saturation effects in coherent anti-stokes raman scattering spectroscopy of hydrogen

Vol. 6, No. 12/December 1989/J. Opt. Soc. Am. B 2313

Saturation effects in coherent anti-Stokes Raman scatteringspectroscopy of hydrogen

Robert P. Lucht and Roger L. Farrow

Combustion Research Facility, Sandia National Laboratories, Livermore, California 94551

Received May 2, 1989; accepted August 28, 1989

Saturation of coherent anti-Stokes Raman scattering (CARS) spectra of the Q(1) line of the hydrogen (1, 0)vibrational transition was investigated experimentally by using high-resolution lasers and theoretically by solvingthe time-dependent density matrix equations. The saturation behavior of hydrogen is complicated by the largeDoppler width of the resonance and the high rate of velocity-changing collisions relative to dephasing collisions.Experimentally, CARS line shapes and saturation curves were measured in pure hydrogen at pressures of 100 and3050 Torr. Surprisingly, the measured saturation intensity was found to be less at 3050 Torr than at 100 Torr. Thetime-dependent density matrix equations were numerically integrated to obtain CARS saturation curves and lineshapes. Excellent agreement between calculated and experimental line shapes was obtained at both 100 and 3050Torr, and the predicted saturation intensity was less at 3050 Torr than at 100 Torr. Based on the good agreementbetween theory and experiment obtained at 100 and 3050 Torr, the theoretical results were extended over a muchwider pressure range, from 0.1 to 100,000 Torr. Below 1 Torr the saturation behavior is independent of pressurebecause collision times are long compared with times associated with laser excitation of the resonance, and themolecular response is completely transient. Between 1 and a few hundred Torr the saturation intensity increases asthe rate of velocity-changing collisions increases. Above a few hundred Torr, however, the saturation intensitybegins to decrease because the high rate of velocity-changing collisions ensures that all molecules couple effectivelywith the Raman-pumping lasers. Calculations show a minimum in saturation intensity at 2000 Torr. For evenhigher pressures, saturation is controlled by dephasing collisions, and the saturation intensity increases rapidly withpressure.

1. INTRODUCTION

The usefulness of coherent anti-Stokes Raman scattering(CARS) as a technique for gas-phase temperature and spe-cies concentration measurements has been demonstrated innumerous experiments.1- 3 At low laser powers CARS lineshapes and spectral profiles are independent of the intensi-ties of the pump and Stokes laser beams. As laser intensi-ties increase, however, CARS spectra are perturbed by Starkeffects4'5 and by saturation 6-19 of the Raman transition.Saturation of the Raman transition can significantly affectthe accuracy of CARS temperature and concentration mea-surements. Saturation in CARS spectra is not always readi-ly apparent, however, especially when the pump and Stokeslasers have frequency bandwidths that are larger than theRamnan linewidths.

In this paper we describe an experimental and theoreticalinvestigation of saturation of CARS spectra of hydrogen.CARS measurements of hydrogen have been performed overa wide range of pressure, ranging from low-pressure plas-mas,20 free jets, 21 and reaction vessels22 to high-pressurepropellant flames2 3 and engines.24 In our experimentshigh-resolution, nearly Fourier-transform-limited pumpand Stokes lasers were used to measure saturated CARSspectra. The experimental measurements were comparedwith the theoretical line shapes and saturation intensities;the density matrix equations for the CARS interaction werenumerically integrated, using the measured pump andStokes laser pulse shapes. Saturation intensities and lineshapes of hydrogen CARS spectra were calculated over apressure range from 0.1 to 100,000 Torr.

We previously reported experimental measurements andtheoretical predictions of saturation line shapes and intensi-ties for nitrogen CARS.1 6'18 Modeling coherent Raman in-teractions for the hydrogen Q(1) line presents different chal-lenges than for nitrogen Q-branch lines because the colli-sional dynamics are quite different (the physics of collisionsfor Q-branch Raman transitions is discussed by Kozlov etal.

2 5). Collisions that change the velocity of the molecule

Without changing the phase of the Raman polarization werenot included in the nitrogen calculations, but such collisionsare important for Raman spectroscopy of hydrogen. Forpure hydrogen the rate of these velocity-changing collisionsis almost 50 times greater than the combined rate of rota-tional transfer and pure dephasing collisions. Consequent-ly, collisional transfer between velocity groups must be con-sidered in modeling Raman spectra of hydrogen.

As with the nitrogen calculations, dephasing of the polar-ization owing to rotational and vibrational transfer collisionsis considered. The rate of vibrational transfer is much lessthan the rate of rotational transfer for pure hydrogen andnitrogen.

Because the rates of various collisional processes varywidely in pure hydrogen, saturation of the Raman transitionis controlled by different collisional mechanisms at differentpressures. For low pressures, the saturation intensity isindependent of the collision rate. At pressures near 100Torr, the saturation intensity is determined by the rate ofvelocity-changing collisions. At pressures above a few thou-sand Torr, the Doppler profile of hydrogen is nearly com-pletely narrowed, and dephasing collisions determine thesaturation intensity.

0740-3224/89/122313-13$02.00 © 1989 Optical Society of America

R. P. Lucht and R. L. Farrow

2314 J. Opt. Soc. Am. B/Vol. 6, No. 12/December 1989

The approach that we have adopted for analysis of thesaturation process is to model the interaction on the com-puter rather than to make simplifying assumptions thatwould allow us to obtain analytical solutions in limitingcases. This approach allows us to solve the density matrixequations for the CARS interaction over a wide range ofpressure without a priori consideration of the validity ofsuch assumptions in each pressure range; by examining thenumerical solution in detail, however, we are able to assesstheir validity.

