02142-502-M 2142-502-M = RL-2042 THE UNIVERSITY OF MICHIGAN COLLEGE OF ENGINEERING DEPARTMENT OF ELECTRICAL ENGINEERING Radiation Laboratory THE MONOPULSE POINTING ERROR ASSOCIATED WITH A TWODIMENSIONAL CONICAL OR OGIVAL RADOME WITH OR WITHOUT A SURROUNDING (WEAK) PLASMA By T. B. A. Senior and V. V. Liepa February 1970 Prepared for Harry Diamond Laboratories I"I Microwave Branch, 250 Attn: Dr. Irvin Pollin, AMXDO-RDB Washington, DC 20438 Ann Arbor, Michigan, 48108 201 East Catherine Street

THE UNIVERSITY OF MICHIGAN TABLE OF CONTENTS I INTRODUCTION 1 II RAY OPTICS FOR A DIELECTRIC SLAB 6 II APPLICATION TO A BARE RADOME 13 IV MONOPULSE RESPONSE 21 V EFFECT OF A SURROUNDIWG %WBkI) PLASMA 29 VI NUMERICAL PROCEDURES AND RESULTS 42 APPENDIX: THE COMPUTER PROGRAM 58 (By Mrs. P. Larsen) i

THE UNIVERSITY OF MICHIGAN I. INTRODUCTION A problem of oontliulaW praotloal concern is that of assessing the effect of a radome on ti performa oe of a radar antenna placed within it. If the antenna is required only to radiate a signal and then to detect the time of arrival of an echo from seme distant object, the electrical constraints plaoed upon the radome are very light indeed. Unfortunaly, however, such a simple situation is the oexception rather than the rule, and more generally the task is to design a radome which will maximize the transmitted power at some frequency (or over a rage of frequencies), or will minimise the phase distortions over a range of look angles, and which also satisfies those oonstraints which are imposed by aerodynamic considerations or by the environment. An approximate but versatile tool which has been in general use in radome deMpi for many years is geometrical optics, or ray tracing. Taking, for example, the reception problem in which a wave (usually a plans wave) is incident on the outer surface, the incident wave is sampled by means of rays drawn normal to the wave front. Each is traced to the outer surface, and followed as it undergoes refraction at that surface and transmission at the inner one. From a knowledge of the transmission coefficients, which depend on refractive index, polarization and angles of incidence relative to the local normals, and a computation of the electrical distance, an approximate sampling of the field within the radome is obtained. Depending upon the design requirement, the shape and (perhap) the refractive index of the radome are now adjusted, and to increase the flexibility, multi-layer (sandwibt radomes can be considered. This is th sse essence o te oretial techniques in common use, and though they have proved adequate for many purposes, it should be emphasized that the sampling of the interior field is approximate not only by virtue of the approximations inherent in ray tracing (namely, the assumption that the surfaces all have radii "large" compared with the wavelength), but because the rays reflected at each surface are neglected. On the other hand, to include any reflected 1

THE UNIVERSITY OF MICHIGAN rays, many of which would ultimately produce additional ray contributions to the field within the radome, greatly increases the magnitude and complexity of the computation, and was not feasible until the advent of the present generation of high speed, large capacity computers. The particular problem of interest to us here is one in which the greatest posible accuracy of estimation of the field within the radome is necessary, and In which the inclusion of all possible ray contributions Is mandatory. The problem is conoerned with the operation of a monopulse radar mounted inside the nose radome of a high speed missile. The radome is either conical or ogival in shape and of single layer construction; its material )erglss is effectively lossless at the C-band frequencies of operation, and for purposes of a8lysis can be treated as a pure dielectric. The relevant feature of the monopulse system is a gimballed plane containing slots mounted within the radome at a given distance back from the apex. From a comparison of the signals induced in the slots, the monopulse plane is made to take up a position parallel to the effective phase front of the field impinging upon them. Were this field Indeed a plane wave, the precise interconnections of the slots through which the comparison is made would be of little relevance inasmuch as any "reasonable" design of monopulse would produce the required alignment. Because of the perturbing effect of the radome, however, the field is not a homogeneous plane wave inside the radome even when the field outside is. The method of comparison of the slot signals will then influence the position taken up by the monopulse plane, and it becomes desirable to record the signals induced in the Individual slots. In addition, however, it is convenient to conceive of a simple mode of operation, which is essentially a phase comparison scheme, and which permits a straightforward calculation of the monopulse orientation, and this we shall do. Since the field inside the radome is a perturbed version of the plane wave incident on its outer surface, the monopulse plane will not in general align itself parallel to the external wavefront. The angle between these two 2

THE UNIVERSITY OF MICHIGAN planes is the pointing error of the system, and its determination is one objective of this study. Inasmuch as the system is desired to have, and is presumably designed to have, a pointing error not exceeding 2 milliradians for all polarizations of the incident field, and for all angles from 00 to 550 from axial incidence, the reason why we must include all possible ray contributions, in order to achieve the greatest possible accuracy in the estimation of the field distribution over the monopulse plane, is now apparent. Indeed, it is not without question whether ray tracing can provide this sort of accuracy, but it certainly cannot without taking account of reflections from the inner surfaces of the radome w-lls, as well as from within the radome layer itself. Unfortunately, there is a further complication. Because of the speed and altitude at which the missile is required to operate, a plasma layer will be formed just outside the radome wall. In penetrating this layer, the field will suffer a perturbation additional to-that produced by the radome itself, and this will in turn produce a change in the pointing error of the monopulse system. Such a change will depend on the nature of the plasma and, hence, upon the altitude, and could negate any attempt (by, for example, shaving or blocking portions of the radome) to minimize the operational pointing error. In consequence, any change in error produced by the plasma is even more serious than the error associated with the bare radome, and the determination of this change is our prime objective. On setting out to develop and assemble the formulae for three-dimensional ray tracing with even the bare radome, it was at once apparent that the computation of the interior field was an intricate task involving large amounts of time even for a high speed computer. To find, for example, the pointing error for one monopulse system within a specific radome for a wave with a single angle of incidence and polarization, it seemed possible that a running time of as much as one hour on a high speed (IBM 360 ) computer would be necessary. Subsequent events have shown that this estimate is not unrealistic. Rt therefore seemed essential to start out with something more simple than the general case, 3

THE UNIVERSITY OF MICHIGAN and the two-dimensional analogue of the actual problem was a natural one to choose. In the two-dimensional problem a conical radome appears as a wedge and an ogive (whose surfaces are arcs of circles) is replaced by one having surfaces which are arcs of circular cylinders. The field is assumed incident in a plane perpendicular to the generators of the surfaces (i.e. in the xy plane), and it is sufficient to consider only the two principal polarization cases in which the electric vector is entirely in the z direction (E-polarization, or TE) or the magnetic vector is so aligned (H-polarization, or TM). Since the entire problem is now two-dimensional and can be expressed in terms of either Ez or Hz, the visualization and details of the analysis are greatly simplified. The present Memorandum is concerned entirely with this two-dimensional problem. The conception and development of the analyses are described, and the limitations which are imposed. by the use of ray tracing are discussed, as are the steps necessary to derive an expression for the monopulse pointing error ufter with or without a plasma sheath about the radome. Full details of a computer program (in FORTRAN IV) including flow diagram, input data and program listing, are given in the Appendix. Specific results for the pointing error with and without the plasma are presented and discussed. The analogous procedures for the more general three-dimensional problem are described in a companion Memorandum. Although the two-dimensional radome is, of course, a mathematical idealization, it should be emphasized that we do not consider the case treated here as one having academic interest only. The practical purpose for our study is to see whether it is realistic to expect a maximum pointing error of 2 milliradians for a wide range of aspect angles, with or without a (weak) plasma sheath. The pointing error arises because of the field distortion produced by radome reflections. A three-dimensional geometry will produce more reflections than does the two dimensional, and perturb the wave front in three directions rather than two. It is therefore only natural to expect that the 4

THE UNIVERSITY OF MICHIGAN results for the two-dimensional geometry will constitute a lower bound on the pointing errors that will occur in the three-dimensional case. As will be shown, the errors found using the two-dimensional geometry in general exceed the 2 milliradian requirement. 5

THE UNIVERSITY OF MICHIGAN n. RAY OPTICS FOR A DLEXCTtIC SLAB X Is a St bat iutruotive probem to examine the transmission of a p-ba eleetrmM lo waVS through a homogeneos and isotropic dielectric sd*. AMMi- tibs topio is treated in many electromagnetic theory tents and is a-met ell book devoted to opttis, the reults and their interpretation 3pir sio a vital role in the treatment of the raome problem that a brief dsmuson is desirable. Consider a pla-e deletr slab of thickness d and Infinite extent occupying a < y < a + d, where, y, a) ae rectangular Cartesian oerdinate. It is sufficient to take the permeability of the dielectric to be the same as that of free space, i.e. = AO, but the permittivity differs from to, and we define the refactive index n of the dielectric relative to free space to be n ~= /co. The regions above and below the slab are occupied by free space. A plane electromagnetic wave is incident on the lower face of the slab. We treat first he case in which the wave is H-polarized, i.e. TM, and write the inoident field as Hi A iklin a+y coo a) - (1) I_ ^ lk(xsinct+yoosa) E = - Z x oos-yslna) e where Z i*fo/c is the chr acteristic impedance of free space, and a time factor ea has been suppressed. As evidentkrom Fig. 1, a is the angle whioh the propagation vetor makes with the normal to the slab. To find the field transmtted through the slab, we postulate the following field expressions: Figures are placed at the end of each section. 6

THE UNIVERSITY OF MICHIGAN y < a: A eik(x sin o+y cos )+A ik(x sin a-y cos a) | H=z e +Ae E= -Z[ Osa {ikxsinycoosa)Aik(xs lna-ycoa)} (2) -y sinlk(x sin +y cos )+Aeik(x sin o-y oosa ] a y - a-d: H=A finke c sfty * cofft) ink sin * -y coo l) E - o c08n {Beink i Ycos einkin8-yeo (3) AC rBink(xsin+ycos(3) ink(xsinp-ycosP) -ycosop jBe +C0 J a+d <y: H Deik(x sin a +y cos a) -- 0 The unknown coefficients, A, B, C and D can be determined from the boundary conditions at the two faces of the slab, which conditions require that H- z and E - x be continuous there. Applying these conditions, we obtain ika cosaA -ika coso= inka cos+C -inka cos, e +Ae Be e ikacosa -ika cosa 1 r inka coso3 -inka coso e -Ae d-o Be -Ce De ik(a4d)cosca= Beink(a4d)cos3+-Cink(a-d)cos3 De - Be +ce9 r"7 7

THE UNIVERSITY OF MICHIGAN where r ncos (5) cos3 with sin = in (&ell's law), (6) n and hence 1 ik(a4d)(oos an+ cos 3) C = 2(1- Dr) ^ o B -= (1+P) Deik(ad)(c~ oo —n os) A = - ( - )D sin(nkdcos3)eik(2a+d)cos a with 4e ikd(n cos8-cos a) D 2 2 2Inkdcos 0 (r l-ar i)2.2"da * f7) D represents the transmission coefficient of the slab and is the quantity of most interest to us. Its exact expression is given in Eq. (7), but for future purposes an alternative representation is more convenient, viz. 00 D D (8) / m where D = 4 r ikd(nsg-cocosa) -P 12(m-l) 2i(m-l)nkd ose +1) 2'V+l (9) Each term D in (8) can be associated with a partial transmitted field m ^(m) e ik sinn yycoss ), E (m -Z ( coss08 in )D eiksnay s - o m 8

THE UNIVERSITY OF MICHIGAN and to appreciate the origin of this field, consider for the moment the simpler problem shown in Fig. 2. The dielectric now occupies the entire half-space y > a, and if the incident field is again that given by the Eqs. (1), the reflection and transmission coefficients are, respectively, -1 2ikasin 12 r +1 (11) T 2r eka(cosa-n coso) 12 rF+l as shown in many standard texts. The first suffix refers to the medium in which the wave is incident ( 1 denotes free space) and the second refers to the medium at which it is reflected or into which it is transmitted ( 2 denotes the dielectric). Alternatively, if the dielectric occupies the half-space y < a + d (see Fig. 3 ) so that the interface is illuminated by a wave propagating in the denser medium and of the form given by the leading terms in the Eqs. (3), the reflection and transmission coefficients are R -1 2ink(a+d)cos3 R21 r +1 (12) 2 ik(a+d) (n cos3-cos a) T21 r +1 where r and 3 are again as defined by Eqs. (5) and (6) respectively. (m) With the aid of (11) and (12), the interpretation of the partial fields H, (m) E is now obvious. As evident from the moduli and phases of the coefficients D, ml, 2,3,... (see Eq. 9), H(, E is the field produced by refraction at the lower interface of the slab as though the upper interface were not present, followed by a refraction of this wave at the upper interface as though the lower interface were not present. Indeed, = T12 T21 9

THE UNIVERSITY OF MICHIGAN (2) (2) Likewise, H, E) arises from a refraction at the lower interface, reflection at the upper interface, reflection at the lower interface and a refraction at the upper interface, so that lD21 = T1R21R21T211 with the phase of D2 being that appropriate to the zig-zag path; and so on. We can therefore build up the exact field transmitted through the slab by considering each interface separately, and by superposing the partial fields resulting from all possible bounces within the layer. Each partial field is such that the boundary conditions at the isolated interfaces are exactly satisfied, but it is only through the superposition of all these fields that the boundary conditions at both interfaces are jointly satisfied regardless of d, and of the absence of losses within the dielectric. If, instead of an H-polarized (or TM) wave, the field incident on the slab is an E-polarized (or TE) plane wave, such that i A ik(x sinQ ycos) E ze (13) Hi y cosa- sin a) eik sin y cosa) - 0 with Y = 1/Zo, the analysis goes through just as before with the sole difference that n is now replaced by 1/n. Hence r n = r t, (14) n cos0 and the reflection and transmission coefficients (based, of course, on E ) for single interfaces are as given in Eqs. (11) and (12) with n replaced by l/n and therefore r replaced by P'. Although the above description has been phrased in terms of partial (plane wave) fields, it is obvious that the picture that has evolved is identical to that 10

THE UNIVERSITY OF MICHIGAN which is provided by geometrical optics, i.e. ray theory. Starting from any point in the region below the slab, we trace the ray through this point and perpendicular to the incident wave front (i.e. in the direction of the propagation vector of the incident field) until the ray meets the lower interface of the slab. Here it undergoes reflection and refraction. The refracted ray now makes an angle B with the normal to the slab and proceeds with the decreased velocity c/n until it hits the upper interface, where reflection and refraction takes place. The refracted ray provides a direct sample of the field above the slab. The reflected ray is followed back to the lower interface, thence to the upper interface, to provide another sampling of the field in y > a + d; and so on. This one single incident ray therefore provides an infinite sequence of discrete samplings of the field in y > a + d. Moving now to an adjacent point on the same incident wavefront, we repeat the process to provide yet another sequence of samples, but because of the planar nature of the geometry, it is apparent that the second sequence differs from the first only by a linear shift. It is this fact which permitted a discussion in terms of partial fields, thereby obviating the need for sampling the incoming field. For other than a planar geometry the partial fields would not be plane waves and could not be easily obtained. We then have no alternative but to resort to ray theory and to sample the incoming wavefront over that portion that produces a significant contribution to the field beyond the dielectric in the region of space of interest to us. Clearly, the samples must be sufficiently close (<< X) for the rays to be reasonably dense throughout this spatial region, and in particular, if there are several different categories of rays, we must ensure that several rays of each category are included. Nevertheless, it should go without saying that the solution obtained in this more general case is only approximate no matter how many reflections within the layer are included, and no matter how closely the incident wavefront is sampled, but if the lateral dimensions (includitngradii of curvature) of the interfaces are large compared with the wavelength, and if all caustic or focussing effects (where aninfinity of rays come together) can be ignored, the solution should reproduce the dominant features of the true transmitted field, and be accurate enough for most practical purposes. These conditions would appear to be fulfilled in the radome problem to which this technique will beS applied. 11

THE UNIVERSITY OF MICHIGAN free space I y = a4d y=a free space 3 IX IX Fig. 1: Slab Geometry I Medium 2 (dielectric) -----—!/ ---St -- By Medium 1 // (free space) Fig. 2: Geometry for Single Interface (a). Medium 1 I (free space) y=a+ Medium 2 (dielectric) / Fig. 3: Geometry for Single Interface (b):a d 12

