Geometrical optics and Imaging theory EE 231 Introduction to Optics Geometrical optics and Imaging theory lesson 1 Andrea Fratalocchi www.primalight.org

Nature of light: Rays vs. Waves Before the nineteenth century, light was considered to be a stream of particles. Newton and Fermat were the main architect of this vision. Newton believed that light ‘particles’ left the object and stimulated the ‘sense of light’ upon entering in the eyes. The modern theory of images is still mainly based on such a ‘particle’ approach, based on Fermat principle Christian Huygens, conversely, argued that light is a wave and light propagation is described by a wave motion and not by a particle. Thomas Young in 1801 provides the first clear experiment showing the wave nature of light. Young experiments shown interferences effects that could not be explained by particles. Christiaan Huygens (1629-1695) Thomas Young (1773-1829) Sir Isaac Newton (1642-1726) Pierre de Fermat (1607-1665)

Nature of light: Rays vs. Waves Who is right?

Nature of light: Rays vs. Waves Who is right? All of them: to every wave it can be associated a particle behavior, which holds true in approximate conditions. Wave Optics Ray optics

Nature of light: Rays Plane waves How do I associate rays to waves? Rays are constructed from surfaces of constant phase, which define the wavefront. Rays are orthogonal to the wavefront of a wave. Spherical waves

Nature of light: Rays How do light rays propagate? Fermat Principle. Optical rays traveling between two points A and B follow a path such that the time of travel (or the optical pathlength) between the two points is minimum The refractive index of a material is n=c0/c, i.e., the ratio between the speed of light in vacuum and the speed of light in the medium. Therefore, the time T taken by light to travel a distance L is T=L/c=nL/c0. The time T is proportional to the optical pathlength, or optical path, nL

Nature of light: Rays Propagation of rays in an homogeneous medium In an homogeneous medium the refractive index is constant everywhere. The path of minimum distance between to points A and B is a straight line, which implies that: In an homogeneous medium light rays travel in straight lines

Nature of light: Rays What path would you choose? Propagation of rays at the interface between two different media What path would you choose?

Nature of light: Rays What path would you choose? Propagation of rays at the interface between two different media What path would you choose?

Nature of light: Rays What path would you choose? Propagation of rays at the interface between two different media What path would you choose? The optical path is:

Nature of light: Rays What path would you choose? Propagation of rays at the interface between two different media What path would you choose? The optical path is: Snell Law, demonstrated via Fermat principle

Nature of light: Rays Propagation of rays at the interface between two different media Snell law

Nature of light: Rays Propagation of rays at the interface between two different media Snell law Snell law is nonlinear, however for small angles and it becomes: Paraxial Snell law

Nature of light: Rays and Maxwell Equations Fermat principle

Nature of light: Rays and Maxwell Equations Fermat principle Path of minimum time

Nature of light: Rays and Maxwell Equations Fermat principle Equation of rays

Nature of light: Rays and Maxwell Equations Fermat principle Equation of rays Eikonal equation

Nature of light: Rays and Maxwell Equations Fermat principle Equation of rays Eikonal equation

Nature of light: Rays and Maxwell Equations Fermat principle Equation of rays Equation of rays Maxwell equations Wave Optics Ray optics

Nature of light: Rays vs. Waves Who is right? All of them: Fermat and Newton were probing a specific limit of waves, which under short wavelengths conditions become equivalent to a stream of particles. In optics, this limit is called geometrical optics or ray optics Wave Optics Ray optics

Nature of light: Ray optics Fermat principle Propagation of straight lines in homogeneous media Law of reflection and refraction at discontinuous interfaces between materials of different refractive index

Nature of light: Ray tracing examples

Ray and Geometrical Optics References A. Yariv, P. Yeh, Photonics, 6th Ed., Chapter 2 M. Born and E. Wolf, Principle of Optics, 6th Ed., Chapter 3.