Abstract

The mathematical methods required to model simple stochastic processes are reviewed briefly. These methods are used to determine the probability-density function (PDF) for noise-induced energy perturbations of isolated (solitary) optical pulses in fiber communication systems. The analytical formula is consistent with the numerical solution of the energy-moment equation. System failures are caused by large energy perturbations. For such perturbations the actual PDF differs significantly from the (ideal-ized) Gauss PDF that is often used to predict system performance.

© 2003 Optical Society of America

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References

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  1. L. F. Mollenauer, J. P. Gordon and P. V. Mamyshev, �??Solitons in high bit-rate, long-distance transmission,�?? in Optical Fiber Telecommunications IIIA, edited by I. P. Kaminow and T. L. Koch (Academic, San Diego, 1997), pp. 373�??460.
  2. E. Iannone, F. Matera, A. Mecozzi and M. Settembre, Nonlinear Optical Communications Networks (Wiley, New York, 1998).
  3. K. Ishida, T. Kobayashi, J. Abe, K. Kinjo, S. Kuroda and T. Misuochi, �??A comparative study of 10 Gb/s RZ-DPSK and RZ-ASK WDM transmission over transoceanic distances,�?? OFC (Optical Society of America, Washington, D.C., 2003) paper ThE2.
  4. J. X. Cai, D. G. Foursa, C. R. Davidson, Y. Cai, G. Domagala, H. Li, L. Liu, W. W. Patterson, A. N. Pilipetskii, M. Nissov and N. S. Bergano, �??A DWDM demonstration of 3.73 Tb/s over 11,000 km using 373 RZ-DPSK channels at 10 Gb/s,�?? OFC (Optical Society of America, Washington, D.C., 2003) paper PD22.
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    [CrossRef] [PubMed]
  6. C. J. McKinstrie, �??Gordon�??Haus timing jitter in dispersion-managed systems with distributed amplification,�?? Opt. Commun. 200, 165�??177 (2001) and references therein.
    [CrossRef]
  7. J. P. Gordon and L. F. Mollenauer, �??Phase noise in photonic communication systems using linear amplifiers,�?? Opt. Lett. 23, 1351�??1353 (1990).
    [CrossRef]
  8. C. J. McKinstrie, C. Xie and T. I. Lakoba, �??Efficient modeling of phase jitter in dispersion-managed soliton systems,�?? Opt. Lett. 27, 1887�??1889 (2002) and references therein.
    [CrossRef]
  9. S. N. Vlasov, V. A. Petrishchev and V. I. Talanov, �??Averaged dersciption of wave beams in linear and nonlinear media,�?? Radiophys. Quantum Electron. 14, 1062�??1070 (1971).
    [CrossRef]
  10. W. J. Firth, �??Propagation of laser beams through inhomogeneous media,�?? Opt. Commun. 22, 226�??230 (1977).
    [CrossRef]
  11. D. Anderson, �??Variational approach to pulse propagation in optical fibers,�?? Phys. Rev. A 27, 3135�??3145 (1983).
    [CrossRef]
  12. D. J. Kaup, �??Perturbation theory for solitons in optical fibers,�?? Phys. Rev. A 42, 5689�??5694 (1990).
    [CrossRef] [PubMed]
  13. H. A. Haus and Y. Lai, �??Quantum theory of soliton squeezing: a linearized approach,�?? J. Opt. Soc. Am. B 7, 386�??392 (1990).
    [CrossRef]
  14. D. Marcuse, �??Derivation of analytical expressions for the bit-error probability in lightwave systems with optical amplifiers,�?? J. Lightwave Technol. 8, 1816�??1823 (1990).
    [CrossRef]
  15. J. P. Gordon and L. F. Mollenauer, �??Effects of fiber nonlinearities and amplifier spacing on ultra-long distance transmission,�?? J. Lightwave Technol. 9, 170�??173 (1991).
    [CrossRef]
  16. P. A. Humblet and M. Azizoglu, �??On the bit error rate of lightwave systems with optical amplifiers,�?? J. Lightwave Technol. 9, 1576�??1582 (1991).
    [CrossRef]
  17. J. S. Lee and C. S. Shim, �??Bit-error-rate analysis of optically preamplified receivers using an eigenfunction expansion method in optical frequency domain,�?? J. Lightwave Technol. 12, 1224�??1229 (1994).
    [CrossRef]
  18. T. Yoshino and G. P. Agrawal, �??Photoelectron statistics of solitons corrupted by amplified spontaneous emission,�?? Phys. Rev. A 51, 1662�??1668 (1995).
    [CrossRef] [PubMed]
  19. B. A. Malomed and N. Flytzanis, �??Fluctuational distribution function of solitons in the nonlinear Schrodinger system,�?? Phys. Rev. E 48, R5�??R8 (1993).
    [CrossRef]
  20. G. E. Falkovich, I. Kolokolov, V. Lebedev and S. K. Turitsyn, �??Statistics of soliton-bearing systems with additive noise,�?? Phys. Rev. E 63, 25601R (2001).
    [CrossRef]
  21. R. Holzlohner, V. S. Grigoryan, C. R. Menyuk and W. L. Kath, �??Accurate calculation of eye diagrams and bit error rates in optical transmission systems using linearization,�?? J. Lightwave Technol. 20, 389�??400 (2002).
    [CrossRef]
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    [CrossRef] [PubMed]
  23. C. J. McKinstrie and P. J. Winzer, �??How to apply importance-sampling techniques to simulations of optical systems,�?? <a href="http://arxiv.org/physics/0309002">http://arxiv.org/physics/0309002</a>.
  24. H. Kim and A. H. Gnauck, �??Experimental investigation of the performance limitation of DPSK systems due to nonlinear phase noise,�?? IEEE Photon. Technol. Lett. 15, 320�??322 (2003).
    [CrossRef]
  25. C. W. Gardiner, Handbook of Stochastic Methods, 2nd Ed. (Springer, Berlin, 2002).
  26. J. W. Goodman, Statistical Optics (Wiley, New York, 1985).
  27. J. G. Proakis, Digital Communications, 3rd Ed. (McGraw-Hill, New York, 1995).
  28. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), result 29.3.81.
  29. W. Horsthemke and R. Lefever, Noise-Induced Transitions (Springer, Berlin, 1983), Sections 5.5 and 6.7.
  30. R. Graham, �??Hopf bifurcation with fluctuating control parameter,�?? Phys. Rev. A 25, 3234�??3258 (1982).
    [CrossRef]
  31. R. Graham and A. Schenzle, �??Carleman imbedding of multiplicative stochastic processes,�?? Phys. Rev. A 25, 1731�??1754 (1982).
    [CrossRef]
  32. I. S. Gradsteyn and I. M. Rhyzhik, Table of Integrals, Series and Products, 5th Ed. (Academic, San Diego, 1994), result 6.633.2.
  33. P. J.Winzer, S. Chandrasekhar and H. Kim, �??Impact of filtering on RZ-DPSK reception,�?? IEEE Photon. Technol. Lett. 15, 840�??842 (2003) and references therein.
    [CrossRef]
  34. J. D. Hoffman, Numerical Methods for Scientists and Engineers (McGraw-Hill, New York, 1992).

