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- THE CALCULATION OFELECTROMAGNETIC FIELDS IN THE

FRESNEL AND FRAUNHOFER REGIONSUSING NUMERICAL INTEGRATION METHODS

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zIRICHARD F. SCHMIDT'

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N72-146 9 7 (NASA-TI4-X-65 7 7 9

) THE CALCULATION OF

ELECTROMAGNETIC FIELDS IN THE FRESNEL AND

FRAUNHOFER REGIONS USING NUMERICAL

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X-811-71-392

THE CALCULATION OF ELECTROMAGNETIC FIELDSIN THE FRESNEL AND FRAUNHOFER REGIONS

USING NUMERICAL INTEGRATION METHODS

Richard F. Schmidt

July 1971

Goddard Space Flight CenterGreenbelt, Maryland

A L kAL" .6£, LA!i NW By

THE CALCULATION OF ELECTROMAGNETIC FIELDSIN THE FRESNEL AND FRAUNHOFER REGIONS

USING NUMERICAL INTEGRATION METHODS

Richard F. SchmidtNetwork Engineering Division

ABSTRACT

This document presents some results obtained with a digital com-puter program written at Goddard Space Flight Center to obtain electro-magnetic fields scattered by perfectly reflecting surfaces. For purposesof illustration a paraboloidal reflector is illuminated at radio frequenciesin the simulation for both receiving and transmitting modes of operation.Fields are computed in the Fresnel and Fraunhofer regions. A dual-reflector system (Cassegrain)is also simulated for the transmitting case,and fields are computed in the Fraunhofer region.

Appended results include derivations which show that the vectorKirchhoff-Kottler formulation has an equivalent form requiring only inci-dent magnetic fields as a driving function. Satisfaction of the radiationconditions at infinity by the equivalent form is demonstrated by a conver-sion from Cartesian to sphericalvector operators. A subsequent develop-ment presents the formulation by which Fresnel or Fraunhofer patternsare obtainable for dual-reflector systems. A discussion of the time-average Poynting vector is also appended.

iii

CONTENTS

Page

GLOSSARY OF NOTATION .............................. vii

INTRODUCTION ..................................... 1

FORMULATION OF THE SCATTERED FIELDS ................ 4

SINGLE-REFLECTOR SYSTEMS .......................... 10

DUAL-REFLECTOR SYSTEMS ........................... 26

SUMMARY ....... . . ............................... 29

ACKNOWLE DGMENTS ......................... 30

REFERENCES ....................................... 31

APPENDIX A - THE FORMULATION OF THE SCATTERED FIELDSE (x', y', z') H (x', y', z') BY EQUALENT METHODS.. 33

APPENDIX B -THE TIME-AVERAGE POYNTING VECTOR ...... 35

APPENDIX C - FRAUNHOFER FIELDS FOR DUAL-REFLECTORSYSTEMS .................. ....... 37

ILLUSTRATIONS

Figure Page

1 NASA Tracking Station, Rosman, N. C .......... 9

2 Electric Field vs. Angle ................... ..... 11

3 Electric Field vs. Angle O ........................ 12

4 Direction of the time-average Pointing vector .......... 14

v

ILLUSTRATIONS (Continued)

Figure Page

5 Amplitude of magnetic field near focal-plane (axialreception) ................................... 15

6 Airy disc ana rings (axial reception) .................. 17

7 Phase of electric field near focal plane (axial reception) ..... 18

8 Direction of the time-average Pointing vector (axialreception) ................................... 19

9 Direction of the time-average Pointing vector in the vicinityof a null (axial reception) ........... ............. 20

10 Amplitude of electric field near focal plane (paraxialreception) .................................... 22

11 Airy disc and rings (paraxial reception) ................ 23

12 Phase of magnetic field near focal plane (paraxial reception).. 24

13 Direction of the time-average Pointing vector (paraxial reception) 25

14 Dual-reflector system (Cassegrain) .................. 27

C-1 Abstract Dual-reflector system ........................ .. 37

vi

GLOSSARY OF NOTATION

Meaning

E(x', y', z')H(x

El, HI;

irI

K', y', z') backscattered electric and magnetic fields at observerposition (x', y',z')

A, E, a constitutive parameters: magnetic permeability, inductivecapacity, electric conductivity

o angular frequency

j, i imaginary operator = TI

E2 , H2

electric and magnetic illumination distribution fields

q solution to the wave equation

V q' vector gradient of q

d s differential area

d e differential arc-length

n unit normal to a surface

ei, hi electric and magnetic polarization vectors

p vector from a feed point to a surface

v a vector used to generate e¥ and h. for a spherical wave

O order, origin

k wave number = 27T/x

X wavelength

r radial distance in spherical system

O polar angle in spherical system

azimuthal angle in spherical system

unit vectors in spherical system

D aperture diameter

<P> time-average Poynting vector

R distance from origin O

F focal length of a paraboloid

vii

Symbol

A feed displacement in focal-plane

K beam deviation factor

0 s beam squint angle

K electric sheet current

qS, qt electric surface and line charges

a, C radial and angular variables for surfaces in parametricform

viii

THE CALCULATION OF ELECTROMAGNETIC FIELDSIN THE FRESNEL AND FRAUNHOFER REGIONS

USING NUMERICAL INTEGRATION METHODS

INTRODUCTION

The purpose of this document is (1) to present some computed results ob-tained with the Kirchhoff-Kottler formulation in the Fresnel and Fraunhoferregions, (2) to discuss equivalent scattering formulations, (3) to introduce asimplification in the procedure for obtaining Fresnel and Fraunhofer patternsfor dual-reflector systems, and (4) to derive the expression used for the time-average Poynting vector in the digital computer program.

The results obtained here are based on the contributions of many persons.It is impossible to acknowledge all of them properly. Most of the fundamentalideas were well-developed many years ago, but the advent of the fast digitalcomputer has, within the past 5 years, made possible the numerical evaluationof existing formulations for practical geometries, feeds etc. Since this docu-ment contains representative results in Fresnel as well as Fraunhofer regions,some of the language and concepts of optics are introduced. The bright diffrac-tion disc and rings of Airy, for example, are especially significant in the recep-tion mode of operation (image space). A brief historical outline is included.Most of the dates and comments in this introduction were taken from textbooks,journal articles, etc.

