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Chapter 3 Electromagnetic Theory, Photonsand Light
September 5,8 Electromagnetic waves
3.1 Basic laws of electromagnetic theoryLights are electromagnetic waves.Electric fields are generated by electric charges or time-varying magnetic fields.Magnetic fields are generated by electric currents or time-varying electric fields.Maxwell’s wave equation is derived from the following four laws (Maxwell’s equations).
3.1.1 Faraday’s induction law
SB
lE dt
dAC
Electromotive force (old term, actually a voltage):
SBlE ddt
d
dt
dd
A
M
C
emf
B
d Sd lE
A time-varying magnetic field produces an electric field.
2
3.1.2 Gauss’s law - electricFlux of electric field: SE d
AE For a point charge: , 0 is the permittivity of free space.
0q
E
Generally, VA
dVd 0
1SE
For general material, the permittivity , where KE is the relative permittivity (dielectric constant).
0 EK
3.1.3 Gauss’s law- magneticThere is no isolated magnetic monopoles:
0 SB dAM
3
3.1.4 Ampere’s circuital law
For electric currents:
0 is the permeability of free space.
For general materials, the permeability , where KM is the relative permeability.
SJlB ddAC 0
J
d Sd lB
0 MK
i
B
ECA1 A2
Moving charges are not the only source for a magnetic field. Example: in a charging capacitor, there is no current across area A2 (bounded by C).
A
i
t
E
A
QE
tD
EJ
Ampere’s law: SE
JlB dt
dAC
A time-varying electric field produces a magnetic field.
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3.1.5 Maxwell’s equations
Gaussian’s divergence theorem:
SFlF ddAC
Stokes’s theorem:
VA
dVd FSF
zyx
zyx
FFFzyx
kji
Fz
Fy
Fx
kz
jy
ix
ˆˆˆ
ˆˆˆ
F
F
Maxwell’s equations in differential form:(integrals in finite regions derivatives at individual points)
00
1
B
E
EJB
BE
SB
SE
SE
JlB
SB
lE
t
t
d
dVd
dt
d
dt
d
A
VA
AC
AC
In free space,.0 0, , , 00 J
0
0
00
B
E
EB
BE
t
t
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3.2 Electromagnetic waves
Applying to free space Maxwell’s equations, we have the 3D wave equations:
2)()(
2
2
002
2
2
002
t
tB
B
EE
triumphralinterllectgreat A
m/s 109979.21 8
00
c
3.2.1 Transverse waves
For a plane EM wave propagating in vacuum in the x direction: ),( txEE
000
xx E
x
EE
For linearly polarized wave ),(ˆ txEj yE
),(ˆ0
0
ˆˆˆ
0),(0
ˆˆˆ
txBk
x
E
t
Bt
Bt
B
t
Bk
t
Bj
t
Bi
txEzyx
kji
t z
yz
y
x
zyx
y
BB
E
In free space the plane EM waves are transverse.
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Harmonic waves:
)(0
)(0
),(
),(
tkxiyyz
tkxiyy
ec
Edt
x
EtxB
eEtxE
medium)a (in ),(),(),(
vacuum)(in ),(),(
txBn
ctxvBtxE
txcBtxE
zzy
zy
Characteristics of the electromagnetic fields of a harmonic wave:1)E and B are in phase, and are interdependent.2)E and B are mutually perpendicular.3)E × B points to the wave propagation direction.
x
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Read: Ch3: 1-2Homework: Ch3: 1,3,7Due: September 12
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September 10 Energy and momentum
3.3 Energy and momentum
3.3.1 Poynting vectorE-field and B-field store energy:Energy density (energy per unit volume) of any E- and B-field in free space:
2
0
20
2
12
1
Bu
Eu
B
E
(The first equation can be obtained from a charging capacitor:E=q/A, dW=El·dq).
For light, applying E=cB, we have
The energy stream of light is shared equally between its E-field and B-field.
The energy transport per unit time across per unit area:
Assuming energy flows along the light propagation direction,
Poynting vector: is the power across a unit area whose
normal is parallel to S.
022
0 / , BEuuuuu BEBE
EBcuctA
tAcuS 0
2
00
2
BE
BES
c
9
For a harmonic, linearly polarized plane wave:
)(cos
)cos(
)cos(
2000
2
0
0
ωtc
ωt
ωt
rkBES
rkBB
rkEE
Time averaging: )when ( 2
1cos2 Tt
T
3.3.2 IrradianceIrradiance (intensity): The average energy transport across a unit area in a unit time.
||2
1)(cos|| 000
22000
2 BErkBE cωtcSITT
2002
1EcI In a medium
TEvI 2 Note
The inverse square law: The irradiance from a point source is proportional to 1/r2.Total power I·4r2 = constant, I E0
2 E01/r.
.20EI
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3.3.3 PhotonsThe electromagnetic wave theory explains many things (propagation, interaction with matter, etc.). However, it cannot explain the emission and absorption of light by atoms (black body radiation, photoelectric effect, etc.).
Planck’s assumption: Each oscillator could absorb and emit energy of hwhere is the oscillatory frequency.
