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1 Lecture 3 Date: Incidence, Reflectn, and Transmissn of Plane Waves

2 Wave Incidence For many applicatns, [such as fiber optics, wire line transmissn, wireless transmissn], it s necessary to know what happens to a wave when it meets a different medium. How much is transmitted? How much is reflected back? Normal incidence: Wave arrives at 0 o from normal Oblique incidence: Wave arrives at another angle

3 Reflectn at Normal Incidence Medium i e, s, m x Medium e, m, s H i a k t Incident wave r H t a k a kr H r Reflected wave y z=0 Transmitted wave z

4 Incident wave Reflected wave ( z) e a z is x z z H ( z) H e a e a is y y ( z) e a z rs ro x z ( ) z H z H e a e a rs ro y y i H i a k Incident wave r a kr H r Reflected wave Transmitted wave ( z) e a z ts to x z to z H ( z) H e a e a ts to y y t a H k t Transmitted wave

5 The total waves in medium : i r The total waves in medium : t H H i H r H Ht At the interface z = 0, the boundary conditns require that the tangential components of and H fields must be continuous. Since the waves are transverse, and H fields are entirely tangential to the surface. Therefore, at z = 0: tan = tan and H tan = H tan imply that - (0) (0) (0) is rs ts H (0) H (0) H (0) is rs ts ro to to ro

6 Simplificatn results in: ro to These expressns aid us in the definitns of reflectn coefficient Γ and transmissn coefficient τ. ro ro to to It is important to note that: + Γ = τ Both Γ and τ are dimensnless and may be complex 0 Γ

7 Therefore the total fields in the two medium are: z z e e a s x z z H s e e a z s e a s x z H e a Special Cases: η = η Γ = 0 τ = (total transmissn, no reflectn) η = 0 Γ = τ = (total reflectn, no inversn of ) η = 0 Γ = τ = 0 (total reflectn, inversn of ) y y

8 Special Case I Medium : perfect dielectric (lossless): σ = 0, η = μ ε, α = 0, γ = jβ Medium : perfect conductor: σ =, η =0, α = β = η = 0 Γ = τ = 0 (total reflectn, inversn of s is rs jz jz s x e e a j sin za s x z z H s e e a y H s cos za y e a z 0 s x z H s e a 0 y

9 The instantaneous electric field: j t Similar steps result in: z t e z ta (, ) Re s sin sin x H ( z, t) cos z costa y Note that the positn dependence of the instantaneous electric and magnetic fields is not a functn of time standing wave!!! It is expected considering that there is total reflectn and in a lossless dielectric the waves consist of two travelling waves ( i and r ) of equal amplitudes but in opposite directns.

10 z t e z ta (, ) Re j s t sin sin x The wave doesn t travel but oscillate Standing waves = sinβ zsinωta x. The curves 0,,, 3, 4,..., are, respectively, at times t 0, T/8, T/4, 3T/8, T/,... ; l p/.

11 The locatns of the minimums (nulls) and maximums (peaks) in the standing wave electric field pattern are found by: z t min (, ) 0 when sin z 0 ( z) np z np nl z t max (, ) when sin z ( ) ( ) z n p z (n) p (n) l p 4 l

12 Special Case II: Two Perfect Dielectrics Medium : perfect dielectric (lossless): σ = 0, η = μ ε, α = 0, γ = jβ Medium : perfect dielectric (lossless): σ = 0, η = μ ε, α = 0, γ = jβ If η > η 0 < Γ < < τ < If η < η < Γ < 0 0 < τ <

13 Therefore: ( e e ) a e ( e ) a j z j z j z j z s x x jz jz jz H ( e e ) a ( e ) a s y y jz jz jz jz s e ax e ax H e a e a s y y Standing wave exists only in medium. The magnitude of the electric field in medium can be analyzed to determine the locatns of the maximum and minimum values of the electric field standing wave pattern. jz jz s ( e ) ( e ) 0 z This can be described in the complex plane using crank diagram

14 The distance from the origin to the respective point on the circle in the crank diagram represents the magnitude of: e j z If η > η, (Γ is positive), then the maximum and minimum of the functn are: jz ( e ) max jz ( e ) when ( z) n p when min ( z) n np nl z (n) p (n) p z l 4 s min s max

15 If η > η then Γ is negative. The positns of the maximums and minimums are reversed, but the equatns for the maximum and minimum electric field magnitude in terms of Γ are the same. s max s min

16 The standing wave rat (s) in a medium where standing waves exist is defined as the rat of the maximum electric field magnitude to the minimum electric field magnitude. s s max s min The standing wave rat (purely real) ranges from a minimum value of (no reflectn, Γ = 0) to (total reflectn, Γ =). The standing wave rat is sometimes defined in db as: s( db) 0log 0 s

17 xample A uniform plane wave in air is normally incident on an infinite lossless dielectric material having ε = 3ε 0 and μ = μ 0. If the incident wave is is = 0cos ωt z a y V/m, find (a) ω and λ of the waves in both the mediums, (b)h is, (c) Γ and τ, (d) the total electric field and time-average power in both mediums.

18 xample (contd.) Medium [z < 0] : Air (μ = μ 0, ε = ε 0, σ = 0) Medium [z > 0] : Dielectric (μ = μ 0, ε = 3ε 0, σ = 0) α = 0, β = ω μ 0 ε 0 = ω c α = 0, β = ω 3μ 0 ε 0 = 3 ω c m j 0 0 e 0 j m 0 0 e 0 u c i j H i

19 xample (contd.) (a) is 0cos( t z) a y jz jz 0e a e a is y o y 0 p rad/m p 3 3 l u c l u c rad/m p p p l p 6.8 m l 3.63 m u u rad/sec 47.8 MHz j z 0 jz jz H e ( a ) e a 0.066e a 377 (b) is x x x 0 H Re 0.066e jz a 0.066cos( t z) a A/m is x x

20 xample (contd.) (c) (d) e e a s ( j z j z ) y 0cos( t z).68cos( t z) a y V/m jz e a 7.3cos( t 3 z) a y s o y V/m The time average power density in medium is due to the +z directed incident wave and the z directed reflected wave. The time-average power density in medium is due to the +z directed transmitted wave. is rs P a ( a ) ave, z z

Module 5 : Plane Waves at Media Interface. Lecture 39 : Electro Magnetic Waves at Conducting Boundaries. Objectives

Module 5 : Plane Waves at Media Interface. Lecture 39 : Electro Magnetic Waves at Conducting Boundaries. Objectives Objectives In this course you will learn the following Reflection from a Conducting Boundary. Normal Incidence at Conducting Boundary. Reflection from a Conducting Boundary Let us consider a dielectric