Abstract

A set of strong-field Maxwell–Bloch equations is derived for a multimode, standing-wave, homogeneously broadened laser such as a dye laser, which obeys the atomic rate-equation limit. Numerical solutions show that the energy in the cavity oscillates between the field and the gain medium at multiples of the cavity mode spacing. By using these Maxwell–Bloch equations, intensity autocorrelation functions of a dye-laser system under varying conditions are calculated. We find that the numerical results qualitatively agree with recent experimental data.

© 1988 Optical Society of America

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References

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  1. S. M. Curry, R. Cubeddu, and T. W. Hänsen, “Intensity stabilization of dye laser radiation by saturated amplification,” Appl. Phys. 1, 3923 (1964).
  2. D. S. King and R. R. Cavanagh, “Streak-camera analysis of XeCl- and N2-pumped dye-laser outputs,” Opt. Lett. 8, 18 (1983).
    [Crossref] [PubMed]
  3. L. A. Westling, M. G. Raymer, and J. J. Snyder, “Single-shot spectral measurements and mode correlations in a multimode pulsed dye laser,” J. Opt. Soc. Am. B 1, 150 (1984).
    [Crossref]
  4. L. A. Westling and M. G. Raymer, “Intensity autocorrelation measurements and spontaneous FM phase locking in a multi-mode pulsed dye laser,” J. Opt. Soc. Am. B 3, 911 (1986).
    [Crossref]
  5. S. A. Kovalenko, “Quantum intensity fluctuations in multimode cw lasers and maximum sensitivity of intracavity-laser spectroscopy,” Sov. J. Quantum Electron. 11, 759 (1981).
    [Crossref]
  6. H. Haken, Laser Theory (Springer-Verlag, Berlin, 1970;Laser Theory2nd corr. ed.1984);M. Sargent, M. O. Scully, and W. E. Lamb, Laser Physics (Addison-Wesley, Reading, Mass., 1974).
  7. W. Brunner, R. Fisher, and H. Paul, “Regular and chaotic behavior of multimode lasers,” J. Opt. Soc. Am. B 2, 202 (1985).
    [Crossref]
  8. C. L. Tang and H. Statz, “Large-signal effects in self-locked lasers,” J. Appl. Phys. 39, 31 (1968).
    [Crossref]
  9. H. Risken and K. Nummedal, “Self-pulsing in lasers,” J. Appl. Phys. 39, 4662 (1968).
    [Crossref]
  10. J. B. Hambenne and M. Sargent, “Strong-signal laser operation. I. General theory,” Phys. Rev. A 13, 784 (1976).
    [Crossref]
  11. L. W. Hillman, J. Krasinski, K. Koch, and C. R. Stroud, “Dynamics of homogeneously broadened lasers: higher-order bichromatic states of operation,” J. Opt. Soc. Am. B 2, 211 (1985).
    [Crossref]
  12. The derivation leading to Eqs. (10) is due to P. W. Milonni, Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 (personal communication, 1985).
  13. F. Aronowitz, “Theory of a traveling-wave optical maser,” Phys. Rev. 139, 635 (1965).
    [Crossref]
  14. Ya. I. Khanin, “Theory and experiments on bidirectional ring lasers,” in Optical Instabilities, R. W. Boyd, M. G. Raymer, and L. M. Narducci, eds. (Cambridge U. Press, Cambridge, 1986), p. 212.
  15. R. Graham and H. Haken, “Quantum theory of light propagation in a fluctuating laser-active medium,” Z. Phys. 213, 420 (1968).
    [Crossref]
  16. L. W. Hillman, J. Krasinski, R. W. Boyd, and C. R. Stroud, “Observation of higher order dynamical states of a homogeneously broadened laser,” Phys. Rev. Lett. 52, 1605 (1984).
    [Crossref]
  17. I. McMackin, C. Radzewicz, M. Beck, and M. G. Raymer, “Instabilities and chaos in a multimode, standing-wave, cw dye laser,” Phys. Rev. A (to be published).
  18. L. A. Lugiato, L. M. Narducci, E. V. Eschenazi, D. K. Bandy, and N. B. Abraham, “Multimode instabilities in a homogeneously broadened ring laser,” Phys. Rev. A 32, 1563 (1985).
    [Crossref] [PubMed]
  19. L. A. Lugiato, F. Prati, D. K. Bandy, L. M. Narducci, P. Ru, and J. R. Tredicce, “Low threshold instabilities in unidirectional ring lasers,” Opt. Commun. 64, 167 (1987).
    [Crossref]
  20. Hong Fu and H. Haken, “Semiclassical dye laser equations and the unidirectional single-frequency operation,” Phys. Rev. A 36, 4802 (1987).
    [Crossref] [PubMed]

