## Nonlinear pulse propagation in optical fibers using second order moments

Optics Express, Vol. 15, Issue 16, pp. 10075-10090 (2007)

http://dx.doi.org/10.1364/OE.15.010075

Acrobat PDF (1602 KB)

### 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

## 1. Introduction

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]

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]

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]

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

## 2. Definition of the moments

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]

*𝓜*

_{jk}*t*and the frequency operator

*ω≡id/dt*. This operator is used to express the moments as

*A(t)*is the complex representation of the amplitude of the scalar electric field, normalized such as |

*A(t)*|

^{2}represents power. The moments are normalized with respect to the energy

*n=j*+

*k*there are

*n*+1 moments, among which only 〈

*t*〉 and 〈

^{n}*ω*〉 are real. The other moments are complex and crossed; they contain information pertaining to both time and frequency domains. These moments thus describe how the frequency changes in time, i.e. the temporal phase of the field. More insight is gained on the relations between the different moments if Eq. (2) is integrated by parts. One then obtains

^{n}*t*〉* is then easily found to be

^{j}ω^{k}*r*and

*i*denote the real and imaginary part respectively. An important result is found from Eq. (5); the imaginary part of the moments can be expressed as a sum of lower order moments. It can even be expressed solely in terms of the real part of lower order moments if the imaginary part is recursively replaced in Eq. (5). In other words, the imaginary part of the moments is redundant with the real part since it does not contain new information. Another way of obtaining the imaginary part is through the commutator [

*t*].

^{j},ω^{k}*j*=0 or

*k*=0, i.e. when the moments are real. In a similar way, the anticommutator {

*t*} yields the real part of the moment 〈

^{j},ω^{k}*t*〉.

^{j}ω^{k}*t*〉 can be expressed in terms of the real part of lower order moments, only the latter are considered in the following analysis.

^{j}ω^{k}*t*〉 and 〈

*ω*〉 are the propagation time and average frequency respectively. The real second order moments 〈

*t*

^{2}〉 and 〈

*ω*

^{2}〉 are a measure of the temporal width of the intensity and spectral width of the spectral density. These moments, when centered around 〈t〉 and 〈

*ω*〉 respectively, yield the variances of the intensity

*σ*and spectral density

^{2}_{t}*σ*.

^{2}_{ω}*φ(t)*of the field

*A(t)*=|

*A(t)*|exp[

*iφ(t)*], considering only its real part.

*j*times and the instantaneous frequency. It represents to what extent the frequency

*ω*changes in time as

*t*. For instance, 〈

_{j}*tω*〉

_{r}indicates how linear in time is the instantaneous frequency. When centered, we obtain the covariance which is proportional to the chirp parameter C as defined in Ref. [10].

*σ*and the first order moments are sufficient to describe the instantaneous frequency

^{2}_{tω}, σ^{2}_{t}*ω*

_{inst}were quadratic in time, the moment 〈

*t*〉

^{2}ω_{r}would be required to describe it. Due to the time-frequency duality, the crossed moment 〈

*tω*〉

^{j}_{r}yields the same information about the spectral phase as 〈

*t*〉

^{j}ω_{r}does about the temporal phase. More precisely, let us first define the frequency delay, that is the delay at which a frequency can be found inside the pulse, by

*ω*) the phase of the field in the frequency domain. The moment 〈

*tω*〉

^{k}_{r}can then be seen as how the delay

*t*

_{freq}depends on

*ω*. The general crossed moments 〈

^{k}*t*〉

^{j}ω^{k}_{r}(with

*j*≠0 and

*k*≠0) represents the high order temporal dependence of the instantaneous frequency

*ω*

_{inst}, or from another point of view, the high order spectral dependence of the frequency delay

*t*

_{freq}. Another important operator is required to deal with third order nonlinear effects. We define the power operator as

*P*〉 represents the effective power of the pulse, which is proportional to the peak power. The moments 〈

*Pt*〉 give the same information about the square of the intensity as the moments 〈

^{j}*t*〉 do about the intensity. The frequency moments 〈

^{j}*P*〉 are in general complex and their imaginary part may contain relevant information about the phase of the field. In the remainder of the article, only the second order moments are considered for the analysis.

