Perturbative Theory and Modeling of Electronic-Resonance-Enhanced
Coherent Anti-Stokes Raman Scattering Spectroscopy of Nitric Oxide
Joel P. Kuehner∗
Department of Physics and Engineering,
Washington and Lee University
Lexington, VA 24450
Sameer V. Naik, Waruna D. Kulatilaka, Ning Chai,
Normand M. Laurendeau, and Robert P. Lucht∗∗
School of Mechanical Engineering,
Purdue University
West Lafayette, IN 47907
∗∗
The Institute for Quantum Studies
Texas A&M University
College Station, TX 77843
Anil Patnaik and Marlan O. Scully
Department of Physics
Texas A&M University
College Station, TX 77843
Sukesh Roy
Innovative Scientific Solutions, Inc.
2766 Indian Ripple Road
Dayton, OH 45440
James R. Gord
Air Force Research Laboratory, Propulsion Directorate
Wright Patterson AFB, OH 45433
Submitted to the Journal of Chemical Physics, 2007
∗
Author to whom correspondence should be addressed. Phone: (540) 458-8153 Fax: (540)
458-8884 Email: kuehnerj@wlu.edu
1
Abstract
A theory is developed for three-laser electronic-resonance-enhanced (ERE) coherent antiStokes Raman scattering (CARS) spectroscopy of nitric oxide (NO). A vibrational Q-branch
Raman polarization is excited in the NO molecule by the frequency difference between visible
Raman pump and Stokes beams. An ultraviolet probe beam is scattered from the induced
Raman polarization to produce an ultraviolet ERE-CARS signal. The frequency of the
ultraviolet probe beam is selected to be in electronic resonance with rotational transitions
in the A2 Σ+ ← X 2 Π (1, 0) band of NO. This choice results in a resonance between the
frequency of the ERE-CARS signal and transitions in the (0, 0) band. The theoretical model
for ERE-CARS NO spectra has been developed in the perturbative limit. Comparisons
to experimental spectra are presented where either the probe laser was scanned with fixed
Stokes frequency or the Stokes laser was scanned with fixed probe frequency. At atmospheric
pressure and an NO concentration of 100 ppm, good agreement is found between theoretical
and experimental spectral peak locations and relative intensities for both types of spectra.
Factors relating to saturation in the experiments are discussed, including implications for
the theoretical predictions.
1
Introduction
Electronic-resonance-enhanced (ERE) coherent anti-Stokes Raman scattering (CARS)
spectra of nitric oxide (NO) have been investigated theoretically and experimentally by
employing a three-laser or dual-pump CARS method.1 This paper explores the details and
capabilities of a developed theory for the ERE-CARS process. With increased attention
on combustion emissions and pollutant formation2 and the use of NO tracers in flowfield
mixing studies,3 the need has arisen for a diagnostic technique capable of probing low number
density levels (∼ 10 ppm) of radical species such as NO. Moreover, strong interest exists in
studying thermally generated NO in hypersonic flowfields, as knowledge of NO concentration
at elevated temperature levels would provide insight into the reaction rates required for
modeling these flowfields.4 Beyond the effects on the flowfield, NO formation has been
2
demonstrated as an excellent flowfield tracer in hypersonic flows.5 Finally, developments
in the area of national security have prompted a need for techniques that not only can
detect minor species, pollutants, and toxins, but that can do so rapidly, and ERE-CARS is
a possible solution to this requirement.6, 7
Figure 1 represents the energy-level diagram for the three-laser process utilized in this
CARS method. Visible Raman pump and Stokes beams induce a Raman transition through
a virtual state that is far from resonance with the A2Σ+ ← X 2 Π electronic transition of NO.
The third laser beam, termed the ultraviolet “probe” beam, has a frequency that probes the
Raman resonance through the first excited electronic state of NO. Usually, this third beam
is termed a “pump” beam. This nomenclature arose in the CARS literature because, in
most CARS experiments, the ω1 and ω3 beams are produced from the same laser beam, i.e.,
ω1 = ω3 , and each beam serves as a probe for the Raman polarization pumped by the other.
However, for the case shown in Fig. 1, the ω3 beam does not participate in the generation of
a Raman polarization in the medium and is thus referred to as the “probe” beam. In this
manner, the CARS method employed in this study is a modified version of the dual-pump
CARS process previously investigated for the simultaneous detection of two species.8, 9
Typically, the application of CARS for minor species detection and measurement is limited by coherent interference of the nonresonant background and the CARS signal from the
Raman resonance of interest. The inherent nonresonant background susceptibility from major species, such as diatomic nitrogen (N2 ), dominates the susceptibility when the Raman
species is present in low concentration. This problem is magnified for molecules such as NO
that possess relatively weak Raman cross-sections, resulting in an additional reduction in
signal level compared to N2 .10 The use of polarization techniques to suppress the nonresonant background has been demonstrated; however, the application of such techniques results
in a reduction in signal strength by at least a factor of 16.11 Therefore, polarization techniques are beneficial, but increases in the Raman resonant signal strength must be performed
simultaneously.
3
Since the late 1970’s, a few groups have investigated the potential of ERE-CARS for
minor species measurements in flames or plasmas.12 The term electronic-resonance-enhanced
is used to indicate that the frequency of the pump, probe, and/or Stokes beam is tuned near
resonance with an electronic transition of the molecule. The electronic resonance can increase
the resonant CARS signal considerably, minimizing concerns associated with the nonresonant
background. The current technique differs from previous ERE-CARS investigations13–15 in
that the frequency of the Raman pump (ω1 ) and ultraviolet probe (ω3 ) beams are well
separated. As a result, the Raman process and electronic process are, in a sense, uncoupled.
