Laser Physics- lecture notes

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6/14/20151
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 
2
1 12 2 21( ) ( ).........(3)
dN
N 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 c
I
A t A t n
 
 
 


   
 
They are related as:
Rate equation can be rewritten as:
2 2
21 2 1
1
( )( )......(4)
I ndN g
B g N N
dt 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
2
1 2( )abs SE
IdN
N N
dt h

 

 
2 2
2
21 212 2
1
( ) ( ) ( ) ( )
8 8
SE abs
g
A 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: h
h



  

   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 N
N 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
N
dt 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
I
N
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 I
dt 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 P
I
A

 


 NI
dz
dI
 )( E/V
P=E/t
If we define the gain coefficient as
we obtain
0 ( ) ( ) N    
0 ( )
dI
I
dz

 
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 2
0
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
( )l
e  
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
2
l R R
L
    
At the threshold
1 2
1
( ) ln
2
th 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 R
R 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 2
1 / (2 ln )
2
c l
c
L R R
nL
  
But we know that
2
2
0 21 2 12
1
( ) ( )( ) ( )
8
g
A g N N N
n g

    

   
Therefore, we can write the population inversion as 2 3
3
8 1
( )
sp
c
n t
N
c g
 
 
 
Hence, the threshold of population
inversion is given by 2 3 2 3
3 3
4 81 1
( ) ( )
sp sp th
th
c c
n t n t
N
c t g c t g
   
  
   