1. Introduction

Mathematical models for polarized Rayleigh-Brillouin scattering (RBS) spectra from molecular gases are commonly used in many aerospace and atmospheric applications, where parameters such as gas temperature can be measured through quantitative comparison between a set of data and RBS spectra models[1–6]. In some applications, a more accurate RBS model is necessary to improve the measurement accuracies of parameters of interest. For example, for LIDAR measurements of atmospheric wind, while Doppler shifts of the elastic light backscattering of air molecules are of interest and measured, an accurate RBS model describing broadening effects of the elastic light backscattering of air molecules is indispensable during data processing in order to achieve better wind estimation accuracies[7,8] and to improve temperature and optical turbulence measurements. However, it is difficult to obtain a satisfying mathematical model for RBS spectra in gases because the corresponding model equations are linearized Boltzmann or Boltzmann-like equations, which are mathematically intractable. Currently, Tenti’s S6 model is generally considered the best model to describe the RBS spectra in molecular gases. It depends on two dimensionless parameters, x and y, given that the transport properties of the scattering gases are known, where $x=\frac{\omega }{\sqrt{k{v}_0}}$ —ω is light frequency, k the wave number and v₀ the most probable speed of scattering gas molecules—the scaled light frequency and y can be thought of as the ratio between the incident light wavelength and the mean free path of the scattering gases. Nonetheless, the simplified Gaussian model is still occasionally seen in some atmospheric applications. Using a crude Gaussian model inevitably leads to large measurement errors, particularly when the parameters to be measured are changing, as we can tell from Fig. 1 where the S6 model and the Gaussian model are compared. For atmospheric applications, the differences between S6 and the Gaussian model are much greater than the differences caused by a 10 K temperature change, with all other parameters staying unchanged.

 

Fig. 1. Comparison of S6 model (solid line) and Gaussian (dashed line) for Polarized Rayleigh-Brillouin scattering spectra in the atmosphere where pressure is set to 1atm and temperature 283 K.

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However, the S6 model also suffers its own limitations: the model equation for S6 is the primitive kinetic Wang Chang-Uhlenbeck (WCU) equation instead of a more accurate Waldmann-Snider (WS) equation, in which molecular internal degrees of freedom are included and treated quantum mechanically [9]. On physical grounds, the WCU equation loses its validity for molecules with degenerate internal states. For example, it cannot be applied to light scattering where the scatters are non-spherical molecules at room temperature because the rotational degrees of freedom have to be considered. The rotational energy levels are highly degenerate. For gases with spherical molecules, the WCU equation is plausible and can be used to calculate the RBS spectra. In addition, due to the natural complexity of the real world, all RBS models are approximate; even for models that are based on the WS equation. Furthermore, although most of the light photons involved in the scattering process have polarization unchanged during the scattering, there is always a small portion of scattered light that becomes depolarized. Therefore, models for both polarized and depolarized RBS spectra are needed in real applications.

There were efforts to pursue more accurate models for both polarized and depolarized RBS spectra in molecular gases[9,10, 11]. A general model equation for both polarized and depolarized RBS spectra based on the correct WS equation was successfully obtained. In this paper, we have developed a Rayleigh-Brillouin scattering model based on the linearized WS equation which can be used to model both the polarized and depolarized RBS spectra in molecular gases. Its explicit model equation can be derived from the general model equation in reference[10].

2. Rayleigh-Brillouin scattering

RBS is an elastic scattering process where the Brillouin scattering spectral component is frequency-shifted from the central Cabannes line. The Brillouin double peaks in an RBS spectrum are shifted by plus and minus the speed of sound in the scattering medium, as we shall see later on. The shape of an RBS spectrum is totally dependent on the gas/fluid dynamics and is irrelevant to one single photon-molecule interaction. This claim is true for both the polarized and depolarized RBS cases. To obtain an accurate model for RBS spectra in gases, one needs to solve an appropriate kinetic model equation which, for general local equilibrium statistics, should be the Boltzmann equation. Even so, people have often managed to construct and solve a linearized version of a Boltzmann-like equation instead of solving the notoriously complicated and mathematically intractable Boltzmann equation itself.

