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1^

CONCERNING THE THEORY OF RADIATION

FROM A SLOTTED CONDUCTING PLANE

IN A PLASMA ENVIRONMENT

lames R. Wait

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
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NATIONAL BUREAU OF STANDARDS

technical *^ote 223

Issued September 28, 1964

CONCERNING THE THEORY OF RADIATION FROM A SLOTTED

CONDOCTING PLANE IN A PLASMA ENVIRONMENT

James R. Wait

Central Radio Propagation Laboratory

Notional Bureau of Standards

Boulder, Colorado

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: 20 cents

ii

CONTENTS Page

Abstract , 1

1. Introduction 1

2. Formulation of the Homogeneous Plasma Problem 2

3. Transform Method of Solution , 3

4. Asymptotic Eocpressions for the Field. ...... ..o ...... . 7

5. Some Extensions of the Theory 10

6. Concluding Remarks , 13

7. Acknowledgement 13

8. References 13

111

CONCERNING THE THEORY OF RADIATION FROM A SLOTTED

CONDUCTING PLANE IN A PLASMA ENVIRONMENT

James R. Wait

A preliminary analysis is given for the radiation into

a plasma half- space from an infinitely long slot in a perfectly conducting ground plane. The plasma is anisotropic

by virtue of a d-c magnetic field which is parallel to the

slot. For a homogeneous plasma, it is found that for such

a configuration the radiation pattern will be symmetrical

but the excited surface wave is highly asymmetrical. The

extension of the theory to an inhomogeneous magneto plasma

is outlined briefly.

1. Introduction

The characteristics of an antenna immersed in an ionized medium

has been the subject of numerous recent papers such as those presented at a recent symposium on electromagnetic theory [Jordan, 1963],

In most of these, the surrounding plasma has been assumed to be

homogeneous. More realistically, the plasma properties w^ill vary

significantly in a direction normal to the conducting surface of the

antenna and to its supporting structure.

In this note, the formulation is given for a slot antenna radiating

into a stratified plasma medium. To simplify matters, the problem

is idealized by assuming that the slot is of infinite length and the

associated ground plane is perfectly conducting and is of infinite

extent. Furthermore, it is assumed that the plasma may be described

in terms of conventional magneto -ionic theory. In other words, thermal

effects are ignored but collisions between electrons and the immobile

heavy particles are accounted for by a constant collision frequency v.

k
2. Formulation of the Homogeneous Plasma Problem

A relatively simple preliminary problem is considered first.

An infinitely long slot in a perfectly conducting ground plane, at

z = 0, is assumed to be excited by a voltage V which is constant

for all values of y. The region, z > 0, above the slot is a cold

electron plasma characterized by an angular plasma frequency 03 and a constant collision frequency v. To achieve some kind of

correspondence with existing configurations used in plasma diagnostic

studies, the d-c magnetic field of strength H is applied uniformly

in the y direction. The resulting angular gyrofrequency is denoted

CO . The problem is to calculate the fields in the plasma which are

produced by the prescribed slot voltage V. A subsidiary problem is

the determination of the driving point admittance of the slot. In what

follows, the implied factor is exp (icot).

The details of the mathematical derivation will be suppressed

since closely related problems have been treated elsewhere [e. g-. , Wait, 1962], For the situation described, the magnetic field of the

waves has only a y component, H . Then, on using the appropriate

tensor representation for the dielectric constant of the plasma, it may be shown that H satisfies the Helmholtz equation of the form

(V2 - r^)H =0 , (1)

y

where V^ = r—5 + g is the two-dimensional Laplace operator,

O X

' o z and where F is the appropriate propagation constant. It was shown

previously [Wait, 1962], that for transverse propagation

r^ = -kVM . (2) o p

where k = 2 TT/(free space wavelength), and

-2-
co^

1 + M = '-^^^ , (3)