THE UNIVERSITY OF MICHIGAN III APPLICATION TO A BARE RADOME Consider a two dimensional radome symmetrical with respect to the plane y = 0 of a rectangular Cartesian coordinate system (x, y, z) and whose outer and inner surfaces have generators parallel to the z axis. A plane electromagnetic wave is assumed incident on the outer surface of the radome, and by taking Its direction of propagation to be perpendicular to the z-axis, the entire problem becomes two dimensional. It is then sufficient to confine attention to the plane z = 0, and each surface of the radome can be defined by an equation of the form y = f (x). Two particular* radome configurations are considered, namely, ogival and conical. In either instance it is assumed that the outer and inner surfaces are both ogival or both conical, and whilst this still permits the radome thickness to be non-uniform, it will be appreciated that the type of thickness variation that can be considered is quite restricted. For computational purposes it is convenient to choose the origin of the coordinates at a small but non-zero distance I to the left of the 'nose' of the radome (see Fig. 4). In the ogival case, the outer surface can then be defined by the equation Youter + (A+B2-( A2-) (15) with the upper (lower) sign referring to the upper (lower) surface of the radome, and where A, B and C are positive real numerical constants in terms of which the maximum diameter of the radome is 2 (A2+B2A) occurring at x = C, the radius of the curvature is R, where R = A2 +B2" * The extension to any radome configuration that can be analytically defined is entirely trivial. 13

THE UNIVERSITY OF MICHIGAN and I = C-B. The overall length of the radome is 2B, but in practice interest is confined to that portion of the interior extending to at most the position of the maximum diameter, i.e. to x < C. The inner surface of the radome is defined in a similar manner. For the conical case, the definitions of the radome surfaces are more straightforward, and for the outer surface we have y ta (x..) (x >I) (16) youter ^ -) c> (16) where the upper (lower) sign again refers to the upper (lower) surface of the radome. The inner surface is defined in a similar manner, but with a different value for I. Regardless of the configuration, the radome is assumed to be of single layer construction and formed from a material which can be treated as a homogeneous, isotropic and lossless dielectric whose permeability is the same as that of free space. The electromagnetic properties of the material can therefore be represented by the (real) refractive index n. A typical value is n = 2.5 appropriate to fiberglass at a frequency in the GHz range. Before detailing the various steps in the ray tracing procedure, a few words about the overall objective of the program are desirable. Although ray tracing could be used to determine the field characteristics anywhere within the radome, the particular objective is to assess the performance of a (receiving) monopulse system. The monopulse plane pivots about an axis parallel to the z axis and located at a point x = D < C, its orientation being determined from a comparison of the signals induced in a number of slots located in its plane. If the field Inside the radome were indeed a plane wave propagating in the same direction as the external field incident on the radome, the monopulse would align its plane parallel to the external wave front. Because of the existence of the radome, however, the field inside differs from that 14

THE UNIVERSITY OF MICHIGAN outside, and the monopulse can be expected to take up a position which is not quite parallel to the external wavefront, but differs by some small angle E which is then the pointing error of the system. On the assumption that both the radome and monopulse systems are well designed, e will be of the order of a few milliradlans or less, and it is then sufficient to assume that the monopulse plane is actually parallel to the external wavefront, and to deduce a pointing error in the manner described in Section IV. Only in the event that the error so obtained was measured in several tens of milliradians would it be necessary to re-align the plane over which the field distribution was being sought. No such base has been found, and we can therefore state the intent of the ray tracing as the determination of the field distribution in amplitude and phase over a plane parallel to the incident wavefront and centered on a line parallel to the z axis at a distance D -1 back from the front of the radome. Consider a plane TE or TM wave incident on the radome at an angle* 13 to the plane y = 0, i. e. the x axis, as shown in Fig. 5, and take as basis the zero phase wavefront passing through the origin. Choose a point (xo, yo) on this wavefront (clearly, x0o -yo tan 3 ), and follow the ray through this point (and normal to the wavefront) until it strikes the radome. Obviously this will be the outer surface, and we can ensure that it is also the lower surface by taking yo sufficiently large and negative. Find the point of intersection (x1, yl) and record the distance do = {(l-x )2+(yl-y o)2 1 v2 Compute the direction of the outward normal to the surface at this point and hence determine a, the angle which the ray makes with the normal. The angle which the refracted ray makes with the normal can now be found from Snell's law (Eq. 6) and the amplitude of the ray is T12. Follow this ray until it strikes the inner surface of the radome and compute the point (x2, Y2) of intersection. Record the 'optical' distance n d21 = n {(x2-xl)2 + (y2y)2)/2 and add to d01. Compute the direction of the normal to the surface at (x2,y2), thereby finding the direction which the ray makes with the normal, and allowing the This is the angle between the direction of propagation and the x axis. 15

THE UNIVERSITY OF MICHIGAN determination of the direction and strength T12T21 of the transmitted ray. Follow this ray until it (i) strikes the monopulse plate at a point (p, y p) in which case the distance d2p = (x-x2) +(yp-yp) 1/2 is computed and added to d01 nd21, and the result recorded along with the strength T 12T21 of the ray and the direction which it makes with the horizotal; or (ii) passes beyond the monopulse plate between the extremities of this plate and the inner radome surface, in which case the ray contribution is ignored, and attention reverts to the previous intercept of the ray with an inner radome surface; or (iii) the ray strikes an inner (upper) radome surface. In this event the point of intercept (x3,y3) and the normal direction are computed, along with the accumulated (optical) ray distance to this point. The direction and amplitude T12T21Rl2 of the reflected ray are found, and the ray followed until it strikes the monopulse plate (and is recorded), or passes beyond the plate (and is ignored), or strikes the radome again. If it does strike the radome, the process is continued, but ultimately the amplitude of such a ray will fall below a pre-set level, and can then be ignored on this account. Having followed a 'dominant' ray to a conclusion, attention is transferred to the previous intercept of this ray with a radome surface, and the reflected or refracted ray that was omitted is now considered, and this also followed to a conclusion. But in the course of its path, this ray may also have spawned further rays by reflection and refraction, and these too must be picked up and traced through to a conclusion; and so on, arriving ultimately at the stage at which all significant contributions generated by the original ray through the point (x, yo) have been considered. We now return to the incident wave front and step a distance A along it, where A is some pre-set value <<~. The above process is then repeated with the ray originating at the point (-0 + Acos3 tan, y +A cos8 ) and so on until, with further stepping, no rays can be found to intercept the monopulse plate. 16

THE UNIVERSITY OF MICHIGAN The procedure should now be apparent. Each individual computation is rather simple, requiring only the calculation of an intercept point, a normal, a distance and an angle, and from these, r (or rP ) and transmission and reflection coefficients; but because of the considerable number of such computations entailed in following one ray through the radome and on to the monopulse plate (one ray may produce many tens of significant contributions to the monopulse field), the process would be extremely tedious to carry out by hand. Nevertheless, it is well suited to a digital computer, and the only complioated aspect of the programming is the ordering of the sequence in which the ray contributions are computed to ensure that no significant rays are omitted. For each polarization (TM or TE) there is just one form of reflection coefficient that must be computed, but two forms of transmission coefficient. Thus, for a TM wave, we have* R -R (17) 12 21 r i+1 2r 12 r '+1 2!' -(18) 2 21 r + T21 +1 where n cos a r = (20) with sin = i, and the notation is as shown in Figs. 2 and 3. Since n n > 1 (typically, 2. 5), any (real) a gives rise to a real value of 8, and r is a monotonically decreasing function of a, ranging from a maximum of n for a = 0, through unity for a = tan'" n (Brewster angle), to zero for a = '/2. As a consequence of this, R12 is also a monotonically decreasing function for 0 < a< tan 1 n, and is quite small for most angles of interest to us, but for increasing a> tan-1 n, -R12 increases rapidly to unity. The situation The phase factors appearing in Eqs. (11) and (12) are here omitted since they are picked up in the computation of the (normalized) distances along rays. 17

THE UNIVERSITY OF MICHIGAN is, however, somewhat different as a function of,, i. e. if the ray starts within the denser medium. A real (3 gives rise to a real a only if I sin | -, and as a function /3, r varies from n for j3 = 0 down to zero for /3 = sin- l. For larger (3, Pis pure imaginary and total internal reflection occurs. For an incident TE wave, -R = r (21) 12= -2 1= -(21 T 2! (22) 12 r +1 22' 21 -, (23) where rp = Cos.a (24) n cos p ' Since r1 = l/n2, rP is a monotonically decreasing function of a varying from a maximum of 1/n for a = 0 to 0 fbr a = c/2. Since it is always less than unity, R21 is a monotonically increasing function, and for most angles of interest to us, the reflection coefficient for TE waves is substantially greater than for TM waves. This implies a larger number of significant ray contributions in the former case. There are two final comments that should be made, one pertaining to both radomes and the other to the ogival one only. We have noted that the determination of a ray path requires the calculation (or knowledge) of the direction of the local normals to the radome surfaces. With both radomes, the normal is undefined at the very apices of the inner and outer surfaces, and to avoid difficulty in this regard, any ray which hits these points is automatically terminated. This is equivalent to assuming the radome to have infinitesimal opaque 'plugs' here. The second comment applies only to the ogival radome whose surfaces (3 and a are here the angles used in Section II. 18

THE UNIVERSITY OF MICHIGAN in the xy plane are curved. Because of this curvature and, in consequence, the slight difference in the directions of the normals at different distances from the apex, it is possible that a ray striking the outer surface at an angle very close to grazing may, on reaching the inner surface, find itself within the critical angle and be entirely reflected. The reflection coefficient is then complex and would require a modification to the program. It is fortunate, however, that for convex surfaces ( as we are dealing with ) such a ray can never provide a non-attenuated ray contribution within the radome: no matter how many more reflections it undergoes within the layer, it will continue to be critically reflected at the inner radome surface. Any ray that is critically reflected can therefore be abandoned. 19

THE UNIVERSITY y OF MICHIGAN x I Fig. 4: Coordinate System. y Ek x Fig. 5: Ray Tracing. 20

THE UNIVERSITY OF MICHIGAN IV MONOPULSE RESPONSE The end result of the ray tracing procedure described in the previous section and implemented in Section VI is a sampling of the field distribution over a monopulse plane aligned parallel to the external wave front and centered at a distance D - I back from the front of the radome. Each sample is an individual ray stribution and consists of an amplitude (which may be positive or negative), accumulated (optical) ray distaaoe (in inches) and a direction of arrival measured with respect to the x axis. The distance can be converted to a phase by multiplying by k = 2ir/X, where X is the wavelength (in halts), and we can express the direction of arrival as an angle 0 measured from the normal to the monopulse plane by subtracting. Each sample now takes the form implying a contribution A n e.On } n where An is a real (positive or negative) amplitude and &n is a phase. If the radome were not present, each sampling of the incident wavefront would provide a single sample in the monopulse plane. The spatial distributions of the two samplings would then be the same and we should have A = 1, I = constant, = 0, n n n appropriate to a plane wave incident on the monopulse plane. In this case, the signal in the difference channel of the monopulse would be zero for its plane oriented as chosen: the chosen plane would therefore be the actual monopulse plane, and the pointing error would be zero. Because of the radome, however, the field inside will not be identical to that outside and will not, in fact, be a plane wave. Each ray drawn from the Incident wavefront will generate an infinity of rays within the radome, a finite number of which will intercept the monopulse plate in a spatially nonuniform pattern. The composite of all such rays obtained by sampling the 21

THE UNIVERSITY OF MICHIGAN wavefront at a uniformly-spaced set of points constitutes our sampling of the field distribution over the monopulse plane from which we have to deduce the monopulse response. It is to be expected tthat the difference channel of the monopulse system will contain a non-zero signal which would produce a re-alignment of the monopulse plane through a small angle e which is then the (angular) pointing error. The determination of is our objective. Since the monopulse is required to operate only on reception, it is sufficient to regard it merely as a 'split beam' system in which the fields induced in identical slots symmetrically placed on the two halves of the monopulse plane are compared, and from the difference signal, the effective direction of the excitation field is deduced. In the simplest version of this system, we have just two slots, one on either side of the center of the monopulse plane, as shown in Fig. 6. With e a running variable on the surface of the monopulse plate in the xy plane, let e = + 1 be the coordinates of the centers of the two slots, and let 2d be the width of each slot. The upper slot therefore extends from e = g1-d to e1+ d andthe lower from e = -91+d to - 1-d, and only if a ray strikes the monopulse plate inside one of these apertures can it contribute to the signal induced in that slot. The corollary to this last statement is that a ray which strikes the monopulse plane outside a slot does not contribute, and this is certainly reasonable as regards any immediate contribution. But if the monopulse plate outside the slots is metallic; or, more generally, if it is not absorbing to an extent which is complete for all practical purposes, a ray striking this portion of the plae will be reflected and will thereby generate a whole series of new ray families, some of whose members may return to the monopulse plate and strike it within the slots. The directions of these further rays will bear no direct relation to the direction of the wave normal (or rays) of the incident field outside the radome, and will serve, in general, to increase the pointing error of the system. Since we are concerned to keep the pointing error as small as possible, it is desirable to suppress these rays, and this we can do by the mere 22

THE UNIVERSITY OF MICHIGAN process of placing a layer of appropriate absorbing material on the monopulse plate outside the slots. At the frequency of interest the absorber could be quite thin, and would not appear to have any deleterious effect on the performance of the system: indeed, it could be advat-ageous in decreasing the far side lobe levels of the individual slots. For these reasons, we shall neglect any contributions from rays which do not strike a slot, and will presume that the monopulse plate is so treated as to suppress any reflections if these would otherwise produce more unwanted ray contributions of significant magnitude within the slots. We note that such suppression would probably occur naturally if the monopulse consisted of two (or more) horn antennas rather than slots in a base plate. Although all rays that strike a slot contribute to the signal induced, it would be unreasonable to assume that the magnitudes are the same regardless of the directions at which the rays impinge. In order to take this effect into account, a polar diagram is associated with each slot, and for convenience this is taken to be sin(M,/sin I P() 2kd sin ' where 0 is measured from the normal to the plane of the slot (see Fig. 6). Although such a factor is generally used for plane wave incidence, it will be assumed that this same factor obtains for the non-planar, inhomogeneous field that is actually present, implying a reduction in the magnitude of each ray contribution with increasing angle from the normal; and though the actual factor appropriate to a particular practical system may differ somewhat from (25), the differences are unlikely to be significant for our purposes. Were it necessary to do so, any function capable of analytic representation could be used in place of (25) in the digital program. Combining Eq. (25) with the form of each ray contribution previously arrivedat, the signal induced in the slot centered on 5 = 1 can be written as 23

THE UNIVERSITY OF MICHIGAN V e AnP(n)e (26) n where the summation extends over those rays which strike the aperture. It will be observed that the angle of arrival of each ray affects the output of the slot only through the polar diagram factor P()). Likewise, for the slot centered on 1 = -I, the output is V e = AnP()e (27) n and from a comparison of these outputs the pointing error of the monopulse must be deduced. Since the field within the radome is at least approximately a plane wave, V+ and V. will be very close in magnitude, and the most natural method of comparison is based on phase, i.e. + and._, alone. We also note that this pseudo plane wave is incident in a direction which is almost normal to the plane of the slots, and in expectation that the average direction of propagation makes only a small angle e with the normal to the monopulse plate', which angle is the pointing error of the system, we can now proceed as follows. If a plane wave travelling in a direction 0 - e (see Fig. 7) with respect to the positive x axis were incident on the monopulse, the phase of the signal in the upper slot with respect to the pivot point would take the form etk e sin e and for sufficiently small (, the dominant effect of this aperture distribution on the radiation polar diagram is to displace the effective phase center a distance C1 sin behind the slot. For the lower slot the phase center is brought a distance sin forward, and from a comparison of Eqs. (26) and (27) it now follows that 24

THE UNIVERSITY OF MICHIGAN 2k isine = + I-, (28) implying a pointing error = sin-1 e (+- _. (29) The above description has been phrased in terms of two slots, but a monopulse system will in general have more. The extension to any (even) number 2M of slots is quite straightforward, and the computer program (see Appendix) has been written to permit up to 6 equal-width slots symmetrically placed with respect to the middle of the monopulse plane. The resulting expression for the pointing error now depends on the manner in which the outputs from the various slots are combined. If the centers of the slots are located at 2= +,,..'., and the corresponding outputs are - - e, V+ e V,..., one approach is to combine all the outputs from the slots on the upper half of the plane to give (m) V + W V(m)i + m and similarly combine the outputs from the lower slots, givtng V e W V(m) e m where the Wm are the appropriate (amplitude) weighting factors of the slots. The pointing error c can then be obtained from the expression 25

