Abstract

The theoretical basis for photoacoustic Raman spectroscopy (PARS) is developed. An expression for the Raman gain coefficient is derived. This gain coefficient is used to determine the change in the internal translational energy of a gaseous sample that is produced by illumination with two laser beams whose frequency difference corresponds to a Raman frequency shift. The magnitude of the pressure change associated with this nonlinear Raman process was deduced for a simple quasi-equilibrium model appropriate for modulated cw laser excitation and for a kinetic model applicable to pulsed laser excitation. The kinetic model explicitly accounts for the various pumping and relaxation rates associated with the PARS process.

© 1981 Optical Society of America

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References

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  1. J. J. Barrett, “Photoacoustic Raman spectroscopy of gases,” in Chemical Applications of Nonlinear Raman Spectroscopy, A. B. Harvey, ed. (Academic, New York, 1981), pp. 89–169.
  2. J. J. Barrett and M. J. Berry, “Photoacoustic Raman scattering in gases,” in Proceedings of the Sixth International Conference on Raman Spectroscopy, E. Schmid, R. Krishnan, W. Kiefer, and H. Schrötter, eds. (Heyden, London, 1978), Vol. 1, pp. 466–467; “Photoacoustic Raman spectroscopy using cw laser sources,” Appl. Phys. Lett. 34, 144–146 (1979).
  3. G. A. West, D. R. Siebert, and J. J. Barrett, “Gas phase photoacoustic Raman spectroscopy using pulsed laser excitation,” J. Appl. Phys. 51, 2823–2828 (1980).
    [CrossRef]
  4. D. R. Siebert, G. A. West, and J. J. Barrett, “Gaseous trace analysis using pulsed photoacoustic Raman spectroscopy,” Appl. Opt. 19, 53–60 (1980).
    [CrossRef] [PubMed]
  5. Preliminary results were reported by J. J. Barrett at the Second Chemical Congress of the North American Continent, Las Vegas, Nev., August 24–29, 1980, and at the 1980 Federation of Analytical Chemistry and Spectroscopy Societies (FACSS) Meeting, Philadelphia, Pa., September 28–October 3, 1980.
  6. G. A. West and J. J. Barrett, “Pure rotational stimulated Raman photoacoustic spectroscopy,” Opt. Lett. 4, 395–397 (1979).
    [CrossRef] [PubMed]
  7. G. A. West and J. J. Barrett, “Photoacoustic pure rotational Raman spectroscopy,” in The Proceedings of the VII International Conference on Raman Spectroscopy (North-Holland, Amsterdam, 1980), pp. 696–697.
  8. A. G. Bell, “On the production and reproduction of sound by light,” Proc. Am. Assoc. Adv. Sci. 29, 115–136 (1880); “Upon the production of sound by radiant energy,” Philos. Mag. 11, 510–528 (1881).
  9. J. Tyndall, “Action of an intermittent beam of radiant heat upon gaseous matter,” Proc. R. Soc. London 31, 307–317 (1881).
  10. E. L. Kerr and J. G. Atwood, “The laser illuminated absorptivity spectrophone: a method for measurement of weak absorptivity in gases at laser wavelengths,” Appl. Opt. 7, 915–921 (1968).
    [CrossRef] [PubMed]
  11. L. B. Kreuzer, “Ultralow gas concentration infrared absorption spectroscopy,” J. Appl. Phys. 42, 2934–2943 (1971).
    [CrossRef]
  12. L. B. Kreuzer and C. K. N. Patel, “Nitric oxide air pollution: detection by optoacoustic spectroscopy,” Science 173, 45–47 (1971).
    [CrossRef] [PubMed]
  13. L. B. Kreuzer, N. D. Kenyon, and C. K. N. Patel, “Air pollution: sensitive detection of ten pollutant gases by carbon monoxide and carbon dioxide lasers,” Science 177, 347–349 (1972).
    [CrossRef] [PubMed]
  14. L. B. Kreuzer, “Laser optoacoustic spectroscopy—A new technique of gas analysis,” Anal. Chem. 46, 239A–244A (1974).
  15. C. K. N. Patel, E. G. Burkhardt, and C. A. Lambert, “Spectroscopic measurements of stratospheric nitric oxide and water vapor,” Science 184, 1173–1176 (1974).
    [CrossRef] [PubMed]
  16. E. Kritchman, S. Shtrikman, and M. Slatkine, “Resonant optoacoustic cells for trace gas analysis,” J. Opt. Soc. Am. 68, 1257–1271 (1978).
    [CrossRef]
  17. K. V. Reddy, R. G. Bray, and M. J. Berry, “Dye laser-induced photochemistry,” in Advances in Laser Chemistry, A. H. Zewail, ed. (Springer-Verlag, Berlin, 1978), pp. 48–61.
  18. M. W. Tolles, J. W. Nibler, J. R. McDonald, and A. B. Harvey, “A review of the theory and application of coherent anti-Stokes Raman spectroscopy (CARS),” Appl. Spectrosc. 31, 253–271 (1977).
    [CrossRef]
  19. D. Heiman, R. W. Hellwarth, M. D. Levenson, and G. Martin, “Raman-induced Kerr effect,” Phys. Rev. Lett. 36, 189–192 (1976).
    [CrossRef]
  20. A. Owyoung, “Coherent Raman gain spectroscopy using cw laser sources,” IEEE J. Quantum Electron. QE-14, 192–203 (1978).
    [CrossRef]
  21. M. Maier, W. Kaiser, and J. A. Giordmaine, “Backward stimulated Raman scattering,” Phys. Rev. 177, 580–599 (1969).
    [CrossRef]
  22. M. Maier, “Applications of stimulated Raman scattering,” Appl. Phys. 11, 209–231 (1976).
    [CrossRef]

