Frequency measurements and self-broadening of sub-Doppler transitions in the v 1 + v 3 band of C 2 H 2
FFrequency measurements and self-broadening of sub-Doppler transitions in the v + v band of C H Sylvestre Twagirayezu, a) Gregory E. Hall, b) and Trevor J. Sears
1, 2, c)1)
Division of Chemistry, Department of Energy and Photon Sciences,Brookhaven National Laboratory, Upton, NY 11973-5000,USA Chemistry Department, Stony Brook University, Stony Brook, NY 11794-3400,USA (Dated: 10 March 2020)
Frequency comb-referenced measurements of sub-Doppler laser saturation dip absorp-tion lines in the v + v band of acetylene near 1 . µ m are reported. These measure-ments include transitions involving higher rotational levels than previously frequencymeasured in this band. The accuracy of the measured frequencies is typically betterthan 10 kHz. Measurements of the observed sub-Doppler line widths as a func-tion of pressure showed that the self pressure-broadening coefficients are about 3.5times larger than those derived from conventional pressure broadening of unsaturatedDoppler-limited spectra. This is attributed to the contribution of velocity-changingcollisions to the total dephasing rate in the low pressure sub-Doppler measurements.At higher pressures, when the homogeneous broadening becomes comparable to thetypical Doppler shift per elastic collision, the velocity changing collisions cease tocontribute significantly to the incremental pressure broadening. A time-dependentsoft collision model is developed to illustrate the transition between low and highpressure regimes of sub-Doppler pressure-broadening. a) Electronic mail: [email protected]; Present Address: Department of Chemistry & Biochemistry,Lamar University, Beaumont, TX 77710 b) Electronic mail: [email protected] c) Electronic mail: [email protected] a r X i v : . [ phy s i c s . c h e m - ph ] M a r . INTRODUCTION The v + v band of acetylene, C H , is a strong vibrational combination band in the1.5 µ m region of the spectrum. It is a parallel band with a simple P- (∆ J = −
1) and R-(∆ J = +1) branch rotational structure and has long been used as a secondary wavelength,and more recently frequency, standard. Techniques for the routine frequency-measurementof optical spectra have opened the way to far more precise determinations of line positionsand shapes than were previously available. We have previously frequency-measured hotband lines in this spectrum for the purposes of calibrating Doppler- and pressure-broadenedmeasurements. These measurements were made using laser saturation dip spectroscopy witha fiber laser-amplified c.w. extended cavity diode laser referenced to a frequency comb asthe source.Here, we report rest frequency measurements for some lines in the spectrum that havenot previously been frequency measured. During the frequency measurements, it was notedthat the width of the saturation resonance varied with pressure, in a similar way to thatseen in a conventional absorption spectrum. The pressure and power dependence of thesaturation dips have been measured and analyzed. The pressure broadening coefficients aretypically about 3.5 times larger than those previously reported from conventional, Doppler-limited, spectroscopic measurements at much higher pressures. Previous measurements ofsub-Doppler saturation dip spectra in atoms such as Xe and others discussed in sectionIV also report self-pressure broadening coefficients larger than seen in Doppler broadenedspectra. Multiple observations, see for example references, of sub-Doppler spectra ofmethane in the 3.39 µ m lasing transition have also been reported and these also exhibitlarger self-pressure-broadening coefficients. Measurements of sub-Doppler CO vibration-rotation transitions also show larger self-broadening than observed in Doppler-limitedspectraThe interpretation of these and other similar observations has generally focussed on theeffects of velocity-changing collisions. Physically, this is reasonable as sub-Doppler spec-troscopy, in which a single velocity group of molecules is selected, might be expected to bemore sensitive to weak, large cross section, collisions compared to Doppler-limited spectrawhere the line width is determined by the ensemble average of all velocities in the sam-ple. In the spirit of these ideas, we describe a simple collisional model based on a soft2ollision model and the interplay between damping and dephasing, which mimics theobservations. II. EXPERIMENTAL
