A 1 Elec-troma 'neic Scatterting at;rdl Abs-orpt.ion by h.in \Vr Wa i:d ele.tr. C.y ai de:rs with Applieations ito Ice Crystals Thomas B.A. Senior and Herschel Wei.l RL-632 = RL-632 The authors ar. with the University of Michigan, Department of Electrical and Computer Engineerinig Radiation Laboratory, Ann Arbor, Michigan 48109. Dr. Well is presently on leave at the National Center for Atmospheric Research, Boulder, Colorado 80307.

i - J. Tu- J () i1t) o Abstract Integral equations are developed to determine the scattering and absorption of electromagnetic radiation by thin walled cylinders of arbitrary cross-section and refractive index. Numerical data are presented at wavelengths in the infrared for hollow, circular and hexagonal cross-section cylinders which sinmulate columnar sheath ice crystals. The numerical procedures are economical for cylinders whose perimeters are less than about 15 free-space wavelengths. i

1. Introduction The reflection, transmission and absorption of visible and infrared radiation by clouds and by pollutted atmrospheres are of considlerable?pr-actical importance. Cirrus clouds, in particular, are found all over the globe and have a profound effect on the atmospheric heat balance. The ice crystals wvhich compose them scatter and absorb- the predominantly ir radiation emanating from the earth and the lower regions of the atmosphere, as well as the mainly sho(rter wavelengthl solar radiation incident from above. The difference in tle scattering arid absorption properties of ice crystals in these t\wo wavelength regions play a major part in the atmospheric heat; balance wuich governs the global location of energy sources and sinks and, hence, the atmospheric 1 2 circulation patterns. ' Techniques for calculating the transfer of electromagnetic radiation through clouds of particles have been summarized by Plass et al. 3 All of them require a Imowledge of the scattering and labsorption pro-perties of the individual particles either for single scattering or, in the case of optically thickS clouds, multiple scatte:ring. Since each scattering event can affect the polarization by producing an electric field orthogonal to the incident field as well as parallel to it, an accurate treatment of the transfer problem mrust involve the complete scattering and absorption matrices for a single particle. When the cloud particles are roughly spherical as they are for water droplets, it is not unreasonable to model them as spheres, and results obtained from the classical 4 Mic theory are then in good agreement with measured data. With ice crystals, however, a wide variety of shapes and sizes have been observed in clouds (see, for exam5 6 pie, Aufm Kampe and Weickman; Ono ). Plate crystals, i. e., cylinders of length much less than their diameter, and columnar (long thin) crystals, hollow as well as solid and of hexagonal cross-section, are quite commonly found, and for shapes as varied as these a sphere cannot provide an accurate simulation of the scattering. Nevertheless, for lack of any other applicable method, it has been customary to model these crystals using spheres of some equivalent radius, thereby introducing unknown 1

and;ossibJy iar;e errors in the valu;s for t:e x i-tion transfer, which are the end 7 produc.is of extensive and expensive computations. The importance of using the proper scattering matrix when the particles are irregular is clear from the data presented by Holland and Gagne. They measured the matrix elements for clouds of irregularly shaped, random-ly oriented silicon flakes, and found results quite different from-i those predicted by Mie thecry, particularly for back and forward scatter. The last few years have seen several attempts at calculating the scattering from more realistically-shaped crystals, and it is appropriate to mention here the work of 9 10-12 Jacobowitz and Liou directed at the scattering properties of columnar ice crystals. Jacobowitz's data were obtained for infinitely long hexagonal crystals using ray tracing. Diffraction produced, for example, by the six longitudinal edges of the cylinder, was necessarily omitted, as well as the effect of polarization, and the calcu — lations were limited to cylinders not less than 40 1jm in diameter (at a wavelength of 0. 55 jim) with the apparent objective of assuring the reasonable validity of geometrical optics. Finally, no account was taken of internal absorption by the ice in spite of the fact that the appreciable imaginary part of the refractive index at some infrared wavelengths suggests that absorption may not be negligible. Liou's analyses are based on the assumption that the crystal can be modelled by an infinitely long homogeneous dielectric cylinder of circular cross section. Although this obviously suppresses those features of the scattering which are peculiar to the hexagonal cross section of an actual ice crystal, there is now a mathematically exact expression for the scattered field as a sum over orthogonal functicns analogous to the Mie series for a sphere. It is therefore possible to compute the scattering precisely with all polarization information present a,nd internal absorption taken into account. The retention of the hexagonal geometry is a key feature of the present work, and in contrast to a circular cylinder, there is no exact expansion available for this shape. However, an integral equation approach has been developed with which to comlnpute 2

