Ray Tracing in Spherical Interfaces Using Geometric Algebra

In this chapter, we have shown how geometric algebra may be used to derive the spherical-polar ray tracing equations for finite skew, paraxial skew, and finite meridional rays in spherical interfaces. In Section II, we summarized the state of the art in the vector formulations of the laws of geometric optics using geometric algebra. This section was divided into four parts. In the first part, we introduced geometric algebra and its two fundamental theorems: the Pauli identity for vector multiplication and the Euler theorem for vector rotation. In the next three parts, we reformulated the laws of geometric optics. For propagation, we derived the propagation distance of the ray before it hits a spherical interface, which is either concave or convex depending on the sign of the concavity function. And for refraction and reflection, we discussed three known vector formulations: the asymmetric exponential form of Sugon and McNamara, the product forms of Hestenes and of Born and Wolf, and the addition form of Klein and Furtak. We also discussed a new one: the symmetric exponential form. We mentioned that this form is the vector form of the Bessel-Conrady refraction invariant and we showed that this invariant is related to the normal vector. In Section III, we derived the ray tracing equations for finite skew rays in spherical coordinates. In the first part, we reformulated the spherical coordinate system in terms of exponential rotation operators or rotors. In the next three parts, we used the vector addition form of the laws in Klein and Furtak to derive the corresponding ray tracing equations for finite skew rays. We argued that compared to the Cartesian coordinate system used in the geometric optics literature, the spherical coordinate system is more appropriate for describing finite skew rays: the polar and azimuthal angles of the rays immediately define their paraxiality and skewness, respectively. Also, though our recasting of the spherical coordinate system is already known in the geometric algebra literature, what may be new is our rotor product derivation of cross and dot product expressions in terms of sums and differences of spherical coordinate angles, expressed in compact form using a uniform subscripting system. These compact forms greatly simplify the ray tracing equations for finite skew rays. In Section IV, we took the paraxial limits of the ray tracing equations derived in the previous section. To do this, we defined sign functions, such as the axial direction function and the relative direction function, that generalize the concavity function in Section II. These two sign functions are related by the following theorem: the relative direction of two vectors is equal to the product of the axial directions of the two vectors. These sign functions provide us with a consistent and transparent sign convention system. They also enable us to express the sines and cosines of paraxial polar angles as step functions, making a clear distinction between polar angles that are close to 0 and those that are close to π. In this way, we were able to derive some useful results, such as the approximate expression for the paraxial angle of incidence in terms of the polar and azimuthal angles of the incident ray and the normal vector to the interface. And in Section V, we reformulated the polar coordinate system using plane exponential rotation operators. We then used this formalism to derive the ray tracing equations for finite meridional rays. In particular, we derived the Bessel-Conrady invariant and other useful relations for refraction and reflection, by using the exponential and product form of the laws given in Sugon and McNamara and in Hestenes, respectively. Considering that Conrady derived his invariant from a geometric analysis of a ray-tracing diagram, our algebraic derivation that equates the arguments of exponential functions provides a more straightforward approach, with sign conventions handled by sign functions such as the concavity and cross-functions that are explicitly embedded on the ray tracing equations. We hope that the ray tracing equations in polar and spherical coordinates derived in this chapter may provide new insights on classical optics problems such as image formation and aberration theory.