IEEE Photon. Technol. Lett. (2)

H. Kim and A. H. Gnauck, �??Experimental investigation of the performance limitation of DPSK systems due to nonlinear phase noise,�?? IEEE Photon. Technol. Lett. 15, 320�??322 (2003).
[CrossRef]

P. J.Winzer, S. Chandrasekhar and H. Kim, �??Impact of filtering on RZ-DPSK reception,�?? IEEE Photon. Technol. Lett. 15, 840�??842 (2003) and references therein.
[CrossRef]

J. Lightwave Technol. (5)

R. Holzlohner, V. S. Grigoryan, C. R. Menyuk and W. L. Kath, �??Accurate calculation of eye diagrams and bit error rates in optical transmission systems using linearization,�?? J. Lightwave Technol. 20, 389�??400 (2002).
[CrossRef]

D. Marcuse, �??Derivation of analytical expressions for the bit-error probability in lightwave systems with optical amplifiers,�?? J. Lightwave Technol. 8, 1816�??1823 (1990).
[CrossRef]

J. P. Gordon and L. F. Mollenauer, �??Effects of fiber nonlinearities and amplifier spacing on ultra-long distance transmission,�?? J. Lightwave Technol. 9, 170�??173 (1991).
[CrossRef]

P. A. Humblet and M. Azizoglu, �??On the bit error rate of lightwave systems with optical amplifiers,�?? J. Lightwave Technol. 9, 1576�??1582 (1991).
[CrossRef]

