013381-1-F 13381-1-F = RL-2261 THE UNIVERSITY OF MICHIGAN COLLEGE OF ENGINEERING DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING Radiation Laboratory INVESTIGATIONS OF ELECTROMAGNETIC SCATTERING BY COLUMNAR ICE CRYSTALS Herschel Weil and Thomas B. A. Senior 5 March 1976 k Final Report Grant N8G 5044 Prepared for NASA Goddard Space Flight Center Greenbelt, Maryland 20771 Ann Arbor, Michigan

Abstract An integral equation approach is developed to determine the scattering and absorption of electromagnetic radiation by thin walled cylinders of arbitrary cross-section and refractive index. Based on this method extensive numerical data are presented at wavelengths in the infrared for hollow hexagonal cross section cylinders which simulate columnar sheath ice crystals.

TABLE OF CONTENTS 1. Introduction 1 2. Numerical Results 5 2.1 Angular Data 6 2.2 Spectral Data 8 3. Conclusions and Suggestions for Future Work 10 References 11 Figures 12 Appendix: Scattering by a Cylindrical Resistive Shell 48 A. 1 Formulation 48 A. 2 Scattering by a Resistive Membrane 49 A. 3 Scattering by a Cylindrical Resistive Shell 54 A. 4 Computer Programs 65

1. Introduction This is the Final Report on NASA Grant NSG 5044 and describes the work carried out in computing the scattering and absorption of electromagnetic radiation by columnar sheath ice crystals. In this first section we begin by citing some of the important geophysical problems which require such data, and note some of the consequences of the lack of sufficiently accurate values for the scattering and absorption properties of single ice crystals. After surveying the work done in recent years to remedy this deficiency, we then summarise our own contributions to the subject. The detailed results are given in the remaining sections of the report. The reflection, transmission and absorption of visible and infrared radiation by clouds and by polluted atmospheres is of considerable importance in many practical areas. Cirrus clouds are composed of ice crystals. They are found over the entire globe and their infrared and optical scattering properties have a profound effect on the atmospheric heat balance. The clouds scatter and absorb the primarily infrared radiation resulting from the earth and the lower regions of the atmosphere as well as the mainly shorter wavelength solar radiation incident from above. The difference then plays a major part in the atmospheric heat balance which governs the global location of energy sources and sinks and, hence, the atmospheric circulation patterns (Cox, 1971; AFWG, 1972). Similarly, but on a much smaller scale, the cirrus clouds created artificially by jet contrails have been observed to markedly affect local weather (Appleman, 1966; Reinking, 1968); and a knowledge of the scattering by clouds is also needed in using satellite measurements of the IR emission of water vapor to estimate the relative humidity of regions above the clouds. Finally, we remark that the use of LIDAR as an atmospheric probling tool depends on the difference in the reradiation of the aerosols and the much smaller background molecules (Grams, 1975). 1

Techniques for calculating the transfer of electromagnetic radiation through clouds of particles have been summarized by Plass et al. (1973), and one of their most basic ingredients is a knowledge of the scattering and absorption properties of the individual particles. This may involve either single scattering or multiple scattering in the case of optically thick clouds. Since each scattering event can affect the polarisation by producing an electric field having a component orthogonal to the incident vector as well as parallel to it, and this field is in turn incident on another particle, an accurate treatment of the transfer problem will involve the complete scattering and absorption matrices for a single particle. In an atmospheric cloud, the particles are either water droplets or ice crystals. The water drops are close to spherical and it is not unreasonable to model them as spheres. This enables Mie theory to be applied and the results obtained are reasonably accurate. Ice crystals, however, are another story. The shapes, sizes, concentrations and fall patterns which have been observed in clouds have been discussed by Mossop and Ono, 1969; Ono, 1969; Aufm Kampe and Weickman, 1957; and Heymsfield and Knollenberg, 1972. Both plate crystals, i. e. cylinders of length much less than their diameter, and columnar crystals, i. e. long thin cylinders, hollow as well as solid, are commonly found, and for shapes as varied as this, a sphere cannot provide an accurate simulation of the scattering behavior. Nevertheless, for lack of a more accurate method for calculating the scattering properties, the Mie theory has been widely used even for ice crystals, thereby introducing unknown and possibly large errors in the values for the radiation transfer, which are the end products of extensive and expensive computations (Kattewar and Plass, 1972). The importance of using the proper scattering matrix when the particles are irregular is clear from the data presented by Holland and Gagne (1970). They measured the elements of the scattering matrix for clouds of irregularly shaped, randomly oriented silicon flakes. The results were quite different from the matrix elements predicted by Mie theory, particularly for back and forward scatter, and the depolarization was also poorly predicted by the theory. 2

The last few years have seen several attempts to calculate the scattering of more realistically-shaped crystals, and it is appropriate to mention here the work of Jacobowitz (1971) and Liou (1972 a and b; 1973) which has been directed at the scattering properties of columnar ice crystals. Jacobowitz's data were obtained for infinitely long crystals hexagonal in cross section using ray tracing. All end effects were necessarily omitted, including the 45 deviation of the rays passing through the end faces which contributes to the large halo observed about ice clouds (see Minnaert, 1954, section 104). The method also excludes all diffraction effects produced, for example, by the six longitudinal edges of the cylinder, as well as polarization effects, and the calculations were limited to cylinders not less than 40s in diameter (for a wavelength of 0.55 M) 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 ice crystal can be modelled by an infinitely long, homogeneous dielectric cylinder of circular cross section. For this simplified geometry there is a mathematically exact expression for the scattered field in the form of a series of Bessel and Hankel functions analogous to the Mie series for a sphere. It is therefore possible to compute the scattering matrix precisely, with all polarization information present, and with internal absorption taken into account. Nevertheless, end effects are omitted by virtue of the model chosen, and the assumption of a circular cylinder necessarily suppresses those features of the scattering which are peculiar to the hexagonal cross section of an actual ice crystal. The retention of the hexagonal geometry is one of the key features of the work carried out under the present Grant. Based on an integral equation approach, a numerical technique has been developed to compute the scattering patterns, and the scattering and absorption spectra for cylindrical dielectric shells of arbitrary cross sectional shape when illuminated by a plane wave of either principal polarization. The dielectric can be lossy, and by applying the method to infinitely long cylinders 3

hexagonal in cross section, scattering and absorption data have been generated applicable to hollow columnar (sheath) ice crystals in the infrared. The method originated from a study of the scattering properties of resistive sheets and membranes (Knott and Senior, 1974) in which the non-zero thickness sheets were simulated by infinitesimally thin sheets of appropriate electromagnetic properties. Accordingly, a hexagonal shell cylinder whose actual walls are composed of a material of (complex) dielectric constant n, having thickness T(small compared to the free space wavelength X), is replaced by a hexagonal membrane having a complex relative resistivity R iX Z (n 2-1)22T ohms per square. It is then possible to derive an integral equation for the current which an incident plane wave of either principal polarization induces in the membrane, and the integral equation is quite amenable to solution by digital techniques. The formulation of the equation and its subsequent solution constitute a significant contribution to the theory and application of integral equation methods to electromagnetic scattering and absorption problems. The mathematical details are given in the Appendix along with a listing of the computer programs employed in generating the data in this Report. For the most part the formulas are applicable for arbitrary angles of incidence, but the numerical results presented here are for broadside incidence only. The problem is then two-dimensional, and we now turn to a presentation and discussion of the data obtained. 4

2. Numerical Results Our numerical results are given in the form of cross sections which are defined as follows. For a power density S incident on the cylinder, the bistatic scattering cross section is o(0) = 2TI/S where I is the power scattered per unit length of the cylinder per unit angle about the direction 0 and measured in the far field of the cylinder. The angle 0 is defined so that 0 = 0 is in the backscattering direction and 0 = X is in the forward. The total (or integrated) scattering cross section is then 2r aT = ()d. T 2 0 The absorbed power is measured by the absorption cross section aA = - (power absorbed) A S and the extinction cross section is the sum aext = T +A Formulas relating these two dimensional cross sections to the currents which are computed are given in the Appendix, eqs. (37) through (41). The computations were carried out using the refractive indices n = n + in. r 1 for ice in the infrared wavelength range given by Irvine and Pollack (1968; hereafter referred to as IP) and Schaaf and Williams (1973; referred to as SW). Their data are plotted in Figs. 1 and 2 and show significant discrepancies in certain wavelength ranges. To obtain some idea of how sensitive the scattering is to the particular refractive index chosen, computations have been made at two wavelengths using the values from each reference. For a given wavelength and perimeter of the hexagonal sheath, the scattering has been computed for the incident plane wave polarized with its electric 5

vector parallel to the axis of the cylinder (E polarization) and also with its magnetic vector parallel to the axis (H polarization). The directions of incidence and observation are always in a plane perpendicular to the axis, but for each polarization we have considered two directions of incidence corresponding to 'edge-on' and 'face-on' as regards the hexagon, viz. edge-on face-on Most of the calculations have been for a hexagonal cylinder 3 pm on a side with a wall thickness T = 0. 1 pm. Only these data are presented here though we have carried out some exploratory calculations for other parameter values. Table 1 lists the wavelength, the corresponding refractive index and its source, the appropriate resistivity value and the Figure numbers where the computer-generated plot of the bistatic scattering versus 0 can be found. Each of these Figures shows the data for edge-on and face-on incidence on the left and right respectively, with the intensity on top and the phase below. The phase is that of a scattered field component at a large (constant) distance from the axis of the cylinder and is shown relative to that of a line source on the axis. The intensity plotted is actually the dimensionless quantity u(6)/X in dB. This particular normalization is convenient for computation and presentation purposes, but since a(0) is a very complicated function of A. (through, for example, the refractive index), it must be borne in mind that u(0)/X is not a wavelength- independent quantity. Spectral information is presented in Figs. 31 through 35 where v(0), o(r), aT' sA and aext are plotteddn dB pm) versus X. Note that the explicit factor X has been removed, so that here the 'normalization' is relative to a micron (pm). 2.1 Angular Data The curves in Figs. 3 through 30 are self explanatory. They clearly show 6

Table 1: Computed Data for Hexagonal (Shell) Cylinder 3 um on side, 0.1 um thick X(um) Pol. n Ref. R/Z Fig. No. 0.76 1.61 2.0 2.25 2.6 2.8 3.0 3.1 3.3 3.5 8.0 9.0 11.0 12.5 E H E H E H E H E H E H E H E H E H E H E H E H E H E H E H E H 1.307 1.293 + i0.000365 1.291 + i0.00161 1.254 + iO.001 1.278 + iO.000213 1.206 + iO.00080 1.152 + iO.0123 1.130 + iO.2273 1.045 + i0.429 1.280 + i0.3252 1.530 + iO.0625 1.422 + iO.0163 1.312 + i0.045 1.269 + iO.043 1.093+ iO.242 1.387 + iO.422 IP IP IP SW IP IP IP IP SW IP IP IP SW SW SW SW 0.00536 + 0.02977 + 0. 02740 + 0.00486 + 0.03876 + 13.5260 + 7.79740 + 5.26973 - 4.20510 + 0.55090 + 0.24680 + 2.82939 + 4.09006 + 31.03981 + 12.08921 + il. 70784 i3. 81395 i4.77438 i6.25466 i5.65462 i9.10650 il. 17250 i3.41910 iO. 54082 i2. 69040 i3.84930 i5.44060 i17.23610 i22.80586 i7.98514 i7.70071 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 7

how the number of maxima and minima in 0 < 0 < ir increases with decreasing X, and at wavelengths which are much longer than the side length of the hexagon, i. e. X > 8 Mm, the cross section has almost no angular structure. Changing the incidence from edge-on to face-on has most effect in the directions closest to backscattering, and we also note the substantial differences between the results for E and H polarizations. 2.2 Spectral Data The particular cross sections or(O), a), UT, A and ae are shown as functions of A in the infrared range in Figs. 31 through 35. Each Figure has two parts, covering the ranges 0. 76 to 3.5 pm and 8 to 12.5 pm. Separate curves have been included wherever the results for edge-on and face-on incidence are clearly distinguishable. For the most part this is only with the backscattering cross section or for X S 1.5 pm, and in other cases the differences are confined to the immediate vicinity of local maxima or minima in the data. In the shorter wavelength range most of the data were computed using the IP values for the refractive index. This range covers the main absorption band centered on X = 3 pm and a secondary one at X = 2,um as seen in the IP data plotted in Fig. 2. The wavelengths close to these show the main discrepancies between the IP and SW data, and at X = 2.25 and 3 um we therefore ran computations using both sets. The different refractive indices at X = 2.25 pm do indeed produce substantial differences in the cross sections, and because of this sensitivity, we have not extended our detailed computations through the third absorption band centered at X n 1.25 pm in the IP data. For X < 1.61 pm the scattering has been computed only at the single wavelength X = 0.76 um of interest for a particular application. Since the IP data are given only for X > 0.95 um, the necessary refractive index was obtained by extrapolation. The SW data for the refractive index were used in the longer wavelength range 8.0 < X < 12.5 pm. The geometrical effects are particularly pronounced for 0.76 < X < 3.5 pm. This is not surprising since the dimensions of the cylinder are then comparable to 8

the wavelength, or are low multiples thereof, and this is the region where resonance effects and other interactions between the various contributors to the scattering are most important. As an example, while aA has a strong local maximum near the maximum in n. at X = 3.075 Mm, the shape and overall width of the maximum in aA is apparently affected by the side length of the hexagon being close to this wavelength. a(0) and a(7r) both show a corresponding drop in this absorption region. The behavior is quite different near the secondary maximum in n. at X = 2 pm. For H-polarization but not for E, aA is large as expected, while ou0) and oir) have local maxima for X just above 2 Mm 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 pm computed from the IP and SW data. At this wavelength the SW value for n. is roughly five times the IP value, with n almost equal in both listings, but the SW value produced an absorption cross section which is about eleven times less than that given by the IP value for the refractive index. It is therefore obvious that any predictions of absorption and scattering based only on the properties of the material of which the scatterer is composed may be considerably in error. 9

