Physics

1. 2. Laser physics - basics
Spontaneous and stimulated processes
Einstein A and B coefficients
Rate equation analysis
Gain saturation

2. What is a laser?
LASER: Light Amplification by Stimulated Emission of Radiation
"light" could mean anything from microwaves to x-rays
Essential elements:
1. A laser medium - a collection of atoms, molecules, etc.
2. A pumping process - puts energy into the laser medium
3. Optical feedback - provides a mechanism for the light to interact
(possibly many times) with the laser medium

3. ground state
excited state
Quantum
energy levels
Absorption: promotes an electron from the ground to the excited state
Emission: drops the electron back to the ground state
Conservation of energy: Eexcited - Eground = Ephoton
"spontaneous emission" - the decay of an excited state to the ground state
with the corresponding emission of a photon
The two-level atom

4. Three things can occur
•Promotes molecule to a higher energy state
•Decreases the number of photons
•Molecule drops from a high energy state to a lower state
•Increases the number of photons
•This is the only one that does NOT require a photon in
the initial state
•Molecule drops from a high energy state to a lower state
•The presence of one photon stimulates the emission of a
second one
Absorption
Spontaneous Emission
Stimulated Emission

5. An atom in the excited state can relax to the ground state by:
 spontaneous emission: rate is rad
 any of a variety of non-radiative pathways: rate = nr
All of these processes are single-atom processes;
each atom acts independently of all the others.
Thus, evolution of the excited state population only depends
on the number of atoms in the excited state:
10 = total spontaneous relaxation rate from state 1 to state 0
Relaxation of the two-level atom
10
    
e
rad e nr e e
dN
N N N
dt
  

6. "Stimulated transitions" - a collective process involving many two-level atoms
stimulated absorption: light induces a transition from 0 to 1
stimulated emission: light induces a transition from 1 to 0
In the emission process, the emitted photon is
identical to the photon that caused the emission!
Stimulated transitions: likelihood depends on the number of photons around
A collection of two-level atoms

7. Rayleigh-Jeans law (circa 1900):
energy density of a radiation field u() = 82kT/c3
Total energy radiated from a black body:  
u d
   

uh-oh… the "ultraviolet catastrophe"
Solution: quantum mechanics
Time-dependent perturbation theory
As a result of a perturbation h(t), a system in quantum state 1 makes a
transition to quantum state 2 with probability given by:
 
21
2
'
1 2 2
1
' '


 


t
iE t
P e h t dt
How did it all begin?
21
21
:


Notation
E

Note: the units of this expression are correct. Strictly speaking, u()
is an energy density per unit bandwidth, such that the integral
gives an answer with units of energy/volume.
 
u d
 


8. Key example: suppose we subject a two-level system, initially in state 1,
to a harmonic perturbation, of the form:
 
0
0 0
2 sin 0


 


t
h t
A t t

 
21
2
2
' ' '
0
1 2 2
0
'

  


t
i t i t i t
A
P e e e dt
  
Transition probability to state 2 is:
 
 
2
2
21
0
2
2
21
sin / 2
4 
 
 
 


t
A  
 
Note that 1 2 2 1
 

P P
Absorption and stimulated
emission are equally likely!
 
 
 
 
 
 
21 21
2
/2 /2
2
21 21
0
2
21 21
sin / 2 sin / 2
4
 
   
 
 
   
   
 
   

i t i t
e t e t
A
   
   
   
Harmonic perturbation
(and suppose that the frequency of
the perturbation, , is close to 21)
RWA

9. Note: this is the
same as rad
Quantum mechanics says that these two coefficients must be equal!
Consider a radiation field and a collection of two-level
systems, in thermal equilibrium with each other.
stimulated emission probability: proportional to the number of atoms in
upper state N2, and also to the number of photons
spontaneous emission probability: proportional to N2, but does not depend
on the photon density!
 
