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^ecUnlcal ^iote 256

MICROWAVE REFLECTION TECHNIQUES

FOR DENSE PLASMA DIAGNOSTICS

S. TAKEDA AND T. TSUKISHIMA

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
THE NATIONAL BUREAU OF STANDARDS

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NATIONAL BUREAU OF STANDARDS

Technical Note 256

ISSUED MARCH 10, 1965

MICROWAVE REFLECTION TECHNIQUES

FOR DENSE PLASMA DIAGNOSTICS

S. Takeda and T. Tsukishima

NBS Technical Notes are designed to supplement the Bureau's regular publications program. They provide a means for making available scientific data that are of transient or limited interest. Technical Notes may be

listed or referred to in the open literature.

For sale by the Superintendent of Documents, U.S. Government Printing Office Washington, D.C., 20402 Price 25 cents.
Contents

page

1. Introduction 1

2. Principle of Reflection Method . 3

3. Some Considerations for Reducing Experimental Errors .... 8

3.1 Difference Method 8

3.2 Effect of Power Losses Along the Path Between the

Plasma Boundary and the Standing Wave Detectors .... 10

4-. Further Extension of Measurable Density Range . 11

5. Microwave Reflection Probe 14

6. Reflection by Inhomogeneous Plasma ....... 17

7. Reflection by Boundary Layer with Density Gradient 20

8. Concluding Remarks 22

9- Acknowledgment 23

10. References 24-

li
Microwave Reflection Techniques for

Dense Plasma Diagnostics

Susumu Takeda* and Takashige Tsukishima

The microwave reflection method for measuring

the electron density whose plasma frequency is

higher than the probing frequency is described in

detail. Various expressions and formulas which

are useful for a variety of experimental conditions

are given. A method to extend further the measurable density range is proposed. The fringing field

effects are considered when a waveguide is used as

a probe. References are given to the experimental

works which substantiate the theoretical analyses.

Also included are the analyses of reflection by a non-uniform boundary and also by an inhomogeneous

plasma.

1. Introduction

The determination of the electron density, including its spatial

distribution, has been one of the first requisites in plasma researches.

Among other methods, microwave techniques have been extensively used

because of great simplicity in handling and interpreting the results.

The microwave methods hitherto developed may be divided into l) the

cavity method and 2) the propagation method which is applicable to plasmas in a waveguide as well as in free space. The cavity method and

the waveguide propagation method are mainly used for fundamental

studies of small scale laboratory plasmas, while the free space propagation method has been widely used in various high-temperature devices.

A shortcoming of the conventional transmission method is the basic

difficulty of investigating the density distribution, as well as a

certain cumbersomeness of the arrangements. Another difficulty resides

Present address, Nagoya University, Nagoya, Japan
in the fact that an electron density whose plasma frequency exceeds the

probing frequency is not measurable. Thus, even with 3 mm waves the

maximum measurable density is 1014 /cc. In order to overcome these difficulties, a new reflection method was proposed by one of the authors. 1

While the conventional reflection method or cutoff method is only for

locating plasma, the present method admits measuring an electron density higher than critical. The method is thus different also from that

of S. C. Lin et al., who determined sub-critical densities by measuring

reflected power. The principle is based on the fact that the electron

density' of a uniform plasma bounded by a plane boundary is related, in

an extremely simple manner, to the phase angle of the reflected wave at

the boundary. Since the interaction between the incident microwave and

plasma is limited within a narrow region near the boundary, in other

words, the penetration distance is very small when the density is high,

the measured value by the present method is essentially local.

The proposed method was first applied successfully to a uniform

afterglow plasma produced in an X-band waveguide. 4 Later, an electron

density as high as 1015 /cc of a shock produced plasma was also measured

with the same X-band (9000 Mc/sec.) microwave. 5 A K-band (35 Gc/sec.)

microwave reflection probe has been used to obtain the density profile

of CANDLE plasma (A magnetically confined argon arc) at the National

Bureau of Standards, Washington, D. C. The values obtained by this

microwave probe have fitted remarkably with those obtained by other

techniques such as the Stark broadening of Hq and Langmuir probe.

