Advanced Radant Studies - II B. L. J. Rao This document is subject to special export controls and each transmittal to foreign governments or foreign nationals may be made only with prior approval of AFAL(AVWE-3) Wright-Patterson AFB, Ohio 45433. i

AFAL-TR-67- 309 FOREWORD This report was prepared by the University of Michigan Radiation Laboratory, Ann Arbor, Michigan 48108, under Contract No. F 33615-67-C-1444, Task 4161, Project 416102. The work was administered under the direction of the Air Force Avionics Laboratory (AVWE-3), Research and Technology Division, Air Force Systems Command: E.M. Turner, Technical Monitor, O.E. Horton, AVWE, Project Engineer. This Final Technical Report covers the period from 1 February 1967 to 1 November 1967. The University of Michigan Report Number is 08664-1-F. This report was submitted by the authors October 1967. t is tl porthas bn viewed and is approved. />SEPH A. DOMBROWSKI Lt Colonel, USAF / C ief, Electronic Warfare Division 11ii

ABSTRACT The transmission characteristics of an anisotropic panel formed by conducting discs are investigated theoretically, for different orientation of the principal axis of the medium. The results indicate that the best transmission efficiency is obtained, for high incident angles, when the principal axis is oriented in such a way that it is perpendicular to the plane of the panel. It is shown by adjusting the orientation of the principal axis according to the incident angles, that better and more uniform transmission could be obtained through radomes of conical structure. iii

TABLE OF CONTENTS I. INTRODUCTION 1:I. TENSOR PERMITTIVITY AND PERMEABILITY OF AN ARTIFICIAL ANISOTROPIC MEDIUM 3 III. TRANSMISSION OF A PLANE ELECTROMAGNETIC WAVE THROUGH AN ANISOTROPIC PANEL 7 3.1 Incident E-field Perpendicular to the Plane of Incidence (Perpendicular Polarization) 7 3.2 Incident E-field Parallel to the Plane of Incidence (Parallel Polarization) 15 3.3 Numerical Results 18 IV. CONCLUSIONS AND RECOMMENDATIONS 56 REFERENCES 57 DD 1473 v

LIST OF ILLUSTRATIONS Fig. No. Caption Page 2-1 A Three-dimensional Array of Discs as an Artificial Anisotropic Medium. 3 3-1 Incident E-field Perpendicular to the Plane of Incidence 7 3-2 Incident E-field Parallel to the Plane of Incidence. 15 3-3 Reflection Coefficient (r( for Er=2, Elr=3.3, r=.755 20 3-4 Brewster Angle eBvs a for plr =755, r= 1.65, e =2 e =4 21 1Ir'ir r r 3-5 Transmission Coefficient ITJ2 for = 2, =3. 3, =. 755, (d/X r ~ir ir (d/k =.04) 22 3-6 Transmission Coefficient TI2 for E =2, E =3.3, =.755, (d/X=.05). 23 3-7 Transmission Coefficient T12 for Er=4, Elr=6.6, lr=.755, (d/ =.04). 24 3-8 Transmission Coefficient IT'i for Er=2, 1r=3.3, = 1 755 (d/X=.04) 25 3-9 Transmission Coefficient IT'2 for E =2, l=3.3, P =.755 (d/X =.05) 26 3-10 Transmission Coefficient (Tti for er=4, Elr=6 6,3 1r=. 755 (d/X =.05) 27 3-11 Transmission Coefficient TI2 for Er=2, elr=3.3, l =. 755 a =900 28 3-12 Transmission Coefficient (T( for r=4, Er =6. 6, lr=.755 a = 90~. 29 3-13 Transmission CoefficientlT' 2 for C =2, =3. 3 =.755 a=90~. 30 3-14 Transmission Coefficient r12 for Er=4, E1r=6.6, lr=. 755, a= 90~ 31 3-15 Conditions for Perfect Transmission ( T12 = 1) Through an Equivalent Half-wave Panel for er =2, r=3. 3, r. 755 33 (continued) vi

3-16 Conditions for Perfect Transmission (ITIZ = 1) through an Equivalent Half-wave Panel for er=4, lr=6. 6, ulr = 755. 34 3-17 Conditions for Perfect Transmission (ITt'2 = 1) Through an Equivalent Half-wave Panel for e =2, Elr=3 3, lr= 755 35 3-18 Conditions for Perfect Transmission (ITt/2 = 1) Through an Equivalent Half-wave Panel for r=4, Clr=6. 6, lr=.755 36 3-19 Transmission Coefficient IT| 2 for t=60~ (d/ X =. 329) er=2, lr=3.3, lr=. 755 38 3-20 Transmission CoefficientlTf2 for 0t=70~(dt/ =.342) Er=2, lt= 3 3, -1 lr=.755 39 3-21 Transmission Coefficient iTI2 for 0 t80~ (dt/ X =. 352), r=2, lr 3. 3 lr=.755. 40 3-22 Transmission Coefficient!TI2 for 0t= 600 (dt/X =.211), E =4, =6.6, p =.755 41 r ir ir 3-23 Transmission CoefficientlTl2 for 0 =700 (dt/X=.215), r=4, lr= 6.6, ulr = 755 42 3-24 Transmission CoefficientfTl2 for 0 = 800~ (dt/l.217), r=4, e lr= 6.6, t lr 755 43 3-25 Transmission CoefficientIT' 2 for t = 60~ (dt/X =. 329), r =2, r= 3 3, l=.755 44 3-26 Transmission CoefficientIT'j2 for t = 700 (dt/X =. 342), er=2, 1 3. 3, r=.755 45 3-27 Transmission Coefficient IT112 for 0t= 800 (dt/X =. 352), r =2, lr=3.3, l= 755 46 3-28 Transmission Coefficient I T' 2 for 0t = 60 (dt/ =.211), er 4 1r=6 6, =. 755 47 (continued) vii