2. EXPERIMENTAL SYSTEM AND RESULTS

We employed a high-resolution scanning CARS system toobtain spectra of the Q(1) transition of hydrogen in a staticcell. A single-frequency Nd:YAG laser (Molectron MY-32with an MY-SAM attachment) after frequency doublingproduced a smooth, nearly Gaussian pulse with a duration of18 nsec and a FWHM linewidth of less than 45 MHz. Forthese saturation studies, pulse energies of 0.05-5.0 mJ wereemployed for each of two CARS pump beams. Approxi-mately 70 mJ of the frequency-doubled Nd:YAG laser radia-tion was used to pump a pulse-amplifier chain, consisting oftwo side-pumped preamplifiers using LDS-698 dye and alongitudinally pumped final amplifier using DCM dye. Thering dye laser was operated with LD-688 dye. The outputafter pulse amplification was as much as 5 mJ per pulse at682 nm, with a 60-MHz linewidth and a pulse duration of 12nsec. The Stokes beam energy was 0.05-5.0 mJ per pulse.A typical scan consisted of 200-300 Stokes frequency stepsof 10-3 cm-', with the results of 10 laser pulses averaged foreach step. All laser polarizations were parallel and vertical-ly aligned. The relative beam angles were sufficiently small,less than 1.5 deg, that essentially forward-scattering Ramanline shapes were measured.

A monochromator was used to filter stray light from theanti-Stokes beam, which was detected with a photomultipli-er tube. A computer synchronized the tracking of themonochromator with the anti-Stokes frequency, scannedthe Stokes laser, and acquired the CARS signal, pump, andStokes laser intensities (Ia, IP, and I,, respectively) afterdetector integration and digitization.

The experimental laser intensities were measured byscanning a pinhole across the spatial profile of the beam,recording the pulse shape by using a fast diode and 1-GHzoscilloscope, and measuring the pulse energy by using acalibrated energy meter. We estimate that our measuredvalue of III is uncertain by a factor of 3. The measuredvalues quoted in this paper correspond to values at the peakof both the spatial and temporal profiles.

VELOCITY-CHANGINGCOLLISIONS

U

- .

Pr - I P la

I @

Aw CO k

V=1, J=1, k

N k

Qk (t)

DEPHASING AND ROTATIONALTRANSFER COLLISIONS

N4

V=1, J=3

N3

v=O, J=3

v=O, J=1, Uk

Fig. 1. Schematic diagram of the CARS interaction for the Q(1)line and for a single-velocity group.

velocity-changing collisions. It is assumed that the velocity-changing collisions do not interrupt the phase or affect theamplitude of the vibrational polarization. Rotational trans-fer is treated by including the (v = 0, J = 3) and (v = 1, J = 3)level populations in the calculations. A major advantage inthe modeling of saturation of the Q(1) line of hydrogen isthat the rates of velocity changing, rotational transfer, andpure dephasing collisions are well known.26 2 7

The equations that describe the coherent Raman interac-tion of the pump and Stokes beams with the molecules invelocity group k were derived in detail in a previous paper.' 8

The derivation is appropriate for homogeneously broadenedvibrational Raman lines and follows those of Giordmaineand Kaiser28 and Pennzkofer et al.

2 9 closely. The responseof the molecule is described in terms of the expectation valueof the internuclear coordinate, (qk) (X, t) (cm). The electricfield E(x, t) (g'/ 2/cml/2 sec) is given by

E(x, t) = /2[Ap(x, t)exp(ikpx - icopt)

+ A8(x, t)exp(ikx - it) + Aa(x, t)

X exp(ikax - iwt) + c.c.], (1)

where Ap, A8, and Aa (g 1/2

/cml/2 sec) are slowly varying am-

plitude functions for the pump, Stokes, and anti-Stokesfields, respectively. The central frequencies of the pump,Stokes, and anti-Stokes fields are wp, w,, and a (sec'1),respectively, and the wave vectors are kp, k,, and ka (cm-'),respectively. The expectation value (qk) (X, t) is driven atthe difference frequency of the pump and Stokes fields,

(qk)(x, t) = 1 [Qk(X, t)exp(ikx - it) + c.c.]. (2)2 Nk

3. DENSITY MATRIX FORMULATION OFTHE CARS INTERACTION

The interaction of a velocity group with the pump andStokes laser radiation is schematically illustrated in Fig. 1.The (J = 1, v = 0) and (J = 1, v = 1) levels are directlycoupled by the laser radiation. The laser radiation estab-lishes a time-dependent vibrational polarization for mole-cules in velocity group k. Population transfer occurs be-tween velocity group k and other velocity groups by means of

The slowly varying amplitude function Qk(x, t) (cm- 2 ) is

defined not on a per-molecule basis but as a macroscopicquantity for velocity group k with total number density Nk

(cm- 3). The quantities w, and kV are given by

(UV = oP-, kv = kp - k.

For velocity group k, the slowly varying envelope function Qhof the vibrational polarization is given by



aQk

t (Kvel + Kdeph + iAk)Qk+ irApA,*(Nk-2nd

+ E QiKv,1(i, k), (3)

the excited state population nk (cm-3) by

- (KVib + Krot + Kvei)nk + E Kv,,(i, k)at

+ ir2 (APA8 *Qk*- AP*AsQ) + N4Kro (4)

and the total (ground-plus excited-state) population Nk by

atk =_Kronk- Krlot(Nk - nk) -KvelNk

+ NKv(i, k) + (N4Kr4o2t + N3Kr3t)fk. (5)

In Eqs. (3)-(5), Kvel, Kdeph, Krot, and KVib (sec-) are thevelocity-changing, dephasing, rotational transfer, and vibra-tional transfer collision rates, respectively. A hard-collisionmodel was used for the calculations discussed in this paper;i.e., it is assumed that the distribution of final velocities aftera velocity-changing collision is Boltzmann, and the velocity-changing collision rate from group i to group k is

Kvel(i, k) = Kvejfk, (6)

where

Kvel = Z Kvel(i, k) (7)k

and is assumed to be the same for all groups i. For a Max-well-Boltzmann velocity distribution, the population frac-tion fk is given by

2 (In 2)1/2 exF4 n ( WOOk\2fk = - _) exp[-4 n ) Ak, (8)

where ACOD = (2wo/c) (2kT ln 2/m)'12 is the FWHM for spon-taneous Raman scattering in the forward direction, WOk is thecentral resonance frequency for velocity group k, k is Boltz-mann's constant, and Aak is the width of the velocity group.Equation (8) is an approximate expression for fk and as-sumes that ACOk is small compared with AWD.

