Transfer matrix analysis of backscattering and reflection effects on WDM-PON systems

Joni Welman Simatupang; San-Liang Lee

doi:10.1364/OE.21.027565

1. Introduction

Rayleigh backscattering (RB) and Fresnel reflection (FR) can be major impairments in the deployment of wavelength division multiplexing-passive optical network (WDM-PON) technology, which utilizes the loop-back technique [1–9] as shown in Fig. 1. The loop-back technique is frequently applied in WDM-PONs to avoid the use of colored upstream transmitters. Seeding lights are transmitted from the optical line terminal (OLT) at the central office (CO) to the optical network units (ONUs), where upstream signals are encoded on the seeding lights with modulation devices, such as injection-locked Fabry-Perot (IL-FP) lasers, reflective semiconductor optical amplifiers (RSOAs), reflective electro-absorption modulators (REAMs), and integrated REAM with the semiconductor optical amplifier (REAM-SOA) [10–13]. When a single-fiber is used to transmit the seeding light and the upstream signal, FR and RB can cause optical beat interferometric (OBI) noise. The level of FR in such optical systems can be mitigated by choosing optical components with appropriate return loss and carefully handling fiber splicing and connection. RB, on the other hand, is an intrinsic impairment in fiber propagation, and its level is determined by the fiber type and configuration used [14, 15]. In these systems, RB can cause severe degradations to upstream transmission when signals are transmitted along the full-duplex ðber configuration.

Fig. 1 Mechanism of a loop-back type basic WDM-PON system with Rayleigh backscattering (RB_f,d) and Fresnel reflection (R_f,d) events for single-fiber transmission.

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The effects of RB and FR on WDM-PON systems have been intensively investigated using the analytical methods [1–9, 15–21]. However, the existing approaches may encounter some difficulties when dealing with complex system structures. Such complex cases include the long-reach hybrid WDM on top of time division multiplexing (TDM) PON structures, where a combination of optical gain and optical power splitters or demultiplexers is employed at the remote node (RN), and multiple stages of optical gain are needed in the systems [22–24]. To our knowledge, this paper is the first to demonstrate a simple and systematic transfer matrix (T-matrix) method for analyzing the accumulated effects of RB and FR crosstalk on signal transmission in various types of single-fiber bidirectional WDM-PON systems.

The rest of this paper is organized as follows. Section 2 describes T-matrix analysis for systems with back-reflections. Section 3 verifies that for basic WDM-PON systems with loop-back schemes, the total crosstalk-to-signal (C/S) ratio which is derived with the transfer matrices can lead to similar results that are frequently reported in the literature. In Section 4 we present the plot results for a simple architecture where the analytical solution can be derived and investigate the accuracy of the T-matrix method. The T-matrix method is then applied to analyze more complicated hybrid WDM/TDM PON systems. Lastly, conclusions are drawn in Section 5.

2. Transfer matrix analysis for systems with back-reflections

T-matrices have been applied to model the transmission and reflection characteristics of optical devices and systems [25]. The power T-matrix method was adopted here to directly relate the bidirectional input and output powers of a system block to matrix elements. Referring to Fig. 2, the transfer matrix of a two-port block is defined as:

[\begin{matrix} A_{1} \\ B_{1} \end{matrix}] = [\begin{matrix} T {}_{11} & T {}_{12} \\ T {}_{21} & T {}_{22} \end{matrix}] [\begin{matrix} A_{2} \\ B_{2} \end{matrix}] .

Here, A_i and B_i represent the forward-going and backward-going powers at the i-th port, where i = 1 and 2, respectively. The transmission network structure can be divided into a number of sections, where the structure and material parameters are assumed to be homogeneous throughout each section. For structures consisting of several concatenated sections, the total power T-matrix for the complete structure is simply the matrix product of the individual matrices for all of the sections. With this method, the transmission and reflection characteristics of WDM-PON networks can be determined.

Fig. 2 Typical model of a single transfer matrix with input and output powers.