History of Diffraction

1452-1519 da Vinci: first reference to diffraction

1601-1665 Fermat: principle of least time which implied rectilinearpropagation, independent rays, law of reflection,law of refraction

1621 Snell: sometimes credited with laws of reflection andrefraction

1663 Gregory: "Gregorian" dual-reflector system

1663 Cassegrain: "Cassegrainian" dual-reflector system

1665 Grimaldi: first accurate descriptions of diffraction

1690 Huygens: first proponent of wave theory without periodicity

1

Newton:

Maraldi:

Young:

1818-1827 Fresnel:

Pois son:

Arago:

Fraunhofer:

Hamilton:

Airy:

Stokes:

Maxwell:

Kirchhoff:

Poynting:

Maggi:

corpuscular theory (received acceptance for 100 years)to the exclusion of Huygens wave theory

found the "Arago" spot in the umbral region

suggested transverse rather than longitudinal wavemotion and the interference principle

extended Huygens' theory by adding periodicity intime and space to the wavefronts, but several argu-ments had no physical significance. Accounted forrectilinear propagation as k- 0, reflection, refraction,and diffraction effects. Predicted bright spot in discumbra

reviewed Fresnel's prize memoir, and refuted conclu-sion of bright spot in shadow of disc

verified Fresnel's bright spot by experiment

diffraction formulations, measurements

geometrical optics (ignored Huygens' and Newton'sformulation; indifferent to physical interpretationand interference concepts)

calculation of diffraction disc and rings

limited version of general treatment later made byKirchhoff

wrote the fundamental equations of electromagnetictheory V x E = - B/at and V x H = J + 3D/ t

wrote the Kirchhoff-Helmholtz Integral Theorem,

4x yn ' r I

[r dn

Kirchhoff started with the scaler wave equation inthree dimensions, and deduced a formulation whichin many respects embodies the basic ideas of theHuygens-Fresnel principle.

theorem on energy flow

provided additional insight on Kirchhoff theory, in-cluding a transformation for an unclosed surfacewith rim F.

2

1642-1727

1718

1817

1818

1818

1827

1824-1844

1834

1849

1865

1882

1883

1888

1893 Heaviside: studied the energy flow problem and physical inter-

pretation of formulations

1909 Debye: analytical work on scattering

1909 Reiche: investigated axial non- linear phase shift

1911 Sommerfeld: rigorous analytical solutions, used boundary value

approach

1917 Rubinowicz: preceded Kottler in studies of unclosed surfaces withrim

1923 Kottler: showed that Kirchhoff's solution to the black-screen

diffraction problem was a first approximation to aboundary value problem (actually a solution to a"saltus" problem). Kottler annexed a contour inte-gral, for unclosed surfaces, to the Kirchoff formulation.

1939 Stratton and further analysis of Kottler's contour integral and its

Chu: relationship to Maxwell's equations

1957 Keller: presented an extension of the theory of geometrical

optics

1967 Sancer: showed that the vector Kirchhoff equations can bederived using the free-space dyadic Green's function

and the appropriate Green's theorem. The Kottlercontour integral was obtained without appeal to physi-cal intuition, and was inherently contained in thederivation rather than appended at the close (thehistorical development.

The preceding partial list of contributors is indicative of the broad field ofanalysis and research associated with the subject of diffraction. In this documentthe Kirchhoff-Kottler formulation has been stressed with a view toward obtainingmeaningful solutions for practical feed-reflector configurations. Exploration of

the bounds of validity of the selected theory is predicated on objective compari-sons with other analyses and direct physical measurements where practicable.The interested reader may wish to review the mathematical and physical rea-

soning which underlie the Kirchhoff-Kottler theory of diffraction. Problemsassociated with the definition of a black screen, the introduction of polarizationin a physical 3-space, and finite surfaces are discussed in detail in numerous

textbooks 1 and will not be considered here. The extrapolation from the original

1 Ref. 1, Chapter II.Ref. 2, pp. 460-470Ref. 3, Chapter 5Ref. 4, Chapter VIII

3

black-screen derivations to present day applications, such as those involvingmultimode waveguide feeds in dual-reflector systems, provides a sharp contrast.A review of the analytic difficulties encountered in the theory, the departurefrom the assumption of continuity required by the underlying Green's theorem,and the additional compromises due to numerical evaluation should be weighedagainst nearly a century of useful predictions resulting from the Kirchhoff theory.Barakat', in paraphrasing a remark of Poincarg, said "The theoreticians believein the Kirchhoff theory because they hold it to be an experimental fact, while theexperimentalists think it to be a mathematical theorem." The statement is atribute to the theory considering that scatterer size, observer range, and radiiof curvature of both scatterer and wavefront must be large in terms of a wave-length. Further, the effect of mutual coupling on the scatterer, and betweensource and scatterer, are traditionally neglected altogether.

FORMULATION OF THE SCATTERED FIELDS

The formulation for the scattered electric and magnetic fields from per-fectly conducting surfaces is taken as 2

E(x', ', -j X e 47r 1 [(n x H1) ' V] -V d s - j w / 1 (n xH1) d s

H(x', y', z')4 l ( x H1) x V q d s'

where

e-ikr

Previously, the equivalent electric-field formulation

Previously, the equivalent electric-field formulation

)E (x', yC, ')p - 1 f H i dT 9X( [pp.6i (nx)H) 16+(iE)V+]ds

1 Ref. 5 Chapter 9, pp. 6-162 Appendix A provides background for this formulation from several sources.

4

was employed. 1 The latter formulation for E(x',y',z') could be termed the"historical" approach since the Kottler contour integral appears explicitly asan amendment to the Kirchhoff formulation. It has been shown that this line-charge integral can be obtained along with the Kirchhoff formulation withoutappeal to physical intuition via a dyadic Green's function,2 and further, the twoforms for E(x',y',z') yield identical results.

From a programmer's viewpoint the organization of the problem is compli-cated by the "historical" formulation for the electric field E(x',y',z'). It is notedthat a mixture of contour and surface integrals requires separate quantizing orsampling intervals for numerical evaluation. In addition, the surface-chargeintegral appearing in the "historical" formulation involves the incident electricfield, so that the polarization vector e. must also be developed and retained inthe numerical evaluation. This should be contrasted with the formulation usedherein, which proceeds simply with surface integrals, the incident magnetic field,and the associated magnetic polarization vector hi for the incident wave.

_P x (v x 7)e.

1 x (Vx P'

h. = ___pV

Unless there is a specific reason for distinguishing the Kottler line-chargeintegral and Kirchhoff surface-charge integral contributions in a problem, theircombined influence can be computed via the surface integral

j oe 4 J (x [ H) XH) v 7 d s

for the total bound electric charge distribution. It is easy to verify that theformulation for E(x',y',z') and H(x',y',z') satisfy the radiation condition at in-finity. A slight complication is encountered with the operator del, written

V=i-+j +k-Yax yy ¢z

in Cartesian coordinates.

