UNCLAsSI FIED 404514 AD - Defense Technical … analysis is reported in Technical Scientific Note N....
Transcript of UNCLAsSI FIED 404514 AD - Defense Technical … analysis is reported in Technical Scientific Note N....
UNCLAsSI FIED
AD 404514
DEFENSE DOCUMENTATION CENTERFOR
SCIENTIFIC AND TECHNICAL INFORMATION
CAMERON STATION, ALEXAtKDRIA, VIRGINIA
UNCLASSIFIED
NOTICE: Ihen goverment or other drawings, speci-fications or other data are used for any purposeother than in connection with a definitely relatedgovernment procurement operation, the U. S.Government thereby incurs no responsibility, nor anyobligation whatsoever; and the fact that the Govern-ment may have formulated, furnished, or in any waysupplied the said drawings, specifications, or otherdata is not to be regarded by implication or other-wise as in any manner licensing the holder or anyother person or corporation, or conveying any rightsor permission to manufacture, use or sell anypatented invention that may in any way be relatedthereto.
AFCRL- 63-81
SAD
AV 61 (052) - 145
Ju 1962
9 4= TECHNICAL SUMM(ARY REPORT N.2
z 1 April 1960 - 31 December 1961
MICROWAVE INVESTIGATION OF THE DIELECTRIC WAVEGUIDE PROPAGATION BY
MAGNETO-IONIC DUCTS
lug. A. GILARDINI
SELENIA S.p.A.SVIA TIBURTINA Km 12,4
S*ROME - ITALY
The research reported in this document hasbeen sponsored by the Cambridge ResearchLaboratories, OAR through the 1kropean Offioe,Aerospace Research, United States Air Force.
mom
I
SUMMARY
The work performed under this Contract during the indicated
two years period is described. Theoretica] work includes : completion
of the basic theory for propagation along plasma columns in magnetic
fields, derivation of Brillouin diagrams, discussion of particularly
significant limits and preliminary analyses of the non-uniform plasma
case. Experimental work includes measurements of basic propagation
parameters : transmitted signal, wavelength and group velocity, and
of physical plasma parameters as the electron density.
TABLE OF CONTEN;TS
1. - Introduction ............... ................ pag. 2
2. - Comparison with beam waves ........ ............. pag. -
3. - Whistler propagation ......... ................ .. pat. 11
4. - Propagation in non-uniform colimns ..... ......... paC. 14
5. - Electron density measurements ..... ............ .. pa.,. 22
6. - Conclusions .... ................... .. ....... pag. 279
7. - References .. .......... ..................... ... . 30
33S1 d36
a i i m . . . • ,
INTRODUCTION
During the two years covered by the present report extensive
theoretical and experimental work has been performed on the propagation
of electromagnetic waves along plasma columns in longitudinal magnetic
fields.
In the theorj the most substantial contributions to the under-
standing of the propagation cLaracteristics are the discussion of the
dispersion equation solutions in the W6 e(L region, the evaluation of the
power ratios in the same region and the derivation of the Brillouin
diagrams. This analysis is reported in Technical Scientific Note N. 2.
A more complete discussion of the propagation characteristics of circular
ly symmetrical modes along an uniform plasma column of circular cross-
section will appear on the September issue of the Joarnal of Research,
edited by the Nationnl Bureau of Standards.
In the present report we describe other theoretical coroiier-
ations, which are of importance for understanding the ýro-agation theo-
retical and experimental results.
The Brillouin diagrams in the limits of a very thin hsnma
(d-_.o) and fora large cross-section plasma (d-p o) are discussed. 3y
means of a Doppler shift they become the corresponding diagrams for an
electron beam; it is interesting to rediscover in tLis woy the well-
known space-charge, cyclotron and syncronous waves.
As another special case, the Brillouin diaj-rams are discussed,
when the following typical ionospheric conditions for whistler propagatioa
are satisfied
Purpose of it is to find whether the delay versus frequency character-
istic of whistler atmosphericimay be explained also assuming propagation
raglio .2 di 36
"•SL S.p.A.
along well-defined plasma columns.
All the previous theoretical analyses assume an uniform density
plasma, an assumption which is not satisfied in the experiments. The ne-
cessity to explain the observed experimental data has led us to consider
also theoreticaly the effects of disuniformities.Results have been obtained
on simplified aspects of this problem; they are reported in this document.
A very long report would be necessary to describe fully all the
experiments and tentatives performed during these two years. Many
technical difficulties have been encountered, due to the RF high-power
involved in producing the discharge (gas purityand transfer of power
are the most important), to the difficulty of exciting selectively the
various propagation modes, to the necessary limitations in the dimensiona
of the experiment (which would have to be the scaled laboratory model
of an ionospheric duct) and finally to the complexity of some of the
me:asuring electronic equipment.
These experiments can be grouped in the followin~j cla•ses
a) Transmission measurements. Purpose of these measurements was to
collect more data over a wider range of experimental conditions, in
order to confirm the previous experim'ntal results and the general
correctness of their qualitative interpretation.
b) Propagation w:'velenght measurements. Previous data were insignificant,
due to the presence at the same time of various modes and of undesired
stray fields.
c) Electron density measurements. This is a basic quantity, which is
necessary to know in order to compare propagation experiments with
theory. Conventional methods had failed in aur geometrical and physical
conditions; new special methods had then to be used.
d) Group velocity measurements. From these data, knowing electron density
and the other physical parameters, we can derive experimentally
Brillouin diagrams. The same could be obtained in a more simple way
0 3 di 36
30- 3.-= S.p.A.
from the propagation wavelength measurements, but , as it is later
explained in this report, we can measure these wavelengths only for
the first mode, which is the less interesting one.
In the report detailed information is given on these experiments
and on their results. In all the experiments the plasma has been produced/
using a new RF generator capable of delivering up to 4 KW of power at
frequencies between 1.5 and 3 Mc.'fhen specific data of the operating
conditions will not be given, reference has to be made to the analogous
cases reported in Technical Summary Report N.1.
p
4 36fogioe d
SpLA.
COXPARISON WITH BEAU WAVYS
It is enlightening to relate our propagation modes to the
well-known space-charge, cyclotron and synchronous waves,which can be
excited in electron beams.
This relation is best shown on a /1, versus d/A diagram,
6, and E3 being the fix parameters for each curve. These curves can
be simply derived from the AI/ Xo versus d/ke plots, previously
obtained from the theoretical analysis. The usual beam waves are derived
in the microwave tube theory under the simplifying assumption of a phase
velocity much smaller than the light velocity, or <,/ K 1 1, which is
equivalent to our quasi-static approximation.
The result is that all the A •A, curves, near to the d/1
axis, which is the region where the quasi-static aprroximation holds, are
vertical and'given by the relation
where )/(")is the m - th root of the modified characteristic •quation:
The above vertical curves exist only when
- 6) i)e 3 ) , 1 f eO (an infinite set of curves exist4
b) e L 0 F, 2: 4 (an infinite set of curves exists)
c) "3 £ 0 , a&A/E,(only one curve exists m -
In an infinitely extended plasma d/x-tw. Then )/,, approaches
infinity and the second member of the characteristic equation becomes
equal to- 1.
5 ., 36
--- '•-•Sp.A.
Let us consider first the cases a) and b) above. The two
dielectric components e and E. have opposite signs and the equation
becomes s
('V' E3 14 (3)
Two cases exist, in which this equation is satisfiedas - aId Vv 0 asa where
is the m - th root of the Bessel function Jn.
When E and E$ are both negative, I and I, functions replace
Jo and J1 and another poesibility exists. This is ,'j3- 1, a condition
which satisfies the characteristic equation~because the function ratio
Ij/Io equals unity when its argument becomes infinity.
