.AN nIDOOR l-fB:l'nOD FOR liiEP.sURE1·iEtJT OF BACK-.sCATTERHm COEFFICIENT,s

by

David Clarence Roee

A thesis s'J.bmi tted in partial fulfilment of the

requirements of Doctor of Philosophy in the

Faculty of Graduate Studies and Research

Ea.ton Electronics Research Laborat.ory

He Gill Uni ver si ty, Hontreal

April, 1953 TABLE OF CONT&"'IJTS Pace ABSTRliCT Ci)

J.CKHOHLEDm~S Cii)

INTRODUCTION l l A LABORATORY 11EI'HOD l"OR Iv:1EliSUREHENT OF BACK-SGATTERING COEFFICI:E:f-.ITS ••••••••••••••••••••••••••••••••••••••••••• 5

2 LENS INVESTIG~~TIOl'JS 2.1 Lens Requirements •••••••••••••••••••••••••••••••••••••• 9 2.2 Lens Construction •••••••••••••••••••••••••••••••••••••• 10 2.3 Design of the Lens Feed •••••••••••••••••••••••••••••••• 14 2.L.,. Incident Fields and Aperture Fields •••••••••••••••••••• 16 2.5 Heur Fields of Lenses: Theory and l'1easurement •••••••••• 21

3 THEORY OF BACX-SCli.TTERDm THROUGH ~~ LENS 3.1 Statement of the Problem ••••••••••••••••••••••••••••••• 31 3.2 Errors in the Eack-scattering Coefficient due to Diffraction •••••••••••••••••••••••••••••••••••••••••• 31 a. Nuti1atior: and Lens &rors •••••••••••••••••••••••••• 32 b. IJon-uniform Il1wnination •••••••••••••••••••••••••••• 41

4 EXPERIHENTAL ARRAHGEl'Œl'lT .t'tm PREIJIHINARY l>1Ej~URE1vlENTS 4.1 l\.pparatus •••••••••••••••••••••••••••••••••••••••••••••• ~.5 /." .• 2 Preliminary Heasurements ••••••••••••••••••••••••••••••• 49

5 HEASUREHENTS .4JID IYIEASUREivjJ!!NT ~l.NALYSIS •••••••••••••••••••••• 55

6 SUl·'1HA.-qy iùID CONCLUSIONS •••••••• ~ •• • • • • • • •• • • • • • • • • •• • • • • • • • 65

llPPENDIX 1:

DEVELOPt"lENT OF AN .iffiTIFICIAL DIELECTRIC FOR l-'lICROHAVE L~\ SES ••••••••••••••••••••••••••••••••••••••••••••••• 67

BIBLIOGR~œHY ••••••••••••••••••••••••••••••••••••••••••••••• 77 (; \ \~ ,

ABSTR.il.CT

An indoor method for measurement of the back-scatterin,:; of short radio waves is described. The scatterer is illuminated by a quasi-plane

\oJave generated by collimating the field froID a point source by a micro- loJave lens. It is 5hO\-;'}1 that the back-scattering coefficients of metal discs of radii one to six \-Iavelensths can be ffieasurec1 w"Ï th considerable accuracy using 8. lens thirty-three Havelengths in d.i.ameter. Baclc- scatterinz by circl.ùar cylinders also is discussed. Perturbations of the baclc-scattered fields, due to diffraction by the fini te lens aperture are evaluCl.ted theoretice.lly.

T",O neu features of the diffraction fields associated ,·Ii th lenses are explained: the field in the plane of a c.ircular aperture, due ta incident spherical ",aves, is predicted b:{ an empirical eCiuationj the field neé~r the surface of a dielectric lens is explained qualito.tively.

Investizations on an isotropie artificial dielectric, and on a le115 construct.ed from it, are described. The author is indebted to 11.1.3 director, Professor 8- •.ù .• ::oonton for expert guidance throushout t.he investigation. DiscussioDc Hi th

Drs H.E.

]'-fr'. G.C. HcCor:rnick, have he1ped to c1éxify the mat.heI'l.a.tical l,·Jork.

Hany of the 8.ri thr.18tica1 computations Here madA by J1iss E. l1ajor.

Br. V. Avar1aid

The research l,-JaS financed, in part, by the Defence Research Board of Canada through Contract D.R.B. No. 176. A studentship from the

National Research COlIDcil 0f Camlc~ a for the 1952-53 session is e;rate- f ully aclmo',-Iled eed. -1-

INTRODUCTION The investigation of back-scattering coefficients, or radar cross- sections, occupies one part of the broad field of diffraction studies.

When electromagnetic energy irradiates an object, sorne of the energy is absorbed and dissipated as heat, the remainder being distributed in space in a manner that depends on the geometry and electrical constants of the object, and the characteristics of the incident radiation. This modi- fication of the incident energy is referred to as scattering, or diffraction by the obstacle. Since the advent of radar, considerable interest has been stimulated in the nature of scattered fields at that point in space where the incident energy originates. These are called the back-scattered fields; their magnitude, for steady state conditions, is specified by a constant, 0', characteristic of the obstacle. The back-scattered energy in such a measurement is often interpreted as being set up by specular reflection from the scatterer; in general, this interpretation is incorrect.

The back-scattered field which results from a plane incident wave is the type that is of practical interest and in addition, i t is of great theoretical interest. The plane wave field can be obtained experimentally by placing the scatterer at a large distance from the radiating source since the phase variation over a given scattering area is inversely pro- portional to the source distance. Investigations, utilizing this fact have been carried out at YcGill University1,2 and elsewhereJ ,4. For a given size of scatterer, the condition on phase variation also stipulates

1. H.D. Griffiths, "Br.ck Scatterine of HicroltTaves by a Conducting Cylinder", Thesis, McGill Uni versi ty, 1950. 2. G.A. Woonton, D.R. Hay and D.C. Hogg, Report No. 1 on Contract D.R.B. X-27 to the Defence Research Board of Canada; Baton Electronics Research Laboratory, 1951. 3. Ohio State Research Foundation Report No. 302-5. 4. Naval Research Laboratory Reports Nos. C-3460-94A, 1951 and C-3460-lJ8A, 1951. -2- that distance from scatterer to source increase as the YTavelength decreases, therefore, at microwave frequencies, the scattering sites, of necessity, are out of doors. Experience has shown that many difficulties are inherent in outdoor back-scattering experiments. For example, high towers are required in order to avoid multiple path transmission due to reflections from the earth. The towers, in turn, produce undesirable secondary fields that are often dependent on wind and weather conditions. Rain, snow and wind also prevent measura~ent because of their effect on the antennas and apparatus, and because of transmission and reflection difficulties. These objectionable features led to consideration of the possibility of performing back-scattering measurements within the laboratory.

It \.n.ll be seen in the text that an indoor measurement involves several fundamental problems in microwave optics. Briefly, the quasi-plane wave that illuminates the scatterer is produced by the Fresnel field of a lens. Of necessity, the lens must be isotropic wi th regard to polarization and have 10\-1 reflection properties; it must therefore be a dielectric lens. This requirement prompted an extensive investigation of an artificial dielectric that also possessed other desirable properties from the point of view of the back-scattering experiment. A new fonn of artificial dielectric has been tested and a lens const~Qcted from it.

It is necessary to know the nature of the field in \-lhich the scatterer is placed, therefore the problem of obtaining a mathematical expression for the diffraction field of a lens arises. The magnitude of this problem will be appreciated when it is recalled that relatively few rigorous solutions of electromagnetic diffraction problems exist even when no -3- lens is involved. It is important, nevertheless, to have workable solutions, not necessarily rigorous, which ,dll predict diffraction fields wi th an accuracy sufficient that experiments in microwave optics can be perfonned with confidence. During the present investigation, two new features of diffraction fields have been observed. Bath are of importance in deter- mining the behaviour of a lens. One of them has been predicted wi th con- siderable accuracy by an empirical forffiula. The influence of lens action on back-scattering measurements will be evident throughout the thesis.

It is also necessary to have sorne theoretical information on back- scattered fields, considered as a distinct class of diffraction problem.

Scattering by a sphere and an infinite cylinder, for example, can be evaluated rigorously. Moreover, since one is concerned with the distant field in evaluating scattering coefficients, optical approximations are fairly reliable in certain instances. The indoor method was tested by comparing experimental results with these theories.

Errors in the scattering coefficient, introduced by the lens diffraction have been evaluated. Other errors, due to limitations in the technique of microwave optics are also discussed. It is found that agreement between indoor and outdoor measurements exists, for scatterers with simple geometrical shapes, over a limited range of size of scatterer and over a limited angle of back-scattering.

The lens method was suggested by G.A. Uoonton5 several years ago. Recently6, an outdoor back-scattering experiment, utilizing a 1ens of very low refractive index to partial1y correct a spherical wavefront,

5. G.A. Woonton, J.A. Carruthers, A.R. Elliot and E.C. Rigby, J. App1. Phys., ~, pp 390-397, 1951.

6. J.R. Mentzer, Proc. I.R.E., ~, pp 252-256, 1953. -4-

was described by J.R. Mentzer; this will be referred to in section 4.2 of the texte The indoor method to be described has proven superior in sensitivity and stab11ity to equivalent outdoor methods within the knowledge and experience of the author. -5-

1. A LABORATORY NEI'HOD FOR MEASUREMENT OF BACK-SCATTERING COEFFICIENTS.

The back-scattering coefficient is defined in terms of a plane incident field, produced by a source at a large distance from the scatterer. Reciprocally, the scatterer acts like a reradiating element at a large distance from the source. On the basis of geometrical optics, both of these conditions can be met by the method shown in figure la.

A lens is mounted in a thin metallic screen in the plane ZI = O. The screen is assumed infini te in extent so that only radiation which passes through the lens occupies the region z 1 > O. A coherent point source, placed at the focal point, F, will produce a field that is constant in phase over any plane zl = Zo since, to the geometrical optics approxima­ tion, the optical path lengths of all rays from F to this plane are equal. If a scattering object, Sl, is placed in this field, the incident energy will appear to originate froID a large distance zt_~-oO • Reciprocally, only back-scattered energy represented by rays parallel to the optic axis will return to the focus, therefore, viewed from F, the scatterer appears to be at a large distance z t~+oo. One could conclude frOID this reasoning that the back-scattered field, at F, is equivalent to that obtained in an ideal outdoor measurement. This interpretation is complicated by the physical restriction on the ratio, c/À, of lens radius to wavelength at microwave frequencies. The field in the region z 1 >0 is not a plane parallel beam, it is a diffraction field that varies frOID point to point in both amplitude and phase. It is evident from these considerations that the back-scattered energy, as measured by the system of figure la, will not be identical to that obtained in an outdoor measurement.

'1'0 illustrate the laboratory method, a derivation of the back-scattering coefficient will be given on the basis of geometrical optics. The surface FIG. 10

BACK-SCATTERING THROUGH A LENS

METAL SCREEN --t----- c

------:~:::±:::n===z=~-ZlF

1 1 20

FIG.lb

EXPERIMENTAL ARTIFICIAL DIELECTRIC LENS

P3 ~-____-Zl

no 1= 1-016

n02 =1'21 =1 -43 n03

1 1 1 1 1 1 1 1 1 1 1 :~(------f =100)" ._- ~ 1 -7-

power density at the entrance to the lens is given by St ;:: G(Q, ~)Ptl47Tf2 where G(Q, ~) is the gain function and Pt is the power developed in the small antenna at F. The lens changes the direction of power flow and thereby modifies the surface power density. In general, reflection at the lens surfaces and losses in the lens material will be present also.