J. S. Lee and C. S. Shim, �??Bit-error-rate analysis of optically preamplified receivers using an eigenfunction expansion method in optical frequency domain,�?? J. Lightwave Technol. 12, 1224�??1229 (1994).
[CrossRef]

J. Opt. Soc. Am. B (1)

OFC (1)

J. X. Cai, D. G. Foursa, C. R. Davidson, Y. Cai, G. Domagala, H. Li, L. Liu, W. W. Patterson, A. N. Pilipetskii, M. Nissov and N. S. Bergano, �??A DWDM demonstration of 3.73 Tb/s over 11,000 km using 373 RZ-DPSK channels at 10 Gb/s,�?? OFC (Optical Society of America, Washington, D.C., 2003) paper PD22.

Opt. Commun. (2)

W. J. Firth, �??Propagation of laser beams through inhomogeneous media,�?? Opt. Commun. 22, 226�??230 (1977).
[CrossRef]

C. J. McKinstrie, �??Gordon�??Haus timing jitter in dispersion-managed systems with distributed amplification,�?? Opt. Commun. 200, 165�??177 (2001) and references therein.
[CrossRef]

Opt. Lett. (4)

Phys. Rev. A (5)

D. Anderson, �??Variational approach to pulse propagation in optical fibers,�?? Phys. Rev. A 27, 3135�??3145 (1983).
[CrossRef]

D. J. Kaup, �??Perturbation theory for solitons in optical fibers,�?? Phys. Rev. A 42, 5689�??5694 (1990).
[CrossRef] [PubMed]

R. Graham, �??Hopf bifurcation with fluctuating control parameter,�?? Phys. Rev. A 25, 3234�??3258 (1982).
[CrossRef]

R. Graham and A. Schenzle, �??Carleman imbedding of multiplicative stochastic processes,�?? Phys. Rev. A 25, 1731�??1754 (1982).
[CrossRef]

T. Yoshino and G. P. Agrawal, �??Photoelectron statistics of solitons corrupted by amplified spontaneous emission,�?? Phys. Rev. A 51, 1662�??1668 (1995).
[CrossRef] [PubMed]

Phys. Rev. E (2)

B. A. Malomed and N. Flytzanis, �??Fluctuational distribution function of solitons in the nonlinear Schrodinger system,�?? Phys. Rev. E 48, R5�??R8 (1993).
[CrossRef]

G. E. Falkovich, I. Kolokolov, V. Lebedev and S. K. Turitsyn, �??Statistics of soliton-bearing systems with additive noise,�?? Phys. Rev. E 63, 25601R (2001).
[CrossRef]

Radiophys. Quantum Electron. (1)

S. N. Vlasov, V. A. Petrishchev and V. I. Talanov, �??Averaged dersciption of wave beams in linear and nonlinear media,�?? Radiophys. Quantum Electron. 14, 1062�??1070 (1971).
[CrossRef]

Other (11)

L. F. Mollenauer, J. P. Gordon and P. V. Mamyshev, �??Solitons in high bit-rate, long-distance transmission,�?? in Optical Fiber Telecommunications IIIA, edited by I. P. Kaminow and T. L. Koch (Academic, San Diego, 1997), pp. 373�??460.

E. Iannone, F. Matera, A. Mecozzi and M. Settembre, Nonlinear Optical Communications Networks (Wiley, New York, 1998).

K. Ishida, T. Kobayashi, J. Abe, K. Kinjo, S. Kuroda and T. Misuochi, �??A comparative study of 10 Gb/s RZ-DPSK and RZ-ASK WDM transmission over transoceanic distances,�?? OFC (Optical Society of America, Washington, D.C., 2003) paper ThE2.

C. W. Gardiner, Handbook of Stochastic Methods, 2nd Ed. (Springer, Berlin, 2002).

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

J. G. Proakis, Digital Communications, 3rd Ed. (McGraw-Hill, New York, 1995).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), result 29.3.81.

W. Horsthemke and R. Lefever, Noise-Induced Transitions (Springer, Berlin, 1983), Sections 5.5 and 6.7.

C. J. McKinstrie and P. J. Winzer, �??How to apply importance-sampling techniques to simulations of optical systems,�?? <a href="http://arxiv.org/physics/0309002">http://arxiv.org/physics/0309002</a>.

I. S. Gradsteyn and I. M. Rhyzhik, Table of Integrals, Series and Products, 5th Ed. (Academic, San Diego, 1994), result 6.633.2.