3. Conclusions and Suggestions for Future Work In this Report we have derived the theoretical basis for determining the scattering and absorption of electromagnetic radiation by thin-walled cylinders of infinite length and arbitrary cross sectional shape. Numerical procedures have been developed and have been used to obtain data for hexagonal cylinders which model sheath crystals of the type found in cirrus clouds. The procedures are economical for cylinders whose cross sections are not more than about 15 wavelengths in perimeter, with the cost decreasing rapidly with decreasing size. For wavelengths comparable to or less than the face length of the hexagon, the results (particularly for the back scattering) are quite sensitive to the polarization and direction of the incident plane wave, i. e. on whether the field is incident edge-on or face-on as shown on p. 6. At any given wavelength, the results can also be very sensitive to the refractive index employed. Geometric effects can so influence the absorption that it is not at all safe to assume that the absorption versus wavelength curve will follow that of the imaginary part of the refractive index. The time available for the present study did not permit an adequate investigation of the effect of wall thickness, nor allow us to do a detailed comparison of the results with those of the Mie-type series for hollow cylinders circular in cross section. Although our data for the bistatic scattering cross section versus angle are somewhat similar to those previously published (Liou, 1972a) for solid circular cylinders at near broadside incidence, data showing the precise role played by the geometry should have a high priority in any future continuation of the study. With only minor modifications our computer programs can also handle irregular additions to the hexagonal surface, thereby simulating rimed crystals, and allowing us to compute the effects of riming. Since the theory for non-broadside incidence has been derived, we would also like to develop the computer programs necessary to obtain numerical data in this more general case, and once these types of data are in hand, it would be possible and desirable to examine the forms of averaging that could simplify the practical applicability of the data without losing its essential properties. 10

REFERENCES ARWG, (1972), "Major problems in atmospheric radiation: An evaluation and recommendations for future efforts, " Bull. Amer. Meteo. Soc., 53, 950-956. Aufm Kampe, H.J. and Weickman, H.K., (1957), "Physics of clouds," in Meteorological Research Reviews, Summaries of Progress from 1951 -1955, 3 Appleman, H.S., (1966), "Effect of supersonic aircraft on cirrus formation and climate, " AMS/AIAA Conf. on Aerospace Meteorology, Paper No. 66-369. Bateman, H., (1915), Electrical and Optical Wave Motion, Cambridge University Press, p. 19. Cox, Stephen K., (1971), "Cirrus clouds and the climate, " J. Atmos. Sci., 28, 1513-1515. Fritz, S. and Krishna, P. Rao, (1971), "On the infrared transmission through cirrus clouds and the estimation of relative humidity from satellites," J. Appl. Meteo., 6, 1088-1096. Grams, G.W., (1975), "Lidar: Some current uses and potential applications in the atmospheric sciences, " Atmospheric Technology (NCAR), Winter 1974 -1975. Heymsfield, A.J. and Knollenberg, R.G., (1972), "Properties of cirrus generating cells, " J. Atmos. Sciences, 29, 1358-1366. Holland, A.C. and Gagne, G., (1970), "The scattering of polarized light by polydisperse systems of irregular particles, " Applied Optics, 9, 1113-1173. Irvine, Wm.M. and Pollack, J.B., (1968), "Infrared optical properties of water and ice spheres, " Icarus, 8, 324-360. Jacobowitz, H., (1971), "A method for computing the transfer of solar radiation through clouds of hexagonal ice crystals, " J. Quant. Spectrosc. Radiat. Transfer, 11, 691-695. Kattawar, George W. and Plass, Gilbert N., (1972), "Degree and direction of polarization of multiple scattered light. 1. Homogeneous cloud layers," Applied Optics, 11, 2851-2865. 11

Knott, E.F. and T.B.A. Senior (1974), "Non-specular radar cross section study,)" University of Michigan Radiation Laboratory Report 0110764-1-T (AFALTR-73-422). Liepa, Valdis V. et al., (1974), "Scattering from two-dimensional bodies with absorber sheets, " University of Michigan Radiation Laboratory Report 0110764-2-T (AFAL-TR-74-119). Liou, Kuo-Nan, (1972a), "Electromagnetic scattering by arbitrarily oriented ice cylinders, " Appl. Optics, 11, 667-674. Liou, Kuo-Nan, (1972b), "Light scattering by ice clouds in the visible and infrared: A theoretical study, " J. Atmos. Sci., 29, 524-536. Mossop, S. C. and Ono, A., (1969), "Measurements of ice crystal concentration in clouds, " J. Atmos. Sci., 26, 130-137. Ono, A., (1969), "The shape and riming properties of ice crystals in natural clouds,"J. Atmos. Sci., 26, 138-147. Plass, Gilbert N., Kattawar, George W., and Catchings, Frances E., (1973), "Matrix operator theory of radiative transfer Rayleigh scattering, " Applied Optics, 12 314-328. Poggio, A. J. and E. K. Miller (1973), "Integration equation solutions of threedimensional scattering problems, " in Computer Techniques for Electromagnetics (ed. R. Mittra), Pergamon Press, New York. Reinking, R., (1968), "Insolation reduction by contrails, " Weather, 23, 171-173. Schaaf, Joel W. and Williams, Dudley, (1973), "Optical constants of ice in the infrared, " J. Optical Soc. of America, 63, 726-732. Senior, T.B.A., (1975), "The spherical cavity problem, " the Air Force Weapons Laboratory Interaction Notes, Note 220. Stratton, J.A., Electromagnetic Theory Mc-Graw-Hill Book Co., New York, 1941. 12

1.0 I 0.8 0 0.6 0- o nr-I 0.4 00 000 00000 0 0 I I o X( pLm) Fig. 1: Data for the (reduced) real part of the refractive index of ice: (Irvine and Pollack,1968),. (Schaaf and Williams, 1973). 13

0 0 0 0 164 1-5 i06 0 2 4 6 8 10 12 \(/,m) Fig. 2: Data for the imaginary part of the refractive index of ice: (Irvine and Pollack, 1968), *.. (Schaaf and Williams, 1973). 14

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W GM-ON, THICVI-ESS - 0.1 Mal=Y4 163. 120. 140. 060. M0. 0. THETA (DECREES) ICE CRY5TPL. 3 MICRONS ON P SIDE. LP~1,9R= 2.P0 MICRON (H-POL) SIM(T/(~~~0.00363 -20.61I CB SIGi1H(H)/LWRi3DR= 0.21783 -6.62 08 0o. 20. 42. 60. eo. I 'O. 120. 1140. 08. 8. THEIR (DEGREES) ICE CRTSTPL. 3 MICRONS ON P SIDE. LRMRRR= 2.80 MICRON (H-POLl SICMfl(I)/LRl'3Oil 0.00776 -21.10 08 SI GMA (HI/LRMBOfP 0. 21939 -6.S8 06 F ig. 12

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Du-im, T"tcvmss - 0.1 mtcFor. FACE-o", rmicouicss - 0.1 micqms I i i i i 4 0 4 I ~D0E-~I, ThIC614CS1 - 0.1 MICF6*~ FC6-P.mi ~ -01MC0 mc 6c a:9 a:) cc s 0. 2. 40 60. 6~1. Is). 20. 140. 1,60. 0. CIO THETR (DEGREES) (Jo 14 L ~ a_ _ _ _ _ _ _ _ _ _ _ _ _ _ 0 20 4060. 60. e 00. 120. (40. 160. 180. THETA (DEGREES) ICE CRYSTn!. 3 MICRONS ON A SIDE. LFPh97F)l 3.39 MICRON (H-POLl SIGMAl(T)/LftM~U-!hl1' O. 10J~31 -9.61 0U SIlGVfl(f0/LfiMtU9)A 0.094811 -10.23 08 S i 6.I 20 4. 0 80 i0. (0 4. 10 THETA (DEGREES) 1140. 160. 11 W. I0. 20. t40. 60. P0. 100. 120. 1140. 16O. 19J. THETA iDOEGRIES) ICE CRYSTPL. 3 MICRONS TiN A 5IDE. LPMROni= 3.30 MICRON IH-POLl IAhLA{4 0.11204 -9.51 Oal SGt&/AhA 0.08622 -10.64 08 Fig. 20

In t i 7 w 1 4 f w w.0 c 3SU,48o co CZ) Xcr) a7; x -) I- CI:"3 to I CD Ur) cc 0 CD Cr) CIO cr. Ct) T: C')z Lii a: LD Li L C), r —q cq b.r-j Pr., IC 9 r C).1I w 2!t! i-C 33

EtXM-M,4 MIMCVESS 0.1 MIC FC-C.?1C! -q MC2 F ACE -014, Di: -1 - 9. 1 MlCt" -I-, I i 0 1 i 6- -- i a~i, (LI 0. 4.. 60 IC). 12g. V640. ISO I 0. o 0 '1. 60. 0) 2. 12. 6. 160 160. ICE CBrYFTPL. 3 M~FFi:CUS Ct; P SIDE. LFh'Sr,5P 3.50 ti[CRCrIS,(t-POLl SIGM1R 11LF:) 0. 607J -12 16 [D3 SIGCilH~) 'LFM~BD- 0.06207b -16.83 06 '0. - 20. '10. 60. W0. 100.. 1.00. 140. 16FO. 180. THEM (DEGREES) ICE CRYSTAL, 3 MICRONS CTi A SIDE. LPM9rP= 3.S0 MICROlN (H-POLl SlGMfl(T)/LF,.6btiS 0.041656 -13.32 CE3 SIGMR(ARI)/LfM3DP0- 0.02126 -16.72 06 Fig. 22

I I -i, C: 11 V. IV se a f I.ol I I 9 x c::p.111 i i 9, 0~ IW-~.2. C12 CC) tif 0 C9~ ~ A 0 r (S33Y~) ~C., co C\l b. F.4 44 - I Ir 0 *&WI *ORt *Q 35

. EDrA-oN, THICOMSS - 0.1 MICROUS 2 - -.FXE-M, THICVJJESS - 0.1 MICRUIS 9 - - *r ~ 0. i J I._.. Qr a: orF!,. 9 I 1 I i. i i C; -J X:..,0 20. 40. o 6. 60. 12.. 1. 120.. 60. THETl (DEGREES) 8.,,,,,-,-,-,. -- 'A IL IL a 0. 20. '0. 60. 63. 100. 120. 140. 160. 180 THETA (OEGFEES) J. J 'i J -. -=,.::.,,..,....... 0. 2 C-') ui U-i m Lai e C; W (n cr 3.:. CL c. If 9 I I 2 I - = I a 0 i -V. 2. 1U. o0. J. I. 120. 12. 40. t60. 100. THETR (DEGREES) ICE CRYSTRL. 3 MICRONS ON A SIOE. LRngR9= 8.00 MICRON (H-POLl SIGMR(T)/LfMBC~f= 0.00233 -26.33 08 SIGHlA(R)/LRMBHR= 0.01007 -19.97 OB '0. 20. 20. 60. 60. o0. 120.. 140. 160. 10. THETR (CEG:EES) ICE CRTSYTRL. 3 MICPCN.5 ON P SIDE. LRMBnR= 8.00 MICRON (H-POLl SIGMR(T)/LF;i5z-b= 0.00234 -26.32 GB SIGMRR) /LFiBOR= 0.01009 -19.96 08 Fig. 24

