Impact of modulation bandwidth on multiplexing using principal modes in MMF links

Rohan Prasad; Kumar Appaiah

doi:10.1364/OE.26.001779

1. Introduction

Optical fibers offer a large amount of bandwidth, making them ideally suited for large data rate communication systems. Based on the their propagation characteristics, we can classify them as single mode fiber (SMF) and multimode fiber (MMF). While SMFs are primarily used owing to the absence of modal dispersion, whereas large-core MMFs offer several benefits, such as higher offset tolerance and the possibility of spatial multiplexing to enhance bandwidths. However, modal dispersion in large-core MMFs limits the fiber bandwidth. To overcome this limit, one could use so called “principal modes” (PMs), which offer a set of orthonormal modes to communicate through multimode fibers without modal dispersion for small modulation bandwidths [1]. The key benefit of using these modes is that the dispersion free property obviates the need for complex signal processing at the receiver to eliminate cross-talk for multiplexing and additional dispersion compensation, thereby simplifying receiver designs significantly. However, for larger modulation bandwidth, these advantages diminish [2]. In this paper, we characterize the performance of few-mode and large-core graded-index MMFs of various lengths to gauge the impact of modulation bandwidth on the performance of principal modes. In particular, we implement a matrix based fiber model to predict the performance of PMs for various modulation bandwidths, in terms of the modal dispersion and cross-talk when using PMs for multiplexing, and predict for what duration of bandwidth the same PMs can be used without incurring too much performance loss due to modal dispersion.

The larger core diameter of MMFs causes propagation of the signal through multiple spatial modes. Not only do these modes possess different spatial amplitude distributions, but they also travel at different speeds through the fiber [3]. Ideally, if one were to launch a signal in a particular mode of an ideal large-core MMFs, the signal would propagate without any modal dispersion in the absence of intermodal coupling. However, practical fibers undergo intermodal coupling due to physical characteristics and non-uniformity [4]. Thus, launching even in ideal fiber modes could result in dispersion due to differential mode delay, thus limiting bandwidth and imposing the requirement of dispersion compensation at the receiver. One way to mitigate this issue is using the concept of PMs, which are special linear combination of the ideal fiber modes that do not undergo any modal dispersion for small modulation bandwidths [5]. In particular, for low MDL, these principal modes are also orthogonal, thus permitting multiplexing. Experimental results have revealed that PMs are effective in combating dispersion in large-core MMFs [6–8]. However, the impact of using larger modulation bandwidths and how the performance of PMs is affected, in terms of dispersion and cross-talk control, has not been characterized. In this paper, we build a propagation model for optical fibers based on the framework in [5], and numerically characterize the propagation property of PMs in large-core MMFs without and with MDL; under different modulation bandwidths without using typical equalization at the receiver. In particular, we use simulation to study how the PMs of few-mode and large-core MMFs of various lengths perform when the modulation bandwidth becomes large. Since our interest is in analyzing the impact of propagation characteristics of large-core MMFs, we restrict our consideration to modal dispersion and ignore the impact of material and waveguide dispersion. The key benefit of using principal modes over large bandwidths is that the amount of processing required to compensate for dispersion at the receiver can be reduced significantly. When compared to conventional approaches that use the ideal fiber modes for signaling and multiplexing, the PMs offer superior performance in terms of dispersion resistance while still retaining their orthogonal nature.

The concept of PMs is motivated by the initial study of the principal states of polarization in single mode fiber systems [9]. This was extended for general large-core MMFs as principal modes in [1]. The characterization of higher order PMs in the case of polarization was discussed in [9], and a treatment for conventional large-core MMFs was presented in [2]. One practical approach, to realize PMs, is by using a swept wavelength interferometer (SWI) to measure propagation properties of the fiber for different wavelengths and find the PMs using these measurements [10, 11]. Several discussions of mode division multiplexing (MDM) and mode-group diversity multiplexing (MGDM) have been presented in the past, including [12, 13]. An initial discussion of PMs in large-core MMFs was provided in [5]. The use of multiple-input multiple-output (MIMO) techniques to mitigate dispersion in MDM links [12]. However, the efficiency of MIMO techniques is limited by fiber dispersion [14]. The model described in [15] considers propagation fibers with MDL using the eigenvalue distribution of zero trace Gaussian unitary ensemble. However, this analysis is restricted to studying the performance over a single frequency. In our work, we build and extend a model for the fiber based on the field propagation based study described in [5] to account for the performance of the system MDL. The experimental performance of principal modes and their bandwidth has been evaluated under various coupling regimes and loss conditions in [16]. This work used step-index fibers and altered the coupling conditions by varying the application of stress on a fiber of constant length. In our work we extend this concept to model to graded index fibers of various lengths and core diameters, and study the impact of these parameters on the performance of principal modes. In addition, we also quantify the impact of mode-dependent losses (MDL) on cross-talk across the PMs. Our simulation results closely match the trends that have been observed experimentally as well, cf. [11, 16, 17].

