instead of R, (kr) Rcj)(kF) it is necessary to write R, (k?) R (j)( kr) .
The functions R, and R,(l)
satisfy the equation d”R,/d(kr)2+q(kr,
v)R,=O,
(2.4)
where q(kr, v)=&(r)-(v2--‘II)
(kr)-“,
(2.5)
and the supplementary conditions lim R, (kr) =O,
,ti~~ $$(R.‘“)
-l=i
kr-.o
(2.6)
and R
dR?
dR, R,“’ = 2
Y---z dkr
(2.7)
n’
where the first condition of (2.6) is connected with the finiteness of Green’s function for r = 0, the second condition of (2.6) with the radiation condition (1.4), and in condition (2.7) the Wronskian of the functions R, and R,(l) is identified with the Wronskian of the functions (kr) ‘A,/,(Jcr) and (kr) ‘I’H$ (kr) in the expansion (2.2), which is connected with the same singularity of the functions (2.2) and (2.3) for kR = 0. The same formal expansion for Green’s function of Eq. (13) differs from (2.3) by the fact that the factor [ii (r?) ‘“I e-i is replaced by [E (r) E (F) 1’”[ Ic (rI;) ‘“1 --i, the radial functions R, and R,(l) satisfy Eq. (2.4), in which p(kr,v)=c(r)+$(rz)
[2k2re(r)]-’
(2.8) -3
(
$
1
’ [2kc(r)]-“--(v2-‘l,)
and the conditions (2.6) and (2.7) remain unchanged.
(kr)-2,
146
E. S. Birger, L. A. Vainshteinand N. B. Konyukhova
However, we are interested not in the Green’s function determining the field at the point -of observation r, 9 from the point source with coordinates r, 9, but in the field radiated by the transmitting antenna 1 and received by the receiving antenna 2. If both antennas are situated outside the plasma (E (r) =e (F) =I>, then without going beyond the range of the two-dimensional problem, we can replace the antenna 1 by some distribution gI (x-q) of sources of the field, and the antenna 2 by a distribution .g? (cp-rpO) of “sinks” of the field; the sources and sinks are situated on the circle r=F=r,%z, E (rt) =l. We consider that the sources corresponding to a transmitting antenna are situated at ‘p=“n, so that the axis produced by its wave bunch is parallel to the z-axis, and the sinks corresponding to the receiving antenna are situated near cp=(p,,, and in the operation of this antenna on transmission its wave bunch intersects the negative x-axis at an angle 90 (see Fig. 1).
FIG. 1 The field received by the antenna 2 from the antenna 1 in the presence of the plasma cylinder is defmed by the double integral n vo+n g= j f G(r,,r. --n %-II
cp (2.9)
which we will compare with the field go received by the antenna 2 from the antenna 1 in the absence of the plasma cylinder and in the absence of the rotation of the antenna 2 through the angle 9. (that is, in the case where the antenna 2 is directed straight at the antenna 1). The quantity go is (2.10) and the transmission coefficient
f
(along the field) is defined as the ratio
f=g/go,
(2.11)
Diffraction of a wave bunch
and &I2is the power tr~~~~on
147
coefficient.
We choose the functions gI($) and g*(p) in the form g, (9) =exp t- ~~-~~)~~(2~,) 1m&I-“‘. g, (fp)=exp [ - (ip-cp?) ‘i (25:) ] (2n52)-‘“. c,=i(l/kr,fl/kx,),
~2=i(l/kr*-l/kx2), x,=X,-i& $,=-b/x,, -?
x,=X,+id,,
(2.12) (pz= b/xa,
where X,, X2, c&SO, d,>O and b are real parameters, possessing the tendons of length; we assume that kr,Bl, kd,Bl and k&Bl, as a consequence of which the complex parameters <1 and f2 are small. In order to explain the physical significance of the functionsgl and82 and the parameters occurring in (2.12), we calculate the field (3 =E=II, produced by the antenna 1 in the absence of the plasma; it is defmed by the integral
which is most simply calculated in Cartesian coordinates, replacing the Hankel function by the approximate expression
considering that k (Z-X) B 1 and ) y-5 1
b--b)2
(Do (z-x,)‘J
2(x--x,)
I> ’
where the factor @, is independent of x and y. For x=--r* this expression is propo~ion~ to gl($) and (if kd,>l ) it is applicable for practically any x)--r. andy, and is a Gaussian wave bunch [ 121. Indeed, writing
l@,I’ b/--b)’ l@l”= [ (x_xi)2+dlz]~!2exp kdt 2(x-X,)2+d,2 f
I
’ 11
we see that on any straight line x = const the intensity of the distribution is Gaussian. The least efficient width of a Gaussian ~st~bution is realized for x = X1 (the “neck” of the Gaussian wave bunch is situated there, see Fig. l), the axis of the bunch (where the intensity is a maximum) is the straight line y = b, on moving away from the point x=X,, y= b in the direction of the the intensity falls by a factor e. The antenna 2 when y-axis through the distance 6y= (r&/k) ” working on tr~smis~on produces a similar wave bunch (Fig. 1): the axis of this Gaussian bunch passes at a distance b the axis of the plasma cylinder (b is usually called the limiting state), the paramter X2 determines the position of the neck of the bunch, the parameter d2 the effective half-width 6y = (c&/k) lh of the bunch in the neck. Therefore, in the description of formulas (2.12) it is assumed that the target distances of the wave bunches 1 and 2 are the same, that is, that on varying b the plasma cylinder is displaced along the bisector of the angle (equal to n-rp&> , formed by the axes of the bunches. It is easy to release ourselves from this constraint, but we will not do this; moreover, in the numerical calculations we will put X,=X,=0, that is, we will
148
E. S. Birger, L. A. Vainshtein and N. B. Konyukhova
assume that for b = 0 the necks of the wave bunches he on the diameters of the plasma cylinder. This choice of the functionsgl and g2 leads to excellently converging series for the quantities g and go (see section 4) and is justified by the fact that in an experimental set-up [7] the wave bunches are close to Gaussian. When necessary it is also possible to calculate the transmission coefficient for wave bunches of a different type.
3. Calculation of the radial functions It follows from the formal considerations advanced in section 2, and the computing formulas (see section 4), that to calculate the quantity g occurring in (2.11) it is necessary to calculate a sufficient number of products of the functions R, (kr)XR$ (kr) , defined by the relations (2.4)-(2.8). We have to know the products of the functions for r=r,>!‘, where the value of r is defined by the condition e(r)=1
for
r>r._
(3.1)
In this section we write for brevity kr = x. From the normalization condition (2.7) we have
[n{dR;x’x’ [R?' (x) I-’
R,
(x)Ry(‘)
(x) =
-
““‘r)
[R,(X)]-‘}jl
(3.2) 2i.
