#### Transcript Saturation

```Saturation
Roi Levy
Motivation
•
•
To show the deference between linear and
non linear spectroscopy
To understand how saturation spectroscopy is
been applied
Motivation
Outline
•
•
•
•
Widths and Profiles of Spectral Lines theory
Nonlinear Spectroscopy theory
Experimental Schemes
Some papers
Widths and Profiles of
Spectral Lines
•Natural Line width
•Doppler width
Natural Line width
• An excited atom can emit its excitation energy
• Describing it as dumped harmonic oscillator
with dumping constant
• As a result the amplitude decreases (in time)
and the emitted radiation is no longer
monochromatic
Natural Line width
The square of amplitude A(w)
as function of w we can see
it has the shape of a Lorentzian.
The normalized intensity is
1
L(   0 ) 
2


(   0 )  ( ) 2
2
2
Where the full halfwidth at half-maximum is

Natural Line width
• Using the uncertainty principle we can relate the
mean life time of an exited level i to it’s energy,
Ei 1
 i  i 
  Aik

i
• Where A is Einstein coefficient for spontaneous
emission.
ik
Doppler Width
• One of the major contributions to the spectral
line width in gases.
• Due to the thermal motion of the absorbing or
emitting molecules.
• Because Doppler Width is of our main interest
we shall present the outline of the derivation.
Doppler Width
Consider an excited molecule with a velocity v
relative to the rest fame of the observer.
The central frequency of the emission is w
At the rest frame the frequency is
0
e  0  k  v
The same relation holds for the absorption
frequency
a  0  k  v
Doppler Width
At thermal equilibrium the molecule of a gas
ni (vz )dvz 
Ni
(
e
vi 2
)
vp
vp 
n - density of molecules in level E
i
2k b T
vp 
m
dvz
i
- the most probable velocity
Doppler Width
c
d and that the
Using the relation

absorbed radiant power is proportional to the
Density we get the intensity of the DopplerBroadened spectral line
dv z 
0
  c(   )  2 
0  
I ( )  I 0 exp  
   0 v p  


Gaussian profile with full halfwidth
D 
0 8k bT
c
m
ln 2
Doppler Width
More detailed consideration will have to include
the natural Line width for every molecule.
Doppler Width
The spectral intensity distribution of the total
Absorption or emission is
I ( )  I 0  n( ' ) L(   ' )d '
This intensity profile is called Voigt profile.
• At sufficiently large laser intensities the optical
pumping rate on an absorbing transition
become larger than the relaxation rates.
• This saturation causes additional line
Saturation of Level Population by
Optical pumping
For two-level system with population N and N
The rate equations are,
1
dN1
dN
  2   PN1  R1 N1  PN 2  R2 N 2
dt
dt
With P  B12  () the rate for stimulated
emission (absorption), and Ri the relaxation
Probability for level i.
2
Saturation of Level Population by
Optical pumping
Solving for the steady state we get for N1
N1  N
P  R1
2 P  R1  R2
And for the difference between the population of
the levels is
N 0
N 0
N 

1  2P /( R1  R2 ) 1  S
Where S the saturation parameter is the ratio of the
Pumping rate to the average relaxation rate.
Saturation of Level Population by
Optical pumping
The pump rate due to monochromatic wave with
intensity I ( ) is P   12 () I () /  so that
2 12 I ( )
S
 A12
And the saturated absorption coefficient is
 ( )   12 N 
0
1 S
Homogeneous Line Profiles
Since the absorption profile of a homogeneously
broadened line is Lorentzian, the induced
absorption probability is B12  ( ) L(  0 )
And the saturation parameter is
B12  ( )
S 
L(   0 )  S ( 0 )
 R

 2 


2
(   0 )    
 2
2
2
Homogeneous Line Profiles
The absorption coefficient will be
 S ( ) 
With  s
  1  S0
 0 ( )
1  S
  0 ( 0 )
  
 2
2

(   0 )   s 
 2
2
2
Nonlinear Spectroscopy
•Linear and Nonlinear Absorption
•Saturation of Inhomogeneous Line Profiles
•Saturation Spectroscopy
Linear and Nonlinear Absorption
Assume that a monochromatic plane lightwave
E  E0 cos(t  kz)
With the mean intensity
1
I  c 0 E 02
2
passes through a sample of molecules.