3-29 Transmission Coefficient Ir'12 for 0t=70~ (dt/X =. 215), er=4, Elr=6.6, 1r= 755 48 3-30 Transmission Coefficient JT'J2 for 0t = 800(dt/X=.217), Er=4, Er= 6. 6,1 r= 755 49 3-31 Transmission CoefficientIT12 for r=2, elr=3. 3, ulr=. 755 = 90~ and Various Values of 0t 50 3-32 Transmission CoefficiendT T2 for r,=4, lr=6. 6, llr=. 755 a = 90~ and Vaious Values of t. 51 3-33 Transmission Coefficient IT'12 for er=2, elr=3. 3, r=.755, a = 90~ and Various Values of Ot. 52 3-34 Transmission Coefficient iT'2 for fe =4, 1r=6. 6, lr =. 755, a = 90~ and Various Values of 0t. 53 3-35 Anisotropic Conical Radome with Variable Principal Axis Orientation (a ) 54 3-36 Transmission Coefficients IT2 and IT'(2 for er=4, lr=6. 6, lr=. 755, and 0t = 800 with a depending on 0. 55 viii

I INTRODUCTION The problem to be discussed here is an extension of the work previously reported.'' Our aim is to find the effect of the orientation of the principal axis of an anisotropic panel on the transmission coefficients. The problem discussed previously is the transmission of a plane electromagnetic wave through an artificial anisotropic panel. The artificial dielectric which was under consideration consisted of arrays of small conducting discs whose planes are perpendicular to the interface of air and the panel. The resulting medium was double anisotropic (both permittivity and permeability are tensors) with the principal axis being parallel to one of the coordinate axes. The results of the analysis indicated that by properly adjusting the lattice parameters it is possible to change the transmission coefficient of the panel which cannot be accomplished by an isotropic panel. It was also noted that the double anisotropic panel offers the possibility of high transmission efficiency at incident angles near grazing. In the present study the above mentioned work is extended by considering the anisotropic medium consisting of arrays of discs which are arranged in such a way that the principal axis lies in a plane which is perpendicular to the plane of incidence. The problem studied previously becomes a special case of this more general problem. Specifically, in S ection II tensor permittivity and permeability are determined as a function of the orientation of the principal axis for the anisotropic medium under consideration. In the following section, the propagation constants in the anisotropic panel are first derived and then the power transmission coefficients are determined for the incident electric field is either perpendicular or parallel to the plane of incidence. In the final section, detailed numerical results are presented. * See list of References for Radant Analysis Reports on the previous contract. 1

According to this study, it appears that the best transmission efficiency is obtained for high incident angles when the principal axis is oriented such that it is perpendicular to the plane of the panel. It is shown that by adjusting the orientation of the principal axis according to the incident angles, better and more uniform transmission could be obtained through radomes of conical structure. 2

II TENSOR PERMITTIVITY AND PERMEABILITY OF AN ARTIFICIAL ANISOTROPIC MEDIUM The artificial anisotropic medium is assumed to be made of arrays of disks embedded in a medium with permitivity and permeability denoted respectively by e and u. The arrays of discs are arranged in such a way that the principal axis is pointed in x' direction, which makes an angle a with x-axis, as shown in Fig. 2-1. z. /X zt / / /// ////// ////!/// // / / / / / / / FG2-/'A hreimn/7oa/7lo/ d FIG. 2-1: A three dimensional array of disbs as an artificial anisotropic medium. In the previous work (C-T Tai, et al, 1965), the principal axis is assumed to be in x-direction (which becomes a special case (a = 0) of the more general case considered here). In the primed coordinates, the constitutive relationships of the field vector are described by 3

D' = e'. E' (2.1) B' H' (2.2) where the permittivity and permeability tensors are defined by e 0 0 et = 0 e 0 (2.3) 0 0 e p1 0 0 t _ 0 po 0 (2. 4) o o p0 0 0 Pu Formulae for e~ and p1 are found in the previous work (Tai, C-T, et al 1965). Since the field vectors in the two systems transform according to the rule A' R. A (2.5) where cos a 0 -sin a R = 1 0 (2.6) sin a 0 cos a one finds D = R. D' =[R1 ~ R. -'. E' - R -1 e. R. E (2.7) 4

which shows that D=. E (2.8) where E [|.. R 2 2 e cos a+ El sin a 0 (l - ) sin a cos a 0 ~ 0 2 2 (El - E) sin a+ cos e (2.9) For later use it is convenient to represent |11 0 913 E~ it 1-e"2 0 0 0 io 022 931 ~0 33 +\ C sin cosa e sin a+ E cos a - sin a cos a I r ir _ r 1r, E E C ~ lr r lr r 0 C 0 E lr 2 2 / ~ -e \ sin a cos C E cos a+ e sin ~r~r 0, r ir I (er lr) 0 r Elr r lr r j (2.10) E 1 where r and E o o 5