In a stationary plasma, 6.0 when W , 4 5 W when w &it
and 64 3 = 1 when i = I )/L provided Cdj > W In -n
electron beam moving with velocity V& along the magnetic field axis
t
-4 (4)
(, (3 • (5)
Then e o when
di d 36
; p.A.
This is the well-known phase propagation charaoteristio of space charge
waves.
The 9,. oe condition is satisfied when t
(.,,_ o,-, ' o
This is the phase propagation characteristic of cyclotron waves.
The E1- 1 -condition implies t
Here too this case exists only if W tj For this reason and bectuse
such waves are of the surface type, they are not considered in the usual
microwave tube theory. They may however be of importance when a magnetic
beam focusing is not used and when proper excitation is provided.
Let us consider now the other limit case d/j-0 a •
The, JVý approaches zero and the KiKO ratio goes to infinity. One case,
which satisfies this limit, is 63 'O ( I being positive). Anothier
set of solutions is given by the equation
0(9
.Iis implies + *0 , beside the previous 1,+ -40 case.
0 u0_7... di 36
~7LSp.A.
In a stationary plasma, . - -. only when & - 0 and
e .0 when tj-
In an electron beam, E. = -a when
This is the phase propagation characteristic of synchronous waves.
II In an electron beam, 61--0 when:
z 2
P V& ) - t =(ii)
In the theory of transverse - field microwave tubes these waves are also
called cyclotron waves. In fact, we have usually t; a Ls t , so that the
phase characteristic at this limit coincides with the previously given
characteristic for these waves in an infinitely extended beam.
It is interesting to see how the above picture of beam waves is
modified, according to our analysis, when the complete theory is used
in -lace of the results of the quasi-static approximation.
In the infinitely thin column ( d -. o ) the complete theory
does not contribute any further solution, beyond those given by the
quasi-static analysis. In fact, the /X. versus d/A. curves
show clearly that, when d -+ o, A,/. 0 always, so that this case
implies always the validity of the quasi-static approximation.
8. d, 36
inA
The same )i//o •urves indicate instead more possibilities
for the infinitely extended plasma. In fact, if , I and eoae E Y
all the investigated circularly symmetrical modeeoexcept the first one)
approach, when 4 '+ to , the common ratio /a I/so . It is
worth of mention that this same ratio is attained also in the case of
a TEX plane uniform wave with a right hand polarization; however, fields
are different and in our case we obtain
E F 1___S. . . . .( 1 2 )
whereas in the TEM case one has t
0
When ~ 4and 63 -CEN the common ~/ .limit as d -P w is
I/o , for all modes except the first one ; in this limit A2 = -A, and
the total field then vanishes in the infinite plasma.
The propagation characteristics for a plasma column or for a
beam with an extremely large cross-section can be easily derived from
the above considerations (the first mode will be negleotedadue to its
low capacity of carrying microwave power inside).In order to have ,
we assume W 4 L46 . Starting from cj@ , 1% is a large negative number,
certainly less than • In this region, substituting for I and 3
" 9. d136
their values in the e formula, we obtain i
t "(14)
There will be a frequency at which this becomes equal to the value
similarly derived from the a( formula after substitution of the £ and
E 3 values
It
For &j values larger than this one, the versus 0 curve follows the
last formula up to w = Cj•
Waen ar(WL, as in most microwave tubes, the curve intersects
the W - W e horizontal line, given previously by the quasi-static
analysis; the diagram P versus W for very large d values shows the
continuity behaviour sketched in fig.1.
Wb
S~FIG 1
The corresponding diagram for a beam is obtained as usuallyby substitu-
ting the doppler shifted frequency (-Ptr, to signal frequency
10 36j m ,,, g, o. , i
WHISTLER PROPAGATION
The possibility of finding a region, on the kd versus Pdagram, where the propagation shows the typical f - I--.behaviour of
whistler propagation in an uniform ionosphere, has been reexamined on the
basis of our most recent results for uniform cylindrical plasme.,olumns.
We are assuming here the following conditions, which are satiefieý
along the ionospheric path of propagation of whistler atmospherics
< W4 a (16)
This means that E3 is negative, whereas 1 is positive; both, however,
are much larger than unity.
Recalling the shape of the previously discussed c( )f/A, versus
kd or P curves for constant E4 and ' 3 , we see that each of them
consistsbasically of two straight line parts. It is reasonable to expect
that over any significantly wide frequency range the shale of the re-
sulting kd versus ýk curves are correctly predicted by the elaboration
of one alone of the two sets of straight lines.
Let us consider first the constant *( part of the ('(versus kd)
curves, which is typical of the infinite plasma condition. We have q I
for the first solution ( &4 and Jej I), %(.or 0(ý for all the other
solutions. Then E4 is very large i
E0a - (17)
Then ft condition : 4 : - 3 • , for which the constant o value
I..1.1 . di 36
3il n1I
¶~3~9 p.A.
is es( , is attained when t
•. (18)
In our case ( £, and It],'.
c.J&L(19)
f t (20)
Correspondingly we have i
X &,/r first solution
p • other solutions,when w 4 ()/2- = •"1,(22)
.. r other solutionswhen C/a itJ (23)
The classical whistler propagation behaviour may thus be found
also in plasma columns, but only over the frequency region botweon the
cyclotron value and its half.Over this region whistlcr delay versus
frequency curves show a characteristic rising behaviour.
Let us consider now the set of straight lines diverging from
the origin in the O( versus k4 plane (quasi-static approximation
region).
For discussion simplicity it is' convenient to solve this problem
in the similar case of a plane geometry, where the basic equation of
IbIo12 .6
these lines is i
~ - (24)
Sbeing half the plasma thickness.
The solution is approximately t
P3 L F 3-S 'MWii + %I I3(25)
When m~
4 t!- ! (26)
"When m# # 0
the whistler characteristic, which is found in uniform plasmauis here
never attained.
It seems then possible to conclude that in general propagation
of circularly symmetrical fields along ionospheric plasma columns does
not explain whistler typical characteristics. In fact, we may justify
at most only the rising part of the whistler delay versus frequency
curve near cyclotron resonance. ,Moreover, our theoretical curve must
join, at frequencies lower than half the cyclotron value, a constant
delay instead of showing the typical behaviour(frequency drops as the
square of time delay).
rge. 13 di 36
LPA
PROPAGATION 3 NON-UNIORN COLDMNS
Various investigations on the chanjeo of the propagation oharac-
terietics due to a non - uniform electron density in the plasma column
have been performed for the purpose to explain some experimental results.
A general method for taking into account longitudinal disuniformi
ties has been worked out by Prof. Cambi of the Rome University, on a
consultant basis for this Contract. His analysis is reported in Appendix A
More detailed analyses have been performed on the effects of
transverse disuniformities. ?or analytical simplicity we have always
considered a plane geometry, instead of our cylindrical one. In this case
we have assumed density to be dependent only on the coordinate of the
axis perpendicular to the plasma boundary.
As a simple case we have first investigated the propagation
conditions when the electron density drops linearly from a maximum at
the center to a finite value at the plasma boundary and when no magnetic
field is present. This same case has been investigated and similar resailts
reported at about the same time by Prof. Schumann. For this reason and
because of the scarce interest of this case for the interpretation of our
experiments, no further discussion will be presented here.
Details will instead be given of' a simple, but here more inte-
resting case.
