I -

. P ?) _ ~ ~ ~ - p - ~ - - ~ - _ ~ . _

THE INFLUENCE O F CONDUCTING FLAPS ON I THE REFLECTION COEFFICIENT OF A PARALLEL-

PLATE WAVEGUIDE ILLUMINATING A CONDUCTING SHEET

L . L . Tsai and R.C. Rudduck

!

I Hard copy (HC) 5k ' Microfiche (MF) # I

https://ntrs.nasa.gov/search.jsp?R=19680014009 2018-08-19T16:10:39+00:00Z

REPORT 2143-5

R E P O R T

The Ohio State University ElectroScience Laboratory (Former ly The Antenna Laboratory)

Columbus, Ohio 43212

by

Sponsor

Grant No.

Investigation of

Subject of Report

Subrni t t e d by

Date

National Aeronautics and Space Administration Office of Grants and Research Contracts Washington, D. C. 20546

NGR- 3 6 - 008- U4 8

Theoret ical and Experimental Studies of Antennas f o r Re f 1 e c tom e t e r Ap pl i c at i on

The Influence of Conducting Flaps on the Reflection Coefficient of a Para l le l -P la te Waveguide Illuminating a Conducting Sheet

L .L . Tsa i and R.C. Rudduck Ele c t r os c ienc e Lab o r a t o r y Department of Elec t r ica l Engineering

9 A p r i l 1968

i

ABSTRACT

The influence of conducting flaps on the reflection coefficient of

The backscatter f rom the guide a ground-plane-mounted TEol mode parallel-plate waveguide illumi- nating a reflecting sheet is analyzed. i s determined by use of wedge diffraction and surface integration tech - niques. The reflection coefficient of the guide is then obtained through an i terative procedure that describes the multiple interactions between the guide and reflector as bouncing cylindrical waves. flap length produces a significant reduction in the backscatter f rom the guide and consequently a significant reduction in the oscillations of the reflection coefficient.

The optimum

ii

! -

TABLE OF CONTENTS

Page

I. INTRODUCTION

A. Statement of the Problem

1

1 3 B. Background

11. BACKSCATTERING O F GUIDE WITH FLAPS 7

7 10 15 it3 24

A. Direct Diffraction Contributions B . Reflected Diffraction Contributions C . Reflected Geometrical ODtics Contributions

~

D. Results E. Equivalent Line Source Representations

30 111 REFLECTION COEFFICIENT ANALYSIS

A. F i r s t Bounce 31 33 35 37

B. Second Bounce C. Multiple Bounces D. Results

3 9 IV. CONCLUSIONS

ACKNOWLEDGEMENTS

REFERENCES

4 1

42

APPENDIX I 44

APPENDIX I1 54

iii

!.

THE INFLUENCE O F CONDUCTING FLAPS ON

PLATE WAVEGUIDE ILLUMINATING A CONDUCTING SHEET THE REFLECTION COEFFICIENT OF A PARALLEL-

I. INTRODUCTION

Recent advances in space exploration has given impetus to the need fo r bet ter understanding of spacecraft antenna problems. important phenomenon is communications blackout encountered during reent ry due to p lasma formation. In o rde r to overcome this problem, a knowledge of the character is t ics of the plasma formed around the space- c ra f t i s necessary . A widely used technique f o r plasma diagnostics is a flight measurement of antenna impedances o r r e f l e c t i ~ n coefficients . Improved designs a r e being sought f o r ref lectometer antennas useful f o r this purpose. havior of reflectometer antennas is to analyze the aperture reflection coefficient of a waveguide illuminating a conducting sheet which approxi- ma tes a cutoff plasma. and the ground-plane-mounted waveguide a r e s imi la r to those which occur between a cutoff plasma and the vehicle skin around the reflecto- m e t e r antenna. the performance of flush mounted reflectometer antennas.

An

One approach being taken to gain insight into the be-

The interactions between the conducting sheet

Consequently this type of analysis gives a measu re of

A. Statement of the Problem

The problem being t rea ted in this repor t is the analysis of the ref lect ion coefficient of a ground-plane-mounted TEel mode paral le l - plate waveguide with flaps attached a t the aperture and illuminating a conducting shee t . The geometry of the problem is as shown in F ig . 1 .

A s imi l a r problem which has been previously treated”’ is the TEM mode ground-plane-mounted guide (without flaps) illuminating a conducting sheet . f r o m the conducting sheet was given in t e r m s of successive contributions o r bounces that descr ibe the interacting waves between the waveguide and the re f lec tor . into component cylindrical waves. obtained by summing these i terative contributions. F o r the TEM guide without flaps, the reflection coefficients when plotted as a function of d, the ref lector spacing, exhibited strong oscil latory behavior. f o r values of d equal to an integral multiple of k/2, complete reflection, o r a unity reflection coefficient, w a s observed

By using the wedge diffraction method the reflection

Each of these bounce waves was subsequently resolved The reflection coefficient was then

In fact,

1

GROUND PLANE

RECEIVED TRANSMITTED c - O Q POWER J. POWER

CONDUCT I NG ' SHEET \:

/ /

/ / /

/ / / /

G /

\ \ \ \ \

,- -4-d TEo, MODE Y"\-\: f - . PARALLEL - PLATE

WAVEGUIDE \ /

/

/ /

\ \ \

ONDUCTIN FLAPS

Fig. 1 . TEol mode parallel-plate waveguide with flaps illuminating a conducting sheet .

In order to accurately determine the distance d of a perfectly reflecting surface from an antenna in a ground plane, however, it is desirable to have the reflection magnitude from the antenna be a mono- tone function of d. The ground plane mounted TEM guide studied in Ref. 2 is therefore unsuitable for reflectometer applications without fur ther modifications.

The problem considered in this. report is motivated by the work of R. Lentz' where he experimentally observed that the reflected power received by an absorber-backed pyramidal horn illuminating a reflecting sheet may be made a monotone function of the distance to the reflector by the attachment of conducting flaps t o the E-plane edges of the horn. However, the important question remaining unanswered is: can a ground plane mounted aperture antenna achieve this monotone reception, since the ground plane is the pr imary sca t t e re r? multiple interactions can occur between the ground plane and ref lector so as to yield oscillatory resul ts . The purpose of this investigation is then to examine the effect of conducting flaps in reducing these multiple interactions.

