IMPLearn PROPAGATION OF EM WAVE IN A CONDUCTING MEDIUM (LOSSY DIELECTRIC) Consider a linear (J=0), isotropic (ρ=0), homogeneous( σ=0), lossy dielectric medium (ρv=0). Electric filed and magnetic fields are assumed to be varying sinusoidally with time. The wave equation in phaser form for conducting medium γ๐ = ๐๐๐๐ − ๐๐ ๐ε 2 ๐ป ๐ฌ๐ = ๐๐๐ ๐ + ๐๐ε ๐๐ฌ ---------- (2) Comparing real & imaginary values of (1) & (2) Lets substitute γ๐ = ๐๐๐ ๐ + ๐๐ε ๐ผ 2 − ๐ฝ 2 = −๐๐ ๐ε ------ (3) 2๐ผ๐ฝ = ๐๐๐ ------ (4) From (3) & (4) we get ๐ป 2 ๐ฌ๐ = γ๐ ๐๐ฌ ๐ป 2 ๐ฏ๐ = γ๐ ๐ฏ๐ฌ ๐ε ๐ผ=๐ 2 ๐ 1+ ๐ε 2 −1 Propagation constant is a complex value so, γ = ๐ผ + ๐๐ฝ γ๐ = ๐ผ + ๐๐ฝ ๐ε ๐ฝ=๐ 2 2 γ๐ = ๐ผ 2 − ๐ฝ 2 + 2๐๐ผ๐ฝ ๐พ = ± ๐๐๐ ๐ + ๐๐ε ---------- (1) ๐ 1+ ๐ε 2 +1 www.iammanuprasad.com IMPLearn PROPAGATION OF EM WAVE IN A CONDUCTING MEDIUM (LOSSY DIELECTRIC) If we assume that the wave propagates along z direction & it has only x components ๐ธ๐ = ๐ธ๐ฅ๐ ๐ง ๐เท๐ฅ We know ๐ป 2 ๐ฌ๐ = γ๐ ๐๐ฌ ๐ป 2 ๐ธ๐ฅ๐ ๐ง − γ๐ ๐ธ๐ฅ๐ ๐ง = 0 ๐2 ๐2 ๐2 ๐ธ๐ฅ๐ ๐ง + 2 ๐ธ๐ฅ๐ ๐ง + 2 ๐ธ๐ฅ๐ ๐ง − γ๐ ๐ธ๐ฅ๐ ๐ง = 0 2 ๐๐ฅ ๐๐ฆ ๐๐ง ๐2 ๐ธ๐ฅ๐ ๐ง − γ๐ ๐ธ๐ฅ๐ ๐ง = 0 2 ๐๐ง ๐2 − γ๐ ๐ธ๐ฅ๐ ๐ง = 0 2 ๐๐ง This a scalar wave equation www.iammanuprasad.com IMPLearn PROPAGATION OF EM WAVE IN A CONDUCTING MEDIUM (LOSSY DIELECTRIC) Let ๐ธ๐ฅ๐ ๐ง = ๐ธ0 ๐ −γ๐ง + ๐ธ0′ ๐ γ๐ง Where ๐ธ0 =η ๐ป0 Assuming E0’=0, and including the time factor ejωt in the above equation ๐ธ ๐ง, ๐ก = ๐ ๐[๐ธ๐ฅ๐ ๐ง ๐ ๐ω๐ก ๐เท๐ฅ ] η= = ๐ ๐[๐ธ0 ๐ −γ๐ง ๐ ๐ω๐ก ๐เท๐ฅ ] (intrinsic impedance of medium) ๐๐๐ = ๐ ๐ ๐๐๐ ๐ + ๐๐ε = ๐ ๐[๐ธ0 ๐ −(๐ผ+๐๐ฝ)๐ง ๐ ๐ω๐ก ๐เท๐ฅ ] = ๐ ๐[๐ธ0 ๐ −๐ผ๐ง −๐๐ฝ๐ง ๐ω๐ก ๐ ๐ ๐เท๐ฅ ] For perfect dielectric (σ=0) = ๐ ๐[๐ธ0 ๐ −๐ผ๐ง ๐ −๐(ω๐ก−๐ฝ๐ง) ๐เท๐ฅ ] ๐ธ ๐ง, ๐ก = ๐ธ0 ๐ −๐ผ๐ง ๐๐๐ (ω๐ก − ๐ฝ๐ง)๐เท ๐ฅ ๐ป ๐ง, ๐ก = ๐ป0 ๐ −๐ผ๐ง ๐๐๐ (ω๐ก − ๐ฝ๐ง − ๐๐ )๐เท๐ฆ η = ๐ ε ๐ 1 + ๐ε 1, 2 4 η= ๐ ε tan 2๐๐ = ๐ ๐ε www.iammanuprasad.com IMPLearn SUMMERY Propagation ๐ธ = constant Attenuation constant Phase shift constant ๐๐บ ๐ถ=๐ ๐ ๐ ๐+ ๐๐บ ๐ ๐๐บ ๐ท=๐ ๐ ๐ ๐+ ๐๐บ ๐ Intrinsic impedance ๐ผ = Wave length ๐๐๐ ๐ + ๐๐๐บ −๐ +๐ ๐๐๐ ๐ + ๐๐๐บ ๐๐ ๐= ๐ท Velocity ๐ฝ = ๐ ๐ท www.iammanuprasad.com IMPLearn PROPAGATION OF EM WAVE IN PERFECT DIELECTRIC (LOSSLESS MEDIUM) Consider an isotropic homogeneous perfect dielectric of permittivity ε and permeability μ Wave equation γ = ๐ผ + ๐๐ฝ ๐ป 2 ๐ฌ๐ = γ๐ ๐๐ฌ Since there is no real part Where γ๐ = ๐๐๐๐ − ๐๐ ๐ε For a homogeneous medium (σ=0) γ๐ = −๐๐ ๐ε γ = ๐๐ ๐ε (i) Attenuation constant ๐ผ=0 (ii) Phase shift constant ๐ฝ = ๐ ๐๐ ๐= (iii) Intrinsic impedance (iv) Wave length (v) Velocity ๐= ๐ ๐ 2๐ ๐ ๐๐ ๐= ๐ ๐ ๐๐ = 1 ๐ ๐๐ = 1 ๐๐ www.iammanuprasad.com IMPLearn PROPAGATION OF PLANE EM WAVE IN GOOD CONDUCTOR In a good conductor σ ≈ ∞ , ε = ε0 μ = μ0μr σ โซ ๐ε σ โซ1 ๐ε ๐๐ ๐ผ, ๐ฝ = ๐ 2 ๐ 1+ ๐๐ ๐๐ =๐ 2 ๐ ๐๐ (i) Attenuation constant ๐ผ= (ii) Phase shift constant ๐ฝ= (iii) Intrinsic impedance ๐= 2 ±1 2 ±1 (iv) Wave length =๐ ๐ผ, ๐ฝ = ๐๐ ๐ 2 ๐๐ ๐๐๐ 2 (v) Velocity ๐= ๐= ๐๐๐ 2 ๐๐๐ 2 ๐๐๐ = ๐ 2๐ ๐๐๐ 2 ๐ ๐๐๐ 2 ๐๐ ∠450 ๐ 2๐ = ๐๐๐๐ = 2๐ ๐๐ www.iammanuprasad.com IMPLearn Q) Derive the expression for attenuation constant and phase shift constant in a lossy dielectric medium? Solution: Propagation constant is a complex value so, γ๐ = ๐๐๐ ๐ + ๐๐ε γ = ๐ผ + ๐๐ฝ ------ (1) γ๐ = ๐ผ 2 − ๐ฝ 2 + 2๐๐ผ๐ฝ ------ (2) γ๐ = ๐๐๐๐ − ๐๐ ๐ε Comparing real & imaginary values of (1) & (2) ๐ผ 2 − ๐ฝ 2 = −๐๐ ๐ε ------ (3) ๐๐๐ ๐ผ − 2๐ผ 2 2 2๐ผ๐ฝ = ๐๐๐ ๐ฝ= ๐๐๐ ------ (4) 2๐ผ = −๐๐ ๐ε Multiply by α2 in both sides ๐๐๐ (๐ผ ) − 2๐ผ 2 2 2 ๐ผ 2 = −๐๐ ๐ε ๐ถ๐ ๐๐๐ (๐ผ ) +๐ ๐ε ๐ถ = 2 2 2 ๐ ๐ 2 www.iammanuprasad.com ๐๐๐ (๐ผ ) +๐ถ ๐ ๐ε = 2 IMPLearn 2 2 ๐ ๐ 2 Converting LHS into (a + b)2 format (๐ = ๐ผ 2 , ๐ = a2 ) 2๐ผ 2 ๐2 ๐๐ ๐2 ๐๐ 2 2 (๐ผ ) + + 2 2 2 2 ๐2 ๐๐ − 2 ๐2 ๐๐ = 2 2 2 ๐๐๐ = 2 ๐๐๐ + 2 ๐4 ๐ 2 ๐ 2 ๐ 2 ๐2 ๐2 = + 4 4 ๐4 ๐2 ๐ 2 ๐2 = 1+ 2 2 4 ๐ ๐ Take square root on both sides 2 ๐2 ๐๐ ๐2 ๐2 ๐๐ ๐ผ = 1+ 2 2− 2 ๐ ๐ 2 2 b2 2ab ๐2 ๐๐ 2 ๐ผ + 2 ๐2 ๐๐ 2 ------ (5) 2 ๐2 ๐๐ = 