Similarly, B -. H (2.11) with p - R. ~' R 2 2 |p1 cos a+ o sin 0 (o - L1) sin a cos a i o oo _0 p 0 2 2 |o^ 1") )sina cos a 0 p1 sin2 a + cos 2 (2.12) It is convenient, for later use, to represent 0 l13 l-V 0 v2 ~ 131 0 V33 P p sin a+p0 cos 0a { - uo) Sin a cos a 1 --- 0-22 1 - 3= 0 1 0V 2 2 (L 1 -/ P osin c cos cos a + sin a - l P1 (2.13) 6

HII TRANSMISSION OF PLANE ELECTROMAGNETIC WAVE THROUGH AN ANISOTROPIC PANEL 3.1 Incident E-field Perpendicular to the Plane of Incidence (Perpendicular Polarization) We consider now the transmission of a plane electromagnetic wave through an anisotropic slab of thickness d as shown in Fig. 3-1. z T d T R e /\ R FIG. 3-1: Incident E-field Perpendicular to the Plane of Incidence The Poynting vector of the plane incident wave makes an arbitrary angle 0 with the normal to the interface. The tensor permittivity and permeability of the panel are the same as the ones defined by equations (2. 9) and (2. 12). The E-field of the incident wave is assumed to be in the y-direction. 7

Inside the slab, the fields satisfy the Maxwell's equation for harmonically varying field ( ejt), which are given in terms of tensor permittivity and permeability, as V. ( E) 0 (3. 1) 07. (i. H) 0 (3.2) Vx E - -j ts H (3.3) Vx i H jw. E (3.4) The last two Maxwell's equations lead to the wave equations Vx([. VX E - e E- (3. 5) Vx [.7 X. -x H) - w ) H 0 (3.6) The sharp boundaries of the slab (see Fig. 3-1) imposes the following two constraints on the solution of the wave equation. Within the slab, the fields must be related to the fields on the interface so they must not vary as a function of y (i.e. a/ay=0). In addition, the phase velocity of the wave in the x-direction must be the same as the phase velocity in the x-direction at the interface. Because of these two constraints, the y and x dependence of the wave equation is specified. It is only necessary to solve for the remaining z dependence. The fields in the anisotropic slab can Lnus be written in the form (for E-vector polarized in the y-direction) E E1 e - (K ) (3.7) 8

where K-xK + K x z and K =K sin 0 x o with K being the usual free space wave number. By substituting equation (3.7) in equation (3. 5) one finds z 22 o - x x z 22 -i E =0 (3.8) (KZv22-K2 1l E ( + KK K2 -K2 1 (38 0/ 0 K2 +K v -K K (v +V )-K2 22\ E ~ (3 9) 33 z 11 x z 31 13 o y K Kz 22 -K 3 E + 2K V -K 2 33 E 0 (3.10) o / 0 where the E's are the coefficients of the dielectric constant tensor e as given in equation (2. 9). The non-trivial solution of the above three equations is evidently given by 2 2 / + 2 22 K v +K v - K K ( + v13 - K - 0 (3.11) x 33 z 11 x z 31 13 y E 0 C22 O o Noting that 31 13 and e =c e, we obtain from equation (3.11) the following o characteristic equation which mustbe satisfiedby K and K 9

2 2 2 v K 2v K K +v3 K -K lr0 (3.12) 11 z 13 x z 33 x o lr =~1,-1 where vll, "13 and v3 are the elements of the tensor and are given ill, lo 3o 33 by equation (2.13). Equation (3.12) represents a quadratic equation in K, the corresponding two solutions are given by K K2 v 2- v v + K 2 v (3. 13) 13.x - - x k13 11 33 K 0 11 l lr K - Z'11V The value of K with positive sign before the square root represents the propagation constant in the positive z-direction and is represented by 13 K +P K 1 x (3.14) z+ Yl1 Z+ Vll where v ex (^13 11- v33 o 1 lr The value of K with negative sign before the square root represents the negative of propagation constant in the negative z-direction. Thus, if k represents z-." the propagation constant in the negative z - direction, it is given by -v K+P K = 13 x (3.15) Z- V 1 Having determined the propagation constants in the anisotropic slab, the general expressions for the fields in the three regions are given by: 10

Region 1, z < 0 i A' j x + K z. 16) E = Eo y e x zo (3.16) E LO Z i 0 ex+K H= ~- K +' K ) e K x Kzo (3. 17) where K K sin 0, K K cos 0 and K 2 x 0 Zo 0 0 X r. A -jf xX- Kz\ E RE y e \ zo (3.18) 0LLP \ zo o Region 2, 0 < z < d Ed=E ) T e-JKx 1x+ z-j x-K Z\ E T e (x+ Kz+ R1 e( x z- )I (3.20) L IJ Z K + x/ + R1 (Z X+ KX eJ (K x Kz- z) (3.21) where Kz+ and K are given by equations (3. 14) and (3. 15). is the inverse Zwh K+ LZ ith of p and is given by equation (2. 13). 11

