Fiber parameters.
Abstract
Firstly, based on the small-signal analysis theory, the nonlinear Schrodinger equation (NLSE) with fiber loss is solved. It is also adapted to the NLSE with the high-order dispersion terms. Furthermore, a general theory on cross-phase modulation (XPM) intensity fluctuation which adapted to all kinds of modulation formats (continuous wave, non-return-to-zero wave, and return-zero pulse wave) is presented. Secondly, by the Green function method, the NLSE is directly solved in the time domain. It does not bring any spurious effect compared with the split-step method in which the step size has to be carefully controlled. Additionally, the fourth-order dispersion coefficient of fibers can be estimated by the Green function solution of NLSE. The fourth-order dispersion coefficient varies with distance slightly and is about 0.002 ps4/km, 0.003 ps4/nm, and 0.00032 ps4/nm for SMF, NZDSF, and DCF, respectively. In the zero-dispersion regime, the higher-order nonlinear effect (higher than self-steepening) has a strong impact on the short pulse shape, but this effect degrades rapidly with the increase of β 2. Finally, based on the traveling wave solution of NLSE for ASE noise, the probability density function of ASE by solving the Fokker-Planck equation including the dispersion effect is presented.
Keywords
- small-signal analysis
- Green function
- traveling wave solution
- Fokker-Planck equation
- nonlinear Schrodinger equation
1. Introduction
The numerical simulation and analytical models of nonlinear Schrödinger equation (NLSE) play important roles in the design optimization of optical communication systems. They help to understand the underlying physics phenomena of the ultrashort pulses in the nonlinear and dispersion medium.
The inverse scattering [1], variation, and perturbation methods [2] could obtain the analytical solutions under some special conditions. These included the inverse scattering method for classical solitons [3], the dam-break approximation for the non-return-to-zero pulses with the extremely small chromatic dispersion [4], and the perturbation theory for the multidimensional NLSE in the field of molecular physics [5]. When a large nonlinear phase was accumulated, the Volterra series approach was adopted [6]. With the assumption of the perturbations, the NLSE with varying dispersion, nonlinearity, and gain or absorption parameters was solved in [7]. In [8], the generalized Kantorovitch method was introduced in the extended NLSE. By introducing Rayleigh’s dissipation function in Euler-Lagrange equation, the algebraic modification projected the extended NLSE as a frictional problem and successfully solved the soliton transmission problems [9].
Since the numerical computation of solving NLSE is a huge time-consuming process, the fast algorithms and efficient implementations, focusing on (i) an accurate numerical integration scheme and (ii) an intelligent control of the longitudinal spatial step size, are required.
The finite differential method [10] and the pseudo-spectral method [11] were adopted to increase accuracy and efficiency and suppress numerically induced spurious effects. The adaptive spatial step size-controlling method [12] and the predictor-corrector method [13] were proposed to speed up the implementation of split-step Fourier method (SSFM). The cubic (or higher order) B-splines were used to handle nonuniformly sampled optical pulse profiles in the time domain [14]. The Runge-Kutta method in the interaction picture was applied to calculate the effective refractive index, effective area, dispersion, and nonlinear coefficients [15].
Recently, the generalized NLSE, taking into account the dispersion of the transverse field distribution, is derived [16]. By an inhomogeneous quasi-linear first-order hyperbolic system, the accurate simulations of the intensity and phase for the Schrödinger-type pulse propagation were obtained [17]. It has been demonstrated that modulation instability (MI) can exist in the normal GVD regime in the higher-order NLSE in the presence of non-Kerr quintic nonlinearities [18].
In this chapter, several methods to solve the NLSE will be presented: (1) The small-signal analysis theory and split-step Fourier method to solve the coupled NLSE problem, the MI intensity fluctuation caused by SPM and XPM, can be derived. Furthermore, this procedure is also adapted to NLSE with high-order dispersion terms. The impacts of fiber loss on MI gain spectrum can be discussed. The initial stage of MI can be described, and then the whole evolution of MI can also be discussed in this way; (2) the Green function to solve NLSE in the time domain. By this solution, the second-, third-, and fourth-order dispersion coefficients is discussed; and (3) the traveling wave solution to solve NLSE for ASE noise and its probability density function.
