A comparative examination of the crosstalk induced by XPM as a result of the 5OD coefficient in SCM-WDM communication system

A comparative examination of the crosstalk induced by XPM as a result of the 5OD coefficient in SCM-WDM communication system | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article A comparative examination of the crosstalk induced by XPM as a result of the 5OD coefficient in SCM-WDM communication system Vikram Singh, Naveen Chandra Joshi, Shikha Gupta, Yash Pathak This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7106552/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract This research investigates the impact of cross phase modulation (XPM) caused crosstalk due to higher order dispersion (HOD) within subcarrier multiplexed wavelength-division multiplexed (SCM–WDM) communication systems across different optical power levels, modulation frequencies and transmission lengths. We carried out an analytical comparison of XPM induced crosstalk due to 5 th order dispersion (5OD) coefficient using nonlinear Schrödinger equation (NLSE) and couple equations in SCM–WDM communication links. It is noted that when transmission lengths, optical powers, and modulation frequencies grow, XPM-induced crosstalk has been shown to increase exponentially. Though less significant than 2OD, the effects of 3OD, 4OD, and 5OD coefficients are nonetheless present. It can be further noted that the analysis of XPM crosstalk through NLSE is reliable compared to couple equations, as only a single approximation method has been utilized to reach this conclusion using NLSE. Consequently, the results derived from NLSE are definitive and precise. The XPM crosstalk associated with the 5OD parameter ranges from (-280dB to -255.09dB) when using couple equations, while it fluctuates between (-315dB to -270dB) when employing NLSE as the modulation frequency increases from 0.4 to 5GHz. Additionally, XPM crosstalk is observed to be between (-320dB to -281dB) with couple equations, whereas it escalates from (-346.03dB to -297dB) when using NLSE as the transmission length extends from 3 to 50 km. Furthermore, XPM crosstalk varies from (-270.1dB to -240.2dB) with couple equations, while it is found within the range of (-295.1dB to -255.06dB) using NLSE as the optical power changes from 0.1 to 2mW. Cross phase modulation Group velocity dispersion Subcarrier multiplexing Wavelength division multiplexing Higher order dispersion Nonlinear Schrödinger equation Figures Figure 1 Figure 2 Figure 3 1. INTRODUCTION The enormous demand for wireless communication services, driven by the increasing consumption of multimedia content, has created significant challenges for network operators. The combination of SCM-WDM is an economically viable method employed to enhance the bandwidth efficiency of optical fibers in communication systems. Network operators are finding it extremely challenging to handle the growing traffic as a result of wireless communication’s remarkable grow in recent years. A mature service with a high proportion of consumer use, a reduced and consistent access charge, full-time connectivity to service provider and increased bandwidth is required as the demand for multimedia services including data, voice and video continue to rise. A huge capacity is necessary for future wireless communication systems to meet the diverse needs. Micro cellular systems have been proposed as a solution to the convergent demands for high bandwidths and subscriber mobility. These systems can boost system capacity by improving the efficiency of reusing limited radio resources [ 1 ]. The microcellular system presents challenges since it takes time and money to install new radio base stations. Since it uses microwave photonics techniques to provide the so-called radio over fiber link, the combination of SCM and WDM is viewed as a potential solution to the issues presented by a microcellular system. However, nonlinear effects in fiber plague SCM–WDM systems. Crosstalk between subcarriers on different wavelengths can occur when numerous wavelengths carrying SCM signals propagate in a single cable due to fiber nonlinearities. XPM induced crosstalk is the predominant fiber nonlinearity that generates crosstalk in a dispersive fiber. Due to their tight spacing, adjacent SCM channels may experience large quantities of nonlinear crosstalk when subjected to cross phase modulation (XPM) [ 2 – 4 ]. The optical Kerr effect causes non-linear crosstalk in a dispersive optical fibre. One of the most prevalent non-linear effects is XPM. One of the most common uses of SCM-WDM technology in the optical realm is the transmission and distribution of analog video. The investigation of XPM crosstalk in SCM-WDM video links XPM crosstalk depends on the channel spacing, according to the [ 5 ] finding. Researchers looked into an enhanced model for determining XPM crosstalk in SCM-WDM links that takes impact of higher order dispersion (HOD) into account [ 6 ]. To evaluate XPM crosstalk in SCM-WDM links, the non-linear Schrodinger equation (NLSE) is utilized. It has been concluded that HOD significantly affects XPM crosstalk. As the order of dispersion term lowers, the impact of HOD parameters on XPM crosstalk grows. It has been examined how 2OD and 3OD cause SRS and XPM crosstalk in SCM-WDM communication systems [ 7 ]. The 2OD and 3OD coefficients are found to significantly affect crosstalk. It has been presented the manner in which HOD coefficients affect XPM crosstalk in SCM-WDM communication systems [ 8 , 9 ]. When the cumulative influence of all three HOD coefficients is taken into account, it is found that the 3OD, 4OD, and 5OD coefficients have a substantial impact, however the impact of the 2OD coefficient is considerable. In the SCM-WDM system, a comparative study of XPM crosstalk caused by the 2OD parameter is estimated [ 10 , 11 , 13 ]. A comparative analysis of the 3OD and 4OD coefficients-induced XPM crosstalk has been published [ 12 , 14 ]. The current study extends the work published by [ 14 ] and attempts to examine XPM crosstalk induced by XPM due to 5OD parameter. 2. MATHEMATICAL INVESTIGATIONS 2.1 Examination of XPM crosstalk through the use of NLSE Starting with the nonlinear wave propagation equation, the theoretical analysis begins [ 10 – 12 ]. Let us consider probe A j (t,z) and pump A k (t,z) waves are concurrently conveyed through the same fiber. $$\:\frac{\partial\:{\text{A}}_{\text{j}}\left(\text{z},\text{t}\right)}{\partial\:\text{z}}+\frac{{\alpha\:}\:}{2\:}{\text{A}}_{\text{j}}\left(\text{z},\text{t}\right)+\frac{1\:}{{\text{V}}_{\text{j}}}\frac{\partial\:{\text{A}}_{\text{j}}\left(\text{z},\text{t}\right)}{\partial\:\text{z}}+\frac{\text{i}{{\beta\:}}_{2}}{2}\frac{{\partial\:}^{2}{\text{A}}_{\text{j}}\left(\text{z},\text{t}\right)\:\:\:}{\partial\:{\text{t}}^{2}}-\frac{\text{i}{{\beta\:}}_{3}}{6}\frac{{\partial\:}^{3}{\text{A}}_{\text{j}}\left(\text{z},\text{t}\right)\:\:\:}{\partial\:{\text{t}}^{3}}-\frac{\text{i}{{\beta\:}}_{4}}{24}\frac{{\partial\:}^{4}{\text{A}}_{\text{j}}\left(\text{z},\text{t}\right)\:\:\:}{\partial\:{\text{t}}^{4}}+\frac{\text{i}{{\beta\:}}_{5}}{120}\frac{{\partial\:}^{5}{\text{A}}_{\text{j}}\left(\text{z},\text{t}\right)\:\:\:}{\partial\:{\text{t}}^{5}}=\:2i\gamma\:{P}_{k\:}\{t-{d}_{jk}\}{A}_{j}\left(\text{z},\text{t}\right)$$ 1 Where \(\:{\:d}_{jk}\) is relative walk off parameter \(\:{{\beta\:}}_{2},\:{{\beta\:}}_{3}\:{{\beta\:}}_{4}\:\text{a}\text{n}\text{d}\:\:{{\beta\:}}_{5}\:\) are 2OD, 3OD, 4OD and 5OD coefficients respectively, \(\:{\alpha\:}\) and \(\:\gamma\:\:\) are attenuation and coupling coefficients respectively here \(\:\:\gamma\:=\frac{2\pi\:{n}_{2}}{{\lambda\:}_{m}{A}_{eff}}\) , where symbols mean what they usually mean. In order to obtain XPM induced cross talk due 5OD coefficient put \(\:{{\beta\:}}_{2}={{\beta\:}}_{3}={{\beta\:}}_{4}=0\) in Eq. ( 1 ) we get $$\:\frac{\partial\:{\text{A}}_{\text{j}}\left(\text{z},\text{t}\right)}{\partial\:\text{z}}=-\frac{{\alpha\:}\:}{2\:}{\text{A}}_{\text{j}}\left(\text{z},\text{t}\right)\:-\:\frac{1\:}{{\text{V}}_{\text{j}}}\frac{\partial\:{\text{A}}_{\text{j}}\left(\text{z},\text{t}\right)}{\partial\:\text{z}}\:+\:\frac{{\text{i}{\beta\:}}_{4}}{24}\frac{{\partial\:}^{4}{\text{A}}_{\text{j}}\left(\text{z},\text{t}\right)\:\:\:}{\partial\:{\text{t}}^{4}}-\:\frac{\text{i}{{\beta\:}}_{5}}{120}\frac{{\partial\:}^{5}{\text{A}}_{\text{j}}\left(\text{z},\text{t}\right)\:\:\:}{\partial\:{\text{t}}^{5}}\:\:+\:2\:i\gamma\:{P}_{k\:}\{t-{d}_{jk}\}{A}_{j}\left(\text{z},\text{t}\right)$$ 2 In order to focus on inter channel cross talk Eq. ( 2 ) needs to be converted into frequency domain, therefore $$\:\frac{\partial\:{\text{A}}_{\text{j}}\left({\omega\:},\text{z}\right)}{\partial\:\text{z}}=[-\:\frac{{\alpha\:}\:}{2\:}-\:\frac{i{\omega\:}\:}{{\text{V}}_{\text{j}}\:}-\:\frac{\text{i}{{\omega\:}}^{5}{{\beta\:}}_{5}}{120}\:+\:i2\gamma\:{P}_{k\:}({\omega\:},0\left){\text{e}}^{\text{i}{\omega\:}\text{z}{d}_{jk}}{\text{e}}^{-{\alpha\:}\text{z}}\right]\:{\text{A}}_{\text{j}}\left({\omega\:},\text{z}\right)$$ 3 The last term in Eq. ( 3 ) converts phase noise into intensity noise in the probe signal. The phase noise in an incredibly tiny fiber segment, say dz, is provided as \(\:{d\varPhi\:}_{jk}\left(\omega\:,z\right)=2\gamma\:{P}_{k\:}({\omega\:},0){\text{e}}^{\left(-{\alpha\:}+\text{i}{\omega\:}{d}_{jk}\right)z}\text{d}\text{z}\) (4) Due to chromatic dispersion, phase noise generated at the fiber's output changes into intensity noise at the fiber's end, z = L. The overall phase noise at fiber output can be expressed as [ 10 ]. $$\:{\Delta\:}{\text{A}}_{\text{j}}\left({\omega\:},\text{L}\right)\:\:=\:2{P}_{j}\left(0\right){\text{e}}^{-\left({\alpha\:}-\frac{\text{i}\omega\:}{{\text{V}}_{\text{j}}}\right)L}\left[\underset{0}{\overset{\text{L}}{\int\:}}2\gamma\:{P}_{k\:}({\omega\:},0){\text{e}}^{\left(-{\alpha\:}+\text{i}{\omega\:}{d}_{jk}\right)z}\text{sin}\left\{\frac{{\omega\:}^{5}{{\beta\:}}_{5}\left(\text{L}-\text{z}\right)}{120}\right\}\}\right]\text{d}\text{z}$$ 5 Now assessing the integral \(\:\underset{0}{\overset{\text{L}}{\int\:}}\gamma\:{P}_{k\:}({\omega\:},0){\text{e}}^{\left(-{\alpha\:}+\text{i}{\omega\:}{d}_{jk}\right)z}\text{sin}\left\{\frac{{\omega\:}^{5}{{\beta\:}}_{5}\left(\text{L}-\text{z}\right)}{120}\right\}\}\text{d}\text{z}\) = \(\:\:{\gamma\:}{\text{P}}_{\text{k}\:}\left({\omega\:},0\right)[\frac{{\text{e}}^{\left\{{\text{i}{\omega\:}}^{5}{{\beta\:}}_{5}\frac{\text{L}}{120}\:\right\}}-{\text{e}}^{-\left({\alpha\:}-\text{i}{\omega\:}{\text{d}}_{\text{j}\text{k}}\right)\text{L}}}{\left\{\text{i}\left({\alpha\:}-\text{i}{\omega\:}{\text{d}}_{\text{j}\text{k}}\right)+(\text{i}.