1987 (2)

L. A. Lugiato, F. Prati, D. K. Bandy, L. M. Narducci, P. Ru, and J. R. Tredicce, “Low threshold instabilities in unidirectional ring lasers,” Opt. Commun. 64, 167 (1987).
[Crossref]

Hong Fu and H. Haken, “Semiclassical dye laser equations and the unidirectional single-frequency operation,” Phys. Rev. A 36, 4802 (1987).
[Crossref] [PubMed]

1986 (1)

1985 (3)

1984 (2)

L. W. Hillman, J. Krasinski, R. W. Boyd, and C. R. Stroud, “Observation of higher order dynamical states of a homogeneously broadened laser,” Phys. Rev. Lett. 52, 1605 (1984).
[Crossref]

L. A. Westling, M. G. Raymer, and J. J. Snyder, “Single-shot spectral measurements and mode correlations in a multimode pulsed dye laser,” J. Opt. Soc. Am. B 1, 150 (1984).
[Crossref]

1983 (1)

1981 (1)

S. A. Kovalenko, “Quantum intensity fluctuations in multimode cw lasers and maximum sensitivity of intracavity-laser spectroscopy,” Sov. J. Quantum Electron. 11, 759 (1981).
[Crossref]

1976 (1)

J. B. Hambenne and M. Sargent, “Strong-signal laser operation. I. General theory,” Phys. Rev. A 13, 784 (1976).
[Crossref]

1968 (3)

C. L. Tang and H. Statz, “Large-signal effects in self-locked lasers,” J. Appl. Phys. 39, 31 (1968).
[Crossref]

H. Risken and K. Nummedal, “Self-pulsing in lasers,” J. Appl. Phys. 39, 4662 (1968).
[Crossref]

R. Graham and H. Haken, “Quantum theory of light propagation in a fluctuating laser-active medium,” Z. Phys. 213, 420 (1968).
[Crossref]

1965 (1)

F. Aronowitz, “Theory of a traveling-wave optical maser,” Phys. Rev. 139, 635 (1965).
[Crossref]

1964 (1)

S. M. Curry, R. Cubeddu, and T. W. Hänsen, “Intensity stabilization of dye laser radiation by saturated amplification,” Appl. Phys. 1, 3923 (1964).

Abraham, N. B.

L. A. Lugiato, L. M. Narducci, E. V. Eschenazi, D. K. Bandy, and N. B. Abraham, “Multimode instabilities in a homogeneously broadened ring laser,” Phys. Rev. A 32, 1563 (1985).
[Crossref] [PubMed]

Aronowitz, F.

F. Aronowitz, “Theory of a traveling-wave optical maser,” Phys. Rev. 139, 635 (1965).
[Crossref]

Bandy, D. K.

L. A. Lugiato, F. Prati, D. K. Bandy, L. M. Narducci, P. Ru, and J. R. Tredicce, “Low threshold instabilities in unidirectional ring lasers,” Opt. Commun. 64, 167 (1987).
[Crossref]

L. A. Lugiato, L. M. Narducci, E. V. Eschenazi, D. K. Bandy, and N. B. Abraham, “Multimode instabilities in a homogeneously broadened ring laser,” Phys. Rev. A 32, 1563 (1985).
[Crossref] [PubMed]

Beck, M.

I. McMackin, C. Radzewicz, M. Beck, and M. G. Raymer, “Instabilities and chaos in a multimode, standing-wave, cw dye laser,” Phys. Rev. A (to be published).

Boyd, R. W.

L. W. Hillman, J. Krasinski, R. W. Boyd, and C. R. Stroud, “Observation of higher order dynamical states of a homogeneously broadened laser,” Phys. Rev. Lett. 52, 1605 (1984).
[Crossref]

Brunner, W.

Cavanagh, R. R.

Cubeddu, R.

S. M. Curry, R. Cubeddu, and T. W. Hänsen, “Intensity stabilization of dye laser radiation by saturated amplification,” Appl. Phys. 1, 3923 (1964).

