1 Introduction

The growing number and expectations of end users which require high-bandwidth services have given rise to the evolution of access networks (Raghavan et al. 2005; Chamberland 2010). Passive optical networks (PONs) are popular options in modern access networks. xPON standards, i.e. APON, BPON, GPON, EPON, 10G-EPON, XG-PON, NG-PON2, XGS-PON and NG-EPON, have been emerged to overcome rapidly increasing demands of end-users for high transmission rate and low latency (Wey and Zhang 2019). With its advantage in boosting system capacity as well as enhancing the network design flexibility, wavelength division multiplexing (WDM) is a crucial technology in optical communications (Wagner and Kobrinsky 1989). Since its invention at Bellcore (Wagner et al. 1988), WDM-PON has always attracted the intention of researchers over the last three decades (Iannone et al. 1996; Han et al. 2004; Zhang and Ansari 2010; Choudhury and Khan 2016; Fazea 2019; Roka 2021; Verma and Garg 2023). International Telecommunication Union (ITU) released G.9802.1 recommendation for WDM-PON in 2021 (ITU-T 2021). The releasing of G.9802.1 has raised up the usage of WDM-PON in both promising fronthaul structures for 5G and beyond networks (Honda et al. 2021; Effenberger and Zhang 2022; Akhtar et al. 2023; Jaffer et al. 2023; Kumari 2023a; Yücel and Acıkgöz 2023) and hybrid fiber-free space optics PONs (FFSO-PONs) or full free space optics PONs (FSO-PONs) (Hayal et al. 2021; Kumari et al. 2021a; Elsayed et al. 2022; Elsayed et al. 2024; Hsu et al. 2022; Yeh et al. 2022; Chen et al. 2023; El-Nahal et al. 2023). In particular, the usage of hybrid time and wavelength division multiplexing PON (TWDM-PON) has gained an emerging importance for next generation PON (NG-PON) structures (Kaur et al. 2021; Garg et al. 2022; Kumari 2023b, 2024; Kumari et al. 2023; Abdulla et al. 2024).

With the widespread usage of dense WDM (DWDM) technology, where channel spacings are narrower than 100 GHz and requirements of high numbers of end-users can be fulfilled, DWDM-PON has also attracted the intention of researchers. The main research topics in the literature about DWDM-PONs can be listed as short- or medium-reach DWDM-PON configurations employing various injection locking techniques (Tsai et. al 2006; Lin et al. 2010; Li et al. 2013), long-reach DWDM-PON architectures with various light sources (Lee et al. 2007a; Zhou et al. 2016; Usman 2020), usage of reflective semiconductor optical amplifiers (RSOAs) in DWDM-PONs (Zhou et al. 2013; Batalha et al. 2018), novel colorless technologies and remodulation schemes for DWDM-PONs (Yu et al. 2007; Chow 2008; Saliou et al. 2015; Chen et al. 2017), usage of arrayed waveguide gratings (AWGs) in DWDM-PON architectures (Lin et al. 2009; Husein and El Nahal 2014), channel-reuse bidirectional transmission in DWDM-PON systems (Zhang et al. 2014, 2015), usage of different modulation methods in DWDM-PONs (Giorgi et al. 2009; Huszanik et al. 2018; Lubana and Kaur 2024) and DWDM-PON proposals for various application areas (Lee et al. 2007b; Lu et al. 2007; Sarmiento et al. 2019; Braunfelds et al. 2023).

However, with usage of channel spacings below 100 GHz in DWDM technology, optical nonlinear phenomena occurring due to Kerr effect, i.e. self phase modulation (SPM), cross phase modulation (XPM) and four-wave mixing (FWM), and scattering effects, in particular stimulated Raman scattering (SRS), have begun to show significant impacts on DWDM-based communication networks (Tang et al. 2003; Schneider 2004; Kaur et al. 2010; Sabapathi and Sundaravadivelu 2011; Agrawal 2019). Thus, DWDM-PON system performance under single or combined impacts of nonlinear impairments has also been analyzed in the literature with various papers.

In Abdul-Rashid et al. (2006), (2007) and Abbou and Choong (2015) subcarrier multiplexing WDM-PON (SCM-WDM PON) performance limitations due to the presence of XPM and group velocity dispersion (GVD) were analyzed with semi-analytical approach in systems with various channel numbers, 500 GHz channel spacings and 50 km standard single mode fiber (SSMF) lengths. In Singh et al. (2016) and Singh et al. (2019), considering second-order and third-order dispersion effects, respectively, XPM-induced nonlinear crosstalk in 2-channel SCM-WDM PONs with 250 GHz and 500 GHz channel spacings was comparatively analyzed with coupled and Schrödinger equations. In Acharya and Raja (2007), SRS mitigation in a 5-channel WDM-PON implemented with 15 km SSMF and 1000 GHz spacings was achieved with the proposed method. In Kim et al. (2014), SRS-induced power depletions due to a high-power optical time domain reflectometry (OTDR) signal in a 32-channel DWDM system used as a mobile fronthaul with 100 GHz spacings and 20 km total fiber length were simulated. In Gupta et al. (2015), reduction of SRS crosstalk in a WDM-time division multiplexing (WDM-TDM) PON where transmitted signals in a 5-channel architecture were splitted to 63 subscribers in a system with 20 km fiber length was accomplished by Dicode coding. In Neto et al. (2017), SRS induced power depletion in downlink channels of GPON systems occurring as a result of coexistence of ultra-dense WDM-PON (UDWDM-PON) and GPON systems was investigated experimentally and with simulations. In Bi et al. (2014), FWM mitigation was provided with a novel channel allocation method and related simulation results in a 12-channel WDM-PON with a channel spacing range of 3–100 GHz were presented. In Manzoor et al. (2015) and Manzoor et al. (2020), FWM mitigation was performed by using alternative circular polarizers and PMD emulator, respectively, in an 8-channel DWDM-PON. In Vardanyan (2017), a wide guarding band between uplink and downlink channels preventing the crosstalk among original signals and FWM products was proposed for DWDM-PONs and results of Q-factor simulations for 32-, 64- and 128-channel systems using the proposed architecture were presented. In Ahmed and Sahu (2018), FWM suppression based on unequal channel spacing system architectures was examined and related simulation results in 12-channel UDWDM-PONs were presented. In Song et al. (2021), unbounded from each other, XPM and FWM impacts on DWDM-PON performance were observed experimentally for 3- and 4-channel systems and numerically for a 12-channel system considering 100 GHz, 200 GHz and 400 GHz spacings and 25 km SSMFs. In Saleem et al. (2023) the FWM-induced and XPM-induced sensitivity penalties of WDM-PON based 5G fronthauls using various modulation formats were investigated. In Reis et al. (2012), SPM, XPM and FWM impacts on the performance of a 32-channel UDWDM-PON were researched with Volterra series independently from one another for system architectures with 3.125 GHz channel spacings and SSMF lengths of 25 km, 60 km and 100 km. In Reza et al. (2023) semiconductor optical amplifier (SOA)-induced nonlinearities such as SPM, XPM, FWM and amplified spontaneous emission (ASE) noise on a 4-channel DWDM-PON with 50 GHz, 100 GHz and 200 GHz spacings were focused on both experimentally and numerically and recurrent neural network based mitigation of those nonlinearities was proposed.

The above-listed milestone papers show that general trend in the literature in analyzing effects of nonlinear impairments on WDM-PON is to focus on a single impact or examine a group of nonlinearities but independently from one another or concentrate on a common optical fiber parameter that is important for a group of nonlinear phenomena. The combined dual and triple impacts of nonlinearities on the performance of WDM-PONs have rarely been studied and to the best of our knowledge no paper exists in the literature dealing with combined quadruple impact of nonlinearities on WDM-PON performance. This is because of the rather complicated nature of nonlinear optical impairments and enhanced modelling complexity when combined multiple impacts are considered.

Furthermore, fiber nonlinearities begin to attract the intention of researchers that focus on proposing novel architectures and hybrid multiplexing techniques for next generation optical access networks.