I. 4. 0 4 i 0 Oo M4 oL.j a;Li L C; C; VI) 0r c (u a;C., I x -i Cp I11 w w I.c la 9 x C=0 s w 3 1. X.Del.0(S3q Ozl) 3odld (f) bfl.-1 - n, -~ cr~ r 4 -U, m uj w, ICC JLi u a;C v-c' I-4~-L tr, 0V *0.44 i.:0 -i I la~% 0ozi.09 *O *0)p- *OZI- WCj:o (S33~333) 3&81Nd 37

I~o I o i co *oi oo, Ln.. t,9, tsi U: L:) w p cc , P CD -0 U:r.J C-D CD) -JJ Ci) Ci3 C-a Li C) ~cc a: 7; S2 9 x I-! T w;z U i-C I IS3360J0) 3SUNId ~; oe 0? 09- 0~I- L0A. O) bfl 0 0L zg Li C] CAic Cr) 0 aI-i Li.(at.WI LS30 W JO J f- ct.9 d I 38

01 C,, ~ c U"U, a:) Lij u, a: ~C)m Cno 0 6W Ds clo- CIcI )0 - -(SJ3~3430 3SiBR44 3StU.4J tC\],4 to 2 IJ, 14 LL L:) ul p ;E., a" v I.C; I to 10 9 x -4 9 C=; a k i-r co Li CtL ci ccO 39

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0 i i 0 i Lf) a) 0 0 * CC * rOo.7 a:) C-i Lcc L-n (~a: N.00) CL.0 0 09 oat- o9L~ t&3~93Q 3SBC4 1sr I c) LI) CD. c-J cr_ ciu JLUl t CNI CL 2::;7co 41)."I (SJ3030) 3SB84d,061-:1 42

20 10 o (0) 0 (d Bm) - lo I I -I I I I I I I I( - 1 2 3 4'S E-pol. I I 1 I I I I I 1 9 10 II 12 13 -201 -301 -40 ( X(/ m) 201-.0 H-pol. 10 0 o-(0) (dB. m) -10 - 20 - 30 -40 & -. - O r I I I II I I I I I I I I I I 1 - ~ I.. '. I = = I I I a " __a 0 1 2 3 41 8 9 10 11 12 13 X(Hfm) Fig. 31: Computed backscattering cross sections of hexagonal (shell) cylinders. In this and the following four figures, -and e show edge-on and face-on results respectively for IP data; similarly x and O are for SW data. 43

6 20 10 r-(r) 0 (dBbm) -10 -2( -3( -4C 2C 10 (7r) 0 (dB.Lm) -10 ) ) i E-pol. I a I I I I I ) )l i I a I I I I I * 0 1 2 3 4 8 9 10 II 12 13 X(Hm) )H-pol. o ~1 H-pol. I ) -20 -30 -40 I I I I I I I I ') I a a I I a I I I I 0 I 2 3 4 8 9 10 II 12 13 X(/m) Fig. 32: Computed forward scattering cross sections of hexagonal (shell) cylinders. 44

20 10 aT 0 (dBeLm) -10 -20 - 30 -40 E- pol. I I I I I I I IA. 4 I I I I I I I I S \ y - V ~^ ^ — ^^~-r - 1^-^^ ~ ~ - -.. 0 I 2 3 4 8 X(HLm) 9 10 II 12 13 20 10 0 bT (dBm m) - 10 -20 -30 -40 0 H-pol. I I I I I I I, I ~ I i I I, I I I I I 1 2 3 4 8 9 10 II 12 13 X(~m) Fig. 33: Computed total scattering cross sections of hexagonal (shell) cylinders. 45

20 I0 0 E -pol. (dBIm) -20 -30~ -40 S t I I I I 0 -TI.o 0 I 2 3 4 >X(dbm) 9 10 11 12 20 10 0 -10) H-pot. (dB/Lm) -20 -30 -40 p I I I I I I I I I 0 I 2 3 4' X (di bin) 9 10 II 12 Fig. 34: Computed absorption cross sections of hexagonal (shell) cylinders. 46

20 E- pol. 10 e xt (dBFm) - 0 -20 - - 30 - 40 I I I I I 1. 1 I I I I I I I i 0 I 2 3 4'8 9 10 II 12 13 X\(Lm) 20 H- pol. 0 0 ext (dBLm) -10 -20 -30 -40 I I I I I i i ] I I i i f I I i i 1 0 I 2 3 4 8 9 10 II 12 13 X(Mm) Fig. 35: Computed extinction cross sections of hexagonal (shell) cylinders. 47

Appendix: Scattering by a Cylindrical Resistive Shell A-1 Formulation The shell crystal is simulated by a hexagonal cylindrical shell of uniform thickness and constitution, which is in turn treated as an infinitesimally thin, electrically resistive membrane. The concept of such a membrane arises naturally from a consideration of a thin sheet of highly conducting material whose permeability A is that of free space. If a is the conductivity and r is the thickness, we can define a surface resistivity R as 1= -- (1) CTT Sie-Y T u"e where X is the electric susceptibility, e0 is the permittivity of free space and a time factor e has been assumed; and as T ->0 we can imagine a to be increased in such a manner that R is finite and non-zero in the limit. The result is an idealized (infinitesimally thin) electrically resistive sheet whose electromagnetic properties are specified by the single quantity R. In terms of the (complex) refractive index n of the layer material i, (2) R = _ -------- (n - l)kT where k and Z are the propagation constant and intrinsic impedance of free space, respectively. Mathematically at least, the membrane is simply an electric current sheet 2 whose strength is related to the tangential electric field via the resistivity R ohms/m, which may be a function of position. Since p = p0, there is no magnetic current present and +- ) (3) A(E -E 48

where the affices + refer to the positive (upper or outer) and negative (lower or inner) sides of the sheet and fi is the outward normal to the positive side. If J is the total current A + n (HA -H )- =J (4) and from the definition of the surface resistivity A + A ( E -) - J (5) The tangential components of the electric field are therefore continuous across the sheet, whereas the tangential components of the magnetic field have a jump discontinuity J directly related to the electric field via the resistivity R. With R specified, the conditions (3) - (5) define a transition problem for the electromagnetic field and were first used by Levi-Civita (see Bateman, 1915) in studies of charges and currents close to a conducting sheet. A-2 Scattering by a Resistive Membrane It is convenient to start by considering a general resistive membrane in three space dimensions illuminated by an arbitrary electromagnetic field. We treat first an open sheet for which a representation of the scattered field was obtained by Knott and Senior (1974) and then examine the case of a closed resistive shell. If S is an open electrically resistive sheet and we surround it by a closed surface S1, the scattered field at any point r outside S1 can be written as ES(r) = V^+ ikZ ^_\L'l E- - (6) H (r) = ^ VA^ "- Ai. 49

-1 where Y = Z and _r and _r'* are the electric and magnetic Hertz vectors defined as *~ r W k * Sn^HgdS-' (7) si 1 rJr) -— f Y, E d S' and ikd g = lg(r') (8) 4wd with d = |r-r'| is the free space Green function. On oollapsing S1 to the two sides of 8, the expression for w(r) reduoes to iZ A r(r) - 5 5 (H+ — )gdS' l.e., ) I iZ I\J(r')dS (9) 8 where the integration is m a one-sided integration along S. Since n (E+ - E 0), we have similarly S. h )- O, (10) and eqa. and lO1)are preoisely those which would have been obtained by starting with the concept of an electric current sheet. Hence E(r) = Ei(r) + VAVA (11) H(r) = Hi(r)-lkYVAr where Ei, H are the incident field vectors. 50

If r is net on S, the derivative operations can be applied directly to the integrand ofd ) giving { g d Sr ')',}dS E(r) Ei(r)+ ikZ f.r')+ J(r) -V~V dS (12) HI(r) H r+ r') ^ VgdS'. (13) In the particular case when the idat field and the surface S are both independent of a Cartesaan coordinate z, eq.(12)was the starting point for the analysis of resistive sheets by Knott and enior (1974). Integral equations r the tangential components of J were developed by taking the limit r->S, and similar results can be obtained even in three dimensions. On the other hand, a somewhat different expression for the electric field is also possible. By applying standard vector identities to the second term in the integrad of(12), we have (Senior, 1975) E(r) = Ei(r)+ilkZ J(')g k (V -)V'g}dS' 2 * - V'g lr') A~ 'h (14) k where V is the surface divergence operator and the line integral is in the positive direction around the edge L of S. For a perfectly conducting surface the line integral vanishes by virtue of the edge condition. The electric field expression (14) is then identical to that given by Poggio and Miller (1973) for a closed surface S, and since V' J = ikYPs/0 -I kYn'.(E -E ), (15) S - S "" 51

it is also identical to that produced by application of the appropriate Stratton-Chu formula (Stratton, 1941) to the closed surface consisting of the two sides of S. For a nonmetallic surface, however, the line integral is in general nonzero. The Stratton-Chu formulas are no longer valid and to avoid the occurrence of a line integral it is necessary to use the representation (12). If S is a closed resistive shell, the Stratton-Chu formulas can be used to give the following expressions for the total field at a point r outside S: I ZAi ]as +(1A) E(r) E(r)+ \ {ikZ(n'^H ) +(n'^ )V' ) g(&'.E 'g} dS' (16) H(r) - Hr)+ {-ikY(n' E ) +(' ) g+(n' H )V'g d8' (17) where the unit vector normal is directed into the space containing r. In contrast to (12) and (13), the surface integrals now involve the fields on one side of S alone, rather than their jump discontinuities across S. We recall, however, that(16) and (17) are obtained by application of the vector Green's theorem to the volume exterior to S containing the observation point. If, instead, we apply the theorem to the volume interior to S with the observation point still outside S, g will be regular throughout the entire volume of integration, implying O 5 fi{kZ(n'H g+(n'^AE )A^V'ig+ (i' E)Vg}dS' (18) S o0 {- ikY(t'AE )g + (^h) ^V + (A1^'- )V'g}dS' (19) Byr ohoosing r the same as in eqs. (16) and (17), subtracton of d(18) from (16) gives E(r) E () W+ {kZ(' ^ a )g + ('A L i) V'g + (n' * [J)Vg dS A A [E) AV'9 + I f O where [.] = E - E and WI =- H.-H. Similarly 52