Sub-Doppler spectra of selected ro-vibrational lines in the v + v band of acetylene weremeasured by the transmission of an external cavity diode laser beam through a resonantcavity containing low pressures of acetylene. The probe laser beam was locked to a cavityresonance using the Pound-Drever-Hall method, with a DC-300 kHz error correction signaladded directly to the diode injection current. The resonant cavity consisted of two highly-reflective mirrors, each mounted on a PZT piezo-electric actuator. One actuator with a slowresponse, but longer travel was used for coarse positioning of the cavity modes, the secondwith short travel, but higher frequency response executed an 830 Hz dither in additionto a stepwise ramp scan, generated by a second feedback loop that controlled the offsetfrequency between the probe laser and a reference tooth of the frequency comb. With bothfeedback loops locked, the repetition rate of the frequency comb was stepped to changethe average optical frequency of the synchronously dithered probe laser and sample cavitytransmission maximum through the sub-Doppler resonance. Data were recorded using anoptical frequency step size of approximately 70 kHz through the center of each line, withsteps three times larger elsewhere, to accelerate the data acquisition. Each data point wastypically averaged for 3s with a 300ms output time constant for the lock-in amplifier. Aprogrammed delay was included after each frequency step to allow the system to stabilizebefore data acquisition began. More complete details have been previously reported. As detailed below, the data analysis was carried out assuming a Voigt-type profile toaccount for a combination of transit-time and pressure-broadening effects. The (narrow)transit-time contribution to the observations is most important for the lower pressure mea-surements and an analysis of the effect of changing the width of the Gaussian component isincluded in Supplementary Information for this paper.We found the line shape modeling required a pressure-independent Lorentzian compo-nent to the line shape function, that could not be attributed to the effect of wavelengthmodulation broadening.
Instead, this residual broadening is mostly due to fluctuationsof the probe laser frequency derived from mechanical and acoustical noise sources. The av-3rage laser frequency was accurately maintained by reference to the frequency comb, but thebandwidth of the phase lock loop holding the absolute comb frequency was slow enough thatmeasured data points were an average over a Lorentzian distribution of frequencies centeredabout the exact center one. The 830 Hz component of the fluctuations was demodulated bythe detection system. We have modeled the effect quantitatively by assuming a Lorentziandistribution of frequency noise on top of 1-f wavelength modulation of a Voigt line shape.The convolution of two Lorentzian functions, i.e. frequency noise plus a pressure-dependentbroadening, leads to a new Lorentzian function with width parameter, γ , equal to the sum ofthe widths of the two constituents. The modeling also showed that he modulation-inducedbroadening was small, and its effects were confined to the Gaussian, transit-time, contri-bution to the Voigt line shape, implying the actual transit time broadening was somewhatsmaller than derived below. III. RESULTSA. Pressure-dependent line profiles
Sub-Doppler resonances for the P(31) line at several pressures are shown in Figure 1. Ina low intensity limit, where the power broadening is negligible, the observed saturation dipline shape can be treated as the convolution of a Gaussian component due to transit-timebroadening and a Lorentzian component due to multiple damping and dephasing mech-anisms. The transit-time broadening depends on the laser beam size and the distribution oftransverse velocities in the molecular sample. Most models of the effect just include the mostprobable Maxwell-Boltzmann velocity, u p = (cid:112) ( k B Tm ), although a more realistic descriptionof the interaction would include the fact the slower molecules will contribute differently tofaster ones and a description of the 2-D velocity distribution perpendicular to the laseraxis. The convolution of the Gaussian and Lorentzian gives the well-known Voigt profile.With an estimated beam waist radius in the cavity of ≈ . ∆ ν t t ≈ u p w we estimate a Gaussianhalf-width-half-maximum (HWHM) of 0.112 MHz. In fact, an initial analysis of the P(31)data showed the Gaussian contribution to be constant over the pressure range studied, atapproximately 0.105 MHz HWHM, which is very close to the estimate. We note that other4 S i g n a l ( a r b ) MHz - 194018374.0Data and Fit Function P(31)