From the boundary condition (2.6) for RL’) (x) and Eq. (3.1) it is obvious that for Z> 2 =I& the function R,(l) (x) is proportional to $‘zH(t) (z>. Hence, using the recurrence formula for the Bessel functions, we obtain (3.3)
x+0
We will now determine (dR,ldx)lR,. Since e(r) is an even analytic function of r, then as the function (2.5) behaves as follows: 4(X,
Y)X”=-((Y2-1/~)+E(o)X2+O(X2).
At a point x0 sufficiently close to zero this permits us to replace the first condition of (2.6) by the linear relation (3 -4) To transfer this relation to the point x+=kr* we use a version of the stable pivotal condensation proposed in [ 131, namely we introduce the auxiliary functions av
where we require for all
ZE [x0, x~]
22x0,
the satisfaction of the normalisation condition a,‘a,+b,‘b,=l.
(3.6)
Diffraction of a wave bunch
149
Using (2.4) (3.5), (3.6), we obtain for a&x) and by(x) the system of equations -
da,
dx
-
= - L--K, (5, a,, b,) a,, (3.7)
dbv dx
= q(x, v’)~~,--Kv(x,
G, b,) bv,
where
K(x, arrb,) =
Re[b,'a,q
(.r, v) -a,*b,]
(3.8)
a,*a,+b,‘b,
is a real function. Ilere the satisfaction of condition (3.6) guarantees the stable integration of Esq. (3.7) over the whole interval {x0, x*] (the index * in Eqs. (3X+-(3.8) denotes the complex conjugate). Integrating system (3.7) from x0 to x* with the initial values +(x0) and b,(xo), satisfying by (3.4), (3.5) the condition s&b,(x*) -
Ev+*iz-
E (0)
2 (vfl)
I
202 a, (cc,) =o
and the normalisation condition (3.6) we obtain
This completes the calculation of the product R, (.L) R(Vi)(x,) . We note that in practice (see section 4) the point x* is so chosen that the tr~~ission coefficient (2.11) is unchanged when x* is increased. The calculation of the product of the radial functions for a wave polarized perpendicular to the axis of the cylinder is, by Eq. (2.8), comp~cated for e (0)~0 by the fact that in Eq. (2.4) there occurs an additional singular point r+O, at which E (rk)=O. In thiscase the function R,(x) is subject to an additional condition: it must be analytic in the upper half-plane Im 3~0; this follows from the fact that for Im e>O the singular point xk = krk lies in the lower h~f-plane (see section 1). At a point x0 sufficiently close to zero, instead of (3.4) we obtain by Eq. (1 .l) for the function e(r) the condition (3.9) For 0C1 the quantities (3.2) are calculated just as in the previous case. But if 8> 1, then in the nei~bo~hood of the singular point ~~=tka (In 8) v* we have to integrate the system (3.7) along a complex path in the upper half-plane Im r>O, for example along the path shown in Fig. 2. The remainder of the problem is solved just as before.
1-t 0
x0
“u FIG. 2
t
x* Re,
E. S. Birger, L. A. Vainshteinand N. B. Konyukhova
150
The ad~tion~ singular point xk leads to the result that in the asymptotic expression 2’00,
(3.10)
the ratio L--Y&,
(3.11)
which can be called the reflection coefficient of the cylindrical wave (of the azimuthal harmonic with index V)from the plasma cylinder, is less than unity in absolute value. For waves polarized along the z-axis we always obtain 1I?,] = 1, so that the radial function R,(x) is a purely standing wave (as X-+m ) The same will hold also for waves polarized perpendicular to the z-axis, if E (r) >O; everywhere; but if there is a singular point Xk, then for some v (see section 5) we obtain ] f, ] (1. Physically this is connected with plasma resonance, which even for infinitely small losses in the plasma leads to a fmite value of the absorbed power and incomplete reflection. hr the plane-layer plasma plasma resonance has been studied in some detail (see, for example, [14-161); here we have encountered it in a cylindrically layered plasma. We note that on replacing the cylinder by a cylinder with a continuous distribution of e(r), consisting of a large number of layers with constant E in each layer [3, 4] , this effect can be extended. The calculation of rr, proceeds as follows. Taking into account relation (3.5), we obtain r
_
y-
1-?nv2--nv2+2inv (Itm,)
2+n,2
exp (-Ziz.) ,
-
b” (X*) a, (cr.)
= nv+inzp,
whence, in particular (3.12)
We solve the recurrence equations (3.7) for every fixed Y,starting from the initial condition (3.9), and calculate 1I’, 1 by Eq. (3.12).
4. Derivation,justification and summation of the series for g and go In order to derive a computing formula for the quantity (2.9), we first find (for a wave polarized along the z-axis) the function El@, cp),satisfying Eq. (1,2) and the radiation condition (1.4), bounded in all space and possessing on the circle r =T a jump of the derivative (4.1) and cont~uous and cont~uo~ly d~fe~ntiable for r-G and r>? . Such a function Er(r, (p) corresponds physically to the field of the antenna 1 (see section 2), and mathematically to the solution of the inhomogeneous equation (1.2), on whose right side there is the product of the delta function 6 (T-F) by R, (Z-Q). &me the function gl(J/) is non-zero only for small J, (for the function gl by Eq. (2.12) this follows from the smallness of Re 51 and Im t;J , ), we can approximate it by a segment of a Fourier series
Diffraction of a wave bunch
gl’N4C$)=$ 2 yi(v)e”“‘,
151
rl(v)= j gl($)e-i’*d$,
v=-N
-
1
replacing the limits by -m and m in the integral. For large values of N the functions gt ($) and g!“)(q) are close, therefore, replacingg1 byglcN) on the right side of (4.1), we obtain the corresponding function E, W) in the form Ey)
fJ ~l(~)RV(kr)R,(l)(kr)ei”“-“,
(r,cp)=[k(r~)l’~]-l
r< F,
v=-N
where, because of the existence of Green’s function [l] , this expression for all r and cpand for large N is close to the exact function El(r, cp). Since the quantity (2.9) of interest to us can be represented in the form ‘P.l+n
we
arrive at the expression
g(N)
=-
N
I kr. c
r,(~)~~(v)R,(kr,)R,(”
(kr.)exp[ivh-93)
I,
v=-N
close tog. If we take into account the fact that R-vR~!,‘=R,R~l’ (this follows from the fact that only v2 occurs in Fq. (2.4)), we obtain for g(N) the final expression 1
CN) _
B
--z
N 7 (Y, cpO)R, (kr.) Rv(‘) (kr.) , c
(4.2)
v=ll
in which
y (v, 90)=yi (v) y2(y)exp [ivkc-90) 1 +rt(-y)y,(-v)
exp [iv (cpo--n)],
v=l,
2. . . . .
and for the Gaussian wave bunches (2.12) we obtain y, (Y) y2 (Y) ==exp [ - (f;+2ilkr*)
v2/2+Spv],
p=kb.
g=i(llkz,-l/kz!J,
‘I’hesame expression (4.2) is obtained for a wave polarized perpendicular to the z-axis: the function y (Y, qpo)remains the same, and the functions R, andR,(l) are calculated rather differently (see section 3). In the absence of the plasma and with ~0 = 0 expression (4.2) assumes the form
cy N
(N)
go =
(Y, O)J, (kr.)Hi”
W-0
(kr.)
