calculation of saturation line shapes and intensities in coherent anti-stokes raman scattering...

Vol. 5, No. 6/June 1988/J. Opt. Soc. Am. B 1243

Calculation of saturation line shapes and intensities incoherent anti-Stokes Raman scattering spectra of nitrogen

Robert P. Lucht and Roger L. Farrow

Combustion Research Facility, Sandia National Laboratories, Livermore, California 94550

Received June 8, 1987; accepted January 20, 1988

Line shapes and intensities for coherent anti-Stokes Raman scattering (CARS) at saturation laser intensities aredetermined for nitrogen vibrational Q-branch lines by solving the time-dependent density-matrix equations nu-merically. We have previously performed measurements of saturated CARS line shapes in pure nitrogen by usingnearly Fourier-transform-limited pump and Stokes lasers. The experimental laser pulse shapes, Stark effects,Doppler broadening, and the nonresonant background are incorporated in the numerical calculations. Thenumerical results show good agreement with the high-resolution measurements of saturated CARS line shapes.The lines show prominent saturation dips, and agreement between theory and experiment is excellent in terms ofthe depth and the width of the dips. The numerical results indicate that the Doppler effect tends to broaden and todeepen the dip in highly saturated lines, an effect that cannot be explained by a steady-state theory.

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.1-8 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 Starkeffects9 10 and by saturationl-21 of the Raman transition.Saturation effects become more important in perturbing lineshapes compared with Stark effects as laser linewidths de-crease and/or the pressure of the medium being probed de-creases. Thus, when narrow-linewidth lasers7 are used orwhen low-pressure media such as plasmas8 or supersonicjets" are probed, saturation becomes the main factor thatlimits laser intensity. Because the onset of saturation is notobvious in CARS spectra, the ability to accurately calculatesaturation intensities for CARS lines would be useful inallowing the experimentalist to restrict laser intensities tovalues low enough to avoid saturation. In some cases thepressure of the gas or plasma may be so low that saturationcannot be avoided,8 and it is important to understand howsaturation affects the relative intensities of resonance lines.

Saturation effects in CARS spectra were first clearly dem-onstrated experimentally by Duncan et al.11 Their mea-surements were performed with high-resolution lasers withpulse lengths of the order of 10 nsec. These lasers were usedto probe acetylene in a molecular beam. As pointed out byWilson-Gordon and Friedmann,12 acetylene relaxationtimes in the experiment were much longer than the laserpulse length, and thus solution of the transient density-matrix (Bloch) equations was required to calculate the satu-ration line shapes and intensities correctly. Wilson-Gordonand Friedmann' 2 presented analytical solutions of the densi-ty-matrix equations for monochromatic laser excitation.They considered both steady-state conditions (opticalpumping times much longer than characteristic relaxationtimes) and fully transient conditions (optical pumping times

much shorter than relaxation times). Steady-state solu-tions for saturation by monochromatic lasers were also pre-sented by Zadkov et al.13 and by Zadkov and Koroteev.14Wilson-Gordon et al.15 have discussed the validity of thesteady-state approximation in detail. Agarwal and Singh16obtained steady-state, analytical solutions for the case ofexcitation by lasers with fluctuating electric fields and Lor-entzian line shapes.

In this paper we present numerical calculations of satura-tion line shapes and intensities in CARS spectra of the Q(12)line in the nitrogen (0, 1) vibrational Raman transition. Wehave previously reported experimental measurements ofsaturation line shapes and intensities for nitrogen CARS2 0

using high-resolution, nearly Fourier-transform-limitedpump and Stokes lasers, which have electric fields that arewell-characterized functions of time. The density-matrixequations for the CARS interaction are numerically inte-grated using the measured pump and Stokes laser pulseshapes. The anti-Stokes electric-field amplitude is calcu-lated as a function of time, and the intensity is then integrat-ed over the laser pulse to give anti-Stokes energy.

This treatment differs from earlier work in that assump-tions of either steady-state or fully transient conditions areunnecessary. The equations are integrated for arbitrarylaser pulse shapes and frequency structures. Furthermore,Doppler effects are included in the calculations by summingCARS amplitudes from different velocity groups at eachtime step during the numerical integration. It was neces-sary to treat Doppler broadening at this fundamental level,rather than by performing a frequency convolution, becauseof the complicated dependence of CARS amplitudes on timeand detuning. Stark shifting of the nitrogen Q-branchCARS resonance is also included. Stark splitting of theresonance is neglected.

The numerical solutions contain a wealth of informationon the time dependence of level populations and CARSamplitudes. They also provide a framework for evaluatingdiagnostic techniques that use the stimulated Raman pro-cess to pump an excited vibrational level selectively. 22 23

0740-3224/88/061243-10$02.00 © 1988 Optical Society of America

R. P Lucht and R. L. Farrow

1244 J. Opt. Soc. Am. B/Vol. 5, No. 6/June 1988

2. DENSITY-MATRIX FORMULATION OF THECOHERENT ANTI-STOKES RAMANSCATTERING INTERACTION

A. Calculation of the Homogeneous Line ShapeThe equations that describe coherent Raman interactionsfor a homogeneously broadened, vibrational Q-branch lineof nitrogen are derived below. The derivation follows thoseof Giordmaine and Kaiser24 and Penzkofer et al.

