Abstract

Chemical lasers pumped by the reaction of atomic fluorine with molecular hydrogen emit power from rotation–vibration transitions of excited HF with upper levels as high as v = 3. Collisional processes compete with stimulated emission for the energy of the excited HF. A simplified analysis is presented here for intensity, energy, and chemical efficiency of a class of such lasers. Results are obtained in closed form. A comparison with more exact computer solutions establishes the validity of the analysis despite its simplifications. A comprehensive parametric study examines the relative importance of initial conditions, optical cavity parameters, and rate coefficients for pumping and deactivation reactions.

© 1972 Optical Society of America

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References

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  1. J. R. Airey, J. Chem. Phys. 52, 156 (1970).
    [CrossRef]
  2. G. Emanuel, W. D. Adams, E. B. Turner, “RESALE-1: A Chemical Laser Computer Program,” TR-0172(2776)-1 (The Aerospace Corporation, El Segundo, Calif., 1972).
  3. R. L. Kerber, G. Emanuel, J. S. Whittier, Appl. Opt. 11, 1112 (1972).
    [CrossRef] [PubMed]
  4. G. Emanuel, N. Cohen, T. A. Jacobs, “Theoretical Performance of an HF Chemical cw Laser,” TR-0172(2776)-2 (The Aerospace Corporation, El Segundo, Calif., 1972).
  5. G. Emanuel, J. Quant. Spectrosc. Radiative Transfer 11, 1481 (1971).
    [CrossRef]
  6. W. C. Marlow, J. Appl. Phys. 41, 4019 (1970).
    [CrossRef]
  7. J. R. Airey, S. F. Fried, Chem. Phys. Lett. 8, 23 (1971).
    [CrossRef]

1972 (1)

1971 (2)

G. Emanuel, J. Quant. Spectrosc. Radiative Transfer 11, 1481 (1971).
[CrossRef]

J. R. Airey, S. F. Fried, Chem. Phys. Lett. 8, 23 (1971).
[CrossRef]

1970 (2)

J. R. Airey, J. Chem. Phys. 52, 156 (1970).
[CrossRef]

W. C. Marlow, J. Appl. Phys. 41, 4019 (1970).
[CrossRef]

Adams, W. D.

G. Emanuel, W. D. Adams, E. B. Turner, “RESALE-1: A Chemical Laser Computer Program,” TR-0172(2776)-1 (The Aerospace Corporation, El Segundo, Calif., 1972).

Airey, J. R.

J. R. Airey, S. F. Fried, Chem. Phys. Lett. 8, 23 (1971).
[CrossRef]

J. R. Airey, J. Chem. Phys. 52, 156 (1970).
[CrossRef]

Cohen, N.

G. Emanuel, N. Cohen, T. A. Jacobs, “Theoretical Performance of an HF Chemical cw Laser,” TR-0172(2776)-2 (The Aerospace Corporation, El Segundo, Calif., 1972).

Emanuel, G.

R. L. Kerber, G. Emanuel, J. S. Whittier, Appl. Opt. 11, 1112 (1972).
[CrossRef] [PubMed]

G. Emanuel, J. Quant. Spectrosc. Radiative Transfer 11, 1481 (1971).
[CrossRef]

G. Emanuel, N. Cohen, T. A. Jacobs, “Theoretical Performance of an HF Chemical cw Laser,” TR-0172(2776)-2 (The Aerospace Corporation, El Segundo, Calif., 1972).

G. Emanuel, W. D. Adams, E. B. Turner, “RESALE-1: A Chemical Laser Computer Program,” TR-0172(2776)-1 (The Aerospace Corporation, El Segundo, Calif., 1972).

Fried, S. F.

J. R. Airey, S. F. Fried, Chem. Phys. Lett. 8, 23 (1971).
[CrossRef]

Jacobs, T. A.

G. Emanuel, N. Cohen, T. A. Jacobs, “Theoretical Performance of an HF Chemical cw Laser,” TR-0172(2776)-2 (The Aerospace Corporation, El Segundo, Calif., 1972).

Kerber, R. L.

Marlow, W. C.

W. C. Marlow, J. Appl. Phys. 41, 4019 (1970).
[CrossRef]

Turner, E. B.

G. Emanuel, W. D. Adams, E. B. Turner, “RESALE-1: A Chemical Laser Computer Program,” TR-0172(2776)-1 (The Aerospace Corporation, El Segundo, Calif., 1972).

Whittier, J. S.

