Scattered radiation from microwave antennas and the design of a paraboloid-plane reflector antenna

SCATTERED RADIATION

FROM MICROWAVE ANTENNAS

AND THE DESIGN OF

SCATTERED RADIATION

FROM MICROWAVE ANTENNAS

AND THE DESIGN OF

A PARABOLOID-PLANE REFLECTOR ANTENNA

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE HOGESCHOOL DELFT , OP GEZAG VAN DE RECTOR MAGNIFICUS IR. H . J . DE WIJS, HOOGLERAAR IN DE AFDELING DER MIJNBOUW-KUNDE, VOOR EEN COMMISSIE UIT DE SENAAT TE VERDEDIGEN OP DONDERDAG 6 JULI 1967

TE 11.00 UUR

DOOR

MOSTAFA SAVED ABD-EL-RAHMAN AFIFI

\ ,

DIT PROEFSCHRIFT IS GOEDGEKEURD DOOR DE PROMOTOR P R O F . D R . m . J . P . SCHOUTEN

To my Father To Raga, Khaled

C O N T E N T S

t

Chapter I. GENERAL REVIEW AND INTRODUCTION

1. General review of the literature 1

2. Introductory remarks 5

Chapter II. SCATTERED RADIATION FROM A PARABOLOID OF REVOLUTION

3. Introduction 9 4. Formulation of the problem 9

5. Incident field from the primary radiator 11 6. Currents induced on the surface of the reflector 13

7. Radiated field 15 8. Cross-polarization 18 9. Effect of reflector gain on the radiation pattern 19

10. Local effect of the illumination-function on the scattered

radiation 21 11. Scattered field around the penumbra (spill-over lobe) . . . 25

Theorem I 25 12. Corollaries 26

Corollary I, "cross-polarization" 26 Corollary II, "position of spill-over lobe" 27

13. First order term in the contribution of a stationary edge

point 28 14. Theorem II, scattered field in the backward direction

(6Q=TI) 29

15. Some illustrative applications 32 16. Experimental setup and practical results 36

Radiation pattern measurements 42

Conclusion 45 17. Design considerations and techniques to reduce the scattered

radiation 46 a. The diffraction ring 51

b. Cassegrainian systems 53

t

Chapter III. SCATTERED RADIATION FROM A PART OF THE PARABOLOID AND ATTAINMENT OF HIGH QUALITY RADIATION PATTERNS

18. Introduction and formulation of the problem 55

19. Scattered radiation in the penumbra 59 20. Scattered radiation in the backward direction 61

21. Some illustrative applications 63 Radiation pattern in a plane of maximum scattered radiation 63

a. Horizontal polarization 64 b. Vertical polarization 65 Radiation pattern in a plane of minimum scattered radiation 69

22. The illumination function of horn-paraboloid antennas . . . 71 23. Configurations of the radiation pattern of horn-paraboloid

antennas 75 24. Measured radiation patterns 78

25. Design considerations 82

Chapter IV. A PARABOLOID-PLANE REFLECTOR ANTENNA

26. Introduction 83 27. Formulation of the problem 84

Spill-over radiation 86 28. Some illustrative applications and comparison with a part of

a paraboloid 87 Plane of maximum scattered radiation 88

Plane of minimum scattered radiation (Superdirective

pattern) 92 29. The illumination function of the reflector 96

30. The performance of the antenna 98 A rectangular construction 98 31. Constructional effects of a circular antenna on the radiation

32. Design considerations 107

APPENDIXES 110 Appendix A. Contribution of a stationary point 110

Appendix B. Contribution of a stationary edge point (Theorem I).111

Appendix C.. Aperture efficiency and gain 114 Appendix C . Spill-over lobe level 115

REFERENCES 116

SAMENVATTING 121

C h a p t e r I

GENERAL REVIEW AND INTRODUCTION

7.

General review of the literature

The microwave region (from 1.2 to 300 GHz) is a transition between the ordinary radio region, in which the wavelengths are very large com-pared with the dimensions of the system components and the optical re-gion in which the wavelengths are excessively small. Lenses and mir-rors, used in the optioal region are employed, as antennas, to he used in the microwave region. These antennas are highly directive, small in size in comparison with those used in long wave practice and can be easily installed and manipulated in a restricted space. These factors contribute greatly to the potential uses of microwaves.

In modern microwave-systems, reflector antennas find wide-spread applications. They are more easy to be constructed and can handle wide range of frequencies by changing the primary feed of the system. This primary feed is a source of a spherical wave generated in the focal point with a given radiation pattern, and transformed by the reflector

surface to a specified wavefront in its aperture. In case of using a paraboloidal reflector, this wavefront is getting the form of a plane wave. Following the simple geometrical picture, this plane wave con-sists of a family of parallel rays'* (Chapt.4) . The main beam would have then a "zero" width. However, this simple picture is modified by the diffraction phenomenon due to the finiteness of the reflector, and the main beam in this case gets a finite minimum widtA'*(Chapt.6) of (X/D),

with a characteristic side-lobe structure. The attainment of the mini-mum main beamuidth, which is concomitant with high gain, is always in-compatible with minimum side-lobe structure. The fundamental approach to an understanding of the radiation pattern, formed by the main beam and side-lobe structure is necessarily based on electromagnetic theory

(Chapts.3,4,5,6)'13'l'*(Chapt.8)'36. The field in the far-zone of the reflector can be formulated with knowledge of the field distribution on any surface surrounding the sources of radiation. This surface can be

Sect.l

taken to be the aperture of the reflector over which the field can be estimated from geometrical optics relations, with rays and wave-fronts'*

(Chapts.4,12). This method is easy to formulate, due to existence of excessive literature^^'^^'23 and known solutions of resulting inte-grals'* (Chapt. 6)'^'' . However, it is not suitable for calculating the far-side and backward radiation due to the approximations used in estimat-ing the aperture field. A better approach, in computestimat-ing the radiation pattern, could be obtained by performing the integration over the re-flector surface^'"* . The use of the metallic surface of the reflec-tor as boundary surface defines more accurately the induced currents, as sources of radiation, and permitting account to be taken of the ra-tio of diameter to focal distance of the reflector. Little deviara-tions from the assumed distribution of currents may affect the amplitude of scattered radiation with small values in some regions of the space, but not the general shape of the radiation pattern. This method agrees with the aperture-method in the forward direction and around the main beam only"*'^. The region of coincidence between the two methods is defined in Sect.9 of this thesis. Induced surface-current density j7 can be ob-tained approximately from the incident magnetic field of the feeder using the simple relation

J = 2n X H, (1.1)

where n is a unit vector normal to the reflector surface. This method makes it possible to introduce corrections to this value due to curva-ture of the surface, to the near field of the exciter or to edge-cur-rents^ . These corrections have been proved to be of minor impor-tance^'^.