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The use of power T-matrix is good for analyzing incoherent interferences in an optical network. The Rayleigh backscattering originating from distributed reflections along the fiber is typically treated as incoherent interference [4, 5]. On the other hand, the Fresnel reflection effect might interfere with the signal coherently, depending on the relative path difference between the reflections and signals. The power T-matrix deals with only those interferences out of coherent length from the received signal. This assumption is frequently adopted to deal with the reflections in fiber transmission systems, including WDM-PON systems [3, 5, 16]. However, this approach can be generalized to deal with coherent interference by using the field-component T-matrix instead of power T-matrix method.

Figure 3 shows the model for the loop-back type basic WDM-PON system shown in Fig. 1 when the total crosstalk problems caused by the interferences of RB and FR in upstream transmission are considered. To deal with the reflective-type upstream transmitters, the loop-back path is unfolded at the ONU as a forward-propagating one. The downstream transmission is terminated at ONU; and the upstream receiver is located at the right end of the transmission link. The RB effect can be included in the modeling of fibers, while the FR effect can be modeled as a discrete section or a portion of passive/active components. This approach can handle those systems with many cascaded sections as well as multiple backscattering and reflection effects, which would be hard to deal with using the analytical methods. For the sake of clarity and simplicity, the FR effects are not shown in either Fig. 3 or Fig. 4, but can be easily added into the calculations when they do occur.

Fig. 3 Building blocks of a loop-back type WDM-PON showing all RB interferences that affecting the upstream transmission.

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Fig. 4 T-matrix model for the upstream scenario of a WDM-PON system with the loop-back scheme.

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Figure 4 shows the cascade T-matrix modeling of the system depicted in Fig. 3. In Fig. 4, P_s is the initial seeding power sent from the light source at CO, and P_R is the total received power of the upstream transmission at CO. P_BR-UL is the total crosstalk power which comes from the accumulation of RB and FR effects on the loop-back upstream transmission. T_f, T_RN, T_d, and T_ONU are the power T-matrices for the feeder fiber, remote node, drop fiber, and optical network unit, respectively. T_u is the T-matrix of the entire uplink direction. Each T-matrix shown in Fig. 4 can represent a series of passive or active components, and T_u can be obtained by multiplying the matrices of all building blocks for the uplink.

P_BR-UL accounts for both the back-reflections of the downstream injected seeding light at CO (Type I) and the modulated upstream signal at ONU (Type II) [2, 4]. The latter passes through the ONU twice and can be dominant when the ONU provides high gain to the upstream transmission. Therefore, both Type I and II effects are included in this modeling.

The input/output relation of the whole system can be expressed by:

[\begin{matrix} P_{S} \\ P_{B R - U L} \end{matrix}] = T_{u} [\begin{matrix} P_{R} \\ 0 \end{matrix}] = [\begin{matrix} T_{u 11} & T_{u 12} \\ T_{u 21} & T_{u 22} \end{matrix}] [\begin{matrix} P_{R} \\ 0 \end{matrix}] .

The received signal power,

P_{R 0}

, is obtained by ignoring the RB and FR effects in Eq. (2) as:

[\begin{matrix} P_{S} \\ 0 \end{matrix}] = T_{u}^{*} [\begin{matrix} P_{R 0} \\ 0 \end{matrix}] = [\begin{matrix} T_{u 11}^{*} & T_{u 12}^{*} \\ T_{u 21}^{*} & T_{u 22}^{*} \end{matrix}] [\begin{matrix} P_{R 0} \\ 0 \end{matrix}],

where

T_{u}^{*}

is the total transfer matrix calculated by neglecting all of the RB and FR effects.

The C/S ratio for the upstream transmission is defined as the ratio of the crosstalk power, $P_{B R - U L}$ , to the received signal power, $P_{R 0}$ . From Eqs. (2) and (2a), the C/S ratio is given by:

{(\frac{C}{S})}_{u} = \frac{P_{B R - U L}}{P_{R 0}} = T_{u 11}^{*} (\frac{T_{u 21}}{T_{u 11}}) .

The T-matrix modeling accounts for the multiple reflection effects [5], which occur when applying a similar approach to calculate the optical properties of Fabry-Perot etalons [25]. It can be shown that when the cyclic reflection effects are much smaller than the single-pass effects,

T_{u 11} \approx T_{u 11}^{*}

; then, the crosstalk-to-signal ratio is simplified as:

{(\frac{C}{S})}_{u} \approx T_{u 21} .

