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THE PROPAGATION OF LIGHT
How to View LightHow to View Light
As a ParticleAs a Particle
As a RayAs a Ray As a WaveAs a Wave
Theories on nature of light:Light as a particle vs. Light as a wave
• Only corpuscular theory of light prevalent until 1660
• Francesco Maria Grimaldi (Bologna) described diffraction in 1660
Light as a particleSir Isaac Newton (1642-1727)• Embraces corpuscular theory of
light because of inability to explain rectilinear propagation in terms of waves
• Demonstrates that white light is mixture of a range of independent colors
• Different colors excite ether into characteristic vibrations---sensation of red corresponds to longer ether vibration
Light as a wave
Christiaan Huygens (1629-1695)Huygens’ principle (Traite de la
Lumière, 1678):Every point on a primary wavefront
serves as the source of secondary spherical wavelets, such that the primary wavefront at some later time is the envelope of these wavelets. Wavelets advance with speed and frequency of primary wave at each point in space
http://id.mind.net/~zona/mstm/physics/waves/propagation/huygens1.html
Light as a wave
Thomas Young (1773-1829)
1801-1803: double slit experiment, showing interference by light from a single source passing though two thin closely spaced slits projected on a screen far away from the slitshttp://vsg.quasihome.com/interfer.htm
Light as a waveAugustine Fresnel (1788-
1827)1818: Developed
mathematical wave theory combining concepts from Huygens’ wave propagation and wave interference to describe diffraction effects from slits and small apertures
Electromagnetic wave nature of light
• Michael Faraday (1791-1865)
• 1845: demonstrated electromagnetic nature of light by showing that you can change the polarization direction of light using a strong magnetic field
Electromagnetic theory
• James Clerk Maxwell (1831-1879)
• 1873: Theory for electromagnetic wave propagation
• Light is an electromagnetic disturbance in the form of waves propagated through the ether
Quantum mechanics• 1900: Max Planck postulates that
oscillating electric system imparts its energy to the EM field in quanta
• 1905: Einstein-photoelectric effect– Light consists of individual energy quanta, photons, that
interact with electrons like particle• 1900-1930 it becomes obvious that concepts of
wave and particle must merge in submicroscopic domain
• Photons, protons, electrons, neutrons have both particle and wave manifestations– Particle with momentum p has associated wavelength
given by p=h/l• QM treats the manner in which light is absorbed
and emitted by atoms
Max Planck
Niels Bohr
Louis de Broglie
Schrödinger
Heisenberg
Reflection and Refraction
vph = c/n1
vph = c/n2
qiqr
qt
normal
ri
1
2
sin
sin
nn
t
i
All of Geometrical optics boils down to…
Law of Reflection:
Snell’s Law:
The Snell’s Laws
Willebrordus Snellius
Fundamental Rules for Reflection and Refractionin the limit of Ray Optics
Fundamental Rules for Reflection and Refractionin the limit of Ray Optics
1. Huygens’s Principle2. Fermat’s Principle3. Electromagnetic Wave Boundary
Conditions
1. Huygens’s Principle2. Fermat’s Principle3. Electromagnetic Wave Boundary
Conditions
Huygen’s Principle
• Huygen assumed that light is a form of wave motion rather than a stream of particles
• Huygen’s Principle is a geometric construction for determining the position of a new wave at some point based on the knowledge of the wave front that preceded it
Christian Huygens (1629-1695)
Huygen’s Principle, cont.
• All points on a given wave front are taken as point sources for the production of spherical secondary waves, called wavelets, which propagate in the forward direction with speeds characteristic of waves in that medium– After some time has elapsed, the new position
of the wave front is the surface tangent to the wavelets
Huygen’s Construction for a Plane Wave
• At t = 0, the wave front is indicated by the plane AA’
• The points are representative sources for the wavelets
• After the wavelets have moved a distance cΔt, a new plane BB’ can be drawn, which is the tangent to the wavefronts
Huygen’s Construction for a Spherical Wave
• The points are representative sources for the wavelets
• The new wavefront is tangent at each point to the wavelet
Huygen’s Principle and the Law of Reflection
• The Law of Reflection can be derived from Huygen’s Principle
• AA’ is a wave front of incident light
• The reflected wave front is CD
Reflection According to Huygens
Reflection According to Huygens
Side-Side-SideDAA’C ADC1 = 1’
Side-Side-SideDAA’C ADC1 = 1’
Incoming ray Outgoing ray
Huygen’s Principle and the Law of Reflection, cont.
