Download Optical techniques for molecular manipulation

Light
and
Matter
Classical electrodynamics
Tim Freegarde
School of Physics & Astronomy
University of Southampton
Electromagnetic waves
• electrostatic force acts through vacuum
• retardation due to finite speed of light, enhanced by inertia of any charged particles
Q1
a
F
Q1Q2
40 r 2

Q2
• net force due to oscillating dipole
F t   Q2
Q1
40 r
3
at  r c 
2
Maxwell’s equations
• Gauss
• no monopoles
• Faraday
• Ampère

dv

 E.dDS  
0
 B.dS  0

 E.ds   t  B.dS
DE 

B
.
d
s


J


 0 
H J  0
.dS

t t 


 D
E   0E  P
0
 B  
0 0 H  Μ 
Β
E
E
J 
t
E 

B
 J0 J 00


t

t


constitutive equations
3
Constitutive equations
D  DE, B
D   0 r E
D   0E  P
H  HΕ, B
B  0  r H
B  0 H  Μ 
J  JE, B
• conservation of charge
J  E

J 
0
t
constitutive equations
4
Maxwell’s equations
D  
D   0 r E
D   0E  P
B  0
B  0  r H
B  0 H  Μ 
J  E
constitutive equations
Β
E  
t
D
H  J 
t
conservation of charge

J 
0
t
5
Electromagnetic wave equation
use constitutive equations to
reduce electric
& magnetic
0
fields to single functions
D  
D   0 r E
B  0
B  0  r H
B  0 H  Μ 
J  E
differentiate equations to
allow electric or magnetic
constitutive
equations
field to be eliminated
Β
E  
t
D
H  J 
t
conservation of charge
D  EP


 Jvector
 relations
 0 to
apply
t equation
produce wave
6
Sinusoidal plane wave solutions
D  
B  0
Β
E  
t
D
H  J 
t
Er, t   E0 cost  kz
E y
Bx
E z


k
y
t
z
B  E
H y H z
Dx x , y  y , x


y
t
z
7
Maxwell’s equations
D  
D  D0Er ,EB
D   0E  P
B  0
BH
0Εr,H
H
B
B  0 H  Μ 
J 
JE, B
constitutive equations
Β
E  
t
D
H  J 
t
conservation of charge

J 
0
t
8
Electromagnetic wave equations
use constitutive equations to
reduce electric
& magnetic
0
fields to single functions
D  
D   0 r E
B  0
B  0  r H
B  0 H  Μ 
J  E
differentiate equations to
allow electric or magnetic
constitutive
equations
field to be eliminated
Β
E  
t
D
H  J 
t
conservation of charge
D  EP


 Jvector
 relations
 0 to
apply
t equation
produce wave
9
Constitutive equations
D  
D   0 r E
B  0
B  0  r H
Β
E  
t
D
H  J 
t
J  E

J 
0
t
use constitutive equations to
reduce electric & magnetic
fields to single functions
differentiate equations to
allow electric or magnetic
field to be eliminated
apply vector relations to
produce wave equation
10
Electromagnetic waves in isotropic media
• atoms and molecules are polarized by applied fields
• induced polarization
• alignment of permanent
dipole moment
• polarization modifies field propagation: refractive index; absorption
11
Constitutive equations
• define polarization P
and magnetization M
• governed by properties
of the optical medium
• vapours, dielectrics,
plasmas, metals
D   0 r E
apply Newtonian mechanics
to determine response of
medium to applied field
B  0  r H
J  E
• magnetization usually too slow to have effect
at optical frequencies
• assume (for now) D[E] to be linear and scalar
use result to write (complex)
conductivity, dielectric
constant etc.
insert into constitutive
equations and hence derive
wave equation as usual
12
Vapours and dielectrics
m02 2
x
2
• bound or massive nuclei
• electrons confined in harmonic potential
• restoring force proportional to displacement
• Newtonian dynamics
• frequency dependence (dispersion)
Ne2 m
P 2
E
2
0  
13
Metals and conductors
• free charges
• diffusion in response to applied field
• equilibrium velocity characterized by conductivity
v
• frequency dependence (dispersion)
E
• damped solutions (absorption)
• dissipation through resistive heating
14
Plasmas and the ionosphere
• independent, free charges
• inertia in response to applied field
• Newtonian dynamics
• frequency dependence (dispersion)
Ne2
P
E
2
m
15
Electromagnetic energy density & flow
D  
D   0 r E
D   0E  P
B  0
B  0  r H
B  0 H  Μ 
J  E
constitutive equations
Β
E  
t
D
H  J 
t
conservation of charge

