solving the NLSE 非线性薛定谔方程

SOLVING THE NONLINEARSCHRÖDINGEREQUATIONEnricoForestieri and Marco SecondiniScuola Superiore Sant’Anna di Studi Universitari e Perfezionamento, Pisa, Italy, and PhotonicNetworks National Laboratory, CNIT, Pisa, Italy.forestieri@sssup.itAbstract:Some simplerecursive methods are described for constructing asymptoticallyexactsolutions of the nonlinear Schrödinger equation (NLSE). It is shown thatthe NLSE solution can be expressed analytically by two recurrence relationscorresponding to two different perturbation methods.optical Kerr effect; opticalfiber nonlinearity; nonlinear distortion; opticalfibertheory.Key words:1.INTRODUCTIONThe nonlinear Schrödinger equation governs the propagation of the opticalfield complex envelope in a single-mode fiber [1]. Accounting for groupvelocity dispersion (GVD), self-phase modulation (SPM), and loss, in a timeframemovingwith the signal group velocity, the NLSE can be written aswhere is the Kerr nonlinear coefficient [1], is the power attenuation con-stant, and is the GVD parameter being the referencewavelength,the light speed, and D the fiber dispersion parameter at Let-ting we can get rid of the last term in (1), whichbecomesExactsolutions of this equation are typically not known in analytical form,except for soliton solutions when[2–4]. Given an input condition4E. Forestieri and M. Secondinithe solution of (2) is then to be found numerically, the most widelyused method being the Split-Step Fourier Method (SSFM) [1]. Analyticalapproximations to the solution of (2) can be obtained by linearization tech-niques [5–12], such as perturbation methods taylored for modulation instabil-ity (or parametric gain) [5–8] or of more general validity [9,10], small-signalanalysis [11], and the variational method [12]. An approach based on Volterraseries [13] was recently shown to be equivalent to the regular perturbationmethod[9]. However, all methods able to deal with an arbitrarily modulatedinput signal, provide accurate approximations either only for very small inputpowers or only for very small fiber losses, with the exception of the enhancedregular perturbation methodpresented in [9] and the multiplicative approxi-mation introduced in [10], whose results are valid for input powers as high asabout 10 dBm. We present here two recursiveexpressionsthat, starting fromthe linear solution of (2) for asymptotically converge to the exact solu-tion for and revisit the multiplicative approximation in [10], relating itto the regular perturbation method.2.AN INTEGRAL EXPRESSION OF THE NLSEIn this Section we willobtain an integralexpression of the NLSE which, toour knowledge, is not found in the literature. Letting(2), we obtainand taking the Fourier transform1 of which, by the positionbecomesIntegrating (6) from 0 to leads toand, taking into account (5), we have1The Fourier transform with respect to time of a function letter such that andwill be denoted by the same but capitalSolving the Nonlinear Schrödinger Equation5where is theFourier transform of the solutionof (2) forLetting now so thatand antitransforming (8) by taking into account (3), giveswhere denotes temporal convolution, and signal at in a linear and lossless fiber.is the3.A FIRST RECURRENCE RELATIONCORRESPONDING TO A REGULARPERTURBATION METHODAccording to the regular perturbation(RP) method [9], expanding the opti-cal field complex envelope in power series inand substituting (10) in (9), after some algebra we obtainwhere we omitted the arguments for the appearing on the left side,and for those on the rightside. By equating the powers in with thesame exponent, we can recursively evaluate all theAs an example, the first three turn out to be6E. Forestieri and M. SecondiniTurning again our attention to (9), we note that it suggests the following recur-rence relationand it is easy to see thatas it can be shown thatThis means that the rate of convergence of (13) is not greaterthan that of (10)when using the same number of terms as the recurrence steps, i.e., it is poor [9].We will now seek an improved recurrence relation with an accelerated rate ofconvergence to the solution of (2).4.AN IMPROVED RECURRENCE RELATIONCORRESPONDING TO A LOGARITHMICPERTURBATION METHODAs shown in [10], a faster convergence rate is obtained when expanding inpower series in the log of ratherthan itself as done in (10). So,we try to recast (9) in terms of log and to this end rewrite it asUsing now the expansionwe replace the term in (16), obtainingwhere, for simplicity, we omitted all the function arguments. So, the soughtinproved recurrence relation suggested by (18) isSolving the Nonlinear Schrödinger Equation7

