Abstract

Analytic solutions are presented for inhomogeneous broadening effects in multimode cw chemical lasers. A Fabry-Perot (F.P.) resonator and a saturated amplifier are considered in the limits Δνh ≪ Δνd and Δνc ≪ Δνh, where Δνh, Δνd, and Δνc are homogeneous, Doppler, and longitudinal mode separation widths, respectively. The former inequality requires p(Torr) ⩽ 0(10). The results are believed valid for Δνcνh ⩽ 0(1) and apply for resonator mirror separation lengths and amplifier lengths of the order of 10 m or more. The normalized frequency difference from line center is denoted X, and the value of X corresponding to the largest longitudinal mode frequency is denoted Xf. The quantity Xf is a measure of laser frequency bandwidth and the number of active longitudinal modes. For the case of an F.P. resonator, Xf varies with streamwise distance. The local value of Xf is independent of upstream conditions for the case of a saturated F.P. resonator. The variation of lasing intensity with X at each streamwise station is found to be a truncated Gaussian. The slope of the curve of η − 1 (anomalous index of refraction) vs X is positive in the lasing region |X| < Xf. The magnitude of η − 1 is proportional to the threshold gain. For a typical saturated cw chemical laser oscillator, the anomalous index of refraction is shown to be small, compared with the regular index, for medium pressures in the range p(Torr) ⩾ 0(1). The present analytic results are in good agreement with the numerical results of Bullock and Lipkis.

© 1981 Optical Society of America

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References

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  1. W. R. Bennett, Phys. Rev. 126, 580 (1961).
    [CrossRef]
  2. M. Sargent, M. O. Scully, W. E. Lamb, Laser Physics (Addison-Wesley, Reading, Mass., 1974), pp. 144–155.
  3. T. Kan, C. J. Wolga, IEEE J. Quantum Electron. QE-7, 141 (1971).
  4. A. Y. Cabezas, R. P. Treat, J. Appl. Phys. 37, 3556 (1966).
    [CrossRef]
  5. H. Mirels, AIAA J. 17, 478 (1979).
    [CrossRef]
  6. D. L. Bullock, R. S. Lipkis, “Saturation of the Gain and Resonant Dispersion in Chemical Lasers,” at Fourth Annual Tri-Service Chemical Laser Conference, Albuquerque, New Mex., 22 Aug. 1979.
  7. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, AMS 55 (National Bureau of Standards, Washington, D.C., June1964), pp. 297–303.
  8. In the derivation of Eq. (12a) it was noted that the present solution is in error in a region of order Δνh about νjf and 2ν0 − νjf. The latter region corresponds to |X − Xf| = 0.04 in the present example.

1979 (1)

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

1971 (1)

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

1966 (1)

A. Y. Cabezas, R. P. Treat, J. Appl. Phys. 37, 3556 (1966).
[CrossRef]

1961 (1)

W. R. Bennett, Phys. Rev. 126, 580 (1961).
[CrossRef]

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, AMS 55 (National Bureau of Standards, Washington, D.C., June1964), pp. 297–303.

Bennett, W. R.

W. R. Bennett, Phys. Rev. 126, 580 (1961).
[CrossRef]

Bullock, D. L.

D. L. Bullock, R. S. Lipkis, “Saturation of the Gain and Resonant Dispersion in Chemical Lasers,” at Fourth Annual Tri-Service Chemical Laser Conference, Albuquerque, New Mex., 22 Aug. 1979.

Cabezas, A. Y.

A. Y. Cabezas, R. P. Treat, J. Appl. Phys. 37, 3556 (1966).
[CrossRef]

Kan, T.

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

Lamb, W. E.

M. Sargent, M. O. Scully, W. E. Lamb, Laser Physics (Addison-Wesley, Reading, Mass., 1974), pp. 144–155.

Lipkis, R. S.

D. L. Bullock, R. S. Lipkis, “Saturation of the Gain and Resonant Dispersion in Chemical Lasers,” at Fourth Annual Tri-Service Chemical Laser Conference, Albuquerque, New Mex., 22 Aug. 1979.

Mirels, H.

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

Sargent, M.

M. Sargent, M. O. Scully, W. E. Lamb, Laser Physics (Addison-Wesley, Reading, Mass., 1974), pp. 144–155.

Scully, M. O.