J. D. Hoffman, Numerical Methods for Scientists and Engineers (McGraw-Hill, New York, 1992).

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Figures (4)

Fig. 1.
Fig. 1.

PDFs for the normalized energy E/E 0. The solid, dot-dashed and dashed curves represent the exact PDF (86), the approximate PDF (78) and Gauss PDF (75), respectively. The differences between the exact and approximate PDFs are barely perceptible. Because the noise-induced energy perturbations depend on the pulse energy, the exact and approximate PDFs (which account for this dependence) have enhanced high-energy tails and diminished low-energy tails relative to the Gauss PDF (which does not).

Fig. 2.
Fig. 2.

PDFs for the normalized energy E/E 0. The solid curve represents the analytical solution (86), whereas the dashed curve represents the numerical solution of the FPE (79) and the dots represent the results of importance-sampled simulations based on the FDE (113).

Fig. 3.
Fig. 3.

Mean and variance of the normalized energy E/E(0) plotted as functions of distance. The curves were obtained by solving Eqs. (124) and (125) numerically.

Fig. 4.
Fig. 4.

Relative energy variance plotted as a function of distance. The solid curve was obtained by solving Eqs. (124) and (125) numerically, whereas the dashed line was obtained from an approximate analytical formula.

Equations (128)

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X . = a ( X , z ) + b ( X , z ) r ( z ) ,
r ( z ) = 0 ,
r ( z ) r ( z ) = δ ( z z ) ,
W ( z ) = 0 z r ( z ) d z
W ( z ) = 0 ,
W ( z ) W ( z ) = min ( z , z ) .
G z ( ζ ) = lim L L 2 L 2 r ( z + ζ ) r * ( z ) d z L
G e ( z , z ) = r ( z ) r * ( z ) .
r ¯ ( k ) = L 2 L 2 r ( z ) exp ( i 2 π k z ) d z
S ( k ) = lim L r ¯ ( k ) 2 L .
S ( k ) = G ( ζ ) exp ( i 2 π k ζ ) d ζ ,
G ( ζ ) = S ( k ) exp ( i 2 π k ζ ) d k .
S ( k ) = rect ( k K ) .
G ( ζ ) = K sinc ( π K ζ )
δ X = 0 δ z { a [ X ( z ) ] + b [ X ( z ) ] r ( z ) } d z ,
0 δ z a [ X ( z ) ] d z a 0 δ z ,
0 δ z b [ X ( z ) ] r ( z ) d z 0 δ z [ b 0 + b 0 b 0 0 z r ( z ) d z ] r ( z ) d z ,
δ X a 0 δ z + b 0 0 δ z r ( z ) d z + b 0 b 0 0 δ z 0 z r ( z ) r ( z ) d z d z .
δ X a 0 δ z + b 0 b 0 0 δ z 0 z r ( z ) r ( z ) d z d z .
δ X = 0 δ z { a [ X ( z ) ] + b [ X ( z ) ] r ( z + l ) } d z + O ( l ) .
δ X a 0 δ z .
δ X 2 b 0 2 0 δ z 0 δ z r ( z ) r ( z ) d z d z ,
= b 0 2 δ z .
δ X ( a 0 + b 0 b 0 2 ) δ z ,
δ X 2 b 0 2 δ z .