\text{i})\frac{{{\omega\:}}^{5}{{\beta\:}}_{5}}{120}\right\}}\) - \(\:\:\frac{{\text{e}}^{\left\{-{\text{i}{\omega\:}}^{5}{{\beta\:}}_{5}\frac{\text{L}}{120}\:\right\}}-{\text{e}}^{-\left({\alpha\:}-\text{i}{\omega\:}{\text{d}}_{\text{j}\text{k}}\right)\text{L}}}{\left\{\text{i}\left({\alpha\:}-\text{i}{\omega\:}{\text{d}}_{\text{j}\text{k}}\right)-(\text{i}.\text{i})\frac{{{\omega\:}}^{5}{{\beta\:}}_{5}}{120}\right\}}\) ] (6) Using Eqs. ( 5 ) & (6), we have Δ \(\:{\text{A}}_{\text{j}}\left({\omega\:},\text{L}\right)\:\:\) = \(\:\:2{P}_{j}\left(0\right){\text{e}}^{-\left({\alpha\:}-\frac{\text{i}\omega\:}{{\text{V}}_{\text{j}}}\right)L}{\gamma\:}{\text{P}}_{\text{k}\:}\left({\omega\:},0\right)[\frac{{\text{e}}^{\left\{{\text{i}{\omega\:}}^{5}{{\beta\:}}_{5}\frac{\text{L}}{120}\:\right\}}-{\text{e}}^{-\left({\alpha\:}-\text{i}{\omega\:}{\text{d}}_{\text{j}\text{k}}\right)\text{L}}}{\left\{\text{i}\left({\alpha\:}-\text{i}{\omega\:}{\text{d}}_{\text{j}\text{k}}\right)+(\text{i}.\text{i})\frac{{{\omega\:}}^{5}{{\beta\:}}_{5}}{120}\right\}}\) - \(\:\:\frac{{\text{e}}^{\left\{-{\text{i}{\omega\:}}^{5}{{\beta\:}}_{5}\frac{\text{L}}{120}\:\right\}}-{\text{e}}^{-\left({\alpha\:}-\text{i}{\omega\:}{\text{d}}_{\text{j}\text{k}}\right)\text{L}}}{\left\{\text{i}\left({\alpha\:}-\text{i}{\omega\:}{\text{d}}_{\text{j}\text{k}}\right)-(\text{i}.\text{i})\frac{{{\omega\:}}^{5}{{\beta\:}}_{5}}{120}\right\}}\) ] (7) Utilizing the subsequent relation now $$\:{\:P}_{j}\left(L\right)={P}_{j}\left(0\right){\text{e}}^{-{\alpha\:}\text{L}}$$ 8 From Eqs. (7) and ( 8 ), we have Δ \(\:{\text{A}}_{\text{j}}\left({\omega\:},\text{L}\right)\:\:\) = \(\:\:2{P}_{j}\left(L\right){\text{e}}^{\left(\frac{\text{i}\omega\:}{{\text{V}}_{\text{j}}}\right)L}{\gamma\:}{\text{P}}_{\text{k}\:}\left({\omega\:},0\right)[\frac{{\text{e}}^{\left\{{\text{i}{\omega\:}}^{5}{{\beta\:}}_{5}\frac{\text{L}}{120}\:\right\}}-{\text{e}}^{-\left({\alpha\:}-\text{i}{\omega\:}{\text{d}}_{\text{j}\text{k}}\right)\text{L}}}{\left\{\text{i}\left({\alpha\:}-\text{i}{\omega\:}{\text{d}}_{\text{j}\text{k}}\right)+(\text{i}.\text{i})\frac{{{\omega\:}}^{5}{{\beta\:}}_{5}}{120}\right\}}\) - \(\:\:\frac{{\text{e}}^{\left\{-{\text{i}{\omega\:}}^{5}{{\beta\:}}_{5}\frac{\text{L}}{120}\:\right\}}-{\text{e}}^{-\left({\alpha\:}-\text{i}{\omega\:}{\text{d}}_{\text{j}\text{k}}\right)\text{L}}}{\left\{\text{i}\left({\alpha\:}-\text{i}{\omega\:}{\text{d}}_{\text{j}\text{k}}\right)-(\text{i}.\text{i})\frac{{{\omega\:}}^{5}{{\beta\:}}_{5}}{120}\right\}}\) ] (9) $$\:\left|{\Delta\:}{\text{A}}_{\text{j}}\left({\omega\:},\text{L}\right)\right|=\left|2{P}_{j}\left(L\right){\gamma\:}{\text{P}}_{\text{k}\:}\left({\omega\:},0\right)[\frac{{\text{e}}^{\left\{{\text{i}{\omega\:}}^{5}{{\beta\:}}_{5}\frac{\text{L}}{120}\:\right\}}-{\text{e}}^{-\left({\alpha\:}-\text{i}{\omega\:}{\text{d}}_{\text{j}\text{k}}\right)\text{L}}}{\left\{\text{i}\left({\alpha\:}-\text{i}{\omega\:}{\text{d}}_{\text{j}\text{k}}\right)+(\text{i}.\text{i})\frac{{{\omega\:}}^{5}{{\beta\:}}_{5}}{120}\right\}}\:-\:\frac{{\text{e}}^{\left\{-{\text{i}{\omega\:}}^{5}{{\beta\:}}_{5}\frac{\text{L}}{120}\:\right\}}-{\text{e}}^{-\left({\alpha\:}-\text{i}{\omega\:}{\text{d}}_{\text{j}\text{k}}\right)\text{L}}}{\left\{\text{i}\left({\alpha\:}-\text{i}{\omega\:}{\text{d}}_{\text{j}\text{k}}\right)-(\text{i}.\text{i})\frac{{{\omega\:}}^{5}{{\beta\:}}_{5}}{120}\right\}}]\:\right|$$ A simple expression that characterizes the relative amplitude variation caused by XPM is [ 10 – 14 ] (10) $$\:{\Delta\:}{\text{P}}_{\text{j}}\left({\omega\:},\text{L}\right)={\left|\frac{{\text{A}}_{\text{j}}\left({\omega\:},\text{L}\right)}{{P}_{j}\left(L\right)}\right|}^{2}$$ 11 The final 5OD coefficient-related XPM crosstalk expression can be written as [ 10 ] $$\:{\Delta\:}{\text{P}}_{\text{j}}\left({\omega\:},\text{L}\right)={\left|2{\gamma\:}{\text{P}}_{\text{k}\:}\left({\omega\:},0\right)[\frac{{\text{e}}^{\left\{{\text{i}{\omega\:}}^{5}{{\beta\:}}_{5}\frac{\text{L}}{120}\:\right\}}-{\text{e}}^{-\left({\alpha\:}-\text{i}{\omega\:}{\text{d}}_{\text{j}\text{k}}\right)\text{L}}}{\left\{\text{i}\left({\alpha\:}-\text{i}{\omega\:}{\text{d}}_{\text{j}\text{k}}\right)+(\text{i}.\text{i})\frac{{{\omega\:}}^{5}{{\beta\:}}_{5}}{120}\right\}}\:-\:\frac{{\text{e}}^{\left\{-{\text{i}{\omega\:}}^{5}{{\beta\:}}_{5}\frac{\text{L}}{120}\:\right\}}-{\text{e}}^{-\left({\alpha\:}-\text{i}{\omega\:}{\text{d}}_{\text{j}\text{k}}\right)\text{L}}}{\left\{\text{i}\left({\alpha\:}-\text{i}{\omega\:}{\text{d}}_{\text{j}\text{k}}\right)-(\text{i}.\text{i})\frac{{{\omega\:}}^{5}{{\beta\:}}_{5}}{120}\right\}}]\right|}^{2}$$ 12 2.2 Examination of XPM crosstalk through the use of couple equations Assume that two wavelengths are concurrently transmitted over an optical fiber. The subsequent two couple equations illustrate XPM within the framework of a slowly varying envelope, as referenced in [ 8 – 14 ]. In this context, the higher-order dispersion term, specifically the fifth-order dispersion (5OD) coefficient, has been accounted for in the results pertaining to XPM-induced crosstalk. $$\:\frac{\partial\:{\text{A}}_{1}}{\partial\:\text{Z}}+\frac{1}{{\text{V}}_{{\text{g}}_{1}}}\frac{\partial\:{\text{A}}_{1}}{\partial\:\text{t}}=(-\text{j}\gamma\:{P}_{2\:}-\frac{{\alpha\:}\:}{2\:}){\text{A}}_{1}$$ 13 $$\:\frac{\partial\:{\text{A}}_{2}}{\partial\:\text{Z}}+\frac{1}{{\text{V}}_{{\text{g}}_{2}}}\frac{\partial\:{\text{A}}_{2}}{\partial\:\text{t}}=(-\text{j}\gamma\:{P}_{1\:}-\frac{{\alpha\:}\:}{2\:}){\text{A}}_{2}$$ 14 Where the slowly changing complex field envelops of each wave is shown by \(\:\:{\text{A}}_{\text{k}\:}\left(\text{t},\text{z}\right)(\text{k}=\text{1,2})\) , \(\:{\text{V}}_{{\text{g}}_{\text{i}},}(\text{i}=\text{1,2})\) is the carrier wave's group velocity, γ and α are nonlinearity & absorption coefficients. The power entering the fiber is expressed as [ 10 – 14 ] $$\:{P}_{i}={P}_{c}(1+mcos{{\omega\:}}_{1}t)$$ 15 Here \(\:{\text{P}}_{\text{i}}\) is input optical power at fiber’s input, m is modulation index, \(\:{\text{P}}_{\text{c}}\) is average optical power and \(\:{{\omega\:}}_{1}\) is optical frequency. The solution of Eq. ( 13 ) may be written as $$\:{\text{A}}_{1}\left(\text{z},\text{t}\right)={\text{A}}_{1}\left(0,{{\tau\:}}_{1}\right){\text{e}}^{\frac{-{\alpha\:}\text{z}\:}{2\:}}$$ 16 We used initial conditions as \(\:\text{z}=0,\:\text{t}=\) \(\:{{\tau\:}}_{1}\) and ignored γ in Eq. ( 13 ) to arrive at the aforementioned Eq. ( 16 ). Substituting the value of \(\:{\text{A}}_{1}\left(\text{z},\text{t}\right)\) from Eq. ( 16 ) to Eq. ( 14 ), we get $$\:{\text{A}}_{2}\left(\text{z},\text{t}\right)={\text{A}}_{2}\left(0,{{\tau\:}}_{2}\right){\text{e}}^{-\text{j}{\psi\:}}{\text{e}}^{-{\alpha\:}\text{z}/2}$$ 17 $$\:\text{W}\text{h}\text{e}\text{r}\text{e}\:\:{\psi\:}\:=-2{\gamma\:}\int\:{\text{P}}_{1}\{{0,({\tau\:}}_{2}+{\text{d}}_{21}\text{z})\}{\text{e}}^{-{\alpha\:}\text{z}}\text{d}\text{z}\:\text{a}\text{n}\text{d}\:{{\tau\:}}_{1}={{\tau\:}}_{2}+{\text{d}}_{21}\text{z},\:\text{w}\text{h}\text{e}\text{r}\text{e}\:{\:\text{d}}_{21}=\frac{1}{{\text{V}}_{{\text{g}}_{2}}}-\frac{1}{{\text{V}}_{{\text{g}}_{1}}}$$ We now consider that, as a result of GVD, phase modulation gets converted into intensity modulation through following relationship [ 11 – 14 ]. $$\:{\:P}_{2}\left(z,{\tau\:}_{2}\right)={P}_{2}\left(0,{\tau\:}_{2}\right){\left|[1+\text{j}\left\{\frac{{\partial\:}^{2}\psi\:\:\:}{\partial\:{\text{t}}^{2}}+\text{j}{\left(\frac{\partial\:\psi\:\:\:}{\partial\:\text{t}}\right)}^{2}\right\}{F}_{1}+1+\text{j}\{\frac{{\partial\:}^{3}\psi\:\:\:}{\partial\:{\text{t}}^{3}}-3\frac{{\partial\:}^{2}\psi\:\:\:}{\partial\:{\text{t}}^{2}}\frac{\partial\:\psi\:\:\:}{\partial\:\text{t}}-\text{j}{\left(\frac{\partial\:\psi\:\:\:}{\partial\:\text{t}}\right)}^{3}{\}F}_{2}+1+j\left\{-j\:\frac{{\partial\:}^{4}\psi\:\:\:}{\partial\:4}-4\frac{{\partial\:}^{3}\psi\:\:\:}{\partial\:{\text{t}}^{3}}\frac{\partial\:\psi\:\:\:}{\partial\:\text{t}}-3{\left(\frac{{\partial\:}^{2}\psi\:\:\:}{\partial\:{\text{t}}^{2}}\right)}^{2}+6j\frac{{\partial\:}^{2}\psi\:\:\:}{\partial\:{\text{t}}^{2}}\frac{\partial\:\psi\:\:\:}{\partial\:\text{t}}+j{\left(\frac{\partial\:\psi\:\:\:}{\partial\:\text{t}}\right)}^{4}\right\}{F}_{3\:}+1+j\{-j\frac{{\partial\:}^{5}\psi\:\:\:}{\partial\:{\text{t}}^{5}}-3\frac{{\partial\:}^{4}\psi\:\:\:}{\partial\:{\text{t}}^{4}}\frac{\partial\:\psi\:\:\:}{\partial\:\text{t}}-10\frac{{\partial\:}^{3}\psi\:\:\:}{\partial\:{\text{t}}^{3}}\frac{\partial\:\psi\:\:\:}{\partial\:\text{t}}+\:3\frac{{\partial\:}^{2}\psi\:\:\:}{\partial\:{\text{t}}^{2}}\frac{\partial\:\psi\:\:\:}{\partial\:\text{t}}-j{\left(\frac{\partial\:\psi\:\:\:}{\partial\:\text{t}}\right)}^{5}{\}F}_{4}]\right|}^{2}\:\:$$ 18 $$\:Here\:{F}_{1}=-{{\beta\:}}_{2}\frac{\text{Z}}{2}\:,\:{F}_{2}={-{\beta\:}}_{3}\frac{\text{Z}}{6}\:,\:\:{F}_{3}={-{\beta\:}}_{4}\frac{\text{Z}}{24}\:,\:{\:F}_{4}={-{\beta\:}}_{5}\frac{\text{Z}}{120}\:,\:{{\beta\:}}_{\text{j}}\:=\frac{{\partial\:}^{\text{j}}{\beta\:}\:}{\partial\:{{\omega\:}}^{\text{j}}}\:,\:\text{w}\text{h}\text{e}\text{r}\text{e}\:\text{j}=\text{2,3},\text{4,5}$$ Where β is phase constant. Eq. ( 18 ) further can be simplified as [11–14,] $$\:{P}_{2}\left(z,{\tau\:}_{2}\right)={P}_{2}\left(0,{\tau\:}_{2}\right){\text{e}}^{-{\alpha\:}\text{z}}\left\{\right(1-2{F}_{1}-6{F}_{2}\frac{\partial\:\psi\:\:\:}{\partial\:\text{t}}-24{F}_{3}\frac{{\partial\:}^{2}\psi\:\:\:}{\partial\:{\text{t}}^{2}}-120{F}_{4}\frac{{\partial\:}^{3}\psi\:\:\:}{\partial\:{\text{t}}^{3}}\left)\frac{{\partial\:}^{2}\psi\:\:\:}{\partial\:{\text{t}}^{2}}\right\}$$ 19 Now substituting the value of \(\:{\text{F}}_{1},\) \(\:{\text{F}}_{2}\:,\) \(\:{\text{F}}_{3}\text{a}\text{n}\text{d}\:{\:\text{F}}_{4}\) in Eq. ( 19 ) and differentiating with respect to z, one gets subsequent Eq. [ 9 ]. $$\:\frac{\partial\:{\text{P}}_{2}(\text{z},{\tau\:}_{2})}{\partial\:\text{z}}={P}_{2}\left(0,{\tau\:}_{2}\right){\text{e}}^{-{\alpha\:}\text{z}}\{{{\beta\:}}_{2}+{{\beta\:}}_{3}\frac{\partial\:\psi\:\:\:}{\partial\:\text{t}}+{{\beta\:}}_{4}\frac{{\partial\:}^{2}\psi\:\:\:}{\partial\:{\text{t}}^{2}}+{{\beta\:}}_{5}\frac{{\partial\:}^{3}\psi\:\:\:}{\partial\:{\text{t}}^{3}}\}\frac{{\partial\:}^{2}\psi\:\:\:}{\partial\:{\text{t}}^{2}}$$ 20 Here, our main concern is to determine XPM produced crosstalk caused by 5OD coefficient in SCM-WDM optical link, therefore we need to put \(\:{{\beta\:}}_{2}={{\beta\:}}_{3}={{\beta\:}}_{4}=0\) in Eq. ( 20 ), therefore Eq. ( 20 ) becomes $$\:\frac{\partial\:{\text{P}}_{2}(\text{z},{\tau\:}_{2})}{\partial\:\text{z}}={P}_{2}\left(0,{\tau\:}_{2}\right){\text{e}}^{-{\alpha\:}\text{z}}{{\beta\:}}_{5}\frac{{\partial\:}^{3}\psi\:\:\:}{\partial\:{\text{t}}^{3}}\frac{{\partial\:}^{2}\psi\:\:\:}{\partial\:{\text{t}}^{2}}$$ 21 The impact of \(\:{{\beta\:}}_{5}\) in \(\:\frac{\partial\:{\text{P}}_{2}(\text{z},{\tau\:}_{2})}{\partial\:\text{z}}\) is given by Y therefore The XPM crosstalk resulting from the 5OD coefficient is expressed as [ 9 ] $$\:\text{Y}={P}_{2}\left(0,{\tau\:}_{2}\right){\text{e}}^{-{\alpha\:}\text{z}}{{\beta\:}}_{5}\frac{{\partial\:}^{3}\psi\:\:\:}{\partial\:{{\tau\:}_{2}}^{3}}\frac{{\partial\:}^{2}\psi\:\:\:}{\partial\:{{\tau\:}_{2}}^{2}}$$ 22 Now using Eqs. ( 15 ) & ( 17 ), we have $$\:{P}_{1}(0,{\tau\:}_{1})={P}_{c}\{1+mcos({{\omega\:}}_{1}{\tau\:}_{1}\left)\right\}$$ \(\:{P}_{1}\{0,{(\tau\:}_{2}+{d}_{21}z)\}={P}_{c}[1+mcos{\{{\omega\:}}_{1}{(\tau\:}_{2}+{d}_{21}z\left)\right\}]\) And $$\:\psi\:=2\gamma\:{\int\:}_{0}^{z}{P}_{c}[1+mcos\left\{{{\omega\:}}_{1}{(\tau\:}_{2}+{d}_{21}z\right)\}]{\text{e}}^{-{\alpha\:}\text{z}}\text{d}\text{z}$$ 23 Differentiating \(\:{\psi\:}\) thrice with respect to \(\:\:{{\tau\:}}_{2}\) and putting value of \(\:\frac{{\partial\:}^{2}{\psi\:}\:\:}{\partial\:{{{\tau\:}}_{2}}^{2}}\) and \(\:\frac{{\partial\:}^{3}{\psi\:}\:\:}{\partial\:{{{\tau\:}}_{2}}^{3}}\) in Eq. ( 22 ), we get $$\:\text{Y}\:=\:{P}_{c}{\text{e}}^{-{\alpha\:}\text{z}}{{\beta\:}}_{5}\:\frac{{\left(2\gamma\:{P}_{c}m{\text{e}}^{\text{i}{{\omega\:}}_{1}{\tau\:}_{2}}\right)}^{4}{{{\omega\:}}_{1}}^{5}}{{(\text{i}{{\omega\:}}_{1}{d}_{21}-{\alpha\:})}^{2}}{\{{\text{e}}^{\left(\text{i}{{\omega\:}}_{1}{d}_{21}-{\alpha\:}\right)\text{z}}-1\}}^{4}$$ 24 This gradual change is lessened by a factor \(\:{\text{e}}^{-{\alpha\:}(\text{L}-\text{z})}\) due to attenuation. The modulation is accomplished at the fibre’s end by distributing attenuated power along a fibre of length L $$\:{\int\:}_{0}^{\text{L}}\text{Y}{\text{e}}^{-{\alpha\:}(\text{L}-\text{z})}\text{d}\text{z}\:=\:\:{-{\beta\:}}_{5}{P}_{c}\:\frac{{\left(2\gamma\:{P}_{c}m\right)}^{4}{{{\omega\:}}_{1}}^{5}{\text{e}}^{4\text{i}{{\omega\:}}_{1}{\tau\:}_{2}}}{12{\left(\text{i}{{\omega\:}}_{1}{d}_{21}-{\alpha\:}\right)}^{4}\:\left(\text{i}{{\omega\:}}_{1}{d}_{21}-{\alpha\:}\right)}{\text{e}}^{-{\alpha\:}\text{L}}[{12{(\text{i}{\omega\:}}_{1}{d}_{21})\text{L}-12{\alpha\:}\text{L}-48{(\text{e}}^{\left(\text{i}{{\omega\:}}_{1}{d}_{21}-{\alpha\:}\right)\text{L}}-1)+36\left({\text{e}}^{2\left(\text{i}{{\omega\:}}_{1}{d}_{21}-{\alpha\:}\right)\text{L}}-1\right)-16\left({\text{e}}^{3\left(\text{i}{{\omega\:}}_{1}{d}_{21}-{\alpha\:}\right)\text{L}}-1\right)+3(\text{e}}^{4\left(\text{i}{{\omega\:}}_{1}{d}_{21}-{\alpha\:}\right)\text{L}}-1)]$$ 25 The process of normalizing Eq. ( 25 ) yields the crosstalk in phasor form. One can obtain crosstalk in phasor form after multiplying Eq. ( 23 ) by ( \(\:\frac{1}{{\text{m}\text{P}}_{\text{C}}{\text{e}}^{-{\alpha\:}\text{L}}}\) ) factor. XPM crosstalk resulting from 5OD is provided by $$\:{XT}_{{XPM}_{5OD}}=\:{-{\beta\:}}_{5}\:\frac{4{\left(\gamma\:{P}_{c}\right)}^{4}{{{\omega\:}}_{1}}^{5}{\text{m}}^{3}{\text{e}}^{4\text{i}{{\omega\:}}_{1}{\tau\:}_{2}}}{3{\left(\text{i}{{\omega\:}}_{1}{d}_{21}-{\alpha\:}\right)}^{4}\:\left(\text{i}{{\omega\:}}_{1}{d}_{21}-{\alpha\:}\right)}[{12{(\text{i}{\omega\:}}_{1}{d}_{21})\text{L}-12{\alpha\:}\text{L}-48{(\text{e}}^{\left(\text{i}{{\omega\:}}_{1}{d}_{21}-{\alpha\:}\right)\text{L}}-1)+36\left({\text{e}}^{2\left(\text{i}{{\omega\:}}_{1}{d}_{21}-{\alpha\:}\right)\text{L}}-1\right)-16\left({\text{e}}^{3\left(\text{i}{{\omega\:}}_{1}{d}_{21}-{\alpha\:}\right)\text{L}}-1\right)+3(\text{e}}^{4\left(\text{i}{{\omega\:}}_{1}{d}_{21}-{\alpha\:}\right)\text{L}}-1)]$$ 26 Simplifying further Eq. ( 26 ) we get $$\:{XT}_{{XPM}_{5OD}}=\:{-{\beta\:}}_{5}\:\frac{4{\left(\gamma\:{P}_{c}\right)}^{4}{{{\omega\:}}_{1}}^{5}{\text{m}}^{3}{\text{e}}^{4\text{i}{{\omega\:}}_{1}{\tau\:}_{2}}}{3{\left(\text{i}{{\omega\:}}_{1}{d}_{21}-{\alpha\:}\right)}^{5}\:}[-12{\alpha\:}\text{L}+25-48{\text{e}}^{-{\alpha\:}\text{L}}\text{cos}\left({{\omega\:}}_{1}{d}_{21}\text{L}\right){+36\text{e}}^{-2{\alpha\:}\text{L}}\text{cos}\left(2{{\omega\:}}_{1}{d}_{21}\text{L}\right)-16{\text{e}}^{-3{\alpha\:}\text{L}}\text{cos}\left(3{{\omega\:}}_{1}{d}_{21}\text{L}\right)+3{\text{e}}^{-4{\alpha\:}\text{L}}\text{cos}\left(4{{\omega\:}}_{1}{d}_{21}\text{L}\right)\:+\:\:\:\:\text{i}\left\{12{{\omega\:}}_{1}{d}_{21}\text{L}{-4{8\text{e}}^{-{\alpha\:}\text{L}}\text{s}\text{i}\text{n}{({\omega\:}}_{1}{d}_{21}\text{L})+36\text{e}}^{-2{\alpha\:}\text{L}}\text{sin}\left(2{{\omega\:}}_{1}{d}_{21}\text{L}\right)-16{\text{e}}^{-3{\alpha\:}\text{L}}\text{s}\text{i}\text{n}{(3{\omega\:}}_{1}{d}_{21}\text{L}\right)+3{\text{e}}^{-4{\alpha\:}\text{L}}\text{s}\text{i}\text{n}{(4{\omega\:}}_{1}{d}_{21}\text{L}\:\}]$$ The fifth-order dispersion parameter is defined here as [ 8 ] \(\:{{\beta\:}}_{5}=\frac{{\lambda\:}^{4}}{({2{\pi\:}\text{c})}^{4}}({\lambda\:}^{4}{\text{D}}_{3}+{12\lambda\:}^{3}{\text{D}}_{2}+{36\lambda\:}^{2}{\text{D}}_{1}+24\lambda\:D\) ) (27) Here letters, \(\:\text{D},\:{\lambda\:}\:\&\:\text{c}\:\) represent chromatic dispersion, wavelength of light and velocity of light respectively and \(\:\:{\text{D}}_{1}\) , \(\:{\text{D}}_{2}\) & \(\:{\text{D}}_{3}\) are first, second and third order GVD parameters. 3. RESULTS AND DISCUSSION The values of various parameters have been considered as: \(\:\:{\Delta\:}{\lambda\:}=2\text{n}\text{m}\) , \(\:\text{L}=50\text{k}\text{m}\) , \(\:\:{\alpha\:}=0.25\text{d}\text{B}/\text{k}\text{m}\) , \(\:{\lambda\:}_{1}=1550\text{n}\text{m}\) , \(\:{\lambda\:}_{2}=1552\text{n}\text{m},\:\:\text{D}=2\text{p}\text{s}/(\text{n}\text{m}.\:\text{k}\text{m})\) , \(\:{D}_{1}=0.084\text{p}\text{s}/(\text{n}\text{m}.\:\text{k}\text{m})\) , \(\:{D}_{2}=0.00023\text{p}\text{s}/(\text{n}\text{m}.\:\text{k}\text{m})\) , \(\:{D}_{3}=0.0000092\text{p}\text{s}/(\text{n}\text{m}.\:\text{k}\text{m})\) , \(\:{n}_{2}=2.67\text{x}{10}^{-20}\frac{{\text{m}}^{2}}{\text{W}}\) , m = 0.7 and \(\:{\text{A}}_{\text{e}\text{f}\text{f}}=81{{\mu\:}\text{m}}^{2}\) . From Fig. 1 , it is clear that the XPM crosstalk attributed to the 5OD coefficient changes from (-315dB to -270dB) when the modulation frequency ranges from 0.4GHz to 5GHz, as determined by the NLSE. In contrast, the XPM induced crosstalk caused by 5OD coefficient fluctuates from (-280dB to -255.09dB) when the modulation frequency increases from 0.2GHz and 5GHz, utilizing couple equations. At a modulation frequency of 3GHz, XPM crosstalk caused by 5OD coefficient is measured at -274dB and − 256.08dB for the NLSE and couple equations, respectively. XPM crosstalk fluctuates between − 295.1dB to -255.06dB using NLSE when optical power changes from 0.1mW to 2mW. In contrast, the XPM crosstalk caused by 5OD coefficient changes from (-270.1dB to -240dB) when optical power varies from 0.1mW to 2mW using couple equations. At an optical power of 1mW, the XPM crosstalk values are − 244.5dB and − 265.01dB when calculated using couple equations and NLSE respectively. From Fig. 3 it is deduced that the XPM crosstalk attributed to the 5OD coefficient escalates between (-346.03 dB to -297 dB) as the transmission length varies between 3 to 50 km when employing the NLSE. Conversely, the XPM crosstalk related to the 5OD coefficient fluctuates between (-320 dB to -281 dB) when the transmission length ranges from 3 km to 50 km employing couple equations. At a distance of 30 km, the XPM crosstalk caused by 5OD coefficient is measured at -285dB and − 302dB when utilizing couple equations and NLSE, respectively. 4. Conclusion This research paper gives comprehensive analysis of the effects of higher-order dispersion, specifically 5OD, on XPM crosstalk. It has also been noted that the HOD terms significantly affect crosstalk induced by XPM. Furthermore, the The impact of the HOD decreases as the order of the dispersion term increases. It has been deduced that the examination of induced crosstalk by XPM resulting from the 5OD coefficient through the use of NLSE yields a lower level of crosstalk when compared to the couple equations. The XPM crosstalk related to the 5OD parameter spans from (-280dB to -255.09dB) when utilizing couple equations, whereas it varies between (-315dB to -270dB) when applying NLSE as the modulation frequency rises from 0.4 to 5GHz. Moreover, XPM crosstalk is noted to be between (-320dB to -281dB) with couple equations, while it increases from (-346.03dB to -297dB) when using NLSE as the transmission distance varies between 3 to 50 km. In addition, XPM crosstalk fluctuates between (-270.1dB to -240.2dB) with couple equations, while it is observed within the range of (-295.1dB to -255.06dB) using NLSE as the optical power varies from 0.1 to 2mW. Declarations Author Contribution The authors confirm their contribution to the paper as follows: VS has made a substantial contribution to the concept and design of the article and performed work of validation and wrote the article. NCJ, SG and YP did an analysis and interpretation of the results. We all authors reviewed the results and approved the final version of the manuscript. References Way, W.I., Wagner, S.S., Choy, M.M., Lin, C., Menendez, R.C., Tohme, H., Yi-Yan, A., Von Lehman, A.C., Spicer, R.E., Andrejco, M., Saifi, M.A.: Simultaneous distribution of multichannel analog and digital video channels to multiple terminals using high density WDM and a broad-band in-line Erbium doped fiber amplifier. IEEE Photonics Technol. Lett. 2 (9), 665–668 (1990) Lee, C.C., Chi, S.: Three-wavelength-division-multiplexed multichannel subcarrier-multiplexing transmission over multimode fiber with potential capacity of 12Gbps. IEEE Photonics Technol. Lett. 11 (8), 1066–1068 (1999) Hui, R., Zhu, B., Huang, R., Allen, C., Demarest, K., Richards, D.: 10Gbps SCM fiber system using optical SSB modulation. IEEE Photonics Technol. Lett. 13 (8), 896–898 (2001) Woodward, S.L., Lu, X., Darcie, T.E., Bodeep, G.E.: Reduction of optical-beat interference in subcarrier networks. IEEE Photonics Technol. Lett. 8 (5), 694–696 (1996) Phillips, M.R., Ott, D.M.: Crosstalk due to optical fiber non linearities in WDM CATV lightwave systems. J. Lightwave Technol. 17 (10), 1782–1792 (1999) Yang, F.S., Marhic, M.E., Kazovsky, L.G.: Nonlinear crosstalk and two countermeasures in SCM–WDM optical communication systems. IEEE Journal Lightwave Technology. 18 (4), 512–520 (2000) Wang, Z., Bodtker, E., Jacobsen, G.: Effects of cross phase modulation in wavelength multiplexed SCM video transmission systems. Electron. Lett. 31 , 1591–1592 (1995) Kumar, N., Sharma, A.K., Kapoor, V.: XPM-induced crosstalk with higher-order dispersion in SCM–WDM optical transmission link. Optik Int. J. Light Electron. Opt. 123 (22), 2056–2061 (2012) Arya, S.K., Sharma, A.K., Agarwala, R.A.: Impact of 2OD and 3OD on SRS-and XPM-induced crosstalk in SCM-WDM optical transmission link. Optik. 120 (8), 364–369 (2009) Kumar, N., Sharma, A.K., Kapoor, V.: Improved XPM-induced crosstalk with higher-order dispersion in SCM–WDM optical transmission link. Optik. 124 , 941–944 (2013) Singh, V., Kumar, N., Kumar, S.: Comparative analysis of XPM-induced crosstalk in SCM-WDM transmission links. Opt. Quant. Electron. 