Plane EM Wave: Oblique Incidence 𝐸 ⊥ 𝑟 = 𝐸 0⊥ 𝑟 𝑒 −𝑗 𝛽 𝑟 ∙ 𝑟 𝐸 ⊥ 𝑖 = 𝐸 0⊥ 𝑖 𝑒 −𝑗 𝛽 𝑖 ∙ 𝑟 𝐸 || 𝑖 = 𝐸 0|| 𝑖 𝑒 −𝑗 𝛽 𝑖 ∙ 𝑟 𝐸 || 𝑟 = 𝐸 0|| 𝑟 𝑒 −𝑗 𝛽 𝑟 ∙ 𝑟 Plane of Incidence 𝐸 ⊥ 𝑡 = 𝐸 0⊥ 𝑡 𝑒 −𝑗 𝛽 𝑡 ∙ 𝑟 0°≤ 𝜃 𝑖 , 𝜃 𝑟 , 𝜃 𝑡 ≤90° 𝐸 || 𝑡 = 𝐸 0|| 𝑡 𝑒 −𝑗 𝛽 𝑡 ∙ 𝑟

Propagation Directions and Magnitude Relations 𝐸 0⊥ 𝑟 𝐸 0⊥ 𝑖 = 𝜂 𝑡 cos 𝜃 𝑖 − 𝜂 𝑖 cos 𝜃 𝑡 𝜂 𝑡 cos 𝜃 𝑖 + 𝜂 𝑖 cos 𝜃 𝑡 𝐸 0⊥ 𝑡 𝐸 0⊥ 𝑖 = 2 𝜂 𝑡 cos 𝜃 𝑖 𝜂 𝑡 cos 𝜃 𝑖 + 𝜂 𝑖 cos 𝜃 𝑡 𝐸 0|| 𝑟 𝐸 0|| 𝑖 = 𝜂 𝑖 cos 𝜃 𝑖 − 𝜂 𝑡 cos 𝜃 𝑡 𝜂 𝑖 cos 𝜃 𝑖 + 𝜂 𝑡 cos 𝜃 𝑡 𝐸 0|| 𝑡 𝐸 0|| 𝑖 = 2 𝜂 𝑡 cos 𝜃 𝑖 𝜂 𝑖 cos 𝜃 𝑖 + 𝜂 𝑡 cos 𝜃 𝑡 𝜃 𝑖 = 𝜃 𝑟 𝑛= 𝜀 𝑟 𝑛: index of refraction 𝜀 𝑖 𝜇 𝑖 sin 𝜃 𝑖 = 𝜀 𝑡 𝜇 𝑡 sin 𝜃 𝑡 𝜀 𝑟 : relative permittivity 𝑛 𝑖 sin 𝜃 𝑖 = 𝑛 𝑡 sin 𝜃 𝑡

Reflection and Transmission Coefficients Γ || =− 𝐸 0|| 𝑟 𝐸 0|| 𝑖 Γ ⊥ = 𝐸 0⊥ 𝑟 𝐸 0⊥ 𝑖 𝑇 || = 𝐸 0|| 𝑡 𝐸 0|| 𝑖 𝑇 ⊥ = 𝐸 0⊥ 𝑡 𝐸 0⊥ 𝑖

Reflection and Transmission Coefficients Γ ⊥ = 𝐸 0⊥ 𝑟 𝐸 0⊥ 𝑖 = 𝜂 𝑡 cos 𝜃 𝑖 − 𝜂 𝑖 cos 𝜃 𝑡 𝜂 𝑡 cos 𝜃 𝑖 + 𝜂 𝑖 cos 𝜃 𝑡 Γ || =− 𝐸 0|| 𝑟 𝐸 0|| 𝑖 =− 𝜂 𝑖 cos 𝜃 𝑖 − 𝜂 𝑡 cos 𝜃 𝑡 𝜂 𝑖 cos 𝜃 𝑖 + 𝜂 𝑡 cos 𝜃 𝑡 𝑇 ⊥ = 𝐸 0⊥ 𝑡 𝐸 0⊥ 𝑖 = 2 𝜂 𝑡 cos 𝜃 𝑖 𝜂 𝑡 cos 𝜃 𝑖 + 𝜂 𝑖 cos 𝜃 𝑡 𝑇 || = 𝐸 0|| 𝑡 𝐸 0|| 𝑖 = 2 𝜂 𝑡 cos 𝜃 𝑖 𝜂 𝑖 cos 𝜃 𝑖 + 𝜂 𝑡 cos 𝜃 𝑡

