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THE CHARACTERISTICS OF THE TEo WAVES IN A CIRCULAR WAVEGUIDE FILLED WITH A PLASMA HAVING A PARABOLIC RADIAL DISTRIBUTION OF DENSITY
Mitsugu KONOMI Yasushi SATO
and Mitsuyoshi MATSUMOTO
Summary : The authors give a description of the characteristics of the TEo wave in a waveguide of circular cross section filled with a plasma whose density assumes parabolic distribution toward the guide wall where it falls to zero.
The solution of the wave equation for the TEo wave for such a distribution of plasma density is given by the Coulomb wave functions of the zeroth order,
Making full use of a table of the functions published, eigenvalues and propagation constants of several lower radial modes of the wave are computed.
The values of the definite integrals having Coulomb wave functions as integrands, which are necessary for calculation of the power transmitted through the waveguide, are also given.
1. Wave equation and its so,lution , in vacuum, So the plasma electron angular A straight waveguide of circularcross section freqUenCy on the waveguide axis normalized is filied with aLorentz plasma whosedensity is by the angular frequency of the wave tu, x
uniform along, and symmetrical around the the radial coordi pate normalized by the inner Waveguide axis, while assumes parabolic radial radiUS Of the guide R. Hence,
g:.krg'bX/r?P.fgZM,, t?8, g8,'S,e,,SXifugegrS,,th.e so =- -l;-i/eiiii\s (2)
When the number of collisions per Unit tiMe and x=,ft (3)
between electrons and gas molecules of the
plasma is much lower than the angular eand m are electric charge and mass of an frequency of the waves, the medium can be electron respectively, Go the dielectric constant regarded as nondissipative. The permittivity of vacuum and No the density of electron on is the only parameter of the medium which the guide axis.
depends upon the radial ccordinate of the space The authors have reported on the general
within the guide. treatment of (1)(i)•(2).
The equation which the electric field vector For the symmetrical configuration of the
Emust satisfy is fields around the guide axis, the components
AE+ko211 --So2(1-x2)}E.vv.E.= o a) Of the electric field for the TE wave are
where leo iis the phase cons' tant of the wave Er=Ez=o, Eth="g(r)el'(BZ-cat) Åq4)
80
,W ,R:•:,eff•6.9,.Z",g,•:,,gZ ,tY,e,, IOPr,gzateS,,OS,gh,g po{( `,h,o)2, Z2 }-o a2)
constant of the wave'travelling along the guide
axis•V(r),isascalarfunction tobedetermined. The eigenvalue which corresponds to an Substituting (4) into (1), we have eaCh given value of ho can be determined as / 'a root of (12).
:}- ddx--(x ddVx)+(le2+ho2x2- i,)V== o (s) ' The table of Coulomb wave functions (3)
where, Which the authors made use of, covers the
following range of the parameter and the le2= {leo2 (1-So2) --B2}R2 1 variable :
and ho=leosoR J (6) parameter aoto 2.owithastep of o.2,
By the following transformation of variables, Varia ljle P O to 10. 0 with a step of o. 1.
The eigenvalues are plotted in Fig. 1 against
p ==-IIiO-x2 and w=Vp-th (7) the total number of electrons contained within the unit axial length of the wavegtiide, Nt.
(5) can be reduced to
Nt has the following relation with No and
d-dtiff'i+(a+-7)w=:o (s) ho:
gNo =rrM82o Ce;ho2 a3) a=( 4leigo )2 (g) wherecis the light velocity in vacuum•
The solution of (8) is well known as Coulomb
wave function of the zeroth order, i• e• 3. The phase constant.
w-CPo (a,p) (10)
The phase constant B can be obtained by
Cisaconstant. the first formula of (6)
Omitting the factor expj(Bz-wt), the , -
components of the fieids of the TEo wave can BR= v! (tucR)2-z2mep2olXs--le2
be written down as follows :
Er== E, =o Fig•2shows the curves of BR against toR
R Po (a, p)
Edi=CL E- rr
BR Po(a,p)
Hr =C tupa,le ' -- r7'L
HÅë == O
1 dPo(a,p)
H2==-jC4. R d
(11)
for several !ower radial modes with Nt as parameters.
pao p roax
pao is the magnetic permeability of vacuum. Jto X
4. The Value of
le-2
8p2( , )
dx
2. The eigenvalue. The powerPtransmitted through the
The eigenvalues k can be obtained from WaVeguide is given by
iY,2h,Z,Cg."S,'t;O,".l,h,at.th9,gi',Cthgre."gifl'?leg,triC p-rr,R2fgeEÅëHrrdr-=,%B.5Åí,c2A
81
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5
4
3
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v EIL
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