2 5 closely.The inclusion of Doppler broadening and the nonresonantbackground in the line shape is discussed in Subsection 2.B.

The molecular system is illustrated schematically in Fig.1. The wave function A(t) for the system formed by levels 1and 2 can be expressed in terms of the eigenfunctions of thestates,

I(t) = l(t)'l1 + 2(t)t 2. (1)

The time evolution of the system is described by the density-matrix equation

dt = [ HI) (2)

where h (erg sec) is Planck's constant, and the density ma-trix is given by

FP1 P12 Cll* ClC2*1

P21 P22 C2C1* C2C2*(3)

The Hamiltonian Hint (ergs) that describes the Raman inter-action between levels 1 and 2 is given by

Hint 2 q\aq/E a i(t), (4)

where q (in centimeters) is the vibrational amplitude, (/aq)ij (cm2 ) is the polarizability derivative for the Ramantransition, Ej and Ej (g1/ 2/cm 1/2 sec) are electric-field compo-nents, and summation over repeated indices is assumed.

The Hamiltonian matrix for the system is given by

R = W1 V12 (t01L V2(t) w2 J

(5)

where relaxation terms have been neglected for the moment,W and W2 (ergs) are the energies of the unperturbed states,and

V12(t) = - q12 (d) E(t)E(t). (6)

We have assumed in writing Eq. (6) that all laser fields arelinearly polarized and parallel. For a harmonic oscillator,the matrix element q12 is given by

q12 = (11q12) = (21qll) = (h/2mrw0)'/ 2(n + 1)1/2, (7)

where mr (g) is the reduced mass, wo (sec- 1) is the angularfrequency of the resonance, and n is the vibrational quantumnumber for the lower level 1.

Substituting Eq. (5) into Eq. (2), we obtain the density-matrix equations in the absence of relaxation as

dt = (P12 - P21) V1 2,

dt_ =h [(p11 - P22) V12 + P12(W2 -W1 I

at = -- [(P1 - P22)V 1 2 + P21(W2 -WI)b

ap22 idt = - (P12 - P2 1)V12.

(8a)

(8b)

(8c)

(8d)

With the inclusion of phenomenological decay terms thatare schematically illustrated in Fig. 1, the density-matrixequations become

apt = Iat h [(p12 - P21) V121 - pl 1 krot + R3 1, (9a)'

aP12 = [P1 - P2 2)V12 + P12(W2 - W1 )] - P1 2 /T2 , (9b)

dt =- (P1 - P 22)V12 + P21(W 2 W1 )] -P 21/T2, (9c)

aP22 =- (P12 - P21)V12 - P22(krot + kvib) + R42,

2

P2 2 krot

R2

4

R31

3

Plikrot

Fig. 1. Energy-level schematic for the Raman pumping process.

(9d)

where krot (sec- 1) is the rotational transfer rate coefficient,which is assumed to be equal in the upper and lower levels,kvib (sec-1) is the vibrational transfer rate coefficient (kvib <<krot for nitrogen), and R31 and R42 (sec-1) are transfer ratesfrom the vibration-rotation manifolds into the lower andupper levels, which are coupled by the laser radiation.

Before laser excitation, rotational transfer out of level 1 isbalanced by rotational transfer from the rest of the vibra-tion-rotation manifold, pll0krot = R31, and the upper levelsare initially unpopulated. After laser excitation begins, lev-el 2 is populated by the stimulated Raman process. Upperrotational levels that are not directly excited by the laser alsostart to become populated on a time scale determined by therotational transfer rate. If the laser pulse is shorter than orcomparable with the rotational transfer rate, then R42 <<

P22krot. Therefore, if rotational transfer rate coefficients are

R. P. Lucht and R. L. Farrow

1


nearly equal in the upper and lower vibration-rotation man-ifolds,2 6 if kvib << krot, and if R42 << P22krot, then the totalpopulation P11 + P22 will be approximately constant duringthe laser pulse, and we can treat levels 1 and 2 as a two-levelsystem. For the two-level system, characteristic times forpopulation transfer, T, (sec), and dephasing, T2 (sec), arethen given by12,27

T1 = 1/krot, (10)

tude and the pump, Stokes, and anti-Stokes fields, respec-tively. The central frequencies of the pump, Stokes, andanti-Stokes fields are cop (sec-'), sz and a, respectively.

