Download Laser and its applications

An expression for the Gain taking into
consideration Doppler broadening :
In the case of broadening due to thermal motion, the
kinetic theory given the fraction of atoms whose
component of velocity lies between vx and vx as
N (  )

N
m
e
2KT
(
mv x 2
)
2 KT
v x
where m is the atomic mass, K is the Boltzman’s constant and T
is the absolute temperature
As previously explained due to the Doppler effect, these atoms
will emit or absorb radiation propagating in the x direction of
frequency
   o (1 
vx
)
c
 vx 
c
(   o )
o
)1(
where  o is the frequency of the line center. It follows that the
fraction of atoms in a given level that can absorb or emit in the
frequency range
N 
 to    is given by
m
mc 2    o 2 c
exp[
(
) ]
 N
2KT
2KT  o
o
where dv x 
N 
where
c
d
o
)from eqn. (1)
m  (    o ) 2 c
e
 N
2KT
o
  mc 2 / 2KT o
2
The rate of upward transition is
I
m
 (    o ) 2 c
B12 U  N1  B12 ( )
e
 N 1
c
2KT
o
The rate of stimulated or induced downward transitions
I
m  (    o ) 2 c
B 21U v N 2  B 21 ( )
e
 N 2
c 2KT
o
The net time rate change of the spectral energy density
is given by in the interval
2 
d
m
( U   )  h o (I  )
e  (    o )
(B 21 N 2  B12 N1 )
dt
2KT
o
2
d  I 
m
e  (   o ) ( N 2  N1 )B
   h(I  )
dt  c 
2KT
where B21=B12=B


d  I 
m
 (    o ) 2
e
( N 2  N1 )B
   hI 
dx / c  c 
2KT


2

dI  
m
 h
e  (   o ) ( N 2  N1 )B dx
I
 2KT

I   I o , e   . x
where
m  (    o ) 2

e
( N 2  N1 )hB
2KT
 max 
m
( N 2  N 1 )hB
2KT
at
  o
 max
m
c3

(N 2  N1 )
A 21
3
2KT
8
where
B 21
c3

A 21
3
8
 is positive if N2 <N1which is the condition for amplification .
other wise if N2<N1 which in the normal equilibrium condition
) then  is negative , we have absorption .
Population Inversion
In order to invert population of atomic levels the atoms must
be excited by depositing energy in the medium using such
method as to decrease the number of atoms at the lower level
NL and to increase the number of atoms at the upper level Nu .
This process is called pumping since the atoms are
redistributed as if pumped from the lower level to the upper
level. The methods of pumping are i) optical pumping, where
the atoms are excited by illumination of light ii) excitation by
electric discharge in the case of gases iii) Injection of carriers
by a forward current through a p-n junction in the case of
semi-conductors iv) excitation by irradiation with electron
beams v) excitation by chemical reaction. Historically, in 1954,
Townes succeeded in realizing population inversion with a
molecular beam of ammonia to make a maser at 1.25 cm
wavelength.
As the ammonia molecular are distributed among
energy levels in thermal equilibrium, the molecules at
the upper level were collected and those in the lower
level
were
eliminated
by
the
action
of
an
inhomogeneous electric field, so that population
inversion can be achieved. However, such a method
where
population
inversion
is
established
by
decreasing the number of atoms in the lower level
cannot be applied successfully to optical transition.
This is because the number of atoms Nu and NL as
related by Boltzmann’s formula namely
N u  N L exp(  h
K BT
)
KB Boltzmann const
yields Nu  NL in the microwave case, since h << KBT at
the microwave frequency , while the population of the
upper level Nu in the optical case is very small, since h >>
KBT at the optical frequency  . Therefore, it is not
sufficient by merely eliminating atoms at the lower level,
but it is necessary to increase the number of atoms at the
upper level by a process of pumping.
For a two - level system, when its atoms are exited by
irradiation or by electron collision, the number of atoms at
the upper level will increase, but at the same time the
probability of de-excitation that brings these excited
atoms back to the lower level will increase with incident
light or electrons.
Consequently no matter how strong the atoms may be
excited, population inversion cannot be obtained.
Therefore, three or four atomic systems must be used to
achieve population inversion.
It is not always necessary that the energy levels
concerned should be discrete and sharp. Band levels
may be used. Thus, dye lasers and semiconductor
lasers can be considered as four -level lasers whose
description follows.
Population Inversion in a Threelevel Laser:
There are many three – level lasers such as ruby
laser and the optically pumped gas laser. Let the
energies and populations of the relevant three levels of
laser atomic system be denoted respectively by w1, w2,
w 3 and N1, N2, N3. If w3> w2> w1 as shown in figure, then
N1>N2>N3 in the three – level system in thermal
equilibrium. Here the lowest state 1 is not necessarily
the ground state of the atom. Atoms in level 1 will be
excited atoms of appropriate energy. We denote by  the
probability of exciting the atoms from level 1 to level 3
by any such method of pumping.
Fig. (6(
When the pumping is removed, the excited atoms will
in general gradually return to the state of thermal
equilibrium. This is termed relaxation. If we consider the
atoms individually the relaxation process takes place at
the same time as other atoms are excited.
Besides the radiative process, where the excited
atoms make a transition to the lower state by emitting
a photon, there are non-radiative processes such as
collision of molecules in gases or the atom lattice
interaction in solids, where the excited atom makes a
transition to the lower state by releasing its energy in
the form of molecular kinetic energy or vibrational
energy of the lattice. Since relaxation is the results of
such statistical processes, the relaxation rate or the
relaxation constant is defined as a statistical average
of the relaxation probabilities of the excited atoms per
unit time. The reciprocal of the relaxation rate is the
average life time of the excited atoms:
Now, the probability 
Lu
of an atom being thermally
excited from the lower state wL to the upper state wu is
related to the probability uL of the reveres process
from wu to wL by thermal relaxation. This relation in
thermal equilibrium . is Nu uL = NL  Lu
Nu = N L
where

