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

2. 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 ( )
Thomas Young
( )
Sir Isaac Newton ( )
Pierre de Fermat ( )

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

4. 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

5. 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

6. 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

7. 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

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

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

10. 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:

11. 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

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

13. 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

14. Nature of light: Rays and Maxwell Equations
Fermat principle

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

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

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

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

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

20. 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

21. 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

22. Nature of light: Ray tracing examples

23. 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.