Coherent Rayleigh-Brillouin Scattering in gases in the kinetic regime

Xingguo Pan, Mikhail N. Shneider, and Richard B. Miles

Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544

(Dated: June 19, 2003)

We present an experimental and theoretical study of the coherent Rayleigh-Brillouin scattering ingases in the kinetic regime. Gas density perturbation waves were generated by two crossing pumplaser beams through optical dipole forces. A probe laser beam was then coherently scattered fromthe perturbation waves. The line shape of the scattered light was modeled using kinetic theory. Themodel takes into account the internal energy modes of the gas particles and is applicable to bothmolecular and atomic gases. We discuss the implication of coherent Rayleigh-Brillouin scatteringon kinetic theory and photon matter interaction.

PACS numbers: 42.65.Es, 33.20.Fb, 42.50.Vk, 51.10.+y

INTRODUCTION

Nonlinear light scattering processes at nonresonant fre-quencies have been widely studied in liquids and in gases.In such a process, the medium is first perturbed by cross-ing laser beams, and a probe beam is then scattered fromthe perturbed medium. Two examples in liquids are Bril-louin enchanced four wave mixing [1] and the observationof the complex Brillouin spectrum [2]. In gases, the prob-lems can be treated in two regimes: the hydrodynamicsregime and the kinetic regime. They are defined usingthe y parameter, which represents the ratio between thescattering wavelength and the gas particles mean freepath. When y is large, the gas can be treated as a con-tinuum and the scattering process can be modeled usingthe Navier-Stokes equations coupled with the Maxwellequations. The kinetic regime is defined where the scat-tering wavelength is comparable or smaller than the meanfree path, with 0 y . 5. In this regime, kinetic equa-tions and the Maxwell equations are generally used tostudy the scattering problem. She et al. [3, 4] first ob-served the stimulated Rayleigh-Brillouin scattering in ar-gon and SF6. The differential detection technique theyused can only partially resolve the Rayleigh peak. It isinteresting that She et al. envisioned coherent Rayleigh-Brillouin scattering (CRBS) as it is performed today inthis paper. In the last decade, the laser induced thermalacoustics (LITA) and laser induced electrostrictive grat-ings (LIEG) have been studied [57]. These techniqueshave been widely used in optical diagnostics. The theo-retical framework of above research efforts are generallybased on the Navier-Stokes equations and the Maxwellequations. In the collisionless limit of the kinetic regime,Grinstead and Barker [8] observed the line shape of co-herent Rayleigh scattering. A theoretical line shape wasobtained by solving a collisionless kinetic equation in thespace time domain. In a previous Letter [9], we reportedcoherent Rayleigh-Brillouin scattering (CRBS) in gasesin the kinetic regime. Data in argon and krypton anda single parameter kinetic model were presented. CRBScontains coherent Rayleigh scattering and can be con-

nected to phenomena in the hydrodynamic regime suchas LIEG.

Previous works on spontaneous Rayleigh-Brillouinscattering provide a substantial foundation for the cur-rent work. In the kinetic regime, Yip and Nelkin [10] firstdeveloped a kinetic model with the linearized Bhatnagar-Gross-Krook collision term [11]. Boley et al. [12] andTenti et al. [13] started from the linearized Wang-Chang-Ulenbeck equation [14] and developed models for molecu-lar gases. Their six moment model, popularly called thes6 model, has been considered the best model by sev-eral authors [1517]. The modeling of the spontaneousRayleigh-Brillouin line shape in the kinetic regime con-tinues to attract interests. Recently, Marques Jr. andcoworkers presented extended kinetic models [18, 19]and reported good agreement with experimental data.Rangel-Huerta and Velasco [20] extended the hydrody-namic equations into the kinetic regime to study theRayleigh-Brillouin line shape in monotomic gases. Mostof these works can be extended to study nonlinear lightscattering processes at nonresonant frequencies by incor-porating the optical dipole force into the model.

In this paper, we present experimental data in N2, O2,and CO2. The details of the experiment are described.We also present a theoretical model of the CRBS lineshape that takes into acount the internal energy modesof the gas particles. This model is based on Wang-Chang-Ulenbeck equation and follows the work of [12, 13].

OVERVIEW

The physical process of CRBS is illustrated in Fig. 1.Two pump beams, both polarized perpendicular to thepage, are focused and crossed at their foci. They forman interference pattern and generate a wavelike densityperturbation field in the gas. A probe beam is then co-herently scattered from the perturbation and forms theCRBS signal, which is also a beam.

It is through the optical dipole force that the pumpbeams generate the gas density perturbation. As dis-

2

probe

pump 2pump 1

signal : = 1 2 k = k1 - k2

FIG. 1: Coherent Rayleigh-Brillouin scattering in gases.

cussed by Boyd [21], a molecule with polarizability in an electric field E obtains a potential energy U = 12|E|2. If the electric field is inhomogeneous, the ten-dency for the system to approach minimum potential en-ergy manifests itself as the optical dipole force on themolecule F = U . Therefore, a molecule is pushedby the optical dipole force to the region where the fieldintensity is stronger.

The generated gas density perturbation has a coher-ent wave structure following the pump beams interfer-ence pattern, as illustrated in FIG. (1). When the probebeam, counterpropagated against pump beam 2, is shoneupon the density wave, it will be coherently scattered.The signal beam maintains the probe beams polariza-tion and follows a path determined by the phase match-ing condition,

ksignal kprobe = k = k1 k2 , (1)

where ksignal and kprobe are the wave vectors of the signaland the probe beams, respectively; k is the gas densitywave vector; k1 and k2 are the wave vectors of the pumpbeams. The phase matching condition states the mo-mentum conservation in this process. The process alsomaintains energy conservation, which is,

signal probe = = 1 2 , (2)

where 1 and 2 represent the frequencies of two singlemode pump beams. The frequency of the CRBS signalsignal is shifted from that of the probe beam signal bythe frequency of the density perturbation wave . In thiswork, the gas density perturbation wave has a wavelengthof about 256 nm and the frequency spans about 6 GHz.

The intensity of light scattered by a collection of scat-terers is derived from the Maxwell equations [22]. Thescattering results from the dielectric constant fluctua-tions of the medium, and the intensity of the scatteredlight is proportional to 2. In fluids, the dielectric con-stant is generally a function of density and temperature[23]. In dilute gases, its dependency on temperature canbe ignored and the dielectric constant fluctuation is pro-portional to gas density fluctuation. Thus, we have theintensity of the scattered light,

Isc 2Iinc ,

where is the gas density fluctuation, and Iinc is theincident light intensity.

Based on the results in [9], one finds that 2 is pro-portional to the product of the pump lasers intensity inCRBS. Therefore, the signal intensity in CRBS is

Icrbs Ipump1Ipump2Iprobe , (3)

which is a general formula for a third order nonlinearoptical process.

For an incident light whose power spectrum is a func-tion, the power spectrum of the scattered light S(k, )is proportional to the spectral density of the dielectricconstant fluctuations S(k, ) [23]

S(k, ) S(k, ) = F{< (k, 0)(k, t) >} , (4)

where the sign F{} denotes Fourier transform and thetime-correlation of the dielectric constant fluctuation isdefined as

< (k, 0)(k, t) >=

(k, )(k, + t)d .

We will look for a steady state solution to our problem.For such a solution, in which the direction of time doesnot play a role, the time correlation function equals itstime convolution

(k, t) (k, t) =

(k, )(k, t )d ,

Using the convolution theorem of Fourier transform onEqn. (4) and the relation , we conclude that thecoherent Rayleigh-Brillouin scattering power spectrumS(k, ) is related to the gas density perturbation by

S(k, ) (k, )(k, ) , (5)

where

(k, ) =1

2

ei(krt)(r, t)d3rdt (6)

is the space-time Fourier transform of the gas densityperturbation. Our task to find the power spectrum ofthe scattered light then becomes finding the gas densityperturbation.

EXPERIMENT

The experimental setup is shown in Figure 2. AQ-switched, frequency doubled, broad band, pulsedNd:YAG laser was used to produce the two pump beams.The laser beam was first passed through a half-wave plateand a Glan polarizer. The polarizer was used to setthe polarization of the pump beams. Turning the halfwave plate allows one to control the beam power passing

3

Tunable Narrow Band Q-switched Nd:YAG Laser

Broad Band Q-switched Nd:YAG Laser

etalonPD 1

PD 2

gas cell

50/50 beam splitter

TFP pump 1pump 2

probe

signal

/2 plate

Glan polarizer

monitor etalon

FIG. 2: The experimental setup of coherent Rayleigh-Brillouin scattering in gases.

through the polarizer. In this way, the beams pulse en-ergy could be controlled while the pulses temporal profilewas maintained.

