Fringe width formula: $\beta = \lambda D/d$. The fringe width is directly proportional to $\lambda$ and $D$, inversely proportional to $d$. All fringes in YDSE have equal width; the central fringe is bright (zero path difference for all wavelengths).

Bright fringe condition: path difference $\Delta = n\lambda$ ($n = 0, 1, 2, \ldots$). Dark fringe condition: $\Delta = (2n-1)\lambda/2$ ($n = 1, 2, 3, \ldots$). The $n$th dark fringe lies between the $(n-1)$th and $n$th bright fringes.

Intensity in YDSE: $I = 4I_0\cos^2(\phi/2)$ for identical slits. $I_{\max} = 4I_0$; $I_{\min} = 0$. The ratio $I_{\max}/I_{\min}$ is infinite for equal slits (completely dark minima). For unequal slits, $I_{\min} > 0$.

Effect of medium: In a medium of refractive index $n$, the wavelength changes to $\lambda/n$. Fringe width becomes $\beta/n$ (decreases). Slit separation $d$ and screen distance $D$ are unchanged.

Single slit central maximum: Full width $= 2\lambda D/a$; angular full width $= 2\lambda/a$. First minimum at $a\sin\theta = \lambda$. Secondary maxima have half the width of the central maximum.

Coherence requirement: For sustained visible interference, sources must have the same frequency and a constant phase difference. Independent sources (even lasers pointed at the same screen) produce rapidly shifting patterns that appear uniform to the eye.

Brewster's law: $\tan\theta_p = n$. At the Brewster angle: (i) reflected light is 100% plane-polarized with $\vec{E}$ perpendicular to the plane of incidence; (ii) reflected and refracted rays are at $90°$ to each other; (iii) no refracted light is polarized in that same plane.

Malus's law: $I = I_0\cos^2\theta$. First polaroid halves unpolarized intensity regardless of orientation ($I_0 \to I_0/2$). This is because $\langle\cos^2\theta\rangle = 1/2$ averaged over all angles.

Three-polaroid result: Unpolarized $I_0 \xrightarrow{P_1} I_0/2 \xrightarrow{P_2 \text{ at }45°} I_0/4 \xrightarrow{P_3 \text{ at }45° \text{ to }P_2} I_0/8$.

Transverse wave proof: Polarization is uniquely possible for transverse waves. Sound (longitudinal) cannot be polarized. The success of Brewster's law and Malus's law in describing light confirms its transverse electromagnetic nature.$