Prblma dl smipian Cas i cs ω µ Cas sin cs i cs

Cas fas [ ] [ ] ] [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ttal S S J L J J L A d I I A d I d I V d d V V d d V n J n J ˆ 0 ˆ ˆ ˆ 0 ˆ 0 ˆ ˆ ˆ 0 S S S S i i Z sin

0 0 0 0 0 0 ω µ V V Z I I A V Z V Z I V Z V Z I c c c c c quain W- mgna in frma nn cannica rnl ω µ V A S 0 0 < < A nn standard 0 < < V S nn standard

0 0 < < 0 0 0 d g d S g S s A A A J J J 0 0 sin 0 0 cs g s g s J Z J g Z d Z A cs sin sin 0 cs d V A ω µ cs sin

d V A ω µ cs sin fas: afattriain d V A ω µ cs sin cs cs sin sin cs cs sin cs sin bdcmpsiin

0 cs sin cs sin sin ω µ w V A d csparain dll piu dll mn cs d Sluin A ω µ V i A V V ω µ sign A I I i

3 fas. Discussin sign d i ω µ i i Z sin sin cs } cs cs { cs cs 4 i F F d Fu: funin di Frsnl

Andamnt vicin l spigl: / A J S / divrg

Dcmpsiin in camp gmtric camp scattrrat s g s g D u u >> cs cs sin sin 4 / 4 / cs cs

ω ε d A V V divrg i i s / 0 } cs cs { cs cs 4 F F

J S / 0 0 pr 0

Dcmpsiin in camp gmtric camp scattrrat s g s g D u u >> cs cs sin sin 4 / 4 / cs cs

Dcmpsiin in camp gmtric camp scattrrat s g s g D u u >> cs cs cs cs 4 / 4 / cs cs

Far fild prsnt in a hmgnus angular rgin Th surc fild is cnstitutd b a plan wav having th fllwing lngitudinal cmpnnts:. i cs } i cs whr b indicating with β and th nithal and aimuthal angl f th dirctin f th plan wav nˆ i it is: ω µ ε cs β sin β. In an arbitrar hmgnus angular rgin s fr ampl fig. th lngitudinal cmpnnts f an lctrmagntic fild ar prssd b th Smmrfld rprsntatin:..3 s [ w ] c c s [ w ] c c cs[ w ] cs[ w ] dw dw

whr C and C cnstituts th Smmrfld cntur and s w and s w Smmrfld functins rlvant t th lngitudinal filds and : W cnsidr nl tim harmnic lctrmagntic filds with a tim dpndnc spcifid t b th factr ω which is mittd. Lt s cnsidr a hmgnus angular rgin. Withut lss f gnralit w intrduc a plar sstm f crdinats whr th dirctin 0 blngs t this rgin. fig. hmgnus angular rgin. In gnral in this rgin thr is th prsnc f all th cmpnnts and fr th ttal fild th factr is mittd. Th Winr-pf tchniqu fr wdg prblms is basd n th intrductin f th fllwing Laplac transfrms:.4 η η V d I η η d 0 0 η η.5 V η d I η d 0 0 Lt s us supps t nw th fllwing Laplac transfrm rlvant t th tangntial cmpnnts f th lctrmagntic fild n th dirctin 0 Lt s us supps t nw th fllwing Laplac transfrm rlvant t th tangntial cmpnnts f th lctrmagntic fild n th dirctin 0.6.7 0 0 η V η d 0 0 V 0 0 η η d 0 0 η I η d 0 0 I 0 0 η η d

B putting.8 η cs w.9 η sin w w hav th fllwing prssins f th Smmrfld functins rlvant t th lngitudinal filds and :.0 cs w s w sin wv cs w0 I cs w0 I cs w0 ω ε ω ε. cs w s w sin wi cs w0 V cs w0 V cs w0 ω µ ω µ Th Smmrfld functins prvid th diffractd filds prsnt in th angular rgin: 4 d. [ s s ] d.3 [ s s ] 4 An ampl: Diffractin b a prfctl cnducting half-plan b an incidnt plan wav Figur : Gmtr f th prblm sw angl β

Fr th sa f simplicit w will cnsidr nl th cas whr th wdg rduc t a half-plan Φ fig. PC half-plan prblm B indicating with.4 t ˆ ˆ t ˆ ˆ lt s intrduc th bilatral Laplac transfrms: V ˆ t d.5 I d t Frm th thr f th stratifid mdia w hav:.6 V 0 Z I 0 c.7 V 0 Z c I 0

O.8 Zc P ωεξ ξ whr: O O ξ ξ O O O P plnmial matri ωε B rprsnting th filds in.4 and.5 in plar crdinats and taing int accunt th dfinitins. and.3 ilds th Winr-pf quatins:.9.0 ξ V 0 I 0 I 0 ωε ωε I I ωε ωε ξ I 0 V 0 V 0 ωµ ωµ ξ I. ξ V 0 I 0 I 0 ωε ωε I I ωε ωε. η ξ I 0 V 0 V 0 ωµ ωµ ξ I Th sstm dcupls in fur indpndnt W- quatins b summing and subtracting

quatins.7 and.9 and b summing and subtracting quatins.8 and.0. It ilds:.3 ξ V 0 I I I I ωε ωε.4 I 0 I 0 ωε ωε I I I I ωε ωε.5 I 0 I I.6 V 0 V 0 I I ξ ωµ ωµ In th fur uncupld W- quatins th plus functins ar V 0 I 0 I 0 I 0 and V 0 V 0 ωε ωε ωµ ωµ and th minus functins ar: I I I I I I and I I W cnsidr as surc th gmtrical ptic cntributin n th half plan. Taing int accunt that th in th cas f fig. th fac is nt illuminatd w hav:.7 whr: g g Z cs sin I ± I I Z g g g I ± I I cs p Th nl rquird factriatin is that f th scalar: p p

.8 ξ Th slutin f th fur W- quatins can b dn immdiatl and it ilds in th w- plan th slutin Φ : cs sin Φ Φ.9 V cs w0 Φsin w cs w cs Φ Φ.30 V cs w0 cs ct wct βsin w Zcs w csc βsin Φ Φ Φ Φ Φ cs w cs Φ Φ sin w Φ.3 I cs w0 Φsin w cs w cs Φ Φ I cs w0.3 Z ct wct β sin w csc βsin Φ Φ ZΦ cs w cs Φ Φ Substituting in.8 and.9 ild th Smmrfld functins:.33 s 55 cs Φ w Φ sin w sin Φ Φ

.34 s cs w Φ w Φ sin w sin Φ Φ Substituting in.0 and. ilds th lngitudinal diffractd filds:.35.36 4 cs cs d cs cs 4 sin sin d cs cs quatins.33 and.34 hld fr arbitrar valu f < < Th diffractd ras cnstituts a gnratri f th Kllr cn and it is dfind b th angular sphrical crdinats β and fig.. Taing int accunt that th incidnt fild has angular sphrical crdinats β and it is cnvnint t rlat th transvrsal d d cmpnnt β f th diffractd ra t th transvrsal cmpnnt i i β f th incidnt ra. B gmtrical cnsidratin w hav : d d d d β Z.37 sin β sin β.38 i sin β i β i Z sin β i Th prvius quatins cannt b usd whn th bsrvatin pint apprachs shadw bundaris f th incidnt and th rflctd wavs.