where we used again (17) to obtain the rightside of the second equation. Alsoin this case as it can be shown that the power seriesin of log coincides with that of log in the first terms.Notice that evaluated from (19) coincides with the first-order multi-plicative approximation in [10], there obtained with a different approach. Themethod in [10] is really a logarithmicperturbation (LP) method as the solutionis written asand are evaluated by analytically approximating the SSFMsolution. The calculation of becomes progressively more involved forincreasingvalues of but that method is useful because it can provide ananalytical expression for the SSFM errors due to a finite step size [10].We now follow anotherapproach. Lettingsuch thatfor every can be easily evaluated in the following manner. Thepower series expansion of in (22) iswhereEquating(10) to (23), and taking into account that we can recur-sively evaluate the as8E. Forestieri and M. SecondiniThus, from the order regular perturbationapproximation we can constructthe order logarithmicperturbation approximation. As an example we haveSo, once evaluated the from (12), we can evaluate the from (25)and then through (22), unless is zero (or very small), in whichcase we simply use (10) as in this case is also small and (10) is equallyaccurate.5.COMPUTATIONAL ISSUESThe computational complexity of (12), (13) and (19) is the same, and at firstglance it may seem that a order integral must be computed for theorder approximation. However, it is not so and the complexity only increaseslinearly withIndeed, the terms depending on can be taken out of the2integration and so all the integrals can be computed in parallel. However,only for these methods turn out to be faster than the SSFM becauseof the possibility to exploit efficient quadrature rules for the outer integral,whereas the inner ones are to be evaluated through the trapezoidal rule as, toevaluate them in parallel, we are forced to use the nodes imposed by the outerquadraturerule.Although(12), (13), (19), and (22) hold for a single fiber span, they canalso be used in the case of many spans with given dispersionmaps and per-span amplification. Indeed, one simply considers the output signal at the endof each span as the input signal to the next span [9, 10]. We would like to pointout that even if the propagation in the compensatingfiber is considered to belinear,(19) or (22) shouldstill be used for the total span length L, by simplyreplacing with the length of the transmissionfiber in the upper limit ofintegration and with L in all other places.This is apparent when performing the integrals in thefrequencydomain, but is also true in the time domainas when is the impulse response of a linear fiber of lengthsimply corresponds to a fiber of length and opposite sign of dispersion parameter).2Solving the Nonlinear Schrödinger Equation96.SOME RESULTSTo illustrate the results obtainable by the RP and LP methods, we considereda link, composed of100 km spans of transmission fiber followedby a compensating fiber and per span amplification recovering all the span loss.The transmission fiber is a standard single-mode fiber withD = 17 ps/nm/km, whereas the compensating fiber hasD = –100 ps/nm/km, and a length suchthat the residual dispersion per span is zero.In Table 1 we report the minimum order of the RP and LP methods necessaryto have a normalized square deviation (NSD) less than The NSD isdefined aswhere is the solution obtained by the SSFM with a step size of100 m,is either the RP or LP approximation, and the integrals extendto the whole transmission period, which in our case is that corresponding to apseudorandom bit sequence of length 64 bits. The input signal format is NRZat 10 Gb/s, filtered by a Gaussian filter with bandwidth equal to 20 GHz.It can be seen that the LP method requires a lower order than the RP methodto achieve the same accuracy when the input peak power increases beyond6 dBm and the number of spans execeeds 4. As an example, Fig. 1 shows theoutput intensity for an isolated “1” in the pseudorandom sequence when theinput peak power is 12 dBm and the number of spans is 5, showing that, inthiscase, 3rd-order is required for the RP method, whereas only 2nd-order forthe LP method. As a matter of fact, until 12 dBm and 8 spans, the 2nd-orderLP method suffices for a However, for higher values of10E. Forestieri and M. SecondiniFigure 1. Output intensity for an isolated “1” with and 5 spans.and number of spans, i.e., when moving form left to right along a diagonal inTable 1, the two methods tend to become equivalent, in the sense that they tendto require the same order to achieve a given accuracy.This can be explained by making the analytical form(19) of the NLSE so-lution explicit. Indeed, doing so we can see that the nonlinear parameterappears at the exponent, and then at the exponent of the exponent, and then atthe exponent of the exponent of the exponent, and so on. So, the LP approxi-mation has an initial advantage over the RP one, but when orders higher than 3or 4 are needed, this initial advantage is lost and the two approximations tendto coincide.7.CONCLUSIONSWe presented two recurrence relationsthat asymptotically approach the so-lution of the NLSE. 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