F o r this case significant

I

The method of solution for the f lap guide problem is outlined below. First, the backscatter of the guide s t ruc ture with flaps is determined

2

for plane wave incidence by applications of wedge diffraction and surface integration techniques. at a constant distance away from the guide aper ture fo r different flap lengths ( f ) . The optimum flap length is that which yields the deepest null in the sca t te red field since this means the lowest re turn from the guide s t ruc ture and hence lowest multiple interactions between the guide and ref lector . Secondly, the scat tered field is resolved by superposition to be the sum of two equivalent line source fields plus the reflected geo- me t r i ca l optics field. Finally, the reflection coefficient is analyzed by a multiple bounce procedure s imi la r to that employed in Ref. 2 .

The scat tered field is then observed on a plane

B. Background

Following the method in Refs. 4 and 5, the incident field in a TEol mode para l le l plate waveguide may be expressed in t e r m s of plane - - -^-_^- ~a~~~ (as sbzv.m In F i g - 2 ) where the angle of propagation i s given by

The incident modal power flow is

(2) Po = Za Yo cos A,

where

Y 0 = W o ,

and the associated modal voltage is

(3) cos A,

where yg i s the guide admittance f o r the TEol mode. F o r this polar i - zation the relationship between the electr ic field E, r ay R, diffraction coefficient D, and modal voltage V is given by

- j k r e - jk r t srI4 e - - j (kr t r / 4 )

- D - = V E = R ~ dmz 6 G (4)

3

'3,

I

V I . . . . . . . . .

./- I

F i g . 2. TEol mode in a parallel-plate waveguide.

4

The diffraction at the aperture of the parallel-plate waveguide m a y be t rea ted as in Ref. 4 by summing the singxe and double diffraction contributions. The guide s t ructure .in this analysis may be approximated by a symmetr ic half-plane guide when considering radiation in the on- axis region. The singly diffracted r ays f rom edges 1 and 2 a r e thus given respectively by

and

( 6 )

where nl = n2 = 2 . 0 .

The r ay f r o m edge 1 (o r 2 ) which illuminates edge 2 ( o r 1 ) is given by

The doubly diffracted rays f rom the two edges a r e given by

5

and

where V g ( r , +, n) is the wedge diffraction function employed in previous analyses. Calculations have shown6 that the field radiated by the guide near the on-axis region may be approximated by a line source field given by

where

The equivalent line source with modal voltage V1 which approximates the guide radiation i s then given by

A primary resul t used in this analysis i s the response of the TEol mode parallel-plate waveguide to an incident cylindrical wave. This i s given by the line source to waveguide coupling relationship presented in Ref. 5 as

where V is the modal voltage induced in the guide, Ei is the incident field at the guide aperture center , and DT(8) is the diffraction coeffici- ent of the guide given through Eq. ( 9 ) . a form useful for computing the reflection coefficient r , we find

Rewriting this relationship in

6

I .

and the constant C i s given in t e r m s of the on-axis r a y RT(B = 0) as

where Vi is the modal voltage of the line source generating the incident field Ei given by

I

Ei = vi e - jkr f j r/4

rn ( 1 3 ) I

11. BACKSCATTERING O F GUIDE -.*.. m-1

W l l H

In this section the backscattering propert ies of the guide s t ruc ture shown in Fig. 1 are investigated as a function of flap length. a unit amplitude plane wave incident on the guide as shown in F i g . 3, the sca t te red field m a y be analyzed by a combination of wedge diffraction and surface integration techniques. The sca t te red field is then observed on a plane normal to the guide axis and located at a distance d ' away f rom the guide aperture . in the total sca t te red field may then be considered optimal for minimi- zing multiple interactions between the guide and the ref lector .

Assuming

The flap length which gives the deepest null

The sca t te red field for this problem m a y be thought of as being composed of the following components: tions or the diffracted rays f rom the flap edges which a r e not influenced by the presence of the ground plane; (2) reflected diffraction contributions 01- the diffracted rays f r o m the flaps which reflect off the ground plane and contribute to the total scattered field, and ( 3 ) reflected geometr ical optics contributions f r o m the ground plane. Detailed analyses of these components a r e given in the following sections.

(1) d i rec t diffraction contribu-

A. Direct Diffraction Contributions

The analysis of the direct diffraction contributions from the flap These edges is analogous to that employed in previous problems .IJ7>

contributions m a y be obtained by simply summing the single and double diffraction contributions f rom the flap edges as shown in Fig. 4.

7

\ \ \ \

Fig. 3 . Unity amplitude plane wave normally incident on the guide.

I I\ OBSERVATION Y I PLANE I

I

I I I

OB5 E R V AT ION PLANE

Fig. 4 . Direct diffraction contributions to the scat tered field f rom the flap edges.

8

The singly diffracted field at the observation point P(x, y) f rom edges 1 and 2 are given, respectively, by

rl =

and

a1 = tan'l (y/d')

and

where

r 2 = dd" t (y t a)2

and

The single diffraction rays which originate at one edge and illumi- nate the opposite edge a r e given by

(17) R1(-jl) = R ~ G 1 1 1

n IT 90" IT 270' cos - - cos - n n n n

1 n

(1) = - sin -

Hence the doubly diffracted field f r o m edges 1 and 2 a r e given, respectively, by

9

and

i

The total d i r e c t diffraction contribution to the sca t te red field may then be expressed as

B . Reflected Diffraction Contributions

The diffracted rays f rom the edges of the flaps which reflect off the ground plane and thus contribute to the total sca t te red field may be analyzed to a f i r s t o rde r approximation by the same ray tracing techni- que employed in the previous section. If this method were applied the geometry and diffraction components involved would be as shown in F ig . 5 where ERD represents the diffracted wave which emanates f rom edge 1, is reflected by the ground plane, and contributes to the scat tered field at P(x, y). A s imi la r process , of course, resu l t s from edge 2 f o r y < -a . Preliminary calculations have shown that this technique would be adequate i f one were interested only in finding the scat tered field f o r Iyl > > 0 . However, for the regions of in te res t in the problem of this report further considerations must be given.

The reflected diffraction wave, ERD, can be seen in Fig. 5 to actually reilluminate edge 1 as j3 - 0. The subsequent diffracted wave

10

OBSERVATION I PLANE \I

F i g . 5. Ray t racing technique fo r finding the reflected diffraction contributions f rom the flaps.

which resu l t s cannot be adequately descr ibed by conventional wedge diffraction techniques, which inherently assumes cylindrical wave incidence, because the magnitude of ERD rapidly approaches a sha rp peak with a s teep slope near the surface of the flap. Consequently, a sur face integration technique is applied to include the effects of this nonuniform wave.