1980 (2)

G. A. West, D. R. Siebert, and J. J. Barrett, “Gas phase photoacoustic Raman spectroscopy using pulsed laser excitation,” J. Appl. Phys. 51, 2823–2828 (1980).
[CrossRef]

D. R. Siebert, G. A. West, and J. J. Barrett, “Gaseous trace analysis using pulsed photoacoustic Raman spectroscopy,” Appl. Opt. 19, 53–60 (1980).
[CrossRef] [PubMed]

1979 (1)

1978 (2)

E. Kritchman, S. Shtrikman, and M. Slatkine, “Resonant optoacoustic cells for trace gas analysis,” J. Opt. Soc. Am. 68, 1257–1271 (1978).
[CrossRef]

A. Owyoung, “Coherent Raman gain spectroscopy using cw laser sources,” IEEE J. Quantum Electron. QE-14, 192–203 (1978).
[CrossRef]

1977 (1)

1976 (2)

D. Heiman, R. W. Hellwarth, M. D. Levenson, and G. Martin, “Raman-induced Kerr effect,” Phys. Rev. Lett. 36, 189–192 (1976).
[CrossRef]

M. Maier, “Applications of stimulated Raman scattering,” Appl. Phys. 11, 209–231 (1976).
[CrossRef]

1974 (2)

L. B. Kreuzer, “Laser optoacoustic spectroscopy—A new technique of gas analysis,” Anal. Chem. 46, 239A–244A (1974).

C. K. N. Patel, E. G. Burkhardt, and C. A. Lambert, “Spectroscopic measurements of stratospheric nitric oxide and water vapor,” Science 184, 1173–1176 (1974).
[CrossRef] [PubMed]

1972 (1)

L. B. Kreuzer, N. D. Kenyon, and C. K. N. Patel, “Air pollution: sensitive detection of ten pollutant gases by carbon monoxide and carbon dioxide lasers,” Science 177, 347–349 (1972).
[CrossRef] [PubMed]

1971 (2)

L. B. Kreuzer, “Ultralow gas concentration infrared absorption spectroscopy,” J. Appl. Phys. 42, 2934–2943 (1971).
[CrossRef]

L. B. Kreuzer and C. K. N. Patel, “Nitric oxide air pollution: detection by optoacoustic spectroscopy,” Science 173, 45–47 (1971).
[CrossRef] [PubMed]

1969 (1)

M. Maier, W. Kaiser, and J. A. Giordmaine, “Backward stimulated Raman scattering,” Phys. Rev. 177, 580–599 (1969).
[CrossRef]

1968 (1)

1881 (1)

J. Tyndall, “Action of an intermittent beam of radiant heat upon gaseous matter,” Proc. R. Soc. London 31, 307–317 (1881).

1880 (1)

A. G. Bell, “On the production and reproduction of sound by light,” Proc. Am. Assoc. Adv. Sci. 29, 115–136 (1880); “Upon the production of sound by radiant energy,” Philos. Mag. 11, 510–528 (1881).