The velocity-changing collision rates were taken from Far-row and Palmer.26 They used a hard-collision model to fitunsaturated CARS spectra over a wide range of pressure.The fitted collision rate was not strictly proportional topressure; the collision rate Kvej was equal to 9.0 X 103 and 4.0X 10's at 100 and 3050 Torr, respectively. A soft-collisionmodel did give a fitted collision rate that was almost directlyproportional to pressure, but incorporation of the soft-colli-sion model into the saturation calculations was significantlymore complicated. For the hard-collision model, the veloci-ty-changing collision rate was given by KveI = 9.0 X 106 P forP < 100 Torr, Kvei = 9.0 X 106 P + 4.25 X 106 (P - 100) for100 Torr < P < 3000 Torr, and Kvej = 13.0 X 106 P for P 23000 Torr.

The dephasing and rotational transfer rates were takenfrom Farrow and Chandler.2 7 The rates were given by Kdeph

= 2.2 X 105 P and K,24t = 8.9 X 104 P. Level 1 is defined as (v= 0, J = 1) and level 2 as (v = 1, J = 1) (see Fig. 1), and the su-perscript ij in Krt refers to transfer from level i to level j.Rotational transfer rates are assumed to be independent ofvelocity and to be equal for the upper and lower rotationallevels, Kro3t = K rot and Kr4 2t = Kr3 t. For rotational transferback into levels 1 and 2 from levels 3 and 4, the distributionof final velocities is assumed to be Maxwell-Boltzmann.

The detuning parameter Ak is given by

2 2

Ak k 2cv (9

where

k °Ok + CStark[Ip(t) + I,(t)]. (10)

The resonance frequency c0k for group k is shifted from itsunperturbed value by the Stark effect, which is proportionalto the total electric field intensity I(t) (ergs/cm2 sec). Forthe hydrogen Q(1) transition, CStark = 4.0 X 10-20 cm 2/erg. 3 0

The constants r, and r 2 are given by

= 4mrr (aq)(11)

where mr (g) is the reduced mass and (a/aq) (cm-') is thepolarizability derivative, and by

r 2 = WV laa2-8hco tdqJ'

(12)

where h (erg sec) is Planck's constant.The solution to equations similar to Eqs. (3) and (4) is

discussed in detail in our previous paper on nitrogenCARS.18 Separate equations are derived for the real andimaginary parts of Qk(t), and Eqs. (3) and (4) are modified toaccount for the use of two pump beams in the three-dimen-sional phase-matching scheme. Equations (3)-(5) weresolved by using a variable-order Adams code (routineDEABM in the FORTRAN SLATEC library). The accuracy ofthe solution was checked by varying the step size and thenumber of velocity groups until convergence was achieved.

Assuming perfect phase matching and negligible absorp-tion in the medium, the resonant CARS amplitude A0

4(t)(the term resonant CARS here refers to vibrational and notelectronic resonance) from the velocity group k is given by

Aa k(t) = i 2 a [( ) 1 tPQ ]tj (13)

where ,a is the refractive index at the anti-Stokes frequencyand c (cm/sec) is the velocity of light. Equation (13) mustalso be modified to account for the use of two pump beams inthe three-dimensional phase-matching scheme.'8 The totalCARS amplitude A0(t) is the sum of the resonant CARSamplitude over all velocity groups, plus the contributionfrom the nonresonant susceptibility Xnres (cm3 /erg),

A,(t) = Y A k(t) + i XnresAp2(t)As*(t), (14)

and the total CARS signal is the time integral over the laserpulse lengths,


(9)


Wa = J Aa(t)Aa*(t)dt.1.25

(15)

4. COMPARISON OF THEORY ANDEXPERIMENT

Saturation of the Q(1) CARS transition was investigatedexperimentally for pure hydrogen at pressures of 100 and3050 Torr. In this section the experimental results are com-pared with the results of numerical solution of the time-dependent density matrix equations derived in Section 3.The comparison serves to validate the numerical calcula-tions over a wide range of pressure.

Experimental spectra for the Q(1) line and the results ofnumerical calculations are shown for pure hydrogen at 100and 3050 Torr in Figs. 2 and 3, respectively. Spectra werecalculated numerically by integrating the density matrixequations over the laser pulse lengths for 50-100 successivevalues of the detuning Aw = cwp - WS- o, where w0is the non-Stark-shifted resonance frequency for hydrogen moleculeswith zero velocity. For these calculations, only the Stokeslaser intensity was varied to obtain good agreement betweenexperimental and theoretical spectra.

H

Cl)

zZH

z

L '

U)

C/)z

w

z

0

HCl)

0

1.25

0.76

0.50

0.25

0

1.25

-0.05 0.00 0.05

RAMAN SHIFT (cm")Fig., 2. Comparison of experimental and theoretical line shapes forthe Q(1) line of hydrogen at 100 Torr at three different laser intensi-ties.

H

z

zCl)

zz

nl

0

us

H

Wlz_wHcc

Ul

05

0.75

0.50

0.25

0.000 0.025 0.050

-0.050 -0.025 0.000 0.025RAMAN SHIFT (cm-')

0.050

0.050

Fig. 3. Comparison of experimental and theoretical line shapes forthe Q(1) line of hydrogen at 3050 Torr at three different laserintensities.

For the spectra at 100 Torr, the best agreement betweentheory and experiment was attained when the theoreticalproduct of pump and Stokes peak laser intensities was ap-proximately eight times less than the measured product.The theoretical product IpI, is less than the measured prod-uct because the spatial profile of the laser beam intensities isneglected in the calculations. The factor of 8 is consistentwith the results of our modeling of CARS saturation innitrogen.'8 For the highest-intensity spectrum at 3050Torr, shown in Fig. 3(a), the measured to theoretical ratio isapproximately 6, consistent with the 100-Torr results andwith the nitrogen results. For the spectrum shown in Fig.3(b), however, the ratio is less than 3; we do not have anexplanation for the large difference in ratio for Figs. 3(a) and3(b).

