Abstract

The nature of optical scattering from laser-induced thermal gratings created in the gas phase is investigated. Thermal gratings are produced with the illumination geometry used to perform degenerate four-wave mixing (DFWM) measurements. Such scattering from thermal gratings can act as a phase-matched interference signal. A solution to the linearized hydrodynamic equations is developed to model the dynamics of the thermal grating. Predictions of this model that uses realistic gas properties are shown to compare favorably with laboratory measurements. The model includes the effects of finite-rate energy deposition, damping by viscosity and thermal conduction, mass diffusion of the excited-state grating, and electrostrictive compression.

© 1995 Optical Society of America

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  1. D. C. Auth, Appl. Phys. Lett. 16, 512 (1970).
    [CrossRef]
  2. M. A. Buntine, D. W. Chandler, and C. C. Hayden, "Detection of vibrational-overtone excitation in water via laser-induced grating spectroscopy," J. Chem. Phys. (to be published).
  3. S. Williams, L. A. Rahn, P. H. Paul, J. W. Forsman, and R. N. Zare, "Laser-induced thermal-grating effects in flames," Opt. Lett. 19, 1681–1683 (1994).
    [CrossRef] [PubMed]
  4. P. M. Danehy, E. J. Friedman-Hill, P. H. Paul, and R. L. Farrow, "Thermal-grating contributions to degenerate fourwave mixing in nitric oxide," submitted to J. Opt. Soc. Am. B.
  5. R. L. Vander Wal, B. E. Holmes, J. B. Jefferies, P. M. Danehy, R. L. Farrow, and D. J. Rakestraw, Chem. Phys. Lett. 191, 251 (1992).
    [CrossRef]
  6. L. Landau and G. Placzek, Z. Phys. 5, 172 (1933).
  7. M. E. Mack, Ann. N.Y. Acad. Sci. 168, 419 (1970).
    [CrossRef]
  8. R. D. Mountain, Rev. Mod. Phys. 38, 205 (1966).
    [CrossRef]
  9. R. M. Herman and M. A. Gray, Phys. Rev. Lett. 19, 324 (1967).
    [CrossRef]
  10. D. Pohl, I. Reinhold, and W. Kaiser, Phys. Rev. Lett. 20, 1141 (1968).
    [CrossRef]
  11. H. Eichler and H. Stahl, Opt. Commun. 6, 239 (1972); J. Appl. Phys. 44, 3429 (1973).
    [CrossRef]
  12. J. R. Salcedo and A. E. Seigman, IEEE J. Quantum Electron. QE-15, 250 (1979).
    [CrossRef]
  13. K. A. Nelson, D. R. Lutz, and M. D. Fayer, Phys. Rev. A 24, 3261 (1981); K. A. Nelson, R. J. D. Miller, D. R. Lutz, and M. D. Fayer, J. Appl. Phys. 53, 1144 (1982); R. J. D. Miller, R. Casalegno, K. A. Nelson, and M. D. Fayer, Chem. Phys. 72, 371 (1982).
    [CrossRef]
  14. E. B. Cummings, Opt. Lett. 19, 1361 (1994).
    [CrossRef] [PubMed]
  15. W. G. Vincenti and C. K. Kruger, Introduction to Physical Gas Dynamics (Krieger, New York, 1967), Chap. 12.
  16. H. J. Eichler, P. Günter, and D. W. Pohl, Laser-Induced Dynamic Gratings (Springer-Verlag, Berlin, 1986), pp. 29.
  17. D. E. Gray, ed., American Institute of Physics Handbook (McGraw-Hill, New York, 1972), Chap. 6e.
  18. R. L. Farrow and D. J. Rakestraw, Science 257, 1984 (1992).
    [CrossRef]
  19. J. W. Thom, Jr., J. A. Gray, J. L. Durant, Jr., and P. H. Paul, J. Chem. Phys. 97, 8156 (1992); J. A. Gray, P. H. Paul, and J. L. Durant, Chem. Phys. Lett. 190, 266 (1992).
    [CrossRef]
  20. R. J. Kee, G. Dixon-Lewis, J. Warnatz, M. E. Coltrin, and J. A. Miller, "A Fortran computer code package for the evaluation of gas-phase multicomponent transport properties," Rep. SAND86-8246.UC-32, Sandia National Laboratories, Livermore, Calif., 1986.
  21. R. H. Perry and C. H. Chilton, eds., Chemical Engineering Handbook, 5th ed. (McGraw-Hill, New York, 1973), pp. 3–231.
  22. R. Cambi, D. Cappelletti, G. Luiti, and R. Pirani, J. Chem. Phys. 95, 1852 (1991).
    [CrossRef]
  23. Y. R. Shen, Principles of Nonlinear Optics (Wiley, New York, 1984), Chap. 11.
  24. D. E. Govoni, J. A. Booze, A. Sinha, and F. F. Crim, Chem. Phys. Lett. 216, 525 (1993).
    [CrossRef]

