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Performance analysis of an SDM-WDM multicore optical fiber link

Satya Prasad Majumder and Afzal Hossain

Open Access Open Access

Abstract

Analytical evaluation of inter-channel and inter-core coupling induced crosstalk including Amplified Spontaneous Emission (ASE) noise in a single mode 7-core homogeneous multi-core optical fiber (MCF) based Space Division Multiplexed Wavelength Division Multiplexing (SDM-WDM) communication system with distributed Raman amplifier is presented. The Stimulated Raman Scattering (SRS) induced crosstalk impairment including ASE noise and inter-core and inter-channel coupling induced crosstalk at the output of any core with excitation into other core of the MCF is investigated. Signal to Crosstalk plus Noise Ratio (SCNR) is determined at the output of photo detector of the excited core. Finally, Bit Error Rate (BER) performance results are evaluated for different system parameters, viz. WDM channel number, channel spacing, propagation distance, pump power etc. It is noticed that BER deteriorates with SRS induced crosstalk and ASE noise, which is worsened further by the effect of inter-core crosstalk (IC-XT). The maximum acceptable propagation distance for a desired value of BER and other system parameters considering inter-channel crosstalk (Ich-XT), IC-XT including SRS induced crosstalk and ASE noise is also evaluated analytically. For example, to maintain a particular BER (10E-9, say) we need to increase the input signal power from 2.5dBm to 5.5dBm while increasing the WDM channels from 12 to 16 considering only the SRS and ASE noise. On the other hand, we need to increase the input signal power from 5.5dBm to 9.0dBm to increase the WDM channels from 12 to 16 when the IC-XT is added up with the SRS and ASE noise. The maximum allowable pump power considering the collective impact of Ich-XT and IC-XT including SRS and ASE noise is determined for a given value of BER (10E-9). Power penalty versus WDM channel spacing at a given BER value (10E-10), link distance and pump power at different WDM channels is also evaluated and presented.

© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Exponential growth in the demand of data traffic in access and backbone networks has been experienced by researchers and network engineers during the last couple of years [1]. With this sharp increasing trend of traffic demand, particularly with the evolution of streaming transmissions and distributed and cloud computing—increasing by a factor of 10 in 5 years, it is forecasted that the capacity growth of single core optical fiber system will result in capacity crunch in the very near future [2,3]. The prevailing COVID19 pandemic has forced the traditional physical classroom-based education system to change dramatically, with the distinctive rise of e-learning, whereby teaching is undertaken remotely and on digital platforms thereby demanding huge bandwidth. As a result of 30 years of systems research, the experimental capacity has reached more than a hundred terabits per second (Tb/s). This is considered to be almost the capacity for a single mode fiber (SMF) because of the maximum input power and nonlinear Shannon limits. To meet the demand for higher capacity, new multiplexing technologies are needed that can offer an additional multiplicity of around ten to a hundred times and full compatibility with current Time Division Multiplexing (TDM), Wavelength Division Multiplexing (WDM), and digital coherent transmission technologies [4]. In the recent years, the introduction of WDM and dense wavelength-division multiplexing (DWDM) has been considered to be the main approach to meet the increasing traffic demand across different optical networks with single-core single mode (SCSM) transmission systems [5]. By employing the four domains of multiplexing, namely time, wavelength (frequency), polarization, and multilevel modulations could achieve bandwidth capacities up to about 100 Tb/s per fiber. Researches indicate that the WDM and DWDM systems are rapidly approaching their Shannon capacity limit [2], [6]. The use of space division multiplexing (SDM) with multicore fiber (MCF) or multimode fiber (MMF) was proposed as the potential next generation multiplexing technology for optical fiber communications. In the recent years, extensive research works have been done in SDM with a view to overcome the Shannon capacity limit of WDM networks using single mode-single core fibers (SM-SCFs) [2], [ 4–7].

In SDM systems comprising MCF and/or MMF can induce coupling between signals in different cores and/or different modes resulting in random evolution of propagating fields. This coupling of signal between the cores and/or modes of the MCF and/or MMFs generates undesired crosstalk impairment. Thus, suppression of coupling induced crosstalk becomes the main challenge in design and implementation of MCF and MMF based transmission systems [8–11]. To estimate and characterize crosstalk in various MCFs and MMFs, coupled-mode theory (CMT) and coupled-power theory (CPT) are introduced and investigated in [12–15]. Employing the CMT and CPT analytical evaluation of coupling coefficient and its characterization are carried out in [16–18]. Based on this analysis the mode coupling dynamics in seven core single mode MCF based transmission system are reported in [19–21]. BER performance limitations due to inter-core crosstalk in a single mode multicore fiber optic transmission system are presented in [22]. SDM concepts, based on new types of optical fibers, like, single mode multicore fiber (SM-MCFs), multimode single core fibers (MM-SCFs) and multimode multicore fibers (MM-MCFs) have been proposed in [23–25].

More recently, interest in fibers with 125 µm cladding diameter as standard single-mode fibers has grown due to their compatibility with conventional cabling infrastructure and concerns over the mechanical reliability of larger fibers [26]. Particularly with multi-core fibers (MCFs), reducing the cladding diameter limits the number of spatial channels, leading to increasing interest in combining such fibers with wider transmission bandwidths to meet the expected growth in transmission capacity expected in SDM fibers. The SDM system exploits WDM and a combination of optical amplification technology to enable long distance transmission WDM channels in a wide range of optical bandwidth spectrum [27–29]. Raman amplification is based on stimulated Raman scattering (SRS), which is a nonlinear effect. In practice, a Raman amplifier uses multiple pump lasers through WDM coupler to realize high gain and flatness in WDM transmission system. This process, as with other stimulated emission processes, allows all-optical amplification in optical fibers with the gain depending on the material of the fiber core [30–32]. Use of Raman amplification as an enabling technology for long-haul coherent transmission systems is presented [33], whereas modeling Raman amplification in multimode and multicore fibers is reported in [34]. BER performance of a single core single mode wavelength division multiplexing (SCSM-WDM) fiber-optic transmission link with distributed Raman amplifier considering the combined effect of ASE and SRS induced crosstalk for direct detection receiver is presented in [35]. Analysis of wavelength sensitivity of coupling coefficients and inter-core crosstalk in a 9 core Fiber is presented in [36].

In this paper we present an analytical model to evaluate the effect of Raman induced crosstalk, ASE power and inter core coupling induced crosstalk on the BER performance of an SDM-WDM transmission system comprising a 7 core single mode MCF with distributed Raman fiber amplifier using direct detection receiver. Results are evaluated and presented in terms of ASE power, Raman scattering and inter core crosstalk and BER for different WDM system parameters.

2. Proposed fiber structure

Figure 1 shows schematic of the cross section of a 7-core homogeneous single mode MCF, which is assumed to be consist of a centre core (core-1) plus 6 outer cores (cores 2 through 6) spaced at equal angular distance (spatially 60o apart). Each core is considered to have identical fiber parameters (core pitch, index profile, relative refractive index difference, radius etc.). Table-2 presents the details of the multicore fiber parameters.

 figure: Fig. 1.

Fig. 1. Cross-section of a homogeneous 7-core single mode MCF

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3. System model and analysis

An optical fiber communication link comprising of single mode homogeneous 7 core MCF with distributed Raman amplifier is used as the system model (Fig. 2).

 figure: Fig. 2.

Fig. 2. Schematic illustration of a 7 core MCF based SDM-WDM communication system with distributed Raman amplifier

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Figure 2 shows the schematic illustration of a 7 core MCF based SDM-WDM communication system with distributed Raman amplifier. The multiplexed WDM signal is transmitted through single mode 7 core fiber with distributed Raman amplifier with 20 mW of pump power following the transmission scheme as shown in Table 1.

Tables Icon

Table 1. SDM-WDM Transmission Scheme

View Table | View all tables in this article

4. Coupled mode equations

Referring to Fig. 1, suppose that the pth core is identified by its radius and refractive index, ap and n1p respectively, while the cladding has a refractive index of n2. In order to maintain well-enough isolation, the cores are geometrically positioned in a way so that the electric fields of the cores are well-separated and perturbation methods with CMT could be used for analyzing the mode coupling dynamics [19].

