Abstract

The high gain and narrow linewidth of HF chemical lasers lead to a large resonant component of the refractive index. Calculations indicate that this (unsaturated) anomalous dispersion is at least as large as the nonresonant Gladstone-Dale dispersion. Possible deleterious effects of anomalous dispersion are briefly reviewed. These effects in a high-power device are mitigated by multimode operation, which acts to saturate homogeneously the anomalous dispersion at intensities considerably lower than those required for power broadening of the inhomogeneous Doppler line shape over to a homogeneous line.

© 1981 Optical Society of America

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References

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  1. The effects of the nonresonant dispersion are well understood and are not difficult to include in theoretical investigations. See, for example, P. W. Milonni, A. H. Paxton, J. Appl. Phys. 49, 1012 (1978).
    [CrossRef]
  2. L. Casperson, A. Yariv, Appl. Phys. Lett. 17, 259 (1970).
    [CrossRef]
  3. See, for example, D. H. Close, Phys. Rev. 153, 360 (1967). The equations in this paper are often used in discussions of anomalous dispersion.
    [CrossRef]
  4. D. L. Bullock, paper delivered at Tri-Service Chemical Laser Symposium, Air Force Weapons Laboratory, 28–30 Aug. 1979.
  5. T. Kan, G. J. Wolga, IEEE J. Quantum Electron. QE-7, 141 (1971).
  6. H. Mirels, AIAA J. 17, 478 (1979).
    [CrossRef]
  7. S. Stenholm, “The Semiclassical Theory of the Gas Laser,” in Progress in Quantum Electronics, J. H. Sanders, K. W. H. Stevens, Eds., (Pergamon, Oxford, 1971), pp. 187–271.
  8. W. W. Rigrod, J. Appl. Phys. 36, 2487 (1965).
    [CrossRef]
  9. J. C. Polanyi, K. B. Woodall, J. Chem. Phys. 56, 1563 (1972).
    [CrossRef]
  10. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (U.S. GPO, Washington, D.C., 1964), pp. 297–304, 325–328.
  11. W. E. Lamb, Phys. Rev. 134, 1429 (1964).
    [CrossRef]
  12. L. Allen, J. H. Eberly, Optical Resonance and Two-Level Atoms (Wiley, New York, 1975).
  13. See, for example, R. Ladenburg, H. Kopfermann, Nature Paris 122, 438 (1928).
    [CrossRef]
  14. W. R. Bennett, Appl. Opt. Suppl. 1, 24 (1962).
    [CrossRef]
  15. L. W. Casperson, A. Yariv, Appl. Opt. 11, 462 (1972).
    [CrossRef] [PubMed]
  16. R. E. Meredith, T. S. Chang, F. G. Smith, D. R. Woods, SAI-73-004-AA.
  17. J. A. Glaze, Appl. Phys. Lett. 23, 300 (1973).
    [CrossRef]
  18. G. Emanuel, in Handbook of Chemical Lasers, R. W. F. Gross, J. F. Bott, Eds. (Wiley, New York, 1976), pp. 469–549.
  19. J. Jarecki, R. Herman, J. Quant. Spectrosc. Radiat. Transfer 15, 707 (1975).
    [CrossRef]
  20. L. Casperson, IEEE J. Quantum Electron. QE-9, 250 (1973).
    [CrossRef]

1979 (1)

H. Mirels, AIAA J. 17, 478 (1979).
[CrossRef]

1978 (1)

The effects of the nonresonant dispersion are well understood and are not difficult to include in theoretical investigations. See, for example, P. W. Milonni, A. H. Paxton, J. Appl. Phys. 49, 1012 (1978).
[CrossRef]

1975 (1)

J. Jarecki, R. Herman, J. Quant. Spectrosc. Radiat. Transfer 15, 707 (1975).
[CrossRef]

1973 (2)

L. Casperson, IEEE J. Quantum Electron. QE-9, 250 (1973).
[CrossRef]

J. A. Glaze, Appl. Phys. Lett. 23, 300 (1973).
[CrossRef]

1972 (2)

L. W. Casperson, A. Yariv, Appl. Opt. 11, 462 (1972).
[CrossRef] [PubMed]

J. C. Polanyi, K. B. Woodall, J. Chem. Phys. 56, 1563 (1972).
[CrossRef]

1971 (1)

