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412 PHYS

Lasers and their

Applications

Department of Physics

Faculty of Science

Jazan University

KSA

Lecture-2

Laser Linewidth

• When observing the laser radiation emitted

from atoms, It is found to be spread out over

a certain but narrow range of frequencies or

wavelengths forming a spectral curve

Therefore, the energy levels have certain width and can not be represented by lines (i.e., they are bands according to Heisenberg uncertainty principle in determining energy levels and their life time).

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Lineshape function )(g

So energy levels may not be represented by lines. They can

be represented by lineshape function )(g

The lineshape function is normalized as ( ) 1g d

From the total number of atoms per unit volume in lower and upper

energy level are N1 ، N2

Only contribute to the interaction in the frequency

range between and 2 1( ) & ( )N g d N g d

d

21 12 2 21( ) ( ).........(3)

dNN B N B g

dt

The rate equation becomes

The total intensity is related to spectral intensity with

the total energy density ( )

d

I I d

Intensity is defined as: The power per unit area

Total energy density is defined as: Energy per unit volume

I

1V A z cI

A t A t n

They are related as:

Rate equation can be rewritten as:

2 221 2 1

1

( )( )......(4)I ndN g

B g N Ndt c g

Rate equation in terms of total intensity

In many systems there are multiple excited states that have identical energy These

levels are called degenerate. 12 2 1 21( / )B g g B

if level 2 has three allowed

states and level 1 has one

allowed state, transitions from

1→2 are more likely than

from 2→1

/I P A

/E V

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Rate equation in terms of absorption & stimulated emission cross sections

21 2( )abs SE

IdNN N

dt h

2 2

221 212 2

1

( ) ( ) ( ) ( )8 8

SE abs

gA g ،Ag

n n g

Where;

Example: A He-Ne laser beam with a wavelength of 633nm and power of 5mW.

Find the photon flux density inside the laser tube if the beam diameter is (1 mm)

and the gas refractive index is (n=1)

Hint: hh

2( )

n I n P nP

c c A c r

h hc

Relating population inversion to Gain

Assume that the transition is occurring between tow levels

Now we know relations between A, B12 and B21, and the rate

equation is written which can be written in terms of cross-

sections as 2 2

1 2

21

( ) ( )abs SE

I IdN NN N

dt h h

If we assume that the spontaneous emission is small and can be

neglected and when the transition is considered between two

levels only, we have 2 1 ( ) ( ) ( )abs SEg g

2 ( )IdN

Ndt h

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Therefore, decrease in N2 corresponds to light emission*

relate ( dN2 /dt )to light amplification

Let us Define as the photon density (number of photons per unit volume)

change in photon density should equal the change in ‘excitation density’ N2

h

IN

dt

dN

dt

d

stim

)(2

* note: we assume that the only relaxation process for the atom is either spontaneous emission or

stimulated emission. In many realistic systems there is also non-radiative relaxation

This equation relate the change in the photon density to N2 and

irradiance I.

To convert the change in photon density to a change in (optical) energy

density as light travels through a material, we express the energy density as:

The total energy density = photon density × photon energy

h

( ) ( )I dd d

N h N Idt h dt dt

Therefore, the rate equation can now be rewritten as

When the population difference (inversion) N>0

optical energy density increases

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If the energy per unit volume gained or lost per unit time is /d dt

Then the Power P gained or lost in thin slice of material is

( )dP I Adz N

Since the irradiance is PI

A

NI

dz

dI )( E/V

P=E/t

If we define the gain coefficient as

we obtain

0( ) ( ) N

0 ( )dI

Idz

with solutions of the form

( )( ) (0) ( ) (0)o z

I z I e G I

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.5

1

1.5

2

2.5

3

3.5

4

Z [cm]

I(z) /

I(0

)

N <0

N >0

Gain is possible if N2 > N1

population inversion

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Condition of laser gain threshold

For any laser medium to be suitable for stimulated emission and

hence laser amplification, population inversion should occur.

To make the necessary oscillation, the medium should be placed inside an

optical resonator

The oscillating photons between mirrors inside the oscillator get amplified and

at the same time some losses are occurred due to, absorption, scattering,

mirrors reflectivities, optical inhomogeneity and diffraction at the mirror edges

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Oscillation is sustained and laser emission becomes possible when

gain= losses

For the laser emission to start, gain must overcome losses

(i.e., becomes greater that the losses

This means that there is a minimum value of the population i inversion

to overcome losses

Let’s consider a resonator with two mirrors’ of reflectivities R1 and R2 as shown

in last figure

Let is the average losses per unit length of laser medium

(for all losses except mirror’s reflectivities)

( )l

0 ( ) And is the gain of the active medium.

If oscillated photons are incident from the side of

M1 with intensity I0, they will be passed through

the medium and get amplified by

And reduced due to losses by

)(0 e

( )le

The net gain for each oscillation round-trip is given by

0exp[ ( ) ( )]lG L

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The intensity becomes I0G and after reflection from M2 becomes I0G R2

Upon passing through the medium again it will be amplified and becomes I0GG

R2 and finally after reflection from M1 it becomes I0G2 R1 R2

Therefore, the photons get amplified by GG due to passing through the

active medium twice (one complete round-trip)

The necessary condition for the oscillator to act as an amplifier is:

2 2

0 1 2 1 2

1 2 0

1

exp 2[ ( ) ( )] 1

o

l

I G R R I G R R

R R L

When oscillation reaches the steady state, the equality in the above equation is

satisfied and the oscillation continues in a continuous wave

When the population inversion increases, the left hand side of the last

equation becomes greater than one.

When reaching the saturation, population inversion will decrease until it

comes to its threshold value, Therefore, the gain coefficient can be

expressed as

0 1 2

1( ) ( ) ln

2l R R

L

At the threshold 1 2

1( ) ln

2th l R R

L

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In the last equation that expresses the threshold gain coefficient, the right

hand side depends only on the passive parameters of the cavity (resonator)

and hence it is related to the life time of photons inside the cavity, c

c Is the cavity effective decay time: it is the time at

which the energy of the oscillating photons inside the

cavity is reduced to 1/e from its initial value.

In case of no amplification the intensity will be reduced by a factor

1 22 (2 ln )

1 2 ( 1)l lL L R RR R e e a

This occurs in a round trip time duration given by 2 / ( / ) 2 /t L c n Ln c

Hence, the intensity reduction will be in a factor of

/ 2 /( 2)c ct Ln c

e e a

From a1 and a2 we get

1 21 / (2 ln )

2c l

cL R R

nL

But we know that

2

20 21 2 12

1

( ) ( )( ) ( )8

gA g N N N

n g

Therefore, we can write the population inversion as 2 3

3

8 1

( )

sp

c

n tN

c g

Hence, the threshold of population

inversion is given by 2 3 2 3

3 3

4 81 1

( ) ( )

sp sp thth

c c

n t n tN

c t g c t g