Curry, S. M.

S. M. Curry, R. Cubeddu, and T. W. Hänsen, “Intensity stabilization of dye laser radiation by saturated amplification,” Appl. Phys. 1, 3923 (1964).

Eschenazi, E. V.

L. A. Lugiato, L. M. Narducci, E. V. Eschenazi, D. K. Bandy, and N. B. Abraham, “Multimode instabilities in a homogeneously broadened ring laser,” Phys. Rev. A 32, 1563 (1985).
[Crossref] [PubMed]

Fisher, R.

Fu, Hong

Hong Fu and H. Haken, “Semiclassical dye laser equations and the unidirectional single-frequency operation,” Phys. Rev. A 36, 4802 (1987).
[Crossref] [PubMed]

Graham, R.

R. Graham and H. Haken, “Quantum theory of light propagation in a fluctuating laser-active medium,” Z. Phys. 213, 420 (1968).
[Crossref]

Haken, H.

Hong Fu and H. Haken, “Semiclassical dye laser equations and the unidirectional single-frequency operation,” Phys. Rev. A 36, 4802 (1987).
[Crossref] [PubMed]

R. Graham and H. Haken, “Quantum theory of light propagation in a fluctuating laser-active medium,” Z. Phys. 213, 420 (1968).
[Crossref]

H. Haken, Laser Theory (Springer-Verlag, Berlin, 1970;Laser Theory2nd corr. ed.1984);M. Sargent, M. O. Scully, and W. E. Lamb, Laser Physics (Addison-Wesley, Reading, Mass., 1974).

Hambenne, J. B.

J. B. Hambenne and M. Sargent, “Strong-signal laser operation. I. General theory,” Phys. Rev. A 13, 784 (1976).
[Crossref]

Hänsen, T. W.

S. M. Curry, R. Cubeddu, and T. W. Hänsen, “Intensity stabilization of dye laser radiation by saturated amplification,” Appl. Phys. 1, 3923 (1964).

Hillman, L. W.

L. W. Hillman, J. Krasinski, K. Koch, and C. R. Stroud, “Dynamics of homogeneously broadened lasers: higher-order bichromatic states of operation,” J. Opt. Soc. Am. B 2, 211 (1985).
[Crossref]

L. W. Hillman, J. Krasinski, R. W. Boyd, and C. R. Stroud, “Observation of higher order dynamical states of a homogeneously broadened laser,” Phys. Rev. Lett. 52, 1605 (1984).
[Crossref]

Khanin, Ya. I.

Ya. I. Khanin, “Theory and experiments on bidirectional ring lasers,” in Optical Instabilities, R. W. Boyd, M. G. Raymer, and L. M. Narducci, eds. (Cambridge U. Press, Cambridge, 1986), p. 212.

King, D. S.

Koch, K.

Kovalenko, S. A.

S. A. Kovalenko, “Quantum intensity fluctuations in multimode cw lasers and maximum sensitivity of intracavity-laser spectroscopy,” Sov. J. Quantum Electron. 11, 759 (1981).
[Crossref]

Krasinski, J.

L. W. Hillman, J. Krasinski, K. Koch, and C. R. Stroud, “Dynamics of homogeneously broadened lasers: higher-order bichromatic states of operation,” J. Opt. Soc. Am. B 2, 211 (1985).
[Crossref]

L. W. Hillman, J. Krasinski, R. W. Boyd, and C. R. Stroud, “Observation of higher order dynamical states of a homogeneously broadened laser,” Phys. Rev. Lett. 52, 1605 (1984).
[Crossref]

Lugiato, L. A.

L. A. Lugiato, F. Prati, D. K. Bandy, L. M. Narducci, P. Ru, and J. R. Tredicce, “Low threshold instabilities in unidirectional ring lasers,” Opt. Commun. 64, 167 (1987).
[Crossref]

L. A. Lugiato, L. M. Narducci, E. V. Eschenazi, D. K. Bandy, and N. B. Abraham, “Multimode instabilities in a homogeneously broadened ring laser,” Phys. Rev. A 32, 1563 (1985).
[Crossref] [PubMed]

McMackin, I.

I. McMackin, C. Radzewicz, M. Beck, and M. G. Raymer, “Instabilities and chaos in a multimode, standing-wave, cw dye laser,” Phys. Rev. A (to be published).

