LECTURE 12: Aperture Antennas Part I Introduction 1. Uniqueness theorem

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1 LECTURE 1: Apetue Antennas Pat I (The uniqueness theem. The equivalence pinciple. The applicatin f the equivalence pinciple t apetue pblem. The unifm ectangula apetue. The tapeed ectangula apetue.) Intductin Apetue antennas cnstitute a lage class f antennas, which emit electmagnetic wave thugh an pening ( apetue). These antennas have clse analgs in acustics: the megaphne and the paablic micphne. The pupil f the human ee is a tpical apetue eceive f ptical EM adiatin. At adi and micwave fequencies, hns, waveguide apetues and eflects ae eamples f apetue antennas. Apetue antennas ae f cmmn use at UHF and abve. It is because apetue antennas have thei gain incease as f. F an apetue antenna t be efficient and have high diectivit, it has t have an aea cmpaable lage than λ. Obviusl, these antennas wuld be impactical at lw fequencies. Anthe psitive featue f the apetue antennas is thei nea-eal valued input impedance and gemet cmpatibilit with waveguide feeds. T facilitate the analsis f these antennas, the equivalence pinciple is applied. This allws us t ca ut the fa-field analsis in the ute (unbunded) egin nl, which is etenal t the adiating apetue and the antenna. This equies the knwledge f the tangential field cmpnents at the apetue, as it fllws fm the equivalence pinciple. 1. Uniqueness theem A slutin is said t be unique if it is the nl ne pssible amng a given class f slutins. The EM field in a given egin V [ ] is uniquel defined if - all suces ae given; - eithe the tangential E τ cmpnents the tangential Hτ cmpnents ae specified at the bunda. 1

2 The uniqueness theem is pven b making use f the Pnting s theem in integal fm: * E H ds + jω µ H ε E dv + σ E dv = ( ) ( ) V V E J + H M dv V [ ] i* * i ( ) [ ] [ ] (1.1) Pnting s theem states the cnsevatin f eneg law in EM sstems. One stats with the suppsitin that a given EM pblem has tw slutins (due t the same suces and the same bunda cnditins): ( a a E, H ) and ( b b E, H ). The diffeence field is then fmed: a b δ E = E E (1.) a b δ H = H H ince the diffeence field has n suces, it will satisf the suce-fee fm f (1.1): * ( δe δh ) ds + jω ( µ δh ε δe ) dv+ σ δe dv= 0 (1.3) V[ ] V[ ] ince bth fields satisf the same bunda cnditins at,thenδ E = 0 and δ H = 0 ve. This leaves us with jω µ δh ε δe dv+ σ δe dv= 0, (1.4) which is tue nl if V ( ) [ ] [ ] ω µ δh ε δe dv= 0 V V [ ] [ ] ( ) E dv= 0 V (1.5) σ δ If we assume sme dissipatin, hweve slight, equatins (1.5) ae satisfied nl if δe = δh = 0 evewhee in the vlume V [ ].This implies the uniqueness f the slutin. If σ = 0, which is a phsical impssibilit, but is ften used appimatin, multiple slutins ( δe, δh) ma eist in the fm f self-esnant mdes f the stuctue

3 unde cnsideatin. In pen pblems, esnance is impssible in the whle egin. Ntice that the uniqueness theem hlds if eithe δ E = 0 δ H = 0 is tue n an pat f the bunda.. Equivalence pinciples The equivalence pinciple fllws fm the uniqueness theem. It allws us t build simple t slve pblems. As lng as the equivalent pblem peseves the bunda cnditins f the iginal pblem f the field at, it is ging t pduce the nl ne pssible slutin f the egin utside V [ ]. ( E, H ) ( E, H ) V[ ] ( E, H) suces V[ ] J s e ( Ee, He) n suces M s e ˆn ˆn J ˆ s = n H e He (1.6) M = E E n e ( ) ( ) ˆ s e J ˆ s = n H (1.7) M = E nˆ s (a) Oiginal pblem The ze-field fmulatin is ften efeed t as Lve s equivalence pinciple. (b) Geneal equivalent pblem ( E, H ) V[ ] n fields n suces ˆn (c) Equivalent pblem with ze fields J s M s 3