The pump laser beam was then split into two by a50/50 beam splitter. The two pump beams were directedinto a gas cell from opposite ends. They were focused bytwo plano-convex lens and were crossed at their foci in thegas cell. The path lengths of the two beams were carefullymatched so that the two pulses reached the interactionvolume simultaneously. The pump beam pulse had aduration of about 10 ns and the pulse energy directedinto the interaction region was around 6 mJ/pulse foreach beam.

The pump lasers power spectrum was measured usinga plane-parallel Fabry-Perot etalon to have a full widthat half maximum (FWHM) of 24.8 2.5 GHz. Therewas also a 250 MHz longitudinal mode structure in thepump lasers power spectrum, which matched the laserscavity length. This mode structure manifested itself inthe raw profile of the scattered light. The pump beamsbroad band power spectrum indicated that the gas den-sity waves generated by them also had a broad bandpower spectrum, i.e, a continuous superposition of sinu-soidal wave modes.

The probe laser was an injection seeded, frequencydoubled, pulsed Nd:YAG laser. Its pulse duration was7 ns. The laser was operated in a single longitudinalmode with a 150 MHz band width.

The probe beam was also passed through a half-waveplate and a Glan polarizer, so that the beams powercould be controlled while its temporal profile was main-tained. The probe beams polarization was set perpen-

dicular to that of the pump beams. Therefore, the probebeam would not form an interference pattern with thepump beams, and complexities associated with this wereavoided. In addition, since we used a counter propagatingscheme for pump beam 2 and the probe beam, a perpen-dicular polarization prevented the probe beam from theentering pump laser and vice versa.

The probe beam was then directed and focused ontothe pump beams crossing volume in the gas cell. Two500 mm focal-length plano-convex lenses were used to fo-cus the beams. The pump beams passed off-axis throughthe lenses and intersected at their foci at a 178 angle.The diameter of the focal region was about 200 m. Theprobe beam was counterpropagated against pump beam2. The arrival time of the probe pulse was adjusted rela-tive to that of the pump pulses so that maximum signalwas obtained. The optimal delay was about 1 ns. Thejitter of the two lasers was each about 2 ns.

In the experiment, the overlap of the probe beam andpump beam 2 was achieved by aligning them so that bothpassed through two appertures located at different placesalong the path. Pump beam 1 was then fine tuned so thata strong signal was obtained. When the room was dark,the signal generated by a 1 mJ/pulse probe beam and6 mJ/pulse pump beams in room air could be clearlyseen by eye on a white card. The measured signal beamintensity at these conditions was less than 5 J. To avoidthe complexity of gas density fluctuations generated bythe probe and signal beams interaction, we reduced thepulse energy of the probe beam to minimum level in theexperiment.

The gas cell was a 5 cm diameter and 0.8 m long stain-less steel tube with windows on both ends. Before eachscan, the cell was evacuated and purged with the work-ing gas several times before being charged to the desiredpressure.

By the phase matching condition, the signal beam fol-lowed, in reverse, the path of pump beam 1. A thinfilm polarizer was used to separate the signal beam frompump beam 1. The signal beam was then propagated for7 meters in the lab to be detected. At this distance, thebackground ambient scattering of laser light from opti-cal elements was greatly attenuated. Still, mirrors andlenses were carefully turned to keep surface refections oflaser beams away from the signal beams direction.

The detection system was composed of two photodec-tectors and an air spaced plane-parallel Fabry-Perotetalon. The inward facing surfaces of the two etalon mir-rors had a 99.6% reflectivity at 532 nm. As calibrated bythe manufacturer, the mirror set yielded a finesse of 215at a test wavelength of 543 nm. One of the etalon mirrorswas mounted on a piezoelectric mount. The three piezostacks were independently controlled by a ramp genera-tor. The other mirror was mounted on a high precisionhand adjustable mount. The entire etalon was put inan enclosure whose temperature was stabilized to within

4

0.1K.In the laboratory, the etalon was aligned using a HeNe

laser at 632.8 nm. The HeNe beam was collimated andexpanded to a 20 mm diameter. Coarse alignment wasdone by adjusting the high precision hand adjustablemounts. Fine alignment was performed using the rampgenerator. The etalon mirror sparation could be adjustedto fit the needs of the measurement task. For most scansof CRBS data in this paper, the free spectral range wasset at 11.85 GHz.

The photodectectors were photomultiplier tubes(PMT). About 4% of the signal beam was directed toa photodetector for normalizing purpose. We call it thenormalizing PMT, and it corresponds to PD 2 in Fig. 2.The majority of the signal was sent into the etalon andthe passed light was received by another PMT. This PMTcorresponds to PD 1 in Fig. 2, and will be called the signalPMT. The output from both PMTs were sent to a gatedaverager and the data were collected by a computer.

In the experiment, the probe lasers frequency wasscanned by scanning the voltage applied to the seed laser.For each frequency step, the ratio of the signal intensitydetected by the signal PMT (the one after the etalon) tothat by the normalizing PMT made a data point of theprofile. Thirty shots were averaged for each frequencystep.

The instrument function of the system had a full widthat half maximum of 150 MHz. It was measured by send-ing the probe laser beam directly to the detector systemand scanning the laser frequency.

Two approaches may be used to resolve the power spec-trum of the signal beam. In the first approach, the etalonis scanned while fixing the probe lasers frequency. Toscan the etalon, one changes the mirror separation byvarying the electric voltage applied onto the piezo-electricstacks on which one of the mirrors is mounted. An alter-native approach is to scan the probe laser while fixing theetalon mirror separation. This is the approach used formost scans in this paper, mainly because the probe lasercould be scanned at a higher resolution. Both approachesyield the same profile, which is a convolution of the sig-nal beams power spectrum and the etalons transmissionfunction.

As shown in Figure 2, during each scan, part of theprobe beam was directed to a monitor etalon. This mon-itor etalon was also an air spaced plane-parallel Fabry-Perot etalon, and its free spectral range was set to be900.0 0.2 MHz. When the probe laser frequency wasscanned, there is a passband at every 900 MHz. Thesedata were used to monitor the linearity of the frequencyscan and to convert the control voltage applied on theseed laser to the probe beams freqency shift.

The voltage applied to the seed laser V was convertedto the frequency difference using the monitor etalontransmission profile of the probe beam. A programwas written to identify the center of each passband. A

3 2 1 0 1 2 30

0.5

1

1.5

Frequency difference (GHz)

Rel

ativ

e in

tens

ity (

arb.

uni

t)

0 5 10 150

100

200

300

400

500

Reciprocal frequency (GHz1)Raw

dat

a po

wer

spe

ctru

m (

arb.

u.)

pump laser mode structure

FIG. 3: (a). A raw CRBS data is shown by the upper curve,offset vertically by 0.5. The lower curve is the data after themode structure is filtered. (b). The power spectrum of theraw experimental data.

quadratic relation between the frequency difference andthe control voltage, = 0 + aV + bV

2, was fitted to thedata. In general, the coefficient of the linear term a ismuch larger than the quadratic coefficient b, indicatinggood linearity of the scan. Still, a quadratic fitting of(V ) is necessary to take care of the long term driftingof the frequency tuning.

The raw data of the coherent Rayleigh-Brillouin scat-tering profile is shown by the green curve in Fig. 3(a).A periodic, small amplitude modulation is clearly no-ticeable. It is due to the longitudinal mode structure ofthe pump laser. This mode structure can be seen moreclearly in the datas power spectrum, shown in Fig. 3(b).The pump lasers mode structure is shown as the isolated,well defined peak around 4.0 GHz1. It shows that thepump lasers longitudinal modes are 250 MHz apart, avalue that matched the pump lasers cavity length.

In the following section, we will compare the theoret-ical model and the experimental data for a pump forcefield that is constant in frequency. To accomplish this,we need to filter out the pump lasers mode structurefrom the raw data.

