Nonlinear pulse propagation in optical fiber

Nonlinear pulse propagation in opticalfibers using second order moments

Bryan Burgoyne, Nicolas Godbout, and Suzanne LacroixCentre d’optique, photonique et laser, Laboratoire des fibres optiques,Engineering Physics Department,Ecole Polytechnique de Montreal,

P.O. Box 6079, station Centre-ville, Montreal, Quebec H3C 3A7, Canada

[email protected]

Abstract: We present simple yet efficient formulae for the propagationofthe second order moments of a pulse in a nonlinear and dispersive opticalfiber over many dispersion and nonlinear lengths. The propagation of thetemporal and spectral widths, chirp and power of pulses are very preciselyapproximated and quickly calculated in both dispersion regimes as long asthe pulses are not high order solitons.

© 2007 Optical Society of America

OCIS codes:(190.4370) nonlinear optics, fibers; (060.5530) pulse propagation and solitons

References1. V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-

modulation of waves in nonlinear media,” Soviet physics JETP34,62–69 (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. B10,1185–1190 (1993).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. B3, 205–211 (1992).4. D. Marcuse, “RMS Width of Pulses in Nonlinear Dispersive Fibers,” J. Lightwave Technol.10,17–21 (1992).5. P.-A. Belanger and N. Belanger, “RMS characteristics of pulses in nonlinear dispersive lossy fibers,” Opt. Com-

mun.117,56–60 (1995).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).7. R. Martınez-Herrero, P. M. Mejıas, M. Sanchez and J. L. H. Neira, “Third- and fourth-order parametric charac-

terization of partially coherent beams propagating throughABCD optical systems,” Opt. and Quantum Electron.24,S1021–S1026 (1992).

8. H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. and Quantum Elec-tron.24,1027–1049 (1992).

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. A38,1618–1620 (1988).12. D. Anderson, M. Desaix, M. Karlsson, M. Lisak and M. L. Quiroga-Teixeiro, “Wave breaking in nonlinear-optical

fibers,” J. Opt. Soc. Am. B9, 1358–1361 (1992).13. J. F. Geer, “Rational trigonometric approximations usingFourier Series Partial Sums,” J. Sci. Comput.10,325–

356 (1995).

1. Introduction

The nonlinearity of silica is more than ever relevant in designing optical fiber components andsystems. The nonlinear propagation of pulses in optical fibers is described by the nonlinearSchrodinger equation (NLSE) which can only be solved analytically for the solitonic [1] and

#83197 - $15.00 USD Received 18 May 2007; revised 11 Jul 2007; accepted 12 Jul 2007; published 26 Jul 2007

(C) 2007 OSA 6 August 2007 / Vol. 15, No. 16 / OPTICS EXPRESS 10075

asymptotic [2] cases. To consider the general case, the moments method have been used to sim-plify the problem [3]–[6]. In this method, only the main characteristics of the pulse (position,width, etc.) are propagated. While the method does not describe the whole field, it highlightsthe evolution of the meaningful physical parameters. Approximate solutions to this problemreported in literature are either valid over short propagation distances, low power or for specificpulse shapes [3]–[5].

We present here an approximate solution of the NLSE using themoments method. While thismethod has been previously used to solve the NLSE in particular cases, the formulae shown inthis article allow an adequate description of the evolutionof the temporal and spectral widths,chirp and peak power of the pulse, over short or long distances, for arbitrary pulse shapes.The only assumption in the calculation is that the phase is quadratic in time, which, while notexact, provides good results as long as the pulse shape remains bell-like along propagation.The calculation is done for both the normal and anomalous dispersion regimes and equationsdescribing the propagation in both cases are presented.

2. Definition of the moments

The complex electric field of the pulse can be represented by aset of moments. Since the fieldsconsidered here are complex, a special set of moments is required to describe them properly.These moments have been defined before through Wigner transforms [7], but we present here asystematic operator representation, similar to one developed for spatial mode fields [8], which issimple and flexible. The raw (as opposed to centered) momentsare defined in the time domainthrough the moment operatorM jk

M jk ≡ t jωk ≡ t j(

iddt

)k

= t j ikdk

dtk, (1)

which includes the time operatort and the frequency operatorω ≡ id/dt. This operator is usedto express the moments as

〈t jωk〉 =〈M jk〉〈M00〉

=

∫ +∞−∞ A∗(t)M jkA(t)dt∫ +∞−∞ A∗(t)M00A(t)dt

, (2)

whereA(t) is the complex representation of the amplitude of the scalarelectric field, normalizedsuch as|A(t)|2 represents power. The moments are normalized with respect to the energy

E = 〈M00〉 =∫ +∞

−∞|A(t)|2dt, (3)

which is the zeroth order moment.For a given ordern = j + k there aren+ 1 moments, among which only〈tn〉 and〈ωn〉 are

real. The other moments are complex and crossed; they contain information pertaining to bothtime and frequency domains. These moments thus describe howthe frequency changes in time,i.e. the temporal phase of the field. More insight is gained onthe relations between the differentmoments if Eq. (2) is integrated by parts. One then obtains

〈t jωk〉 =k

∑p=0p≤ j

(−i)p j!k!p!( j − p)!(k− p)!