4 One can appl Lve s equivalence pinciple in thee diffeent was: (a) One can assume that the bunda is a pefect cnduct. This eliminates the suface electic cuents, i.e. J s = 0, and leaves just suface magnetic cuents M s, which adiate in the pesence f a pefect electic suface. (b) One can assume that the bunda is a pefect magnetic cnduct. This eliminates the suface magnetic cuents, i.e. M s = 0, and leaves just suface electic cuents J s,which adiate in the pesence f a pefect magnetic suface. (c) Make n assumptins abut the mateials inside, and define bth J s and M s cuents, which ae adiating in fee space (n fictitius cnducts behind them). It can be shwn that these equivalent cuents ceate ze fields inside V [ ]. All thee appaches lead t the same field slutin accding t the uniqueness theem. The fist tw appaches ae nt ve useful in the geneal case f cuvilinea bunda suface. Hweve, in the case f flat infinite planes (walls), the image the can be used t educe the pblem t an pen ne. Image the can be successfull applied t cuved sufaces pvided the cuvatue s adius is lage cmpaed t the wavelength. Hee is hw ne can implement Lve s equivalence pinciples in cnjunctin with image the. ( E, H ) ( E, H ) n fields ( E, H ) n fields ( E, H ) V[ ] ˆn ˆn ˆn suces n suces M s 0 J s = n suces s sm J = 0 (a) Oiginal pblem (b) Equivalent pblem - electic wall (c) Equivalent pblem -images 4

5 ( E, H ) ( E, H ) n fields ( E, H ) n fields ( E, H ) V[ ] ˆn ˆn ˆn suces n suces J s 0 M s = n suces J s J s = 0 (a) Oiginal pblem (b) Equivalent pblem - magnetic wall (c) Equivalent pblem -images The abve appach is used t evaluate fields in half-space as ecited b apetues. The field behind is assumed knwn. This is enugh t define equivalent suface cuents. Using image the, the pen-egin fa-zne slutins f the vect ptentials, A (esulting fm J )and F (esulting fm M s ), ae fund fm: jβ e jβˆ AP ( ) = µ Js( ) e ds 4 π (1.8) jβ e jβ ˆ FP ( ) = ε Ms( e ) ds 4π (1.9) Hee, ˆ dentes the unit vect pinting fm the igin f the cdinate sstem t the pint f bsevatin P. The integatin pint Q is specified thugh the adius-vect. In the fa zne, it is assumed that the field ppagates adiall awa fm the antenna. It is cnvenient t intduce the s-called ppagatin vect: β = βˆ, (1.10) which chaacteizes bth the phase cnstant and the diectin f ppagatin f the wave. The vect ptentials can then be witten as: jβ e jβ AP ( ) = µ Js( ) e ds 4 π (1.11) s 5

6 jβ e jβ FP ( ) = ε Ms( e ) ds 4π (1.1) The elatins between the fa-zne fields and the vect ptentials ae athe simple. fa E ( ˆ ˆ A = jω Aθθ + Aϕϕ ) (1.13) fa H ( ˆ ˆ F = jω Fθθ + Fϕϕ ) (1.14) ince fa fa EF = ηhf, (1.15) the ttal fa-zne electic field is fund as: fa fa fa E = E ( ) ˆ ( ) ˆ A + EF = jω Aθ ηfϕ θ + Aϕ + ηfθ ϕ (1.16) Equatin (1.16) invlves bth vect ptentials as aising fm bth tpes f suface cuents. Cmputatins ae educed in half if image the is used in cnjunctin with an electic magnetic wall assumptin. 3. Applicatin f the equivalence pinciple t apetue pblems The equivalence pinciple is widel used in the analsis f apetue antennas. T calculate eactl the fa fields, the eact field distibutin at the apetue is needed. In the case f eact knwledge f the apetue field distibutin, all thee appaches given abve will pduce the same esults. Hweve, such eact knwledge f the apetue field distibutin is usuall impssible, and cetain appimatins ae used. Then, the thee equivalence-pinciple appaches pduce slightl diffeent esults, the cnsistenc being dependent n hw accuate u knwledge abut the apetue field is. Usuall, it is assumed that the field is t be detemined in half-space, leaving the feed and the antenna behind a infinite wall (electic magnetic). The apetue f the antenna A is this ptin f whee we have an appimate knwledge f the field distibutin based n the tpe f the feed line the incident wave illuminating the apetue. This is the s-called phsical ptics appimatin, which cetainl is me accuate than the gemetical ptics appach f a tacing. The lage the apetue (as cmpaed t 6