Basically, the surface integration technique t r ea t s multiple inter- actions which occur in diffraction problems by integrating the diffracted fields of an interaction wave of a specific o rde r over a sur face , to obtain what corresponds to the subsequent o rde r wedge-diffracted wave. This method has been successfully used in Refs. 9 and 10 and have been shown to provide higher accuracies than the wedge diffraction method while p re - cluding the limitations of nonuniform wave interactions

Formulating the analysis of the reflected diffraction components by the sur face integration technique, the pertinent parameters are as shown in F i g . 6 . aper ture plane. s u r f a c e of integration will be determined.

The surface of integration is chosen to Pe at the guide First, the reflected diffraction fields, ERD, on the

Then the reflected diffraction

11

h Y 9 Y '

/ I / / / / / / / / / / /

/ / / / / / / / / / I '

4 a -

. . -. P - P -

I ( 1 ) RIG J., .

\ \ \ \ \ \ \ \ \ \ \ 01 leJ0 I

\ I I \ OBSERVATION I I

\ I \+ \ f - d ' d

PLANE

SURFACE OF INTEGRATION S I

1

Fig . 6 . Applications of surface integration techniques to analyze the reflected diffraction contributions from the flaps.

12

!.

where

i

contributions, ERD, at the observation plane w i l l be determined using Green 's Second Identity f o r planar surfaces" given by

Go(kr) = - Hi2) (k r ) 4

with

A A A ' = p - p ' .

Noting that r = Gx-x'f t (y-yI)L" Eq. (21) may be simplified by the following relationship

- 1 ( 2 ) kx - Hi (k r ) - 4J r -

I

where H1 ( 2 ) (k r ) is the Hankel function of the second kind of first o r d e r .

The reflected diffraction field on the surface of integration S is given by summing the single and double diffraction contributions. only edge 1 will contribute and f o r yl < -a only edge 2 will contribute while for 0 > yl > -a the reflected diffraction field is zero on S . be recognized that due to the symmetry about the guide axis, identical f ie ld distributions exist fo r y ' > U and for y ' < -a, thus only the upper half of S needs to be t reated. The singly diffracted field f rom edge 1 is thus given by

F o r yl > 0

It may

13

where the minus sign results f rom the reflection from the ground plane and the geometrical quantities a r e

p = tan-'($) ,

and

The doubly diffracted field from- edge 1 i s similarly given by -

8 2 7 0 ° - p , 2.0) - VB(& , 4 5 O 0 - P , 2 . 0 I I 1

p ' t a

The total reflected diffraction field on S f o r y ' > 0 is thus given by

The symmetry property of the field on S will now be invoked to simplify Eq. ( 2 1 ) . Let

where

EI(x, y) = field at P(x, y) by integration over the upper half of S

and

14

I

I

I

EII(x, y) = field at P(x, y) by integration over the lower half of s

Then, by symmetry ,

Rewriting Eq. (21), the total reflected diffract sca t te red field at P(Y, y ) is thus given by

on contribution to the

with the integrand given through Eqs. ( 2 2 ) and (25).

It may be noted in passing that the d i rec t diffraction contributions obtained by ray techniques in Section A may also be analyzed by surface integration techniques. However, when interaction waves a r e uniform to a good approximation, this more tedious technique is not necessary and, in fact , can be shown to yield values directly corresponding to the r ay technique .9

C . Reflected Geometrical Optics Contributions

Analogous to the reflected diffraction contribution in the previous section, the reflected geometrical optics component may be seen to a l so reil luminate the f lap edges. Conventional wedge diffraction theory again cannot be applied directly since this Component is actually discontinuous along the sur face of the flap. with superposit ion is therefore applied to analyze this component.

Surface integration techniques together

15

It may be noted here that both the reflected geometrical optics field, ERG, and the reflected diffracted field, ERD, a r e discontinuous at the surface of the flaps. ditions for this polarization their sum, i . e . , ERD -t ERG, mus t be continuous and approach zero along the flap surface.

But in o rde r to satisfy the boundary con-

The geometry involved f o r ERG is the same as that in Fig. 6 with the surface of integration S at the aperture plane. geometrical optics field on S is given by

The reflected

- J k ( 2 f ) f o r yl > 0 o r yl < - a ,

for 0 , y') - a ( 2 9 ) E k ~ ( y ' ) = [l

I I I I

I I I I

I IEbl

I

where the minus si n a r i s e s f rom the reflection by the ground plane. By superposition ERG may be resolved as shown in Fig. 7 into two components: Eb, o r the reflected plane wave from a ground plane without an aperture , and Eks, the negative of the reflection from a thick wall or s t r i p (reflected geometrical optics s t r i p ) . these components on S a r e given by

F

The values of

THICK WALL

- - -

EASl FIELD

E'c FIELD PLOT PLOT

GUIDE /1 4

9

\ 3 I

GROUNDA PLANE 2

1

1 - -

1 E'RG A

9 FIELD PLOT

I

I

F ig . 7 . Applications of superposition to the reflected geometrical optics components from the guide.

16

I

otherwise I"

Applying Eq. (21) to these components and integrating over S, it is seen that the plane wave component EG, will still remain a plane wave at the observation plane located a distance d ' away. The contri- butions of the reflected geometrical optics s t r ip , Eks, is seen to be given by

( 3 3 )

kd - jk( 2 f ) 0

2 j e

- - 1 H1(')(kr) dy' . -a

Thus the total contribution from the reflected geometrical optics com- ponent at P(x, y) is given by

where

(35) -jk[2f tx]

E ~ ( x , y ) = - e

17

D. Results

From Eqs. (20), (28), and (34), the total scattered field on the observation plane may be determined by

ETS = ED * ERD + ERG

where

ETS = total scattered field,

ED = direct diffraction contribution from the f laps ,

ERD = reflected diffraction contribution from the f laps ,

ERG = reflected geometrical optics component from the guide,

EG = reflected plane wave from a ground plane without an aperture , and

ERS = reflected geometrical optics strip or the negative of the reflection from a conducting wall.

A computer program in For t r an IV presented in Appendix I has been written to aid in the calculation of the scat tered field. thus obtained a r e shown in F igs . 8 through 12 as a function of flap length for various values of f and d ' . scat tered field on an observation plane located 3.0X from the guide aperture for various flap lengths shor te r than U . 6 X . It can be seen that f = U.3205h yields the deepest dip in 1 ETSI . Figure 9 gives the s a m e data for f l e s s than one wavelength while Fig. 10 supplies the cases for which f > 1 . O X . It can be seen from these resul ts that optimum flap lengths, which give the deepest null, occur approximately once every h / 2 , an observation in line with physical intuition. f o r relatively short flaps a r e thus concluded to be f = 0.3205X, 0.8205X and 1.3205X.

