Scattering at an Interface: Oblique Incidence

Transcription

1 Course Insrucor Dr. Raymond C. Rumpf Offce: A 337 Phone: (915) E Mal: rcrumpf@uep.edu EE 4347 Appled Elecromagnecs Topc 3g Scaerng a an Inerface: Oblque Incdence Scaerng These Oblque noes may Incdence conan copyrghed maeral obaned under far use rules. Dsrbuon of hese maerals s srcly prohbed Slde 1 Lecure Oulne Plane Wave a Oblque Incdence Reflecance and Transmance Eample Plo of Reflecance and Transmance Scaerng Oblque Incdence Slde 1

2 Plane Wave a Oblque Incdence Scaerng Oblque Incdence Slde 3 Geomery for Oblque Incdence (1 of 6) We sar wh a perfecly fla nerface beween wo maerals. For mahemacal convenence, we le he nerface le eacly n he y plane. We mus draw our coordnaes so s a rgh handed sysem. aˆ aˆ aˆ y Scaerng Oblque Incdence Slde 4

3 Geomery for Oblque Incdence ( of 6) Le here be a wave descrbed by he wave vecor be ncden nc ono he nerface from above. Recall he wave vecor s: ˆ ˆ ˆ nc a a a nc y y y Scaerng Oblque Incdence Slde 5 Geomery for Oblque Incdence (3 of 6) The wave vecor nc and he surface normal aˆ defne a plane. Boh of hese vecors le whn hs plane. Ths s called he plane of ncdence. Scaerng Oblque Incdence Slde 6 3

4 Geomery for Oblque Incdence (4 of 6) The wave vecor s defned by wo angles, and. elevaon angle amuhal angle The componens of he ncden wave vecor can be calculaed accordng o ˆ ˆ ˆ nc a yay a y 0n1cos sn 0n1sn sn n cos 0 1 Scaerng Oblque Incdence Slde 7 Geomery for Oblque Incdence (5 of 6) The choce of drecons for polaraon becomes mporan when a devce s nvolved. The polaraon s defned o be perpendcular o he plane of ncdence. aˆ nc aˆ aˆ polaraon s polaraon polaraon nc Scaerng Oblque Incdence Slde 8 4

5 Geomery for Oblque Incdence (6 of 6) The polaraon s defned o be parallel o he plane of ncdence. aˆ nc aˆ aˆ polaraon p polaraon polaraon nc Scaerng Oblque Incdence Slde 9 Convenen Choce for Plane of Incdence (1 of ) The ncden wave, refleced wave, and ransmed wave all le whn he plane of ncdence. Scaerng Oblque Incdence Slde 10 5

6 Convenen Choce for Plane of Incdence ( of ) Snce everyhng happens whn he plane of ncdence, we can roae he plane of ncdence o somehng more convenen o analye. Le he plane of ncdence le n he plane. aˆ aˆ, 0 Scaerng Oblque Incdence Slde 11 y Ths roaon s vald for calculang angle of reflecon, angle of refracon, and amplude of he refleced and ransmed waves. However, vecor quanes le, nc â, â, and P wll be dfferen n he roaed sysem. y Boundary Condon for (1 of 3) Whou mang any assumpons, our waves can be wren as E r E e E e e E r E e E e e E r E e E e e j r 0, 0, j, j, jr r r 0,r 0,r j,r j,r j r 0, 0, j, j, However, we only care abou wha s happenng eacly on he nerface so we se = 0. E E e e E e E E e e E e E E e e E e j, 0 j, 0 0, 0, j,r 0 j,r r 0 0,r 0,r j, 0 j, 0 0, 0, Remember y = 0 for our analyss. Scaerng Oblque Incdence Slde 1 6

7 Boundary Condon for ( of 3) Boundary condons requre he angenal componens of E o be connuous across he nerface.,1,,,r, j, j,r j,,,r, E E E E E E e E e E e Scaerng Oblque Incdence Slde 13 Boundary Condon for (3 of 3) j, j,r j,,,r, E e E e E e j, j,r j, y, y,r y, E e E e E e The only possble way hese equaons can be sasfed s f,,r, s he angenal componen of he wave vecor. We generale hs o any orenaon of he plane of ncdence a The angenal componens of are connuous across he nerface. 1,an,an We can also conclude from hs ha he ncden wave, refleced wave, and ransmed wave all le whn he plane he ncdence. Scaerng Oblque Incdence Slde 14 7

8 Wha Abou,r and,? The vecor componens of a plane wave mus sasfy he dsperson relaon of he medum he wave s n. The dsperson relaons for he ncden, refleced, and ransmed waves are 0n1,, r 0n1,r,r n 0,, However we now ha, =,r =, so we jus call all of hese. 0n1,, 0n1 r 0n1,r,r 0n1 n n 0,, 0 Scaerng Oblque Incdence Slde 15 Refleced Wave,,r From he prevous slde, he dsperson relaon for he ncden and refleced waves were From hese, we see ha. Therefore,r, We conclude ha,r,, 0 1,r 0 1 n n,r, We resolve he sgn by recognng ha he refleced wave mus be propagang n he drecon. Scaerng Oblque Incdence Slde 16 8