T. Kan, G. J. Wolga, IEEE J. Quantum Electron. QE-7, 141 (1971).

1970 (1)

L. Casperson, A. Yariv, Appl. Phys. Lett. 17, 259 (1970).
[CrossRef]

1967 (1)

See, for example, D. H. Close, Phys. Rev. 153, 360 (1967). The equations in this paper are often used in discussions of anomalous dispersion.
[CrossRef]

1965 (1)

W. W. Rigrod, J. Appl. Phys. 36, 2487 (1965).
[CrossRef]

1964 (1)

W. E. Lamb, Phys. Rev. 134, 1429 (1964).
[CrossRef]

1962 (1)

W. R. Bennett, Appl. Opt. Suppl. 1, 24 (1962).
[CrossRef]

1928 (1)

See, for example, R. Ladenburg, H. Kopfermann, Nature Paris 122, 438 (1928).
[CrossRef]

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (U.S. GPO, Washington, D.C., 1964), pp. 297–304, 325–328.

Allen, L.

L. Allen, J. H. Eberly, Optical Resonance and Two-Level Atoms (Wiley, New York, 1975).

Bennett, W. R.

W. R. Bennett, Appl. Opt. Suppl. 1, 24 (1962).
[CrossRef]

Bullock, D. L.

D. L. Bullock, paper delivered at Tri-Service Chemical Laser Symposium, Air Force Weapons Laboratory, 28–30 Aug. 1979.

Casperson, L.

L. Casperson, IEEE J. Quantum Electron. QE-9, 250 (1973).
[CrossRef]

L. Casperson, A. Yariv, Appl. Phys. Lett. 17, 259 (1970).
[CrossRef]

Casperson, L. W.

Chang, T. S.

R. E. Meredith, T. S. Chang, F. G. Smith, D. R. Woods, SAI-73-004-AA.

Close, D. H.

See, for example, D. H. Close, Phys. Rev. 153, 360 (1967). The equations in this paper are often used in discussions of anomalous dispersion.
[CrossRef]

Eberly, J. H.

L. Allen, J. H. Eberly, Optical Resonance and Two-Level Atoms (Wiley, New York, 1975).

Emanuel, G.

G. Emanuel, in Handbook of Chemical Lasers, R. W. F. Gross, J. F. Bott, Eds. (Wiley, New York, 1976), pp. 469–549.

Glaze, J. A.

J. A. Glaze, Appl. Phys. Lett. 23, 300 (1973).
[CrossRef]

Herman, R.

J. Jarecki, R. Herman, J. Quant. Spectrosc. Radiat. Transfer 15, 707 (1975).
[CrossRef]

Jarecki, J.

J. Jarecki, R. Herman, J. Quant. Spectrosc. Radiat. Transfer 15, 707 (1975).
[CrossRef]

Kan, T.

T. Kan, G. J. Wolga, IEEE J. Quantum Electron. QE-7, 141 (1971).

Kopfermann, H.

See, for example, R. Ladenburg, H. Kopfermann, Nature Paris 122, 438 (1928).
[CrossRef]

Ladenburg, R.

See, for example, R. Ladenburg, H. Kopfermann, Nature Paris 122, 438 (1928).
[CrossRef]

Lamb, W. E.

W. E. Lamb, Phys. Rev. 134, 1429 (1964).
[CrossRef]

Meredith, R. E.

R. E. Meredith, T. S. Chang, F. G. Smith, D. R. Woods, SAI-73-004-AA.

Milonni, P. W.

The effects of the nonresonant dispersion are well understood and are not difficult to include in theoretical investigations. See, for example, P. W. Milonni, A. H. Paxton, J. Appl. Phys. 49, 1012 (1978).
[CrossRef]

Mirels, H.

H. Mirels, AIAA J. 17, 478 (1979).
[CrossRef]

Paxton, A. H.

The effects of the nonresonant dispersion are well understood and are not difficult to include in theoretical investigations. See, for example, P. W. Milonni, A. H. Paxton, J. Appl. Phys. 49, 1012 (1978).
[CrossRef]

Polanyi, J. C.

J. C. Polanyi, K. B. Woodall, J. Chem. Phys. 56, 1563 (1972).
[CrossRef]

Rigrod, W. W.

W. W. Rigrod, J. Appl. Phys. 36, 2487 (1965).
[CrossRef]

Smith, F. G.