1994

Auth, D. C.

D. C. Auth, Appl. Phys. Lett. 16, 512 (1970).
[CrossRef]

Booze, J. A.

D. E. Govoni, J. A. Booze, A. Sinha, and F. F. Crim, Chem. Phys. Lett. 216, 525 (1993).
[CrossRef]

Buntine, M. A.

M. A. Buntine, D. W. Chandler, and C. C. Hayden, "Detection of vibrational-overtone excitation in water via laser-induced grating spectroscopy," J. Chem. Phys. (to be published).

Cambi, R.

R. Cambi, D. Cappelletti, G. Luiti, and R. Pirani, J. Chem. Phys. 95, 1852 (1991).
[CrossRef]

Cappelletti, D.

R. Cambi, D. Cappelletti, G. Luiti, and R. Pirani, J. Chem. Phys. 95, 1852 (1991).
[CrossRef]

Chandler, D. W.

M. A. Buntine, D. W. Chandler, and C. C. Hayden, "Detection of vibrational-overtone excitation in water via laser-induced grating spectroscopy," J. Chem. Phys. (to be published).

Coltrin, M. E.

R. J. Kee, G. Dixon-Lewis, J. Warnatz, M. E. Coltrin, and J. A. Miller, "A Fortran computer code package for the evaluation of gas-phase multicomponent transport properties," Rep. SAND86-8246.UC-32, Sandia National Laboratories, Livermore, Calif., 1986.

Crim, F. F.

D. E. Govoni, J. A. Booze, A. Sinha, and F. F. Crim, Chem. Phys. Lett. 216, 525 (1993).
[CrossRef]

Cummings, E. B.

E. B. Cummings, Opt. Lett. 19, 1361 (1994).
[CrossRef] [PubMed]

Danehy, P. M.

P. M. Danehy, E. J. Friedman-Hill, P. H. Paul, and R. L. Farrow, "Thermal-grating contributions to degenerate fourwave mixing in nitric oxide," submitted to J. Opt. Soc. Am. B.

R. L. Vander Wal, B. E. Holmes, J. B. Jefferies, P. M. Danehy, R. L. Farrow, and D. J. Rakestraw, Chem. Phys. Lett. 191, 251 (1992).
[CrossRef]

Dixon-Lewis, G.

R. J. Kee, G. Dixon-Lewis, J. Warnatz, M. E. Coltrin, and J. A. Miller, "A Fortran computer code package for the evaluation of gas-phase multicomponent transport properties," Rep. SAND86-8246.UC-32, Sandia National Laboratories, Livermore, Calif., 1986.

Durant, J. L.

J. W. Thom, Jr., J. A. Gray, J. L. Durant, Jr., and P. H. Paul, J. Chem. Phys. 97, 8156 (1992); J. A. Gray, P. H. Paul, and J. L. Durant, Chem. Phys. Lett. 190, 266 (1992).
[CrossRef]

Eichler, H.

H. Eichler and H. Stahl, Opt. Commun. 6, 239 (1972); J. Appl. Phys. 44, 3429 (1973).
[CrossRef]

Eichler, H. J.

H. J. Eichler, P. Günter, and D. W. Pohl, Laser-Induced Dynamic Gratings (Springer-Verlag, Berlin, 1986), pp. 29.

Farrow, R. L.

R. L. Vander Wal, B. E. Holmes, J. B. Jefferies, P. M. Danehy, R. L. Farrow, and D. J. Rakestraw, Chem. Phys. Lett. 191, 251 (1992).
[CrossRef]

R. L. Farrow and D. J. Rakestraw, Science 257, 1984 (1992).
[CrossRef]

P. M. Danehy, E. J. Friedman-Hill, P. H. Paul, and R. L. Farrow, "Thermal-grating contributions to degenerate fourwave mixing in nitric oxide," submitted to J. Opt. Soc. Am. B.