48 , 8 (2016) Singh, V., Kumar, S., Dimri, P.K.: Comparative study of XPM-induced crosstalk due to 3OD parameter in SCM-WDM transmission system, Optik 186C 177–181. (2019) Singh, V., Kumar, S., Dimri, P.K.: Performance evaluation of SCM–WDM-HAN communication link using millimeter waves in the presence of XPM, Optik 164580. (2020) Singh, V., Kumar, S., Dimri, P.K.: Comparative analysis of XPM induced crosstalk due to 4OD coefficient in SCM-WDM link. Optik. 241 , 166924 (2021) Additional Declarations No competing interests reported. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-7106552","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":485161137,"identity":"6b788424-51b7-4dfc-b963-9d4a101b468d","order_by":0,"name":"Vikram Singh","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA3UlEQVRIiWNgGAWjYDACZjYwyWDAwNz4AMji4SNBC2OzAUgLG2FrEFraJOB8fMDgOFvah5851nLm7AfbKr/m2MmwMTA/fHQDn5bDbIdn9m5LN7bsSWy7LbstGegwNmPjHDxaJJvZmxl4tx1O3HAAqEVyGzNQCw+bNCEtjH9BWs4/bCuW3FZPWAs/M9thZrAtNxLbGD9uO0yUlmRmWZBfZjxslmbcdpyHjZmAX9j4jxkzvt0GDDH+5IMff26rtudnb374GJ8WFMDMAyaJVQ4CjD9IUT0KRsEoGAUjBgAA0a5CgZcddrkAAAAASUVORK5CYII=","orcid":"","institution":"Lingaya’s Vidyapeeth","correspondingAuthor":true,"prefix":"","firstName":"Vikram","middleName":"","lastName":"Singh","suffix":""},{"id":485161138,"identity":"1646b52d-0e45-416d-8109-4ee19e568b70","order_by":1,"name":"Naveen Chandra Joshi","email":"","orcid":"","institution":"Graphic Era Deemed to be University","correspondingAuthor":false,"prefix":"","firstName":"Naveen","middleName":"Chandra","lastName":"Joshi","suffix":""},{"id":485161140,"identity":"e4275640-dacc-456d-a20d-e96f1ddb0f85","order_by":2,"name":"Shikha Gupta","email":"","orcid":"","institution":"Lingaya’s Vidyapeeth","correspondingAuthor":false,"prefix":"","firstName":"Shikha","middleName":"","lastName":"Gupta","suffix":""},{"id":485161144,"identity":"7e355731-ba03-4591-8d2e-7420dfb2561f","order_by":3,"name":"Yash Pathak","email":"","orcid":"","institution":"Lingaya’s Vidyapeeth","correspondingAuthor":false,"prefix":"","firstName":"Yash","middleName":"","lastName":"Pathak","suffix":""}],"badges":[],"createdAt":"2025-07-12 07:53:06","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-7106552/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-7106552/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":86731903,"identity":"c805b0b8-30c9-4792-ac01-09e5b4691a38","added_by":"auto","created_at":"2025-07-15 04:29:44","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":41208,"visible":true,"origin":"","legend":"\u003cp\u003emodulation frequency vs. crosstalk\u003c/p\u003e","description":"","filename":"1.png","url":"https://assets-eu.researchsquare.com/files/rs-7106552/v1/c7ae4c4056f774371b8e7915.png"},{"id":86731998,"identity":"6f972358-a592-40d4-8d06-73efd824736d","added_by":"auto","created_at":"2025-07-15 04:37:44","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":38336,"visible":true,"origin":"","legend":"\u003cp\u003eoptical power vs. crosstalk\u003c/p\u003e","description":"","filename":"2.png","url":"https://assets-eu.researchsquare.com/files/rs-7106552/v1/b9e65875f12ad9b96ab08d29.png"},{"id":86731904,"identity":"9eb2649c-db64-45d5-98a0-77fc89b0be66","added_by":"auto","created_at":"2025-07-15 04:29:44","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":38846,"visible":true,"origin":"","legend":"\u003cp\u003etransmission distance vs. crosstalk\u003c/p\u003e","description":"","filename":"3.png","url":"https://assets-eu.researchsquare.com/files/rs-7106552/v1/388d35a1cf0e8cddb3a03ff2.png"},{"id":93905930,"identity":"81da6f6e-b40f-4f7e-b356-917b4cf0a029","added_by":"auto","created_at":"2025-10-20 07:08:50","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1565160,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7106552/v1/a1640326-f99a-49b6-84ef-d9090600a171.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"A comparative examination of the crosstalk induced by XPM as a result of the 5OD coefficient in SCM-WDM communication system","fulltext":[{"header":"1. INTRODUCTION","content":"\u003cp\u003eThe enormous demand for wireless communication services, driven by the increasing consumption of multimedia content, has created significant challenges for network operators. The combination of SCM-WDM is an economically viable method employed to enhance the bandwidth efficiency of optical fibers in communication systems. Network operators are finding it extremely challenging to handle the growing traffic as a result of wireless communication\u0026rsquo;s remarkable grow in recent years. A mature service with a high proportion of consumer use, a reduced and consistent access charge, full-time connectivity to service provider and increased bandwidth is required as the demand for multimedia services including data, voice and video continue to rise. A huge capacity is necessary for future wireless communication systems to meet the diverse needs. Micro cellular systems have been proposed as a solution to the convergent demands for high bandwidths and subscriber mobility. These systems can boost system capacity by improving the efficiency of reusing limited radio resources [\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e]. The microcellular system presents challenges since it takes time and money to install new radio base stations. Since it uses microwave photonics techniques to provide the so-called radio over fiber link, the combination of SCM and WDM is viewed as a potential solution to the issues presented by a microcellular system. However, nonlinear effects in fiber plague SCM\u0026ndash;WDM systems.\u003c/p\u003e\u003cp\u003eCrosstalk between subcarriers on different wavelengths can occur when numerous wavelengths carrying SCM signals propagate in a single cable due to fiber nonlinearities. XPM induced crosstalk is the predominant fiber nonlinearity that generates crosstalk in a dispersive fiber. Due to their tight spacing, adjacent SCM channels may experience large quantities of nonlinear crosstalk when subjected to cross phase modulation (XPM) [\u003cspan additionalcitationids=\"CR3\" citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e]. The optical Kerr effect causes non-linear crosstalk in a dispersive optical fibre. One of the most prevalent non-linear effects is XPM. One of the most common uses of SCM-WDM technology in the optical realm is the transmission and distribution of analog video. The investigation of XPM crosstalk in SCM-WDM video links XPM crosstalk depends on the channel spacing, according to the [\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e] finding. Researchers looked into an enhanced model for determining XPM crosstalk in SCM-WDM links that takes impact of higher order dispersion (HOD) into account [\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e]. To evaluate XPM crosstalk in SCM-WDM links, the non-linear Schrodinger equation (NLSE) is utilized. It has been concluded that HOD significantly affects XPM crosstalk. As the order of dispersion term lowers, the impact of HOD parameters on XPM crosstalk grows. It has been examined how 2OD and 3OD cause SRS and XPM crosstalk in SCM-WDM communication systems [\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e]. The 2OD and 3OD coefficients are found to significantly affect crosstalk. It has been presented the manner in which HOD coefficients affect XPM crosstalk in SCM-WDM communication systems [\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e, \u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e]. When the cumulative influence of all three HOD coefficients is taken into account, it is found that the 3OD, 4OD, and 5OD coefficients have a substantial impact, however the impact of the 2OD coefficient is considerable. In the SCM-WDM system, a comparative study of XPM crosstalk caused by the 2OD parameter is estimated [\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e, \u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e, \u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e]. A comparative analysis of the 3OD and 4OD coefficients-induced XPM crosstalk has been published [\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e, \u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e]. The current study extends the work published by [\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e] and attempts to examine XPM crosstalk induced by XPM due to 5OD parameter.\u003c/p\u003e"},{"header":"2. MATHEMATICAL INVESTIGATIONS","content":"\u003cdiv id=\"Sec3\" class=\"Section2\"\u003e\u003ch2\u003e2.1 Examination of XPM crosstalk through the use of NLSE\u003c/h2\u003e\u003cp\u003eStarting with the nonlinear wave propagation equation, the theoretical analysis begins [\u003cspan additionalcitationids=\"CR11\" citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e]. Let us consider probe A\u003csub\u003ej\u003c/sub\u003e(t,z) and pump A\u003csub\u003ek\u003c/sub\u003e(t,z) waves are concurrently conveyed through the same fiber.\u003cdiv id=\"Equ1\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e\n$$\\:\\frac{\\partial\\:{\\text{A}}_{\\text{j}}\\left(\\text{z},\\text{t}\\right)}{\\partial\\:\\text{z}}+\\frac{{\\alpha\\:}\\:}{2\\:}{\\text{A}}_{\\text{j}}\\left(\\text{z},\\text{t}\\right)+\\frac{1\\:}{{\\text{V}}_{\\text{j}}}\\frac{\\partial\\:{\\text{A}}_{\\text{j}}\\left(\\text{z},\\text{t}\\right)}{\\partial\\:\\text{z}}+\\frac{\\text{i}{{\\beta\\:}}_{2}}{2}\\frac{{\\partial\\:}^{2}{\\text{A}}_{\\text{j}}\\left(\\text{z},\\text{t}\\right)\\:\\:\\:}{\\partial\\:{\\text{t}}^{2}}-\\frac{\\text{i}{{\\beta\\:}}_{3}}{6}\\frac{{\\partial\\:}^{3}{\\text{A}}_{\\text{j}}\\left(\\text{z},\\text{t}\\right)\\:\\:\\:}{\\partial\\:{\\text{t}}^{3}}-\\frac{\\text{i}{{\\beta\\:}}_{4}}{24}\\frac{{\\partial\\:}^{4}{\\text{A}}_{\\text{j}}\\left(\\text{z},\\text{t}\\right)\\:\\:\\:}{\\partial\\:{\\text{t}}^{4}}+\\frac{\\text{i}{{\\beta\\:}}_{5}}{120}\\frac{{\\partial\\:}^{5}{\\text{A}}_{\\text{j}}\\left(\\text{z},\\text{t}\\right)\\:\\:\\:}{\\partial\\:{\\text{t}}^{5}}=\\:2i\\gamma\\:{P}_{k\\:}\\{t-{d}_{jk}\\}{A}_{j}\\left(\\text{z},\\text{t}\\right)$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e1\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eWhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\:d}_{jk}\\)\u003c/span\u003e\u003c/span\u003eis relative walk off parameter \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{{\\beta\\:}}_{2},\\:{{\\beta\\:}}_{3}\\:{{\\beta\\:}}_{4}\\:\\text{a}\\text{n}\\text{d}\\:\\:{{\\beta\\:}}_{5}\\:\\)\u003c/span\u003e\u003c/span\u003e are 2OD, 3OD, 4OD and 5OD coefficients respectively, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\alpha\\:}\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\gamma\\:\\:\\)\u003c/span\u003e\u003c/span\u003eare attenuation and coupling coefficients respectively here\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\:\\gamma\\:=\\frac{2\\pi\\:{n}_{2}}{{\\lambda\\:}_{m}{A}_{eff}}\\)\u003c/span\u003e\u003c/span\u003e, where symbols mean what they usually mean.