YDSE Formulas

$\Delta = d\sin\theta \approx \frac{dy}{D} \quad [\text{L}] \quad \text{(path difference)}$

$y_n^\text{bright} = \frac{n\lambda D}{d} \quad [\text{L}] \quad \text{(nth bright fringe position)}$

$y_n^\text{dark} = \frac{(2n-1)\lambda D}{2d} \quad [\text{L}] \quad \text{(nth dark fringe position)}$

$\beta = \frac{\lambda D}{d} \quad [\text{L}] \quad \text{(fringe width, SI: m)}$

$\phi = \frac{2\pi}{\lambda}\Delta \quad [\text{dimensionless}] \quad \text{(phase difference, SI: rad)}$

$I = 4I_0\cos^2\!\left(\frac{\phi}{2}\right) \quad [\text{M T}^{-3}] \quad \text{(intensity, equal slits, SI: W\,m}^{-2})$

$I = I_1 + I_2 + 2\sqrt{I_1 I_2}\cos\phi \quad [\text{M T}^{-3}] \quad \text{(intensity, unequal slits)}$

$I_\text{max} = \left(\sqrt{I_1}+\sqrt{I_2}\right)^2, \quad I_\text{min} = \left(\sqrt{I_1}-\sqrt{I_2}\right)^2$

Single Slit Diffraction

$a\sin\theta = n\lambda \quad [\text{dimensionless}] \quad \text{(nth minimum condition, } n = 1,2,\ldots\text{)}$

$a\sin\theta = \frac{(2n+1)\lambda}{2} \quad \text{(nth secondary maximum, approximation)}$

$W_\text{central} = \frac{2\lambda D}{a} \quad [\text{L}] \quad \text{(linear width of central max)}$

$\theta_\text{angular,half} = \frac{\lambda}{a} \quad [\text{rad}] \quad \text{(angular half-width of central max)}$

Polarization

$\tan\theta_p = n \quad [\text{dimensionless}] \quad \text{(Brewster's law)}$

$\theta_p + \theta_r = 90° \quad \text{(at Brewster's angle)}$

$I = I_0\cos^2\theta \quad [\text{M T}^{-3}] \quad \text{(Malus's law, SI: W\,m}^{-2})$

$I_\text{after polaroid 1} = \frac{I_0}{2} \quad \text{(unpolarized through first polaroid)}$

$I_\text{three polaroids, 45°} = \frac{I_0}{8} \quad \text{(crossed ends + middle at 45°)}$

Fringe Shift (Glass Slab)

$y_\text{shift} = \frac{(n-1)t\,D}{d} \quad [\text{L}] \quad \text{(fringe shift due to slab)}$

$N = \frac{(n-1)t}{\lambda} \quad \text{(number of fringes shifted)}$

Medium Effect

$\lambda' = \frac{\lambda}{n}, \quad \beta' = \frac{\beta}{n} \quad \text{(wavelength and fringe width in medium of refractive index } n)$