If these effects and G(Q, ~) are collected into a function K(Q, ~), the 2 surface power density at Sl is St ;:: K(Q, ~) Pt /47Tf , since geometrical optics is assumed. To the srune approximation, the scattered field from

Sl' in the direction -z' through the lens is Sr ;::O"(a)K(Q, ~) St. This energy flow is convergent toward F, resulting in a received power Pr ;:: 4TIf2 Sr ;:: (i't(a)K2(Q, ~) Pt. It will be shown in section 2.3 that, by a suitable choice of constants, K(Q, ~) can be made sensibly independent of~. One then obtains

(1)

If this result is compared with the free space radar equation7, 2 3 Pr/Pt ;:: G~ G' 0 (a)/(4'Tr) R4, where R, the distance from source to scatterer is necessarily large and G is limited by the maximum permissible amplitude variation over the scatterer, it becomes evident that the laboratory method has possibilities of resolving relatively small 00- efficients. The coefficient is written ~'(a) in (1) since it is only roughly the value a'(a) obtained in experiment; the actual scattering process will be discussed in detail in section 3. At this point, a question as to choice of method may arise since it is weIl known that paraboloids are capable of producing quasi-plane fields

7. S. Silver, "Microwave Antenna Theory and Design", Radiation Laboratory Series, No. 12, McGraw-Hill Co., New York, 1949, p. 5. i Y) the Fresnel re::;ion. The reaSOè\ for Tej ectinz them ','las bAcause considerEèble

•• :1;1: interaction bet'..reen them and the scatterer \-ras ant:,cl.pateo-; !!1areover, the fields 8.re di 3tortecl hy the paraboloid feed. ~'~i th the system nf ficure l, interaction \'.rill be substantie..lly mW.l1er since that ~ortion of the back-sca.t teree energ~T that (loes not reach the sma.1l antenm1,. e.t F can be dir3sipatecl in the reIl!nnder of the volu.r:1e Zl < -f • .1.120, the p03sibility of reducin;3 interactio:él by tb.8 fl. ~plic0.tio D of non-reflectinc 1ayers t.o lens surfaces exists.

The uavelen[3t h a t uhich t.he experiment is performed 8houlc1 be a s short as pos sible in the interest of laboré:.toTy s~ Q ce E.no. of practicnl consider5.tions rela,ted ta lons size. The y,!e Eè SUTe!'lents ta be descri bed

'-·rere carried out at El \-lé1vel!:mgtll of 1.25 cm.

8. C.G. Lrmtgomer:r , R.H. Dicke, B. ii. P1.lTCEül, "PrinciplAsof Eicro\!ave Circuits", Re.di::.tion. I.aborc.br2T 88ri88 fTO • [), Ii icGrr.l.\ ;-Hill Co., l:m'l'forl:, 194 8, p. 317. -9-

2. LENS II~TIGATIONS

2.1 Lens Requirements The design of a lens for general purpose experiments in microwave optics and for back-scattering measurements in particular is governed by the following considerations:

1. The lens should be isotropic so that measurement in any plane of polarization is possible. 2. The weight should be small enough to allow the lens to be rigidly mounted in a thin screen. J. Aberrations should be minimized. The rigour of this stipulation depends on the proposed application since odd order aberrations do not influence certain axial measurements. 4. Collimation of the beam in the Fresnel field should be as complete as possible. This ls consistent with a constant amplitude distribution

over the exit pupil when the lens is illuminated by a source of spherical waves at the focus. 5. Reflection at the lens surfaces should be small, that is, the lens should perform its required function of altering the phase of a wave without appreciably disturbing the amplitude. Since condition 1 eliminates the use of metallic path length and parallel plate delay lenses, dielectric types appear most suitable. Condition J implies that the lens surfaces be aspherical, therefore the physical properties of the dielectric should be such that i t can be machined by a lathe, a procedure which can be tolerated because of the relatively large wavelength. Polystyrene and other synthetic resins can be readily machined and in addition, because of their relatively low refractive index, reflections from the surfaces are not too high. -10-

Before the baek-scattering experiment was undertaken, considerable researeh had been done in the Eaton Electronics Research Laboratory on a low-densi ty artifieial dieleetric. This die1ectrie is discussed in detail in Appendix 1. It is sufficient to state here that development reached a stage where incorporation of the dielectric into a lens was feasible. In its final form, the die1ectric constituted a random, but compressible granular substance, the individual granules being composed of microscopic aluminum flakes embedded in a vehiele of Alkyd resin foam.

The fact that the material was light (specifie gravi ty ~ 0.2), had a contro1lab1e refractive index (1 to 1.5) and a re1ative1y low loss tangent

(~ 0.02), suggested that a lens, suitable for the back-scattering measure­ ments could be constructed fro~ it.

Two lenses were used for the investigation. The first, of di&~eter

40À and focal length lOOÀ, was constructed from the artificial die1ectric; a second, 33À in diameter with a focal 1ength of 80À was cut from bulk po1ystyrene. Details of the lenses and measurements in the fields associated wi th them are gi ven in the remainder of section 2.

2.2 Lens Construction. a. Artificial Die1ectric Lens. The granular foma particles of the artificial die1ectric, described in Appendix l, constituted a substance which was analagous to a compressible fluide For this reason it was necessary to provide a container to support the dielectric. The container was a shell i.Jhose inner surfaces had been maehined to the required lens curvature. This arrangement is Shoiffi in figure lb, and forms a meniscus lens. Design of the lens surface depends -11- on the index of refraction, nol' of the material of the shell. StyrofoamÛ was used for the shell container because of i ts 10\.J' refracti ve index, 10,,1 specifie gravity (~ 0.025), and because it had suitable physical properties. The surfaces shovrn in figure lb form one of the simplest aspherical a lenses? This type was chosen in preference to more highly corrected types because the concave form was most suitable for purposes of construction and because the properties of large volumes of the granular dielectric

'.J'ere then unlmovm. :t-'Joreover, i twill be shovrn that thi s type of lens most nearly fulfills condition 4 of section 2.1. The surfaces of the first half of the she11 are concentric about F; the inner surface of the second half is elliptical to correct for spherical aberration at F. One advantage of the granular dielectric is that application of non­ reflecting layers to the lens surfaces is possible. It is weIl knovrnlO thata layer of dielectric, whose index of refraction is the geometrical mean of the indices of the media on opposite sides of th~ayer, that is, n02 = (nolnO))1/2 will reduce reflections at the surface if the volumes occupied by nol and no) are appreciable. The layer must be (2p + 1)À2/4 in "lidth, where À2 is the wavelength in llo2 and p is an integer. ~loreover, a layer of constant thickness does not affect the refracting pro perty of the surface if the radius of curvature is large and the layer is not tao thick. For if il, rI' and i2' r 2 refer to angles of incidence and refraction at the surfaces separating Ua), n02 and no2' Ual respectively then

11: The Dow Chemical Co., Midland, Mich. 9. S. Silver, "Iv'd.crowave Antenna Theory and Design", ibid, p. )92. 10. J .A. Stratton, "Electromagnetic Theory", liT..cGraw-Hill Co., New York, 1941, p. 511. -12-

sin rI = ~ sin il' sin r2 = ~ sin i 2 %2 %1 and since rI '::::t i 2' sin r 2 '::::1 no3 sin il. %1

Layers, 3~2/4 in width, were incorporated into the lens shown in figure lb. Their function is to reduce reaction of the first surface on the antenna at F, and interaction between the scatterer and the second surface.

The most important operation in the construction of a lens from a compressible dielectric is the filling process. Evidence of anisotropy and high sensitivity of the dielectric to compression had been obtained from experiments given in Appendix 1. These results made i t clear that the dielectric must be uniformly distributed in the shel1, without excessive packing, and must be mechanically stabalized with a binder. The completed lens is shovm in figure 2a, rotated 900 clockwise with respect to the schematic of figure lb. The lens was filled in the former position to utilise the concave section as a base. Consistent thickness of the layers was obtained by means of plane templates that were adjusted to the required distance from the surface of the base, and rotated about the axis of the lens. The layer was then baked to the melting point of the binder and allowed to set. This process is described in more detail in Appendix 1. b. Polystyrene Lens. The other lens used for back-scattering measurements is shown in figure 2b. It was designed by 11.H. Chapmanll, using a method due to

L.C. 11artin12, and cut from solid polystyrene; the index of refraction

Il. M.H. Chapman, IIDesign and Experimental Investigation of a Radio Lens", thesis, !1cGill University, 1951 12. L.G. Martin, Phys. Soc. Proc., jQ, pp. 104-113, 1944. -1)-

FIG. '2-a

!~tificial Dielect le Lp.ns

FIG. 2-b

?olystyrenp. Lens -14- of the polystyrene was 1.61. Both surfaces of the lens are aspherical to correct for spherical aberration and coma. 2.3 Design of the Lens Feed. According to condition 4 of section 2.1, the amplitude of the field over the exit pupil of the lens should be constant for maximum collimation of the beam. The most important factors in obtaining this are: the shape of the lens surfaces, losses in the dielectric, and the radiation pattern of the feed. In particular, the single refracting surface of the lans in figure lb, defined by the equation r = f(n13 - 1)/(n13 - cos 0), is such that the power flow, in the +zl direction, increases with O. The relation between the surface power density, at the exit pupil, and Q, is 13 Fl(Q) =

(n13 - cos 0)3/(n13 - 1)2 f2(n13 cos Q - 1). Fl(O) is plotted in figure 3 using the constants given in figure lb. The loss, per free space wavelength of a plane .lave in a dissipative medium is given by!4 27.3 n tan b decibels, where tan b = E1 lE If is the ratio of the imaginary to the real part of the dielectric constant. tan b, the loss tangent of the medium, was about 0.016 for the artificial dielectric. Assuming that each ray incident on the lens behaves like a local plane wave, a 10ss function F2(Q) = 27.3 n t (O),/À. tan D can be formulated. The variable thickness of the

1ens, t(Q), is known. Fl(Q) and F2(Q), expressed in decibels, are added in figure 3, to give the total increase in surface power density with increasing Q. The radiation pattern of the feed must compensate Fl(O) +

F2(9) to produce a constant distribution over the exit pupil. It will be shown in section 4.1 that a horn antenna ls the most suitab1e feed for the back-scatterlng experiment. Nevertheless, since

13. S. Silver, "Microwave Antenna Theory and Design", ibid: p. 393. 14. C. G. Hontgomery, "Technique of Micro'lave Measurements", Radiation Laboratory Series No. Il, l"bGraw-Hill Co., New York, 1947, p. 561. 4

SURFACE 3 POWER DENSITY- DECIBELS ,-'1 2 '0"1 ------1

--j oJ : i H o 4 6 tO '" 12 ::") ~ .. O ',--.) APERTURE ANGLE- e

EQUALIZATION OF POWER FLUX PER UNIT AREA OVER A LENS -16-

FI and F2 are independent of ~, the radiation pattern of the horn must be also if the entire aperture is to be uniformly illuminated. The independence can be closely approximated by increasing the dimension of the horn aperture in the magnetic plane over that in the electric plane unti1 the radiation patterns in the two planes of po1arization are the same. The experimental

H plane power pattern of a horn of length 18.5~ and aperture dimensions

J~ x 2.4À is p10tted as G(Q) in figure J. The E plane pattern agreed to within ± 0.25 decibe1s. The constant K = F1(Q) + F2 (Q) + G(Q) is indicated by a broken line; this is the constant referred to in equation (1).