From the results of [9], and as will be confirmed laterin this paper, the power spectrum of the scattering signalcan be written as

Sraw() = F ()`crbs() , (7)

where is a constant coefficient, F () is related to thepower spectrum of the dipole force field, and `crbs()is the coherent Rayleigh-Brillouin scattering line shapewith a force field that is constant in the frequency do-main. Ideally, one would like to have a force field uniformin frequency, i.e., F () = const. In our experiment, theforce can be considered as a superposition of two com-ponents: one is a broad Gaussian profile with 24.8 GHzFWHM; the other is a periodic component representingthe 250 MHz mode structure.

With a 24.8 GHz FWHM of the pump beam, the slowly

5

changing component of the force varies less than 2% overthe 6 GHz range of interest. We will therefore treat it asa constant over this range. We decompose the force as

F () = F0 + F1() , (8)

where F0 is a constant and F1() represents the modestructure.

We apply Fourier transform to Eqn. (7),

F{Sraw()}=F{(

F0 + F1())

`crbs()}

= F0F {`crbs()}+F{

F1()}

F {`crbs()} . (9)

The convolution term F{F1()} F{`crbs()} is wellseparated from the first term in the Fourier transform ofthe data, and is filtered out numerically. The filtered datais shown in Fig. 3(a) by the blue curve. The profile is aconvolution of the coherent Rayleigh-Brillouin scatteringline shape and the instrument function, and it will becompared with the theoretical line shape.

We have so far obtained data in argon, krypton, nitro-gen, oxygen, and carbon dioxide. The argon and kryptondata have been reported in [9, 24]. The nitrogen, oxygenand carbon dioxide data are shown in Figures 4, 5, and7 with the blue curve. The data present common char-acteristics. From y 0 to y & 1, the profile evolves froma near Gaussian curve to a curve with three distinctivepeaks. For the line shape with a large y parameter, thecentral, unshifted, peak is due to the non-propagatingentropy fluctuations, and can be identified as the coher-ent Rayleigh peak. The position of the two side peakscorresponds to the speed of sound in the gas. The sidebands are due to the coherent sound waves generated bythe optical dipole force field, and can be identified as thecoherent Brillouin peaks. At larger y, the intensity ratiobetween the two Brillouin peaks and the Rayleigh peakkeeps growing. This is different from the spontaneousRayleigh-Brillouin scattering, where the ratio tends to alimit cv/(cp cv) given by Landau and Placzek [25].

A MODEL FOR COHERENT

RAYLEIGH-BRILLOUIN SCATTERING IN

MOLECULAR GASES

In [9], we presented a model for CRBS line shape ofatomic gases. The perturbations on mass, momentum,and energy were solved from a linearized kinetic equa-tion. The source of the perturbations is the optical dipoleforce field and the perturbations relax through collisions.The CRBS line shape of atomic gases was obtained an-alytically. It is found that the line shape in differentgases scales with the y parameter and the match be-tween the model and the data is satisfactory. A step-by-step derivation of the model can be found in Chap-ter 3 of [26]. In this paper, we discuss a model of co-herent Rayleigh-Brillouin scattering for molecular gases.

We compare it with experimental data in nitrogen, oxy-gen, and carbon dioxide. Supressing the internal energymodes, this model can be used for atomic gases as well.

The important difference between atomic gases andmolecular gases is in a collision event. In an atomicgas, we only need to consider translational energy con-servation. In a molecular gas, rotational and vibrationalenergy also need to be considered. We collectively callthem internal energy. A collision event can be elastic,in which the total translational energy of the collidingparticles is conserved, or it can be inelastic, in whichtranslational energy is transferred into internal energy,or the other way around. We therefore need to find a ki-netic equation that properly takes the internal energylevels into account. Wang-Chang, Uhlenbeck and DeBoer first proposed such a kinetic equation [14], whichwas soon successfully applied to problems such as trans-port coefficients [27, 28], sound dispersion [29] and lightscattering [12] in molecular gases. Following [14, 2830], Boley et al. [12] presented an eigentheory for the lin-earized Wang-Chang Uhlenbeck equation. A model col-lision term was constructed using the eigentheory. Theyobtained a model for the spontaneous Rayleigh-Brillouinscattering line shape.

Our model for coherent Rayleigh-Brillouin scatteringin molecular gases is an extension of the spontaneousRayleigh-Brillouin scattering model by Boley et al. [12].We add to the equation a source term due to the op-tical dipole force field and solve for the steady statesolution. The solution procedure greatly resembles theatomic CRBS model presented in [9]. We will first lin-earize the WCU equation and use the model collisionterm. By taking the inner product of the linearized WCUequation and the eigenfunctions of the elastic collision op-erator, we obtain a linear equation set for the perturbedquantities (e.g., density, flow velocity, and temperature)in the Fourier domain. The perturbed quantities are thensolved from this linear equation set. The line shape ofthe coherent Rayleigh-Brillouin scattering can thereforebe calculated from the density perturbation.

We consider a gas whose molecular mass is M at tem-perature T and has N internal energy levels. The dis-tribution function of the ith energy level is denoted byfi(v, r, t). In equilibrium, fi follows the Boltzmann dis-tribution, i.e,

fi, eq(v, r, t) = n0xi(v) , (10)

where n0 is the average number denisty, xi is the fractionof molecules with internal energy Ei at the equilibriumstate,

xi =gie

Ei/kbT

j gjeEj/kbT

, (11)

where gi is the degeneracy of state i. (v) is the

6

Maxwellian distribution

(v) = (1

v20)3/2 exp(v

2

v20) , (12)

where

v0 =

2kbT/M . (13)

For such a gas, the full WCU equation is

(

t+ v + a v)fi(v, r, t) =

jkl

(f kfl fifj)|v v1|klij dd3v1 , (14)

This is a set of equations that describe the dynamics ofthe distribution function of each internal state. In thispaper, we assume that the acceleration a of each moleculedue to the optical dipole force does not depend on theinternal state. The collision integral on the right handside describes a collision between two particles. One isin the ith state before the collision, and has velocity v.It transits to the kth state after the collision. The sec-ond particle is in the jth state before the collision andhas velocity v1. It transits to the lth state after the colli-sion. The collision cross section klij (v,v1;v

,v1) dependson the internal energy levels involved in the collision, aswell as the relative velocity of the two particles. Sucha collision term includes the effects of elastic collisions,where k = i and l = j, as well as inelastic collisions.

The derivation of WCU equation follows the same ar-guments used in the derivation of Boltzmann equation.It is based on the assumptions that only binary collisionsare important and that previous collisions do not influ-ence subsequent collision probabilities (molecular chaos).As a result, the WCU equation satisfies the conseravtionsof mass, momentum, and total energy. It also satisfies theH-theorem. In addition, the principle of detailed balanc-ing is assumed in the WCU equation. A possible weak-ness of the WCU equation is the assumption that thereis no degeneracy in the internal energy levels. This setsgi = 1 in Eqn. (11). The spatial degeneracy of rotationalstates (due to the orientation of the rotor) seems to posea serious limit, however, for scalar problems such as den-sity perturbation, the average effect of many collisionsof the same kind is spherically symmetric. In this sense,the degeneracy of rotational states does not compromisethe usefulness of WCU equations in scalar problems, asshown by its many successful applications.

The full, nonlinear, WCU equation is hard to handle.We now linearize it and use a model collision term toreplace the collision integral. The material from here toEqn. (36) closely follows Boley et al. [12]. Notice that ourdefinition of v0 and several eigen functions are different.We write the distribution function of the ith internal levelwith a linearized deviation from the overall equalibrium

as

fi(v, r, t) = n0xi(v)[1 + hi(v, r, t)] , (15)

where the deviation hi(v, r, t) is dimensionless and xiand (v) have been given above.

The perturbation quantities that we are interested incan be expressed by the distribution function. Thesequantities include the deviation from equilibrium of thenumber densities of each internal state,

n0xii(r, t) = n0xi

(v)hi(v, r, t)d3v , (16)

the deviation of the total number density,

n0(r, t) = n0

i

xi

hid3v , (17)

the macroscopic flow velocity,

v0u(r, t) =

i

xi

vhid3v , (18)

the deviation of the translational temperature,

T0tr(r, t) =

i

xi

(

Mv2

3kb T0

)

hid3v , (19)

and the deviation of the internal temperature,

T0int(r, t) =1

kbcint

i

xi

(Ei E)hid3v , (20)

where the angle bracket means average of all internalstates, i.e.,

E =

i

xiEi , (21)

and the dimensionless internal specific heat capacity cintis defined as

cint =1

kb

dEdT0

=

i xi(Ei E)2(kbT0)2

. (22)

The deviation in total temperature T0 is then equal tothe deviation of local average energy divided by (3/2 +cint)kb,

T0(r, t) =T0

3/2 + cint(3

2tr + cintint) , (23)

The translation and internal heat flux are given by, re-spectively,

n0kbT0v0qtr = n0

i

xi

(Mv2

2 5kbT0

2)vhid

3v ,(24)

n0kbT0v0qint = n0

i

xi

(Ei E)vhid3v . (25)

7

In addition, the deviation of the traceless pressure tensorfrom equilibrium is given by

n0kbT0 = n0M

i

xi

(vv 1

3v

2)hid3v .