The Absorption in volume dV is
Linear and Nonlinear Absorption
In case the incident wave with spectral energy
density  ( )  I ( ) c
And spectral width  L
Which is large compared to the halfwidth of the
Absorption profile  L
The total intensity becomes
I   I ( )d  I ( 0 ) L
Linear and Nonlinear Absorption
The absorbed power is then

dP  N  dV  I   ik ( 0 )
 L
Remembering that the absorbed power is
proportional to the number of absorbed photons
n ph
We can obtain
P

 Bik  ( )NdV
h
c
Bik 
 ik ( )d

h
Linear and Nonlinear Absorption
Let us discuss what happens in open systems
Many relaxation channels
Also Molecules can diffuse in and out of the
excitation volume
Linear and Nonlinear Absorption
The rate equations
dN1
 B12  ( N 2  N1 )  R1 N1  C1
dt
dN 2
 B12  ( N1  N 2 )  R2 N 2  C2
dt
Where
Ci   Rik N k  Di
k
Is the contribution of other levels to the population of
level i , Di is the diffusion rate of the molecules in level
i into the excitation volume
Linear and Nonlinear Absorption
Solving these equations under stationary
Conditions (dN/dt=0) we get for   0
C 2 R1  C1 R2
N  N  N 
R1 R2
0
And for
0
2
0
1
0
N 0
N 0
N 

1  B12  ( 1  1 ) 1  S
R1
R2
Linear and Nonlinear Absorption
The saturation parameter
B I
S  12* 
R c
R1 R2
R 
R1  R2
*
,
The power decrease of the incident light wave
from absorption along the length dz
N 0 a
dP   A  I   12
dz
1  S  L
Linear and Nonlinear Absorption
In case of incoherent light sources S<<1
a
dP   P   12N
dz
 L
0
And P is
P  P0 exp(  12 N z )  P0 e
0
z
This is the Lambert-Beer law of linear absorption
Linear and Nonlinear Absorption
In open systems the saturated population density N
Can be very small.
1
I
(C1  C 2 ) B12
 R2 C1
c
N1 
I
( R1  R2 ) B12
 R2 R1
c
C1  C2
N1 ( I  ) 
R1  R2
Where C and C are small compared to R and R
1
2
For close systems
1
N
N1 
2
2
Saturation of Inhomogeneous
Inhomogeneous broadened line profiles such as
• Hole Burning
• Lamb Dip
Hole Burning
When a monochromatic light wave passes
through a gas with Maxwell-Boltzmann velocity
distribution the laser frequency in the frame of
the molecule is
'    k  v z
With k  v z fall within the linewidth 
 '  0  
Hole Burning
The absorption cross section for the molecule
( 2) 2
 12 ( , vz )   0
(  0  kvz ) 2  ( 2) 2
Due to the saturation the population N (v )dv
Decreases within the velocity range dvz   k
and the population N (v )dv increases.
Let us right the equations for N and N
1
2
z
z
1
2
z
z
Hole Burning
2
0


S
(

2
)

N
0
0
N1 (, v z )  N1 (v z ) 


 1  (  0  kvz ) 2  ( s 2) 2 

S0 ( 2) 2
N 0 
N 2 (, vz )  N (vz ) 
 2  (  0  kvz ) 2  ( s 2) 2 
0
2
where    1   2 is the homogeneous width of the
transition.
And  s   1  S 0
Hole Burning
Bennet hole (peak)
For  1   2 the depth of
the hole in N1 is different
from the height of the
peak in N2.
Hole Burning
The saturated population difference
2


S
(

2
)
0
0
N (, v z )  N (v z )1 
2
2
(




kv
)

(

2
)
0
z
s


The absorption coefficient
 ( )   N (v z ) 12 ( , v z )dv z
Hole Burning
Solving the integral we get
   
 (0 )
0
 ( ) 
exp  
1  S0
  0.6D
0



2



• We can not detect Bennet hole by tuning the
laser through the absorption profile.