;
(4.3)
it can be obtained by formal integration of a finite sum corresponding to the series (2.2); in exactly the same way from the series (2.3) we can obtain the expression (4.2).
152
E. S. Birger, L. A. Vainshteinand N. 8. Konyukhova
The decrease of the successive terms in the expressions (4.2) and (4.3) is determined by the fundamental product y1(Y) yZ(Y) , that is, it is exponential, unlike the series (2.2) and (2.3), whose terms decrease like 1Iv. In order to avoid the accumulation of large quantities, we multiply the expressions for gtN) and gOcN) by kr. esp(-gl.~“/Z!) , then Jyl(v)r2(~) Iyi(-v)y2(-v)
I=exp
[-i/2Re
I=exp
5(+~)“1,
[-‘/~ReI;(v+~)~l,
from which it is obvious that 1y (Y, cpo)1~6, if V> [ 2 In (2/6)/Re possible to take N=m+ I p 1, m = 10 to 20.
G]‘“+ I p) , therefore it is
The functions J,(x) and H(*j (x) were calculated using the algorithms proposed by L. N. Karmazina and E. A. Chistova.
5. Numerical results and conclusions Calculations were performed as follows: all the parameters were fixed except for the limiting distance b or the quantity /_t= kb proportional to it, and the quantitiesgcN) and gocN) and their ratio flN) were calculated as a function of /J from the value /J = 0 to values of /J (positive or negative) yielding with the desired accuracy flN) = 1 (for q. = 0) or another constant value (for p. = n12 this value was practically zero). Such values of p can be estimated both from Fig. 1 and also directly in the computing process. The values of N and x* = km were selected empirically to be such that on further increasing them the transmission coefficient fno longer changed. These values are indicated on the graphs given below. We present only some numerical results illustrating the principal physical phenomena: 1) refraction in a transparent plasma, where G-0; 2) the shading of the wave bunch by the non-transparent plasma, for which e -=ZO (diffraction); 3) polarisation effects arising at the boundary of the transparent and non-transparent plasma (for EFO, that is for Frk). Refraction is obvious in Fig. 3, diffraction in Fig. 4; both graphs correspond to the value q. = 0, for which the antennas are aimed at one another and j@) is an even function. In the first caseforb=Othereisalocalmaximum If(b) 1’) in the second case a flat minimum where Curves of both types are observed experimentally and can be interpreted practically If(b) 1’=0. in the spirit of geometrical optics [7]. It is interesting to note that for p. = 0 the polarisation effects are extremely weak: for ku = 10 the curves for both polarisations differ negligibly, and for ka = 20 they are practically identical (E is the full-line curves, His dashed). LO
Power transmission coefficient for 0 = 0.5; 1 is for ka = 10, 5 = l/30, X* = 80,N= 60; 2 is for ka = 20,{ = 1/20,x* = 12O,N= 90.
Diffraction of a wave bunch
153
FIG. 4 Power transmission coefficient for 8 = 1.5: 1 is for ka = 10, { = 1/30,x* = 80, N = 60; 2 is for ka = 20, *:{ = 1/20,x* = 160, N= 90. Stronger polarisation effects are observed for (p,=n/2. Figure 5 shows the corresponding curves for If(b) 1” for various values of 6, showing that for perpendicular antennas 1 and 2 (cpo=n/2) and 821 we,obtain a noticeable transmission coefficient, different for different polarizations, the intensity ratio for b = 0 being approximately 0.5, and at the maximum 1f(b) 1z approximately 0.7. In the conditions we have considered (kaBl) the wave transmission coefficient polarised perpendicular to the axis of the cylinder is less than the wave transmission coefficient polarised along the z-axis, the cause of this being plasmic resonance (see the end of section 3). We note that at long waves (kacl) plasma resonance of another kind appears, in which a wave polarized perpendicular to the cylinder is quite strongly scattered [2] .
--0.4
0
L7.4 0.6
0
0.4
,x,, -0.4
0.6
c 0.2 , / AI -0.4
/
0
0.4
0.6
FIG. 5 Power transmission coefficients for ka = 20,{ = 1/20,x* = 160, N = 40:. a is for 6 = 0.95; b is for 8=O.99;cisfor6=1.05,krk=4.4;disfore=1.5,krk=12.7;eisfore=2,krk=16.6.
154
I!?.S Birger, L. A. Vatishrein and N. B. Konyukhova
FIG. 6 A measure of plasma resonance is the quantity (3.12) shown in Fig. 6 (for convenience the points are .connected by dashed curves and correspond to the parameteis indicated in Fig. 5). We see [ 16]), a noticeable see that I I’, 1 falls to about 0.6 (for a plane-layered plasma 1rv 1sO.7, difference of 1F, 1 from unity being observed only for comparatively small values of (YG&) 1 and when v increases the plasma resonance does not decrease. The latter fact explains the strong curves of E andHin Fig. 5 for large values of 1~1,when terms with large values of v existed in the expression (4.6) (see the end of section 4). The calculations were performed on the BESMd computer, the computing time for one curve in Figs. 3-5 was from 20 to 40 min. Experimental results have been obtained for larger values of ka and smaller values of s [7], and for such values calculations by the method developed above become difficult. Therefore this work does not enable us to perform a qu~tit~tive analysis of the results of mea~remen~. Its aim was different: it permits the determination of the accuracy of the approximate (asymptotic) formulas used for large values of ka and arbitrary values of 5, enabling the converse problem to be solved - the reconstruction of the function e(r) from the measured function fib), These topics are discussed in [ 171. We thank P. L. Kapitsa for supporting this work and E. A. Tishchenko for useful discussions. ~uns~red by J. Berry REFERENCES 1.
DANILOVA, I. A. The third exterior boundary value problem for a second-order elliptic equation. I. Diffeerennts utwiya, 7,4,661-669,197l; II. 8,3,486-493,1972.
2.
KAISER, T. R. and CLOSS, R. L. Theory of radio reflections from meteor trails. Phil. Msg., 43,336, 1952.
3.
