A NOTE ON THE EXACT AND APPROXIMATE SURFACE WAVE THEORY by John L. Volakis The Radiation Laboratory Department of Electrical Engineering and Computer Science The University of Michigan Ann Arbor, Ml 48109-2122 July 1987 389492-1-T = RL-2565

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Abstract: The computation of the surface wave field parameters on a grounded and an ungrounded permeable dielectric slab is discussed. A summary of the exact surface wave field theory is first presented. This is then followed by an approximate theory based on sheet and impedance boundary conditions. For the last case explicit expressions are derived for the propagation constant. 3

I. EXACT SURFACE WAVE THEORY 1. Dielectric Slab j i —% II (er,,m) |nl =-2 2 ko =- WmoeO e, = relative permittivity m = relative permeability Figure 1. Geometry of the dielectric slab. (a) Ez case (even mode) Assume the following surface wave fields exist in regions I and II satisfying the wave equation (an ejwt time dependence is assumed and suppressed throughout): Ez= Be cosh (g1y) (1) II -fx J2y Ez =Ae e (2) with 4

2 2 2 2 2 f2 g = - ko er g2=-ko-. (3) From Maxwell's equations we now find that the corresponding tangential magnetic fields are:, 1 ~Ez 4x H = - a = - gBe sinh(gy) /(jwrr) (4) L" 1 ~"z A 4x J2Y / x jw=n w ay 1g2 Ae e (jwo) (5) The propagation constants f, g1 and g2 and the ratio A/B can be found via the application of the boundary conditions requiring continuity of Ez and Hx at y = t. We find that g2t Ae =Bcosh(g1t) (6) and g2t Ang2 e =-gB sinh(g1t). (7) By dividing equations (6) and (7) we further find that P = g1 tanh(g1t)+ mrg2 = 0 (8) which is the characteristic equation for the Ez case even mode. We may now use equation (3) to eliminate gl and g2 so that P is a function of f only. In so doing, we find that 5

P= /-knqer-f tanh( -kne,-f)+r/ - =0 (9) which can be solved numerically to find the propagation constant f. If we write f as f=a +jb, (10) where a and b are real, then a can be identified as the attenuation constant of the surface wave (1). In addition, if we let v denote the phase velocity of the surface wave then b=k ) (11) where c is the speed of light. The computer program [1] in Figure 2 can be used to find the roots of P and thus determining the propagation constant f. (b) Hz case (even mode) We now assume that the following Hz field exists in regions I and II: H= B e cosh(gy) (12) II -fx 92y Hz =Ae e (13) with g1, g2 and f again satisfying equation (3). Using Maxwell's equations we also find that the corresponding tangential E field is given by 6

_ 1 aHz -fx /I E a- =gBe sinh(g1y)/(jwee) (14) x jweoer ay ~ II 1 aH 4x 92Y! Ex -we y =-g2Ae e /we) (15) The determination of the propagation constants can be again accomplished via the application of the boundary conditions requiring continuity of the tangential electric and magnetic fields. We find that g2t Ae =Bcosh(gt) (16) and g2t Aeg2e =-gB sinh(g1t), (18) giving P = gltanh(gt) + erg2 = 0 (19) which is the dual of (8). By using (3), (19) can be written as a function of f only. We note that the computer program given in Figure 2 is still applicable for the solution of (19) by simply interchanging the values of mr and er. (c) Ez case (odd mode) Assume the fields 7

Ez = Be sinh(gly) (20) II -fx <J2y Ez =Ae e (21) with g1, 92 and f again satisfying equation (3). The Hx field is found by Hx = jw- ay = g1B e cosh(g ly) /wn) (22) aE 1 aE A -fx gY/io Hx= — ~ o =2Ae e 2 23 A e = B sinh(glt) (24) A mrg2e = +B glcosh(glt) (25) giving P = g1coth(g1t) - mrg2 = 0 (26) which can be solved numerically to find the propagation constant f in conjunction with (3). (d) Hz case (odd mode) Assume the fields Hz = B e sinh(g1y) (27) 8