2 ๐2 1+ 2 2−1 ๐ ๐ Taking square root ๐ε ๐ผ=๐ 2 ๐ 1+ ๐ε 2 −1 Attenuation constant ๐2 ๐๐ ๐2 ๐๐ ๐2 ๐ผ + = 1+ 2 2 2 2 ๐ ๐ 2 www.iammanuprasad.com IMPLearn From equation (3) ๐ผ 2 − ๐ฝ 2 = −๐๐ ๐ε ------ (3) ๐ฝ 2 = ๐ผ 2 + ๐๐ ๐ε ๐2 ๐๐ ๐2 ๐2 ๐๐ ๐ผ = 1+ 2 2− 2 ๐ ๐ 2 2 From equation (5) ------ (5) ๐2 ๐๐ ๐2 ๐2 ๐๐ ๐ฝ = 1+ 2 2− + ๐๐ ๐ε 2 ๐ ๐ 2 2 ๐2 ๐๐ ๐2 ๐2 ๐๐ = 1+ 2 2+ 2 ๐ ๐ 2 ๐2 ๐๐ ๐ฝ = 2 2 ๐2 1+ 2 2+1 ๐ ๐ Taking square root ๐ε ๐ฝ=๐ 2 ๐ 1+ ๐ε 2 +1 Phase shift constant www.iammanuprasad.com IMPLearn SKIN DEPTH & SKIN EFFECT We saw the equation intrinsic impedance ๐= ๐ธ0 ๐ป0 ๐ธ = ๐ธ0 ๐ −๐ผ๐ง ๐๐๐ (ω๐ก − ๐ฝ๐ง)๐เท๐ฅ ๐ธ ๐ง, ๐ก = ๐ธ0 ๐ −๐ผ๐ง ๐๐๐ (ω๐ก − ๐ฝ๐ง)๐เท ๐ฅ ๐ธ0 ๐ป = ๐ −๐ผ๐ง ๐๐๐ (ω๐ก − ๐ฝ๐ง)๐เท๐ฆ ๐ ๐ป ๐ง, ๐ก = ๐ป0 ๐ −๐ผ๐ง ๐๐๐ (ω๐ก − ๐ฝ๐ง)๐เท ๐ฆ An E (or H) was travels in a conducting medium, its amplitude is attenuated by a factor e-αz x The distance δ through which the wave amplitude decrease to a factor e-1 (about 37% of the original value) is called skin depth or penetration depth of the medium (δ). The skin depth is the measure of the depth to which an EM wave can penetrate the medium ๐ธ0 0.368๐ธ0 z For a good conductor ๐ผ = ๐๐๐ 2 δ 1 δ= = ๐ผ 2 = ๐๐๐ 1 ๐๐๐๐ ๐ธ0 ๐ −๐ผδ = ๐ธ0 ๐ −1 → ๐ผδ = 1 δ= 1 ๐ผ The phenomenon when by field in a conductor rapidly decrease is known as skin effect. It is the tendency of charge to migrate from the bulk of the conducting material to the surface, resulting in higher resistance, As frequency increases skin depth decreases www.iammanuprasad.com IMPLearn PHASE VELOCITY & GROUP VELOCITY Phase velocity Group velocity • Phase velocity is the rate at which the phase and the wave propagates in space • Group velocity is the velocity with the overall shape of the wave amplitude also known as modulation or envelop of wave propagates through space • This is the velocity at which the phase of any one frequency component of the wave travels. • It is represented in terms of wave length (λ) & time