YAKIMENKO, I. P. Scattering of sound by an inhomogeneous
4.
FONG, K. Computation of ~~ks~~e~g cross section of pktsma-clad metal cylinders by the “Layer Method”. IEEE IIznns,AP-16, No. 1,138-140,1968.
5.
NOZAKA, M., SATO, G. and TAKAKU, K. Scattering of electromagnetic inhombgeneous plasma.J. Hays Sot. Japan, 24,1,172-184,1968.
cylinder.Akust.
l-32,
zh., 16,1,112-121,1968.
waves by a cylindrically
Beam equations for randomly curved cylinders
155
waves with cylindrical plasmas. IEEE Trans.,
6.
GAL, G. and GIBSON, W. E. Interaction of electromagnetic AP-16, No. 4,468-475, 1968.
1.
TISHCHENKO, E. A. and ZATSEPIN, V. G. The active submilliieter diagnostics of a plasma fiient microwave discharge at high pressure. Z/r. eksp. teor. fiz., 68,2,547-561, 1975.
8.
ASKOLI-BARTOLI, U., BAD’UAMI, M., F. De MARKO, DUSHIN, L. A. and P’ERONI, L. The probing of an inhomogeneous plasma by electromagnetic waves (Zondirovanie neodnorodnoi plazmy elektromagnitnymi vohtami), Atomizdat, Moscow, 1973.
9.
DUSHIN, L. A. Microwave interferometry for measuring the density of plasma in a pulsed gas discharge (SVCh-interferometry dlya izmereniya plotnosti plazmy v impul’snom gazovom razryade), Atomizdat, Moscow, 1973.
10. SHMOYS, J. Proposed diagnostic method for cylindrical plasmas. J. Appl. Phys., 32,4,689-695,196l. 11. WATSON, G. N. Theory of Bessel functions (Teoriya besselevykh funktsii), Izd-vo in. lit., Moscow, 1949. 12. DESCHAMPS, G. A. The Guassian beam as a bundle of complex rays. Electron. Letters, 7,23,684-685, 1971.
13. BIRGER, E. S. and KONYUKHOVA, N. B. Numerical calculation of the propagation of radio waves in a vertically-inhomogeneous troposphere. Radiotekhn. elektronika, 14,7,1147-1156,1969. 14. DENISOV, N. G. On a feature of the field of an electromagnetic plasma. Zh. eksp. teor. fiz., 31,4 (IO), 609-619. 1956. 15. GINZBURG, V. L. Propagation of electromagnetic vom v p&me), “Nauka”, Moscow, 1967.
wave propagating in an inhomogeneous
waves in a plasma (Rasprostranenie
elektromagnitnykh
16. BUDDEN, K. G. Radio waves in the ionosphere. Univ. Press, Cambridge, 1967. 17. VAINSHTEIN, L. A., BIRGER, E. S., KONYUKHOVA, N. B., KOSARFV, E. L. and PRUDKOVSKII, G. P. Short-wave diagnostics of a plasma filament. Fiz. plazmy, 2,4,658-671, 1976.
BEAM EQUATIONS FOR RANDOMLY CURVED CYLINDERS WITH A REFRACTIVE INDEX THAT VARIES OVER THE CROSS-SECTION AND SOME CASES OF THEIR INTEGRATION* R. P.TARASOV
Moscow (Received 18 March 1975) IT IS SHOWN that the problem of the integration of the beam equations for cylinders with a refractive index that varies over the cross-section curved along the arc of a circle (helical line), is equivalent to the problem of integrating the differential equations of the one-parameter group of transformations, realized with all permissible trajectories within a torus (a cylinder bent along a helical line), of the set consisting of the points of the transverse section and the unit directional vectors. The use of exponential representations for the operator of the group enables us to consider efficient methods of integrating the beam equations in such media. For arbitrary cylinders curved in one plane (space) the beam equations can be integrated numerically with arbitrary accuracy, if the solutions for cylinders curved along the arc of a circle (helical line) have been determined.
1. INTRODUCTION
The problem of the propagation of waves in inhomogeneous media is associated with the difficulties of solving second-order differential equations with variable coefficients. The number of cases for which the analytic solution is known does not include a whole set of inhomogeneous media *Zh v%hisL Mat. mat. Fiz., 16,6,1539-1550,1976.
143
3.
TIKHONOV, N. A. Invariantsof the non-linear equation of oscillations. Zh. eks. teor. fiz., 66,1,109-115, 1974.
4.
KELDYSH, M. V. On the completeness of the e~enfunctions of some classes of non-selfconjugate linear operators. Usp. mot. rim&, t&4,15-41,197l.
D~FRA~ION
OF A WAVE BUNCH ON A PLASMA CYLINDER*
E. S. BIRGER, L. A. VAINSHTEIN and N. B. KONWKHOVA
Moscow (Received I 1 April 1975) A METHOD for the numerical solution of the twodimensional problem of the diffraction of an electroma~etic wave by a plasma cylinder, whose dielectric const~t is a fiction of the radius-vector r, being unity for large values of I, is described. In this problem the fields are represented by series of azimuthal harmonics, the corresponding radial functions being found by numerical integration of second-order linear differential equations with singular points. A calculation is made of the result of this averaging of the Green’s function over the coordinates of the source and point of observation, which corresponds to the irradiation of the plasma by an antenna creating Gaussian wave bunches, and to the reception of the diffraction field by a similar receive antenna. Numerical results are given, and the principal physical phenomena arising on the diffraction of a wave bunch by a plasma cylinder are discussed.
1. Statement of the problem
We consider the two-dimensional problem of the diffraction of an electromagnetic wave by a plasma cylinder, in which the dielectric constant e(r) is a monotonic~ increasing function of the distance from the axis of the cylinder, the cylinder being assumed of infinite length, and the field of the incident wave independent of the coordinate z along the axis of the cylinder. In numerical cal$ulations we describe a plasma cylinder (with the time relation a distribution of the complex dielectric constant E
(f-1=1-0
e-I”‘)
exp (4/a’>,
by
wo
where f is the distance from the axis of the cylinder, II is the effective radius of the cylinder, and 8 is a positive parameter. For 6> $ an important value is the critical radius rk for which E (tx) =O, for r>rA and e(r)
If a wave polarized along the z-axis falls on the plasma cylinder, then in the cylindrical coordinate system r, cp,z the field has components E,=E,
fj,=---
and the function E(r, ip) satisfies the equation Vh. vfihhisl.Mat. mat. Fiz., 16,6,1526-1538,1976.