Appl. Opt. (1)

Chem. Phys. Lett. (1)

J. R. Airey, S. F. Fried, Chem. Phys. Lett. 8, 23 (1971).
[CrossRef]

J. Appl. Phys. (1)

W. C. Marlow, J. Appl. Phys. 41, 4019 (1970).
[CrossRef]

J. Chem. Phys. (1)

J. R. Airey, J. Chem. Phys. 52, 156 (1970).
[CrossRef]

J. Quant. Spectrosc. Radiative Transfer (1)

G. Emanuel, J. Quant. Spectrosc. Radiative Transfer 11, 1481 (1971).
[CrossRef]

Other (2)

G. Emanuel, N. Cohen, T. A. Jacobs, “Theoretical Performance of an HF Chemical cw Laser,” TR-0172(2776)-2 (The Aerospace Corporation, El Segundo, Calif., 1972).

G. Emanuel, W. D. Adams, E. B. Turner, “RESALE-1: A Chemical Laser Computer Program,” TR-0172(2776)-1 (The Aerospace Corporation, El Segundo, Calif., 1972).

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Figures (5)

Fig. 1
Fig. 1

Parameters mi vs JΘr/T. See Eqs. (26) and (30).

Fig. 2
Fig. 2

Rate coefficient ratios k ¯ F , k ¯ HF, and k ¯ H 2, vs temperature. See Table II for expressions for the rate coefficients.

Fig. 3
Fig. 3

Efficiency vs rotational quantum number J. Maximum efficiency occurs for J = 7 for all three cases.

Fig. 4
Fig. 4

Output intensity vs time for Case A of Table III Assumptions for the computer model are listed in Table I.

Fig. 5
Fig. 5

Efficiency and pulse length vs pressure.

Tables (4)

Tables Icon

Table I Comparison of Theoretical Models

Tables Icon

Table II Chemical Model for the F + H2 Laser

Tables Icon

Table III Predictions of Theory Compared with Those of Computer Model

Tables Icon

Table IV Parametric Study

Equations (53)