Atwood, J. G.

Barrett, J. J.

D. R. Siebert, G. A. West, and J. J. Barrett, “Gaseous trace analysis using pulsed photoacoustic Raman spectroscopy,” Appl. Opt. 19, 53–60 (1980).
[CrossRef] [PubMed]

G. A. West, D. R. Siebert, and J. J. Barrett, “Gas phase photoacoustic Raman spectroscopy using pulsed laser excitation,” J. Appl. Phys. 51, 2823–2828 (1980).
[CrossRef]

G. A. West and J. J. Barrett, “Pure rotational stimulated Raman photoacoustic spectroscopy,” Opt. Lett. 4, 395–397 (1979).
[CrossRef] [PubMed]

Preliminary results were reported by J. J. Barrett at the Second Chemical Congress of the North American Continent, Las Vegas, Nev., August 24–29, 1980, and at the 1980 Federation of Analytical Chemistry and Spectroscopy Societies (FACSS) Meeting, Philadelphia, Pa., September 28–October 3, 1980.

G. A. West and J. J. Barrett, “Photoacoustic pure rotational Raman spectroscopy,” in The Proceedings of the VII International Conference on Raman Spectroscopy (North-Holland, Amsterdam, 1980), pp. 696–697.

J. J. Barrett, “Photoacoustic Raman spectroscopy of gases,” in Chemical Applications of Nonlinear Raman Spectroscopy, A. B. Harvey, ed. (Academic, New York, 1981), pp. 89–169.

J. J. Barrett and M. J. Berry, “Photoacoustic Raman scattering in gases,” in Proceedings of the Sixth International Conference on Raman Spectroscopy, E. Schmid, R. Krishnan, W. Kiefer, and H. Schrötter, eds. (Heyden, London, 1978), Vol. 1, pp. 466–467; “Photoacoustic Raman spectroscopy using cw laser sources,” Appl. Phys. Lett. 34, 144–146 (1979).

Bell, A. G.

A. G. Bell, “On the production and reproduction of sound by light,” Proc. Am. Assoc. Adv. Sci. 29, 115–136 (1880); “Upon the production of sound by radiant energy,” Philos. Mag. 11, 510–528 (1881).

Berry, M. J.

K. V. Reddy, R. G. Bray, and M. J. Berry, “Dye laser-induced photochemistry,” in Advances in Laser Chemistry, A. H. Zewail, ed. (Springer-Verlag, Berlin, 1978), pp. 48–61.

J. J. Barrett and M. J. Berry, “Photoacoustic Raman scattering in gases,” in Proceedings of the Sixth International Conference on Raman Spectroscopy, E. Schmid, R. Krishnan, W. Kiefer, and H. Schrötter, eds. (Heyden, London, 1978), Vol. 1, pp. 466–467; “Photoacoustic Raman spectroscopy using cw laser sources,” Appl. Phys. Lett. 34, 144–146 (1979).

Bray, R. G.

K. V. Reddy, R. G. Bray, and M. J. Berry, “Dye laser-induced photochemistry,” in Advances in Laser Chemistry, A. H. Zewail, ed. (Springer-Verlag, Berlin, 1978), pp. 48–61.

Burkhardt, E. G.

C. K. N. Patel, E. G. Burkhardt, and C. A. Lambert, “Spectroscopic measurements of stratospheric nitric oxide and water vapor,” Science 184, 1173–1176 (1974).
[CrossRef] [PubMed]

Giordmaine, J. A.

M. Maier, W. Kaiser, and J. A. Giordmaine, “Backward stimulated Raman scattering,” Phys. Rev. 177, 580–599 (1969).
[CrossRef]

Harvey, A. B.

Heiman, D.

D. Heiman, R. W. Hellwarth, M. D. Levenson, and G. Martin, “Raman-induced Kerr effect,” Phys. Rev. Lett. 36, 189–192 (1976).
[CrossRef]

Hellwarth, R. W.

D. Heiman, R. W. Hellwarth, M. D. Levenson, and G. Martin, “Raman-induced Kerr effect,” Phys. Rev. Lett. 36, 189–192 (1976).
[CrossRef]

Kaiser, W.

M. Maier, W. Kaiser, and J. A. Giordmaine, “Backward stimulated Raman scattering,” Phys. Rev. 177, 580–599 (1969).
[CrossRef]

Kenyon, N. D.