The resul ts

Figure 8 presents the magnitude of the

These optimum lengths

Figure 11 presents l E ~ s 1 f o r f = 0.3205A observed at dl = 2 . o X , From this result the dip obtained by the attachment 5 . O X , and 10. OX .

of the f laps onto the guide aperture can thus be concluded to diminish as the observation distance d ' is increased. At d ' = u,, of course, the

18

19

2 0

I

21

k 0 w

4 Q) c a cdc

“ ( d a

w w 0 -

0 4

1-1 - i U

E: 0

d - w

II 0

M .+ ctr

2 2

23

presence of the aper ture should have rio effect on the sca t te red field which then becomes the reflected plane wave from a ground plane.

The phase of the sca t te red field is exemplified by the cases for which f = 0.32051, and 0.571, and d ' = 3 .0 as presented in F ig . 12

E. Eauivalent Line Source Representations

The sca t te red field f rom the guide due to an incident plane wave may be seen both theoretically and numerically to be resolvable into cylindrical component waves . sections can be t reated as follows:

The components obtained in the previous

(1) EG, the reflected plane wave from a ground plane without an aperture , can be identified as the geometr ical optics reflection of the incident field by the ground plane

( 2 ) ED, the d i rec t diffraction contribution from the flaps, actually seems to eminate f rom edges 1 and 2 and hence can be r e p r e - sented by an equivalent line source, VD, located at the guide aper ture center .

(3) ERD t ERS, the sum of the reflected diffraction contribution from the flaps and the reflected georr.etrica1 optics s t r i p component, as seen by image theory, can be represented by an equivalent line source , VR located at a distance Z f behind the guide aper ture . The modal voltages of these equivalent sources a r e then obtained f rom Eq. (4 ) as

(37 1

and

VD = eJkd' ED(d ' , - a /2 ) ,

Numerical verification of these equivalent line source representa- tions for the sca t te red field i s given in Tables I and I1 f o r the cases in which a = 0.76iX, f = 0.32051, and d ' = 3 . 0 1 and 10.01.

24

I

M .+ cr

2 5

k Q,

E: 0,

u

u 0 CD

-0 w Q, -0

ul Q) 9 -r

26

t

I

Q) M '0 w

co 9 N 4

0

0 0 a 0

0

Ln 0 ca m 0

m P 0 Ln

0

N * cr) Q

0

0

9 I'

0

4

27

k 0)

c e,

Y

u e, M ‘D w

2z,A - 0

I Q) M -a w Q) a .+

3

It m a y be noted that the equivalent line source modal voltages, VR and VD, depend only on the amplitude of the incident field (unity amplitude plane wave assumed incident for E q s . (37) and (38)) and is independent of the observation distance d ' . in the reflection coefficient analysis discussed in the next section, where the same property is assumed for cylindrical wave incidence. The scat tered field components associated with the presence of the aperture, i.e., those representable by the equivalent line sources VR and VD, may be thus expressed in general as

This property will be useful

and

where Ei may be an incident plane wave of a rb i t r a ry amplitude and ro is the observation distance. contribution and ER is the equivalent reflected contribution from the aperture . cients of a par t icular guide s t ruc ture with their values obtainable through Eqs. (39) and (40) by making the substitutions Ei = 1 eJo, ro = d ' , and the values of ED and ER as exemplied in Tables I o r 11. The cases to be considered in the next section are: (1) f = 0.3205X, a = 0.7611; KD = 0.254/108.2" ; KR = 1 . 1 6 5 / 2 0 . 4 " ; and (2) f = 0.57X9 a = 0.7611; K D = U.254 / & & ; KR = 1.337 / 17.7".

ED is the equivalent d i rec t diffraction

KR and KD may be regarded as constant scat ter ing coeffi-

111. REFLECTION COEFFICIENT ANALYSIS

. In this section the reflection coefficient of the ground-plane-mounted TEoi mode paral le l plate waveguide with conducting flaps attached to the aperture and illuminating a conducting shee t as shown in Fig. 1 is analyzed in a manner similar to that employed in Ref . 2 . method the reflection coefficient of the waveguide is the superposit ion of the f ree space reflection coefficient, rs, and the reflection coefficient caused by the presence of the conducting sheet , rp . reflection coefficient fo r this guide s t ruc tu re may be approximated by the solution for the thin walled case given in Ref. 5. general is quite small and in prac t ice can be matched out;* therefore ,

By the wedge diffraction

The f r e e space

However, rs in

30

I

it w i l l not be included in the subsequent discussions. The reflection due to the sheet i s analyzed in t e r m s of successive contributions o r bounces which descr ibe the interaction of the fields between the guide and the sheet .

Formulating the reflection from the sheet in t e r m s of successive bounces, the first bounce wave is the f r ee space radiation f r o m the waveguide, Eq. ( 9 ) , which reflects f rom the sheet back onto the waveguide. The first bounce wave then sca t te rs f rom the guide producing a second bounce wave which propagates toward the reflecting sheet . bounce wave in turn reflects f rom the sheet back onto the waveguide giving r i s e to a third bounce wave, and s o on to higher o r d e r bounces. Each bounce produces a contribution to the reflected TEot mode in the wave guide .

The second

A. First Bounce

As stated by Eq. ( l o ) , calculations show that in the region of the projected guide c r o s s section the f ree space wave radiated from the guide may be represented by an isotropic cylindrical wave f rom a line source located at the center of the guide aperture . This and subsequent

F i g . 14. The reflection coefficient of a TEol mode ground plane mounted guide with aperture flaps i l lumi- nating a conducting sheet.

31

I

'1

SECOND ,- r2 BOUNCE

TH IRD BOUNCE --r3

I I I I v3 v2 l o o I I I I

V I 0

( 2 d ) ( 2 d + 2 f ) 2 d + ( 2 d + 2 f )

I I I

I I (2:) (2.d+2f) f' I 2d + ( 2d + 2f ) 2( 2d+2f 1 I I

VI 0

vs v4 v 3 v2

'\ 2d+2(2d+2f)

- r4 FOURTH BOUNCE

I '7 "6 vs v4 v3 z 2 V I

{ 3(2d+2f 1). 0 . ' (2d) ( 2 d + 2 f ) f

I I 2d +(2d +2f 1 j 2d+2(2d+2f 1

I

F i g . 15. Equivalent line source locations for the n;ultiple bounce d iagram.