Milonni, P. W.

The derivation leading to Eqs. (10) is due to P. W. Milonni, Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 (personal communication, 1985).

Narducci, L. M.

L. A. Lugiato, F. Prati, D. K. Bandy, L. M. Narducci, P. Ru, and J. R. Tredicce, “Low threshold instabilities in unidirectional ring lasers,” Opt. Commun. 64, 167 (1987).
[Crossref]

L. A. Lugiato, L. M. Narducci, E. V. Eschenazi, D. K. Bandy, and N. B. Abraham, “Multimode instabilities in a homogeneously broadened ring laser,” Phys. Rev. A 32, 1563 (1985).
[Crossref] [PubMed]

Nummedal, K.

H. Risken and K. Nummedal, “Self-pulsing in lasers,” J. Appl. Phys. 39, 4662 (1968).
[Crossref]

Paul, H.

Prati, F.

L. A. Lugiato, F. Prati, D. K. Bandy, L. M. Narducci, P. Ru, and J. R. Tredicce, “Low threshold instabilities in unidirectional ring lasers,” Opt. Commun. 64, 167 (1987).
[Crossref]

Radzewicz, C.

I. McMackin, C. Radzewicz, M. Beck, and M. G. Raymer, “Instabilities and chaos in a multimode, standing-wave, cw dye laser,” Phys. Rev. A (to be published).

Raymer, M. G.

Risken, H.

H. Risken and K. Nummedal, “Self-pulsing in lasers,” J. Appl. Phys. 39, 4662 (1968).
[Crossref]

Ru, P.

L. A. Lugiato, F. Prati, D. K. Bandy, L. M. Narducci, P. Ru, and J. R. Tredicce, “Low threshold instabilities in unidirectional ring lasers,” Opt. Commun. 64, 167 (1987).
[Crossref]

Sargent, M.

J. B. Hambenne and M. Sargent, “Strong-signal laser operation. I. General theory,” Phys. Rev. A 13, 784 (1976).
[Crossref]

Snyder, J. J.

Statz, H.

C. L. Tang and H. Statz, “Large-signal effects in self-locked lasers,” J. Appl. Phys. 39, 31 (1968).
[Crossref]

Stroud, C. R.

L. W. Hillman, J. Krasinski, K. Koch, and C. R. Stroud, “Dynamics of homogeneously broadened lasers: higher-order bichromatic states of operation,” J. Opt. Soc. Am. B 2, 211 (1985).
[Crossref]

L. W. Hillman, J. Krasinski, R. W. Boyd, and C. R. Stroud, “Observation of higher order dynamical states of a homogeneously broadened laser,” Phys. Rev. Lett. 52, 1605 (1984).
[Crossref]

Tang, C. L.

C. L. Tang and H. Statz, “Large-signal effects in self-locked lasers,” J. Appl. Phys. 39, 31 (1968).
[Crossref]

Tredicce, J. R.

L. A. Lugiato, F. Prati, D. K. Bandy, L. M. Narducci, P. Ru, and J. R. Tredicce, “Low threshold instabilities in unidirectional ring lasers,” Opt. Commun. 64, 167 (1987).
[Crossref]

Westling, L. A.

Appl. Phys. (1)

S. M. Curry, R. Cubeddu, and T. W. Hänsen, “Intensity stabilization of dye laser radiation by saturated amplification,” Appl. Phys. 1, 3923 (1964).

J. Appl. Phys. (2)

C. L. Tang and H. Statz, “Large-signal effects in self-locked lasers,” J. Appl. Phys. 39, 31 (1968).
[Crossref]

H. Risken and K. Nummedal, “Self-pulsing in lasers,” J. Appl. Phys. 39, 4662 (1968).
[Crossref]

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

Opt. Commun. (1)

L. A. Lugiato, F. Prati, D. K. Bandy, L. M. Narducci, P. Ru, and J. R. Tredicce, “Low threshold instabilities in unidirectional ring lasers,” Opt. Commun. 64, 167 (1987).
[Crossref]

Opt. Lett. (1)

Phys. Rev. (1)

F. Aronowitz, “Theory of a traveling-wave optical maser,” Phys. Rev. 139, 635 (1965).
[Crossref]

Phys. Rev. A (3)