Region 3, z > d T A -tTEA -j x zoK (3. 22) K x + K / eJ (Kx+ zo (3.23) W P zo x By matching the tangential components of the E and H fields at the two interfaces, corresponding to z = 0 and z = d, one can determine unknown coefficients T1, R1, T and R using the equations (3. 16) to (3. 23). The results are given below. 2 K K +v K -v K\ zo/ zo 11 z + 13 T /(K+ -1 (3.24) 1 r2 e-J (Kz++K z) d l-r r2 e - z.+ z-) R -Trl e (Kz+ K )d (3.25) R -1+T1 1-r e j(K + + z j (3.26) and TT1 1-r1) e (Kz+ Kzo)d (3.27) where -V1l K+ +V1 K +K 11 z + 13 x zO r = 1 +K ~~~~~~~~~~~~(3. 28) 1 v11 K +3 K +K 3 11 z - 13 x zo 12

-V11 K -V13 K +K ~r 1 1 13X ZO (3.29) 2 11 K + V13 x Kzo From equations (3. 14) and (3. 15) it is noted that V11 K + -v 13 K = P (3.30) ll Kz+- 13K 310 Vll K + 3Kx-P (3.31) 1ll Kz- 13 x Adding the above two equations results in K +K 2P (3.32) z+ z- V11 Using the relations in equations (3. 30) to (3. 32), the expressions for the coefficients T1, R1, R and T could be expressed in a compact form, the results are summarized below T..1+ r (3.33) 1 2 -j2 P d 1 -r e r(l+r)e-j 2P d r (1 t r) e J 2 d R V11 (3.34) 2 -j 2 P 1-re - d V11 21-j2P r (l-e j 2 P d) R= vll 1 - r2 e j 2 d (3.35) V 11 13

and r 2 e-j (K -K )d (1- r ) z + zo T - 2 -j 2 P (3.36) 1-r e d V11 K -P Z~ where r=rl r2 K - +P (3.37) zo P =IO ll v - K (3.38) llr and. 1 / r Since the interest is mainly in the magnitude of power transmission coefficient 1T| 2 it can easily be verified that T 2 21 L 2r sin(i d2 (3.39)!+!2r sin 1 - r2 From equation (3. 39), it is evident that perfect transmission (IT = 1) is P obtained when -d is equal to an integral multiple of ir. We consider later the "11 case when - d = r. It is also noted from equation (3. 39) that the transmission efficiency increases as the value of r decreases. For given E1 and pl, it can easily be verified that r will have the least value when a = 900, for any given value of 0. These comments will be explored in more detail in the final section. 14

3.2 Incident E-field Parallel to the Plane of Incidence (Parallel Polarization) We consider now the transmission of a plane electromagnetic wave through an anisotropic slab of thickness d, when the incident E-field is parallel to the plane of incidence, as shown in Fig. 3-2. z T' I -_ E -2 i FIG. 3-2: Incident E-field parallel to the plane of incidence. The tensor permittivity and permeability of the panel are the same as the ones defined by equations (2. 9) and (2. 12). In the present case the plane wave is characterized by a H component such y that HH H e - (K ) (3.40) y o 15

where K'-K x+K' Z x z By substituting (3. 40) into the corresponding wave equation given by equation (3. 6) which is repeated below for convenience, Vx([j- 1.V xH - O one finds that K and Kt must satisfy the following characteristic equation, x z 2 2 2 1 Kt -2 13 KK K K33K -K =0 (3.41) By solving equation (3.41) it is not difficult to show that the propagation constants in the positive and negative z-direction are respectively given by S K +P' Kt (3. 42) z + 11 -3 K +P' K13 x Xt' ~ (3. 43) w11 where P'Ko2 11- x r lr and 9's are the coefficients of the ~o [ -1 and are given by equation (2.10). Having determined the propagation constant, it is convenient to define the reflection and the transmission coefficients with reference to the incident magnetic field, and the general expressions for the electromagnetic field in the different regions are given by, 16

Region 1, z < 0 /,\ i + K zH H y ej X zo) (3.44) H 0E= (K O x-KeX ei ( x+ zoZ (3.45) WOE zo x) 0 where K = K sin, K K cos 0, and K 2-. x o zo o0 o Hr INR u X." x X- z5 Hr' H e j(Kxx zo ) (3.46) R'H ( -r o zo X w e ( Kzo~Kx (Kxxezo (3.47) 0 \ Region 2, 0 < z d -d K ~ x+ Kt z' X e H yH TeX(+RI e- z- (3.48) E x-K i x +K + z) HL l~+ x -R -K^ -K_ / j ZX e i z- x (3.49) Region 3, z > d H = T' Hy e x -Kzo (3.50) 17

H (K K e K x Kzo (3.51) W zo x / 0 By matching the tangential components of the E and H fields at the two interfaces, corresponding to z - 0 and z = d, one can determine the unknown coefficients T1, Rt, T1 and R' as we did in the case of the perpendicular polarization. The results have the same algebraic forms as those given by equations (3. 33) to (3. 36) except that the coefficients K +, K, P, r and v's are replaced, respectively by K't K' P', r' and's. The expressions for K, Kt, Pt and i's are Z+ zo-'z+ z* — given in equations (3.42), (3.43) and (2.10). The expression for r' is as defined below r' = zo (3.52) K + P zo Thus, the expression for T' 2, which is our main interest, is given by TI =. -- - - 2 (3.53) 1 2 r' sin P'd\ l-r rJ 3. 3 Numerical Results As stated in the introduction, our aim is to study the effect of a (which represents the orientation of the principal axis of the anisotropic panel) on the transmission of a plane wave through an anisotropic panel. To this end, we note, from the expression of power transmissions coefficients given by equation (3. 39) and (3. 53), that the transmission efficiency increases as the values of r and r' decreases. From the expressions of r, it could be noted that the value of r decreases with the increase 18

