Notes 2 on Nonlinear Spectroscopy
Review
Diagram for CARS
Diagram for CSRS
When two laser beams of frequencies 1 and 2 (1>2) are incident in time and space
in a sample, if the beam intensity in 1 is greater than in 2, and the beam in 2 is
typically a very stable beam and vib = 1-2, then there can be a net gain in the intensity
of the beam 2. (SRG)
If on the other hand 1 is a lower intensity stable beam is 1 and the higher intensity
beam is 2 then the intensity of the 1 beam can be lowered (SRL)
Diagram for SRG
Diagram for SRL or TIRE (The Inverse Raman Effect)
Which third order susceptibility is accessed by a particular geometry?
For SRG
P1
E1
E2
Also Last time I alluded to the fact that Maxwell's eqns can be used to determine the
electromagnetic field emitted due to the polarization wave which resulted from the high
intensity electric fields.
We found the Gain Eqn. for plane waves
2 (w3)/dz2 + 2ik3 dE(w3)/dz = - 432 / c2 cars Ew12 Ew22 exp ikz
k = 2k1 - k2 - k3
k3 = n3 3
Neglecting the 2nd order term and using I(w3) = n3c/(8) E(w3)2 
one can get:
I(w3) = 256 4 32 / (n12 n2 n3 c4 ) cars 2 I12 I2 L2 [(sin (kL/2)) /kL/2)]2
L is the coherence length or the interaction length
Important points:
1) Shows that the intensity of the electric filed generated by the Polarization for the
CARS beam is proportional to I12 and is linear in I2 --- at high powers
2) I3 depends on the square of the bulk susceptibility
3) The maximum signal occurs when the beams are phase matched. That is the beams
are phase matched. This means that the sum of the wave vectors, k = 2k1 - k2 -k3 = 0.
sin(k) / k in limit as k -> 0 is equal to
So for k  0 the Intensity of the Electromagnetic Radiation due to the CARS Beam
becomes periodic and reaches a maximum when L = /k. The intensity at L = /k is
reduced by a factor of (2/)2 compared to the phase matched case.
The coherence length depends on the relative indicies of refraction of the media for the
different wavelengths involved
It turns out that Lc (gases) ~ 100 cm at atmospheric pressure and the Lc  1/Pressure
And in liquids the index changes more between wavelengths so Lc ~ 5mm
To increase Lc k = 2k1 -k2 - k3 is reduced in liquids by crossing the beams at a phase
matching angle typically of o . Cars thimited by the beam walk off
This then gives an exact sampling volume!
What is still missing here. The relationship to the Raman spectrum!
For the CARS beam at 3, 21-2 = 3
SO if (1 - 2) corresponds to the fundamental vibrational frequency v or vib (that is
vib = 1 - 2 then there should be some SPECIAL ENHANCEMENT of the
POLARIZATION for the case where E1 and E2 have these frequencies. AND THUS the
CARS signal should be frequency dependent.
In fact the total 3rd order susceptibility might include a part that is resonant (res) when
the above condition is met, and a non resonant (nr) part
( 3 ) cars = r e s + n r
Expressions for the frequency dependence of 3 or more precisely for r e s can be
developed classically or in Q.M.
We'll look at the classical case and express the r e s in terms of the normal Raman cross
section. In the QM approach the various resonances present in  are explicitly given.
We start with the harmonic oscillator as with the normal Raman for the classical
approach, and relate the polarizability to the bond stretching coordinate
 = o + (q)o q + …….
the force produces on the oscillator by the field due to the polarizability is
F = ½ (/q)o E2
The eqn. of motion for a damped oscillator can be written
2 q/dt2 +  q/dt + v2 q + 1/(2m) ((q))
is the damping constant, m = effective or reduced mass, and v is the vibrational
frequency of interest
The solution of the damped oscillator eqn. for q gives rise to the polarization at the antiStokes frequency due to the relation: P = q)o q E where N is the molecular
number density.
The driving force arises from a NONLINEAR interaction of the fields present and their
combination, E1 E2
ie. E() = E(1) E* (2) exp(i(2k1-k2) + other terms
Plugging in for E2 into the the damped oscillator eqn. and solution of the resulting 2nd
order differential eqn gives
q = 1/(8m) (/q)o E1(1)E2(2) exp(i(k1-k2)
Now remember that
P ~ ( 3 ) E2* E12 exp(i(2k1-k2) ~ q)o q E1exp(i(k1)
Plugging in q and simplifying yields
( 3 ) ~ n/(8m) (/q)2 1 / [ v2 - (1-2)2 - i (1-2) ]
this is pretty good but it actually should depend on the relative populations of the ground
and excited state, so let N j be the difference between the upper and lower state.
based on Boltzmann distribution.
