Ni: >V YORK UMIVERSI fY iNSTITUTE C F MATHEMATICAL SCIENCES LIBRARY AFCRC-TN-60-392 25 Waverly Place, Ncnv York 3, N. Y. A^». -rx r^. NEW YORK UNIVERSITY Institute of Mathematical Sciences Division of Electromagnetic Researcli ^-occcxx+ RESEARCH REPORT No. EM-154 Radio Propagation Past a Pair of Dielectric Interfaces JULIUS KANE and SAMUEL N. KARP Contract No. AF 19(604)5238 MAY, 1 960 NPW YORK UNIVERSITY INflTITUTlE. OF MATHEMATICAL SCIENCES Library fif y^averly Pl*t*. NiW Ybfk 3> N. Y. ftCW' VOOR BJNJV^SITY ^. INSTITUTE OF MATiHiEfaATiCAL SCIETC® UBRAHY Nr.V Y0R:< INIVERSITY INSTITUTE O/ M Vr. !C.,-ATJCAL SCIENCES AFCRC-TN -60-392 pi y/avcrly Place, New York ?, N, X, NEW YORK UNIVERSITT Institute of Mathematical Sciences Division of Electromagnetic Research Research Report No. EM-1$U RADIO PROPAGATION PAST A PAIR OF DIEIfiCTRIC INTEHFAjCES Julius Kane and Samuel N. Karp Samuel N. Karp Morris Kline, Director Dr« Werner Gert*s Contract Monitcfr The research reported in tMs dociiment has been sponsored by the Electronics Research Directorate of the Air Force Cambridge Research Center, Air itesearch and Development Command, under Contract No. AF 19(6CU)5238, and by the American Petroleum Institute. Requests for additional copies by Agencies of the Departinent of Defense, their contractor^ and other Government agencies should be directed to that ARMED SERVICES TECHNICAL INFORMATION AGENCY DOCUMENTS SERVICE CENTER ARLINGTON HALL STATION ARLINGTON 12, VIRGINIA Department of Defense contractors must be established for ASTIA services or have their 'need-to-know' certified by the cognizant military agency of their project or contract. All other persons and organizations should apply to the: U.S. DEPARWENT OF CO>MERCE OFFICE OF TECHNICAL SERVICES WASHINGTON 2$, D. C. Abstract In a previovis report we have introduced a linear boundary condition that serves to accurately replace transition conditions at dielectric- dielectric interfaces. In this work we apply this procedure to obtain an approximate solution for an otherv/ise mathematically intractable problem. The original geometry of the problem is that of a dielectric half space above two dielectric quarter spaces. After we apply our techniqvie of re- formulation, the problem reduces to one of obtaining a solution of a two- part boimdary value problem, in the upper half space. This problem is solved exactly by the method of Wiener and Hopf . Physically reasonable results are obtained in a form suitable for numerical computation. Table of Contents Page Introduction ii 1. ForTDulation 1 2. Solution 5 3. The Near Field U h. The Far Field I3 $. Conclusion 22a Appendix A 23 References 27 Introduction In F?rt I we have discussed a procedure which allows one to replace a dielectric-dielectric interface by a linear boundary condition. We have made this approach plausible by: (1) proving that it guarantees at most a smal] error in the far field of a line source above a dielectric half space, (2) proving reciprocity and unig'jeness theorems for this eeometry, and (3) obtaining excellent agreement in a comparison of the use of this formulation with an exact solution in a problem involving diffraction. Howe^/er, we have not demonstrated the use of this apDroach in a hitherto unsolved problem. In this work we seek to find the field of the following problem: A plane wave is incident in a dielectric half space above two dielectric wedges (cf. Figure l). /.s the problem stands it is not amenable to available methematical techniques. However, we have made plausible a procedure which replaces a dielectric-dielectric interface by a linear boundary condition. This then allows us to replace two of the dielectric interfaces shown in Figure 1 by two different linear boundary conditions of the form described in Part I (cf. Figure 2). We neglect the phenomena arising at the interface between the two wedges in the lower half space. In Section 1 we explicitly formulate the problem shown in Figure 2, and obtain an exact solution in Section 2. We devote Section 3 to an analysis of the field in a vicinity of the origin. We find that the solution has reasonable behavior in that