In the course of these experiments, a number of formulas and
expression have been derived to account for a variety of boundary

conditions. It is the purpose of the present paper to collect these

results in one place, although some of them have been published elsewhere. Also included are recent analyses of reflection by a non-uniform

boundary and also by an inhomogeneous plasma.

The idea described in Chapter U has been proposed by S. Takeda,

and the calculations in Chapters 6 and 7 have been made by T. Tsukishima.

2. Principle of Reflection Method

A plane electromagnetic wave, e^ ^ , propagating along the

z-direction in vacuum with angular frequency w and propagation constant k , is reflected by a plane boundary, z = 0, separating a homogeneous plasma (z > o) and vacuum. The reflection coefficient, R, is given

by

O

where

K = a - jB (2)

0i and B are the phase and attenuation constants in the plasma. They

are related to plasma parameters,

U) 2 /,.,/uo2 E T|,

P

(3)

v/oo = 6,

where cti and w are the plasma frequency and the frequency of the

incident wave, v the collision frequency of electrons in the plasma,

by
K

o rf/. Tl n2 . , 716 n»1

a = ^2

id- j£ri + (A) )

+ (1- ^)J , <*)

when there is no magnetic field.

Equations (l) , (2), (4a) and (4b) give the complex reflection

coefficient R as a function of 7] and 6. They can be inverted to

give T] and 6 as functions of the modulus, |r|, and phase angle, 6,

i i 19 of the reflection coefficient R = |R|e .

For example, let us consider first the case for 6-0. Eqs. (4a)

and (4b) then reduce to

a = k */l-T] ,. o

\ for T) < 1, ($a)

P =0, J

a = 0, ^} for T| > 1. (5b)

o

Substitution of Eq. (5b) into Eq. Cl) gives

R

^^^ . (6)

. 1-j/il-i

The phase angle, 6, is immediately obtained from Eq. (6):

= 2 tan Vl|-1 , which indicates that tends to tt as T] goes to °°. Thus, a plasma of

infinite density is equivalent to a perfect conductor as far as the

reflection coefficient is concerned. It is sometimes convenient to use 4
cp E tt - 9 rather than 9. Eq. (7) is then written as,

1

cp = 2 tan

-1 —— (8)

Eq. (8), solved for T], yields,

Tl = ^~7~ (9a)

sin2 9/2

« —~— for T| » 1 (9b)

tan2 £

2

The Eq. (9) shows that the normalized density T) is readily obtained

by measuring the phase angle cp. The value of cp is conventionally

calculated from the shift of the standing wave appearing in the vacuum

side. The standing wave E is expressed as,

-

3 k

o

z „ J k

o z E = e + Re

jfi... = 2e 2

C o S (k

Q z+ |). (10)

Accordingly, the shift of the standing wave minimum, S , as referred to

£ tt the position for the perfect conductor, is obtained by putting k S+ p = 7

Then,

k S = J = tan

-1 —z= (ll)

2

y-n-i

This shift of the standing wave can be expressed in terms of the attenuation or penetration distance, d, defined by

dE^, (12a)

= L_ for 11 > 1 and 6 = (12b)

k^Tl-l

o

'

From Eqs. (8), (ll) and (l2b) , we obtain the following relation:

k S = tan"

1

(k d) (13)

o o
Thus the shift of the standing wave is equal to the penetration depth of

the incident wave into the plasma, provided k d « 1.

For plasmas with 5 / 0, things are somewhat involved as is seen in

Eq. (4.). However the expressions for Tl and 5 in terms of |R| and 6 have

been derived for T] » 1 + § 2

: they are, 1

-n = —-

u . ,-— , d4a) 1 tan2

cp - (l-|R|)'2

2 1 + (l-lRl) 2 /tan2

cp 1 - (l-|R|) 2 Aan2

cp J

- 1. (Hb)

Noting k S = cp/2 and tan 2^2" ^an CP> one can rewri"ke Eqs. (l4a) and

(l4b) in the following forms,

T] = : — (15a)

tan2 kn S - (1- R|) 2 A

6

2 =

1 + (1-|R|) 2 A tan2 k

Q

S

2

- 1 - (1-|R|) 2 A tan2 kQ S

-J - 1. (15b)

The values of T] against tan cp are shown in Fig. 1 for various values of

§. It is seen that the correction to T| due to finite 6 is relatively

small. For time varying plasmas, it is more convenient to utilize a

suitable couple of the standing wave signal. The normalized amplitude,

h , of the standing wave detector which is assumed to have an ideal

z

to

square characteristic, is given by,

!,...„.

