Geometrical Optics

1
Geometrical Optics

• Refraction, reflection at a spherical/planar
  interface
• Hecht, Chapter 5
• Wednesday Sept. 11, 2002

2
Independence of path
For any rays traveling from point S to another
point P in an optical system the optical path
lengths are identical!!
3
Reflection by plane surfaces
r1 (x,y,z)
r2 (-x,y,z)
r1 (x,y,z)
r3(-x,-y,z)
r4(-x-y,-z)
r2 (x,-y,z)
Law of Reflection r1 (x,y,z) ? r2
(x,-y,z) Reflecting through (x,z) plane
4
Refraction by plane interface Total internal
reflection
n1 gt n2
P
Snells law n1sin?1n2sin?2
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Examples of prisms and total internal reflection
45o
45o
45o
Totally reflecting prism
45o
Porro Prism
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Fermats principle

• Light, in going from point S to P, traverses the
  route having the smallest optical path length
• More generally, there may be many paths with the
  same minimum transit time, e.g. locus of a
  cartesian surface

7
Imaging by an optical system
O and I are conjugate points any pair of
object-image points - which by the principle of
reversibility can be interchanged
O
I
Fermats principle optical path length of every
ray passing through I must be the same
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Cartesian Surfaces

• A Cartesian surface those which form perfect
  images of a point object
• E.g. ellipsoid and hyperboloid

O
I
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Cartesian refracting surface
ngtn
P(x,y)
n
n
y
x
I
O
s
s
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Cartesian refracting surface

• Surface ƒ(x,y) will be cartesian for points
  points O and I if
• ___________________________________
• The equation defines an ovoid of revolution for a
  given s, s
• Equality means all paths are equal (i.e. for all
  x,y)
• We then have perfect imaging by Fermats
  principle
• But we can see that the surface will be cartesian
  for one set of s, s (no too useful)

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Paraxial ray approximation

• We would like a single surface to provide imaging
  for all s, s.
• This will be true if we place certain
  restrictions on the bundle of rays collected by
  the optical system
• Make the PARAXIAL RAY APPROXIMATION
• Assume y ltlt s,s (i.e. all angles are small)
• x ltlt s, s (of course)

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Paraxial ray approximation

• All distances measured from V (i.e. assume x0)
• All angles are small
• sina tan a a cos a 1
• Snells law
• n? n?

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n
n
V
O
I
C
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Refraction at spherical interfaces

1. Light travels left to right
2. V origin measure all distances from here
3. R positive to the right of V, negative to the
   left
4. S positive for real objects (i.e. one to the
   left of V), negative for virtual
5. S positive for real image (to right of V),
   negative for virtual images
6. Heights y,y positive up, negative down

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Refraction at a spherical interface Paraxial ray
approximation
y
C
Note small angles means that s x s
a ? ?1
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Refraction at a spherical interface Paraxial ray
approximation
I
C
a ?2 ?
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Refraction at a spherical interface Paraxial ray
approximation

• Snells law
• ____________________________
• Leads to