Abstract

We present simple yet efficient formulae for the propagation of the second order moments of a pulse in a nonlinear and dispersive optical fiber over many dispersion and nonlinear lengths. The propagation of the temporal and spectral widths, chirp and power of pulses are very precisely approximated and quickly calculated in both dispersion regimes as long as the pulses are not high order solitons.

© 2007 Optical Society of America

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References

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  1. V. E. Zakharov and A. B. Shabat, "Exact theory of two-dimensional self-focusing and one-dimensional selfmodulation of waves in nonlinear media," Soviet Physics JETP 34,62-69 (1972)
  2. D. Anderson, M. Desaix, M. Karlsson, M. Lisak and M. L. Quiroga-Teixeiro, "Wave-breaking-free pulses in nonlinear-optical fibers," J. Opt. Soc. Am. B 10,1185-1190 (1993).
    [CrossRef]
  3. M. Potasek, G. P. Agrawal and S. C. Pinault, "Analytic and numerical study of pulse broadening in nonlinear dispersive optical fibers," J. Opt. Soc. Am. B 3,205-211 (1992).
    [CrossRef]
  4. D. Marcuse, "RMS Width of Pulses in Nonlinear Dispersive Fibers," J. Lightwave Technol. 10,17-21 (1992).
    [CrossRef]
  5. P.-A. Bélanger and N. Bélanger, "RMS characteristics of pulses in nonlinear dispersive lossy fibers," Opt. Commun. 117,56-60 (1995).
    [CrossRef]
  6. J. Santhanam and G. P. Agrawal, "Raman-induced spectral shifts in optical fibers: general theory based on the moment method," Opt. Commun. 222,413-420 (2003).
    [CrossRef]
  7. R. Martínez-Herrero, P. M. Mejías, M. Sánchez and J. L. H. Neira, "Third- and fourth-order parametric characterization of partially coherent beams propagating through ABCD optical systems," Opt. and Quantum Electron. 24,S1021-S1026 (1992).
    [CrossRef]
  8. H. Weber, "Propagation of higher-order intensity moments in quadratic-index media," Opt. and Quantum Electron. 24,1027-1049 (1992).
    [CrossRef]
  9. J. F. Kenney and E. S. Keeping, Mathematics of statistics, 2nd ed., (D. Van Nostrand Company Inc., 1951).
  10. G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed., (Academic Press, 2001).
  11. D. Anderson, M. Lisak and T. Reichel, "Approximate analytical approaches to nonlinear pulse propagation in optical fibers: A comparison," Phys. Rev. A 38,1618-1620 (1988).
    [CrossRef] [PubMed]
  12. D. Anderson, M. Desaix, M. Karlsson, M. Lisak and M. L. Quiroga-Teixeiro, "Wave breaking in nonlinear-optical fibers," J. Opt. Soc. Am. B 9,1358-1361 (1992).
    [CrossRef]
  13. J. F. Geer, "Rational trigonometric approximations using Fourier Series Partial Sums," J. Sci. Comput. 10,325-356 (1995).
    [CrossRef]

2003 (1)

J. Santhanam and G. P. Agrawal, "Raman-induced spectral shifts in optical fibers: general theory based on the moment method," Opt. Commun. 222,413-420 (2003).
[CrossRef]

1995 (2)

P.-A. Bélanger and N. Bélanger, "RMS characteristics of pulses in nonlinear dispersive lossy fibers," Opt. Commun. 117,56-60 (1995).
[CrossRef]

J. F. Geer, "Rational trigonometric approximations using Fourier Series Partial Sums," J. Sci. Comput. 10,325-356 (1995).
[CrossRef]

1993 (1)

1992 (5)

M. Potasek, G. P. Agrawal and S. C. Pinault, "Analytic and numerical study of pulse broadening in nonlinear dispersive optical fibers," J. Opt. Soc. Am. B 3,205-211 (1992).
[CrossRef]

D. Anderson, M. Desaix, M. Karlsson, M. Lisak and M. L. Quiroga-Teixeiro, "Wave breaking in nonlinear-optical fibers," J. Opt. Soc. Am. B 9,1358-1361 (1992).
[CrossRef]

D. Marcuse, "RMS Width of Pulses in Nonlinear Dispersive Fibers," J. Lightwave Technol. 10,17-21 (1992).
[CrossRef]

R. Martínez-Herrero, P. M. Mejías, M. Sánchez and J. L. H. Neira, "Third- and fourth-order parametric characterization of partially coherent beams propagating through ABCD optical systems," Opt. and Quantum Electron. 24,S1021-S1026 (1992).
[CrossRef]