2. Small-signal analysis solution of NLSE for MI generation
2.1 Theory for continuous wave
The NLSE governing the field in nonlinear and dispersion medium is
where
where
Usually, the field amplitudes can be written as
Assuming:
The amplitude
The small-signal theory implies that the frequency modulation or noise
The operation
The modulation or noise
So
And
When only intensity modulation is present and no phase modulation exists, the transfer function
This is in very good agreement with [24] for small-phase modulation index. Even for large modulation index
Obviously, the above process can be used to treat NLSE with higher-order dispersion (
The corresponding MI gain
Figure 2
shows a comparison of the gain spectra between Eq. (11) and [6] for the case
2.2 The general theory on cross-phase modulation (XPM) intensity fluctuation
For the general case of two channels, the input optical powers are denoted by
This phase shift is converted to an intensity fluctuation through the group velocity dispersion (GVD) from
The walk-off between co-propagating waves is regulated by the convolution operation.
3. Green function method for the time domain solution of NLSE
3.1 NLSE including the resonant and nonresonant cubic susceptibility tensors
From Maxwell’s equation, the field in fibers satisfies
where
There are
Repeating the process of [3]
where
3.2 The solution by Green function
The solution has the form
Then, there is
Let
and taking the operator
Assuming
Its characteristic roots are
where
By the construction method, it is
At the point
Let
Finally, the solution of (27) can be written with the eigen function and Green function:
The accuracy can be estimated by the last item of (40). The algorithm is plotted in Figure 3 .
3.3 Estimation of the fourth-order dispersion coefficient
β
4
The NLSE governing the wave’s transmission in fibers is
where
where
Constructing the iteration
where
The minimum value of
Next, we take the higher-order nonlinear effect into account. Constructing another iteration related to
Now, we can simulate the pulse shape affected by high-order dispersive and nonlinear effects. Assume
Firstly, we see what will be induced by the above items
Is the pulse split in
Figure 4(a)
caused by
From the deviation between the red and black lines in
Figure 5
, we can also detect the impact of
Generally, we do not take
So, we can utilize
|
|
|
|
|
|
---|---|---|---|---|---|
DCF | 0.59 | 5.5 | 0.01 | 110 | 0.1381 |
NZDSF | 0.21 | 2.2 | 0.01 | −5.6 | 0.115 |
SMF | 0.21 | 1.3 | 0.01 | −21.7 | −0.5 |
Table 2
is the average of
|
|
|
|
---|---|---|---|
DCF | 0.0003 | 0.00035 | 0.00032 |
NZDSF | 0.0022 | 0.003 | 0.0032 |
SMF | 0.0012 | 0.002 | 0.0025 |
4. Traveling wave solution of NLSE for ASE noise
4.1 The in-phase and quadrature components of ASE noise
The field including the complex envelopes of signal and ASE noise is:
where
Substituting Eq. (47) into (1), we can get the equation that
So, the in-phase and quadrature components of ASE noise obey:
We now seek their traveling wave solution by taking [37]
Then, (49) and (50) are converted into
(52) is differentiated to
Replacing
From (51) and (54), we can easily obtain
and
In the above calculation process,
4.2 Probability density function of ASE noise
Because
Here,
Now, they can be regarded as the stationary equations, and we can gain their probabilities according to Sections (7.3) and (7.4) in [39]. By solving the corresponding Fokker-Planck equations of (60) and (61), the probabilities of ASE noise are
(66) and (67) are efficient in the models of Gaussian and correlated non-Gaussian processes as our (49) and (50). Obviously, the Gaussian distribution has been distorted. They are no longer symmetrical distributions, and both have phase shifts consistent with [40], and as its authors have expected that “if the dispersion effect was taken into account, the asymmetric modulation side bands occur.” The reasons are that item
5. Conclusion
NLSE is solved with small-signal analyses for the analyses of MI, and it can be broadened to all signal formats. The equation can be solved by introducing the Green function in the time domain, and it is used as the tool for the estimations of high-order dispersion and nonlinear coefficients. For the conventional fibers, SMF, NZDSF, and DCF, the higher-order nonlinear effect contribution to
By the traveling wave methods, the p.d.f. of ASE noise can be obtained, and it provides a method for the calculation of ASE noise in WDM systems. So, the properties of MI, pulse fission, coefficient value, and ASE noise’s probability density function are also discussed for demonstrations of the theories.
References
- 1.
Hasegawa A, Matsumoto M, Kattan PI. Optical Solitons in Fibers. 3rd ed. New York: Springer-Verlag; 2000 - 2.
Brandt-Pearce M, Jacobs I, Shaw JK. Optimal input Gaussian pulse width for transmission in dispersive nonlinear fiber. Journal of the Optical Society of America B. 1999; 16 :1189-1196. DOI: 10.1364/JOSAB.16.001189 - 3.