M. Sargent, M. O. Scully, W. E. Lamb, Laser Physics (Addison-Wesley, Reading, Mass., 1974), pp. 144–155.

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, AMS 55 (National Bureau of Standards, Washington, D.C., June1964), pp. 297–303.

Treat, R. P.

A. Y. Cabezas, R. P. Treat, J. Appl. Phys. 37, 3556 (1966).
[CrossRef]

Wolga, C. J.

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

AIAA J. (1)

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

IEEE J. Quantum Electron. (1)

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

J. Appl. Phys. (1)

A. Y. Cabezas, R. P. Treat, J. Appl. Phys. 37, 3556 (1966).
[CrossRef]

Phys. Rev. (1)

W. R. Bennett, Phys. Rev. 126, 580 (1961).
[CrossRef]

Other (4)

M. Sargent, M. O. Scully, W. E. Lamb, Laser Physics (Addison-Wesley, Reading, Mass., 1974), pp. 144–155.

D. L. Bullock, R. S. Lipkis, “Saturation of the Gain and Resonant Dispersion in Chemical Lasers,” at Fourth Annual Tri-Service Chemical Laser Conference, Albuquerque, New Mex., 22 Aug. 1979.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, AMS 55 (National Bureau of Standards, Washington, D.C., June1964), pp. 297–303.

In the derivation of Eq. (12a) it was noted that the present solution is in error in a region of order Δνh about νjf and 2ν0 − νjf. The latter region corresponds to |X − Xf| = 0.04 in the present example.

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Figures (8)

Fig. 1
Fig. 1

Continuous wave chemical laser: (a) with F.P. resonator; (b) simplifier model. (A single semichannel is shown.)

Fig. 2
Fig. 2

Characteristic frequencies and frequency dependent variables for cases Δνh ≪ Δνd and Δνc ≪ Δνh: (a) nonlasing case; (b) lasing case (expanded view).

Fig. 3
Fig. 3

Variation of number density with frequency at two stream-wise stations: (a) start of lasing (ζ = ζi, ΔN = ΔNi); (b) lasing region; (ζ > ζi).

Fig. 4
Fig. 4

Frequency width variation with streamwise distance in F.P. resonator for laminar flame sheet in limit R ≫ 1, ΔNi ≪ 1, and RΔNi = 0(1) [see Eq. (28)].

Fig. 5
Fig. 5

Variation of index of refraction with frequency for number density distributions illustrated in Figs. 3 and 7 [see Eqs. (31) and (46)]: (a) A = 0.00; (b) A = 0.25; (c) A = 0.50; (d) A = 0.75; (e) A = 1.00.

Fig. 6
Fig. 6

Saturated amplifier.

Fig. 7
Fig. 7

Number density profile for case of saturated amplifier.

Fig. 8
Fig. 8

Comparison with numerical solution of Ref. 6 for case of saturated cw chemical laser with F.P. resonator: (a) gain; (b) intensity; (c) index of refraction.

Equations (108)