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F + H 2 HF ( v ) + H ,             v = 0 , 1 , 2 , 3.
ρ [ d n ( 0 ) / d t ] = χ ch ( 0 ) + χ rad ( 0 ) , ρ [ d n ( 1 ) / d t ] = χ ch ( 1 ) + χ rad ( 1 ) - χ rad ( 0 ) , ρ [ d n ( v f ) / d t ] = χ ch ( v f ) - χ rad ( v f - 1 ) . }
χ rad ( 0 ) = ρ ( d / d t ) n ( 0 ) - χ ch ( 0 ) , χ rad ( 1 ) = ρ ( d / d t ) [ n ( 0 ) + n ( 1 ) ] - [ χ ch ( 0 ) + χ ch ( 1 ) ] , χ rad ( v f - 1 ) = ρ ( d / d t ) [ n ( 0 ) + + n ( v f - 1 ) ] - [ χ ch ( 0 ) + + χ ch ( v f - 1 ) ] . }
χ rad v = 0 v f - 1 χ rad ( v ) = A 1 - A 2 ,
A 1 ρ v = 0 v f - 1 ( v f - v ) d n ( v ) d t ,
A 2 v = 0 v f - 1 ( v f - v ) χ ch ( v ) .
E ( t ) = h c N A ω t 0 t χ rad d t = 11.96 ω t 0 t ξ rad d t .
η = 11.96 × 10 2 ω 1.4 × 10 5 ρ ( η F ) 0 t 0 t c χ rad d t ,
r 1 r 2 e 2 L g ( v , J ) = 1 ,             v = 0 , , v f - 1 ,
g ( v , J ) = g r J e - ( J 2 - J ) δ [ n ( v + 1 ) - n ( v ) e - 2 J δ ] .
g r 2.74 × 10 47 σ Θ r W 1 2 T - / 2 3 ρ M 2 ,             δ Θ r / T ,
I = 11.96 ω [ 1 - ( r 1 r 2 ) 1 2 ] χ rad / g .
n ( v ) = { ( 1 - e - 2 J δ ) / [ 1 - e - 2 J δ ( v f + 1 ) ] } e - 2 J δ v n tot + ( g / g r ) × [ e ( J 2 - J ) δ / J ] × ( { ( 1 - e - 2 J δ v ) [ 1 - e - 2 J δ ( v f + 1 ) ] - v f e - 2 J δ v ( 1 - e - 2 J δ ) + e - 2 J δ ( v + 1 ) × ( 1 - e - 2 J δ v f ) ) } / { ( 1 - e - 2 J δ ) [ 1 - e - 2 J δ ( v f + 1 ) ] } ) , v = 0 , 1 , , v f ,
n tot = v = 0 v f n ( v ) .
( d / d t ) n ( v ) = { ( 1 - e - 2 J δ ) / [ 1 - e - 2 J δ ( v f + 1 ) ] } e - 2 J δ v ( d / d t ) n tot , v = 0 , 1 , , v f .
v = 0 v f a ( v ) = 1.
χ ch ( v ) = ρ 2 { a ( v ) k n F n H 2 + k HF n HF [ a HF ( v + 1 ) n ( v + 1 ) - a HF ( v ) n ( v ) ] + k F n F [ a F ( v + 1 ) n ( v + 1 ) - a F ( v ) n ( v ) ] + k H 2 n H 2 [ a H 2 ( v + 1 ) n ( v + 1 ) - a H 2 ( v ) n ( v ) ] + k f HF [ a f ( v + 1 ) n 2 ( v + 1 ) - 2 a f ( v ) n 2 ( v ) + a f ( v - 1 ) n 2 ( v - 1 ) ] + k b HF [ - a b ( v + 1 ) n ( v + 2 ) n ( v ) + 2 a b ( v ) n ( v + 1 ) n ( v - 1 ) - a b ( v - 1 ) n ( v ) n ( v - 2 ) ] } ,
n H + n HF + 2 n H 2 = ( n H + n HF + 2 n H 2 ) 0 , n F + n HF = ( n F + n HF ) 0 ,
( d / d t ) n F = - ( d / d t ) n H = - k ρ n F n H 2 .
n F + n H = ( n F + n H ) 0 .
( d / d t ) n F = - k ρ n F [ n F + ( n H 2 - n F ) 0 ] ,
n F / ( n F ) 0 = ( 1 - κ 1 ) e - t / τ / ( 1 - κ 1 e - t / τ ) ,
κ 1 ( n F / n H 2 ) 0 ,             τ - 1 k ρ ( n H 2 - n F ) 0 .
n F / ( n F ) 0 = [ 1 + k ρ ( n F ) 0 t ] - 1 ,             κ 1 = 1.
ξ = [ n HF - ( n HF ) 0 ] / ( n F ) 0 .
n HF = ( n F ) 0 ξ + ( n HF ) 0 , n F = - ( n F ) 0 ξ + ( n F ) 0 , n H = ( n F ) 0 ξ + ( n H ) 0 , n H 2 = - ( n F ) 0 ξ + ( n H 2 ) 0 , }
ξ = ( 1 - e - t / τ ) / ( 1 - κ 1 e - t / τ ) .
A 1 = m 1 ρ ( n F ) 0 ( d ξ / d t ) ,
m 1 = ( 3 + 2 e - 2 J δ + e - 4 J δ ) / [ ( 1 + e - 2 J δ ) ( 1 + e - 4 J δ ) ] .
A 1 = m 1 k ρ 2 ( n F n H 2 ) 0 ( 1 - ξ ) ( 1 - κ 1 ξ ) .
