Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 23 March 2021 doi:10.20944/preprints202103.0558.v1 Article Parallel simulation of a multi-span DWDM system limited by FWM Rafael Sanchez-Lara 1 , Jose Luis Lopez-Martinez 2 , Joel Antonio Trejo-Sanchez 3 ,Herman Leonard Offerhaus 4 and Jose Alfredo Alvarez-Chavez 4* 1 Universidad Autonoma del Carmen (UNACAR), Facultad de Ingenieria, C.56, No. 4, C.P. 24180, Cd. Del Carmen, Campeche, Mexico; [email protected] 2 Facultad de Matematicas, Universidad Autonoma de Yucatan,Merida, Mexico; [email protected] 3 CONACYT-Centro de Investigación en Matemáticas; [email protected] 4 Optical Sciences Group, University of Twente, Enschede, The Netherlands;[email protected]; [email protected] * Correspondence: [email protected] 1 Abstract: One of the non-linear phenomena affecting high bandwidth and long reach communica- 2 tion systems is Four-Wave Mixing (FWM). Unfortunately, the simulation of systems for parameter 3 optimization requires more time as the number of channels increases. In this paper, we propose a 4 new high-performance computational model to obtain the optimal design parameters in a multi- 5 span Dense Wavelength Division Multiplexing (DWDM) system, limited by FWM and the intrinsic 6 Amplified Spontaneous Emission (ASE) noise of optical amplifiers employed in each segment. The 7 simulation in this work provides a complete optical design characterization and compares the effi- 8 ciency and speed improvement of the proposed parallelization model versus an earlier sequential 9 model. Additionally, an analysis of the computational complexity of parallel models is presented, 10 in which two parallel implementations are used; firstly, Open MultiProcessing (OpenMP) which is 11 based on the use of a central multi-core processing unit and secondly, the Compute Unified Device 12 Architecture (CUDA), which is based on the use of a Graphics Processing Unit (GPU). Results 13 show that parallelism improves by up to 40 times the performance of a simulation when nested 14 parallelization with CUDA is used over a sequential method and up to 6 times compared with 15 the implementation with OpenMP using 12 processors. Within our parallel implementation, it Citation: Sanchez-Lara, R.; 16 is possible to simulate with an increased number of channels that which was impractical in the Lopez-Martinez, J.L.; Trejo-Sanchez, 17 sequential simulation. J. A.; Offerhaus, H.;Alvarez-Chavez,J.A. Parallel 18 Keywords: Computer Simulation; DWDM; FWM; GPU programming; Parallel Architectures; simulation of a multi-span DWDM 19 Parallel Programming system limited by FWM. Appl. Sci. 2021, 1, 0. https://doi.org/ Received: Accepted: 20 1. Introduction Published: 21 Non-linear phenomena occur in optical fibre systems especially when they are 22 used at their maximum . It is a limitation for multi-segment, high-bandwidth and long- Publisher’s Note: MDPI stays neu- 23 range Dense Wavelength Division Multiplexing (DWDM) systems. DWDM is used to tral with regard to jurisdictional 24 maximize the bandwidth capabilities of optical fibres where their segments are connected claims in published maps and insti- 25 by optical repeaters. Such communication systems are found in land or transoceanic tutional affiliations. 26 communication systems. Five non-linear effects that play a role in such systems are 27 known so far: Four-Wave Mixing (FWM), Stimulated Raman Scattering (SRS), Stimulated 28 Copyright: © 2021 by the authors. Brillouin Scattering (SBS), Self-Phase Modulation (SPM) and Cross-Phase Modulation Submitted to Appl. Sci. for possible 29 (XPM); each appears under certain design circumstances. In any design analysis, basic open access publication under the 30 linear effects will also be present such as fibre attenuation and dispersion, defined by terms and conditions of the Cre- 31 the parameter a and D, respectively. Erbium Doped Fiber Amplifiers (EDFAs) have ative Commons Attribution (CC 32 been used to compensate for fibre losses and to increase the transmission distance, BY) license (https://creativecom- 33 causing at the same time an increase on amplified spontaneous emission (ASE) noise mons.org/licenses/by/ 4.0/). 34 and nonlinearities [1,2].Different kinds of optical fibres have been designed to reduce the Version March 20, 2021 submitted to Appl. Sci. https://www.mdpi.com/journal/applsci © 2021 by the author(s). Distributed under a Creative Commons CC BY license. Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 23 March 2021 doi:10.20944/preprints202103.0558.v1 Version March 20, 2021 submitted to Appl. Sci. 2 of 13 35 effect of dispersion and nonlinear effects [3–6]. There are numerous studies on non-linear 36 impairments showing the importance and complexity of such phenomena for DWDM 37 [7–12]. 