Reflection and Transmission Coefficients Γ ⊥ = 𝐸 0⊥ 𝑟 𝐸 0⊥ 𝑖 = 𝑛 𝑖 cos 𝜃 𝑖 − 𝑛 𝑡 cos 𝜃 𝑡 𝑛 𝑖 cos 𝜃 𝑖 + 𝑛 𝑡 cos 𝜃 𝑡 Γ || =− 𝐸 0|| 𝑟 𝐸 0|| 𝑖 =− 𝑛 𝑡 cos 𝜃 𝑖 − 𝑛 𝑖 cos 𝜃 𝑡 𝑛 𝑡 cos 𝜃 𝑖 + 𝑛 𝑖 cos 𝜃 𝑡 𝑇 ⊥ = 𝐸 0⊥ 𝑡 𝐸 0⊥ 𝑖 = 2 𝑛 𝑖 cos 𝜃 𝑖 𝑛 𝑖 cos 𝜃 𝑖 + 𝑛 𝑡 cos 𝜃 𝑡 𝑇 || = 𝐸 0|| 𝑡 𝐸 0|| 𝑖 = 2 𝑛 𝑖 cos 𝜃 𝑖 𝑛 𝑡 cos 𝜃 𝑖 + 𝑛 𝑖 cos 𝜃 𝑡

Reflection and Transmission Coefficients Example 1: air to water 𝑛 𝑖 =1, 𝑛 𝑡 =4/3 Brewster angle 𝜃 𝐵

Reflection and Transmission Coefficients Example 2: water to air 𝑛 𝑖 =4/3, 𝑛 𝑡 =1 𝜃 𝑖 > 𝜃 𝑐 𝜃 𝑐 𝜃 𝐵 Brewster angle Critical angle

Total (Internal) Reflection: Critical Angle Γ ⊥ = 𝐸 0⊥ 𝑟 𝐸 0⊥ 𝑖 = 𝑛 𝑖 cos 𝜃 𝑖 − 𝑛 𝑡 cos 𝜃 𝑡 𝑛 𝑖 cos 𝜃 𝑖 + 𝑛 𝑡 cos 𝜃 𝑡 Γ || =− 𝐸 0|| 𝑟 𝐸 0|| 𝑖 =− 𝑛 𝑡 cos 𝜃 𝑖 − 𝑛 𝑖 cos 𝜃 𝑡 𝑛 𝑡 cos 𝜃 𝑖 + 𝑛 𝑖 cos 𝜃 𝑡 𝜃 𝑖 = 𝜃 𝑐 Critical angle 𝜃 𝑡 =90° Γ ⊥ =1 Γ || =−1 𝑛 𝑖 sin 𝜃 𝑖 = 𝑛 𝑡 sin 𝜃 𝑡 sin 𝜃 𝑐 = 𝑛 𝑡 / 𝑛 𝑖 𝑛 𝑖 > 𝑛 𝑡

Total Transmission: Brewster Angle Γ ⊥ = 𝐸 0⊥ 𝑟 𝐸 0⊥ 𝑖 = 𝑛 𝑖 cos 𝜃 𝑖 − 𝑛 𝑡 cos 𝜃 𝑡 𝑛 𝑖 cos 𝜃 𝑖 + 𝑛 𝑡 cos 𝜃 𝑡 Γ || =− 𝐸 0|| 𝑟 𝐸 0|| 𝑖 =− 𝑛 𝑡 cos 𝜃 𝑖 − 𝑛 𝑖 cos 𝜃 𝑡 𝑛 𝑡 cos 𝜃 𝑖 + 𝑛 𝑖 cos 𝜃 𝑡 𝜃 𝑖 = 𝜃 𝐵 Brewster angle (Polarization angle) 𝜃 𝐵 + 𝜃 𝑡 =90° Γ ⊥ ≠0 Γ || =0 Parallel component only 𝑛 𝑖 sin 𝜃 𝑖 = 𝑛 𝑡 sin 𝜃 𝑡 tan 𝜃 𝐵 = 𝑛 𝑡 / 𝑛 𝑖 𝑛 𝑖 sin 𝜃 𝐵 = 𝑛 𝑡 cos 𝜃 𝐵

Total Reflection/Transmission: Water 𝑛 𝑤𝑎𝑡𝑒𝑟 = 𝜀 𝑤𝑎𝑡𝑒𝑟 𝑟𝑒𝑙 ≈ 4 3 tan 𝜃 𝐵 = 𝑛 𝑡 𝑛 𝑖 𝑛 𝑎𝑖𝑟 = 𝜀 𝑎𝑖𝑟 𝑟𝑒𝑙 ≈1 sin 𝜃 𝑐 = 𝑛 𝑡 𝑛 𝑖 Ray travels from Air to water Water to air 𝜃 𝑐 N/A 48.59° 𝜃 𝐵 53.13° 36.87°