1 Ref. 2, page 4692 Ref. 6

5

The radial component, at infinity, of the integral

JW/z 4-1 f (-a x Hi) ¢d s

1

can be set down immediately as

1 s1- U 4ct t J ( x H[ ); i(lr] ir d s

The integral

1 -Vj Vqjds1

J [( X x H1) Vvd s

is entirely radial at infinity, as is anticipated for charges, however, manipulationof the integrand in terms of the Cartesian operator V is undesirable since

V = - (j k +- ik

is radial.' An interpretation of (n x H,) · V in spherical coordinates is, therefore,needed to establish the vanishing of the radial part of the electric field at infinity.

A vector dot product in orthogonal curvilinear coordinates can be definedas 2

P' = Pu q + Pv qv + Pw qw

and the dot product (ff x H1 ) ' V should follow directly except for the fact thatp - (n x H ,) is a vector, and q - Vis an operator. Furthermore, the form of Vin spherical coordinates depends on the operation to be performed, which isunlike the Cartesian case where V is the same for grad, div, curl, cross-product,and dot-product.

The numerical evaluation of [(Ci x H 1 ) -V] 7,V balso needs to be considered. A conversion ofV qfrom spherical to Cartesian coordinates was used previously in the program, posing noproblems, but in the present formulation

(xHY +Yy a Z zNx H~1 *=Kx- +K + K·a

is a differential operator, and Va is a numerical result for each differential area ds.2 Ref. 7, page 155.

6

The form of V to be used is suggested by the quantity on which scalar form[(n x ) V] operates. Here

is a function of (r) alone, and is a vector. The spherical basis vector lr isinvariant with radial changes. From orthogonal curvilinear coordinate relation-ships,1 the gradient of a scalar (5) in spherical coordinate is

V0- 1•i +1 i- +-a ar r z aa -- + a'

and

a = r ) = 1

= |a-(r lr) r

y |-(r 1) |r sin 0.

Then

[(nx "i) V] V, = [(nx Hl)"lr -r ]-rr V b= [(nx l)'1,] a2 q lr

(k + 2 +) b [(x H) ]r( +j r2

which shows (1/r) in orders O 1, 02, 0 3 since p = e-ikr /r. The highest orderof the term (1/r) in the formulation for E(x',y',z') containing the contour integralis O2 .

1 Ref. 7, page 152

7

Vanishing the radial fields at infinity is now made very simple.' Summingthe radial parts of

1 1E (x', y, z') = j E 4 7T

[(nx H1) V] V d s - j a)L 1 f (n x Hj) d sL~Xi/JYJW/I~J~flX11i·YU

at infinity, terms of 02, 03 in (1/r) are neglected. Remaining terms of 0 1 in(l/r) annihilate.

1 k29- i 2)(-' -i =;,ULJW 6 I()/, (_xH)---i,=

and

1c= f = 1,HFIe

an identity, is recovered.

The magnetic field is inherently simpler duenetic charges.

H (x', y', z') -47T T ( i x

SI

to the absence of bound mag-

H1) x Vd s.

This field is transverse since V9 is radial at infinity and the cross-product oftwo vectors is orthogonal to the vectors entering into that cross-product.

1 The numerical evaluation can also be carried out now since the integrand [(n x H1) V] V 9has been rewritten as 1r (-k2 + 2 j k/r + 2/r2 )t which is convertibile to Cartesian componentsand free of troublesome operators. V g=- (i k + l/r) 411r is assumed to be reduced to theCartesian or "working notation" of the program already.

8

Figure 1. NASA Track ing Stat ion, Rosman, N.C.

9

SINGLE-REFLECTOR SYSTEMS

The vector Kirchhoff-Kottler diffraction theory is applied here to the studyof paraboloidal systems, initially, then to the conics, and subsequently to dis-torted conics and more general surfaces exhibiting aberrations of various kinds.Results presented in this document are principally for the paraboloid in bothtransmit and receive modes of operationl The paraboloid is singular in itsfocussing properties, has found wide application in optics and microwave tech-nology, and extensive data exists for it in the literature. Particularly interestingare those paraboloids with small f numbers (f = F/D < 0.5) as these are encounteredoften in radio-frequency applications and are somewhat more difficult to analyze 2

than the large F/D structures.

A typical RAD HAZ environment exists in the installation at the NASA track-ing station (Rosman, N. C.) shown in Fig. 1. Some years ago (circa 1964) a studyof the possible radiation hazard existing on one of the access roads was under-taken. The only methods available at that time were awkard and empirical?3 Ingeneral, the field-strength and energy-density levels at a set of points in thevicinity of a high-gain paraboloid energized by a high-power transmitter weresought. Without trying to re-create the precise set of conditions at Rosman, anattempt will be made to illustrate how the Diffraction Program developed atGoddard Space Flight Center might be used, not only in a RAD-HAZ application,but also in a more general way to design feeds for scanned beams, explore otherreflector geometries, and capture energy from the Airy-type diffractions rings,etc.

To illustrate the RAD-HAZ application: Fig. 2 and Fig. 3 show the trans-mitted electric field versus angle 0 as the radius of the observer is reducedfrom 100 miles through 2D 2 /k, 1D2/k, and D2/2k. Coordinates in the fieldexplored by these four arcs are representative of the access road mentionedpreviously. The object is to determine amplitude, and to a lesser degree polar-ization and phase in a RAD-HAZ study. It can be seen that the field pattern shapeis only slightly degraded in the Fraunhofer region between 100 miles and 2D 2 /X.

1 The results presented in this document formed the basis of a presentation give at the 1971USNC-URSI Spring Meeting, Washington, D. C. under Commission I, Near-Field Measurementsand Radiation Hazards Probing (RAD HAZ).

2 Ref. 11, page 485Ref. 12, page 353

3 Ref. 13, see section on antennas (power density multiplication)

10

F=12.99 FT., D=30.0 FT., f=1.11520 GHZ

8- DEGREES

0 1 2 3 4 5 6 7 88-DEGREES ( = 0

Figure 2. Electric Field vs. Angle 0

11

db

4DEGREES

400

300

200

100

TO

db

4DEGREES

400

300

200

100

0

9 10

= 0°

F =12.99 FT., D=30.0 FT., f=1.11520 GHZ

0 1 2 3 4 5 6- DEGREES

7 8= 0°

9

4DEGREES

400

300

200

100

0

10

0 1 2 3 4 5 6 7 8 9 108- DEGREES = 0°

Figure 3. Electric-Field vs. Angle 0

12

+2.051-db \E

-_~ Ro =-FROM 0 _

0

-10

-20

-30db

-40

-50

-60

-70

0

-10

-20

-30db

-40

-50

-60

-70 I I

Some null-filling is in evidence, the 7T-radian phase jumps at pattern nulls havebecome more gradual transitions at pattern minima and signal level is nearlyproportional to 1/R. Within the 2D 2 /X arc the pattern is unique for each radius,null-filling is pronounced, beamwidth is increased, and the signal level no longervaries as the inverse first-power of distance.

