Chapter 27: Geometrical Optics – Reflection & Refraction

The theory is rooted in advanced concepts from Hamiltonian mechanics, specifically the principle that light always follows the path requiring the minimum transit time. The mathematical framework begins by analyzing light refraction at a single spherical surface. To manage the intricate geometry required for perfect focusing, the analysis employs the crucial simplification of paraxial rays, assuming all light paths remain very close to the central axis. This geometric constraint enables the derivation of the foundational relationship linking the object distance, the image distance, the medium's refractive index, and the radius of curvature of the surface. The discussion then establishes necessary sign conventions to distinguish between real images (where light physically converges) and virtual images (where light rays only appear to diverge from a point). Expanding this concept to lenses, the full lens maker's rule is developed, which defines the dual focal lengths of the system. The chapter explores magnification, deriving geometric proportionality relationships for thin lenses that connect object and image heights to their positions. For multi-element or compound lens systems, the entire assembly can be analyzed by using the image of one element as the object for the next, or simplified by defining the overall system using principal planes. Finally, the limitations of geometrical optics are examined, first through aberrations like spherical aberration (where non-paraxial rays fail to focus perfectly) and chromatic aberration (where light color causes focus differences), and ultimately by the physical constraint of resolving power, which is governed by the wave phenomenon of diffraction and sets the absolute theoretical limit on optical clarity.