7T in a for any given angle of incidence 0, and has the minimum value when a t. This can easily be noted from the computed values of r as shown in Fig. 3-3. From the expressions of r', it can be noted that it becomes zero for a certain incident angle 0B, which is usually called the Brewster angle and is given by r 1 j cos 0 (I-i) sin2 a + (e1-.1}cos a (3.54) \B lr ir For this incident angle T' 2 =1, indicating a perfect transmission. It can easily be noted either from equation (3. 54) or from Fig. 3-4, that 0 increases with a and has maximum value for a =, provided e and e remains the same. This r l r indicates that better transmission is obtained for high angles of incidence when a. 2 2 Figures 3-5 to 3-7 represent the computed transmission coefficient JTI for different values of a, for several thicknesses of the panel < < and for two sets of values for e and. Similarly the Figs. 3-8 to 3-10 represent TT as a function a. From these computed results for thin panel, it is evident that better transmission is obtained when a, especially for high angles of incidence which 2i is of interest for radomes characterized by high fineness ratio. Having noted that a=T gives better transmission, the transmission coefficients are computed for 2 df r \ several thicknesses of the panel - << 1 and are plotted in Figs. 3-11 to 3-14. These results will be of interest to see how the transmission changes with frequency. Now, it is of some interest to point out the special characteristics of transparent panels at oblique incidence. From equations (3.49) and (3. 53) we see that T2 becomes unity when Pd N7T (3.55) /11 19

.0.9.8 -.7.6 Irl -.5 0 a 30 -.4.L~- - ~ ~ —~-=- 60.2 10 20 30 40 50 60 70 80 90 0 (in degrees FIG. 3-3: REFLECTION COEFFICIENT rl for Er=2, Elr=3.3, /,lr=.755. 20

r-100 -900 800 -70~ 860 0B Er =2 400 30~ 20~ 10~ _10 20 3080 o FIG.3-4 — BREWSTERANGLE (in degrees) FIG. BREWSTER ANGLE B VS a FOR lr= 755, er= 1. 65, r= 2, er=4. 21

-1.0.9 goo- =900.8 —.7. 2 -.1 10 20 30 40 50 60 70 80 90 0 (in degrees) FIG. 3-5: TRANSMISSION COEFFICIENT IT 2 FOR r=2, elr=3. 3',1rr. 755, (d/X=. 04). 22

(.9 3 900 L ~~~~.3~~300.2 2~~~~~~~2.1 6o 70 10 (in degrees) 755, FIo 3-6-. TBANSNg~SSION