i $13 kr dip
Hi=--,
i SE k dr
144
E. S. Birger, L. A. Vainshteinand N. B. Konyukhova
+-
1 d’E
r2
-
+ k% (r) E=O
dcp”
(1.2)
everywhere with the exception of the sources (where a right side is present in (1.2)): k = w/c is the wave number, c is the velocity of light in a vacuum, where e = 1. If the wave is polarized perpendicular to the z-axis, the field will have components H,=H,
E,=LdH kera(p’
E,=-
i ,aH ke dr ’
where H(r, 9) is the solution of the equation -I- -
1 r’
-
d2H
+ kZE (r) H=O,
w
(1.3)
which is rather more complicated than Eq. (1.2), and also applicable only outside the sources of the field. Equations (1.2) and (1.3) follow from Maxwell’s equations for a space with cylindrical plasma. It is known that for these equations, supplemented by the radiation condition
(1.4) Green’s function exists [l] . As we shall see below this enables us to calculate the diffraction of a directed wave bunch by the plasma and the action of the diffracted bunch on the receive antenna, which is also directional. It must be pointed out that the two-dimensional problem of diffraction by a cylinder, inhomogeneous in the radial direction, has been discussed in many papers (see, for example, [2-6]), but even in the : of them directed towards numerical methods [3,4,6], the directivity of the receiving and transmitting antennas was not considered. Out statement of the problem corresponds to the experimental method [7] and in the limiting case of extremely narrow wave bunches and extremely short waves (ka-+ 00) it degenerates into the geometrical optics problem of the transillumination of a cylindrical plasma [8-lo]. However, we are interested in not too narrow bunches (the effective width of the bunch is, for example, a third to a quarter of the effective diameter 2a of the plasma) and not too short waves (ka = 10 to 20), when the numerical method explained below is most suitable for the solution of the problem.
2. Green’s function and the transmission coefficient In the absence of the plasma (E = I) Green’s function for Eqs. (1.2) and (1.3) is proportional to Hci) 0 (kd) , where H dl) is the Hankel function of the first kind, and R=[r’+F’-2rFcos
(cp-q)]“’
(2.1)
is the distance between the point of observation with the polar coordinates r, and the source (the -luminous filament) with coordinates r, 9. The function Hy) (kR) has at the origin (for R = 0)
Diffraction of a wave bunch
145
a logarithmic singularity and the following expansion in terms of the azimuthal harmonics:
I+”
x,.ly (kr) II,“’ (kr) I!2 v=*
(kR) =
cos v (cp-cp)
)
x0=1, (2.2)
x,=?.Cz=. . . =2, applicable for r
lcvRv (kr)R,“’
(kF)cos v(p-9).
(2.3)
v-0
in
which for r>F
instead of R, (kr) Rcj)(kF) it is necessary to write R, (k?) R (j)( kr) .
The functions R, and R,(l)
satisfy the equation d”R,/d(kr)2+q(kr,
v)R,=O,
(2.4)
where q(kr, v)=&(r)-(v2--‘II)
(kr)-“,
(2.5)
and the supplementary conditions lim R, (kr) =O,
,ti~~ $$(R.‘“)
-l=i
kr-.o
(2.6)
and R
dR?
dR, R,“’ = 2
Y---z dkr
(2.7)
n’
where the first condition of (2.6) is connected with the finiteness of Green’s function for r = 0, the second condition of (2.6) with the radiation condition (1.4), and in condition (2.7) the Wronskian of the functions R, and R,(l) is identified with the Wronskian of the functions (kr) ‘A,/,(Jcr) and (kr) ‘I’H$ (kr) in the expansion (2.2), which is connected with the same singularity of the functions (2.2) and (2.3) for kR = 0. The same formal expansion for Green’s function of Eq. (13) differs from (2.3) by the fact that the factor [ii (r?) ‘“I e-i is replaced by [E (r) E (F) 1’”[ Ic (rI;) ‘“1 --i, the radial functions R, and R,(l) satisfy Eq. (2.4), in which p(kr,v)=c(r)+$(rz)
[2k2re(r)]-’
(2.8) -3
(
$
1
’ [2kc(r)]-“--(v2-‘l,)
and the conditions (2.6) and (2.7) remain unchanged.
(kr)-2,
146
E. S. Birger, L. A. Vainshteinand N. B. Konyukhova
However, we are interested not in the Green’s function determining the field at the point -of observation r, 9 from the point source with coordinates r, 9, but in the field radiated by the transmitting antenna 1 and received by the receiving antenna 2. If both antennas are situated outside the plasma (E (r) =e (F) =I>, then without going beyond the range of the two-dimensional problem, we can replace the antenna 1 by some distribution gI (x-q) of sources of the field, and the antenna 2 by a distribution .g? (cp-rpO) of “sinks” of the field; the sources and sinks are situated on the circle r=F=r,%z, E (rt) =l. We consider that the sources corresponding to a transmitting antenna are situated at ‘p=“n, so that the axis produced by its wave bunch is parallel to the z-axis, and the sinks corresponding to the receiving antenna are situated near cp=(p,,, and in the operation of this antenna on transmission its wave bunch intersects the negative x-axis at an angle 90 (see Fig. 1).
FIG. 1 The field received by the antenna 2 from the antenna 1 in the presence of the plasma cylinder is defmed by the double integral n vo+n g= j f G(r,,r. --n %-II
cp (2.9)
which we will compare with the field go received by the antenna 2 from the antenna 1 in the absence of the plasma cylinder and in the absence of the rotation of the antenna 2 through the angle 9. (that is, in the case where the antenna 2 is directed straight at the antenna 1). The quantity go is (2.10) and the transmission coefficient
f
(along the field) is defined as the ratio
f=g/go,
(2.11)
Diffraction of a wave bunch
and &I2is the power tr~~~~on
147
coefficient.
We choose the functions gI($) and g*(p) in the form g, (9) =exp t- ~~-~~)~~(2~,) 1m&I-“‘. g, (fp)=exp [ - (ip-cp?) ‘i (25:) ] (2n52)-‘“. c,=i(l/kr,fl/kx,),
~2=i(l/kr*-l/kx2), x,=X,-i& $,=-b/x,, -?