δ X = a δ z + b δ W .
δ Y f X δ X + f XX δ X 2 2 ,
Y . = f X X . + f XX b 2 2 .
δ X = ( a + b b X 2 ) δ z + b δ W ,
δ Y = f X ( a + b b X 2 ) δ z + f X b δ W + f XX b 2 δ z 2 .
δ Y = [ ( a g Y ) + ( b g Y ) ( b g Y ) Y 2 ] δ z + ( b g Y ) δ W .
Y . = ( a + b r ) g Y ,
= f X X . .
P ( x , δ z ) = T ( δ x , δ z | x δ x , 0 ) P ( x δ x , 0 ) d ( δ x ) ,
P + δ z z P P T ( δ x , δ z ) d ( δ x )
x [ P δ x T ( δ x , δ z ) d ( δ x ) ]
+ xx [ P δ x 2 T ( δ x , δ z ) d ( δ x ) 2 ] ,
T ( δ x , δ z ) = exp [ ( δ x a δ z ) 2 2 b 2 δ z ] ( 2 π b 2 δ z ) 1 2 .
z P = x ( a P ) + xx ( b 2 P ) 2 .
z P = x [ ( a + b x b 2 ) P ] + xx ( b 2 P ) 2 ,
z P = x ( a P ) + x [ b x ( b P ) ] 2 .
f ( X ) r = 0
X ( 0 ) X ( δ z ) + b [ X ( δ z ) ] δ z 0 r ( z ) d z ,
f [ X ( 0 ) ] r ( 0 ) f [ X ( δ z ) ] r ( 0 ) + f [ X ( δ z ) ] b [ X ( δ z ) ] δ z 0 r ( z ) r ( 0 ) d z .
f ( X ) r = f ( X ) b ( X ) 2 .
r x ( z ) = a ( X ) ,
r x ( z ) = a ( X ) + b ( X ) b ( X ) 2 .
δ r x ( z ) δ r x ( z ) = b 2 ( X ) δ ( z z ) .
z A = [ g ( z ) α ] A 2 i β tt 2 A 2 + i γ A 2 A + R ( z , t ) ,
R ( z , t ) = 0 ,
R ( z , t ) R ( z , t ) = 0 ,
R * ( z , t ) R ( z , t ) = S ( z ) δ ( z z ) δ ( t t ) ,
δ A = 0 δ z R ( z , t ) d z + O ( δ z 3 2 ) .
δ E = T 2 T 2 δ A ( t ) 2 d t .
δ E = T 2 T 2 0 δ z 0 δ z R * ( z , t ) R ( z , t ) d z d z d t .
δ E = S δ z F T ,
E = T 2 T 2 A ( t ) 2 d t .
d z E = [ g ( z ) α ] E + S ( z ) F T + T 2 T 2 [ A * ( z , t ) R ( z , t ) + A ( z , t ) R * ( z , t ) ] d t .
d z E = a ( E , z ) + b ( E , z ) r ( z ) ,
r e ( z ) = a ( E ) ,
δ r e ( z ) δ r e ( z ) = b 2 ( E ) δ ( z z ) .
A * ( z , t ) R ( z , t ) = 0 .
r e ( z ) = ( g α ) E + S F T ,
δ r e ( z ) δ r e ( z ) = 2 S E δ ( z z ) .
d z E = ( g α ) E + S F T + ( 2 S E ) 1 2 r ( z ) ,
z P = ( α g ) e ( e P ) S F T e P + S ee 2 ( e P ) .
d z E = [ g ( z ) α ] E + T 2 T 2 [ A * ( z , t ) R ( z , t ) + A ( z , t ) R * ( z , t ) ] d t .
r e ( z ) = a ( E ) + b ( E ) b ( E ) 2 ,
δ r e ( z ) δ r e ( z ) = b 2 ( E ) δ ( z z ) ,
A * ( z , t ) R ( z , t ) = S F 2 .
r e ( z ) = ( g α ) E + S F T ,
δ r e ( z ) δ r e ( z ) = 2 S E δ ( z z ) .
d z E = ( g α ) E + S ( F T 1 2 ) + ( 2 S E ) 1 2 r ( z ) ,
z P = ( α g ) e ( e P ) S F T e P + S ee 2 ( e P ) .
d ζ X = μ + ( 2 X ) 1 2 r ( ζ ) ,
d ζ X μ + 2 1 2 r ( ζ ) .
P ( x , ζ ) exp [ ( x m n ) 2 2 v n ] ( 2 π v n ) 1 2 .
d ζ Y = μ ¯ Y + r ( ζ ) 2 1 2 ,
d ζ Y μ ¯ + r ( ζ ) 2 1 2 .
P ( x , ζ ) exp [ ( x 1 2 m n ) 2 2 v n ] ( 8 π x v n ) 1 2 .