Pi fj-C3 - the scattering from thin cyl i ndri.cal dielectrric shells of arbiltrary cross section when irradiated by a plane wave of ainy polarzatiorn ina-dent in a plane perpendi.cular to the generators. By applying the method to infini-ely long Lxagonal cylinders, scattering and absorption datca have been generated applicable to hollow columnar (sheath) ice crystals in the infrared. To display the role played by the h'-xagonal geometry, these data have also been compared with the analogous results for hollow circular cylinders of equivalent dimensions. 2. Analysis The method that we shall use originated from a study3 of the scattering by dielectric and absorbing layers in which the layers were approximated by impedance sheets or membranes of infinitesimal thickness. According to this approximation, a layer of thickness c composed of a material whose complex relative permittivity is E and whose permeability is the same as that of the surrounding free space medium is represented as an infinitesimally thin mermbrane having complex impedance r iZX (l= 1)27r7 (1) ohms per square, where Z is the intrinsic impedance of free space and X is the free space wavelength. Mks units are employed and a time factor e i t suppressed. The resulting membrane is just an electric current sheet subject to the conditions A ] + [AEI = J0 [AH =J n A (A E -J (2) at its surface, where n is a unit vector (outward) normal to the side indicated by the plus sign and J is the total electric current supported. 3

R L- 63 This type of sheet has proved useful in a. variety of problems, but it is only 14 an approximation to a layer of non-zero thickness. It ignores componentLs of volume currents perpendicular to a layer as well as variations of the tangential components within the layer, but from analytical and numerical comparison of the results obtained with those for specific layers of non-zero thickness. it has been found accurate pro — vided T ~ 0. 03X and the sheet is located at the middle of the layer. Its great merit is that it allows us to simulate the scattering from a layer using a single surface current distribution, thereby making the problem of a sheath cryslstal relatively straightforward. For generality, we consider a cylindrical membrane of infinite length with generators parallel to the z axis of a cylindrical polar coordinate system. The membrane has complex impedance 77 ohms per square and is illuminated by a plane electromagnetic wave incident in a plane perpendicular to the z axis. Since there is no dependence on the z coordinate, the scattering problem is two dimensional and can be expressed as two scalar ones for the field components E and H. It is therefore z z convenient to consider separately the cases of E and H polarizations in which the incident electric or magnetic field respectively is in the z direction. Since the membrane supports no magnetic current, the scattered electric field can be represented in terms of the electric Hertz vector II(P) = f J(s- 4)I (kR)ds (3) as ES(p) = VAVA^(p), (4) where ds' is an element of arclength on the closed contour C constituting the perimeter of the cylinder in the plane z = 0. R is the distance between the field point and the point of integration, k = 27r/Ak and H (kR) is the zero order Hankel function of the first kind. For E polarization E = zE implying J = zJ, and when the vector differentiations z Z are performed, it is found that 4

