Summary

This chapter developed the ray (geometrical) picture of how light forms images, from the simplest single reflecting or refracting surface up to complete multi-element optical instruments. Reflection at a spherical mirror obeys the mirror formula $\frac1v+\frac1u=\frac1f$ (magnification $m=-v/u$), with the New Cartesian sign convention fixing the sign of every distance so that one formula covers both concave and convex mirrors and every object position. Refraction at a plane boundary obeys Snell's law, $n_1\sin i=n_2\sin r$; when light travels from a denser to a rarer medium beyond the critical angle ($\sin C=1/n$), it undergoes total internal reflection, the working principle of the totally-reflecting prism and, most importantly, of the optical fibre (light guided by repeated TIR at a core-cladding boundary of stepped refractive index). Refraction at a single spherical surface, $\frac{n_2}{v}-\frac{n_1}{u}=\frac{n_2-n_1}{R}$, is the building block for the thin lens formula $\frac1v-\frac1u=\frac1f$ and for the Lens-Maker's formula $\frac1f=(n-1)\left(\frac1{R_1}-\frac1{R_2}\right)$; a lens's power $P=1/f$ (in dioptres) adds directly for lenses in contact ($P=P_1+P_2+\cdots$), and a silvered lens combines lens and mirror powers as $\frac1F=\frac2{f_{\text{lens}}}+\frac1{f_{\text{mirror}}}$. The displacement method finds a convex lens's focal length from two conjugate lens positions as $f=(D^2-d^2)/4D$. A prism refracts twice and, because refractive index depends slightly on wavelength, disperses white light into its spectrum, with $n=\sin\!\left(\frac{A+D_m}{2}\right)/\sin\!\left(\frac A2\right)$ at minimum deviation. Rayleigh scattering ($\propto1/\lambda^4$) explains the blue sky and the red sunrise/sunset. The human eye focuses light via the cornea and the accommodating crystalline lens onto the retina between its near point ($D\approx25\ \text{cm}$) and far point (infinity); myopia (corrected by a concave lens) and hypermetropia (corrected by a convex lens) are opposite shifts of this range. Finally, the simple microscope ($M=1+D/f$ or $D/f$), the compound microscope ($M …