Resulting expressions due to this current method are complicated and need involved computation of integrals. SCHOUTEN and BEUKELMAN^ have given an expression for the radiated field from a paraboloid illu-minated by a dipole feed in terms of an infinite series of Bessel func-tions, with a simplified form for the approximation obtained by the aperture method. One year later, CARTER^^ used the computer for the first time to calculate the scattered radiation (formulated also by the

n

Sect.1 3

6 is the angle with the main axis), between the main beam and the pen-umbra. This could give a rough idea about the relation between the rel-ative value of this level and the dimensions of the reflector for dif-ferent values of n. These calculations have been performed at wide in-tervals of the radiation pattern, so that far side-lobe structure could not be cleared out. Four years later, TARTAKOVSKII^ could give better results making use also of the computer. His radiation patterns have been calculated with steps of three degrees, in a trial to investigate the effect of illumination near the reflector edge on the scattered ra-diation. This trial succeeded to illustrate the general character of the radiation pattern but did not deal with the problem of optimization of gain from the reflector for given requirements of the scattered ra-diation. This limitation of the trial was due to the involved

computa-tions which were restricted to special choice of the illumination

func-tion. This work is followed by excessive investigations about

correc-tions to geometrical optics currents''^^ as given by (1.1), without

dinite conclusion about the limitations of the paraboloid. Further ef-forts have been made by KINBER-'^, using the diffracted rays given by KELLER^^, to calculate the scattered field from the reflector. It is interesting that Kinber has notified good agreement between the scat-tered field obtained from the rays of Keller and that computed from the current-distribution method by Tartakovskii^, when the edge of the re-flector is intensively illuminated. If the current method is used also to calculate the diffracted field from a simple metallic disc, making use of the asymptotic expansions of Bessel coefficients, it can be seen that the radiated field, in first order of approximation, is equivalent to rays radiated from the rim of the disc. This current method, used in case of the disc, has shown good agreement with BRAUNBEK s solution and the exact solution for this problem, in most parts of the space, even with a disc of nearly two wavelengths in diameter^^. The applicability of these diffracted rays of Keller has to be restricted to intensive edge illumination. Kinber^^ estimated the minimum relative value of edge illumination, compared with the value at the centre of the reflec-tor, for which Keller's rays could be applied with good approximation, to be 0.3. The author has shown in a previous work^ that this value is controlled by two factors

4

(ii) the steepness of the illumination function (the value of the first derivative) at the edge of the reflector.

The techniques of asymptotic expansions for integrals containing fast oscillating functions, due to the existence of a big multiplica-tion-factor in the exponent, in particular the contribution of

station-ary points, were introduced to this subject to give better

understand-ing of the radiation pattern. KONTOROVICH and MORAV'EV^^ found an ex-pression for the contribution of the stationary point in the shadow of a reflector. They have shown that it cancels completely the direct ra-diator field in the shadow (a similar method is used in Sect.10 of this thesis using the theory of ERDELYI^^(Sect.2.9)), and the scattered ra-diation in this region is, in the first order of approximation, a con-tribution from currents flowing at the rim of the reflector. This tech-nique is used also in unpublished work^^''*""'*'^''*^, as proposed by SCHOUTEN and DE HOOP, to show the effect of the reflector dimensions on the relative level of the scattered radiation. VAN GILS^^ could get a general expression for the scattered field from a paraboloid with di-pole feed using the asymptotic form of Hankel functions, when the inte-gration is transformed to the complementary paraboloid. This general expression is calculated making use of a fast computer^ and compared with the exact expression (as shown also in Fig.15.1). This method is also used by KRITICOS^^'^^ as a powerful tool to get an estimate for the scattered radiation.

All the previous efforts could give some light on the problem of finding a solution^* ^5''*2, for the radiation pattern, in special cases of illumination, without showing limitations for minimizing the scat-tered radiation from the paraboloid. New fields for application of the paraboloid introduced more requirements for higher quality of the radi-ation patterns. Rapid growth for line of sight communicradi-ations demanded the use of different parabolic dishes on one tower to deal with growing traffic. The cross-talk between these dishes necessitates the suppres-sion of far side and backward radiation to more than 30 db below the

isotropic level^^. This requirement is not to be obtained by normal

dishes. Some techniques are used to approach this level using a long diffract ion-ring"* 3''*'* around the rim of the reflector. This

diffrac-Sect.1,2 ₅

tion-ring causes an increase in level of the scattered radiation be-tween the main-beam and the penumbra^^. More improvement has been also obtained by the invention of the horn-paraboloid antenna which gives enough small side and backward radiation. This device has been widely applied in microwave communication systems without enough theoretical background to understand its performance. The diffraction phenomenon from the horn-paraboloid is handled in the literature making use of the aperture method to get an idea about the level of nearby side-lobes to the main beam'*^''*^. The first trial to estimate the spill-over lobe level is done by CRAWFORD, HOGG and HUNT'*^. Their calculations have been performed using the field distribution at the aperture of the feeding horn in the plane of symmetry, with modifications supposed to be caused by the curvature of the reflector edge in a plane perpendicu-lar to this symmetry plane.

The horn-paraboloid has excellent electrical performance regarding the gain, the band w i d t h ^ and the scattered radiation. However it suf-fers from mechanical disadvantages caused by the bulky construction of the feeding horn, especially when it is used as a ground station anten-na for satellite communications'*^. Trials are running to reduce the size of this antenna, without spoiling its electrical characteristics, either by folding the horn'*^ or it can be totally removed and replaced

by a co-ssegrain sj/stem^"'^^ (pp.896). These trials are based only on

me-chanical modifications of the construction maintaining the original form of the reflector. It is interesting to note that the tendency in these trials is to reduce the angle of the feeding horn to get better band-width. This gives mechanical complications because of the required long construction of the feeding horn. These difficulties with the horn-paraboloid are reviewed by SHIN^^ and HUTCHINSON^1, who emphasized the requirement of more developments to find a new system which can re-place the horn-paraboloid with more simple mechanical construction and without spoiling its excellent electrical characteristics.

2.

Introductory remarks

develop-6 Sect.2

ments, in design of microwave antennas, is going faster than the

theo-retical explanations of its electrical performance. This thesis is a

trial to fill out this gap, to formulate theoretical tools which can be used in a handy form for designing microwave antennas and to introduce a new type of these antennas, designed basically on theoretical analy-sis.

The influence of the illumination of the reflector on the radia-tion pattern is described in terms of currents induced on the surface of the reflector caused by the incident field of the primary radiator. The general form of the illumination function can include corrections to the current given by (l.I). This current is composed of two compo-nents, one transversal to the main axis of the reflector and the other

is parallel to this axis. The influence of these two components on the

scattered radiation, in different portions of the space will be ex-plained in detail (Chapt.II). The space is divided into five different regions. These regions are determined by the configurations of the re-flector and its dimensions with respect to the wavelength. The concept of expanding the integral, representing the scattered radiation, in such a way that the term determined by the edge illumination of the re-flector is separated as a zero order term in an expansion followed by higher order terms, determined by the derivatives of the illumination function at the edge of the reflector, which is used in a previous pub-lication , will be explained in more detail to specify the scattered radiation in different directions. An important phenomenon, called the

local effect of the illumination function on the scattered radiation

(even nearby the main beam) is explained in Sect.10. SILVER (pp.194-195) has given also an explanation for such an effect around the main beam of the reflector (including the main beam itself). This phenomenon comprises simply that the scattered radiation in any plane passing through the main axis of the reflector is mainly determined by the il-lumination distribution on the reflector surface at the intersection line with this plane. The scattered radiation at position of maximum

radiation around the penumbra and in the backward direction is also

ex-plained in detail (Sects.11,12,13,14).