THE UNIVERSITY OF MICHIGAN e = nl sin-1 ) (30) where e is a 'mean' position of the combined slots, given by z Wm5m z Wm rn m m Although the above procedure is the one that was actually adopted, it should be noted that it is by no means the only way of finding an expression for the pointing error. We could, for example, compare the outputs of corresponding slots, and then average the individual pointing errors to produce a value for e, viz. M 1 1(m) V(m) il sin)_ which is approximately M: e i 1 X ({I+ sin t (32) m4i on the assumption that each individual error is small. In this form, e is independent of any amplitude weighting applied to the slots. When the original approach was programmed, it was found that a slight displacement of the initial (and, hence, all subsequent) point(s) at which the incident phase front was sampled led to a small but detectable change in e. Such a displacement produces, in turn,a shift in the position at which the rays hit the monopulse plate, and when this causes a dominant ray to strike just outside (instead of just inside) the slot, a discontinuity in the induced signal results. With the sampling frequency that was used, the maximum 26

THE UNIVERSITY OF MICHIGAN discontinuity observed was no more than (abbut) 10 percent, and was not therefore a severe problem. Nvert,es,, it seemed desirable to seek a reduction in the discontinuity, particularly because of our ultimate aim of comparing pointing errors with and without a plasma present. An obvious way of reducing the effect is to decrease significantly the distance between successive sampling points, thereby decreasing the relative weight attached to any one ray contribution. This would, however, markedly increase the length of an already-long computation, and since the main objective was to provide a smooth transition as any one ray traverses the boundary of a slot, it is sufficient to assign an amplitude taper to each slot. Each ray contribution is then weighted according to the position at which the ray strikes the slot, the weighting varying from unity at the center of the slot to zero at the boundary. The same taper was applied to each slot, and the particular taper assumed was T() = cos2{ -( m), (33) where e is the position at which the ray impinges and m is the coordinate of the midpoint of the (m th) slot. Each term in the summands in Eqs. (26) and (27) was modified by multiplication by this additional taper factor T(g), and though it may be claimed that the expression for T (e) is not entirely in accordance with the assumed polar diagram factor (25), the discrepancy is not regarded as significant. 27

k Radome Axis 1 Fig. 6: Monopulse Geometry. Position monopulse plane /Would take (e > 0 ) / / A k Phase Front x 6 Assumed position of monopulse plane. Fig. 7: Pointing Error Geometry. 28

THE UNIVERSITY OF MICHIGAN V EFFECT OF A SURROUNDING (WEAK) PLASMA Under some conditions of operation the radome may be surrounded by a weak plasma that can be treated as lossless, and this modification to the external environment can affect the pointing error of the system. Any change of this type will be a function of the plasma characteristics and, hence, of the height and velocity of the vehicle, and may constitute a more severe problem than the pointing error for the bare radome. It is feasible (and, indeed, common practice) to attempt to compensate for the latter error by either electronic or physical means (for example, by shaving or blocking-off portions of the surface in a manner determined by experiment), but this would be ineffective for an error which was a variable function of position along a trajectory of the vehicle. Moreover, were the plasma-induced effect to serve to decrease the pointing error when the radome were bare, complete compensation for the latter error could be undesirable. It is therefore appropriate to examine the change in pointing error produced by a plasma sheath or layer. We first seek an expression for the equivalent refractive index of a plasma. In terms of the polarization vector P, the displacement vector D is D = E+P 0 - The movement of an electromagnetic wave through a plasma leads to a displacement of the electrons, and to the creation of effective dipoles. If there are N electrons per unit volume and if all move through the same distance r (parallel to E), the equivalent dipole moment per unit volume is P= Ner where e is the charge on an electron. 29

THE UNIVERSITY OF MICHIGAN The equation of motion of an individual electron in the absence of an imposed magnetic field, but taking account of collisions, is a2 r m + my =eE t2 =t where v is the average frequency of collisions between electrons and heavy particles. Hence, with a time dependence ewt, e E r' = m -H i giving Ne2 P mw()+v) E_ so that 2 \ D= om N2) E The equivalent permittivity is therefore /E Ne2 e o \1 me oW(W+iv)7 and taking -=,o the equivalent refractive index is / Ne2 2 n = - me N +i v) (34) me 0u(a+i VI If, as we shall assume, any losses in the plasma can be neglected, v = 0. It is then convenient to define a radian plasma frequency c such that 2 Ne2 2 = N (35) p me o 30

THE UNIVERSITY OF MICHIGAN in terms of which (2)1l/2 n = 1- ). (36) (36) It will be observed that for u> p the refractive index is real (as expected) and 0 < n < 1. Some idea of its magnitude can be obtained by inserting the magnitudes of e, m and E0 into (35). With e:-1. 6021 x 1019 coulombs, m = 9.108 x 10-21 kgm., -9 10 E -36 faids/m., o 36w we have w = 5.64 x 104 1 radians, where N is here the number of electrons per cc. When expressed in terms of cycles/sec ( wp = 2r fp), f/2 1'/2 n = (37) with f =8.98x 103 /N Hz, (38) and hence, at C-band ( f = 5 GHz), n = (1- 3.22 x 1012 N)/2 (39) Some typical values of 1-n are as follows: 31

THE UNIVERSITY OF MICHIGAN N= 108 1-n 1.61x 10-4 5x 108 8.05x 104 109 1.61x 103 5x 109 8.08x 10'3 1010 1.62x 102 5x 1010 8.40x 102 1011 1.77 x 10 1 We note that for N < 5. 9 x 0, n>0.9 andthat for N < 2 x 110, n can be approximated by the expression n 1- 1.61x 10O12N (40) with at least three digit accuracy. The largest value of N in the electron density profiles which have been furnished us is 1010, and accordingly, from the above Table, n > 0. 984. The presence of the plasma outside the radome will modify the rays which would have impinged on the radome in the absence of the plasma, and in order to extend the bare-radome treatment to this case, we must now examine the perturbation of the rays on passing through the plasma. Since the electron density and, hence, refractive index is a function of distance normal bt the radome surface throughout the sheath, the only practical method of ray tracing is to assume that the plasma is locally stratified parallel to the surface and to ignore the reflection from the stratification layers. This is the usual technique for ray tracing through a region of variable refractive index (e.g. the atmosphere), and because of the relatively small maximum variation in n, it might be thought that the procedure is unquestionably valid. In the present problem, however, the variation in n takes place within a layer whose thickness is no more than X/6 and can be as small as X/60. The criterion for neglecting subsidiary 32

THE UNIVERSITY OF MICHIGAN reflections, i.e. for assuming each ray to proceed undiminished, is co5s c ~ 1n.oo a << 1, (41) k ay where y is measured normal to the stuatifioation and a is the angle which the incident ray makes with respect to the normal. For the data furnished to us, the maximum values of an/ay occurs near to the nose of the radome where the electron density is largest and the layer is thinnest. We have max i (3 xx.61x 12) /(min. lyer thickness) x ~3 (3 X10 - 1m l 2.9 -1 inch and hence 1 an max. o 0.46 k ay In view of the fact that over most of eathe sheath, n/y is a great deal less than its maximum value, and that under most circumstances a wave will strike the region of the maximum at a rather oblique angle, it is legitimate to conclude that the criterion (41) is fulfilled, albe to a somewhat less degree than might have been expected from a consideration of the change in refractive index alone. More to the point, perhaps, is the fact that to proceed on any other basis would produce a problem of forbiddable complexity. As a result of traversing a slowly varying medium of this type, a ray will reach the surface of the radome with its amplitude undiminished, but with its phase, impact point and (possibly) direction changed from what they would have been had the plasma not been present. To compute these modifications, we postulate a planar stratification of the region traversed by any given ray, as shown in Fig. 8. Let y be the coordinate normal to the stratification (and to the local radome surface), and n = n(y) be the refractive index 33

THE UNIVERSITY OF MICHIGAN throughout the plasma layer of thickness t. For the moment we assume that the region y < 0, as well as y > t, is free space with refractive index unity, and choose origin of coordinates at that point on the surface y = 0 which a ray incident the oer -surface of the layer at an angle a to the normal would have reached in the absence of the layer. The coordinates of the point (x1, y1) at which the ray strikes the lower surface of the layer are therefore = -ttan a y t. (42) From Snell's law n(y) sin A = sin a, (43) where, is the local inclination to the vertical of the ray path within the layer. If ds is an element of distance along the ray path, the horizontal distance of travel within the layer is y=O 0 t f ds sin 3 = dy tan 3 sif p a..i. dy, — t and the x-coordinate of the point at which the ray strikes the plane y = 0 is now t x sinaf -seca dy. (44) The ( optical ) distance traversed by the ray within the layer is likewise y=O 0 t 2 = ndy 2, a2 sind dy, =(Xo-xl)sina+ /2-sin20 dy 0 34

THE UNIVERSITY OF MICHIGAN and thus the excess of optical distance over that of the ray in the absence of the plasma is 6S = xsin + /n-n2 -cos ndy. (45) If the quantities x0 and 61 were computed as functions of a, t and n(y), the most obvious procedure for extending the bare radome procedure to the case in which the plasma was present would be to follow any individual ray from the wavefront to the outer surface of the radome ignoring the plasma, and then displace the ray laterally by the (small) amount xo and increase its phase by k 6&. Since the plasma is actually in contact with the surface of the radome, there is also a slight change in direction of the impinging ray from a to a', where af = sin l(sina/n(O)), and an associated change in the reflection and transmission coefficients. For convenience, and because the electron density in general decreases in the immediate vicinity of the radome surface, these effects will be ignored: this is tantamount to conceiving of a small air gap between the inner plasma surface and the radome, and allows us to make full use of the original computation procedures for a bare radome, providing the quantities x0 and 61 can be determined. In their 'exact' forms given in Eqs. (44) and (45), the evaluation of the expressions for xo and 61 requires a knowledge of the variation of the refractive index n as a function of y, leading to numerical integrations, but if a is not too close to 7r/2, the two formulae can be simpliftsil.to a considerable extent. Since n2-sin2a = cos2a - (1-n2) we have (n2-sin2a)V 2. sec 1+ 1 (1-n2)sec2a p2 providing cos2>> 1-n2, 35

THE UNIVERSITY OF MICHIGAN and hence t t x 0 -2 tan a sec2a (1-n2)dy = 1.61x 10-12 tana sec2| N dy, * (46) (46) which involves only the integrated electron density through the layer. The fact that is positive is consistent wit h a deviation of the wave away from the aormal (n being less than unity). Smilarly, 6 t x xinoa t -1.61x 10'12so al-ta2*) Ndy 0 (47) and thus 6 1 l - cos 2* cosec a x. (48) Since xo > 0, 61 is t e or Pe according as a is smaller or greater than ir/4, respectively. To make use of the formulae (46) and (47), we now turn to the data for plasma layer thickness and electron density that Ueve been furnishid us. The boundary layer thickness t measured normal to the outer surface of the radome at an axial distance xl from the nose is represented by. 0 1227 x0' + 0.0012 t = 0. 012x1 -0.08 0. 0224 x 8 0.0958 I 0 < xl< 10 -x1 -10 < x < 15 15 <xI -48 (49) I where all dimensions are in inches. In terms of the coordinate system of Fig. 4, x1x -x I 36

THE UNIVERSITY OF MICHIGAN Note that t(10)=0.04, t(15) = 0. 10, t(48)= 0.40 The electron density varies as a function of distance y normal to the radome surface throughout 0 < y < t, but does so in a manner that depends on x1. Data curves* for N=N(y) (electrons/co) as functions of y = y/t are shown in Fig. 9. From Eq. (46) it is observed that the approximate expression for the lateral displacement x0 of the intercept point with the radome surface is proportional to t | N(y) dy t (50) 0 where 1 I= Nv)dy. (51) Since N(y) depends on the particular range of xl (see Fig. 9), I likewise differs in the four ranges, but its value for each can be obtained by numerical integration of the corresponding curve in Fig. 9. But fitting log N to a polyk nomial form and then integrating numerically, it is found that I = 2.039x109 for 0x1< 5 2.223x 108 for 5<x1< 10 1. 147x 108 for 10< xl < 20 6. 962x 108 for 20 x < 48 Knowing I and t (see Eq.49) for the various ranges of x1, the integrated electron density is obtained from Eq. (50) and the lateral displacement x0 for any given a then follows from Eq. (46). The change in owtical dis-' tance is trivially related to xo through Eq. (48). Private communication with Dr. I. Pollin, 13 November 1968. 37

THE UNIVERSITY OF MICHIGAN To assess the accuracy of the approximate formulae (44) and (48), we considered the actual electron density profile in Fig. 9 for the range 0 < xl < 5, and evaluated numerically the precise integral expressions for xo and 61 in Eqs. (44) and (45) for a series of incidence angles a. The results are shown in the following Table along with the approximate values obtained from Eqs. (46) and (48). TABLE a(~) Exact x0/t 10 6. 082xl0-3 20 1.381x10-2 30 2. 590x10-2 40 4.845x10-2 50 9.911x10-2 60 2.458x10 1 70 9.257x10-1 Approx. xo/t Exact 6 t/t 5. 967x10-3 -3. 248x10-2 1. 353x10-2 -3.045x10-2 2. 527x10-2 -2. 526x10-2 4.693x10-2 - 1. 216x10-2 9.467x10-2 2.408x10-2 2. 274x10-1 1. 456x10-1 7.709x10-1 7.682x10- 1 Approx. 61/t -3. 22910-2 -3.030x10-2 -2. 527x10-2 - 1. 268x10-2 2. 146x10-2 1. 313x10-1 6. 284x10 1 Certainly for a < 60~ the agreement is rather good, and bearing in mind that we have here considered the case in which N achieves its maximum possible value 1010 (electrons/cc) and for which the discrepancies between the exact and approximate expressions are greatest, the results provide reasonable confidence in using (46) and (48). Only the integrated electron density is then relevant. For a = 70~, the approximate expressions underestimate both xo and 61, and the discrepancies become more apparent as a increases. This is due to the considerable bending which a ray now undergoes. From Eq. (43) it is seen that the local inclination to the vertical will reach 90~ at a depth within the sheath such that n(y) = sin a; and having become parallel to the radome surface, the ray will now emerge from the sheath without ever 38

THE UNIVERSITY OF MICHIGAN intercepting the radome. In the case considered above where the maximum value of N is 1010 (corresponding to n=. 9838), failure to intercept will occur for all a> 79.70. Thus, for example, for a sheathed conical radome of halfangle 9.5^ illuminated at an angle within 0.80 from axial incidence, no rays can penetrate the forward part where 0 < x < 5, but rays which strike further back where the peak electron density is less will be able to get through. Although the accuracy of the approximate formulae (46) and (48) is not entirely adequate for a> 65~ (say), the time that would be involved in performing the numerical integrations demanded by (44) and (45) for each and every ray makes the use of the simplified expressions most desirable, if not mandatory. Having said this, it should be noted that for any given (large) angle a, the 'error' implied by (46) and (48) decreases with the peak electron density and could in any case be removed by employing an integrated electron density (or layer thickness) somewhat greater than is implied by the curves in Fig. 9. Since it is presumed that such an 'adjustment' is small compared with the inherent uncertainty associated with the data in Fig. 9, Eqs. (46) and (48) will be employed without any modification. It is now a rather straightforward matter to take Into account the presence of the plasma. Any given ray is first traced to an intercept with the radome as though the plasma were not there. Knowing the value of x (and hence xl) appropriate to this intercept, xo and 6 are computed. The intercept point is then displaced by a distance xO along the surface of the radome and the optical distance increased by 61, and the ray is traced through the radome in the same manner as before. It should be emphasized that xO is a displacement along the surface, rather than in the x direction as such. This is trivial to execute for a conical radome; for the ogival one, xo is treated as an arc length, and the intercept is formed by constructing a circle of radius xO about the point at which the ray strikes the radome in the absence of the plasma. Of the two possible intercepts of this circle with the radome, that for which x is larger i s selected. 39

THE UNIVERSITY OF MICHIGAN x y=o y t n=l nn(y) n=l Fig. 8: Local Geometry of the Plasma Layer. 40

10 10 10 l8 y20 < xl < 4B 5<xl<10 A\\ \ \ \ \ 5 < 1 10< -x < 20 4 o6 02\ 0 0.2 0.4 I- --------- 0 0.2 0.4 y 0.6 0.8 1.0 41