L. B. Kreuzer, N. D. Kenyon, and C. K. N. Patel, “Air pollution: sensitive detection of ten pollutant gases by carbon monoxide and carbon dioxide lasers,” Science 177, 347–349 (1972).
[CrossRef] [PubMed]

Kerr, E. L.

Kreuzer, L. B.

L. B. Kreuzer, “Laser optoacoustic spectroscopy—A new technique of gas analysis,” Anal. Chem. 46, 239A–244A (1974).

L. B. Kreuzer, N. D. Kenyon, and C. K. N. Patel, “Air pollution: sensitive detection of ten pollutant gases by carbon monoxide and carbon dioxide lasers,” Science 177, 347–349 (1972).
[CrossRef] [PubMed]

L. B. Kreuzer, “Ultralow gas concentration infrared absorption spectroscopy,” J. Appl. Phys. 42, 2934–2943 (1971).
[CrossRef]

L. B. Kreuzer and C. K. N. Patel, “Nitric oxide air pollution: detection by optoacoustic spectroscopy,” Science 173, 45–47 (1971).
[CrossRef] [PubMed]

Kritchman, E.

Lambert, C. A.

C. K. N. Patel, E. G. Burkhardt, and C. A. Lambert, “Spectroscopic measurements of stratospheric nitric oxide and water vapor,” Science 184, 1173–1176 (1974).
[CrossRef] [PubMed]

Levenson, M. D.

D. Heiman, R. W. Hellwarth, M. D. Levenson, and G. Martin, “Raman-induced Kerr effect,” Phys. Rev. Lett. 36, 189–192 (1976).
[CrossRef]

Maier, M.

M. Maier, “Applications of stimulated Raman scattering,” Appl. Phys. 11, 209–231 (1976).
[CrossRef]

M. Maier, W. Kaiser, and J. A. Giordmaine, “Backward stimulated Raman scattering,” Phys. Rev. 177, 580–599 (1969).
[CrossRef]

Martin, G.

D. Heiman, R. W. Hellwarth, M. D. Levenson, and G. Martin, “Raman-induced Kerr effect,” Phys. Rev. Lett. 36, 189–192 (1976).
[CrossRef]

McDonald, J. R.

Nibler, J. W.

Owyoung, A.

A. Owyoung, “Coherent Raman gain spectroscopy using cw laser sources,” IEEE J. Quantum Electron. QE-14, 192–203 (1978).
[CrossRef]

Patel, C. K. N.

C. K. N. Patel, E. G. Burkhardt, and C. A. Lambert, “Spectroscopic measurements of stratospheric nitric oxide and water vapor,” Science 184, 1173–1176 (1974).
[CrossRef] [PubMed]

L. B. Kreuzer, N. D. Kenyon, and C. K. N. Patel, “Air pollution: sensitive detection of ten pollutant gases by carbon monoxide and carbon dioxide lasers,” Science 177, 347–349 (1972).
[CrossRef] [PubMed]

L. B. Kreuzer and C. K. N. Patel, “Nitric oxide air pollution: detection by optoacoustic spectroscopy,” Science 173, 45–47 (1971).
[CrossRef] [PubMed]

Reddy, K. V.

K. V. Reddy, R. G. Bray, and M. J. Berry, “Dye laser-induced photochemistry,” in Advances in Laser Chemistry, A. H. Zewail, ed. (Springer-Verlag, Berlin, 1978), pp. 48–61.

Shtrikman, S.

Siebert, D. R.

D. R. Siebert, G. A. West, and J. J. Barrett, “Gaseous trace analysis using pulsed photoacoustic Raman spectroscopy,” Appl. Opt. 19, 53–60 (1980).
[CrossRef] [PubMed]

G. A. West, D. R. Siebert, and J. J. Barrett, “Gas phase photoacoustic Raman spectroscopy using pulsed laser excitation,” J. Appl. Phys. 51, 2823–2828 (1980).
[CrossRef]

Slatkine, M.

Tolles, M. W.

Tyndall, J.

J. Tyndall, “Action of an intermittent beam of radiant heat upon gaseous matter,” Proc. R. Soc. London 31, 307–317 (1881).

West, G. A.

