16 Reflection and transmission, TE mode

Transcription

1 16 Reflecton transmsson TE mode Last lecture we learned how to represent plane-tem waves propagatng n a drecton ˆ n terms of feld phasors such that η = Ẽ = E o e j r H = ˆ Ẽ η µ ɛ = ˆ = ω µɛ E o =0. Such waves are only permtted n homogeneous propagaton meda wth constant µ ɛ ero σ. The condton of ero σ can be relaed easly n that case the above relatons would stll hold f we were to replace ɛ by ɛ + σ jω as we wll see later on. In ths lecture we wll eamne the propagaton of plane-tem waves across two dstnct homogeneous meda havng a planar nterface between them. Wth no loss of generalty we can choose unt vector ˆ be the untnormal of the nterface plane separatng medum 1 n the regon <0 from medum n the regon >0. In 1808 Etenne-Lous Malus dscovered that lght reflected from a surface at an oblque angle wll n general be polared dfferently than the ncdent wave on the reflectng surface. Ths s caused by the dfference of the reflecton coeffcents of TE TM components of the ncdent wave as we wll learn n ths lecture. Practcal mplementaton of the phenomenon nclude polarers polarng flters used n optcal nstruments photography LCD dsplays. H r t θ 1 sn 1 cos Medum 1 Medum sn θ 1

2 Aplane-TEMwavencdentontothenterfacefrommedum1sassgned a wavevector = 1 (ˆ cos +ẑ sn ) by tang ŷ to be orthogonal to (see margn). Ths maes the -plane the plane of ncdence the angle of ncdence furthermore f we were to consder the case of Ẽ =ŷe o e j r we would call the problem a TE mode problem where TE s short for Transverse Electrc feld transverse s wth respect to the plane of ncdence. f we were to consder the case of H =ŷh o e j r we would call the problem a TM mode problem where TM s short for Transverse Magnetc feld transverse s once agan wth respect to the plane of ncdence. Ths lecture we wll eamne the TE mode net lecture the TM mode. These dfferent modes have dfferent transmsson reflecton propertes. They are easy to study one at a tme suffcent n general snce all cases can be represented as a superposton of TE TM cases. r t θ H 1 sn 1 cos Medum 1 Medum 1 sn 1 cos E Medum 1 Medum r t θ sn θ sn θ

3 TE mode reflecton problem: In TE mode reflecton problem the plane-wave feld phasors ncdent on the nterface between medum 1 =0plane are specfed as where j Ẽ =ŷe o e r H = Ẽ 1 = 1 (ˆ cos +ẑ sn ) 1 = ω v 1 v 1 = 1 µ1 ɛ 1 = µ1 ɛ 1. r t θ H 1 sn 1 cos Medum 1 Medum sn θ The plane-wave feld specfed above satsfes Mawell s equatons n the homogeneous medum 1 occupyng the regon <0 but f there were no other felds n meda 1 Mawell s boundary condton equatons requrng the contnuty of tangental Ẽ H across any boundary not supportng a surface current would be volated. In order to comply wth the boundary condton equatons we postulate asetofreflected transmtted wave felds n meda 1 as follows: In medum 1 we postulate a reflected plane-wave wth feld phasors j Ẽ r =ŷe o Γ e r r H r = r Ẽr 1

4 where r = 1 ( ˆ cos +ẑ sn ). In medum we postulate a transmtted plane-wave wth feld phasors where j Ẽ t =ŷe o τ e t r H t = t Ẽt η t = (ˆ cos θ +ẑ sn θ ). = ω v v = 1 µ ɛ η = µ ɛ. To justfy our postulates determne a set of reflecton transmsson coeffcents Γ τ defnedntermsofelectrcfeldcomponents we wll net apply the boundary condtons on =0surface where (usng =0) Ẽ =ŷe o e j 1 sn Ẽ r =ŷe o Γ e j 1 sn Ẽ t =ŷe o τ e j sn θ. Clearly wth these feld components tangental contnuty of the total feld phasor Ẽ across =0surface wll be satsfed for all f only f e j 1 sn +Γ e j 1 sn = τ e j sn θ whch s only possble (non-trvally) f r t θ H 1 sn 1 cos Medum 1 Medum sn θ 4

5 1. A phase matchng condton 1 1 sn = sn θ r t nown as Snell s law s satsfed. Γ τ satsfy 1+Γ = τ. H θ 1 sn 1 cos sn θ Tangental components of H H r H t on =0plane are obtaned from Medum 1 Medum Ẽ =ŷe o e j 1 sn Ẽ r =ŷe o Γ e j 1 sn Ẽ t =ŷe o τ e j sn θ. as ẑ H = E o cos e j 1 sn ẑ H r = E oγ cos e j 1 sn ẑ H t = E oτ cos θ η e j sn θ. Clearly gven Snell s law tangental contnuty of the total feld phasor H across =0surface wll then be satsfed for all f only f cos cos Γ = cos θ η τ. Combnng ths wth 1+Γ = τ 1 Note that Snell s law can also be nterpreted as havng the components of wavevectors t equal along the nterface between meda 1. 5

6 we fnd that cos cos Γ = cos θ η (1 + Γ ) Γ = η cos cos θ η cos + cos θ τ =1+Γ = η cos η cos + cos θ. r t H θ 1 sn 1 cos sn θ Medum 1 Medum Concluson: In TE reflecton problem the Fresnel reflecton transmsson coeffcents are Γ E yr E y = η cos cos θ η cos + cos θ τ = E yt E y = η cos η cos + cos θ respectvely. The coeffcents enable us to epress the reflected transmtted wave phasors n terms of the ncdent-wave electrc feld phasor at the orgn (.e. E y ). 6

7 Eample 1: Medum s vacuum whle medum 1 has µ 1 = µ o ɛ 1 =ɛ o. Gven that determne Ẽr Ẽt H t. Soluton: We have = Also accordng to Snell s law Ẽ =ŷ5e j 1(cos 0 +sn 0 ) V m µ1 µo = = η o η = η o. ɛ 1 ɛ o 1 sn = sn θ sn θ = 1 sn = ɛ1 µ 1 ɛ µ sn = sn0 = 1 ndcatng that Now the reflecton coeffcent s θ =45. Γ E yr = η cos cos θ = η o E y η cos + cos θ The transmsson coeffcent s η o 1 η o + η o 1 = τ = E yt E y = η cos η cos + cos θ = η o η o + η o 1 Consequently the reflected transmtted wave phasors are Ẽ r =ŷ(5 0.68)e j 1( cos 0 +sn 0 ) V m 7

8 Ẽ t =ŷ(5 1.68)e j (cos 45 +sn 45 ) V m. Fnally H t = Ẽt η = (cos 45 ˆ +sn45 j(cos 45 +sn 45 ) ẑ) ŷ(5 1.68)e = (ẑ ˆ)(5 1.68)e j (cos 45 +sn 45 ) 10π η o A m. 8