Chapter 2: Geometrical optics

Table of Opticks, from the 1728 Cyclopaedia, Volume 2

All of geometrical optics boils down to… normal Law of reflection: qi qr n1 n2 qt Law of refraction “Snell’s Law”: Incident, reflected, refracted, and normal in same plane Easy to prove by two concepts: Huygens’ principle Fermat’s principle

Huygens’ principle every point on a wavefront may be regarded as a secondary source of wavelets curved wavefront: planar wavefront: obstructed wavefront: In geometrical optics, this region should be dark (rectilinear propagation). Ignore the peripheral and back propagating parts! cDt

Huygens’ proof of law of reflection

Huygens’ proof of law of refraction vi Dt qi vi = c/ni vt = c/nt qt vt Dt L

shortest path between 2 points “Economy of nature” shortest path between 2 points Hero—least distance: Fermat—least time: Fermat’s principle the path a beam of light takes between two points is the one which is traversed in the least time

Fermat’s proof of law of refraction normal A qi a n1 O n2 x b qt B c

Huygens’- and Fermat’s principles: provide qualitative (and quantitative) proof of the law of reflection and refraction within the limit of geometrical optics.

Principle of reversibility In life If you don’t use it, you lose it (i.e. fitness; calculus) If you can take it apart you should be able to put it back together Do unto others as you would have them do to you … In optics Rays in optics take the same path backward or forwards

Reflections from plane surfaces retroreflector

Image formation in plane mirrors point object extended object image point; SN = SN′ Note: virtual images (cannot be projected on screen) object displaced from mirror multiple images in perperdicular mirrors

Imaging by an optical system conjugate points Fermat’s principle: every ray from O to I has same transit time (isochronous) Principle of reversibility: I and O are interchangeable (conjugate) Perfect imaging: Cartesian surfaces (i.e. ellipsoid; hyperbolic lens) Practical imaging: Spherical surfaces

Reflections from spherical surfaces virtual image Chicago focal length: mirror equation: magnification:

Ray tracing three principle rays determine image location Starting from object point P: (1) parallel—focal point (2) focal point—parallel (3) center of curavature—same Image at point of intersection P′ Concave: real (for objects outside focal point) Convex: virtual

Ray tracing for (thin) lenses converging lens diverging lens magnification:

Simple lens systems

Is geometrical optics the whole story? No. -neglects the phase -implies that we could focus a beam to a point with zero diameter and so obtain infinite intensity and infinitely good spatial resolution. The smallest possible focal spot is ~l. Same for the best spatial resolution of an image. This is fundamentally due to the wave nature of light. To be continued… ~0 > ~l

Exercises M.C. Escher You are encouraged to solve all problems in the textbook (Pedrotti3). The following may be covered in the werkcollege on 1 September 2010: Chapter 1 2, 10, 17 Chapter 2 4, 6, 9, 25, 27, 31