2076 THE BELL SYSTEM TECHNICAL JOURNAL,SEPTEMBER 1969 Ey OR Ha ↑Ey OR Ha cos kz CONSTANT e-2/传 CONSTANT e-yA cos kyy Ey OR Hz E29 4↑40↓+ h+10 Ey OR Ha (a) E OR Hy Ee OR Hy cos ka CONSTANT e-2/E CONSTANT e-y/ cos kyy E Ex OR Hy E Ez OR Hy (b) ELECTRIC FIELD --◆MAGNETIC FIELD Fig.5-(a)Field configuration of E"modes.(b)Field configuration of E modes. in which ks ka=k2=k (④) and k,=k1=ka=k8… (5) This means that the fields in media 1,2,and 4 have the same x
2076 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 196» COS kj, i Eu OR H j CONSTANT e Ey OR Hj T " b «—a— » . t . . 1 . • t . .1 . . j . .ft (a) CONSTANT e- / OR R «A c O R H b COS kxX . EG O R Hy CONSTANT e-*/< Ee OR Hy CONSJANT e-y/ ' y ^co s ky y / OR H „ - c -12 -22 O R H l ( b ) • ELECTRIC FIELD • MAGNETIC FIELD Fig. 5 — (a) Field configuratíon of modes, (b) Field configuration of E„' modes. in which and (4) fc, = A;,i — Ä.s — fc,e · (5) This means tha t the fields in media 1, 2, and 4 have the same χ
DIELECTRIC WAVEGUIDE 2077 dependence and similarly those in media 1,3,and 5 have identical y dependence.These transverse propagation constants are solutions of the transcendental equations: k,a pr -tan-1 kta -tankts (6) ,b=g红-tan1生k,n-tan, (7) 1 in which (8) 4 (9) and A.8=-4.=20m-.. (10) In the transcendental equations (6)and (7),a and b are the trans- verse dimensions of the guiding rod,and the tan-1 functions are to be taken in the first quadrant. What are the physical meanings of,,and A...?The amplitude of each field component in medium 3(Fig.4)decreases exponentially along z.It decays by 1/e in a distance=1/k l.Similarly,, andn measure the "penetration depths"of the field components in media 5,2,and 4,respectively. The meaning of Aa is the following.Consider a symmetric slab derived from Fig.4 by choosing a oo and na n.The maximum thickness for which the slab supports only the fundamental mode is A,. Expressions(3),(8),and (9)contain k,and k,,which are solutions of the transcendental equations(6)and(7).These cannot be solved exactly in closed form.Nevertheless,for well-guided modes,most of the power travels within medium 1,implying k:As <1 and K1. (11) It is possible then to solve those transcendental equations in closed
DIELECTRIC WAVBGXnD E 2077 dependence and similarly those in media 1, 3, and 5 have identical y dependence. These transverse propagation constants are solutions of the transcendental equations: in which k,a = ρπ - tan ' ^,{ 3 — tan ' k,^t fc,6 = q-ir - tan" -'^Μ ' - | . -tan->^fc., . n, (6) (7) s V2 * « x 3 S 2 'i r - k l 5 k,2 π A, L I 4J - K (8) (9) and A a ,3,4 ,s — (10) (fc ? - fc».3...5)* " 2R^^"<;i¡? ' In the transcendental equations (6) and (7), a and b are the transverse dimensions of the guiding rod, and the tan"* functions are to be taken in the first quadrant. Wha t are the physical meanings of f3 , »íj , and Aj.3.4,i ? The amplitude S 4 of each field component in medium 3 (Fig. 4) decreases exponentially along X. It decays by 1/e in a distance ¿ 3 = 1/ | fc>3 | . Similarly ξ», v» 1 and 174 measure the "penetration depths " of the field components in media 5, 2, and 4, respectively. T h e meaning of A] is the following. Consider a symmetric slab derived from Fig. 4 by choosing a = » and nj = n« . The maximum thickness for which the slab supports only the fundamental mode is A,. Expressions (3), (8), and (9) contain fc, and fc, , which are solutions of the transcendental equations (6) and (7). These cannot be solved exactly in closed form. Nevertheless, for well-guided modes, most of the power travels \vithin medium 1, implying k,A « 1 and k,A « 1. (11) It is possible then to solve those transcendental equations in closed
2078 THE BELL SYSTEM TECHNICAL JOURNAL,BEPTEMBER 1969 though approximate,form.Their solutions are =(+4t4 (12) =答(+贴+4 xn b (13) For large a and b,the electrical width,ka,and the electrical height, k,b,of the guide are close to pa and g,respectively. Substituting equations(12)and (13)in equations(3),(8),and (9), we obtain explicit expressions for k=,s,s,n,and: =[-(+4结4-(gl+4门 xnib (14) + (15) (16) 32 The Ei Modes Except for the fact that the main transverse components are E.and H.,the E modes are qualitatively similar to the E modes (Fig.5b); they differ quantitatively.Distinguishing with bold-face type thesymbols corresponding to E modes,the axial propagation constant and the "penetration depth"in media 2,3,4,and 5 are,according to equations (60),(63),and(64), k,=(好-经-) (17) (18) 南因 (19)