The above mentioned items could lead to better understanding of the relation between the scattered radiation from the reflector and its

Sect.2 7 «

illumination function. It could make it possible to prescribe a given illumination function for the attainment of a specified level of the scattered radiation, with a minimum limit determined by the physical

configurations of the system, for a required aperture efficiency of the

reflector. These results, obtained in the form of asymptotic expansions have shown good agreement with those obtained by numerical computation of the exact expressions as illustrated in Sects.15,21. Fast computers,

in the Technological University of Delft and Dr. Neher Laboratory of

the P.T.T., have been used to calculate different radiation patterns,

shown in the above mentioned illustrations, with steps of half a

de-gree, in most of the cases, to study thoroughly the influence of edge

illumination and its steepness, to the edge of tbe reflector, on the radiation pattern. In this way clear understanding for the limitations of the paraboloid could be obtained, especially when the practical mea-surements have shown good agreement with the theoretical expectations. This study is extended further to inspect the radiation pattern, caused by diffraction, from a part of a paraboloid. It is found that using a part from a big paraboloid is better, as far as the scattered radiation is concerned, than a complete paraboloid giving the same gain

(this is notified also in the work of PETERS and KILCOYNE^" about radi-ation from an offset parabola). The theoretical radiation patterns, computed for this part of a paraboloid are compared with measurements making use of a model for a horn-paraboloid antenna. The construction of the horn eliminates spill-over radiation lobes in some directions and increases the level of scattered radiation in other directions (cf. Sect.25). Modifications of the horn construction are performed to mini-mize this scattered radiation, which led to better agreement between theoretical and measured radiation patterns in most of the directions

(some of these modifications have been found later by RUBER and LAUB^^). In this way a better insight in the radiation pattern from this device could be obtained.

It could be seen also from the expression representing the scat-tered radiation from such a part of a paraboloid (cf. Sect.21), that this scattered radiation is decreasing with decrease of the feeding angle for the reflector. The limiting condition is achieved when this feeding angle is reduced to zero. In this case the reflecting surface

8 Sect.2

becomes a plane-reflector, when the calculated radiation pattern of this plane-reflector is compared with that of a part from a paraboloid (giving nearly the same gain),with feeding angle of 20 , the influence of the surface curvature of the reflector on the scattered radiation pattern is clearly brought out (cf. Sect.28). This plane-reflector could be illuminated by a paraboloid, making use of a primary radiator extending from behind the middle point of the plane-reflector. Both re-flectors are combined with special side construction preventing spill-over radiation from the paraboloid edge. This spill-spill-over radiation is compensated by induced currents, having fast phase variations, along the side construction. Most of the radiation caused by these currents is concentrated near the main beam in a plane, nearly perpendicular to the axis of the feeding-paraboloid.

Using this construction high aperture efficiency can be obtained by increasing the illumination to the edge of the feeding-paraboloid. This high efficiency could in this case be combined with smaller level for the scattered radiation in most directions, than in case of the horn-paraboloid antenna. The first side lobe level can also be control-led by appropriate design of the primary radiator. This new construc-tion is flexible in complying with different modern requirements (cf. Sect.32) for antennas to be used in line of sight and satellite commu-nications. It is much smaller in size than the horn-paraboloid antenna and has a simpler construction than the so called "Cassfejrn"^"'^'(pp. 8 9 6 ) ' " .

C h a p t e r II

SCATTERED RADIATION FROM A PARABOLOID OF REVOLUTION

S. Introduction

The paraboloid of revolution, as a reflecting surface, is a famous device with wide-spread applications. It transforms the spherical wave-front generated by a point source in its focal point into a nearly plane wavefront in its aperture plane. The control of the resulting ra-diation pattern, caused by diffraction phenomenon due to the finiteness of the reflector, is a difficult problem treated partially by many au-thors in the last few years.

For modern applications of this device, interest is chiefly di-rected to the far side and backward radiation (around the penumbra and

in the shadow of the reflector). The design of a feeding device for the

reflector to achieve low level requirements, for the energy scattered in these directions was a subject of empirical trials. The problem is always to comply with the contradictory requirements for high gain and comparatively law scattered radiation. No theoretical background was existing to show, in sufficient detail, the limitations of this device. In experimental installations, it was not possible to reduce the scat-tered radiation to the low prescribed levels, required in some special applications, with reasonable modifications in the construction of the antenna^®'^^. The purpose of this chapter is to derive approximate mathematical equations, which describe in a handy form the diffraction phenomenon associated with this important device, and which can easily

be handled.

4. Formulation of the problem

The wavefront for the field launched away from a reflector, de-pends on two factors

10 Sect.4

(ii) the configuration of the reflecting surface.

The incident wave from the primary radiator on the reflecting surface is assumed to have a spherical wavefront, with a radiation pattern de-scribed by a general illumination function. The influence of the re-flecting surface on the radiation pattern can be calculated from knowl-edge of the field distribution on any surface surrounding the sources of radiation. This surface will be divided here into two parts. The first part exists at the opening S of the primary radiator (at the fo-cal plane of the reflector). The resulting radiation due to this part is the previously assumed radiation from the primary feed. The second part is taken to be the surface of the paraboloid itself, because it is easier, due to the boundary conditions, to determine the field compo-nents on the conducting surface of the reflector by knowledge of the incident wave from the primary radiator than on any other surface. The focal point of the reflector is in the centre of coordinates. The focal length and diameter are assumed to be much greater than the wave

length. The points Q on the surface of the paraboloid are considered to be in the far field of the feeding source whose phase centre coincides with the focal point 0, as shown in Fig.4.1. The meaning of symbols used in the following can be seen by inspection in this figure.

Fig.4.1. Configurations of the paraboloid.

inci-Sects.4,5

₁₁

dent field from the primary radiator, the induced currents on the re-flecting surface and the scattered radiation from the system.

5.

Incident field from the primary radiator

The wave launched from the aperture S. of the primary radiator, has predominant polarization parallel to the x-axis. The radiated field is given by the following representation theorem^, at any point Q on the reflector surface

mht) =

(1/4TI) rot{(3/3t) //^ {(nxfl*)/p }

ds\ +

1

(5.1)

- (iMiry) rot{rot //^ {(n>^E*)/p2] ds] ,

where n is a unit vector normal to the surface, which can be replaced

by (-i_ ) in this case, and p. is the distance between Q and points of

integration on S.. Assuming time dependence to be e , tangential components of electric and magnetic field strengths to the surface 5 related by

E

=

Hiv/c)K

(5.2)

and using far field approximations in performing the vector operations, we get

H

=

(3/2X)e^''^^(e~^^^/p){{ixi)

+ i x(i

xi )]

•

— p u; p — p t/

(5.3)

• IJc ^(^.J/)

exp{jkg(x,y,e,,p)} dx dy.

1

The integral in (5.3) is a function of 6 and (f, defining the position of point Q on the reflecting surface, which will be denoted by F.{%,^).

As spherical coordinates are used, (5.3) can be written in the form (dropping the time dependence factor)

12

For simplicity we introduce

F(B,0 = U/2X)(v/e)^

F,(e,*). (5.5)

As p for a paraboloid of revolution is given by

p = 2//(1-cose),

where ƒ is the focal length, then the magnetic field at Q gets the form

H = F(6,.t.)(l-cose)^(i:„sin4, - i^cos,),) (e/y)^ (e""^'^P/2f) . (5.6) — —b —(J)

The factor representing the unperturbed magnetic field strength in (5.6), is contributing to the induced current density on the surface of the reflector. This factor will be denoted by

f(e,<t.)S^(e,<|.)(i-cose)^, (5.7)

which will be referred to as the illumination function of the reflec-tor. It describes, to a certain extent, the field distribution in the aperture, on the assumption that the reflected rays from the paraboloid are all parallel to the s-axis, and the amplitude of the field strength remains constant along the reflected ray.