wu wL
exp

K BT





Where T is the temperature of the medium .
There for

 Lu
wu wL
 exp


 uL
K BT





(1 (
Fig .( 7 )
This last relation holds generally, even if Nu and NL do
not represent populations in thermal equilibrium .
If these probabilities are constant under the conditions
considered the rate equations expressing the rate of
change at the number of atoms in each level of the three –
level system under pumping are given as follows.
dN 1
 (   12   13 )N1   21N 2   31N 3
dt
(2)
dN 2
  12 N 1  (  21   23 )N 2   32 N 3
dt
(3)
dN 3
 (   13 )N1   23N 2  (  31   32 )N 3
dt
(4)
Where N1+N2+N3 = const. = N the total number of
atoms in the three – level system.
In the steady state, the distribution of the number of
atoms under constant pumping can be obtained by putting
the left – hand side of equations 2,3&4 equal to zero.
Although the solutions giving N1, N2&N3 can be readily
calculated, yet we shall assume that the separations
between the level are sufficiently greater than the thermal
energy KBT, so that when applying equation (1) we find that
 Lu
 e
 uL
(
wu  w L
)
KB T
  Lu   uL
So that ,
 12   21
 23   32
 13   31
, wu - wL>>KBT
We can thus neglect 12, 13, 23 and equations 2,3&4 yield
in the steady state
 N1   21N 2   31N 3  0
  21 N 2   32 N 3
0
N1  (  32   31 )N 3  0
 N1   21N 2   31N 3
(5)
  21 N 2   32 N 3
(6)
 N1  (  32   31 )N 3
(7)
 N  N1  N 2  N 3
(8)
Therefore
 21 ( 31   32 ) N   21 ( 31   32 )( N1  N 2  N 3 )
 21 ( 31   32 )( N1  N 2  N 3 )
  21 ( 31   32 ) N1   21 ( 31   32 ) N 2   21 ( 31   32 ) N 3
(9)
(10)
Therefore
 21 (  31   32 )N 
(11)
 21 (  31   32 )N 1   21 (  31   32 )N 2   21 (  31   32 )N 3
From equations 5, 6, 7 and 11 we can write
 21 (  31   32 )N   21 (  31   32 )N 1   32 N 3 (  31   32 )   21N 1
  21 (  31   32 )N 1   32 N 1   21N 1
 N 1 {  21 (  31   32 )  (  21   32 )}
Thus we obtain the steady – state solution
 21 (  31   32 )
N1 
N
 21 (  31   32 )  (  21   32 )
(12)
  21 N 2   32 N 3
 32
 32
N 1
 N2 
N3 
(
)
 21
 21  32   31
 32
( N 1 )
 N2 
 21 (  32   31 )
(13)
From equation 12
N2 
 32
 21 (  31   32 )
N
 21 (  32   31 )  21 (  31   32 )  (  21   32 )
 32 
 N2 
N
 21 (  31   32 )  (  21   32 )
(14)
from equations (12, 14)
 32 
N2