1.0. 90. 1 40 30J-5^' ^' 750 G.3d3l 4.224 g4

-.7 10 20 30 40 50 60 70 80 90 6 (d =. 4 TT125.5 -4.3.2 *1 10 20 30 40 50 60 70 80 90 0 (in degrees FIG. 3-8: TRANSMISSION COEFFICIENT I TI 2 FOR er=2, Elr 3. 3, M lr= 755' (d/X =. 04). 25

~~~c10 20 30 40 50 60= 730 8 -- =0 (in degrees) 0 0~ (/ = 900 626 -.5 T'.4.3 10 20 30 40 50 60 70 80 90 o (in degrees) FIG. 3-9: TRANSMISSION COEFFICIENT T' 12 FOR Er=2, Elr=3. 3, lr= 755, (d/X -. 0)2. 26

11.0 a =00 = 90(d/X.627! T11I.5.3 B.2 10 20 30 40 50 60 70 80 90 0 (in degrees) FIG. 3-10: TRANSMISSION COEFFICIENT ITf 1 FOR E =4, Elr 6. 6,,lr=.755, (d/ =.05). 27

1.0... d/X=.01 10 20 30 40 50 60 70 80 T 0 (in degrees) FIG. 3-11: TRANSMISSION COEFFICIENT IT12 FOR ~j=2, el= 3,,u-. 755, fM=90r..28.9.~..~.02.04 -5.2 10 20 30 40 50 60 70 80 ~0 ( (in degrees) FIG. 3-11: TRANSMISSION COEFFICIENT IT12 FOR %r=2, clr= 3. 3, lr 755, J lr= 28

1. 0 d/X=. 01 10 20 30 40 50 60 70 80 90 0 (in degrees) FIG. 3-12: TRANSMISSION COEFFICIENT | 1f2 FOR %;.=4, elr=6. 6, ulr=. 755, c=90~. 29.02 ~04.6 ~05 -.4.3 10 20 30 40 50 60 70 80 90 e (in degrees) FIG. 3-12: TRANSMISSION COEFFICIENT I ~2 FOR %=4, Elr=6.6, 6 lr=.755, =90o. 29~

0 d/=. 01 2 025.8 7. -.6 FIG 313 TRANSS COEFFICIENTI Tj2 FOR =2,E 3. 755 30

_1.0 d/X=. 01. 02.2.9.025.04 ~05'.5 -.4 10 20 30 40 50 60 70 80 90 0 (in degrees) FIG. 3-14: TRANSMISSION COEFFICIENT | T'2 FOR er=4, lr= 6. 6, lr=. 755, a = 90~. 31

and T'2 becomes unity when -= N7r (3.56) ~11 Usually N = 1 (which corresponds to equivalent half wave panel) is of prime interest. Letting d and dt represent the corresponding thickness of the panel for which perfect transmission occurs for perpendicular and parallel polarization respectively, it can be shown from the equations (3. 55) and (3. 56) that 2 -- 2 d =r sin a + \ir cos a d -0.5... (3.57) I 2 2 2,\jr sin a+ cos a El -sin 0 2 2 E sin a+ e cos a d' O.5 r ir (3.58) Lt 9__2 2 2 ~r rle sin c + cos - sin 0 These conditions for perfect transmission are plotted in Figs. (3-15) to (3-18). From these plots it is noted that the necessary thicknesses are smallest for a =, which may be an advantage in some cases where weight is a major factor in a radome design. In what follows, we discuss the design of an equivalent half wave panel and present the numerical results. The interest is to obtain high transmission for both polarizations for all angles of incidence up to 85. As noted before, perfect transmission is obtained for parallel polarization when the incident angle 0 is equal to the Brewster angle 0B. It was also noted that 0B has maximum value, and better transmission is obtained for perpendicular polarization 32

.4 a 0 = o. dt/X 3 e 1~~0.2 10 20 30 40 50 60 70 80 90 I I i I l i l l 0 (in degrees) FIG. 3-15: CONDITIONS FOR PERFECT TRANSMISSION (IT12=1) THROUGH EQUIVALENT HALF-WAVE PANEL FOR Ec=2, lr=3. 3, lr=. 755. 33

w )4.3 dt/X a0= 0~ 40~ 600 900 10 20 30 40 50 60 70 80 90 I I I I I I I 0 (in degrees) FIG. 3-16: CONDITIONS FOR PERFECT TRANSMISSION (ITI2=1) THROUGH AN EQUIVALENT HALF-WAVE PANEL FOR er=4, er=6. 6, Alr=. 755. 34

.2 10 20 30 40 50 60 70 80 90 0 (in degrees) FIG. 3-17: CONDITIONS FOR PERFECT TRANSMISSION (IT'I 2=1) THROUGH AN EQUIVALENT HALF-WAVE PANEL FOR Cr=2, Clr=3. 3, r=. 755 35

_.4.3 dt /X c =00 40~ i 40~I I I I i 0 (in degrees) FIG. 3-18: CONDITIONS FOR PERFECT TRANSMISSION (JT'12=1) THROUGH AN EQUIVALENT HALF-WAVE PANEL FOR cr=4, elr=6. 6, Plr=. 755. 36

when a=. Considering these observations, it is appropriate to design the equivalent half wave panel in such a way that perfect transmission is obtained for perpendicular polarization at a desired angle t, which is related to the required thickness dt as given in equation (3. 57) for a =. Figures 3-19 to 3-24 represent the transmission coefficient T 2 as a function of 0 and a for several assumed values of 0t. Similarly, 2 2 2 Figs. 3-25 to 3-30 represent T'. Figures 3-31 to 3-34 represent T2 or T as a function of 0 and 0 for a= 2. From these figures it is noted that higher the value t 2 of Ot, the better is the transmission at high incident angles, for the case of a =. However as 0t is increased to higher value>, the transmission efficiency at normal incidence is degraded for a-. For a <, the transmission efficiency is better 2 " 2 for normal incidence. This dependence of the transmission coefficients on a can be taken as an advantage in order to obtain high transmission throughout the range of incident angles. We take the case of e = 4, e = 6. 6 and 0 = 80, as it has the r ir' t 0?7T better transmission for high incident angles (up to 0 = 85 ) for a 2 to illustrate how one could design a panel which has power transmission efficiencies better than 95 per cent over the range of incident angles up to 0 = 85. To do this, draw a line at T =. 95 (and T' =.95) in Fig. 3-24 (and Fig. 3-30). From these figures, it is noted that the curves for a = 30 can be used up to 0 = 350, a= 600 can be used from 350 to 65 and a = 90 can be used for higher values of 0. To apply these results to a cone shaped radome, one has to vary a along the radome depending on the angle of incidence as shown in Fig. 3-35. The corresponding transmission coefficients are plotted in Fig. 3-36. It is expected that by varying a continuously along the radome the transmission efficiencies could be improved further. 37