(26)Assuming the perturbation is small, we plug Eqn. (15)

into Eqn. (14) to obtain the linearized WCU equation

( t +v )hi 2

v20a v =

n0

jkl

xj

d3v1d(v1)|v v1|

klij [hk(v) hl(v1) hi(v) hj(v1)] . (27)

We have used the total energy conservation relation in-side the collision integral and ignored higher order smallterms O(h2i ). We have also applied the small perturba-tion approximation to the force term, which is,

1

(v)a v[(v)(1 + hi)]

1

(v)a v(v) =

2

v20a v .(28)

It is convenient to work on the problem in a Hilbertspace. We choose to use the Dirac notations and writethe column vector hi as

|h =

h1h2...

, (29)

The Hermitian conjugate of |hi is then a row vector

hi| = (h1, h2, . . . , hn) .

The Hilbert space is based on the following definition ofthe inner product of vectors h| and |h,

h|h

i

xi

(v)hi (v)hi(v)d

3v . (30)

Using Dirac notation, Eqn. (27) can be written as

(

t+ v )|h 2

v20a v = n0J |h . (31)

The collision operator J is a N N matrix, and eachelement of this matrix is an integral operator. One canseparate the operator J into its elastic and inelastic partsJ = J +J . The elastic collision operator J comes fromthe terms in Eqn. (31) that have k = i and l = j, and theinelastic operator consists of all other terms. Argumentbased on the symmetry of the elastic collision processleads to the conclusion that J is a hermitian operatorwith

h|J |h = h|J |h ,

and the H-theorem requires that

h|J |h 0, h|J |h 0 .

Wang-Chang and Uhlenbeck have found the eigenvec-tors and eigenvalues for the elastic collision operator J

for molecules that interact by the Maxwell force lawFM = /r

5 , where r is the separation between the col-liding molecules and the force constant is the same forall colliding pairs. In this work, we assume that the gasmolecules are Maxwellian molecules. The Maxwell forcelaw is spherically symmetric, but molecules in generalare not spherically symmetric. This seemingly contra-dictary assumption is again justified by the fact that theaverage cross section of numerous collisions is sphericallysymmetric.

The eigen theory of J has been clearly explained byBoley et al. [12]. The reader is refered to [12, 26] fordetail. The eigen vectors are denoted by |rlm,n. Theyare orthogonal to one another, i.e.,

rlm,n|rlm,n = rrllmmnn . (32)

The eigenvectors rlm,n span the entire Hilbert space,giving the completeness condition,

rlmn

|rlm,nrlm,n| = 1 . (33)

Explicitely, the eigenvector rlm,n is given by

rlm,n =

rlmPn(1)rlmPn(2)

...

, (34)

where rlms are the eigenfunctions of the collision oper-ator for a atomic gas with a Maxwell force law, or a gaswith only one internal energy state. Pn(i) is a polyno-mial of the dimensionless internal energy i = Ei/kbT0.The first two Pns are given by

P0(i) = 1 , P1(i) =i

( )2.

Using the dimensionless velocity c = v/v0, the func-tion rlm is written explicitly as,

rlm =

23/2r!

(r + l + 1/2)!S

(r)l+1/2(c

2)clYlm(c) , (35)

where, S(m)n (x) is the Sonine polynomial and Ylm(c) is

the spherical harmonics [31] given on the direction of thedimensionless velocity c.

The first several rlms, which we will use, are

000 = 1 , 010 =2cz ,

100 =

2

3(3

2 c2) , 110 =

25(5

2 c2)cz ,

020 =13(3c2z c2) ,

8

where the dimensionless velocity along z-axis is cz =c cos . The eigenvectors of J that we will be using in-clude, with the subscripts denoting (rl, n), 00,0, 00,1, , 00,(N1) , 01,0, 10,0, 11,0, 01,1, 02,0. Theseterms have clear physical meanings. The first N eigen-vectors denotes the fraction of particles in a specific in-ternal state. 01,0 is the momentum, 10,0 is the trans-lational energy term, 11,0 corresponds to translationalheat flux, 00,1 is the internal energy, 01,1 is the internalheat flux, and 02,0 is the traceless pressure tensor. Onlyvery limited knowledge of the eigen theory of the inelastic

collision operator J is available. The five known eigen-vectors all have an eigenvalue of zero. They are 000,0,

01m,0 and 100,0 +

2cint/3000,1, corresponding tothe quantities that are conserved in an inelastic collision,i.e., mass, momentum and total energy, respectively.

We use the collision model developed by Hanson andMorse [32] and Boley et al. [12]. It is constructed by themethod of Gross and Jackson [33]. The collision modelis the summation of the elastic and the inelastic models,given by

(J (7)|h)i =J 030i J 030 2c u + tr[

(J030 + J100)(c2 3

2) 3

2cintJ100(i )

]

+ int

[

J100(c2 3

2) + (

3

2cintJ100 J 030)(i )

]

+ 2c qtr[

(J110 J030)2

5(c2 5

2) J 110011

2

5cint(i )

]

+ 2c qint[

J 110011

2

5cint(c2 5

2) + (J011 J030)

i cint

]

+ (J020 J030)(cc 1

3c

2) + J030hi . (36)

where Jrlns, with or without primes and upperscripts,are related to the eigen values. They are related to gasdy-namics parameters through a Chapman-Enskog analysis.Details about Jrlns can be found in [12, 29].

Now we proceed to solve for the gas density perturba-tion from the linearized Wang-Chang-Uhlenbeck with amodel collision term

(

t+ v )|h 2

v20a v = n0J |h . (37)

We seek for a steady state wavelike perturbation alongthe z axis of the form exp[i(kz t)]. The z axis isperpendicular to the fringes shown in Fig. 1. It is par-allel to the direction of the optical dipole force and thedensity perturbation wave vector k. We have a v = avzand k v = kvz. We will first apply a space time Fouriertransform to Eqn. (37), and then take the inner prod-uct, as defined by Eqn. (30), of the transformed equationand the corresponding eigenvectors. We therefore obtaina system of seven linear equations of the physical quan-tities: (k, ), uz(k, ), qtr,z(k, ), zz(k, ), tr(k, ),int(k, ), qint,z(k, ).

The Fourier transform pair of gas density perturbationis

(z, t) =1

2

(k, ) exp [i(kz t)] dkd , (38)

(k, ) =

(z, t) exp [i(kz t)] dzdt . (39)

The same transform is applied to the other six perturbedquantities and the acceleration a. For simplicity, thesame symbol is used for each Fourier pair, but they havedifferent dimensions. For example, while (z, t) is di-mensionless, (k, ) is of dimension [Length Time]; andwhile a(z, t) is of dimension [Length Time2], a(k, ) isof dimension [Length2 Time1].

In the following, we use the dimensionless variables

=

kv0, y7 =

nJ030kv0

. (40)

Notice that, since J030 0, we have y7 0. It alsoproves convenient to define

I(, y7; cz) =1

+ iy7 cz. (41)

Taking Fourier transform of Eqn. (37) and multiplyingboth sides by I(, y7; cz), we obtain the kinetic equationfor |h in the frequency domain as

9

kv0in0

|h= I(, y7; cz){

J 030| J 030|000,0 J0302uz|010,0+

[

J100

3

2int + (J030 J100)

3

2tr

]

|100,0

+

[

J1003

2cint

(tr int) J 030cintint

]

|000,1+[

(J030 + J011)

2

cintqint,z J110011

25qtr,z

]

|010,1

+

[

(J110 J030)25qtr,z + J

110011

2

cintqint,z

]

|110,0+ (J020 J030)13zz|020,0+

2a

n0v0|010,0

}

. (42)

where, the gas perturbation vector | = col{1, 2, . . . }.The equation has been sorted using the eigenvectors ofthe elastic collision operator.