Based on the model equation derived from the linearized WCU equation, Boley et al [9] first calculated a 7-moment model for RBS spectra in gases. However, the computed model didn’t agree with the existing experimental results: there was a salient dip in the middle of the calculated S7 model, while there was no such dip in the measured RBS spectra. In 1974, Tenti et al [13] reduced the 7-moment model equation to 6 moments by removing the pressure tensor moment. It was found that the result of the 6-moment equation (S6 model) agreed well with then-existing measured RBS spectra. Later on, the S6 model was verified by more experimental results and was gradually accepted as the best existing model to describe RBS spectra in molecular gases [14].

The linearized Boltzmann and Boltzmann-like model equations mentioned above have this fundamental form:

where f is the distribution function, which is a function of position r, time t, velocity ν , and sometimes the rotational angular momentum J (e.g. in linearized WS equation). Î is the linearized collision operator of f and all the physical moments are included in it. J does not appear in the linearized WCU equation. Assuming the system is not far from the equilibrium, we can write,

Here h (t rνJ) is the dimensionless derivation of the distribution function from thermal dynamic equilibrium, and f ₀ is the distribution function where all scattering particles are in thermal dynamic equilibrium. The RBS spectrum can be acquired after h is solved from Eq. (1) and (2). The RBS spectrum is proportional to the double Fourier transform of the density correlation function, which has been proven to be proportional to h .

In order to calculate a model for molecular RBS spectra from a Boltzmann-type model equation like the kinetic WS equations, which takes a form like Eq. (2) above, one first linearizes the model equation and writes the right-hand side explicitly using the Gross-Jackson-Sirovich (GJS) procedure to linearize the kinetic WS equation. From this linearization process, one can construct a general model equation for RBS spectra with all physical moments, such as velocity and density, being written explicitly. Then, for each physical moment, one can obtain an equation for in terms of the other moments and appropriate initial conditions. Therefore, there will be n equations if the model equation is an n-moment model equation. From linear algebra and equation theory, we can solve this group of equations through matrix methods. At the end, we obtained a form like below,

Here A is an n by n matrix, B is an n-variable vector which contains all the physical moments, and C is an n-element vector. The model can then be calculated once the necessary physical moment is obtained. Solving for the RBS model from the kinetic WCU equation is similar to this.

The linearization process of the kinetic Boltzmann-type equations, like WCU and WS, can be considered a Taylor expansion, since the gas kinetics can be treated as a perturbation problem: the distribution function f is fluctuating and the system is not far from equilibrium. Each physical moment in the model equation corresponds to either a standing or a propagated mode in the gas. The mode corresponding to the first physical moment—the instantaneous density—is the dominating term. The second physical moment, the velocity vector, which corresponds to a damping propagated wave mode and therefore corresponding to the Brillouin doublets in the RBS spectra, is the second most important term. The other physical moments, representing energy dissipation and pressure tensorial effects, are less important and can be reckoned as high-order perturbation terms to the first two.

We only write down the linearized version of the 9-moment model equation here. Other model equations for 6-, 7-, and 8-moment models can be derived through the procedure described above. It is:

with

with

all S’s are expansion coefficients and

where c = v⃗/v ₀ is the scaled velocity, a dimensionless parameter (v₀ is the average thermal speed of gas molecules); Z_int the internal partition function, c_int the internal specific heat per molecule, j the angular momentum of molecules, ε _j the scaled internal energy.