3 2

i

approximations in this report a r e valid provided the observation distances a r e sufiiciently removed f rom the aper ture . would be to keep the conducting sheet distance d always grea te r than a, the guide width, for a < X .

In general, a rule of thumb

By image theory the equivalent line source representing the first bounce wave m a y thus be seen from Fig . 15a to be located a t a distance 2d f r o m the guide aperture and with modal voltage Vi given by

where RT(e= 0 ) is the on axis ray from Eq. ( 9 ) . f r o m the reflection by the conducting shee t . froiii the first b o u ~ c e ecplvalent line source is then given by

The minus sign a r i s e s The field incident on the guide

Using the line source to waveguide coupling expression given in Eq. (12), the first bounce reflection coefficient is given by

I

-jk(2d) e

m = c v1

B. Second Bounce

The first bounce equivalent line source field sca t t e r s f rom the guide producing a second bounce wave. It was seen in Section I1 that the sca t te red field f rom the guide fo r plane wave incidence may be resolved into the geornetrical optics component and two cylindrical component waves associated with the presence of the aperture . ponents of the scat tered field resulting from an incident cylindrical wave, however, depends only on the value of the incident field and is independent of the source location provided the source is sufficiently

The aper ture com-

33

removed f rom the guide. the aperture component of the scat tered wave is, therefore , the s a m e as that for plane wave incidence with the plane wave field equal to the incident field of the cylindrical wave at the waveguide aper ture .

F o r the case of cylindrical wave incidence,

The geometrical optics component of the second bounce wave reflects f r o m the sheet back onto the waveguide such that it may be represented by the line source VI located at a distance [ 2d t (2d t 2 f ) ] f rom the guide aperture , as shown in F ig . 15b. The aper ture components of the second bounce wave reflect onto the waveguide as descr ibed by the line sources Vt and V3 in F ig . 15b The values of V2 and V3 a r e ob- tained by equating the radiated fields with those of the aper ture components in Eqs . ( 3 9 ) and (40) . Thus

- j k r o t j ~ r / 4 - jkr, e e = - Ei K D J

G-

where the minus s gn again resul ts f rom the reflection off the conducting sheet and E’ is the incident field of the illuminating line source Vi at the guide aper ture given by Eq. (42) . a r e given, respectively, by

Hence the value of Vt and V3

-jk(2d) e ( 4 5 4 V2 = - VI KR 9

and

-jk(Zd)

The corresponding second bounce reflection coefficient is then given by the modal voltage induced by Vi, V2, and V3 as shown in Fig . 15b

34

I

I

- jk( 2d) e - jk( 2dt2f) e

4- vz m v3 6 d

C . Multiple Bounces

The generation of the third bounce is similar to the generation of the second bounce with the line source locations as shown in Fig. 15c and modal voltages given by

- jk( 2d) e -jk(2dt2f)

(47) 2dt ( 2 dt2f j + v 2 d m > + vs ai- e 2dt(2dt2f) l

-jk[ 2dt(2d+2f)] - jk( 2d+2f) -jk(2d) e e m + v 3 ai- f v2

The third bounce reflection coefficient contribution i s given by

-jk[2(2dt2f)] e -jk[ 2dt2(2d+2f)]

2dt2(2d+Zf) dqzzz) f v2 (49)

-jk[ 2dt2fl e - jk[ 2dt(2df2f)J e c d t ( 2 d+2f) + v 4 IIZd+Zf f v3

1 - jk( 2d) e

m- i- v5

Generalizing, the n-th bounce wave is given by (2n-1) cylindrical wave components with sources Vi at [2dt(n- l ) (2dt2f)] a Vz at [(n-1) (2dt2f)] , v3 at [2dt(n-2)(2d+2f)], V4 at ((n-2)(2d+2f)], - - - - - V( 2n-2) at [ZdSZf], and V(tn-1) a t [2d] . The sources associated with this bounce are given by

3 5

and

n- 2 - - jk[(n-m- 1)(2d+2f)]

(n-m- 1)(2d+2f) m= 1

The n-th bounce contribution to the reflection coefficient is thus given by

n e -jk[2dt(n-m)(2dtZf)]

2dt2f)

1 n -,1 e -jk[ (n-m)(2dt2f)] ' V'2m) d(n-m)(2dt2f ) m= 1

The total reflection coefficient due to the reflecting sheet o r plate is given by

36

I .

D. Results

I

I

The total reflection coefficient, rp due to the conducting sheet, as given in Eq. (53) , was calculated as a function of the reflector spacing d fo r two different flap lengths with the aid of the For t r an IV computer program presented in Appendix IT. The f lap lengths chosen were 0.32051, an optimum length determined from the scat ter ing program to give the deepest null, and 0.57h, a non-optimal value. F o r both cases , a guide width of 0.7611 was used. The r e s u l t s thus obtained a r e as presented in F igs . 16 through 18 with the inclusions of u p to 300 bounces.

F igure 16 compares the magnitude of rp f o r the two flap lengths along with the reflection coefficient from the first bounce wave I'l . F o r both cases , the behavior of 1 rPl is quite like that observed in Ref. 2 f o r a TEM mode ground plane mounted guide (without flaps). seen t e escillate about rl with a period of X/2 in d. At values of d fo r which f t d = multiples of X / 2 , the reflection coefficient is seen to rise to a s h a r p peak. This is exactly analogous to the resonance behavior observed in Ref. 2 for cavity spacings equal t o nX / 2 , where unity magni- tude reflection coefficients were observed. F r o m the basic nature of the problern of a TEol ground-plane-mounted guide without flaps illu- minating a conducting sheet, it is expected that the reflection coefficient fo r this problem will have the same fundamental behavior near cr i t ical values of ref lector spacing d = n1: 2 as that fo r the TEM mode. is expected that complete reflection will occur a t d = nX/2 for the TEol mode problem without flaps. complete reflection as can be seen from the resul ts in Fig. 16. fact , the optimum flap length of 0.32051 can be seen to yield smaller oscil lations in 1 r I than that f rom f = U.571 , with a significant reduction P in the peak values. i nc reases , an observation in line with that observed in Ref 2 .

II',I is

Thus i t

The piesence of the flaps eliminates In

F o r both cases , the peaks remain constant as d

The phase of rp f o r the two flap lengths is shown in Fig. 17 with the phase of I ' l . rp f o r the f = 0.32051 case with large values of d . value is observed in the sha rp peak of lrPl at d = 1U.17951 as was ob- s e r v e d at sma l l e r values of d . not diminish i r r ega rd le s s of the size of d due to the idealized geometry assumed, namely, infinite ground planes and conducting sheets.