Neglecting terms that oscillate at optical frequencies andassuming small conversion (Aa << AP, As), we obtain for thedensity-matrix equations13 25

at + 1T+2 2 W2 Qat T2 2w /

T2 = [2 (krot)i + (krot)2 + 1 *) X (11)

where the T2* (sec-') is the characteristic time for puredephasing collisions. For nitrogen Q-branch transitions,pure dephasing collisions are negligible.2 8 29

Giordmaine and Kaiser24 showed that, for a two-level sys-tem with constant population P11 + P22, the coherent Raman-scattering process can be described in terms of the expecta-tion value (q) of the vibrational amplitude and the excited-state population n = P22. The nonlinear polarization, whichacts as a source term for the generation of anti-Stokes light,is given by

P(t) = N(a (q)E(t) + Pnres(t),\aq ) (12)

where P(t) (gl/2/cml' 2 sec) is the total macroscopic polariza-tion, N(aa/q) (q)E(t) is the polarization due to the Ramanresonance, and Pnres(t) is the nonresonant polarization. Theexpectation value (q) (cm) for the vibrational amplitude isgiven by

(q) = q12(P12 + P21), (13)

and N (cm-3 ) is the total number density. Substituting Eq.(13) into Eqs. (9a)-(9d), and solving for (q) and n gives

a2 (q) 2 a ( + 2 = woql 22aa\

at + T2a + WO' (q) = it ( EE(t)(l -2n),at2 T2 at h (aq ((14)

an n 1 IaE(tE(t (a(q) +(q) .at T, 2hw0 aq/ k at T2 J (15)

Slowly varying amplitude functions are then defined for

- 4m I (la) (ApAS*) (1 -2n),

an +n neq (d (ApA*)* A*AC)at T, 8hwo aq )

(19)

(20)

Rewriting Eq. (19) in terms of the real and imaginary partsof Q (Q = Q. + iQi), we obtain

aQr Qr

at T2

= Q.+

2

aQi _ Qi

at T2

Qi

T 2

where

Qj- y(ApiAsr -AprAsd) (21a)

5Qr + y(ApAs*),

'Qr + (AprAsr + ApiAsi) (21b)

WO2 -W,

=

2w,(22)

and

(23)= (alaq) (1-2n4m,wv

Substituting Eq. (11) into Maxwell's equation and solvingfor the anti-Stokes electric field gives25

aAa a aAa

ax c at

(q) = 1/2 Q exp[ikvx - ivt] + c.c.j,

and for the electric field E(x, t):

E(x, t) = 12 1Ap(x, t)exp(ikx - iwPt)

+ A,(x, t)exp(ikx - is t)

+ Aa(X, t)exp(ikax - iWat) + c.c.-,

where cov and kv are given by

(A = p-Ws, kv = - k,.

(16)= AaC exp(-iAkaX) [N( q )ApQ + XnresAp 2As *]

where negligible absorption in the medium is assumed, Ma isthe refractive index at the anti-Stokes frequency, c (cm/sec)is the velocity of light, and Xnres (cm3/erg) is the nonresonantsusceptibility. Equation (24) is transformed by substitut-

(17) ing t' = t-x/v and x' = x to give

(18)

The spatial dependencies of the electric fields and vibration-al polarization are explicitly acknowledged for the first timein Eqs. (16) and (17). The slowly varying envelope func-tions Q (cm), Ap (g1/2/cm1/ 2 sec), A, (g1/2 /cM1/ 2 sec), and Aa(g1/ 2/cml/2 sec) have been defined for the vibrational ampli-

aa = aexp(-iAkax)[N(_)ApQ + XnresA As ]-

ax AaC -aqP

(25)

Assuming that the interaction length lint is much less thanthe coherence length of the pump or Stokes laser and thatphase matching is perfect, we can integrate Eq. (25) to ob-tain

(24)


6Qj - -y(APA,*)j


a ilintWa a\ 2Aaflint, t) = j I N 11q ApQ + XnresA P As*]I (26)

In many CARS experiments three-dimensional phasematching is employed. Thus the vibrational amplitude Q isexcited by two different pump beams, Apl and Ap2. TheCARS signal Aares is generated by interaction of pump beam2, AP2, with the vibrational amplitude Q, generated by pumpbeam 1, AP1, and vice versa. Thus Eqs. (21) must be solvedfor the interaction of both AP1 and Ap2 with the Stokes laserAs to give Qri, Qi1, Qr2, and Qi2. The anti-Stokes amplitudeAaires due to the interaction of Ap2 with the vibrationalamplitude Q, is given by

Aaires = iAresAp 2Ql = iAres(Apr2 + iApi 2)(Qrl + iQ 1), (27)

Ares = 4aCnt qrs, ac , )

Similarly, Aa2res is given by

(28)

Aa2re,= iAresAp1Q 2 = iAres(Aprl + iApil)(Qr2 + iQi2)- (29)

The real and imaginary parts of the resonant anti-Stokesamplitudes are given by

A re, -A A[A l LA rlAarl =- reA pr2 ii+ A pi2r1J (30a)

Aailres = Ares[Apr2Qrl Apj2Qi], (30b)