38 The sequential model presented in [13] can be used on a multi-span DWDM system 39 limited by FWM/ASE to obtain the optimal power (Popt) and maximum link length (Lmax) 40 with respect to the dispersion D. The model is highly parallelizable so that different 41 parallelization models can be used to reduce the computational complexity. Paralleliza- 42 tion can significantly reduce the processing or execution time of a sequential algorithm 43 [14],including the simulation of Non-Linear phenomena [15]. To implement paralleliza- 44 tion, several tools exist, such as MPI (Message Passing Interface) [16], OpenMP[17], and 45 recently GPU cards [18]. The latter are programmable using the libraries provided by 46 NVIDIA, in particular the CUDA-C language. Other examples using GPU cards are 47 mentioned in [19,20], where the use of this tool significantly improves the execution of 48 algorithms. 49 50 In this work we use two schemes to significantly reduce the time complexity of 51 DWDM and simulate more channels than obtained in [13].First, we show a multipro- 52 cessing parallel paradigm using OpenMP since it only uses the processors located in the 53 Central Processing Unit (CPU) and does not require a distributed system like MPI. The 54 efficiency has been proven [21], but it is limited by the number the cores in the processor. 55 Secondly,we use of Graphic Processing Units for the cost-benefit ratio [22–24].For CUDA, 56 we use Dynamic Parallelism (DM) [22]. Dynamic parallelism consists of hierarchical 57 kernels in a tree structure, where a thread is defined as a father kernel. This father kernel 58 can generate new child kernels to execute tasks in the GPU context. The height of the 59 tree depends on the characteristics of the GPU card. Finally, an analysis of both imple- 60 mentations using metrics such as speedup, efficiency and performance ratio is presented. 61 62 The paper is divided into the following sections: Section 2 describes the theory of a 63 multi-span DWDM system limited by FWM; Section 3 describes the proposed sequential 64 model of computation; in Section 4 architectural models are explained, in Section 5, 65 experimental, and simulated results are presented in terms of execution time, efficiency, 66 performance ratio, and speedup. Finally concluding remarks are presented in Section 6. 67 2. Theoretical Analysis 3 68 The power dependence of the refractive index is denoted by c and FWM, is de- 3 69 scribed using c . If three optical fields with carrier frequencies f1, f2, and f3 copropagate 70 inside the fiber simultaneously, FWM generates a fourth field whose frequency f4 is 71 related to other frequencies by a relation f4 = f1 ± f2 ± f3. All frequencies correspond- 72 ing to different plus and minus sign combinations are possible. In practice, most do 73 not grow to significance because of a phase-mis-matching. Frequency combinations of 74 the form f4 = f1 + f2 − f3 are troublesome for multichannel communication systems 75 since they can become nearly phase-matched when channel wavelengths lie close to the 76 zero-dispersion wavelength [1]. 77 A multi-span DWDM system refers to the multiplexing of multiple information 78 frequency channels ( f1, f2, ... , fN) in a single fibre with the aim of transmitting the 79 largest possible bandwidth. To achieve long ranges, the optical fibre is segmented 80 and connected by optical amplifiers as shown in Figure1. The operating bandwidth 81 of the amplifiers must match the spectral bandwidth of the combined channels to be 82 transmitted. The C band is the most commonly used for its low lossesand the EDFA fits 83 this system [2]. Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 23 March 2021 doi:10.20944/preprints202103.0558.v1 Version March 20, 2021 submitted to Appl. Sci. 3 of 13 Figure 1. Multi-span DWDM System setup 84 Figure1 represents the multi-span DWDM system setup. fi represents the i − th 85 frequency channel, LS is the length of a segment or the space between amplifiers (typical 86 values between 50 km and 100 km), N is the total number of channels, LT is the total 87 length of the system and M is the total number of segments. 88 The total spectral width in band C is W = fN − f1 = 3.75 THz (30nm). The multi- 89 channel density D f is a function of the total number of N channels and can be expressed 90 as: W D f = f − f − = . (1) i i 1 N − 1 91 The central frequency can be expressed as: f + f f = 1 N = 193THz[1550nm]. (2) 2 92 An FWM signal appears by mixing three frequency channels fi, fj and fk, so that is, 93 fn = fi + fj − fk(i, j 6= k). The power at fn is given by [11,12,25]: MLe f f 2 2 2 −aLS Pi,j,k = k PiPjPkhi,j,kdi,j,k( ) e , (3) Ae f f 32p3c3 94 k = c for h2cl , where l is wavelength, is the vacuum light speed, h is the core 95 refractive index, a is the linear loss coefficient,Pi, Pj, Pk are launched input power levels 96 per channel, Le f f = (1 − exp(−aLS)/a is the fibre effective length, Ae f f is the effective 97 area of the guided mode, d is the degeneracy factor (d = 3 for i = j, d = 6 in case i 6= j), 3 −14 3 −1 98 c is the third-order nonlinear electric susceptibility (6 × 10 m W ), and hi,j,k is the 99 FWM efficiency, which is given by: 2 −aLS a 4e 2 Dbi,j,k LS hi,j,k = ( )(1 + ) sin ( ), (4) 2 + 2 ( − −aLS )2 a Dbi,j,k 1 e 2 100 where Dbi,j,k is the phase mismatch that is expressed as: 2pl2 Dbi,j,k = ( )(j fi − fkjj fj − fkj) c (5) dD (D + (( f − f ) + ( f − f ))).