J. B. Hambenne and M. Sargent, “Strong-signal laser operation. I. General theory,” Phys. Rev. A 13, 784 (1976).
[Crossref]

Hong Fu and H. Haken, “Semiclassical dye laser equations and the unidirectional single-frequency operation,” Phys. Rev. A 36, 4802 (1987).
[Crossref] [PubMed]

L. A. Lugiato, L. M. Narducci, E. V. Eschenazi, D. K. Bandy, and N. B. Abraham, “Multimode instabilities in a homogeneously broadened ring laser,” Phys. Rev. A 32, 1563 (1985).
[Crossref] [PubMed]

Phys. Rev. Lett. (1)

L. W. Hillman, J. Krasinski, R. W. Boyd, and C. R. Stroud, “Observation of higher order dynamical states of a homogeneously broadened laser,” Phys. Rev. Lett. 52, 1605 (1984).
[Crossref]

Sov. J. Quantum Electron. (1)

S. A. Kovalenko, “Quantum intensity fluctuations in multimode cw lasers and maximum sensitivity of intracavity-laser spectroscopy,” Sov. J. Quantum Electron. 11, 759 (1981).
[Crossref]

Z. Phys. (1)

R. Graham and H. Haken, “Quantum theory of light propagation in a fluctuating laser-active medium,” Z. Phys. 213, 420 (1968).
[Crossref]

Other (4)

I. McMackin, C. Radzewicz, M. Beck, and M. G. Raymer, “Instabilities and chaos in a multimode, standing-wave, cw dye laser,” Phys. Rev. A (to be published).

Ya. I. Khanin, “Theory and experiments on bidirectional ring lasers,” in Optical Instabilities, R. W. Boyd, M. G. Raymer, and L. M. Narducci, eds. (Cambridge U. Press, Cambridge, 1986), p. 212.

The derivation leading to Eqs. (10) is due to P. W. Milonni, Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 (personal communication, 1985).

H. Haken, Laser Theory (Springer-Verlag, Berlin, 1970;Laser Theory2nd corr. ed.1984);M. Sargent, M. O. Scully, and W. E. Lamb, Laser Physics (Addison-Wesley, Reading, Mass., 1974).

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Figures (3)

Fig. 1
Fig. 1

Intensities of a, the different spatial modes and b, the spatial components U and V of the population inversion plotted as function of time. Vr and Vi are, respectively, the real and imaginary parts of V. The equilibrium inversion W is chosen so that after time t = 0 the laser is 20 times above threshold. The gain medium thickness ΔZ is 0.05 cm. The other parameters are L = 15 cm, a = 1.1 × 105 cm3/(erg-sec), b = 15.2 × 1010 (cm-sec)−1, Z1 = 4 cm, γ = 5 × 108 sec−1, and γ0 = 22 × 108 sec−1. The unit of intensity is 1024 photons/(cm2-sec).

Fig. 2
Fig. 2

a, Energy distribution of the spatial modes and b, the temporal power spectrum (energy distribution of temporal modes) with the same parameters used in Fig. 1. The horizontal frequency scale is labeled by the corresponding mode number l, i.e., ω = lΔ.

Fig. 3
Fig. 3

The intensity autocorrelation function of five different cavity lengths L plotted for (a)–(e) the experimental data of Ref. 4, (f)–(j) for the numerical results at 80 times above threshold, and (k)–(o) for the numerical results at 40 times above threshold. In each row, the cavity lengths are the same [i.e., (a), (f), and (k) each correspond to the same cavity length]. Each row is identified by the corresponding cavity mode spacing, Δνc = 1/2L. Time τ is given in units of the cavity round-trip time 2 L/c.

Equations (34)

