Download Huygens` and Fermat`s Principles – Application to reflection

Huygens’ and Fermat’s
principles (Hecht 4.4, 4.5)
Application to reflection & refraction at an
interface
Monday Sept. 9, 2002
1
Huygens’ wave front construction
Construct the
New wavefront
wave front tangent
to the wavelets
r = c Δt ≈ λ
Given wavefront at t
Allow wavelets to evolve
for time Δt
What about –r direction?
See Bruno Rossi Optics. Reading, Mass:
Addison-Wesley Publishing Company, 1957, Ch. 1,2
for mathematical explanation
2
Plane wave propagation
New wave front is
still a plane as long
as dimensions of
wave front are >>
λ
If not, edge effects
become important
Note: no such
thing as a perfect
plane wave, or
collimated beam
3
Geometric Optics
As long as apertures
are much larger than
a wavelength of light
(and thus wave fronts
are much larger than
λ) the light wave
front propagates
without distortion (or
with a negligible
amount)
i.e. light travels in
straight lines
4
Physical Optics
If, however,
apertures,
obstacles etc have
dimensions
comparable to λ
(e.g. < 103 λ) then
wave front
becomes distorted
5
Let’s reflect for a moment
6
Hero’s 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
7
Hero’s principle and reflection
A
B
R
O’
O
A’
8
Let’s refract for a moment
9
Speed of light in a medium
c
v
n
Light slows on entering a medium – Huygens
Also, if n → ∞  = 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)
10
Snel’s 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
11
Huygens’ (1629-1695) Principle:
Reflection and Refraction of light
Light slows on entering a medium
Reflection and Refraction of Waves
Click on the link above
12
Total internal reflection
1611 – Discovered by Kepler
θC
n1
n2
n1 > n2
13
Pierre de Fermat’s 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”
14
Fermat’s principle
A
n1
θi
h
O
n1 < n2
x
n2
b
What geometry gives the
shortest time between
the points A and B?
θr
a
B
15
Optical path length
S
n1
n2
n3
n4
n5
P
nm
16
Optical path length
Transit time from S to P
m
1
t   ni si
c i 1
m
OPL   ni si
i 1
P
OPL   n(s)ds
S
P
c
OPL   ds
v
S
Same for all rays
17
Fermat’s principle
t = OPL/c
Light, in going
from point S to P,
traverses the route
having the smallest
optical path length
OPL
t
c
18
Optical effects
Looming
Mirages
19
Reflection
by plane surfaces
y
r1 = (x,y,z)
z
r2= (-x,y,z)
r1 = (x,y,z)
x
r3=(-x,-y,z)
x
y
r4=(-x-y,-z)
r2 = (x,-y,z)
Law of Reflection
r1 = (x,y,z) → r2 = (x,-y,z)
Reflecting through ((x,z) plane
20
Refraction by plane interface
& Total internal reflection
n2
θ2
θ2
n1 > n2
θ1
θ1 θ
C
θ1
θ1
n1
P
Snell’s law
n1sinθ1=n2sinθ2
21
Examples of prisms and total
internal reflection
45o
45o
45o
Totally reflecting prism
45o
Porro Prism
22
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
Optical
O
System
Fermat’s principle – optical path length of every ray passing
through I must be the same
I
23
Cartesian Surfaces
Cartesian surfaces – those surfaces
which form perfect images of a point
object
E.g. ellipsoid and hyperboloid
24