THE UNIVERSITY OF MICHIGAN ON NEAR ZONE ANTENNAS by R. F. Goodrich and R. E. Hiatt June 1959 Reporkt:No. 2861 -I-F Final Report on Purchase Order 211-500-8668 Prepared for GENERAL ELECTRIC COMPANY DEFENSE ELECTRONICS DIVISION SYRACUSE, NEW YORK

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THE UNIVERSITY OF MICHIGAN 2861-1-F I INTRODUCTION Given a finite source distribution, say a circular aperture of radius a, we wish to find the behavior of the resulting field at some finite distance R from the source... _,.. Detector Source FIGURE I-1 We wish to determine the excitation which will lead to a focussing at Ro More explicitly we wish to find the dependence of the strength of the field and the size of the focus on the parameters of the problem. The focussing will obtain provided the phase of the excitation is proportional to the distance from a source element to the focus. The origin of this phase control of the excitation can be a phase controlled array or an elliptical reflector. In the latter case the source would be placed at one focus of the ellipse, the focussing will occur at the other focus of the ellipse. 1

THE UNIVERSITY OF MICHIGAN 2861-1-F Source 1 efle to Detector Elliptical Reflector FIGURE 1-2 Accepting this requirement on the phase distribution the problem is then that of finding the dependence of the strength and spread of the field at the focus on the amplitude of the distribution. To this end we deal with the scalar problem. This is no essential restriction since we will be most interested in the case in which the aperture size is considerably smaller than the range, a << R. Under this assumption, a<<R the scalar and vector problems are sensibly equivalent. 2

THE UNIVERSITY OF MICHIGAN 2861-1 -F II THE SCALAR PROBLEM We consider a scalar source distribution f( p) on an aperture of radius a. The field due to this source is given by 1 (R)= J f() dS (2.1) where R is the vector to the field point and p is the vector to the source point with the origin at the center of the aperture as in Figure 2-1. r P 0 Ro FIGURE 2-1 We now require the phase.of the source distribution to be such that the field is focussed at a point Ro on the axis. We put f(p) ikro g(p) (2.2) where p + Ro = r. Hence, 3

THE UNIVERSITY OF MICHIGAN 2861-1-F _ ____p -ikro (R)- ~ ~e g(p) dS (2.3) We now refer the field point to the point of focus and put r= R-R R (2. 4) and take Ro >r and p. Making use of this inequality we use the approximation 2 r ~{1 r o... (2.6) Substituting, we have k R( -- r* (r=de Ro 24 P (2. 7) 2 R 0 2 R~ Subsin the exponential. Furtherg, we omit phase dependences independent of p. 4 1r p 1 rdRo (R)= oe 0 g(p) (2.7) where we use the approximation R- pK Ro (2. 8) in the denominator but the higher order approximations, equaltions (2.5) and (2. 6), in the exponential. Further, we omit phase dependences independent of p.

THE UNIVERSITY OF MICHIGAN 2861-1-F We express the vectors, p, r, and Ro in the forms p = p (cos 0, sin 0, 0) r = r (sin e 0, cos 0) (2.9) Ro =Ro(, O, 1) and equation (2. 7) becomes a 2r -ikI r psin cos+1 rcos p2 O Icos 0 2 R p T(r) = dp do e I d ~ g(p) (2.10) 0 0 Under the assumption that the excitation g(p) is independent of the angle 0 the 0-integration can be performed at once giving a _ik rcos 0 2 r0) = -p dp e kr sin P g(P) (2.11) Ro J o ). 0 where Jo is the 0'th order Bessel function. The problem is now to characterize the transformation of equation (2. 11) leading from the excitation g(p) to the field 4 (r,a). We start by considering 2 the distribution g(p) which will maximize 4 (0,0) with the restriction that g(p) pdp= 1 (2.1Z) 5

