## Model for polarized and depolarized Rayleigh Brillouin scattering spectra in molecular gases

Optics Express, Vol. 15, Issue 21, pp. 14257-14265 (2007)

http://dx.doi.org/10.1364/OE.15.014257

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### Abstract

Numerical models for Rayleigh-Brillouin scattering (RBS) spectra from molecular gases are obtained and discussed in this paper. The current publicly-available S6 model is for polarized RBS spectra only, despite the existence of both polarized and depolarized RBS light in many real applications. One of the new models (Q9) can be used to calculate both polarized and depolarized RBS spectra. In addition, this model has a solid physical ground because it is based on the correct Waldmann-Snider equation in which molecular internal energy is treated quantum-mechanically.

© 2007 Optical Society of America

## 1. Introduction

1. R. Schwiesow and L. Lading, “Temperature profiling by Rayleigh scattering lidar,” Appl. Opt. **20**, 1972 (1981). [CrossRef] [PubMed]

7. C. Flesia and C. L. Korb, “Theory of the double-edge molecular technique for doppler lidar wind measurement,” Appl. Opt. **38**, 432 (1999). [CrossRef]

8. M. L. Chanin, A. Garnier, A. Hauchecorne, and J. Porteneuve, “A Doppler lidar for measuring winds in the middle atmosphere,” Geophys. Res. Lett. **16**, 1273 (1989). [CrossRef]

*x*and

*y*, given that the transport properties of the scattering gases are known, where

*ω*is light frequency, k the wave number and v

_{0}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.

10. G. Tenti and R. C. Desai, “Kinetic theory of molecular gases i: Models of the linear Waldmann-Snider collision operator,” Can. J. Phys. **53**, 1266 (1975). [CrossRef]

11. G. Tenti and R. C. Desai, “Kinetic theory of molecular gases ii: Calculation of time dependent correlation functions,” Can. J. Phys. **53**, 1279–1291 (1975). [CrossRef]

10. G. Tenti and R. C. Desai, “Kinetic theory of molecular gases i: Models of the linear Waldmann-Snider collision operator,” Can. J. Phys. **53**, 1266 (1975). [CrossRef]

## 2. Rayleigh-Brillouin scattering

*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

14. A. T. Young and G. W. Kattawar, ”Rayleigh-scattering line profiles,” Appl. Opt. **22**, 3668 (1983). [CrossRef] [PubMed]

*is the distribution function, which is a function of position*

**f***, time*

**r***t*, velocity

*, and sometimes the rotational angular momentum*

**ν***(e.g. in linearized WS equation).*

**J****is the linearized collision operator of**

*Î***and all the physical moments are included in it.**

*f***does not appear in the linearized WCU equation. Assuming the system is not far from the equilibrium, we can write,**

*J**h*(

**t**) is the dimensionless derivation of the distribution function from thermal dynamic equilibrium, and

*rνJ*

*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*.

*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,*

**n***is an*

**A***by*

**n***matrix,*

**n***is an*

**B***-variable vector which contains all the physical moments, and*

**n***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.*

**C***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.*

**f****c**=

**v**⃗/

**v**

_{0}is the scaled velocity, a dimensionless parameter (v

_{0}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

*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.

*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.

*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.

14. A. T. Young and G. W. Kattawar, ”Rayleigh-scattering line profiles,” Appl. Opt. **22**, 3668 (1983). [CrossRef] [PubMed]

*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

15. K. Rah and B. C. Eu
, “Density and temperature dependence of the bulk viscosity of molecular liquids: Carbon dioxide and nitrogen,” J. Chem. Phys. **114**, 10436 (2001). [CrossRef]

*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.

*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).

## 4. Conclusion and comments

16. H. Chen, S. Kandasamy, S. Orszag, R. Shock, S. Succi, and Victor Yakhot
, “Extended Boltzmann Kinetic Equation for Turbulent Flows,” Science **301**, 633 (2003). [CrossRef] [PubMed]

## Acknowledgments

## References and links

1. | R. Schwiesow and L. Lading, “Temperature profiling by Rayleigh scattering lidar,” Appl. Opt. |

2. | R. B. Miles, Lempert R. Walter, and J. N. Forkey, “Laser Rayleigh Scattering,” Meas. Sci. Technol. |

3. | J. N. Forkey, N. D. Finkelstein, W. R. Lempert, and R. B. Miles, “Demonstration and characterization of filtered Rayleigh scattering for planar velocity measurements,” AIAA Journal |

4. | G. Fiocco and J. B. DeWolf, “Frequency spectrum of laser echoes from atmospheric constituents and determination of the aerosol content of air,” J. Atmos. Sci. |

5. | G. Fiocco, G. Benedetti-Michelangeli, K. Maischberger, and E. Madonna, “Measurement of temperature and aerosol to molecule ratio in the troposphere by optical radar,” Nature (London) Phys. Sci. |

6. | J. Fielding, |

7. | C. Flesia and C. L. Korb, “Theory of the double-edge molecular technique for doppler lidar wind measurement,” Appl. Opt. |

8. | M. L. Chanin, A. Garnier, A. Hauchecorne, and J. Porteneuve, “A Doppler lidar for measuring winds in the middle atmosphere,” Geophys. Res. Lett. |