"

J

' 2kol z

2

h =7 ll + R'e ° I . (16)

z 4

'

The numerical factor 1/4 is introduced in order to normalize all standing wave amplitudes to unit maximum for R = 1.
Eq. (l6) can be solved for T] and 6 with two h ' s observed at two

z different positions. While the choice of these points is to some extent

arbitrary, the four points shown in Fig. 2 are preferred to get concise

expressions for 1| and 6. After some calculations and suitable approximations, the following relations are deduced:

a) For Tl » 1 + 62

,

T| =

7 s*

4

f

, TT^ (17a)

(h

3

-h^) 2 - [l-(h

3+h^)}2

r(ho-hy ) 2 + fi-(ho+hy )}

2 >

2

, , 6 2 - 3 V l 3 £±_l - 1. (17b)

Mh

3

-h

4

) 2 - {l-(h

3

+h^)]2

J

The above equations reduce to

'3 ""4

11 ~~ (n^TF ' (l8a)

(h

3

-h

4

) 3 4

for 6 « 1. Eq. (l8a) can be slightly modified to give,

3£- = 7 * , (18c)

11-1 (h

3

-h^) 2

which is still applicable even for T| > 1, provided 6 <<1.

b) For T| > 1 and 62 « 1,

Tl - 1 + rr (19a) n

l

a «V^,{1 "

Chi+h2» (l9b)

In deriving Eqs. (19a) and (19b) , a and |3 have been approximated by

a =

Q

T|6/2 yi'l-1 (20a)

7
P - k

Q

VtFT (20b)

Eqs. (l8c) and (19a) simplify respectively to

T12 _

1 1 (21a)

H-1 (l/2-h^) 2 (h

3 -l/2) 2

and

T\=r- (21b)

hi

when' 6=0. Eq. (21b) will be seen to be exactly equivalent to Eq. (9a).

Since the phase and attenuation constants in a plasma filled waveguide through which the microwave excited to TE mode propagates can be

obtained by merely replacing T| in Eqs. (4ai ) and (4-b) by T|/(l-X2 /x2

)

where X and X are the wavelength in free space and the cutoff wavec length of the waveguide, the formulas in this chapter and in the following chapters can be applied for a waveguide system provided T) is replaced

by T/(l-X2 A2 c

) • Eqs. (19a) and (19b) have been successfully used to

determine T) and § in a recombination controlled afterglow plasma which

was produced in an X-band rectangular waveguide. In this experiment an

electron density as high as 1013 /cc has been measured. 4

3« Some Considerations for Reducing Experimental Errors

3.1 Difference Method

In order to be able to measure a density as high as T| = 104 one has

to measure within a reasonable error range a value of (ho-hy) as small

as 2xl0~Ji or 1.1 degree in the phase angle. 7) = 104 also corresponds to

the shift of the standing wave minimum of S - (x/2tt) x 10~2 cm - 5xl0~3 cm

for an X-band microwave of x = 3 cm. Thus the correct location of the

standing wave detectors is of the primary importance.

8
However, the errors in T| and 6 due to the slight mis-location of the

detectors can be to some extent avoided by a sort of difference method

when it is possible to get an electron density much higher than those

which are to be measured. Such a condition as above is often fulfilled

in shock experiments.

Let ho' and h

,

' be the standing wave amplitudes for a plasma whose

density is much higher than those of interest j then Tj and 6 can be expressed in terms of

Ah-3 = h^ - hi,

a. = h - n;,

(22)

as follows;

(Ah

3

-Ab^) 2 - (Ah

3

+£h.) 2 (23a)

,2 2

f (Ah

3

-Ah^ + (g£g^ r x> (23b)

*.(Ah

3

-Ah

4

) 2 - (Ah

3

+Ah

4

) 2 J

provided 6/£/T| » 6' /A]' and (2ttA)(a X3 + A X/) << 1» where A x 3 and

A Xj are the errors in the positions of the standing wave detectors for

h„ and h. respectively, and 6' and T]' are the values for the reference

3 4

high density plasma.