2.4 Incident Fields and Aperture Fields. In order to determine the action of a lens, the field that illumina tes it must be known. For this reason, G(Q) of figure J was measured at a distance of about 100~ from the feed. The field, Ey a Gl / 2 (Q) is given approximate1y by Ey = F(x) = cos rrx/4c (2) in terms of the co-ordinate x along the magnetic diameter. The amplitude has been normalized to unity. Corresponding phase measurements, shown as open dots in figure 4, are p10tted against x. The theoretical prediction for a point source is shawn on the same graphe The horn approximates a point source and therefore is a suitable lens feed. Recently, it has been established that the type of measuring probe has a marked effect on the result that is obtained when fields near diffracting obstacles are being investigated. The theoretical investiga­ tions of G.A. Woonton15, and the experiments of R.B. Borts16, have shawn

15. G.A. Woonton, "The Probe Antenna and the Diffraction Field", Eaton Electronics Research Laboratory Technical Report No. 17, 1952. 16. R.B. Borts, "The Effect of the Directi vi ty of the Probe Upon the Measurement of Near Field Diffraction Patterns", thesis, McGi11 University, 1952. -:! '7-

FIG. :~

PHASE CORRECTION OF LENS

... ' ...... • • • .' •• • • ...... ,. .", ... • • •• • • • /0> 0 \ jO 0\ 0 o '1f

PHASE - RADIANS \

\ 0 2TT

/ \0

3TT 000. .. EXPERIMENTAL / - THI!ORE TlCAL

411

20 16 12 8 4 4 8 12 16 20 -l&- that a thin wire antenna, equal to or less than one-half wavelength in length, is necessary in order to obtain a reasonably accurate result.

Accordingly, a 1.25 cm half-wave electric dipole was oonstructed to permit examination of the type of field in which the lens was to be placed.

Amplitude measurements were then taken in the plane of the 40À lens aper- ture, wi th the lens removed, and the horn at 100À from the plane of the aperture; the result is shown by the solid line of figure 5a.

It is not surprising that variations appear in the aperture since

C.L. Andrews17 had shown that fluctuations, fairly constant in amplitude, and of spatial period about lÀ in x,existed in the magnetic plane of a circular aperture illuminated by a plane wave. He devised an empirical relation

cos ~ cos y dl (3) which was successful in predicting this field. Here z' and ~ have the significance gi ven them in figure lb; (l is the distance from any point x on the magnetic diameter to an increment dl of the aperture rime y is the angle between l' and x, and k == ZrrA i s the propagation constant. The first term of (3) is the geometrical optics field, that is, the contri- bution of the incident plane wave. The second term can be interpreted in terms of a wave scattered from the aperture edge. Evidently (3) can be written

The significant feature of the measurement in figure 5a i8 the envelope of the fluctuations. In an attempt to establish the physical

17. C.L. Andrews, J. Appl. Phys. 21, p. 761, 1950. -l ~-

APERTURE FIELDS

RADIUS = 20 >..

H PLANE

3 SOURCE DISTANCE- R 0--100 >.

10 Il '2 · '3 00 lJ ~ -1 1 1·~PH~~~7~~89~V\~...... r..~ "" i'\ V J\\.J ~~-:j. .•.:... V -. \

XI>'

'0 Il - 12, '3! b. ~ i'.' • 3 .r..... 4 ~ 5 rVAJ". .. ,".' 'A;'\ ft," _ 2 '·.. V\ -1 ~~~~~~t",~~~Jf~Jf~ï'~~8~~9~~~ri!~;;;rv"r '" v""..., · ~(~\.IXi -2

If) ...J W CD U 3 t.J o 2 XI>. 7 8 9 10 '2 13 " .t!• ••...... :/(\ V r-

Ro= 80>' xI>'

. \0 Il . 12 '3 7~ · .A~ ~-;T-o - J'y V 'rJ V'-" V V ' -20-

significance of this field, it was noticed that a correlation existed between phase measurements such as that of figure 4 and the minima of the envelope. Comparison showed that the minima occurred at intervals

(X2~ - XI~) over which the phase of the incident radiation changed by rr. Andrew's reasoning was therefore extended as follows: Ei of (4) was replaced by F(x) e-jkR where R is the distance from the point source to any point x on the magnetic diameter and F(x) is given by (2). Es becomes

VIL -jkf - F(c) e-J~~. ~ cos ~ cos y dt where F(c) and kRc are the l..../2ir , amplitude and phase of the incident field at the aperture rim, respectively. The physical implication is that the excitation of the aperture edge is a function of the complex value of the" incident field at the edge. Super- position of the resulting reradiated field, Es, with Bi then results in constructive and destructive interference. The solid dots of figure 5a are points predicted by the relation Ey ;;: F(x) e-jkR - LfF(c) ~-jk('p+Rc)cos~ cos y dt (5) 2rr P The result is entirely e~pirical. It is evident that, in general, F(c) sbould be replaced by F(c, ~). In order to check the validity of (5), aperture fields were measured for various distances, Ro, of the source fro~ the aperture plane. Two of these, figures 5b and 5c, show the dependence of the field on Ra, which is proportional to Rc. The values of Ra were chosen such that kRc differs by rr in figures 5a, band c. In figure 5d, the dependence on F(x) and

F(c) is tested. In this case, the incident field was produced by a wave­ guide feed, consequently F(x) had a smaller taper than previously and F(c) was of greater magnitude. On comparison with 5c, an increase in the amplitude of the fluctuations is evident whereas the positions of the -21-

minima remain unchanged. The result has been published by the Journal of Applied Physics18• Recently, S. Silver19 reported measurements similar to those just discussed. A subsequent theoretical investigation by G. Bekefi20 has shown equation (5) to be a limi ting case of a boundary value solution. This solution has also proven to he in good agreement with Fresnel fields, measured by the author, due to incident spherical waves.

The above fields are representative of the type of energy distribution in which a microwave lens is immersed. The relationship of the field to

lens design is as yet undetermined. However, the mechanism by which the aperture fields have been explained has proved useful in discussion of the Fresnel fields of lenses.

2.5 Near Fields of Lenses: Theory and Measurement. To a first approximation, the field produced by an ideal lens, fed by a source of spherical waves at its focal point, is equivalent to the field produced by a plane wave incident on a circular aperture. This approximation is a consequence of Kirchhoffls scalar diffraction theory. It has proven adequate in predicting Fraunhoffer fields at points not too far removed from the optic axis. The case of line source antennas has been treated by R.C. Spencer2l on this basis. Nearer the aperture, in the region Zl approximately equal to four aperture radii, the theory of 22 E. Lommel has had a degree of success. At distances less than this, Lommells theor-J will not predict the field.

18. D.C. Hogg, J. Appl. Phys. ~, p. 110, 1953. 19. S. Silver, Antenna Laboratory Report No. 185, 1952. 20. G. Bekefi and G.A. Woonton, "Microwave Diffraction Measurements on Circular Apertures", Eaton Electronics Research Laboratory Report No. 24/52 . 21. R.C. Spencer and P.M. Austin, "Tables and t~thods for Calculation of Line Sources", Radiation Laboratory Report No. 762-2, 1946. 22. E. Lommel, "Theoretical and Experimental Investigations of Diffraction Phenomen~ at a Circular Aperture and Obstacle'~ English Translation by G. Bekef~ and G.A. Woonton, Eaton Electronlcs ~esearch Lab. Technicil Report No. 4, 1950. -22-

Recently, approximate electromagnetic solutions by G.A. Woontonl5, H.E.J. Neugebauer23 and G. Bekefi24 have been successful in predicting the electromagnetic diffraction field, caused by a plane wave incident on a circular aperture, with considerable accuracy for aIl points, Zl ~O, not too far removed from the optic axis. For example, their theories reduce to Andrew's empirical form (3), in the aperture proper. These theories will be utilised to interpret the Fresnel field of a lens. It is evident that the field must be known since the energy back-scattered froID an object is a function of the field incident upon it. Sorne practical considerations of lens field measurements wi~l be discussed before detailed comparison with theory is given.

An electromagnetic field is specified by two characteristics, amplitude and phase; since both can be measured with good precision, it is useful to examine both properties.

The phase correction to the incident field, produced by the artificial dielectric lens, was measured first. At the exit pupil, geometrical optics predicts a plane phase front as indicated by the solid line at the top of figure 4. It is evident frOID figure lb that the lens shell is too thick to permit measurement very near the dielectric proper. The phase, measured along the surface of the styrofoam shell, is shown by the solid dots in figure 4. In spite of diffraction effects, the average value of the phase approximates the geometrical optics prediction. It will be noted, however, that the phase is asym~etrical with respect to the optic axis.

23. H.E.J. Neugebauer, "A Method for Solving Diffraction Problems by Approximation", Eaton Electron1cs Research Laboratory Technical Report No. 23, 1952. 24. G. Bekefi, '~iffraction of Electromagnetic Waves by an Aperture in an Infinite Conducting Screen", Eaton Electronics Research Laboratory Technical Report No. 22, 1952. -2)-

Further out in the Fresnel field, measurements of phase and amplitude were taken in both planes of polarization. Figure 6 shows a typical measure- ment taken in a plane 48À from the plane of the screen. Asymmetries are marked in this region; a departure of as much as ) deci bels from an average of the values at ± x/f.. occurs for the case shown. The asymmetries are due to inhomogeneities in the lens structure. These, in turn, are a function of the technique of filling the lens and the sensitivity of the dielectric to compression (see Appendix 1). Several different methods of filling were tried to overcome this difficulty. It was established that the asymmetries were not caused by the lens shell.

At still larger distances, Zo ~ llOÀ, the Fresnel field varies less rapidly. Heasurements in thi s region were in good agreement wi th Lommel' s theory. Moreover, asymmetries in amplitude were less than one decibel from an average of the experimental values. This tendency is in agreement with a theoretical treatment by J. Ruze25 concerning the effect of random phase deviations over an aperture on the Fraunhoffer field. One concludes that the near Fresnel field is highly sensitive to the aperture distribution. Symmetry in the Fresnel field requires a higher degree of randomness than does symmetry in the Fraunhoffer field. The effect of these asymmetries on back-scattering measurements will be discussed in section 4.2.

Heasurements had been made previously by H.H. Chapmanll on the field of the polystyrene lens. Ag ai n, good agreement with the scalar theory of Lommel was obtained in the far Fresnel region.

25. J. Ruze, "Effect of Aperture Distribution Errors on the Radiation Pattern ", Antenna Laboratory Memorandum, Air Force Cambridge Research Centre, 1952. FRESNEL FIELD - H -PLANE

Zo = 48 >..