period (T)(or angular frequency (ω) and phase constant (β)) ๐ฃ๐ = ๐ ๐ ๐๐ ๐ฃ๐ = ๐ ๐ฝ • It is represented in terms of angular frequency (ω) and angular wave number (k) ๐ฃ๐ = ๐๐ ๐๐ • If ω is directly proportional to k then group velocity is equal to phase velocity (wave traves undistorted) • If ω is a linear function of K but not directly proportional (ω=ak+b), then both velocities are different • If ω is not a linear function of k, then the envelop of the wave packet will distorted as it travels which is related to group velocity The โ red square moves with the phase velocity, and the โ green circles propagate with the group velocity. • Since wave packet contains a range of different frequencies / values the envelop does not move at a single velocity, so the envelop distorted www.iammanuprasad.com IMPLearn REFLECTION OF A PLANE WAVE AT NORMAL INCIDENCE When a plane wave from one medium meets a different medium it is partly reflected and partly transmitted. The proportion of the incident wave that is reflected or transmitted depends on the constitutive parameters off the two medium involved here we will assume that the incident wave plane is normal to the boundary between the media x Medium 1 Medium 1 Medium 2 Medium 2 Ei ak Et Hi (Incident wave) ak Ht (Transmitted wave) Er ak z Hr (Reflected wave) y z=0 www.iammanuprasad.com IMPLearn ๐ธ ๐ง, ๐ก = ๐ธ๐ฅ๐ ๐ง ๐ ๐ω๐ก ๐เท๐ฅ Incident wave (Ei, Hi) Reflected wave (Er, Hr) Transmitted wave (Et, Ht) If we suppress the time factor ejωt and wave us traveling along +az Travels along −๐เท๐ง in medium 1 Travels along +๐เท๐ง in medium 2 ๐ธ๐๐ ๐ง = ๐ธ๐0 ๐ −γ1 ๐ง ๐เท๐ฅ ๐ธ๐๐ ๐ง = ๐ธ๐0 ๐ γ1 ๐ง ๐เท๐ฅ ๐ธ๐ก๐ ๐ง = ๐ธ๐ก0 ๐ −γ2 ๐ง ๐เท๐ฅ ๐ป๐๐ ๐ง = ๐ป๐0 ๐ −γ1 ๐ง ๐เท๐ฆ ๐ป๐๐ ๐ง = ๐ป๐0 ๐ γ1 ๐ง −๐เท๐ฆ ๐ป๐ก๐ ๐ง = ๐ป๐ก0 ๐ −γ2 ๐ง ๐เท๐ฆ ๐ธ๐0 −γ ๐ง ๐ง = ๐ 1 ๐เท๐ฆ ๐1 ๐ธ๐0 γ ๐ง ๐ง =− ๐ 1 ๐เท๐ฆ ๐1 ๐ป๐๐ ๐ป๐๐ ๐ป๐ก๐ ๐ง = ๐ธ๐ก0 −γ ๐ง ๐ 2 ๐เท๐ฆ ๐2 Also at z=0 ๐ธ๐๐ 0 = ๐ธ๐0 ๐ธ๐๐ 0 = ๐ธ๐0 ๐ธ๐ก๐ 0 = ๐ธ๐ก0 