Eqs. (23a) and (23b) reduce to

4

Ah

3

-Ah^)

; (24a)

Ah +Ah,

(24b)

Ah

3

-Ah

2 • 3 ""4
for 6 2 « 1. Note that (Ah Q+Ah,) < (Ah -Ah,) because of &h„ > and

j 4 5 4 3

Ah, < 0.

4

Eq. (24-a) has been used to evaluate the electron densities from 1013

to 3xl015 /cc in a plasma produced by an electromagnetically driven shock

wave into argon. 5

3-2 Effect of Power Losses Along the Path Between the Plasma Boundary

and the Standing Wave Detectors

The power losses, such as the circuit loss due to the finite conductivity of the waveguide or some scattering loss between a horn antenna

and the plasma boundary, have to be taken into account to get the correct

values of T] and 5. When these losses are of pure dissipative type and

accordingly give rise to no phase shift, the standing wave signal, h , z given by Eq. (l6) is modified as follows:

2

i -j 2k I z

h = 7 1 + pRe . (25)

z 4

Here, a correction factor, p, which is a real number less than one, is

introduced to account for these power losses. The similar calculations

as in the preceding chapter result in,

^p2 _^__ , (26a)

(Vh4

}

•[l + \ [1 - 2(ho+h,)}], (26b)

h3"h

4

P

provided T] > 1 and 6

2 « 1.

Alternate expressions for T| and 6 in terms of h, and h are

possible: they are,

10
*10

with

T| = 1 + h

20/h1Q , (27a)

hog

6 = ^ -^ - (hio+h20»' (27b)

h

20 = 2? l^VV^ + (V^lWhen 6=0, Eq. (27a) reduces to

(28)

71= # , (29)

h, - J(l-P)

2

by noting that \+^2 = |{l+p3

) for 6 =0.

Eqs. (26a) and (26b) can again be rewritten in terms of Afcu and Ah,

defined in Eq. (22);

71 = p2 U

- (30a)

(Ah

3

-Ah

4

) 2

g =£ . _J i ( 30b ) P Ah

3

-Ah^

Eqs. (26), (29) and (30) reduce to Eqs. (18), 21b and (24.) respectively

when p = 1.

Note that the introduction of the correction factor p does not result in any change in the shift of the standing wave minimum S, as is

clear from Eq. (25) and the definition of p.

4. Further Extension of Measurable Density Range

As is well known in transmission theory, a small phase shift can be

expanded by making use of a suitable buffer dielectric plate of \ M

11
thickness. This idea is here utilized to extend further the measurable

density range. The reflection coefficient R at the surface of the

dielectric plate, whose other side is in contact with plasma as is shown

in Fig. 3, can be easily calculated to give,

k 3/k - (k2/kQ ) 2 -

k

3

/k

o + (k/k^

(32)

where

k

3 = -jkyihl

k = k JT

2 o

and £ is the specific dielectric constant of the plate.

Substitution of Eq. (32) into (31) gives

R = .

1 + yg , ( 33)

1 - yri-i/e

which, when re-evaluated at a point off the dielectric boundary by a

distance \/U, turns out to be

T

1 1 - yi'i-l/e

he shift of the standing wave minimum, St, is now given by

kS = tan"

1

- -7= (35)

1

*/f\-l

«^ for T]»l

Eq. (35) shows that S is nearly £ times S of Eq. (ll). Accordingly the

maximum, measurable value of T| is risen up by a factor £

2

. Eq. (35) can

be easily generalized to the case where n dielectric plates are placed

in front of the plasma boundary, each plate being separated by a distance

12
\/U from its neighbors. Then the shift of the standing wave minimum.

S , observed in front of the n-th plate is given by

n

n

k S = tan

X -= . (36) n /n-i

It is noted that the first dielectric plate does not necessarily have to

be in contact with the plasma; the plate located at distance \/2 from

the plasma boundary gives the same result.

