THE CHARACTERISTICS OF THE TE, MODE WAVES IN A CIRCULAR WAVEGUIDE FILLED WITH RADIALLY NONUNIFORM PLASMA. (THE CASE WHERE THE PLASMA ELECTRON FREQUENCY ON THE
AXIS EXCEEDS THE WAVE FREQUENCY)
by
Mitsugu KoNoMi and Yoshio KAGE
t
SUMMARY
One of the authors has dealt with the characteristics of the TE waves in a waveguide of circular cross section fi11ed with dissipationless Lorentz plasma whose radial distribution is parabolic. The author's view has hitherto been restricted to the case where the plasma electron frequency on the waveguide axis is always lower than the wave frequency. In present paper, however, we shall treat the waves which are not subjected to such restriction on frequency.
The function that satisfies the field equation is determined by the aid of an electronic computer. Some results are shown in graphs.
1. FIELD EQUATIONS
The equation that must be satisfied by both the electric and the magnetic field in a dissipationless medium are
VXVX.E-ko2rceE=O (1)
H=,-J•17xE (2)
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where E and II are respectively the electric and the magnetic field vectors, k, the wave number in the vacuum and w the radian frequency of the wave. rc, is the specific inductive capacity. pt is the magnetic permeability of the medium.
Eq. (1) yields e and z components of the field vector for the cylindrical coor- dinates system (r, e, z) respectively
-l- -Qa-bEi- -Qe2117-/ -t-a, {-IL- a(5ee))+ aa, (-l- {l3Ee') -k,2rc,E,-o (3)
il- (-8T, (r aaEi) - 8, (r aal;2) -il- aa2E,,z+aae2gs)- k,2rc,E. =-o (4)
Hereafter, we shall discuss only the transverse electric waves of axially
symmetrical configuration, viz. TEo waves. We have now
;- -i;.T (" ddÅë,)+ (ko2rce-B2-",)Åë ==O (6)
where Åë is the function to be determined which is defined by
Ee=Åë(r)ei`BZ-tot' (7)
B is the phase constant.
As previously done{iÅr' (2)' (3), we put
rce==1- So2(1-X2) (8)
where S, is the plasma electron frequency on the waveguide axis normalized by the wave frequency, and x the radial coordinate normalized by the inner radius of the waveguide, say, R.
Substituting Eq. (8) into Eq. (6), we have
litdi(x-dd9-)+(-k2+h,2x2-f,)Åë-o (g)
k2 and ho2 are respectively
k2 =- {k,2(S,2-1)+B"}R2 (10)
and
ho2= (ko So R)2 (11)
As noted above, S,;}iil is the present case, so k2 in Eq. (10) has always a positive sign within the transmission band of the wave. One is reminded that the solu- tion of Eq. (9) was discussed in the previous reports on the premise that the sign of k2 is always negative.
2. SOLUTION OF THE FIELD EQUATIONS By the transformation
and
k2
Eq. (9) can be converted to
dipur,. + (a - {) va-o (14)
where
a-(tdeh)2 (ls)
(2)
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The sign of p in Eq. (14) is now positive, whereas it was not in the former reports.
In order to determine the function ur (a, p), we expand it into an ascending series of p, viz.,
W(a, p) -= :i]b.pM"i (16)
m=o
Substituting Eq. (16) into Eq. (14), we have
bo=1, bi=-1 (17)
The coefficient of higher order can be obtained from the following recurrence relation
m(m+1) b.-2b. -i+ab.-2 =-O (18)
Fig. 1-a and Fig. 1-b show VV(a, p), and Fig. 2-a and Fig. 2-b the first derivatives
of M7(a, p) with respect to p. These functions are determined by the aid of an ' electronic computer.
The condition that must be satisfied by the electric field on the inner wall of the waveguide is
ur(a, e')-o ag)
For a given value of a the numerable roots of Eq. (14) can successively be found from the curves in Figs. 1-a and 1-b, say, ki2/8, k22/8 and so on. Combining these values with Eq. (15), we have eigenvalues ki, k2, etc. for a given value of
ho•
Fig.3 shows the results for the TEoi and TEo2 modes.
The phase constant of the wave along the waveguide axis is given by
(BR) 2- (k,R) 2-r2 (20)
where
r2 === h,2- k2 (21)
Fig. 4 shows r2 plotted against ho.
Each component of the field can be written down as follows:
Er=Ez=O NI
E, --cW(a., P)i (22)
H.--c.B pt M7(a ., p)
He=O
.1 2 dor(a, p)
Hz =" ==J- d CtuptR p
(23)
C is a constant.
REIERENcaS
(1) M. Konomi, Y. Tokumitsu, H. Nishino and Y. Mieno : Bulletin of the Kyushu Inst. of Tech. (Technology), No. 18, p 101, March 1968. (in Japanese)
(2) M. Konomi, Y. Mieno, K. Miyamoto and S.Sato:ibid., No. 19, p123, March 1969. (in Japanese)
(3) M. Konomi, Y. Sato and M. Matsumoto: ibid., No. 20, p 79, March 1970.
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