^{n}t^{j}ω^{k}## 3. Propagation of the moments

*T*=

*t*-

*β*

_{1}

*z*=

*t*-〈

*t*〉 is a local time in the reference frame of the pulse,

*β*

_{1}is the inverse of the group velocity, β2 represents chromatic dispersion, and

*γ*is the nonlinearity coefficient [10]. In the following, we only consider symmetric pulse shapes, so that the first and third order moments are zero. By combining Eqs. (2) and (14), the propagation equations of the second order moments of the amplitude of the pulse and its effective power are found to be

*ω*-〈

*ω*〉 is the frequency offset from the carrier frequency 〈

*ω*〉. The moments 〈

*T*

^{2}〉 and 〈Ω

^{2}〉 are thus the variance of the intensity and spectral density respectively. The chirp moment 〈

*T*Ω〉 is the cross variance of Eq. (10). It is proportional to the instantaneous frequency or chirp. The moment 〈

*P*Ω

^{2}〉 contains information about the phase and power of the field. These equations cannot be solved exactly because of the moment 〈

*P*Ω

^{2}〉

_{i}, whose propagation equation contains higher order moments. This moment plays a very important role in the propagation since all the second order moments and the effective power depend on its evolution.

*φ(t)*of the field

*A(t)*=|

*A(t)*|exp[

*iφ(t)*] is assumed to be parabolic and centered on

*T*=0. In this case, the chirp moment becomes

*P*Ω

^{2}〉

_{i}can then be approximated as

*A(t)*=|

*A(t)*|exp[

*iφ(t)*] in the operator definition and integrating by parts. Approximation (17) is good as long as the shape of the pulse changes slowly along propagation. In fact, (16) is exact if the shape of the pulse is invariant and the phase parabolic, which happens only for dispersive Gaussian pulses, first order solitons and self-similar parabolic pulses. In general, the approximation holds as long as the pulse maintains a bell-like shape. The system of Eqs. (15) can now be closed using Eq. (17) and rewritten as

*T*Ω〉

_{r}and chromatic dispersion

*β*

_{2}have different signs. In a similar way Eq. (18c) shows that spectral compression can only occur if the pulse has a negative chirp, since the nonlinear coefficient g is positive in silica. The compression is also more important for high peak powers and short pulse durations. The evolution of the effective power is inversely proportional to the pulse duration from Eq. (18d), so that the narrower the pulse is, the more rapidly does the peak power change along propagation. It is straightforward to see form Eq. (18b) that there are two different contributions to the chirp : the chromatic dispersion, which is proportional to the square of the bandwidth of the pulse and the nonlinearity which is proportional to the effective power. The usual solitonic condition can be obtained from Eq. (18b) by setting the derivative equal to zero, leading to

*L*

_{D}and nonlinear length

*L*

_{NL}are redefined in terms of the moments as

*T*is the full RMS width of the intensity. The first invariant is a well-known invariant of the NLSE expressed in the moments formalism [11

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]

*β*

_{2}>0), the dispersion effects increase, the nonlinear effects decrease. In other words, the bandwidth becomes broader through nonlinearity, which increases the effect of dispersion, leading to a decrease in peak power. In the anomalous dispersion regime, both effects increase or decrease together. An example of this behavior is the second order soliton temporal compression, which increases both the bandwidth and peak power at the same time.

*γ*=0, then

*I*

_{1}becomes strictly invariant. The third invariant indirectly expresses the conservation of energy. Since it has been normalized by the energy

*E*, which is also an invariant, the third invariant also describes the pulse shape. Note that if the propagation is purely nonlinear, i.e.

*β*

_{2}=0, then

*I*

_{2}is strictly invariant. Including the energy in the third invariant keeps it valid if the propagation is lossy [5

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

*I*

_{1}but not

*I*

_{0}). While the values of both

*I*

_{1}and

*I*

_{2}depend solely on the pulse shape, they do not give the same information about it. Note that, even though we consider only the second order moments, some information about the shape is still present, information which is usually described by higher order moments. The reason for that is the use of information in the time and frequency domains simultaneously. Values of

*I*

_{1}and

*I*

_{2}for typical pulse shapes are given in Table 1. Note that 〈Ω

^{2}〉 cannot be calculated for parabolic and square unchirped pulses since the relevant integrals diverge in both cases.

*T*

^{2}〉.