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d d t S = β S i κ W ,
d d t W = γ ( W W ) + i 2 κ S * i 2 κ * S * ,
S i κ W / β .
d d t W = γ ( W W ) κ 2 β W | | 2 .
E = 1 2 exp ( i ω ¯ t ) l A l ( t ) u l ( z ) exp ( i Δ l t ) + c.c. ,
= l A l ( t ) u l ( z ) exp ( i Δ l t ) .
d 2 d 2 z u l ( z ) + k l 2 u l ( z ) = 0 ,
u l ( 0 ) = u l ( L ) = 0 .
k l = ( N + l ) δ ,
( 2 z 2 1 c 2 2 t 2 ) exp ( i ω ¯ t ) = 2 t 2 4 π c 2 Ñ d S exp ( i ω ¯ t ) ,
d d t A l = i 2 π ω ¯ Ñ d exp ( i Δ l t ) 2 L Z 1 Z 1 + Δ Z u l ( z ) S ( z , t ) d z ,
d d t A l = γ l A l + b n A n exp ( i Δ n l t ) × Z 1 Z 1 + Δ Z u l ( z ) u n ( z ) W ( z , t ) d z ,
d d t W = γ ( W W ) a W ln A l A n * u l ( z ) u n ( z ) exp ( i Δ n l t ) ,
W ( z , t ) = j [ U j exp ( i j δ z ) + c.c. ] + k = 1 j { V k , j exp [ i ( k N + j ) δ z ] + c.c. } ,
d d t A l = γ l A l b 4 n A n exp ( i Δ n l t ) × Z 1 Z 1 + Δ Z d z { exp [ i ( k l + k n ) z ] + exp { i ( k l + k n ) z ] exp [ i ( k l k n ) z ] exp [ i ( k l k n ) z ] } ( j [ U j exp ( i j δ z ) + c.c. ] + k j { V k , j exp [ i ( k N + j ) δ z ] + c.c. } ) .
d d t A l = γ l A l b 4 n A n exp ( i Δ n l t ) ( j V 2 j * i ( l + n j ) δ × { exp [ i ( n + l j ) δ ( Z 1 + Δ Z ) ] exp [ i ( l + n j ) δ Z 1 ] } U j i ( l n + j ) δ { exp [ i ( l n + j ) δ ( Z 1 + Δ Z ) ] exp [ i ( l n + j ] δ Z 1 ] } U j i ( n l + j ) δ × { exp [ i ( n l + j ) δ ( Z 1 + Δ Z ) ] exp [ i ( n l + j ) δ Z 1 ] } + c.c. ) .
Δ Z P L ,
d d t A l = γ l A l + b Δ Z 4 n A n exp ( i Δ n l t ) × [ 2 U f ln V g ln * V * g ln ] + ξ l ( t ) ,
W ( z , t ) U ( t ) + V ( t ) exp ( i 2 N δ z ) + V * ( t ) exp ( i 2 N δ z ) .
d d t U = γ ( U W ) a 4 ln A l A n * exp ( i Δ n l t ) × [ 2 f ln U g ln * V g ln V * ]
d d t V = γ V a 2 ln A l A n * exp ( i Δ n l t ) [ f ln V ( 1 / 2 ) g ln U ] .
d d t A l = γ l A l + b n A n exp ( i Δ n l t ) × m = 1 M Z m Z m + Δ Z / M u l ( z ) u n ( z ) W ( m ) ( z , t ) d z ,
W ( m ) ( z , t ) = U ( m ) + V ( m ) exp [ i 2 N δ ( z + Z m ) ] + V ( m ) * exp [ i 2 N δ ( z + Z m ) ] .
d d t A l = γ l A l + b Δ Z 4 n A n exp ( i Δ n l t ) m [ 2 U ( m ) f l n ( m ) V ( m ) g l n ( m ) * V ( m ) * g l n ( m ) ] + ξ l ( t ) ,
d d t U ( m ) = γ [ U ( m ) W ] a 4 ln A l A n * exp ( i Δ n l t ) × m [ 2 f ln ( m ) U ( m ) g ln ( m ) * V ( m ) g ln ( m ) V ( m ) * ] ,
d d t V ( m ) = γ V ( m ) a 2 ln A l A n * exp ( i Δ n l t ) × [ f ln ( m ) V ( m ) ( 1 / 2 ) g ln ( m ) U ( m ) ] ,
γ l = γ 0 exp [ l 2 / ( 2 Γ 2 ) ] ,
F ( τ ) = I ( t ) I ( t + τ ) I ( t ) I ( t + τ ) ,
I ( t ) = | l E l ( t ) | 2 ,
E l ( t ) = A l ( t ) exp ( i Δ l t ) .
I ( t ) = 1 T 0 T I ( t ) d t .
γ l = γ 0 L L 0 exp [ l 2 / ( Γ L / L 0 ) 2 ] ,
P ( x l ) = exp [ x l 2 / ( 2 s 2 ) ] ,
P ( y l ) = exp [ y l 2 / ( 2 s 2 ) ] .

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