Figures 18a and 18b give the magnitude and phase of The same maximum

It is believed that the peak values will

F r o m the data thus obtained, it can be seen that the presence of the flaps causes a considerable reduction in the amplitude of the oscil- la t ions in I rpl . F r o m the unity reflection case observed fo r the guide without f laps , the new structure i s seen to offer lower peaks, e s ecially at optimum flap lengths. Even though the monotone curve of IrPT vs d,

37

I

tn W

! I m W u z 3 0 rn

a > 2

I x b

0 L?

11

Y-

I I I I

-4

cn %

5 4 ru

.+ 3 cn Q) -a 3 M k 0

Q) -a 2

.+

W

c, .+ 5l

E d

c, d a, V r= w a, 0 V

r: 0

V Q) r( W a k

Q)

.+

.+ 4 4

4z t-c

9 d

M .+ c.l

I

38

c

v) W W a

I80 f = 0.3205 X

f 8 0.57 X

W 0 Y MANY BOUNCES

MANY BOUNCES ----

F I R S T BOUNCE ( F O R BOTH 1

LL

w 0 0 - 6 0 z

+ \ g-140-

a I .4 I .6 I .0

-I h-It30 I I I I I I

2.0 2.2 2.4 2.6 2.8 3.0

REFLECTING SHEET DISTANCE d (WAVELENGTHS 1

Fig . 17. Phase of Fp.

I

suitable for reflectometer antennas, was not obtained, this analysis nevertheless quantitatively predicts the bes t resul t that can be obtained with simple flaps.

IV. CONCLUSIONS

The influence of conducting flaps on the reflection coefficient of a ground-plane mounted TEcr mode parallel-plate waveguide illuminating a conducting sheet has been analyzed. The backscat ter f rom the guide s t r u c t u r e was obtained by applications of wedge diffraction and surface integration techniques. The reflection coefficient was then obtained through

39

a = 0.761 X f = 0.3205X MANY BOUNCES FIRST BOUNCE ---

0 10.0 (a 1 10.5

1 220

REFLECTING SHEET DISTANCE d (WAVELENGTHS)

Fig . 18. (a) (b )

IrPl fo r l a r g e r d Phase of rp for l a r g e r d .

an iterative multiple bounce procedure that descr ibes the intera.ctions between the guide and the ref lector .

This analysis was motivated by the need to improve the design of reflectometer antennas which a r e used in p lasma diagnostic measurements . Previous analyses and measurements have shown that a reflecting su r face in front of the antenna will usually produce l a rge interactions between the surface and the antenna; this resul ts in la rge oscil lations in the reflection coefficient as a function of the spacing between the two s t ruc tu res .

The resu l t s of this analysis show that the presence of flaps at edges of the antenna aperture can significantly reduce the oscillations in the

40

reflection coefficient. Optimum flap lengths can a l so be determined as those lengths which produce the greatest reduction in the on-axis back- sca t t e r f rom the ground-plane-mounted guide.

Though only thin planar flaps a r e used in this analysis, extensions may be easi ly made to other f lap geometries which may yield lower backscat ter and hence a m o r e monotone response.

One possible application of the resul ts f rom this analysis would be in reducing r ada r echo a r e a s of s lot a r r a y s .

ACKNOWLEDGEMENTS

The authors wish to thank Prof . J .H. Richmond for providing the Hankei I'unciioil subrzliitine a d L$r. E - Le Pelton for siupplying the integration subroutine.

41

REFERENCES

1 .

2.

3 .

4.

5.

6.

7.

8 .

Tsai, L . L . , "The Reflection Coefficient of a TEM Mode Para l le l - Plate Waveguide Illuminating a Perfectly Reflecting Sheet, ' ' Report 2143-1, 25 August 1966, Antenna Laboratory, The Ohio State University Research Foundation; prepared under Grant NGR-36- 008-048, National Aeronautics and Space Administration, Office of Grants and Research Contracts, Washington, D. C.

Tsa i , L. L . , and Rudduck, R . C. , "The Reflection Coefficient of a Ground-Plane Mounted TEM Mode Para l le l -P la te Waveguide Illuminating a Conducting Sheet, ' I (ElectroScience Laboratory repor t in p rocess ) .

Semiannual Status Report , Report 2 143 - 3 , 9 June 1967, Electro- Science Laboratory, The Ohio State University Research Founda- tion; prepared under Grant NGR-36-008-048, National Aeronautics and Space Administration, Washington, D. C .

Dybdal, R . B . , Rudduck, R. C. , and Tsai , L. L., "Mutual Coupling Between TEM and TEol Para l le l -P la te Waveguide Aperture , IEEE Transactions on Antennas and Propagation, Vol. AP- 14, No. 5, (September 1966), pp. 574-580.

I I

Rudduck, R. C., and Tsai , L. L., "Aperture Reflection Coeffici- ent of TEM and TEol Mode Para l le l -P la te Waveguides, I ' IEEE Transactions on Antennas and Propagation, Vol. AP-16, No. 1, (January 1968) pp. 83-89.

W u , D.C.F . , Tsai , L .L . , and Rudduck, R . C . , "Broadside Radiation of Parallel- Plate Waveguides, ' I (ElectroScience Laboratory repor t in p rocess ) .

Rudduck, R . C., "Application of Wedge Diffraction to Antenna Theory, ' I Report 1691-13, 30 June 1965, Antenna Laboratory, The Ohio State University Research Foundation; prepared under Grant NsG-448, National Aeronautics and Space Administration, Office of Grants and Research Contracts , Washington, D. C.

Burnside, W . D., "The Reflection Coefficient of a TEM Mode Symrr,etric Para l le l - Plate Waveguide Illuminating a Loss less Dielectric Layer , ' I Report 1691 - 2 5 , March 1968, ElectroScience Laboratory, The Ohio State University Research Foundation; prepared under Grant NsG-448, National Aeronautics and Space Administration, Office of Grants and Resea rch Contracts, Washington, D. C.

42

9 Wu, D. C. F., "The TEM Radiation Pa t te rn of a Thin- Walled Para l le l - Plate Waveguide Analyzed by a Surface Integration Technique, I ' Report 169 1 - 2 3 , 20 September 1967, ElectroScience Laboratory, The Ohio State University Research Foundation; prepared under Grant NsG-448, National Aeronautics and Space Administration, Washington, D. C.

10. Pelton, E. L., Wu, D. C. F., and Rudduck, R. C. , "The TEM Radiation Pa t te rn of a Ground- Plane Mounted Para l le l - Plate Waveguide Analyzed by a Surface Integration Technique, I '

(ElectroScience Laboratory report in p rocess ) .

