1. Introduction

The nonlinearity of silica is more than ever relevant in designing optical fiber components and systems. The nonlinear propagation of pulses in optical fibers is described by the nonlinear Schrödinger equation (NLSE) which can only be solved analytically for the solitonic [1] and asymptotic [2] cases. To consider the general case, the moments method have been used to simplify 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 highlights the evolution of the meaningful physical parameters. Approximate solutions to this problem reported in literature are either valid over short propagation distances, low power or for specific pulse shapes [3]–[5].

We present here an approximate solution of the NLSE using the moments method. While this method has been previously used to solve the NLSE in particular cases, the formulae shown in this article allow an adequate description of the evolution of 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 not exact, 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 equations describing the propagation in both cases are presented.

2. Definition of the moments

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

which includes the time operator t and the frequency operator ω≡id/dt. This operator is used to express the moments as

where 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

which is the zeroth order moment.

For a given order n=j+k there are n+1 moments, among which only 〈tⁿ〉 and 〈ωⁿ〉 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

The imaginary part of 〈t^jω^k〉* is then easily found to be

where the subscripts 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].

The time and frequency operator commute only when j=0 or k=0, i.e. when the moments are real. In a similar way, the anticommutator {t^j,ω^k} yields the real part of the moment 〈t^jω^k〉.

Since the information given by the imaginary part of the moments 〈t^jω^k〉 can be expressed in terms of the real part of lower order moments, only the latter are considered in the following analysis.

Let us first consider in detail the moments that are real and their physical interpretation. The first order moments 〈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 σ²_t and spectral density σ²_ω.

The real third order moments 〈t ³〉 and 〈ω ³〉 are linked to the skewness or asymmetry of the intensity and spectral density respectively, whereas the fourth order moments 〈t ⁴〉 and 〈ω ⁴〉 are related to their kurtosis or degree of peakedness [9]. These moments are well known and extensively used in probability theory and statistics. Higher order moments give additional information on the shape of the intensity and spectral density.

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

This moment is thus a correlation between the time operator t applied j times and the instantaneous frequency. It represents to what extent the frequency ω changes in time as t_j. For instance, 〈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].

If the phase is parabolic, the actual instantaneous frequency is linear in time. In this case σ²_tω, σ²_t and the first order moments are sufficient to describe the instantaneous frequency

If ω _inst were quadratic in time, the moment 〈t²ω〉_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^jω〉_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

with θ(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^k. The general crossed moments 〈t^jω^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

The moment 〈P〉 represents the effective power of the pulse, which is proportional to the peak power. The moments 〈Pt^j〉 give the same information about the square of the intensity as the moments 〈t^j〉 do about the intensity. The frequency moments 〈Pⁿt ^jω^k〉 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.

3. Propagation of the moments

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

where 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

where the frequency Ω=ω-〈ω〉 is the frequency offset from the carrier frequency 〈ω〉. The moments 〈T ²〉 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 instantaneous frequency or chirp. The moment 〈PΩ²〉 contains information about the phase and power of the 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 role in the propagation since all the second order moments and the effective power depend on its evolution.

An approximate solution can be found if the phase 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

where the second derivative of the phase is constant in time. The troublesome moment 〈PΩ²〉_i can then be approximated as

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 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

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 that there can only be temporal pulse compression if the chirp 〈TΩ〉_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

where the dispersion length L _D and nonlinear length L _NL are redefined in terms of the moments as

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

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

where ΔT 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] and the only one to be a true invariant; the other ones come from the approximation of Eq. (17). I0 can be written in a more enlightening form as

This invariant states that if, during propagation in the normal dispersion regime (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.

The second and the third invariants are a direct consequence of Eq. (17); they are only invariant within that approximation. The second invariant is a direct formulation of the uncertainty principle and can be expressed as

where the equality occurs for Fourier limited (unchirped) pulses. It characterizes the pulse shape which is assumed to be unchanging along propagation. If the propagation is linear, i.e. 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] (and so is 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 〈Ω²〉 cannot be calculated for parabolic and square unchirped 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 written as

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

Table 1. Invariants I₁ and I₂ for typical pulse shapes

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

We are now ready to solve Eqs. (18). By using the expression of the chirp moment Eq. (24) and substituting in Eq. (18a), we have a differential equation depending solely on 〈T ²〉.

Eq. (27) can be integrated analytically but the solution depends on the sign of β ₂, I ₀ and the sign of 〈TΩ〉_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

If 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.

Table 2. Sign of β₂I₀

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3.1. β₂I₀>0

The first step to integrate Eq. (27) is to determine the sign of the chirp moment 〈TΩ〉_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_c 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

where the integration constant K is defined as

and |〈TΩ〉_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

The parameter 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_c at which the chirp changes sign (corresponding to a Fourier limited pulse) is easily found by replacing Eq. (25) in Eq. (29) and setting |〈TΩ〉_r|=0.

A negative value of z_c 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 z_c if the sign of the initial chirp and dispersion are opposed. It can be written as

The propagation of the the moment 〈T ²〉 can now calculated if Eq. (27) is integrated while considering the sign of the chirp

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

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

3.2. β₂I₀<0

The case where β₂I₀<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_c at which the pulse width is minimum because of the oscillations. To find those distances, we integrate Eq. (27), while assuming that sgn(〈TΩ〉_r) is constant

with the integration constant K defined as

where the parameter 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_cm at which the pulse duration is minimum, are found by replacing Eq. (25) in Eq. (36) and by setting |〈TΩ〉_r|=0.

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

The period of oscillation is approximately inversely proportional to 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

where ⌊ and ⌋ denotes the floor function and 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.