*β*

_{2},

*I*

_{0}and the sign of 〈

*T*Ω〉

_{r}. Two cases are to be considered depending on the sign of

*β*

_{2}

*I*

_{0}. These cases are examined in details in the subsequent subsections. If

*β*

_{2}

*I*

_{0}>0, the pulse width monotonically broadens or, if it is initially chirped, it may show one minimum along propagation. This is always the case in the normal dispersion regime (

*β*

_{2}>0). In the anomalous dispersion regime, the dispersion and nonlinearity are opposed; so in order to have

*β*

_{2}

*I*

_{0}>0, we must have

*I*

_{0}<0. This only occurs if

*N*

^{2}≤1/2, where

*N*

^{2}>1/2 in the anomalous dispersion regime (

*β*

^{2}<0), the pulse exhibit a periodic behavior along propagtion. Table 2 shows the different cases that can occur depending on the dispersion regime and the the effective power through

*N*.

### 3.1. β_{2}I_{0}>0

*T*Ω〉

_{r}, which may depend on

*z*. The sign of the chirp can only change once along propagation since the pulse behaves monotonically along propagation. The sign of the chirp moment, which is lost in the invariant

*I*

_{1}, can be deduced from three quantities; the sign of the initial chirp, the dispersion coefficient

*β*

_{2}and the position

*z*at which the pulse width is minimum. This position is found by integrating Eq. (27) while assuming that sgn(〈

_{c}*T*Ω〉

_{r}) is constant, which yields

*K*is defined as

*T*Ω〉

_{r}| is defined by Eq. (24). The 0 subscript on the moments refers to initial values. The dimensionless parameter a indicates the regime of propagation and is defined as

*a*is roughly proportional to

*N*in the normal dispersion regime (

*β*

_{2}>0). This means that the regime is highly nonlinear if

*a*≫1 while it is dominated by dispersion effects if

*a*≪1. In the anomalous dispersion regime (

*β*

_{2}<0)

*a*→∞ for the limit value of

*N*

^{2}=1/2. However, in this case, it does not mean that the propagation is highly nonlinear. The distance

*z*at which the chirp changes sign (corresponding to a Fourier limited pulse) is easily found by replacing Eq. (25) in Eq. (29) and setting |〈

_{c}*T*Ω〉

_{r}|=0.

*z*means that the pulse monotonically broadens during propagation. A positive value indicates the propagation distance at which the pulse duration is minimum. The sign of the chirp moment can now be reconstructed; it has the same sign as the initial chirp moment and changes sign at

_{c}*z*if the sign of the initial chirp and dispersion are opposed. It can be written as

_{c}*T*

^{2}〉 can now calculated if Eq. (27) is integrated while considering the sign of the chirp

*T*Ω〉

_{r}is known, the other moments are easily found through the different invariants.

### 3.2. β_{2}I_{0}<0

_{2}I

_{0}<0 occurs only in the anomalous dispersion regime when

*N*

^{2}>1/2. In this regime, the pulse moments show oscillations along propagation. This is the general case for high order solitons. Once again, to solve Eq. (27), the sign of the chirp must be known; in this case however, there are several distances

*z*at which the pulse width is minimum because of the oscillations. To find those distances, we integrate Eq. (27), while assuming that sgn(〈

_{c}*T*Ω〉

_{r}) is constant

*K*defined as

*a*is defined by Eq. (31). The propagation is now described through an inverse tangent function which confirms the periodic behavior of the pulse moments observed when

*N*

^{2}>1/2 for some specific parameters. Note that the argument of the inverse tangent becomes zero for solitons (

*N*=1). The distances

*z*at which the pulse duration is minimum, are found by replacing Eq. (25) in Eq. (36) and by setting |〈

_{cm}*T*Ω〉

_{r}|=0.

*T*Ω〉

_{r}| is zero twice per oscillation.

*N*if

*N*≫1. This fits well the solitonic behavior where the soliton width oscillates more rapidly along propagation with increasing peak power (soliton order). The sign of the chirp is thus

*mod*calculates the modulus of congruence. Knowing the sign of the chirp, Eq. (27) can now be integrated properly using the sign of the chirp.

*T*

^{2}〉 alone if |〈

*T*Ω〉

_{r}| is replaced by Eq. (24). Solving numerically this equation yields 〈

*T*

^{2}〉.

*P*〉 and 〈Ω

^{2}〉 are found through Eqs. (35b). Finally, to obtain the chirp moment 〈

*T*Ω〉

_{r}, we use Eq. (24) and (40).