This is related to the Raman differential cross section by Scattering Theory
2
4 4
Differential Raman Cross section = d/d = h/(2m
/
v) (q) 3 /c
Plugging this in gives:
q)2 = 2c4 mv/(34 h)
/ d/d
so
χcars ~ {[(2c4 / h
/ 34 ) (d/d) vib Nj ] / [(ωvib2 – (ω1 – ω2)2 – iΓvib(1-2)] } + NR
Note that the resonant part is a complex quantity
And we now have a resonance condition
If v2 - (1-2)2 ->
THEN ( 3 ) res gets very large
Relate to CARS figure
Now we have a relation for the contribution to the CARS signal from the resonance part,
but we have to realize that there is also a nonresonance part
The nonresonance part is much smaller by definition, but it can contribute to the overall
signal and its contribution can be large if the sample that we are probing has a low
analyte concentration. With no NR contribution, a plot of ( 3 ) 2 vs cm- 1 shift has a
Lorentzian line shape similar to a normal Raman resonance.
Morover, the real part is related to the nonlinear refractive index
The imaginary part is related to the normal Raman transition
But ( 3 ) 2 vs cm- 1 still looks pretty much like a stunted Raman signal
This was with out nr but as soon as a significant amount of nonresonant susceptibility
(from solvent or the resonant material itself) is introduced or when another resonance is
nearby the bandshapes become distorted due to the nonresonant part.
( 3 ) 2 = (' + nr)2 + "2 = '2 + 2'nr + nr2 + "2
This adds a non-symmetric part to the band contour.
Plot of CARS intensity vs frequency for varying amounts of a resonant material while
maintaining a constant background susceptibility
The Asymmetry is apparent even when the resonant material is the main component.
Note also that the frequency of the maximum shifts slightly and a minimum develops
The resonant component dcreases
In the limit that the resonant - nonresonant -> zero, the CARS output takes on the shape
of the real part of the susceptibility
The CARS signal (maximum) does not follow an N2 dependence when the resonant part
of the signal < nonresonant part of the signal
Interferences can also occur amoung neighboring resonances which add a frequency
dependent background to the susceptibility
This mixing allow the two overlapping peaks to be better "resolved" by CARS
than by normal Raman spectroscopy.
Because the negative part of one resonance cancels or partially cancels or partially
cancels the positive part of the other creating less overlap between the adjacent peaks.
Other unusual features may be present where overlapping bands are of unequal intensity.
Normal Raman cross sections are in a ration of 2:1. Because of a squared dependence on
cross sections, expect a 4:1 intensity in CARS but cross terms cause it to look like a
regular spectrum
What about other combinations based on the 3rd order susceptibility including SRG and
SRL?
For 2<1
ISRG = I2/I2 (intensity gain of 2 Stokes wave)
= -(32 2 2/(n1n2c2 )) "SRG I1 L ("SRG < 0)
For 1<2
ISRL = I2/I2 (intensity loss of 2 anti-Stokes wave)
= -(32 2 2/(n1n2c2 )) "SRL I1 L ("SRL > 0)
where
"SRG = is the imaginary part of SRG
Re  = {[(c4 / h
/ 24 ) (d/d) vib Nj ] [(ωvib – (ω1 – ω2)] / [(ωvib2 – (ω1 – ω2)2 – Γvib(12)] }
Im " = {[(c4 / h
/ 24 ) (d/d) vib Nj ] i vib / [(ωvib2 – (ω1 – ω2)2 – Γvib(1-2)] }
Using a linear focused Gaussian pump and probe beams having identical beam
parameters
Gain = P2/P2 = -961 2 / c2 Im SRG P1
or P2 = = -961 2 / c2 Im SRG P1P2
Compare ICARS to ISRG
ICARS
Two color and Three colors
ISRG
two or three colors out
Others Four wave mixing
Three different colors(frequencies) incident on the sample at o, 1, and 2
The third order non-linear susceptibility produces polarization to produce and
electromagnetic wave at 3 = o + 1 - 2 and 3 = 2 + 0 - 1
Raman Resonances can occur at both outputs.
Experimental Layouts
CARS
SRG/SRL
Applications
Coherent Raman methods have been applied to a variety of problems. One very active
area involves remote sensing of temperature and species in hostile environments such as
flames, internal combustion engines, exhaust from jet engines, and plasmas.
Consider a plasma
One study by Pealat, Taran et al. studied the vibrational excitation of hydrogen
molecules. The impetus for this was theoretical an experimental work which showed that
vibrational excitation of hydrogen molecules leads to an important increase in the
crosssection for negative hydrogen ion formation by dissociative electron attachment.