We now have transcendental expressions describing the evolution of the pulse duration through 〈T ²〉 alone if |〈TΩ〉_r| is replaced by Eq. (24). Solving numerically this equation yields 〈T ²〉.

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

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

4. Comparison with numerical simulations

Several numerical simulations were carried out to validate Eqs. (29) and (36). Pulses were propagated numerically using a typical split-step Fourier method and their moments calculated. Those moments were then compared to the ones found by solving the transcendental equations (29) and (36) for different values of 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.

The propagation of the second order moments for different values of 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.

The pulse shaping becomes more pronounced with increasing pulse power, since the dispersion induced chirp and nonlinear chirp are opposed in the anomalous dispersion regime (leading eventually to wavebreaking [12]). This explains why the discrepancies between the analytical and numerical model are large for high values of 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.

Numerical simulations were also performed to verify the behavior of the analytical solution when the pulse has an initial chirp. Gaussian pulses having 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 〈Ω²〉 for the reasons mentioned above. In the anomalous dispersion regime, the behavior of the moments is similar, except that the chirp moment 〈TΩ〉 has an opposite slope.

 

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.

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We finish this discussion with the invariants 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

So, the evolution of both invariants are almost the same, up to the pulse width. It is obvious now from Eq. (43) that 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.

 

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.

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5. Approximation of transcendental equations

Although the transcendental equations give very good results, they are not user friendly. So, we present here simpler approximate formulae based on the transcendental equations in the cases where the propagation distance is either very short or very long.

 

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.

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For short propagation distances, we first linearize the chirp moment of Eq. (18b)

where the subscript L stands for “linearized” To calculate 〈T ²〉, we substitute Eq. (44) in Eq. (18a) and carry out the integration.

The effective power 〈P〉 and bandwidth 〈Ω²〉 are calculated via Eq. (35b). The evolution of the moment 〈T ²〉L is similar to the one previously derived by Ref. [4] where the effect of the bandwidth was not included. This approximation is also good over long distances if the propagation is dominated by dispersion, i.e., if N ²<1. When the nonlinear effects are weak, the pulse bandwidth undergoes very little change during propagation, so that 〈Ω²〉=〈Ω²〉₀ in Eq. (44) is justified. As a matter of fact, when γ=0, Eqs. (44)–(45) exactly describe the purely dispersive case.

For long propagation distances, two different cases must be considered, depending on the sign of β ₂ I ₀.

5.1. β₂I₀>0

Let us now consider the asymptotic case for monotonically broadening pulses, i.e. β ² I ₀>0. When z→∞, the logarithm in Eq. (29) becomes negligible; we thus find the asymptotic expression for 〈TΩ〉, once the sign of the chirp is taken into account.

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

This result is comparable to the asymptotic instantaneous frequency of a parabolic pulse previously reported [2], where a variation on 1/z was found. Note that both 〈TΩ〉_rL and 〈TΩ〉_r∞ are linear with z and that their slopes differ only by γ〈P〉₀/2.

Another approximation is needed to describe the transition region between the linearized and asymptotic cases. To do so, we write Eq. (24)

when 〈T ²〉≫〈T ²〉0, i.e. for highly chirped pulses. Using Eq. (24) and (34), the moment 〈TΩ〉 can be written as

where the expression s|〈TΩ〉_r| in the logarithm has been replaced by 〈TΩ〉_rL. Note that it could also have been replaced by 〈TΩ〉_r∞, but since we already supposed large chirps to derive Eq. (49), using 〈TΩ〉_rL actually improves the performance for short propagation lengths. The other moments are found using Eqs. (35a) and (35b).

The transition point between the linearized and transition regimes is approximately found by calculating the distance z_M that minimizes 〈TΩ〉_rL-〈TΩ〉_rM.

If the propagation distance is shorter than z_M, 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 〈TΩ〉_0r.

To show the validity of Eqs. (44) and (49), they are compared to numerical simulations in Fig. 4 for different values of 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_M. 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 z_M.

 

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< z_M and the long distance model if z>z_M. The transition points are respectively z_M/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.

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5.2. β2I0<0

The high power anomalous dispersion regime is much more difficult to handle since there are no asymptotic solutions. If the pulse has an initial chirp, the moments are well approximated by Eq. (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 way to solve the transcendence of Eq. (41) in order to obtain a simple form. It is however well approximated by

which is similar to a Pade-Fourier expansion [13]. The period in 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).

The transcendental Eq. (41) and the approximate Eq. (51) are compared in Fig. 5 for several integer values of 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

We developed analytical formulae describing the propagation of a pulse in a nonlinear dispersive optical fiber using the moments method. We first presented the different moments through an operator formalism, which enabled to write the propagation equations in terms of second order moments alone. Second order moments describe the temporal and spectral width and the chirp of the pulse. To solve these equations, a single approximation was made, which is that 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 linear Schrödinger equation and one is an invariant when the propagation is non-dispersive. These invariants characterize the shape of the pulse, which is usually characterized by fourth order moments. The invariants allowed us to find simple relations between the different moments of the pulse.

 

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.

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The solutions of the propagation equations take the form of transcendental equations, which is not surprising since the propagation is nonlinear. When solved numerically, these transcendental equations compare excellently with numerical simulations in the normal dispersion regime and in the low power anomalous dispersion regime. The agreement is good for short as well as long propagation distances. In the high power anomalous dispersion regime, the solution to the transcendental equations describe more qualitatively than quantitatively the propagation. While not much is gained on calculation speed using the transcendental equations instead of the numerical propagation over a single step, only a single step is needed using the transcendental equation to know the moments over arbitrary long propagation distances. The main advantage though of the moments method is the physical insight it brings on the behavior of the pulse along propagation in terms of intuitive physical quantities.

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