THE UNIVERSITY OF MICHIGAN 2861-1-F Now a (0 0)- r pdp g(p). (2.13) o 0 But by Schwartz' inequality a a a JOp dp g(p) K< g p dp p (2. 14) 0 0o hence, a (.(0,0)< TJr p dp (2.15) 0 where we note that the equality in (2. 15) holds provided Y2 g(P). (2.16) a Having shown that the uniform amplitude distribution leads to a "maximum gain" configuration we now investigate this case further. We deal with an integral of the form a -i x I(c,)= xdxe Jo(ax). (2.17) 0 The integral of equation (2.17) is just a combination of Lommel functions of two variables (Ref. 1) 6

THE UNIVERSITY OF MICHIGAN 2861-1-F i a2 I(a,)= e U2 [Ula2, a a)+iU2(pa2, aa). (2.18) Putting this in the form of a Bessel function series of the Neumann type (Ref. 1) we have - a2 co m l((s3)=1- e y >i ) Jm a). (2.19) In our application of this representation to our problem we will require the values of I (a,3) for a= 0, = 0, and a =3 = 0. These are - i —a2 o m (a)m _a2(i3 )a I(,l)=i i 22 I(O,3)= - - ei a)m 2 2 (3O.(!C)M- - e -el 1 (2.20) 9 sin 2-a I(a, ~)= 2 J(a a) (2 21) I(0,0)-=. (2.22) Returning to the representation of the field 4/(r,) we have 2-i. a2 oo / m ~ (rcR)= - Ro3 2 Jm(a a) (2.23) R, p a, m

THE UNIVERSITY OF MICHIGAN 2861-1 -F with kr sinO kr cos 0 Ro R2 hence, i krcos0 a2 m ~(rO')= -Zri kr eos e Cos0 a (kr sin ) (2. 24) The special values are i kra2 47Ros i r4Ro (. 25) ) (ao (2. 6) (r(, ))= 27a J (a) (2. 26) k2 kr / 2 (0, )=. (2.27) R 0 We now investigate the zeros of ~ for the special values 0 = 0, 7r/2 of its argument, First we consider r1 such that 4(r, 0)= 0. The first of these zeros occurs for 4ia 2XR 0 r= 7T=2~. (2.28) ka2 a2 8

THE UNIVERSITY OF MICHIGAN 2861 -1-F This indicates that for our assumption a<<RR the first zero along the axis will occur many wavelengths from the focus r = 0. For the second case r = r2 such that / (r2, ) = 0, the first zero occurs at Ro X Ro r2= ka jl 2ra J where j1 is the first zero of the first order Bessel function. Now j1 3. 83; hence 3.83 RX (2.29) r2 = (2 29) 27r a and the first zero occurs somewhat closer to the focus r= 0 than in the preceding case. This result, equation (2. 29), indicates the limitation on the sharpness of the focus imposed by the physical parameters. The form of equation (2. 17) indicates that with no more difficulty we can develop the case of a Gaussian amplitude distribution. In fact, this simply involves taking the quantity 3 to be complex. We write 2 g(p)= e /2 (2.30) and put kr cos 0 2i 2P = ~^2~ ~ ^T ~(2.31) Rco t This complex 3 then leads to the following results for the special values 9

THE U NIVERS ITY OF MICHIGA'N 2861-1-F 0=, r= 0: ikr a 2 0^' ) _kr Zi (le (2.32) M2 P2 2 1 (0, 0)= i 2 l e (2.33) We find that a certain approximate formulation will make the effect of the Gaussian excitation more transparent. We start with equation (2.17) as before and remark that for complex 13 as defined in equation (2. 31) and for!A>~a we can use the approximation 00 i x I() x dxe 2 Jo(ax).(2.34) 0 This integration can be performed in finite terms giving (Ref. 1).2 ia 1(0e3)^~-^0 e~. 2(2.35) The transformation under this approximation takes a Gaussian into a Gaussian and we find (kr) cos 0 sin2 0 R R 1kr cos 2 4, y 27ri __ R2 4 C(r 0) - I e 1 u - R krcos 2 (2.36) 2 (kr sin 0 102'Ro (krcos \2 4 * e 0F) +-7410