V 1 cog G^ Vg^ ojcy

with

g = V + i CO and a = 1 + —-^

The electric field components are found from a direct application

of Maxwell' s. equations. Thus,

a H an

ox Z + i K ^ p3z p9x i e CO E = - M ^ + i K —-^ , (4)

and

where

and

a H B H

i e ozcoE = +M ^ + iK ^ , (5) p?^x p?^z

-12

e = 8. 854 X 10 o

CO^ 60 o c K = 5—

p

p p

• '"/

p CO^ ^ CO^ CO ^ V icog cog cya Vg^' COG y

3. Transform Method of Solution

An important step in the solution is now to express the fields in

a Fourier transform representation. Thus,

oo H = rA(X) e'^'^'^^'^dX . (7)

y -L

-3-
where A(X) and u are functions of X. Since H satisfies (1), it

is not difficult to see that

u = (X^ + r^)2 ,

which is chosen to have a positive real part. The corresponding

integral represent-ations for the electric fields are found from (4)

and (5). In particular, the x coraponent is given by

+ 00

i e wE = \A(X)rM u + K X] e' ^ ^"^ e" ^ ^ dX . (8) o X J p p

- oo A formal solution of the problem is obtained by noting that

E = at the ground plane z = except over the slot. Thus, the

requirement is

E =0 for |x| > b/2 and z = , (9a)

while

E = e(x) for |x| < b/2 and z = , (9b)

where e(x) is the electric field in the aperture or slot. Integrating

e(x) on the width of the slot, we obtain + b/2

\ e(x) dx = V . (10)

- b/2

Now, it follows from (8), and the prescribed conditions at z = 0, that

b N „^b i,^^e(x)U(^A.,)u(|-+ x)

+ 00

= \ A(X)[M u + K X] e'^^'^dX , (Ua)

J p p

-4-
where U(x) = 1 for x > 0, =0 for x < 0, is the positive step

function. By the well-known property of the Fourier integral, (11a)

may be inverted to give

b/2

A(X)[M u + K X] = ^ \ e(x)e^'^''dx , (lib)

-b/2

which, in essence, is the formal solution of the problem.

To carry out the integration in (lib) requires that e(x) be

specified. The simplest assumption is a uniform field across the slot.

For example,

if e(x) = V/b , (12)

it is seen that

b/2

I

/ \ i^^j ,, sin(Xb/2) e(x)e dx = ^f—^^—J^ , (13)

-b/2

A second and somewhat more realistic assumption is

e(x) =

'^

-I , (14)

n[(b/2)5 . xS]2

which both satisfies (9) and has the singular behavior at |xj = b/2

which is appropriate for a conducting half-plane or knife edge.

Furthermore, the field distribution given by (14) is exact for the

electrostatic problem of a slot in a conducting plane [Smythe, 1950].

Thus, using (14), we find that

b/2

Je(x)e'^''dx = VJ^(Xb/2) , (15)

- b/2

-5-
where J (Z) is the Bessel function of zero order and argument Z.

o On arriving at (l5), we make use of the well-known identity

TT nMZ) = Je^^^°'^d9 . (16) o

Using the A(X)'s deduced from (11), it is now found that (7) may

be written

+ 0O Vi scOr* -uz-i Xx

y Zn J Mu + KX (17)

- oo p p

where

f(X) = ^i;y^f^

when e(x) = V/b , (18)

and

1. - 2

In general,

f(X) = J^(Xb/2) when e(x) = (V/n)[(b/2)2 - x^ ] .

(19)

b/2

^'

-b/2

f(X) = -^ ^e(x) e'^'^dx . (20)

From any of the above representations, it readily follows that f(X)

approaches unity when b tends to zero. Thus, in many applications

when dealing with excitation by a narrow slot, it is permissible to

replace f(X) by unity. However, for the sake of generality, the

function will be retained in the integral of (17). Corresponding integral

expressions for the electric field components are obtained by differentiation of H according to (4) and (5).

y

.6
4. Asymptotic Expressions for the Field

Of principal interest is the radiation pattern of the slot. This

may be obtained by applying the saddle -point method to evaluate (17).

The saddle point is located at X = ^- which is located where

c» [u z + i X x]/b X = 0. In the resulting deformation of the contour

to the steepest descent path, one must account for any poles which

are crossed. Under most conditions the integrand of (17) has a pole

in the fourth quadrant at X = X which is a solution of

g

M u + K X = . (21)

P P

Thus, in an asymptotic sense, (17) becomes

H s

y

Vi e CO ^ f (X ) u , 7 ^2 T-D o r a a r Z nN -PR

2n Lm u + K X VrRy ®

pa p a- f(X )6 . . _, - (2ni) i e-^g^ e-^^g"] , (22)

r-^(M u + K X)| LBX p P J

x=x 11. g

1

where u = (X^ +r^)^, u =(X^+r2)^, and R=(x2+z2)2.

a a g g

The saddle -point condition, mentioned above, leads readily to the

relations

^ = -i r sin and u = P cos , a a

where sin © = x/R and cos 6 = z/R. The factor, 6, occurring in

(22) is unity or zero depending on whether or not the pole at X is swe

over in the deformation of the contour to the steepest descent path.

