In geometrical optics, light is assumed to consist of rays that propagate according tom three laws: rectilinear propagation, refraction, and reflection. We begin this chapter with a statement of Fermat's principle and the derivation of these laws from it. We consider the refraction of two neighboring rays and show that their optical path lengths between planes that are perpendicular to one or both of them are equal to each other. The Malus-Dupin theorem, which states that rays are normal to a wavefront and remain so after refraction and/or reflection, is discussed. Hamilton's point characteristic function representing the optical path length of a ray from one point to another is introduced and a relationship between it and unit vectors along the ray through these points is obtained.
An optical imaging system consists of a series of refracting and/or reflecting surfaces that generally have a common axis of rotational symmetry called the optical axis. The surfaces bend light rays from an object according to the laws of geometrical optics to form its image. Gaussian optics or imaging, which is the subject of this chapter, relates the object distance and size to the image distance and size through the parameters of the imaging system such as the radii of curvature of the surfaces and refractive indices of the media between them. Throughout this book, a Cartesian sign convention is used for the object and image distances and their heights, the radii of curvature of surfaces, angles of incidence and refraction or reflection, and the slope angles of the rays. This convention offers the simplicity of few rules to remember, namely, those of a universally known right-handed coordinate system, regardless of whether an object or its image is real or virtual. Any quantities that are numerically negative are indicated with a parenthetical negative sign (â) in the figures.
In Gaussian optics, the angle that a ray from a point object makes with the optical axis or a surface normal is treated as a small quantity so that its sine or tangent is replaced by the angle itself and, as a result, the law of refraction takes a simple form. This assumption or approximation is referred to as the Gaussian or the paraxial (meaning near the optical axis) approximation. Sometimes a distinction is made between Gaussian and paraxial optics in that paraxial optics is a limiting case of Gaussian optics in which the angles are infinitesimal quantities. The rays traced in this approximation are called paraxial rays and the corresponding method of ray tracing is referred to as paraxial ray tracing. The refraction or reflection of a ray incident on a surface takes place at a plane that is tangent to it and passes through its vertex. All of the rays diverging from a point object and propagating through the imaging system converge to a point called the Gaussian image point.
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