Chapter1_The Propagation of Light
-
Upload
maeya-pratama -
Category
Documents
-
view
249 -
download
3
description
Transcript of Chapter1_The Propagation of Light
![Page 1: Chapter1_The Propagation of Light](https://reader037.fdocuments.us/reader037/viewer/2022102513/55cf8ef3550346703b975438/html5/thumbnails/1.jpg)
THE PROPAGATION OF LIGHT
![Page 2: Chapter1_The Propagation of Light](https://reader037.fdocuments.us/reader037/viewer/2022102513/55cf8ef3550346703b975438/html5/thumbnails/2.jpg)
How to View LightHow to View Light
As a ParticleAs a Particle
As a RayAs a Ray As a WaveAs a Wave
![Page 3: Chapter1_The Propagation of Light](https://reader037.fdocuments.us/reader037/viewer/2022102513/55cf8ef3550346703b975438/html5/thumbnails/3.jpg)
![Page 4: Chapter1_The Propagation of Light](https://reader037.fdocuments.us/reader037/viewer/2022102513/55cf8ef3550346703b975438/html5/thumbnails/4.jpg)
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
![Page 5: Chapter1_The Propagation of Light](https://reader037.fdocuments.us/reader037/viewer/2022102513/55cf8ef3550346703b975438/html5/thumbnails/5.jpg)
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
![Page 6: Chapter1_The Propagation of Light](https://reader037.fdocuments.us/reader037/viewer/2022102513/55cf8ef3550346703b975438/html5/thumbnails/6.jpg)
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
![Page 7: Chapter1_The Propagation of Light](https://reader037.fdocuments.us/reader037/viewer/2022102513/55cf8ef3550346703b975438/html5/thumbnails/7.jpg)
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
![Page 8: Chapter1_The Propagation of Light](https://reader037.fdocuments.us/reader037/viewer/2022102513/55cf8ef3550346703b975438/html5/thumbnails/8.jpg)
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
![Page 9: Chapter1_The Propagation of Light](https://reader037.fdocuments.us/reader037/viewer/2022102513/55cf8ef3550346703b975438/html5/thumbnails/9.jpg)
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
![Page 10: Chapter1_The Propagation of Light](https://reader037.fdocuments.us/reader037/viewer/2022102513/55cf8ef3550346703b975438/html5/thumbnails/10.jpg)
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
![Page 11: Chapter1_The Propagation of Light](https://reader037.fdocuments.us/reader037/viewer/2022102513/55cf8ef3550346703b975438/html5/thumbnails/11.jpg)
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
![Page 12: Chapter1_The Propagation of Light](https://reader037.fdocuments.us/reader037/viewer/2022102513/55cf8ef3550346703b975438/html5/thumbnails/12.jpg)
Reflection and Refraction
![Page 13: Chapter1_The Propagation of Light](https://reader037.fdocuments.us/reader037/viewer/2022102513/55cf8ef3550346703b975438/html5/thumbnails/13.jpg)
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
![Page 14: Chapter1_The Propagation of Light](https://reader037.fdocuments.us/reader037/viewer/2022102513/55cf8ef3550346703b975438/html5/thumbnails/14.jpg)
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
![Page 15: Chapter1_The Propagation of Light](https://reader037.fdocuments.us/reader037/viewer/2022102513/55cf8ef3550346703b975438/html5/thumbnails/15.jpg)
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)
![Page 16: Chapter1_The Propagation of Light](https://reader037.fdocuments.us/reader037/viewer/2022102513/55cf8ef3550346703b975438/html5/thumbnails/16.jpg)
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
![Page 17: Chapter1_The Propagation of Light](https://reader037.fdocuments.us/reader037/viewer/2022102513/55cf8ef3550346703b975438/html5/thumbnails/17.jpg)
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
![Page 18: Chapter1_The Propagation of Light](https://reader037.fdocuments.us/reader037/viewer/2022102513/55cf8ef3550346703b975438/html5/thumbnails/18.jpg)
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
![Page 19: Chapter1_The Propagation of Light](https://reader037.fdocuments.us/reader037/viewer/2022102513/55cf8ef3550346703b975438/html5/thumbnails/19.jpg)
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
![Page 20: Chapter1_The Propagation of Light](https://reader037.fdocuments.us/reader037/viewer/2022102513/55cf8ef3550346703b975438/html5/thumbnails/20.jpg)
Reflection According to Huygens
Reflection According to Huygens
Side-Side-SideDAA’C ADC1 = 1’
Side-Side-SideDAA’C ADC1 = 1’
Incoming ray Outgoing ray
![Page 21: Chapter1_The Propagation of Light](https://reader037.fdocuments.us/reader037/viewer/2022102513/55cf8ef3550346703b975438/html5/thumbnails/21.jpg)
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