This simplification can be applied to most of the cases in which the reflection and backscattering effects in various WDM-PON systems are considered.

Similarly, modeling for the downstream transmission can be written as:

[\begin{matrix} P_{T} \\ P_{B R 1} \end{matrix}] = T_{d n} [\begin{matrix} P_{R d n} \\ 0 \end{matrix}] = [\begin{matrix} T_{d n 11} & T_{d n 12} \\ T_{d n 21} & T_{d n 22} \end{matrix}] [\begin{matrix} P_{R d n} \\ 0 \end{matrix}],

where P_T, P_Rdn, and P_BR1 stand for the transmitted downstream power, received downstream power, and back-reflected power, respectively. The T-matrix for the downstream transmission, T_dn, includes the one-way building blocks from the CO transmitter to the ONU receiver.

The crosstalk power resulting from the reflections and backscattering as well as the downstream signals are received by the ONU receiver, so the crosstalk contribution can be obtained by subtracting the received signal power without considering the impairments, P_Rdn0, from P_Rdn. Therefore, the C/S ratio for the downstream transmission is given by:

{(\frac{C}{S})}_{d} = (\frac{P_{R d n} - P_{R d n 0}}{P_{R d n 0}}) = (\frac{T_{d n 11}^{*}}{T_{d n 11}} - 1),

where the superscript “*” refers to the matrix element for the downstream transmission without considering the RB and FR effects. Equation (5) will be applied in Section 4, Case B, for analyzing the upstream transmission case using tunable lasers as ONU transmitters.

Typically, major impairments caused by the RB and FR effects in single-fiber loop-back type WDM-PONs occur in or by the upstream transmission [15]. Thus, following discussions and calculations will focus on the uplink direction. Models and transfer matrices for key components of basic WDM-PON systems are described and then summarized in Table 1.

Table 1. Transfer matrices for WDM-PON components.

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2.1. Discontinuity (Fresnel reflection)

Discrete Fresnel reflections in optical fiber systems arise from the refractive-index discontinuities that occur in, for example, fusion mechanical splices and bad or dirty connectors. The reflection is often specified as a return loss. The discontinuity can be modeled by the reflection (R) and transmission (Γ) power coefficients as (Fig. 5(a)):

Fig. 5 Model for (a) an optical discontinuity and (b) feeder or drop fiber section in WDM-PON systems.

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T_{F R} = \frac{1}{Γ} [\begin{matrix} 1 & - R \\ R & Γ^{2} - R^{2} \end{matrix}]; Γ = 1 - R .

2.2. Feeder or drop fiber

A uniform fiber section is modeled as a matrix that accounts for the transmission loss and RB noise power. Take a feeder fiber section as an example: it can be modeled with the RB of the feeder fiber, γ_f, and the loss of the feeder fiber section, l_f, as shown in Fig. 5(b). Similar modeling can be applied for a drop fiber section. The loss of a feeder or drop fiber can be written as $l_{f, d} = \exp (- α_{f, d} L_{f, d})$ with α_f,d being the fiber loss coefficients and L_f,d the lengths. The subscripts f and d denote the feeder fiber and the drop fiber, respectively, and the relative RB power can be written as $γ_{f, d} = B (1 - l_{f, d}^{2})$ , where B = Sα_s/2α is the RB coefficient [5, 9], with S being the fiber recapture coefficient (dimensionless) and α_s [km⁻¹] the fiber scattering coefficient.

2.2. Remote node (RN): passive and/or active components

Passive and/or active components which are used in optical networks can be simply modeled with their return loss (denoted as R_p for a passive device and R_a for an active device) and either insertion loss (l_RN) for a passive device or optical gain (G_RN) for an active device, as shown in Fig. 6(a). For clarity, return losses are not shown in this figure. In a WDM-PON, an array waveguide grating (AWG) module is usually used at the RN as a multiplexer (MUX) or demultiplexer (DEMUX). The RN may include optical amplifiers and optical splitters for long-reach WDM-PON systems.