5. PRESSURE DEPENDENCE OF HYDROGENCARS SATURATION

Having obtained good agreement between experimental andcalculated saturation line shapes at 100 and 3050 Torr, weinvestigated the saturation behavior of hydrogen theoreti-

-0.060 -0.025

-0.050 -0.025 0.000 0.025



cally over a much wider pressure range. Calculations wereperformed from 0.1 Torr, where the response of the moleculeis in the transient regime and steady-state solutions of theCARS equations do not apply, to 100,000 Torr, where theQ(1) line is predominantly homogeneously broadened andsteady-state solutions are accurate. In the intermediateregion, velocity-changing collisions are important in deter-mining saturation intensities at lower pressures, but, at pres-sures above a few thousand Torr, dephasing collisions con-trol the saturation process.

Saturation curves were generated for a given pressure byintegrating the density matrix equations for a sequence ofStokes intensities, assuming zero detuning and no Starkshifting. The saturation intensity (IpIs)sat was defined whenthe normalized CARS intensity had fallen to half its low-intensity value, i.e.,

,ps =(I sat when W. = 05 Ip2 Is Is low intensity

The results of the calculations are shown in Fig. 4. Thesaturation intensity reaches a limiting value at low pressurebecause the response of the molecule is transient and inde-pendent of collision rates. As pressure and the rate of veloc-ity-changing collisions increase, so does the saturation in-tensity. Eventually, however, as the rate of velocity-chang-ing collisions increases further, the saturation intensitydrops because of the rapid transfer of population and vibra-tional amplitude among velocity groups. Above a few thou-sand Torr, the saturation intensity is determined by thedephasing collision rate and increases rapidly with pressure.

Low-Pressure Region: Transient Molecular ResponseIn the low-pressure regime collisional times are long com-pared with the laser pulse lengths of 12-15 nsec, and satura-tion behavior is controlled by the characteristics of the laserradiation rather than by the collisional environment of themolecule. Numerical calculations of hydrogen CARS lineshapes at low pressures are complicated by the narrownessof the homogeneous response for each velocity group; at 1.0Torr the homogeneous width (FWHM) is 2.3 X 10-6 cm'1.The calculated unsaturated line shape for a single-velocity

2-

o THEORY

* EXPERIMENT x 0.17

ei 1.5 Ol' E

0 X

PRESSURE (Torr)

Fig. 4. Pressure dependence of the CARS saturation intensity forthe Q(1) line of hydrogen. The experimental saturation intensitiesare multiplied by 0.17 so that the experimental and theoreticalsaturation intensities are equal atl100Torr. The pressures at whichthe velocity-changing and total dephasing (pure dephasing plusrotational transfer) collision rates are equal to 109 sec-1 are indicat-ed by vertical lines.

-j

z0.50-

0.25-

0--0.010 -0.005 0.000 0.005 0.010

RAMAN SHIFT (cm')Fig. 5. Calculated unsaturated CARS line shape for a single-veloc-ity group for the Q(1) line of hydrogen at 1.0 Torr. The width ofthecurve corresponds to the convolved width of the pump and Stokeslasers. The peak pump and Stokes laser intensities were equal to10-4 GW/cm 2.

-jz

0

v I I 1 1 1

-0.050 -0.025 0.000 0.025 0.050RAMAN SHIFT (cm-')

Fig. 6. Calculated unsaturated CARS line shape for the Q(1) line ofhydrogen at 1.0 Torr. Calculations were performed for 31, 51, 71,and 101 velocity groups. The peak pump and Stokes laser intensi-ties were equal to 10-4 GW/cm2. Calculation of the spectra re-quired 5, 15, 25, and 52 h of CPU time on a Vax 8650 computer for31, 51, 71, and 101 velocity groups, respectively.

group is shown in Fig. 5; the FWHM of the line is approxi-mately 0.0027 cm-1 and is almost entirely due to the pumpand Stokes laser linewidths. Because of the narrowness ofthe line shape for a single-velocity group, numerous velocitygroups must be included for an accurate calculation of theinhomogeneous line shape. Figure 6 shows calculated un-saturated line shapes for 31, 51, 71, and 101 velocity groups,spaced by 0.00357, 0.00215, 0.00153, and 0.00107 cm-', re-spectively. The unsaturated solution converges for 71 and101 velocity groups; the solutions for 71 and 101 velocitygroups were also identical for high laser intensities. Forhydrogen pressures less than 100 Torr, 71 velocity grdupswere used in the calculations.

The intensity dependence of the integrated CARS signalfor hydrogen at 1.0 Torr is plotted in Fig. 7. The CARSsignal increases smoothly with increasing laser intensity atlow Stokes laser intensities but exhibits numerous local min-ima and maxima at higher Stokes laser intensities becaus ofRabi oscillations. Furthermore, the CARS signal continues

3 9


ni


' r,,,,,`,t ' 7 ' ' ' '. l r

IPIS

. _~~~~~~~~~~~~~

(G 10 0' 10

(GW2/cm4 )

Fig. 7. Dependence of integrated CARS signal on the peak intensi-ty product of the pump and Stokes lasers for hydrogen at 1.0 Torr.The frequency difference Aw, of the pump and Stokes lasers wastuned to the center of the Doppler profile (Aw = 0).

(L0a-

x

U2z

wH

LU

z

W

U)

n

0

U)

U)0r

1.00

0.75

0.50

0.25

0.00

0 10 20 30

TIME (nsec)Fig. 8. Temporal dependence of the CARS signal and excited-statepopulation fraction for hydrogen at 1.0 Torr for two differentpump-Stokes intensity products. The peak laser intensity productIp1, was equal to (a) 1.58 and (b) 3.16 GW2/cm 4. The frequencydifference of the pump and Stokes lasers was tuned to the center ofthe Doppler profile.

to increase with increasing Stokes laser intensity rather thandecreasing at high intensity as is the case for steady state.' 8

The low-pressure CARS saturation curve can be under-stood by examining the time-dependent behavior of theCARS pulse. The time dependence of the CARS intensity is

-LJ

-J

z0

!R

J.,

0.