S (1 ) f1 Ek - J (s')I- ( )( k)ds'. (5) C The total field is then Ei + E and on allowing to lie on C and applying the boundary Z Z condition (2), we have i kZ r (1) E(s) = J((s) — J J (s')li (kR)ds'. (6) z Z 4 z o This is a sinudar integral equation from which J (s) can be obt0ained numerically by 15 7 t the moment method. As always the contribution from the self cell containing the field point must be determined analytically, so that in effect (6) is replaced by i 'kZ (1 ~ n2+.... z E (s) - [ Z 1 {+ rl + 0. 028798...))1 (s) z L 4 7Tr X z 1 kZ, (1)( + 4-j J (s')H ()kR)ds (7) C-z where Ais the self cell of size A. Having found J (s), (5) can be used to compute the z scattered field at any point in the plane z = 0. The case of H polarization (H = zH ) is more complicated because the current z is now circumferential (J = sJ ), but by a process similar to the above we find ES(P) -kf f(J (sS)(-S) -2 J- T (S') -S THH W(kI) ds. (8) s 4 o/ kZ c s( s o i s The total circumrferential electric field is then E + E and in the limit when the field S S point lies on the surface application of the boundary condition gives E (s) s=r7 J(s) J (s ' sH ( KR) k}s4 s 1 S k at 's ( A R)"H0 (kR)} ds (9) A where R is a unit vector directed from the point of integration to the field point. This is a valid integral equation for the circumferential current J (s) but is complicated by 5 5

L-G63 the presence of the sirface derivative of t;le current. On -the other hand, by a process equivalent to an inrAegration by parts, (S) can be wriithtJ:nm as S f( f A2 (1.) E () = - (s'),(;S ) -— H ( (kR)ds' (10) 4 sk2 ^St2) o leading to the integral equation i k Z (1) E (s) =rJ J(s) + J J4(s')( ')S ( (kR)ds S 0 Z lim A^A.4c d s' 1 )11 in which only the undnown current J (s) appears. S This simplification has its price. Because of the high order singularity of the second integrand in (11), it is no longer possible to reverse the order of the limiti and integral operations unless we segment the range of integration prior to taking the limltit and then maintain the segment or cell size A: 0. This is, of course, no restriction as regards a numerical solution. If A is the self cell, an analytical evaluation then shows limds, = 8 1 2i k lim f... 7ds=- k + + A 0. 528798+. SJ (s). and rT) s' = s. +A/2 lim fC.. ds= J (s.) (s.R)Hl ( c j=l LI.s -A/2 j/i where we have assumed the current constant over each of the N cells into which the profile C has been divided. By also evaluating the self cell contribution of the first integral in (11) in the same manner as we did for (6), our integral equation becomes 6

'i- T Er (s) + -n 2 4-.;: L. 4.471202...1. (s) 1., 47(1) tkZ ': (sf)(s.)lI (kR) ds' C-Z - E J (s) '(s.R)Il )(k (12) 4,- s " L, /2-^ - S, S - jfi J Although the coefficient of J (s) in the first termr on the right hand side becomes infinite as L 0, the equation is quite amenable to numerical solution, and the results obtained are (perhaps surprisingly) insensitive to i for a wide range of cell sizes. We have, in fact, used this equation in a nu.mber of applications, and it will also be used here. For an E-polarized incident- plane wave, E' - exp -i(xcosO + ysjnm)} (13) Z 0 P where Tr + 0 is the angle which the direction of propagation makes with respect to the 0 x axis. Once J (s) has been found from (7), the scattered electric field in the far zone z of the cylinder is z ( -kp i / 4) g 71- P e 0 where the complex scattering amplitude A is given by kZ A (f4, 0 ) = (s)e ds', (14) and p = xcosS + ysinyj 7

R1L-6 I. is a unit vector in the direction of observatio-:. The backscattering direction is therefore. In teirmns of A,thl two diimens-nal b istatic scattering cross section or pthase function is (v0,) |0 0. j2 (15) If the incident plane wave is HI- polarized, IZ now has the form shown in (13), and z having obtained J (s) from (12), the scattered magnetic field in tie far zone is given by is. i (.- '/4) ( ) z = eA h(' h o where Ah(6 o): - f (p.')J(s')eikP2 ds t (16) CS The Hl-polarized cross section can be found using (15) with Ah replacing A. e Two quantities of particular interest are the total (integrated) scattering cross section o'T and the absorption cross section oA. The former is given by 27r T(17) = T(0 ~) 2 0 ( )d, w (17) 0 and if this integral is confined to the ranges-0 - '- or n - 2r we have the forward or backward power fluxes. From the foward scattering theorem, the absorption cross section is Ao o - T(o) - I Re. (T 0 * (18) The extinction cross section is then the sum aE(o) = oT( ) + GA(Oo) (19) We remark that for a cylinder of finite length,AX, each three-dimensional cross section a( for incidence in a plane perpendicular to the length, computed on the 8

