Einstein coefficients for quantum process.

1. Lecture-2
• Einstein’s Coefficients

2. Basic concepts for a laser
• Absorption
• Spontaneous Emission
• Stimulated Emission
• Population inversion

3. Absorption
• Energy is absorbed by an atom, the electrons are
excited into vacant energy shells.

4. Spontaneous Emission
• The atom decays from level 2 to level 1 through the
emission of a photon with the energy hv. It is a
completely random process.

5. Stimulated Emission
atoms in an upper energy level can be triggered or
stimulated in phase by an incoming photon of a
specific energy.

6. Stimulated Emission
The stimulated photons have unique properties:
– In phase with the incident photon
– Same wavelength as the incident photon
– Travel in same direction as incident photon

7. Relation between Einstein’s A, B
Coefficients
• Einstein in 1917 first introduced the concept of stimulated or induced
emission of radiation by atomic systems. He showed that in order to
describe completely the interaction of matter and radiative, it is necessary
to include that process in which an excited atom may be induced by the
presence of radiation emit a photon and decay to lower energy state.
An atom in level E2 can decay to level E1 by emission of photon. Let us call A21 the
transition probability per unit time for spontaneous emission from level E2 to level
E1. Then the number of spontaneous decays per second is N2A21, i.e. the number
of spontaneous decays per second=N2A21.
In addition to these spontaneous transitions, there will induced or stimulated
transitions. The total rate to these induced transitions between level 2 and level 1
is proportional to the density (U) of radiation of frequency , where
 = ( E2-E1 )/h , h Planck's const.

8. Let B21 and B12 denote the proportionality constants for
stimulated emission and absorption. Then number of stimulated
downward transition in stimulated emission per second = N2 B21 U
similarly , the number of stimulated upward transitions per second =
N1 B12 U
The proportionality constants A and B are known as the Einstein A
and B coefficients. Under equilibrium conditions we have

9. by solving for U (density of the radiation) we obtain
U [N1 B12- N2 B21 ] = A21 N2
21
2
12
1
21
2
B
N
B
N
A
N
)
(
U




N2 A21 + N2 B21 U =N1 B12 U
SP ST
A b

10. 








1
)
(
2
1
21
12
21
21
N
N
B
B
B
A
U 
KT
/
h
KT
/
)
E
E
(
1
2
e
e
N
N 1
2 

















1
e
B
B
B
A
)
(
U
KT
/
h
21
12
21
21
According to Planck’s formula of radiation
 
1
e
1
c
h
8
)
(
U KT
/
h
3
3




  )2)
)1)

11. from equations 1 and 2 we have
B12=B21 (3)
21
3
3
21 B
c
h
8
A



equation 3 and 4 are Einstein’s relations. Thus for
atoms in equilibrium with thermal radiation.
)4 (
21
21
21
2
21
2
A
)
(
U
B
A
N
)
(
U
B
N
emission
eous
tan
spon
emission
stimulate 



from equation 2 and 4

12.  
1
e
1
c
h
8
h
8
c
)
(
U
h
8
c
emission
.
spon
emission
.
stim
KT
/
h
3
3
3
3
3
3











 
1
e
1
emission
.
spon
emission
.
stim
KT
/
h

 
5)
Accordingly, the rate of induced emission is extremely small in
the visible region of the spectrum with ordinary optical sources (
T10 3 K (.

13. Hence in such sources, most of the radiation is emitted through
spontaneous transitions. Since these transitions occur in a random
manner, ordinary sources of visible radiation are incoherent.
On the other hand, in a laser the induced transitions become
completely dominant. One result is that the emitted radiation is
highly coherent. Another is that the spectral intensity at the
operating frequency of the laser is much greater than the spectral
intensities of ordinary light sources .