2078 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 196» a \ πα / κ πα ηΐΑ, + rri[b l·nlAλ-' (12) (13) For large a and b, the electrical width, k^a, and the electrical height, kyb, of the guide are close to ρπ and qw, respectively. Substituting equations (12) and (13) inequations (3), (8), and (9), we obtain explicit expressions for fc«, es, is, i?2, and ij4: k. = As + A ira ·Γ-(?)'(-=^Γ] ' (14) A, 9 Τ A, 1 - 1 - ρΑϊ 5 . α ^ _|_ A3 + A5 ira gAj 4 b n|A^jfnjA . 1 Η —:5t; (15) (16) 3Λ TAe .β;. Modes Except for the fact tha t the main transverse components are and H, , the E'„ modes are qualitatively similar to the .EJ, modes (Fig. 5b); they differ quantitatively. Distinguishing with bold-face type the symbols corresponding to E'„ modes, the axial propagation constant and the "penetration depth" in media 2, 3, 4, and 5 are, according to equations (60), (63), and (64), k . = {k\ - - «3 = 5 nj 4 k k . , 4 (17) (18) (19) though approximate, form. Their solutions are
DIELECTRIC WAVEGUIDE 2079 in which k.and k,are solutions of the transcendental equations ka=pm-tan4光k,5-tan1元k6 (20) k,b gx-tan"k,ma -tan"ik,n. (21) The approximate closed form solutions of these equations are k,=(+贴4+4) xnja (22) and -+4者4 (23) Substituting these expressions in equations (17),(18),and (19),we derive the explicit results: -[-(l+44-(gl+44] (24) 在-1-。，+ (25) A 94 61+4+4 1 1- (26) If 《1, these results coincide with those in equations (14),(15),and (16), indicating that the E3 and E modes become degenerate. 8.3 Examples The axial propagation constants k.and k.,given in equations(3) and(17)and properly normalized,have been plotted in Figs.6a through k as a function of the normalized height of the guide 名=尖阳-
DIELECTRIC WAVEGUIDE 2079 in which k. and k, are solutions of the transcendental equations t o = pir - tan" ' ^k . ? s - tan" ' ^k,e . k,6 = ς* — tan" ' k^nj — tan" ' k^n« . The approximate closed form solutions of these equations are and a \ vn'a / (20) (21) (22) (23) Substituting these expressions in equations (17), (18), and (19), we derive the explicit results: (24) γ γ \ ^ (25) k. = A, A^ 1 - 1 - pAs s α J _|_ nlAi + nlAt — πη,α qAi 4 1 6 1 , A, + A, 1-J (26) If n, I 4 5 « 1 , these results coincide with those in equations (14), (15), and (16), indicating tha t the E'^ and E',, modes become degenerate. 3 .3 Examples The axial propagation constants k, and k., given in equations (3) and (17) and properly normalized, have been plotted in Figs. 6a through k as a function of the normalized height of th e guide b _2b ,2 „a^J
2080 THE BELL SYSTEM TECHNICAL JOURNAL,SEPTEMBER 1969 1.2 n4 1.0 E,Ei2 E、En9、 0.8 1.05 <n4<n E2,E2IV e,E22 0.6 w1 E,E1 0.4 E3,Ear 也（起 0.2 E,E29 (a) e,e327 E5,E3 n4 E开，En 1.0 E2i,E2V 105 <n4<n E3,E3. 0.8 信=2 0.8 =n E话，E2 0.4 -E五，e22 0.2 (b) 0 04 08 1.2 1.6 2.0 2.4 2.8 3.2 3.8 4.0 贵=驶(6h-n） Fig.6-Propagation constant for different modes and guides. tran- scendental equation solutions;- 一clo8 ed form solutions;一·-·一Goell's computer solutions of the boundary value problem. for several geometries and surrounding media.*The ordinate in each of these figures is 发=当 it varies between 0 and 1.It is 0 when k.=k,that is,when the guide +In these figures we use the same symbol k,for both the E and the E modes
2080 THE BELL SYSTEM TECHNICAL JOUBNAL, SEPTEMBER 1969 I.Zr Fig. 6 — Propagation constant for different mode s and guides. transcendental equation solutions; closed form solutions; — Goell's compute r solutions of the boundary value problem. for several geometries and surrounding media.* The ordinate in each of these figures is fc^ - kl it varies between 0 and 1. It is 0 when k, = kt, tha t is, when the guide • I n these figures we use the same symbo l k, for both the E,,' and the E„' modes