Abstract

In describing the excitation of atomic and molecular species with lasers in spectroscopic applications, only the density matrix formulation is exactly correct. Many workers, however, have used the conventional rate equation formulation. The range of application of the conventional rate equations is examined and, for flames, shown to be valid for sufficiently slow laser pulse rise times under single-mode excitation and for certain special cases of multimode excitation.

© 1977 Optical Society of America

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References

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  1. E. H. Piepmeir, Spectrochim. Acta Part B 27, 431 (1972).
    [CrossRef]
  2. E. H. Piepmeir, Spectrochim. Acta Part B 27, 445 (1972).
    [CrossRef]
  3. N. Omenetto, P. Benetti, L. P. Hart, J. D. Winefordner, C. T. Alkemade, Spectrochim. Acta Part B 28, 289 (1973).
    [CrossRef]
  4. A. Yariv, Quantum Electronics (Wiley, New York, 1975).
  5. T. J. McIlrath, J. L. Carsten, Phys. Rev. A. 6, 1091 (1972).
    [CrossRef]
  6. A. C. G. Mitchell, M. W. Zermansky, Resonance Radiation and Excited Atoms (Cambridge U.P., London, 1971).
  7. W. G. Vincent, C. H. Kruger, Introduction to Physical Gas Dynamics (Wiley, New York, 1967).
  8. P. J. Lynse, J. Quant. Spectrosc. Radiat. Transfer 14, 1143 (1974).
    [CrossRef]
  9. J. L. Steinfeld, Acc. Chem. Res. 3, 313 (1970).
    [CrossRef]
  10. C. Van Frigt, T. J. Hollander, C. T. J. Alkemade, J. Quant. Spectrosc. Radiat. Transfer 5, 813 (1965).
    [CrossRef]
  11. B. R. Mollow, “Elastic and Inelastic Collisional and Radiative Damping Effects on Saturated Lineshapes in the Limit of Well-Separated Spectral Lines,” Department of Physics, University of Massachusetts (1976).

1974 (1)

P. J. Lynse, J. Quant. Spectrosc. Radiat. Transfer 14, 1143 (1974).
[CrossRef]

1973 (1)

N. Omenetto, P. Benetti, L. P. Hart, J. D. Winefordner, C. T. Alkemade, Spectrochim. Acta Part B 28, 289 (1973).
[CrossRef]

1972 (3)

T. J. McIlrath, J. L. Carsten, Phys. Rev. A. 6, 1091 (1972).
[CrossRef]

E. H. Piepmeir, Spectrochim. Acta Part B 27, 431 (1972).
[CrossRef]

E. H. Piepmeir, Spectrochim. Acta Part B 27, 445 (1972).
[CrossRef]

1970 (1)

J. L. Steinfeld, Acc. Chem. Res. 3, 313 (1970).
[CrossRef]

1965 (1)

C. Van Frigt, T. J. Hollander, C. T. J. Alkemade, J. Quant. Spectrosc. Radiat. Transfer 5, 813 (1965).
[CrossRef]

Alkemade, C. T.

N. Omenetto, P. Benetti, L. P. Hart, J. D. Winefordner, C. T. Alkemade, Spectrochim. Acta Part B 28, 289 (1973).
[CrossRef]

Alkemade, C. T. J.

C. Van Frigt, T. J. Hollander, C. T. J. Alkemade, J. Quant. Spectrosc. Radiat. Transfer 5, 813 (1965).
[CrossRef]

Benetti, P.

N. Omenetto, P. Benetti, L. P. Hart, J. D. Winefordner, C. T. Alkemade, Spectrochim. Acta Part B 28, 289 (1973).
[CrossRef]

Carsten, J. L.

T. J. McIlrath, J. L. Carsten, Phys. Rev. A. 6, 1091 (1972).
[CrossRef]

Hart, L. P.

N. Omenetto, P. Benetti, L. P. Hart, J. D. Winefordner, C. T. Alkemade, Spectrochim. Acta Part B 28, 289 (1973).
[CrossRef]

Hollander, T. J.

