Abstract

A theoretical model is reported for spontaneous mode locking in mixed-broadened laser oscillators. Experimental observations of this effect have been available for many years, but no rigorous interpretation has been given. Numerical calculations emphasize the case of a high-gain xenon laser, for which extensive experimental data have been published. Complex pulsation characteristics are observed as the cavity length and pumping rate are varied, and the theoretical results are in good agreement with the experimental data.

© 2007 Optical Society of America

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References

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  1. M. F. H. Tarroja, M. Sharafi, and L. W. Casperson, 'Spontaneous mode locking in long-cavity lasers,' J. Opt. Soc. Am. B 6, 1564-1573 (1989).
    [CrossRef]
  2. H. Risken and K. Nummedal, 'Self-pulsing in lasers,' J. Appl. Phys. 39, 4662-4672 (1968).
    [CrossRef]
  3. H. Haken, 'Analogy between higher instabilities in fluids and lasers,' Phys. Lett. A 53, 77-78 (1975).
    [CrossRef]
  4. U. Hubner, N. B. Abraham, and C. O. Weiss, 'Dimensions and entropies of chaotic intensity pulsations in a single-mode far-infrared NH3 laser,' Phys. Rev. A 40, 6354-6365 (1989).
    [CrossRef] [PubMed]
  5. T. Voigt, M. O. Lenz, F. Mitschke, E. Roldan, and G. J. De Valcarcel, 'Experimental investigation of Risken-Nummedal-Graham-Haken laser instability in fiber ring lasers,' Appl. Phys. B: Lasers Opt. 79, 175-183 (2004).
    [CrossRef]
  6. H. H. Kim and H. Marantz, 'Continuous self-mode locking of infrared gas lasers,' IEEE J. Quantum Electron. QE-6, 749-750 (1970).
    [CrossRef]
  7. L. W. Casperson and A. Yariv, 'Pulse propagation in a high-gain medium,' Phys. Rev. Lett. 26, 293-295 (1971).
    [CrossRef]
  8. L. W. Casperson, 'Spontaneous coherent pulsations in ring-laser oscillators,' J. Opt. Soc. Am. B 2, 62-72 (1985).
    [CrossRef]
  9. E. I. Gordon, A. D. White, and J. D. Rigden, 'Gain saturation at 3.39 microns in the He-Ne maser,' in Proceedings of the Symposium on Optical Masers, Polytechnic Institute of Brooklyn, New York, 309-317 (Wiley, 1964).
  10. L. W. Casperson, 'Spontaneous coherent pulsations in standing-wave laser oscillators,' J. Opt. Soc. Am. B 5, 958-969 (1988).
    [CrossRef]
  11. L. W. Casperson and M. F. H. Tarroja, 'Spontaneous coherent pulsations in standing-wave laser oscillators: simplified models,' J. Opt. Soc. Am. B 8, 250-261 (1991).
    [CrossRef]
  12. L. M. Narducci, J. R. Tredicce, L. A. Lugiato, N. B. Abraham, and D. K. Bandy, 'Mode-mode competition and unstable behavior in a homogeneously broadened ring laser,' Phys. Rev. A 33, 1842-1854 (1986).
    [CrossRef] [PubMed]
  13. P. Chenkosol, 'Spontaneous pulsations in laser oscillators: effects of spatial field distributions; self-mode-locking dynamics; pulsations in 3.39 μm He-Ne standing-wave lasers,' Ph.D. dissertation (Portland State University, Portland, Oregon, USA, 2003), Appendix A, pp. 196-200.

2004 (1)

T. Voigt, M. O. Lenz, F. Mitschke, E. Roldan, and G. J. De Valcarcel, 'Experimental investigation of Risken-Nummedal-Graham-Haken laser instability in fiber ring lasers,' Appl. Phys. B: Lasers Opt. 79, 175-183 (2004).
[CrossRef]

1991 (1)

1989 (2)

M. F. H. Tarroja, M. Sharafi, and L. W. Casperson, 'Spontaneous mode locking in long-cavity lasers,' J. Opt. Soc. Am. B 6, 1564-1573 (1989).
[CrossRef]

U. Hubner, N. B. Abraham, and C. O. Weiss, 'Dimensions and entropies of chaotic intensity pulsations in a single-mode far-infrared NH3 laser,' Phys. Rev. A 40, 6354-6365 (1989).
[CrossRef] [PubMed]

1988 (1)

1986 (1)