As it is well-known, in the quasi-static approximation retar-
dation effects are neglected and the L.C. electric field is derived from
a scalar potential:
r -E (28)
W0.1 . Idi36
whereas a.o. magnetio fields are entirely neglected. We consider a slab
of plasma in free space; a coordinate axis system is chosen as shown
in figure 2. The constant magnetic field is directed along the poesitive
L. axis. Plasma electron density is a function of the transverse x ooordi
nate only, and so are e, Eand E3"
FIG. 2
Maxwell divergence equation provides then the following
differential equation for 0 :
t0- * ,~ ) ,~x(29)
In the plane geometry the modes, which correspond to the circularly
symmetrical modes of the cylindrical geometry, require 'O/ = 0 , and
then the differential equation for 0 becomes
4 4 E_ A + (30)
Let us assume now that typical whistler conditions (16) are
15 c, 36
satisfied, so that :
4 •(31)
3 -(32)
T heir ratio is thus density independent and is not function of position.
Eq. (30) becomes
- ej .~ e4. . (33)CD' ef x a t Ef Ox Ox
A simple solution is obtained if the density varies esponential-
ly
I 1f (34)
ie may call I/a the characteristic length of the density disuniformity.
Fq. (33) becomes
~J~L !Ix (35)
which has as a suitable solution
A Li[ IJ: e/ j3 (36)
16 36NOW ,. . di
For continuity E. (z - o) must vanish, and then )4/xrnO.Thi.leads to the following relation for I I
06A (37)
Outside the plasma a suitable solution for the potential in
* a PILo (38)
At the plasma boundary (x - b) Dx and E. must be continuous.
This leads to two relations between A and B, which are satisfied only
if the propagation constant P and the dielectric constant components
are related by the following dispersion relation
where I is evaluated at the plasma boundary
The second member of (39) can be rewritten as
1'3)P t (40)
S44
ýhe first of these two terms, as we shall show later, becomes negligible
for all modes except the lowest one, and eq. (39) has then the following
solutions s
S1 36
from which i
(I [(T4~' W i&~r*~ (42)
The first term of (40) is then :
4 F(11 I __ 4_6__.)_
which is obviously negligible in whistler conditions and provided k 4O.
Eq. (42) has to be compared with (27). The result is that in
the presence of an exponential density decay, the propagation oonstant
P given by the uniform plasma theory has to be multiplied by the factor
"• --- -- IT,, When the plasma thickness is a few times the
characteristic length of the disuniformity the effect may thus be large,
particularly for the lower ft modes.
18 36fgo di
PROPAGATION AND WAVELUGTH KEASURDIRITS
The experimental results reported in Technical Summary Report M.1
can be explained in a much more clear way with reference to the Brillouin
diagrams derived during this period and reported in Technical Scientific
Note n.2.
We recall that the transmitted signal is detected by an internal
probe and that propagation is observed only when Wd > &4.
Reference to figs. 3 and 4 indicates clearly that when UJ 4 cj
propagation may take place only t
a) In the first mode, when CJr> eg API. This is a surface wave and
as such carries very low microwave power along the central axial region
of the plasma column. For this reason this mode is not detected by the
probe.
b) In the upper branch modes, when YZC&4( r C In most cases these
waves are backward, and then are not properly launched by the helical
couplers. These waves may carry also forward power, but only at fre-
quencies very near to cyclotron resonance; near this resonance,
however, waves are very strongly attenuated by electron collisions.
All these facts may explain,why such waves have never clearly detected
in our experimental arrangement.
"When WC. Wj propagation takes place
a) when CJr .4 , over a limited range of densities and provided the plasma
diameter is sufficiently large
b) when (J always.
In Technical Summay Report n.1 we have given reasons to believe that
experimentally observed propagations are related to conditions b) above,
namely (•. L•,W!
.19 di 36
JI Ill A
One of the experiments, initiated during the first year of the
Contract and continued during the successive months of 1960, had the
purpose to measure wavelengths, when propagation was taking place along
the plasma column.These wavelengths are determined from the standing wave
patterns measured with a pick-up probe, which slides along the column
very closely to the outside surface of the plasma container.
To obtain significant data it was immediately apparent the necessity to
reduce the intensity of some undesired, stray fields, which were definite
ly found to be present outside the plasma.
S... The situation was improved by a better choice and arrangement of absorb-
ing materials around the tube. These materials reduce propagation
between input and output couplers, as well as spurious coaxial and
waveguide modes. However, it was found that residual propagation due
direct coupling and to dielectric modes along the container glass walls
could be effectively reduced only feeding the microwave signal directly
into the plasma, instead of injecting it through the glass walls by means
of an external helical coupler, as done in the previous experiments.
Many different types of inside couplers were tested for this purpose;
finally a plane spiral antenna of small dimensions was chosen for its
good coupling and mode selection properties.
Using thin couplers, properly placed absorbing materials and a
new accurate driving mechanism for the probe carriage, it was possible
to perform ainificant wavelength measurements. The result is that the
measured wavelengths are only slightly less than the corresponding free
apace wavelengths.
This is in agreement with our theoretical analysis, provided the
signal detected by the sliding external probe is propagating according
to the first circularly symmetrical mode. Such a possibility is highly
plausibile, because the first mode represents a surface wave; in this case
2C 36g di
fields just outside the plasma are strong and may mask easily higher
order mode fields.
The opposite situation has to be found inside the plasma* where
the transmission detecting probe is placed. For this reason we believe
that the observed transmitted signal characteristics are in most oases
due to modes higher than the first, whereas the externally measured
wavelengths refer to first mode propagation.
During the two years covered by this Report further propagation
experiments have been performed. We shall mention here only two of them.
In one experiment, we have used our typical set-up but a different
frequency range, namely the 1300 Nc/s range. Basically, we have observed
the same phenomena as in the 5000 Mc/s range, the only difference being
an higher pumber of transmission peaks. This is in agreement with the
assumption that propagatioA is observed when &t, •
In a second experiment we have investigated propagation character-
istics using different tube diameters. We have found that the detected
signal power dependes on the tube diameter and that this signal practi-
cally disappears when the diameter is less than 4 cm. This experiment
was performed in pure Neon and in Neon contamined with Mercury, but no
significant difference was observed changing the gas. To explainwith
our physical picture, the observed behaviour we must suppose that the
electron density decreases with the tube diameter (for instance, this
may be due to larger diffusion losses); so that when d 4 4 cm the con-
dition Wt W.> is no longer satisfied.
21 i 36
ELECTRON DENSI7T MEASUMU=
In order to proceed from the previous qualitative considerations
into quantitative verifications of the theoretical analysis, it is neces-
sary to know the values of all basic physical parameters which charac-
terize the problem. Geometrical data, signal frequency and magnetic field
intensity are easily measured according to standard techniques. Diffioul-
ties arise, when we try to measure electron densities with conventional
methods; the failures of these tentatives have already been described
in Technical Summary Report N.1
A new approach was then chosen. The value of the average electron
density in the discharge tube is derived from measurements of the guided
wavelengths of microwave signals propagating along the plasma, when
metallic walls are placed around the tube in a waveguide arrangement.
Except for the presence of the external cylindrical metallic wall, con-
centricaly placed near to the discharge tube, all the other geometrical
and physical conditions are unchanged.
A.W. Trivelpiece and R.W. Gould 2), who first proposed this
method, compared it 4ith other microwave measuring techniques, as the
cavity and the scattering methods, and found that it provides results
in good agreement with those given by the other methods.
The measuring set-up consists of a cylindrical waveguide, 7.0 cm
I.D., with a longitudinal slot. A screw-drived carriage, traveling along
the guide, carries the probe. The cut-off frequency of the empty wave-
guide is at 2500 Mc/s; measurements have been taken at lower frequencies,
so that propagation modes derived from conventional waveguide modes do
not exist. At these frequencies and when the density becomes infinitely
large, the field configuration and the propagation characterietiec ap-
proach those of the classical T&' coaxial mode.
Measurementsare conveniently platted as Brillouin diagrams
2/. X3 versus k - 21/ Pomparing these curves with a set of
22 36
S-PA
theoretically evalusted diagrams for various plasma frequeneise the
experimental densities are determined. Theory necessarily assumes uniform
plasma, so that the described data must be regarded as average density
values.
