Geometric Optics: Lenses & Imaging
Snell's law, thin lenses, and ray tracing through optical systems.
Geometric optics treats light as rays — valid when wavelengths are tiny compared to features. At an interface, Snell's law applies:
$$n_1 \sin\theta_1 = n_2 \sin\theta_2.$$Total internal reflection occurs when $\sin\theta_1 > n_2/n_1$ (critical angle).
A thin lens in air with refractive index $n$ and radii of curvature $R_1, R_2$ has focal length
$$\frac{1}{f} = (n - 1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right) \quad \text{(lensmaker)}.$$For object distance $s_o$ and image distance $s_i$, the thin-lens equation is
$$\frac{1}{f} = \frac{1}{s_o} + \frac{1}{s_i}, \qquad m = -\frac{s_i}{s_o}.$$Sign conventions: $f > 0$ for converging lenses; $s_o, s_i > 0$ on opposite sides. Three principal rays — through the centre, parallel→through focal point, focal point→parallel — locate the image.