The direct radiator field, due to the opening S., can be obtained using (5.6) at any point of observation P to be written in the form

E^ = {/(eQ,((iQ)/(l-coseQ)}(-igCos(|.Q - i^sin<^Q) exp(-jfefl)//?. (5.8)

This form, for the direct radiation field, covers most of the practi-cal cases^'1"'11'^3. If the field in the opening of the feeder, has another polarization component along the zy-direction, similar terms can be introduced by superposition to (5.8). A general expression for the radiated field due to x and !/-components is given by SILVER'*(Chapt. 10, equation 5 ) . The term corresponding to j/-components in our notations

is

Sects.5,6 13

E^= {ƒ, (9Q,<t>g)/(l-coseQ)}(-igSin*Q+i^cos(t)g) exp(-jfefl)/i?, (5.9)

where ƒ, (6 , ()i ) is a function similar to that given by (5.7), but it corresponds to the distribution of y-components on S .

8. Currents induced on the surface of the reflector

Calculations based on using the current-density distribution on the surface of the reflector, to be obtained on basis of geometrical optics, are expected to yield good results taking into account that the dimensions of the reflector are large compared with the wavelength. If the incident field at Q is assumed to be reflected locally as though an infinite plane wave along OQ were incident on and reflected by the tan-gent plane at Q, we find for the induced linear current density J on such a reflecting surface of infinite conductivity^

J" = 2n X ^, (6.1)

where ^J is a unit vector normal to the surface of the reflector as shown in Fig.6.1., which is given by

• - a

14 Sect.6

n = -i sin(|9) - i^ cos(56), (6.2)

— —p —e

and H_ is the magnetic field strength of the incident wave at Q.

Substitution of (6.2) and (5.6) in (6.1) yields

£ = 2(e/y)^ /(9,*){i sin(5e) +i cos(i9) cos*} (T^^'^I2f. (6.3)

We notice that this current-density vector on the surface is al-ways parallel to the xos plane. This is of practical importance, be-cause the absence of transversal components parallel to the i/-direction minimizes radiation with cross-polarization.

Corrections for the current-density are due to the curvature of the reflecting surface, to near-zone field of the feeder and to cur-rents flowing at the edge due to the finiteness of the reflector. These corrections have been treated by different authors. TARTAKOVSKII and TANDIT^ have shown that corrections due to curvature of a parabolic cy-linder are equivalent to a constant phase shift of ±(l/4fe/) radians in the current-distribution on the surface. KINBER^ used the condition of shadowing behind an infinite reflector to get an expression for these correction currents. This "condition of shadowing" is given by

-s " ^ ^ " ('7''2^)(i: =* (ii^Z)} = 0, (6.4)

where JI is the Hertzian vector-potential due to the correction current-densities given by

n, -

!jg£^(.e~^'^/r) dS.

(6.5)

E is the scattered field due to the current-density given by (6.1) and

Ej is the direct radiator field. The expression obtained for the cor-rection current £ may be interpreted as follows. The geometircal-op-tics current-density compensates the field E-, in the shadow region to

-1

within (fep) . The rest of (E_ +Ej) is compensated by the current J to within (fep) . This means that these correction-currents are in the or-der of the currents induced by the second oror-der terms of the exciter

_2 field, which are proportional to (fep)

Sects.6,7 15

Edge correction-currents due to the finiteness of the reflector, have been estimated also by Tartakovskii and Tandit^. They have found that the effective zone of action of the edge effect, bordering the re-flector contour, is of the order of tenths of a wavelength. The distri-bution of the edge current along the reflector contour basically

fol-lows the distribution of unperturbed currents, differing locally, how-ever, in phase and polarization.

SILVER^ has given a survey for the problems of induced currents on big reflectors with reference to the work of PLONSEY^. The analysis done by Plonsey, supported by his experiments, leads to a final conclu-sion for this problem. He has shown clearly that geometrical-optics current-densities given by (6.1) represent with good approximation the induced currents on the surface of big reflectors.

The only remaining factor to be taken in consideration is the

cur-rent distribution on the side of the reflecting surface in the shadow.

This current is excited by the currents bordering the reflector contour over the rim, and must be smaller in amplitude than the edge currents at this rim. The existence of these currents at the shadow will not af-fect the basic first order term of the expansion for the scattered field , which is determined by the value of the edge current itself, Practical observations, as will be shown in Sect.16, have given com-plete satisfaction concerning the negligible influence of these cur-rents.

7. Radiated field

The Hertzian vector-potential due to the current-density e7 at a point of observation P is given by'^'*(Chapt.VIII)

ü = -j(4ire(.)"' j j ^ Jie'^'^""/r) dS, (7,1)

where dS is the elementary area of the reflecting surface of the para-boloid. The radiated electric field at P will then be^'*(Chapt. I)

16 Sect.7

Using (6.3), (7.1) and performing the vector operations of (7.2) in the far field, retaining only the terms containing the factor (1/i?), we get

4 = (-j72X) jj^ /(e,4.)(e"'^'^^P"'''V i?){sin(i9){ig COS<|.Q C O S 9 Q +

- i sinifiQ} -jig sine^ cos(Je) cos<t>} dS. (7.3)

The first term in brackets is due to the x-components of the current-density J. It has precisely the polarization of the scattered field, in the far zone, due to an electric dipole along the x-axis. A similar term would exist if field components along i^-direction were existing in the opening of the primary radiator. The second term in brackets is due to the 2-components of the current density J.

In the following we use the relations

cot(ie^^_) = (0/4/) i ^ < 7 , ' (7.4)

which defines the depth of the reflector,

t

dsf ^-1 cot(ie), (7.5)

defining the radial distance of the projection of point Q in the aper-ture ix-y plane), normalized to the radius of the rim,

ds = fDq[sin(e/2)}~^ t dt d<\,, (7.6)

is the elementary surface area,

p + r = i ? + p - p • ^ - , = — — X T = R + 2/{t;^^^sin^(j0Q) - tq s i n S ^ COS((|)-(),Q) + C O S ^ ( J 9 Q ) } , ( 7 . 7 ) and G^^ (wD/A), ( 7 . 8 )

Sect.7 17

is the field-strength gain of the reflector in the forward direction (with 100% efficiency), which will be used as a large constant in the

integrand and the exponent.

Introducing these parameters into (7.3) yields

E^= -JGq(2i^)-\e~^^/R) J^j^^''fit,i)li^(.coa^Q cosö^ +

2 2 2

- tq cos((i sine^) - i sirnfi^} eKp{(.-3G/q){q t s i n ( i S ^ ) +

- tq s i n e ^ COS(<!)-()|Q) + cos'^(JeQ)}} t dt d*. ( 7 . 9 )

In (7.9) f(t,^) has replaced /(9,i(.) of (7.3), where t is related to 9 in (7.5). In this case the illumination function, as represented by

f(.^>^)t gives directly the approximate field-strength distribution in the aperture of the reflector. The total field can be obtained by add-ing the direct radiator field, as given by (5.8), to the scattered field from the reflector, given by (7.9), then

E = E + E, - (7.10)

— — s —d'

The second term (£,) in this form is a function determined by the de-sign of the primary radiator. The subject of this study is to prescribe this function to satisfy special requirements for the total field E. An important part of this function is included also implicitly in the in-tegrand representing the first term of (7.10). This leads us, in combi-nation with the form of (7.9), to inspect an integration having the general form (ignoring the factor exp (-j'fei?)/if)

Eg = (-JGq/2-n) J Q ' J Q ^ ' I ^ * ' * ) exp{(-jG/<7) *(t,*)} t dt d^, (7.11)

where

*(*,*) = ƒ(*,*) {ig(cos<(.Q C O S 9 Q - tq cos4> sine^) - i sin^.^}, (7. 12)

is a vector function determined by the distribution of current-density on the reflecting surface, induced due to direct radiation from the

18 Sects.7,8

primary source (this function can take any other form due to the exis-tence of other polarization components in the field of the primary ra-diator or deviations of the current-density from the value assumed in

(6.3)), and

*(t,4>) = q^t'^ sir^Vi^Q) - tq sing^ cos(4.-i)>g) + C O S ^ ( | 9 Q ) , (7.13)

is only related to the configuration of the reflector.