〈t j−pωk−p〉∗ = 〈ωkt j〉∗. (4)



The imaginary part of〈t jωk〉∗ is then easily found to be

〈t jωk〉i =− i2

(

〈t jωk〉−〈t jωk〉∗)

=k

∑p=1p≤ j

p impair

(−1)p−1

2 p!

(

jp

)(

kp

)

〈t j−pωk−p〉r −k

∑p=0p≤ j

p pair

(−1)p2 p!

(

jp

)(

kp

)

〈t j−pωk−p〉i (5)

where the subscriptsr andi denote the real and imaginary part respectively. An important resultis found from Eq. (5); the imaginary part of the moments can beexpressed as a sum of lowerorder moments. It can even be expressed solely in terms of thereal part of lower order momentsif the imaginary part is recursively replaced in Eq. (5). In other words, the imaginary part ofthe moments is redundant with the real part since it does not contain new information. Anotherway of obtaining the imaginary part is through the commutator [t j ,ωk].

[t j ,ωk] = 〈t jωk〉−〈ωkt j〉 = 〈t jωk〉−〈t jωk〉∗ = 2i〈t jωk〉i . (6)

The time and frequency operator commute only whenj = 0 ork= 0, i.e. when the moments arereal. In a similar way, the anticommutatort j ,ωk yields the real part of the moment〈t jωk〉.

t j ,ωk = 〈t jωk〉+ 〈ωkt j〉 = 〈t jωk〉+ 〈t jωk〉∗ = 2〈t jωk〉r . (7)

Since the information given by the imaginary part of the moments〈t jωk〉 can be expressed interms of the real part of lower order moments, only the latterare considered in the followinganalysis.

Let us first consider in detail the moments that are real and their physical interpretation. Thefirst order moments〈t〉 and〈ω〉 are the propagation time and average frequency respectively.The real second order moments〈t2〉 and 〈ω2〉 are a measure of the temporal width of theintensity and spectral width of the spectral density. Thesemoments, when centered around〈t〉and〈ω〉 respectively, yield the variances of the intensityσ2

t and spectral densityσ2ω .

σ2t = 〈t2〉−〈t〉2 σ2

ω = 〈ω2〉−〈ω〉2. (8)

The real third order moments〈t3〉 and〈ω3〉 are linked to the skewness or asymmetry of theintensity and spectral density respectively, whereas the fourth order moments〈t4〉 and 〈ω4〉are related to their kurtosis or degree of peakedness [9]. These moments are well known andextensively used in probability theory and statistics. Higher order moments give additionalinformation on the shape of the intensity and spectral density.

Whereas the real moments yield information on the intensity and spectrum, the crossed mo-ments characterize the phase of the field. Let us rewrite the definition of those moments con-taining a first order derivative as a function of the phaseφ(t) of the fieldA(t) = |A(t)|exp[iφ(t)],considering only its real part.

〈t jω〉r = − 1E

∫ ∞

−∞|A|2t j dφ

dtdt. (9)

This moment is thus a correlation between the time operatort applied j times and the instan-taneous frequency. It represents to what extent the frequency ω changes in time ast j . Forinstance,〈tω〉r indicates how linear in time is the instantaneous frequency. When centered, weobtain the covariance which is proportional to the chirp parameterC as defined in Ref. [10].

σ2tω = 〈tω〉r −〈t〉〈ω〉. (10)



If the phase is parabolic, the actual instantaneous frequency is linear in time. In this caseσ2tω ,

σ2t and the first order moments are sufficient to describe the instantaneous frequency

ωinst(t) = −dφdt

= 〈ω〉+ σ2tω

σ2t

(t −〈t〉) . (11)

If ωinst were quadratic in time, the moment〈t2ω〉r would be required to describe it. Due tothe time-frequency duality, the crossed moment〈tω j〉r yields the same information about thespectral phase as〈t jω〉r does about the temporal phase. More precisely, let us first define thefrequency delay, that is the delay at which a frequency can befound inside the pulse,by

tfreq(ω) =dθdω

, (12)

with θ(ω) the phase of the field in the frequency domain. The moment〈tωk〉r can then be seenas how the delaytfreq depends onωk. The general crossed moments〈t jωk〉r (with j 6= 0 andk 6= 0) represents the high order temporal dependence of the instantaneous frequencyωinst, orfrom another point of view, the high order spectral dependence of the frequency delaytfreq.Another important operator is required to deal with third order nonlinear effects. We define thepower operator as

P≡ |A(t)|2 (13)

The moment〈P〉 represents the effective power of the pulse, which is proportional to the peakpower. The moments〈Pt j〉 give the same information about the square of the intensity as themoments〈t j〉 do about the intensity. The frequency moments〈Pnt jωk〉 are in general complexand their imaginary part may contain relevant information about the phase of the field. In theremainder of the article, only the second order moments are considered for the analysis.

3. Propagation of the moments

The lossless propagation of pulses in an optical fiber is given by the nonlinear Schrodingerequation, if the third order dispersion and high order nonlinear effects are neglected.