The rest of the paper is organized as follows: Section II describes the system model, calculation of the propagation operator that characterizes the fiber and the effect of modulation bandwidth on the PMs. Section III describes the simulations and summarize the results. Finally, Section IV provides concluding remarks and future directions.

2. System model

In this section, we discuss the model we utilize to describe the large-core MMFs propagation characteristics, using which dispersion and cross-talk can be analyzed. We use this model to describe the impact of large modulation bandwidth on the performance of PMs with respect to dispersion and cross-talk.

2.1. Multimode optical fiber model

The spatial characteristic of any propagating signal within the large-core MMF can be described as a linear combination of the fiber modes. The input column matrix a and the output column matrix b of the fiber are related by the Eq. (1). Both input and output column matrices are size of 2M × 1, where M represents number of spatial modes in the fiber, and can be considered as an extension of the Jones vector to the multimode fiber case. The factor of 2 arises because of the two polarizations. The element a_i of matrix a represents the amplitude component of the i-th input ideal mode at the fiber input, and the element b_i of matrix b represents the amplitude of the i-th fiber mode at the fiber output, as shown in Fig. 1. The cause for modal dispersion is that, even if we fix the value of a for all frequencies, the value of b changes with ω, since U(ω) is a frequency dependent transformation.

b = U (ω) a

Fig. 1 Input column matrix a projected on graded index fiber using spatial light modulator (SLM) and get output as a column matrix b.

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To create a model for evaluating large-core MMF performance, we utilize the multisection field propagation approach described in [5]. This model takes into account fiber inhomogeneities and random coupling that affect dispersion in large-core MMFs. Here, the fiber is modeled as the concatenation of several sections as depicted in Fig. 2(a). Each section of the fiber is assumed to be smaller than the “coherence length”, within which no significant inter-modal coupling occurs. Then, geometric perturbations within the section alter the mode propagation characteristics within the fiber, thereby resulting in mode coupling. In particular, the model considers minor variations in the fiber due to random twists across sections (denoted by κ_i in Fig. 2(a), the inverse of the radius of curvature) and rotation across sections (denoted by θ_i in the same fig.). The propagation within the k-th section of the fiber is described by a matrix U^k(ω), referred to in the sequel as the propagation operator. The geometric perturbation is considered to consist of a “curvature”, wherein the radius of curvature. In our implementation, we extend this model to account for mode-dependent losses that arise within sections of the fiber, based on the analysis in [18], so as to study the impact of MDL and on the frequency response and cross-talk performance of PMs.

Fig. 2 Schematic of the multi-section model of the fiber.

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U(ω) can be found by evaluating the product of the propagation operators of each section as follows:

U (ω) = U^{N} (ω) U^{(N - 1)} (ω) \dots U^{(i)} (ω) \dots U^{(1)} (ω)

where ω is the angular frequency. Note that the matrices U⁽ⁱ⁾(ω) are not commutative, since the mode transformation effected depends on the order in which the sections affect them.

To develop a model for the large-core MMF that accounts for MDL, we use a multisection field propagation model with splices that cause losses, much like the model described in [18]. Small sections of multisection lossless fiber are joined with lossy splices.

From Figure. 2(b), it is clear that k-th lossy splice is present between (k − 1)-th and k-th lossless fiber segment. The k-th lossy splice has an offset of a_x and a_y in x and y directions with respect to the previous segment. It is also clear that x-y axes in k-th fiber segment are rotated by an angle Φ with respect to previous section. We model these offsets as uniform random variables that are independent of the modulation bandwidth, since these effects largely stem from to be the physical properties of the fiber, as opposed to the signaling. The mode propagation through k-th lossy splice is modeled using C⁽^k⁾ matrix and it is given by:

C^{(k)} = R^{(k)} M_{l}^{(k)}

where R⁽^k⁾ is a rotation matrix as defined in [18]

$M_{l}^{(k)}$ is the modal projection matrix that accounts for the physical displacement and rotation of the mode pattern at the interface of two sections. For offsets a_x and a_y and rotation angle Φ, the modal projection matrix is:

M_{l}^{(k)} = {[\begin{matrix} B_{M \times M} & 0 \\ 0 & B_{M \times M} \end{matrix}]}_{2 M \times 2 M}

where each element b₍_mn₎ in B_M_×_M matrix is the overlap integral between the spatial pattern f_m(x, y) of the m-th ideal mode and the rotated and translated version of mode pattern f_n(x, y) of n-th ideal mode, i.e.

b_{(m, n)} = \int \int_{- \infty}^{\infty} f_{m} (x, y) f_{n}^{'} (x, y) d x d y

where, from Fig. 2(b), we have:

f_{n}^{'} (x, y) = f_{n} (x \cos Φ + y \sin Φ + a_{x}, x \sin Φ + y \cos Φ + a_{y})

From Fig. 2(b), we can observe that, due to the presence of offsets a_x and a_y, the modes do not perfectly couple with the other modes of the fiber. So, there is a loss of power as propagation occurs between these segments of the fiber, thus making $M_{l}^{(k)}$ a non-unitary matrix, unlike the treatment in [5]. If the offset and rotation angles of lossy splices is zero, the fiber becomes lossless, i.e. b₍_mn₎ = δ_mn, where δ_ij is the Kronecker delta function. From Fig. 2(b), we can represent total propagation operator matrix of the fiber as follows:

U (ω) = U^{(S)} (ω) C^{(S - 1)} U^{(S - 1)} (ω) \dots C^{(k)} (ω) U^{(k)} (ω) U^{(k - 1)} (ω) \dots C^{(1)} U^{(1)} (ω)

where S is the number of lossless segments.