3. Results and discussions

We calculated four different models for polarized RBS spectra of molecular gases from four different model equations, with 6, 7, 8 or 9 physical moments. The first two (6- and 7-moment) were derived from linearized WCU equations, the other two from linearized WS equations. The 6-moment model is exactly the same as Tenti’s S6 model. The 7-moment model has one additional term—the pressure tensorial moment—in its equation compared to the 6-moment model. The 8- and 9-moment model equations were derived from linearizing the WS equation. Terms reflecting molecular rotational effects are included in the last two model equations. Again, the 9-moment model has one more physical moment than the 8-moment one. It is not surprising that, after being linearized, the first 7 moments of the WS equation with Î(f) as the right-hand side of Eq. (1) are the same as the 7 moments in the linearized WCU equation [see Tenti’s papers reference]. We call the new 6-, 7-, 8- and 9-moment models Q6, Q7, Q8 and Q9, respectively.

In Fig. 2, we plot the four models against S6 with air temperature and pressure at 300K and 1 atm respectively. Transprot properties like viscosities of Nitrogen gases were used in all the plots. Because RBS spectrum is symmetric over the incident light frequency, we only show the right part of each RBS spectrum in Fig 2. It is expected that there are not salient differences between these models in this case; in fact, one can hardly tell the differences between them in Fig. 2. It seems that the degenerate internal rotational energy of gas molecules does not affect the polarized RBS spectra. That is consistent with our knowledge that they are mostly responsible for the depolarized RBS spectra of gas molecules (the incoherent Rayleigh-Brillouin scattering part). The 8^th and 9^th moments are therefore weakly coupled to the other 7 moments for the polarized RBS spectra. The Q6 model shares the same model equation with the S6 model, so there should be no difference between the two. Nonetheless, given the complexity of the calculations of these models and the possibility that different approaches might have been employed, it is reasonable to present both of them here. The Q7 model differs from Q6 by an additional pressure tensorial moment, which results in Q7 being different from Q6, which might not be trivial for some applications. In the range of atmospheric interest, i.e. when the parameter y is in the range 0 to 0.6, the differences between models S6/Q6, Q7, Q8 and Q9 are very small. However, their shape response to the varying temperature in the atmosphere may be different. In other words, for example, from one lidar data set one may retrieve different temperature profiles using these models.

 

Fig. 2. Comparisons between newly calculated polarized RBS models (dashed line) and the S6 (solid line) model. The air temperature was set to 300K and pressure 1 atm for all plots.

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As explained before, the RBS model should be based on the more appropriate WS equation in which degenerate internal energy of molecules is accounted for. From now on, we will focus on the Q9 model for further comparison with S6, since it is based on the WS equation and has one more moment than Q8. The Q9 model for polarized RBS spectra is plotted against S6 in Fig. 3. The RBS spectrum is close to a Gaussian when the parameter y is small (0.1) and gradually becomes more structured as y increases. The Brillouin peaks become more pronounced when y = 1.5, and continue to grow as the central part of the Rayleigh scattering spectrum—the Cabannes line—decreases as y increases to 5.0. The ratio of intensity of the Brillouin lines and the Cabannes line is called the Landau-Placzek ratio, which is a constant when y is greater than a threshold value. For light scattering in the atmosphere, y (see the definition above) spans from around 0.5 at the boundary layer (0~3 km altitude) to close to 0 in the stratosphere (12~50 km). From Fig. 2 and Fig. 3, we can tell differences between the Q9 model and Q6/S6 are more obvious when y grows larger.

 

Fig. 3. Comparison of Q9 model (dashed line) and S6 model (solid line) at different y values using transport properties of Nitrogen gas. All of the models shown here were normalized (integrated to 1).

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The polarized RBS spectrum is determined by the scalar part of the molecular polarizability α. The molecular rotating movement, causing the reorientation of molecules and taking them from one degenerate state to another, contributes little to the scalar part of α. Therefore, the polarized Rayleigh-Brillouin scattering spectrum sees little difference between Q9 and S6. However, as we shall see later, molecular rotation is the major source of the tensorial part of molecular polarizability, which is responsible for the depolarized RBS.