Aar2 res =-AreJAPrQi2 + ApiQr2 ] (30c)

Aai2res = Ares[AplQr2 - AP 1 Qi2J- (30d)

The equation for the time dependence of the excited-statepopulation n [Eq. (20)] must also be modified to account forthe two pump fields:

_t T1 8hW0 (w a [(AP + A 2)As*(Ql + Q2)*

- (AP1 + Ap2)*A,(Q1 + Q2)]- (31)

Stark shifting of the nitrogen vibrational Q-branch lines isincluded by calculating the time-dependent shift in the reso-nant frequency 91 2:

AWO(t) = KStark[Ip(t) + Is(t)]. (32)

Stark splitting of the Q-branch lines is not considered.Equations (22) and (32) were solved using a variable-order

Adams code (routine DEABM in the FORTRAN SLATEC li-brary). The accuracy of the computer solution was checkedby comparing nonresonant and resonant amplitudes for Ra-man lines that were broad relative to the laser bandwidth.In this case the Raman polarization will respond with a timeconstant fast enough to follow the fluctuations in the laserradiation. If the laser intensities are low enough that nosaturation occurs, the ratio of the resonant to nonresonantamplitude should be constant throughout the laser pulse.After appropriate adjustment of the initial step size, correctconvergence was observed.

B. Calculation of the Line Shape Including DopplerEffects and the Nonresonant BackgroundThe Doppler width is significant relative to the homoge-neous width at room temperature and at pressures less than

100 Torr. The Doppler effect is incorporated by consideringthe anti-Stokes electric-field amplitude generated by mole-cules grouped in a Doppler velocity interval, which has afrequency width that is small relative to the homogeneouslinewidth. The anti-Stokes field amplitude Aa(t, AWH) iscalculated, as outlined above, as a function of the detuningAWH, where

AWH = P WS - Wo (33)

Each Doppler interval is characterized by a different reso-nant frequency wo. The total amplitude Aa(t) is calculatedfor given values of detuning AWL from the center of thehomogeneously and inhomogeneously broadened line by in-tegrating over the Doppler intervals:

Aares(t, AWL) = J A,(t, AWH)(AWL - AWH)dAWH, (34)

where the Doppler population distribution f(Aw) is given by

2 /1n 2 1 /2 /AW 21f(AO) = A- ( ) expL- 4 n2 (35)

where AWD = (2wo/c)(2kT ln 2/m)1/2 is the FWHM for spon-taneous Raman scattering in the forward direction. It isimportant to note that the summation of anti-Stokes ampli-tudes must be performed for each time t because, as will beshown, Aa(t, AWH) can be a complicated function of bothtime and detuning for a saturated line. Thus the predictionof Doppler-broadened line shapes by frequency convolu-tion30 will not give correct results in this case.

For an unsaturated, purely Doppler-broadened CARSline, Henesian and Byer30 calculate that the FWHM is 1.20times the FWHM for spontaneous Raman. Our numericalcalculations are in agreement with this result. For an unsat-urated line, as the Doppler width is increased relative to thehomogeneous width, the FWHM of the CARS line ap-proaches 1.2 times the spontaneous Raman FWHM.

The anti-Stokes amplitude Aanres due to the interaction ofthe pump and Stokes fields with the nonresonant suscepti-bility Xnres is

Aanres(t, AWL) = iAnres(AWL)(AplAp 2 As* + Ap2AplA *), (36)

where

Anres(AWL) = 2 lintca nres(A (37)nresL MAacXre(L)

The real and imaginary parts of the nonresonant anti-Stokesamplitude Aanres are given by

Aarnres= Anres(-Api 2 APiAsi- Api 2AprAsr

-Apr2ApilAsr + Apr2AprAsi) (38a)

* Aainres = Anres[Apr2AprlAsr + Apr2ApilAsi

+ Api2 Apr1Asi - Api 2 ApjiAsrJ. (38b)

The total CARS amplitude is the sum of the resonant andnonresonant amplitudes,

Aa(t, AWL) = Aares + Aa res, (39)

and the observed CARS intensity Ia(t) is given by



Ia(t, AWL) = CA AA,*a a

(40)

3. COMPARISON WITH EXPERIMENT

In this section theoretical calculations of CARS line shapesand intensities are compared with experimental measure-ments. The experimental system and data were describedin a previous publication.20 For the purposes of comparisonwith theory, it is important to note that the pump andStokes lasers were nearly Fourier-transform limited withpulse lengths of 18 and 12 nsec, respectively. The temporalsmoothness of the pulses permits a direct comparison oftime-dependent theory and experiment. The frequencywidths of the pump and Stokes lasers were 0.0015 and 0.0023cm'1, respectively. Laser pulse energies were modest. Thespectra discussed below were acquired with 9 mJ in eachpump beam and from 0.25 to 6 mJ in the Stokes beam.