![Page 22: Chapter1_The Propagation of Light](https://reader037.fdocuments.us/reader037/viewer/2022102513/55cf8ef3550346703b975438/html5/thumbnails/22.jpg)
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
![Page 23: Chapter1_The Propagation of Light](https://reader037.fdocuments.us/reader037/viewer/2022102513/55cf8ef3550346703b975438/html5/thumbnails/23.jpg)
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
![Page 24: Chapter1_The Propagation of Light](https://reader037.fdocuments.us/reader037/viewer/2022102513/55cf8ef3550346703b975438/html5/thumbnails/24.jpg)
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
![Page 25: Chapter1_The Propagation of Light](https://reader037.fdocuments.us/reader037/viewer/2022102513/55cf8ef3550346703b975438/html5/thumbnails/25.jpg)
Huygens’s Principle and the Law of Refraction, final
• The preceding equation can be simplified to
• This is Snell’s law of refraction
![Page 26: Chapter1_The Propagation of Light](https://reader037.fdocuments.us/reader037/viewer/2022102513/55cf8ef3550346703b975438/html5/thumbnails/26.jpg)
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”
![Page 27: Chapter1_The Propagation of Light](https://reader037.fdocuments.us/reader037/viewer/2022102513/55cf8ef3550346703b975438/html5/thumbnails/27.jpg)
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.
![Page 28: Chapter1_The Propagation of Light](https://reader037.fdocuments.us/reader037/viewer/2022102513/55cf8ef3550346703b975438/html5/thumbnails/28.jpg)
28
Optical path length
n1
n4
n2
n5
nm
n3
S
P
![Page 29: Chapter1_The Propagation of Light](https://reader037.fdocuments.us/reader037/viewer/2022102513/55cf8ef3550346703b975438/html5/thumbnails/29.jpg)
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
![Page 30: Chapter1_The Propagation of Light](https://reader037.fdocuments.us/reader037/viewer/2022102513/55cf8ef3550346703b975438/html5/thumbnails/30.jpg)
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
![Page 31: Chapter1_The Propagation of Light](https://reader037.fdocuments.us/reader037/viewer/2022102513/55cf8ef3550346703b975438/html5/thumbnails/31.jpg)
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
![Page 32: Chapter1_The Propagation of Light](https://reader037.fdocuments.us/reader037/viewer/2022102513/55cf8ef3550346703b975438/html5/thumbnails/32.jpg)
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?
![Page 33: Chapter1_The Propagation of Light](https://reader037.fdocuments.us/reader037/viewer/2022102513/55cf8ef3550346703b975438/html5/thumbnails/33.jpg)
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
![Page 34: Chapter1_The Propagation of Light](https://reader037.fdocuments.us/reader037/viewer/2022102513/55cf8ef3550346703b975438/html5/thumbnails/34.jpg)
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
![Page 35: Chapter1_The Propagation of Light](https://reader037.fdocuments.us/reader037/viewer/2022102513/55cf8ef3550346703b975438/html5/thumbnails/35.jpg)
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.
![Page 36: Chapter1_The Propagation of Light](https://reader037.fdocuments.us/reader037/viewer/2022102513/55cf8ef3550346703b975438/html5/thumbnails/36.jpg)
![Page 37: Chapter1_The Propagation of Light](https://reader037.fdocuments.us/reader037/viewer/2022102513/55cf8ef3550346703b975438/html5/thumbnails/37.jpg)
![Page 38: Chapter1_The Propagation of Light](https://reader037.fdocuments.us/reader037/viewer/2022102513/55cf8ef3550346703b975438/html5/thumbnails/38.jpg)
Electromagnetic Waves
![Page 39: Chapter1_The Propagation of Light](https://reader037.fdocuments.us/reader037/viewer/2022102513/55cf8ef3550346703b975438/html5/thumbnails/39.jpg)
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
![Page 40: Chapter1_The Propagation of Light](https://reader037.fdocuments.us/reader037/viewer/2022102513/55cf8ef3550346703b975438/html5/thumbnails/40.jpg)
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)
![Page 41: Chapter1_The Propagation of Light](https://reader037.fdocuments.us/reader037/viewer/2022102513/55cf8ef3550346703b975438/html5/thumbnails/41.jpg)
Its general solutions (plane wave) :
trkie
B
E
B
E
0
0
![Page 42: Chapter1_The Propagation of Light](https://reader037.fdocuments.us/reader037/viewer/2022102513/55cf8ef3550346703b975438/html5/thumbnails/42.jpg)
TE TM
Electromagnetic Wave Boundary Conditions
(E fields)
Light at a Plane Dielectric Interface
![Page 43: Chapter1_The Propagation of Light](https://reader037.fdocuments.us/reader037/viewer/2022102513/55cf8ef3550346703b975438/html5/thumbnails/43.jpg)
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?