H(r) = Hi(r) + + '-IkY( + ( t and when the boundary oonditiom(3), (4) and the relation (15) are used, we have E(r) E (r) + ikZ J(r')g - (V7. J)Vvg d (20) H(r) Hi(r) + J(rt') A VgdS'. (21) S The above results could also have been obtained using a Hertz vector representation of the scattered field, and are identical to those for an open resistive sheet when the line integral contribution to the electric field is ignored. More to the point, when the steps leading from (12) to (14) are reversed and applied to (20), eq. (12) for an open sheet is recovered, thereby validating this integral representation for both open and closed resistive sheets. For a closed shell either (12) or (20) can be used to generate an integral equation for J, but the two equations are significantly different. On selecting the tangential components of (20) and then allowing the observation point to lie on either side of S, we have n E(r) = ^E (r) + ikZ rS n J(r)g - (J) - J) g (22) A-r A — 2J1 k k S and since flnV'g = ( - fil),V7'g+ 't,^V'g, the more singular term in the integrand contains only tangential derivations of the kernel at the self point r' = r. It follows that the integral in (22) is continuous as r approaches S allowing us to apply the limit directly to thei integrand, and imposition of the boundary condition (5) then gives 53

n^E (r) = RnAJ(r) - ikZ i^J(r')g- 2(V J) ^V dS1 (23) S k for r on S. This is a valid integral equation whose only disadvantage is the occurrence of surface derivatives of J. This difficulty can be overcome if we use the integral representation (12). On paralleling the steps leading from (20 to (22), we have nA E(r) = nAE (r) + ikZ i H{AJ(r ')g +- J(r')V']fnAV' gdS' (24) - r-S S and because of the higher order, non-integrable singularity of the integrand at the self point, it is no longer possible to apply the limit directly to the integrand. Since the contribution of the 'self cell' tends to infinity as the cell size tends to zero, even the limit shown in (24) does not exist, and though (24) differs from (22) only through an integration by parts applied to the second term in the integrand, it does Mot constitute an analytically valid integral equation. Nevertheless, it can be used as the basis of a numerical solution provided the segmentation of the integral which is inherent in such a method is performed prior to the limiting operation, with the segment size remaining non-zero. The resulting equation is in some respects preferable to (22) and has been found more convenient for our present purposes. A-3 Scattering by a Cylindrical Resistive Shell We now turn to the problem of a closed cylindrical shell illuminated by a plane wave at oblique incidence, and consider the form which the above integral equations take in this particular' case. Since the shell is independent of the z coordinate, the entire dependence on z is that produced by the incident field. If, therefore, E (r) = E(p)exp(ik z), H (r) = H (p)exp(ik z) (25)..-.- Z v - --- Z 54

where r = e + z z, the total fields admit the same decomposition and, in particular, J(r) = J(s)exp(ik z) (26) z where s is the circumferential distance around the shell in the plane z = 0. Thus JS (r'))dS' = z exp(ik z) J(,)H (Kd where (now) dun and (27) z The remaining integration Is with respect to the ciroumbrential distanoe s around the (olosed) perimeter of the shell in the plane z = 0. Also V.J -exp(ikz ') a J(.)+lk ^(.) and sinoe A ') exp(ik z')V'g dS' - exp(lkcz) S0 ')(V - ik z) H (Kd) da' z 4 z t z 0 where V is the two-dimensional (transverse) del operator in the plane z = 0, t eq. (23) becomes n ^ E() Rn J(s) + k ) + ik(t ) k (V- ik ^)H ()^ic}d' (28) The 8 oompowt is simply Y1(s) - YRJ() + ^S <.+ } H4 d ( JK ' -- 8SdD '.d (29) C s 55

which is a coupled integral equation involving the longitudinal and circumferential components of the current. Likewise, from the z component of (28) we obtain the second coupled equation YE() YRJ (a) + } J(m'X. 5 L )o: ) k *{a?.u )]t d 1* jd) d }dj'. * (30) and though it is not in general possible to decouple (29) and (30), we can eliminate J from the integral portion of (30). Differentiating (29) with respect to s, we z have 5, )H.)1)(Kd)da' R(Y 8 (} J (')(. a RKd)d ' 3- O { -Ez() C K 2 as, 1 -i~ C ' (d)HuI )dt ' K and when this is substituted into (30), the integral equation beoomes kz E k + ' ~ m:mm'XI E~ = Y eRJK(: )-at2m @ (I 'd)H } + k (la) ( dE)Ja'. i((31) I an zt. (231 which now involves J only as a pseudo excitation term. z Formally at least, the corresponding equations resulting from (24) can be obtained from (20) and (30) by integrating by parts the terms involving 3J /as'. Equation (29) is then replaced by 56

2 (1) k(1) YE (s) = YRJ (s) +- (s')H (Kd) - i- J (s')('' * )H (Kd dS' (32) Z Z 4k z L K S 1 C and (30) by i k A ) H(1d) ' YE (s) = YRJ (s)+ J (s')(k.s')H d)ds' S S 4 0 C + imC J (S (sd) H (K d) ds' (33) 4k p —) C s as' C Kk (1) J (s')(s d)H (d)ds'. 4k J z 1 C The latter is meaningful only if the second integral on the right hand side is handled in the manner described earlier. With this proviso, however, (32) and (33) are adequate for a numerical determination of the currents and we can, of course, eliminate J (s') from the integral portion of (33) if we so desire. z The case considered now is that in which the plane wave is incident in a plane perpendicular to the z axis of the cylindrical shell. Then k = 0 implying K = k, and (32) and (33) reduce to i k (1) YE (s) = YRJ( (s)+)H )(kd)ds' (34) z z 40 C i k C + p —0C J (s')- ) ds', - as iJ ' C 57

Vi Ul v i G e o X Fig. A-1. Geometry. 58

which are two uncoupled integral equations for the longitudinal and transverse current components J (s) and J (s) respectively. z S If the incident field is E - polarized, i.e. Ei = -ik(xcos00 + ysin0) E ze 0 where 00 is the angle of incidence with respect to the negative x axis (see Fig. A-), only the component J (s) is excited and we shall refer to (34) as the E - polarized equation. Once J (s) has been found, the scattered electric field in the far zone is z Es A i(kp-7r/4)p (: ) E z r kp where the complex scattering amplitude PE is given by kZ -ik ~ p' (36) P(0 - J (s3e 6ds' C with p = xcos + ysine and in terms of P, the two dimensional scattering cross section is c(0, 0O) = 4 |P(0, o) 2 (37) If, on the other hand, the incident field is H- polarized, i.e. Hi ^ -ik(xcos0 +ysin 0) H ze the only component excited is J (s) and this can be obtained from the H - polarized s equation (35). In the far zone Hs A2 i(kp - r/4) HS z e PH(O P 0) (38) k A A -ik~-~' where P( 0) = - 1 p ~ n'J (s )e ikPds', H 0 4 pn S C from which the H-polarized cross section can be found using (37). Two quantities of particular interest are the total (integrated) scattering and the absorption cross sections aT and rA respectively. The former is given by 59

T 0) 27r 0p ^T 0O cr(O\0)d (39) 0 and from the forward scattering theorem the absorption cross section is uA(0) C T( o0) k Re. P( 0 + i00r (40) AO TO k -^ ^'V (40) We remark that for a finite shell of length I (~>>X) each three dimensional cross (3) section a() for incidence in a plane perpendicular to the length and computed on the assumption that the surface field is the same as for the infinite shell is related to the corresponding two dimensional cross section by (3) 2 (41) a = -- a(41) where X is the free space wavelength. A-4 Computer Programs The integral equations (34) and (35) are special cases of those solved by computer program RAMVS (Liepa et al., 1974), and because of the availability of this general program we chose to concentrate on these equations rather than the ones involving the derivatives of the currents. While seeking to refine RAMVS and to make it more efficient for a polygonal shell, we became aware of certain errors and/or deficiencies in the program which are most apparent when the resistivity is small. These had not shown up in the testing and verification done earlier, and it proved quite time consuming to locate the errors and rectify them. We believe this has now been done. In addition, the program has been extended to compute the normalized two dimensional cross sections and to plot the normalized bistatic scattering cross section, where the normalization is with respect to the free space wavelength to make the quantities dimensionless. The two programs that resulted are designated RICE and RICH and are based on eqs. (34) and (35) for E- and H- polarization respectively. 60

Both equations are solved by breaking up the integrals into N equal segments or cells within each of which the current is assumed constant, but because the kernels are infinite when the integration and observation points coincide, the self cells must be treated analytically. This is a rather trivial matter in the case of eq. (34) whose kernel has an integrable singularity. From the small argument expansion of the the Hankel function we have ( 2i kd (kd 212 i kdi 2 H)(kd) — 1 +((In +y) 1 +-1 -i1 0 7L 2 2/J r21 where y = 0.577215... is Euler's constant, and hence, for the self cell A of width 26, z 0 z 7T... s J (s')H(l)(kd)ds 2dJ(){ 4 (en-+0.028798...)J } Equation (34) now becomes YE (s) = + r i 6 2+-i In- + 0.028798... J (s) z.X z (42) + T? J (s' )H (kd)ds' 2X C-A and this is the equation used in program RICE. The same reasoning applied to the first integral in (35) gives J (s ')(S* s ')H( (kd) ds' I^ 2WJ (s) l1 + 2i (n + 0. 028798...)}, s 0 s L 7r.XIJ A but the second integral is more difficult to treat. When the self cell is excluded from the range of integration, the limiting operation can be applied to the integrand directly, and since J (s') is assumed constant over each cell, S 61

Cim o-C-AC -A N J as') (1)kd ds' = J (s.) [(.)((kd) s) a- ("'' -)H1 s I' =S.j=1 p'=pj-6 jfi where the subscripts j and i denote the integration and observation points respectively. To evaluate the self cell contribution we first use the fact that as a (1) A 1)' as (S)H1 (kd)J = k (n.a)(n.) - (s d.)(s d)} H (kd) - k(n" - )(n-I)H( 1)(kd) 0 with the prime attached to the Hankel function denoting differentiation to express the contribution as Jimr J (s)_Am I s P-. where I = k ((n ')(n-d) - (~ d)(-d) H(1 (kd)ds'. A For a locally plane element with the observation point a small distance y above its midpoint, 6 2 2 y -t (1) 2 2k 2 2 H+ (k +t 2)dt 2 2 1 0 y +t and on inserting the small argument expansion of the Hankel function, we have 4i6 rk 2 62 y + 6 2 2 y t2 2 0 y +t {+2(n k + y ]+y ) dt ~ + k2 t 2 ]i 62

Hence -im I 4i- k f+ 2i(n kt 1+ )dt 0 4i - k 1 2in 2 + 0.528798... 7rk6 1 7r X the first term of which becomes infinite as the cell size tends to zero. Equation (35) now has the form i f 2 YE (S) YR + i {6 + + 0. 028798.. 5)} J (S) s 2 26 2X 7r X s 2X J (s')(s s')H0 (kd)ds' (43) c-A N + E J (s.) [( d)H()(kd) 3 j=1 jwi whose solution is computed by program RICH. Of the two programs it is evident that RICH is the more complicated, and to obtain a feeling for the rapidity of convergence as a function of the cell size, the program was run for edge-on incidence ona hexagonal shell 3p on the side at a wavelength 2. 6. The normalised resistivity employed was YR = 0. 03876 + i9. 10650 corresponding to a shell 0. lp thick and the refractive index n = 1.206 + iO.00080 for ice at 2. 6p quoted by Irvine and Pollack (1968). The results of decreasing the total number N of cells from 72 down to 24 are listed in Table A-l, and we observe that the total scattering and absorption cross sections are relatively insensitive to 63

TABLE A - 1: Effect of Cell Size cells per time used a( 0 )/ | 7r+0, O)/ cX ~ )/k (o)/\ 0 0 0 ' e0)/) wavelength (sec. ) () (dB) A 0 72 10.4 10.911 -44.83 -6.45 0.02229 0.00158 60 8.7 6.645 -44.70 -6.45 0.02230 0.00157 48 6.9 3.768 -44.48 -6.45 0.02233 0.00155 36 5.2 1.948 -44.04 -6.46 0.02241 0.00153 30 4.3 1.332 -43.63 -6.46 0.02250 0. 00151 24 3.5 0.895 -42.91 -6.46 0.02272 0.00149,,, l, 0m 4^1

N, as is the forward scattering cross section, but the backscattering changes noticeably on decreasing the sampling rate below (about) 9 cells per wavelength. In running the program we have therefore used 9 or more cells per wavelength whenever possible consistent with the maximum of N = 100 allowed by the matrix inversion routine used. Listings of the two programs follow. *' IV * * * *.. V. V V. V ' '..'.'.'.".'...C C INPUT FORMAT FOR PROGRAM RICF SFPT, 1975 C C —,J. A A J* ** * s *** * * I * 444 ' 4 4' C C CARn I FnRMAT (18A4) TITLF CARD: IISF UP TA 72 CnLIJMNS C. rC C CARD 2 FFRMAT ( I,I 5 F 10.5) MORF,KnnF,ZFAC,WAV/F,FIRST,LAST,INK C C MnRF=O THIS WIIL RF THF LAST RJUN FnR THIS DATA SFT C C MORF=I THERF ARF MIRE DATA TO RE RFAD AFTFR THIS SFT C C KnnF=O COMPIITFS RISTATIC SCATTERING PATTERN C C KnAF=l CAMPIITES RACKSCATTFRING PATTFRN C C ZFAC A CNMPLFX FACTFAR MULTTPLYIING ALL FLFMFNT r C W A V/ F A \/ F L F r T H C C FIRST INITIAL SCATTFRING AND INCIDENCE ANGLE C C LAST FINAL ANGF_ C C INK ANIGULAR INlCRFMFNIT C, C C C CARD 3 FnRMAT (I 2,X,7F10.5) N,Z,XA,YA,XR,YR,ANG C C N NIMRFR OF SAMPLING, PFINTS An THIS SFGMENT C C 7- NAiRMALIZ7F IMPFDANCE nF THF SFGMFNT C C XA,YA,XB,YR FSGMFNT FN)PIINITS C C ANG ANGLF SIIUTTENDED BY THE SFGMFNIT C C ZFRn IN COL? SHITS OFF C C READINIG FF SFGMFN!TS C C C C CARD 4 FORMAT (I 2,I3,F1..5) MnRF,KnDE,ZFAC,FIRST,LAST,INK C C THIS CARD IS UISEDF ONLY IF, C C nn CARr 2, MnRF=] r,A, *, **~ *, *c ***** **** *: **** ** ** **J4 *8a ** ****** 4 *** **. 4* ** ** 'S. 44 ** ** 4,4 C C C C iT SPFCIFICATInNS: C C 5:INPIT DATA: 6:Ol)TPUT(PRIN TFR): 7:OITPIUT(GRAPHICS) C fC C * * * $4 * *: * * ** * * ^,^: * * * * * * * * * 4L: 4 ' 4 4 ' * * 4,,, C 65