Figure 2 shows experimental and theoretical CARS spec-tra for the Q(12) line of nitrogen at 25 Torr at relatively lowStokes laser intensity. The calculated homogeneous lineshape is shown in Fig. 2(a). The homogeneous, collision-broadened FWHM,3 ,3

2 F, is 0.0027 cm-' (T2 = 4 nsec), andthe calculated Stark shift9 is approximately -0.01 cm-'.The polarizability derivative for the line is 1.78 X 10-16 cm 2

(details of this calculation are given by Farrow et al.4 ). Oncethe homogeneous line shape is calculated, the Doppler effect

1 (a) I= 0.04 GW/cm 2

U 0.75-z

Z 0.50

< 0.25

0--0.10 -0.05 0.00 0.05 0.10

1 (b) I (exp) = 0.4H~~~~~~~~~~~

U, 0.75- 5Is(the) = 0.04

<) 0.7 -

zLU

Z 0.50U,

< 0.25

0--0.10 -0.05 0.00 0.05 0.10

RAMAN SHIFT (cm')Fig. 2. CARS line shapes at low-Stokes laser intensity. The solidlines are theoretical calculations of (a) the homogeneous line shapeand (b) the line shape including Doppler effects and the nonreso-nant background. The experimental data in (b) are indicated byfilled circles. The nitrogen pressure was 25 Torr, and the Q(12) linewas probed. The peak intensity of the Stokes laser was 0.4 (experi-mental) and 0.04 (theoretical) GW/cm2. The Stokes pulse energywas 0.25 mJ. The peak intensity and pulse energy of each pumplaser beam were 37 GW/cm 2 and 9 mJ, respectively. The nonreso-nant background susceptibility was 1.2 x 10-18 cm3/erg at linecenter with a slope of +2.8 X 10-18 (cm3/erg)/cm-1 at line center.The Stark-shifting coefficient Kstark was equal to -0.6 X 10-13cm1/W.

(/2zLU

zU)

()

0.75

0.50

0.25

0

zLU

z

U,

0.75

0.50

0.25

0-0.10 -0.05 0.00 0.05 0.10

RAMAN SHIFT (cm-')Fig. 3. CARS line shapes at high Stokes laser intensity. The solidlines are theoretical calculations of (a) the homogeneous line shapeand (b) the line shape including Doppler effects and the nonreso-nant background. The experimental data in (b) are indicated byfilled circles. The nitrogen pressure was 25 Torr, and the Q(12) linewas probed. The peak intensity of the Stokes laser was 8.3 (experi-mental) and 0.7 (theoretical) GW/cm2. The peak intensity of eachpump laser beam was 37 GW/cm2 . The Stokes pulse energy was 5.2mJ. The pulse energy for each pump beam was 9 mJ. The nonreso-nant background susceptibility and Stark-shifting coefficient werethe same as for Fig. 2.

and the nonresonant background are included in the calcula-tion, as outlined above. The nonresonant background sus-ceptibility was determined from the Sandia CARS code4 andexhibited a significant slope in the region of the Q(12) linedue to off-resonant contributions from neighboring Q-branch lines. Comparison between theory and experimentis shown in Fig. 2(b). Agreement between theory and ex-periment is good, especially in terms of the FWHM of theline, but the experimental line exhibits some asymmetry,which is not predicted theoretically. This is possibly be-cause we have not included Stark splitting in the calcula-tions.

As the Stokes laser intensity increases, the CARS lineshape broadens, and a dip appears near line center, wherestimulated Raman pumping is most efficient. The CARSspectrum of the Q(12) line at high Stokes laser intensity isshown in Fig. 3. The calculated homogeneous line shapeshown in Fig. 3(a) exhibits a deep dip near line center andsignificant asymmetry due to Stark shifting [note that theStark shift is nearly the same as in Fig. 2(a) because thepump laser intensity is the same in both cases]. The peak onthe high-frequency side of the dip is narrower and moreintense for the homogeneous line shape. Inclusion of thenonresonant background shifts peak intensities, so that inFig. 3(b) the peak on the low-frequency side is higher. Notethat the Doppler effect does not wipe out the saturation dip,as might be expected, because the calculated Doppler widthis 0.0065 cm-'.

0.00 0.05 0.10-0.10 -0.05



U)LU0D

-J

0-

Ona:

U

U)LUC:D

-0-

n

U)c:U

U)LU0

-J(L

U

1.0

0.5

0.0

-0.5

-1.0

1.0

0.5

0.0

-0.5

-1.0

1.0

c

cI

0.5-

0.0

-0.5-

-1.0-0 10 20

TIME (nsec)30

temporal shape of the real amplitude remains approximate-ly unchanged, but the magnitude of the real amplitude de-creases relative to that of the imaginary amplitude. Theimaginary CARS amplitude peaks slightly later than the realamplitude. At large detunings the peaks of both the imagi-nary amplitude and the resonant CARS intensity shift closerto the peak of the laser pulses.