THE UNIVERSITY OF MICHIGAN 2861-1-F We now use equation (2. 36) to approximate the case 0 =-. We have 2 l (kr2 2 ___ 2 { R0) 2 (r2)^ —e. (2.37) The use of the approximation in equation (2. 34) for the Gaussian excitation while of no great interest to the central problem does illustrate the important result of this analysis. The behavior of the field as a function of the excitation is essentially the same as for the far field cases. A uniform amplitude gives a maximum gain configuration while a Gaussian amplitude gives a zero side-lobe configuration. In conclusion, we see the restriction imposed on the focussing effect by the physical parameters may be summed up by equation (2.29). 11

THE UNIVERSITY OF MICHIGAN 2861-1 -F HIII PATTERN CALCULATIONS We now turn to a more detailed consideration of the fields arising from a uniform or Gaussian excitation of the aperture. To this end we will find it convenient to express the fields in terms of cylindrical coordinates having their origin at the focus and the z-axis along the axis of aperture as in Figure 3-1. (p,z) X:^ (PI P z z +Z, FIGURE 3-1 We will find that the field pattern is expressible as a linear combination of tabulated basic patterns. More precisely, the field pattern is an infinite series of these basic patterns the coefficients of which are given as powers of certain "small" parameters. We will truncate the series of basic patterns and compute various patterns for the uniform amplitude configuration. Finally, we will compare the theoretical results with those of a series of experiments using an elliptical reflector. Recapitulating the previous section, we have = 2- I(a,f3) (3.1) Ro 12

THE UNIVERSITY OF MICHIGAN 2861-1-F where -ia 2/2 I(ao, 3) = e U( a2, aa) + iU2( a2, a a -i^a2/2 oo 2 m 2 z (p^ a a) m (J (a). (3.2) 1 If we rewrite the series in terms of the function (Ref. 2, p. 128) m In \m(X) = (2/x) ml J(x), (3.3) equation (3.2) becomes 1 -ioa2/2 / [i(3/2)a ] I(a,f3) = - B e 2ml A m(a a). (3.4) 1 Recalling that for the Gaussian excitation kr sin 0 R 0R~~~~~~~o ~(3.5) kr cos e 2i 3= " 2 ".2 we make the identification of the cylindrical coordinates z =r cos 0 (3.6) p = r sin 0;

THE UNIVERSITY OF MICHIGAN 2861 -1-F hence, 2 aa2 =kz R) -2i (a/ )2 (3.7) a a =kp (a/R ). 0 Now we have assumed a<<R and anticipating that the interesting Gaussian excitations will be relatively broad, i. e., a<< I, we define the'small" parameters r7 - a/RO (3.8) 4 = alp. Substituting in equation (3. 1) we have an expansion explicitly exhibiting these "small" parameters, m 2 exp[ +(ikz/) +(ikz2) ~(p z) 7ra (kp19). 0o + (ikz/) 71 1 (3. 9) We now need to consider the properties of the functions. (x). These are tabulated and graphed in Reference 2, pp. 180-189, so we restrict our consideration to a qualitative description. We take m as positive integer and x 0. 14

T HE U N IVE R S I TY OF M I CH IGAN 2861-1 -F Each of the - (x) show a damped oscillatory behavior with a maximum at x = 0, -\ (0) = 1. This is illustrated in Figure 3-2 where we use m In -A l(x) and-iV2(x) as prototypes. 0.8 0.6 0.4 2........ __ ~ ~~ 0.2..... 0 4. -0.2 _ _ -0.4 ~~ 0 5 10 FIGURE 3-2 J\1 (x) and 2(x) As the order m increases the main lobe becomes broader. In fact, if we call Xm p 0 the first non-vanishing zero of the Bessel function Jm(x), J (x ) = 0, we have x > x i for all m > 0 and these values give a measure of the width of the main lobe. We normalize the field to unit amplitude at p = z = 0, 4 = 0, and split off the first term of the series -F ({) [~ +(ikz/2)12].\ (kpr) (,z;1)= A e(kp7]) +) (m + 1)1 m+l1 (3.10) 15