RL-6(2 assumption that the surface field is the same as for the infinite cylinder, is related to the corresponding two-dimensional cross section a by (3) 21 U ='47. (20) 3. Computed Data Computer programs have been written to solve the integral equations (7) and (12) respectively and, hence, determine the various cross sections for any cylindrical membrane whose profile is made up of straight line and circular arc segments. Data have been generated for circular and hexagonal cylindrical shells having a variety of dimensions (including thickness) and for the refractive index of ice over a wide range of frequencies in the infrared. In the case of the hexagon, two directions of plane wave incidence were considered: edge-on and face-on, i. e., along a bisector of the angle at an edge and normal to a face respectively. The majority of the data were for a hexagon 3pm on a face with a wall thickness 0. Ijm and we present only these data here. The number N of sampling points used in the computer program ranged from 96 at the shorter 16 wavelengths to 30 at the longer. The program is described in Weil and Senior where additional data can be found. We have also used the program to generate results for hollow circular cylinders of the same thickness r= 0. l [m and with radii equal to (i) 3/jm, the length of a face of the hexagon, and (ii) 9/7r jm, so that the hexagonal and circular cylinders have the same circumference. At the wavelengths X= 3. 1 and 12. 5,pm the data for the circular cylinder have been compared with those for a cylindrical shell of outer and inner radii 9/7r ~ 0. 05pm computed using the orthogonal function (or Mie series) expansion of the scattered field. For the refractive indices used, the bistatic scattering cross sections agreed to within 0.2 dB for E-polarization, but showed somewhat larger (e.1. 0 dB) discrepancies for H-polarization. 17 The refractive indices were taken from Irvine and Pollock (IP) and Schaaf 1 8 and Williams (SW), and are plotted in Figure 1. There are noticable differences at certain wavelengths and to see the effect that they produce, computations at X = 2.25 and 3. Opm were carried out for the refractive indices from both references. 9

IRL — 3 Angular cross sec'tinon data tfor X = 3j an.d 12. 5 i.n are presented in Figure 2. As expected, the number of maxirma and minima ini 0 /_ ZZ wT increases with decreasing X, and at wavelengths which are much longer than the face length of the hexagon, e.g., 8jum, the cross section has almost no ango;lar structure. Changing the incidence from edge-on to face-on has most effect in directions close to backscattering, and we note the substantial differences between the results for -i and H polarizations. The results for corresponding circular cylin ders lhave been included in the figures to demTonstrate the importance of using the exact cross sectional shape. The locations and magnitudes of the extrema differ for the two types of body, and though the curves are similar in certain angular ranges, they are far apart in others. The values of oT, cA and yE for the cases shown in Figure 2 are listed in Table 1. For a fixed wavelength, it wculd appear that the absorption is primarily volume rather than shape dependent. Spectral data for the cross sections r(0), o(7i), UT and A of the hexagonal cylinder are given in Figures 3 and 4. Each figure has two parts covering the ranges 1. 6 to 3. 51in and 8 to 12. 5rm. Separate curves are shown wherever the results for edge-on and face-on are clearly distinguishable. For the most part this is only true of the backscattering cross section o(0), and in the other figures the differences are confined to the immediate vicinity of local maxima and minima. In the longer wavelength range the SW values for the refractive index were employed, but most of the data at the shorter wavelengths were computed using the IP values. The latter range spans the main absorption band centered on X = 3JLm and the secondary one at X = 2 inm. These wavelengths show the main discrepancies between the IP and SW values and here we ran the data for both sets of refractive index. As Figures 3 and 4 show, the discrepancies do produce substantial differences in the cross sections. The geometrical effects are particularly pronounced for X 3.5 jLm. This is not surprising since the dimensions of the cylinder are now comparable to a wavelength, or a low multiple thereof, and this is the region where resonance effects and other 10

