Huygens

1
Huygens and Fermats principles (Hecht 4.4, 4.5)

• Application to reflection refraction at an
  interface
• Monday Sept. 9, 2002

2
Huygens wave front construction
Construct the wave front tangent to the wavelets
What about r direction? See Bruno Rossi Optics.
Reading, Mass Addison-Wesley Publishing Company,
1957, Ch. 1,2 for mathematical explanation
3
Plane wave propagation

• New wave front is still a plane as long as
  dimensions of wave front are gtgt ?
• If not, edge effects become important
• Note no such thing as a perfect plane wave, or
  collimated beam

4
Geometric Optics

• As long as apertures are much larger than a
  wavelength of light (and thus wave fronts are
  much larger than ?) the light wave front
  propagates without distortion (or with a
  negligible amount)
• i.e. light travels in straight lines

5
Physical Optics

• If, however, apertures, obstacles etc have
  dimensions comparable to ? (e.g. lt 103 ?) then
  wave front becomes distorted

6
Lets reflect for a moment
7
Heros principle

• Hero (150BC-250AD) asserted that the path taken
  by light in going from some point A to a point B
  via a reflecting surface is the shortest possible
  one

8
Heros principle and reflection
R
O
O
A
9
Lets refract for a moment
10
Speed of light in a medium
Light slows on entering a medium Huygens Also,
if n ? 8 ? 0 i.e. light stops in its track
!!!!! See P. Ball, Nature, January 8, 2002 D.
Philips et al. Nature 409, 490-493 (2001) C. Liu
et al. Physical Review Letters 88, 23602 (2002)
11
Snels law

• 1621 - Willebrord Snel (1591-1626) discovers the
  law of refraction
• 1637 - Descartes (1596-1650) publish the, now
  familiar, form of the law (viewed light as
  pressure transmitted by an elastic medium)
• n1sin?1 n2sin?2

12
Huygens (1629-1695) PrincipleReflection and
Refraction of light

• Light slows on entering a medium
• Reflection and Refraction of Waves
• Click on the link above

13
Total internal reflection
1611 Discovered by Kepler
n1 gt n2
14
Pierre de Fermats principle

• 1657 Fermat (1601-1665) proposed a Principle of
  Least Time encompassing both reflection and
  refraction
• The actual path between two points taken by a
  beam of light is the one that is traversed in the
  least time

15
Fermats principle
O
n1 lt n2
What geometry gives the shortest time between the
points A and B?
16
Optical path length
S
17
Optical path length

• Transit time from S to P

18
Fermats principle

• t OPL/c
• Light, in going from point S to P, traverses the
  route having the smallest optical path length

19
Optical effects

• Looming
• Mirages

20
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
21
Refraction by plane interface Total internal
reflection
n1 gt n2
P
Snells law n1sin?1n2sin?2
22
Examples of prisms and total internal reflection
45o
45o
45o
Totally reflecting prism
45o
Porro Prism
23
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
24
Cartesian Surfaces

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