In the past, the pump, Stokes, and probe beam frequencies were all in or near electronic
resonance, a case termed “triple resonance.”16, 17 Triple resonance spectra are difficult to
interpret, because weaker double resonances appear near the major peaks.18 In almost all of
these investigations, the intensity of any ultraviolet beam is restricted so as to maintain both
Raman and electronic transitions within the perturbative regime; i.e, so that laser-induced
population transfer is negligible for both the Raman and electronic transitions. However,
the intensities of the visible pump and Stokes beams in this experiment are high enough
that stimulated Raman pumping is expected to be a significant effect, and the Raman
polarization that is established in the medium by the visible pump and Stokes beams is
close to the maximum that can be attained without saturating the transition. The use of
visible pump and Stokes beams simplifies the experimental implementation and theoretical
treatment of ERE-CARS without sacrificing detection sensitivity. Additional benefits of
Raman excitation with visible beams are that it permits easier interpretation of the features
in the spectra and more transitions can be probed simultaneously.
Other techniques, such as laser-induced fluorescence (LIF), degenerate four-wave mixing
(DFWM), and laser-induced polarization spectroscopy (PS) have proven useful for probing
minor species; however, ERE-CARS has advantages over these techniques when the medium
is at high pressure or when multiple species are present. For instance, to interpret an
experimental LIF spectrum for concentration measurements, the collision partners must
4
be identified.2 DFWM can be independent of quenching rates,19 but it loses sensitivity
compared to ERE-CARS at increased pressure levels.20 Similar to DFWM, PS can also be
affected by collision rates.12 Hence, ERE-CARS is a preferable technique at high pressure
or when quenching is problematic.
Additionally, at elevated temperatures and/or pressures, the LIF or DFWM spectra of
various species can overlap, resulting in uncertainty as to which molecule is being probed.21, 22
As implemented in these experiments, ERE-CARS has enhanced selectivity compared to LIF
and DFWM because of the requirement for simultaneous Raman and electronic resonance
for signal generation. This benefit in conjunction with minimized effects owing to pressure
and varying collision partners make ERE-CARS a very attractive prospect.
2
Experimental System
The experimental system for NO ERE-CARS measurements is shown schematically in
Fig. 2. A Q-switched Nd:YAG laser (Spectra-Physics GCR-4) with a repetition rate of 10 Hz
and a pulse duration of approximately 7 ns was used to pump a narrowband, tunable dye
laser (Lumonics), with Rhodamine 610 dye as the lasing medium. The dye laser output was
centered near 590 nm with a frequency bandwidth of approximately 0.08 cm−1 and supplied
the source for the Stokes beam (ω2 ). The 532-nm pump beam (ω1 ) for the CARS process
was generated using a second injection-seeded, Q-switched Nd:YAG laser (Spectra-Physics
290-10). A narrowband dye laser (Continuum ND6000) was also pumped using this 532-nm
output to produce tunable radiation in the vicinity of 704 nm; the laser dye used here was
LDS 698. This radiation was mixed with the 355-nm third-harmonic output of the injectionseeded Nd:YAG laser to generate the probe beam (ω3 ) near 236 nm. A feedback-controlled
frequency conversion system (INRAD Autotracker III) with a β-BBO nonlinear crystal was
used for the sum-frequency-mixing process. The full-width at half-maximum (FWHM) frequency bandwidth of the single-axial-mode pump beam (ω1 = 532 nm) was 0.003 cm−1 while
the bandwidth of the ultraviolet probe beam (ω3 = 236 nm) was approximately 0.1 cm−1 .
5
A three-dimensional phase-matching scheme was employed to generate the ultraviolet,
226-nm ERE-CARS signal. The BOXCARS geometry23 is slightly modified from the case
of CARS performed with all visible beams. The patterns of the three laser beams on the
focusing lens and of the laser beams and the CARS signal beam on the collimating lens are
shown in Fig. 3. As a consequence of the large difference in momentum of the ultraviolet
and visible photons interacting in the CARS process, the ultraviolet probe beam is much
closer to the axis joining the centers of the input and output lenses than the visible pump
and Stokes beams. The ultraviolet CARS signal beam is also generated much closer to this
axis.
During a typical experiment, pulse energies of 5 mJ were used for the visible pump and
Stokes beams. The maximum pulse energy of the ultraviolet probe beam was 0.25 mJ. The
beams were focused at the probe volume and recollimated after interaction, along with the
ERE-CARS signal, using a pair of 500-mm, focal-length lenses. At this point, beam dumps
were used to capture the visible pump and Stokes beams. A joulemeter (Molectron J3-05)
monitored the pulse energy of the ultraviolet probe beam after interaction. Four 215-nm
dielectric mirrors, designed to be used at 45◦ incidence, and a set of apertures filtered the
ERE-CARS signal. By utilizing the 215-nm mirrors at 0◦ incidence, a low-pass filter is
formed that transmits 70% at 226 nm but only 1% at 236 nm, thereby significantly reducing background interference from the ultraviolet probe beam. The ERE-CARS signal was
further isolated by a 1-m spectrometer (SPEX). The resulting signal intensity was collected
using a solar-blind-photomultiplier tube (Hamamatsu R166) and oscilloscope (Tektronix
TDS5054B). The joulemeter and photomultiplier output were recorded on a shot-by-shot
basis during which time either the Stokes or the probe-beam frequency was scanned under
computer control.
In addition to interference caused by scattered background from the ultraviolet probe
beam, the ERE-CARS signal contained significant contributions from the nonresonant fourwave-mixing background signal, as discussed earlier. Therefore, once the ERE-CARS signal
6
was located and optimal system alignment obtained, a polarization technique1 was utilized
to suppress the nonresonant background. This polarization arrangement is demonstrated in
Fig. 4. The ultraviolet probe beam was vertically polarized, while the axes of the visible
pump and Stokes beams were rotated 60◦ to the vertical. This generates an almost vertically
polarized ERE-CARS signal, and at the same time, caused the polarization of the nonresonant background to be rotated 30◦ to the vertical. To take advantage of this technique,
an α-BBO analyzer was placed in the signal channel such that the transmission axis was
perpendicular to the nonresonant background polarization. This significantly reduced the
nonresonant background signal. While this diminishes the ERE-CARS signal as well, the
overall effect is a considerable increase in signal-to-noise ratio (SNR).