nei^borhood. In ''.ection h we obtain asyiiptotic expressions which describe the field for large distances from the origin. We conclude by compiling the results in Section 5 in a form suitable for computation. - 1 - 1. Formulation In this report we seek an explicit approximate solution to the following non-separable problem: ik(x cos e -ysin&p) A plane wave e is incident upon two dielectric quarter-spaces (cf . FLi^ure l) at some angle & . IkU cos e^-y sin e^ ) ^j?y:^to Figure 1 We seek solutions of the wave equations (1) (V^ + k^) u(x,y) = (V^ + k^) u(x,y) - (V^ + kg) u(x,y) - y > i y < 0, X < y <0, X > where (2) k < k. k < k. and appropriate continuity conditions are to be satisfied at each interface. These boundary conditions are derived from the physics of the problem; In our context u(x,y) represents one of the transverse components of the elect romsgietic field, either H or E . If u = H and H = H = then we say that we have transverse magnetic or TM excitation. Corres- pondingly, if u = E , E^ = E = then we shall speak of transverse z X y electric or TE excitation. In either case solving the problem for u(x,y) yields the remaining components by use of the source-free Maxwell equations (3) (U) 7 >< E - iccuxH ^ VXH - ia)eE -icot with a suppressed time factor of e . - 3 - The boundary cxinditions referred to above are determined by the continuity of the follovring components V" X "E*" and v • B of the electromsgnetic field across a discontinuity in k^ ' P* ^ -1 . The vectors v and s are the unit normal and tancjent vector at each interface. The results of Part I of this work lends plausibility to the conclusion that we can replace these continuity conditions by a linear boundary condition of the form (5) 1 4- + A + ^ -4 u = . ^ 3V k^ ds In the sequel we shall illustrate the procedure for a transverse magnetic (TM) excitation. For this problem we choose the coefficients A and B in the boundary condition to give an exact match at normal incidence and Brewster's angle incidence. It is convenient to note that (6) A and B are real and positive and (7) A > B . Both of these requirements are consistent with the results of Part I. With this information we can replace the three-media problem (l) for y > by the following two-part boundary-value problem which can be solved explicitly by the nethod of Wiener and Hopf (cf. Figure 2). li- gik(x cos«^-y sin 6^) \«k dy ,j« ^x*/ \ \ ik dy k* dx* / Figure 2 (8) u = e^^^"" ''°^ ®o - 7 3in 9^) inc (9) (V^ » k^) ti(T,7) - 0, (10) C^A«*.B 8 )^.„_ ic ox (H) (-Tt^*^' ♦£i-4)'i = 0, 'lE^ T — 2 y > y - 0, X < y = 0, X > - 5 - We seek a solution of the form (12) u(x,y) = u. (x,y) + u (x,y) and add the condition that u (x,y) and its first derivatives be bounded and continuous in any finite nei ::hborhood of the origin. In Section 3 this condition will be shown to be sufficient to guarantee that the origin behaves neither as a source nor a sink. The solution of this problem will occupy our attention for the balance of the report. 2. Solution Introduce the complex v-plane 19 V » p e in which we define the radical (1) R(v) = v4^-v^ as follows: Assume k to have a vanishingly small ima?;inary part e |k| ie ind draw the branch cuts from ♦k and -k to infinity along the rays • = e and 9 = n + e respectively (cf. Figure 3)» If we choose v-k-pe , -2n+e<94?77. (8) a = z'(k cos © ) = sin ©^ + A ' - b' cos^ 9 2, o Since for k » |k|e 'we have 00 + iO ivx e dv ._ V -k cos e^ 00 + lO o the expression (6) will vanish by Jordan's lemma if G^Cv) (9) G*(v) = °^^- ^'^ + K(v) A-(v) V -k cos ^ ^2 is an analy-tic function for Im v>0 of al^braic order 0(v ) where e is any nee^ative mimber. The problem will be solved once the required A"(v) and G*(v) are found by an appeal to Liouville's theorem For this purpose we show in appendix A that K(v) can be expressed as (10) K(v) ^ P*(v) P*(-v) where F*"(v) is analytic, and zeroless fbr Im v > -Im k and 0(1) at infinity. 