In Kachhatiya and Prince (2018a, b), focusing on wavelength-routed, power-split and wavelength-selected optical distributed network (ODN) architectures, performance optimization of NG-PON2 structures which use various commercially available optical fibers under the impact of fiber nonlinearities, in particular SPM and XPM, was accomplished by optimizing the launch power. In Kumari et al. (2020), a novel PON system using hybrid multiplexing methods, i.e. backward compatible TWDM with wavelength division multiplexing optical code division multiple access (WDM-OCDMA), and being able to provide a ‘smooth up-gradation to NG-PON3 standards’ was proposed. The performance of the proposed system was evaluated considering the impact of FWM noise on the system. In Xu et al. (2021), the impact of FWM on lost budget of 4 × 25 Gbps and 4 × 50 Gbps NG-PON architectures using non-return to zero (NRZ) and four-level pulse amplitude modulation (PAM4), respectively, in O-band transmission range with 800 GHz channel spacings was analyzed both experimentally and with simulations. In Kumari et al. (2021b), a novel TWDM-PON architecture incorporating various polarization multiplexed modulation techniques was proposed. The performance of the proposed system was evaluated with simulations and the impact of FWM on transmitted signals was shown and interpreted. In Kumari et al. (2022), 2 × 8-channel WDM-PON, TDM-PON and optical time division multiplexing PON (OTDM-PON) systems with a total transmission capacity of 2 × 20 Gbps were proposed for long reach NG-PON2. Comparative analysis of systems was performed and results were interpreted with taking fiber nonlinearities, i.e. both Kerr and scattering nonlinearities, into account in computing total loss. In Xu and Wang (2022), the impact of XPM on three different NG-PON2 system architectures was analyzed with simulations. In Kumari (2022), 16 × 10 Gbps super-PON architecture using TWDM/DWDM for 5G fiber-wireless access networks was proposed. In performance evaluation, the impact of residual chromatic dispersion, FWM and fiber nonlinearities on the system performance was also interpreted. In Pagare et al. (2021, 2022a, b) modelling, analysis and optimization of ODN and derivation of extended power budgets were performed for various implementations of NG-PON2 systems by paying a particular attention to modeling, analysis and estimation of optical nonlinearity coefficient γ, which has an important role in SPM, XPM and FWM Kerr nonlinearities.

In the outstanding and inspiring papers given above for next generation and future optical access networks, a similar case to the one previously emphasized for mile-stone papers of WDM-PON in the literature has been observed. Besides the rather complicated nature of nonlinear optical impairments, the enhanced modelling complexity of next generation and future optical access networks depending on the usage of hybrid multiplexing techniques and modulation formats makes it difficult to observe the combined impact of nonlinearities on those systems. Therefore, in the above-listed papers focusing on next generation and future optical access networks, although the importance of nonlinear impacts on the system performance has been considered, generally the dominant single impact, in particular FWM, or double impact of SPM and XPM or a common optical fiber parameter that is important for a group of nonlinear phenomena has been taken into account in analyses. To the best of our knowledge, there is yet no paper in the literature presenting an in-depth analysis of the combined quadruple impact of nonlinearities on the system performance of NG-PON2 and other future optical access network proposals.

In one of our previous works, benefiting from SSMF’s ability of operating in both wavelength regions of 1310 nm and 1550 nm, a novel WDM-PON architecture was proposed (Karlık 2023). The dual impact of FWM and SRS on both central uplink and central downlink channels of the proposed architecture was analyzed to show the feasibility of the proposed system in real applications (Karlık 2023). In another previous work, we dealt with combined impact of Kerr nonlinearities on long-haul DWDM network performance (Yıldırım and Karlık 2023). Benefiting from those former experiences, quadruple impact of SPM, XPM, FWM and SRS on the system performance of DWDM-PONs has been investigated in this paper for both uplink channels (ULCs) and downlink channels (DLCs).

The detailed literature analysis given above for both conventional WDM-PONs and next generation/future PON proposals shows the lack of an in-depth analysis of the combined quadruple impact of SPM, XPM, FWM and SRS on both ULCs and DLCs of WDM-based PONs. The novelty and contribution of this paper is its performing that analysis for transmission lengths up to 80 km on bidirectional DWDM-PON system structures with both moderate and high numbers of channels and a channel spacing range of 12.5–100 GHz. Therefore, it is thought that with this paper an obvious gap in the performance analysis of WDM-PONs is eliminated and an inspiring reference is presented for system analysis of next generation/future PON proposals.

In Sect. 2, theoretical background for SPM, XPM, FWM and SRS is presented. The model, important system and fiber parameters and conditions used in simulations are described in Sect. 3. Simulation results and related comments are given in Sect. 4.

2 Theoretical background for SPM, XPM, FWM and SRS in WDM systems

The variation in the optical fiber refractive index due to the intensity of the pulse propagating in the optical fiber is called the optical Kerr effect. Mathematical description of this phenomenon can be given as (Schneider 2004; Agrawal 2019)

$${\text{n}} = {\text{n}}_{0} + {\text{n}}_{2} {\text{I}} = {\text{n}}_{0} + {\text{n}}_{2} \left( {\frac{{\text{P}}}{{{\text{A}}_{{{\text{eff}}}} }}} \right)$$
(1)

where n represents refractive index of the optical fiber occurring due to optical Kerr effect, n0 is linear refractive index of the optical fiber, n2 describes nonlinear refractive index of the optical fiber, I denotes intensity of the light pulse propagating in the optical fiber, P is the power of the light pulse and Aeff represents effective area of the optical fiber.

Nonlinear impairments occurring as a result of the optical Kerr effect are SPM, XPM and FWM.

A nonlinear phase shift φNL occurs in the light pulse while propagating through the optical fiber due to the refractive index variation caused by Kerr effect. That phase shift can be expressed as (Schneider 2004; Agrawal 2019)

$$\varphi_{{{\text{NL}}}} = \gamma {\text{PL}}_{{{\text{eff}}}}$$
(2)

where γ and Leff are nonlinearity coefficient and effective length of the fiber, respectively.

γ can be computed with (Schneider 2004; Agrawal 2019)

$$\gamma = \frac{{2\pi {\text{n}}_{2} }}{{\lambda {\text{A}}_{{{\text{eff}}}} }}$$
(3)

where λ is the operating wavelength.

A shift in the frequency of the light pulse and spectral broadening which can be expressed with (Schneider 2004; Agrawal 2019)

$$\Delta \omega = - \frac{{{\text{d}}\varphi_{{{\text{NL}}}} }}{{{\text{dt}}}} = - \gamma {\text{L}}_{{{\text{eff}}}} \frac{{{\text{dP}}}}{{{\text{dt}}}}$$
(4)

occur  due to φNL. As given in (4), the main cause of the spectral broadening is the power of the light pulse varying with the time. When the spectral broadening occurs due to the pulse itself traveling in an optical channel, without the impact of any external impairment, SPM exists.

Contrary to SPM, XPM occurs when at least two distinct pulses interact with each other, i.e. the phase modulation occurring in a pulse is due to the intensity variation in another pulse in XPM. Thus, since two or generally much more optical channels are present in WDM systems, XPM can be a crucial nonlinear impact for those systems.

Total nonlinear phase shift in a specific kth channel of a WDM system, i.e. φNLk can be given as (Schneider 2004; Agrawal 2019)

$$\varphi_{{{\text{NLk}}}} = \gamma {\text{L}}_{{{\text{eff}}}} \left( {{\text{P}}_{{\text{k}}} + 2\mathop \sum \limits_{{{\text{i}} \ne {\text{k}}}} {\text{P}}_{{\text{i}}} } \right)$$
(5)

where light pulses propagating in all channels are considered in the same polarization. In (5), the term \(\gamma {\text{L}}_{{{\text{eff}}}} {\text{P}}_{{\text{k}}}\) defines SPM impact and the term \(2\gamma {\text{L}}_{{{\text{eff}}}} \mathop \sum \limits_{{{\text{i}} \ne {\text{k}}}} {\text{P}}_{{\text{i}}}\) is the total XPM impact in the specific kth channel.

Using (5), the total spectral broadening caused by the total XPM in the specific kth channel will be (Schneider 2004; Agrawal 2019)

$$\Delta \omega_{{{\text{XPM}}}} = - 2\gamma {\text{L}}_{{{\text{eff}}}} \frac{{{\text{d}}\left( {\mathop \sum \nolimits_{{{\text{m}} \ne {\text{j}}}} {\text{P}}_{{\text{m}}} } \right)}}{{{\text{dt}}}}$$
(6)

As obvious in both (5) and (6), XPM has a much more significant impact on nonlinear phase shift and spectral pulse broadening than does SPM in WDM systems, in particular for increasing channel numbers.

FWM is an extremely important nonlinear impairment for WDM-based systems since it causes generation of a new wave, i.e. the FWM product, due to the interaction among three distinct waves. The frequency of the FWM product in the optical fiber can be computed with (Schneider 2004; Agrawal 2019)

$${\text{f}}_{{{\text{ijk}}}} = {\text{f}}_{{\text{i}}} + {\text{f}}_{{\text{j}}} - {\text{f}}_{{\text{k}}}$$
(7)

where fi, fj and fk denote frequencies of three waves that propagate in ith, jth and kth channels of an N-channel WDM system. In (7), in all triple combinations of channels among N channels, fk ≠ fi and fk ≠ fj inequalities must be provided to guarantee the generation of a novel FWM product.

When the frequency of an FWM product is equal to the frequency of an original signal transmitted through any channel of the WDM system, FWM crosstalk occurs and the FWM product interferes with that original signal. FWM products with different frequencies than frequencies of original channels fall into the channel spacings of the system and contribute to the total system noise. However, FWM crosstalk can have much more significant impacts on the system performance than FWM products falling into the channel spacings. In WDM systems with equal channel spacings much more numbers of FWM products can possibly be generated when compared to the case in WDM systems owning unequally spaced channels. However, since usage of the total optical bandwidth is more efficient in systems with equal channel spacings compared to systems with unequal channel spacings, architectures with equal channel spacings are generally preferred in forming WDM channels. Therefore, determining and mitigating FWM impacts on WDM systems are two crucial topics.