THE UNIVERSITY OF MICHIGAN VI NUMERICAL PROCEDURES AND RESULTS Before going on to present the specific numerical results that have so far been obtained with our numerical program, there are a few points sonoerning the manner in which the incident wave front is sampled that should be enlarged upon. In Section IV we discussed the need to 'smooth' the change in the induced signal in a slot when a primary ray just hits a slot as opposed to when it just misses, and we noted that a reasonably effective (and realistic) procedure is to incorporate an amplitude taper in each slot response function. This has the effect of enabling us to sample the incident wave front at a lower rate than would otherwise be the case, but still leaves us with the task of determining what is an 'optimum' sampling rate; by 'optimum' we here mean one that will minimize the computation time (i. e use a minimum number of sampling points) and still yield values for the pointing error of sufficient accuracy. In order to obtain data from which to estimate the optimum rate, we chose to consider a bare ogival radome for an H-polarized (or TM) wave incident at 30~ from nose-on. To judge from a variety of data sets then available, this case appeared satisfactory for test purposes. Pointing errors were now computed for various stepping distances, A (in inches) along the wave front as a function of the position of the starting point for the first (lowest*) ray. Thus, for a given and a given starting point, c was computed. The starting point was then displaced a distance A/10 upwards, and the process repeated; andso on until, after 10 such displacements, the starting point in the 10th case coincided with that of the second ray in the first case. It was found that the pointing error was a smooth oscillatory function of the displacement, with one complete cycle correspoing to 10 displacements. For A = 0.05 (inches), for example, E varied from In the program, sampl* starts at the lowest point works up the wavefrot. In the program, sampling starts at the lowest point and works up the wavefront. 42

THE UNIVERSITY OF MICHIGAN 9.82 to 9. 99 milliradians, whereas for A = 0.2, the variation was from 9.85 to 10. 14 milliradians. It was felt that an uncertainty of t 0. 15 milliradians was an acceptable accuracy, and since such factors as the weighting applied to the slots had no real effect on the variation of E, we therefore settled on a rate of sampling of the incident wave front corresponding to a stepping distance of 0.2 inches along it. This same value of Ahas been employed in the 3-dimensional program, and whereas in the present case the increase in Afrom 0.05 to 0.2 inches reduces the number of rays (and hence the running time of the program) by a factor 4 - an improvement which is certainly not negligible, the improvement in the 3-dimensional program is by a factor 16 and is considerable. The next topic to be discussed is the choice of the initial or starting point at which the incident wave front is sampled. The ray tracing program is designed to start with the lowest ray which can produce an intercept with the monopulse plate, and then to step up along the wave front through a chosen distance A a specified number of times. The starting point is taken to be the projection of the lowest point of the monopulse plate on to the incident wave front (see Fig. 10): consideration of the refractive effect of a convex radome shows that no lower ray could conceivably provide an intercept. We can similarly specify an uppermost sampling point by projecting the top of the monopulse plate and by finding that point from which a ray is tangent to the upper surface of the radome (in the case of a conical radome, the latter point is replaced by the projection of the radome tip). The upper of these two points consitutes the final sampling point, and the number, N, of increments is then obtained by dividing the wavefront distance (in inches) by A. Another matter of some importance is the cut-off or tolerance criterion that is adopted. As each ray undergoes successive reflections and transmissions, its amplitude decreases and ultimately falls to a level at which its contribution can be neglected without significant error. Although one would prefer to set this level as low as possible, to do so could increase the computation time without any marked improvement in accuracy. A rather wide variety of test cases were 43

THE UNIVERSITY OF MICHIGAN run in which the tolerance was sett anything between 0.001 and 0.05, i.e. in which rays were abandoned when their amplitude had decreased to less than 0. 1 percent and 5 percent, respectively, of their initial value (unity). it was found that decreasing the tolerance from 0. 01 to 0.001 affected the pointing error by no more than 0. 2 or 0.3 milliradians, and though the tolerance criterion was preserved as one of the input parameters, it was set at 0.01 in all subsequent runs. It may be noted here that this wame level has been adopted in the 3-dimensional program. In the description of the monopulse system given in Section IV, we allowed for the possibility of differential weighting factors applied to the various slots. The weighting factors that are appropriate depend, of course, on the manner in which the slots are connected, but to get a general feeling for the extent to which they can affect the pointing error, a series of tests was run using four symmetrical slots with either equal weighting or with the outer slots weighted by a factor 1/2 relative to the inner ones. It was found that such a decrease in the sensitivity of the outer slots increased the pointing error, the change being of order 10 peroent. The final form taken by the 2-dimensional computer program is rather general, and permits the computation of the pointing error for either a TE or TM plane wave at any angle of incidence on a radome whose surfaces are specified by either linear or quadratic equations, with or without a surrounding (weak) plasma, and for a variety of monopulse plate configurations. A complete list of the input variables and their format is given in the Appendix I "Evn though this list may seem quite lengthy, only 4 cards are needed for each set of data, and for a typical modification of input parameters such as a change in the Incidence angle of the wave, only one of these cards has to be altered. Numerical Results In order to test out the programs and, at the same time, to obtain explicit values for the pointing errors in cases of some practical interest, complete runs 44

THE UNIVERSITY OF MICHIGAN were carried out for two particular radome and slot configurations with and without a plasma present. The incident field was (as always) assumed to be a plane wave having either its electric or magnetic vector in the z-direction (TE or TM polarized waves, respectively) and incident at an angle 3 to the radome axis varying from 0~ (axial incidence) to 55~. The (free space) wavelength was taken as 2.4 inches, corresponding to a frequency 4.918 GHz. The radome material was treated as a homogeneous isotropic dielectric with permeability p = A,. Thus, n,. =2.5, which is typical of the refractive index of fiberglass in the microwave range. The monopulse plate was assumed to consist of 4 symmetrical slots centered at t 2.4 inches and ~ 4. 8 inches from its mid-line, and having widths 2d 1.2 inches. The slots were weighted equally and had a cosine amplitude taper associated with each. For simplicity, the individual slot voltages were not printed out, and the output therefore consisted of only a single pointing error e (in radians) arrived at in the manner described in Section IV. The plasma (when present) was taken to have the characteristics provided by Dr. I. Pollin (personal communication) and listed in Section V. The two particular radomes that were considered are the two-dimensional analogues of those described by Dr. Pollin (loc. cit.). The first is a twodimensional conical (or wedge) radome (see Fig. 1la) whose outer and inner surfaces are given in terms of the coordinates of Fig. 4 by the equations y 6 + y+ 6 _-) y =+(x - 4.191), respectively, where all dimensions are in inches and the upper (lower) sign refers to the upper (lower) surface. The half-angle of the cone is approximately 9.5~, and thus, in Eq. (16), 1 a, = 1 (outer) 4.191 (er) =4. 191 (inner). 45

THE UNIVERSITY OF MICHIGAN The mid point of the monopulse plate is at x = 43. The second radome considered is a two-dimensional ofve each of whose surfaces is formed by the rotation of a cylindrical arc about a chord (Fig. lb) Each surface is defined by an equation of the form (15) with A = 140, B = 48, C = 49 for the ater surface, and A = 140, B = 45.832, C = 49 for the inner surface. The half angle of the (outer surface of the) radome at its tip is approximately 18.40~, and the monopulse plate is situated at x = 39. It should be noted that for neither radome have we postulated any tip rounding. The program was run first for the bare conical radome for each of the two polarizations, and with the angle of incidence 0 varying from 0~ to 550 in 5~ steps. When it was found that the pointing error, e, varied rather rapidly in certain ranges of j, computations were carried out for additional values of, in order to pinpoint the oscillations more precisely, and we ultimately ran the program at increments of 10 in 3 for portions of the entire range. Analogous computations were then performed with the plasma present. A complete listing of the pointing errors (in milliradians) for the conical radome, with and without the plasma sheath, are given in Table 1, and the results for the bare conical radome are plotted as functions of 0 in Fig. 12. Attention was then turned to the ogival radome, and the pointing errors computed without the plasma present. Because rapid variations in e now existed throughout the entire range of B (the source of these variations will be discussed in a moment), it was felt desirable to compute c for no more than 1~ increments in B over most of the range, and even smaller increments were employed over a limited region. The results for the bare radome with TM polarization are plotted in Fig. 13, and a partial listing of these data and of the analogous results in the presence of the plasma, is given in Table 2. Due to lack of time and money, no comparable runs were carried out for the ogival radome with TE polarization. 46

Table 1: Conical Radome Pointing Errors (Milliradians) Bare e(deg.) TE TM... I JI [.. I LI With Plasma TE TM _ - l[ II JII I - '[[i I Plasma-Bare TE TM...I 0 1 2 3 4 5 6 7 8 10 11 12 13 14 15 16 0.00 2.13 4.47 6.20 4.36 5.28 6.25 -2.70 0.39 0.36 0.42 3.77 10.72 10.57 -11.62 -14.67 0.00 2.42 4.80 7.08 5.16 6.01 7.25 -4.96 0.00 0.00 0.00 -0.13 -1.63 -10. 93 -18.97 -15.05 0.35 10.85 -0.51 -0.19 0.00 0.02 -0.80 -8.21 -16.96 -0.80 11.52 18.31 0.95 -0.56 0.00 -2.26 -6.10 -17. 13 -23.33 -20.33 -5.90 13.55 -0.90 -0.55 0.00 -2.40 -5.60 -15.29 -22.12 -6.81 4.27 23.27 0.95 -0.56 -0.07 -0.86 10.34 -11.54 -0.23 0.08 -0.75 0.11 17 -10.22 18 19 20 25 30 35 40 45 50 55 -7.18 -4.46 -2.53 -0.24 -1.42 -0.48 -3.40 0.12 0.96 -0.13 0.23 0.11 0.00 -0.15 0,43 0.07 0.17 0.05 -2. 13 -0.25 -1.42 -0.48 -3.42 0.11 0.95 -0.05 0.23 0.11 0.00 -0.14 0.41 0.07 0.17 0.11 0.40 -0.01 0.00 0.00 -0.02 -0.01 -0.01 0.08 0.00 0.00 0.00 0.01 -0.02 0.00 0.00 0.06 47

THE UNIVERSITY OF MICHIGAN Table 2: Oglwl Radome Pointing Errors (Milliradians) 3(deg.) 0 5 10 15 20 25 30 35 45 TM Bare 0.00 1.20 -10.21 4.98 5.90 5.17 -9.99 5.03 4.43 TM Plasma 0.00 0.44 -9.15 5.64 5.65 5.27 -9.97 5.08 4.66 Plasma-Bare 0.00 -0.76 1.06 0.66 -0.25 0.10 0.02 0.05 0.23 48

THE UNIVERSITY OF MICHIGAN Discussion of Results Inspection of the results that have been obtained shows that in all instances the pointing error is a rather rapidly varying function of the incidence angle B and except for the conical radome with 1,3 200 the peak excursions are much larger than the 2 milliradian limit that was hoped for. This is true independently of the presence (or otherwise) of the plasma sheath. It is our belief that these features are real and that the results are not a reflection of the approximation inherent in using ray tracing. In support of this belief, we examined in some detail the pointing errors that were obtained for the bare conical radome (see Fig. 12). Since the geometry is rather straightforward in this case, it was possible to trace manually the paths followed by the primary rays (which are the source of the dominant portion of the monopulse excitation) over various portions of the angular range. As expected from symmetry considerations, the pointing error is zero for nose-on incidence. As 1 increases from zero, the difference in path length between rays which have passed through the upper and lower radome walls also increases, and this in turn leads to an increasing phase difference between the excitations which the upper and lower slots receive. The pointing error therefore increases, and does so in a manner almost independent of polarization. With increasing 13, however, even the upper slots begin to receive the bulk of their excitation from rays that have passed through the lower radome wall, and c now decreases* from a peak value of 6~ 7 milliradians to (effectively) zero for 3B = 9 10~ when the rays are at glancing incidence on the upper wall. As 3 increases still further, we start receiving reflections off the upper (interior) radome surface, The 'overshoot' occurring in the 7 to 80 range is attributable to internal reflections within the lower radome wall. Geometrical considerations show that the reflected waves will affect the slots differentially, leading to a rather localized pointing error which is polarization dependent (as observed). 49

THE UNIVERSITY OF MICHIGAN and since the reflection coefficients for the TE polarization greatly exceed those for TM, significant differences between the two polarizations now occur. Indeed, for B > 90, the pointing error for the TM case remains less than 1 milliradian and, as such, is not much greater than the expected accuracy attainable from the computational procedure. In contrast, however, the strong wall reflections in the TE case produce a pointing error which rapidly increases for 3 > 110 and reaches a maximum of about 11 milliradians for 3 = 130~. To begin with, only the upper slot(s) receive this perturbation signal, and because of the relatively small path difference between the direct and reflected 'waves', the pointing error is positive (see Fig. 7). But with increasing Af the path difference becomes significant and the pointing error changes rapidly to a negative value. It is the phase (or path) difference which is primarily responsible for this, and the effect is analogous to a 'hunting' action on the part of the monopulse. The peak (negative) c is nearly -15 milliradians, and occurs for B = 160. As B increases still further, the pointing error decreases, partly due to the increasingly uniform illumination of the monopulse plane by the reflected wave, but more importantly because of the decreasing effect of the perturbation arising from the progressive reduction in the reflection coefficients and the suppresive action of the polar diagram associated with the slot response. Although C continues to oscillate even for 3 > 20~, it now does so with a much reduced amplitude, and it is not possible to pinpoint any single source for each individual peak. The above interpretation of the dominant features of the curves in Fig. 12 was arrived at by a detailed examination of the ray contributions to the monopulse excitation, and by a few exercises with a ruler, protractor and a slide rule. The same general picture continues to hold when the plasma is present, and whilst the plasma can be expected to change some of the details as a result of its lateral non-uniformity and its modification to the rays that enter the radome, we should expect to see the same principal features in the pointing error curves. Inspection of Table 1 shows that this is the case. Note, however, that some of the

THE UNIVERSITY OF MICHIGAN pointing error peaks are reversed in sign ( a phasing effect primarily), and because all tend to be slightly displaced in angle, the change in pointing error produced by the radome can be quite large particularly in the critical region near to nose on where the non-zero values of e are almost completely due to a phase (or path difference) effect. For B >O 15~, the plasma produces no significant change in e. The understanding resulting from the above dissection' of the results for the conical radome enables us to appreciate why it is that the ogival radome displays the even more complicated behavior shown in Fig. 13. Since the half angle of the radome is now almost 190, we might expect that in this range the pointing error will oscillate in a similar manner to what it did for a conical radome when 0 < 3 < 9.5~. To at least some degree, this is indeed the case, but because of the curved geometry implying variations in reflection and transmission coefficients over different portions of the walls for even a fixed value of 3, the detailed structure of the c pattern is a great deal more complex than for the conical radome. Were it in isolation, the lower radome could no longer transmit a uniform plane wave, so that the monopulse plate receives a field of rather complicated structure even from this part of the radome alone. The situation is still worse for the upper wall. At any given angle of incidence, a single slot will see a dominant reflected wave coming from only that small portion of the wall which is appropriately aligned; and in consequence, both the phase and the direction of the dominant reflected signals will vary markedly from slot to slot. Under these circumstances it is not surprising that E shows large and violent changes as a function of 3, with each peak being attributable to the cumulative effect of many small contributions which themselves change rapidly from one value of 3 to the next. Un-physical as the results in Fig. 13 may appear, we have no reason to doubt them. 51

THE UNIVERSITY OF MICHIGAN A comparison between the pointing errors with and without the plasma present is given in Table 2 for 9 isolated values of 1. It is interesting to note that based on this small sample alone one would conclude that the plasma has less effect on e for an ogival radome than it did for a conical one. There does not seem any obvious explanation for this. Conclusions In this Memorandum we have given a complete description of the numerical approach that we have adopted in determining the behavior of a monopulse system mounted within a 'two-dimensional' radome with or without a weak plasma sheath surrounding it. The theoretical foundations have been presented in some detail, as have the approximations which are necessary to permit the computations. Much of this is also appropriate to a treatment of a three dimensional radome, and we shall rely heavily on the present material when we come to describe the three-dimensional work. The Appendix to this Memorandum contains a complete listing, flow chart and operating instructions for the two-dimensional numerical program, and we have given (and discussed) some of the results that have been obtained by applying it to two particulars radome-monopulse configurations. These raise considerable doubt whether a 2 milliradian maximum pointing error for incidence angles out to 55~ is an achievable objective regardless of the presence of a plasma sheath. Although a two-dimensional geometry is, of course, a mathematical idealization, we believe that the program presented here is a valuable one in its own right and that the values of the pointing error obtained with it do represent a lower bound on those that would be found for the corresponding three-dimensional geometry. Certainly the program given here permits a much more rapid computation of e than does the three-dimensional one, and whilst the present program was not written specifically for economy of operation (rather did we aim to permit the pointing out of all intermediate data that might facilitate the understanding of the 52

