# Numerical Method for Coupled Nonlinear Schrödinger Equations in Few-Mode Fiber

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Computing Method for Complex Envelopes of Optical Wave Calculation

#### 2.1. CNSES for Few Modes in Dimensionless Form

_{i}—the complex envelope of the optical wave of the i-th mode, α

^{(i)}—attenuation of the i-th mode; β

^{(i)}

_{1}, β

^{(i)}

_{2}, β

^{(i)}

_{3}—the first, second and third order dispersion of the i-th mode respectively; γ

^{(i)}—parameter of nonlinearity for the i-th mode; C

_{i}

_{,m}, B

_{i}

_{,m}—coupling coefficients between the i-th and m-th modes; T

_{R}—Raman scattering parameter; ω

^{(i)}

_{0}—angular frequency of the i-th mode; z—coordinate along an optical fiber; t—time, j—imaginary one, T

_{N}—final time, and f

^{(i)}(t)—known functions at the initial time.

_{N}— time, and P—power, by relations:

#### 2.2. The Finite-Difference Scheme and Computing Scheme

^{(i)}(

**x**) terms to denote the right parts of Equation (4):

_{k}= Δξ⋅(k − 1) and n is used to denote the dimensionless time mesh points τ

_{n}= Δτ⋅(n − 1). The dimensionless parameter θ allows to attribute the value of right part of Equation (8) to an arbitrary mesh point with fractional indexes (k + θ, n − ½). The parameter θ ∈ [0, 1], defines the explicit θ = 0 or implicit θ ∈ (0, 1] finite-difference computing scheme. The recommendation for θ parameter choosing was given in [1]. There, it was approved that θ = ½, which leads to stable and reasonable results.

^{(i)}are attributed to middleware virtual mesh points between (k+ θ, n − ½).

^{(i)}and Φ2

^{(i)}, and a third-order partial derivative by time in Equation (4), require modification of Crank–Nicolson’s method [1,8]. The aim is to use implicit form for linear terms, and explicit form for nonlinear ones. Thus, we separate the F

^{(i)}on the sum of linear and nonlinear terms:

^{(i)}and non-linear N

^{(i)}terms at the k-layer:

^{(i)}on the (k + 1)-th layer, we can just replace k on (k + 1) in Equation (10). For the nonlinear terms on (k + 1)-th layer, we must use finite-difference definition in explicit form:

^{(i)}and Φ2

^{(i)}, while the implicit form is used for the linear terms. The nonlinear terms Φ1

^{(i)}and Φ2

^{(i)}in virtual grid point (k, n − ½) are written as:

^{(i)}

_{k}

_{+1,n}unknowns and does not depend on unknowns from other wave modes. It allows solving the linear equation system for the each i-th mode independently. The index n for each linear equation system for the i-th mode begins from 2 up to the N − 1, because the values of x

_{k}

_{+1,1}, x

_{k}

_{+1,2}, and x

_{k}

_{+1,N}are known due to Equation (2).

^{(i)}

_{k}

_{+1,n}(where n begins from 3 and increases to N − 1). Each equation system can be solved separately. These linear equation systems have four-diagonal matrix form, where in addition to the main diagonal, there are one “upper” and two “sub” diagonals. We use the modified Thomas tridiagonal algorithm, which allows us to transform this matrix to triangular form. The described refinement algorithm is used at each integration step. The definition of nonlinear terms (from previous k-th integration layer) explicitly contributes to the error in calculating the values of the new (k + 1)-th layer. The main goal of the proposed refinement algorithm is to introduce iteration of the computation process at each integration step, which corrects the explicit form of nonlinear terms [1].