Agrawal GP. Nonlinear Fiber Optics. 4th ed. San Diego, CA: Academic; 2007 - 4.
Kodama Y, Wabnitz S. Analytical theory of guiding center nonreturn to zero and return to zero signal transmission in normally dispersive nonlinear optical fibers. Optics Letters. 1995; 20 :2291-2293. DOI: 10.1364/OL.20.002291 - 5.
Surján PR, Ángyán J. Perturbation theory for nonlinear time-independent Schrödinger equations. Physical Review A–Physical Review Journals. 1983; 28 :45-48. DOI: 10.1103/PhysRevA.28.45 - 6.
Peddanarappagari KV, Brandt-Pearce M. Volterra series transfer function of single-mode fibers. Journal of Lightwave Technology. 1997; 15 :2232-2241. DOI: 10.1109/50.643545 - 7.
Serkin VN, Hasegawa A. Novel soliton solutions of the nonlinear Schrödinger equation model. Physical Review Letters. 2000; 85 :4502-4505. DOI: 10.1103/PhysRevLett.85.4502 - 8.
Cerda SC, Cavalcanti SB, Hickmann JM. A variational approach of nonlinear dissipative pulse propagation. European Physical Journal D. 1998; 1 :313-316. DOI: 10.1007/s100530050 - 9.
Roy S, Bhadra SK. Solving soliton perturbation problems by introducing Rayleigh’s dissipation function. Journal of Lightwave Technology. 2008; 26 :2301-2322. DOI: 10.1109/JLT.2008.922305 - 10.
Chang Q, Jia E, Suny W. Difference schemes for solving the generalized nonlinear Schrodinger equation. Journal of Computational Physics. 1999; 148 :397-415. DOI: 10.1006/jcph.1998.6120 - 11.
Bosco G, Carena A, Curri V, Gaudino R, Poggiolini P, Bendedetto S. Suppression of spurious tones induced by the split-step method in fiber systems simulation. IEEE Photonics Technology Letters. 2000; 12 :489-491. DOI: 10.1109/68.841262 - 12.
Sinkin V, Holzlohner R, Zweck J, Menyuk CR. Optimization of the split-step Fourier method in modeling optical-fiber communication systems. Journal of Lightwave Technology. 2003; 21 :61-68. DOI: 10.1109/JLT.2003.808628 - 13.
Liu X, Lee B. A fast method for nonlinear Schrodinger equation. IEEE Photonics Technology Letters. 2003; 15 :1549-1551. DOI: 10.1109/LPT.2003.818679 - 14.
Premaratne M. Numerical simulation of nonuniformly time-sampled pulse propagation in nonlinear fiber. Journal of Lightwave Technology. 2005; 23 :2434-2442. DOI: 10.1109/JLT.2005.850770 - 15.
Dabas B, Kaushal J, Rajput M, Sinha RK. Nonlinear pulse propagation in chalcogenide As2Se3 glass photonic crystal fiber using RK4IP method. Applied Optics. 2011; 50 :5803-5811. DOI: 10.1364/AO.50.005803 - 16.
Pedersen MEV, Ji C, Chris X, Rottwitt K. Transverse field dispersion in the generalized nonlinear Schrodinger equation: Four wave mixing in a higher order mode fiber. Journal of Lightwave Technology. 2013; 31 :3425-3431. DOI: 10.1109/JLT.2013.2283423 - 17.
Deiterding R, Glowinski R, Oliver H, Poole S. A reliable split-step Fourier method for the propagation equation of ultra-fast pulses in single-mode optical fibers. Journal of Lightwave Technology. 2013; 31 :2008-2017. DOI: 10.1109/JLT.2013.2262654 - 18.
Choudhuri A, Porsezian K. Impact of dispersion and non-Kerr nonlinearity on the modulational instability of the higher-order nonlinear Schrodinger equation. Physical Review A. 2012; 85 :033820. DOI: 10.1103/PhysRevA.85.033820 - 19.
Huang J, Yao J. Small-signal analysis of cross-phase modulation instability in lossy fibres. Journal of Modern Optics. 2005; 52 :1947-1955. DOI: 10.1080/09500340500106717 - 20.
Wang J, Petermann K. Small signal analysis for dispersive optical fiber communication systems. Journal of Lightwave Technology. 1992; 10 :96. DOI: 10.1109/50.108743 - 21.
Huang W, Hong J. A coupled-mode analysis of modulation instability in optical fibers. Journal of Lightwave Technology. 1992; 10 :156-162. DOI: 10.1109/50.120570 - 22.