Taking inner product of Eqn. (42) and |000,0 =col (1, 1, ), we obtain the mass equation. Taking innerproduct of Eqn. (42) and |010,0 =

2czcol (1, 1, ),

we obtain the momentum equation. Similarly, one canwork out the other five equations. The result is shownbelow by the matrix equation Eqn. (43).

We have used the relation,

J100 =

2cint3

J001100 =

2cint3

J100001 =2cint3

J001 , (44)

to write the matrix elements A(6, 5), A(7, 5) and the sixthcolumn of A in a concise form. This relation comes fromthe fact that the collision operator is hermitian and thetotal energy is conserved in a collision.

It is interesting to compare our equation with Eqn. (43)of Boley et al. [12]. Our matrix A is essentially the sameas Boley et al.s. The sign difference in columns 3 and5 and element A(4, 4) is due to different choices of theeigenvectors 100, 110 and 020. Boleys choices re-sulted in a concise formula, while ours follow Eqn. (35)and preserve the orthornormalilty of the eigen vectors.As long as the calculation of the Ir

l

rl is consistent withones choice of the eigenfunctions, the results about theperturbation quantaties such as (, k) are the same.

Using our notation, the spontaneous Rayleigh-Brillouin scattering would have a B vector as

Bspt = 1

k2v0

I0000I0100I1100I0200I100000

. (45)

This will be used later, when we compare our calcu-lation of the spontaneous Rayleigh-Brillouin line shapewith Tenti et al.s.

In order to solve from Eqn. (43) the gas density per-turbation (, k) and other physical quantities of inter-est, we need to calculate Jr

ln

rln s and Irl

rl s. Jrln

rln s aredetermined by four gas parameters through a Chapman-Enskog analysis. This has been done by Hanson andMorse [32] and Boley et al. [12]. The gas parametersinclude the shear viscosity , bulk viscosity b, heat con-ductivity and the dimensionless internal specific heatcapacity cint as defined in Eqn. (22).

For computation purposes, it is convenient to use di-mensionless quantities: the dimensionless internal heatcapacity cint, the y paramter, the internal relaxation

10

AX

=B

,(43)

with,

A=

n0

kv 0

J

030I00

00

kv0

in0J

030I00

01

(J030

J110)I

00

11

(J020

J030)I

00

02

(J030

J100)I

00

10

J001

100I00

10

J110

011I00

11

J

030I01

00

J

030I01

01

kv0

in0(J

030

J110)I

01

11

(J020

J030)I

01

02

(J030

J100)I

01

10

J001

100I01

10

J110

011I01

11

J

030I11

00

J

030I11

01

(J030

J110)I

11

11+

kv0

in0(J

020

J030)I

11

02

(J030

J100)I

11

10

J001

100I11

10

J110

011I11

11

J

030I02

00

J

030I02

01

(J030

J110)I

02

11

(J020

J030)I

02

02

3k

v0

i2n0(J

030

J100)I

02

10

J001

100I02

10

J110

011I02

11

J

030I10

00

J

030I10

01

(J030

J110)I

10

11

(J020

J030)I

10

02

(J030

J100)I

10

10+

kv0

in0J

001

100I10

10

J110

011I10

11

00

J

110

011I00

01

0J

001

100I00

00

(J001

J030)I

00

00

kv0

in0(J

011

J030)I

00

01

00

J

110

011I01

01

0J

001

100I01

00

(J001

J030)I

01

00

(J011

J030)I

01

01

kv0

in0

,

X=

2u

z

(2/5)q

tr,z

(1/3)

zz

3/2 t

rc i

nt i

nt

(2/c i

nt)q

int,z

,B=

2a

kv2 0

I00

01

I01

01

I11

01

I02

01

I10

01 0 0

.

number Rint and the Eucken factor fu. The latter threeare given by

y =n0kbT0kv0

, Rint =3b2

(3/2 + cint)

cint,

fu =M

kb(3/2 + cint). (46)

In terms of the dimensionless numbers, we rewrite therelations between Jr

ln

rln s and the gasdynamic quantitiesas

n0kv0

J020 =y ,n0kv0

J030 = 3

2y ,

n0kv0

J100 =y

Rint

cint3/2 + cint

,

n0kv0

J110 =2

3y 5

6

y

Rint

cint3/2 + cint

, (47)

n0kv0

J110011 =

5

8cint

y

Rint

cint3/2 + cint

,

n0kv0

J011 =2

3y

cint3/2 + cint

25 (

32 + cint) +

3+cint2Rint

+ 9fu16R2int

1 + 415fu( 32 + cint) +cintfu3Rint

.

From these relations, one can see that y7 = 3y/2.

Irl

rl (, y7)s in Eqn. (43) are defined by

Irlrl(, y7)

(v)I(, y7; cz)rl0(v)rl0(v)d3v .

(48)

Before calculating Irl

rl (, y7)s, we discuss the integra-tion

wj(Z) =

+

tj exp(t2)Z t dt, j = 0, 1, 2, . . . (49)

where Z is a complex variable in the first or second quad-rant. w0(Z) is the plasma dispersion function multipliedby () [34]. Simple relations exist for wj s, and thefollowing were used in the computation.

w1 =Zw0 , w2 = Zw1 ,

w3 =

2+ Zw2 , w4 = Zw3 ,

w5 =3

4

+ Zw4 , w6 = Zw5 .

After completing the integration over the radial ve-locity components, Ir

l

rl (, y7)s are represented with

11

wj(Z = + iy7) as

I0000 =1w0 , I

0202 =

2

3(w0 2w2 + 2w4) ,

I0100 =

2

w1 , I

1002 =

1

32

(w0 + 4w2 4w4) ,

I0200 =13

(w0 + 2w2) , I1000 =16

(w0 + 2w2) ,

I0101 =2w2 , I

1102 =

115

(w1 + 8w3 4w5) ,

I1010 =1

6(5w0 4w2 + 4w4) , (50)

I1100 =15

(3w1 + 2w3) , I0201 =

2

3(w1 + 2w3) ,

I1110 =130

(7w1 8w3 + 4w5) ,

I1001 =13

(w1 + 2w3) , I1101 =

2

5(3w2 + 2w4) ,

I1111 =1

5(13w2 12w4 + 4w6) .

With Eqn. (47) and (50), we can solve the matrixEqn. (43). The solution is

X = A1B , (51)

where A1 is the inverse matrix of A and both are dimen-sionless. We see that the gas density perturbation (, y)is a product of a/kv20 and a part independent of the force.Using the relation between the gas density perturbationand the power spectrum of the scattered light, we obtainthe CRBS line shape

S(, y) n20(, y)(, y)

(

n0a

kv20

)2

`crbs(, y; cint, Rint, fu) . (52)

This general relation is similar to that obtained in theatomic gas model [9]. It is a product of two fac-tors: the first factor containing a(, y)2 is related tothe power spectrum of the pump field; the second factor`crbs(, y; cint, Rint, fu) is a function of the y parameteras well as other gasdynamics parameters, but is inde-pendent of the optical dipole force. For a pump beamconstant in the frequency range of interest, the CRBSsignal power spectrum reveals the force independent part`crbs(, y; cint, Rint, fu). This property was used in lastsection to filter out the pump beams mode structure fromthe raw data. We conclude that, in the perturbationregime, the CRBS line shape is independent of the pumpbeams intensity.

Because we have kept seven physical quantities inabove derivations, we call the model given by Eqn. (43),(47), (50), and (51) the seven moment model, or, the s7model. We are also interested in a six moment model,

called the s6 model, which does not include the trace-less pressure tensor . We can obtain the s6 modelby repeating the derivation as with the s7 model. Or wecan obtain it by reducing Eqn. (43) and (45) directly. Wedelete the fourth row and the fourth column of matrix A,and change all J030 into J020 in matrix A. The deletedelements are associated with the traceless pressure ten-sor element zz. We also delete the fourth elements invector X and B. The calculation of Ir

l

rl will be changed

accordingly. In the s7 model, we have Irl

rl (, y7) as de-fined by Eqn. (41) and (48). In the six moment model,we have Ir

l

rl (, y), replacing y7 with y in Eqn. (41).

We now compare the model with experimental data innitrogen (N2), oxygen (O2), and carbon dioxide (CO2).The gas dynamic parameters needed by the model arelisted in Table. I for T0 = 292

K. The shear viscosityand heat conductivity data was found from CRC hand-book [35] and interpolated at the temperature. The bulkviscosity data of nitrogen was found in [13, 3638], ofoxygen in [39, 40], of carbon dioxide in [37].