\u003c/p\u003e\u003cp\u003eIn order to obtain XPM induced cross talk due 5OD coefficient put \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{{\\beta\\:}}_{2}={{\\beta\\:}}_{3}={{\\beta\\:}}_{4}=0\\)\u003c/span\u003e\u003c/span\u003e in Eq.\u0026nbsp;(\u003cspan refid=\"Equ1\" class=\"InternalRef\"\u003e1\u003c/span\u003e) we get\u003cdiv id=\"Equ2\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ2\" name=\"EquationSource\"\u003e\n$$\\:\\frac{\\partial\\:{\\text{A}}_{\\text{j}}\\left(\\text{z},\\text{t}\\right)}{\\partial\\:\\text{z}}=-\\frac{{\\alpha\\:}\\:}{2\\:}{\\text{A}}_{\\text{j}}\\left(\\text{z},\\text{t}\\right)\\:-\\:\\frac{1\\:}{{\\text{V}}_{\\text{j}}}\\frac{\\partial\\:{\\text{A}}_{\\text{j}}\\left(\\text{z},\\text{t}\\right)}{\\partial\\:\\text{z}}\\:+\\:\\frac{{\\text{i}{\\beta\\:}}_{4}}{24}\\frac{{\\partial\\:}^{4}{\\text{A}}_{\\text{j}}\\left(\\text{z},\\text{t}\\right)\\:\\:\\:}{\\partial\\:{\\text{t}}^{4}}-\\:\\frac{\\text{i}{{\\beta\\:}}_{5}}{120}\\frac{{\\partial\\:}^{5}{\\text{A}}_{\\text{j}}\\left(\\text{z},\\text{t}\\right)\\:\\:\\:}{\\partial\\:{\\text{t}}^{5}}\\:\\:+\\:2\\:i\\gamma\\:{P}_{k\\:}\\{t-{d}_{jk}\\}{A}_{j}\\left(\\text{z},\\text{t}\\right)$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e2\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eIn order to focus on inter channel cross talk Eq.\u0026nbsp;(\u003cspan refid=\"Equ2\" class=\"InternalRef\"\u003e2\u003c/span\u003e) needs to be converted into frequency domain, therefore\u003cdiv id=\"Equ3\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ3\" name=\"EquationSource\"\u003e\n$$\\:\\frac{\\partial\\:{\\text{A}}_{\\text{j}}\\left({\\omega\\:},\\text{z}\\right)}{\\partial\\:\\text{z}}=[-\\:\\frac{{\\alpha\\:}\\:}{2\\:}-\\:\\frac{i{\\omega\\:}\\:}{{\\text{V}}_{\\text{j}}\\:}-\\:\\frac{\\text{i}{{\\omega\\:}}^{5}{{\\beta\\:}}_{5}}{120}\\:+\\:i2\\gamma\\:{P}_{k\\:}({\\omega\\:},0\\left){\\text{e}}^{\\text{i}{\\omega\\:}\\text{z}{d}_{jk}}{\\text{e}}^{-{\\alpha\\:}\\text{z}}\\right]\\:{\\text{A}}_{\\text{j}}\\left({\\omega\\:},\\text{z}\\right)$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e3\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eThe last term in Eq.\u0026nbsp;(\u003cspan refid=\"Equ3\" class=\"InternalRef\"\u003e3\u003c/span\u003e) converts phase noise into intensity noise in the probe signal. The phase noise in an incredibly tiny fiber segment, say dz, is provided as\u003c/p\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{d\\varPhi\\:}_{jk}\\left(\\omega\\:,z\\right)=2\\gamma\\:{P}_{k\\:}({\\omega\\:},0){\\text{e}}^{\\left(-{\\alpha\\:}+\\text{i}{\\omega\\:}{d}_{jk}\\right)z}\\text{d}\\text{z}\\)\u003c/span\u003e\u003c/span\u003e (4) Due to chromatic dispersion, phase noise generated at the fiber's output changes into intensity noise at the fiber's end, z\u0026thinsp;=\u0026thinsp;L. The overall phase noise at fiber output can be expressed as [\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e].\u003cdiv id=\"Equ4\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ4\" name=\"EquationSource\"\u003e\n$$\\:{\\Delta\\:}{\\text{A}}_{\\text{j}}\\left({\\omega\\:},\\text{L}\\right)\\:\\:=\\:2{P}_{j}\\left(0\\right){\\text{e}}^{-\\left({\\alpha\\:}-\\frac{\\text{i}\\omega\\:}{{\\text{V}}_{\\text{j}}}\\right)L}\\left[\\underset{0}{\\overset{\\text{L}}{\\int\\:}}2\\gamma\\:{P}_{k\\:}({\\omega\\:},0){\\text{e}}^{\\left(-{\\alpha\\:}+\\text{i}{\\omega\\:}{d}_{jk}\\right)z}\\text{sin}\\left\\{\\frac{{\\omega\\:}^{5}{{\\beta\\:}}_{5}\\left(\\text{L}-\\text{z}\\right)}{120}\\right\\}\\}\\right]\\text{d}\\text{z}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e5\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eNow assessing the integral\u003c/p\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\underset{0}{\\overset{\\text{L}}{\\int\\:}}\\gamma\\:{P}_{k\\:}({\\omega\\:},0){\\text{e}}^{\\left(-{\\alpha\\:}+\\text{i}{\\omega\\:}{d}_{jk}\\right)z}\\text{sin}\\left\\{\\frac{{\\omega\\:}^{5}{{\\beta\\:}}_{5}\\left(\\text{L}-\\text{z}\\right)}{120}\\right\\}\\}\\text{d}\\text{z}\\)\u003c/span\u003e\u003c/span\u003e=\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\:{\\gamma\\:}{\\text{P}}_{\\text{k}\\:}\\left({\\omega\\:},0\\right)[\\frac{{\\text{e}}^{\\left\\{{\\text{i}{\\omega\\:}}^{5}{{\\beta\\:}}_{5}\\frac{\\text{L}}{120}\\:\\right\\}}-{\\text{e}}^{-\\left({\\alpha\\:}-\\text{i}{\\omega\\:}{\\text{d}}_{\\text{j}\\text{k}}\\right)\\text{L}}}{\\left\\{\\text{i}\\left({\\alpha\\:}-\\text{i}{\\omega\\:}{\\text{d}}_{\\text{j}\\text{k}}\\right)+(\\text{i}.\\text{i})\\frac{{{\\omega\\:}}^{5}{{\\beta\\:}}_{5}}{120}\\right\\}}\\)\u003c/span\u003e\u003c/span\u003e -\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\:\\frac{{\\text{e}}^{\\left\\{-{\\text{i}{\\omega\\:}}^{5}{{\\beta\\:}}_{5}\\frac{\\text{L}}{120}\\:\\right\\}}-{\\text{e}}^{-\\left({\\alpha\\:}-\\text{i}{\\omega\\:}{\\text{d}}_{\\text{j}\\text{k}}\\right)\\text{L}}}{\\left\\{\\text{i}\\left({\\alpha\\:}-\\text{i}{\\omega\\:}{\\text{d}}_{\\text{j}\\text{k}}\\right)-(\\text{i}.\\text{i})\\frac{{{\\omega\\:}}^{5}{{\\beta\\:}}_{5}}{120}\\right\\}}\\)\u003c/span\u003e\u003c/span\u003e] (6)\u003c/p\u003e\u003cp\u003eUsing Eqs.\u0026nbsp;(\u003cspan refid=\"Equ4\" class=\"InternalRef\"\u003e5\u003c/span\u003e) \u0026amp; (6), we have\u003c/p\u003e\u003cp\u003eΔ\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\text{A}}_{\\text{j}}\\left({\\omega\\:},\\text{L}\\right)\\:\\:\\)\u003c/span\u003e\u003c/span\u003e=\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\:2{P}_{j}\\left(0\\right){\\text{e}}^{-\\left({\\alpha\\:}-\\frac{\\text{i}\\omega\\:}{{\\text{V}}_{\\text{j}}}\\right)L}{\\gamma\\:}{\\text{P}}_{\\text{k}\\:}\\left({\\omega\\:},0\\right)[\\frac{{\\text{e}}^{\\left\\{{\\text{i}{\\omega\\:}}^{5}{{\\beta\\:}}_{5}\\frac{\\text{L}}{120}\\:\\right\\}}-{\\text{e}}^{-\\left({\\alpha\\:}-\\text{i}{\\omega\\:}{\\text{d}}_{\\text{j}\\text{k}}\\right)\\text{L}}}{\\left\\{\\text{i}\\left({\\alpha\\:}-\\text{i}{\\omega\\:}{\\text{d}}_{\\text{j}\\text{k}}\\right)+(\\text{i}.\\text{i})\\frac{{{\\omega\\:}}^{5}{{\\beta\\:}}_{5}}{120}\\right\\}}\\)\u003c/span\u003e\u003c/span\u003e -\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\:\\frac{{\\text{e}}^{\\left\\{-{\\text{i}{\\omega\\:}}^{5}{{\\beta\\:}}_{5}\\frac{\\text{L}}{120}\\:\\right\\}}-{\\text{e}}^{-\\left({\\alpha\\:}-\\text{i}{\\omega\\:}{\\text{d}}_{\\text{j}\\text{k}}\\right)\\text{L}}}{\\left\\{\\text{i}\\left({\\alpha\\:}-\\text{i}{\\omega\\:}{\\text{d}}_{\\text{j}\\text{k}}\\right)-(\\text{i}.\\text{i})\\frac{{{\\omega\\:}}^{5}{{\\beta\\:}}_{5}}{120}\\right\\}}\\)\u003c/span\u003e\u003c/span\u003e] (7)\u003c/p\u003e\u003cp\u003eUtilizing the subsequent relation now\u003cdiv id=\"Equ5\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ5\" name=\"EquationSource\"\u003e\n$$\\:{\\:P}_{j}\\left(L\\right)={P}_{j}\\left(0\\right){\\text{e}}^{-{\\alpha\\:}\\text{L}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e8\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eFrom Eqs.\u0026nbsp;(7) and (\u003cspan refid=\"Equ5\" class=\"InternalRef\"\u003e8\u003c/span\u003e), we have\u003c/p\u003e\u003cp\u003eΔ\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\text{A}}_{\\text{j}}\\left({\\omega\\:},\\text{L}\\right)\\:\\:\\)\u003c/span\u003e\u003c/span\u003e=\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\:2{P}_{j}\\left(L\\right){\\text{e}}^{\\left(\\frac{\\text{i}\\omega\\:}{{\\text{V}}_{\\text{j}}}\\right)L}{\\gamma\\:}{\\text{P}}_{\\text{k}\\:}\\left({\\omega\\:},0\\right)[\\frac{{\\text{e}}^{\\left\\{{\\text{i}{\\omega\\:}}^{5}{{\\beta\\:}}_{5}\\frac{\\text{L}}{120}\\:\\right\\}}-{\\text{e}}^{-\\left({\\alpha\\:}-\\text{i}{\\omega\\:}{\\text{d}}_{\\text{j}\\text{k}}\\right)\\text{L}}}{\\left\\{\\text{i}\\left({\\alpha\\:}-\\text{i}{\\omega\\:}{\\text{d}}_{\\text{j}\\text{k}}\\right)+(\\text{i}.\\text{i})\\frac{{{\\omega\\:}}^{5}{{\\beta\\:}}_{5}}{120}\\right\\}}\\)\u003c/span\u003e\u003c/span\u003e -\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\:\\frac{{\\text{e}}^{\\left\\{-{\\text{i}{\\omega\\:}}^{5}{{\\beta\\:}}_{5}\\frac{\\text{L}}{120}\\:\\right\\}}-{\\text{e}}^{-\\left({\\alpha\\:}-\\text{i}{\\omega\\:}{\\text{d}}_{\\text{j}\\text{k}}\\right)\\text{L}}}{\\left\\{\\text{i}\\left({\\alpha\\:}-\\text{i}{\\omega\\:}{\\text{d}}_{\\text{j}\\text{k}}\\right)-(\\text{i}.\\text{i})\\frac{{{\\omega\\:}}^{5}{{\\beta\\:}}_{5}}{120}\\right\\}}\\)\u003c/span\u003e\u003c/span\u003e] (9)\u003cdiv id=\"Equa\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equa\" name=\"EquationSource\"\u003e\n$$\\:\\left|{\\Delta\\:}{\\text{A}}_{\\text{j}}\\left({\\omega\\:},\\text{L}\\right)\\right|=\\left|2{P}_{j}\\left(L\\right){\\gamma\\:}{\\text{P}}_{\\text{k}\\:}\\left({\\omega\\:},0\\right)[\\frac{{\\text{e}}^{\\left\\{{\\text{i}{\\omega\\:}}^{5}{{\\beta\\:}}_{5}\\frac{\\text{L}}{120}\\:\\right\\}}-{\\text{e}}^{-\\left({\\alpha\\:}-\\text{i}{\\omega\\:}{\\text{d}}_{\\text{j}\\text{k}}\\right)\\text{L}}}{\\left\\{\\text{i}\\left({\\alpha\\:}-\\text{i}{\\omega\\:}{\\text{d}}_{\\text{j}\\text{k}}\\right)+(\\text{i}.\\text{i})\\frac{{{\\omega\\:}}^{5}{{\\beta\\:}}_{5}}{120}\\right\\}}\\:-\\:\\frac{{\\text{e}}^{\\left\\{-{\\text{i}{\\omega\\:}}^{5}{{\\beta\\:}}_{5}\\frac{\\text{L}}{120}\\:\\right\\}}-{\\text{e}}^{-\\left({\\alpha\\:}-\\text{i}{\\omega\\:}{\\text{d}}_{\\text{j}\\text{k}}\\right)\\text{L}}}{\\left\\{\\text{i}\\left({\\alpha\\:}-\\text{i}{\\omega\\:}{\\text{d}}_{\\text{j}\\text{k}}\\right)-(\\text{i}.\\text{i})\\frac{{{\\omega\\:}}^{5}{{\\beta\\:}}_{5}}{120}\\right\\}}]\\:\\right|$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eA simple expression that characterizes the relative amplitude variation caused by XPM is [\u003cspan additionalcitationids=\"CR11 CR12 CR13\" citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e] (10)\u003cdiv id=\"Equ6\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ6\" name=\"EquationSource\"\u003e\n$$\\:{\\Delta\\:}{\\text{P}}_{\\text{j}}\\left({\\omega\\:},\\text{L}\\right)={\\left|\\frac{{\\text{A}}_{\\text{j}}\\left({\\omega\\:},\\text{L}\\right)}{{P}_{j}\\left(L\\right)}\\right|}^{2}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e11\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eThe final 5OD coefficient-related XPM crosstalk expression can be written as [\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e]\u003cdiv id=\"Equ7\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ7\" name=\"EquationSource\"\u003e\n$$\\:{\\Delta\\:}{\\text{P}}_{\\text{j}}\\left({\\omega\\:},\\text{L}\\right)={\\left|2{\\gamma\\:}{\\text{P}}_{\\text{k}\\:}\\left({\\omega\\:},0\\right)[\\frac{{\\text{e}}^{\\left\\{{\\text{i}{\\omega\\:}}^{5}{{\\beta\\:}}_{5}\\frac{\\text{L}}{120}\\:\\right\\}}-{\\text{e}}^{-\\left({\\alpha\\:}-\\text{i}{\\omega\\:}{\\text{d}}_{\\text{j}\\text{k}}\\right)\\text{L}}}{\\left\\{\\text{i}\\left({\\alpha\\:}-\\text{i}{\\omega\\:}{\\text{d}}_{\\text{j}\\text{k}}\\right)+(\\text{i}.