Huygens’ and Fermat’s principles
(Textbook 4.4, 4.5)
Application to reflection & refraction at an
interface
1
Propagation of light: Sources
E.g. point source – a fundamental source
Light is emitted in all directions – series of
crests and troughs (stone dropped in water)
Rays – lines
perpendicular to
wave fronts
λ
Wave front - Surface of
constant phase
2
Propagation of light: Sources
Incandescent light bulb (blackbody radiation source)
B
I lamp = ∫ I p ( s )ds
A
A
B
1. Glass bulb
2. Low pressure inert gas
3. Tungsten filament
4. Contact wire (goes out of stem)
5. Contact wire (goes into stem)
6. Support wires
7. Stem (Glass mount)
8. Contact wire (goes out of stem)
9. Cap (Sleeve)
10. Insulation (Vitrite)
11. Electrical contact
3
Two ideal light waves
Spherical waves (from a point source) – wave fronts
are spherical
Plane waves (from a point source at infinite) – wave
fronts are planes
Rays – lines perpendicular to wave fronts in the
direction of propagation
x
Planes parallel to y-z plane
4
Huygens’ principle
In the 17th Century, Christiaan Huygens (1629–
1695) proposed what we now know as Huygens’
Principle, one of the fundamental concepts of waves
and wave optics.
A typical statement of the principle is “every point on
a wavefront acts as a source of a new wavefront,
propagating radially outward.”
5
Huygen’s principle
Every point on a wave front is a source of
secondary wavelets.
i.e. particles in a medium excited by electric
field (E) re-radiate in all directions
i.e. in vacuum, E, B fields associated with
wave act as sources of additional fields
6
Huygens’ wave front construction
New wavefront
Construct the
wave front tangent
to the wavelet
r = c ∆t ≈ λ
Given wave-front at t
What about –r direction
(focusing wave or beams)?
Allow wavelets to evolve
for time ∆t
7
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
8
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
9
Physical Optics
If, however, apertures,
obstacles etc have
dimensions comparable
to λ (e.g. < 103 λ) then
wave front becomes
distorted
10
Geometric Optics
Let’s reflect for a moment
11
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.
12
Hero’s principle and reflection
A
B
R
O’
O O”
A’
13
Law of reflection
θi = θ r
14
Reflection by plane surfaces
Reflecting through (x, z) plane
y
z
r2= (-x,y,z)
r1 = (x,y,z)
x
r1 = (x,y,z)
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)
15
Geometric Optics
Let’s refract for a moment
16
Speed of light in a medium
c
v=
n
n – refractive index
Light slows on entering a medium – Huygens
Also, if n → ∞ ν = 0
i.e. light stops in its track !!!!! See:
P. Ball, Nature, January 8, 2002
D. Philips et al. Nature 409,
409 490-493 (2001)
C. Liu et al. Physical Review Letters 88,
88 23602 (2002)
17
Index of refraction
A property of a material that changes the
speed of light.
speed of light in vacuum
c
nα =
=
speed of light in medium α v
nα = ε r µ r
where εr is the material's relative permittivity, and µr is its
relative permeability. For a non-magnetic material, µr is very
close to 1, therefore n is approximately ε r .
18
Speed of light in a medium
c
v=
n
Light slows on entering a medium – Huygens
Also, if n → ∞ ν = 0
i.e. light stops in its track !!!!! See:
P. Ball, Nature, January 8, 2002
D. Philips et al. Nature 409,
409 490-493 (2001)
C. Liu et al. Physical Review Letters 88,
88 23602 (2002)
19
Refraction: Bending light
20
Refraction: Bending light
As light passes from one transparent medium to another, it
changes speed, and bends. How much this happens depends
on the refractive index of the mediums and the angle
between the light ray and the line perpendicular (normal) to
the surface separating the two mediums (medium/medium
interface).
Each medium has a different refractive index. The angle
between the light ray and the normal as it leaves a medium
is called the angle of incidence. The angle between the light
ray and the normal as it enters a medium is called the angle
of refraction.
21
Refraction: Bending light
n1 is the refractive index of the medium the light is leaving,
θ1 is the incident angle between the light ray and the normal
to the medium to medium interface,
n2 is the refractive index of the medium the light is entering,
θ2 is the refractive angle between the light ray and the
normal to the medium to medium interface.