∂A∂z

= − iβ2

2∂ 2A∂T2 + iγ|A|2A (14)

whereT = t −β1z= t −〈t〉 is a local time in the reference frame of the pulse,β1 is the inverseof the group velocity,β2 represents chromatic dispersion, andγ is the nonlinearity coefficient[10]. In the following, we only consider symmetric pulse shapes, so that the first and third ordermoments are zero. By combining Eqs. (2) and (14), the propagation equations of the secondorder moments of the amplitude of the pulse and its effectivepower are found to be

ddz

〈T2〉 = 2β2〈TΩ〉rddz

〈TΩ〉r = β2〈Ω2〉+ γ2〈P〉

ddz

〈Ω2〉 = 2γ〈PΩ2〉iddz

〈P〉 = −2β2〈PΩ2〉i , (15)

where the frequencyΩ = ω −〈ω〉 is the frequency offset from the carrier frequency〈ω〉. Themoments〈T2〉 and〈Ω2〉 are thus the variance of the intensity and spectral density respectively.The chirp moment〈TΩ〉 is the cross variance of Eq. (10). It is proportional to the instantaneousfrequency or chirp. The moment〈PΩ2〉 contains information about the phase and power ofthe field. These equations cannot be solved exactly because of the moment〈PΩ2〉i , whose



propagation equation contains higher order moments. This moment plays a very important rolein the propagation since all the second order moments and theeffective power depend on itsevolution.

An approximate solution can be found if the phaseφ(t) of the fieldA(t) = |A(t)|exp[iφ(t)]is assumed to be parabolic and centered onT = 0. In this case, the chirp moment becomes

〈TΩ〉r = − 1E

∫ +∞

−∞|A(T)|2T

dφdT

dT = − 1E

∫ +∞

−∞|A(T)|2T2 d2φ

dT2 dT = −〈T2〉d2φdT2 (16)

where the second derivative of the phase is constant in time.The troublesome moment〈PΩ2〉i

can then be approximated as

〈PΩ2〉i = − 12E

∫ +∞

−∞|A(T)|4 d2φ

dT2 dT ≈ 〈P〉〈TΩ〉r

2〈T2〉 (17)

Note that the integral expression in (17) is found by replacing the field by A(t) =|A(t)|exp[iφ(t)] in the operator definition and integrating by parts. Approximation (17) is goodas long as the shape of the pulse changes slowly along propagation. In fact, (16) is exact ifthe shape of the pulse is invariant and the phase parabolic, which happens only for dispersiveGaussian pulses, first order solitons and self-similar parabolic pulses. In general, the approxi-mation holds as long as the pulse maintains a bell-like shape. The system of Eqs. (15) can nowbe closed using Eq. (17) and rewritten as

ddz

〈T2〉 = 2β2〈TΩ〉r (18a)

ddz

〈TΩ〉r = β2〈Ω2〉+ γ2〈P〉 (18b)

ddz

〈Ω2〉 = γ〈P〉〈TΩ〉r

〈T2〉 (18c)

ddz

〈P〉 = −β2〈P〉〈TΩ〉r

〈T2〉 . (18d)

One advantage of using the moments is to access more easily the physics of the propagation.For instance, we find from Eq. (18a) the well-know result thatthere can only be temporal pulsecompression if the chirp〈TΩ〉r and chromatic dispersionβ2 have different signs. In a similarway Eq. (18c) shows that spectral compression can only occurif the pulse has a negative chirp,since the nonlinear coefficientγ is positive in silica. The compression is also more importantfor high peak powers and short pulse durations. The evolution of the effective power is in-versely proportional to the pulse duration from Eq. (18d), so that the narrower the pulse is, themore rapidly does the peak power change along propagation. It is straightforward to see formEq. (18b) that there are two different contributions to the chirp : the chromatic dispersion, whichis proportional to the square of the bandwidth of the pulse and the nonlinearity which is pro-portional to the effective power. The usual solitonic condition can be obtained from Eq. (18b)by setting the derivative equal to zero, leading to

LD

LNL= 1 =

γ〈P〉2|β2|〈Ω2〉 (19)

where the dispersion lengthLD and nonlinear lengthLNL are redefined in terms of the momentsas

LD =1

|β2|〈Ω2〉 , LNL =2

γ〈P〉 . (20)



A opposed to Ref. [10], the dispersion length is now defined interms of the bandwidth ratherthan the pulse duration, which makes more physical sense, asthe chromatic dispersion effectdepends on the bandwidth of the pulse. While these lengths represent the same concepts asthose in Ref. [10], they do not give the same numerical values; this discrepancy has no physicalsignificance as those lengths are arbitrarily defined.

There are several different techniques to solve Eqs. (18), one of which being finding enoughinvariant quantities to easily integrate the system. By combining the propagation equations ofthe moments, three of these invariants are found to be

I0 = β2〈Ω2〉+ γ〈P〉 I1 = 〈Ω2〉〈T2〉−〈TΩ〉2r I2 =

〈P〉√

〈T2〉E

=〈P〉∆T

2E(21)

where∆T is the full RMS width of the intensity. The first invariant is awell-known invariant ofthe NLSE expressed in the moments formalism [11] and the onlyone to be a true invariant; theother ones come from the approximation of Eq. (17).I0 can be written in a more enlighteningform as

I0 =sgn(β2)

LD+

2LNL

. (22)

This invariant states that if, during propagation in the normal dispersion regime (β2 > 0), thedispersion effects increase, the nonlinear effects decrease. In other words, the bandwidth be-comes broader through nonlinearity, which increases the effect of dispersion, leading to a de-crease in peak power. In the anomalous dispersion regime, both effects increase or decreasetogether. An example of this behavior is the second order soliton temporal compression, whichincreases both the bandwidth and peak power at the same time.