We consider the fiber as cascaded system of first order system and higher order system as shown in Fig. 3 under the same geometric perturbation conditions described above. For narrow band signals, the impact of the higher order system is very small and we can neglect it. Let ω₀ be the center angular frequency of the laser used for communication and Ω be the modulation bandwidth of the signal being transmitted over the fiber. For a fiber system that does not have MDL, the propagation operator can be defined as [1]:

U (ω) = \exp (j Θ (ω))

where Θ(ω) is a Hermitian matrix which describes the propagation property of the fiber. For narrow band signals, we will see below that Θ(ω) varies linearly with ω, and this is the regime where the use of first order PMs enables dispersion-free propagation [5]. It is worth mentioning that, if there is no intermodal coupling and there are no losses within the fiber, Θ(ω) would be a diagonal matrix whose entries correspond to the propagation constants β_m, m = 0, 1,… M − 1 of the fiber M fiber modes. In the general case, Θ(ω) captures the frequency dependent intermodal coupling through the fiber. For smaller modulation bandwidths, Θ(ω) may be assumed to vary linearly with the modulation bandwidth Δω. However, as the modulation bandwidth increases, higher order effects can no longer be ignored. Here, the phase response starts to depend on higher order terms of frequency, viz. ω², ω³, The dependence of Θ on higher order terms of ω encapsulates the effects of chromatic dispersion (CD) and polarization mode dispersion (PMD) in the system, which are intramodal dispersive effects (seen in single mode fiber links as well). Since the propagation operator U(ω) depends on higher powers of ω, the use of first order PMs no longer provides dispersion-free transmission, and their performance degrades due to dispersion. Thus, in our model, we do not restrict the expansion of U(ω₀) to only the first order group delay term, as in [5] and relax the assumption that Θ(ω) is Hermitian, so that we may account for the higher order group delay terms as well as losses in the fiber respectively.

Fig. 3 The fiber can be considered to be a cascade of a first order system, consisting only of F⁽¹⁾, and a higher order system, as described in Eq. (10). The input x(t) is projected on to the complete system. If the input to the first order system is in a principal mode, it experiences no distortion.

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One key difference that manifests between the models of the fiber with and without MDL is the nature matrices U(ω) and Θ(ω). In the case where there is no MDL, the PMs become orthonormal, and all signals transmitted on the PMs can be assumed to propagate without losses over narrow bandwidths. However, in the presence of MDL, the PMs no longer remain orthonormal, and thus, this leads to cross-talk across PMs, thus making the detection problem more difficult. We have summarized the effects of introducing MDL on various model parameters in the Table 1.

Table 1. Differences between fiber without MDL and with MDL.

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To study how the higher order terms of ω affect the performance of the first order PMs in large-core MMF links, we can expand Θ(ω) around the center frequency ω₀ using the Taylor series [19]:

\begin{array}{l} U (ω_{0} + Ω) = \exp (j \sum_{n = 0}^{\infty} {\frac{Ω^{n}}{n!} \frac{d^{n} Θ (ω_{0} + Ω)}{d Ω^{n}} |}_{Ω = 0}) \\ = U (ω_{0}) \hat{U} (Ω) \\ = U_{0} \exp (- j Ω F^{(1)}) \exp (\frac{1}{2} Ω F^{(2)}) \dots \\ = U_{0} \exp (\sum_{k = 1}^{\infty} \frac{- {(j Ω)}^{k} F^{(k)}}{k!}) \end{array}

where

\hat{U} (Ω)

captures the Ω dependence, and we have denoted U(ω₀) as U₀ for convenience and it is defined as propagation operator matrix at the centre frequency. F⁽^k⁾ is k-th order group delay operator matrix. F⁽^k⁾ can be found using U₀ and successive differentiation of U(ω₀ + Ω).