In PhaseVoigt functionData - Fit (a) 5 . × − mbar -0.05-0.04-0.03-0.02-0.01 0 0.01 0.02 0.03 0.04 0.05-6 -4 -2 0 2 4 6 S i g n a l ( a r b ) MHz - 194018374.0Data and Fit Function P(31)
In PhaseVoigt functionData - Fit (b) 14 . × − mbar -0.03-0.02-0.01 0 0.01 0.02 0.03 0.04-6 -4 -2 0 2 4 6 S i g n a l ( a r b ) MHz - 194018374.0Data and Fit Function P(31)
In PhaseVoigt functionData - Fit (c) 26 . × − mbar -0.03-0.02-0.01 0 0.01 0.02 0.03 0.04-6 -4 -2 0 2 4 6 S i g n a l ( a r b ) MHz - 194018374.0Data and Fit Function P(31)
In PhaseVoigt functionData - Fit (d) 39 . × − mbar FIG. 1:
Saturation dip spectra for the P(31) line in the v + v band of C H at a series of samplepressures, shown in the subcaptions, 1 mbar=0.750062 Torr=100 Pa. Scans were recorded with a moredense sampling near line center. The increase in observed width with sample pressure is clear. All datawere recorded at approximately 11 mW intracavity power. The line shapes were fit to a derivative of aVoigt profile consisting of a fixed Gaussian transit-time, and an adjustable Lorentzian component. derived approximate forms for ∆ ν t t such as Demtroeder’s √ ln π u p w do not agree as closelywith the experiment. Since the data for transitions other than P(31) were less extensive andgenerally of poorer signal-to-noise, in all subsequent analysis the Gaussian contribution tothe measurements was assumed to be constant and fixed to 0.105 MHz HWHM.Voigt fits to the pressure-dependent P(31) line profiles with fixed Gaussian HWHM of0.105 MHz lead to the Lorentzian broadening as plotted in Figure 2. The intercept in5igure 2 is the extrapolated zero-pressure Lorentzian width, which corresponds to residualbroadening due mainly to the mechanisms discussed above. The natural (fluorescence)lifetime for the upper level, J = 30 in the v + v level can be estimated from known EinsteinA-factors. It is approximately 50 Hz, too small to contribute significantly to the line widths.Power broadening was characterized in separate experiments in which data for P(31) wererecorded at a pressure of 14 . × − mbar at a series of intracavity laser powers. Over therange 11-40 mW, the HWHM increased linearly with a slope of γ pwr = 3 . slope = 9.35 +/- 0.13 MHz/mbarintercept = 0.2705 +/- 0.0024 MHz Γ ( H W H M ) / M H z Pressure / mbar*10 Pressure v. Lorentzian Width, P(31), 11 mW
Γ v. p DataFitted line
FIG. 2:
Plot of Lorentzian width Γ for the P(31) saturation dip spectra in a Voigt fit with GaussianHWHM fixed at 0.105 MHz. The values are well described by a linear model. The error bars on the pointsare three standard deviations of the fitted widths for the individual measurements.
The slope of the line in Figure 2, i.e. γ self (HWHM): the self-broadening coefficient for the sub-Doppler P(31) transition at aone-way intracavity power of 11 mW. The cavity-defined, counter-propagating, single laserbeam geometry in the present measurements means that they may be compared directly to γ self ( Doppler ) derived from single-pass measurements of Doppler-broadened gaseous sam-ples. The sub-Doppler γ self determined here is about 3.5 times larger than γ self ( Doppler ) forthe same line. Forward- and backward-going beams address separate velocity groups, eachlifetime broadened. The convolution of a tunable probe with a fixed-frequency depletionwould lead to feature that is double the width of either one alone. However, this effect isexactly compensated in a single beam cavity measurement, since the relative Doppler de-6uning of the forward and backward beams changes at twice the scan rate. There was someconfusion on this topic in the early saturation dip literature, when comparing pressurebroadening of sub-Doppler features and Doppler-broadened spectra at higher pressures.A series of saturation dip spectra at pressures from 2.9 to 29.4 mTorr ((3.9 - 39.2) × − mbar) for the P(31) - P(37) lines in the v + v band spectrum of acetylene were measuredand analyzed. Table I summarizes the self pressure-broadening coefficients for sub-Dopplerresonances in these transitions, along with literature values of the pressure broadening coeffi-cients derived from Doppler-broadened spectra. All sub-Doppler coeffcients are significantlylarger than their Doppler-broadened counterparts and we also note that the change in γ self measured here ( ≈ Unfortunately, the estimated error in the self-broadening coefficient for the only para- linefor which pressure broadening has been measured here, P(34), is too large to distinguishsmall ortho-/para- differences.