As the paraboloid is approached more closely, the moving spherical triad ofbasis vectors conforms poorly to the transmitted wave which, in the focal plane

(R = 0), resembles a plane wave more than a spherical wave. For this reasonthe Cartesian basis is used to resolve the fields. In addition, the time-averagedPoynting vector is now displayed.I See Fig. 4. The region considered is centered

about the axial point D2 /2k from the focus of the paraboloid. To the far-left of

the area shown by Fig. 4, the more intense time-average Poynting vectors < P >form a nearly parallel bundle whose diameter is roughly that of the paraboloid.To the far-right of the area shown, the vectors < P> diverge and appear to comefrom origin 0, which is also the focal-point F of the paraboloid. That is, theset of vectors <P > in the Fraunhofer region appear to flow radially from a pointorigin, but their intensity is modulated in accordance with the envelope shown byFig. 2. Fig. 4 is interesting because it shows that transition region where thePoynting vectors < P > have not quite aligned themselves with the lines tracingback to origin 0. No attempt was made to scale the length of the power flowvectors < P>, but it was noted that those vectors which departed considerablyfrom the lines intersecting at O were also reduced in intensity.

A measured result, obtained for a region between the focal point and a pointapproximately 17 further away from the transmitting paraboloid, is shown inthe text by Collin and Zucker.2 It is noted that although the discussion has beenlimited to paraboloids with focal-point excitation, the program is written in ageneral way to accept other surfaces, including distorted conics, etc. and thereis no restriction on the number of feeds, method of excitation, position of feedsetc., other than those restrictions inherently imposed by the Kirchhoff theoryitself.

At this point it is convenient to invoke reciprocity and study some of thefields obtained from reflectors in the receiving mode of operation. The receivedfields are less intense than the transmitted fields, above, and do not constitute a

RAD-HAZ problem in the preceding context, however, an element of RAD-HAZ isstill present since, for the case of paraboloids, spheres and other reflectors,energy is being concentrated in a relatively small region. From a standpoint offeed design, the received fields provide a means of matching feeds to fields (fre-quently in a conjugate sense). The literature of optics abounds in both experi-mental and theoretical studies of the image space.3 As before, the paraboloid

1 Appendix B2 Ref. 14, page 503 Ref. 4, pp. 435-480

13

f=1.11520 GHzD=30.0 ft.F=12.99 ft.

D2RANGE= -X 510ft.2X

250X

/ box

'-8X 6 -40X -2 O 50XA-80 -60X - -40 -0 - 0 -

Figure 4. Direction of the time-average Pointing vector

14

f=1.11520 GHz X/2=SAMPLING INTERVAL

-2.0 XI I I) I < )) ) _

. I I II. . .~~~~~~~~~~~~~~~~~~~~~~~~~~.OX -1.OX .OX 2.

U a

Figure 5. Amplitude of magnetic fied near focal-plane (axial reception)

.OX 3.OX

F=12.99 ft. D=30.0 ft.

2.0X

1.OX

Cn

2.OX

+Z

REFLECTORAXIS

-I.0X

-2.OX '-

-3.OX

I+X

-2

is utilized here for convenience and because its properties are well-known. Anobjective of the program, however, is to obtain insight for those scattering sys-tems that are difficult to treat analytically and for which only meager data canbe found in the literature.

The constant field intensity plots (isophotes) shown in Fig. 5 are for thefocal-region of an ideal paraboloid for the axial reception of a plane-wave. Thesource in this case is t o the right, the scatterer to the left, and the convergingspherical wave is progressing from left to right. This convention will be re-tained in all of the subsequent examples. Results are symmetrical about they-z plane, but asymmetrical about the focal plane. The F/D of this system isonly 0.433, therefore, energy flows into the image space at a steep angle. Fig. 6shows the Airy disc and rings in the x-y on focal plane.

The phase of the electric field is shown in Fig. 7 for the preceding set ofisophotes (Fig. 5) over a restricted portion of the z-axis (±1 k re 0). Power-flow in this region is orthogonal to these wave fronts. It is also apparent thatthere is no rational definition of phase for axial nulls, and zero-intensity ringsof the Airy pattern. Energy flow is around such singular points.

A plot of the direction of the time-average Poynting vector <P> for thefocal region being discussed is given as Fig. 8. The crossed lines are the boundsof the geometrical shadow. Airy discs, bright rings, and dark rings can be seenwithout difficulty. No attempt was made here to scale the length of vector < P >in accordance with its magnitude, however, this capability has now been auto-mated in a Calcomp plot program. The turbulent regions to the left and right ofthe geometrical focus correspond to the limits of the depth of field at which theintensity of field vanishes. It is noted that the time-average Poynting vectorsdo not cross the system axis, which is distinct from a ray-optics presentation,but rather diverge as if they originated from F for large z.

Especially interesting is the region surrounding a null. The null betweenthe Airy bright disc and the first bright ring is shown in Fig. 9. A circulationor vortex-like flow can be seen about the null and an examination of the magni-tude of <P > shows that the intensity of the time-average Poynting vector dimin-ishes in a continuous manner as the null in the vortex is approached. Theexistence of this counter-flow of energy has been reported previously, and wascomputed by Minnett and Thomas 1 who used axial-wave theory to investigate thecharacteristics of fields in the focal region of a paraboloidal reflector.

1 Ref. 15

16/

ELECTRIC FIELD (Ey)F=12. 99ft. D=30.0 ft. f=1.11520 GHz

2.00-/ 7=7--=- *82.9.,

sB ~~~~~~~~~~~~~~-1.00

oIo- /\

,., ~ ~~~-~--~----~

1.5 -.5.

.-95.7--

1.00 -74.5 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~-1.00

.00 1 _ ,

I 240

x

1.00-~~~~~~~~~~~~~~~~~~6.