-7-
1. In (22), the term proportional to (F R) ^ exp (- PR) is the

cylindrical space wave. By definition, the dependence of its

multiplier is the radiation pattern of the source. Because of the

- 3/2 asymptotic nature of (22), terms which contain factors like (PR) , -5/2

(PR) , . . . etc. , have been neglected. The second term, in the

square brackets of (22), has the characteristics of a surface wave.

Under some conditions the imaginary part of X is very small in

which case the attenuation in the x direction is very small. However,

in these cases, the real part of u is appreciable, so the attenuation

in the z direction (i. e. , normal to the ground plane) is quite large.

The situation discussed here bears a close resemblance to one

considered by Seshadri [1962]. He dealt with the fields of a magnetic

line source in a lossless anisotropic plasma with a perfectly conducting

boundary. To compare our results with his, we set v = and note

that

where

M = P

-rr- and K = - K p

^2

K

•^1 -

2 -1

60 ^ CO ^

f

•^2 =

CO^ CO "^

/' 60 C N

05^ VCO ^ CO y

c

(23)

(24)

2 2

and K = K - K . The corresponding integral representation for

H is given by

+ 00 Ve 60K /-, -ivz -iXx

H = —^

\ f(X) ~ .^^ , dX , (25)

y 2 n J K V + 1 K X

- 00

where v = (k^ - X^)^ with k^ = k^ k/K,. The pole now occurs

e e o 1 at X = X , where

g

-8-
1 l_2 X = k kVk^ = k k

g el o 1

As indicated by Seshadri [1962], this pole is either on the real or the

negative imaginary axis depending on whether K is positive or

negative, respectively. However, if the sign of K^ is changed by

replacing OJ by - CO , this pole is the negative real or the positive

imaginary axis, depending on whether K is positive or negative,

respectively.

The asymptotic evaluation of (2 5) shows that, for large positive

X,

V€ coKf(k sine) cose -i(k^R-n/4)

H - - ^ .1. . .^. ... . ^ r— (26)

y K^cose + iK^sin e ^2^^ j,^3 e

1 1 -2 + Vf(k K2) 6e K_ expf-ik K2 x - k K_ K ol o2' ^ ' ol o2'l

1. where 5 = 1 if tan 6 = (x/z ) > (K) Vk^ and = otherwise.

The latter inequality is the explicit condition that the surface wave

pole be swept over in the steepest descent deformation of the contour. It

is important to remember, as indicated above, that the excitation of the

surface wave for x > requires K_ to be positive. Also, in order

that the surface wave be non-attenuating in the x direction, it is

understood that Im X = 0. This requires, in addition to other

g

restrictions, that co/oj < 1 and

°

1 -

CO ^2^2 CO ^ ^ CO ^ ^ -[^-(-i)]<-^<[-(l^)] •

It is evident from (26) and the conditions imposed on it that if

the direction of the d-c magnetic field is reversed, K_ becomes negative and the surface wave is not present for x > 0. However, in this

-9-
situation, the surface wave is still excited but it propagates in the

negative x direction. Thus, the solution given by (26) also holds

for negative x if these conventions are used.

5. Some Extensions of the Theory

In the foregoing discussion, the plasma has been assumed homogeneous for the whole half- space z > 0. The theory may be extended

to a horizontally stratified plasma w^ithout too much difficulty. In

certain plasma configurations this may be a more realistic assumption.

An outline of the theoretical approach is given here for the case when

M and K , as defined by (3) and (6), are functions of z. P P

Maxw^ell's equations for the horizontally stratified plasma may

be written

i e oxWE = -M ^-^ + K XH , (27)

p 9 z p

ieojE =-iXMH + iK ^-^ , (28) o z p P a z

;^ E

-i a coU = -—^ + iXE , (29)

'^o y S z z

where X stands for the operator i'd/^x. Using (27) and (28) to

eliminate E and E from (29), it readily follows that

x z 5^H ^_o_ gX _1_ £ _BH _X £„ _

^ z^ Vm ^ J^ M Sz Sz " M Sz " ~ ^* P P P (30)