www.iammanuprasad.com IMPLearn ๐ธ1 = ๐ธ๐ + ๐ธ๐ ๐ธ2 = ๐ธ๐ก ๐ป1 = ๐ป๐ + ๐ป๐ ๐ป2 = ๐ป๐ก Applying boundary conditions the tangential components of E & H are continuous at z=0 ๐ธ๐ 0 + ๐ธ๐ 0 = ๐ธ๐ก (0) ๐ธ๐ + ๐ธ๐ = ๐ธ๐ก Substitute the value of Et ๐ป๐ 0 + ๐ป๐ 0 = ๐ป๐ก (0) ๐ป๐ − ๐ป๐ = ๐ป๐ก 1 ๐ธ๐ก ๐ธ − ๐ธ๐ = ๐1 ๐ ๐2 ๐ธ๐ − ๐ธ๐ ๐ธ๐ + ๐ธ๐ = ๐1 ๐2 ๐2 (๐ธ๐ − ๐ธ๐ ) = ๐1 (๐ธ๐ + ๐ธ๐ ) ๐ธ๐ ๐2 + ๐1 = ๐ธ๐ ๐2 − ๐1 ๐ธ๐ ๐2 − ๐1 = =Γ ๐ธ๐ ๐2 + ๐1 Γ → ๐ช๐๐๐๐๐๐ ๐ฎ๐๐๐๐ Where ๐ is the reflection coefficient www.iammanuprasad.com IMPLearn ๐ธ๐ − ๐ธ๐ ๐ธ๐ก = ๐1 ๐2 ๐ธ๐ + ๐ธ๐ = ๐ธ๐ก ๐2 ๐ธ๐ − ๐2 ๐ธ๐ = ๐1 ๐ธ๐ก ๐ธ๐ = ๐ธ๐ก − ๐ธ๐ Note : ๐2 ๐ธ๐ − ๐2 (๐ธ๐ก − ๐ธ๐ ) = ๐1 ๐ธ๐ก 1. 1+Γ=τ ๐2 ๐ธ๐ − ๐2 ๐ธ๐ก + ๐2 ๐ธ๐ = ๐1 ๐ธ๐ก 2. Both Γ and τ are dimensionless and may be complex 3. 0 ≤ | Γ | ≤ 1 2๐2 ๐ธ๐ − ๐2 ๐ธ๐ก = ๐1 ๐ธ๐ก 2๐2 ๐ธ๐ = (๐2 + ๐1 )๐ธ๐ก ๐ธ๐ก 2๐2 = ๐ธ๐ ๐1 + ๐2 =τ Where ๐ is the transmission coefficient www.iammanuprasad.com IMPLearn REFLECTION OF A PLANE WAVE AT OBLIQUE INCIDENCE To simplify the analysis the analysis we will assume that we are dealing with lossless media. We may extend our analysis to that of lossy media. It can be shown that a uniform plane wave taken the general form of Assume a time dependent field ๐ธ = ๐ ๐[๐ธ0 ๐ ๐ ๐.๐−๐๐ก ] ๐ธ = ๐ธ0 cos ๐. ๐ − ๐๐ก and ๐ป = ๐ ๐[๐ป0 ๐ ๐ ๐.๐−๐๐ก ๐ = ๐๐๐ฅ + ๐๐๐ฆ + ๐๐๐ง ๐ = ๐๐ฅ ๐๐ฅ + ๐๐ฆ ๐๐ฆ + ๐๐ง ๐๐ง The magnitude of k is related to ω ] Position vector Propagation vector (in the direction of wave propagation) ๐๐ = ๐๐ฅ2 + ๐๐ฆ2 + ๐๐ง2 = ๐2 ๐ε and ๐ × ๐ธ = ๐๐๐ป ๐ × ๐ป = −๐ε๐ธ ๐ธ⊥๐ป⊥๐ ๐. ๐ธ = 0 ๐. ๐ป = 0 ๐. ๐ = ๐๐ฅ ๐ฅ + ๐๐ฆ ๐ฆ + ๐๐ง ๐ง = ๐๐๐๐ ๐ก๐๐๐ก www.iammanuprasad.com IMPLearn Medium 2 Medium 1 kr Incident wave ๐ธ๐ = ๐ธ๐0 cos ๐๐๐ฅ ๐ฅ + ๐๐๐ฆ ๐ฆ + ๐๐๐ง ๐ง − ๐๐ ๐ก kt ๐๐ Reflected wave ๐๐ก ๐๐ z ๐ธ๐ = ๐ธ๐0 cos ๐๐๐ฅ ๐ฅ + ๐๐๐ฆ ๐ฆ + ๐๐๐ง ๐ง − ๐๐ ๐ก Transmitted wave ki ๐ธ๐ก = ๐ธ๐ก0 cos ๐๐ก๐ฅ ๐ฅ + ๐๐ก๐ฆ ๐ฆ + ๐๐ก๐ง ๐ง − ๐๐ก ๐ก Since the tangential component of E must be continuous across the boundary z=0 ๐ธ๐ ๐ง = 0 + ๐ธ๐ ๐ง = 0 = ๐ธ๐ก (๐ง = 0) www.iammanuprasad.com This boundary condition can be satisfied by the waves for all x and y only if IMPLearn 1. ω๐ = ω๐ = ω๐ก = ๐ Frequency is unchanged 2. ๐๐๐ฅ = ๐๐๐ฅ = ๐๐ก๐ฅ = ๐๐ฅ 3. ๐๐๐ฆ = ๐๐๐ฆ = ๐๐ก๐ฆ = ๐๐ฆ Phase-matching conditions Under these conditions ๐๐ sin ๐๐ = ๐๐ก sin ๐๐ก ๐๐ sin ๐๐ = ๐๐ sin ๐๐ β1 sin θi ๐๐ But for lossless medium ๐ฝ = ๐ ๐๐ ๐๐ก = ๐ฝ2 = ๐ ๐2 ๐2 ๐๐ = ๐๐ = ๐ฝ1 = ๐ ๐1 ๐1 kiz = β1 cos θi krz = β1 cos θr sin ๐๐ก ๐๐ = sin ๐๐ ๐๐ก = ๐๐ ๐1 ๐1 ๐2 ๐2 ๐1 sin ๐๐ = ๐2 sin ๐๐ก β1 sin θr Snell’s law ๐1 = ๐ ๐1 ๐1 ๐2 = ๐ ๐2 ๐2 ๐๐ก Refractive indices β2 sin θt ktz = β2 cos θt www.iammanuprasad.com IMPLearn PLANE WAVE AT OBLIQUE INCIDENCE - CASE 1: PARALLEL POLARIZATION Medium 1 E field lies in the xz – plane, the plane of incident, illustrates the case of parallel polarization, Medium 2 Er kr Et In medium 1 ๐ธ๐๐ = ๐ธ๐0 cos ๐๐ ๐๐ฅ − sin ๐๐ ๐๐ง ๐ ๐ป๐๐ = Ht Hr −๐๐ฝ1 (๐ฅ ๐ ๐๐๐๐ +๐ง cos ๐๐ ) ๐ธ๐0 −๐๐ฝ (๐ฅ ๐ ๐๐๐ +๐ง cos ๐ ) ๐ ๐ ๐ ๐ 1 ๐ฆ ๐1 ๐๐ ๐ธ๐๐ = ๐ธ๐0 cos ๐๐ ๐๐ฅ + sin ๐๐ ๐๐ง ๐ −๐๐ฝ1 (๐ฅ ๐ ๐๐๐๐ −๐ง cos ๐๐ ) Where ๐ธ๐0 −๐๐ฝ (๐ฅ ๐ ๐๐๐ −๐ง cos ๐ ) ๐ ๐ ๐ ๐ป๐๐ = − ๐ 1 ๐ฆ ๐ฝ1 = ๐ ๐1 ๐1 ๐1 ๐๐ก z ๐๐ Ei Hi In medium 2 ๐ธ๐ก๐ = ๐ธ๐ก0 cos ๐๐ก ๐๐ฅ − sin ๐๐ก ๐๐ง ๐ −๐๐ฝ2 (๐ฅ ๐ ๐๐๐๐ก +๐ง cos ๐๐ก ) ๐ป๐ก๐ = ๐ธ๐ก0 −๐๐ฝ (๐ฅ ๐ ๐๐๐ +๐ง cos ๐ ) ๐ก ๐ก ๐ ๐ 2 ๐ฆ ๐2 Where ๐ฝ2 = ๐ ๐2 ๐2 www.iammanuprasad.com IMPLearn Requiring that θr = θi and that the tangential components of E & H be continuous at the boundaries z=0, we get ๐ธ๐๐ + ๐ธ๐๐ = ๐ธ๐ก๐ ๐ธ๐0 + ๐ธ๐0 cos ๐๐ = ๐ธ๐ก0 cos ๐๐ก ๐ป๐๐ + ๐ป๐๐ = ๐ป๐ก๐ 1 1 ๐ธ๐0 − ๐ธ๐0 = ๐ธ๐ก0 ๐1 ๐2 Expressing Er0 and Et0 in terms of Ei0 Γ|| = ๐ธ๐0 ๐2 cos ๐๐ก − ๐1 cos ๐๐ = ๐ธ๐0 ๐2 cos ๐๐ก + ๐1 cos ๐๐ ๐ธ๐0 = Γ|| ๐ธ๐0 τ|| = ๐ธ๐ก0 2๐2 cos ๐๐ = ๐ธ๐0 ๐2 cos ๐๐ก + ๐1 cos ๐๐ ๐ธ๐ก0 = ๐|| ๐ธ๐0 and These equations are called Fresnel’s equations for parallel polarization www.iammanuprasad.com IMPLearn In this equation, it is possible that Γ|| =0, under this condition there is