J 
0
t
16
Electromagnetic energy density & flow
• BBC Radio 4 long wave transmitter, Droitwich
frequency: 198 kHz
l = 1515 m
power:
400 kW
• MSF clock transmitter, Rugby
frequency: 60 kHz
power:
60 kW
l = 5000 m
17
Constitutive equations
• define polarization P
and magnetization M
• governed by properties
of the optical medium
• vapours, dielectrics,
plasmas, metals
D   0 r E
apply Newtonian mechanics
to determine response of
medium to applied field
B  0  r H
J  E
• magnetization usually too slow to have effect
at optical frequencies
• assume (for now) D[E] to be linear and scalar
use result to write (complex)
conductivity, dielectric
constant etc.
insert into constitutive
equations and hence derive
wave equation as usual
18
Continuity conditions
• transverse waves on a guitar string
T
T
2 y
2 y
 2 T 2
t
x
x
• continuity of
y
… finite extension
• conservation of energy
• continuity of
y
x
… finite acceleration
• conservation of momentum
19
Continuity conditions
• electromagnetic fields
• parallel components
E//,1  E//,2
• conservation of energy
H //,1  H //,2
• perpendicular components
D,1  D, 2
• conservation of momentum
B,1  B, 2
E//
1
2
D
1
2
20
Reflection at metal and dielectric interfaces
• electromagnetic fields
• parallel components
• conservation of energy
E//,1  E//,2
E1 H
t  k
1 z HE2 t  k1
2 z
//,1
//,2
D,1  D, 2
x
• conservation of momentum y B ,1
A
 B, 2
2
apply continuity conditions
for separate components
E3 t  k1 z 
• perpendicular components
combine forward and
reflected waves to give total
E//
fields for each region
D
B
1
z
hence derive fractional
transmission and reflection
2
21
Reflection at metal and dielectric interfaces
E//,1  E//,2
H //,1  H //,2
E1 t  k1 z  E2 t  k2 z 
D,1  D, 2
E3 t  k1 z 
B,1  B, 2
x
y
A
combine forward and
reflected waves to give total
fields for each region
apply continuity conditions
for separate components
B
z
hence derive fractional
transmission and reflection
22
Reflection at multiple dielectric interfaces
combine forward and
reflected waves to give total
fields for each region
E//,1  E//,2
H //,1  H //,2
D,1  D, 2
B,1  B, 2
E1 t  k1 z  E23 t  k2 z  E5 t  k3 z 
E23 t  k1 z  E4 t  k2 z 
x
y
A
B
z
apply continuity conditions
for separate components
hence derive fractional
C
transmission and reflection
23
Reflection at multiple dielectric interfaces
E//,1  E//,2
H //,1  H //,2
D,1  D, 2
B,1  B, 2
i  t  k3 z 
i  t  k 2 z 
i  t  k1 z 
t  k3 z 
t  k2 z  E5e
E1e
t  k1 z  E3e
i  t  k1 z 
i  t  k 2 z 
E2e
t  k1 z  E4e
t  k2 z 
A
0
x
y
B
l
C
z
E1ei  t k1 0   E2eEi 1 t  kE1 02  E3ei Et 4 k2 0   E4ei  t  k2 0 
k1
1
E e 
i  t  k1 0 
1

1 i  t  k1 0 
eE  E2  
E
Z12 1
k12
Z2

E3ei Et 4k 0   E4ei  t k 0  
2
2
iik
3tl k3l 
i  t  kik22l l
2l
E3ei  t Ek23l e ikE
e

E
e

E
e
4
4
5
k2
2
E e  E e  E e E e   
i 1 t  k 2l  ik2l
i  t  kik22l l
3
4
3 Z2
4
k13
Z3

E5eiik3tlk3l 
24
Reflection at multiple dielectric interfaces
E//,1  E//,2
H //,1  H //,2
D,1  D, 2
B,1  B, 2
E1ei  t k1z 
E3ei  t k2 z  E5ei  t k3 z 
E2ei  t  k1z 
E4ei  t  k2 z 
x
y
A
0
B
l
C
z
25