N 1  21 (  31   32 )  21
(  31   32 )
 32


(15)
 31
 21 (1 
)
 32
If the excitation is so strong such that
 31
   21 (1 
)
 32
we have N2>N1
(15\)
This is the condition of population inversion. Thus to obtain
population inversion with moderate pumping 21 should be
small and 32 should be large compared with 31 . This means
that it is desirable that the relaxation from the upper laser
level to the lower laser level should be slow, while the
relaxation from the upper most level 3 to which the atoms was
initially excited to the upper laser level 2 should be fast .
Fig. (8)
The population inversion as defined by N=N2-N1 is
calculated from 12 & 14 as a function of the excitation
intensity  to be
 32    21 (  31   32 )
N
N 
 21 (  31   32 )  (  21   32 )
 32    21 (  31   32 )
N

N
 21 (  31   32 )  (  21   32 )
 32
 1
 21 (  31   32 )

(  21   32 )
1

 21 (  31   32 )
 21 (  31   32 )
put o =
(  21   32 )

 21 (  31   32 ) = (  21   32 ) 
 32

1
N (  21   32 ) o

N


 1  
o 

(16)
N
Let us represent graphically the dependence of
as
N
a function of excitation intensity  expressed in terms
of o . Consider the two cases when
)i) 32 = 21. Where 21 is the laser transition
)ii ( 32 = 9 21
 1 


 1


2 o
N



N

 


1





o 
(i) In the first case

0
2
10
15
-1
0
4/11
13/32
o
N
N
(ii) In the second case

o
N
N





0
.
9

1



N


o

N

 


1


o 


0
10/9
4
9
19
24
-1
0
0.52
0.71
0.81
0.82
Fig. (9(
Population inversion in a four- level laser
Since the lower level of the laser transition is the lowest
level in a there - level laser , the majority of atoms ( N1  N ) are
in this level at thermal equilibrium thus in order to invert the
population , the number of atoms in the lowest level must be
reduced to less than half by intense pumping. This demand is
much reduced in a four - level system.
Let us consider an atom, which has four energy levels as
shown in fig (10) . It is required to invert the population
between levels 2 and 1. Since the lower level 1 lies at an
energy higher than KBT above the ground level , the number
of thermally excited atoms in the lower laser level 1 is so small
that the population can be easily inverted by pumping a
relatively small number of atoms into the upper level 2. The
conditions for population inversion in this case are as follows.
Although separations between levels 1, 2 & 3 are
assumed to be much greater than KBT as in the case of
a three level laser, the number of thermally excited
atoms go, No from the most population ground level O
to level 1 are not neglected. The rate equations for
atomic populations in the four-levels, than become.
dN 1
  o1 N o   10 N 1   21N 2   31 N 3
dt
dN 2
   2 N 2   32 N 3
dt
dN 3
 N o   3 N 3
dt
dN dN o dN 1 dN 2 dN 3




dt dt dt dt dt
(17)
Since N=No+N1+N2+N3
Laser Emission
Fig. (11)Energy-level diagram of a four-level laser
where 2 =  20 +  21
&
3 =  3o +  31 + 
32
The steady –state solution is obtained as before
 o1No - 1oN1 + 21N2 +31N 3= 0
- 2N2 + 32N3 = 0
No - 3N3 = 0
Therefore,