3
Experimental Demonstration of Electronic Resonance Enhancement
The initial NO ERE-CARS experiments were performed with a room-temperature, sub-
atmospheric mixture of 1% NO in a buffer gas of N2 (10, 000 ppm). The NO ERE-CARS
spectrum shown in Fig. 5(a) was recorded by scanning the Stokes dye laser beam with a fixed
ultraviolet probe-beam frequency ω3 of 42140.75 cm−1 (237.3 nm). For this probe frequency,
there is some electronic resonance enhancement, but the probe frequency is below the electronic transition frequency for the (vb = 1, vd = 0) vibrational band. The spectrum shown
in Fig. 5(b) was recorded by scanning the Stokes dye laser beam with a fixed ultraviolet
probe-beam frequency closer to resonance (42194.09 cm−1 or 237.0 nm). The enhancement
upon electronic resonance is clearly evident.
The ERE-CARS signal frequency, ω4 , is readily absorbed by 1% NO. Therefore, as the
frequency ω3 is scanned further into electronic resonance, the NO concentration in the gas
cell was decreased by a factor of ten to 1000 ppm. The results of this change are shown in
Fig. 5(c), which represents a spectrum recorded for a fixed ultraviolet probe-beam frequency
of 42337.00 cm−1 (236.2 nm). The spectrum shown in Fig. 5(c) was acquired with all beams
vertically polarized. The Stokes beam was blocked at the beginning of the scan for a Raman
shift range of 1871.1 to 1871.6 cm−1 ; hence, the increase in signal level at 1871.6 cm−1 is
7
due to the nonresonant background. For subsequent recording of ERE-CARS spectra, the
polarization scheme shown in Fig. 4 was used to suppress the nonresonant background. A
typical spectrum recorded with polarization suppression is shown in Fig. 5(d) for 1000 ppm
NO and a fixed probe-beam frequency of 42342.9 cm−1 .
4
Perturbative Theory for ERE-CARS
Theoretical studies of the third-order polarization susceptibility in the perturbative limit
(low input laser powers) have been previously developed and tested for ERE-CARS.17, 24–27
Bloembergen et al.24 and Oudar and Shen25 both performed a perturbation expansion of
the density matrix for the system. Eesley26 employed Hellwarth diagrams as a basis for
the derivation, and Druet et al.27 used a time-ordered diagrammatic approach. A review of
these theories is found in Attal et al.17 Line strength calculations for various Hund’s coupling
cases for OH, C2, and CH18 and for I2 28 have also been performed. While significant work
has been done to model the aforementioned molecular species, applying these models to NO
requires some additional considerations. Because NO is closer to a pure Hund’s case (a),
the satellite branches in the absorption spectrum are much stronger for higher values of the
rotational quantum number, J , as opposed to OH.18 This introduces new transitions into
the spectrum, which must be accounted for in the model.
The form of the polarization susceptibility modeled here is derived from the general
third-order susceptibility given by Prior for all four-wave mixing processes.29 The doublesided Feynman diagrams concerning Raman resonances between levels a and b (see Fig. 1) are
displayed in Fig. 6. These diagrams are similar to previous time-ordered diagrams;13 however,
this presentation accounts for nondegenerate pump beams. In particular, Figs. 6(b), (d), (e),
and (g) correspond to Fig. 1(a), 1(b), 2(a), and 2(b) in Attal et al.,13 respectively. Assuming
levels c and d are initially unpopulated, the diagrams of Fig. 6 result in a reduction of the
general form to the following (in units of m2 /V2):
8
N X
χCARS (ω4 : ω1 , −ω2, ω3 ) = χN R +
4πεoh̄3 a,b
×
X
×
X
ρ(0)
aa
c
−
1
ωba − (ω1 − ω2 ) − iΓba
µ3da µ4bd
µ4da µ3bd
+
ωda − ω4 − iΓda ωdb + ω4 + iΓdb
d
"
(
X
(0)
ρbb
c
!
!
!
µ1cb µ2ac
µ2cb µ1ac
+
ωca + ω2 − iΓca ωca − ω1 − iΓca
!#)
µ1cb µ2ac
µ2cb µ1ac
+
ωcb − ω2 + iΓcb ωcb + ω1 + iΓcb
(1)
where N is the total population being probed, εo is the permittivity of free space, h̄ is Planck’s
(0)
constant, ρii is the fractional population of the state i, ωij is defined as (Ei − Ej )/h̄ with
Ei being the energy associated with state i, ωk is the frequency of laser beam k, Γij is the
dephasing rate for the electric dipole transition between states i and j, and µkij is defined
as:
µkij = êk · ~µij
(2)
where ~µij is the dipole matrix element for the transition i → j, and êk is the normalized
polarization vector for laser beam k. The form of Eq. 1 is in agreement with others in
the literature.17 The remaining susceptibility terms not shown in Eq. 1 arise from Raman
resonances between levels other than a and b and one- and two-photon transitions. These
excluded terms are small in magnitude, largely frequency-independent as compared to the
terms shown, and can be accounted for in χN R, the nonresonant susceptibility.27
Considering the case shown in Fig. 1, it is evident that ωcb ∼
= ωca − ω1 + ω2 and ωca , ωcb ω1 , ω2 . Therefore, Eq. 1 can be reduced to:
N X
χCARS (ω4 : ω1 , −ω2, ω3 ) = χN R +
4πεoh̄3 a,b
×
X
d
×
"
X
(
1
ωba − (ω1 − ω2 ) − iΓba
µ3da µ4bd
µ4da µ3bd
+
ωda − ω4 − iΓda ωdb + ω4 + iΓdb
(ρ(0)
aa
c
9
−
(0)
ρbb )
µ µ
µ1cb µ2ac
+ 2cb 1ac
ωcb + ω1 ωca − ω1
!