7 the wavelength), the me accuate the appimatin based n the incident wave. Let us assume that the fields at the apetue ae knwn: Ea, Ha,and the ae ze evewhee else at. The equivalent cuent densities ae: J ˆ s = n Ha (1.17) M ˆ s = Ea n Using (1.17) in (1.11) and (1.1) pduces: jβ e jβ AP ( ) = µ nˆ He a ds 4 π (1.18) jβ e jβ FP ( ) = ε nˆ Ee a ds 4π (1.19) The adiatin integals in (1.18) and (1.19) will be dented shtl as: H jβ I He ds (1.0) = a E jβ = Ee a ds I (1.1) One can find geneal vect epessin f the fa-field E vect making use f equatin (1.16) witten as: fa E = jωa jωηf ˆ, (1.) whee the lngitudinal A cmpnent is t be neglected. ubstituting (1.18) and (1.19) ields: jβ fa e jβ E = jβ ˆ nˆ E ˆ ( ˆ a η n Ha) e ds 4π (1.3) A This is the full vect fm f the adiated field esulting fm the apetue field, and it is efeed t as the vect diffactin integal ( vect Kichhff integal). 7

8 We shall nw cnside a pactical case f a flat apetue ling in the plane with nˆ zˆ. Then: jβ e H H A= µ ( I ˆ ˆ + I ) 4 π (1.4) jβ e E E F = ε ( I ˆ ˆ + I ) 4π (1.5) The integals in the abve epessins can be eplicitl witten f this case in which = ˆ + ˆ : E I E jβ( sin cs sin sin ) (, ) e θ ϕ+ θ ϕ d d (1.6) I I I = a A E jβ( sin cs sin sin ) (, ) θ ϕ+ θ ϕ = a A H jβ( sin cs sin sin ) (, ) θ ϕ+ θ ϕ = a A H jβ( sin cs sin sin ) a (, ) θ ϕ+ θ ϕ = A E e dd H e d d H e dd Nte that the abve integals ae eactl the duble invese Fuie tansfms f the apetue field s cmpnents. The vect ptentials in spheical tems ae: (1.7) (1.8) (1.9) jβ e ˆ H H H H A = µ θcsθ( sin cs ) ˆ ϕ ϕ + ϕ( csϕ+ sinϕ) 4π I I I I (1.30) jβ e ˆ E E E E F = ε θcsθ( sin cs ) ˆ ϕ ϕ + ϕ( csϕ + sinϕ) 4π I I I I (1.31) B substituting the abve epessins in (1.16), ne btains the fa E field cmpnents as: jβ e E E Eθ = jβ [ I csϕ + I sinϕ + 4π (1.3) H H ηcs θ( I csϕ I sin ϕ)] jβ e H H Eϕ = jβ [-η ( I csϕ + I sin ϕ) + 4π (1.33) E E cs θ( I csϕ I sin ϕ)] 8

9 F apetues munted n a cnducting plane, the pefeed equivalent mdel is the ne with electic wall with magnetic cuent densit M ( ˆ s = Ea n) (1.34) adiating in pen space. The slutin, f cuse, is valid nl f z 0. In this case, I H = 0. F apetues in pen space, the dual cuent fmulatin is used. Then, a usual assumptin is that the apetue fields ae elated as in the TEM-wave case: 1 Ha = z ˆ Ea (1.35) η This implies that E 1 E H ˆ E H I, H I I = z I I = I = (1.36) η η η This assumptin is valid f mdeate and high-gain apetues; theefe, the apetues shuld be at least a cuple f wavelengths in etent. The abve assumptins educe (1.3)-(1.33) t: jβ e ( 1+ csθ ) E E Eθ = jβη csϕ + sinϕ 4π I I (1.37) jβ e ( 1+ csθ ) E E Eϕ = jβη csϕ sinϕ 4π I I (1.38) 9

10 4. The unifm ectangula apetue n an infinite gund plane A ectangula apetue is defined in the plane as shwn belw. L L If the fields ae unifm in amplitude and phase acss the apetue, it is efeed t as a unifm ectangula apetue. Let us assume that the apetue field is -plaized. L L Ea = E0 ˆ, and (1.39) Accding t the equivalence pinciple, we assume an electic wall at z = 0, whee the equivalent magnetic cuent densit is given b M ˆ s e = E n. Appling image the, ne can find the equivalent suces adiating in pen space as: M = M = E ˆ zˆ = E ˆ (1.40) s s e