The technique described here can be also applied to plasmas whose

densities are less than critical. When the plasma shown in Fig. 3 is

replaced by a plasma layer of thickness L, of density 1] < 1 and whose

other boundary is made in contact with a perfect conductor, the relection

coefficient evaluated at a distance \/4 off the dielectric plate is given

by

! 1 + yi-T/e-tan k

QW]P)i

R. = ~= *—= • (37)

^o 1 - yi-T/e-tan k 1L/l-T]

If we choose kL = 2mrr where m is an interger, we have tan k L i/l-T\ = mrrT| o for T| « 1. Thus the shift of the standing wave minimum, S , is given by

k

Q

S^ « mnel], (38)

which is e times larger than that without the dielectric plate. The minimum, measurable value of Tj is therefore lowered by a factor e. With

n plates, being separated by a distance \/4 from each other, the shift of

minimum observed in front of the n-th plate is,

kS 1 = mrren

7]. (39)

o n

The other boundary of the plasma layer needs not necessarily be in contact with the metallic conductor; the standing wave can be constructed

from two signals in a bridge type circuit. One signal is taken directly

13
from the microwave oscillator, while the other is taken through the

plasma layer. Thus the present technique covers a wide range of measurable density, for instance, from T| = 10~4 to 10s when two dielectric

plates of e = 3*7 are used for a plasma layer whose thickness is \/2.

In a waveguide, e has also to be replaced by ( e-X

2 /xc

2 )/(l-X3 Ac2

) •

5. Microwave Reflection Probe

Hitherto we have been mainly concerned about plane wave analyses.

However additional problems arise when a microwave probe, as is illustrated in Fig. U, is used to obtain the local electron density of a

plasma in free space.

First of all, a part of incident microwave power is reflected at

the probe end even when there is no plasma outside the probe, because of

the abrupt change of geometry. Of course this type of reflection can be

eliminated by using a suitable matching device such as three stubs or

E-H tuner. The problem is whether the tuner, once matched for free

space, has to be re-adjusted when the probe is inserted into plasmas.

An analysis has been made of the equivalent circuit based on the theory

of the transmission lines; the result shows that the overall reflection

coefficient is given by a product of a constant phase factor and the

reflection coefficient of the plasma boundary, provided the system is

matched initially for free space. This constant phase factor can

experimentally be corrected for by referring phase to the metallic

short case after the matching for free space is achieved.

The second and more serious problem is how to handle the distortion

of the electric field outside the probe. Galejs has calculated the

H
admittance of a waveguide radiating into plasma, 7 and Wait has given a

formal solution to the radiation field outside a slotted conducting

plane in a plasma environment. 8 In both cases, the electromagnetic

wave propagates into the plasma through a slot antenna which is located

in a perfectly conducting infinite plane.

Instead of having recourse to the rigorous, but cumbersome mathematics, we develop here an heuristic and semi -empirical method. 9 We

first assume that the fringing field in vacuum can be expressed approximately by the field produced by a line dipole placed at distance Z

behind the probe end. Then the field strength at distance z from the

probe end is given by E Z

2 /(z+Z ) 2

. This has to be multiplied by the

attenuation factor exp (-z/d) in order to give the field when plasma is

present . Thus, to a first approximation, we obtain

for the fringing field in plasma, where E is the value at z =0, and Z

depends presumably on the ratio of the wavelength to the characteristic

probe dimension. On the other hand, we can also express E(z) formally

in terms of an effective penetration depth, d' , as follows:

E(z) = E

Q exp (- |,). (a)

To define d' in terms of d and Z , the two expressions for E(z) are set

equal at some point z = Z-. . While the choice of such a point seems

arbitrary, we may argue on physical grounds that the best choice of Z

would be Z = d, where the field is roughly e

-1 times E . The result

is:

15
d' = d[l + 2 in(l+d/Z )]

1

. (42)