Fayer, M. D.

K. A. Nelson, D. R. Lutz, and M. D. Fayer, Phys. Rev. A 24, 3261 (1981); K. A. Nelson, R. J. D. Miller, D. R. Lutz, and M. D. Fayer, J. Appl. Phys. 53, 1144 (1982); R. J. D. Miller, R. Casalegno, K. A. Nelson, and M. D. Fayer, Chem. Phys. 72, 371 (1982).
[CrossRef]

Forsman, J. W.

Friedman-Hill, E. J.

P. M. Danehy, E. J. Friedman-Hill, P. H. Paul, and R. L. Farrow, "Thermal-grating contributions to degenerate fourwave mixing in nitric oxide," submitted to J. Opt. Soc. Am. B.

Govoni, D. E.

D. E. Govoni, J. A. Booze, A. Sinha, and F. F. Crim, Chem. Phys. Lett. 216, 525 (1993).
[CrossRef]

Gray, J. A.

J. W. Thom, Jr., J. A. Gray, J. L. Durant, Jr., and P. H. Paul, J. Chem. Phys. 97, 8156 (1992); J. A. Gray, P. H. Paul, and J. L. Durant, Chem. Phys. Lett. 190, 266 (1992).
[CrossRef]

Gray, M. A.

R. M. Herman and M. A. Gray, Phys. Rev. Lett. 19, 324 (1967).
[CrossRef]

Günter, P.

H. J. Eichler, P. Günter, and D. W. Pohl, Laser-Induced Dynamic Gratings (Springer-Verlag, Berlin, 1986), pp. 29.

Hayden, C. C.

M. A. Buntine, D. W. Chandler, and C. C. Hayden, "Detection of vibrational-overtone excitation in water via laser-induced grating spectroscopy," J. Chem. Phys. (to be published).

Herman, R. M.

R. M. Herman and M. A. Gray, Phys. Rev. Lett. 19, 324 (1967).
[CrossRef]

Holmes, B. E.

R. L. Vander Wal, B. E. Holmes, J. B. Jefferies, P. M. Danehy, R. L. Farrow, and D. J. Rakestraw, Chem. Phys. Lett. 191, 251 (1992).
[CrossRef]

Jefferies, J. B.

R. L. Vander Wal, B. E. Holmes, J. B. Jefferies, P. M. Danehy, R. L. Farrow, and D. J. Rakestraw, Chem. Phys. Lett. 191, 251 (1992).
[CrossRef]

Kaiser, W.

D. Pohl, I. Reinhold, and W. Kaiser, Phys. Rev. Lett. 20, 1141 (1968).
[CrossRef]

Kee, R. J.

R. J. Kee, G. Dixon-Lewis, J. Warnatz, M. E. Coltrin, and J. A. Miller, "A Fortran computer code package for the evaluation of gas-phase multicomponent transport properties," Rep. SAND86-8246.UC-32, Sandia National Laboratories, Livermore, Calif., 1986.

Kruger, C. K.

W. G. Vincenti and C. K. Kruger, Introduction to Physical Gas Dynamics (Krieger, New York, 1967), Chap. 12.

Landau, L.

L. Landau and G. Placzek, Z. Phys. 5, 172 (1933).

Luiti, G.

R. Cambi, D. Cappelletti, G. Luiti, and R. Pirani, J. Chem. Phys. 95, 1852 (1991).
[CrossRef]

Lutz, D. R.

K. A. Nelson, D. R. Lutz, and M. D. Fayer, Phys. Rev. A 24, 3261 (1981); K. A. Nelson, R. J. D. Miller, D. R. Lutz, and M. D. Fayer, J. Appl. Phys. 53, 1144 (1982); R. J. D. Miller, R. Casalegno, K. A. Nelson, and M. D. Fayer, Chem. Phys. 72, 371 (1982).
[CrossRef]

Mack, M. E.

M. E. Mack, Ann. N.Y. Acad. Sci. 168, 419 (1970).
[CrossRef]

Miller, J. A.

R. J. Kee, G. Dixon-Lewis, J. Warnatz, M. E. Coltrin, and J. A. Miller, "A Fortran computer code package for the evaluation of gas-phase multicomponent transport properties," Rep. SAND86-8246.UC-32, Sandia National Laboratories, Livermore, Calif., 1986.