11 The nl nn-ze adiatin integal is: I L / L / E jβ sinθcsϕ jβ sinθsinϕ 0 L / L / = E e d e d = β L β L sin sin cs sin sinθ sinϕ θ ϕ = ELL 0 βl βl sinθcsϕ sinθ sinϕ (1.41) It is apppiate t intduce the patten vaiables: β L u = sinθ csϕ (1.4) β L v = sinθ sinϕ The cmplete adiatin fields ae fund b substituting (1.41) in (1.3) and (1.33): jβ e sinu sin v Eθ = jβ E0LLsinϕ π u v (1.43) jβ e sin u sin v Eϕ = jβ E0LLcsθcsϕ π u v The ttal-field amplitude patten is, theefe: sinu sin v E = F( θϕ, ) = sin ϕ+ cs θcs ϕ = u v (1.44) sinu sin v = 1 sin θcs ϕ u v 11

12 The pincipal plane pattens ae: E-plane patten ( ϕ = π /) β L sin sinθ Eθ = β L sinθ H-plane patten ( ϕ = 0) β L sin sinθ Eϕ = csθ β L sinθ (1.45) (1.46) Pinciple pattens f apetue f size: L = 3λ, L = λ 0 E-plane H-plane

13 F electicall lage apetues, the main beam is naw and the 1 sin θ cs ϕ in (1.44) is negligible, i.e. it is ughl equal t 1 f all bsevatin angles within the main beam. That is wh, in the the f lage aas, it is assumed that the amplitude patten f a ectangula apetue is: sinu sin v f( u, v) u v (1.47) β L whee β L u = sinθ csϕ and v = sinθ sinϕ. Hee is a view f the sin u/ u functin f L = 0λ and ϕ = 0 (Hplane patten): 1 sin[0*pi*sin(theta)]/[0*pi*sin(theta)] sin(theta) 13

14 Hee is a view f the sin v/ v functin f L = 10λ and ϕ = 90 (Eplane patten): 1 sin(10 π sin(theta))/(10 π sin(theta)) sin(theta) Beamwidths (a) fist-null beamwidth One needs the lcatin f the fist nulls in the patten in de t calculate the FNBW. The nulls f the E-plane patten ae detemined fm (1.45) as: β L sin θ/ θ= θ = nπ, n= 1,, (1.48) n nλ θn = acsin, ad (1.49) L The fist null ccus at n = 1. 14

15 λ FNBWE = θ n = acsin, ad (1.50) L In a simila fashin, FNBWH is detemined t be: λ FNBWH = acsin, ad (1.51) L (b) half-pwe beamwidth The half-pwe pint in the E-plane ccus when β L sin sinθ 1 = (1.5) β L sinθ β L sinθ = = (1.53) / θ θ h 0.443λ θh = acsin, ad L (1.54) 0.443λ HPBWE = acsin L (1.55) A fist-de appimatin is pssible f ve small aguments in (1.55), i.e. when L 0.443λ (lage apetue): λ HPBWE (1.56) L The half-pwe beamwidth in the H-plane is analgus: 0.443λ HPBWH = acsin (1.57) L 15

16 ide-lbe level It is bvius fm the ppeties f the sin / functin that the fist side lbe has the lagest maimum f all side lbes, and it is: sin E θ ( θ = θ s ) = = 0.17 = 13.6, db (1.58) When evaluating side-lbe levels and beamwidths in the H-plane, ne has t include the csθ fact, t. The smalle the apetue, the less imptant this fact is. Diectivit In a geneal appach t the calculatin f the diectivit, the ttal adiated pwe Π has t be calculated fist using the fa-field patten epessin (1.44). 4π Uma D0 = = 4π (1.59) ΩA Πad Hee, 1 U( θ, ϕ) = Eθ Eϕ Uma F( θ, ϕ) η + = (1.60) ππ 0 0 Ω A = F( θ, ϕ) sinθdθdϕ (1.61) Hweve, in the case f an apetue illuminated b a TEM wave, ne can use a simple appach. Geneall, f all apetue antennas, the assumptin f a unifm TEM wave at the apetue ( E = E ˆ 0), E0 Ha = ˆ, (1.6) η is quite accuate (althugh η is nt necessail the intinsic impedance f vacuum). The fa-field cmpnents in this case wee alead deived in (1.37) and (1.38). The lead t the fllwing epessin f the adiatin intensit: β E E U ( θϕ, ) = (1+ cs θ) + 3πη J J (1.63) 16

