Guided Wave Foulation of Maxwell's Equations I. Geneal Theoy: Recapitulation -- fequency doain foulation of the acoscopic Maxwel l equations in a souce-fee egion: cul E H cul H ( ) jω µ ( ) [ I-1a ] ( ) + jωε E ( ) [ I-1b ] div ε E div µ H ( ) 0 [ I-1c ] ( ) 0 [ I-1d ] Ipotant Review: Let us once again deive the electoagnetic wave equations (o oe pecisely, the electo agnetic Helholtz equations) by stating with Eq. [ I-1a ]. as we have shown any tie s, the tick is to opeate on this equation with the cul opeato -- viz. cul cul E ( ) jωcul µ H ( ) and then use the "abc bac - cab" ule - i.e. gad [ div E ( )] div gad Finally, we ake use of Eq. [ I-1c ] and obtain whee k ω µε. [ div gad] E Siilaly, stating fo Eq. [ I-1b ] we obtain [ ] jω µ [ + jω ε E ( )] [ I- ] [ ] E ( ) ω µε E ( ) [ I-3 ] ( ) + k E ( ) E ( ) + k E ( ) 0 [ I-4a ] ( ) That is a b c ( ) b a c ( ) c a b
Tansission Line Theoy Page [ div gad] H ( ) + k H (,ω) H ( ) + k H ( ) 0 [ I-4b ] These ae the Helholtz equations which ae the appopiate stating point fo the analysis of any pobles in electoagnetic popagation. We have aleady used the to study pla ne wave popagation in unifo dielectics. Teatent of Guided Waves: While the Helholtz appoach is a valuable stating point in any instances, we take a slig htly diffeent tack in analyzing guided wave pobles. The ipotant point of depatue is that we assue the waves ae popagating in the z-diection along a unifo guided wave s tuctue and, thus, we ay wite in all diffeential opeatos in tes of tansvese and long itudinal coponents by taking { any field vaiable} γ any field vaiable z { } [ I-5a ] so that { any field vaiable } t { any field vaiable } + ˆ z γ any field vaiable { } [ I-5b ] whee, fo exaple and in paticula, t x ˆ x + y ˆ y. [ I-6 ] Futheoe, it is also useful to esolve the fields into tansvese and longitudinal copone nts -- viz. E ( ) H E t ( ) H t ( ) + ˆ ( ) + ˆ z z ( ) [ I-7a ] ( ) [ I-7b ] Thus, we ay ewite Maxwell's equations and esolve the into tansvese and longitudin al coponents -- viz. Eq. [ I-1a ] becoes { + ˆ z γ} t E { t ( ) + z ˆ E z ( )} j ω µ H t (,ω { ) + z ˆ H z ( )} [ I-8a ]
Tansission Line Theoy Page 3 which has tansvese coponents γ z ˆ E t ( ) ˆ z t ( ) jω µ H t ( ) [ I-8b ] and longitudinal coponent t E t ( ) j ω µ ( ) [ I-8c ] and Eq. [ I-1a ] becoes { + ˆ z γ} t which has tansvese coponents H { t ( ) + z ˆ H z ( )} +j ωε E t γ z ˆ H t ( ) ˆ z t ( ) +j ωε ( { ) + ˆ z E z ( )} [ I-9a ] E t ( ) [ I-9b ] and longitudinal coponent t H t ( ) + jω ε ( ) [ I-9c ] Now ultiply the fist tansvese equation -- i.e. Eq. [ I-8b ] -- by jω ε γ z ˆ jωε E t ( ) z t ( ) ω εµ H t ( ) { } jω ε ˆ and substitute fo the second tansvese equation -- i.e. Eq. [ I-9b ] γ z ˆ γ ˆ z H t ( ) ˆ z t H z ( ) z t { } j ωε ˆ Finally, using the "abc bac - cab" ule we find { k +γ } H t ( ) γ t ( ) jω ε ˆ ( ) ω ε µ z t H t ( ) ( ) [ I-10a ] Siilaly, ultiplying the second tansvese equation -- i.e. Eq. [ I-9b ] -- by jω µ ( ) That is a b c ( ) b a c ( ) c a b