L. M. Narducci, J. R. Tredicce, L. A. Lugiato, N. B. Abraham, and D. K. Bandy, 'Mode-mode competition and unstable behavior in a homogeneously broadened ring laser,' Phys. Rev. A 33, 1842-1854 (1986).
[CrossRef] [PubMed]

1985 (1)

1975 (1)

H. Haken, 'Analogy between higher instabilities in fluids and lasers,' Phys. Lett. A 53, 77-78 (1975).
[CrossRef]

1971 (1)

L. W. Casperson and A. Yariv, 'Pulse propagation in a high-gain medium,' Phys. Rev. Lett. 26, 293-295 (1971).
[CrossRef]

1970 (1)

H. H. Kim and H. Marantz, 'Continuous self-mode locking of infrared gas lasers,' IEEE J. Quantum Electron. QE-6, 749-750 (1970).
[CrossRef]

1968 (1)

H. Risken and K. Nummedal, 'Self-pulsing in lasers,' J. Appl. Phys. 39, 4662-4672 (1968).
[CrossRef]

Abraham, N. B.

U. Hubner, N. B. Abraham, and C. O. Weiss, 'Dimensions and entropies of chaotic intensity pulsations in a single-mode far-infrared NH3 laser,' Phys. Rev. A 40, 6354-6365 (1989).
[CrossRef] [PubMed]

L. M. Narducci, J. R. Tredicce, L. A. Lugiato, N. B. Abraham, and D. K. Bandy, 'Mode-mode competition and unstable behavior in a homogeneously broadened ring laser,' Phys. Rev. A 33, 1842-1854 (1986).
[CrossRef] [PubMed]

Bandy, D. K.

L. M. Narducci, J. R. Tredicce, L. A. Lugiato, N. B. Abraham, and D. K. Bandy, 'Mode-mode competition and unstable behavior in a homogeneously broadened ring laser,' Phys. Rev. A 33, 1842-1854 (1986).
[CrossRef] [PubMed]

Casperson, L. W.

Chenkosol, P.

P. Chenkosol, 'Spontaneous pulsations in laser oscillators: effects of spatial field distributions; self-mode-locking dynamics; pulsations in 3.39 μm He-Ne standing-wave lasers,' Ph.D. dissertation (Portland State University, Portland, Oregon, USA, 2003), Appendix A, pp. 196-200.

De Valcarcel, G. J.

T. Voigt, M. O. Lenz, F. Mitschke, E. Roldan, and G. J. De Valcarcel, 'Experimental investigation of Risken-Nummedal-Graham-Haken laser instability in fiber ring lasers,' Appl. Phys. B: Lasers Opt. 79, 175-183 (2004).
[CrossRef]

Gordon, E. I.

E. I. Gordon, A. D. White, and J. D. Rigden, 'Gain saturation at 3.39 microns in the He-Ne maser,' in Proceedings of the Symposium on Optical Masers, Polytechnic Institute of Brooklyn, New York, 309-317 (Wiley, 1964).

Haken, H.

H. Haken, 'Analogy between higher instabilities in fluids and lasers,' Phys. Lett. A 53, 77-78 (1975).
[CrossRef]

Hubner, U.

U. Hubner, N. B. Abraham, and C. O. Weiss, 'Dimensions and entropies of chaotic intensity pulsations in a single-mode far-infrared NH3 laser,' Phys. Rev. A 40, 6354-6365 (1989).
[CrossRef] [PubMed]

Kim, H. H.

H. H. Kim and H. Marantz, 'Continuous self-mode locking of infrared gas lasers,' IEEE J. Quantum Electron. QE-6, 749-750 (1970).
[CrossRef]

Lenz, M. O.

T. Voigt, M. O. Lenz, F. Mitschke, E. Roldan, and G. J. De Valcarcel, 'Experimental investigation of Risken-Nummedal-Graham-Haken laser instability in fiber ring lasers,' Appl. Phys. B: Lasers Opt. 79, 175-183 (2004).
[CrossRef]

Lugiato, L. A.

L. M. Narducci, J. R. Tredicce, L. A. Lugiato, N. B. Abraham, and D. K. Bandy, 'Mode-mode competition and unstable behavior in a homogeneously broadened ring laser,' Phys. Rev. A 33, 1842-1854 (1986).
[CrossRef] [PubMed]

Marantz, H.

H. H. Kim and H. Marantz, 'Continuous self-mode locking of infrared gas lasers,' IEEE J. Quantum Electron. QE-6, 749-750 (1970).
[CrossRef]

Mitschke, F.