11 . Hansen, R. C. , Microwave Scanning Antennas, Vol. I, Academic P r e s s Inc. (1966), pp . 5-9.

43

APPENDIX I I

The computation of the backscatter f rom the guide for plane wave incidence given in Section 11 was aided by the For t ran IV computer program given below. The parameters used in the program a r e as shown in Fig. 19. The scat tered field contributions are: ETD, the d i rec t diffraction component; EG, the reflected geometrical optics s t r ip component; and ETP, the reflected diffraction component.

I

~

I

~

1 't I ETP ETD 7:

EG ' I I I

+ X

I I -/--_____,- - 7- A

I

I 1 I I

1 1 SURFACE OF INTEGRATION S

I I I I 4

0 B S ER VAT I ON PLANE

Fig. 19. Backscatter f rom the guide with plane wave incidence.

44

I

I

I

I

CTE~P2=CMPLX(ETDR(I).ETDIo) E A M = C A B S ( C T E M P 2 ) E A P = 1 8 ~ . O / P I * A T A N 2 ( A I M A ; o r R t A L ( C T E M P 2 ~ ) RTE=SQRT(D**2+(Y+A/2.0)++2) C T E M P 2 = C T E ~ P l * S Q R T ( D / R T E ) * C E X P ( C M P L X ( 3 . 3 . T P I * ( ~ - R T E ) ) ) E Q M = C A B S ( C T E M P E ) E Q P = 1 8 C . O / P I * A T A N 2 ( A l M A G ( C T E M P 2 ) ~ R E A L ( C T E P ’ l P 2 ) ) W R I T E ( 6 9 4 0 5 ) Y ~ E A M * E A P I E Q M ~ E Q P Y =Y +DA

850 C O N T I N U E CTEMP=CTEMPl*SGRT(D)*CEX?~CMPLX~OOO.TPI*D~~ W R I T E (6.8’55) C T E M P

855 F O R M A T : 5 u X r 4 H X A 2 = ~ 2 E ! 5 0 7 ) ! W R I T E ( 6 9 9 ~ 3 )

900 F O R M A T ( / / / l H r 5 3 t i W i i F L E C T E C G I F F R A C T E D PLUS G E O M E T R I C A L O P T I C S S T R 2 1 P / / 7 X i l H Y r 1 5 X 1 6 H A C T U A L I 1 6 X I 2 2 H E Q U I V & L E N T L I N E S O U Q C E / )

Y = - A / 2 0 b

C T E M P l = C M P L X ( E T P R ( 1 1 9 E T P I ( 1 ) 1 D O 950 I = l r N F C T E M P 2 = t M P L X ( E T P R ( I ) r E T P I ( I ) )

E A M = C A B S ( C T E M P 2 ) E A P = 1 8 O o 3 / P I * A T A N 2 ( A I M A G ( C T E M P 2 ) , R E C L ( t f E M P 2 ) ) R T E = S Q R T ( ( D + 2 0 O * F ) * * 2 + ( Y + A / 2 0 3 ) + + 2 ) C T E M P E = C T E M P l * S C R T ( ( ~ + ~ O U * F ) / R T E ) * C E X F ’ ( C M P L X ( ~ ~ ~ ~ T P I * ( D + ~ ~ O * F - R T E )

2 ) 1 E Q M = C A U S ( C T E M P 2 ) E Q P = ~ ~ C O O / P I * A T A N ~ ( A I ~ A G ( C T E M P ~ ) ~ R € & L ( C T E M P ~ ) ) 1 d R I T E (6 .4d5) YIEAMIEAPIEQYIEOP Y =Y+DA

953 C O N T I N U E C T E M P ~ C T E M ~ 1 * S Q R T ~ ~ + 2 o ~ * F ~ * C ~ X P ~ C M P L X ~ 3 . O ~ T P l * ~ D + 2 o O * ~ ) ) 1 I N R I T E ( 6 r Y 5 5 ) C T E M P

T E M P A = A T E M P D = D G O T O 10

500 C O N T I N U E S T O P E N D

955 F O R M A T ( 5 5 X * 4 H X A l = r 2 E 1 5 . 7 )

47

I

114

118

124

132

133

20

135

136

33 137

138

48

I '

S U d R O U T I N E F N C T N ( F F Q R * F F ; U . Y I N C * N F * Y P I F I D I A ) C O M P L E X E D ~ P I E D D ~ P * C T E M P * E R D ~ * C T ~ I ~ P ~ q C T E V P 2 D I M E N S I ON P 1 = 3 . 1 4 1 5 9 2 7 T P I = Z . O * P I I F ( Y P . L T o 3 . 3 ) GO T O 1 3 3 ~ E T A = I ~ O . ~ / P I + A T A N ~ ( Y P * Z O ~ ~ F ) RHOP=SQRT(YP*YP+4 .0*F*F) T E M P = d E T A - 1 8 0 . 0 C A L L V U ( R V a l r U V B l * R H O P * T E ~ P 1 2 . 0 ) T E M P = d E T A + 1 8 0 . 0 C A L L V B ( R V B 2 ~ U V R 2 * W H O P * T E M P 1 2 . 0 ) E D l P = - C ~ ~ P L X ( 9 V B I - ~ V B 2 * U V d 2 )

F F Q R ( 1 0 0 ) F F Q U ( 1 dnJ )

T E M P l = R H O P * A / ( R H O P + A ) C T E M P = - 1 . 4 1 4 2 1 3 5 6 * C ~ P L X ( ~ o 7 0 7 1 ~ 6 7 ~ ~ - ~ o 7 ~ 7 1 ~ 6 7 8 ) / T P I * C E X P ( C ~ P L X ( O ~ O

2 r T P I * ( T E M P l - R H O P - A ) ) ) / S Q R T ( R H O P + A ) T E M P = Z ~ ~ O J - H E ~ A C A L L V B ( R V B 1 * U V E l * T E N P I tTEi.1Pv2.C) T E M P = 4 5 2 . 3 - 3 E T A C A L L V b ( R V d 2 * J V B 2 * T t M P 1 * T E M P * 2 3 ) E D D l P = - C T E ~ P * C M P L X ( R V l 3 l - R V E 2 * U V B l - L J V t j ~ ~ E R D P = E D l P + E D D l P

D O 50 I = l t N F R l = S Q R T ( D * D + ( Y - Y P ) * ( Y - Y P ) ) R2=SQRT(D*D+(-Y-A-YP)w(-Y-A-Y-A-Y?)