0 (in degrees) FIG. 3-19: TRANSMISSION COEFFICIENT |T12 FOR 0=60~ (dt/x=. 329), e~=2, eCr3. 3, 1.0=. 755. 600 38' —908.7. 6 iT 2.5.4.3 10 20 30 40 50 60 70 80 90 e (in degrees) FIG. 3-19: TRANSMISSION COEFFICIENT IT12 FOR Ot=60~ (dt/X=. 329), cr=2, Elr=3.3, lr=.755. 38

0 (in degrees)00 e=0 3. 3,.-r =. 755.' 303 -9 600.8.7.6 [Ti2.5.3.2.1 10 20 30 40 50 60 70 80 90 0 (in degrees) FIG. 3-20: TRANSMISSION COEFFICIENT IT 2 FOR 0t = 700 (dt/ =. 342), %=2, Elr= 3. 3, lr=755. 39

ri.o'-'1.0~ ~40 300 600.8 900 7e 6,.4 1O 2P 30 40 50 60 FIG. 3-21: TRAN ~YSSN 0 (in degrees) F: SMSSIO COEFFICIENT IT12 FOR et-80o (dt/;t. 32), %-2, Elr= 3.3, /Ulr = ~ 755. 40

1.0!"-,...-'~e =6 90~ 60.9 30~ 00.8 -.7 -.6 1~i2 —.5.4.3.2 10 20 30 40 50 60 70 80 90 0 (in degrees FIG. 3-22: TRANSMISSION COEFFICIENT IT12 FOR Ot=60~ (dt/X=. 211), %r=4, Elr=6. 6, /lr=.755. 41

1.0 = = 90~ 60 30 0o 42.2.1 10 20 30 40 50 60 70 80 90 e (in degrees) FIG. 3-23: TRANSMISSION COEFFICIENT IT12 FOR 0t= 700 (dt/X =.215), Er =4, 1r 6.6, lr =.755. 42

1. 0 a = 60~ 10 20 30 40 50 60 70 80 Ah\ \ 90 0 (in degrees) FIG. 3-24: TRANSMISSION COEFFICIENT | TI2 FOR Ot=80~ (dt/x=. 217), er=4, elr^ 6. 6, Alr =. 755.._a430 ITi2.2 10 20 30 40 50 60 70 80 90 0 (in degrees) FIG. 3-24: TRANSMISSION COEFFICIENT I T12 FOR at=80~ (dt/X=. 217), %=4, Elr 6.'6, PAr =:.755. 43

1.0 a= 00 = a =900 10 20 30 40 50 60 70 80 90.8 6.=. 7 —.6.5 [T 12.4.3.2.1 10 20 30 40 50 60 70 80 90 0 (in degrees FIG. 3-25: TRANSMISSION COEFFICIENT IT'12 FOR Ot= 600(dt/X =. 329), E=2, Elr= 3.3, lr = 755 44

1. 0 0~ and 30~ 900.7 10 20 30 40 50 60 70 80 190 0 (in degrees) FIG. 3-26: TRANSMISSION COEFFICIENT \T'\2 FOR Bt= 70~ (dt/h =. 342) eP=2, lr=3. 3, ^ ~lr~ = 97550.645.5 IT,' 10 20 30 40 50 60 70 80 90 0 (in degrees) FIG. 3-26: TRANSMISSION COEFFICIENT T' 2 FOR 0t= 700 (dt/X =.342) Er=2, elr=3.3, -.r = 755.

1. 0 ac=0~ 30~ / 90~'~.84 (in degrees) 303.7 FIG. 3-27: TRANSMISSION COEFFICIENT |T12 FOR 0t=80~ (dt/X =. 352), e=2, Clr= 3 3, ar=. 755. 46

1.0O 1 ~ a = 600 90 90 0 00 600.8.7 303.6 lT 112 15 ~~~~.4~47.3 10 20 30 4 O 6 0 8 PIG. 3-28: a ti~0n degrees) 6.6 US ONr' 75COEFFICIENT IT? 12 FOP, 600 (tX.1) r4 P /ai.755 tEr 4.

1. 0 = 60~ 9go 10 20 30 40 50 60 70 80 600 300~0 0 900.8.7.6 i Ti2.5.4 0 (in degres.2 10 20 30 40 50 60 70 80 & (in degrees) FIG. 3-29: TRANSMISSION COEFFICIENT [T'12 FOR Ot=70~ (dt/X=. 215), er=4, lr=6. 6, lr=. 755. 48

30 60 90 -.8 — 6,7 T,1249.3 10 20 30 40 50 60 70 80 a (in degrees) FIG. - 3-30: TRANSMISSION COEFFICIENT I T'2 FOR t:80~(dt/X=.217), %r=4, elr=6. 6, Alr=. 755. 49

I-1.0.7,= 500.9 20 30 40 80.7.6 1T12.4.3 30 (in degrees)50 60 70 0 (in degrees) FIG. 3-31: TRANSMISSION COEFFICIENT I TI2 FOR c=2, 1. 755 900 AND VARIOUS VALUES OF r r 3 r= 50

-1.5 Ot=50.9 ot=7. - t=80 ot=90 -.8.7 -.5.4.3 -.2 10 20 30 40 50 60 70 80 0 (in degrees) FIG. 3-32: TRANSMISSION COEFFICIENTIT12 FOR Er=4, Elr=6. 6, lr= 755 a 900 AND VARIOUS VALUES OF et. 51

-1.0 Or=50 g oo.8a = 90~ AND VARIOUS VALUES OF 0.'-=.6.5.4.3.2.1 10 20 30 40 50 60 70 80 0 (in degrees) FIG. 3-33: TRANSMISSION COEFFICIENT I T'12 FOR Er=2, Elr=3.3, lr 755, a = 900 AND VARIOUS VALUES OF et. 52

10 20 30 40 50 60 70 80 90 -- 1i~~~. ~0 0(in degrees).8 a'= 900 AND VARIOUS VALUES OF 9 -.7 -.6.5.4 -.3.2 10 20 30 40 50 60 70 80 910 0 (in degrees) FIG. 3-34: TRANSMISSION COEFFICIENT ITI12 FOR r =4, 1r=6. 6, lr= 755, a= 90g AND VARIOUS VALUES OF Ot 53

a=300 _ a=600 =350= FIG. 3-35: ANISOTROPIC CONICAL RADOME WITH VARIABLE PRINCIPAL AXIS ORIENTATION ( a ), 54

1.0 T T, 2.9 1T12.4.3.2 I _.1 10 20 30 40 50 60 70 80 0 (in degrees) FIG. 3-36: TRANSMISSION COEFFICIENTS ITI2 AND IT'12 FOR cE=4, elr=6. 6, l'r=. 755 AND Ot=800 WITH a DEPENDING ON 0. 55