1.5~~~~~~~~~~~~~~~~~~ 0-5

Y IN WAVELENGTHS

Figure 6. Airy disc and rings (axial reception)

\ //

1.so- \ /-s

I.ooso Io2,00~~~~~ -- --2,OO-

1,5 ~ 0 0 0.5 0 OO 0 . 0 ~.0 1.50 20Y -- INWAELNGH

Figur`----e~.Ar ic=dig oilrcpin

17

F = 12.99 ft.D = 30.0 ft.

f = 1.11520 GHz

m m olum en en en 04040404990- 2.OX

130

179

217

238

2542

271

296

335

14

38- 1.OX

54

68

84

106

136

164

181

191

1963 51 298 2145 191 1 7 83 29 334 280 25 197 - Z

324 271 218 164 110 56 2 307 252egrees 196 REFLECT(

1i91 AXIS

181

164136

U I12 ~~~~~~~~10684

68

54

38---1.OX14

335

296

271

254

238217

179

130

... 99- -2.OX

-1.OX

Figure 7. Phase of electric field

OR

OX 1.OXFOCAL PLANE

-X

near focal plane (axial reception)

18

,~ _ _

_ a 2"

., _ 4 4

_ _ 4/

_ '4 _

I4 ' _

4 4 '4

4 '4 '4

4 _ '_ 4

_ 4 _ 4

_4 _ _ '4

_ _4 _. _

_ _ _. -

4 _4 _. _

4 _4 _. _

4 ,4 _4 _

.4 _. _

,4 _4 _

4 ., .,4 -4

Af J

. ~ -4 A

A A

- A AA

', a

_ / _/

r -- _r / /

r

4 '4

/ / /

..· /~ '

A

-2.0X

-4-4 - -4

A A A

A A A

A A A

A A AA A AA A r

A A A

A A /

-2.0A

f=1.11520 GHz, D=30.0 ft., F=12.99 ft._~A.4 _ A A ,, /60° A A A r r -2.OX

'~9d~ f ' f ! >' / / A . A

\ ' ' a a J//f!f ! ! /A / / / / / / x

,Z

' '' "4 4 ' . . . . ,

. . _ _ REFLECTORAXIS

4 ~, A A _ A A _; _ _ r, r-r- - - - A 4 - A -

' " " " " ... ...- ' x \ A ', _ . r, ,, A, __.,' - t -2.

I4 AI I *~60 I

A r___ \/ A r r _ AXIS

1.0X -X 1.0X 2.OX 3.OXFOCAL PLANE

A ~ ~ ~ ~ OA APL i/II44ANE 4' 4'4'"' - 4'

Figure 8. Direction of the time-average Pointing vector (axial reception)

co

f=1.11520 GHz, D=30.0 ft., F=12.99 ft.

__ - - -- - - - --- ff ff f f - A

f f`/ f !-0.2X -.1 i X Z=O _ / 1 0. / / 2

X X -a - ' A A / / / t t f

> \ ' ' A P ' A/// / A A A A A t

4 A fi s i i e h t f t t f

I

Figure 9. Direction of the time-average Pointing vector in the vicinity of a nul (axial reception)f

\\ \\, _ _' -

A' A A A' / / / -0 .6A

-0.2A -0.1X Z=OX 0.1X 0.2X

FOCAL PLANEFigure 9. Direction of the time-average Pointing vector in the vicinity of a null (axiol reception)

Additional information concerning the calculation and measurement of am-plitude and phase in the image space (of a microwave lens) has been given byFarnell,', Bachynski, and Bekefi, 2 who applied scalar theory. Emphasis is placedon the fact that vector Kirchhoff-Kottler theory was used to obtain results pre-sented in this document and that a numerical approach has been stressed toavoid many of the mathematical difficulties encountered in analysis, particularlyfor those cases where the diffracting system was complicated by intricate feeds,or illuminating sources, and non-conic or distorted surfaces. The latter areaccommodated by the program without additional effort on the part of the pro-gram user, who simply modifies the input parameters, and the penalty in in-creased computer running-time (CPU time) is usually moderate.

A very simple illustration of the flexibility of the program is afforded bydisplacing the illuminating source so that a plane-wave (effectively) arrives withan arbitrary angle of inclination relative to the system axis (z-axis). The simu-lation produces this change for the source by calling Euler-angles in a rigid-bodytransformation and the components of a displacement vector. An inclinationangle of 4-degrees was chosen, together with the same input parameters usedpreviously for axial reception, based on the anticipated displacement of the in-tense field region associated with a focussing paraboloid. The equation

sinO KAS F

relates beam squint-angle es through beam-deviation factor K3 , feed displacementA, and focal length F. Under reciprocity then, for F/D = 0.433,

sin 4°0 (.89) (A)(12.99)

and the energy can be expected to concentrate near

AZ 1.02 ft. - 1.14k (a 1.11520 GHz,

on the opposite side of the z-axis.

1 Ref. 16Ref. 17

2 Ref. 183 Ref. 3, page 488

21

F=12.99 ft. D0=30.0 ft. f=1.11520 GHz A/2=SAMPLING INTERVAL

T T T T

I I 1I I I-2.OX -1.Ox I, -x 1.0x 2.0X

Figure 10. Amplitude of electric field near focal plane (paraxial reception)

MAGNETIC FIELD (Hy)F=12. 99 ft. D=30. 0 ft. f=l. 11520 GHz

~·4146.7 -71.00

-0.50

__________ - 0.00O

1. 12.3.116.82 -131.4 \\\137.5 -134.9 141.6 Z\ d'22/ ////\,0'l a 1 3.1 I 1 - / ]

Y IN WAVELENGTHS

Figure 11. Airy disc and oings (paraxial reception)

23

zx

I.C

zx

F = 12.99 FT. D = 30.0 FT. f = 1.11520 SAMPLING INTERVAL = X/2

"CD 0OW2 M . N °W a CoD N" t V at-rN ! T _- " - Mz r Iq W) M a) t W , =

290 5133

134234 98134211188 223160 283

130 102305 283

AXIS357 334 64

314 29982

282806106

276 125281 136 ~~~~~~~~~~~~~~~~~~~~~~~~1.0>X

329 4' ~~~~~~~~~~~~~~~~~~~~~~~~~~~151329 2 1 112 3 193 137 1 24 326 6~~~~~~27\205,17115

35 7 341 25I29 322 58 5 232232 322 332 ~~~~~~~~~~~~~~~~~~57 59 -2.0X

1 41 52

287 1293 319

119 252 ,75158 280 276243281~~~~~~~~1 88 243 1~~~~~~~~~-3.0>X

CO C~~~~~ - I~~~ -.- rC -rC c CD co- C D CO CD In Z.J It coC-2.o x / 131 ' CD C.-x ,23 D o0 ) ! 0 ' C n 124 It\

-2.0. -1.0x X .ox 2.o0X,-,C

Figure 12. rPhase of magnetic field near focal plane (paraxial reception)

F 12.99 FT. D0 30.0 FT. f 1.11520 GHz

\ \ \ \\\o\ \ _- -~-~-A A / / / / / A A / A

\ \ \ \ \ \ \ \ \ \ \ \ \ \\\ \','..' "J//'/ / / / / / A / / ~~.~AAAAAA A ~~/ A/-1.0A A A,

N N N \s \ N \ N \ \ N \ _ ____ , ,///,// / / / / / / / / / AN ... \N \\\--J/./-f, / / / / / / / .'/

. N \ \ \ \ \ \ \ \ \ \\\\ \ _ I I \ A AA / / / / / / / A,_ " N N N ' N \ N\ \N \ \ / / / / / / / A

. _ -' \ \ . \'\ \ N N//AA__- / -\, ~ ~ ~ \ , I ~ I _ _ _ _ _I_ _ - -- -- _. / - / , / / /

N_~~I _ /-. / / /A /A _ - X

'. " -'- '- 'x _- '- '. _ _ _ \ \ \ _

\\ / N /i tI

/ / / _/ / / _

_ _ _ , / __/

_ / N N N N \ N \ \N\\ t\\\\ \\\ \\ \ \\N\ \

NN N N N N \ N...-"-"-.".'-/- / / Ar / A / / A .AX _ C

_ " / " ' "- '. "- "-'-- "- " ' "\\\ " ' " / / /

N N N ' N N N N N////// // / A A A A - X A A A

-. N. N .. A -. -1.0N

A, -A .. N N N N -

Af /

..,, /~~ / /\\\\\~\ ~ / //.'",.".-,'%..