Y = - A / 2 0

T E MP 1 = T P I *g 1 C A L L H A N K t L ( ? t S L l * Y N t U l * T E M P l ) T E M P E = T P I * R 2 C A L L H A N K E L ( S E S L 2 * Y N E U 2 t T E M P Z ) C T E M P = T P I * D / C M P L X ( O o O t 4 . 0 ) C T E M P l = C T E V P / R l * C P ' P L X ( t 3 E S L l * - Y N E U l ) C T E M P 2 = C T E ~ ~ ; P / R 2 * C ~ P L X ( B E S L 2 q - Y N E U 2 C T E M P = 2 . L * ( L T E M P l + C T t M P 2 )*ENdP F F Q R ( I ) = R E A L ( C T E M P )

Y =Y+Y I N C 50 C O N T I N U E

GO T O 2 5 3 1 0 0 C T E M P = C ~ , I P L X ( 3 . Y ~ - , . 5 ) * C ~ X ~ ( C , ~ ~ P L X ( ~ O ~ * - T P I * ~ O J * F ) 1

F F Q U ( I ) = A I M A G ( C T E M P )

Y = - A / 2 LJ

D O 1 5 0 I = l * N F R I = s Q R T ( D * D + ( Y - Y P ) * ( Y ' Y P ) ) T E M P I = T P I * W l C A L L H A N K E L ( E E S L l * Y N E L I l * T E M P 1 ) C T E M P ~ = T P I * D / R ~ + C T ~ M P * C M ? L X ( ~ ~ S L ~ * - Y N ~ ~ ~ )

F F Q R ( I ) = R t A L ( C T E M P l ) F F Q U ( I ) = A I M A C ( C T E ~ P l ) Y = Y + Y I N C

153 C O N T I N U E 203 C O N T I N U E

R E T U R N END

4 9

I

J C ~ I C W ~ O N D SULRCGRCII: F X F I R S T ORDER CYLINDRICAL HANKEL FUNCTIONS

5 5

6 3

70

83

I

S U B R O U T I N E V R (RVBIUVHIRIANGIFN)

D O U B L E P R E C I L I O N R A G I D P I T S I N P I = 3 . 1 4 1 5 9 2 6 5 T P I = 6 * 2 8 3 1 U 5 3 0 A N G = A N G * P I / l 8 C . C DEM=CMPLX(3.OtFN*SQRT(TPI 1 ) T O P = C E X P ( C M P L X ( 0 . C I - ( T P I *R+P I /4.3 ) ) )

C 0 M = T O P / D E M N = I F I X ( ( P I + A N G ) / ( 2 . 3 s F N + P 1 ) + 0 . 5 )

C O M P L E X DEN I T O P I C0i.l * E X P I U P P I I UNP I

D N = F L O A T ( N )

B O T L = S Q R T ( T P I + R * A )

C A L L F R N E L S ( C e S I i j O T L ) C = S Q R T ( P I / 2 . 3 ) * ( 3 . 5 - C ) S = S Q H T ( P I / 2 . C ) * ( S - O . 5 ) R A G = ( P I + A N G ) / ( 2 . ~ * F N )

A = 1 . 3 + C O S ( 4 N G - 2 . O * F N * P I * ~ N )

EXP=CEXP(CKPLX(S.S*TPI*R*A))

T S I N = D S I N ( R A G ) T S = A d S ( S N G L ( T S I N ) ) X = 1 0 . 0 Y = 1 O / X * * 5 I F ( T S . G T . Y ) GO T O 442 C O M P = - S Q R T ( 2 . O ) * F N * S I N ( A N G / 2 . O - F N w P I + D N ) l F ( C O S ( A N G / 2 . O - F N * P I * D N ) ~ L T ~ ~ e O ) COMP=-COMP G O T O 443

C O M P = S N C L ( D P ) UPP I = C O M * E X P * C O M P * C ~ P L X ( C I s 1

D N = F L O A T ( N )

B O T L = S G R T ( T P I * R * A ) E X P = C E X P (CiViPLX (U. 3 1 T P I * R * A )

442 DP=SQRT(A)*DCOS(RAG)/TSIN

443 N= I F I X ( ( - P I + L I N G ) / ( 2 .b*FN*P I ) + 3 . 5 )

A=l.O+COS(ANC-2.O*FN*PI*DN)

C A L L F R N E L S ( C I S I B O T L ) C = S Q R T ( P I / 2 . 0 ) * ( 3 . 5 - C ) S = S Q R T ( P 1 / 2 . 3 ) * ( S - n . 5 ) R A G = ( P I - A N G ) / ( ~ . ~ * F N ) T S I ‘\]=OS I N ( R A G ) T S = A O S ( S N G L ( T S I N ) ) I F ( T S 0 G T . Y ) GO TO 542 C O M P = S O R T ( 2 . 3 ) * F N * S I N ( A N G / 2 * ~ ~ - F N * P I * D N ) IF(COS(ANG/~.O-FN*FI*DN)OLTOC.C) C 3 M P = - C 3 M P GO T O 1 2 3

C O M P = S N G L ( D P ) UNP I =COM*EXP*COMP*CMPLX ( CI 5 ) A N G = 4 N G * 1 8 S 3 /P I R V B = R E A L ( U P P I + U N P I 1 U V Y = A I M A G ( ~ J P P I + C I N P I )

R E T U R N E N D

542 D P = S Q R T ( A ) * D C O S ( R A C ) / T S I N

1 2 3

51

5 2

C = ( F R * c O S ( X )+F I *51 N ( X )+3 ,QRT ( Y ) S = ( F R * S I N ( X ) - F I * C O S ( X ) ) w S 3 R T ( Y ) R E T U R N

4c Y=4 .Z /X 50 K = K - 1

F R = ( F R + C C ( K ) ) + Y F I = ( F I +D ( K ) ) * Y I F ( K - 2 ) 6 2 , 6 3 9 5 3

60 F R = F R + C C ( 1 1 F I = F I + D ( l ) C=3*5+(FR*COS(X)+FI*bIN(X) ) * C i r i ? T ( Y ) S = ~ O ~ + ( F R * ; I N ( X ) - F I * C ~ S ( X ) ) ~ ~ ~ ~ T ( Y ) R E T U R N

4 1 4 C=-3.0 5=-3 . 2 R E T U R N E N D

I

53

APPENDIX I1

The For t r an IV computer program used in the computation of the reflection coefficient is as presented below.

SEXELUTE PJFFT SPUFFT 4 L i C C

9

1 1

1 r:

54

5 5