Fig. 6 (a) Model for passive (insertion loss) or active (optical gain) components; (b) Reflective ONU is modified to obtain the transmissive ONU.

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2.3. Reflective ONU (RONU) → Transmissive ONU

Reflective ONU in loop-back type WDM-PONs typically employs a reflective type modulation device, e.g., an RSOA or REAM. To simplify the modeling of this type of reflective device, we assume zero reflectivity at the input facet and total reflection at the reflected end. Then, the one-port reflective ONU is modeled as a two-port transmissive ONU with the same gain (G_ONU) by unfolding the device into one with double the original length and assuming zero reflectivity at both end facets as shown in Fig. 6(b). Thus, the loop-back upstream transmission (bidirectional link) is simplified as a unidirectional link, as illustrated in Fig. 3. Therefore, an RSOA can be modeled as a traveling-type SOA, and an REAM as an EAM.

3. C/S ratio of basic WDM-PON systems with loop-back scheme

In this section, we will show that, for basic WDM-PON systems, the C/S ratio derived by the transfer matrix method leads to results that are similar to those obtained with analytical approaches [2, 4, 5].

A basic WDM-PON system is defined here as one that employs an AWG at the RN to connect to multiple RONUs of moderate span (Fig. 1), where no additional optical amplification is needed at the RN. With the transfer matrix method, the relationships among the seeding power, received power, FR, and RB are given by Eq. (1). The total power transfer matrix is the result of two matrix multiplication, i.e., $T_{u} = T_{g} T_{h}$ , where T_g is the overall transmission matrix from CO to ONU, and T_h is the overall transmission matrix from the ONU back to the CO. We assume that $γ_{f, d}^{2} < < l_{f, d}^{2}$ . For simplicity, the FR and return losses of passive and active components are neglected. Under these conditions, T_g can be written as:

T_{g} = \frac{1}{l_{f}} [\begin{matrix} 1 & - γ_{f} \\ γ_{f} & l_{f}^{2} \end{matrix}] \frac{1}{l_{R N}} [\begin{matrix} 1 & 0 \\ 0 & l_{R N}^{2} \end{matrix}] \frac{1}{l_{d}} [\begin{matrix} 1 & - γ_{d} \\ γ_{d} & l_{d}^{2} \end{matrix}] \frac{1}{G_{O N U}} [\begin{matrix} 1 & 0 \\ 0 & G_{O N U}^{2} \end{matrix}] .

We can compose this as:

T_{g} = \frac{1}{G_{O N U} l_{f} l_{R N} l_{d}} [\begin{matrix} T_{A} & T_{B} \\ T_{C} & T_{D} \end{matrix}],

where

\begin{array}{l} T_{A} = 1 - l_{R N}^{2} γ_{f} γ_{d} \approx 1 \\ T_{B} = G_{O N U}^{2} [- l_{R N}^{2} l_{d}^{2} γ_{f} - γ_{d}] \\ T_{C} = γ_{f} + l_{f}^{2} l_{R N}^{2} γ_{d} \\ T_{D} = G_{O N U}^{2} [l_{f}^{2} l_{R N}^{2} l_{d}^{2} - γ_{f} γ_{d}] \approx G_{O N U}^{2} l_{f}^{2} l_{R N}^{2} l_{d}^{2} \end{array}

The approximation in Eq. (9) is based on the assumption that;

γ_{f} γ_{d} ≪ l_{f}^{2} l_{R N}^{2} l_{d}^{2}

i.e., the product of the RB of the feeder and drop fiber is much smaller than the square of the total single-pass link loss. In a basic WDM-PON in which the total fiber length is less than 50 km and the RN loss is less than 10 dB, this assumption is typically true if the relative RB power is less than −30 dB (e.g., −35 dB) in each section of the feeder or drop fiber. Similarly,