Let the mode field of each core be expressed as

$${{\rm E}_{\rm p}}(\rm z) = {{\rm A}_{\rm p}}({\rm z})\exp( - {\rm j}{\mathrm{\beta} _{\rm p}}{\rm z})$$
where Ap is the amplitude of the field at the pth core and ${\beta _p}$ is the propagation constant of the single core in the absence of other cores.

The propagation of the fields in each core can be described by using a set of Coupled Mode Equations (CME) which can be written in matrix form as in [19]

$$\frac{{{\rm dA}({\rm z})}}{{{\rm dz}}} ={-} {\rm CA}({\rm z})$$
where, ${\rm A}({\rm z}) = {[{{{A}_1}({\rm z}){{\rm A}_2}({\rm z}){{\rm A}_3}({\rm z})\ldots \ldots .{{\rm A}_{n}}({\rm z})} ]^P{\rm T}}$ is a column vector and T denotes the transpose, z is the direction of propagation, and C is a n × n matrix. Coupling coefficient between pth and qth core, Cpq denotes the spatial overlap of the mode fields of core p and q over the cross-sectional area of core q. From the eigenvalue equation, the addition theorem allows us to determine Cpq in a step-index optical fiber expressing the mode field of core p in terms of the local coordinate system of core q, as in [19].
$$\begin{array}{l} {C_{pq}} = \left\{ \begin{array}{l} j{C_{pq}}\exp [{j({{\beta_p} - {\beta_q}} )z} ]\quad \quad ;\quad p \ne q\\ 0\quad \quad ;\quad p = q \end{array} \right.\\ \end{array}$$
where βp is the propagation constant of the LP01 mode for core P. Cpq can be analytically determined as in [20]
$${C_{pq}} = \sqrt {2{\Delta _q}} {W_q}{U_p}{K_0}\left( {\left( {\frac{{{W_p}{d_{pq}}}}{{{a_p}}}} \right)} \right)\left[ \begin{array}{l} {a_p}{U_q}{J_1}({{U_q}} ){I_0}\left( {\frac{{{W_p}{a_q}}}{{{a_p}}}} \right) + \\ {a_q}{W_p}{J_0}({{U_q}} ){I_1}\left( {\frac{{{W_p}{a_q}}}{{{a_p}}}} \right) \end{array} \right]{\left[ \begin{array}{l} {V_p}{J_1}({{U_q}} ){K_1}({{W_p}} )\\ ({{a_p}^2{U_q}^2 + {a_q}^2{W_p}^2} )\end{array} \right]^{ - 1}}$$
where, Jl is the Bessel function of the first kind, Il and Kl are the modified Bessel functions of the first and second kinds of order 1 respectively, and $\Lambda_{pq}$ is the pitch between core p and core q. Up, Vp and Wp are the normalized fiber parameters defined as Up = ap[(2πn1p/λ)2 - βp2]1/2, Vp = ap[(2π/λ)(n1p2 - n22)1/2, Wp = ap[(βp2 - (2πn2/λ)2 ]1/2 with the free space wavelength, λ. The relative core-cladding refractive index difference is given by, Δp = (n1p2 – n22)/(2n1p2).

Equation (1) can be translated to an eigenvalue problem as

$$\frac{{dE(z)}}{{dz}} ={-} {\mathbf{RE}}(\mathbf{z})$$
where, $E(z) = {[{{E_1}(z)\;{E_2}(z)\;{E_3}(z)\ldots \ldots .{E_n}(z)} ]^T}$ and R contains z-independent elements rpq given by
$${r_{pq}} = \left\{ \begin{array}{l} j{C_{pq}}\quad \quad \quad ;\quad p \ne q\\ j{\beta_p}\quad \quad \quad \;;\quad p = q \end{array} \right.$$

The general solution for n-core MCF describing power exchange between the cores as the light propagates can be obtained by substituting ${\rm E}({\rm z}) = [{\exp ( - {{\rm Rz}}){\rm E}(0)} ]$ in Eq. (5). The solution is

$${\rm E}({\rm z}) = {\rm V}[{\exp ( - {\gamma_{\rm p}}{\rm z}){\delta_{{\rm pq}}}} ]{{\rm V}^{ - 1}}{\rm E}(0)$$
$${\rm V} = [{{v_1}\;{v_2}\;{v_3}\ldots \ldots .{{\rm v}_n}} ]$$
where, $\delta_{pq}$ denotes Kronecker delta function, ${\gamma_p}$ stands for an eigenvalue of R, and vp denotes the corresponding eigenvector.

In the subsequent sections of this paper we numerically evaluate the inter-core coupling coefficients, and analytically investigate the impact of inter-core crosstalk on the performance of WDM system in addition to the influence of SRS and ASE in 7-core homogeneous MCF arranged in a hexagonal form as shown in Fig. 1.

5. Evaluation of coupling coefficients of homogeneous 7-core MCF

Let us consider the homogeneous 7 core MCF with Δp = Δq = Δ, where the CMT can be used [19] to evaluate the coupling coefficients between the cores. The coupled-mode equations given in Eq. (1) with the matrix C for the 7-core MCF would be as in [19]

$$C = j \times \;\left|{\begin{array}{ccccccc} 0&{{C_{12}}}&{{C_{13}}}&{{C_{14}}}&{{C_{15}}}&{{C_{16}}}&{{C_{17}}}\\ {{C_{21}}}&0&{{C_{23}}}&{{C_{24}}}&{{C_{25}}}&{{C_{26}}}&{{C_{27}}}\\ {{C_{31}}}&{{C_{32}}}&0&{{C_{34}}}&{{C_{35}}}&{{C_{36}}}&{{C_{37}}}\\ {{C_{41}}}&{{C_{42}}}&{{C_{43}}}&0&{{C_{45}}}&{{C_{46}}}&{{C_{47}}}\\ {{C_{51}}}&{{C_{52}}}&{{C_{53}}}&{{C_{54}}}&0&{{C_{56}}}&{{C_{57}}}\\ {{C_{61}}}&{{C_{62}}}&{{C_{63}}}&{{C_{64}}}&{{C_{65}}}&0&{{C_{67}}}\\ {{C_{71}}}&{{C_{72}}}&{{C_{73}}}&{{C_{74}}}&{{C_{75}}}&{{C_{76}}}&0 \end{array}} \right|$$
where, the diagonal zeros correspond to negligible mutual (self) coupling of the core itself when the same is excited. As shown in Fig. 1, the coupling coefficient between the closest adjacent cores are denoted as: C12, C13, C14,…..C17; C23, C34, C45—C67, termed as the “C12 group” in which the distance between the cores is ${\varLambda}$; the next adjacent cores are: C24, C35, …C57, C62, and termed as “C24” group in which the distance between the cores is $1.732\varLambda$.; the last group of cores are those which are located just opposite to each other in the arrangement: C25, C36, …..C47 termed as the “C25 group” in which the distance between the cores is $2\varLambda$.

All the coupling coefficients between adjacent cores and non-adjacent cores (except self-coupling) are evaluated in our analysis. Now, according to the grouping of the cores based on their spatial distance the coupling coefficient, Cpq matrix of Eq. (9) can be re-written below as Eq. (10):

$$C = j \times \;\left|{\begin{array}{ccccccc} 0&{{C_{12}}}&{{C_{12}}}&{{C_{12}}}&{{C_{12}}}&{{C_{12}}}&{{C_{12}}}\\ {{C_{12}}}&0&{{C_{12}}}&{{C_{24}}}&{{C_{25}}}&{{C_{24}}}&{{C_{12}}}\\ {{C_{12}}}&{{C_{12}}}&0&{{C_{12}}}&{{C_{24}}}&{{C_{25}}}&{{C_{24}}}\\ {{C_{12}}}&{{C_{24}}}&{{C_{12}}}&0&{{C_{12}}}&{{C_{24}}}&{{C_{25}}}\\ {{C_{12}}}&{{C_{25}}}&{{C_{24}}}&{{C_{12}}}&0&{{C_{12}}}&{{C_{24}}}\\ {{C_{12}}}&{{C_{24}}}&{{C_{25}}}&{{C_{24}}}&{{C_{12}}}&0&{{C_{12}}}\\ {{C_{12}}}&{{C_{12}}}&{{C_{24}}}&{{C_{25}}}&{{C_{24}}}&{{C_{12}}}&0 \end{array}} \right|$$

Taking the values of the MCF parameters as shown in Table 1 and solving the Eq. (4) with MATLAB software we evaluate all the coupling coefficients, Cpq and plot the same against relative refractive index contrast, Δ with different values of core pitch, ${\varLambda}$ (Fig. 3). However, in our analysis while evaluating the inter-core crosstalk power, we have considered the coupling coefficients of the closest adjacent cores as worst case scenario, as the coupling coefficients between the non-adjacent cores being negligible.

 figure: Fig. 3.

