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

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  1. R. Schwiesow and L. Lading, "Temperature profiling by Rayleigh scattering lidar," Appl. Opt. 20, 1972 (1981).
    [CrossRef] [PubMed]
  2. R. B. Miles, Lempert R. Walter, and J. N. Forkey, "Laser Rayleigh Scattering," Meas. Sci. Technol. 12, R33-R51 (2001).
    [CrossRef]
  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 34, 442-448 (1996).
    [CrossRef]
  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. 25, 488 (1968).
    [CrossRef]
  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. 229, 78-79 (1971).
  6. J. Fielding,  et al, "Polarized/depolarized Rayleigh scattering for determining fuel concentrations in flames," P. Conbustion Inst. 29, 2703-2709 (2002).
    [CrossRef]
  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]
  9. 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).
  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]
  12. 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]
  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. 52, 285 (1974).
  14. A. T. Young and G. W. Kattawar, "Rayleigh-scattering line profiles," Appl. Opt. 22, 3668 (1983).
    [CrossRef] [PubMed]
  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]
  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]

2003

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]

2002

J. Fielding,  et al, "Polarized/depolarized Rayleigh scattering for determining fuel concentrations in flames," P. Conbustion Inst. 29, 2703-2709 (2002).
[CrossRef]

2001

R. B. Miles, Lempert R. Walter, and J. N. Forkey, "Laser Rayleigh Scattering," Meas. Sci. Technol. 12, R33-R51 (2001).
[CrossRef]

R. B. Miles, Lempert R. Walter, and J. N. Forkey, "Laser Rayleigh Scattering," Meas. Sci. Technol. 12, R33-R51 (2001).
[CrossRef]

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]

1999

1996

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]

1989

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]

1983

1981

1975

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]

1974

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).

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).

1972

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]

1971

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).

1968

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]

Benedetti-Michelangeli, G.

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).

Boley, C. D.

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).

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).

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]

Chanin, M. L.

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]

Chen, H.

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]

Desai, R. C.

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]

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).

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).

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]

DeWolf, J. B.

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]

Eu, B. C.

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]

Fielding, J.

J. Fielding,  et al, "Polarized/depolarized Rayleigh scattering for determining fuel concentrations in flames," P. Conbustion Inst. 29, 2703-2709 (2002).
[CrossRef]

Finkelstein, N. D.

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]

Fiocco, G.

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).

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]

Flesia, C.

Forkey, J. N.

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]

Garnier, A.

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]

Hauchecorne, A.

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]

Kandasamy, S.

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]

Kattawar, G. W.

Korb, C. L.

Lading, L.

Lempert, R. B.

R. B. Miles, Lempert R. Walter, and J. N. Forkey, "Laser Rayleigh Scattering," Meas. Sci. Technol. 12, R33-R51 (2001).
[CrossRef]

Lempert, W. R.

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]

Madonna, E.

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).

Maischberger, K.

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).

Miles, R. B.

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]

Orszag, S.

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]

Porteneuve, J.

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]

Rah, K.

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]

Schwiesow, R.

Shock, R.

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]

Succi, S.

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]

Tenti, G.

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]

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).

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).

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]

Young, A. T.

AIAA Journal

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]

Appl. Opt.

Can. J. Phys.

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).

Geophys. Res. Lett.

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]

J. Atmos. Sci.

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]

J. Chem. Phys.

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]

Meas. Sci. Technol.

R. B. Miles, Lempert R. Walter, and J. N. Forkey, "Laser Rayleigh Scattering," Meas. Sci. Technol. 12, R33-R51 (2001).
[CrossRef]

P. Conbustion Inst.

J. Fielding,  et al, "Polarized/depolarized Rayleigh scattering for determining fuel concentrations in flames," P. Conbustion Inst. 29, 2703-2709 (2002).
[CrossRef]

Phys. Sci.

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).

Science

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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Figures (5)

Fig. 1.
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.

Fig. 2.
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.

Fig. 3.
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).

Fig. 4.
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.

Fig. 5.
Fig. 5.

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

Equations (14)

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x = ω k v 0
( ∂t + v · ) f = Î ( f )
f ( rtvJ ) = f 0 ( 1 + h ( trvJ ) ) .
AB = C
( ∂t + v · ) h = Q ̂ ( 9 ) h
Q ̂ ( 9 ) h = S 993 ( b 1 ϕ 1 + b 2 ϕ 2 + b 3 ϕ 3 + b 4 ϕ 4 + b 5 ϕ 5 + b 6 ϕ 6 + b 7 ϕ 7 + b 8 ϕ 8 + b 9 ϕ 9 )
+ S 33 b 3 ϕ 3 + S 34 b 4 ϕ 3 + S 43 b 3 ϕ 3 + S 44 b 4 ϕ 4 + S 55 b 5 ϕ 5 + S 56 b 6 ϕ 5
+ S 65 b 5 ϕ 6 + S 66 b 6 ϕ 6 + S 77 b 7 ϕ 7 + S 44 b 4 ϕ 4 + S 78 b 8 ϕ 7 + S 87 b 7 ϕ 8
ϕ 1 = 1 , ϕ 2 = c z , ϕ 3 = ( c 2 3 ) 6 , ϕ 4 = ( ε j ε ) c int ,
ϕ 5 = ( c 2 5 ) c z 10 , ϕ 6 = ( ε j ε ) c z c int ,
ϕ 6 = 15 2 ( 1 2 ( J m J n + J n J m ) j 2 ( j + 1 ) 2 3 4 j ( j + 1 ) ) ,
ϕ 7 = ( c m c n c 2 3 δ mn ) 2 ,
ϕ 9 = ( c z 2 3 5 c 2 c z ) 90 ,
b k = Tr ( dc 1 ( 2 π ) 3 Z int exp ( ( c 2 2 + ε j ) ) ϕ k )

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