T_{h} = \frac{1}{l_{d}} [\begin{matrix} 1 & - γ_{d} \\ γ_{d} & l_{d}^{2} \end{matrix}] \frac{1}{l_{R N}} [\begin{matrix} 1 & 0 \\ 0 & l_{R N}^{2} \end{matrix}] \frac{1}{l_{f}} [\begin{matrix} 1 & - γ_{f} \\ γ_{f} & l_{f}^{2} \end{matrix}] = \frac{1}{l_{f} l_{R N} l_{d}} [\begin{matrix} T_{P} & T_{Q} \\ T_{R} & T_{S} \end{matrix}],

where

\begin{array}{l} T_{P} = 1 - l_{R N}^{2} γ_{f} γ_{d} \approx 1 \\ T_{Q} = - l_{R N}^{2} l_{d}^{2} γ_{f} - γ_{f} \\ T_{R} = γ_{d} + l_{R N}^{2} l_{d}^{2} γ_{f} \\ T_{S} = l_{f}^{2} l_{R N}^{2} l_{d}^{2} - γ_{f} γ_{d} \approx l_{f}^{2} l_{R N}^{2} l_{d}^{2} \end{array}

From Eqs. (9) and (11),

T_{u 11} \approx M [1 - G_{O N U}^{2} {(γ_{d} + γ_{f} l_{R N}^{2} l_{d}^{2})}^{2}] \approx M = T_{u 11}^{*},

where

M = \frac{1}{G_{O N U} l_{f}^{2} l_{R N}^{2} l_{d}^{2}} .

The second approximation in Eq. (12) applies when the cyclic term

G_{O N U}^{2} {(γ_{d} + γ_{f} l_{R N}^{2} l_{d}^{2})}^{2}

is much smaller than 1. The approximation is typically good because

γ_{f}

(

γ_{d}

) is usually less than −30 dB and G_ONU less than 20 dB. Under such conditions and from Eq. (3),

T_{u 11} \approx T_{u 11}^{*}

. Thus, C/S ratio can be obtained as:

{(\frac{C}{S})}_{u} \approx T_{u}_{21} = M (T_{C} T_{P} + T_{D} T_{R}) .

From Eqs. (9) and (11), the C/S ratio is expressed by:

\begin{array}{l} {(\frac{C}{S})}_{u} = \frac{1}{G_{O N U} l_{f}^{2} l_{R N}^{2} l_{d}^{2}} [γ_{f} + l_{f}^{2} l_{R N}^{2} γ_{d} + G_{O N U}^{2} (l_{f}^{2} l_{R N}^{4} l_{d}^{4} γ_{f} + l_{f}^{2} l_{R N}^{2} l_{d}^{2} γ_{d})] \\ = \frac{γ_{f}}{G_{O N U} l_{f}^{2} l_{R N}^{2} l_{d}^{2}} + \frac{γ_{d}}{G_{O N U} l_{d}^{2}} + G_{O N U} γ_{f} l_{R N}^{2} l_{d}^{2} + G_{O N U} γ_{d} \end{array}

This result is similar to that obtained with the approximate analytical method for the basic WDM-PON system [4]. In Section 4, this finding will be validated by our calculated results.

4. Calculated results and discussion

In order to verify the accuracy achieved using T-matrix to analyze the RB and FR effects in simple WDM-PON systems with Fresnel reflections, the results will be first compared with those calculated using the analytical formula. Then, this method will be applied to analyze the long-reach hybrid PON systems where the analytical formula is difficult to derive. In the calculations, B = −34.6 dB, obtained from S = 0.001, α_s = 0.032/km, and α = 0.046/km at λ = 1550 nm [4, 5]. The C/S ratio is calculated with MATLAB programs.

Case A. WDM-PON systems with Fresnel reflections

To validate the accuracy of analysis achieved using T-matrix, we calculate the simple system structure of a single-fiber loop-back WDM-PON network where the reflections and multiple Rayleigh backscatterings have been thoroughly analyzed with the analytical formula [5]. In this network, the OLT and ONU are connected with a fiber in which a discontinuity with Fresnel reflection occurs. To match our notations, the discontinuity is assumed to be the remote node, and the two fiber sections divided by the discontinuity correspond to the feeder fiber and drop fiber (see the inset in Fig. 7). For comparison, the analytical equation of the C/S ratio for all interferences affecting the upstream signal is rewritten as [5]:

{(\frac{C}{S})}_{u} = \frac{γ_{t}}{G_{O N U} l_{t}^{2}} + \frac{R}{G_{O N U} l_{d}^{2}} + \frac{(1 - R) R [γ_{d} + G_{O N U}]}{1 - R l_{d}^{2} G_{O N U}} + \frac{γ_{t} G_{O N U}}{1 - γ_{t} G_{O N U}} + γ_{f} R,

where l_t and γ_t stand for the loss and RB effect of the whole fiber, respectively, i.e.,

l_{t} = l_{f} l_{d}

.