Fig. 3. Plots of coupling coefficients vs. relative refractive index contrast, Δ with different values of core pitch, ${\varLambda}=30{\mathrm{\mu} \mathrm{m}}$, radius, r = 4.0 µm of a SDM-WDM system with 7 core homogeneous MCF

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6. System analysis

The equations that govern the Raman process for a single co-polarized pump wavelength and signal wavelength travelling in the same direction are (neglecting spontaneous noise) are expressed as in [32]

$$\frac{{{\textrm d}{{\textrm P}_{\textrm{sig}\_\textrm{n}}}}}{\textrm{dz}} = \frac{{\textrm{g}^{\prime}}}{{{\textrm{A}_{\textrm{eff}}}}}{{\textrm P}_{\textrm{sig}\_{\textrm n}}}{\textrm{P}_{\textrm{pump}\_{\textrm n}}} - {\mathrm{\alpha}_{\textrm s}}{{\textrm P}_{\textrm{sig}\_{\textrm n}}}$$
$$\frac{{{\rm d}{{\rm P}_{{\rm pump}\_{\rm n}}}}}{{\rm dz}} ={-} \frac{{{\nu _{\rm p}}}}{{{\nu _{\rm s}}}}\,\frac{{{\rm g}^{\prime}}}{{{{\rm A}_{{\rm eff}}}}}{{\rm P}_{{\rm pump}\_{\rm n}}}{{\rm P}_{{\rm sig}\_{\rm n}}} - {\mathrm{\alpha} _{\rm p}}{{\rm P}_{{\rm sig}\_{\rm n}}}$$
where Psig_n and Ppump_n are the signal and pump powers in n-th channel, respectively; $g^{\prime}$ is the Raman gain coefficient; Aeff is the effective area of the modes in the fiber; αs and αp are the fiber attenuation coefficients for the signal and pump, respectively; νs and νp are the signal and pump frequencies, respectively; and z is the propagation length variable. The term νp/νs accounts for the fact that one pump photon is depleted for each signal photon created (i.e., the quantum efficiency). In Eq. (11) and Eq. (12) Aeff is explicitly shown such that the scaling with fiber core area may be seen. The quantity ($g^{\prime}$/Aeff) (which includes the effects of signal-pump mode overlap) has units of 1/(W·km). Raman gain is polarization dependent with very low interaction between orthogonally polarized signal and pump.

Upon solving the differential Eq. (12), the expression for the n-th channel input signal power at propagation distance z, Psig_n (z) becomes as in [35]

$${P_{sig\_n}}(z )= \; {P_{is\_{n_\_}}}\; {P_{pump\_n}}{e^{ - \alpha z}}exp [{G{P_{pump\_n}}({n - 1} ){Z_e}} ].$$
$${\left[ {{P_{pump\_n}}\mathop \sum \nolimits_{m = 1}^{{N_{ch}}} {e^{G{P_{pump\_n}}({m - 1} ){Z_e}}}} \right]^{ - 1}}$$
where Pis_n is the input signal power to any n-th channel of the Nch channel WDM system and Ppump_n is the input pump power to the Raman amplifier. $G = \frac{{g^{\prime}\Delta f}}{{2{A_{eff}}}},$ G is the Raman amplifier gain (divided by 2 to account for polarization averaging), and Δf is the inter-channel optical frequency spacing which is also termed as the Optical Bandwidth of the system, B0, when added the guard band Bg with it. ${Z_e} = \frac{{1 - {e^{ - \alpha z}}}}{\alpha },$ is the effective propagation distance, α is the fiber attenuation coefficient. Equation (13) can be re-written as (n-th channel power at z distance)
$${P_{sig\_n}}(z) = \frac{{{P_{is\_n}}{e^{ - \alpha z}}{e^{G{P_{pump\_n}}n{Z_e}}}}}{{\mathop \sum \nolimits_{m = 1}^{{N_{ch}}} {e^{mG{P_{pump\_n}}{Z_e}}}}}$$

Let, ${J_0} = \mathop \sum \nolimits_{m = 1}^{{N_{ch}}} {P_{pump\_m}}$ is the total input pump power (per core) of the multi pump system to the amplifiers. Let’s consider each of the Nch WDM system is fed with equal input power i.e, Pis_0. Then the total input power to the Nch channel WDM system at a propagation distance z can be expressed as in [35]

$${P_{si{g_{\_n}}}}(z )= {N_{ch}}{P_{is{\__0}}}{e^{ - \alpha z}}.\,exp \left[ {\left( {\frac{{G{J_0}{Z_e}}}{2}} \right)({2n - {N_{ch}} - 1} )} \right].\left[ {\frac{{sinh \left( {\frac{{G{J_0}{Z_e}}}{2}} \right)}}{{sinh \left( {\frac{{{N_{ch}}G{J_0}{Z_e}}}{2}} \right)}}} \right]. $$

6.1 Crosstalk evaluation

Maximum SRS induced crosstalk impairment in WDM channels can be expressed as in [35]

$${P_{CT}}(z )= \frac{{[{{P_{is{\__0}}}{e^{ - \alpha z}} - {P_1}(z)} ]}}{{({{P_{is{\__0}}}{e^{ - \alpha z}}} )}}\;$$
which is the fractional power lost through SRS in the lowest channel

Now, in case of equal channel loading (each channel by Pis_0) the total inter-channel crosstalk, PCT_ich (z) (fractional crosstalk multiplied by Ppump_0) at a propagation distance, z in core 1 (centre core in Fig. 1) can be obtained from Eq. (15) and (16) as

$${P_{C{T_{ich}}}}(z )\; = \left\{ {\begin{array}{*{20}{c}} {1 - {N_{ch}}exp \left[ { - G{N_{ch}}{P_{is{\__0}}}{Z_e}\left( {\frac{{{N_{ch}} - 1}}{2}} \right)} \right].}\\ {\left[ {\frac{{sinh \left( {\frac{{G{N_{ch}}{P_{is{\__0}}}{Z_e}}}{2}} \right)}}{{sinh \left( {\frac{{GN_{ch}^2{P_{is{\__0}}}{Z_e}}}{2}} \right)}}} \right]} \end{array}} \right\}.{P_{pump\_0}}$$

Now, the inter-core crosstalk, PCT_ico of core 1 (centre core in Fig. 1) due to overlapping of power from all WDM channels of the other cores can be obtained as (ignoring the imaginary terms, and self-coupling in core 1).

$$\begin{array}{l}{P_{C{T_{ico}}}}(z )= \;{P_{C{T_{ich}}}}(z ).[{cos({{C_{12}}.z} )+ cos + cos({{C_{14}}.z} )+ cos({{C_{15}}.z} )+ cos({{C_{16}}.z} )+ cos({{C_{17}}.z} )} ]\; \\ Or,\quad {P_{C{T_{ico}}}}(z )= \;{P_{C{T_{ich}}}}(z ).\mathop \sum \limits_{j = 2}^7 cos({{C_{ij}}.z} )\;\;i = 1,\quad i \ne j \end{array}$$

Here, C12, C13, …, C17 are the coupling coefficients between the cores: 1-2, 1-3, ….1-7. The coupling coefficients ${C_{ij}}$ are evaluated as in Eq. (10)

Now, as the worst-case scenario, the highest value of the coupling coefficients (between centre core and each of the 6 surrounding cores) is considered to determine the crosstalk power of core 1.