Now, viewing Fig. 3 in the context of this model, we see that, when a signal x(t) is launched into the modes of the fiber using a transmit pattern designed using a SLM, we can model its propagation as being in two phases, one through the higher order system, denoted by $\prod_{n = 2}^{\infty} \exp (\frac{1}{n!} {(- j Ω)}^{n} F^{(n)})$ , and then through the first order system, denoted by exp (−jΩF⁽¹⁾) followed by a frequency independent transmission. The impact of the higher order system causes dispersion of the signal x(t), thus resulting in a y₁(t) that is not representative of the transmitted signal. However, the first order system merely produces a delay if the transmission occurs in a principal mode, as will be discussed in the following subsection. Now, since the impact of the higher order system is minimal in the low bandwidth regime, if the SLM is used to launch a small bandwidth signal into the fiber over a first order principal mode, the signal can be expected to propagate without dispersion induced pulse spreading, as is the case in the discussions in [5]. However, across what bandwidths limits this dispersion-free assumptions hold, and how MDL affect this performance based on the fiber characteristics is something we focus on in the following sections.

An important point to note is that we have made the assumption that it is possible to launch into the precise PM of the fiber without any leakage into other PMs. Since the PMs are a linear combination of the ideal fiber modes, this is equivalent to assuming that we can accurately launch into the ideal modes of the fiber. The assumption that one can launch signals into the ideal modes can be assumed to hold, for few-mode fibers (cf. [12, 20]), though the same may not be true for large-core MMFs. However, to a significant extent, some ideal modes and PMs can be utilized in large-core fibers also [11, 13, 17], and the study we perform can be assumed to provide a trend or a limit of the maximum performance that can be achieved in these situations.

2.2. Principal modes

In large-core MMFs, each mode has a different velocity and different group delay, due to which there is modal dispersion. In single mode fibers, the presence of two polarizations was observed to cause PMD. To reduce the impact of PMD in SMF links, the principal states of polarization (PSP) model was used to show that there exists a pair of “principal states” of polarization that allow dispersion-free signalling for narrow-band signals [21]. The field based coupling model described in [5] generalizes this concept to modal dispersion and mode coupling in large-core MMFs, and shows the existence of principal modes (PMs) that are orthogonal and dispersion-free to the first order of frequency, thereby obviating the need to perform additional equalization for dispersion compensation at the receiver. Although both PMs and ideal fiber modes are both orthogonal, the PM amplitude and group delay(GD) are independent of frequency to the first order. Thus, over a reasonable bandwidth, the same PMs can be used to transmit the signal without incurring too much of performance loss due to modal dispersion.

2.2.1. First order propagation operator

The first order propagation operator is appropriate for narrow band signals. For such signals, the higher order frequency components in Eq. (9) can be omitted without incurring significant error. The first order propagation operator is given by:

\begin{matrix} U (ω_{0} + Ω) = U (ω_{0}) \hat{U} (Ω) \\ = U (ω_{0}) \exp (- j Ω F^{(1)}) \end{matrix}

where F⁽¹⁾, the first order group delay operator, centered at central frequency ω₀, is calculated from the propagation operator as:

{F^{(1)} = j U {(ω_{0})}^{H} \frac{\partial U (ω_{0} + Ω)}{\partial Ω} |}_{Ω = 0}

Eigen vectors and eigen values of F⁽¹⁾ are the first order PMs, and their corresponding GDs respectively. The first order PMs are orthonormal at the input and the output of the fiber and they are permit dispersion free propagation in large-core MMFs. However, this approximation breaks when the modulation bandwidth increases.

2.2.2. Higher order propagation operator

For narrow band signals, the higher order component of propagation operator may be neglected without much error, but when the modulation bandwidth is large, we cannot ignore the higher order components in Eq. (9). To find the GD matrices F⁽^k⁾ in Eq. (9), we need to find the derivatives of the propagation operator with respect to the modulation bandwidth Ω. From Eq. (1), the propagation operator of the fiber is expressed as the product of matrices that constitute the propagation operator within each section of the fiber. The propagation operator of each section is a function of Ω. Calculation of the derivative of the complete fiber propagation operator U(ω₀ + Ω) is cumbersome to implement, since the derivative depends on the number of sections we consider in the fiber. Instead, we take the approach of finding the derivative of the propagation operator within each section with respect to Ω, and use it to calculate propagation operator of the fiber using the chain rule of differentiation. Now, when viewed as an analytical function of Ω, evaluating the derivative is not easy, since the dependence of $\frac{\partial U^{(i)}}{\partial Ω}$ on Ω is quite intricate [5]. Thus, we resort to the use of numerical methods [22] to evaluate the derivative of each fiber section. We use the forward difference of approximation to calculate: $\frac{\partial U^{(i)}}{\partial Ω}$ as

\frac{\partial U^{(i)}}{\partial Ω} = \frac{U^{(i)} (ω_{0} + Ω + h) - U^{(i)} (ω_{0} + Ω)}{h}

Once we have the derivative of each section, we use the chain rule to calculate the derivative of the total propagation operator of the fiber. The chain rule of differentiation is easy to apply when number of sections is small for the lower order GD matrices F⁽^k⁾. However, to find the higher order GD matrices for fibers that contain large number of sections, applying the chain rule directly increases the complexity of the implementation. To overcome this issue, we calculate the derivative recursively, by adding one section at a time in each step. Here, we evaluate the derivative of the propagation operator of a single section, and add the next section along with chain rule to calculate the derivative of the total propagation operator U(ω) of these two sections combined. After this we consider this U(ω) as a section and add new section one by one recursively to evaluate the complete derivative. In other words, the first order derivative of the propagation operator was evaluated by recursive differentiation [23] as follows:

U_{2} = U^{(2)} U^{(1)}, U_{3} = U^{(3)} U_{2}, \dots U_{n + 1} = U_{(n + 1)} = U^{(n + 1)} U_{n},

\frac{\partial U_{n + 1}}{\partial Ω} = U^{(n + 1)} \frac{\partial U_{n}}{\partial Ω} + \frac{\partial U^{(n + 1)}}{\partial Ω} U_{n}

where U_n₊₁ is the total propagation operator of the fiber of n + 1 sections and U⁽ⁿ⁺¹⁾ is the propagation operator of the (n + 1)-th section. For all the cases where the derivative could be evaluated without using the chain rule, the numerical values matched those that were obtained using the approach in 14 while greatly reducing the complexity of evaluation. Thus, using this approach, we are able to obtain the derivative of the propagation operator efficiently, and we can now use Eq. (11) to calculate the group delay matrix.

Now that we have a efficient method to calculate the group delay matrix, we can now calculate the PMs of the fiber, and examine the frequency response that we would obtain if we used the PMs to communicate through the fiber. To calculate the frequency response of the fiber, we assume that the transmission occurs in a particular PM, say PM-k, and observe the frequency response of the resulting system. We now consider two representations of the system: the first is an approximation where only the first order group delay matrix is present, as in Eq. (10), and the other where the complete propagation operator U(ω₀ + Ω) is present. We assume that square matrix whose columns are the input PMs is P and the corresponding output PM modes is Q, defined as follows:

P = [p_{1} p_{2} \dots p_{k} \dots p_{M}], Q = [q_{1} q_{2} \dots q_{k} q_{M}]

Without loss of generality, we assume that the transmission is performed on the k-th input PM corresponding to the vector p_k in the ideal mode basis. Then the output PM q_k corresponding to the input PM p_k is given by:

q_{k} = U (ω_{0}) p_{k}

Since the condition with which p_k has been derived ensures that there is no dispersion, if we detect the signal along q_k, we expect no dispersion for narrow band signals; i.e., we have

q_{k}^{H} q_{k} = 1

for all values of Ω, corresponding to flat frequency domain response, or a time-domain impulse response that is the Dirac impulse. However, when used for larger bandwidths, the dispersion-free assumption fails to hold. Thus, if we let the channel response of fiber without MDL that we would observe if we projected the incoming signal onto q_k be represented by h_k(Ω), we have:

\begin{array}{l} H_{MDL} (Ω) = {[h_{MDL, 1} (Ω), h_{MDL, 2} (Ω), \dots h_{MDL, M} (Ω)]}^{T} = Q_{inv} U (ω_{0} + Ω) P \\ = Q_{inv} [U_{0} \exp (- j Ω F^{(1)}) \exp (\frac{1}{2} Ω^{2} F^{(2)}) \dots] P \end{array}

In Eq. (17), we observe that if all terms in the exponential after the term containing F⁽¹⁾ are ignored, we get back the flat response that we obtained for the narrow band signal. However, if we detect along the output principal mode q_k, the wide band nature of the signal violates the small Ω assumption that was used to derive the p_k, q_k pair for zero modal dispersion, resulting in modal dispersion. We wish to evaluate the values of the modulation bandwidth Ω for which the frequency response deviates significantly from the flat response seen in the conventional narrow band signal case, and analyze how various fiber parameters such as length, diameter and coupling characteristics affect the fiber performance in this bandwidth regime. For the case with MDL, the output PMs are no longer orthogonal. Thus, we canot use the Eq. (17) to calculate the frequency response of the fiber with MDL. We defined the frequency response of the fiber with MDL as H_MDL(Ω) and for calculation of the frequency response we use the following equation:

\begin{array}{l} H_{MDL} (Ω) = {[h_{MDL, 1} (Ω), h_{MDL, 2} (Ω), \dots h_{MDL, M} (Ω)]}^{T} = Q_{inv} U (ω_{0} + Ω) P \\ = Q_{inv} [U_{0} \exp (- j Ω F^{(1)}) \exp (\frac{1}{2} Ω^{2} F^{(2)}) \dots] P \end{array}

where Q_inv is a matrix that satisfies Q_invQ = I, as defined in [18].

To evaluate the variation of the cross-talk with modulation bandwidth, we multiplex the input signals over the PMs. The cross-talk is given by:

cross-talk for PM i = 10 \log_{10} (\frac{\sum_{k = 1, k \neq i}^{M} | h_{k} |^{2}}{| h_{i} |^{2}}) (dB)

where h_k is the element of frequency response of the output PMs of the optical fiber and h_i is the frequency response of the of the desired output PM at which we calculate the cross-talk.