0.'

-0.

-1.

0 Q.(b) 0\ -Qreal,k

/ ,~~ .... Qimag,k

(a) ."

5-

.0~

0 10 2

TIME (nsec)20 30

Fig. 9. Temporal dependence of the vibrational amplitude and theexcited-state population fraction for the central velocity group forhydrogen at 1.0 Torr. The quantity IAQkl2 shown in (a) would bethe CARS intensity if the contribution from velocity group k couldbe separated from the contribution from other velocity groups. Thepeak laser intensity product 1pI was equal to 3.16 GW2/cm 4. Thefrequency difference of the pump and Stokes lasers was tuned to thecenter of the Doppler profile.

-J

zUCnz0Nz000I

a-Ci

0c

1.25

0.75

0.50

0.25

0-0.050 -0.026 0.000 0.025

RAMAN SHIFT (cm-')0.050

Fig. 10. Calculated CARS and photoionization (PI) line shapes forthe Q(1) line of hydrogen at 1.0 Torr. The number of velocitygroups was 71 for the calculations shown. The peak laser intensityproduct IpI, was equal to 1.0 GW2/cm 4 for calculations of the satu-rated CARS and photoionization spectra and was 10-1 GW2/cm4 forthe unsaturated CARS spectrum.

5000-

4000--a

0=3000 -

2000-

1000-

0 -

1 3

(L0

CL)X>-

zzU)

0



-4

O 10 20 311.-0 - (

0.5

D.0

05Awk

0.5 -. ' ' .' .

0.000cm. .

1.0- I0 10 20 31

10 20

TIME (nsec)

are plotted in Fig. 9(b). Here fexc,k is the excited-state popu-lation fraction for the central velocity group rather than forall velocity groups; nearly complete population inversion isattained during the laser pulses. Whenfexck exceeds 0.5, thesign of the laser pumping term in Eq. (3) changes, driving theimaginary part of Q negative. This is the origin of thebeating observed in the resonant CARS intensity, and thephasing between the laser pulse and the molecular responseis responsible for the local minima and maxima in the satu-ration curve.

The spectra shown in Fig. 6 were calculated for peak pumpand Stokes laser intensities of 10-4 GW/cm2 . In Fig. 10 theCARS spectrum for Ip = I = 1.0 GW/cm2 is shown; thespectrum is broadened and exhibits a marked saturationdip. The unsaturated CARS spectrum is shown for compar-ison. The dip at the center of the CARS line occurs becausethe real part of Qk has opposite sign for velocity groups with

2

C]

a

-JzIs0

5:

Fig. 11. Temporal dependence of the vibrational amplitude Q forvelocity groups with detunings of (a) -0.0031, (b) 0.000, and (c)0.0031 cm-' for hydrogen at 1.0 Torr. The real and imaginary partsof the vibrational amplitude are shown. The peak laser intensityproduct II, was equal to 3.16 GW2/cm4. The frequency differenceof the pump and Stokes lasers was tuned to the center of the Dopp-ler profile.

1.

0.

0.

-0.

-1.

Io (b)X

10 - U ." '-.

0.000 cm 1

0 -06 10 20 341

1.0-

shown in Fig. 8 for two different values of the laser intensityproduct IpI, corresponding to a local minimum (1.58 GW2/cm4) and maximum (3.16 GW2/cm4) in Fig. 7. The nonreso-nant CARS signal is proportional to Ip21p; an unsaturatedresonant CARS pulse would follow this curve closely. Theresonant CARS pulse has three lobes for the low-intensitycase and four for the high-intensity case. The excited-statepopulation fraction fexc averaged over all velocity groups isalso shown and reaches a maximum value of approximately0.15 in both cases.

The calculated resonant CARS pulse has a complicatedtemporal structure because of the nonlinear coupling of thevibrational amplitude Qk and the excited-state populationfraction fexc,k for each velocity group. The temporal re-sponse of the central velocity group (zero detuning) is shownin Fig. 9. The normalized product ApQkI2 and fexc,k areplotted in Fig. 9(a), and the imaginary and real parts of Qk

0.5 -

0.0 -

-0.5 -

-1.00 10

TIME (nsec)Fig. 12. Temporal dependence of the vibrational amplitude forvelocity groups with detunings of (a) -0.0031, (b) 0.000, and (c)0.0031 cm 1 for hydrogen at 1.0 Torr. The real and imaginary partsof the vibrational amplitude are shown. The peak laser intensityproductI ~was equal to 0.316 GW2/cm4. The frequency differenceof the pump and Stokes lasers was tuned to the center of the Dopp-ler profile.

0-J

-Jz

5

0 10 20 30

(C) Auk = 0.0031 cm-1

. ~~..-- .. '

0

20 30



positive and negative detunings, and therefore the contribu-tion of the real part of Qk to the CARS signals cancels at linecenter. This is illustrated in Figs. 11 and 12, where the realand imaginary parts of Qk are plotted for single-velocitygroups with positive, negative, and zero detuning and for twodifferent laser intensity products II,. The real part of Qkhas opposite sign for opposite detuning, and the magnitudeof the real part becomes larger relative to the magnitude ofthe imaginary part of Qk as the line becomes more saturated.

When the difference frequency of the pump and Stokeslasers is tuned to the resonance frequency of the centralvelocity group, the velocity groups on either side of thedirectly pumped central velocity group have equal popula-tion, and there is no net contribution to the CARS signalwhen the real part of Qk is summed over all velocity groups.However, when the difference frequency is tuned off linecenter, the velocity groups on either side of the directlypumped velocity group no longer have equal population, andthe real part of Qk summed over all velocity groups is non-zero (remember that Qk has been defined as a macroscopicquantity and not on a per-molecule basis). The contribu-tions to the real part of Qk when the laser difference frequen-cy is tuned on and off line center are schematically illustrat-ed in Fig. 13. As IpI, increases, the real part of Q becomesmore significant compared with the imaginary part, and theCARS signal starts to peak on either side of line center. Thesaturation dip for this inhomogeneously broadened line thusoccurs for a different reason than for a homogeneouslybroadened line. In a homogeneously broadened line, thesaturation dip occurs because population pumping is mostefficient near line center.