8. Cross-polarization

The main polarization of the radiated field from the reflector, due to the configurations of Fig.4.1, is parallel to the a^-direction. Components of the scattered field in the transversal y-direction give undesirable contribution to radiation from the system. These components contribute to so-called aross-polarization which is given by the fol-lowing form

^a "^s'^y = (•?'^/^^) /o'/o^" fit,<^){sinhQ sin2*Q +

+ tq sin29 sinij)- cos*} exp{(-jG/^)*(t,<|))) t dt d<^. (8.1)

This field, in (8.1), vanishes along the main axis of the reflector (9_=0). The position of the first big lobe, nearby the main beam at an angle 9^, , is partially determined by the illumination function ƒ(*,<()). However, this lobe is wider than the main beam and its position can be determined approximately from the following formulation.

In case of uniform illumination {/(t,i(i) = l }, when the quadratic term in t, in the exponent of the integral of (8.1), is neglected for small values of 9-, the cross-polarization will be given by

E^ = UGq/2) sin24)Q{sin 9 Q ( G sinSp)"' J^(G sing^) +

+ (Jq/2) sin2e.(G s i n e j " ' JAG sin9„)}. (8.2)

Sects.8,9 19

It can be seen in (8.2) (also due to SILVER'*(Sect. 12.5) , JONES^" and KINBER^^) that maximum cross-polarization exists in the planes given by (j).=±45 . For small values of 6-, the first term in (8.2) can be ne-glected in comparison with the second term because the multiplication

2 2

factor in the former (sin 9.'^'9_ ) is very small with respect to that (sin29f,'\'2e_) of the latter. The position of the first big cross-polar-ization lobe can be determined then from the maximum value of the sec-ond order Bessel coefficient. In this case

^Om '^ ^^''^^' ^^-^^

which means that big cross-polarization lobes are around the main beam at an angle approximately equals to (X/D) (main beam width). The main contribution to these lobes is due to the second term of (8,1) which is caused by the s-components of the current distribution on the surface of the reflector. This term contains the variable t , which means that

currents nearer to the reflector boundary contribute more to these cross-polarization lobes. It can be seen also that this radiation is proportional to the depth of the reflector through the factor q in this

second term.

9. Effect of reflector gain on the radiation pattexm

The scattered field as given by (7.9) is influenced by the big constant G existing simultaneously in the integrand and exponent. The form of the radiation pattern is mainly determined by the value of *(t,((i) which is a function of 9- and ()>_ for the point of observation P.

The relative level of the scattered radiation with respect to the maxi-mum radiated field along the main axis of the reflector is mainly de-termined by the illumination function as given by f(.t,<^). The subject of this section and the following sections is to describe the radiation pattern in different portions of space (as indicated in Fig.9.1).

Around the main axis of the reflector (in region 7) *(t,((i) gets the following approximate form

20 Sect.9

*(t,((i) = l-t(7sin9 cos(.(>-<jif.). (9.1)

Fig.9.1. Space regions.

If the condition for the applicability of this relation is stated so that the maximum phase error in the exponent of (7.9) is (TI/2), then

sin(9Q/2) = a/D)(2f/X)^

(9.2) = (main beam width)(2f/X)S

For small values of 9. this condition will lead to an approximate width for region 7 which can be given by

e J = 4(main beam width)(2//X)^ (9.3)

This means that this simplification can be applied in a wide range around the main beam in case of shallow reflectors (large f/X, for a given diameter).

The function _y(t,((i) can also be simplified in this region 7 by ne-glecting the influence of 2-components for the current-distribution. In this case the radiation pattern around the main beam (region 7) is

de-Sects,9,10 21

scribed by

S.S = (-Ja7/2TT) JQ'JO^'' /(*.'l'){-i^sin,(.Q + iQCOs<|.QCos9Q} •

(9.4) • exp{(jG/q'){tsineQCos(({i-<tiQ) - \)] t dt d^ .

This form can be easily used in calculating the broadness of the main beam and the relative level of the nearby side-lobes'*(Chapt.6 and Sect.

12.3)'^'^1. These simplified calculations have been referred to, in the literature, as the aperture method. The broadness of the main beam'* (Chapt.6) is proportional to (G) , and the relative level of the near-by side-lobes is independent on G.

A simple relation between the scattered field away from the main beam (especially around the penimibra and deep in the shadow) and the illumination-function is difficult to be attained. In a previous work for the author^, this scattered radiation is described as a function of the illumination at the edge of the reflector. In the following sec-tions, this study will be extended to explain in more detail the rela-tion between the scattered field and condirela-tion of illuminarela-tion at the edge.

10. Local effect of the illumination-function on the scattered

radia-tion

To explain this local effect we will use the formulation, follow-ing our work^ in which the function _y(t,i))) is expanded in a Fourier series for one of the variables (<>). This technique is powerful in per-forming complicated integrations of the type we are dealing with. In recent publications this method is now used in evaluating integrals representing the radiation-field in the Fresnel region^^.

The scattered field disappears behind an infinite paraboloid

4 o o • ' ^ = 0- (10.1)

22

E = (JGC?/2TT) S^'"JQ^'' HtA) exp{(-jG/c?) <J>(t,<^)} t dt d^. (10.2)

A Fourier expansion of ^(t,.))) is given by

!(*.*) = Coo C;(*) e-^'"*- ('0-3)

Substituting (10.3) in (10.2) and integrating with respect to * yields

E = jGq exp{(-iG/c7) cos^(i9Q)} J f ï„^_„ (i)" exp(jw<(.Q) •

(10.4)

• C (t) J (Gt sin9„) expl-jGot^ sin^(en/2) } t dt.

—M n u ^ u

The Bessel coefficients can be replaced by an asymptotic form3'*(which is not applicable near the main axis of the reflector when sine« is ap-proaching zero) given by

1 y

^„(a;) = (2/TTa:)'{cos(x-Jmr-iTT)-sin(x-Jn -JTT) { ( 4 M -1)/8x}}. (10.5)

For an illumination function slowly varying with if), the number of terms in the Fourier expansion is restricted to a few important terms. If the

2

smallest term is the N-th so that N << 2x., (x,=G sine^ and x=Gt sing,-.) then

E = (j&?){TTG(sineQ)/2}"^ exp(-JG COS^(|9Q)/^) \"^^=-n '^'^^ '

• exp(jn(tiQ) C (t) cos(Gt sineQ-mr/2-ir/4) • (10.6)

2 2 i • exp{-JGi7t sin (ie^)} t' dt.