10 - We re-write (9) as (11) G*(v) ^ a(R - r') P^v) V -k cos &, P (k cos e) P (v) = tt(R- P') 1 v-k cos & t>+/, a \ P (k cos ^) * A-(v) P*(v) . Owing to the assumed behavior of A~(v) and G (v) the left side of (11) is analytic for Im v > and the right side analytic for Im v < 0, thus each is the analytic continuation of the other and hence, defines an entire function. Furthermore, each side is of order 0(v -^), where e. » max {t^^tJ) is less than zero, so that each side of (11) is equal to zero by an application of Liouville's theorem^ this allows one to solve for both A"(v) and G*(v). (12) A-(v) a(R - R) P*(v) (v -k cos &q) P*'(k cos e) K(v) and (^3) O^v) = ^(H^^ P*(v) p (k cos e) The complete solution to the problem can then be displayed as /CO a(R' - R) (11;) u. + Ru , ref 2iuP^kcose) P^v)e^^" iV£^r7y^^ (v .eos.X#^.A'.V^) 11 3. The Near Field We consider the integral representing the scattered field (1) u,(x,y) • T — '-' ,. °^ « B 2> ;__ /T— 2 CO (^^. (v-k COS 9 J (^^^4^* A' -%vO where (2) (r' - R)a 2ni P*(k cos 6^) and observe that apart from the exponential factor the integrand is 0(v~ ) and consequently the integral continues to converge for x » y = 0. Indeed, any linear combination of first derivatives with respect to x and y of u (x,y) will also be bounded and continuous in any neighborhood of the origin. The implication of these 1 ast remarks is that the origin neither absorbs nor emits radiation. To clarify this point we observe that if we identify u(x,y) as the transverse magnetic component of the electromagnetic field u = H,H =»H =0, then Maxwell's equation* 7 X H = - icoe B in cylindrical coordinates (p, 9, z) X - p cos 9 y » p sin & *Having obtained a solution, we maJ^?17 sin ») E^ . ^ I (V cos e*y^^ sin e)N(v)e^P^^ ^°^ 9*yK7 sin 6) dv Both E and E_ are bounded and continuous in any neighborhood of the origin. Consequently, if we form the surface integrals over the Poynting flux P - J (^>^^) • d^ = J E^ uld^l - 13 about the surface S consisting of a cylinder of unit height along the z-axis whose base is a semicricle of radius p in the xy-plane Figure U then &3 p —*■ 0, P -> 0, since each term in the integrand of P is bounded about p - 0, U. The Far Field In this section we will obtain an asymptotic development of the integral (1) Ug(x,y) - Y °° P^v) e^^ ' ^ >^^^^ ^ ^ (v -k cos e^) Z'(v) dv -lU - where we recall the definitions (2) r- ^^'-P') 2ni P*(k cos e^) (3) a = z'(k cos 9) - sin e + A* - b' cos^ 9 5 2 ' 2 It is convenient to make the transformations (5) V » k cos ^ and (6) X = p cos 9 y «= p sin © so that (l) becomes P^kcog0) e^^P^°^ ^'^-^^ sin^d) (7) u (p,9) = -Y ^ y^ (cos - cos e^) z'(k cos 0) where the contour C is taken as shown in Figure 5. Note that the transformation (5) has the effect of removing the branch cuts at v =» ♦k. Iha - ORIGINAL CONTOUR C Figure 5 1? Following standard operating procedure we introduce the steepest descents contour S(9). This path is defined as the locus of points such that (8) cos (0-9) = 1 + is^ (9) 3 - * /2 e"^/'^ sin(^) where s is aiiy real number -oo < s < +oo. For fixed 6 it originates at = (6 - n/2, +icx)), crosses the real axis at 8 at an angle n/U, and terminates at = (9 + n/2, -ioo). The virtue of this contour is that tiie exponential e V J t)ecomes ^ikp . ^-kps^ along this path. Hence if we can transform C to S(e) then the major contribution to the integral (?) arises from a neighborhood of s =0. If there are no singularities of the integrand of (?) for a sufficiently - 16 large neighborhood of s « 0, or - 9 we can obtain a good approximation to u by expanding (10) K(0) = ^ ^^ ^°^ ^^ (cos -cos 9^) Z(9) as a power series about (^-6, ors-O and integrating (?) term by term. We see from (ID) that the pertinent singularities are at •■ 9 and any zeros of Z(e). The procedure of Van der Waerden is to isolate these poles and discuss them separately. The singularity at » will be shown to generate a function which affords the transition across the shadow boundary. Roots of Z(v) will be shown to be important if our direction of observation 9 is near the interface. We shall illustrate our remarks for the TM case for which u(x,y) represents the ma me tic component H . If u » H , then from (U) z z (n) z'(-0) - - cos + (a' - b' sin^ 9^) and by the discussion of Part I we have chosen A and B so that (12) Z'(- ©„,) - where 9„i is the Brewster's angle of the k -k- interface -1 ^2 (13) ©g. - tan -^ ^ 17 - If we define Z(0) = sin ?