B. Rest frequencies
The rest frequencies determined for the observed rotational transitions are also summa-rized in Table I. The cited frequency uncertainties are the statistical (1 σ ) errors from mul-tiple measurements of each line at various pressures between approximately 3 and 40 × − mbar. We noted a slight tendency towards a negative pressure-frequency shift coefficientfor the P(31) data, ∆ = − . i.e. slightly larger than quoted in Table I, but within the combinedestimated 1- σ errors. For P(11), we previously determined ∆ = − . If the present rotational transitions had a similarpressure-dependent shift coefficient, the low-pressure shifts would be close to the accuracyand precision limits of our current measurements, leading to no conclusive comparison ofhigh and low pressure shifting coefficients. 7ABLE I: Frequencies and broadening parameters of selected transitions in the v + v band of C H . Transition Frequency a γ self b γ self ( Doppler ) c MHz MHz/mbar MHz/mbarP(31) 194018374.101(3) d a Measured rest frequency from sub-Doppler spectroscopy (this work). b Sub-Doppler self-broadening coefficient (this work). c Doppler self-broadening coefficient, average of literature values.
Uncertainties in parenthesis are one standard deviation in units of the last quoted significant figures.
There have been relatively few direct comparisons of self-pressure shifts for the sametransitions between Doppler-limited and sub-Doppler resonances. Recent very high precisionmeasurements on sub-Doppler lines in the (3-0) band of carbon monoxide found an upperbound to the pressure shift of a few tenths kHz/mbar. A significantly larger CO self-shiftof 0.16 to 0.31 MHz/Torr (120-230 kHz/mbar) was reported in Doppler and pressure broad-ened FTIR spectra. Sub-Doppler measurements on a methane transition at 3.39 µm foundan unmeasurably small (75 ±
150 Hz/Torr) self-pressure shift coefficient at low pressures,while conventional IR absorption methane spectra have been reported to have much largerself-pressure shifts of -0.017 cm − atm − (-670 kHz/Torr) for transitions in the 6000 cm − region. Other sub-Doppler IR measurements of carbon monoxide and water similarlyled to upper bounds smaller than the high-pressure self-pressure shift. Precise measurementsof saturation dip spectra of ethylene in the 10 µ m region and iodine in the visible showeddetectable, small self-shifts, but with a nonlinear pressure dependence. Self-collisional fre-quency shift coefficients for sub-Doppler HD overtone lines were, however, found to bemuch larger at -900 kHz/mbar, some 10 × larger than found in Doppler-broadened spectraof the normal isotopomer, H . Overall then, there is little or no pressure-induced shift seenin most of the sub-Doppler experiments and theoretical consideration of this phenomenonin the literature is sparse or non-existent. 8 . Time-Dependent Collisional Model
The influence of velocity-changing collisions (VCCs) on the widths of low-pressure sub-Doppler line profiles has been investigated by numerical evaluation of model electric fieldcorrelation functions, which detail the damping and dephasing that are responsible for thebroadening of a resonant absorption, in the spirit of Galatry and Rautian and Sobel’man. The line profile is given by the real Fourier transform of the correlation function Φ( τ ): I ( ω − ω ) = 1 π (cid:60) (cid:90) ∞ e i ( ω − ω ) τ Φ( τ ) dτ. (1)The total correlation function can be approximated as the