• Triangle ADC is congruent to triangle AA’C
• θ1 = θ1’• This is the Snell’s Law
of Reflection
Huygen’s Principle and the Law of Refraction
• Every point on a wave front can be considered to be a source of secondary waves. The figure explains the refraction at an interface between media with different optical densities.
Air
Huygens’s Principle and the Law of Refraction, cont.
• Ray 1 strikes the surface and at a time interval ∆t later, ray 2 strikes the surface– During this time interval, the wave
at A sends out a wavelet, centered at A, toward D
– The wave at B sends out a wavelet, centered at B, toward C
Huygens’s Principle and the Law of Refraction, cont.
• The two wavelets travel in different media, therefore their radii are different
• From triangles ABC and ADC, we find
Huygens’s Principle and the Law of Refraction, final
• The preceding equation can be simplified to
• This is Snell’s law of refraction
26
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”
Fermat’s Principle
The path a beam of light takes between two points is the one which is traversed in the least time.
A B
Isotropic medium: constant velocity.
Minimum time = minimum path length.
28
Optical path length
n1
n4
n2
n5
nm
n3
S
P
Optical Path Length (OPL)
When n constant, OPL = n geometric length.
nvac vac
LL
n > 1n = 1
For n = 1.5, OPL is 50% larger than L
For n = 1.5, OPL is 50% larger than L
P
SdxxnOPL )(
S P
30
Fermat’s principle
• t = OPL/c• Light, in going from
point S to P, traverses the route having the smallest optical path length
c
OPLt
31
Optical path length
• Transit time from S to P
m
iiisn
ct
1
1
m
iiisnOPL
1
P
SdssnOPL )(
P
S
dsv
cOPL
Same for all rays
32
n1
n2
Fermat’s principle
n1 < n2
A
O
B
θi
θr
x
a
h
bWhat geometry gives the shortest time between the points A and B?
n1
n2
qi
qt
normalA
B
O
Method 1
a
b
c
ti vOB
vAO
t
x
ti v
xcbv
xat
2222
2222 xcbv
xc
xav
xdxdt
ti
0
sinsin t
t
i
i
vvdxdt
ttii nn sinsin
Method 2
Minimizing the time (optical path length) between points Q and Q’ yields Snell’s Law:
2
2/122
1
2/122
21
)'())((
'
v
xh
v
xpht
V
AQ
v
QAt
'sin'sin
''
)'('
])([
,
02)'(
2/')22(
)([
2/
:
)'('])([
:
)'('
))((
'
2/1222/122
2/1222/12
2/1222/122
2/122
2/122
nn
andd
xn
d
xpn
or
xh
xn
xph
xpn
thus
xxh
nxp
xph
n
dx
d
atingdifferenti
xhnxphn
ngsubstituti
xhd
xphd
dnndOPL
Fermat’s Principle and ReflectionFermat’s Principle and Reflection
A light ray traveling from one fixed point to another will followa path such that the time required is an extreme point – either amaximum or a minimum.
Electromagnetic Waves
Maxwell’s Equations for time varying electric and magnetic fields in free space
0
E
t
BE
0 B
t
EIB
000
(where r is the charge density)
Simple interpretation
Divergence of electric field is a functionof charge density
A closed loop of E field lines will exist whenthe magnetic field varies with time
Divergence of magnetic field =0(closed loops)
A closed loop of B field lines will exist inThe presence of a current and/or time varying electric field
Description of Light
Wave Equation (derived from Maxwell’s equations)Any function that satisfies this eqn is a waveIt describes light propagation in free space and in time
operatorLaplacian
fieldinductionmagnetic
fieldelectricE
lightofspeedc
wheretc
t
E
cE
2
2
2
22
2
2
22
,
1
1
B
BB
(see calculus review handout)
Its general solutions (plane wave) :
trkie
B
E
B
E
0
0
TE TM
Electromagnetic Wave Boundary Conditions
(E fields)
Light at a Plane Dielectric Interface
TE TM
trkjjoiinc
ii eeE
E
tωrkjjφorref
rrr eeE
E tωrkjjφ
ottransttt eeE
E
ki kr
kt
ki kr
kt
n
Assume:
A plane wave is incident:
A plane wave is reflected:
A plane wave is transmitted:
What are the relative amplitudes, wave numbers, frequencies, and phases?