17 The maimum value f the functin in (1.63) is easil deived afte substituting the adiatin integals fm (1.6) and (1.7): β Uma = E ads 8πη (1.64) A The integatin f the adiatin intensit (1.63) ve a clsed sphee is in geneal nt eas. It can be avided b bseving that the ttal pwe eaching the fa zne must have passed thugh the apetue in the fist place. In the geneal apetue case, this pwe is detemined as: 1 Π ad = Pav ds = Ea ds η (1.65) A ubstituting (1.64) and (1.65) in (1.59) finall ields: D a 4π A 0 = λ Ea A Eds (1.66) ds In the case f a unifm ectangula apetue, E0 Π= LL (1.67) η E0 LL Uma = (1.68) λ η Thus, the diectivit is fund t be: Uma D0 = 4π = π L L = π A π p = A eff (1.69) Π λ λ λ The phsical and the effective aeas f a unifm apetue ae equal. 17

18 5. The unifm ectangula apetue in pen space Nw, we shall eamine the same apetue when it is nt munted n a gund plane. The field distibutin is the same as in (1.39) but nw the H field must be defined, t, in de t appl the geneal fm f the equivalence pinciple with bth tpes f suface cuents. Ea = E ˆ 0 L/ L/ E, ˆ 0 (1.70) H L/ L/ a = η Abve, again an assumptin was made that thee is a diect elatin between the electic and the magnetic field cmpnents. T fm the equivalent pblem, an infinite suface is chsen again t etend in the z = 0 plane. Ove the entie suface, the equivalent J s and M s must be defined. Bth J s and M s ae nt ze utside the apetue in the z = 0 plane because the field is nt ze thee. Meve, the field is nt knwn a pii utside the apetue. Thus, the eact equivalent pblem cannt be built in pactice (at least, nt b making use f the infinite plane mdel). The usual assumptin made is that E a and H a ae ze utside the apetue in the z = 0 plane, and, theefe, s ae the equivalent cuents J s and M s : Ms = nˆ Ea = ˆ ˆ z E 0 ˆ L/ L/ E, (1.71) ˆ ˆ 0 J ˆ ( ) L/ L/ s = n Ha = z η ˆ ince the equivalent cuents ae elated via the TEM-wave assumptin, E nl the integal J is needed f substitutin in the fa field epessins deived in (1.37) and (1.38). 18

19 I L / L / E jβ sinθcsϕ jβ sinθsinϕ 0 L / L / = E e d e d = β L β L (1.7) sin sin cs sin sinθ sinϕ θ ϕ = ELL 0 βl βl sinθcsϕ sinθ sinϕ Nw, the fa-field cmpnents ae btained b substituting in (1.37) and (1.38): ( 1+ csθ ) sin u sin v Eθ = Csinϕ u v (1.73) ( 1+ csθ ) sin u sin v Eϕ = Ccsϕ u v whee: jβ e C = jβ LLE 0 ; π β L u = sinθ csϕ ; β L v = sinθ sinϕ. The fa-field epessins in (1.73) wuld be identical t thse f the apetue munted n a gund plane if csθ wee eplaced b 1. Thus, f small values f θ, the pattens f bth apetues ae pacticall identical. An eact analtical evaluatin f the diectivit is difficult. Hweve, accding t the appimatins made, the diectivit fmula deived in (1.66) shuld pvide accuate enugh value. Accding t (1.66), the diectivit is the same as in the case f the apetue munted n a gund plane. 19

20 6. The tapeed ectangula apetue n a gund plane The unifm ectangula apetue has the maimum pssible effective aea (f an apetue-tpe antenna) equal t its phsical aea. This als implies that it has the highest pssible diectivit f all cnstant-phase ecitatins f a ectangula apetue. Hweve, diectivit is nt the nl imptant fact in the design f an antenna. A fact that fequentl cmes int a cnflict with the diectivit is the side-lbe level (LL). The unifm distibutin ecitatin pduces the highest LL f all cnstant-phase ecitatins f a ectangula apetue. It will be shwn that the eductin f LL can be achieved b tapeing the equivalent suces distibutin fm a maimum at the apetue s cente t ze values at its edges. One ve pactical apetue f tapeed suce distibutin is the pen ectangula waveguide. The dminant (TE 10 ) mde has the fllwing distibutin: π L/ L/ Ea = E ˆ 0 cs, L L/ L/ (1.74) E The geneal pcedue f the fa-field analsis is the same as befe (in ectin 4). The nl diffeence is in the field distibutin. Again, nl the integal J is t be evaluated. E 0