TABLE I: A List of Gas Dynamic Quantities of Gases atT0 = 292

K

Gas (Pa sec) b/ (watt m1 K1) cint

N2 17.63 106 0.73 25.2 103 1.0

O2 20.21 106 0.4 25.76 103 1.0

CO2 14.6 106 1000 16.2 103 2.0

Among these, the bulk viscosity b is a controversialparameter [37, 41]. Bulk viscosity originates from theequilibration of internal energy and translational energy.It is well defined only when the time scale of the re-laxation process is much shorter than the characteristictime of the system. In atomic gases, due to the decou-pling of internal and translational energy modes, the bulkviscosity is zero. In molecular gases, the bulk viscosityis nonzero, but there are only a limited number of ex-perimental measurements [36, 39]. These measurementswere based on sound wave dissipation and are in the fre-quency range of several megahertz. Partially because ofthis shortage of b data, the Stokes hypothesis, b = 0, iswidely used in fluid dynamics calculations. Our experi-ment corresponds to a GHz frequency regime. The defi-nition and the value of the bulk viscosity in this regimemay need further study.

In Fig. (4) , we compare the seven moment model withthe experimental data for nitrogen. The blue curve is theexperimental data. The red curve is a convolution of thetheoretical line shape and the instrument function. Theonly adjustment was to match the height of the curves.The model matches with the experimental data reason-ably well for the 1 atm and 4 atm data, but underesti-mated the relative height of the Rayleigh peak for the 2

12

3 2 1 0 1 2 30

0.5

1

1.5

Frequency difference (GHz)

Rel

ativ

e in

tens

ity (

arb.

uni

t) N2 S7 modelp = 1.0 atmy = 0.58

3 2 1 0 1 2 30

0.5

1

1.5

Frequency difference (GHz)

Rel

ativ

e in

tens

ity (

arb.

uni

t) N2 S7 modelp = 2.0 atmy = 1.17

3 2 1 0 1 2 30

0.5

1

1.5

Frequency difference (GHz)

Rel

ativ

e in

tens

ity (

arb.

uni

t) N2 S7 modelp = 4.0 atmy = 2.34

3 2 1 0 1 2 30

0.5

1

1.5

Frequency difference (GHz)

Rel

ativ

e in

tens

ity (

arb.

uni

t) N2 S7 modelp = 5.0 atmy = 2.92

FIG. 4: (color) Coherent Rayleigh-Brillouin scattering in ni-trogen. A comparison between the seven moment moleculargas model with the experimental data. The blue curves arethe experimental data, the red curves are the convolution ofthe instrument function (shown by the dashed red curve inthe 1 atm panel) and the theoretical line shape.

and 3 atm data.

In Fig. (5), experimental data in oxygen is comparedwith the model results. b/ = 1.0 is used to calcu-late the model curve. The value found in [39, 40] isb/ = 0.4. The model curve calculated with this valuedoes not match the experimental data. In Fig. (6), themodel line shapes with different b are compared for oxy-gen at y = 2.18. We see that the model is reasonablysensitive to the bulk viscosity. This opens the possibilityto measure bulk viscosity using the coherent Rayleigh-Brillouin scattering.

In Fig. (7), the experimental data in carbon dioxide iscompared with the seven moment molecular model. Thecomparison is not satisfactory. For the 1 atm data, themodel gives a higher Rayleigh peak. The most seriousproblem is that at larger y parameters, the p = 3.0 atmand p = 4.0 atm panels, the model failed to predict thecorrect position of the Brillouin peaks. The peak separa-tion of the experimental data at p = 4.0 atm correspondsto a speed of sound at 283.0 m/s. Available CO2 dataat 4 and 5 atm gives a value of the speed of sound as280.0 5.8 m/s. This value is larger than the values( 259 m/s at 273 K) listed in reference books [35].The model gives a speed of 305 m/s, which is signifi-cantly larger than the experimental data. Here, a simplefitting by varying the bulk viscosity is insufficient. For abulk viscosity b that is 103 times larger than the shearviscosity , the position of the Brillouin peak calculatedby the model is no longer sensitive to the internal specificheat cint. If one wants to match the Brillouin peaks sep-

3 2 1 0 1 2 30

0.5

1

1.5

Frequency difference (GHz)

Rel

ativ

e in

tens

ity (

arb.

uni

t) O2 S7 modelp = 1.0 atmy = 0.55

3 2 1 0 1 2 30

0.5

1

1.5

Frequency difference (GHz)

Rel

ativ

e in

tens

ity (

arb.

uni

t) O2 S7 modelp = 2.0 atmy = 1.09

3 2 1 0 1 2 30

0.5

1

1.5

Frequency difference (GHz)

Rel

ativ

e in

tens

ity (

arb.

uni

t) O2 S7 modelp = 4.0 atmy = 2.18

3 2 1 0 1 2 30

0.5

1

1.5

Frequency difference (GHz)

Rel

ativ

e in

tens

ity (

arb.

uni

t) O2 S7 modelp = 5.0 atmy = 2.73

FIG. 5: (color) Coherent Rayleigh-Brillouin scattering in oxy-gen. A comparison between the seven moment molecular gasmodel with the experimental data. The blue curves are theexperimental data, the red curves are the convolution of theinstrument function and the theoretical line shape. A bulkviscosity b = 1.0 is used to calculate the theoretical lineshape.

0 0.5 1 1.5 2 2.50

0.2

0.4

0.6

0.8

/ k v0

Nor

mal

ized

inte

nsity

b/ = 0.0

b/ = 0.4

b/ = 1.0

FIG. 6: Coherent Rayleigh-Brillouin scattering line shape inoxygen with different bulk viscosity at y = 2.18.

aration by using a small b/ 1, the calculated relativeintensity of the Brillouin peaks to the Rayleigh peak ismuch smaller than the corresponding experimental data.

The mismatch between the model and the experimentin CO2 indicates that the model needs further develop-ment. In CO2, the vibrational modes come into play.In light diatomic molecules, the vibrational quantum ismuch higher than room temperature; for example, it isabout 3000 K in N2. The vibrational quantum of poly-atomic molecules such as CO2 is closer to the room tem-perature (960 K in CO2) [37]. It would be of interestto obtain the CRBS data in a wide temperature range invarious gases. The data would be useful in the studies ofkinetic theory.

13

3 2 1 0 1 2 30

0.5

1

1.5

Frequency difference (GHz)

Rel

ativ

e in

tens

ity (

arb.

uni

t) CO2 S7 modelp = 0.5 atmy = 0.44

3 2 1 0 1 2 30

0.5

1

1.5

Frequency difference (GHz)

Rel

ativ

e in

tens

ity (

arb.

uni

t) CO2 S7 modelp = 1.0 atmy = 0.88

3 2 1 0 1 2 30

0.5

1

1.5

Frequency difference (GHz)

Rel

ativ

e in

tens

ity (

arb.

uni

t) CO2 S7 modelp = 3.0 atmy = 2.65

3 2 1 0 1 2 30

0.5

1

1.5

Frequency difference (GHz)

Rel

ativ

e in

tens

ity (

arb.

uni

t) CO2 S7 modelp = 4.0 atmy = 3.54

FIG. 7: (color) Coherent Rayleigh-Brillouin scattering in car-bon dioxide. A comparison between the seven moment molec-ular gas model with the experimental data. The blue curvesare the experimental data, the red curves are the convolutionof the instrument function (shown in chapter 3 figure 1.) andthe theoretical line shape.

MODEL COMPARISON

The molecular CRBS s7 and s6 models we developedin the last section are also applicable to atomic gases.We suppress the internal modes by setting b = 0 andcint = 0. In the calculation, we actually set b = 10

3and cint = 10

3 to avoid numerical singularity.We previously presented an atomic model [9] that

matches well with the CRBS data of atomic gases. Wenow compare the molecular s7 model with this atomicmodel. The two models have different definitions of they paramter. The way to compare is to use the same op-tical configuration data and the same gas pressure. Forexample, with our experimental setup, at p = 1 atm,the atomic model y paramter is ymono = 0.47 and themolecular model y parameter is y = 0.55. The atomicmodel requires shear viscosity for the calculation, andthe molecular s7 model needs both shear viscosity andthe thermal conductivity.