!
!#)
(3)
The final summation in brackets has the form of a spontaneous Raman cross-section (in
units of m2/sr) for excitation at ω1 :30
∂α
∂Ω
!
ab
ω24
=
(4πεo )2c4
X
µ2cb µ1ac
µ1cb µ2ac
+
h̄(ωcb + ω1 ) h̄(ωca − ω1 )
c
!2
(4)
Eq. 3 can then be recast as:
N X
χCARS (ω4 : ω1 , −ω2, ω3 ) = χN R +
4πεoh̄3 a,b
×
X
1
ωba − (ω1 − ω2 ) − iΓba
µ3da µ4bd
µ4da µ3bd
+
ωda − ω4 − iΓda ωdb + ω4 + iΓdb
d
v
u
u (4πεo )2h̄2 c4
(0)
(0)
(ρ − ρ )t

×
(
aa
bb
ω24
∂α
∂Ω
!
!
! 



(5)
ab
where c is the speed of light. Observing for the electronic resonance case in Fig. 1 that
ωda ≈ ω4 , the second term in the second line of Eq. 5 is nonresonant and can be absorbed
into χN R. Rearranging, Eq. 5 now becomes:

Nc2 X  ∂α
χCARS (ω4 : ω1 , −ω2 , ω3 ) = χN R + 2 2
h̄ ω2 a,b,d  ∂Ω
×
µ4da µ3bd
ωda − ω4 − iΓda
!)
!1/2 
ab
(0)

(ρ(0)
aa − ρbb )


ωba − (ω1 − ω2 ) − iΓba
(6)
This equation is employed in modeling the ERE-CARS spectra presented herein. To compare
the theoretical spectra with the acquired data, convolutions must be performed to account
for the finite laser linewidths used in the experiment. Three convolutions were conducted:
one each for the pump, the Stokes, and the probe beams. Gaussian profiles were assumed
for each laser beam. The third convolution is unique to this ERE-CARS process, and has
not been applied in previous models, in which a convolution for the Stokes beam is typically
applied.31 Once the convolutions are completed, the magnitude of the complex susceptibility
can be compared to the data.
As seen from Fig. 1, if ω3 is tuned into resonance with ωbd , then ω4 will be resonant with
ωda . The enhancement resulting from this electronic resonance enables detection of weak
Raman transitions of molecules such as NO, even at low concentrations. Improvements in
10
the detection limit on the order of 102 to 108 can be achieved.27 In addition, species selectivity
for the CARS process is strengthened by this electronic resonance. For the standard CARS
process, the Raman resonance criterion is:
ω1 − ω2 = ωba
(7)
For the ERE-CARS process under consideration in this study, the electronic resonance criterion is:
ω4 = ωda
(8)
The extra selectivity provided by two resonance requirements can eliminate interferences
from overlapping spectra of molecules, such as the A ← X fluorescence band of NO with the
Schumann-Runge system of O2 .21
Equation 6 is now considered in light of the specific energy-level diagram for the Π3/2
ground state of NO shown in Fig. 7. A similar schematic could be presented for the X 2 Π1/2
state. Only the Raman Q-branch was probed during the experiments; therefore, the S- and
O-branches are not included in the diagram. For the case under consideration in Fig. 7,
vd = 0. In accordance with the selection rules for Raman transitions, two distinct Raman
Q-branch transitions can occur, corresponding to either parity of the lower rotational levels.
To distinguish between parity levels, the transitions starting from negative parity ground
states are shown with solid lines, whereas those that originate from positive parity states
are shown with dashed lines. The Raman transition occurs through a virtual state, as only
the ultraviolet probe beam is in electronic resonance. Once the molecule is excited to its
first vibrational level (vb = 1, Jb ), three electronic transitions can couple the molecule to
the A2Σ state for a specific rotational level and parity. Each of the electronic transitions is
matched with a corresponding electronic transition that returns the molecule to the ground
state. The large difference in frequency between the pump and probe beams separates the
Raman process from the electronic process; thus, the enhancement of the Raman process is
well defined, which assists in reducing the complexity of the model.
11
To apply the model, the Raman cross-section was calculated from values provided by
Schrötter and Klöckner.10 The same cross-section was used for both electronic ground states
of NO, as a previous study of NO CARS determined the values to be essentially the same.32
Resonance effects between ω4 and ωda were evaluated using the dipole matrix elements µ3bd
and µ4da . These matrix elements were calculated following the formulae in Hilborn33 incorporating spectroscopic data from LIFBASE.34 The spontaneous emission coefficient A21, used
to calculate the matrix elements, was divided by a factor of three to account for isotropic
emission and the fact that the Zeeman effect was not explicitly taken into account, as used
previously by Siegman.35 The Raman shift (ωba ) was based on the constants of Huber and
Herzberg36 and of Laane and Kiefer.37 The electronic transition frequencies, ωda , were taken
from LIFBASE.34
The Raman linewidths (cm−1 ), Γba (FWHM), were obtained from Doerk et al. as follows:38
Γba
Tref
= 0.12
T
!
P
Pref
!