THE UNIVERSITY OF MICHIGAN 2861-1 -F Now in the focal plane z = 0 and for a uniform amplitude excitation, r = 0, the field is given by -o (P, 0) =\ (kp1). (3.11) Hence, our expansion, equation (3. 10), is of the form V(p, z; ) = e _ o(p, 0)+- (p, z; (3.12) where we have written ~ for 0 2 2]m Pp, z;t) (m+k1)! _m+l (kp 7) (3.13) 1 This term (3.13) is the correction to the focal plane pattern expressed as a series expansion in powers of the small parameters 4 and 1. For uniform illumination, 0 = 0, and in the focal plane, z = 0, the pattern is given by equation (3.11). Furthermore, as long as the requirements 77< < 1 and p~R are met this single expression holds for these various values of the dimensions a and R. o Suppose we fix the wavelength and examine the behavior of the pattern as a function of the parameter r). If we take some criterion for the size of the region of focus, say the distance out to the half-power points, we can express the size of this spot as a function of 7r. For example, for the halfpower criterion we have 16

THE UNIVERSITY OF MICHIGAN 2861-1-F kp 7 1. 6 k-~ ~' 1.6: (3.14) = 0.8X o Ir a More generally we can say that whatever our criterion, the size of the spot is proportional to the range Ro. As we move out of the focal plane, again for uniform amplitude in the aperture, we note that for kz so small that only two terms of the series (3.10) are required the energy in the pattern is given by f12 i2 (1z4112)2L nI2 (3.15) where we note that to this order the pattern is symmetric in z up to the inverse square dependence on the distance from the source. We will give examples of the uniform amplitude patterns for various values of z and 77. We will find it convenient to re-define our parameters somewhat. We write R = a /MX, (3.16) z = ERo ll <<l. The arguments of the field function then become kz 2 kz -7 =r M (3.17) kp nr = 2r M -. 17

THE UNIVERSITY OF MICHIGAN 2861-1 -F We substitute the expressions (3. 17) in equation (3.10) for' = 0 ao m (P, z;0) = (2z M + (+ l M) 7 (2 M P) (3.18) a 1(m+1)1 m+1 a 1 Including the first two terms and explicitly introducing the dependence on the distance from the source we have the pattern function (3.19) 1+0 2 2 2 ~-~ M/za) _(2 M a) We plot f(p, ( ) in db below f(p, 0) = 1 in Figures 3-3, 3-4, 3-5, and 3-6 for (1+6)2 L 2 a where M = 1.2; 6 = 0, t0.12, + 0.24 ~=z/R% leOi<< M = 0. 5; 6 - 0, + 0.10, +0.25 (3.21) M =a /R X We plot f(p, 6 ) in db below f(0o, 0) = I in Figures 3-3, 3-4, 3-5, and 3-6 for the values M= 1.2; 6 =0o +_0.12, + 0.24 M=0.5; 6 =0o + 0.10, +0.25 (3.21) M =0.1; 6 =0, +0.10, +0.25. ( = O; M = 1.2, 0.5, 0.1. 18

Relative Power in db 0 0 o oo o o 0-I I I. ^ ^ ~~H',/."?~"~~ / *Z^ ~~~~~~~~~~~~~~~~~~~~./!/ 1-' b- w P Ui ~~/. ~~~~~ 03 0:C OJ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ o'^ I ^ ~ ~ ~ ~ ~ ~ ~ ~ ~~~ I,^ \. z,'/ // 0 01 I:!! O I ^^^*^^i^ ^^ i I " ii ii ~~^^ m ~~ ^ ^ -r~ / /'1 - iI)II n, rt. ^ i —, r

THE UNIVERSITY OF MICHIGAN 2861-1-F ~~~~~~~~~-10~~~E - 0=,1 --—, - — ___ -zo'd ~,1 1 id\ *5,~,.,'\.40 A // -30 l' I'I 0 1.0 p/a 2.0 3.0 FIG. 3-4: f(p,)VS. p/a FOR M =:0.5 20