## 4. Comparison with numerical simulations

*N*

^{2}defined by Eq. (28). The results are shown in Fig. 1 for the normal dispersion regime (

*β*

_{2}

*I*

_{0}>0). In the normal dispersion regime, the agreement between the analytical model and the numerical simulations is excellent in general. The small discrepancies occur mostly for 〈

*P*〉. They are more pronounced at the beginning of the propagation and for low peak power. These discrepancies are caused by pulse shaping, which is more pronounced at the beginning of the propagation in the normal dispersion regime; in this case the approximate invariant

*I*

_{2}changes along propagation, as seen in Fig. 1, which explains the difference between the numerical and analytical moment 〈

*P*〉. The difference decreases with increasing peak power because the pulse shaping process is mainly caused by dispersion in this regime. Also, with increasing peak power, the pulse reaches the asymptotic parabolic regime more rapidly; the pulse shapes thus remains invariant.

*N*

^{2}in the anomalous dispersion regime (

*β*

_{2}

*I*

_{0}<0) is shown in Fig. 2. While the agreement is good for low values of

*N*

^{2}, it becomes increasingly worse with increasing peak power. While the analytical model predicts relatively well the periodic behavior of the propagation, the period of the analytical model drifts compared to the numerical simulation at high power. It can be noticed however that the period of oscillation does become shorter with increasing soliton order

*N*as predicted from Eq. (39). The evolution of the moments obtained through Eq. (41) is always periodic, which is obviously not the case for the numerical simulations when

*N*is not an integer or if the pulse shape is not an hyperbolic secant. Note that even when

*N*is integer, the period of oscillation is not accurately predicted.

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]

*N*

^{2}; the pulse shape invariance hypothesis is no longer valid so that the quantities

*I*

_{1}and

*I*

_{2}are no longer invariant. These “invariants” are plotted in Fig. 2 where it can be clearly seen that they change significantly along propagation. The period of oscillation is also affected since it depends indirectly on

*I*

_{1}and

*I*

_{2}, which explains the discrepancies of the periods at high power, even when

*N*is an integer.

*N*

^{2}=1 when Fourier-limited were propagated in the normal dispersion regime for different values of initial chirp while assuming that the initial chirp comes from a dispersive propagation. The comparison with the analytical model is shown in Fig. 3. Once again the agreement is very good. The error increases with the initial chirp because the pulse shaping incidentally increases. The error is also greater for 〈

*P*〉 and 〈Ω

^{2}〉 for the reasons mentioned above. In the anomalous dispersion regime, the behavior of the moments is similar, except that the chirp moment 〈

*T*Ω〉 has an opposite slope.

*I*

_{1}et

*I*

_{2}. A close inspection of

*I*

_{1}and

*I*

_{2}reveals that both approximate invariants changes similarly along propagation in both propagation regimes. The evolution of these two approximate invariants are linked together by

*I*

_{1}and

*I*

_{2}stay invariant along propagation when

*γ*=0 and

*β*

_{2}=0 respectively. The invariant

*I*

_{1}in the regime

*β*

_{2}

*I*

_{0}>0 shows oscillations at high power. These oscillations are errors coming from the numerical calculation of the moments. They occur because two terms of comparable magnitudes are subtracted in the definition of

*I*

_{1}. The errors are larger on the moments involving the frequency operator

*id/dω*because of the finite difference scheme used in calculating the derivative.

## 5. Approximation of transcendental equations

*L*stands for “linearized” To calculate 〈

*T*

^{2}〉, we substitute Eq. (44) in Eq. (18a) and carry out the integration.

*P*〉 and bandwidth 〈Ω

^{2}〉 are calculated via Eq. (35b). The evolution of the moment 〈

*T*

^{2}〉

*L*is similar to the one previously derived by Ref. [4

4. D. Marcuse, “RMS Width of Pulses in Nonlinear Dispersive Fibers,” J. Lightwave Technol. **10**, 17–21 (1992). [CrossRef]

*N*

^{2}<1. When the nonlinear effects are weak, the pulse bandwidth undergoes very little change during propagation, so that 〈Ω

^{2}〉=〈Ω

^{2}〉

_{0}in Eq. (44) is justified. As a matter of fact, when

*γ*=0, Eqs. (44)–(45) exactly describe the purely dispersive case.

*β*

_{2}

*I*

_{0}.

### 5.1. β_{2}I_{0}>0

*β*

^{2}

*I*

_{0}>0. When

*z*→∞, the logarithm in Eq. (29) becomes negligible; we thus find the asymptotic expression for 〈

*T*Ω〉, once the sign of the chirp is taken into account.