CnMPLFX*8 A(1.00,101),PHI (100),PTINIK(100),ZS(I10) COM PLI-F X*% DEFL,.9Si JM,~7F A C P FA L-4 L A STTIN K P F A J_4 X ( 20 0) Y(2 00),XN(?0 0)YNI(2 0 0),S ( 20,DnS0 ( IO)A SI m(3 6 1)R StM(3 61) DATA MH/100/ commON /PIFS/ PI,TPL, PIT, PTP1, Y7,PFD,DIC DATA IPnOL /'FFFF',9 'HHHHl/ C.....PRFAD KINPI T DATA AND GFNERAIF RODy PROFILE PFRAD (9,100) ID) REFADn (5,200) MnRFKnODF,7FACWA\fF,FIP\ST,,LAST, NINK 6AIRITF(7 1.00) IDn IORITTF(7 101I)I A\V/F WPRI TF( 7,199) F IRS Tv LA ST, I N K 1 P P 101I FFORMA T(I L AMRDA=', F5 o2, M ICRONiS' l 99 FnRMAT(1IX v3F10.*5,1 3) I WdH I CH= I I F ( RFA L IZFAC F) *F. 0. AND. A IMAG ( ZFAC ). FO.. WARITET ( 6,p150) IDF WRITF (6,300) CALL Z-tIM(LtJ1-1MP, X, YXN, 9YN1,SD fS,ZS M) MT=M/2 IA t =LtJmp (M) 20 IF (Kn1DF.NF.0) GFn TO 25 N\1I N NC1= N\BIT=1+IFIX((L-AST-FIRST)/INIK) (;n TO 30?~ NBIRT=0 N\ITINC(=] 1+FIX( (LAST-EIRST )/INK) 30 AR I TE ( 6,r150 ) IDF WRITE (6,9400) I POL (I P P ),L LqMTN1I NCNR IT~w 4A VF,97 FAC, W4RITET ( 6,425) F IRST 38p DO 35 I=1,vLL 7S(,I I)=S( )*ZAC, 35 DSo( I)=Dso( I) /W.AVfF XK=TP I /WAVF C....Cni S TRIJIC T M AT RI X EL F MFNTS IF (I WH ICH.FO.,2 ) o TO 37 C AL L M TX ( Ml MH,9 XK,9 X, Y, XNl YN, DS O,v Ll JMP,7 S, A) C......CnMPtTF INCIDENIT FIFLD AND ltINVERT MATRIX 37 TFTA=RFD*FIRST C',T=Cns ( TFTA ) ST=SINl(TETA) DO 60 I=2,M.2 Hn[ D=-XK*(CT*X( I )+ST*-Y( I) DFL-=CMPLX(COS( HnLD) SNI(HnLD)) 60PINK(/12 )=DFL CALL FLIP( AMTMH4,tLV\MMwPTINK,~PHI,IWHICH) 66

C.....PR INIT OUi T CI JR RFNITS AND EL FM FNT PPO PFRTIFS FOR FIRST ANGrLF W~fRI TF (6,v500) I T=O DO 6)5 I=?,vMv? I T=TI T.1 AMP=CAAS( PHI (IT)) PHASE=DIG*"ATAN2?(AIMAG,(PHI(IT)),RFAL-(PHI(IT))) I SFG=LII)MP ( I ) 65 WARITE (6,P250) ITISFG,,X(I),tY(I),PS(I ),DSO,(ISFG-),Z7S(ISFG-),vAMPIPHA\SF C7'.....DnPF PUlT THE APPROPRIATE FIELD FACTORS THF=FIRPST-INIK K<=0 IF (KnDFE0.EO.) GO TO 70 WARITE (6,P800) FIRST GO TO 75 70 WARI TE (6,)i600) 75 THF=THF+INK I F ( THF.GT.LI-AST) GO TniO 15 IF (THF.EO.FTRST) GO TO 85 C,...IN' THE FOL LOnWI NIG LoOP, P I NK I S NOFT N FC FSSAR I LY THE I NiCIDnFNT F IF L f TF TA=RFD*,THF CT=Cns( TFTA) ST=SINI(TFTA) DO 80 J=2,pM,2 HnLn=-X K*(C T* X (J)+ST"'Y( J) DFL-=CMPLX(cns( HnLD).S I Ki(HOl- )) 80 P I NK ( J/2 )=DF L IF (KODE.EO.0) GO TO 85 C A LL F LI P (A, M T MH, L V, MM, P1 ~INKPH I 2) R 9s I I M=C M PL X 0.0,90.10) C,...ADD UiP THE CUJRRE-NTS 00 95 J=2pMv2 JT=J/2 95 SIIMA=SIIM+PHI(J)T)*-PINIK(J.T)*4S50(LUIMP(,)) SI im=-stJM SU MR=R FAL( SU M) SIIM I=A I MAG ( SUiM) Si IM SO=SIJMR* StIM R+SIJM I *SU M I SC AT=l0. * ALOGIO( StIJMS0)+I1.9 612 K=K+l A SUM(K )SUMSOn R SU M(K )=REAL (StIM) PHA SE=D iG*ATAN2 ( SIM I, SIJIMR) WAR ITE( 6,p 01) THEvSC AT,PHAS F W~RITE (7,9q00) THE, SCAT, PHASE GOl TO 75 1-05 DIFE=L-AST-EIRST iF(DIFF.NF. 1890.0.AND. 01IFF.NIF. 360.0) GO TO 205 iODD= mOD ( K,? 2) IF(DIFF FEQ. 360.0.AND. IPnD NIF. 1) GO TO 205 IF(DIFF EO,. 180.0) FAC=2.0 I F (DIFF EO,. 360.0) FAC=1.0 67

C*, SIMPSON INTFGRA4TI'ON FIF CRnSSFCTInNS KI AST=K IF(nnnD.EO. 0) KLAST=KLAST-3 SIGT=ASM( 1 )+ASIJM(KLAST) SI G=(. 0 On?03 I=2,KLAST,2 203 SIG=SIG+AStM( I ) SI G T=S I GT+4.0 OS I G SI =0.0 KLAST=K-2 nn 210 I=3,KLAST,2?10 STIG=SIG +ASIIM( I ) SIGT=SIGT+2.0*SITG SIGT=SIGT* INK/3.0 IF( InDO.EO. 0) SIGT=SIGT+3./8A.*INK".&(ASl)M(K-3)+ASItM(K)+3.*( A SuM(K-2 ) +ASIJM(K-1 ) ) ) SI GT=S I GT*FAC*RF E/4. C*** FnI SIMPSnN INTFGRATinN KF=K IF(FAC.FO. 1) KF=(K+I )/2 SIGF=-RSlIM( KF ) SIGA=-SIGT+S IGF I F ( KnOF) 220,220,240 220 SICTDR=10.0*ALOG10( SIGT ) IF ( S IGA.LF. 0.0) Gn T 225 SIGADB=10.0*ALnG10( SIGA) GO Tn 230 225 WRITE ( 6,825) S I GTSI GA,S IGF WRITE(7,825) SIGT,SIGA,SIGF Gn TO 205 230 SIGTDR=10.O*ALnG10( S IGT) WRITE( 6, 850) SIGT,SIGTD,SIGA,SIGADR WRITE(7,851) SIGT,SIGTDR,SIGA,SIGADR Gn Tn 205 240 SIGTDR=10.O*ALnG1O(SIGT) WRITF(6, 875) SIGT.SIGTDR 825 FnRMAT( ///, 5X,4OH*FntJL* NFGATIVF ARSnRPTIFN CRFISSFCTInN,//, ~15X, 5HS I GT=,. F85,5X,5HS IGA=, FR.5,5X, 5HS IGF=, F8.5) 850 FnRMAT( ///,15X,5HSIGT=,F8.5,FR.2,3H DR,5X, 5HSIGA=, F8. 5,FR.?, &F3H DR ) 851 F RMAT(///,15X, ' S I GMA(T)/LAMRDA=',F8.5,F.2,' DR',5X, &'SIGMA(A)/LAMBDA=',F8.5,F8.2,' DR' 875 FnRMAT(///,15X,7HSIGAVF=,F8.5,FR.2,3H DR) 205 IF (MnRF.EO.O) Gn Tn 5 nn 103 I=1,LL 7S( I)=7S( I )/ZFAC 103 fSO( I )=DSO(I)*WA\/F DFL=ZFAC P A V\/ F = WAVE 68

PFAn (5,200) MORF,,KnDE,Z F AC, IWA\/F9,FJIRSTqLAST, INK WRITE(7,91 00) In WAR I TE( 7,199) F I R ST L AST, I NK, I PP ITWH ICH= I IF (RF AL( ZF AC,).*FO. 0. AND. AIM AG( -F AC ). Fn.O.). IF(RFAL(ZFAC).EO.,RFAL(DE L)AND. A I M C',( 7FAC.FO.A I MAG(DFL) E.ANF). WA\/F.,FO.PWA\/F) IWH ICH=? (~n TO 20 100 FORMAT (18A4) 150 FORMAT (1H1,18A4) 200 FOR MA T (129,13,6FI0.5) 250 FORMAT (?I 15,7 F 10. 5,F I0. 3 300 FORMAT (IOHOSFCO NIIM,11X,?4HFNDPOIlNIS OF THF SFGMFNlTtl9Xv FJ8PHSFCGMFNT PARAMFTFRS/11H NItIM C FL L S,6X,? HX Aq8Xv2HY A,8Xv?HXB, 98X, F~?HYR,q6X,924HANGLF R A DI tIS LEFNCTH,4X,14HRF-7 TM-7l 400 FORMAT (//31X,14HKFY PARAMFTFRS// F,1 6X,21HINCiDFNT PnLARIZATION.22X,lA1l/ fJ I6X,23HN\IJMBFR OF SFGMFNITS lJSFDqIe?l/ fAl6~X,,33HTnTAL N\uIMBER OF CFLLS ON THF FBODytili/ f~16-X,35HNtJMRFR OF INiciDFNT FIFLn DIRFCTIPNS;,o9/ F, 1 6X,2 9HN\JiM B R OF RISTATIC DIRFC-,TIDNis,15/ &1 6X, l0HWAVELFNGTH, F34.o5/ &I Xv 1 6HI MPFDAr\CF FATOR,,F 16.,5,' 9,'F IO,5) 425 FORMAT (////,36X,18HSI.JRFACF FIELD DATA/,27X, F%9HFnR INICIDFNIT FIELD- DIRFCTIOlN=,,F7.?) 500o FORMAT (11HO I SFEG,4X,4HX(I),6x,4HYU1),6X,4HS(I),5X,6Hr)Srj(J), &4X, 6HRS( I),4X,,6HXS( I),4X,,6HMODn( j),4X,,6HARG-( j) )/ 600 FORMAT (lH1,27X,28HRACKSCATTFRING CROSS SFCTIFDNI//23X, F,3 6H TH FTA S IGM A /LAMBDA. DR PHASF nDF G) 800 FORMAT (1IH1,23X,33HBISTATIC SCATTFRING CROSS SFCTIPNl/23X, F,?9H FnR INCIDFN\T FIELD DI RFC TiON = vF 7,2 / /23 X F,36HTHF TA S IGM A /LAMBDA, D )R PHASE,nDFc) 900 FORMAT (15XFl3.2,Fl4.2,Fl6.1) 901 FORMAT (15XIF13.2,FI4,2,F1,5.1) F tinD SUBROU TI NE M T X (M MH9X K IXq Y X N, Y N,9D SOLIIM P,7S A) C RICE \/FRSION,, F-POLARIZATION F A L*4 X ( 1) Y ( 1 ), XN ( I) YN ( 1)DS 00(1 INTFCFR*4 LtIMP(1) COnm PL FX *RAA ( MH,9 1 ), Z S( 1) HZ,D i im COMMON /PIES/ PITPI, PIT, PIPI, Y7,RFD,DIG I H= 0 DO 10 1=2,M.2 TH=IH+l XI=X( I) YI=Y( I),JH= 0 nn 10,i=2?,M,,? j H =,-J H + 1 I S F G=L I JM P ( J) Ds=Ds)( ISEG) I F (IH. F O.JH ) GO TO g0 69