C. Van Frigt, T. J. Hollander, C. T. J. Alkemade, J. Quant. Spectrosc. Radiat. Transfer 5, 813 (1965).
[CrossRef]

Kruger, C. H.

W. G. Vincent, C. H. Kruger, Introduction to Physical Gas Dynamics (Wiley, New York, 1967).

Lynse, P. J.

P. J. Lynse, J. Quant. Spectrosc. Radiat. Transfer 14, 1143 (1974).
[CrossRef]

McIlrath, T. J.

T. J. McIlrath, J. L. Carsten, Phys. Rev. A. 6, 1091 (1972).
[CrossRef]

Mitchell, A. C. G.

A. C. G. Mitchell, M. W. Zermansky, Resonance Radiation and Excited Atoms (Cambridge U.P., London, 1971).

Mollow, B. R.

B. R. Mollow, “Elastic and Inelastic Collisional and Radiative Damping Effects on Saturated Lineshapes in the Limit of Well-Separated Spectral Lines,” Department of Physics, University of Massachusetts (1976).

Omenetto, N.

N. Omenetto, P. Benetti, L. P. Hart, J. D. Winefordner, C. T. Alkemade, Spectrochim. Acta Part B 28, 289 (1973).
[CrossRef]

Piepmeir, E. H.

E. H. Piepmeir, Spectrochim. Acta Part B 27, 431 (1972).
[CrossRef]

E. H. Piepmeir, Spectrochim. Acta Part B 27, 445 (1972).
[CrossRef]

Steinfeld, J. L.

J. L. Steinfeld, Acc. Chem. Res. 3, 313 (1970).
[CrossRef]

Van Frigt, C.

C. Van Frigt, T. J. Hollander, C. T. J. Alkemade, J. Quant. Spectrosc. Radiat. Transfer 5, 813 (1965).
[CrossRef]

Vincent, W. G.

W. G. Vincent, C. H. Kruger, Introduction to Physical Gas Dynamics (Wiley, New York, 1967).

Winefordner, J. D.

N. Omenetto, P. Benetti, L. P. Hart, J. D. Winefordner, C. T. Alkemade, Spectrochim. Acta Part B 28, 289 (1973).
[CrossRef]

Yariv, A.

A. Yariv, Quantum Electronics (Wiley, New York, 1975).

Zermansky, M. W.

A. C. G. Mitchell, M. W. Zermansky, Resonance Radiation and Excited Atoms (Cambridge U.P., London, 1971).

Acc. Chem. Res. (1)

J. L. Steinfeld, Acc. Chem. Res. 3, 313 (1970).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer (2)

C. Van Frigt, T. J. Hollander, C. T. J. Alkemade, J. Quant. Spectrosc. Radiat. Transfer 5, 813 (1965).
[CrossRef]

P. J. Lynse, J. Quant. Spectrosc. Radiat. Transfer 14, 1143 (1974).
[CrossRef]

Phys. Rev. A. (1)

T. J. McIlrath, J. L. Carsten, Phys. Rev. A. 6, 1091 (1972).
[CrossRef]

Spectrochim. Acta Part B (3)

E. H. Piepmeir, Spectrochim. Acta Part B 27, 431 (1972).
[CrossRef]

E. H. Piepmeir, Spectrochim. Acta Part B 27, 445 (1972).
[CrossRef]

N. Omenetto, P. Benetti, L. P. Hart, J. D. Winefordner, C. T. Alkemade, Spectrochim. Acta Part B 28, 289 (1973).
[CrossRef]

Other (4)

A. Yariv, Quantum Electronics (Wiley, New York, 1975).

A. C. G. Mitchell, M. W. Zermansky, Resonance Radiation and Excited Atoms (Cambridge U.P., London, 1971).

W. G. Vincent, C. H. Kruger, Introduction to Physical Gas Dynamics (Wiley, New York, 1967).

B. R. Mollow, “Elastic and Inelastic Collisional and Radiative Damping Effects on Saturated Lineshapes in the Limit of Well-Separated Spectral Lines,” Department of Physics, University of Massachusetts (1976).