In recent rotational energy transfer experiments, Sitz andFarrow3" used stimulated Raman pumping to populate thefirst vibrational level of nitrogen and monitored rotationaltransfer in the excited state by photoionization. They alsomonitored the CARS spectrum as they tuned the Stokeslaser through Q-branch resonances. They noticed that atlow pressure the CARS spectrum broadened and exhibited asignificant saturation dip, but the shape of the photoioniza-tion spectrum was insensitive to laser power. These resultsare consistent with our calculations and our explanation ofsaturation for inhomogeneous CARS lines. The curve inFig. 10 marked with triangles is a calculated photoionizationspectrum for the same laser intensities as for the saturatedCARS lines. For the photoionization spectrum it is as-sumed that there is a constant rate of photoionization fromthe excited vibrational level during the laser pulse,

(16)Wpi = Kpj E nk(t)dt,k

and that the rate of photoionization is small enough that theeffect on the excited-state population is negligible. Even atthese high laser intensities, the photoionization spectrum isapproximately 20% narrower than the unsaturated CARSspectrum (the unsaturated CARS spectrum is also approxi-mately 20% broader than the spontaneous Ramanwidth'8'3 2 ). The photoionization spectrum maps out theDoppler population distribution even when significant Ra-man pumping occurs. The photoionization spectrum doesnot broaden because the rate of velocity-changing collisions

is low at 1.0 Torr and there is little exchange of populationamong velocity groups.

Figure 14 illustrates that collision rates have little effecton the molecular response in this pressure regime. Theresonant CARS pulses from hydrogen at 1.0 and 0.1 Torr areplotted; the temporal behavior and signal intensity are simi-lar at the two pressures. Below 1 Torr the saturation behav-ior of the Q(1) hydrogen CARS resonance is nearly indepen-dent of pressure. The response of the molecule can becalculated to a good approximation by deleting the collision-al terms in Eqs. (3) and (4):

Qreal i

i k

I ,

zLasers tunedto line center

zLasers tunedoff line center

,e, i

'

'l

AcoJ L

real, i

JK\

Aco

Fig. 13. Schematic illustration of the contributions to the real partof the vibrational amplitude for velocity groups detuned from thelaser excitation frequency. The cases where the lasers are tuned onand off line center are illustrated.

7.50-

X IpS = 3.16 _ P = 10 Torr. 6.25 -

c .....2 - P = 0.1 Torr

4 5 5.00

3.75z

2.50

U,1.25

0.00-0 10 20 30

TIME (nsec)

Fig. 14. CARS pulse shapes for hydrogen at 0.1 and 1.0 Torr. Thepeak laser intensity product I1 I, was equal to 3.16 GW2/cm4. Thefrequency difference of the pump and Stokes lasers was tuned to thecenter of the Doppler profile.



0.76

- 0.50 ,

0.25

0 ,J, o1 1 0 2

II (GW2/cm4)p I

Fig. 15. Calculated saturation curves for hydrogen at 100 Torr,with and without velocity-changing collisions included in the calcu-lations. The frequency difference of the pump and Stokes laserswas tuned to the center of the Doppler profile.

these levels (kKvel and QkKvel). Consequently, when therate of velocity-changing collisions is low, these collisionsaffect the saturation process in the same manner as rotation-al transfer or pure dephasing collisions. As pressure in-creases, however, the rates for transfer of population andpolarization back into levels that are strongly pumped by thelasers become comparable with transfer rates out of theselevels, and their effect on saturation line shapes and intensi-ties is no longer straightforward.

The resonant CARS signal and excited state-populationfraction for hydrogen at 100 Torr are shown in Fig. 16(a). At100 Torr the velocity-changing collision rate is approximate-ly 109 sec'1. At the beginning of the laser pulses, the CARSintensity and excited-state population fraction both start torise. As fexc increases, the CARS signal generation efficien-

1.00

aQkt= _AkQk + iAPA,(Nk - 2nk)at (17) 0.75

and

t = ir2(APAS*Q* - AP*AQk).

0.50

0.25(18)

These equations are solved analytically by Wilson-Gordonand Friedmann.7

Intermediate-Pressure Region: Importance of Velocity-Changing CollisionsAs shown in Fig. 4, the saturation intensity increases gradu-ally above 1 Torr until it reaches a peak near 100 Torr andbegins to decrease again. Because the rate of velocity-changing collisions is approximately 50 times that of de-phasing collisions, it is reasonable to assume that saturationbehavior is determined by velocity-changing collisions inthis pressure regime. Figure 15 shows calculated saturationcurves for hydrogen at 100 Torr. Velocity-changing colli-sions were included for the calculation of the solid curve; thevelocity-changing collision rate was set equal to zero for thecalculation of the dashed curve. It is evident that saturationoccurs at a much lower intensity product when velocity-changing collisions are not included in the calculations; thesaturation intensity is nearly equal to the limiting value of0.16 GW2 /cm4 reached at low pressure. At higher intensi-ties, the curve for Kve = 0 is not smooth, which is anotherstrong indication that transient effects would be importantif there were no velocity-changing collisions. Dephasingcollisions have little influence on saturation behavior at 100Torr.

