1. 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

2. 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

3. 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

4. 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
Using 𝜂= 𝜇/𝜀
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
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
𝑇 || = 𝐸 0|| 𝑡 𝐸 0|| 𝑖 = 2 𝑛 𝑖 cos 𝜃 𝑖 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
Using 𝑛= 𝜀 𝑟

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