THE UNIVERSITY OF MICHIGAN pointing errors found), the running (CPU) time in any one case (i.e. one angle of incidence, one polarization and one radome with or without plasma) varied from about 5 sec. to 20 sec. at most, depending on the particular circumstances. In contrast, the time for the three-dimensional program is two or more orders of magnitude greater. Finally, it should be.mpbaiedwso a that the entire approach has been based on ray theory and, in consequence, the results obtained are only approximate. Though it is our belief that the values found for the pointing error are accurate to within 1 milliradian, prudence would dictate that before complete reliance is placed on these data, some attempt be made to verify the conclusions experimentally. To do so for just the bare radome would be a valuable test, and it would not be hard to perform such an experiment using a simulated twodimensional structure. 53

THE UNIVERSITY OF MICHIGAN Final Sampling Point Plate Incident Phase Initial Sampling Point Fig. 10: Choice of Initial and Final Sampling Points. 54

THE UNIVERSITY OF MICHIGAN y x z I I I I I 1.4 43 am I Fig. h a:- Conical Radome Geometry. y x z I I I 39 W-.' Ogival Radome Geometry 55 Fig. 1 lb:

k "4 &E on 0 0 O 10 5 0 --5 -10' I II U Fig. 12: Pointing Error for the Bare Conical Radome for TM ( - - -*) and TE (x —x) Polarizations. 20 30 3 (degrees) 40 50 60

10 5 -4 V-I | U' I 0 0 S3 w 0 94 )L - o 0 -5 V ^ ---'I Fig. 13: Pointing Error for the Bare Ogival Radome, TM Polarization. -10 -15 10 30 0 (degrees) 50 60

THE UNIVERSITY OF MICHIGAN APPENDIX FORTRAN COMPUTER PROGRAM This description of the Fortran computer program consists of the following six parts: (1) Usage hints and a complete description of the if-erences betwee the version for a conical surface and the version for a surface described by a certain quadratic function, (2) Diagram showing terminology used in plaing the program, (3) List of input variables with their program code names and input card formats, (4) List of Fortran source program, (5) List of all variables used in program with code names and their meanings and usage, and (6) Semi-detailed chart of logic flow in the program. (1) Usage Hints The program has been written to process an unlimited number of data sets, each data set consisting in form of data cards and resulting in a single pointing error, for a given aspect angle, frequency, polarization, geometry, and so forth. As a precautionary measure against incorrect data inputs a trap has been incorporated in the program to terminate the program when more than three errors are encountered in the input data. The program, as it now exists, is by no means in its most efficient state. Storage requirements can be reduced by making changes to parts of the program which were included for flexibility and to aid in the program checkout. The dimensioning of variables ANGLE, AMPL, and DIST serves no purpose during current usage of the program. Also, some of the statements beginning near the statement numbered 370 may be eliminated by rearranging statements near the statement number 243 and then looping back sooner. This loop is performed each time a ray reflects inIRegion II. 58

THE UNIVERSITY OF MICHIGAN Differences Between Quadratic and Conical Versions Conversion of the computer program from one that handles the quadratic case where the surfaces are described by Fl and F2, segments of circles, to one which computes for the conical case where the surfaces are cones described by F1 and F2, straight lines, consists of the following steps (for the most part indicated in the card deck by comments). 1) The function FX = ISIGN ( A2(X —B)2' ) becomes FX I8IGN*(X-A)*B The function =ISIGN*(B-X) FDERX = J ^ \ 4 A2-(X-B)2 becomes FDERX = ISIGN*B. 2) In the format number 171 the word "quadratic" becomes "conical". 3) In the plasma calculations the statement XB=XB+DELX(coslarctan(DX)I ) becomes XB-XB+DELXlcos(B)1 and the statement XB=FZ(ZB) is eliminated. 4) The subroutine XPT is replaced with one which computes the point of intersection of straight lines. Notice also that input variables A, B and C necessarily have different meanings. 59

THE UNIVERSITY OF MICHIGAN (2) Diagram Showing Terminology Used in the Program. Fl F2 Region Fl Phase front B is any point on Fl. C is point where a ray from point B hits F2. D is a point of reflection on Ft; it becomes point -B after reflection calculations are made. ISIGN = + 1 when intersection with top function is being sought, = - 1 for bottom part. IBELO is argument of call to subroutine XPT; indicates which point of intersection of surface with the circle is to be used (quadratic case); 2 means point used is upper if ISIGN is + and lower if ISIGN is -; - 2: other point is used. On return IBELO = 0 if intersection with aoerble is foundand if it is left of center of the circle determined by F. 60

THE UNIVERSITY OF MICHIGAN (3}) mu Variable List for W,RTRAN PrepMam.-.Hlmee Columnn Card 1 - q a-muartc urhfce A 1-10 B 11-20 C 21-30 SMA 31'40 Card 1 - coical surface A B C 8MA Card 2 DELTA 1-10 TWON 11-20 XZERO 21-30 radius of circle described by radome (Fl) X-coordinate *of center of above circle Zrcoordtnate of oenter of above circle difference between radii of Fl and F2 X-coordinate of vertex of Fl: outer surface slope of top surface of cone = 0 indicates case is conical A- (X-coordlnate of vertex (F2 = outer surface)) increment used in stepping up radiating plan refractive index of Riegon I X-coordinate of center (on X-axis) of receiving plane angle (in degrees) of radiating plane to X-axis angle (degrees) of receiving plane to X-axis indicator which results in intermediate printig when set = 1 Indicator = 1 when plasma present, 0 otherwise indicator = 0 results in printing output whenever ray hits a slot; when set x 1 only pointing error is printed, BETA BETA2 ITEST IPLAS NPRINT 31-40 41-50 51-54 55-58 59-62 *ts All linear measurements are in inches. 61

THE UNIVERSITY OF MICHIGAN - -- 7 qXRMOMO Columns Card 3 Zi 1-10 Zeotdlmte of fist point on radltlq as.-In ZZERO I1-30 mxM d f center of rceivng -ck d w ays will r -qister (.usuall to Om.P GI top SWt) TOL 21-30 am$ims ~e amij*u. of a ray Ml. below this value, -M& theel - fr tbt ray NUMDKC 31-34 a maximum limitof tw aumber o xemets which will be stepped alsq diatiq W plas TM 35-38 sgt 1 fIr uasgutlc case; 0 otherwIse TE 39-42.t =1 f alestric case; 0 othrwise Card 4 WIDTH ~~~1-10 h*l &~~~s~v~rln WAVEL 11-20 watl Igth APER(1 -~ 3) 21-30... distaus alqraeltving plain of centers of slot;r bottom Is symmetric WEIGHT((1-.w3) 51-60... weightingfactors for up to3 pairs of slots 62

(4) FORTRAN Source Program F rCQlT 'Jl I V C, COMOTPFP NA IN I?~~ 17.?29 PAGE 0001 Oct ) REAt N\,l?-VC),MvA(;_ or)0? T\NT r-C-~ T MTc - 0f 1)3 1'A F SN 7( C P FP (3WFITCI-4T() _____C 4 I VF jSTLEN (v;L F(3 ), 3L 3),IST( 30)C 0101) 5 fI NA F S I0J y ~ Si~Y 6) C C T?-E F I -LL0'D TIN G S TflrAF I S 1 SEP1 T~- WOC"u.uLATE nATA O7F PErLECTION POINTS C CIflATI I- T CAN RE LS~FT Ff3P Cf7`PlT"TIPN OF FINAL INTERCEPT POIN.T RESULT5 0 0 IS 6 IESI C SY(5 r(5)SV),PI5),P51S T( 0),S R(5 0) CF U "JC TIUN?E 1FINIT I C N S _______ _ 00087APAvPPK r 3010 Io F Z (7)-7~+ S'> T (A A 7- ZI SI -C7) (7-TSIGN*C)) C _F JNC TI CiN flF S CPI PT IONS F-CR, (C N IC Al C ASE _ CNCTE: ANOT-FP VP-RSICN CF SUF5PPI.TIN,\F XPT IS ALSO NEEOEDC C FDc-px( I S'I G), tPRX ~I SI GN C =DISPLACFYFt'T OF VERE l-' LP OF FUNCTION C 0C C (NJPUT) SMA= VEFPTFX(F1)-VERTEY(F2); Nr'TF: A NE'`ATTVF VALUE 0011 F rTS T (Y1 2,XY 1,9Y2)SQ T( (X?- X)I(X 2- X1) +(Y?-Y1)I(Y2-Yl) I 3012 ~22(CV=1. CA) I(. +GA1 O r0 ) 1 3 Q 3 r ) ~ ( 1 - A ) /. + t? )_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _C 001)15?3(GA)=?.f 1.+ GyAN C C Df1SC IPT I Ct! iF I NPU TC 0Dj1 7 7) f: C,-T ( I UT Ac C RS' ' 3F01F1I'I. 3f I___ `)T-LT', T I-`',_tX7F It W:TA, c~T-,ITFST, PLMo?QINT:lq9F1j~3,3T4 /'Z177r'r, -U.AVT1 I T F-@,3(i.4,l A4;/ C AF - IncT! cTS ' PCFU:A LT VA LIJ F C I %TTM7iF ) I1L0 )0120 )0130 )0140 ')0150 0160;170 )0180 )0300 )0310 )0320 "035 0 )0360 )0370 )0380 )0390 )0400 )0410 "0,420 )0430 )0440 )0450 )0460 )0500 )0O5 10 )05 20n )09530 )0560

o ~ 1 '3 0 I) 20 - 00?21 002 2 00 24 00 i 25 -00 26 00 27 0028 0029 00 30 -0-031 -00-343 _____ A TO)L VtiY/C*/, FLEC/'Ti WI /. /~M~ 00570 ~' X SLUCT= 00580 L VI I N C=40O INU"'1 NC=C-0' 00600 P =1 =*141593 -~00610 I ','OF P =0 00620 I S=0, 00630 C CfCNS4"TArTS FPU; CIALCULATINCU Ar~jIJRTMFRNTSFr, PLASMA 00650 rD hL S T 1 = * 0 _ _ _ _ _ _ __ _ _ _ _ _ _ _ _ _ _ _ -- DFLST2=. I DEL5T3=.4 FFN S 1= 2* 21 23 IE10 G3 C Tr.147Eioi C FAD ANO PRINT INIPUT CATA C 'WHE"4 %TA?=Ql TFEN XO PLAINE IS TaKF-,. PFR~P;NDTCUJLAR Tn X AXIS. C IF N4 VALUE 1 QF~n IN' FM~ PrFA2, THE~J IT C'ONTAINS ZERO. 00660 00670 00680 00690 00700 00710 00720 00730 00740 00750 00760 a) 003 5 003 — o ---6 U 044 -')0 4 7 0 C) 4 P 00 —4 7' On 5f 1 R FTA2=0. V U='). XL=0.) F~ 75 ~,SC J=I+N'XSLCT Y L YL + 1 V ( j ) 4v J 00770

O C Uc 0 V C, 0 O C 00 C, 0o 0,.0 i I 00,00.. o.4 -. - - -. 0 OCOO 00 00 0 C 0 00:0 LI) I '1 I L-. I; I!, Ic,.!! * 0 4!; l |- I,, I r 1i..:: u:' ' | |! r I. <' - I < 1. 1: ' I i — - ~ <t;! I - " -!!Z |C ' '-,, — f -- < * i L t i- 0:, -'"._.I * -: i ' U I I LL*,-. <.., -c.. - - J xI em em '. 0 Lj + i C 1- 1.- - i.. e:,,!- l ' * I im. ^ r. - LI, m i U L |- LI U 1 a i o i; l u I,- L - a | 4 i '0 10 It! * t j L' u'.-. < U-.. I II |: 1, ~.:...J t' IJ. L C _. 'tX1 L.. t-. iI + I- c - a S r? a I- a. C- ' > i > a. U LU Z i S!;- 2r ~-!-c <; * i -X I oD L|Z L C 0) I J L * - a C- L i. ' cc JC1 ' '^ lt/-l L. LL U I \ 0 C I '> - (~. IC,- ' t L 'LU ( 1)V, ' r IL, Im,-C I"- - 'l"1 -r r',- c - KI 1f. *- 1 C, - j (l 1 LL - -J < <- *l i; V)- ) > <*'LU < 1 - *.!* 114 a. <: I L-, 0 0 I XC C " (\I _ 0 C.L0 1.0.: - 4 x,- t <- a. C J 4 | V - 1l. u - < | r - I. --. L. - i- - - 11 C t L t, U-::D U -J LP It L o a. 11 CL LL U_ v- L ) i >;> )!. - x x - u(. C * - cL ) JC -. - C -,, |u u C) C ~ c u.1 u~ - r' u: I + i a / - CI co I C vP >- I i' P- CWl CC C CrLc U J, i I.47 a. a: C o LQU ir cr. ( - >ii Lr 'tr Lr In trc X x Lr cr Ou a' —, I 0 0 < ~ I is Ci 0, O C ' | o G) C C: O o 0 0 C o ) G C C' 0 C c O C,C C I I 65 65

cn 033 __IFI'l=4-II 01220 C C (,'EC TI\6LT'zFCP VAIl INTY 01 290 Go082 TF (APEFH)) I9-q,l1.3qln-2_ __ _ 19313~T TE (ER112 (jI )I Il,"XYSL nT)-_ __O0l4]30F2MT 'Mb FIST APEP~Tgf` (C7\TF)VAJI UST f3F' 2 C-PL:A'TP TWAAJ Z[ff- ANP~ -VY QF'"AINING, VALI-IE~r MWUT ' 3 CtfL-~ INC CQDEIR OF INC-PFASPJGJ 'k4AG-NITIDF',3fE20,6) _______ 00395 GO TO 1, 0 0 8 6 1 0 N\SL VlT = 1_ __ _ _ __ __ __ _ ___ _ _ _ 00R8 I F (A P T:()I) -1 Q 116 10 5 00389 1D0 51 F-(/PEF(I)-/\PFRPI-1)) 10,-39,133-Q9,104 0 0 9 0 1 914 N S Oi T = 1 I__ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 0911% o VtLf1)1'',l,1 01300 0 092 1 09 ~RIT F__6 I 08 ______ _ _ __01310 031 08p FOR'lVT Vt-LUE FnPR EFRACT IVF It~,nEX N IS LF"ZS THAN OR 1.*) 01320 01094 W. T~TF (6,1 I ) 01330 ('095 1 99 FO`Z~14T CkUTA SET I S NOT USED') 01340 0096 I N FR=IU!P FP+1I 01350 I - I OC, q7 GC T[ q9l C 00 9 3 01 '7 I o 1 6- 1 -01 1;) 3 C, 1- flj /) 01 U) 11 0 IF ( u Ir C iL 1 1 4,1) 1 20 —_ _ __0_ ___14_ IIF3 r R&t T (3\'A[ LF FOrP THE L I v'IT QIF THF NUIARF O'F INCREM' 1',, F'T I VS UNrPFAS0NtlAPLFq ASUR~EO VI lF 14) 0360 01370 01380 01390 01400QT 01410 -01420 014 30 01440 01460 01470 01480 01490 01 500 015 10 C 12 0 124?1 1 19I I 1 9 IF II A) l1,?4,121__ FV~' ALL~ T A R T A2 I S NJFT T WF FN 7ZEPO', l4rJF F EGF S r