H. Weber, "Propagation of higher-order intensity moments in quadratic-index media," Opt. and Quantum Electron. 24,1027-1049 (1992).
[CrossRef]

1988 (1)

D. Anderson, M. Lisak and T. Reichel, "Approximate analytical approaches to nonlinear pulse propagation in optical fibers: A comparison," Phys. Rev. A 38,1618-1620 (1988).
[CrossRef] [PubMed]

1972 (1)

V. E. Zakharov and A. B. Shabat, "Exact theory of two-dimensional self-focusing and one-dimensional selfmodulation of waves in nonlinear media," Soviet Physics JETP 34,62-69 (1972)

J. Lightwave Technol. (1)

D. Marcuse, "RMS Width of Pulses in Nonlinear Dispersive Fibers," J. Lightwave Technol. 10,17-21 (1992).
[CrossRef]

J. Opt. Soc. Am. B (3)

J. Sci. Comput. (1)

J. F. Geer, "Rational trigonometric approximations using Fourier Series Partial Sums," J. Sci. Comput. 10,325-356 (1995).
[CrossRef]

Opt. and Quantum Electron. (2)

R. Martínez-Herrero, P. M. Mejías, M. Sánchez and J. L. H. Neira, "Third- and fourth-order parametric characterization of partially coherent beams propagating through ABCD optical systems," Opt. and Quantum Electron. 24,S1021-S1026 (1992).
[CrossRef]

H. Weber, "Propagation of higher-order intensity moments in quadratic-index media," Opt. and Quantum Electron. 24,1027-1049 (1992).
[CrossRef]

Opt. Commun. (2)

P.-A. Bélanger and N. Bélanger, "RMS characteristics of pulses in nonlinear dispersive lossy fibers," Opt. Commun. 117,56-60 (1995).
[CrossRef]

J. Santhanam and G. P. Agrawal, "Raman-induced spectral shifts in optical fibers: general theory based on the moment method," Opt. Commun. 222,413-420 (2003).
[CrossRef]

Phys. Rev. A (1)

D. Anderson, M. Lisak and T. Reichel, "Approximate analytical approaches to nonlinear pulse propagation in optical fibers: A comparison," Phys. Rev. A 38,1618-1620 (1988).
[CrossRef] [PubMed]

Soviet Physics JETP (1)

V. E. Zakharov and A. B. Shabat, "Exact theory of two-dimensional self-focusing and one-dimensional selfmodulation of waves in nonlinear media," Soviet Physics JETP 34,62-69 (1972)

Other (2)

J. F. Kenney and E. S. Keeping, Mathematics of statistics, 2nd ed., (D. Van Nostrand Company Inc., 1951).

G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed., (Academic Press, 2001).

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Figures (5)

Figure 1.
Figure 1.

Propagation of the second order moments of a super Gaussian pulse in the normal dispersion regime (β 2>0). The different curves represent different peak power; starting from the top curve N 2=10,5,2,1,0.5,0.1,0.01. The plain lines show numerically found simulations while the dotted lines with diamonds are the numerical solutions of the analytical transcendental equations. Both approximate invariants are also plotted. Note that the curves are in reverse order for I 2, with the low values of N 2 at the top and the high values at the bottom.

Figure 2.
Figure 2.

Propagation of the second order moments of an hyperbolic secant pulse in the anomalous dispersion regime (β 2<0). The different curves represent different peak power; starting from the top curve N 2=9,5,4,3,2,1,0.5. The plain lines show numerical simulations while the dotted lines with diamonds are the numerically found solutions of the analytical transcendental equations. Each curve has been shifted by two units (100 in graphs with log scale) for the sake of clarity. The approximate invariants are also plotted; they show large variations along propagation.

Figure 3.
Figure 3.

Propagation of the second order moments of a Gaussian pulse in the normal dispersion regime. The different curves represent different initial chirp 〈TΩ〉 r0=-10,-5,-2,-1,0,1,2,5,10. The plain lines show numerical simulations while the dotted lines with diamonds are the numerically found solutions of the analytical transcendental equations. The hollow diamonds represent positive initial chirps while black diamonds represent negative chirp. The moments of the unchirped pulse are plotted with circle. The moments have been normalized by their Fourier-limited values, denoted by the subscript FL. All pulses have N=1 when Fourier-limited.