The temporal histories of the resonant and nonresonantCARS intensities and the excited-state population areshown in Fig. 5 for the same values of the detuning. Thenonresonant response of the medium is extremely fast, andthe nonresonant CARS intensity is proportional to the in-stantaneous product of the Stokes intensity and the squareof the pump laser intensity. The resonant CARS intensitypeaks nearly 10 nsec before the peak of the nonresonantintensity. The excited-state population n exceeds 0.5 forbrief periods. Furthermore, when n is 0.5 at t = 12 nsec, theresonant CARS intensity is just past its peak. At steadystate the resonant CARS intensity is zero when n = 0.5.Clearly, in this intensity regime and at this pressure thesteady-state solution would be inaccurate. The damping ofthe Rabi oscillations as the laser pulse proceeds indicatesthat a fully transient solution would not apply in this caseeither. As detuning increases, the pumping of the excited

40

40

Fig. 4. The real and imaginary parts of the CARS amplitude forhomogeneous detunings A (cm-') of (a) 0.000, (b) -0.002, and (c)-0.004 cm-'. The ratio of the detuning to the Raman linewidth(HWHM), 2Aw/r, is (a) 0.0, (b) 1.48, and (c) 2.96. The parametersof the calculation are identical to those of Fig. 2(a) except that Starkshifting is neglected.

Agreement between the theoretical spectrum (after inclu-sion of the Doppler effect and nonresonant background) andthe experimental spectrum is good, as shown in Fig. 3(b).The intensity ratio between the peaks on either side of thesaturation dip is slightly different in the theoretical andexperimental spectra, but agreement is excellent in terms ofthe depth of the dip, the overall width of the line, and therelative widths of the spectral features on each side of thedip. The experimental and theoretical Stokes laser intensi-ties differ by a factor of 10, which we attribute to the effectsof the spatial profiles of the laser beams and discuss in moredetail below.

The real and imaginary parts of the calculated CARSelectric-field amplitude are shown in Fig. 4 for differentvalues of the homogeneous detuning, Ac = xv - co, at thesame laser intensities used to calculate the spectra in Fig. 3.The real and imaginary amplitudes are generated by theimaginary and real parts, respectively, of the CARS suscep-tibility. The Stokes and pump laser pulses have a FWHMof 12 and 18 nsec, respectively, and both peak at 20 nsec inFig. 4. At line center, Ac = 0, the real CARS amplitudepeaks well before the peak of the laser pulses and exhibitsoscillations due to population transfer. The imaginary am-plitude is zero at line center. As the detuning increases, the

0-0C)LUX

LU

LU

z

1.00 -

0.75 -

0.50-

0.25-

Q.0

LUH

LU

zLU

CL

xLU

U)

X

LUH_

zZ

Lu

F_

40

0 10 20 30 40TIME (nsec)

Fig. 5. Resonant and nonresonant CARS intensities and the excit-cd-stato population fraction at homogeneous detuninga Aw (cm-')of (a) 0.000, (b) -0.002, and (c) -0.004 cm-'. The ratio of thedetuning to the Raman linewidth (HWHM), 2wx/r, is (a) 0.0, (b)1.48, and (c) 2.96. The parameters of the calculation are identical tothose of Fig. 2(a) except that Stark shifting is neglected.

(a) Acw = 0.000 cm-'

- Aar..... 1 A2

A ~~~~~~~a

I I I . I

Aw = -0.002. ....

10 20 30

(c) ..... Aco = -0.004

(a) Aco = 0.000 cm-'- I res

......nres

.n

10 20 30


I

I

10 20 30 40

Ub

......

Il


0.75

0.50

0.25

U)z

z

U

0

-0.050 -0.025 0.000 0.025 0.050

-0.050 -0.025 0.000 0.025 0.050

RAMAN SHIFT (cm-1)Fig. 6. Calculated homogeneous (dashed line) and Doppler-broad-ened (solid line) CARS line shapes at Stokes laser intensities of (a)0.001 and (b) 0.7 GW/cm 2. The pump laser intensity was 37 GW/cm2, the nitrogen pressure was 25 Torr, and the calculations wereperformed for the Q(12) line. Stark shifting and the nonresonantbackground were neglected.

state becomes less efficient, and the resonant intensity pro-file shifts closer to the nonresonant intensity profile. Notethat even at a detuning (-0.004 cm-') that is three times thehomogeneous HWHM, nearly 35% of the ground-state popu-lation is pumped to the excited state.