Mountain, R. D.

R. D. Mountain, Rev. Mod. Phys. 38, 205 (1966).
[CrossRef]

Nelson, K. A.

K. A. Nelson, D. R. Lutz, and M. D. Fayer, Phys. Rev. A 24, 3261 (1981); K. A. Nelson, R. J. D. Miller, D. R. Lutz, and M. D. Fayer, J. Appl. Phys. 53, 1144 (1982); R. J. D. Miller, R. Casalegno, K. A. Nelson, and M. D. Fayer, Chem. Phys. 72, 371 (1982).
[CrossRef]

Paul, P. H.

S. Williams, L. A. Rahn, P. H. Paul, J. W. Forsman, and R. N. Zare, "Laser-induced thermal-grating effects in flames," Opt. Lett. 19, 1681–1683 (1994).
[CrossRef] [PubMed]

P. M. Danehy, E. J. Friedman-Hill, P. H. Paul, and R. L. Farrow, "Thermal-grating contributions to degenerate fourwave mixing in nitric oxide," submitted to J. Opt. Soc. Am. B.

J. W. Thom, Jr., J. A. Gray, J. L. Durant, Jr., and P. H. Paul, J. Chem. Phys. 97, 8156 (1992); J. A. Gray, P. H. Paul, and J. L. Durant, Chem. Phys. Lett. 190, 266 (1992).
[CrossRef]

Pirani, R.

R. Cambi, D. Cappelletti, G. Luiti, and R. Pirani, J. Chem. Phys. 95, 1852 (1991).
[CrossRef]

Placzek, G.

L. Landau and G. Placzek, Z. Phys. 5, 172 (1933).

Pohl, D.

D. Pohl, I. Reinhold, and W. Kaiser, Phys. Rev. Lett. 20, 1141 (1968).
[CrossRef]

Pohl, D. W.

H. J. Eichler, P. Günter, and D. W. Pohl, Laser-Induced Dynamic Gratings (Springer-Verlag, Berlin, 1986), pp. 29.

Rahn, L. A.

Rakestraw, D. J.

R. L. Farrow and D. J. Rakestraw, Science 257, 1984 (1992).
[CrossRef]

R. L. Vander Wal, B. E. Holmes, J. B. Jefferies, P. M. Danehy, R. L. Farrow, and D. J. Rakestraw, Chem. Phys. Lett. 191, 251 (1992).
[CrossRef]

Reinhold, I.

D. Pohl, I. Reinhold, and W. Kaiser, Phys. Rev. Lett. 20, 1141 (1968).
[CrossRef]

Salcedo, J. R.

J. R. Salcedo and A. E. Seigman, IEEE J. Quantum Electron. QE-15, 250 (1979).
[CrossRef]

Seigman, A. E.

J. R. Salcedo and A. E. Seigman, IEEE J. Quantum Electron. QE-15, 250 (1979).
[CrossRef]

Shen, Y. R.

Y. R. Shen, Principles of Nonlinear Optics (Wiley, New York, 1984), Chap. 11.

Sinha, A.

D. E. Govoni, J. A. Booze, A. Sinha, and F. F. Crim, Chem. Phys. Lett. 216, 525 (1993).
[CrossRef]

Stahl, H.

H. Eichler and H. Stahl, Opt. Commun. 6, 239 (1972); J. Appl. Phys. 44, 3429 (1973).
[CrossRef]

Thom, J. W.

J. W. Thom, Jr., J. A. Gray, J. L. Durant, Jr., and P. H. Paul, J. Chem. Phys. 97, 8156 (1992); J. A. Gray, P. H. Paul, and J. L. Durant, Chem. Phys. Lett. 190, 266 (1992).
[CrossRef]

Vincenti, W. G.

W. G. Vincenti and C. K. Kruger, Introduction to Physical Gas Dynamics (Krieger, New York, 1967), Chap. 12.

Wal, R. L. Vander

R. L. Vander Wal, B. E. Holmes, J. B. Jefferies, P. M. Danehy, R. L. Farrow, and D. J. Rakestraw, Chem. Phys. Lett. 191, 251 (1992).
[CrossRef]

Warnatz, J.

