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

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.

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.

Figure )2(: four types of end mirrors in common use for lasers. )Mirror curvatures are exaggerated (