This equation reduces to the correct limit, d'/d = 1, for large T| or

small d, for it is just when d « Z that fringing field effects are

expected to be negligible. For d large, d'/d < 1 expresses the fact

that a smaller effective penetration depth is here used to compensate

for the field decrease caused by the effect of geometry. With Eq. (4-2),

one can easily obtain the reflection coefficient, noting d' = l/(3' and

d = l/j3. Or one may make use of Eq. (l2b) to compare T|' with 7], where

T| is the actual density just outside the probe and T|' includes the

fringing field effect. In Fig. 5 are shown the values of d'/d and T)'/T|

against ° d/Z for various values of Z /\. The value of Z^ remains to be

o o o determined experimentally. And this was done with a uniform argon afterglow plasma produced in a cylindrical waveguide. The phase shift was

simultaneously measured with an X-band microwave at the two ends of the

chamber. At one end, the chamber was attached to a cylindrical waveguide of the same diameter as the discharge tube so that fringing fields

are absent. The microwave probe was used at the other end. It consisted

of a standard RG-52/U waveguide tapered to a 3 mm height so that the

plasma in the cylindrical chamber is- considered as a plasma in free space

for the probe. The calculated and observed values of T]' showed good

agreement within the experimental range of Tl < 10 when Z /x_ was chosen o&

equal to 0.63*

A K-band microwave probe of the cross section 7x1 mm

2 was successfully used to determine the density profile of an argon arc plasma. The

16
densities measured by the probe fitted quite well with those obtained by

other techniques such as the Stark broadening of Hg line and Langmuir

probe. 6

6. Reflection by Inhomogeneous Plasma

Need sometimes arises to take account of inhomogeneity of the

electron density along the direction normal to the boundary surface,

when one wants, for instance, to measure the density profile of a

cylindrical plasma whose characteristic diameter is of a comparable

order of magnitude as the wavelength of the probing signal.

To choose a simple case for illustration, let us consider a linear

density ramp with a sharp cut at z = where the reflection probe is

located, as is depicted in Fig. 6. Since the gradient of the electron

density is parallel to the direction of the propagation of the incident

wave, the wave equation for the electric field strength, E, is simply

given by10

^+ k

2 (1-71) E = 0, U3)

dz 3 ° with

T] = for z < 0,

Tl = 11 + w~ for z > 0.

M)

The solutions for the region I and II, in Fig. 6, is given by

-jk z jk z E

I =e °+Re °, (4-5)

En =A^I

l/3 (0 +B^I_l/3 (d, (4-6)

17
where

X 3 1) - 1,

r = £ 3/2

^ _

3 Vox »

and I , Aj(0 are the modified Bessel functions of orders ±1/3.

boundary condition thatETT has to vanish at z -n» requires A =

the asymptotic form of l±-i/o(0 is given by1

(47a)

(4.7b)

The

-B, since

(48)

The reflection coefficient R is obtained from Eqs. (45) and (4-6) by

dK dE

II putting Ej. = E at z = 0.

1X

and — = ^

After some calculations, we obtain

1+jK

(49a)

where

K = I_o/o(0 + XI o/o(0 -2/3 2/3.

Uxi z = 0. l/3 (0 -Ai.l/3 (0 J

When k Z (1) -l) » 1, in other words, when the density gradient is

very small and/or T| is very high, Eq. (49b) reduces to

(49b)

- yVl { 1 +

1 1

^oZ

o

" (\-D 3/2

'

' (50)

Hence, we have

R =

1 + t -/\-l { 1 +

1 - j V^T {

1 +

41c Z (Tln-1) -i^3/2 }

^ko Z

o (TL-1) 77372 }

(51)

18
As is expected, Eq. (5l) reduces to Eq. (6) when Z

""" °°, that is, when

the density gradient approaches zero.

It can be shown that a WKB-type approximation method leads to the

same result as Eq. (5l)« When the change in the electron density over

a range considered is very small compared to the value at the boundary,

the electric field can be approximated by

r, Z -k j «/T|-l dz

En~Ce ° , (52)

where C is a constant. The effective attenuation distance, d, is then

defined by

k I //Tl-1 • dz = 1. (53)

o J

The integrand can be expanded as follows,

^i-1 = A + 1

- i

o

« s/'\\ -1 { 1 + S -V.

l (54)

2Z Tj -1 J o o

Inserting Eq. (54-) into Eq. (53) and solving for d, we obtain,

7«k ^tT3! I 1 + —

7-\, (55) d o 'o 1 4k

oZQ (Vi)3/2/'

provided 4k Z (T) -l) 3' 2 » 1. Since d = l/p and R = (k +jg)/(k -j p) for

a loss-less plasma, Eq. (55) gives the same reflection coefficient as

Eq. (51).