The second and the third invariants are a direct consequenceof Eq. (17); they are only invari-ant within that approximation. The second invariant is a direct formulation of the uncertaintyprinciple and can be expressed as

〈Ω2〉〈T2〉 ≥ I1 (23)

where the equality occurs for Fourier limited (unchirped) pulses. It characterizes the pulse shapewhich is assumed to be unchanging along propagation. If the propagation is linear, i.e.γ = 0,thenI1 becomes strictly invariant. The third invariant indirectly expresses the conservation ofenergy. Since it has been normalized by the energyE, which is also an invariant, the thirdinvariant also describes the pulse shape. Note that if the propagation is purely nonlinear, i.e.β2 = 0, thenI2 is strictly invariant. Including the energy in the third invariant keeps it valid ifthe propagation is lossy [5] (and so isI1 but notI0). While the values of bothI1 andI2 dependsolely on the pulse shape, they do not give the same information about it. Note that, even thoughwe consider only the second order moments, some informationabout the shape is still present,information which is usually described by higher order moments. The reason for that is the useof information in the time and frequency domains simultaneously. Values ofI1 andI2 for typicalpulse shapes are given in Table 1. Note that〈Ω2〉 cannot be calculated for parabolic and squareunchirped pulses since the relevant integrals diverge in both cases.

By using the three invariants Eqs. (21) and Eq. (17), the chirp moment〈TΩ〉r can be writtenas

|〈TΩ〉r | =[

I0β2

〈T2〉− γI2Eβ2

√

〈T2〉− I1

]12

. (24)

By setting Eq. (24) to zero, the Fourier limited pulse full RMS width∆Tmin can be calculated.

∆Tmin = 2√

〈T2〉min

= 2σt =γI2E

I0+

sgn(β2)

I0

√

γ2I22E2 +4β2I0I1. (25)



Table 1. InvariantsI1 andI2 for typical pulse shapes

I1 I2Gaussian 1

41

2√

πHyperbolic secant π

12π

6√

3Super Gaussian 0.3427 0.2697

Parabolic — 35√

5Square — 1

2√

3

The Fourier limited bandwidth can then be found using Eq. (21).

∆Ωmin = 2

√

I1〈T2〉min

= 4

√I1

∆Tmin(26)

We are now ready to solve Eqs. (18). By using the expression ofthe chirp moment Eq. (24)and substituting in Eq. (18a), we have a differential equation depending solely on〈T2〉.

d〈T2〉dz

= 2β2〈TΩ〉r = 2β2sgn(〈TΩ〉r)

[

I0β2


√

〈T2〉− I1

]12

. (27)

Eq. (27) can be integrated analytically but the solution depends on the sign ofβ2, I0 and thesign of〈TΩ〉r . Two cases are to be considered depending on the sign ofβ2I0. These cases areexamined in details in the subsequent subsections. Ifβ2I0 > 0, the pulse width monotonicallybroadens or, if it is initially chirped, it may show one minimum along propagation. This isalways the case in the normal dispersion regime (β2 > 0). In the anomalous dispersion regime,the dispersion and nonlinearity are opposed; so in order to haveβ2I0 > 0, we must haveI0 < 0.This only occurs ifN2 ≤ 1/2, where

N2 =LD

LNL=

γ〈P〉2|β2|〈Ω2〉 . (28)

If N2 > 1/2 in the anomalous dispersion regime (β2 < 0), the pulse exhibit a periodic behavioralong propagtion. Table 2 shows the different cases that canoccur depending on the dispersionregime and the the effective power throughN.

Table 2. Sign ofβ2I0N2 ≤ 1/2 N2 ≥ 1/2

β2 > 0 + +β2 < 0 + −

3.1. β2I0 > 0

The first step to integrate Eq. (27) is to determine the sign ofthe chirp moment〈TΩ〉r , whichmay depend onz. The sign of the chirp can only change once along propagationsince the pulsebehaves monotonically along propagation. The sign of the chirp moment, which is lost in theinvariant I1, can be deduced from three quantities; the sign of the initial chirp, the dispersioncoefficientβ2 and the positionzc at which the pulse width is minimum. This position is found

by integrating Eq. (27) while assuming that sgn(〈TΩ〉r) is constant, which yields

|〈TΩ〉r | = K +sgn(β2〈TΩ〉r) I0z−sgn(β2)a2

ln

[

2|〈TΩ〉r |+2

√

I0〈T2〉β2

−sgn(β2)a

]

(29)

where the integration constantK is defined as

K = |〈TΩ〉0r |+sgn(β2)a2

ln

[

2|〈TΩ〉0r |+2

√

I0〈T2〉0

β2−sgn(β2)a

]

(30)

and|〈TΩ〉r | is defined by Eq. (24). The 0 subscript on the moments refers toinitial values. Thedimensionless parametera indicates the regime of propagation and is defined as

a =γI2E√

|β2I0|= N2

[

2(I1 + 〈TΩ〉20r)

|1/2+sgn(β2)N2|

]

12

. (31)

The parametera is roughly proportional toN in the normal dispersion regime (β2 > 0). Thismeans that the regime is highly nonlinear ifa≫ 1 while it is dominated by dispersion effectsif a≪ 1. In the anomalous dispersion regime (β2 < 0) a→ ∞ for the limit value ofN2 = 1/2.However, in this case, it does not mean that the propagation is highly nonlinear. The distancezc at which the chirp changes sign (corresponding to a Fourier limited pulse) is easily found byreplacing Eq. (25) in Eq. (29) and setting|〈TΩ〉r | = 0.