\\text{i})\\frac{{{\\omega\\:}}^{5}{{\\beta\\:}}_{5}}{120}\\right\\}}\\:-\\:\\frac{{\\text{e}}^{\\left\\{-{\\text{i}{\\omega\\:}}^{5}{{\\beta\\:}}_{5}\\frac{\\text{L}}{120}\\:\\right\\}}-{\\text{e}}^{-\\left({\\alpha\\:}-\\text{i}{\\omega\\:}{\\text{d}}_{\\text{j}\\text{k}}\\right)\\text{L}}}{\\left\\{\\text{i}\\left({\\alpha\\:}-\\text{i}{\\omega\\:}{\\text{d}}_{\\text{j}\\text{k}}\\right)-(\\text{i}.\\text{i})\\frac{{{\\omega\\:}}^{5}{{\\beta\\:}}_{5}}{120}\\right\\}}]\\right|}^{2}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e12\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec4\" class=\"Section2\"\u003e\u003ch2\u003e2.2 Examination of XPM crosstalk through the use of couple equations\u003c/h2\u003e\u003cp\u003eAssume that two wavelengths are concurrently transmitted over an optical fiber. The subsequent two couple equations illustrate XPM within the framework of a slowly varying envelope, as referenced in [\u003cspan additionalcitationids=\"CR9 CR10 CR11 CR12 CR13\" citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e]. In this context, the higher-order dispersion term, specifically the fifth-order dispersion (5OD) coefficient, has been accounted for in the results pertaining to XPM-induced crosstalk.\u003cdiv id=\"Equ8\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ8\" name=\"EquationSource\"\u003e\n$$\\:\\frac{\\partial\\:{\\text{A}}_{1}}{\\partial\\:\\text{Z}}+\\frac{1}{{\\text{V}}_{{\\text{g}}_{1}}}\\frac{\\partial\\:{\\text{A}}_{1}}{\\partial\\:\\text{t}}=(-\\text{j}\\gamma\\:{P}_{2\\:}-\\frac{{\\alpha\\:}\\:}{2\\:}){\\text{A}}_{1}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e13\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ9\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ9\" name=\"EquationSource\"\u003e\n$$\\:\\frac{\\partial\\:{\\text{A}}_{2}}{\\partial\\:\\text{Z}}+\\frac{1}{{\\text{V}}_{{\\text{g}}_{2}}}\\frac{\\partial\\:{\\text{A}}_{2}}{\\partial\\:\\text{t}}=(-\\text{j}\\gamma\\:{P}_{1\\:}-\\frac{{\\alpha\\:}\\:}{2\\:}){\\text{A}}_{2}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e14\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eWhere the slowly changing complex field envelops of each wave is shown by\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\:{\\text{A}}_{\\text{k}\\:}\\left(\\text{t},\\text{z}\\right)(\\text{k}=\\text{1,2})\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\text{V}}_{{\\text{g}}_{\\text{i}},}(\\text{i}=\\text{1,2})\\)\u003c/span\u003e\u003c/span\u003e is the carrier wave's group velocity, γ and α are nonlinearity \u0026amp; absorption coefficients.\u003c/p\u003e\u003cp\u003eThe power entering the fiber is expressed as [\u003cspan additionalcitationids=\"CR11 CR12 CR13\" citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e]\u003cdiv id=\"Equ10\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ10\" name=\"EquationSource\"\u003e\n$$\\:{P}_{i}={P}_{c}(1+mcos{{\\omega\\:}}_{1}t)$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e15\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eHere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\text{P}}_{\\text{i}}\\)\u003c/span\u003e\u003c/span\u003e is input optical power at fiber\u0026rsquo;s input, m is modulation index, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\text{P}}_{\\text{c}}\\)\u003c/span\u003e\u003c/span\u003e is average optical power and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{{\\omega\\:}}_{1}\\)\u003c/span\u003e\u003c/span\u003e is optical frequency.\u003c/p\u003e\u003cp\u003eThe solution of Eq.\u0026nbsp;(\u003cspan refid=\"Equ8\" class=\"InternalRef\"\u003e13\u003c/span\u003e) may be written as\u003cdiv id=\"Equ11\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ11\" name=\"EquationSource\"\u003e\n$$\\:{\\text{A}}_{1}\\left(\\text{z},\\text{t}\\right)={\\text{A}}_{1}\\left(0,{{\\tau\\:}}_{1}\\right){\\text{e}}^{\\frac{-{\\alpha\\:}\\text{z}\\:}{2\\:}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e16\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eWe used initial conditions as \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\text{z}=0,\\:\\text{t}=\\)\u003c/span\u003e\u003c/span\u003e \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{{\\tau\\:}}_{1}\\)\u003c/span\u003e\u003c/span\u003e and ignored γ in Eq.\u0026nbsp;(\u003cspan refid=\"Equ8\" class=\"InternalRef\"\u003e13\u003c/span\u003e) to arrive at the aforementioned Eq.\u0026nbsp;(\u003cspan refid=\"Equ11\" class=\"InternalRef\"\u003e16\u003c/span\u003e).\u003c/p\u003e\u003cp\u003eSubstituting the value of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\text{A}}_{1}\\left(\\text{z},\\text{t}\\right)\\)\u003c/span\u003e\u003c/span\u003e from Eq.\u0026nbsp;(\u003cspan refid=\"Equ11\" class=\"InternalRef\"\u003e16\u003c/span\u003e) to Eq.\u0026nbsp;(\u003cspan refid=\"Equ9\" class=\"InternalRef\"\u003e14\u003c/span\u003e), we get\u003cdiv id=\"Equ12\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ12\" name=\"EquationSource\"\u003e\n$$\\:{\\text{A}}_{2}\\left(\\text{z},\\text{t}\\right)={\\text{A}}_{2}\\left(0,{{\\tau\\:}}_{2}\\right){\\text{e}}^{-\\text{j}{\\psi\\:}}{\\text{e}}^{-{\\alpha\\:}\\text{z}/2}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e17\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equb\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equb\" name=\"EquationSource\"\u003e\n$$\\:\\text{W}\\text{h}\\text{e}\\text{r}\\text{e}\\:\\:{\\psi\\:}\\:=-2{\\gamma\\:}\\int\\:{\\text{P}}_{1}\\{{0,({\\tau\\:}}_{2}+{\\text{d}}_{21}\\text{z})\\}{\\text{e}}^{-{\\alpha\\:}\\text{z}}\\text{d}\\text{z}\\:\\text{a}\\text{n}\\text{d}\\:{{\\tau\\:}}_{1}={{\\tau\\:}}_{2}+{\\text{d}}_{21}\\text{z},\\:\\text{w}\\text{h}\\text{e}\\text{r}\\text{e}\\:{\\:\\text{d}}_{21}=\\frac{1}{{\\text{V}}_{{\\text{g}}_{2}}}-\\frac{1}{{\\text{V}}_{{\\text{g}}_{1}}}$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eWe now consider that, as a result of GVD, phase modulation gets converted into intensity modulation through following relationship [\u003cspan additionalcitationids=\"CR12 CR13\" citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e].\u003cdiv id=\"Equ13\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ13\" name=\"EquationSource\"\u003e\n$$\\:{\\:P}_{2}\\left(z,{\\tau\\:}_{2}\\right)={P}_{2}\\left(0,{\\tau\\:}_{2}\\right){\\left|[1+\\text{j}\\left\\{\\frac{{\\partial\\:}^{2}\\psi\\:\\:\\:}{\\partial\\:{\\text{t}}^{2}}+\\text{j}{\\left(\\frac{\\partial\\:\\psi\\:\\:\\:}{\\partial\\:\\text{t}}\\right)}^{2}\\right\\}{F}_{1}+1+\\text{j}\\{\\frac{{\\partial\\:}^{3}\\psi\\:\\:\\:}{\\partial\\:{\\text{t}}^{3}}-3\\frac{{\\partial\\:}^{2}\\psi\\:\\:\\:}{\\partial\\:{\\text{t}}^{2}}\\frac{\\partial\\:\\psi\\:\\:\\:}{\\partial\\:\\text{t}}-\\text{j}{\\left(\\frac{\\partial\\:\\psi\\:\\:\\:}{\\partial\\:\\text{t}}\\right)}^{3}{\\}F}_{2}+1+j\\left\\{-j\\:\\frac{{\\partial\\:}^{4}\\psi\\:\\:\\:}{\\partial\\:4}-4\\frac{{\\partial\\:}^{3}\\psi\\:\\:\\:}{\\partial\\:{\\text{t}}^{3}}\\frac{\\partial\\:\\psi\\:\\:\\:}{\\partial\\:\\text{t}}-3{\\left(\\frac{{\\partial\\:}^{2}\\psi\\:\\:\\:}{\\partial\\:{\\text{t}}^{2}}\\right)}^{2}+6j\\frac{{\\partial\\:}^{2}\\psi\\:\\:\\:}{\\partial\\:{\\text{t}}^{2}}\\frac{\\partial\\:\\psi\\:\\:\\:}{\\partial\\:\\text{t}}+j{\\left(\\frac{\\partial\\:\\psi\\:\\:\\:}{\\partial\\:\\text{t}}\\right)}^{4}\\right\\}{F}_{3\\:}+1+j\\{-j\\frac{{\\partial\\:}^{5}\\psi\\:\\:\\:}{\\partial\\:{\\text{t}}^{5}}-3\\frac{{\\partial\\:}^{4}\\psi\\:\\:\\:}{\\partial\\:{\\text{t}}^{4}}\\frac{\\partial\\:\\psi\\:\\:\\:}{\\partial\\:\\text{t}}-10\\frac{{\\partial\\:}^{3}\\psi\\:\\:\\:}{\\partial\\:{\\text{t}}^{3}}\\frac{\\partial\\:\\psi\\:\\:\\:}{\\partial\\:\\text{t}}+\\:3\\frac{{\\partial\\:}^{2}\\psi\\:\\:\\:}{\\partial\\:{\\text{t}}^{2}}\\frac{\\partial\\:\\psi\\:\\:\\:}{\\partial\\:\\text{t}}-j{\\left(\\frac{\\partial\\:\\psi\\:\\:\\:}{\\partial\\:\\text{t}}\\right)}^{5}{\\}F}_{4}]\\right|}^{2}\\:\\:$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e18\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equc\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equc\" name=\"EquationSource\"\u003e\n$$\\:Here\\:{F}_{1}=-{{\\beta\\:}}_{2}\\frac{\\text{Z}}{2}\\:,\\:{F}_{2}={-{\\beta\\:}}_{3}\\frac{\\text{Z}}{6}\\:,\\:\\:{F}_{3}={-{\\beta\\:}}_{4}\\frac{\\text{Z}}{24}\\:,\\:{\\:F}_{4}={-{\\beta\\:}}_{5}\\frac{\\text{Z}}{120}\\:,\\:{{\\beta\\:}}_{\\text{j}}\\:=\\frac{{\\partial\\:}^{\\text{j}}{\\beta\\:}\\:}{\\partial\\:{{\\omega\\:}}^{\\text{j}}}\\:,\\:\\text{w}\\text{h}\\text{e}\\text{r}\\text{e}\\:\\text{j}=\\text{2,3},\\text{4,5}$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eWhere β is phase constant. Eq.\u0026nbsp;(\u003cspan refid=\"Equ13\" class=\"InternalRef\"\u003e18\u003c/span\u003e) further can be simplified as [11\u0026ndash;14,]\u003cdiv id=\"Equ14\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ14\" name=\"EquationSource\"\u003e\n$$\\:{P}_{2}\\left(z,{\\tau\\:}_{2}\\right)={P}_{2}\\left(0,{\\tau\\:}_{2}\\right){\\text{e}}^{-{\\alpha\\:}\\text{z}}\\left\\{\\right(1-2{F}_{1}-6{F}_{2}\\frac{\\partial\\:\\psi\\:\\:\\:}{\\partial\\:\\text{t}}-24{F}_{3}\\frac{{\\partial\\:}^{2}\\psi\\:\\:\\:}{\\partial\\:{\\text{t}}^{2}}-120{F}_{4}\\frac{{\\partial\\:}^{3}\\psi\\:\\:\\:}{\\partial\\:{\\text{t}}^{3}}\\left)\\frac{{\\partial\\:}^{2}\\psi\\:\\:\\:}{\\partial\\:{\\text{t}}^{2}}\\right\\}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e19\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eNow substituting the value of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\text{F}}_{1},\\)\u003c/span\u003e\u003c/span\u003e \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\text{F}}_{2}\\:,\\)\u003c/span\u003e\u003c/span\u003e \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\text{F}}_{3}\\text{a}\\text{n}\\text{d}\\:{\\:\\text{F}}_{4}\\)\u003c/span\u003e\u003c/span\u003e in Eq.\u0026nbsp;(\u003cspan refid=\"Equ14\" class=\"InternalRef\"\u003e19\u003c/span\u003e) and differentiating with respect to z, one gets subsequent Eq.\u0026nbsp;[\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e].\u003cdiv id=\"Equ15\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ15\" name=\"EquationSource\"\u003e\n$$\\:\\frac{\\partial\\:{\\text{P}}_{2}(\\text{z},{\\tau\\:}_{2})}{\\partial\\:\\text{z}}={P}_{2}\\left(0,{\\tau\\:}_{2}\\right){\\text{e}}^{-{\\alpha\\:}\\text{z}}\\{{{\\beta\\:}}_{2}+{{\\beta\\:}}_{3}\\frac{\\partial\\:\\psi\\:\\:\\:}{\\partial\\:\\text{t}}+{{\\beta\\:}}_{4}\\frac{{\\partial\\:}^{2}\\psi\\:\\:\\:}{\\partial\\:{\\text{t}}^{2}}+{{\\beta\\:}}_{5}\\frac{{\\partial\\:}^{3}\\psi\\:\\:\\:}{\\partial\\:{\\text{t}}^{3}}\\}\\frac{{\\partial\\:}^{2}\\psi\\:\\:\\:}{\\partial\\:{\\text{t}}^{2}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e20\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eHere, our main concern is to determine XPM produced crosstalk caused by 5OD coefficient in SCM-WDM optical link, therefore we need to put\u003c/p\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{{\\beta\\:}}_{2}={{\\beta\\:}}_{3}={{\\beta\\:}}_{4}=0\\)\u003c/span\u003e\u003c/span\u003e in Eq.