01) ID cQ 1) ) 12)4 I F (%I2 1~ llql3C,125 1 2 5 IFr (TP2-5-. ) 130,130, 1119 c 0153n 041101W 1 30 I F (71) 140,13!,129 101I11 1 31 P I T (6127) 01?12 7 Fl'T(T HF IrIT TEU V A LL' F- l~ 71IIS s VT A4 Ifl') 011.3 GOC TOC 19Q 01 14 1 29 V.':"T (6,12 P)_ ______ 0 11'5 1 28 F~t~c-'T('-EGtTIVF OF GIVE-N! V'\LUF FOR 71 IS ASSUM~EP') 0 1 1 6 7 I - 7 1_ __ ___ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ C 01540 01 550~ 01560 01570 01580 - - 01590 — _ 01600 01610 01 17 140 IF (0OEIT-.TA) 140q,14rc,,141 0162( 0118 141 IF( PcLTA+f 71f2)) 144,14491409013 011Q 1409 WRI0,TP (6,14n'R) __ __________ 0164( 1T40 k FOP~ ('CU RO r~ELTA IS UNP.AS1?NAFRLF ) 0165( 01 21 G 0 TO 19 0166( c017 0122 1IL4 4 IF (C) 146,9145,w146 0168( 0123 1 45 VErTFI=A 0169( ) 5-' ) 5 - t cn 0124 __VERTF2=t F2_______ - -- l -.0-17 00 012 5 _ _ Gn 1O 152 _ _ _ _ _ _ _ 01 4' I?('-C) 1476N,i5150 0127 146c ~%1TF (6,146P) _ _ __ 0120 J4~F'P'' T Pr OrITTIN. OF FUNrCTMN\ W-(-F-S N19T GTV7 Pr"'IFP SHAPF' F-~,~,A APE N\E F )E D. 0 1 '2I r,)I 33 -C" 4 01 -1 a C' I1I 013Io,-. 12 JIr TF 1~ V(7-'TF],vVERTP? 151T (YV (,F 13 *6) 01710 01720 017 30 01740 0-1750 01 760 0177C 0178 0 0 1790 0 1900 T F VAII 1~.C,15 4 V ( F7 T C V F2TX f- V FlI Sv14 'I I F r'c- OlI'HT rlF 71pr' ( T I A T,F I.3?TO 1'

01 39 I~ ' I41F ( PFTA,2 ) 16 f91,6 0,j9 1 61 C 0119qo 0I14C) 1 60 UF2XO=FX (+1,Ar?,fACX7E-Rn) 0 14 1 f FX 0=-UF 2 X0 02010 01l42 XU0'=X7Pt 02020 0143 YLr'=xZEPC- 02030 ')I 44 X '=X P02040 0145 GO TO, 163__ _ __ _ _ __ ___ _ 02050 0146 1 61 ~xDi-l1. /TA!(0FTA2'PI/10.) 0147 CA''1 XPT(+11,\'X~,)X7E,0q~.,tF?vr,FCXUPUFX0,IFP1=R) __ 0143 C A LL Y nT-lVX0?X 7 PL —vO F 2,v vC rX L oP F 2 Y0 IE R 02070 0149 XLO=YZFPOr+ZZ&PC4-'SI N( 3ETA2*0DI/P0.) 0 150O 16 3 F?X'=UJF2XO 02080 01 51 1WITF (,1_XLCPFP2XC),PXtIPUF2XO _ _ _ 01 52 COSqF=CC-S (ETA2'F I/1 PO.) 0153 IF (7iO'r,,Cnlsp9-P2Xl) 157,157,1639 _____ _ __ ___ 0154 16393?Z E7 —,I:= 'f(F2X0)1CCSAF-10.E-6 0155 VP IT E (6,163P) ZZERC 02110 0156 1 63 8 FOPAAT ('(V A LUF FCOP Z ZE r-p rlF S NOT L IE W ITH IN R EGITON 3/ 02120 1' V\L F FCRP- F2 ( XO,) I S-A-S S-UM'F 0'F 10, 6) __ __ 0230 017 1 57 IF (PT 1 55164,155 -01 58 1 55 S =-1./IT~t(QFTA*9I13I 0.)__.01 I ERRP=2 _-_ cn 00 01560 0161 0 162? 01 IS3 01 64 01 o05 0 166 -O ) 7 0 1 X9Q 0169 01 7)b017 1 CALL XPT(-19S,C., alpAt RICItX, Y,?IFRR) IF (TP7"P) 1A4,1,164 151)8 I F (Y) 1 59,13 9,lv64 159 IF (X-XLC7) 1q5PQi5F99,164 I59c) 4Q TT (4,15FF) XY ISPS FJQIAT ( ICID&\-,T DL!~Nc JITF%&CT-S FUNrTIO A '?E.2 r r T n- 1 r _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ C T ET I F X) P L A\F TS T CF/.FIT 164 JR (c) l~17S9~ 7,153 1 53 I F Y -L C) l~,rOa, 15 C `,1 ~7 1' r FCA T (I P rTV IV P LAN T IS, T2 F A R I;H T __ 2TO l3 -C' "TH!fS IS FKV I I PJT MTA C i K IN0J T N F S C 01960 01970 __ 0 19 80 -_ - 0 1 F 50 0 1 820 01 830 01 840 02 14 0 02 1 50

M co C ( rFONV E \T A TA I NJ!T C QC PF R LIN IT F r C A LCUL A TION S 02160 01 72 1 67.3 -PT -E-TA _ 31 73 3 F TA ='3 ET A PT/I I _ _ 02180 0174 A P'~= TA t, (FTA) 02190 01 75 F ACTK=?.'PI fWAVFL 02200 01 76 DK=F AC TK *i ITF 02210 01 77 Pf-ETA2P=FETA? _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 01 78 Z 7E:R r Z7 E RC *C C S - F_ _ _ _ _ _ _ _ _ _ _ _ 01 79 ~I 1T H='V,1) T H *C V'S RE _ __ _ 180 CO 1638 1=1,3 011 1 s8 A P(~I-oE- ( I )*CC!~P E ________ 013R2 GN=TWON 02,220 01 83 CASE=MAG _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _02230 014 —F(P-) 111,7C,17C 02240 01395 1 55 I F (TE -1) 1 7 C 16 6.1 7C __ _ _ _ _ _ _ _ _ _ _02250 01 1 66f GN=.I TWCN 02260 01 87 CASF=ELEC 02270 01398 170 WRITF (6,171) 02280 C109 1 71 F rV.A T ('iV ALir IP"NPU T FlATA TS c-rAD~Py-.-CrlMOUTATIPNS BEG.IN'1, 02290 1 ' ( U A CLMRPATIC CASE)'/) -_ (110-0 ~RITF (e,173) 3 0T A Pt-T A?2,X ZFPON nF-L TA CASE 02310 001 1 73 FOBMAT(' FT \=lF6.2, 3XvWPcTA2= I vF6.2, 02320 1 7X, 'XYO='I,F.2,7XINJ=W,5.?,v7Xv 02330 -71' DE =,F 6 3,7X, CASF=',A4) 02340 Ol' 3 1T 7 Cc; 'A (1,? X,'FINAL lV,'TFQfr:DTwf - 02360 1 ICrET',2 XIS T A PTT( I Mr TF'' ~,3X AY I',SY9X,'X'7X,K Z I, 02370 2 5y,'A~ DI TT t.FFI',Xv lI ST AirWFO,9Q'I A N 0,L-F ) 02380 0 1 9q4 7 F'A 7 1 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ __02390 C1~5 ~IT=302400 I 0 19 6,-I1 97 GC Tnl?rA 02 41 0 024 20

C rI\PF1'E4T F CP F [4A l\AT N; PCiT v.AVE FRONT 0 24 1159 1% PITF (6,179) X m', P TU fALL 024 1 7 t. 'T ( FATIVE 17ISCR1~v1NAJT i, 02 4 I' WAS BE'T AI\EFC IN SUBROUT INE XPT WITH ARG,-UMFNTS_' 024 22(/(3 F 1),4) 024 0 2 00 20 0 Z F-A=7 F;.1+ D/LFIT A *C 0S (B E Tf A__ _ 024 0 2.31 IF t H-I T-SLV-) 20C-4, 9204,031- - 0 20~2 203 I F (IS~) 203<9,2239,204 02 033 23~ IvITE (6,2038) 020(4 2 0339 F CP YAT ( A QN ING: FIPST R AY D I NOT P AS S B[LOW 70' 025 I.' OiN XZ[Po DLANEW) 026 0 2 ) 2 34 I S = I S + 1 _ _ _ _ __ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 0 2 6 0236- IF (NUMINC-IS) 20419,206,206 026 02 2n7 2049 WPITE (6,2048) NUP1NC 026 02204R FORMAT ('-SPECIFIED NIJM8ER OF lNCREMEN-TS-APF M~ADEI,14) 026 0209 GO TO ioo__ _ 026 021 2 t- ~6 T 14-FLv= 026 C P EGIN CALCULATIONS _ _ _ _ _ _ _ _ _ _ _ _ _ _ 026 021 -IHOT=0 026 0 212 I NC)P= 0 2 02 1 1 PRlC00 27 31214 ALPHA=0. 027 0 2 15 __PHI=0. _____0 27 21C >MA= 0. 02 7 0 2 17 T C= 0. 2 2 1 3IS T =0. 027 0)2 19 P HT?=O 0 27 3??0 ~~kM=O. 027 C ~028 '22 21 I' F T71 2 21,16,2 142 028 30 40 60 7 0 8 0 90 0 0 1 0 2 0 4 0 5 0 7 0 8 0 190 1 0 20 3 0 4 0 5 0 60P 7 0 8 0 9 0 0 0 1 0 20O 4 0

0 227 0 2 21?14C WRITE (6,2?1491 TS 2'-14,0 FC-P'AAT (lJH 1 4,' ITH INC REiMFNT 7 71 HAS REA-H~Fl X AIXI SI') C FIlfly WHFPE PAY TKTrIQSEC,,-(TS Fl, (YP, 791R 0 2R86-01 02 8701 02 89011 0229G 00 TOn 2 17 0 2 02?30 216 S I GN=-l __02~ 0 231 IF (AF 21v,26,52160 0 23 2 21 60 IF (xr7EfM)v rF)217,216],P21P________ _ _ 02~ 02 13?Il-,, LVERT=l 02C 0?34 G G Tn '2 330 0 2 0 23-51 2 1 f I F ( 7 E M) 2-18,?21 61,v? 17 02~ 02 36 2 17 I SI1G = + 1 ___ _ _ _ _ _ _ _ _ __02~ 0237 -I PEL U-=2 - - _ _ _ _ _ _ _ 02 38 I F I VEP T -I) 2 179, 2-IP,2 17 0 02 39 21I70 L VF-R T =1 0 2 0 240 T VERT= I 0 241 vR ITEF (~2 61U~; ISLVEFPT 03( 024? 2 1 60 FORMAT (10R~yl,I5, PASEn LEFT OF VERTEX OF F',12) _03( 043 1 CL XPESG,~EZ~A,,~BzRLO) 03( 0244 C A LL=1 __ _ _ _ _ _ _ __ 03( CAIF LINE ODO NUT INTFERSECT WITH F THEN A NnN-OVALUE IS - 03( C PETURN'FD IN IRFLO 03C 0245 I F I5BEL C) 2 1`3,9l2JIIv2 19 C (NnTE: NOT CGPPFCT IF AL-PHA IS \JFGA4TIVE-I.E.wHEN REFLECTIONS GO TOWARDS V031 C - VEPTE -03] 0246 2 19 IF (THITr) 2CO.200,7C0 03] 0247 1-219 q IFr (Zr- P'1S 7 79,20,20-._ C CHECK IF kA~y CFSSEr YC PLANF 03] ~0i?300?40?50' ~60?70 )O00 )10! )20 )30 )40 )50 L1 5 1230 j I i i i I i I f,% -% 'f -1 - - - -. 1.- - -.. - - - - -. - - - 024 9 01259?I' I F ( ISOr~.) 221,?2?1,2?-2 -221 IF (YX-f-XQ)?O,202?4 222 IF (XIUP-YR) 7'C,"7CO22?4 C CF I W Al- PHt (M IF NIC IDEN'CE) 0)L2?4 Tj=1./FUIFPX(IST IARXP) u2~ -- IF (IPLA'~)?23,122'c,?23 92 53 223 IF THT) I?2?5, 2 - r 42~ AAHT= 0'1 -~j,'' ( -I T9 E 2HII T-FlTST(S ~FP,7PA7_ C AY AFLST' HAT F>C'S PL/\INI 03140 03150 03160 03170 03180 0.O3 19 0 03 21 0 032 20 032 30 0.3240 032 60

0 259 02 6 D 02 6' 3 02 Yi 026tj7 02 6 0 271 0272 027 027 6 0277 -027 8 02890 7Y 2 R 2 IF (IO L4S)I 1?-,?r26',,1220 1 2?,)I F XP) lflC,91222, 1222 - 1 22? CfIN r'ST=r3FL ST14CENSI ____ I F (XP-'3. 1 240,r 1240, 1224 1 224 C0.rjST=f-lF[ST1*nFlNS2 IF (Xf?-10l.) 1?4),?124C,11?26 1226 C C4-NST= D~ELST2*DENJS3__ _ _ ___ _ _ _ __ ___ _ IF (XP-1~r-.) 124C',12-40,9122P 1228 CCNS-T=F)'EN-S3__ ______ _ I F( XR-20.) 1236,p1236,p1230- _ - 12310 C-tI'NST=nENS4 IF (X'9-48.) 123-46,1236,,100 1 236- C ON-j ST-= CCNST'DELST3 _ _ _ _ _ _ _ _ _ _ _ _ _ _ 124(" Y=TAN( ALPI-A) Y= 1 * /CCS ( ALPH A ) __ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ __ ___ _ U=( 1.61E-12)tAY*C CNST ID EL X=U'-X*Y 'DPLAS=IJ*(l1.-X*X) C 9C1\ TCA CA~S E P L tSMNA -AICJ USPAMF N TF0, X COO 0R DTINA T F: 7~X=Fflc- X (T1S171NAt 6,XP) X f =YP+U.ELX*(C0S(APS(ATAN(DX))))__ ___ __ _ -- - -r) X= F nrlFPX ( I SI CN A,A B,XE3 _ __ __6 DI STO-=n1 SrO+DPL-AS I H3T=1 r END- CF PLASMA CALCULATIONS 2 226 IOUTP=0 I WIT=l Jr- (IDLA-S)?20,9230,v220 C TrQ RF rIF I27N T hi PFGI] fftm1: t 2 ifR vAY c" FI?:ST D!X~s sFO C G FT c f I T f PF C ICN 2. C F!>D C, I T nI I-TE- S COF_-C I tT C (WHITC H IS rN rQ2nR nE P, ~F cI9nNS '3) C F SL(CP LF LI NEFF PC~ Y TOn C P P P P P P P P __ P P 03280 03290 032 95 03 310 0332 0 033 30 -03 340 03 350 — I w 0 28B3 028-I4 02 85 0? 356 02 37

0 -11 W,02B8 2 30:IC = T AN( A TANI(RMN) +ISIN*PF1-1)036 02 89 KLCC_;-P=KLCOP+l C2 QO I~ KL00,P-F0) 2 3 1,231,9 231 _ 02 9 21 9T' (,2 1) KCP 0 2 2 FOjRMAT t'LO1( PEISN TMT 0O.?3fl HAS RCFFNl PERFORME ' 0 2 9 3 IN9'F RP= INIPE R4-+ 1 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 02 04 -' GO TC 100 029;5 231 IF (PCY*I'~IGN)?12,1235,1234 ______ _ 03600 029?6 -2 2 I F ((YEP-ZB./PCM)-VEP.TF2) 233,v2329,r1235 03610 02917 2 q2 9 LVERT=2 03620 0 2Q P 3 339 WRITE (6,2?38) ISJLVEPT 03630 029I?9 2 33R' F CP "AAT__('OP AVY,1 5, ~IT D I REC TLY O'N POII NT A T S UR FAC E, _ 0640 1 1 2, SVP-1 Y WAS IGNORED') 03650 0 3 90 GO1 T O 200 __ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _0 3 6 6 0 0301 2 33 I SIGiN=-TSIGN 03670 0 30-2 LVFRT =2 03680 0393 WR%,ITE (6,92169 —) ISLVERT 03690 03-)4 1 2 34 IBFFLO= 2 I *03 05 1-C3 ~ 06 030-u-7- -, C3 r8 0 3 09 0 3 19' 0)31 1 0 31 2 0 3 13 1 23r- -,C ALL XPT ( ISITCNRC"V,,lP,9 F'A,2,,r~,C.XC ZCv IBELO) - IF (T-F-L C) 8009?3 C, r8C0 C ClA4ECK IF PAY CF'CSSEF ) CPL.ANE __________ -X>9 XX= X L -_ — IF (TSTCr. ) 23t,,- 236, 235 - - 2 35 X'X= YIIP 26 IF (XY-XC) 48C,4P C,?3 C rALCULA1FPITAC T[ C 2 7 I F (ZC#I SI&N R02 3,3R 2 33 IC- IST ( XR,~ XC 7R 7 C 01ST'=I 5 TC0+R.;C IN _____ r VU S!L(-PF tE cf '" FRDI NT C (ON F?) 24 "'l I~N= fF1 (SIOL?,C C.L>JL A T S LOrF F L! F F~V C T H U R IfCUN 3 Tf X X I P! A NE I-t2 IS I O",~':' ( A Tr (i (Y )-A - 1.ITfN(CN1) 03710 03720 03730 03750 03760 037 7 0 03780 03790 03800 038 10 -038 —A-2 0 -03 830 03840 03 8 5r' 0 ~1 5