3. Simulation and discussion

In this section, we quantify the performance of PMs on large-core MMF links as the bandwidth increases by simulating the performance based on the model discussed in Section 2.1, and discuss the implication of our observations on both few-mode and large-core MMF based links. For our simulations, we consider few-mode MMFs having 6 modes, with core diameter of 10 μm, and large-core MMFs having 110 modes with diameter 50 μm. The numerical aperture was assumed to be 0.14 for the few-mode fiber and 0.19 for the large-core multimode fiber. The laser carrier is assumed to have a wavelength of 1550 nm and the fiber has a parabolic refractive index profile with refractive index at the center of the fiber being n₀ = 1.444. The curvature of each section is an independent and identically distributed (i.i.d.) random variable κ, whose probability density function (pdf) is the positive side of a normal pdf, and has variance $σ_{κ}^{2}$ . As $σ_{κ}^{2}$ is increased, the model transitions from the low-coupling to the medium and high-coupling regimes. Differences between few-mode fiber and large-core MMFs are given in Table 2. The rotation between successive sections is captured by another random variable θ with a pdf of a normal random variable with zero mean and $σ_{θ}^{2} = 0.36$ rad². The propagation matrix of each section was calculated using the techniques described in [5].

Table 2. Criterion of few mode and large-core MMFs.

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The overall propagation matrix was then calculated as product of all these matrices of all the sections, ranging from 10² to 10⁴, with the length of each section kept fixed at 1 m (coherence length). That is, for example, a 100 m fiber would have 100 sections. The first order and higher order group delay propagation operator matrices are calculated using Eq. (11) and 9. In each case, the eigenvalue decomposition of the first order group delay operator gives us the first order PMs. We assume that the transmitter possesses the accurate PM as measured at the center frequency of transmission (corresponding to Ω = 0).

To simulate the frequency response of the fiber, we divide the spectrum between 0 Hz and 100 GHz into 100 different modulation frequencies and perform the simulation for each of these frequencies and interpolate the results, averaged over several fiber realizations.

Figures. 4–5 shows the variation of the frequency response of the first PM of the optical fiber without MDL, as the modulation bandwidth increases for different coupling regimes and different fiber lengths. We observe that if we assume that only the first order approximation of the fiber as in Eq. (10), is used, we see no change in the frequency response of the fiber as the modulation bandwidth increases in all cases. This is expected, since the PMs have been derived with the assumption of not encountering any dispersion. However, when the more realistic fiber model that accounts for the higher order group delays (as in Eq. (9)) is used, there is a clear degradation in the performance of the link, and dispersion causes the frequency response to degrade significantly. We can see that for few-mode fibers, if the modulation bandwidth is less than 10 GHz, then the effect of the higher order group delay matrices on the frequency response of the fiber is quite small, and these effects become significant at higher bandwidths. In Fig. 4(a), we see that the frequency response of a 10 km long few-mode fiber with 10 μm diameter does not show significant changes with increase in modulation bandwidths in the low coupling regime. For higher intermodal coupling, the frequency response of the fiber shows a 2.5 dB reduction in range of 40 – 100 GHz of modulation bandwidth. Fig. 4(b) shows that if we increase length of few-mode fiber to 100 km, the frequency response of the fiber shows a 5 dB degradation in high intermodal coupling regime. Moreover, for lengths of 100 km and above, the impact of chromatic and waveguide dispersion (not considered here) may cause the performance to degrade even further. Similar simulations performed for a 100 m length large-core MMF shows the similar frequency response degradation with increase in modulation bandwidth, as shown in Fig. 5(a).

Fig. 4 Frequency response of few-mode fiber with different σ_κ.

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Fig. 5 Frequency response of large-core multimode fiber with different σ_κ.

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However, when we consider large-core MMFs of 1 km length, the frequency response of the fiber degrades by around 0.8 dB for low modulation bandwidths, as is clearly visible in Fig. 5(b). This can be attributed to the fact that difference of propagation constants of the first mode and last mode in large-core MMFs is much higher than the difference between the propagation constant of first mode and last mode of few-mode fibers. Hence dispersion in the large-core MMFs is much more than the few-mode fibers. These are largely in agreement with the experimental results reported in [11], although the focus there was restricted only to 100 m fibers.

We study the cross-talk of the multiplexed signal in optical fibers without MDL, with lengths ranging between 100 m and 10 km for modulation bandwidths up to 10 GHz.

Figures 6–7, we plot the performance in the presence of cross-talk in the low and high coupling regimes. Assuming a cross-talk limitation threshold of −20 dB, in the Fig. 6(a), we observe that when the modulation bandwidth is 10 GHz, the cross-talk is still below −30 dB. However, for a 10 km few-mode fiber, the cross-talk reaches the −20 dB threshold at for Ω = 10 GHz, as seen in Fig. 6(b) This can be attributed to the fact that as the fiber length increases, mode coupling causes the cross-talk of the multiplexed signal to rise. We observe that cross-talk is dominant in the strong coupling regime (i.e. for high fiber curvature σ_κ), and large-core MMFs display a larger amount of cross-talk in the multiplexed signal (as in Fig. 7(b)) than the few-mode fibers of comparable length. This is because the difference between the largest and smallest propagation constants of the fiber modes is much larger in large-core MMFs than in few-mode fibers, thus making them more vulnerable to cross-talk impairments.