19
7. Reflection by Boundary Layer with Density Gradient

In the preceding chapters, it has been assumed that the reflection

always takes place at a well defined boundary surface across which the

electron density changes stepwise. However, there exists always a

finite boundary layer in an actual plasma, at least, of the order of

the Debye length. The boundary layer may become still thicker due to

diffusion into the waveguide, when a waveguide probe is inserted into

a plasma and when it is impossible, due to heat damage or charge accumulation which might cause some undesirable effect on the plasma, to use

a thin insulator to divide the plasma and vacuum.

Microwave reflection by a non-uniform boundary has been treated

by many authors: Taylor has calculated reflections at linear density

ramps12 and also at an inverse parabolic density profile, 13 Wait has

given a formal solution for a stratified plasma, 14 Albini and Jahn have

made numerical calculations of reflection and transmission coefficients

for linear and "kinked" ramps, connecting uniform plasmas and for trapezoid geometries, 1 and recently an exponential electron distribution has

been attacked by Yen. 15

Our problem here is to seek the phase shift of the reflection

coefficient caused by the existence of a relatively thin boundary layer.

Jahn has already given an answer for a special case. He has derived

the following formula for the geometry shown in Fig. 7,

lS

20
~* , k r/ ^

2

-i 3^3 / k \ J kO

Z l r % k

/ * ? i R T = ?5 f°!i ^ S2s kg

s

, 2 nl+i-ti-d-ol)] -^-3 (i+ ff-) -^[l-Q+^d-o)]

2k| k Zl k2 2Q 2 k3

where

U0p 2

/uj2

_ /n , _. v,

(57)

"2 1+V

2

/U)2 <"

k3 = o: - jp.

In deriving Eq. (56), the Bessel functions of the forms J+-i/<a(0 have

been approximated by the first few terms of their series expansions.

Jahn has also shown that, for a special case of T) = 1 and 6=0,

R T turns out to be

VH)iV (58)

provided k z, << 1* We see from Eq. (58) that the boundary layer with

z

i linear ramp over z^ is equivalent to a step boundary located at z = ~

as far as the reflection is concerned.

Although Eq. (56) was obtained for T| < 1, it can be shown that

Eq. (56) is also valid for T| > 1. Accordingly the statement made under

Eq. (58) can be generalized for T| > 1. To show this, we replace Q 2 and

k2 in Eq. (56) by 7] and -jk v7]-l respectively, assuming 6=0, then

o

A + jB , , RJ=I^i. (59)

with

(1-T1)

2 A = T| + Tl a/71-1 .kQZl +

' *(koZl ) 2

, (60a)

21
B = T)VT]-1 + | (71-2) kQZi + |J • ^Tl-l '(koZl ) 2

. (60b)

The third term in Eqs. (60a) and (60b) can be neglected when k z 1 « 1.

Hence we have

R

-

1 + ^' k

o

z

i +

j jj^j

+ ( 2 -1 ) k

o

Z

i i (6l)

i +/rhi • koZl - j &/th + (| -l) koZl }

The Eq. (61) divided by Eq. (6) yields,

knZl k z1 Rj =

i + -^-Vn-i - j -fR

i + SST^rTT^p

(62)

which reduces to

when

R/R = e"

Jk°Zl

, (63)

k z^Tl-l « 1. (64)

Eq. (63) implies that a boundary layer with linear ramp over z x connecting homogeneous plasma of density T] is equivalent to a step boundary

located at z = z

1 /2 of a uniform plasma having the same density.

8. Concluding Remarks

One of the advantages of the reflection method lies in the fact

that the ratio of the sheath thickness, which is supposed to be of the

order of the Debye length X , to the penetration distance d does not

depend on the electron density, but only a function of the electron

temperature T : since ^ = vm

/ou and d = c/uo , we have xVd - vVc,

e D T p P v T

where c and v^ are the light velocity and the mean thermal velocity of

22
electrons. Thus, the thickness of the boundary layer can always be

neglected compared to the penetration distance provided v^/c « 1.

Although we have mainly dealt with plasmas in the absence of a

magnetic field, it is clear that the formulas can be used as they stand

for magnetized plasmas if geometry is chosen such that k _[_ B and E |j B,

where B is the magnetic field.

