# Fundamentals of geometrical optics by Virendra N. Mahajan

By Virendra N. Mahajan

Optical imaging begins with geometrical optics, and ray tracing lies at its leading edge. This booklet starts off with Fermat's precept and derives the 3 legislation of geometrical optics from it. After discussing imaging via refracting and reflecting structures, paraxial ray tracing is used to figure out the scale of imaging parts and obscuration in replicate structures. Stops, students, radiometry, and optical tools also are mentioned. The chromatic and monochromatic aberrations are addressed intimately, by way of spot sizes and see diagrams of aberrated photos of element gadgets. each one bankruptcy ends with a precis and a collection of difficulties. The booklet ends with an epilogue that summarizes the imaging method and descriptions the following steps inside of and past geometrical optics

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6 Conic Surface and Surface Normal The equation of a conic surface (conicoid) of eccentricity e and vertex radius of curvature R is given by [1] ( ) x 2 + y 2 - 2Rz + 1 - e 2 z 2 = 0 , (1-44) where ( x , y , z ) are the coordinates of a point on its surface with its origin at its vertex. The sag of the surface, namely, the z coordinate, is given by z = (x 2 + y 2) R 1 + [1 - (1 - e 2 ) ( x 2 + y 2 ) R2 ] 1/ 2 . (1-45) If we let e = 0, as for a sphere, Eqs. (1-44) and (1-45) reduce to the corresponding equations (1-30) and (1-31) for a spherical surface, respectively.

1-23) into Eq. (1-22) yields the value of eˆ¢ according to n ¢eˆ¢ = neˆ + ÈÍ n ¢ 2 - n 2 sin 2 q Î ( 12 )1 2 - n cos q ˘˙˚ vˆ . 5 Reflection in 3D (1-24) r Consider a ray originating at a point P( r ) , as in Figure 1-10, incident on a reflecting r surface at a point A0 and passing through a point P ¢ ( r ¢ ) after reflection by the surface, with O as an arbitrary origin of coordinates. This figure, like Figure 1-9, is also not in a plane. Given the incident ray PA0 , we want to determine the reflected ray A0 P ¢ .

5 Paraxial Ray Tracing When the rays make small angles with the optical axis and surface normals, we can approximate their sines and tangents with the angles themselves. Similarly, if the transverse coordinates ( x , y ) of a point on a surface are much smaller than its radius of curvature, we can neglect the sag of a refracting or reflecting surface and approximate the diagonal distance between two points by the corresponding axial distance. Such assumptions yield equations for the transverse coordinates that are no longer coupled.