2 1 2 2
    
W A N B u N

stimulated absorption probability: proportional to the number of atoms in
lower state N1, and also to the number of photons
spontaneous absorption: there is no such thing
 
1 2 1
  
W B u N

But: in thermal equilibrium, the upward and
downward transition rates must balance: 1 2 2 1
 

W W
Einstein A and B coefficients

10. Equate these two rates:
 
 
1
2
A B u
N
N B u


 


But Boltzmann's Law tells us that
(in equilibrium)
 
2 1
1
2
E E kT
N
e
N


Recognizing that E2  E1 = h, we solve for u():
 
1
h
kT
A B
u
e

 

This must correspond to the Rayleigh-Jeans result in the
classical limit (h  0), which implies: 3
3
8
A h
B c
 

Einstein A and B coefficients
Since A = rad, we can now solve for B also:
3
3
8
 rad c
B
h

 
Also has units of energy
density per unit bandwidth

11. Bose-Einstein
distribution
 
 
2 1 2 2
2 1
    
 
  
 
 
W A N B u N
B
A N u
A


Transition rates
Our expression for the downward transition rate is now:
 
1


h
kT
A B
u
e

 2 1 2
1
1
1

 
  
 

 
h kT
W A N
e 
But since we therefore have
In other words, W21 is proportional to: 1 + the number of photons.
 
1 2 1 1
1
1

 
    

 
h kT
W B u N AN
e 

It is easy to see that the upward transition rate, W12, is
proportional to the number of photons:

12. g
e
stimulated spontaneous
spontaneous emission:
proportional to initial state population
stimulated transitions:
proportional to initial state population
proportional to photon density np
the same for upwards, downwards transitions
Note: the constant K is simply
given by h·B, where B is the
Einstein B coefficient
Rate equation analysis

 

e
P P g
eg e
e
d
K
N
d
N
N n Kn
t
N

 
   
e
eg e P g e
dN
N Kn N N
dt

 
g
dN
dt

13. emitted photons go
in all directions
emitted photons go only into the
direction of the incident light
Number of photons grows exponentially if Ne > Ng
A LASER!
Rate equation analysis, continued
So, photon number varies according to:
 
     
P
P g e P
dn
Kn N N Kn N
dt
 
   
e
eg e P g e
dN
N Kn N N
dt


14. Population inversion is impossible in equilibrium.
Population inversion is impossible in steady-state.
Rate equation analysis, part 3
In thermal equilibrium:

  
E kT
e g g
N e N N
In a steady-state situation:
  0
    
e
eg e P g e
dN
N Kn N N
dt

 

P
e g g
eg P
Kn
N N N
Kn


15. Non-radiative
decay
Non-radiative
decay
Pump
Four-level system
1
0
3
2
Steps 1 and 2:
Combine to give an effective
pumping rate for level 2: Rp
Step 3:
stimulated transitions due to np
spontaneous decay rate: 21
Lasing transition
Step 4:
spontaneous decay rate: 10
p
p
R
R

N
 1
10
N

 1
10
N
N



2
21
2
21
p N
N
Kn 
 2
1 )
(
p N
N
Kn 
+ 2
1 )
(
So how do you make a laser?
2
1
0



dN
dt
dN
dt
dN
dt

16. Steady-state solution:
A necessary condition for lasing
Population inversion (i.e., N < 0) is assured if 10 > 21
(even if np = 0, and even if Rp is small)
Other necessary conditions:
• a resonant cavity - provides feedback
• net gain per round trip > net loss per round trip - “threshold”
The four-level model
21 10
1 2
10 21
 

     

 
P
P
R
N N N
Kn
 
 

17. N0
• population inversion when 2 > 1
• small signal inversion is proportional to
the pump rate
• inversion level drops when Wsig > 21
• the characteristic intensity for this effect is
independent of pump rate Rp
1/21
Knp
0 2 4 6 8 10
0
0.2
0.4
0.6
0.8
1
Wsig 21
N/N
0
"gain saturation"
Saturation in the four-level atom
21 10
10 21
 

   

 
P
P
R
N
Kn
 
 
21 10
21 10 21
1
1
 

   

 
 
P
sig
R
W
 
  
Note: Wsig is proportional to nP and therefore to the
intensity of the light in the medium. Thus, we can
define a saturation intensity Isat such that:
21 
sig sat
W I I