THE UNIVERSITY' OF MICHIGAN 2861 -1 -F E =.25 ^: ^ \ C6-.1 ~ -.-\ E —-. I ----- - -10 / 1 -70 -''. /.' I^\I i I': PI- v ^ 0 - 1\ -40 -50 i FIG. 3-5: f(p,) VS. p/a FOR M 0.1 21

THE UNIVERSITY OF MICHIGAN 2861-1-F a \ \ M= 1.2 \~\ M:WM =.5 I | \ M=.1 *.-10i \ \ FIG. 3-6- f(p, ) VS. p/a FOR = 0 ~~~~~, Ip/a 10 ^ ^ v\ ^ I' 4 -40 Ii ii -50 I 0 5 p/a 10 15 22

THE UNIVERSITY OF MICHIGAN 2861-1-F Finally, we will make a comparison with certain of the experiments described in Section IV which follows. We consider the H plane pattern using the modified waveguide feed. Now in the experiment the aperture illumination is tapered to a value 6 db below the center. This implies for the field a reduction of one-half at the edge of the aperture. Hence we take ~ = 0.7 and compute the pattern using equation (3. 10). The other necessary parameter appears in the first case of the uniform amplitude pattern calculations, M= 1.2. From equation (3.10) we have - 2 2 2 ^(0,0; )e- e -l (3.22) = e (2/ 2) sinh (2/2). Hence, in order to normalize to unity at z = p = 0 we write, including the dependence on the distance from the source, the pattern function (p, z;) 2 f(P, Z;r) f(P. 6e;) f (1+6) JI(0, 0; 4) ) = (3.23) f(P'6;V (1+6) (0, 0;) series and write f(p,6;t)= (1 6)2 (2)1 J\(27r M + 46 _3/ 6 [\ M /\A_ + Gr6M _A \ 2 2 3 \ (3.24) 23

THE UN IVE R S I TY OF M I C HI G A N 2861-1-F This expression (3.24) we compare with the experimental results in Figures 3-7 and 3-8 for the values =0, +0.06, + 0.12 The comparison with the experimental results in Figures 3-7 and 3-8 indicates that we can predict at least the main lobe whenever the aperture illumination is able to be approximated by a Gaussian distribution. That this form of the aperture illumination is not an essential restriction follows from the fact that an arbitrary distribution is expressible as a series of Gaussians. That is, we can expand a general aperture function g(p) in a series, 2 2 nP / g(p) = a e (3.25) n n This then leads to the field as a series of terms of the form of equation (3.10). In closing we mention an extensive study of microwave lenses and image formation by G. Bekefi (Ref. 4). This work was brought to our attention as this report was being written. It considers in great detail the properties of the image formed by a uniformly illuminated lens or aperture taking into account the various optical aberrations. 24

T HE UNIVERSITY OF M ICHIGAN 2861-1-F --- =-. 12 f = -0. 06 FIG. 3-7: COMPARISON WITH EXPERIMENT \ =0 o FOR = 0, -0.06, -0.12 Theoretical Values * — =0 o - ~ = -0.06 a -- E= -0.12 -10 8 4 0 4 8 5p;~~ -20 9~O25 o 0 C 0 -30 8 4 0 4 8 Angle Measured From the Aperture Axis (Degrees) 25

THE UNIVERSITY OF MICHIGAN 2861-1 -F -_., — ~ ~0 ~= 0 E = 0.06 FIG. 3-8: COMPARISON WITH E 0.12 EXPERIMENT FOR E=0, 0.06, 0.12 Theoretical Values ~ -e =0 0- E = 0.06 -- = 0.12 -10 0 k~~~~~~~~ I70 8 4 0 4 8, —-, -20 ~0~~ ~~ *. I0 0 -30 V ~ 8 4 0 4 8 Angle Measured From the Aperture Axis (Degrees) 26