Figure 4.
Figure 4.

Propagation of the second order moments of a super Gaussian pulse in the normal dispersion regime (β 2>0). The different curves represent different peak powers; starting from the top curve N 2=10,5,2,1,0.5,0.1,0.01. The plain lines show numerically found simulations while the dotted lines with diamonds represent the short distance model if z< zM and the long distance model if z>zM . The transition points are respectively zM/L D=0.13,0.18,0.26,0.34,0.41,0.53,0.58. The figure on the right is a zoom of the left figure.

Figure 5.
Figure 5.

Propagation of the moments of an hyperbolic secant pulse in the anomalous dispersion regime (β 2<0). The approximate model of Eq. (51) (dashed line with diamonds) and the solutions of the transcendental Eq. (41) (plain lines) are compared. The curves are plotted for N 2=2,3,4. The curves are shifted by three units for 〈TΩ〉 and two units for 〈T 2〉 for the sake of clarity.

Tables (2)

Tables Icon

Table 1. Invariants I 1 and I 2 for typical pulse shapes

Tables Icon

Table 2. Sign of β 2 I 0

Equations (60)

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𝓜 jk t j ω k t j ( i d dt ) k = t j i k d k d t k ,
t j ω k = 𝓜 jk 𝓜 00 = + A * ( t ) 𝓜 jk A ( t ) dt + A * ( t ) 𝓜 00 A ( t ) dt ,
E = 𝓜 00 = + A ( t ) 2 dt ,
t j ω k = p = 0 p j k ( i ) p j ! K ! p ! ( j p ) ! ( k p ) ! t j p ω k p * = ω k t j * .
t j ω k i = i 2 ( t j ω k t j ω k * )
= p = 1 p j p impair k ( 1 ) p 1 2 p ! ( j p ) ( k p ) t j p ω k p r p = 0 p j p pair k ( 1 ) p 2 p ! ( j p ) ( k p ) t j p ω k p i
[ t j , ω k ] = t j ω k ω k t j = t j ω k t j ω k * = 2 i t j ω k i .
{ t j , ω k } = t j ω k + ω k t j = t j ω k + t j ω k * = 2 t j ω k r .
σ t 2 = t 2 t 2 σ ω 2 = ω 2 ω 2 .
t j ω r = 1 E A 2 t j d ϕ dt dt .
σ t ω 2 = t ω r t ω .
ω inst ( t ) = d ϕ dt = ω + σ t ω 2 σ t 2 ( t t ) .
t freq ( ω ) = d θ d ω ,
P A ( t ) 2
A z = i β 2 2 2 A T 2 + i γ A 2 A
d dz T 2 = 2 β 2 T Ω r d dz T Ω r = β 2 Ω 2 + γ 2 P d dz Ω 2 = 2 γ P Ω 2 i d dz P = 2 β 2 P Ω 2 i ,
T Ω r = 1 E + A ( T ) 2 T d ϕ dT dT = 1 E + A ( T ) 2 T 2 d 2 ϕ d T 2 dT = T 2 d 2 ϕ d T 2
P Ω 2 i = 1 2 E + A ( T ) 4 d 2 ϕ d T 2 dT P T Ω r 2 T 2
d dz T 2 = 2 β 2 T Ω r
d dz T Ω r = β 2 Ω 2 + γ 2 P
d dz Ω 2 = γ P T Ω r T 2
d dz P = β 2 P T Ω r T 2 .
L D L NL = 1 = γ P 2 β 2 Ω 2
L D = 1 β 2 Ω 2 , L NL = 2 γ P .