So the total crosstalk of core 1 is obtained as

$${{\rm P}_{{\rm CT}\_{{\rm Total}}}}({\rm z} )\,{\kern 1pt} \; = \;\;{{\rm P}_{{\rm CT}\_{{\rm ich}}}}({\rm z} )\, + \,{{\rm P}_{{{\rm CT}}\_{{\rm ico}}}}({\rm z} )$$
$$\scalebox{0.93}{$\begin{aligned} &= {P_{CT\_ich}}(z )+ {P_{CT\_ich}}(z )[{cos({{C_{12}}z} )+ cos({{C_{13}}z} )+ cos({{C_{14}}z} )+ cos({{C_{15}}z} )+ cos({{C_{16}}z} )+ cos({{C_{17}}z} )} ]\\ &= {P_{CT\_ich}}(z )\{{1 + [{cos({{C_{12}}z} )+ cos({{C_{13}}z} )+ cos({{C_{14}}z} )+ cos({{C_{15}}z} )+ cos({{C_{16}}z} )+ cos({{C_{17}}z} )} ]} \}\end{aligned}$}$$

The electric field due to the total crosstalk may be expressed as in [35]

$${e_{CT\_n}}({t,\;z} )= \sqrt {2{P_{CT\_Total}}(z)} cos({{\omega_n}ct} )$$

The optical signal to crosstalk noise ratio, OSCNR at a propagation distance, z can be expressed as in [35]

$$OSCNR = \frac{{{P_{sig\_n}}(z)}}{{P{{(z )}_{CT\_Total}}}}$$
$$=\frac{{\frac{{{P_{is\_n}}{e^{ - \alpha z}}{e^{G{P_{pump\_n}}n{Z_e}}}}}{{\mathop \sum \nolimits_{m = 1}^{{N_{ch}}} {e^{mG{P_{pump\_n}}{Z_e}}}}}}}{{{P_{CT\_ich}}(z )\left\{ {1 + \left[ {\begin{array}{*{20}{c}} {cos({{C_{12}}z} )+ cos({{C_{13}}z} )+ cos({{C_{14}}z} )+ }\\ {cos({{C_{15}}z} )+ cos({{C_{16}}z} )+ cos({{C_{17}}z} )} \end{array}} \right]} \right\}}}$$
$$=\frac{{\frac{{{{\boldsymbol P}_{{\boldsymbol{is}}\_{\boldsymbol n}}}{{\boldsymbol e}^{ - {\boldsymbol \alpha z}}}{{\boldsymbol e}^{{\boldsymbol G}{{\boldsymbol P}_{{\boldsymbol{pump}}\_{\boldsymbol n}}}{\boldsymbol n}{{\boldsymbol Z}_{\boldsymbol e}}}}}}{{\mathop \sum \nolimits_{{\boldsymbol m} = 1}^{{{\boldsymbol N}_{{\boldsymbol{ch}}}}} {{\boldsymbol e}^{{\boldsymbol{mG}}{{\boldsymbol P}_{{\boldsymbol{pump}}\_{\boldsymbol n}}}{{\boldsymbol Z}_{\boldsymbol e}}}}}}}}{{{{\boldsymbol P}_{{\boldsymbol{CT}}\_{\boldsymbol{ich}}}}({\boldsymbol z} )\;.\;\left[ {1 + \;\mathop \sum \nolimits_{{\boldsymbol j} = 2}^7 {\boldsymbol{cos}}({{{\boldsymbol C}_{{\boldsymbol i}{\boldsymbol j}}}.{\boldsymbol z}} )\quad {\boldsymbol i} = 1,{\boldsymbol \; }{\boldsymbol i} \ne {\boldsymbol j}} \right]}}$$

6.2 ASE noise evaluation

ASE power is expressed as in [35]

$${P_{ASE}} = \left[ {{G_0} - 1 + \frac{{{G_0}\alpha }}{{G{P_{pump\_n}}}}\left( {{e^{\alpha z}} - \frac{1}{{{G_0}}}} \right)} \right]{h_\nu }{B_0}\eta T$$
where, ηT is the thermal equilibrium phonon number, ${G_0} = \exp ({G{Z_e}{P_{pump\_n}}} )$ is the Gain hυ = 6.63X10−34 J is the photon energy, in Joule, B0 = 2Be = 2B is the optical bandwidth

At the receiver end, the output current of the photodiode is expressed as in [35]

$$\begin{array}{l} i(t )= {R_d}\;{|{{e_{sig\_n}}({t,z} )+ {e_{CT}}({t,z} )+ {e_{ASE}}({t,z} )} |^2}\\ = {R_d}\;{\left|{\sqrt {2{P_{si{g_n}}}} cos\; {\omega_{si{g_n}}}({t,z} )+ \sqrt {2{P_{CT\_n}}} cos\; {\omega_{CT\_n}}({t,z} )+ \sqrt {2{P_{ASE}}} cos\; {\omega_{ASE}}({t,z} )} \right|^2}\; \end{array}$$
Where, Rd is the photodiode Responsivity
$$\scalebox{0.9}{$\begin{array}{l} = {R_d}\;\left|\begin{array}{l} {P_{sig\_n}} + {P_{sig\_n}}cos{\kern 1pt} {\kern 1pt} {\omega_{sig\_n}}({t,\,z} )+ 2\sqrt {{P_{sig\_n}}{P_{CT\_n}}} cos{\kern 1pt} {\kern 1pt} [{{\omega_{sig\_n}}({t,\,z} )+ {\omega_{CT\_n}}({t,\,z} )} ]+ \\ 2\sqrt {{P_{sig\_n}}{P_{CT\_n}}} cos{\kern 1pt} {\kern 1pt} [{{\omega_{sig\_n}}({t,\,z} )- {\omega_{CT\_n}}({t,\,z} )} ]+ {P_{CT\_n}} + {P_{CT\_n}}cos{\kern 1pt} {\kern 1pt} 2{\omega_{CT\_n}}({t,\,z} )+ \\ 2\sqrt {{P_{sig\_n}}{P_{ASE}}} cos{\kern 1pt} {\kern 1pt} [{{\omega_{sig\_n}}({t,\,z} )+ {\omega_{ASE}}({t,\,z} )} ]+ 2\sqrt {{P_{sig\_n}}{P_{ASE}}} cos{\kern 1pt} {\kern 1pt} [{{\omega_{sig\_n}}({t,\,z} )- {\omega_{ASE}}({t,\,z} )} ]+ \\ 2\sqrt {{P_{CT\_n}}{P_{ASE}}} cos{\kern 1pt} {\kern 1pt} [{{\omega_{CT\_n}}({t,\,z} )+ {\omega_{ASE}}({t,\,z} )} ]+ 2\sqrt {{P_{CT\_n}}{P_{ASE}}} cos{\kern 1pt} {\kern 1pt} [{{\omega_{CT\_n}}({t,\,z} )- {\omega_{ASE}}({t,\,z} )} ]+ \\ {P_{ASE}} + {P_{ASE}}cos{\kern 1pt} {\kern 1pt} 2{\omega_{ASE\_n}}({t,\,z} )\end{array} \right|\end{array}$}$$

Now, in the WDM system input signal and crosstalk are in the same wavelength. Thus, ${\omega _{sig\_n}} = {\omega _{CT\_n}}$

Removing the higher frequency terms from Eq. (26) we get the photocurrent at the output of the band pass filter

$${i_0}(t )= {R_d}\left|{\begin{array}{*{20}{c}} {{P_{sig\_n}} + {P_{CT\_n}} + {P_{ASE}} + 2\sqrt {{P_{sig\_n}}{P_{CT\_n}}} }\\ { + 2\sqrt {{P_{sig\_n}}{P_{ASE}}} cos[{{\omega_{sig\_n}}({t,z} )- {\omega_{ASE}}({t,z} )} ]}\\ { + 2\sqrt {{P_{CT\_n}}{P_{ASE}}} cos[{{\omega_{CT\_n}}({t,z} )- {\omega_{ASE}}({t,z} )} ]} \end{array}} \right|$$

The average signal current can be written as in [35]

$${I_{sig\_n}} = \overline {{i_{sig\_n,j}}} (t )= {R_d}.\overline {{P_{sig\_n}}} ({t,z} )$$

The variance of signal power is expressed as

$$I_{sig\_n}^2 = {[{{R_d}{P_{sig\_n}}({t,z} )} ]^2}\,({\textrm{it is the signal power at the output of the photodiode}})$$