A single FWM product causing crosstalk in a specific kth channel has a power at the output of that channel which can be given with (Schneider 2004; Agrawal 2019)

$${\text{P}}_{{{\text{FWM}}}} \left( {{\text{f}}_{{{\text{ijk}}}} } \right) = \left( {\frac{{{\text{d}}_{{{\text{ijk}}}} \gamma {\text{L}}_{{{\text{eff}}}} }}{3}} \right)^{2} {\text{P}}_{{\text{i}}} {\text{P}}_{{\text{j}}} {\text{P}}_{{\text{k}}} {\text{e}}^{{ - \alpha {\text{L}}}} \eta_{{{\text{ijk}}}}$$
(8)

where dijk is the factor of degeneration; Pi, Pj and Pk imply input powers of interacting channels that form the FWM product; α denotes the attenuation coefficient; L implies the fiber length; ηijk is called the FWM efficiency that can be expressed with (Schneider 2004; Agrawal 2019)

$$\eta_{{{\text{ijk}}}} = \frac{{\alpha^{2} }}{{\alpha^{2} + \Delta \beta_{{{\text{ijk}}}}^{2} }}\left[ {1 + \frac{{4{\text{e}}^{{ - \alpha {\text{L}}}} }}{{\left( {1 - {\text{e}}^{{ - \alpha {\text{L}}}} } \right)^{2} }}\sin^{2} \left( {\frac{{\Delta \beta_{{{\text{ijk}}}} {\text{L}}}}{2}} \right)} \right]$$
(9)

where Δβijk is the phase mismatching factor among interacting channels.

Δβijk can be described with (Schneider 2004; Agrawal 2019)

$$\Delta \beta_{{{\text{ijk}}}} = \frac{{2\pi \lambda_{{\text{k}}}^{2} }}{{\text{c}}} \left( {\left| {{\text{f}}_{{\text{i}}} - {\text{f}}_{{\text{k}}} } \right|\left| {{\text{f}}_{{\text{j}}} - {\text{f}}_{{\text{k}}} } \right|} \right)\left[ {{\text{D}}_{{\text{c}}} + \frac{{\lambda_{{\text{k}}}^{2} }}{{2{\text{c}}}} \frac{{{\text{dD}}_{{\text{c}}} \left( {\lambda_{{\text{k}}} } \right)}}{{{\text{d}}\lambda }}\left( {\left| {{\text{f}}_{{\text{i}}} - {\text{f}}_{{\text{k}}} } \right| + \left| {{\text{f}}_{{\text{j}}} - {\text{f}}_{{\text{k}}} } \right|} \right)} \right]$$
(10)

where λk is the kth channel wavelength; c is the speed of light in vacuum; Dc is the chromatic dispersion of the optical fiber; dDc/dλ is the slope of the chromatic dispersion.

The total power of FWM crosstalk generated in a specific channel can be computed with (Schneider 2004; Agrawal 2019)

$${\text{P}}_{{{\text{FWM}}}} \left( {{\text{f}}_{{\text{s}}} } \right) = \mathop \sum \limits_{{{\text{f}}_{{\text{k}}} = {\text{f}}_{{\text{i}}} + {\text{f}}_{{\text{j}}} - {\text{f}}_{{\text{s}}} }} \mathop \sum \limits_{{{\text{f}}_{{\text{j}}} }} \mathop \sum \limits_{{{\text{f}}_{{\text{i}}} }} {\text{P}}_{{{\text{FWM}}}} \left( {{\text{f}}_{{\text{i}}} + {\text{f}}_{{\text{j}}} - {\text{f}}_{{\text{k}}} } \right)$$
(11)

where fs denotes the frequency of the specific channel.

In Song (1998) and Song et al. (1999), considering the dual impact of SPM and XPM on FWM, it was proven that Δβijk given in (10) should take the form

$$\Delta \beta^{\prime }_{{{\text{ijk}}}} = \Delta \beta_{{{\text{ijk}}}} - \gamma \left( {{\text{P}}_{{\text{i}}} + {\text{P}}_{{\text{j}}} - {\text{P}}_{{\text{k}}} } \right)\left[ {\frac{{\left\{ {1 - \exp \left( { - \alpha {\text{L}}_{{{\text{eff}}}} } \right)} \right\}}}{{\alpha {\text{L}}_{{{\text{eff}}}} }}} \right]$$
(12)

Then, taking into account the interaction among SPM, XPM and FWM and using (12), (8), (9) and (11) should be re-written, respectively, as follows.

$${\text{P}}^{\prime }_{{{\text{FWM}}}} \left( {{\text{f}}_{{{\text{ijk}}}} } \right) = \left( {\frac{{{\text{d}}_{{{\text{ijk}}}} \gamma {\text{L}}_{{{\text{eff}}}} }}{3}} \right)^{2} {\text{P}}_{{\text{i}}} {\text{P}}_{{\text{j}}} {\text{P}}_{{\text{k}}} {\text{e}}^{{ - \alpha {\text{L}}}} \eta^{\prime }_{{{\text{ijk}}}}$$
(13)
$${\upeta }^{\prime }_{{{\text{ijk}}}} = \frac{{{\upalpha }^{2} }}{{{\upalpha }^{{2}} + \left( {\Delta {\upbeta }_{{{\text{ijk}}}}^{\prime } } \right)^{2} }}\left[ {1 + \frac{{4{\text{e}}^{{ - {\alpha L}}} }}{{\left( {1 - {\text{e}}^{{ - {\alpha L}}} } \right)^{2} }}\sin^{2} \left( {\frac{{\Delta {\upbeta }_{{{\text{ijk}}}}^{\prime } {\text{ L}}}}{2}} \right)} \right]$$
(14)
$${\text{P}}_{{{\text{FWM}}}}^{\prime } \left( {f_{s} } \right) = \mathop \sum \limits_{{{\text{f}}_{{\text{k}}} = {\text{f}}_{{\text{i}}} + {\text{f}}_{{\text{j}}} - {\text{f}}_{{\text{s}}} }} \mathop \sum \limits_{{{\text{f}}_{{\text{j}}} }} \mathop \sum \limits_{{{\text{f}}_{{\text{i}}} }} P_{{{\text{FWM}}}}^{\prime } \left( {f_{i} + f_{j} - f_{k} } \right)$$
(15)

Since SRS is an inelastic scattering occurring due to light interaction with molecular vibrations of material, novel waves having frequencies higher and lower than the frequency of the original wave will be generated as an outcome of SRS while the light wave is propagating through an optical channel. Therefore, SRS results in signal power transfer from higher-frequency channels to lower-frequency ones in WDM systems.

In an N-channel WDM system, the signal output power obtained at a specific sth channel due to the overall SRS impact in the system can be given as (Singh and Hudiara 2004; Kılınçarslan and Karlık 2023)

$$\begin{array}{*{20}c} {{\text{P}}_{{\text{M}}} \left[ {\text{s}} \right] = \left\{ {{\text{P}}_{{\text{T}}} \left[ {\text{s}} \right] - {\text{P}}_{{\text{T}}} \left[ {\text{s}} \right].\mathop \sum \limits_{{{\text{i}} = {\text{s}} + 1}}^{{\text{N}}} {\text{D}}\left[ {{\text{s}},{\text{i}}} \right] + \mathop \sum \limits_{{{\text{j}} = 1}}^{{{\text{s}} - 1}} {\text{P}}_{{\text{T}}} \left[ {\text{j}} \right] \cdot {\text{D}}\left[ {j,s} \right]} \right\} \cdot {\text{e}}^{{ - \alpha {\text{L}}}} , \quad {\text{s}} = 1,2, \ldots ,{\text{N}}} \\ \end{array}$$
(16)

where PT[s] is the input signal power of the sth channel, PT[s]\(\mathop \sum \limits_{{{\text{i}} = {\text{s}} + 1}}^{{\text{N}}} {\text{D}}\left[ {{\text{s}},{\text{i}}} \right]\) denotes the signal power that is transferred from the sth channel to all other lower-frequency channels of the N-channel WDM system and \(\mathop \sum \limits_{{{\text{j}} = 1}}^{{{\text{s}} - 1}} {\text{P}}_{{\text{T}}} \left[ {\text{j}} \right].{\text{D}}\left[ {{\text{j}},{\text{s}}} \right]\) describes signal power transferred by the sth channel from all other higher-frequency channels of the N-channel WDM system. In (16), if i > N, D[s,i] = 0 and if s = 1, D[j,s] = 0.