G. A. West, D. R. Siebert, and J. J. Barrett, “Gas phase photoacoustic Raman spectroscopy using pulsed laser excitation,” J. Appl. Phys. 51, 2823–2828 (1980).
[CrossRef]

D. R. Siebert, G. A. West, and J. J. Barrett, “Gaseous trace analysis using pulsed photoacoustic Raman spectroscopy,” Appl. Opt. 19, 53–60 (1980).
[CrossRef] [PubMed]

G. A. West and J. J. Barrett, “Pure rotational stimulated Raman photoacoustic spectroscopy,” Opt. Lett. 4, 395–397 (1979).
[CrossRef] [PubMed]

G. A. West and J. J. Barrett, “Photoacoustic pure rotational Raman spectroscopy,” in The Proceedings of the VII International Conference on Raman Spectroscopy (North-Holland, Amsterdam, 1980), pp. 696–697.

Anal. Chem. (1)

L. B. Kreuzer, “Laser optoacoustic spectroscopy—A new technique of gas analysis,” Anal. Chem. 46, 239A–244A (1974).

Appl. Opt. (2)

Appl. Phys. (1)

M. Maier, “Applications of stimulated Raman scattering,” Appl. Phys. 11, 209–231 (1976).
[CrossRef]

Appl. Spectrosc. (1)

IEEE J. Quantum Electron. (1)

A. Owyoung, “Coherent Raman gain spectroscopy using cw laser sources,” IEEE J. Quantum Electron. QE-14, 192–203 (1978).
[CrossRef]

J. Appl. Phys. (2)

G. A. West, D. R. Siebert, and J. J. Barrett, “Gas phase photoacoustic Raman spectroscopy using pulsed laser excitation,” J. Appl. Phys. 51, 2823–2828 (1980).
[CrossRef]

L. B. Kreuzer, “Ultralow gas concentration infrared absorption spectroscopy,” J. Appl. Phys. 42, 2934–2943 (1971).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Lett. (1)

Phys. Rev. (1)

M. Maier, W. Kaiser, and J. A. Giordmaine, “Backward stimulated Raman scattering,” Phys. Rev. 177, 580–599 (1969).
[CrossRef]

Phys. Rev. Lett. (1)

D. Heiman, R. W. Hellwarth, M. D. Levenson, and G. Martin, “Raman-induced Kerr effect,” Phys. Rev. Lett. 36, 189–192 (1976).
[CrossRef]

Proc. Am. Assoc. Adv. Sci. (1)

A. G. Bell, “On the production and reproduction of sound by light,” Proc. Am. Assoc. Adv. Sci. 29, 115–136 (1880); “Upon the production of sound by radiant energy,” Philos. Mag. 11, 510–528 (1881).

Proc. R. Soc. London (1)

J. Tyndall, “Action of an intermittent beam of radiant heat upon gaseous matter,” Proc. R. Soc. London 31, 307–317 (1881).

Science (3)

L. B. Kreuzer and C. K. N. Patel, “Nitric oxide air pollution: detection by optoacoustic spectroscopy,” Science 173, 45–47 (1971).
[CrossRef] [PubMed]

L. B. Kreuzer, N. D. Kenyon, and C. K. N. Patel, “Air pollution: sensitive detection of ten pollutant gases by carbon monoxide and carbon dioxide lasers,” Science 177, 347–349 (1972).
[CrossRef] [PubMed]

C. K. N. Patel, E. G. Burkhardt, and C. A. Lambert, “Spectroscopic measurements of stratospheric nitric oxide and water vapor,” Science 184, 1173–1176 (1974).
[CrossRef] [PubMed]

Other (5)

Preliminary results were reported by J. J. Barrett at the Second Chemical Congress of the North American Continent, Las Vegas, Nev., August 24–29, 1980, and at the 1980 Federation of Analytical Chemistry and Spectroscopy Societies (FACSS) Meeting, Philadelphia, Pa., September 28–October 3, 1980.

K. V. Reddy, R. G. Bray, and M. J. Berry, “Dye laser-induced photochemistry,” in Advances in Laser Chemistry, A. H. Zewail, ed. (Springer-Verlag, Berlin, 1978), pp. 48–61.

G. A. West and J. J. Barrett, “Photoacoustic pure rotational Raman spectroscopy,” in The Proceedings of the VII International Conference on Raman Spectroscopy (North-Holland, Amsterdam, 1980), pp. 696–697.

