Effects of polarization on the reflected wave
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1 Lecture Notes. L Ros PPLIED OPTICS Effects of polrzton on the reflected wve Ref: The Feynmn Lectures on Physcs, Vol-I, Secton 33-6 Plne of ncdence Z Plne of nterfce Fg. 1 Y Y r 1 Glss r 1 Glss Fg. Reflecton nd refrcton wth s-polrzed rdton Fg. 3 Reflecton nd refrcton wth p-polrzed rdton E perpendculr to the plne of ncdence Trnsverse Electrc (TE) rdton lso clled s- polrzed rdton E prllel to the plne of ncdence Trnsverse Mgnetc rdton, lso clled p- polrzed rdton : mpltude of the refrcted wve : mpltude of the refrcted wve : mpltude of the reflected wve : mpltude of the r reflected wve
2 Ojectve: To fgure out the reflecton coeffcents ncdence nd refrcton nd s functon of the ngles of Prncple: The currents tht re generted n the glss produce two wves ) They produce the reflected wve ) If there were no currents generted n the glss, the ncdent wve would contnue strght nto the glss;.e. producng feld of unt mpltude, whch would go long the dotted lne s shown n the fgure. Ths feld s not oserved. Therefore the currents generted n the glss must produce feld of mpltude tht moves long the dotted lne. Usng these prncples we wll fnd the reflecton coeffcents In fgure : feld of mpltude s rdted y the moton of chrges nsde the glss whch re respondng to feld nsde the glss. Therefore t s plusle to ssume tht, s proportonl to. (1) In fgure 3: Except for the polrzton of the felds, ths fgure s dentcl to Fg.. Cn we sy then tht the rto / s the sme s /? The nswer s no. The reson s ecuse n Fgure 3 the drecton of the polrztons (of the reflected nd trnsmtted wves) re not prllel to ech other. How to fnd /? To fnd proper reltonshp etween / nd / let s tke look t Fg 4. Notce only the component of long the drecton SN s effectve n genertng the reflected wve. Such component s ). The fgure lso revels tht, ccordngly, = ( +. (ttned from Fg. nd Fg. 4) ()
3 Y R 90 o S N M SN = MN cos N r M 1 Only ts component long SN s effectve n genertng the reflected wve. Fg. 4 Reflecton nd refrcton from p-polrzed rdton. On the other hnd, n oth cses t hppens tht the osclltons of chrges n the glss generte feld of mpltude (so tht when dded to the ncdent feld of mpltude 1 oth wll cncel out). In Fgure-5 we notce tht the component of long the PR drecton s the effectve component n genertng trnsmtted wve long the dotted lne. Tht component s ). The fgure lso revels tht, ccordngly =( -. From () nd (3) we elmnte the rto / nd otn, (from Fg. nd Fg. 5) (3) (4) o Verfcton of the formul: Notce when r 90 expresson (4) gves =0, s expected (rewster condton). No reflected p-polrzed reflected wve.
4 Y Q PR = PQ cos R Q P P r Only ts component long PR s effectve n genertng trnsmtted wve of mpltude Fg. 5 Fndng the reflecton coeffcents nd. Expresson (4) so fr cn gve us only the rto of the reflecton coeffcents. To fnd ech coeffcent ndvdully we resort to the conservton of energy. The energy n ech wve (ncdent, reflected, or trnsmtted) s proportonl to the correspondng mpltude. In fgure : Conservton of energy mples tht, the energy of the trnsmtted wve (5) must e proportonl to 1 -. [Whtever coeffcent tht must hve een ncluded n front of the (mpltude) t wll e the sme for the ncdent nd reflected wves, snce oth re n the sme medum. For tht reson the energy of the trnsmtted wve s smply proportonl to 1 - ]
5 In fgure 3: Conservton of energy mples tht, the energy of the trnsmtted wve (6) must e proportonl to 1 -. Notce tht the constnt of proportonlty n (5) nd (6) must e the sme. [sclly, we re syng tht the sme postve constnt h plys n 1 = + h nd 1 = + h ] On the other hnd, the rto of the energes of the trnsmtted wves n (7) Fg. nd Fg. 3 must e equl to / From (5), (6), nd (7), 1 1 (8) Usng (3),, nd (4),, n expresson (8) one otns, 1 1 cos ( 1 cos ( cos ( 1 cos ( cos ( sn ( sn ( sn ( (9) sn ( Usng (4),, one otns,
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