The ratio of the signal power that is transferred by a specific jth channel from a specific ith channel of the WDM system, i.e. D[i,j], may be computed as (Singh and Hudiara 2004; Kılınçarslan and Karlık 2023)

$${\text{D}}\left[ {{\text{i}},{\text{j}}} \right] = \left( {\frac{{\lambda_{{\text{j}}} }}{{\lambda_{{\text{i}}} }}} \right) \cdot {\text{P}}_{{\text{T}}} \left[ {\text{j}} \right] \cdot \left\{ {\frac{{\left( {{\text{f}}_{{\text{i}}} - {\text{f}}_{{\text{j}}} } \right)}}{{1.5 \times 10^{13} }}} \right\} \cdot {\text{g}}_{{{\text{Rmax}}}} \cdot \left\{ {\frac{{{\text{L}}_{{{\text{eff}}}} .10^{5} }}{{{\text{b}}.{\text{A}}_{{{\text{eff}}}} }}} \right\} , \quad {\text{for}} \left( {{\text{f}}_{{\text{i}}} - {\text{f}}_{{\text{j}}} } \right) \le 1.5 \times 10^{13} {\text{Hz and j}} > {\text{i}}$$
(17)

where λi and λj are the wavelengths of channels taking part in the power transfer; \({\text{g}}_{{{\text{Rmax}}}}\) is the peak Raman gain coefficient; b is a constant varying between 1 and 2 according to the polarization. In (17), if (fi-fj) > 1.5 × 1013 Hz and j ≤ i, D[i,j] = 0.

Considering (15) and (16), the signal-to-crosstalk ratio (SXR) obtained at the channel output for a specific sth channel under the quadruple impact of SPM, XPM, FWM and SRS can be described as

$${\text{SXR}} = 10{\text{log}}_{10} \frac{{{\text{P}}_{{\text{M}}} \left[ {\text{s}} \right]}}{{{\text{P}}_{{{\text{FWM}}}}^{\prime } \left( {{\text{f}}_{{\text{s}}} } \right)}}$$
(18)

3 The DWDM-PON model

Contrary to the conventional method of using the C-band for both uplink and downlink transmission in bidirectional DWDM-PONs proposed in the literature (Lu et al. 2007; Usman 2020), a bidirectional DWDM-PON model whose ULCs use 1310 nm region while DLCs operate in 1550 nm region was proposed in Karlık (2023) benefiting from the ability of G.652 SSMFs operating in using both wavelength regions. Thus, not only full-duplex transmission via a single optical fiber can be maintained in 2xN-channel architecture but also the wide band gap between ULCs and DLCs prevents the occurrence of nonlinear interactions between upstream and downstream transmissions. However, the combined impacts of nonlinearities should be carefully analyzed separately for ULCs and DLCs to achieve a reliable communication. In this research, that requirement has been fulfilled for the quadruple impact of SPM, XPM, FWM and SRS on both ULCs and DLCs of the DWDM-PON having the system model given in Fig. 1 with channel spacings from 12.5 to 100 GHz.

Fig.1
figure 1

2xN-channel DWDM-PON model

Full size image

As shown in Fig. 1, in this paper, we are dealing with the bidirectional DWDM transmission on the main G.652 SSMF located between the multiplexer/demultuplixer (MUX/DEMUX) at the optical line termination (OLT) side in the DWDM-PON central office and the other MUX/DEMUX at optical network units (ONUs) side of end-users. Transceivers at both sides, i.e. T/Rk k = 1, 2,…., N, are capable of transmitting and receiving ULCs and DLCs at specific wavelengths. Both MUX/DEMUX can be implemented with arrayed waveguide grating (AWG) routers in application to perform multiplexing in one direction while demultiplexing in the other for bidirectional transmission. λDLi (i = 1, 2,….,N) and λULj (j = 1, 2,…., N) denote wavelengths of signals transmitted through DLCs and ULCs, respectively, in bidirectional DWDM transmission. 15-channel and 63-channel architectures have been considered in simulations where wavelengths of central channels, i.e. the 8th channel for 15-channel DWDM-PON and the 32nd channel for 63-channel DWDM-PON, have been selected as 1310 nm for ULCs and 1550 nm for DLCs. The essential parameters of commercially-available G.652 SSMF, which are used in simulations, are presented in Table 1.

Table 1 G.652 SSMF parameters used in simulations
Full size table

Since simulations have been performed to observe the bidirectional DWDM transmission on the main G.652 SSMF link between two MUX/DEMUX structures of the whole system, we are not dealing with the up/down ‘side link’ transmissions between transceivers and the MUX/DEMUX structures at both central office and ONUs sides in this paper. Transmission lengths and transmission media types may differ from end-user to end-user due to locations of end-users and cost constraints in side link transmission. Therefore, possible limitations about design parameters of transceivers or power budget concerns can easily be overcome by using convenient optical attenuators or amplifiers at inputs and outputs of MUX/DEMUX structures in real world system applications.

In DWDM systems having equal channel spacings and implemented with SSMFs or non-zero dispersion shifted fibers (NZDSFs), FWM products mostly fall into central channels and form FWM crosstalk (Harboe et al. 2008; Karlık 2016). Thus, in this paper, SXR values at central channels of ULCs and DLCs have been analyzed as the system performance parameter in order to obtain the worst-case performance values under the quadruple impact of SPM, XPM, FWM and SRS. Furthermore, the zero-dispersion wavelength for the commercially-available G.652 SSMF fiber, whose parameters have been used in simulations, is in the wavelength range of 1302–1322 nm. This means that the most severe conditions for FWM have been observed for ULCs in simulations since the zero-dispersion wavelength is in the transmission wavelength region of ULCs. This is also obvious from the chromatic dispersion value of − 0.26 ps/(nm km) given in Table 1 for the central uplink channel operating at 1310 nm wavelength.

Total number of FWM products, i.e. M, generated in 15-channel and 63-channel DWDM-PONs may be computed with M = N2(N-1)/2 as 1575 and 123,039, respectively. In equally-spaced channels architecture, 73 among 1575 FWM products for 15-channel DWDM-PON and 1441 among 123,039 FWM products for 63-channel DWDM-PON fall into the central channel. The channel combinations that generate those products in central channels are given in Fig. 2 and Fig. 3 for 15-channel and 63-channel DWDM-PONs, respectively. Those combinations are valid for central channels of both ULCs and DLCs, separately.

Fig. 2
figure 2

All channel combinations generating FWM products in the 8th (the central) channel of 15-channel DWDM-PON

Full size image
Fig. 3
figure 3

All channel combinations generating FWM products in the 32nd (the central) channel of 63-channel DWDM-PON

Full size image

In Fig. 2, when the highlighted cell in green is considered, it can be mentioned that the 4th, the 11th and the 7th channels interact and generate a FWM product in the 8th (the central) channel of 15-channel DWDM-PON.

In the literature, various reference values have been selected for minimum SXR value, e.g. 20 dB, 23 dB and 25 dB, to interpret the performance of communication systems under nonlinear impairments (Nakajima et al. 1997; Bogoni and Poti 2004; Harboe et al. 2008; Karlık 2016, 2023; Yıldırım and Karlık 2023). In this paper, the reference value for minimum SXR is chosen as 23 dB averaging the related values given in the literature.

In the literature, channel lengths of DWM-PONs have been generally considered as 20 km (Neto et al. 2017; Vardanyan 2017; Garg et al. 2019), 25 km (Reis et al. 2012; Guo et al. 2021; Song et al. 2021; Karlık 2023), 60 km (Reis et al. 2012; Li et al. 2014; Fazea 2019; Manzoor et al. 2020) or 80 km (Bi et al. 2014; Reis et al. 2015; Shahpari et al. 2015). In SubSect. 4.1, where variations in SXR with channel input power variations have been observed under combined quadruple impact of SPM, XPM, FWM and SRS in central channels of ULCs and DLCs, an average channel length value of 25 km has been considered for DWDM-PON. In SubSect. 4.2, where variations in SXR with varying channel lengths have been observed under combined quadruple impact of SPM, XPM, FWM and SRS in central channels of ULCs and DLCs, channel lengths have been changed in the range of 1–80 km.

Considering aforementioned 2 × 15-channel and 2 × 63-channel DWDM-PON models, simulations have been performed with the help of MATLAB R2021a. In writing m-files, computation of SXR values given with (18) under the quadruple impact of SPM, XPM, FWM and SRS is rather difficult. To compute (15), three distinct equations, i.e. (10), (12) and (14), should be written for each of 2 × 73 and 2 × 1441 FWM products generated in central channels of 2 × 15-channel and 2 × 63-channel systems, respectively, due to varying values of channel spacings and varying wavelengths among channels for each of the triple channel combinations. Moreover, due to same reasons, 2 × 15 and 2 × 63 additional equations should be written for D[i,j] given with (17) to compute (16) in central channels of 2 × 15-channel and 2 × 63-channel systems, respectively. Therefore in order to prevent further complexity enhancements in computations, input powers of all channels have been selected as equal and comparative performance analysis with different modulation types has been left to future works.

