Waveguide Guide: A and V. Ross L. Spencer


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1 Wveguide Guide: A nd V Ross L. Spencer I relly think tht wveguide fields re esier to understnd using the potentils A nd V thn they re using the electric nd mgnetic fields. Since Griffiths doesn t do it this wy, I will hve to show you wht I men. The strting point is Mxwell s equtions for the potentils, which re just the wve equtions (in the Lorentz guge). Wve Equtions: 2 A 1 c 2 2 A 2 = 0 ; 2 V 1 c 2 2 V 2 = 0 (1) Lorentz Condition: A = 1 c 2 V (2) And, of course, when we need to find E nd B tht s esy too. E nd B: E = V A ; B = A (3) To find out how the fields ehve in wveguide we use not only the wve equtions, ut lso the oundry conditions. Boundry Conditions: E = 0 ; B = 0 (4) Even though the discussion in the ook out wveguides looks complicted, it relly isn t; the electromgnetic fields inside wveguides re just superpositions of wves. They re trveling wves prllel to the xis of the guide nd stnding wves trnsverse to it. In rectngulr guide the fields re just superpositions of fields tht re proportionl to the usul complex wve function e i(kxx+kyy+kzz ωt) (5) nd hence the dispersion reltion is exctly the sme s in infinite spce: 1
2 Dispersion Reltion in Rectngulr Coordintes: It is the sme for trveling, stnding, or comined stnding/trveling (wveguide) wves: ω 2 = (k 2 x + k 2 y + k 2 z)c 2 (6) This dispersion reltion comes from the wve equtions, so now we hve to worry out the oundry conditions. The oundry conditions impose restrictions on the vlues of the components of k trnsverse to the guide, so we need now to specify the geometry we will e working in. Rectngulr Wveguide: The wve guide is infinitely long in the zdirection nd goes from 0 to in the xdirection nd from 0 to in the ydirection. Since the x nd y directions re the trnsverse ones, we use wve functions tht re trveling wves in z nd hve stnding wve forms in x nd y. Wveguide Wveform: A(z, x, y, t) = A(x, y)e i(kzz ωt) ; V (z, x, y, t) = V (x, y)e i(kzz ωt) z = ik z nd = iω where A(x, y) nd V (x, y) will turn out to involve sines nd cosines. (7) Applying the oundry conditions now nd doing the lger is rther complicted, so I will just tell you tht it turns out tht there re two distinct types of solutions. One type hs E z = 0, i.e., the mode electric field only hs components perpendiculr (trnsverse) to the long xis of the wve guide. These re the TE (Trnsverse Electric field only) modes. The second type hs B z = 0, i.e., the mode mgnetic field only hs components perpendiculr (trnsverse) to the long xis of the wve guide. These re the TM (Trnsverse Mgnetic field only) modes. If you work with E nd B, this minor mircle reduces the numer of vector components you need to find from 6 to 5; not relly ig del. But if you work with A nd V, s we re doing here, it reduces the numer from 4 to 2, which is ig help. Once we know tht there is seprtion into two mode types the detils re not hrd to work out. 1 TE Modes TE mn Modes (A z = 0 nd V = 0): Trnsverse Electric mens tht E z = 0, nd since E z = A z V z = iωa z ikv (8) 2
3 n esy wy to get E z = 0 is to tke A z = 0 nd V = 0. This turns out to e exctly right. Hence, we only hve to find A x (x, y) nd A y (x, y). Applying oth the oundry conditions nd the Lorentz guge condition leds to nd with k x = mπ A x = A x0 cos sin ; k y = nπ ; A y = A y0 sin cos m A x0 + A n y0 = 0 (11) This condition reltes the two mplitudes, ut does not determine the overll mgnitude. This is determined y the person who shoots the energy into the wveguide. TE 0n nd TE m0 Specil Cses: If you set either m = 0 or n = 0 in the mplitude reltion ove you will see tht only one of the vector field components survives. m = 0 k x = 0 ; k y = nπ ( ) nπy ; A x = A 0 sin ; A y = 0 (12) n = 0 k x = mπ ( ) mπx ; k y = 0 ; A x = 0 ; A y = A 0 sin (13) (9) (10) 2 TM Modes TM mn Modes (A x = 0 nd A y = 0): Trnsverse Mgnetic mens tht B z = 0, nd since B z = A y x A x y n esy wy to get B z = 0 is to tke A x = 0 nd A y = 0. This turns out to e exctly right. Hence, we only hve to find A z (x, y) nd V (x, y). Applying oth the oundry conditions nd the Lorentz guge condition leds to TM 0n nd TM m0 Specil Cses: (14) k x = mπ ; k y = nπ (15) A z = A 0 sin sin ; V = k zc 2 ω A z (16) Setting m = 0 or n = 0 in the formul for A z mkes the fields vnish, so there re no such modes. This mens tht m 1 nd n 1 for TM modes. 3
4 3 TEM Modes In completely hollow wveguide wves with oth E = 0 nd B = 0 prllel to the xis of the guide re impossile. But with conductor long the xis these wves re possile. Their dispersion reltion is simply ω = kc (17) with k prllel to the xis of the guide. The electric field points outwrd from the centrl conductor nd termintes on the outer surfce of the guide while the mgnetic field circultes round the centrl conductor nd runs prllel to the outer conductor. Hence, they re just out like free spce wves with E, B, nd k mutully perpendiculr. In cylindricl guide ( coxil cle) of rdius these wves re descried in cylindricl coordintes y k = kẑ ; E r = E 0 e i(kz ωt) s ; B θ = E 0 c e i(kz ωt) where E 0 is the electric field mplitude t s = nd where s is the rdil coordinte in cylindricl coordintes. s (18) 4
5 4 Wveguide Prolems 1. (Griffiths Prolem 9.27) Show tht the mode T E 00 cnnot occur in rectngulr wveguide y working with the equtions for A x nd A y given in the Wveguide Guide. (Note: this is rel prolem the wy Griffiths does wveguides, ut here it is trivil.) 2. (Griffiths Prolem 9.30) Using the wve equtions for A nd V, the Lorentz guge condition, nd the oundry conditions, derive the properties of TM modes in rectngulr wveguide, i.e., derive Eqs. (6), (15), nd (16) in the Wveguide Guide. Just follow the procedure we followed in clss for the TE modes. (You my tke s given tht A x = 0 nd A y = 0 nd tht A z nd V re products of sines nd/or cosines of the rguments k x x nd k y y.) In prticulr, show the following: () nd k x = mπ ; k y = nπ ω 2 = (k 2 x + k 2 y + k 2 z)c 2 () A z = A 0 sin sin ; V = k zc 2 ω A z (c) Verify the sttement in the section on TM modes tht TM 0n nd TM m0 modes cnnot exist. (d) Mke some kind of rough 3d sketch of the electric nd mgnetic fields of TM 12 mode in rectngulr wveguide. I think the est wy to do this is to mke sketches of oth E nd B in the xyplne for the x nd y components of the fields, nd lso contour plot of E z in the xyplne. Then mke sideview sketch of the Elines in the xz plne. Mple nd Mtl cn help here. 5
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