0316 031 7 I0TA1a 0319 0320 0321 0322 S TINP 2= S I ( PHI 2) - 03860 IF (N 'SI 1 N?-l.) 241,9241, c241 03870 q241 IF (NPPINT) 20),7241,200 7241 WRITF (6,P241) IS,XEF,ZFM 9241 FOPM/ t T (LH-,3x,I,4,'( ',F6.3,1X,F7.3,') ',2X, 03890 1 I P IY ^AS NCT TPANSS'ITTFD TO TIRPD MEDTUM AFTFP PE rCHING IT') 03900 GC Tr 23C 241 ALPHC=tkS I ( N*S IN (PH I 2 ) ) 03910 03920 I I I I 0T; 3X= ( T CMN ) + I GN*ALPHC) 03930 0324 I T=1 ___ ___ -- -_- - 0325 - CALL TUN! ( X; I T } 0326 IF (IT) 245, _5,245__ ___ ---- 0327 245 CZM'- =X -KJ 0328 GA"'MA= GNCES (ALPHC)/COS ( P- 1? ) 03940 0329 T2= 2./(1.+GAMMA) 03950 0330 RTEMP=P?2( (GAVMA) ______ 03960 0331 I - IF (AFeS(TO9T2)-TOL ) 480,480243 0332 243 I.UTP= I.UTP+1 _ 03970 0333 K P2=0 -C CALCULATF 7 CCOPr) CF PCINT CN XZERO PLANE 03980 0334 IF (BETA2) 244,242,244 03990 0335 242 Z ( IOtlTD)=ZC+CZM,^(X7FRG-XC) 04000 0336 lCQ TO 246 04010 0337 244 XI = ( * X* XZEP + 7C-C Z* XC )/( X-C 7';) 04020 03389 7 ( IFUT ) =vXC^^( X IM-XZ FRO) - 04030 0339 246 IF (ITEST-1) 250,2509t2509 04040 0340 25C9 I T T F (6,259) ISIG\,7(IOUTP) 04050 0341 2538 FD3!''1AT ( 1X, I,E13.6) 04060 C TFST!,hIFTH[R PAY FALLS O-h XW7Fsr Pl _&AE WITHIN RFGION.3 04070 0342 250 I P5'= 1 04n80 0343 AS^7=^S 7 ( I CLTP} ) (344!F (A:)q (?7LCr-) B-SZ ) 2?o,4o, ___ ___ Ir I L= I,'tr- t,,j y ASStf r^R Fl, t 'r ';ATI7vF 7 -I -- -T. 04100?045 2.F~ I F Z ( ITP ) )7n 40OC,?.RO 04110 0:346 27. IF (t Q S(FX()-_\PS7) 3 (),?,2?9n 0o47 2 F" I M-('c (L r?>" )-Z( uITD )Q 3)C,?Q 2QO 04140 034q 2C, I ELC,I(;N ( 1.,7( ICtTP) ) 0349 rnU Tn 4,C 0 ( TI;:F I-. F:.;, T I(" F:; LA F ( iC!. I T I (' I F FF CTINI GIN S41 80 0- 50 3r ' i '4P= I '.CP+ 1 I I I

0 351 P3 52 0 353 03 54 03 5 03 56 - - --- 03 5 8- ------ - 0 359 0 36,50 0361 -— 0 3 6-2 - - - -- 03 63 -0 3 6-4: C C C IFIIQP-P) 340,1320,309 __ Vrnc I T F (6,30P 3.08 Frfdl~rAT (R~~ REFL EC TION ST;R Af-7E ExCEEDED,' IIt U ' FINAL CUTPUT MAY NOT HAVE COMPLETE SIET OF POINTS') I NDP=INOP-1 CC TO 350 3 2 0 I T E ( 6 3 I e _ _ _ _ __ _ _ _ _ _ 3 1 F(lR?,-AT (0*'4-ARN ING: R EFL ECTFION STOR AG3E F ILLED, PROGPAM MAY NEED TO I PF PEC( YPI LED WI TH A LARGER Ml')___ 340 SK X(NO P)=X _____ _ SZ( INJ)P)=7C, SV( INOP)=CVN SPHI (I NCIP)-=PH H2_____ SD( Irop)=rCTSO ST( INOP)=TO __ __ _ _ _ __ _ _ _ _ _ S ( I N Q, P ) P T F M P - -- _ _ _ _ _ _ _ _ _ _ _ _ FEG~IN' LDUP FOr.R RAY B3OUNCING ~'ACK AND FORTH IN PFGION 3 ISIGN I4JIC-ATFS 'ThE SIGN VT7 THE F? THAT THE RAY IS DIPFCTED TOWARD (TO THE TOP nR F3OTTOm OF WEDGE) -S4200 S421 0 S 4220 S 42 30 S4240 S4250 S4260 S4270 S42 80 S42 90 S4300 s410 S4330~6 S4340 S 4350 S4360 54370 -S438 0 04400 04410 c-I 0 365 03 66 03 6 7 0370, 03 71 C -)'7 OY~ 778 0 -378 350 IS1I-7N= ST GN (I. C7M ) -_-_____ CALL XPT( IS I Gtf,CZMXC,9ZCAFP2,qf~CXFIZE, IFPR) __ - if (EPP) 30,36SQ359 3%q ITAT=3519 WR ITE (6,3%)6 IS TATq IELRRIS I GN,iPC 7N.,XC,ZC 9PF2,8,vC 9Y, ZE 3 59,' FA T ('G ADS: NO I NTE% SFC T ION- - SE S T AT FPAFNT I,1I4,p G 14,214,8F14.5) - - -- - 3 ~X X= XL C I F (I SIT0C -3 vI6 2,3 I I~ Y X= YUP ~-2 TI y (X -YE 4 9C,4RP0,3tf5 ALP HA=~ '1+ IS1 A TAN, EvFM)-AT AI\ Oh7M) IC=TO*T? 044 50 044 70 044 80 04490

0 380 0 381 0 3 9? 03 93 03 84 0339'5 03 86 -03 87 03~-88' 03 89 03906 C4:A'lA=CYN'C1S (AL PH A) /COS( P1- 12) T2=0 33( GAYAMA) P TF VP=?. GA A/( I+GAMMA) -__ _ DI1 STO=CI STC-+Fn I1ST (XC I XF, ZC 9 7) F IF (Aq8S (TO*2)TePL) 490,37037 3 70 X= ( A T A (AN)+ I SI GN* A LPH A 04500 04510 04520 -- --- - --— 04530 04~540 v l04550 0391 0392 0393 _ 0394 0395 0397 I I'~2 C ALL TUN ( X, I T ) _ _ __ _ _ _ _ _ _ _ _ _ _ _ _ IF (IT) 373, 915,373 3 7 3 E Z M = X_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ -- F (9E T A2 372,371,372 3 71 Z ( I DU T P=Z + F 7M*XZ7ERC- XF) GO TO 374 372 X I M(M X 0XZ7ERC0+ Z E-E Z M *XE/IM X0- E ZM) Z (Ig13U T P)=r'X (X I M - X ZF RI 374 IF (ITEST)_369,376,9369 _ _ _ _ _ _ _ _ _ _ _ _ _ -6 WRITE (6944P) lSIGNEMNXCZCCMNALPHA, PHI1,7AMMiAT2, 1 PI ST 0,1T CIC Z I X E,9ZE9P H IA LDPHC,9EZ7M 3 76 XC=X E C =Z F 04560 04570 04580 045900 04630 04640 I -.3 0398 0 399 C Z -=EZ-M __ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _04650 04001M'=1'~- 04660 0401__ IELC= _ _ —__ _ -04670 404 KLP2=KLP2"+1 0 4 03 IF (VLP?-10) 295910,3O399 0404 396 F{8 ('0 LCOP STN'T 390.') 0 43'l5 -31~9 i-ITF (61?09) K~ KL P2- ------- - 04%6 - GOT~20 04743~ GL F(I UTr= ~FT k2 —AT-AN,~(r 7 N') I1C. p 04740 0408 AYPI ( I fIT P) Tn*T045 0439~ 43I,4 I F_(A (f 1( ICrU T P))Trl 4R0,436,406 __-04760 A 4FP~T4V XC7IOUT0 17C 04770 C, FS LT v RTPIX~3I YOSFDA(V l FL"-W 71 LTMTTl~ 04780 CIC~t, F:U& 1H[ AC P A SINCEL THERF I S r-)SS I 1I-ITY 04790 C T~ —4 T ~F LFLC T IrS ~ILL FALA ' —1'THI LIMIT,;* 04800

04 S T0 -_-048 04 12 H SZ =A 1S (Z (I OUTP)) 048 04131- DCk 4115 1=19dXSLOT 048 0414 IF (APS(APFP(T))-VTUTP.-AUSZ) 410-),440,,440 _____048 04 15 410) IF (tSt~H)VrT-eZ 4159415,412 -048' 0416 412 I SLOT= I _ __ _________ _ _08 04 17 CENTP=APFR(I) 04 18 IF' (7(IVUTP)) 414,t420,420 048 0419 414 lSLTI=ISLf7I+MXSL0T 048 0420 CENTP=-CErNTR______ 0421 GC TO0420- 048' '0422_ 415 CONTINUF _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 049 0423 GC TOl 440 049 0424 420 ANG=A'NGLF (IGUTP) *P I/1P0. 049 0425 DAMP=CC'_((P I *CGS FE/(2.*W inTH)) (7 (I 0UTP)-CFNTR)f/cnSBE) 0426 T~DI(CUTP)=AMPL(ICUTP) -D-ANP*DAMP _ ______ _ _ 04 27 PL R=AAP (,AM~APL( ICUTP)-_______ _- 049 0428 POLTHCrIST( TOUTP)*FACTK _ _______ __049 0429 R FC T Y= P C L *COS P CL TH -_ __-049 0430, Q, CTY= PtR 4SIN( fPflLTH) 049 _Q 0431 S UMX (ITSLL.~T)=SU~X (I SLC T) +RECTX 049 0,432 SUAv ( I sl CT) SUMY (I SLCT)+P EC TY _ _ _____049 0433 A L PHAP = A L PHA * 1 8.P I 049 0434 1 Vo=Tl, +I 050 0"'435 KHITS.L=K<HITSL+l 0436 IF. (VHTTSL-1) 421,422,421 0437 421 IAF- (rJPPINT) 440,422,440 0438 422 IF ILW I1 434,424,434 050 14 39 424 W 1TT[(s 4?6 T ISX F 7,ZMI TJQT PXITMZ(I O0T P),A MP L(IfUTP), 090 1CI ST( IOUTP),9ANGLE(I CUTP) 050 0440 426 %1Rt.T IH-,2X,q1 4,p4 X, (',6. 3,IXqF 7. 3,p ) X,2,T2,p3 X F 7. 4 050 1 IX, F 7.4,?xvFq. 6,3x, vF 1 6,9 X 9 F. 5) o_____ 09 -0,441 60 Tr '440 050 0442 4& 14L 1Tr (6,,4Th) T IUT P X T 'Av7(IOUTP),l 0 50 1A VPL(I XLT P)) v IS T (I 9U-T P), 4 NrI..F(IUT P) Q44 3 435 F 7?A"RlT (IH ql,?2(-12,,3YF7.4q 050 I 1 X Fr7. 4,,e YtF~. 6, 3X,0E1 3.6,4XF 9. 5 051 (-CCC (7 AL"iL(VUT P '2PL T 4 4 1,44)9,4 44 091 04 45!+444 IF (IT%7 T) 449~,450,449 0915 20 30 40 — 50 60 - 70 86 0 -90U 0 0 i o 2 0 30c 4 0 50-~ 6 0 7 0 8 0 190 00 10 20 30 40 50 60 70 8 0 90 00 1 0 2"~ 3 0

0446 0447 0443 4 49 0450 449 WRITE (6,44R) ISIfl\,A3MZXPtZtPHPNALPHAPHT11,GAMMAT2 1 -)IS TO,teC, XCZC IPHI2 ALPHCIC ZM 448 FCv4T (I6,2(2X,4F15,6f)) 450 IF (lOUTP-29)50lv47,4479 470 WRITE (6,472) 47? F0? R'AAT ( ' wA NINTNG: PPOGRAM' LIMIT OF 29 PAIRS OF REFLE' 1'CTTCNS/I (IN REGiICN 2)-PER EACH OR IGINAL RAY —', 21HAS QUEFEN REACHED') GO TO 500 479 WRITE (6,474) AMPLT 47 3 FOnRf AAT ('I PROGRAM L IV IT OF NO. OF PEFLECT IONS' 05140 05150 05160 05170 05180 05190 05200 05210 05220 05230 05240 - 0451 0452 0453 1' FflP RAY HAS REEt EXCEEDED', 05250 1/' *iITH AMPLITUDE CFAROUT ',E13.6) 05260 O454- G 0T0 20 0 — _ __ _______ - ___- 05270 C * 05280 C 0455 0 4 '5 8> 0456 o~ 0457 0458 0459 o, 4 60 0462 0463 I0464 0 435 PR9OCESS PAYS FPOM ANY REMAINING RFFLFCTION POINTS 480 TF (INOP) 2C0,v200,490 490 IF (!PCC-INCP) 494,200,2C0 493 4I PRDC=IPRCC I PP_______+ 1 I UTP=0 XC=SX(I~P~0C)_____ ZC=S7(IPR0C) CM\V11=S2~'( I D Flfl( ) PHISPH I (I POC) OISTC=S( rIPPOC) TO=ST(IPpCC) P TF.'-'P= S r- ( I PPT C) 05290 05300 05310 015320 05330 05340 05 350 05360 05370 05380 05390 C5400 05410 05420 05430 05-440, 054501 C A ( IJ CULAT F ITfMS FOP FFL CTIOi rQn' POINT TC 'f (IN PEGION 2) 5r12 ISIGN~=SICFhFLrtT(ISIG~liZC) 4h7 L= ( r T ~T HF N: il T F 51r~J P I- - G P O( 34F-3 040c r, 04 70 11=3 CALL TUNIXIT) IF (IT)_SO?,__ 5,52 _2 502 CFP-=V