Fig. 7 Upstream C/S ratio as a function of ONU gain for different values of link loss and reflections when B = 0 The curves appear overlapped and the insert figure shows the observed architecture.

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The denominator of the third and fourth terms of Eq. (16) indicates the cyclic effects caused by multiple reflections and backscatterings. These two terms can become dominant when the ONU gain is increased to about the same order of magnitude as the return loss or RB coefficient. However, in a well-managed system, this situation hardly ever happens. From Eq. (3), the cyclic effect of T-matrix modeling is accounted in the $T_{u 11}$ term, as shown in Eq. (12) for the case without Fresnel reflection. A similar formula can be derived for the case in which the FR effect is considered.

Figure 7 shows similar results obtained with Eq. (16), which is denoted as Arellano’s model in the following plots, and the T-Matrix model for two different reflection losses and three different ODN losses (l_t). In the calculation, the reflection is assumed to occur at the ONU side, i.e., l_d = 1; in addition, the RB effect is neglected. The two methods generate the same results for the low-reflection case (R = 60 dB), revealing a small discrepancy at the higher-gain regime for the R = 30 dB case. Both methods predict the steep increase in the C/S ratio as the gain approaches the return loss value, where the cyclic effects make the denominator of the third term in Eq. (16) close to zero.

For the T-matrix modeling, due to the unfolding of a reflective ONU into a transmission type, the cyclic reflection path involving the ONU is doubled. Therefore, only the even orders of cyclic terms in the multiple reflections are included in the modeling, as shown in Eq. (12). This is the reason for the discrepancy for R = 30 dB when the gain and reflection are large enough to reach a lasing (oscillating) condition. Under such an extreme condition, the T-matrix can still predict the severe degradation of system performance.

Figure 8 compares the calculated C/S ratios for different values of link loss and reflections when B ≠ 0. Again, the two methods give the same results except at the extreme cases where the C/S ratio rises steeply. The calculated C/S is about the same for up to 25 dB of ONU gain. With Arellano’s model, the C/S curves reach infinity when G = 30 dB and R = 30 dB, as can be seen from Eq. (16). With T-Matrix model, the infinite C/S ratio occurs when G_ONU = 28.7 dB. For similar derivation of Eqs. (7) to (12) in this particular case (l_d = 1) but including the FR effect, it can be proved that $T_{u 11} \approx 1 - G_{O N U} {(R + γ_{f})}^{2}$ . Therefore, the discrepancy can be clearly explained by comparing T_u₁₁ with the denominator of the third and fourth terms of Eq. (16). In Eq. (16), the cyclic effects of the RB and FR effects are treated separately whereas they are combined in T-matrix modeling. Therefore, when both effects are considered, the T-matrix predicts an earlier breakdown of the system performance when the ONU gain increases. In spite of the discrepancy at the extremely high-gain regime, the two methods give the same results for typical operation conditions.

Fig. 8 C/S ratio as a function of ONU gain for different values of link loss and reflections situated at the ONU (l_d = 1).

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The T-matrix is then applied to calculate the effect of RB on a basic WDM-PON system, and the results are compared with the approximate analytical solution in Eq. (15). The WDM-PON system covers 50 km of distance with 5 dB insertion loss at the remote node (RN). Figure 9(a) shows the calculated C/S versus the location of RN for different ONU gains. The results indicate that the approximation shown in Eq. (15) and the T-matrix modeling generate the same C/S ratios. This is due to the fact that under the calculated gain and RB coefficient, the conditions leading to the approximation in Eq. (15) are all met.