J. J. Barrett, “Photoacoustic Raman spectroscopy of gases,” in Chemical Applications of Nonlinear Raman Spectroscopy, A. B. Harvey, ed. (Academic, New York, 1981), pp. 89–169.

J. J. Barrett and M. J. Berry, “Photoacoustic Raman scattering in gases,” in Proceedings of the Sixth International Conference on Raman Spectroscopy, E. Schmid, R. Krishnan, W. Kiefer, and H. Schrötter, eds. (Heyden, London, 1978), Vol. 1, pp. 466–467; “Photoacoustic Raman spectroscopy using cw laser sources,” Appl. Phys. Lett. 34, 144–146 (1979).

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Figures (4)

Fig. 1
Fig. 1

Schematic representation of the PARS process. (a) An energy level diagram for a Raman-type process involving the interaction of pump and Stokes photons at frequencies ωp and ωs, respectively, with the energy states |a〉 and |b〉 of a molecular system. The horizontal dashed line represents an intermediate virtual state of the interaction. (b) Schematic representation of the arrangement for generating a PARS signal in a gas. The pump and Stokes beams are overlapped spatially and temporally in the gas sample. When the frequency difference ωpωs equals a Raman frequency of the gas, amplification of the Stokes beam and attenuation of the pump beam occurs, and the molecular population of the upper energy level |b〉 is increased. Relaxation of these excited molecules by VT processes results in the generation of an acoustic wave that is detected by a microphone.

Fig. 2
Fig. 2

An energy-level diagram for a two-level molecular system depicting the transition rates for various processes. The quantities kr, kc, and ke represent the transition probabilities (in units of sec−1) for Raman, collisional, and emission processes, respectively.

Fig. 3
Fig. 3

Schematic representations of focused Gaussian beams. (a) Focal region for a single Gaussian beam. The beam radius at the focus (z = 0) is denoted by w0. The confocal parameter b is the distance between the two points along the beam axis at which the cross-sectional area of the beam is twice the minimum cross-sectional area at the focus (z = 0). The focusing half angle is denoted by θ. (b) The focal regions for two Gaussian beams with different focusing half angles. The beam axis and focal plane for the two beams are coincident. The shaded area indicates the region in which the two beams are overlapped. The acoustic signal that is detected by the PARS technique is produced mainly in the shaded region in the vicinity of the focal plane (z = 0).

Fig. 4
Fig. 4

Plot of the calculated integrated Raman transition rate kr [as defined by Eqs. (63) and (71a)] as a function of location z along the laser beam axes. The parameter z is plotted in units of the confocal parameter for the pump laser beam bp. The quantity R is defined as the ratio of the minimum-spot-sized radii for the Stokes and pump beams (i.e., Rws0/wp0). Therefore the curves for different values of R correspond to various degrees of focusing mismatch between the pump and the Stokes beams. These curves are normalized to the curve for R = 1 for which the pump and the Stokes beams have the same focusing parameters.

Equations (83)

