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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.

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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

5

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.

6

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

10

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