Fig. 9 C/S ratio versus RN position for different ONU gain of a basic WDM-PON when considering (a) RB only and (b) both RB and FR. The data is calculated by T-matrix and (a) Eq. (15) and (b) effective reflection.

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We also apply the T-matrix to calculate the performance of a basic WDM-PON system when FR occurs and compare the results with those obtained using effective reflection approximation [16]. The Fresnel reflections are accounted for by adding the effective contribution of the discrete reflection from a fiber section to one end of the fiber onto the RB term. By replacing the RB terms in Eqs. (7) to (12) with the effective reflections, the C/S ratio can be obtained. In the calculation, we assume that the FR is located at both edges of the RN. Figure 9(b) shows the calculated upstream C/S ratio as a function of the RN position for a basic WDM-PON system with RN loss equal to 5 dB at different return losses and ONU gain values. The total fiber length (L_f + L_d) is 50 km. We observe that both methods obtain the same curves for every condition. With a very tiny reflection (R = 70 dB), the crosstalk is dominated by RB, while for a high reflection (R = 40 dB), the crosstalk by the discontinuity (FR) can be comparable or even larger than RB.

The above results indicate that T-matrix modeling can account for the combined contributions of multiple reflections and backscatterings, and generate accurate results. The advantages of using T-matrices can be clearly found when applying them to more complicated systems where it can be difficult to determine all of the possible crosstalk paths for multiple reflections and/or backscatterings. In addition, the building equations are very lengthy when the analytical approach is used.

Case B. Long-reach hybrid WDM/TDM PON systems

After verifying the feasibility of using the T-matrix approach, we will investigate the effects of RB on long-reach hybrid PON systems. These systems usually support hundreds to thousands of users and extend the coverage span to 100 km or beyond [23–25], where optical amplifiers are needed in the intermediate stages to compensate for the high splitting ratio loss and fiber propagation loss. Figure 10 depicts a long-reach hybrid WDM/TDM PON system with three sections: feeder, distribution, and drop fibers. Two different cases will be considered and compared: one uses a reflective modulator while the other uses a tunable laser as an upstream transmitter at the ONU. In both cases, only RB effects are considered.

Fig. 10 Long-reach hybrid WDM/TDM-PON system that can serve thousands of users.

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The system employs two in-line optical amplifiers (OAs), such as bidirectional EDFAs, connected by circulators at both sides [2, 3, 9]. At the ONU, a reflective modulator is used to modulate and re-amplify the data for upstream transmission for loop-back architecture. The second optical amplifier (OA-2) mainly compensates for the splitting loss at RN-2. Rayleigh backscattering can have a more severe impact on the system performance of a long-reach PON than on a basic one due to the three fiber sections and multiple gain stages involved. The optimal C/S ratio depends on the lengths of the feeder, distribution, and drop fibers as well as the optical gain of each node.

Use of the T-matrix method eliminates the need to derive lengthy equations. We calculate the reflective ONU case by using Eq. (3), and the tunable laser case by using an equation that is similar to Eq. (5) but forms a T-matrix by using building blocks running from ONU to OLT. In the calculations, the splitting ratios of AWG and power splitter (PS) are assumed to be 1:32 and 1:64 with insertion loss of 5 dB and 20.5 dB, respectively. The length of the drop fiber is assumed to be 10 km. The OA-2 gain is fixed at 20 dB to compensate for the optical splitting loss of RN-2. The OA-1 and ONU gains vary between 10, 15, and 20 dB for loop-back type systems. Figure 11 shows the calculated C/S ratio for various RN-1 locations in long-reach hybrid WDM/TDM PON systems that use reflective modulators at ONUs. The results show that, at the optimal position, a combination of 15 dB ONU and 15 dB OA-1 gains can provide the best C/S ratio down to −19 dB at 50 km feeder fiber. With larger gain at OA-1, the C/S ratio is more sensitive to the location of RN-1 because the higher OA-1 gain will increase the impact of RB on upstream transmission.

Fig. 11 C/S ratio versus RN-1 locations for different combinations of OA-1 and ONU gains in the loop-back scheme, and different OA-1 gains when a tunable laser is used to transmit the upstream signals.