x,=X,+id,,
(2.12) (pz= b/xa,
where X,, X2, c&SO, d,>O and b are real parameters, possessing the tendons of length; we assume that kr,Bl, kd,Bl and k&Bl, as a consequence of which the complex parameters <1 and f2 are small. In order to explain the physical significance of the functionsgl and82 and the parameters occurring in (2.12), we calculate the field (3 =E=II, produced by the antenna 1 in the absence of the plasma; it is defmed by the integral
which is most simply calculated in Cartesian coordinates, replacing the Hankel function by the approximate expression
considering that k (Z-X) B 1 and ) y-5 1
b--b)2
(Do (z-x,)‘J
2(x--x,)
I> ’
where the factor @, is independent of x and y. For x=--r* this expression is propo~ion~ to gl($) and (if kd,>l ) it is applicable for practically any x)--r. andy, and is a Gaussian wave bunch [ 121. Indeed, writing
l@,I’ b/--b)’ l@l”= [ (x_xi)2+dlz]~!2exp kdt 2(x-X,)2+d,2 f
I
’ 11
we see that on any straight line x = const the intensity of the distribution is Gaussian. The least efficient width of a Gaussian ~st~bution is realized for x = X1 (the “neck” of the Gaussian wave bunch is situated there, see Fig. l), the axis of the bunch (where the intensity is a maximum) is the straight line y = b, on moving away from the point x=X,, y= b in the direction of the the intensity falls by a factor e. The antenna 2 when y-axis through the distance 6y= (r&/k) ” working on tr~smis~on produces a similar wave bunch (Fig. 1): the axis of this Gaussian bunch passes at a distance b the axis of the plasma cylinder (b is usually called the limiting state), the paramter X2 determines the position of the neck of the bunch, the parameter d2 the effective half-width 6y = (c&/k) lh of the bunch in the neck. Therefore, in the description of formulas (2.12) it is assumed that the target distances of the wave bunches 1 and 2 are the same, that is, that on varying b the plasma cylinder is displaced along the bisector of the angle (equal to n-rp&> , formed by the axes of the bunches. It is easy to release ourselves from this constraint, but we will not do this; moreover, in the numerical calculations we will put X,=X,=0, that is, we will
148
E. S. Birger, L. A. Vainshtein and N. B. Konyukhova
assume that for b = 0 the necks of the wave bunches he on the diameters of the plasma cylinder. This choice of the functionsgl and g2 leads to excellently converging series for the quantities g and go (see section 4) and is justified by the fact that in an experimental set-up [7] the wave bunches are close to Gaussian. When necessary it is also possible to calculate the transmission coefficient for wave bunches of a different type.
3. Calculation of the radial functions It follows from the formal considerations advanced in section 2, and the computing formulas (see section 4), that to calculate the quantity g occurring in (2.11) it is necessary to calculate a sufficient number of products of the functions R, (kr)XR$ (kr) , defined by the relations (2.4)-(2.8). We have to know the products of the functions for r=r,>!‘, where the value of r is defined by the condition e(r)=1
for
r>r._
(3.1)
In this section we write for brevity kr = x. From the normalization condition (2.7) we have
[n{dR;x’x’ [R?' (x) I-’
R,
(x)Ry(‘)
(x) =
-
““‘r)
[R,(X)]-‘}jl
(3.2) 2i.
From the boundary condition (2.6) for RL’) (x) and Eq. (3.1) it is obvious that for Z> 2 =I& the function R,(l) (x) is proportional to $‘zH(t) (z>. Hence, using the recurrence formula for the Bessel functions, we obtain (3.3)
x+0
We will now determine (dR,ldx)lR,. Since e(r) is an even analytic function of r, then as the function (2.5) behaves as follows: 4(X,
Y)X”=-((Y2-1/~)+E(o)X2+O(X2).
At a point x0 sufficiently close to zero this permits us to replace the first condition of (2.6) by the linear relation (3 -4) To transfer this relation to the point x+=kr* we use a version of the stable pivotal condensation proposed in [ 131, namely we introduce the auxiliary functions av
where we require for all
ZE [x0, x~]
22x0,
the satisfaction of the normalisation condition a,‘a,+b,‘b,=l.
(3.6)
Diffraction of a wave bunch
149
Using (2.4) (3.5), (3.6), we obtain for a&x) and by(x) the system of equations -
da,
dx
-
= - L--K, (5, a,, b,) a,, (3.7)
dbv dx
= q(x, v’)~~,--Kv(x,
G, b,) bv,
where
K(x, arrb,) =
Re[b,'a,q
(.r, v) -a,*b,]
(3.8)
a,*a,+b,‘b,
is a real function. Ilere the satisfaction of condition (3.6) guarantees the stable integration of Esq. (3.7) over the whole interval {x0, x*] (the index * in Eqs. (3X+-(3.8) denotes the complex conjugate). Integrating system (3.7) from x0 to x* with the initial values +(x0) and b,(xo), satisfying by (3.4), (3.5) the condition s&b,(x*) -
Ev+*iz-
E (0)
2 (vfl)
I
202 a, (cc,) =o
and the normalisation condition (3.6) we obtain
This completes the calculation of the product R, (.L) R(Vi)(x,) . We note that in practice (see section 4) the point x* is so chosen that the tr~~ission coefficient (2.11) is unchanged when x* is increased. The calculation of the product of the radial functions for a wave polarized perpendicular to the axis of the cylinder is, by Eq. (2.8), comp~cated for e (0)~0 by the fact that in Eq. (2.4) there occurs an additional singular point r+O, at which E (rk)=O. In thiscase the function R,(x) is subject to an additional condition: it must be analytic in the upper half-plane Im 3~0; this follows from the fact that for Im e>O the singular point xk = krk lies in the lower h~f-plane (see section 1). At a point x0 sufficiently close to zero, instead of (3.4) we obtain by Eq. (1 .l) for the function e(r) the condition (3.9) For 0C1 the quantities (3.2) are calculated just as in the previous case. But if 8> 1, then in the nei~bo~hood of the singular point ~~=tka (In 8) v* we have to integrate the system (3.7) along a complex path in the upper half-plane Im r>O, for example along the path shown in Fig. 2. The remainder of the problem is solved just as before.
1-t 0
x0
“u FIG. 2
t
x* Re,
E. S. Birger, L. A. Vainshteinand N. B. Konyukhova
150
The ad~tion~ singular point xk leads to the result that in the asymptotic expression 2’00,
(3.10)
the ratio L--Y&,
(3.11)
which can be called the reflection coefficient of the cylindrical wave (of the azimuthal harmonic with index V)from the plasma cylinder, is less than unity in absolute value. For waves polarized along the z-axis we always obtain 1I?,] = 1, so that the radial function R,(x) is a purely standing wave (as X-+m ) The same will hold also for waves polarized perpendicular to the z-axis, if E (r) >O; everywhere; but if there is a singular point Xk, then for some v (see section 5) we obtain ] f, ] (1. Physically this is connected with plasma resonance, which even for infinitely small losses in the plasma leads to a fmite value of the absorbed power and incomplete reflection. hr the plane-layer plasma plasma resonance has been studied in some detail (see, for example, [14-161); here we have encountered it in a cylindrically layered plasma. We note that on replacing the cylinder by a cylinder with a continuous distribution of e(r), consisting of a large number of layers with constant E in each layer [3, 4] , this effect can be extended. The calculation of rr, proceeds as follows. Taking into account relation (3.5), we obtain r
_
y-
1-?nv2--nv2+2inv (Itm,)
2+n,2
exp (-Ziz.) ,
-
b” (X*) a, (cr.)