6.3 Noise variances

The crosstalk variance is expressed as

$$\begin{aligned} \overline {i_{CT}^2} &= {[{{R_d}{P_{CT}}({t,z} )} ]^2}\\ \sigma _{CT}^2 &= \overline {i_{CT}^2} \end{aligned}$$

The shot noise of the receiver can be expressed as

$$\begin{aligned} \sigma _{shot}^2 &= 2eB[{{i_{si{g_n}}}({t,z} )+ {i_{CT}}({t,z} )+ {i_{ASE}}({t,z} )} ]\;\\ \sigma _{shot}^2 &= 2eB{R_d}[{{P_{sig\_n}}({t,z} )+ {P_{CT}}({t,z} )+ {P_{ASE}}({t,z} )} ]\end{aligned}$$

The thermal noise variance of the receiver is expressed as

$$\sigma _{th}^2 = \frac{{4KT}}{{{R_L}}}B$$
where, Ith is the thermal noise current spectral density in $\frac{A}{{\sqrt {Hz} }}$, K is Boltzmann constant (1.3806X10−23 J/0 K), B is the receiver bandwidth (Hz), RL is receiver equivalent load resistance (Ohm), T = Standard temp (300 0 K)

6.4 Beat noises

Signal-crosstalk beat noise is expressed as

$$\sigma _{sig\_CT}^2 = {\left( {2{R_d}\sqrt {{P_{sig\_n}}{P_{CT\_n}}} } \right)^2} = 4R_d^2{P_{sig\_n}}{P_{CT\_n}}$$

ASE-crosstalk beat noise is expressed as

$$\sigma _{ASE\_CT}^2 = {\left( {2{R_d}\frac{{\sqrt {{P_{ASE}}{P_{CT\_n}}} }}{2}} \right)^2} = 2R_d^2{P_{ASE}}{P_{CT\_n}}$$

Signal-ASE beat noise is expressed as

$$\sigma \,_{{\boldsymbol{sig}}\_{\boldsymbol{ASE}}}^2 = {\left( {2{{\boldsymbol R}_{\boldsymbol d}}\,\frac{{\sqrt {{{\boldsymbol P}_{{\boldsymbol{sig}}\_{\boldsymbol n}}}{\boldsymbol{P}_{\boldsymbol{ASE}}}} }}{2}} \right)^2} = 2\boldsymbol{R}\,_{\boldsymbol{d}}^2\,{\boldsymbol{P}_{{\boldsymbol{sig}}\_{\boldsymbol{n}}}}\,{\boldsymbol{P}_{\boldsymbol{ASE}}}$$

Electrical signal to crosstalk plus noise ratio at the output of the receiver low pass filter (LPF) is expressed as

$$SCNR = \frac{{{{[{{R_d}\,{P_{sig\_n}}({t,\,\,z} )} ]}^2}}}{{\sigma _{CT}^2 + \sigma _{shot}^2 + \sigma _{th}^2 + \sigma _{sig\_CT}^2 + \sigma _{sig\_{\textrm{ASE}}}^2 + \sigma _{{{\rm ASE}}\_{{\rm CT}}}^2}}$$

The bit error rate (BER) can be evaluated as

$$BER = \frac{1}{2}erfc\; \left( {\frac{{\sqrt {SCNR} }}{{2\sqrt 2 }}} \right)$$

7. Results and discussions

In this section as per the analytical approach presented in section III, the performance of the 7 core MCF based SDM-WDM communication system considering the inter-channel and inter-core crosstalk and ASE noise power of distributed Raman Scattering optical amplifier is evaluated and presented. The system parameters used are given in Table 2

Tables Icon

Table 2. Parameter values used for the system

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In Fig. 3, we observe that the coupling coefficients decrease with the increase of relative refractive index contrast, and it is higher for lower values of core pitch. For our analysis we consider the highest value (0.0052) of coupling coefficient as the worst case scenario.

Figure 4 presents the relationship between BER and input power at different WDM channel numbers with a given channel bandwidth (BW), amplifier pump power and propagation distance. It is noticed that the BER decreases (system performance improves) with the increase of input power, but to maintain a particular BER value at higher number of WDM channels it is required to increase the input power while considering only SRS and ASE. Again, to increase the same number of WDM channels, the requirement of increasing the input power becomes higher while considering the IC-XT. For example, to maintain a particular BER (10E-9, say) we need to increase the input power from 2.5dBm to 5.5dBm while increasing the WDM channels from 12 to 16 considering only the SRS and ASE. On the other hand, we need to increase the input power from 5.5dBm to 9.0dBm to increase the WDM channels from 12 to 16 when the IC-XT is added up with the SRS and ASE.

 figure: Fig. 4.

Fig. 4. Plots of BER Vs. input signal power at a given pump power, channel bandwidth, link length and different channel numbers considering SRS, ASE and IC-XT of a SDM-WDM system with 7 core homogeneous MCF.

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Figure 5 depicts the relationship between the power penalty and WDM channel spacing at BER value of 10E-9 with link distance z = 50 km, pump power = 10 mW at different number of WDM channels. It is observed that with the increase of channel spacing and inclusion of inter-core coupling induced crosstalk (PCT_ico) power penalty is increased irrespective of the number of WDM channels. It is also noticed that at a particular channel spacing to increase the number of WDM channels the power penalty is also increased. For example, at channel spacing 50 GHz, to increase the number of WDM channels from 8 to 16 considering the inter-core crosstalk, the power penalty is 1.4 dB.

 figure: Fig. 5.

Fig. 5. Pots of power penalty and WDM channel spacing at BER value of 10E-9 with link distance z = 50 km, pump power =10 mW at different WDM channels.

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Figure 6 depicts the relationship between BER and WDM channel spacing at various pump power at given WDM channel numbers, input signal power and link length. It is observed that with the increase of channel spacing, BER increases (system performance deteriorates) irrespective of the value of pump power. It is also noticed that at a particular value of channel spacing, the BER deteriorates further with the incorporation of inter-core crosstalk along with SRS and ASE noises. For example, at channel spacing of 20 GHz and pump power 15 mW, the BER without IC-XT is 10E-8, and the same with IC-XT is 10E-4. Again, at a channel spacing of 25 GHz and pump power 20 mW, the BER without IC-XT is 10E-6, and the same with IC-XT is 10E-2.

 figure: Fig. 6.

Fig. 6. Plots of BER Vs. channel spacing at different pump power of Raman Amplifier considering SRS, ASE and IC-XT in a SDM-WDM system with 7 core homogeneous MCF.

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The length dependence of BER is presented in Fig. 7. For direct detection SDM-WDM communication system with 7 core homogeneous MCF considering SRS, ASE as well as inter-core coupling induced crosstalk noise. It is noticed that the BER increases (system performance deteriorates) with the increase of propagation distance. The figure also depicts that the BER increases with the increase of number of WDM channels. It is further noticed that the system performance is severely affected due to addition of IC-XT to the SRS and ASE. For example, at a propagation distance, z = 30 km and number of WDM channels, Nch = 8, the BER without IC-XT is 10E-5, and it is 10E-3 with IC-XT.

 figure: Fig. 7.

Fig. 7. Plots of BER Vs. propagation distance at different number of WDM channels considering SRS, ASE and IC-XT of a SDM-WDM system with 7 core homogeneous MCF at a given input signal power, pump power and channel BW.

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The relationship between BER and pump power with a given propagation distance, z = 50 km, Pin_s = 4 mW and various WDM channel BW considering SRS, ASE as well as IC-XT noise is shown in Fig. 8. It is observed that BER deteriorates SRS and ASE which is further worsened by the coupling induced IC-XT. For example, to maintain BER value of 10E-9 the allowable pump power is reduced from 8dBm to 1.0 dBm to increase the number of WDM channels from 8 to 16 at a channel spacing of 50 GHz without considering the IC-XT. On the other hand, taking into account of the IC-XT, for maintaining the same value of BER (10E-9) and increase the same number of WDM channels (8 to 16) at the same channel spacing (50 GHz) the allowable pump power is reduced from 6dBm to 0dBm.

 figure: Fig. 8.