21 L / L / E π j sin cs j sin sin = β θ ϕ β θ ϕ 0 cs L L / L / I E e d e d (1.75) The integal f the vaiable was alead encunteed in (1.41): β L L / sin sin sin θ ϕ j sin sin β θ ϕ I( ) = e d L = (1.76) β L L / sinθsinϕ The integal f the vaiable is easil slved: L / π j sin cs I( ) = β θ ϕ cs e d = L L / L / π = cs cs( sin cs ) jsin ( sin cs ) d L β θ ϕ + β θ ϕ = L / L / 1 π π = cs βsinθcsϕ cs βsinθ csϕ d L L / L L / j π π + sin βsinθcsϕ cs βsinθcsϕ d + + L L / L β L cs sinθ csϕ π L I( ) = π βl sinθ csϕ β L β L cs sinθcsϕ sin sinθ sinϕ E J = π ELL 0 (1.77) β L π βl sinθ sinϕ sinθcsϕ v u T deive the fa-field cmpnents, (1.77) is substituted in (1.3) and (1.33). 1

22 π csu sin v Eθ = Csinϕ π v u π csu Eϕ = Ccsθcsϕ π u whee: jβ e C = jβ LLE 0 ; π β L u = sinθ csϕ ; β L v = sinθ sinϕ. sin v v (1.78) Pinciple plane pattens In the E-plane, the apetue is nt tapeed. As epected, the E-plane pincipal patten is the same as that f a unifm apetue. E-plane ( ϕ = 90 ): H-plane ( ϕ = 0 ): E ϕ E θ β L sin sinθ = β L sinθ β L cs sinθ = csθ β L π sinθ (1.79) (1.80)

23 H-plane patten unifm vs. tapeed illuminatin ( L = 3λ ): unifm tapeed The lwe LL f the tapeed-suce fa field is bvius. It is bette seen in the ectangula plt given belw. The pice t pa f the lwe LL is the decease in diectivit (the beamwidth f the maj lbe inceases). 3

24 1 tapeed unifm H-plane amplitude patten sin(theta) The abve eample f L = 3λ is illustative n the effect f suce distibutin n the fa-field patten. Hweve, a me pactical eample is the ectangula-waveguide pen-end apetue, whee the waveguide peates in a dminant mde, i.e. λ0 /< L < λ0. Hee, λ0 is the wavelength in pen space ( λ 0 = c/ f 0 ). Cnside the case L = 0.75λ. The pincipal-plane pattens f an apetue n a gund plane lk like this: 4

25 30 0 H-plane 30 E-plane In the abve eample, a pactical X-band waveguide was cnsideed whse css-sectin has the fllwing sizes: L =.86 cm, L = cm. Obviusl, λ 0 = cm, and f 0 = 9.84 GHz. The case f a dminant-mde pen-end waveguide adiating in fee space can be analzed fllwing the appaches utlined in this ectin andinectin5. The calculatin f beamwidths and diectivit is analgus t pevius cases. Onl the final esults will be given hee f the case f the -tapeed apetue n a gund plane. 8 4π Diectivit: D0 = L L (1.81) π λ 8 Effective aea: Aeff = L 0.81 L = Ap (1.8) π Nte the decease in the effective aea. 5

26 Half-pwe beamwidths: 50.6 HPBWE =,deg.(= HPBWE f the unifm apetue) (1.83) L / λ 68.8 HPBWH =,deg.(> HPBWH f the unifm apetue) (1.84) L / λ The abve esults ae appimate. Bette esults wuld be btained if the fllwing facts wee taken int accunt: the phase cnstant f the waveguide β is nt equal t the feespace phase cnstant β ω µ ε = ;itisdispesive; the abupt teminatin at the waveguide pen end intduces eflectin, which affects the field at the apetue; thee ae stng finge cuents at the waveguide walls, which cntibute t the veall adiatin. g 6