Replacing the cosine term in the integrand by exponential terms we get

E

=

(jGq/2)Qi,G

sine^)"^

ST^Jl-N

e^P(j"*o) ^(*> * *

• { exp{ (-jG/q) (c?t sin^eQ-cos^Bg)^ - j JTT } + e'?'"''} • (10.7)

Sect.10 23

We notice that the summation

5:„!_^exp(jn*Q) £ ^ ( t ) , (10.8)

occurring in the integrand of (10.7) defines the illumination function, as given by (10.3), on the surface of the reflector in the plane, 4'°=<l'/-i» corresponding to the point of observation P. Then it can be seen clear-ly that the radiation-pattern at points of observation in a plane <)>p^ is mainly determined by the illumination function on the surface of the reflector at the line of intersection with this plane. The integration in (10.7) is composed of two terms. For positive values of 9^,, the plane defined by 4). of the first term cuts the paraboloid in the upper half of the space. This term is representing, then, the influence of illumination at the upper half of the paraboloid on radiation in the corresponding portion of space. The exponent of this term has station-ary points for values of 6„ determined by the relation

t^ = {cot(.iiÖ^)]/q > I, (10.9)

which defines points of observation in region 2 of Fig.9.1, The second term of (10.7) is representing the influence of illumination on the lower half (ct)=(|>_+Ti) of the paraboloid on radiation in the upper half of space. This can be seen from the existance of the summation

^N J « ( V ^ > C (t), (10.10)

n=-N — n

in the integrand of the second term. The exponent for this term does not get stationary points, because e„ is positive (by definition of coordinates). As the contribution of a stationary point is more impor-tant than an ordinary point, the contribution of the lower part of the reflector to radiation in the upper part of space is of second order of importance.

This formulation has given an idea about the local effect, which means more contribution to scattered radiation from illuminated

por-tions at the nearest part of the reflector surface, to the point of ob-servation. In the following this phenomenon will be discussed in more

24 Sect.10

detail.

The main contribution to the scattered field in region 2 is given by the stationary point, for a value of t given by (10.9). Applying the formula of Erdélyi^(Appendix A ) , for the contribution of this sta-tionary point, to (10.7) we obtain

K = {^n=-/l? ^^P^'^'^^O^ C;^(*g)}/2sin^(ieg) . (10.11)

= {/(eQ,(t.Q)/2sin^(i9Q)}{-_i^sin(t.Q-jtgCOS(t>Q} . (10.12)

The negative value of (10.12) can be obtained if this method of sta-tionary phase is applied for the two variables t and ^ in (7.11) to evaluate this first order term in the shadow (region 4). The asymptot-ic expansion of Bessel coeffasymptot-icients has been used to explain the local

effect because it is easier in this way to evaluate the higher order

term, as given in (10.5). From (10.12), it can be seen that the scat-tered field in region 2, in first approximation, is the direct radia-tion from the feeding source. The higher order terms then represent the scattered radiation from the reflector in this region. These higher or-der terms originate from the edge points of the interval of integra-tion^ ^(pp.5 I ) . It has been shown^, by expanding (10.2) asymptotically, using partial integration, that this radiation is in first approxima-tion, proportional to the edge illumination of the reflector. This for-mulation is applicable to the scattered field in regions 2 and 4. The only difference between these two regions is that the contribution of the stationary point in region 4 cancels the direct radiator field while in region 2 it represents the direct radiator field itself. For this reason much attention was always given to minimize direct radia-tion from the feed in region 2. This direct radiaradia-tion is the first or-der term for the first part of (10.7). The second oror-der term is, then,

of the same order as the first order term of the second part of the equation (both are contribution of the end points of the integration interval). This indicates that the scattered radiation from the reflec-tor, in regions 2 and 4, is caused by edge waves propagating from the two halfs of the reflector. In the concept of Keller, these two edge waves are rays propagating locally from the edges of the reflector in

Sects.10,1 1 25

the plane of interest, connecting the point of observation with the main axis of the reflector. The interaction between these two waves is

the cause of big fluctuations in the radiation pattern in 2 and 4. This interaction gives rise to peaks of radiation with relative level of 6 db. above the level of the scattered wave due to one term only of (10.7). This effect of interaction is clearly shown in Sect.23. The scattered radiation in regions 3 and 5 will be described exciplicitly in the following two sections.

7 7. Scattered field around the penumbra (spill-over lobe)

The local effect of the illumination-function on scattered radia-tion in regions 2 and 4 has been explained in the previous section. In region 2 the exponent of the first term in (10.7) is stationary at t=l (when 9|^=9 ) , which is an end point for the interval of integration. The contribution of this stationary end point (on the contrary to the contribution of a stationary interior point as explained for regions 2

and 4) do not yield the direct radiator field or its negative value. It forms a transition between these two values. The resulting radiation has a maximum in this region caused by interaction between the scatter-ed and direct fields which are of the same order. The field around this maximum is changing slowly over a wide angle due to slow variation of

the exponent with 9„. The following theorem will describe the relation between scattered field in this direction and the illumination function at the edge of the reflector.

Theorem I

The scattered field in the penumbra (9„=e ) of a paraboloidal reflector, in first approximation, is half of the direct radiator field in this direction. Higher orders of approximation are proportional to the derivatives of the illumination function with respect to the vari-able t at the edge of the reflector (t=l), divided by the square root of the field-strength-gain (G) raised to a power equal to the order of the derivative, and multiplied by an asymptotic series, the first-order

26 Sect.11

term of which is unity. An explicit form for this field is given by

^r, = ^ £ . + .S^^Uje '''*'') (2TiGsin9_) h +

(11.1) + (21) '_E;^^{8Gc7sin^(^eQ)} ' + 0(G ^^h

where £,, is given by

E^^ = {f(tQ,((.Q)/(l-cos6Q)}(-igCos<j>Q-i_j^sin<).Q)e""''^/ff, (11.2)

and t„ is given by (7.5) for points of observation P having 9=9„. For the proof of this theorem the contribution of a stationary end point as given by Erdélyi (pp.52) is used as shown in Appendix B. It can be no-ticed in (11.1) that the first order term is independent of the reflec-tor gain (G), so it contributes to a fixed value for the scattered ra-diation with respect to the isotropic level. The first derivative of the illumination function at the edge is divided by (G) . In the case of scattered field in regions 2 and 4, this first derivative is divided by G. This means that in case of tapered illumination to the edge of the reflector, with high steepness, the influence of increasing the gain (G) on reducing the scattered radiation in regions 2 and 4 is more than its influence in region S.

12. Corollaries

Corollary J, "cross-polarization":

Scattered radiation at the penumbra in cross-polarization, for fo-cal plane reflectors (at ^^=-1*5 ) , is in first order of approximation a quarter of the direct radiator field-strength in this direction. This value is proportional to the square of the depth q of the reflector and given by

%o " -J/(l.<l'o)<?^sin2*. (12.1)

Sect.12 27

The contribution of stationary point at it'=i|'rv (Appendix A) to (8.1) yields

E^ = (j<?/47T)(2TrGsineQ)^e'''^''sin2*Q ƒ J /(t,.}.^) (sineQ+t<7cos9g) •

. ^-jc7Gsin2(i9Q)(t-<7-'cotJeQ)2 ^J ^^_ ^^2.2)

At the penumbra, the first term in (12.2) (which is due to transversal

components of the current distribution) is of more importance than the

second term. Making use of Appendix B to evaluate (12.2) at 9_=9 , yields (12.1) as first order term in an expansion similar to that of theorem I,

Evaluating (12.2), making use also of Appendix A for the variable

t in region 2, we get for cross-polarization, in first approximation,

ff^g = -Jsin^eQsin2(l>Q|È'^| . (12.3)

£• „ is maximum when i)'n=45 . When 9„ is also 45 , the scattered

radia-tion in cross-polarizaradia-tion in this direcradia-tion is

From (12.1) and (12.4) in combination with Sect.8, an idea about the

cross-polarization pattern can be easily obtained.