no reflection (Er0=0), and the incident angle at which this takes place is called Brewster angle θB|| (polarizing angle) The Brewster angle is obtained by settling θi= θB|| when Γ|| =0 ๐2 cos ๐๐ก = ๐1 cos ๐๐ต|| ๐22 (1 − sin2 ๐๐ก ) = ๐12 (1 − sin2 ๐๐ต|| ) sin2 ๐๐ต|| ๐2 ε1 ๐1 ε2 = ε1 2 1− ε2 1− When μ1= μ2= μ0 ๐ก๐๐ ๐๐ต|| = ε2 ε1 www.iammanuprasad.com IMPLearn PLANE WAVE AT OBLIQUE INCIDENCE - CASE 2 : PERPENDICULAR POLARIZATION Medium 2 Medium 1 When the E field is perpendicular to the plane of incidence which is perpendicular polarization Hr kr Et In medium 1 ๐ธ๐๐ = ๐ธ๐0 ๐ −๐๐ฝ1 (๐ฅ ๐ ๐๐๐๐ +๐ง cos ๐๐ ) ๐๐ฆ ๐ป๐๐ = Er Ht ๐ธ๐0 − cos ๐๐ ๐๐ฅ + sin ๐๐ ๐๐ง ๐ −๐๐ฝ1 (๐ฅ ๐ ๐๐๐๐ +๐ง cos ๐๐ ) ๐1 ๐๐ ๐ธ๐๐ = ๐ธ๐0 ๐ −๐๐ฝ1 (๐ฅ ๐ ๐๐๐๐ −๐ง cos ๐๐ ) ๐๐ฆ ๐ป๐๐ = ๐๐ก z ๐๐ ๐ธ๐0 cos ๐๐ ๐๐ฅ + sin ๐๐ ๐๐ง ๐ −๐๐ฝ1 (๐ฅ ๐ ๐๐๐๐ −๐ง cos ๐๐ ) ๐1 Ei In medium 2 Hi ๐ธ๐ก๐ = ๐ธ๐ก0 ๐ −๐๐ฝ2 (๐ฅ ๐ ๐๐๐๐ก +๐ง cos ๐๐ก ) ๐๐ฆ ๐ป๐ก๐ = ๐ธ๐ก0 − cos ๐๐ก ๐๐ฅ + sin ๐๐ก ๐๐ง ๐ −๐๐ฝ2 (๐ฅ ๐ ๐๐๐๐ก +๐ง cos ๐๐ก ) ๐2 www.iammanuprasad.com IMPLearn Requiring that θr = θi and that the tangential components of E & H be continuous at the boundaries z=0, we get ๐ธ๐๐ + ๐ธ๐๐ = ๐ธ๐ก๐ ๐ธ๐0 + ๐ธ๐0 = ๐ธ๐ก0 ๐ป๐๐ + ๐ป๐๐ = ๐ป๐ก๐ 1 1 ๐ธ๐0 − ๐ธ๐0 cos ๐๐ = ๐ธ๐ก0 cos ๐๐ก ๐1 ๐2 Expressing Er0 and Et0 in terms of Ei0 Γ⊥ = ๐ธ๐0 ๐2 cos ๐๐ − ๐1 cos ๐๐ก = ๐ธ๐0 ๐2 cos ๐๐ + ๐1 cos ๐๐ก ๐ธ๐0 = Γ⊥ ๐ธ๐0 and τ⊥ = ๐ธ๐ก0 2๐2 cos ๐๐ = ๐ธ๐0 ๐2 cos ๐๐ + ๐1 cos ๐๐ก ๐ธ๐ก0 = ๐⊥ ๐ธ๐0 These equations are called Fresnel’s equations for perpendicular polarization www.iammanuprasad.com IMPLearn In this equation, it is possible that Γ⊥ =0, under this condition there is no reflection (Er=0), and the incident angle at which this takes place is called Brewster angle θB⊥ (polarizing angle) The Brewster angle is obtained by settling θi= θB⊥ when Γ⊥ =0 ๐2 cos ๐๐ต⊥ = ๐1 cos ๐๐ก ๐22 (1 − sin2 ๐๐ต⊥ ) = ๐12 (1 − sin2 ๐๐ก ) sin2 ๐๐ต⊥ ๐1 ε2 ๐2 ε1 = μ1 2 1− μ2 1− When ε1= ε2 = ε0 ๐ก๐๐ ๐๐ต⊥ = ๐2 ๐1 www.iammanuprasad.com