31 L- 63O. interactions betw.-;een the various contributors to the scattering are most important. As an example, while oA has a strong local rmaximum near the maximum in n. at 1 X = 3.0'75ln,;the shape and overall width of the maximum in A are apparently affected by the fact that the side lenoth of the hexagon is Lnow almost a Awavelength. o(0) and oa(r) both show a corresponding drop in this absorption region. The behavior is quite different near the secondary maximumn in n. at X = 2 pm. For HI-polarization 1 but not for E, oA is large as expected, while a(0) and -(v7) have local maxima for X just above 2 pm with both polarizations. Further evidence for the way in which a geometrical effect can dominate a material absorption effect can be found by comparing the absorption cross sections at 2. 25 iml computed from the IP and SW data. At this wavelength the SW value for n. 1 is rcughly five timethe IP value, with n almost equal in both sets of data, but the SW value produces an asorption cross section which is an order of magnitude less than that given by the IP value. It is therefore obvious that predictions of absorption and scattering based only on the properties of the material of which the scatterer is composed may be ccnsiderably in error. Aclnowledgenents This work was supported by NASA Grant 5044 and in part by AFC)SR Grant 72 -2262. 11

Cylinder Type and Irradiation waTelen/gt /A E/ hex, 3Lmn side, edge-on, E polarization 3. -7.74 -1.29 -0.40 H polarization 3.1 -12.16 — 4.10 -3,47 E polarization 12.5 -23.76 -10.97 -10.75 H polarization 12.5 -29.09 — 13.45 -3.82 hex, 3,um side, face-on, E polarization 3.1 -7.97 -1.22 -0.37 Hpolarization 3.1 -12.88 -4.06 -3.52 E polarization 12.5 -23.77 -10.97 -10.75 H polarization 12.5 -29.12 -13.95 -13. S2 I -- -- I — l circular, rad = hex side, E polarization 3.1 0.00 2.27 4.30 H polarization 3.1 -3.75 0.65.99 E polarization 12.5 -23.59 -10.76 -10.55 H polarization 12.5 -27.55 -13.74 -13,57 -- _______ 1 ___ _ -,1-I — i circular, E polarization 3.1 -8.11 -1.16 -0.36 perim = hex perim, H polarization 3.1 -12.77 -3.95 -3.42 E polarization.2.5 -23.89 -10.96 -10.74 H polarization 12.5 -28.19 -13.72 -13.76 Table 1. Total, absorption and extinction cross sections of cylindrical shell ice cylinders 0. 1 uam thick for two infrared wavelengths: X= 3. l/rm with 17 18 n = 1. 280 + i 0.3252 and X = 12.5[tm with n = 1.387 + i 0422. i,O G0) C.k

,R efe:e' resn," $ 1. S. K. C;x, J. Atmos. Sci., 28, 1513 (1971). 2. ARWG, Bull. Amer. Meteo. Soc., 53, 950 (12t72). 3. G. N. Plass, G.W. Kattawar and F.E. Catchings, App!. Opt., 1_2, 314 (1973). 4. J. A. Stratton. Electromagnetic Theory (McGraw-Hill, New York, 1941). 5. H. J. AurnfKampe and H.K. Weickman, Meteo. Res. Revs., Summaries of Progress from 1951 - 1955, 3 (1955). 6. A. Ono, J. Atmos. Sci., 26, 138 (1969). 7. G. W. Kattawar and G.N. Plass, Appl. Opt., 11, 2851 (1972). 8. A. C. Holland and G. Gagne, Appl. Opt., 9, 1.113 (1970). 9. H. Jacobowitz, J. Quant. Spectrosc. Radiat. Transfer, 11, 691 (1971). J0. K -N. LioLt, Appl. Opt., 11 667 (1972). 11. K. -N. Liou, J. Atmos. Sci., 29, 524 (1972). 12. K.-N. Liou, J. Geophys. Res., 78, 1409 (1973). 13. E. F. Knott, and T. B.A. Senior, University of Michigan Radiation Laboratory Report No. 0110764-1-T, Ann Arbor, MI. (1974). 14. R. F. Harrington and J.R. Mautz, IEEE Trans. AP-23, 531 (1975). 15. R. F. Harrington, Field Computations by Moment Methods (Macmillan, New York, 1968). 16. H. Well and T. B.A. Senior, University of Michigan Radiation Laboratory Report No. 013381-1-F, Ann Arbor, MI. (1976). 17. W. M. Irvine, and J. B. Pollk, Icarus, 8, 324 (1968). 18. J. W. Schaff and D. Williams, J. Opt. Soc. Amer., 63, 726 (1973). 13