22
Snell’s law
1621 - Willebrord Snell (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)
n1sinθ1 = n2sinθ2
23
Snell’s law
n1sinθ1 = n2sinθ2
24
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”
25
Pierre de Fermat’s principle
"Light, in going between two points, traverses the
route having the smallest optical path length."
c
vα =
nα
d 1
1
t=
= × (nα ⋅ d ) = × OPL
vα c
c
OPL = nαd
= index of refraction × distance traveled
26
Fermat’s principle
“The actual path between
two points taken by a
beam of light is the one
that is traversed in the
least time”
Light, in going from point
S to P, traverses the route
having the smallest
optical path length
OPL
t=
c
27
Application of Fermat’s principle
o
28
Application of Fermat’s principle
o
c
(v = )
n
29
Optical path length
S
n1
n2
n3
n4
n5
P
nm
30
Optical path length
Transit time from S to P
P
m
1
t = ∑ ni s i
c i =1
m
OPL = ∑ ni si
i =1
OPL = ∫ n( s )ds
S
P
c
OPL = ∫ ds
v
S
Same for all rays
31
Optical path length
In an inhomogeneous medium the refractive index n(r) is a
function of the position. The optical path length along a given
path between two points A and B is therefore
where ds is the differential element of the length along the
path (Could be more than one path).
r(x,y,z)
A
ds
B
n(r)
32
Nature of light
Law of Reflection:
θi = θ r
Law of Refraction
Snell’s Law:
ni sin θi = nr sin θ r
33
Refraction by plane interface
& Total internal reflection
n2
θ2
θ2
n1 > n2
θ1
n1
θ1 θ
C
P
Snell’s law
n1sinθ1=n2sinθ2
34
Total internal reflection (TIR)
1611 – Discovered by Kepler
θC
n1
θc – Critical angle
n1 sin θ c = n2 sin 90°
n2
 n2 
θ c = sin  
 n1 
−1
n1 > n2
35
TIR Critical Angle
Light bends toward the normal when the light enters a
medium of greater refractive index, and away from
the normal when entering a medium of lesser
refractive index.
As you approach the critical angle the refracted light
approaches 90° and, at the critical angle, the angle of
refraction becomes 90° and the light is no longer
transmitted across the medium/medium interface. For
angles greater in absolute value than the critical
angle, all the light is reflected. This is called total
internal reflection.
36
Refraction by plane interface
& Total internal reflection
n2
 n2 
θ c = sin  
 n1 
−1
n1 > n2
θC
θ1
θ1
n1
P
Snell’s law
n1sinθ1=n2sinθ2
37
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
38
Examples of prisms and total internal
reflection
45o
45o
45o
Totally reflecting prism
45o
Porro Prism
39
Porro prism
A Porro prism, named for its inventor Ignazio Porro, is a type
of reflection prism used in optical instruments to alter the
orientation of an image.
An image travelling through a Porro prism is rotated by 180°
and exits in the opposite direction offset from its entrance point.
Since the image is reflected twice, the handedness of the image
is unchanged.
40
Porro prism
An image travelling through a Porro prism is rotated by 180°
and exits in the opposite direction offset from its entrance point.
Since the image is reflected twice, the handedness of the image
is unchanged.
41
Porro prism
An image travelling through a Porro prism is rotated by 180°
and exits in the opposite direction offset from its entrance point.
Since the image is reflected twice, the handedness of the image
is unchanged.
Porro prism binoculars
42
Image inversion and reversion
43
Optical waveguide
Many devices take advantage of the total internal reflection,
including optical waveguides (like optical fiber). A waveguide is
a length of transparent material that is surrounded by material
that has a lower index of refraction. Rays that intersect the
interface between the waveguide material and the surrounding
material at angles equal to or larger than the critical angle are
trapped in the waveguide and travel losslessly along it.
Rays Can be bound (trapped) in a waveguide through total internal reflection.
44
Optical waveguide
45
Optical waveguide
Many devices take advantage of the total internal reflection,
including optical waveguides (like optical fiber). A waveguide is
a length of transparent material that is surrounded by material
that has a lower index of refraction. Rays that intersect the
interface between the waveguide material and the surrounding
material at angles equal to or larger than the critical angle are
trapped in the waveguide and travel losslessly along it.
Rays Can be bound (trapped) in a waveguide through total internal reflection.
46
Natural phenomenon
47