IEVl2 _ODECIBELS

;", l

6 .~ 1

8

PHASE l DEGREES 0 20 0

0 0 0 0 0 0 0 0 0 0 0 ,) 0 0 r.~ 0 0 0 0 -j 0 i

Il 10 9 8 7 6 !5 4 3 2 0 2 3 4 5 6 7 8 9 10 Il XI). MEASURED POWER AVERAGED POWER 000 MEASURED PHASE -25-

It will be shown in section 3.2 that an error, proportional to zo, appears in the fields back-scattered through the lens aperture from a scatterer. For this reason, the near Fresnel field was examined in detail; the measurements are shown in figures 7 and 8. In order to interpret these results, it is necessary to discuss sorne formaI theory of diffraction of a

plane wave by a circular aperture in a plane, conducting screen. The physical mechanism of diffraction by a finite aperture in a con-

ducting screen is not clearly understood. For instance, an aperture in

, a screen sets up a perturbation field in the half spaces on both sides

of the screen due to the fact that the surface of the screen is not com­

plete26• On the other hand, it was shown in section 2.4 that fields can be predicted with considerable accuracy froID the physical interpretation of waves scattered froID the edge of the aperture. It is evident that diffraction of a plane wave by a circular aperture is only a first approx- imation to diffraction by a lens. In the latter case, the symmetry on

the two sides of the sere en no longer exists; moreover, discontinuities

are introduced by the lens medium, and the lens is by no means of the

same order of thickness as the screen. Uorking from the diffraction problem as formulated by Stratton and Chu27, ~{oontonI5 deri ved an empirical solution for the diffraction field of a circular aperture with plane wave incidence. Subsequently, Neugebauer23 ,

and Bekefi 24 arri ved at similar approximate solutions by independent methods •

.lUI three solutions agree weIl wi th experiment20 at all points z 1 ~ 0,

not too far removed from the optic axis. Neugebauer and Bekefils solution

26. B. B. Balcer and E. T. Copson, "The Mathematical Theory of Huygens Principle", Clarendon Press, Oxford, 1950, p. 159.

27. J.A. Stratton, ibid: p. 467. - -: ( -

rI~. 7

POLYSTYRENE LENS FRESNEL FIELD

IEl- OB.

lEi -OB.

4

E.I-OB• .

. . E.I -DII. 5 4 " ,'. - ~- -.-- ,-,_ . _ ------"_ .._ ----_. .- ~ .. _------,--- -- _.-- --- ,- _., "- _. _. .. - _.-

POL YSTYRENE lEI', 'l FR ESNEL FIELD

)( l( J( EQUATION 8 C ;j '- ' MOD'FlfD EOU~TION 8 EXPERIMENTAL

IE./ 08.6 x 4 '\ O~ )( z./)~ = 37 2 ." .>: 0 \ l -2 P ,x ~ -4 X

2 o -2

4 Z./)'· 44 2 o -2

-4

-8

-IO~ __-, ____ -. ____ -, ____ ~ ____~~ __~~ __~~ ____ o 2 4 8 10 12 14 XI). -28-

can be derived from the potential

- L { -jkz 1 l.... f2rr -jkr (c - P'COS i6) cdl

for an incident plane field polarized in the y direction. r is the dis­

tance from a field point (l'" z 1) expressed in cylindrical co-ordinates to

an element cd~ of the aperture rim. The field co:nponents can be obtained

by the relations ! = {j (E IlL) 1/2Ik} V X li y (7)

2 ~;;:: k ]I Y +'QV.I!y (8) }fuasurements in the Fresnel field of the polystyrene lens are given in figure 8 for three values of Zo that are sui table from the point of

view of the back-scattering experiment. They lie wi thin the fourth axia.l

Fresnel zone. The horn feed, described previously, was placed at the focal

point of the lens. The values predicted by (8) are plotted as crosses in

aIl cases.

Two discrepancies are noticeable: first, the amplitude of oscillation in the predicted values exceeds that of the measured field; secondly, the measured values undergo a "phase reversalll in x at about four wavelengths frOID the axis. The first effect was accounted for empirically by weighting

the contour integral in (6) by F(c), the value of the incident field at

the contour, in the same manner as equation (5). The values predicted

by the modified equation (8) are indicated by open dots. The E plane

fields were aL~ost identical to those shown.

In an attempt to interpret the departure from theory for larger values of x/'A, the fini te size of the lens was considered. In figure lb, two -29-

rays R2, RJ' have been drmm to two arbi trary field points P2, Py It is clear that the optical path lengths of R2 and RJ differ, not only because of the relative po si tians of P2, PJ in space but also because the wavelets have passed different distances through the dielectric. Such phase corrections, with refraction at the lens surface taken into account \iere applied to the integre.nd of (6), which is assumed to represent wave­ lets from the contour. Evaluation of the field on this basis did not produce a correction of the required order of magnitude.

\vi th the purpose of e:h.'Plaining the deviation of (8) from experiment still in mind, measurements were taken closer to the lens in the H and E planes. These are shown by the successive pairs in figure 7. It is observed that the two planes of polarization differ markedly for small values of Zo but converge rapidly to the same form as Zo increases. The

H plane pattern of the first pdr, taken at Zo = J.51\, or about l mm. from the lens surface, is of particular interest. A series of rapid fluctuations appear near the optic axis. A detailed analysis of this field showed that the average period of the fluctuations in the x direction was Àm = 0.751\; the wavelength in the dielectric was ~ = 0.7~. ~breover, a wavelet proceeding from the aperture edge, through the lens, toward the optic axis, will have exceeded the angle of total reflection at the lens surface. The field due to such a wavelet is not zero immediately outside the dielectric, for it can be shown28 that a local plane wave, vdth a period Àd in x should appear just outside the surface of a dielectric under these conditions. The observed fluctuations could therefore be a superposition of wavelets scattered from the lens contour.

28. J.A. Stratton, ibid: p. 516 -)0-

The concept of waves proceeding from the diffracting contour only in part resolves the problem of the lens field since many internal and external reflections will modify the field distribution over the surface of the apert1lre. The premise is that the wave scattered directly from the edge is of first order importance.

Despite the fact that the modified fornl of equation (8) does not describe the entire field, it will be used to evaluate one of the per­ turbations on the back-scattering coefficient. This procedure will be valid over the range in wr~ch agreement between (8) and experiment holds. -Jl-

J. THEORY OF BACK-SCATTERING TIffiOUGH A LENS.

J.l Statement of the problem.

The data, presented in section 2, shows that the interpretation of back-scattering through a lens on the basis of geometrical optics as given in section l, is a rough approximation only; the problem is, in fact, one

for microwave optics. The relatively large ratio of lens aperture to wavelength allows one to consider effects, additional to those of geometrical optics as errors, or perturbations. The electromagnetic nature of the

problem must, of course, be taken into account also. Two perturbations

that lend themselves to analytic treatment are: the distortion of the back-

scattered fields by the finite lens aperture, and the effect of the non- uniform field that illuminates the scatterer. It is evident that the

entire system - lens, diffraction screen and scatterer should be considered

as a single diffracting system. Since the present state of diffraction theory by no means permits this, the lens system and scatterer are considered to be divorced. The perturbations are evaluated on this basis. Errors due to limi tations in the technique of microwave optics will be discussed in section 4.2. J.2 Errors in the Back-scattering Coefficient due to Diffraction. The lens method was proposed originally by G.A. -~'loonton29 for measure- ment of antenna patterns indoors. The same author made initial theoretical and experimental investigations on the distortion introduced in the distant field of an antenna byobstructing the Fresnel field of the antenna with e:. a second aperture ~. The second aperture in this case represents a lens

5. (See page J) 29. G.!. Woonton, R.B. Borts and J.A. Carruthers, J. Appl. ?hys., 2l pp 428 - 4JO, 1950. -32-

of infini te focal length. Briefly, the experiments showed that an antenna wi th aperture dimensions of the order lOi. x 10/~ to 20't" x 20i. co1.Ù.d be measured, over a limited angular range, through a second aperture of dimensions 40) .. x L~OÀ wi thout appreciable error. The mutilation of the original pattern was treated theoretically by an interesting application of Fourier transforms.30 The back-scattering experiment is, in some respects, equivalent to the antenna experiment in that the lens aperture, in General, is in the Fresnel field of the scatterer. It will be seen that mutilation is more serious in the scattering problem than in the equivalent antenna problem because the effective phase distribution across the reradiating element is hdce as great. The investigations on mutilation of antenna fields are, at present, being extended by G.C. HcCormick"* to systems \v:i.. th circular symmetry. Recently, Jakes31 has solved a similar problem in connection \v:i..th periscope antennas.

In addition to mutilation, an error is caused by non-uniform illumina- tion of the scatterer. Tt is possible, subject to rather severe approxima- tions, to account for non-uniform illlmdnation and mutilation analytically for scatterers of simple geometry. The mutilation theory for a circ1.Ù.arly symmetric system will be gi ven first. 1-11ch of the formal theory is due to :1oonton and HcCormick; in particular, the approximation for small angles of scattering and the proposed inclusion of lens aberrations into the theory are due to the author. The effect of non-uniform illumination, re1ated to equation (8) of the last section ~dll be given in section 3.2b.

a. Hutilation and Lens Errors.

Referring to figure 9, the surface SI represents a plane metal1ic

30. G.A. ~oonton, J. App1. Phys. 2l, pp 577-580, 1950 t Eaton Electronics Research Laboratory, MCGill University 31. w.C. Jruces, Jr., Proc. I.R.E., kl, p. 272, 1953. -]]- scatterer rotated about the Yo axis by an angle a to the XC axis. The scatterer, assumed to be irradiated by energy from the _Zl direction, produces a scattered field at sorne arbitrary but distant point P(Rl,Q1'~). If the normal to SI is -Y\ , the surface current excited in SI is 2n- x -H where -H is the incident magnetic field at SI. Stratton and Chu32, by a vector formulation of Greenls theorem have shown that the field at any point within a closed surface S that inc1udes no sources is given by

~Rl,Q1,~) = - Zrr f{j CD f-lo (:!! x !!:) 'lr + (~x !:)X~ S + ( 'Y\- • -EI)V'lr- dS -jkr where 'lr = ~ is the Greenls function and the primes indicate the value r of H and -E on S. With reservations as to the size of SI and satisfaction of the boundary conditions at the edge of SI' (9) becomes

(la)

(la) assumes that the scattered field is due sole1y ta the induced current ...... " X HI = 2~x...... H on SI if SI is not too sma.11. Imen wri tten in terms of the co-ordinates (.p" ~,) on SI, (la) becomes

jkRl El = - x _ (R1,Ql,~,a) AÀ seRI f (!! ~

(11)

32. J.A. Stratton, ibid: p. 467. y Y2 Y p o --- ......

1 ' ....l .i-' 1

• d>" l -A:: 1 Z'

-'=:J H •':1 CO"ORDINATE SYSTEM FOR CALCULATION OF BACK-SCATTERING ERRORS '0 -35-

\·[here

It i·!ill be seen that the sicnificant vD.l1J.es :)f a. and Q are fairly s~nall in experiment_ If this approximation is ussd in (11), &nd Yl defineo. by "1'1 =k sin Q" (11) can be '{Tri tten - (12)

- 1~

(13) is

sina. cos .01

The vector symbol has been dropped for the sake of clari ty. Flis independent of YI- In El. similar manner, ~ ce.n be ... rri tten in terms of a set of co-ordinates (R2' Q2) centred at 0 and (1'2' .02) in the plane z '=0_

Thus

uhere F 2 i S l.J.nlcno\m and 32 extends over the entire plane z' = 0 _ The swne approxiP1.atio~s have been useel in deri vinG (14) as \'lere used in deri vine (12).

Both RI and R2 are assQmed larze, that is, P is in the Fraunhoffer reeion of both SI and 32- One can therefore "Tri t.e Q2~ QI =Q and 2 R1:= R2 + Zo cos Q, or R2 = RI - 2 0 (1 - sin2Q)1/~ Rl-zo+zo/2 sin Q. -36-

Substitution in (14) results in 2 jk El (R1,Q,)3',a) = -j 't,IÀ. e- (R1 - zo) - j Zo y /2k G2 (y,a,)3') (15)

sinee Yl ~ Y2 :: y. F2 is independent of y.