ζ P = μ x P + xx 2 ( x P )
x n ( ζ ) = 0 x n P ( x , ζ ) d ζ .
d ζ x = μ ,
d ζ x 2 = 2 ( 1 + μ ) x ,
P ¯ ( s , ζ ) = exp [ s ( 1 + s ζ ) ] ( 1 + s ζ ) μ .
x n ( ζ ) = ( 1 ) n lim s 0 d n P ¯ ( s , ζ ) d s n .
exp ( k s ) s μ ( x k ) ( μ 1 ) 2 I μ 1 [ 2 ( k x ) 1 2 ] ,
P ( x , ζ ) = [ x ( μ 1 ) 2 ζ ] exp [ ( 1 + x ) ζ ] I μ 1 ( 2 x 1 2 ζ ) .
P ( x , t ) x μ 2 exp [ ( x 1 2 1 ) 2 ζ ] ( 4 π x 3 2 ζ ) 1 2 .
ζ P = ( μ 2 ) x 0 P + x 0 x 0 2 ( x 0 P ) .
P ( x , 0 | x 0 , 0 ) = δ ( x x 0 ) .
0 P ( x , ζ | x 0 , 0 ) d x = 1 = 0 P ( x , ζ | x 0 , 0 ) d x 0 .
P ( x , ζ | x 0 , 0 ) = 0 exp ( λ ζ ) p ( x , λ ) p ˜ ( x 0 , λ ) d λ ,
x d xx 2 P ( μ 2 ) d x p = λ p ,
d xx 2 ( x p ˜ ) + ( μ 2 ) d x p ˜ = λ p ˜ .
0 P ( x , λ ) p ˜ ( x , λ ) d x = δ ( λ λ ) .
p ( x , λ ) = q ( x ) p ˜ ( x , λ ) ,
d x ( x q ) μ q = 0 .
0 p ( x , λ ) p ˜ ( x , λ ) d λ = δ ( x x ) .
f ( x ) = 0 δ ( x x ) f ( x ) d x ,
= 0 c ( λ ) p ( x , λ ) d λ , where c ( λ ) = 0 f ( x ) p ˜ ( x , λ ) d x ,
= 0 c ˜ ( λ ) p ˜ ( x , λ ) d λ , where c ˜ ( λ ) = 0 f ( x ) p ( x , λ ) d x .
p ( x , λ ) = x ( μ 1 ) 2 J μ 1 [ 2 ( λ x ) 1 2 ] ,
p ˜ ( x , λ ) = x ( 1 μ ) 2 J μ 1 [ 2 ( λ x ) 1 2 ] ,
0 exp ( γ 2 u 2 ) J n ( α u ) J n ( β u ) u d u = exp [ ( α 2 + β 2 ) 4 γ 2 ] I n ( α β 2 γ 2 ) 2 γ 2 ,
P ( x , ζ | x 0 , 0 ) = ( x x 0 ) ( μ 1 ) 2 exp [ ( x 0 + x ) ζ ] I μ 1 [ 2 ( x 0 x ) 1 2 ζ ] ζ .
μ P + x ( x P ) = 0 ,
X n + 1 = X n + a ( X n ) δ z + b ( X n ) δ W n + 1 ,
δ W m δ W n = δ z δ mn ,
X n + 1 p = X n + a ( X n ) δ z + b ( X n ) δ W n + 1 ,
X n + 1 c = X n + [ a ( X n ) + a ( X n + 1 p ) ] δ z 2
+ [ b ( X n ) + b ( X n + 1 p ) ] δ W n + 1 2 .
X n + 1 c X n + a n δ z + ( b n + b n b n δ W n + 1 2 ) δ W n + 1 .
δ X n + 1 ( a n + b n b n 2 ) δ z ,
δ X n + 1 2 b n 2 δ z .
X n + 1 = X n + ( a n + b n b n 2 ) δ z + b n δ W n + 1 .
d z X = μ σ x ( z ) + ν ( z ) X + [ 2 σ x ( z ) X ] 1 2 r ( z ) ,
σ x ( z ) = n sp ω g ( z ) E ( 0 )
d z Y = μ σ y ( z ) + [ 2 σ y ( z ) Y ] 1 2 r ( z ) ,
σ y ( z ) = [ n sp ω E ( 0 ) ] g ( z ) e 0 z ν ( z ) d z .
ζ ( z ) = 0 z σ y ( z ) d z .
d ζ Y = μ + ( 2 Y ) 1 2 r ( ζ ) .
E a = E ( 0 ) 0 l e 0 z ν ( z ) dz dz l .
ζ ( l ) = σ a 0 l e 0 l ν ( z ) dz dz 0 l e 0 z ν ( z ) dz dz l .
ζ ( z ) ρ σ a z ,
g ( z ) α p α l exp [ α p mod ( z , l ) ] exp ( α p l ) 1 ,
d z m x = ν ( z ) m x + μ σ x ( z ) ,
d z v x = 2 ν ( z ) v x + 2 σ x ( z ) m x ,

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