Fig. 8. Plots of BER vs. pump power at different number of WDM channels and channel BW considering SRS, ASE and IC-XT of a SDM-WDM system with 7 core homogeneous MCF at a given input signal power and propagation distance.

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The plots of allowable pump power vs. number of WDM channels and channel BW considering SRS, ASE and IC-XT of a SDM-WDM system with 7 core homogeneous MCF is shown in Fig. 9. It is observed that with the increase of channel bandwidth the allowable pump power is reduced for maintaining same BER value with same number of WDM channels in both the cases (without and with IC-XT). For example, to maintain BER value of 10E-9 the allowable pump power is reduced from 6.5dBm to 1.0dBm to increase the number of WDM channels from 8 to 16 at a channel spacing of 25 GHz without considering the IC-XT. On the other hand, taking into account of the IC-XT, for maintaining the same value of BER (10E-9) and increase the same number of WDM channel (8 to 16) the allowable pump power is reduced from 4.8dBm to -1.5dBm.

 figure: Fig. 9.

Fig. 9. Plots of allowable pump power vs. number of WDM channels and channel BW considering SRS, ASE and IC-XT in a SDM-WDM system with 7 core homogeneous MCF at a given input signal power and propagation distance for maintaining BER value of 10E-9.

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The allowable propagation distance vs. number of WDM channels is depicted in Fig. 10 with input signal power, Pin-s = 4 mW, pump power =15 mW, and different values of WDM channel bandwidth. It is noticed that allowable transmission distance is reduced by Raman amplifier induced crosstalk and ASE noise for increased number of WDM channels which is further reduced by the effect of IC-XT noise. For example, at number of WDM channels 8, it is noticed that the maximum allowable propagation distance without IC-XT are: 42 km, 37 km, 31 km and 29 km against the optical channel bandwidth of 25 GHz, 50 GHz, 75 GHz and 100 GHz respectively. On the other hand, the maximum allowable propagation distance with IC-XT is: 34 km, 33 km, 24 km and 20 km against the optical channel bandwidth of 25 GHz, 50 GHz, 75 GHz and 100 GHz respectively.

 figure: Fig. 10.

Fig. 10. Plots of allowable propagation distance vs. number of WDM Channels and channel BW considering Raman scattering, ASE and IC-XT of a SDM-WDM system with 7 core homogeneous MCF at a given input signal power for BER value of 10E-6.

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8. Conclusion

The inter-channel and inter-core coupling induced crosstalk including SRS and ASE noise in a single mode 7-core homogeneous MCF based SDM-WDM communication system with distributed Raman amplifier is analytically evaluated and presented in this paper. The SRS induced crosstalk including ASE noise and IC-XT at the output of any core with light launched into other core is investigated. The Signal to Crosstalk plus Noise Ratio (SCNR) is evaluated analytically at the output of photo detector of the excited core. The Bit Error Rate (BER) performance results of the link against various system parameters like, input signal power, WDM channel spacing, propagation distance, pump power etc. are determined and presented. The maximum allowable pump power considering both inter-channel and inter-core crosstalk including SRS impairment and ASE noise is evaluated for a given value of BER, input signal power, data rate and propagation distance at various WDM channel bandwidth and number of WDM channels. The analysis indicates that the BER deteriorates with SRS induced crosstalk in addition to the ASE noise which is further affected by the effect of inter-core crosstalk. Power penalty versus WDM channel spacing at a given BER value (10E-9), link distance, and pump power with different WDM channels is also evaluated. Furthermore, the maximum allowable propagation distance of the SDM-WDM communication system considering both inter-channel and inter-core crosstalk including SRS and ASE noise for a given value of BER and other system parameters are evaluated analytically.

Acknowledgments

The authors acknowledge the laboratory and technical support provided by the department of EEE, BUET, Bangladesh to conduct this research as a part of the PhD dissertation of the corresponding author.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (10)
Fig. 1.
Fig. 1. Cross-section of a homogeneous 7-core single mode MCF

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Fig. 2.
Fig. 2. Schematic illustration of a 7 core MCF based SDM-WDM communication system with distributed Raman amplifier

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Fig. 3.
Fig. 3. Plots of coupling coefficients vs. relative refractive index contrast, Δ with different values of core pitch, ${\varLambda}=30{\mathrm{\mu} \mathrm{m}}$, radius, r = 4.0 µm of a SDM-WDM system with 7 core homogeneous MCF

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Fig. 4.
Fig. 4. Plots of BER Vs. input signal power at a given pump power, channel bandwidth, link length and different channel numbers considering SRS, ASE and IC-XT of a SDM-WDM system with 7 core homogeneous MCF.

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Fig. 5.
Fig. 5. Pots of power penalty and WDM channel spacing at BER value of 10E-9 with link distance z = 50 km, pump power =10 mW at different WDM channels.

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Fig. 6.
Fig. 6. Plots of BER Vs. channel spacing at different pump power of Raman Amplifier considering SRS, ASE and IC-XT in a SDM-WDM system with 7 core homogeneous MCF.

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Fig. 7.
Fig. 7. Plots of BER Vs. propagation distance at different number of WDM channels considering SRS, ASE and IC-XT of a SDM-WDM system with 7 core homogeneous MCF at a given input signal power, pump power and channel BW.

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Fig. 8.
Fig. 8. Plots of BER vs. pump power at different number of WDM channels and channel BW considering SRS, ASE and IC-XT of a SDM-WDM system with 7 core homogeneous MCF at a given input signal power and propagation distance.

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Fig. 9.
Fig. 9. Plots of allowable pump power vs. number of WDM channels and channel BW considering SRS, ASE and IC-XT in a SDM-WDM system with 7 core homogeneous MCF at a given input signal power and propagation distance for maintaining BER value of 10E-9.

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Fig. 10.
Fig. 10. Plots of allowable propagation distance vs. number of WDM Channels and channel BW considering Raman scattering, ASE and IC-XT of a SDM-WDM system with 7 core homogeneous MCF at a given input signal power for BER value of 10E-6.

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Tables (2)

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Table 1. SDM-WDM Transmission Scheme

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Tables Icon

Table 2. Parameter values used for the system

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Equations (39)