R. J. Kee, G. Dixon-Lewis, J. Warnatz, M. E. Coltrin, and J. A. Miller, "A Fortran computer code package for the evaluation of gas-phase multicomponent transport properties," Rep. SAND86-8246.UC-32, Sandia National Laboratories, Livermore, Calif., 1986.

Williams, S.

Zare, R. N.

Opt. Lett.

Other

P. M. Danehy, E. J. Friedman-Hill, P. H. Paul, and R. L. Farrow, "Thermal-grating contributions to degenerate fourwave mixing in nitric oxide," submitted to J. Opt. Soc. Am. B.

R. L. Vander Wal, B. E. Holmes, J. B. Jefferies, P. M. Danehy, R. L. Farrow, and D. J. Rakestraw, Chem. Phys. Lett. 191, 251 (1992).
[CrossRef]

L. Landau and G. Placzek, Z. Phys. 5, 172 (1933).

M. E. Mack, Ann. N.Y. Acad. Sci. 168, 419 (1970).
[CrossRef]

R. D. Mountain, Rev. Mod. Phys. 38, 205 (1966).
[CrossRef]

R. M. Herman and M. A. Gray, Phys. Rev. Lett. 19, 324 (1967).
[CrossRef]

D. Pohl, I. Reinhold, and W. Kaiser, Phys. Rev. Lett. 20, 1141 (1968).
[CrossRef]

H. Eichler and H. Stahl, Opt. Commun. 6, 239 (1972); J. Appl. Phys. 44, 3429 (1973).
[CrossRef]

J. R. Salcedo and A. E. Seigman, IEEE J. Quantum Electron. QE-15, 250 (1979).
[CrossRef]

K. A. Nelson, D. R. Lutz, and M. D. Fayer, Phys. Rev. A 24, 3261 (1981); K. A. Nelson, R. J. D. Miller, D. R. Lutz, and M. D. Fayer, J. Appl. Phys. 53, 1144 (1982); R. J. D. Miller, R. Casalegno, K. A. Nelson, and M. D. Fayer, Chem. Phys. 72, 371 (1982).
[CrossRef]

E. B. Cummings, Opt. Lett. 19, 1361 (1994).
[CrossRef] [PubMed]

W. G. Vincenti and C. K. Kruger, Introduction to Physical Gas Dynamics (Krieger, New York, 1967), Chap. 12.

H. J. Eichler, P. Günter, and D. W. Pohl, Laser-Induced Dynamic Gratings (Springer-Verlag, Berlin, 1986), pp. 29.

D. E. Gray, ed., American Institute of Physics Handbook (McGraw-Hill, New York, 1972), Chap. 6e.

R. L. Farrow and D. J. Rakestraw, Science 257, 1984 (1992).
[CrossRef]

J. W. Thom, Jr., J. A. Gray, J. L. Durant, Jr., and P. H. Paul, J. Chem. Phys. 97, 8156 (1992); J. A. Gray, P. H. Paul, and J. L. Durant, Chem. Phys. Lett. 190, 266 (1992).
[CrossRef]

R. J. Kee, G. Dixon-Lewis, J. Warnatz, M. E. Coltrin, and J. A. Miller, "A Fortran computer code package for the evaluation of gas-phase multicomponent transport properties," Rep. SAND86-8246.UC-32, Sandia National Laboratories, Livermore, Calif., 1986.

R. H. Perry and C. H. Chilton, eds., Chemical Engineering Handbook, 5th ed. (McGraw-Hill, New York, 1973), pp. 3–231.

R. Cambi, D. Cappelletti, G. Luiti, and R. Pirani, J. Chem. Phys. 95, 1852 (1991).
[CrossRef]

Y. R. Shen, Principles of Nonlinear Optics (Wiley, New York, 1984), Chap. 11.

D. E. Govoni, J. A. Booze, A. Sinha, and F. F. Crim, Chem. Phys. Lett. 216, 525 (1993).
[CrossRef]

D. C. Auth, Appl. Phys. Lett. 16, 512 (1970).
[CrossRef]

M. A. Buntine, D. W. Chandler, and C. C. Hayden, "Detection of vibrational-overtone excitation in water via laser-induced grating spectroscopy," J. Chem. Phys. (to be published).

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Figures (7)

Fig. 1
Fig. 1

Density, temperature, and pressure transients predicted for a spatially infinite volume grating in air at 295 K and 760 Torr with a fringe spacing of 5 μm.