zc =−sgn(β2〈TΩ〉0r)

∣

∣

∣

∣

a4I0

ln

(

γ2I22E2

β2I0+4I1

)

− K|I0|

∣

∣

∣

∣

= sgn(〈TΩ〉0r)a

2|I0|ln

(

a2 +4I1)

12

2|〈TΩ〉0r |+(a2 +4I1 +4〈TΩ〉0r)12

− 〈TΩ〉0r

I0. (32)

A negative value ofzc means that the pulse monotonically broadens during propagation. Apositive value indicates the propagation distance at whichthe pulse duration is minimum. Thesign of the chirp moment can now be reconstructed; it has the same sign as the initial chirpmoment and changes sign atzc if the sign of the initial chirp and dispersion are opposed. It canbe written as

s= sgn(〈TΩ〉r) = sgn(β2)sgn[1+sgn(β2〈TΩ〉0r)+sgn(z−zc)] (33)

The propagation of the the moment〈T2〉 can now calculated if Eq. (27) is integrated whileconsidering the sign of the chirp

〈TΩ〉r = s|〈TΩ〉r | = 〈TΩ〉0r + I0z−sgn(β2)sa2

ln

〈TΩ〉r + s2

(

a2 +4I1 +4〈TΩ〉r)

12

〈TΩ〉0r + s2 (a2 +4I1 +4〈TΩ〉0r)

12

.

(34)

Unfortunately Eq. (34) is transcendental; this is not surprising since the propagation is nonlin-ear. While it cannot be solved analytically, it is however quickly solved numerically. Once themoment〈TΩ〉r is known, the other moments are easily found through the different invariants.

√

〈T2〉 =γ〈P〉0

√

〈T2〉0

2I0+

[

γ2〈P〉20〈T2〉0

4I20

+β2

I0

(

I1 + 〈TΩ〉2r

)

]

12

(35a)

〈P〉 = 〈P〉0

√

〈T2〉0

〈T2〉〈ω2〉 = 〈ω2〉0 +γβ2

〈P〉 (35b)

Comparison of these solutions with the full numerical solutions are shown in Section 4. Wenow turn to the regime where the pulse no longer monotonically broadens along propagation.

3.2. β2I0 < 0

The case whereβ2I0 < 0 occurs only in the anomalous dispersion regime whenN2 > 1/2. Inthis regime, the pulse moments show oscillations along propagation. This is the general case forhigh order solitons. Once again, to solve Eq. (27), the sign of the chirp must be known; in thiscase however, there are several distanceszc at which the pulse width is minimum because of theoscillations. To find those distances, we integrate Eq. (27), while assuming that sgn(〈TΩ〉r) isconstant

|〈TΩ〉r | = sgn(〈TΩ〉r) I0z+K +a2

arctan

[

1|〈TΩ〉r |

(

√

I0〈T2〉|β2|

− a2

)]

(36)

with the integration constantK defined as

K = |〈TΩ〉0r |−a2

arctan

[

1|〈TΩ〉0r |

(

√

I0〈T2〉0

|β2|− a

2

)]

(37)

where the parametera is defined by Eq. (31). The propagation is now described through aninverse tangent function which confirms the periodic behavior of the pulse moments observedwhenN2 > 1/2 for some specific parameters. Note that the argument of the inverse tangentbecomes zero for solitons (N = 1). The distanceszcm at which the pulse duration is minimum,are found by replacing Eq. (25) in Eq. (36) and by setting|〈TΩ〉r | = 0.

zcm =sgn(〈TΩ〉0r)

I0

∣

∣

∣

aπ4

−K∣

∣

∣+

amπ2I0

m∈ Z (38)

The period of oscillation can then be found since|〈TΩ〉r | is zero twice per oscillation.

Tosc=aπI0

∝1N

(39)

The period of oscillation is approximately inversely proportional toN if N ≫ 1. This fits wellthe solitonic behavior where the soliton width oscillates more rapidly along propagation withincreasing peak power (soliton order). The sign of the chirpis thus

sm =sgn(〈TΩ〉r) = 1−2

∣

∣

∣

∣

⌊

2(z−zcm)

Tosc

⌋

mod 2

∣

∣

∣

∣

(40)

where⌊ and⌋ denotes the floor function andmodcalculates the modulus of congruence. Know-ing the sign of the chirp, Eq. (27) can now be integrated properly using the sign of the chirp.

sm|〈TΩ〉r | = I0z+ 〈TΩ〉0r −aπ2

⌊

2(z−zcm)

Tosc

⌋

(41)

+a2

arctan

[

1s|〈TΩ〉r |

(

√

I0〈T2〉|β2|

− a2

)]

− a2

arctan

[

1〈TΩ〉0r

(

√

I0〈T2〉0

|β2|− a

2

)]

We now have transcendental expressions describing the evolution of the pulse duration through〈T2〉 alone if |〈TΩ〉r | is replaced by Eq. (24). Solving numerically this equation yields 〈T2〉.

The moments〈P〉 and〈Ω2〉 are found through Eqs. (35b). Finally, to obtain the chirp moment〈TΩ〉r , we use Eq. (24) and (40).