R. E. Meredith, T. S. Chang, F. G. Smith, D. R. Woods, SAI-73-004-AA.

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (U.S. GPO, Washington, D.C., 1964), pp. 297–304, 325–328.

Stenholm, S.

S. Stenholm, “The Semiclassical Theory of the Gas Laser,” in Progress in Quantum Electronics, J. H. Sanders, K. W. H. Stevens, Eds., (Pergamon, Oxford, 1971), pp. 187–271.

Wolga, G. J.

T. Kan, G. J. Wolga, IEEE J. Quantum Electron. QE-7, 141 (1971).

Woodall, K. B.

J. C. Polanyi, K. B. Woodall, J. Chem. Phys. 56, 1563 (1972).
[CrossRef]

Woods, D. R.

R. E. Meredith, T. S. Chang, F. G. Smith, D. R. Woods, SAI-73-004-AA.

Yariv, A.

L. W. Casperson, A. Yariv, Appl. Opt. 11, 462 (1972).
[CrossRef] [PubMed]

L. Casperson, A. Yariv, Appl. Phys. Lett. 17, 259 (1970).
[CrossRef]

AIAA J. (1)

H. Mirels, AIAA J. 17, 478 (1979).
[CrossRef]

Appl. Opt. (1)

Appl. Opt. Suppl. (1)

W. R. Bennett, Appl. Opt. Suppl. 1, 24 (1962).
[CrossRef]

Appl. Phys. Lett. (2)

J. A. Glaze, Appl. Phys. Lett. 23, 300 (1973).
[CrossRef]

L. Casperson, A. Yariv, Appl. Phys. Lett. 17, 259 (1970).
[CrossRef]

IEEE J. Quantum Electron. (2)

T. Kan, G. J. Wolga, IEEE J. Quantum Electron. QE-7, 141 (1971).

L. Casperson, IEEE J. Quantum Electron. QE-9, 250 (1973).
[CrossRef]

J. Appl. Phys. (2)

The effects of the nonresonant dispersion are well understood and are not difficult to include in theoretical investigations. See, for example, P. W. Milonni, A. H. Paxton, J. Appl. Phys. 49, 1012 (1978).
[CrossRef]

W. W. Rigrod, J. Appl. Phys. 36, 2487 (1965).
[CrossRef]

J. Chem. Phys. (1)

J. C. Polanyi, K. B. Woodall, J. Chem. Phys. 56, 1563 (1972).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer (1)

J. Jarecki, R. Herman, J. Quant. Spectrosc. Radiat. Transfer 15, 707 (1975).
[CrossRef]

Nature Paris (1)

See, for example, R. Ladenburg, H. Kopfermann, Nature Paris 122, 438 (1928).
[CrossRef]

Phys. Rev. (2)

W. E. Lamb, Phys. Rev. 134, 1429 (1964).
[CrossRef]

See, for example, D. H. Close, Phys. Rev. 153, 360 (1967). The equations in this paper are often used in discussions of anomalous dispersion.
[CrossRef]

Other (6)

D. L. Bullock, paper delivered at Tri-Service Chemical Laser Symposium, Air Force Weapons Laboratory, 28–30 Aug. 1979.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (U.S. GPO, Washington, D.C., 1964), pp. 297–304, 325–328.

S. Stenholm, “The Semiclassical Theory of the Gas Laser,” in Progress in Quantum Electronics, J. H. Sanders, K. W. H. Stevens, Eds., (Pergamon, Oxford, 1971), pp. 187–271.

L. Allen, J. H. Eberly, Optical Resonance and Two-Level Atoms (Wiley, New York, 1975).

G. Emanuel, in Handbook of Chemical Lasers, R. W. F. Gross, J. F. Bott, Eds. (Wiley, New York, 1976), pp. 469–549.

R. E. Meredith, T. S. Chang, F. G. Smith, D. R. Woods, SAI-73-004-AA.

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Figures (8)

Fig. 1
Fig. 1

Broadening of HF υ = 1 ↔ 0 transition by HF (υ) T = 300 K.

Fig. 2
Fig. 2

Broadening of HF υ = 2 ↔ 1 transition by HF (υ) T = 300 K.

Fig. 3
Fig. 3

Broadening of HF υ = 3 ↔ 2 transition by HF (υ) T = 300 K.