The solution of the propagation problem in our real geometry
(plasma bounded by a glass tube, air and metallic waveguide) was first
carried out and solved with the aid of the so-called quasi-static approxi
mation and with the assumption of uniform plasma density. Whereas the
experimental conditions of other authors were such that the results of
the quasi - static approximation are sufficiently accurate, our compu-
tations have shown that this is not the case in our experiment, so that
we have faced the necessity of more complete computations based on the
rigorous characteristic equation of the plasma waveguile with metallic
walls.
Experimental P versus k data fall on a straight line, passing
through the origin; theoretically then, we must evaluate only the value
of the Brillouin diagram slope k/p. i/e( at the limit k - 0.
Assuming no glass walls around the plasma, the dispersion
equation is derived along the same lines used in the free space case
discussed in Technical Scientific Notesni1 and 2. For circularly symmetri
cal modes we obtain :
4 F/_
° Xf y;) J. X X N)
23 36fogho. d
lpAl
where
= .k, I., , (X,-9/ N~'.') * (44)
F" IF'<,<'. , ~~)1F ~ -___
"L L.(x.) .(.i) - (.)
b - metallic waveguide to plasma diameter ratio
XoXltx 2, - see Technical (Scientific) Note n. 1
No are interested in the low frequency range of each Brillouin
diagram, where the following approximations can be made
xo. • aL ,-4 ,--t-o
Under these conditions, the above defined parameters become
F•' ,,-. (&, >)"
F_ X £(e . 4).
('4) -(.4) A Vl(k4
It - V'4I- Ie.C•,-,)/(AI)
f +. ,
24 36No-di
When these expressions are substituted into (43), the dispersion
equation can be regarded as a relation between kpd and .4 (which is now
equal to the initial elope of the Brillouin diagram), provided we assume
that a- /'l1(JiS a constant parameter (kb a ZIN) The o( versus " d
curves can be found rather easily by graphical methods.
From these curves kpd - ((, 4. ) and the equation kpd =K1JAIr
a new set of curves kpd - Irb, KJL ) can be obtained.
Using these curvesfrom the measured 0( and kbd values the corre
sponding average plasma density is determined.
Following the above discussed procedure the X versus kPd curvei
for a constant a (fig. 5) and for a constant kbd (fig. 6) have been
computed, assuming for the diameter ratio b the experimental value 1.47.
The curves are those corresponding to the lowest x° solution of the dis-
persion equation. In fact we believe that our experimental data refer to
this solution, because the fields outside the plasma are in this cace
much larger than those of the higher order solutions.
Technically we had to face the problem of insuring a well defi-
ned standing wave pattern by using good reflective terminations at the
discharge ends. The best results were obtained with a mesh of tungsten
wires placed transversally inside the discharge tube together with an
external metallic iris.
Using this type of highly reflective terminations, measurements
have been performed using several frequencies in the 1 - 2 K.Mcis range.
To characterize discharge conditions the RF voltage applied to the plasma
between the electrodes was monitored. Typical results are as follows 2
RF discharge voltage 600 V peak, kbd - 5.86, 0( - 1.6 and then
nm3.4.10 1 1 ele.,/cm3 ; RF discharge voltage 1500 V peak, kbd - 5.86,
S- 1.3 and then nm 5.5.1011 elec./cm3 .
25 36
Sp.A.
UBOUP-V~LWCITY NEASUMOWETS
Wavelength measurements using a sliding antenna probe have pro-
vided information only on the first mode propagation. To obtain experi-
mentally Brillouin diagrams for the other modes a different approach must
be used. For this purpose, instead of measuring the phase c6nstant A
versus the signal frequency, we measure the derivative.&j / , that
is the group velocity.
This velocity is determined from measurements of the trans-
mission time delay using free space propagation as reference. The signal
is detected by the same probe used for the transmission experiments, so
that we are certainly measuring the properties of those signals, which.
propagate through the main plasma body (second and higher order modes).
For measuring the delay the 5 k Mc/s signal is amplitude modu-
lated from a lower frequency sinusoidal signal (30 Mo/a) and the phase
variations of this modulating signal at the receiver are measured. To
achieve the derived accuracy, the modulating sigal is converted in the
receiver to a lower frequency (30 Kc/s ), so that the delay, which is
displayed on a CRT is larger than the propagation delay "s the frequency
ratio (103 in this case).
Group-velocity vg is given by the formula
CS= (45)
where A(iniuseC)is the variation of the displayed time delay passing
from free space to plasma guided propagation and 1 (in meters) is the
length of the changed path, which is set equal to the discharge tube
length.
26 36
The block diagram is shown in fig. 7.
The electrical oharacteristios of each main block are as follows t
a) !'ýrowave varactor modulator.- Its purpose is to amplitude modulate
at 30 Mc the microwave signal. The index of modulation is 1, up
to 100 mW of microwave power, and reduces to 0.5 at I W.
The levels of sideband harmonics are always at least 25 db lower than
the 30 Mc modulation sideband.
b) 30 Mc Amplif. and A.G.C. - This is a 120 db gain, 0,5 Mc band-widtb,
30 Mc center frequency amplifier; the output is held constant within
10 + 0,1 Volt over 90 db signal input va2.atf.ow.
3. 0 Mc quarts oscillator and O Mo + 30 Kc quartz L.O. - The two quartz
oscillators are built in the same thermostatically controlled containr,
so that the 30 Kc beating frequency can be held constant within 30/,
peak to peak frequency v~nriation over long time intervals.
The method and the equipment have been satisfactorily tested
measuring the propagation time delays in coaxial cables of known lengths
and in free space varying the path lengths. & maximum errors are of
the order of .1/usec, that is sufficient for our experiment.
The plane spiral antenna, placed inside the plasma, which was
chosen for the wavelength measurements, was no longer used, because of
its extremely chort life in the Iresence of strong discharges. A plane
iisk was used in some of the earlier tests, but it was also abandoned
because its design was found to be critical, the coupling characteristics
being different for each tube we have constructed.
Then a new design was chosen for both the transmitting and the
receiving antennas. It consists of a dielectric antenna, incorporated
into the end plates of the discharge tube (see fig.8). These new antennas
not only show more constant coupling characteristics, but, due to the
27 36S di
LpA
absence of metallio parts inside the discharge, allow to attain very
good vacuum, to maintain gas purity and t avoid the presence of sputtered
surfaces.
Using these antennas, however, the transmitted signals have
always been at low level. The detected power was found to increase with
the magnetic field up to the maximum attainable field (- 3000 Gauss). To
improve the signal to noise ratios at the receiver, a new larger power
supply for the solenoid was then built (50 V, 1000 A, so that fields
around 4500 Gauss are attained). Larger signals were detected at these
fields.
A large amount of data was taken using the curved and the
straight solenoids, discharge powers from a few hundred watts up to 4 IN,
neon gas at pressures in the millimeter range. Time delays from I to over
20!usec have been measured. Inconsistencies of these data, however, have
shown that operating conditiont are not uniquely determined by our
measured data : signal frequency, magnetic field intensity and RF voltage
across the discharge.
This implies that gas and glass surface conditions ire changing
..... with time and from one experiment to another. Consequently density and
its distribution are also changing. Ireviously discussed theories predict
that these variations may largely influence propagation characteristics.
This problem deserves then more work both theoretical and
experimental.
N6 28 d 36
CONCLUSIONS
The main results obtained during the two years period, covered
by the present report, have been discussed.
In the theory they are
1) The basic theory has been completed, including the C• c. region and
deriving Brillouin diagrams.