product of an exponential dampingfunction Φ γ ( τ ), a fly-out or transit time damping function Φ t t ( τ ), and a velocity-dependentdephasing function Φ v ( τ ) Φ( τ ) = Φ γ ( τ )Φ t t ( τ )Φ v ( τ ) (2)Φ γ ( τ ) = e − γτ (3)Φ t t ( τ ) = e − ( τ/t t ) (4)Φ v ( τ ) = (cid:10) e − i k · r ( τ ) (cid:11) (5)with r ( τ ) = (cid:90) t + τt v ( t (cid:48) ) dt (cid:48) . (6)The damping rate γ is primarily a pressure-dependent inelastic damping rate but may alsoinclude a zero-pressure radiative contribution. The transit time contribution is approximatedas a Gaussian function of time, with typical collisionless transit time, t t . The Doppler effectsdue to a static or fluctuating velocity distribution are contained in the correlation functionΦ v ( τ ), where the angle brackets in Eq.(5) denote an ensemble average. The motion of theabsorber along the propagation direction of the light introduces a time-dependent phase, k · r ( τ ). For time τ shorter than the time interval between collisions, the velocity v isconstant, and r ( τ ) = v τ . The phase, k · r ( τ ), is then a linear function of time, correspondingto a fixed Doppler frequency offset of v/λ , with v the component of velocity along thewavevector k . For a thermal sample with a low collision rate, the ensemble average ofvelocities gives a Gaussian correlation function as the low pressure limit of Φ v ( τ ). The9ourier transform gives the usual Gaussian Doppler line profile, I ( ω − ω ), for which thetransit time damping is typically negligible compared to the rapid dephasing from multipleDoppler shifts. Including an additional exponential damping, through Φ γ ( τ ), leads to aVoigt profile. As the collision rate increases with pressure, each absorber may experiencemore than one Doppler-shifted frequency before suffering an inelastic collision or leavingthe interaction region. In this case, the correlation function Φ v ( τ ) will dephase more slowlythan the inhomogeneous average over multiple Doppler shifts, characterizing the narrowingassociated with the velocity diffusion. The confinement at higher pressures makes Eq. (4)an increasingly inaccurate representation of transit time contributions, although a pressure-dependent increase in effective transit time only renders this slow damping an even morenegligible contribution to the total correlation function at higher pressures.For application to sub-Doppler line profiles, the inhomogeneous averaging over the ther-mal velocity distribution is left out, and only the resonantly selected velocity class ofmolecules with a specific initial Doppler shift is considered, while retaining the same treat-ment of velocity diffusion, inelastic loss of population, and the transit time.The central point of the model is to depict the variable effect of velocity changing collisionswith pressure, as the pressure-dependent homogeneous line width changes from small to largecompared to the typical Doppler shifts accompanying an elastic collision. Relevant previousdiscussions of these questions can be found, for example, in several references from the earlydays of saturation spectroscopy. Neglected in this model are any pressure shifts, collision effects in the excited state,properly treated with a density matrix approach, power broadening and the variation ofthe saturation parameter on the pressure, and any effects of radially variable intensities,wavefront curvature and recoil effects, all important in more quantitative treatments.