![Page 44: Chapter1_The Propagation of Light](https://reader037.fdocuments.us/reader037/viewer/2022102513/55cf8ef3550346703b975438/html5/thumbnails/44.jpg)
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:
![Page 45: Chapter1_The Propagation of Light](https://reader037.fdocuments.us/reader037/viewer/2022102513/55cf8ef3550346703b975438/html5/thumbnails/45.jpg)
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.
![Page 46: Chapter1_The Propagation of Light](https://reader037.fdocuments.us/reader037/viewer/2022102513/55cf8ef3550346703b975438/html5/thumbnails/46.jpg)
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!
![Page 47: Chapter1_The Propagation of Light](https://reader037.fdocuments.us/reader037/viewer/2022102513/55cf8ef3550346703b975438/html5/thumbnails/47.jpg)
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
![Page 48: Chapter1_The Propagation of Light](https://reader037.fdocuments.us/reader037/viewer/2022102513/55cf8ef3550346703b975438/html5/thumbnails/48.jpg)
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
![Page 49: Chapter1_The Propagation of Light](https://reader037.fdocuments.us/reader037/viewer/2022102513/55cf8ef3550346703b975438/html5/thumbnails/49.jpg)
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
![Page 50: Chapter1_The Propagation of Light](https://reader037.fdocuments.us/reader037/viewer/2022102513/55cf8ef3550346703b975438/html5/thumbnails/50.jpg)
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:
![Page 51: Chapter1_The Propagation of Light](https://reader037.fdocuments.us/reader037/viewer/2022102513/55cf8ef3550346703b975438/html5/thumbnails/51.jpg)
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
![Page 52: Chapter1_The Propagation of Light](https://reader037.fdocuments.us/reader037/viewer/2022102513/55cf8ef3550346703b975438/html5/thumbnails/52.jpg)
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
![Page 53: Chapter1_The Propagation of Light](https://reader037.fdocuments.us/reader037/viewer/2022102513/55cf8ef3550346703b975438/html5/thumbnails/53.jpg)
![Page 54: Chapter1_The Propagation of Light](https://reader037.fdocuments.us/reader037/viewer/2022102513/55cf8ef3550346703b975438/html5/thumbnails/54.jpg)
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
![Page 55: Chapter1_The Propagation of Light](https://reader037.fdocuments.us/reader037/viewer/2022102513/55cf8ef3550346703b975438/html5/thumbnails/55.jpg)
![Page 56: Chapter1_The Propagation of Light](https://reader037.fdocuments.us/reader037/viewer/2022102513/55cf8ef3550346703b975438/html5/thumbnails/56.jpg)
![Page 57: Chapter1_The Propagation of Light](https://reader037.fdocuments.us/reader037/viewer/2022102513/55cf8ef3550346703b975438/html5/thumbnails/57.jpg)
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
![Page 58: Chapter1_The Propagation of Light](https://reader037.fdocuments.us/reader037/viewer/2022102513/55cf8ef3550346703b975438/html5/thumbnails/58.jpg)
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
![Page 59: Chapter1_The Propagation of Light](https://reader037.fdocuments.us/reader037/viewer/2022102513/55cf8ef3550346703b975438/html5/thumbnails/59.jpg)
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.
![Page 60: Chapter1_The Propagation of Light](https://reader037.fdocuments.us/reader037/viewer/2022102513/55cf8ef3550346703b975438/html5/thumbnails/60.jpg)
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
![Page 61: Chapter1_The Propagation of Light](https://reader037.fdocuments.us/reader037/viewer/2022102513/55cf8ef3550346703b975438/html5/thumbnails/61.jpg)
Photons and The Laws of Reflection and Refraction