The dip in the spectra in Fig. 3 seems to have about thesame width and depth in the homogeneous line shape as inthe line shape after inclusion of Doppler broadening. Forthis high-intensity case, the result of a frequency convolu-tion of the Gaussian Doppler profile with the saturated ho-mogeneous line shape would be to decrease the depth of thesaturation dip. Figures 6(a) and 6(b) show the effect ofDoppler broadening on the Q(12) line shape for an unsatu-rated case and a saturated case, respectively. When the lineis not saturated, the time dependence of the CARS ampli-tude from each Doppler is similar, and the line simply broad-ens. However, for a highly saturated CARS line such as thatshown in Fig. 6(b), the temporal dependence of the electric-field amplitudes of the CARS signal is a complicated func-tion of detuning, and CARS signals from neighboring Dopp-ler velocity groups can constructively or destructively inter-fere. Therefore the Doppler effect can actually cause thedip to broaden and to deepen, as shown in Fig. 6(b).

Figure 7 shows experimental and theoretical CARS spec-tra of the Q(12) line of nitrogen at 100 Torr for three differ-ent Stokes laser intensities. Figure 8 shows the temporaldependence of the CARS intensity and excited-state popu-lation for the spectrum shown in Fig. 7(c), which is the mostsaturated spectrum at 100 Torr. The steady-state approxi-mation would be nearly valid for this case. The asymmetryaround the peak laser intensity, which is apparent in boththe CARS intensity and excited-state population curves,indicates that the steady-state solution would not be strictlyvalid.

H

IIzU)

zul

ccH

zU

C-)

c-U

-0.10 -0.05 0.00 0.05 0.10RAMAN SHIFT (cm-')

Fig. 7. CARS line shapes for the Q(12) line of nitrogen at a pressureof 100 Torr. The experimental Stokes laser intensities were (a) 0.8,(b) 3.3, and (c) 10 GW/cm2, corresponding to pulse energies of (a)0.5, (b) 2.0, and (c) 6.0 mJ. The theoretical Stokes laser intensitieswere (a) 0.12, (b) 0.55, and (c) 1.8 GW/cm2. The solid lines aretheoretical calculations of the line shape, including Doppler effects,Stark shifting, and the nonresonant background. The experimen-tal data are indicated by filled circles. The nitrogen pressure was100 Torr, and the Q(12) line was probed. The peak intensity andpulse energy of each pump laser beam were 37 GW/cm2 and 9 mJ,respectively. The nonresonant background susceptibility was 5.4 X10-18 cm3/erg at line center with a slope of +11 X 10-8 (cm3/erg)/cm'1 at line center. The Stark-shifting coefficient Kstark was equalto -0.6 X 10-13 cm-1/W.

EL0

XLL

(-

zU)z

U)

1.00

0.75

0.50

0.25

0.000 10 20 30 40

TIME (nsec)

Fig. 8. Resonant and nonresonant CARS intensities and the excit-ed-state population fraction at a homogeneous detuning A (cm-')of 0.000 cm-'. The parameters of the calculation are identical tothose of Fig. 6(c) except that Stark shifting is neglected.

U)zUlzU)

cc

0.75

0.50

0.25

0

HU)zLuzU)

CCU

-0.10 -0.05 0.00 0.05 0.10



The best agreement between experiment and theory forthe nitrogen spectra at 25 and 100 Torr was obtained withtheoretical Stokes laser intensities approximately 10 and 5times lower than the measured experimental Stokes laserintensity. The theoretical pump laser intensity was setequal to the experimental value. The experimental uncer-tainty in the measured Stokes-pump intensity product was200-300% because of difficulties in determining the overlap-ping temporal and spatial intensity profiles of the lasers.20

The geometric profile of the pump and Stokes lasers was notincluded in the time-dependent calculations because itwould have greatly increased the computational time re-quired and because the transverse-mode structures of thepump and Stokes lasers are not well characterized at thefocus.

To estimate the effect of the beam spatial profiles on themeasured spectra, we assumed that the pump and Stokeslaser beams had Gaussian profiles. The steady-state solu-tion of the CARS density-matrix equations for a homoge-neously broadened line pumped by monochromatic laserswas then integrated radially using the formula' 9

Ia(A~) = [1 + (AWT 2)2]I 2(r)Il(r) _rdr, (41)

1 [1/T 2 + A 2T2 + Ip(r)Is(r)/(II)r J2

where

hmc2oupi~sa 82(Oa/aq)2 TIT 2 (42)

Figure 9 shows the result of the calculation for nitrogen at 25Torr. The solid line was calculated assuming a top-hatconstant intensity profile for the laser beams, and thedashed line was calculated assuming Gaussian profiles withthe measured experimental FWHM's for the pump andStokes lasers, 50 and 75 Am, respectively. The peak Stokeslaser intensity used for the Gaussian profile beams was a

LU

Z

'C-

0.75

0.50

0.25

0-0.050 -0.025 0.000 0.025 0.050

RAMAN SHIFTFig. 9. Calculated homogeneous line shapes assuming top-hat (sol-id line) and Gaussian (dashed line) intensity profiles. For theGaussian intensity profile calculation, the FWHM's of the pumpand Stokes beams were 50 and 75 ,um, respectively. The peakintensity of each pump beam was 75 GW/cm 2 in both cases. Thepeak intensity of the Stokes beam was 0.64 GW/cm 2 for the top-hatprofile and 1.28 GW/cm2 for the Gaussian profile. The dephasingtime T2 was 4 X 10-9 sec. The Doppler effect and Stark shiftingwere neglected.

factor of 2 higher than for the top-hat profile beams. Thedepth of the saturation dip is slightly less when the beamsare assumed to have Gaussian profiles, but the widths of thedip and of the line as a whole are nearly the same in bothcases.