In WDM systems, due to the fiber birefringence, the states of polarization (SOPs) for input waves remain unchanged for only a few meters (Agrawal 2019; Schneider 2004). In Schneider (2004), where a detailed analysis about the dependence of XPM and FWM products on polarization is found, analytical results express that pulse broadening due to XPM is weaker in case of perpendicular polarization of waves with respect to the case of parallel polarization of waves. Furthermore, it was also shown that FWM products reach their maximum intensities for the interaction of three waves being in the same polarization while in all other SOPs of waves FWM product intensities are much weaker with respect to the case of maximum intensity (Schneider 2004). Since effects of arbitrary altering polarization on XPM and FWM can fully be computed with much more complicated simulations and since our main aim is to observe the worst-case SXR results, waves propagating in all channels have been assumed to be in the same polarization for both DWDM-PONs in this paper.

4 Simulations

Results of SXR simulations performed to observe the quadruple impact of SPM, XPM, FWM and SRS on DWDM-PONs using the system model and conditions described in Sect. 3 and related interpretations are given in SubSect. 4.1 and SubSect. 4.2, where variations in channel input powers and channel lengths are considered, respectively.

4.1 SXR simulations for varying channel input powers

Results of SXR simulations obtained for channel input power variations in the range of 0.1–10 mW at the uplink central channels (ULCCs) and downlink central channels (DLCCs) of 2 × 15-channel DWDM-PONs are shown in Figs. 4, 5, 6, 7 while those of 2 × 63-channel ones are given in Figs. 8–11. In Figs. 4–11, the line given in cyan color implies the reference value of 23 dB for the targeted minimum SXR.

Fig. 4
figure 4

SXR versus channel input power variations at ULCC and DLCC of 2 × 15-channel DWDM-PON having 12.5 GHz equally-spaced channels

Full size image
Fig. 5
figure 5

SXR versus channel input power variations at ULCC and DLCC of 2 × 15-channel DWDM-PON having 25 GHz equally-spaced channels

Full size image
Fig. 6
figure 6

SXR versus channel input power variations at ULCC and DLCC of 2 × 15-channel DWDM-PON having 50 GHz equally-spaced channels

Full size image
Fig. 7
figure 7

SXR versus channel input power variations at ULCC and DLCC of 2 × 15-channel DWDM-PON having 100 GHz equally-spaced channels

Full size image
Fig. 8
figure 8

SXR versus channel input power variations at ULCC and DLCC of 2 × 63-channel DWDM-PON having 12.5 GHz equally-spaced channels

Full size image

In Figs. 4, 5, 6, for ULCC of 2 × 15-channel DWDM-PON, the SXR value remains always above the 23 dB level in channel input power range of 0.1–10 mW, but shows oscillatory variations in ranges of 31.11–34.8 dB, 31.51–35.59 dB and 31.85–39.18 dB for channel spacings of 12.5 GHz, 25 GHz and 50 GHz, respectively. Those oscillations are due to the decrease in Δβꞌ given with (12) occurring as a result of increasing channel input powers. That decrement of Δβꞌ causes oscillatory variations in ηꞌ given with (14) and subsequently in PꞌFWM given with (15). With the increment in channel spacings the peak-to-peak amplitude of those oscillations decreases. This is due to the increase in Δβ of (10) occurring as a result of increasing channel spacings. Thus the effect of input power variations on Δβꞌ decreases and so decrements in the peak-to-peak amplitude of SXR oscillations exist with increasing channel spacings.

In Fig. 7, for ULCC of 15-channel DWDM-PON, the SXR value remains above 23 dB throughout the channel input power range of 0.1–10 mW. However, SXR first exhibits oscillatory variations with increasing input power values until 1.4 mW, then, it increases with the input power increment from 1.4 to 2 mW and remains almost constant at an average value of 32 dB for input power values greater than 2 mW. The oscillatory variations observed for input powers less than 1.4 mW are again due the decrement in Δβꞌ explained above. As it is obvious from (16) and (17) that the overall SRS impact on the output signal power of a specific channel in an N-channel DWDM system can be in the form of power loss or power gain depending on input powers, wavelengths and channel spacings of interacting channels that contribute to SRS. The increment in SXR values in 1.4–2 mW power range shows that the impact of SRS on the output signal power of ULCC occurs as power gain in that range and this power gain due to SRS is dominant to the crosstalk power due to SPM, XPM and FWM. Therefore, SXR increases in the power range of 1.4–2 mW. Almost constant SXR level obtained for input powers greater than 2 mW indicates that the power amplification due to the SRS gain and the increment in the crosstalk power caused by SPM, XPM and FWM are nearly equal and therefore SXR value remains nearly constant.

In Fig. 4, for DLCC of 2 × 15-channel DWDM-PON, when input powers smaller than 0.9 mW are considered, it is obvious that dominant nonlinear phenomena determining SXR is FWM due to narrow channel spacings of 12.5 GHz. It is clear in (8) that PFWM increases proportionally with \({\text{P}}_{{{\text{in}}}}^{3}\) for equal input powers and therefore causes an exponential decay in SXR, which is just the case in the related region of Fig. 4. Then, as it can be easily understood from the oscillatory behavior of SXR and the reason of that behavior which is shortly stated above, the triple impact of SPM, XPM and FWM begins to be dominant factor on SXR and this impact increases with increasing input powers in 0.9–6.6 mW region. For a detailed analysis of the oscillatory behavior of SXR in 0.9–6.6 mW region and also to understand for why the triple impact of SPM, XPM and FWM begins to be the dominant factor on SXR in that region, one should focus on Eqs. (10), (12) and (14). It is clear from (12) that for sufficiently high values of phase mismatching factor Δβijk and low input power values such as the case of Pin < 0.9 mW for DLCC in Fig. 4, the modified phase mismatching factor \(\Delta \beta^{\prime }_{{{\text{ijk}}}}\) will be \(\Delta \beta^{\prime }_{{{\text{ijk}}}} \cong \Delta \beta_{{{\text{ijk}}}}\). This means that in those cases the triple impact of SPM, XPM and FWM can be neglected and the single impact of FWM can be taken into account. However, with increasing channel input powers, \(\Delta \beta^{\prime }_{{{\text{ijk}}}}\) begins to significantly vary from \(\Delta \beta_{{{\text{ijk}}}}\) and should be taken into account. Thus, the FWM efficiency should be computed in the modified form, i.e. with \(\eta_{{{\text{ijk}}}}^{\prime }\) given in (14). With varying input powers, \({\upeta }_{{{\text{ijk}}}}^{{^{\prime} }}\) can take high, because of decreasing \(\Delta \beta^{\prime }_{{{\text{ijk}}}}\) values with increasing input powers, but oscillating values, because of variation of the term \(\sin^{2} \left( {\frac{{\Delta \beta_{{{\text{ijk}}}}^{\prime } {\text{L}}}}{2}} \right)\) in (14) between 0 and 1. Those oscillations in \(\eta_{{{\text{ijk}}}}^{\prime }\) result in SXR oscillations occuring with increasing input powers. The reason why similar oscillations do not occur under the single FWM impact with increasing input powers is that for sufficiently large values of \(\Delta \beta_{{{\text{ijk}}}}\) and low input powers below a threshold, the FWM efficiency computed with (9), i.e. \(\eta_{{{\text{ijk}}}}\), is much lower than the modified FWM efficiency so the oscillating impact of the term \(\sin^{2} \left( {\frac{{\Delta \beta_{{{\text{ijk}}}} {\text{L}}}}{2}} \right)\) in (9) on those lower \(\eta_{{{\text{ijk}}}}\) values is hardly visible. Therefore, the existence or lack of the oscillatory behavior of SXR with increasing input powers gives clues for when the triple impact of SPM, XPM and FWM should be taken into account or not. For all input powers greater than 6.6 mW, SRS shows a dominant positive impact on SXR with a negative triple impact of SPM, XPM and FWM slightly accompanying to SRS dominance for input powers greater than 6.9 mW. As it is obvious in Fig. 4, SXR at DLCC remains mostly below 23 dB level in 0.59–7.03 mW range due to negative impacts of nonlinearities. As shown in Table 1, chromatic dispersion value at DLCC is significantly higher than that at ULCC. Since it is obvious from (8)–(10), high values of chromatic dispersion have a suppression effect on FWM impact. This is the reason why SXR at DLCC is greater than SXR at ULCC for input powers below 0.22 mW when SXR curves of ULCC and DLCC in Fig. 4 are comparatively analyzed. However, for input powers greater than 0.22 mW, the combined impact of nonlinearities on DLCC is worse than that on ULCC causing lower SXR values.

In Fig. 5, at DLCC of 2 × 15-channel DWDM-PON having 25 GHz equally-spaced channels, for input powers lower than 0.6 mW FWM has dominant impact on SXR while for input powers greater than 0.6 mW, the triple impact of SPM, XPM and FWM is the dominant cause for the oscillatory variation of SXR. SXR takes values greater than 23 dB for input powers lower than 1.02 mW, falls down and rises over 23 dB level due to oscillations in 1.02–1.96 mW range and remains below 23 dB for all input powers greater than 1.96 mW. Comparatively analyzing SXR curves of DLCC and ULCC, it is clear that SXR at DLCC is higher than that at ULCC for input powers lower than 0.5 mW. Due to the suppression effect of higher channel spacings on FWM, which is obvious in (8)–(10), that upper limit for the power is greater than the value, i.e. 0.22 mW, in DWDM-PON having 12.5 GHz equally-spaced channels. SXR at ULCC is mostly greater than that at DLCC for input powers exceeding 0.5 mW.

