A vessel contains two immiscible liquids of refractive indices $n_1 = 1.5$ and $n_2 = 1.25$. A coin lies at the bottom of the vessel. The height of liquid 1 (bottom, $n_1$) is 3 cm and liquid 2 (top, $n_2$) is 2.5 cm. The apparent depth of the coin as seen from above is:

Refraction: bending of light at interface. Snell\'s Law: $n_1\sin\theta_1 = n_2\sin\theta_2$. Refractive index $n = c/v = \lambda_{vacuum}/\lambda_{medium}$. Denser medium ($n$ larger): light slows, bends toward normal. Rarer medium ($n$ smaller): light speeds up, bends away from normal. Critical angle: $\sin\theta_c = n_2/n_1$, TIR occurs when $\theta > \theta_c$ going from denser to rarer.

For an object at real depth $d$ in medium of index $n$, seen from air: by Snell\'s law for small angles: $n\sin\theta_r \approx n\theta_r = 1 \cdot \sin\theta_i \approx \theta_i$ where $\theta_r$ = angle in medium, $\theta_i$ = angle in air. $\tan\theta_r \approx \theta_r = x/d$ (real), $\tan\theta_i \approx \theta_i = x/d_{app}$ (apparent). So $n/d = 1/d_{app}$ → $d_{app} = d/n$. For multiple layers: each layer shifts image; apparent depth adds: $d_{app} = \sum d_i/n_i$.

Mirage: hot air near ground has lower $n$; light from sky undergoes TIR → appears as water. Looming: cold air near surface has higher $n$; objects appear elevated. Twinkling of stars: atmospheric refraction and turbulence. Prism: $\delta = (n-1)A$ for thin prism; dispersion because $n$ varies with $\lambda$. Optical fibre: TIR keeps light inside fibre core; used in telecommunications (internet, cable TV) and medical endoscopes.

Mirror: $\frac{1}{v} + \frac{1}{u} = \frac{1}{f} = \frac{2}{R}$, magnification $m = -v/u$. Thin lens: $\frac{1}{v} - \frac{1}{u} = \frac{1}{f}$ (Gaussian form), magnification $m = v/u$. Lens maker\'s equation: $\frac{1}{f} = (n-1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$. Lens power: $P = 1/f$ (in metres), unit = Dioptre (D). Lenses in contact: $P_{total} = P_1 + P_2$. Sign convention: distances measured from optical centre, positive in direction of incident light.

When light travels from a denser to a rarer medium and the angle of incidence exceeds the critical angle, it is entirely reflected back. The critical angle satisfies $\sin C = 1/\mu$, so a denser medium has a smaller critical angle — about 49° for water and 24° for diamond, which is why diamonds sparkle so strongly. Two conditions are needed and both must be stated: denser to rarer, and incidence beyond the critical angle.

For a prism, $A + \delta = i + e$, where A is the prism angle and $\delta$ the deviation. Deviation is minimum when the ray passes symmetrically, with $i = e$, and at that point $\mu = \frac{\sin\left(\frac{A+\delta_m}{2}\right)}{\sin(A/2)}$ — the standard method for measuring refractive index. Because $\mu$ differs slightly for each colour, violet deviates most and red least, producing dispersion.

An object under water appears raised because refraction bends the emerging rays, giving apparent depth = real depth / $\mu$. This is why a pool looks shallower than it is and a stick appears bent at the surface. A glass slab instead produces no deviation but a lateral shift, because the two surfaces are parallel and the second refraction undoes the angular change of the first while displacing the ray sideways.