The finite duration of the collisions is neglected by assuming a negligible phase shift duringthe collision. The speed-dependence of collision rates has also been neglected, as has anyexplicit treatment of residual broadening due to instrumental effects discussed above.It is worth noting that the velocity change associated with inelastic collisions, whilecontributing substantially to the conventional diffusion constant and the overall translationalthermalization rate, is irrelevant to pressure broadening, since the state change immediatelyremoves the molecule from the near-resonantly driven ensemble under consideration. Itis only the differential cross sections for elastic (state-preserving) collisions that need be10onsidered in the evaluation of Φ v ( τ ). When expressed as a laboratory-frame collision kernel,the forward peak in the elastic differential cross section leads to a cusp-like distribution ofvelocity change along a viewing direction. In the present numerical illustration, thecollision kernel is approximated as a symmetric exponential function of the velocity changealong the viewing direction: A ( v, v (cid:48) ) ∝ e −| v − v (cid:48) | / ∆ vcc (7)with ∆ vcc the average absolute value of the velocity change following an elastic collision. Thisform of a collision kernel does not have the correct asymptotic form required to converge atlong times to a Maxwell-Boltzmann velocity distribution, but it can quite accurately repre-sent the first few elastic collisions with small total displacements in velocity space comparedto the thermal velocity. Inelastic collisions truncate the sequence of elastic velocity changingcollision long before thermalization, so that this failure to satisfy asymptotic reversibilityis not a serious problem. Realistic calculations of elastic differential cross sections peak atscattering angles less than 2 degrees, which, when combined with the kinematic averagingof center of mass to laboratory frame velocity changes find typical Doppler shifts of a fewMHz per collision for C H transitions in the near infrared at room temperature.With these simplifications, a numerical simulation based on Eq. (1-7) for the pressure-dependent sub-Doppler line profiles can be made with only three relevant parameters: theratio of elastic to inelastic collision rates γ e /γ , the typical Doppler shift per elastic collision∆ vcc /λ , and a transit time t t . In numerical calculations designed to mimic the conditions ofthe present acetylene measurements, the correlation function is evaluated with 100 ps timesteps from 0 to 50 µ s. The transit time contribution Φ t t ( τ ) is evaluated with t t = 5 µ s. Theinelastic contribution Φ γ ( τ ) is evaluated with the damping rate γ = 2 πp · . × µ s − mbar − , with pressure p in mbar, chosen to match the experimental 2.65 MHz/mbar high-pressure self-broadening coefficient (HWHM) for a J = 31 line of C H . The velocity-dependent correlation function Φ v ( τ ) is evaluated by a Monte Carlo procedure with oneparameter for the elastic to inelastic collision ratio, γ e /γ , and another to parameterize thetypical magnitude of the velocity change per collision, ∆ vcc . For each time array in the en-semble average, the time array is partitioned into a sequence of collisionless intervals, { ∆ t i } ,with a random sample of times between elastic collisions generated from an exponential dis-tribution with a mean value < ∆ t > = 1 /γ e . For each molecular trajectory in the ensembleaverage, then, the phase factor k · r ( τ ) in Eq (5) is a continuous, piecewise linear function of11IG. 3: Simulated pressure dependence of sub-Doppler line broadening. At pressures above 3 mbar, theHWHM increases with a slope characteristic of the inelastic rate. The steeper dashed line at lowerpressures has a slope characteristic of the total elastic + inelastic rate. The low pressure inset shows thetransit time limit as a horizontal dashed line, and the steeper sloped line from the main figure. time, with a local slope in each collisionless time interval equal to the current detuning v i /λ .The molecule follows a random walk in velocity { v i } starting at zero displacement fromthe laser-selected initial value. The velocity steps v i +1 − v i are sampled from the collisionkernel, A ( v, v (cid:48) ) of Eq (7), with ∆ vcc /λ set to 3 MHz. For the illustrations in Figures (3) and(4), γ e /γ is set to 2.5. Demonstration code implementing this algorithm is available in theSupplemental Material for this paper in the form of a stand-alone Windows executable fileand a LabView 2011 library.Figure 3 shows the resulting trend in HWHM as a function of pressure, based on theFourier transform of the simulated correlation function, using the model parameters above.Two limiting pressure regimes can be seen, with local slopes corresponding to the totalcollision rate at low pressure, and just the inelastic collision rate at high pressures. Thetransition between high and low pressure regimes occurs when the total pressure broadeningis comparable to the Doppler shift per collision.Figure 