Path Difference and Fringe Positions

$\Delta = \frac{dy}{D} \approx d\sin\theta \qquad [L] = \frac{[L][L]}{[L]}$

$y_n^{\text{bright}} = \frac{n\lambda D}{d}, \quad n = 0, \pm1, \pm2, \ldots \qquad [L]$

$y_n^{\text{dark}} = \frac{(2n-1)\lambda D}{2d}, \quad n = 1, 2, 3, \ldots \qquad [L]$

$\boxed{\beta = \frac{\lambda D}{d}} \qquad [L] = \frac{[L][L]}{[L]}$

Intensity

$I = 4I_0\cos^2\!\left(\frac{\phi}{2}\right), \quad \phi = \frac{2\pi}{\lambda}\Delta \qquad [MT^{-3}] = \text{W/m}^2$

$I = I_1 + I_2 + 2\sqrt{I_1 I_2}\cos\phi \quad \text{(general, unequal slits)}$

$I_{\max} = 4I_0 \; (\phi = 2m\pi), \quad I_{\min} = 0 \; (\phi = (2m-1)\pi)$

Single Slit Diffraction

$\text{Minima: } a\sin\theta = n\lambda, \quad n = 1, 2, 3, \ldots$

$\text{Secondary maxima: } a\sin\theta = \frac{(2n+1)\lambda}{2}$

$\text{Central maximum full width} = \frac{2\lambda D}{a} \qquad [L]$

Snell's Law (Huygens' derivation)

$\frac{\sin\theta_1}{\sin\theta_2} = \frac{v_1}{v_2} = \frac{n_2}{n_1}$

Brewster's Law

$\tan\theta_p = n \quad (\theta_p + \theta_r = 90°) \qquad [\text{dimensionless}]$

Malus's Law

$\boxed{I = I_0\cos^2\theta} \qquad [MT^{-3}] = \text{W/m}^2$

Effect of Medium

$\lambda_{\text{medium}} = \frac{\lambda_{\text{vacuum}}}{n}, \quad \beta_{\text{medium}} = \frac{\beta_{\text{air}}}{n}$

Three-Polaroid Chain

$I_0 \xrightarrow{\div 2} \frac{I_0}{2} \xrightarrow{\times\cos^2 45°} \frac{I_0}{4} \xrightarrow{\times\cos^2 45°} \frac{I_0}{8}$

Key Constants

• Visible light: $\lambda \approx 400\text{ nm (violet) to }700\text{ nm (red)}$
• $1\text{ nm} = 10^{-9}\text{ m}$; $1\text{ Å} = 10^{-10}\text{ m}$