H =Ae e2Y (28) with gl, g2 and f satisfying (3). The tangential E fields are given by E =- g 1B e cosh(g1y) /(jweOe) (29) II -fx g2YI Ex=-g2Ae e /jwe). (30) Following the same procedure as before, we obtain the characteristic equation P = g1 coth(glt)- erg2 = 0 (31) which can be used to compute f in conjunction with (3). A graphical solution of equations (9), (19), (26) or (31) is illustrated in Figure 3 [2]. As seen a solution of equations (9) and (19) always exists for t E 0. However, a solution of (26) and (31) can only be possible for larger values of t. Thus, when we refer to surface waves one generally assumes the existance of even modes. 2. Grounded Dielectric Slab Y y / e//Y s /E 9

Figure 4. Geometry of the grounded dielectric slab. (a) Ez case Only the odd mode can be supported by this geometry since they are the only modes satisfying the boundary condition Ez = 0 at y = 0. Thus, the solution given in section 1 (c) is applicable to this case since d = 2t, where d is the thickness of the ungrounded slab. (b) Hz case Only the even mode can be supported in this case since it produces a vanishing tangential E field (Ex) at y = 0. Thus, the solution given in section 1 (b) is applicable here. 10

II. APPROXIMATE SURFACE WAVE THEORY 1. Dielectric Slab If we assume the dielectric slab shown in Figure 1 has a very small thickness d = 2t we can then model it by coincident resistive and conductive sheets associated with a resisitivity R = - (32) kd(er - 1) and a conductivity R*= -— ~- (33) kd(n7- 1) respectively. In the above ZO = 1N/Y is the free space intrinsic impedance. It is know that these sheets can support a surface wave field [3] of the form (C is a constant) EHSW = C exp(-kox cosqe) (34) Hz m depending on whether an Ez or Hz excitation is assumed. The parameters qe,m are found from diffraction theory to be given by 1 e=22RY (35) qe=sin ); he=2R/Zo=2RYo (35) 11

and qm=sin (hm); hm2R*/Y 2R* Z (36) Clearly, the resistive sheet supports a surface wave only with Ez excitation, whereas the conductive sheet suports a surface wave only with Hz excitation. From (34), one easily identifies the propagation constant g = jkocosqe, of the surface waves associated with the resistive and conductive sheets. As before, if we write g = a + jb, then a is the attenuation constant of the surface wave and is easily found from a knowledge of qem. A computer program [10] for evaluating qe given the parameters he and hm is shown in Figure 5. 2. Grounded Dielectric Slab Assuming that the ground plane is coated with a very thin dielectric layer of material, it can then be modeled as an impedance surface. Employing transmission line theory, we find that the surface impedance of this plane is given by 12

Zs=j ZO / tan(ko, t) =Zoh (37) where t is the thickness of the coating and h is the normalized impedance relative to the free space intrinsic impedance ZO. The impedance plane can support a surface wave of the form Hsw = C exp(-j koX ) (38) Z m depending on whether an Ez or Hz excitation is assumed. The parameters qem are found from diffraction theory [3] to be given by qe = sin-1 (1/h) (39) and qm = sin-1 (h). (40) The similarity of (35) - (36) with (39) - (40) should be noted. It should be further noted that a surface wave cannot be supported on an impedance surface for all values of h. The condition that a surface wave is supported by the impedance surface is [5] - Re(q) + gd( lm(q)) sgn(lm(q)) > 0 (41) 13

where gd(x) = cos-1{1/cosh(x)} is the Gudermann function and q can denotes qe or qm as given in (39) - (40). Thus, in the case of Ez excitation it is necesary (but not sufficient) that h be capacitive whereas in the case of Hz excitation it is necessary that h be inductive. We remark that he in (35) is always capacitive and hm in (36) is always inductive. The propagation constant of the surface wave is again given by g= a +jb =j ko cosqe (42) m and from (39) j ko 1 - (1/h) for E case g= (43) j k1 -h2 for Hz case If we write /(Zh) Ez case Rs+jXs case( (44) s s zh Hz case we find that a=Re( g)=l(k1 + X2 -R- i2XsRs) (45) and b = lm(g) = Rek + Xs- Rs- j 2XsRs) (46) 14