9. | G. Tenti, C. D. Boley, and R. C. Desai, “On the kinetic model description of RBS from molecular gases,” Can. J. Phys. |

10. | G. Tenti and R. C. Desai, “Kinetic theory of molecular gases i: Models of the linear Waldmann-Snider collision operator,” Can. J. Phys. |

11. | G. Tenti and R. C. Desai, “Kinetic theory of molecular gases ii: Calculation of time dependent correlation functions,” Can. J. Phys. |

12. | C. D. Boley, R. C. Desai, and G. Tenti, “Kinetic models and brillouin scattering in a molecular gas,” Can. J. Phys. |

13. | G. Tenti, C. D. Boley, and R. C. Desai
, “On the kinetic model description of Rayleigh-Brillouin scattering from molecular gases,“ Can. J. Phys. |

14. | A. T. Young and G. W. Kattawar, ”Rayleigh-scattering line profiles,” Appl. Opt. |

15. | K. Rah and B. C. Eu
, “Density and temperature dependence of the bulk viscosity of molecular liquids: Carbon dioxide and nitrogen,” J. Chem. Phys. |

16. | H. Chen, S. Kandasamy, S. Orszag, R. Shock, S. Succi, and Victor Yakhot
, “Extended Boltzmann Kinetic Equation for Turbulent Flows,” Science |

**OCIS Codes**

(010.1310) Atmospheric and oceanic optics : Atmospheric scattering

(020.3690) Atomic and molecular physics : Line shapes and shifts

(300.6390) Spectroscopy : Spectroscopy, molecular

**ToC Category:**

Atmospheric and oceanic optics

**History**

Original Manuscript: August 23, 2007

Revised Manuscript: October 3, 2007

Manuscript Accepted: October 5, 2007

Published: October 12, 2007

**Citation**

Qiuhua Zheng, "Model for polarized and depolarized Rayleigh Brillouin scattering spectra in molecular gases," Opt. Express **15**, 14257-14265 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-21-14257

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### References

- R. Schwiesow and L. Lading, "Temperature profiling by Rayleigh scattering lidar," Appl. Opt. 20, 1972 (1981). [CrossRef] [PubMed]
- R. B. Miles, Lempert R. Walter, and J. N. Forkey, "Laser Rayleigh Scattering," Meas. Sci. Technol. 12, R33-R51 (2001). [CrossRef]
- J. N. Forkey, N. D. Finkelstein, W. R. Lempert and R. B. Miles, "Demonstration and characterization of filtered Rayleigh scattering for planar velocity measurements," AIAA Journal 34, 442-448 (1996). [CrossRef]
- G. Fiocco and J. B. DeWolf, "Frequency spectrum of laser echoes from atmospheric constituents and determination of the aerosol content of air," J. Atmos. Sci. 25, 488 (1968). [CrossRef]
- G. Fiocco, G. Benedetti-Michelangeli, K. Maischberger, and E. Madonna, "Measurement of temperature and aerosol to molecule ratio in the troposphere by optical radar," Nature (London) Phys. Sci. 229, 78-79 (1971).
- J. Fielding, et al, "Polarized/depolarized Rayleigh scattering for determining fuel concentrations in flames," P. Conbustion Inst. 29, 2703-2709 (2002). [CrossRef]
- C. Flesia and C. L. Korb, "Theory of the double-edge molecular technique for doppler lidar wind measurement," Appl. Opt. 38, 432 (1999). [CrossRef]
- M. L. Chanin, A. Garnier, A. Hauchecorne, and J. Porteneuve, "A Doppler lidar for measuring winds in the middle atmosphere," Geophys. Res. Lett. 16, 1273 (1989). [CrossRef]
- G. Tenti, C. D. Boley, and R. C. Desai, "On the kinetic model description of RBS from molecular gases,"Can. J. Phys. 52, 285 (1974).
- G. Tenti and R. C. Desai, "Kinetic theory of molecular gases i: Models of the linear Waldmann-Snider collision operator," Can. J. Phys. 53, 1266 (1975). [CrossRef]
- G. Tenti and R. C. Desai, "Kinetic theory of molecular gases ii: Calculation of time dependent correlation functions," Can. J. Phys. 53, 1279-1291 (1975). [CrossRef]
- C. D. Boley, R. C. Desai and G. Tenti, "Kinetic models and brillouin scattering in a molecular gas," Can. J. Phys. 50, 2158 (1972). [CrossRef]
- G. Tenti, C. D. Boley and R. C. Desai, "On the kinetic model description of Rayleigh-Brillouin scattering from molecular gases," Can. J. Phys. 52, 285 (1974).
- A. T. Young and G. W. Kattawar, "Rayleigh-scattering line profiles," Appl. Opt. 22, 3668 (1983). [CrossRef] [PubMed]
- K. Rah and B. C. Eu, "Density and temperature dependence of the bulk viscosity of molecular liquids: Carbon dioxide and nitrogen," J. Chem. Phys. 114, 10436 (2001). [CrossRef]
- H. Chen, S. Kandasamy, S. Orszag, R. Shock, S. Succi, and Victor Yakhot, "Extended Boltzmann Kinetic Equation for Turbulent Flows," Science 301, 633 (2003). [CrossRef] [PubMed]

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