I 0 = β 2 Ω 2 + γ P I 1 = Ω 2 T 2 T Ω r 2 I 2 = P T 2 E = P Δ T 2 E
I 0 = sgn ( β 2 ) L D + 2 L NL .
Ω 2 T 2 I 1
T Ω r = [ I 0 β 2 T 2 γ I 2 E β 2 T 2 I 1 ] 1 2 .
Δ T min = 2 T 2 min = 2 σ t = γ I 2 E I 0 + sgn ( β 2 ) I 0 γ 2 I 2 2 E 2 + 4 β 2 I 0 I 1 .
Δ Ω min = 2 I 1 T 2 min = 4 I 1 Δ T min
d T 2 dz = 2 β 2 T Ω r = 2 β 2 sgn ( T Ω r ) [ I 0 β 2 T 2 γ I 2 E β 2 T 2 I 1 ] 1 2 .
N 2 = L D L NL = γ P 2 β 2 Ω 2 .
T Ω r = K + sgn ( β 2 T Ω r ) I 0 z sgn ( β 2 ) a 2 ln [ 2 T Ω r + 2 I 0 T 2 β 2 sgn ( β 2 ) a ]
K = T Ω 0 r + sgn ( β 2 ) a 2 ln [ 2 T Ω 0 r + 2 I 0 T 2 0 β 2 sgn ( β 2 ) a ]
a = γ I 2 E β 2 I 0 = N 2 [ 2 ( I 1 + T Ω 0 r 2 ) 1 2 + sgn ( β 2 ) N 2 ] 1 2 .
z c = sgn ( β 2 T Ω 0 r ) a 4 I 0 ln ( γ 2 I 2 2 E 2 β 2 I 0 + 4 I 1 ) K I 0
= sgn ( T Ω 0 r ) a 2 I 0 ln [ ( a 2 + 4 I 1 ) 1 2 2 T Ω 0 r + ( a 2 + 4 I 1 + 4 T Ω 0 r ) 1 2 ] T Ω 0 r I 0 .
s = sgn ( T Ω r ) = sgn ( β 2 ) sgn [ 1 + sgn ( β 2 T Ω 0 r ) + sgn ( z z c ) ]
T Ω r = s T Ω r = T Ω 0 r + I 0 z sgn ( β 2 ) s a 2 ln [ T Ω r + s 2 ( a 2 + 4 I 1 + 4 T Ω r ) 1 2 T Ω 0 r + s 2 ( a 2 + 4 I 1 + 4 T Ω 0 r ) 1 2 ] .
T 2 = γ P 0 T 2 0 2 I 0 + [ γ 2 P 0 2 T 2 0 4 I 0 2 + β 2 I 0 ( I 1 + T Ω r 2 ) ] 1 2
P = P 0 T 2 0 T 2 ω 2 = ω 2 0 + γ β 2 P
T Ω r = sgn ( T Ω r ) I 0 z + K + a 2 arctan [ 1 T Ω r ( I 0 T 2 β 2 a 2 ) ]
K = T Ω 0 r a 2 arctan [ 1 T Ω 0 r ( I 0 T 2 0 β 2 a 2 ) ]
z cm = sgn ( T Ω 0 r ) I 0 a π 4 K + am π 2 I 0 m Z
𝓣 osc = a π I 0 1 N
s m = sgn ( T Ω r ) = 1 2 2 ( z z cm ) 𝓣 osc mod 2
s m T Ω r = I 0 z + T Ω 0 r a π 2 2 ( z z cm ) 𝓣 osc
+ a 2 arctan [ 1 s T Ω r ( I 0 T 2 β 2 a 2 ) ] a 2 arctan [ 1 T Ω 0 r ( I 0 T 2 0 β 2 a 2 ) ]
T Ω r = s [ T Ω 0 r 2 + Ω 2 0 ( T 2 T 2 0 ) + γ P 0 β 2 ( T 2 T 2 T 2 0 ) ] 1 2
β 2 d I 1 dz + γ T 2 d I 2 dz = 0 .
T Ω rL = T Ω 0 r + ( β 2 Ω 2 0 + γ 2 P 0 ) z
T 2 L = T 2 0 + 2 β 2 T Ω 0 r z + β 2 ( β 2 Ω 2 0 + γ 2 P 0 ) z 2
T Ω r = T Ω 0 r + I 0 z
T 2 = T 2 0 + 2 β 2 T Ω 0 r z + β 2 I 0 z 2
Ω i = T Ω r T 2 T T β 2 z
T Ω r = [ I 0 β 2 T 2 γ I 2 E β 2 T 2 I 1 ] 1 2 I 0 T 2 β 2 sgn ( β 2 ) a 2
T Ω rM = T Ω 0 r + I 0 z sgn ( β 2 ) s a 2 ln [ 2 T Ω rL T Ω 0 r + s I 0 T 2 0 β 2 sgn ( β 2 ) s a 2 ]
z M = T 2 0 β 2 I 0 T Ω 0 r I 0 γ P 0 2
T Ω rF = ( I 0 T 2 0 β 2 a 2 ) sin ( 2 I 0 z a ) 1 + ( 1 1 N ) cos ( 2 I 0 z a )
T 2 F = T 2 0 a β 2 I 0 ( 1 1 N ) ln [ 1 + ( 1 1 N ) cos ( 2 I 0 z a ) 2 1 N ]

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