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Δ ν h Δ ν d ,
Δ ν c Δ ν h ,
Δ N N 2 - N 1 = ( n 2 - n 1 ) y f / ( n r w ) ,
Δ N ν N 2 ( ν ) - N 1 ( ν ) = [ n 2 ( ν ) - n 1 ( ν ) ] y f / ( n r p ¯ 0 w ) ,
p ¯ 0 = [ 4 ( ln 2 ) / π ] 1 / 2 / Δ ν d .
Δ N = p ¯ 0 - Δ N ν d ν .
G ( ν ) g ( ν ) y f σ 0 n r w p ¯ 0 Δ ν h = - L ( ν - ν ) Δ N ν d ν Δ ν h ,
η ¯ ( ν ) ( 2 π / λ ) [ η ( ν ) - 1 ] y f σ 0 n r w p ¯ 0 Δ ν h = - ν - ν Δ ν h L ( ν - ν ) Δ N ν d ν Δ ν h ,
L ( ν - ν ) σ ( ν , ν ) σ 0 = [ 1 + 4 ( ν - ν Δ ν h ) 2 ] - 1 .
[ g ( ν ) ] av = g ( ν ) y f / w ,
[ η ( ν ) - 1 ] av = [ η ( ν ) - 1 ] y f / w .
Δ N ν / Δ N = exp ( - X 2 ) ,
G ( ν ) / Δ N = ( π / 2 ) exp ( - X 2 ) [ 1 + 0 ( Y ) ] ,
η ¯ ( ν ) / Δ N = ( π 1 / 2 / 2 ) D ( X ) [ 1 + 0 ( Y ) ] ,
X = 2 ( ln 2 ) 1 / 2 ( ν - ν 0 ) / Δ ν d ,
Y = ( ln 2 ) 1 / 2 Δ ν h / Δ ν d ,
D ( X ) = exp ( - X 2 ) 0 X exp ( X 0 2 ) d X 0
= n = 0 { ( - 2 ) n X 2 n + 1 / [ 1 · 3 · 5 ( 2 n + 1 ) ] }
= ( 2 X ) - 1 { 1 + n = 1 [ 1 · 3 · 5 ( 2 n - 1 ) / ( 2 X 2 ) n } .
d Δ N ν d ζ = exp ( - X 2 ) d N T d ζ - [ exp ( - X 2 ) N T + Δ N ν ] + R [ exp ( - X 2 ) Δ N - Δ N ν ] - 2 Δ N ν - I ( ν ) L ( ν - ν ) d ν ,
d Δ N d ζ = d N T d ζ - N T - Δ N - 2 p ¯ 0 Δ ν h - I ( ν ) G ( ν ) d ν .
ν j = ν 0 ± j Δ ν c ,             j = 0 , 1 , 2 j f ,
I j = ν j - ( Δ ν c / 2 ) ν j + ( Δ ν c / 2 ) I ( ν ) d ν
I ¯ ( ν ) = I j / Δ ν c ,
- I ( ν ) L ( ν - ν ) d ν = ( π / 2 ) Δ ν h I ˜ ( ν ) .
d Δ N ν d ζ = exp ( - X 2 ) ( d N T d ζ - N T + R Δ N ) - ( 1 + R ) Δ N ν - π Δ ν h Δ N ν I ˜ ( ν ) ,
d Δ N d ζ = d N T d ζ - N T - Δ N - 2 d P d ζ ,
d P d ζ p ¯ 0 Δ ν h 2 ν 0 - ν j f ν j f G ( ν ) I ˜ ( ν ) d ν .
P P ¯ ( ζ ) u n r w n s c = 0 ζ d P d ζ d ζ .
2 P = [ N T - Δ N ] ζ i ζ - ζ i ζ ( N T + Δ N ) d ζ ,
G c = - ( ln R m ) / ( σ 0 n r Δ ν h ρ ¯ 0 w η s c ) ,
( g c ) av = - ( ln R m ) / ( w η s c ) ,
Δ N i = ( 2 / π ) G c ,
Δ N ν / Δ N i = 1             X < X f
= exp ( X f 2 - X 2 )             X > X f ,
X f = 2 ( ln 2 ) 1 / 2 ( ν j f - ν 0 ) / Δ ν d .
Δ N / Δ N i = 2 X f / ( π ) 1 / 2 + exp ( X f 2 ) erfc X f ,
d X f 2 d ζ = exp ( - X f 2 ) [ 1 Δ N i ( d N T d ζ - N T ) + 2 R X f ( π ) 1 / 2 ] - R erf X f - 1 ,
π Δ ν h I ˜ ( ν ) = exp ( - X 2 ) [ 1 Δ N i ( d N T d ζ - N T ) + R Δ N Δ N i ] - R - 1.