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For comparison, Fig. 11 also shows the calculated C/S ratio as a function of the RN-1 location for the same hybrid WDM/TDM PON system but using tunable lasers as upstream transmitters. In this case, no seeding light and ONU gain are needed for upstream transmission as in reflective modulator case. The upstream C/S ratio calculation shows that the RB contribution on the upstream signal happens due to the double Rayleigh backscattering effects (RB which is twice amplified by the OA-1 gain). The calculated results indicate a negligible effect of RB except at the condition of 30 dB OA-1 gain.

5. Conclusions

In this work, the power T-matrix method has been applied to model the WDM-PON systems. With this approach, a reflective ONU for upstream transmission can be modeled as a two-port travelling-wave component by unfolding the device into one with double the original length. The C/S ratio for upstream transmission can be simply calculated by using the matrix elements of the whole system. This approach has been applied to analyze the C/S ratio of a simple loop-back link, and the results have been compared with those calculated using the analytical approach. Both methods give about the same results except in extreme cases, and the reason for this discrepancy has been identified. T-matrix method has also been applied to verify the accuracy of using approximation equations in a basic WDM-PON system. The findings show that approximation methods can lead to correct results under most conditions.

T-matrix method has, in addition, been applied to corroborate the system performance of bidirectional long-reach hybrid WDM/TDM-PONs, where more stages of optical gains and more fiber sections are involved. It demonstrates the feasibility of using this approach to calculate a relatively complicated system. Also, T-matrix was used to calculate the long-reach hybrid PON system using tunable wavelength ONU transmitters. As expected, the RB effect is typically negligible for this type of system, but may be a concern when the gain of the intermediate stage is high due to the influence of double Rayleigh backscattering (DRB) effects on the upstream signals.

T-matrix modeling can be a powerful way or method for analyzing the optical transmission systems where RB and/or FR interferences affect the signal transmission. Furthermore, the method is a robust tool for analyzing systems with many sections and combinations of multiple reflections and backscatterings.

Acknowledgment

The authors are grateful to D. Ludwig for proof-reading this manuscript and also to R. B. Sarean and M.-H. Chuang for helping with the plots. This research was supported by National Science Council of Taiwan under project number NSC 101-3113-P-011-003.

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Component	Transfer Matrix
Feeder or drop fiber	$Τ_{f, d} = \frac{1}{l_{f, d}} [\begin{matrix} 1 & - γ_{f, d} \\ γ_{f, d} & l_{f, d}^{2} - γ_{f, d}^{2} \end{matrix}]$
RN (passive)	$T_{R N} = \frac{1}{l_{R N}} [\begin{matrix} 1 & - R_{p} \\ R_{p} & l_{R N}^{2} - R_{p}^{2} \end{matrix}]$
RN (active)	$T_{G_{R N}} = \frac{1}{G_{R N}} [\begin{matrix} 1 & - R_{a} \\ R_{a} & G_{R N}^{2} - R_{a}^{2} \end{matrix}]$
Reflective ONU (RONU)	$T_{O N U} = \frac{1}{G_{O N U}} [\begin{matrix} 1 & 0 \\ 0 & G_{O N U}^{2} \end{matrix}]$
Discontinuity (Fresnel reflection)	$T_{F R} = \frac{1}{Γ} [\begin{matrix} 1 & - R \\ R & Γ^{2} - R^{2} \end{matrix}]; Γ = 1 - R .$

Transfer matrix analysis of backscattering and reflection effects on WDM-PON systems

Abstract

1. Introduction

2. Transfer matrix analysis for systems with back-reflections

2.1. Discontinuity (Fresnel reflection)

2.2. Feeder or drop fiber

2.2. Remote node (RN): passive and/or active components

2.3. Reflective ONU (RONU) → Transmissive ONU

3. C/S ratio of basic WDM-PON systems with loop-back scheme

4. Calculated results and discussion

Case A. WDM-PON systems with Fresnel reflections

Case B. Long-reach hybrid WDM/TDM PON systems

5. Conclusions

Acknowledgment

References and links

Cited By

Figures (11)

Tables (1)

Equations (18)

Optics Express