The attenuation of the surface wave power per unit length can now be written as L =20 iog(ea) = 8.69a dBtnelar (47) and Figure 6 shows the constant L (loss) contours as a function of Rs and Xs [6]. The definition (44) can also be employed for the parameters he and hm appearing in (35) and (36). In that case (47) will also be applicable for the computation of the surface wave power loss in an ungrounded dielectric slab. 15

REFERENCES 1. J. H. Richmond, "Scattering by Thin Dielectric Strips," The Ohio State University Electro Science Lab., Report 711930-7, August 1983. 2. R. F. Harrington, Time Harmonic Electromagnetic Fields, McGraw-Hill, 1961, pp. 163-168. 3. M. I. Herman and J. L. Volakis, "High Frequency Scattering by a Resistive Strip and Extensions to Conductive and Impedance Strips," Radio Science, May-June 1987, pp. 335-349. 4. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, Appl. Math Series 55, 10th printing, 1972, pp. 80-81. 5. G. D. Maliuzhinets, "Excitation, Reflection and Emission of Surface Waves from a Wedge with Given Face Impedances," Soc. Phy. Dokl, Engl. Transl., 3, pp. 752-755. 6. R. Stratton, personal notes. 16

100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400 2500 2600 2700 2800 2900 3000 3100 3200 3300 3400 3500 3600 3700 3800 3900 4000 4100 4200 4300 4400 4500 4600 4700 4800 4900 5000 C SJRFACE WAVES ClN LOSSY DIELCIRIC SLAB, (XlMM~EX C3T, 33T, EPR, EP2,F, FE, FFEGG, ES COMPFLEX GilG2,G12,cG41,M2, GS1 G2 G!Il,~1 (X)MPLEX MURZvU2 P, PP, SGT DATA EQO,U0/8. 8541 8533677E-12,19.25663706144E-6/ DATA PI,TP/3.14159265359,76,28318530718/ C ft4 = SLAB TEHICKNE~S, ME~JlS C UR, ER= RELATIVE PERMEABILT AND PERMI'ITIVITY OF SLAB. C TDE,Tfl4 = EECIIC AND M4AGNETIC LOSS TANGENTS, C EGG, FTM = FREQUENCY IN GIGAHERTLZ, MEGAHERTZ. C Ml = SLAB TIHICKNESS / SKIN DEPT~H, C NE = NUM'BER OF NEWMflN-RAPHSCN ITERATIONS, C I5EVCE = ATTEN ODNST AND PHASE VEL FOR SURF WAVE WIL PERP POL. 2 FOR1fwT (1.XI5r8Fl2.5) 5 FORMAT (IHiO) DM4=.025 TIM=DMv/20 ER=4.0 UR=1. NAX=20 FMC=-300. FGC=FMC/1000. WAVO=300./FMC DL=Et4/WAVO BETO=PAqAV0 OMEG —TP*FMC*1. E6 EPR=-ER*CMPLX (1.,-TDE) EP2=EO*EPR MlUR —UR'*aMPLX (1.-Tf14) MUJ241JR*tUO GS1=D.-OMElG*0MElG*U0*E0 GS2=-OMBG*OMEG*MrJ2*EP2 GM1 =GaPLX(. 0 iBETIO) GM2=CSQRr (GS2) ALP2=REAL (cGM2) DEL=.0 DD=.0 IF (ALP2,.LE..o0) GO MI 12 DEL=1.,/ALP2 DIDEl4/DEL 12 GDNTINUE BET2 =AIMAG (GMb2) TK —*IP*7I/WAVO Gl1= (1AUR*E2PR-1.) *TK*TK/ (MUR*'1T4) FF=-GM* (1.-.5*G1*G1/GS1L) F=FF C D0 60 N=1,NAX Figure 2. Computer program for surface wave parameters.