Equating (12) and (15), ~(y,a,)3') = ej(kzo - zoy2/2k) G2(y,a,)3') (16)

The explici t exponential factors in the integrands of ~ and G2 ean be expressed as a Bessel series33, thus,

which has a non-zero value for p = n. One can therefore write 00 jn G2n (y,a,.0) = 21Tjn 1 F2n(f2,a) Jn (YI'2) e .01'2 0/' 2 (19) ,p,. =0 since dS2 = ~ 2 df' 2 d )3'2·

33. J.A. Stratton, ibid: p. 372. -37-

Moreover from (16), (17) and (19), l Fl{,Pl' f31' a) Jn(y'pl) e-jn,01 dSl = ej(kzo - Zo-(2/2k) x SI 00

2rT J F2n (f 2,a) Jn (Yf 2)f2 df2 (20), Pa=O by equating coefficients of eim$, einf3•

(20) is in the form of a Fourier-Bessel transform34, F2n can therefore be o btai ned from 00

F2n (1' 2' a) = ~ 1e - j (kzo - p=O x J n (0,P 2) pdB where p =y is a durnmy variable. Finally, from (16), (18) and (19), one obtains

eo r.-.( ft) = e-jzol/2k < "'.L y,a,iU ~--

(21)

If one proceeds to the limi t y-> 0, which is equi valent to a measurement of Elon the z 1 axis, then GJ.. (O,a) = 0 if n1 ° and (21.) reduces to

~ (O,a) =J "'J"" ejzo~2/2k j FI (1' 1,i61, a) Jo (PI' 1) dSl ~:O #=a S,

At this point, the physics of the system is introduced. The function

F2' as represented by the first SUffi of equation (18) is, in fact, the

34. J.A. Stratton, ibid: p. 371. -38-

Fresnel field of the scatterer in the plane z' = O. F2 can be obtained in teTIns of FI' which is essentially the field distribution over the scattering surface SI as has just been shown. The distant scattered field of SI is then re-expressed in terms of F2 on the entire plane 32 at z' = O. If the plane 32, .Thich, up to this point, has been simply a geometrical surface, is replaced by an infini te conducting screen pierced by a circular aperture of radius "e Il about 0, the Hmi ts off 2 are now restricted ta 0 ~1'2 ~ c, since the field F2 is essentially zero on the negati ve side of S2 for f 2 > c if c is not too small. The value of GI(O, cr) obtained by this process is called the mutilated value Glm(O,a).

Upon integration \.;i th respect to p 35, and substitution of (22) into (12), one obtains -jkR ! Co! I Elm (RI,O,a) = k ~/'Mo îl:ï (l'I x H) P2.=o S, -- 2 x sina eik(fl2 +f2 )/2z J (kP1P2) e-jkfICOS~1 o o (23) ~zb for the mutilated scattered field. 22 The integral .ri. th respect to ~ 2 \{as evaluated by Lommel ; (23) reduces to

(24)

35. G.N. Watson liA Treatise on the Theory of Bessel Functions", University Press, Cambridge, p. 395, 1952. -39-

'i-lhere Vo and VI are the Lommel functions. In particular, if SI is a disc scatterer of radius "a", and -H :: Ho- e-jk(Rl + ~ lCOS .01 sin a) is the magnetic field on SI due to a plane wave field e-jkz ', then Elm (Rl,O,a) :: -Elo + EJ.m- (25) where

_~o=-jY).!f.. fj2kRl1- ( nxI1o) A~A,(2kasina).

At:: rra2 is the area of the disc and~, (2ka sin a) :: 2 Jl(2ka sin a). (2ka sin a) E:t.o is the unrnutilated field. The error term

Efm :: j "lA ejk(c2/2zo-2Rl)

x [Vo + j Vll,Pl dPI must be evaluated numerically.

Although the vector nature of the fields has been preserved to a certain extent in the derivation of (25), the basis of the argument is

founded on Kirchhoff' s theory. Thi s means that each compone nt of El.i'U

in (23) is related to a single vector component of current in SI, due solely to the incident plane field. O:ne now introduces a perfect lens into the aperture of radius "c"

in the plane z' ;:; O. The point of observation P(Rl,O) is thereby trans- formed to the focal point F of the lens. As assQmed in section l, the

two systems are equivalent on the basis of geometrical optics.

The practical experiment must deal with imperfect lenses whose aberra-

tions will distort the back-scattered field. In contrast to light optics,

the phase changes introduced by aberrations have an added physical significance at radio wavelengths since the sources are coherent, and the -40-

phase can be measured. L. vo~ Seide136 developed a general series 4 3 w<,p 2' ~2', ) = c40? 2 + c31' f 2 cos ~2

2 2 2 ~ 2 2 + C22) /12 cos "'2 + c20 ' P 2 for the difference in optical path length, due to aberrations, between

a ray passing through the point (/,2' ~2) on the lens aperture and the ray along the optic axis. The Cij go vern the magnitude of the ab3rrations

and' is a function of the field point. 'l'he function~ 'ltl{p 2' ~2', ) can be interpreted as a phase change over the lens aperture21, for instance,

the term inf'23 cos ~2' for a given value of ~2' represents an antisym­ metrical cubic phase variation over the lens aperture diameter ~2 = a constant. TIns is the coma tenn. The other aberrations can be interpreted in a similar manner; the phase variation can be related to the usual optical interpretation.

The following reasoning was used in developing (23): a function F2

over 32 was expressed in terms of FI over 31; 32 was then limited to an aperture of radius "c". A perfeet lens Vias then introduced to transform the point P(Rl,O) to F, the focal point of the lens. If an imperfect lens is used, the funetion F2' which, as previously explained, i5

actually the Fresnel field of SI over 32, is modified in phase by the aberration. The process of limiting the surface 32 to an aperture of radius "c" and introducing an imperfect lens into this aperture should therefore be accompanied by the addition of a phase error 2lr Vi over the À. aperture. }hthematically, this means that F2 should be replaced by

F2' ;:; F2 ej~ VI in equation (19). By this procedure, the characteristics of a practical lens can be introduced into the theory of measurement by means of lenses.

36. H.H. Hopldns, tI~.jave Theory of Aberrations", Clarendon Press, Oxford, p. 53, 1950. -41-

(b) Non-uniform Illumination. The error in the measured scattering cross-section, as opposed to the true free spaee value, due to variation in the field incident on the scatterer ean be evaluated approximately. To do this, the effect of variation in the scattered field

E1. = - j"A ~jkRll (~x~) e-jkflcos.0lsina dSl - RI S, caused by non-planeness in ! will be determined. In section 2.5, one component of a field, derived from the potential

TTy as gi ven by (6), was shown to be in agreement wi th measured values of a lens field over a limited range. Upon integration, the modified form of (6) becomes

Jl(kl'C) J)

(26) 2 2 1/2 where A = (c + z') and the Ji are Bessel functions of the first kind. For vertical polarization,.! can be obtained from l!.. = jk/"Yt Vxrr~~, the result being applicable over the region in which agreement with experiment was reaehed. The veetor operation results in

pie

- ~-jkA eos .0' ,Ç [ p'le (27) .{2 A where te~ms in l/A2 have been neglected. -42-

The co-ordinates,P', l1', z, refer to any point in the space z' ,>0; the unit vectors are defined in figure 9.

In particular, the field at any point (J'l' ~l, zl) on Sl can be obtained by the transformation

Z' = zl = Zo + f> 1 cos Ji]. sin a 2 2 2 1/2 p' == fl [cos ~l cos a + sin ~l 1 tan ~I = tan ~,cos a

If Zo is chosen such that variation of the axial field is not too

great, and the discussion restricted to directional scatterers for v/hich

the significant values of a are small, these relations can be reduced to

Z 1 = Zo + l' l co s ~l sin a

To this approximation, A reduces to 2 2 1/2 - A = (c · + Z1) ~ A + ~ cos ~ sin a where A = (c2 + z02)1/2.

Further, if A is replaced by A in the argument of the Bessel functions which vary slowly "Ji th fi' if Zo is not too small, and also in the ampli­ tude terms of (27), the magnetic field incident on SI becomes

!. : _~ 1{ e-jk(zo + fl cos ~ sin a) _e-jk(A + zo~l cos it1 sin a)

(28) -43-

The validity of the approximations in (28) increases with Zoe (28) is of the form

(29)

where Ha is the geometrical optics field; Hol and HD3 are the diffraction terms.

Strictly, the field given by (29) is that which should be substituted in the mutilation integral (23). If the substitution is made, it will

be seen that the numerical integration becomes prohibitive for practical purposes. The following procedure is therefore adopted: Ho.i-l of (29) is considered an average value of H and is used in (23) to calcuJate

the mutilation effect. This has been done already in obtaining (25).

Conversely, the remaining terms of (29) are treated as though no mutilation were present for calculating the error due to field variation. The error in the scattered field due to non-uniform illumination is obtained from

E:!.f = -j 'h/À. CjkRll (VIX H) e-jk'pl cos ~ sin cr (JO) L Rl S, -- From figure 9, 't1 = - il sin cr + i~ cos cr. Substitution of (28) into (30) - - ~ and completion of the vector product results in

E:!.f = j/À.Rl e-jk(Rl + zo) cos cr {Il - ~ (12 + IJ1} i2 (JI) - ~A- where Il = 1 e-j2kfl cos Wl sin cr dSl is the integral obtained in the S, case of plane wave incidence, 1 e-jkfl cos Wl sin cr 2 =1 S, and -44-

A term in &. sin cr has been dropped in (:31). The effective propagation A constant in the correction tenus is k' = k (1 + zo/[).

The integrals can be evaluated most readily if the scatterer is a

circular disc. For this shape of scatterer, Il = À'~l(2ka sin cr) where A'is the area of the disc,

12 = 2rr f~o (k~fl sin cr) Jo (kelc) 1'1 dfl o A and CI, 13 = 2rrj/c J Jo (k',.ol sin cr) o 12 is a standard integral and 13 must be integrated numerically. (31) can be uritten , , 1 E:I.f = ~O + Eif + ~f (32) ,

1 Il Elf and Eif being the errors due to non-uniform illumination. A similar

analysis for horizontal polarization showed that the error is approximately the sarne as that just derived.

To sum up the mutilation and field variation errors, (25) and (32) are added, discarding the first term, Elo, of (32). The back-scattered field is then given by , , , 1 9. = E:Lo + E:Lm + ~f + Elf

The back-scattering coefficient can be obtained from (33) by complex addition of (33) and the relation37

section 5.

37. K.M. Siegel, Project H.I.R.O., Contract No. Ar 30(602)-9, Willow Run Research Centre, University of Michigan, p. 42, 1952. -45-

4. EXPERIMENTAL ARRANGEï·1ENT AND PRELIMINARY MEASUREi\1ENTS

4.1 Apparatus. The back-scattering experiment was carried out in a laboratory of dimensions 40' x 12' X 9'. The room was bisected by a vertical screen of transverse dimensions 12' x 9', made of 3/4" plywood backed by a double layer of fine mesh copper screen. The screen must be solid since vibrations destroy the resolution of the measurement. A 3' x 3' square hole was cut in the centre of the screen. The screen was recessed along the contour of the hole to allow mounting of the 1/8" thick apertures that supported the lenses. The arrangement, viewed from behind the scatterer is shown in figure 10.

A portion of the styrofoam column on which the scatterers were mounted is also shown in the illustration. The column was accurately machined to prevent eccentricity. The low density foam is desirable, in general, for reducing interaction between the scatterer and the column. MOreover, slightdeviations in a shaft of low dielectric con­ stant will not create an additional error in measurement. Of equal importance is low reflection from the styrofoambase in which the scatterer is roounted. This is the narrow portion at the top of the collli~ in figure 10.