〈TΩ〉r =s[

〈TΩ〉20r + 〈Ω2〉0

(

〈T2〉−〈T2〉0)

+γ〈P〉0

β2

(

〈T2〉−√

〈T2〉〈T2〉0

)

]12

(42)

Let us now compare the predictions from these expressions with numerical simulations.

4. Comparison with numerical simulations

Several numerical simulations were carried out to validateEqs. (29) and (36). Pulses werepropagated numerically using a typical split-step Fouriermethod and their moments calculated.Those moments were then compared to the ones found by solvingthe transcendental equations(29) and (36) for different values ofN2 defined by Eq. (28). The results are shown in Fig. 1for the normal dispersion regime (β2I0 > 0). In the normal dispersion regime, the agreementbetween the analytical model and the numerical simulationsis excellent in general. The smalldiscrepancies occur mostly for〈P〉. They are more pronounced at the beginning of the propaga-tion and for low peak power. These discrepancies are caused by pulse shaping, which is morepronounced at the beginning of the propagation in the normaldispersion regime; in this casethe approximate invariantI2 changes along propagation, as seen in Fig. 1, which explainsthedifference between the numerical and analytical moment〈P〉. The difference decreases withincreasing peak power because the pulse shaping process is mainly caused by dispersion in thisregime. Also, with increasing peak power, the pulse reachesthe asymptotic parabolic regimemore rapidly; the pulse shapes thus remains invariant.

The propagation of the second order moments for different values ofN2 in the anomalousdispersion regime (β2I0 < 0) is shown in Fig. 2. While the agreement is good for low valuesof N2, it becomes increasingly worse with increasing peak power.While the analytical modelpredicts relatively well the periodic behavior of the propagation, the period of the analyticalmodel drifts compared to the numerical simulation at high power. It can be noticed howeverthat the period of oscillation does become shorter with increasing soliton orderN as predictedfrom Eq. (39). The evolution of the moments obtained throughEq. (41) is always periodic,which is obviously not the case for the numerical simulations whenN is not an integer or ifthe pulse shape is not an hyperbolic secant. Note that even when N is integer, the period ofoscillation is not accurately predicted.

The pulse shaping becomes more pronounced with increasing pulse power, since the disper-sion induced chirp and nonlinear chirp are opposed in the anomalous dispersion regime (leadingeventually to wavebreaking [12]). This explains why the discrepancies between the analyticaland numerical model are large for high values ofN2; the pulse shape invariance hypothesis isno longer valid so that the quantitiesI1 and I2 are no longer invariant. These “invariants” areplotted in Fig. 2 where it can be clearly seen that they changesignificantly along propagation.The period of oscillation is also affected since it depends indirectly onI1 andI2, which explainsthe discrepancies of the periods at high power, even whenN is an integer.

Numerical simulations were also performed to verify the behavior of the analytical solutionwhen the pulse has an initial chirp. Gaussian pulses havingN2 = 1 when Fourier-limited werepropagated in the normal dispersion regime for different values of initial chirp while assumingthat the initial chirp comes from a dispersive propagation.The comparison with the analyticalmodel is shown in Fig. 3. Once again the agreement is very good. The error increases with theinitial chirp because the pulse shaping incidentally increases. The error is also greater for〈P〉and〈Ω2〉 for the reasons mentioned above. In the anomalous dispersion regime, the behaviorof the moments is similar, except that the chirp moment〈TΩ〉 has an opposite slope.

0 1 2 3 4 5100

101

102

103

<T

2 >/<

T2 >

0

z/LD

0 1 2 3 4 510-2

10-1

100

101

102

<T

Ω>

r

z/LD

0 1 2 3 4 510-3

10-2

10-1

100

101

/

0,

N=

1

z/LD

0 1 2 3 4 50

5

10

15

20

25

<Ω

2 >/<

Ω2 >

0

z/LD

0 1 2 3 4 50.6

1.0

1.4

1.8

I 1/I

10

z/LD

N2=0.01

N2=10

0 1 2 3 4 5

1.0

1.1

1.2

I 2 /

I 20

z/LD

N2=0.01

N2=10

Figure 1. Propagation of the second order moments of a super Gaussian pulse in the normaldispersion regime (β2 > 0). The different curves represent different peak power; startingfrom the top curveN2 = 10,5,2,1,0.5,0.1,0.01. The plain lines show numerically foundsimulations while the dotted lines with diamonds are the numerical solutions of theanalyt-ical transcendental equations. Both approximate invariants are also plotted. Note that thecurves are in reverse order forI2, with the low values ofN2 at the top and the high valuesat the bottom.