Tansission Line Theoy Page 4 γ z ˆ jωε E t ( ) z t ( ) ω εµ H t ( ) { } jω ε ˆ and substituting fo the fist tansvese equation -- i.e. Eq. [ I-8b ] -- we obtain E t ( ) γ t ( ) + j ω µ z ˆ t { k +γ } ( ) [ I-10b ] In suay, these equations say: If we know the longitudinal electic and agnet ic fields, we know eveything these is to know! But how ae we to know the lon gitudinal fields? Answe: The longitudinal fields ust, of couse, obey the Helholtz equ ations, which ay now be witten in the fo t t ( ) + k + γ whee, fo exaple and in paticula, { } { } ( ) + k + γ ( ) 0 [ I-11a ] ( ) 0 [ I-11b ] t x + y 1 ρ ρ ρ + 1 ρ ρ θ. [ I-1 ] II. TEM Waveguide Modes: The aguent above is incoplete in the sense that one ipotant case has been iplicitly o velooked - i.e. the case when both longitudinal fields ae zeo. When does this happen? Answe: when k +γ 0 o γ ±j k [ II-1 ] Theefoe, TEM waves always popagate at the speed of light in the dielectic and satisfy the equations t E t ( ) 0 [ II-a ]
Tansission Line Theoy Page 5 t H t ( ) 0 [ II-b ] An ipotant paticula exaple of such TEM waveguide odes ae those deived fo the logaithic potential - viz. ( ) A ln ( x + a) + ( y + b) φ which includes the TEM odes on coaxial and two-wie lines. [ ] [ II-3 ] III. Conducting Wall Tube Waveguides: Fo the nealy ealizable case of pefectly conducting walls we have the siplest possi ble situation the longitudinal fields satisfy the Helholtz equations -- viz. Eqs [ I-11 ] -- in t he intenal dielectic egion and the following the bounday conditions on the walls: A. Paallel Plate Waveguides wall,ω z ˆ t ( ) 0 [ III-1a ] wall,ω TM-Modes (E-Waves) ae deived fo ( ) 0 [ III-1b ] ( ) A sin π x d exp γ z [ ] [ III-a ] whee k +γ π d and thus γ j β π d k. TE-Modes (H-Waves) ae deived fo n ( ) B n cos nπ x d exp γ z n [ ] [ III-b ] whee k +γ n nπ d and thus γ n j β n nπ d k.
Tansission Line Theoy Page 6 Dispesion Cuves fo a Paallel Plate Waveguide "β ω Diaga" Dispesion of phase velocity Fo Eqs. [ I-10 ], we ay wite the coplete tansvese fields fo a given ode E t H t [ ] 1 γ ( ) k +γ ( ) k +γ which siplify to t [ ] 1 γ t A sin π x d + j ω µ z ˆ t B cos π x d j ωε z ˆ t B cos π x d exp γ z A sin π x d exp γ z [ ] [ III-3a ] [ ] [ III-3a ] E t ( ) d π j β A π x cos d x ˆ jω µ B sin π x d y ˆ exp j β z [ ] [ III-4a ]
Tansission Line Theoy Page 7 H t ( ) B. Rectangula Waveguides d π j β B π x sin d x ˆ jωε A cos π x d y ˆ exp j β z [ ] [ III-4b ] TM-Modes (E-Waves) ae deived fo n ( ) A n sin π x sin nπ y a b exp γ z n [ ] [ III-5a ] whee k +γ π a + nπ b and thus γ n j β n π a + nπ b k. TE-Modes (H-Waves) ae deived fo n ( ) B n cos π x cos nπ y a b exp γ z n [ ] [ III-5b ] whee, again, k +γ π a + nπ b
Tansission Line Theoy Page 8 and γ n j β n π a + nπ b k [ III-6 ] Dispesion Cuves fo a Rectangula Waveguide (Most coon o standad configuation whee a b )