Einstein’s assumption: Light is a stream of photons, each photon has an energy of
./ hch sJ10626.6 34 h
3.3.3 Radiation pressure and momentumMaxwell’s theory shows radiation pressure P = energy density: (Work done = PAct = uAct P=u)
c
tStPBEuuP BE
)()(or ,
2
1
2
1 2
0
20
For lightc
I
c
tStP
TT
)(
)(
11
Momentum density of radiation ( pV):
2
c
SpA
c
S
t
tAcp
t
pPA V
V
c
up
c
Sp
c
Su
V
V 2
Momentum of a photon (p):
h
c
hp
Vector momentum: kp
The energy and momentum of photons are confirmed by Compton scattering.
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Read: Ch3: 3Homework: Ch3: 8,14,16,19,27Due: September 19
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September 12 Radiation
3.4 Radiation
3.4.1 Linearly accelerating charges
0 t1 t2
ct2
c(t2-t1)
Constant speed
Field lines of a moving charge
With acceleration
Assuming the E-field information propagates at speed c.Gauss’s law suggests that the field lines are curved when the charge is accelerated. The transverse component of the electric field will propagate outward. A non-uniformly moving charge produces electromagnetic waves.
0 t1 t2
ct2
c(t2-t1)
Analogy: A train emits smokes at speed c from 8 chimneys over 360º. What do the trajectories of the smoke look like when the train is:1)still,2)moving at a constant speed,3)moving at a constant acceleration.
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Examples:1. Synchrotron radiation. Electromagnetic radiation
emitted by relativistic charged particles curving in magnetic or electric fields. Energy is mostly radiated perpendicular to the acceleration.
2. Electric dipole radiation.tqdtpp coscos 00
Far from the dipole (radiation zone):
cEB
r
tkrkpE
/
)cos(
4
sin
0
20
Irradiance:2
2
032
420 sin
32)(
rc
pI
1) Inverse square law,2) Angular distribution (toroidal). 3) Frequency dependence.4) Directions of E, B, and S.
+
-
+B
E
S
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3.4.4 The emission of light from atoms
Bohr’s model of H atom:
a0
E0 (Ground state)
E1
(Excited states)
Pump Relaxation (E = h)
E∞
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Read: Ch3: 4Homework: Ch3: 37Due: September 19
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September 15,17 Dispersion
3.5 Light in bulk matter
Phase speed in a dielectric (non-conducting material):
Index of refraction (refractive index): .
KE and KM are the relative permittivity and relative permeability.
For nonmagnetic materials
Dispersion: The phenomenon that the index of refraction is wavelength dependent.
3.5.1 Dispersion
How do we get ()?
./1 v
ME KKv
cn
00
. ,1 0 EM KnK
.)()( 0 n
Lorentz model of determining n (): The behavior of a dielectric medium in an external field can be represented by the averaged contributions of a large number of molecules.
Electric polarization: The electric dipole moment per unit volume induced by an external electric field.
For most materials
Examples: Orientational polarization, electronic polarization, ionic polarization.
.)( 0 EP
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Atom = electron cloud + nucleus. How is an atom polarized ?
Restoring force:
Natural (resonant) frequency:
Forced oscillator:
Damping force: (does negative work)
Newton’s second law of motion:
Solution:
Electric polarization (= dipole moment density):
xmxkF E20
eE mk0
)exp()( 0 tiEqtEqF eeE
2
2200 )exp(
dt
xdm
dt
dxmxmtiEq eeee
)(/
)exp(/
)exp()(22
022
0
00 tE
i
mqti
i
mEqtixtx eeee
)(/
)()(22
0
2
tEi
mNqtxNqtP ee
e
i
mNq
tE
tP ee
220
2
00
/
)(
)(
+
E
Dispersion equation:
Frequency dependent frequency dependent n ():
im
Nqn
e
e22
00
2
0
2 11)(
)(0 x
dt
dxme
)()()( 0 xPn
Nn 12
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For a material with several transition frequencies:
Oscillator strength:
j jj
j
e
e
i
f
m
Nqn
2200
22 1)(
j
jf 1
Quantum theory: 0 is the transition frequency.
Re (n)
-Im (n)
Normal dispersion: n increases with frequency.Anomalous dispersion: n decreases with frequency.
xnxniikx
innn
''2
'2
exp)exp(
"'
n' Phase velocityn" Absorption (or amplification)
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Sellmeier equation: An empirical relationship between refractive index n and wavelength for a particular transparent medium:
Coefficient Value
B1 1.03961212
B2 2.31792344×10−1
B3 1.01046945
C1 6.00069867×10−3 μm2
C2 2.00179144×10−2 μm2
C3 1.03560653×102 μm2
Example: BK7 glass
• Sellmeier equations work fine when the wavelength range of interests is far from the absorption of the material.
• Beauty of Sellmeier equations: are obtained analytically.
• Sellmeier equations are extremely helpful in designing various optics. Examples: 1) Control the polarization of lasers. 2) Control the phase and pulse duration of ultra-short laser pulses. 3) Phase-match in nonlinear optical processes.
.1)( Refer to22
00
22
j jj
j
e
e
i
f
m
Nqn
, , ),(2
2
d
nd
d
dnn
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Read: Ch3: 5-7Homework: Ch3: 45,46,48,57Due: September 26