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
x
)KT2
mv(
veKT2
m
N
)(N2
x
where m is the atomic mass, is the Boltzman’s constant and 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
)c
v1( x
o )(c
v oo
x
(1)
where 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 to is given by
o
Nc
])(KT2
mc[exp
KT2
mN
o
2
o
o2
where
dc
dvo
x from eqn. )1()
Nc
eKT2
mN
o
)( 2o
where 2
o2 KT2/mc
The rate of upward transition is
1o
)(12112 N
ce
KT2
m)
c
I(BNUB
2o
The rate of stimulated or induced downward transitions
2o
)(212v21 N
ce
KT2
m)
c
I(BNUB
2o
The net time rate change of the spectral energy density in the interval is given by
)NBNB(eKT2
m)I(h)U(
dt
d112221
o
)(o
2o
B)NN(eKT2
m)I(h
c
I
dt
d12
)( 2o
where B21=B12=B
B)NN(eKT2
mhI
c
I
c/dx
d12
)( 2o
dxB)NN(eKT2
mh
I
dI12
)( 2o
x.,o eII
where
hB)NN(eKT2
m12
2)o(
hB)NN(KT2
m12max
oat
213
3
12max A8
c)NN(
KT2
m
213
3
21 A8
cB
where
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
)TKhexp(NN
BLu
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 Three- level 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,
w3 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 where
Nu=
TK
wwexpN
B
LuL
Where T is the temperature of the medium.
There for
TK
wwexp
B
Lu
uL
Lu )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.
331221113121 NNN)(
dt
dN (2)
332223211122 NN)(N
dt
dN (3)
332312231133 N)(NN)(
dt
dN (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
)TK
ww(
uL
Lu B
Lu
e
, wu - wL>>KBT
uLLu
So that ,
2112
3223
3113
We can thus neglect 12, 13, 23 and equations 2,3&4 yield
in the steady state
33122113312211 NNN0NNN (5)
332221332221 NN0NN (6)
331321331321 N)(N0N)(N (7)
321 NNNN (8)
Therefore
))(()( 321323121323121 NNNN (9)
332312123231211323121
321323121
)()()())((
NNNNNN
(10)
Therefore
332312123231211323121
323121
N)(N)(N)(
N)(
(11)
From equations 5, 6, 7 and 11 we can write
12132313321323121323121 N)(NN)(N)(
1211321323121 NNN)(
})()({N 32213231211
Thus we obtain the steady – state solution
N)()(
)(N
3221323121
3231211
(12)
332221 NN
)N
(NN3132
1
21
323
21
322
)(
)N(N
3132
1
21
322
(13)
From equation 12
N)()(
)(
)(N
3221323121
323121
313221
322
N)()(
N3221323121
322
(14)
from equations )12, 14(
)()(N
N
323132
21323121
32
1
2
)1(32
3121
(15)
If the excitation is so strong such that
)1(32
3121
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
)()(
)(N
3221323121
32312132 N
)()(
)(
N
N
3221323121
32312132
)(
)(1
1)(
323121
3221
323121
32
put o= )(
)(
3221
323121
)( 323121 = )( 3221
o
o3221
32
1
1)(
N
N )16(
Let us represent graphically the dependence of as
a function of excitation intensity expressed in terms
of o . Consider the two cases when
N
N
(i( 32 =21. Where 21 is the laser transition
(ii) 32 = 9 21
)i( In the first case
o
o
1
12
1
N
N
0 2 10 15
-1 0 4/11 13/32 o
NN
(ii) In the second case
o
o
1
19.0
N
N
0 10/9 4 9 19 24
-1 0 0.52 0.71 0.81 0.82
o
NN
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.
331221110o1o1 NNNN
dt
dN
332222 NN
dt
dN
33o3 NN
dt
dN
dt
dN
dt
dN
dt
dN
dt
dN
dt
dN 321o
Since N=No+N1+N2+N3
Laser Emission
Fig. )11(Energy-level diagram of a four-level laser
)17(
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
o3
3 NN
Therefore,
)18(
o32
323
2
322 NNN
o3
31
32
32211o
o11 N)(
1N
o32o1
3123221
o1
011 N)(N
from equation 19, 20 N2 is > N1 when
o32o1
3123221
o1
1oo
32
32 NN
o1
1o
32o1
3123221
32
32
(19)
(20)
o1
1o
32o1
3123221o132
3123221o132
321o (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
TK
wexp
B1001
therefore, the excitation intensity necessary for
population inversion is lowered. Since
323130331
2021221 &
Then equation )21( can be approximated
32
o3312
TK
w
32o1
321o 1e B (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
Since the four-level system has an extra level O, it is
obvious that we have instead of
instead of . Here it is the 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.
TK
wexp
B
o221 o33121 &
TK
wexp
B
31
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.