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$${{\rm E}_{\rm p}}(\rm z) = {{\rm A}_{\rm p}}({\rm z})\exp( - {\rm j}{\mathrm{\beta} _{\rm p}}{\rm z})$$
$$\frac{{{\rm dA}({\rm z})}}{{{\rm dz}}} ={-} {\rm CA}({\rm z})$$
$$\begin{array}{l} {C_{pq}} = \left\{ \begin{array}{l} j{C_{pq}}\exp [{j({{\beta_p} - {\beta_q}} )z} ]\quad \quad ;\quad p \ne q\\ 0\quad \quad ;\quad p = q \end{array} \right.\\ \end{array}$$
$${C_{pq}} = \sqrt {2{\Delta _q}} {W_q}{U_p}{K_0}\left( {\left( {\frac{{{W_p}{d_{pq}}}}{{{a_p}}}} \right)} \right)\left[ \begin{array}{l} {a_p}{U_q}{J_1}({{U_q}} ){I_0}\left( {\frac{{{W_p}{a_q}}}{{{a_p}}}} \right) + \\ {a_q}{W_p}{J_0}({{U_q}} ){I_1}\left( {\frac{{{W_p}{a_q}}}{{{a_p}}}} \right) \end{array} \right]{\left[ \begin{array}{l} {V_p}{J_1}({{U_q}} ){K_1}({{W_p}} )\\ ({{a_p}^2{U_q}^2 + {a_q}^2{W_p}^2} )\end{array} \right]^{ - 1}}$$
$$\frac{{dE(z)}}{{dz}} ={-} {\mathbf{RE}}(\mathbf{z})$$
$${r_{pq}} = \left\{ \begin{array}{l} j{C_{pq}}\quad \quad \quad ;\quad p \ne q\\ j{\beta_p}\quad \quad \quad \;;\quad p = q \end{array} \right.$$
$${\rm E}({\rm z}) = {\rm V}[{\exp ( - {\gamma_{\rm p}}{\rm z}){\delta_{{\rm pq}}}} ]{{\rm V}^{ - 1}}{\rm E}(0)$$
$${\rm V} = [{{v_1}\;{v_2}\;{v_3}\ldots \ldots .{{\rm v}_n}} ]$$
$$C = j \times \;\left|{\begin{array}{ccccccc} 0&{{C_{12}}}&{{C_{13}}}&{{C_{14}}}&{{C_{15}}}&{{C_{16}}}&{{C_{17}}}\\ {{C_{21}}}&0&{{C_{23}}}&{{C_{24}}}&{{C_{25}}}&{{C_{26}}}&{{C_{27}}}\\ {{C_{31}}}&{{C_{32}}}&0&{{C_{34}}}&{{C_{35}}}&{{C_{36}}}&{{C_{37}}}\\ {{C_{41}}}&{{C_{42}}}&{{C_{43}}}&0&{{C_{45}}}&{{C_{46}}}&{{C_{47}}}\\ {{C_{51}}}&{{C_{52}}}&{{C_{53}}}&{{C_{54}}}&0&{{C_{56}}}&{{C_{57}}}\\ {{C_{61}}}&{{C_{62}}}&{{C_{63}}}&{{C_{64}}}&{{C_{65}}}&0&{{C_{67}}}\\ {{C_{71}}}&{{C_{72}}}&{{C_{73}}}&{{C_{74}}}&{{C_{75}}}&{{C_{76}}}&0 \end{array}} \right|$$
$$C = j \times \;\left|{\begin{array}{ccccccc} 0&{{C_{12}}}&{{C_{12}}}&{{C_{12}}}&{{C_{12}}}&{{C_{12}}}&{{C_{12}}}\\ {{C_{12}}}&0&{{C_{12}}}&{{C_{24}}}&{{C_{25}}}&{{C_{24}}}&{{C_{12}}}\\ {{C_{12}}}&{{C_{12}}}&0&{{C_{12}}}&{{C_{24}}}&{{C_{25}}}&{{C_{24}}}\\ {{C_{12}}}&{{C_{24}}}&{{C_{12}}}&0&{{C_{12}}}&{{C_{24}}}&{{C_{25}}}\\ {{C_{12}}}&{{C_{25}}}&{{C_{24}}}&{{C_{12}}}&0&{{C_{12}}}&{{C_{24}}}\\ {{C_{12}}}&{{C_{24}}}&{{C_{25}}}&{{C_{24}}}&{{C_{12}}}&0&{{C_{12}}}\\ {{C_{12}}}&{{C_{12}}}&{{C_{24}}}&{{C_{25}}}&{{C_{24}}}&{{C_{12}}}&0 \end{array}} \right|$$
$$\frac{{{\textrm d}{{\textrm P}_{\textrm{sig}\_\textrm{n}}}}}{\textrm{dz}} = \frac{{\textrm{g}^{\prime}}}{{{\textrm{A}_{\textrm{eff}}}}}{{\textrm P}_{\textrm{sig}\_{\textrm n}}}{\textrm{P}_{\textrm{pump}\_{\textrm n}}} - {\mathrm{\alpha}_{\textrm s}}{{\textrm P}_{\textrm{sig}\_{\textrm n}}}$$
$$\frac{{{\rm d}{{\rm P}_{{\rm pump}\_{\rm n}}}}}{{\rm dz}} ={-} \frac{{{\nu _{\rm p}}}}{{{\nu _{\rm s}}}}\,\frac{{{\rm g}^{\prime}}}{{{{\rm A}_{{\rm eff}}}}}{{\rm P}_{{\rm pump}\_{\rm n}}}{{\rm P}_{{\rm sig}\_{\rm n}}} - {\mathrm{\alpha} _{\rm p}}{{\rm P}_{{\rm sig}\_{\rm n}}}$$
$${P_{sig\_n}}(z )= \; {P_{is\_{n_\_}}}\; {P_{pump\_n}}{e^{ - \alpha z}}exp [{G{P_{pump\_n}}({n - 1} ){Z_e}} ].$$
$${\left[ {{P_{pump\_n}}\mathop \sum \nolimits_{m = 1}^{{N_{ch}}} {e^{G{P_{pump\_n}}({m - 1} ){Z_e}}}} \right]^{ - 1}}$$
$${P_{sig\_n}}(z) = \frac{{{P_{is\_n}}{e^{ - \alpha z}}{e^{G{P_{pump\_n}}n{Z_e}}}}}{{\mathop \sum \nolimits_{m = 1}^{{N_{ch}}} {e^{mG{P_{pump\_n}}{Z_e}}}}}$$
$${P_{si{g_{\_n}}}}(z )= {N_{ch}}{P_{is{\__0}}}{e^{ - \alpha z}}.\,exp \left[ {\left( {\frac{{G{J_0}{Z_e}}}{2}} \right)({2n - {N_{ch}} - 1} )} \right].\left[ {\frac{{sinh \left( {\frac{{G{J_0}{Z_e}}}{2}} \right)}}{{sinh \left( {\frac{{{N_{ch}}G{J_0}{Z_e}}}{2}} \right)}}} \right]. $$
$${P_{CT}}(z )= \frac{{[{{P_{is{\__0}}}{e^{ - \alpha z}} - {P_1}(z)} ]}}{{({{P_{is{\__0}}}{e^{ - \alpha z}}} )}}\;$$
$${P_{C{T_{ich}}}}(z )\; = \left\{ {\begin{array}{*{20}{c}} {1 - {N_{ch}}exp \left[ { - G{N_{ch}}{P_{is{\__0}}}{Z_e}\left( {\frac{{{N_{ch}} - 1}}{2}} \right)} \right].}\\ {\left[ {\frac{{sinh \left( {\frac{{G{N_{ch}}{P_{is{\__0}}}{Z_e}}}{2}} \right)}}{{sinh \left( {\frac{{GN_{ch}^2{P_{is{\__0}}}{Z_e}}}{2}} \right)}}} \right]} \end{array}} \right\}.{P_{pump\_0}}$$
$$\begin{array}{l}{P_{C{T_{ico}}}}(z )= \;{P_{C{T_{ich}}}}(z ).[{cos({{C_{12}}.z} )+ cos + cos({{C_{14}}.z} )+ cos({{C_{15}}.z} )+ cos({{C_{16}}.z} )+ cos({{C_{17}}.z} )} ]\; \\ Or,\quad {P_{C{T_{ico}}}}(z )= \;{P_{C{T_{ich}}}}(z ).\mathop \sum \limits_{j = 2}^7 cos({{C_{ij}}.z} )\;\;i = 1,\quad i \ne j \end{array}$$
$${{\rm P}_{{\rm CT}\_{{\rm Total}}}}({\rm z} )\,{\kern 1pt} \; = \;\;{{\rm P}_{{\rm CT}\_{{\rm ich}}}}({\rm z} )\, + \,{{\rm P}_{{{\rm CT}}\_{{\rm ico}}}}({\rm z} )$$
$$\scalebox{0.93}{$\begin{aligned} &= {P_{CT\_ich}}(z )+ {P_{CT\_ich}}(z )[{cos({{C_{12}}z} )+ cos({{C_{13}}z} )+ cos({{C_{14}}z} )+ cos({{C_{15}}z} )+ cos({{C_{16}}z} )+ cos({{C_{17}}z} )} ]\\ &= {P_{CT\_ich}}(z )\{{1 + [{cos({{C_{12}}z} )+ cos({{C_{13}}z} )+ cos({{C_{14}}z} )+ cos({{C_{15}}z} )+ cos({{C_{16}}z} )+ cos({{C_{17}}z} )} ]} \}\end{aligned}$}$$