04 72 CALL XPT( ISTGN,COMXC,ZCARCXD,7DIFRR) _ __ 05460 0473 -IC A L L 05470 0474 IF (IEFfZ) 1P0,l;04,1l8C _________ 05.480 0-475- 5704 X X=XLO 05490 0476 IF (ISIGN) 5-10,510,505 05500 0477 5 05 XX=XUP 05510 04798 510 IF I(XX -XD) 480,480,520______________ 05520 -047 520 DiMN=-1I/F 7FR X (I SI GN, A 9 3,XD) 05530 0480 _P___ PH3= I SI GN(AT 4NU3YN - AT AN (CflM) 05540 0481 IF (CD&'4*IS IGN) 5409540953C -05550 0482 530 PHI3=PI+PHI3 05560 0483 540 SINP3=SIN(PHI3) 0-5570 0484 ____ IF (SINP3) 9549,549,514?2________ 0485 549 WRITE (6,548) 0486 548 F ORMM (' PH I IS E G AT I VE fEEK)' 047WPI TE (6,448) I SI Gr~PHI 3,9XCvZCiX09,ZDCDMDMN,9XB, Z8 0488 GC TO 100 0489 542 IF (TU~~31)545,58C,580 0490 545 A LPH A = A RSI N (T CN S INP 3) 05590 I (0 ----- -- - -, - ------ - - 0491 GAMkIA= CN""C CS (ALPHA) /COS( PHI 3) __ 0492 T0=T0*PTE'AP*P22 (G GM4'A) 040-3__ 5 55 IF (.ABS(T0)-TCL) 4-83-~480,550____ _____ 550 nISTC=DISTC+FDiST(XC,XD,ZC,7n1*TWON 0495 PHII=PHI3 J496 BMN=nMN 0497 X R=X 0__ ___ __ 049R -- P=70 D_ __ __ 0499 IF (ITFST) 5'60 3C,, 60 %0560 WRPITE (6,44R)lIGiNA9BMX8,Zv, pALPHAvPHI,GAMMAT2, I1 TlIS,T CvF C~X CZ C,9PHI1 2A ALPHC, 7 M D~01 GO TO 230 C,HJ Ay IS NJT T PA:SI TT Fnr INT O) PFION 1 RMQFGION 2 -3JT PEFLECTS C CGhLY. 0 5)? 5 ~3C Y=oNiSQ7 T( ~T NP3*SA INP3-l.) --— 05600 -05610 05620 05630 05640 05650 05660 05670 056 80 056190 _ 05 700 05 710

ds 503 f'lISTO=DIS~TO+2.*4TAK(Y) C AMPLITUDE IS UNCHtNGEFD 0504 TO=TO'RTEMP*l. 05 05 GO TO 555__ _ _ _ _ _ _ __C WHENRAY PASSES ABOVE -fF1 05720~ 0567 r)0W ~TE F(6,710) IS 057301 0507 710 FORM-AT ('-PAY PASSEC ABOVE FUNCTION ***' 05740 1 10* THIS CCNPLETES PROCESSING OF DATA SET"'/ 05750 2 fv*',5v'__INCREMENTS HAVE SEEN ADrOFf) TO Z11') _ _ 05760 058GO TO 100 05770 __C WHE —N PAY PASSES ABOVE +F2 OR BELOW -F2 ______ 05780 05 ~ 80 IBELC=0 59 3510 IF (ISIGN*PCM) 4FC,480,802 0511 802 CALL XPT( IS[Gtf,3~CMXBZBvA,%.CX,XD-,ZDIBELO) 0512 _____ IF (IBELC) 48C,805,480 _ 05810 053905 CDM=PBCM 05820 0 5 1 4 _ _ _ X C ~= X B _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 0 515 ZC=ZRB 0516 RTEMP=l. _ _____05830 00 0 0517 05 18 -o511 9 05 20 GOl TO 504 900 WPITE_(69910) B-CMXRBZBrXO,pZDBELO _____ 910 FPM~T(' I CES IT?~'-ATTER?',p5F14.6,12) — ___ G O T O 2 00 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ C** * * * 052 1 -052 2 0 —5-231 052 4 052 5 05 26 052 7 052 8 05 29 05 30 C EXITS FPflm PROGRAM 993( WPITE (6,991) INPER 991 F C7 Y AT (12,'1 INDUT EPR OR S R FSUL TED IN PROnGRA M ENDO GO TO 990 094 WQITE (6,9'c5) 991;5 F CR~`Jd'AT 'E~P CR CCUP PED DUR ING REFAD') GP Tr 999 05840 05850 _0586 05870 0588o 05890 05900 05910 05920.05930 05940 -05950 05960 0 5 97 0 05qPO 05990 992 '.RITE (6,9q929) 99?2 F:C1I4AT ('Al D nITA HAS REEN- PprrCESSFD' I; -J9 S TO11P E N) TO0T AL N E MCP Y R -Q'- J T R -,'F NT S OO04 I-C A F3 YT ES,

o'!() I 01 02 Of) 0 3 o0 005 0006 0007 0008 0009 0010 0011 SU~fRCUTINF TI!N(XIX) INTF~~EP LIST(2)/1925,10 CALL GETIHC(I FPRLIST,&99) X=TAI( X) C AL L PUT I __ RFTURN 99 WRITE (6,98) XIX CALL PUT IHO -. I X=O R ET LU RN 98 FORMAT ('OCOMPUTATICN FOP DATA SET IS TERMINATED 2B ECAUSE TANrENT ARGUMENT,E14.6, 3' IS TOO CLOSE TO SINGULAR ITY FQ_ FORTRAN ROUTINES (rI1O)' 4F'%GGEStTHATFR ST STAlRTIG PTE ESLIGHTL END ___ 0012 TOTAL MEMORY REQUIREMENTS 00029E BYTES 0001 C C C SUBROUTINE XPT(ISIONSLMSLXSLZABCXZVR-P) 08000 SUBROUTINE TO FIND WHERE A GIVEN STRAIGHT LINE (StD) 0801C INTERSECTS FL OR F2, WHICH ARE REGMFNTS OF CONCENTRIC CIRCLES WITH RADIUS A A4C CENTER (BC). INP'UT IS SLUDE ANL A POINT OF LINE, 0803C C AND COEFFICIENTS OF Fl CR F2. 0804( C OUTPUT IS X ANO Z CCCRDINATES OF INTERSECTION POINT. 0805( C AND ERROR INDICMfOR WHICH WILL=O WHEN 0806( C A REAL VALUED INTERSECTION POINT HAS BEEN OTAINED 0807( - -TSTGN= -1 INDICATES UPPER CIRCLE, C I.E., THE PART OF FL OR F? WHICH IS IN 4TH QUADRANT. C THE BOTTOM PCINT IS SOVED FOR EXCEPT WHEN IERPR=2(=TPT)L 0 C ISIGN= +1 INDICATES LOWE? CIRCLE, -c- F.VTWPk (-~-E r1YOP -)1-7-71J F~R F2~ WHIC}V[-fS I~:[ FIRST QUA ANT C THE TOP POINT IS SOLVED FOR EXCEPT WHEN IERR=?(=IPTI. c IfiPT=2 nNILY WHEN RAY PASSES LEFT OF VERTEX OF EITHER Fl OR F2.

-u 08 —2. [ T rF — 082 0004 IF (IPT-?) 5,3,c 082; 0005 3. CTHFR=-1 - 08 2 0006 5 IFOR=0 082 0007 A SL -SL_ 082 0008 IF (ASL) 7,6,7 082 C LINE I S HCRIt ZONT —L 082' 000O 6 Z=SLZ Z7 082 -0010.... -.Q. —. —... 082 0011 O -B- _ 083 0012 -Z --- —-ZIZ+ISIGN*T(C+T)+l 8*+C*C-+)+ ~ ' C A-A 083 0013 DISCR = RO*BQ-4.^AG*COQ 083 -0014 IF (DISCR) 9,16,16 083 0015 16 X=-.5*( RC+SQRT(DISCR) ) 083 00I' 1- -4 083 0017 7 8SL=I. 083 0018. CSL=-( SLZ+SLX*ASL ) 083 001l9 AQ=BSL*BSL+ASL*ASL 083 0020 BQ=2.*t BSL*CSL+ISIGtN*ASL*JSL*C+3B*8SL*ASL) 083 0021 CQ=CSL*CSL+2.*R,*A SL C CSL-ASL*A SL* ( A*A-B*B-C*C) 084 0022 DISCR = BQ*BQ-4. *AC*CQ 084 0023 IF (DISCR) 9,10,10 084 0024 9 IERR= — 084 0025 RETURN 084.0026 10 Z=(-B+ I S IGN*CTHER*SQRT(DISCR) )/(2.*AO) 084 -0027 X=-( BSL*Z+CSL)/ASL 084 C Fl AND F2 HAVE RIGHT-FaND L UIT OF X=B 084 0028 14 IF (B-X) 9,15,15 084 0029 15 — RETURN- 084 -0030 END 085 1C' 20 30 40 50 60 70 80 90 00 10 20 30 40 50 60 70 80 90 00 10 20 -Rb30 40 50 60 70 80 90 00 TOTAL MEMORY REOUIREMENTS 00040A PYTES EXECUTION TERMINATED

66 I 6T6 673 67 1 C C C 67 3 9 67 4 6 75 10 676 67 7 67 8 67 J TH-iSf5N VPI rl r FFXP I~ TLCF ES E- f~T[ CONICAL CAS F YIFLS, THE Ir"TEPSFCTICNN PCINT I-F TWfl STPAT,(HT [P1%IF G IVFNtH SLnPFS AN) t PVP1IT CN EACH. PEFA L MdllV? N,1= M2 * IS! 7NJ IF (ML-M4) 10.09,10 I E P R= 1 R E T U R N__ _ __ _ _ __ _ _ _ _ _ _ IEP~= X= ( ~I1 *X 1-'I X2 +Y?2- Y1 / ( M1-! __ _ Y=M1* (X-Xl )+YI. RETURN FND

THE UNIVERSITY OF MICHIGAN (5) Variable List (partial) for FORTRAN Computer Program with Meaning and Use** ABM Slope of ray from point A (where ray originates) to point B. AF2 The A constant for function F2. ALPHA Angle between ray approaching F1 in Region I and slope of normal to F1 at intersection point B ALPHC Angle between ray leaving F2 in Region mI and slope of normal to F2 at point C. BC Geometric distance between point B (on Fl) and point C (on F2). BCM Slope of ray from point B to point C. BETAP The angle,, in degrees for printing purposes. BETA2P The angle 32, in degrees for printing purposes. BF2XO Z-coordinate of the bottom intersection point that the receiving plane, centered at Xo, makes with the radome. BMN Slope of normal to F1 at point B. CASE Contains the literal 'MAG' when data for magnetic case is being processed and 'ELEC' for the electrical case; used in print-out. CMN Slope of normal F2 at poinc C. COSBE Result of cosine function applied to 3. CZM Slope of ray transmitted at point C into Region Im. CAMP Result of function which damps the amplitude. DELX Used in plasma calculations; represents the change in distance along the function Fl. DK (2r/wavelength)*(width of slot). DPLAS The change in optical distance caused by presence of plasma. DX Used for results of taking derivatives. Input variables are included in another section See diagram in Section (2) for aid in understanding symbols used in the program. 84

THE UNIVERSITY OF MICHIGAN Variable List (continued) EMN Slope of normal to F2 at a point E. E ZM Slope of ray reflected at a point E in region towards receiving plane. FACTK 2 /wavelength GAMMA The factor used to compute reflection and transmission coefficients. GN Contains the index of refraction during calculations for the magnetic polarization and its inverse for the elect rical polarizations. IBELO Is argument of call to subroutine XPT. Indicates which point of intersection of surface with the circle is to be used (quadratic case). IERR Usually used in call to subroutine XPT, in which case value 0 means a satisfactory intersection was found. IHIT Registers 0 during computations for a data set until the first ray hits the radome. IHOT Registers 0 during computations for a data set until plasma adjustments are made (if any). INOP The count of the number of points in sbrage to be processed for reflections. INPER The count of the number of times input data errors have been detected. IOUTP The count of the number of output points. IPROC The count of the number of points in reflection storage that have been processed. IS The number of increments that have been added to the original starting point on incoming plane. ISIGN + 1 when intersection with top function is being sought; - 1 for bottom part. IVERT Is set to 1 for remainder of data set computation when some originating ray first passes left of vertex of Fl. KHITSL Indicator to allow printing of more information when the first component of a ray hits any slot. 85

THE UNIVERSITY OF MICHIGAN Variable List (continued) KLOOP Count of the number of times for a particular ray that the main loop has been performed. KLP2 Counts number of times a ray reflects wholly within Regin MI. LIMINC Program limit to the number of increments that can be made, i.e. the number of points used on incoming plane. LVERT Contains the number of vertices for which rays have passed to the left within a given data set (e.g. 0, 1, 2). M Contains number of points for which reflections storage is capable of holding information. MXO Slope of receiving plane, which is centered at (X, 0). MXSLOT The maximum number of pairs of slots program can process. PHI1 Angle in Region II between ray and normal to Fl, outer surface. PHI2 Angle in Rlqftn I between ray and normal to F2, inner surface. PHI3 Angle in Region i between a reflected ray and the normal to F1, outer surface. POLR, POLTH Contains polar radius and angle for impact angle, amplitude and electrical distance of a ray impacting on receiving plane. PSIL, PSIV Difference is pointing error. RECTX, RECTY Information converted from polar information (POLR, POLTH) before summation. RTEMP Temporary storage for amplitude of ray segment just split from primary ray being followed; segment is RegionI reflection on bottom half of wedge and transmission segment to Region I1 on radome section in first quadrant. SUMX(6), SUMY(6) Sum for each slot of angle, amplitude and distance information for - individual rays. T2 Latest amplitude factor of primary ray being followed. 86

THE UNIVERSITY OF MICHIGAN Variable List (continued) UF2XO Value of function representing the upper helf (1st quadrant) of the inner surface (F2) at the point where the receiving plane (centered at (X, 0)) intersects. VERTF1 X-oordinate of vertex of outer surface i.e. F1; Z-coordnate is zero. VERTF2 X-coordinate of vertex of inner surface i.e. F2; Z-coordinate is zero. XB, XC, XE X-coordinates of points B, C, and E respectively; see diagrams. XEM X-coordinate of point where ray originates on emanating plane front. XIM X-ooordinate of point where ray impacts on receiving plane. XL, XV Used for combiing factors from rectangular sums: L for information from lower slots, (4th quadrant), U for information from upper slots (1st quadrant). XLW X-ooordinate of point of intersection between receiving plane and lower part of F2 (4th quadrant). XUP X-coordinate of point of intersection between receiving plane and upper part of F2 (1st quadrant). XX Used in test of whether ray has crossed receiving plane (within Region r; contains XLO when ray is directed towards bittom and XUP when ray is directed towards top of radome. YL, YU See XL, XV ZB, ZC, ZE Z-eoordriates of points B, C, and E respectively; see diagrams. ZEM Z-coordinate of point where ray originates, which is on emanating plane front. Intermediate storage, which are variables, with no particular meaning are: X, Y, S, U, LIM, CONST, Ad IT. 87

THE UNIVERSITY OF MICHIGAN (6) Flow Chart - A Broad Outline of Logic Flow of FORTRAN Program Dimemakesihatemeuia START 101 Read and print tn~out dataI 100 -19 i L Calce1ats as sad Pe1ti -.0 _a set Idesto zero: IAl EERR, IOUTPD, iVERT,, LVERT,, also SUMX, SUMY No- T WON Z.0 I z0 AF2 = A-a Check Inputs for yhlldct (c4t1-gt coordiate Of A-of Da~tog of mouI vpaze 200 Add inorembnt~ to obtain new Plint, of o01igm f" aftw Convert dt h pro~' Unto Write headn ff"fi, s ray; Jid njotTpj~ik;S A'af~jj E ciii n~rm1nt! _ r Zbero- stome vgtrtigh)A LMML -A 149 No 88

149 Find intersection of ray With + F1 (point B) it exists Calculate normal to Fl at B; Angles of Fay to normal: a, 0; Distance and amplitude Make adjustments for /lasma, if Wanted 226 Set indices IOUTP, IW, KLOOP = 0 begin main 1 op (1) 230 ind slope of line from B to C Has loop ) Yes been done 100 50 times No 231 Positive O0 IGINGM t.Negatlve 23 Ray's X nte pt 0 Print message that L >0 X coord. of F2 "ray hit vertex 2" go to next increment 233 < 0 ISKmN - ISIGi - print message "ray passed left of vertex of F2" 234 IBELO =- 2 235 Find interesction point C lith F2, (XC, ZC) 236 89

236 238 Add to total distance: distace BC * N 240 Find iJormal to F2 at C (CMN) -l Calculate angles of ray with normal: 02 and a if exists c pray t No Print message "ray was not pass thru to transmittedto third medium" Calculate slope of ray from C thru Region m 90

246 4 Yes as amplitude beomii smaller than "totlranoe&' 243 l No Calculate w-re ray hits receiving plan (extended) 300 41 0 Store Information so ray passig nto Reglon II may be followed later! I 350 1 ISION = sign of aope of ray from C v. Find intersetioa of ray with F2 (XE, ZE) 359 nter- No section exist Os Print message "gads no intersection see statement 359" I Calculate normal to F2 from intersection E, angle", amplitude, distance amplitde Yes < smller than - "tolerance" 4 No 91 370

370 Find slqs of ray fom E 371 l Fld 'Awre ray ts ret cohivug plan Mow iormatIon from variablos Into C varlablhs of ray at mAop amplate Of ary at mmnopuls lane, 420 w Convert angle to doroesa oompute damnps Aotor for ampltude 421 92

421 Add certain functions of angle and amplitude, diest. to accumulating sum 421 - 0 Print indiator 422 =0 Print line with impct coordinates, amplitude, distance, angle I I 450 y — W as ray been~ reflected more an 29 times 480 Transfer information from previous reflection points to pt. C variables for processing., 4 500 ISIGN = + 1 * (sign of Z ooord.) Fine slope of ray CD Find where ray (CD) intersects F1, i.e. Point D Was therNo < interseotion ^"" point Yes T 93 504 Print message: neg. discrim. in subroutine 200 XPT with arguments

504 Was intersection left of receiving plane 549 542 580 Distance distance + 2. * atan (nn2 sinz '- 1) 545 Amplitude = amplitude T. *RTEMP*R23( ) 550 94

Then ray passes above +F2 or below -F2 800 <0 802 Move information from B variables into C variables RTEMP = 1 990 95