Corollary II, "position of spill-over lobe":

The position of maximum for the spill-over lobe in region 2 is close to region 2, at an angle

'V' 2sin(j9Q) (Ac?/D)^ radians, (12.5)

from the penumbra (6„=9 ) . This value is affected by two factors, (i) deviation of the wavefront of the primary radiator from the

assumed spherical wave around the penumbra,

(ii) decay of the direct radiator field away from the reflector edge.

28 Sects.12,13

Proof:

Referring to Appendix B, the exponent of (B. 1 ) , at t=\ and 9_=9 , is zero. This exponent changes slowly for values of 9-, nearby 9 , and on the assumption that the integrand changes also slowly around this value, then the phase incremental variation of the scattered field around this direction is given by

Y = G(A9Q)^/4(7sin^(j9Q), (12,6)

which yields

A 9 Q = 2sin(JeQ)(A<?'Y/ÖTr)^. (12.7)

Taking in consideration also that the scattered field in this direction has a reversed phase with respect to the direct radiator field, then the position of maximum radiation in front of the penumbra can approxi-mately be determined by the position at which y is equal to ii, then A9^ at this position will be given by

Ü9^ = 2sin(i9Q)(A<7/ö)^ (12.8)

At this position, the peak of the spill-over lobe is nearly three times the scattered radiation at 9„=9 . When using big reflectors, this

0 cr. 6 6 > value is less affected by the decay of direct radiation away from the reflector edge. This decay will be slow in comparison with the fast phase variation of the scattered field.

75.

First order term in the contribution of stationary edge point

The contribution of stationary edge point is given in Appendix B. The expansion of theorem I is obtained by neglecting higher order terms in the expansions representing different derivatives of the illumina-tion funcillumina-tion at the edge of the reflector. In this secillumina-tion a compari-son is made to get an idea about the difference between the value of this first order term and the exact value of the integral, for

differ-Sects.13,14

29

ent values of (D/A). For this purpose the form (B.14), given by

|j| = I J^l ^Ja(t-l)%i ^^1^ (13_j)

is calculated by a fast computer and compared with the asymptotic value

| j | = (Tr/4a)^ (13.2)

in Fig.13.1. It can be seen in this figure that the error in using the

first order term is periodic in a. It is maximum when a is an even

mul-tiple of T[ ('I' 11% at a=4Tr) and minimum when a is an odd multiple of it

(2.5% at a=5TT) . This error is smaller for bigger values of a (a:=5Tt

cor-responds to a value of i^'^'lOA),

J . 1 . L _

2, 3v 4w 5* 6v a

Fig.13.1. Comparison between the exact form and the first order term in the expansion.

14. Theorem II, Scattered field in the backward direction (9Q=T)

The scattered field in the backward direction, on the axis of a paraboloid of revolution is proportional, in the first approximation, to half of the average value of incident field at the rim of the re-flector. The second higher order term is proportional to the inverse of

o.t

\t\

0.3

30 Sect. 14

the square root of the gain multiplied by

C i t'"(-)'"/'":>t'^o^'"^^*^^t=i' ^'^-'^

Proof:

Using (5.8), (7.9) and (7.10), the scattered radiation at 9 =180° is

2

E = i^{-jGq/2T,) IQ' JQ^" ƒ(*,,)>) e~^^^ t dt d^ *

(14.3)

where /(tfiji^r,) is the illumination function at 9„=ii. In (14.3), it can be seen that the scattered radiation is only due to the transversal components of the current distribution (cf. Sect.7).

Making use of (10.3), (14.2) and disregarding the polarization, (14.3) gets the form

2

E = -ÓGq JQ C^{t) e~^^^ t dt + {CQ(Ö) }/2. (14.4)

Using Taylor-expansion for G-.(t) around t=l as given by

C^it) = G Q ( 1 ) + (t-DG^^'^l) + {(t-l)^/2;}CQ^^^ + ...etc., (14.5)

and integrating by parts yields

E = jGpd)

e'^''^'^

+

iC"/2Gq)h{(2GqAi')h{-C^^^\\)+CQ^^\])

+

- G Q ^ ^ \ I ) / 2 : + +(-)'"mCQ^'"\l)/mI + .,,} + 0(l/&?),

(14.6)

S e c t , 1 4 31

F{aGqh)h = / Q ( 2 a 7 / T T ) ^ ^ - j h x ^ ^ ^ (1^,7)

being the Fresnel integral,

In case of weak illumination at the edge of the reflector (small values of G^(l)), the second order term vanishes if the derivatives of

{C„(t;) },_. are so chosen that the series (14.1) is zero. This is the case of illumination having the form

d-ylt"), (14.8)

as will be explained in Sect.17.

The first term in (14.6) is changing rapidly in phase with fre-quency due to the big exponent Gq, By special choice of the illumina-tion funcillumina-tion, compensaillumina-tion can be attained for the second term by the first one, in a band width of ^ iX/D) 100%.

For practical applications, with optimum illumination, it is pos-sible to get an estimate for the level of the backward lobe due to the first term of (14.6). In case of a reflector with 9 =115 ,

illumi-or. ' nated by a source giving an illumination to the edge which is 5 db less than the illumination at the centre of the reflector, the edge illumi-nation is expected to be about 6 db above the isotropic level. In this case the peak of the backward lobe will be nearly at the isotropic lev-el.

To get an estimate about the width of the backward lobe, it is to be expected that the contribution of the second term of the exponent of

(7.9) begins to be recognizable when maximum phase shift due to this term is approaching JTI. In this case

JTT = - G C O S 9 Q . ABg, (14.9)

and the total width will be given by

2 A 9 Q = (A/Ö), (14.10)

32

75.

Some illustrative applications

A general relation is available to describe the scattered radia-tion from the paraboloid as a funcradia-tion of the field distriburadia-tion near its rim^. This relation is complicated, but clear enough to describe the main factors which relate the radiation in regions 2 and 4 to edge illumination. Even so, it was essential to get simpler expressions in-cluding the field around the penumbra and in the backward direction. These are given in the previous three sections. The purpose of this section is to consider some examples to demonstrate the applicability of the previous expansions,

Regarding the local effect of the illumination function on the ra-diation pattern, as described in Sect.10, an illustration is given in Fig. 15.1. In this figure we compare the numerical calculations (based on (7.2), following the formulation of Schouten and Beukelman^), with the first order term of the expansion. This is obtained (the formula-tion is due to Van Gils^^) by retaining the first order term in an as-ymptotic expansion of Hankel functions which resulted from integration to <t>, followed by partial integration to retain the term corresponding to the edge illumination (neglecting the derivatives) in the plane of interest (fl-plane). This was computed in a previous work^. An electric dipole, perpendicular to the 2-axis, is used in this case as a feeder. The first order term, as can be seen in Fig.15.1, is sufficient to rep-resent the scattered radiation from the reflector in case of strong edge illumination. The lack of higher order terms, representing the de-rivatives at the edge, causes deviations from the complete form in po-sitions of minima, at a level of 45 db below the main beam level (in case of Ö=10A).