(9)
where Tref = 298 K, and Pref = 1 atm. The electronic transition linewidth, Γda , was
calculated from a combination of the Doppler and collisional widths. The collisional width
was approximated as:
Γda,c = 0.6P
(10)
where Γda,c has units of cm−1 and P has units of atm. The above relation is an approximation
to the work of Chang et al.39 and is in agreement with other studies in the literature.19, 40
The Doppler width for the electronic resonance transitions is based on standard theory12 and
at room temperature is approximately40 Γda,D = 0.1 cm−1 . The total linewidth was then
approximated by:
Γda =
5
q
Γ2da,c + Γ2da,D
(11)
Definition of an Effective Intermediate Raman Level
To address the contribution to the susceptibility in Eq. 6 from the numerous excited
electronic states of NO, we define an effective electronic level for the calculations. We use
12
the general transition polarizability to determine the characteristics of the effective electronic
level. The transition polarizability can be related to the Raman cross-section of Eq. 4 by:
∂α
∂Ω
!
=
ab
ω24
(α )2
(4πεo )2 c4 zz ab
(12)
Therefore, the transition polarizability is found to be:
(αzz )2ab
=
X
c
µ2cb µ1ac
µ1cb µ2ac
+
h̄(ωcb + ω1 ) h̄(ωca − ω1 )
!2
(13)
The summation over c includes only the possible intermediate states that have single-photon
allowed transitions with both states a and b. The polarizability for a transition between an
initial rotational level A and final rotational level B is found by summing over all possible
Zeeman states a and b:
(αzz )2AB
X X
µ2cb µ1ac
1
µ1cb µ2ac
+
=
(2JA + 1) a,b c h̄(ωcb + ω1 ) h̄(ωca − ω1 )
!2
(14)
In general, the Raman cross-section or transition polarizability can not be calculated
directly, especially for molecules such as NO which possesses numerous excited levels that
are connected with the X 2 Π ground state through single-photon transitions. In addition,
many vibrational levels in each of these electronic levels are connected with the v 00 = 0 and
v 00 = 1 vibrational levels in the X 2 Π state by transitions with significant oscillator strengths.
Beyond the sheer number of transitions that would have to be accounted for, the oscillator
strengths for most of these transitions are not well known. This is especially the case for
electronic levels above the E 2 Σ+ level of NO.
Consequently, we chose one effective electronic level (C 2Π), aside from the A2 Σ+ state, to
act as an intermediate level in the Raman transition by representing all of the nonresonant
excited electronic levels. The oscillator strengths, and thus the dipole matrix elements, of
the (0,0) and (0,1) bands in the C 2Π ← X 2 Π electronic manifold were artificially enhanced
so as to obtain the literature value10 of the transition polarizability for the fundamental
(1,0) band of the Q-branch of NO in the X 2 Π state. In particular, the values of the dipole
matrix elements for this band were increased by a factor of 2.2 from the values of Luque and
13
Crosley.34 The calculation for Eq. 14 was performed by summing only over states c in the
v = 0 vibration-rotation bands of the A2 Σ+ and C 2 Σ+ levels.
6
Enhancement Factor
Because a primary advantage of employing ERE-CARS over CARS is the enhancement
in signal level, a parameter that quantifies this increase was developed as part of the study.
The resulting enhancement factor, or δ(ω4), is defined as:
δ(ω4) =
χCARS (ω4 : ω1 , −ω2 , ω3 )
χCARS ((2ω1 − ω2 ) : ω1 , −ω2 , ω1 )
(15)
where the denominator represents the degenerate pump-beam case, ω1 = ω3 . The enhancement factor is used to evaluate the reduction in detection limit resulting from electronicresonance enhancement. This factor is proportional to the increase in CARS susceptibility
and thus inversely proportional to the NO detection limit. The CARS signal is proportional
to the square of the enhancement factor.
7
Results
During the experiments, either the Stokes or the ultraviolet probe-beam frequency was
scanned to produce a CARS spectrum. These two spectrum types were acquired experimentally and theoretically modeled and are referred to as “Stokes scans” or “probe scans” to
indicate which laser beam was tuned across the spectrum. Because the spectral features of
these scan types differ, the results will be covered in separate sections.
For clarity, a notation is introduced here, similar to that found in the literature,17 to
denote the overall ERE-CARS transition. The notation begins by listing the Raman transition with a subscript (1 for Π1/2 and 2 for Π3/2)41 to indicate the Π state of molecule.
The final electronic transition is identified next following the notation of Mavrodineanu and
Boiteux.42 The value of Ja in the ground state is then listed in parentheses. Therefore,
processes involving transitions on the far right and second from far right in Fig. 7 would be
denoted by Q2R2 (2.5) and Q2P Q12(2.5), respectively.
14
7.1
Results for Probe Scans
Figure 8 shows comparisons between theoretical and experimental spectra for six atmo-
spheric probe-scan cases. For these scans, the probe frequency was tuned over the approximate range of 42280 to 42400 cm−1 (236.5 to 237.0 nm). The spectra encompass fixed
Stokes frequencies that correlate to Raman shifts between 1872.74 and 1875.95 cm−1 . The
data were acquired at room temperature using a concentration of 100 ppm NO in N2.
As shown in Fig. 8, good agreement exists between the theoretical and experimental
spectra for all fixed Raman shifts. Spectral peak locations and relative intensities are well
represented. While a good match generally occurs in linewidths, we should note that the
theoretical spectra were generated using Stokes and probe linewidths of 0.08 cm−1 and
1.0 cm−1 , respectively. This probe linewidth is wider than that estimated experimentally.
The enhanced linewidth indicates the presence of saturation in the electronic process b → d
(see Fig. 7) and highlights the necessity of a triple convolution in generating the theoretical
spectra. These results confirm that the model is capable of calculating theoretical spectra in
good agreement with experimental spectra, and, more importantly, that the essential physics
of the ERE-CARS process are captured by the model.