#### 2.3. The Ultra-Short Pulse Evolution in Fiber

^{(1)}= α

^{(2)}= 0.2 dB∙m/km, β

^{(1)}= 4.294 × 10

^{−9}s/m, β

^{(2)}

_{1}= 4.290 × 10

^{−9}s/m, β

^{(1)}

_{2}= 3.600 × 10

^{−26}s

^{2}/m, β

^{(2)}

_{2}= 3.250 × 10

^{−26}s

^{2}/m, β

^{(1)}

_{3}= β

^{(2)}= 2.750 × 10

^{−41}s

^{3}/m, γ

^{(1)}= γ

^{(2)}= 3.600 × 10

^{−2}(m⋅W)

^{−1}, T

_{R}= 4.000 × 10

^{−15}s, ω

^{(1)}

_{0}= ω

^{(2)}

_{0}=2.3612 × 10

^{−15}s

^{−1}(λ = 798 nm). The single chirped Gauss pulse (chirp C = −0.4579) was put into the fiber−s input; pulse duration is 12 fs, peak power is P = 1.75 × 10

^{5}W. The pulse form is described as:

^{−4}d.u. were made. Usage of the automatic integration step correction algorithm was included, allowing to calculate the pulse evolution length up to ~2.5 mm. The maximum error was set as 10

^{−30}d.u. All calculations were made using a processor with double precision and 64-bit architecture.

## 3. The Phase Velocity or Phase Delay Calculation during the Wave Propagation

^{−1}[] denote direct and inverse Fourier transform operator.

_{p}and group delay τ

_{g}(fast time) terms are involved into consideration, which are defined from phase-frequency characteristics of the system [20]:

_{ch}with average frequency ϖ, and amplitude-frequency and phase-frequency characteristics (Equation (15)) are presented by Taylor power series of deferential frequencies (ω − ϖ) = Ω at the ϖ average frequency point:

_{g}(ϖ, L) parameter is commonly referred to as Group-Delay Dispersion (GDD), and the τ″

_{g}(ϖ, L) = φ′′′(ϖ, L) = TOD parameter is commonly referred to as Third-Oder Dispersion (TOD).

_{ef}in the time domain corresponds to the function exp(–jφ″(ϖ, L)Ω

^{2}/2) in the frequency domain at Ω

_{ch}T

_{ef}>> 1 (where T

_{ef}≈ φ″(ϖ,L)⋅Ω

_{ch}), according to Fourier transform property:

_{ch}T

_{ef}>> 1. The low frequencies are shown by red lines, and high frequencies are shown by blue lines.

_{0}= const, and:

_{ch}T

_{ef}>> 1 and other than zero at delays τ ∈ [τ

_{g}(ϖ, L) − T

_{ef}/2; τ

_{g}(ϖ, L) + T

_{ef}/2).

^{2}/2. Therefore, if the condition Ω

^{2}/2ω′ + φ″(ϖ, L

_{a})·Ω

^{2}/2 = 0 is fulfilled at given communication line length L = L

_{a}, the linear frequency modulated pulse is matched with dispersal radio channel and it compresses over the fast time. This condition can be rewritten in the form ω′·φ″(ϖ, L

_{a}) = −1. It means that the submitted derivatives must be reciprocal in magnitude and opposite in signs. Thus, for fiber with normal dispersion φ″(ϖ, L) > 0, the frequency changing rate of the optical pulse must be negative ω′ < 0, while for the fiber with anomalous dispersion, it must have a positive sign.

_{a}.

_{c2}with second-order dispersion from the equivalence |φ″(ϖ, L)(Ω

_{c2}/2)

^{2}| = 1, choosing as the line length L = L

_{e2}the value, for which Ω

_{c2}= Ω

_{ch}. Therefore, the communication line length can be evaluated by the relation:

_{e2}, dimensionless second-order dispersion coefficient, and dimensionless frequency η = 2Ω/Ω

_{ch}), where second-order dispersion is described by the following relation:

_{C3}= Ω

_{ch}: L

_{e3}= 6c/|(ϖn(ϖ))″′|(Ω

_{ch}/2)

^{3}. In this case, the dimensionless communication line length is m = L/L

_{e3}, the dimensionless dispersion coefficient is p

_{e3}= Ω

_{ch}/Ω

_{C3}= 1, and dimensionless frequency is η = 2Ω/Ω

_{ch}∈ [−1, 1]. The third-order nonlinear component at given dimensionless values for phase equals:

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Sakhabutdinov, A.Z.; Anfinogentov, V.I.; Morozov, O.G.; Burdin, V.A.; Bourdine, A.V.; Gabdulkhakov, I.M.; Kuznetsov, A.A. Original Solution of Coupled Non-linear Schrödinger Equations for Simulation of Ultrashort Optical Pulse Propagation in a Birefringent Fiber. Fibers
**2020**, 8, 34. [Google Scholar] [CrossRef] - Samad, R.; Courrol, L.; Baldochi, S.; Vieira, N. Ultrashort Laser Pulses Applications, Coherence and Ultrashort Pulse Laser Emission; Duarte, F.J., Ed.; IntechOpen: London, UK, 2010; pp. 663–688. [Google Scholar] [CrossRef] [Green Version]
- Sugioka, K.; Cheng, Y. Ultrafast lasers—Reliable tools for advanced materials processing. Light Sci. Appl.
**2014**, 3, e149. [Google Scholar] [CrossRef] - Sugioka, K. Progress in ultrafast laser processing and future prospects. Nanophotonics
**2017**, 6, 393–413. [Google Scholar] [CrossRef] [Green Version] - Hodgson, N.; Laha, M. Industrial Femtosecond Lasers and Material Processing; Industrial Laser Solutions, PennWell Publishing: Tulsa, OK, USA, 2019. [Google Scholar]
- Göbel, W.; Nimmerjahn, A.; Helmchen, F. Distortion-free delivery of nanojoule femtosecond pulses from a Ti:sapphire laser through a hollow-core photonic crystal fiber. Opt. Lett.
**2004**, 29, 1285–1287. [Google Scholar] [CrossRef] [PubMed] - Michieletto, M.; Lyngsø, J.K.; Jakobsen, C.; Lægsgaard, J.; Bang, O.; Alkeskjold, T.T. Hollow-core fibers for high power pulse delivery. Opt. Express
**2016**, 24, 7103–7119. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Sakhabutdinov, A.Z.; Anfinogentov, V.I.; Morozov, O.G.; Gubaidullin, R.R. Numerical approaches to solving a nonlinear system of Schrödinger equations for wave propagation in an optical fiber. Comput. Technol.
**2020**, 25, 42–54. [Google Scholar] [CrossRef] - Karasawa, N.; Nakamura, S.; Morita, R.; Shigekawa, H.; Yamashita, M. Comparison between theory and experiment of non-linear propagation for 4.5-cycle optical pulses in a fused-silica fiber. Nonlinear Opt.
**2000**, 24, 133–138. [Google Scholar] - Nakamura, S.; Li, L.; Karasawa, N.; Morita, R.; Shigekawa, H.; Yamashita, M. Measurements of Third-Order Dispersion Effects for Generation of High-Repetition-Rate, Sub-Three-Cycle Transform-Limited Pulses from a Glass Fiber. Jpn. J. Appl. Phys.
**2002**, 41, 1369–1373. [Google Scholar] [CrossRef] [Green Version] - Nakamura, S.; Koyamada, Y.; Yoshida, N.; Karasawa, N.; Sone, H.; Ohtani, M.; Mizuta, Y.; Morita, R.; Shigekawa, H.; Yamashita, M. Finite-difference time-domain calculation with all parameters of Sellmeier’s fitting equation for 12-fs laser pulse propagation in a silica fiber. IEEE Photonics Technol. Lett.
**2002**, 14, 480–482. [Google Scholar] [CrossRef] [Green Version] - Nakamura, S.; Takasawa, N.; Koyamada, Y. Comparison between finite-difference time-domain calculation with all parameters of Sellmeier’s fitting equation and experimental results for slightly chirped 12-fs laser pulse propagation in a silica fiber. J. Light. Technol.
**2005**, 23, 855–863. [Google Scholar] [CrossRef] - Nakamura, S.; Takasawa, N.; Koyamada, Y.; Sone, H.; Xu, L.; Morita, R.; Yamashita, M. Extended Finite Difference Time Domain Analysis of Induced Phase Modulation and Four-Wave Mixing between Two-Color Femtosecond Laser Pulses in a Silica Fiber with Different Initial Delays. Jpn. J. Appl. Phys.
**2005**, 44, 7453–7459. [Google Scholar] [CrossRef] - Nakamura, S. Comparison between Finite-Difference Time-Domain Method and Experimental Results for Femtosecond Laser Pulse Propagation. Coherence Ultrashort Pulse Laser Emiss.
**2010**, 442–449. [Google Scholar] [CrossRef] [Green Version] - Burdin, V.A.; Bourdine, A.V. Simulation results of optical pulse non-linear few-mode propagation over optical fiber. Appl. Photonics
**2016**, 3, 309–320. (In Russian) [Google Scholar] [CrossRef] - Burdin, V.A.; Bourdine, A.V. Model for a few-mode nonlinear propagation of optical pulse in multimode optical fiber. In Proceedings of the OWTNM, Warsaw, Poland, 20–21 May 2016. [Google Scholar]
- Ivanov, V.A.; Ivanov, D.V.; Ryabova, N.V.; Ryabova, M.I.; Chernov, A.A.; Ovchinnikov, V.V. Studying the Parameters of Frequency Dispersion for Radio Links of Different Length Using Software-Defined Radio Based Sounding System. Radio Sci.
**2019**, 54, 34–43. [Google Scholar] [CrossRef] [Green Version] - Agrawal, G.P. Nonlinear Fiber Optics, 4th ed.; Academic Press: San Diego, CA, USA, 2006. [Google Scholar]
- Xiao, Y.; Maywar, D.N.; Agrawal, G.P. New approach to pulse propagation in nonlinear dispersive optical media. J. Opt. Soc. Am. B
**2012**, 29, 2958–2963. [Google Scholar] [CrossRef] - Ivanov, D.V. Methods and Mathematical Models for Studying Propagation of Spread Spectrum Signals in the Ionosphere and Correction for Their Dispersion Distortions: Monograph; MarSTU: Yoshkar-Ola, Russia, 2006; 268p. (In Russian) [Google Scholar]