2) The relations with well-known aspects of the propagation in electron
beams have been discussed and understood; these results can be usefully
applied in microwave tube theory.
3) The differences between whistler characteristics for propagation in an
uniform ionosphere and in a well-defined ionospheric duct have been
reconsidered on more general grounds. These results are definitive for
the case of uniform plasma ducts, but it seems interesting to examine
from the same point of view the case of non-uniform ducts.
4) Preliminary theories on the propagation characteristics in non-uniform
plasma-columns have been worked out. In spite of the obvious analytical
difficulties of this problem, more work along these lines is highly
recommended.
£-xperiments have provided the following results z
1) Previous interpretation of transmission experiments has been confirmed
by further data, which cover a wiler range of experimental conditions.
2) Propagation wavelength measurements have been successfully performed
but they concern only first mode propagation.
3) A satisfactory method for electron density measurements has been found.
This method ia based on wavelength measurements in a circular waveguide
beyond out-offt which contains the plasma column concentrically around
the axis.
29 36NOO di
4) Group velocity measurements have been performed by measuring the
propagation time delays of the microwave signal in the plasma.
5) Operating conditions are not uniquely determined by the assumed funda-
mental parameters : signal frequency, magnetic intensity and RF voltage
across the discharge. There are reasons to believe that gas and glass
surface conditions are changing with time and from experiment to experi
ment. More work is required to insure known and repeatable conditions
in the future experiments.
These conclusions indicate also some of the basic work which next year
research program will have to accomplish. In particular we mention the
further development of non-uniform plasma theory and the quantitative
analysis of experiments performed under known and repeatable conditions.
REFERENCES
1) W.O. Schumann - Z. Angew. Phys. 12, 145 (1960)
2) A.4. Trivelpiece and R.N. Could - J. Appl. Fhys. 30, 1784 (1959)
30 36ml i di
0 0
CYW
-duU I I
CV,
In
1=1 0
0
9-. OD (a.
met
N.-HETI Vill fifftof 1:RM1141
iflifif ta v 11 Ill -P V 11 mflw, lfi v i1filim ml111111111111 W IN M M to fill. irtit., t 1 14
A M -IMUIRfill 1 T Itit i I
:jfmif '11112 3, 1111114 114 fl' ,, M lit I;f I H olf G R IV , t
,lit; tit. 1;1l" - q* A 1441fin 10 1FIN ION Vvinfillill-UPPIN Me It it,I'll'it it -i tM DOiF ITTill W i tt liflil. itliqV i 11,11MUMN11111110 i i 0: Ill 11+11 it
OAH11f "".. till* 111M,11 1 '1 I'M 111113 M IflIT 1111111 114111113M 11111 M 111111 litfiflimil ItS ,HIT O U R 11,911 it ill _1111111 I 'if Rai tifffitHIM1111111, 11110 1111113111,41110111"If 1111TO I H IM 111H iff,I-f 14 Iffifinifill fag JUM111411111,111110 fal 01111111-;h.-iffliallITH 4 -1, , ý I is]!I I; it It: 6 1 IT IT 1, fl iffiflIff 111 111F, 1 411111111 1 1 ACOM
flIKIVIii'll" + Q ilITH11111111 1111f, 11441 tit 011111 W, fix I JiMit ''JO I, li'litflifli f flu l O ff if M t, -flit l1w ittRal
f ;it fitlit . ý11
IT 1, ItIlIMIll it+lit .,,I , I if IllI, i I Ili T, H IMM10- H,if P1,11i, 11, 1ý1
I, H ýiit ikl it P it f fl; fl t If It 10 1 V` 111111H :10 141'111111M fit IM I
i. 1 it I;i! I'M tIltil'', a rt it
fI4I liffil, I
II lit flfIt 4;r it I I, ." 1111114 ft,IT W;4j _4,+itit it, M tfl !!if 141 It 1 1114 'till IIt 4; i VU It", fulfltfljý IM I t p1lifl 11M, 1 4 1;,,I ll:i S H I It f I. IV if I' MI i litI:i: Jp+ 1 . I j I T 'I:;ý Iý ; II I! I m, iii! 41flvtit 3 11, 411 9 1", 11W Lift V-311 I I
it iýiý fillfit ii1i t-T Tq1fHR iif_1ý11Tfýf Tit T;41 "fil lIf 11 If M IN
;if 4;1: ;1!m ;1I; I:i: i:Ii:j;: tit I., ;It li + I It: i M fl ill +
it ::1- j; It. ch4-It ti I I,H;; '141 111 fýt fl- if*, 10 H O N tit It V II I it M WA
IT :::i till:;;:1!t H lfl* U: 1t ill, liflllllý if it Tit 1 It I It, W ifil I H,
71-7 77: T:1ý! !:it iiýi ýiij iii. 1 IT: 1:
flifilif:ii! I it44 1 1 ; 'ITT l it
I Pill
qi4ý It I till l!;!
;lit !;Ii;! HIT 1: 11;: +111 111 Iii ;,:: , I IN; 11131fli All 1;i" 1114 IW.
fl4t :I
:lit I:1 fý!V ;l till
tit:t: :14 T !:!' Tit: tw;:: Tit;i . I� .; ý it
lit _'I'l,.. .... .... i I::; , ii;:i: T:A +ii till 1:i;
it!
1:t ::T! I; T'fl-111ii I; t
itITill it 1;: 1;
lit l4
littill it I ý;4Pa l '1! oi; ii! i;;; i;;i 11A.II
till 4!
lit 1, !!if t!li :!I: ýil it IT ::1i tillTit; it III mtla illý iýi:lilff , ifl;tiý IIH IN `1 1 Ill lit I filt dtý; lit! f!ii I liflill ';tflfl4PHI illi; I1114 4.3 it-11 -t.; I It; T Tim. IT fl][011111
NI 11 1 H o 1111fill iii, flýi IRM4111441111 ;it ti, i ffl III:- I III,If i MR 1 tal 1111114 "IV -Ili M illi"Pull tj 1 1 fill :]if Ri. flfl; fl i "111A
W., 313 WE flif fnififf Rt flit fillifll
itHfl !fl if I lit 01 It flill' i f ii It it" 114 111111HIRM I'll If I ill F. I I v 111 Ill i I I I f MAIN 11 4 M : I I
I It , qi .
I'll Hill * it ill 1111111 11,11 11 t I I 'All I a m ! V 11411414liT O w lit I
1 '114
II :24 H So f4al m won
ýl fill it I i iý I ii, 1HIIARIIT: .!I 1-1111NANiNil I'll H il;ýTfll 1 711; Iiii it I flit, I if!, Hiallill 3111111141M V111*111 lit; !if ;:l 14
O il :t 1; I l;, i I I I I týtlij ý .1 1 "if 111111H
'r- 1 Jý It ; ; :!I: :il: 11:ý!l I'l lif Nff o; t.:f I I: ;I W I 1 1; 1 -1: NI ii i idLilt it IrITIM ;ill I'll lilt lili ii. I It 1. 1:"44till Ill, it! :1 1 it if:fill t r"' it i I
If ii:7 ::!I jllý i:i! it I t IIII', Wl iii;1,11111, 41ýi IP: 1: Ili. I :Ili ;it ýii, 41 1;; WWI!It I
it i Ntit
T I i t4j
it
H.", 11T il l 41 W t''. t':' it ;I
lit
i FT i it
ý4 4
iýi2 lilt It I" it tZO i :t
I fill
Hit 1::: 1 [,1 1
It 1;1; 1it. f :'il I H !'' it. till l1l;
Ili!, if till I I i i4il I j! 'I i ff
I rl I It
i 1 14 if li it 11 If A t It it 1 lit ii- !;'itf l,:f ilf I it i li jt J;: I tilt 14, it
Till 1:3 !11 1 :i:t ;If 1 li: I
itif W ýý itiij.'ji 1 114 lit
Ofi H ilii:i.1i Hit Nit ii1f I!; i I q1;
ill if M 1 Ifil 11 !it 11! i'M IM I !l+ 1 .!