As in the previous section, H is written in the form of a Fourier

y

integral. Thus,

H = yB(X)F(X,z) e'^^'^dX , (31)

-10-
where B(X) is some undetermined function of X and F(X., z) is a

function of X and z which satisfies (30), and is well behaved at

z _ + oo. Using (27) and (30), it follows that + 00

i e wE = \ B(X)[-M (z)F'(X,z) + K (z) \ F (X, z)] e" ^ ^ "^ dX

o X J

• p p

- oo (32)

where the prime indicates a differentiation with respect to z. The formal solution for initial conditions (9a) and (9b) are obtained

in almost the same manner as for the homogeneous plasma. In terms

of the prescribed electric field, e(x), in the slot, it is readily shown

that

b/2

B(X)[-M (0)F'(X,0) + K (0)XF(X,0)] = -y^ \e(x)e'^''dx . p P 2n J

-b/2 (33)

It is then found that the resultant magnetic field anywhere in the halfspace is given by

+ 0O Vi e CO p „.. . - i X X

"y - 2n J' ^^' [-M (O)F' (X,0) + K (0)XF(X,0)] ' ^ ' -00 P P

where

b/2

-b/2

_l_ C , . iXx

~ V ) f(X) = -^ \ e(x) e'^^dx . (35)

in terms of the voltage V across the slot.

Equation (34) is a formal solution of the problem. In the present

form, it is not very useful since, among other things, it requires

that a solution of (30) be found before the integration in (34) may be

effected. However, some simplification is achieved when the

properties of the plasma are slowly varying in the z direction.

-11-
In this case, a WKB-type solution is applicable.

To obtain a WKB solution, it is desirable to introduce a new

function $(\, z), which is defined by

1 $ = M^ F , (36)

Now, since F satisfies (30) it is not difficult to show that $ satisfies

a wave equation of the form

^ ^^ + Q^] $-= , (37)

;^ z? where

Q

^ O ,2AP.i/'P"\ 1 1 M[ MBz 4M2V^z^ " 4M2V^zy 2M Qz^

P P P

(3 8)

The required WKB solution of (37) is given by

z $ = —T- exp -i \ Q dz , (39)

and

F(\,z) s —T ^T exp f-i \ Q(\,z) dz] , (40)

M2(z)QMX.z) '- ^

where the explicit dependence on \ and z is noted. This approximate

solution is only valid when Q(X, z) does not have any singularities

over the significant range of \ and z. In particular, a zero of Q

at some value of z will give rise to a turning point, and the solution

must be modified [Budden, 1961]. The matter will not be pursued

further here.

-12-
6. Concluding Remarks

Although the present note contains only a mathematical derivation,

some important physical principles are evident. In particular, it is

seen that the fields produced by the slot are asymmetrical about the

broadside direction (i. e. , 9 = 90 ). However, for the lossless homogeneous plasma, the radiation pattern is symmetrical insofar as

amplitude of the field is concerned. On the other hand, the surface

wave which may be excited in the lossless plasma is uni-directional.

Under more general conditions such as in the presence of collisions

and stratification in the plasma, the situation becomes much more

complicated and further work is obviously required. Recent progress

on related problems for isotropic plasma has been reported by

Galejs [1964], It is believed that the methods which he develops are

necessary for considering the impedance of the slot.

7. Acknowledgement

I am indebted to Dr. C. K. McLane for providing motivation

and to J. R. Johler for his constructive comments.

8. References

Budden, K. G. (1961), Radio waves in the ionosphere (Cambridge

Univ. Press, Cambridge).

Galejs, J. (Mar. 1934), Admittance of annular slot antennas radiating

into a plasma layer. Radio Sci. J. Res. NBS/USNC-URSI 68D , No. 3, 317-324.

Jordan, E.C., editor (1963), Electromagnetic theory and antennas

(Pergamon Press, Oxford).

Seshadri, S. R. (Nov. 1962), Excitation of surface waves on a perfectly

conducting screen covered with anisotropic plasnna, IRE Trans.

MTT-10, 573-578.

Smythe, W. R. (1950), Static and dynamic electricity, 2nd ed. , pg. 513

(McGraw-Hill Pub. Co., New York).

-13-
Wait, J. R. (1962), Electromagnetic waves in stratified media, pg. 232

(Pergamon Press, Oxford).

GPO 844-352 -14-