T. Voigt, M. O. Lenz, F. Mitschke, E. Roldan, and G. J. De Valcarcel, 'Experimental investigation of Risken-Nummedal-Graham-Haken laser instability in fiber ring lasers,' Appl. Phys. B: Lasers Opt. 79, 175-183 (2004).
[CrossRef]

Narducci, L. M.

L. M. Narducci, J. R. Tredicce, L. A. Lugiato, N. B. Abraham, and D. K. Bandy, 'Mode-mode competition and unstable behavior in a homogeneously broadened ring laser,' Phys. Rev. A 33, 1842-1854 (1986).
[CrossRef] [PubMed]

Nummedal, K.

H. Risken and K. Nummedal, 'Self-pulsing in lasers,' J. Appl. Phys. 39, 4662-4672 (1968).
[CrossRef]

Rigden, J. D.

E. I. Gordon, A. D. White, and J. D. Rigden, 'Gain saturation at 3.39 microns in the He-Ne maser,' in Proceedings of the Symposium on Optical Masers, Polytechnic Institute of Brooklyn, New York, 309-317 (Wiley, 1964).

Risken, H.

H. Risken and K. Nummedal, 'Self-pulsing in lasers,' J. Appl. Phys. 39, 4662-4672 (1968).
[CrossRef]

Roldan, E.

T. Voigt, M. O. Lenz, F. Mitschke, E. Roldan, and G. J. De Valcarcel, 'Experimental investigation of Risken-Nummedal-Graham-Haken laser instability in fiber ring lasers,' Appl. Phys. B: Lasers Opt. 79, 175-183 (2004).
[CrossRef]

Sharafi, M.

Tarroja, M. F. H.

Tredicce, J. R.

L. M. Narducci, J. R. Tredicce, L. A. Lugiato, N. B. Abraham, and D. K. Bandy, 'Mode-mode competition and unstable behavior in a homogeneously broadened ring laser,' Phys. Rev. A 33, 1842-1854 (1986).
[CrossRef] [PubMed]

Voigt, T.

T. Voigt, M. O. Lenz, F. Mitschke, E. Roldan, and G. J. De Valcarcel, 'Experimental investigation of Risken-Nummedal-Graham-Haken laser instability in fiber ring lasers,' Appl. Phys. B: Lasers Opt. 79, 175-183 (2004).
[CrossRef]

Weiss, C. O.

U. Hubner, N. B. Abraham, and C. O. Weiss, 'Dimensions and entropies of chaotic intensity pulsations in a single-mode far-infrared NH3 laser,' Phys. Rev. A 40, 6354-6365 (1989).
[CrossRef] [PubMed]

White, A. D.

E. I. Gordon, A. D. White, and J. D. Rigden, 'Gain saturation at 3.39 microns in the He-Ne maser,' in Proceedings of the Symposium on Optical Masers, Polytechnic Institute of Brooklyn, New York, 309-317 (Wiley, 1964).

Yariv, A.

L. W. Casperson and A. Yariv, 'Pulse propagation in a high-gain medium,' Phys. Rev. Lett. 26, 293-295 (1971).
[CrossRef]

Appl. Phys. B: Lasers Opt. (1)

T. Voigt, M. O. Lenz, F. Mitschke, E. Roldan, and G. J. De Valcarcel, 'Experimental investigation of Risken-Nummedal-Graham-Haken laser instability in fiber ring lasers,' Appl. Phys. B: Lasers Opt. 79, 175-183 (2004).
[CrossRef]

IEEE J. Quantum Electron. (1)

H. H. Kim and H. Marantz, 'Continuous self-mode locking of infrared gas lasers,' IEEE J. Quantum Electron. QE-6, 749-750 (1970).
[CrossRef]

J. Appl. Phys. (1)

H. Risken and K. Nummedal, 'Self-pulsing in lasers,' J. Appl. Phys. 39, 4662-4672 (1968).
[CrossRef]

J. Opt. Soc. Am. B (4)

Phys. Lett. A (1)

H. Haken, 'Analogy between higher instabilities in fluids and lasers,' Phys. Lett. A 53, 77-78 (1975).
[CrossRef]

Phys. Rev. A (2)

U. Hubner, N. B. Abraham, and C. O. Weiss, 'Dimensions and entropies of chaotic intensity pulsations in a single-mode far-infrared NH3 laser,' Phys. Rev. A 40, 6354-6365 (1989).
[CrossRef] [PubMed]