16 ; '1 f wili lit W414 ifiLý,11 411t 11, iIf i !j MII ,14, Ili, It t 11 f, '; itlifill" 111111 11 'li Mililfil 11 - 11fill"i ill" 111:1,I 1011ill 41i111141 if 11 "Will; -,fill +1 it;,fl- 1,111N " Rif ililfN '1" :14 1.,; 4 ill ij, if it it T! 4 -Ill: Ill ill T, ill flulit 11111ilIfI iftilil: ýýi fll[fli llil illitil'i tiil'UR if '11i"All "41 lit 11112 11111
1; 1;, 1111 d 4T1 l .1" tf 111 'Iflifill IN I11T11I l3llii I I itI lit It I ljjjttjtjiIý Tif It I li 1 i I *
liltW ++ if "11111, 4 ..ll i;; I I f I if It 11't1t I fit it it
fit H Nit I'l!IT11111 I ",'Iilil I if 4ý 1 1; 1; 1 1 Pli -fill 'I :U ." "I
I I I I lit' .41, T flF
f i 1 1! "It t ill r" 1' 1 ik PH liij f tilit IMP il 41111
-H 'if 'Ill Off if 1 flil III lim it IfY I, fI"'ift 1 ttTi lit iit t.4 liftilif Hi fillil 111114W1,111111: ;,ill 14114 !MPH it t4fi*'T' ;,It t ýt#'[ffft J fit f, i4likif iitH I i ill f 51i 14 MIR 41 ff" lit I lillfiliffl-": 1141i CIH 'fix : flIfli it f .11 Imi N, 91 Hl it, I Hil 11 it
MN't i14 Fii 1 11 qT 11 q 411 If 41, I'M 111111i 1111111M ifl ilil i I i VVVII I 1 111 Ili I'll 1 1111,1110 11 it I Iff I,14
11 V " 11 t i I I i I 'fill 11, 1 1 1,13411 1 1 lit fill. IM T
if a II :ý I, If Milli I it ill I + i 1 4 IM!, .9 , till if ifie llfilli
9 all 1#1 1 MAN41 11 1 MOMM Willf:
"M 11114114 fl it Ifni -414: 1 1 :10 4 Ilt Mi it UMMI 14: MUMS off 14, 1
Mqwpl 1 19 ý ffllq 19 11114 THH1111111 W ME INT 10walllip, IS, I+ 1111441 IM It 11111101111111 f I :fit olat If in I O VUM4f fill 491' Iniwwwwww 'ZIA; i1s;
IM:pp I= =1 ZU ii..... .....mo s; p Xii J :1q: 77 sonsW7,
............ .....
It -.
0
a -,3-
109
Ai
36
SINDEL&p. A
"PWPAGATION IN nIfM)OGSMOUS PILSA.S(By Prof. Cambi 1960)
1- Introduc tory
The presenoe of free charges in a plasma can notoriously be taken
into account by regarding the medium as possesuing a tensorial per-
mittivity as defined, in oartesian coordinates for instanoe, by
D.. a [a, E, +j,, E7 ]D,. 0 Z[-j, a& ., E +ti1
where 4E is the absolute dielectric constant of free space and
C, t. 6, are relative tensorial components defined by
W2 -• aj W (i( w
In these formulas, &jl is the square of the "plasma fre-
quency" related to the electron density n by
while W,, is the cyclotron frequency, related to the maSnetic
field by
44=eB/in.-
In an inhomogeneous plasma duct, the electron density
and/or the magnetic intensity are, generally speaking, functions
of the p*ints in t,.rms of the quantities
WP (2)
the components of the permittivity tensor becomei +b2 k ($)
Ifthe coordinates arc measured in torms of -- '
that is, if I & l- times the physical abscissa, and so on;
and if, further,t is written for the vector 49H if ,/to.
(homogeneous with 1), the adimensional Maxwell equations become
foglio 1 di 19
M"d. 8 • 210 ."S
SINDELp . A
C i- s -' " E., +ja E)by Zz
'a x 'a Y' a 31. -
As stated above, go, EIt, & are functions of the point through the
quantities b and k appearing in (3)s but it is quite obvious
that the analytical problem cannot be solved in the present ge=
nerality. Actuallythe problem is quite complicated even in the
most schematic asumptions: so that it is necessary to discuss
separately the most elementary laws of variability.
2- T,.M mde in a uniform plasma
ihe simplest conceivable case is that of a plasma where the elec-
tron density, or the magnetic induction, or both, are functions
of P qingle coordinate, say x. In this assumption, a TEM mode
independent of x and y is obviously possible, satisfying the
reduced system
'Z z "62
S{s)~
0 :•z 0 •,
P, and L. arc to be regarded as known functions of z. Elimina-
ting Kx and Ky, the system
j~ +,L (6)
dZt t E
is written at once.
foglio 2 di 19Me. 10 -210 * 295
SINDELiLp. A
In spite of the apparent simplicity, the ysteip in
ozoeedingly complicated, even for the simplest laws of varia-
bility of &I , and C.. In view of this, a rather extended
discussion is in order.
In a first instance, we consider the apparently trivial (but
still delicate enough) case where a, and &,are constant, that
is, the case of a TEX propagation in a uniform plasma. When
I, and 1. are constant, system (6) is solved by making E.
and E of the respective forms e, tipsI e J. •j provided
Sis a root of ( )' o.The possible exponents are, accordingly
From the general expression of E.,F x k , J 'e J k * " ) a & + Lk , + "e'4 ,
that of Ey is found at once to be
E7 jke, e6w" *,k, e _j ')ceýP& k epThe first Maxwell equations also give
g, . - a kit , + w k a t k. 3 e,•k 3 , lP, "'Ipk.*J P
If *t and P are taken positive (assuming that 9, > It, )
the terms with negative exponents represent a forward propagations
assuming that the field originates "from the left", and that
no discontinuities occur, these are the only components of in-
terest.
Of course, i:, t to excite,say at z-0, such a progrossive
field, the initial E and K should match the local characteristic
impedance; that is, they are not simultaneously arbitrary.
Taking as x- direction that of I at z-0 and denoting
by E the (constant) modulus, a progressive field is qbtained by
making k• •kj, o , ki , k4 s L: the propagating field is then
+ jF a C-, 0"' ;#a
joglio 3 di 19Med. - 210 .2908
SINDELLp. A
This can be regarded as resulting from the superposition of an
"alfa field"
EM' F. C-Of ore"
2E: .. i} e,;ig• /y rE °"
with a "beta field" similarly defined. The velocity of propa-
gation of the a-field is c/o( , that of the beta field o/.
In each of the two fields the vector K is perpendicular
to 1, but the Poynting vector is zeros in other words, the fields,
isolated, do not convey any power. Power propagates by virtue
of the superposition of the two fields. actually
-01 -.0 9
E
In the alfa field, the plane of polarization rotates ne-
gatively: actually, denoting by V the angle of E with x, we ha-
vsI
while in the beta field'f
When the two fields are superimposed the polarization is
fixed with time, but depends on as that is, the polarization
continuously rotates in the course of the propagation.
Actuallys. -( a, +. su". ( ,w-P)- ,2*±-,
1, w.. COs6.,f-p,/+CoJ(G.t.-o,, "V •.:.2
3- TFAI modes in a lona.tudinally variable plasma
The more general system (6) where A. and &.are generic functions
of s alone can be attacked in substantially the same way.