A 2 = ρ 2 { 0.945 k n F n H 2 + 1 6 ( k HF n HF + k F n F ) [ n ( 1 ) + 2 n ( 2 ) + 3 n ( 3 ) ] + k H 2 n H 2 [ 0.965 n ( 1 ) + 0.035 n ( 2 ) ] } .
A 2 = ρ 2 { k n F n H 2 + ( n HF / m 3 ) [ m 2 ( k HF n HF + k F n F ) + k H 2 n H 2 ] + ( g / g r ) [ e ( J 2 - J ) δ / J ] [ m 4 ( k HF n HF + k F n F ) + m 5 k H 2 n H 2 ] } ,
m 2 = 1 6 ( 1 + 2 e - 2 J δ + 3 e - 4 J δ ) ,             m 3 = ( 1 + e 2 J δ ) ( 1 + e - 4 J δ ) , m 4 = [ e 2 J δ + e - 2 J δ + ( 4 3 ) ] / m 3 ,             m 5 = [ e 2 J δ - e - 2 J δ - 2 ] / m 3 .
A 2 = k ρ 2 ( n F n H 2 ) 0 ( ( 1 - ξ ) ( 1 - κ 1 ξ ) + { [ ( ξ + κ 2 ) / m 3 ] + m 5 α } ( 1 - κ 1 ξ ) k ¯ H 2 + κ 1 [ ( m 2 / m 3 ) ( ξ + κ 2 ) + m 4 α ] × [ ( ξ + κ 2 ) k ¯ HF + ( 1 - ξ ) k ¯ F ] ) ,
κ 2 = ( n HF / n F ) 0 ,             α = ( g / g r ) [ e ( J 2 - J ) δ / J ] [ 1 / ( n F ) 0 ]
k ¯ H 2 = ( k H 2 / k ) ,             k ¯ F = ( k F / k ) ,             k ¯ HF = ( k HF / k ) .
χ rad = k ρ 2 ( n F n H 2 ) 0 ( ( m 1 - 1 ) ( 1 - ξ ) ( 1 - κ 1 ξ ) - { [ ( ξ + κ 2 ) / m 3 ] + m 5 α } ( 1 - κ 1 ξ ) k ¯ H 2 - κ 1 [ ( m 2 / m 3 ) ( ξ + κ 2 ) + m 4 α ) [ ( ξ + κ 2 ) k ¯ HF + ( 1 - ξ ) k ¯ F ] ) .
t 0 = 0 t χ rad d t = 1 k ρ ( n H 2 ) 0 0 ξ χ rad ( ξ ) d ξ ( 1 - ξ ) ( 1 - κ 1 ξ ) = ρ ( n F ) 0 [ ( m 1 - 1 ) ξ + ( k ¯ H 2 / m 3 ) × [ ξ + ( 1 + κ 2 + m 3 m 5 α ) ln ( 1 - ξ ) ] + ( k ¯ F / m 3 ) ( m 2 ξ + { [ ( 1 + κ 1 κ 2 ) m 2 + κ 1 m 3 m 4 α ] / κ 1 } ln ( 1 - κ 1 ξ ) ) + ( k ¯ HF / m 3 ) ( - m 2 ξ + { κ 1 ( 1 + κ 2 ) × [ ( 1 + κ 2 ) m 2 + m 3 m 4 α ] / ( 1 - κ 1 ) } × ln ( 1 - ξ ) - { ( 1 + κ 1 κ 2 ) [ ( 1 + κ 1 κ 2 ) m 2 + κ 1 m 3 m 4 α ] / κ 1 ( 1 - κ ) } ln ( 1 - κ 1 ξ ) ) ] .
χ rad = k ρ 2 ( n F n H 2 ) 0 ( m 1 - 1 ) ( 1 - ξ ) ( 1 - κ 1 ξ ) ,
ξ c = { 1 / κ 1 for κ 1 1 , 1 for κ 1 < 1 ,
E ( t ) = 11.96 ω ρ ( n F ) 0 ( m 1 - 1 ) ξ ,
η = [ ( 11.96 × 10 2 ) / ( 1.4 × 10 4 ) ] ω ( m 1 - 1 ) ξ c .
χ rad ~ [ k ρ 2 ( n F ) 0 2 / κ 1 ] ( ( m 1 - 1 ) ( 1 - ξ ) - { [ ξ + κ 2 ) / m 3 ] + m 5 α } k ¯ H 2 ) ,
η = 8.55 × 10 - 3 ω { ( m 1 - 1 ) ξ + ( k ¯ H 2 / m 3 ) × [ ξ + ( 1 + κ 2 + m 3 m 5 α ) ln ( 1 - ξ ) ] } .
σ = 1 ,             Θ r = 30.16 K ,             W = 20 g / mol , M 2 = 2.8 × 10 - 38 erg / cm 3 ,             ω = 3790 cm - 1 .
d p = ( m 1 - 1 ) ξ ,
d H 2 = ( k ¯ H 2 / m 3 ) [ ξ + ( 1 + κ 2 + m 3 m 5 α ) ln ( 1 - ξ ) ] ,
d F = ( k ¯ F / m 3 ) { m 2 ξ + κ 1 - 1 [ ( 1 + κ 1 κ 2 ) m 2 + κ 1 m 3 m 4 α ] ln ( 1 - κ 1 ξ ) } .
d HF ( κ 1 1 ) = ( k ¯ HF / m 3 ) { - m 2 ξ + ( 1 - κ 1 ) - 1 κ 1 ( 1 + κ 2 ) × [ ( 1 + κ 2 ) m 2 + m 3 m 4 α ) ln ( 1 - ξ ) - ( κ 1 - κ 1 2 ) - 1 ( 1 + κ 1 κ 2 ) × [ ( 1 + κ 1 κ 2 ) m 2 + κ 1 m 3 m 4 α ] ln ( 1 - κ 1 ξ ) } ,
d HF ( κ 1 = 1 ) = ( k ¯ HF / m 3 ) { - m 2 ξ - ( 1 + κ 2 ) [ m 2 ( 1 + κ 2 ) + m 3 m 4 α ] [ ξ / ( 1 - ξ ) ] - [ 2 ( 1 - κ 2 2 ) m 2 + m 3 m 4 α ] ln ( 1 - ξ ) } .
E ( t ) = 11.96 ω ρ ( n F ) 0 ( d p + d H 2 + d F + d HF ) ,
η = 8.55 × 10 - 3 ω ( d p + d H 2 + d F + d HF ) c .
P ( t ) = 11.96 ω ( A u ) t 0 t χ rad d t ,

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