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Formulas

Snell’s Law:

\[ Φ_L = arcsin( \frac{sin(Φ_i) ⋅ n_air}{n_L} ) \]

\[ Φ_S = arcsin( \frac{sin(Φ_L) ⋅ n_L}{n_S} ) \]

Fresnel Equation:

air/layer:

\[ r_s,al = \frac{n_air ⋅ cos(φ_i) - n_L ⋅ cos(φ_L)}{n_air ⋅ cos(φ_i) + n_L ⋅ cos(φ_L)} \]

\[ r_p,al = \frac{n_L ⋅ cos(φ_i) - n_air ⋅ cos(φ_L)}{n_L ⋅ cos(φ_i) + n_air ⋅ cos(φ_L)} \]

layer/substrate:

\[ r_s,ls = \frac{n_L ⋅ cos(φ_L) - n_S ⋅ cos(φ_S)}{n_L ⋅ cos(φ_L) + n_S ⋅ cos(φ_S)} \]

\[ r_p,ls = \frac{n_S ⋅ cos(φ_L) - n_L ⋅ cos(φ_S)}{n_S ⋅ cos(φ_L) + n_L ⋅ cos(φ_S)} \]

Fujiwara ellipsometer

\[ β = \frac{2π ⋅ d}{λ} ⋅ N_L ⋅ cos(φ_L) \]

\[ ρ = \frac{r_p}{r_s} = \frac{r_p,al + r_p,ls ⋅ e^-2iβ}{1 + r_p,al ⋅ r_p,ls ⋅ e^-2iβ} ⋅ \frac{1 + r_s,al ⋅ r_s,ls ⋅ e^-2iβ}{r_s,al + r_s,ls ⋅ e^-2iβ} \] (Kehrwert zweiter Bruch = r_s)

\[ ρ = tan(Ψ) ⋅ e^iΔ \]