In Fig. 6, at DLCC of 2 × 15-channel DWDM-PON having 50 GHz equally-spaced channels, for input powers lower than 0.5 mW FWM is the dominant impact on SXR while for input powers greater than 0.5 mW, the triple impact of SPM, XPM and FWM begins to appear and becomes the dominant impact with increasing input powers. SXR is greater than 23 dB for input powers lower than 2.06 mW, falls down and rises over 23 dB level due to oscillations in 2.06–7.92 mW range and remains below 23 dB for all input powers greater than 7.92 mW. SXR at DLCC is higher than that at ULCC for input powers lower than 1.93 mW and in the input power region of 2.63–3.47 mW. SXR at ULCC is mostly higher than that at DLCC for input powers greater than 3.47 mW.

In Fig. 7, at DLCC of 2 × 15-channel DWDM-PON having 100 GHz equally-spaced channels, for input powers lower than 0.8 mW FWM is the dominant impact on SXR while for input powers greater than 0.8 mW the triple impact of SPM, XPM and FWM impact begins to appear slightly and enhances its impact with increasing input powers. However, for the most part of the power range, this impact is not as significant as the ones observed in previous systems with narrower channel spacings since Δβ given with (10) takes its highest value for 100 GHz channel spacing compared to narrower channel spacings. Therefore, Δβꞌ given with (12) approximately equals to Δβ for small input powers and has a slightly different value from Δβ for large input powers, which limits the dual impact of SPM and XPM on FWM. Excluding 8.25–9.14 mW input power range, SXR is greater than 23 dB at all input powers. Comparing SXR values of DLCC and ULCC, one can say that SXR at DLCC is greater than SXR at ULCC for input powers below 7.46 mW. For input powers greater than 7.46 mW, SXR at ULCC is generally higher than SXR at DLCC.

In Figs. 8, 9, 10, 11, at ULCCs of 2 × 63-channel DWDM-PONs, SXR value falls below 23 dB level with powers exceeding 0.21 mW, 0.17 mW, 0.17 mW and 0.37 mW for channel spacings of 12.5 GHz, 25 GHz, 50 GHz and 100 GHz, respectively. This is because of the enhancing number of FWM products that fall into the central channel compared to the case in 2 × 15-channel DWDM-PON, which results in a significant decrement in SXR.

Fig. 9
figure 9

SXR versus channel input power variations at ULCC and DLCC of 2 × 63-channel DWDM-PON having 25 GHz equally-spaced channels

Full size image
Fig. 10
figure 10

SXR versus channel input power variations at ULCC and DLCC of 2 × 63-channel DWDM-PON having 50 GHz equally-spaced channels

Full size image
Fig. 11
figure 11

SXR versus channel input power variations at ULCC and DLCC of 2 × 63-channel DWDM-PON having 100 GHz equally-spaced channels

Full size image

In Fig. 8, at ULCC of 2 × 63-channel DWDM-PON having 12.5 GHz equally-spaced channels, FWM is the dominant impact on SXR for input powers lower than 0.3 mW while the triple impact of SPM, XPM and FWM slightly appears for input powers greater than 0.3 mW due to strong phase matching conditions for 12.5 GHz channel spacings.

In Fig. 9, at ULCC of 2 × 63-channel DWDM-PON having 25 GHz equally-spaced channels, FWM is the dominant impact on SXR for input powers lower than 1.4 mW while SRS shows a dominant positive impact on SXR for input powers in the range of 1.4–2.7 mW. For input powers exceeding 2.7 mW the positive impact of SRS on SXR is being almost compensated by the triple impact of SPM, XPM and FWM and SXR level is nearly kept constant at an average value of 18.85 dB with slight variations.

In Fig. 10, at ULCC of 2 × 63-channel DWDM-PON having 50 GHz equally-spaced channels, FWM is the dominant impact on SXR for input powers lower than 2.3 mW while for input powers exceeding 2.3 mW the triple impact of SPM, XPM and FWM begins to appear and due to the enhancing phase mismatching conditions of 50 GHz channel spacings it shows a more obvious effect on SXR for increasing input powers compared to the case in narrower channel spacings.

Similar to the case in Fig. 10 and with similar reasons, in Fig. 11, at ULCC of 2 × 63-channel DWDM-PON having 100 GHz equally-spaced channels, FWM is the dominant impact on SXR for input powers lower than 2.1 mW while for input powers exceeding 2.1 mW the triple impact of SPM, XPM and FWM begins to appear and shows a more obvious effect on SXR for increasing input powers.

In Fig. 8, at DLCC of 2 × 63-channel DWDM-PON having 12.5 GHz equally-spaced channels, SXR falls below 23 dB level for 0.58 mW channel input power and remains there for all greater input powers. It is clear that the dominant nonlinear phenomenon determining SXR is FWM for input powers below 1.7 mW due to narrow channel spacings of 12.5 GHz. Then, the triple impact of SPM, XPM and FWM begins to appear slightly and becomes the dominant factor on SXR for 4–10 mW power range. SXR value in DLCC is greater than the value in ULCC for input powers below 0.73 mW in Fig. 8 since the chromatic dispersion at 1550 nm has a higher value than the one at 1310 nm. For powers exceeding 0.73 mW, SXR value in ULCC is above the value in DLCC implying that the impact of nonlinearities on DLCC is more severe than the one on ULCC at those input powers.

In Fig. 9, at DLCC of 2 × 63-channel DWDM-PON having 25 GHz equally-spaced channels, the dominant nonlinear impairment on SXR is FWM for input powers smaller than 0.6 mW. Then, the triple impact of SPM, XPM and FWM becomes dominant and causes strong SXR oscillations for input powers exceeding 0.6 mW. During oscillations, SXR falls below and raises above the 23 dB level in 1.01–1.95 mW power range. For input powers greater than 1.95 mW, SXR remains below 23 dB with continuing oscillations. Comparatively analyzing SXR curves of DLCC and ULCC in Fig. 9, it is obvious that SXR at DLCC is higher than that at ULCC for input powers smaller than 2.13 mW. In the input power region of 2.13–3.52 mW, SXR at DLCC is sometimes higher and sometimes lower than SXR at ULCC due to oscillations. However, for all input powers greater than 3.52 mW SXR at DLCC remains below SXR at ULCC.

In Fig. 10, at DLCC of 2 × 63-channel DWDM-PON having 50 GHz equally-spaced channels, FWM is the dominant nonlinearity on SXR for input powers smaller than 0.5 mW. For powers exceeding 0.5 mW, the triple impact of SPM, XPM and FWM begins to appear and becomes the dominant impact with increasing input powers. SXR is over 23 dB level for powers below 2.06 mW. Due to oscillations in 2.06–7.92 mW power range, SXR falls down 23 dB level and raises over it. For input powers exceeding 7.92 mW, SXR remains below 23 dB level. SXR at DLCC is higher than SXR at ULCC for all input powers outside 8.6–8.72 mW range.

In Fig. 11, at DLCC of 2 × 63-channel DWDM-PON having 100 GHz equally-spaced channels, the dominant impact on SXR is FWM for input powers below 0.8 mW. Then, the triple impact of SPM, XPM and FWM begins to appear slightly and enhances its effect with increasing input powers. This impact is not as significant as the ones observed in previous systems with narrower channel spacings just as the case in Fig. 7 and with similar reasons. For input powers below 8.28 mW and above 9.11 mW, SXR at DLCC is greater than 23 dB. Comparing SXR values at ULCC and DLCC, it is obvious that SXR at DLCC is greater than SXR at ULCC for all input powers.

Comparing SXR variations in Figs. 4, 5, 6, 7 with those given in Figs. 8, 9, 10, 11, differences among SXR graphics at ULCCs for all channel spacing values of 2 × 15-chanel and 2 × 63-channel DWDM-PONs are clear. That is also the case for SXR graphics at DLCCs in 2 × 15-channel and 2 × 63-channel DWDM-PONs having 12.5 GHz equally-spaced channels. However, SXR graphics at DLCCs in 2 × 15-channel and 2 × 63-channel DWDM-PONs having 25 GHz equally-spaced channels are partly similar and SXR graphics at DLCCs in 2 × 15-channel and 2 × 63-channel DWDM-PONs having 50 GHz and 100 GHz equally-spaced channels are very similar to each other. Therefore, to clearly understand and interpret the differences Tables 2 and 3 are given. Using Figs. 4, 5, 6, 7, 8, 9, 10, 11, SXR values observed at ULCCs and DLCCs of 2 × 15-channel and 2 × 63-channel DWDM-PONs under the quadruple impact of SPM, XPM, FWM and SRS for input power values of 0.1 mW, 0.5 mW, 1 mW, 3 mW, 5 mW and 10 mW are presented in Tables 2 and 3, respectively. In Tables 2 and 3, SXR values given in bold show values below the targeted minimum SXR value of 23 dB.  