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Equations (52)

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ψ ( r , t ) = Σ C n ( t ) U n ( r ) ,
C n ( t ) = U n ( r ) ψ ( r , t ) d r
A ( t ) = ψ * ( r , t ) A o p ψ ( r , t ) d r = m n C m * ( t ) U m * ( r ) A o p U n ( r ) d r C n ( t ) = m , n C m * A m n C n .
A ( t ) ¯ = m n C m * ( t ) C n ( t ) A m n ¯ .
ρ n m ( t ) C m * ( t ) C n ( t ) ¯
A ( t ) ¯ = m n ρ m n A m n = t r ( ρ A ) .
H ψ ( r , t ) = i ψ ( r , t ) t .
i [ ( ρ ) / ( t ) ] = [ H , ρ ] = H ρ - ρ H ,
H m n = ψ m * [ - e r · E ( z , t ) ] ψ n d r .
H m n = ψ m * [ - e r · i E i ( z , t ) ] ψ n d r ψ m * ( - e r ) ψ n d r · i E i ( z , t ) - i μ m n E i ( z , t ) ,
H = H 0 + H ,
H = | E 1 - i μ i E i ( t ) - Σ μ i E i ( t ) E 2 | .
[ H , ρ ] 21 = ρ 21 ( E 2 - E 1 ) - i μ i E i ( t ) ( ρ 11 - ρ 22 ) ;
[ H , ρ ] 22 = - i μ i E i ( t ) ( ρ 12 - ρ 21 ) .
ρ 21 t = - i ω o ρ 21 + i i μ i ( ρ 11 - ρ 22 ) - Q 21 ρ 21 ;
ρ 22 t = - i i μ i E i ( t ) ( ρ 21 - ρ 21 * ) - ( A 21 + Q 21 ) ρ 22 ,
E ( t ) = E 0 ( t ) cos ω t = ½ E o ( t ) [ exp ( i ω t ) + exp ( - i ω t ) ] .
ρ 21 t = - ( i ω 0 + Q 21 ) ρ 21 + i μ E o ( t ) 2 [ exp ( i ω t ) + exp ( - i ω t ) ] ( ρ 11 - ρ 22 )
ρ 21 = σ 21 exp ( - i ω t ) .
σ 21 t exp ( - i ω t ) - i ω exp ( - i ω t ) σ 21 = - ( i ω o + Q 21 ) exp ( - i ω t ) + i Ω ( t ) [ exp ( i ω t ) + exp ( - i ω t ) ] ( ρ 11 - ρ 22 ) .
σ 21 t = [ i ( ω - ω o ) + Q 21 ] σ 21 + i Ω ( t ) ( ρ 11 - ρ 22 ) .
ρ 22 t = - i Ω ( σ 21 - σ 21 * ) - ( A 21 + Q 21 ) ρ 22 .
0 = [ i ( ω - ω 0 ) + Q 21 ] σ 21 + i Ω ( t ) ( ρ 11 - ρ 22 ) .
Im ( σ 21 ) = Ω ( t ) ( ρ 11 - ρ 22 ) [ Q 21 ( ω - ω o ) 2 + Q 21 2 ] .
ϕ ( ω ) = 1 π [ Q 21 ( ω - ω o ) 2 + Q 21 2 ] ,
ρ 22 t = 2 π Ω 2 ( t ) ϕ ( ω ) ( ρ 11 - ρ 22 ) - ( A 21 + Q 21 ) ρ 22 .
ρ 21 t = - ( i ω o + Q 21 ) ρ 21 - i ( ρ 22 - ρ 11 ) i Ω i ( t ) × { exp [ i ( ω i t - k i z ) ] + exp [ - i ( ω i t - k i z ) ] } ,
ρ 22 t = i i Ω i ( t ) { exp [ i ( ω i t - k i z ) ] + exp [ - i ( ω i t - k i z ) ] } × ( ρ 21 - ρ 21 * ) - ( A 21 + Q 21 ) ρ 22 .