Meslener GJ. Chromatic dispersion induced distortion of modulated monochromatic light employing direct detection. IEEE Journal of Quantum Electronics. 1984. DOI: QE-20:1208-1216. DOI:10.1109/JQE.1984.1072286 - 23.
Koyama F, Suematsu Y. Analysis of dynamic spectral width of dynamic-single-mode (DSM) lasers and related transmission bandwidth of single-mode fibers. IEEE Journal of Quantum Electronics. 1985. DOI: QE-21:292-297. DOI:10.1109/JQE.1985.1072653 - 24.
Rchraplyvy A, Tkach RW, Buhl LL, Alferness RC. Phase modulation to amplitude modulation conversion of CW laser light in optical fibres. Electronics Letters. 1988; 22 :409-412. DOI: 10.1049/el:19860279 - 25.
Grudihin AB, Dianov EM, Korobkin DV, Prokhorov AM, Serkin VN. Decay of femtosecond pulses in single-mode fiber-optic waveguides. JETP Letters. 1987; 46 :221-225 - 26.
Agrawal GP, Baldeck PL, Alfano RR. Modulation instability induced by cross-phase modulation in optical fibers. Physical Review A. 1989; 39 :3406-3413. DOI: 10.1103/PhysRevA.39.3406 - 27.
Ciaramella E, Tamburrini M. Modulation instability in long amplified links with strong dispersion compensation. IEEE Photonics Technology Letters. 1999; 11 :1608-1610. DOI: 10.1109/ 68.806862 - 28.
Huang J, Yao J. Analysis of cross-phase modulation in WDM systems. Journal of Modern Optics. 2005; 52 :1819-1825. DOI: 10.1080/09500340500092016 - 29.
Cartaxo AVT. Impact of modulation frequency on crossphase modulation effect in intensity modulation-direct detection WDM systems. IEEE Photonics Technology Letters. 1998; 10 :1268-1270. DOI: 10.1109/68.705612 - 30.
Huang J, Yao J, Degang X. Green function method for the time domain simulation of pulse propagation. Applied Optics. 2014; 53 (16):-20 - 31.
Shen YR. The Principles of Nonlinear Optics. Hoboken, NJ: John Wiley & Sons, Inc; 1984 - 32.
Huang J, Yao J. Estimation of the fourth-order dispersion coefficient β4. Chinese Optics Letters. 2012; 10 :101903-101903. DOI: 10.3788/COL201210.101903 - 33.
Capmany J, Pastor D, Sales S, Ortega B. Effects of fourth-order dispersion in very high-speed optical time-division multiplexed transmission. Optics Letters. 2002; 27 :960-962. DOI: 10.1364/OL.27.000960 - 34.
Igarashi K, Saito S, Kishi M, Tsuchiya M. Broad-band and extremely flat super-continuum generation via optical parametric gain extended spectrally by fourth-order dispersion in anomalous-dispersion-flattened fibers. IEEE Journal of Selected Topics in Quantum Electronics. 2002; 8 :521-526. DOI: 10.1109/JSTQE.2002.1016355 - 35.
Gholami F, Chavez Boggio JM, Moro S, Alic N, Radic S. Measurement of ultra-low fourth order dispersion coefficient of nonlinear fiber by distant low-power FWM. IEEE Photonics Society Summer Topical Meeting Series. 2010; 2010 :162-163. DOI: 10.1109/PHOSST.2010.5553630 - 36.
Marhic ME, Wong KK-Y, Kazovsky LG. Wide-band tuning of the gain spectra of one-pump fiber optical parametric amplifiers. IEEE Journal of Selected Topics in Quantum Electronics. 2004; 10 :1133-1141. DOI: 10.1109/JSTQE.2004.835298 - 37.
Gui M, Jing H. Statistical analyses of ASE noise. Optics and Photonics Journal. 2017; 7 :160-169. DOI: 10.4236/opj.2017.710016 - 38.
Nichel J, Schurmann HW, Serov VS. Some elliptic travelling wave solution to the Nonikov-Vesela equation. In: Proceedings of the International Conference on Days on Diffraction, DD2005; 2005, 28–01; 2005. pp. 177-186 - 39.
Primak S, Kontorovich V, Lyandres V. Stochastic Method and Their Applications to Communications. Chichester, UK: John Wiley & Sons; 2004. Chap. 7. DOI:10.1002/0470021187 - 40.
Dlubek MP, Phillips AJ, Larkins EC. Nonlinear evolution of Gaussian ASE noise in ZMNL fiber. Journal of Lightwave Technology. 2008; 26 :891-898. DOI: 10.1109/JLT.2008.917373