Figure 9 displays three probe-scan cases over an extended probe-frequency range. These
three cases correspond to those shown in Fig. 8(a), (b), and (d). By expanding the scale,
the spectra now encompass resonances in the Π3/2 ground state. The amplitude variation
between the spectra of Figs. 8 and 9 occurs because the frequency difference ω1 − ω2 is in
resonance with different rotational levels. For the expanded spectra, the Π1/2 transitions
correspond to the peaks on the righthand side of each spectrum and the Π3/2 transitions are
on the lefthand side. As an example, the major peaks in Fig. 9(a) are marked and correspond to the following transitions from left to right: Q2P2 (1.5) + Q2P Q12(1.5), Q1Q1 (2.5)
+ Q1Q P 21(2.5), and Q1R1 (2.5) + Q1R Q21(2.5). Similar transitions occur in Figs. 9(b) and
9(c),with the transition Q1P 1 appearing in Fig. 9(c) for J1 = 7.5.
15
The relative intensities within a given Π state are generally predicted with good accuracy.
However, the relative intensities between Π states, i.e., on the right and left sides of the
spectra, are not in complete agreement with the experimental spectra. This is primarily due
to a decrease in probe-beam power during the scan, which arises from the short lifetime of the
LDS698 laser dye employed in the experiments. The experimental spectra were normalized
by dividing by the measured probe-beam power, but this procedure might not be the most
accurate normalization approach if the electronic resonance were saturated. In addition, the
two Π states may not have the same Raman cross-section, as assumed for our calculations.
The enhancement factor associated with each spectrum in Fig. 8 is shown in Fig. 10.
Because of the isolated electronic resonances that dominate each spectrum, the spectral behavior of the enhancement factor is very similar to its corresponding spectrum. As can be
seen from Fig. 10, the electronic resonance for the ERE-CARS process provides an enhancement in signal strength by a factor of approximately 1000 compared to a typical CARS
process. This significant enhancement permits the CARS signal to be detected above the
nonresonant background.
7.2
Results for Stokes Scans
Figure 11 displays comparisons between theoretical and experimental spectra for three
atmospheric Stokes-scan cases. The experimental spectra were acquired by scanning the
Stokes frequency over a Raman shift range of 1872.5 to 1877.5 cm−1 for three different
fixed ultraviolet probe-beam frequencies. The experimental spectra were obtained for the
same diagnostic conditions as for the probe-scan spectra shown in Fig. 8 (1.0 atm and
100 ppm NO). The theoretical spectra were generated using a smaller probe-beam linewidth
(0.08 cm−1 ) and an identical Stokes-beam linewidth. This behavior indicates that the effects
of saturation occurring in the electronic process are not significant for Stokes-scan spectra. As
the probe-beam frequency is fixed, the convolution with the Stokes beam linewidth becomes
the dominant factor in determining the shape of the Stokes-scan spectra.
16
The theoretical spectra in Fig. 11 are in good agreement with the experimental results.
Because of the numerous Q-branch transitions probed and the three possible electronic transitions at or near resonance for each case (see Fig. 7), the Stokes scans include a greater
number of enhanced transitions as compared to the probe scans. These transitions overlap
and interfere, and thus appear as a single wide peak. This merging results from the frequency
spacing of the NO Q-branch Raman lines, which is far less than the frequency spacing of
the electronic transitions, and the use of visible pump and Stokes beams. Similar to the
probe scans, the spectral shape of the experimental Stokes scans changes as the ultraviolet
probe-beam frequency is varied. The modeled spectra capture these spectral changes well
for all cases.
The behavior of the Stokes-scan spectra can be explained by considering the dominant
transitions as the probe frequency is changed for each scan. Only transitions arising from
the Π1/2 state are probed using these diagnostic parameters. Starting with Fig. 11(a), the
ERE-CARS process is in strong resonance with the Q1Q1 and the Q1Q P21 transitions for
J = 1.5 − 5.5, with the dominant peak occuring for J = 4.5. As the probe beam is shifted to
higher frequencies, these resonances move to higher values of J , with a maximum intensity
occuring at J = 7.5 and a shift to the left of the dominant peak, as shown in Fig. 11(b).
Simultaneously, a new set of resonances appears on the righthand side of the spectrum,
corresponding to Q1R1 and Q1 R Q21 for J = 1.5 − 2.5. These two sets of resonances continue
to shift to the left as the probe frequency rises, as displayed in Fig. 11(c), with the maximum
intensity occurring for J = 10.5 and J = 4.5 for the left and right peaks, respectively.
The spectral response of NO is clearly detectable over the nonresonant background even
at this low concentration and the model is capable of predicting the spectrum for these
thermodynamic conditions for both types of scans. The SNR for the spectra shown in Fig. 8
and Fig. 11 is approximately 10, and as such, the detection limit is estimated to be less than
10 ppm. In comparison, using polarization CARS, Pott et al.43 report a detection limit of
200 ppm with a SNR of approximately unity.
17
Figure 12 displays the enhancement factors corresponding to the Stokes scan spectra
shown in Fig. 11. As with the probe scans, the spectral shape of the enhancement factor is
similar to the corresponding spectrum. Moreover, the enhancement magnitude is similar to
that of the probe scans, near a factor of 1000.
8
Conclusions
In conclusion, the ERE-CARS process has been demonstrated both experimentally and
theoretically for NO. The good agreement between the experimental and theoretical spectra
for both the probe and Stokes scans highlights the capabilities of the theoretical model
developed and described in this paper. The predicted line positions match closely with
those of the transitions in the experimental spectra. The relative intensities corresponding
to each Π state are approximately correct, and good comparisons occurred between spectra
encompassing both Π states. This theory complements the sensitivity of the experimental
setup, which was able to obtain NO spectra for a concentration of 100 ppm at a SNR of
10. This low detection limit makes the technique viable for many practical applications. In
addition, the enhancement factor illustrates the advantages of employing ERE-CARS over
the degenerate pump-beam case.