Table 2 SXR values at ULCCs and DLCCs of 2 × 15-channel DWDM-PONs having various channel spacings and input powers
Full size table
Table 3 SXR values at ULCCs and DLCCs of 2 × 63-channel DWDM-PONs having various channel spacings and input powers
Full size table

It is obvious in Table 2 that bidirectional transmission providing an SXR over 23 dB level at both ULCCs and DLCCs under the quadruple impact of SPM, XPM, FWM and SRS is possible for all channel spacing values in 15-channel DWDM-PON when 0.1 mW or 0.5 mW channel input power values are chosen. For 1 mW channel input powers, bidirectional transmission providing an SXR over 23 dB level at both ULCCs and DLCCs is possible for channel spacing values greater than 12.5 GHz. For 3 mW and 5 mW input powers, relevant bidirectional transmission is possible for only 50 GHz and 100 GHz channel spacings. For 10 mW input powers, the targeted bidirectional transmission can be achieved with 12.5 GHz and 100 GHz channel spacings under the quadruple impact of SPM, XPM, FWM and SRS.

It is clear in Table 3 that at all channel spacing values of 2 × 63-channel DWDM-PON bidirectional transmission being able to provide an SXR over 23 dB level at both ULCCs and DLCCs under the quadruple impact of SPM, XPM, FWM and SRS is only possible for 0.1 mW input powers. SXR at ULCC is below 23 dB at all channel spacing values for all input powers greater than 0.1 mW. Furthermore, for channel input powers greater than 0.5 mW, as well as SXR at ULCC, SXR at DLCC is also below 23 dB at some channel spacing values.

Comparing SXR values for DLCCs given at same cell locations of Table 2 and Table 3, the minimum and maximum differences of SXR between 12.5 GHz columns of Table 2 and Table 3 are almost 0.21 dB and 27.22 dB, respectively, while the relevant values at columns of 25 GHz, 50 GHz and 100 GHz of Table 2 and Table 3 are almost 0.03 dB and 10.72 dB; 0.03 dB and 0.8 dB; 0.09 dB and 1.54 dB, respectively. Therefore, Figs. 5 and 9 are partly similar while Figs. 6 and 10 and Figs. 7 and 11 are very similar to each other for DLCCs.

Comparatively analyzing Figs. 4, 5, 6, 7, under the quadruple nonlinear impact, the maximum channel input power providing bidirectional transmission with an SXR over 23 dB for both ULCCs and DLCCs in all channel spacing values of 2 × 15-channel DWDM-PON has been obtained as 0.58 mW. The relevant value for all channel spacing values of 2 × 63-channel DWDM-PON has been found as 0.16 mW when Figs. 8, 9, 10, 11 have been comparatively analyzed.

4.2 SXR simulations for varying channel lengths

In 2 × 15-channel and 2 × 63-channel DWDM-PONs, results of SXR simulations at ULCCs and DLCCs obtained for varying channel lengths in the range of 1–80 km are shown in Figs. 12, 13 and Figs. 14, 15, respectively. In Figs. 12, 13, 14, 15, the line given in cyan color implies the minimum reference value of 23 dB for SXR. To obtain SXR values over 23 dB level all channel input powers have been chosen as 0.15 mW for both 2 × 15-channel and 2 × 63-channel DWDM-PONs considering results observed in previous subsection.

Fig. 12
figure 12

SXR variation with varying channel length at ULCCs of 2 × 15-channel DWDM-PONs with various channel spacings (Δf) for 0.15 mW input powers

Full size image
Fig. 13
figure 13

SXR variation with varying channel length at ULCCs of 2 × 63-channel DWDM-PONs with various channel spacings (Δf) for 0.15 mW input powers

Full size image
Fig. 14
figure 14

SXR variation with varying channel length at DLCCs of 2 × 15-channel DWDM-PONs with various channel spacings (Δf) for 0.15 mW input powers

Full size image
Fig. 15
figure 15

SXR variation with varying channel length at DLCCs of 2 × 63-channel DWDM-PONs with various channel spacings (Δf) for 0.15 mW input powers

Full size image

In Fig. 12, under the quadruple impact of SPM, XPM, FWM and SRS it is evident for ULCCs of 2 × 15-channel DWDM-PONs that variation in channel lengths does not have a significant effect on SXR at lengths over 40 km in systems having 50 GHz and 100 GHz equally-spaced channels while at lengths over 50 km in systems having 12.5 GHz and 25 GHz equally-spaced channels. At lengths shorter than 18.62 km, SXR values in ULCCs increase with raising channel spacings at same channel lengths as expected. However, contrary to expectations, at lengths longer than 18.62 km, it is clear that SXR in ULCC of the system with 50 GHz equally-spaced channels is higher than that in the system with 100 GHz equally-spaced channels. This shows that for those channel lengths positive impact of SRS on SXR is higher in the system with 50 GHz equally-spaced channels compared to the case in the system with 100 GHz equally-spaced channels. However, because of the complicated nature of nonlinearities and much more complicated nature of the quadruple impact of SPM, XPM, FWM and SRS on SXR, where various combinations of system parameters can affect the results, we cannot currently give a clear interpretation about why this happens in ULCCs of 2 × 15-channel systems with 50 GHz and 100 GHz spacings at those channel lengths but does not occur in other systems with 12.5 GHz and 25 GHz spacings since this point requires an in-depth analysis that will be handled in future works.

In Fig. 13, under the quadruple impact of SPM, XPM, FWM and SRS it is clear for ULCCs of 2 × 63-channel DWDM-PONs that variation in channel lengths does not have a significant effect on SXR at lengths over 30 km in systems having 50 GHz and 100 GHz equally-spaced channels and at lengths over 40 km and 45 km in systems having 25 GHz and 12.5 GHz equally-spaced channels, respectively. At lengths shorter than 19.65 km, SXR values in ULCCs increase with raising channel spacings for same channel lengths as expected. However, contrary to expectations, at lengths longer than 19.65 km, it is clear that SXR in ULCC of the system having 25 GHz equally-spaced channels is higher than that in the system with 50 GHz equally-spaced channels and remains so until the channel length value of 30.55 km. Furthermore, SXR in ULCC of the system having 12.5 GHz equally-spaced channels exceeds SXR in ULCC of the system having 50 GHz equally-spaced channels at channel length values longer than 20.03 km and it also exceeds SXR in ULCC of the system having 25 GHz equally-spaced channels at channel length values longer than 20.20 km. Similar to the relevant case observed in ULCCs of 2 × 15-channel DWDM-PONs having 50 GHz and 100 GHz equally-spaced channels, this is due to the positive impact of SRS on SXR at those channel lengths and for those channel spacings. This point also requires an in depth analysis in future works. It should also be noted that under the quadruple impact of SPM, XPM, FWM and SRS, SXR falls below 23 dB level at channel length ranges of 5.84–16.48 km and 6.99–10.5 km in ULCCs of 2 × 63-channel DWDM-PONs having 12.5 GHz and 25 GHz equally-spaced channels, respectively.

In Figs. 14 and 15, SXR versus channel length variations in DLCCs of both 2 × 15-channel and 2 × 63-channel DWDM-PONs show oscillations for systems having 25 GHz, 50 GHz and 100 GHz equally-spaced channels. Those oscillations are due to the phase mismatching phenomenon occurring in systems having wider channel spacings than 12.5 GHz. It is clear in (10) and (12) that phase mismatch increases with increasing channel spacings. Due to the increasing phase mismatch and channel lengths, \(\sin^{2} \left( {\frac{{\Delta \beta_{{{\text{ijk}}}}^{\prime } {\text{L}}}}{2}} \right)\) in (14) causes oscillations in FWM efficiency since its value differs in the range from 0 to 1. Therefore, those oscillations cause subsequent oscillations in total FWM crosstalk power and SXR. The fading of SXR oscillations at channel lengths generally over 50 km depends on the decrement in FWM efficiency occurring at those channel lengths. As it is obvious in Figs. 12 and 13, similar oscillations do not occur in ULCCs since phase mismatch in 1310 nm is weaker than in 1550 nm due to lower channel wavelength and chromatic dispersion value. For 1 km increment in channel lengths, maximum SXR variation due to those oscillations approaches to 2.6 dB, 4.89 dB and 4.92 dB in DLCCs of 2 × 15-channel DWDM-PONs having 25 GHz, 50 GHz and 100 GHz equally-spaced channels, respectively. The relevant values in DLCCs of 2 × 63-channel DWDM-PONs having 25 GHz, 50 GHz and 100 GHz equally-spaced channels are 2.47 dB, 3.81 dB and 4.19 dB, respectively. Thus, those oscillations occurring in short variations of channel lengths may threaten the communication reliability and enhance the complexity of performance analysis under the quadruple impact of SPM, XPM, FWM and SRS. As shown in Figs. 14 and 15, contrary to the case in systems with wider channel spacings, SXR oscillations do not occur in DLCCs of both 2 × 15-channel and 2 × 63-channel DWDM-PONs having 12.5 GHz equally-spaced channels depending on strong phase match in narrower channel spacings that annihilates the oscillation impact of \(\sin^{2} \left( {\frac{{\Delta \beta_{{{\text{ijk}}}}^{\prime } {\text{L}}}}{2}} \right)\) in (14) causing oscillations.