L. M. Narducci, J. R. Tredicce, L. A. Lugiato, N. B. Abraham, and D. K. Bandy, 'Mode-mode competition and unstable behavior in a homogeneously broadened ring laser,' Phys. Rev. A 33, 1842-1854 (1986).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

L. W. Casperson and A. Yariv, 'Pulse propagation in a high-gain medium,' Phys. Rev. Lett. 26, 293-295 (1971).
[CrossRef]

Other (2)

E. I. Gordon, A. D. White, and J. D. Rigden, 'Gain saturation at 3.39 microns in the He-Ne maser,' in Proceedings of the Symposium on Optical Masers, Polytechnic Institute of Brooklyn, New York, 309-317 (Wiley, 1964).

P. Chenkosol, 'Spontaneous pulsations in laser oscillators: effects of spatial field distributions; self-mode-locking dynamics; pulsations in 3.39 μm He-Ne standing-wave lasers,' Ph.D. dissertation (Portland State University, Portland, Oregon, USA, 2003), Appendix A, pp. 196-200.

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Figures (5)

Fig. 1
Fig. 1

Theoretical spontaneous-pulsation waveforms for a single-mode xenon laser with line-center tuning and uniformly distributed loss. The threshold parameter values are R = (a) 3.0, (b) 2.5, (c) 2.0, (d) 1.2, (e) 1.1, (f) 1.06. These are the same conditions as those in Fig. 2 of Ref. [5].

Fig. 2
Fig. 2

Theoretical self-locking waveform for a long-cavity xenon laser. The length of the gain medium is 1.1 m . The cavity lengths and the threshold parameter values are: (a) L c a v i t y = 7.4 m , R = 2.8 ; (b) L c a v i t y = 15 m , R = 2.3 ; (c) L c a v i t y = 22.5 m ; R = 2.1 ; (d) L c a v i t y = 30 m ; R = 2.0 ; (e) L c a v i t y = 44.2 m , R = 1.6 ; (f) L c a v i t y = 66 m , R = 1.3 .

Fig. 3
Fig. 3

Frequency spectra of the theoretical self-locking waveforms in Fig. 2. The length of the gain medium is 1.1 m . The cavity lengths and the threshold parameter values are, respectively, the same as in Fig. 2.

Fig. 4
Fig. 4

Theoretical self-locking waveforms and the corresponding frequency spectra for a long-cavity xenon laser. The gain medium and cavity lengths are 1.1 meters and 15 meters, respectively. The threshold parameter values are: R = (a) 1.6, (b) 2.0, (c) 2.3.

Fig. 5
Fig. 5

Theoretical self-locking waveforms and the corresponding frequency spectra for a long-cavity xenon laser. The gain medium and cavity lengths are 1.1 meters and 22.5 meters, respectively. The threshold parameter values are: R = (a) 1.2, (b) 1.5, (c) 2.0, (d) 2.1.

Equations (57)