F;~), Captic Figure 1. Refractive index of ice i. the infrared. The solid line represents data. 17 from Irvine and Poll ck, the O poits represent data from Schaff and 18 Williams: (a) real part of refractive index minus one vs. wavelength; (b) imaginary part of refractive index vs. wavelength. Figulre 2. Angcular distribution of the normalized scattering cross section o(0, 0 )/X vs. 0 for fixed incidence angle 5: hexagonal shell cylinder 3pmm on a side irradiated edge-on;. -.- same cylinder irradiated edge-on; - - -- circular shell cylinder of radius equal to the side length of the hexagon; - - circular shell cylinder of same perimeter length as the hexagon. Each cylinder has a simulated 0.1 pLm wall thickness. (a) wavelength X = 3.1 pm, E-polarization; (b) = 3. 1,m, H-polarization; (c) = 12.51wm, E-polarization; (d) X = 12.5ptm, H-polarization. The 17 refractive index values used are n = 1.280 + i0. 3252 for X = 3.1, and 18 n= 1.387+i 0.422 for X = 12.5. Figure 3. Scattering cross sections of hexagonal shell cylinders vs. wavelength, wall thickness 0. Ium; ~ edge-on, 0 face-on irradiation for Irvine and Pollack7 refractive index data; x edge-on, ) face-on irradiation for 18 Schaaf and Williams refractive index data: (a) backscatter, E-polarization; (b) backscatter, H-polarization; (c) forward scatter, E-polarization; (d) forward scatter, H-polarization. 14

.R iLj <' 3 FiguEre,..ap..tIon..s ("Co '-,ti-,, Figure 4. Total absorption cross sections of hexagonal shell cylinders for data of Figure 3. (a) total scatterEig, E-polarization; (b) total scattering, H-polarization; (c) absorption, E-r-polarization, (d) absorption, Hpolarization. 15

0.68 -Ir 10.4 -N 00 00 40 0 0 024 6 8IC) 12 4

100 0 0, (E) 0Dr, i0-2 ni 1(-4 I0-6 10-0 ---------- J 0 2 4 8 10 12 X (/.m)

20 -I0. 0 i I1 I A 'e 0 -J 0I 000.0 - " 0 #I N 1% ol % N % V, '... N Nll t I I I 1 I I %, % , %%, - p -10 lb.11 0 / / 7 -20 — 30 -.40 I I III f I I I f I I I I I I I I i I f i i /0 711 0.111' I I* I 0 30( d5 ( DEG R EES ) 019r~ I; A ~I ' I. I

,I- K%., r , k - " 4' -; "' " IL-x 'U.,,.., C)Q C) c7) Liu LdJ 10 0.0 0 0 0 0 0 0 1 0i CJ 0 X Cf -o

-I0, -2.0 b:A-< -3"0 0 -J O- AO1.0 -50 -60. / "IN / I I I I ' I 'II A (7 —) 4.& ( D 40, I " " " 4,1

C) C) IL() C) 1c\J Ie-:1* /jO:11 0 0 T M

IM I10 0 cr(0) -10 (dI/.Im) -20 - 40 i E —po!. II I i t: \t 'h YC - I 0 1 2 3 4 8 9 0 II 1 12 t3 X (/Lmn) 20 i0 0 cr (0) -I0 (dB/ m) -20 -30 -40 Ie 0. ( O' IS 10I, Ii 91 H- pol.. I.. - -I - I *! I Iq I *. I I i, -.,ll L]I__ f1 I t. A 0 I 2 3 4 8 9 10 II 12 13 X(p. m)

