Polarization Basics E. Polarization Basics The equations

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1 Plazan Ba he equan [ ω δ ] [ ω δ ] eeen a a f lane wave: he w mnen f he eleal feld f an wave agang n he z den, n neeal mnhma. he amlude, and hae δ, fluuae lwl wh ee he ad llan f he ae ω. z Plazan Ba [ ω δ ] [ ω δ ] z If we elmnae he em ω beween he w equan, and ne δ δ - δ, we fnd he lazan elle vald n geneal a a gven me, whh he lu f n debed b he al feld a agae: fa fa fa δ n δ lw lw lw lw lw Plazan Ba F uel mnhma wave, amlude and hae mu be nan wh me: [ ω δ] [ ω δ] z And he lazan elle al nan: fa fa fa δ n δ Plazan Ba In geneal a beam f lgh ellall lazed. he lazan elle degeneae eal fm f eal value f he amlude and f he hae. Lnea lazed wave: when he elle llae a lne,.e. when δ,π. he den f he ve eman nan. Culal lazed wave: when he elle edue a le,.e. when δπ/, π/ and. he lazan elle efed b he amlude aamee,,δ. Bu an be eeed equvalenl b he ellal aamee: Oenan angle ψ: δ an ψ ll angle b χ : an χ a n δ n χ F lneal lazed lgh χ. Plazan Ba b a ψ ke Paamee Ou dee ae lw fllw he me evlun f he feld. Wha we an meaue ae me aveage, ve ed muh lnge han π/ω. Due he ed f he wave, enugh mue me aveage ve a ngle ed f llan. hee ae eeened b he mbl < >. we ake he me aveage f he lazan elle: δ n δ

2 ke Paamee ullng b 4 we fnd ne and ae ne wave, we an mue he me aveage and ubue abve: δ ne we wan ee h n em f nene, we an add and uba 4 4 : δ n δ n δ We fnd ke Paamee δ We ne he ke Paamee: ha u equan edue n δ δ n δ ke Paamee If lgh n uel mnhma, he amlude and hae fluuae wh me. I an be hwn ha, n geneal, l he gn vald f full lazed P lgh, whle he > gn vald f I al aall lazed unlazed P lgh. Pdegee f lazan: he nen elaed : δ he enan f he lazan an ψ elle elaed and : n δ he ell f he lazan n χ elle elaed : I δ n δ ke Paamee Ne ha, f lnea lazed lgh δ, bh aamee and eeen he dffeene n nen aed b w hgnal mnen: he dffeene n nen beween he mnen alng a and he dffeene n nen beween he mnen alng w a and aed 45 wh ee and. 45 ke Paamee: eamle Unlazed lgh: δandm < >< >I Lneal lazed lgh: Hznal Veal 45 ; δ I I I Cula lazed lgh: I Lef Rgh I I δ nδ I n ke Paamee he wave an be eeened a mle funn: [ ω δ ] [ ω δ ] e[ ω δ ] e[ ω δ ] h hel n he me-aveagng e needed mue he ke Paamee. he an be ewen a fllw ke ve: δ n δ

3 ke Paamee he ke ve an al be eeed n em f, ψ, χ. δ Fm an ψ we an we an ψ n δ And fm n χ we an we n χ Pnae Ung we fnd, we have: χ ψ χn ψ n χ Claal meauemen f he ke Paamee ue eade laze dee he meauemen f he 4 ke Paamee need w al mnen: A eade wave lae: a hae-hfng elemen, whe effe advane he hae f he mnen b / and ead he hae f he mnen b -/. he feld emegng fm he eade e / and e- / A laze. he al feld an a nl alng ne a, he anmn a. he al feld emegng fm he laze n, whee he nden feld and he angle f he anmn a. he beam avng n he dee e / e - / n Claal meauemen f he ke Paamee ue eade laze dee e / e - / n he dee meaue nen,.e. I we ge I ϑ, n e n e n Whh an be ewen ung he half-angle fmula: n n n I ϑ, n n n [ n n n ] I ϑ, Claal meauemen f he ke Paamee ue eade laze dee I ϑ, [ n n n ] h he fmula deved n 85 b Gege I, [ ] Gabel ke. I 45, [ ] he f hee aamee [ ] an be meaued b I9, emvng he eade I 45, 9 [ ] and meaung he nen wh hee enan f he laze,45,9 I, I 9, : he fuh aamee an I, I 9, be meaued b neng a I45, I, I 9, 9 eade quae wave I45, 9 I, I 9, lae: Claal meauemen f he ke Paamee ue eade laze dee I ϑ, [ n n n ] he gea advanage f he ke Paamee ha he ae bevable. he lazan elle n fa. eve, he ke aamee an be ued debe unlazed lgh: lgh whh n affeed b he an f a lazed b he eene f a eade. ke wa he f ne debe mahemaall unlazed and aall lazed lgh. I evden fm ke fmula ha, f unlazed lgh,, whle >. he full lazed lgh had he nemedae ae aall lazed lgh, whee Paall lazed lgh he ke aamee f a mbnan f ndeenden wave ae he um f he eeve ke aamee f he eaae wave. If we mbne a full lazed wave wh an ndeenden, unlazed ne, we fnd aall lazed lgh. P P Il P P Ial h een wll be ueful n he fllwng.