RY=YI-Y(J)) P =SOR T( RX*RX+RY*RY) CALL HANlKZ1( RKq0,HZ.D9IJM) A ( IJHP,JH ) =PIT*H7*r)S (;n TO 10 50 A ( I HJH )=Z S( I SFG),+ F, DS*CM PLX (P I T A LnO(DS)+.O2R79837) 10 CONITINIJF RF N 1 F) SlFRRnhi TINE GFOm( LIJMPwXYXNlYNSSno,Z,7-,M) C THIS VFRSION RFAf)S AND GFNFTATFS SFG-,MFEN.TS TNI CflU-NITFR-CLOCKWISF DIlRFCTlflN)N. C, THE SUIRFACF MUIST BF CLnSFn. C PO INTS ARE GENIFRFTFn AT THE START AinD miDPnINITs OF FACH CELL;-" C, THE START POINT OF THE FIRST CFLL EVENTUIALLY COINCIDES C WI1TH THE FND PONl\ T OF THF L-AST CFLL. CnMPLFX*g ZS(1)97 R FA L*4 X ( 1 ) Y ( 1 ), XNK( 1),YN ( I) 5(1 D SOn( 1) TNITFGFR*4 LI IMP(1I) co~'mmO /PIFS/ PITP1, PIT, PIPIv Y7,RFn,nIG I = 0 L-=0 C..READ NINPUJT PARAMETERS ANDn PREPARE TO GENERATE SAMPLING POINITS 1.0 READ (5,200o) NvZqXAYAXRYRANG, IE(N.L.T.1) GOn TO 120 -I JM= 2*4,N TX=XA-XFB TY=YA-YR n=SOR T( TX*TX+TY*TY) L= L + 1 7 S( L)=Z IF (ANG.EO.0.0) GO TO 20 T=0 *5*RED*ANG, T R X =T X +TY*cTANI(T) TRY=TY-Tx*cOTANI( T) RAD=o.q'5n/sTNI(T) ARC=2 *0*RAD*T AL F = T /N sO( )L )=?.0*RAn* A LE GO TOl 30 20 RAn=999.999 A R C = F) OS)( L )=n/N C'.....START GENERATING Ds=ns0( L )/2. 5(1 )=[,l55 nss=nF)s5+ns 70

II)MP ( T )= L IF (ANG.FO.0.0) COf Tf 40 AR, C = J A L F SI N1 n=STINI( ARC) cns0=Cns ( ARC) X(I)=XA-0.5-(TRX*,(l.o-Cnso) —TRY*~ST NO,) Y(I)=YA+0.5't(TRX*SIN0,+TRY*-(1..o-C~nSo)) XNI(i)=+0.5*(TRX*IC~nSo+TRY-SIr~ln)/RAn) YN(T)I 0. 5* (T RXg*1N 0 - T R Y*CnSO,)/RAn Gn TO g0 40 X( I )=X A-0. 5*J*TX /Ni Y( I)=YA-0.'5*J*TY/N XNI( I =TY/F) YN(I) TX /D 0O COnNIT I NUEII 100 I-R IIEF( 6,9300 ) Lv N XA,YA XB, YPANCRAD,,ARC, 7 o"n Tn Io0 120 =I200 FOR MA T( 12,p3x,7 F Io,5) 300 FOnR MA T( I13,p1 6,94 X,4E1 0. 5,F7. 24F]I0. 5 R FTI IRNK ENDn SU R R OH T I N H ANKZ71 (R,Nv HZ7EROHnNE) C...H A NK E LFlNCTinNiS ARE OF F IRST K inD —~j+ Iy C..... N=0 RETUlRNS HZEFRn C.... N=1 RFTIJRN~S HONE C.... N=? RETURNIS H7ERf1 ANDn HOnKF C....SUlRRniPJTINF RFOUTJRES R>0 C.....*SIJRRnUTINE' ADAM MUST RF SUlPPLIED BY USE';R DIMENSION A(7),BR(7),C(-(7),D(7),E(7),,F(7),C-(7),,H(7) COmPLEX HZFRnHnNF D)ATA ABCD, EFCqH/1.0,-2.2499997l,126562?10.,31638R6)6, F~0,04444,79,-0,0039444,0.000210,o36,7466'9lO.6055q366,-0.743503841 F-0.29300117,.-0,04261?1490*00427916,o.o00024o46,09,-0s562499R5s En.2l093573, -0.03954289,0.00443319,-0.00031761,-0 00001109, E.O. 6366198, 0. 221291, 2. 16827091,-1. 3le~4P27,0. 312395l,-0.0400976I Fi0.0027873t,079788456,-0.00000077,9-0,0055274,-0.O0009512v EO. 00137237, -0.,00072805,90.00014476, -0. 78539816, -0. 04 166397, F,-0.0000395490.00262573,-0.00054125s-,000oo93~3-,0Oooo355sR U). 79788456,0.000O00156,90.01 65966-7,0. 00017105v-0. 00249511? U).00113 653,-0.00020033,-2.35619449,0.12499612,0.0000565, F,-0. 00 637879,90.000743480,0.00079824,-0. 00029166/ IF (R.L-F,0,0) CO TO 50 IF (N.LTo0.ORK.N.CT.2) CO' TO 50 IF (R.GT.3.0) CO TO 20 X=R*R/9. 0 IF (N.EO.1) CO TO 10 CALL ADAM(AXq,Bj) CA LL ADAM ( f9X 9y ) RY=0.A6366199*,AIOC,(-~~ *JH7F-RO=CMPL X( RI,JvRY ) IF (N.FnO.0) RFT1JRN 71

1 0 CALL AnAM(C,X,Y) CALL ADAm(D,X,Y) BY =0. 6366198* AL n(O,.5*R )*R,.,+Y/R Hnr\IF=CMPLX( RJ01Y) R F T1 IRN 20 X=3.0/R IF (NI.F0.1) cO TO 30 CALL ADAM(F,,X,,Y) FnnL=Y/SORT(R) CALI-L ADAM ( F Xv Y) T=R+Y RJ=FnnL*cOns( T) RY=FnnOL*SIN(T) HZ FRn=cMPLX( R~i.BY) IF (N.FO.0) RFTHRN 30 CALL ADAM(GX,X,Y) FnnL=Y/SORT(R) CALL ADAM(H,X,Y) T=R+Y Rj=FOOL*COS ( T) BY=FnnL-*sINI(T) HONF=CMPLX( BJ-,RY) R FT1 RN '; O WR IT ( 6,v90) Nv R go FnRMAT(32H0SICK DATA IN; HANIKZI11*01JIT* N1=q12?Xt2HR=,Fl1.3) CALL SYSTEM F NI Fn SIBROni ITINE ADAM (C,,X,Y) DIMENSION C(7) Y=X*C ( 7 ) DO 10I= 115 1.0 Y=X*( C( 7-I )+Y) Y=Y+C( 1) R F TI IR N F N F SURRO1IIT INE F L I P(Av N1M I LvM,X,9Y I IAT) COMP LEX A (M I,9 )X ( 1 )Y ( 1 ),qDPIG (-AH(DLD D IMENS ION L (1),M( 1 ) IF (IAT.GT.1) (0 TO 150 D=cmpL-x(i1.0.0) DO 80 K=19N L( K )=K ( K )=K RIGA=A( KK) DO 2-0 J=KNl DOn 20 I =K,-PN 10 IF (CABS(BICA).GF.CARS(A(I,,J))) GO nTO 20 I GA= A( 1,1) L (K )=I M(K )=,I 72

2 0 cO\I T IMNH F IF (J-.LF.K) nf TAl 35 HO LOn=-A (K, I) A(KI )=A(J~pi) 30n A(JI,I)=HnLn 35 I =M(K ) IF (I.LF.K) en in 45 DO r 4 0 J=I1,N A (3 K = =A( 3,q I) 40 A j I)=HOCL0n L,5 I F (CARS ( BTICA).NF.0.0) G n in 50 n=C MP L X( 0.0,00) R F T1IR N 50 nn s5 I=lN IF (I.FO..K) en in 55 A ( I, K ) =- A ( I,K )/f I C A 55 CO T I NU1F Dfn 65 =I,N no 65~11, I F ( I.Fn. K.nR J FOK) e-n in 6 A (1,J )=A ( 1, K )A Kq,J ) +A( 1,) 65 cfnr\TINHIIF nn 75 ~=, TF (J).F0..K) CO TO 75 A (K -,J ) =A(K,I)R I GA 7 5 ON T I NJJF D=~*,R I CA P 0 A ( K,9 K) =1 * 0/RI CA K = NJ 100o K= K-1I IF (K.1LF.0) Co, in 150 I =L (K) I F (I.IF.K) COl Tn 120 DO 110 J=l,N~ Hnt[D= A( J K ) A(J,K )=-A4( J,I 11I0 A J, I )=HOLD 1.20.j=M(K) IF (J).L.F.K) CO TO 100 no 130 I =1, N HOlD-=A( K, I) A ( K,9 I ) =-A ( J,1) 130 A(,j,I)=HOLD Cf1 TO 100n 1.50 Dn O 20I= Y( I)=C-MPLX( 0.0,n0.0) no 200j=,I 200 Y ( I ) =A ( I,9J ) 9X ( j)+Y ( 1) R F TI IR NJ F mnD FRLPCK DATA cnimmONp IFS~/P1, TPI,PIT, PJPI,Y7,RFD, DIC D)ATA PITP I,PI TvPIP I Y7 RFDn,D)I,/,3. 14159279 6.2831 853, 1. 5l7079 63,9. 869 6044, 0.0026559824,o0.n01745'329, 57. 29578/ FMnD 73