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]

*z*was found. Note that both 〈

*T*Ω〉

_{rL}and 〈

*T*Ω〉

_{r∞}are linear with

*z*and that their slopes differ only by

*γ*〈

*P*〉

_{0}/2.

*T*

^{2}〉≫〈

*T*

^{2}〉0, i.e. for highly chirped pulses. Using Eq. (24) and (34), the moment 〈

*T*Ω〉 can be written as

*s*|〈

*T*Ω〉

_{r}| in the logarithm has been replaced by 〈

*T*Ω〉

_{rL}. Note that it could also have been replaced by 〈

*T*Ω〉

_{r∞}, but since we already supposed large chirps to derive Eq. (49), using 〈

*T*Ω〉

_{rL}actually improves the performance for short propagation lengths. The other moments are found using Eqs. (35a) and (35b).

*z*that minimizes 〈

_{M}*T*Ω〉

_{rL}-〈

*T*Ω〉

_{rM}.

*z*, Eq. (44) holds. Otherwise, the propagation is best described by Eq. (49). Since the model in the transition region is valid for highly chirped pulses, the transition point depends on 〈

_{M}*T*Ω〉

_{0r}.

*N*

^{2}. The linearized model does fit nicely at the beginning of the propagation and the transition model for long propagation distance. Their discrepancies are obviously larger around the transition point

*z*. While using the minimum difference between both models as the transition point is simple, it is not optimal. We can see in Fig. 4 that the linearized model is still valid beyond

_{M}*z*.

_{M}### 5.2. β2I0<0

13. J. F. Geer, “Rational trigonometric approximations using Fourier Series Partial Sums,” J. Sci. Comput. **10**, 325–356 (1995). [CrossRef]

*z*of the moments is given by Eq. (39). The parameter

*N*in the denominator deforms the sine function into a more sawtooth-like function. Since Eq. (51) is not a valid solution of Eq. (27), Eq. (35a) cannot be used to find the variance 〈

*T*

^{2}〉. The variance is found instead by direct integration of Eq. (51).

*N*. The approximate model represents well the solutions of transcendental equation. The agreement gets better with increasing values of

*N*, especially for 〈

*T*

^{2}〉; unfortunately, the discrepancies between “transcendental model” and the numerical simulations grow larger with increasing values of

*N*, as highlighted in the Fig. 2 comparisons.

## 6. Conclusion

## References

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 |

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 |

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 |

4. | D. Marcuse, “RMS Width of Pulses in Nonlinear Dispersive Fibers,” J. Lightwave Technol. |

5. | P.-A. Bélanger and N. Bélanger, “RMS characteristics of pulses in nonlinear dispersive lossy fibers,” Opt. Commun. |

6. | J. Santhanam and G. P. Agrawal, “Raman-induced spectral shifts in optical fibers: general theory based on the moment method,” Opt. Commun. |

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. |

8. | H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. and Quantum Electron. |

9. | J. F. Kenney and E. S. Keeping, |

10. | G. P. Agrawal, |

11. | D. Anderson, M. Lisak, and T. Reichel, “Approximate analytical approaches to nonlinear pulse propagation in optical fibers: A comparison,” Phys. Rev. A |

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 |

13. | J. F. Geer, “Rational trigonometric approximations using Fourier Series Partial Sums,” J. Sci. Comput. |

**OCIS Codes**

(060.5530) Fiber optics and optical communications : Pulse propagation and temporal solitons

(190.4370) Nonlinear optics : Nonlinear optics, fibers

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: May 18, 2007

Revised Manuscript: July 11, 2007

Manuscript Accepted: July 12, 2007

Published: July 26, 2007

**Citation**

Bryan Burgoyne, Nicolas Godbout, and Suzanne Lacroix, "Nonlinear pulse propagation in optical fibers using second order moments," Opt. Express **15**, 10075-10090 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-16-10075

Sort: Year | Journal | Reset

### References

- 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)
- 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]
- 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. Marcuse, "RMS Width of Pulses in Nonlinear Dispersive Fibers," J. Lightwave Technol. 10,17-21 (1992). [CrossRef]
- 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]
- 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]
- 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).
- 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]
- 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]
- J. F. Geer, "Rational trigonometric approximations using Fourier Series Partial Sums," J. Sci. Comput. 10,325-356 (1995). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.