Chapter 2: Geometrical optics

All of geometrical optics boils down to…

Law of reflection: ri

i

normal

n1

n2

r

t Law of refraction“Snell’s Law”:

1

2

sin

sin

n

n

t

i

Easy to prove by two concepts:Huygens’ principleFermat’s principle

Incident, reflected, refracted, and normal in same plane

Huygens’ principleevery point on a wavefront may be regarded as a secondary source of wavelets

planar wavefront:

ct

curved wavefront:

In geometrical optics, this region should be dark (rectilinear propagation).

Ignore the peripheral and back propagating parts!

obstructed wavefront:

Huygens’ proof of law of reflection

L

L

tci

90cos

r90 i90

L

tcr

90cos

ri

riθ

vt t

vi t

i

L

t

L

tvii

sin

L

tvtt

sin

itti vv sinsin

Huygens’ proof of law of refraction

ttii nn sinsin vi = c/ni

vt = c/nt

“Economy of nature”shortest path between 2 points

Hero—least distance: Fermat—least time:

Fermat’s principlethe path a beam of light takes between two points is the one which is traversed in the least time

n1n2

O

a

b

c

x

Fermat’s proof of law of refraction

ti v

OB

v

AOt

ti v

xcb

v

xat

2222

0

2222

xcbv

xc

xav

x

dx

dt

ti

0sinsin

t

t

i

i

vvdx

dt

A

B

i

normal

t

22sin

xa

xi

22sin

xcb

xct

ttii nn sinsin

provide qualitative (and quantitative) proof of the law of reflection and refraction within the limit of geometrical optics.

Huygens’- and Fermat’s principles:

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

Note: virtual images (cannot be projected on screen)

point object extended object

object displaced from mirror multiple images in perperdicular mirrors

image point; SN = SN′

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

Imaging by an optical system

Reflections from spherical surfaces

virtual image

mirror convex 0,

mirror concave 0,

2

Rffocal length:

mirror equation:fss

1

1

1

s

s

o

i

h

hmmagnification:

Chicago

Ray tracingthree 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: s

s

h

hm

o

i

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 ~. Same for the best spatial resolution of an image. This is fundamentally due to the wave nature of light.

To be continued…

> ~

~0

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

Exercises

M.C. Escher