4 shows correlation functions at several pressures. In the low pressure region, thecurvature in the plots of log Φ( τ ) comes from the pressure-independent quadratic transittime contribution, which becomes negligible at higher pressures, when compared to the in-creasingly fast collision-induced damping. Throughout the pressure range below 0.05 mbar,12IG. 4: Correlation functions computed from 0.01 to 10 mbar. Solid black lines are the full simulation, aproduct of Φ γ ( τ ) , Φ t t ( τ ) , and Φ v ( τ ) At high pressures, the computed correlation function approaches thered dashed line, which neglects all velocity changing elastic collisions. At low pressures, the computedcorrelation function approaches the blue dot-dashed line, representing the limit where all elastic collisionscontribute to the effective damping. where the present acetylene measurements were made, the ensemble average of the elasticcollisions results in nearly full dephasing at the total collision rate, γ e + γ . This is consistentwith good fits to a Voigt line shape, and a linear pressure dependence of the Lorentzianbroadening, as observed experimentally. The slope of HWHM vs. pressure in this region isthe total collision rate/2 π . The line shapes begin to deviate from Voigt forms by 0.1 mbar as13he increasingly fast inelastic damping rate begins to compete with the dephasing betweendifferent velocity groups. Eventually, at high enough pressures, the velocity changing colli-sions have no further incremental effect on the pressure broadening. Our measurement of asub-Doppler pressure broadening with a slope 3.5 times the value obtained by conventionalspectroscopy in Doppler-broadened gaseous samples is consistent with an elastic collisionrate 2.5 times larger than the inelastic contribution. The linearity of the Lorentzian fittingparameter up to 0.05 mbar is consistent with a typical Doppler shift per elastic collision ofat least 2 MHz, according to this simple model. IV. DISCUSSION
Using saturation dip laser absorption spectroscopy, we have frequency measured the po-sitions of a number of rotational lines in the v + v combination band of acetylene. For thestronger transitions, we were able to determine the effect of self-collisional broadening of theline profiles as the sample pressure was increased. The broadening coefficients determinedin the present experiment, see Table I, are all 3 . − × those previously reported in conven-tional, Doppler-broadened spectroscopic measurements of the same transitions. To accountfor and understand this observation, we have developed a sub-Doppler soft collision modelwhich takes into account the effect of large cross-section, VCCs in the sample. The corre-lation function Φ v ( τ ), Equ. (5), actually accounts for several types of velocity-dependenteffects in line shapes. In the absence of VCCs at low pressure it gives the normal GaussianDoppler broadening when averaged over the thermal velocity distribution. Including theeffects of VCCs modifies this ensemble average such that extreme velocity groups in the en-semble tend to become more similar with collisions and slow down dephasing. This is a wayof describing Dicke narrowing. In the sub-Doppler case, the initial ensemble of interestis velocity-selected and VCCs broaden the velocity distribution and cause a correspondingline broadening. The particular model detailed here with a symmetric small-angle scatteringkernel would not show Dicke narrowing however, as this requires significant excursions invelocity space for each molecule prior to inelastic scattering and a kernel with asymmetricprobabilities for decreasing and increasing velocity changes in collisions.Other reported measurements of molecular sub-Doppler lines have also found 2-4 × greater pressure broadening coefficients compared to measurements of Doppler-broadened14ines. The most detailed studies of the effects of VCCs on saturated absorp-tion profiles have been in atoms such as Xe, and Rb with He, Ne, Ar and Xe collisionpartners. The extremely strong atomic transitions permit measurements over a wide rangeof pressures. These studies show the same qualitative behavior as depicted in our model,although pressure-broadening in them is primarily driven by phase interrupting atomiccollisions, rather than the rotationally inelastic collisions that dominate molecular systems.At low pressures, the narrow sub-Doppler saturated resonance is observed to have a largerLorentzian broadening component than calculated, based on the pressure broadening coef-ficient measured in conventional absorption studies. Line widths are also found to increasewith a non-linear pressure dependence: linear at low pressures followed by a region ofdownward curvature, approaching a high pressure linear region with a slope near the valuemeasured for the Doppler-broadened gas-phase absorption data.Similar results were seen in methane in early measurements of the 3.39 µ m lasing tran-sition. The direct effect of VCCs with increasing pressure can also be seen in the formationof a broad pedestal under the narrow saturation feature under appropriate conditions. Mea-surements of vibration-rotation transitions in CO found a γ self for the saturation featuresthat was 2.7 × larger than the Doppler-broadened quantity. A similar result for CO wasreported earlier by Meyer et al. and incorrectly attributed to a combination of transit timebroadening and saturation effects.Recent measurements of the self-broadening of sub-Doppler spectra of CH F in the 3 µ m region are of interest to compare with the present results. Okabayashi et al. report γ self =9.6(1.3) MHz/Torr for the r Q (7 ,