Figure Il ShOHS the continuous wave system used for the back­ scattering measurements. This is one of the moi:f~ practical methods of measuring back-scattering coefficients at short distances if high dis­ crimination is desired between the transmitted and received energy.

The system is built around the waveguide magic tee situated just behind

the horn feed. This device, in reality a waveguide bridge, is capable FIG. 10

JiffraC'tion 3creen and ~atterine Voltune vTi th

Po] ystyrene Lens ann Disc ,:lcatterer in Si tu. 10 NC/S A·F·C· S UNIT

LOCAL OSCILLATOR

POWER SUPPLY 8ALANCING METER

1 KClS 10 Me/S AMlLIfRR RECEIVER J ~ 30 MClS RECTANGULAR POUND RECORDER STA81LIZER .' TRAHSMITTER OSCllLATOR POWER SUPPLY

WAVEGUIDE SYSTEM AND EL.ECTRONIC UNITS FOR 8ACK-SCATTERING MEASUREMENTS -48- of high discrimination bet,.,reen opposite arms. If the bridge is properly matched, discrimination of more than 100 decibels can be obtained between energy proceeding from the transmitting klystron through the horn, and energy in the arm that feeds the receiving detector. This remark applies if the energy is monochromatic since the magic tee is a highly frequency- sensitive element. Thus, in order to obtain high discrimination, the transmitting klystron must be weIl stabilized. The transmitting klystron was controlled by a Pound stabilizer.38 This arrangement afforded dis- crimination of more than 90 decibels for periods of several minutes.

The ultimate controlling element is the reference cavity, Y.

The system is continuous wave in the sense that the transmitted pulse width is long compared to the time of travel of energy over twice the distance from the horn feed to the scatterer; in fact, the energy is square-.-lave modulated at 1 kc/~. In the interest of high sensitivity and low noise figure, a 10 Irc/s recei ver, wi th a band width of about 1 mc/s, was used to detect the back-scattered energy. The receiver feeds a rectangular recorder that is connected by a three speed selsyn system to the rotating mechanism of the styrofoam column. In this manner, a plot of received ener~J, proportional to ~(a), versus angle of rotation, a, i5 obtained.

I.F detection involves the use of a local oscillator. The local osci11ator, in turn, must be highly stabilized to maintain the inter- mediate frequency at the same point on the signal receiver pass band.

The stability must exceed one part in 105 for the receiver ju~t discussed.

38. R. V. Pound, "Frequency Stabilisation of Hicrm.,rave Oscilla.tors", Proc. I.R.E., 12, pp 1405-1415, 1947. -49-

To ensure tbis, a small portion of the transmi tter klystron output is mixed wi th part of the local oscillator output. The resulti:ng inter­ mediéite frequency is fed to an automatic frequency control receiver containing a 10 nefs electronic discriminator. The control voltage, produced by the discrimina.tor, is taken to the reflector of the local oscillator klystron. The local oscillator frequency is thereby locked to the transmitter frequency. The use of a magic tee in this part of the system ensures that none of the A.F.e. reference signal enters the line that feeds the signal detector. The entire system was tested for leakage. A waveguide attenuator '-Jas used to calibrate the system.

Bath klystrons were fed by \.J'ell regulated, 2000 volt power supplies.

A measurement is taken as follows: the waveg~de bridge is balanced

~illtil the null, as indicated by the auxilliéiry amplifier and the oscilloscope, falls below the limit of sensitivity of the detecting system. A scatterer is then placed on the styrofoam column and rotated; the pattern is pro- duced automatically on the rectangular recorder. The sensitivity of the system will be discussed in section 5.

The aperture of the horn vJaS centred in a 6' X 6' layer of absorbing material ta eliminate reflections from the apparatus bebind the horn.

The horn vJaS completely electroformed for good mechanical st abi li ty.

4.2 Preliminary Heasu.rements.

Initial back-scattering measurements were made using the artificial dielectric lens. The first of these, shovm in fig~e 12, is the pattern obtained from a disc of radius 10/.... The ordinates are proportional to

(j' (a). o Aet-c) 1 DEC

6 • Zl F 8 •

12 1 ", :') 1 14 WALL

16-1 PECULAR 18.J EFLECTION 1 !

-::j ~i .:;-) , ., j .' )

324 12 312 336 348 o 24 36 48 ~ _ DEGREES '"'"51--

Some of the difficulties inherent in the indoor measurement are illustrated by this pattern. It had been anticipated that room reflections would influence the measurements, especially in the sense that part of the reflection from the end (+z') wall would be masked by the scatterer. p~ absorbing mediumâ was therefore placed two feet behind the styrofoam col1.Ulln and adjusted for minimum reflection. Figure 12 shows, however, that specular reflection from the side walls of the laboratory plays an important role, as indicated by the prominent peaks at a = ± 450 from the central maximum. An absorbing medium was plaeed along the sides of the scattering volumej this reduced the specular reflections by more than 25 decibels. The specular reflections are not serious in the case of a directional scatterer sueh as the 10/\ dise, sinee they do not influence the cross-section near a = O. Considerable error could be introduced if scatterers of complex geometry were involved. Comparison of subsequent patterns from discs, using the artificial dielectric lens, with predicted values, showed good agreement in the main lobe only.

Asymmetries in the side lobes made quantitative investigation impossible.

In the interest of determining the validity of the theory of section

3, the rest of the back-scattering measurements to be discussed ~~ll be those taken wi th the polystyrene lens. Evidently the diffraction errors will be of a higher order since the lens aperture is smaller (16.5À radius as opposed to 20~ radius for the artificial lens). l~reover, interaction between the lens and scatterer will be more serious.

:K Il Darkflex", Sponge Rubber Products, Shelton, Conn. -52-

It was sbown in section 3 that the back-scattering pattern is a function of Zo, the position of the scatterer. In general, the mutilation effect increases with Zo vThereas the effect of variation of the field decreases over a limited range, due to gradual broadening of the axial

Fresnel zones. In addition, the interaction effect also introduces a dependence on Zoe Evidence from the refractive index measurements of

Appendix 1 indicated that the period of the dependence sbould be 1v'2 in

Zoe Figure 13 ShOHS a set of four patterns, from a disc of radius 1).., taken at successive distances separated by 1v'4. Correlation is exhibited between the first and third, and between the second and fourth patterns. l'bst significant is that the (J' (0) of the two pair differs by about ± l decibel from the average. .~though the functional dependence is maintained to sorne degree, the second side lobes are lOvler in the first and third patterns where ~(O) is highest. It is concluded froID this experiment that 0'(0) can be obtained from the average of such a set. The overall dependence of tr (a) on the position of the scatterer, zo' is discussed in section 5.

The ordinate scale of figure 13 will be used consistently throughout. O'(a) is the back-scattering cross-section relative to the true value of 0'0(0) for a circular disc of radius 5À. This scatterer was used as reference in preference to a sphere or cylinder because the directional properties are such that the predictable errors are not excessive. Mbre­ over, previous experimentsl , 2 had shown that Ki rchhoff theory was acceptable for this size of disc. (7'{a) = y decibels, can be obtained in square metres from the relation

(i" {cr):;; 12.17 antilog y/10 ~') !. .. J

z.·,.,.

211

-12 -10 -8 -6 ,-4 -2 z :3

-12 -10 -8 -6 -4 -2 z 4 6 8 :3

-12 -10 -8 -6 -4 -2 2 oc· -54-

6 Toward the end of the present investigation, J .R. t13ntzer descri bed an outdoor measurement that utilised a styrofoam lens to partially correct for phase. Because of the low refractive index of the dielectric in this case, diffraction effects were ignored. The problem of errors introduced by reaction of the lens on the antenna feed was especially emphasized, and treated in terms of a scattered field from the lens superimposed on the field from the scatterer. It is clear, however, that the effect of the scattered wave from the lens, vievred from the waveguide bridge, is equi valent to a horn wi th a different impedance. The absolute balance of the bridge is affected somewhat39• The author has examined the effect of reaction of the lens on the horn feed by slightly adjusting the position of the feed about the focal point. No significant change in back-scattering patterns \las observed •

.39. C.G. t· bntGO '1le:t'~T , R :~ cli2.tion Laoo!'o.tory, Series lb. 11, ibid: p. 521 -55-

5. NEASUREMENTS AND l4EASURElOOIT ANALYSIS In order to establish the range of scatterer size for vlhich the laboratory method was capable of reliable measurement, a series of dises of radii ]À to 10;"" vIere examined. The first six of these, measured in the H plane, that is, with the H vector of the incident field horizontal, are shown in figure 14. A similar set for the E plane, taken at the same distance Zo = 40'.. , is shown in figure 15. Comparison wi th Kirchhoff theory that had not been corrected for diffraction erram showed fair agreement out to the first side lobe over the range of radii 3;"" to 6;"". For smaller radii, the validity of Kirchhoff's theory is questionable.

It is true, however, that the mutilation error for small scattersrewill be higher since the lens aperture becomes essentially in the Fraunhoffer field of the scatterer. The measurements just discussed were taken at Zo = 40;""; the lens

~eld at this distance, given in figure 8, was fairly constant, especially n ear the opti c ax:i. s.

Figure 16 shows patterns of three dises of radii 4"-, 5;"" and 6;"" placed at Zo = 37;"". The corresponding lens field, gi ven in figure 8, varies more rapidly at this distance. That the total diffraction error fillS decreased is shown in the pattern of the 6À disc; the third side lobe is in evidence. The broken lines are uncorrected theoretical values obtained from Kirchhoff theory. wnere theory agrees with experiment, the broken lines have been merged. A noticeable shift of the side lobes toward a = 0 occurs for the 6;"" dise.

Back-scattering patterns of circular cylinders, 16;"" in length, are gi ven in figures 17 and 18 in the E and H planes. The centres of the IS 25

.,.~De· -50

55

40

-, 1 1 1 ~I. ~12 ~I -4 0 .- il ;2 -II -12 -e -4 0 4 1 .... 12 16 ..... •

O/~o3 . •., " 1

-II -'2 -R - 4 .0 4 8 12 16 -12 -1 -4 o 4 8 0<" 12 do·

5

-'J o" 1 ... :......

~II -'2 ~I -16

BACK-SCATTERING BY C!RCULAR DISCS H PLANE Z.o40). ~ !!

N "- • ~ ~ !! ..... !!! " ; • D • 15 .. ., .. ..