Looking for a function F of z such that EX . C,F and Ey a C2 F
foglio 4 di 19Me.d.18 -210 x295
SINDEL1. p. A
may satisfy the system, we find the conditions
-j• 6,eC, +( F,)C, -o0.
Non zero soia'don C and C2 exist if, and onlp ifS+ (f, -&,)1 F = 0 (7)
In spite of the apparent elementarity, the equation is
emeedingly complicated unless f, and tare constant (in which
case the classic solution is found). To make the problem practical,
we shall accept first the reasonable assumption that the varia-
tion of a and Lwith z be relatively slow (as referred to the
wavelength) so that in a substantial number ,of "wavelengths" the
variation may be regarded as linear. In this case the equation
becomes of the formS+ (A ÷S:) O o0)
and has the complicated solutions
PuTA + - (
where J is the Bessel fumction.
In the present case A + Bz is, alternatively, If, -
or V1,+a. ; so that, extending the value of the definitions
Mg~jiý:and Th'~+~to the present case where 01 and
ari functions of z, the general expression of, say, Ex is
wit b ej-
with bi- i, -i 2 , b2 2,it + is (The f•ormula, of course, is
only valid when &, and 12are linear functions of z). When F sati-
sfies (7) with the upper (minus) sign, the coefficients of Eyare + j times those of E ; the converse is true in the case of
the lower sign.
Accordingly
M21. 16.• 0.295
SINDELLp. A
It is seen that the knowledge of a closed solution of
equation (7) when a,* a& is linear is far from oonetiftking
an advantages in addition to being complioated, solutions of
form (9) are not physically evidents for instance, the way is
b• no means clear by which combinations of functions (9) may
tend to e tjlor e ,as is necessary, when w and P tend to
become constant (b1 and b2 to zero)
The remark suggests the opportunity of looking for a
more practical solution of the 2hysioal problem as reprosented
by the general equation
F U- ON)I
where g (z) - *.ts t a known function of a. The assumption
of a linear behavior of g is only an approximation which obviously
looses interest if it does not lead to a simpliftoation.
Writing F =•, as is natural to do, the equation transforms into
a Riccati equationa" -_o)
The fact that g is normally negative, and real, suggests
for ÷ a complex form a +j ; : this leads to
a .- 2j•; a. -I(%).
For the reality of the l.fthand Uide, b +2U must vanish:
this gives = and
The further position a = In u yields
44--. C •
The physical problem is thus redused to that of assuming
a convenient form of u which, whert substituted into (11), may
yield a reasonable approximation to g(s). This can be done in
many different ways. For example, a function u -e , that is,
42 , oould approximate g(s) by the expression
foglio 6 di 19
MeW. 1. 4 210.29
SINDELLp. A
.I.the parameters of which can be so chosen as to reasonablY
"1matohthe actual g(s). For a match near the orlgin, for instanoe,
the conditions
are written.
The corresponding i is aN"214 s as the components of
the field have been assumed to be of the form +t the
"*einstantaneous wavelength" is 2x/i.(2ry/h) a
From the initial value 2X/h the wavelength increases if is
positives in this case the amplitude also increases.
The complete expression is
that is, neglecting a multiplicative constant
F *xp (yyz - h C'For given g(O) and g'(O), is the real root of the
equation
4 . t - . N '(0) 0.o
As h is real by assumptionj is negative or positive according
as g'(O) is positive or negative.
According to (3), the two physical expressions for the
function g are
,k- 4 k+1i , k-t
where k -4i/4 is proportional to the static magnetic induction
and *l, OlZn/m1t, Wt is proportional to the electron den-
sity. If the variation of g (referred to the wavelength) is slow,
Sis small and h 2 is close to g(o), that is, to ealor P .
"Given the four constants as defined by the
foglio 7 di 19
Mod. 18 - 210 * 295
SINDEL
L.p. A
initial values of I,, ,and their derivatives we thus have
ba
x aC, e. (, z .- p e'") +j e, cw(, (.jh. X 2 v" ) -
if the constants ]', • are relatc4d to t,- an•,od kz , to
L,+ E.2 "
The Maxwell equations -ive.e,( ý, . , () X-£ ( C -z L
•, • je , ( &Z*•.'") C p (•. - -• e•
- ie,(j,-2 e,,"' ) , •. (•:, "
It is easily recojnized that 1 hl and h2 are chosen
positive, the progressive fields are those involving + J inthe exponentials: assuminr that the entering field at t o 0
is matched to the local impedance the conditions for the pro-
&-ressive field where I~x(O) - E, Ey(O) - 0 are
foglio 8 di 19
Med. 21 • 210 x 295
SINDELL p. ASpACI lik/1'.O lhal ,
es alk/l e. h~ie 40r
C, 2,! no.
Accordingly, and writing for brevity
PC = GWp ; = expe a
the field is
EX -81 l CXF1 4...L. ewp'
2
-Ey, E g/. if, ex"p,' T, Lvp).lt,•72 2 , h
-Z 8 1 2
In every expression, the first term represents the "alfa
field", the other the "beta field".As usual, the Poynting vector
pertaining to e:ch separzte field vanishes identically, while4 -•. • J= -• "3
2. 4 woo•,iEp . 2 etc.
The discu'xion of Lhe field: rotation, attenuation, etc.
is carried out without difficulty: but the consideration of a
more ooncrute c se is of greater interest.
4- Propagation in a longitudinally variable laminar plasma duct
An a next case, in order of increasing complication, we
can consider Lhat of a plasma istribution that, in a cartesian
space, may show a variation ir the z-direotion alone, as above,
but is confined in a finite width in the x-di..eotion being unli-
mited, however, along the y-axis. If we look for fields whioh
are independent of y, the Kaxwell equations become
foglio 9 di 19
Mod. 16 2- 0 x 29 3
SINDELS. p. A
'6 -n~jc"b UJIz y* "z"aK E, .U E,.A E:
-9
that is, eliminating K,
V5Y + "a' E Y
V'E2 'b& Ea-A -6 -A z t16 : 3
It is recalled that 9, LA ta are functions of z alone.
The question is spontaneous whether a solution can exist
of he form E - AF where the vector A is function of x and the
scalar F of z only. Writing for brevity F/F -yas before, and
F/F' - -• . .. •i - i the system becomes
.A,(. • +j,,t , A , = ÷11
-AY (2 -A)y ,A,. - 0
ItAX 0 -As + 13 A
faglio 10 di 19
Mod. Is • 210 X 295
SINDELLt. A
(Accents lenote differentiation in respect to x, dots in re-
spect to a).
System (13), where the ooefficients are functions of z,
should admit of solutions A , Ay, Az, functions of x only. The
requirement is very -rostkictive, and for given 1E, at L I cannot
be satisfied by a single function +. This means that the exist-
ence of a solution of type tF is an exceptional event, if even
possible.
The investigation whether of not such a plane propagation
can take place is r,.latively simple. If the first two equations
(13) aro written
AX + A ,, A'
we find, upon differentiation in respect to z:
S .A + A; 0
j. A,.I. AY .0d •X
this requires, first, that
foglio 'I di 19MAd. Sl •210 x 295
SINDELL p. A
Furthermore, the ratio
~(S -gjCOSJIS/dx
must be independent of ma that is, as &, and &.depend on u only,
must be a constant. This means
- e, 1.e 3 (14)
Replacing in the first condition, we find
) AL e, e.. •ii•,o
and
Ax = ,•
The equations are necessary consequences of the assumption
of a form IF(-) for the vector I (plane waves).
Replacing in the first two (13), we have
A" -eIAY -0.