In this pressure regime, the calculated saturation intensi-ty increases when velocity-changing collisions are includedbecause the velocity-changing collisions transfer vibration-ally excited molecules away from the velocity groups that arestrongly pumped by the laser radiation. When the rate ofvelocity-changing collisions is low, the rate of transfer ofexcited-state population [i niKvel(i, k)] and polarization[hi QiKvei(i, k)] back into the strongly pumped velocitygroups (Ak 0) is small relative to the rate of transfer out of

9-09-

U

X

zzLU

Cn

G

0.00

10 21

TIME (nsec)30

Fig. 16. Temporal dependence of the resonant and nonresonantCARS intensities and the excited-state population fraction for hy-drogen at 100 Torr. Calculations are shown for (a) all velocitygroups and for particular velocity groups with detunings of (b) 0.000and (c) -0.0184 cm-'. The quantity ApQk12 shown in (b) and (c)would be the CARS intensity if the contribution from velocity groupk could be separated from the contribution from other velocitygroups. The peak laser intensity product IpI was equal to 1.00GW2/cm4. The frequency difference of the pump and Stokes laserswas tuned to the center of the Doppler profile.



-J

0

!R

1.0

0.5

0.0

-0.5

-1.0

1.0 -

0.5 -

0.0 -

-0.5-

-1.0 -

1.0 -

0.5-

-0.6 -

-1.0 -

(a)

1~~~- Qo".. *

0.000 cm 1 Qimag,k0.000 cm0 - 0 realk

0 10 2'0 3'

0 in InTIME nsec)

0

30

30

Fig. 17. Temporal dependence of the vibrational amplitude forvelocity groups with detunings of (a) 0.000, (b) -0.031, and (c)-0.0184 cm-' for hydrogen at 100 Torr. The real and imaginaryparts of the vibrational amplitude are shown. The peak laser inten-sity product IIS was 1.00 GW2 /cm4 . The frequency difference ofthe pump and Stokes lasers was tuned to the center of the Dopplerprofile.

cy decreases because of saturation. Consequently, the peakof the resonant CARS intensity occurs earlier than the peakof the nonresonant CARS intensity. The temporal responseof velocity groups with detunings of 0.0 and -0.0184 cm-' isshown in Figs. 16(b) and 16(c), respectively. The excited-state population fraction fexc,k for the velocity group withzero detuning rises more quickly than for the groups with adetuning of -0.0184 cm-' but reaches approximately thesame limiting value when laser excitation ceases.

The time dependence of the real and imaginary parts ofthe vibrational amplitude for velocity groups with detuningsof 0.0, -0.0031, and -0.0184 cm-' Torr is shown in Fig. 17.Although the time dependence of fexck is similar for thesegroups, the time dependence of Qreal,k and Qimag,k is quitedifferent. At 100 Torr the dephasing rate is still low, andthe homogeneous FWHM for a single-velocity group, 0.0027cm-', is determined by the laser linewidths. For the veloci-ty group with a detuning of -0.0184 cm-', the magnitude of

Qreia,k is markedly greater than the magnitude of Qimag,k,

which is nearly zero. The low value of Qimag,k for this veloci-ty group indicates that the vibrational amplitude Qk is notsignificantly affected by velocity-changing collisions. Thecollision frequency is low enough that molecules transferredinto velocity group k do not oscillate in phase with oneanother, and thus IQmag,hl remains low in Fig. 17(c) despitethe significant transfer of population indicated by the calcu-lations. Terms such as Ei QiKVel(i, k) remain negligiblecompared with QkKvel throughout the laser pulse.

However, as pressure increases, the effect of velocity-changing collisions on the saturation intensity changes. Asthe rate of collisions increases, the transfer rates into and outof the strongly pumped velocity groups begin to balance.Furthermore, as velocity-changing collisions begin to narrowthe linewidth significantly, all molecules begin to interactstrongly with lasers tuned to line center. Thus the satura-tion intensity is actually reduced by the velocity-changingcollisions. This is illustrated in Fig. 18, where saturationcurves for hydrogen at 2000 Torr, calculated by alternatelyincluding and neglecting velocity-changing collisions, areplotted.

The response of velocity groups with detunings of 0.0,-0.0031, and -0.0184 cm-' are shown in Figs. 19 and 20.The resonant and nonresonant CARS signals and excited-state population fraction are shown in Fig. 19. At 2000 Torr,the curves look similar for all three values of the detuning.For all velocity groups, fexc,k approaches 0.5 at the peak of thelaser pulses, and the resonant CARS intensity drops signifi-cantly as the excited-state population increases. The tem-poral dependence of the vibrational amplitude is also similarfor the three velocity groups. The real part of the vibration-al amplitude is always nearly zero because of the fast trans-fer among velocity groups. The sign of Qrealk will be oppo-site on either side of line center, and therefore IQreal,kl willapproach zero if transfer among velocity groups is rapid.For the velocity group with a detuning of -0.0184 cm-1,IQimagkl is half that at line center, which is to be expectedbecause the population Nh is also half that for the velocitygroup at line center.

The 2000-Torr calculations indicate that velocity-chang-ing collisions ensure that all velocity groups interact stronglywith the laser radiation. Terms that account for transfer ofpolarization and population back into velocity groups thatare strongly coupled to the laser radiation (Ak 0 0) must beincluded in the equations, and the response of the moleculeis described by Eqs. (3)-(5). The terms Eji QiKvel(i, k) and

ji niKvei(i, k) are comparable in magnitude with the terms-KvelQk and Kvelnk, respectively. The effect of increasingthe rate of velocity-changing collisions is to decrease themagnitude of the difference terms -KvelQk + Es QjKvei(i, k)and -Kvelnk + Ej nijKvel(i, k).

The ratio of the experimental to the theoretical saturationintensity is a factor of 3 greater at 100 than at 3050 Torr.Experimentally, it is found that the saturation intensitydrops by a factor of 5 between 100 and 3050 Torr. Theoreti-cally, a drop of 40% is predicted. We attribute this discrep-ancy between theory and experiment to the difficulty ofaccurately determining the experimental value of the laserintensity product IpT,. However, another possibility is thatthe use of the hard-collision model for velocity-changingcollisions, or the assumption that dephasing and velocity-

(b)

Acok-=-0.031 cm-1

O 10 20

(c)

-0.0184 cm1



N

3.