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A ( z , t ) t + υ p A ( z , t ) z = σ 2 ε 1 [ { 1 i δ ( y y 0 ) } A ( z , t ) i P ( V , U , z , t ) d V d U ] ,
P ( V , U , z , t ) t + ε u V P ( V , U , z , t ) z = γ [ { 1 i ( y U V ) } P ( V , U , z , t ) + i A ( z , t ) D ( V , U , z , t ) ] ,
D ( V , U , z , t ) t + ε u V D ( V , U , z , t ) z = λ d ( V , U , z , t ) h 1 D ( V , U , z , t ) h 2 M ( V , U , z , t ) + i γ 1 [ A ( z , t ) P * ( V , U , z , t ) A * ( z , t ) P ( V , U , z , t ) ] + ε Γ a 2 π exp ( ε 2 V 2 ) [ M ( V , U , z , t ) + D ( V , U , z , t ) ] d V ε Γ b 2 π exp ( ε 2 V 2 ) [ M ( V , U , z , t ) D ( V , U , z , t ) ] d V ,
M ( V , U , z , t ) t + ε u V M ( V , U , z , t ) z = λ m ( V , U , z , t ) h 3 M ( V , U , z , t ) h 4 D ( V , U , z , t ) + ε Γ a 2 π exp ( ε 2 V 2 ) [ M ( V , U , z , t ) + D ( V , U , z , t ) ] d V + ε Γ b 2 π exp ( ε 2 V 2 ) [ M ( V , U , z , t ) D ( V , U , z , t ) ] d V ,
V = υ ε u = k υ γ ,
U = ω α ω 0 γ ,
( Δ ν h Δ ν D ) ( ln 2 ) 1 2 ,
A ( z , t ) t + υ p A ( z , t ) z = σ 2 ε 1 [ { 1 i δ ( y y 0 ) } A ( z , t ) i P ( V , z , t ) d V ] ,
P ( V , z , t ) t + ε u V P ( V , z , t ) z = γ [ { 1 i ( y V ) } P ( V , z , t ) + i A ( z , t ) D ( V , z , t ) ] ,
D ( V , z , t ) t + ε u V D ( V , z , t ) z = λ d ( V , z , t ) h 1 D ( V , z , t ) h 2 M ( V , z , t ) + i γ 1 [ A ( z , t ) P * ( V , z , t ) A * ( z , t ) P ( V , z , t ) ] + ε Γ a 2 π exp ( ε 2 V 2 ) [ M ( V , z , t ) + D ( V , z , t ) ] d V ε Γ b 2 π exp ( ε 2 V 2 ) [ M ( V , z , t ) D ( V , z , t ) ] d V ,
M ( V , z , t ) t + ε u V M ( v , z , t ) z = λ m ( V , z , t ) h 3 M ( V , z , t ) h 4 D ( V , z , t ) + ε Γ a 2 π exp ( ε 2 V 2 ) [ M ( V , z , t ) + D ( V , z , t ) ] d V + ε Γ b 2 π exp ( ε 2 V 2 ) [ M ( V , z , t ) D ( V , z , t ) ] d V .
A ( 0 , t ) = R A ( l , t L l υ p ) ,
η = z ,
τ = t + ( L l υ p ) z l .
A ( 0 , τ ) = A ( l , τ ) .
A ( z , t ) t = A ( η , τ ) η η t + A ( η , τ ) τ τ t = A ( η , τ ) τ ,
A ( z , t ) z = A ( η , τ ) η η z + A ( η , τ ) τ τ z
= A ( η , τ ) η + A ( η , τ ) τ ( 1 υ p L l 1 υ p ) .
A ( z , t ) t + υ p A ( z , t ) z = A ( η , τ ) τ + υ p [ A ( η , τ ) η + A ( η , τ ) τ ( 1 υ p L l 1 υ p ) ] = L l A ( η , τ ) τ + υ p A ( η , τ ) η .
A ( η , τ ) τ + υ p l L A ( η , τ ) η = γ c l L [ { 1 i δ ( y y 0 ) } A ( η , τ ) i P ( V , η , τ ) d V ] ,
γ c = 1 2 t c = σ 2 ε 1
P ( V , η , τ ) τ + ε u V χ ( V ) P ( V , η , τ ) η = γ χ ( V ) [ { 1 i ( y V ) } P ( V , η , τ ) + i A ( η , τ ) D ( V , η , τ ) ] ,
D ( V , η , τ ) τ + ε u V χ ( V ) D ( V , η , τ ) η = χ ( V ) [ λ d ( V , η , τ ) h 1 D ( V , η , τ ) h 2 M ( V , η , τ ) + i γ 1 { A ( η , τ ) P * ( V , η , τ ) A * ( η , τ ) P ( V , η , τ ) } + ε Γ a 2 π exp ( ε 2 V 2 ) { M ( V , η , τ ) + D ( V , η , τ ) } d V ε Γ b 2 π exp ( ε 2 V 2 ) { M ( V , η , τ ) D ( V , η , τ ) } d V ] ,