The second relation is perfectly compatible with the ins
dependence of A from sj the first equation requires that they
ratio
D'/' (15)
be a constant. iy is connected to 1, and Eby (14))
Ioglio 12 di 19M"d. 1S - 210 x 295
SINDELLp. A
From A' J- D A. we find, replaoing in the third (13)
-jD (Ag' + S3 Ay)-4
that is
.)D (e, .+ 3)A, j' e,A; CA . Ay,4•,
or finally
D (Ct.+ 9j) + CC 0 o. (16)
(14),(15) and (16) represent the necessary conditions for the
existence of plane wavess 4 can readily be eliminated, writing
from (15)
(e. -( Oas) +e1 ,0. +,,+
that is, replacing into(14),
-(Ha.s+ ) Hi,,s (17)
and,replacing into(16)
D (et * t... + ee, I•,.a +. K] .o.(,8)
The conditions should be satisfied by suitable values
of the constants C1 C2 and D (to which H and K are obviously
connected). As 1j is expressible in terms of E, and 1,, the
two equations, with given constants, completely determine the
plasma configuration. it is possible that the conditions be
less reatrictive thqn they appear at first, owing to the pre.
sence of three arbitrary constants which could possibly match
well enough the actual situation.
For . jiven configuration the problem only imposes two
conditions upon the three constants, one of which could possibly
be left free for a best pos.ible matcht if the configuration
varies rather slowly with z, a solution by steps is then conceivable.
loglio 13 d 19
Mad. 18 • 210.295
SINDELLSpA
- 5-Suggested method for the solution of more meneral oarteaian oases
Cases of wider generality than those that have been cons
sidered hitherto cannot be trateby methods of comparable aim-
plioity. The description of a computational method that may pre-
sumably be valid in a larger variety of cases is thus in order.
We assume, as before, that s, , o, ,o be functions of z only,
but their generality is not restriteed at the moment.
The solutions of Lhe Maxvell equations are dupposed to be
expanded ih series of orthogonal functions of x, having functions
of z as coefficients. If the width of the channel is indicated
by 2w, cos x and sin k can constitute such a set of
orthogonal functions. A simple inspection of equations (12) shows
that E x and Ey, as functions of x, have the same parity, while Ez
is opposite: the same is true of the components of K. Accordingly,
"even" harmonic components can be written as
"(22)E a a os -0-wx E - e 00s-~-WA E - esi=x' E~x = x co y = y co~ z = e Wsn
"odd" components as
E - a sin -r E a e sin±!-!-- E. -- e cos-1=xX X w y y w z W
Similar expression are valid for Kx, K , Kz in terms of harmonic
components k , k y, ke. (These components are actually functions
of the order ns the corresponding index, however, is omitted for
simplicity.) With the notations accepted in (22), the harignic
components appear to be connected by the ordinary system:
W I
.iy r iC 04 (23)j
The equations are valid both the even and for the odd components.
foglio '4 di 19
Me. IS. 210 X 295
SINDELLp. A
Even in the simplest assumptions as to the form of the
funotionse 1,,€s of a, the resolution of (23) ie oxeoeed1ly
complicated. If attention is given to the fact that one system
(23) exists for any positive value of n, the neoessity of soe
standard method of solution, at least approximated, is obvious.
Regarding the variation of the parameters as a relatively
small perturbation in respect to the constancy, we write 0- -
. ,. '...9-0+.t51, etc., where, of course, T, Ta, 2
Assuming that •J, .. ' k• satisfy the static system.
4X nr( F-
'.V W + A, Cy (24)
M Tr J . kZj-'r
We look for solutions of system (23) written in the form
x + f • I kz + z, where it is supposed that products of
small terms of the form. of f times z , are negligible in the
whole useful runge cx" z in comparison to the p•'idriFal ter:.ia of
4forn e times S. It is readily founA that the corrective terms f
and 1 have to satisfy the system
•r~
If the functions - , k, solutions of (23), have been
so constructed as to satisfy the boundary conditions at too, the
correcting functions f and £ can be subject to the simple condition
of having zero initial values.
foglio 15 di 19
Mod. 19 - 210 x 295
SINDELLp. A
Application of the method first requires solution of
system (24) where the ooefficients are constant. (Of course, in
the case n - 0, the field is TM). The constant-coefficient systom
obviously ---its of solutions of the formhA
y 1L ai.. constant v C
provided j has Auch a value as to make zero the determinant of
the algebraic system of the"constants".
Before entering in further detail, we remark that the
equation for r is biquadr.itic and that for every value of r a
solution of the form
is available, with' constant ax, . . b (one of them arbitrary)
System (25) is subs~antially of the same form as (24), but
is complete instead of homogereous, the "forcing terms" buing of
'he form
that is, of the form
constant x Z e
The analytical problem is thus solved as soon ab one
particular solution of the complete system is known. This can
be done with ý1t andqrd rules as soon as "a fundamont 1 system"
of solutions of the homogeneous system is known. In the present
case, the "fundam-n al system" is merely cons 4 Iuted by the set
of four sextuples of functions corresponding to the four
different values of 5.In a somewhat more detailed discussion, we may observe,
first, that the unkno~n functions are actually four rather
than six, since two, out 0f equaions (24), are in fixite terms.
"Eliminating e and kz, (24) is replaced by the :;ystom of four equations:
foglio di
Met. 11 - 210a 295
SINDEL•pA
Y j~ e y(z ,)
ky k. - j' (A ) j,
where • is written for brovity in the place of A11T/*'. Similarly,
system (25) becomes
I4 S= 2-
~~~ý a~(>,.i IF- - Z3, e
I ) Y+.ft.fx 1
The solutions of (26) are of the 'form
(i-1,2,3,4,)wherel: is a root of the biquadratic equation ob-
tained by making zero the determinant of the algebraic system
obtained by replacing the last expression in (26), and BI, C|, D,
arc the corresponding solutions. Writing for brevity
F, . o F .2e2e,, F.%=j5.4
the general solution of the complete system (27) is given by
el r~l [f/A,1 o, l,] + el"EfhA ý4.1a + - e fJai$/,a 'i a*g +
+8e41fA /Az4i+
I = as above with coefficients C1 , C2 , C 3 , 04.
fir as above with coeffioients D1, D2e D3 , D4 19#ogliA di 19
SINDELLp.A
4 is the determinant of the "fundamental system" namely
X g B.s4 lz J) eaz
A as (it
4z 4& X • 4 2
As (,... -4 are solutions of a biquadratio equation, the sum
,+r,4r3 +r* is sere: so that A, reduces to the determinant
of ,he constants. Similarly,
4F 4
4 4 c J) 12
while A,, 4j, £14 are 6imilar in form but have fF,,F3, F4 respectively
at the second, third, fourth row.
Functions F have obviously the form of . sum of termsJ. z
ez where 5 is a known function of z. The determinants Ai
can thus be split in sums of four determinants the first
of which has the first row proportional to ea , the 3econd to
e a•z and so on. Out of these determinants the first is pro-
portional to e + ' )21 the second to e etc. In any
case, the solutions (29) are explicit and can be directly evalu-
ated, for any given set of functions "•t / 5 / 1 '.
Of course, the solution is valid in the limit of the ap-
proximation consisting in regarding the pertubations as first
order quantities. The functions ; + f.,K + 9 which by virtuex x z
of the presence of the arbitrary constants R,9 5 ...,R4 have just
the same forms (29) are solutions of a system of form (23) where,
however, ,, ,,• ar• not the diven functions of z but differ from
them b:! terms of the order of .he square of he perturbations.
joglio i19
Mod. 1. • 210 x 29S
SINDELSp.A
g
The prooedure oan premzmably be iterated, thus inoreasing
the.complioation, but not the difficulty of "he solution.
loglio 19 di 19
MA. 1 - 210 x 29S
.1 1
a.~S 044~t4. . U U
N ~ -.4. * 4 0
0 4.'U - -d
.0 14-.
V wa
00 IN
4. .~ C