N3 
No
3
(18)
 32
 32
N2 
N3 
N o
2
 2 3
(19)
 21  32
 31
1
N1 
(  o1 

 )N o
 1o
 2 3
3
 01  21  32   2  31
N1  (

 )N o
 1o
 1o  2  3
from equation 19, 20 N2 is > N1 when
 32
N o
 2 3
  o1  21  32   2  31  
  
 N o
 1o  2  3
  1o

  32  21  32   2  31   o1
 


 1 o  2  3   1o
  2 3
(20)
  32  1o   21  32   2  31   o1
 

 1o  2  3

  1o


 o1  2  3

  
  32  1o   21  32   2  31 
(21)
This is the condition for population inversion now 01 in
the numerator of this equation is the probability of thermal
excitation from level O to level 1, and is a small quantity as
shown by the relation
  w 
 01   10 exp
 K T



B
therefore, the excitation intensity necessary for
population inversion is lowered. Since
 31   3   30   31   32
 21   2   21   20
&
Then equation (21) can be approximated
 o1  2  3

e
 1o  32
  w 


K
T
 B 
  31   3o 

 2  1 
 32 

(22)
Comparing equation (22) with equation (15 \)for population
inversion in a three level laser, it is seen that they are
similar except for the factor
  w
exp
 K T

B




Since the four-level system has an extra level O, it is
obvious that we have
of
 31
 21   2o
instead of
instead
 21 &  31   3o
  w 
 K BT 

expit is the
. Here
factor
,which is important, because population
inversion can be obtained even with very week pumping if
the lower laser level 1 is above the ground level O by at
least a few times KBT in energy.
Laser Operation
(1) Essential Elements of Laser
The laser device consists of basically of three elements;
External source (pump), Amplifying medium and optical
cavity (resonator (
The pump is an external energy source
that produces a population inversion in
the laser medium. Pumps can be optical,
electrical, chemical or thermal in nature.
For gas lasers (e.g. He-Ne laser), the used
pump is an electrical discharge. The
important parameters governing this type
of pumping are the electron excitation
cross-sections and the lifetimes of the
energy levels.
In
some
lasers,
the
free
electrons
generated in the discharge process collide
with and excite the laser atoms, ions, or
molecules directly.
In others, the excitation occurs by means of
inelastic atom-atom (or molecule – molecule)
collisions. In this case a mixture of two gasses
is used such that the tow different species of
atoms, say A and B, have excited states A* and
B*. Energy may be transferred from one
excited species to the other in a process as
follows relation
A*+B A+B*
e.g. He-Ne laser, where the laser – active neon atoms
are excited by resonant transfer of energy from helium
atoms in metastable state, where the He atoms receive
their energy from free electrons via collisions.
2- Laser medium
The amplifying medium or laser medium is an important
part of the laser device. Many laser are named after the type
of laser medium used (e.g. He-Ne, CO2 and Nd:YAG). This
laser medium may be gas, liquid, or solid, determines the
wavelength of the laser radiation.
In some lasers the amplifying medium consists of two
parts, the laser host medium and the laser atoms. For
example, in Nd: YAG laser, the host medium is a crystal of
yttrium Aluminum Garnet (or YAG), whereas the laser atoms
are the Neodymium ions.
The most important requirement of the amplifying medium
is its ability to support a population inversion between two
energy levels of the laser atoms.
3-The Resonator
The resonator is an optical “feed back device” that
directs photons back and forth through the laser
medium. Resonator or “optical activity” consists of a
pair of carefully aligned plane or curved mirrors (see
figure 2). One of them is chosen with a reflectivity 100%
as possible. The other mirror is selected with a
reflectivity somewhat less than 100% in order to allow
part of the internally reflecting beam to escape and
become the useful laser output beam. The geometry of
the mirrors and their separation distance determine the
structure of the electromagnetic field within the laser
cavity and controlling the emerging laser beam.
Figure (2): four types of end mirrors in common use for
lasers. (Mirror curvatures are exaggerated )