! _. 1 • • <. ... 0 !!! 0 Il .. '" 0 0 0

• •1 .. 1

!!!, III U III !P , 0 • cr ct ..c ..J 0 ~ U • cr .• U N )- CD ~ C C) ..J Z 0.. ir III lU ~ ~ cl !P U • III 1 :w: U ... ., cl CD !! ..... !!! .c. D ... • • 0 11 'IS

.. CD

$ • :!! ~ 6 " b .. ., 0 ., .. !!! N N 0 0 0

! •

• CD

!!! , ~ ~,

~ !' 1 BAO<-SCATTERING BV ClRCULt.R Dises

Z.·37).. "

/ ",\ \

1 1 25 1 1 1 1 1 1 1 1 1

14 4 0 2 4 6 8 10 12 14 0(0

10 12 14 r;!..0 0'- DB·

RADIUS - 0'1>. - 0·040 , 0/>.-0·127

-4 -z o 2 4 -4 -2 o 2 4 c;I.' 0<0

CYLINOER LENGTH - 16>' , 0'1>.- 0·256 0/).- 0·517 o

1 .." ~, , \/ ,II I-I( 'P -16 -14· -,? -10 -8 -6 -4 -2 o 2 4)~ 6 8 10 12 14 16 -~ -~ -10 -8 2 4 6 8 10 12 14 -6 -4 -2 o dO 0<

(j'-08·

, 1 al>' -1·016 01>. - 1·64

.'J H :.;") • ,54 I ..J ,. -J

-'0 -'4 -12 -10 -II -6 -4 -2 o'. 2 4 6 1 10 12 14 • -~ -~ -~ -10 -1 -6 -4 -2 o 2 4 6 8 10 12 14 16 0(. 0(0

?p·~k-~;(',.,t.t(>"'inzr7 C:irc~llnrCyli l"::) Ar'<: ry , r-' E P]):~!!P.';""J :'i·_i .. _ _ ___ . 1• . __ ..

-(-.0-

,~ -1. ' .'

li!

!

0 . ~ li ... ..!:! • .; ..ô \. ! .:. .:. -A 0 -Il

:000- .N :il :;1 ~ :1 :1

r..: ...... 8 c .r·J r- i ~ .~ t.) f-,-:: ~ l C ':.1..... :;- é;. r.- Il A li! " 1 C ! , c.: ! 1

:z: ~ :' l ~ 1 .c Cl ' loi ... ~, ..J !.1 > ,' II: ~: .-1 .r; (-1 • r.. 1 r.. ~ ~ J ...: " ... :. Il :::' t • . • c'-) 'lS 1111 , .!>! l C) . M .. l!i ~ C:- •~ .:. -i .; -Il .; .. CO CO -II ! 1 1 0 1 c.., b j <::. ~ .. 0 • N ... ~ ..... :II i ~ 0

N 1 -61-

cylinders coincided with Zo = 40~ in all cases. The functional dependence on angle of rotation, as calculated from optical theory, is shown by the plotted points. For this purpose, the theory was normalized to the experimental value at cr = 0; the absolute cross-sections (j'(0) will be discussed later. The cylinder radii, "a''', range from 0.01.)... to 1.64À.

The characteristics of the results for the larger cylinders were fairly well maintained in the two planes of polarization. In the case of the smal1est cylinder, which \-las, in fact, a piece of 12 gauge wire, the effect of po1arization becomes extremely noticeab1e. Thus, when the electric vector lies along the \-are, (figure 17), ~ (0) does not differ greatly from that of the cylinder of radius 1v8. The electric vector normal to the wire (figure 18) results in a large departure of (j' (0) from that of the ~ cylinder. Comparison of the two sets indicates the transition from conditions in which polarization plays the most important role to conditions where optical scalar theories can be applied. The cross-section of the 20 gauge wire, in the H plane, was the limit of the sensitivity of the system. It corresponds to a cross- 2 section of about 12 cm •

The solid curves of figure 19 are halves of measured patterns on discs of radius 4À and 5~ placed at a distance Zo = 40À from the lens aperture. The crosses are uncorrected values as predicted by Kirchhoff diffraction theory. Numerical integration was carried out to obtain the error terms of equation (33). The cross-section, computed using (33) is indicated by the open dots in the figure. In general, the predicted errors are of correct sign and of a reasonable order of magnitude. PREDICTED DIFFRACTION ERRORS

x. ~ X KIRCHHOFF THEORY o 0 0 EQUATION ~~ " 0, \ O . 10 \ o

25 ~ j~~- o 2 4 6 8 10 12 14 0 0(

20

o 2 8 10 12 -fJ-

Near the second minimum, in the case of 5À. disc, a considerable discrepancy is evident. This is highly possible at angles whererapid changes in the field are exhibited.

A plot of the absolute cross-sections (/' (0) of the discs and cylinders of figures 14, 17 and 18, is shown in figure 20. The theoretical values for the discs were calculatedusing equation (10); the cross- sections of the cylinders, from a reduced form of the rigorous solution for the infinite cylinder.40 No confusion "Jill develop in reading the

Im'Ter values on the figure if i t is recalled that the dise radii lia", are lÀ., ~, 3À., etc. The calculated theoretical values for the discs are joined by a smooth curve. The theoretical values for the cylinders are joined by straight lines to distinguish between the tvlO planes of polarization. l~tually, these values oscillate. Good agreement is obtained for discs less than 6À. in radius. Large experimental errors are involved for large discs because of the highly directional nature of their scattered fields. The measured values for the cylinders deviate considerably from theory, due in part to the fact that each eylinder occupies a different volume of the field, and in part to a different interaction error in each case. t~reover, the mutilation error for the cylinders will be high due to the non-directional nature of the scattered fields in the vertical plane. No corrections have been applied to the theoretical values of figure 20.

40. D.E. Kerr, "Propagation of Short Radio l';aves", Radiation Laboratory Series No. 13, NcGra"I-Hill Co., NeH York, p.459, 1951 - 6l-

FIr,. 20

MAXIMUM CROSS-SECTIONS OF DISCS AND CYLINDERS

10

0' (0)- 08· + ... , ... ' , ~ .0' ,. ,. J( ,. ,. o ,. ,. ;' ;' * ~ "t' -5 , , /'/~ elReULAR Dises

-10 ,~ ,, /' ~ CIRCULAR CYLlNŒRS-E PLANE ;

1 )( ._ .~. .s -115 }I ~..... ,""",. , _._._._.~~-:.?-_._._._ . _ . - . / + 0-·_·_·-· _.

-20 /,.;;::- . _ . _._~ CI. CUL'. CYLlNOf:RS-H PLANE

,.K,o'i ."". x'· ~ -25 ,·0c,,;,·"'Y- 1 , . 1 1 ,. j+ *' o 0 UNCORRECTED THEORETICAL VALUES -30 1 1 + + E PLANE MEASUREMENTS 1 X X H PLANE MEASUREMENTS Z.-40X -36 1 1 ,1 -40

DISC RAOIUS al>' z 3 4 li 6 7 1 1 1 1 1 1 0 ·2 ·4 ·e ., I.() 1-2 Il.4 Ir.. ,.. CYLINOER RADIUS al>. -65-

6. Sillvrr·L\.RY AND CONCLUSIONS

It has proven possible to measure back-scattering coefficients of plane scatterers of lir~ted size, over a liwited scattering angle, by an indoor method utilising a lens of radius 16.9.... Good agreement "Ii th

Kirchhoff theory has been 0 btained for metal di scs of radius 3/\ to 6/\, measured out the first side lobe of the back-scattering pattern. Agree­ ment is improved at small angles of scattering by taking into account, theoretically, the non-planeness of the incident field and mutilation due to the nnite lens aperture. Comparison of the absolute cross-sections,

(J' (0), of discs measured ut a distance Zo = 40,\ fron the lens aperture

8ho\.[s agreement to vii thin ± 1.5 de ci bels ovel' a rane;e of radii 1/\ to 1A. wi th Kirchhoff theory. Ovel' the ranee of di sc radii 3:'" to 51..., (/' (0) differs by less than ± 0.75 decibels ,Ath l.illcorrected theoretical values.

Scattering coefficients of three dimensional objects, such as circular cylinders, cannot be measured with as good an accuracy as those of plane scatterer8 due to limi tations in the technique of microwave optics. The limitations are caused by the inability to construct suitable dielectric lenses at microwave frequencies and by finite reflections from absol'bers.

Limitations in existing diffraction theories prevent accurate prediction of the diffraction erl'ors associated with three dimensional scatterers.

Evaluation of the indoor method as a practical device for measure­ ment of back-scattering coefficients of scatterers of arbitrary geometry is based on the following considerations. Large lenses with low reflecting properties are required; spherical Luneberg lenses, in which the index of refraction of the lens medi~~ varies continuously, or lenses with non­ reflecting loyers being possibilities. This factor involves renewed -66-

research on dielectrics and microwave lens design. Of almost equal importance are microwave absorbers. Imechoic laboratories that simulate a volume element of free space are a necessity. Sorne simplification may occur if scattering experiments were carried out at millimetre wavelengths.

For a scatterer of arbitrary geometry, the size of the lens aperture is decided by the diffraction errors that are tolerable. It has been shown experimentally, and with sorne theoretical justification in two instances during the investigation, that diffraction effects are pro­ portional to the magnitude of the field that irradiates a discontinuity.

A large lens, illtuninated by a point source feed ,.ri th directi vi ty such that the feed pattern is essentially zero at the lens contour, should produce small diffraction effects in the field that illuminates the scatterer. Reciprocally, a finite scattered field at the edge of the lens aperture would have only a small effect at the feed. This pre­ supposes that the lens design is such that the amplitude of the lens field is essentially constant over the volume occupied by the scatterer.

Reasoning along these lines is essential since the mathematics required to permit prediction of diffraction e1'rors for scatterers of arbitrary geometry is very complicated. -67-

iù>PEIIDIX l

DEVELOPHENT OF AN ARTIFICI.àL DIELEDTRIC FOR f.ITCROWAVE LENSE3

The artificial dielectric that l,las used to construct the microwave 41 lens of section 2.2 lofas proposed by J.A. Carruthers • The dielectric consists of microscopic aluminum flakes, such as those used in metallic paints, mounted in a foa~ of alkyd resin. The dielectric action is due to polarization of the metal particles by the electromagnetic field.

The change in dielectric constant VIi th the amount of alUJ.llinum pel' unit volume was investigated by E.L. Vo gan42 using a waveguide method at X- band vTavelengths. This method gives a result for the dielectric constant L, perme abi li ty /-L, and the loss of a smal1 sample of material. In order to examine the optical properties of large samples of the artificial die1ectric at micro':>/ave frequencies, the author used a K-band interferometer43•

This instrQ~ent dete~mines the optical path 1ength in a samp1e and therefore provides a measut'e of the refractive index n = (;1& )1/2.

1. Experimental Arrangement for Free-space Iv'Jeasurement of Refractive

Index.

The interferometer, shown in figure 21 is essentially a phase measuring device. Energy, at a wavelength of 1.25 cm, is provided by a stabilized klystron, modulated at 1 Kc/s. The output is split into two paths; the reference path, which feeds one arm of a ..,aveguide tee, and the sample path, which includes two pyramidal horns Hl and H2 ..Ji th aperture dimensions 41. J.A. Carruthers, "A New ù>w-density Artificial Dielectric", Eaton Electronics Research Laboratory Report No. 8, 1951. 42. E.L. Vogan, "An Experimental Determination of the Dielectric Properties of an .~tificial Dielectric", Eaton Slectronics Laboratory Technical Report No. 13, 1952. 43. C.G. f-bntgomery, ibid: p. 592. BALAN.CING 1 KC/S METER AMPLIFIER

DETECTOR

R2 1 :.";'. ,) ) STABAUZED 1 KLYSTRON SANPLE li fj '-:Jc,,:mSS9~===:::::--- HI ,..- RI -

STYROFOAM ABSORBER

'.!j H •~J /'.) f-'

FREE-SPACE MEASUREMENT OF REFRACTIVE INDEX -69-

15À x 15À and 10;\ x 101~ respecti vely. Hl and H2 are separated by a distance D in excess of the Rayleigh cri terion, 2 x (15/~) 2/A, for the larger horn. Since changes in ambient temperature affect the guide wavelength, the waveguide was lagged with glass wool, and the tee placed at the centre of the waveguide - free-space loop to reduce drifting of the relative phase between fields froID the two paths. Discrimination of more than fort y decibels can be obtained between the two paths by balancing the waveguide tee. The attenuator R2 is preset such that po\fer reaching the crystal detector from each arm is of the Sfu~e order of magnitude. If the micrometer that co~tro1s the position of H2 in the z direction is adjusted until the phases of the fields froID the two paths differ by 'IT at the mixer, a null will be obtained at the balancing meter.