The laser beams actually used in the experimental werenot diffraction limited, and we have not included the effectof the laser-beam profiles and the finite extent of the CARSprobe volume along the axis of laser-beam propagation.Consequently, it is not unreasonable that there is more thana factor-of-2 difference in the measured and the calculatedCARS saturation intensities. The shape of the CARS satu-ration spectrum, however, appears to be fairly insensitive tothe pump and Stokes laser-beam intensity profiles. It istherefore not surprising that ignoring the laser-beam inten-sity profiles and using the Stokes laser intensity as a scalingfactor to fit the shapes of experimental spectra seems to givegood agreement between theory and experiment.

It is also interesting to note that the ratio of the experi-mentally observed to the calculated saturation intensity is afactor of 2 lower for the 100-Torr data compared with the 25-Torr data. This may be due to the fact that at 100 Torr theupper rotational manifold acquires significant populationduring the laser pulse because of transfer out of the directlypumped rotational level. At 100 Torr, the integral f nkrotdtover the entire laser pulse was -4. Thus, by the end of thepulse, four times the initial population of the laser-pumpedlower rotational level was transferred to other rotationallevels in the upper rotational level manifold. For nitrogenat room temperature, 7% of the population is initially in theJ = 12 rotational level.

At 100 Torr the transfer term R42 [Eq. 9(d)] is starting tobecome significant relative to the term P22krot = flkrot Inthis case the effective characteristic time T, for the systemwill be given by

T = (vib + rot n - (43)

The characteristic time T, is increased by rotational transferback into the directly pumped level, thus lowering the satu-ration intensity. For long pulses the terms krot and R42/nmay be nearly equal, and the saturation intensity will begoverned by the vibrational transfer time.

The characteristic time T, is, in general, a function of thepopulation distribution of the molecule. For an experimentsuch as ours in which level populations are greatly perturbedby laser pumping, T, can vary significantly during the laserpulse. To describe completely the saturation behavior athigher pressure, therefore, the population dynamics of bothrotational manifolds will have to be modeled in detail.

4. SUMMARY

Results of numerical calculations of saturation intensitiesand line shapes in CARS have been presented. The calcula-tions are performed by integrating the time-dependent den-sity-matrix equations for single-axial-mode pump andStokes laser radiation. Agreement between experimentaland calculated saturation line shapes is excellent, althoughpredicted saturation intensities are an order of magnitude

- R. P. Lucht and R. L. Farrow


lower than observed saturation intensities. The discrepan-cy between predicted and measured saturation intensities isprobably due to the radial and axial intensity profiles of thelaser beams.

The framework for the solution of the homogeneous lineshape facilitated the inclusion of Doppler broadening effectsat a fundamental level. CARS amplitudes from differentvelocity groups were summed for each time step in the nu-merical solution. It was found that for a highly saturatedline the effect of Doppler broadening was to deepen and tobroaden the saturation dip, which is a non-steady-state ef-fect that could not have been predicted from a frequencyconvolution of the Doppler profile with the homogeneousline shape. The effect of the nonresonant background wasalso incorporated in the calculations in a straightforwardfashion.

At pressures higher than 25 Torr, population of the upperrotational manifold by collisional transfer out of the laser-pumped upper rotational level may start to reduce the satu-ration intensity significantly. The two-level saturationmodel described in this paper does not take this effect intoaccount. The ratio of measured to predicted saturationintensity decreases by a factor of 2 from 25 to 100 Torr,providing some evidence for a reduction in saturation inten-sity as rotational transfer rates increase. To address thisissue it will be necessary to perform numerical modeling ofpopulation transfer within the rotational manifolds.

The laser pulse energies used to saturate nitrogen lines inthis study were relatively modest; 5-10 mJ per beam wassufficient to attain a high degree of saturation. When laserswith frequency widths larger than Raman linewidths areused in CARS measurements, the onset of saturation will bemuch harder to discern. The solution procedure outlinedabove can be extended in a straightforward fashion to thecase when the pump and/or Stokes laser has numerous axialmodes. For multi-axial-mode laser radiation, computation-al time is drastically increased by the need to average overnumerous laser pulses. Good models of the laser radiationare necessary if accurate theoretical saturation intensitiesfor multi-axial-mode laser radiation are to be acheived.

ACKNOWLEDGMENT

This research was supported by the U.S. Department ofEnergy, Office of Basic Energy Sciences, Division of Chemi-cal Sciences.

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calculation of saturation line shapes and intensities in coherent anti-stokes raman scattering...

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