Using simulation results given in Figs. 12, 13, 14, 15, SXR values at ULCCs and DLCCs of 2 × 15-channel and 2 × 63-channel DWDM-PONs having 12.5 GHz, 25 GHz, 50 GHz and 100 GHz equally-spaced channels for 20 km, 25 km, 60 km and 80 km channel lengths, i.e. the most commonly used channel length values for DWDM-PONs in the literature as stated in Sect. 3, are shown in Table 4.

Table 4 SXR values at ULCCs and DLCCs of 15-channel and 63-channel DWDM-PONs with various channel spacings and commonly selected channel lengths in the literature
Full size table

In Table 4, as also stated while interpreting simulation results presented in Fig. 12, SXR in ULCCs of 2 × 15-channel DWDM-PONs having 50 GHz equally-spaced channels and 20 km, 25 km, 60 km, 80 km channel lengths are higher than those in systems having 100 GHz equally-spaced channels. This is due to positive impact of SRS on SXR in ULCCs of systems having 50 GHz equally-spaced channels at those channel lengths. Similar case occurs for SXRs in ULLCs of 2 × 63-channel DWDM-PONs having 12.5 GHz, 25 GHz and 50 GHz equally-spaced channels, where SXRs in ULCCs of systems having 12.5 GHz equally-spaced channels are higher than those in systems having 25 GHz and 50 GHz equally-spaced channels at channel lengths over 20 km. In both 2 × 15-channel and 2 × 63-channel DWDM-PONs, for all values of channel spacings, there is no notable difference among SXR values of ULCCs with 60 km and 80 km channel lengths. For each of 20 km, 60 km and 80 km channel lengths, there is no distinct difference between SXR values of ULCCs in 2 × 63-channel DWDM-PONs having 25 GHz and 50 GHz equally-spaced channels. Among 2 × 15-channel DWDM-PONs having 12.5 GHz, 25 GHz, 50 GHz, 100 GHz equally-spaced channels and same channel lengths, maximum SXR variations in ULCCs are 5.24 dB, 6.57 dB, 3.97 dB and 3.92 dB at 20 km, 25 km, 60 km and 80 km channel lengths, respectively. In 2 × 63-channel DWDM-PONs, relevant values are 6.78 dB, 6.68 dB, 6.71 dB and 6.72 dB at 20 km, 25 km, 60 km and 80 km channel lengths, respectively. Among 2 × 15-channel DWDM-PONs having 12.5 GHz, 25 GHz, 50 GHz, 100 GHz equally-spaced channels and same channel lengths, maximum SXR variations in DLCCs are 39.81 dB, 42.19 dB, 42.3 dB and 42.34 dB at 20 km, 25 km, 60 km and 80 km channel lengths, respectively. In 63-channel DWDM-PONs, relevant values are 39.61 dB, 41.71 dB, 41.88 dB and 41.95 dB at 20 km, 25 km, 60 km and 80 km channel lengths, respectively. The distinct difference between those maximum SXR variation values of ULCCs and DLCCs in both 2 × 15-channel and 2 × 63-channel DWDM-PONs depends on higher chromatic dispersion value in 1550 nm since high chromatic dispersion has a significant impact on suppressing FWM crosstalk and therefore enhancing SXR values. Among 2 × 15-channel DWDM-PONs having same channel spacings and 20 km, 25 km, 60 km and 80 km channel lengths, maximum SXR variations at ULCCs are 0.87 dB, 0.79 dB, 0.71 dB and 0.93 dB for 12.5 GHz, 25 GHz, 50 GHz and 100 GHz channel spacings, respectively. In 2 × 63-channel DWDM-PONs, relevant values are 0.52 dB, 0.33 dB, 0.24 dB and 0.32 dB for same spacing values, respectively. Among 2 × 15-channel DWDM-PONs with same channel spacings and 20 km, 25 km, 60 km and 80 km channel lengths, maximum SXR variations at DLCCs are 0.64 dB, 2.54 dB, 1.21 dB and 2.1 dB for 12.5 GHz, 25 GHz, 50 GHz and 100 GHz channel spacings, respectively. In 2 × 63-channel DWDM-PONs, relevant values are 0.55 dB, 2.3 dB, 1.12 dB and 1.89 dB for same spacing values, respectively.

Considering values given above, under the quadruple impact of SPM, XPM, FWM and SRS it can be concluded for both ULCCs and DLCCs of 2 × 15-channel and 2 × 63-channel DWDM-PONs that varying the channel spacing value while fixing the channel length has a much more distinct impact on SXR compared to the impact of varying the channel length value while fixing the channel spacing value.

5 Conclusion

Analyzing the quadruple impact of SPM, XPM, FWM and SRS on ULCCs operating at 1310 nm and DLCCs operating at 1550 nm of 2 × 15-channel and 2 × 63-channel DWDM-PONs and investigating appropriate design parameters that provide reliable communication with SXR levels over 23 dB at the highest level of the quadruple impact can be stated as the novelty of this paper.

Results obtained from SXR simulations for varying channel input powers emphasize that when any of 12.5 GHz, 25 GHz, 50 GHz and 100 GHz channel spacings are considered, channel input powers should be selected below 0.58 mW for 2 × 15-channel DWDM-PONs and below 0.16 mW for 2 × 63-channel DWDM-PONs for reliable bidirectional transmission with an SXR over 23 dB under the quadruple impact of SPM, XPM, FWM and SRS.

Results obtained from SXR simulations for varying channel lengths imply that when any of 12.5 GHz, 25 GHz, 50 GHz and 100 GHz channel spacings are considered, variations in channel lengths do not significantly affect SXR in ULCCs of 2 × 15-channel and 2 × 63-channel DWDM-PONs at channel lengths exceeding 50 km and 45 km, respectively, under the quadruple impact of SPM, XPM, FWM and SRS. Furthermore, SXR oscillations due to phase mismatching conditions enhancing for wider channel spacing values than 12.5 GHz have been observed in DLCCs of both 2 × 15-channel and 2 × 63-channel DWDM-PONs at varying channel lengths. Those oscillations, which have a potential in threatening the reliability of communication, become generally insignificant at channel lengths over 50 km in both 2 × 15-channel and 2 × 63-channel DWDM-PONs under the quadruple impact of SPM, XPM, FWM and SRS.

Contrary to the conventional usage of C-band for both uplink and downlink transmission in various bidirectional DWDM-PON proposals in literature, the bidirectional DWDM-PON system architecture focused on in this paper uses O-band for uplink transmission and C-band for downlink transmission. Therefore, the wide gap between two bands highly prevents the occurrence of nonlinear impairments such as XPM, FWM and SRS among ULCs and DLCs. Furthermore, using the O-band for just uplink transmission and C-band for just downlink transmission instead of using C-band for both transmissions provides transmitting much more channels by using this architecture with respect to conventional bidirectional DWDM-PONs. Thus, it can be mentioned that the scalability, cost-effectiveness and the reliability of the system given in this paper is higher than the conventional DWDM-PONs.

It is obvious that the optical modulation technique used in transmission is important for Kerr nonlinearities in optical networks since channel input signal powers depend on the selected modulation technique. To avoid facing further complexities in modelling and simulations, the average modulated signal powers have been considered in this paper. Impacts of conventional and advanced modulation techniques on the performance of optical access and long-haul networks under the combined effects of optical nonlinear impairments will be focused on in our future works. Thus, with comparative analysis of systems using various modulation techniques, the BER-SXR relations that will be derived for each system will help us to compute maximum achievable data rates with those systems under the required quality of service (QoS) parameters. Furthermore, the bidirectional DWDM-PON architecture given in this paper is convenient to transmit upstream and downstream data between central office and low power wide area networks (LPWANs), e.g. sensor networks using LoRaWAN technologies, when radio-over-fiber (RoF) integration is provided on the ONU-side. Therefore, another future aim is to design such an integrated system and evaluate the system performance for rural and urban applications.

Raising numbers of DWDM-PON applications in fronthauls of 5G and beyond networks in the literature, in particular after releasing of the WDM-PON standard in 2021 by ITU-T, imply also to the importance of the requirement for a reliable communication where impacts of nonlinear phenomena in optical fibers are fully considered in 5G and beyond networks achieving rapidly enhancing user demands. The content of this paper presents remarkable point of views and interpretations for design and analysis of such systems.