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I s ( z ) = I s ( 0 ) e g s z ,
g s = - 16 π 2 k s 0 c n s 2 n p χ I p ,
2 q t 2 + Γ q t + ω 0 2 q = 1 m q ( P · E ) ,
P = α ˜ ˜ · E + ½ β ˜ ˜ ˜ · EE + .
2 q t 2 + Γ q q + ω 0 2 q = 1 m ( α q ) E 2 .
E 2 ½ E p E s * exp i [ ( k p - k s ) z - ( ω p - ω s ) t ] .
q ( ω ) = 1 2 π - e i ω t q ( t ) d t ,
- ω 2 q ( ω ) - i ω Γ q ( ω ) + ω 2 q ( ω ) = 1 2 m ( α q ) { E p E s * exp [ i ( k p - k s ) z ] δ ( ω - ω p + ω s ) + E p * E s exp [ i ( k s - k p ) z ] δ ( ω - ω s + ω p ) }
q ( ω ) = ( α q ) E p E s * exp [ i ( k p - k s ) z ] δ ( ω - ω p + ω s ) 2 m ( ω 0 2 - ω 2 - i ω Γ )
q ( t ) = - e - i ω t q ( ω ) d ω ,
q ( t ) = ( α q ) E p E s * exp [ i ( k p - k s ) z ] exp [ - i ( ω p - ω s ) t ] 2 m [ ω 0 2 - ( ω p - ω s ) 2 - i ( ω p - ω s ) Γ ] .
P N L = N Δ q * ( α q ) E ,
P s i N L = N Δ 4 m [ ω 0 2 - ( ω p - ω s ) 2 + i ( ω p - ω s ) Γ ] × j k l ( α q ) i j ( α q ) Kl * E p j E p k * E s l e i k s z .
P s N L = 3 j k l { χ i j k l ( 3 ) ( - ω s , ω p , ω p , - ω A ) × E p j E p k E p l * exp [ i ( 2 k p - k A ) z ] + 2 χ i j k l ( 3 ) ( - ω s , ω p , - ω p , ω s ) E p j E p k * E s l exp ( i k s z ) } ,
χ i j k l ( 3 ) ( - ω s , ω p , - ω p , ω s ) = N Δ ( α q ) i j ( α α q ) k l * 24 m [ ω 0 2 - ( ω p - ω s ) 2 + i ( ω p - ω s ) Γ ] .
( α q ) i j = ( 2 m ω 0 ) 1 / 2 α i j ,
a α i j b = 1 g ( a Q i g g Q j b ω g a + ω s + a Q j g g Q i b ω g a - ω p ) .
χ i j k l ( 3 ) ( - ω s , ω p , - ω p , ω s ) = N Δ 24 [ α i j α k l * ω 0 - ( ω p - ω s ) + i Γ / 2 ] .
χ i j k l ( 3 ) ( - ω s , ω p , - ω p , ω s ) = χ N R + χ
a α i j b 2 ¯ = ( c / ω s ) 4 ( d σ d Ω ) i j ,
χ 1111 ( 3 ) ( - ω s , ω s , ω p , - ω p ) = χ N R + N Δ c 4 24 ω s 4 ( 1 ω 0 - ω p + ω s + i Γ / 2 ) ( d σ d Ω ) ,
χ ( 3 ) ( ω s ) = 6 χ 1111 ( 3 ) ( - ω s , ω s , ω p , - ω p ) = 6 [ χ + i χ ] R + 6 χ N R ,
[ χ ( 3 ) ( ω s ) ] = 6 ( χ + χ N R ) = N Δ c 4 4 ω s 4 [ ω 0 - ω p + ω s ( ω 0 - ω p + ω s ) 2 + Γ 2 / 4 ] ( d σ d Ω ) + 6 χ N R ,
[ χ ( 3 ) ( ω s ) ] = χ = - N Δ c 4 8 ω s 4 [ Γ ( ω 0 - ω p + ω s ) 2 + Γ 2 / 4 ] ( d σ d Ω ) .
[ χ ( 3 ) ( ω s ) ] peak = - i N Δ c 4 2 ω s 4 Γ ( d σ d Ω ) + 6 χ N R .
g s = 4 π 3 N Δ c 2 h n p n s ω s 3 { Γ ( ω 0 - ω p + ω s ) 2 + Γ 2 / 4 } ( d σ d Ω ) I p .
( g s ) peak = 16 π 3 N Δ c 2 h n s n p ω s 3 Γ ( d σ d Ω ) I p .
( g s ) peak = N Δ π h c 2 n s n p ω s 3 Γ ( d σ d Ω ) I p ,
H = H 0 - α ˜ ˜ : EE
C ˙ a ( t ) = - i / V a b C b e - i ω t ,
C ˙ b = - i / V b a C a e + i ω t - ½ Γ b C b ( t ) ,
ω ω b - ω a ,
V a b = V b a * = α ˜ ˜ b a : EE ,
V a b = - α b a E p E s cos [ ( k p - k s ) z - ν t ] .