Another comparison is shown in Fig.15.2. In this case, two differ-ent illumination functions are used, giving the same illumination to the edge of the reflector in the plane of interest (ff-plane). One of these functions is the dipole illumination, for which the calculated radiation pattern is shown in Fig.15.1. The second illumination func-tion is fictifunc-tions. The polarizafunc-tion and intensity distribufunc-tion over the reflector surface are different from that of the dipole, but the polarization is the same for the two cases in the ii/-plane. The

ficti-S e c t . 1 5 33 Farabolic r « f U e c o r v i t h d i p c l a f*«d B-rl— 0 - 1 0 » F i g . 1 5 . 1 . Parabolic taflacwr

(

34

fit)

=

tK

(15.1)

Substitution of (15.1) in (7.9) and making use of Appendix A to perform the integration to ^, in the ff-plane (<()„= 2ii) , with 9 =^11, yields

E.S " iif (G)^e''«''/(2Tr sing^)^ j j {exp{-jG sin^i9Q(t-cot^9g)^} +

- 3 exp{-jG sin^^9Q(t+coti9Q)^}} t dt. (15.2)

The total radiated field is calculated by numerical integration, using a fast computer for (15.2) in addition to the direct radiator field given by

Ed\ = -i4(c°t^9Q)V(l-coseQ), for | 7 I < 9 Q < T T (15.3)

and

E^2 = -i^(tani9Q)V(l+cosBg), for 0 < 9 Q - h . (15.4)

This direct radiation shows continuity of illumination at the edge of the reflector and decreases away from the rim in the forward and back-ward directions. The resulted radiation pattern is shown in Fig.15.2 in comparison with the calculated radiation pattern with a dipole feed. It can be seen that the levels for the peaks of the scattered radiation in the shadow (region 4 and a part of region 3) are nearly the same in the two cases. In the illuminated part of the space (region 2 and the first part of region 2), the two pattern configurations are the same regard-ing the number and positions of peaks. The scattered field in this re-gion fluctuates around an average value which is the direct radiation from the feeding source. It can be seen that the direct radiation from the source is predominant in this region, as explained in Sect.10. The peak of the spill-over lobe exists around the expected position as can be calculated from corollary II (this expected position is '^ 25 and

Sect.15 ' 35

the position in Fig. 15.2 is 23 in front of the penumbra). The ficti-tious direct radiator field is decaying for directions approaching the main beam causing a shift of 2 for the peak of the spill-over lobe to the direction of the penumbra, compared with the case of the dipole feed. For higher values of (D/X) the position of the peak will be less affected by this decay of direct radiation.

It is interesting to notice that more difference between the two patterns is existing around the penumbra than in the shadow, regarding the peaks of the radiation pattern. This deviation is due to the deriv-atives of the illumination function at the edge of the reflector (the contribution of the first derivative in the penumbra is weighted by 1/G , and in the shadow it is weighted by 1/G). It can be seen also in Fig.15.2, that the fluctuations of the scattered field around the value of direct radiation, in region 2, are higher in case of fictitious feed than in the case of dipole feed. This is due to the difference of rela-tive level for direct radiation in this region. Inspection of these levels indicates that the scattered field in region 2, from the re-flecting surface, is nearly the same for the two cases. It is to be no-ticed also that the alternative deep and shallow minima in region 2 are of the same character as those in region 4. These minima are deeper in the direction of the main beam and the backward lobe (9.=TT). This indi-cates that the dependence of the scattered radiation on edge illumina-tion is of the same nature in regions 2 and 4, which emphasizes the

conclusion, that

the features of the scattered radiation pattern are

mainly determined by the configuration of the reflector and, in first

approximation, by the edge illumination.

The local effect of the illumination function can be seen clearly also, when the computed radiation pattern of the dipole feed in the H-plane (((>-=90 ) , is compared with the pattern in the H-plane ^rr^'*'^ (cf. Fig.15.2). The difference between the levels of scattered radiation in the two cases is 3 db in regions 3, 4 and a part of region 2. This cor-responds precisely to the difference of edge illumination in the two planes of consideration. That difference is diminishing for decreasing angles due to the increasing, value of the direct radiator field in this region.

re-36 Sects.15,16

speet to the wavelength is shown in Fig.15.3. When the diameter of the

reflector is doubled, keeping the same opening angle and using the same feed, the relative level of scattered radiation decreases by the same ratio in regions 2 and 5. In region 2 the average level decreases also by the same ratio, but in region 4 the peaks of the pattern decrease by a ratio of about 3 ('v- 9.5 db) due to the splitting effect of the side lobes as will be seen in a more simple way in Sect.21. The number of side lobes in regions 2 and 4 is nearly doubled.

76. Experimental setup and practical results

Measurements have been made on a laboratory model, using the image

plane method, to verify and inspect the previously mentioned

Sect.16 37

a large conducting plate and illuminated by a nearly plane wave-front emitted from a large aperture radiator placed near the turntable. This large aperture is first introduced in the form of an opening of a big horn with correcting lens as shown in Fig.16.1. Improvements to this construction have been performed making use of a 50 cm paraboloid as another form for that large aperture. The measured model is placed at a distance of 70 cm from the opening of this big paraboloid. The configu-.rations of this measuring setup is shown in Fig.16.2. In this case an

approximate plane wave condition is obtained without using a lens. At the same time, the plane wave condition is expected to exist also around the model above the surface of the turntable. The relative am-plitude and phase of the wave propagating along the plane surface is measured by inserting electric probes through small holes in the turn-table. This construction is shown in Fig.16.3.

The phase measuring equipment is shown in Fig.16.4. The original

idea to build this equipment is due to De Ronde (N.V. Philips' Gloei-lampenfabrieken, Eindhoven). Four detection probes have been inserted

38 Sect.16

Fig.16.2. Second setup.

Sect.16 39

Fig.16.4. Phase measuring equipment.

in the four corners of a square cross-section waveguide (cf. Fig.16.5), in which the reference and measured signals, A and B, propagate in op-posite directions, with their polarizations perpendicular to each

[\

/

/

B

\

4

— - ^

H

40 Sect.16

other. The first two probes are separated from the others by a distance preferably equal to an odd multiple of (A /8) (A is the

guide-wave-9 o °

length). This gives rise to an increase of 90 in the relative phase between the two waves A and B as detected by (1 and 2) or (3 and 4 ) . It can be easily seen that the difference between the detected signals in 1 and 2 is proportional to the cosine of the phase angle between A and

B. Similarly, the difference between 3 and 4 is proportional to the co-sine of the phase angle at 1 added to the extra phase deviation due to the path length 1-3-1. If this extra phase deviation is 90 the angle measurement will be easier than other values for this deviation. Exter-nal correction circuits can be ^dded to simplify the measurement of this phase-angle between 4 and B. For example, connecting (1-2) and (3-4) to the x and y deflection plates of an oscilloscope can give di-rectly a visual measure of this phase angle. An example is shown in Fig.16.6 for the visual path of the spot on the screen, caused by

a. Normal detection. b. Detection with correction. Fig.16.6. Oscilloscopic visual path for phase measurements,

changing the phase of B by moving the turntable carrying the phase de-tection probe. The ellipse corresponds to the case when the path length 1-3-1 is different from 90 and the circle to the case of adding an ex-ternal correction for this difference. The overlap in the circle of

Sect. 16 41

Fig.16.7. Measurements with first setup.

• ^ .

phase distribution

Vlitudt distribution

12 10 a