Another advantage of ERE-CARS is increased species selectivity. This feature can be
demonstrated from the results presented for both the probe and Stokes scans. As shown
in Fig. 8, only a few dominant transitions occur for each configuration of probe and Stokes
frequencies. While an increased number of transitions occur for the Stokes-scan spectra
(Fig. 11), it is clear from all cases that not all Raman Q-branch transitions can be probed
and enhanced simultaneously. Therefore, if the entire range of transitions for one molecule
can not be satisfied completely, it is very unlikely that the probe and Stokes frequencies
chosen will satisfy the selection criteria for two molecules simultaneously. This benefit of
ERE-CARS is just as important as the enhancement in signal strength, because a strong
signal arising from multiple species can be just as difficult to analyze as a weak, indiscernible
signal.
18
Acknowledgments
Funding for this research was provided by a Phase II STTR SB00207 from the Air Force
Office of Scientific Research (Dr. Julian Tishkoff, Program Manager), by the U.S. Department of Energy, Division of Chemical Sciences, Geosciences and Biosciences, under Grant
No. DE-FG02-03ER15391, by the Air Force Office of Scientific Research, under Contract
No. FA9550-05-C-0096, and by the Air Force Research Laboratory, Propulsion Directorate,
Wright-Patterson Air Force Base, under Contract No. F33615-03-D-2329. Additional support is gratefully acknowledged from the Defense Advanced Research Project Agency and
the US Army Research Office.
19
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23
Excited Electronic State
d
v =2
v =1
v =0
c
v =2 1
v =1
v =0
2
3
4
a
b
Ground Electronic State
Figure 1: CARS energy-level diagram including electronic resonances
24
Energy
Meter
532 nm 236 nm
Pol
λ/2
λ/2
Narrowband
Dye Laser
590 nm
532 nm
590 nm
Pol
λ/2
532 nm
Gas Cell
NO in N Buffer Gas
2
BBO SFM Crystal
Narrowband
Dye Laser
704 nm
355 nm
Pol
Beam Dumps
Solar-Blind
PMT
Q-switched
Nd:YAG Laser
Aperture
1-Meter
Spectrometer
ER CARS
226 nm
Oscilloscope
Delay
Generator
Q-switched Nd:YAG Laser
Injection-Seeded
Figure 2: Experimental system for the NO ERE-CARS system
25
Filters
70% T 226 nm
1%T 236 nm
Collimating Lens
Focusing Lens
590 nm
236 nm
532 nm 236 nm
590 nm
226 nm
532 nm
Figure 3: Phase-matching geometry employed in the NO ERE-CARS system
26
NO ERE
CARS Signal
236 nm
Nonresonant
Background
532 nm
590 nm
Transmission Axis
for Polarizer in
CARS Signal
Channnel
Figure 4: Polarization arrangement for ERE-CARS detection
27
6
(arb. units)
1.5
1/2
1
Intensity
Intensity
1/2
(arb. units)
2
0.5
0
1871
1873
1875
5
4
3
2
1
0
1871
1877
-1
(b)
4
(arb. units)
1
0.4
0.2
0
1871
3
2
1/2
Stokes beam
blocked
Intensity
1/2
(arb. units)
1.2
Intensity
1877
Raman Shift (cm )
(a)
0.6
1875
-1
Raman Shift (cm )
0.8
1873
1873
1875
1877
1
0
1871
-1
1873
1875
1877
-1
Raman Shift (cm )
Raman Shift (cm )
(c)
(d)
Figure 5: Experimental NO ERE-CARS spectra demonstrating electronic resonance enhancement at room temperature and subatmospheric pressure for the following conditions
(a) ω3 = 42140.75 cm−1 and 1% NO, (b) ω3 = 42194.09 cm−1 and 1% NO, (c) ω3 =
42337.00 cm−1 and 1000 ppm NO, and (d) ω3 = 42342.92 cm−1 and 1000 ppm NO
28
a
d
3
1
2
3
2
b
a
c
1
a
c
a
a
b
(e)
a
a
3
a
2
b
c
3
2
c
1
a
b
3
3
1
c
2
1
b
(f)
b
(g)
3
c
a
a
(d)
a
d
d
b
(c)
d
b
1
c
b
1
2
(b)
d
d
b
b
(a)
b
a
d
2
3
a
d
b
c
b
2
1
(h)
Figure 6: Double-sided Feynman diagrams for the ERE-CARS process corresponding to the
following terms in Prior:29 (a) 3, (b) 4, (c) 21, (d) 24, (e) 9, (f) 10, (g) 39, and (h) 42.