( + A - B k^ cos^ (f and use (2.10) then we have an alternate representation for (7) as ikp cos(0-&)„. ^ ^ (111) u (p,9) . ^ I -^ sd^L^^ - P*(-k cos ^) Z(k cos ff) (cos 0- cos © ) w O The representation (7) will be useful for an asymptotic development in the range < & < n/2 while the representation (ll;) is preferred for n/2 < 9 < n. The same procedure that derived (12) will yield (15) Z(&3 - n) - where ©„ is the Brewster angle for the k-k^ interface -1 ^1 (16) &g - tan -^ ^ Collecting these results we see that - 6 , is in the neighborhood of B JZ5"0 if kp»k and -n+^g is i"" the vicinity of - n for k^»k . If k» k we shall see that the singularity of the integrand in (7) at /5 « & will influence the asymptotic developaent of Ug(p,9) for » in a neighborhood of © « 0. Similarlyif k^ » k the development of Ug(p,&) will be influenced for a vicinity of © ■ n. We can use the preceding ideas in the following manner: Let 18 - G(9^) be analytic in a sufficiently vd.de circle about the saddle point « 9. Then we can expend G((^) - G(s) in the s-plane about 3 - as (17) G(3) - G(e) + g^ s^ ♦ ... + g^^a^"" and the integral 1(9) . f Gm e^P °°«(^-^^ 60 - e^^P f G(s) ^^^Z^ can be integrated term by teiro to yield (18) 1(e) =/|r ei(kp-n/U) ^^^^ ^ o[il:pr^/^] , the conventional saddle point result. However if G(s) has a singularity in the immediate vicinity of a • then the radius of convergence of (1?) may be too small to permit term by term integration, followed by use of the first terra. Let G(s) have a pole at s where s is in a vicinity of s = 0, Then we need to modify the procedure: We can define h(s) by (19) G(s) _ ^ h(s) cos'^^ and then in the manner of Van der Waerden we split off the pole by using the 4f of. pp. 15, 16 above. - 19 identity (20) h(3) h(s^) h(3) - h(3^) h(s^) o o The bracketed term in (20) leads to an integral which can be evaluated without complication by the saddle point method to yield the same result as (l8). The other term then leads to the integral (21) J - ^ ^ikp e-^P^ da Van der Waerden shows that this integral may be evaluated in terms of the error function erf(2). The final result is (22) Yn/kp - 2ni s^ e -kps. -erf(i3^-y1cp)j For large values of |s kp| one may show that the integral J is 0[(kp )J and hence can be neglected in comparison with the leading term of (18). Now G(^) has poles at 9 - 9^, 9 - -9„i, and 9 = -" + ©5. Each of these lead to an s by the relation (9). Consequently for each pole say 9 we will have a region of p,6 space exterior to the parabola (23) |s^ kp| ■ |l - cos (e -e^)\ kp = K » 1 say, for which we can neglect the contribution of (22). The curve 20 (2U) 1 - cosO - &^) = K/kp our later results somewhat), we will have three parabolas in p,6 space in whose interior we need include the tenn (22). One of these parabolas will contain a transition field across the shadow boundary, and the other two will be centered about the rays 9 = 9„ - n, and 9 = - © , where 9„ and B-t are the Brewster's angles (of. Figure 6) for the right and left half -planes. It is now a simple matter to collect these ideas and obtain an explicit asymptotic development of (7). The first step is to deform the original contour C to the steepest descents S(6)j in so doing we must pick up the residues at any pole in the region bounded by C and S(&). Since the observation space y > corresponds to < 9 < n we never pick up the poles at 9- - -^ or 6 , ♦ ^ which both occur outside this range. However if < & < ©^ then we do pick up the residue of the pole at » which corresponds to a transition across the shadow boundary. Using (2), (3), and (7) it is a simple matter to show that the contribution of this residue is ikp cos(0-9„) (r' - R) e ° or in the notation of Section 2 (25) (R - R) u ref 21 - If we introduce this result in (2.1U), we see that this residue ^ives us the correct geometrically reflected plane wave (Figure 7) in the total field for the region Q < 9< 0. The transition across the shadow boundary at e = 9^ is then afforded by a field of the fohn (22). That is, if our kp 1 - C03(& -9^) we must include the field (22) where s is given in terms