C INPUJT F[URPAAT FOR PRAGRAM klICH SFPT9 1975 C CCARD 1 FAIRMAT (l8A4) TITLE CARD: IISE UP TA 7 2 CAL tIJMN'S C C C, C, CARD 2 FARMAT TI2,1 3,5F I0.5) MAnRE, KADnE,7FACqWA\/EFIR\STLASTITN K C C, MnPRF=0 THIS W4ILL RF THE LAST RUNI FAR THIS D)ATA SF1 C C MORE=l THFRF ARE MARE DATA TA FIF RFAD AFTFR THIS SET C C KnDF=0 CAMImJtTES BISTATIC SCATTFRTING PATTERN C CKADnF= 1 CAmpU TFS RACKSCATTFRING PATTERNC C7 FAC, A CAMPLFX FACTAR MUL.TIPLYING, ALL FLFMENTC C AIAV/ W A\1/ LFNI 7GT HC C, FI R ST INITIAL SCATTFRNING A~ND TINCIDENCE ANGLE C,CL A ST FINAL ANGI FC CI NK ANIGUIAR NINCREMEN\TC C C C, CARD 3FnRMAT (I~,5X,7F10.5) N7,,XAwYAXRYRoANG C C N NUM4RFR AF S AMP LING C PAIINTS AIN THIS SFCMFN,\T C C7 NARMALIZED IMPEDAN\CE DE THF SEG MEN T C C XAYAAXBYR SEGMEN' F~nnTENPANTS C C, A NGC ANGLF SIIRTFNDFD RY THF SFEGMFN\T C, C, ZERA IN CAL 2 SHUJTS AFE C, CREADINGr AF SEGMENTS C C CA/RD 4 FARMAT (12,13,E1O.5) mAREKADE,7FAC,FIRST,LAST,TNlKC C, THIS CARD IS USED AN L Y I F, C C, AN CARD 2, mARE=lC -~ -A -, — ' —e-.- 'I,-,-, —, C, C, C iA SPECIEICATIAINS: C C, i: INPI IT DATA: 6:AIIlTPllT(PRINlTER):, 7:AU TPIJT(GRAPHICS) C, C C C*4***4**4*0 -I- -I- ******* 444**4*:***- *-'-J. J —* --- — -- CAnMPLFX*8 A( 100,101),PHI (100),PINIK (1AO) 75(10) CAnMPLEX49R DEL, SUlM,7FAC REFA l_ *'-4 f. AS T, INK REAL1 *4- X (2 00), Y( 0 0 ) XN (?00),Y N(2?0 0), S(0ooDS 1O),AS I M(3 6 1)RSUIIM(3 6 1 DATA MH/lOO/ CAnM MAN / P IES/ P IT PI P IT, P I PI Y 7,vREDD9)I G D A TA pIPL- /'FEFE'i, lHHHH'/ D A TA I PP /2 / C.....READ INpiiT D)ATA Ai ND GFNERATE RADnY PRAE)ILF 9S READ (5,100) ID WRI TF(7,91.00) IDF 61R ITE(7,0I) WAl/\/E WRITE(7,1I99) ER ST,I1 AST,?Ir KI~IPP lTl EAFRMAT(' LAMRDA=',E5.?,' MICRAN~s') I1Qq FARMAT(I1X,3E1A.q. 13) I WH ICH-i TIE( RFAL ( 7AC.EO. 0. AN'. A IMAG( 7EAC)Eo.n.) '.'RITET (6,300 ) C A II. F Fmt EMiUmPXYX- Y -ym-,D-SAnc_7 C M MTI= M I I L =LUM(,P ( PI) 74

20 IFP (KDnF. NF.0) 00 TrI?5 NJ CT NIRIT=l+IFIX((LAS~T-FJRS~T)/INIK) 00 TO 30 251 M FI T=0 NIINIC=I+1IF I X(L A ST -FI RS T/I NK) 30 WRI TE (6 150) IDF WIR ITE ( 6,P400) IpnL (I PP ) L-,vMTN I NIC NBITI A\/VF,97F AC, AR ITE (6, 425) F IRST ~R n0 35 I =1,PL L 7 S (I )= 7 S(I ):*,7 FAC 35 nS n( I ) =f1S( I ) /kfAVF XK=TPTI/W4AVF C...CONrS TRt)JCT MATR IX FLFMEN\TS IF( I WH ICH.oFO.?2) 00 TO 37 CALLI M TX(MP,9MH, XK, X Y, XN19YNi DS0, LI MP q7S, A) C..CnM PtIITF I NCIDENT FIELD AND IN'VFRT MATRIX 3 7 TFTA=RFD*FIRST C T=cnOS( TFTA ) ST=SINI(TFTA) DO 60 I=?,M,2 HnLn=-XK*(CT'X( I )+ST*"Y( I) nFL=CMPLX( COS( HOLD-) pS IN(H) HOLD) 60O P I NK (I/ 2 )=OF.* (XN1( I ) CT+Ytl( I)*S~T CALL FLIP(AMTMHL\f,pMMPINIKPHI pIWHICH) C7...PRINT OUT CUJRREN\TS AND FLFMFNT PROPERTIES FOIR FIRST ANGLF WRIP\ITE (6,9500) I T=0 DO 65 I=2,wMw2 I T= I T+ 1 AMP=CARS( PH I( I T) PHASF=DIG~*ATAN?(AIMAO,(PHI(IT)),PRFAI-(PHI(JT))) I SFCG=[LUmp( I) WR I T E (6,2 50) I T I SF G,,vX (I),Y (I),S( IO I),D(1FG), 7 S(I SE),M PPH A SF C'.....nOPE OUfT THE APPROPRIATF FIELD EACTORs THE=EI RST-INIK K=0 IF (KODF.EO.l) GO TO 70 WRITE (6,800) FIRST 00 TO 75 70 WRI TE ( 6,600) 759 THF=THF+INIK I F (T HF.GT.L A ST) G o TOn 105 TE (THE-.FO,.FIRS;T) GO TO 85 C....I N THEF OnLLOw ING( LOOP, P I NK I S NOT NFPC FSSAR I LY THE I NC IDENT F IEL1-D T F TA =RFD*THF CT=COs( TFTA) ST=SIN1(TFTA) DO 80 1i=?,MV? HOLDF=-XK' (CT' X (J)+ST-" Y(J) D)FL-=CMPL-X ( COnS ( HOLD n) w S I KN( HO,,LD) 75

0 n PI NK ( J/2 )=DFL:- ( XN( )CT+YN( J) ST) TF (KnnF.FO.O) Gn TFt R5 CA l_ FL I P ( A, MT, MH, LV,\ MM, PI NK, PH I 2 85 SlJM=CMPL X ( 0. 0,0 n. 0) C.... ADD UP THE CUIRRFNTS nn 95 J=2,M,2 JtT=,I/2 95 SIIM=SIIM+PHI ( JT )-P I NK ( IT )' DS O ( LIMP(,) ) St UMR=RFAL ( SUIM ) FSI1M I=AIMAG( SlIM) SI JMSO)= S IMR* S IJMR+St IM I * SI JM I SCAT= 10.*AL.n G10( SIIMSO)+1.9612 K= K+1 A SIIM( K ) =SIJMSO RSIIM(K )=REAL( SIIM) PHASF=nDIG*ATAN2( SUIMI,S!UMR) WRITF(6,,901) THF,SCAT,PHASF IWRITF (7,900) THF,SCAT,PHASF c.n Tn 57 5 10 nDIFF=LAST-FIRST IF(DIFF.NF. 180.0.AND. DIFF.NF. 360.0) (G Tr1 205 T nnD=Mn(K, 2) IF(DIFF. FO. 360.0.AND\. iDD.NIE. 1) [r TO 205 I F( DI FF.FO. 180.0) FAC=2.0 IF(DIFF.EO. 360.0) FAC=I.n C**' SIMPSON INITFGRATIONN OF CRnSSFCTInNS KLAST=K IF(inOD.EO. 0) KLAST=KLAST-3 SlGT=ASIJM( 1 )+ASIJM( KLAST ) ST=0.0 Dn 203 I=2,KLAST,2 203 S I G=S JIG+ AS IM ( I ) SIGT=S ICT+4.0*SIG S IG=0.0 KLAST=K-2 nn 21n I=3,KLAST,2 21.0 SIG=STG+ASIJM(I) SI GT=S I GT+2. 0*S I, SIGT=SIGT* INK/3. 0 IF(Innn.EO. 0) SIGT=SIGT+3./!8.*INK* FA( ASIIM( K-3 )+AS1M( K )+3.* ( ASIJM(K-2 )+ASM ( K-1 ) ) ) S I TS I GT*FAC*RFD/4. C,** FNtD SIMPSOnN INTFGR AT I I K F=K IF(FAC.FO. 1) KF=(K+1)/2 SIGF=-RSitM( KF) SI GA=-S I GT+SI GF IF(KnDF) 220,220,240 220 S T GTDR= 10. O* A LnG1 l ( S I GT ) IF(SIGA.LF. 0.0) Gn Tn 225 S I GAR=10.0*ALnG 10( SIGA) Gn Tn?30 76

2 25l, WRI T E ( 5,q2 5) S I GTvSI(;A,S/,qI GF WAR I TE (7 o82 5 S IGTw S~I;A, SI GF cO TO 205 230 SITOP=100*A-O1O( SlIGT) AR I TF ( 6,p8 50 ) S IGCTS IGTnP,S~I GA,9S IGAOP4 WR.ITE (7,R851) STGT.SIGTnRSIGAjSIlGAOP GOn TO 20q 240 STGTnR=io.0*ALnriO(STGT) 1AIRTTF(69,R79) SJGTSICTDR R 25 FnRMAT(///, 5X,40H Fntl*l- N (, I \IF ARSnRPTPITONi C.RnS SF CT ONf// F~19X,5HSIGT=,F8.5,5X,5HSIGA=,FP.5,5Y,5HSTGF=,F8.5) 850 FOPMAT( ///,lSX,5HSICT=,F8.5,F8.?,3H DP,5X,5HSlG-A=,F8.5,E8.?, R3H nD I psi. FORMAT(///,15X,'SIG7MA(T)/LAmRnA=',F8.5,F82.,1 n OP,5X, F IS I GMA (A ) /IAMPnA= I,9FP. 5,F8.2, 2 OP' 8~7 FORMA T(///,5X,7HS IGA\,/F=,9F8.9, FR.?31- OP)' 2 0 5 IF (MORF.EO,.o) GO TO 5 00 103 I1L 7S(1 )=7S( I)/7FAC1.03- nsO( )=O( iIsn. )WA\V/F O FL =7FA C P WAV\1F = 4AJA\/E RFAO (5,200) MORFKOOF,ZFACW4A\/FFTRST,,1.AST,TINK 1WRI TE (7, 100) TOF WAR I TF( 'I,9101) 1 A\/ F 1W\RITE(7,199) FlRST,L-AST,INIK, TPP I WH I CH= 1 TF(RFAL(ZFAC).FO.0O.ANOn.AIMAG,(ZEAC).Fn.o.) F1 7FA CI 1.F -20 I 1.F-2?0) fl iF(REAL(\FAC().PW\FQ.RFL(OE).A.AIMAG(-7FAC,).Fn.A1MAC('F, GOn TO 20 100 FORMAT (18A4) 150n FORMAT (IHI,I8A4) 200 FORMAT ( 12,p1 3,6F1I0.5) 250 FORMAT (?I5,97F10.5,F1O.3) '300 FORMAT (lOHOSFG NIIIM,11X,2-4HE\N~pOINTS OlF THF SFG-MENIT,19x,, F~1RHSEGMFNT PARAMFTER\S/J1H N1IM CEL-LS,6X,2HXARX,?HYA,RX,?HXK,8XI F,2HYR,6Xq24HANGLF RADiiuS LFNIGTH14X, 14HRF-7 I M-7/ 400 FORMAT ( //31X, 14HKFY PARAMETERS/I F,1I6X, 21 HTINCTIOFNIT POLAR IZ7AT iONt22 X, lAlI FI X,2-4HNIJMRFER OF SFGMFNITS I SEOFIT?I FJAXv 33H TOT AL NUMR R O F CEFLL O-~ N T HF ROOny,1 I.16)X, 35HN IMBF FR OF iNr~C iDENT FIEFL D DITREFCT i nms I9/ f,16X,29HNiMBFER OIF BIST AT IC, O IR ECT T OrS, T15/; F, 1 6X, I OHWAVFL FNGTH, F 34. 5 / F16~X,16~HimpEOANC-F FATOR,E1(6.5,'l,',F1O.5) 425 FORMAT (////,v36~X,18HSURFACF FlIELO OATA/,?7X, &29HFnR TN~CIOFNT FIFLI ) OIRFCTIONi=,F7.2) 500 FORMAT (11HO I SEG,,4X f4H X (I )6X,4HY(I),6Xq4HS(I),5x,(H-nHS0I F~4X,A HR S(,4X, 6H X S(,4X,,6tHMnO( j ) 4 X, 6HARG( j/ 77