= nv+inzp,
whence, in particular (3.12)
We solve the recurrence equations (3.7) for every fixed Y,starting from the initial condition (3.9), and calculate 1I’, 1 by Eq. (3.12).
4. Derivation,justification and summation of the series for g and go In order to derive a computing formula for the quantity (2.9), we first find (for a wave polarized along the z-axis) the function El@, cp),satisfying Eq. (1,2) and the radiation condition (1.4), bounded in all space and possessing on the circle r =T a jump of the derivative (4.1) and cont~uous and cont~uo~ly d~fe~ntiable for r-G and r>? . Such a function Er(r, (p) corresponds physically to the field of the antenna 1 (see section 2), and mathematically to the solution of the inhomogeneous equation (1.2), on whose right side there is the product of the delta function 6 (T-F) by R, (Z-Q). &me the function gl(J/) is non-zero only for small J, (for the function gl by Eq. (2.12) this follows from the smallness of Re 51 and Im t;J , ), we can approximate it by a segment of a Fourier series
Diffraction of a wave bunch
gl’N4C$)=$ 2 yi(v)e”“‘,
151
rl(v)= j gl($)e-i’*d$,
v=-N
-
1
replacing the limits by -m and m in the integral. For large values of N the functions gt ($) and g!“)(q) are close, therefore, replacingg1 byglcN) on the right side of (4.1), we obtain the corresponding function E, W) in the form Ey)
fJ ~l(~)RV(kr)R,(l)(kr)ei”“-“,
(r,cp)=[k(r~)l’~]-l
r< F,
v=-N
where, because of the existence of Green’s function [l] , this expression for all r and cpand for large N is close to the exact function El(r, cp). Since the quantity (2.9) of interest to us can be represented in the form ‘P.l+n
we
arrive at the expression
g(N)
=-
N
I kr. c
r,(~)~~(v)R,(kr,)R,(”
(kr.)exp[ivh-93)
I,
v=-N
close tog. If we take into account the fact that R-vR~!,‘=R,R~l’ (this follows from the fact that only v2 occurs in Fq. (2.4)), we obtain for g(N) the final expression 1
CN) _
B
--z
N 7 (Y, cpO)R, (kr.) Rv(‘) (kr.) , c
(4.2)
v=ll
in which
y (v, 90)=yi (v) y2(y)exp [ivkc-90) 1 +rt(-y)y,(-v)
exp [iv (cpo--n)],
v=l,
2. . . . .
and for the Gaussian wave bunches (2.12) we obtain y, (Y) y2 (Y) ==exp [ - (f;+2ilkr*)
v2/2+Spv],
p=kb.
g=i(llkz,-l/kz!J,
‘I’hesame expression (4.2) is obtained for a wave polarized perpendicular to the z-axis: the function y (Y, qpo)remains the same, and the functions R, andR,(l) are calculated rather differently (see section 3). In the absence of the plasma and with ~0 = 0 expression (4.2) assumes the form
cy N
(N)
go =
(Y, O)J, (kr.)Hi”
W-0
(kr.)
;
(4.3)
it can be obtained by formal integration of a finite sum corresponding to the series (2.2); in exactly the same way from the series (2.3) we can obtain the expression (4.2).
152
E. S. Birger, L. A. Vainshteinand N. 8. Konyukhova
The decrease of the successive terms in the expressions (4.2) and (4.3) is determined by the fundamental product y1(Y) yZ(Y) , that is, it is exponential, unlike the series (2.2) and (2.3), whose terms decrease like 1Iv. In order to avoid the accumulation of large quantities, we multiply the expressions for gtN) and gOcN) by kr. esp(-gl.~“/Z!) , then Jyl(v)r2(~) Iyi(-v)y2(-v)
I=exp
[-i/2Re
I=exp
5(+~)“1,
[-‘/~ReI;(v+~)~l,
from which it is obvious that 1y (Y, cpo)1~6, if V> [ 2 In (2/6)/Re possible to take N=m+ I p 1, m = 10 to 20.
G]‘“+ I p) , therefore it is
The functions J,(x) and H(*j (x) were calculated using the algorithms proposed by L. N. Karmazina and E. A. Chistova.
5. Numerical results and conclusions Calculations were performed as follows: all the parameters were fixed except for the limiting distance b or the quantity /_t= kb proportional to it, and the quantitiesgcN) and gocN) and their ratio flN) were calculated as a function of /J from the value /J = 0 to values of /J (positive or negative) yielding with the desired accuracy flN) = 1 (for q. = 0) or another constant value (for p. = n12 this value was practically zero). Such values of p can be estimated both from Fig. 1 and also directly in the computing process. The values of N and x* = km were selected empirically to be such that on further increasing them the transmission coefficient fno longer changed. These values are indicated on the graphs given below. We present only some numerical results illustrating the principal physical phenomena: 1) refraction in a transparent plasma, where G-0; 2) the shading of the wave bunch by the non-transparent plasma, for which e -=ZO (diffraction); 3) polarisation effects arising at the boundary of the transparent and non-transparent plasma (for EFO, that is for Frk). Refraction is obvious in Fig. 3, diffraction in Fig. 4; both graphs correspond to the value q. = 0, for which the antennas are aimed at one another and j@) is an even function. In the first caseforb=Othereisalocalmaximum If(b) 1’) in the second case a flat minimum where Curves of both types are observed experimentally and can be interpreted practically If(b) 1’=0. in the spirit of geometrical optics [7]. It is interesting to note that for p. = 0 the polarisation effects are extremely weak: for ku = 10 the curves for both polarisations differ negligibly, and for ka = 20 they are practically identical (E is the full-line curves, His dashed). LO
Power transmission coefficient for 0 = 0.5; 1 is for ka = 10, 5 = l/30, X* = 80,N= 60; 2 is for ka = 20,{ = 1/20,x* = 12O,N= 90.
Diffraction of a wave bunch
153
FIG. 4 Power transmission coefficient for 8 = 1.5: 1 is for ka = 10, { = 1/30,x* = 80, N = 60; 2 is for ka = 20, *:{ = 1/20,x* = 160, N= 90. Stronger polarisation effects are observed for (p,=n/2. Figure 5 shows the corresponding curves for If(b) 1” for various values of 6, showing that for perpendicular antennas 1 and 2 (cpo=n/2) and 821 we,obtain a noticeable transmission coefficient, different for different polarizations, the intensity ratio for b = 0 being approximately 0.5, and at the maximum 1f(b) 1z approximately 0.7. In the conditions we have considered (kaBl) the wave transmission coefficient polarised perpendicular to the axis of the cylinder is less than the wave transmission coefficient polarised along the z-axis, the cause of this being plasmic resonance (see the end of section 3). We note that at long waves (kacl) plasma resonance of another kind appears, in which a wave polarized perpendicular to the cylinder is quite strongly scattered [2] .