\u0026nbsp;(\u003cspan refid=\"Equ15\" class=\"InternalRef\"\u003e20\u003c/span\u003e), therefore Eq.\u0026nbsp;(\u003cspan refid=\"Equ15\" class=\"InternalRef\"\u003e20\u003c/span\u003e) becomes\u003cdiv id=\"Equ16\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ16\" name=\"EquationSource\"\u003e\n$$\\:\\frac{\\partial\\:{\\text{P}}_{2}(\\text{z},{\\tau\\:}_{2})}{\\partial\\:\\text{z}}={P}_{2}\\left(0,{\\tau\\:}_{2}\\right){\\text{e}}^{-{\\alpha\\:}\\text{z}}{{\\beta\\:}}_{5}\\frac{{\\partial\\:}^{3}\\psi\\:\\:\\:}{\\partial\\:{\\text{t}}^{3}}\\frac{{\\partial\\:}^{2}\\psi\\:\\:\\:}{\\partial\\:{\\text{t}}^{2}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e21\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eThe impact of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{{\\beta\\:}}_{5}\\)\u003c/span\u003e\u003c/span\u003e in \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\frac{\\partial\\:{\\text{P}}_{2}(\\text{z},{\\tau\\:}_{2})}{\\partial\\:\\text{z}}\\)\u003c/span\u003e\u003c/span\u003e is given by Y therefore\u003c/p\u003e\u003cp\u003eThe XPM crosstalk resulting from the 5OD coefficient is expressed as [\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e]\u003cdiv id=\"Equ17\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ17\" name=\"EquationSource\"\u003e\n$$\\:\\text{Y}={P}_{2}\\left(0,{\\tau\\:}_{2}\\right){\\text{e}}^{-{\\alpha\\:}\\text{z}}{{\\beta\\:}}_{5}\\frac{{\\partial\\:}^{3}\\psi\\:\\:\\:}{\\partial\\:{{\\tau\\:}_{2}}^{3}}\\frac{{\\partial\\:}^{2}\\psi\\:\\:\\:}{\\partial\\:{{\\tau\\:}_{2}}^{2}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e22\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eNow using Eqs.\u0026nbsp;(\u003cspan refid=\"Equ10\" class=\"InternalRef\"\u003e15\u003c/span\u003e) \u0026amp; (\u003cspan refid=\"Equ12\" class=\"InternalRef\"\u003e17\u003c/span\u003e), we have\u003cdiv id=\"Equd\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equd\" name=\"EquationSource\"\u003e\n$$\\:{P}_{1}(0,{\\tau\\:}_{1})={P}_{c}\\{1+mcos({{\\omega\\:}}_{1}{\\tau\\:}_{1}\\left)\\right\\}$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{P}_{1}\\{0,{(\\tau\\:}_{2}+{d}_{21}z)\\}={P}_{c}[1+mcos{\\{{\\omega\\:}}_{1}{(\\tau\\:}_{2}+{d}_{21}z\\left)\\right\\}]\\)\u003c/span\u003e\u003c/span\u003e And\u003cdiv id=\"Equ18\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ18\" name=\"EquationSource\"\u003e\n$$\\:\\psi\\:=2\\gamma\\:{\\int\\:}_{0}^{z}{P}_{c}[1+mcos\\left\\{{{\\omega\\:}}_{1}{(\\tau\\:}_{2}+{d}_{21}z\\right)\\}]{\\text{e}}^{-{\\alpha\\:}\\text{z}}\\text{d}\\text{z}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e23\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eDifferentiating \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\psi\\:}\\)\u003c/span\u003e\u003c/span\u003e thrice with respect to\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\:{{\\tau\\:}}_{2}\\)\u003c/span\u003e\u003c/span\u003e and putting value of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\frac{{\\partial\\:}^{2}{\\psi\\:}\\:\\:}{\\partial\\:{{{\\tau\\:}}_{2}}^{2}}\\)\u003c/span\u003e\u003c/span\u003e and\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\frac{{\\partial\\:}^{3}{\\psi\\:}\\:\\:}{\\partial\\:{{{\\tau\\:}}_{2}}^{3}}\\)\u003c/span\u003e\u003c/span\u003e in Eq.\u0026nbsp;(\u003cspan refid=\"Equ17\" class=\"InternalRef\"\u003e22\u003c/span\u003e), we get\u003cdiv id=\"Equ19\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ19\" name=\"EquationSource\"\u003e\n$$\\:\\text{Y}\\:=\\:{P}_{c}{\\text{e}}^{-{\\alpha\\:}\\text{z}}{{\\beta\\:}}_{5}\\:\\frac{{\\left(2\\gamma\\:{P}_{c}m{\\text{e}}^{\\text{i}{{\\omega\\:}}_{1}{\\tau\\:}_{2}}\\right)}^{4}{{{\\omega\\:}}_{1}}^{5}}{{(\\text{i}{{\\omega\\:}}_{1}{d}_{21}-{\\alpha\\:})}^{2}}{\\{{\\text{e}}^{\\left(\\text{i}{{\\omega\\:}}_{1}{d}_{21}-{\\alpha\\:}\\right)\\text{z}}-1\\}}^{4}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e24\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eThis gradual change is lessened by a factor \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\text{e}}^{-{\\alpha\\:}(\\text{L}-\\text{z})}\\)\u003c/span\u003e\u003c/span\u003e due to attenuation. The modulation is accomplished at the fibre\u0026rsquo;s end by distributing attenuated power along a fibre of length L\u003cdiv id=\"Equ20\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ20\" name=\"EquationSource\"\u003e\n$$\\:{\\int\\:}_{0}^{\\text{L}}\\text{Y}{\\text{e}}^{-{\\alpha\\:}(\\text{L}-\\text{z})}\\text{d}\\text{z}\\:=\\:\\:{-{\\beta\\:}}_{5}{P}_{c}\\:\\frac{{\\left(2\\gamma\\:{P}_{c}m\\right)}^{4}{{{\\omega\\:}}_{1}}^{5}{\\text{e}}^{4\\text{i}{{\\omega\\:}}_{1}{\\tau\\:}_{2}}}{12{\\left(\\text{i}{{\\omega\\:}}_{1}{d}_{21}-{\\alpha\\:}\\right)}^{4}\\:\\left(\\text{i}{{\\omega\\:}}_{1}{d}_{21}-{\\alpha\\:}\\right)}{\\text{e}}^{-{\\alpha\\:}\\text{L}}[{12{(\\text{i}{\\omega\\:}}_{1}{d}_{21})\\text{L}-12{\\alpha\\:}\\text{L}-48{(\\text{e}}^{\\left(\\text{i}{{\\omega\\:}}_{1}{d}_{21}-{\\alpha\\:}\\right)\\text{L}}-1)+36\\left({\\text{e}}^{2\\left(\\text{i}{{\\omega\\:}}_{1}{d}_{21}-{\\alpha\\:}\\right)\\text{L}}-1\\right)-16\\left({\\text{e}}^{3\\left(\\text{i}{{\\omega\\:}}_{1}{d}_{21}-{\\alpha\\:}\\right)\\text{L}}-1\\right)+3(\\text{e}}^{4\\left(\\text{i}{{\\omega\\:}}_{1}{d}_{21}-{\\alpha\\:}\\right)\\text{L}}-1)]$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e25\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eThe process of normalizing Eq.\u0026nbsp;(\u003cspan refid=\"Equ20\" class=\"InternalRef\"\u003e25\u003c/span\u003e) yields the crosstalk in phasor form. One can obtain crosstalk in phasor form after multiplying Eq.\u0026nbsp;(\u003cspan refid=\"Equ18\" class=\"InternalRef\"\u003e23\u003c/span\u003e) by (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\frac{1}{{\\text{m}\\text{P}}_{\\text{C}}{\\text{e}}^{-{\\alpha\\:}\\text{L}}}\\)\u003c/span\u003e\u003c/span\u003e) factor.\u003c/p\u003e\u003cp\u003eXPM crosstalk resulting from 5OD is provided by\u003cdiv id=\"Equ21\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ21\" name=\"EquationSource\"\u003e\n$$\\:{XT}_{{XPM}_{5OD}}=\\:{-{\\beta\\:}}_{5}\\:\\frac{4{\\left(\\gamma\\:{P}_{c}\\right)}^{4}{{{\\omega\\:}}_{1}}^{5}{\\text{m}}^{3}{\\text{e}}^{4\\text{i}{{\\omega\\:}}_{1}{\\tau\\:}_{2}}}{3{\\left(\\text{i}{{\\omega\\:}}_{1}{d}_{21}-{\\alpha\\:}\\right)}^{4}\\:\\left(\\text{i}{{\\omega\\:}}_{1}{d}_{21}-{\\alpha\\:}\\right)}[{12{(\\text{i}{\\omega\\:}}_{1}{d}_{21})\\text{L}-12{\\alpha\\:}\\text{L}-48{(\\text{e}}^{\\left(\\text{i}{{\\omega\\:}}_{1}{d}_{21}-{\\alpha\\:}\\right)\\text{L}}-1)+36\\left({\\text{e}}^{2\\left(\\text{i}{{\\omega\\:}}_{1}{d}_{21}-{\\alpha\\:}\\right)\\text{L}}-1\\right)-16\\left({\\text{e}}^{3\\left(\\text{i}{{\\omega\\:}}_{1}{d}_{21}-{\\alpha\\:}\\right)\\text{L}}-1\\right)+3(\\text{e}}^{4\\left(\\text{i}{{\\omega\\:}}_{1}{d}_{21}-{\\alpha\\:}\\right)\\text{L}}-1)]$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e26\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eSimplifying further Eq.\u0026nbsp;(\u003cspan refid=\"Equ21\" class=\"InternalRef\"\u003e26\u003c/span\u003e) we get\u003cdiv id=\"Eque\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Eque\" name=\"EquationSource\"\u003e\n$$\\:{XT}_{{XPM}_{5OD}}=\\:{-{\\beta\\:}}_{5}\\:\\frac{4{\\left(\\gamma\\:{P}_{c}\\right)}^{4}{{{\\omega\\:}}_{1}}^{5}{\\text{m}}^{3}{\\text{e}}^{4\\text{i}{{\\omega\\:}}_{1}{\\tau\\:}_{2}}}{3{\\left(\\text{i}{{\\omega\\:}}_{1}{d}_{21}-{\\alpha\\:}\\right)}^{5}\\:}[-12{\\alpha\\:}\\text{L}+25-48{\\text{e}}^{-{\\alpha\\:}\\text{L}}\\text{cos}\\left({{\\omega\\:}}_{1}{d}_{21}\\text{L}\\right){+36\\text{e}}^{-2{\\alpha\\:}\\text{L}}\\text{cos}\\left(2{{\\omega\\:}}_{1}{d}_{21}\\text{L}\\right)-16{\\text{e}}^{-3{\\alpha\\:}\\text{L}}\\text{cos}\\left(3{{\\omega\\:}}_{1}{d}_{21}\\text{L}\\right)+3{\\text{e}}^{-4{\\alpha\\:}\\text{L}}\\text{cos}\\left(4{{\\omega\\:}}_{1}{d}_{21}\\text{L}\\right)\\:+\\:\\:\\:\\:\\text{i}\\left\\{12{{\\omega\\:}}_{1}{d}_{21}\\text{L}{-4{8\\text{e}}^{-{\\alpha\\:}\\text{L}}\\text{s}\\text{i}\\text{n}{({\\omega\\:}}_{1}{d}_{21}\\text{L})+36\\text{e}}^{-2{\\alpha\\:}\\text{L}}\\text{sin}\\left(2{{\\omega\\:}}_{1}{d}_{21}\\text{L}\\right)-16{\\text{e}}^{-3{\\alpha\\:}\\text{L}}\\text{s}\\text{i}\\text{n}{(3{\\omega\\:}}_{1}{d}_{21}\\text{L}\\right)+3{\\text{e}}^{-4{\\alpha\\:}\\text{L}}\\text{s}\\text{i}\\text{n}{(4{\\omega\\:}}_{1}{d}_{21}\\text{L}\\:\\}]$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eThe fifth-order dispersion parameter is defined here as [\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e]\u003c/p\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{{\\beta\\:}}_{5}=\\frac{{\\lambda\\:}^{4}}{({2{\\pi\\:}\\text{c})}^{4}}({\\lambda\\:}^{4}{\\text{D}}_{3}+{12\\lambda\\:}^{3}{\\text{D}}_{2}+{36\\lambda\\:}^{2}{\\text{D}}_{1}+24\\lambda\\:D\\)\u003c/span\u003e\u003c/span\u003e) (27)\u003c/p\u003e\u003cp\u003eHere letters, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\text{D},\\:{\\lambda\\:}\\:\\\u0026amp;\\:\\text{c}\\:\\)\u003c/span\u003e\u003c/span\u003erepresent chromatic dispersion, wavelength of light and velocity of light respectively and\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\:{\\text{D}}_{1}\\)\u003c/span\u003e\u003c/span\u003e,\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\text{D}}_{2}\\)\u003c/span\u003e\u003c/span\u003e \u0026amp; \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\text{D}}_{3}\\)\u003c/span\u003e\u003c/span\u003e are first, second and third order GVD parameters.\u003c/p\u003e\u003c/div\u003e"},{"header":"3. RESULTS AND DISCUSSION","content":"\u003cp\u003eThe values of various parameters have been considered as:\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\:{\\Delta\\:}{\\lambda\\:}=2\\text{n}\\text{m}\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\text{L}=50\\text{k}\\text{m}\\)\u003c/span\u003e\u003c/span\u003e,\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\:{\\alpha\\:}=0.25\\text{d}\\text{B}/\\text{k}\\text{m}\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\lambda\\:}_{1}=1550\\text{n}\\text{m}\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\lambda\\:}_{2}=1552\\text{n}\\text{m},\\:\\:\\text{D}=2\\text{p}\\text{s}/(\\text{n}\\text{m}.\\:\\text{k}\\text{m})\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{D}_{1}=0.084\\text{p}\\text{s}/(\\text{n}\\text{m}.\\:\\text{k}\\text{m})\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{D}_{2}=0.00023\\text{p}\\text{s}/(\\text{n}\\text{m}.