of 9 by (9). It is worthwhile to observe that the field on the shadow boundary is an elementary function to terms of order o[(kp)~ 'J, namely u(p,9^).H^e^''''*o[(M-^/2] . Observe that for kp » 1 the field on the shadow boundary approaches the arithmetic mean of the two reflected fields. With this information we can draw three parabolas (a) 1 - cos(9-9^) = k^ (b) 1 - co3(9-9„ +Ti) ^ K fixed (c) l-cos(9-.9 ) -^ then for fixed 9 / 9^, we can choose kp so that if This is the result one would expect from experience with related problems. - 21» - , I d X Figure 7 22 (26) kp > K max ( ^ , , ) C03(9 - 9 ) cos(Q-e + n) cos(9 -»- 9-, ) o " we can display an asyit^Jtotic development of (7) which does not require transition fields of the form (22). ik(x cos 9 +y sin 9 ) (27a) Ug(p,&) . (R' - R) e ° ° i(kp*^) P*(k cos 9) Z (k cos 9 ) sin 9 e '^ o cos 9 -cos 9 Z (k cos 9) V2n k (kp) -3/2 ikp>5 , , , , Z (k cos 9) sin 9 e ^ (27b) u (p,9) - ® F*"(-k 003 9) Z(k cos 9)(co8 9 - cos 9^) >/2n kp o[(kp)"^/^], 9g <9 < n The complete solution is then displayed in figure 8 which illustrates the regions of validity. 22a - 5. Conclusion We have found an explicit solution of an approximate formulation of a mat hematic ally intractable problem. The solution is displayed in Figure 8 which illustrates the regions of validity. For convenience we list the symbols and notations that appear in that figure: A = kA-, a'= k/k. = A ../T .-^- A'^+ 1 sin © - (A - B cos^ ) i 2 2_ sin 9 + (A - B cos^ ) o It? ^ sin 9^ - (A - B cos 9^) sin 9 + (a'- b' cos^9 ) Z (k cos &) = sin 9^ ♦ (A - B cos^©^) z'(k cos ©) » sin 9 + (a'- b' cos^e ) P*(k cos ©) = V\ exp -1 "^ B^-^'^ +^^(b-b')] nj^ ^^^^^*'" P.1MA'-B'^2)(^_g^2) xOO 1 1 - 23 - APPEimiX A Factorization of K(v) We wish to show that the function (1) K(v) . " ^^^ " "''■' - S'"' k >/lc-v + k^A - Bv^ can be expressed as (2) K(v) = P^v) P*(-v) in the cut v-plane (of. Fig. U) (where P (v) is analytic and zeroless for Im V > -Im k) by an appeal to the Cauchy Integral Theorem. For this purpose introduce (3) /nD(v) = n^ K(v) which is analytic in the strip -Im k < Lm v < Im k since neither the numerator nor the denominator of ■K(v) vanishes in that strip . Furthermore since li™ J_ K(v) = 1 |v|^oo b' we have , Yl /n D(v) = ofi |vl^"0O \v Cf . Appendix C, Part I. - 2U so that by the Cauchy Integral Theorem we have 1 i n HK) (U) /n D(v) = ^ ^ Y^ °^ where we take the contour C as shown in figure 9 where p < k. Define 00 + ip '-(v,.^j i^^ (5) - -00 -ip which is analytic and zeroless for Im v > p and l/nD*(v)| is bounded above and below - by positive constants in that half plane of regularity. Likewise we can define .+ 00 +ip D-(v) . ^ /"P*^^ ^ '^^'' 2Jti I j: - V ^-00 +i^ dX which shares the regularity properties of Ti*'(v) for Ira v < p. We have (6) /aD(v) - /nD*(v) -/nD-(v) Since D(v) is even in v we have the relation (7) /n D*(v) - -/n D-(-v) - 2Us E ^^ 25 so that we can rewrite (h) as so that /n D(v) -/n D*(v) + /n D*(-v) D(v) - D*(v) D*(-v) (8) K(v) - P*(v) P*(-v) \^ere (9) P*(v) ^exp 2ni ) *co -iP/n|, K(C) dK • 00 -ip V - ^ For purposes of computation it is convenient to transform the contour defining P (v) to obtain a representation which lends itself to numerical integration. For this purpose we can deform the contour Im l^ - p to the path S as shown in figure 9 since K(JS) is analytic and zeroless in the ^-plane except for branch points at J^ - + k . Using the same definitions of the ar-g ">Tc -IC as in the v-plane we have argVTc-^ "*2 ^^°^® the cut at !C - -k and arg Vk^ -K^ " " | ^®1°*' ^* follows then that the integral that appears within the brackets of (9) can be rewritten 26 - 1 ( 1 2Hi I ^ -00 .kiy^^.k^A'-B'^^ B ' ( +ki y^2-k2 + k2A - B 4^ g -ki v^^^~? + k^A* - B*^^ ^ -ki v^^- k^ + k^A - B^^ c^ which in turn can be expressed as li)5238 (ONE copy u"hleS8 otherwise noted) Director Evans Signal laboratory Bolmar, New Jersey Attn: Mr. 0. C, Woodyard V. S. Array Signal Engineering Labs. Evans Signal Laboratory Belmar, New Jersey Attn: Technical Document Center and Electronic Director, Communications Hq. 0. S. Air Force Washington 25, D. C. 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