$${e_{CT\_n}}({t,\;z} )= \sqrt {2{P_{CT\_Total}}(z)} cos({{\omega_n}ct} )$$
$$OSCNR = \frac{{{P_{sig\_n}}(z)}}{{P{{(z )}_{CT\_Total}}}}$$
$$=\frac{{\frac{{{P_{is\_n}}{e^{ - \alpha z}}{e^{G{P_{pump\_n}}n{Z_e}}}}}{{\mathop \sum \nolimits_{m = 1}^{{N_{ch}}} {e^{mG{P_{pump\_n}}{Z_e}}}}}}}{{{P_{CT\_ich}}(z )\left\{ {1 + \left[ {\begin{array}{*{20}{c}} {cos({{C_{12}}z} )+ cos({{C_{13}}z} )+ cos({{C_{14}}z} )+ }\\ {cos({{C_{15}}z} )+ cos({{C_{16}}z} )+ cos({{C_{17}}z} )} \end{array}} \right]} \right\}}}$$
$$=\frac{{\frac{{{{\boldsymbol P}_{{\boldsymbol{is}}\_{\boldsymbol n}}}{{\boldsymbol e}^{ - {\boldsymbol \alpha z}}}{{\boldsymbol e}^{{\boldsymbol G}{{\boldsymbol P}_{{\boldsymbol{pump}}\_{\boldsymbol n}}}{\boldsymbol n}{{\boldsymbol Z}_{\boldsymbol e}}}}}}{{\mathop \sum \nolimits_{{\boldsymbol m} = 1}^{{{\boldsymbol N}_{{\boldsymbol{ch}}}}} {{\boldsymbol e}^{{\boldsymbol{mG}}{{\boldsymbol P}_{{\boldsymbol{pump}}\_{\boldsymbol n}}}{{\boldsymbol Z}_{\boldsymbol e}}}}}}}}{{{{\boldsymbol P}_{{\boldsymbol{CT}}\_{\boldsymbol{ich}}}}({\boldsymbol z} )\;.\;\left[ {1 + \;\mathop \sum \nolimits_{{\boldsymbol j} = 2}^7 {\boldsymbol{cos}}({{{\boldsymbol C}_{{\boldsymbol i}{\boldsymbol j}}}.{\boldsymbol z}} )\quad {\boldsymbol i} = 1,{\boldsymbol \; }{\boldsymbol i} \ne {\boldsymbol j}} \right]}}$$
$${P_{ASE}} = \left[ {{G_0} - 1 + \frac{{{G_0}\alpha }}{{G{P_{pump\_n}}}}\left( {{e^{\alpha z}} - \frac{1}{{{G_0}}}} \right)} \right]{h_\nu }{B_0}\eta T$$
$$\begin{array}{l} i(t )= {R_d}\;{|{{e_{sig\_n}}({t,z} )+ {e_{CT}}({t,z} )+ {e_{ASE}}({t,z} )} |^2}\\ = {R_d}\;{\left|{\sqrt {2{P_{si{g_n}}}} cos\; {\omega_{si{g_n}}}({t,z} )+ \sqrt {2{P_{CT\_n}}} cos\; {\omega_{CT\_n}}({t,z} )+ \sqrt {2{P_{ASE}}} cos\; {\omega_{ASE}}({t,z} )} \right|^2}\; \end{array}$$
$$\scalebox{0.9}{$\begin{array}{l} = {R_d}\;\left|\begin{array}{l} {P_{sig\_n}} + {P_{sig\_n}}cos{\kern 1pt} {\kern 1pt} {\omega_{sig\_n}}({t,\,z} )+ 2\sqrt {{P_{sig\_n}}{P_{CT\_n}}} cos{\kern 1pt} {\kern 1pt} [{{\omega_{sig\_n}}({t,\,z} )+ {\omega_{CT\_n}}({t,\,z} )} ]+ \\ 2\sqrt {{P_{sig\_n}}{P_{CT\_n}}} cos{\kern 1pt} {\kern 1pt} [{{\omega_{sig\_n}}({t,\,z} )- {\omega_{CT\_n}}({t,\,z} )} ]+ {P_{CT\_n}} + {P_{CT\_n}}cos{\kern 1pt} {\kern 1pt} 2{\omega_{CT\_n}}({t,\,z} )+ \\ 2\sqrt {{P_{sig\_n}}{P_{ASE}}} cos{\kern 1pt} {\kern 1pt} [{{\omega_{sig\_n}}({t,\,z} )+ {\omega_{ASE}}({t,\,z} )} ]+ 2\sqrt {{P_{sig\_n}}{P_{ASE}}} cos{\kern 1pt} {\kern 1pt} [{{\omega_{sig\_n}}({t,\,z} )- {\omega_{ASE}}({t,\,z} )} ]+ \\ 2\sqrt {{P_{CT\_n}}{P_{ASE}}} cos{\kern 1pt} {\kern 1pt} [{{\omega_{CT\_n}}({t,\,z} )+ {\omega_{ASE}}({t,\,z} )} ]+ 2\sqrt {{P_{CT\_n}}{P_{ASE}}} cos{\kern 1pt} {\kern 1pt} [{{\omega_{CT\_n}}({t,\,z} )- {\omega_{ASE}}({t,\,z} )} ]+ \\ {P_{ASE}} + {P_{ASE}}cos{\kern 1pt} {\kern 1pt} 2{\omega_{ASE\_n}}({t,\,z} )\end{array} \right|\end{array}$}$$
$${i_0}(t )= {R_d}\left|{\begin{array}{*{20}{c}} {{P_{sig\_n}} + {P_{CT\_n}} + {P_{ASE}} + 2\sqrt {{P_{sig\_n}}{P_{CT\_n}}} }\\ { + 2\sqrt {{P_{sig\_n}}{P_{ASE}}} cos[{{\omega_{sig\_n}}({t,z} )- {\omega_{ASE}}({t,z} )} ]}\\ { + 2\sqrt {{P_{CT\_n}}{P_{ASE}}} cos[{{\omega_{CT\_n}}({t,z} )- {\omega_{ASE}}({t,z} )} ]} \end{array}} \right|$$
$${I_{sig\_n}} = \overline {{i_{sig\_n,j}}} (t )= {R_d}.\overline {{P_{sig\_n}}} ({t,z} )$$
$$I_{sig\_n}^2 = {[{{R_d}{P_{sig\_n}}({t,z} )} ]^2}\,({\textrm{it is the signal power at the output of the photodiode}})$$
$$\begin{aligned} \overline {i_{CT}^2} &= {[{{R_d}{P_{CT}}({t,z} )} ]^2}\\ \sigma _{CT}^2 &= \overline {i_{CT}^2} \end{aligned}$$
$$\begin{aligned} \sigma _{shot}^2 &= 2eB[{{i_{si{g_n}}}({t,z} )+ {i_{CT}}({t,z} )+ {i_{ASE}}({t,z} )} ]\;\\ \sigma _{shot}^2 &= 2eB{R_d}[{{P_{sig\_n}}({t,z} )+ {P_{CT}}({t,z} )+ {P_{ASE}}({t,z} )} ]\end{aligned}$$
$$\sigma _{th}^2 = \frac{{4KT}}{{{R_L}}}B$$
$$\sigma _{sig\_CT}^2 = {\left( {2{R_d}\sqrt {{P_{sig\_n}}{P_{CT\_n}}} } \right)^2} = 4R_d^2{P_{sig\_n}}{P_{CT\_n}}$$
$$\sigma _{ASE\_CT}^2 = {\left( {2{R_d}\frac{{\sqrt {{P_{ASE}}{P_{CT\_n}}} }}{2}} \right)^2} = 2R_d^2{P_{ASE}}{P_{CT\_n}}$$
$$\sigma \,_{{\boldsymbol{sig}}\_{\boldsymbol{ASE}}}^2 = {\left( {2{{\boldsymbol R}_{\boldsymbol d}}\,\frac{{\sqrt {{{\boldsymbol P}_{{\boldsymbol{sig}}\_{\boldsymbol n}}}{\boldsymbol{P}_{\boldsymbol{ASE}}}} }}{2}} \right)^2} = 2\boldsymbol{R}\,_{\boldsymbol{d}}^2\,{\boldsymbol{P}_{{\boldsymbol{sig}}\_{\boldsymbol{n}}}}\,{\boldsymbol{P}_{\boldsymbol{ASE}}}$$
$$SCNR = \frac{{{{[{{R_d}\,{P_{sig\_n}}({t,\,\,z} )} ]}^2}}}{{\sigma _{CT}^2 + \sigma _{shot}^2 + \sigma _{th}^2 + \sigma _{sig\_CT}^2 + \sigma _{sig\_{\textrm{ASE}}}^2 + \sigma _{{{\rm ASE}}\_{{\rm CT}}}^2}}$$
$$BER = \frac{1}{2}erfc\; \left( {\frac{{\sqrt {SCNR} }}{{2\sqrt 2 }}} \right)$$
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