Fig. 6 Cross-talk of few-mode fiber with different σ_κ.

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Fig. 7 Cross-talk of multimode fiber with different σ_κ_.

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In multimode fibers with mode dependent losses, we use different values of offsets ranging between 0.6 μm and 2 μm to simulate the impact on the frequency response of the fiber with respect to modulation bandwidth.

The length of the fiber is varied from 10 m to 1 km, and we consider 5 lossy splices, with the other fiber sections being lossless. From our simulations, we observe that due to the offset between sections, the frequency response of the fiber is poorer than the frequency response of the fiber without mode dependent losses.

Figures 8 through 9, show the frequency response of the optical fiber with MDL. From Fig. 8(a), we observe that for small lengths (100 m) of the large core optical fiber, the frequency response degrades significantly at large modulation bandwidths. As the length of the optical fiber increases (100 m to 1 km), we observe that the frequency response of the fiber degrades further, as shown in Fig. 8(b) to 9(b).

Fig. 8 Frequency response of few-mode lossy fiber with different σ_κ.

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Fig. 9 Frequency response of multimode lossy fiber with different σ_κ.

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Figure 10 shows the frequency response for different core diameters and corresponding numerical apertures of a 1 km long fiber. From the fig., we observe that, for both few-mode and multimode fibers, as the core diameter of the fiber increases, the frequency response degrades further. This is because the difference between the largest and smallest propagation constants of the fiber modes increases as the core diameter of the fiber increases.

Fig. 10 Frequency response of different core diameter and numerical aperture (NA) of length 1 km.

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In Figs. 11–12, we observe the cross-talk of multiplexed signals in the presence of MDL for lengths ranging from 100 m to 1 km. We can see that the cross-talk of the multiplexed signal rises and exceeds 10 dB for modulation bandwidths above 3 GHz for all coupling regimes. From these simulations of frequency response and cross-talk of optical fibers with MDL, we observe that for all coupling regimes, the effect of higher order components with modulation bandwidths is similar, since the offset causes effective mode mixing and reduces the impact of just mode coupling on the frequency response.

Fig. 11 Cross-talk of few-mode fiber with different σ_κ.

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Fig. 12 Cross-talk of lossy multimode fiber with different σ_κ.

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Figure 13 shows the frequency response of the fiber (few-mode and large-core multimode) for different PMs. From the figs., we observe that PM display a similar degradation in frequency response as the modulation bandwidth increases, but seem to be separable in to two sets based on how much the frequency response changes. The reason why the two sets of PMs appear to be “grouped” is that the geometric variations in the fiber cause distinct birefringent effects, thereby causing different polarizations to exhibit different frequency responses. We thus observe that in the few-mode fibers, the performance of the PMs can be grouped in corresponce to their polarization in Fig. 13(a), although the gap in performance between the pair of PMs that correspond to each polarization is small, within 0.2 dB. In the case of large-core MMFs, this gap in performance based on polarization does not exhibit itself significantly.

Fig. 13 Frequency responce of different PM of length 1 km.

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Figures 14(a) and 14(b) show the bit error rate (BER) obtained when a BPSK signal is sent using the PMs of fibers of different lengths (ranging from 100 m to 1 km). Note that there is no additional equalization and cross-talk compensation considered in these figs. From Fig. 14(a), we observe that, in lossy few-mode fibers, if we assume that the fiber is modeled as having only a first order group delay dependent frequency response, then for higher signal-to-noise ratios (SNRs), the BER is low. However, when we account for the higher order group delay components in the fiber response, the performance is significantly poorer. In Fig. 14(b), we observe that in large-core multimode fiber with MDL, intermodal coupling is high for long fibers, since even for very large SNRs, cross-talk among modes causes the performance to diminish. This performance can only be improved by performing additional dispersion and cross-talk compensation at the receiver.

Fig. 14 Comparison of bit error rate of different length (no additional dispersion and cross-talk compensation).

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We have summarized the bandwidth performance of the PMs in Table 3, where the bandwidth of the signal after which the frequency response of the mode diminishes by 1 dB is listed for various fiber types and lengths. This table is for σ_κ = 7 m⁻¹, which is the strong coupling regime, since in the weaker coupling regime, we can expect the 1 dB bandwidth of PMs to be much more than those of the ideal modes, due to enhanced frequency flatness in this regime [24]. We can observe that transmission in the using PMs provides a significant improvement in performance, in some cases, by an order of magnitude. The direct implication of this is that we can multiplex streams of much higher data rates through the fiber, than would be possible using the ideal modes directly. While this would seem to indicate that the increase in bandwidth can be much more if all the PMs are used, one way to benefit this technique could be to use a fewer number of PMs to send data, thereby reducing the number of parallel streams. This could provide a higher data rate without having to multiplex through several modes or mode groups, since launching in specific modes is challenging in large-core multimode fibers.