9 Law of Reflecon Le s relae he angle of he ncden and refleced wave based on wha we now abou he wave vecor componens. n sn y n cos, 0 1 Medum 1, nc r r,r, Medum Scaerng Oblque Incdence Slde 17 Geomery of Reflecon and Refracon Medum 1, nc r,r, n, 0 Medum Scaerng Oblque Incdence Slde 18 9

10 Snell s Law Recall he dsperson relaons for he ncden and ransmed waves. n n 0 1, Solvng boh of hese equaons for The rgh hand sde of hese equaons mus be equal. n n 0 1, 0, ncos 0 1 0, gves n 0 1, n 0, n cos 0 n ncos n n cos n 1cos n 1cos 1 n sn n sn 1 n sn n sn 1 Scaerng Oblque Incdence Slde 19 Summary of Scaerng Angles Snell s Law n sn n sn 1 nc Medum 1 n 1 Medum n r Law of Reflecon ref r Scaerng Oblque Incdence Slde 0 rn 10

11 Anmaon of Reflecon & Refracon Law of Reflecon r Snell s Law n sn n sn 1 Scaerng Oblque Incdence Slde 1 Fresnel Equaons r, s, Polaraon cos 1cos cos cos 1 cos cos cos 1r 1, p, Polaraon r 1r cos 1cos cos cos 1 cos cos cos 1 cos cos Law of reflecon and Snell s law ells us he drecons of he refleced and ransmed waves relave o he ncden wave. The Fresnel equaons ell how much ges refleced and ransmed. Scaerng Oblque Incdence Slde 11

12 Reflecance & Transmance Scaerng Oblque Incdence Slde 3 Defnon of R and T The reflecon and ransmsson coeffcens relae he amplude s of he refleced and ransmed waves relave o he ncden wave. E0,r E0, r E E 0, 0, The reflecance and ransmance descrbes he fracon of power ha s refleced or ransmed from he nerface. Pr R P P T P Scaerng Oblque Incdence Slde 4 1

13 RMS Power Flow In he frequency doman, he Poynng vecor descrbes he RMS power flow. I s calculaed from he elecrc and magnec felds as 1 * avg Re E H Recall ha he elecrc and magnec felds can be epressed as jr E 0 jr Er, E0e Hr, e Subsung hese no he defnon of RMS Poynng vecor gves * Im Re j r E E j r r Ee 0 e e Re 00 r Scaerng Oblque Incdence Slde 5 Wha Carres Power To and From he Inerface? The flow of power s descrbed by he Poynng vecor. However, s only he componens of he Poynng vecor ha are perpendcular o he nerface ha carry power o and from he nerface., r,r Medum 1 Medum,,,r, Scaerng Oblque Incdence Slde 6 13

14 Revsed Defnon of R and T We are now n a poson o revse our defnons of reflecance and ransmance n erms of he Poynng vecors.,r 0, 0 R T 0 0,, Gven our epresson for RMS power flow, he above equaons become Er,r Re 00 E r,r r R E E, Re 00 E r,, Re 0 0 E r, Re, r, T E Re E, r,, Re 00 r, Scaerng Oblque Incdence Slde 7 Reflecance, R Recall ha E re r These reduce our epresson for reflecance o Er re R E E R r Scaerng Oblque Incdence Slde 8 14

15 Transmance, T Recall ha E E These reduces our epresson for ransmance o E Re, r, T Re, r, T E Re, r, Re, r, Flow lossless maerals, he permeably s purely real and n cos, 0 1 n cos, 0 T 1 cos cos n n 1 r, r, r, r, 1 r, r, r, r, Afer some algebra, he ransmance n lossless maerals s Scaerng Oblque Incdence Slde 9 Relaons Beween he Parameers T T 1 cos cos 1 cos cos T T Scaerng Oblque Incdence Slde 30 15

16 Toal Reflecance and Transmance A wave ncden ono a surface may have boh and componens. Power n he source wave s herefore P P P nc,nc,nc I follows ha he oal power refleced and ransmed s Pref R P,nc R P P,nc rn T P,nc T P,nc Overall reflecance R and ransmance T are derved by dvdng hese equaons by he frs equaon. P R P R P ref,nc,nc R P nc P,nc P,nc T P T P T P rn P P P,nc,nc nc,nc,nc Scaerng Oblque Incdence Slde 31 Eample Plo of Fresnel Equaons Scaerng Oblque Incdence Slde 3 16

17 Plos of he Fresnel Equaons Low o Hgh Inde (n 1 = 1.0 and n = 1.5) No crcal angle n c sn n1 sn Brewser s angle 1n 1 B, an an n1 Hgh o Low Inde (n 1 = 1.5 and n = 1.0) Crcal angle 1n 1 c sn sn n1 Brewser s angle 1n 1 B, an an n1 Scaerng Oblque Incdence Slde 33 17