THE U N IVE R S I TY OF M I CH I GAN 2861-1-F IV EXPERIMENTAL NEAR ZONE FOCUSSING OF PARABOLOIDAL AND PROLATE SPHEROIDAL DISHES It is of interest to study experimentally the extent to which electromagnetic energy may be focussed to a point at some finite distance Ro from a circular aperture. As suggested in the introduction, the proper phase distribution to produce focussing is provided in an elliptical reflector. By placing the antenna feed at one focus of such a reflector, energy may be focussed to the other focal point. The degree to which this can be accomplished is dependent on how nearly the feed approximates a point source as well as upon the considerations discussed in the preceding sections. The elliptical reflector chosen to study this phenomena was one whose surface is generated by revolving an ellipse about its major axis. The generating ellipse has a semi-major axis, 2a, of 225 inches and a semi-minor axis, 2b, of 90 inches. The reflector section consisted of the end cap extending to the first latus rectum. This gives an aperture of 36 inches; due to its rolled edges, the reflector procured had an aperture of about 35 inches. Focal point, fl of the reflector is approximately 9.4 inches from the vertex. The distance between focal points 21/2 R= 2(a2 - b = 17.2 feet. To provide a basis of comparison, a second reflector type antenna was 27

THE UNIVERSITY OF MICHIGAN 2861 -1-F also studied. The reflector used was a standard parabolic reflector, 35 inches in diameter and with a focal length of 9 inches. The feed used for these reflectors was a modified waveguide feed. The modification, made as described in Silver (Ref. 3 ) produced a broader primary pattern in the H plane than is achieved with a standard waveguide. Figure 4-2a shows the E and H plane patterns of the feed. Since energy striking the edge of the reflector travels two times as far as energy going to the reflector center, the illumination taper is increased by 6 db making it about 12.5 db in the E plane and about 13.2 db in the H plane. / U 5/1 I/ 16 WAVEGUIDE (Reference 3, p. 381) The location of the center of radiation of this feed was studied by examining the phase of the radiated energy. The results showed the center of phase to be inside the waveguide. In the E plane the phase center was. 16 X from the 28

THE UNIVERSITY OF MICHIGAN 2861 -1 -F pin; in the H plane the phase center was.4k from the pin. The phase, measured on circles whose centers were at these points, was uniform to better than + X/16 in both the E and H plane over the semi-circle occupied by the reflector. In the last set of patterns presented (Fig. 4-9 and 4-10), a dipole feed, fed by a coaxial line, was used to study the effect of a more uniform illumination. Only H plane patterns were taken. In this case, the illumination taper at the edges was down only 6 db and this was due to the factor of two difference in distance to the center and to the edges of the dish. All measurements were made at a frequency of 9450 mcs (X = 1.25 inches). The measurements were made in an indoor antenna range, some in a room 30 x 80 feet and others in the microwave free space room which is 30 x 60 feet. A photograph of the free space room and of some of the equipment used is shown in Figure 4-1. Both reflectors were first "checked out" as standard far-zone antennas. Resulting E and H patterns are shown in Figure 4-2b. In these and in succeeding patterns shown, the feed is positioned in the E and H plane to produce the optimum patterns. As would be expected, these positions differ by 0.3 or 0.4 inches, due to the difference in the effective center of radiation. In studying the near zone behavior, the antenna under test served as the receiver. It was mounted on an antenna pedestal which provided for rotation in 29