""''''"' / / /!!!/ /""~--".---"""','-\ , .,'- ','-,--,--, -. , -, .,--30

·*3.0), -2.0), -1 .0) ` 0.0) ` 1 .0;' 2.0), 3.0),

/~~~ ~PLAN

FOCAL

/ A .., / / AAA .. I N \

/~ ~ / A / ////~/AA. A AA~ ~ / A A ///t/- -NN N NN,\ ,\,

- - A A / A A A A , N N N N N - - / / / / AA'~ / NN NN N NN N N NN

- -0 - -2O A1O A.O A.O /.O / if /A A - .- -. A A A A ,,//,W¶ N\\\N\

/SF x\\\ \\\

Figure 13. Direction of the time-average vector (paraxial reception)

A displaced and distorted version of the isophote field for axial receptionis shown in Fig. 10. It appears that the peak intensity occurs just to the left offocal plane (toward the reflector). Solid lines at 40 are shown through F and F'as a reference for the displaced high intensity focal region. Fig. 11 shows thedisplaced Airy disc and rings in the x-y or focal plane. The wavefronts for theFig. 10 computations are shown in Fig. 12. As before, the time-average Poyntingvector was computed for the paraxial case. See Fig. 13. The Airy bright disc isdisplaced, together with the bright and dark rings. As with previous examples,power flow in the vicinity of dark rings and limits of the depth of field, whichtends to be orthogonal to the general trend from left to right, is accompanied bylarge reductions in magnitude of the power density. The vectors <P> were notscaled at this writing because of the large amount of data reduction involved forthis asymmetric example.

DUAL-REFLECTOR SYSTEMS

An initial effort to obtain the fields scattered from dual-reflector systemswas undertaken by computing the Fraunhofer fields from a Cassegrain configu-ration. See Appendix C. Several idealizations were made, including omission ofmultiple reflections between yl and 72, mutual coupling of the sheet currents,and even first-order blockage effects. The latter effects were taken into accountpreviously in single-reflector systems by deleting the integration over the umbralregion. A polygonal approximation to the shadow boundary was specified foreach observer position.

The radiation pattern scattered by the main (parabolic) reflector was ob-tained as follows. A prime source 3, = S1 cosN ® was selected with N = 35.0 sothat the feed taper plus space taper of the illumination at the edge of the (hyper-bolic) subreflector equaled -9.0 db. The electric sheet current K

1on the hyper-

boloid y1

was then computed with contiguous increments of area not larger than0.3 k by 0.3k. Each value of sheet current K on the paraboloid T2 was thenobtained by integrating over all of K1 on T1 . A sampling grid not larger than3.0 X by 3.0 k was used on the main reflector. Finally the distribution K 2 wasintegrated in the Fraunhofer region for each observer position (r', d', A'). Thedistributions K

1and K

2are invariant with observer position and were there-

fore stored in the computer. For the reflector and feed geometries employedhere, a four-fold symmetry exists for K1 and K , however, this feature wasnot exploited even though cpu time can be conserved in such situations.

26

MAIN REFLECTOR

SUBREFLECTOR

D - 24 8 FTF = 10.74 FT.

SAMPLING INTERVAL 3.0 X

d = 2 4 FTA = 2.138C= 1881SAMPLING INTERVAL 0.3 X

FEED N = 35.0 = > 6 db SPACE TAPER-8.4 db FEED TAPER ON

HYPERBOLOIDFREQUENCY = 8.0 GHZPOLARIZATION: P(i) = O. P(J) I O. P(K) = 0

A D 2PARABOLOID Z:= F + Zlp

HYPERBOLOID z= ClI + 2/A 2)'/ + ZH

-- = M- X = /I + A2 /C2

)'/7

(: = 90 degrees)

.6 .8 1.08- DEGREES

Figure 14. Dual-reflector system (cassegrain)

27

-50

-60 -

-70

-80

db

-90F

-100

-110

-120

4DEGREES

400

300

200

100

0

0 .2 .4

J-

4.

The results for the dual-reflector system, computed by applying Kirchhofftheory redundantly to the surfaces, are shown as Fig. 14. The degraded second-null is probably due to the sampling intervals chosen for integration of the dis-tributions K1 and K

2 . The 0 domain for Fig. 14 was adequately sampled(AO = 0.01 deg.) to detect the null, if it developed in the computations. This canbe seen by an inspection of the phase characteristic b,9 which is also degradedin the vicinity of the second null. It is a more or less general rule that samplingintervals for sheet current distribution must be smaller for large values of 0 .

The topics of sheet-current-distribution sampling and cpu time were alsoexplored from the standpoint of stationary phase. For sufficiently high fre-quencies the geometric optics solution should be approached and, therefore,one might search for stationarity with the idea of reducing cpu time. The radialand azimuthal parameters, cr and C, used to generate the hyperboloid can besuitably restricted on Y, for a point on y2 at which K

2is to be evaluated. By

this method only a portion of y, is integrated - hopefully all of the stationarypart, to the exclusion of the self-annihilating or non-stationary terms. Thesearch for stationary phase regions was restricted by control over the radialvariable (o-) alone in the initial attempt, but will be repeated with control overC, the angular variable, subsequently, to form increments of area. A typicalincrement of area on the hyperbolic subreflector will then be integrated byspecifying o-i , ma , i, m , and the observer point on ?2 (for that regionof K on Tl) will be determined by geometrical optics. Preliminary results forone particular set of surface and feed parameters indicated that stationary con-tributions ranged over most of T2 when the observer position was in the vicinityof the vertex of T2 . When the observer position was near the edge or limbs of

?Y2 , however, only an annular region of ?y appeared stationary.