Fig. (8) shows the comparison using argons shear vis-cosity and heat conductivity. Four pairs of curves werecompared, corresponding to p = 0, 1, 3, and 5 atm. Forp = 0 atm, the two models yield identical results. Thisis in accordance with our expectation, since, in the colli-sionless limit, the collision terms do not play a role. Forthe higher pressure cases, the molecular s7 model pre-dicts stronger Brillouin doublets than the atomic model.We have observed in [9] that the atomic model over es-timates the Brillouin peaks, therefore, the molecular s7models performance has not surpassed the atomic model

3 2 1 0 1 2 30

0.2

0.4

0.6

0.8

1Argon, p = 0 atmy

mono= 0

y = 0

/ k v0

Nor

mal

ized

inte

nsity

3 2 1 0 1 2 30

0.2

0.4

0.6

0.8

1Argon, p = 1 atmy

mono= 0.47

y = 0.55

/ k v0

Nor

mal

ized

inte

nsity

3 2 1 0 1 2 30

0.2

0.4

0.6

0.8

1Argon, p = 3 atmy

mono= 1.40

y = 1.65

/ k v0

Nor

mal

ized

inte

nsity

3 2 1 0 1 2 30

0.2

0.4

0.6

0.8

1Argon, p = 5 atmy

mono= 2.33

y = 2.75

/ k v0

Nor

mal

ized

inte

nsity

FIG. 8: A comparison between the molecular s7 model (solidcurve) and the atomic model (dashed curve) for the coherentRayleigh-Brillouin scattering line shape. The area under eachcurve is normalized to 1. The curves are plotted using argonsshear viscosity and thermal conductivity at 292K.

for atomic gases.

It is a good place to revisit our y parameter given in[9], we now denote by ymono for clarity. Our choice ofthe ymono parameter does not follow a Chaptman-Enskoganalysis, which is the basis for most other authors. Wefollowed a more empirical approach by identifying thecollision frequency in the BGK model literally and con-necting it to the shear viscosity. Such a ymono parameterdefinition was justified by favorable agreement betweenthe experimental and theoretical line shapes. Now, thematch between the s7 model and the atomic gas modelover a wide kinetic range further supports this choice ofymono.

Next, we compare the s7 and s6 models. In a subse-quent paper, Tenti et al. [13] modified their model for thespontaneous Rayleigh-Brillouin scattering. They con-cluded that the six moment model fits the experimentaldata better than the seven moment model. Specifically,for y 1, their seven moment model shows a weakerRayleigh peak than the six moment model, with the lat-ter matching better with the observed line shape in HD[13]. Since we have used the same collision terms, it isnecessary for us to check the relative performance of thes6 and s7 models.

In our derivation, the difference between the sponta-neous and coherent scattering lies only in the B vectorof Eqn. (43). For spontaneous scattering, the right handside of the matrix equation is given by Eqn. (45). In ad-dition, the power spectrum is no longer given by Eqn. (5).Instead, it is proportional to the real part of the sponta-

14

neous density perturbation spt(k, ),

`spt(k, ) Re(spt(k, )) . (53)

The reason for the difference is that, in spontaneous scat-tering, we solve an initial value problem and a Laplacetransform is applied over time; while in coherent scatter-ing, we solve for a steady state solution and a Fouriertransform is applied over time.

We first check the difference of s6 and s7 models forthe spontaneous scattering. This way, we can compareour results with that of Tenti et al. [13]. Tentis FOR-TRAN program of his six moment model is widely cir-culated. We obtained a copy from Forkey [38]. The pro-gram has been slightly modified to adapt to more recentcomputer operating systems, but the core algorithm re-mains original. We have used nitrogens dynamics pa-rameters to calculate the curves in this figure. We alsocompared the model programs using parameters of O2and CO2. Similar difference as shown in Fig. (9) wasobserved. In general, for spontaneous Rayleigh-Brillouinscattering, the s6 and s7 models yield identical line shapefor y . 0.5. In the range of 0.5 . y . 5, the s7 modelpredicts slightly narrower and higher peaks for both theRayleigh and Brillouin peaks, representing a less dampedperturbation. Our observation of the two models differ-ences is the opposite of Tentis [13]. They found a sharperRayleigh peak for HD at y 1. The reason of the oppo-site observation remains unsolved. A possible confusion,which requires special care, is the calculation of Ir

l

rl s.

In s6 model, they are Irl

rl (, y), while in the s7 model,

they are Irl

rl (, y7), with y7 = (3/2)y.We now compare the coherent Rayleigh-Brillouin line

shapes calculated by the s7 and the s6 model. We wouldlike to see which model matches the experimental databetter. In Fig. (10), we plot the line shape calculate bythe seven moment and six moment models. Nitrogensgas dynamics parameters were used in the calculation.

For y . 0.2, the results are indistinguishable. Obiv-ious difference can be seen for y 1. For larger y pa-rameters, the seven moment model in general predicts aslightly sharper Brillouin peaks, indicating slower dissi-pation processes. With the uncertainty in bulk viscosi-ties, the data do not show a definitive preference. Wefeel that more experimental observation is necessary tojudge which model is better.

DISCUSSIONS

The experimental configuration of coherent Rayleigh-Brillouin scattering follows a generic four wave mixingexperiment. The phase matching scheme can be quiteflexible. The data in this paper were collected using acoplanar backward scattering phase matching configura-tion, illustrated in Figure 11(a). With a crossing angle

0 0.5 1 1.5 2 2.50

0.2

0.4

0.6

y = 0.5

/ k v0

Nor

mal

ized

inte

nsity

Pan s7Pan s6Tenti s6

0 0.5 1 1.5 2 2.50

0.2

0.4

0.6

y = 1

/ k v0

Nor

mal

ized

inte

nsity

Pan s7Pan s6Tenti s6

0 0.5 1 1.5 2 2.50

0.2

0.4

0.6

0.8

y = 3

/ k v0

Nor

mal

ized

inte

nsity

Pan s7Pan s6Tenti s6

0 0.5 1 1.5 2 2.50

0.2

0.4

0.6

0.8

1

y = 5

/ k v0

Nor

mal

ized

inte

nsity

Pan s7Pan s6Tenti s6

FIG. 9: A comparison of spontaneous Rayleigh-Brillouin scat-tering line shape by three model programs. The solid curveis by the seven moment model (Pan s7) program, the dashedcurve by the six moment model (Pan s6) program, and thedotted line by Tentis six moment model (Tenti s6) program.The curves by the two six moment model programs completelyoverlap. Area under each curve is normalized to 1/2.

0 0.5 1 1.5 2 2.50

0.2

0.4

0.6

y = 0.5

/ k v0

Nor

mal

ized

inte

nsity

s7s6

0 0.5 1 1.5 2 2.50

0.2

0.4

0.6

y = 1

/ k v0N

orm

aliz

ed in

tens

ity

s7s6

0 0.5 1 1.5 2 2.50

0.2

0.4

0.6

0.8

1

y = 3

/ k v0

Nor

mal

ized

inte

nsity

s7s6

0 0.5 1 1.5 2 2.50

0.5

1

1.5

y = 5

/ k v0

Nor

mal

ized

inte

nsity

s7s6

FIG. 10: A comparison of the coherent Rayleigh-Brillouinscattering line shape calculated by the seven moment (s7) andthe six moment (s6) models. The solid curves are calculatedby the s7 model, the dashed curves by the s6 model. Areaunder each curve is normalized to 1/2.

near 180, the fringe pattern has a large wave numberk, i.e., a short spatial period. Such a configuration has astrong optical dipole force and a small y parameter.

In Fig. 11(b), we also illustrate a backward scattering,a 3 dimensional, and a two-color phase matching scheme.By varying the crossing angle , one can access different

15

k1k2

kprobeksignal

k

k2

k1 ksignal

kprobe

k

!#"$%

k1k2

kprobeksignal

k

x

y

z

&!#"$')(*+,

k2

k1 ksignal

kprobe

k

-."/"'"$0(*+,

FIG. 11: Phase matching schemes for coherent Rayleigh-Brillouin scattering. k1 and k2 are the pump beams wavevectors. k is the wave vector of the density perturbation.

y parameter for a given gas pressure. The backward scat-tering configuration is useful when the speed of sound inthe gas is of interest.