A sample of thickness d introduces a phase change 2Tr(n - l)d/À in a plane ....rave passing through i t. If & = Dl - D2 is the difference in micrometer readings for nulls with the sample removed from, and in the path of the radiation, then

Z,T/À (n - 1) d = Zlr J lA or

n = l + 0 Id (i) d may be large enough to introduce a phase change in excess of 21r; (i) then becomes

n=l+ d +pÀ p = 1,2,.3 (ii) d p cau be determined by measuring various thicknesses of the s~~e material or by a 10\>r frequency bridge method if dispersion is not excessive. -70-

2. Heasurements on the Artificial Dielectric. llhen large samples of the foamed dielectric \o/ere measured, variations in n were observed at various points in the transverse planes of the samples. In addition, vath the sample in a given position, the value of n "laS dependent on the polarization of the incident field. The first effect is caused by variations in density, the second, by anisotropy due to asymmetry in the foam matrix. Experimental results and theoretical aspects of these problems are given by Neugebauer44• The anisotropy was verified by Vogan42• It became clear from these investigations that the dielectric could not be used for construction of a lens in the original foamed state. The brittle foam matrix was pulverized into small foam particles, and the resulting mixture of different sized particles sieved until a limited range of particle size (0.5 ta 1.7 mm.) was obtained. In this state, the dielectric forms a random ~edium45 with a density of about 0.2. The substance has an index of refraction of 1.5 or less, depending on the initial amount of aluminu.rn and the densi ty of the granular material.

The sensi tivi ty of the refracti ve index ta packing is shovm in figure 22.

The plotted points are measured indices of refraction, t~~en at successive time intervals, durine which the material was automatically packed. This experiment also showed that, when equilibrium was reached, a slight anisotropy developed due to orientation of the granules in a preferred

44. H.E.J. Neugebauer, "Properties of a Ne\o/ Low-density Dielectric", Eaton Electronics Research Laboratory Technical Report No. 18, 1952. 45. H.E.J. Neugebauer, "Clausius-11osotti Equation for Dielectrics with Randomly Distributed Dipoles", Eaton Electronics Research Laboratory Technical Report No. 19, 1952. _. '- -. -. _ . .. __ .. •• • --" ~- - -'. ,";'= '-! . • ' . - ••

,..,.. -' .. -

VARIATION OF INDEX ' OF REFRACTION W :H~ MECHANICAL PACKING

~ __ ------o ,.-­ -- / ~ -" / "" / / "" / ./ / 1 1 1 . o' 1·40 1 ,,l' 1

"30 +----.,------T·- - --· 1 o 10 20 30 PACKING TIME - MINUTES -72-

(horizontal) direction. Thus, in order to obtain a useful, uniform, isotropic dielectric, the material must be packed to a predetennined density in a suitable container and be mechanically stabalized by a small amount of binder with good electrical characteristics. Styrofoam was used for the walls of the containers and low melting point formalde- dyde resin for the binder. The resin was mixed, in povlder forro, wi th the dielectric and baked in place in the styrofoam container. The binding process increased the refractive index by less than 3%.

il. dielectric, and in particular a granular dielectric ,d th a rough surface is bounded by a transition region that becomes increasingly important as the thickness of the dielectric decreases. In figure 230. the dielectric is illustrated surrounded by free space. dm and de are the measured and effective thicknesses of the sample. E. = dm - de is :2 the effective decrease in thickness due to surface roughness at the inter- face. The true index of refraction, from (i) is n = l + &Ide = l+cf/<\a-2e.

~. vlhich approaches 1 + 6/dm if dm is large. If several measurements ~ are made on sarnples of thiclmess dm then &i = 1/2 { dm - ~ i/n.o- l} where n is the asymptotic value of n. 00 Measurements were made on the granular dielectric in a set of four styrofoam boxes ,d th plane parallel \-lalls, the result is shown in figure 23b. In this case, the value of E is negative; the dielectric granules have filled the open cells of the styrofoam surfaces.. The value of E is approximately -0.02 inches. The difference between the measured and effecti ve iddth is not negligible when applications such as quarter-wave layers on lens surfaces are considered. -7.3-

THE INTERFACE EFFECT

~------dm ------~

n

f :~(----- d. ------4): : l' ...... !~, 1 .1 "'1 e

FIG. 2.31:>

n

1·41

1·39

1·38

1·37

dm-INCHES -74-

3. Errors in the Free-Space Heasurement. The most disturbing error is an apparent dependence of n on the sample po si tion z :; zo, as shown in figure 24c. This dependence can be explained in terms of standing waves between Hl' H2 and the sample. The magnitude of the standing \-lave depends upon the scat tering cross- section of the antennas, which in turn is a function of the degree to which the antennas are matched to the waveguide.

In figure 24a, the primary field that has propagated through the jkz sample is represented by E:L = tsEo where Eo :; e- and t s is the jkz a~plitude transmission coefficient. The field E2 :; rH2 t s e is jkz reflected from H2. il secondary field E:3 :; rs rH2 t s e- is therefore superimposed on El. This is a multiple process but it is sufficient to terminate with E3 for a first order evaluation of the error. rs and rH2 are complex amplitude reflection coefficients that depend on Zo and the scattering cross-sections of the sample and horn. Figure 24b shows four resultant fields E = E1 + E3 for sample distances z :; zo' Zo - !V8, Zo - N'4 and Zo - 3À./$. A field, proportional to E is produced at the \-laveguide mixer. Since the detector is phase sensitive, the angle G, for an arbitrary sarnple distance, is autœnatically compensated for by adjustment of H2 for a null. Heasurement at an arbitrary distance, zo' will not result in a correct value of n. The true value of n is calculated from the average of the curve in figure

25c; this corresponds to 9 :; 0 in 24b. Computation of the field component E3 is complicated by the fact that, \.!hen a null is obtained for various sample distances Zo , H2 assumes a • •.• ___ • __ • • _ _ _ .• • _ , •• _. :.:.. • • _ • •; , __ ••• :....:. ' . _~ .. _ .•...: _. _~_ ';'''':; ____- ':'' ~ •. ______•• _ __ ,. ,_ ___ '';''''''.'':.;;...;o..: '; __ . _

-75-

THE INTERACTION EFFECT FIG. 2Ml.

, - ,~ ---~ E4*4---- E6 ----__ ----I------~E~o---~D~HI E5--+ ~, H2 : ...... ------0 ------~I-ll----+»o' ,l '1" 1'

lM (E)

Z=Zo Z· Za- ~/8

n 1·47 /0--"'0

'·46 /" o 1·45 / 1-44

1.43 ...... ----,I....-----""I'I----T"i ------,,;-----Ti ------", 4 4·~5 4·5 4·75 5 5·25 5·5 (0 - ZO)- ,CM . -76-

different position relative to the sample. This means that Q is not a linear function of zo. This effect, evident in 24c, produces a distortion of the otherwise sinusoidal variation of n with zo. The average of two values of n, taken at t\./o distances separated by 1../4 is not sufficient for a correct value of n.

There is no doubt that a similar interaction exists bet,.een Hl and the sample. In tbis case, a vector E6 = t s E5 must be added to the terminal of the vector ~ +~. Since the phase of E6 is a linear function of sample distance, the sinusoidal nature of the measurement will not be disturbed.

An error, causedby diffraction of ~ around the sample, depends on the transverse dimensions of the sample and the distance D - zo. The minimum sample size and the maximum trustworthy value of D - Zo can be determined experimentally.

The effect of internaI reflections within the sample is negligible.

Experimental errors, due ta inaccuracies in measuring the distance ~ , and the wavelength, can be obtained in the usual manner from (ii)

I>n = ~{~S + p~;\.} If the values of n and d are such that (i) is valid, the mea8urement i8 independent of vIavelength. In practice, this does not mean that n is independent of variations in \'Iavelength.

The research described in tbis appendix forms part of a general invest:igation, at the Eaton Electronics Research Laboratory, into artificial dielectrics suitable for rrdcrowave lens construction. dielectric, composed of alumin~m flakes in a vehicle of polystyrene foam

(styrofoam) rather then alkyd foam, nm ... appears possible. The techniciues employed here will be used to determine sorne of the properties of this new material. -77-

BI BLIOGKi?HY

1:". Di. ffro.ction

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~irl

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Si1ver, S., iLl1tenna Laboratory Reports Hos 181 (1951), 185 (1952), Uni versi ty of California, Depécrtment of Enzineerine -78-

.spencer, R.C. and Austin, P.H., IITables and IJethods for C3.1culation Il of 1.ine S01JrCeS , Radia.tion Laboratory Report No. 762- 2 (191/))

Strat ton, .T. L., "Electromacnetic Theory", BcGrm-J-Hi11 Co., HeH York, (1941)

~· ;oonto!.l, IIThe Probe :illtenna and the Diff raction Fieldll , :&"lton Electro!Ucs Research Laboratory Techn...ical Repo:-:-t No. 17 (1952)

1ioonton, G• .!!.., Bort.s, R.B. a.nd Carruthers, J.A., IIIndoor l'èasurements of Nicroc.rave Antenna Patterns by Iieans of a Het!;1.l Lens ll, ,T. ~;.pp1. Phys., 2l, 428 (1950)

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\-Joonton, G. A., Carrut hers , J. ~~., Elliot, A.J., and RiZ'by, E.C., ''Diffraction Errors in an Optic·9,1 He a surement at R2,dio ~ ,.raveleneths ", J. J.ppl. Phys., 22, 390 (1951)

R. Scatterinr.

Bril1ouin, L., "The Scattering Cross Section of Spheres for 'Electro­ nar;netic ~iavAslI., J. Appl. Phys., 20, 1110 (1949)

Ha.rlSen, H.!:!. and ,schiff, L.I., "Theoretical Study of Ele~tro!l1agnetic :Iaves Scatterec1 from Shaped Hetal Surfaces", ':;'uarterly Repr.:>rt No. 3, Stanford UIÙ versi t y, Vr.i.cro;,·!ave Laboratory

Kerr, D.E., IIPropasation of 3ho:-:-t Radio ~ ;ave s Il, Radi2.tion Laboratory Series rb . 13, f.lcGrau-Hill Co., l'·!e1·! Yorl<: (1951)

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l''lontcornery, C.G., Dicke, R.H., and Purcell, E.M., "Princip1es of Hicro­ ,·rave Circuits", Radiation Le.box-utory Series Ho. 8, HcGra'H-lli1l Co., Ne\-! York (194,3)

Hontro1l, E.\,~ . and Hart, R.li., "Scatterine; of Plane ~iaves by Soft Obstacles. II'', J. Appl. l-hys., ,22, 1278, (1951)

Norton, K.A. a.Yld Omber e; , A.C., "The HaximuIll Ranee of a Rade.r Set", Proc. I.R.E., 12, 4 (1947)

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Siegel, K. H., and Alperin, H .~~., "Stuclies in Radar Cro ss - .se ~tion s - III SC8.tt erine; by a Cone ", Report Tm:'·}.-37, Uni versi t y of j,n. cbigan, En sineerin ~ ReGearch Insti tut e (195 2)

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