• Something missing ?
Hole Burning
The Bennet hole can be detected if two lasers
are used
• The saturating pump laser with the wave vector
k1 which is kept at the frequency 1 and which
burn the hole
• A weak probe laser with the wave vector k2 and
frequency  tunable across the Voigt profile
Hole Burning
The absorption coefficient for the probe laser
 ( )   N (vz ) 12 (2 , vz )dvz
N 0
 S (1 , 2 ) 
vp 


S0 ( 2) 2
( 2) 2
 1  (  0  k1vz )2  ( s 2) 2  (  0  k2vz )  ( 2)2 dvZ
Hole Burning
The absorption coefficient for the probe laser
2


  


2
S0

0

 S (1 ,  2 )   ( ) 1 
2
 1  S0
2
 S  
(



'
)

 2 


 
 '  0  (1  0 )
k2
k1
S     s   (1  1  S 0 )
 ( ' )   0 ( ' )   S ( ' )   0 ( ' )
S0
1  S 0 (1  1  S 0 )
Lamb Dip
• Pump and probe waves can be generated by a
single laser when the incident beam is reflected
back into the absorption cell
• The absorption profile will be Dopplerbroadened profile with a dip at the center
at   0
• This dip is called Lamb dip after W.E.Lamb
who first explained it theoretically
Lamb Dip
The saturated absorption coefficient in case
of equal intensities (weak filed S 0  1)
 S0
0
 ( )   ( ) 1 
2

 s   1  S0
2





2
S
1 

    2   22 
0
S


Lamb Dip
For strong laser fields
 S ( )   0 ( )
 2
  2(   0 ) 
B 1  

A

B

 


1
2 2
A  (  0 ) 2   2

2



1
2
B  (   0 )   2 (1  2S )
2
2

1
2
Lamb Dip
In case the intensity of the reflected wave is
very small I 2  I1
 S
0
 ( )   ( ) 1  0
2

 
*
S
2





2
S
1 

    2   * 2 2 
0
S



 S
2

Saturation Spectroscopy
• Experimental Schemes
Saturation Spectroscopy
• Experimental Schemes
Saturation Spectroscopy
• Laser-induced fluorescence
Instead of measuring the attenuation of the probe
beam the absorption can be monitored by the
laser-induced fluorescence.
• Advantageous when the density of the
absorbing molecule is low.
Saturation Spectroscopy
Intermodulated fluorescence
the pump beam and the probe beam are chopped
At different frequencies
I1  I10 (1  cos(1t )
I 2  I 20 (1  cos( 2 t )
The fluorescence intensity is
I FL  CN S ( I1  I 2 )
Saturation Spectroscopy
At the center of an absorption line
N S  N 0 [1  a( I 1  I 2 )]
I FL  CN 0 [( I1  I 2 )  a( I1  I 2 ) 2 ]
Only the term with both intensities contribute
to the saturation effect
1
I10 I 20 cos(1t ) cos(2t )  I10 I 20[cos(1  2 )t  cos(1  2 )t ]
2
Saturation Spectroscopy
Saturation Spectroscopy
Saturation Spectroscopy
• Cross over signal
In case of two center frequencies 1 , 2
which fulfill | 1  2 | D
At laser frequency   1  2 the incident
2
Wave saturates the velocity class
vZ  dvZ 
2  1
2k
 k
Saturation Spectroscopy
At that frequency we will see an additional
saturation signal
• Positive for common lower level
• Negative for common upper level
Beer’s law in presence of saturation effect
0
dI
I
 n *
 n ( I ) I
sat
dz
 1  I I ess
And the optical depth
od 0   0  n( x, y, z )dz  f ( x, y;  *)
If
f ( x, y; *)   * ln 
 Ii
 Ii  I f
  sat
I0

Reference
1.
2.
3.
4.
W.Demtroder, Laser Spectroscopy (Springer 1991)
T.W.Hansch, et al. P.R.L 27, 707 (1971)
G.Reinaudi, et al. OPTICS LETTERS 32,21 (2007)