M ( V , η , τ ) τ + ε u V χ ( V ) M ( V , η , τ ) η = χ ( V ) [ λ m ( V , η , τ ) h 3 M ( V , η , τ ) h 4 D ( V , η , τ ) + ε Γ a 2 π exp ( ε 2 V 2 ) { M ( V , η , τ ) + D ( V , η , τ ) } d V + ε Γ b 2 π exp ( ε 2 V 2 ) { M ( V , η , τ ) D ( V , η , τ ) } d V ] ,
χ ( V ) = [ 1 + ε u V υ p ( L l 1 ) ] 1 .
0 = A s , t h + i P s , t h ( V ) d V ,
0 = ( A r s , t h + i A i s , t h ) + i [ P r s , t h ( V ) + i P i s , t h ( V ) ] d V ,
0 = ( 1 + i V ) P s , t h ( V ) + i A s , t h D s , t h ( V ) ,
0 = ( 1 + i V ) [ P r s , t h ( V ) + i P i s , t h ( V ) ] + i ( A r s , t h + i A i s , t h ) D s , t h ( V ) ,
0 = A r s , t h P i s , t h ( V ) d V ,
0 = A i s , t h + P r s , t h ( V ) d V ,
0 = P r s , t h ( V ) V P i s , t h ( V ) A i s , t h D s , t h ( V ) ,
0 = P i s , t h ( V ) + V P r s , t h ( V ) + A r s , t h D s , t h ( V ) .
P r s , t h ( V ) = ( A i s , t h V A r s , t h ) ( 1 + V 2 ) D s , t h ( V ) ,
P i s , t h ( V ) = ( A r s , t h + V A i s , t h ) ( 1 + V 2 ) D s , t h ( V ) ,
0 = ( λ a s , t h ( V ) λ b s , t h ( V ) ) h 5 D s , t h ( V ) h 6 M s , t h ( V ) ,
0 = ( λ a s , t h ( V ) + λ b s , t h ( V ) ) h 7 M s , t h ( V ) h 8 D s , t h ( V ) ,
D s , t h ( V ) = 1 γ a γ b [ ( γ b γ a b ) λ a s , t h ( V ) γ a λ b s , t h ( V ) ] ,
M s , t h ( V ) = 1 γ a γ b [ ( γ b + γ a b ) λ a s , t h ( V ) + γ a λ b s , t h ( V ) ] .
0 = [ 1 1 γ a γ b ( γ b γ a b ) λ a s , t h ( V ) γ a λ b s , t h ( V ) 1 + V 2 d V ] A r s , t h 1 γ a γ b { V 1 + V 2 [ ( γ b γ a b ) λ a s , t h ( V ) γ a λ b s , t h ( V ) ] d V } A i s , t h .
0 = 1 γ a γ b { V 1 + V 2 [ ( γ b γ a b ) λ a s , t h ( V ) γ a λ b s , t h ( V ) ] d V } A r s , t h + { 1 1 γ a γ b ( γ b γ a b ) λ a s , t h ( V ) γ a λ b s , t h ( V ) 1 + V 2 d V } A i s , t h .
0 = [ 1 1 γ a γ b ( γ b γ a b ) λ a s , t h ( V ) γ a λ b s , t h ( V ) 1 + V 2 d V ] 2 ( 1 γ a γ b { V 1 + V 2 [ ( γ b γ a b ) λ a s , t h ( V ) γ a λ b s , t h ( V ) ] d V } ) 2 .
1 = 1 γ a γ b ( γ b γ a b ) λ a s , t h ( V ) γ a λ b s , t h ( V ) 1 + V 2 d V .
λ a ( V , η , τ ) = λ a ( V ) = ε L a π exp ( ε 2 V 2 ) ,
λ b ( V , η , τ ) = λ b ( V ) = ε L b π exp ( ε 2 V 2 ) .
1 = ε π [ ( γ b γ a b ) γ a γ b L a γ a γ a γ b L b ] exp ( ε 2 V 2 ) ( 1 + V 2 ) d V
= ε π [ ( 1 γ a b γ b ) L a γ a L b γ b ] exp ( ε 2 V 2 ) ( 1 + V 2 ) d V
= ε π L a [ ( 1 γ a b γ b ) 1 γ a q γ b ] exp ( ε 2 V 2 ) ( 1 + V 2 ) d V ,
L a = R L a , t h
= R π ε [ { ( 1 γ a b γ b ) 1 γ a q γ b } exp ( ε 2 V 2 ) ( 1 + V 2 ) d V ] 1 .
D i n i t ( V ) = 1 γ a γ b [ ( γ b γ a b ) ε L a π exp ( ε 2 V 2 ) γ a ε L b π exp ( ε 2 V 2 ) ]
= ε π [ ( 1 γ a b γ b ) L a γ a L b γ b ] exp ( ε 2 V 2 ) ,
M i n i t ( V ) = 1 γ a γ b [ ( γ b + γ a b ) ε L a π exp ( ε 2 V 2 ) + γ a ε L b π exp ( ε 2 V 2 ) ]
= ε π [ ( 1 + γ a b γ b ) L a γ a + L b γ b ] exp ( ε 2 V 2 ) .
X ( m , n ) η = X ( m , n ) X ( m 1 , n ) Δ η + 1 2 Δ η 2 X ( m , n ) η 2 ,
X ( m , n ) τ = X ( m , n + 1 ) X ( m , n ) Δ τ 1 2 Δ τ 2 X ( m , n ) τ 2 ,
Δ η = υ p l L Δ τ .

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