Fig. 2
Fig. 2

Refractive-index transient squared for the conditions of Fig. 1 shown as convolved with a probe pulse of various widths.

Fig. 3
Fig. 3

Refractive-index transient squared for the conditions of Fig. 1 with Q = 0.05 ns for Sc ≫ 1 and Sc = 1/3. The result is shown as convolved with an 8-ns probe pulse.

Fig. 4
Fig. 4

Predicted and measured thermal-grating signals obtained in 760 Torr of CO2 at 295 K. The prediction has been convolved with an 8-ns probe pulse.

Fig. 5
Fig. 5

Predicted and measured thermal-grating signals obtained in 100 Torr of CO2 at 300 K. The prediction has been convolved with an 8-ns probe pulse and is shown with and without the action of excited-state grating diffusion.

Fig. 6
Fig. 6

Predicted and measured thermal-grating signals obtained in the products of a laminar premixed H2–O2 flame at atmospheric pressure. The fringe spacing is 44 μm. The predictions are shown for single- and two-step thermalization models.

Fig. 7
Fig. 7

Refractive-index transient squared for conditions of Fig. 1 at a pressure of 1 atm (760 Torr) and an absorption coefficient of 5 × 10−7 cm−1. Lower panel, pure thermal and electrostrictive gratings. Upper panel, combined thermal and electrostrictive gratings.

Equations (40)

Equations on this page are rendered with MathJax. Learn more.