A separate study of Cassegrain and Gregorian subsystems consisting ofprime feed ('' and subreflector (y ) was carried out to explore the geo-metrical optics conclusion that such a subsystem is effectively a virtual pointsource in the limit as A - 0. The feed 5I is at the conjugate focus F* of yland the virtual source should appear at F of yl, which also coincides with Fof 'Y2 in the fully assembled system. Since the parameters used in the studieswere not those of Fig. 14, graphical results are not presented here. For ahyperboloid diameter of 24X and an ellipsoid diameter of 32 k it was found thatthe radius or range of observation relative to focus F could be reduced until,for wide angles, the distribution K or yl was no longer optically visible.That is, the Cassegrain or Gregorian subsystems behaved as virtual pointsources situated at F. Amplitude variations on a spherical locus of observationdeparted from a monotonic descent by about ±1 db typically. Phase variationson the same spherical locus of observation varied by approximately +3 electricaldegrees from the phase of a point source situated at F. In conclusion, as the

28

radius of observation is reduced (R - 0) and the frequency is increased (K - 0)the observer does not lie in the near-zone (2D 2 /X) for the conics studied heresince the scattering approximates that from a virtual source and D is effectivelyzero.

Preliminary calculations have been made for the "receive" mode of opera-tion for dual reflectors, applying the theory of Appendix C with near-field inte-grals present. The Airy disc and rings have been observed in the conjugate focalplane for a coarse sampling (LI). A mapping of the focal region is planned for theexample shown in Fig. 14, taking into account the domain of interest in the x-yand x-z planes for a magnification factor M = (e + 1)/(e - 1). The optimumsampling (LI ,, LI2 ) of the distributions K, and K 2 is not known at this time,and several computer runs will be required to establish a reasonable tradeoffbetween cpu time and accuracy of results. The magnified width of the Airy discis a factor here since LI must usually be decreased as the observer moves offof the system axis. Over-sampling of K1 and K2 eventually establishes the"stability" of the solution with respect to LI, observer position, etc.

SUMMARY

The electric and magnetic fields, their associated phases, and the time-average Poynting vector were computed in the Fresnel and Fraunhofer regionsfor several well-known scatterers. Transmit and receive modes were discussedalthough, from a program standpoint, there is no logical distinction. A dual-reflector system was considered, and it appears that even multiple reflectorsmight be treated by the methods presented, however, the blockage (obscuration)problem requires additional effort.

Present plans call for extensive application of the existing program to ground-based and spacecraft antennas. Measured prime-feed data will be injected intothe simulation and distorted antennas of various types will be treated. A varietyof methods for obtaining directive gain and spillover will be compared and docu-mented. Long-term plans include studies of multibeam polarization-diversityand frequency-diversity systems, shaped reflector and Schwarzschild configura-tions, parabolic cylinder antennas, spherical reflectors, and doubly curvedsurfaces. This document is an interim report.

29

ACKNOWLEDGMENTS

The author acknowledges input from the classical literature, periodicals,and journal papers. In addition, he has received help in the form of discussionsand correspondence with numerous interested persons. Mr. Raymons Miezis

of Programming Methods, Inc. provided most of the programming support over

a period of several years and has succeeded in transforming the theory into auseful, working program. Mrs. Rose Kirkpatrick, of the same company, pro-

vided programming support and simplifications using assembly language. The

author is also indebted to Mr. Anthony F. Durham, Branch Head, Antenna Sys-tems Branch, Network Engineering Division, for helpful advice and discussions,

and to Mr. Carl Riffe who assisted by developing a program for automatic plottingof the time-average Poynting vectors and made various suggestions, includingalternative integration schemes on the current and charge distributions.

30

REFERENCES

1. Baker, B. B. and Copson, E. T., "The Mathematical Theory of Huygens'Principle," Oxford at the Clavendon Press, 1950, 1953

2. Stratton, J. A., "Electromagnetic Theory," McGraw-Hill Book Company,

Inc., 1941

3. Silver, S. (ed.), "Microwave Antenna Theory and Design," McGraw-Hill

Book Company, Inc., 1944

4. Born, M. and Wolf, E., "Principles of Optics," Pergamon Press, 1964

5. Skolnik, M., "Radar Handbook," McGraw-Hill Book Company, Inc., 1970

6. Sancer, M. I., "An Analysis of the Vector Kirchhoff Equations and the

Associated Boundary-Line Charge," Radio Science, Vol. 3 (New Series),

No. 2, 1968

7. Kaplan, W., "Advanced Calculus," Addison-Wesley Publishing Company,

Inc., 1959

8. Schmidt, R. F., "The Calculation of Electromagnetic Fields by the Kirchhoff-

Koffler Method," X-525-70-293, Goddard Space Flight Center, 1970

9. Rusch, W.V.T. and Potter, P. D., "Analysis of Reflector Antennas,"Academic Press, 1970

10. Kraus, J. D., "Electromagnetics," McGraw-Hill Book Company, Inc. 1953

11. Gniss, vH., and Ries, G., "Feldbild um den Brennpunkt von Parabolreflektorenmit Kleinem F/D-Verhaltnis," Archiv der Elektrischen Ubertragung (A.E.U.),

Band 23, Oktober, 1969

12. Rudge, A. W., Davies, D.E.N., "Electronically Controllable Primary Feed

for Profile-Error Compensation of Large Parabolic Reflectors," Proc.

IEEE, Vol. 117, No. 2, February 1970

13. "The Microwave Engineer's Handbook and Buyer's Guide, Horizon House,

1965

14. Collin, R. E., and Zucker, F. J., "Antenna Theory," Part 2, McGraw-Hill

Book Company, 1969

31

15. Minnett, H. C., and Thomas, B. Mac A., "Fields in the Image Space ofSymmetrical Focussing Reflectors," Proc. IEE., Vol. 115, No. 10, October1968

16. Farnell, G. W., "Calculated Intensity and Phase Distribution in the ImageSpace of a Microwave Lens," Canad. J. Phys., 1957, 35, pp. 777-783

17. Farnell, G. W., "Measured Phase Distribution in the Image Space of aMicrowave Lens, Ibid, 1958, 36, pp. 935-943

18. Bachynski, M. P. and Bekefi, G., "Study of Optical Diffraction Images atMicrowave Frequencies," J. Opt. Soc. Amer. 1957, 47, pp. 428-438

32

APPENDIX A

THE FORMULATION OF THE SCATTERED FIELDS

E(x',y',z') H(x',y',z') BY EQUIVALENT METHODS

The following formulation of Stratton' was used, initially, to obtain fields,time-average Poynting vectors, isophotes and wavefronts

E (x', y', z') = -1 1f

jwE 4vrTc

Vpofld;[- - s [ j Co) (n x H1+) b + ( -El) Vib] ds4 1

H (x', y', z') =- 1

1I

(nx H,) x V0 d s

e- jkr 1- = , V¢7= - j k +-) i r .

r r

An immediate conversion to the formulation used subsequently can beeffected by making use of a relationship in one of Stratton's proofs.2

fs [(nx H)-V) V d s = -f Vr o hl'd t - j e ri (n-E) Vq d s

1 c 1S

From the above, the equivalent form of the elctric field is

I) 1 1 1E (x', y', z')- = [(n x )V V d s - j a /

j o4 e 4 s7T1 Si

The operator VV¢ of Stratton can be avoided by the use of dyadic notation.3 It isnoted that the equivalent form for the scattered electric field was obtained byStratton in the course of demonstrating that the "historical" form of E(x',y',z')

satisfies Maxwell's equations.