29
N J
4 4.5
3.5
3 3.5
2.5
2 2.5
1.5
1 1.5
0.5
vd = 0
2
A Σ
N J
4 3.5
3.5
3 2.5
2.5
2 1.5
1.5
vb = 1
X2Π3/2
X2Π3/2
Q
O
Q2
P12
R12
P
P2
R2
Q12
N
J
4 3.5
3.5
3 2.5
2.5
2 1.5
1.5
va = 0
Figure 7: Energy-level diagram of the ERE-CARS process for the X 2 Π3/2 ground electronic
state of NO
30
Q
10
(arb. units)
Q
8
P
21
R
R
4
1
Q
21
(2.5)
2
0
42280
1
Q
8
P
21
R
(3.5)
6
R
0
42320
42360
-1
Probe Frequency (cm )
42400
42280
42320
42360
-1
Probe Frequency (cm )
10
R
1
Q
P
R
21
(4.5)
1
Q
(arb. units)
Q
21
(4.5)
8
4
2
0
8
Q
P
6
21
(7.5)
42320
42360
-1
Probe Frequency (cm )
42280
42400
21
(7.5)
1
(7.5)
42320
42360
-1
Probe Frequency (cm )
10
6
1
Q
R
P
21
(10.5)
8
P
R
1
Q
21
(10.5)
1
1/2
(10.5)
(arb. units)
Q
42400
(d)
4
Intensity
(arb. units)
1/2
Intensity
1
Q
0
12
2
0
42280
R
R
P
2
(c)
6
1
Q
4
1/2
6
-2
42280
42400
(b)
Intensity
Intensity
1/2
(arb. units)
14
10
21
2
(a)
12
1
Q
(3.5)
4
1/2
(2.5)
6
Q
1
Intensity
Intensity
1/2
(arb. units)
10
42320
42360
-1
Probe Frequency (cm )
(e)
42400
5
Q
4
P
21
(13.5)
3
2
1
Q
P
1
(13.5)
1
0
-1
42280
42320
42360
-1
Probe Frequency (cm )
42400
(f)
Figure 8: Comparison of theoretical (black line) and experimental (red line) ERE-CARS
spectra generated for a pressure of 1.0 atm, a concentration of 100 ppm NO, and for Raman shifts and Raman transitions of (a) 1875.95 cm−1 , Q1(2.5); (b) 1875.82 cm−1 , Q1(3.5);
(c) 1875.66 cm−1 , Q1(4.5); (d) 1875.00 cm−1 , Q1(7.5); (e) 1874.02 cm−1 , Q1(10.5); and
(f) 1872.74 cm−1 , Q1(13.5)
31
10
Q
Intensity
1/2
(arb. units)
P
8
P
Q
R
1
Q
2
P
R
21
12
(2.5)
6 (1.5)
1
Q
21
(2.5)
4
2
0
42200
42300
-1
Probe Frequency (cm )
42400
(a)
10
Q
Intensity
1/2
(arb. units)
P
8
P
Q
R
P
21
12
(3.5)
(1.5)
6
R
1
Q
2
1
Q
21
(3.5)
4
2
0
42200
42300
-1
Probe Frequency (cm )
42400
(b)
10
P
Intensity
1/2
(arb. units)
2
8
6
P
Q
Q
12
(6.5)
Q
2
Q
R
1
Q
P
12
(6.5)
21
(7.5)
R
R
1
Q
21
(7.5)
4
P
1
2
(7.5)
0
42200
42300
-1
Probe Frequency (cm )
42400
(c)
Figure 9: Comparison of theoretical (black line) and experimental (red line) ERE-CARS
spectra generated for a pressure of 1.0 atm, a concentration of 100 ppm NO, and for Raman shifts and Raman transitions of (a) 1875.95 cm−1 , Q1(2.5)/Q2 (1.5); (b) 1875.82 cm−1 ,
Q1(3.5)/Q2 (1.5); and (c) 1875.00 cm−1 , Q1(7.5)/Q2 (6.5)
32
1200
1000
1000
Enhancement Factor
Enhancement Factor
1200
800
600
400
200
0
42280
42320
42360
-1
Probe Frequency (cm )
800
600
400
200
0
42280
42400
1200
1200
1000
1000
800
600
400
200
0
42280
42320
42360
-1
Probe Frequency (cm )
800
600
400
200
0
42280
42400
1200
1200
1000
1000
800
600
400
200
42320
42360
-1
Probe Frequency (cm )
42320
42360
-1
Probe Frequency (cm )
42400
(d)
Enhancement Factor
Enhancement Factor
(c)
0
42280
42400
(b)
Enhancement Factor
Enhancement Factor
(a)
42320
42360
-1
Probe Frequency (cm )
800
600
400
200
0
42280
42400
(e)
42320
42360
-1
Probe Frequency (cm )
42400
(f)
Figure 10: Enhancement factor for a pressure of 1.0 atm, a concentration of 100 ppm NO, and
for Raman shifts of (a) 1875.95 cm−1 , (b) 1875.82 cm−1 , (c) 1875.66 cm−1 , (d) 1875.00 cm−1 ,
(e) 1874.02 cm−1 , and (f) 1872.74 cm−1
33
Intensity
1/2
(arb. units)
5
Q
4
1
Q
P
21
(1.5-5.5)
3
2
1
0
1873
1875
1877
-1
Raman Shift (cm )
(a)
Q
2.5
1
Q
P
21
2
(6.5-8.5)
R
1.5
R
Intensity
1/2
(arb. units)
3
1
Q
21
(1.5-2.5)
1
0.5
0
1873
1875
1877
-1
Raman Shift (cm )
(b)
Intensity
1/2
(arb. units)
7
Q
6
R
1
Q
P
5
4
R
21
(9.5-11.5)
1
Q
21
(3.5-5.5)
3
2
1
0
1872
1874
1876
-1
Raman Shift (cm )
1878
(c)
Figure 11: Comparison of theoretical (black line) and experimental (red line) ERE-CARS
spectra for a pressure of 1.0 atm, a concentration of 100 ppm, and probe frequencies of
(a) 42322.61 cm−1 , (b) 42330.41 cm−1 , and (c) 42343.00 cm−1
34
Enhancement Factor
1200
1000
800
600
400
200
0
1873
1875
1877
-1
Raman Shift (cm )
(a)
Intensity
1/2
(arb. units)
1200
1000
800
600
400
200
0
1873
1875
1877
-1
Raman Shift (cm )
(b)
Enhancement Factor
1200
1000
800
600
400
200
0
1872
1874
1876
-1
Raman Shift (cm )
1878
(c)
Figure 12: Enhancement factor for a pressure of 1.0 atm, a concentration of 100 ppm, and
probe frequencies of (a) 42322.61 cm−1 , (b) 42330.41 cm−1 , and (c) 42343.00 cm−1
35