When the extraordinary mode of propagation is utilized, that is,

when k I B and E I B, the condition uo 2 « uo 2 has to be satisfied. ~ J ~ -1- ~ c p

The argument described in Chapter 6 suggests that a waveguide

probe without insulating film at its end could be used for a diffusion

experiment: the diffusion coefficient could be calculated from the

diffusion length at the probe end.

9 • Acknowledgment

The authors wish to express their deep appreciation to Dr. C. K.

McLane of the National Bureau of Standards (NBS), Washington, D. C.

,

for his hospitality extended to them. The discussion with him on the

microwave probe was also helpful.

23
10. References

1. S. Takeda and M. Roux, J. Phys. Soc. Japan 16, 95 (l96l).

2. A. I. Anisimov, N. I. Vinogradov, V. E. Golant, and B. I.

Konstantinov, Zh. Tekhn. Fiz. 30, 1010 (i960).

3. S. C. Lin et al., Phys. Fluids 5_, 1633 (1962).

4. T. Tsukishima and S. Takeda, J. Appl. Phys. 33, 3290 (1962).

5. S. Takeda and T. Tsukishima, J. Phys. Soc. Japan 18, 4-26 (1963).

6. C. K. McLane, S. Takeda, W. E. Thomas, Jr., and R. C. Thompson,

J. Appl. Phys. 35, No. 1 (Jan. 1965).

7. J. Galejs, Res. Rpt. No. 347R, Project No. 125 (1964) Jan., Applied

Res. Lab., Sylvania Electronic System; also Res. Rpt. No. 346,

(July 1963)

•

8. J. R. Wait, NBS Tech. Note No. 223 (1964).

9. S. Takeda and T. Tsukishima, J. Appl. Phys. 35, 2548 (1964).

10. F. A. Albini and R. G. Jahn, J. Appl. Phys. 32, 75 (l96l).

11. L. I. Schiff, Quantum Mechanics (McGraw-Hill, 1955, New York)

p. 189.

12. L. S. Taylor, J. Appl. Phys. 32, 1796 (l96l)

.

13. L. S. Taylor, IRE Trans, on Antennas and Propagation AP-9 , No. 6,

582 (1961).

14. J. R. Wait, J. Research, NBS, 65_B, No. 2, April-June 196l.

15. K. T. Yen, J. Appl. Phys. 35, 290 (1964).

16. R. G. Jahn, Tech. Rpt. No. 9 (1959), Contract No. AFl8(603)-2,

California Institute of Technology, p. 42.

24
io2

t

10'

\o°-

10

10

1.0

/#%=i.o

// j,=3 Jr/ ylti **•>/

r2

10 *~tan* 1.0 IO"' 10

Fig. 1. Relation between T| and tan cp for various values of 6
Q 40

\\\\\\\\^s\\\\\^^ a

cd U

£ o

co <+h cdft

=HO

-Po

H

•H

cd -P

CO

CD XiP

> CO hD O

£ Xi

o

Ch-P

cd X!

-P

coo

Xi

co

01 -P-PO

•H

cd •p

co

CO CD xi

CD P>

cp t5 O -H

cd

(D

p>•HH

ft

H

iH O

O P>

CO oM

T3

^i O

o

CD

cd

cr

1 o

CO r^

cv

•H

• -P

!>> O

M

cd «m M

CO ft cd H cd ft
(I)

VACUUM

Re*"2

.-jk z

(I)

DIELECTRIC

Bei k

2

Z

Ae-i k

i *t/A

(I)

PLASMA

Ce-iM

Fig. 3. Reflection by plasma through a dielectric plate whose

thickness is A

gA, where \ is the wavelength in the

dielectric.

MICROWAVE PROBE

1

PLASMA

Fig. 4-« Sketch of microwave probe.
N*

J 3 *t

J*JT3o

COo

•H

-po

co

cd

cd

t3

<ho

co<D H

cd

t>

<D XiEH

•H

P-4

<0

O*

CMO oo

**
(I)

>KTo

(8=0)

Fig. 6. Density profile of the inhomogeneous plasma having a linear ramp.

LINEAR RAMP

>

Te-Jk2

z

*-e>

z,

Fig. 7. Density profile of the boundary layer with a constant

density gradient.


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