() 0 c (-,T) -10 (d Bp, r) t 17 (C- Pr2;cII 0- i kol--. 0 -30, - 40 L I_ _ I4.I 0 I 2 3 4 8 9 10 II 12 13 "A(/.Lrn) 20 10 0 cr (7T) - 10 (d DLrf) - 20 -30 -40 H- pol. i II I - -j [ J IL ----L 1 0 1 2 3 4 8 9 10 I11 12 13 X/I~m)

20 10 0 [- pot. G\ 1..-~10 (d E~r tn 'Id - 20 - 30 — 40 ii - --- - I I I I - - - I -I- -' ) o 1 2 3 4 8 9 X(LL) I 10 11 12 13 20 10 0 %V (d 8/. Lmi ) H-pot. - 10 - 20 41b, I I I-,. I~ I I II 9 -i — 7 ".1 I 0. - I 2 3 41 a 9 1O I11 12 13 X yIL m),

^. r, :,." r 10.."10 - 3C) -40 E- pol. A"I -fl?% Vy c L JL.-.1... I. ZA jZL0 1 2 3 18W. 9.. tO It 12 13 >X ([L M) ~O-II 10 0 H- pot1. (dBLm -40 I -1? I, I I 1 —4. 0 2 3 4 8 9 tO It1 12.13 X(~Lr)

0.8 0 0.06 0 0 0.2 0 00 o 00 0II 0 2 4 6 8 10 12 X (p m) Figure 1 a.

0 0D 0 0 io- 2 ni i -L -L I 0 2 4 6 8 10 12 X yp.m) Figure lb.

21 I0. 0. bI-< 0D 0 -J 0 m.I0. -20 -— 30 - so 11*.0...OW G- a" N.. I 00000 NNIS, to p0 / / F I I a 700 7100 / I 'I -4 0 30 60 90 Z (DEGREES) 120 150 180 Figure 2a.

I0 0 op 0/ 0.01/ ~-20 10I 00/ dM44 50/ '40609 1201/ 0 (DEREES 18 Figure 2 b.

1I tI 0 NC x 0 - -~- - 7:6 9.'l 0 I OJ

-10j -20 -0 40 -50 - -60 -— " ---.....-,-....... —.....-,' — 0 30 60 90 120 150 Z (DEGREES) 180 Figure 2 d.

20 10 0 a-(O) -10 (dBpm) - 20 - 30 6 E -pol. h I I I II II I I% % 6 1 2 3 4 9 10 11 12 13 X (L m) Figure 3 a. 20 10 0 o-(0) (dBp.Lm I -I0 -20 - 30.-40 H -pol., II I I -AU I I C I I IIf 1 2 3 4 8 9 10 II 12 13 X(p.rn) Figure 3 b.

20 10 0 cT (lr - 10 (dB/.Lm) - 20.30 - 40 E- pol. I I IIIIII. 8 9 IC II 12 13 C 1 2 3 4 Figi 20 10 0 o-(r) -10 (dB M) ' - 20 - 30 ~ 40 H- pot.. S,%I I I III C I I I 1 1 2 3 4 Fi~ ~ 1 8 9 10 II 12 13

20 I0 0 IT -"I010 (dBI.Lm) - 20.-30 -40 E -pol. I I I 0 1 2 3 I0 20 i0 T10 (dBF.Lm) — 20 - 30 - o40 4 8 9.Figure 4 a..10 II 12 13 H-p'ol. — j I -I.7 11 0 1 2 3 4 8 9 X(~Lm) Figure 4 b. II I i 10 II1 12 13

20 I0 0 E- pol. cA (dB/im) -I0 - 201 - vIi I I I I I -30 -40 I -- I. I I LsiZ ) I 2 3 4 8 9 10 II 12 13 X(Fm) Figure 4 c. 20 10 0 I0 H-pot. I (dB/.m) -30 -40 I I I I7 2. I I I I I I 0 I 2 3 4 -8 -X(I.m) Figure 4 9 IC II1 12.13:d.