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Chapter 2: Geometrical optics
All of geometrical optics boils down to…
normal
i
Law of reflection:
i   r
Law of refraction
“Snell’s Law”:
sin  i  n2

sin  t  n1
r
n1
n2
t
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:
c Dt
obstructed
wavefront:
In geometrical optics, this
region should be dark
(rectilinear propagation).
Ignore the peripheral and
back propagating parts!
Huygens’ proof of law of reflection
r
i
90   r
cos90   i  
cDt
L
cos90   r  
cDt
L
90  i
 θi   r
L
Huygens’ proof of law of refraction
vi Dt
sin  i 
L
v Dt
sin  t  t
L
vi sin t  vt sin i
i
ni sin i  nt sin t
vi = c/ni
vt = c/nt
t
L
“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
t
normal
AO OB

vi
vt
b 2  c  x 
a2  x2
t

vi
vt
2
A
i
a
n1
n2
dt
x
cx


0
2
2
2
2
dx vi a  x
vt b  c  x 
sin  i 
O
x
a2  x2
sin t 
cx
b 2  c  x 
dt sin  i sin  t


0
dx
vi
vt
b
t
c
x
B
ni sin i  nt sin t
2
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:
Practical imaging:
Cartesian surfaces (i.e. ellipsoid; hyperbolic lens)
Spherical surfaces
Reflections from spherical surfaces
virtual image
Chicago
focal length:
f 
R   0, concave mirror

2   0, convex mirror
1 1
1


mirror equation:
s s
f
magnification:
m
hi s

ho s
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:
hi
s
m

ho
s
Simple lens systems
Is geometrical optics the whole story?
No.
-neglects the phase
~0
-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…
> ~l
Exercises
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
M.C. Escher
Chapter 2
4, 6, 9, 25, 27, 31