P b ( t ) = C b ( t ) 2 = | α b a E p E s Ω R sin ( Ω R t / 2 ) | 2 e - Γ b t ,
Ω R = [ ( ω a b - ν ) 2 + Γ b / 4 - i ( ω a b - ν ) Γ b + ( α b a E p E s ) 2 ] 1 / 2 .
Γ b = Γ b D + Γ b R ,
E t = - ω a b Γ b R P b ( t ) ,
Δ I s I s ( z ) - I s ( 0 ) = I s ( 0 ) [ e g s z - 1 ] I s ( 0 ) g s z
s = Δ I s A s T = Δ η s ( ω s ) ,
Δ N 1 = Δ η s = Δ I s A s T / ω s .
Δ N 1 = I s ( 0 ) g s z A s T / ω s .
Δ U = Δ N 1 ( ω 0 ) .
Δ p = ( γ - 1 ) Δ U / V
Δ p = [ ( γ - 1 ) Δ N 1 ω 0 ] / V ,
Δ p = [ ( γ - 1 ) I s ( 0 ) g s z A s T ( ω 0 / ω s ) ] / V ,
P a = ( γ - 1 ) ( ω 0 / ω s ) g s z P s ( 0 ) ,
k r = C 0 ω s I s I p ,
C 0 = 16 π 3 c 2 h n s n p ω s 3 Γ d σ d Ω
k c 0 = k c 1 exp ( - ω 0 k T ) ,
d N 1 d t = k r N 0 + k c 0 N 0 - k r N 1 - k c 1 N 1 - k e N 1 .
d N 1 d t = C 1 - C 2 N 1 ,
C 1 = N 0 e [ k r - ( k r + k e ) exp ( - ω 0 k T ) ]
C 2 = 2 k r + k c [ 1 + exp ( - ω 0 k T ) ] + k e ,
N 1 ( t ) = C 1 C 2 ( 1 - e - C 2 t ) ,
N 1 ( t ) = N k r 2 k r + k c { 1 - exp [ - ( 2 k r + k c ) t ] } .
p t = ( γ - 1 V c ) V f H d V ,
H = ω 0 k c N 1 ( t ) .
p t = ( γ - 1 V c ) ω 0 N k c V f ( k r 2 k r + k c ) × { 1 - exp [ - ( 2 k r + k c ) t ] } d V .
p ( t ) = ( γ - 1 V c ) ω 0 N k c V f ( k r 2 k r + k c ) × ( t + 1 2 k r + k c { exp [ - ( 2 k r + k c ) t ] - 1 } ) d V .
p ( t ) ( γ - 1 V c ) ω 0 N t V f k r d V = ( γ - 1 V c ) ω 0 N t K r ,
p ( t ) 1 2 ( γ - 1 V c ) ω 0 N t k c V f ,
N 1 ( t > τ ) = N 1 ( τ ) e - k c t ,
p ( t > t ) 1 2 ( γ - 1 V c ) ω 0 N τ k c V f exp [ - k c ( t - τ ) ] .
p ( t ) = { p 1 ( t ) for t τ p 2 ( t ) for t > τ ,
C 0 = 7.8 × 10 - 38 cm 4 sec / erg .
I p = I s = 10 13 erg sec - 1 cm - 2 .
k r = 2.2 sec - 1
g s = 2.0 × 10 - 5 / cm ,
I p = I s = 3.4 × 10 18 erg sec - 1 cm - 2 .
k r = 2.5 × 10 11 sec - 1 ,
g s = 6.7 / cm .
K r V f k r d V = C 0 ω s V f I s I p d V .
w 0 = λ π θ ,
w ( z ) = w 0 [ 1 + ( λ z π w 0 2 ) 2 ] 1 / 2 .
b = 2 π w 0 2 / λ .
K r = C 0 ω s ( P s π w 0 2 ) ( P p π w 0 2 ) ( π w 0 2 ) ( 2 π w 0 2 λ ) = C 0 ω s P s P p ( 2 / λ ) 2 C 0 P s P p / h c ,
Ψ ( x , y ) = ( 2 π ) 1 / 2 1 w exp [ - ( x 2 + y 2 ) w 2 ] .
P = A I d A = P 0 Ψ ( x , y ) 2 d x d y ,
K r = C 0 w s - Z Z 0 2 π 0 { 2 P s exp [ - 2 r 2 / w s ( z ) 2 ] π w s ( z ) 2 } × { 2 P p exp [ - 2 r 2 / w p ( z ) 2 ] π w p ( z ) 2 } r d r d ϕ d z .
K r = 4 C 0 P s P p π ω s 0 Z d z w s ( z ) 2 + w p ( z ) 2 = 4 C 0 P s P p π ω s 0 Z d z w s 0 2 + w p 0 2 + λ 2 ( w s 0 - 2 + w p 0 - 2 ) π - 2 z 2 ,
K r = 4 C 0 P s P p λ ω s ( w s w p 0 w s 0 2 + w p 0 2 ) tan - 1 ( λ Z π w s 0 w p 0 ) ,
K r = π C 0 P s P p 2 λ ω s π C 0 P s P p / 2 h c ,