6)OO FnPM AT (I1HI,127X, 2HRHACKSC ATTFR I NW( CFU1SS, SFCT IOM//23X, F~36HTHFTA S I(;MA/LAMRnA,nR PHASFDF() 800 FnRMA T I1Hl,?3X v33HB I STAT IC SCATTFR I NI CRASS SFCTLn / 2 3 X F,29HFnR INiciDFNT FIFLOF) IRFCTIf-W=,pF7.?//?3X, F,,3 6~HTH F TA STGM A/LAMFBDA, DR PHASFDFG) 900 FnRMAT ( 15XvFli3.?qFl4.?,FD6.1 901 FnRMAT (15XqF13~.?jFj4.2,Fi-5.1) F- rin si iRR iI T I NE M T X ( WMHXK Xv Y X NI IYNDS(~yL I INPv 75, A) C RICH VFRSiON,. H-POLARIZATIOnt R F AL4 X ( 1 ) Y ( 1.) I XNI( 1) YN ( 1) n,OO( 1) INITFGFR*4 LIJMP(I) CnmPLFX*8S A(MHv1)9,ZS(1 ),vHZ,vHIAH1B,DH)IMH1RLH7AHZPH7RLCOmmONi /PIFS/ PITPI, PIT, PIPT9 Y7,RFn,nDJC I H=C) I L = C 00n1 I=?M, IH=IH+l X I=X ( I) YI =Y( I) XNII=XtN( I) YNIT=YNI( I.J H= 0 n0?5 ~=,,,-H=J)H+1 I SFG=[Iif'AP(,4) nS=FnSn( ISFC,) IF( IH.FO.JH)GfO Tfl 50,J A =J-1I 3B=3i+1 I F ( J.F l. M)I JR X~JA=X(..IA) YJA =Y( IA) XIJB=X( JR) Y JR =Y ( "JRf X NW A=XN ( JA) Y NUA =YN( JA) XNJB=XN (IB) Y NJ)R= YN( JR) R\X=X I-X ( 3) RY=YI-Y( 4) R = S R T( R X *RX +R YR Y) SnSp=XNiI*xNi j) +ym,yNi( j PK = R X K CALL HANKZL1( RK *OH7 iDH)ll H7 =HZ*snsp R X = XI - X \IR RY=Y I-YJBR R=SOR T( RX*"RX+RY*RY) SOR= ( RX*YNI I-PY-',XitI)/R snsp= xNi i*xNiJIA+Ym T~yNij PRBK=R' XK 78

C7,AI H A NKZ I( P RK,2,vH 7 RAH IF HI.R=HI R'SDR H7R= SnDSP ' Hi7R F (X J A.FO),XJB RL.A NJ1. Y~JA F n.YJ RL.A ND. I.FOn IL) (;O T r 15 PRX=XI -XJA P Y =Y I - YJ IA P=SOR T( RX*RX+RY*P''Y) S D R(RX *Y NI - RYy-X N I)/P nSDSP= XN I XNJ1, A +YNI I *'(Y 1\,1A PRA K =RP*X K CAL L H ANKK ZI(RA K, 2,HZAH IA) HI A=H1lA'SDR H/7A =SDnSP4ZA (0 TO?0 15 HIA=HlRL l-Z7A=H7RL 20 A ( IHJH)=PlT*DS/6vX.*(H7A+4.*l~H7+HZFA)-0.254( HlR-Hl1A) XI PL =X JR Y J PR L = Y J R Hi PL=H1 P H7RL=H7R (;O TO 25 570 A(IHv,JH)=ZS(ISFC-,)+ FCMPL-X(PIT*DS/2.,I../DS/PIPI+DS*(AIOC(DSr)')+0.02R7989-7)/2.) 25; Jt=I 10 CO N T I NuE P F TI JR1\ f \ ND S I J RROi)TNINE GFOm ( I-i imP, x, Y,XNY N, s,DnsOn, 7 vM) CTHIS \/FPSInN READS~ ANID (-FNIFTATFS SFCMFNlTS IN' COIJNlTFR-C.LfCKWISlPF DIpFCTIlr)., CTHF SI.IRFAC(F MU.ST RF CtfiSFD. C, POINTS ARF GFNIFRFTFD) AT THF START AND miDPOINITS OF EACH CFLL-: C, THF START POINIT OF THF FIRST CFLI F\/FNITIIAL-LY COINlCIDES C WI1TH THF FNID POINT OF THF I.AST CFLL. CD FM P L F Xy4'8. ZS( 1) 97 PFAL*'#4 X (1),qY (1) XN( 1) *YNI( 1), (1) D sn (1) I NITFG7FR*4 LUMP I) COnMMONI /PI FS/ P IT PI P IT, PITP, Y 7,9R FDDnTI I =0 VL=0 C....RFAD INlPUT PARAMFTFRS, A~ND PPFPAREF TO GFNIFRATF SAMPL ING POINITS I0 RFAD (5,200) lN,7,XAYAXPYP,ANlC I F( N.LtT. 1 cO TO 120 M IM=2* T X =XA -X R TY=YA-YB D=SOR T( TX*TX+TY*TY) L-= J + 1 7 5 (I = 7 IF (ANIGFO.0.0) CO TO 20 T=O. * S4FD*AN; 79

TR X=TX+ TY* COT AMI( T) TRY=TY-Tx*cOnTANI( T),ARC=?.0*PAfl*T AL F = T/ N nfln( L )=2.0*RAD*'Al.F GO TO 30 20 RAfl=9Q9*999 ARC=Dl nfiSO ( L ) =nl/N C.... OSTART GFNFRATTIMG fls=lso ( L ) /2. n(1S=flSSrS I-I MP( I )= L I F ( A NGC.FO.0.0) GO TO 40 SI NO=S I K( ARG) cOns0=COS (ARG,) X(1)=XA-0.5*( TRX* (i.0-COnsO )-TRY*,',STNOQ) Y(I)=YA+0.'5*(TRX*,SINIO+TRY*'(1.0-CnSO)) XN()=+0.5*(TRX*COSO+TRY*-SINIO)/R AD YN(I )= 0.5*(TRX'SINO-TRY*cnsO) /RAD GO TO 50 40 X( I ) =X A-0. 5*J*T X/N Y( I)=YA-0.5*,t*TY/Nl XN( I )=-TY/D) Y N(I )= TX/fl SO CON TINIJF 100o WR ITF ( 61,v300) L, NX A, Y A, X R, YANrJGv R Ar),vAR C,,7 GOn TO 10 120 M=I -LL= L 200 FOR MA T(I 2,3 X,7FlO.5) 300 FOR MA T 1 3, 69 4X,4 F10. 5,F7.2,94 F1I0. 5 R FTI R N F Nfl SIJFROmJTINIF HANIK7.( R,,Ni.H7FRO,HOnrF) C...H A N K L FUIINCT IONiS A RF O F F I PST K I ND —J +I Y C..... 1=0 RFTIJRNS H7ZFRO C*,* r =]. RETURNS HOnF~ C..... =? RFTIIRNIS H7FRfl A~ND HINIF C... SfIJRntiTINF RFOUlIRF7S R\>On C-.....SIJROniITINlE AflAM MUST f~F SUIPPLIE) [AY USFFR 80

DI MENS I UNl A ( 7 ) * F ( 7 ) 9C ( 7 ) N (7, ( 7 )v F ( 7 ), C( 7 ), H ( 7 CC)nMPLFX HZFRUl,HnNF nATA A, FC, U, FF, C,,H/i * ),-2. 2499997,1. 2656208, -C. 3163866,9 FC0.C)444479,-C).0039444,C).000)21,0).36746691,0).60559366,-0.743sC)3849 FC0.?53001~17,-0).C4261214,C0.00C427916,-o).CCon24846,C). 5?-0).56249985, FC).?1093~573,-0).C3954?89,C).00443319,-0.o0003176,19-0.00001109, F,-C).6366-198,C).2?1?O91.9?.16827C)9,-1.3164R27,O.31?3951,-O.O4O0976, ~C0.00C27P7390.79788456,-C).CC00000)C77,9-0.00C55?74,-0i.00009512, FC0.00C13~7237,-0).00C)728C)51C).000144476,-C)7839816,-..-)04166397, R~-0).0000C3954,C0.0 O?(573,-0).00054125,-0).00029333,0.000 13558I FC). 797Pq456,O.OC00000C156,0.0)1 659667,t 0.O00017105,-0).00249-5 11 * FC).00O1l3653,v-0.C000OO33,-2.35649q449 90.12499612*CO. oO0~569, F~-0).0637879,C).)00C74348,o.Coon79824,-0.000C)?g1),,/ IF (R.LF.O.0)) GO TC) gC) TF (N.LT.O.C)R.N.CT.2) GOr TC) C) TIF (R.GT.3.CO) Gnf T) 2C) X =R4R /9. C) IF (NI.FC).1) cn1 TOfloC CALL ADAM(AXq,KI) CALL AnAM(B,,X,,Y) RY=C,63 66198* ALnC)(C)05*R)*-RJ+Y H7FRl=c~mPLX( RI,FY) IFP ( N.FC).O) RETuIRN~ 10n CALL AnAM(C,X,Y) RF-I=R*Y C ALI 4nAm(nl,x,,Y) RY=C).6366198* ALCC (C). 9 -"R )RJ+ YR HC)N F = CMPL X( RJ, RY) R F TI IRM 20 X =3. 0/ R IF (NI.FO,.l) GO TC) '-0 CALL AF)AM(EXY) FC)CL_=y/sCRT(R) CALL An)AM(F,X,Y) T=R+Y Ki.-=FC)CLCC)ns (T) RY=EC)C)L*SIN(T) HZFRC)=CMPLX( FJ,RY) IF (N.FC).O) RETUJRN 30 CALL Af)AM(C,.XY) FC)C)=Y/SC)RT(R) CALL AnAM(H,,X.,Y) T=R+Y RJ= FC)CL*cC)s (T) RY=FC)CL*,SIN(T) HC)ntiF-= C MP LX(Rj,JR Y R F TI IRN SC) IAR ITF (6,v90) N,9R go EC)RMAT(3?HOSICK DATA IN HANIKZ1 *C)I)TT* Nl=v12,2X,2HR=,Fll.3~) CALL SYSTEM E NDF si iRC)i JTINE AnAM( C,,XY) nimFNSJC)N C(7) Y=X*C (7) nn) IC) 1=1.,, 81

10 Y=X*( C( 7-1 )+Y) Y=Y+C( 1 ) R F Tl IR SlIRRnITINIF FLIP( A,N,MI,L,M,X,Y,IAT) CnmPLFX A(MI, 1 ), X 1 ),Y( 1), D,RIA, HnLn fDIMFNSIlO L( ),M( 1 ) IF (IAT.cGT.1) (n TO 150 n=CMPLX(1.0,0.0) Dn RO K=I,N L ( K )=K M( K )=K RIGA=A(K,K) nn 20 I=K,N nn 20 T=K,N 10 IF (CARS(RIGA).GF.CA/RS(A( I,J) )) Tn?0 RI A=A( I,J ) L ( K ) = I M( K )=J 20 nN T INIUIF,1J= (K ) IF (J.LF.K) Cn Tn 35 nn 30 I=IN HnO.nD=-A ( K, I ) A ( K I ) = A ( J, I ) 30 A(J, I )=HnLD 35 I=M(K) IF (1. IF.K) t6n Tn 45 nnO 40,J=I], N Hn.- D=-A( J,K ) A ( J, K ) = A (J, I) 40 ^ ( I A I)=HnL 45 IF (CARS (RI A).NF. 0.) C, Tn 50 r)=CMPL..X(0.0,0.n) R F Tl IR N 50 nn 55 I=l,N IF (I.FO.K) rn Tn 55 A( IK)=-A( I,K)/RIGA 55 CnN TI NIIF nn 65 I=1,l N nn 65 J=1,N IF ( I.FO.K.!R.t. FO.K<) G, Tn 65 A(I,J )=A( I,K )*A( K,I )+A( I,t) 65 C.O TINI F nn 75 J=l,N IF (J.FO.K) CGr Tn 75 A (K,J ) =A( K, J ) /R I GA 75 C ON!TI NIIF D=nD-R I GA HO A(KK)=1.O/RIGA K =-N 82

100 K=K-1 T F (K.LF.0) GA TA 150o I = L ( K) IF (I.1-F.K) GA, TA 120 Hni-n=A(,I,,K) 110A ( Jv K) =HAn(J, 12 0 J=M (K) IF (J).LF-.K) GA TA 100 DO 130 I=1,rNl AKI)n=-A(KI) 1-30 A(j),I)=HnLD (A TA 100, 150 DO 200 I=1,Nj Y(I)=CMPLX(0.o0, 0.0) DO 200 J1=10\N?()0 Y( I)=A( I,JI)*X(,j)-Y( I) RFTIIRNI E iND RLnCK DATA COMMON/P IES/P 1gTP IPIT, PIP I Y79 RFD, DGf) DATA PT 9TPI,PIT,PIPI,Y7,RFD,DIG,/3.1.4159?7,6.2831853, F,1, 5707 6~39,9, s694'6044,,0. 00205?5 R?4, 0O1 7453?9,597. 2957A/ F ND 83