Table 3. Modulation bandwidth at which fiber response is degraded by 1 dB with σ_k = 7 m⁻¹.

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From the above simulations, we can conclude that, for longer lengths and higher bandwidths, the use of principal modes without additional dispersion compensation at the receiver degrades performance significantly. This becomes more acute in the case of large-core MMFs due to the large differences in the mode propagation constants. For σ_κ < 1, we see that the impact of large modulation bandwidths does not significantly degrade performance. That is, the frequency response of the overall system is more immune dispersion in the weak coupling regime.

These results indicate that since the bandwidth over which the PMs may be assumed to be invariant (referred to as coherence bandwidth in [5]), the processing requirements at the receiver can be eased significantly when compared to a system that uses a finer frequency split, such as those that employ orthogonal frequency division multiplexing (OFDM). We also remark that these results are largely consistent with related experimental observations in [11].

4. Conclusion

In large-core MMF links, principal modes offer a method to enable high data rate communication with low intermodal dispersion and cross-talk free multiplexing. However, increasing the modulation bandwidth reduces these advantages, and diminishes system performance. In this paper we have described and analyzed how the dispersion free and orthonormal property of PMs with degrade with increase in the modulation bandwidth. We have used a field propagation based fiber model to evaluate the frequency response of the few-mode and multimode fibers. Simulations reveal that for large-core MMFs without MDL, if we consider the fiber frequency response using the first order group delay operator F⁽¹⁾, with modulation bandwidths less than 10 GHz, then there is not much of an impact on the dispersion free nature of PMs across all coupling regimes. However, this approximation does not hold for larger modulation bandwidths, and simulations reveal that the frequency response of 100 km and 100 m links degrades by over 5 dB for few-mode and large-core multimode fibers respectively, thereby indicating that additional dispersion compensation is required at these high modulation bandwidths. When we consider MDL, simulations reveal that the frequency response degraded by 5 dB when we increase the length of the fiber from 100 m to 1 km at modulation bandwidths in excess of 5 GHz when compared to similar fibers without MDL. Future work would focus on the impact of these effects on space-division multiplexed links and an improved characterization of mode-dependent losses in few-mode fiber based systems. In addition, the impact of material and waveguide dispersion, particularly for few-mode fibers, when PM based transmission is employed, should be studied.

Funding

Department of Science and Technology, Government of India (INSPIRE DST/INSPIRE/04/2014/001392).

References and links

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Parameter	Fiber without MDL	Fiber with MDL
U(ω)	U⁽^N⁾(ω)U⁽^N⁻¹⁾(ω) …	U^(S)(ω)C⁽^S⁻¹⁾U⁽^S⁻¹⁾(ω) …
PMs	Orthonormal	Not orthonormal
Group delay operator	Hermitian	Not hermitian
$M_{l}^{(k)}$	Unitary	Non-unitary

Fiber	Length	Ideal Modes	PMs
Few mode	1 km	~ 100 GHz	> 1000 GHz
Few mode	10 km	< 60 GHz	> 100 GHz
Few mode	100 km	< 40 GHz	~ 60 GHz
Multimode	100 m	~ 60 GHz	> 70 GHz
Multimode	1 km	< 20 GHz	~ 50 GHz

Parameter	Fiber without MDL	Fiber with MDL
U(ω)	U⁽^N⁾(ω)U⁽^N⁻¹⁾(ω) …	U^(S)(ω)C⁽^S⁻¹⁾U⁽^S⁻¹⁾(ω) …
PMs	Orthonormal	Not orthonormal
Group delay operator	Hermitian	Not hermitian
$M_{l}^{(k)}$	Unitary	Non-unitary

Fiber	Length	Ideal Modes	PMs
Few mode	1 km	~ 100 GHz	> 1000 GHz
Few mode	10 km	< 60 GHz	> 100 GHz
Few mode	100 km	< 40 GHz	~ 60 GHz
Multimode	100 m	~ 60 GHz	> 70 GHz
Multimode	1 km	< 20 GHz	~ 50 GHz

Impact of modulation bandwidth on multiplexing using principal modes in MMF links

Abstract

1. Introduction

2. System model

2.1. Multimode optical fiber model

2.2. Principal modes

2.2.1. First order propagation operator

2.2.2. Higher order propagation operator

3. Simulation and discussion

4. Conclusion

Funding

References and links

Cited By

Figures (14)

Tables (3)

Equations (19)

Optics Express

Criterion	Few mode fiber	Large-core MMFs
Number of modes	6	110
Core diameter	10 μm	50 μm
Numerical aperture	0.14	0.19