1Ref. 2, page 4692 Ref. 2, page 4703 Ref. 8, pages 27-28

33

(nx H1) d s.

The equivalent form of the elctric field is also recognized in the generaldyadic derivation of Sancer. 1 Attention is called to the fact that Sancer con-cludes with the forms of E(x',y',z') and H(x',y',z') of Stratton, and does not dis-

cuss the more compact formulation which disposes of the contour integral, theincident electric field Ei , and its polarization vector ei .

In a recent text by Rusch and Potter,2 the equivalent compact form forE(x',y',z') is presented together with a discussion of open surfaces. Sign vari-ations appear between the formulations of Stratton, Sancer, Rusch and Potter,and Silver.3 These appear to be due to conventions pertaining to the Green'sfunction V and its gradient, V q. In practice, for perfectly reflecting surfaces,the E(x',y',z') form has two integrals which must annihilate the radial fields andthe H(x',y',z') form has only one integral. Signs, therefore, are not a vital issuehere even though easy comparisons are occasionally frustrated by the appearanceof what seems to be a spurious minus sign.

The text by Silver (1949) presents the "historical" form of E(x',y',z')4 andthe compact equivalent form 5 as a specialization from volume to surface inte-gration. 6 The "historical" form evolves when a single surface integral is equatedto the sum of a surface-change integral plus a line-change integral. One mightspeculate on the consequences of a complete transformation of E(x',y',z') to line-integral form7 for open and closed surfaces.

1 Ref. 6, page 142, equation (2.17)2 Ref. 9, page 46-473 Ref. 3, page 1604 Ref. 3, page 1605 Ref. 3, page 1326 Ref. 3, page 877 Ref. 1, page 79

34

APPENDIX B

THE TIME-AVERAGE POYNTING VECTOR

The time-average Poynting vector is defined , 2 as

<P> = Re E (x', y', z') x Re H (a', y', z') =Re E (x', y', z') x H* (x', y', z')2

It is noted that either the real part or the imaginary part of a complex solution(such as E or H) may be chosen at the conclusion of a calculation to representa physical state. For squares and products, however, the real part of the resultis not equal to the product of the real parts, of E and H, say. Details of thederivation of < P > can be found in several textbooks.3

The "working notation" of the digital computer program is the Cartesianvector basis, and the complex aspects of the solution are retained by separatingreal and imaginary components. After integration of the reflector surface thecomputer stores

ExR, ExI' EyR, EyI' EzR, EzI

and

HXR, HxI HyR

, H H Hz

the real and imaginary vector components of the electric and magnetic fields.Then the time-average Poynting vector

P> Re E (x', y', z') x* (', y, )

Re [i (EY H*- E H)+j(E H H* H E + H* EY H)]2 y Y x -

2 Re [ {(EYR + j EyI) (HzR - j HZI) - (EZR + j Ez) (HR j Hyi)}

Ref. 2, pp. 131-1372 Ref. 10, pp. 370.3773 Ref. 2, p. 136

Ref. 3, p. 701

35

1Re

2

[(EyR HR + EYI H - ER HYR - Ei HyI) + j (Ey

IHZR - ER Hz

I

- EZI HYR + EzR HYI)] ( + j ( .....)]+ k [(...) + j (...)]

1[i (EY R HzR + EYI HI - EzR HYR - EzI HYI)

+ j (EZR HXR EZI H -ExR HzR -ExI HZ,)

+ k (ER HYR + ExI Hyi - EYR HxR - EYI HxI)]

gives the time-average power flow in watts/meter2.

36

APPENDIX C

FRAUNHOFER FIELDS FOR DUAL-REFLECTOR SYSTEMS1

OBSERVER (X', Y', Z')

/ E(X', Y', z'), H(X', Y', Z')

K2, qS qt K1, qrlqt

E2 , H2 El ,

n2 nI

(x2' Y2 ' z 2 ) (X1 , Y 1 z 1 )

Figure C-1. Abstract Dual-reflector system

Consider a dual-reflector system as in Fig. C-1. The feed is designated31, main-reflector y

2, and subreflector y,, Associated sheet currents, sur-

face changes, line charges, and normals as given by K, qs, 4q, and n appropri-ately subscripted. The fields due to 31 are proportional to E1, H1, the fieldsdue to T1 are proportional to E 2 , H 2 , and the scattered fields due to ?2 aredesignated E(x',y',z'),.H(x' ,y' ,z').

1 Due to Raymons Miezis of Computer Applications, Inc.

37

Assume initially that every point of T2 lies deep in the Fresnel region of y 1 ,and that the observer is in the Fresnel region of ?2. The general formulation isexamined to determine what integrals must be evaluated.

1 1 1 fjE z E 47T i; [( n2 x H2)'V] V - j blo /z (n2 x H2) ~ d s

E ~·(x', y(x )=- (n x H2) x V H d s. 2

2 S2~~~S 2

H(x', y', z') -4 H) d s.

Now only H 2 is required above, and

H2 (x 2 , Y2 , z2 ) (n x HI) x V2)d s.1

Since the observer is assumed to be in the Fraunhofer region of y 2 here,the transverse fields are obtained from

E(co, e', 95) =- j pc 1 2._ (i 2 x H2 ) q d s

S2

projected onto the basis vectors 19, and ¢,,. H(o,,0', 4') is most economicallyobtained from the auxiliary relationship

Z 0I H(co ', ¢')|

In this formulation only H2 is required andonlytwo integrals, H2 (x2 , Y2 , z2 ) andE (co, 0', 4') are evaluated since the observer is inthe Fraunhofer region, com-pared with four integrals for the general case, observer in the Fresnel zone.

The advantages of the compact equivalent form of the electric field can nowbe seen. If E(x',y',z') were written in the "historical" form

E(x', y', z')- J H 2 d 4 [(j t( 2 x 2 ) Vds

2f V- [(j2 (nxH72)b+ 2 'E2) ds

38

for the Fresnel region of y 2 , E2 as well as H2 would be required for the evalu-ation. But

1 1E2 (x 2 ' Y2' z2)= -· -

c,

V i' d 1 - [ji do)(nlx $1) q + (nl - E1) V+] d s,

which is complicated by the appearance of a contour integral and involves a mix-ture of electric and magnetic fields in the integrands.

39