4 4 Plazan-ave al mnen When a beam f lgh nea wh mae lazan ae alm alwa hanged. I an be hanged b hangng he amlude hangng he hae hangng he den f he hgnal feld mnen. he effe an be debed b mean f he uelle mae: a 44 ma uh ha he emegng ke ve. Plaze Daenua Ra Wave-lae Reade Plaze Daenua I aenuae he hgnal mnen f an al beam unequall: Ung he nn f and And neng he een f we ge eal ae If he daenua ml an aenua,.e. f we have a neual den fle: If he Plaze deal and hznal,.e. f we have If he Plaze deal and veal,.e. f we have Plaze: he haae f he laze and an be ewen n em f new aamee and : Wh hee aamee he uelle ma f a laze : An deal laze nve an nmng beam n a lneal lazed beam: P n n n P Reade I ndue a hae hf beween he hgnal mnen f an al beam : Ung he nn f and And neng he een f we ge / / e e n n eal ae If he eade a quaewave lae 9 : uh a eade nve a 45 lneal lazed beam n a gh/lef ulal lazed beam: If he eade a half-wave lae 8 : h evee he ell and enan f he nmn lazan ae. n n

5 5 Ra Hee Ung he nn f and And neng he een f we ge n n n n Raed Oal Cmnen We have aumed ha he al a f he mnen we have ndeed wee algned he dnae em. If he ae n a fen haen, we have. ae he nden beam fm he gnal dnae em he ne algned wh he mnen: R n. ull b he uelle ma C f he al mnen C. Rae he uu beam bak n he gnal dnae em: u R - we have: u R -q C R q n Whee he an n he al mnen C. n n R Raed Plaze P Hee and P C P n Lnea Plamee A lamee a deve able dee lazed lgh and meaue lazan haae. he mle lamee we an magne a lnea lamee, whh an be bul wh a ang laze n fn f an nen dee. An nen dee eeened b a ke ve D,,,. he we deeed b he dee fm an al beam wh ke ve ml wd If we u a laze n fn f he dee, he laze alled analze, and he we deeed wll be w D P ue laze Inen dee Lnea Plamee ue Plaze analze Inen dee,,, D w P n w h lamee n enve ula lazan n. I enve lnea lazan and and unlazed lgh. If he laze deal: n ; ; w Lnea Plamee ω ue Rang analze Inen dee If we ae neeed he lnea lazed mnen nl, we an ae nnuul he laze: ω and lk nl f he AC gnal a fequen ω. h allw eje he unlazed mnen, even f dmnan, and emve all he ne mnen a fequene dffeen han ω nhnu demdulan. w ω ω n [ ] n N R N Rw V ω ω nan gnal DC mdulaed gnal AC ne AC dee env

6 Lnea Plamee Lg Pω ne /RC ue A[RwωNω] Demdulaed gnal gnal ω < > Lg ω ω Dee Rang analze RwN Refω C A ARwN σ P ω d ω ω - AC ω - / R Hw d we eaae and [ ω n ] N V Rw N R ω Negleng he ha effe f ne we negae enugh ha N beme neglgble and f he nan em whh we emve wh he AC deulng V Rw R[ ω n ω ] We meaue V and we wan emae and. We an ue w efeene gnal, u f hae b /8 and nhnul demdulae wh hem: Y Y Hw d we eaae and V n ωd R ω n ωd n ω n ωd R V ωd d ω ω n ω ωd he duble lnea lamee R nenve 8 and ea albae. R 8 I h a ublele numen? N! I neffen fa /8 fm mdulan and demdulan I an be mhn. And, a all lamee, need a elee. 6