4) transition, converted to the HWHM quantity, whichcompares to a value of 10.7(0.5) previously reported by Cartlidge and Butcher from linearspectroscopy in the Doppler-limited regime. Published data for ammonia also find nosignificant differences between the self-broadening coefficients for Doppler or sub-Dopplerbroadening. This implies that, for these molecules, the effects of VCCs in the saturation dipexperiment are minimal, contrary to the case of acetylene, methane, CO or ethylene, above.The obvious difference is the presence of the dipole moment in CH F and NH , causing evenlong-range collisions to be dominated by dipole-dipole interactions more effective at causingstate- or phase-changing transitions than in molecules or atoms without a dipole moment.These observations are in accord with measurements and modeling of the decay of co-herent transients reported by Berman et al. Their work investigated the effects of velocity15hanging collisions by measurement of photon echo signals following rapid Stark switchingof the molecular transition frequency out of resonance with the probe laser frequency. Theconclusions were that, in CH F, the decay of a Doppler hole created by a narrow band laseris dominated by hard, inelastic, collisions. Further, the measurements were interpreted toshow that phase-changing collisions in the absence of state-changing ones, were also of lowprobability, so that the T and T lifetimes were the same. Much more recent work byRubtsova et al. confirmed this result. Stark-shifting measurements are only possible onmolecules possessing a dipole moment and degenerate or nearly degenerate levels of oppo-site parity, so cannot be applied to acetylene, for example, where the present results suggestVCCs play a much more significant role.In related work by the Schwendeman group, collisions in CH F were probed by elegantsub-Doppler IR-IR double resonance methods. They observed collisional transfer of velocityspikes characteristic of resonant rotational energy transfer. After such a collisional rotationalenergy exchange, a new population of a nearby rotational state is formed with the narrowvelocity distribution of the initially pumped one, and a corresponding component of theinitially tagged level appears to be fully thermalized translationally in a single collision. Thisspecial type of collision between indistinguishable particles may of course also be considered,rather than the complete translational thermalization of the tagged quantum state, as asmall-angle soft collision, but with an exchange of rotational state labels. In either view,these special energy-elastic self-collisions will contribute to pressure broadening, despitehaving no observable effect on the energy transfer rates. It is the relative importance ofthese collisions that is at issue in the current interest in possible ortho/para differences inthe pressure broadening of acetylene.
It would be interesting to measure coherent transients in acetylene using a rapidfrequency-switching, or transient double resonance, experiment to compare with the presentinterpretation of the sub-Doppler saturation dip data. Precise understanding of the variouscontributions to sub-Doppler line profiles are crucial to the interpretation of spectroscopicexperiments that attempt to test fundamental theory while information on the rate ofVCCs is required for the modeling precise experimental measurements of pressure-broadenedlines recorded under Doppler-broadened conditions relevant to remote sensing and Dopplerthermometry. . SUPPLEMENTARY MATERIAL Supplementary material consisting of a) compressed files containing LabView c (cid:13) exe-cutable code and source code libraries to permit readers to implement the model for com-puting the effects of velocity-changing collisions on sub-Doppler lineshapes described in thispaper and b) a pdf document showing the effect of a larger width Gaussian transit-timebroadening on the fits to a Voigt profile, may be obtained from the publisher.
VI. ACKNOWLEDGEMENTS
We gratefully acknowledge helpful discussions with Dr Patrick Dupr´e (Universit´e LittoralCˆote d’Opal) and Dr Benoit Darqui´e, (Universit´e Paris-13). Work at Brookhaven NationalLaboratory was carried out under Contract No. DE-SC0012704 with the U.S. Departmentof Energy, Office of Science, and supported by its Division of Chemical Sciences, Geosciencesand Biosciences within the Office of Basic Energy Sciences.
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