In the three dimensional backward scattering schemeshown in Fig. 11(c), the two pump beams (k1 and k2)define a plane. The phonon wave vector lies in thisplane and is perpendicular to the equipartition line ofthe pump beams crossing angle . The probe beam andthe signal beam forms a plane that intersects the pumpbeams plane along the phonon wave vector. This phasematching scheme is similar to that of a folded BOXCARS[42]. In a 3D phase matching scheme, the signal beamno longer follows one of the pump beams. The perpen-dicular polarization of the pump beams and the probebeam is no longer necessary. The scheme improves thesignal-to-noise ratio since the signal beam is separatedin space, however, the alignment of the beams is a lit-tle more challenging than in a coplanar scheme. Severalscans in room air were performed with this 3D phasematching scheme. The two color configuration shown inFig. 11(d) is widely used in LITA and LIEG, usually witha cw probe laser. Many experimental issues have beendiscussed in the literature.

CRBS can be used as a laser diagnostic technique.Together with laser induced thermal acoustics and laserinduced electrostrictive gratings, they provide techniquecapable of making localized and high signal-to-noise ratiomeasurements of gases from the collisionless limit to thehydrodynamic regime. Further development of CRBSis desirable. In our demonstrative experiments, it takeshundreds laser shots to scan a high resolution line shape.This limits the use of CRBS in diagnostics where phys-ical processes change quickly. We note that the CRBSsignal is sufficiently strong, and a single shot scheme is

possible. The challenge would be to resolve the narrow( 6 GHz) line width for the shot. Overall, we expectwide application of CRBS in gas diagnostics.

The CRBS line shape is a result of the balance betweenlaser excitation and collisional relaxation of the gas den-sity perturbation. CRBS data can be a valuable sourcefor the study of kinetic theories. We propose to measurebulk viscosity using coherent Rayleigh-Brillouin scatter-ing in various gases in a wide temperature and pressurerange. The unsatisfactory comparison between the the-ory and the CO2 data indicates that the theory needsrefinement.

In CRBS, the electromagnetic energy of the pumpbeams is coupled with the kinetic and internal energies ofthe gas particles at a non-resonant frequency. In this pa-per, we only considered a small optical dipole force. Thepotential well produced by the crossing pump beams ismuch less than the average thermal energy of the gasparticles. The perturbation to the gas particles energydistribution function is small so that the kinetic equa-tion can be linearized. With more intensive pump laserbeams, a significant amount of neutral particles may betrapped in the optical lattice, as suggested in [43, 44]. Astrong optical dipole force field may be used to perturbor transport an ensemble of gas particles at room tem-perature. When an ensemble of particles is considered,the balanced effects of the optical perturbation and thecollisional relaxation may be of interest in recent effortsof optically manipulating nanoscopic particles [45, 46].

The author would like to thank Professor G. Tenti ofthe University of Waterloo and Professor S. H. Lam ofPrinceton University for useful discussions. This workwas supported by the U.S. Air Force Office of ScientificResearch under the Air Plasma Rampart program.

Electronic address: miles@princeton.edu[1] W. A. Schroeder, M. J. Damzen, and M. H. R. Hutchin-

son, IEEE J. of Quantum Electron. 25, 460 (1989).[2] H. Tanaka, T. Sonehara, and S. Takagi, Phys. Rev. Lett.

79, 881 (1997).[3] C. Y. She, G. C. Herring, H. Moosmuller, and S. A. Lee,

Phys. Rev. Lett. 51, 1648 (1983).[4] C. Y. She, G. C. Herring, H. Moosmuller, and S. A. Lee,

Phys. Rev. A 31, 3733 (1985).[5] E. B. Cummings, Opt. Lett. 19, 1361 (1994).[6] W. Hubschmid, B. Hemmerling, and A. Stampanoni-

Panariello, J. Opt. Soc. Am. B 12, 1850 (1995).[7] A. Stampanoni-Panariello, B. Hemmerling, and W. Hub-

schmid, Phys. Rev. A 51, 655 (1995).[8] J. H. Grinstead and P. F. Barker, Phys. Rev. Lett. 85,

1222 (2000).[9] X. Pan, M. N. Shneider, and R. B. Miles, Phys. Rev.

Lett. 89, 183001 (2002).[10] S. Yip and M. Nelkin, Phys. Rev. 135, A1241 (1964).[11] P. L. Bhatnagar, E. P. Gross, and M. Krook, Phys. Rev.

94, 511 (1954).

16

[12] C. D. Boley, R. C. Desai, and G. Tenti, Can. J. Phys. 50,2158 (1972).

[13] G. Tenti, C. D. Boley, and R. C. Desai, Can. J. Phys. 52,285 (1974).

[14] C. S. Wang-Chang, G. E. Uhlenbeck, and J. de Boer, inStudies in Statistical Mechaincs, vol. II, edited by J. deBoer and G. E. Uhlenbeck, (North-Holland PublishingCo., Amsterdam, 1964), part C. The heat conductivityand viscosity of polyatomic gases.

[15] V. Ghaem-Maghami and A. D. May, Phys. Rev. A 22,692 (1980).

[16] A. T. Young and G. W. Kattawar, Appl. Opt. 22, 3668(1983).

[17] R. B. Miles, W. N. Lempert, and J. Forkey, Meas. Sci.Techol. 12, R33 (2001).

[18] W. M. Jr. and G. M. Kremer, Continuum Mech. Ther-modyn. 10, 319 (1998).

[19] W. M. Jr., Phyica A 264, 40 (1999).[20] A. Rangel-Huerta and R. M. Velasco, Phyica A 264, 52

(1998).[21] R. W. Boyd, Nonlinear Optics (Academic Press, Boston,

1992).[22] J. D. Jackson, Classical Electrodynamics (John Wiley &

Sons, Inc., New York, 2000), 3rd ed.[23] B. J. Berne and R. Pecora, Dynamic Light Scattering

(John Wiley & Sons, New York, 1976), dover reprint2000.

[24] X. Pan, P. F. Barker, A. Meschanov, J. H. Grinstead,M. N. Shneider, and R. B. Miles, Opt. Lett. 27, 161(2002).

[25] L. D. Landau and E. M. Lifshitz, Electrodynamicsof Continuous Media (Addison-Wesley, Reading, MA,1966).

[26] X. Pan, Coherent Rayleigh-Brillouin scattering (Ph.D.Thesis, Princeton University, 2003).

[27] E. A. Mason and L. Monchick, J. Chem. Phys. 36, 1622(1962).

[28] L. Monchick, E. A. Mason, and K. Yun, J. Chem. Phys.

39, 654 (1963).[29] F. B. Hanson, T. F. Morse, and L. Sirovich, Phys. Fluids

12, 84 (1969).[30] T. F. Morse, Phys. Fluids 7, 159 (1964).[31] G. Arfken, Mathematical Methods for Physicists (Aca-

demic Press, Inc., New York, 1985), 3rd ed.[32] F. B. Hanson and T. F. Morse, Phys. Fluids 10, 345

(1967).[33] E. P. Gross and E. A. Jackson, Phys. Fluids 2, 432 (1959).[34] B. D. Fried and S. D. Conte, The Plasma Dispersion

Function (Academic Press, New York, 1961).[35] CRC handbook of chemistry and physics (Knovel, Nor-

wich, N.Y., 2000), 3rd ed., electronic version.[36] G. J. Prangsma, A. H. Alberga, and J. J. M. Beenakker,

Physica 64, 278 (1973).[37] G. Emanuel, Phys. Fluids A 2, 2252 (1990).[38] J. N. Forkey, Development and demonstration of filtered

Rayleigh scattering : a laser based flow diagnostic for

planar measurement of velocity, temperature and pressure

(Ph.D. Thesis, Princeton University, 1996), appendix A.[39] C. Truesdell, J. Rat. Mech. Anal. 2, 643 (1953).[40] P. A. Thompson, Compressible-Fluid Dynamics

(McGraw-Hill, New York, 1972).[41] W. E. Meador, G. A. Miner, and L. W. Townsend, Phys.

Fluids 8, 258 (1996).[42] A. C. Eckbreth, Laser diagnostics for combustion temper-

ature and species (Gordon and Breach Publishers, Ams-terdam, 1996), 2nd ed.

[43] P. F. Barker and M. N. Shneider, Phys. Rev. A 64,033408 (2001).

[44] P. F. Barker and M. N. Shneider, Phys. Rev. A 66,065402 (2002).

[45] T. Iida and H. Ishihara, Phys. Rev. Lett. 90, 057403(2003).

[46] R. R. Agayan, F. Gittes, R. Kopelman, and C. F.Schmidt, Appl. Opt. 41, 2318 (2002).