We finish this discussion with the invariantsI1 etI2. A close inspection ofI1 andI2 reveals thatboth approximate invariants changes similarly along propagation in both propagation regimes.The evolution of these two approximate invariants are linked together by

β2dI1dz

+ γ√

〈T2〉dI2dz

= 0. (43)

So, the evolution of both invariants are almost the same, up to the pulse width. It is obvious nowfrom Eq. (43) thatI1 andI2 stay invariant along propagation whenγ = 0 andβ2 = 0 respectively.The invariantI1 in the regimeβ2I0 > 0 shows oscillations at high power. These oscillations areerrors coming from the numerical calculation of the moments. They occur because two terms

0 1 2 30

4

8

12

<T

2 >/<

T2 >

0

z/LD

N=1

N=2

N=3

0 1 2 3-2

2

6

10

14

z/LD

<T

Ω>

r0 1 2 3

100

104

108

1012

1016

/

0, N

=1

z/LD0 1 2 3

100

104

108

1012

z/LD

<Ω

2 >/<

Ω2 >

0

0 1 2 3

100

101

102

I 1 /

I 10

z/LD

0 1 2 3

1

2

3

4

I 2 /

I 20

z/LD

Figure 2. Propagation of the second order moments of an hyperbolic secant pulse in theanomalous dispersion regime (β2 < 0). The different curves represent different peak power;starting from the top curveN2 = 9,5,4,3,2,1,0.5. The plain lines show numerical simula-tions while the dotted lines with diamonds are the numerically found solutions of the ana-lytical transcendental equations. Each curve has been shifted by two units (100 in graphswith log scale) for the sake of clarity. The approximate invariants are also plotted; theyshow large variations along propagation.

of comparable magnitudes are subtracted in the definition ofI1. The errors are larger on themoments involving the frequency operatorid/dω because of the finite difference scheme usedin calculating the derivative.

5. Approximation of transcendental equations

Although the transcendental equations give very good results, they are not user friendly. So, wepresent here simpler approximate formulae based on the transcendental equations in the cases

0 2 4 6 8 10100

101

102

103

<T

2 >/<

T2 >

FL

z/LD

0 2 4 6 8 10-10

0

10

20

30

<T

Ω>

r

z/LD

0 2 4 6 8 10

10-1

100

/

FL

z/LD

0 2 4 6 8 100

1

2

3

<Ω

2 >/<

Ω2 >

FL

z/LD

Figure 3. Propagation of the second order moments of a Gaussian pulsein the nor-mal dispersion regime. The different curves represent differentinitial chirp 〈TΩ〉r0 =−10,−5,−2,−1,0,1,2,5,10. The plain lines show numerical simulations while the dot-ted lines with diamonds are the numerically found solutions of the analytical transcenden-tal equations. The hollow diamonds represent positive initial chirps while black diamondsrepresent negative chirp. The moments of the unchirped pulse are plotted with circle. Themoments have been normalized by their Fourier-limited values, denoted bythe subscriptFL. All pulses haveN = 1 when Fourier-limited.

where the propagation distance is either very short or very long.For short propagation distances, we first linearize the chirp moment of Eq. (18b)

〈TΩ〉rL = 〈TΩ〉0r +(

β2〈Ω2〉0 +γ2〈P〉0

)

z (44)

where the subscriptL stands for “linearized” To calculate〈T2〉, we substitute Eq. (44) inEq. (18a) and carry out the integration.

〈T2〉L =〈T2〉0 +2β2〈TΩ〉0rz+β2

(

β2〈Ω2〉0 +γ2〈P〉0

)

z2 (45)

The effective power〈P〉 and bandwidth〈Ω2〉 are calculated via Eq. (35b). The evolution ofthe moment〈T2〉L is similar to the one previously derived by Ref. [4] where theeffect ofthe bandwidth was not included. This approximation is also good over long distances if thepropagation is dominated by dispersion, i.e., ifN2 < 1. When the nonlinear effects are weak,the pulse bandwidth undergoes very little change during propagation, so that〈Ω2〉 = 〈Ω2〉0 inEq. (44) is justified. As a matter of fact, whenγ = 0, Eqs. (44)–(45) exactly describe the purelydispersive case.

For long propagation distances, two different cases must beconsidered, depending on thesign ofβ2I0.

5.1. β2I0 > 0

Let us now consider the asymptotic case for monotonically broadening pulses, i.e.β2I0 > 0.Whenz→ ∞, the logarithm in Eq. (29) becomes negligible; we thus find the asymptotic ex-pression for〈TΩ〉, once the sign of the chirp is taken into account.

〈TΩ〉r∞ = 〈TΩ〉0r + I0z (46)

〈T2〉∞ = 〈T2〉0 +2β2〈TΩ〉0rz+β2I0z2

The instantaneous frequency for the asymptotic case is easily calculated using Eq. (11).

Ωi∞ =〈TΩ〉r∞

〈T2〉∞T ≈ T

β2z(47)

This result is comparable to the asymptotic instantaneous frequency of a parabolic pulse previ-ously reported [2], where a variation on 1/zwas found. Note that both〈TΩ〉rL and〈TΩ〉r∞ arelinear withzand that their slopes differ only byγ〈P〉0/2.

Another approximation is needed to describe the transitionregion between the linearized andasymptotic cases. To do so, we write Eq. (24)

|〈TΩ〉r | =[

I0β2


√

〈T2〉− I1

]12

≈√

I0〈T2〉β2

−sgn(β2)a2

(48)

when〈T2〉 ≫ 〈T2〉0, i.e. for highly chirped pulses. Using Eq. (24) and (34), themoment〈TΩ〉can be written as

〈TΩ〉rM =〈TΩ〉0r + I0z−sgn(β2)sa2

ln

2〈TΩ〉rL

〈TΩ〉0r +s√

I0〈T2〉0β2

−sgn(β2)sa2

(49)

where the expressions|〈TΩ〉r | in the logarithm has been replaced by〈TΩ〉rL . Note that itcould also have been replaced by〈TΩ〉r∞, but since we already supposed large chirps to deriveEq. (49), using〈TΩ〉rL actually improves the performance for short propagation lengths. Theother moments are found using Eqs. (35a) and (35b).