5100 20 FS=F*F 5200 Gl=CSQRT(GS1-FS) 5300 G2=CSQRT(GS2-FS) 5400 G12=G1*G2 5500 Al12=CABS (G1 2) 5600 IF(A12.'LE..0O)CX MI 100 5700 GT2=G2*ETM 5800 BGT~D-CEXP(OT2) 5900 ()3~D=(EGT~lW1/EGT) /2. 6000 SGTL'(EGT'-1.o/EGT) /2.0 6100 G -G T 6200PG2STCTMRl 6300 40 FGG=-F/(G12*03T) 6400 PP=-BYGG* (MU*2*CG14*SGG*GTi2/cxT) 6500 ALP=REAL(F) 6600 BE' 1 —AMAG (F 6700 VC=-BE'/BET 6800 AP-CABS (P) 6900 -WRITE(6,2)NALPVCAP 7000 F=F-P/PP 7100 FE=F 7200 NE=N 7300 IF(N.,LT.3)GO ~ID 60 7400 APP=CtABS (P/ (F*PP)) 7500 IF(APP.LT..0001)GO MJ 62 7600 60 CONTINJE 7700 62 CONTINUE 7800 WRTE (6,5) 7900 ALPE=REAL(FE) 8000 VCGE=BE2IO/AJt'G (FE) 8100 DBE=-8.686*ALPE 8200 WRITE(6,2)NEtFt4crrL~DrME,VCE 8300 100 CONTINUE 8400 CALL EXIT 8500 END Figure 2. (continued)

1 COMPLEX FUNCTION HEE(ETA,IUD,SB0) 2 ClII NEW VOLAKIS VERSION 3 COMPLEX ETA,ETA1,CJ 4 DATA SRT2,FPI,CJ/1.414213562,12.56637061,(0.,1.)/ 5 DATA PSIPI2,PI/.9656228,3.14159265/ 6 ETA1=SB0/ETA 7 IF(IUD.EQ. 1)ETA1=SB0*ETA 8 RE=REAL(ETA1) 9 AE=AIMAG(ETA1) 10 REP=RE+1. 11 REM=RE-1. 12 AA=.5* (SQRT (REP*REP+AE*AE)+SQRT (REM*REM+AE*AE)) 13 BB=.5* (SQRT (REP*REP+AE*AE) -SQRT (REM*REM+AE*AE)) 14 SGN=AE/ABS (AE) 15 RAA=AA*AA-1. 16 IF (RAA. LT. 1. E-6) RAA=0. 17 HEE=ARSIN(BB)+CJ*ALOG(AA+SQRT(RAA))*SGN 18 C HEE=.5*PI-HEE 19 GO TO 300 20 ETAM=CABS(ETA1) 21 ETAA=ATAN(AIMAG(ETA1)/REAL(ETA1)) 22 ETAM2=ETAM*ETAM 23 SA=SIN(ETAA) 24 CA=COS(ETAA) 25 F1=ETAM2-1.+SQRT((ETAM2-1.)**2+4.*ETAM2*SA*SA) 26 Fl=F1/(2.*ETAM2) 27 IF(Fl.LT.0.)Fl=0. 28 HEER=ASIN(SQRT(F1)) 29 SHEER=SIN (HEER) 30 CHEER=COS (HEER) 31 IF(CABS(ETA1).GT.1.)GO TO 100 32 HEEI=CA/(ETAM*CHEER) 33 HEEI=ALOG (HEEI+SQRT (ABS (HEEI*HEEI-1.))) 34 IF(ETAA.LT.0.)HEEI=-HEEI 35 GO TO 200 36 100 HEEI=SA/(ETAM*SHEER) 37 HEEI=ALOG(HEEI+SQRT(HEEI*HEEI+1.)) 38 200 HEE=CMPLX(HEER,HEEI) 39 300 RETURN 40 END Figure 5. Computer program for the computation of O = sin'1 (1/r) or o = sin1 (Tn).

1 t tan g t - g t cot(g t) -. - - -/ Ic I I. — 4- -1 -- - - IX *t 2, / I, I..=j I \, -- /T 3 T/2 2T I,. - Figure 3. Graphical solution of the characteristic equation for the slab waveguide.

.5. 1 2 \ \ \ \ CONSTANT \ \ \ \ \ \ \ ATTENUATION \ \ \ \ X \ \ \CONTOURS,3 -Xs Figure 6. Contours of constant surface wave loss and velocity for an impedance plane. // ' ~ ~~WAVE 0.4.5.6 7..7.3 Rs Figure 6. Contours of constant surface wave loss and velocity for an impedance plane.