0.75

0.50

0.26

0

Fig. 18. Calculated saturation curves for hydrogen at 2000 Torr,with and without velocity-changing collisions included in the calcu-lations.

a-0a-

xwLo

I-

U)zwOHza

1.00

0.75

0.50

0.25

0 10 20 30TIME (nsec)

Fig. 19. Temporal dependence of the resonant and nonresonantCARS intensities and the excited-state population fraction for hy-drogen at 2000 Torr. Calculations are shown for (a) all velocitygroups and for particular velocity groups with detunings of (b) 0.000and (c) -0.0184 cm-'. The quantity lApQkl2 shown in (b) and (c)would be the CARS intensity if the contribution from velocity groupk could be separated from the contribution from other velocitygroups. The peak laser intensity product 1I, was equal to 1.00GW2/cm4. The frequency difference of the pump and Stokes laserswas tuned to the center of the Doppler profile.

A nn

uJwQ

I

CJ

z

0

!;:

1.0

0.5-

0.0

-0.5

-1.0

1.'

0.

0.1

-0.

-1.

1.

0.

0.

-0.

-1.

0 -

0- (C)

.5-

01.0- ,, _..k,

-0.0184 cm'

0 - AA0

30

0 20TIME (nsec)

30

Fig. 20. Temporal dependence of the vibrational amplitude forvelocity groups with detunings of (a) 0.000, (b) -0.031, and (c)-0.0184 cm-' for hydrogen at 2000 Torr. The real and imaginaryparts of the vibrational amplitude are shown. The peak laser inten-sity product I1I, was equal to 1.00 GW2/cm4. The frequency differ-ence of the pump and Stokes lasers was tuned to the center of theDoppler profile.

changing collisions are statistically independent, may intro-duce some errors in the predicted saturation intensity. Wewill explore these issues in future calculations.

High Pressure Region: Approach to HomogeneouslyBroadened Line ShapeAt pressures that are high enough that the homogeneouslinewidth significantly exceeds the Doppler inewidth, theresponse of molecules in each velocity group will be thesame, and the contribution of the difference terms KvelQk+ Ei QiKvel(i, k) and -Kvelnk + i ni~vei(i, k) will approachzero. In the high-pressure regime, the response of the mole-cule will then be described by

a =-(Kdeph + iA)Q + ilApA 3*(N - 2n)at

(19)

and

(a)

Awk- .0.

000 Cm1 -I realkI

10 20 30~~---

0

U-(b)

5- .\

-0.031 cm 1

10 200

,


10 20 30

1


an = -(KVib + Krot)n + i 2(ApAs*Qk* - Ap*AsQk) + N3Krot.atP(20)

The solution to Eqs. (19) and (20) was discussed in detailpreviously. 7 '8"14 '18 For a homogeneously broadened line, thesaturation intensity will be proportional to the square of thepressure; the rapid increase of saturation intensity withpressure above 10,000 Torr is evident from Fig. 4.

6. SUMMARY AND CONCLUSIONS

We have investigated the saturation behavior of the Q(1)CARS line of hydrogen over a wide range of pressure. Ex-perimentally, saturation intensities and line shapes wereobtained at 100 and 3050 Torr. Theoretically, the time-dependent density matrix equations were solved numerical-ly to determine saturation intensities and line shapes. Be-cause Doppler broadening and velocity-changing collisionsare significant for pure hydrogen at pressures up to 10,000Torr, the density matrix equations for as many as 71 velocitygroups had to be solved simultaneously. Despite the com-plexity of the numerical scheme, the solution procedure out-lined in this paper is a valuable tool for studying the detailsof the physics of the Raman interaction and in particular fordetermining the influence of various collisional processes atdifferent pressures. The solution is general in that assump-tions of steady-state or completely transient (collisionless)behavior are unnecessary, and the inclusion of collisionalmodels that affect population and polarization in differentways and at different rates is straightforward.

We obtained good agreement between theory and experi-ment at both 100 and 3050 Torr. Numerical solutions werethen obtained at pressures from 0.1 to 100,000 Torr. TheCARS saturation intensity was found to reach a limitingvalue of approximately 0.16 GW2/cm4 at a pressure of 1 Torrfor the particular pump and Stokes pulses used in our ex-periment. Saturation is controlled not by collisions at theselow pressures but rather by the transient dynamics of theCARS process.7 Consequently, experimentalists usingCARS to probe low-pressure molecular beams or plasmascan be confident of avoiding saturation even if saturationintensity thresholds are determined in cells at higher pres-sure, provided that the cell pressure is low enough that theCARS process is in the transient regime.

We also explored the effect of velocity-changing collisionson CARS saturation. We found that velocity-changing col-lisions are responsible for the increase of the saturationintensity for the Q(1) line with pressure up to 100 Torr andfor the subsequent decrease in the saturation intensity untilit reaches a minimum at 2000 Torr. We used a hard-colli-sion model for the velocity-changing collisions, and excellentagreement was obtained between predicted and measuredline shapes. However, there was some discrepancy betweentheory and experiment in the ratio of saturation intensitiesat 100 and 3050 Torr.

In the future, we plan to incorporate different velocity-changing collision models into our computer code for com-parison with the hard-collision model. Because we solve thedensity matrix equations numerically, we can investigate awide range of collision models or incorporate measuredstate-to-state rate data. The effect of velocity-changing

collisions on four-wave mixing spectra was recently studiedby Singh and Agarwal,33 who used a hard-collision model inorder to obtain an analytical solution to the density matrixequations. The effect of velocity-changing collisions on sat-urated absorption spectra of sodium was also consideredpreviously,3 4-36 and several different collision models wereinvestigated. Liao et al.35 compared velocity-changing col-lision models based on Keilson-Storer and hard-sphere ker-nels, both of which are considerably more complicated thanthe hard-collision model used in the research reported here.

ACKNOWLEDGMENTS

This research was supported by the U.S. Department ofEnergy, Office of Basic Energy Sciences, Division of Chemi-cal Sciences. Helpful discussions with Greg Sitz (Universityof Texas at Austin) are gratefully acknowledged.

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saturation effects in coherent anti-stokes raman scattering spectroscopy of hydrogen

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