IV CONCLUSIONS AND RECOMMENDATIONS Theoretical study on the transmission of a plane wave through an anisotropic panel indicates that the transmission efficiency depends on the orientation of the principal axis, and better transmission efficiency is obtained, especially for high incident angles, when the principal axis is perpendicular to the plane of the panel. For this orientation it is noted that the physical thickness of the equivalent half wave panel is smallest which may be an advantage where the weight is a major factor in a radome design. It was also noted that the transmission efficiency is degraded at normal incidence, when the principal axis is oriented perpendicular to the plane of the panel. By proper orientation of the principal axis, which depends on the angle of incidence, it is shown that transmission efficiencies greater than 95 per cent could be obtained. for both polarizations. It is recommended that the future work on anisotropic panels include experimental verification of the theoretical predictions. A suggested approach for the experimental work would be to employ some low density foam as the base material. Metallic discs of appropriate diameter would be placed on the surface of the material and sandwiched together to form an anisotropic panel. Two or more configurations may be considered, for example, one in which metallic discs could be oriented with the plane of the discs parallel to the interface and for a second case, the discs would be inclined at the other required angles. 56

REFERENCE Tai, C-T, et al, (1965), "Radant Analysis Studies - Interim Report No. 1", The University of Michigan, Radiation Laboratory Technical Report No. 07300-1-T. AD 471813 UNCLASSIFIED Other Reports Written by The Radiation Laboratory Associated with this Contract Tai, C-T and E S. Andrade (1965), "Radant Analysis Studies - Interim Report No. 2,"f The University of Michigan Radiation Laboratory Report 7300-2-T, AD 476 698 UNCLASSIFIED Tai, C-T (1966), "Radant Analysis Studies - Interim Report No. 3, " The University of Michigan Radiation Laboratory Report 7300-3-T, AD 488066 UNCLASSIFIED Tai, C-T (1966), "Radant Analysis Studies - Interim Report No. 4, " The University of Michigan Radiation Laboratory Report 7300-4-T, AD 801269 UNCLASSIFIED Tai, C-T, E. S Andrade and M. A Plonus (1966), "Radant Analysis Studies" AFAL-TR-66-186, The University of Michigan Radiation Laboratory Report No. 7300-1-F, AD 486 763 UNCLASSIFIED Tai, C-T and R M Kalafus (1967), "Radant Analysis Studies", AFAL TR 67-62, The University of Michigan Radiation Laboratory Report 7300-2-F, AD 815541 UNCLASSIFIED 57

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UNCLASSIFIED Security Classificntion DOCUMENT CONTROL DATA R & D (Security cinssilicntlon of title, hody of f nhstrtct trol Ihtcl.xhtl lmilottilln nimt he entered wIhen the ovorall report Ie rln.nitfled) 1. ORIGINATING ACTIVITY (Corporlte nrlthor) 2t. REPORT SECURI T CLASSIIC ATION The University of Michigan Radiation Laboratory, Dept. of UNCLASSIFIED Electrical Engineering, 201 Catherine Street, Zb. GROUP Ann Arbor, Michigan 48108 3. REPORT TITLE ADVANCED RADANT STUDIES - n 4. DESCRIPTIVE NOTES (Type of report and Incluitlve dtelo) FINAL REPORT 1 February - 1 November 1967 5. AUTHOR(S) (First name, middle initial, laat name) Boppana, L.J. Rao 6. REPORT DATE ln. TOTAL NO. OF PAGES f7b. NO. OF REFS December 1967 58 1 8a. CONTRACT OR GRANT NO. 90. ORIGINATOR'S REPORT NUMBER(S) F 33615-67-C-1444 866 8664-1-F b. PROJECT NO. 416102 c- Task No. ih. OTHER REPORT NO(S) (Any other numbeor thot many he assiglned I this report) 4161 d. AFAL-TR-67- 309 10. DISTRIBUTION STATEMENT This document is subject to special export controls and each transmittal to foreign governments or foreign nationals may be made only with prior approval of AFAL (AVWE-3), WrightPatterson AFB, Ohio 45433 11. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY Air Force Avionics Laboratory Research and Technology Division, AFSC Wright-Patterson AFB, Ohio 45433 13. ABSTRACT The transmission characteristics of an anisotropic panel formed by conducting discs are investigated theoretically for different orientation of the principal axis of the medium. The results indicate that the best transmission efficiency is obtained for high incident angles when the principal axis is oriented in such a way that it is perpendicular to the plane of the panel. It is shown that by adjusting the orientation of the principal axis according to the incident angle, better and more uniform transmission could be obtained through radomes of conical structure. DD,FORM 1473 DDUU~,NF~~~oR.4/j ~UNCLASSIFIED...,. St'cutrilV. t Ii* l.i s; i' i. i -)n

UNCLASSIFIED Security Classification KEY WORDS tLINK A LINK 113 LINK C ROLE WT ROLE WT ROLE CWT RADOME ANISOTROPIC PANEL ARTIFICIAL DIELECTRIC TRANSMISSION EFFICIENCY UNCLASSIFIED

UNIVERSITY OF MICHIGAN 3 9015 03525 0623