2 Δ N i d P d ζ = [ 1 Δ N i ( d N T d ζ - N T ) + R Δ N Δ N i ] erf X f - 2 π 1 / 2 ( 1 + R ) X f .
2 d P d ζ = d N T d ζ - N T - d Δ N d ζ - Δ N .
exp ( X f 2 ) erf X f - 2 π 1 / 2 X f = 1 R Δ N i ( d N T d ζ - N T ) ,
Δ N Δ N i = exp ( X f 2 ) - 1 R Δ N i ( d N T d ζ - N T ) ,
2 Δ ν h G c I ˜ ( ν ) = Δ N i R [ exp ( X f 2 - X 2 ) - 1 ]             X X f ,
2 d P d ζ = d N T d ζ - N T ,
2 P = N T - 0 ζ N T d ζ ,
ζ D 1 / 2 N T = ζ 1 / 2 ,
ζ D 1 / 2 Δ N ( 2 / π ) ζ D 1 / 2 G ( ν 0 ) = 2 D ( ζ 1 / 2 ) - ζ 1 / 2 ,
ζ D 1 / 2 Δ N mzp = ( 2 / π ) ζ D 1 / 2 [ D ( ν 0 ) ] mzp = 0.3528.
σ 0 n r p ¯ 0 Δ ν h / ζ D 1 / 2 = 1.804 g mzp ,
[ g ( ν ) ] av / g mzp = 1.804 ζ D 1 / 2 G ( ν ) ,
( g c ) av / g mzp = 1.804 ζ D 1 / 2 G c ,
g mzp { [ g ( ν 0 ) ] av } mzp .
ζ D 1 / 2 Δ N i = ( 2 / π ) ζ D 1 / 2 G c = 2 D ( ζ i 1 / 2 ) - ζ i 1 / 2 .
ζ D 1 / 2 Δ N i = ζ i 1 / 2 [ 1 + 0 ( ζ i ) ] .
d exp ( X f 2 ) d z = ζ i 1 / 2 ζ D 1 / 2 Δ N i ( 1 - 2 ζ i z 2 ) - 2 ζ i z × [ exp ( X f 2 ) + 2 R π 1 / 2 n = 1 2 n X f 2 n + 1 1 · 3 · 5 ( 2 n + 1 ) ] ,
exp ( X f 2 ) - 1 = ( z - 1 ) { 1 - 16 15 R ζ i π 1 / 2 ( z - 1 ) 3 / 2 [ 1 + 0 ( X f 2 ) ] + 0 ( ζ i ) } ,
2 ζ 1 / 2 = - β + ( β 2 + 2 ) 1 / 2 ,
β = ζ i 1 / 2 R [ exp ( X f 2 ) erf X f - ( 2 X f / π 1 / 2 ) ] .
X f = [ ln ( 2 R ζ i 1 / 2 ζ 1 / 2 ) - 1 ] 1 / 2             ζ 0 ,
= [ 3 ( 2 π ) 1 / 2 8 R ζ i 1 / 2 ( 1 - 2 ζ ) ] 1 / 3             ζ ½ .
Δ N Δ N i = exp ( X f 2 ) - 1 R ζ i 1 / 2 ( 1 2 ζ 1 / 2 - ζ 1 / 2 ) ,
2 Δ ν h ζ D 1 / 2 G c I ¯ ( ν ) = R ζ i 1 / 2 [ exp ( X f 2 - X 2 ) - 1 ]             X X f ,
2 ζ D 1 / 2 d P d ζ = 1 2 ζ 1 / 2 - ζ 1 / 2 ,
2 ζ D 1 / 2 P e = 2 1 / 2 / 3.
η ¯ Δ N i = 1 2 X X f 1 1 - exp [ - X f 2 ( t 2 - 1 ) ] t 2 - ( X / X f ) 2 d t ,
2 X f X η ¯ Δ N i = π 1 / 2 X f exp ( X f 2 ) erfc ( X f ) [ 1 - ( ) X 2 ] + ( ) X 2 + 0 ( X / X f ) 4 .
exp ( X f 2 ) erfc X f = 1 - ( 2 / π 1 / 2 ) X f + X f 2 + ( X f ) 3
= [ 1 - ( 2 x f ) - 2 + 0 ( X f - 4 ) ] / ( π 1 / 2 X f ) .
( 2 X f / X ) η ¯ / Δ N i = π 1 / 2 X f [ 1 + 0 ( X 2 ) ] .
2 π λ [ η ( ν ) - 1 ] av X / X f = 2 π ( g c ) av [ X f X η ¯ Δ N i ] .
t j ( 2 / c ) 0 L η ( ν j ) d s = ( 2 L / c ) { 1 + ( η s c w / L ) [ η ( ν j ) - 1 ] av } ,