Fig. 4
Fig. 4

Broadening of HF υ = 1 ↔ 0 transition by HF (υ) T = 600 K.

Fig. 5
Fig. 5

Broadening of HF υ = 2 ↔ 1 transition by HF (υ) T = 600 K.

Fig. 6
Fig. 6

Broadening of HF υ = 0 ↔ 1 transition by Ar.

Fig. 7
Fig. 7

Effective saturation of the gain.

Fig. 8
Fig. 8

Saturation of anomalous dispersion.

Equations (44)

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δ n ( ν ) = ( const ) ( ν ν 0 ) ( g 0 δ ν ) ,
g ( ν l ) + 2 i k l δ n ( ν l ) = ( 8 π 2 ν l 3 h c ) ( J μ υ 2 2 J 1 ) × d u [ N υ , J 1 ( u ) 2 J 1 2 J + 1 N υ 1 , J ( u ) ] γ + i ( ν 0 ν l + ν l u / c ) .
g ( ν l ) + 2 i k l δ n ( ν l ) = β υ J d u [ N υ ( u ) N υ 1 ( u ) exp ( 2 B J / k T ) ] γ + i ( ν 0 ν l + ν l u / c ) ,
β υ J = ( 8 π 2 ν l 3 h c ) μ υ 2 J Z R 1 exp [ B J ( J 1 ) / k T ] ,
N υ ( u ) = N υ W ( u ) ,
g ( ν l ) 2 i k l δ n ( ν l ) = 4 π ln 2 δ ν D β υ J w ( x l + i b ) × [ N υ N υ 1 exp ( 2 B J / k T ) ] ,
x l = 4 ln 2 ( ν 0 ν l δ ν D ) ,
b = 4 ln 2 ( γ δ ν D ) ,
δ n ( ν l ) = β AD ( N 0 / N STP ) ,
β AD 0.05 ( ν 0 ν l δ ν D ) ( 1 2 N 1 N 0 )
δ n ( ν l ) = λ g 0 ( ν l ) 4 π Im w ( x l + i b ) Re w ( x l + i b ) .
δ n ( ν l ) λ g 0 ( ν 0 ) 2 π 3 / 2 [ exp ( x l 2 ) x l d t exp ( t 2 ) ] .
δ n GD 3.6 × 10 5 ( P / T ) .
δ n ( ν l ) AD δ n GD 0.067 g 0 ( ν 0 ) T P ( Torr )
δ ν D 1.60 × 10 3 T 1 / 2 / λ cm 1 ,
γ HF ( m ) = c 1 + c 2 exp ( m / 4 ) + c 3 m exp ( m / 4 ) + c 4 m 2 exp ( m / 2 ) + c 5 m exp ( m 2 / 8 ) + c 6 m 2 exp ( m 2 / 16 ) + c 7 exp ( m 2 / 8 ) ,
N υ ( u ) exp ( 2 B J / k T ) N υ 1 ( u ) = [ N ¯ υ exp ( 2 B J / k T ) N ¯ υ 1 ] W ( u ) 1 + σ l ( u ) h ν 0 I l [ 1 λ υ + 1 λ υ 1 exp ( 2 B J / k T ) ] .
σ l ( u ) = γ β υ J [ ( v 0 ν l + ν l u / c ) 2 + γ 2 ] 1 ,
g ( ν l ) + 2 i k l δ n ( ν l ) = β υ J [ N ¯ υ exp ( 2 B J / k T ) N ¯ υ 1 ] × d u [ γ i ( ν 0 ν l + ν l u / c ) ] W ( u ) ( ν 0 ν l + ν l u / c ) 2 + γ 2 ξ l 2 = π c ν 0 β υ J [ N ¯ υ exp ( 2 B J / k T ) N ¯ υ 1 ] × [ ξ l 1 Re w ( x l + i b ξ l ) i Im w ( x l + i b ξ l ) ] ,
ξ l = { 1 + β υ J I l γ h ν 0 [ 1 λ υ + 1 λ υ 1 exp ( 2 B J / k T ) ] } 1 / 2
I s = γ h ν 0 β υ J [ 1 λ υ + 1 λ υ 1 exp ( 2 B J / k T ) ] 1
g ( ν l ) = g 0 ( ν 0 ) exp ( x l 2 ) ( 1 + I / I s ) 1 / 2 ,