ρ 21 ( t ) = - i exp [ - ( i ω o + Q 21 ) t ] i o t Δ ρ Ω i exp ( Q 21 t ) × { exp [ i ( ω o + ω i ) t ] + exp [ i ( ω o - ω i ) t ] } d t .
ρ 21 ( t ) = - i Δ ρ exp [ - ( i ω 0 + Q 21 ) t ] i Ω i ( t ) o t exp ( Q 21 t ) × { exp [ i ( ω o + ω i ) t ] + exp [ i ( ω o - ω i ) t ] } d t .
ρ 21 ( t ) = - i Δ ρ exp [ - ( i ω o + Q 21 ) t ] i Ω i ( t ) × ( exp { [ Q 21 + i ( ω o + ω i ) ] t } - 1 Q 21 + i ( ω o + ω i ) + exp { [ Q 21 + i ( ω o - ω i ) ] t } - 1 Q 21 + i ( ω o - ω i ) ) .
σ 21 ( t ) = ρ 21 ( t ) exp ( i ω o t ) .
σ 21 ( t ) = - i Δ ρ i Ω i exp [ i ( ω o + ω i ) t ] - exp ( - Q 21 t ) Q 21 + i ( ω o + ω i ) + exp [ i ( ω o - ω i ) t ] - exp ( - Q 21 t ) Q 21 + i ( ω o - ω i ) .
σ 21 ( t ) = - i Δ ρ i Ω i exp [ i ( ω o - ω i ) t ] Q 21 + i ( ω o - ω i ) .
σ 21 ( t ) = - Δ ρ i Ω i [ i Q 21 + ( ω o - ω i ) ] ( ω o - ω i ) 2 + Q 21 exp [ i ( ω o - ω i ) t ] .
i Δ ρ i j Ω i Ω j ( ω o - ω j ) 2 + Q 21 2 × ( [ i Q 21 + ( ω o - ω j ) ] { exp ( i ω i j t ) + exp [ - i ( ω i + ω j ) t ] } + [ i Q 21 - ( ω o - ω j ) ] { exp [ i ( ω i + ω j ) t ] + exp ( - i ω i j t } ) ,
- 2 π Δ ρ i j Ω i Ω j ϕ ( ω j ) [ cos ω i j t + ( ω o - ω j Q 21 ) sin ω i j t ] ,
ρ 22 t = - 2 π Δ ρ i j Ω i Ω j ϕ ( ω j ) × [ cos ω i j t + ( ω o - ω j Q 21 ) sin ω i j t ] - ( Q 21 + A 21 ) ρ 22 + Q 12 ρ 11 .
ρ 22 t = 2 π [ i Ω i 2 ( t ) ϕ ( ω i ) ] × ( ρ 11 - ρ 22 ) - ( A 21 + Q 21 ) ρ 22 + Q 12 ρ 11 .
Δ ρ = ρ 22 - ρ 11 = Δ ρ * 2 { 2 π i j Ω i Ω j ϕ ( ω j ) [ cos ω i j t + ( ω o - ω j Q 21 ) sin ω i j t ] - ( A 21 + Q 21 + Q 12 ) } + 1 ,
ϕ ( ω ) = 1 π Q 21 [ ω - ω o ( t ) ] 2 + Q 21 2 .
ϕ ( ν ) = 2 ( ln 2 ) 1 / 2 π Δ ν D V ( a , ξ ) ,
Δ ν D ν o ( 8 k T ln 2 m c 2 ) 1 / 2
V ( a , ξ ) = a π - exp ( - y 2 ) ( ξ - y ) 2 + a 2 d y
a Δ ν L Δ ν D ( ln 2 ) 1 / 2 ;
ξ 2 ( ν - ν o ) ( ln 2 ) 1 / 2 Δ ν ¯ D .
Δ ν L Q 21 / π .
Q 21 = Q 2 E + Q 21 + A 21 ,
τ vel ~ 1 N g σ ~ 10 - 9 sec ,
τ d p 1 N g σ ~ 2 × 10 - 10 sec .
τ coll ~ 1 N n 2 g σ ~ 10 - 10 sec .
τ o p ~ 10 - 15 sec , τ vel ~ 10 - 9 sec , τ d p ~ 10 - 10 sec , τ mode ~ 10 - 9 sec .

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