ν j - ν 0 = ( c / 2 L ) j 1 + w n s c L ν 0 ( X / X f ) ν j - ν 0 [ η ( ν j ) - 1 ] av X / X f ,
I ˜ ( ν ) / y = [ g ( ν ) ] av I ˜ ( ν ) .
[ g ( ν ) ] av ( π / 2 ) σ 0 n r p ¯ 0 Δ ν h Δ N ν = 1 y { 1 - [ I ˜ ( ν ) ] y = 0 I ˜ ( ν ) } .
Δ N ν / ζ = 0 ,
Δ N ν / X = 0.
Δ N ν / Δ N ν f = A 1             X X f ,
= exp ( X f 2 - X 2 )             X > X f ,
Δ N / Δ N ν f = ( 2 / π 1 / 2 ) A X f + exp ( X f 2 ) erfc X f ,
Δ N = B exp ( - ϕ ζ ) 0 ζ exp ( ϕ ζ ) [ ( d N T / d ζ ) - N T ] d ζ ,
B ( Δ N / Δ N ν f ) exp ( - X f 2 ) = ( 2 / π 1 / 2 ) A X f exp ( - X f 2 ) + erfc X f ,
ϕ = 1 + ( 1 - B ) R .
π Δ ν h Δ N ν I ˜ ( ν ) = exp ( - X 2 ) [ ( d N T / d ζ ) - N T + R Δ N ] - ( 1 - R ) A Δ N / [ B exp ( X f 2 ) ] ,
Δ N = erfc X f R erf X f ( d N T d ζ - N T ) ,
π Δ ν h Δ N ν I ˜ ( ν ) = exp ( - X 2 ) erf X f ( d N T d ζ - N T ) .
ζ b 1 / 2 Δ N ν = 0.353 y g mzp { 1 - [ I ˜ ( ν ) ] y = 0 I ˜ ( ν ) } ,
ζ D 1 / 2 Δ N ζ D 1 / 2 Δ N ν f exp ( X f 2 ) erfc X f = erfc X f R erf X f ( 1 2 ζ 1 / 2 - ζ 1 / 2 ) ,
A = 0.353 y g mzp R exp ( X f 2 ) erf X f 1 2 ζ ½ - ζ 1 / 2 ( 1 + 0 { [ I ˜ ( ν ) ] y = 0 I ˜ ( ν ) } ) .
η ¯ Δ N ν f = 1 2 X X f 1 { 1 - exp [ - X f 2 ( t 2 - 1 ) ] } t 2 - ( X / X f ) 2 d t - 1 - A 4 ln ( X f + X X f - X ) .
2 X f X η ¯ Δ N ν f = A - 1 + π 1 / 2 X f exp ( X f 2 ) erfc X f + ( 1 / 3 ) ( X / X f ) 2 { A - 1 + 2 X f 2 [ 1 - π 1 / 2 X f × exp ( X f 2 ) erfc X f ] } + 0 ( X / X f ) 4 .
A = 1 - π 1 / 2 X f exp ( X f 2 ) erfc X f .
OPD = 1.804 ζ D 1 / 2 g mzp [ λ / ( 2 π ) ] 0 w n s c η ¯ d y
Δ ν h , Δ ν c , Δ ν d = 9.9 × 10 6 , 1.1 × 10 7 , 4.3 × 10 8 sec - 1 ,
g mzp = 0 ( 0.10 ) cm - 1 ,
( g c ) av = 1.155 × 10 - 3 cm - 1 ,
λ = 2.832 × 10 - 4 cm .
Δ ν h / Δ ν d = 2.3 × 10 - 2 ,
Δ ν c / Δ ν h = 1.1 ,
Δ ν d = 7.163 × 10 - 7 ν 0 ( T / M ) 1 / 2 sec - 1 ,
Δ ν d = 3.00 × 10 8 ( T / 300 ) 1 / 2 .
Δ ν h = 3.0 × 10 6 p ( Torr ) ,
Δ ν c = c / 2 L = 1.499 × 10 8 / L ( m ) .
Δ ν h / Δ ν d = 0.01 ( 300 / T ) 1 / 2 p ( Torr ) ,
Δ ν c / Δ ν h = 50 / [ p ( Torr ) L ( m ) ] .
( η - 1 ) A = 1.2 × 10 - 8 X [ λ 3 × 10 - 4 ( g c ) av 10 - 3 ( η ¯ / Δ N i ) / X 0.4 ] ,
( η - 1 ) R = 1.3 × 10 - 7 p ( Torr ) [ β 10 - 4 ρ ρ st 760 p ( Torr ) ] ,
( η - 1 ) A / ( η - 1 ) R = 0.1 X / p ( Torr ) .

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