δ n ( ν l ) = g 0 ( ν 0 ) π k l exp ( x l 2 ) 0 x l d t exp ( t 2 ) ,
g ( ν l ) = π 1 / 2 b g 0 ( ν 0 ) x l 2 + b 2 ( 1 + I / I s ) ,
δ n ( ν l ) = ( x l 2 b k l ) g ( ν l ) .
g ( ν l ) I = 0 = ( 8 π 3 μ 1 2 B ν 0 3 h c k T ) υ J { N ¯ υ exp [ B J ( J 1 ) / k T ] N ¯ υ 1 exp [ B J ( J + 1 ) / k T ] } S ( ν l ) ,
P 1.22 λ T 1 / 2 ( j f j γ j ) 1 Torr ,
P 0.42 T 1 / 2 ( f HF + 0.055 f Ar + 0.029 f He + 0.12 f N 2 + 0.065 f H 2 ) 1 Torr .
N υ ( u ) exp ( 2 B J / k T ) N υ 1 ( u ) = [ N ¯ υ exp ( 2 B J / k T ) N ¯ υ 1 ] W ( u ) 1 + 1 h ν 0 [ 1 λ υ + 1 λ υ 1 exp ( 2 B J / k T ] m = 1 N σ m ( u ) I m
g ( ν l ) + 2 i k l δ n ( ν l ) = β υ J [ N ¯ υ exp ( 2 B J / k T ) N ¯ υ 1 ] × d u W ( u ) [ γ + i ( ν 0 ν l + ν 0 u / c ) ] [ 1 + 1 I s m = 1 N γ 2 I m ( ν 0 ν m + ν 0 u / c ) 2 + γ 2 ] .
m = 1 N γ 2 I m ( ν 0 ν m + ν 0 u / c ) 2 + γ 2 γ 2 d ν m I ( ν m ) ( ν 0 ν m + ν 0 u / c ) 2 + γ 2 π γ I ( ν 0 + ν 0 u / c )
g ( ν l ) + 2 i k l δ n ( ν l ) = β υ J [ N ¯ υ exp ( 2 B J / k T ) N ¯ υ 1 ] × d u W ( u ) [ γ + i ( ν 0 ν l ) + ( ν 0 u / c ) ] [ 1 + π γ I s I ( ν 0 + ν 0 u c ) ] .
g ( ν l ) + 2 i k 1 δ n ( ν l ) 4 π ln 2 δ ν D β υ J [ N ¯ υ exp ( 2 B J / k T ) N ¯ υ 1 ] 1 + π γ I ( ν l ) / I s × [ exp ( x l 2 ) 2 i π 1 / 2 exp ( x l 2 ) 0 x l d t exp ( t 2 ) ] ,
g ( ν l ) = g 0 ( ν 0 ) exp ( x l 2 ) 1 + π γ I ( ν l ) / I s ,
δ n ( ν l ) = g 0 ( ν ) π k l exp ( x l 2 ) 0 x l d t exp ( t 2 ) 1 + π γ I ( ν l ) / I s .
I ( ν l ) = I s π γ [ g 0 ( ν 0 ) α exp ( x l 2 ) 1 ] ,
Δ ν d ν l I ( ν l ) = I T ,
π γ I ( ν l ) = π γ I T δ ν D [ g 0 ( ν 0 ) α exp ( x l 2 ) 1 g 0 ( ν 0 ) α π 4 ln 2 1 ] ,
π γ I ( ν l ) 4 π ln 2 ( γ δ ν D ) I T exp ( x l 2 ) = π b I T exp ( x l 2 ) .
g ( ν l ) = g 0 ( ν 0 ) exp ( x l 2 ) 1 + I T / I ˜ s (D) ,
δ n ( ν l ) = g 0 ( ν 0 ) π k l exp ( x l 2 ) 0 x l d t exp ( t 2 ) 1 + I T / I ˜ s ( D ) ,
I ˜ s ( D ) = 1 π b I s exp ( x l 2 ) .
1 1 + ( ν l ) d I + ( ν l ) d z = g 0 ( ν 0 ) exp ( x l 2 ) 1 + π γ I s [ I + ( ν l ) + I ( ν l ) ] ,
1 I ( ν l ) d I ( ν l ) d z = g 0 ( ν 0 ) exp ( x l 2 ) 1 + π γ I s [ I + ( ν l ) + I ( ν l ) ]

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