The differences between the Q9 and S6 models (subtraction of S6 from Q9) for each case in Fig. 3 are plotted in Fig. 4. Transport properties like viscosities, thermal conductivity of Nitrogen gases were used in all the plots. The major differences are at the positions of Brillouin peaks; also, notice that the difference between Q9 and S6 increases as the value of y increases. This agrees well with the hydrodynamic theory for Brillouin scattering, in which small corrections of Brillouin peak intensity in a polarized RBS spectrum are proportional to the tensorial part of molecular polarizability. Therefore, the inclusion of molecular rotation in the Q9 model naturally leads to an enlargement of the Brillouin peak intensities.

 

Fig. 4. Plot of difference between the Q9 model and the S6 model in Fig. 3. In each case, both models were normalized to 1 before their difference was calculated.

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Although the difference between these two models is relatively small, it cannot always be ignored. For example, for temperature measurements of molecular gases, the spectral change from a 1K shift in temperature is much smaller (c.f., [14]) than those plotted in Fig. 4.

The Q9 model is determined by 4 parameters; the y parameter mentioned earlier is one of them. These four parameters depend on temperature, pressure, light frequency, thermal conductivity, and bulk and shear viscosities of the scattering gases. Therefore, theoretically one can apply Q9 to the measured RBS spectra to retrieve temperature, pressure, etc or any of above transport coefficients of the scattering gases, if all the others are known. However, things are more complicated for real applications, since for many molecular gases transport coefficients such as bulk and shear viscosities are hard to measure and usually temperature-dependent[15]. Knowing the exact relationship between viscosity and temperature is thus crucial to temperature retrieval from measured polarized RBS spectra.

So far we have been mostly talking about the polarized RBS spectra of molecular gases. As we have argued in the beginning, one important advantage of the Q9 model equation is that it can also be used to calculate depolarized RBS spectra in molecular gases. Plotted in Fig. 5 are the calculated depolarized RBS spectra at different y values from the Q9 model equation. When the y parameter is close to zero, where the molecules are predominantly free streaming, the depolarized RBS spectrum becomes a Gaussian. This Gaussian is almost identical to the polarized RBS case when the y parameter approaches zero. In other words, when the gas is diluted enough, polarized and depolarized RBS spectra are pratically indistinguishable.

As y increases, the depolarized RBS spectrum approaches a Lorentzian. It is interesting to note that the width of the spectrum becomes smaller until it reaches the minimum when y equals 1; after that, the width increases rapidly with increasing y and the spectral shape remains Lorentzian-like. This behavior explains why the depolarized RBS spectra are different from their polarized counterparts: the width of the polarized RBS stops growing once a critical y value is reached (y = 1) while concurrently the spectral shape becomes more and more structured (c.f. Fig. 3).

 

Fig. 5. Calculated depolarized Rayleigh-Brillouin scattering spectra at different y values.

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4. Conclusion and comments

An accurate model for Rayleigh-Brillouin scattering spectra is important, crucial in some instances, for many atmospheric applications, such as for optical turbulence measurement by direct-detection LIDAR systems. A new model for both polarized and depolarized Rayleigh-Brillouin scattering spectra in molecular gases based on the correct linearized Waldmann-Snider equation was obtained and presented. Compared to the current widely-adopted S6 model for polarized RBS spectra, the new Q9 model is based on the physically more correct Waldmann-Snider (WS) equation in which internal energy degeneracy is treated quantum mechanically. Therefore, the Q9 model is more accurate and should be used in modeling future applications that require better accuracy. Additionally, due to the inherent connection between the photon depolarization and tensorial rotational effects of scattering gas molecules, a model for the depolarized RBS spectra of molecular gases was also obtained using the same model equation. On the other hand, the current kinetic approach (from BGK model to GJS procedure) in solving for Rayleigh scattering with assumption of weak departures from local thermodynamic equilibrium may also shed some light in our understanding turbulence[16].