N * t = E P B N a 0 2 [ 1 + cos ( 2 π x / Λ ) ] g ( t , τ L ) f ( x , d ) ( A + Q ) N * + D 2 N * x 2 ,
g ( t ) = 2 t τ L 2 exp [ ( t τ L ) 2 ] .
ρ ζ + u ξ = 0 ,
u ζ + 1 γ P ξ 4 3 Re 2 u ξ 2 = 0 ,
1 γ P ζ ρ ζ 1 Re Pr ( 2 P ξ 2 2 ρ ξ 2 ) = γ 1 γ h υ a Q Λ c 0 N * ( ζ , ξ ) P 0 .
N ζ * = E p B N a 0 2 [ 1 + cos ( 2 π ξ ) ] F ( ξ ) G ( ζ ) ( A + Q ) τ N * + 1 Re Sc N ξ ξ * .
G ( ζ ) = 2 ( τ / τ L ) 2 ζ exp [ ( ζ τ / τ L ) 2 ] .
ρ ζ ζ ζ ( 4 3 Re + γ Re Pr ) ρ ξ ξ ζ ζ ρ ξ ξ ζ + 4 γ 3 Re 2 Pr ρ ζ ξ ξ ξ ξ + 1 Re Pr ρ ξ ξ ξ ξ = γ 1 γ h υ a Q τ P 0 N ξ ξ * .
N ξ ξ * = ( 2 π ) 2 ( 8 E p / π d 2 ) B N a 0 cos ( 2 π ξ ) h ( ζ ) ,
h ( ζ ) = 0 ζ 2 y ( τ τ L ) 2 exp { [ ( A + Q ) τ + ( 2 π ) 2 Re Sc ] ( y ζ ) ( τ τ L y ) 2 } d y .
ρ = ( 2 π ) 2 γ 1 γ α P 0 8 E p π d 2 cos ( 2 π ξ ) Z ( ζ ) .
Z ζ ζ ζ + [ 4 ( 2 π ) 2 3 Re + ( 2 π ) 2 γ Re Pr ] Z ζ ζ + [ ( 2 π ) 2 + 4 ( 2 π ) 4 γ 3 Re 2 Pr ] Z ζ + ( 2 π ) 4 Re Pr Z = Q τ h ( ζ ) ,
Z ( ζ ) = Q τ 0 ζ W ( y ) h ( ζ y ) d y .
w ( s ) = s 3 + [ 4 ( 2 π ) 2 3 Re + ( 2 π ) 2 γ Re Pr ] s 2 + [ ( 2 π ) 2 + 4 ( 2 π ) 4 γ 3 Re 2 Pr ] s + ( 2 π ) 4 Re Pr .
W ( ζ ) = j = 1 3 exp ( s j ζ ) [ w ( s ) s s = s j ] 1 .
W ( ζ ) = a 1 exp ( s 1 ζ ) 2 a 2 exp ( u ζ ) cos ( υ ζ + ϕ ) .
( γ 1 ) | 3 4 Pr | ( 2 π ) 6 / Re 3 1 ,
s 1 = ( 2 π ) 2 / Re Pr ,
s 2 , 3 = ( γ 1 + 4 Pr / 3 ) ( 2 π ) 2 / 2 Re Pr ± 2 π i .
η I = ( π l / λ ) 2 | Δ n TG + Δ n PG | 2 .
Δ n = ( n R 1 ) β T R ( ρ 0 / ρ R ) × [ ρ ( 1 1 β T R β T 0 ) + T 1 β T R β T 0 ] ,
Δ α = T 0 α T T 0 T + ρ 0 α ρ ρ 0 ρ .
n 1 = n R 1 1 β T R + β T P P R ,
N ζ * = E p B N a 0 2 [ 1 + cos ( 2 π ξ ) ] F ( ξ , d ) G ( ζ , τ L ) ( A + Q ) τ N * + N ξ ξ * / Re Sc ,
M ζ * = q τ N * R τ M * + M ξ ξ * / Re Sc M ,
Ė = Q τ N * + ( 1 ) R Q τ 2 M * .
Ė = Q τ N * + ( 1 ) R ( Q + A ) τ 2 M * .
S = exp [ ( q τ + 4 π 2 / Re Sc ) ζ ] * { Q τ + ( 1 ) R q τ 2 × exp [ ( R τ + 4 π 2 / Re Sc M ) ζ ] } * G ( τ L , ζ ) ,
Z ( ζ ) = exp [ ( q τ + 4 π 2 / Re Sc ) ζ ] * ( W ( ζ ) * { Q τ + ( 1 ) R q τ 2 × exp [ ( R τ + 4 π 2 / Re Sc M ) ζ ] } ) * G ( τ L , ζ ) .
S M = Γ E γ P 0 E p G ( ζ , τ L ) ξ [ F ( ξ , d ) ( 1 + cos ( 2 π ξ ) ]
Γ E = 1 2 π n c n 2 1 2 n 2 + 2 3 .
ρ = ( 2 π ) 2 γ P 0 Λ 8 E p π d 2 cos ( 2 π ξ ) Z ( ζ ) .
S = ( γ 1 ) Q τ α Λ h ( ζ ) c 0 Γ E [ G ζ ( ζ ) + ( 2 π ) 2 Re Pr G ( ζ ) ] ,
S ( γ 1 ) α Λ G ( ζ ) c 0 Γ E [ G ζ ( ζ ) + ( 2 π ) 2 Re Pr G ( ζ ) ] .
s 1 ( 2 π ) 2 Re Pr [ 1 + ( 2 π ) 2 Re 2 Pr 2 ( γ 1 ) ( 1 4 Pr 3 ) ] ,
s 2 , 3 ( 2 π ) 2 2 Re Pr [ 4 Pr 3 + γ 1 ( 2 π ) 2 2 Re 2 Pr 2 ( γ 1 ) ( 1 4 Pr 3 ) ] ± 2 π i { 1 ( 2 π ) 2 Re 2 Pr 2 [ ( γ 1 ) ( 1 4 Pr 3 ) + ( 4 Pr 3 + γ 1 ) 2 / 4 ] } 1 / 2 .
W ( t ) = a 1 4 π 2 exp ( 4 π 2 K t c p ρ 0 Λ 2 ) 2 a 2 4 π 2 × exp { 4 π 2 η t Λ 2 [ 2 3 + ( γ 1 ) K 2 c p ρ 0 η ] } × cos ( 2 π c 0 t / Λ + ϕ ) ,
ϕ = arctan [ 2 π ( 3 γ 4 Pr 3 ) η 2 Pr Λ c 0 ] .
W ( ζ ) * Ė ( ζ ) = a 1 Q τ 4 π 2 exp ( s 1 ζ ) exp ( Θ ζ ) Θ + s 1 2 a 2 Q τ 4 π 2 × exp ( u ζ ) cos ( υ ζ + ϕ + ϕ 2 ) exp ( Θ ζ ) cos ( ϕ + ϕ 2 ) [ ( Θ u ) 2 + υ 2 ] 1 / 2
Θ = ( A + Q ) τ + 4 π 2 / Re Sc , ϕ 2 = arctan [ υ / ( Θ u ) ] .

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