--0.4
0
L7.4 0.6
0
0.4
,x,, -0.4
0.6
c 0.2 , / AI -0.4
/
0
0.4
0.6
FIG. 5 Power transmission coefficients for ka = 20,{ = 1/20,x* = 160, N = 40:. a is for 6 = 0.95; b is for 8=O.99;cisfor6=1.05,krk=4.4;disfore=1.5,krk=12.7;eisfore=2,krk=16.6.
154
I!?.S Birger, L. A. Vatishrein and N. B. Konyukhova
FIG. 6 A measure of plasma resonance is the quantity (3.12) shown in Fig. 6 (for convenience the points are .connected by dashed curves and correspond to the parameteis indicated in Fig. 5). We see [ 16]), a noticeable see that I I’, 1 falls to about 0.6 (for a plane-layered plasma 1rv 1sO.7, difference of 1F, 1 from unity being observed only for comparatively small values of (YG&) 1 and when v increases the plasma resonance does not decrease. The latter fact explains the strong curves of E andHin Fig. 5 for large values of 1~1,when terms with large values of v existed in the expression (4.6) (see the end of section 4). The calculations were performed on the BESMd computer, the computing time for one curve in Figs. 3-5 was from 20 to 40 min. Experimental results have been obtained for larger values of ka and smaller values of s [7], and for such values calculations by the method developed above become difficult. Therefore this work does not enable us to perform a qu~tit~tive analysis of the results of mea~remen~. Its aim was different: it permits the determination of the accuracy of the approximate (asymptotic) formulas used for large values of ka and arbitrary values of 5, enabling the converse problem to be solved - the reconstruction of the function e(r) from the measured function fib), These topics are discussed in [ 171. We thank P. L. Kapitsa for supporting this work and E. A. Tishchenko for useful discussions. ~uns~red by J. Berry REFERENCES 1.
DANILOVA, I. A. The third exterior boundary value problem for a second-order elliptic equation. I. Diffeerennts utwiya, 7,4,661-669,197l; II. 8,3,486-493,1972.
2.
KAISER, T. R. and CLOSS, R. L. Theory of radio reflections from meteor trails. Phil. Msg., 43,336, 1952.
3.
YAKIMENKO, I. P. Scattering of sound by an inhomogeneous
4.
FONG, K. Computation of ~~ks~~e~g cross section of pktsma-clad metal cylinders by the “Layer Method”. IEEE IIznns,AP-16, No. 1,138-140,1968.
5.
NOZAKA, M., SATO, G. and TAKAKU, K. Scattering of electromagnetic inhombgeneous plasma.J. Hays Sot. Japan, 24,1,172-184,1968.
cylinder.Akust.
l-32,
zh., 16,1,112-121,1968.
waves by a cylindrically
Beam equations for randomly curved cylinders
155
waves with cylindrical plasmas. IEEE Trans.,
6.
GAL, G. and GIBSON, W. E. Interaction of electromagnetic AP-16, No. 4,468-475, 1968.
1.
TISHCHENKO, E. A. and ZATSEPIN, V. G. The active submilliieter diagnostics of a plasma fiient microwave discharge at high pressure. Z/r. eksp. teor. fiz., 68,2,547-561, 1975.
8.
ASKOLI-BARTOLI, U., BAD’UAMI, M., F. De MARKO, DUSHIN, L. A. and P’ERONI, L. The probing of an inhomogeneous plasma by electromagnetic waves (Zondirovanie neodnorodnoi plazmy elektromagnitnymi vohtami), Atomizdat, Moscow, 1973.
9.
DUSHIN, L. A. Microwave interferometry for measuring the density of plasma in a pulsed gas discharge (SVCh-interferometry dlya izmereniya plotnosti plazmy v impul’snom gazovom razryade), Atomizdat, Moscow, 1973.
10. SHMOYS, J. Proposed diagnostic method for cylindrical plasmas. J. Appl. Phys., 32,4,689-695,196l. 11. WATSON, G. N. Theory of Bessel functions (Teoriya besselevykh funktsii), Izd-vo in. lit., Moscow, 1949. 12. DESCHAMPS, G. A. The Guassian beam as a bundle of complex rays. Electron. Letters, 7,23,684-685, 1971.
13. BIRGER, E. S. and KONYUKHOVA, N. B. Numerical calculation of the propagation of radio waves in a vertically-inhomogeneous troposphere. Radiotekhn. elektronika, 14,7,1147-1156,1969. 14. DENISOV, N. G. On a feature of the field of an electromagnetic plasma. Zh. eksp. teor. fiz., 31,4 (IO), 609-619. 1956. 15. GINZBURG, V. L. Propagation of electromagnetic vom v p&me), “Nauka”, Moscow, 1967.
wave propagating in an inhomogeneous
waves in a plasma (Rasprostranenie
elektromagnitnykh
16. BUDDEN, K. G. Radio waves in the ionosphere. Univ. Press, Cambridge, 1967. 17. VAINSHTEIN, L. A., BIRGER, E. S., KONYUKHOVA, N. B., KOSARFV, E. L. and PRUDKOVSKII, G. P. Short-wave diagnostics of a plasma filament. Fiz. plazmy, 2,4,658-671, 1976.
BEAM EQUATIONS FOR RANDOMLY CURVED CYLINDERS WITH A REFRACTIVE INDEX THAT VARIES OVER THE CROSS-SECTION AND SOME CASES OF THEIR INTEGRATION* R. P.TARASOV
Moscow (Received 18 March 1975) IT IS SHOWN that the problem of the integration of the beam equations for cylinders with a refractive index that varies over the cross-section curved along the arc of a circle (helical line), is equivalent to the problem of integrating the differential equations of the one-parameter group of transformations, realized with all permissible trajectories within a torus (a cylinder bent along a helical line), of the set consisting of the points of the transverse section and the unit directional vectors. The use of exponential representations for the operator of the group enables us to consider efficient methods of integrating the beam equations in such media. For arbitrary cylinders curved in one plane (space) the beam equations can be integrated numerically with arbitrary accuracy, if the solutions for cylinders curved along the arc of a circle (helical line) have been determined.
1. INTRODUCTION
The problem of the propagation of waves in inhomogeneous media is associated with the difficulties of solving second-order differential equations with variable coefficients. The number of cases for which the analytic solution is known does not include a whole set of inhomogeneous media *Zh v%hisL Mat. mat. Fiz., 16,6,1539-1550,1976.