The transition point between the linearized and transitionregimes is approximately found bycalculating the distancezM that minimizes〈TΩ〉rL −〈TΩ〉rM .

zM =

√

〈T2〉0

β2I0− 〈TΩ〉0r

I0− γ〈P〉02

(50)

If the propagation distance is shorter thanzM, Eq. (44) holds. Otherwise, the propagation isbest described by Eq. (49). Since the model in the transitionregion is valid for highly chirpedpulses, the transition point depends on〈TΩ〉0r .

To show the validity of Eqs. (44) and (49), they are compared to numerical simulations inFig. 4 for different values ofN2. The linearized model does fit nicely at the beginning of thepropagation and the transition model for long propagation distance. Their discrepancies areobviously larger around the transition pointzM. While using the minimum difference betweenboth models as the transition point is simple, it is not optimal. We can see in Fig. 4 that thelinearized model is still valid beyondzM.



0 1 2 310-1

100

101

102

<T

Ω>

r

z/LD

0 0.2 0.4 0.610-1

100

101

<T

Ω>

r

z/LD

Figure 4. Propagation of the second order moments of a super Gaussian pulse in the normaldispersion regime (β2 > 0). The different curves represent different peak powers; startingfrom the top curveN2 = 10,5,2,1,0.5,0.1,0.01. The plain lines show numerically foundsimulations while the dotted lines with diamonds represent the short distance model if z<zM and the long distance model ifz> zM . The transition points are respectivelyzM/LD =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.

5.2. β2I0 < 0

The high power anomalous dispersion regime is much more difficult to handle since there areno asymptotic solutions. If the pulse has an initial chirp, the moments are well approximated byEq. (49) since the pulse behave more or less monotonically over short propagation distances.When the pulse is Fourier-limited, it shows an oscillatory behavior. There is no obvious wayto solve the transcendence of Eq. (41) in order to obtain a simple form. It is however wellapproximated by

〈TΩ〉rF =

(

√

I0〈T2〉0

|β2|− a

2

)

sin(

2I0za

)

1+(

1− 1N

)

cos(

2I0za

) (51)

which is similar to a Pade-Fourier expansion [13]. The period in z of the moments is given byEq. (39). The parameterN 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 findthe variance〈T2〉. The variance is found instead by direct integration of Eq. (51).

〈T2〉F = 〈T2〉0−aβ2

I0(1− 1N )

ln

1+(1− 1N )cos

(

2I0za

)

2− 1N

(52)

The transcendental Eq. (41) and the approximate Eq. (51) arecompared in Fig. 5 for sev-eral integer values ofN. The approximate model represents well the solutions of transcendentalequation. The agreement gets better with increasing valuesof N, especially for〈T2〉; unfortu-nately, the discrepancies between “transcendental model”and the numerical simulations growlarger with increasing values ofN, as highlighted in the Fig. 2 comparisons.

6. Conclusion

We developed analytical formulae describing the propagation of a pulse in a nonlinear disper-sive optical fiber using the moments method. We first presented the different moments throughan operator formalism, which enabled to write the propagation equations in terms of second

0 1 2 3

0

3

6

9

<T

Ω>

r

z/LD

N=4

N=3

N=2

0 1 2 3-1

1

3

5

<T

2 >/<

T2 >

0

z/LD

N=4

N=3

N=2

Figure 5. Propagation of the moments of an hyperbolic secant pulse in theanomalous dis-persion regime (β2 < 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 areplotted forN2 = 2,3,4. The curves are shifted by three units for〈TΩ〉 and two units for〈T2〉 for the sake of clarity.

order moments alone. Second order moments describe the temporal and spectral width andthe chirp of the pulse. To solve these equations, a single approximation was made, which isthat the phase of the pulse is quadratic in time along propagation. Using this approximation,three invariants were found; one is a true invariant of NLSE,one is an invariant of the linearSchrodinger equation and one is an invariant when the propagation is non-dispersive. Theseinvariants characterize the shape of the pulse, which is usually characterized by fourth ordermoments. The invariants allowed us to find simple relations between the different moments ofthe pulse.

The solutions of the propagation equations take the form of transcendental equations, whichis not surprising since the propagation is nonlinear. When solved numerically, these transcen-dental equations compare excellently with numerical simulations in the normal dispersionregime and in the low power anomalous dispersion regime. Theagreement is good for short aswell as long propagation distances. In the high power anomalous dispersion regime, the solutionto the transcendental equations describe more qualitatively than quantitatively the propagation.While not much is gained on calculation speed using the transcendental equations instead of thenumerical propagation over a single step, only a single stepis needed using the transcendentalequation to know the moments over arbitrary long propagation distances. The main advantagethough of the moments method is the physical insight it brings on the behavior of the pulsealong propagation in terms of intuitive physical quantities.

In order to have all analytical propagation formulae, approximate solutions to the transcen-dental were also found. These formulae represent very precisely the numerical solution to thetranscendental equations in both dispersion regimes, regardless of the propagation distance.Using those formulae, we were able to describe the asymptotic propagation in the normal dis-persion regime and the periodic propagation in the anomalous dispersion regime.

Nonlinear pulse propagation in optical fiber

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