These ions are necessary for injection into controlled fusion devices. In this study, the
density of vibrationally excited hydrogen molecules in a plasma at pressure of 0.1 mbar
was measured using CARS
Measuring the temperature depends largely upon modeling the vibrational rotational
peaks present in the CARS spectrum. Remember the O, P, Q, R, and S branches as
alluded to in this diagram.
In the next experiment a diffusion flame burner was probed with CARS. The
experimental and theoretical spectra of the nitrogen Q-Branch bandhead measured 5 cm
above the center of the burner. The best fit yields temperature of 1826K compared with a
thermocouple reading of 1830 ±60K. Some fitting problems very close to the bandhead
could be a result of small errors in molecular constants or Raman linewidths or from a
nonuniform temperature over the experimental probe volume.
In this set of experiments a GE filament lamp was analyzed by CARS investigation at
several precise points. Here again the nitrogen Q branch was analyzed and a temperature
of 2020K was determined from the fit in the hottest position. A plot of the temperature as
a function of the position is also shown.
Next the temperature inside of a single cylinder engine was determined using CARS as a
function of the crank angle. Is this a 2 stroke engine?
One can also use CARS to probe the formation of species formed by cooling of a gas
through expansion into a vacuum through a nozzle. The following figure details that.
In this figure the formation of a dimer, trimer, and polymeric species are realized in the
CARS spectrum after expansion of the gas through a nozzle into vacuum.
CARS can also be used to probe the decomposition of a molecule. In this particular case
CF3NO was photodecomposed by the 2 laser beam and the decomposition products
were observed.
High Laser Intensities
RAMAN GAIN can occur at the Stokes Frequency even WITHOUT than input beam at
the Stokes frequency.
Stimulated Raman or "Raman Shifting" as it is most often called has be seen since the
early 1960s as a downshift in the frequency of the pump laser by an amount of energy
equal to the Raman Mode of the molecule.
A PRACTICAL use of this is for creating laser beams at new frequencies. Gases often
used include H2, D2, CH4 which would result in shifts of several thousand cm- 1 .
Once the threshold of 1-10MW in the laser at p is exceeded, then the following results
1) The Stokes shifted frequency 1 - v, S1, builds up (depletes the pump radiation
intensity.
2) Once this has built up what can happen?
3) What else can happen?
4) AS1 acts as 1 and 2 = p and this gives
Thus the anti-Stokes and higher order Stokes act as inputs in cascades generating
typically 4 orders of Stokes and 9 orders of anti-Stokes each shifted from the laser by the
Raman frequency of the molecule!
Stokes - diffraction limited collinear w/pump
Anti-Stokes - diverging cones of radiation due to phase matching condition
Maximize output by
1) varying the pressure
2) varying the pump power
3) changing the focusing geometry
How far out
Cahen et al Opt. Commun. 21 387 (1977)
50kW pulsed of 16micron radiation were generated by frequency shifting the visible
output of a Ruby pumped dye laser using H2 as the gas.
In the stimulated Raman effects a sample illuminated by two lasers that are coherent may
produce a selective population of a given energy level.
1) The transition involving the given energy levels must be Raman active
2) the frequency difference of the two incident beams must be adjusted to equal the
frequency of the Raman active transiton
3) the population is accomplished by a nonlinear optical process associate with stimulated
Raman scattering
Here now, as the excited molecules relax, in particular, by collisions, a pressure wave can
be generated which can be detected by a sensitive microphone.
If p - s = vib then a nonlinear interaction occurs to amplify s and lessen p AND
the net effect is to put the molecules in to an excited state.
The molecular translational energy of the molecules in the gas sample can then increase
by collisional deactivation of the molecules in the excited vibrational state back to the
ground state. this increase produces a pressure change, which is detected by
Using the gain eqn. and the correct expression for ( 3 ) on finds Is, the irradiance in the
Stokes wave which pumps the molecules into an excited vibrational or rotational state is
Is = cns/8 Es 2 = Iso exp(gsz)
where gs is the Stokes gain coefficient = - 162 kso /(cns2 np) "
Essentially we need to figure out how big the PARS signal will be, need to look at the #
of molecules put into the higher vibrational state.
Simplest Model relates the intensity of the Stokes wave to the pressure change and the
acoustic power
Is = Is(z) - Iso = Iso (exp(gsz) -1)
ex ~ 1 + x + ….
so s = Isogsz
Es = IsAst = ns h
/ s where As is the cross sectional area of the beam, t is the interaction
time of the pump and Stokes beams, and ns is the # of Stokes photons produced
by the amplification process
Continuing to assume the simplest case
Yields Pa = VP/t
= (-1)(o/s) gszPso
Shows the the photoacoustic power depends on the initial power in the probe beam and
the ".