\\:\\text{k}\\text{m})\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{D}_{3}=0.0000092\\text{p}\\text{s}/(\\text{n}\\text{m}.\\:\\text{k}\\text{m})\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{n}_{2}=2.67\\text{x}{10}^{-20}\\frac{{\\text{m}}^{2}}{\\text{W}}\\)\u003c/span\u003e\u003c/span\u003e, m\u0026thinsp;=\u0026thinsp;0.7 and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\text{A}}_{\\text{e}\\text{f}\\text{f}}=81{{\\mu\\:}\\text{m}}^{2}\\)\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\u003cp\u003eFrom Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e, it is clear that the XPM crosstalk attributed to the 5OD coefficient changes from (-315dB to -270dB) when the modulation frequency ranges from 0.4GHz to 5GHz, as determined by the NLSE. In contrast, the XPM induced crosstalk caused by 5OD coefficient fluctuates from (-280dB to -255.09dB) when the modulation frequency increases from 0.2GHz and 5GHz, utilizing couple equations.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eAt a modulation frequency of 3GHz, XPM crosstalk caused by 5OD coefficient is measured at -274dB and \u0026minus;\u0026thinsp;256.08dB for the NLSE and couple equations, respectively.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eXPM crosstalk fluctuates between \u0026minus;\u0026thinsp;295.1dB to -255.06dB using NLSE when optical power changes from 0.1mW to 2mW.\u003c/p\u003e\u003cp\u003eIn contrast, the XPM crosstalk caused by 5OD coefficient changes from (-270.1dB to -240dB) when optical power varies from 0.1mW to 2mW using couple equations. At an optical power of 1mW, the XPM crosstalk values are \u0026minus;\u0026thinsp;244.5dB and \u0026minus;\u0026thinsp;265.01dB when calculated using couple equations and NLSE respectively.\u003c/p\u003e\u003cp\u003eFrom Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e it is deduced that the XPM crosstalk attributed to the 5OD coefficient escalates between (-346.03 dB to -297 dB) as the transmission length varies between 3 to 50 km when employing the NLSE. Conversely, the XPM crosstalk related to the 5OD coefficient fluctuates between (-320 dB to -281 dB) when the transmission length ranges from 3 km to 50 km employing couple equations. At a distance of 30 km, the XPM crosstalk caused by 5OD coefficient is measured at -285dB and \u0026minus;\u0026thinsp;302dB when utilizing couple equations and NLSE, respectively.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e"},{"header":"4. Conclusion","content":"\u003cp\u003eThis research paper gives comprehensive analysis of the effects of higher-order dispersion, specifically 5OD, on XPM crosstalk. It has also been noted that the HOD terms significantly affect crosstalk induced by XPM. Furthermore, the The impact of the HOD decreases as the order of the dispersion term increases. It has been deduced that the examination of induced crosstalk by XPM resulting from the 5OD coefficient through the use of NLSE yields a lower level of crosstalk when compared to the couple equations. The XPM crosstalk related to the 5OD parameter spans from (-280dB to -255.09dB) when utilizing couple equations, whereas it varies between (-315dB to -270dB) when applying NLSE as the modulation frequency rises from 0.4 to 5GHz. Moreover, XPM crosstalk is noted to be between (-320dB to -281dB) with couple equations, while it increases from (-346.03dB to -297dB) when using NLSE as the transmission distance varies between 3 to 50 km. In addition, XPM crosstalk fluctuates between (-270.1dB to -240.2dB) with couple equations, while it is observed within the range of (-295.1dB to -255.06dB) using NLSE as the optical power varies from 0.1 to 2mW.\u003c/p\u003e"},{"header":"Declarations","content":"\u003ch2\u003eAuthor Contribution\u003c/h2\u003e\u003cp\u003eThe authors confirm their contribution to the paper as follows: VS has made a substantial contribution to the concept and design of the article and performed work of validation and wrote the article. NCJ, SG and YP did an analysis and interpretation of the results. We all authors reviewed the results and approved the final version of the manuscript.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\u003cli\u003e\u003cspan\u003eWay, W.I., Wagner, S.S., Choy, M.M., Lin, C., Menendez, R.C., Tohme, H., Yi-Yan, A., Von Lehman, A.C., Spicer, R.E., Andrejco, M., Saifi, M.A.: Simultaneous distribution of multichannel analog and digital video channels to multiple terminals using high density WDM and a broad-band in-line Erbium doped fiber amplifier. IEEE Photonics Technol. Lett. \u003cb\u003e2\u003c/b\u003e(9), 665\u0026ndash;668 (1990)\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eLee, C.C., Chi, S.: Three-wavelength-division-multiplexed multichannel subcarrier-multiplexing transmission over multimode fiber with potential capacity of 12Gbps. IEEE Photonics Technol. Lett. \u003cb\u003e11\u003c/b\u003e(8), 1066\u0026ndash;1068 (1999)\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eHui, R., Zhu, B., Huang, R., Allen, C., Demarest, K., Richards, D.: 10Gbps SCM fiber system using optical SSB modulation. IEEE Photonics Technol. Lett. \u003cb\u003e13\u003c/b\u003e(8), 896\u0026ndash;898 (2001)\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eWoodward, S.L., Lu, X., Darcie, T.E., Bodeep, G.E.: Reduction of optical-beat interference in subcarrier networks. IEEE Photonics Technol. Lett. \u003cb\u003e8\u003c/b\u003e(5), 694\u0026ndash;696 (1996)\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003ePhillips, M.R., Ott, D.M.: Crosstalk due to optical fiber non linearities in WDM CATV lightwave systems. J. Lightwave Technol. \u003cb\u003e17\u003c/b\u003e(10), 1782\u0026ndash;1792 (1999)\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eYang, F.S., Marhic, M.E., Kazovsky, L.G.: Nonlinear crosstalk and two countermeasures in SCM\u0026ndash;WDM optical communication systems. IEEE Journal Lightwave Technology. \u003cb\u003e18\u003c/b\u003e(4), 512\u0026ndash;520 (2000)\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eWang, Z., Bodtker, E., Jacobsen, G.: Effects of cross phase modulation in wavelength multiplexed SCM video transmission systems. Electron. Lett. \u003cb\u003e31\u003c/b\u003e, 1591\u0026ndash;1592 (1995)\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eKumar, N., Sharma, A.K., Kapoor, V.: XPM-induced crosstalk with higher-order dispersion in SCM\u0026ndash;WDM optical transmission link. Optik Int. J. Light Electron. Opt. \u003cb\u003e123\u003c/b\u003e(22), 2056\u0026ndash;2061 (2012)\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eArya, S.K., Sharma, A.K., Agarwala, R.A.: Impact of 2OD and 3OD on SRS-and XPM-induced crosstalk in SCM-WDM optical transmission link. Optik. \u003cb\u003e120\u003c/b\u003e(8), 364\u0026ndash;369 (2009)\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eKumar, N., Sharma, A.K., Kapoor, V.: Improved XPM-induced crosstalk with higher-order dispersion in SCM\u0026ndash;WDM optical transmission link. Optik. \u003cb\u003e124\u003c/b\u003e, 941\u0026ndash;944 (2013)\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eSingh, V., Kumar, N., Kumar, S.: Comparative analysis of XPM-induced crosstalk in SCM-WDM transmission links. Opt. Quant. Electron. \u003cb\u003e48\u003c/b\u003e, 8 (2016)\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eSingh, V., Kumar, S., Dimri, P.K.: Comparative study of XPM-induced crosstalk due to 3OD parameter in SCM-WDM transmission system, Optik \u003cb\u003e186C\u003c/b\u003e 177\u0026ndash;181. (2019)\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eSingh, V., Kumar, S., Dimri, P.K.: Performance evaluation of SCM\u0026ndash;WDM-HAN communication link using millimeter waves in the presence of XPM, Optik 164580. (2020)\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eSingh, V., Kumar, S., Dimri, P.K.: Comparative analysis of XPM induced crosstalk due to 4OD coefficient in SCM-WDM link. Optik. \u003cb\u003e241\u003c/b\u003e, 166924 (2021)\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Cross phase modulation, Group velocity dispersion, Subcarrier multiplexing, Wavelength division multiplexing, Higher order dispersion, Nonlinear Schrödinger equation","lastPublishedDoi":"10.21203/rs.3.rs-7106552/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7106552/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThis research investigates the impact of cross phase modulation (XPM) caused crosstalk due to higher order dispersion (HOD) within subcarrier multiplexed wavelength-division multiplexed (SCM–WDM) communication systems across different optical power levels, modulation frequencies and transmission lengths. We carried out an analytical comparison of XPM induced crosstalk due to 5\u003csup\u003eth\u003c/sup\u003e order dispersion (5OD) coefficient using nonlinear Schrödinger equation (NLSE) and couple equations in SCM–WDM communication links. It is noted that when transmission lengths, optical powers, and modulation frequencies grow, XPM-induced crosstalk has been shown to increase exponentially. Though less significant than 2OD, the effects of 3OD, 4OD, and 5OD coefficients are nonetheless present. It can be further noted that the analysis of XPM crosstalk through NLSE is reliable compared to couple equations, as only a single approximation method has been utilized to reach this conclusion using NLSE. Consequently, the results derived from NLSE are definitive and precise. The XPM crosstalk associated with the 5OD parameter ranges from (-280dB to -255.09dB) when using couple equations, while it fluctuates between (-315dB to -270dB) when employing NLSE as the modulation frequency increases from 0.4 to 5GHz. Additionally, XPM crosstalk is observed to be between (-320dB to -281dB) with couple equations, whereas it escalates from (-346.03dB to -297dB) when using NLSE as the transmission length extends from 3 to 50 km. Furthermore, XPM crosstalk varies from (-270.1dB to -240.2dB) with couple equations, while it is found within the range of (-295.1dB to -255.06dB) using NLSE as the optical power changes from 0.1 to 2mW.\u003c/p\u003e","manuscriptTitle":"A comparative examination of the crosstalk induced by XPM as a result of the 5OD coefficient in SCM-WDM communication system","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-07-15 04:29:39","doi":"10.21203/rs.3.rs-7106552/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"74c76e06-b7fd-4b9e-bf8b-9cfb8d29cf64","owner":[],"postedDate":"July 15th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2025-10-20T07:08:16+00:00","versionOfRecord":[],"versionCreatedAt":"2025-07-15 04:29:39","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-7106552","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-7106552","identity":"rs-7106552","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}