To remain constant at a certain place:
tri
rkrkrk tri
To remain constant at a certain time:
ki, kr, kt are all co-planar
incident, reflected, and refracted all at same frequency.
tωrktωrktωrk ttrrii
Relationship between fields at the interface should not depend on position or time:
vk ''kv
tkx sin
A boundary at one point in space for all time:
The left side shakes the right at frequency w, which creates a wave with a different velocity (different medium) and therefore different wavelength.
ki krn
O
r
(could be in any direction)ki dot r makes this happen… kr dot r makes this happen…
0 rkk ri
ki - kr
ki-kr dot r makes a plane, but it must be the surface since the boundary condition is for r at the surface.
But this can only be true if kr is also in the plane of incidence!
kikr
kt
r
rrii rkrk 22 coscos
Same medium, same velocity, same wavelength, same wavenumber, so:
ttii rkrk 22 coscos
qi qr
ri
ttii nn sinsin
qt
to
ti
o
i nn
sin2
sin2
Law of Reflection
Snell’s Law
SCad
tld
BE
SCad
ttld
PEB 000
0tangentialinside
tangentialoutside lElE
tangentialinside
tangentialoutside EE
0tangentialinside
tangentialoutside lBlB
tangentialinside
tangentialoutside BB
Tangential components of both E and B are continuous at the boundary.
Therefore, for all points on the boundary at all times:
tωrkjjφot
tωrkjjφor
tωrkjjφoi
tttrrriii eeEeeEeeE
Now for the relative amplitudes: reflection and transmission
TETM
tri EEE
ttrrii BBB coscoscos
tri BBB
ttrrii EEE coscoscos
BvBE nc
BiBr
E E
Bt
TE
TM
tri EEE
tttiriiii EnEnEn coscoscos
ttriii EnEnEn
ttirii EEE coscoscos
For reflection: eliminate Et, separate Ei and Er, and get ratio:
TE
tnn
i
tnn
i
i
r
i
t
i
t
EE
r
coscos
coscos
TM
tinn
tinn
i
r
i
t
i
t
EE
r
coscos
coscos
Get all in terms of E, and recall that qi = qr:
Apply Snell’s law (let n = nt/ni)
TE
ii
ii
n
nr
22
22
sincos
sincos
TM
ii
ii
nn
nnr
222
222
sincos
sincos
Coefficient of transmission: t
ii
i
i
t
nE
Et
22 sincos
cos2
TE
ii
i
i
t
nn
n
E
Et
222 sincos
cos2
TM
internal reflection: n = 0.667
-0.2
0
0.2
0.4
0.6
0.8
1
0 20 40 60 80
r
Angle of incidence
TM
TE
-1
-0.5
0
0.5
1
0 20 40 60 80
r
Angle of incidence
TM
TE
external reflection: n = 1.5
TE/TM wave optical reflection• TE (transverse electric) polarization
– Electric field parallel to substrate surface
• TM (transverse magnetic) polarization– Magnetic field parallel to substrate surface
low index high index high index low index
TETM
TETM
RS
the critical angle for total reflection
If i cri, then it is total reflection and no power can be transmitted, these fields are referred as evanescent waves.
1 2critical
1
( ) sini
Brewster’s angle for total transmission
For lossless, non-magnetic media, we have
Total transmission for TM polarization
2 2 21 2 2 1
2 2 2 22 1 1 2
( )sini BA
1
1
2
1sin
1BA
r
r
RS
Ex1 A 2 GHz TE wave is incident at 30 angle of incidence from air on to a thick slab of nonmagnetic, lossless
dielectric with r = 16. Find TE and TE.
RS
Ex2 A uniform plane wave is incident from air onto glass at an angle from the normal of 30. Determine the fraction of the incident power that is reflected and transmitted for a) and b). Glass has refractive index n2 =
1.45.
a) TM polarization
b) TE polarization
Photons and The Laws of Reflection and Refraction