**Figure 2.**The chirplets with rectangular duration window in the case of normal and anomalous dispersions and its pulse characteristics at soliton mode during propagation in the medium.

**Figure 3.**The computing results of the compression of line frequency-modulated pulse with the Gauss form in the fiber with second-order dispersion in dimensionless values. The effect reaches its maximum at communication link length L = 50⋅L

_{e2}, with 10 dB gain.

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**MDPI and ACS Style**

Sakhabutdinov, A.Z.; Anfinogentov, V.I.; Morozov, O.G.; Burdin, V.A.; Bourdine, A.V.; Kuznetsov, A.A.; Ivanov, D.V.; Ivanov, V.A.; Ryabova, M.I.; Ovchinnikov, V.V.;
et al. Numerical Method for Coupled Nonlinear Schrödinger Equations in Few-Mode Fiber. *Fibers* **2021**, *9*, 1.
https://doi.org/10.3390/fib9010001

**AMA Style**

Sakhabutdinov AZ, Anfinogentov VI, Morozov OG, Burdin VA, Bourdine AV, Kuznetsov AA, Ivanov DV, Ivanov VA, Ryabova MI, Ovchinnikov VV,
et al. Numerical Method for Coupled Nonlinear Schrödinger Equations in Few-Mode Fiber. *Fibers*. 2021; 9(1):1.
https://doi.org/10.3390/fib9010001

**Chicago/Turabian Style**

Sakhabutdinov, Airat Zh., Vladimir I. Anfinogentov, Oleg G. Morozov, Vladimir A. Burdin, Anton V. Bourdine, Artem A. Kuznetsov, Dmitry V. Ivanov, Vladimir A. Ivanov, Maria I. Ryabova, Vladimir V. Ovchinnikov,
and et al. 2021. "Numerical Method for Coupled Nonlinear Schrödinger Equations in Few-Mode Fiber" *Fibers* 9, no. 1: 1.
https://doi.org/10.3390/fib9010001