THE UNIVERSITY OF MICHIGAN 2861 -1-F azimuth. A simple open waveguide was used as the transmitting antenna in order to provide as much detail as possible in the area being probed. This antenna and a stable X-band oscillator were mounted on a trolley which permitted the range to be varied over a fourteen-feet distance. This track was carefully aligned with the axis of the test antenna, so that peak signal was obtained at the same angular position of the test antenna as the transmitter was positioned at various points along the track. A series of thirteen patterns were taken for eight conditions of the test antenna. The thirteen patterns were taken as the range was varied from R (17.2 feet) to R - 6 feet and R + 6 feet in steps of one foot. For the 0 0 reflectors used, this represents a change in range from.14 D2/X to 0.28 D2/X. The first four sets of these patterns (Figures 4-3, 4-4, 4-5 and 4-6) are for the parabolic reflector. Figures 4-3 and 4-4 (E and H plane) show that a standard reflector, focussed for far-zone performance, is a very poor antenna for use at such short ranges. Figures 4-5 and 4-6 (E and H plane) show that it is possible to defocus the feed in a parabola so that a very acceptable beam is produced, even at.21 D/X. The patterns of the elliptical reflector are shown in Figures 4-7 through 4-10; the first two sets use the waveguide feed while the last two use the dipole feed. In Figure 4-10, the expanded angle scale is used to present more detailed information on the behavior in the focal region. 30

THE UNIVERSITY OF MICHIGAN 2861-1 -F Summary curves showing 10 db and 3 db beamwidths of the patterns of the parabolic and elliptical reflector are given in Figures 4-11 through 4-14. Figure 4-15 is a plot of the received power vs range as the transmitting antenna was pulled toward the receiver. The signal was first recorded on the pattern recorder and later condensed and replotted as shown. The agreement between predicted performance and experimental results has been discussed in the preceding sections. It may be of interest to note further that the beamwidths for both the defocussed parabola and the elliptical 2 reflector at the 0. 21 D /X range is approximately equal to or less than that expected from a far field pattern of a uniformly illuminated parabola. The patterns with the waveguide feed show beamwidths to be wider in the H plane than in the E plane as would be expected due to the increased illumination taper in the H plane. The sharpest patterns are obtained when the dipole is used as a feed. These, of course, occur in the H plane where the illumination was uniform except for the inverse distance squared loss. The comparison between the linear width of the beam for the far' field and for the near field case is more striking. For the 35 inch elliptical reflector, this width (2R tan 0/2) is 7.5 inches as contrasted to the width of 37.6 inches for the parabola when R = D2/X. Table 4-1 compares the minimum beamwidths obtained experimentally + with theoretical values. +Reference 3, p. 195. 31

TABLE 4-1 m Theoretical With Experimental _ Uniform Illumination Parabola Focussed Parabola Elliptical 1.02 A/D For Far Field Focussed For Reflector R =co R = D2/.21 D2/X R=.21D2X 0 Beamwidth in Degrees 2.09 2.2 2.18 2.08 Beamwidth in Inches -37.6 7.9 7.5 2R tan 0/2 _________________________I C) 0fi z~

THE UNIVERSITY OF MICHIGAN 2861-1-F The gain of the parabolic and elliptical reflector antenna was measured at the range, R = 17.2 feet. With the modified waveguide feed in use in both 0 antennas, the gain, measured by comparison with a gain standard, was 32.3 db for the parabola and 33.3 db for the elliptical reflector. 33

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THIE UN IVERSIT Y OF MICHIGAN 2 861.- -F ACE JrOWLEDGE MENTS Appreciation is expressed to rofesso A. te for suggestions on the experimental progra ar-n d to T, Hon, D,, Pep.pers I Rimawi and R Wolford for assisance in iaking the experiren-el:ital'easur i.at 48 ~ ~ ~ e sreet

THE UNIVERSITY O C G A N 28G61 - -F REFERENC"ES 1, WGatson CG. N. Theoiy of Bessel iun1tiot0s1 2nd EditiEon Camnbridge at the University Press 1952) 2, JaY ke, E. and Emxdet F., Tables of Functions With Formulae aend Curves, Dover Pblications, New York (1945). 3. Silver S icowave Antenna Theosx -and Desi n Massachusetts Insti te of Te c nology Radiation Laboratory Series No 1iZ McGrawHill Book Co, Inc. (l949) 4. ekef, "StudiAes in Microwave Optics," E E R. L. Tecmical Repor No. 38- McGill Universit, Montreal (Canada (March 1957), 49